LMFDB - Embedded newform 8280.2.a.bi.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:	$1$
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:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2760)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
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)

$f(q)$	$=$	$q-1.00000 q^{5} -4.76491 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -4.76491 q^{7} -5.60975 q^{11} +3.60975 q^{13} +4.12489 q^{17} +2.64002 q^{19} +1.00000 q^{23} +1.00000 q^{25} -3.87511 q^{29} +9.79518 q^{31} +4.76491 q^{35} -6.76491 q^{37} +11.3444 q^{41} -2.96972 q^{43} -1.35998 q^{47} +15.7044 q^{49} +3.09461 q^{53} +5.60975 q^{55} -12.4352 q^{59} -10.5795 q^{61} -3.60975 q^{65} +6.82546 q^{67} -11.6547 q^{71} +10.1698 q^{73} +26.7299 q^{77} +12.4995 q^{79} -10.1249 q^{83} -4.12489 q^{85} +8.31032 q^{89} -17.2001 q^{91} -2.64002 q^{95} +16.8099 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) $ 3 * q - 3 * q^5 + 2 * q^7	$ 3 q - 3 q^{5} + 2 q^{7} - 8 q^{11} + 2 q^{13} + 4 q^{17} + 3 q^{23} + 3 q^{25} - 20 q^{29} + 14 q^{31} - 2 q^{35} - 4 q^{37} + 8 q^{41} - 8 q^{43} - 12 q^{47} + 29 q^{49} + 8 q^{55} - 14 q^{59} - 22 q^{61} - 2 q^{65} + 6 q^{67} + 6 q^{71} - 10 q^{73} + 8 q^{77} + 4 q^{79} - 22 q^{83} - 4 q^{85} + 10 q^{89} - 12 q^{91} + 2 q^{97}+O(q^{100}) $ 3 * q - 3 * q^5 + 2 * q^7 - 8 * q^11 + 2 * q^13 + 4 * q^17 + 3 * q^23 + 3 * q^25 - 20 * q^29 + 14 * q^31 - 2 * q^35 - 4 * q^37 + 8 * q^41 - 8 * q^43 - 12 * q^47 + 29 * q^49 + 8 * q^55 - 14 * q^59 - 22 * q^61 - 2 * q^65 + 6 * q^67 + 6 * q^71 - 10 * q^73 + 8 * q^77 + 4 * q^79 - 22 * q^83 - 4 * q^85 + 10 * q^89 - 12 * q^91 + 2 * 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
$7$	−4.76491	−1.80097	−0.900483	−	0.434891i	$-0.856787\pi$
$7$	−4.76491	−1.80097	−0.900483	+	0.434891i	$0.856787\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.60975	−1.69140	−0.845701	−	0.533657i	$-0.820817\pi$
$11$	−5.60975	−1.69140	−0.845701	+	0.533657i	$0.820817\pi$
$12$	0	0
$13$	3.60975	1.00116	0.500582	−	0.865689i	$-0.333119\pi$
$13$	3.60975	1.00116	0.500582	+	0.865689i	$0.333119\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	4.12489	1.00043	0.500216	−	0.865901i	$-0.333254\pi$
$17$	4.12489	1.00043	0.500216	+	0.865901i	$0.333254\pi$
$18$	0	0
$19$	2.64002	0.605663	0.302831	−	0.953044i	$-0.402068\pi$
$19$	2.64002	0.605663	0.302831	+	0.953044i	$0.402068\pi$
$20$	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
$29$	−3.87511	−0.719591	−0.359795	−	0.933031i	$-0.617153\pi$
$29$	−3.87511	−0.719591	−0.359795	+	0.933031i	$0.617153\pi$
$30$	0	0
$31$	9.79518	1.75927	0.879634	−	0.475652i	$-0.157788\pi$
$31$	9.79518	1.75927	0.879634	+	0.475652i	$0.157788\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	4.76491	0.805417
$36$	0	0
$37$	−6.76491	−1.11214	−0.556072	−	0.831134i	$-0.687692\pi$
$37$	−6.76491	−1.11214	−0.556072	+	0.831134i	$0.687692\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	11.3444	1.77169	0.885847	−	0.463977i	$-0.153578\pi$
$41$	11.3444	1.77169	0.885847	+	0.463977i	$0.153578\pi$
$42$	0	0
$43$	−2.96972	−0.452879	−0.226439	−	0.974025i	$-0.572709\pi$
$43$	−2.96972	−0.452879	−0.226439	+	0.974025i	$0.572709\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−1.35998	−0.198373	−0.0991865	−	0.995069i	$-0.531624\pi$
$47$	−1.35998	−0.198373	−0.0991865	+	0.995069i	$0.531624\pi$
$48$	0	0
$49$	15.7044	2.24348
$50$	0	0
$51$	0	0
$52$	0	0
$53$	3.09461	0.425077	0.212539	−	0.977153i	$-0.431827\pi$
$53$	3.09461	0.425077	0.212539	+	0.977153i	$0.431827\pi$
$54$	0	0
$55$	5.60975	0.756418
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.4352	−1.61893	−0.809463	−	0.587171i	$-0.800242\pi$
$59$	−12.4352	−1.61893	−0.809463	+	0.587171i	$0.800242\pi$
$60$	0	0
$61$	−10.5795	−1.35456	−0.677281	−	0.735724i	$-0.736842\pi$
$61$	−10.5795	−1.35456	−0.677281	+	0.735724i	$0.736842\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.60975	−0.447734
$66$	0	0
$67$	6.82546	0.833863	0.416931	−	0.908938i	$-0.363106\pi$
$67$	6.82546	0.833863	0.416931	+	0.908938i	$0.363106\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−11.6547	−1.38316	−0.691579	−	0.722300i	$-0.743085\pi$
$71$	−11.6547	−1.38316	−0.691579	+	0.722300i	$0.743085\pi$
$72$	0	0
$73$	10.1698	1.19029	0.595145	−	0.803618i	$-0.297095\pi$
$73$	10.1698	1.19029	0.595145	+	0.803618i	$0.297095\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	26.7299	3.04616
$78$	0	0
$79$	12.4995	1.40631	0.703154	−	0.711037i	$-0.251774\pi$
$79$	12.4995	1.40631	0.703154	+	0.711037i	$0.251774\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−10.1249	−1.11135	−0.555675	−	0.831399i	$-0.687540\pi$
$83$	−10.1249	−1.11135	−0.555675	+	0.831399i	$0.687540\pi$
$84$	0	0
$85$	−4.12489	−0.447407
$86$	0	0
$87$	0	0
$88$	0	0
$89$	8.31032	0.880892	0.440446	−	0.897779i	$-0.354820\pi$
$89$	8.31032	0.880892	0.440446	+	0.897779i	$0.354820\pi$
$90$	0	0
$91$	−17.2001	−1.80306
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.64002	−0.270861
$96$	0	0
$97$	16.8099	1.70678	0.853392	−	0.521270i	$-0.174542\pi$
$97$	16.8099	1.70678	0.853392	+	0.521270i	$0.174542\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−10.6850	−1.06320	−0.531598	−	0.846997i	$-0.678408\pi$
$101$	−10.6850	−1.06320	−0.531598	+	0.846997i	$0.678408\pi$
$102$	0	0
$103$	−2.31032	−0.227643	−0.113821	−	0.993501i	$-0.536309\pi$
$103$	−2.31032	−0.227643	−0.113821	+	0.993501i	$0.536309\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.6850	1.22630	0.613152	−	0.789965i	$-0.289901\pi$
$107$	12.6850	1.22630	0.613152	+	0.789965i	$0.289901\pi$
$108$	0	0
$109$	10.1698	0.974094	0.487047	−	0.873376i	$-0.338074\pi$
$109$	10.1698	0.974094	0.487047	+	0.873376i	$0.338074\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−11.3444	−1.06719	−0.533595	−	0.845740i	$-0.679159\pi$
$113$	−11.3444	−1.06719	−0.533595	+	0.845740i	$0.679159\pi$
$114$	0	0
$115$	−1.00000	−0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−19.6547	−1.80174
$120$	0	0
$121$	20.4693	1.86084
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	−5.60975	−0.497785	−0.248892	−	0.968531i	$-0.580067\pi$
$127$	−5.60975	−0.497785	−0.248892	+	0.968531i	$0.580067\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−20.4995	−1.79105	−0.895527	−	0.445008i	$-0.853201\pi$
$131$	−20.4995	−1.79105	−0.895527	+	0.445008i	$0.853201\pi$
$132$	0	0
$133$	−12.5795	−1.09078
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−20.8099	−1.77791	−0.888953	−	0.457998i	$-0.848567\pi$
$137$	−20.8099	−1.77791	−0.888953	+	0.457998i	$0.848567\pi$
$138$	0	0
$139$	6.45459	0.547471	0.273735	−	0.961805i	$-0.411741\pi$
$139$	6.45459	0.547471	0.273735	+	0.961805i	$0.411741\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−20.2498	−1.69337
$144$	0	0
$145$	3.87511	0.321811
$146$	0	0
$147$	0	0
$148$	0	0
$149$	8.10929	0.664339	0.332169	−	0.943220i	$-0.392219\pi$
$149$	8.10929	0.664339	0.332169	+	0.943220i	$0.392219\pi$
$150$	0	0
$151$	−11.5904	−0.943211	−0.471605	−	0.881810i	$-0.656325\pi$
$151$	−11.5904	−0.943211	−0.471605	+	0.881810i	$0.656325\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−9.79518	−0.786768
$156$	0	0
$157$	−14.1055	−1.12574	−0.562871	−	0.826545i	$-0.690303\pi$
$157$	−14.1055	−1.12574	−0.562871	+	0.826545i	$0.690303\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−4.76491	−0.375527
$162$	0	0
$163$	5.03028	0.394002	0.197001	−	0.980403i	$-0.436880\pi$
$163$	5.03028	0.394002	0.197001	+	0.980403i	$0.436880\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	11.9201	0.922403	0.461201	−	0.887295i	$-0.347419\pi$
$167$	11.9201	0.922403	0.461201	+	0.887295i	$0.347419\pi$
$168$	0	0
$169$	0.0302761	0.00232893
$170$	0	0
$171$	0	0
$172$	0	0
$173$	21.5904	1.64149	0.820743	−	0.571298i	$-0.193560\pi$
$173$	21.5904	1.64149	0.820743	+	0.571298i	$0.193560\pi$
$174$	0	0
$175$	−4.76491	−0.360193
$176$	0	0
$177$	0	0
$178$	0	0
$179$	3.46927	0.259305	0.129653	−	0.991559i	$-0.458614\pi$
$179$	3.46927	0.259305	0.129653	+	0.991559i	$0.458614\pi$
$180$	0	0
$181$	−6.37088	−0.473543	−0.236772	−	0.971565i	$-0.576089\pi$
$181$	−6.37088	−0.473543	−0.236772	+	0.971565i	$0.576089\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6.76491	0.497366
$186$	0	0
$187$	−23.1396	−1.69213
$188$	0	0
$189$	0	0
$190$	0	0
$191$	14.6400	1.05931	0.529657	−	0.848212i	$-0.322320\pi$
$191$	14.6400	1.05931	0.529657	+	0.848212i	$0.322320\pi$
$192$	0	0
$193$	4.18922	0.301547	0.150773	−	0.988568i	$-0.451824\pi$
$193$	4.18922	0.301547	0.150773	+	0.988568i	$0.451824\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−5.34060	−0.380502	−0.190251	−	0.981735i	$-0.560930\pi$
$197$	−5.34060	−0.380502	−0.190251	+	0.981735i	$0.560930\pi$
$198$	0	0
$199$	−16.7493	−1.18733	−0.593664	−	0.804713i	$-0.702319\pi$
$199$	−16.7493	−1.18733	−0.593664	+	0.804713i	$0.702319\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	18.4646	1.29596
$204$	0	0
$205$	−11.3444	−0.792326
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−14.8099	−1.02442
$210$	0	0
$211$	−14.4546	−0.995095	−0.497547	−	0.867437i	$-0.665766\pi$
$211$	−14.4546	−0.995095	−0.497547	+	0.867437i	$0.665766\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	2.96972	0.202533
$216$	0	0
$217$	−46.6732	−3.16838
$218$	0	0
$219$	0	0
$220$	0	0
$221$	14.8898	1.00160
$222$	0	0
$223$	−6.96972	−0.466727	−0.233364	−	0.972390i	$-0.574973\pi$
$223$	−6.96972	−0.466727	−0.233364	+	0.972390i	$0.574973\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−13.9394	−0.925194	−0.462597	−	0.886569i	$-0.653082\pi$
$227$	−13.9394	−0.925194	−0.462597	+	0.886569i	$0.653082\pi$
$228$	0	0
$229$	−26.7493	−1.76764	−0.883822	−	0.467823i	$-0.845038\pi$
$229$	−26.7493	−1.76764	−0.883822	+	0.467823i	$0.845038\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3.03028	−0.198520	−0.0992600	−	0.995062i	$-0.531648\pi$
$233$	−3.03028	−0.198520	−0.0992600	+	0.995062i	$0.531648\pi$
$234$	0	0
$235$	1.35998	0.0887151
$236$	0	0
$237$	0	0
$238$	0	0
$239$	9.59415	0.620594	0.310297	−	0.950640i	$-0.399571\pi$
$239$	9.59415	0.620594	0.310297	+	0.950640i	$0.399571\pi$
$240$	0	0
$241$	−12.1093	−0.780028	−0.390014	−	0.920809i	$-0.627530\pi$
$241$	−12.1093	−0.780028	−0.390014	+	0.920809i	$0.627530\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−15.7044	−1.00331
$246$	0	0
$247$	9.52982	0.606368
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6.96972	−0.439925	−0.219963	−	0.975508i	$-0.570594\pi$
$251$	−6.96972	−0.439925	−0.219963	+	0.975508i	$0.570594\pi$
$252$	0	0
$253$	−5.60975	−0.352682
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−14.0487	−0.876336	−0.438168	−	0.898893i	$-0.644373\pi$
$257$	−14.0487	−0.876336	−0.438168	+	0.898893i	$0.644373\pi$
$258$	0	0
$259$	32.2342	2.00293
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−27.1240	−1.67254	−0.836268	−	0.548321i	$-0.815267\pi$
$263$	−27.1240	−1.67254	−0.836268	+	0.548321i	$0.815267\pi$
$264$	0	0
$265$	−3.09461	−0.190100
$266$	0	0
$267$	0	0
$268$	0	0
$269$	6.15424	0.375231	0.187615	−	0.982243i	$-0.439924\pi$
$269$	6.15424	0.375231	0.187615	+	0.982243i	$0.439924\pi$
$270$	0	0
$271$	−22.7044	−1.37919	−0.689596	−	0.724195i	$-0.742212\pi$
$271$	−22.7044	−1.37919	−0.689596	+	0.724195i	$0.742212\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−5.60975	−0.338280
$276$	0	0
$277$	−14.6206	−0.878469	−0.439235	−	0.898372i	$-0.644750\pi$
$277$	−14.6206	−0.878469	−0.439235	+	0.898372i	$0.644750\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−5.79897	−0.345937	−0.172969	−	0.984927i	$-0.555336\pi$
$281$	−5.79897	−0.345937	−0.172969	+	0.984927i	$0.555336\pi$
$282$	0	0
$283$	1.04587	0.0621707	0.0310853	−	0.999517i	$-0.490104\pi$
$283$	1.04587	0.0621707	0.0310853	+	0.999517i	$0.490104\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−54.0549	−3.19076
$288$	0	0
$289$	0.0146797	0.000863513 0
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−22.4040	−1.30886	−0.654428	−	0.756124i	$-0.727091\pi$
$293$	−22.4040	−1.30886	−0.654428	+	0.756124i	$0.727091\pi$
$294$	0	0
$295$	12.4352	0.724006
$296$	0	0
$297$	0	0
$298$	0	0
$299$	3.60975	0.208757
$300$	0	0
$301$	14.1505	0.815619
$302$	0	0
$303$	0	0
$304$	0	0
$305$	10.5795	0.605779
$306$	0	0
$307$	8.57947	0.489656	0.244828	−	0.969566i	$-0.421268\pi$
$307$	8.57947	0.489656	0.244828	+	0.969566i	$0.421268\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8.49954	0.481965	0.240982	−	0.970529i	$-0.422530\pi$
$311$	8.49954	0.481965	0.240982	+	0.970529i	$0.422530\pi$
$312$	0	0
$313$	−24.2947	−1.37322	−0.686610	−	0.727026i	$-0.740902\pi$
$313$	−24.2947	−1.37322	−0.686610	+	0.727026i	$0.740902\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	8.26915	0.464442	0.232221	−	0.972663i	$-0.425401\pi$
$317$	8.26915	0.464442	0.232221	+	0.972663i	$0.425401\pi$
$318$	0	0
$319$	21.7384	1.21712
$320$	0	0
$321$	0	0
$322$	0	0
$323$	10.8898	0.605924
$324$	0	0
$325$	3.60975	0.200233
$326$	0	0
$327$	0	0
$328$	0	0
$329$	6.48016	0.357263
$330$	0	0
$331$	−1.54541	−0.0849436	−0.0424718	−	0.999098i	$-0.513523\pi$
$331$	−1.54541	−0.0849436	−0.0424718	+	0.999098i	$0.513523\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.82546	−0.372915
$336$	0	0
$337$	−28.4390	−1.54917	−0.774585	−	0.632470i	$-0.782041\pi$
$337$	−28.4390	−1.54917	−0.774585	+	0.632470i	$0.782041\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−54.9485	−2.97563
$342$	0	0
$343$	−41.4755	−2.23946
$344$	0	0
$345$	0	0
$346$	0	0
$347$	17.7796	0.954458	0.477229	−	0.878779i	$-0.341641\pi$
$347$	17.7796	0.954458	0.477229	+	0.878779i	$0.341641\pi$
$348$	0	0
$349$	−23.0147	−1.23195	−0.615974	−	0.787767i	$-0.711237\pi$
$349$	−23.0147	−1.23195	−0.615974	+	0.787767i	$0.711237\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	14.3297	0.762693	0.381346	−	0.924432i	$-0.375460\pi$
$353$	14.3297	0.762693	0.381346	+	0.924432i	$0.375460\pi$
$354$	0	0
$355$	11.6547	0.618567
$356$	0	0
$357$	0	0
$358$	0	0
$359$	1.85952	0.0981416	0.0490708	−	0.998795i	$-0.484374\pi$
$359$	1.85952	0.0981416	0.0490708	+	0.998795i	$0.484374\pi$
$360$	0	0
$361$	−12.0303	−0.633172
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−10.1698	−0.532314
$366$	0	0
$367$	34.0138	1.77550	0.887752	−	0.460322i	$-0.152266\pi$
$367$	34.0138	1.77550	0.887752	+	0.460322i	$0.152266\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−14.7455	−0.765550
$372$	0	0
$373$	1.46927	0.0760757	0.0380378	−	0.999276i	$-0.487889\pi$
$373$	1.46927	0.0760757	0.0380378	+	0.999276i	$0.487889\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−13.9882	−0.720428
$378$	0	0
$379$	−10.4390	−0.536215	−0.268107	−	0.963389i	$-0.586398\pi$
$379$	−10.4390	−0.536215	−0.268107	+	0.963389i	$0.586398\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−2.65562	−0.135696	−0.0678479	−	0.997696i	$-0.521613\pi$
$383$	−2.65562	−0.135696	−0.0678479	+	0.997696i	$0.521613\pi$
$384$	0	0
$385$	−26.7299	−1.36228
$386$	0	0
$387$	0	0
$388$	0	0
$389$	19.9007	1.00901	0.504503	−	0.863410i	$-0.331676\pi$
$389$	19.9007	1.00901	0.504503	+	0.863410i	$0.331676\pi$
$390$	0	0
$391$	4.12489	0.208604
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−12.4995	−0.628920
$396$	0	0
$397$	32.5289	1.63258	0.816289	−	0.577643i	$-0.196027\pi$
$397$	32.5289	1.63258	0.816289	+	0.577643i	$0.196027\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	5.21949	0.260649	0.130325	−	0.991471i	$-0.458398\pi$
$401$	5.21949	0.260649	0.130325	+	0.991471i	$0.458398\pi$
$402$	0	0
$403$	35.3581	1.76131
$404$	0	0
$405$	0	0
$406$	0	0
$407$	37.9494	1.88108
$408$	0	0
$409$	10.3940	0.513952	0.256976	−	0.966418i	$-0.417274\pi$
$409$	10.3940	0.513952	0.256976	+	0.966418i	$0.417274\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	59.2526	2.91563
$414$	0	0
$415$	10.1249	0.497011
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−6.26915	−0.306268	−0.153134	−	0.988205i	$-0.548937\pi$
$419$	−6.26915	−0.306268	−0.153134	+	0.988205i	$0.548937\pi$
$420$	0	0
$421$	24.1093	1.17501	0.587507	−	0.809219i	$-0.300109\pi$
$421$	24.1093	1.17501	0.587507	+	0.809219i	$0.300109\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	4.12489	0.200086
$426$	0	0
$427$	50.4102	2.43952
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−11.7190	−0.564486	−0.282243	−	0.959343i	$-0.591078\pi$
$431$	−11.7190	−0.564486	−0.282243	+	0.959343i	$0.591078\pi$
$432$	0	0
$433$	28.6656	1.37758	0.688790	−	0.724960i	$-0.258142\pi$
$433$	28.6656	1.37758	0.688790	+	0.724960i	$0.258142\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	2.64002	0.126289
$438$	0	0
$439$	−29.7484	−1.41981	−0.709907	−	0.704296i	$-0.751263\pi$
$439$	−29.7484	−1.41981	−0.709907	+	0.704296i	$0.751263\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−37.2001	−1.76743	−0.883715	−	0.468025i	$-0.844966\pi$
$443$	−37.2001	−1.76743	−0.883715	+	0.468025i	$0.844966\pi$
$444$	0	0
$445$	−8.31032	−0.393947
$446$	0	0
$447$	0	0
$448$	0	0
$449$	0.624427	0.0294685	0.0147343	−	0.999891i	$-0.495310\pi$
$449$	0.624427	0.0294685	0.0147343	+	0.999891i	$0.495310\pi$
$450$	0	0
$451$	−63.6391	−2.99665
$452$	0	0
$453$	0	0
$454$	0	0
$455$	17.2001	0.806354
$456$	0	0
$457$	−12.1736	−0.569458	−0.284729	−	0.958608i	$-0.591904\pi$
$457$	−12.1736	−0.569458	−0.284729	+	0.958608i	$0.591904\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	6.78051	0.315800	0.157900	−	0.987455i	$-0.449528\pi$
$461$	6.78051	0.315800	0.157900	+	0.987455i	$0.449528\pi$
$462$	0	0
$463$	−12.0799	−0.561402	−0.280701	−	0.959795i	$-0.590567\pi$
$463$	−12.0799	−0.561402	−0.280701	+	0.959795i	$0.590567\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	19.1240	0.884952	0.442476	−	0.896780i	$-0.354100\pi$
$467$	19.1240	0.884952	0.442476	+	0.896780i	$0.354100\pi$
$468$	0	0
$469$	−32.5227	−1.50176
$470$	0	0
$471$	0	0
$472$	0	0
$473$	16.6594	0.766000
$474$	0	0
$475$	2.64002	0.121133
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−3.42053	−0.156288	−0.0781440	−	0.996942i	$-0.524899\pi$
$479$	−3.42053	−0.156288	−0.0781440	+	0.996942i	$0.524899\pi$
$480$	0	0
$481$	−24.4196	−1.11344
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16.8099	−0.763297
$486$	0	0
$487$	35.8889	1.62628	0.813140	−	0.582068i	$-0.197756\pi$
$487$	35.8889	1.62628	0.813140	+	0.582068i	$0.197756\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	14.6244	0.659991	0.329996	−	0.943982i	$-0.392953\pi$
$491$	14.6244	0.659991	0.329996	+	0.943982i	$0.392953\pi$
$492$	0	0
$493$	−15.9844	−0.719901
$494$	0	0
$495$	0	0
$496$	0	0
$497$	55.5336	2.49102
$498$	0	0
$499$	13.9239	0.623317	0.311659	−	0.950194i	$-0.399115\pi$
$499$	13.9239	0.623317	0.311659	+	0.950194i	$0.399115\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.7758	−0.703408	−0.351704	−	0.936111i	$-0.614398\pi$
$503$	−15.7758	−0.703408	−0.351704	+	0.936111i	$0.614398\pi$
$504$	0	0
$505$	10.6850	0.475475
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−20.3103	−0.900239	−0.450120	−	0.892968i	$-0.648619\pi$
$509$	−20.3103	−0.900239	−0.450120	+	0.892968i	$0.648619\pi$
$510$	0	0
$511$	−48.4584	−2.14367
$512$	0	0
$513$	0	0
$514$	0	0
$515$	2.31032	0.101805
$516$	0	0
$517$	7.62912	0.335529
$518$	0	0
$519$	0	0
$520$	0	0
$521$	22.9192	1.00411	0.502053	−	0.864837i	$-0.332578\pi$
$521$	22.9192	1.00411	0.502053	+	0.864837i	$0.332578\pi$
$522$	0	0
$523$	−22.5601	−0.986484	−0.493242	−	0.869892i	$-0.664188\pi$
$523$	−22.5601	−0.986484	−0.493242	+	0.869892i	$0.664188\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	40.4040	1.76003
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	40.9503	1.77376
$534$	0	0
$535$	−12.6850	−0.548419
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−88.0975	−3.79463
$540$	0	0
$541$	8.18922	0.352082	0.176041	−	0.984383i	$-0.443671\pi$
$541$	8.18922	0.352082	0.176041	+	0.984383i	$0.443671\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−10.1698	−0.435628
$546$	0	0
$547$	−12.0000	−0.513083	−0.256541	−	0.966533i	$-0.582583\pi$
$547$	−12.0000	−0.513083	−0.256541	+	0.966533i	$0.582583\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−10.2304	−0.435829
$552$	0	0
$553$	−59.5592	−2.53271
$554$	0	0
$555$	0	0
$556$	0	0
$557$	35.0946	1.48701	0.743503	−	0.668732i	$-0.233163\pi$
$557$	35.0946	1.48701	0.743503	+	0.668732i	$0.233163\pi$
$558$	0	0
$559$	−10.7200	−0.453406
$560$	0	0
$561$	0	0
$562$	0	0
$563$	4.55631	0.192026	0.0960128	−	0.995380i	$-0.469391\pi$
$563$	4.55631	0.192026	0.0960128	+	0.995380i	$0.469391\pi$
$564$	0	0
$565$	11.3444	0.477262
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−38.8392	−1.62822	−0.814112	−	0.580707i	$-0.802776\pi$
$569$	−38.8392	−1.62822	−0.814112	+	0.580707i	$0.802776\pi$
$570$	0	0
$571$	−36.6694	−1.53457	−0.767283	−	0.641309i	$-0.778392\pi$
$571$	−36.6694	−1.53457	−0.767283	+	0.641309i	$0.778392\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	1.00000	0.0417029
$576$	0	0
$577$	15.4087	0.641473	0.320737	−	0.947168i	$-0.396070\pi$
$577$	15.4087	0.641473	0.320737	+	0.947168i	$0.396070\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	48.2442	2.00150
$582$	0	0
$583$	−17.3600	−0.718977
$584$	0	0
$585$	0	0
$586$	0	0
$587$	32.5895	1.34511	0.672555	−	0.740047i	$-0.265197\pi$
$587$	32.5895	1.34511	0.672555	+	0.740047i	$0.265197\pi$
$588$	0	0
$589$	25.8595	1.06552
$590$	0	0
$591$	0	0
$592$	0	0
$593$	43.7896	1.79822	0.899111	−	0.437721i	$-0.144214\pi$
$593$	43.7896	1.79822	0.899111	+	0.437721i	$0.144214\pi$
$594$	0	0
$595$	19.6547	0.805764
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−10.7200	−0.438005	−0.219003	−	0.975724i	$-0.570280\pi$
$599$	−10.7200	−0.438005	−0.219003	+	0.975724i	$0.570280\pi$
$600$	0	0
$601$	6.67500	0.272279	0.136139	−	0.990690i	$-0.456530\pi$
$601$	6.67500	0.272279	0.136139	+	0.990690i	$0.456530\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−20.4693	−0.832194
$606$	0	0
$607$	−12.4508	−0.505363	−0.252681	−	0.967550i	$-0.581312\pi$
$607$	−12.4508	−0.505363	−0.252681	+	0.967550i	$0.581312\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−4.90917	−0.198604
$612$	0	0
$613$	−38.0000	−1.53481	−0.767403	−	0.641165i	$-0.778451\pi$
$613$	−38.0000	−1.53481	−0.767403	+	0.641165i	$0.778451\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	41.5336	1.67208	0.836040	−	0.548669i	$-0.184865\pi$
$617$	41.5336	1.67208	0.836040	+	0.548669i	$0.184865\pi$
$618$	0	0
$619$	−4.99908	−0.200930	−0.100465	−	0.994941i	$-0.532033\pi$
$619$	−4.99908	−0.200930	−0.100465	+	0.994941i	$0.532033\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−39.5979	−1.58646
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−27.9045	−1.11262
$630$	0	0
$631$	34.4490	1.37139	0.685696	−	0.727888i	$-0.259498\pi$
$631$	34.4490	1.37139	0.685696	+	0.727888i	$0.259498\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.60975	0.222616
$636$	0	0
$637$	56.6888	2.24609
$638$	0	0
$639$	0	0
$640$	0	0
$641$	13.6779	0.540243	0.270122	−	0.962826i	$-0.412936\pi$
$641$	13.6779	0.540243	0.270122	+	0.962826i	$0.412936\pi$
$642$	0	0
$643$	11.8633	0.467843	0.233921	−	0.972256i	$-0.424844\pi$
$643$	11.8633	0.467843	0.233921	+	0.972256i	$0.424844\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−25.9882	−1.02170	−0.510850	−	0.859670i	$-0.670669\pi$
$647$	−25.9882	−1.02170	−0.510850	+	0.859670i	$0.670669\pi$
$648$	0	0
$649$	69.7584	2.73826
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−10.5795	−0.414007	−0.207003	−	0.978340i	$-0.566371\pi$
$653$	−10.5795	−0.414007	−0.207003	+	0.978340i	$0.566371\pi$
$654$	0	0
$655$	20.4995	0.800983
$656$	0	0
$657$	0	0
$658$	0	0
$659$	5.44989	0.212298	0.106149	−	0.994350i	$-0.466148\pi$
$659$	5.44989	0.212298	0.106149	+	0.994350i	$0.466148\pi$
$660$	0	0
$661$	−24.7200	−0.961495	−0.480747	−	0.876859i	$-0.659635\pi$
$661$	−24.7200	−0.961495	−0.480747	+	0.876859i	$0.659635\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.5795	0.487811
$666$	0	0
$667$	−3.87511	−0.150045
$668$	0	0
$669$	0	0
$670$	0	0
$671$	59.3482	2.29111
$672$	0	0
$673$	33.6391	1.29669	0.648346	−	0.761346i	$-0.275461\pi$
$673$	33.6391	1.29669	0.648346	+	0.761346i	$0.275461\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−24.0937	−0.925996	−0.462998	−	0.886359i	$-0.653226\pi$
$677$	−24.0937	−0.925996	−0.462998	+	0.886359i	$0.653226\pi$
$678$	0	0
$679$	−80.0975	−3.07386
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.6382	1.70803	0.854016	−	0.520246i	$-0.174160\pi$
$683$	44.6382	1.70803	0.854016	+	0.520246i	$0.174160\pi$
$684$	0	0
$685$	20.8099	0.795104
$686$	0	0
$687$	0	0
$688$	0	0
$689$	11.1708	0.425572
$690$	0	0
$691$	−17.9083	−0.681262	−0.340631	−	0.940197i	$-0.610641\pi$
$691$	−17.9083	−0.681262	−0.340631	+	0.940197i	$0.610641\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−6.45459	−0.244836
$696$	0	0
$697$	46.7943	1.77246
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−45.6703	−1.72494	−0.862472	−	0.506105i	$-0.831085\pi$
$701$	−45.6703	−1.72494	−0.862472	+	0.506105i	$0.831085\pi$
$702$	0	0
$703$	−17.8595	−0.673584
$704$	0	0
$705$	0	0
$706$	0	0
$707$	50.9130	1.91478
$708$	0	0
$709$	−8.76113	−0.329031	−0.164516	−	0.986374i	$-0.552606\pi$
$709$	−8.76113	−0.329031	−0.164516	+	0.986374i	$0.552606\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	9.79518	0.366833
$714$	0	0
$715$	20.2498	0.757298
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−50.5932	−1.88681	−0.943405	−	0.331644i	$-0.892397\pi$
$719$	−50.5932	−1.88681	−0.943405	+	0.331644i	$0.892397\pi$
$720$	0	0
$721$	11.0085	0.409977
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−3.87511	−0.143918
$726$	0	0
$727$	13.1046	0.486022	0.243011	−	0.970023i	$-0.421865\pi$
$727$	13.1046	0.486022	0.243011	+	0.970023i	$0.421865\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−12.2498	−0.453074
$732$	0	0
$733$	−14.9248	−0.551259	−0.275629	−	0.961264i	$-0.588886\pi$
$733$	−14.9248	−0.551259	−0.275629	+	0.961264i	$0.588886\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−38.2891	−1.41040
$738$	0	0
$739$	26.5757	0.977603	0.488801	−	0.872395i	$-0.337434\pi$
$739$	26.5757	0.977603	0.488801	+	0.872395i	$0.337434\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	35.0596	1.28621	0.643107	−	0.765777i	$-0.277645\pi$
$743$	35.0596	1.28621	0.643107	+	0.765777i	$0.277645\pi$
$744$	0	0
$745$	−8.10929	−0.297101
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−60.4428	−2.20853
$750$	0	0
$751$	−45.0403	−1.64354	−0.821771	−	0.569818i	$-0.807014\pi$
$751$	−45.0403	−1.64354	−0.821771	+	0.569818i	$0.807014\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.5904	0.421817
$756$	0	0
$757$	−44.1424	−1.60438	−0.802192	−	0.597066i	$-0.796333\pi$
$757$	−44.1424	−1.60438	−0.802192	+	0.597066i	$0.796333\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−27.7228	−1.00495	−0.502476	−	0.864591i	$-0.667577\pi$
$761$	−27.7228	−1.00495	−0.502476	+	0.864591i	$0.667577\pi$
$762$	0	0
$763$	−48.4584	−1.75431
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−44.8880	−1.62081
$768$	0	0
$769$	−48.1992	−1.73811	−0.869054	−	0.494718i	$-0.835272\pi$
$769$	−48.1992	−1.73811	−0.869054	+	0.494718i	$0.835272\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−24.8998	−0.895583	−0.447791	−	0.894138i	$-0.647789\pi$
$773$	−24.8998	−0.895583	−0.447791	+	0.894138i	$0.647789\pi$
$774$	0	0
$775$	9.79518	0.351853
$776$	0	0
$777$	0	0
$778$	0	0
$779$	29.9494	1.07305
$780$	0	0
$781$	65.3799	2.33948
$782$	0	0
$783$	0	0
$784$	0	0
$785$	14.1055	0.503447
$786$	0	0
$787$	−7.19634	−0.256522	−0.128261	−	0.991740i	$-0.540939\pi$
$787$	−7.19634	−0.256522	−0.128261	+	0.991740i	$0.540939\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	54.0549	1.92197
$792$	0	0
$793$	−38.1892	−1.35614
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−13.0341	−0.461690	−0.230845	−	0.972991i	$-0.574149\pi$
$797$	−13.0341	−0.461690	−0.230845	+	0.972991i	$0.574149\pi$
$798$	0	0
$799$	−5.60975	−0.198459
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−57.0502	−2.01326
$804$	0	0
$805$	4.76491	0.167941
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4.53452	0.159425	0.0797125	−	0.996818i	$-0.474600\pi$
$809$	4.53452	0.159425	0.0797125	+	0.996818i	$0.474600\pi$
$810$	0	0
$811$	14.9853	0.526206	0.263103	−	0.964768i	$-0.415254\pi$
$811$	14.9853	0.526206	0.263103	+	0.964768i	$0.415254\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.03028	−0.176203
$816$	0	0
$817$	−7.84014	−0.274292
$818$	0	0
$819$	0	0
$820$	0	0
$821$	0.630041	0.0219886	0.0109943	−	0.999940i	$-0.496500\pi$
$821$	0.630041	0.0219886	0.0109943	+	0.999940i	$0.496500\pi$
$822$	0	0
$823$	16.9991	0.592551	0.296275	−	0.955103i	$-0.404255\pi$
$823$	16.9991	0.592551	0.296275	+	0.955103i	$0.404255\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−3.28383	−0.114190	−0.0570949	−	0.998369i	$-0.518184\pi$
$827$	−3.28383	−0.114190	−0.0570949	+	0.998369i	$0.518184\pi$
$828$	0	0
$829$	−2.63624	−0.0915605	−0.0457802	−	0.998952i	$-0.514577\pi$
$829$	−2.63624	−0.0915605	−0.0457802	+	0.998952i	$0.514577\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	64.7787	2.24445
$834$	0	0
$835$	−11.9201	−0.412511
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−7.96881	−0.275114	−0.137557	−	0.990494i	$-0.543925\pi$
$839$	−7.96881	−0.275114	−0.137557	+	0.990494i	$0.543925\pi$
$840$	0	0
$841$	−13.9835	−0.482189
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−0.0302761	−0.00104153
$846$	0	0
$847$	−97.5342	−3.35131
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−6.76491	−0.231898
$852$	0	0
$853$	−33.0984	−1.13327	−0.566634	−	0.823970i	$-0.691755\pi$
$853$	−33.0984	−1.13327	−0.566634	+	0.823970i	$0.691755\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	27.3775	0.935198	0.467599	−	0.883941i	$-0.345119\pi$
$857$	27.3775	0.935198	0.467599	+	0.883941i	$0.345119\pi$
$858$	0	0
$859$	23.1651	0.790384	0.395192	−	0.918598i	$-0.370678\pi$
$859$	23.1651	0.790384	0.395192	+	0.918598i	$0.370678\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−0.370875	−0.0126247	−0.00631237	−	0.999980i	$-0.502009\pi$
$863$	−0.370875	−0.0126247	−0.00631237	+	0.999980i	$0.502009\pi$
$864$	0	0
$865$	−21.5904	−0.734095
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−70.1193	−2.37863
$870$	0	0
$871$	24.6382	0.834833
$872$	0	0
$873$	0	0
$874$	0	0
$875$	4.76491	0.161083
$876$	0	0
$877$	10.7805	0.364032	0.182016	−	0.983296i	$-0.441738\pi$
$877$	10.7805	0.364032	0.182016	+	0.983296i	$0.441738\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−43.9494	−1.48069	−0.740347	−	0.672225i	$-0.765339\pi$
$881$	−43.9494	−1.48069	−0.740347	+	0.672225i	$0.765339\pi$
$882$	0	0
$883$	−2.92855	−0.0985535	−0.0492768	−	0.998785i	$-0.515692\pi$
$883$	−2.92855	−0.0985535	−0.0492768	+	0.998785i	$0.515692\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−2.68876	−0.0902797	−0.0451399	−	0.998981i	$-0.514373\pi$
$887$	−2.68876	−0.0902797	−0.0451399	+	0.998981i	$0.514373\pi$
$888$	0	0
$889$	26.7299	0.896493
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−3.59037	−0.120147
$894$	0	0
$895$	−3.46927	−0.115965
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−37.9575	−1.26595
$900$	0	0
$901$	12.7649	0.425261
$902$	0	0
$903$	0	0
$904$	0	0
$905$	6.37088	0.211775
$906$	0	0
$907$	−17.3544	−0.576242	−0.288121	−	0.957594i	$-0.593031\pi$
$907$	−17.3544	−0.576242	−0.288121	+	0.957594i	$0.593031\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	20.6206	0.683192	0.341596	−	0.939847i	$-0.389032\pi$
$911$	20.6206	0.683192	0.341596	+	0.939847i	$0.389032\pi$
$912$	0	0
$913$	56.7980	1.87974
$914$	0	0
$915$	0	0
$916$	0	0
$917$	97.6784	3.22563
$918$	0	0
$919$	−41.1202	−1.35643	−0.678215	−	0.734864i	$-0.737246\pi$
$919$	−41.1202	−1.35643	−0.678215	+	0.734864i	$0.737246\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−42.0705	−1.38477
$924$	0	0
$925$	−6.76491	−0.222429
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−47.7153	−1.56549	−0.782743	−	0.622345i	$-0.786180\pi$
$929$	−47.7153	−1.56549	−0.782743	+	0.622345i	$0.786180\pi$
$930$	0	0
$931$	41.4599	1.35879
$932$	0	0
$933$	0	0
$934$	0	0
$935$	23.1396	0.756745
$936$	0	0
$937$	32.3103	1.05553	0.527766	−	0.849390i	$-0.323030\pi$
$937$	32.3103	1.05553	0.527766	+	0.849390i	$0.323030\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−3.60975	−0.117674	−0.0588372	−	0.998268i	$-0.518739\pi$
$941$	−3.60975	−0.117674	−0.0588372	+	0.998268i	$0.518739\pi$
$942$	0	0
$943$	11.3444	0.369424
$944$	0	0
$945$	0	0
$946$	0	0
$947$	37.6585	1.22374	0.611868	−	0.790960i	$-0.290418\pi$
$947$	37.6585	1.22374	0.611868	+	0.790960i	$0.290418\pi$
$948$	0	0
$949$	36.7106	1.19168
$950$	0	0
$951$	0	0
$952$	0	0
$953$	36.0606	1.16812	0.584058	−	0.811712i	$-0.301464\pi$
$953$	36.0606	1.16812	0.584058	+	0.811712i	$0.301464\pi$
$954$	0	0
$955$	−14.6400	−0.473740
$956$	0	0
$957$	0	0
$958$	0	0
$959$	99.1571	3.20195
$960$	0	0
$961$	64.9456	2.09502
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−4.18922	−0.134856
$966$	0	0
$967$	−15.4281	−0.496134	−0.248067	−	0.968743i	$-0.579795\pi$
$967$	−15.4281	−0.496134	−0.248067	+	0.968743i	$0.579795\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	12.4608	0.399886	0.199943	−	0.979808i	$-0.435924\pi$
$971$	12.4608	0.399886	0.199943	+	0.979808i	$0.435924\pi$
$972$	0	0
$973$	−30.7555	−0.985976
$974$	0	0
$975$	0	0
$976$	0	0
$977$	20.4958	0.655717	0.327859	−	0.944727i	$-0.393673\pi$
$977$	20.4958	0.655717	0.327859	+	0.944727i	$0.393673\pi$
$978$	0	0
$979$	−46.6188	−1.48994
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−25.0634	−0.799399	−0.399699	−	0.916646i	$-0.630885\pi$
$983$	−25.0634	−0.799399	−0.399699	+	0.916646i	$0.630885\pi$
$984$	0	0
$985$	5.34060	0.170166
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−2.96972	−0.0944317
$990$	0	0
$991$	5.80275	0.184330	0.0921652	−	0.995744i	$-0.470621\pi$
$991$	5.80275	0.184330	0.0921652	+	0.995744i	$0.470621\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	16.7493	0.530989
$996$	0	0
$997$	44.8099	1.41914	0.709571	−	0.704634i	$-0.248889\pi$
$997$	44.8099	1.41914	0.709571	+	0.704634i	$0.248889\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bi.1.1		3
3.2	odd	2		2760.2.a.u.1.1	✓	3
12.11	even	2		5520.2.a.bz.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2760.2.a.u.1.1	✓	3	3.2	odd	2
5520.2.a.bz.1.3		3	12.11	even	2
8280.2.a.bi.1.1		3	1.1	even	1	trivial

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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -4.76491 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -4.76491 q^{7} -5.60975 q^{11} +3.60975 q^{13} +4.12489 q^{17} +2.64002 q^{19} +1.00000 q^{23} +1.00000 q^{25} -3.87511 q^{29} +9.79518 q^{31} +4.76491 q^{35} -6.76491 q^{37} +11.3444 q^{41} -2.96972 q^{43} -1.35998 q^{47} +15.7044 q^{49} +3.09461 q^{53} +5.60975 q^{55} -12.4352 q^{59} -10.5795 q^{61} -3.60975 q^{65} +6.82546 q^{67} -11.6547 q^{71} +10.1698 q^{73} +26.7299 q^{77} +12.4995 q^{79} -10.1249 q^{83} -4.12489 q^{85} +8.31032 q^{89} -17.2001 q^{91} -2.64002 q^{95} +16.8099 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{5} + 2 q^{7}+O(q^{10}) \) 3 * q - 3 * q^5 + 2 * q^7	\( 3 q - 3 q^{5} + 2 q^{7} - 8 q^{11} + 2 q^{13} + 4 q^{17} + 3 q^{23} + 3 q^{25} - 20 q^{29} + 14 q^{31} - 2 q^{35} - 4 q^{37} + 8 q^{41} - 8 q^{43} - 12 q^{47} + 29 q^{49} + 8 q^{55} - 14 q^{59} - 22 q^{61} - 2 q^{65} + 6 q^{67} + 6 q^{71} - 10 q^{73} + 8 q^{77} + 4 q^{79} - 22 q^{83} - 4 q^{85} + 10 q^{89} - 12 q^{91} + 2 q^{97}+O(q^{100}) \) 3 * q - 3 * q^5 + 2 * q^7 - 8 * q^11 + 2 * q^13 + 4 * q^17 + 3 * q^23 + 3 * q^25 - 20 * q^29 + 14 * q^31 - 2 * q^35 - 4 * q^37 + 8 * q^41 - 8 * q^43 - 12 * q^47 + 29 * q^49 + 8 * q^55 - 14 * q^59 - 22 * q^61 - 2 * q^65 + 6 * q^67 + 6 * q^71 - 10 * q^73 + 8 * q^77 + 4 * q^79 - 22 * q^83 - 4 * q^85 + 10 * q^89 - 12 * q^91 + 2 * q^97

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