LMFDB - Embedded newform 3240.2.a.r.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1,3240)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(3240, 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("3240.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.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:	$25.8715302549$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 360)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	3240.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} +1.43807 q^{7} +O(q^{10})$	$q+1.00000 q^{5} +1.43807 q^{7} -1.35194 q^{11} -5.52420 q^{13} +4.82032 q^{17} -0.648061 q^{19} -8.90645 q^{23} +1.00000 q^{25} -7.17226 q^{29} -4.64806 q^{31} +1.43807 q^{35} +1.35194 q^{37} +0.351939 q^{41} +4.82032 q^{43} -9.49389 q^{47} -4.93196 q^{49} -8.17226 q^{53} -1.35194 q^{55} +1.46838 q^{59} +6.69646 q^{61} -5.52420 q^{65} +12.4307 q^{67} +2.22808 q^{71} -4.34452 q^{73} -1.94418 q^{77} -13.0484 q^{79} -5.26581 q^{83} +4.82032 q^{85} -11.0000 q^{89} -7.94418 q^{91} -0.648061 q^{95} +17.5800 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 5 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 5 * q^7	$ 3 q + 3 q^{5} - 5 q^{7} - 2 q^{11} + 2 q^{17} - 4 q^{19} - 7 q^{23} + 3 q^{25} - 7 q^{29} - 16 q^{31} - 5 q^{35} + 2 q^{37} - q^{41} + 2 q^{43} - 13 q^{47} + 10 q^{49} - 10 q^{53} - 2 q^{55} - 6 q^{59} - 11 q^{61} + q^{67} - 14 q^{71} + 16 q^{73} - 12 q^{77} - 6 q^{79} - 21 q^{83} + 2 q^{85} - 33 q^{89} - 30 q^{91} - 4 q^{95} + 30 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 5 * q^7 - 2 * q^11 + 2 * q^17 - 4 * q^19 - 7 * q^23 + 3 * q^25 - 7 * q^29 - 16 * q^31 - 5 * q^35 + 2 * q^37 - q^41 + 2 * q^43 - 13 * q^47 + 10 * q^49 - 10 * q^53 - 2 * q^55 - 6 * q^59 - 11 * q^61 + q^67 - 14 * q^71 + 16 * q^73 - 12 * q^77 - 6 * q^79 - 21 * q^83 + 2 * q^85 - 33 * q^89 - 30 * q^91 - 4 * q^95 + 30 * 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
$5$	1.00000	0.447214
$5$	1.00000	0.447214
$6$	0	0
$7$	1.43807	0.543539	0.271770	−	0.962362i	$-0.412391\pi$
$7$	1.43807	0.543539	0.271770	+	0.962362i	$0.412391\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−1.35194	−0.407625	−0.203813	−	0.979010i	$-0.565333\pi$
$11$	−1.35194	−0.407625	−0.203813	+	0.979010i	$0.565333\pi$
$12$	0	0
$13$	−5.52420	−1.53214	−0.766069	−	0.642759i	$-0.777790\pi$
$13$	−5.52420	−1.53214	−0.766069	+	0.642759i	$0.777790\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.82032	1.16910	0.584550	−	0.811358i	$-0.301271\pi$
$17$	4.82032	1.16910	0.584550	+	0.811358i	$0.301271\pi$
$18$	0	0
$19$	−0.648061	−0.148675	−0.0743377	−	0.997233i	$-0.523684\pi$
$19$	−0.648061	−0.148675	−0.0743377	+	0.997233i	$0.523684\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−8.90645	−1.85712	−0.928562	−	0.371178i	$-0.878954\pi$
$23$	−8.90645	−1.85712	−0.928562	+	0.371178i	$0.878954\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−7.17226	−1.33186	−0.665928	−	0.746016i	$-0.731964\pi$
$29$	−7.17226	−1.33186	−0.665928	+	0.746016i	$0.731964\pi$
$30$	0	0
$31$	−4.64806	−0.834816	−0.417408	−	0.908719i	$-0.637061\pi$
$31$	−4.64806	−0.834816	−0.417408	+	0.908719i	$0.637061\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.43807	0.243078
$36$	0	0
$37$	1.35194	0.222257	0.111129	−	0.993806i	$-0.464553\pi$
$37$	1.35194	0.222257	0.111129	+	0.993806i	$0.464553\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	0.351939	0.0549637	0.0274818	−	0.999622i	$-0.491251\pi$
$41$	0.351939	0.0549637	0.0274818	+	0.999622i	$0.491251\pi$
$42$	0	0
$43$	4.82032	0.735092	0.367546	−	0.930005i	$-0.380198\pi$
$43$	4.82032	0.735092	0.367546	+	0.930005i	$0.380198\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−9.49389	−1.38483	−0.692413	−	0.721501i	$-0.743452\pi$
$47$	−9.49389	−1.38483	−0.692413	+	0.721501i	$0.743452\pi$
$48$	0	0
$49$	−4.93196	−0.704565
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−8.17226	−1.12255	−0.561273	−	0.827631i	$-0.689688\pi$
$53$	−8.17226	−1.12255	−0.561273	+	0.827631i	$0.689688\pi$
$54$	0	0
$55$	−1.35194	−0.182295
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.46838	0.191167	0.0955835	−	0.995421i	$-0.469528\pi$
$59$	1.46838	0.191167	0.0955835	+	0.995421i	$0.469528\pi$
$60$	0	0
$61$	6.69646	0.857394	0.428697	−	0.903448i	$-0.358973\pi$
$61$	6.69646	0.857394	0.428697	+	0.903448i	$0.358973\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.52420	−0.685193
$66$	0	0
$67$	12.4307	1.51865	0.759323	−	0.650714i	$-0.225530\pi$
$67$	12.4307	1.51865	0.759323	+	0.650714i	$0.225530\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	2.22808	0.264424	0.132212	−	0.991221i	$-0.457792\pi$
$71$	2.22808	0.264424	0.132212	+	0.991221i	$0.457792\pi$
$72$	0	0
$73$	−4.34452	−0.508488	−0.254244	−	0.967140i	$-0.581827\pi$
$73$	−4.34452	−0.508488	−0.254244	+	0.967140i	$0.581827\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−1.94418	−0.221560
$78$	0	0
$79$	−13.0484	−1.46806	−0.734030	−	0.679117i	$-0.762363\pi$
$79$	−13.0484	−1.46806	−0.734030	+	0.679117i	$0.762363\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−5.26581	−0.577998	−0.288999	−	0.957329i	$-0.593322\pi$
$83$	−5.26581	−0.577998	−0.288999	+	0.957329i	$0.593322\pi$
$84$	0	0
$85$	4.82032	0.522837
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−11.0000	−1.16600	−0.582999	−	0.812473i	$-0.698121\pi$
$89$	−11.0000	−1.16600	−0.582999	+	0.812473i	$0.698121\pi$
$90$	0	0
$91$	−7.94418	−0.832777
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−0.648061	−0.0664896
$96$	0	0
$97$	17.5800	1.78498	0.892490	−	0.451067i	$-0.148956\pi$
$97$	17.5800	1.78498	0.892490	+	0.451067i	$0.148956\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	2.87614	0.286187	0.143093	−	0.989709i	$-0.454295\pi$
$101$	2.87614	0.286187	0.143093	+	0.989709i	$0.454295\pi$
$102$	0	0
$103$	−15.8687	−1.56359	−0.781796	−	0.623535i	$-0.785696\pi$
$103$	−15.8687	−1.56359	−0.781796	+	0.623535i	$0.785696\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0.314208	0.0303757	0.0151878	−	0.999885i	$-0.495165\pi$
$107$	0.314208	0.0303757	0.0151878	+	0.999885i	$0.495165\pi$
$108$	0	0
$109$	15.9320	1.52600	0.763002	−	0.646396i	$-0.223724\pi$
$109$	15.9320	1.52600	0.763002	+	0.646396i	$0.223724\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.51678	0.613047	0.306524	−	0.951863i	$-0.400834\pi$
$113$	6.51678	0.613047	0.306524	+	0.951863i	$0.400834\pi$
$114$	0	0
$115$	−8.90645	−0.830531
$116$	0	0
$117$	0	0
$118$	0	0
$119$	6.93196	0.635451
$120$	0	0
$121$	−9.17226	−0.833842
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−8.19777	−0.727434	−0.363717	−	0.931509i	$-0.618492\pi$
$127$	−8.19777	−0.727434	−0.363717	+	0.931509i	$0.618492\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−7.29612	−0.637465	−0.318733	−	0.947845i	$-0.603257\pi$
$131$	−7.29612	−0.637465	−0.318733	+	0.947845i	$0.603257\pi$
$132$	0	0
$133$	−0.931956	−0.0808109
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−12.8761	−1.10008	−0.550041	−	0.835137i	$-0.685388\pi$
$137$	−12.8761	−1.10008	−0.550041	+	0.835137i	$0.685388\pi$
$138$	0	0
$139$	5.40776	0.458680	0.229340	−	0.973346i	$-0.426343\pi$
$139$	5.40776	0.458680	0.229340	+	0.973346i	$0.426343\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	7.46838	0.624537
$144$	0	0
$145$	−7.17226	−0.595624
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−2.88356	−0.236230	−0.118115	−	0.993000i	$-0.537685\pi$
$149$	−2.88356	−0.236230	−0.118115	+	0.993000i	$0.537685\pi$
$150$	0	0
$151$	−10.6481	−0.866527	−0.433263	−	0.901267i	$-0.642638\pi$
$151$	−10.6481	−0.866527	−0.433263	+	0.901267i	$0.642638\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−4.64806	−0.373341
$156$	0	0
$157$	16.2839	1.29960	0.649798	−	0.760107i	$-0.274853\pi$
$157$	16.2839	1.29960	0.649798	+	0.760107i	$0.274853\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−12.8081	−1.00942
$162$	0	0
$163$	14.1164	1.10569	0.552843	−	0.833286i	$-0.313543\pi$
$163$	14.1164	1.10569	0.552843	+	0.833286i	$0.313543\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−18.6029	−1.43954	−0.719768	−	0.694214i	$-0.755752\pi$
$167$	−18.6029	−1.43954	−0.719768	+	0.694214i	$0.755752\pi$
$168$	0	0
$169$	17.5168	1.34744
$170$	0	0
$171$	0	0
$172$	0	0
$173$	24.8007	1.88556	0.942780	−	0.333415i	$-0.108201\pi$
$173$	24.8007	1.88556	0.942780	+	0.333415i	$0.108201\pi$
$174$	0	0
$175$	1.43807	0.108708
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−15.8687	−1.18608	−0.593042	−	0.805172i	$-0.702073\pi$
$179$	−15.8687	−1.18608	−0.593042	+	0.805172i	$0.702073\pi$
$180$	0	0
$181$	11.5168	0.856036	0.428018	−	0.903770i	$-0.359212\pi$
$181$	11.5168	0.856036	0.428018	+	0.903770i	$0.359212\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	1.35194	0.0993965
$186$	0	0
$187$	−6.51678	−0.476554
$188$	0	0
$189$	0	0
$190$	0	0
$191$	0.228078	0.0165031	0.00825157	−	0.999966i	$-0.497373\pi$
$191$	0.228078	0.0165031	0.00825157	+	0.999966i	$0.497373\pi$
$192$	0	0
$193$	25.8687	1.86207	0.931036	−	0.364928i	$-0.118907\pi$
$193$	25.8687	1.86207	0.931036	+	0.364928i	$0.118907\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−0.419983	−0.0299225	−0.0149613	−	0.999888i	$-0.504762\pi$
$197$	−0.419983	−0.0299225	−0.0149613	+	0.999888i	$0.504762\pi$
$198$	0	0
$199$	12.0410	0.853562	0.426781	−	0.904355i	$-0.359647\pi$
$199$	12.0410	0.853562	0.426781	+	0.904355i	$0.359647\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−10.3142	−0.723915
$204$	0	0
$205$	0.351939	0.0245805
$206$	0	0
$207$	0	0
$208$	0	0
$209$	0.876139	0.0606038
$210$	0	0
$211$	6.99258	0.481389	0.240695	−	0.970601i	$-0.422625\pi$
$211$	6.99258	0.481389	0.240695	+	0.970601i	$0.422625\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	4.82032	0.328743
$216$	0	0
$217$	−6.68423	−0.453755
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−26.6284	−1.79122
$222$	0	0
$223$	−14.0861	−0.943277	−0.471639	−	0.881792i	$-0.656337\pi$
$223$	−14.0861	−0.943277	−0.471639	+	0.881792i	$0.656337\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−23.4487	−1.55635	−0.778174	−	0.628049i	$-0.783854\pi$
$227$	−23.4487	−1.55635	−0.778174	+	0.628049i	$0.783854\pi$
$228$	0	0
$229$	−12.8129	−0.846700	−0.423350	−	0.905966i	$-0.639146\pi$
$229$	−12.8129	−0.846700	−0.423350	+	0.905966i	$0.639146\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−28.7449	−1.88314	−0.941569	−	0.336820i	$-0.890649\pi$
$233$	−28.7449	−1.88314	−0.941569	+	0.336820i	$0.890649\pi$
$234$	0	0
$235$	−9.49389	−0.619313
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−0.643253	−0.0416086	−0.0208043	−	0.999784i	$-0.506623\pi$
$239$	−0.643253	−0.0416086	−0.0208043	+	0.999784i	$0.506623\pi$
$240$	0	0
$241$	−20.8687	−1.34427	−0.672136	−	0.740428i	$-0.734623\pi$
$241$	−20.8687	−1.34427	−0.672136	+	0.740428i	$0.734623\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−4.93196	−0.315091
$246$	0	0
$247$	3.58002	0.227791
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−17.8687	−1.12786	−0.563932	−	0.825821i	$-0.690712\pi$
$251$	−17.8687	−1.12786	−0.563932	+	0.825821i	$0.690712\pi$
$252$	0	0
$253$	12.0410	0.757010
$254$	0	0
$255$	0	0
$256$	0	0
$257$	11.4078	0.711596	0.355798	−	0.934563i	$-0.384209\pi$
$257$	11.4078	0.711596	0.355798	+	0.934563i	$0.384209\pi$
$258$	0	0
$259$	1.94418	0.120806
$260$	0	0
$261$	0	0
$262$	0	0
$263$	0.475800	0.0293391	0.0146696	−	0.999892i	$-0.495330\pi$
$263$	0.475800	0.0293391	0.0146696	+	0.999892i	$0.495330\pi$
$264$	0	0
$265$	−8.17226	−0.502018
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−7.57260	−0.461709	−0.230855	−	0.972988i	$-0.574152\pi$
$269$	−7.57260	−0.461709	−0.230855	+	0.972988i	$0.574152\pi$
$270$	0	0
$271$	−19.6965	−1.19647	−0.598237	−	0.801319i	$-0.704132\pi$
$271$	−19.6965	−1.19647	−0.598237	+	0.801319i	$0.704132\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1.35194	−0.0815250
$276$	0	0
$277$	−21.5094	−1.29237	−0.646186	−	0.763180i	$-0.723637\pi$
$277$	−21.5094	−1.29237	−0.646186	+	0.763180i	$0.723637\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−10.1648	−0.606384	−0.303192	−	0.952930i	$-0.598052\pi$
$281$	−10.1648	−0.606384	−0.303192	+	0.952930i	$0.598052\pi$
$282$	0	0
$283$	−22.4865	−1.33668	−0.668341	−	0.743855i	$-0.732995\pi$
$283$	−22.4865	−1.33668	−0.668341	+	0.743855i	$0.732995\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	0.506113	0.0298749
$288$	0	0
$289$	6.23550	0.366794
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5.99519	−0.350243	−0.175121	−	0.984547i	$-0.556032\pi$
$293$	−5.99519	−0.350243	−0.175121	+	0.984547i	$0.556032\pi$
$294$	0	0
$295$	1.46838	0.0854925
$296$	0	0
$297$	0	0
$298$	0	0
$299$	49.2010	2.84537
$300$	0	0
$301$	6.93196	0.399551
$302$	0	0
$303$	0	0
$304$	0	0
$305$	6.69646	0.383438
$306$	0	0
$307$	−23.9549	−1.36718	−0.683588	−	0.729868i	$-0.739581\pi$
$307$	−23.9549	−1.36718	−0.683588	+	0.729868i	$0.739581\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	27.5094	1.55991	0.779956	−	0.625834i	$-0.215241\pi$
$311$	27.5094	1.55991	0.779956	+	0.625834i	$0.215241\pi$
$312$	0	0
$313$	3.00742	0.169989	0.0849947	−	0.996381i	$-0.472913\pi$
$313$	3.00742	0.169989	0.0849947	+	0.996381i	$0.472913\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	10.4003	0.584141	0.292071	−	0.956397i	$-0.405656\pi$
$317$	10.4003	0.584141	0.292071	+	0.956397i	$0.405656\pi$
$318$	0	0
$319$	9.69646	0.542898
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−3.12386	−0.173816
$324$	0	0
$325$	−5.52420	−0.306427
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−13.6529	−0.752707
$330$	0	0
$331$	−9.23550	−0.507629	−0.253814	−	0.967253i	$-0.581685\pi$
$331$	−9.23550	−0.507629	−0.253814	+	0.967253i	$0.581685\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	12.4307	0.679159
$336$	0	0
$337$	34.7497	1.89293	0.946467	−	0.322799i	$-0.104624\pi$
$337$	34.7497	1.89293	0.946467	+	0.322799i	$0.104624\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	6.28390	0.340292
$342$	0	0
$343$	−17.1590	−0.926498
$344$	0	0
$345$	0	0
$346$	0	0
$347$	21.6210	1.16068	0.580338	−	0.814376i	$-0.302920\pi$
$347$	21.6210	1.16068	0.580338	+	0.814376i	$0.302920\pi$
$348$	0	0
$349$	7.23289	0.387167	0.193584	−	0.981084i	$-0.437989\pi$
$349$	7.23289	0.387167	0.193584	+	0.981084i	$0.437989\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−23.2813	−1.23914	−0.619569	−	0.784942i	$-0.712693\pi$
$353$	−23.2813	−1.23914	−0.619569	+	0.784942i	$0.712693\pi$
$354$	0	0
$355$	2.22808	0.118254
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−31.9655	−1.68708	−0.843538	−	0.537070i	$-0.819531\pi$
$359$	−31.9655	−1.68708	−0.843538	+	0.537070i	$0.819531\pi$
$360$	0	0
$361$	−18.5800	−0.977896
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−4.34452	−0.227403
$366$	0	0
$367$	15.5242	0.810357	0.405178	−	0.914238i	$-0.367209\pi$
$367$	15.5242	0.810357	0.405178	+	0.914238i	$0.367209\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−11.7523	−0.610148
$372$	0	0
$373$	31.3323	1.62232	0.811162	−	0.584821i	$-0.198835\pi$
$373$	31.3323	1.62232	0.811162	+	0.584821i	$0.198835\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	39.6210	2.04059
$378$	0	0
$379$	22.0000	1.13006	0.565032	−	0.825069i	$-0.308864\pi$
$379$	22.0000	1.13006	0.565032	+	0.825069i	$0.308864\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	29.2765	1.49596	0.747979	−	0.663722i	$-0.231024\pi$
$383$	29.2765	1.49596	0.747979	+	0.663722i	$0.231024\pi$
$384$	0	0
$385$	−1.94418	−0.0990847
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−2.00742	−0.101780	−0.0508901	−	0.998704i	$-0.516206\pi$
$389$	−2.00742	−0.101780	−0.0508901	+	0.998704i	$0.516206\pi$
$390$	0	0
$391$	−42.9320	−2.17116
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−13.0484	−0.656536
$396$	0	0
$397$	16.9368	0.850032	0.425016	−	0.905186i	$-0.360268\pi$
$397$	16.9368	0.850032	0.425016	+	0.905186i	$0.360268\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	21.8129	1.08928	0.544642	−	0.838669i	$-0.316665\pi$
$401$	21.8129	1.08928	0.544642	+	0.838669i	$0.316665\pi$
$402$	0	0
$403$	25.6768	1.27905
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−1.82774	−0.0905977
$408$	0	0
$409$	−14.8007	−0.731846	−0.365923	−	0.930645i	$-0.619247\pi$
$409$	−14.8007	−0.731846	−0.365923	+	0.930645i	$0.619247\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	2.11164	0.103907
$414$	0	0
$415$	−5.26581	−0.258488
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−12.2887	−0.600342	−0.300171	−	0.953885i	$-0.597044\pi$
$419$	−12.2887	−0.600342	−0.300171	+	0.953885i	$0.597044\pi$
$420$	0	0
$421$	−13.3929	−0.652731	−0.326365	−	0.945244i	$-0.605824\pi$
$421$	−13.3929	−0.652731	−0.326365	+	0.945244i	$0.605824\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.82032	0.233820
$426$	0	0
$427$	9.62997	0.466027
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12.0968	0.582682	0.291341	−	0.956619i	$-0.405899\pi$
$431$	12.0968	0.582682	0.291341	+	0.956619i	$0.405899\pi$
$432$	0	0
$433$	21.1648	1.01712	0.508559	−	0.861027i	$-0.330178\pi$
$433$	21.1648	1.01712	0.508559	+	0.861027i	$0.330178\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	5.77192	0.276108
$438$	0	0
$439$	−1.48322	−0.0707902	−0.0353951	−	0.999373i	$-0.511269\pi$
$439$	−1.48322	−0.0707902	−0.0353951	+	0.999373i	$0.511269\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−3.55451	−0.168880	−0.0844400	−	0.996429i	$-0.526910\pi$
$443$	−3.55451	−0.168880	−0.0844400	+	0.996429i	$0.526910\pi$
$444$	0	0
$445$	−11.0000	−0.521450
$446$	0	0
$447$	0	0
$448$	0	0
$449$	40.3659	1.90498	0.952491	−	0.304566i	$-0.0985114\pi$
$449$	40.3659	1.90498	0.952491	+	0.304566i	$0.0985114\pi$
$450$	0	0
$451$	−0.475800	−0.0224046
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−7.94418	−0.372429
$456$	0	0
$457$	28.4003	1.32851	0.664256	−	0.747505i	$-0.268749\pi$
$457$	28.4003	1.32851	0.664256	+	0.747505i	$0.268749\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	19.7497	0.919834	0.459917	−	0.887962i	$-0.347879\pi$
$461$	19.7497	0.919834	0.459917	+	0.887962i	$0.347879\pi$
$462$	0	0
$463$	0.177068	0.00822905	0.00411452	−	0.999992i	$-0.498690\pi$
$463$	0.177068	0.00822905	0.00411452	+	0.999992i	$0.498690\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	21.5094	0.995335	0.497667	−	0.867368i	$-0.334190\pi$
$467$	21.5094	0.995335	0.497667	+	0.867368i	$0.334190\pi$
$468$	0	0
$469$	17.8761	0.825443
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−6.51678	−0.299642
$474$	0	0
$475$	−0.648061	−0.0297351
$476$	0	0
$477$	0	0
$478$	0	0
$479$	1.31096	0.0598992	0.0299496	−	0.999551i	$-0.490465\pi$
$479$	1.31096	0.0598992	0.0299496	+	0.999551i	$0.490465\pi$
$480$	0	0
$481$	−7.46838	−0.340529
$482$	0	0
$483$	0	0
$484$	0	0
$485$	17.5800	0.798267
$486$	0	0
$487$	10.4152	0.471957	0.235978	−	0.971758i	$-0.424171\pi$
$487$	10.4152	0.471957	0.235978	+	0.971758i	$0.424171\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	0.187097	0.00844358	0.00422179	−	0.999991i	$-0.498656\pi$
$491$	0.187097	0.00844358	0.00422179	+	0.999991i	$0.498656\pi$
$492$	0	0
$493$	−34.5726	−1.55707
$494$	0	0
$495$	0	0
$496$	0	0
$497$	3.20413	0.143725
$498$	0	0
$499$	−33.7981	−1.51301	−0.756505	−	0.653988i	$-0.773095\pi$
$499$	−33.7981	−1.51301	−0.756505	+	0.653988i	$0.773095\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	20.9623	0.934661	0.467331	−	0.884083i	$-0.345216\pi$
$503$	20.9623	0.934661	0.467331	+	0.884083i	$0.345216\pi$
$504$	0	0
$505$	2.87614	0.127986
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−24.7374	−1.09647	−0.548234	−	0.836325i	$-0.684700\pi$
$509$	−24.7374	−1.09647	−0.548234	+	0.836325i	$0.684700\pi$
$510$	0	0
$511$	−6.24772	−0.276383
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15.8687	−0.699259
$516$	0	0
$517$	12.8352	0.564490
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−7.28390	−0.319113	−0.159557	−	0.987189i	$-0.551006\pi$
$521$	−7.28390	−0.319113	−0.159557	+	0.987189i	$0.551006\pi$
$522$	0	0
$523$	12.0255	0.525839	0.262919	−	0.964818i	$-0.415315\pi$
$523$	12.0255	0.525839	0.262919	+	0.964818i	$0.415315\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−22.4051	−0.975983
$528$	0	0
$529$	56.3249	2.44891
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−1.94418	−0.0842119
$534$	0	0
$535$	0.314208	0.0135844
$536$	0	0
$537$	0	0
$538$	0	0
$539$	6.66771	0.287198
$540$	0	0
$541$	−23.6890	−1.01847	−0.509236	−	0.860627i	$-0.670072\pi$
$541$	−23.6890	−1.01847	−0.509236	+	0.860627i	$0.670072\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	15.9320	0.682450
$546$	0	0
$547$	15.5497	0.664857	0.332429	−	0.943128i	$-0.392132\pi$
$547$	15.5497	0.664857	0.332429	+	0.943128i	$0.392132\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	4.64806	0.198014
$552$	0	0
$553$	−18.7645	−0.797948
$554$	0	0
$555$	0	0
$556$	0	0
$557$	12.2477	0.518953	0.259476	−	0.965749i	$-0.416450\pi$
$557$	12.2477	0.518953	0.259476	+	0.965749i	$0.416450\pi$
$558$	0	0
$559$	−26.6284	−1.12626
$560$	0	0
$561$	0	0
$562$	0	0
$563$	22.1978	0.935524	0.467762	−	0.883854i	$-0.345060\pi$
$563$	22.1978	0.935524	0.467762	+	0.883854i	$0.345060\pi$
$564$	0	0
$565$	6.51678	0.274163
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−20.8155	−0.872632	−0.436316	−	0.899794i	$-0.643717\pi$
$569$	−20.8155	−0.872632	−0.436316	+	0.899794i	$0.643717\pi$
$570$	0	0
$571$	−45.6014	−1.90836	−0.954179	−	0.299238i	$-0.903268\pi$
$571$	−45.6014	−1.90836	−0.954179	+	0.299238i	$0.903268\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−8.90645	−0.371425
$576$	0	0
$577$	−0.172260	−0.00717129	−0.00358565	−	0.999994i	$-0.501141\pi$
$577$	−0.172260	−0.00717129	−0.00358565	+	0.999994i	$0.501141\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−7.57260	−0.314164
$582$	0	0
$583$	11.0484	0.457578
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−12.4503	−0.513879	−0.256939	−	0.966428i	$-0.582714\pi$
$587$	−12.4503	−0.513879	−0.256939	+	0.966428i	$0.582714\pi$
$588$	0	0
$589$	3.01223	0.124117
$590$	0	0
$591$	0	0
$592$	0	0
$593$	13.2403	0.543714	0.271857	−	0.962338i	$-0.412362\pi$
$593$	13.2403	0.543714	0.271857	+	0.962338i	$0.412362\pi$
$594$	0	0
$595$	6.93196	0.284183
$596$	0	0
$597$	0	0
$598$	0	0
$599$	45.2813	1.85014	0.925072	−	0.379793i	$-0.124005\pi$
$599$	45.2813	1.85014	0.925072	+	0.379793i	$0.124005\pi$
$600$	0	0
$601$	−30.1116	−1.22828	−0.614140	−	0.789197i	$-0.710497\pi$
$601$	−30.1116	−1.22828	−0.614140	+	0.789197i	$0.710497\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−9.17226	−0.372905
$606$	0	0
$607$	−28.4307	−1.15396	−0.576982	−	0.816757i	$-0.695770\pi$
$607$	−28.4307	−1.15396	−0.576982	+	0.816757i	$0.695770\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	52.4461	2.12174
$612$	0	0
$613$	3.06804	0.123917	0.0619586	−	0.998079i	$-0.480265\pi$
$613$	3.06804	0.123917	0.0619586	+	0.998079i	$0.480265\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1.48322	0.0597121	0.0298561	−	0.999554i	$-0.490495\pi$
$617$	1.48322	0.0597121	0.0298561	+	0.999554i	$0.490495\pi$
$618$	0	0
$619$	−1.10422	−0.0443822	−0.0221911	−	0.999754i	$-0.507064\pi$
$619$	−1.10422	−0.0443822	−0.0221911	+	0.999754i	$0.507064\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−15.8188	−0.633765
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.51678	0.259841
$630$	0	0
$631$	−8.07546	−0.321479	−0.160740	−	0.986997i	$-0.551388\pi$
$631$	−8.07546	−0.321479	−0.160740	+	0.986997i	$0.551388\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−8.19777	−0.325318
$636$	0	0
$637$	27.2451	1.07949
$638$	0	0
$639$	0	0
$640$	0	0
$641$	0.539036	0.0212907	0.0106453	−	0.999943i	$-0.496611\pi$
$641$	0.539036	0.0212907	0.0106453	+	0.999943i	$0.496611\pi$
$642$	0	0
$643$	17.0229	0.671317	0.335659	−	0.941984i	$-0.391041\pi$
$643$	17.0229	0.671317	0.335659	+	0.941984i	$0.391041\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−45.1952	−1.77680	−0.888402	−	0.459065i	$-0.848184\pi$
$647$	−45.1952	−1.77680	−0.888402	+	0.459065i	$0.848184\pi$
$648$	0	0
$649$	−1.98516	−0.0779245
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−17.2403	−0.674665	−0.337333	−	0.941386i	$-0.609525\pi$
$653$	−17.2403	−0.674665	−0.337333	+	0.941386i	$0.609525\pi$
$654$	0	0
$655$	−7.29612	−0.285083
$656$	0	0
$657$	0	0
$658$	0	0
$659$	10.5168	0.409676	0.204838	−	0.978796i	$-0.434333\pi$
$659$	10.5168	0.409676	0.204838	+	0.978796i	$0.434333\pi$
$660$	0	0
$661$	−4.47099	−0.173901	−0.0869507	−	0.996213i	$-0.527712\pi$
$661$	−4.47099	−0.173901	−0.0869507	+	0.996213i	$0.527712\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−0.931956	−0.0361397
$666$	0	0
$667$	63.8794	2.47342
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.05321	−0.349495
$672$	0	0
$673$	4.87133	0.187776	0.0938880	−	0.995583i	$-0.470070\pi$
$673$	4.87133	0.187776	0.0938880	+	0.995583i	$0.470070\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	13.7113	0.526968	0.263484	−	0.964664i	$-0.415128\pi$
$677$	13.7113	0.526968	0.263484	+	0.964664i	$0.415128\pi$
$678$	0	0
$679$	25.2813	0.970207
$680$	0	0
$681$	0	0
$682$	0	0
$683$	31.8687	1.21942	0.609711	−	0.792624i	$-0.291285\pi$
$683$	31.8687	1.21942	0.609711	+	0.792624i	$0.291285\pi$
$684$	0	0
$685$	−12.8761	−0.491972
$686$	0	0
$687$	0	0
$688$	0	0
$689$	45.1452	1.71990
$690$	0	0
$691$	32.0968	1.22102	0.610510	−	0.792009i	$-0.290965\pi$
$691$	32.0968	1.22102	0.610510	+	0.792009i	$0.290965\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.40776	0.205128
$696$	0	0
$697$	1.69646	0.0642580
$698$	0	0
$699$	0	0
$700$	0	0
$701$	11.5268	0.435362	0.217681	−	0.976020i	$-0.430151\pi$
$701$	11.5268	0.435362	0.217681	+	0.976020i	$0.430151\pi$
$702$	0	0
$703$	−0.876139	−0.0330442
$704$	0	0
$705$	0	0
$706$	0	0
$707$	4.13609	0.155554
$708$	0	0
$709$	−25.2691	−0.948999	−0.474500	−	0.880256i	$-0.657371\pi$
$709$	−25.2691	−0.948999	−0.474500	+	0.880256i	$0.657371\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	41.3977	1.55036
$714$	0	0
$715$	7.46838	0.279302
$716$	0	0
$717$	0	0
$718$	0	0
$719$	20.6284	0.769310	0.384655	−	0.923060i	$-0.374320\pi$
$719$	20.6284	0.769310	0.384655	+	0.923060i	$0.374320\pi$
$720$	0	0
$721$	−22.8203	−0.849873
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−7.17226	−0.266371
$726$	0	0
$727$	23.9958	0.889956	0.444978	−	0.895541i	$-0.353211\pi$
$727$	23.9958	0.889956	0.444978	+	0.895541i	$0.353211\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	23.2355	0.859396
$732$	0	0
$733$	−9.28128	−0.342812	−0.171406	−	0.985200i	$-0.554831\pi$
$733$	−9.28128	−0.342812	−0.171406	+	0.985200i	$0.554831\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−16.8055	−0.619038
$738$	0	0
$739$	−31.3323	−1.15258	−0.576289	−	0.817246i	$-0.695500\pi$
$739$	−31.3323	−1.15258	−0.576289	+	0.817246i	$0.695500\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	25.8432	0.948096	0.474048	−	0.880499i	$-0.342792\pi$
$743$	25.8432	0.948096	0.474048	+	0.880499i	$0.342792\pi$
$744$	0	0
$745$	−2.88356	−0.105645
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0.451853	0.0165104
$750$	0	0
$751$	3.17968	0.116028	0.0580141	−	0.998316i	$-0.481523\pi$
$751$	3.17968	0.116028	0.0580141	+	0.998316i	$0.481523\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10.6481	−0.387523
$756$	0	0
$757$	45.0288	1.63660	0.818299	−	0.574793i	$-0.194917\pi$
$757$	45.0288	1.63660	0.818299	+	0.574793i	$0.194917\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	29.0213	1.05202	0.526011	−	0.850478i	$-0.323687\pi$
$761$	29.0213	1.05202	0.526011	+	0.850478i	$0.323687\pi$
$762$	0	0
$763$	22.9113	0.829443
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−8.11164	−0.292894
$768$	0	0
$769$	13.6136	0.490918	0.245459	−	0.969407i	$-0.421061\pi$
$769$	13.6136	0.490918	0.245459	+	0.969407i	$0.421061\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−17.7523	−0.638505	−0.319253	−	0.947670i	$-0.603432\pi$
$773$	−17.7523	−0.638505	−0.319253	+	0.947670i	$0.603432\pi$
$774$	0	0
$775$	−4.64806	−0.166963
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−0.228078	−0.00817174
$780$	0	0
$781$	−3.01223	−0.107786
$782$	0	0
$783$	0	0
$784$	0	0
$785$	16.2839	0.581197
$786$	0	0
$787$	55.8687	1.99150	0.995752	−	0.0920715i	$-0.0293488\pi$
$787$	55.8687	1.99150	0.995752	+	0.0920715i	$0.0293488\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9.37158	0.333215
$792$	0	0
$793$	−36.9926	−1.31365
$794$	0	0
$795$	0	0
$796$	0	0
$797$	41.5700	1.47248	0.736242	−	0.676718i	$-0.236598\pi$
$797$	41.5700	1.47248	0.736242	+	0.676718i	$0.236598\pi$
$798$	0	0
$799$	−45.7636	−1.61900
$800$	0	0
$801$	0	0
$802$	0	0
$803$	5.87353	0.207272
$804$	0	0
$805$	−12.8081	−0.451426
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17.6917	0.622005	0.311003	−	0.950409i	$-0.399335\pi$
$809$	17.6917	0.622005	0.311003	+	0.950409i	$0.399335\pi$
$810$	0	0
$811$	13.5438	0.475589	0.237794	−	0.971316i	$-0.423576\pi$
$811$	13.5438	0.475589	0.237794	+	0.971316i	$0.423576\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	14.1164	0.494477
$816$	0	0
$817$	−3.12386	−0.109290
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1.53643	−0.0536216	−0.0268108	−	0.999641i	$-0.508535\pi$
$821$	−1.53643	−0.0536216	−0.0268108	+	0.999641i	$0.508535\pi$
$822$	0	0
$823$	−8.86547	−0.309031	−0.154515	−	0.987990i	$-0.549382\pi$
$823$	−8.86547	−0.309031	−0.154515	+	0.987990i	$0.549382\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−14.4307	−0.501803	−0.250901	−	0.968013i	$-0.580727\pi$
$827$	−14.4307	−0.501803	−0.250901	+	0.968013i	$0.580727\pi$
$828$	0	0
$829$	−3.30354	−0.114737	−0.0573683	−	0.998353i	$-0.518271\pi$
$829$	−3.30354	−0.114737	−0.0573683	+	0.998353i	$0.518271\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−23.7736	−0.823707
$834$	0	0
$835$	−18.6029	−0.643780
$836$	0	0
$837$	0	0
$838$	0	0
$839$	0.0706544	0.00243926	0.00121963	−	0.999999i	$-0.499612\pi$
$839$	0.0706544	0.00243926	0.00121963	+	0.999999i	$0.499612\pi$
$840$	0	0
$841$	22.4413	0.773839
$842$	0	0
$843$	0	0
$844$	0	0
$845$	17.5168	0.602596
$846$	0	0
$847$	−13.1903	−0.453226
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−12.0410	−0.412760
$852$	0	0
$853$	15.2403	0.521818	0.260909	−	0.965363i	$-0.415978\pi$
$853$	15.2403	0.521818	0.260909	+	0.965363i	$0.415978\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	45.1404	1.54197	0.770983	−	0.636856i	$-0.219765\pi$
$857$	45.1404	1.54197	0.770983	+	0.636856i	$0.219765\pi$
$858$	0	0
$859$	51.0288	1.74108	0.870539	−	0.492099i	$-0.163770\pi$
$859$	51.0288	1.74108	0.870539	+	0.492099i	$0.163770\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.6784	−0.431577	−0.215788	−	0.976440i	$-0.569232\pi$
$863$	−12.6784	−0.431577	−0.215788	+	0.976440i	$0.569232\pi$
$864$	0	0
$865$	24.8007	0.843248
$866$	0	0
$867$	0	0
$868$	0	0
$869$	17.6406	0.598418
$870$	0	0
$871$	−68.6694	−2.32677
$872$	0	0
$873$	0	0
$874$	0	0
$875$	1.43807	0.0486156
$876$	0	0
$877$	−32.3493	−1.09236	−0.546180	−	0.837668i	$-0.683918\pi$
$877$	−32.3493	−1.09236	−0.546180	+	0.837668i	$0.683918\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−40.6816	−1.37060	−0.685299	−	0.728261i	$-0.740329\pi$
$881$	−40.6816	−1.37060	−0.685299	+	0.728261i	$0.740329\pi$
$882$	0	0
$883$	−10.3700	−0.348979	−0.174490	−	0.984659i	$-0.555828\pi$
$883$	−10.3700	−0.348979	−0.174490	+	0.984659i	$0.555828\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−10.4906	−0.352241	−0.176121	−	0.984369i	$-0.556355\pi$
$887$	−10.4906	−0.352241	−0.176121	+	0.984369i	$0.556355\pi$
$888$	0	0
$889$	−11.7890	−0.395389
$890$	0	0
$891$	0	0
$892$	0	0
$893$	6.15262	0.205889
$894$	0	0
$895$	−15.8687	−0.530433
$896$	0	0
$897$	0	0
$898$	0	0
$899$	33.3371	1.11185
$900$	0	0
$901$	−39.3929	−1.31237
$902$	0	0
$903$	0	0
$904$	0	0
$905$	11.5168	0.382831
$906$	0	0
$907$	1.11319	0.0369630	0.0184815	−	0.999829i	$-0.494117\pi$
$907$	1.11319	0.0369630	0.0184815	+	0.999829i	$0.494117\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−17.2355	−0.571037	−0.285519	−	0.958373i	$-0.592166\pi$
$911$	−17.2355	−0.571037	−0.285519	+	0.958373i	$0.592166\pi$
$912$	0	0
$913$	7.11905	0.235606
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−10.4923	−0.346487
$918$	0	0
$919$	28.4413	0.938193	0.469096	−	0.883147i	$-0.344580\pi$
$919$	28.4413	0.938193	0.469096	+	0.883147i	$0.344580\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−12.3083	−0.405134
$924$	0	0
$925$	1.35194	0.0444515
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−58.9219	−1.93317	−0.966583	−	0.256354i	$-0.917479\pi$
$929$	−58.9219	−1.93317	−0.966583	+	0.256354i	$0.917479\pi$
$930$	0	0
$931$	3.19621	0.104751
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−6.51678	−0.213122
$936$	0	0
$937$	−32.0458	−1.04689	−0.523445	−	0.852059i	$-0.675353\pi$
$937$	−32.0458	−1.04689	−0.523445	+	0.852059i	$0.675353\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	22.3323	0.728012	0.364006	−	0.931397i	$-0.381409\pi$
$941$	22.3323	0.728012	0.364006	+	0.931397i	$0.381409\pi$
$942$	0	0
$943$	−3.13453	−0.102074
$944$	0	0
$945$	0	0
$946$	0	0
$947$	38.1829	1.24078	0.620389	−	0.784294i	$-0.286975\pi$
$947$	38.1829	1.24078	0.620389	+	0.784294i	$0.286975\pi$
$948$	0	0
$949$	24.0000	0.779073
$950$	0	0
$951$	0	0
$952$	0	0
$953$	14.1771	0.459240	0.229620	−	0.973280i	$-0.426252\pi$
$953$	14.1771	0.459240	0.229620	+	0.973280i	$0.426252\pi$
$954$	0	0
$955$	0.228078	0.00738043
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−18.5168	−0.597938
$960$	0	0
$961$	−9.39553	−0.303082
$962$	0	0
$963$	0	0
$964$	0	0
$965$	25.8687	0.832744
$966$	0	0
$967$	−23.4084	−0.752763	−0.376382	−	0.926465i	$-0.622832\pi$
$967$	−23.4084	−0.752763	−0.376382	+	0.926465i	$0.622832\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−32.7300	−1.05036	−0.525178	−	0.850992i	$-0.676001\pi$
$971$	−32.7300	−1.05036	−0.525178	+	0.850992i	$0.676001\pi$
$972$	0	0
$973$	7.77673	0.249311
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−29.5604	−0.945720	−0.472860	−	0.881138i	$-0.656778\pi$
$977$	−29.5604	−0.945720	−0.472860	+	0.881138i	$0.656778\pi$
$978$	0	0
$979$	14.8713	0.475290
$980$	0	0
$981$	0	0
$982$	0	0
$983$	45.6465	1.45590	0.727949	−	0.685632i	$-0.240474\pi$
$983$	45.6465	1.45590	0.727949	+	0.685632i	$0.240474\pi$
$984$	0	0
$985$	−0.419983	−0.0133818
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−42.9320	−1.36516
$990$	0	0
$991$	−45.9245	−1.45884	−0.729421	−	0.684066i	$-0.760210\pi$
$991$	−45.9245	−1.45884	−0.729421	+	0.684066i	$0.760210\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	12.0410	0.381725
$996$	0	0
$997$	−28.7401	−0.910207	−0.455103	−	0.890439i	$-0.650398\pi$
$997$	−28.7401	−0.910207	−0.455103	+	0.890439i	$0.650398\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.a.r.1.3		3
3.2	odd	2		3240.2.a.q.1.3		3
4.3	odd	2		6480.2.a.bx.1.1		3
9.2	odd	6		1080.2.q.d.361.1		6
9.4	even	3		360.2.q.d.241.1	yes	6
9.5	odd	6		1080.2.q.d.721.1		6
9.7	even	3		360.2.q.d.121.1	✓	6
12.11	even	2		6480.2.a.bu.1.1		3
36.7	odd	6		720.2.q.j.481.3		6
36.11	even	6		2160.2.q.j.1441.3		6
36.23	even	6		2160.2.q.j.721.3		6
36.31	odd	6		720.2.q.j.241.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.d.121.1	✓	6	9.7	even	3
360.2.q.d.241.1	yes	6	9.4	even	3
720.2.q.j.241.3		6	36.31	odd	6
720.2.q.j.481.3		6	36.7	odd	6
1080.2.q.d.361.1		6	9.2	odd	6
1080.2.q.d.721.1		6	9.5	odd	6
2160.2.q.j.721.3		6	36.23	even	6
2160.2.q.j.1441.3		6	36.11	even	6
3240.2.a.q.1.3		3	3.2	odd	2
3240.2.a.r.1.3		3	1.1	even	1	trivial
6480.2.a.bu.1.1		3	12.11	even	2
6480.2.a.bx.1.1		3	4.3	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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} +1.43807 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} +1.43807 q^{7} -1.35194 q^{11} -5.52420 q^{13} +4.82032 q^{17} -0.648061 q^{19} -8.90645 q^{23} +1.00000 q^{25} -7.17226 q^{29} -4.64806 q^{31} +1.43807 q^{35} +1.35194 q^{37} +0.351939 q^{41} +4.82032 q^{43} -9.49389 q^{47} -4.93196 q^{49} -8.17226 q^{53} -1.35194 q^{55} +1.46838 q^{59} +6.69646 q^{61} -5.52420 q^{65} +12.4307 q^{67} +2.22808 q^{71} -4.34452 q^{73} -1.94418 q^{77} -13.0484 q^{79} -5.26581 q^{83} +4.82032 q^{85} -11.0000 q^{89} -7.94418 q^{91} -0.648061 q^{95} +17.5800 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 5 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 5 * q^7	\( 3 q + 3 q^{5} - 5 q^{7} - 2 q^{11} + 2 q^{17} - 4 q^{19} - 7 q^{23} + 3 q^{25} - 7 q^{29} - 16 q^{31} - 5 q^{35} + 2 q^{37} - q^{41} + 2 q^{43} - 13 q^{47} + 10 q^{49} - 10 q^{53} - 2 q^{55} - 6 q^{59} - 11 q^{61} + q^{67} - 14 q^{71} + 16 q^{73} - 12 q^{77} - 6 q^{79} - 21 q^{83} + 2 q^{85} - 33 q^{89} - 30 q^{91} - 4 q^{95} + 30 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 5 * q^7 - 2 * q^11 + 2 * q^17 - 4 * q^19 - 7 * q^23 + 3 * q^25 - 7 * q^29 - 16 * q^31 - 5 * q^35 + 2 * q^37 - q^41 + 2 * q^43 - 13 * q^47 + 10 * q^49 - 10 * q^53 - 2 * q^55 - 6 * q^59 - 11 * q^61 + q^67 - 14 * q^71 + 16 * q^73 - 12 * q^77 - 6 * q^79 - 21 * q^83 + 2 * q^85 - 33 * q^89 - 30 * q^91 - 4 * q^95 + 30 * q^97

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