LMFDB - Embedded newform 1440.4.a.o.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1440,4,Mod(1,1440)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1440, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1440.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1440 = 2^{5} \cdot 3^{2} \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1440.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:	$84.9627504083$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 160)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1440.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q+5.00000 q^{5} +6.00000 q^{7} -60.0000 q^{11} +50.0000 q^{13} +30.0000 q^{17} +40.0000 q^{19} -178.000 q^{23} +25.0000 q^{25} -166.000 q^{29} +20.0000 q^{31} +30.0000 q^{35} +10.0000 q^{37} +250.000 q^{41} +142.000 q^{43} -214.000 q^{47} -307.000 q^{49} -490.000 q^{53} -300.000 q^{55} +800.000 q^{59} +250.000 q^{61} +250.000 q^{65} -774.000 q^{67} -100.000 q^{71} -230.000 q^{73} -360.000 q^{77} -1320.00 q^{79} -982.000 q^{83} +150.000 q^{85} -874.000 q^{89} +300.000 q^{91} +200.000 q^{95} -310.000 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
$5$	5.00000	0.447214
$5$	5.00000	0.447214
$6$	0	0
$7$	6.00000	0.323970	0.161985	−	0.986793i	$-0.448210\pi$
$7$	6.00000	0.323970	0.161985	+	0.986793i	$0.448210\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−60.0000	−1.64461	−0.822304	−	0.569049i	$-0.807311\pi$
$11$	−60.0000	−1.64461	−0.822304	+	0.569049i	$0.807311\pi$
$12$	0	0
$13$	50.0000	1.06673	0.533366	−	0.845885i	$-0.320927\pi$
$13$	50.0000	1.06673	0.533366	+	0.845885i	$0.320927\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	30.0000	0.428004	0.214002	−	0.976833i	$-0.431350\pi$
$17$	30.0000	0.428004	0.214002	+	0.976833i	$0.431350\pi$
$18$	0	0
$19$	40.0000	0.482980	0.241490	−	0.970403i	$-0.422364\pi$
$19$	40.0000	0.482980	0.241490	+	0.970403i	$0.422364\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−178.000	−1.61372	−0.806860	−	0.590743i	$-0.798835\pi$
$23$	−178.000	−1.61372	−0.806860	+	0.590743i	$0.798835\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	−166.000	−1.06295	−0.531473	−	0.847075i	$-0.678361\pi$
$29$	−166.000	−1.06295	−0.531473	+	0.847075i	$0.678361\pi$
$30$	0	0
$31$	20.0000	0.115874	0.0579372	−	0.998320i	$-0.481548\pi$
$31$	20.0000	0.115874	0.0579372	+	0.998320i	$0.481548\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	30.0000	0.144884
$36$	0	0
$37$	10.0000	0.0444322	0.0222161	−	0.999753i	$-0.492928\pi$
$37$	10.0000	0.0444322	0.0222161	+	0.999753i	$0.492928\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	250.000	0.952279	0.476140	−	0.879370i	$-0.342036\pi$
$41$	250.000	0.952279	0.476140	+	0.879370i	$0.342036\pi$
$42$	0	0
$43$	142.000	0.503600	0.251800	−	0.967779i	$-0.418977\pi$
$43$	142.000	0.503600	0.251800	+	0.967779i	$0.418977\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−214.000	−0.664151	−0.332076	−	0.943253i	$-0.607749\pi$
$47$	−214.000	−0.664151	−0.332076	+	0.943253i	$0.607749\pi$
$48$	0	0
$49$	−307.000	−0.895044
$50$	0	0
$51$	0	0
$52$	0	0
$53$	−490.000	−1.26994	−0.634969	−	0.772538i	$-0.718987\pi$
$53$	−490.000	−1.26994	−0.634969	+	0.772538i	$0.718987\pi$
$54$	0	0
$55$	−300.000	−0.735491
$56$	0	0
$57$	0	0
$58$	0	0
$59$	800.000	1.76527	0.882637	−	0.470056i	$-0.155766\pi$
$59$	800.000	1.76527	0.882637	+	0.470056i	$0.155766\pi$
$60$	0	0
$61$	250.000	0.524741	0.262371	−	0.964967i	$-0.415496\pi$
$61$	250.000	0.524741	0.262371	+	0.964967i	$0.415496\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	250.000	0.477057
$66$	0	0
$67$	−774.000	−1.41133	−0.705665	−	0.708545i	$-0.749352\pi$
$67$	−774.000	−1.41133	−0.705665	+	0.708545i	$0.749352\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−100.000	−0.167152	−0.0835762	−	0.996501i	$-0.526634\pi$
$71$	−100.000	−0.167152	−0.0835762	+	0.996501i	$0.526634\pi$
$72$	0	0
$73$	−230.000	−0.368760	−0.184380	−	0.982855i	$-0.559028\pi$
$73$	−230.000	−0.368760	−0.184380	+	0.982855i	$0.559028\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	−360.000	−0.532803
$78$	0	0
$79$	−1320.00	−1.87989	−0.939947	−	0.341321i	$-0.889126\pi$
$79$	−1320.00	−1.87989	−0.939947	+	0.341321i	$0.889126\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	−982.000	−1.29866	−0.649328	−	0.760508i	$-0.724950\pi$
$83$	−982.000	−1.29866	−0.649328	+	0.760508i	$0.724950\pi$
$84$	0	0
$85$	150.000	0.191409
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−874.000	−1.04094	−0.520471	−	0.853879i	$-0.674244\pi$
$89$	−874.000	−1.04094	−0.520471	+	0.853879i	$0.674244\pi$
$90$	0	0
$91$	300.000	0.345588
$92$	0	0
$93$	0	0
$94$	0	0
$95$	200.000	0.215995
$96$	0	0
$97$	−310.000	−0.324492	−0.162246	−	0.986750i	$-0.551874\pi$
$97$	−310.000	−0.324492	−0.162246	+	0.986750i	$0.551874\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	1498.00	1.47581	0.737904	−	0.674906i	$-0.235816\pi$
$101$	1498.00	1.47581	0.737904	+	0.674906i	$0.235816\pi$
$102$	0	0
$103$	1402.00	1.34120	0.670598	−	0.741821i	$-0.266038\pi$
$103$	1402.00	1.34120	0.670598	+	0.741821i	$0.266038\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−1194.00	−1.07877	−0.539385	−	0.842059i	$-0.681343\pi$
$107$	−1194.00	−1.07877	−0.539385	+	0.842059i	$0.681343\pi$
$108$	0	0
$109$	650.000	0.571181	0.285590	−	0.958352i	$-0.407810\pi$
$109$	650.000	0.571181	0.285590	+	0.958352i	$0.407810\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	1510.00	1.25707	0.628535	−	0.777782i	$-0.283655\pi$
$113$	1510.00	1.25707	0.628535	+	0.777782i	$0.283655\pi$
$114$	0	0
$115$	−890.000	−0.721678
$116$	0	0
$117$	0	0
$118$	0	0
$119$	180.000	0.138660
$120$	0	0
$121$	2269.00	1.70473
$122$	0	0
$123$	0	0
$124$	0	0
$125$	125.000	0.0894427
$126$	0	0
$127$	1246.00	0.870588	0.435294	−	0.900288i	$-0.356645\pi$
$127$	1246.00	0.870588	0.435294	+	0.900288i	$0.356645\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−2660.00	−1.77409	−0.887043	−	0.461687i	$-0.847244\pi$
$131$	−2660.00	−1.77409	−0.887043	+	0.461687i	$0.847244\pi$
$132$	0	0
$133$	240.000	0.156471
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−2770.00	−1.72742	−0.863712	−	0.503986i	$-0.831866\pi$
$137$	−2770.00	−1.72742	−0.863712	+	0.503986i	$0.831866\pi$
$138$	0	0
$139$	−560.000	−0.341716	−0.170858	−	0.985296i	$-0.554654\pi$
$139$	−560.000	−0.341716	−0.170858	+	0.985296i	$0.554654\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−3000.00	−1.75435
$144$	0	0
$145$	−830.000	−0.475364
$146$	0	0
$147$	0	0
$148$	0	0
$149$	2350.00	1.29208	0.646039	−	0.763305i	$-0.276424\pi$
$149$	2350.00	1.29208	0.646039	+	0.763305i	$0.276424\pi$
$150$	0	0
$151$	580.000	0.312581	0.156290	−	0.987711i	$-0.450046\pi$
$151$	580.000	0.312581	0.156290	+	0.987711i	$0.450046\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	100.000	0.0518206
$156$	0	0
$157$	−1310.00	−0.665920	−0.332960	−	0.942941i	$-0.608047\pi$
$157$	−1310.00	−0.665920	−0.332960	+	0.942941i	$0.608047\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−1068.00	−0.522796
$162$	0	0
$163$	1862.00	0.894743	0.447371	−	0.894348i	$-0.352360\pi$
$163$	1862.00	0.894743	0.447371	+	0.894348i	$0.352360\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−726.000	−0.336405	−0.168202	−	0.985752i	$-0.553796\pi$
$167$	−726.000	−0.336405	−0.168202	+	0.985752i	$0.553796\pi$
$168$	0	0
$169$	303.000	0.137915
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−3250.00	−1.42828	−0.714141	−	0.700001i	$-0.753183\pi$
$173$	−3250.00	−1.42828	−0.714141	+	0.700001i	$0.753183\pi$
$174$	0	0
$175$	150.000	0.0647939
$176$	0	0
$177$	0	0
$178$	0	0
$179$	1120.00	0.467669	0.233834	−	0.972276i	$-0.424873\pi$
$179$	1120.00	0.467669	0.233834	+	0.972276i	$0.424873\pi$
$180$	0	0
$181$	−2842.00	−1.16710	−0.583548	−	0.812079i	$-0.698336\pi$
$181$	−2842.00	−1.16710	−0.583548	+	0.812079i	$0.698336\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	50.0000	0.0198707
$186$	0	0
$187$	−1800.00	−0.703899
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−3180.00	−1.20469	−0.602347	−	0.798234i	$-0.705768\pi$
$191$	−3180.00	−1.20469	−0.602347	+	0.798234i	$0.705768\pi$
$192$	0	0
$193$	−4670.00	−1.74173	−0.870865	−	0.491522i	$-0.836441\pi$
$193$	−4670.00	−1.74173	−0.870865	+	0.491522i	$0.836441\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	2990.00	1.08136	0.540682	−	0.841227i	$-0.318166\pi$
$197$	2990.00	1.08136	0.540682	+	0.841227i	$0.318166\pi$
$198$	0	0
$199$	−4240.00	−1.51038	−0.755190	−	0.655506i	$-0.772455\pi$
$199$	−4240.00	−1.51038	−0.755190	+	0.655506i	$0.772455\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	−996.000	−0.344362
$204$	0	0
$205$	1250.00	0.425872
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−2400.00	−0.794313
$210$	0	0
$211$	−4060.00	−1.32465	−0.662327	−	0.749215i	$-0.730431\pi$
$211$	−4060.00	−1.32465	−0.662327	+	0.749215i	$0.730431\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	710.000	0.225217
$216$	0	0
$217$	120.000	0.0375398
$218$	0	0
$219$	0	0
$220$	0	0
$221$	1500.00	0.456565
$222$	0	0
$223$	−5622.00	−1.68824	−0.844119	−	0.536156i	$-0.819876\pi$
$223$	−5622.00	−1.68824	−0.844119	+	0.536156i	$0.819876\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−1554.00	−0.454373	−0.227186	−	0.973851i	$-0.572953\pi$
$227$	−1554.00	−0.454373	−0.227186	+	0.973851i	$0.572953\pi$
$228$	0	0
$229$	1134.00	0.327235	0.163618	−	0.986524i	$-0.447684\pi$
$229$	1134.00	0.327235	0.163618	+	0.986524i	$0.447684\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	1710.00	0.480798	0.240399	−	0.970674i	$-0.422722\pi$
$233$	1710.00	0.480798	0.240399	+	0.970674i	$0.422722\pi$
$234$	0	0
$235$	−1070.00	−0.297017
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−4440.00	−1.20167	−0.600836	−	0.799372i	$-0.705166\pi$
$239$	−4440.00	−1.20167	−0.600836	+	0.799372i	$0.705166\pi$
$240$	0	0
$241$	−850.000	−0.227192	−0.113596	−	0.993527i	$-0.536237\pi$
$241$	−850.000	−0.227192	−0.113596	+	0.993527i	$0.536237\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−1535.00	−0.400276
$246$	0	0
$247$	2000.00	0.515210
$248$	0	0
$249$	0	0
$250$	0	0
$251$	660.000	0.165971	0.0829857	−	0.996551i	$-0.473554\pi$
$251$	660.000	0.165971	0.0829857	+	0.996551i	$0.473554\pi$
$252$	0	0
$253$	10680.0	2.65394
$254$	0	0
$255$	0	0
$256$	0	0
$257$	7590.00	1.84222	0.921111	−	0.389299i	$-0.127283\pi$
$257$	7590.00	1.84222	0.921111	+	0.389299i	$0.127283\pi$
$258$	0	0
$259$	60.0000	0.0143947
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−762.000	−0.178658	−0.0893288	−	0.996002i	$-0.528472\pi$
$263$	−762.000	−0.178658	−0.0893288	+	0.996002i	$0.528472\pi$
$264$	0	0
$265$	−2450.00	−0.567933
$266$	0	0
$267$	0	0
$268$	0	0
$269$	150.000	0.0339987	0.0169994	−	0.999856i	$-0.494589\pi$
$269$	150.000	0.0339987	0.0169994	+	0.999856i	$0.494589\pi$
$270$	0	0
$271$	−6580.00	−1.47493	−0.737466	−	0.675384i	$-0.763978\pi$
$271$	−6580.00	−1.47493	−0.737466	+	0.675384i	$0.763978\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−1500.00	−0.328921
$276$	0	0
$277$	4530.00	0.982604	0.491302	−	0.870989i	$-0.336521\pi$
$277$	4530.00	0.982604	0.491302	+	0.870989i	$0.336521\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	−6950.00	−1.47545	−0.737726	−	0.675100i	$-0.764101\pi$
$281$	−6950.00	−1.47545	−0.737726	+	0.675100i	$0.764101\pi$
$282$	0	0
$283$	−3882.00	−0.815410	−0.407705	−	0.913114i	$-0.633671\pi$
$283$	−3882.00	−0.815410	−0.407705	+	0.913114i	$0.633671\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	1500.00	0.308509
$288$	0	0
$289$	−4013.00	−0.816813
$290$	0	0
$291$	0	0
$292$	0	0
$293$	−1370.00	−0.273161	−0.136581	−	0.990629i	$-0.543611\pi$
$293$	−1370.00	−0.273161	−0.136581	+	0.990629i	$0.543611\pi$
$294$	0	0
$295$	4000.00	0.789454
$296$	0	0
$297$	0	0
$298$	0	0
$299$	−8900.00	−1.72141
$300$	0	0
$301$	852.000	0.163151
$302$	0	0
$303$	0	0
$304$	0	0
$305$	1250.00	0.234671
$306$	0	0
$307$	4106.00	0.763328	0.381664	−	0.924301i	$-0.375351\pi$
$307$	4106.00	0.763328	0.381664	+	0.924301i	$0.375351\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−2220.00	−0.404774	−0.202387	−	0.979306i	$-0.564870\pi$
$311$	−2220.00	−0.404774	−0.202387	+	0.979306i	$0.564870\pi$
$312$	0	0
$313$	−9430.00	−1.70292	−0.851462	−	0.524417i	$-0.824283\pi$
$313$	−9430.00	−1.70292	−0.851462	+	0.524417i	$0.824283\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	6470.00	1.14635	0.573173	−	0.819435i	$-0.305712\pi$
$317$	6470.00	1.14635	0.573173	+	0.819435i	$0.305712\pi$
$318$	0	0
$319$	9960.00	1.74813
$320$	0	0
$321$	0	0
$322$	0	0
$323$	1200.00	0.206718
$324$	0	0
$325$	1250.00	0.213346
$326$	0	0
$327$	0	0
$328$	0	0
$329$	−1284.00	−0.215165
$330$	0	0
$331$	900.000	0.149452	0.0747258	−	0.997204i	$-0.476192\pi$
$331$	900.000	0.149452	0.0747258	+	0.997204i	$0.476192\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−3870.00	−0.631166
$336$	0	0
$337$	530.000	0.0856704	0.0428352	−	0.999082i	$-0.486361\pi$
$337$	530.000	0.0856704	0.0428352	+	0.999082i	$0.486361\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	−1200.00	−0.190568
$342$	0	0
$343$	−3900.00	−0.613936
$344$	0	0
$345$	0	0
$346$	0	0
$347$	414.000	0.0640481	0.0320240	−	0.999487i	$-0.489805\pi$
$347$	414.000	0.0640481	0.0320240	+	0.999487i	$0.489805\pi$
$348$	0	0
$349$	8614.00	1.32119	0.660597	−	0.750741i	$-0.270303\pi$
$349$	8614.00	1.32119	0.660597	+	0.750741i	$0.270303\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	2270.00	0.342266	0.171133	−	0.985248i	$-0.445257\pi$
$353$	2270.00	0.342266	0.171133	+	0.985248i	$0.445257\pi$
$354$	0	0
$355$	−500.000	−0.0747528
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−8080.00	−1.18787	−0.593936	−	0.804512i	$-0.702427\pi$
$359$	−8080.00	−1.18787	−0.593936	+	0.804512i	$0.702427\pi$
$360$	0	0
$361$	−5259.00	−0.766730
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−1150.00	−0.164914
$366$	0	0
$367$	2374.00	0.337662	0.168831	−	0.985645i	$-0.446001\pi$
$367$	2374.00	0.337662	0.168831	+	0.985645i	$0.446001\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	−2940.00	−0.411421
$372$	0	0
$373$	1810.00	0.251255	0.125628	−	0.992077i	$-0.459906\pi$
$373$	1810.00	0.251255	0.125628	+	0.992077i	$0.459906\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−8300.00	−1.13388
$378$	0	0
$379$	8120.00	1.10052	0.550259	−	0.834994i	$-0.314529\pi$
$379$	8120.00	1.10052	0.550259	+	0.834994i	$0.314529\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	11782.0	1.57189	0.785943	−	0.618299i	$-0.212178\pi$
$383$	11782.0	1.57189	0.785943	+	0.618299i	$0.212178\pi$
$384$	0	0
$385$	−1800.00	−0.238277
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4350.00	0.566976	0.283488	−	0.958976i	$-0.408508\pi$
$389$	4350.00	0.566976	0.283488	+	0.958976i	$0.408508\pi$
$390$	0	0
$391$	−5340.00	−0.690679
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−6600.00	−0.840714
$396$	0	0
$397$	−7470.00	−0.944354	−0.472177	−	0.881504i	$-0.656532\pi$
$397$	−7470.00	−0.944354	−0.472177	+	0.881504i	$0.656532\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−11698.0	−1.45678	−0.728392	−	0.685161i	$-0.759732\pi$
$401$	−11698.0	−1.45678	−0.728392	+	0.685161i	$0.759732\pi$
$402$	0	0
$403$	1000.00	0.123607
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−600.000	−0.0730735
$408$	0	0
$409$	−3650.00	−0.441274	−0.220637	−	0.975356i	$-0.570814\pi$
$409$	−3650.00	−0.441274	−0.220637	+	0.975356i	$0.570814\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4800.00	0.571895
$414$	0	0
$415$	−4910.00	−0.580777
$416$	0	0
$417$	0	0
$418$	0	0
$419$	1120.00	0.130586	0.0652931	−	0.997866i	$-0.479202\pi$
$419$	1120.00	0.130586	0.0652931	+	0.997866i	$0.479202\pi$
$420$	0	0
$421$	4850.00	0.561460	0.280730	−	0.959787i	$-0.409424\pi$
$421$	4850.00	0.561460	0.280730	+	0.959787i	$0.409424\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	750.000	0.0856008
$426$	0	0
$427$	1500.00	0.170000
$428$	0	0
$429$	0	0
$430$	0	0
$431$	12580.0	1.40593	0.702967	−	0.711223i	$-0.251858\pi$
$431$	12580.0	1.40593	0.702967	+	0.711223i	$0.251858\pi$
$432$	0	0
$433$	13130.0	1.45725	0.728623	−	0.684915i	$-0.240161\pi$
$433$	13130.0	1.45725	0.728623	+	0.684915i	$0.240161\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7120.00	−0.779395
$438$	0	0
$439$	8560.00	0.930630	0.465315	−	0.885145i	$-0.345941\pi$
$439$	8560.00	0.930630	0.465315	+	0.885145i	$0.345941\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	4258.00	0.456667	0.228334	−	0.973583i	$-0.426672\pi$
$443$	4258.00	0.456667	0.228334	+	0.973583i	$0.426672\pi$
$444$	0	0
$445$	−4370.00	−0.465523
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−2550.00	−0.268022	−0.134011	−	0.990980i	$-0.542786\pi$
$449$	−2550.00	−0.268022	−0.134011	+	0.990980i	$0.542786\pi$
$450$	0	0
$451$	−15000.0	−1.56613
$452$	0	0
$453$	0	0
$454$	0	0
$455$	1500.00	0.154552
$456$	0	0
$457$	−6710.00	−0.686828	−0.343414	−	0.939184i	$-0.611583\pi$
$457$	−6710.00	−0.686828	−0.343414	+	0.939184i	$0.611583\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	14482.0	1.46311	0.731555	−	0.681782i	$-0.238795\pi$
$461$	14482.0	1.46311	0.731555	+	0.681782i	$0.238795\pi$
$462$	0	0
$463$	162.000	0.0162609	0.00813043	−	0.999967i	$-0.497412\pi$
$463$	162.000	0.0162609	0.00813043	+	0.999967i	$0.497412\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	15974.0	1.58284	0.791422	−	0.611270i	$-0.209341\pi$
$467$	15974.0	1.58284	0.791422	+	0.611270i	$0.209341\pi$
$468$	0	0
$469$	−4644.00	−0.457228
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−8520.00	−0.828224
$474$	0	0
$475$	1000.00	0.0965961
$476$	0	0
$477$	0	0
$478$	0	0
$479$	10760.0	1.02638	0.513191	−	0.858274i	$-0.328463\pi$
$479$	10760.0	1.02638	0.513191	+	0.858274i	$0.328463\pi$
$480$	0	0
$481$	500.000	0.0473972
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−1550.00	−0.145117
$486$	0	0
$487$	−9266.00	−0.862182	−0.431091	−	0.902309i	$-0.641871\pi$
$487$	−9266.00	−0.862182	−0.431091	+	0.902309i	$0.641871\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	−2860.00	−0.262872	−0.131436	−	0.991325i	$-0.541959\pi$
$491$	−2860.00	−0.262872	−0.131436	+	0.991325i	$0.541959\pi$
$492$	0	0
$493$	−4980.00	−0.454945
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−600.000	−0.0541523
$498$	0	0
$499$	7160.00	0.642336	0.321168	−	0.947022i	$-0.395925\pi$
$499$	7160.00	0.642336	0.321168	+	0.947022i	$0.395925\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	1398.00	0.123924	0.0619620	−	0.998079i	$-0.480264\pi$
$503$	1398.00	0.123924	0.0619620	+	0.998079i	$0.480264\pi$
$504$	0	0
$505$	7490.00	0.660001
$506$	0	0
$507$	0	0
$508$	0	0
$509$	−7446.00	−0.648405	−0.324203	−	0.945988i	$-0.605096\pi$
$509$	−7446.00	−0.648405	−0.324203	+	0.945988i	$0.605096\pi$
$510$	0	0
$511$	−1380.00	−0.119467
$512$	0	0
$513$	0	0
$514$	0	0
$515$	7010.00	0.599801
$516$	0	0
$517$	12840.0	1.09227
$518$	0	0
$519$	0	0
$520$	0	0
$521$	16438.0	1.38227	0.691134	−	0.722726i	$-0.257111\pi$
$521$	16438.0	1.38227	0.691134	+	0.722726i	$0.257111\pi$
$522$	0	0
$523$	−7322.00	−0.612177	−0.306089	−	0.952003i	$-0.599020\pi$
$523$	−7322.00	−0.612177	−0.306089	+	0.952003i	$0.599020\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	600.000	0.0495947
$528$	0	0
$529$	19517.0	1.60409
$530$	0	0
$531$	0	0
$532$	0	0
$533$	12500.0	1.01583
$534$	0	0
$535$	−5970.00	−0.482440
$536$	0	0
$537$	0	0
$538$	0	0
$539$	18420.0	1.47200
$540$	0	0
$541$	10878.0	0.864476	0.432238	−	0.901759i	$-0.357724\pi$
$541$	10878.0	0.864476	0.432238	+	0.901759i	$0.357724\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	3250.00	0.255440
$546$	0	0
$547$	16114.0	1.25957	0.629785	−	0.776769i	$-0.283143\pi$
$547$	16114.0	1.25957	0.629785	+	0.776769i	$0.283143\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−6640.00	−0.513382
$552$	0	0
$553$	−7920.00	−0.609028
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−3690.00	−0.280701	−0.140350	−	0.990102i	$-0.544823\pi$
$557$	−3690.00	−0.280701	−0.140350	+	0.990102i	$0.544823\pi$
$558$	0	0
$559$	7100.00	0.537206
$560$	0	0
$561$	0	0
$562$	0	0
$563$	2562.00	0.191786	0.0958929	−	0.995392i	$-0.469429\pi$
$563$	2562.00	0.191786	0.0958929	+	0.995392i	$0.469429\pi$
$564$	0	0
$565$	7550.00	0.562179
$566$	0	0
$567$	0	0
$568$	0	0
$569$	6050.00	0.445746	0.222873	−	0.974848i	$-0.428457\pi$
$569$	6050.00	0.445746	0.222873	+	0.974848i	$0.428457\pi$
$570$	0	0
$571$	8260.00	0.605377	0.302688	−	0.953090i	$-0.402116\pi$
$571$	8260.00	0.605377	0.302688	+	0.953090i	$0.402116\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−4450.00	−0.322744
$576$	0	0
$577$	−16870.0	−1.21717	−0.608585	−	0.793489i	$-0.708263\pi$
$577$	−16870.0	−1.21717	−0.608585	+	0.793489i	$0.708263\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−5892.00	−0.420725
$582$	0	0
$583$	29400.0	2.08855
$584$	0	0
$585$	0	0
$586$	0	0
$587$	966.000	0.0679235	0.0339617	−	0.999423i	$-0.489188\pi$
$587$	966.000	0.0679235	0.0339617	+	0.999423i	$0.489188\pi$
$588$	0	0
$589$	800.000	0.0559651
$590$	0	0
$591$	0	0
$592$	0	0
$593$	−26290.0	−1.82057	−0.910287	−	0.413977i	$-0.864139\pi$
$593$	−26290.0	−1.82057	−0.910287	+	0.413977i	$0.864139\pi$
$594$	0	0
$595$	900.000	0.0620108
$596$	0	0
$597$	0	0
$598$	0	0
$599$	11640.0	0.793986	0.396993	−	0.917822i	$-0.370054\pi$
$599$	11640.0	0.793986	0.396993	+	0.917822i	$0.370054\pi$
$600$	0	0
$601$	−25450.0	−1.72733	−0.863667	−	0.504064i	$-0.831838\pi$
$601$	−25450.0	−1.72733	−0.863667	+	0.504064i	$0.831838\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	11345.0	0.762380
$606$	0	0
$607$	16694.0	1.11629	0.558145	−	0.829743i	$-0.311513\pi$
$607$	16694.0	1.11629	0.558145	+	0.829743i	$0.311513\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10700.0	−0.708471
$612$	0	0
$613$	15890.0	1.04697	0.523484	−	0.852036i	$-0.324632\pi$
$613$	15890.0	1.04697	0.523484	+	0.852036i	$0.324632\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	1230.00	0.0802560	0.0401280	−	0.999195i	$-0.487223\pi$
$617$	1230.00	0.0802560	0.0401280	+	0.999195i	$0.487223\pi$
$618$	0	0
$619$	−10840.0	−0.703871	−0.351936	−	0.936024i	$-0.614476\pi$
$619$	−10840.0	−0.703871	−0.351936	+	0.936024i	$0.614476\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−5244.00	−0.337233
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	300.000	0.0190171
$630$	0	0
$631$	−14060.0	−0.887036	−0.443518	−	0.896265i	$-0.646270\pi$
$631$	−14060.0	−0.887036	−0.443518	+	0.896265i	$0.646270\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	6230.00	0.389339
$636$	0	0
$637$	−15350.0	−0.954771
$638$	0	0
$639$	0	0
$640$	0	0
$641$	17650.0	1.08757	0.543786	−	0.839224i	$-0.316990\pi$
$641$	17650.0	1.08757	0.543786	+	0.839224i	$0.316990\pi$
$642$	0	0
$643$	27358.0	1.67791	0.838953	−	0.544203i	$-0.183168\pi$
$643$	27358.0	1.67791	0.838953	+	0.544203i	$0.183168\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	6786.00	0.412342	0.206171	−	0.978516i	$-0.433900\pi$
$647$	6786.00	0.412342	0.206171	+	0.978516i	$0.433900\pi$
$648$	0	0
$649$	−48000.0	−2.90318
$650$	0	0
$651$	0	0
$652$	0	0
$653$	9030.00	0.541150	0.270575	−	0.962699i	$-0.412786\pi$
$653$	9030.00	0.541150	0.270575	+	0.962699i	$0.412786\pi$
$654$	0	0
$655$	−13300.0	−0.793395
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15600.0	0.922139	0.461070	−	0.887364i	$-0.347466\pi$
$659$	15600.0	0.922139	0.461070	+	0.887364i	$0.347466\pi$
$660$	0	0
$661$	16850.0	0.991511	0.495756	−	0.868462i	$-0.334891\pi$
$661$	16850.0	0.991511	0.495756	+	0.868462i	$0.334891\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1200.00	0.0699759
$666$	0	0
$667$	29548.0	1.71530
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−15000.0	−0.862993
$672$	0	0
$673$	−7990.00	−0.457640	−0.228820	−	0.973469i	$-0.573487\pi$
$673$	−7990.00	−0.457640	−0.228820	+	0.973469i	$0.573487\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−18690.0	−1.06103	−0.530513	−	0.847677i	$-0.678001\pi$
$677$	−18690.0	−1.06103	−0.530513	+	0.847677i	$0.678001\pi$
$678$	0	0
$679$	−1860.00	−0.105126
$680$	0	0
$681$	0	0
$682$	0	0
$683$	−19182.0	−1.07464	−0.537320	−	0.843379i	$-0.680563\pi$
$683$	−19182.0	−1.07464	−0.537320	+	0.843379i	$0.680563\pi$
$684$	0	0
$685$	−13850.0	−0.772527
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−24500.0	−1.35468
$690$	0	0
$691$	−23380.0	−1.28714	−0.643572	−	0.765385i	$-0.722548\pi$
$691$	−23380.0	−1.28714	−0.643572	+	0.765385i	$0.722548\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−2800.00	−0.152820
$696$	0	0
$697$	7500.00	0.407579
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−11850.0	−0.638471	−0.319236	−	0.947675i	$-0.603426\pi$
$701$	−11850.0	−0.638471	−0.319236	+	0.947675i	$0.603426\pi$
$702$	0	0
$703$	400.000	0.0214599
$704$	0	0
$705$	0	0
$706$	0	0
$707$	8988.00	0.478117
$708$	0	0
$709$	25646.0	1.35847	0.679235	−	0.733921i	$-0.262312\pi$
$709$	25646.0	1.35847	0.679235	+	0.733921i	$0.262312\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−3560.00	−0.186989
$714$	0	0
$715$	−15000.0	−0.784571
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30280.0	1.57059	0.785294	−	0.619122i	$-0.212512\pi$
$719$	30280.0	1.57059	0.785294	+	0.619122i	$0.212512\pi$
$720$	0	0
$721$	8412.00	0.434507
$722$	0	0
$723$	0	0
$724$	0	0
$725$	−4150.00	−0.212589
$726$	0	0
$727$	17446.0	0.890009	0.445004	−	0.895528i	$-0.353202\pi$
$727$	17446.0	0.890009	0.445004	+	0.895528i	$0.353202\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	4260.00	0.215543
$732$	0	0
$733$	−16750.0	−0.844032	−0.422016	−	0.906588i	$-0.638677\pi$
$733$	−16750.0	−0.844032	−0.422016	+	0.906588i	$0.638677\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46440.0	2.32108
$738$	0	0
$739$	−36560.0	−1.81987	−0.909933	−	0.414755i	$-0.863867\pi$
$739$	−36560.0	−1.81987	−0.909933	+	0.414755i	$0.863867\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	30142.0	1.48829	0.744147	−	0.668016i	$-0.232856\pi$
$743$	30142.0	1.48829	0.744147	+	0.668016i	$0.232856\pi$
$744$	0	0
$745$	11750.0	0.577834
$746$	0	0
$747$	0	0
$748$	0	0
$749$	−7164.00	−0.349488
$750$	0	0
$751$	11860.0	0.576268	0.288134	−	0.957590i	$-0.406965\pi$
$751$	11860.0	0.576268	0.288134	+	0.957590i	$0.406965\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	2900.00	0.139790
$756$	0	0
$757$	37010.0	1.77695	0.888475	−	0.458925i	$-0.151765\pi$
$757$	37010.0	1.77695	0.888475	+	0.458925i	$0.151765\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	11718.0	0.558183	0.279091	−	0.960265i	$-0.409967\pi$
$761$	11718.0	0.558183	0.279091	+	0.960265i	$0.409967\pi$
$762$	0	0
$763$	3900.00	0.185045
$764$	0	0
$765$	0	0
$766$	0	0
$767$	40000.0	1.88307
$768$	0	0
$769$	4706.00	0.220680	0.110340	−	0.993894i	$-0.464806\pi$
$769$	4706.00	0.220680	0.110340	+	0.993894i	$0.464806\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	28670.0	1.33401	0.667004	−	0.745054i	$-0.267576\pi$
$773$	28670.0	1.33401	0.667004	+	0.745054i	$0.267576\pi$
$774$	0	0
$775$	500.000	0.0231749
$776$	0	0
$777$	0	0
$778$	0	0
$779$	10000.0	0.459932
$780$	0	0
$781$	6000.00	0.274900
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−6550.00	−0.297808
$786$	0	0
$787$	20434.0	0.925532	0.462766	−	0.886481i	$-0.346857\pi$
$787$	20434.0	0.925532	0.462766	+	0.886481i	$0.346857\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	9060.00	0.407252
$792$	0	0
$793$	12500.0	0.559758
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−3930.00	−0.174665	−0.0873323	−	0.996179i	$-0.527834\pi$
$797$	−3930.00	−0.174665	−0.0873323	+	0.996179i	$0.527834\pi$
$798$	0	0
$799$	−6420.00	−0.284259
$800$	0	0
$801$	0	0
$802$	0	0
$803$	13800.0	0.606465
$804$	0	0
$805$	−5340.00	−0.233802
$806$	0	0
$807$	0	0
$808$	0	0
$809$	4854.00	0.210949	0.105474	−	0.994422i	$-0.466364\pi$
$809$	4854.00	0.210949	0.105474	+	0.994422i	$0.466364\pi$
$810$	0	0
$811$	−13140.0	−0.568937	−0.284468	−	0.958685i	$-0.591817\pi$
$811$	−13140.0	−0.568937	−0.284468	+	0.958685i	$0.591817\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	9310.00	0.400141
$816$	0	0
$817$	5680.00	0.243229
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−22050.0	−0.937333	−0.468666	−	0.883375i	$-0.655265\pi$
$821$	−22050.0	−0.937333	−0.468666	+	0.883375i	$0.655265\pi$
$822$	0	0
$823$	14578.0	0.617445	0.308722	−	0.951152i	$-0.400099\pi$
$823$	14578.0	0.617445	0.308722	+	0.951152i	$0.400099\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	37054.0	1.55803	0.779017	−	0.627003i	$-0.215719\pi$
$827$	37054.0	1.55803	0.779017	+	0.627003i	$0.215719\pi$
$828$	0	0
$829$	−6150.00	−0.257658	−0.128829	−	0.991667i	$-0.541122\pi$
$829$	−6150.00	−0.257658	−0.128829	+	0.991667i	$0.541122\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	−9210.00	−0.383082
$834$	0	0
$835$	−3630.00	−0.150445
$836$	0	0
$837$	0	0
$838$	0	0
$839$	8200.00	0.337420	0.168710	−	0.985666i	$-0.446040\pi$
$839$	8200.00	0.337420	0.168710	+	0.985666i	$0.446040\pi$
$840$	0	0
$841$	3167.00	0.129854
$842$	0	0
$843$	0	0
$844$	0	0
$845$	1515.00	0.0616776
$846$	0	0
$847$	13614.0	0.552282
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−1780.00	−0.0717011
$852$	0	0
$853$	−42990.0	−1.72561	−0.862807	−	0.505533i	$-0.831296\pi$
$853$	−42990.0	−1.72561	−0.862807	+	0.505533i	$0.831296\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−32130.0	−1.28068	−0.640338	−	0.768093i	$-0.721206\pi$
$857$	−32130.0	−1.28068	−0.640338	+	0.768093i	$0.721206\pi$
$858$	0	0
$859$	15440.0	0.613278	0.306639	−	0.951826i	$-0.400796\pi$
$859$	15440.0	0.613278	0.306639	+	0.951826i	$0.400796\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−46938.0	−1.85143	−0.925717	−	0.378216i	$-0.876538\pi$
$863$	−46938.0	−1.85143	−0.925717	+	0.378216i	$0.876538\pi$
$864$	0	0
$865$	−16250.0	−0.638747
$866$	0	0
$867$	0	0
$868$	0	0
$869$	79200.0	3.09169
$870$	0	0
$871$	−38700.0	−1.50551
$872$	0	0
$873$	0	0
$874$	0	0
$875$	750.000	0.0289767
$876$	0	0
$877$	−31230.0	−1.20247	−0.601233	−	0.799074i	$-0.705324\pi$
$877$	−31230.0	−1.20247	−0.601233	+	0.799074i	$0.705324\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−25550.0	−0.977073	−0.488537	−	0.872543i	$-0.662469\pi$
$881$	−25550.0	−0.977073	−0.488537	+	0.872543i	$0.662469\pi$
$882$	0	0
$883$	4318.00	0.164567	0.0822833	−	0.996609i	$-0.473779\pi$
$883$	4318.00	0.164567	0.0822833	+	0.996609i	$0.473779\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−1766.00	−0.0668506	−0.0334253	−	0.999441i	$-0.510642\pi$
$887$	−1766.00	−0.0668506	−0.0334253	+	0.999441i	$0.510642\pi$
$888$	0	0
$889$	7476.00	0.282044
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−8560.00	−0.320772
$894$	0	0
$895$	5600.00	0.209148
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−3320.00	−0.123168
$900$	0	0
$901$	−14700.0	−0.543538
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−14210.0	−0.521941
$906$	0	0
$907$	41906.0	1.53414	0.767071	−	0.641563i	$-0.221714\pi$
$907$	41906.0	1.53414	0.767071	+	0.641563i	$0.221714\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−25140.0	−0.914298	−0.457149	−	0.889390i	$-0.651129\pi$
$911$	−25140.0	−0.914298	−0.457149	+	0.889390i	$0.651129\pi$
$912$	0	0
$913$	58920.0	2.13578
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−15960.0	−0.574750
$918$	0	0
$919$	32920.0	1.18164	0.590822	−	0.806802i	$-0.298804\pi$
$919$	32920.0	1.18164	0.590822	+	0.806802i	$0.298804\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−5000.00	−0.178307
$924$	0	0
$925$	250.000	0.00888643
$926$	0	0
$927$	0	0
$928$	0	0
$929$	−10150.0	−0.358461	−0.179231	−	0.983807i	$-0.557361\pi$
$929$	−10150.0	−0.358461	−0.179231	+	0.983807i	$0.557361\pi$
$930$	0	0
$931$	−12280.0	−0.432289
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−9000.00	−0.314793
$936$	0	0
$937$	28530.0	0.994701	0.497350	−	0.867550i	$-0.334306\pi$
$937$	28530.0	0.994701	0.497350	+	0.867550i	$0.334306\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	−9678.00	−0.335275	−0.167638	−	0.985849i	$-0.553614\pi$
$941$	−9678.00	−0.335275	−0.167638	+	0.985849i	$0.553614\pi$
$942$	0	0
$943$	−44500.0	−1.53671
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−36986.0	−1.26915	−0.634574	−	0.772862i	$-0.718824\pi$
$947$	−36986.0	−1.26915	−0.634574	+	0.772862i	$0.718824\pi$
$948$	0	0
$949$	−11500.0	−0.393368
$950$	0	0
$951$	0	0
$952$	0	0
$953$	3350.00	0.113869	0.0569345	−	0.998378i	$-0.481867\pi$
$953$	3350.00	0.113869	0.0569345	+	0.998378i	$0.481867\pi$
$954$	0	0
$955$	−15900.0	−0.538756
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−16620.0	−0.559633
$960$	0	0
$961$	−29391.0	−0.986573
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−23350.0	−0.778925
$966$	0	0
$967$	43774.0	1.45572	0.727858	−	0.685728i	$-0.240516\pi$
$967$	43774.0	1.45572	0.727858	+	0.685728i	$0.240516\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−8740.00	−0.288857	−0.144428	−	0.989515i	$-0.546134\pi$
$971$	−8740.00	−0.288857	−0.144428	+	0.989515i	$0.546134\pi$
$972$	0	0
$973$	−3360.00	−0.110706
$974$	0	0
$975$	0	0
$976$	0	0
$977$	48310.0	1.58196	0.790979	−	0.611843i	$-0.209571\pi$
$977$	48310.0	1.58196	0.790979	+	0.611843i	$0.209571\pi$
$978$	0	0
$979$	52440.0	1.71194
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−2282.00	−0.0740432	−0.0370216	−	0.999314i	$-0.511787\pi$
$983$	−2282.00	−0.0740432	−0.0370216	+	0.999314i	$0.511787\pi$
$984$	0	0
$985$	14950.0	0.483601
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−25276.0	−0.812669
$990$	0	0
$991$	−31580.0	−1.01228	−0.506141	−	0.862451i	$-0.668929\pi$
$991$	−31580.0	−1.01228	−0.506141	+	0.862451i	$0.668929\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−21200.0	−0.675462
$996$	0	0
$997$	−2790.00	−0.0886261	−0.0443130	−	0.999018i	$-0.514110\pi$
$997$	−2790.00	−0.0886261	−0.0443130	+	0.999018i	$0.514110\pi$
$998$	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	1440.4.a.o.1.1		1
3.2	odd	2		160.4.a.a.1.1	✓	1
4.3	odd	2		1440.4.a.n.1.1		1
12.11	even	2		160.4.a.b.1.1	yes	1
15.2	even	4		800.4.c.f.449.2		2
15.8	even	4		800.4.c.f.449.1		2
15.14	odd	2		800.4.a.h.1.1		1
24.5	odd	2		320.4.a.i.1.1		1
24.11	even	2		320.4.a.f.1.1		1
48.5	odd	4		1280.4.d.f.641.2		2
48.11	even	4		1280.4.d.k.641.1		2
48.29	odd	4		1280.4.d.f.641.1		2
48.35	even	4		1280.4.d.k.641.2		2
60.23	odd	4		800.4.c.e.449.2		2
60.47	odd	4		800.4.c.e.449.1		2
60.59	even	2		800.4.a.d.1.1		1
120.29	odd	2		1600.4.a.r.1.1		1
120.59	even	2		1600.4.a.bj.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
160.4.a.a.1.1	✓	1	3.2	odd	2
160.4.a.b.1.1	yes	1	12.11	even	2
320.4.a.f.1.1		1	24.11	even	2
320.4.a.i.1.1		1	24.5	odd	2
800.4.a.d.1.1		1	60.59	even	2
800.4.a.h.1.1		1	15.14	odd	2
800.4.c.e.449.1		2	60.47	odd	4
800.4.c.e.449.2		2	60.23	odd	4
800.4.c.f.449.1		2	15.8	even	4
800.4.c.f.449.2		2	15.2	even	4
1280.4.d.f.641.1		2	48.29	odd	4
1280.4.d.f.641.2		2	48.5	odd	4
1280.4.d.k.641.1		2	48.11	even	4
1280.4.d.k.641.2		2	48.35	even	4
1440.4.a.n.1.1		1	4.3	odd	2
1440.4.a.o.1.1		1	1.1	even	1	trivial
1600.4.a.r.1.1		1	120.29	odd	2
1600.4.a.bj.1.1		1	120.59	even	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

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	1440.4.a.o.1.1

Level	$1440$
Weight	$4$
Character	1440.1
Self dual	yes
Analytic conductor	$84.963$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$