LMFDB - Embedded newform 6480.2.a.br.1.2

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:	$0$
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 405)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
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)

$f(q)$	$=$	$q+1.00000 q^{5} +4.73205 q^{7} +5.73205 q^{11} +1.46410 q^{13} -2.73205 q^{17} -4.46410 q^{19} +3.46410 q^{23} +1.00000 q^{25} +3.19615 q^{29} +3.00000 q^{31} +4.73205 q^{35} -2.73205 q^{37} -7.19615 q^{41} -0.196152 q^{43} +8.73205 q^{47} +15.3923 q^{49} +6.73205 q^{53} +5.73205 q^{55} +8.26795 q^{59} +4.00000 q^{61} +1.46410 q^{65} -3.46410 q^{67} +3.73205 q^{71} -7.66025 q^{73} +27.1244 q^{77} -15.4641 q^{79} -2.19615 q^{83} -2.73205 q^{85} +5.19615 q^{89} +6.92820 q^{91} -4.46410 q^{95} -9.66025 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{5} + 6 q^{7} + 8 q^{11} - 4 q^{13} - 2 q^{17} - 2 q^{19} + 2 q^{25} - 4 q^{29} + 6 q^{31} + 6 q^{35} - 2 q^{37} - 4 q^{41} + 10 q^{43} + 14 q^{47} + 10 q^{49} + 10 q^{53} + 8 q^{55} + 20 q^{59}+ \cdots - 2 q^{97}+O(q^{100}) $ 2 * q + 2 * q^5 + 6 * q^7 + 8 * q^11 - 4 * q^13 - 2 * q^17 - 2 * q^19 + 2 * q^25 - 4 * q^29 + 6 * q^31 + 6 * q^35 - 2 * q^37 - 4 * q^41 + 10 * q^43 + 14 * q^47 + 10 * q^49 + 10 * q^53 + 8 * q^55 + 20 * q^59 + 8 * q^61 - 4 * q^65 + 4 * q^71 + 2 * q^73 + 30 * q^77 - 24 * q^79 + 6 * q^83 - 2 * q^85 - 2 * q^95 - 2 * 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
$7$	4.73205	1.78855	0.894274	−	0.447521i	$-0.147693\pi$
$7$	4.73205	1.78855	0.894274	+	0.447521i	$0.147693\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.73205	1.72828	0.864139	−	0.503253i	$-0.167864\pi$
$11$	5.73205	1.72828	0.864139	+	0.503253i	$0.167864\pi$
$12$	0	0
$13$	1.46410	0.406069	0.203034	−	0.979172i	$-0.434920\pi$
$13$	1.46410	0.406069	0.203034	+	0.979172i	$0.434920\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−2.73205	−0.662620	−0.331310	−	0.943522i	$-0.607491\pi$
$17$	−2.73205	−0.662620	−0.331310	+	0.943522i	$0.607491\pi$
$18$	0	0
$19$	−4.46410	−1.02414	−0.512068	−	0.858945i	$-0.671120\pi$
$19$	−4.46410	−1.02414	−0.512068	+	0.858945i	$0.671120\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	3.46410	0.722315	0.361158	−	0.932505i	$-0.382382\pi$
$23$	3.46410	0.722315	0.361158	+	0.932505i	$0.382382\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	3.19615	0.593511	0.296755	−	0.954954i	$-0.404095\pi$
$29$	3.19615	0.593511	0.296755	+	0.954954i	$0.404095\pi$
$30$	0	0
$31$	3.00000	0.538816	0.269408	−	0.963026i	$-0.413172\pi$
$31$	3.00000	0.538816	0.269408	+	0.963026i	$0.413172\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.73205	0.799863
$36$	0	0
$37$	−2.73205	−0.449146	−0.224573	−	0.974457i	$-0.572099\pi$
$37$	−2.73205	−0.449146	−0.224573	+	0.974457i	$0.572099\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−7.19615	−1.12385	−0.561925	−	0.827188i	$-0.689939\pi$
$41$	−7.19615	−1.12385	−0.561925	+	0.827188i	$0.689939\pi$
$42$	0	0
$43$	−0.196152	−0.0299130	−0.0149565	−	0.999888i	$-0.504761\pi$
$43$	−0.196152	−0.0299130	−0.0149565	+	0.999888i	$0.504761\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	8.73205	1.27370	0.636850	−	0.770988i	$-0.280237\pi$
$47$	8.73205	1.27370	0.636850	+	0.770988i	$0.280237\pi$
$48$	0	0
$49$	15.3923	2.19890
$50$	0	0
$51$	0	0
$52$	0	0
$53$	6.73205	0.924718	0.462359	−	0.886693i	$-0.347003\pi$
$53$	6.73205	0.924718	0.462359	+	0.886693i	$0.347003\pi$
$54$	0	0
$55$	5.73205	0.772910
$56$	0	0
$57$	0	0
$58$	0	0
$59$	8.26795	1.07640	0.538198	−	0.842819i	$-0.319105\pi$
$59$	8.26795	1.07640	0.538198	+	0.842819i	$0.319105\pi$
$60$	0	0
$61$	4.00000	0.512148	0.256074	−	0.966657i	$-0.417571\pi$
$61$	4.00000	0.512148	0.256074	+	0.966657i	$0.417571\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.46410	0.181599
$66$	0	0
$67$	−3.46410	−0.423207	−0.211604	−	0.977356i	$-0.567869\pi$
$67$	−3.46410	−0.423207	−0.211604	+	0.977356i	$0.567869\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	3.73205	0.442913	0.221456	−	0.975170i	$-0.428919\pi$
$71$	3.73205	0.442913	0.221456	+	0.975170i	$0.428919\pi$
$72$	0	0
$73$	−7.66025	−0.896565	−0.448282	−	0.893892i	$-0.647964\pi$
$73$	−7.66025	−0.896565	−0.448282	+	0.893892i	$0.647964\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	27.1244	3.09111
$78$	0	0
$79$	−15.4641	−1.73985	−0.869924	−	0.493186i	$-0.835832\pi$
$79$	−15.4641	−1.73985	−0.869924	+	0.493186i	$0.835832\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−2.19615	−0.241059	−0.120530	−	0.992710i	$-0.538459\pi$
$83$	−2.19615	−0.241059	−0.120530	+	0.992710i	$0.538459\pi$
$84$	0	0
$85$	−2.73205	−0.296333
$86$	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
$91$	6.92820	0.726273
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.46410	−0.458007
$96$	0	0
$97$	−9.66025	−0.980850	−0.490425	−	0.871483i	$-0.663158\pi$
$97$	−9.66025	−0.980850	−0.490425	+	0.871483i	$0.663158\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−2.66025	−0.264705	−0.132353	−	0.991203i	$-0.542253\pi$
$101$	−2.66025	−0.264705	−0.132353	+	0.991203i	$0.542253\pi$
$102$	0	0
$103$	0.535898	0.0528036	0.0264018	−	0.999651i	$-0.491595\pi$
$103$	0.535898	0.0528036	0.0264018	+	0.999651i	$0.491595\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.53590	0.825196	0.412598	−	0.910913i	$-0.364621\pi$
$107$	8.53590	0.825196	0.412598	+	0.910913i	$0.364621\pi$
$108$	0	0
$109$	−6.07180	−0.581573	−0.290786	−	0.956788i	$-0.593917\pi$
$109$	−6.07180	−0.581573	−0.290786	+	0.956788i	$0.593917\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−19.1244	−1.79907	−0.899534	−	0.436851i	$-0.856094\pi$
$113$	−19.1244	−1.79907	−0.899534	+	0.436851i	$0.856094\pi$
$114$	0	0
$115$	3.46410	0.323029
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−12.9282	−1.18513
$120$	0	0
$121$	21.8564	1.98695
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	14.5885	1.29452	0.647258	−	0.762271i	$-0.275916\pi$
$127$	14.5885	1.29452	0.647258	+	0.762271i	$0.275916\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−15.5885	−1.36197	−0.680985	−	0.732297i	$-0.738448\pi$
$131$	−15.5885	−1.36197	−0.680985	+	0.732297i	$0.738448\pi$
$132$	0	0
$133$	−21.1244	−1.83171
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2.53590	0.216656	0.108328	−	0.994115i	$-0.465450\pi$
$137$	2.53590	0.216656	0.108328	+	0.994115i	$0.465450\pi$
$138$	0	0
$139$	−0.607695	−0.0515440	−0.0257720	−	0.999668i	$-0.508204\pi$
$139$	−0.607695	−0.0515440	−0.0257720	+	0.999668i	$0.508204\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	8.39230	0.701800
$144$	0	0
$145$	3.19615	0.265426
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−8.00000	−0.655386	−0.327693	−	0.944784i	$-0.606271\pi$
$149$	−8.00000	−0.655386	−0.327693	+	0.944784i	$0.606271\pi$
$150$	0	0
$151$	−5.39230	−0.438820	−0.219410	−	0.975633i	$-0.570413\pi$
$151$	−5.39230	−0.438820	−0.219410	+	0.975633i	$0.570413\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	3.00000	0.240966
$156$	0	0
$157$	−19.1244	−1.52629	−0.763145	−	0.646227i	$-0.776346\pi$
$157$	−19.1244	−1.52629	−0.763145	+	0.646227i	$0.776346\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	16.3923	1.29189
$162$	0	0
$163$	−12.7321	−0.997251	−0.498626	−	0.866817i	$-0.666162\pi$
$163$	−12.7321	−0.997251	−0.498626	+	0.866817i	$0.666162\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	17.6603	1.36659	0.683296	−	0.730142i	$-0.260546\pi$
$167$	17.6603	1.36659	0.683296	+	0.730142i	$0.260546\pi$
$168$	0	0
$169$	−10.8564	−0.835108
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.53590	0.648972	0.324486	−	0.945890i	$-0.394809\pi$
$173$	8.53590	0.648972	0.324486	+	0.945890i	$0.394809\pi$
$174$	0	0
$175$	4.73205	0.357709
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−8.12436	−0.607243	−0.303621	−	0.952793i	$-0.598196\pi$
$179$	−8.12436	−0.607243	−0.303621	+	0.952793i	$0.598196\pi$
$180$	0	0
$181$	26.4641	1.96706	0.983531	−	0.180742i	$-0.0578498\pi$
$181$	26.4641	1.96706	0.983531	+	0.180742i	$0.0578498\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−2.73205	−0.200864
$186$	0	0
$187$	−15.6603	−1.14519
$188$	0	0
$189$	0	0
$190$	0	0
$191$	8.12436	0.587858	0.293929	−	0.955827i	$-0.405037\pi$
$191$	8.12436	0.587858	0.293929	+	0.955827i	$0.405037\pi$
$192$	0	0
$193$	−5.26795	−0.379195	−0.189598	−	0.981862i	$-0.560718\pi$
$193$	−5.26795	−0.379195	−0.189598	+	0.981862i	$0.560718\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−13.8564	−0.987228	−0.493614	−	0.869681i	$-0.664324\pi$
$197$	−13.8564	−0.987228	−0.493614	+	0.869681i	$0.664324\pi$
$198$	0	0
$199$	2.00000	0.141776	0.0708881	−	0.997484i	$-0.477417\pi$
$199$	2.00000	0.141776	0.0708881	+	0.997484i	$0.477417\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	15.1244	1.06152
$204$	0	0
$205$	−7.19615	−0.502601
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−25.5885	−1.76999
$210$	0	0
$211$	8.85641	0.609700	0.304850	−	0.952400i	$-0.401394\pi$
$211$	8.85641	0.609700	0.304850	+	0.952400i	$0.401394\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−0.196152	−0.0133775
$216$	0	0
$217$	14.1962	0.963698
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−4.00000	−0.269069
$222$	0	0
$223$	−16.7846	−1.12398	−0.561990	−	0.827144i	$-0.689964\pi$
$223$	−16.7846	−1.12398	−0.561990	+	0.827144i	$0.689964\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	18.0526	1.19819	0.599095	−	0.800678i	$-0.295527\pi$
$227$	18.0526	1.19819	0.599095	+	0.800678i	$0.295527\pi$
$228$	0	0
$229$	−12.0000	−0.792982	−0.396491	−	0.918039i	$-0.629772\pi$
$229$	−12.0000	−0.792982	−0.396491	+	0.918039i	$0.629772\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−28.0526	−1.83778	−0.918892	−	0.394509i	$-0.870915\pi$
$233$	−28.0526	−1.83778	−0.918892	+	0.394509i	$0.870915\pi$
$234$	0	0
$235$	8.73205	0.569616
$236$	0	0
$237$	0	0
$238$	0	0
$239$	0.535898	0.0346644	0.0173322	−	0.999850i	$-0.494483\pi$
$239$	0.535898	0.0346644	0.0173322	+	0.999850i	$0.494483\pi$
$240$	0	0
$241$	−16.3205	−1.05130	−0.525648	−	0.850702i	$-0.676177\pi$
$241$	−16.3205	−1.05130	−0.525648	+	0.850702i	$0.676177\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	15.3923	0.983378
$246$	0	0
$247$	−6.53590	−0.415869
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.3923	0.655956	0.327978	−	0.944685i	$-0.393633\pi$
$251$	10.3923	0.655956	0.327978	+	0.944685i	$0.393633\pi$
$252$	0	0
$253$	19.8564	1.24836
$254$	0	0
$255$	0	0
$256$	0	0
$257$	10.3923	0.648254	0.324127	−	0.946014i	$-0.394929\pi$
$257$	10.3923	0.648254	0.324127	+	0.946014i	$0.394929\pi$
$258$	0	0
$259$	−12.9282	−0.803319
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−21.3205	−1.31468	−0.657339	−	0.753595i	$-0.728318\pi$
$263$	−21.3205	−1.31468	−0.657339	+	0.753595i	$0.728318\pi$
$264$	0	0
$265$	6.73205	0.413547
$266$	0	0
$267$	0	0
$268$	0	0
$269$	10.6603	0.649967	0.324984	−	0.945720i	$-0.394641\pi$
$269$	10.6603	0.649967	0.324984	+	0.945720i	$0.394641\pi$
$270$	0	0
$271$	2.92820	0.177876	0.0889378	−	0.996037i	$-0.471653\pi$
$271$	2.92820	0.177876	0.0889378	+	0.996037i	$0.471653\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	5.73205	0.345656
$276$	0	0
$277$	3.80385	0.228551	0.114276	−	0.993449i	$-0.463545\pi$
$277$	3.80385	0.228551	0.114276	+	0.993449i	$0.463545\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.4641	0.922511	0.461255	−	0.887267i	$-0.347399\pi$
$281$	15.4641	0.922511	0.461255	+	0.887267i	$0.347399\pi$
$282$	0	0
$283$	29.3205	1.74292	0.871462	−	0.490464i	$-0.163173\pi$
$283$	29.3205	1.74292	0.871462	+	0.490464i	$0.163173\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−34.0526	−2.01006
$288$	0	0
$289$	−9.53590	−0.560935
$290$	0	0
$291$	0	0
$292$	0	0
$293$	28.7321	1.67854	0.839272	−	0.543712i	$-0.182981\pi$
$293$	28.7321	1.67854	0.839272	+	0.543712i	$0.182981\pi$
$294$	0	0
$295$	8.26795	0.481379
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.07180	0.293310
$300$	0	0
$301$	−0.928203	−0.0535007
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.00000	0.229039
$306$	0	0
$307$	−14.0526	−0.802022	−0.401011	−	0.916073i	$-0.631341\pi$
$307$	−14.0526	−0.802022	−0.401011	+	0.916073i	$0.631341\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.7321	−1.11890	−0.559451	−	0.828863i	$-0.688988\pi$
$311$	−19.7321	−1.11890	−0.559451	+	0.828863i	$0.688988\pi$
$312$	0	0
$313$	−9.07180	−0.512768	−0.256384	−	0.966575i	$-0.582531\pi$
$313$	−9.07180	−0.512768	−0.256384	+	0.966575i	$0.582531\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6.19615	0.348011	0.174005	−	0.984745i	$-0.444329\pi$
$317$	6.19615	0.348011	0.174005	+	0.984745i	$0.444329\pi$
$318$	0	0
$319$	18.3205	1.02575
$320$	0	0
$321$	0	0
$322$	0	0
$323$	12.1962	0.678612
$324$	0	0
$325$	1.46410	0.0812137
$326$	0	0
$327$	0	0
$328$	0	0
$329$	41.3205	2.27807
$330$	0	0
$331$	0.464102	0.0255093	0.0127547	−	0.999919i	$-0.495940\pi$
$331$	0.464102	0.0255093	0.0127547	+	0.999919i	$0.495940\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3.46410	−0.189264
$336$	0	0
$337$	27.3205	1.48824	0.744121	−	0.668044i	$-0.232868\pi$
$337$	27.3205	1.48824	0.744121	+	0.668044i	$0.232868\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	17.1962	0.931224
$342$	0	0
$343$	39.7128	2.14429
$344$	0	0
$345$	0	0
$346$	0	0
$347$	28.5885	1.53471	0.767354	−	0.641223i	$-0.221573\pi$
$347$	28.5885	1.53471	0.767354	+	0.641223i	$0.221573\pi$
$348$	0	0
$349$	−18.8564	−1.00936	−0.504680	−	0.863306i	$-0.668390\pi$
$349$	−18.8564	−1.00936	−0.504680	+	0.863306i	$0.668390\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	25.5167	1.35811	0.679057	−	0.734085i	$-0.262389\pi$
$353$	25.5167	1.35811	0.679057	+	0.734085i	$0.262389\pi$
$354$	0	0
$355$	3.73205	0.198077
$356$	0	0
$357$	0	0
$358$	0	0
$359$	6.12436	0.323231	0.161616	−	0.986854i	$-0.448330\pi$
$359$	6.12436	0.323231	0.161616	+	0.986854i	$0.448330\pi$
$360$	0	0
$361$	0.928203	0.0488528
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.66025	−0.400956
$366$	0	0
$367$	−31.1769	−1.62742	−0.813711	−	0.581270i	$-0.802556\pi$
$367$	−31.1769	−1.62742	−0.813711	+	0.581270i	$0.802556\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	31.8564	1.65390
$372$	0	0
$373$	20.0526	1.03828	0.519141	−	0.854689i	$-0.326252\pi$
$373$	20.0526	1.03828	0.519141	+	0.854689i	$0.326252\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.67949	0.241006
$378$	0	0
$379$	−2.39230	−0.122884	−0.0614422	−	0.998111i	$-0.519570\pi$
$379$	−2.39230	−0.122884	−0.0614422	+	0.998111i	$0.519570\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.53590	−0.129578	−0.0647892	−	0.997899i	$-0.520637\pi$
$383$	−2.53590	−0.129578	−0.0647892	+	0.997899i	$0.520637\pi$
$384$	0	0
$385$	27.1244	1.38239
$386$	0	0
$387$	0	0
$388$	0	0
$389$	27.4641	1.39249	0.696243	−	0.717807i	$-0.254854\pi$
$389$	27.4641	1.39249	0.696243	+	0.717807i	$0.254854\pi$
$390$	0	0
$391$	−9.46410	−0.478620
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−15.4641	−0.778083
$396$	0	0
$397$	14.3923	0.722329	0.361165	−	0.932502i	$-0.382379\pi$
$397$	14.3923	0.722329	0.361165	+	0.932502i	$0.382379\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	24.9282	1.24486	0.622428	−	0.782677i	$-0.286147\pi$
$401$	24.9282	1.24486	0.622428	+	0.782677i	$0.286147\pi$
$402$	0	0
$403$	4.39230	0.218796
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.6603	−0.776250
$408$	0	0
$409$	−17.8564	−0.882942	−0.441471	−	0.897275i	$-0.645543\pi$
$409$	−17.8564	−0.882942	−0.441471	+	0.897275i	$0.645543\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	39.1244	1.92518
$414$	0	0
$415$	−2.19615	−0.107805
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−20.3923	−0.996229	−0.498115	−	0.867111i	$-0.665974\pi$
$419$	−20.3923	−0.996229	−0.498115	+	0.867111i	$0.665974\pi$
$420$	0	0
$421$	33.7846	1.64656	0.823281	−	0.567635i	$-0.192141\pi$
$421$	33.7846	1.64656	0.823281	+	0.567635i	$0.192141\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−2.73205	−0.132524
$426$	0	0
$427$	18.9282	0.916000
$428$	0	0
$429$	0	0
$430$	0	0
$431$	21.3397	1.02790	0.513950	−	0.857820i	$-0.328182\pi$
$431$	21.3397	1.02790	0.513950	+	0.857820i	$0.328182\pi$
$432$	0	0
$433$	−35.4641	−1.70430	−0.852148	−	0.523301i	$-0.824700\pi$
$433$	−35.4641	−1.70430	−0.852148	+	0.523301i	$0.824700\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−15.4641	−0.739748
$438$	0	0
$439$	−5.39230	−0.257361	−0.128680	−	0.991686i	$-0.541074\pi$
$439$	−5.39230	−0.257361	−0.128680	+	0.991686i	$0.541074\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	0.339746	0.0161418	0.00807091	−	0.999967i	$-0.497431\pi$
$443$	0.339746	0.0161418	0.00807091	+	0.999967i	$0.497431\pi$
$444$	0	0
$445$	5.19615	0.246321
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8.12436	0.383412	0.191706	−	0.981452i	$-0.438598\pi$
$449$	8.12436	0.383412	0.191706	+	0.981452i	$0.438598\pi$
$450$	0	0
$451$	−41.2487	−1.94233
$452$	0	0
$453$	0	0
$454$	0	0
$455$	6.92820	0.324799
$456$	0	0
$457$	2.73205	0.127800	0.0639000	−	0.997956i	$-0.479646\pi$
$457$	2.73205	0.127800	0.0639000	+	0.997956i	$0.479646\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−37.0526	−1.72571	−0.862855	−	0.505452i	$-0.831326\pi$
$461$	−37.0526	−1.72571	−0.862855	+	0.505452i	$0.831326\pi$
$462$	0	0
$463$	10.3923	0.482971	0.241486	−	0.970404i	$-0.422365\pi$
$463$	10.3923	0.482971	0.241486	+	0.970404i	$0.422365\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−37.3731	−1.72942	−0.864710	−	0.502272i	$-0.832498\pi$
$467$	−37.3731	−1.72942	−0.864710	+	0.502272i	$0.832498\pi$
$468$	0	0
$469$	−16.3923	−0.756926
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−1.12436	−0.0516979
$474$	0	0
$475$	−4.46410	−0.204827
$476$	0	0
$477$	0	0
$478$	0	0
$479$	11.8756	0.542612	0.271306	−	0.962493i	$-0.412544\pi$
$479$	11.8756	0.542612	0.271306	+	0.962493i	$0.412544\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−9.66025	−0.438650
$486$	0	0
$487$	−24.3923	−1.10532	−0.552660	−	0.833407i	$-0.686387\pi$
$487$	−24.3923	−1.10532	−0.552660	+	0.833407i	$0.686387\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	13.8756	0.626199	0.313100	−	0.949720i	$-0.398633\pi$
$491$	13.8756	0.626199	0.313100	+	0.949720i	$0.398633\pi$
$492$	0	0
$493$	−8.73205	−0.393272
$494$	0	0
$495$	0	0
$496$	0	0
$497$	17.6603	0.792171
$498$	0	0
$499$	−24.3205	−1.08874	−0.544368	−	0.838847i	$-0.683230\pi$
$499$	−24.3205	−1.08874	−0.544368	+	0.838847i	$0.683230\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.32051	0.326405	0.163203	−	0.986593i	$-0.447818\pi$
$503$	7.32051	0.326405	0.163203	+	0.986593i	$0.447818\pi$
$504$	0	0
$505$	−2.66025	−0.118380
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−6.78461	−0.300723	−0.150361	−	0.988631i	$-0.548044\pi$
$509$	−6.78461	−0.300723	−0.150361	+	0.988631i	$0.548044\pi$
$510$	0	0
$511$	−36.2487	−1.60355
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.535898	0.0236145
$516$	0	0
$517$	50.0526	2.20131
$518$	0	0
$519$	0	0
$520$	0	0
$521$	19.4641	0.852738	0.426369	−	0.904549i	$-0.359793\pi$
$521$	19.4641	0.852738	0.426369	+	0.904549i	$0.359793\pi$
$522$	0	0
$523$	−22.2487	−0.972868	−0.486434	−	0.873717i	$-0.661703\pi$
$523$	−22.2487	−0.972868	−0.486434	+	0.873717i	$0.661703\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−8.19615	−0.357030
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−10.5359	−0.456360
$534$	0	0
$535$	8.53590	0.369039
$536$	0	0
$537$	0	0
$538$	0	0
$539$	88.2295	3.80031
$540$	0	0
$541$	−24.4641	−1.05179	−0.525897	−	0.850548i	$-0.676270\pi$
$541$	−24.4641	−1.05179	−0.525897	+	0.850548i	$0.676270\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−6.07180	−0.260087
$546$	0	0
$547$	33.8564	1.44760	0.723798	−	0.690012i	$-0.242395\pi$
$547$	33.8564	1.44760	0.723798	+	0.690012i	$0.242395\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−14.2679	−0.607835
$552$	0	0
$553$	−73.1769	−3.11180
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−2.53590	−0.107449	−0.0537247	−	0.998556i	$-0.517109\pi$
$557$	−2.53590	−0.107449	−0.0537247	+	0.998556i	$0.517109\pi$
$558$	0	0
$559$	−0.287187	−0.0121467
$560$	0	0
$561$	0	0
$562$	0	0
$563$	16.7321	0.705172	0.352586	−	0.935779i	$-0.385302\pi$
$563$	16.7321	0.705172	0.352586	+	0.935779i	$0.385302\pi$
$564$	0	0
$565$	−19.1244	−0.804568
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−32.9090	−1.37962	−0.689808	−	0.723993i	$-0.742305\pi$
$569$	−32.9090	−1.37962	−0.689808	+	0.723993i	$0.742305\pi$
$570$	0	0
$571$	39.7846	1.66493	0.832467	−	0.554075i	$-0.186928\pi$
$571$	39.7846	1.66493	0.832467	+	0.554075i	$0.186928\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	3.46410	0.144463
$576$	0	0
$577$	15.2679	0.635613	0.317807	−	0.948156i	$-0.397054\pi$
$577$	15.2679	0.635613	0.317807	+	0.948156i	$0.397054\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−10.3923	−0.431145
$582$	0	0
$583$	38.5885	1.59817
$584$	0	0
$585$	0	0
$586$	0	0
$587$	11.6603	0.481270	0.240635	−	0.970616i	$-0.422644\pi$
$587$	11.6603	0.481270	0.240635	+	0.970616i	$0.422644\pi$
$588$	0	0
$589$	−13.3923	−0.551820
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−0.143594	−0.00589668	−0.00294834	−	0.999996i	$-0.500938\pi$
$593$	−0.143594	−0.00589668	−0.00294834	+	0.999996i	$0.500938\pi$
$594$	0	0
$595$	−12.9282	−0.530005
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−27.1962	−1.11120	−0.555602	−	0.831448i	$-0.687512\pi$
$599$	−27.1962	−1.11120	−0.555602	+	0.831448i	$0.687512\pi$
$600$	0	0
$601$	−31.2487	−1.27466	−0.637331	−	0.770590i	$-0.719961\pi$
$601$	−31.2487	−1.27466	−0.637331	+	0.770590i	$0.719961\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	21.8564	0.888589
$606$	0	0
$607$	16.1962	0.657382	0.328691	−	0.944438i	$-0.393393\pi$
$607$	16.1962	0.657382	0.328691	+	0.944438i	$0.393393\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	12.7846	0.517210
$612$	0	0
$613$	1.46410	0.0591345	0.0295673	−	0.999563i	$-0.490587\pi$
$613$	1.46410	0.0591345	0.0295673	+	0.999563i	$0.490587\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−6.92820	−0.278919	−0.139459	−	0.990228i	$-0.544536\pi$
$617$	−6.92820	−0.278919	−0.139459	+	0.990228i	$0.544536\pi$
$618$	0	0
$619$	11.8564	0.476549	0.238275	−	0.971198i	$-0.423418\pi$
$619$	11.8564	0.476549	0.238275	+	0.971198i	$0.423418\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	24.5885	0.985116
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	7.46410	0.297613
$630$	0	0
$631$	32.7128	1.30228	0.651138	−	0.758959i	$-0.274292\pi$
$631$	32.7128	1.30228	0.651138	+	0.758959i	$0.274292\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	14.5885	0.578925
$636$	0	0
$637$	22.5359	0.892905
$638$	0	0
$639$	0	0
$640$	0	0
$641$	9.33975	0.368898	0.184449	−	0.982842i	$-0.440950\pi$
$641$	9.33975	0.368898	0.184449	+	0.982842i	$0.440950\pi$
$642$	0	0
$643$	−41.6603	−1.64292	−0.821460	−	0.570266i	$-0.806840\pi$
$643$	−41.6603	−1.64292	−0.821460	+	0.570266i	$0.806840\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−11.4641	−0.450700	−0.225350	−	0.974278i	$-0.572353\pi$
$647$	−11.4641	−0.450700	−0.225350	+	0.974278i	$0.572353\pi$
$648$	0	0
$649$	47.3923	1.86031
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17.4641	−0.683423	−0.341712	−	0.939805i	$-0.611007\pi$
$653$	−17.4641	−0.683423	−0.341712	+	0.939805i	$0.611007\pi$
$654$	0	0
$655$	−15.5885	−0.609091
$656$	0	0
$657$	0	0
$658$	0	0
$659$	1.46410	0.0570333	0.0285167	−	0.999593i	$-0.490922\pi$
$659$	1.46410	0.0570333	0.0285167	+	0.999593i	$0.490922\pi$
$660$	0	0
$661$	18.3205	0.712585	0.356293	−	0.934374i	$-0.384041\pi$
$661$	18.3205	0.712585	0.356293	+	0.934374i	$0.384041\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−21.1244	−0.819167
$666$	0	0
$667$	11.0718	0.428702
$668$	0	0
$669$	0	0
$670$	0	0
$671$	22.9282	0.885133
$672$	0	0
$673$	−10.3923	−0.400594	−0.200297	−	0.979735i	$-0.564191\pi$
$673$	−10.3923	−0.400594	−0.200297	+	0.979735i	$0.564191\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18.0000	−0.691796	−0.345898	−	0.938272i	$-0.612426\pi$
$677$	−18.0000	−0.691796	−0.345898	+	0.938272i	$0.612426\pi$
$678$	0	0
$679$	−45.7128	−1.75430
$680$	0	0
$681$	0	0
$682$	0	0
$683$	19.6077	0.750268	0.375134	−	0.926971i	$-0.377597\pi$
$683$	19.6077	0.750268	0.375134	+	0.926971i	$0.377597\pi$
$684$	0	0
$685$	2.53590	0.0968917
$686$	0	0
$687$	0	0
$688$	0	0
$689$	9.85641	0.375499
$690$	0	0
$691$	17.7128	0.673827	0.336914	−	0.941536i	$-0.390617\pi$
$691$	17.7128	0.673827	0.336914	+	0.941536i	$0.390617\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−0.607695	−0.0230512
$696$	0	0
$697$	19.6603	0.744685
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−20.8038	−0.785750	−0.392875	−	0.919592i	$-0.628520\pi$
$701$	−20.8038	−0.785750	−0.392875	+	0.919592i	$0.628520\pi$
$702$	0	0
$703$	12.1962	0.459987
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−12.5885	−0.473438
$708$	0	0
$709$	−22.5359	−0.846353	−0.423177	−	0.906047i	$-0.639085\pi$
$709$	−22.5359	−0.846353	−0.423177	+	0.906047i	$0.639085\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	10.3923	0.389195
$714$	0	0
$715$	8.39230	0.313854
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−8.41154	−0.313698	−0.156849	−	0.987623i	$-0.550134\pi$
$719$	−8.41154	−0.313698	−0.156849	+	0.987623i	$0.550134\pi$
$720$	0	0
$721$	2.53590	0.0944418
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3.19615	0.118702
$726$	0	0
$727$	8.39230	0.311253	0.155627	−	0.987816i	$-0.450260\pi$
$727$	8.39230	0.311253	0.155627	+	0.987816i	$0.450260\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0.535898	0.0198209
$732$	0	0
$733$	34.7846	1.28480	0.642399	−	0.766370i	$-0.277939\pi$
$733$	34.7846	1.28480	0.642399	+	0.766370i	$0.277939\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.8564	−0.731420
$738$	0	0
$739$	−22.4641	−0.826355	−0.413178	−	0.910650i	$-0.635581\pi$
$739$	−22.4641	−0.826355	−0.413178	+	0.910650i	$0.635581\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−19.9090	−0.730389	−0.365195	−	0.930931i	$-0.618998\pi$
$743$	−19.9090	−0.730389	−0.365195	+	0.930931i	$0.618998\pi$
$744$	0	0
$745$	−8.00000	−0.293097
$746$	0	0
$747$	0	0
$748$	0	0
$749$	40.3923	1.47590
$750$	0	0
$751$	−48.7846	−1.78018	−0.890088	−	0.455789i	$-0.849357\pi$
$751$	−48.7846	−1.78018	−0.890088	+	0.455789i	$0.849357\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−5.39230	−0.196246
$756$	0	0
$757$	−9.17691	−0.333541	−0.166770	−	0.985996i	$-0.553334\pi$
$757$	−9.17691	−0.333541	−0.166770	+	0.985996i	$0.553334\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	23.4449	0.849876	0.424938	−	0.905223i	$-0.360296\pi$
$761$	23.4449	0.849876	0.424938	+	0.905223i	$0.360296\pi$
$762$	0	0
$763$	−28.7321	−1.04017
$764$	0	0
$765$	0	0
$766$	0	0
$767$	12.1051	0.437090
$768$	0	0
$769$	−9.53590	−0.343873	−0.171937	−	0.985108i	$-0.555002\pi$
$769$	−9.53590	−0.343873	−0.171937	+	0.985108i	$0.555002\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	1.51666	0.0545505	0.0272752	−	0.999628i	$-0.491317\pi$
$773$	1.51666	0.0545505	0.0272752	+	0.999628i	$0.491317\pi$
$774$	0	0
$775$	3.00000	0.107763
$776$	0	0
$777$	0	0
$778$	0	0
$779$	32.1244	1.15097
$780$	0	0
$781$	21.3923	0.765477
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−19.1244	−0.682578
$786$	0	0
$787$	48.0526	1.71289	0.856444	−	0.516239i	$-0.172668\pi$
$787$	48.0526	1.71289	0.856444	+	0.516239i	$0.172668\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−90.4974	−3.21772
$792$	0	0
$793$	5.85641	0.207967
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15.6077	−0.552853	−0.276426	−	0.961035i	$-0.589150\pi$
$797$	−15.6077	−0.552853	−0.276426	+	0.961035i	$0.589150\pi$
$798$	0	0
$799$	−23.8564	−0.843979
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−43.9090	−1.54951
$804$	0	0
$805$	16.3923	0.577753
$806$	0	0
$807$	0	0
$808$	0	0
$809$	45.4449	1.59776	0.798878	−	0.601493i	$-0.205427\pi$
$809$	45.4449	1.59776	0.798878	+	0.601493i	$0.205427\pi$
$810$	0	0
$811$	18.4641	0.648362	0.324181	−	0.945995i	$-0.394911\pi$
$811$	18.4641	0.648362	0.324181	+	0.945995i	$0.394911\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−12.7321	−0.445984
$816$	0	0
$817$	0.875644	0.0306349
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−37.7321	−1.31686	−0.658429	−	0.752643i	$-0.728779\pi$
$821$	−37.7321	−1.31686	−0.658429	+	0.752643i	$0.728779\pi$
$822$	0	0
$823$	−3.85641	−0.134426	−0.0672129	−	0.997739i	$-0.521411\pi$
$823$	−3.85641	−0.134426	−0.0672129	+	0.997739i	$0.521411\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	11.6077	0.403639	0.201820	−	0.979423i	$-0.435315\pi$
$827$	11.6077	0.403639	0.201820	+	0.979423i	$0.435315\pi$
$828$	0	0
$829$	−23.7846	−0.826074	−0.413037	−	0.910714i	$-0.635532\pi$
$829$	−23.7846	−0.826074	−0.413037	+	0.910714i	$0.635532\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−42.0526	−1.45703
$834$	0	0
$835$	17.6603	0.611158
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−33.1962	−1.14606	−0.573029	−	0.819535i	$-0.694232\pi$
$839$	−33.1962	−1.14606	−0.573029	+	0.819535i	$0.694232\pi$
$840$	0	0
$841$	−18.7846	−0.647745
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−10.8564	−0.373472
$846$	0	0
$847$	103.426	3.55375
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−9.46410	−0.324425
$852$	0	0
$853$	10.4833	0.358943	0.179471	−	0.983763i	$-0.442561\pi$
$853$	10.4833	0.358943	0.179471	+	0.983763i	$0.442561\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	44.4974	1.52000	0.760002	−	0.649921i	$-0.225198\pi$
$857$	44.4974	1.52000	0.760002	+	0.649921i	$0.225198\pi$
$858$	0	0
$859$	11.0000	0.375315	0.187658	−	0.982235i	$-0.439910\pi$
$859$	11.0000	0.375315	0.187658	+	0.982235i	$0.439910\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	23.1244	0.787162	0.393581	−	0.919290i	$-0.371236\pi$
$863$	23.1244	0.787162	0.393581	+	0.919290i	$0.371236\pi$
$864$	0	0
$865$	8.53590	0.290229
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−88.6410	−3.00694
$870$	0	0
$871$	−5.07180	−0.171851
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.73205	0.159973
$876$	0	0
$877$	33.4641	1.13000	0.565001	−	0.825090i	$-0.308876\pi$
$877$	33.4641	1.13000	0.565001	+	0.825090i	$0.308876\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−47.0526	−1.58524	−0.792620	−	0.609715i	$-0.791284\pi$
$881$	−47.0526	−1.58524	−0.792620	+	0.609715i	$0.791284\pi$
$882$	0	0
$883$	30.1962	1.01618	0.508091	−	0.861304i	$-0.330351\pi$
$883$	30.1962	1.01618	0.508091	+	0.861304i	$0.330351\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−33.8038	−1.13502	−0.567511	−	0.823366i	$-0.692094\pi$
$887$	−33.8038	−1.13502	−0.567511	+	0.823366i	$0.692094\pi$
$888$	0	0
$889$	69.0333	2.31530
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−38.9808	−1.30444
$894$	0	0
$895$	−8.12436	−0.271567
$896$	0	0
$897$	0	0
$898$	0	0
$899$	9.58846	0.319793
$900$	0	0
$901$	−18.3923	−0.612737
$902$	0	0
$903$	0	0
$904$	0	0
$905$	26.4641	0.879697
$906$	0	0
$907$	30.0000	0.996134	0.498067	−	0.867139i	$-0.334043\pi$
$907$	30.0000	0.996134	0.498067	+	0.867139i	$0.334043\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−23.5885	−0.781520	−0.390760	−	0.920493i	$-0.627788\pi$
$911$	−23.5885	−0.781520	−0.390760	+	0.920493i	$0.627788\pi$
$912$	0	0
$913$	−12.5885	−0.416617
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−73.7654	−2.43595
$918$	0	0
$919$	56.9615	1.87899	0.939494	−	0.342566i	$-0.111296\pi$
$919$	56.9615	1.87899	0.939494	+	0.342566i	$0.111296\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	5.46410	0.179853
$924$	0	0
$925$	−2.73205	−0.0898293
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−44.3731	−1.45583	−0.727917	−	0.685666i	$-0.759511\pi$
$929$	−44.3731	−1.45583	−0.727917	+	0.685666i	$0.759511\pi$
$930$	0	0
$931$	−68.7128	−2.25197
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15.6603	−0.512145
$936$	0	0
$937$	−4.14359	−0.135365	−0.0676827	−	0.997707i	$-0.521561\pi$
$937$	−4.14359	−0.135365	−0.0676827	+	0.997707i	$0.521561\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	35.1769	1.14673	0.573367	−	0.819298i	$-0.305637\pi$
$941$	35.1769	1.14673	0.573367	+	0.819298i	$0.305637\pi$
$942$	0	0
$943$	−24.9282	−0.811774
$944$	0	0
$945$	0	0
$946$	0	0
$947$	57.7128	1.87541	0.937707	−	0.347427i	$-0.112944\pi$
$947$	57.7128	1.87541	0.937707	+	0.347427i	$0.112944\pi$
$948$	0	0
$949$	−11.2154	−0.364067
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.3923	1.17886	0.589431	−	0.807819i	$-0.299352\pi$
$953$	36.3923	1.17886	0.589431	+	0.807819i	$0.299352\pi$
$954$	0	0
$955$	8.12436	0.262898
$956$	0	0
$957$	0	0
$958$	0	0
$959$	12.0000	0.387500
$960$	0	0
$961$	−22.0000	−0.709677
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−5.26795	−0.169581
$966$	0	0
$967$	25.8038	0.829796	0.414898	−	0.909868i	$-0.363817\pi$
$967$	25.8038	0.829796	0.414898	+	0.909868i	$0.363817\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−17.4449	−0.559832	−0.279916	−	0.960024i	$-0.590307\pi$
$971$	−17.4449	−0.559832	−0.279916	+	0.960024i	$0.590307\pi$
$972$	0	0
$973$	−2.87564	−0.0921889
$974$	0	0
$975$	0	0
$976$	0	0
$977$	5.46410	0.174812	0.0874060	−	0.996173i	$-0.472142\pi$
$977$	5.46410	0.174812	0.0874060	+	0.996173i	$0.472142\pi$
$978$	0	0
$979$	29.7846	0.951920
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−48.5885	−1.54973	−0.774866	−	0.632126i	$-0.782182\pi$
$983$	−48.5885	−1.54973	−0.774866	+	0.632126i	$0.782182\pi$
$984$	0	0
$985$	−13.8564	−0.441502
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−0.679492	−0.0216066
$990$	0	0
$991$	−30.8564	−0.980186	−0.490093	−	0.871670i	$-0.663037\pi$
$991$	−30.8564	−0.980186	−0.490093	+	0.871670i	$0.663037\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	2.00000	0.0634043
$996$	0	0
$997$	25.5167	0.808121	0.404060	−	0.914732i	$-0.367599\pi$
$997$	25.5167	0.808121	0.404060	+	0.914732i	$0.367599\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6480.2.a.br.1.2		2
3.2	odd	2		6480.2.a.bi.1.2		2
4.3	odd	2		405.2.a.g.1.2	✓	2
12.11	even	2		405.2.a.h.1.1	yes	2
20.3	even	4		2025.2.b.g.649.2		4
20.7	even	4		2025.2.b.g.649.3		4
20.19	odd	2		2025.2.a.m.1.1		2
36.7	odd	6		405.2.e.l.271.1		4
36.11	even	6		405.2.e.i.271.2		4
36.23	even	6		405.2.e.i.136.2		4
36.31	odd	6		405.2.e.l.136.1		4
60.23	odd	4		2025.2.b.h.649.3		4
60.47	odd	4		2025.2.b.h.649.2		4
60.59	even	2		2025.2.a.g.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
405.2.a.g.1.2	✓	2	4.3	odd	2
405.2.a.h.1.1	yes	2	12.11	even	2
405.2.e.i.136.2		4	36.23	even	6
405.2.e.i.271.2		4	36.11	even	6
405.2.e.l.136.1		4	36.31	odd	6
405.2.e.l.271.1		4	36.7	odd	6
2025.2.a.g.1.2		2	60.59	even	2
2025.2.a.m.1.1		2	20.19	odd	2
2025.2.b.g.649.2		4	20.3	even	4
2025.2.b.g.649.3		4	20.7	even	4
2025.2.b.h.649.2		4	60.47	odd	4
2025.2.b.h.649.3		4	60.23	odd	4
6480.2.a.bi.1.2		2	3.2	odd	2
6480.2.a.br.1.2		2	1.1	even	1	trivial

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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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +4.73205 q^{7} +5.73205 q^{11} +1.46410 q^{13} -2.73205 q^{17} -4.46410 q^{19} +3.46410 q^{23} +1.00000 q^{25} +3.19615 q^{29} +3.00000 q^{31} +4.73205 q^{35} -2.73205 q^{37} -7.19615 q^{41} -0.196152 q^{43} +8.73205 q^{47} +15.3923 q^{49} +6.73205 q^{53} +5.73205 q^{55} +8.26795 q^{59} +4.00000 q^{61} +1.46410 q^{65} -3.46410 q^{67} +3.73205 q^{71} -7.66025 q^{73} +27.1244 q^{77} -15.4641 q^{79} -2.19615 q^{83} -2.73205 q^{85} +5.19615 q^{89} +6.92820 q^{91} -4.46410 q^{95} -9.66025 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{5} + 6 q^{7} + 8 q^{11} - 4 q^{13} - 2 q^{17} - 2 q^{19} + 2 q^{25} - 4 q^{29} + 6 q^{31} + 6 q^{35} - 2 q^{37} - 4 q^{41} + 10 q^{43} + 14 q^{47} + 10 q^{49} + 10 q^{53} + 8 q^{55} + 20 q^{59}+ \cdots - 2 q^{97}+O(q^{100}) \) 2 * q + 2 * q^5 + 6 * q^7 + 8 * q^11 - 4 * q^13 - 2 * q^17 - 2 * q^19 + 2 * q^25 - 4 * q^29 + 6 * q^31 + 6 * q^35 - 2 * q^37 - 4 * q^41 + 10 * q^43 + 14 * q^47 + 10 * q^49 + 10 * q^53 + 8 * q^55 + 20 * q^59 + 8 * q^61 - 4 * q^65 + 4 * q^71 + 2 * q^73 + 30 * q^77 - 24 * q^79 + 6 * q^83 - 2 * q^85 - 2 * q^95 - 2 * q^97

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