LMFDB - Embedded newform 441.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [441,4,Mod(1,441)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(441, 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("441.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 441 = 3^{2} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	441.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:	$26.0198423125$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 7)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	441.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-2.00000 q^{2} -4.00000 q^{4} +7.00000 q^{5} +24.0000 q^{8} +O(q^{10})$

$q-2.00000 q^{2} -4.00000 q^{4} +7.00000 q^{5} +24.0000 q^{8} -14.0000 q^{10} +5.00000 q^{11} +14.0000 q^{13} -16.0000 q^{16} -21.0000 q^{17} -49.0000 q^{19} -28.0000 q^{20} -10.0000 q^{22} +159.000 q^{23} -76.0000 q^{25} -28.0000 q^{26} -58.0000 q^{29} -147.000 q^{31} -160.000 q^{32} +42.0000 q^{34} +219.000 q^{37} +98.0000 q^{38} +168.000 q^{40} +350.000 q^{41} -124.000 q^{43} -20.0000 q^{44} -318.000 q^{46} +525.000 q^{47} +152.000 q^{50} -56.0000 q^{52} -303.000 q^{53} +35.0000 q^{55} +116.000 q^{58} -105.000 q^{59} +413.000 q^{61} +294.000 q^{62} +448.000 q^{64} +98.0000 q^{65} +415.000 q^{67} +84.0000 q^{68} +432.000 q^{71} +1113.00 q^{73} -438.000 q^{74} +196.000 q^{76} -103.000 q^{79} -112.000 q^{80} -700.000 q^{82} +1092.00 q^{83} -147.000 q^{85} +248.000 q^{86} +120.000 q^{88} -329.000 q^{89} -636.000 q^{92} -1050.00 q^{94} -343.000 q^{95} +882.000 q^{97} +O(q^{100})$

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 $

$2$	−2.00000	−0.707107	−0.353553	−	0.935414i	$-0.615027\pi$
$2$	−2.00000	−0.707107	−0.353553	+	0.935414i	$0.615027\pi$
$3$	0	0
$3$	0	0
$4$	−4.00000	−0.500000
$5$	7.00000	0.626099	0.313050	−	0.949737i	$-0.398649\pi$
$5$	7.00000	0.626099	0.313050	+	0.949737i	$0.398649\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	24.0000	1.06066
$9$	0	0
$10$	−14.0000	−0.442719
$11$	5.00000	0.137051	0.0685253	−	0.997649i	$-0.478171\pi$
$11$	5.00000	0.137051	0.0685253	+	0.997649i	$0.478171\pi$
$12$	0	0
$13$	14.0000	0.298685	0.149342	−	0.988786i	$-0.452284\pi$
$13$	14.0000	0.298685	0.149342	+	0.988786i	$0.452284\pi$
$14$	0	0
$15$	0	0
$16$	−16.0000	−0.250000
$17$	−21.0000	−0.299603	−0.149801	−	0.988716i	$-0.547863\pi$
$17$	−21.0000	−0.299603	−0.149801	+	0.988716i	$0.547863\pi$
$18$	0	0
$19$	−49.0000	−0.591651	−0.295826	−	0.955242i	$-0.595595\pi$
$19$	−49.0000	−0.591651	−0.295826	+	0.955242i	$0.595595\pi$
$20$	−28.0000	−0.313050
$21$	0	0
$22$	−10.0000	−0.0969094
$23$	159.000	1.44147	0.720735	−	0.693211i	$-0.243805\pi$
$23$	159.000	1.44147	0.720735	+	0.693211i	$0.243805\pi$
$24$	0	0
$25$	−76.0000	−0.608000
$26$	−28.0000	−0.211202
$27$	0	0
$28$	0	0
$29$	−58.0000	−0.371391	−0.185695	−	0.982607i	$-0.559454\pi$
$29$	−58.0000	−0.371391	−0.185695	+	0.982607i	$0.559454\pi$
$30$	0	0
$31$	−147.000	−0.851677	−0.425838	−	0.904799i	$-0.640021\pi$
$31$	−147.000	−0.851677	−0.425838	+	0.904799i	$0.640021\pi$
$32$	−160.000	−0.883883
$33$	0	0
$34$	42.0000	0.211851
$35$	0	0
$36$	0	0
$37$	219.000	0.973064	0.486532	−	0.873663i	$-0.338262\pi$
$37$	219.000	0.973064	0.486532	+	0.873663i	$0.338262\pi$
$38$	98.0000	0.418361
$39$	0	0
$40$	168.000	0.664078
$41$	350.000	1.33319	0.666595	−	0.745420i	$-0.267751\pi$
$41$	350.000	1.33319	0.666595	+	0.745420i	$0.267751\pi$
$42$	0	0
$43$	−124.000	−0.439763	−0.219882	−	0.975527i	$-0.570567\pi$
$43$	−124.000	−0.439763	−0.219882	+	0.975527i	$0.570567\pi$
$44$	−20.0000	−0.0685253
$45$	0	0
$46$	−318.000	−1.01927
$47$	525.000	1.62934	0.814671	−	0.579923i	$-0.196917\pi$
$47$	525.000	1.62934	0.814671	+	0.579923i	$0.196917\pi$
$48$	0	0
$49$	0	0
$50$	152.000	0.429921
$51$	0	0
$52$	−56.0000	−0.149342
$53$	−303.000	−0.785288	−0.392644	−	0.919691i	$-0.628439\pi$
$53$	−303.000	−0.785288	−0.392644	+	0.919691i	$0.628439\pi$
$54$	0	0
$55$	35.0000	0.0858073
$56$	0	0
$57$	0	0
$58$	116.000	0.262613
$59$	−105.000	−0.231692	−0.115846	−	0.993267i	$-0.536958\pi$
$59$	−105.000	−0.231692	−0.115846	+	0.993267i	$0.536958\pi$
$60$	0	0
$61$	413.000	0.866873	0.433436	−	0.901184i	$-0.357301\pi$
$61$	413.000	0.866873	0.433436	+	0.901184i	$0.357301\pi$
$62$	294.000	0.602226
$63$	0	0
$64$	448.000	0.875000
$65$	98.0000	0.187006
$66$	0	0
$67$	415.000	0.756721	0.378361	−	0.925658i	$-0.376488\pi$
$67$	415.000	0.756721	0.378361	+	0.925658i	$0.376488\pi$
$68$	84.0000	0.149801
$69$	0	0
$70$	0	0
$71$	432.000	0.722098	0.361049	−	0.932547i	$-0.382419\pi$
$71$	432.000	0.722098	0.361049	+	0.932547i	$0.382419\pi$
$72$	0	0
$73$	1113.00	1.78448	0.892238	−	0.451565i	$-0.149134\pi$
$73$	1113.00	1.78448	0.892238	+	0.451565i	$0.149134\pi$
$74$	−438.000	−0.688060
$75$	0	0
$76$	196.000	0.295826
$77$	0	0
$78$	0	0
$79$	−103.000	−0.146689	−0.0733443	−	0.997307i	$-0.523367\pi$
$79$	−103.000	−0.146689	−0.0733443	+	0.997307i	$0.523367\pi$
$80$	−112.000	−0.156525
$81$	0	0
$82$	−700.000	−0.942708
$83$	1092.00	1.44413	0.722064	−	0.691827i	$-0.243194\pi$
$83$	1092.00	1.44413	0.722064	+	0.691827i	$0.243194\pi$
$84$	0	0
$85$	−147.000	−0.187581
$86$	248.000	0.310960
$87$	0	0
$88$	120.000	0.145364
$89$	−329.000	−0.391842	−0.195921	−	0.980620i	$-0.562770\pi$
$89$	−329.000	−0.391842	−0.195921	+	0.980620i	$0.562770\pi$
$90$	0	0
$91$	0	0
$92$	−636.000	−0.720735
$93$	0	0
$94$	−1050.00	−1.15212
$95$	−343.000	−0.370432
$96$	0	0
$97$	882.000	0.923232	0.461616	−	0.887080i	$-0.347270\pi$
$97$	882.000	0.923232	0.461616	+	0.887080i	$0.347270\pi$
$98$	0	0
$99$	0	0
$100$	304.000	0.304000
$101$	1379.00	1.35857	0.679285	−	0.733874i	$-0.262290\pi$
$101$	1379.00	1.35857	0.679285	+	0.733874i	$0.262290\pi$
$102$	0	0
$103$	679.000	0.649552	0.324776	−	0.945791i	$-0.394711\pi$
$103$	679.000	0.649552	0.324776	+	0.945791i	$0.394711\pi$
$104$	336.000	0.316803
$105$	0	0
$106$	606.000	0.555282
$107$	−457.000	−0.412896	−0.206448	−	0.978458i	$-0.566190\pi$
$107$	−457.000	−0.412896	−0.206448	+	0.978458i	$0.566190\pi$
$108$	0	0
$109$	−1125.00	−0.988582	−0.494291	−	0.869296i	$-0.664572\pi$
$109$	−1125.00	−0.988582	−0.494291	+	0.869296i	$0.664572\pi$
$110$	−70.0000	−0.0606749
$111$	0	0
$112$	0	0
$113$	1538.00	1.28038	0.640190	−	0.768217i	$-0.278856\pi$
$113$	1538.00	1.28038	0.640190	+	0.768217i	$0.278856\pi$
$114$	0	0
$115$	1113.00	0.902502
$116$	232.000	0.185695
$117$	0	0
$118$	210.000	0.163831
$119$	0	0
$120$	0	0
$121$	−1306.00	−0.981217
$122$	−826.000	−0.612972
$123$	0	0
$124$	588.000	0.425838
$125$	−1407.00	−1.00677
$126$	0	0
$127$	72.0000	0.0503068	0.0251534	−	0.999684i	$-0.491993\pi$
$127$	72.0000	0.0503068	0.0251534	+	0.999684i	$0.491993\pi$
$128$	384.000	0.265165
$129$	0	0
$130$	−196.000	−0.132233
$131$	2149.00	1.43327	0.716637	−	0.697446i	$-0.245680\pi$
$131$	2149.00	1.43327	0.716637	+	0.697446i	$0.245680\pi$
$132$	0	0
$133$	0	0
$134$	−830.000	−0.535083
$135$	0	0
$136$	−504.000	−0.317777
$137$	1125.00	0.701571	0.350786	−	0.936456i	$-0.385915\pi$
$137$	1125.00	0.701571	0.350786	+	0.936456i	$0.385915\pi$
$138$	0	0
$139$	−252.000	−0.153772	−0.0768862	−	0.997040i	$-0.524498\pi$
$139$	−252.000	−0.153772	−0.0768862	+	0.997040i	$0.524498\pi$
$140$	0	0
$141$	0	0
$142$	−864.000	−0.510600
$143$	70.0000	0.0409349
$144$	0	0
$145$	−406.000	−0.232527
$146$	−2226.00	−1.26182
$147$	0	0
$148$	−876.000	−0.486532
$149$	201.000	0.110514	0.0552569	−	0.998472i	$-0.482402\pi$
$149$	201.000	0.110514	0.0552569	+	0.998472i	$0.482402\pi$
$150$	0	0
$151$	1619.00	0.872532	0.436266	−	0.899818i	$-0.356301\pi$
$151$	1619.00	0.872532	0.436266	+	0.899818i	$0.356301\pi$
$152$	−1176.00	−0.627541
$153$	0	0
$154$	0	0
$155$	−1029.00	−0.533234
$156$	0	0
$157$	−679.000	−0.345160	−0.172580	−	0.984996i	$-0.555210\pi$
$157$	−679.000	−0.345160	−0.172580	+	0.984996i	$0.555210\pi$
$158$	206.000	0.103725
$159$	0	0
$160$	−1120.00	−0.553399
$161$	0	0
$162$	0	0
$163$	−467.000	−0.224407	−0.112203	−	0.993685i	$-0.535791\pi$
$163$	−467.000	−0.224407	−0.112203	+	0.993685i	$0.535791\pi$
$164$	−1400.00	−0.666595
$165$	0	0
$166$	−2184.00	−1.02115
$167$	1204.00	0.557894	0.278947	−	0.960306i	$-0.410015\pi$
$167$	1204.00	0.557894	0.278947	+	0.960306i	$0.410015\pi$
$168$	0	0
$169$	−2001.00	−0.910787
$170$	294.000	0.132640
$171$	0	0
$172$	496.000	0.219882
$173$	−2821.00	−1.23975	−0.619875	−	0.784701i	$-0.712817\pi$
$173$	−2821.00	−1.23975	−0.619875	+	0.784701i	$0.712817\pi$
$174$	0	0
$175$	0	0
$176$	−80.0000	−0.0342627
$177$	0	0
$178$	658.000	0.277074
$179$	3253.00	1.35833	0.679164	−	0.733987i	$-0.262343\pi$
$179$	3253.00	1.35833	0.679164	+	0.733987i	$0.262343\pi$
$180$	0	0
$181$	−1582.00	−0.649664	−0.324832	−	0.945772i	$-0.605308\pi$
$181$	−1582.00	−0.649664	−0.324832	+	0.945772i	$0.605308\pi$
$182$	0	0
$183$	0	0
$184$	3816.00	1.52891
$185$	1533.00	0.609235
$186$	0	0
$187$	−105.000	−0.0410608
$188$	−2100.00	−0.814671
$189$	0	0
$190$	686.000	0.261935
$191$	−2557.00	−0.968681	−0.484340	−	0.874880i	$-0.660940\pi$
$191$	−2557.00	−0.968681	−0.484340	+	0.874880i	$0.660940\pi$
$192$	0	0
$193$	−397.000	−0.148066	−0.0740329	−	0.997256i	$-0.523587\pi$
$193$	−397.000	−0.148066	−0.0740329	+	0.997256i	$0.523587\pi$
$194$	−1764.00	−0.652824
$195$	0	0
$196$	0	0
$197$	−2914.00	−1.05388	−0.526939	−	0.849903i	$-0.676660\pi$
$197$	−2914.00	−1.05388	−0.526939	+	0.849903i	$0.676660\pi$
$198$	0	0
$199$	−3339.00	−1.18942	−0.594712	−	0.803939i	$-0.702734\pi$
$199$	−3339.00	−1.18942	−0.594712	+	0.803939i	$0.702734\pi$
$200$	−1824.00	−0.644881
$201$	0	0
$202$	−2758.00	−0.960654
$203$	0	0
$204$	0	0
$205$	2450.00	0.834709
$206$	−1358.00	−0.459303
$207$	0	0
$208$	−224.000	−0.0746712
$209$	−245.000	−0.0810861
$210$	0	0
$211$	1780.00	0.580759	0.290380	−	0.956911i	$-0.406218\pi$
$211$	1780.00	0.580759	0.290380	+	0.956911i	$0.406218\pi$
$212$	1212.00	0.392644
$213$	0	0
$214$	914.000	0.291961
$215$	−868.000	−0.275335
$216$	0	0
$217$	0	0
$218$	2250.00	0.699033
$219$	0	0
$220$	−140.000	−0.0429036
$221$	−294.000	−0.0894868
$222$	0	0
$223$	1400.00	0.420408	0.210204	−	0.977658i	$-0.432587\pi$
$223$	1400.00	0.420408	0.210204	+	0.977658i	$0.432587\pi$
$224$	0	0
$225$	0	0
$226$	−3076.00	−0.905365
$227$	−2205.00	−0.644718	−0.322359	−	0.946617i	$-0.604476\pi$
$227$	−2205.00	−0.644718	−0.322359	+	0.946617i	$0.604476\pi$
$228$	0	0
$229$	−287.000	−0.0828188	−0.0414094	−	0.999142i	$-0.513185\pi$
$229$	−287.000	−0.0828188	−0.0414094	+	0.999142i	$0.513185\pi$
$230$	−2226.00	−0.638166
$231$	0	0
$232$	−1392.00	−0.393919
$233$	−4587.00	−1.28972	−0.644859	−	0.764301i	$-0.723084\pi$
$233$	−4587.00	−1.28972	−0.644859	+	0.764301i	$0.723084\pi$
$234$	0	0
$235$	3675.00	1.02013
$236$	420.000	0.115846
$237$	0	0
$238$	0	0
$239$	−1668.00	−0.451439	−0.225720	−	0.974192i	$-0.572473\pi$
$239$	−1668.00	−0.451439	−0.225720	+	0.974192i	$0.572473\pi$
$240$	0	0
$241$	3409.00	0.911174	0.455587	−	0.890191i	$-0.349429\pi$
$241$	3409.00	0.911174	0.455587	+	0.890191i	$0.349429\pi$
$242$	2612.00	0.693825
$243$	0	0
$244$	−1652.00	−0.433436
$245$	0	0
$246$	0	0
$247$	−686.000	−0.176717
$248$	−3528.00	−0.903340
$249$	0	0
$250$	2814.00	0.711892
$251$	−4760.00	−1.19701	−0.598503	−	0.801121i	$-0.704238\pi$
$251$	−4760.00	−1.19701	−0.598503	+	0.801121i	$0.704238\pi$
$252$	0	0
$253$	795.000	0.197554
$254$	−144.000	−0.0355723
$255$	0	0
$256$	−4352.00	−1.06250
$257$	−805.000	−0.195387	−0.0976936	−	0.995217i	$-0.531147\pi$
$257$	−805.000	−0.195387	−0.0976936	+	0.995217i	$0.531147\pi$
$258$	0	0
$259$	0	0
$260$	−392.000	−0.0935031
$261$	0	0
$262$	−4298.00	−1.01348
$263$	257.000	0.0602559	0.0301279	−	0.999546i	$-0.490409\pi$
$263$	257.000	0.0602559	0.0301279	+	0.999546i	$0.490409\pi$
$264$	0	0
$265$	−2121.00	−0.491668
$266$	0	0
$267$	0	0
$268$	−1660.00	−0.378361
$269$	3591.00	0.813930	0.406965	−	0.913444i	$-0.366587\pi$
$269$	3591.00	0.813930	0.406965	+	0.913444i	$0.366587\pi$
$270$	0	0
$271$	−1393.00	−0.312246	−0.156123	−	0.987738i	$-0.549900\pi$
$271$	−1393.00	−0.312246	−0.156123	+	0.987738i	$0.549900\pi$
$272$	336.000	0.0749007
$273$	0	0
$274$	−2250.00	−0.496086
$275$	−380.000	−0.0833268
$276$	0	0
$277$	415.000	0.0900178	0.0450089	−	0.998987i	$-0.485668\pi$
$277$	415.000	0.0900178	0.0450089	+	0.998987i	$0.485668\pi$
$278$	504.000	0.108733
$279$	0	0
$280$	0	0
$281$	4954.00	1.05171	0.525856	−	0.850574i	$-0.323745\pi$
$281$	4954.00	1.05171	0.525856	+	0.850574i	$0.323745\pi$
$282$	0	0
$283$	4277.00	0.898379	0.449190	−	0.893437i	$-0.351713\pi$
$283$	4277.00	0.898379	0.449190	+	0.893437i	$0.351713\pi$
$284$	−1728.00	−0.361049
$285$	0	0
$286$	−140.000	−0.0289454
$287$	0	0
$288$	0	0
$289$	−4472.00	−0.910238
$290$	812.000	0.164422
$291$	0	0
$292$	−4452.00	−0.892238
$293$	7742.00	1.54366	0.771830	−	0.635829i	$-0.219342\pi$
$293$	7742.00	1.54366	0.771830	+	0.635829i	$0.219342\pi$
$294$	0	0
$295$	−735.000	−0.145062
$296$	5256.00	1.03209
$297$	0	0
$298$	−402.000	−0.0781451
$299$	2226.00	0.430545
$300$	0	0
$301$	0	0
$302$	−3238.00	−0.616973
$303$	0	0
$304$	784.000	0.147913
$305$	2891.00	0.542748
$306$	0	0
$307$	7364.00	1.36901	0.684504	−	0.729009i	$-0.260019\pi$
$307$	7364.00	1.36901	0.684504	+	0.729009i	$0.260019\pi$
$308$	0	0
$309$	0	0
$310$	2058.00	0.377053
$311$	9975.00	1.81875	0.909374	−	0.415980i	$-0.136562\pi$
$311$	9975.00	1.81875	0.909374	+	0.415980i	$0.136562\pi$
$312$	0	0
$313$	4753.00	0.858324	0.429162	−	0.903228i	$-0.358809\pi$
$313$	4753.00	0.858324	0.429162	+	0.903228i	$0.358809\pi$
$314$	1358.00	0.244065
$315$	0	0
$316$	412.000	0.0733443
$317$	3477.00	0.616050	0.308025	−	0.951378i	$-0.400332\pi$
$317$	3477.00	0.616050	0.308025	+	0.951378i	$0.400332\pi$
$318$	0	0
$319$	−290.000	−0.0508993
$320$	3136.00	0.547837
$321$	0	0
$322$	0	0
$323$	1029.00	0.177260
$324$	0	0
$325$	−1064.00	−0.181600
$326$	934.000	0.158679
$327$	0	0
$328$	8400.00	1.41406
$329$	0	0
$330$	0	0
$331$	3341.00	0.554797	0.277399	−	0.960755i	$-0.410528\pi$
$331$	3341.00	0.554797	0.277399	+	0.960755i	$0.410528\pi$
$332$	−4368.00	−0.722064
$333$	0	0
$334$	−2408.00	−0.394491
$335$	2905.00	0.473782
$336$	0	0
$337$	7366.00	1.19066	0.595329	−	0.803482i	$-0.297022\pi$
$337$	7366.00	1.19066	0.595329	+	0.803482i	$0.297022\pi$
$338$	4002.00	0.644024
$339$	0	0
$340$	588.000	0.0937905
$341$	−735.000	−0.116723
$342$	0	0
$343$	0	0
$344$	−2976.00	−0.466439
$345$	0	0
$346$	5642.00	0.876635
$347$	−7415.00	−1.14714	−0.573571	−	0.819156i	$-0.694442\pi$
$347$	−7415.00	−1.14714	−0.573571	+	0.819156i	$0.694442\pi$
$348$	0	0
$349$	3878.00	0.594798	0.297399	−	0.954753i	$-0.403881\pi$
$349$	3878.00	0.594798	0.297399	+	0.954753i	$0.403881\pi$
$350$	0	0
$351$	0	0
$352$	−800.000	−0.121137
$353$	1267.00	0.191036	0.0955179	−	0.995428i	$-0.469549\pi$
$353$	1267.00	0.191036	0.0955179	+	0.995428i	$0.469549\pi$
$354$	0	0
$355$	3024.00	0.452105
$356$	1316.00	0.195921
$357$	0	0
$358$	−6506.00	−0.960483
$359$	−4685.00	−0.688760	−0.344380	−	0.938830i	$-0.611911\pi$
$359$	−4685.00	−0.688760	−0.344380	+	0.938830i	$0.611911\pi$
$360$	0	0
$361$	−4458.00	−0.649949
$362$	3164.00	0.459382
$363$	0	0
$364$	0	0
$365$	7791.00	1.11726
$366$	0	0
$367$	4641.00	0.660104	0.330052	−	0.943963i	$-0.392934\pi$
$367$	4641.00	0.660104	0.330052	+	0.943963i	$0.392934\pi$
$368$	−2544.00	−0.360367
$369$	0	0
$370$	−3066.00	−0.430794
$371$	0	0
$372$	0	0
$373$	−8797.00	−1.22116	−0.610578	−	0.791956i	$-0.709063\pi$
$373$	−8797.00	−1.22116	−0.610578	+	0.791956i	$0.709063\pi$
$374$	210.000	0.0290343
$375$	0	0
$376$	12600.0	1.72818
$377$	−812.000	−0.110929
$378$	0	0
$379$	13680.0	1.85407	0.927037	−	0.374969i	$-0.122347\pi$
$379$	13680.0	1.85407	0.927037	+	0.374969i	$0.122347\pi$
$380$	1372.00	0.185216
$381$	0	0
$382$	5114.00	0.684961
$383$	9765.00	1.30279	0.651395	−	0.758739i	$-0.274184\pi$
$383$	9765.00	1.30279	0.651395	+	0.758739i	$0.274184\pi$
$384$	0	0
$385$	0	0
$386$	794.000	0.104698
$387$	0	0
$388$	−3528.00	−0.461616
$389$	−1731.00	−0.225617	−0.112809	−	0.993617i	$-0.535985\pi$
$389$	−1731.00	−0.225617	−0.112809	+	0.993617i	$0.535985\pi$
$390$	0	0
$391$	−3339.00	−0.431868
$392$	0	0
$393$	0	0
$394$	5828.00	0.745204
$395$	−721.000	−0.0918416
$396$	0	0
$397$	−10983.0	−1.38847	−0.694233	−	0.719750i	$-0.744256\pi$
$397$	−10983.0	−1.38847	−0.694233	+	0.719750i	$0.744256\pi$
$398$	6678.00	0.841050
$399$	0	0
$400$	1216.00	0.152000
$401$	−6603.00	−0.822289	−0.411145	−	0.911570i	$-0.634871\pi$
$401$	−6603.00	−0.822289	−0.411145	+	0.911570i	$0.634871\pi$
$402$	0	0
$403$	−2058.00	−0.254383
$404$	−5516.00	−0.679285
$405$	0	0
$406$	0	0
$407$	1095.00	0.133359
$408$	0	0
$409$	−10955.0	−1.32443	−0.662213	−	0.749316i	$-0.730382\pi$
$409$	−10955.0	−1.32443	−0.662213	+	0.749316i	$0.730382\pi$
$410$	−4900.00	−0.590229
$411$	0	0
$412$	−2716.00	−0.324776
$413$	0	0
$414$	0	0
$415$	7644.00	0.904167
$416$	−2240.00	−0.264002
$417$	0	0
$418$	490.000	0.0573366
$419$	6636.00	0.773723	0.386861	−	0.922138i	$-0.373559\pi$
$419$	6636.00	0.773723	0.386861	+	0.922138i	$0.373559\pi$
$420$	0	0
$421$	−16630.0	−1.92517	−0.962585	−	0.270980i	$-0.912652\pi$
$421$	−16630.0	−1.92517	−0.962585	+	0.270980i	$0.912652\pi$
$422$	−3560.00	−0.410659
$423$	0	0
$424$	−7272.00	−0.832923
$425$	1596.00	0.182159
$426$	0	0
$427$	0	0
$428$	1828.00	0.206448
$429$	0	0
$430$	1736.00	0.194692
$431$	−4923.00	−0.550192	−0.275096	−	0.961417i	$-0.588710\pi$
$431$	−4923.00	−0.550192	−0.275096	+	0.961417i	$0.588710\pi$
$432$	0	0
$433$	−8974.00	−0.995988	−0.497994	−	0.867180i	$-0.665930\pi$
$433$	−8974.00	−0.995988	−0.497994	+	0.867180i	$0.665930\pi$
$434$	0	0
$435$	0	0
$436$	4500.00	0.494291
$437$	−7791.00	−0.852847
$438$	0	0
$439$	4179.00	0.454334	0.227167	−	0.973856i	$-0.427054\pi$
$439$	4179.00	0.454334	0.227167	+	0.973856i	$0.427054\pi$
$440$	840.000	0.0910123
$441$	0	0
$442$	588.000	0.0632767
$443$	12927.0	1.38641	0.693206	−	0.720740i	$-0.256198\pi$
$443$	12927.0	1.38641	0.693206	+	0.720740i	$0.256198\pi$
$444$	0	0
$445$	−2303.00	−0.245332
$446$	−2800.00	−0.297273
$447$	0	0
$448$	0	0
$449$	2826.00	0.297032	0.148516	−	0.988910i	$-0.452550\pi$
$449$	2826.00	0.297032	0.148516	+	0.988910i	$0.452550\pi$
$450$	0	0
$451$	1750.00	0.182715
$452$	−6152.00	−0.640190
$453$	0	0
$454$	4410.00	0.455884
$455$	0	0
$456$	0	0
$457$	8479.00	0.867901	0.433951	−	0.900937i	$-0.357119\pi$
$457$	8479.00	0.867901	0.433951	+	0.900937i	$0.357119\pi$
$458$	574.000	0.0585617
$459$	0	0
$460$	−4452.00	−0.451251
$461$	9338.00	0.943414	0.471707	−	0.881755i	$-0.343638\pi$
$461$	9338.00	0.943414	0.471707	+	0.881755i	$0.343638\pi$
$462$	0	0
$463$	−4016.00	−0.403109	−0.201554	−	0.979477i	$-0.564599\pi$
$463$	−4016.00	−0.403109	−0.201554	+	0.979477i	$0.564599\pi$
$464$	928.000	0.0928477
$465$	0	0
$466$	9174.00	0.911969
$467$	−5859.00	−0.580561	−0.290281	−	0.956942i	$-0.593749\pi$
$467$	−5859.00	−0.580561	−0.290281	+	0.956942i	$0.593749\pi$
$468$	0	0
$469$	0	0
$470$	−7350.00	−0.721341
$471$	0	0
$472$	−2520.00	−0.245747
$473$	−620.000	−0.0602698
$474$	0	0
$475$	3724.00	0.359724
$476$	0	0
$477$	0	0
$478$	3336.00	0.319216
$479$	6503.00	0.620312	0.310156	−	0.950686i	$-0.399619\pi$
$479$	6503.00	0.620312	0.310156	+	0.950686i	$0.399619\pi$
$480$	0	0
$481$	3066.00	0.290639
$482$	−6818.00	−0.644297
$483$	0	0
$484$	5224.00	0.490609
$485$	6174.00	0.578035
$486$	0	0
$487$	−16049.0	−1.49333	−0.746663	−	0.665203i	$-0.768345\pi$
$487$	−16049.0	−1.49333	−0.746663	+	0.665203i	$0.768345\pi$
$488$	9912.00	0.919457
$489$	0	0
$490$	0	0
$491$	−8864.00	−0.814718	−0.407359	−	0.913268i	$-0.633550\pi$
$491$	−8864.00	−0.814718	−0.407359	+	0.913268i	$0.633550\pi$
$492$	0	0
$493$	1218.00	0.111270
$494$	1372.00	0.124958
$495$	0	0
$496$	2352.00	0.212919
$497$	0	0
$498$	0	0
$499$	−10211.0	−0.916046	−0.458023	−	0.888940i	$-0.651442\pi$
$499$	−10211.0	−0.916046	−0.458023	+	0.888940i	$0.651442\pi$
$500$	5628.00	0.503384
$501$	0	0
$502$	9520.00	0.846411
$503$	−1680.00	−0.148921	−0.0744607	−	0.997224i	$-0.523724\pi$
$503$	−1680.00	−0.148921	−0.0744607	+	0.997224i	$0.523724\pi$
$504$	0	0
$505$	9653.00	0.850600
$506$	−1590.00	−0.139692
$507$	0	0
$508$	−288.000	−0.0251534
$509$	−9457.00	−0.823525	−0.411762	−	0.911291i	$-0.635087\pi$
$509$	−9457.00	−0.823525	−0.411762	+	0.911291i	$0.635087\pi$
$510$	0	0
$511$	0	0
$512$	5632.00	0.486136
$513$	0	0
$514$	1610.00	0.138160
$515$	4753.00	0.406684
$516$	0	0
$517$	2625.00	0.223302
$518$	0	0
$519$	0	0
$520$	2352.00	0.198350
$521$	−18081.0	−1.52043	−0.760214	−	0.649673i	$-0.774906\pi$
$521$	−18081.0	−1.52043	−0.760214	+	0.649673i	$0.774906\pi$
$522$	0	0
$523$	−20377.0	−1.70368	−0.851839	−	0.523803i	$-0.824513\pi$
$523$	−20377.0	−1.70368	−0.851839	+	0.523803i	$0.824513\pi$
$524$	−8596.00	−0.716637
$525$	0	0
$526$	−514.000	−0.0426073
$527$	3087.00	0.255165
$528$	0	0
$529$	13114.0	1.07783
$530$	4242.00	0.347662
$531$	0	0
$532$	0	0
$533$	4900.00	0.398204
$534$	0	0
$535$	−3199.00	−0.258514
$536$	9960.00	0.802624
$537$	0	0
$538$	−7182.00	−0.575535
$539$	0	0
$540$	0	0
$541$	−6193.00	−0.492159	−0.246079	−	0.969250i	$-0.579142\pi$
$541$	−6193.00	−0.492159	−0.246079	+	0.969250i	$0.579142\pi$
$542$	2786.00	0.220791
$543$	0	0
$544$	3360.00	0.264814
$545$	−7875.00	−0.618950
$546$	0	0
$547$	−18464.0	−1.44326	−0.721630	−	0.692279i	$-0.756607\pi$
$547$	−18464.0	−1.44326	−0.721630	+	0.692279i	$0.756607\pi$
$548$	−4500.00	−0.350786
$549$	0	0
$550$	760.000	0.0589209
$551$	2842.00	0.219734
$552$	0	0
$553$	0	0
$554$	−830.000	−0.0636522
$555$	0	0
$556$	1008.00	0.0768862
$557$	9413.00	0.716053	0.358027	−	0.933711i	$-0.383450\pi$
$557$	9413.00	0.716053	0.358027	+	0.933711i	$0.383450\pi$
$558$	0	0
$559$	−1736.00	−0.131351
$560$	0	0
$561$	0	0
$562$	−9908.00	−0.743672
$563$	3199.00	0.239470	0.119735	−	0.992806i	$-0.461795\pi$
$563$	3199.00	0.239470	0.119735	+	0.992806i	$0.461795\pi$
$564$	0	0
$565$	10766.0	0.801644
$566$	−8554.00	−0.635250
$567$	0	0
$568$	10368.0	0.765901
$569$	−21583.0	−1.59017	−0.795085	−	0.606498i	$-0.792574\pi$
$569$	−21583.0	−1.59017	−0.795085	+	0.606498i	$0.792574\pi$
$570$	0	0
$571$	20267.0	1.48537	0.742686	−	0.669640i	$-0.233551\pi$
$571$	20267.0	1.48537	0.742686	+	0.669640i	$0.233551\pi$
$572$	−280.000	−0.0204675
$573$	0	0
$574$	0	0
$575$	−12084.0	−0.876413
$576$	0	0
$577$	−13951.0	−1.00656	−0.503282	−	0.864122i	$-0.667874\pi$
$577$	−13951.0	−1.00656	−0.503282	+	0.864122i	$0.667874\pi$
$578$	8944.00	0.643636
$579$	0	0
$580$	1624.00	0.116264
$581$	0	0
$582$	0	0
$583$	−1515.00	−0.107624
$584$	26712.0	1.89272
$585$	0	0
$586$	−15484.0	−1.09153
$587$	−20972.0	−1.47463	−0.737314	−	0.675550i	$-0.763906\pi$
$587$	−20972.0	−1.47463	−0.737314	+	0.675550i	$0.763906\pi$
$588$	0	0
$589$	7203.00	0.503895
$590$	1470.00	0.102574
$591$	0	0
$592$	−3504.00	−0.243266
$593$	−189.000	−0.0130882	−0.00654410	−	0.999979i	$-0.502083\pi$
$593$	−189.000	−0.0130882	−0.00654410	+	0.999979i	$0.502083\pi$
$594$	0	0
$595$	0	0
$596$	−804.000	−0.0552569
$597$	0	0
$598$	−4452.00	−0.304441
$599$	10281.0	0.701286	0.350643	−	0.936509i	$-0.385963\pi$
$599$	10281.0	0.701286	0.350643	+	0.936509i	$0.385963\pi$
$600$	0	0
$601$	6090.00	0.413338	0.206669	−	0.978411i	$-0.433738\pi$
$601$	6090.00	0.413338	0.206669	+	0.978411i	$0.433738\pi$
$602$	0	0
$603$	0	0
$604$	−6476.00	−0.436266
$605$	−9142.00	−0.614339
$606$	0	0
$607$	−4949.00	−0.330929	−0.165464	−	0.986216i	$-0.552912\pi$
$607$	−4949.00	−0.330929	−0.165464	+	0.986216i	$0.552912\pi$
$608$	7840.00	0.522951
$609$	0	0
$610$	−5782.00	−0.383781
$611$	7350.00	0.486660
$612$	0	0
$613$	−15797.0	−1.04084	−0.520420	−	0.853910i	$-0.674225\pi$
$613$	−15797.0	−1.04084	−0.520420	+	0.853910i	$0.674225\pi$
$614$	−14728.0	−0.968035
$615$	0	0
$616$	0	0
$617$	9378.00	0.611903	0.305951	−	0.952047i	$-0.401025\pi$
$617$	9378.00	0.611903	0.305951	+	0.952047i	$0.401025\pi$
$618$	0	0
$619$	24353.0	1.58131	0.790654	−	0.612263i	$-0.209741\pi$
$619$	24353.0	1.58131	0.790654	+	0.612263i	$0.209741\pi$
$620$	4116.00	0.266617
$621$	0	0
$622$	−19950.0	−1.28605
$623$	0	0
$624$	0	0
$625$	−349.000	−0.0223360
$626$	−9506.00	−0.606927
$627$	0	0
$628$	2716.00	0.172580
$629$	−4599.00	−0.291533
$630$	0	0
$631$	−12640.0	−0.797449	−0.398725	−	0.917071i	$-0.630547\pi$
$631$	−12640.0	−0.797449	−0.398725	+	0.917071i	$0.630547\pi$
$632$	−2472.00	−0.155587
$633$	0	0
$634$	−6954.00	−0.435613
$635$	504.000	0.0314971
$636$	0	0
$637$	0	0
$638$	580.000	0.0359913
$639$	0	0
$640$	2688.00	0.166020
$641$	1041.00	0.0641451	0.0320726	−	0.999486i	$-0.489789\pi$
$641$	1041.00	0.0641451	0.0320726	+	0.999486i	$0.489789\pi$
$642$	0	0
$643$	−9548.00	−0.585593	−0.292797	−	0.956175i	$-0.594586\pi$
$643$	−9548.00	−0.585593	−0.292797	+	0.956175i	$0.594586\pi$
$644$	0	0
$645$	0	0
$646$	−2058.00	−0.125342
$647$	−3241.00	−0.196935	−0.0984674	−	0.995140i	$-0.531394\pi$
$647$	−3241.00	−0.196935	−0.0984674	+	0.995140i	$0.531394\pi$
$648$	0	0
$649$	−525.000	−0.0317535
$650$	2128.00	0.128411
$651$	0	0
$652$	1868.00	0.112203
$653$	8853.00	0.530543	0.265272	−	0.964174i	$-0.414538\pi$
$653$	8853.00	0.530543	0.265272	+	0.964174i	$0.414538\pi$
$654$	0	0
$655$	15043.0	0.897372
$656$	−5600.00	−0.333298
$657$	0	0
$658$	0	0
$659$	−7044.00	−0.416381	−0.208191	−	0.978088i	$-0.566757\pi$
$659$	−7044.00	−0.416381	−0.208191	+	0.978088i	$0.566757\pi$
$660$	0	0
$661$	12089.0	0.711358	0.355679	−	0.934608i	$-0.384250\pi$
$661$	12089.0	0.711358	0.355679	+	0.934608i	$0.384250\pi$
$662$	−6682.00	−0.392301
$663$	0	0
$664$	26208.0	1.53173
$665$	0	0
$666$	0	0
$667$	−9222.00	−0.535348
$668$	−4816.00	−0.278947
$669$	0	0
$670$	−5810.00	−0.335015
$671$	2065.00	0.118805
$672$	0	0
$673$	982.000	0.0562456	0.0281228	−	0.999604i	$-0.491047\pi$
$673$	982.000	0.0562456	0.0281228	+	0.999604i	$0.491047\pi$
$674$	−14732.0	−0.841922
$675$	0	0
$676$	8004.00	0.455394
$677$	−30513.0	−1.73222	−0.866108	−	0.499857i	$-0.833386\pi$
$677$	−30513.0	−1.73222	−0.866108	+	0.499857i	$0.833386\pi$
$678$	0	0
$679$	0	0
$680$	−3528.00	−0.198960
$681$	0	0
$682$	1470.00	0.0825355
$683$	−11475.0	−0.642868	−0.321434	−	0.946932i	$-0.604165\pi$
$683$	−11475.0	−0.642868	−0.321434	+	0.946932i	$0.604165\pi$
$684$	0	0
$685$	7875.00	0.439253
$686$	0	0
$687$	0	0
$688$	1984.00	0.109941
$689$	−4242.00	−0.234553
$690$	0	0
$691$	28315.0	1.55883	0.779416	−	0.626506i	$-0.215516\pi$
$691$	28315.0	1.55883	0.779416	+	0.626506i	$0.215516\pi$
$692$	11284.0	0.619875
$693$	0	0
$694$	14830.0	0.811151
$695$	−1764.00	−0.0962767
$696$	0	0
$697$	−7350.00	−0.399428
$698$	−7756.00	−0.420586
$699$	0	0
$700$	0	0
$701$	−10614.0	−0.571876	−0.285938	−	0.958248i	$-0.592305\pi$
$701$	−10614.0	−0.571876	−0.285938	+	0.958248i	$0.592305\pi$
$702$	0	0
$703$	−10731.0	−0.575715
$704$	2240.00	0.119919
$705$	0	0
$706$	−2534.00	−0.135083
$707$	0	0
$708$	0	0
$709$	10299.0	0.545539	0.272769	−	0.962079i	$-0.412060\pi$
$709$	10299.0	0.545539	0.272769	+	0.962079i	$0.412060\pi$
$710$	−6048.00	−0.319686
$711$	0	0
$712$	−7896.00	−0.415611
$713$	−23373.0	−1.22767
$714$	0	0
$715$	490.000	0.0256293
$716$	−13012.0	−0.679164
$717$	0	0
$718$	9370.00	0.487027
$719$	32529.0	1.68724	0.843621	−	0.536939i	$-0.180420\pi$
$719$	32529.0	1.68724	0.843621	+	0.536939i	$0.180420\pi$
$720$	0	0
$721$	0	0
$722$	8916.00	0.459583
$723$	0	0
$724$	6328.00	0.324832
$725$	4408.00	0.225806
$726$	0	0
$727$	−29456.0	−1.50270	−0.751350	−	0.659904i	$-0.770597\pi$
$727$	−29456.0	−1.50270	−0.751350	+	0.659904i	$0.770597\pi$
$728$	0	0
$729$	0	0
$730$	−15582.0	−0.790021
$731$	2604.00	0.131754
$732$	0	0
$733$	−27867.0	−1.40422	−0.702109	−	0.712070i	$-0.747758\pi$
$733$	−27867.0	−1.40422	−0.702109	+	0.712070i	$0.747758\pi$
$734$	−9282.00	−0.466764
$735$	0	0
$736$	−25440.0	−1.27409
$737$	2075.00	0.103709
$738$	0	0
$739$	19539.0	0.972603	0.486302	−	0.873791i	$-0.338346\pi$
$739$	19539.0	0.972603	0.486302	+	0.873791i	$0.338346\pi$
$740$	−6132.00	−0.304617
$741$	0	0
$742$	0	0
$743$	−1248.00	−0.0616214	−0.0308107	−	0.999525i	$-0.509809\pi$
$743$	−1248.00	−0.0616214	−0.0308107	+	0.999525i	$0.509809\pi$
$744$	0	0
$745$	1407.00	0.0691926
$746$	17594.0	0.863488
$747$	0	0
$748$	420.000	0.0205304
$749$	0	0
$750$	0	0
$751$	28093.0	1.36502	0.682509	−	0.730877i	$-0.260889\pi$
$751$	28093.0	1.36502	0.682509	+	0.730877i	$0.260889\pi$
$752$	−8400.00	−0.407336
$753$	0	0
$754$	1624.00	0.0784385
$755$	11333.0	0.546292
$756$	0	0
$757$	35954.0	1.72625	0.863124	−	0.504991i	$-0.168504\pi$
$757$	35954.0	1.72625	0.863124	+	0.504991i	$0.168504\pi$
$758$	−27360.0	−1.31103
$759$	0	0
$760$	−8232.00	−0.392903
$761$	−861.000	−0.0410134	−0.0205067	−	0.999790i	$-0.506528\pi$
$761$	−861.000	−0.0410134	−0.0205067	+	0.999790i	$0.506528\pi$
$762$	0	0
$763$	0	0
$764$	10228.0	0.484340
$765$	0	0
$766$	−19530.0	−0.921211
$767$	−1470.00	−0.0692029
$768$	0	0
$769$	−24710.0	−1.15873	−0.579366	−	0.815067i	$-0.696700\pi$
$769$	−24710.0	−1.15873	−0.579366	+	0.815067i	$0.696700\pi$
$770$	0	0
$771$	0	0
$772$	1588.00	0.0740329
$773$	16499.0	0.767694	0.383847	−	0.923397i	$-0.374599\pi$
$773$	16499.0	0.767694	0.383847	+	0.923397i	$0.374599\pi$
$774$	0	0
$775$	11172.0	0.517819
$776$	21168.0	0.979236
$777$	0	0
$778$	3462.00	0.159536
$779$	−17150.0	−0.788784
$780$	0	0
$781$	2160.00	0.0989640
$782$	6678.00	0.305377
$783$	0	0
$784$	0	0
$785$	−4753.00	−0.216104
$786$	0	0
$787$	−16471.0	−0.746033	−0.373016	−	0.927825i	$-0.621676\pi$
$787$	−16471.0	−0.746033	−0.373016	+	0.927825i	$0.621676\pi$
$788$	11656.0	0.526939
$789$	0	0
$790$	1442.00	0.0649418
$791$	0	0
$792$	0	0
$793$	5782.00	0.258922
$794$	21966.0	0.981794
$795$	0	0
$796$	13356.0	0.594712
$797$	−36470.0	−1.62087	−0.810435	−	0.585828i	$-0.800769\pi$
$797$	−36470.0	−1.62087	−0.810435	+	0.585828i	$0.800769\pi$
$798$	0	0
$799$	−11025.0	−0.488156
$800$	12160.0	0.537401
$801$	0	0
$802$	13206.0	0.581446
$803$	5565.00	0.244564
$804$	0	0
$805$	0	0
$806$	4116.00	0.179876
$807$	0	0
$808$	33096.0	1.44098
$809$	−35751.0	−1.55369	−0.776847	−	0.629690i	$-0.783182\pi$
$809$	−35751.0	−1.55369	−0.776847	+	0.629690i	$0.783182\pi$
$810$	0	0
$811$	16492.0	0.714072	0.357036	−	0.934091i	$-0.383787\pi$
$811$	16492.0	0.714072	0.357036	+	0.934091i	$0.383787\pi$
$812$	0	0
$813$	0	0
$814$	−2190.00	−0.0942991
$815$	−3269.00	−0.140501
$816$	0	0
$817$	6076.00	0.260186
$818$	21910.0	0.936510
$819$	0	0
$820$	−9800.00	−0.417355
$821$	41473.0	1.76299	0.881497	−	0.472190i	$-0.156536\pi$
$821$	41473.0	1.76299	0.881497	+	0.472190i	$0.156536\pi$
$822$	0	0
$823$	−25065.0	−1.06162	−0.530809	−	0.847492i	$-0.678112\pi$
$823$	−25065.0	−1.06162	−0.530809	+	0.847492i	$0.678112\pi$
$824$	16296.0	0.688954
$825$	0	0
$826$	0	0
$827$	−9732.00	−0.409208	−0.204604	−	0.978845i	$-0.565591\pi$
$827$	−9732.00	−0.409208	−0.204604	+	0.978845i	$0.565591\pi$
$828$	0	0
$829$	−27755.0	−1.16281	−0.581406	−	0.813614i	$-0.697497\pi$
$829$	−27755.0	−1.16281	−0.581406	+	0.813614i	$0.697497\pi$
$830$	−15288.0	−0.639342
$831$	0	0
$832$	6272.00	0.261349
$833$	0	0
$834$	0	0
$835$	8428.00	0.349297
$836$	980.000	0.0405431
$837$	0	0
$838$	−13272.0	−0.547105
$839$	21112.0	0.868733	0.434367	−	0.900736i	$-0.356972\pi$
$839$	21112.0	0.868733	0.434367	+	0.900736i	$0.356972\pi$
$840$	0	0
$841$	−21025.0	−0.862069
$842$	33260.0	1.36130
$843$	0	0
$844$	−7120.00	−0.290380
$845$	−14007.0	−0.570243
$846$	0	0
$847$	0	0
$848$	4848.00	0.196322
$849$	0	0
$850$	−3192.00	−0.128806
$851$	34821.0	1.40264
$852$	0	0
$853$	21238.0	0.852492	0.426246	−	0.904607i	$-0.359836\pi$
$853$	21238.0	0.852492	0.426246	+	0.904607i	$0.359836\pi$
$854$	0	0
$855$	0	0
$856$	−10968.0	−0.437942
$857$	−35609.0	−1.41935	−0.709673	−	0.704531i	$-0.751158\pi$
$857$	−35609.0	−1.41935	−0.709673	+	0.704531i	$0.751158\pi$
$858$	0	0
$859$	−2177.00	−0.0864706	−0.0432353	−	0.999065i	$-0.513767\pi$
$859$	−2177.00	−0.0864706	−0.0432353	+	0.999065i	$0.513767\pi$
$860$	3472.00	0.137668
$861$	0	0
$862$	9846.00	0.389044
$863$	32247.0	1.27196	0.635980	−	0.771706i	$-0.280596\pi$
$863$	32247.0	1.27196	0.635980	+	0.771706i	$0.280596\pi$
$864$	0	0
$865$	−19747.0	−0.776206
$866$	17948.0	0.704270
$867$	0	0
$868$	0	0
$869$	−515.000	−0.0201038
$870$	0	0
$871$	5810.00	0.226021
$872$	−27000.0	−1.04855
$873$	0	0
$874$	15582.0	0.603054
$875$	0	0
$876$	0	0
$877$	27631.0	1.06389	0.531946	−	0.846779i	$-0.321461\pi$
$877$	27631.0	1.06389	0.531946	+	0.846779i	$0.321461\pi$
$878$	−8358.00	−0.321263
$879$	0	0
$880$	−560.000	−0.0214518
$881$	24402.0	0.933172	0.466586	−	0.884476i	$-0.345484\pi$
$881$	24402.0	0.933172	0.466586	+	0.884476i	$0.345484\pi$
$882$	0	0
$883$	−19612.0	−0.747448	−0.373724	−	0.927540i	$-0.621919\pi$
$883$	−19612.0	−0.747448	−0.373724	+	0.927540i	$0.621919\pi$
$884$	1176.00	0.0447434
$885$	0	0
$886$	−25854.0	−0.980341
$887$	2261.00	0.0855884	0.0427942	−	0.999084i	$-0.486374\pi$
$887$	2261.00	0.0855884	0.0427942	+	0.999084i	$0.486374\pi$
$888$	0	0
$889$	0	0
$890$	4606.00	0.173476
$891$	0	0
$892$	−5600.00	−0.210204
$893$	−25725.0	−0.964003
$894$	0	0
$895$	22771.0	0.850448
$896$	0	0
$897$	0	0
$898$	−5652.00	−0.210033
$899$	8526.00	0.316305
$900$	0	0
$901$	6363.00	0.235274
$902$	−3500.00	−0.129199
$903$	0	0
$904$	36912.0	1.35805
$905$	−11074.0	−0.406754
$906$	0	0
$907$	−23833.0	−0.872505	−0.436252	−	0.899824i	$-0.643695\pi$
$907$	−23833.0	−0.872505	−0.436252	+	0.899824i	$0.643695\pi$
$908$	8820.00	0.322359
$909$	0	0
$910$	0	0
$911$	−31824.0	−1.15738	−0.578692	−	0.815546i	$-0.696437\pi$
$911$	−31824.0	−1.15738	−0.578692	+	0.815546i	$0.696437\pi$
$912$	0	0
$913$	5460.00	0.197919
$914$	−16958.0	−0.613699
$915$	0	0
$916$	1148.00	0.0414094
$917$	0	0
$918$	0	0
$919$	−16819.0	−0.603708	−0.301854	−	0.953354i	$-0.597606\pi$
$919$	−16819.0	−0.603708	−0.301854	+	0.953354i	$0.597606\pi$
$920$	26712.0	0.957248
$921$	0	0
$922$	−18676.0	−0.667095
$923$	6048.00	0.215680
$924$	0	0
$925$	−16644.0	−0.591623
$926$	8032.00	0.285041
$927$	0	0
$928$	9280.00	0.328266
$929$	1799.00	0.0635342	0.0317671	−	0.999495i	$-0.489887\pi$
$929$	1799.00	0.0635342	0.0317671	+	0.999495i	$0.489887\pi$
$930$	0	0
$931$	0	0
$932$	18348.0	0.644859
$933$	0	0
$934$	11718.0	0.410519
$935$	−735.000	−0.0257081
$936$	0	0
$937$	−14154.0	−0.493480	−0.246740	−	0.969082i	$-0.579359\pi$
$937$	−14154.0	−0.493480	−0.246740	+	0.969082i	$0.579359\pi$
$938$	0	0
$939$	0	0
$940$	−14700.0	−0.510065
$941$	12047.0	0.417344	0.208672	−	0.977986i	$-0.433086\pi$
$941$	12047.0	0.417344	0.208672	+	0.977986i	$0.433086\pi$
$942$	0	0
$943$	55650.0	1.92175
$944$	1680.00	0.0579230
$945$	0	0
$946$	1240.00	0.0426172
$947$	24379.0	0.836548	0.418274	−	0.908321i	$-0.362635\pi$
$947$	24379.0	0.836548	0.418274	+	0.908321i	$0.362635\pi$
$948$	0	0
$949$	15582.0	0.532996
$950$	−7448.00	−0.254363
$951$	0	0
$952$	0	0
$953$	52330.0	1.77874	0.889368	−	0.457192i	$-0.151145\pi$
$953$	52330.0	1.77874	0.889368	+	0.457192i	$0.151145\pi$
$954$	0	0
$955$	−17899.0	−0.606490
$956$	6672.00	0.225720
$957$	0	0
$958$	−13006.0	−0.438627
$959$	0	0
$960$	0	0
$961$	−8182.00	−0.274647
$962$	−6132.00	−0.205513
$963$	0	0
$964$	−13636.0	−0.455587
$965$	−2779.00	−0.0927038
$966$	0	0
$967$	−12416.0	−0.412897	−0.206449	−	0.978457i	$-0.566191\pi$
$967$	−12416.0	−0.412897	−0.206449	+	0.978457i	$0.566191\pi$
$968$	−31344.0	−1.04074
$969$	0	0
$970$	−12348.0	−0.408732
$971$	36813.0	1.21667	0.608334	−	0.793681i	$-0.291838\pi$
$971$	36813.0	1.21667	0.608334	+	0.793681i	$0.291838\pi$
$972$	0	0
$973$	0	0
$974$	32098.0	1.05594
$975$	0	0
$976$	−6608.00	−0.216718
$977$	−34995.0	−1.14595	−0.572973	−	0.819574i	$-0.694210\pi$
$977$	−34995.0	−1.14595	−0.572973	+	0.819574i	$0.694210\pi$
$978$	0	0
$979$	−1645.00	−0.0537022
$980$	0	0
$981$	0	0
$982$	17728.0	0.576093
$983$	−14301.0	−0.464019	−0.232010	−	0.972713i	$-0.574530\pi$
$983$	−14301.0	−0.464019	−0.232010	+	0.972713i	$0.574530\pi$
$984$	0	0
$985$	−20398.0	−0.659832
$986$	−2436.00	−0.0786796
$987$	0	0
$988$	2744.00	0.0883586
$989$	−19716.0	−0.633905
$990$	0	0
$991$	−2665.00	−0.0854253	−0.0427127	−	0.999087i	$-0.513600\pi$
$991$	−2665.00	−0.0854253	−0.0427127	+	0.999087i	$0.513600\pi$
$992$	23520.0	0.752783
$993$	0	0
$994$	0	0
$995$	−23373.0	−0.744697
$996$	0	0
$997$	−24871.0	−0.790043	−0.395021	−	0.918672i	$-0.629263\pi$
$997$	−24871.0	−0.790043	−0.395021	+	0.918672i	$0.629263\pi$
$998$	20422.0	0.647743
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	441.4.a.e.1.1		1
3.2	odd	2		49.4.a.c.1.1		1
7.2	even	3		441.4.e.k.361.1		2
7.3	odd	6		63.4.e.b.37.1		2
7.4	even	3		441.4.e.k.226.1		2
7.5	odd	6		63.4.e.b.46.1		2
7.6	odd	2		441.4.a.d.1.1		1
12.11	even	2		784.4.a.r.1.1		1
15.14	odd	2		1225.4.a.d.1.1		1
21.2	odd	6		49.4.c.a.18.1		2
21.5	even	6		7.4.c.a.4.1	yes	2
21.11	odd	6		49.4.c.a.30.1		2
21.17	even	6		7.4.c.a.2.1	✓	2
21.20	even	2		49.4.a.d.1.1		1
84.47	odd	6		112.4.i.c.81.1		2
84.59	odd	6		112.4.i.c.65.1		2
84.83	odd	2		784.4.a.b.1.1		1
105.17	odd	12		175.4.k.a.149.1		4
105.38	odd	12		175.4.k.a.149.2		4
105.47	odd	12		175.4.k.a.74.2		4
105.59	even	6		175.4.e.a.51.1		2
105.68	odd	12		175.4.k.a.74.1		4
105.89	even	6		175.4.e.a.151.1		2
105.104	even	2		1225.4.a.c.1.1		1
168.5	even	6		448.4.i.f.193.1		2
168.59	odd	6		448.4.i.a.65.1		2
168.101	even	6		448.4.i.f.65.1		2
168.131	odd	6		448.4.i.a.193.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
7.4.c.a.2.1	✓	2	21.17	even	6
7.4.c.a.4.1	yes	2	21.5	even	6
49.4.a.c.1.1		1	3.2	odd	2
49.4.a.d.1.1		1	21.20	even	2
49.4.c.a.18.1		2	21.2	odd	6
49.4.c.a.30.1		2	21.11	odd	6
63.4.e.b.37.1		2	7.3	odd	6
63.4.e.b.46.1		2	7.5	odd	6
112.4.i.c.65.1		2	84.59	odd	6
112.4.i.c.81.1		2	84.47	odd	6
175.4.e.a.51.1		2	105.59	even	6
175.4.e.a.151.1		2	105.89	even	6
175.4.k.a.74.1		4	105.68	odd	12
175.4.k.a.74.2		4	105.47	odd	12
175.4.k.a.149.1		4	105.17	odd	12
175.4.k.a.149.2		4	105.38	odd	12
441.4.a.d.1.1		1	7.6	odd	2
441.4.a.e.1.1		1	1.1	even	1	trivial
441.4.e.k.226.1		2	7.4	even	3
441.4.e.k.361.1		2	7.2	even	3
448.4.i.a.65.1		2	168.59	odd	6
448.4.i.a.193.1		2	168.131	odd	6
448.4.i.f.65.1		2	168.101	even	6
448.4.i.f.193.1		2	168.5	even	6
784.4.a.b.1.1		1	84.83	odd	2
784.4.a.r.1.1		1	12.11	even	2
1225.4.a.c.1.1		1	105.104	even	2
1225.4.a.d.1.1		1	15.14	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:	\( N \)	\(=\)	\( 441 = 3^{2} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	441.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more