LMFDB - Embedded newform 2088.4.a.b.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2088,4,Mod(1,2088)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2088, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2088.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 2088 = 2^{3} \cdot 3^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	2088.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:	$123.195988092$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 232)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.86081$ of defining polynomial
Character	$\chi$	$=$	2088.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+12.4882 q^{5} -28.6773 q^{7} +O(q^{10})$	$q+12.4882 q^{5} -28.6773 q^{7} +18.2099 q^{11} +37.9978 q^{13} +3.95162 q^{17} -36.8211 q^{19} -42.7480 q^{23} +30.9561 q^{25} +29.0000 q^{29} -160.731 q^{31} -358.129 q^{35} +313.040 q^{37} -496.787 q^{41} -139.195 q^{43} +417.656 q^{47} +479.386 q^{49} +137.116 q^{53} +227.409 q^{55} -190.033 q^{59} +161.072 q^{61} +474.525 q^{65} +125.259 q^{67} +165.110 q^{71} -938.243 q^{73} -522.209 q^{77} -1315.60 q^{79} -505.294 q^{83} +49.3487 q^{85} +769.587 q^{89} -1089.67 q^{91} -459.830 q^{95} +1333.29 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 4 q^{5} + 16 q^{7}+O(q^{10}) $ 3 * q - 4 * q^5 + 16 * q^7	$ 3 q - 4 q^{5} + 16 q^{7} + 2 q^{11} + 28 q^{13} + 66 q^{17} - 66 q^{19} - 176 q^{23} - 9 q^{25} + 87 q^{29} - 190 q^{31} - 660 q^{35} + 442 q^{37} - 1162 q^{41} + 30 q^{43} + 738 q^{47} + 851 q^{49} - 312 q^{53} + 464 q^{55} - 44 q^{59} + 54 q^{61} - 178 q^{65} - 116 q^{67} + 1200 q^{71} - 1118 q^{73} - 792 q^{77} - 2262 q^{79} + 1804 q^{83} - 8 q^{85} - 1578 q^{89} - 1972 q^{91} + 1052 q^{95} + 1450 q^{97}+O(q^{100}) $ 3 * q - 4 * q^5 + 16 * q^7 + 2 * q^11 + 28 * q^13 + 66 * q^17 - 66 * q^19 - 176 * q^23 - 9 * q^25 + 87 * q^29 - 190 * q^31 - 660 * q^35 + 442 * q^37 - 1162 * q^41 + 30 * q^43 + 738 * q^47 + 851 * q^49 - 312 * q^53 + 464 * q^55 - 44 * q^59 + 54 * q^61 - 178 * q^65 - 116 * q^67 + 1200 * q^71 - 1118 * q^73 - 792 * q^77 - 2262 * q^79 + 1804 * q^83 - 8 * q^85 - 1578 * q^89 - 1972 * q^91 + 1052 * q^95 + 1450 * 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
$5$	12.4882	1.11698	0.558491	−	0.829511i	$-0.311381\pi$
$5$	12.4882	1.11698	0.558491	+	0.829511i	$0.311381\pi$
$6$	0	0
$7$	−28.6773	−1.54843	−0.774213	−	0.632925i	$-0.781854\pi$
$7$	−28.6773	−1.54843	−0.774213	+	0.632925i	$0.781854\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	18.2099	0.499134	0.249567	−	0.968358i	$-0.419712\pi$
$11$	18.2099	0.499134	0.249567	+	0.968358i	$0.419712\pi$
$12$	0	0
$13$	37.9978	0.810668	0.405334	−	0.914169i	$-0.367155\pi$
$13$	37.9978	0.810668	0.405334	+	0.914169i	$0.367155\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.95162	0.0563769	0.0281885	−	0.999603i	$-0.491026\pi$
$17$	3.95162	0.0563769	0.0281885	+	0.999603i	$0.491026\pi$
$18$	0	0
$19$	−36.8211	−0.444597	−0.222298	−	0.974979i	$-0.571356\pi$
$19$	−36.8211	−0.444597	−0.222298	+	0.974979i	$0.571356\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−42.7480	−0.387547	−0.193773	−	0.981046i	$-0.562073\pi$
$23$	−42.7480	−0.387547	−0.193773	+	0.981046i	$0.562073\pi$
$24$	0	0
$25$	30.9561	0.247649
$26$	0	0
$27$	0	0
$28$	0	0
$29$	29.0000	0.185695
$29$	29.0000	0.185695
$30$	0	0
$31$	−160.731	−0.931231	−0.465616	−	0.884987i	$-0.654167\pi$
$31$	−160.731	−0.931231	−0.465616	+	0.884987i	$0.654167\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−358.129	−1.72957
$36$	0	0
$37$	313.040	1.39090	0.695451	−	0.718573i	$-0.255204\pi$
$37$	313.040	1.39090	0.695451	+	0.718573i	$0.255204\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−496.787	−1.89232	−0.946159	−	0.323703i	$-0.895072\pi$
$41$	−496.787	−1.89232	−0.946159	+	0.323703i	$0.895072\pi$
$42$	0	0
$43$	−139.195	−0.493653	−0.246827	−	0.969060i	$-0.579388\pi$
$43$	−139.195	−0.493653	−0.246827	+	0.969060i	$0.579388\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	417.656	1.29620	0.648100	−	0.761556i	$-0.275564\pi$
$47$	417.656	1.29620	0.648100	+	0.761556i	$0.275564\pi$
$48$	0	0
$49$	479.386	1.39763
$50$	0	0
$51$	0	0
$52$	0	0
$53$	137.116	0.355365	0.177683	−	0.984088i	$-0.443140\pi$
$53$	137.116	0.355365	0.177683	+	0.984088i	$0.443140\pi$
$54$	0	0
$55$	227.409	0.557524
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−190.033	−0.419326	−0.209663	−	0.977774i	$-0.567237\pi$
$59$	−190.033	−0.419326	−0.209663	+	0.977774i	$0.567237\pi$
$60$	0	0
$61$	161.072	0.338084	0.169042	−	0.985609i	$-0.445933\pi$
$61$	161.072	0.338084	0.169042	+	0.985609i	$0.445933\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	474.525	0.905502
$66$	0	0
$67$	125.259	0.228400	0.114200	−	0.993458i	$-0.463570\pi$
$67$	125.259	0.228400	0.114200	+	0.993458i	$0.463570\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	165.110	0.275984	0.137992	−	0.990433i	$-0.455935\pi$
$71$	165.110	0.275984	0.137992	+	0.990433i	$0.455935\pi$
$72$	0	0
$73$	−938.243	−1.50429	−0.752144	−	0.658999i	$-0.770980\pi$
$73$	−938.243	−1.50429	−0.752144	+	0.658999i	$0.770980\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−522.209	−0.772873
$78$	0	0
$79$	−1315.60	−1.87363	−0.936816	−	0.349821i	$-0.886242\pi$
$79$	−1315.60	−1.87363	−0.936816	+	0.349821i	$0.886242\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−505.294	−0.668231	−0.334116	−	0.942532i	$-0.608438\pi$
$83$	−505.294	−0.668231	−0.334116	+	0.942532i	$0.608438\pi$
$84$	0	0
$85$	49.3487	0.0629720
$86$	0	0
$87$	0	0
$88$	0	0
$89$	769.587	0.916584	0.458292	−	0.888802i	$-0.348461\pi$
$89$	769.587	0.916584	0.458292	+	0.888802i	$0.348461\pi$
$90$	0	0
$91$	−1089.67	−1.25526
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−459.830	−0.496606
$96$	0	0
$97$	1333.29	1.39561	0.697807	−	0.716286i	$-0.254159\pi$
$97$	1333.29	1.39561	0.697807	+	0.716286i	$0.254159\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	−51.3094	−0.0505493	−0.0252746	−	0.999681i	$-0.508046\pi$
$101$	−51.3094	−0.0505493	−0.0252746	+	0.999681i	$0.508046\pi$
$102$	0	0
$103$	1062.26	1.01619	0.508097	−	0.861300i	$-0.330349\pi$
$103$	1062.26	1.01619	0.508097	+	0.861300i	$0.330349\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1414.85	−1.27831	−0.639155	−	0.769078i	$-0.720716\pi$
$107$	−1414.85	−1.27831	−0.639155	+	0.769078i	$0.720716\pi$
$108$	0	0
$109$	495.330	0.435266	0.217633	−	0.976031i	$-0.430166\pi$
$109$	495.330	0.435266	0.217633	+	0.976031i	$0.430166\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	603.969	0.502802	0.251401	−	0.967883i	$-0.419109\pi$
$113$	603.969	0.502802	0.251401	+	0.967883i	$0.419109\pi$
$114$	0	0
$115$	−533.848	−0.432883
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−113.322	−0.0872956
$120$	0	0
$121$	−999.401	−0.750865
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1174.44	−0.840363
$126$	0	0
$127$	−1282.23	−0.895902	−0.447951	−	0.894058i	$-0.647846\pi$
$127$	−1282.23	−0.895902	−0.447951	+	0.894058i	$0.647846\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2446.25	−1.63152	−0.815762	−	0.578388i	$-0.803682\pi$
$131$	−2446.25	−1.63152	−0.815762	+	0.578388i	$0.803682\pi$
$132$	0	0
$133$	1055.93	0.688425
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1259.91	−0.785703	−0.392852	−	0.919602i	$-0.628511\pi$
$137$	−1259.91	−0.785703	−0.392852	+	0.919602i	$0.628511\pi$
$138$	0	0
$139$	−822.786	−0.502070	−0.251035	−	0.967978i	$-0.580771\pi$
$139$	−822.786	−0.502070	−0.251035	+	0.967978i	$0.580771\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	691.934	0.404632
$144$	0	0
$145$	362.159	0.207418
$146$	0	0
$147$	0	0
$148$	0	0
$149$	477.366	0.262466	0.131233	−	0.991352i	$-0.458106\pi$
$149$	477.366	0.262466	0.131233	+	0.991352i	$0.458106\pi$
$150$	0	0
$151$	−1740.92	−0.938240	−0.469120	−	0.883135i	$-0.655429\pi$
$151$	−1740.92	−0.938240	−0.469120	+	0.883135i	$0.655429\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−2007.25	−1.04017
$156$	0	0
$157$	−1372.90	−0.697892	−0.348946	−	0.937143i	$-0.613460\pi$
$157$	−1372.90	−0.697892	−0.348946	+	0.937143i	$0.613460\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1225.90	0.600088
$162$	0	0
$163$	3688.03	1.77220	0.886100	−	0.463495i	$-0.153405\pi$
$163$	3688.03	1.77220	0.886100	+	0.463495i	$0.153405\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−4059.56	−1.88107	−0.940534	−	0.339701i	$-0.889674\pi$
$167$	−4059.56	−1.88107	−0.940534	+	0.339701i	$0.889674\pi$
$168$	0	0
$169$	−753.169	−0.342817
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2811.92	−1.23576	−0.617879	−	0.786274i	$-0.712008\pi$
$173$	−2811.92	−1.23576	−0.617879	+	0.786274i	$0.712008\pi$
$174$	0	0
$175$	−887.737	−0.383466
$176$	0	0
$177$	0	0
$178$	0	0
$179$	−163.787	−0.0683913	−0.0341956	−	0.999415i	$-0.510887\pi$
$179$	−163.787	−0.0683913	−0.0341956	+	0.999415i	$0.510887\pi$
$180$	0	0
$181$	−3540.42	−1.45391	−0.726954	−	0.686686i	$-0.759065\pi$
$181$	−3540.42	−1.45391	−0.726954	+	0.686686i	$0.759065\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	3909.31	1.55361
$186$	0	0
$187$	71.9584	0.0281397
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2163.65	−0.819665	−0.409833	−	0.912161i	$-0.634413\pi$
$191$	−2163.65	−0.819665	−0.409833	+	0.912161i	$0.634413\pi$
$192$	0	0
$193$	4632.75	1.72784	0.863919	−	0.503630i	$-0.168003\pi$
$193$	4632.75	1.72784	0.863919	+	0.503630i	$0.168003\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	1239.91	0.448427	0.224214	−	0.974540i	$-0.428019\pi$
$197$	1239.91	0.448427	0.224214	+	0.974540i	$0.428019\pi$
$198$	0	0
$199$	4651.45	1.65695	0.828474	−	0.560028i	$-0.189210\pi$
$199$	4651.45	1.65695	0.828474	+	0.560028i	$0.189210\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−831.641	−0.287536
$204$	0	0
$205$	−6203.99	−2.11369
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−670.506	−0.221913
$210$	0	0
$211$	3516.90	1.14746	0.573728	−	0.819046i	$-0.305497\pi$
$211$	3516.90	1.14746	0.573728	+	0.819046i	$0.305497\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1738.30	−0.551402
$216$	0	0
$217$	4609.33	1.44194
$218$	0	0
$219$	0	0
$220$	0	0
$221$	150.153	0.0457030
$222$	0	0
$223$	3480.17	1.04506	0.522532	−	0.852620i	$-0.324988\pi$
$223$	3480.17	1.04506	0.522532	+	0.852620i	$0.324988\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1259.51	−0.368266	−0.184133	−	0.982901i	$-0.558948\pi$
$227$	−1259.51	−0.368266	−0.184133	+	0.982901i	$0.558948\pi$
$228$	0	0
$229$	−274.295	−0.0791524	−0.0395762	−	0.999217i	$-0.512601\pi$
$229$	−274.295	−0.0791524	−0.0395762	+	0.999217i	$0.512601\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−6517.70	−1.83257	−0.916285	−	0.400526i	$-0.868827\pi$
$233$	−6517.70	−1.83257	−0.916285	+	0.400526i	$0.868827\pi$
$234$	0	0
$235$	5215.79	1.44783
$236$	0	0
$237$	0	0
$238$	0	0
$239$	6052.89	1.63820	0.819098	−	0.573654i	$-0.194474\pi$
$239$	6052.89	1.63820	0.819098	+	0.573654i	$0.194474\pi$
$240$	0	0
$241$	2265.19	0.605452	0.302726	−	0.953078i	$-0.402103\pi$
$241$	2265.19	0.605452	0.302726	+	0.953078i	$0.402103\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	5986.68	1.56112
$246$	0	0
$247$	−1399.12	−0.360420
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−6653.27	−1.67311	−0.836555	−	0.547882i	$-0.815434\pi$
$251$	−6653.27	−1.67311	−0.836555	+	0.547882i	$0.815434\pi$
$252$	0	0
$253$	−778.435	−0.193438
$254$	0	0
$255$	0	0
$256$	0	0
$257$	3144.58	0.763244	0.381622	−	0.924319i	$-0.375366\pi$
$257$	3144.58	0.763244	0.381622	+	0.924319i	$0.375366\pi$
$258$	0	0
$259$	−8977.12	−2.15371
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−2409.66	−0.564965	−0.282483	−	0.959272i	$-0.591158\pi$
$263$	−2409.66	−0.564965	−0.282483	+	0.959272i	$0.591158\pi$
$264$	0	0
$265$	1712.34	0.396937
$266$	0	0
$267$	0	0
$268$	0	0
$269$	4542.85	1.02968	0.514838	−	0.857288i	$-0.327852\pi$
$269$	4542.85	1.02968	0.514838	+	0.857288i	$0.327852\pi$
$270$	0	0
$271$	−6519.35	−1.46134	−0.730668	−	0.682733i	$-0.760791\pi$
$271$	−6519.35	−1.46134	−0.730668	+	0.682733i	$0.760791\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	563.706	0.123610
$276$	0	0
$277$	−7440.91	−1.61401	−0.807005	−	0.590544i	$-0.798913\pi$
$277$	−7440.91	−1.61401	−0.807005	+	0.590544i	$0.798913\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−272.086	−0.0577625	−0.0288813	−	0.999583i	$-0.509194\pi$
$281$	−272.086	−0.0577625	−0.0288813	+	0.999583i	$0.509194\pi$
$282$	0	0
$283$	−1716.92	−0.360636	−0.180318	−	0.983608i	$-0.557713\pi$
$283$	−1716.92	−0.360636	−0.180318	+	0.983608i	$0.557713\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	14246.5	2.93012
$288$	0	0
$289$	−4897.38	−0.996822
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−5903.51	−1.17709	−0.588544	−	0.808465i	$-0.700299\pi$
$293$	−5903.51	−1.17709	−0.588544	+	0.808465i	$0.700299\pi$
$294$	0	0
$295$	−2373.18	−0.468379
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−1624.33	−0.314172
$300$	0	0
$301$	3991.74	0.764386
$302$	0	0
$303$	0	0
$304$	0	0
$305$	2011.50	0.377633
$306$	0	0
$307$	−3769.05	−0.700688	−0.350344	−	0.936621i	$-0.613935\pi$
$307$	−3769.05	−0.700688	−0.350344	+	0.936621i	$0.613935\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−3141.76	−0.572839	−0.286419	−	0.958104i	$-0.592465\pi$
$311$	−3141.76	−0.572839	−0.286419	+	0.958104i	$0.592465\pi$
$312$	0	0
$313$	−1743.33	−0.314820	−0.157410	−	0.987533i	$-0.550314\pi$
$313$	−1743.33	−0.314820	−0.157410	+	0.987533i	$0.550314\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1836.62	0.325410	0.162705	−	0.986675i	$-0.447978\pi$
$317$	1836.62	0.325410	0.162705	+	0.986675i	$0.447978\pi$
$318$	0	0
$319$	528.086	0.0926869
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−145.503	−0.0250650
$324$	0	0
$325$	1176.26	0.200761
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−11977.2	−2.00707
$330$	0	0
$331$	−106.340	−0.0176585	−0.00882924	−	0.999961i	$-0.502810\pi$
$331$	−106.340	−0.0176585	−0.00882924	+	0.999961i	$0.502810\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	1564.26	0.255118
$336$	0	0
$337$	347.401	0.0561547	0.0280774	−	0.999606i	$-0.491062\pi$
$337$	347.401	0.0561547	0.0280774	+	0.999606i	$0.491062\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−2926.89	−0.464809
$342$	0	0
$343$	−3911.17	−0.615694
$344$	0	0
$345$	0	0
$346$	0	0
$347$	4319.94	0.668319	0.334159	−	0.942517i	$-0.391548\pi$
$347$	4319.94	0.668319	0.334159	+	0.942517i	$0.391548\pi$
$348$	0	0
$349$	−9732.19	−1.49270	−0.746350	−	0.665554i	$-0.768195\pi$
$349$	−9732.19	−1.49270	−0.746350	+	0.665554i	$0.768195\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−8396.92	−1.26607	−0.633035	−	0.774123i	$-0.718191\pi$
$353$	−8396.92	−1.26607	−0.633035	+	0.774123i	$0.718191\pi$
$354$	0	0
$355$	2061.93	0.308270
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1306.89	−0.192130	−0.0960652	−	0.995375i	$-0.530626\pi$
$359$	−1306.89	−0.192130	−0.0960652	+	0.995375i	$0.530626\pi$
$360$	0	0
$361$	−5503.21	−0.802334
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−11717.0	−1.68026
$366$	0	0
$367$	−9859.59	−1.40236	−0.701181	−	0.712984i	$-0.747343\pi$
$367$	−9859.59	−1.40236	−0.701181	+	0.712984i	$0.747343\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−3932.12	−0.550257
$372$	0	0
$373$	−3393.01	−0.471001	−0.235500	−	0.971874i	$-0.575673\pi$
$373$	−3393.01	−0.471001	−0.235500	+	0.971874i	$0.575673\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	1101.94	0.150537
$378$	0	0
$379$	−2030.82	−0.275240	−0.137620	−	0.990485i	$-0.543945\pi$
$379$	−2030.82	−0.275240	−0.137620	+	0.990485i	$0.543945\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−4.81248	−0.000642052 0	−0.000321026	−	1.00000i	$-0.500102\pi$
$383$	−4.81248	−0.000642052 0	−0.000321026		1.00000i	$0.500102\pi$
$384$	0	0
$385$	−6521.47	−0.863285
$386$	0	0
$387$	0	0
$388$	0	0
$389$	−9031.93	−1.17722	−0.588608	−	0.808418i	$-0.700324\pi$
$389$	−9031.93	−1.17722	−0.588608	+	0.808418i	$0.700324\pi$
$390$	0	0
$391$	−168.924	−0.0218487
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−16429.6	−2.09281
$396$	0	0
$397$	−1443.16	−0.182443	−0.0912217	−	0.995831i	$-0.529077\pi$
$397$	−1443.16	−0.182443	−0.0912217	+	0.995831i	$0.529077\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	9877.59	1.23008	0.615042	−	0.788495i	$-0.289139\pi$
$401$	9877.59	1.23008	0.615042	+	0.788495i	$0.289139\pi$
$402$	0	0
$403$	−6107.43	−0.754920
$404$	0	0
$405$	0	0
$406$	0	0
$407$	5700.41	0.694247
$408$	0	0
$409$	14530.7	1.75672	0.878358	−	0.478003i	$-0.158639\pi$
$409$	14530.7	1.75672	0.878358	+	0.478003i	$0.158639\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	5449.63	0.649295
$414$	0	0
$415$	−6310.23	−0.746403
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−3312.84	−0.386259	−0.193130	−	0.981173i	$-0.561864\pi$
$419$	−3312.84	−0.386259	−0.193130	+	0.981173i	$0.561864\pi$
$420$	0	0
$421$	−6708.67	−0.776629	−0.388314	−	0.921527i	$-0.626943\pi$
$421$	−6708.67	−0.776629	−0.388314	+	0.921527i	$0.626943\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	122.327	0.0139617
$426$	0	0
$427$	−4619.09	−0.523498
$428$	0	0
$429$	0	0
$430$	0	0
$431$	14687.7	1.64148	0.820742	−	0.571299i	$-0.193560\pi$
$431$	14687.7	1.64148	0.820742	+	0.571299i	$0.193560\pi$
$432$	0	0
$433$	759.822	0.0843296	0.0421648	−	0.999111i	$-0.486575\pi$
$433$	759.822	0.0843296	0.0421648	+	0.999111i	$0.486575\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	1574.03	0.172302
$438$	0	0
$439$	4803.78	0.522260	0.261130	−	0.965304i	$-0.415905\pi$
$439$	4803.78	0.522260	0.261130	+	0.965304i	$0.415905\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−10123.3	−1.08572	−0.542858	−	0.839824i	$-0.682658\pi$
$443$	−10123.3	−1.08572	−0.542858	+	0.839824i	$0.682658\pi$
$444$	0	0
$445$	9610.78	1.02381
$446$	0	0
$447$	0	0
$448$	0	0
$449$	636.749	0.0669266	0.0334633	−	0.999440i	$-0.489346\pi$
$449$	636.749	0.0669266	0.0334633	+	0.999440i	$0.489346\pi$
$450$	0	0
$451$	−9046.41	−0.944521
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−13608.1	−1.40210
$456$	0	0
$457$	12598.2	1.28954	0.644770	−	0.764376i	$-0.276953\pi$
$457$	12598.2	1.28954	0.644770	+	0.764376i	$0.276953\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−15728.3	−1.58902	−0.794512	−	0.607249i	$-0.792273\pi$
$461$	−15728.3	−1.58902	−0.794512	+	0.607249i	$0.792273\pi$
$462$	0	0
$463$	5504.55	0.552523	0.276262	−	0.961083i	$-0.410904\pi$
$463$	5504.55	0.552523	0.276262	+	0.961083i	$0.410904\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−14851.2	−1.47159	−0.735796	−	0.677203i	$-0.763192\pi$
$467$	−14851.2	−1.47159	−0.735796	+	0.677203i	$0.763192\pi$
$468$	0	0
$469$	−3592.07	−0.353660
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−2534.73	−0.246399
$474$	0	0
$475$	−1139.84	−0.110104
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−5778.20	−0.551175	−0.275587	−	0.961276i	$-0.588872\pi$
$479$	−5778.20	−0.551175	−0.275587	+	0.961276i	$0.588872\pi$
$480$	0	0
$481$	11894.8	1.12756
$482$	0	0
$483$	0	0
$484$	0	0
$485$	16650.4	1.55888
$486$	0	0
$487$	−12639.2	−1.17605	−0.588027	−	0.808841i	$-0.700095\pi$
$487$	−12639.2	−1.17605	−0.588027	+	0.808841i	$0.700095\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	363.554	0.0334154	0.0167077	−	0.999860i	$-0.494682\pi$
$491$	363.554	0.0334154	0.0167077	+	0.999860i	$0.494682\pi$
$492$	0	0
$493$	114.597	0.0104689
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−4734.89	−0.427342
$498$	0	0
$499$	−4327.18	−0.388198	−0.194099	−	0.980982i	$-0.562178\pi$
$499$	−4327.18	−0.388198	−0.194099	+	0.980982i	$0.562178\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−21621.8	−1.91664	−0.958318	−	0.285704i	$-0.907773\pi$
$503$	−21621.8	−1.91664	−0.958318	+	0.285704i	$0.907773\pi$
$504$	0	0
$505$	−640.764	−0.0564626
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−12903.3	−1.12363	−0.561817	−	0.827261i	$-0.689898\pi$
$509$	−12903.3	−1.12363	−0.561817	+	0.827261i	$0.689898\pi$
$510$	0	0
$511$	26906.2	2.32928
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13265.8	1.13507
$516$	0	0
$517$	7605.45	0.646977
$518$	0	0
$519$	0	0
$520$	0	0
$521$	5015.65	0.421765	0.210883	−	0.977511i	$-0.432366\pi$
$521$	5015.65	0.421765	0.210883	+	0.977511i	$0.432366\pi$
$522$	0	0
$523$	−10454.3	−0.874059	−0.437030	−	0.899447i	$-0.643970\pi$
$523$	−10454.3	−0.874059	−0.437030	+	0.899447i	$0.643970\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−635.148	−0.0525000
$528$	0	0
$529$	−10339.6	−0.849807
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−18876.8	−1.53404
$534$	0	0
$535$	−17669.0	−1.42785
$536$	0	0
$537$	0	0
$538$	0	0
$539$	8729.54	0.697603
$540$	0	0
$541$	16028.3	1.27377	0.636886	−	0.770958i	$-0.280222\pi$
$541$	16028.3	1.27377	0.636886	+	0.770958i	$0.280222\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6185.80	0.486184
$546$	0	0
$547$	−7590.17	−0.593295	−0.296647	−	0.954987i	$-0.595869\pi$
$547$	−7590.17	−0.593295	−0.296647	+	0.954987i	$0.595869\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−1067.81	−0.0825595
$552$	0	0
$553$	37727.9	2.90118
$554$	0	0
$555$	0	0
$556$	0	0
$557$	15598.4	1.18658	0.593291	−	0.804988i	$-0.297828\pi$
$557$	15598.4	1.18658	0.593291	+	0.804988i	$0.297828\pi$
$558$	0	0
$559$	−5289.11	−0.400189
$560$	0	0
$561$	0	0
$562$	0	0
$563$	18275.4	1.36806	0.684030	−	0.729454i	$-0.260226\pi$
$563$	18275.4	1.36806	0.684030	+	0.729454i	$0.260226\pi$
$564$	0	0
$565$	7542.51	0.561621
$566$	0	0
$567$	0	0
$568$	0	0
$569$	11103.6	0.818081	0.409040	−	0.912516i	$-0.365864\pi$
$569$	11103.6	0.818081	0.409040	+	0.912516i	$0.365864\pi$
$570$	0	0
$571$	12505.8	0.916550	0.458275	−	0.888810i	$-0.348467\pi$
$571$	12505.8	0.916550	0.458275	+	0.888810i	$0.348467\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−1323.31	−0.0959756
$576$	0	0
$577$	16122.3	1.16322	0.581612	−	0.813466i	$-0.302422\pi$
$577$	16122.3	1.16322	0.581612	+	0.813466i	$0.302422\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	14490.4	1.03471
$582$	0	0
$583$	2496.87	0.177375
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−82.7289	−0.00581701	−0.00290851	−	0.999996i	$-0.500926\pi$
$587$	−82.7289	−0.00581701	−0.00290851	+	0.999996i	$0.500926\pi$
$588$	0	0
$589$	5918.29	0.414022
$590$	0	0
$591$	0	0
$592$	0	0
$593$	5049.72	0.349692	0.174846	−	0.984596i	$-0.444057\pi$
$593$	5049.72	0.349692	0.174846	+	0.984596i	$0.444057\pi$
$594$	0	0
$595$	−1415.19	−0.0975076
$596$	0	0
$597$	0	0
$598$	0	0
$599$	16393.1	1.11820	0.559101	−	0.829100i	$-0.311147\pi$
$599$	16393.1	1.11820	0.559101	+	0.829100i	$0.311147\pi$
$600$	0	0
$601$	22564.2	1.53147	0.765733	−	0.643158i	$-0.222376\pi$
$601$	22564.2	1.53147	0.765733	+	0.643158i	$0.222376\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−12480.8	−0.838703
$606$	0	0
$607$	−10441.9	−0.698230	−0.349115	−	0.937080i	$-0.613518\pi$
$607$	−10441.9	−0.698230	−0.349115	+	0.937080i	$0.613518\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	15870.0	1.05079
$612$	0	0
$613$	−5153.31	−0.339544	−0.169772	−	0.985483i	$-0.554303\pi$
$613$	−5153.31	−0.339544	−0.169772	+	0.985483i	$0.554303\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−22893.5	−1.49377	−0.746886	−	0.664952i	$-0.768452\pi$
$617$	−22893.5	−1.49377	−0.746886	+	0.664952i	$0.768452\pi$
$618$	0	0
$619$	1872.55	0.121590	0.0607949	−	0.998150i	$-0.480636\pi$
$619$	1872.55	0.121590	0.0607949	+	0.998150i	$0.480636\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−22069.6	−1.41926
$624$	0	0
$625$	−18536.2	−1.18632
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1237.01	0.0784148
$630$	0	0
$631$	12644.7	0.797745	0.398872	−	0.917006i	$-0.369402\pi$
$631$	12644.7	0.797745	0.398872	+	0.917006i	$0.369402\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	−16012.8	−1.00071
$636$	0	0
$637$	18215.6	1.13301
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17232.7	1.06186	0.530929	−	0.847416i	$-0.321843\pi$
$641$	17232.7	1.06186	0.530929	+	0.847416i	$0.321843\pi$
$642$	0	0
$643$	4194.66	0.257265	0.128632	−	0.991692i	$-0.458941\pi$
$643$	4194.66	0.257265	0.128632	+	0.991692i	$0.458941\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	29338.5	1.78271	0.891355	−	0.453305i	$-0.149755\pi$
$647$	29338.5	1.78271	0.891355	+	0.453305i	$0.149755\pi$
$648$	0	0
$649$	−3460.48	−0.209300
$650$	0	0
$651$	0	0
$652$	0	0
$653$	30988.6	1.85709	0.928543	−	0.371225i	$-0.121062\pi$
$653$	30988.6	1.85709	0.928543	+	0.371225i	$0.121062\pi$
$654$	0	0
$655$	−30549.3	−1.82238
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−8159.70	−0.482332	−0.241166	−	0.970484i	$-0.577530\pi$
$659$	−8159.70	−0.482332	−0.241166	+	0.970484i	$0.577530\pi$
$660$	0	0
$661$	24409.7	1.43635	0.718174	−	0.695864i	$-0.244978\pi$
$661$	24409.7	1.43635	0.718174	+	0.695864i	$0.244978\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	13186.7	0.768959
$666$	0	0
$667$	−1239.69	−0.0719657
$668$	0	0
$669$	0	0
$670$	0	0
$671$	2933.09	0.168749
$672$	0	0
$673$	371.652	0.0212870	0.0106435	−	0.999943i	$-0.496612\pi$
$673$	371.652	0.0212870	0.0106435	+	0.999943i	$0.496612\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−8788.32	−0.498911	−0.249455	−	0.968386i	$-0.580252\pi$
$677$	−8788.32	−0.498911	−0.249455	+	0.968386i	$0.580252\pi$
$678$	0	0
$679$	−38235.0	−2.16101
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−8840.50	−0.495274	−0.247637	−	0.968853i	$-0.579654\pi$
$683$	−8840.50	−0.495274	−0.247637	+	0.968853i	$0.579654\pi$
$684$	0	0
$685$	−15734.0	−0.877616
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5210.11	0.288083
$690$	0	0
$691$	34252.5	1.88571	0.942856	−	0.333199i	$-0.108128\pi$
$691$	34252.5	1.88571	0.942856	+	0.333199i	$0.108128\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10275.1	−0.560804
$696$	0	0
$697$	−1963.11	−0.106683
$698$	0	0
$699$	0	0
$700$	0	0
$701$	27978.5	1.50747	0.753733	−	0.657180i	$-0.228251\pi$
$701$	27978.5	1.50747	0.753733	+	0.657180i	$0.228251\pi$
$702$	0	0
$703$	−11526.5	−0.618391
$704$	0	0
$705$	0	0
$706$	0	0
$707$	1471.41	0.0782719
$708$	0	0
$709$	−378.764	−0.0200632	−0.0100316	−	0.999950i	$-0.503193\pi$
$709$	−378.764	−0.0200632	−0.0100316	+	0.999950i	$0.503193\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	6870.94	0.360896
$714$	0	0
$715$	8641.03	0.451967
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−9369.50	−0.485985	−0.242993	−	0.970028i	$-0.578129\pi$
$719$	−9369.50	−0.485985	−0.242993	+	0.970028i	$0.578129\pi$
$720$	0	0
$721$	−30462.8	−1.57350
$722$	0	0
$723$	0	0
$724$	0	0
$725$	897.728	0.0459873
$726$	0	0
$727$	14672.0	0.748494	0.374247	−	0.927329i	$-0.377901\pi$
$727$	14672.0	0.748494	0.374247	+	0.927329i	$0.377901\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	−550.047	−0.0278307
$732$	0	0
$733$	−9764.90	−0.492053	−0.246027	−	0.969263i	$-0.579125\pi$
$733$	−9764.90	−0.492053	−0.246027	+	0.969263i	$0.579125\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	2280.94	0.114002
$738$	0	0
$739$	11150.5	0.555043	0.277521	−	0.960719i	$-0.410487\pi$
$739$	11150.5	0.555043	0.277521	+	0.960719i	$0.410487\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−17461.6	−0.862186	−0.431093	−	0.902307i	$-0.641872\pi$
$743$	−17461.6	−0.862186	−0.431093	+	0.902307i	$0.641872\pi$
$744$	0	0
$745$	5961.47	0.293169
$746$	0	0
$747$	0	0
$748$	0	0
$749$	40574.2	1.97937
$750$	0	0
$751$	12692.5	0.616717	0.308358	−	0.951270i	$-0.400220\pi$
$751$	12692.5	0.616717	0.308358	+	0.951270i	$0.400220\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−21741.0	−1.04800
$756$	0	0
$757$	35923.3	1.72477	0.862387	−	0.506250i	$-0.168969\pi$
$757$	35923.3	1.72477	0.862387	+	0.506250i	$0.168969\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	27302.4	1.30054	0.650270	−	0.759703i	$-0.274656\pi$
$761$	27302.4	1.30054	0.650270	+	0.759703i	$0.274656\pi$
$762$	0	0
$763$	−14204.7	−0.673978
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−7220.84	−0.339934
$768$	0	0
$769$	17941.0	0.841314	0.420657	−	0.907220i	$-0.361800\pi$
$769$	17941.0	0.841314	0.420657	+	0.907220i	$0.361800\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−6566.84	−0.305553	−0.152777	−	0.988261i	$-0.548822\pi$
$773$	−6566.84	−0.305553	−0.152777	+	0.988261i	$0.548822\pi$
$774$	0	0
$775$	−4975.61	−0.230619
$776$	0	0
$777$	0	0
$778$	0	0
$779$	18292.2	0.841318
$780$	0	0
$781$	3006.62	0.137753
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−17145.1	−0.779533
$786$	0	0
$787$	−28734.0	−1.30147	−0.650736	−	0.759304i	$-0.725539\pi$
$787$	−28734.0	−1.30147	−0.650736	+	0.759304i	$0.725539\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−17320.2	−0.778552
$792$	0	0
$793$	6120.36	0.274074
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22553.9	1.00238	0.501192	−	0.865336i	$-0.332895\pi$
$797$	22553.9	1.00238	0.501192	+	0.865336i	$0.332895\pi$
$798$	0	0
$799$	1650.42	0.0730757
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−17085.3	−0.750841
$804$	0	0
$805$	15309.3	0.670288
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−24921.7	−1.08307	−0.541533	−	0.840680i	$-0.682156\pi$
$809$	−24921.7	−1.08307	−0.541533	+	0.840680i	$0.682156\pi$
$810$	0	0
$811$	41652.2	1.80346	0.901730	−	0.432300i	$-0.142298\pi$
$811$	41652.2	1.80346	0.901730	+	0.432300i	$0.142298\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	46057.0	1.97952
$816$	0	0
$817$	5125.32	0.219477
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10969.1	0.466288	0.233144	−	0.972442i	$-0.425099\pi$
$821$	10969.1	0.466288	0.233144	+	0.972442i	$0.425099\pi$
$822$	0	0
$823$	16669.2	0.706015	0.353008	−	0.935620i	$-0.385159\pi$
$823$	16669.2	0.706015	0.353008	+	0.935620i	$0.385159\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9424.07	0.396260	0.198130	−	0.980176i	$-0.436513\pi$
$827$	9424.07	0.396260	0.198130	+	0.980176i	$0.436513\pi$
$828$	0	0
$829$	−23359.5	−0.978658	−0.489329	−	0.872099i	$-0.662758\pi$
$829$	−23359.5	−0.978658	−0.489329	+	0.872099i	$0.662758\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	1894.35	0.0787938
$834$	0	0
$835$	−50696.8	−2.10112
$836$	0	0
$837$	0	0
$838$	0	0
$839$	9260.03	0.381039	0.190520	−	0.981683i	$-0.438983\pi$
$839$	9260.03	0.381039	0.190520	+	0.981683i	$0.438983\pi$
$840$	0	0
$841$	841.000	0.0344828
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−9405.76	−0.382921
$846$	0	0
$847$	28660.1	1.16266
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−13381.8	−0.539040
$852$	0	0
$853$	−43938.1	−1.76367	−0.881836	−	0.471556i	$-0.843693\pi$
$853$	−43938.1	−1.76367	−0.881836	+	0.471556i	$0.843693\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−44540.6	−1.77535	−0.887676	−	0.460469i	$-0.847681\pi$
$857$	−44540.6	−1.77535	−0.887676	+	0.460469i	$0.847681\pi$
$858$	0	0
$859$	−5126.48	−0.203624	−0.101812	−	0.994804i	$-0.532464\pi$
$859$	−5126.48	−0.203624	−0.101812	+	0.994804i	$0.532464\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−29711.0	−1.17193	−0.585964	−	0.810337i	$-0.699284\pi$
$863$	−29711.0	−1.17193	−0.585964	+	0.810337i	$0.699284\pi$
$864$	0	0
$865$	−35115.9	−1.38032
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−23956.9	−0.935194
$870$	0	0
$871$	4759.55	0.185156
$872$	0	0
$873$	0	0
$874$	0	0
$875$	33679.8	1.30124
$876$	0	0
$877$	−24925.9	−0.959735	−0.479867	−	0.877341i	$-0.659315\pi$
$877$	−24925.9	−0.959735	−0.479867	+	0.877341i	$0.659315\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−13514.7	−0.516825	−0.258413	−	0.966035i	$-0.583199\pi$
$881$	−13514.7	−0.516825	−0.258413	+	0.966035i	$0.583199\pi$
$882$	0	0
$883$	21759.9	0.829309	0.414654	−	0.909979i	$-0.363903\pi$
$883$	21759.9	0.829309	0.414654	+	0.909979i	$0.363903\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15556.4	0.588877	0.294439	−	0.955670i	$-0.404867\pi$
$887$	15556.4	0.588877	0.294439	+	0.955670i	$0.404867\pi$
$888$	0	0
$889$	36770.9	1.38724
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−15378.5	−0.576286
$894$	0	0
$895$	−2045.42	−0.0763919
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−4661.20	−0.172925
$900$	0	0
$901$	541.831	0.0200344
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−44213.6	−1.62399
$906$	0	0
$907$	−36707.3	−1.34382	−0.671910	−	0.740633i	$-0.734526\pi$
$907$	−36707.3	−1.34382	−0.671910	+	0.740633i	$0.734526\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	21608.1	0.785849	0.392924	−	0.919571i	$-0.371463\pi$
$911$	21608.1	0.785849	0.392924	+	0.919571i	$0.371463\pi$
$912$	0	0
$913$	−9201.33	−0.333537
$914$	0	0
$915$	0	0
$916$	0	0
$917$	70151.7	2.52630
$918$	0	0
$919$	−40069.5	−1.43827	−0.719135	−	0.694870i	$-0.755462\pi$
$919$	−40069.5	−1.43827	−0.719135	+	0.694870i	$0.755462\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	6273.79	0.223732
$924$	0	0
$925$	9690.49	0.344456
$926$	0	0
$927$	0	0
$928$	0	0
$929$	18663.0	0.659112	0.329556	−	0.944136i	$-0.393101\pi$
$929$	18663.0	0.659112	0.329556	+	0.944136i	$0.393101\pi$
$930$	0	0
$931$	−17651.5	−0.621380
$932$	0	0
$933$	0	0
$934$	0	0
$935$	898.633	0.0314315
$936$	0	0
$937$	13742.0	0.479115	0.239557	−	0.970882i	$-0.422998\pi$
$937$	13742.0	0.479115	0.239557	+	0.970882i	$0.422998\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−31953.9	−1.10698	−0.553489	−	0.832856i	$-0.686704\pi$
$941$	−31953.9	−1.10698	−0.553489	+	0.832856i	$0.686704\pi$
$942$	0	0
$943$	21236.6	0.733362
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11988.8	−0.411386	−0.205693	−	0.978617i	$-0.565945\pi$
$947$	−11988.8	−0.411386	−0.205693	+	0.978617i	$0.565945\pi$
$948$	0	0
$949$	−35651.1	−1.21948
$950$	0	0
$951$	0	0
$952$	0	0
$953$	6567.87	0.223247	0.111623	−	0.993751i	$-0.464395\pi$
$953$	6567.87	0.223247	0.111623	+	0.993751i	$0.464395\pi$
$954$	0	0
$955$	−27020.2	−0.915552
$956$	0	0
$957$	0	0
$958$	0	0
$959$	36130.8	1.21660
$960$	0	0
$961$	−3956.49	−0.132808
$962$	0	0
$963$	0	0
$964$	0	0
$965$	57854.9	1.92997
$966$	0	0
$967$	40868.4	1.35909	0.679545	−	0.733634i	$-0.262177\pi$
$967$	40868.4	1.35909	0.679545	+	0.733634i	$0.262177\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	53372.2	1.76395	0.881976	−	0.471295i	$-0.156213\pi$
$971$	53372.2	1.76395	0.881976	+	0.471295i	$0.156213\pi$
$972$	0	0
$973$	23595.3	0.777419
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−32609.8	−1.06784	−0.533920	−	0.845535i	$-0.679282\pi$
$977$	−32609.8	−1.06784	−0.533920	+	0.845535i	$0.679282\pi$
$978$	0	0
$979$	14014.1	0.457499
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−23719.0	−0.769602	−0.384801	−	0.923000i	$-0.625730\pi$
$983$	−23719.0	−0.769602	−0.384801	+	0.923000i	$0.625730\pi$
$984$	0	0
$985$	15484.3	0.500885
$986$	0	0
$987$	0	0
$988$	0	0
$989$	5950.33	0.191314
$990$	0	0
$991$	−12328.2	−0.395176	−0.197588	−	0.980285i	$-0.563311\pi$
$991$	−12328.2	−0.395176	−0.197588	+	0.980285i	$0.563311\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	58088.4	1.85078
$996$	0	0
$997$	55140.7	1.75158	0.875790	−	0.482693i	$-0.160341\pi$
$997$	55140.7	1.75158	0.875790	+	0.482693i	$0.160341\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2088.4.a.b.1.3		3
3.2	odd	2		232.4.a.b.1.1	✓	3
12.11	even	2		464.4.a.g.1.3		3
24.5	odd	2		1856.4.a.p.1.3		3
24.11	even	2		1856.4.a.u.1.1		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
232.4.a.b.1.1	✓	3	3.2	odd	2
464.4.a.g.1.3		3	12.11	even	2
1856.4.a.p.1.3		3	24.5	odd	2
1856.4.a.u.1.1		3	24.11	even	2
2088.4.a.b.1.3		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:	\( N \)	\(=\)	\( 2088 = 2^{3} \cdot 3^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	2088.a (trivial)

\(f(q)\)	\(=\)	\(q+12.4882 q^{5} -28.6773 q^{7} +O(q^{10})\)	\(q+12.4882 q^{5} -28.6773 q^{7} +18.2099 q^{11} +37.9978 q^{13} +3.95162 q^{17} -36.8211 q^{19} -42.7480 q^{23} +30.9561 q^{25} +29.0000 q^{29} -160.731 q^{31} -358.129 q^{35} +313.040 q^{37} -496.787 q^{41} -139.195 q^{43} +417.656 q^{47} +479.386 q^{49} +137.116 q^{53} +227.409 q^{55} -190.033 q^{59} +161.072 q^{61} +474.525 q^{65} +125.259 q^{67} +165.110 q^{71} -938.243 q^{73} -522.209 q^{77} -1315.60 q^{79} -505.294 q^{83} +49.3487 q^{85} +769.587 q^{89} -1089.67 q^{91} -459.830 q^{95} +1333.29 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 4 q^{5} + 16 q^{7}+O(q^{10}) \) 3 * q - 4 * q^5 + 16 * q^7	\( 3 q - 4 q^{5} + 16 q^{7} + 2 q^{11} + 28 q^{13} + 66 q^{17} - 66 q^{19} - 176 q^{23} - 9 q^{25} + 87 q^{29} - 190 q^{31} - 660 q^{35} + 442 q^{37} - 1162 q^{41} + 30 q^{43} + 738 q^{47} + 851 q^{49} - 312 q^{53} + 464 q^{55} - 44 q^{59} + 54 q^{61} - 178 q^{65} - 116 q^{67} + 1200 q^{71} - 1118 q^{73} - 792 q^{77} - 2262 q^{79} + 1804 q^{83} - 8 q^{85} - 1578 q^{89} - 1972 q^{91} + 1052 q^{95} + 1450 q^{97}+O(q^{100}) \) 3 * q - 4 * q^5 + 16 * q^7 + 2 * q^11 + 28 * q^13 + 66 * q^17 - 66 * q^19 - 176 * q^23 - 9 * q^25 + 87 * q^29 - 190 * q^31 - 660 * q^35 + 442 * q^37 - 1162 * q^41 + 30 * q^43 + 738 * q^47 + 851 * q^49 - 312 * q^53 + 464 * q^55 - 44 * q^59 + 54 * q^61 - 178 * q^65 - 116 * q^67 + 1200 * q^71 - 1118 * q^73 - 792 * q^77 - 2262 * q^79 + 1804 * q^83 - 8 * q^85 - 1578 * q^89 - 1972 * q^91 + 1052 * q^95 + 1450 * q^97

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