LMFDB - Embedded newform 8280.2.a.bk.1.3

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:	$1$
Dimension:	$3$
Coefficient field:	3.3.568.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 6x - 2 $ x^3 - x^2 - 6*x - 2
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$3.12489$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} +4.12489 q^{7} +O(q^{10})$	$q-1.00000 q^{5} +4.12489 q^{7} -4.64002 q^{11} +2.64002 q^{13} -4.12489 q^{17} -1.60975 q^{19} +1.00000 q^{23} +1.00000 q^{25} +1.48486 q^{29} -0.515138 q^{31} -4.12489 q^{35} -2.12489 q^{37} -4.51514 q^{41} +5.28005 q^{43} -2.39025 q^{47} +10.0147 q^{49} -4.12489 q^{53} +4.64002 q^{55} +3.79518 q^{59} +2.32970 q^{61} -2.64002 q^{65} -0.185438 q^{67} -14.9541 q^{71} -10.1698 q^{73} -19.1396 q^{77} -1.93945 q^{79} -0.844838 q^{83} +4.12489 q^{85} +0.780505 q^{89} +10.8898 q^{91} +1.60975 q^{95} -9.21949 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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} - 6 q^{11} - 4 q^{17} + 4 q^{19} + 3 q^{23} + 3 q^{25} + 4 q^{29} - 2 q^{31} - 4 q^{35} + 2 q^{37} - 14 q^{41} - 16 q^{47} - 3 q^{49} - 4 q^{53} + 6 q^{55} - 4 q^{59} + 14 q^{61} + 6 q^{67} - 10 q^{71} + 10 q^{73} - 16 q^{77} - 4 q^{79} - 10 q^{83} + 4 q^{85} + 20 q^{89} + 8 q^{91} - 4 q^{95} - 10 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 4 * q^7 - 6 * q^11 - 4 * q^17 + 4 * q^19 + 3 * q^23 + 3 * q^25 + 4 * q^29 - 2 * q^31 - 4 * q^35 + 2 * q^37 - 14 * q^41 - 16 * q^47 - 3 * q^49 - 4 * q^53 + 6 * q^55 - 4 * q^59 + 14 * q^61 + 6 * q^67 - 10 * q^71 + 10 * q^73 - 16 * q^77 - 4 * q^79 - 10 * q^83 + 4 * q^85 + 20 * q^89 + 8 * q^91 - 4 * q^95 - 10 * 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
$7$	4.12489	1.55906	0.779530	−	0.626365i	$-0.215458\pi$
$7$	4.12489	1.55906	0.779530	+	0.626365i	$0.215458\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−4.64002	−1.39902	−0.699510	−	0.714623i	$-0.746598\pi$
$11$	−4.64002	−1.39902	−0.699510	+	0.714623i	$0.746598\pi$
$12$	0	0
$13$	2.64002	0.732211	0.366105	−	0.930573i	$-0.380691\pi$
$13$	2.64002	0.732211	0.366105	+	0.930573i	$0.380691\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−4.12489	−1.00043	−0.500216	−	0.865901i	$-0.666746\pi$
$17$	−4.12489	−1.00043	−0.500216	+	0.865901i	$0.666746\pi$
$18$	0	0
$19$	−1.60975	−0.369301	−0.184651	−	0.982804i	$-0.559115\pi$
$19$	−1.60975	−0.369301	−0.184651	+	0.982804i	$0.559115\pi$
$20$	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
$29$	1.48486	0.275732	0.137866	−	0.990451i	$-0.455976\pi$
$29$	1.48486	0.275732	0.137866	+	0.990451i	$0.455976\pi$
$30$	0	0
$31$	−0.515138	−0.0925215	−0.0462608	−	0.998929i	$-0.514731\pi$
$31$	−0.515138	−0.0925215	−0.0462608	+	0.998929i	$0.514731\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−4.12489	−0.697233
$36$	0	0
$37$	−2.12489	−0.349329	−0.174665	−	0.984628i	$-0.555884\pi$
$37$	−2.12489	−0.349329	−0.174665	+	0.984628i	$0.555884\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.51514	−0.705146	−0.352573	−	0.935784i	$-0.614693\pi$
$41$	−4.51514	−0.705146	−0.352573	+	0.935784i	$0.614693\pi$
$42$	0	0
$43$	5.28005	0.805200	0.402600	−	0.915376i	$-0.368107\pi$
$43$	5.28005	0.805200	0.402600	+	0.915376i	$0.368107\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−2.39025	−0.348654	−0.174327	−	0.984688i	$-0.555775\pi$
$47$	−2.39025	−0.348654	−0.174327	+	0.984688i	$0.555775\pi$
$48$	0	0
$49$	10.0147	1.43067
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−4.12489	−0.566597	−0.283298	−	0.959032i	$-0.591429\pi$
$53$	−4.12489	−0.566597	−0.283298	+	0.959032i	$0.591429\pi$
$54$	0	0
$55$	4.64002	0.625661
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.79518	0.494091	0.247045	−	0.969004i	$-0.420540\pi$
$59$	3.79518	0.494091	0.247045	+	0.969004i	$0.420540\pi$
$60$	0	0
$61$	2.32970	0.298288	0.149144	−	0.988816i	$-0.452348\pi$
$61$	2.32970	0.298288	0.149144	+	0.988816i	$0.452348\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−2.64002	−0.327455
$66$	0	0
$67$	−0.185438	−0.0226548	−0.0113274	−	0.999936i	$-0.503606\pi$
$67$	−0.185438	−0.0226548	−0.0113274	+	0.999936i	$0.503606\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−14.9541	−1.77473	−0.887364	−	0.461069i	$-0.847466\pi$
$71$	−14.9541	−1.77473	−0.887364	+	0.461069i	$0.847466\pi$
$72$	0	0
$73$	−10.1698	−1.19029	−0.595145	−	0.803618i	$-0.702905\pi$
$73$	−10.1698	−1.19029	−0.595145	+	0.803618i	$0.702905\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−19.1396	−2.18116
$78$	0	0
$79$	−1.93945	−0.218205	−0.109102	−	0.994031i	$-0.534798\pi$
$79$	−1.93945	−0.218205	−0.109102	+	0.994031i	$0.534798\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−0.844838	−0.0927331	−0.0463665	−	0.998924i	$-0.514764\pi$
$83$	−0.844838	−0.0927331	−0.0463665	+	0.998924i	$0.514764\pi$
$84$	0	0
$85$	4.12489	0.447407
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.780505	0.0827334	0.0413667	−	0.999144i	$-0.486829\pi$
$89$	0.780505	0.0827334	0.0413667	+	0.999144i	$0.486829\pi$
$90$	0	0
$91$	10.8898	1.14156
$92$	0	0
$93$	0	0
$94$	0	0
$95$	1.60975	0.165157
$96$	0	0
$97$	−9.21949	−0.936098	−0.468049	−	0.883703i	$-0.655043\pi$
$97$	−9.21949	−0.936098	−0.468049	+	0.883703i	$0.655043\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−1.23509	−0.122896	−0.0614481	−	0.998110i	$-0.519572\pi$
$101$	−1.23509	−0.122896	−0.0614481	+	0.998110i	$0.519572\pi$
$102$	0	0
$103$	−16.0294	−1.57942	−0.789710	−	0.613481i	$-0.789769\pi$
$103$	−16.0294	−1.57942	−0.789710	+	0.613481i	$0.789769\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−0.185438	−0.0179269	−0.00896347	−	0.999960i	$-0.502853\pi$
$107$	−0.185438	−0.0179269	−0.00896347	+	0.999960i	$0.502853\pi$
$108$	0	0
$109$	5.92007	0.567040	0.283520	−	0.958966i	$-0.408498\pi$
$109$	5.92007	0.567040	0.283520	+	0.958966i	$0.408498\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	6.31410	0.593981	0.296990	−	0.954880i	$-0.404017\pi$
$113$	6.31410	0.593981	0.296990	+	0.954880i	$0.404017\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−17.0147	−1.55973
$120$	0	0
$121$	10.5298	0.957256
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−19.0790	−1.69299	−0.846494	−	0.532398i	$-0.821291\pi$
$127$	−19.0790	−1.69299	−0.846494	+	0.532398i	$0.821291\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	11.2800	0.985542	0.492771	−	0.870159i	$-0.335984\pi$
$131$	11.2800	0.985542	0.492771	+	0.870159i	$0.335984\pi$
$132$	0	0
$133$	−6.64002	−0.575763
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0.310323	0.0265127	0.0132563	−	0.999912i	$-0.495780\pi$
$137$	0.310323	0.0265127	0.0132563	+	0.999912i	$0.495780\pi$
$138$	0	0
$139$	−7.85574	−0.666315	−0.333157	−	0.942871i	$-0.608114\pi$
$139$	−7.85574	−0.666315	−0.333157	+	0.942871i	$0.608114\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−12.2498	−1.02438
$144$	0	0
$145$	−1.48486	−0.123311
$146$	0	0
$147$	0	0
$148$	0	0
$149$	10.1698	0.833146	0.416573	−	0.909102i	$-0.363231\pi$
$149$	10.1698	0.833146	0.416573	+	0.909102i	$0.363231\pi$
$150$	0	0
$151$	1.03028	0.0838427	0.0419213	−	0.999121i	$-0.486652\pi$
$151$	1.03028	0.0838427	0.0419213	+	0.999121i	$0.486652\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	0.515138	0.0413769
$156$	0	0
$157$	4.06433	0.324369	0.162185	−	0.986760i	$-0.448146\pi$
$157$	4.06433	0.324369	0.162185	+	0.986760i	$0.448146\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	4.12489	0.325087
$162$	0	0
$163$	6.18922	0.484777	0.242389	−	0.970179i	$-0.422069\pi$
$163$	6.18922	0.484777	0.242389	+	0.970179i	$0.422069\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.3893	0.881333	0.440667	−	0.897671i	$-0.354742\pi$
$167$	11.3893	0.881333	0.440667	+	0.897671i	$0.354742\pi$
$168$	0	0
$169$	−6.03028	−0.463867
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−6.90917	−0.525295	−0.262647	−	0.964892i	$-0.584596\pi$
$173$	−6.90917	−0.525295	−0.262647	+	0.964892i	$0.584596\pi$
$174$	0	0
$175$	4.12489	0.311812
$176$	0	0
$177$	0	0
$178$	0	0
$179$	12.0294	0.899117	0.449558	−	0.893251i	$-0.351581\pi$
$179$	12.0294	0.899117	0.449558	+	0.893251i	$0.351581\pi$
$180$	0	0
$181$	−3.81078	−0.283253	−0.141627	−	0.989920i	$-0.545233\pi$
$181$	−3.81078	−0.283253	−0.141627	+	0.989920i	$0.545233\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	2.12489	0.156225
$186$	0	0
$187$	19.1396	1.39962
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−0.0799296	−0.00578350	−0.00289175	−	0.999996i	$-0.500920\pi$
$191$	−0.0799296	−0.00578350	−0.00289175	+	0.999996i	$0.500920\pi$
$192$	0	0
$193$	−16.9697	−1.22151	−0.610754	−	0.791821i	$-0.709133\pi$
$193$	−16.9697	−1.22151	−0.610754	+	0.791821i	$0.709133\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−10.6206	−0.756690	−0.378345	−	0.925665i	$-0.623507\pi$
$197$	−10.6206	−0.756690	−0.378345	+	0.925665i	$0.623507\pi$
$198$	0	0
$199$	−17.5298	−1.24266	−0.621328	−	0.783551i	$-0.713407\pi$
$199$	−17.5298	−1.24266	−0.621328	+	0.783551i	$0.713407\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.12489	0.429883
$204$	0	0
$205$	4.51514	0.315351
$206$	0	0
$207$	0	0
$208$	0	0
$209$	7.46927	0.516660
$210$	0	0
$211$	−2.70436	−0.186176	−0.0930878	−	0.995658i	$-0.529674\pi$
$211$	−2.70436	−0.186176	−0.0930878	+	0.995658i	$0.529674\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.28005	−0.360096
$216$	0	0
$217$	−2.12489	−0.144247
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−10.8898	−0.732527
$222$	0	0
$223$	−1.21949	−0.0816634	−0.0408317	−	0.999166i	$-0.513001\pi$
$223$	−1.21949	−0.0816634	−0.0408317	+	0.999166i	$0.513001\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2.56009	0.169919	0.0849597	−	0.996384i	$-0.472924\pi$
$227$	2.56009	0.169919	0.0849597	+	0.996384i	$0.472924\pi$
$228$	0	0
$229$	−17.9688	−1.18741	−0.593706	−	0.804682i	$-0.702336\pi$
$229$	−17.9688	−1.18741	−0.593706	+	0.804682i	$0.702336\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−4.18922	−0.274445	−0.137222	−	0.990540i	$-0.543818\pi$
$233$	−4.18922	−0.274445	−0.137222	+	0.990540i	$0.543818\pi$
$234$	0	0
$235$	2.39025	0.155923
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−16.3553	−1.05794	−0.528968	−	0.848642i	$-0.677421\pi$
$239$	−16.3553	−1.05794	−0.528968	+	0.848642i	$0.677421\pi$
$240$	0	0
$241$	11.0790	0.713662	0.356831	−	0.934169i	$-0.383857\pi$
$241$	11.0790	0.713662	0.356831	+	0.934169i	$0.383857\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−10.0147	−0.639814
$246$	0	0
$247$	−4.24977	−0.270406
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−15.2800	−0.964468	−0.482234	−	0.876042i	$-0.660175\pi$
$251$	−15.2800	−0.964468	−0.482234	+	0.876042i	$0.660175\pi$
$252$	0	0
$253$	−4.64002	−0.291716
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−4.14048	−0.258276	−0.129138	−	0.991627i	$-0.541221\pi$
$257$	−4.14048	−0.258276	−0.129138	+	0.991627i	$0.541221\pi$
$258$	0	0
$259$	−8.76491	−0.544625
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−11.5260	−0.710726	−0.355363	−	0.934728i	$-0.615643\pi$
$263$	−11.5260	−0.710726	−0.355363	+	0.934728i	$0.615643\pi$
$264$	0	0
$265$	4.12489	0.253390
$266$	0	0
$267$	0	0
$268$	0	0
$269$	28.5445	1.74039	0.870194	−	0.492709i	$-0.163993\pi$
$269$	28.5445	1.74039	0.870194	+	0.492709i	$0.163993\pi$
$270$	0	0
$271$	22.1736	1.34695	0.673476	−	0.739209i	$-0.264801\pi$
$271$	22.1736	1.34695	0.673476	+	0.739209i	$0.264801\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−4.64002	−0.279804
$276$	0	0
$277$	−4.24977	−0.255344	−0.127672	−	0.991816i	$-0.540750\pi$
$277$	−4.24977	−0.255344	−0.127672	+	0.991816i	$0.540750\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−11.0109	−0.656855	−0.328428	−	0.944529i	$-0.606519\pi$
$281$	−11.0109	−0.656855	−0.328428	+	0.944529i	$0.606519\pi$
$282$	0	0
$283$	1.21571	0.0722667	0.0361333	−	0.999347i	$-0.488496\pi$
$283$	1.21571	0.0722667	0.0361333	+	0.999347i	$0.488496\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−18.6244	−1.09937
$288$	0	0
$289$	0.0146797	0.000863513 0
$290$	0	0
$291$	0	0
$292$	0	0
$293$	15.8439	0.925612	0.462806	−	0.886460i	$-0.346843\pi$
$293$	15.8439	0.925612	0.462806	+	0.886460i	$0.346843\pi$
$294$	0	0
$295$	−3.79518	−0.220964
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2.64002	0.152676
$300$	0	0
$301$	21.7796	1.25535
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.32970	−0.133398
$306$	0	0
$307$	8.86043	0.505692	0.252846	−	0.967507i	$-0.418633\pi$
$307$	8.86043	0.505692	0.252846	+	0.967507i	$0.418633\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	22.6888	1.28656	0.643281	−	0.765630i	$-0.277573\pi$
$311$	22.6888	1.28656	0.643281	+	0.765630i	$0.277573\pi$
$312$	0	0
$313$	27.6841	1.56480	0.782398	−	0.622779i	$-0.213996\pi$
$313$	27.6841	1.56480	0.782398	+	0.622779i	$0.213996\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.9192	−1.51193	−0.755965	−	0.654612i	$-0.772832\pi$
$317$	−26.9192	−1.51193	−0.755965	+	0.654612i	$0.772832\pi$
$318$	0	0
$319$	−6.88979	−0.385754
$320$	0	0
$321$	0	0
$322$	0	0
$323$	6.64002	0.369461
$324$	0	0
$325$	2.64002	0.146442
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−9.85952	−0.543573
$330$	0	0
$331$	30.9541	1.70139	0.850696	−	0.525657i	$-0.176181\pi$
$331$	30.9541	1.70139	0.850696	+	0.525657i	$0.176181\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0.185438	0.0101315
$336$	0	0
$337$	−20.1892	−1.09978	−0.549888	−	0.835238i	$-0.685330\pi$
$337$	−20.1892	−1.09978	−0.549888	+	0.835238i	$0.685330\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.39025	0.129439
$342$	0	0
$343$	12.4352	0.671438
$344$	0	0
$345$	0	0
$346$	0	0
$347$	0	0		−	1.00000i	$-0.5\pi$
$347$	0	0			1.00000i	$0.5\pi$
$348$	0	0
$349$	0.0449558	0.00240643	0.00120321	−	0.999999i	$-0.499617\pi$
$349$	0.0449558	0.00240643	0.00120321	+	0.999999i	$0.499617\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	20.9797	1.11664	0.558319	−	0.829627i	$-0.311447\pi$
$353$	20.9797	1.11664	0.558319	+	0.829627i	$0.311447\pi$
$354$	0	0
$355$	14.9541	0.793683
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−14.1405	−0.746306	−0.373153	−	0.927770i	$-0.621723\pi$
$359$	−14.1405	−0.746306	−0.373153	+	0.927770i	$0.621723\pi$
$360$	0	0
$361$	−16.4087	−0.863616
$362$	0	0
$363$	0	0
$364$	0	0
$365$	10.1698	0.532314
$366$	0	0
$367$	9.93567	0.518638	0.259319	−	0.965792i	$-0.416502\pi$
$367$	9.93567	0.518638	0.259319	+	0.965792i	$0.416502\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−17.0147	−0.883358
$372$	0	0
$373$	−28.2110	−1.46071	−0.730356	−	0.683067i	$-0.760646\pi$
$373$	−28.2110	−1.46071	−0.730356	+	0.683067i	$0.760646\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	3.92007	0.201894
$378$	0	0
$379$	23.5592	1.21015	0.605077	−	0.796167i	$-0.293142\pi$
$379$	23.5592	1.21015	0.605077	+	0.796167i	$0.293142\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−29.3444	−1.49943	−0.749714	−	0.661762i	$-0.769809\pi$
$383$	−29.3444	−1.49943	−0.749714	+	0.661762i	$0.769809\pi$
$384$	0	0
$385$	19.1396	0.975443
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−5.09839	−0.258499	−0.129249	−	0.991612i	$-0.541257\pi$
$389$	−5.09839	−0.258499	−0.129249	+	0.991612i	$0.541257\pi$
$390$	0	0
$391$	−4.12489	−0.208604
$392$	0	0
$393$	0	0
$394$	0	0
$395$	1.93945	0.0975842
$396$	0	0
$397$	−22.9385	−1.15125	−0.575626	−	0.817713i	$-0.695242\pi$
$397$	−22.9385	−1.15125	−0.575626	+	0.817713i	$0.695242\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.1892	−1.10808	−0.554038	−	0.832491i	$-0.686914\pi$
$401$	−22.1892	−1.10808	−0.554038	+	0.832491i	$0.686914\pi$
$402$	0	0
$403$	−1.35998	−0.0677453
$404$	0	0
$405$	0	0
$406$	0	0
$407$	9.85952	0.488718
$408$	0	0
$409$	−19.6741	−0.972821	−0.486410	−	0.873730i	$-0.661694\pi$
$409$	−19.6741	−0.972821	−0.486410	+	0.873730i	$0.661694\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	15.6547	0.770318
$414$	0	0
$415$	0.844838	0.0414715
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−30.4196	−1.48610	−0.743048	−	0.669239i	$-0.766620\pi$
$419$	−30.4196	−1.48610	−0.743048	+	0.669239i	$0.766620\pi$
$420$	0	0
$421$	−36.8586	−1.79638	−0.898189	−	0.439609i	$-0.855117\pi$
$421$	−36.8586	−1.79638	−0.898189	+	0.439609i	$0.855117\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−4.12489	−0.200086
$426$	0	0
$427$	9.60975	0.465048
$428$	0	0
$429$	0	0
$430$	0	0
$431$	19.0596	0.918070	0.459035	−	0.888418i	$-0.348195\pi$
$431$	19.0596	0.918070	0.459035	+	0.888418i	$0.348195\pi$
$432$	0	0
$433$	−28.6244	−1.37560	−0.687801	−	0.725899i	$-0.741424\pi$
$433$	−28.6244	−1.37560	−0.687801	+	0.725899i	$0.741424\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−1.60975	−0.0770047
$438$	0	0
$439$	−20.2498	−0.966469	−0.483234	−	0.875491i	$-0.660538\pi$
$439$	−20.2498	−0.966469	−0.483234	+	0.875491i	$0.660538\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−6.76869	−0.321590	−0.160795	−	0.986988i	$-0.551406\pi$
$443$	−6.76869	−0.321590	−0.160795	+	0.986988i	$0.551406\pi$
$444$	0	0
$445$	−0.780505	−0.0369995
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−16.3553	−0.771853	−0.385927	−	0.922529i	$-0.626118\pi$
$449$	−16.3553	−0.771853	−0.385927	+	0.922529i	$0.626118\pi$
$450$	0	0
$451$	20.9503	0.986513
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.8898	−0.510521
$456$	0	0
$457$	−15.4655	−0.723445	−0.361722	−	0.932286i	$-0.617811\pi$
$457$	−15.4655	−0.723445	−0.361722	+	0.932286i	$0.617811\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−4.31032	−0.200752	−0.100376	−	0.994950i	$-0.532005\pi$
$461$	−4.31032	−0.200752	−0.100376	+	0.994950i	$0.532005\pi$
$462$	0	0
$463$	18.5483	0.862012	0.431006	−	0.902349i	$-0.358159\pi$
$463$	18.5483	0.862012	0.431006	+	0.902349i	$0.358159\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−18.2148	−0.842880	−0.421440	−	0.906856i	$-0.638475\pi$
$467$	−18.2148	−0.842880	−0.421440	+	0.906856i	$0.638475\pi$
$468$	0	0
$469$	−0.764909	−0.0353202
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−24.4995	−1.12649
$474$	0	0
$475$	−1.60975	−0.0738603
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−33.5786	−1.53424	−0.767122	−	0.641502i	$-0.778312\pi$
$479$	−33.5786	−1.53424	−0.767122	+	0.641502i	$0.778312\pi$
$480$	0	0
$481$	−5.60975	−0.255782
$482$	0	0
$483$	0	0
$484$	0	0
$485$	9.21949	0.418636
$486$	0	0
$487$	40.3591	1.82884	0.914422	−	0.404763i	$-0.132646\pi$
$487$	40.3591	1.82884	0.914422	+	0.404763i	$0.132646\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	8.16606	0.368529	0.184265	−	0.982877i	$-0.441010\pi$
$491$	8.16606	0.368529	0.184265	+	0.982877i	$0.441010\pi$
$492$	0	0
$493$	−6.12489	−0.275851
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−61.6841	−2.76691
$498$	0	0
$499$	−25.0147	−1.11981	−0.559905	−	0.828557i	$-0.689163\pi$
$499$	−25.0147	−1.11981	−0.559905	+	0.828557i	$0.689163\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−35.9944	−1.60491	−0.802455	−	0.596712i	$-0.796473\pi$
$503$	−35.9944	−1.60491	−0.802455	+	0.596712i	$0.796473\pi$
$504$	0	0
$505$	1.23509	0.0549608
$506$	0	0
$507$	0	0
$508$	0	0
$509$	16.3179	0.723278	0.361639	−	0.932318i	$-0.382217\pi$
$509$	16.3179	0.723278	0.361639	+	0.932318i	$0.382217\pi$
$510$	0	0
$511$	−41.9494	−1.85573
$512$	0	0
$513$	0	0
$514$	0	0
$515$	16.0294	0.706338
$516$	0	0
$517$	11.0908	0.487774
$518$	0	0
$519$	0	0
$520$	0	0
$521$	29.8283	1.30680	0.653401	−	0.757012i	$-0.273341\pi$
$521$	29.8283	1.30680	0.653401	+	0.757012i	$0.273341\pi$
$522$	0	0
$523$	35.9688	1.57281	0.786403	−	0.617714i	$-0.211941\pi$
$523$	35.9688	1.57281	0.786403	+	0.617714i	$0.211941\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	2.12489	0.0925615
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−11.9201	−0.516316
$534$	0	0
$535$	0.185438	0.00801717
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−46.4683	−2.00153
$540$	0	0
$541$	−22.1287	−0.951386	−0.475693	−	0.879611i	$-0.657803\pi$
$541$	−22.1287	−0.951386	−0.475693	+	0.879611i	$0.657803\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.92007	−0.253588
$546$	0	0
$547$	20.6206	0.881675	0.440838	−	0.897587i	$-0.354681\pi$
$547$	20.6206	0.881675	0.440838	+	0.897587i	$0.354681\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−2.39025	−0.101828
$552$	0	0
$553$	−8.00000	−0.340195
$554$	0	0
$555$	0	0
$556$	0	0
$557$	18.5251	0.784935	0.392467	−	0.919766i	$-0.371622\pi$
$557$	18.5251	0.784935	0.392467	+	0.919766i	$0.371622\pi$
$558$	0	0
$559$	13.9394	0.589576
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−5.87511	−0.247607	−0.123803	−	0.992307i	$-0.539509\pi$
$563$	−5.87511	−0.247607	−0.123803	+	0.992307i	$0.539509\pi$
$564$	0	0
$565$	−6.31410	−0.265636
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−5.03028	−0.210880	−0.105440	−	0.994426i	$-0.533625\pi$
$569$	−5.03028	−0.210880	−0.105440	+	0.994426i	$0.533625\pi$
$570$	0	0
$571$	41.8208	1.75014	0.875072	−	0.483992i	$-0.160814\pi$
$571$	41.8208	1.75014	0.875072	+	0.483992i	$0.160814\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	−16.4314	−0.684049	−0.342025	−	0.939691i	$-0.611113\pi$
$577$	−16.4314	−0.684049	−0.342025	+	0.939691i	$0.611113\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−3.48486	−0.144576
$582$	0	0
$583$	19.1396	0.792680
$584$	0	0
$585$	0	0
$586$	0	0
$587$	14.0294	0.579054	0.289527	−	0.957170i	$-0.406502\pi$
$587$	14.0294	0.579054	0.289527	+	0.957170i	$0.406502\pi$
$588$	0	0
$589$	0.829242	0.0341683
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−29.1396	−1.19662	−0.598309	−	0.801265i	$-0.704161\pi$
$593$	−29.1396	−1.19662	−0.598309	+	0.801265i	$0.704161\pi$
$594$	0	0
$595$	17.0147	0.697534
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.7106	1.33652	0.668259	−	0.743929i	$-0.267040\pi$
$599$	32.7106	1.33652	0.668259	+	0.743929i	$0.267040\pi$
$600$	0	0
$601$	−16.7044	−0.681385	−0.340692	−	0.940175i	$-0.610661\pi$
$601$	−16.7044	−0.681385	−0.340692	+	0.940175i	$0.610661\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.5298	−0.428098
$606$	0	0
$607$	−0.480164	−0.0194893	−0.00974463	−	0.999953i	$-0.503102\pi$
$607$	−0.480164	−0.0194893	−0.00974463	+	0.999953i	$0.503102\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6.31032	−0.255288
$612$	0	0
$613$	−8.65940	−0.349750	−0.174875	−	0.984591i	$-0.555952\pi$
$613$	−8.65940	−0.349750	−0.174875	+	0.984591i	$0.555952\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11.0559	0.445092	0.222546	−	0.974922i	$-0.428563\pi$
$617$	11.0559	0.445092	0.222546	+	0.974922i	$0.428563\pi$
$618$	0	0
$619$	−33.6197	−1.35129	−0.675646	−	0.737227i	$-0.736135\pi$
$619$	−33.6197	−1.35129	−0.675646	+	0.737227i	$0.736135\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	3.21949	0.128986
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	8.76491	0.349480
$630$	0	0
$631$	3.95883	0.157598	0.0787992	−	0.996891i	$-0.474891\pi$
$631$	3.95883	0.157598	0.0787992	+	0.996891i	$0.474891\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	19.0790	0.757128
$636$	0	0
$637$	26.4390	1.04755
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−6.26915	−0.247616	−0.123808	−	0.992306i	$-0.539511\pi$
$641$	−6.26915	−0.247616	−0.123808	+	0.992306i	$0.539511\pi$
$642$	0	0
$643$	−12.9348	−0.510097	−0.255048	−	0.966928i	$-0.582091\pi$
$643$	−12.9348	−0.510097	−0.255048	+	0.966928i	$0.582091\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	36.9192	1.45144	0.725721	−	0.687989i	$-0.241506\pi$
$647$	36.9192	1.45144	0.725721	+	0.687989i	$0.241506\pi$
$648$	0	0
$649$	−17.6097	−0.691243
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−16.8898	−0.660949	−0.330474	−	0.943815i	$-0.607209\pi$
$653$	−16.8898	−0.660949	−0.330474	+	0.943815i	$0.607209\pi$
$654$	0	0
$655$	−11.2800	−0.440748
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−21.4205	−0.834425	−0.417213	−	0.908809i	$-0.636993\pi$
$659$	−21.4205	−0.834425	−0.417213	+	0.908809i	$0.636993\pi$
$660$	0	0
$661$	3.81834	0.148516	0.0742582	−	0.997239i	$-0.476341\pi$
$661$	3.81834	0.148516	0.0742582	+	0.997239i	$0.476341\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	6.64002	0.257489
$666$	0	0
$667$	1.48486	0.0574941
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−10.8099	−0.417310
$672$	0	0
$673$	−40.4802	−1.56040	−0.780198	−	0.625533i	$-0.784882\pi$
$673$	−40.4802	−1.56040	−0.780198	+	0.625533i	$0.784882\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−27.8751	−1.07133	−0.535664	−	0.844431i	$-0.679939\pi$
$677$	−27.8751	−1.07133	−0.535664	+	0.844431i	$0.679939\pi$
$678$	0	0
$679$	−38.0294	−1.45943
$680$	0	0
$681$	0	0
$682$	0	0
$683$	12.3297	0.471783	0.235891	−	0.971779i	$-0.424199\pi$
$683$	12.3297	0.471783	0.235891	+	0.971779i	$0.424199\pi$
$684$	0	0
$685$	−0.310323	−0.0118568
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−10.8898	−0.414868
$690$	0	0
$691$	49.0890	1.86743	0.933717	−	0.358013i	$-0.116546\pi$
$691$	49.0890	1.86743	0.933717	+	0.358013i	$0.116546\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	7.85574	0.297985
$696$	0	0
$697$	18.6244	0.705450
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−36.7299	−1.38727	−0.693635	−	0.720326i	$-0.743992\pi$
$701$	−36.7299	−1.38727	−0.693635	+	0.720326i	$0.743992\pi$
$702$	0	0
$703$	3.42053	0.129008
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−5.09461	−0.191603
$708$	0	0
$709$	−1.33062	−0.0499724	−0.0249862	−	0.999688i	$-0.507954\pi$
$709$	−1.33062	−0.0499724	−0.0249862	+	0.999688i	$0.507954\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−0.515138	−0.0192921
$714$	0	0
$715$	12.2498	0.458115
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−31.7034	−1.18234	−0.591169	−	0.806547i	$-0.701334\pi$
$719$	−31.7034	−1.18234	−0.591169	+	0.806547i	$0.701334\pi$
$720$	0	0
$721$	−66.1193	−2.46241
$722$	0	0
$723$	0	0
$724$	0	0
$725$	1.48486	0.0551464
$726$	0	0
$727$	20.7455	0.769409	0.384705	−	0.923040i	$-0.374303\pi$
$727$	20.7455	0.769409	0.384705	+	0.923040i	$0.374303\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−21.7796	−0.805547
$732$	0	0
$733$	−3.81456	−0.140894	−0.0704470	−	0.997516i	$-0.522443\pi$
$733$	−3.81456	−0.140894	−0.0704470	+	0.997516i	$0.522443\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0.860435	0.0316945
$738$	0	0
$739$	12.2654	0.451189	0.225594	−	0.974221i	$-0.427568\pi$
$739$	12.2654	0.451189	0.225594	+	0.974221i	$0.427568\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.9991	0.917127	0.458564	−	0.888662i	$-0.348364\pi$
$743$	24.9991	0.917127	0.458564	+	0.888662i	$0.348364\pi$
$744$	0	0
$745$	−10.1698	−0.372594
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−0.764909	−0.0279492
$750$	0	0
$751$	−10.7687	−0.392955	−0.196478	−	0.980508i	$-0.562950\pi$
$751$	−10.7687	−0.392955	−0.196478	+	0.980508i	$0.562950\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−1.03028	−0.0374956
$756$	0	0
$757$	24.4352	0.888113	0.444056	−	0.895999i	$-0.353539\pi$
$757$	24.4352	0.888113	0.444056	+	0.895999i	$0.353539\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−6.98532	−0.253218	−0.126609	−	0.991953i	$-0.540409\pi$
$761$	−6.98532	−0.253218	−0.126609	+	0.991953i	$0.540409\pi$
$762$	0	0
$763$	24.4196	0.884049
$764$	0	0
$765$	0	0
$766$	0	0
$767$	10.0194	0.361779
$768$	0	0
$769$	−28.0705	−1.01225	−0.506125	−	0.862460i	$-0.668922\pi$
$769$	−28.0705	−1.01225	−0.506125	+	0.862460i	$0.668922\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−20.2791	−0.729390	−0.364695	−	0.931127i	$-0.618827\pi$
$773$	−20.2791	−0.729390	−0.364695	+	0.931127i	$0.618827\pi$
$774$	0	0
$775$	−0.515138	−0.0185043
$776$	0	0
$777$	0	0
$778$	0	0
$779$	7.26823	0.260411
$780$	0	0
$781$	69.3875	2.48288
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−4.06433	−0.145062
$786$	0	0
$787$	1.50424	0.0536203	0.0268102	−	0.999641i	$-0.491465\pi$
$787$	1.50424	0.0536203	0.0268102	+	0.999641i	$0.491465\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	26.0450	0.926052
$792$	0	0
$793$	6.15046	0.218409
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−55.0247	−1.94907	−0.974537	−	0.224228i	$-0.928014\pi$
$797$	−55.0247	−1.94907	−0.974537	+	0.224228i	$0.928014\pi$
$798$	0	0
$799$	9.85952	0.348805
$800$	0	0
$801$	0	0
$802$	0	0
$803$	47.1883	1.66524
$804$	0	0
$805$	−4.12489	−0.145383
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−4.35528	−0.153123	−0.0765617	−	0.997065i	$-0.524394\pi$
$809$	−4.35528	−0.153123	−0.0765617	+	0.997065i	$0.524394\pi$
$810$	0	0
$811$	−1.51422	−0.0531715	−0.0265858	−	0.999647i	$-0.508464\pi$
$811$	−1.51422	−0.0531715	−0.0265858	+	0.999647i	$0.508464\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−6.18922	−0.216799
$816$	0	0
$817$	−8.49954	−0.297361
$818$	0	0
$819$	0	0
$820$	0	0
$821$	21.1807	0.739213	0.369606	−	0.929188i	$-0.379493\pi$
$821$	21.1807	0.739213	0.369606	+	0.929188i	$0.379493\pi$
$822$	0	0
$823$	−2.12867	−0.0742006	−0.0371003	−	0.999312i	$-0.511812\pi$
$823$	−2.12867	−0.0742006	−0.0371003	+	0.999312i	$0.511812\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	29.0559	1.01037	0.505186	−	0.863011i	$-0.331424\pi$
$827$	29.0559	1.01037	0.505186	+	0.863011i	$0.331424\pi$
$828$	0	0
$829$	3.55298	0.123400	0.0617000	−	0.998095i	$-0.480348\pi$
$829$	3.55298	0.123400	0.0617000	+	0.998095i	$0.480348\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−41.3094	−1.43129
$834$	0	0
$835$	−11.3893	−0.394144
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−9.03028	−0.311760	−0.155880	−	0.987776i	$-0.549821\pi$
$839$	−9.03028	−0.311760	−0.155880	+	0.987776i	$0.549821\pi$
$840$	0	0
$841$	−26.7952	−0.923972
$842$	0	0
$843$	0	0
$844$	0	0
$845$	6.03028	0.207448
$846$	0	0
$847$	43.4343	1.49242
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−2.12489	−0.0728401
$852$	0	0
$853$	0.530734	0.0181720	0.00908600	−	0.999959i	$-0.497108\pi$
$853$	0.530734	0.0181720	0.00908600	+	0.999959i	$0.497108\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.2800	0.931869	0.465934	−	0.884819i	$-0.345718\pi$
$857$	27.2800	0.931869	0.465934	+	0.884819i	$0.345718\pi$
$858$	0	0
$859$	11.6060	0.395990	0.197995	−	0.980203i	$-0.436557\pi$
$859$	11.6060	0.395990	0.197995	+	0.980203i	$0.436557\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	36.5895	1.24552	0.622760	−	0.782413i	$-0.286011\pi$
$863$	36.5895	1.24552	0.622760	+	0.782413i	$0.286011\pi$
$864$	0	0
$865$	6.90917	0.234919
$866$	0	0
$867$	0	0
$868$	0	0
$869$	8.99908	0.305273
$870$	0	0
$871$	−0.489560	−0.0165881
$872$	0	0
$873$	0	0
$874$	0	0
$875$	−4.12489	−0.139447
$876$	0	0
$877$	51.9688	1.75486	0.877431	−	0.479703i	$-0.159256\pi$
$877$	51.9688	1.75486	0.877431	+	0.479703i	$0.159256\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−25.7384	−0.867149	−0.433575	−	0.901118i	$-0.642748\pi$
$881$	−25.7384	−0.867149	−0.433575	+	0.901118i	$0.642748\pi$
$882$	0	0
$883$	−39.8889	−1.34237	−0.671184	−	0.741291i	$-0.734214\pi$
$883$	−39.8889	−1.34237	−0.671184	+	0.741291i	$0.734214\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	35.0908	1.17823	0.589117	−	0.808047i	$-0.299476\pi$
$887$	35.0908	1.17823	0.589117	+	0.808047i	$0.299476\pi$
$888$	0	0
$889$	−78.6987	−2.63947
$890$	0	0
$891$	0	0
$892$	0	0
$893$	3.84770	0.128758
$894$	0	0
$895$	−12.0294	−0.402097
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−0.764909	−0.0255111
$900$	0	0
$901$	17.0147	0.566841
$902$	0	0
$903$	0	0
$904$	0	0
$905$	3.81078	0.126675
$906$	0	0
$907$	−18.5932	−0.617378	−0.308689	−	0.951163i	$-0.599890\pi$
$907$	−18.5932	−0.617378	−0.308689	+	0.951163i	$0.599890\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−24.8392	−0.822960	−0.411480	−	0.911419i	$-0.634988\pi$
$911$	−24.8392	−0.822960	−0.411480	+	0.911419i	$0.634988\pi$
$912$	0	0
$913$	3.92007	0.129735
$914$	0	0
$915$	0	0
$916$	0	0
$917$	46.5289	1.53652
$918$	0	0
$919$	55.8989	1.84393	0.921967	−	0.387269i	$-0.126582\pi$
$919$	55.8989	1.84393	0.921967	+	0.387269i	$0.126582\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−39.4792	−1.29948
$924$	0	0
$925$	−2.12489	−0.0698658
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−17.8851	−0.586791	−0.293395	−	0.955991i	$-0.594785\pi$
$929$	−17.8851	−0.586791	−0.293395	+	0.955991i	$0.594785\pi$
$930$	0	0
$931$	−16.1211	−0.528348
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−19.1396	−0.625931
$936$	0	0
$937$	46.4608	1.51781	0.758904	−	0.651203i	$-0.225735\pi$
$937$	46.4608	1.51781	0.758904	+	0.651203i	$0.225735\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	17.6703	0.576035	0.288018	−	0.957625i	$-0.407004\pi$
$941$	17.6703	0.576035	0.288018	+	0.957625i	$0.407004\pi$
$942$	0	0
$943$	−4.51514	−0.147033
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−21.5374	−0.699871	−0.349935	−	0.936774i	$-0.613796\pi$
$947$	−21.5374	−0.699871	−0.349935	+	0.936774i	$0.613796\pi$
$948$	0	0
$949$	−26.8486	−0.871543
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−9.62156	−0.311673	−0.155836	−	0.987783i	$-0.549807\pi$
$953$	−9.62156	−0.311673	−0.155836	+	0.987783i	$0.549807\pi$
$954$	0	0
$955$	0.0799296	0.00258646
$956$	0	0
$957$	0	0
$958$	0	0
$959$	1.28005	0.0413349
$960$	0	0
$961$	−30.7346	−0.991440
$962$	0	0
$963$	0	0
$964$	0	0
$965$	16.9697	0.546275
$966$	0	0
$967$	44.8586	1.44256	0.721278	−	0.692646i	$-0.243555\pi$
$967$	44.8586	1.44256	0.721278	+	0.692646i	$0.243555\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	7.19014	0.230742	0.115371	−	0.993322i	$-0.463194\pi$
$971$	7.19014	0.230742	0.115371	+	0.993322i	$0.463194\pi$
$972$	0	0
$973$	−32.4040	−1.03883
$974$	0	0
$975$	0	0
$976$	0	0
$977$	18.4428	0.590037	0.295018	−	0.955492i	$-0.404674\pi$
$977$	18.4428	0.590037	0.295018	+	0.955492i	$0.404674\pi$
$978$	0	0
$979$	−3.62156	−0.115746
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−19.2526	−0.614064	−0.307032	−	0.951699i	$-0.599336\pi$
$983$	−19.2526	−0.614064	−0.307032	+	0.951699i	$0.599336\pi$
$984$	0	0
$985$	10.6206	0.338402
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5.28005	0.167896
$990$	0	0
$991$	−14.3846	−0.456943	−0.228472	−	0.973551i	$-0.573373\pi$
$991$	−14.3846	−0.456943	−0.228472	+	0.973551i	$0.573373\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	17.5298	0.555733
$996$	0	0
$997$	22.0606	0.698665	0.349332	−	0.936999i	$-0.386408\pi$
$997$	22.0606	0.698665	0.349332	+	0.936999i	$0.386408\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bk.1.3	✓	3
3.2	odd	2		8280.2.a.bp.1.3	yes	3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8280.2.a.bk.1.3	✓	3	1.1	even	1	trivial
8280.2.a.bp.1.3	yes	3	3.2	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:	\( N \)	\(=\)	\( 8280 = 2^{3} \cdot 3^{2} \cdot 5 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8280.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} +4.12489 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} +4.12489 q^{7} -4.64002 q^{11} +2.64002 q^{13} -4.12489 q^{17} -1.60975 q^{19} +1.00000 q^{23} +1.00000 q^{25} +1.48486 q^{29} -0.515138 q^{31} -4.12489 q^{35} -2.12489 q^{37} -4.51514 q^{41} +5.28005 q^{43} -2.39025 q^{47} +10.0147 q^{49} -4.12489 q^{53} +4.64002 q^{55} +3.79518 q^{59} +2.32970 q^{61} -2.64002 q^{65} -0.185438 q^{67} -14.9541 q^{71} -10.1698 q^{73} -19.1396 q^{77} -1.93945 q^{79} -0.844838 q^{83} +4.12489 q^{85} +0.780505 q^{89} +10.8898 q^{91} +1.60975 q^{95} -9.21949 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 4 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 4 * q^7	\( 3 q - 3 q^{5} + 4 q^{7} - 6 q^{11} - 4 q^{17} + 4 q^{19} + 3 q^{23} + 3 q^{25} + 4 q^{29} - 2 q^{31} - 4 q^{35} + 2 q^{37} - 14 q^{41} - 16 q^{47} - 3 q^{49} - 4 q^{53} + 6 q^{55} - 4 q^{59} + 14 q^{61} + 6 q^{67} - 10 q^{71} + 10 q^{73} - 16 q^{77} - 4 q^{79} - 10 q^{83} + 4 q^{85} + 20 q^{89} + 8 q^{91} - 4 q^{95} - 10 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 4 * q^7 - 6 * q^11 - 4 * q^17 + 4 * q^19 + 3 * q^23 + 3 * q^25 + 4 * q^29 - 2 * q^31 - 4 * q^35 + 2 * q^37 - 14 * q^41 - 16 * q^47 - 3 * q^49 - 4 * q^53 + 6 * q^55 - 4 * q^59 + 14 * q^61 + 6 * q^67 - 10 * q^71 + 10 * q^73 - 16 * q^77 - 4 * q^79 - 10 * q^83 + 4 * q^85 + 20 * q^89 + 8 * q^91 - 4 * q^95 - 10 * q^97

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