LMFDB - Embedded newform 8280.2.a.bb.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:	$2$
Coefficient field:	$\Q(\sqrt{17}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 4 $ x^2 - x - 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
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			$-1.56155$ 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)

$f(q)$	$=$	$q-1.00000 q^{5} -3.12311 q^{7} +O(q^{10})$	$q-1.00000 q^{5} -3.12311 q^{7} +4.00000 q^{11} +3.56155 q^{13} -5.12311 q^{17} +4.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} +4.43845 q^{29} +5.56155 q^{31} +3.12311 q^{35} +1.12311 q^{37} +3.56155 q^{41} -0.876894 q^{43} -8.68466 q^{47} +2.75379 q^{49} -12.2462 q^{53} -4.00000 q^{55} -10.2462 q^{59} +2.87689 q^{61} -3.56155 q^{65} -10.2462 q^{67} +8.68466 q^{71} +12.4384 q^{73} -12.4924 q^{77} +6.24621 q^{79} -12.0000 q^{83} +5.12311 q^{85} -10.0000 q^{89} -11.1231 q^{91} -4.00000 q^{95} +0.246211 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) $ 2 * q - 2 * q^5 + 2 * q^7	$ 2 q - 2 q^{5} + 2 q^{7} + 8 q^{11} + 3 q^{13} - 2 q^{17} + 8 q^{19} - 2 q^{23} + 2 q^{25} + 13 q^{29} + 7 q^{31} - 2 q^{35} - 6 q^{37} + 3 q^{41} - 10 q^{43} - 5 q^{47} + 22 q^{49} - 8 q^{53} - 8 q^{55} - 4 q^{59} + 14 q^{61} - 3 q^{65} - 4 q^{67} + 5 q^{71} + 29 q^{73} + 8 q^{77} - 4 q^{79} - 24 q^{83} + 2 q^{85} - 20 q^{89} - 14 q^{91} - 8 q^{95} - 16 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 + 2 * q^7 + 8 * q^11 + 3 * q^13 - 2 * q^17 + 8 * q^19 - 2 * q^23 + 2 * q^25 + 13 * q^29 + 7 * q^31 - 2 * q^35 - 6 * q^37 + 3 * q^41 - 10 * q^43 - 5 * q^47 + 22 * q^49 - 8 * q^53 - 8 * q^55 - 4 * q^59 + 14 * q^61 - 3 * q^65 - 4 * q^67 + 5 * q^71 + 29 * q^73 + 8 * q^77 - 4 * q^79 - 24 * q^83 + 2 * q^85 - 20 * q^89 - 14 * q^91 - 8 * q^95 - 16 * 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
$7$	−3.12311	−1.18042	−0.590211	−	0.807249i	$-0.700956\pi$
$7$	−3.12311	−1.18042	−0.590211	+	0.807249i	$0.700956\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	4.00000	1.20605	0.603023	−	0.797724i	$-0.293963\pi$
$11$	4.00000	1.20605	0.603023	+	0.797724i	$0.293963\pi$
$12$	0	0
$13$	3.56155	0.987797	0.493899	−	0.869520i	$-0.335571\pi$
$13$	3.56155	0.987797	0.493899	+	0.869520i	$0.335571\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.12311	−1.24254	−0.621268	−	0.783598i	$-0.713382\pi$
$17$	−5.12311	−1.24254	−0.621268	+	0.783598i	$0.713382\pi$
$18$	0	0
$19$	4.00000	0.917663	0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000	0.917663	0.458831	+	0.888523i	$0.348268\pi$
$20$	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
$29$	4.43845	0.824199	0.412099	−	0.911139i	$-0.364796\pi$
$29$	4.43845	0.824199	0.412099	+	0.911139i	$0.364796\pi$
$30$	0	0
$31$	5.56155	0.998884	0.499442	−	0.866347i	$-0.333538\pi$
$31$	5.56155	0.998884	0.499442	+	0.866347i	$0.333538\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	3.12311	0.527901
$36$	0	0
$37$	1.12311	0.184637	0.0923187	−	0.995730i	$-0.470572\pi$
$37$	1.12311	0.184637	0.0923187	+	0.995730i	$0.470572\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	3.56155	0.556221	0.278111	−	0.960549i	$-0.410292\pi$
$41$	3.56155	0.556221	0.278111	+	0.960549i	$0.410292\pi$
$42$	0	0
$43$	−0.876894	−0.133725	−0.0668626	−	0.997762i	$-0.521299\pi$
$43$	−0.876894	−0.133725	−0.0668626	+	0.997762i	$0.521299\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−8.68466	−1.26679	−0.633394	−	0.773830i	$-0.718339\pi$
$47$	−8.68466	−1.26679	−0.633394	+	0.773830i	$0.718339\pi$
$48$	0	0
$49$	2.75379	0.393398
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−12.2462	−1.68215	−0.841073	−	0.540921i	$-0.818076\pi$
$53$	−12.2462	−1.68215	−0.841073	+	0.540921i	$0.818076\pi$
$54$	0	0
$55$	−4.00000	−0.539360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−10.2462	−1.33394	−0.666972	−	0.745083i	$-0.732410\pi$
$59$	−10.2462	−1.33394	−0.666972	+	0.745083i	$0.732410\pi$
$60$	0	0
$61$	2.87689	0.368349	0.184174	−	0.982894i	$-0.441039\pi$
$61$	2.87689	0.368349	0.184174	+	0.982894i	$0.441039\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−3.56155	−0.441756
$66$	0	0
$67$	−10.2462	−1.25177	−0.625887	−	0.779914i	$-0.715263\pi$
$67$	−10.2462	−1.25177	−0.625887	+	0.779914i	$0.715263\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	8.68466	1.03068	0.515340	−	0.856986i	$-0.327666\pi$
$71$	8.68466	1.03068	0.515340	+	0.856986i	$0.327666\pi$
$72$	0	0
$73$	12.4384	1.45581	0.727905	−	0.685678i	$-0.240494\pi$
$73$	12.4384	1.45581	0.727905	+	0.685678i	$0.240494\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−12.4924	−1.42364
$78$	0	0
$79$	6.24621	0.702754	0.351377	−	0.936234i	$-0.385714\pi$
$79$	6.24621	0.702754	0.351377	+	0.936234i	$0.385714\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−12.0000	−1.31717	−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000	−1.31717	−0.658586	+	0.752506i	$0.728845\pi$
$84$	0	0
$85$	5.12311	0.555679
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.0000	−1.06000	−0.529999	−	0.847998i	$-0.677808\pi$
$89$	−10.0000	−1.06000	−0.529999	+	0.847998i	$0.677808\pi$
$90$	0	0
$91$	−11.1231	−1.16602
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−4.00000	−0.410391
$96$	0	0
$97$	0.246211	0.0249990	0.0124995	−	0.999922i	$-0.496021\pi$
$97$	0.246211	0.0249990	0.0124995	+	0.999922i	$0.496021\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	16.2462	1.61656	0.808279	−	0.588799i	$-0.200399\pi$
$101$	16.2462	1.61656	0.808279	+	0.588799i	$0.200399\pi$
$102$	0	0
$103$	14.2462	1.40372	0.701860	−	0.712314i	$-0.252353\pi$
$103$	14.2462	1.40372	0.701860	+	0.712314i	$0.252353\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	12.0000	1.16008	0.580042	−	0.814587i	$-0.303036\pi$
$107$	12.0000	1.16008	0.580042	+	0.814587i	$0.303036\pi$
$108$	0	0
$109$	−8.24621	−0.789844	−0.394922	−	0.918715i	$-0.629228\pi$
$109$	−8.24621	−0.789844	−0.394922	+	0.918715i	$0.629228\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−3.75379	−0.353127	−0.176563	−	0.984289i	$-0.556498\pi$
$113$	−3.75379	−0.353127	−0.176563	+	0.984289i	$0.556498\pi$
$114$	0	0
$115$	1.00000	0.0932505
$116$	0	0
$117$	0	0
$118$	0	0
$119$	16.0000	1.46672
$120$	0	0
$121$	5.00000	0.454545
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	3.80776	0.337884	0.168942	−	0.985626i	$-0.445965\pi$
$127$	3.80776	0.337884	0.168942	+	0.985626i	$0.445965\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	18.9309	1.65400	0.826999	−	0.562204i	$-0.190046\pi$
$131$	18.9309	1.65400	0.826999	+	0.562204i	$0.190046\pi$
$132$	0	0
$133$	−12.4924	−1.08323
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−6.87689	−0.587533	−0.293766	−	0.955877i	$-0.594909\pi$
$137$	−6.87689	−0.587533	−0.293766	+	0.955877i	$0.594909\pi$
$138$	0	0
$139$	12.6847	1.07590	0.537949	−	0.842977i	$-0.319199\pi$
$139$	12.6847	1.07590	0.537949	+	0.842977i	$0.319199\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	14.2462	1.19133
$144$	0	0
$145$	−4.43845	−0.368593
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−12.2462	−1.00325	−0.501624	−	0.865086i	$-0.667264\pi$
$149$	−12.2462	−1.00325	−0.501624	+	0.865086i	$0.667264\pi$
$150$	0	0
$151$	3.80776	0.309871	0.154936	−	0.987925i	$-0.450483\pi$
$151$	3.80776	0.309871	0.154936	+	0.987925i	$0.450483\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.56155	−0.446715
$156$	0	0
$157$	−2.00000	−0.159617	−0.0798087	−	0.996810i	$-0.525431\pi$
$157$	−2.00000	−0.159617	−0.0798087	+	0.996810i	$0.525431\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	3.12311	0.246135
$162$	0	0
$163$	−12.6847	−0.993539	−0.496770	−	0.867882i	$-0.665481\pi$
$163$	−12.6847	−0.993539	−0.496770	+	0.867882i	$0.665481\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	8.00000	0.619059	0.309529	−	0.950890i	$-0.399829\pi$
$167$	8.00000	0.619059	0.309529	+	0.950890i	$0.399829\pi$
$168$	0	0
$169$	−0.315342	−0.0242570
$170$	0	0
$171$	0	0
$172$	0	0
$173$	22.4924	1.71007	0.855034	−	0.518573i	$-0.173536\pi$
$173$	22.4924	1.71007	0.855034	+	0.518573i	$0.173536\pi$
$174$	0	0
$175$	−3.12311	−0.236085
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−4.68466	−0.350148	−0.175074	−	0.984555i	$-0.556016\pi$
$179$	−4.68466	−0.350148	−0.175074	+	0.984555i	$0.556016\pi$
$180$	0	0
$181$	−5.12311	−0.380797	−0.190399	−	0.981707i	$-0.560978\pi$
$181$	−5.12311	−0.380797	−0.190399	+	0.981707i	$0.560978\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	−1.12311	−0.0825724
$186$	0	0
$187$	−20.4924	−1.49855
$188$	0	0
$189$	0	0
$190$	0	0
$191$	11.1231	0.804840	0.402420	−	0.915455i	$-0.368169\pi$
$191$	11.1231	0.804840	0.402420	+	0.915455i	$0.368169\pi$
$192$	0	0
$193$	20.4384	1.47119	0.735596	−	0.677421i	$-0.236902\pi$
$193$	20.4384	1.47119	0.735596	+	0.677421i	$0.236902\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	15.5616	1.10871	0.554357	−	0.832279i	$-0.312964\pi$
$197$	15.5616	1.10871	0.554357	+	0.832279i	$0.312964\pi$
$198$	0	0
$199$	24.0000	1.70131	0.850657	−	0.525720i	$-0.176204\pi$
$199$	24.0000	1.70131	0.850657	+	0.525720i	$0.176204\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−13.8617	−0.972903
$204$	0	0
$205$	−3.56155	−0.248750
$206$	0	0
$207$	0	0
$208$	0	0
$209$	16.0000	1.10674
$210$	0	0
$211$	2.24621	0.154636	0.0773178	−	0.997006i	$-0.475364\pi$
$211$	2.24621	0.154636	0.0773178	+	0.997006i	$0.475364\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0.876894	0.0598037
$216$	0	0
$217$	−17.3693	−1.17911
$218$	0	0
$219$	0	0
$220$	0	0
$221$	−18.2462	−1.22737
$222$	0	0
$223$	12.4924	0.836554	0.418277	−	0.908319i	$-0.362634\pi$
$223$	12.4924	0.836554	0.418277	+	0.908319i	$0.362634\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−0.876894	−0.0582015	−0.0291008	−	0.999576i	$-0.509264\pi$
$227$	−0.876894	−0.0582015	−0.0291008	+	0.999576i	$0.509264\pi$
$228$	0	0
$229$	−16.2462	−1.07358	−0.536790	−	0.843716i	$-0.680363\pi$
$229$	−16.2462	−1.07358	−0.536790	+	0.843716i	$0.680363\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	8.43845	0.552821	0.276411	−	0.961040i	$-0.410855\pi$
$233$	8.43845	0.552821	0.276411	+	0.961040i	$0.410855\pi$
$234$	0	0
$235$	8.68466	0.566525
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−11.8078	−0.763781	−0.381890	−	0.924208i	$-0.624727\pi$
$239$	−11.8078	−0.763781	−0.381890	+	0.924208i	$0.624727\pi$
$240$	0	0
$241$	14.4924	0.933539	0.466769	−	0.884379i	$-0.345418\pi$
$241$	14.4924	0.933539	0.466769	+	0.884379i	$0.345418\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−2.75379	−0.175933
$246$	0	0
$247$	14.2462	0.906465
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.2462	0.646735	0.323368	−	0.946273i	$-0.395185\pi$
$251$	10.2462	0.646735	0.323368	+	0.946273i	$0.395185\pi$
$252$	0	0
$253$	−4.00000	−0.251478
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−28.0540	−1.74996	−0.874979	−	0.484160i	$-0.839125\pi$
$257$	−28.0540	−1.74996	−0.874979	+	0.484160i	$0.839125\pi$
$258$	0	0
$259$	−3.50758	−0.217950
$260$	0	0
$261$	0	0
$262$	0	0
$263$	8.00000	0.493301	0.246651	−	0.969104i	$-0.420670\pi$
$263$	8.00000	0.493301	0.246651	+	0.969104i	$0.420670\pi$
$264$	0	0
$265$	12.2462	0.752279
$266$	0	0
$267$	0	0
$268$	0	0
$269$	−30.3002	−1.84743	−0.923717	−	0.383074i	$-0.874865\pi$
$269$	−30.3002	−1.84743	−0.923717	+	0.383074i	$0.874865\pi$
$270$	0	0
$271$	14.2462	0.865396	0.432698	−	0.901539i	$-0.357562\pi$
$271$	14.2462	0.865396	0.432698	+	0.901539i	$0.357562\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	4.00000	0.241209
$276$	0	0
$277$	−15.5616	−0.935003	−0.467502	−	0.883992i	$-0.654846\pi$
$277$	−15.5616	−0.935003	−0.467502	+	0.883992i	$0.654846\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	2.49242	0.148685	0.0743427	−	0.997233i	$-0.476314\pi$
$281$	2.49242	0.148685	0.0743427	+	0.997233i	$0.476314\pi$
$282$	0	0
$283$	31.1231	1.85008	0.925038	−	0.379874i	$-0.124033\pi$
$283$	31.1231	1.85008	0.925038	+	0.379874i	$0.124033\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−11.1231	−0.656576
$288$	0	0
$289$	9.24621	0.543895
$290$	0	0
$291$	0	0
$292$	0	0
$293$	21.1231	1.23403	0.617013	−	0.786953i	$-0.288343\pi$
$293$	21.1231	1.23403	0.617013	+	0.786953i	$0.288343\pi$
$294$	0	0
$295$	10.2462	0.596557
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−3.56155	−0.205970
$300$	0	0
$301$	2.73863	0.157852
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−2.87689	−0.164730
$306$	0	0
$307$	12.0000	0.684876	0.342438	−	0.939540i	$-0.388747\pi$
$307$	12.0000	0.684876	0.342438	+	0.939540i	$0.388747\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	14.9309	0.846652	0.423326	−	0.905977i	$-0.360863\pi$
$311$	14.9309	0.846652	0.423326	+	0.905977i	$0.360863\pi$
$312$	0	0
$313$	6.87689	0.388705	0.194353	−	0.980932i	$-0.437739\pi$
$313$	6.87689	0.388705	0.194353	+	0.980932i	$0.437739\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	28.7386	1.61412	0.807061	−	0.590468i	$-0.201057\pi$
$317$	28.7386	1.61412	0.807061	+	0.590468i	$0.201057\pi$
$318$	0	0
$319$	17.7538	0.994021
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.4924	−1.14023
$324$	0	0
$325$	3.56155	0.197559
$326$	0	0
$327$	0	0
$328$	0	0
$329$	27.1231	1.49535
$330$	0	0
$331$	6.43845	0.353889	0.176945	−	0.984221i	$-0.443379\pi$
$331$	6.43845	0.353889	0.176945	+	0.984221i	$0.443379\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	10.2462	0.559810
$336$	0	0
$337$	32.2462	1.75656	0.878282	−	0.478144i	$-0.158690\pi$
$337$	32.2462	1.75656	0.878282	+	0.478144i	$0.158690\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	22.2462	1.20470
$342$	0	0
$343$	13.2614	0.716046
$344$	0	0
$345$	0	0
$346$	0	0
$347$	−16.4924	−0.885360	−0.442680	−	0.896680i	$-0.645972\pi$
$347$	−16.4924	−0.885360	−0.442680	+	0.896680i	$0.645972\pi$
$348$	0	0
$349$	17.8078	0.953228	0.476614	−	0.879113i	$-0.341864\pi$
$349$	17.8078	0.953228	0.476614	+	0.879113i	$0.341864\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−14.1922	−0.755377	−0.377688	−	0.925933i	$-0.623281\pi$
$353$	−14.1922	−0.755377	−0.377688	+	0.925933i	$0.623281\pi$
$354$	0	0
$355$	−8.68466	−0.460934
$356$	0	0
$357$	0	0
$358$	0	0
$359$	30.2462	1.59633	0.798167	−	0.602436i	$-0.205803\pi$
$359$	30.2462	1.59633	0.798167	+	0.602436i	$0.205803\pi$
$360$	0	0
$361$	−3.00000	−0.157895
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−12.4384	−0.651058
$366$	0	0
$367$	−12.4924	−0.652099	−0.326050	−	0.945353i	$-0.605718\pi$
$367$	−12.4924	−0.652099	−0.326050	+	0.945353i	$0.605718\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	38.2462	1.98564
$372$	0	0
$373$	24.7386	1.28092	0.640459	−	0.767992i	$-0.278744\pi$
$373$	24.7386	1.28092	0.640459	+	0.767992i	$0.278744\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	15.8078	0.814141
$378$	0	0
$379$	2.24621	0.115380	0.0576901	−	0.998335i	$-0.481626\pi$
$379$	2.24621	0.115380	0.0576901	+	0.998335i	$0.481626\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	7.61553	0.389135	0.194568	−	0.980889i	$-0.437670\pi$
$383$	7.61553	0.389135	0.194568	+	0.980889i	$0.437670\pi$
$384$	0	0
$385$	12.4924	0.636673
$386$	0	0
$387$	0	0
$388$	0	0
$389$	33.6155	1.70437	0.852187	−	0.523237i	$-0.175276\pi$
$389$	33.6155	1.70437	0.852187	+	0.523237i	$0.175276\pi$
$390$	0	0
$391$	5.12311	0.259087
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6.24621	−0.314281
$396$	0	0
$397$	−24.9309	−1.25124	−0.625622	−	0.780126i	$-0.715155\pi$
$397$	−24.9309	−1.25124	−0.625622	+	0.780126i	$0.715155\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.7386	−1.03564	−0.517819	−	0.855490i	$-0.673256\pi$
$401$	−20.7386	−1.03564	−0.517819	+	0.855490i	$0.673256\pi$
$402$	0	0
$403$	19.8078	0.986695
$404$	0	0
$405$	0	0
$406$	0	0
$407$	4.49242	0.222681
$408$	0	0
$409$	−14.6847	−0.726110	−0.363055	−	0.931768i	$-0.618266\pi$
$409$	−14.6847	−0.726110	−0.363055	+	0.931768i	$0.618266\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	32.0000	1.57462
$414$	0	0
$415$	12.0000	0.589057
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0.876894	0.0428391	0.0214195	−	0.999771i	$-0.493181\pi$
$419$	0.876894	0.0428391	0.0214195	+	0.999771i	$0.493181\pi$
$420$	0	0
$421$	−6.49242	−0.316421	−0.158211	−	0.987405i	$-0.550573\pi$
$421$	−6.49242	−0.316421	−0.158211	+	0.987405i	$0.550573\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−5.12311	−0.248507
$426$	0	0
$427$	−8.98485	−0.434807
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−7.61553	−0.366827	−0.183414	−	0.983036i	$-0.558715\pi$
$431$	−7.61553	−0.366827	−0.183414	+	0.983036i	$0.558715\pi$
$432$	0	0
$433$	−24.7386	−1.18886	−0.594431	−	0.804146i	$-0.702623\pi$
$433$	−24.7386	−1.18886	−0.594431	+	0.804146i	$0.702623\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−4.00000	−0.191346
$438$	0	0
$439$	−32.3002	−1.54160	−0.770802	−	0.637075i	$-0.780144\pi$
$439$	−32.3002	−1.54160	−0.770802	+	0.637075i	$0.780144\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	3.31534	0.157517	0.0787583	−	0.996894i	$-0.474904\pi$
$443$	3.31534	0.157517	0.0787583	+	0.996894i	$0.474904\pi$
$444$	0	0
$445$	10.0000	0.474045
$446$	0	0
$447$	0	0
$448$	0	0
$449$	32.7386	1.54503	0.772516	−	0.634996i	$-0.218998\pi$
$449$	32.7386	1.54503	0.772516	+	0.634996i	$0.218998\pi$
$450$	0	0
$451$	14.2462	0.670828
$452$	0	0
$453$	0	0
$454$	0	0
$455$	11.1231	0.521459
$456$	0	0
$457$	−9.12311	−0.426761	−0.213380	−	0.976969i	$-0.568447\pi$
$457$	−9.12311	−0.426761	−0.213380	+	0.976969i	$0.568447\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−27.5616	−1.28367	−0.641835	−	0.766843i	$-0.721826\pi$
$461$	−27.5616	−1.28367	−0.641835	+	0.766843i	$0.721826\pi$
$462$	0	0
$463$	−6.24621	−0.290286	−0.145143	−	0.989411i	$-0.546364\pi$
$463$	−6.24621	−0.290286	−0.145143	+	0.989411i	$0.546364\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−10.6307	−0.491929	−0.245965	−	0.969279i	$-0.579105\pi$
$467$	−10.6307	−0.491929	−0.245965	+	0.969279i	$0.579105\pi$
$468$	0	0
$469$	32.0000	1.47762
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−3.50758	−0.161279
$474$	0	0
$475$	4.00000	0.183533
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16.0000	−0.731059	−0.365529	−	0.930800i	$-0.619112\pi$
$479$	−16.0000	−0.731059	−0.365529	+	0.930800i	$0.619112\pi$
$480$	0	0
$481$	4.00000	0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−0.246211	−0.0111799
$486$	0	0
$487$	34.0540	1.54313	0.771566	−	0.636149i	$-0.219474\pi$
$487$	34.0540	1.54313	0.771566	+	0.636149i	$0.219474\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−6.43845	−0.290563	−0.145282	−	0.989390i	$-0.546409\pi$
$491$	−6.43845	−0.290563	−0.145282	+	0.989390i	$0.546409\pi$
$492$	0	0
$493$	−22.7386	−1.02410
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−27.1231	−1.21664
$498$	0	0
$499$	38.0540	1.70353	0.851765	−	0.523924i	$-0.175532\pi$
$499$	38.0540	1.70353	0.851765	+	0.523924i	$0.175532\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	15.6155	0.696262	0.348131	−	0.937446i	$-0.386816\pi$
$503$	15.6155	0.696262	0.348131	+	0.937446i	$0.386816\pi$
$504$	0	0
$505$	−16.2462	−0.722947
$506$	0	0
$507$	0	0
$508$	0	0
$509$	4.43845	0.196731	0.0983654	−	0.995150i	$-0.468639\pi$
$509$	4.43845	0.196731	0.0983654	+	0.995150i	$0.468639\pi$
$510$	0	0
$511$	−38.8466	−1.71847
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.2462	−0.627763
$516$	0	0
$517$	−34.7386	−1.52780
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−39.8617	−1.74637	−0.873187	−	0.487385i	$-0.837951\pi$
$521$	−39.8617	−1.74637	−0.873187	+	0.487385i	$0.837951\pi$
$522$	0	0
$523$	33.8617	1.48067	0.740335	−	0.672238i	$-0.234667\pi$
$523$	33.8617	1.48067	0.740335	+	0.672238i	$0.234667\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−28.4924	−1.24115
$528$	0	0
$529$	1.00000	0.0434783
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12.6847	0.549434
$534$	0	0
$535$	−12.0000	−0.518805
$536$	0	0
$537$	0	0
$538$	0	0
$539$	11.0152	0.474456
$540$	0	0
$541$	21.3153	0.916418	0.458209	−	0.888844i	$-0.348491\pi$
$541$	21.3153	0.916418	0.458209	+	0.888844i	$0.348491\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	8.24621	0.353229
$546$	0	0
$547$	38.0540	1.62707	0.813535	−	0.581516i	$-0.197540\pi$
$547$	38.0540	1.62707	0.813535	+	0.581516i	$0.197540\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	17.7538	0.756337
$552$	0	0
$553$	−19.5076	−0.829547
$554$	0	0
$555$	0	0
$556$	0	0
$557$	24.2462	1.02734	0.513672	−	0.857986i	$-0.328285\pi$
$557$	24.2462	1.02734	0.513672	+	0.857986i	$0.328285\pi$
$558$	0	0
$559$	−3.12311	−0.132093
$560$	0	0
$561$	0	0
$562$	0	0
$563$	−43.2311	−1.82197	−0.910986	−	0.412438i	$-0.864678\pi$
$563$	−43.2311	−1.82197	−0.910986	+	0.412438i	$0.864678\pi$
$564$	0	0
$565$	3.75379	0.157923
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−28.7386	−1.20479	−0.602393	−	0.798200i	$-0.705786\pi$
$569$	−28.7386	−1.20479	−0.602393	+	0.798200i	$0.705786\pi$
$570$	0	0
$571$	12.0000	0.502184	0.251092	−	0.967963i	$-0.419210\pi$
$571$	12.0000	0.502184	0.251092	+	0.967963i	$0.419210\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1.00000	−0.0417029
$576$	0	0
$577$	29.4233	1.22491	0.612454	−	0.790506i	$-0.290183\pi$
$577$	29.4233	1.22491	0.612454	+	0.790506i	$0.290183\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	37.4773	1.55482
$582$	0	0
$583$	−48.9848	−2.02874
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−9.17708	−0.378779	−0.189389	−	0.981902i	$-0.560651\pi$
$587$	−9.17708	−0.378779	−0.189389	+	0.981902i	$0.560651\pi$
$588$	0	0
$589$	22.2462	0.916639
$590$	0	0
$591$	0	0
$592$	0	0
$593$	38.9848	1.60092	0.800458	−	0.599389i	$-0.204590\pi$
$593$	38.9848	1.60092	0.800458	+	0.599389i	$0.204590\pi$
$594$	0	0
$595$	−16.0000	−0.655936
$596$	0	0
$597$	0	0
$598$	0	0
$599$	32.0000	1.30748	0.653742	−	0.756717i	$-0.273198\pi$
$599$	32.0000	1.30748	0.653742	+	0.756717i	$0.273198\pi$
$600$	0	0
$601$	−27.1771	−1.10858	−0.554288	−	0.832325i	$-0.687009\pi$
$601$	−27.1771	−1.10858	−0.554288	+	0.832325i	$0.687009\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−5.00000	−0.203279
$606$	0	0
$607$	0	0		−	1.00000i	$-0.5\pi$
$607$	0	0			1.00000i	$0.5\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−30.9309	−1.25133
$612$	0	0
$613$	−5.12311	−0.206920	−0.103460	−	0.994634i	$-0.532991\pi$
$613$	−5.12311	−0.206920	−0.103460	+	0.994634i	$0.532991\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	5.61553	0.226073	0.113036	−	0.993591i	$-0.463942\pi$
$617$	5.61553	0.226073	0.113036	+	0.993591i	$0.463942\pi$
$618$	0	0
$619$	36.9848	1.48655	0.743273	−	0.668988i	$-0.233272\pi$
$619$	36.9848	1.48655	0.743273	+	0.668988i	$0.233272\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	31.2311	1.25125
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−5.75379	−0.229419
$630$	0	0
$631$	−6.63068	−0.263963	−0.131982	−	0.991252i	$-0.542134\pi$
$631$	−6.63068	−0.263963	−0.131982	+	0.991252i	$0.542134\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−3.80776	−0.151107
$636$	0	0
$637$	9.80776	0.388598
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−40.2462	−1.58963	−0.794815	−	0.606852i	$-0.792432\pi$
$641$	−40.2462	−1.58963	−0.794815	+	0.606852i	$0.792432\pi$
$642$	0	0
$643$	−46.3542	−1.82803	−0.914015	−	0.405681i	$-0.867034\pi$
$643$	−46.3542	−1.82803	−0.914015	+	0.405681i	$0.867034\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	5.56155	0.218647	0.109324	−	0.994006i	$-0.465132\pi$
$647$	5.56155	0.218647	0.109324	+	0.994006i	$0.465132\pi$
$648$	0	0
$649$	−40.9848	−1.60880
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−50.7926	−1.98767	−0.993834	−	0.110876i	$-0.964634\pi$
$653$	−50.7926	−1.98767	−0.993834	+	0.110876i	$0.964634\pi$
$654$	0	0
$655$	−18.9309	−0.739690
$656$	0	0
$657$	0	0
$658$	0	0
$659$	2.24621	0.0875000	0.0437500	−	0.999043i	$-0.486070\pi$
$659$	2.24621	0.0875000	0.0437500	+	0.999043i	$0.486070\pi$
$660$	0	0
$661$	−22.4924	−0.874854	−0.437427	−	0.899254i	$-0.644110\pi$
$661$	−22.4924	−0.874854	−0.437427	+	0.899254i	$0.644110\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	12.4924	0.484435
$666$	0	0
$667$	−4.43845	−0.171857
$668$	0	0
$669$	0	0
$670$	0	0
$671$	11.5076	0.444245
$672$	0	0
$673$	−0.438447	−0.0169009	−0.00845045	−	0.999964i	$-0.502690\pi$
$673$	−0.438447	−0.0169009	−0.00845045	+	0.999964i	$0.502690\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	28.7386	1.10452	0.552258	−	0.833673i	$-0.313766\pi$
$677$	28.7386	1.10452	0.552258	+	0.833673i	$0.313766\pi$
$678$	0	0
$679$	−0.768944	−0.0295094
$680$	0	0
$681$	0	0
$682$	0	0
$683$	14.4384	0.552472	0.276236	−	0.961090i	$-0.410913\pi$
$683$	14.4384	0.552472	0.276236	+	0.961090i	$0.410913\pi$
$684$	0	0
$685$	6.87689	0.262753
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−43.6155	−1.66162
$690$	0	0
$691$	−10.2462	−0.389784	−0.194892	−	0.980825i	$-0.562436\pi$
$691$	−10.2462	−0.389784	−0.194892	+	0.980825i	$0.562436\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−12.6847	−0.481157
$696$	0	0
$697$	−18.2462	−0.691125
$698$	0	0
$699$	0	0
$700$	0	0
$701$	26.9848	1.01920	0.509602	−	0.860410i	$-0.329793\pi$
$701$	26.9848	1.01920	0.509602	+	0.860410i	$0.329793\pi$
$702$	0	0
$703$	4.49242	0.169435
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−50.7386	−1.90822
$708$	0	0
$709$	8.73863	0.328186	0.164093	−	0.986445i	$-0.447530\pi$
$709$	8.73863	0.328186	0.164093	+	0.986445i	$0.447530\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−5.56155	−0.208282
$714$	0	0
$715$	−14.2462	−0.532778
$716$	0	0
$717$	0	0
$718$	0	0
$719$	10.7386	0.400483	0.200242	−	0.979747i	$-0.435827\pi$
$719$	10.7386	0.400483	0.200242	+	0.979747i	$0.435827\pi$
$720$	0	0
$721$	−44.4924	−1.65698
$722$	0	0
$723$	0	0
$724$	0	0
$725$	4.43845	0.164840
$726$	0	0
$727$	28.8769	1.07098	0.535492	−	0.844540i	$-0.320126\pi$
$727$	28.8769	1.07098	0.535492	+	0.844540i	$0.320126\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4.49242	0.166158
$732$	0	0
$733$	2.87689	0.106261	0.0531303	−	0.998588i	$-0.483080\pi$
$733$	2.87689	0.106261	0.0531303	+	0.998588i	$0.483080\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−40.9848	−1.50970
$738$	0	0
$739$	−42.5464	−1.56509	−0.782547	−	0.622591i	$-0.786080\pi$
$739$	−42.5464	−1.56509	−0.782547	+	0.622591i	$0.786080\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	30.2462	1.10963	0.554813	−	0.831975i	$-0.312790\pi$
$743$	30.2462	1.10963	0.554813	+	0.831975i	$0.312790\pi$
$744$	0	0
$745$	12.2462	0.448666
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−37.4773	−1.36939
$750$	0	0
$751$	8.38447	0.305954	0.152977	−	0.988230i	$-0.451114\pi$
$751$	8.38447	0.305954	0.152977	+	0.988230i	$0.451114\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−3.80776	−0.138579
$756$	0	0
$757$	33.1231	1.20388	0.601940	−	0.798541i	$-0.294395\pi$
$757$	33.1231	1.20388	0.601940	+	0.798541i	$0.294395\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	−42.3002	−1.53338	−0.766690	−	0.642017i	$-0.778098\pi$
$761$	−42.3002	−1.53338	−0.766690	+	0.642017i	$0.778098\pi$
$762$	0	0
$763$	25.7538	0.932350
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−36.4924	−1.31767
$768$	0	0
$769$	5.50758	0.198608	0.0993042	−	0.995057i	$-0.468338\pi$
$769$	5.50758	0.198608	0.0993042	+	0.995057i	$0.468338\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−12.2462	−0.440466	−0.220233	−	0.975447i	$-0.570682\pi$
$773$	−12.2462	−0.440466	−0.220233	+	0.975447i	$0.570682\pi$
$774$	0	0
$775$	5.56155	0.199777
$776$	0	0
$777$	0	0
$778$	0	0
$779$	14.2462	0.510423
$780$	0	0
$781$	34.7386	1.24305
$782$	0	0
$783$	0	0
$784$	0	0
$785$	2.00000	0.0713831
$786$	0	0
$787$	−29.7538	−1.06061	−0.530304	−	0.847808i	$-0.677922\pi$
$787$	−29.7538	−1.06061	−0.530304	+	0.847808i	$0.677922\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	11.7235	0.416839
$792$	0	0
$793$	10.2462	0.363854
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−18.8769	−0.668654	−0.334327	−	0.942457i	$-0.608509\pi$
$797$	−18.8769	−0.668654	−0.334327	+	0.942457i	$0.608509\pi$
$798$	0	0
$799$	44.4924	1.57403
$800$	0	0
$801$	0	0
$802$	0	0
$803$	49.7538	1.75577
$804$	0	0
$805$	−3.12311	−0.110075
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−22.4924	−0.790791	−0.395396	−	0.918511i	$-0.629393\pi$
$809$	−22.4924	−0.790791	−0.395396	+	0.918511i	$0.629393\pi$
$810$	0	0
$811$	−20.6847	−0.726337	−0.363168	−	0.931724i	$-0.618305\pi$
$811$	−20.6847	−0.726337	−0.363168	+	0.931724i	$0.618305\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	12.6847	0.444324
$816$	0	0
$817$	−3.50758	−0.122715
$818$	0	0
$819$	0	0
$820$	0	0
$821$	48.2462	1.68380	0.841902	−	0.539630i	$-0.181436\pi$
$821$	48.2462	1.68380	0.841902	+	0.539630i	$0.181436\pi$
$822$	0	0
$823$	−21.5616	−0.751588	−0.375794	−	0.926703i	$-0.622630\pi$
$823$	−21.5616	−0.751588	−0.375794	+	0.926703i	$0.622630\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	12.0000	0.417281	0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000	0.417281	0.208640	+	0.977992i	$0.433096\pi$
$828$	0	0
$829$	17.5076	0.608063	0.304032	−	0.952662i	$-0.401667\pi$
$829$	17.5076	0.608063	0.304032	+	0.952662i	$0.401667\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−14.1080	−0.488812
$834$	0	0
$835$	−8.00000	−0.276851
$836$	0	0
$837$	0	0
$838$	0	0
$839$	31.6155	1.09149	0.545745	−	0.837952i	$-0.316247\pi$
$839$	31.6155	1.09149	0.545745	+	0.837952i	$0.316247\pi$
$840$	0	0
$841$	−9.30019	−0.320696
$842$	0	0
$843$	0	0
$844$	0	0
$845$	0.315342	0.0108481
$846$	0	0
$847$	−15.6155	−0.536556
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1.12311	−0.0384996
$852$	0	0
$853$	−42.9848	−1.47177	−0.735887	−	0.677105i	$-0.763234\pi$
$853$	−42.9848	−1.47177	−0.735887	+	0.677105i	$0.763234\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	11.1771	0.381802	0.190901	−	0.981609i	$-0.438859\pi$
$857$	11.1771	0.381802	0.190901	+	0.981609i	$0.438859\pi$
$858$	0	0
$859$	−23.4233	−0.799192	−0.399596	−	0.916691i	$-0.630850\pi$
$859$	−23.4233	−0.799192	−0.399596	+	0.916691i	$0.630850\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	43.4233	1.47815	0.739073	−	0.673625i	$-0.235264\pi$
$863$	43.4233	1.47815	0.739073	+	0.673625i	$0.235264\pi$
$864$	0	0
$865$	−22.4924	−0.764765
$866$	0	0
$867$	0	0
$868$	0	0
$869$	24.9848	0.847553
$870$	0	0
$871$	−36.4924	−1.23650
$872$	0	0
$873$	0	0
$874$	0	0
$875$	3.12311	0.105580
$876$	0	0
$877$	8.73863	0.295083	0.147541	−	0.989056i	$-0.452864\pi$
$877$	8.73863	0.295083	0.147541	+	0.989056i	$0.452864\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	50.1080	1.68818	0.844090	−	0.536202i	$-0.180141\pi$
$881$	50.1080	1.68818	0.844090	+	0.536202i	$0.180141\pi$
$882$	0	0
$883$	−7.50758	−0.252650	−0.126325	−	0.991989i	$-0.540318\pi$
$883$	−7.50758	−0.252650	−0.126325	+	0.991989i	$0.540318\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.5616	1.26119	0.630597	−	0.776111i	$-0.282810\pi$
$887$	37.5616	1.26119	0.630597	+	0.776111i	$0.282810\pi$
$888$	0	0
$889$	−11.8920	−0.398847
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−34.7386	−1.16248
$894$	0	0
$895$	4.68466	0.156591
$896$	0	0
$897$	0	0
$898$	0	0
$899$	24.6847	0.823279
$900$	0	0
$901$	62.7386	2.09013
$902$	0	0
$903$	0	0
$904$	0	0
$905$	5.12311	0.170298
$906$	0	0
$907$	20.0000	0.664089	0.332045	−	0.943264i	$-0.392262\pi$
$907$	20.0000	0.664089	0.332045	+	0.943264i	$0.392262\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−12.4924	−0.413892	−0.206946	−	0.978352i	$-0.566353\pi$
$911$	−12.4924	−0.413892	−0.206946	+	0.978352i	$0.566353\pi$
$912$	0	0
$913$	−48.0000	−1.58857
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−59.1231	−1.95242
$918$	0	0
$919$	21.8617	0.721152	0.360576	−	0.932730i	$-0.382580\pi$
$919$	21.8617	0.721152	0.360576	+	0.932730i	$0.382580\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	30.9309	1.01810
$924$	0	0
$925$	1.12311	0.0369275
$926$	0	0
$927$	0	0
$928$	0	0
$929$	24.0540	0.789185	0.394593	−	0.918856i	$-0.370886\pi$
$929$	24.0540	0.789185	0.394593	+	0.918856i	$0.370886\pi$
$930$	0	0
$931$	11.0152	0.361007
$932$	0	0
$933$	0	0
$934$	0	0
$935$	20.4924	0.670174
$936$	0	0
$937$	16.6307	0.543301	0.271650	−	0.962396i	$-0.412431\pi$
$937$	16.6307	0.543301	0.271650	+	0.962396i	$0.412431\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−46.6004	−1.51913	−0.759564	−	0.650432i	$-0.774588\pi$
$941$	−46.6004	−1.51913	−0.759564	+	0.650432i	$0.774588\pi$
$942$	0	0
$943$	−3.56155	−0.115980
$944$	0	0
$945$	0	0
$946$	0	0
$947$	55.8078	1.81351	0.906754	−	0.421659i	$-0.138552\pi$
$947$	55.8078	1.81351	0.906754	+	0.421659i	$0.138552\pi$
$948$	0	0
$949$	44.3002	1.43804
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−46.4924	−1.50604	−0.753019	−	0.657999i	$-0.771403\pi$
$953$	−46.4924	−1.50604	−0.753019	+	0.657999i	$0.771403\pi$
$954$	0	0
$955$	−11.1231	−0.359935
$956$	0	0
$957$	0	0
$958$	0	0
$959$	21.4773	0.693537
$960$	0	0
$961$	−0.0691303	−0.00223001
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−20.4384	−0.657937
$966$	0	0
$967$	−45.1771	−1.45280	−0.726398	−	0.687274i	$-0.758807\pi$
$967$	−45.1771	−1.45280	−0.726398	+	0.687274i	$0.758807\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	21.3693	0.685774	0.342887	−	0.939377i	$-0.388595\pi$
$971$	21.3693	0.685774	0.342887	+	0.939377i	$0.388595\pi$
$972$	0	0
$973$	−39.6155	−1.27002
$974$	0	0
$975$	0	0
$976$	0	0
$977$	26.1080	0.835267	0.417634	−	0.908615i	$-0.362860\pi$
$977$	26.1080	0.835267	0.417634	+	0.908615i	$0.362860\pi$
$978$	0	0
$979$	−40.0000	−1.27841
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−28.8769	−0.921030	−0.460515	−	0.887652i	$-0.652335\pi$
$983$	−28.8769	−0.921030	−0.460515	+	0.887652i	$0.652335\pi$
$984$	0	0
$985$	−15.5616	−0.495832
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0.876894	0.0278836
$990$	0	0
$991$	20.4924	0.650963	0.325482	−	0.945548i	$-0.394474\pi$
$991$	20.4924	0.650963	0.325482	+	0.945548i	$0.394474\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−24.0000	−0.760851
$996$	0	0
$997$	0.738634	0.0233928	0.0116964	−	0.999932i	$-0.496277\pi$
$997$	0.738634	0.0233928	0.0116964	+	0.999932i	$0.496277\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8280.2.a.bb.1.1		2
3.2	odd	2		920.2.a.f.1.1	✓	2
12.11	even	2		1840.2.a.k.1.2		2
15.2	even	4		4600.2.e.m.4049.3		4
15.8	even	4		4600.2.e.m.4049.2		4
15.14	odd	2		4600.2.a.r.1.2		2
24.5	odd	2		7360.2.a.bj.1.2		2
24.11	even	2		7360.2.a.bm.1.1		2
60.59	even	2		9200.2.a.bx.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
920.2.a.f.1.1	✓	2	3.2	odd	2
1840.2.a.k.1.2		2	12.11	even	2
4600.2.a.r.1.2		2	15.14	odd	2
4600.2.e.m.4049.2		4	15.8	even	4
4600.2.e.m.4049.3		4	15.2	even	4
7360.2.a.bj.1.2		2	24.5	odd	2
7360.2.a.bm.1.1		2	24.11	even	2
8280.2.a.bb.1.1		2	1.1	even	1	trivial
9200.2.a.bx.1.1		2	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -3.12311 q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -3.12311 q^{7} +4.00000 q^{11} +3.56155 q^{13} -5.12311 q^{17} +4.00000 q^{19} -1.00000 q^{23} +1.00000 q^{25} +4.43845 q^{29} +5.56155 q^{31} +3.12311 q^{35} +1.12311 q^{37} +3.56155 q^{41} -0.876894 q^{43} -8.68466 q^{47} +2.75379 q^{49} -12.2462 q^{53} -4.00000 q^{55} -10.2462 q^{59} +2.87689 q^{61} -3.56155 q^{65} -10.2462 q^{67} +8.68466 q^{71} +12.4384 q^{73} -12.4924 q^{77} +6.24621 q^{79} -12.0000 q^{83} +5.12311 q^{85} -10.0000 q^{89} -11.1231 q^{91} -4.00000 q^{95} +0.246211 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} + 2 q^{7}+O(q^{10}) \) 2 * q - 2 * q^5 + 2 * q^7	\( 2 q - 2 q^{5} + 2 q^{7} + 8 q^{11} + 3 q^{13} - 2 q^{17} + 8 q^{19} - 2 q^{23} + 2 q^{25} + 13 q^{29} + 7 q^{31} - 2 q^{35} - 6 q^{37} + 3 q^{41} - 10 q^{43} - 5 q^{47} + 22 q^{49} - 8 q^{53} - 8 q^{55} - 4 q^{59} + 14 q^{61} - 3 q^{65} - 4 q^{67} + 5 q^{71} + 29 q^{73} + 8 q^{77} - 4 q^{79} - 24 q^{83} + 2 q^{85} - 20 q^{89} - 14 q^{91} - 8 q^{95} - 16 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 + 2 * q^7 + 8 * q^11 + 3 * q^13 - 2 * q^17 + 8 * q^19 - 2 * q^23 + 2 * q^25 + 13 * q^29 + 7 * q^31 - 2 * q^35 - 6 * q^37 + 3 * q^41 - 10 * q^43 - 5 * q^47 + 22 * q^49 - 8 * q^53 - 8 * q^55 - 4 * q^59 + 14 * q^61 - 3 * q^65 - 4 * q^67 + 5 * q^71 + 29 * q^73 + 8 * q^77 - 4 * q^79 - 24 * q^83 + 2 * q^85 - 20 * q^89 - 14 * q^91 - 8 * q^95 - 16 * q^97

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