LMFDB - Embedded newform 8280.2.a.bs.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8280,2,Mod(1,8280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8280, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8280.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8280.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:	$66.1161328736$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.13955077.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 14x^{3} - x^{2} + 32x + 16 $ x^5 - 14x^3 - x^2 + 32x + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 920)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.31091$ of defining polynomial
Character	$\chi$	$=$	8280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{5} -4.66212 q^{7} +O(q^{10})$	$q+1.00000 q^{5} -4.66212 q^{7} -2.23020 q^{11} -2.80072 q^{13} -7.63271 q^{17} -1.36222 q^{19} +1.00000 q^{23} +1.00000 q^{25} -8.94362 q^{29} -1.58140 q^{31} -4.66212 q^{35} -1.40251 q^{37} -10.7134 q^{41} +7.26391 q^{47} +14.7353 q^{49} +8.38212 q^{53} -2.23020 q^{55} +4.88331 q^{59} +4.33282 q^{61} -2.80072 q^{65} +8.54355 q^{67} -8.81604 q^{71} -5.26391 q^{73} +10.3974 q^{77} +7.08222 q^{79} +4.59749 q^{83} -7.63271 q^{85} +4.70241 q^{89} +13.0573 q^{91} -1.36222 q^{95} +16.5160 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 5 q^{5} - 2 q^{7}+O(q^{10}) $ 5 * q + 5 * q^5 - 2 * q^7	$ 5 q + 5 q^{5} - 2 q^{7} + q^{11} + 4 q^{13} - 4 q^{17} + 7 q^{19} + 5 q^{23} + 5 q^{25} - 4 q^{29} + 19 q^{31} - 2 q^{35} + 15 q^{37} - 25 q^{41} + 11 q^{47} + 25 q^{49} - 3 q^{53} + q^{55} + q^{59} - 5 q^{61} + 4 q^{65} + 9 q^{67} - q^{71} - q^{73} - 18 q^{77} - 2 q^{79} + 45 q^{83} - 4 q^{85} - 6 q^{89} + 11 q^{91} + 7 q^{95} + 25 q^{97}+O(q^{100}) $ 5 * q + 5 * q^5 - 2 * q^7 + q^11 + 4 * q^13 - 4 * q^17 + 7 * q^19 + 5 * q^23 + 5 * q^25 - 4 * q^29 + 19 * q^31 - 2 * q^35 + 15 * q^37 - 25 * q^41 + 11 * q^47 + 25 * q^49 - 3 * q^53 + q^55 + q^59 - 5 * q^61 + 4 * q^65 + 9 * q^67 - q^71 - q^73 - 18 * q^77 - 2 * q^79 + 45 * q^83 - 4 * q^85 - 6 * q^89 + 11 * q^91 + 7 * q^95 + 25 * 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
$7$	−4.66212	−1.76211	−0.881057	−	0.473010i	$-0.843168\pi$
$7$	−4.66212	−1.76211	−0.881057	+	0.473010i	$0.843168\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−2.23020	−0.672430	−0.336215	−	0.941785i	$-0.609147\pi$
$11$	−2.23020	−0.672430	−0.336215	+	0.941785i	$0.609147\pi$
$12$	0	0
$13$	−2.80072	−0.776779	−0.388389	−	0.921495i	$-0.626968\pi$
$13$	−2.80072	−0.776779	−0.388389	+	0.921495i	$0.626968\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−7.63271	−1.85120	−0.925602	−	0.378498i	$-0.876441\pi$
$17$	−7.63271	−1.85120	−0.925602	+	0.378498i	$0.876441\pi$
$18$	0	0
$19$	−1.36222	−0.312515	−0.156258	−	0.987716i	$-0.549943\pi$
$19$	−1.36222	−0.312515	−0.156258	+	0.987716i	$0.549943\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−8.94362	−1.66079	−0.830395	−	0.557176i	$-0.811885\pi$
$29$	−8.94362	−1.66079	−0.830395	+	0.557176i	$0.811885\pi$
$30$	0	0
$31$	−1.58140	−0.284028	−0.142014	−	0.989865i	$-0.545358\pi$
$31$	−1.58140	−0.284028	−0.142014	+	0.989865i	$0.545358\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.66212	−0.788042
$36$	0	0
$37$	−1.40251	−0.230572	−0.115286	−	0.993332i	$-0.536778\pi$
$37$	−1.40251	−0.230572	−0.115286	+	0.993332i	$0.536778\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−10.7134	−1.67316	−0.836578	−	0.547848i	$-0.815447\pi$
$41$	−10.7134	−1.67316	−0.836578	+	0.547848i	$0.815447\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	7.26391	1.05955	0.529775	−	0.848138i	$-0.322276\pi$
$47$	7.26391	1.05955	0.529775	+	0.848138i	$0.322276\pi$
$48$	0	0
$49$	14.7353	2.10505
$50$	0	0
$51$	0	0
$52$	0	0
$53$	8.38212	1.15137	0.575686	−	0.817671i	$-0.304735\pi$
$53$	8.38212	1.15137	0.575686	+	0.817671i	$0.304735\pi$
$54$	0	0
$55$	−2.23020	−0.300720
$56$	0	0
$57$	0	0
$58$	0	0
$59$	4.88331	0.635752	0.317876	−	0.948132i	$-0.397030\pi$
$59$	4.88331	0.635752	0.317876	+	0.948132i	$0.397030\pi$
$60$	0	0
$61$	4.33282	0.554760	0.277380	−	0.960760i	$-0.410534\pi$
$61$	4.33282	0.554760	0.277380	+	0.960760i	$0.410534\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.80072	−0.347386
$66$	0	0
$67$	8.54355	1.04376	0.521880	−	0.853019i	$-0.325231\pi$
$67$	8.54355	1.04376	0.521880	+	0.853019i	$0.325231\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−8.81604	−1.04627	−0.523136	−	0.852249i	$-0.675238\pi$
$71$	−8.81604	−1.04627	−0.523136	+	0.852249i	$0.675238\pi$
$72$	0	0
$73$	−5.26391	−0.616095	−0.308047	−	0.951371i	$-0.599675\pi$
$73$	−5.26391	−0.616095	−0.308047	+	0.951371i	$0.599675\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.3974	1.18490
$78$	0	0
$79$	7.08222	0.796812	0.398406	−	0.917209i	$-0.369563\pi$
$79$	7.08222	0.796812	0.398406	+	0.917209i	$0.369563\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	4.59749	0.504640	0.252320	−	0.967644i	$-0.418806\pi$
$83$	4.59749	0.504640	0.252320	+	0.967644i	$0.418806\pi$
$84$	0	0
$85$	−7.63271	−0.827884
$86$	0	0
$87$	0	0
$88$	0	0
$89$	4.70241	0.498454	0.249227	−	0.968445i	$-0.419823\pi$
$89$	4.70241	0.498454	0.249227	+	0.968445i	$0.419823\pi$
$90$	0	0
$91$	13.0573	1.36877
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−1.36222	−0.139761
$96$	0	0
$97$	16.5160	1.67695	0.838474	−	0.544942i	$-0.183448\pi$
$97$	16.5160	1.67695	0.838474	+	0.544942i	$0.183448\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.7267	−1.06735	−0.533676	−	0.845689i	$-0.679190\pi$
$101$	−10.7267	−1.06735	−0.533676	+	0.845689i	$0.679190\pi$
$102$	0	0
$103$	1.95300	0.192435	0.0962175	−	0.995360i	$-0.469326\pi$
$103$	1.95300	0.192435	0.0962175	+	0.995360i	$0.469326\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1.97566	0.190994	0.0954972	−	0.995430i	$-0.469556\pi$
$107$	1.97566	0.190994	0.0954972	+	0.995430i	$0.469556\pi$
$108$	0	0
$109$	−8.92360	−0.854725	−0.427363	−	0.904080i	$-0.640557\pi$
$109$	−8.92360	−0.854725	−0.427363	+	0.904080i	$0.640557\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−17.3486	−1.63202	−0.816008	−	0.578040i	$-0.803818\pi$
$113$	−17.3486	−1.63202	−0.816008	+	0.578040i	$0.803818\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	35.5846	3.26203
$120$	0	0
$121$	−6.02621	−0.547838
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000	0.0894427
$126$	0	0
$127$	19.4872	1.72921	0.864603	−	0.502455i	$-0.167570\pi$
$127$	19.4872	1.72921	0.864603	+	0.502455i	$0.167570\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.68050	−0.496308	−0.248154	−	0.968721i	$-0.579824\pi$
$131$	−5.68050	−0.496308	−0.248154	+	0.968721i	$0.579824\pi$
$132$	0	0
$133$	6.35084	0.550687
$134$	0	0
$135$	0	0
$136$	0	0
$137$	8.68645	0.742134	0.371067	−	0.928606i	$-0.378992\pi$
$137$	8.68645	0.742134	0.371067	+	0.928606i	$0.378992\pi$
$138$	0	0
$139$	−9.22569	−0.782513	−0.391256	−	0.920282i	$-0.627959\pi$
$139$	−9.22569	−0.782513	−0.391256	+	0.920282i	$0.627959\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	6.24615	0.522329
$144$	0	0
$145$	−8.94362	−0.742728
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−18.6056	−1.52423	−0.762115	−	0.647441i	$-0.775839\pi$
$149$	−18.6056	−1.52423	−0.762115	+	0.647441i	$0.775839\pi$
$150$	0	0
$151$	17.4316	1.41856	0.709280	−	0.704927i	$-0.249020\pi$
$151$	17.4316	1.41856	0.709280	+	0.704927i	$0.249020\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−1.58140	−0.127021
$156$	0	0
$157$	0.839497	0.0669992	0.0334996	−	0.999439i	$-0.489335\pi$
$157$	0.839497	0.0669992	0.0334996	+	0.999439i	$0.489335\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.66212	−0.367426
$162$	0	0
$163$	−14.5673	−1.14100	−0.570500	−	0.821297i	$-0.693251\pi$
$163$	−14.5673	−1.14100	−0.570500	+	0.821297i	$0.693251\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	5.03842	0.389884	0.194942	−	0.980815i	$-0.437548\pi$
$167$	5.03842	0.389884	0.194942	+	0.980815i	$0.437548\pi$
$168$	0	0
$169$	−5.15599	−0.396615
$170$	0	0
$171$	0	0
$172$	0	0
$173$	11.3124	0.860067	0.430034	−	0.902813i	$-0.358502\pi$
$173$	11.3124	0.860067	0.430034	+	0.902813i	$0.358502\pi$
$174$	0	0
$175$	−4.66212	−0.352423
$176$	0	0
$177$	0	0
$178$	0	0
$179$	24.3053	1.81667	0.908333	−	0.418247i	$-0.137355\pi$
$179$	24.3053	1.81667	0.908333	+	0.418247i	$0.137355\pi$
$180$	0	0
$181$	−19.4829	−1.44815	−0.724075	−	0.689721i	$-0.757733\pi$
$181$	−19.4829	−1.44815	−0.724075	+	0.689721i	$0.757733\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.40251	−0.103115
$186$	0	0
$187$	17.0225	1.24481
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2.62183	−0.189709	−0.0948543	−	0.995491i	$-0.530239\pi$
$191$	−2.62183	−0.189709	−0.0948543	+	0.995491i	$0.530239\pi$
$192$	0	0
$193$	17.4332	1.25487	0.627436	−	0.778668i	$-0.284105\pi$
$193$	17.4332	1.25487	0.627436	+	0.778668i	$0.284105\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	23.4402	1.67004	0.835022	−	0.550217i	$-0.185455\pi$
$197$	23.4402	1.67004	0.835022	+	0.550217i	$0.185455\pi$
$198$	0	0
$199$	−1.29759	−0.0919839	−0.0459920	−	0.998942i	$-0.514645\pi$
$199$	−1.29759	−0.0919839	−0.0459920	+	0.998942i	$0.514645\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	41.6962	2.92650
$204$	0	0
$205$	−10.7134	−0.748258
$206$	0	0
$207$	0	0
$208$	0	0
$209$	3.03803	0.210145
$210$	0	0
$211$	14.7619	1.01625	0.508127	−	0.861282i	$-0.330338\pi$
$211$	14.7619	1.01625	0.508127	+	0.861282i	$0.330338\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	7.37268	0.500490
$218$	0	0
$219$	0	0
$220$	0	0
$221$	21.3771	1.43798
$222$	0	0
$223$	11.8434	0.793095	0.396548	−	0.918014i	$-0.370208\pi$
$223$	11.8434	0.793095	0.396548	+	0.918014i	$0.370208\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	5.27556	0.350151	0.175075	−	0.984555i	$-0.443983\pi$
$227$	5.27556	0.350151	0.175075	+	0.984555i	$0.443983\pi$
$228$	0	0
$229$	−1.23878	−0.0818611	−0.0409305	−	0.999162i	$-0.513032\pi$
$229$	−1.23878	−0.0818611	−0.0409305	+	0.999162i	$0.513032\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−2.66412	−0.174533	−0.0872663	−	0.996185i	$-0.527813\pi$
$233$	−2.66412	−0.174533	−0.0872663	+	0.996185i	$0.527813\pi$
$234$	0	0
$235$	7.26391	0.473845
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−26.2577	−1.69847	−0.849235	−	0.528014i	$-0.822937\pi$
$239$	−26.2577	−1.69847	−0.849235	+	0.528014i	$0.822937\pi$
$240$	0	0
$241$	−2.22326	−0.143213	−0.0716063	−	0.997433i	$-0.522813\pi$
$241$	−2.22326	−0.143213	−0.0716063	+	0.997433i	$0.522813\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	14.7353	0.941406
$246$	0	0
$247$	3.81520	0.242755
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−13.4829	−0.851031	−0.425515	−	0.904951i	$-0.639907\pi$
$251$	−13.4829	−0.851031	−0.425515	+	0.904951i	$0.639907\pi$
$252$	0	0
$253$	−2.23020	−0.140211
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−22.0281	−1.37408	−0.687039	−	0.726620i	$-0.741090\pi$
$257$	−22.0281	−1.37408	−0.687039	+	0.726620i	$0.741090\pi$
$258$	0	0
$259$	6.53868	0.406294
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−22.3164	−1.37609	−0.688043	−	0.725670i	$-0.741530\pi$
$263$	−22.3164	−1.37609	−0.688043	+	0.725670i	$0.741530\pi$
$264$	0	0
$265$	8.38212	0.514909
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−10.5050	−0.640501	−0.320251	−	0.947333i	$-0.603767\pi$
$269$	−10.5050	−0.640501	−0.320251	+	0.947333i	$0.603767\pi$
$270$	0	0
$271$	−3.84696	−0.233686	−0.116843	−	0.993150i	$-0.537277\pi$
$271$	−3.84696	−0.233686	−0.116843	+	0.993150i	$0.537277\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−2.23020	−0.134486
$276$	0	0
$277$	14.8653	0.893172	0.446586	−	0.894741i	$-0.352640\pi$
$277$	14.8653	0.893172	0.446586	+	0.894741i	$0.352640\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−4.41659	−0.263472	−0.131736	−	0.991285i	$-0.542055\pi$
$281$	−4.41659	−0.263472	−0.131736	+	0.991285i	$0.542055\pi$
$282$	0	0
$283$	3.88495	0.230936	0.115468	−	0.993311i	$-0.463163\pi$
$283$	3.88495	0.230936	0.115468	+	0.993311i	$0.463163\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	49.9472	2.94829
$288$	0	0
$289$	41.2583	2.42696
$290$	0	0
$291$	0	0
$292$	0	0
$293$	25.0257	1.46202	0.731009	−	0.682368i	$-0.239050\pi$
$293$	25.0257	1.46202	0.731009	+	0.682368i	$0.239050\pi$
$294$	0	0
$295$	4.88331	0.284317
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−2.80072	−0.161970
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	4.33282	0.248096
$306$	0	0
$307$	22.0197	1.25673	0.628365	−	0.777918i	$-0.283724\pi$
$307$	22.0197	1.25673	0.628365	+	0.777918i	$0.283724\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−19.6217	−1.11264	−0.556322	−	0.830967i	$-0.687788\pi$
$311$	−19.6217	−1.11264	−0.556322	+	0.830967i	$0.687788\pi$
$312$	0	0
$313$	−17.9496	−1.01457	−0.507285	−	0.861778i	$-0.669351\pi$
$313$	−17.9496	−1.01457	−0.507285	+	0.861778i	$0.669351\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−34.3185	−1.92752	−0.963761	−	0.266768i	$-0.914044\pi$
$317$	−34.3185	−1.92752	−0.963761	+	0.266768i	$0.914044\pi$
$318$	0	0
$319$	19.9461	1.11676
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.3974	0.578529
$324$	0	0
$325$	−2.80072	−0.155356
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−33.8652	−1.86705
$330$	0	0
$331$	−0.299762	−0.0164764	−0.00823821	−	0.999966i	$-0.502622\pi$
$331$	−0.299762	−0.0164764	−0.00823821	+	0.999966i	$0.502622\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	8.54355	0.466784
$336$	0	0
$337$	−22.2965	−1.21457	−0.607284	−	0.794485i	$-0.707741\pi$
$337$	−22.2965	−1.21457	−0.607284	+	0.794485i	$0.707741\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	3.52684	0.190989
$342$	0	0
$343$	−36.0630	−1.94722
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−33.1240	−1.77819	−0.889094	−	0.457725i	$-0.848664\pi$
$347$	−33.1240	−1.77819	−0.889094	+	0.457725i	$0.848664\pi$
$348$	0	0
$349$	−22.0041	−1.17785	−0.588926	−	0.808187i	$-0.700449\pi$
$349$	−22.0041	−1.17785	−0.588926	+	0.808187i	$0.700449\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.6085	0.830759	0.415379	−	0.909648i	$-0.363649\pi$
$353$	15.6085	0.830759	0.415379	+	0.909648i	$0.363649\pi$
$354$	0	0
$355$	−8.81604	−0.467907
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−0.263781	−0.0139218	−0.00696092	−	0.999976i	$-0.502216\pi$
$359$	−0.263781	−0.0139218	−0.00696092	+	0.999976i	$0.502216\pi$
$360$	0	0
$361$	−17.1444	−0.902334
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−5.26391	−0.275526
$366$	0	0
$367$	10.8615	0.566967	0.283484	−	0.958977i	$-0.408510\pi$
$367$	10.8615	0.566967	0.283484	+	0.958977i	$0.408510\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−39.0784	−2.02885
$372$	0	0
$373$	26.8889	1.39225	0.696127	−	0.717919i	$-0.254905\pi$
$373$	26.8889	1.39225	0.696127	+	0.717919i	$0.254905\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	25.0485	1.29007
$378$	0	0
$379$	−2.15824	−0.110861	−0.0554306	−	0.998463i	$-0.517653\pi$
$379$	−2.15824	−0.110861	−0.0554306	+	0.998463i	$0.517653\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	4.62814	0.236487	0.118244	−	0.992985i	$-0.462274\pi$
$383$	4.62814	0.236487	0.118244	+	0.992985i	$0.462274\pi$
$384$	0	0
$385$	10.3974	0.529903
$386$	0	0
$387$	0	0
$388$	0	0
$389$	18.4988	0.937929	0.468964	−	0.883217i	$-0.344627\pi$
$389$	18.4988	0.937929	0.468964	+	0.883217i	$0.344627\pi$
$390$	0	0
$391$	−7.63271	−0.386003
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7.08222	0.356345
$396$	0	0
$397$	−10.2685	−0.515360	−0.257680	−	0.966230i	$-0.582958\pi$
$397$	−10.2685	−0.515360	−0.257680	+	0.966230i	$0.582958\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0.885607	0.0442251	0.0221125	−	0.999755i	$-0.492961\pi$
$401$	0.885607	0.0442251	0.0221125	+	0.999755i	$0.492961\pi$
$402$	0	0
$403$	4.42906	0.220627
$404$	0	0
$405$	0	0
$406$	0	0
$407$	3.12788	0.155043
$408$	0	0
$409$	24.3497	1.20401	0.602006	−	0.798491i	$-0.294368\pi$
$409$	24.3497	1.20401	0.602006	+	0.798491i	$0.294368\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−22.7665	−1.12027
$414$	0	0
$415$	4.59749	0.225682
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−25.4535	−1.24348	−0.621742	−	0.783222i	$-0.713575\pi$
$419$	−25.4535	−1.24348	−0.621742	+	0.783222i	$0.713575\pi$
$420$	0	0
$421$	15.2376	0.742634	0.371317	−	0.928506i	$-0.378906\pi$
$421$	15.2376	0.742634	0.371317	+	0.928506i	$0.378906\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7.63271	−0.370241
$426$	0	0
$427$	−20.2001	−0.977551
$428$	0	0
$429$	0	0
$430$	0	0
$431$	34.6166	1.66742	0.833711	−	0.552202i	$-0.186212\pi$
$431$	34.6166	1.66742	0.833711	+	0.552202i	$0.186212\pi$
$432$	0	0
$433$	28.4774	1.36854	0.684268	−	0.729230i	$-0.260122\pi$
$433$	28.4774	1.36854	0.684268	+	0.729230i	$0.260122\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.36222	−0.0651639
$438$	0	0
$439$	21.6009	1.03095	0.515477	−	0.856904i	$-0.327615\pi$
$439$	21.6009	1.03095	0.515477	+	0.856904i	$0.327615\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−40.3700	−1.91804	−0.959018	−	0.283346i	$-0.908555\pi$
$443$	−40.3700	−1.91804	−0.959018	+	0.283346i	$0.908555\pi$
$444$	0	0
$445$	4.70241	0.222915
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−5.95484	−0.281026	−0.140513	−	0.990079i	$-0.544875\pi$
$449$	−5.95484	−0.281026	−0.140513	+	0.990079i	$0.544875\pi$
$450$	0	0
$451$	23.8931	1.12508
$452$	0	0
$453$	0	0
$454$	0	0
$455$	13.0573	0.612134
$456$	0	0
$457$	−5.66169	−0.264843	−0.132421	−	0.991194i	$-0.542275\pi$
$457$	−5.66169	−0.264843	−0.132421	+	0.991194i	$0.542275\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.26391	0.431463	0.215732	−	0.976453i	$-0.430786\pi$
$461$	9.26391	0.431463	0.215732	+	0.976453i	$0.430786\pi$
$462$	0	0
$463$	37.9899	1.76554	0.882769	−	0.469807i	$-0.155676\pi$
$463$	37.9899	1.76554	0.882769	+	0.469807i	$0.155676\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	26.2865	1.21640	0.608198	−	0.793785i	$-0.291893\pi$
$467$	26.2865	1.21640	0.608198	+	0.793785i	$0.291893\pi$
$468$	0	0
$469$	−39.8310	−1.83922
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−1.36222	−0.0625030
$476$	0	0
$477$	0	0
$478$	0	0
$479$	31.0770	1.41994	0.709971	−	0.704231i	$-0.248708\pi$
$479$	31.0770	1.41994	0.709971	+	0.704231i	$0.248708\pi$
$480$	0	0
$481$	3.92804	0.179103
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16.5160	0.749954
$486$	0	0
$487$	14.9591	0.677860	0.338930	−	0.940812i	$-0.389935\pi$
$487$	14.9591	0.677860	0.338930	+	0.940812i	$0.389935\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−24.1333	−1.08912	−0.544561	−	0.838721i	$-0.683304\pi$
$491$	−24.1333	−1.08912	−0.544561	+	0.838721i	$0.683304\pi$
$492$	0	0
$493$	68.2641	3.07446
$494$	0	0
$495$	0	0
$496$	0	0
$497$	41.1014	1.84365
$498$	0	0
$499$	−17.7266	−0.793552	−0.396776	−	0.917915i	$-0.629871\pi$
$499$	−17.7266	−0.793552	−0.396776	+	0.917915i	$0.629871\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7.69566	0.343133	0.171566	−	0.985173i	$-0.445117\pi$
$503$	7.69566	0.343133	0.171566	+	0.985173i	$0.445117\pi$
$504$	0	0
$505$	−10.7267	−0.477334
$506$	0	0
$507$	0	0
$508$	0	0
$509$	33.0564	1.46520	0.732600	−	0.680659i	$-0.238307\pi$
$509$	33.0564	1.46520	0.732600	+	0.680659i	$0.238307\pi$
$510$	0	0
$511$	24.5410	1.08563
$512$	0	0
$513$	0	0
$514$	0	0
$515$	1.95300	0.0860595
$516$	0	0
$517$	−16.2000	−0.712474
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−10.6047	−0.464598	−0.232299	−	0.972644i	$-0.574625\pi$
$521$	−10.6047	−0.464598	−0.232299	+	0.972644i	$0.574625\pi$
$522$	0	0
$523$	−34.5128	−1.50914	−0.754570	−	0.656219i	$-0.772155\pi$
$523$	−34.5128	−1.50914	−0.754570	+	0.656219i	$0.772155\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	12.0704	0.525794
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	30.0053	1.29967
$534$	0	0
$535$	1.97566	0.0854153
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−32.8627	−1.41550
$540$	0	0
$541$	−27.3344	−1.17520	−0.587598	−	0.809153i	$-0.699926\pi$
$541$	−27.3344	−1.17520	−0.587598	+	0.809153i	$0.699926\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−8.92360	−0.382245
$546$	0	0
$547$	34.6190	1.48020	0.740101	−	0.672496i	$-0.234778\pi$
$547$	34.6190	1.48020	0.740101	+	0.672496i	$0.234778\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	12.1832	0.519022
$552$	0	0
$553$	−33.0181	−1.40407
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−25.2490	−1.06983	−0.534917	−	0.844905i	$-0.679657\pi$
$557$	−25.2490	−1.06983	−0.534917	+	0.844905i	$0.679657\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−41.0793	−1.73128	−0.865642	−	0.500663i	$-0.833090\pi$
$563$	−41.0793	−1.73128	−0.865642	+	0.500663i	$0.833090\pi$
$564$	0	0
$565$	−17.3486	−0.729860
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.4230	1.19155	0.595776	−	0.803150i	$-0.296844\pi$
$569$	28.4230	1.19155	0.595776	+	0.803150i	$0.296844\pi$
$570$	0	0
$571$	13.4690	0.563659	0.281830	−	0.959464i	$-0.409059\pi$
$571$	13.4690	0.563659	0.281830	+	0.959464i	$0.409059\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	3.69392	0.153780	0.0768899	−	0.997040i	$-0.475501\pi$
$577$	3.69392	0.153780	0.0768899	+	0.997040i	$0.475501\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−21.4340	−0.889233
$582$	0	0
$583$	−18.6938	−0.774218
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−19.1042	−0.788513	−0.394257	−	0.919000i	$-0.628998\pi$
$587$	−19.1042	−0.788513	−0.394257	+	0.919000i	$0.628998\pi$
$588$	0	0
$589$	2.15422	0.0887631
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−13.9194	−0.571602	−0.285801	−	0.958289i	$-0.592260\pi$
$593$	−13.9194	−0.571602	−0.285801	+	0.958289i	$0.592260\pi$
$594$	0	0
$595$	35.5846	1.45883
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10.9498	−0.447396	−0.223698	−	0.974659i	$-0.571813\pi$
$599$	−10.9498	−0.447396	−0.223698	+	0.974659i	$0.571813\pi$
$600$	0	0
$601$	28.7034	1.17084	0.585418	−	0.810731i	$-0.300930\pi$
$601$	28.7034	1.17084	0.585418	+	0.810731i	$0.300930\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−6.02621	−0.245000
$606$	0	0
$607$	−43.7745	−1.77675	−0.888376	−	0.459117i	$-0.848166\pi$
$607$	−43.7745	−1.77675	−0.888376	+	0.459117i	$0.848166\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−20.3442	−0.823036
$612$	0	0
$613$	6.59492	0.266366	0.133183	−	0.991091i	$-0.457480\pi$
$613$	6.59492	0.266366	0.133183	+	0.991091i	$0.457480\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−20.3939	−0.821029	−0.410514	−	0.911854i	$-0.634651\pi$
$617$	−20.3939	−0.821029	−0.410514	+	0.911854i	$0.634651\pi$
$618$	0	0
$619$	0.513389	0.0206348	0.0103174	−	0.999947i	$-0.496716\pi$
$619$	0.513389	0.0206348	0.0103174	+	0.999947i	$0.496716\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−21.9232	−0.878333
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.7050	0.426835
$630$	0	0
$631$	16.3428	0.650596	0.325298	−	0.945612i	$-0.394535\pi$
$631$	16.3428	0.650596	0.325298	+	0.945612i	$0.394535\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	19.4872	0.773325
$636$	0	0
$637$	−41.2695	−1.63516
$638$	0	0
$639$	0	0
$640$	0	0
$641$	33.7798	1.33422	0.667110	−	0.744959i	$-0.267531\pi$
$641$	33.7798	1.33422	0.667110	+	0.744959i	$0.267531\pi$
$642$	0	0
$643$	−2.18090	−0.0860062	−0.0430031	−	0.999075i	$-0.513693\pi$
$643$	−2.18090	−0.0860062	−0.0430031	+	0.999075i	$0.513693\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	34.7346	1.36556	0.682778	−	0.730625i	$-0.260771\pi$
$647$	34.7346	1.36556	0.682778	+	0.730625i	$0.260771\pi$
$648$	0	0
$649$	−10.8907	−0.427499
$650$	0	0
$651$	0	0
$652$	0	0
$653$	6.68309	0.261530	0.130765	−	0.991413i	$-0.458257\pi$
$653$	6.68309	0.261530	0.130765	+	0.991413i	$0.458257\pi$
$654$	0	0
$655$	−5.68050	−0.221956
$656$	0	0
$657$	0	0
$658$	0	0
$659$	45.8903	1.78763	0.893815	−	0.448435i	$-0.148018\pi$
$659$	45.8903	1.78763	0.893815	+	0.448435i	$0.148018\pi$
$660$	0	0
$661$	13.6251	0.529957	0.264978	−	0.964254i	$-0.414635\pi$
$661$	13.6251	0.529957	0.264978	+	0.964254i	$0.414635\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.35084	0.246275
$666$	0	0
$667$	−8.94362	−0.346299
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−9.66304	−0.373038
$672$	0	0
$673$	−25.3475	−0.977075	−0.488537	−	0.872543i	$-0.662469\pi$
$673$	−25.3475	−0.977075	−0.488537	+	0.872543i	$0.662469\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	17.2279	0.662123	0.331062	−	0.943609i	$-0.392593\pi$
$677$	17.2279	0.662123	0.331062	+	0.943609i	$0.392593\pi$
$678$	0	0
$679$	−76.9996	−2.95497
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−39.5454	−1.51316	−0.756582	−	0.653899i	$-0.773132\pi$
$683$	−39.5454	−1.51316	−0.756582	+	0.653899i	$0.773132\pi$
$684$	0	0
$685$	8.68645	0.331892
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−23.4759	−0.894361
$690$	0	0
$691$	−29.0501	−1.10512	−0.552558	−	0.833474i	$-0.686348\pi$
$691$	−29.0501	−1.10512	−0.552558	+	0.833474i	$0.686348\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9.22569	−0.349950
$696$	0	0
$697$	81.7725	3.09735
$698$	0	0
$699$	0	0
$700$	0	0
$701$	17.7735	0.671298	0.335649	−	0.941987i	$-0.391044\pi$
$701$	17.7735	0.671298	0.335649	+	0.941987i	$0.391044\pi$
$702$	0	0
$703$	1.91053	0.0720571
$704$	0	0
$705$	0	0
$706$	0	0
$707$	50.0093	1.88079
$708$	0	0
$709$	−14.8606	−0.558103	−0.279052	−	0.960276i	$-0.590020\pi$
$709$	−14.8606	−0.558103	−0.279052	+	0.960276i	$0.590020\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−1.58140	−0.0592240
$714$	0	0
$715$	6.24615	0.233593
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−27.4365	−1.02321	−0.511604	−	0.859221i	$-0.670949\pi$
$719$	−27.4365	−1.02321	−0.511604	+	0.859221i	$0.670949\pi$
$720$	0	0
$721$	−9.10512	−0.339092
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−8.94362	−0.332158
$726$	0	0
$727$	−23.3674	−0.866649	−0.433325	−	0.901238i	$-0.642660\pi$
$727$	−23.3674	−0.866649	−0.433325	+	0.901238i	$0.642660\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	−2.53381	−0.0935884	−0.0467942	−	0.998905i	$-0.514900\pi$
$733$	−2.53381	−0.0935884	−0.0467942	+	0.998905i	$0.514900\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−19.0538	−0.701856
$738$	0	0
$739$	25.9318	0.953918	0.476959	−	0.878925i	$-0.341739\pi$
$739$	25.9318	0.953918	0.476959	+	0.878925i	$0.341739\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	2.11370	0.0775442	0.0387721	−	0.999248i	$-0.487655\pi$
$743$	2.11370	0.0775442	0.0387721	+	0.999248i	$0.487655\pi$
$744$	0	0
$745$	−18.6056	−0.681657
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−9.21077	−0.336554
$750$	0	0
$751$	2.54986	0.0930459	0.0465229	−	0.998917i	$-0.485186\pi$
$751$	2.54986	0.0930459	0.0465229	+	0.998917i	$0.485186\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	17.4316	0.634399
$756$	0	0
$757$	30.6897	1.11544	0.557718	−	0.830030i	$-0.311677\pi$
$757$	30.6897	1.11544	0.557718	+	0.830030i	$0.311677\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−19.3112	−0.700032	−0.350016	−	0.936744i	$-0.613824\pi$
$761$	−19.3112	−0.700032	−0.350016	+	0.936744i	$0.613824\pi$
$762$	0	0
$763$	41.6028	1.50612
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−13.6767	−0.493839
$768$	0	0
$769$	−8.52624	−0.307464	−0.153732	−	0.988113i	$-0.549129\pi$
$769$	−8.52624	−0.307464	−0.153732	+	0.988113i	$0.549129\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−29.5107	−1.06143	−0.530713	−	0.847551i	$-0.678076\pi$
$773$	−29.5107	−1.06143	−0.530713	+	0.847551i	$0.678076\pi$
$774$	0	0
$775$	−1.58140	−0.0568056
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.5941	0.522887
$780$	0	0
$781$	19.6615	0.703545
$782$	0	0
$783$	0	0
$784$	0	0
$785$	0.839497	0.0299629
$786$	0	0
$787$	−37.1733	−1.32508	−0.662542	−	0.749025i	$-0.730522\pi$
$787$	−37.1733	−1.32508	−0.662542	+	0.749025i	$0.730522\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	80.8811	2.87580
$792$	0	0
$793$	−12.1350	−0.430926
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−16.9954	−0.602008	−0.301004	−	0.953623i	$-0.597322\pi$
$797$	−16.9954	−0.602008	−0.301004	+	0.953623i	$0.597322\pi$
$798$	0	0
$799$	−55.4434	−1.96144
$800$	0	0
$801$	0	0
$802$	0	0
$803$	11.7396	0.414281
$804$	0	0
$805$	−4.66212	−0.164318
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−16.8409	−0.592095	−0.296047	−	0.955173i	$-0.595669\pi$
$809$	−16.8409	−0.592095	−0.296047	+	0.955173i	$0.595669\pi$
$810$	0	0
$811$	4.44804	0.156192	0.0780959	−	0.996946i	$-0.475116\pi$
$811$	4.44804	0.156192	0.0780959	+	0.996946i	$0.475116\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−14.5673	−0.510271
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	16.7361	0.584093	0.292047	−	0.956404i	$-0.405664\pi$
$821$	16.7361	0.584093	0.292047	+	0.956404i	$0.405664\pi$
$822$	0	0
$823$	−43.6605	−1.52191	−0.760955	−	0.648805i	$-0.775269\pi$
$823$	−43.6605	−1.52191	−0.760955	+	0.648805i	$0.775269\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37.9750	1.32052	0.660260	−	0.751037i	$-0.270446\pi$
$827$	37.9750	1.32052	0.660260	+	0.751037i	$0.270446\pi$
$828$	0	0
$829$	17.9046	0.621851	0.310925	−	0.950434i	$-0.399361\pi$
$829$	17.9046	0.621851	0.310925	+	0.950434i	$0.399361\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−112.471	−3.89687
$834$	0	0
$835$	5.03842	0.174362
$836$	0	0
$837$	0	0
$838$	0	0
$839$	42.4503	1.46555	0.732773	−	0.680473i	$-0.238226\pi$
$839$	42.4503	1.46555	0.732773	+	0.680473i	$0.238226\pi$
$840$	0	0
$841$	50.9884	1.75822
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−5.15599	−0.177372
$846$	0	0
$847$	28.0949	0.965353
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.40251	−0.0480775
$852$	0	0
$853$	−22.9952	−0.787339	−0.393670	−	0.919252i	$-0.628795\pi$
$853$	−22.9952	−0.787339	−0.393670	+	0.919252i	$0.628795\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	9.46053	0.323166	0.161583	−	0.986859i	$-0.448340\pi$
$857$	9.46053	0.323166	0.161583	+	0.986859i	$0.448340\pi$
$858$	0	0
$859$	−6.20043	−0.211556	−0.105778	−	0.994390i	$-0.533733\pi$
$859$	−6.20043	−0.211556	−0.105778	+	0.994390i	$0.533733\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	9.68754	0.329768	0.164884	−	0.986313i	$-0.447275\pi$
$863$	9.68754	0.329768	0.164884	+	0.986313i	$0.447275\pi$
$864$	0	0
$865$	11.3124	0.384634
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−15.7948	−0.535801
$870$	0	0
$871$	−23.9280	−0.810771
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.66212	−0.157608
$876$	0	0
$877$	51.5663	1.74127	0.870636	−	0.491928i	$-0.163708\pi$
$877$	51.5663	1.74127	0.870636	+	0.491928i	$0.163708\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	33.5969	1.13191	0.565954	−	0.824437i	$-0.308508\pi$
$881$	33.5969	1.13191	0.565954	+	0.824437i	$0.308508\pi$
$882$	0	0
$883$	11.7914	0.396814	0.198407	−	0.980120i	$-0.436423\pi$
$883$	11.7914	0.396814	0.198407	+	0.980120i	$0.436423\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	54.9578	1.84530	0.922652	−	0.385634i	$-0.126017\pi$
$887$	54.9578	1.84530	0.922652	+	0.385634i	$0.126017\pi$
$888$	0	0
$889$	−90.8515	−3.04706
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−9.89506	−0.331126
$894$	0	0
$895$	24.3053	0.812438
$896$	0	0
$897$	0	0
$898$	0	0
$899$	14.1435	0.471711
$900$	0	0
$901$	−63.9783	−2.13143
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−19.4829	−0.647632
$906$	0	0
$907$	8.41429	0.279392	0.139696	−	0.990194i	$-0.455387\pi$
$907$	8.41429	0.279392	0.139696	+	0.990194i	$0.455387\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	26.4143	0.875146	0.437573	−	0.899183i	$-0.355838\pi$
$911$	26.4143	0.875146	0.437573	+	0.899183i	$0.355838\pi$
$912$	0	0
$913$	−10.2533	−0.339335
$914$	0	0
$915$	0	0
$916$	0	0
$917$	26.4832	0.874551
$918$	0	0
$919$	59.2734	1.95525	0.977624	−	0.210359i	$-0.0674633\pi$
$919$	59.2734	1.95525	0.977624	+	0.210359i	$0.0674633\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	24.6912	0.812722
$924$	0	0
$925$	−1.40251	−0.0461143
$926$	0	0
$927$	0	0
$928$	0	0
$929$	17.2475	0.565871	0.282935	−	0.959139i	$-0.408692\pi$
$929$	17.2475	0.565871	0.282935	+	0.959139i	$0.408692\pi$
$930$	0	0
$931$	−20.0728	−0.657859
$932$	0	0
$933$	0	0
$934$	0	0
$935$	17.0225	0.556694
$936$	0	0
$937$	−21.6918	−0.708642	−0.354321	−	0.935124i	$-0.615288\pi$
$937$	−21.6918	−0.708642	−0.354321	+	0.935124i	$0.615288\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	11.1158	0.362365	0.181182	−	0.983449i	$-0.442008\pi$
$941$	11.1158	0.362365	0.181182	+	0.983449i	$0.442008\pi$
$942$	0	0
$943$	−10.7134	−0.348877
$944$	0	0
$945$	0	0
$946$	0	0
$947$	40.8093	1.32613	0.663063	−	0.748564i	$-0.269256\pi$
$947$	40.8093	1.32613	0.663063	+	0.748564i	$0.269256\pi$
$948$	0	0
$949$	14.7427	0.478569
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−11.8237	−0.383009	−0.191504	−	0.981492i	$-0.561337\pi$
$953$	−11.8237	−0.383009	−0.191504	+	0.981492i	$0.561337\pi$
$954$	0	0
$955$	−2.62183	−0.0848403
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−40.4973	−1.30772
$960$	0	0
$961$	−28.4992	−0.919328
$962$	0	0
$963$	0	0
$964$	0	0
$965$	17.4332	0.561195
$966$	0	0
$967$	27.8536	0.895710	0.447855	−	0.894106i	$-0.352188\pi$
$967$	27.8536	0.895710	0.447855	+	0.894106i	$0.352188\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16.5249	0.530309	0.265155	−	0.964206i	$-0.414577\pi$
$971$	16.5249	0.530309	0.265155	+	0.964206i	$0.414577\pi$
$972$	0	0
$973$	43.0112	1.37888
$974$	0	0
$975$	0	0
$976$	0	0
$977$	50.0059	1.59983	0.799915	−	0.600114i	$-0.204878\pi$
$977$	50.0059	1.59983	0.799915	+	0.600114i	$0.204878\pi$
$978$	0	0
$979$	−10.4873	−0.335176
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−22.7894	−0.726869	−0.363434	−	0.931620i	$-0.618396\pi$
$983$	−22.7894	−0.726869	−0.363434	+	0.931620i	$0.618396\pi$
$984$	0	0
$985$	23.4402	0.746866
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	33.9822	1.07948	0.539740	−	0.841832i	$-0.318522\pi$
$991$	33.9822	1.07948	0.539740	+	0.841832i	$0.318522\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−1.29759	−0.0411365
$996$	0	0
$997$	−17.7258	−0.561382	−0.280691	−	0.959798i	$-0.590564\pi$
$997$	−17.7258	−0.561382	−0.280691	+	0.959798i	$0.590564\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bs.1.1		5
3.2	odd	2		920.2.a.j.1.2	✓	5
12.11	even	2		1840.2.a.v.1.4		5
15.2	even	4		4600.2.e.u.4049.7		10
15.8	even	4		4600.2.e.u.4049.4		10
15.14	odd	2		4600.2.a.be.1.4		5
24.5	odd	2		7360.2.a.co.1.4		5
24.11	even	2		7360.2.a.cp.1.2		5
60.59	even	2		9200.2.a.cu.1.2		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.j.1.2	✓	5	3.2	odd	2
1840.2.a.v.1.4		5	12.11	even	2
4600.2.a.be.1.4		5	15.14	odd	2
4600.2.e.u.4049.4		10	15.8	even	4
4600.2.e.u.4049.7		10	15.2	even	4
7360.2.a.co.1.4		5	24.5	odd	2
7360.2.a.cp.1.2		5	24.11	even	2
8280.2.a.bs.1.1		5	1.1	even	1	trivial
9200.2.a.cu.1.2		5	60.59	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{5} -4.66212 q^{7} +O(q^{10})\)	\(q+1.00000 q^{5} -4.66212 q^{7} -2.23020 q^{11} -2.80072 q^{13} -7.63271 q^{17} -1.36222 q^{19} +1.00000 q^{23} +1.00000 q^{25} -8.94362 q^{29} -1.58140 q^{31} -4.66212 q^{35} -1.40251 q^{37} -10.7134 q^{41} +7.26391 q^{47} +14.7353 q^{49} +8.38212 q^{53} -2.23020 q^{55} +4.88331 q^{59} +4.33282 q^{61} -2.80072 q^{65} +8.54355 q^{67} -8.81604 q^{71} -5.26391 q^{73} +10.3974 q^{77} +7.08222 q^{79} +4.59749 q^{83} -7.63271 q^{85} +4.70241 q^{89} +13.0573 q^{91} -1.36222 q^{95} +16.5160 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 5 q^{5} - 2 q^{7}+O(q^{10}) \) 5 * q + 5 * q^5 - 2 * q^7	\( 5 q + 5 q^{5} - 2 q^{7} + q^{11} + 4 q^{13} - 4 q^{17} + 7 q^{19} + 5 q^{23} + 5 q^{25} - 4 q^{29} + 19 q^{31} - 2 q^{35} + 15 q^{37} - 25 q^{41} + 11 q^{47} + 25 q^{49} - 3 q^{53} + q^{55} + q^{59} - 5 q^{61} + 4 q^{65} + 9 q^{67} - q^{71} - q^{73} - 18 q^{77} - 2 q^{79} + 45 q^{83} - 4 q^{85} - 6 q^{89} + 11 q^{91} + 7 q^{95} + 25 q^{97}+O(q^{100}) \) 5 * q + 5 * q^5 - 2 * q^7 + q^11 + 4 * q^13 - 4 * q^17 + 7 * q^19 + 5 * q^23 + 5 * q^25 - 4 * q^29 + 19 * q^31 - 2 * q^35 + 15 * q^37 - 25 * q^41 + 11 * q^47 + 25 * q^49 - 3 * q^53 + q^55 + q^59 - 5 * q^61 + 4 * q^65 + 9 * q^67 - q^71 - q^73 - 18 * q^77 - 2 * q^79 + 45 * q^83 - 4 * q^85 - 6 * q^89 + 11 * q^91 + 7 * q^95 + 25 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more