LMFDB - Embedded newform 54.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [54,4,Mod(1,54)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(54, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("54.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 54 = 2 \cdot 3^{3} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	54.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:	$3.18610314031$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	54.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +3.00000 q^{5} +29.0000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} +3.00000 q^{5} +29.0000 q^{7} +8.00000 q^{8} +6.00000 q^{10} -57.0000 q^{11} +20.0000 q^{13} +58.0000 q^{14} +16.0000 q^{16} -72.0000 q^{17} -106.000 q^{19} +12.0000 q^{20} -114.000 q^{22} +174.000 q^{23} -116.000 q^{25} +40.0000 q^{26} +116.000 q^{28} -210.000 q^{29} +47.0000 q^{31} +32.0000 q^{32} -144.000 q^{34} +87.0000 q^{35} +2.00000 q^{37} -212.000 q^{38} +24.0000 q^{40} -6.00000 q^{41} +218.000 q^{43} -228.000 q^{44} +348.000 q^{46} +474.000 q^{47} +498.000 q^{49} -232.000 q^{50} +80.0000 q^{52} +81.0000 q^{53} -171.000 q^{55} +232.000 q^{56} -420.000 q^{58} +84.0000 q^{59} +56.0000 q^{61} +94.0000 q^{62} +64.0000 q^{64} +60.0000 q^{65} -142.000 q^{67} -288.000 q^{68} +174.000 q^{70} +360.000 q^{71} -1159.00 q^{73} +4.00000 q^{74} -424.000 q^{76} -1653.00 q^{77} -160.000 q^{79} +48.0000 q^{80} -12.0000 q^{82} +735.000 q^{83} -216.000 q^{85} +436.000 q^{86} -456.000 q^{88} -954.000 q^{89} +580.000 q^{91} +696.000 q^{92} +948.000 q^{94} -318.000 q^{95} +191.000 q^{97} +996.000 q^{98} +O(q^{100})\)

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	3.00000	0.268328	0.134164	−	0.990959i	$-0.457165\pi$
$5$	3.00000	0.268328	0.134164	+	0.990959i	$0.457165\pi$
$6$	0	0
$7$	29.0000	1.56585	0.782926	−	0.622114i	$-0.213726\pi$
$7$	29.0000	1.56585	0.782926	+	0.622114i	$0.213726\pi$
$8$	8.00000	0.353553
$9$	0	0
$10$	6.00000	0.189737
$11$	−57.0000	−1.56238	−0.781188	−	0.624295i	$-0.785386\pi$
$11$	−57.0000	−1.56238	−0.781188	+	0.624295i	$0.785386\pi$
$12$	0	0
$13$	20.0000	0.426692	0.213346	−	0.976977i	$-0.431564\pi$
$13$	20.0000	0.426692	0.213346	+	0.976977i	$0.431564\pi$
$14$	58.0000	1.10723
$15$	0	0
$16$	16.0000	0.250000
$17$	−72.0000	−1.02721	−0.513605	−	0.858027i	$-0.671690\pi$
$17$	−72.0000	−1.02721	−0.513605	+	0.858027i	$0.671690\pi$
$18$	0	0
$19$	−106.000	−1.27990	−0.639949	−	0.768417i	$-0.721045\pi$
$19$	−106.000	−1.27990	−0.639949	+	0.768417i	$0.721045\pi$
$20$	12.0000	0.134164
$21$	0	0
$22$	−114.000	−1.10477
$23$	174.000	1.57746	0.788728	−	0.614742i	$-0.210740\pi$
$23$	174.000	1.57746	0.788728	+	0.614742i	$0.210740\pi$
$24$	0	0
$25$	−116.000	−0.928000
$26$	40.0000	0.301717
$27$	0	0
$28$	116.000	0.782926
$29$	−210.000	−1.34469	−0.672345	−	0.740238i	$-0.734713\pi$
$29$	−210.000	−1.34469	−0.672345	+	0.740238i	$0.734713\pi$
$30$	0	0
$31$	47.0000	0.272305	0.136152	−	0.990688i	$-0.456526\pi$
$31$	47.0000	0.272305	0.136152	+	0.990688i	$0.456526\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−144.000	−0.726347
$35$	87.0000	0.420162
$36$	0	0
$37$	2.00000	0.00888643	0.00444322	−	0.999990i	$-0.498586\pi$
$37$	2.00000	0.00888643	0.00444322	+	0.999990i	$0.498586\pi$
$38$	−212.000	−0.905025
$39$	0	0
$40$	24.0000	0.0948683
$41$	−6.00000	−0.0228547	−0.0114273	−	0.999935i	$-0.503638\pi$
$41$	−6.00000	−0.0228547	−0.0114273	+	0.999935i	$0.503638\pi$
$42$	0	0
$43$	218.000	0.773132	0.386566	−	0.922262i	$-0.373661\pi$
$43$	218.000	0.773132	0.386566	+	0.922262i	$0.373661\pi$
$44$	−228.000	−0.781188
$45$	0	0
$46$	348.000	1.11543
$47$	474.000	1.47106	0.735532	−	0.677490i	$-0.236932\pi$
$47$	474.000	1.47106	0.735532	+	0.677490i	$0.236932\pi$
$48$	0	0
$49$	498.000	1.45190
$50$	−232.000	−0.656195
$51$	0	0
$52$	80.0000	0.213346
$53$	81.0000	0.209928	0.104964	−	0.994476i	$-0.466527\pi$
$53$	81.0000	0.209928	0.104964	+	0.994476i	$0.466527\pi$
$54$	0	0
$55$	−171.000	−0.419230
$56$	232.000	0.553613
$57$	0	0
$58$	−420.000	−0.950840
$59$	84.0000	0.185354	0.0926769	−	0.995696i	$-0.470458\pi$
$59$	84.0000	0.185354	0.0926769	+	0.995696i	$0.470458\pi$
$60$	0	0
$61$	56.0000	0.117542	0.0587710	−	0.998271i	$-0.481282\pi$
$61$	56.0000	0.117542	0.0587710	+	0.998271i	$0.481282\pi$
$62$	94.0000	0.192549
$63$	0	0
$64$	64.0000	0.125000
$65$	60.0000	0.114494
$66$	0	0
$67$	−142.000	−0.258926	−0.129463	−	0.991584i	$-0.541325\pi$
$67$	−142.000	−0.258926	−0.129463	+	0.991584i	$0.541325\pi$
$68$	−288.000	−0.513605
$69$	0	0
$70$	174.000	0.297100
$71$	360.000	0.601748	0.300874	−	0.953664i	$-0.402722\pi$
$71$	360.000	0.601748	0.300874	+	0.953664i	$0.402722\pi$
$72$	0	0
$73$	−1159.00	−1.85823	−0.929114	−	0.369793i	$-0.879429\pi$
$73$	−1159.00	−1.85823	−0.929114	+	0.369793i	$0.879429\pi$
$74$	4.00000	0.00628366
$75$	0	0
$76$	−424.000	−0.639949
$77$	−1653.00	−2.44645
$78$	0	0
$79$	−160.000	−0.227866	−0.113933	−	0.993488i	$-0.536345\pi$
$79$	−160.000	−0.227866	−0.113933	+	0.993488i	$0.536345\pi$
$80$	48.0000	0.0670820
$81$	0	0
$82$	−12.0000	−0.0161607
$83$	735.000	0.972009	0.486004	−	0.873956i	$-0.338454\pi$
$83$	735.000	0.972009	0.486004	+	0.873956i	$0.338454\pi$
$84$	0	0
$85$	−216.000	−0.275629
$86$	436.000	0.546687
$87$	0	0
$88$	−456.000	−0.552384
$89$	−954.000	−1.13622	−0.568111	−	0.822952i	$-0.692326\pi$
$89$	−954.000	−1.13622	−0.568111	+	0.822952i	$0.692326\pi$
$90$	0	0
$91$	580.000	0.668138
$92$	696.000	0.788728
$93$	0	0
$94$	948.000	1.04020
$95$	−318.000	−0.343433
$96$	0	0
$97$	191.000	0.199929	0.0999645	−	0.994991i	$-0.468127\pi$
$97$	191.000	0.199929	0.0999645	+	0.994991i	$0.468127\pi$
$98$	996.000	1.02664
$99$	0	0
$100$	−464.000	−0.464000
$101$	−363.000	−0.357622	−0.178811	−	0.983883i	$-0.557225\pi$
$101$	−363.000	−0.357622	−0.178811	+	0.983883i	$0.557225\pi$
$102$	0	0
$103$	−628.000	−0.600764	−0.300382	−	0.953819i	$-0.597114\pi$
$103$	−628.000	−0.600764	−0.300382	+	0.953819i	$0.597114\pi$
$104$	160.000	0.150859
$105$	0	0
$106$	162.000	0.148442
$107$	675.000	0.609857	0.304929	−	0.952375i	$-0.401367\pi$
$107$	675.000	0.609857	0.304929	+	0.952375i	$0.401367\pi$
$108$	0	0
$109$	1730.00	1.52022	0.760110	−	0.649795i	$-0.225145\pi$
$109$	1730.00	1.52022	0.760110	+	0.649795i	$0.225145\pi$
$110$	−342.000	−0.296440
$111$	0	0
$112$	464.000	0.391463
$113$	1866.00	1.55344	0.776719	−	0.629847i	$-0.216882\pi$
$113$	1866.00	1.55344	0.776719	+	0.629847i	$0.216882\pi$
$114$	0	0
$115$	522.000	0.423276
$116$	−840.000	−0.672345
$117$	0	0
$118$	168.000	0.131065
$119$	−2088.00	−1.60846
$120$	0	0
$121$	1918.00	1.44102
$122$	112.000	0.0831148
$123$	0	0
$124$	188.000	0.136152
$125$	−723.000	−0.517337
$126$	0	0
$127$	1379.00	0.963515	0.481758	−	0.876304i	$-0.339999\pi$
$127$	1379.00	0.963515	0.481758	+	0.876304i	$0.339999\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	120.000	0.0809592
$131$	579.000	0.386164	0.193082	−	0.981183i	$-0.438152\pi$
$131$	579.000	0.386164	0.193082	+	0.981183i	$0.438152\pi$
$132$	0	0
$133$	−3074.00	−2.00413
$134$	−284.000	−0.183089
$135$	0	0
$136$	−576.000	−0.363173
$137$	654.000	0.407847	0.203923	−	0.978987i	$-0.434631\pi$
$137$	654.000	0.407847	0.203923	+	0.978987i	$0.434631\pi$
$138$	0	0
$139$	−3004.00	−1.83306	−0.916532	−	0.399961i	$-0.869024\pi$
$139$	−3004.00	−1.83306	−0.916532	+	0.399961i	$0.869024\pi$
$140$	348.000	0.210081
$141$	0	0
$142$	720.000	0.425500
$143$	−1140.00	−0.666654
$144$	0	0
$145$	−630.000	−0.360818
$146$	−2318.00	−1.31397
$147$	0	0
$148$	8.00000	0.00444322
$149$	1803.00	0.991326	0.495663	−	0.868515i	$-0.334925\pi$
$149$	1803.00	0.991326	0.495663	+	0.868515i	$0.334925\pi$
$150$	0	0
$151$	2459.00	1.32524	0.662618	−	0.748958i	$-0.269445\pi$
$151$	2459.00	1.32524	0.662618	+	0.748958i	$0.269445\pi$
$152$	−848.000	−0.452512
$153$	0	0
$154$	−3306.00	−1.72990
$155$	141.000	0.0730670
$156$	0	0
$157$	−196.000	−0.0996338	−0.0498169	−	0.998758i	$-0.515864\pi$
$157$	−196.000	−0.0996338	−0.0498169	+	0.998758i	$0.515864\pi$
$158$	−320.000	−0.161126
$159$	0	0
$160$	96.0000	0.0474342
$161$	5046.00	2.47007
$162$	0	0
$163$	−1564.00	−0.751546	−0.375773	−	0.926712i	$-0.622623\pi$
$163$	−1564.00	−0.751546	−0.375773	+	0.926712i	$0.622623\pi$
$164$	−24.0000	−0.0114273
$165$	0	0
$166$	1470.00	0.687314
$167$	1974.00	0.914687	0.457343	−	0.889290i	$-0.348801\pi$
$167$	1974.00	0.914687	0.457343	+	0.889290i	$0.348801\pi$
$168$	0	0
$169$	−1797.00	−0.817934
$170$	−432.000	−0.194899
$171$	0	0
$172$	872.000	0.386566
$173$	−2217.00	−0.974309	−0.487154	−	0.873316i	$-0.661965\pi$
$173$	−2217.00	−0.974309	−0.487154	+	0.873316i	$0.661965\pi$
$174$	0	0
$175$	−3364.00	−1.45311
$176$	−912.000	−0.390594
$177$	0	0
$178$	−1908.00	−0.803431
$179$	−2475.00	−1.03346	−0.516732	−	0.856147i	$-0.672852\pi$
$179$	−2475.00	−1.03346	−0.516732	+	0.856147i	$0.672852\pi$
$180$	0	0
$181$	1568.00	0.643914	0.321957	−	0.946754i	$-0.395659\pi$
$181$	1568.00	0.643914	0.321957	+	0.946754i	$0.395659\pi$
$182$	1160.00	0.472445
$183$	0	0
$184$	1392.00	0.557715
$185$	6.00000	0.00238448
$186$	0	0
$187$	4104.00	1.60489
$188$	1896.00	0.735532
$189$	0	0
$190$	−636.000	−0.242844
$191$	1140.00	0.431872	0.215936	−	0.976408i	$-0.430720\pi$
$191$	1140.00	0.431872	0.215936	+	0.976408i	$0.430720\pi$
$192$	0	0
$193$	2045.00	0.762706	0.381353	−	0.924429i	$-0.375458\pi$
$193$	2045.00	0.762706	0.381353	+	0.924429i	$0.375458\pi$
$194$	382.000	0.141371
$195$	0	0
$196$	1992.00	0.725948
$197$	−3735.00	−1.35080	−0.675400	−	0.737451i	$-0.736029\pi$
$197$	−3735.00	−1.35080	−0.675400	+	0.737451i	$0.736029\pi$
$198$	0	0
$199$	1163.00	0.414286	0.207143	−	0.978311i	$-0.433583\pi$
$199$	1163.00	0.414286	0.207143	+	0.978311i	$0.433583\pi$
$200$	−928.000	−0.328098
$201$	0	0
$202$	−726.000	−0.252877
$203$	−6090.00	−2.10559
$204$	0	0
$205$	−18.0000	−0.00613256
$206$	−1256.00	−0.424804
$207$	0	0
$208$	320.000	0.106673
$209$	6042.00	1.99968
$210$	0	0
$211$	2126.00	0.693649	0.346824	−	0.937930i	$-0.387260\pi$
$211$	2126.00	0.693649	0.346824	+	0.937930i	$0.387260\pi$
$212$	324.000	0.104964
$213$	0	0
$214$	1350.00	0.431234
$215$	654.000	0.207453
$216$	0	0
$217$	1363.00	0.426389
$218$	3460.00	1.07496
$219$	0	0
$220$	−684.000	−0.209615
$221$	−1440.00	−0.438303
$222$	0	0
$223$	−2752.00	−0.826402	−0.413201	−	0.910640i	$-0.635589\pi$
$223$	−2752.00	−0.826402	−0.413201	+	0.910640i	$0.635589\pi$
$224$	928.000	0.276806
$225$	0	0
$226$	3732.00	1.09845
$227$	−3972.00	−1.16137	−0.580685	−	0.814128i	$-0.697215\pi$
$227$	−3972.00	−1.16137	−0.580685	+	0.814128i	$0.697215\pi$
$228$	0	0
$229$	4502.00	1.29913	0.649564	−	0.760307i	$-0.274951\pi$
$229$	4502.00	1.29913	0.649564	+	0.760307i	$0.274951\pi$
$230$	1044.00	0.299301
$231$	0	0
$232$	−1680.00	−0.475420
$233$	4842.00	1.36142	0.680708	−	0.732555i	$-0.261672\pi$
$233$	4842.00	1.36142	0.680708	+	0.732555i	$0.261672\pi$
$234$	0	0
$235$	1422.00	0.394728
$236$	336.000	0.0926769
$237$	0	0
$238$	−4176.00	−1.13735
$239$	−5334.00	−1.44363	−0.721815	−	0.692086i	$-0.756692\pi$
$239$	−5334.00	−1.44363	−0.721815	+	0.692086i	$0.756692\pi$
$240$	0	0
$241$	−3994.00	−1.06754	−0.533768	−	0.845631i	$-0.679224\pi$
$241$	−3994.00	−1.06754	−0.533768	+	0.845631i	$0.679224\pi$
$242$	3836.00	1.01896
$243$	0	0
$244$	224.000	0.0587710
$245$	1494.00	0.389584
$246$	0	0
$247$	−2120.00	−0.546123
$248$	376.000	0.0962743
$249$	0	0
$250$	−1446.00	−0.365812
$251$	−1008.00	−0.253484	−0.126742	−	0.991936i	$-0.540452\pi$
$251$	−1008.00	−0.253484	−0.126742	+	0.991936i	$0.540452\pi$
$252$	0	0
$253$	−9918.00	−2.46458
$254$	2758.00	0.681308
$255$	0	0
$256$	256.000	0.0625000
$257$	−924.000	−0.224271	−0.112135	−	0.993693i	$-0.535769\pi$
$257$	−924.000	−0.224271	−0.112135	+	0.993693i	$0.535769\pi$
$258$	0	0
$259$	58.0000	0.0139148
$260$	240.000	0.0572468
$261$	0	0
$262$	1158.00	0.273059
$263$	1014.00	0.237741	0.118871	−	0.992910i	$-0.462073\pi$
$263$	1014.00	0.237741	0.118871	+	0.992910i	$0.462073\pi$
$264$	0	0
$265$	243.000	0.0563297
$266$	−6148.00	−1.41714
$267$	0	0
$268$	−568.000	−0.129463
$269$	−2970.00	−0.673175	−0.336588	−	0.941652i	$-0.609273\pi$
$269$	−2970.00	−0.673175	−0.336588	+	0.941652i	$0.609273\pi$
$270$	0	0
$271$	245.000	0.0549177	0.0274588	−	0.999623i	$-0.491258\pi$
$271$	245.000	0.0549177	0.0274588	+	0.999623i	$0.491258\pi$
$272$	−1152.00	−0.256802
$273$	0	0
$274$	1308.00	0.288391
$275$	6612.00	1.44989
$276$	0	0
$277$	4376.00	0.949200	0.474600	−	0.880202i	$-0.342593\pi$
$277$	4376.00	0.949200	0.474600	+	0.880202i	$0.342593\pi$
$278$	−6008.00	−1.29617
$279$	0	0
$280$	696.000	0.148550
$281$	240.000	0.0509509	0.0254754	−	0.999675i	$-0.491890\pi$
$281$	240.000	0.0509509	0.0254754	+	0.999675i	$0.491890\pi$
$282$	0	0
$283$	−6838.00	−1.43631	−0.718157	−	0.695881i	$-0.755014\pi$
$283$	−6838.00	−1.43631	−0.718157	+	0.695881i	$0.755014\pi$
$284$	1440.00	0.300874
$285$	0	0
$286$	−2280.00	−0.471396
$287$	−174.000	−0.0357871
$288$	0	0
$289$	271.000	0.0551598
$290$	−1260.00	−0.255137
$291$	0	0
$292$	−4636.00	−0.929114
$293$	−5118.00	−1.02047	−0.510233	−	0.860036i	$-0.670441\pi$
$293$	−5118.00	−1.02047	−0.510233	+	0.860036i	$0.670441\pi$
$294$	0	0
$295$	252.000	0.0497356
$296$	16.0000	0.00314183
$297$	0	0
$298$	3606.00	0.700973
$299$	3480.00	0.673089
$300$	0	0
$301$	6322.00	1.21061
$302$	4918.00	0.937083
$303$	0	0
$304$	−1696.00	−0.319975
$305$	168.000	0.0315398
$306$	0	0
$307$	−5560.00	−1.03364	−0.516818	−	0.856096i	$-0.672883\pi$
$307$	−5560.00	−1.03364	−0.516818	+	0.856096i	$0.672883\pi$
$308$	−6612.00	−1.22323
$309$	0	0
$310$	282.000	0.0516662
$311$	7662.00	1.39702	0.698508	−	0.715602i	$-0.253848\pi$
$311$	7662.00	1.39702	0.698508	+	0.715602i	$0.253848\pi$
$312$	0	0
$313$	3485.00	0.629341	0.314671	−	0.949201i	$-0.398106\pi$
$313$	3485.00	0.629341	0.314671	+	0.949201i	$0.398106\pi$
$314$	−392.000	−0.0704517
$315$	0	0
$316$	−640.000	−0.113933
$317$	−7059.00	−1.25070	−0.625352	−	0.780343i	$-0.715044\pi$
$317$	−7059.00	−1.25070	−0.625352	+	0.780343i	$0.715044\pi$
$318$	0	0
$319$	11970.0	2.10091
$320$	192.000	0.0335410
$321$	0	0
$322$	10092.0	1.74660
$323$	7632.00	1.31472
$324$	0	0
$325$	−2320.00	−0.395971
$326$	−3128.00	−0.531423
$327$	0	0
$328$	−48.0000	−0.00808036
$329$	13746.0	2.30347
$330$	0	0
$331$	9290.00	1.54267	0.771336	−	0.636428i	$-0.219589\pi$
$331$	9290.00	1.54267	0.771336	+	0.636428i	$0.219589\pi$
$332$	2940.00	0.486004
$333$	0	0
$334$	3948.00	0.646781
$335$	−426.000	−0.0694772
$336$	0	0
$337$	−3814.00	−0.616504	−0.308252	−	0.951305i	$-0.599744\pi$
$337$	−3814.00	−0.616504	−0.308252	+	0.951305i	$0.599744\pi$
$338$	−3594.00	−0.578366
$339$	0	0
$340$	−864.000	−0.137815
$341$	−2679.00	−0.425443
$342$	0	0
$343$	4495.00	0.707601
$344$	1744.00	0.273344
$345$	0	0
$346$	−4434.00	−0.688940
$347$	1929.00	0.298427	0.149213	−	0.988805i	$-0.452326\pi$
$347$	1929.00	0.298427	0.149213	+	0.988805i	$0.452326\pi$
$348$	0	0
$349$	−6586.00	−1.01014	−0.505072	−	0.863077i	$-0.668534\pi$
$349$	−6586.00	−1.01014	−0.505072	+	0.863077i	$0.668534\pi$
$350$	−6728.00	−1.02750
$351$	0	0
$352$	−1824.00	−0.276192
$353$	−6042.00	−0.911001	−0.455500	−	0.890236i	$-0.650540\pi$
$353$	−6042.00	−0.911001	−0.455500	+	0.890236i	$0.650540\pi$
$354$	0	0
$355$	1080.00	0.161466
$356$	−3816.00	−0.568111
$357$	0	0
$358$	−4950.00	−0.730770
$359$	−3762.00	−0.553066	−0.276533	−	0.961004i	$-0.589186\pi$
$359$	−3762.00	−0.553066	−0.276533	+	0.961004i	$0.589186\pi$
$360$	0	0
$361$	4377.00	0.638140
$362$	3136.00	0.455316
$363$	0	0
$364$	2320.00	0.334069
$365$	−3477.00	−0.498615
$366$	0	0
$367$	−7261.00	−1.03276	−0.516378	−	0.856361i	$-0.672720\pi$
$367$	−7261.00	−1.03276	−0.516378	+	0.856361i	$0.672720\pi$
$368$	2784.00	0.394364
$369$	0	0
$370$	12.0000	0.00168608
$371$	2349.00	0.328717
$372$	0	0
$373$	1640.00	0.227657	0.113828	−	0.993500i	$-0.463689\pi$
$373$	1640.00	0.227657	0.113828	+	0.993500i	$0.463689\pi$
$374$	8208.00	1.13483
$375$	0	0
$376$	3792.00	0.520100
$377$	−4200.00	−0.573769
$378$	0	0
$379$	−7396.00	−1.00239	−0.501197	−	0.865333i	$-0.667107\pi$
$379$	−7396.00	−1.00239	−0.501197	+	0.865333i	$0.667107\pi$
$380$	−1272.00	−0.171716
$381$	0	0
$382$	2280.00	0.305379
$383$	−4992.00	−0.666003	−0.333002	−	0.942926i	$-0.608061\pi$
$383$	−4992.00	−0.666003	−0.333002	+	0.942926i	$0.608061\pi$
$384$	0	0
$385$	−4959.00	−0.656452
$386$	4090.00	0.539315
$387$	0	0
$388$	764.000	0.0999645
$389$	−9453.00	−1.23210	−0.616049	−	0.787708i	$-0.711268\pi$
$389$	−9453.00	−1.23210	−0.616049	+	0.787708i	$0.711268\pi$
$390$	0	0
$391$	−12528.0	−1.62038
$392$	3984.00	0.513322
$393$	0	0
$394$	−7470.00	−0.955160
$395$	−480.000	−0.0611428
$396$	0	0
$397$	8588.00	1.08569	0.542846	−	0.839833i	$-0.317347\pi$
$397$	8588.00	1.08569	0.542846	+	0.839833i	$0.317347\pi$
$398$	2326.00	0.292944
$399$	0	0
$400$	−1856.00	−0.232000
$401$	−1716.00	−0.213698	−0.106849	−	0.994275i	$-0.534076\pi$
$401$	−1716.00	−0.213698	−0.106849	+	0.994275i	$0.534076\pi$
$402$	0	0
$403$	940.000	0.116190
$404$	−1452.00	−0.178811
$405$	0	0
$406$	−12180.0	−1.48888
$407$	−114.000	−0.0138840
$408$	0	0
$409$	−9889.00	−1.19555	−0.597775	−	0.801664i	$-0.703948\pi$
$409$	−9889.00	−1.19555	−0.597775	+	0.801664i	$0.703948\pi$
$410$	−36.0000	−0.00433637
$411$	0	0
$412$	−2512.00	−0.300382
$413$	2436.00	0.290237
$414$	0	0
$415$	2205.00	0.260817
$416$	640.000	0.0754293
$417$	0	0
$418$	12084.0	1.41399
$419$	5556.00	0.647800	0.323900	−	0.946091i	$-0.395006\pi$
$419$	5556.00	0.647800	0.323900	+	0.946091i	$0.395006\pi$
$420$	0	0
$421$	−2104.00	−0.243569	−0.121785	−	0.992557i	$-0.538862\pi$
$421$	−2104.00	−0.243569	−0.121785	+	0.992557i	$0.538862\pi$
$422$	4252.00	0.490484
$423$	0	0
$424$	648.000	0.0742209
$425$	8352.00	0.953251
$426$	0	0
$427$	1624.00	0.184054
$428$	2700.00	0.304929
$429$	0	0
$430$	1308.00	0.146692
$431$	7614.00	0.850936	0.425468	−	0.904973i	$-0.360109\pi$
$431$	7614.00	0.850936	0.425468	+	0.904973i	$0.360109\pi$
$432$	0	0
$433$	7805.00	0.866246	0.433123	−	0.901335i	$-0.357412\pi$
$433$	7805.00	0.866246	0.433123	+	0.901335i	$0.357412\pi$
$434$	2726.00	0.301503
$435$	0	0
$436$	6920.00	0.760110
$437$	−18444.0	−2.01898
$438$	0	0
$439$	−5209.00	−0.566314	−0.283157	−	0.959074i	$-0.591382\pi$
$439$	−5209.00	−0.566314	−0.283157	+	0.959074i	$0.591382\pi$
$440$	−1368.00	−0.148220
$441$	0	0
$442$	−2880.00	−0.309927
$443$	4236.00	0.454308	0.227154	−	0.973859i	$-0.427058\pi$
$443$	4236.00	0.454308	0.227154	+	0.973859i	$0.427058\pi$
$444$	0	0
$445$	−2862.00	−0.304880
$446$	−5504.00	−0.584354
$447$	0	0
$448$	1856.00	0.195732
$449$	−16002.0	−1.68192	−0.840959	−	0.541099i	$-0.818008\pi$
$449$	−16002.0	−1.68192	−0.840959	+	0.541099i	$0.818008\pi$
$450$	0	0
$451$	342.000	0.0357077
$452$	7464.00	0.776719
$453$	0	0
$454$	−7944.00	−0.821212
$455$	1740.00	0.179280
$456$	0	0
$457$	7319.00	0.749165	0.374582	−	0.927194i	$-0.377786\pi$
$457$	7319.00	0.749165	0.374582	+	0.927194i	$0.377786\pi$
$458$	9004.00	0.918623
$459$	0	0
$460$	2088.00	0.211638
$461$	9483.00	0.958064	0.479032	−	0.877798i	$-0.340988\pi$
$461$	9483.00	0.958064	0.479032	+	0.877798i	$0.340988\pi$
$462$	0	0
$463$	10793.0	1.08335	0.541677	−	0.840586i	$-0.317789\pi$
$463$	10793.0	1.08335	0.541677	+	0.840586i	$0.317789\pi$
$464$	−3360.00	−0.336173
$465$	0	0
$466$	9684.00	0.962667
$467$	2583.00	0.255946	0.127973	−	0.991778i	$-0.459153\pi$
$467$	2583.00	0.255946	0.127973	+	0.991778i	$0.459153\pi$
$468$	0	0
$469$	−4118.00	−0.405440
$470$	2844.00	0.279115
$471$	0	0
$472$	672.000	0.0655324
$473$	−12426.0	−1.20792
$474$	0	0
$475$	12296.0	1.18775
$476$	−8352.00	−0.804230
$477$	0	0
$478$	−10668.0	−1.02080
$479$	−1254.00	−0.119617	−0.0598087	−	0.998210i	$-0.519049\pi$
$479$	−1254.00	−0.119617	−0.0598087	+	0.998210i	$0.519049\pi$
$480$	0	0
$481$	40.0000	0.00379177
$482$	−7988.00	−0.754862
$483$	0	0
$484$	7672.00	0.720511
$485$	573.000	0.0536466
$486$	0	0
$487$	17336.0	1.61308	0.806539	−	0.591181i	$-0.201338\pi$
$487$	17336.0	1.61308	0.806539	+	0.591181i	$0.201338\pi$
$488$	448.000	0.0415574
$489$	0	0
$490$	2988.00	0.275478
$491$	−15171.0	−1.39441	−0.697207	−	0.716869i	$-0.745574\pi$
$491$	−15171.0	−1.39441	−0.697207	+	0.716869i	$0.745574\pi$
$492$	0	0
$493$	15120.0	1.38128
$494$	−4240.00	−0.386167
$495$	0	0
$496$	752.000	0.0680762
$497$	10440.0	0.942249
$498$	0	0
$499$	8930.00	0.801126	0.400563	−	0.916269i	$-0.368815\pi$
$499$	8930.00	0.801126	0.400563	+	0.916269i	$0.368815\pi$
$500$	−2892.00	−0.258668
$501$	0	0
$502$	−2016.00	−0.179240
$503$	15210.0	1.34827	0.674136	−	0.738608i	$-0.264516\pi$
$503$	15210.0	1.34827	0.674136	+	0.738608i	$0.264516\pi$
$504$	0	0
$505$	−1089.00	−0.0959601
$506$	−19836.0	−1.74272
$507$	0	0
$508$	5516.00	0.481758
$509$	19641.0	1.71036	0.855179	−	0.518333i	$-0.173447\pi$
$509$	19641.0	1.71036	0.855179	+	0.518333i	$0.173447\pi$
$510$	0	0
$511$	−33611.0	−2.90971
$512$	512.000	0.0441942
$513$	0	0
$514$	−1848.00	−0.158583
$515$	−1884.00	−0.161202
$516$	0	0
$517$	−27018.0	−2.29836
$518$	116.000	0.00983928
$519$	0	0
$520$	480.000	0.0404796
$521$	22428.0	1.88597	0.942983	−	0.332840i	$-0.108007\pi$
$521$	22428.0	1.88597	0.942983	+	0.332840i	$0.108007\pi$
$522$	0	0
$523$	−8152.00	−0.681572	−0.340786	−	0.940141i	$-0.610693\pi$
$523$	−8152.00	−0.681572	−0.340786	+	0.940141i	$0.610693\pi$
$524$	2316.00	0.193082
$525$	0	0
$526$	2028.00	0.168108
$527$	−3384.00	−0.279714
$528$	0	0
$529$	18109.0	1.48837
$530$	486.000	0.0398311
$531$	0	0
$532$	−12296.0	−1.00207
$533$	−120.000	−0.00975193
$534$	0	0
$535$	2025.00	0.163642
$536$	−1136.00	−0.0915443
$537$	0	0
$538$	−5940.00	−0.476007
$539$	−28386.0	−2.26841
$540$	0	0
$541$	−2860.00	−0.227285	−0.113642	−	0.993522i	$-0.536252\pi$
$541$	−2860.00	−0.227285	−0.113642	+	0.993522i	$0.536252\pi$
$542$	490.000	0.0388327
$543$	0	0
$544$	−2304.00	−0.181587
$545$	5190.00	0.407918
$546$	0	0
$547$	−9664.00	−0.755398	−0.377699	−	0.925928i	$-0.623285\pi$
$547$	−9664.00	−0.755398	−0.377699	+	0.925928i	$0.623285\pi$
$548$	2616.00	0.203923
$549$	0	0
$550$	13224.0	1.02522
$551$	22260.0	1.72107
$552$	0	0
$553$	−4640.00	−0.356804
$554$	8752.00	0.671186
$555$	0	0
$556$	−12016.0	−0.916532
$557$	14859.0	1.13033	0.565167	−	0.824977i	$-0.308812\pi$
$557$	14859.0	1.13033	0.565167	+	0.824977i	$0.308812\pi$
$558$	0	0
$559$	4360.00	0.329890
$560$	1392.00	0.105041
$561$	0	0
$562$	480.000	0.0360277
$563$	8193.00	0.613310	0.306655	−	0.951821i	$-0.400790\pi$
$563$	8193.00	0.613310	0.306655	+	0.951821i	$0.400790\pi$
$564$	0	0
$565$	5598.00	0.416831
$566$	−13676.0	−1.01563
$567$	0	0
$568$	2880.00	0.212750
$569$	−16572.0	−1.22097	−0.610487	−	0.792026i	$-0.709026\pi$
$569$	−16572.0	−1.22097	−0.610487	+	0.792026i	$0.709026\pi$
$570$	0	0
$571$	−6244.00	−0.457624	−0.228812	−	0.973471i	$-0.573484\pi$
$571$	−6244.00	−0.457624	−0.228812	+	0.973471i	$0.573484\pi$
$572$	−4560.00	−0.333327
$573$	0	0
$574$	−348.000	−0.0253053
$575$	−20184.0	−1.46388
$576$	0	0
$577$	−14794.0	−1.06739	−0.533693	−	0.845678i	$-0.679196\pi$
$577$	−14794.0	−1.06739	−0.533693	+	0.845678i	$0.679196\pi$
$578$	542.000	0.0390039
$579$	0	0
$580$	−2520.00	−0.180409
$581$	21315.0	1.52202
$582$	0	0
$583$	−4617.00	−0.327987
$584$	−9272.00	−0.656983
$585$	0	0
$586$	−10236.0	−0.721579
$587$	−26769.0	−1.88224	−0.941120	−	0.338073i	$-0.890225\pi$
$587$	−26769.0	−1.88224	−0.941120	+	0.338073i	$0.890225\pi$
$588$	0	0
$589$	−4982.00	−0.348522
$590$	504.000	0.0351684
$591$	0	0
$592$	32.0000	0.00222161
$593$	−3078.00	−0.213151	−0.106575	−	0.994305i	$-0.533989\pi$
$593$	−3078.00	−0.213151	−0.106575	+	0.994305i	$0.533989\pi$
$594$	0	0
$595$	−6264.00	−0.431595
$596$	7212.00	0.495663
$597$	0	0
$598$	6960.00	0.475946
$599$	1002.00	0.0683483	0.0341741	−	0.999416i	$-0.489120\pi$
$599$	1002.00	0.0683483	0.0341741	+	0.999416i	$0.489120\pi$
$600$	0	0
$601$	−20653.0	−1.40175	−0.700876	−	0.713283i	$-0.747208\pi$
$601$	−20653.0	−1.40175	−0.700876	+	0.713283i	$0.747208\pi$
$602$	12644.0	0.856032
$603$	0	0
$604$	9836.00	0.662618
$605$	5754.00	0.386667
$606$	0	0
$607$	27128.0	1.81399	0.906995	−	0.421142i	$-0.138371\pi$
$607$	27128.0	1.81399	0.906995	+	0.421142i	$0.138371\pi$
$608$	−3392.00	−0.226256
$609$	0	0
$610$	336.000	0.0223020
$611$	9480.00	0.627692
$612$	0	0
$613$	24518.0	1.61545	0.807727	−	0.589557i	$-0.200698\pi$
$613$	24518.0	1.61545	0.807727	+	0.589557i	$0.200698\pi$
$614$	−11120.0	−0.730890
$615$	0	0
$616$	−13224.0	−0.864952
$617$	−474.000	−0.0309279	−0.0154640	−	0.999880i	$-0.504923\pi$
$617$	−474.000	−0.0309279	−0.0154640	+	0.999880i	$0.504923\pi$
$618$	0	0
$619$	−1132.00	−0.0735039	−0.0367520	−	0.999324i	$-0.511701\pi$
$619$	−1132.00	−0.0735039	−0.0367520	+	0.999324i	$0.511701\pi$
$620$	564.000	0.0365335
$621$	0	0
$622$	15324.0	0.987840
$623$	−27666.0	−1.77916
$624$	0	0
$625$	12331.0	0.789184
$626$	6970.00	0.445012
$627$	0	0
$628$	−784.000	−0.0498169
$629$	−144.000	−0.00912823
$630$	0	0
$631$	6725.00	0.424276	0.212138	−	0.977240i	$-0.431957\pi$
$631$	6725.00	0.424276	0.212138	+	0.977240i	$0.431957\pi$
$632$	−1280.00	−0.0805628
$633$	0	0
$634$	−14118.0	−0.884381
$635$	4137.00	0.258538
$636$	0	0
$637$	9960.00	0.619513
$638$	23940.0	1.48557
$639$	0	0
$640$	384.000	0.0237171
$641$	−21126.0	−1.30176	−0.650879	−	0.759182i	$-0.725599\pi$
$641$	−21126.0	−1.30176	−0.650879	+	0.759182i	$0.725599\pi$
$642$	0	0
$643$	19460.0	1.19351	0.596755	−	0.802423i	$-0.296456\pi$
$643$	19460.0	1.19351	0.596755	+	0.802423i	$0.296456\pi$
$644$	20184.0	1.23503
$645$	0	0
$646$	15264.0	0.929650
$647$	−11664.0	−0.708747	−0.354373	−	0.935104i	$-0.615306\pi$
$647$	−11664.0	−0.708747	−0.354373	+	0.935104i	$0.615306\pi$
$648$	0	0
$649$	−4788.00	−0.289592
$650$	−4640.00	−0.279993
$651$	0	0
$652$	−6256.00	−0.375773
$653$	−3345.00	−0.200459	−0.100230	−	0.994964i	$-0.531958\pi$
$653$	−3345.00	−0.200459	−0.100230	+	0.994964i	$0.531958\pi$
$654$	0	0
$655$	1737.00	0.103619
$656$	−96.0000	−0.00571367
$657$	0	0
$658$	27492.0	1.62880
$659$	9393.00	0.555234	0.277617	−	0.960692i	$-0.410455\pi$
$659$	9393.00	0.555234	0.277617	+	0.960692i	$0.410455\pi$
$660$	0	0
$661$	−1762.00	−0.103682	−0.0518410	−	0.998655i	$-0.516509\pi$
$661$	−1762.00	−0.103682	−0.0518410	+	0.998655i	$0.516509\pi$
$662$	18580.0	1.09083
$663$	0	0
$664$	5880.00	0.343657
$665$	−9222.00	−0.537765
$666$	0	0
$667$	−36540.0	−2.12119
$668$	7896.00	0.457343
$669$	0	0
$670$	−852.000	−0.0491278
$671$	−3192.00	−0.183645
$672$	0	0
$673$	25517.0	1.46153	0.730764	−	0.682630i	$-0.239164\pi$
$673$	25517.0	1.46153	0.730764	+	0.682630i	$0.239164\pi$
$674$	−7628.00	−0.435934
$675$	0	0
$676$	−7188.00	−0.408967
$677$	26898.0	1.52699	0.763496	−	0.645812i	$-0.223481\pi$
$677$	26898.0	1.52699	0.763496	+	0.645812i	$0.223481\pi$
$678$	0	0
$679$	5539.00	0.313059
$680$	−1728.00	−0.0974497
$681$	0	0
$682$	−5358.00	−0.300833
$683$	23940.0	1.34120	0.670599	−	0.741820i	$-0.266037\pi$
$683$	23940.0	1.34120	0.670599	+	0.741820i	$0.266037\pi$
$684$	0	0
$685$	1962.00	0.109437
$686$	8990.00	0.500350
$687$	0	0
$688$	3488.00	0.193283
$689$	1620.00	0.0895749
$690$	0	0
$691$	23060.0	1.26953	0.634764	−	0.772706i	$-0.281097\pi$
$691$	23060.0	1.26953	0.634764	+	0.772706i	$0.281097\pi$
$692$	−8868.00	−0.487154
$693$	0	0
$694$	3858.00	0.211020
$695$	−9012.00	−0.491863
$696$	0	0
$697$	432.000	0.0234766
$698$	−13172.0	−0.714280
$699$	0	0
$700$	−13456.0	−0.726556
$701$	−14175.0	−0.763741	−0.381870	−	0.924216i	$-0.624720\pi$
$701$	−14175.0	−0.763741	−0.381870	+	0.924216i	$0.624720\pi$
$702$	0	0
$703$	−212.000	−0.0113737
$704$	−3648.00	−0.195297
$705$	0	0
$706$	−12084.0	−0.644175
$707$	−10527.0	−0.559984
$708$	0	0
$709$	−8692.00	−0.460416	−0.230208	−	0.973141i	$-0.573941\pi$
$709$	−8692.00	−0.460416	−0.230208	+	0.973141i	$0.573941\pi$
$710$	2160.00	0.114174
$711$	0	0
$712$	−7632.00	−0.401715
$713$	8178.00	0.429549
$714$	0	0
$715$	−3420.00	−0.178882
$716$	−9900.00	−0.516732
$717$	0	0
$718$	−7524.00	−0.391077
$719$	−29556.0	−1.53304	−0.766518	−	0.642223i	$-0.778012\pi$
$719$	−29556.0	−1.53304	−0.766518	+	0.642223i	$0.778012\pi$
$720$	0	0
$721$	−18212.0	−0.940708
$722$	8754.00	0.451233
$723$	0	0
$724$	6272.00	0.321957
$725$	24360.0	1.24787
$726$	0	0
$727$	−36691.0	−1.87179	−0.935897	−	0.352274i	$-0.885408\pi$
$727$	−36691.0	−1.87179	−0.935897	+	0.352274i	$0.885408\pi$
$728$	4640.00	0.236222
$729$	0	0
$730$	−6954.00	−0.352574
$731$	−15696.0	−0.794169
$732$	0	0
$733$	−19798.0	−0.997620	−0.498810	−	0.866711i	$-0.666230\pi$
$733$	−19798.0	−0.997620	−0.498810	+	0.866711i	$0.666230\pi$
$734$	−14522.0	−0.730268
$735$	0	0
$736$	5568.00	0.278858
$737$	8094.00	0.404540
$738$	0	0
$739$	−21976.0	−1.09391	−0.546955	−	0.837162i	$-0.684213\pi$
$739$	−21976.0	−1.09391	−0.546955	+	0.837162i	$0.684213\pi$
$740$	24.0000	0.00119224
$741$	0	0
$742$	4698.00	0.232438
$743$	−13236.0	−0.653542	−0.326771	−	0.945104i	$-0.605961\pi$
$743$	−13236.0	−0.653542	−0.326771	+	0.945104i	$0.605961\pi$
$744$	0	0
$745$	5409.00	0.266001
$746$	3280.00	0.160978
$747$	0	0
$748$	16416.0	0.802444
$749$	19575.0	0.954947
$750$	0	0
$751$	−6325.00	−0.307327	−0.153663	−	0.988123i	$-0.549107\pi$
$751$	−6325.00	−0.307327	−0.153663	+	0.988123i	$0.549107\pi$
$752$	7584.00	0.367766
$753$	0	0
$754$	−8400.00	−0.405716
$755$	7377.00	0.355598
$756$	0	0
$757$	−3238.00	−0.155465	−0.0777326	−	0.996974i	$-0.524768\pi$
$757$	−3238.00	−0.155465	−0.0777326	+	0.996974i	$0.524768\pi$
$758$	−14792.0	−0.708799
$759$	0	0
$760$	−2544.00	−0.121422
$761$	−40416.0	−1.92520	−0.962601	−	0.270923i	$-0.912671\pi$
$761$	−40416.0	−1.92520	−0.962601	+	0.270923i	$0.912671\pi$
$762$	0	0
$763$	50170.0	2.38044
$764$	4560.00	0.215936
$765$	0	0
$766$	−9984.00	−0.470935
$767$	1680.00	0.0790890
$768$	0	0
$769$	−4759.00	−0.223165	−0.111583	−	0.993755i	$-0.535592\pi$
$769$	−4759.00	−0.223165	−0.111583	+	0.993755i	$0.535592\pi$
$770$	−9918.00	−0.464182
$771$	0	0
$772$	8180.00	0.381353
$773$	27414.0	1.27557	0.637783	−	0.770216i	$-0.279852\pi$
$773$	27414.0	1.27557	0.637783	+	0.770216i	$0.279852\pi$
$774$	0	0
$775$	−5452.00	−0.252699
$776$	1528.00	0.0706856
$777$	0	0
$778$	−18906.0	−0.871225
$779$	636.000	0.0292517
$780$	0	0
$781$	−20520.0	−0.940158
$782$	−25056.0	−1.14578
$783$	0	0
$784$	7968.00	0.362974
$785$	−588.000	−0.0267345
$786$	0	0
$787$	6176.00	0.279734	0.139867	−	0.990170i	$-0.455333\pi$
$787$	6176.00	0.279734	0.139867	+	0.990170i	$0.455333\pi$
$788$	−14940.0	−0.675400
$789$	0	0
$790$	−960.000	−0.0432345
$791$	54114.0	2.43246
$792$	0	0
$793$	1120.00	0.0501543
$794$	17176.0	0.767700
$795$	0	0
$796$	4652.00	0.207143
$797$	6879.00	0.305730	0.152865	−	0.988247i	$-0.451150\pi$
$797$	6879.00	0.305730	0.152865	+	0.988247i	$0.451150\pi$
$798$	0	0
$799$	−34128.0	−1.51109
$800$	−3712.00	−0.164049
$801$	0	0
$802$	−3432.00	−0.151107
$803$	66063.0	2.90325
$804$	0	0
$805$	15138.0	0.662788
$806$	1880.00	0.0821590
$807$	0	0
$808$	−2904.00	−0.126439
$809$	−16902.0	−0.734540	−0.367270	−	0.930114i	$-0.619707\pi$
$809$	−16902.0	−0.734540	−0.367270	+	0.930114i	$0.619707\pi$
$810$	0	0
$811$	24086.0	1.04288	0.521439	−	0.853289i	$-0.325395\pi$
$811$	24086.0	1.04288	0.521439	+	0.853289i	$0.325395\pi$
$812$	−24360.0	−1.05279
$813$	0	0
$814$	−228.000	−0.00981744
$815$	−4692.00	−0.201661
$816$	0	0
$817$	−23108.0	−0.989531
$818$	−19778.0	−0.845381
$819$	0	0
$820$	−72.0000	−0.00306628
$821$	7854.00	0.333869	0.166935	−	0.985968i	$-0.446613\pi$
$821$	7854.00	0.333869	0.166935	+	0.985968i	$0.446613\pi$
$822$	0	0
$823$	5771.00	0.244428	0.122214	−	0.992504i	$-0.461001\pi$
$823$	5771.00	0.244428	0.122214	+	0.992504i	$0.461001\pi$
$824$	−5024.00	−0.212402
$825$	0	0
$826$	4872.00	0.205228
$827$	−17568.0	−0.738693	−0.369347	−	0.929292i	$-0.620418\pi$
$827$	−17568.0	−0.738693	−0.369347	+	0.929292i	$0.620418\pi$
$828$	0	0
$829$	31322.0	1.31225	0.656127	−	0.754651i	$-0.272194\pi$
$829$	31322.0	1.31225	0.656127	+	0.754651i	$0.272194\pi$
$830$	4410.00	0.184426
$831$	0	0
$832$	1280.00	0.0533366
$833$	−35856.0	−1.49140
$834$	0	0
$835$	5922.00	0.245436
$836$	24168.0	0.999842
$837$	0	0
$838$	11112.0	0.458064
$839$	41856.0	1.72232	0.861162	−	0.508331i	$-0.169737\pi$
$839$	41856.0	1.72232	0.861162	+	0.508331i	$0.169737\pi$
$840$	0	0
$841$	19711.0	0.808192
$842$	−4208.00	−0.172230
$843$	0	0
$844$	8504.00	0.346824
$845$	−5391.00	−0.219475
$846$	0	0
$847$	55622.0	2.25643
$848$	1296.00	0.0524821
$849$	0	0
$850$	16704.0	0.674050
$851$	348.000	0.0140180
$852$	0	0
$853$	15662.0	0.628671	0.314336	−	0.949312i	$-0.398218\pi$
$853$	15662.0	0.628671	0.314336	+	0.949312i	$0.398218\pi$
$854$	3248.00	0.130146
$855$	0	0
$856$	5400.00	0.215617
$857$	−39864.0	−1.58895	−0.794474	−	0.607298i	$-0.792253\pi$
$857$	−39864.0	−1.58895	−0.794474	+	0.607298i	$0.792253\pi$
$858$	0	0
$859$	−9160.00	−0.363836	−0.181918	−	0.983314i	$-0.558231\pi$
$859$	−9160.00	−0.363836	−0.181918	+	0.983314i	$0.558231\pi$
$860$	2616.00	0.103727
$861$	0	0
$862$	15228.0	0.601703
$863$	5076.00	0.200219	0.100110	−	0.994976i	$-0.468081\pi$
$863$	5076.00	0.200219	0.100110	+	0.994976i	$0.468081\pi$
$864$	0	0
$865$	−6651.00	−0.261434
$866$	15610.0	0.612528
$867$	0	0
$868$	5452.00	0.213195
$869$	9120.00	0.356012
$870$	0	0
$871$	−2840.00	−0.110482
$872$	13840.0	0.537479
$873$	0	0
$874$	−36888.0	−1.42764
$875$	−20967.0	−0.810073
$876$	0	0
$877$	14978.0	0.576706	0.288353	−	0.957524i	$-0.406892\pi$
$877$	14978.0	0.576706	0.288353	+	0.957524i	$0.406892\pi$
$878$	−10418.0	−0.400445
$879$	0	0
$880$	−2736.00	−0.104807
$881$	−22860.0	−0.874203	−0.437102	−	0.899412i	$-0.643995\pi$
$881$	−22860.0	−0.874203	−0.437102	+	0.899412i	$0.643995\pi$
$882$	0	0
$883$	−32506.0	−1.23886	−0.619430	−	0.785052i	$-0.712636\pi$
$883$	−32506.0	−1.23886	−0.619430	+	0.785052i	$0.712636\pi$
$884$	−5760.00	−0.219151
$885$	0	0
$886$	8472.00	0.321244
$887$	35868.0	1.35776	0.678878	−	0.734251i	$-0.262467\pi$
$887$	35868.0	1.35776	0.678878	+	0.734251i	$0.262467\pi$
$888$	0	0
$889$	39991.0	1.50872
$890$	−5724.00	−0.215583
$891$	0	0
$892$	−11008.0	−0.413201
$893$	−50244.0	−1.88281
$894$	0	0
$895$	−7425.00	−0.277308
$896$	3712.00	0.138403
$897$	0	0
$898$	−32004.0	−1.18930
$899$	−9870.00	−0.366166
$900$	0	0
$901$	−5832.00	−0.215640
$902$	684.000	0.0252491
$903$	0	0
$904$	14928.0	0.549223
$905$	4704.00	0.172780
$906$	0	0
$907$	−33586.0	−1.22955	−0.614777	−	0.788701i	$-0.710754\pi$
$907$	−33586.0	−1.22955	−0.614777	+	0.788701i	$0.710754\pi$
$908$	−15888.0	−0.580685
$909$	0	0
$910$	3480.00	0.126770
$911$	−28902.0	−1.05112	−0.525558	−	0.850758i	$-0.676143\pi$
$911$	−28902.0	−1.05112	−0.525558	+	0.850758i	$0.676143\pi$
$912$	0	0
$913$	−41895.0	−1.51864
$914$	14638.0	0.529740
$915$	0	0
$916$	18008.0	0.649564
$917$	16791.0	0.604676
$918$	0	0
$919$	28271.0	1.01477	0.507385	−	0.861719i	$-0.330612\pi$
$919$	28271.0	1.01477	0.507385	+	0.861719i	$0.330612\pi$
$920$	4176.00	0.149651
$921$	0	0
$922$	18966.0	0.677453
$923$	7200.00	0.256762
$924$	0	0
$925$	−232.000	−0.00824661
$926$	21586.0	0.766047
$927$	0	0
$928$	−6720.00	−0.237710
$929$	19140.0	0.675956	0.337978	−	0.941154i	$-0.390257\pi$
$929$	19140.0	0.675956	0.337978	+	0.941154i	$0.390257\pi$
$930$	0	0
$931$	−52788.0	−1.85828
$932$	19368.0	0.680708
$933$	0	0
$934$	5166.00	0.180981
$935$	12312.0	0.430637
$936$	0	0
$937$	31619.0	1.10240	0.551199	−	0.834374i	$-0.314170\pi$
$937$	31619.0	1.10240	0.551199	+	0.834374i	$0.314170\pi$
$938$	−8236.00	−0.286690
$939$	0	0
$940$	5688.00	0.197364
$941$	−20913.0	−0.724489	−0.362245	−	0.932083i	$-0.617989\pi$
$941$	−20913.0	−0.724489	−0.362245	+	0.932083i	$0.617989\pi$
$942$	0	0
$943$	−1044.00	−0.0360523
$944$	1344.00	0.0463384
$945$	0	0
$946$	−24852.0	−0.854131
$947$	17529.0	0.601495	0.300748	−	0.953704i	$-0.402764\pi$
$947$	17529.0	0.601495	0.300748	+	0.953704i	$0.402764\pi$
$948$	0	0
$949$	−23180.0	−0.792892
$950$	24592.0	0.839863
$951$	0	0
$952$	−16704.0	−0.568676
$953$	−53604.0	−1.82204	−0.911020	−	0.412362i	$-0.864704\pi$
$953$	−53604.0	−1.82204	−0.911020	+	0.412362i	$0.864704\pi$
$954$	0	0
$955$	3420.00	0.115883
$956$	−21336.0	−0.721815
$957$	0	0
$958$	−2508.00	−0.0845823
$959$	18966.0	0.638628
$960$	0	0
$961$	−27582.0	−0.925850
$962$	80.0000	0.00268119
$963$	0	0
$964$	−15976.0	−0.533768
$965$	6135.00	0.204656
$966$	0	0
$967$	11117.0	0.369699	0.184849	−	0.982767i	$-0.440820\pi$
$967$	11117.0	0.369699	0.184849	+	0.982767i	$0.440820\pi$
$968$	15344.0	0.509478
$969$	0	0
$970$	1146.00	0.0379339
$971$	−27297.0	−0.902165	−0.451083	−	0.892482i	$-0.648962\pi$
$971$	−27297.0	−0.902165	−0.451083	+	0.892482i	$0.648962\pi$
$972$	0	0
$973$	−87116.0	−2.87031
$974$	34672.0	1.14062
$975$	0	0
$976$	896.000	0.0293855
$977$	25086.0	0.821466	0.410733	−	0.911756i	$-0.365273\pi$
$977$	25086.0	0.821466	0.410733	+	0.911756i	$0.365273\pi$
$978$	0	0
$979$	54378.0	1.77521
$980$	5976.00	0.194792
$981$	0	0
$982$	−30342.0	−0.986000
$983$	−20982.0	−0.680795	−0.340398	−	0.940282i	$-0.610562\pi$
$983$	−20982.0	−0.680795	−0.340398	+	0.940282i	$0.610562\pi$
$984$	0	0
$985$	−11205.0	−0.362458
$986$	30240.0	0.976712
$987$	0	0
$988$	−8480.00	−0.273061
$989$	37932.0	1.21958
$990$	0	0
$991$	11477.0	0.367890	0.183945	−	0.982937i	$-0.441113\pi$
$991$	11477.0	0.367890	0.183945	+	0.982937i	$0.441113\pi$
$992$	1504.00	0.0481371
$993$	0	0
$994$	20880.0	0.666271
$995$	3489.00	0.111165
$996$	0	0
$997$	8588.00	0.272803	0.136402	−	0.990654i	$-0.456446\pi$
$997$	8588.00	0.272803	0.136402	+	0.990654i	$0.456446\pi$
$998$	17860.0	0.566481
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	54.4.a.c.1.1	yes	1
3.2	odd	2		54.4.a.b.1.1	✓	1
4.3	odd	2		432.4.a.j.1.1		1
5.2	odd	4		1350.4.c.b.649.2		2
5.3	odd	4		1350.4.c.b.649.1		2
5.4	even	2		1350.4.a.a.1.1		1
8.3	odd	2		1728.4.a.k.1.1		1
8.5	even	2		1728.4.a.l.1.1		1
9.2	odd	6		162.4.c.g.109.1		2
9.4	even	3		162.4.c.b.55.1		2
9.5	odd	6		162.4.c.g.55.1		2
9.7	even	3		162.4.c.b.109.1		2
12.11	even	2		432.4.a.e.1.1		1
15.2	even	4		1350.4.c.s.649.1		2
15.8	even	4		1350.4.c.s.649.2		2
15.14	odd	2		1350.4.a.o.1.1		1
24.5	odd	2		1728.4.a.v.1.1		1
24.11	even	2		1728.4.a.u.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
54.4.a.b.1.1	✓	1	3.2	odd	2
54.4.a.c.1.1	yes	1	1.1	even	1	trivial
162.4.c.b.55.1		2	9.4	even	3
162.4.c.b.109.1		2	9.7	even	3
162.4.c.g.55.1		2	9.5	odd	6
162.4.c.g.109.1		2	9.2	odd	6
432.4.a.e.1.1		1	12.11	even	2
432.4.a.j.1.1		1	4.3	odd	2
1350.4.a.a.1.1		1	5.4	even	2
1350.4.a.o.1.1		1	15.14	odd	2
1350.4.c.b.649.1		2	5.3	odd	4
1350.4.c.b.649.2		2	5.2	odd	4
1350.4.c.s.649.1		2	15.2	even	4
1350.4.c.s.649.2		2	15.8	even	4
1728.4.a.k.1.1		1	8.3	odd	2
1728.4.a.l.1.1		1	8.5	even	2
1728.4.a.u.1.1		1	24.11	even	2
1728.4.a.v.1.1		1	24.5	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:	\( N \)	\(=\)	\( 54 = 2 \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	54.a (trivial)

Introduction

L-functions

Modular forms

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Fields

Representations

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