LMFDB - Embedded newform 1600.4.a.s.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1600,4,Mod(1,1600)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1600, 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("1600.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1600 = 2^{6} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1600.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:	$94.4030560092$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 5)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1600.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-2.00000 q^{3} +6.00000 q^{7} -23.0000 q^{9} +32.0000 q^{11} -38.0000 q^{13} -26.0000 q^{17} +100.000 q^{19} -12.0000 q^{21} -78.0000 q^{23} +100.000 q^{27} +50.0000 q^{29} +108.000 q^{31} -64.0000 q^{33} +266.000 q^{37} +76.0000 q^{39} +22.0000 q^{41} -442.000 q^{43} -514.000 q^{47} -307.000 q^{49} +52.0000 q^{51} +2.00000 q^{53} -200.000 q^{57} +500.000 q^{59} +518.000 q^{61} -138.000 q^{63} -126.000 q^{67} +156.000 q^{69} -412.000 q^{71} +878.000 q^{73} +192.000 q^{77} -600.000 q^{79} +421.000 q^{81} -282.000 q^{83} -100.000 q^{87} -150.000 q^{89} -228.000 q^{91} -216.000 q^{93} -386.000 q^{97} -736.000 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
$3$	−2.00000	−0.384900	−0.192450	−	0.981307i	$-0.561643\pi$
$3$	−2.00000	−0.384900	−0.192450	+	0.981307i	$0.561643\pi$
$4$	0	0
$5$	0	0
$5$	0	0
$6$	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
$9$	−23.0000	−0.851852
$10$	0	0
$11$	32.0000	0.877124	0.438562	−	0.898701i	$-0.355488\pi$
$11$	32.0000	0.877124	0.438562	+	0.898701i	$0.355488\pi$
$12$	0	0
$13$	−38.0000	−0.810716	−0.405358	−	0.914158i	$-0.632853\pi$
$13$	−38.0000	−0.810716	−0.405358	+	0.914158i	$0.632853\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−26.0000	−0.370937	−0.185468	−	0.982650i	$-0.559380\pi$
$17$	−26.0000	−0.370937	−0.185468	+	0.982650i	$0.559380\pi$
$18$	0	0
$19$	100.000	1.20745	0.603726	−	0.797192i	$-0.293682\pi$
$19$	100.000	1.20745	0.603726	+	0.797192i	$0.293682\pi$
$20$	0	0
$21$	−12.0000	−0.124696
$22$	0	0
$23$	−78.0000	−0.707136	−0.353568	−	0.935409i	$-0.615032\pi$
$23$	−78.0000	−0.707136	−0.353568	+	0.935409i	$0.615032\pi$
$24$	0	0
$25$	0	0
$26$	0	0
$27$	100.000	0.712778
$28$	0	0
$29$	50.0000	0.320164	0.160082	−	0.987104i	$-0.448824\pi$
$29$	50.0000	0.320164	0.160082	+	0.987104i	$0.448824\pi$
$30$	0	0
$31$	108.000	0.625722	0.312861	−	0.949799i	$-0.398713\pi$
$31$	108.000	0.625722	0.312861	+	0.949799i	$0.398713\pi$
$32$	0	0
$33$	−64.0000	−0.337605
$34$	0	0
$35$	0	0
$36$	0	0
$37$	266.000	1.18190	0.590948	−	0.806710i	$-0.298754\pi$
$37$	266.000	1.18190	0.590948	+	0.806710i	$0.298754\pi$
$38$	0	0
$39$	76.0000	0.312045
$40$	0	0
$41$	22.0000	0.0838006	0.0419003	−	0.999122i	$-0.486659\pi$
$41$	22.0000	0.0838006	0.0419003	+	0.999122i	$0.486659\pi$
$42$	0	0
$43$	−442.000	−1.56754	−0.783772	−	0.621049i	$-0.786707\pi$
$43$	−442.000	−1.56754	−0.783772	+	0.621049i	$0.786707\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−514.000	−1.59520	−0.797602	−	0.603184i	$-0.793899\pi$
$47$	−514.000	−1.59520	−0.797602	+	0.603184i	$0.793899\pi$
$48$	0	0
$49$	−307.000	−0.895044
$50$	0	0
$51$	52.0000	0.142774
$52$	0	0
$53$	2.00000	0.00518342	0.00259171	−	0.999997i	$-0.499175\pi$
$53$	2.00000	0.00518342	0.00259171	+	0.999997i	$0.499175\pi$
$54$	0	0
$55$	0	0
$56$	0	0
$57$	−200.000	−0.464748
$58$	0	0
$59$	500.000	1.10330	0.551648	−	0.834077i	$-0.313999\pi$
$59$	500.000	1.10330	0.551648	+	0.834077i	$0.313999\pi$
$60$	0	0
$61$	518.000	1.08726	0.543632	−	0.839324i	$-0.317049\pi$
$61$	518.000	1.08726	0.543632	+	0.839324i	$0.317049\pi$
$62$	0	0
$63$	−138.000	−0.275974
$64$	0	0
$65$	0	0
$66$	0	0
$67$	−126.000	−0.229751	−0.114876	−	0.993380i	$-0.536647\pi$
$67$	−126.000	−0.229751	−0.114876	+	0.993380i	$0.536647\pi$
$68$	0	0
$69$	156.000	0.272177
$70$	0	0
$71$	−412.000	−0.688668	−0.344334	−	0.938847i	$-0.611895\pi$
$71$	−412.000	−0.688668	−0.344334	+	0.938847i	$0.611895\pi$
$72$	0	0
$73$	878.000	1.40770	0.703850	−	0.710348i	$-0.251463\pi$
$73$	878.000	1.40770	0.703850	+	0.710348i	$0.251463\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	192.000	0.284161
$78$	0	0
$79$	−600.000	−0.854497	−0.427249	−	0.904134i	$-0.640517\pi$
$79$	−600.000	−0.854497	−0.427249	+	0.904134i	$0.640517\pi$
$80$	0	0
$81$	421.000	0.577503
$82$	0	0
$83$	−282.000	−0.372934	−0.186467	−	0.982461i	$-0.559704\pi$
$83$	−282.000	−0.372934	−0.186467	+	0.982461i	$0.559704\pi$
$84$	0	0
$85$	0	0
$86$	0	0
$87$	−100.000	−0.123231
$88$	0	0
$89$	−150.000	−0.178651	−0.0893257	−	0.996002i	$-0.528471\pi$
$89$	−150.000	−0.178651	−0.0893257	+	0.996002i	$0.528471\pi$
$90$	0	0
$91$	−228.000	−0.262647
$92$	0	0
$93$	−216.000	−0.240840
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−386.000	−0.404045	−0.202022	−	0.979381i	$-0.564751\pi$
$97$	−386.000	−0.404045	−0.202022	+	0.979381i	$0.564751\pi$
$98$	0	0
$99$	−736.000	−0.747180
$100$	0	0
$101$	−702.000	−0.691600	−0.345800	−	0.938308i	$-0.612392\pi$
$101$	−702.000	−0.691600	−0.345800	+	0.938308i	$0.612392\pi$
$102$	0	0
$103$	−598.000	−0.572065	−0.286032	−	0.958220i	$-0.592337\pi$
$103$	−598.000	−0.572065	−0.286032	+	0.958220i	$0.592337\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	1194.00	1.07877	0.539385	−	0.842059i	$-0.318657\pi$
$107$	1194.00	1.07877	0.539385	+	0.842059i	$0.318657\pi$
$108$	0	0
$109$	550.000	0.483307	0.241653	−	0.970363i	$-0.422310\pi$
$109$	550.000	0.483307	0.241653	+	0.970363i	$0.422310\pi$
$110$	0	0
$111$	−532.000	−0.454912
$112$	0	0
$113$	−1562.00	−1.30036	−0.650180	−	0.759781i	$-0.725306\pi$
$113$	−1562.00	−1.30036	−0.650180	+	0.759781i	$0.725306\pi$
$114$	0	0
$115$	0	0
$116$	0	0
$117$	874.000	0.690610
$118$	0	0
$119$	−156.000	−0.120172
$120$	0	0
$121$	−307.000	−0.230654
$122$	0	0
$123$	−44.0000	−0.0322548
$124$	0	0
$125$	0	0
$126$	0	0
$127$	1846.00	1.28981	0.644906	−	0.764262i	$-0.276897\pi$
$127$	1846.00	1.28981	0.644906	+	0.764262i	$0.276897\pi$
$128$	0	0
$129$	884.000	0.603348
$130$	0	0
$131$	−2208.00	−1.47262	−0.736312	−	0.676642i	$-0.763435\pi$
$131$	−2208.00	−1.47262	−0.736312	+	0.676642i	$0.763435\pi$
$132$	0	0
$133$	600.000	0.391177
$134$	0	0
$135$	0	0
$136$	0	0
$137$	2334.00	1.45553	0.727763	−	0.685829i	$-0.240560\pi$
$137$	2334.00	1.45553	0.727763	+	0.685829i	$0.240560\pi$
$138$	0	0
$139$	−700.000	−0.427146	−0.213573	−	0.976927i	$-0.568510\pi$
$139$	−700.000	−0.427146	−0.213573	+	0.976927i	$0.568510\pi$
$140$	0	0
$141$	1028.00	0.613994
$142$	0	0
$143$	−1216.00	−0.711098
$144$	0	0
$145$	0	0
$146$	0	0
$147$	614.000	0.344502
$148$	0	0
$149$	−2050.00	−1.12713	−0.563566	−	0.826071i	$-0.690571\pi$
$149$	−2050.00	−1.12713	−0.563566	+	0.826071i	$0.690571\pi$
$150$	0	0
$151$	−1852.00	−0.998103	−0.499052	−	0.866572i	$-0.666318\pi$
$151$	−1852.00	−0.998103	−0.499052	+	0.866572i	$0.666318\pi$
$152$	0	0
$153$	598.000	0.315983
$154$	0	0
$155$	0	0
$156$	0	0
$157$	−2494.00	−1.26779	−0.633894	−	0.773420i	$-0.718545\pi$
$157$	−2494.00	−1.26779	−0.633894	+	0.773420i	$0.718545\pi$
$158$	0	0
$159$	−4.00000	−0.00199510
$160$	0	0
$161$	−468.000	−0.229090
$162$	0	0
$163$	−2762.00	−1.32722	−0.663609	−	0.748080i	$-0.730976\pi$
$163$	−2762.00	−1.32722	−0.663609	+	0.748080i	$0.730976\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	3126.00	1.44849	0.724243	−	0.689545i	$-0.242189\pi$
$167$	3126.00	1.44849	0.724243	+	0.689545i	$0.242189\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	0	0
$171$	−2300.00	−1.02857
$172$	0	0
$173$	−78.0000	−0.0342788	−0.0171394	−	0.999853i	$-0.505456\pi$
$173$	−78.0000	−0.0342788	−0.0171394	+	0.999853i	$0.505456\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−1000.00	−0.424659
$178$	0	0
$179$	−1300.00	−0.542830	−0.271415	−	0.962462i	$-0.587492\pi$
$179$	−1300.00	−0.542830	−0.271415	+	0.962462i	$0.587492\pi$
$180$	0	0
$181$	−1742.00	−0.715369	−0.357685	−	0.933842i	$-0.616434\pi$
$181$	−1742.00	−0.715369	−0.357685	+	0.933842i	$0.616434\pi$
$182$	0	0
$183$	−1036.00	−0.418488
$184$	0	0
$185$	0	0
$186$	0	0
$187$	−832.000	−0.325358
$188$	0	0
$189$	600.000	0.230918
$190$	0	0
$191$	−3772.00	−1.42897	−0.714483	−	0.699653i	$-0.753338\pi$
$191$	−3772.00	−1.42897	−0.714483	+	0.699653i	$0.753338\pi$
$192$	0	0
$193$	358.000	0.133520	0.0667601	−	0.997769i	$-0.478734\pi$
$193$	358.000	0.133520	0.0667601	+	0.997769i	$0.478734\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−2214.00	−0.800716	−0.400358	−	0.916359i	$-0.631114\pi$
$197$	−2214.00	−0.800716	−0.400358	+	0.916359i	$0.631114\pi$
$198$	0	0
$199$	2600.00	0.926176	0.463088	−	0.886312i	$-0.346741\pi$
$199$	2600.00	0.926176	0.463088	+	0.886312i	$0.346741\pi$
$200$	0	0
$201$	252.000	0.0884314
$202$	0	0
$203$	300.000	0.103724
$204$	0	0
$205$	0	0
$206$	0	0
$207$	1794.00	0.602375
$208$	0	0
$209$	3200.00	1.05908
$210$	0	0
$211$	−1168.00	−0.381083	−0.190541	−	0.981679i	$-0.561024\pi$
$211$	−1168.00	−0.381083	−0.190541	+	0.981679i	$0.561024\pi$
$212$	0	0
$213$	824.000	0.265068
$214$	0	0
$215$	0	0
$216$	0	0
$217$	648.000	0.202715
$218$	0	0
$219$	−1756.00	−0.541824
$220$	0	0
$221$	988.000	0.300724
$222$	0	0
$223$	−6478.00	−1.94529	−0.972643	−	0.232303i	$-0.925374\pi$
$223$	−6478.00	−1.94529	−0.972643	+	0.232303i	$0.925374\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−646.000	−0.188883	−0.0944417	−	0.995530i	$-0.530107\pi$
$227$	−646.000	−0.188883	−0.0944417	+	0.995530i	$0.530107\pi$
$228$	0	0
$229$	−3750.00	−1.08213	−0.541063	−	0.840982i	$-0.681978\pi$
$229$	−3750.00	−1.08213	−0.541063	+	0.840982i	$0.681978\pi$
$230$	0	0
$231$	−384.000	−0.109374
$232$	0	0
$233$	−1482.00	−0.416691	−0.208346	−	0.978055i	$-0.566808\pi$
$233$	−1482.00	−0.416691	−0.208346	+	0.978055i	$0.566808\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	1200.00	0.328896
$238$	0	0
$239$	−1400.00	−0.378906	−0.189453	−	0.981890i	$-0.560671\pi$
$239$	−1400.00	−0.378906	−0.189453	+	0.981890i	$0.560671\pi$
$240$	0	0
$241$	3022.00	0.807735	0.403867	−	0.914817i	$-0.367666\pi$
$241$	3022.00	0.807735	0.403867	+	0.914817i	$0.367666\pi$
$242$	0	0
$243$	−3542.00	−0.935059
$244$	0	0
$245$	0	0
$246$	0	0
$247$	−3800.00	−0.978900
$248$	0	0
$249$	564.000	0.143542
$250$	0	0
$251$	−1248.00	−0.313837	−0.156918	−	0.987612i	$-0.550156\pi$
$251$	−1248.00	−0.313837	−0.156918	+	0.987612i	$0.550156\pi$
$252$	0	0
$253$	−2496.00	−0.620246
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−2106.00	−0.511162	−0.255581	−	0.966788i	$-0.582267\pi$
$257$	−2106.00	−0.511162	−0.255581	+	0.966788i	$0.582267\pi$
$258$	0	0
$259$	1596.00	0.382898
$260$	0	0
$261$	−1150.00	−0.272733
$262$	0	0
$263$	−3638.00	−0.852961	−0.426480	−	0.904497i	$-0.640247\pi$
$263$	−3638.00	−0.852961	−0.426480	+	0.904497i	$0.640247\pi$
$264$	0	0
$265$	0	0
$266$	0	0
$267$	300.000	0.0687629
$268$	0	0
$269$	6550.00	1.48461	0.742306	−	0.670061i	$-0.233732\pi$
$269$	6550.00	1.48461	0.742306	+	0.670061i	$0.233732\pi$
$270$	0	0
$271$	4388.00	0.983587	0.491793	−	0.870712i	$-0.336342\pi$
$271$	4388.00	0.983587	0.491793	+	0.870712i	$0.336342\pi$
$272$	0	0
$273$	456.000	0.101093
$274$	0	0
$275$	0	0
$276$	0	0
$277$	546.000	0.118433	0.0592165	−	0.998245i	$-0.481140\pi$
$277$	546.000	0.118433	0.0592165	+	0.998245i	$0.481140\pi$
$278$	0	0
$279$	−2484.00	−0.533022
$280$	0	0
$281$	−6858.00	−1.45592	−0.727961	−	0.685619i	$-0.759532\pi$
$281$	−6858.00	−1.45592	−0.727961	+	0.685619i	$0.759532\pi$
$282$	0	0
$283$	−9282.00	−1.94967	−0.974837	−	0.222920i	$-0.928441\pi$
$283$	−9282.00	−1.94967	−0.974837	+	0.222920i	$0.928441\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	132.000	0.0271488
$288$	0	0
$289$	−4237.00	−0.862406
$290$	0	0
$291$	772.000	0.155517
$292$	0	0
$293$	4842.00	0.965436	0.482718	−	0.875776i	$-0.339650\pi$
$293$	4842.00	0.965436	0.482718	+	0.875776i	$0.339650\pi$
$294$	0	0
$295$	0	0
$296$	0	0
$297$	3200.00	0.625195
$298$	0	0
$299$	2964.00	0.573286
$300$	0	0
$301$	−2652.00	−0.507836
$302$	0	0
$303$	1404.00	0.266197
$304$	0	0
$305$	0	0
$306$	0	0
$307$	2594.00	0.482239	0.241120	−	0.970495i	$-0.422485\pi$
$307$	2594.00	0.482239	0.241120	+	0.970495i	$0.422485\pi$
$308$	0	0
$309$	1196.00	0.220188
$310$	0	0
$311$	−7332.00	−1.33685	−0.668424	−	0.743781i	$-0.733031\pi$
$311$	−7332.00	−1.33685	−0.668424	+	0.743781i	$0.733031\pi$
$312$	0	0
$313$	−1562.00	−0.282075	−0.141037	−	0.990004i	$-0.545044\pi$
$313$	−1562.00	−0.282075	−0.141037	+	0.990004i	$0.545044\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	1426.00	0.252657	0.126328	−	0.991988i	$-0.459681\pi$
$317$	1426.00	0.252657	0.126328	+	0.991988i	$0.459681\pi$
$318$	0	0
$319$	1600.00	0.280824
$320$	0	0
$321$	−2388.00	−0.415219
$322$	0	0
$323$	−2600.00	−0.447888
$324$	0	0
$325$	0	0
$326$	0	0
$327$	−1100.00	−0.186025
$328$	0	0
$329$	−3084.00	−0.516798
$330$	0	0
$331$	−4008.00	−0.665558	−0.332779	−	0.943005i	$-0.607986\pi$
$331$	−4008.00	−0.665558	−0.332779	+	0.943005i	$0.607986\pi$
$332$	0	0
$333$	−6118.00	−1.00680
$334$	0	0
$335$	0	0
$336$	0	0
$337$	−8866.00	−1.43312	−0.716561	−	0.697525i	$-0.754285\pi$
$337$	−8866.00	−1.43312	−0.716561	+	0.697525i	$0.754285\pi$
$338$	0	0
$339$	3124.00	0.500509
$340$	0	0
$341$	3456.00	0.548835
$342$	0	0
$343$	−3900.00	−0.613936
$344$	0	0
$345$	0	0
$346$	0	0
$347$	1714.00	0.265165	0.132583	−	0.991172i	$-0.457673\pi$
$347$	1714.00	0.265165	0.132583	+	0.991172i	$0.457673\pi$
$348$	0	0
$349$	−1150.00	−0.176384	−0.0881921	−	0.996103i	$-0.528109\pi$
$349$	−1150.00	−0.176384	−0.0881921	+	0.996103i	$0.528109\pi$
$350$	0	0
$351$	−3800.00	−0.577860
$352$	0	0
$353$	4398.00	0.663122	0.331561	−	0.943434i	$-0.392425\pi$
$353$	4398.00	0.663122	0.331561	+	0.943434i	$0.392425\pi$
$354$	0	0
$355$	0	0
$356$	0	0
$357$	312.000	0.0462543
$358$	0	0
$359$	−1800.00	−0.264625	−0.132312	−	0.991208i	$-0.542240\pi$
$359$	−1800.00	−0.264625	−0.132312	+	0.991208i	$0.542240\pi$
$360$	0	0
$361$	3141.00	0.457938
$362$	0	0
$363$	614.000	0.0887786
$364$	0	0
$365$	0	0
$366$	0	0
$367$	−5874.00	−0.835478	−0.417739	−	0.908567i	$-0.637177\pi$
$367$	−5874.00	−0.835478	−0.417739	+	0.908567i	$0.637177\pi$
$368$	0	0
$369$	−506.000	−0.0713857
$370$	0	0
$371$	12.0000	0.00167927
$372$	0	0
$373$	−2078.00	−0.288458	−0.144229	−	0.989544i	$-0.546070\pi$
$373$	−2078.00	−0.288458	−0.144229	+	0.989544i	$0.546070\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−1900.00	−0.259562
$378$	0	0
$379$	7900.00	1.07070	0.535351	−	0.844630i	$-0.320179\pi$
$379$	7900.00	1.07070	0.535351	+	0.844630i	$0.320179\pi$
$380$	0	0
$381$	−3692.00	−0.496449
$382$	0	0
$383$	−7518.00	−1.00301	−0.501504	−	0.865155i	$-0.667220\pi$
$383$	−7518.00	−1.00301	−0.501504	+	0.865155i	$0.667220\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	10166.0	1.33531
$388$	0	0
$389$	1950.00	0.254162	0.127081	−	0.991892i	$-0.459439\pi$
$389$	1950.00	0.254162	0.127081	+	0.991892i	$0.459439\pi$
$390$	0	0
$391$	2028.00	0.262303
$392$	0	0
$393$	4416.00	0.566814
$394$	0	0
$395$	0	0
$396$	0	0
$397$	13786.0	1.74282	0.871410	−	0.490555i	$-0.163206\pi$
$397$	13786.0	1.74282	0.871410	+	0.490555i	$0.163206\pi$
$398$	0	0
$399$	−1200.00	−0.150564
$400$	0	0
$401$	6402.00	0.797258	0.398629	−	0.917112i	$-0.369486\pi$
$401$	6402.00	0.797258	0.398629	+	0.917112i	$0.369486\pi$
$402$	0	0
$403$	−4104.00	−0.507282
$404$	0	0
$405$	0	0
$406$	0	0
$407$	8512.00	1.03667
$408$	0	0
$409$	11150.0	1.34800	0.674000	−	0.738731i	$-0.264575\pi$
$409$	11150.0	1.34800	0.674000	+	0.738731i	$0.264575\pi$
$410$	0	0
$411$	−4668.00	−0.560232
$412$	0	0
$413$	3000.00	0.357434
$414$	0	0
$415$	0	0
$416$	0	0
$417$	1400.00	0.164408
$418$	0	0
$419$	−13700.0	−1.59735	−0.798674	−	0.601764i	$-0.794465\pi$
$419$	−13700.0	−1.59735	−0.798674	+	0.601764i	$0.794465\pi$
$420$	0	0
$421$	5438.00	0.629529	0.314765	−	0.949170i	$-0.398074\pi$
$421$	5438.00	0.629529	0.314765	+	0.949170i	$0.398074\pi$
$422$	0	0
$423$	11822.0	1.35888
$424$	0	0
$425$	0	0
$426$	0	0
$427$	3108.00	0.352240
$428$	0	0
$429$	2432.00	0.273702
$430$	0	0
$431$	−7692.00	−0.859653	−0.429827	−	0.902911i	$-0.641425\pi$
$431$	−7692.00	−0.859653	−0.429827	+	0.902911i	$0.641425\pi$
$432$	0	0
$433$	1118.00	0.124082	0.0620412	−	0.998074i	$-0.480239\pi$
$433$	1118.00	0.124082	0.0620412	+	0.998074i	$0.480239\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−7800.00	−0.853832
$438$	0	0
$439$	2600.00	0.282668	0.141334	−	0.989962i	$-0.454861\pi$
$439$	2600.00	0.282668	0.141334	+	0.989962i	$0.454861\pi$
$440$	0	0
$441$	7061.00	0.762445
$442$	0	0
$443$	11958.0	1.28249	0.641243	−	0.767337i	$-0.278419\pi$
$443$	11958.0	1.28249	0.641243	+	0.767337i	$0.278419\pi$
$444$	0	0
$445$	0	0
$446$	0	0
$447$	4100.00	0.433833
$448$	0	0
$449$	−17050.0	−1.79207	−0.896035	−	0.443984i	$-0.853565\pi$
$449$	−17050.0	−1.79207	−0.896035	+	0.443984i	$0.853565\pi$
$450$	0	0
$451$	704.000	0.0735035
$452$	0	0
$453$	3704.00	0.384170
$454$	0	0
$455$	0	0
$456$	0	0
$457$	9494.00	0.971796	0.485898	−	0.874016i	$-0.338493\pi$
$457$	9494.00	0.971796	0.485898	+	0.874016i	$0.338493\pi$
$458$	0	0
$459$	−2600.00	−0.264396
$460$	0	0
$461$	11418.0	1.15356	0.576778	−	0.816901i	$-0.304310\pi$
$461$	11418.0	1.15356	0.576778	+	0.816901i	$0.304310\pi$
$462$	0	0
$463$	7962.00	0.799191	0.399596	−	0.916692i	$-0.369151\pi$
$463$	7962.00	0.799191	0.399596	+	0.916692i	$0.369151\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	−6526.00	−0.646654	−0.323327	−	0.946287i	$-0.604801\pi$
$467$	−6526.00	−0.646654	−0.323327	+	0.946287i	$0.604801\pi$
$468$	0	0
$469$	−756.000	−0.0744325
$470$	0	0
$471$	4988.00	0.487972
$472$	0	0
$473$	−14144.0	−1.37493
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−46.0000	−0.00441550
$478$	0	0
$479$	−17400.0	−1.65976	−0.829881	−	0.557940i	$-0.811592\pi$
$479$	−17400.0	−1.65976	−0.829881	+	0.557940i	$0.811592\pi$
$480$	0	0
$481$	−10108.0	−0.958181
$482$	0	0
$483$	936.000	0.0881770
$484$	0	0
$485$	0	0
$486$	0	0
$487$	1166.00	0.108494	0.0542469	−	0.998528i	$-0.482724\pi$
$487$	1166.00	0.108494	0.0542469	+	0.998528i	$0.482724\pi$
$488$	0	0
$489$	5524.00	0.510846
$490$	0	0
$491$	7072.00	0.650010	0.325005	−	0.945712i	$-0.394634\pi$
$491$	7072.00	0.650010	0.325005	+	0.945712i	$0.394634\pi$
$492$	0	0
$493$	−1300.00	−0.118761
$494$	0	0
$495$	0	0
$496$	0	0
$497$	−2472.00	−0.223107
$498$	0	0
$499$	100.000	0.00897117	0.00448559	−	0.999990i	$-0.498572\pi$
$499$	100.000	0.00897117	0.00448559	+	0.999990i	$0.498572\pi$
$500$	0	0
$501$	−6252.00	−0.557522
$502$	0	0
$503$	2602.00	0.230651	0.115325	−	0.993328i	$-0.463209\pi$
$503$	2602.00	0.230651	0.115325	+	0.993328i	$0.463209\pi$
$504$	0	0
$505$	0	0
$506$	0	0
$507$	1506.00	0.131921
$508$	0	0
$509$	−11150.0	−0.970953	−0.485476	−	0.874250i	$-0.661354\pi$
$509$	−11150.0	−0.970953	−0.485476	+	0.874250i	$0.661354\pi$
$510$	0	0
$511$	5268.00	0.456052
$512$	0	0
$513$	10000.0	0.860645
$514$	0	0
$515$	0	0
$516$	0	0
$517$	−16448.0	−1.39919
$518$	0	0
$519$	156.000	0.0131939
$520$	0	0
$521$	−3638.00	−0.305919	−0.152959	−	0.988232i	$-0.548880\pi$
$521$	−3638.00	−0.305919	−0.152959	+	0.988232i	$0.548880\pi$
$522$	0	0
$523$	2078.00	0.173737	0.0868686	−	0.996220i	$-0.472314\pi$
$523$	2078.00	0.173737	0.0868686	+	0.996220i	$0.472314\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−2808.00	−0.232103
$528$	0	0
$529$	−6083.00	−0.499959
$530$	0	0
$531$	−11500.0	−0.939845
$532$	0	0
$533$	−836.000	−0.0679384
$534$	0	0
$535$	0	0
$536$	0	0
$537$	2600.00	0.208935
$538$	0	0
$539$	−9824.00	−0.785064
$540$	0	0
$541$	−5622.00	−0.446781	−0.223391	−	0.974729i	$-0.571713\pi$
$541$	−5622.00	−0.446781	−0.223391	+	0.974729i	$0.571713\pi$
$542$	0	0
$543$	3484.00	0.275346
$544$	0	0
$545$	0	0
$546$	0	0
$547$	−16486.0	−1.28865	−0.644324	−	0.764753i	$-0.722861\pi$
$547$	−16486.0	−1.28865	−0.644324	+	0.764753i	$0.722861\pi$
$548$	0	0
$549$	−11914.0	−0.926188
$550$	0	0
$551$	5000.00	0.386583
$552$	0	0
$553$	−3600.00	−0.276831
$554$	0	0
$555$	0	0
$556$	0	0
$557$	11706.0	0.890483	0.445242	−	0.895410i	$-0.353118\pi$
$557$	11706.0	0.890483	0.445242	+	0.895410i	$0.353118\pi$
$558$	0	0
$559$	16796.0	1.27083
$560$	0	0
$561$	1664.00	0.125230
$562$	0	0
$563$	25038.0	1.87429	0.937146	−	0.348939i	$-0.113458\pi$
$563$	25038.0	1.87429	0.937146	+	0.348939i	$0.113458\pi$
$564$	0	0
$565$	0	0
$566$	0	0
$567$	2526.00	0.187094
$568$	0	0
$569$	17550.0	1.29303	0.646515	−	0.762901i	$-0.276226\pi$
$569$	17550.0	1.29303	0.646515	+	0.762901i	$0.276226\pi$
$570$	0	0
$571$	10712.0	0.785084	0.392542	−	0.919734i	$-0.371596\pi$
$571$	10712.0	0.785084	0.392542	+	0.919734i	$0.371596\pi$
$572$	0	0
$573$	7544.00	0.550009
$574$	0	0
$575$	0	0
$576$	0	0
$577$	13654.0	0.985136	0.492568	−	0.870274i	$-0.336058\pi$
$577$	13654.0	0.985136	0.492568	+	0.870274i	$0.336058\pi$
$578$	0	0
$579$	−716.000	−0.0513920
$580$	0	0
$581$	−1692.00	−0.120819
$582$	0	0
$583$	64.0000	0.00454650
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−14166.0	−0.996071	−0.498035	−	0.867157i	$-0.665945\pi$
$587$	−14166.0	−0.996071	−0.498035	+	0.867157i	$0.665945\pi$
$588$	0	0
$589$	10800.0	0.755528
$590$	0	0
$591$	4428.00	0.308196
$592$	0	0
$593$	−17842.0	−1.23555	−0.617777	−	0.786354i	$-0.711966\pi$
$593$	−17842.0	−1.23555	−0.617777	+	0.786354i	$0.711966\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	−5200.00	−0.356485
$598$	0	0
$599$	17600.0	1.20053	0.600264	−	0.799802i	$-0.295062\pi$
$599$	17600.0	1.20053	0.600264	+	0.799802i	$0.295062\pi$
$600$	0	0
$601$	27302.0	1.85303	0.926516	−	0.376256i	$-0.122789\pi$
$601$	27302.0	1.85303	0.926516	+	0.376256i	$0.122789\pi$
$602$	0	0
$603$	2898.00	0.195714
$604$	0	0
$605$	0	0
$606$	0	0
$607$	−3794.00	−0.253696	−0.126848	−	0.991922i	$-0.540486\pi$
$607$	−3794.00	−0.253696	−0.126848	+	0.991922i	$0.540486\pi$
$608$	0	0
$609$	−600.000	−0.0399232
$610$	0	0
$611$	19532.0	1.29326
$612$	0	0
$613$	−13238.0	−0.872231	−0.436116	−	0.899891i	$-0.643646\pi$
$613$	−13238.0	−0.872231	−0.436116	+	0.899891i	$0.643646\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	11574.0	0.755189	0.377595	−	0.925971i	$-0.376751\pi$
$617$	11574.0	0.755189	0.377595	+	0.925971i	$0.376751\pi$
$618$	0	0
$619$	8300.00	0.538942	0.269471	−	0.963008i	$-0.413151\pi$
$619$	8300.00	0.538942	0.269471	+	0.963008i	$0.413151\pi$
$620$	0	0
$621$	−7800.00	−0.504031
$622$	0	0
$623$	−900.000	−0.0578776
$624$	0	0
$625$	0	0
$626$	0	0
$627$	−6400.00	−0.407642
$628$	0	0
$629$	−6916.00	−0.438409
$630$	0	0
$631$	7508.00	0.473675	0.236837	−	0.971549i	$-0.423889\pi$
$631$	7508.00	0.473675	0.236837	+	0.971549i	$0.423889\pi$
$632$	0	0
$633$	2336.00	0.146679
$634$	0	0
$635$	0	0
$636$	0	0
$637$	11666.0	0.725626
$638$	0	0
$639$	9476.00	0.586643
$640$	0	0
$641$	−27378.0	−1.68700	−0.843499	−	0.537130i	$-0.819508\pi$
$641$	−27378.0	−1.68700	−0.843499	+	0.537130i	$0.819508\pi$
$642$	0	0
$643$	−1842.00	−0.112973	−0.0564863	−	0.998403i	$-0.517990\pi$
$643$	−1842.00	−0.112973	−0.0564863	+	0.998403i	$0.517990\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−10114.0	−0.614563	−0.307282	−	0.951619i	$-0.599419\pi$
$647$	−10114.0	−0.614563	−0.307282	+	0.951619i	$0.599419\pi$
$648$	0	0
$649$	16000.0	0.967727
$650$	0	0
$651$	−1296.00	−0.0780250
$652$	0	0
$653$	10402.0	0.623372	0.311686	−	0.950185i	$-0.399106\pi$
$653$	10402.0	0.623372	0.311686	+	0.950185i	$0.399106\pi$
$654$	0	0
$655$	0	0
$656$	0	0
$657$	−20194.0	−1.19915
$658$	0	0
$659$	7100.00	0.419692	0.209846	−	0.977734i	$-0.432704\pi$
$659$	7100.00	0.419692	0.209846	+	0.977734i	$0.432704\pi$
$660$	0	0
$661$	7118.00	0.418847	0.209424	−	0.977825i	$-0.432841\pi$
$661$	7118.00	0.418847	0.209424	+	0.977825i	$0.432841\pi$
$662$	0	0
$663$	−1976.00	−0.115749
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3900.00	−0.226400
$668$	0	0
$669$	12956.0	0.748741
$670$	0	0
$671$	16576.0	0.953665
$672$	0	0
$673$	31278.0	1.79150	0.895749	−	0.444560i	$-0.146640\pi$
$673$	31278.0	1.79150	0.895749	+	0.444560i	$0.146640\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−30054.0	−1.70616	−0.853079	−	0.521782i	$-0.825268\pi$
$677$	−30054.0	−1.70616	−0.853079	+	0.521782i	$0.825268\pi$
$678$	0	0
$679$	−2316.00	−0.130898
$680$	0	0
$681$	1292.00	0.0727012
$682$	0	0
$683$	4518.00	0.253113	0.126557	−	0.991959i	$-0.459607\pi$
$683$	4518.00	0.253113	0.126557	+	0.991959i	$0.459607\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	7500.00	0.416511
$688$	0	0
$689$	−76.0000	−0.00420228
$690$	0	0
$691$	29272.0	1.61152	0.805759	−	0.592243i	$-0.201758\pi$
$691$	29272.0	1.61152	0.805759	+	0.592243i	$0.201758\pi$
$692$	0	0
$693$	−4416.00	−0.242063
$694$	0	0
$695$	0	0
$696$	0	0
$697$	−572.000	−0.0310847
$698$	0	0
$699$	2964.00	0.160385
$700$	0	0
$701$	5798.00	0.312393	0.156196	−	0.987726i	$-0.450077\pi$
$701$	5798.00	0.312393	0.156196	+	0.987726i	$0.450077\pi$
$702$	0	0
$703$	26600.0	1.42708
$704$	0	0
$705$	0	0
$706$	0	0
$707$	−4212.00	−0.224057
$708$	0	0
$709$	−8950.00	−0.474082	−0.237041	−	0.971500i	$-0.576178\pi$
$709$	−8950.00	−0.474082	−0.237041	+	0.971500i	$0.576178\pi$
$710$	0	0
$711$	13800.0	0.727905
$712$	0	0
$713$	−8424.00	−0.442470
$714$	0	0
$715$	0	0
$716$	0	0
$717$	2800.00	0.145841
$718$	0	0
$719$	−7800.00	−0.404577	−0.202289	−	0.979326i	$-0.564838\pi$
$719$	−7800.00	−0.404577	−0.202289	+	0.979326i	$0.564838\pi$
$720$	0	0
$721$	−3588.00	−0.185332
$722$	0	0
$723$	−6044.00	−0.310897
$724$	0	0
$725$	0	0
$726$	0	0
$727$	−8554.00	−0.436383	−0.218191	−	0.975906i	$-0.570016\pi$
$727$	−8554.00	−0.436383	−0.218191	+	0.975906i	$0.570016\pi$
$728$	0	0
$729$	−4283.00	−0.217599
$730$	0	0
$731$	11492.0	0.581460
$732$	0	0
$733$	2882.00	0.145224	0.0726119	−	0.997360i	$-0.476867\pi$
$733$	2882.00	0.145224	0.0726119	+	0.997360i	$0.476867\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	−4032.00	−0.201521
$738$	0	0
$739$	18700.0	0.930840	0.465420	−	0.885090i	$-0.345903\pi$
$739$	18700.0	0.930840	0.465420	+	0.885090i	$0.345903\pi$
$740$	0	0
$741$	7600.00	0.376779
$742$	0	0
$743$	12242.0	0.604462	0.302231	−	0.953235i	$-0.402269\pi$
$743$	12242.0	0.604462	0.302231	+	0.953235i	$0.402269\pi$
$744$	0	0
$745$	0	0
$746$	0	0
$747$	6486.00	0.317685
$748$	0	0
$749$	7164.00	0.349488
$750$	0	0
$751$	31148.0	1.51346	0.756729	−	0.653729i	$-0.226796\pi$
$751$	31148.0	1.51346	0.756729	+	0.653729i	$0.226796\pi$
$752$	0	0
$753$	2496.00	0.120796
$754$	0	0
$755$	0	0
$756$	0	0
$757$	−7694.00	−0.369410	−0.184705	−	0.982794i	$-0.559133\pi$
$757$	−7694.00	−0.369410	−0.184705	+	0.982794i	$0.559133\pi$
$758$	0	0
$759$	4992.00	0.238733
$760$	0	0
$761$	−4518.00	−0.215213	−0.107607	−	0.994194i	$-0.534319\pi$
$761$	−4518.00	−0.215213	−0.107607	+	0.994194i	$0.534319\pi$
$762$	0	0
$763$	3300.00	0.156577
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−19000.0	−0.894459
$768$	0	0
$769$	−39550.0	−1.85463	−0.927314	−	0.374283i	$-0.877889\pi$
$769$	−39550.0	−1.85463	−0.927314	+	0.374283i	$0.877889\pi$
$770$	0	0
$771$	4212.00	0.196746
$772$	0	0
$773$	22122.0	1.02933	0.514666	−	0.857391i	$-0.327916\pi$
$773$	22122.0	1.02933	0.514666	+	0.857391i	$0.327916\pi$
$774$	0	0
$775$	0	0
$776$	0	0
$777$	−3192.00	−0.147378
$778$	0	0
$779$	2200.00	0.101185
$780$	0	0
$781$	−13184.0	−0.604047
$782$	0	0
$783$	5000.00	0.228206
$784$	0	0
$785$	0	0
$786$	0	0
$787$	16634.0	0.753416	0.376708	−	0.926332i	$-0.377056\pi$
$787$	16634.0	0.753416	0.376708	+	0.926332i	$0.377056\pi$
$788$	0	0
$789$	7276.00	0.328305
$790$	0	0
$791$	−9372.00	−0.421277
$792$	0	0
$793$	−19684.0	−0.881462
$794$	0	0
$795$	0	0
$796$	0	0
$797$	27586.0	1.22603	0.613015	−	0.790071i	$-0.289956\pi$
$797$	27586.0	1.22603	0.613015	+	0.790071i	$0.289956\pi$
$798$	0	0
$799$	13364.0	0.591720
$800$	0	0
$801$	3450.00	0.152184
$802$	0	0
$803$	28096.0	1.23473
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−13100.0	−0.571427
$808$	0	0
$809$	3850.00	0.167316	0.0836581	−	0.996495i	$-0.473340\pi$
$809$	3850.00	0.167316	0.0836581	+	0.996495i	$0.473340\pi$
$810$	0	0
$811$	10032.0	0.434366	0.217183	−	0.976131i	$-0.430313\pi$
$811$	10032.0	0.434366	0.217183	+	0.976131i	$0.430313\pi$
$812$	0	0
$813$	−8776.00	−0.378583
$814$	0	0
$815$	0	0
$816$	0	0
$817$	−44200.0	−1.89273
$818$	0	0
$819$	5244.00	0.223736
$820$	0	0
$821$	−20562.0	−0.874079	−0.437039	−	0.899442i	$-0.643973\pi$
$821$	−20562.0	−0.874079	−0.437039	+	0.899442i	$0.643973\pi$
$822$	0	0
$823$	10322.0	0.437184	0.218592	−	0.975816i	$-0.429854\pi$
$823$	10322.0	0.437184	0.218592	+	0.975816i	$0.429854\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	−8846.00	−0.371954	−0.185977	−	0.982554i	$-0.559545\pi$
$827$	−8846.00	−0.371954	−0.185977	+	0.982554i	$0.559545\pi$
$828$	0	0
$829$	25350.0	1.06205	0.531026	−	0.847355i	$-0.321806\pi$
$829$	25350.0	1.06205	0.531026	+	0.847355i	$0.321806\pi$
$830$	0	0
$831$	−1092.00	−0.0455849
$832$	0	0
$833$	7982.00	0.332005
$834$	0	0
$835$	0	0
$836$	0	0
$837$	10800.0	0.446001
$838$	0	0
$839$	−46000.0	−1.89284	−0.946422	−	0.322932i	$-0.895331\pi$
$839$	−46000.0	−1.89284	−0.946422	+	0.322932i	$0.895331\pi$
$840$	0	0
$841$	−21889.0	−0.897495
$842$	0	0
$843$	13716.0	0.560385
$844$	0	0
$845$	0	0
$846$	0	0
$847$	−1842.00	−0.0747248
$848$	0	0
$849$	18564.0	0.750430
$850$	0	0
$851$	−20748.0	−0.835761
$852$	0	0
$853$	−16998.0	−0.682298	−0.341149	−	0.940009i	$-0.610816\pi$
$853$	−16998.0	−0.682298	−0.341149	+	0.940009i	$0.610816\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	26494.0	1.05603	0.528015	−	0.849235i	$-0.322936\pi$
$857$	26494.0	1.05603	0.528015	+	0.849235i	$0.322936\pi$
$858$	0	0
$859$	−21500.0	−0.853982	−0.426991	−	0.904256i	$-0.640426\pi$
$859$	−21500.0	−0.853982	−0.426991	+	0.904256i	$0.640426\pi$
$860$	0	0
$861$	−264.000	−0.0104496
$862$	0	0
$863$	25762.0	1.01616	0.508082	−	0.861309i	$-0.330355\pi$
$863$	25762.0	1.01616	0.508082	+	0.861309i	$0.330355\pi$
$864$	0	0
$865$	0	0
$866$	0	0
$867$	8474.00	0.331940
$868$	0	0
$869$	−19200.0	−0.749500
$870$	0	0
$871$	4788.00	0.186263
$872$	0	0
$873$	8878.00	0.344186
$874$	0	0
$875$	0	0
$876$	0	0
$877$	30546.0	1.17613	0.588064	−	0.808814i	$-0.299890\pi$
$877$	30546.0	1.17613	0.588064	+	0.808814i	$0.299890\pi$
$878$	0	0
$879$	−9684.00	−0.371596
$880$	0	0
$881$	32942.0	1.25976	0.629878	−	0.776694i	$-0.283105\pi$
$881$	32942.0	1.25976	0.629878	+	0.776694i	$0.283105\pi$
$882$	0	0
$883$	27118.0	1.03351	0.516757	−	0.856132i	$-0.327139\pi$
$883$	27118.0	1.03351	0.516757	+	0.856132i	$0.327139\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−38634.0	−1.46246	−0.731230	−	0.682131i	$-0.761054\pi$
$887$	−38634.0	−1.46246	−0.731230	+	0.682131i	$0.761054\pi$
$888$	0	0
$889$	11076.0	0.417860
$890$	0	0
$891$	13472.0	0.506542
$892$	0	0
$893$	−51400.0	−1.92613
$894$	0	0
$895$	0	0
$896$	0	0
$897$	−5928.00	−0.220658
$898$	0	0
$899$	5400.00	0.200334
$900$	0	0
$901$	−52.0000	−0.00192272
$902$	0	0
$903$	5304.00	0.195466
$904$	0	0
$905$	0	0
$906$	0	0
$907$	1794.00	0.0656767	0.0328384	−	0.999461i	$-0.489545\pi$
$907$	1794.00	0.0656767	0.0328384	+	0.999461i	$0.489545\pi$
$908$	0	0
$909$	16146.0	0.589141
$910$	0	0
$911$	−41732.0	−1.51772	−0.758860	−	0.651254i	$-0.774243\pi$
$911$	−41732.0	−1.51772	−0.758860	+	0.651254i	$0.774243\pi$
$912$	0	0
$913$	−9024.00	−0.327109
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−13248.0	−0.477086
$918$	0	0
$919$	−29200.0	−1.04812	−0.524058	−	0.851682i	$-0.675583\pi$
$919$	−29200.0	−1.04812	−0.524058	+	0.851682i	$0.675583\pi$
$920$	0	0
$921$	−5188.00	−0.185614
$922$	0	0
$923$	15656.0	0.558314
$924$	0	0
$925$	0	0
$926$	0	0
$927$	13754.0	0.487315
$928$	0	0
$929$	−48650.0	−1.71814	−0.859071	−	0.511856i	$-0.828958\pi$
$929$	−48650.0	−1.71814	−0.859071	+	0.511856i	$0.828958\pi$
$930$	0	0
$931$	−30700.0	−1.08072
$932$	0	0
$933$	14664.0	0.514553
$934$	0	0
$935$	0	0
$936$	0	0
$937$	11334.0	0.395161	0.197580	−	0.980287i	$-0.436692\pi$
$937$	11334.0	0.395161	0.197580	+	0.980287i	$0.436692\pi$
$938$	0	0
$939$	3124.00	0.108571
$940$	0	0
$941$	31178.0	1.08010	0.540050	−	0.841633i	$-0.318405\pi$
$941$	31178.0	1.08010	0.540050	+	0.841633i	$0.318405\pi$
$942$	0	0
$943$	−1716.00	−0.0592584
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−4686.00	−0.160797	−0.0803984	−	0.996763i	$-0.525619\pi$
$947$	−4686.00	−0.160797	−0.0803984	+	0.996763i	$0.525619\pi$
$948$	0	0
$949$	−33364.0	−1.14124
$950$	0	0
$951$	−2852.00	−0.0972476
$952$	0	0
$953$	598.000	0.0203265	0.0101632	−	0.999948i	$-0.496765\pi$
$953$	598.000	0.0203265	0.0101632	+	0.999948i	$0.496765\pi$
$954$	0	0
$955$	0	0
$956$	0	0
$957$	−3200.00	−0.108089
$958$	0	0
$959$	14004.0	0.471546
$960$	0	0
$961$	−18127.0	−0.608472
$962$	0	0
$963$	−27462.0	−0.918952
$964$	0	0
$965$	0	0
$966$	0	0
$967$	41726.0	1.38761	0.693804	−	0.720163i	$-0.255933\pi$
$967$	41726.0	1.38761	0.693804	+	0.720163i	$0.255933\pi$
$968$	0	0
$969$	5200.00	0.172392
$970$	0	0
$971$	24312.0	0.803511	0.401756	−	0.915747i	$-0.368400\pi$
$971$	24312.0	0.803511	0.401756	+	0.915747i	$0.368400\pi$
$972$	0	0
$973$	−4200.00	−0.138382
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−40946.0	−1.34082	−0.670409	−	0.741992i	$-0.733881\pi$
$977$	−40946.0	−1.34082	−0.670409	+	0.741992i	$0.733881\pi$
$978$	0	0
$979$	−4800.00	−0.156699
$980$	0	0
$981$	−12650.0	−0.411706
$982$	0	0
$983$	42282.0	1.37191	0.685954	−	0.727645i	$-0.259385\pi$
$983$	42282.0	1.37191	0.685954	+	0.727645i	$0.259385\pi$
$984$	0	0
$985$	0	0
$986$	0	0
$987$	6168.00	0.198916
$988$	0	0
$989$	34476.0	1.10847
$990$	0	0
$991$	−1172.00	−0.0375679	−0.0187840	−	0.999824i	$-0.505979\pi$
$991$	−1172.00	−0.0375679	−0.0187840	+	0.999824i	$0.505979\pi$
$992$	0	0
$993$	8016.00	0.256173
$994$	0	0
$995$	0	0
$996$	0	0
$997$	−31614.0	−1.00424	−0.502119	−	0.864798i	$-0.667446\pi$
$997$	−31614.0	−1.00424	−0.502119	+	0.864798i	$0.667446\pi$
$998$	0	0
$999$	26600.0	0.842429

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	1600.4.a.s.1.1		1
4.3	odd	2		1600.4.a.bi.1.1		1
5.4	even	2		320.4.a.h.1.1		1
8.3	odd	2		25.4.a.c.1.1		1
8.5	even	2		400.4.a.m.1.1		1
20.19	odd	2		320.4.a.g.1.1		1
24.11	even	2		225.4.a.b.1.1		1
40.3	even	4		25.4.b.a.24.1		2
40.13	odd	4		400.4.c.k.49.2		2
40.19	odd	2		5.4.a.a.1.1	✓	1
40.27	even	4		25.4.b.a.24.2		2
40.29	even	2		80.4.a.d.1.1		1
40.37	odd	4		400.4.c.k.49.1		2
56.27	even	2		1225.4.a.k.1.1		1
80.19	odd	4		1280.4.d.e.641.1		2
80.29	even	4		1280.4.d.l.641.2		2
80.59	odd	4		1280.4.d.e.641.2		2
80.69	even	4		1280.4.d.l.641.1		2
120.29	odd	2		720.4.a.u.1.1		1
120.59	even	2		45.4.a.d.1.1		1
120.83	odd	4		225.4.b.c.199.2		2
120.107	odd	4		225.4.b.c.199.1		2
280.19	even	6		245.4.e.g.116.1		2
280.59	even	6		245.4.e.g.226.1		2
280.139	even	2		245.4.a.a.1.1		1
280.179	odd	6		245.4.e.f.226.1		2
280.219	odd	6		245.4.e.f.116.1		2
360.59	even	6		405.4.e.c.136.1		2
360.139	odd	6		405.4.e.l.136.1		2
360.259	odd	6		405.4.e.l.271.1		2
360.299	even	6		405.4.e.c.271.1		2
440.219	even	2		605.4.a.d.1.1		1
520.259	odd	2		845.4.a.b.1.1		1
680.339	odd	2		1445.4.a.a.1.1		1
760.379	even	2		1805.4.a.h.1.1		1
840.419	odd	2		2205.4.a.q.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
5.4.a.a.1.1	✓	1	40.19	odd	2
25.4.a.c.1.1		1	8.3	odd	2
25.4.b.a.24.1		2	40.3	even	4
25.4.b.a.24.2		2	40.27	even	4
45.4.a.d.1.1		1	120.59	even	2
80.4.a.d.1.1		1	40.29	even	2
225.4.a.b.1.1		1	24.11	even	2
225.4.b.c.199.1		2	120.107	odd	4
225.4.b.c.199.2		2	120.83	odd	4
245.4.a.a.1.1		1	280.139	even	2
245.4.e.f.116.1		2	280.219	odd	6
245.4.e.f.226.1		2	280.179	odd	6
245.4.e.g.116.1		2	280.19	even	6
245.4.e.g.226.1		2	280.59	even	6
320.4.a.g.1.1		1	20.19	odd	2
320.4.a.h.1.1		1	5.4	even	2
400.4.a.m.1.1		1	8.5	even	2
400.4.c.k.49.1		2	40.37	odd	4
400.4.c.k.49.2		2	40.13	odd	4
405.4.e.c.136.1		2	360.59	even	6
405.4.e.c.271.1		2	360.299	even	6
405.4.e.l.136.1		2	360.139	odd	6
405.4.e.l.271.1		2	360.259	odd	6
605.4.a.d.1.1		1	440.219	even	2
720.4.a.u.1.1		1	120.29	odd	2
845.4.a.b.1.1		1	520.259	odd	2
1225.4.a.k.1.1		1	56.27	even	2
1280.4.d.e.641.1		2	80.19	odd	4
1280.4.d.e.641.2		2	80.59	odd	4
1280.4.d.l.641.1		2	80.69	even	4
1280.4.d.l.641.2		2	80.29	even	4
1445.4.a.a.1.1		1	680.339	odd	2
1600.4.a.s.1.1		1	1.1	even	1	trivial
1600.4.a.bi.1.1		1	4.3	odd	2
1805.4.a.h.1.1		1	760.379	even	2
2205.4.a.q.1.1		1	840.419	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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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	1600.4.a.s.1.1

Level	$1600$
Weight	$4$
Character	1600.1
Self dual	yes
Analytic conductor	$94.403$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$