LMFDB - Embedded newform 1089.4.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1089,4,Mod(1,1089)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1089, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1089.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1089 = 3^{2} \cdot 11^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1089.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:	$64.2530799963$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 363)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1089.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-4.00000 q^{2} +8.00000 q^{4} +13.0000 q^{5} +26.0000 q^{7} +O(q^{10})$

$q-4.00000 q^{2} +8.00000 q^{4} +13.0000 q^{5} +26.0000 q^{7} -52.0000 q^{10} -73.0000 q^{13} -104.000 q^{14} -64.0000 q^{16} -31.0000 q^{17} -108.000 q^{19} +104.000 q^{20} +86.0000 q^{23} +44.0000 q^{25} +292.000 q^{26} +208.000 q^{28} +207.000 q^{29} +208.000 q^{31} +256.000 q^{32} +124.000 q^{34} +338.000 q^{35} +45.0000 q^{37} +432.000 q^{38} -247.000 q^{41} -450.000 q^{43} -344.000 q^{46} +500.000 q^{47} +333.000 q^{49} -176.000 q^{50} -584.000 q^{52} +441.000 q^{53} -828.000 q^{58} -598.000 q^{59} +378.000 q^{61} -832.000 q^{62} -512.000 q^{64} -949.000 q^{65} +494.000 q^{67} -248.000 q^{68} -1352.00 q^{70} +594.000 q^{71} +1034.00 q^{73} -180.000 q^{74} -864.000 q^{76} +352.000 q^{79} -832.000 q^{80} +988.000 q^{82} -360.000 q^{83} -403.000 q^{85} +1800.00 q^{86} +351.000 q^{89} -1898.00 q^{91} +688.000 q^{92} -2000.00 q^{94} -1404.00 q^{95} +1079.00 q^{97} -1332.00 q^{98} +O(q^{100})$

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 $

$2$	−4.00000	−1.41421	−0.707107	−	0.707107i	$-0.750000\pi$
$2$	−4.00000	−1.41421	−0.707107	+	0.707107i	$0.750000\pi$
$3$	0	0
$3$	0	0
$4$	8.00000	1.00000
$5$	13.0000	1.16276	0.581378	−	0.813634i	$-0.302514\pi$
$5$	13.0000	1.16276	0.581378	+	0.813634i	$0.302514\pi$
$6$	0	0
$7$	26.0000	1.40387	0.701934	−	0.712242i	$-0.252320\pi$
$7$	26.0000	1.40387	0.701934	+	0.712242i	$0.252320\pi$
$8$	0	0
$9$	0	0
$10$	−52.0000	−1.64438
$11$	0	0
$11$	0	0
$12$	0	0
$13$	−73.0000	−1.55743	−0.778714	−	0.627379i	$-0.784128\pi$
$13$	−73.0000	−1.55743	−0.778714	+	0.627379i	$0.784128\pi$
$14$	−104.000	−1.98537
$15$	0	0
$16$	−64.0000	−1.00000
$17$	−31.0000	−0.442271	−0.221135	−	0.975243i	$-0.570976\pi$
$17$	−31.0000	−0.442271	−0.221135	+	0.975243i	$0.570976\pi$
$18$	0	0
$19$	−108.000	−1.30405	−0.652024	−	0.758199i	$-0.726080\pi$
$19$	−108.000	−1.30405	−0.652024	+	0.758199i	$0.726080\pi$
$20$	104.000	1.16276
$21$	0	0
$22$	0	0
$23$	86.0000	0.779663	0.389831	−	0.920886i	$-0.372533\pi$
$23$	86.0000	0.779663	0.389831	+	0.920886i	$0.372533\pi$
$24$	0	0
$25$	44.0000	0.352000
$26$	292.000	2.20254
$27$	0	0
$28$	208.000	1.40387
$29$	207.000	1.32548	0.662740	−	0.748849i	$-0.269393\pi$
$29$	207.000	1.32548	0.662740	+	0.748849i	$0.269393\pi$
$30$	0	0
$31$	208.000	1.20509	0.602547	−	0.798084i	$-0.294153\pi$
$31$	208.000	1.20509	0.602547	+	0.798084i	$0.294153\pi$
$32$	256.000	1.41421
$33$	0	0
$34$	124.000	0.625465
$35$	338.000	1.63236
$36$	0	0
$37$	45.0000	0.199945	0.0999724	−	0.994990i	$-0.468125\pi$
$37$	45.0000	0.199945	0.0999724	+	0.994990i	$0.468125\pi$
$38$	432.000	1.84420
$39$	0	0
$40$	0	0
$41$	−247.000	−0.940852	−0.470426	−	0.882440i	$-0.655900\pi$
$41$	−247.000	−0.940852	−0.470426	+	0.882440i	$0.655900\pi$
$42$	0	0
$43$	−450.000	−1.59592	−0.797958	−	0.602714i	$-0.794086\pi$
$43$	−450.000	−1.59592	−0.797958	+	0.602714i	$0.794086\pi$
$44$	0	0
$45$	0	0
$46$	−344.000	−1.10261
$47$	500.000	1.55176	0.775878	−	0.630883i	$-0.217307\pi$
$47$	500.000	1.55176	0.775878	+	0.630883i	$0.217307\pi$
$48$	0	0
$49$	333.000	0.970845
$50$	−176.000	−0.497803
$51$	0	0
$52$	−584.000	−1.55743
$53$	441.000	1.14294	0.571472	−	0.820622i	$-0.306373\pi$
$53$	441.000	1.14294	0.571472	+	0.820622i	$0.306373\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	0	0
$58$	−828.000	−1.87451
$59$	−598.000	−1.31954	−0.659771	−	0.751467i	$-0.729347\pi$
$59$	−598.000	−1.31954	−0.659771	+	0.751467i	$0.729347\pi$
$60$	0	0
$61$	378.000	0.793409	0.396704	−	0.917946i	$-0.370154\pi$
$61$	378.000	0.793409	0.396704	+	0.917946i	$0.370154\pi$
$62$	−832.000	−1.70426
$63$	0	0
$64$	−512.000	−1.00000
$65$	−949.000	−1.81091
$66$	0	0
$67$	494.000	0.900772	0.450386	−	0.892834i	$-0.351286\pi$
$67$	494.000	0.900772	0.450386	+	0.892834i	$0.351286\pi$
$68$	−248.000	−0.442271
$69$	0	0
$70$	−1352.00	−2.30850
$71$	594.000	0.992885	0.496442	−	0.868070i	$-0.334639\pi$
$71$	594.000	0.992885	0.496442	+	0.868070i	$0.334639\pi$
$72$	0	0
$73$	1034.00	1.65782	0.828908	−	0.559385i	$-0.188963\pi$
$73$	1034.00	1.65782	0.828908	+	0.559385i	$0.188963\pi$
$74$	−180.000	−0.282765
$75$	0	0
$76$	−864.000	−1.30405
$77$	0	0
$78$	0	0
$79$	352.000	0.501305	0.250652	−	0.968077i	$-0.419355\pi$
$79$	352.000	0.501305	0.250652	+	0.968077i	$0.419355\pi$
$80$	−832.000	−1.16276
$81$	0	0
$82$	988.000	1.33057
$83$	−360.000	−0.476086	−0.238043	−	0.971255i	$-0.576506\pi$
$83$	−360.000	−0.476086	−0.238043	+	0.971255i	$0.576506\pi$
$84$	0	0
$85$	−403.000	−0.514253
$86$	1800.00	2.25697
$87$	0	0
$88$	0	0
$89$	351.000	0.418044	0.209022	−	0.977911i	$-0.432972\pi$
$89$	351.000	0.418044	0.209022	+	0.977911i	$0.432972\pi$
$90$	0	0
$91$	−1898.00	−2.18642
$92$	688.000	0.779663
$93$	0	0
$94$	−2000.00	−2.19451
$95$	−1404.00	−1.51629
$96$	0	0
$97$	1079.00	1.12944	0.564721	−	0.825282i	$-0.308984\pi$
$97$	1079.00	1.12944	0.564721	+	0.825282i	$0.308984\pi$
$98$	−1332.00	−1.37298
$99$	0	0
$100$	352.000	0.352000
$101$	−86.0000	−0.0847259	−0.0423630	−	0.999102i	$-0.513489\pi$
$101$	−86.0000	−0.0847259	−0.0423630	+	0.999102i	$0.513489\pi$
$102$	0	0
$103$	−226.000	−0.216198	−0.108099	−	0.994140i	$-0.534476\pi$
$103$	−226.000	−0.216198	−0.108099	+	0.994140i	$0.534476\pi$
$104$	0	0
$105$	0	0
$106$	−1764.00	−1.61637
$107$	846.000	0.764354	0.382177	−	0.924089i	$-0.375174\pi$
$107$	846.000	0.764354	0.382177	+	0.924089i	$0.375174\pi$
$108$	0	0
$109$	1079.00	0.948160	0.474080	−	0.880482i	$-0.342781\pi$
$109$	1079.00	0.948160	0.474080	+	0.880482i	$0.342781\pi$
$110$	0	0
$111$	0	0
$112$	−1664.00	−1.40387
$113$	−117.000	−0.0974021	−0.0487010	−	0.998813i	$-0.515508\pi$
$113$	−117.000	−0.0974021	−0.0487010	+	0.998813i	$0.515508\pi$
$114$	0	0
$115$	1118.00	0.906557
$116$	1656.00	1.32548
$117$	0	0
$118$	2392.00	1.86611
$119$	−806.000	−0.620890
$120$	0	0
$121$	0	0
$122$	−1512.00	−1.12205
$123$	0	0
$124$	1664.00	1.20509
$125$	−1053.00	−0.753465
$126$	0	0
$127$	892.000	0.623246	0.311623	−	0.950206i	$-0.399127\pi$
$127$	892.000	0.623246	0.311623	+	0.950206i	$0.399127\pi$
$128$	0	0
$129$	0	0
$130$	3796.00	2.56101
$131$	2250.00	1.50064	0.750318	−	0.661077i	$-0.229900\pi$
$131$	2250.00	1.50064	0.750318	+	0.661077i	$0.229900\pi$
$132$	0	0
$133$	−2808.00	−1.83071
$134$	−1976.00	−1.27388
$135$	0	0
$136$	0	0
$137$	2138.00	1.33330	0.666648	−	0.745372i	$-0.267728\pi$
$137$	2138.00	1.33330	0.666648	+	0.745372i	$0.267728\pi$
$138$	0	0
$139$	350.000	0.213573	0.106786	−	0.994282i	$-0.465944\pi$
$139$	350.000	0.213573	0.106786	+	0.994282i	$0.465944\pi$
$140$	2704.00	1.63236
$141$	0	0
$142$	−2376.00	−1.40415
$143$	0	0
$144$	0	0
$145$	2691.00	1.54121
$146$	−4136.00	−2.34451
$147$	0	0
$148$	360.000	0.199945
$149$	1543.00	0.848372	0.424186	−	0.905575i	$-0.360560\pi$
$149$	1543.00	0.848372	0.424186	+	0.905575i	$0.360560\pi$
$150$	0	0
$151$	−2600.00	−1.40123	−0.700613	−	0.713542i	$-0.747090\pi$
$151$	−2600.00	−1.40123	−0.700613	+	0.713542i	$0.747090\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	2704.00	1.40123
$156$	0	0
$157$	−586.000	−0.297885	−0.148942	−	0.988846i	$-0.547587\pi$
$157$	−586.000	−0.297885	−0.148942	+	0.988846i	$0.547587\pi$
$158$	−1408.00	−0.708952
$159$	0	0
$160$	3328.00	1.64438
$161$	2236.00	1.09454
$162$	0	0
$163$	−1502.00	−0.721753	−0.360876	−	0.932614i	$-0.617522\pi$
$163$	−1502.00	−0.721753	−0.360876	+	0.932614i	$0.617522\pi$
$164$	−1976.00	−0.940852
$165$	0	0
$166$	1440.00	0.673287
$167$	−234.000	−0.108428	−0.0542140	−	0.998529i	$-0.517265\pi$
$167$	−234.000	−0.108428	−0.0542140	+	0.998529i	$0.517265\pi$
$168$	0	0
$169$	3132.00	1.42558
$170$	1612.00	0.727263
$171$	0	0
$172$	−3600.00	−1.59592
$173$	918.000	0.403435	0.201717	−	0.979444i	$-0.435348\pi$
$173$	918.000	0.403435	0.201717	+	0.979444i	$0.435348\pi$
$174$	0	0
$175$	1144.00	0.494162
$176$	0	0
$177$	0	0
$178$	−1404.00	−0.591204
$179$	−216.000	−0.0901933	−0.0450966	−	0.998983i	$-0.514360\pi$
$179$	−216.000	−0.0901933	−0.0450966	+	0.998983i	$0.514360\pi$
$180$	0	0
$181$	−863.000	−0.354399	−0.177200	−	0.984175i	$-0.556704\pi$
$181$	−863.000	−0.354399	−0.177200	+	0.984175i	$0.556704\pi$
$182$	7592.00	3.09207
$183$	0	0
$184$	0	0
$185$	585.000	0.232487
$186$	0	0
$187$	0	0
$188$	4000.00	1.55176
$189$	0	0
$190$	5616.00	2.14436
$191$	1508.00	0.571283	0.285641	−	0.958337i	$-0.407793\pi$
$191$	1508.00	0.571283	0.285641	+	0.958337i	$0.407793\pi$
$192$	0	0
$193$	−2203.00	−0.821634	−0.410817	−	0.911718i	$-0.634757\pi$
$193$	−2203.00	−0.821634	−0.410817	+	0.911718i	$0.634757\pi$
$194$	−4316.00	−1.59727
$195$	0	0
$196$	2664.00	0.970845
$197$	−949.000	−0.343215	−0.171608	−	0.985165i	$-0.554896\pi$
$197$	−949.000	−0.343215	−0.171608	+	0.985165i	$0.554896\pi$
$198$	0	0
$199$	2322.00	0.827147	0.413573	−	0.910471i	$-0.364281\pi$
$199$	2322.00	0.827147	0.413573	+	0.910471i	$0.364281\pi$
$200$	0	0
$201$	0	0
$202$	344.000	0.119821
$203$	5382.00	1.86080
$204$	0	0
$205$	−3211.00	−1.09398
$206$	904.000	0.305751
$207$	0	0
$208$	4672.00	1.55743
$209$	0	0
$210$	0	0
$211$	702.000	0.229041	0.114521	−	0.993421i	$-0.463467\pi$
$211$	702.000	0.229041	0.114521	+	0.993421i	$0.463467\pi$
$212$	3528.00	1.14294
$213$	0	0
$214$	−3384.00	−1.08096
$215$	−5850.00	−1.85566
$216$	0	0
$217$	5408.00	1.69179
$218$	−4316.00	−1.34090
$219$	0	0
$220$	0	0
$221$	2263.00	0.688805
$222$	0	0
$223$	−2240.00	−0.672652	−0.336326	−	0.941746i	$-0.609184\pi$
$223$	−2240.00	−0.672652	−0.336326	+	0.941746i	$0.609184\pi$
$224$	6656.00	1.98537
$225$	0	0
$226$	468.000	0.137747
$227$	−2574.00	−0.752610	−0.376305	−	0.926496i	$-0.622805\pi$
$227$	−2574.00	−0.752610	−0.376305	+	0.926496i	$0.622805\pi$
$228$	0	0
$229$	−3231.00	−0.932360	−0.466180	−	0.884690i	$-0.654370\pi$
$229$	−3231.00	−0.932360	−0.466180	+	0.884690i	$0.654370\pi$
$230$	−4472.00	−1.28206
$231$	0	0
$232$	0	0
$233$	−855.000	−0.240399	−0.120199	−	0.992750i	$-0.538353\pi$
$233$	−855.000	−0.240399	−0.120199	+	0.992750i	$0.538353\pi$
$234$	0	0
$235$	6500.00	1.80431
$236$	−4784.00	−1.31954
$237$	0	0
$238$	3224.00	0.878071
$239$	576.000	0.155893	0.0779463	−	0.996958i	$-0.475164\pi$
$239$	576.000	0.155893	0.0779463	+	0.996958i	$0.475164\pi$
$240$	0	0
$241$	3770.00	1.00766	0.503832	−	0.863802i	$-0.331923\pi$
$241$	3770.00	1.00766	0.503832	+	0.863802i	$0.331923\pi$
$242$	0	0
$243$	0	0
$244$	3024.00	0.793409
$245$	4329.00	1.12886
$246$	0	0
$247$	7884.00	2.03096
$248$	0	0
$249$	0	0
$250$	4212.00	1.06556
$251$	−554.000	−0.139315	−0.0696577	−	0.997571i	$-0.522191\pi$
$251$	−554.000	−0.139315	−0.0696577	+	0.997571i	$0.522191\pi$
$252$	0	0
$253$	0	0
$254$	−3568.00	−0.881402
$255$	0	0
$256$	4096.00	1.00000
$257$	6799.00	1.65023	0.825117	−	0.564962i	$-0.191109\pi$
$257$	6799.00	1.65023	0.825117	+	0.564962i	$0.191109\pi$
$258$	0	0
$259$	1170.00	0.280696
$260$	−7592.00	−1.81091
$261$	0	0
$262$	−9000.00	−2.12222
$263$	−716.000	−0.167872	−0.0839362	−	0.996471i	$-0.526749\pi$
$263$	−716.000	−0.167872	−0.0839362	+	0.996471i	$0.526749\pi$
$264$	0	0
$265$	5733.00	1.32896
$266$	11232.0	2.58902
$267$	0	0
$268$	3952.00	0.900772
$269$	−8411.00	−1.90642	−0.953211	−	0.302305i	$-0.902244\pi$
$269$	−8411.00	−1.90642	−0.953211	+	0.302305i	$0.902244\pi$
$270$	0	0
$271$	−2366.00	−0.530348	−0.265174	−	0.964201i	$-0.585429\pi$
$271$	−2366.00	−0.530348	−0.265174	+	0.964201i	$0.585429\pi$
$272$	1984.00	0.442271
$273$	0	0
$274$	−8552.00	−1.88557
$275$	0	0
$276$	0	0
$277$	5291.00	1.14767	0.573836	−	0.818970i	$-0.305455\pi$
$277$	5291.00	1.14767	0.573836	+	0.818970i	$0.305455\pi$
$278$	−1400.00	−0.302037
$279$	0	0
$280$	0	0
$281$	2106.00	0.447094	0.223547	−	0.974693i	$-0.428236\pi$
$281$	2106.00	0.447094	0.223547	+	0.974693i	$0.428236\pi$
$282$	0	0
$283$	6578.00	1.38170	0.690851	−	0.722997i	$-0.257236\pi$
$283$	6578.00	1.38170	0.690851	+	0.722997i	$0.257236\pi$
$284$	4752.00	0.992885
$285$	0	0
$286$	0	0
$287$	−6422.00	−1.32083
$288$	0	0
$289$	−3952.00	−0.804396
$290$	−10764.0	−2.17960
$291$	0	0
$292$	8272.00	1.65782
$293$	7443.00	1.48404	0.742022	−	0.670376i	$-0.233867\pi$
$293$	7443.00	1.48404	0.742022	+	0.670376i	$0.233867\pi$
$294$	0	0
$295$	−7774.00	−1.53430
$296$	0	0
$297$	0	0
$298$	−6172.00	−1.19978
$299$	−6278.00	−1.21427
$300$	0	0
$301$	−11700.0	−2.24045
$302$	10400.0	1.98163
$303$	0	0
$304$	6912.00	1.30405
$305$	4914.00	0.922540
$306$	0	0
$307$	982.000	0.182559	0.0912796	−	0.995825i	$-0.470904\pi$
$307$	982.000	0.182559	0.0912796	+	0.995825i	$0.470904\pi$
$308$	0	0
$309$	0	0
$310$	−10816.0	−1.98164
$311$	5868.00	1.06992	0.534958	−	0.844879i	$-0.320328\pi$
$311$	5868.00	1.06992	0.534958	+	0.844879i	$0.320328\pi$
$312$	0	0
$313$	−6265.00	−1.13137	−0.565685	−	0.824621i	$-0.691388\pi$
$313$	−6265.00	−1.13137	−0.565685	+	0.824621i	$0.691388\pi$
$314$	2344.00	0.421273
$315$	0	0
$316$	2816.00	0.501305
$317$	7726.00	1.36888	0.684441	−	0.729069i	$-0.260046\pi$
$317$	7726.00	1.36888	0.684441	+	0.729069i	$0.260046\pi$
$318$	0	0
$319$	0	0
$320$	−6656.00	−1.16276
$321$	0	0
$322$	−8944.00	−1.54792
$323$	3348.00	0.576742
$324$	0	0
$325$	−3212.00	−0.548214
$326$	6008.00	1.02071
$327$	0	0
$328$	0	0
$329$	13000.0	2.17846
$330$	0	0
$331$	1144.00	0.189970	0.0949848	−	0.995479i	$-0.469720\pi$
$331$	1144.00	0.189970	0.0949848	+	0.995479i	$0.469720\pi$
$332$	−2880.00	−0.476086
$333$	0	0
$334$	936.000	0.153340
$335$	6422.00	1.04738
$336$	0	0
$337$	4849.00	0.783804	0.391902	−	0.920007i	$-0.371817\pi$
$337$	4849.00	0.783804	0.391902	+	0.920007i	$0.371817\pi$
$338$	−12528.0	−2.01608
$339$	0	0
$340$	−3224.00	−0.514253
$341$	0	0
$342$	0	0
$343$	−260.000	−0.0409291
$344$	0	0
$345$	0	0
$346$	−3672.00	−0.570543
$347$	−5512.00	−0.852737	−0.426368	−	0.904550i	$-0.640207\pi$
$347$	−5512.00	−0.852737	−0.426368	+	0.904550i	$0.640207\pi$
$348$	0	0
$349$	−8073.00	−1.23822	−0.619109	−	0.785305i	$-0.712506\pi$
$349$	−8073.00	−1.23822	−0.619109	+	0.785305i	$0.712506\pi$
$350$	−4576.00	−0.698850
$351$	0	0
$352$	0	0
$353$	4203.00	0.633720	0.316860	−	0.948472i	$-0.397371\pi$
$353$	4203.00	0.633720	0.316860	+	0.948472i	$0.397371\pi$
$354$	0	0
$355$	7722.00	1.15448
$356$	2808.00	0.418044
$357$	0	0
$358$	864.000	0.127553
$359$	−1930.00	−0.283737	−0.141868	−	0.989886i	$-0.545311\pi$
$359$	−1930.00	−0.283737	−0.141868	+	0.989886i	$0.545311\pi$
$360$	0	0
$361$	4805.00	0.700539
$362$	3452.00	0.501196
$363$	0	0
$364$	−15184.0	−2.18642
$365$	13442.0	1.92763
$366$	0	0
$367$	−494.000	−0.0702632	−0.0351316	−	0.999383i	$-0.511185\pi$
$367$	−494.000	−0.0702632	−0.0351316	+	0.999383i	$0.511185\pi$
$368$	−5504.00	−0.779663
$369$	0	0
$370$	−2340.00	−0.328786
$371$	11466.0	1.60454
$372$	0	0
$373$	7982.00	1.10802	0.554011	−	0.832509i	$-0.313096\pi$
$373$	7982.00	1.10802	0.554011	+	0.832509i	$0.313096\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−15111.0	−2.06434
$378$	0	0
$379$	−6938.00	−0.940320	−0.470160	−	0.882581i	$-0.655804\pi$
$379$	−6938.00	−0.940320	−0.470160	+	0.882581i	$0.655804\pi$
$380$	−11232.0	−1.51629
$381$	0	0
$382$	−6032.00	−0.807916
$383$	8654.00	1.15457	0.577283	−	0.816544i	$-0.304113\pi$
$383$	8654.00	1.15457	0.577283	+	0.816544i	$0.304113\pi$
$384$	0	0
$385$	0	0
$386$	8812.00	1.16197
$387$	0	0
$388$	8632.00	1.12944
$389$	−5715.00	−0.744889	−0.372445	−	0.928054i	$-0.621480\pi$
$389$	−5715.00	−0.744889	−0.372445	+	0.928054i	$0.621480\pi$
$390$	0	0
$391$	−2666.00	−0.344822
$392$	0	0
$393$	0	0
$394$	3796.00	0.485380
$395$	4576.00	0.582895
$396$	0	0
$397$	−7163.00	−0.905543	−0.452772	−	0.891627i	$-0.649565\pi$
$397$	−7163.00	−0.905543	−0.452772	+	0.891627i	$0.649565\pi$
$398$	−9288.00	−1.16976
$399$	0	0
$400$	−2816.00	−0.352000
$401$	−6493.00	−0.808591	−0.404295	−	0.914628i	$-0.632483\pi$
$401$	−6493.00	−0.808591	−0.404295	+	0.914628i	$0.632483\pi$
$402$	0	0
$403$	−15184.0	−1.87685
$404$	−688.000	−0.0847259
$405$	0	0
$406$	−21528.0	−2.63157
$407$	0	0
$408$	0	0
$409$	585.000	0.0707247	0.0353623	−	0.999375i	$-0.488741\pi$
$409$	585.000	0.0707247	0.0353623	+	0.999375i	$0.488741\pi$
$410$	12844.0	1.54712
$411$	0	0
$412$	−1808.00	−0.216198
$413$	−15548.0	−1.85246
$414$	0	0
$415$	−4680.00	−0.553571
$416$	−18688.0	−2.20254
$417$	0	0
$418$	0	0
$419$	−7330.00	−0.854639	−0.427320	−	0.904101i	$-0.640542\pi$
$419$	−7330.00	−0.854639	−0.427320	+	0.904101i	$0.640542\pi$
$420$	0	0
$421$	−16055.0	−1.85861	−0.929303	−	0.369319i	$-0.879591\pi$
$421$	−16055.0	−1.85861	−0.929303	+	0.369319i	$0.879591\pi$
$422$	−2808.00	−0.323913
$423$	0	0
$424$	0	0
$425$	−1364.00	−0.155679
$426$	0	0
$427$	9828.00	1.11384
$428$	6768.00	0.764354
$429$	0	0
$430$	23400.0	2.62430
$431$	−2934.00	−0.327902	−0.163951	−	0.986468i	$-0.552424\pi$
$431$	−2934.00	−0.327902	−0.163951	+	0.986468i	$0.552424\pi$
$432$	0	0
$433$	−8549.00	−0.948819	−0.474410	−	0.880304i	$-0.657338\pi$
$433$	−8549.00	−0.948819	−0.474410	+	0.880304i	$0.657338\pi$
$434$	−21632.0	−2.39256
$435$	0	0
$436$	8632.00	0.948160
$437$	−9288.00	−1.01672
$438$	0	0
$439$	14084.0	1.53119	0.765595	−	0.643323i	$-0.222445\pi$
$439$	14084.0	1.53119	0.765595	+	0.643323i	$0.222445\pi$
$440$	0	0
$441$	0	0
$442$	−9052.00	−0.974117
$443$	−13586.0	−1.45709	−0.728544	−	0.684999i	$-0.759803\pi$
$443$	−13586.0	−1.45709	−0.728544	+	0.684999i	$0.759803\pi$
$444$	0	0
$445$	4563.00	0.486083
$446$	8960.00	0.951274
$447$	0	0
$448$	−13312.0	−1.40387
$449$	3659.00	0.384585	0.192293	−	0.981338i	$-0.438408\pi$
$449$	3659.00	0.384585	0.192293	+	0.981338i	$0.438408\pi$
$450$	0	0
$451$	0	0
$452$	−936.000	−0.0974021
$453$	0	0
$454$	10296.0	1.06435
$455$	−24674.0	−2.54227
$456$	0	0
$457$	−9711.00	−0.994007	−0.497004	−	0.867748i	$-0.665566\pi$
$457$	−9711.00	−0.994007	−0.497004	+	0.867748i	$0.665566\pi$
$458$	12924.0	1.31856
$459$	0	0
$460$	8944.00	0.906557
$461$	6903.00	0.697407	0.348704	−	0.937233i	$-0.386622\pi$
$461$	6903.00	0.697407	0.348704	+	0.937233i	$0.386622\pi$
$462$	0	0
$463$	18694.0	1.87642	0.938212	−	0.346062i	$-0.112481\pi$
$463$	18694.0	1.87642	0.938212	+	0.346062i	$0.112481\pi$
$464$	−13248.0	−1.32548
$465$	0	0
$466$	3420.00	0.339975
$467$	−450.000	−0.0445900	−0.0222950	−	0.999751i	$-0.507097\pi$
$467$	−450.000	−0.0445900	−0.0222950	+	0.999751i	$0.507097\pi$
$468$	0	0
$469$	12844.0	1.26456
$470$	−26000.0	−2.55168
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	−4752.00	−0.459025
$476$	−6448.00	−0.620890
$477$	0	0
$478$	−2304.00	−0.220465
$479$	12298.0	1.17309	0.586545	−	0.809917i	$-0.300488\pi$
$479$	12298.0	1.17309	0.586545	+	0.809917i	$0.300488\pi$
$480$	0	0
$481$	−3285.00	−0.311399
$482$	−15080.0	−1.42505
$483$	0	0
$484$	0	0
$485$	14027.0	1.31326
$486$	0	0
$487$	−11098.0	−1.03265	−0.516323	−	0.856394i	$-0.672700\pi$
$487$	−11098.0	−1.03265	−0.516323	+	0.856394i	$0.672700\pi$
$488$	0	0
$489$	0	0
$490$	−17316.0	−1.59644
$491$	2696.00	0.247798	0.123899	−	0.992295i	$-0.460460\pi$
$491$	2696.00	0.247798	0.123899	+	0.992295i	$0.460460\pi$
$492$	0	0
$493$	−6417.00	−0.586221
$494$	−31536.0	−2.87221
$495$	0	0
$496$	−13312.0	−1.20509
$497$	15444.0	1.39388
$498$	0	0
$499$	26.0000	0.00233250	0.00116625	−	0.999999i	$-0.499629\pi$
$499$	26.0000	0.00233250	0.00116625	+	0.999999i	$0.499629\pi$
$500$	−8424.00	−0.753465
$501$	0	0
$502$	2216.00	0.197022
$503$	−19462.0	−1.72518	−0.862592	−	0.505900i	$-0.831160\pi$
$503$	−19462.0	−1.72518	−0.862592	+	0.505900i	$0.831160\pi$
$504$	0	0
$505$	−1118.00	−0.0985155
$506$	0	0
$507$	0	0
$508$	7136.00	0.623246
$509$	19782.0	1.72264	0.861318	−	0.508066i	$-0.169639\pi$
$509$	19782.0	1.72264	0.861318	+	0.508066i	$0.169639\pi$
$510$	0	0
$511$	26884.0	2.32735
$512$	−16384.0	−1.41421
$513$	0	0
$514$	−27196.0	−2.33378
$515$	−2938.00	−0.251386
$516$	0	0
$517$	0	0
$518$	−4680.00	−0.396964
$519$	0	0
$520$	0	0
$521$	15534.0	1.30625	0.653126	−	0.757250i	$-0.273457\pi$
$521$	15534.0	1.30625	0.653126	+	0.757250i	$0.273457\pi$
$522$	0	0
$523$	376.000	0.0314366	0.0157183	−	0.999876i	$-0.494997\pi$
$523$	376.000	0.0314366	0.0157183	+	0.999876i	$0.494997\pi$
$524$	18000.0	1.50064
$525$	0	0
$526$	2864.00	0.237407
$527$	−6448.00	−0.532978
$528$	0	0
$529$	−4771.00	−0.392126
$530$	−22932.0	−1.87944
$531$	0	0
$532$	−22464.0	−1.83071
$533$	18031.0	1.46531
$534$	0	0
$535$	10998.0	0.888757
$536$	0	0
$537$	0	0
$538$	33644.0	2.69609
$539$	0	0
$540$	0	0
$541$	10322.0	0.820291	0.410146	−	0.912020i	$-0.365478\pi$
$541$	10322.0	0.820291	0.410146	+	0.912020i	$0.365478\pi$
$542$	9464.00	0.750025
$543$	0	0
$544$	−7936.00	−0.625465
$545$	14027.0	1.10248
$546$	0	0
$547$	−13976.0	−1.09245	−0.546225	−	0.837638i	$-0.683936\pi$
$547$	−13976.0	−1.09245	−0.546225	+	0.837638i	$0.683936\pi$
$548$	17104.0	1.33330
$549$	0	0
$550$	0	0
$551$	−22356.0	−1.72849
$552$	0	0
$553$	9152.00	0.703766
$554$	−21164.0	−1.62305
$555$	0	0
$556$	2800.00	0.213573
$557$	−8086.00	−0.615107	−0.307554	−	0.951531i	$-0.599510\pi$
$557$	−8086.00	−0.615107	−0.307554	+	0.951531i	$0.599510\pi$
$558$	0	0
$559$	32850.0	2.48552
$560$	−21632.0	−1.63236
$561$	0	0
$562$	−8424.00	−0.632286
$563$	10418.0	0.779869	0.389935	−	0.920843i	$-0.372498\pi$
$563$	10418.0	0.779869	0.389935	+	0.920843i	$0.372498\pi$
$564$	0	0
$565$	−1521.00	−0.113255
$566$	−26312.0	−1.95402
$567$	0	0
$568$	0	0
$569$	−20826.0	−1.53440	−0.767198	−	0.641410i	$-0.778350\pi$
$569$	−20826.0	−1.53440	−0.767198	+	0.641410i	$0.778350\pi$
$570$	0	0
$571$	10018.0	0.734221	0.367111	−	0.930177i	$-0.380347\pi$
$571$	10018.0	0.734221	0.367111	+	0.930177i	$0.380347\pi$
$572$	0	0
$573$	0	0
$574$	25688.0	1.86794
$575$	3784.00	0.274441
$576$	0	0
$577$	1395.00	0.100649	0.0503246	−	0.998733i	$-0.483974\pi$
$577$	1395.00	0.100649	0.0503246	+	0.998733i	$0.483974\pi$
$578$	15808.0	1.13759
$579$	0	0
$580$	21528.0	1.54121
$581$	−9360.00	−0.668362
$582$	0	0
$583$	0	0
$584$	0	0
$585$	0	0
$586$	−29772.0	−2.09875
$587$	17320.0	1.21784	0.608921	−	0.793231i	$-0.291603\pi$
$587$	17320.0	1.21784	0.608921	+	0.793231i	$0.291603\pi$
$588$	0	0
$589$	−22464.0	−1.57150
$590$	31096.0	2.16983
$591$	0	0
$592$	−2880.00	−0.199945
$593$	10309.0	0.713895	0.356948	−	0.934124i	$-0.383817\pi$
$593$	10309.0	0.713895	0.356948	+	0.934124i	$0.383817\pi$
$594$	0	0
$595$	−10478.0	−0.721943
$596$	12344.0	0.848372
$597$	0	0
$598$	25112.0	1.71723
$599$	11128.0	0.759061	0.379531	−	0.925179i	$-0.376085\pi$
$599$	11128.0	0.759061	0.379531	+	0.925179i	$0.376085\pi$
$600$	0	0
$601$	12349.0	0.838147	0.419073	−	0.907952i	$-0.362355\pi$
$601$	12349.0	0.838147	0.419073	+	0.907952i	$0.362355\pi$
$602$	46800.0	3.16848
$603$	0	0
$604$	−20800.0	−1.40123
$605$	0	0
$606$	0	0
$607$	−2744.00	−0.183485	−0.0917426	−	0.995783i	$-0.529244\pi$
$607$	−2744.00	−0.183485	−0.0917426	+	0.995783i	$0.529244\pi$
$608$	−27648.0	−1.84420
$609$	0	0
$610$	−19656.0	−1.30467
$611$	−36500.0	−2.41675
$612$	0	0
$613$	19387.0	1.27738	0.638690	−	0.769464i	$-0.279477\pi$
$613$	19387.0	1.27738	0.638690	+	0.769464i	$0.279477\pi$
$614$	−3928.00	−0.258178
$615$	0	0
$616$	0	0
$617$	−14265.0	−0.930774	−0.465387	−	0.885107i	$-0.654085\pi$
$617$	−14265.0	−0.930774	−0.465387	+	0.885107i	$0.654085\pi$
$618$	0	0
$619$	26630.0	1.72916	0.864580	−	0.502495i	$-0.167585\pi$
$619$	26630.0	1.72916	0.864580	+	0.502495i	$0.167585\pi$
$620$	21632.0	1.40123
$621$	0	0
$622$	−23472.0	−1.51309
$623$	9126.00	0.586879
$624$	0	0
$625$	−19189.0	−1.22810
$626$	25060.0	1.60000
$627$	0	0
$628$	−4688.00	−0.297885
$629$	−1395.00	−0.0884297
$630$	0	0
$631$	12610.0	0.795557	0.397778	−	0.917482i	$-0.369781\pi$
$631$	12610.0	0.795557	0.397778	+	0.917482i	$0.369781\pi$
$632$	0	0
$633$	0	0
$634$	−30904.0	−1.93589
$635$	11596.0	0.724682
$636$	0	0
$637$	−24309.0	−1.51202
$638$	0	0
$639$	0	0
$640$	0	0
$641$	23535.0	1.45020	0.725099	−	0.688645i	$-0.241794\pi$
$641$	23535.0	1.45020	0.725099	+	0.688645i	$0.241794\pi$
$642$	0	0
$643$	728.000	0.0446493	0.0223247	−	0.999751i	$-0.492893\pi$
$643$	728.000	0.0446493	0.0223247	+	0.999751i	$0.492893\pi$
$644$	17888.0	1.09454
$645$	0	0
$646$	−13392.0	−0.815637
$647$	−18400.0	−1.11805	−0.559025	−	0.829151i	$-0.688825\pi$
$647$	−18400.0	−1.11805	−0.559025	+	0.829151i	$0.688825\pi$
$648$	0	0
$649$	0	0
$650$	12848.0	0.775292
$651$	0	0
$652$	−12016.0	−0.721753
$653$	−19202.0	−1.15074	−0.575369	−	0.817894i	$-0.695142\pi$
$653$	−19202.0	−1.15074	−0.575369	+	0.817894i	$0.695142\pi$
$654$	0	0
$655$	29250.0	1.74487
$656$	15808.0	0.940852
$657$	0	0
$658$	−52000.0	−3.08081
$659$	4640.00	0.274277	0.137139	−	0.990552i	$-0.456209\pi$
$659$	4640.00	0.274277	0.137139	+	0.990552i	$0.456209\pi$
$660$	0	0
$661$	−10367.0	−0.610030	−0.305015	−	0.952348i	$-0.598661\pi$
$661$	−10367.0	−0.610030	−0.305015	+	0.952348i	$0.598661\pi$
$662$	−4576.00	−0.268658
$663$	0	0
$664$	0	0
$665$	−36504.0	−2.12867
$666$	0	0
$667$	17802.0	1.03343
$668$	−1872.00	−0.108428
$669$	0	0
$670$	−25688.0	−1.48121
$671$	0	0
$672$	0	0
$673$	−18278.0	−1.04690	−0.523451	−	0.852056i	$-0.675356\pi$
$673$	−18278.0	−1.04690	−0.523451	+	0.852056i	$0.675356\pi$
$674$	−19396.0	−1.10847
$675$	0	0
$676$	25056.0	1.42558
$677$	−27549.0	−1.56395	−0.781975	−	0.623310i	$-0.785788\pi$
$677$	−27549.0	−1.56395	−0.781975	+	0.623310i	$0.785788\pi$
$678$	0	0
$679$	28054.0	1.58559
$680$	0	0
$681$	0	0
$682$	0	0
$683$	9626.00	0.539281	0.269640	−	0.962961i	$-0.413095\pi$
$683$	9626.00	0.539281	0.269640	+	0.962961i	$0.413095\pi$
$684$	0	0
$685$	27794.0	1.55030
$686$	1040.00	0.0578825
$687$	0	0
$688$	28800.0	1.59592
$689$	−32193.0	−1.78005
$690$	0	0
$691$	27170.0	1.49580	0.747898	−	0.663813i	$-0.231063\pi$
$691$	27170.0	1.49580	0.747898	+	0.663813i	$0.231063\pi$
$692$	7344.00	0.403435
$693$	0	0
$694$	22048.0	1.20595
$695$	4550.00	0.248333
$696$	0	0
$697$	7657.00	0.416111
$698$	32292.0	1.75110
$699$	0	0
$700$	9152.00	0.494162
$701$	−31577.0	−1.70135	−0.850675	−	0.525691i	$-0.823807\pi$
$701$	−31577.0	−1.70135	−0.850675	+	0.525691i	$0.823807\pi$
$702$	0	0
$703$	−4860.00	−0.260737
$704$	0	0
$705$	0	0
$706$	−16812.0	−0.896215
$707$	−2236.00	−0.118944
$708$	0	0
$709$	−31330.0	−1.65955	−0.829776	−	0.558096i	$-0.811532\pi$
$709$	−31330.0	−1.65955	−0.829776	+	0.558096i	$0.811532\pi$
$710$	−30888.0	−1.63268
$711$	0	0
$712$	0	0
$713$	17888.0	0.939566
$714$	0	0
$715$	0	0
$716$	−1728.00	−0.0901933
$717$	0	0
$718$	7720.00	0.401264
$719$	34146.0	1.77111	0.885557	−	0.464531i	$-0.153777\pi$
$719$	34146.0	1.77111	0.885557	+	0.464531i	$0.153777\pi$
$720$	0	0
$721$	−5876.00	−0.303514
$722$	−19220.0	−0.990712
$723$	0	0
$724$	−6904.00	−0.354399
$725$	9108.00	0.466569
$726$	0	0
$727$	−8658.00	−0.441688	−0.220844	−	0.975309i	$-0.570881\pi$
$727$	−8658.00	−0.441688	−0.220844	+	0.975309i	$0.570881\pi$
$728$	0	0
$729$	0	0
$730$	−53768.0	−2.72609
$731$	13950.0	0.705827
$732$	0	0
$733$	−20349.0	−1.02539	−0.512693	−	0.858572i	$-0.671352\pi$
$733$	−20349.0	−1.02539	−0.512693	+	0.858572i	$0.671352\pi$
$734$	1976.00	0.0993672
$735$	0	0
$736$	22016.0	1.10261
$737$	0	0
$738$	0	0
$739$	−9080.00	−0.451980	−0.225990	−	0.974130i	$-0.572562\pi$
$739$	−9080.00	−0.451980	−0.225990	+	0.974130i	$0.572562\pi$
$740$	4680.00	0.232487
$741$	0	0
$742$	−45864.0	−2.26916
$743$	24674.0	1.21831	0.609153	−	0.793053i	$-0.291510\pi$
$743$	24674.0	1.21831	0.609153	+	0.793053i	$0.291510\pi$
$744$	0	0
$745$	20059.0	0.986450
$746$	−31928.0	−1.56698
$747$	0	0
$748$	0	0
$749$	21996.0	1.07305
$750$	0	0
$751$	442.000	0.0214764	0.0107382	−	0.999942i	$-0.496582\pi$
$751$	442.000	0.0214764	0.0107382	+	0.999942i	$0.496582\pi$
$752$	−32000.0	−1.55176
$753$	0	0
$754$	60444.0	2.91942
$755$	−33800.0	−1.62928
$756$	0	0
$757$	−10999.0	−0.528092	−0.264046	−	0.964510i	$-0.585057\pi$
$757$	−10999.0	−0.528092	−0.264046	+	0.964510i	$0.585057\pi$
$758$	27752.0	1.32981
$759$	0	0
$760$	0	0
$761$	20709.0	0.986466	0.493233	−	0.869897i	$-0.335815\pi$
$761$	20709.0	0.986466	0.493233	+	0.869897i	$0.335815\pi$
$762$	0	0
$763$	28054.0	1.33109
$764$	12064.0	0.571283
$765$	0	0
$766$	−34616.0	−1.63280
$767$	43654.0	2.05509
$768$	0	0
$769$	6985.00	0.327549	0.163775	−	0.986498i	$-0.447633\pi$
$769$	6985.00	0.327549	0.163775	+	0.986498i	$0.447633\pi$
$770$	0	0
$771$	0	0
$772$	−17624.0	−0.821634
$773$	−26442.0	−1.23034	−0.615170	−	0.788395i	$-0.710913\pi$
$773$	−26442.0	−1.23034	−0.615170	+	0.788395i	$0.710913\pi$
$774$	0	0
$775$	9152.00	0.424193
$776$	0	0
$777$	0	0
$778$	22860.0	1.05343
$779$	26676.0	1.22692
$780$	0	0
$781$	0	0
$782$	10664.0	0.487652
$783$	0	0
$784$	−21312.0	−0.970845
$785$	−7618.00	−0.346367
$786$	0	0
$787$	−13312.0	−0.602950	−0.301475	−	0.953474i	$-0.597479\pi$
$787$	−13312.0	−0.602950	−0.301475	+	0.953474i	$0.597479\pi$
$788$	−7592.00	−0.343215
$789$	0	0
$790$	−18304.0	−0.824338
$791$	−3042.00	−0.136740
$792$	0	0
$793$	−27594.0	−1.23568
$794$	28652.0	1.28063
$795$	0	0
$796$	18576.0	0.827147
$797$	3614.00	0.160620	0.0803102	−	0.996770i	$-0.474409\pi$
$797$	3614.00	0.160620	0.0803102	+	0.996770i	$0.474409\pi$
$798$	0	0
$799$	−15500.0	−0.686296
$800$	11264.0	0.497803
$801$	0	0
$802$	25972.0	1.14352
$803$	0	0
$804$	0	0
$805$	29068.0	1.27269
$806$	60736.0	2.65426
$807$	0	0
$808$	0	0
$809$	25650.0	1.11472	0.557358	−	0.830272i	$-0.311815\pi$
$809$	25650.0	1.11472	0.557358	+	0.830272i	$0.311815\pi$
$810$	0	0
$811$	−9676.00	−0.418952	−0.209476	−	0.977814i	$-0.567176\pi$
$811$	−9676.00	−0.418952	−0.209476	+	0.977814i	$0.567176\pi$
$812$	43056.0	1.86080
$813$	0	0
$814$	0	0
$815$	−19526.0	−0.839222
$816$	0	0
$817$	48600.0	2.08115
$818$	−2340.00	−0.100020
$819$	0	0
$820$	−25688.0	−1.09398
$821$	−29614.0	−1.25887	−0.629437	−	0.777051i	$-0.716714\pi$
$821$	−29614.0	−1.25887	−0.629437	+	0.777051i	$0.716714\pi$
$822$	0	0
$823$	−38358.0	−1.62464	−0.812318	−	0.583214i	$-0.801795\pi$
$823$	−38358.0	−1.62464	−0.812318	+	0.583214i	$0.801795\pi$
$824$	0	0
$825$	0	0
$826$	62192.0	2.61978
$827$	24862.0	1.04539	0.522694	−	0.852520i	$-0.324927\pi$
$827$	24862.0	1.04539	0.522694	+	0.852520i	$0.324927\pi$
$828$	0	0
$829$	5533.00	0.231808	0.115904	−	0.993260i	$-0.463023\pi$
$829$	5533.00	0.231808	0.115904	+	0.993260i	$0.463023\pi$
$830$	18720.0	0.782868
$831$	0	0
$832$	37376.0	1.55743
$833$	−10323.0	−0.429377
$834$	0	0
$835$	−3042.00	−0.126075
$836$	0	0
$837$	0	0
$838$	29320.0	1.20864
$839$	42016.0	1.72891	0.864454	−	0.502712i	$-0.167664\pi$
$839$	42016.0	1.72891	0.864454	+	0.502712i	$0.167664\pi$
$840$	0	0
$841$	18460.0	0.756899
$842$	64220.0	2.62846
$843$	0	0
$844$	5616.00	0.229041
$845$	40716.0	1.65760
$846$	0	0
$847$	0	0
$848$	−28224.0	−1.14294
$849$	0	0
$850$	5456.00	0.220164
$851$	3870.00	0.155889
$852$	0	0
$853$	7579.00	0.304220	0.152110	−	0.988364i	$-0.451393\pi$
$853$	7579.00	0.304220	0.152110	+	0.988364i	$0.451393\pi$
$854$	−39312.0	−1.57521
$855$	0	0
$856$	0	0
$857$	14854.0	0.592069	0.296034	−	0.955177i	$-0.404336\pi$
$857$	14854.0	0.592069	0.296034	+	0.955177i	$0.404336\pi$
$858$	0	0
$859$	47178.0	1.87391	0.936957	−	0.349444i	$-0.113629\pi$
$859$	47178.0	1.87391	0.936957	+	0.349444i	$0.113629\pi$
$860$	−46800.0	−1.85566
$861$	0	0
$862$	11736.0	0.463724
$863$	−17442.0	−0.687987	−0.343993	−	0.938972i	$-0.611780\pi$
$863$	−17442.0	−0.687987	−0.343993	+	0.938972i	$0.611780\pi$
$864$	0	0
$865$	11934.0	0.469096
$866$	34196.0	1.34183
$867$	0	0
$868$	43264.0	1.69179
$869$	0	0
$870$	0	0
$871$	−36062.0	−1.40289
$872$	0	0
$873$	0	0
$874$	37152.0	1.43785
$875$	−27378.0	−1.05777
$876$	0	0
$877$	−27261.0	−1.04964	−0.524822	−	0.851212i	$-0.675868\pi$
$877$	−27261.0	−1.04964	−0.524822	+	0.851212i	$0.675868\pi$
$878$	−56336.0	−2.16543
$879$	0	0
$880$	0	0
$881$	−15457.0	−0.591101	−0.295550	−	0.955327i	$-0.595503\pi$
$881$	−15457.0	−0.591101	−0.295550	+	0.955327i	$0.595503\pi$
$882$	0	0
$883$	−20852.0	−0.794706	−0.397353	−	0.917666i	$-0.630071\pi$
$883$	−20852.0	−0.794706	−0.397353	+	0.917666i	$0.630071\pi$
$884$	18104.0	0.688805
$885$	0	0
$886$	54344.0	2.06063
$887$	22594.0	0.855279	0.427639	−	0.903949i	$-0.359345\pi$
$887$	22594.0	0.855279	0.427639	+	0.903949i	$0.359345\pi$
$888$	0	0
$889$	23192.0	0.874955
$890$	−18252.0	−0.687425
$891$	0	0
$892$	−17920.0	−0.672652
$893$	−54000.0	−2.02356
$894$	0	0
$895$	−2808.00	−0.104873
$896$	0	0
$897$	0	0
$898$	−14636.0	−0.543886
$899$	43056.0	1.59733
$900$	0	0
$901$	−13671.0	−0.505491
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−11219.0	−0.412080
$906$	0	0
$907$	−47718.0	−1.74691	−0.873457	−	0.486902i	$-0.838127\pi$
$907$	−47718.0	−1.74691	−0.873457	+	0.486902i	$0.838127\pi$
$908$	−20592.0	−0.752610
$909$	0	0
$910$	98696.0	3.59532
$911$	8982.00	0.326660	0.163330	−	0.986572i	$-0.447777\pi$
$911$	8982.00	0.326660	0.163330	+	0.986572i	$0.447777\pi$
$912$	0	0
$913$	0	0
$914$	38844.0	1.40574
$915$	0	0
$916$	−25848.0	−0.932360
$917$	58500.0	2.10670
$918$	0	0
$919$	25154.0	0.902888	0.451444	−	0.892300i	$-0.350909\pi$
$919$	25154.0	0.902888	0.451444	+	0.892300i	$0.350909\pi$
$920$	0	0
$921$	0	0
$922$	−27612.0	−0.986283
$923$	−43362.0	−1.54635
$924$	0	0
$925$	1980.00	0.0703805
$926$	−74776.0	−2.65366
$927$	0	0
$928$	52992.0	1.87451
$929$	−27729.0	−0.979288	−0.489644	−	0.871922i	$-0.662873\pi$
$929$	−27729.0	−0.979288	−0.489644	+	0.871922i	$0.662873\pi$
$930$	0	0
$931$	−35964.0	−1.26603
$932$	−6840.00	−0.240399
$933$	0	0
$934$	1800.00	0.0630597
$935$	0	0
$936$	0	0
$937$	43525.0	1.51750	0.758751	−	0.651381i	$-0.225810\pi$
$937$	43525.0	1.51750	0.758751	+	0.651381i	$0.225810\pi$
$938$	−51376.0	−1.78836
$939$	0	0
$940$	52000.0	1.80431
$941$	34407.0	1.19196	0.595981	−	0.802999i	$-0.296763\pi$
$941$	34407.0	1.19196	0.595981	+	0.802999i	$0.296763\pi$
$942$	0	0
$943$	−21242.0	−0.733547
$944$	38272.0	1.31954
$945$	0	0
$946$	0	0
$947$	−6786.00	−0.232857	−0.116428	−	0.993199i	$-0.537145\pi$
$947$	−6786.00	−0.232857	−0.116428	+	0.993199i	$0.537145\pi$
$948$	0	0
$949$	−75482.0	−2.58193
$950$	19008.0	0.649159
$951$	0	0
$952$	0	0
$953$	−13567.0	−0.461152	−0.230576	−	0.973054i	$-0.574061\pi$
$953$	−13567.0	−0.461152	−0.230576	+	0.973054i	$0.574061\pi$
$954$	0	0
$955$	19604.0	0.664262
$956$	4608.00	0.155893
$957$	0	0
$958$	−49192.0	−1.65900
$959$	55588.0	1.87177
$960$	0	0
$961$	13473.0	0.452251
$962$	13140.0	0.440385
$963$	0	0
$964$	30160.0	1.00766
$965$	−28639.0	−0.955360
$966$	0	0
$967$	−8710.00	−0.289653	−0.144827	−	0.989457i	$-0.546262\pi$
$967$	−8710.00	−0.289653	−0.144827	+	0.989457i	$0.546262\pi$
$968$	0	0
$969$	0	0
$970$	−56108.0	−1.85724
$971$	−32566.0	−1.07631	−0.538153	−	0.842847i	$-0.680878\pi$
$971$	−32566.0	−1.07631	−0.538153	+	0.842847i	$0.680878\pi$
$972$	0	0
$973$	9100.00	0.299828
$974$	44392.0	1.46038
$975$	0	0
$976$	−24192.0	−0.793409
$977$	−22689.0	−0.742974	−0.371487	−	0.928438i	$-0.621152\pi$
$977$	−22689.0	−0.742974	−0.371487	+	0.928438i	$0.621152\pi$
$978$	0	0
$979$	0	0
$980$	34632.0	1.12886
$981$	0	0
$982$	−10784.0	−0.350439
$983$	−37908.0	−1.22999	−0.614994	−	0.788532i	$-0.710841\pi$
$983$	−37908.0	−1.22999	−0.614994	+	0.788532i	$0.710841\pi$
$984$	0	0
$985$	−12337.0	−0.399076
$986$	25668.0	0.829042
$987$	0	0
$988$	63072.0	2.03096
$989$	−38700.0	−1.24428
$990$	0	0
$991$	19540.0	0.626346	0.313173	−	0.949696i	$-0.398608\pi$
$991$	19540.0	0.626346	0.313173	+	0.949696i	$0.398608\pi$
$992$	53248.0	1.70426
$993$	0	0
$994$	−61776.0	−1.97124
$995$	30186.0	0.961769
$996$	0	0
$997$	−26585.0	−0.844489	−0.422244	−	0.906482i	$-0.638758\pi$
$997$	−26585.0	−0.844489	−0.422244	+	0.906482i	$0.638758\pi$
$998$	−104.000	−0.00329866
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1089.4.a.b.1.1		1
3.2	odd	2		363.4.a.g.1.1	yes	1
11.10	odd	2		1089.4.a.j.1.1		1
33.32	even	2		363.4.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
363.4.a.a.1.1	✓	1	33.32	even	2
363.4.a.g.1.1	yes	1	3.2	odd	2
1089.4.a.b.1.1		1	1.1	even	1	trivial
1089.4.a.j.1.1		1	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1089 = 3^{2} \cdot 11^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1089.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more