LMFDB - Embedded newform 1728.4.a.bc.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,4,Mod(1,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1728.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1728 = 2^{6} \cdot 3^{3}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1728.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:	$101.955300490$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 27)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1728.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+15.0000 q^{5} -25.0000 q^{7} -15.0000 q^{11} -20.0000 q^{13} -72.0000 q^{17} -2.00000 q^{19} -114.000 q^{23} +100.000 q^{25} +30.0000 q^{29} +101.000 q^{31} -375.000 q^{35} +430.000 q^{37} +30.0000 q^{41} -110.000 q^{43} +330.000 q^{47} +282.000 q^{49} +621.000 q^{53} -225.000 q^{55} -660.000 q^{59} +376.000 q^{61} -300.000 q^{65} +250.000 q^{67} +360.000 q^{71} +785.000 q^{73} +375.000 q^{77} +488.000 q^{79} +489.000 q^{83} -1080.00 q^{85} +450.000 q^{89} +500.000 q^{91} -30.0000 q^{95} -1105.00 q^{97} +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
$3$	0	0
$3$	0	0
$4$	0	0
$5$	15.0000	1.34164	0.670820	−	0.741620i	$-0.265942\pi$
$5$	15.0000	1.34164	0.670820	+	0.741620i	$0.265942\pi$
$6$	0	0
$7$	−25.0000	−1.34987	−0.674937	−	0.737876i	$-0.735829\pi$
$7$	−25.0000	−1.34987	−0.674937	+	0.737876i	$0.735829\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−15.0000	−0.411152	−0.205576	−	0.978641i	$-0.565907\pi$
$11$	−15.0000	−0.411152	−0.205576	+	0.978641i	$0.565907\pi$
$12$	0	0
$13$	−20.0000	−0.426692	−0.213346	−	0.976977i	$-0.568436\pi$
$13$	−20.0000	−0.426692	−0.213346	+	0.976977i	$0.568436\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−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
$19$	−2.00000	−0.0241490	−0.0120745	−	0.999927i	$-0.503844\pi$
$19$	−2.00000	−0.0241490	−0.0120745	+	0.999927i	$0.503844\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−114.000	−1.03351	−0.516753	−	0.856134i	$-0.672859\pi$
$23$	−114.000	−1.03351	−0.516753	+	0.856134i	$0.672859\pi$
$24$	0	0
$25$	100.000	0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	30.0000	0.192099	0.0960493	−	0.995377i	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	0	0
$31$	101.000	0.585166	0.292583	−	0.956240i	$-0.405485\pi$
$31$	101.000	0.585166	0.292583	+	0.956240i	$0.405485\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	−375.000	−1.81104
$36$	0	0
$37$	430.000	1.91058	0.955291	−	0.295666i	$-0.0955415\pi$
$37$	430.000	1.91058	0.955291	+	0.295666i	$0.0955415\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	30.0000	0.114273	0.0571367	−	0.998366i	$-0.481803\pi$
$41$	30.0000	0.114273	0.0571367	+	0.998366i	$0.481803\pi$
$42$	0	0
$43$	−110.000	−0.390113	−0.195056	−	0.980792i	$-0.562489\pi$
$43$	−110.000	−0.390113	−0.195056	+	0.980792i	$0.562489\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	330.000	1.02416	0.512079	−	0.858938i	$-0.328875\pi$
$47$	330.000	1.02416	0.512079	+	0.858938i	$0.328875\pi$
$48$	0	0
$49$	282.000	0.822157
$50$	0	0
$51$	0	0
$52$	0	0
$53$	621.000	1.60945	0.804726	−	0.593647i	$-0.202312\pi$
$53$	621.000	1.60945	0.804726	+	0.593647i	$0.202312\pi$
$54$	0	0
$55$	−225.000	−0.551618
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−660.000	−1.45635	−0.728175	−	0.685391i	$-0.759631\pi$
$59$	−660.000	−1.45635	−0.728175	+	0.685391i	$0.759631\pi$
$60$	0	0
$61$	376.000	0.789211	0.394605	−	0.918851i	$-0.370881\pi$
$61$	376.000	0.789211	0.394605	+	0.918851i	$0.370881\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−300.000	−0.572468
$66$	0	0
$67$	250.000	0.455856	0.227928	−	0.973678i	$-0.426805\pi$
$67$	250.000	0.455856	0.227928	+	0.973678i	$0.426805\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	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
$73$	785.000	1.25859	0.629297	−	0.777165i	$-0.283343\pi$
$73$	785.000	1.25859	0.629297	+	0.777165i	$0.283343\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	375.000	0.555003
$78$	0	0
$79$	488.000	0.694991	0.347496	−	0.937682i	$-0.387032\pi$
$79$	488.000	0.694991	0.347496	+	0.937682i	$0.387032\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	489.000	0.646683	0.323342	−	0.946282i	$-0.395194\pi$
$83$	489.000	0.646683	0.323342	+	0.946282i	$0.395194\pi$
$84$	0	0
$85$	−1080.00	−1.37815
$86$	0	0
$87$	0	0
$88$	0	0
$89$	450.000	0.535954	0.267977	−	0.963425i	$-0.413645\pi$
$89$	450.000	0.535954	0.267977	+	0.963425i	$0.413645\pi$
$90$	0	0
$91$	500.000	0.575981
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−30.0000	−0.0323993
$96$	0	0
$97$	−1105.00	−1.15666	−0.578329	−	0.815804i	$-0.696295\pi$
$97$	−1105.00	−1.15666	−0.578329	+	0.815804i	$0.696295\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1425.00	1.40389	0.701945	−	0.712232i	$-0.252315\pi$
$101$	1425.00	1.40389	0.701945	+	0.712232i	$0.252315\pi$
$102$	0	0
$103$	−1060.00	−1.01403	−0.507014	−	0.861938i	$-0.669251\pi$
$103$	−1060.00	−1.01403	−0.507014	+	0.861938i	$0.669251\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1485.00	1.34169	0.670843	−	0.741600i	$-0.265933\pi$
$107$	1485.00	1.34169	0.670843	+	0.741600i	$0.265933\pi$
$108$	0	0
$109$	862.000	0.757474	0.378737	−	0.925504i	$-0.376359\pi$
$109$	862.000	0.757474	0.378737	+	0.925504i	$0.376359\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−690.000	−0.574422	−0.287211	−	0.957867i	$-0.592728\pi$
$113$	−690.000	−0.574422	−0.287211	+	0.957867i	$0.592728\pi$
$114$	0	0
$115$	−1710.00	−1.38659
$116$	0	0
$117$	0	0
$118$	0	0
$119$	1800.00	1.38660
$120$	0	0
$121$	−1106.00	−0.830954
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−375.000	−0.268328
$126$	0	0
$127$	1865.00	1.30309	0.651543	−	0.758611i	$-0.274122\pi$
$127$	1865.00	1.30309	0.651543	+	0.758611i	$0.274122\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−1155.00	−0.770327	−0.385163	−	0.922848i	$-0.625855\pi$
$131$	−1155.00	−0.770327	−0.385163	+	0.922848i	$0.625855\pi$
$132$	0	0
$133$	50.0000	0.0325981
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2778.00	1.73241	0.866206	−	0.499686i	$-0.166551\pi$
$137$	2778.00	1.73241	0.866206	+	0.499686i	$0.166551\pi$
$138$	0	0
$139$	1924.00	1.17404	0.587020	−	0.809572i	$-0.300301\pi$
$139$	1924.00	1.17404	0.587020	+	0.809572i	$0.300301\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	300.000	0.175435
$144$	0	0
$145$	450.000	0.257727
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1455.00	0.799988	0.399994	−	0.916518i	$-0.369012\pi$
$149$	1455.00	0.799988	0.399994	+	0.916518i	$0.369012\pi$
$150$	0	0
$151$	−727.000	−0.391804	−0.195902	−	0.980623i	$-0.562763\pi$
$151$	−727.000	−0.391804	−0.195902	+	0.980623i	$0.562763\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	1515.00	0.785082
$156$	0	0
$157$	−3260.00	−1.65717	−0.828587	−	0.559860i	$-0.810855\pi$
$157$	−3260.00	−1.65717	−0.828587	+	0.559860i	$0.810855\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	2850.00	1.39510
$162$	0	0
$163$	−2540.00	−1.22054	−0.610270	−	0.792193i	$-0.708939\pi$
$163$	−2540.00	−1.22054	−0.610270	+	0.792193i	$0.708939\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−3498.00	−1.62086	−0.810429	−	0.585837i	$-0.800766\pi$
$167$	−3498.00	−1.62086	−0.810429	+	0.585837i	$0.800766\pi$
$168$	0	0
$169$	−1797.00	−0.817934
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1149.00	−0.504953	−0.252476	−	0.967603i	$-0.581245\pi$
$173$	−1149.00	−0.504953	−0.252476	+	0.967603i	$0.581245\pi$
$174$	0	0
$175$	−2500.00	−1.07990
$176$	0	0
$177$	0	0
$178$	0	0
$179$	315.000	0.131532	0.0657659	−	0.997835i	$-0.479051\pi$
$179$	315.000	0.131532	0.0657659	+	0.997835i	$0.479051\pi$
$180$	0	0
$181$	−1136.00	−0.466509	−0.233255	−	0.972416i	$-0.574938\pi$
$181$	−1136.00	−0.466509	−0.233255	+	0.972416i	$0.574938\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	6450.00	2.56332
$186$	0	0
$187$	1080.00	0.422339
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−2460.00	−0.931934	−0.465967	−	0.884802i	$-0.654293\pi$
$191$	−2460.00	−0.931934	−0.465967	+	0.884802i	$0.654293\pi$
$192$	0	0
$193$	965.000	0.359908	0.179954	−	0.983675i	$-0.442405\pi$
$193$	965.000	0.359908	0.179954	+	0.983675i	$0.442405\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2493.00	0.901619	0.450809	−	0.892620i	$-0.351135\pi$
$197$	2493.00	0.901619	0.450809	+	0.892620i	$0.351135\pi$
$198$	0	0
$199$	−511.000	−0.182029	−0.0910146	−	0.995850i	$-0.529011\pi$
$199$	−511.000	−0.182029	−0.0910146	+	0.995850i	$0.529011\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−750.000	−0.259309
$204$	0	0
$205$	450.000	0.153314
$206$	0	0
$207$	0	0
$208$	0	0
$209$	30.0000	0.00992892
$210$	0	0
$211$	2086.00	0.680598	0.340299	−	0.940317i	$-0.389472\pi$
$211$	2086.00	0.680598	0.340299	+	0.940317i	$0.389472\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−1650.00	−0.523391
$216$	0	0
$217$	−2525.00	−0.789899
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1440.00	0.438303
$222$	0	0
$223$	5240.00	1.57353	0.786763	−	0.617255i	$-0.211755\pi$
$223$	5240.00	1.57353	0.786763	+	0.617255i	$0.211755\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	2388.00	0.698225	0.349113	−	0.937081i	$-0.386483\pi$
$227$	2388.00	0.698225	0.349113	+	0.937081i	$0.386483\pi$
$228$	0	0
$229$	−182.000	−0.0525192	−0.0262596	−	0.999655i	$-0.508360\pi$
$229$	−182.000	−0.0525192	−0.0262596	+	0.999655i	$0.508360\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−450.000	−0.126526	−0.0632628	−	0.997997i	$-0.520151\pi$
$233$	−450.000	−0.126526	−0.0632628	+	0.997997i	$0.520151\pi$
$234$	0	0
$235$	4950.00	1.37405
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−5190.00	−1.40466	−0.702329	−	0.711853i	$-0.747856\pi$
$239$	−5190.00	−1.40466	−0.702329	+	0.711853i	$0.747856\pi$
$240$	0	0
$241$	−2266.00	−0.605668	−0.302834	−	0.953043i	$-0.597933\pi$
$241$	−2266.00	−0.605668	−0.302834	+	0.953043i	$0.597933\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	4230.00	1.10304
$246$	0	0
$247$	40.0000	0.0103042
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−2880.00	−0.724239	−0.362119	−	0.932132i	$-0.617947\pi$
$251$	−2880.00	−0.724239	−0.362119	+	0.932132i	$0.617947\pi$
$252$	0	0
$253$	1710.00	0.424928
$254$	0	0
$255$	0	0
$256$	0	0
$257$	4188.00	1.01650	0.508250	−	0.861210i	$-0.330293\pi$
$257$	4188.00	1.01650	0.508250	+	0.861210i	$0.330293\pi$
$258$	0	0
$259$	−10750.0	−2.57904
$260$	0	0
$261$	0	0
$262$	0	0
$263$	3030.00	0.710410	0.355205	−	0.934788i	$-0.384411\pi$
$263$	3030.00	0.710410	0.355205	+	0.934788i	$0.384411\pi$
$264$	0	0
$265$	9315.00	2.15931
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3510.00	0.795571	0.397785	−	0.917479i	$-0.369779\pi$
$269$	3510.00	0.795571	0.397785	+	0.917479i	$0.369779\pi$
$270$	0	0
$271$	2999.00	0.672237	0.336119	−	0.941820i	$-0.390886\pi$
$271$	2999.00	0.672237	0.336119	+	0.941820i	$0.390886\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1500.00	−0.328921
$276$	0	0
$277$	7720.00	1.67455	0.837274	−	0.546783i	$-0.184148\pi$
$277$	7720.00	1.67455	0.837274	+	0.546783i	$0.184148\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	7440.00	1.57948	0.789739	−	0.613443i	$-0.210216\pi$
$281$	7440.00	1.57948	0.789739	+	0.613443i	$0.210216\pi$
$282$	0	0
$283$	−830.000	−0.174341	−0.0871703	−	0.996193i	$-0.527782\pi$
$283$	−830.000	−0.174341	−0.0871703	+	0.996193i	$0.527782\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−750.000	−0.154255
$288$	0	0
$289$	271.000	0.0551598
$290$	0	0
$291$	0	0
$292$	0	0
$293$	546.000	0.108866	0.0544329	−	0.998517i	$-0.482665\pi$
$293$	546.000	0.108866	0.0544329	+	0.998517i	$0.482665\pi$
$294$	0	0
$295$	−9900.00	−1.95390
$296$	0	0
$297$	0	0
$298$	0	0
$299$	2280.00	0.440989
$300$	0	0
$301$	2750.00	0.526603
$302$	0	0
$303$	0	0
$304$	0	0
$305$	5640.00	1.05884
$306$	0	0
$307$	5560.00	1.03364	0.516818	−	0.856096i	$-0.327117\pi$
$307$	5560.00	1.03364	0.516818	+	0.856096i	$0.327117\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	8670.00	1.58081	0.790403	−	0.612587i	$-0.209871\pi$
$311$	8670.00	1.58081	0.790403	+	0.612587i	$0.209871\pi$
$312$	0	0
$313$	4565.00	0.824374	0.412187	−	0.911099i	$-0.364765\pi$
$313$	4565.00	0.824374	0.412187	+	0.911099i	$0.364765\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	4233.00	0.749997	0.374998	−	0.927025i	$-0.377643\pi$
$317$	4233.00	0.749997	0.374998	+	0.927025i	$0.377643\pi$
$318$	0	0
$319$	−450.000	−0.0789817
$320$	0	0
$321$	0	0
$322$	0	0
$323$	144.000	0.0248061
$324$	0	0
$325$	−2000.00	−0.341354
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−8250.00	−1.38248
$330$	0	0
$331$	−542.000	−0.0900031	−0.0450015	−	0.998987i	$-0.514329\pi$
$331$	−542.000	−0.0900031	−0.0450015	+	0.998987i	$0.514329\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	3750.00	0.611595
$336$	0	0
$337$	5690.00	0.919745	0.459872	−	0.887985i	$-0.347895\pi$
$337$	5690.00	0.919745	0.459872	+	0.887985i	$0.347895\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1515.00	−0.240592
$342$	0	0
$343$	1525.00	0.240065
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5055.00	0.782036	0.391018	−	0.920383i	$-0.372123\pi$
$347$	5055.00	0.782036	0.391018	+	0.920383i	$0.372123\pi$
$348$	0	0
$349$	−1622.00	−0.248778	−0.124389	−	0.992234i	$-0.539697\pi$
$349$	−1622.00	−0.248778	−0.124389	+	0.992234i	$0.539697\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	−30.0000	−0.00452334	−0.00226167	−	0.999997i	$-0.500720\pi$
$353$	−30.0000	−0.00452334	−0.00226167	+	0.999997i	$0.500720\pi$
$354$	0	0
$355$	5400.00	0.807330
$356$	0	0
$357$	0	0
$358$	0	0
$359$	7470.00	1.09819	0.549097	−	0.835759i	$-0.314972\pi$
$359$	7470.00	1.09819	0.549097	+	0.835759i	$0.314972\pi$
$360$	0	0
$361$	−6855.00	−0.999417
$362$	0	0
$363$	0	0
$364$	0	0
$365$	11775.0	1.68858
$366$	0	0
$367$	−1375.00	−0.195571	−0.0977853	−	0.995208i	$-0.531176\pi$
$367$	−1375.00	−0.195571	−0.0977853	+	0.995208i	$0.531176\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−15525.0	−2.17255
$372$	0	0
$373$	4840.00	0.671865	0.335933	−	0.941886i	$-0.390949\pi$
$373$	4840.00	0.671865	0.335933	+	0.941886i	$0.390949\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−600.000	−0.0819670
$378$	0	0
$379$	−1892.00	−0.256426	−0.128213	−	0.991747i	$-0.540924\pi$
$379$	−1892.00	−0.256426	−0.128213	+	0.991747i	$0.540924\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	10704.0	1.42806	0.714032	−	0.700113i	$-0.246867\pi$
$383$	10704.0	1.42806	0.714032	+	0.700113i	$0.246867\pi$
$384$	0	0
$385$	5625.00	0.744614
$386$	0	0
$387$	0	0
$388$	0	0
$389$	7815.00	1.01860	0.509301	−	0.860588i	$-0.329904\pi$
$389$	7815.00	1.01860	0.509301	+	0.860588i	$0.329904\pi$
$390$	0	0
$391$	8208.00	1.06163
$392$	0	0
$393$	0	0
$394$	0	0
$395$	7320.00	0.932428
$396$	0	0
$397$	−4700.00	−0.594172	−0.297086	−	0.954851i	$-0.596015\pi$
$397$	−4700.00	−0.594172	−0.297086	+	0.954851i	$0.596015\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2100.00	0.261519	0.130759	−	0.991414i	$-0.458258\pi$
$401$	2100.00	0.261519	0.130759	+	0.991414i	$0.458258\pi$
$402$	0	0
$403$	−2020.00	−0.249686
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−6450.00	−0.785540
$408$	0	0
$409$	−10753.0	−1.30000	−0.650002	−	0.759933i	$-0.725232\pi$
$409$	−10753.0	−1.30000	−0.650002	+	0.759933i	$0.725232\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	16500.0	1.96589
$414$	0	0
$415$	7335.00	0.867617
$416$	0	0
$417$	0	0
$418$	0	0
$419$	2940.00	0.342789	0.171394	−	0.985203i	$-0.445173\pi$
$419$	2940.00	0.342789	0.171394	+	0.985203i	$0.445173\pi$
$420$	0	0
$421$	−8696.00	−1.00669	−0.503346	−	0.864085i	$-0.667898\pi$
$421$	−8696.00	−1.00669	−0.503346	+	0.864085i	$0.667898\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−7200.00	−0.821768
$426$	0	0
$427$	−9400.00	−1.06533
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−8370.00	−0.935426	−0.467713	−	0.883880i	$-0.654922\pi$
$431$	−8370.00	−0.935426	−0.467713	+	0.883880i	$0.654922\pi$
$432$	0	0
$433$	−5155.00	−0.572133	−0.286066	−	0.958210i	$-0.592348\pi$
$433$	−5155.00	−0.572133	−0.286066	+	0.958210i	$0.592348\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	228.000	0.0249582
$438$	0	0
$439$	−10987.0	−1.19449	−0.597245	−	0.802059i	$-0.703738\pi$
$439$	−10987.0	−1.19449	−0.597245	+	0.802059i	$0.703738\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	1956.00	0.209780	0.104890	−	0.994484i	$-0.466551\pi$
$443$	1956.00	0.209780	0.104890	+	0.994484i	$0.466551\pi$
$444$	0	0
$445$	6750.00	0.719058
$446$	0	0
$447$	0	0
$448$	0	0
$449$	8730.00	0.917582	0.458791	−	0.888544i	$-0.348283\pi$
$449$	8730.00	0.917582	0.458791	+	0.888544i	$0.348283\pi$
$450$	0	0
$451$	−450.000	−0.0469838
$452$	0	0
$453$	0	0
$454$	0	0
$455$	7500.00	0.772759
$456$	0	0
$457$	−8665.00	−0.886940	−0.443470	−	0.896289i	$-0.646253\pi$
$457$	−8665.00	−0.886940	−0.443470	+	0.896289i	$0.646253\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	−9825.00	−0.992616	−0.496308	−	0.868147i	$-0.665311\pi$
$461$	−9825.00	−0.992616	−0.496308	+	0.868147i	$0.665311\pi$
$462$	0	0
$463$	−5245.00	−0.526470	−0.263235	−	0.964732i	$-0.584790\pi$
$463$	−5245.00	−0.526470	−0.263235	+	0.964732i	$0.584790\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−11007.0	−1.09067	−0.545335	−	0.838218i	$-0.683598\pi$
$467$	−11007.0	−1.09067	−0.545335	+	0.838218i	$0.683598\pi$
$468$	0	0
$469$	−6250.00	−0.615348
$470$	0	0
$471$	0	0
$472$	0	0
$473$	1650.00	0.160396
$474$	0	0
$475$	−200.000	−0.0193192
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−16950.0	−1.61684	−0.808419	−	0.588608i	$-0.799676\pi$
$479$	−16950.0	−1.61684	−0.808419	+	0.588608i	$0.799676\pi$
$480$	0	0
$481$	−8600.00	−0.815231
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−16575.0	−1.55182
$486$	0	0
$487$	10640.0	0.990030	0.495015	−	0.868885i	$-0.335163\pi$
$487$	10640.0	0.990030	0.495015	+	0.868885i	$0.335163\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	1635.00	0.150278	0.0751390	−	0.997173i	$-0.476060\pi$
$491$	1635.00	0.150278	0.0751390	+	0.997173i	$0.476060\pi$
$492$	0	0
$493$	−2160.00	−0.197326
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−9000.00	−0.812284
$498$	0	0
$499$	15802.0	1.41762	0.708812	−	0.705397i	$-0.249231\pi$
$499$	15802.0	1.41762	0.708812	+	0.705397i	$0.249231\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	7866.00	0.697272	0.348636	−	0.937258i	$-0.386645\pi$
$503$	7866.00	0.697272	0.348636	+	0.937258i	$0.386645\pi$
$504$	0	0
$505$	21375.0	1.88351
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−11955.0	−1.04105	−0.520527	−	0.853845i	$-0.674264\pi$
$509$	−11955.0	−1.04105	−0.520527	+	0.853845i	$0.674264\pi$
$510$	0	0
$511$	−19625.0	−1.69894
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15900.0	−1.36046
$516$	0	0
$517$	−4950.00	−0.421085
$518$	0	0
$519$	0	0
$520$	0	0
$521$	−19260.0	−1.61957	−0.809785	−	0.586727i	$-0.800416\pi$
$521$	−19260.0	−1.61957	−0.809785	+	0.586727i	$0.800416\pi$
$522$	0	0
$523$	18520.0	1.54842	0.774209	−	0.632930i	$-0.218148\pi$
$523$	18520.0	1.54842	0.774209	+	0.632930i	$0.218148\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−7272.00	−0.601088
$528$	0	0
$529$	829.000	0.0681351
$530$	0	0
$531$	0	0
$532$	0	0
$533$	−600.000	−0.0487596
$534$	0	0
$535$	22275.0	1.80006
$536$	0	0
$537$	0	0
$538$	0	0
$539$	−4230.00	−0.338032
$540$	0	0
$541$	−8372.00	−0.665324	−0.332662	−	0.943046i	$-0.607947\pi$
$541$	−8372.00	−0.665324	−0.332662	+	0.943046i	$0.607947\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	12930.0	1.01626
$546$	0	0
$547$	−17120.0	−1.33821	−0.669103	−	0.743170i	$-0.733321\pi$
$547$	−17120.0	−1.33821	−0.669103	+	0.743170i	$0.733321\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−60.0000	−0.00463899
$552$	0	0
$553$	−12200.0	−0.938150
$554$	0	0
$555$	0	0
$556$	0	0
$557$	10575.0	0.804447	0.402224	−	0.915541i	$-0.368237\pi$
$557$	10575.0	0.804447	0.402224	+	0.915541i	$0.368237\pi$
$558$	0	0
$559$	2200.00	0.166458
$560$	0	0
$561$	0	0
$562$	0	0
$563$	10455.0	0.782639	0.391319	−	0.920255i	$-0.372019\pi$
$563$	10455.0	0.782639	0.391319	+	0.920255i	$0.372019\pi$
$564$	0	0
$565$	−10350.0	−0.770669
$566$	0	0
$567$	0	0
$568$	0	0
$569$	24540.0	1.80803	0.904016	−	0.427498i	$-0.140605\pi$
$569$	24540.0	1.80803	0.904016	+	0.427498i	$0.140605\pi$
$570$	0	0
$571$	−24644.0	−1.80616	−0.903082	−	0.429469i	$-0.858701\pi$
$571$	−24644.0	−1.80616	−0.903082	+	0.429469i	$0.858701\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−11400.0	−0.826805
$576$	0	0
$577$	−9610.00	−0.693361	−0.346681	−	0.937983i	$-0.612691\pi$
$577$	−9610.00	−0.693361	−0.346681	+	0.937983i	$0.612691\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−12225.0	−0.872941
$582$	0	0
$583$	−9315.00	−0.661729
$584$	0	0
$585$	0	0
$586$	0	0
$587$	4017.00	0.282452	0.141226	−	0.989977i	$-0.454896\pi$
$587$	4017.00	0.282452	0.141226	+	0.989977i	$0.454896\pi$
$588$	0	0
$589$	−202.000	−0.0141312
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−594.000	−0.0411343	−0.0205672	−	0.999788i	$-0.506547\pi$
$593$	−594.000	−0.0411343	−0.0205672	+	0.999788i	$0.506547\pi$
$594$	0	0
$595$	27000.0	1.86032
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−8790.00	−0.599582	−0.299791	−	0.954005i	$-0.596917\pi$
$599$	−8790.00	−0.599582	−0.299791	+	0.954005i	$0.596917\pi$
$600$	0	0
$601$	9371.00	0.636025	0.318013	−	0.948087i	$-0.396985\pi$
$601$	9371.00	0.636025	0.318013	+	0.948087i	$0.396985\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−16590.0	−1.11484
$606$	0	0
$607$	−14560.0	−0.973595	−0.486798	−	0.873515i	$-0.661835\pi$
$607$	−14560.0	−0.973595	−0.486798	+	0.873515i	$0.661835\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−6600.00	−0.437001
$612$	0	0
$613$	18250.0	1.20246	0.601232	−	0.799074i	$-0.294677\pi$
$613$	18250.0	1.20246	0.601232	+	0.799074i	$0.294677\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−19662.0	−1.28292	−0.641461	−	0.767156i	$-0.721671\pi$
$617$	−19662.0	−1.28292	−0.641461	+	0.767156i	$0.721671\pi$
$618$	0	0
$619$	−12044.0	−0.782050	−0.391025	−	0.920380i	$-0.627879\pi$
$619$	−12044.0	−0.782050	−0.391025	+	0.920380i	$0.627879\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−11250.0	−0.723470
$624$	0	0
$625$	−18125.0	−1.16000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−30960.0	−1.96257
$630$	0	0
$631$	14879.0	0.938706	0.469353	−	0.883011i	$-0.344487\pi$
$631$	14879.0	0.938706	0.469353	+	0.883011i	$0.344487\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	27975.0	1.74827
$636$	0	0
$637$	−5640.00	−0.350808
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−8850.00	−0.545326	−0.272663	−	0.962110i	$-0.587904\pi$
$641$	−8850.00	−0.545326	−0.272663	+	0.962110i	$0.587904\pi$
$642$	0	0
$643$	−18380.0	−1.12727	−0.563636	−	0.826023i	$-0.690598\pi$
$643$	−18380.0	−1.12727	−0.563636	+	0.826023i	$0.690598\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−3888.00	−0.236249	−0.118124	−	0.992999i	$-0.537688\pi$
$647$	−3888.00	−0.236249	−0.118124	+	0.992999i	$0.537688\pi$
$648$	0	0
$649$	9900.00	0.598781
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−6789.00	−0.406852	−0.203426	−	0.979090i	$-0.565208\pi$
$653$	−6789.00	−0.406852	−0.203426	+	0.979090i	$0.565208\pi$
$654$	0	0
$655$	−17325.0	−1.03350
$656$	0	0
$657$	0	0
$658$	0	0
$659$	28335.0	1.67492	0.837462	−	0.546496i	$-0.184038\pi$
$659$	28335.0	1.67492	0.837462	+	0.546496i	$0.184038\pi$
$660$	0	0
$661$	6082.00	0.357886	0.178943	−	0.983859i	$-0.442732\pi$
$661$	6082.00	0.357886	0.178943	+	0.983859i	$0.442732\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	750.000	0.0437350
$666$	0	0
$667$	−3420.00	−0.198535
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−5640.00	−0.324486
$672$	0	0
$673$	9965.00	0.570762	0.285381	−	0.958414i	$-0.407880\pi$
$673$	9965.00	0.570762	0.285381	+	0.958414i	$0.407880\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	8130.00	0.461538	0.230769	−	0.973009i	$-0.425876\pi$
$677$	8130.00	0.461538	0.230769	+	0.973009i	$0.425876\pi$
$678$	0	0
$679$	27625.0	1.56134
$680$	0	0
$681$	0	0
$682$	0	0
$683$	33516.0	1.87768	0.938839	−	0.344356i	$-0.111903\pi$
$683$	33516.0	1.87768	0.938839	+	0.344356i	$0.111903\pi$
$684$	0	0
$685$	41670.0	2.32428
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−12420.0	−0.686741
$690$	0	0
$691$	22084.0	1.21580	0.607898	−	0.794015i	$-0.292013\pi$
$691$	22084.0	1.21580	0.607898	+	0.794015i	$0.292013\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	28860.0	1.57514
$696$	0	0
$697$	−2160.00	−0.117383
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−10395.0	−0.560077	−0.280038	−	0.959989i	$-0.590347\pi$
$701$	−10395.0	−0.560077	−0.280038	+	0.959989i	$0.590347\pi$
$702$	0	0
$703$	−860.000	−0.0461387
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−35625.0	−1.89507
$708$	0	0
$709$	4804.00	0.254468	0.127234	−	0.991873i	$-0.459390\pi$
$709$	4804.00	0.254468	0.127234	+	0.991873i	$0.459390\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−11514.0	−0.604772
$714$	0	0
$715$	4500.00	0.235371
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−10980.0	−0.569520	−0.284760	−	0.958599i	$-0.591914\pi$
$719$	−10980.0	−0.569520	−0.284760	+	0.958599i	$0.591914\pi$
$720$	0	0
$721$	26500.0	1.36881
$722$	0	0
$723$	0	0
$724$	0	0
$725$	3000.00	0.153679
$726$	0	0
$727$	−25945.0	−1.32359	−0.661793	−	0.749687i	$-0.730204\pi$
$727$	−25945.0	−1.32359	−0.661793	+	0.749687i	$0.730204\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	7920.00	0.400727
$732$	0	0
$733$	−18650.0	−0.939773	−0.469886	−	0.882727i	$-0.655705\pi$
$733$	−18650.0	−0.939773	−0.469886	+	0.882727i	$0.655705\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−3750.00	−0.187426
$738$	0	0
$739$	5128.00	0.255259	0.127630	−	0.991822i	$-0.459263\pi$
$739$	5128.00	0.255259	0.127630	+	0.991822i	$0.459263\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	32700.0	1.61460	0.807299	−	0.590142i	$-0.200928\pi$
$743$	32700.0	1.61460	0.807299	+	0.590142i	$0.200928\pi$
$744$	0	0
$745$	21825.0	1.07330
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−37125.0	−1.81111
$750$	0	0
$751$	21161.0	1.02820	0.514098	−	0.857731i	$-0.328127\pi$
$751$	21161.0	1.02820	0.514098	+	0.857731i	$0.328127\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	−10905.0	−0.525660
$756$	0	0
$757$	−7130.00	−0.342331	−0.171165	−	0.985242i	$-0.554753\pi$
$757$	−7130.00	−0.342331	−0.171165	+	0.985242i	$0.554753\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	3360.00	0.160052	0.0800262	−	0.996793i	$-0.474500\pi$
$761$	3360.00	0.160052	0.0800262	+	0.996793i	$0.474500\pi$
$762$	0	0
$763$	−21550.0	−1.02249
$764$	0	0
$765$	0	0
$766$	0	0
$767$	13200.0	0.621414
$768$	0	0
$769$	33473.0	1.56966	0.784829	−	0.619712i	$-0.212751\pi$
$769$	33473.0	1.56966	0.784829	+	0.619712i	$0.212751\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−3546.00	−0.164995	−0.0824973	−	0.996591i	$-0.526290\pi$
$773$	−3546.00	−0.164995	−0.0824973	+	0.996591i	$0.526290\pi$
$774$	0	0
$775$	10100.0	0.468133
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−60.0000	−0.00275959
$780$	0	0
$781$	−5400.00	−0.247410
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−48900.0	−2.22333
$786$	0	0
$787$	31840.0	1.44215	0.721076	−	0.692856i	$-0.243648\pi$
$787$	31840.0	1.44215	0.721076	+	0.692856i	$0.243648\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	17250.0	0.775397
$792$	0	0
$793$	−7520.00	−0.336750
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−15717.0	−0.698525	−0.349263	−	0.937025i	$-0.613568\pi$
$797$	−15717.0	−0.698525	−0.349263	+	0.937025i	$0.613568\pi$
$798$	0	0
$799$	−23760.0	−1.05203
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−11775.0	−0.517473
$804$	0	0
$805$	42750.0	1.87173
$806$	0	0
$807$	0	0
$808$	0	0
$809$	−10530.0	−0.457621	−0.228810	−	0.973471i	$-0.573484\pi$
$809$	−10530.0	−0.457621	−0.228810	+	0.973471i	$0.573484\pi$
$810$	0	0
$811$	26782.0	1.15961	0.579805	−	0.814755i	$-0.303129\pi$
$811$	26782.0	1.15961	0.579805	+	0.814755i	$0.303129\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−38100.0	−1.63753
$816$	0	0
$817$	220.000	0.00942084
$818$	0	0
$819$	0	0
$820$	0	0
$821$	10110.0	0.429770	0.214885	−	0.976639i	$-0.431062\pi$
$821$	10110.0	0.429770	0.214885	+	0.976639i	$0.431062\pi$
$822$	0	0
$823$	−12535.0	−0.530914	−0.265457	−	0.964123i	$-0.585523\pi$
$823$	−12535.0	−0.530914	−0.265457	+	0.964123i	$0.585523\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	9792.00	0.411731	0.205865	−	0.978580i	$-0.433999\pi$
$827$	9792.00	0.411731	0.205865	+	0.978580i	$0.433999\pi$
$828$	0	0
$829$	4534.00	0.189955	0.0949773	−	0.995479i	$-0.469722\pi$
$829$	4534.00	0.189955	0.0949773	+	0.995479i	$0.469722\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−20304.0	−0.844528
$834$	0	0
$835$	−52470.0	−2.17461
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8880.00	0.365401	0.182701	−	0.983169i	$-0.441516\pi$
$839$	8880.00	0.365401	0.182701	+	0.983169i	$0.441516\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−26955.0	−1.09737
$846$	0	0
$847$	27650.0	1.12168
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−49020.0	−1.97460
$852$	0	0
$853$	−2270.00	−0.0911176	−0.0455588	−	0.998962i	$-0.514507\pi$
$853$	−2270.00	−0.0911176	−0.0455588	+	0.998962i	$0.514507\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	19608.0	0.781560	0.390780	−	0.920484i	$-0.372205\pi$
$857$	19608.0	0.781560	0.390780	+	0.920484i	$0.372205\pi$
$858$	0	0
$859$	952.000	0.0378135	0.0189068	−	0.999821i	$-0.493981\pi$
$859$	952.000	0.0378135	0.0189068	+	0.999821i	$0.493981\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	17604.0	0.694377	0.347188	−	0.937795i	$-0.387136\pi$
$863$	17604.0	0.694377	0.347188	+	0.937795i	$0.387136\pi$
$864$	0	0
$865$	−17235.0	−0.677465
$866$	0	0
$867$	0	0
$868$	0	0
$869$	−7320.00	−0.285747
$870$	0	0
$871$	−5000.00	−0.194510
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9375.00	0.362209
$876$	0	0
$877$	−21890.0	−0.842842	−0.421421	−	0.906865i	$-0.638469\pi$
$877$	−21890.0	−0.842842	−0.421421	+	0.906865i	$0.638469\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−23940.0	−0.915504	−0.457752	−	0.889080i	$-0.651345\pi$
$881$	−23940.0	−0.915504	−0.457752	+	0.889080i	$0.651345\pi$
$882$	0	0
$883$	34990.0	1.33353	0.666765	−	0.745268i	$-0.267678\pi$
$883$	34990.0	1.33353	0.666765	+	0.745268i	$0.267678\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	22188.0	0.839910	0.419955	−	0.907545i	$-0.362046\pi$
$887$	22188.0	0.839910	0.419955	+	0.907545i	$0.362046\pi$
$888$	0	0
$889$	−46625.0	−1.75900
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−660.000	−0.0247324
$894$	0	0
$895$	4725.00	0.176469
$896$	0	0
$897$	0	0
$898$	0	0
$899$	3030.00	0.112410
$900$	0	0
$901$	−44712.0	−1.65324
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−17040.0	−0.625888
$906$	0	0
$907$	−37370.0	−1.36808	−0.684041	−	0.729444i	$-0.739779\pi$
$907$	−37370.0	−1.36808	−0.684041	+	0.729444i	$0.739779\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−40710.0	−1.48055	−0.740276	−	0.672303i	$-0.765305\pi$
$911$	−40710.0	−1.48055	−0.740276	+	0.672303i	$0.765305\pi$
$912$	0	0
$913$	−7335.00	−0.265885
$914$	0	0
$915$	0	0
$916$	0	0
$917$	28875.0	1.03984
$918$	0	0
$919$	20981.0	0.753100	0.376550	−	0.926396i	$-0.377110\pi$
$919$	20981.0	0.753100	0.376550	+	0.926396i	$0.377110\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−7200.00	−0.256762
$924$	0	0
$925$	43000.0	1.52847
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−20100.0	−0.709860	−0.354930	−	0.934893i	$-0.615495\pi$
$929$	−20100.0	−0.709860	−0.354930	+	0.934893i	$0.615495\pi$
$930$	0	0
$931$	−564.000	−0.0198543
$932$	0	0
$933$	0	0
$934$	0	0
$935$	16200.0	0.566627
$936$	0	0
$937$	15635.0	0.545115	0.272558	−	0.962139i	$-0.412130\pi$
$937$	15635.0	0.545115	0.272558	+	0.962139i	$0.412130\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	23955.0	0.829873	0.414937	−	0.909850i	$-0.363804\pi$
$941$	23955.0	0.829873	0.414937	+	0.909850i	$0.363804\pi$
$942$	0	0
$943$	−3420.00	−0.118102
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36393.0	−1.24880	−0.624400	−	0.781105i	$-0.714656\pi$
$947$	−36393.0	−1.24880	−0.624400	+	0.781105i	$0.714656\pi$
$948$	0	0
$949$	−15700.0	−0.537032
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−43020.0	−1.46228	−0.731141	−	0.682227i	$-0.761012\pi$
$953$	−43020.0	−1.46228	−0.731141	+	0.682227i	$0.761012\pi$
$954$	0	0
$955$	−36900.0	−1.25032
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−69450.0	−2.33854
$960$	0	0
$961$	−19590.0	−0.657581
$962$	0	0
$963$	0	0
$964$	0	0
$965$	14475.0	0.482867
$966$	0	0
$967$	−43585.0	−1.44943	−0.724715	−	0.689049i	$-0.758029\pi$
$967$	−43585.0	−1.44943	−0.724715	+	0.689049i	$0.758029\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−43335.0	−1.43222	−0.716110	−	0.697987i	$-0.754079\pi$
$971$	−43335.0	−1.43222	−0.716110	+	0.697987i	$0.754079\pi$
$972$	0	0
$973$	−48100.0	−1.58480
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−30390.0	−0.995151	−0.497575	−	0.867421i	$-0.665776\pi$
$977$	−30390.0	−0.995151	−0.497575	+	0.867421i	$0.665776\pi$
$978$	0	0
$979$	−6750.00	−0.220358
$980$	0	0
$981$	0	0
$982$	0	0
$983$	59226.0	1.92168	0.960842	−	0.277096i	$-0.0893719\pi$
$983$	59226.0	1.92168	0.960842	+	0.277096i	$0.0893719\pi$
$984$	0	0
$985$	37395.0	1.20965
$986$	0	0
$987$	0	0
$988$	0	0
$989$	12540.0	0.403184
$990$	0	0
$991$	8399.00	0.269226	0.134613	−	0.990898i	$-0.457021\pi$
$991$	8399.00	0.269226	0.134613	+	0.990898i	$0.457021\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−7665.00	−0.244218
$996$	0	0
$997$	−13340.0	−0.423753	−0.211877	−	0.977296i	$-0.567958\pi$
$997$	−13340.0	−0.423753	−0.211877	+	0.977296i	$0.567958\pi$
$998$	0	0
$999$	0	0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.4.a.bc.1.1		1
3.2	odd	2		1728.4.a.c.1.1		1
4.3	odd	2		1728.4.a.bd.1.1		1
8.3	odd	2		432.4.a.a.1.1		1
8.5	even	2		27.4.a.a.1.1	✓	1
12.11	even	2		1728.4.a.d.1.1		1
24.5	odd	2		27.4.a.b.1.1	yes	1
24.11	even	2		432.4.a.n.1.1		1
40.13	odd	4		675.4.b.b.649.2		2
40.29	even	2		675.4.a.j.1.1		1
40.37	odd	4		675.4.b.b.649.1		2
56.13	odd	2		1323.4.a.d.1.1		1
72.5	odd	6		81.4.c.a.55.1		2
72.13	even	6		81.4.c.c.55.1		2
72.29	odd	6		81.4.c.a.28.1		2
72.61	even	6		81.4.c.c.28.1		2
120.29	odd	2		675.4.a.a.1.1		1
120.53	even	4		675.4.b.a.649.1		2
120.77	even	4		675.4.b.a.649.2		2
168.125	even	2		1323.4.a.k.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
27.4.a.a.1.1	✓	1	8.5	even	2
27.4.a.b.1.1	yes	1	24.5	odd	2
81.4.c.a.28.1		2	72.29	odd	6
81.4.c.a.55.1		2	72.5	odd	6
81.4.c.c.28.1		2	72.61	even	6
81.4.c.c.55.1		2	72.13	even	6
432.4.a.a.1.1		1	8.3	odd	2
432.4.a.n.1.1		1	24.11	even	2
675.4.a.a.1.1		1	120.29	odd	2
675.4.a.j.1.1		1	40.29	even	2
675.4.b.a.649.1		2	120.53	even	4
675.4.b.a.649.2		2	120.77	even	4
675.4.b.b.649.1		2	40.37	odd	4
675.4.b.b.649.2		2	40.13	odd	4
1323.4.a.d.1.1		1	56.13	odd	2
1323.4.a.k.1.1		1	168.125	even	2
1728.4.a.c.1.1		1	3.2	odd	2
1728.4.a.d.1.1		1	12.11	even	2
1728.4.a.bc.1.1		1	1.1	even	1	trivial
1728.4.a.bd.1.1		1	4.3	odd	2

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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

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Newspace parameters

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This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	1728.4.a.bc.1.1

Level	$1728$
Weight	$4$
Character	1728.1
Self dual	yes
Analytic conductor	$101.955$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$