LMFDB - Embedded newform 60.4.a.a.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [60,4,Mod(1,60)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("60.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(60, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 60 = 2^{2} \cdot 3 \cdot 5 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	60.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-3,0,-5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$3.54011460034$
Analytic rank:	$1$
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	$\chi$	$=$	60.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-3.00000 q^{3} -5.00000 q^{5} -28.0000 q^{7} +9.00000 q^{9} -24.0000 q^{11} -70.0000 q^{13} +15.0000 q^{15} +102.000 q^{17} +20.0000 q^{19} +84.0000 q^{21} -72.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} +306.000 q^{29} -136.000 q^{31} +72.0000 q^{33} +140.000 q^{35} -214.000 q^{37} +210.000 q^{39} -150.000 q^{41} -292.000 q^{43} -45.0000 q^{45} -72.0000 q^{47} +441.000 q^{49} -306.000 q^{51} -414.000 q^{53} +120.000 q^{55} -60.0000 q^{57} -744.000 q^{59} -418.000 q^{61} -252.000 q^{63} +350.000 q^{65} +188.000 q^{67} +216.000 q^{69} +480.000 q^{71} +434.000 q^{73} -75.0000 q^{75} +672.000 q^{77} +1352.00 q^{79} +81.0000 q^{81} -612.000 q^{83} -510.000 q^{85} -918.000 q^{87} -30.0000 q^{89} +1960.00 q^{91} +408.000 q^{93} -100.000 q^{95} -286.000 q^{97} -216.000 q^{99} +O(q^{100})$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	−28.0000	−1.51186	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	−28.0000	−1.51186	−0.755929	+	0.654654i	$0.772814\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−24.0000	−0.657843	−0.328921	−	0.944357i	$-0.606685\pi$
$11$	−24.0000	−0.657843	−0.328921	+	0.944357i	$0.606685\pi$
$12$	0	0
$13$	−70.0000	−1.49342	−0.746712	−	0.665148i	$-0.768369\pi$
$13$	−70.0000	−1.49342	−0.746712	+	0.665148i	$0.768369\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	102.000	1.45521	0.727607	−	0.685994i	$-0.240633\pi$
$17$	102.000	1.45521	0.727607	+	0.685994i	$0.240633\pi$
$18$	0	0
$19$	20.0000	0.241490	0.120745	−	0.992684i	$-0.461472\pi$
$19$	20.0000	0.241490	0.120745	+	0.992684i	$0.461472\pi$
$20$	0	0
$21$	84.0000	0.872872
$22$	0	0
$23$	−72.0000	−0.652741	−0.326370	−	0.945242i	$-0.605826\pi$
$23$	−72.0000	−0.652741	−0.326370	+	0.945242i	$0.605826\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	306.000	1.95941	0.979703	−	0.200455i	$-0.0642419\pi$
$29$	306.000	1.95941	0.979703	+	0.200455i	$0.0642419\pi$
$30$	0	0
$31$	−136.000	−0.787946	−0.393973	−	0.919122i	$-0.628900\pi$
$31$	−136.000	−0.787946	−0.393973	+	0.919122i	$0.628900\pi$
$32$	0	0
$33$	72.0000	0.379806
$34$	0	0
$35$	140.000	0.676123
$36$	0	0
$37$	−214.000	−0.950848	−0.475424	−	0.879757i	$-0.657705\pi$
$37$	−214.000	−0.950848	−0.475424	+	0.879757i	$0.657705\pi$
$38$	0	0
$39$	210.000	0.862229
$40$	0	0
$41$	−150.000	−0.571367	−0.285684	−	0.958324i	$-0.592221\pi$
$41$	−150.000	−0.571367	−0.285684	+	0.958324i	$0.592221\pi$
$42$	0	0
$43$	−292.000	−1.03557	−0.517786	−	0.855510i	$-0.673244\pi$
$43$	−292.000	−1.03557	−0.517786	+	0.855510i	$0.673244\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	−72.0000	−0.223453	−0.111726	−	0.993739i	$-0.535638\pi$
$47$	−72.0000	−0.223453	−0.111726	+	0.993739i	$0.535638\pi$
$48$	0	0
$49$	441.000	1.28571
$50$	0	0
$51$	−306.000	−0.840168
$52$	0	0
$53$	−414.000	−1.07297	−0.536484	−	0.843911i	$-0.680248\pi$
$53$	−414.000	−1.07297	−0.536484	+	0.843911i	$0.680248\pi$
$54$	0	0
$55$	120.000	0.294196
$56$	0	0
$57$	−60.0000	−0.139424
$58$	0	0
$59$	−744.000	−1.64170	−0.820852	−	0.571141i	$-0.806501\pi$
$59$	−744.000	−1.64170	−0.820852	+	0.571141i	$0.806501\pi$
$60$	0	0
$61$	−418.000	−0.877367	−0.438684	−	0.898642i	$-0.644555\pi$
$61$	−418.000	−0.877367	−0.438684	+	0.898642i	$0.644555\pi$
$62$	0	0
$63$	−252.000	−0.503953
$64$	0	0
$65$	350.000	0.667879
$66$	0	0
$67$	188.000	0.342804	0.171402	−	0.985201i	$-0.445170\pi$
$67$	188.000	0.342804	0.171402	+	0.985201i	$0.445170\pi$
$68$	0	0
$69$	216.000	0.376860
$70$	0	0
$71$	480.000	0.802331	0.401166	−	0.916006i	$-0.368605\pi$
$71$	480.000	0.802331	0.401166	+	0.916006i	$0.368605\pi$
$72$	0	0
$73$	434.000	0.695834	0.347917	−	0.937525i	$-0.386889\pi$
$73$	434.000	0.695834	0.347917	+	0.937525i	$0.386889\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	672.000	0.994565
$78$	0	0
$79$	1352.00	1.92547	0.962733	−	0.270452i	$-0.0871732\pi$
$79$	1352.00	1.92547	0.962733	+	0.270452i	$0.0871732\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−612.000	−0.809346	−0.404673	−	0.914461i	$-0.632615\pi$
$83$	−612.000	−0.809346	−0.404673	+	0.914461i	$0.632615\pi$
$84$	0	0
$85$	−510.000	−0.650791
$86$	0	0
$87$	−918.000	−1.13126
$88$	0	0
$89$	−30.0000	−0.0357303	−0.0178651	−	0.999840i	$-0.505687\pi$
$89$	−30.0000	−0.0357303	−0.0178651	+	0.999840i	$0.505687\pi$
$90$	0	0
$91$	1960.00	2.25784
$92$	0	0
$93$	408.000	0.454921
$94$	0	0
$95$	−100.000	−0.107998
$96$	0	0
$97$	−286.000	−0.299370	−0.149685	−	0.988734i	$-0.547826\pi$
$97$	−286.000	−0.299370	−0.149685	+	0.988734i	$0.547826\pi$
$98$	0	0
$99$	−216.000	−0.219281
$100$	0	0
$101$	−1542.00	−1.51916	−0.759578	−	0.650416i	$-0.774594\pi$
$101$	−1542.00	−1.51916	−0.759578	+	0.650416i	$0.774594\pi$
$102$	0	0
$103$	1172.00	1.12117	0.560585	−	0.828097i	$-0.310576\pi$
$103$	1172.00	1.12117	0.560585	+	0.828097i	$0.310576\pi$
$104$	0	0
$105$	−420.000	−0.390360
$106$	0	0
$107$	1956.00	1.76723	0.883615	−	0.468214i	$-0.155102\pi$
$107$	1956.00	1.76723	0.883615	+	0.468214i	$0.155102\pi$
$108$	0	0
$109$	−1858.00	−1.63270	−0.816349	−	0.577559i	$-0.804005\pi$
$109$	−1858.00	−1.63270	−0.816349	+	0.577559i	$0.804005\pi$
$110$	0	0
$111$	642.000	0.548972
$112$	0	0
$113$	174.000	0.144854	0.0724272	−	0.997374i	$-0.476926\pi$
$113$	174.000	0.144854	0.0724272	+	0.997374i	$0.476926\pi$
$114$	0	0
$115$	360.000	0.291915
$116$	0	0
$117$	−630.000	−0.497808
$118$	0	0
$119$	−2856.00	−2.20008
$120$	0	0
$121$	−755.000	−0.567243
$122$	0	0
$123$	450.000	0.329879
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2068.00	−1.44492	−0.722462	−	0.691411i	$-0.756990\pi$
$127$	−2068.00	−1.44492	−0.722462	+	0.691411i	$0.756990\pi$
$128$	0	0
$129$	876.000	0.597888
$130$	0	0
$131$	312.000	0.208088	0.104044	−	0.994573i	$-0.466822\pi$
$131$	312.000	0.208088	0.104044	+	0.994573i	$0.466822\pi$
$132$	0	0
$133$	−560.000	−0.365099
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	2646.00	1.65010	0.825048	−	0.565063i	$-0.191148\pi$
$137$	2646.00	1.65010	0.825048	+	0.565063i	$0.191148\pi$
$138$	0	0
$139$	−1276.00	−0.778625	−0.389313	−	0.921106i	$-0.627287\pi$
$139$	−1276.00	−0.778625	−0.389313	+	0.921106i	$0.627287\pi$
$140$	0	0
$141$	216.000	0.129011
$142$	0	0
$143$	1680.00	0.982438
$144$	0	0
$145$	−1530.00	−0.876273
$146$	0	0
$147$	−1323.00	−0.742307
$148$	0	0
$149$	−3198.00	−1.75832	−0.879162	−	0.476522i	$-0.841897\pi$
$149$	−3198.00	−1.75832	−0.879162	+	0.476522i	$0.841897\pi$
$150$	0	0
$151$	−760.000	−0.409589	−0.204794	−	0.978805i	$-0.565653\pi$
$151$	−760.000	−0.409589	−0.204794	+	0.978805i	$0.565653\pi$
$152$	0	0
$153$	918.000	0.485071
$154$	0	0
$155$	680.000	0.352380
$156$	0	0
$157$	−166.000	−0.0843837	−0.0421919	−	0.999110i	$-0.513434\pi$
$157$	−166.000	−0.0843837	−0.0421919	+	0.999110i	$0.513434\pi$
$158$	0	0
$159$	1242.00	0.619478
$160$	0	0
$161$	2016.00	0.986851
$162$	0	0
$163$	3020.00	1.45119	0.725597	−	0.688120i	$-0.241564\pi$
$163$	3020.00	1.45119	0.725597	+	0.688120i	$0.241564\pi$
$164$	0	0
$165$	−360.000	−0.169854
$166$	0	0
$167$	−984.000	−0.455953	−0.227977	−	0.973667i	$-0.573211\pi$
$167$	−984.000	−0.455953	−0.227977	+	0.973667i	$0.573211\pi$
$168$	0	0
$169$	2703.00	1.23031
$170$	0	0
$171$	180.000	0.0804967
$172$	0	0
$173$	1962.00	0.862243	0.431122	−	0.902294i	$-0.358118\pi$
$173$	1962.00	0.862243	0.431122	+	0.902294i	$0.358118\pi$
$174$	0	0
$175$	−700.000	−0.302372
$176$	0	0
$177$	2232.00	0.947838
$178$	0	0
$179$	576.000	0.240515	0.120258	−	0.992743i	$-0.461628\pi$
$179$	576.000	0.240515	0.120258	+	0.992743i	$0.461628\pi$
$180$	0	0
$181$	−1210.00	−0.496898	−0.248449	−	0.968645i	$-0.579921\pi$
$181$	−1210.00	−0.496898	−0.248449	+	0.968645i	$0.579921\pi$
$182$	0	0
$183$	1254.00	0.506548
$184$	0	0
$185$	1070.00	0.425232
$186$	0	0
$187$	−2448.00	−0.957302
$188$	0	0
$189$	756.000	0.290957
$190$	0	0
$191$	3384.00	1.28198	0.640989	−	0.767550i	$-0.278525\pi$
$191$	3384.00	1.28198	0.640989	+	0.767550i	$0.278525\pi$
$192$	0	0
$193$	−2038.00	−0.760096	−0.380048	−	0.924967i	$-0.624092\pi$
$193$	−2038.00	−0.760096	−0.380048	+	0.924967i	$0.624092\pi$
$194$	0	0
$195$	−1050.00	−0.385600
$196$	0	0
$197$	4098.00	1.48208	0.741042	−	0.671459i	$-0.234332\pi$
$197$	4098.00	1.48208	0.741042	+	0.671459i	$0.234332\pi$
$198$	0	0
$199$	−2248.00	−0.800786	−0.400393	−	0.916343i	$-0.631126\pi$
$199$	−2248.00	−0.800786	−0.400393	+	0.916343i	$0.631126\pi$
$200$	0	0
$201$	−564.000	−0.197918
$202$	0	0
$203$	−8568.00	−2.96234
$204$	0	0
$205$	750.000	0.255523
$206$	0	0
$207$	−648.000	−0.217580
$208$	0	0
$209$	−480.000	−0.158863
$210$	0	0
$211$	3260.00	1.06364	0.531819	−	0.846858i	$-0.321509\pi$
$211$	3260.00	1.06364	0.531819	+	0.846858i	$0.321509\pi$
$212$	0	0
$213$	−1440.00	−0.463226
$214$	0	0
$215$	1460.00	0.463122
$216$	0	0
$217$	3808.00	1.19126
$218$	0	0
$219$	−1302.00	−0.401740
$220$	0	0
$221$	−7140.00	−2.17325
$222$	0	0
$223$	−2980.00	−0.894868	−0.447434	−	0.894317i	$-0.647662\pi$
$223$	−2980.00	−0.894868	−0.447434	+	0.894317i	$0.647662\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	−3180.00	−0.929797	−0.464899	−	0.885364i	$-0.653909\pi$
$227$	−3180.00	−0.929797	−0.464899	+	0.885364i	$0.653909\pi$
$228$	0	0
$229$	3374.00	0.973625	0.486813	−	0.873506i	$-0.338159\pi$
$229$	3374.00	0.973625	0.486813	+	0.873506i	$0.338159\pi$
$230$	0	0
$231$	−2016.00	−0.574212
$232$	0	0
$233$	1950.00	0.548278	0.274139	−	0.961690i	$-0.411607\pi$
$233$	1950.00	0.548278	0.274139	+	0.961690i	$0.411607\pi$
$234$	0	0
$235$	360.000	0.0999311
$236$	0	0
$237$	−4056.00	−1.11167
$238$	0	0
$239$	2232.00	0.604084	0.302042	−	0.953295i	$-0.402332\pi$
$239$	2232.00	0.604084	0.302042	+	0.953295i	$0.402332\pi$
$240$	0	0
$241$	−1822.00	−0.486993	−0.243497	−	0.969902i	$-0.578294\pi$
$241$	−1822.00	−0.486993	−0.243497	+	0.969902i	$0.578294\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	−2205.00	−0.574989
$246$	0	0
$247$	−1400.00	−0.360647
$248$	0	0
$249$	1836.00	0.467276
$250$	0	0
$251$	1488.00	0.374190	0.187095	−	0.982342i	$-0.440093\pi$
$251$	1488.00	0.374190	0.187095	+	0.982342i	$0.440093\pi$
$252$	0	0
$253$	1728.00	0.429401
$254$	0	0
$255$	1530.00	0.375735
$256$	0	0
$257$	−2994.00	−0.726695	−0.363347	−	0.931654i	$-0.618366\pi$
$257$	−2994.00	−0.726695	−0.363347	+	0.931654i	$0.618366\pi$
$258$	0	0
$259$	5992.00	1.43755
$260$	0	0
$261$	2754.00	0.653135
$262$	0	0
$263$	−2472.00	−0.579582	−0.289791	−	0.957090i	$-0.593586\pi$
$263$	−2472.00	−0.579582	−0.289791	+	0.957090i	$0.593586\pi$
$264$	0	0
$265$	2070.00	0.479846
$266$	0	0
$267$	90.0000	0.0206289
$268$	0	0
$269$	3954.00	0.896207	0.448103	−	0.893982i	$-0.352100\pi$
$269$	3954.00	0.896207	0.448103	+	0.893982i	$0.352100\pi$
$270$	0	0
$271$	−2176.00	−0.487759	−0.243879	−	0.969806i	$-0.578420\pi$
$271$	−2176.00	−0.487759	−0.243879	+	0.969806i	$0.578420\pi$
$272$	0	0
$273$	−5880.00	−1.30357
$274$	0	0
$275$	−600.000	−0.131569
$276$	0	0
$277$	1034.00	0.224285	0.112143	−	0.993692i	$-0.464229\pi$
$277$	1034.00	0.224285	0.112143	+	0.993692i	$0.464229\pi$
$278$	0	0
$279$	−1224.00	−0.262649
$280$	0	0
$281$	−6654.00	−1.41261	−0.706307	−	0.707906i	$-0.749640\pi$
$281$	−6654.00	−1.41261	−0.706307	+	0.707906i	$0.749640\pi$
$282$	0	0
$283$	−1756.00	−0.368846	−0.184423	−	0.982847i	$-0.559042\pi$
$283$	−1756.00	−0.368846	−0.184423	+	0.982847i	$0.559042\pi$
$284$	0	0
$285$	300.000	0.0623525
$286$	0	0
$287$	4200.00	0.863826
$288$	0	0
$289$	5491.00	1.11765
$290$	0	0
$291$	858.000	0.172841
$292$	0	0
$293$	3234.00	0.644820	0.322410	−	0.946600i	$-0.395507\pi$
$293$	3234.00	0.644820	0.322410	+	0.946600i	$0.395507\pi$
$294$	0	0
$295$	3720.00	0.734192
$296$	0	0
$297$	648.000	0.126602
$298$	0	0
$299$	5040.00	0.974818
$300$	0	0
$301$	8176.00	1.56564
$302$	0	0
$303$	4626.00	0.877085
$304$	0	0
$305$	2090.00	0.392371
$306$	0	0
$307$	2036.00	0.378504	0.189252	−	0.981929i	$-0.439394\pi$
$307$	2036.00	0.378504	0.189252	+	0.981929i	$0.439394\pi$
$308$	0	0
$309$	−3516.00	−0.647308
$310$	0	0
$311$	−96.0000	−0.0175037	−0.00875187	−	0.999962i	$-0.502786\pi$
$311$	−96.0000	−0.0175037	−0.00875187	+	0.999962i	$0.502786\pi$
$312$	0	0
$313$	1202.00	0.217064	0.108532	−	0.994093i	$-0.465385\pi$
$313$	1202.00	0.217064	0.108532	+	0.994093i	$0.465385\pi$
$314$	0	0
$315$	1260.00	0.225374
$316$	0	0
$317$	−3798.00	−0.672924	−0.336462	−	0.941697i	$-0.609230\pi$
$317$	−3798.00	−0.672924	−0.336462	+	0.941697i	$0.609230\pi$
$318$	0	0
$319$	−7344.00	−1.28898
$320$	0	0
$321$	−5868.00	−1.02031
$322$	0	0
$323$	2040.00	0.351420
$324$	0	0
$325$	−1750.00	−0.298685
$326$	0	0
$327$	5574.00	0.942639
$328$	0	0
$329$	2016.00	0.337829
$330$	0	0
$331$	−5668.00	−0.941213	−0.470606	−	0.882343i	$-0.655965\pi$
$331$	−5668.00	−0.941213	−0.470606	+	0.882343i	$0.655965\pi$
$332$	0	0
$333$	−1926.00	−0.316949
$334$	0	0
$335$	−940.000	−0.153307
$336$	0	0
$337$	−454.000	−0.0733856	−0.0366928	−	0.999327i	$-0.511682\pi$
$337$	−454.000	−0.0733856	−0.0366928	+	0.999327i	$0.511682\pi$
$338$	0	0
$339$	−522.000	−0.0836317
$340$	0	0
$341$	3264.00	0.518345
$342$	0	0
$343$	−2744.00	−0.431959
$344$	0	0
$345$	−1080.00	−0.168537
$346$	0	0
$347$	−5604.00	−0.866970	−0.433485	−	0.901161i	$-0.642716\pi$
$347$	−5604.00	−0.866970	−0.433485	+	0.901161i	$0.642716\pi$
$348$	0	0
$349$	−11266.0	−1.72795	−0.863976	−	0.503533i	$-0.832033\pi$
$349$	−11266.0	−1.72795	−0.863976	+	0.503533i	$0.832033\pi$
$350$	0	0
$351$	1890.00	0.287410
$352$	0	0
$353$	−6426.00	−0.968899	−0.484450	−	0.874819i	$-0.660980\pi$
$353$	−6426.00	−0.968899	−0.484450	+	0.874819i	$0.660980\pi$
$354$	0	0
$355$	−2400.00	−0.358813
$356$	0	0
$357$	8568.00	1.27021
$358$	0	0
$359$	6936.00	1.01969	0.509844	−	0.860267i	$-0.329703\pi$
$359$	6936.00	1.01969	0.509844	+	0.860267i	$0.329703\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	2265.00	0.327498
$364$	0	0
$365$	−2170.00	−0.311186
$366$	0	0
$367$	−388.000	−0.0551865	−0.0275932	−	0.999619i	$-0.508784\pi$
$367$	−388.000	−0.0551865	−0.0275932	+	0.999619i	$0.508784\pi$
$368$	0	0
$369$	−1350.00	−0.190456
$370$	0	0
$371$	11592.0	1.62217
$372$	0	0
$373$	−8062.00	−1.11913	−0.559564	−	0.828787i	$-0.689031\pi$
$373$	−8062.00	−1.11913	−0.559564	+	0.828787i	$0.689031\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	−21420.0	−2.92622
$378$	0	0
$379$	−3388.00	−0.459182	−0.229591	−	0.973287i	$-0.573739\pi$
$379$	−3388.00	−0.459182	−0.229591	+	0.973287i	$0.573739\pi$
$380$	0	0
$381$	6204.00	0.834227
$382$	0	0
$383$	6984.00	0.931764	0.465882	−	0.884847i	$-0.345737\pi$
$383$	6984.00	0.931764	0.465882	+	0.884847i	$0.345737\pi$
$384$	0	0
$385$	−3360.00	−0.444783
$386$	0	0
$387$	−2628.00	−0.345191
$388$	0	0
$389$	−2526.00	−0.329237	−0.164619	−	0.986357i	$-0.552639\pi$
$389$	−2526.00	−0.329237	−0.164619	+	0.986357i	$0.552639\pi$
$390$	0	0
$391$	−7344.00	−0.949877
$392$	0	0
$393$	−936.000	−0.120140
$394$	0	0
$395$	−6760.00	−0.861095
$396$	0	0
$397$	6146.00	0.776975	0.388487	−	0.921454i	$-0.372998\pi$
$397$	6146.00	0.776975	0.388487	+	0.921454i	$0.372998\pi$
$398$	0	0
$399$	1680.00	0.210790
$400$	0	0
$401$	9786.00	1.21868	0.609339	−	0.792910i	$-0.291435\pi$
$401$	9786.00	1.21868	0.609339	+	0.792910i	$0.291435\pi$
$402$	0	0
$403$	9520.00	1.17674
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	5136.00	0.625509
$408$	0	0
$409$	−886.000	−0.107115	−0.0535573	−	0.998565i	$-0.517056\pi$
$409$	−886.000	−0.107115	−0.0535573	+	0.998565i	$0.517056\pi$
$410$	0	0
$411$	−7938.00	−0.952683
$412$	0	0
$413$	20832.0	2.48202
$414$	0	0
$415$	3060.00	0.361951
$416$	0	0
$417$	3828.00	0.449539
$418$	0	0
$419$	−11352.0	−1.32358	−0.661792	−	0.749688i	$-0.730204\pi$
$419$	−11352.0	−1.32358	−0.661792	+	0.749688i	$0.730204\pi$
$420$	0	0
$421$	10190.0	1.17964	0.589822	−	0.807533i	$-0.299198\pi$
$421$	10190.0	1.17964	0.589822	+	0.807533i	$0.299198\pi$
$422$	0	0
$423$	−648.000	−0.0744843
$424$	0	0
$425$	2550.00	0.291043
$426$	0	0
$427$	11704.0	1.32645
$428$	0	0
$429$	−5040.00	−0.567211
$430$	0	0
$431$	2448.00	0.273587	0.136794	−	0.990600i	$-0.456320\pi$
$431$	2448.00	0.273587	0.136794	+	0.990600i	$0.456320\pi$
$432$	0	0
$433$	−7078.00	−0.785559	−0.392779	−	0.919633i	$-0.628486\pi$
$433$	−7078.00	−0.785559	−0.392779	+	0.919633i	$0.628486\pi$
$434$	0	0
$435$	4590.00	0.505916
$436$	0	0
$437$	−1440.00	−0.157631
$438$	0	0
$439$	−18088.0	−1.96650	−0.983250	−	0.182264i	$-0.941657\pi$
$439$	−18088.0	−1.96650	−0.983250	+	0.182264i	$0.941657\pi$
$440$	0	0
$441$	3969.00	0.428571
$442$	0	0
$443$	3852.00	0.413124	0.206562	−	0.978433i	$-0.433772\pi$
$443$	3852.00	0.413124	0.206562	+	0.978433i	$0.433772\pi$
$444$	0	0
$445$	150.000	0.0159791
$446$	0	0
$447$	9594.00	1.01517
$448$	0	0
$449$	6522.00	0.685506	0.342753	−	0.939426i	$-0.388641\pi$
$449$	6522.00	0.685506	0.342753	+	0.939426i	$0.388641\pi$
$450$	0	0
$451$	3600.00	0.375870
$452$	0	0
$453$	2280.00	0.236476
$454$	0	0
$455$	−9800.00	−1.00974
$456$	0	0
$457$	2090.00	0.213930	0.106965	−	0.994263i	$-0.465887\pi$
$457$	2090.00	0.213930	0.106965	+	0.994263i	$0.465887\pi$
$458$	0	0
$459$	−2754.00	−0.280056
$460$	0	0
$461$	−9894.00	−0.999587	−0.499793	−	0.866145i	$-0.666591\pi$
$461$	−9894.00	−0.999587	−0.499793	+	0.866145i	$0.666591\pi$
$462$	0	0
$463$	3044.00	0.305544	0.152772	−	0.988261i	$-0.451180\pi$
$463$	3044.00	0.305544	0.152772	+	0.988261i	$0.451180\pi$
$464$	0	0
$465$	−2040.00	−0.203447
$466$	0	0
$467$	10236.0	1.01427	0.507137	−	0.861866i	$-0.330704\pi$
$467$	10236.0	1.01427	0.507137	+	0.861866i	$0.330704\pi$
$468$	0	0
$469$	−5264.00	−0.518271
$470$	0	0
$471$	498.000	0.0487190
$472$	0	0
$473$	7008.00	0.681244
$474$	0	0
$475$	500.000	0.0482980
$476$	0	0
$477$	−3726.00	−0.357656
$478$	0	0
$479$	11496.0	1.09659	0.548294	−	0.836286i	$-0.315277\pi$
$479$	11496.0	1.09659	0.548294	+	0.836286i	$0.315277\pi$
$480$	0	0
$481$	14980.0	1.42002
$482$	0	0
$483$	−6048.00	−0.569759
$484$	0	0
$485$	1430.00	0.133882
$486$	0	0
$487$	−15316.0	−1.42512	−0.712561	−	0.701610i	$-0.752465\pi$
$487$	−15316.0	−1.42512	−0.712561	+	0.701610i	$0.752465\pi$
$488$	0	0
$489$	−9060.00	−0.837847
$490$	0	0
$491$	−11616.0	−1.06766	−0.533832	−	0.845591i	$-0.679248\pi$
$491$	−11616.0	−1.06766	−0.533832	+	0.845591i	$0.679248\pi$
$492$	0	0
$493$	31212.0	2.85135
$494$	0	0
$495$	1080.00	0.0980654
$496$	0	0
$497$	−13440.0	−1.21301
$498$	0	0
$499$	14996.0	1.34532	0.672658	−	0.739953i	$-0.265152\pi$
$499$	14996.0	1.34532	0.672658	+	0.739953i	$0.265152\pi$
$500$	0	0
$501$	2952.00	0.263245
$502$	0	0
$503$	−21648.0	−1.91896	−0.959480	−	0.281778i	$-0.909076\pi$
$503$	−21648.0	−1.91896	−0.959480	+	0.281778i	$0.909076\pi$
$504$	0	0
$505$	7710.00	0.679387
$506$	0	0
$507$	−8109.00	−0.710322
$508$	0	0
$509$	3378.00	0.294160	0.147080	−	0.989125i	$-0.453013\pi$
$509$	3378.00	0.294160	0.147080	+	0.989125i	$0.453013\pi$
$510$	0	0
$511$	−12152.0	−1.05200
$512$	0	0
$513$	−540.000	−0.0464748
$514$	0	0
$515$	−5860.00	−0.501403
$516$	0	0
$517$	1728.00	0.146997
$518$	0	0
$519$	−5886.00	−0.497816
$520$	0	0
$521$	−16158.0	−1.35872	−0.679362	−	0.733804i	$-0.737743\pi$
$521$	−16158.0	−1.35872	−0.679362	+	0.733804i	$0.737743\pi$
$522$	0	0
$523$	−76.0000	−0.00635420	−0.00317710	−	0.999995i	$-0.501011\pi$
$523$	−76.0000	−0.00635420	−0.00317710	+	0.999995i	$0.501011\pi$
$524$	0	0
$525$	2100.00	0.174574
$526$	0	0
$527$	−13872.0	−1.14663
$528$	0	0
$529$	−6983.00	−0.573929
$530$	0	0
$531$	−6696.00	−0.547235
$532$	0	0
$533$	10500.0	0.853294
$534$	0	0
$535$	−9780.00	−0.790329
$536$	0	0
$537$	−1728.00	−0.138862
$538$	0	0
$539$	−10584.0	−0.845798
$540$	0	0
$541$	9278.00	0.737324	0.368662	−	0.929563i	$-0.379816\pi$
$541$	9278.00	0.737324	0.368662	+	0.929563i	$0.379816\pi$
$542$	0	0
$543$	3630.00	0.286884
$544$	0	0
$545$	9290.00	0.730165
$546$	0	0
$547$	14564.0	1.13841	0.569206	−	0.822195i	$-0.307251\pi$
$547$	14564.0	1.13841	0.569206	+	0.822195i	$0.307251\pi$
$548$	0	0
$549$	−3762.00	−0.292456
$550$	0	0
$551$	6120.00	0.473177
$552$	0	0
$553$	−37856.0	−2.91103
$554$	0	0
$555$	−3210.00	−0.245508
$556$	0	0
$557$	2154.00	0.163856	0.0819281	−	0.996638i	$-0.473892\pi$
$557$	2154.00	0.163856	0.0819281	+	0.996638i	$0.473892\pi$
$558$	0	0
$559$	20440.0	1.54655
$560$	0	0
$561$	7344.00	0.552699
$562$	0	0
$563$	−8700.00	−0.651263	−0.325632	−	0.945497i	$-0.605577\pi$
$563$	−8700.00	−0.651263	−0.325632	+	0.945497i	$0.605577\pi$
$564$	0	0
$565$	−870.000	−0.0647808
$566$	0	0
$567$	−2268.00	−0.167984
$568$	0	0
$569$	4194.00	0.309001	0.154501	−	0.987993i	$-0.450623\pi$
$569$	4194.00	0.309001	0.154501	+	0.987993i	$0.450623\pi$
$570$	0	0
$571$	−8020.00	−0.587787	−0.293894	−	0.955838i	$-0.594951\pi$
$571$	−8020.00	−0.587787	−0.293894	+	0.955838i	$0.594951\pi$
$572$	0	0
$573$	−10152.0	−0.740150
$574$	0	0
$575$	−1800.00	−0.130548
$576$	0	0
$577$	−2686.00	−0.193795	−0.0968974	−	0.995294i	$-0.530892\pi$
$577$	−2686.00	−0.193795	−0.0968974	+	0.995294i	$0.530892\pi$
$578$	0	0
$579$	6114.00	0.438841
$580$	0	0
$581$	17136.0	1.22362
$582$	0	0
$583$	9936.00	0.705844
$584$	0	0
$585$	3150.00	0.222626
$586$	0	0
$587$	3012.00	0.211786	0.105893	−	0.994378i	$-0.466230\pi$
$587$	3012.00	0.211786	0.105893	+	0.994378i	$0.466230\pi$
$588$	0	0
$589$	−2720.00	−0.190281
$590$	0	0
$591$	−12294.0	−0.855681
$592$	0	0
$593$	−15522.0	−1.07489	−0.537447	−	0.843298i	$-0.680611\pi$
$593$	−15522.0	−1.07489	−0.537447	+	0.843298i	$0.680611\pi$
$594$	0	0
$595$	14280.0	0.983904
$596$	0	0
$597$	6744.00	0.462334
$598$	0	0
$599$	19224.0	1.31130	0.655652	−	0.755063i	$-0.272394\pi$
$599$	19224.0	1.31130	0.655652	+	0.755063i	$0.272394\pi$
$600$	0	0
$601$	−6502.00	−0.441301	−0.220651	−	0.975353i	$-0.570818\pi$
$601$	−6502.00	−0.441301	−0.220651	+	0.975353i	$0.570818\pi$
$602$	0	0
$603$	1692.00	0.114268
$604$	0	0
$605$	3775.00	0.253679
$606$	0	0
$607$	29396.0	1.96565	0.982823	−	0.184552i	$-0.0590834\pi$
$607$	29396.0	1.96565	0.982823	+	0.184552i	$0.0590834\pi$
$608$	0	0
$609$	25704.0	1.71031
$610$	0	0
$611$	5040.00	0.333710
$612$	0	0
$613$	−10006.0	−0.659280	−0.329640	−	0.944107i	$-0.606927\pi$
$613$	−10006.0	−0.659280	−0.329640	+	0.944107i	$0.606927\pi$
$614$	0	0
$615$	−2250.00	−0.147526
$616$	0	0
$617$	23118.0	1.50842	0.754210	−	0.656633i	$-0.228020\pi$
$617$	23118.0	1.50842	0.754210	+	0.656633i	$0.228020\pi$
$618$	0	0
$619$	14036.0	0.911397	0.455698	−	0.890134i	$-0.349390\pi$
$619$	14036.0	0.911397	0.455698	+	0.890134i	$0.349390\pi$
$620$	0	0
$621$	1944.00	0.125620
$622$	0	0
$623$	840.000	0.0540191
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	1440.00	0.0917194
$628$	0	0
$629$	−21828.0	−1.38369
$630$	0	0
$631$	−4288.00	−0.270527	−0.135264	−	0.990810i	$-0.543188\pi$
$631$	−4288.00	−0.270527	−0.135264	+	0.990810i	$0.543188\pi$
$632$	0	0
$633$	−9780.00	−0.614092
$634$	0	0
$635$	10340.0	0.646190
$636$	0	0
$637$	−30870.0	−1.92012
$638$	0	0
$639$	4320.00	0.267444
$640$	0	0
$641$	1314.00	0.0809671	0.0404835	−	0.999180i	$-0.487110\pi$
$641$	1314.00	0.0809671	0.0404835	+	0.999180i	$0.487110\pi$
$642$	0	0
$643$	−628.000	−0.0385162	−0.0192581	−	0.999815i	$-0.506130\pi$
$643$	−628.000	−0.0385162	−0.0192581	+	0.999815i	$0.506130\pi$
$644$	0	0
$645$	−4380.00	−0.267383
$646$	0	0
$647$	−10944.0	−0.664997	−0.332498	−	0.943104i	$-0.607892\pi$
$647$	−10944.0	−0.664997	−0.332498	+	0.943104i	$0.607892\pi$
$648$	0	0
$649$	17856.0	1.07998
$650$	0	0
$651$	−11424.0	−0.687776
$652$	0	0
$653$	1098.00	0.0658010	0.0329005	−	0.999459i	$-0.489526\pi$
$653$	1098.00	0.0658010	0.0329005	+	0.999459i	$0.489526\pi$
$654$	0	0
$655$	−1560.00	−0.0930599
$656$	0	0
$657$	3906.00	0.231945
$658$	0	0
$659$	312.000	0.0184428	0.00922139	−	0.999957i	$-0.497065\pi$
$659$	312.000	0.0184428	0.00922139	+	0.999957i	$0.497065\pi$
$660$	0	0
$661$	8678.00	0.510643	0.255322	−	0.966856i	$-0.417819\pi$
$661$	8678.00	0.510643	0.255322	+	0.966856i	$0.417819\pi$
$662$	0	0
$663$	21420.0	1.25473
$664$	0	0
$665$	2800.00	0.163277
$666$	0	0
$667$	−22032.0	−1.27898
$668$	0	0
$669$	8940.00	0.516652
$670$	0	0
$671$	10032.0	0.577170
$672$	0	0
$673$	−14470.0	−0.828793	−0.414396	−	0.910097i	$-0.636007\pi$
$673$	−14470.0	−0.828793	−0.414396	+	0.910097i	$0.636007\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−11838.0	−0.672040	−0.336020	−	0.941855i	$-0.609081\pi$
$677$	−11838.0	−0.672040	−0.336020	+	0.941855i	$0.609081\pi$
$678$	0	0
$679$	8008.00	0.452605
$680$	0	0
$681$	9540.00	0.536819
$682$	0	0
$683$	−25548.0	−1.43128	−0.715642	−	0.698467i	$-0.753866\pi$
$683$	−25548.0	−1.43128	−0.715642	+	0.698467i	$0.753866\pi$
$684$	0	0
$685$	−13230.0	−0.737945
$686$	0	0
$687$	−10122.0	−0.562123
$688$	0	0
$689$	28980.0	1.60239
$690$	0	0
$691$	−18412.0	−1.01364	−0.506820	−	0.862052i	$-0.669179\pi$
$691$	−18412.0	−1.01364	−0.506820	+	0.862052i	$0.669179\pi$
$692$	0	0
$693$	6048.00	0.331522
$694$	0	0
$695$	6380.00	0.348212
$696$	0	0
$697$	−15300.0	−0.831462
$698$	0	0
$699$	−5850.00	−0.316548
$700$	0	0
$701$	−8814.00	−0.474893	−0.237447	−	0.971401i	$-0.576310\pi$
$701$	−8814.00	−0.474893	−0.237447	+	0.971401i	$0.576310\pi$
$702$	0	0
$703$	−4280.00	−0.229621
$704$	0	0
$705$	−1080.00	−0.0576953
$706$	0	0
$707$	43176.0	2.29675
$708$	0	0
$709$	−17314.0	−0.917124	−0.458562	−	0.888662i	$-0.651635\pi$
$709$	−17314.0	−0.917124	−0.458562	+	0.888662i	$0.651635\pi$
$710$	0	0
$711$	12168.0	0.641822
$712$	0	0
$713$	9792.00	0.514324
$714$	0	0
$715$	−8400.00	−0.439360
$716$	0	0
$717$	−6696.00	−0.348768
$718$	0	0
$719$	−768.000	−0.0398353	−0.0199176	−	0.999802i	$-0.506340\pi$
$719$	−768.000	−0.0398353	−0.0199176	+	0.999802i	$0.506340\pi$
$720$	0	0
$721$	−32816.0	−1.69505
$722$	0	0
$723$	5466.00	0.281166
$724$	0	0
$725$	7650.00	0.391881
$726$	0	0
$727$	−18196.0	−0.928270	−0.464135	−	0.885764i	$-0.653635\pi$
$727$	−18196.0	−0.928270	−0.464135	+	0.885764i	$0.653635\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−29784.0	−1.50698
$732$	0	0
$733$	−18142.0	−0.914175	−0.457087	−	0.889422i	$-0.651107\pi$
$733$	−18142.0	−0.914175	−0.457087	+	0.889422i	$0.651107\pi$
$734$	0	0
$735$	6615.00	0.331970
$736$	0	0
$737$	−4512.00	−0.225511
$738$	0	0
$739$	−13660.0	−0.679961	−0.339981	−	0.940432i	$-0.610420\pi$
$739$	−13660.0	−0.679961	−0.339981	+	0.940432i	$0.610420\pi$
$740$	0	0
$741$	4200.00	0.208220
$742$	0	0
$743$	−12768.0	−0.630434	−0.315217	−	0.949020i	$-0.602077\pi$
$743$	−12768.0	−0.630434	−0.315217	+	0.949020i	$0.602077\pi$
$744$	0	0
$745$	15990.0	0.786347
$746$	0	0
$747$	−5508.00	−0.269782
$748$	0	0
$749$	−54768.0	−2.67180
$750$	0	0
$751$	22952.0	1.11522	0.557610	−	0.830103i	$-0.311718\pi$
$751$	22952.0	1.11522	0.557610	+	0.830103i	$0.311718\pi$
$752$	0	0
$753$	−4464.00	−0.216039
$754$	0	0
$755$	3800.00	0.183174
$756$	0	0
$757$	15818.0	0.759465	0.379732	−	0.925096i	$-0.376016\pi$
$757$	15818.0	0.759465	0.379732	+	0.925096i	$0.376016\pi$
$758$	0	0
$759$	−5184.00	−0.247915
$760$	0	0
$761$	−18558.0	−0.884004	−0.442002	−	0.897014i	$-0.645732\pi$
$761$	−18558.0	−0.884004	−0.442002	+	0.897014i	$0.645732\pi$
$762$	0	0
$763$	52024.0	2.46841
$764$	0	0
$765$	−4590.00	−0.216930
$766$	0	0
$767$	52080.0	2.45176
$768$	0	0
$769$	14978.0	0.702367	0.351184	−	0.936307i	$-0.385779\pi$
$769$	14978.0	0.702367	0.351184	+	0.936307i	$0.385779\pi$
$770$	0	0
$771$	8982.00	0.419557
$772$	0	0
$773$	8946.00	0.416255	0.208128	−	0.978102i	$-0.433263\pi$
$773$	8946.00	0.416255	0.208128	+	0.978102i	$0.433263\pi$
$774$	0	0
$775$	−3400.00	−0.157589
$776$	0	0
$777$	−17976.0	−0.829968
$778$	0	0
$779$	−3000.00	−0.137980
$780$	0	0
$781$	−11520.0	−0.527808
$782$	0	0
$783$	−8262.00	−0.377088
$784$	0	0
$785$	830.000	0.0377375
$786$	0	0
$787$	−18436.0	−0.835035	−0.417517	−	0.908669i	$-0.637100\pi$
$787$	−18436.0	−0.835035	−0.417517	+	0.908669i	$0.637100\pi$
$788$	0	0
$789$	7416.00	0.334622
$790$	0	0
$791$	−4872.00	−0.218999
$792$	0	0
$793$	29260.0	1.31028
$794$	0	0
$795$	−6210.00	−0.277039
$796$	0	0
$797$	16314.0	0.725058	0.362529	−	0.931972i	$-0.381913\pi$
$797$	16314.0	0.725058	0.362529	+	0.931972i	$0.381913\pi$
$798$	0	0
$799$	−7344.00	−0.325172
$800$	0	0
$801$	−270.000	−0.0119101
$802$	0	0
$803$	−10416.0	−0.457749
$804$	0	0
$805$	−10080.0	−0.441333
$806$	0	0
$807$	−11862.0	−0.517425
$808$	0	0
$809$	−25446.0	−1.10585	−0.552926	−	0.833231i	$-0.686489\pi$
$809$	−25446.0	−1.10585	−0.552926	+	0.833231i	$0.686489\pi$
$810$	0	0
$811$	42740.0	1.85056	0.925280	−	0.379284i	$-0.123830\pi$
$811$	42740.0	1.85056	0.925280	+	0.379284i	$0.123830\pi$
$812$	0	0
$813$	6528.00	0.281608
$814$	0	0
$815$	−15100.0	−0.648994
$816$	0	0
$817$	−5840.00	−0.250080
$818$	0	0
$819$	17640.0	0.752615
$820$	0	0
$821$	29946.0	1.27299	0.636494	−	0.771282i	$-0.280384\pi$
$821$	29946.0	1.27299	0.636494	+	0.771282i	$0.280384\pi$
$822$	0	0
$823$	−32596.0	−1.38059	−0.690295	−	0.723528i	$-0.742519\pi$
$823$	−32596.0	−1.38059	−0.690295	+	0.723528i	$0.742519\pi$
$824$	0	0
$825$	1800.00	0.0759612
$826$	0	0
$827$	3804.00	0.159949	0.0799746	−	0.996797i	$-0.474516\pi$
$827$	3804.00	0.159949	0.0799746	+	0.996797i	$0.474516\pi$
$828$	0	0
$829$	3278.00	0.137334	0.0686669	−	0.997640i	$-0.478125\pi$
$829$	3278.00	0.137334	0.0686669	+	0.997640i	$0.478125\pi$
$830$	0	0
$831$	−3102.00	−0.129491
$832$	0	0
$833$	44982.0	1.87099
$834$	0	0
$835$	4920.00	0.203909
$836$	0	0
$837$	3672.00	0.151640
$838$	0	0
$839$	−5784.00	−0.238005	−0.119002	−	0.992894i	$-0.537970\pi$
$839$	−5784.00	−0.238005	−0.119002	+	0.992894i	$0.537970\pi$
$840$	0	0
$841$	69247.0	2.83927
$842$	0	0
$843$	19962.0	0.815573
$844$	0	0
$845$	−13515.0	−0.550213
$846$	0	0
$847$	21140.0	0.857590
$848$	0	0
$849$	5268.00	0.212953
$850$	0	0
$851$	15408.0	0.620657
$852$	0	0
$853$	17306.0	0.694661	0.347331	−	0.937743i	$-0.387088\pi$
$853$	17306.0	0.694661	0.347331	+	0.937743i	$0.387088\pi$
$854$	0	0
$855$	−900.000	−0.0359992
$856$	0	0
$857$	31134.0	1.24098	0.620488	−	0.784216i	$-0.286934\pi$
$857$	31134.0	1.24098	0.620488	+	0.784216i	$0.286934\pi$
$858$	0	0
$859$	−10780.0	−0.428183	−0.214091	−	0.976814i	$-0.568679\pi$
$859$	−10780.0	−0.428183	−0.214091	+	0.976814i	$0.568679\pi$
$860$	0	0
$861$	−12600.0	−0.498730
$862$	0	0
$863$	3456.00	0.136319	0.0681597	−	0.997674i	$-0.478287\pi$
$863$	3456.00	0.136319	0.0681597	+	0.997674i	$0.478287\pi$
$864$	0	0
$865$	−9810.00	−0.385607
$866$	0	0
$867$	−16473.0	−0.645274
$868$	0	0
$869$	−32448.0	−1.26665
$870$	0	0
$871$	−13160.0	−0.511951
$872$	0	0
$873$	−2574.00	−0.0997900
$874$	0	0
$875$	3500.00	0.135225
$876$	0	0
$877$	2618.00	0.100802	0.0504011	−	0.998729i	$-0.483950\pi$
$877$	2618.00	0.100802	0.0504011	+	0.998729i	$0.483950\pi$
$878$	0	0
$879$	−9702.00	−0.372287
$880$	0	0
$881$	−26550.0	−1.01531	−0.507657	−	0.861559i	$-0.669488\pi$
$881$	−26550.0	−1.01531	−0.507657	+	0.861559i	$0.669488\pi$
$882$	0	0
$883$	27596.0	1.05173	0.525866	−	0.850567i	$-0.323741\pi$
$883$	27596.0	1.05173	0.525866	+	0.850567i	$0.323741\pi$
$884$	0	0
$885$	−11160.0	−0.423886
$886$	0	0
$887$	−37848.0	−1.43271	−0.716354	−	0.697737i	$-0.754190\pi$
$887$	−37848.0	−1.43271	−0.716354	+	0.697737i	$0.754190\pi$
$888$	0	0
$889$	57904.0	2.18452
$890$	0	0
$891$	−1944.00	−0.0730937
$892$	0	0
$893$	−1440.00	−0.0539617
$894$	0	0
$895$	−2880.00	−0.107562
$896$	0	0
$897$	−15120.0	−0.562812
$898$	0	0
$899$	−41616.0	−1.54391
$900$	0	0
$901$	−42228.0	−1.56140
$902$	0	0
$903$	−24528.0	−0.903921
$904$	0	0
$905$	6050.00	0.222220
$906$	0	0
$907$	−4804.00	−0.175870	−0.0879351	−	0.996126i	$-0.528027\pi$
$907$	−4804.00	−0.175870	−0.0879351	+	0.996126i	$0.528027\pi$
$908$	0	0
$909$	−13878.0	−0.506385
$910$	0	0
$911$	28608.0	1.04042	0.520211	−	0.854037i	$-0.325853\pi$
$911$	28608.0	1.04042	0.520211	+	0.854037i	$0.325853\pi$
$912$	0	0
$913$	14688.0	0.532423
$914$	0	0
$915$	−6270.00	−0.226535
$916$	0	0
$917$	−8736.00	−0.314600
$918$	0	0
$919$	−40768.0	−1.46334	−0.731672	−	0.681657i	$-0.761259\pi$
$919$	−40768.0	−1.46334	−0.731672	+	0.681657i	$0.761259\pi$
$920$	0	0
$921$	−6108.00	−0.218529
$922$	0	0
$923$	−33600.0	−1.19822
$924$	0	0
$925$	−5350.00	−0.190170
$926$	0	0
$927$	10548.0	0.373724
$928$	0	0
$929$	27642.0	0.976216	0.488108	−	0.872783i	$-0.337687\pi$
$929$	27642.0	0.976216	0.488108	+	0.872783i	$0.337687\pi$
$930$	0	0
$931$	8820.00	0.310487
$932$	0	0
$933$	288.000	0.0101058
$934$	0	0
$935$	12240.0	0.428119
$936$	0	0
$937$	28106.0	0.979918	0.489959	−	0.871746i	$-0.337012\pi$
$937$	28106.0	0.979918	0.489959	+	0.871746i	$0.337012\pi$
$938$	0	0
$939$	−3606.00	−0.125322
$940$	0	0
$941$	14730.0	0.510291	0.255146	−	0.966903i	$-0.417877\pi$
$941$	14730.0	0.510291	0.255146	+	0.966903i	$0.417877\pi$
$942$	0	0
$943$	10800.0	0.372955
$944$	0	0
$945$	−3780.00	−0.130120
$946$	0	0
$947$	−9564.00	−0.328182	−0.164091	−	0.986445i	$-0.552469\pi$
$947$	−9564.00	−0.328182	−0.164091	+	0.986445i	$0.552469\pi$
$948$	0	0
$949$	−30380.0	−1.03917
$950$	0	0
$951$	11394.0	0.388513
$952$	0	0
$953$	−53898.0	−1.83203	−0.916017	−	0.401141i	$-0.868614\pi$
$953$	−53898.0	−1.83203	−0.916017	+	0.401141i	$0.868614\pi$
$954$	0	0
$955$	−16920.0	−0.573318
$956$	0	0
$957$	22032.0	0.744194
$958$	0	0
$959$	−74088.0	−2.49471
$960$	0	0
$961$	−11295.0	−0.379141
$962$	0	0
$963$	17604.0	0.589077
$964$	0	0
$965$	10190.0	0.339925
$966$	0	0
$967$	15140.0	0.503485	0.251742	−	0.967794i	$-0.418996\pi$
$967$	15140.0	0.503485	0.251742	+	0.967794i	$0.418996\pi$
$968$	0	0
$969$	−6120.00	−0.202892
$970$	0	0
$971$	23808.0	0.786854	0.393427	−	0.919356i	$-0.371289\pi$
$971$	23808.0	0.786854	0.393427	+	0.919356i	$0.371289\pi$
$972$	0	0
$973$	35728.0	1.17717
$974$	0	0
$975$	5250.00	0.172446
$976$	0	0
$977$	23094.0	0.756236	0.378118	−	0.925757i	$-0.376571\pi$
$977$	23094.0	0.756236	0.378118	+	0.925757i	$0.376571\pi$
$978$	0	0
$979$	720.000	0.0235049
$980$	0	0
$981$	−16722.0	−0.544233
$982$	0	0
$983$	7584.00	0.246075	0.123038	−	0.992402i	$-0.460736\pi$
$983$	7584.00	0.246075	0.123038	+	0.992402i	$0.460736\pi$
$984$	0	0
$985$	−20490.0	−0.662808
$986$	0	0
$987$	−6048.00	−0.195046
$988$	0	0
$989$	21024.0	0.675960
$990$	0	0
$991$	−26752.0	−0.857523	−0.428761	−	0.903418i	$-0.641050\pi$
$991$	−26752.0	−0.857523	−0.428761	+	0.903418i	$0.641050\pi$
$992$	0	0
$993$	17004.0	0.543409
$994$	0	0
$995$	11240.0	0.358123
$996$	0	0
$997$	7778.00	0.247073	0.123536	−	0.992340i	$-0.460576\pi$
$997$	7778.00	0.247073	0.123536	+	0.992340i	$0.460576\pi$
$998$	0	0
$999$	5778.00	0.182991

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	60.4.a.a.1.1	✓	1
3.2	odd	2		180.4.a.d.1.1		1
4.3	odd	2		240.4.a.i.1.1		1
5.2	odd	4		300.4.d.b.49.2		2
5.3	odd	4		300.4.d.b.49.1		2
5.4	even	2		300.4.a.i.1.1		1
8.3	odd	2		960.4.a.r.1.1		1
8.5	even	2		960.4.a.bc.1.1		1
9.2	odd	6		1620.4.i.f.1081.1		2
9.4	even	3		1620.4.i.l.541.1		2
9.5	odd	6		1620.4.i.f.541.1		2
9.7	even	3		1620.4.i.l.1081.1		2
12.11	even	2		720.4.a.bb.1.1		1
15.2	even	4		900.4.d.h.649.1		2
15.8	even	4		900.4.d.h.649.2		2
15.14	odd	2		900.4.a.q.1.1		1
20.3	even	4		1200.4.f.n.49.2		2
20.7	even	4		1200.4.f.n.49.1		2
20.19	odd	2		1200.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
60.4.a.a.1.1	✓	1	1.1	even	1	trivial
180.4.a.d.1.1		1	3.2	odd	2
240.4.a.i.1.1		1	4.3	odd	2
300.4.a.i.1.1		1	5.4	even	2
300.4.d.b.49.1		2	5.3	odd	4
300.4.d.b.49.2		2	5.2	odd	4
720.4.a.bb.1.1		1	12.11	even	2
900.4.a.q.1.1		1	15.14	odd	2
900.4.d.h.649.1		2	15.2	even	4
900.4.d.h.649.2		2	15.8	even	4
960.4.a.r.1.1		1	8.3	odd	2
960.4.a.bc.1.1		1	8.5	even	2
1200.4.a.a.1.1		1	20.19	odd	2
1200.4.f.n.49.1		2	20.7	even	4
1200.4.f.n.49.2		2	20.3	even	4
1620.4.i.f.541.1		2	9.5	odd	6
1620.4.i.f.1081.1		2	9.2	odd	6
1620.4.i.l.541.1		2	9.4	even	3
1620.4.i.l.1081.1		2	9.7	even	3

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 60 = 2^{2} \cdot 3 \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	60.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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