LMFDB - Embedded newform 960.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [960,4,Mod(1,960)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(960, 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("960.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$960 = 2^{6} \cdot 3 \cdot 5$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	960.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:	$56.6418336055$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 120)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	960.1

$q$ -expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

f(q)

=

q-3.00000 q^{3} -5.00000 q^{5} +9.00000 q^{9} +4.00000 q^{11} -54.0000 q^{13} +15.0000 q^{15} +114.000 q^{17} +44.0000 q^{19} -96.0000 q^{23} +25.0000 q^{25} -27.0000 q^{27} -134.000 q^{29} +272.000 q^{31} -12.0000 q^{33} +98.0000 q^{37} +162.000 q^{39} -6.00000 q^{41} +12.0000 q^{43} -45.0000 q^{45} +200.000 q^{47} -343.000 q^{49} -342.000 q^{51} -654.000 q^{53} -20.0000 q^{55} -132.000 q^{57} +36.0000 q^{59} +442.000 q^{61} +270.000 q^{65} -188.000 q^{67} +288.000 q^{69} +632.000 q^{71} -390.000 q^{73} -75.0000 q^{75} -688.000 q^{79} +81.0000 q^{81} +1188.00 q^{83} -570.000 q^{85} +402.000 q^{87} -694.000 q^{89} -816.000 q^{93} -220.000 q^{95} -1726.00 q^{97} +36.0000 q^{99} +O(q^{100})

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0	0
$2$	0	0
$3$	−3.00000	−0.577350
$3$	−3.00000	−0.577350
$4$	0	0
$5$	−5.00000	−0.447214
$5$	−5.00000	−0.447214
$6$	0	0
$7$	0	0		−	1.00000i	$-0.5\pi$
$7$	0	0			1.00000i	$0.5\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	4.00000	0.109640	0.0548202	−	0.998496i	$-0.482541\pi$
$11$	4.00000	0.109640	0.0548202	+	0.998496i	$0.482541\pi$
$12$	0	0
$13$	−54.0000	−1.15207	−0.576035	−	0.817425i	$-0.695401\pi$
$13$	−54.0000	−1.15207	−0.576035	+	0.817425i	$0.695401\pi$
$14$	0	0
$15$	15.0000	0.258199
$16$	0	0
$17$	114.000	1.62642	0.813208	−	0.581974i	$-0.197719\pi$
$17$	114.000	1.62642	0.813208	+	0.581974i	$0.197719\pi$
$18$	0	0
$19$	44.0000	0.531279	0.265639	−	0.964072i	$-0.414417\pi$
$19$	44.0000	0.531279	0.265639	+	0.964072i	$0.414417\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−96.0000	−0.870321	−0.435161	−	0.900353i	$-0.643308\pi$
$23$	−96.0000	−0.870321	−0.435161	+	0.900353i	$0.643308\pi$
$24$	0	0
$25$	25.0000	0.200000
$26$	0	0
$27$	−27.0000	−0.192450
$28$	0	0
$29$	−134.000	−0.858041	−0.429020	−	0.903295i	$-0.641141\pi$
$29$	−134.000	−0.858041	−0.429020	+	0.903295i	$0.641141\pi$
$30$	0	0
$31$	272.000	1.57589	0.787946	−	0.615745i	$-0.211145\pi$
$31$	272.000	1.57589	0.787946	+	0.615745i	$0.211145\pi$
$32$	0	0
$33$	−12.0000	−0.0633010
$34$	0	0
$35$	0	0
$36$	0	0
$37$	98.0000	0.435435	0.217718	−	0.976012i	$-0.430139\pi$
$37$	98.0000	0.435435	0.217718	+	0.976012i	$0.430139\pi$
$38$	0	0
$39$	162.000	0.665148
$40$	0	0
$41$	−6.00000	−0.0228547	−0.0114273	−	0.999935i	$-0.503638\pi$
$41$	−6.00000	−0.0228547	−0.0114273	+	0.999935i	$0.503638\pi$
$42$	0	0
$43$	12.0000	0.0425577	0.0212789	−	0.999774i	$-0.493226\pi$
$43$	12.0000	0.0425577	0.0212789	+	0.999774i	$0.493226\pi$
$44$	0	0
$45$	−45.0000	−0.149071
$46$	0	0
$47$	200.000	0.620702	0.310351	−	0.950622i	$-0.399553\pi$
$47$	200.000	0.620702	0.310351	+	0.950622i	$0.399553\pi$
$48$	0	0
$49$	−343.000	−1.00000
$50$	0	0
$51$	−342.000	−0.939011
$52$	0	0
$53$	−654.000	−1.69498	−0.847489	−	0.530813i	$-0.821887\pi$
$53$	−654.000	−1.69498	−0.847489	+	0.530813i	$0.821887\pi$
$54$	0	0
$55$	−20.0000	−0.0490327
$56$	0	0
$57$	−132.000	−0.306734
$58$	0	0
$59$	36.0000	0.0794373	0.0397187	−	0.999211i	$-0.487354\pi$
$59$	36.0000	0.0794373	0.0397187	+	0.999211i	$0.487354\pi$
$60$	0	0
$61$	442.000	0.927743	0.463871	−	0.885903i	$-0.346460\pi$
$61$	442.000	0.927743	0.463871	+	0.885903i	$0.346460\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	270.000	0.515221
$66$	0	0
$67$	−188.000	−0.342804	−0.171402	−	0.985201i	$-0.554830\pi$
$67$	−188.000	−0.342804	−0.171402	+	0.985201i	$0.554830\pi$
$68$	0	0
$69$	288.000	0.502480
$70$	0	0
$71$	632.000	1.05640	0.528201	−	0.849119i	$-0.322867\pi$
$71$	632.000	1.05640	0.528201	+	0.849119i	$0.322867\pi$
$72$	0	0
$73$	−390.000	−0.625288	−0.312644	−	0.949870i	$-0.601215\pi$
$73$	−390.000	−0.625288	−0.312644	+	0.949870i	$0.601215\pi$
$74$	0	0
$75$	−75.0000	−0.115470
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−688.000	−0.979823	−0.489912	−	0.871772i	$-0.662971\pi$
$79$	−688.000	−0.979823	−0.489912	+	0.871772i	$0.662971\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	1188.00	1.57108	0.785542	−	0.618809i	$-0.212384\pi$
$83$	1188.00	1.57108	0.785542	+	0.618809i	$0.212384\pi$
$84$	0	0
$85$	−570.000	−0.727355
$86$	0	0
$87$	402.000	0.495390
$88$	0	0
$89$	−694.000	−0.826560	−0.413280	−	0.910604i	$-0.635617\pi$
$89$	−694.000	−0.826560	−0.413280	+	0.910604i	$0.635617\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−816.000	−0.909841
$94$	0	0
$95$	−220.000	−0.237595
$96$	0	0
$97$	−1726.00	−1.80669	−0.903344	−	0.428917i	$-0.858895\pi$
$97$	−1726.00	−1.80669	−0.903344	+	0.428917i	$0.858895\pi$
$98$	0	0
$99$	36.0000	0.0365468
$100$	0	0
$101$	−1182.00	−1.16449	−0.582245	−	0.813014i	$-0.697825\pi$
$101$	−1182.00	−1.16449	−0.582245	+	0.813014i	$0.697825\pi$
$102$	0	0
$103$	−1968.00	−1.88265	−0.941324	−	0.337503i	$-0.890418\pi$
$103$	−1968.00	−1.88265	−0.941324	+	0.337503i	$0.890418\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	796.000	0.719180	0.359590	−	0.933110i	$-0.382917\pi$
$107$	796.000	0.719180	0.359590	+	0.933110i	$0.382917\pi$
$108$	0	0
$109$	−342.000	−0.300529	−0.150264	−	0.988646i	$-0.548013\pi$
$109$	−342.000	−0.300529	−0.150264	+	0.988646i	$0.548013\pi$
$110$	0	0
$111$	−294.000	−0.251399
$112$	0	0
$113$	114.000	0.0949046	0.0474523	−	0.998874i	$-0.484890\pi$
$113$	114.000	0.0949046	0.0474523	+	0.998874i	$0.484890\pi$
$114$	0	0
$115$	480.000	0.389219
$116$	0	0
$117$	−486.000	−0.384023
$118$	0	0
$119$	0	0
$120$	0	0
$121$	−1315.00	−0.987979
$122$	0	0
$123$	18.0000	0.0131952
$124$	0	0
$125$	−125.000	−0.0894427
$126$	0	0
$127$	−2344.00	−1.63777	−0.818883	−	0.573960i	$-0.805406\pi$
$127$	−2344.00	−1.63777	−0.818883	+	0.573960i	$0.805406\pi$
$128$	0	0
$129$	−36.0000	−0.0245707
$130$	0	0
$131$	−2164.00	−1.44328	−0.721640	−	0.692269i	$-0.756611\pi$
$131$	−2164.00	−1.44328	−0.721640	+	0.692269i	$0.756611\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	135.000	0.0860663
$136$	0	0
$137$	−2822.00	−1.75985	−0.879926	−	0.475111i	$-0.842408\pi$
$137$	−2822.00	−1.75985	−0.879926	+	0.475111i	$0.842408\pi$
$138$	0	0
$139$	1972.00	1.20333	0.601665	−	0.798749i	$-0.294504\pi$
$139$	1972.00	1.20333	0.601665	+	0.798749i	$0.294504\pi$
$140$	0	0
$141$	−600.000	−0.358363
$142$	0	0
$143$	−216.000	−0.126313
$144$	0	0
$145$	670.000	0.383727
$146$	0	0
$147$	1029.00	0.577350
$148$	0	0
$149$	1394.00	0.766449	0.383225	−	0.923655i	$-0.374814\pi$
$149$	1394.00	0.766449	0.383225	+	0.923655i	$0.374814\pi$
$150$	0	0
$151$	2216.00	1.19427	0.597137	−	0.802139i	$-0.296305\pi$
$151$	2216.00	1.19427	0.597137	+	0.802139i	$0.296305\pi$
$152$	0	0
$153$	1026.00	0.542138
$154$	0	0
$155$	−1360.00	−0.704760
$156$	0	0
$157$	954.000	0.484952	0.242476	−	0.970157i	$-0.422040\pi$
$157$	954.000	0.484952	0.242476	+	0.970157i	$0.422040\pi$
$158$	0	0
$159$	1962.00	0.978596
$160$	0	0
$161$	0	0
$162$	0	0
$163$	−3404.00	−1.63572	−0.817858	−	0.575419i	$-0.804839\pi$
$163$	−3404.00	−1.63572	−0.817858	+	0.575419i	$0.804839\pi$
$164$	0	0
$165$	60.0000	0.0283091
$166$	0	0
$167$	832.000	0.385522	0.192761	−	0.981246i	$-0.438256\pi$
$167$	832.000	0.385522	0.192761	+	0.981246i	$0.438256\pi$
$168$	0	0
$169$	719.000	0.327264
$170$	0	0
$171$	396.000	0.177093
$172$	0	0
$173$	362.000	0.159089	0.0795444	−	0.996831i	$-0.474653\pi$
$173$	362.000	0.159089	0.0795444	+	0.996831i	$0.474653\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−108.000	−0.0458631
$178$	0	0
$179$	−3252.00	−1.35791	−0.678955	−	0.734180i	$-0.737567\pi$
$179$	−3252.00	−1.35791	−0.678955	+	0.734180i	$0.737567\pi$
$180$	0	0
$181$	−3086.00	−1.26730	−0.633648	−	0.773621i	$-0.718443\pi$
$181$	−3086.00	−1.26730	−0.633648	+	0.773621i	$0.718443\pi$
$182$	0	0
$183$	−1326.00	−0.535632
$184$	0	0
$185$	−490.000	−0.194733
$186$	0	0
$187$	456.000	0.178321
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4080.00	1.54565	0.772823	−	0.634621i	$-0.218844\pi$
$191$	4080.00	1.54565	0.772823	+	0.634621i	$0.218844\pi$
$192$	0	0
$193$	−2654.00	−0.989840	−0.494920	−	0.868939i	$-0.664803\pi$
$193$	−2654.00	−0.989840	−0.494920	+	0.868939i	$0.664803\pi$
$194$	0	0
$195$	−810.000	−0.297463
$196$	0	0
$197$	−1534.00	−0.554787	−0.277393	−	0.960756i	$-0.589471\pi$
$197$	−1534.00	−0.554787	−0.277393	+	0.960756i	$0.589471\pi$
$198$	0	0
$199$	−4344.00	−1.54743	−0.773714	−	0.633536i	$-0.781603\pi$
$199$	−4344.00	−1.54743	−0.773714	+	0.633536i	$0.781603\pi$
$200$	0	0
$201$	564.000	0.197918
$202$	0	0
$203$	0	0
$204$	0	0
$205$	30.0000	0.0102209
$206$	0	0
$207$	−864.000	−0.290107
$208$	0	0
$209$	176.000	0.0582496
$210$	0	0
$211$	−1380.00	−0.450252	−0.225126	−	0.974330i	$-0.572279\pi$
$211$	−1380.00	−0.450252	−0.225126	+	0.974330i	$0.572279\pi$
$212$	0	0
$213$	−1896.00	−0.609914
$214$	0	0
$215$	−60.0000	−0.0190324
$216$	0	0
$217$	0	0
$218$	0	0
$219$	1170.00	0.361010
$220$	0	0
$221$	−6156.00	−1.87374
$222$	0	0
$223$	5224.00	1.56872	0.784361	−	0.620305i	$-0.212991\pi$
$223$	5224.00	1.56872	0.784361	+	0.620305i	$0.212991\pi$
$224$	0	0
$225$	225.000	0.0666667
$226$	0	0
$227$	3364.00	0.983597	0.491799	−	0.870709i	$-0.336340\pi$
$227$	3364.00	0.983597	0.491799	+	0.870709i	$0.336340\pi$
$228$	0	0
$229$	−3998.00	−1.15369	−0.576846	−	0.816853i	$-0.695717\pi$
$229$	−3998.00	−1.15369	−0.576846	+	0.816853i	$0.695717\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−3590.00	−1.00939	−0.504697	−	0.863297i	$-0.668396\pi$
$233$	−3590.00	−1.00939	−0.504697	+	0.863297i	$0.668396\pi$
$234$	0	0
$235$	−1000.00	−0.277586
$236$	0	0
$237$	2064.00	0.565701
$238$	0	0
$239$	1104.00	0.298794	0.149397	−	0.988777i	$-0.452267\pi$
$239$	1104.00	0.298794	0.149397	+	0.988777i	$0.452267\pi$
$240$	0	0
$241$	1618.00	0.432467	0.216233	−	0.976342i	$-0.430623\pi$
$241$	1618.00	0.432467	0.216233	+	0.976342i	$0.430623\pi$
$242$	0	0
$243$	−243.000	−0.0641500
$244$	0	0
$245$	1715.00	0.447214
$246$	0	0
$247$	−2376.00	−0.612070
$248$	0	0
$249$	−3564.00	−0.907066
$250$	0	0
$251$	5780.00	1.45351	0.726754	−	0.686898i	$-0.241028\pi$
$251$	5780.00	1.45351	0.726754	+	0.686898i	$0.241028\pi$
$252$	0	0
$253$	−384.000	−0.0954224
$254$	0	0
$255$	1710.00	0.419939
$256$	0	0
$257$	2594.00	0.629608	0.314804	−	0.949157i	$-0.398061\pi$
$257$	2594.00	0.629608	0.314804	+	0.949157i	$0.398061\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−1206.00	−0.286014
$262$	0	0
$263$	−3696.00	−0.866559	−0.433280	−	0.901260i	$-0.642644\pi$
$263$	−3696.00	−0.866559	−0.433280	+	0.901260i	$0.642644\pi$
$264$	0	0
$265$	3270.00	0.758017
$266$	0	0
$267$	2082.00	0.477215
$268$	0	0
$269$	2250.00	0.509981	0.254991	−	0.966944i	$-0.417928\pi$
$269$	2250.00	0.509981	0.254991	+	0.966944i	$0.417928\pi$
$270$	0	0
$271$	−2208.00	−0.494932	−0.247466	−	0.968897i	$-0.579598\pi$
$271$	−2208.00	−0.494932	−0.247466	+	0.968897i	$0.579598\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	100.000	0.0219281
$276$	0	0
$277$	1682.00	0.364843	0.182422	−	0.983220i	$-0.441606\pi$
$277$	1682.00	0.364843	0.182422	+	0.983220i	$0.441606\pi$
$278$	0	0
$279$	2448.00	0.525297
$280$	0	0
$281$	7306.00	1.55103	0.775515	−	0.631329i	$-0.217490\pi$
$281$	7306.00	1.55103	0.775515	+	0.631329i	$0.217490\pi$
$282$	0	0
$283$	−8164.00	−1.71484	−0.857419	−	0.514618i	$-0.827934\pi$
$283$	−8164.00	−1.71484	−0.857419	+	0.514618i	$0.827934\pi$
$284$	0	0
$285$	660.000	0.137176
$286$	0	0
$287$	0	0
$288$	0	0
$289$	8083.00	1.64523
$290$	0	0
$291$	5178.00	1.04309
$292$	0	0
$293$	514.000	0.102485	0.0512427	−	0.998686i	$-0.483682\pi$
$293$	514.000	0.102485	0.0512427	+	0.998686i	$0.483682\pi$
$294$	0	0
$295$	−180.000	−0.0355254
$296$	0	0
$297$	−108.000	−0.0211003
$298$	0	0
$299$	5184.00	1.00267
$300$	0	0
$301$	0	0
$302$	0	0
$303$	3546.00	0.672318
$304$	0	0
$305$	−2210.00	−0.414899
$306$	0	0
$307$	−2476.00	−0.460302	−0.230151	−	0.973155i	$-0.573922\pi$
$307$	−2476.00	−0.460302	−0.230151	+	0.973155i	$0.573922\pi$
$308$	0	0
$309$	5904.00	1.08695
$310$	0	0
$311$	−2296.00	−0.418631	−0.209315	−	0.977848i	$-0.567124\pi$
$311$	−2296.00	−0.418631	−0.209315	+	0.977848i	$0.567124\pi$
$312$	0	0
$313$	−9878.00	−1.78383	−0.891913	−	0.452207i	$-0.850637\pi$
$313$	−9878.00	−1.78383	−0.891913	+	0.452207i	$0.850637\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	2138.00	0.378808	0.189404	−	0.981899i	$-0.439344\pi$
$317$	2138.00	0.378808	0.189404	+	0.981899i	$0.439344\pi$
$318$	0	0
$319$	−536.000	−0.0940760
$320$	0	0
$321$	−2388.00	−0.415219
$322$	0	0
$323$	5016.00	0.864080
$324$	0	0
$325$	−1350.00	−0.230414
$326$	0	0
$327$	1026.00	0.173510
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−6460.00	−1.07273	−0.536365	−	0.843986i	$-0.680203\pi$
$331$	−6460.00	−1.07273	−0.536365	+	0.843986i	$0.680203\pi$
$332$	0	0
$333$	882.000	0.145145
$334$	0	0
$335$	940.000	0.153307
$336$	0	0
$337$	626.000	0.101188	0.0505941	−	0.998719i	$-0.483889\pi$
$337$	626.000	0.101188	0.0505941	+	0.998719i	$0.483889\pi$
$338$	0	0
$339$	−342.000	−0.0547932
$340$	0	0
$341$	1088.00	0.172782
$342$	0	0
$343$	0	0
$344$	0	0
$345$	−1440.00	−0.224716
$346$	0	0
$347$	876.000	0.135522	0.0677610	−	0.997702i	$-0.478414\pi$
$347$	876.000	0.135522	0.0677610	+	0.997702i	$0.478414\pi$
$348$	0	0
$349$	9850.00	1.51077	0.755385	−	0.655282i	$-0.227450\pi$
$349$	9850.00	1.51077	0.755385	+	0.655282i	$0.227450\pi$
$350$	0	0
$351$	1458.00	0.221716
$352$	0	0
$353$	−8894.00	−1.34102	−0.670510	−	0.741901i	$-0.733925\pi$
$353$	−8894.00	−1.34102	−0.670510	+	0.741901i	$0.733925\pi$
$354$	0	0
$355$	−3160.00	−0.472438
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−1464.00	−0.215228	−0.107614	−	0.994193i	$-0.534321\pi$
$359$	−1464.00	−0.215228	−0.107614	+	0.994193i	$0.534321\pi$
$360$	0	0
$361$	−4923.00	−0.717743
$362$	0	0
$363$	3945.00	0.570410
$364$	0	0
$365$	1950.00	0.279637
$366$	0	0
$367$	7016.00	0.997908	0.498954	−	0.866628i	$-0.333718\pi$
$367$	7016.00	0.997908	0.498954	+	0.866628i	$0.333718\pi$
$368$	0	0
$369$	−54.0000	−0.00761823
$370$	0	0
$371$	0	0
$372$	0	0
$373$	1010.00	0.140203	0.0701016	−	0.997540i	$-0.477668\pi$
$373$	1010.00	0.140203	0.0701016	+	0.997540i	$0.477668\pi$
$374$	0	0
$375$	375.000	0.0516398
$376$	0	0
$377$	7236.00	0.988522
$378$	0	0
$379$	4900.00	0.664106	0.332053	−	0.943261i	$-0.392259\pi$
$379$	4900.00	0.664106	0.332053	+	0.943261i	$0.392259\pi$
$380$	0	0
$381$	7032.00	0.945565
$382$	0	0
$383$	−7800.00	−1.04063	−0.520315	−	0.853974i	$-0.674186\pi$
$383$	−7800.00	−1.04063	−0.520315	+	0.853974i	$0.674186\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	108.000	0.0141859
$388$	0	0
$389$	12258.0	1.59770	0.798850	−	0.601530i	$-0.205442\pi$
$389$	12258.0	1.59770	0.798850	+	0.601530i	$0.205442\pi$
$390$	0	0
$391$	−10944.0	−1.41550
$392$	0	0
$393$	6492.00	0.833278
$394$	0	0
$395$	3440.00	0.438190
$396$	0	0
$397$	−5558.00	−0.702640	−0.351320	−	0.936255i	$-0.614267\pi$
$397$	−5558.00	−0.702640	−0.351320	+	0.936255i	$0.614267\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	1970.00	0.245329	0.122665	−	0.992448i	$-0.460856\pi$
$401$	1970.00	0.245329	0.122665	+	0.992448i	$0.460856\pi$
$402$	0	0
$403$	−14688.0	−1.81554
$404$	0	0
$405$	−405.000	−0.0496904
$406$	0	0
$407$	392.000	0.0477413
$408$	0	0
$409$	15626.0	1.88913	0.944567	−	0.328318i	$-0.106482\pi$
$409$	15626.0	1.88913	0.944567	+	0.328318i	$0.106482\pi$
$410$	0	0
$411$	8466.00	1.01605
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−5940.00	−0.702610
$416$	0	0
$417$	−5916.00	−0.694743
$418$	0	0
$419$	−5412.00	−0.631011	−0.315505	−	0.948924i	$-0.602174\pi$
$419$	−5412.00	−0.631011	−0.315505	+	0.948924i	$0.602174\pi$
$420$	0	0
$421$	10690.0	1.23753	0.618763	−	0.785577i	$-0.287634\pi$
$421$	10690.0	1.23753	0.618763	+	0.785577i	$0.287634\pi$
$422$	0	0
$423$	1800.00	0.206901
$424$	0	0
$425$	2850.00	0.325283
$426$	0	0
$427$	0	0
$428$	0	0
$429$	648.000	0.0729271
$430$	0	0
$431$	14048.0	1.57000	0.784998	−	0.619498i	$-0.212664\pi$
$431$	14048.0	1.57000	0.784998	+	0.619498i	$0.212664\pi$
$432$	0	0
$433$	17778.0	1.97311	0.986554	−	0.163433i	$-0.0522567\pi$
$433$	17778.0	1.97311	0.986554	+	0.163433i	$0.0522567\pi$
$434$	0	0
$435$	−2010.00	−0.221545
$436$	0	0
$437$	−4224.00	−0.462383
$438$	0	0
$439$	−7240.00	−0.787122	−0.393561	−	0.919299i	$-0.628757\pi$
$439$	−7240.00	−0.787122	−0.393561	+	0.919299i	$0.628757\pi$
$440$	0	0
$441$	−3087.00	−0.333333
$442$	0	0
$443$	11740.0	1.25911	0.629553	−	0.776957i	$-0.283238\pi$
$443$	11740.0	1.25911	0.629553	+	0.776957i	$0.283238\pi$
$444$	0	0
$445$	3470.00	0.369649
$446$	0	0
$447$	−4182.00	−0.442510
$448$	0	0
$449$	15234.0	1.60120	0.800598	−	0.599202i	$-0.204515\pi$
$449$	15234.0	1.60120	0.800598	+	0.599202i	$0.204515\pi$
$450$	0	0
$451$	−24.0000	−0.00250580
$452$	0	0
$453$	−6648.00	−0.689515
$454$	0	0
$455$	0	0
$456$	0	0
$457$	3866.00	0.395720	0.197860	−	0.980230i	$-0.436601\pi$
$457$	3866.00	0.395720	0.197860	+	0.980230i	$0.436601\pi$
$458$	0	0
$459$	−3078.00	−0.313004
$460$	0	0
$461$	1706.00	0.172356	0.0861782	−	0.996280i	$-0.472535\pi$
$461$	1706.00	0.172356	0.0861782	+	0.996280i	$0.472535\pi$
$462$	0	0
$463$	−3944.00	−0.395882	−0.197941	−	0.980214i	$-0.563425\pi$
$463$	−3944.00	−0.395882	−0.197941	+	0.980214i	$0.563425\pi$
$464$	0	0
$465$	4080.00	0.406893
$466$	0	0
$467$	−9452.00	−0.936588	−0.468294	−	0.883573i	$-0.655131\pi$
$467$	−9452.00	−0.936588	−0.468294	+	0.883573i	$0.655131\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	−2862.00	−0.279987
$472$	0	0
$473$	48.0000	0.00466605
$474$	0	0
$475$	1100.00	0.106256
$476$	0	0
$477$	−5886.00	−0.564993
$478$	0	0
$479$	12544.0	1.19656	0.598278	−	0.801289i	$-0.295852\pi$
$479$	12544.0	1.19656	0.598278	+	0.801289i	$0.295852\pi$
$480$	0	0
$481$	−5292.00	−0.501652
$482$	0	0
$483$	0	0
$484$	0	0
$485$	8630.00	0.807975
$486$	0	0
$487$	−7936.00	−0.738428	−0.369214	−	0.929344i	$-0.620373\pi$
$487$	−7936.00	−0.738428	−0.369214	+	0.929344i	$0.620373\pi$
$488$	0	0
$489$	10212.0	0.944382
$490$	0	0
$491$	−8412.00	−0.773174	−0.386587	−	0.922253i	$-0.626346\pi$
$491$	−8412.00	−0.773174	−0.386587	+	0.922253i	$0.626346\pi$
$492$	0	0
$493$	−15276.0	−1.39553
$494$	0	0
$495$	−180.000	−0.0163442
$496$	0	0
$497$	0	0
$498$	0	0
$499$	−15092.0	−1.35393	−0.676965	−	0.736016i	$-0.736705\pi$
$499$	−15092.0	−1.35393	−0.676965	+	0.736016i	$0.736705\pi$
$500$	0	0
$501$	−2496.00	−0.222581
$502$	0	0
$503$	−6112.00	−0.541790	−0.270895	−	0.962609i	$-0.587320\pi$
$503$	−6112.00	−0.541790	−0.270895	+	0.962609i	$0.587320\pi$
$504$	0	0
$505$	5910.00	0.520775
$506$	0	0
$507$	−2157.00	−0.188946
$508$	0	0
$509$	−2534.00	−0.220663	−0.110332	−	0.993895i	$-0.535191\pi$
$509$	−2534.00	−0.220663	−0.110332	+	0.993895i	$0.535191\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−1188.00	−0.102245
$514$	0	0
$515$	9840.00	0.841946
$516$	0	0
$517$	800.000	0.0680541
$518$	0	0
$519$	−1086.00	−0.0918499
$520$	0	0
$521$	−9894.00	−0.831985	−0.415992	−	0.909368i	$-0.636566\pi$
$521$	−9894.00	−0.831985	−0.415992	+	0.909368i	$0.636566\pi$
$522$	0	0
$523$	16172.0	1.35211	0.676054	−	0.736852i	$-0.263689\pi$
$523$	16172.0	1.35211	0.676054	+	0.736852i	$0.263689\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	31008.0	2.56305
$528$	0	0
$529$	−2951.00	−0.242541
$530$	0	0
$531$	324.000	0.0264791
$532$	0	0
$533$	324.000	0.0263302
$534$	0	0
$535$	−3980.00	−0.321627
$536$	0	0
$537$	9756.00	0.783990
$538$	0	0
$539$	−1372.00	−0.109640
$540$	0	0
$541$	6138.00	0.487788	0.243894	−	0.969802i	$-0.421575\pi$
$541$	6138.00	0.487788	0.243894	+	0.969802i	$0.421575\pi$
$542$	0	0
$543$	9258.00	0.731674
$544$	0	0
$545$	1710.00	0.134401
$546$	0	0
$547$	−21852.0	−1.70809	−0.854044	−	0.520201i	$-0.825857\pi$
$547$	−21852.0	−1.70809	−0.854044	+	0.520201i	$0.825857\pi$
$548$	0	0
$549$	3978.00	0.309248
$550$	0	0
$551$	−5896.00	−0.455859
$552$	0	0
$553$	0	0
$554$	0	0
$555$	1470.00	0.112429
$556$	0	0
$557$	1962.00	0.149251	0.0746253	−	0.997212i	$-0.476224\pi$
$557$	1962.00	0.149251	0.0746253	+	0.997212i	$0.476224\pi$
$558$	0	0
$559$	−648.000	−0.0490295
$560$	0	0
$561$	−1368.00	−0.102954
$562$	0	0
$563$	−10876.0	−0.814154	−0.407077	−	0.913394i	$-0.633452\pi$
$563$	−10876.0	−0.814154	−0.407077	+	0.913394i	$0.633452\pi$
$564$	0	0
$565$	−570.000	−0.0424426
$566$	0	0
$567$	0	0
$568$	0	0
$569$	5610.00	0.413328	0.206664	−	0.978412i	$-0.433739\pi$
$569$	5610.00	0.413328	0.206664	+	0.978412i	$0.433739\pi$
$570$	0	0
$571$	5076.00	0.372021	0.186010	−	0.982548i	$-0.440444\pi$
$571$	5076.00	0.372021	0.186010	+	0.982548i	$0.440444\pi$
$572$	0	0
$573$	−12240.0	−0.892379
$574$	0	0
$575$	−2400.00	−0.174064
$576$	0	0
$577$	−6526.00	−0.470851	−0.235425	−	0.971892i	$-0.575648\pi$
$577$	−6526.00	−0.470851	−0.235425	+	0.971892i	$0.575648\pi$
$578$	0	0
$579$	7962.00	0.571484
$580$	0	0
$581$	0	0
$582$	0	0
$583$	−2616.00	−0.185838
$584$	0	0
$585$	2430.00	0.171740
$586$	0	0
$587$	2332.00	0.163973	0.0819863	−	0.996633i	$-0.473874\pi$
$587$	2332.00	0.163973	0.0819863	+	0.996633i	$0.473874\pi$
$588$	0	0
$589$	11968.0	0.837237
$590$	0	0
$591$	4602.00	0.320306
$592$	0	0
$593$	−9582.00	−0.663551	−0.331775	−	0.943358i	$-0.607648\pi$
$593$	−9582.00	−0.663551	−0.331775	+	0.943358i	$0.607648\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	13032.0	0.893407
$598$	0	0
$599$	17624.0	1.20217	0.601083	−	0.799187i	$-0.294736\pi$
$599$	17624.0	1.20217	0.601083	+	0.799187i	$0.294736\pi$
$600$	0	0
$601$	−21238.0	−1.44146	−0.720729	−	0.693217i	$-0.756193\pi$
$601$	−21238.0	−1.44146	−0.720729	+	0.693217i	$0.756193\pi$
$602$	0	0
$603$	−1692.00	−0.114268
$604$	0	0
$605$	6575.00	0.441838
$606$	0	0
$607$	−13000.0	−0.869281	−0.434641	−	0.900604i	$-0.643125\pi$
$607$	−13000.0	−0.869281	−0.434641	+	0.900604i	$0.643125\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−10800.0	−0.715092
$612$	0	0
$613$	−9214.00	−0.607096	−0.303548	−	0.952816i	$-0.598171\pi$
$613$	−9214.00	−0.607096	−0.303548	+	0.952816i	$0.598171\pi$
$614$	0	0
$615$	−90.0000	−0.00590106
$616$	0	0
$617$	4474.00	0.291923	0.145961	−	0.989290i	$-0.453372\pi$
$617$	4474.00	0.291923	0.145961	+	0.989290i	$0.453372\pi$
$618$	0	0
$619$	−12556.0	−0.815296	−0.407648	−	0.913139i	$-0.633651\pi$
$619$	−12556.0	−0.815296	−0.407648	+	0.913139i	$0.633651\pi$
$620$	0	0
$621$	2592.00	0.167493
$622$	0	0
$623$	0	0
$624$	0	0
$625$	625.000	0.0400000
$626$	0	0
$627$	−528.000	−0.0336304
$628$	0	0
$629$	11172.0	0.708198
$630$	0	0
$631$	−26936.0	−1.69937	−0.849687	−	0.527287i	$-0.823209\pi$
$631$	−26936.0	−1.69937	−0.849687	+	0.527287i	$0.823209\pi$
$632$	0	0
$633$	4140.00	0.259953
$634$	0	0
$635$	11720.0	0.732432
$636$	0	0
$637$	18522.0	1.15207
$638$	0	0
$639$	5688.00	0.352134
$640$	0	0
$641$	−19134.0	−1.17901	−0.589507	−	0.807764i	$-0.700678\pi$
$641$	−19134.0	−1.17901	−0.589507	+	0.807764i	$0.700678\pi$
$642$	0	0
$643$	12436.0	0.762718	0.381359	−	0.924427i	$-0.375456\pi$
$643$	12436.0	0.762718	0.381359	+	0.924427i	$0.375456\pi$
$644$	0	0
$645$	180.000	0.0109884
$646$	0	0
$647$	2784.00	0.169166	0.0845829	−	0.996416i	$-0.473044\pi$
$647$	2784.00	0.169166	0.0845829	+	0.996416i	$0.473044\pi$
$648$	0	0
$649$	144.000	0.00870954
$650$	0	0
$651$	0	0
$652$	0	0
$653$	−7318.00	−0.438554	−0.219277	−	0.975663i	$-0.570370\pi$
$653$	−7318.00	−0.438554	−0.219277	+	0.975663i	$0.570370\pi$
$654$	0	0
$655$	10820.0	0.645454
$656$	0	0
$657$	−3510.00	−0.208429
$658$	0	0
$659$	8108.00	0.479276	0.239638	−	0.970862i	$-0.422971\pi$
$659$	8108.00	0.479276	0.239638	+	0.970862i	$0.422971\pi$
$660$	0	0
$661$	−1230.00	−0.0723774	−0.0361887	−	0.999345i	$-0.511522\pi$
$661$	−1230.00	−0.0723774	−0.0361887	+	0.999345i	$0.511522\pi$
$662$	0	0
$663$	18468.0	1.08181
$664$	0	0
$665$	0	0
$666$	0	0
$667$	12864.0	0.746771
$668$	0	0
$669$	−15672.0	−0.905702
$670$	0	0
$671$	1768.00	0.101718
$672$	0	0
$673$	−14078.0	−0.806340	−0.403170	−	0.915125i	$-0.632092\pi$
$673$	−14078.0	−0.806340	−0.403170	+	0.915125i	$0.632092\pi$
$674$	0	0
$675$	−675.000	−0.0384900
$676$	0	0
$677$	−25246.0	−1.43321	−0.716605	−	0.697480i	$-0.754305\pi$
$677$	−25246.0	−1.43321	−0.716605	+	0.697480i	$0.754305\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	−10092.0	−0.567880
$682$	0	0
$683$	24332.0	1.36316	0.681580	−	0.731744i	$-0.261293\pi$
$683$	24332.0	1.36316	0.681580	+	0.731744i	$0.261293\pi$
$684$	0	0
$685$	14110.0	0.787030
$686$	0	0
$687$	11994.0	0.666084
$688$	0	0
$689$	35316.0	1.95273
$690$	0	0
$691$	19036.0	1.04799	0.523997	−	0.851720i	$-0.324440\pi$
$691$	19036.0	1.04799	0.523997	+	0.851720i	$0.324440\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−9860.00	−0.538145
$696$	0	0
$697$	−684.000	−0.0371712
$698$	0	0
$699$	10770.0	0.582774
$700$	0	0
$701$	−28806.0	−1.55205	−0.776025	−	0.630702i	$-0.782767\pi$
$701$	−28806.0	−1.55205	−0.776025	+	0.630702i	$0.782767\pi$
$702$	0	0
$703$	4312.00	0.231337
$704$	0	0
$705$	3000.00	0.160265
$706$	0	0
$707$	0	0
$708$	0	0
$709$	25090.0	1.32902	0.664510	−	0.747280i	$-0.268640\pi$
$709$	25090.0	1.32902	0.664510	+	0.747280i	$0.268640\pi$
$710$	0	0
$711$	−6192.00	−0.326608
$712$	0	0
$713$	−26112.0	−1.37153
$714$	0	0
$715$	1080.00	0.0564891
$716$	0	0
$717$	−3312.00	−0.172509
$718$	0	0
$719$	−36432.0	−1.88969	−0.944843	−	0.327523i	$-0.893786\pi$
$719$	−36432.0	−1.88969	−0.944843	+	0.327523i	$0.893786\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−4854.00	−0.249685
$724$	0	0
$725$	−3350.00	−0.171608
$726$	0	0
$727$	−21616.0	−1.10274	−0.551371	−	0.834260i	$-0.685895\pi$
$727$	−21616.0	−1.10274	−0.551371	+	0.834260i	$0.685895\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	1368.00	0.0692166
$732$	0	0
$733$	−28102.0	−1.41606	−0.708029	−	0.706183i	$-0.750416\pi$
$733$	−28102.0	−1.41606	−0.708029	+	0.706183i	$0.750416\pi$
$734$	0	0
$735$	−5145.00	−0.258199
$736$	0	0
$737$	−752.000	−0.0375852
$738$	0	0
$739$	764.000	0.0380300	0.0190150	−	0.999819i	$-0.493947\pi$
$739$	764.000	0.0380300	0.0190150	+	0.999819i	$0.493947\pi$
$740$	0	0
$741$	7128.00	0.353379
$742$	0	0
$743$	−6256.00	−0.308897	−0.154448	−	0.988001i	$-0.549360\pi$
$743$	−6256.00	−0.308897	−0.154448	+	0.988001i	$0.549360\pi$
$744$	0	0
$745$	−6970.00	−0.342766
$746$	0	0
$747$	10692.0	0.523695
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−1184.00	−0.0575297	−0.0287648	−	0.999586i	$-0.509157\pi$
$751$	−1184.00	−0.0575297	−0.0287648	+	0.999586i	$0.509157\pi$
$752$	0	0
$753$	−17340.0	−0.839183
$754$	0	0
$755$	−11080.0	−0.534096
$756$	0	0
$757$	−26446.0	−1.26974	−0.634872	−	0.772617i	$-0.718947\pi$
$757$	−26446.0	−1.26974	−0.634872	+	0.772617i	$0.718947\pi$
$758$	0	0
$759$	1152.00	0.0550922
$760$	0	0
$761$	36778.0	1.75191	0.875954	−	0.482395i	$-0.160233\pi$
$761$	36778.0	1.75191	0.875954	+	0.482395i	$0.160233\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−5130.00	−0.242452
$766$	0	0
$767$	−1944.00	−0.0915173
$768$	0	0
$769$	−10302.0	−0.483094	−0.241547	−	0.970389i	$-0.577655\pi$
$769$	−10302.0	−0.483094	−0.241547	+	0.970389i	$0.577655\pi$
$770$	0	0
$771$	−7782.00	−0.363504
$772$	0	0
$773$	4674.00	0.217480	0.108740	−	0.994070i	$-0.465318\pi$
$773$	4674.00	0.217480	0.108740	+	0.994070i	$0.465318\pi$
$774$	0	0
$775$	6800.00	0.315178
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−264.000	−0.0121422
$780$	0	0
$781$	2528.00	0.115825
$782$	0	0
$783$	3618.00	0.165130
$784$	0	0
$785$	−4770.00	−0.216877
$786$	0	0
$787$	−23084.0	−1.04556	−0.522780	−	0.852468i	$-0.675105\pi$
$787$	−23084.0	−1.04556	−0.522780	+	0.852468i	$0.675105\pi$
$788$	0	0
$789$	11088.0	0.500308
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−23868.0	−1.06882
$794$	0	0
$795$	−9810.00	−0.437641
$796$	0	0
$797$	−10694.0	−0.475283	−0.237642	−	0.971353i	$-0.576374\pi$
$797$	−10694.0	−0.475283	−0.237642	+	0.971353i	$0.576374\pi$
$798$	0	0
$799$	22800.0	1.00952
$800$	0	0
$801$	−6246.00	−0.275520
$802$	0	0
$803$	−1560.00	−0.0685569
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−6750.00	−0.294438
$808$	0	0
$809$	9594.00	0.416943	0.208472	−	0.978028i	$-0.433151\pi$
$809$	9594.00	0.416943	0.208472	+	0.978028i	$0.433151\pi$
$810$	0	0
$811$	10244.0	0.443546	0.221773	−	0.975098i	$-0.428816\pi$
$811$	10244.0	0.443546	0.221773	+	0.975098i	$0.428816\pi$
$812$	0	0
$813$	6624.00	0.285749
$814$	0	0
$815$	17020.0	0.731515
$816$	0	0
$817$	528.000	0.0226100
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−1390.00	−0.0590881	−0.0295441	−	0.999563i	$-0.509406\pi$
$821$	−1390.00	−0.0590881	−0.0295441	+	0.999563i	$0.509406\pi$
$822$	0	0
$823$	8448.00	0.357811	0.178906	−	0.983866i	$-0.442744\pi$
$823$	8448.00	0.357811	0.178906	+	0.983866i	$0.442744\pi$
$824$	0	0
$825$	−300.000	−0.0126602
$826$	0	0
$827$	41484.0	1.74430	0.872152	−	0.489234i	$-0.162724\pi$
$827$	41484.0	1.74430	0.872152	+	0.489234i	$0.162724\pi$
$828$	0	0
$829$	31610.0	1.32432	0.662160	−	0.749363i	$-0.269640\pi$
$829$	31610.0	1.32432	0.662160	+	0.749363i	$0.269640\pi$
$830$	0	0
$831$	−5046.00	−0.210642
$832$	0	0
$833$	−39102.0	−1.62642
$834$	0	0
$835$	−4160.00	−0.172410
$836$	0	0
$837$	−7344.00	−0.303280
$838$	0	0
$839$	−38264.0	−1.57452	−0.787259	−	0.616623i	$-0.788500\pi$
$839$	−38264.0	−1.57452	−0.787259	+	0.616623i	$0.788500\pi$
$840$	0	0
$841$	−6433.00	−0.263766
$842$	0	0
$843$	−21918.0	−0.895488
$844$	0	0
$845$	−3595.00	−0.146357
$846$	0	0
$847$	0	0
$848$	0	0
$849$	24492.0	0.990063
$850$	0	0
$851$	−9408.00	−0.378968
$852$	0	0
$853$	−30350.0	−1.21825	−0.609123	−	0.793076i	$-0.708479\pi$
$853$	−30350.0	−1.21825	−0.609123	+	0.793076i	$0.708479\pi$
$854$	0	0
$855$	−1980.00	−0.0791983
$856$	0	0
$857$	−12566.0	−0.500871	−0.250435	−	0.968133i	$-0.580574\pi$
$857$	−12566.0	−0.500871	−0.250435	+	0.968133i	$0.580574\pi$
$858$	0	0
$859$	11812.0	0.469174	0.234587	−	0.972095i	$-0.424626\pi$
$859$	11812.0	0.469174	0.234587	+	0.972095i	$0.424626\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	31496.0	1.24234	0.621168	−	0.783677i	$-0.286658\pi$
$863$	31496.0	1.24234	0.621168	+	0.783677i	$0.286658\pi$
$864$	0	0
$865$	−1810.00	−0.0711466
$866$	0	0
$867$	−24249.0	−0.949872
$868$	0	0
$869$	−2752.00	−0.107428
$870$	0	0
$871$	10152.0	0.394934
$872$	0	0
$873$	−15534.0	−0.602229
$874$	0	0
$875$	0	0
$876$	0	0
$877$	−7414.00	−0.285465	−0.142733	−	0.989761i	$-0.545589\pi$
$877$	−7414.00	−0.285465	−0.142733	+	0.989761i	$0.545589\pi$
$878$	0	0
$879$	−1542.00	−0.0591699
$880$	0	0
$881$	−22190.0	−0.848581	−0.424291	−	0.905526i	$-0.639477\pi$
$881$	−22190.0	−0.848581	−0.424291	+	0.905526i	$0.639477\pi$
$882$	0	0
$883$	−10172.0	−0.387673	−0.193836	−	0.981034i	$-0.562093\pi$
$883$	−10172.0	−0.387673	−0.193836	+	0.981034i	$0.562093\pi$
$884$	0	0
$885$	540.000	0.0205106
$886$	0	0
$887$	20784.0	0.786763	0.393381	−	0.919375i	$-0.371305\pi$
$887$	20784.0	0.786763	0.393381	+	0.919375i	$0.371305\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	324.000	0.0121823
$892$	0	0
$893$	8800.00	0.329766
$894$	0	0
$895$	16260.0	0.607276
$896$	0	0
$897$	−15552.0	−0.578892
$898$	0	0
$899$	−36448.0	−1.35218
$900$	0	0
$901$	−74556.0	−2.75674
$902$	0	0
$903$	0	0
$904$	0	0
$905$	15430.0	0.566752
$906$	0	0
$907$	−7652.00	−0.280133	−0.140066	−	0.990142i	$-0.544732\pi$
$907$	−7652.00	−0.280133	−0.140066	+	0.990142i	$0.544732\pi$
$908$	0	0
$909$	−10638.0	−0.388163
$910$	0	0
$911$	−19296.0	−0.701762	−0.350881	−	0.936420i	$-0.614118\pi$
$911$	−19296.0	−0.701762	−0.350881	+	0.936420i	$0.614118\pi$
$912$	0	0
$913$	4752.00	0.172254
$914$	0	0
$915$	6630.00	0.239542
$916$	0	0
$917$	0	0
$918$	0	0
$919$	35896.0	1.28847	0.644233	−	0.764830i	$-0.277177\pi$
$919$	35896.0	1.28847	0.644233	+	0.764830i	$0.277177\pi$
$920$	0	0
$921$	7428.00	0.265756
$922$	0	0
$923$	−34128.0	−1.21705
$924$	0	0
$925$	2450.00	0.0870870
$926$	0	0
$927$	−17712.0	−0.627550
$928$	0	0
$929$	−16350.0	−0.577423	−0.288712	−	0.957416i	$-0.593227\pi$
$929$	−16350.0	−0.577423	−0.288712	+	0.957416i	$0.593227\pi$
$930$	0	0
$931$	−15092.0	−0.531279
$932$	0	0
$933$	6888.00	0.241697
$934$	0	0
$935$	−2280.00	−0.0797476
$936$	0	0
$937$	−19686.0	−0.686354	−0.343177	−	0.939271i	$-0.611503\pi$
$937$	−19686.0	−0.686354	−0.343177	+	0.939271i	$0.611503\pi$
$938$	0	0
$939$	29634.0	1.02989
$940$	0	0
$941$	−56246.0	−1.94853	−0.974265	−	0.225405i	$-0.927630\pi$
$941$	−56246.0	−1.94853	−0.974265	+	0.225405i	$0.927630\pi$
$942$	0	0
$943$	576.000	0.0198909
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−11436.0	−0.392418	−0.196209	−	0.980562i	$-0.562863\pi$
$947$	−11436.0	−0.392418	−0.196209	+	0.980562i	$0.562863\pi$
$948$	0	0
$949$	21060.0	0.720376
$950$	0	0
$951$	−6414.00	−0.218705
$952$	0	0
$953$	−22582.0	−0.767579	−0.383789	−	0.923421i	$-0.625381\pi$
$953$	−22582.0	−0.767579	−0.383789	+	0.923421i	$0.625381\pi$
$954$	0	0
$955$	−20400.0	−0.691234
$956$	0	0
$957$	1608.00	0.0543148
$958$	0	0
$959$	0	0
$960$	0	0
$961$	44193.0	1.48343
$962$	0	0
$963$	7164.00	0.239727
$964$	0	0
$965$	13270.0	0.442670
$966$	0	0
$967$	2112.00	0.0702351	0.0351175	−	0.999383i	$-0.488819\pi$
$967$	2112.00	0.0702351	0.0351175	+	0.999383i	$0.488819\pi$
$968$	0	0
$969$	−15048.0	−0.498877
$970$	0	0
$971$	−47964.0	−1.58521	−0.792605	−	0.609736i	$-0.791275\pi$
$971$	−47964.0	−1.58521	−0.792605	+	0.609736i	$0.791275\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	4050.00	0.133030
$976$	0	0
$977$	−10510.0	−0.344160	−0.172080	−	0.985083i	$-0.555049\pi$
$977$	−10510.0	−0.344160	−0.172080	+	0.985083i	$0.555049\pi$
$978$	0	0
$979$	−2776.00	−0.0906245
$980$	0	0
$981$	−3078.00	−0.100176
$982$	0	0
$983$	−11488.0	−0.372747	−0.186373	−	0.982479i	$-0.559673\pi$
$983$	−11488.0	−0.372747	−0.186373	+	0.982479i	$0.559673\pi$
$984$	0	0
$985$	7670.00	0.248108
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1152.00	−0.0370389
$990$	0	0
$991$	23120.0	0.741101	0.370550	−	0.928812i	$-0.379169\pi$
$991$	23120.0	0.741101	0.370550	+	0.928812i	$0.379169\pi$
$992$	0	0
$993$	19380.0	0.619341
$994$	0	0
$995$	21720.0	0.692030
$996$	0	0
$997$	−30078.0	−0.955446	−0.477723	−	0.878510i	$-0.658538\pi$
$997$	−30078.0	−0.955446	−0.477723	+	0.878510i	$0.658538\pi$
$998$	0	0
$999$	−2646.00	−0.0837995

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	960.4.a.e.1.1		1
4.3	odd	2		960.4.a.x.1.1		1
8.3	odd	2		120.4.a.d.1.1	✓	1
8.5	even	2		240.4.a.k.1.1		1
24.5	odd	2		720.4.a.i.1.1		1
24.11	even	2		360.4.a.c.1.1		1
40.3	even	4		600.4.f.d.49.1		2
40.13	odd	4		1200.4.f.l.49.2		2
40.19	odd	2		600.4.a.m.1.1		1
40.27	even	4		600.4.f.d.49.2		2
40.29	even	2		1200.4.a.j.1.1		1
40.37	odd	4		1200.4.f.l.49.1		2
120.59	even	2		1800.4.a.s.1.1		1
120.83	odd	4		1800.4.f.m.649.2		2
120.107	odd	4		1800.4.f.m.649.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.4.a.d.1.1	✓	1	8.3	odd	2
240.4.a.k.1.1		1	8.5	even	2
360.4.a.c.1.1		1	24.11	even	2
600.4.a.m.1.1		1	40.19	odd	2
600.4.f.d.49.1		2	40.3	even	4
600.4.f.d.49.2		2	40.27	even	4
720.4.a.i.1.1		1	24.5	odd	2
960.4.a.e.1.1		1	1.1	even	1	trivial
960.4.a.x.1.1		1	4.3	odd	2
1200.4.a.j.1.1		1	40.29	even	2
1200.4.f.l.49.1		2	40.37	odd	4
1200.4.f.l.49.2		2	40.13	odd	4
1800.4.a.s.1.1		1	120.59	even	2
1800.4.f.m.649.1		2	120.107	odd	4
1800.4.f.m.649.2		2	120.83	odd	4

Overview	Random
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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 $\F_{q}$

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Newspace parameters

Newform invariants

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$q$ -expansion

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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	960.4.a.e.1.1

Level	$960$
Weight	$4$
Character	960.1
Self dual	yes
Analytic conductor	$56.642$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$