LMFDB - Embedded newform 5220.2.a.x.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5220,2,Mod(1,5220)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5220, 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("5220.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5220.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:	$41.6819098551$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 580)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-2.08613$ of defining polynomial
Character	$\chi$	$=$	5220.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -0.648061 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -0.648061 q^{7} -3.35194 q^{11} +4.17226 q^{13} -4.82032 q^{17} +6.82032 q^{19} -5.52420 q^{23} +1.00000 q^{25} +1.00000 q^{29} -2.82032 q^{31} -0.648061 q^{35} -10.2281 q^{37} -8.17226 q^{41} -5.69646 q^{43} +2.64806 q^{47} -6.58002 q^{49} +2.87614 q^{53} -3.35194 q^{55} +13.2207 q^{59} -1.12386 q^{61} +4.17226 q^{65} +1.52420 q^{67} +8.87614 q^{71} +9.69646 q^{73} +2.17226 q^{77} +8.99258 q^{79} -1.94418 q^{83} -4.82032 q^{85} -17.0484 q^{89} -2.70388 q^{91} +6.82032 q^{95} -13.3519 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{5} - 4 q^{7}+O(q^{10}) $ 3 * q + 3 * q^5 - 4 * q^7	$ 3 q + 3 q^{5} - 4 q^{7} - 8 q^{11} - 2 q^{13} - 2 q^{17} + 8 q^{19} + 3 q^{25} + 3 q^{29} + 4 q^{31} - 4 q^{35} - 10 q^{37} - 10 q^{41} + 14 q^{43} + 10 q^{47} + 3 q^{49} - 10 q^{53} - 8 q^{55} - 8 q^{59} - 22 q^{61} - 2 q^{65} - 12 q^{67} + 8 q^{71} - 2 q^{73} - 8 q^{77} - 12 q^{83} - 2 q^{85} - 18 q^{89} - 4 q^{91} + 8 q^{95} - 38 q^{97}+O(q^{100}) $ 3 * q + 3 * q^5 - 4 * q^7 - 8 * q^11 - 2 * q^13 - 2 * q^17 + 8 * q^19 + 3 * q^25 + 3 * q^29 + 4 * q^31 - 4 * q^35 - 10 * q^37 - 10 * q^41 + 14 * q^43 + 10 * q^47 + 3 * q^49 - 10 * q^53 - 8 * q^55 - 8 * q^59 - 22 * q^61 - 2 * q^65 - 12 * q^67 + 8 * q^71 - 2 * q^73 - 8 * q^77 - 12 * q^83 - 2 * q^85 - 18 * q^89 - 4 * q^91 + 8 * q^95 - 38 * 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
$7$	−0.648061	−0.244944	−0.122472	−	0.992472i	$-0.539082\pi$
$7$	−0.648061	−0.244944	−0.122472	+	0.992472i	$0.539082\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−3.35194	−1.01065	−0.505324	−	0.862930i	$-0.668627\pi$
$11$	−3.35194	−1.01065	−0.505324	+	0.862930i	$0.668627\pi$
$12$	0	0
$13$	4.17226	1.15718	0.578588	−	0.815620i	$-0.303604\pi$
$13$	4.17226	1.15718	0.578588	+	0.815620i	$0.303604\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.82032	−1.16910	−0.584550	−	0.811358i	$-0.698729\pi$
$17$	−4.82032	−1.16910	−0.584550	+	0.811358i	$0.698729\pi$
$18$	0	0
$19$	6.82032	1.56469	0.782344	−	0.622846i	$-0.214024\pi$
$19$	6.82032	1.56469	0.782344	+	0.622846i	$0.214024\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.52420	−1.15188	−0.575938	−	0.817494i	$-0.695363\pi$
$23$	−5.52420	−1.15188	−0.575938	+	0.817494i	$0.695363\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−2.82032	−0.506545	−0.253272	−	0.967395i	$-0.581507\pi$
$31$	−2.82032	−0.506545	−0.253272	+	0.967395i	$0.581507\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−0.648061	−0.109542
$36$	0	0
$37$	−10.2281	−1.68149	−0.840743	−	0.541435i	$-0.817881\pi$
$37$	−10.2281	−1.68149	−0.840743	+	0.541435i	$0.817881\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−8.17226	−1.27629	−0.638146	−	0.769915i	$-0.720299\pi$
$41$	−8.17226	−1.27629	−0.638146	+	0.769915i	$0.720299\pi$
$42$	0	0
$43$	−5.69646	−0.868702	−0.434351	−	0.900744i	$-0.643022\pi$
$43$	−5.69646	−0.868702	−0.434351	+	0.900744i	$0.643022\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	2.64806	0.386259	0.193130	−	0.981173i	$-0.438136\pi$
$47$	2.64806	0.386259	0.193130	+	0.981173i	$0.438136\pi$
$48$	0	0
$49$	−6.58002	−0.940002
$50$	0	0
$51$	0	0
$52$	0	0
$53$	2.87614	0.395068	0.197534	−	0.980296i	$-0.436707\pi$
$53$	2.87614	0.395068	0.197534	+	0.980296i	$0.436707\pi$
$54$	0	0
$55$	−3.35194	−0.451975
$56$	0	0
$57$	0	0
$58$	0	0
$59$	13.2207	1.72118	0.860592	−	0.509296i	$-0.170094\pi$
$59$	13.2207	1.72118	0.860592	+	0.509296i	$0.170094\pi$
$60$	0	0
$61$	−1.12386	−0.143896	−0.0719478	−	0.997408i	$-0.522922\pi$
$61$	−1.12386	−0.143896	−0.0719478	+	0.997408i	$0.522922\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	4.17226	0.517505
$66$	0	0
$67$	1.52420	0.186211	0.0931053	−	0.995656i	$-0.470321\pi$
$67$	1.52420	0.186211	0.0931053	+	0.995656i	$0.470321\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.87614	1.05340	0.526702	−	0.850050i	$-0.323428\pi$
$71$	8.87614	1.05340	0.526702	+	0.850050i	$0.323428\pi$
$72$	0	0
$73$	9.69646	1.13488	0.567442	−	0.823413i	$-0.307933\pi$
$73$	9.69646	1.13488	0.567442	+	0.823413i	$0.307933\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	2.17226	0.247552
$78$	0	0
$79$	8.99258	1.01174	0.505872	−	0.862608i	$-0.331171\pi$
$79$	8.99258	1.01174	0.505872	+	0.862608i	$0.331171\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−1.94418	−0.213402	−0.106701	−	0.994291i	$-0.534029\pi$
$83$	−1.94418	−0.213402	−0.106701	+	0.994291i	$0.534029\pi$
$84$	0	0
$85$	−4.82032	−0.522837
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−17.0484	−1.80713	−0.903563	−	0.428455i	$-0.859058\pi$
$89$	−17.0484	−1.80713	−0.903563	+	0.428455i	$0.859058\pi$
$90$	0	0
$91$	−2.70388	−0.283443
$92$	0	0
$93$	0	0
$94$	0	0
$95$	6.82032	0.699750
$96$	0	0
$97$	−13.3519	−1.35568	−0.677842	−	0.735208i	$-0.737085\pi$
$97$	−13.3519	−1.35568	−0.677842	+	0.735208i	$0.737085\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−18.4562	−1.83646	−0.918228	−	0.396052i	$-0.870380\pi$
$101$	−18.4562	−1.83646	−0.918228	+	0.396052i	$0.870380\pi$
$102$	0	0
$103$	−14.8203	−1.46029	−0.730145	−	0.683292i	$-0.760547\pi$
$103$	−14.8203	−1.46029	−0.730145	+	0.683292i	$0.760547\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.35194	0.324044	0.162022	−	0.986787i	$-0.448198\pi$
$107$	3.35194	0.324044	0.162022	+	0.986787i	$0.448198\pi$
$108$	0	0
$109$	2.76450	0.264791	0.132396	−	0.991197i	$-0.457733\pi$
$109$	2.76450	0.264791	0.132396	+	0.991197i	$0.457733\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.94418	−0.747326	−0.373663	−	0.927565i	$-0.621898\pi$
$113$	−7.94418	−0.747326	−0.373663	+	0.927565i	$0.621898\pi$
$114$	0	0
$115$	−5.52420	−0.515134
$116$	0	0
$117$	0	0
$118$	0	0
$119$	3.12386	0.286364
$120$	0	0
$121$	0.235496	0.0214088
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	−17.6965	−1.57031	−0.785153	−	0.619301i	$-0.787416\pi$
$127$	−17.6965	−1.57031	−0.785153	+	0.619301i	$0.787416\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.52420	−0.482652	−0.241326	−	0.970444i	$-0.577582\pi$
$131$	−5.52420	−0.482652	−0.241326	+	0.970444i	$0.577582\pi$
$132$	0	0
$133$	−4.41998	−0.383261
$134$	0	0
$135$	0	0
$136$	0	0
$137$	3.86872	0.330527	0.165264	−	0.986249i	$-0.447153\pi$
$137$	3.86872	0.330527	0.165264	+	0.986249i	$0.447153\pi$
$138$	0	0
$139$	5.64064	0.478433	0.239217	−	0.970966i	$-0.423109\pi$
$139$	5.64064	0.478433	0.239217	+	0.970966i	$0.423109\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−13.9852	−1.16950
$144$	0	0
$145$	1.00000	0.0830455
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.0000	−1.47462	−0.737309	−	0.675556i	$-0.763904\pi$
$149$	−18.0000	−1.47462	−0.737309	+	0.675556i	$0.763904\pi$
$150$	0	0
$151$	−17.2207	−1.40140	−0.700699	−	0.713457i	$-0.747128\pi$
$151$	−17.2207	−1.40140	−0.700699	+	0.713457i	$0.747128\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2.82032	−0.226534
$156$	0	0
$157$	−5.69646	−0.454627	−0.227314	−	0.973822i	$-0.572994\pi$
$157$	−5.69646	−0.454627	−0.227314	+	0.973822i	$0.572994\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.58002	0.282145
$162$	0	0
$163$	16.4003	1.28457	0.642287	−	0.766464i	$-0.277986\pi$
$163$	16.4003	1.28457	0.642287	+	0.766464i	$0.277986\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	23.2765	1.80119	0.900594	−	0.434661i	$-0.143132\pi$
$167$	23.2765	1.80119	0.900594	+	0.434661i	$0.143132\pi$
$168$	0	0
$169$	4.40776	0.339058
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.9245	1.05866	0.529332	−	0.848415i	$-0.322443\pi$
$173$	13.9245	1.05866	0.529332	+	0.848415i	$0.322443\pi$
$174$	0	0
$175$	−0.648061	−0.0489888
$176$	0	0
$177$	0	0
$178$	0	0
$179$	4.34452	0.324725	0.162362	−	0.986731i	$-0.448089\pi$
$179$	4.34452	0.324725	0.162362	+	0.986731i	$0.448089\pi$
$180$	0	0
$181$	20.5168	1.52500	0.762500	−	0.646988i	$-0.223972\pi$
$181$	20.5168	1.52500	0.762500	+	0.646988i	$0.223972\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−10.2281	−0.751983
$186$	0	0
$187$	16.1574	1.18155
$188$	0	0
$189$	0	0
$190$	0	0
$191$	9.10422	0.658758	0.329379	−	0.944198i	$-0.393161\pi$
$191$	9.10422	0.658758	0.329379	+	0.944198i	$0.393161\pi$
$192$	0	0
$193$	−25.1648	−1.81140	−0.905702	−	0.423914i	$-0.860656\pi$
$193$	−25.1648	−1.81140	−0.905702	+	0.423914i	$0.860656\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	0.703878	0.0501493	0.0250746	−	0.999686i	$-0.492018\pi$
$197$	0.703878	0.0501493	0.0250746	+	0.999686i	$0.492018\pi$
$198$	0	0
$199$	−5.22066	−0.370083	−0.185041	−	0.982731i	$-0.559242\pi$
$199$	−5.22066	−0.370083	−0.185041	+	0.982731i	$0.559242\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−0.648061	−0.0454850
$204$	0	0
$205$	−8.17226	−0.570775
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−22.8613	−1.58135
$210$	0	0
$211$	−13.1042	−0.902131	−0.451066	−	0.892491i	$-0.648956\pi$
$211$	−13.1042	−0.902131	−0.451066	+	0.892491i	$0.648956\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.69646	−0.388495
$216$	0	0
$217$	1.82774	0.124075
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−20.1116	−1.35285
$222$	0	0
$223$	−20.5726	−1.37764	−0.688822	−	0.724931i	$-0.741872\pi$
$223$	−20.5726	−1.37764	−0.688822	+	0.724931i	$0.741872\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	9.41256	0.624734	0.312367	−	0.949962i	$-0.398878\pi$
$227$	9.41256	0.624734	0.312367	+	0.949962i	$0.398878\pi$
$228$	0	0
$229$	2.00000	0.132164	0.0660819	−	0.997814i	$-0.478950\pi$
$229$	2.00000	0.132164	0.0660819	+	0.997814i	$0.478950\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.3445	−0.677692	−0.338846	−	0.940842i	$-0.610037\pi$
$233$	−10.3445	−0.677692	−0.338846	+	0.940842i	$0.610037\pi$
$234$	0	0
$235$	2.64806	0.172740
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−20.8761	−1.35037	−0.675183	−	0.737651i	$-0.735935\pi$
$239$	−20.8761	−1.35037	−0.675183	+	0.737651i	$0.735935\pi$
$240$	0	0
$241$	21.1600	1.36304	0.681519	−	0.731801i	$-0.261320\pi$
$241$	21.1600	1.36304	0.681519	+	0.731801i	$0.261320\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.58002	−0.420382
$246$	0	0
$247$	28.4562	1.81062
$248$	0	0
$249$	0	0
$250$	0	0
$251$	8.64806	0.545861	0.272930	−	0.962034i	$-0.412007\pi$
$251$	8.64806	0.545861	0.272930	+	0.962034i	$0.412007\pi$
$252$	0	0
$253$	18.5168	1.16414
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−27.4535	−1.71251	−0.856253	−	0.516557i	$-0.827213\pi$
$257$	−27.4535	−1.71251	−0.856253	+	0.516557i	$0.827213\pi$
$258$	0	0
$259$	6.62842	0.411870
$260$	0	0
$261$	0	0
$262$	0	0
$263$	11.3371	0.699076	0.349538	−	0.936922i	$-0.386339\pi$
$263$	11.3371	0.699076	0.349538	+	0.936922i	$0.386339\pi$
$264$	0	0
$265$	2.87614	0.176680
$266$	0	0
$267$	0	0
$268$	0	0
$269$	20.1723	1.22992	0.614962	−	0.788557i	$-0.289171\pi$
$269$	20.1723	1.22992	0.614962	+	0.788557i	$0.289171\pi$
$270$	0	0
$271$	−10.7449	−0.652704	−0.326352	−	0.945248i	$-0.605819\pi$
$271$	−10.7449	−0.652704	−0.326352	+	0.945248i	$0.605819\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−3.35194	−0.202130
$276$	0	0
$277$	−12.6284	−0.758768	−0.379384	−	0.925239i	$-0.623864\pi$
$277$	−12.6284	−0.758768	−0.379384	+	0.925239i	$0.623864\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6.76450	−0.403536	−0.201768	−	0.979433i	$-0.564669\pi$
$281$	−6.76450	−0.403536	−0.201768	+	0.979433i	$0.564669\pi$
$282$	0	0
$283$	1.06804	0.0634886	0.0317443	−	0.999496i	$-0.489894\pi$
$283$	1.06804	0.0634886	0.0317443	+	0.999496i	$0.489894\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	5.29612	0.312620
$288$	0	0
$289$	6.23550	0.366794
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5.46357	−0.319185	−0.159593	−	0.987183i	$-0.551018\pi$
$293$	−5.46357	−0.319185	−0.159593	+	0.987183i	$0.551018\pi$
$294$	0	0
$295$	13.2207	0.769737
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−23.0484	−1.33292
$300$	0	0
$301$	3.69165	0.212783
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−1.12386	−0.0643521
$306$	0	0
$307$	6.30354	0.359762	0.179881	−	0.983688i	$-0.442429\pi$
$307$	6.30354	0.359762	0.179881	+	0.983688i	$0.442429\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−24.6842	−1.39971	−0.699857	−	0.714283i	$-0.746753\pi$
$311$	−24.6842	−1.39971	−0.699857	+	0.714283i	$0.746753\pi$
$312$	0	0
$313$	−24.1723	−1.36630	−0.683148	−	0.730280i	$-0.739390\pi$
$313$	−24.1723	−1.36630	−0.683148	+	0.730280i	$0.739390\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−1.42740	−0.0801708	−0.0400854	−	0.999196i	$-0.512763\pi$
$317$	−1.42740	−0.0801708	−0.0400854	+	0.999196i	$0.512763\pi$
$318$	0	0
$319$	−3.35194	−0.187673
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−32.8761	−1.82928
$324$	0	0
$325$	4.17226	0.231435
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1.71610	−0.0946119
$330$	0	0
$331$	34.2132	1.88053	0.940265	−	0.340444i	$-0.110577\pi$
$331$	34.2132	1.88053	0.940265	+	0.340444i	$0.110577\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1.52420	0.0832759
$336$	0	0
$337$	4.40034	0.239702	0.119851	−	0.992792i	$-0.461758\pi$
$337$	4.40034	0.239702	0.119851	+	0.992792i	$0.461758\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	9.45355	0.511938
$342$	0	0
$343$	8.80068	0.475192
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−12.9926	−0.697478	−0.348739	−	0.937220i	$-0.613390\pi$
$347$	−12.9926	−0.697478	−0.348739	+	0.937220i	$0.613390\pi$
$348$	0	0
$349$	−20.0968	−1.07576	−0.537878	−	0.843022i	$-0.680774\pi$
$349$	−20.0968	−1.07576	−0.537878	+	0.843022i	$0.680774\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	17.8129	0.948085	0.474043	−	0.880502i	$-0.342794\pi$
$353$	17.8129	0.948085	0.474043	+	0.880502i	$0.342794\pi$
$354$	0	0
$355$	8.87614	0.471097
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−5.52420	−0.291556	−0.145778	−	0.989317i	$-0.546569\pi$
$359$	−5.52420	−0.291556	−0.145778	+	0.989317i	$0.546569\pi$
$360$	0	0
$361$	27.5168	1.44825
$362$	0	0
$363$	0	0
$364$	0	0
$365$	9.69646	0.507536
$366$	0	0
$367$	−20.0558	−1.04691	−0.523453	−	0.852055i	$-0.675356\pi$
$367$	−20.0558	−1.04691	−0.523453	+	0.852055i	$0.675356\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−1.86391	−0.0967695
$372$	0	0
$373$	−1.88836	−0.0977758	−0.0488879	−	0.998804i	$-0.515568\pi$
$373$	−1.88836	−0.0977758	−0.0488879	+	0.998804i	$0.515568\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	4.17226	0.214882
$378$	0	0
$379$	−15.0894	−0.775089	−0.387545	−	0.921851i	$-0.626677\pi$
$379$	−15.0894	−0.775089	−0.387545	+	0.921851i	$0.626677\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−17.3371	−0.885885	−0.442942	−	0.896550i	$-0.646065\pi$
$383$	−17.3371	−0.885885	−0.442942	+	0.896550i	$0.646065\pi$
$384$	0	0
$385$	2.17226	0.110709
$386$	0	0
$387$	0	0
$388$	0	0
$389$	0.890976	0.0451743	0.0225871	−	0.999745i	$-0.492810\pi$
$389$	0.890976	0.0451743	0.0225871	+	0.999745i	$0.492810\pi$
$390$	0	0
$391$	26.6284	1.34666
$392$	0	0
$393$	0	0
$394$	0	0
$395$	8.99258	0.452466
$396$	0	0
$397$	5.39292	0.270663	0.135331	−	0.990800i	$-0.456790\pi$
$397$	5.39292	0.270663	0.135331	+	0.990800i	$0.456790\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−5.23550	−0.261448	−0.130724	−	0.991419i	$-0.541730\pi$
$401$	−5.23550	−0.261448	−0.130724	+	0.991419i	$0.541730\pi$
$402$	0	0
$403$	−11.7671	−0.586162
$404$	0	0
$405$	0	0
$406$	0	0
$407$	34.2839	1.69939
$408$	0	0
$409$	−16.0606	−0.794147	−0.397073	−	0.917787i	$-0.629974\pi$
$409$	−16.0606	−0.794147	−0.397073	+	0.917787i	$0.629974\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−8.56779	−0.421593
$414$	0	0
$415$	−1.94418	−0.0954362
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.9245	0.973377	0.486689	−	0.873575i	$-0.338205\pi$
$419$	19.9245	0.973377	0.486689	+	0.873575i	$0.338205\pi$
$420$	0	0
$421$	−15.5652	−0.758600	−0.379300	−	0.925274i	$-0.623835\pi$
$421$	−15.5652	−0.758600	−0.379300	+	0.925274i	$0.623835\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.82032	−0.233820
$426$	0	0
$427$	0.728330	0.0352464
$428$	0	0
$429$	0	0
$430$	0	0
$431$	13.4078	0.645829	0.322914	−	0.946428i	$-0.395337\pi$
$431$	13.4078	0.645829	0.322914	+	0.946428i	$0.395337\pi$
$432$	0	0
$433$	−34.6842	−1.66682	−0.833409	−	0.552657i	$-0.813614\pi$
$433$	−34.6842	−1.66682	−0.833409	+	0.552657i	$0.813614\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−37.6768	−1.80233
$438$	0	0
$439$	−15.2355	−0.727151	−0.363575	−	0.931565i	$-0.618444\pi$
$439$	−15.2355	−0.727151	−0.363575	+	0.931565i	$0.618444\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−5.00742	−0.237910	−0.118955	−	0.992900i	$-0.537954\pi$
$443$	−5.00742	−0.237910	−0.118955	+	0.992900i	$0.537954\pi$
$444$	0	0
$445$	−17.0484	−0.808172
$446$	0	0
$447$	0	0
$448$	0	0
$449$	17.8129	0.840643	0.420321	−	0.907375i	$-0.361917\pi$
$449$	17.8129	0.840643	0.420321	+	0.907375i	$0.361917\pi$
$450$	0	0
$451$	27.3929	1.28988
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−2.70388	−0.126760
$456$	0	0
$457$	24.4051	1.14162	0.570812	−	0.821081i	$-0.306628\pi$
$457$	24.4051	1.14162	0.570812	+	0.821081i	$0.306628\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	7.98516	0.371906	0.185953	−	0.982559i	$-0.440463\pi$
$461$	7.98516	0.371906	0.185953	+	0.982559i	$0.440463\pi$
$462$	0	0
$463$	11.6210	0.540074	0.270037	−	0.962850i	$-0.412964\pi$
$463$	11.6210	0.540074	0.270037	+	0.962850i	$0.412964\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−1.24030	−0.0573944	−0.0286972	−	0.999588i	$-0.509136\pi$
$467$	−1.24030	−0.0573944	−0.0286972	+	0.999588i	$0.509136\pi$
$468$	0	0
$469$	−0.987774	−0.0456112
$470$	0	0
$471$	0	0
$472$	0	0
$473$	19.0942	0.877952
$474$	0	0
$475$	6.82032	0.312938
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−19.6965	−0.899954	−0.449977	−	0.893040i	$-0.648568\pi$
$479$	−19.6965	−0.899954	−0.449977	+	0.893040i	$0.648568\pi$
$480$	0	0
$481$	−42.6742	−1.94578
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−13.3519	−0.606280
$486$	0	0
$487$	−15.0894	−0.683765	−0.341883	−	0.939743i	$-0.611064\pi$
$487$	−15.0894	−0.683765	−0.341883	+	0.939743i	$0.611064\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	12.2281	0.551845	0.275923	−	0.961180i	$-0.411017\pi$
$491$	12.2281	0.551845	0.275923	+	0.961180i	$0.411017\pi$
$492$	0	0
$493$	−4.82032	−0.217096
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−5.75228	−0.258025
$498$	0	0
$499$	−5.52901	−0.247512	−0.123756	−	0.992313i	$-0.539494\pi$
$499$	−5.52901	−0.247512	−0.123756	+	0.992313i	$0.539494\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−24.2887	−1.08298	−0.541490	−	0.840707i	$-0.682140\pi$
$503$	−24.2887	−1.08298	−0.541490	+	0.840707i	$0.682140\pi$
$504$	0	0
$505$	−18.4562	−0.821288
$506$	0	0
$507$	0	0
$508$	0	0
$509$	9.58002	0.424627	0.212313	−	0.977202i	$-0.431900\pi$
$509$	9.58002	0.424627	0.212313	+	0.977202i	$0.431900\pi$
$510$	0	0
$511$	−6.28390	−0.277983
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.8203	−0.653061
$516$	0	0
$517$	−8.87614	−0.390372
$518$	0	0
$519$	0	0
$520$	0	0
$521$	21.5800	0.945438	0.472719	−	0.881213i	$-0.343273\pi$
$521$	21.5800	0.945438	0.472719	+	0.881213i	$0.343273\pi$
$522$	0	0
$523$	−42.2132	−1.84586	−0.922928	−	0.384972i	$-0.874211\pi$
$523$	−42.2132	−1.84586	−0.922928	+	0.384972i	$0.874211\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	13.5949	0.592201
$528$	0	0
$529$	7.51678	0.326817
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−34.0968	−1.47690
$534$	0	0
$535$	3.35194	0.144917
$536$	0	0
$537$	0	0
$538$	0	0
$539$	22.0558	0.950011
$540$	0	0
$541$	36.5530	1.57153	0.785767	−	0.618522i	$-0.212268\pi$
$541$	36.5530	1.57153	0.785767	+	0.618522i	$0.212268\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	2.76450	0.118418
$546$	0	0
$547$	26.6236	1.13834	0.569172	−	0.822219i	$-0.307264\pi$
$547$	26.6236	1.13834	0.569172	+	0.822219i	$0.307264\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	6.82032	0.290555
$552$	0	0
$553$	−5.82774	−0.247821
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−22.2233	−0.941630	−0.470815	−	0.882232i	$-0.656040\pi$
$557$	−22.2233	−0.941630	−0.470815	+	0.882232i	$0.656040\pi$
$558$	0	0
$559$	−23.7671	−1.00524
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−1.11905	−0.0471625	−0.0235812	−	0.999722i	$-0.507507\pi$
$563$	−1.11905	−0.0471625	−0.0235812	+	0.999722i	$0.507507\pi$
$564$	0	0
$565$	−7.94418	−0.334214
$566$	0	0
$567$	0	0
$568$	0	0
$569$	10.5316	0.441508	0.220754	−	0.975329i	$-0.429148\pi$
$569$	10.5316	0.441508	0.220754	+	0.975329i	$0.429148\pi$
$570$	0	0
$571$	−1.18449	−0.0495692	−0.0247846	−	0.999693i	$-0.507890\pi$
$571$	−1.18449	−0.0495692	−0.0247846	+	0.999693i	$0.507890\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−5.52420	−0.230375
$576$	0	0
$577$	11.8687	0.494101	0.247051	−	0.969003i	$-0.420539\pi$
$577$	11.8687	0.494101	0.247051	+	0.969003i	$0.420539\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	1.25995	0.0522715
$582$	0	0
$583$	−9.64064	−0.399275
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−32.4610	−1.33981	−0.669904	−	0.742448i	$-0.733665\pi$
$587$	−32.4610	−1.33981	−0.669904	+	0.742448i	$0.733665\pi$
$588$	0	0
$589$	−19.2355	−0.792585
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−9.12386	−0.374672	−0.187336	−	0.982296i	$-0.559985\pi$
$593$	−9.12386	−0.374672	−0.187336	+	0.982296i	$0.559985\pi$
$594$	0	0
$595$	3.12386	0.128066
$596$	0	0
$597$	0	0
$598$	0	0
$599$	30.2887	1.23756	0.618781	−	0.785563i	$-0.287627\pi$
$599$	30.2887	1.23756	0.618781	+	0.785563i	$0.287627\pi$
$600$	0	0
$601$	20.5168	0.836897	0.418448	−	0.908241i	$-0.362574\pi$
$601$	20.5168	0.836897	0.418448	+	0.908241i	$0.362574\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	0.235496	0.00957429
$606$	0	0
$607$	−13.9293	−0.565375	−0.282687	−	0.959212i	$-0.591226\pi$
$607$	−13.9293	−0.565375	−0.282687	+	0.959212i	$0.591226\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	11.0484	0.446970
$612$	0	0
$613$	19.7523	0.797787	0.398893	−	0.916997i	$-0.369394\pi$
$613$	19.7523	0.797787	0.398893	+	0.916997i	$0.369394\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	37.2010	1.49766	0.748828	−	0.662764i	$-0.230617\pi$
$617$	37.2010	1.49766	0.748828	+	0.662764i	$0.230617\pi$
$618$	0	0
$619$	7.46357	0.299986	0.149993	−	0.988687i	$-0.452075\pi$
$619$	7.46357	0.299986	0.149993	+	0.988687i	$0.452075\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	11.0484	0.442645
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	49.3026	1.96582
$630$	0	0
$631$	44.4562	1.76977	0.884886	−	0.465808i	$-0.154236\pi$
$631$	44.4562	1.76977	0.884886	+	0.465808i	$0.154236\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−17.6965	−0.702263
$636$	0	0
$637$	−27.4535	−1.08775
$638$	0	0
$639$	0	0
$640$	0	0
$641$	40.7401	1.60914	0.804568	−	0.593861i	$-0.202397\pi$
$641$	40.7401	1.60914	0.804568	+	0.593861i	$0.202397\pi$
$642$	0	0
$643$	20.3493	0.802499	0.401250	−	0.915969i	$-0.368576\pi$
$643$	20.3493	0.802499	0.401250	+	0.915969i	$0.368576\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−41.7933	−1.64306	−0.821531	−	0.570163i	$-0.806880\pi$
$647$	−41.7933	−1.64306	−0.821531	+	0.570163i	$0.806880\pi$
$648$	0	0
$649$	−44.3148	−1.73951
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−23.7933	−0.931102	−0.465551	−	0.885021i	$-0.654144\pi$
$653$	−23.7933	−0.931102	−0.465551	+	0.885021i	$0.654144\pi$
$654$	0	0
$655$	−5.52420	−0.215848
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−7.50936	−0.292523	−0.146262	−	0.989246i	$-0.546724\pi$
$659$	−7.50936	−0.292523	−0.146262	+	0.989246i	$0.546724\pi$
$660$	0	0
$661$	45.9245	1.78626	0.893129	−	0.449801i	$-0.148505\pi$
$661$	45.9245	1.78626	0.893129	+	0.449801i	$0.148505\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−4.41998	−0.171400
$666$	0	0
$667$	−5.52420	−0.213898
$668$	0	0
$669$	0	0
$670$	0	0
$671$	3.76711	0.145428
$672$	0	0
$673$	7.29612	0.281245	0.140622	−	0.990063i	$-0.455090\pi$
$673$	7.29612	0.281245	0.140622	+	0.990063i	$0.455090\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−36.2494	−1.39318	−0.696589	−	0.717470i	$-0.745300\pi$
$677$	−36.2494	−1.39318	−0.696589	+	0.717470i	$0.745300\pi$
$678$	0	0
$679$	8.65287	0.332067
$680$	0	0
$681$	0	0
$682$	0	0
$683$	5.40295	0.206738	0.103369	−	0.994643i	$-0.467038\pi$
$683$	5.40295	0.206738	0.103369	+	0.994643i	$0.467038\pi$
$684$	0	0
$685$	3.86872	0.147816
$686$	0	0
$687$	0	0
$688$	0	0
$689$	12.0000	0.457164
$690$	0	0
$691$	−36.0458	−1.37125	−0.685623	−	0.727957i	$-0.740470\pi$
$691$	−36.0458	−1.37125	−0.685623	+	0.727957i	$0.740470\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	5.64064	0.213962
$696$	0	0
$697$	39.3929	1.49211
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−3.33229	−0.125859	−0.0629295	−	0.998018i	$-0.520044\pi$
$701$	−3.33229	−0.125859	−0.0629295	+	0.998018i	$0.520044\pi$
$702$	0	0
$703$	−69.7588	−2.63100
$704$	0	0
$705$	0	0
$706$	0	0
$707$	11.9607	0.449829
$708$	0	0
$709$	7.48322	0.281038	0.140519	−	0.990078i	$-0.455123\pi$
$709$	7.48322	0.281038	0.140519	+	0.990078i	$0.455123\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15.5800	0.583476
$714$	0	0
$715$	−13.9852	−0.523015
$716$	0	0
$717$	0	0
$718$	0	0
$719$	39.6620	1.47914	0.739571	−	0.673078i	$-0.235028\pi$
$719$	39.6620	1.47914	0.739571	+	0.673078i	$0.235028\pi$
$720$	0	0
$721$	9.60447	0.357689
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.00000	0.0371391
$726$	0	0
$727$	−8.05582	−0.298774	−0.149387	−	0.988779i	$-0.547730\pi$
$727$	−8.05582	−0.298774	−0.149387	+	0.988779i	$0.547730\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	27.4588	1.01560
$732$	0	0
$733$	−28.9320	−1.06863	−0.534313	−	0.845287i	$-0.679430\pi$
$733$	−28.9320	−1.06863	−0.534313	+	0.845287i	$0.679430\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−5.10902	−0.188193
$738$	0	0
$739$	36.0772	1.32712	0.663560	−	0.748123i	$-0.269045\pi$
$739$	36.0772	1.32712	0.663560	+	0.748123i	$0.269045\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	40.9681	1.50297	0.751487	−	0.659747i	$-0.229337\pi$
$743$	40.9681	1.50297	0.751487	+	0.659747i	$0.229337\pi$
$744$	0	0
$745$	−18.0000	−0.659469
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−2.17226	−0.0793727
$750$	0	0
$751$	23.2861	0.849722	0.424861	−	0.905259i	$-0.360323\pi$
$751$	23.2861	0.849722	0.424861	+	0.905259i	$0.360323\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−17.2207	−0.626724
$756$	0	0
$757$	27.3275	0.993234	0.496617	−	0.867970i	$-0.334575\pi$
$757$	27.3275	0.993234	0.496617	+	0.867970i	$0.334575\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−44.5626	−1.61539	−0.807696	−	0.589599i	$-0.799286\pi$
$761$	−44.5626	−1.61539	−0.807696	+	0.589599i	$0.799286\pi$
$762$	0	0
$763$	−1.79157	−0.0648591
$764$	0	0
$765$	0	0
$766$	0	0
$767$	55.1600	1.99171
$768$	0	0
$769$	51.6110	1.86114	0.930570	−	0.366115i	$-0.119312\pi$
$769$	51.6110	1.86114	0.930570	+	0.366115i	$0.119312\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	19.6358	0.706252	0.353126	−	0.935576i	$-0.385119\pi$
$773$	19.6358	0.706252	0.353126	+	0.935576i	$0.385119\pi$
$774$	0	0
$775$	−2.82032	−0.101309
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−55.7374	−1.99700
$780$	0	0
$781$	−29.7523	−1.06462
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−5.69646	−0.203315
$786$	0	0
$787$	37.3371	1.33092	0.665462	−	0.746432i	$-0.268235\pi$
$787$	37.3371	1.33092	0.665462	+	0.746432i	$0.268235\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	5.14831	0.183053
$792$	0	0
$793$	−4.68904	−0.166513
$794$	0	0
$795$	0	0
$796$	0	0
$797$	11.2159	0.397286	0.198643	−	0.980072i	$-0.436347\pi$
$797$	11.2159	0.397286	0.198643	+	0.980072i	$0.436347\pi$
$798$	0	0
$799$	−12.7645	−0.451576
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−32.5019	−1.14697
$804$	0	0
$805$	3.58002	0.126179
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−11.1749	−0.392888	−0.196444	−	0.980515i	$-0.562939\pi$
$809$	−11.1749	−0.392888	−0.196444	+	0.980515i	$0.562939\pi$
$810$	0	0
$811$	17.6406	0.619447	0.309723	−	0.950827i	$-0.399764\pi$
$811$	17.6406	0.619447	0.309723	+	0.950827i	$0.399764\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	16.4003	0.574479
$816$	0	0
$817$	−38.8517	−1.35925
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−24.2477	−0.846251	−0.423126	−	0.906071i	$-0.639067\pi$
$821$	−24.2477	−0.846251	−0.423126	+	0.906071i	$0.639067\pi$
$822$	0	0
$823$	16.9777	0.591807	0.295903	−	0.955218i	$-0.404379\pi$
$823$	16.9777	0.591807	0.295903	+	0.955218i	$0.404379\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	5.58482	0.194203	0.0971017	−	0.995274i	$-0.469043\pi$
$827$	5.58482	0.194203	0.0971017	+	0.995274i	$0.469043\pi$
$828$	0	0
$829$	18.6136	0.646476	0.323238	−	0.946318i	$-0.395229\pi$
$829$	18.6136	0.646476	0.323238	+	0.946318i	$0.395229\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31.7178	1.09896
$834$	0	0
$835$	23.2765	0.805516
$836$	0	0
$837$	0	0
$838$	0	0
$839$	15.9293	0.549942	0.274971	−	0.961453i	$-0.411332\pi$
$839$	15.9293	0.549942	0.274971	+	0.961453i	$0.411332\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	4.40776	0.151631
$846$	0	0
$847$	−0.152616	−0.00524395
$848$	0	0
$849$	0	0
$850$	0	0
$851$	56.5019	1.93686
$852$	0	0
$853$	−27.0288	−0.925447	−0.462723	−	0.886503i	$-0.653128\pi$
$853$	−27.0288	−0.925447	−0.462723	+	0.886503i	$0.653128\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−30.2691	−1.03397	−0.516986	−	0.855994i	$-0.672946\pi$
$857$	−30.2691	−1.03397	−0.516986	+	0.855994i	$0.672946\pi$
$858$	0	0
$859$	−41.4487	−1.41421	−0.707106	−	0.707107i	$-0.750000\pi$
$859$	−41.4487	−1.41421	−0.707106	+	0.707107i	$0.750000\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−42.2494	−1.43819	−0.719093	−	0.694913i	$-0.755443\pi$
$863$	−42.2494	−1.43819	−0.719093	+	0.694913i	$0.755443\pi$
$864$	0	0
$865$	13.9245	0.473448
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−30.1426	−1.02252
$870$	0	0
$871$	6.35936	0.215479
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−0.648061	−0.0219085
$876$	0	0
$877$	37.8129	1.27685	0.638426	−	0.769684i	$-0.279586\pi$
$877$	37.8129	1.27685	0.638426	+	0.769684i	$0.279586\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	5.35675	0.180473	0.0902367	−	0.995920i	$-0.471238\pi$
$881$	5.35675	0.180473	0.0902367	+	0.995920i	$0.471238\pi$
$882$	0	0
$883$	8.53643	0.287274	0.143637	−	0.989630i	$-0.454120\pi$
$883$	8.53643	0.287274	0.143637	+	0.989630i	$0.454120\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22.7597	0.764196	0.382098	−	0.924122i	$-0.375202\pi$
$887$	22.7597	0.764196	0.382098	+	0.924122i	$0.375202\pi$
$888$	0	0
$889$	11.4684	0.384637
$890$	0	0
$891$	0	0
$892$	0	0
$893$	18.0606	0.604376
$894$	0	0
$895$	4.34452	0.145221
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−2.82032	−0.0940630
$900$	0	0
$901$	−13.8639	−0.461874
$902$	0	0
$903$	0	0
$904$	0	0
$905$	20.5168	0.682001
$906$	0	0
$907$	3.94418	0.130964	0.0654822	−	0.997854i	$-0.479141\pi$
$907$	3.94418	0.130964	0.0654822	+	0.997854i	$0.479141\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	40.6546	1.34695	0.673473	−	0.739212i	$-0.264802\pi$
$911$	40.6546	1.34695	0.673473	+	0.739212i	$0.264802\pi$
$912$	0	0
$913$	6.51678	0.215674
$914$	0	0
$915$	0	0
$916$	0	0
$917$	3.58002	0.118223
$918$	0	0
$919$	27.3471	0.902099	0.451049	−	0.892499i	$-0.351050\pi$
$919$	27.3471	0.902099	0.451049	+	0.892499i	$0.351050\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	37.0336	1.21897
$924$	0	0
$925$	−10.2281	−0.336297
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20.9368	−0.686913	−0.343456	−	0.939169i	$-0.611598\pi$
$929$	−20.9368	−0.686913	−0.343456	+	0.939169i	$0.611598\pi$
$930$	0	0
$931$	−44.8778	−1.47081
$932$	0	0
$933$	0	0
$934$	0	0
$935$	16.1574	0.528404
$936$	0	0
$937$	36.4051	1.18930	0.594652	−	0.803983i	$-0.297290\pi$
$937$	36.4051	1.18930	0.594652	+	0.803983i	$0.297290\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−28.0968	−0.915929	−0.457965	−	0.888970i	$-0.651421\pi$
$941$	−28.0968	−0.915929	−0.457965	+	0.888970i	$0.651421\pi$
$942$	0	0
$943$	45.1452	1.47013
$944$	0	0
$945$	0	0
$946$	0	0
$947$	10.8809	0.353583	0.176792	−	0.984248i	$-0.443428\pi$
$947$	10.8809	0.353583	0.176792	+	0.984248i	$0.443428\pi$
$948$	0	0
$949$	40.4562	1.31326
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−38.8103	−1.25719	−0.628594	−	0.777733i	$-0.716369\pi$
$953$	−38.8103	−1.25719	−0.628594	+	0.777733i	$0.716369\pi$
$954$	0	0
$955$	9.10422	0.294606
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−2.50717	−0.0809606
$960$	0	0
$961$	−23.0458	−0.743413
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−25.1648	−0.810085
$966$	0	0
$967$	47.1862	1.51741	0.758703	−	0.651437i	$-0.225834\pi$
$967$	47.1862	1.51741	0.758703	+	0.651437i	$0.225834\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−40.0772	−1.28614	−0.643069	−	0.765809i	$-0.722339\pi$
$971$	−40.0772	−1.28614	−0.643069	+	0.765809i	$0.722339\pi$
$972$	0	0
$973$	−3.65548	−0.117189
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−2.29873	−0.0735430	−0.0367715	−	0.999324i	$-0.511707\pi$
$977$	−2.29873	−0.0735430	−0.0367715	+	0.999324i	$0.511707\pi$
$978$	0	0
$979$	57.1452	1.82637
$980$	0	0
$981$	0	0
$982$	0	0
$983$	8.89578	0.283731	0.141866	−	0.989886i	$-0.454690\pi$
$983$	8.89578	0.283731	0.141866	+	0.989886i	$0.454690\pi$
$984$	0	0
$985$	0.703878	0.0224274
$986$	0	0
$987$	0	0
$988$	0	0
$989$	31.4684	1.00064
$990$	0	0
$991$	37.5652	1.19330	0.596649	−	0.802503i	$-0.296499\pi$
$991$	37.5652	1.19330	0.596649	+	0.802503i	$0.296499\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−5.22066	−0.165506
$996$	0	0
$997$	−25.8081	−0.817351	−0.408675	−	0.912680i	$-0.634009\pi$
$997$	−25.8081	−0.817351	−0.408675	+	0.912680i	$0.634009\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5220.2.a.x.1.2		3
3.2	odd	2		580.2.a.c.1.2	✓	3
12.11	even	2		2320.2.a.m.1.2		3
15.2	even	4		2900.2.c.f.349.3		6
15.8	even	4		2900.2.c.f.349.4		6
15.14	odd	2		2900.2.a.g.1.2		3
24.5	odd	2		9280.2.a.bk.1.2		3
24.11	even	2		9280.2.a.bw.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
580.2.a.c.1.2	✓	3	3.2	odd	2
2320.2.a.m.1.2		3	12.11	even	2
2900.2.a.g.1.2		3	15.14	odd	2
2900.2.c.f.349.3		6	15.2	even	4
2900.2.c.f.349.4		6	15.8	even	4
5220.2.a.x.1.2		3	1.1	even	1	trivial
9280.2.a.bk.1.2		3	24.5	odd	2
9280.2.a.bw.1.2		3	24.11	even	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:	\( N \)	\(=\)	\( 5220 = 2^{2} \cdot 3^{2} \cdot 5 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5220.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -0.648061 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -0.648061 q^{7} -3.35194 q^{11} +4.17226 q^{13} -4.82032 q^{17} +6.82032 q^{19} -5.52420 q^{23} +1.00000 q^{25} +1.00000 q^{29} -2.82032 q^{31} -0.648061 q^{35} -10.2281 q^{37} -8.17226 q^{41} -5.69646 q^{43} +2.64806 q^{47} -6.58002 q^{49} +2.87614 q^{53} -3.35194 q^{55} +13.2207 q^{59} -1.12386 q^{61} +4.17226 q^{65} +1.52420 q^{67} +8.87614 q^{71} +9.69646 q^{73} +2.17226 q^{77} +8.99258 q^{79} -1.94418 q^{83} -4.82032 q^{85} -17.0484 q^{89} -2.70388 q^{91} +6.82032 q^{95} -13.3519 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{5} - 4 q^{7}+O(q^{10}) \) 3 * q + 3 * q^5 - 4 * q^7	\( 3 q + 3 q^{5} - 4 q^{7} - 8 q^{11} - 2 q^{13} - 2 q^{17} + 8 q^{19} + 3 q^{25} + 3 q^{29} + 4 q^{31} - 4 q^{35} - 10 q^{37} - 10 q^{41} + 14 q^{43} + 10 q^{47} + 3 q^{49} - 10 q^{53} - 8 q^{55} - 8 q^{59} - 22 q^{61} - 2 q^{65} - 12 q^{67} + 8 q^{71} - 2 q^{73} - 8 q^{77} - 12 q^{83} - 2 q^{85} - 18 q^{89} - 4 q^{91} + 8 q^{95} - 38 q^{97}+O(q^{100}) \) 3 * q + 3 * q^5 - 4 * q^7 - 8 * q^11 - 2 * q^13 - 2 * q^17 + 8 * q^19 + 3 * q^25 + 3 * q^29 + 4 * q^31 - 4 * q^35 - 10 * q^37 - 10 * q^41 + 14 * q^43 + 10 * q^47 + 3 * q^49 - 10 * q^53 - 8 * q^55 - 8 * q^59 - 22 * q^61 - 2 * q^65 - 12 * q^67 + 8 * q^71 - 2 * q^73 - 8 * q^77 - 12 * q^83 - 2 * q^85 - 18 * q^89 - 4 * q^91 + 8 * q^95 - 38 * q^97

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