LMFDB - Embedded newform 1350.4.a.f.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1350,4,Mod(1,1350)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1350, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("1350.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1350 = 2 \cdot 3^{3} \cdot 5^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1350.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:	$79.6525785077$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 270)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1350.1

$q$ -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^{2} +4.00000 q^{4} -8.00000 q^{7} -8.00000 q^{8} +18.0000 q^{11} -8.00000 q^{13} +16.0000 q^{14} +16.0000 q^{16} -15.0000 q^{17} +23.0000 q^{19} -36.0000 q^{22} -63.0000 q^{23} +16.0000 q^{26} -32.0000 q^{28} +156.000 q^{29} -85.0000 q^{31} -32.0000 q^{32} +30.0000 q^{34} -74.0000 q^{37} -46.0000 q^{38} +246.000 q^{41} +190.000 q^{43} +72.0000 q^{44} +126.000 q^{46} -288.000 q^{47} -279.000 q^{49} -32.0000 q^{52} +177.000 q^{53} +64.0000 q^{56} -312.000 q^{58} +792.000 q^{59} -907.000 q^{61} +170.000 q^{62} +64.0000 q^{64} +322.000 q^{67} -60.0000 q^{68} -270.000 q^{71} -254.000 q^{73} +148.000 q^{74} +92.0000 q^{76} -144.000 q^{77} -1123.00 q^{79} -492.000 q^{82} +771.000 q^{83} -380.000 q^{86} -144.000 q^{88} -198.000 q^{89} +64.0000 q^{91} -252.000 q^{92} +576.000 q^{94} +1192.00 q^{97} +558.000 q^{98} +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

)

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

$2$	−2.00000	−0.707107
$2$	−2.00000	−0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0	0
$5$	0	0
$6$	0	0
$7$	−8.00000	−0.431959	−0.215980	−	0.976398i	$-0.569295\pi$
$7$	−8.00000	−0.431959	−0.215980	+	0.976398i	$0.569295\pi$
$8$	−8.00000	−0.353553
$9$	0	0
$10$	0	0
$11$	18.0000	0.493382	0.246691	−	0.969094i	$-0.420657\pi$
$11$	18.0000	0.493382	0.246691	+	0.969094i	$0.420657\pi$
$12$	0	0
$13$	−8.00000	−0.170677	−0.0853385	−	0.996352i	$-0.527197\pi$
$13$	−8.00000	−0.170677	−0.0853385	+	0.996352i	$0.527197\pi$
$14$	16.0000	0.305441
$15$	0	0
$16$	16.0000	0.250000
$17$	−15.0000	−0.214002	−0.107001	−	0.994259i	$-0.534125\pi$
$17$	−15.0000	−0.214002	−0.107001	+	0.994259i	$0.534125\pi$
$18$	0	0
$19$	23.0000	0.277714	0.138857	−	0.990312i	$-0.455657\pi$
$19$	23.0000	0.277714	0.138857	+	0.990312i	$0.455657\pi$
$20$	0	0
$21$	0	0
$22$	−36.0000	−0.348874
$23$	−63.0000	−0.571148	−0.285574	−	0.958357i	$-0.592184\pi$
$23$	−63.0000	−0.571148	−0.285574	+	0.958357i	$0.592184\pi$
$24$	0	0
$25$	0	0
$26$	16.0000	0.120687
$27$	0	0
$28$	−32.0000	−0.215980
$29$	156.000	0.998913	0.499456	−	0.866339i	$-0.333533\pi$
$29$	156.000	0.998913	0.499456	+	0.866339i	$0.333533\pi$
$30$	0	0
$31$	−85.0000	−0.492466	−0.246233	−	0.969211i	$-0.579193\pi$
$31$	−85.0000	−0.492466	−0.246233	+	0.969211i	$0.579193\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	30.0000	0.151322
$35$	0	0
$36$	0	0
$37$	−74.0000	−0.328798	−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−74.0000	−0.328798	−0.164399	+	0.986394i	$0.552568\pi$
$38$	−46.0000	−0.196373
$39$	0	0
$40$	0	0
$41$	246.000	0.937043	0.468521	−	0.883452i	$-0.344787\pi$
$41$	246.000	0.937043	0.468521	+	0.883452i	$0.344787\pi$
$42$	0	0
$43$	190.000	0.673831	0.336915	−	0.941535i	$-0.390616\pi$
$43$	190.000	0.673831	0.336915	+	0.941535i	$0.390616\pi$
$44$	72.0000	0.246691
$45$	0	0
$46$	126.000	0.403863
$47$	−288.000	−0.893811	−0.446906	−	0.894581i	$-0.647474\pi$
$47$	−288.000	−0.893811	−0.446906	+	0.894581i	$0.647474\pi$
$48$	0	0
$49$	−279.000	−0.813411
$50$	0	0
$51$	0	0
$52$	−32.0000	−0.0853385
$53$	177.000	0.458732	0.229366	−	0.973340i	$-0.426335\pi$
$53$	177.000	0.458732	0.229366	+	0.973340i	$0.426335\pi$
$54$	0	0
$55$	0	0
$56$	64.0000	0.152721
$57$	0	0
$58$	−312.000	−0.706338
$59$	792.000	1.74762	0.873810	−	0.486267i	$-0.161642\pi$
$59$	792.000	1.74762	0.873810	+	0.486267i	$0.161642\pi$
$60$	0	0
$61$	−907.000	−1.90376	−0.951881	−	0.306469i	$-0.900853\pi$
$61$	−907.000	−1.90376	−0.951881	+	0.306469i	$0.900853\pi$
$62$	170.000	0.348226
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	322.000	0.587143	0.293571	−	0.955937i	$-0.405156\pi$
$67$	322.000	0.587143	0.293571	+	0.955937i	$0.405156\pi$
$68$	−60.0000	−0.107001
$69$	0	0
$70$	0	0
$71$	−270.000	−0.451311	−0.225656	−	0.974207i	$-0.572452\pi$
$71$	−270.000	−0.451311	−0.225656	+	0.974207i	$0.572452\pi$
$72$	0	0
$73$	−254.000	−0.407239	−0.203620	−	0.979050i	$-0.565271\pi$
$73$	−254.000	−0.407239	−0.203620	+	0.979050i	$0.565271\pi$
$74$	148.000	0.232495
$75$	0	0
$76$	92.0000	0.138857
$77$	−144.000	−0.213121
$78$	0	0
$79$	−1123.00	−1.59933	−0.799667	−	0.600444i	$-0.794991\pi$
$79$	−1123.00	−1.59933	−0.799667	+	0.600444i	$0.794991\pi$
$80$	0	0
$81$	0	0
$82$	−492.000	−0.662589
$83$	771.000	1.01962	0.509809	−	0.860288i	$-0.329716\pi$
$83$	771.000	1.01962	0.509809	+	0.860288i	$0.329716\pi$
$84$	0	0
$85$	0	0
$86$	−380.000	−0.476470
$87$	0	0
$88$	−144.000	−0.174437
$89$	−198.000	−0.235820	−0.117910	−	0.993024i	$-0.537619\pi$
$89$	−198.000	−0.235820	−0.117910	+	0.993024i	$0.537619\pi$
$90$	0	0
$91$	64.0000	0.0737255
$92$	−252.000	−0.285574
$93$	0	0
$94$	576.000	0.632020
$95$	0	0
$96$	0	0
$97$	1192.00	1.24772	0.623862	−	0.781534i	$-0.285563\pi$
$97$	1192.00	1.24772	0.623862	+	0.781534i	$0.285563\pi$
$98$	558.000	0.575168
$99$	0	0
$100$	0	0
$101$	−1692.00	−1.66693	−0.833467	−	0.552570i	$-0.813647\pi$
$101$	−1692.00	−1.66693	−0.833467	+	0.552570i	$0.813647\pi$
$102$	0	0
$103$	−1748.00	−1.67219	−0.836095	−	0.548585i	$-0.815167\pi$
$103$	−1748.00	−1.67219	−0.836095	+	0.548585i	$0.815167\pi$
$104$	64.0000	0.0603434
$105$	0	0
$106$	−354.000	−0.324373
$107$	948.000	0.856510	0.428255	−	0.903658i	$-0.359128\pi$
$107$	948.000	0.856510	0.428255	+	0.903658i	$0.359128\pi$
$108$	0	0
$109$	593.000	0.521093	0.260546	−	0.965461i	$-0.416097\pi$
$109$	593.000	0.521093	0.260546	+	0.965461i	$0.416097\pi$
$110$	0	0
$111$	0	0
$112$	−128.000	−0.107990
$113$	1062.00	0.884111	0.442056	−	0.896988i	$-0.354249\pi$
$113$	1062.00	0.884111	0.442056	+	0.896988i	$0.354249\pi$
$114$	0	0
$115$	0	0
$116$	624.000	0.499456
$117$	0	0
$118$	−1584.00	−1.23575
$119$	120.000	0.0924402
$120$	0	0
$121$	−1007.00	−0.756574
$122$	1814.00	1.34616
$123$	0	0
$124$	−340.000	−0.246233
$125$	0	0
$126$	0	0
$127$	−326.000	−0.227778	−0.113889	−	0.993493i	$-0.536331\pi$
$127$	−326.000	−0.227778	−0.113889	+	0.993493i	$0.536331\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	0	0
$131$	−990.000	−0.660280	−0.330140	−	0.943932i	$-0.607096\pi$
$131$	−990.000	−0.660280	−0.330140	+	0.943932i	$0.607096\pi$
$132$	0	0
$133$	−184.000	−0.119961
$134$	−644.000	−0.415173
$135$	0	0
$136$	120.000	0.0756611
$137$	147.000	0.0916720	0.0458360	−	0.998949i	$-0.485405\pi$
$137$	147.000	0.0916720	0.0458360	+	0.998949i	$0.485405\pi$
$138$	0	0
$139$	1604.00	0.978773	0.489387	−	0.872067i	$-0.337221\pi$
$139$	1604.00	0.978773	0.489387	+	0.872067i	$0.337221\pi$
$140$	0	0
$141$	0	0
$142$	540.000	0.319125
$143$	−144.000	−0.0842090
$144$	0	0
$145$	0	0
$146$	508.000	0.287962
$147$	0	0
$148$	−296.000	−0.164399
$149$	1218.00	0.669681	0.334840	−	0.942275i	$-0.391318\pi$
$149$	1218.00	0.669681	0.334840	+	0.942275i	$0.391318\pi$
$150$	0	0
$151$	−2248.00	−1.21152	−0.605760	−	0.795647i	$-0.707131\pi$
$151$	−2248.00	−1.21152	−0.605760	+	0.795647i	$0.707131\pi$
$152$	−184.000	−0.0981866
$153$	0	0
$154$	288.000	0.150699
$155$	0	0
$156$	0	0
$157$	2998.00	1.52399	0.761995	−	0.647583i	$-0.224220\pi$
$157$	2998.00	1.52399	0.761995	+	0.647583i	$0.224220\pi$
$158$	2246.00	1.13090
$159$	0	0
$160$	0	0
$161$	504.000	0.246713
$162$	0	0
$163$	−3470.00	−1.66743	−0.833716	−	0.552194i	$-0.813791\pi$
$163$	−3470.00	−1.66743	−0.833716	+	0.552194i	$0.813791\pi$
$164$	984.000	0.468521
$165$	0	0
$166$	−1542.00	−0.720978
$167$	−387.000	−0.179323	−0.0896616	−	0.995972i	$-0.528579\pi$
$167$	−387.000	−0.179323	−0.0896616	+	0.995972i	$0.528579\pi$
$168$	0	0
$169$	−2133.00	−0.970869
$170$	0	0
$171$	0	0
$172$	760.000	0.336915
$173$	855.000	0.375748	0.187874	−	0.982193i	$-0.439840\pi$
$173$	855.000	0.375748	0.187874	+	0.982193i	$0.439840\pi$
$174$	0	0
$175$	0	0
$176$	288.000	0.123346
$177$	0	0
$178$	396.000	0.166750
$179$	−264.000	−0.110236	−0.0551181	−	0.998480i	$-0.517554\pi$
$179$	−264.000	−0.110236	−0.0551181	+	0.998480i	$0.517554\pi$
$180$	0	0
$181$	−2551.00	−1.04759	−0.523797	−	0.851843i	$-0.675485\pi$
$181$	−2551.00	−1.04759	−0.523797	+	0.851843i	$0.675485\pi$
$182$	−128.000	−0.0521318
$183$	0	0
$184$	504.000	0.201931
$185$	0	0
$186$	0	0
$187$	−270.000	−0.105585
$188$	−1152.00	−0.446906
$189$	0	0
$190$	0	0
$191$	−2238.00	−0.847832	−0.423916	−	0.905701i	$-0.639345\pi$
$191$	−2238.00	−0.847832	−0.423916	+	0.905701i	$0.639345\pi$
$192$	0	0
$193$	−2180.00	−0.813056	−0.406528	−	0.913638i	$-0.633261\pi$
$193$	−2180.00	−0.813056	−0.406528	+	0.913638i	$0.633261\pi$
$194$	−2384.00	−0.882274
$195$	0	0
$196$	−1116.00	−0.406706
$197$	−2577.00	−0.931998	−0.465999	−	0.884785i	$-0.654305\pi$
$197$	−2577.00	−0.931998	−0.465999	+	0.884785i	$0.654305\pi$
$198$	0	0
$199$	1412.00	0.502985	0.251493	−	0.967859i	$-0.419079\pi$
$199$	1412.00	0.502985	0.251493	+	0.967859i	$0.419079\pi$
$200$	0	0
$201$	0	0
$202$	3384.00	1.17870
$203$	−1248.00	−0.431490
$204$	0	0
$205$	0	0
$206$	3496.00	1.18242
$207$	0	0
$208$	−128.000	−0.0426692
$209$	414.000	0.137019
$210$	0	0
$211$	−307.000	−0.100165	−0.0500823	−	0.998745i	$-0.515948\pi$
$211$	−307.000	−0.100165	−0.0500823	+	0.998745i	$0.515948\pi$
$212$	708.000	0.229366
$213$	0	0
$214$	−1896.00	−0.605644
$215$	0	0
$216$	0	0
$217$	680.000	0.212725
$218$	−1186.00	−0.368468
$219$	0	0
$220$	0	0
$221$	120.000	0.0365252
$222$	0	0
$223$	−5234.00	−1.57172	−0.785862	−	0.618402i	$-0.787781\pi$
$223$	−5234.00	−1.57172	−0.785862	+	0.618402i	$0.787781\pi$
$224$	256.000	0.0763604
$225$	0	0
$226$	−2124.00	−0.625161
$227$	−1509.00	−0.441215	−0.220608	−	0.975363i	$-0.570804\pi$
$227$	−1509.00	−0.441215	−0.220608	+	0.975363i	$0.570804\pi$
$228$	0	0
$229$	1211.00	0.349455	0.174727	−	0.984617i	$-0.444096\pi$
$229$	1211.00	0.349455	0.174727	+	0.984617i	$0.444096\pi$
$230$	0	0
$231$	0	0
$232$	−1248.00	−0.353169
$233$	−6246.00	−1.75618	−0.878088	−	0.478499i	$-0.841181\pi$
$233$	−6246.00	−1.75618	−0.878088	+	0.478499i	$0.841181\pi$
$234$	0	0
$235$	0	0
$236$	3168.00	0.873810
$237$	0	0
$238$	−240.000	−0.0653651
$239$	4650.00	1.25851	0.629254	−	0.777200i	$-0.283360\pi$
$239$	4650.00	1.25851	0.629254	+	0.777200i	$0.283360\pi$
$240$	0	0
$241$	−3145.00	−0.840611	−0.420306	−	0.907383i	$-0.638077\pi$
$241$	−3145.00	−0.840611	−0.420306	+	0.907383i	$0.638077\pi$
$242$	2014.00	0.534979
$243$	0	0
$244$	−3628.00	−0.951881
$245$	0	0
$246$	0	0
$247$	−184.000	−0.0473994
$248$	680.000	0.174113
$249$	0	0
$250$	0	0
$251$	1020.00	0.256501	0.128251	−	0.991742i	$-0.459064\pi$
$251$	1020.00	0.256501	0.128251	+	0.991742i	$0.459064\pi$
$252$	0	0
$253$	−1134.00	−0.281794
$254$	652.000	0.161063
$255$	0	0
$256$	256.000	0.0625000
$257$	−6741.00	−1.63616	−0.818078	−	0.575107i	$-0.804960\pi$
$257$	−6741.00	−1.63616	−0.818078	+	0.575107i	$0.804960\pi$
$258$	0	0
$259$	592.000	0.142027
$260$	0	0
$261$	0	0
$262$	1980.00	0.466889
$263$	−2340.00	−0.548633	−0.274317	−	0.961639i	$-0.588452\pi$
$263$	−2340.00	−0.548633	−0.274317	+	0.961639i	$0.588452\pi$
$264$	0	0
$265$	0	0
$266$	368.000	0.0848253
$267$	0	0
$268$	1288.00	0.293571
$269$	−6198.00	−1.40483	−0.702414	−	0.711769i	$-0.747894\pi$
$269$	−6198.00	−1.40483	−0.702414	+	0.711769i	$0.747894\pi$
$270$	0	0
$271$	875.000	0.196135	0.0980673	−	0.995180i	$-0.468734\pi$
$271$	875.000	0.196135	0.0980673	+	0.995180i	$0.468734\pi$
$272$	−240.000	−0.0535005
$273$	0	0
$274$	−294.000	−0.0648219
$275$	0	0
$276$	0	0
$277$	−5486.00	−1.18997	−0.594985	−	0.803737i	$-0.702842\pi$
$277$	−5486.00	−1.18997	−0.594985	+	0.803737i	$0.702842\pi$
$278$	−3208.00	−0.692097
$279$	0	0
$280$	0	0
$281$	−3204.00	−0.680194	−0.340097	−	0.940390i	$-0.610460\pi$
$281$	−3204.00	−0.680194	−0.340097	+	0.940390i	$0.610460\pi$
$282$	0	0
$283$	−7322.00	−1.53798	−0.768989	−	0.639262i	$-0.779240\pi$
$283$	−7322.00	−1.53798	−0.768989	+	0.639262i	$0.779240\pi$
$284$	−1080.00	−0.225656
$285$	0	0
$286$	288.000	0.0595447
$287$	−1968.00	−0.404764
$288$	0	0
$289$	−4688.00	−0.954203
$290$	0	0
$291$	0	0
$292$	−1016.00	−0.203620
$293$	−1353.00	−0.269772	−0.134886	−	0.990861i	$-0.543067\pi$
$293$	−1353.00	−0.269772	−0.134886	+	0.990861i	$0.543067\pi$
$294$	0	0
$295$	0	0
$296$	592.000	0.116248
$297$	0	0
$298$	−2436.00	−0.473536
$299$	504.000	0.0974818
$300$	0	0
$301$	−1520.00	−0.291068
$302$	4496.00	0.856675
$303$	0	0
$304$	368.000	0.0694284
$305$	0	0
$306$	0	0
$307$	−1658.00	−0.308231	−0.154116	−	0.988053i	$-0.549253\pi$
$307$	−1658.00	−0.308231	−0.154116	+	0.988053i	$0.549253\pi$
$308$	−576.000	−0.106561
$309$	0	0
$310$	0	0
$311$	−1044.00	−0.190353	−0.0951765	−	0.995460i	$-0.530342\pi$
$311$	−1044.00	−0.190353	−0.0951765	+	0.995460i	$0.530342\pi$
$312$	0	0
$313$	−2588.00	−0.467356	−0.233678	−	0.972314i	$-0.575076\pi$
$313$	−2588.00	−0.467356	−0.233678	+	0.972314i	$0.575076\pi$
$314$	−5996.00	−1.07762
$315$	0	0
$316$	−4492.00	−0.799667
$317$	1449.00	0.256732	0.128366	−	0.991727i	$-0.459027\pi$
$317$	1449.00	0.256732	0.128366	+	0.991727i	$0.459027\pi$
$318$	0	0
$319$	2808.00	0.492846
$320$	0	0
$321$	0	0
$322$	−1008.00	−0.174452
$323$	−345.000	−0.0594313
$324$	0	0
$325$	0	0
$326$	6940.00	1.17905
$327$	0	0
$328$	−1968.00	−0.331295
$329$	2304.00	0.386090
$330$	0	0
$331$	4880.00	0.810360	0.405180	−	0.914237i	$-0.367209\pi$
$331$	4880.00	0.810360	0.405180	+	0.914237i	$0.367209\pi$
$332$	3084.00	0.509809
$333$	0	0
$334$	774.000	0.126801
$335$	0	0
$336$	0	0
$337$	7744.00	1.25176	0.625879	−	0.779920i	$-0.284740\pi$
$337$	7744.00	1.25176	0.625879	+	0.779920i	$0.284740\pi$
$338$	4266.00	0.686508
$339$	0	0
$340$	0	0
$341$	−1530.00	−0.242974
$342$	0	0
$343$	4976.00	0.783320
$344$	−1520.00	−0.238235
$345$	0	0
$346$	−1710.00	−0.265694
$347$	−804.000	−0.124383	−0.0621916	−	0.998064i	$-0.519809\pi$
$347$	−804.000	−0.124383	−0.0621916	+	0.998064i	$0.519809\pi$
$348$	0	0
$349$	−2815.00	−0.431758	−0.215879	−	0.976420i	$-0.569262\pi$
$349$	−2815.00	−0.431758	−0.215879	+	0.976420i	$0.569262\pi$
$350$	0	0
$351$	0	0
$352$	−576.000	−0.0872185
$353$	3738.00	0.563608	0.281804	−	0.959472i	$-0.409067\pi$
$353$	3738.00	0.563608	0.281804	+	0.959472i	$0.409067\pi$
$354$	0	0
$355$	0	0
$356$	−792.000	−0.117910
$357$	0	0
$358$	528.000	0.0779488
$359$	−11022.0	−1.62039	−0.810193	−	0.586163i	$-0.800638\pi$
$359$	−11022.0	−1.62039	−0.810193	+	0.586163i	$0.800638\pi$
$360$	0	0
$361$	−6330.00	−0.922875
$362$	5102.00	0.740760
$363$	0	0
$364$	256.000	0.0368628
$365$	0	0
$366$	0	0
$367$	−7544.00	−1.07301	−0.536504	−	0.843898i	$-0.680255\pi$
$367$	−7544.00	−1.07301	−0.536504	+	0.843898i	$0.680255\pi$
$368$	−1008.00	−0.142787
$369$	0	0
$370$	0	0
$371$	−1416.00	−0.198154
$372$	0	0
$373$	5404.00	0.750157	0.375078	−	0.926993i	$-0.377616\pi$
$373$	5404.00	0.750157	0.375078	+	0.926993i	$0.377616\pi$
$374$	540.000	0.0746597
$375$	0	0
$376$	2304.00	0.316010
$377$	−1248.00	−0.170491
$378$	0	0
$379$	−2335.00	−0.316467	−0.158233	−	0.987402i	$-0.550580\pi$
$379$	−2335.00	−0.316467	−0.158233	+	0.987402i	$0.550580\pi$
$380$	0	0
$381$	0	0
$382$	4476.00	0.599508
$383$	6633.00	0.884936	0.442468	−	0.896784i	$-0.354103\pi$
$383$	6633.00	0.884936	0.442468	+	0.896784i	$0.354103\pi$
$384$	0	0
$385$	0	0
$386$	4360.00	0.574918
$387$	0	0
$388$	4768.00	0.623862
$389$	−7566.00	−0.986148	−0.493074	−	0.869987i	$-0.664127\pi$
$389$	−7566.00	−0.986148	−0.493074	+	0.869987i	$0.664127\pi$
$390$	0	0
$391$	945.000	0.122227
$392$	2232.00	0.287584
$393$	0	0
$394$	5154.00	0.659022
$395$	0	0
$396$	0	0
$397$	7420.00	0.938033	0.469017	−	0.883189i	$-0.344608\pi$
$397$	7420.00	0.938033	0.469017	+	0.883189i	$0.344608\pi$
$398$	−2824.00	−0.355664
$399$	0	0
$400$	0	0
$401$	−8502.00	−1.05878	−0.529389	−	0.848379i	$-0.677579\pi$
$401$	−8502.00	−1.05878	−0.529389	+	0.848379i	$0.677579\pi$
$402$	0	0
$403$	680.000	0.0840526
$404$	−6768.00	−0.833467
$405$	0	0
$406$	2496.00	0.305109
$407$	−1332.00	−0.162223
$408$	0	0
$409$	−1903.00	−0.230067	−0.115033	−	0.993362i	$-0.536697\pi$
$409$	−1903.00	−0.230067	−0.115033	+	0.993362i	$0.536697\pi$
$410$	0	0
$411$	0	0
$412$	−6992.00	−0.836095
$413$	−6336.00	−0.754901
$414$	0	0
$415$	0	0
$416$	256.000	0.0301717
$417$	0	0
$418$	−828.000	−0.0968871
$419$	13482.0	1.57193	0.785965	−	0.618271i	$-0.212166\pi$
$419$	13482.0	1.57193	0.785965	+	0.618271i	$0.212166\pi$
$420$	0	0
$421$	−1537.00	−0.177931	−0.0889653	−	0.996035i	$-0.528356\pi$
$421$	−1537.00	−0.177931	−0.0889653	+	0.996035i	$0.528356\pi$
$422$	614.000	0.0708271
$423$	0	0
$424$	−1416.00	−0.162186
$425$	0	0
$426$	0	0
$427$	7256.00	0.822348
$428$	3792.00	0.428255
$429$	0	0
$430$	0	0
$431$	−10368.0	−1.15872	−0.579361	−	0.815071i	$-0.696698\pi$
$431$	−10368.0	−1.15872	−0.579361	+	0.815071i	$0.696698\pi$
$432$	0	0
$433$	13168.0	1.46146	0.730732	−	0.682665i	$-0.239179\pi$
$433$	13168.0	1.46146	0.730732	+	0.682665i	$0.239179\pi$
$434$	−1360.00	−0.150420
$435$	0	0
$436$	2372.00	0.260546
$437$	−1449.00	−0.158616
$438$	0	0
$439$	7319.00	0.795710	0.397855	−	0.917448i	$-0.369755\pi$
$439$	7319.00	0.795710	0.397855	+	0.917448i	$0.369755\pi$
$440$	0	0
$441$	0	0
$442$	−240.000	−0.0258272
$443$	4119.00	0.441760	0.220880	−	0.975301i	$-0.429107\pi$
$443$	4119.00	0.441760	0.220880	+	0.975301i	$0.429107\pi$
$444$	0	0
$445$	0	0
$446$	10468.0	1.11138
$447$	0	0
$448$	−512.000	−0.0539949
$449$	−5388.00	−0.566315	−0.283158	−	0.959073i	$-0.591382\pi$
$449$	−5388.00	−0.566315	−0.283158	+	0.959073i	$0.591382\pi$
$450$	0	0
$451$	4428.00	0.462320
$452$	4248.00	0.442056
$453$	0	0
$454$	3018.00	0.311986
$455$	0	0
$456$	0	0
$457$	2752.00	0.281692	0.140846	−	0.990032i	$-0.455018\pi$
$457$	2752.00	0.281692	0.140846	+	0.990032i	$0.455018\pi$
$458$	−2422.00	−0.247102
$459$	0	0
$460$	0	0
$461$	−4314.00	−0.435842	−0.217921	−	0.975966i	$-0.569927\pi$
$461$	−4314.00	−0.435842	−0.217921	+	0.975966i	$0.569927\pi$
$462$	0	0
$463$	5794.00	0.581577	0.290788	−	0.956787i	$-0.406082\pi$
$463$	5794.00	0.581577	0.290788	+	0.956787i	$0.406082\pi$
$464$	2496.00	0.249728
$465$	0	0
$466$	12492.0	1.24180
$467$	−6309.00	−0.625151	−0.312576	−	0.949893i	$-0.601192\pi$
$467$	−6309.00	−0.625151	−0.312576	+	0.949893i	$0.601192\pi$
$468$	0	0
$469$	−2576.00	−0.253622
$470$	0	0
$471$	0	0
$472$	−6336.00	−0.617877
$473$	3420.00	0.332456
$474$	0	0
$475$	0	0
$476$	480.000	0.0462201
$477$	0	0
$478$	−9300.00	−0.889900
$479$	14826.0	1.41423	0.707116	−	0.707097i	$-0.249996\pi$
$479$	14826.0	1.41423	0.707116	+	0.707097i	$0.249996\pi$
$480$	0	0
$481$	592.000	0.0561182
$482$	6290.00	0.594402
$483$	0	0
$484$	−4028.00	−0.378287
$485$	0	0
$486$	0	0
$487$	−6758.00	−0.628818	−0.314409	−	0.949288i	$-0.601806\pi$
$487$	−6758.00	−0.628818	−0.314409	+	0.949288i	$0.601806\pi$
$488$	7256.00	0.673081
$489$	0	0
$490$	0	0
$491$	−14574.0	−1.33954	−0.669771	−	0.742567i	$-0.733608\pi$
$491$	−14574.0	−1.33954	−0.669771	+	0.742567i	$0.733608\pi$
$492$	0	0
$493$	−2340.00	−0.213769
$494$	368.000	0.0335164
$495$	0	0
$496$	−1360.00	−0.123117
$497$	2160.00	0.194948
$498$	0	0
$499$	12611.0	1.13135	0.565677	−	0.824627i	$-0.308615\pi$
$499$	12611.0	1.13135	0.565677	+	0.824627i	$0.308615\pi$
$500$	0	0
$501$	0	0
$502$	−2040.00	−0.181374
$503$	−15639.0	−1.38630	−0.693150	−	0.720794i	$-0.743778\pi$
$503$	−15639.0	−1.38630	−0.693150	+	0.720794i	$0.743778\pi$
$504$	0	0
$505$	0	0
$506$	2268.00	0.199259
$507$	0	0
$508$	−1304.00	−0.113889
$509$	15420.0	1.34279	0.671394	−	0.741100i	$-0.265696\pi$
$509$	15420.0	1.34279	0.671394	+	0.741100i	$0.265696\pi$
$510$	0	0
$511$	2032.00	0.175911
$512$	−512.000	−0.0441942
$513$	0	0
$514$	13482.0	1.15694
$515$	0	0
$516$	0	0
$517$	−5184.00	−0.440990
$518$	−1184.00	−0.100429
$519$	0	0
$520$	0	0
$521$	10494.0	0.882439	0.441219	−	0.897399i	$-0.354546\pi$
$521$	10494.0	0.882439	0.441219	+	0.897399i	$0.354546\pi$
$522$	0	0
$523$	10708.0	0.895274	0.447637	−	0.894215i	$-0.352266\pi$
$523$	10708.0	0.895274	0.447637	+	0.894215i	$0.352266\pi$
$524$	−3960.00	−0.330140
$525$	0	0
$526$	4680.00	0.387942
$527$	1275.00	0.105389
$528$	0	0
$529$	−8198.00	−0.673790
$530$	0	0
$531$	0	0
$532$	−736.000	−0.0599805
$533$	−1968.00	−0.159932
$534$	0	0
$535$	0	0
$536$	−2576.00	−0.207586
$537$	0	0
$538$	12396.0	0.993363
$539$	−5022.00	−0.401323
$540$	0	0
$541$	23030.0	1.83020	0.915099	−	0.403229i	$-0.132112\pi$
$541$	23030.0	1.83020	0.915099	+	0.403229i	$0.132112\pi$
$542$	−1750.00	−0.138688
$543$	0	0
$544$	480.000	0.0378306
$545$	0	0
$546$	0	0
$547$	3814.00	0.298126	0.149063	−	0.988828i	$-0.452374\pi$
$547$	3814.00	0.298126	0.149063	+	0.988828i	$0.452374\pi$
$548$	588.000	0.0458360
$549$	0	0
$550$	0	0
$551$	3588.00	0.277412
$552$	0	0
$553$	8984.00	0.690847
$554$	10972.0	0.841436
$555$	0	0
$556$	6416.00	0.489387
$557$	−22266.0	−1.69379	−0.846895	−	0.531761i	$-0.821531\pi$
$557$	−22266.0	−1.69379	−0.846895	+	0.531761i	$0.821531\pi$
$558$	0	0
$559$	−1520.00	−0.115007
$560$	0	0
$561$	0	0
$562$	6408.00	0.480970
$563$	23844.0	1.78491	0.892455	−	0.451136i	$-0.148981\pi$
$563$	23844.0	1.78491	0.892455	+	0.451136i	$0.148981\pi$
$564$	0	0
$565$	0	0
$566$	14644.0	1.08751
$567$	0	0
$568$	2160.00	0.159563
$569$	7488.00	0.551693	0.275846	−	0.961202i	$-0.411042\pi$
$569$	7488.00	0.551693	0.275846	+	0.961202i	$0.411042\pi$
$570$	0	0
$571$	5111.00	0.374586	0.187293	−	0.982304i	$-0.440029\pi$
$571$	5111.00	0.374586	0.187293	+	0.982304i	$0.440029\pi$
$572$	−576.000	−0.0421045
$573$	0	0
$574$	3936.00	0.286212
$575$	0	0
$576$	0	0
$577$	−6986.00	−0.504040	−0.252020	−	0.967722i	$-0.581095\pi$
$577$	−6986.00	−0.504040	−0.252020	+	0.967722i	$0.581095\pi$
$578$	9376.00	0.674724
$579$	0	0
$580$	0	0
$581$	−6168.00	−0.440433
$582$	0	0
$583$	3186.00	0.226330
$584$	2032.00	0.143981
$585$	0	0
$586$	2706.00	0.190757
$587$	−20571.0	−1.44643	−0.723216	−	0.690622i	$-0.757337\pi$
$587$	−20571.0	−1.44643	−0.723216	+	0.690622i	$0.757337\pi$
$588$	0	0
$589$	−1955.00	−0.136765
$590$	0	0
$591$	0	0
$592$	−1184.00	−0.0821995
$593$	−23241.0	−1.60943	−0.804716	−	0.593660i	$-0.797683\pi$
$593$	−23241.0	−1.60943	−0.804716	+	0.593660i	$0.797683\pi$
$594$	0	0
$595$	0	0
$596$	4872.00	0.334840
$597$	0	0
$598$	−1008.00	−0.0689301
$599$	20208.0	1.37842	0.689212	−	0.724559i	$-0.257957\pi$
$599$	20208.0	1.37842	0.689212	+	0.724559i	$0.257957\pi$
$600$	0	0
$601$	−9055.00	−0.614578	−0.307289	−	0.951616i	$-0.599422\pi$
$601$	−9055.00	−0.614578	−0.307289	+	0.951616i	$0.599422\pi$
$602$	3040.00	0.205816
$603$	0	0
$604$	−8992.00	−0.605760
$605$	0	0
$606$	0	0
$607$	−15554.0	−1.04006	−0.520031	−	0.854148i	$-0.674080\pi$
$607$	−15554.0	−1.04006	−0.520031	+	0.854148i	$0.674080\pi$
$608$	−736.000	−0.0490933
$609$	0	0
$610$	0	0
$611$	2304.00	0.152553
$612$	0	0
$613$	5632.00	0.371084	0.185542	−	0.982636i	$-0.440596\pi$
$613$	5632.00	0.371084	0.185542	+	0.982636i	$0.440596\pi$
$614$	3316.00	0.217953
$615$	0	0
$616$	1152.00	0.0753497
$617$	9141.00	0.596439	0.298219	−	0.954497i	$-0.403607\pi$
$617$	9141.00	0.596439	0.298219	+	0.954497i	$0.403607\pi$
$618$	0	0
$619$	−13372.0	−0.868281	−0.434141	−	0.900845i	$-0.642948\pi$
$619$	−13372.0	−0.868281	−0.434141	+	0.900845i	$0.642948\pi$
$620$	0	0
$621$	0	0
$622$	2088.00	0.134600
$623$	1584.00	0.101865
$624$	0	0
$625$	0	0
$626$	5176.00	0.330471
$627$	0	0
$628$	11992.0	0.761995
$629$	1110.00	0.0703634
$630$	0	0
$631$	11165.0	0.704392	0.352196	−	0.935926i	$-0.385435\pi$
$631$	11165.0	0.704392	0.352196	+	0.935926i	$0.385435\pi$
$632$	8984.00	0.565450
$633$	0	0
$634$	−2898.00	−0.181537
$635$	0	0
$636$	0	0
$637$	2232.00	0.138831
$638$	−5616.00	−0.348495
$639$	0	0
$640$	0	0
$641$	−912.000	−0.0561963	−0.0280982	−	0.999605i	$-0.508945\pi$
$641$	−912.000	−0.0561963	−0.0280982	+	0.999605i	$0.508945\pi$
$642$	0	0
$643$	27952.0	1.71434	0.857169	−	0.515035i	$-0.172221\pi$
$643$	27952.0	1.71434	0.857169	+	0.515035i	$0.172221\pi$
$644$	2016.00	0.123356
$645$	0	0
$646$	690.000	0.0420243
$647$	6285.00	0.381899	0.190950	−	0.981600i	$-0.438843\pi$
$647$	6285.00	0.381899	0.190950	+	0.981600i	$0.438843\pi$
$648$	0	0
$649$	14256.0	0.862245
$650$	0	0
$651$	0	0
$652$	−13880.0	−0.833716
$653$	16497.0	0.988633	0.494317	−	0.869282i	$-0.335418\pi$
$653$	16497.0	0.988633	0.494317	+	0.869282i	$0.335418\pi$
$654$	0	0
$655$	0	0
$656$	3936.00	0.234261
$657$	0	0
$658$	−4608.00	−0.273007
$659$	14844.0	0.877451	0.438725	−	0.898621i	$-0.355430\pi$
$659$	14844.0	0.877451	0.438725	+	0.898621i	$0.355430\pi$
$660$	0	0
$661$	31934.0	1.87911	0.939553	−	0.342404i	$-0.111241\pi$
$661$	31934.0	1.87911	0.939553	+	0.342404i	$0.111241\pi$
$662$	−9760.00	−0.573011
$663$	0	0
$664$	−6168.00	−0.360489
$665$	0	0
$666$	0	0
$667$	−9828.00	−0.570527
$668$	−1548.00	−0.0896616
$669$	0	0
$670$	0	0
$671$	−16326.0	−0.939282
$672$	0	0
$673$	24352.0	1.39480	0.697400	−	0.716682i	$-0.254340\pi$
$673$	24352.0	1.39480	0.697400	+	0.716682i	$0.254340\pi$
$674$	−15488.0	−0.885127
$675$	0	0
$676$	−8532.00	−0.485435
$677$	10374.0	0.588929	0.294465	−	0.955662i	$-0.404859\pi$
$677$	10374.0	0.588929	0.294465	+	0.955662i	$0.404859\pi$
$678$	0	0
$679$	−9536.00	−0.538966
$680$	0	0
$681$	0	0
$682$	3060.00	0.171809
$683$	7347.00	0.411603	0.205802	−	0.978594i	$-0.434020\pi$
$683$	7347.00	0.411603	0.205802	+	0.978594i	$0.434020\pi$
$684$	0	0
$685$	0	0
$686$	−9952.00	−0.553891
$687$	0	0
$688$	3040.00	0.168458
$689$	−1416.00	−0.0782951
$690$	0	0
$691$	−5371.00	−0.295691	−0.147845	−	0.989010i	$-0.547234\pi$
$691$	−5371.00	−0.295691	−0.147845	+	0.989010i	$0.547234\pi$
$692$	3420.00	0.187874
$693$	0	0
$694$	1608.00	0.0879522
$695$	0	0
$696$	0	0
$697$	−3690.00	−0.200529
$698$	5630.00	0.305299
$699$	0	0
$700$	0	0
$701$	7086.00	0.381790	0.190895	−	0.981610i	$-0.438861\pi$
$701$	7086.00	0.381790	0.190895	+	0.981610i	$0.438861\pi$
$702$	0	0
$703$	−1702.00	−0.0913117
$704$	1152.00	0.0616728
$705$	0	0
$706$	−7476.00	−0.398531
$707$	13536.0	0.720048
$708$	0	0
$709$	17186.0	0.910344	0.455172	−	0.890404i	$-0.349578\pi$
$709$	17186.0	0.910344	0.455172	+	0.890404i	$0.349578\pi$
$710$	0	0
$711$	0	0
$712$	1584.00	0.0833749
$713$	5355.00	0.281271
$714$	0	0
$715$	0	0
$716$	−1056.00	−0.0551181
$717$	0	0
$718$	22044.0	1.14579
$719$	−23814.0	−1.23520	−0.617602	−	0.786490i	$-0.711896\pi$
$719$	−23814.0	−1.23520	−0.617602	+	0.786490i	$0.711896\pi$
$720$	0	0
$721$	13984.0	0.722318
$722$	12660.0	0.652571
$723$	0	0
$724$	−10204.0	−0.523797
$725$	0	0
$726$	0	0
$727$	22732.0	1.15967	0.579837	−	0.814732i	$-0.303116\pi$
$727$	22732.0	1.15967	0.579837	+	0.814732i	$0.303116\pi$
$728$	−512.000	−0.0260659
$729$	0	0
$730$	0	0
$731$	−2850.00	−0.144201
$732$	0	0
$733$	−4664.00	−0.235019	−0.117509	−	0.993072i	$-0.537491\pi$
$733$	−4664.00	−0.235019	−0.117509	+	0.993072i	$0.537491\pi$
$734$	15088.0	0.758731
$735$	0	0
$736$	2016.00	0.100966
$737$	5796.00	0.289686
$738$	0	0
$739$	5501.00	0.273826	0.136913	−	0.990583i	$-0.456282\pi$
$739$	5501.00	0.273826	0.136913	+	0.990583i	$0.456282\pi$
$740$	0	0
$741$	0	0
$742$	2832.00	0.140116
$743$	27096.0	1.33789	0.668947	−	0.743310i	$-0.266745\pi$
$743$	27096.0	1.33789	0.668947	+	0.743310i	$0.266745\pi$
$744$	0	0
$745$	0	0
$746$	−10808.0	−0.530441
$747$	0	0
$748$	−1080.00	−0.0527924
$749$	−7584.00	−0.369978
$750$	0	0
$751$	−5659.00	−0.274967	−0.137483	−	0.990504i	$-0.543901\pi$
$751$	−5659.00	−0.274967	−0.137483	+	0.990504i	$0.543901\pi$
$752$	−4608.00	−0.223453
$753$	0	0
$754$	2496.00	0.120556
$755$	0	0
$756$	0	0
$757$	−37694.0	−1.80979	−0.904895	−	0.425634i	$-0.860051\pi$
$757$	−37694.0	−1.80979	−0.904895	+	0.425634i	$0.860051\pi$
$758$	4670.00	0.223776
$759$	0	0
$760$	0	0
$761$	−6588.00	−0.313817	−0.156909	−	0.987613i	$-0.550153\pi$
$761$	−6588.00	−0.313817	−0.156909	+	0.987613i	$0.550153\pi$
$762$	0	0
$763$	−4744.00	−0.225091
$764$	−8952.00	−0.423916
$765$	0	0
$766$	−13266.0	−0.625744
$767$	−6336.00	−0.298279
$768$	0	0
$769$	−19.0000	−0.000890972 0	−0.000445486	−	1.00000i	$-0.500142\pi$
$769$	−19.0000	−0.000890972 0	−0.000445486		1.00000i	$0.500142\pi$
$770$	0	0
$771$	0	0
$772$	−8720.00	−0.406528
$773$	−33639.0	−1.56521	−0.782607	−	0.622516i	$-0.786111\pi$
$773$	−33639.0	−1.56521	−0.782607	+	0.622516i	$0.786111\pi$
$774$	0	0
$775$	0	0
$776$	−9536.00	−0.441137
$777$	0	0
$778$	15132.0	0.697312
$779$	5658.00	0.260230
$780$	0	0
$781$	−4860.00	−0.222669
$782$	−1890.00	−0.0864274
$783$	0	0
$784$	−4464.00	−0.203353
$785$	0	0
$786$	0	0
$787$	−23474.0	−1.06322	−0.531612	−	0.846988i	$-0.678414\pi$
$787$	−23474.0	−1.06322	−0.531612	+	0.846988i	$0.678414\pi$
$788$	−10308.0	−0.465999
$789$	0	0
$790$	0	0
$791$	−8496.00	−0.381900
$792$	0	0
$793$	7256.00	0.324928
$794$	−14840.0	−0.663290
$795$	0	0
$796$	5648.00	0.251493
$797$	−7917.00	−0.351863	−0.175931	−	0.984402i	$-0.556294\pi$
$797$	−7917.00	−0.351863	−0.175931	+	0.984402i	$0.556294\pi$
$798$	0	0
$799$	4320.00	0.191277
$800$	0	0
$801$	0	0
$802$	17004.0	0.748668
$803$	−4572.00	−0.200925
$804$	0	0
$805$	0	0
$806$	−1360.00	−0.0594342
$807$	0	0
$808$	13536.0	0.589350
$809$	−41202.0	−1.79059	−0.895294	−	0.445476i	$-0.853034\pi$
$809$	−41202.0	−1.79059	−0.895294	+	0.445476i	$0.853034\pi$
$810$	0	0
$811$	35492.0	1.53674	0.768368	−	0.640008i	$-0.221069\pi$
$811$	35492.0	1.53674	0.768368	+	0.640008i	$0.221069\pi$
$812$	−4992.00	−0.215745
$813$	0	0
$814$	2664.00	0.114709
$815$	0	0
$816$	0	0
$817$	4370.00	0.187132
$818$	3806.00	0.162682
$819$	0	0
$820$	0	0
$821$	−7146.00	−0.303772	−0.151886	−	0.988398i	$-0.548535\pi$
$821$	−7146.00	−0.303772	−0.151886	+	0.988398i	$0.548535\pi$
$822$	0	0
$823$	−8882.00	−0.376193	−0.188097	−	0.982151i	$-0.560232\pi$
$823$	−8882.00	−0.376193	−0.188097	+	0.982151i	$0.560232\pi$
$824$	13984.0	0.591208
$825$	0	0
$826$	12672.0	0.533796
$827$	21705.0	0.912644	0.456322	−	0.889815i	$-0.349166\pi$
$827$	21705.0	0.912644	0.456322	+	0.889815i	$0.349166\pi$
$828$	0	0
$829$	29018.0	1.21573	0.607863	−	0.794042i	$-0.292027\pi$
$829$	29018.0	1.21573	0.607863	+	0.794042i	$0.292027\pi$
$830$	0	0
$831$	0	0
$832$	−512.000	−0.0213346
$833$	4185.00	0.174072
$834$	0	0
$835$	0	0
$836$	1656.00	0.0685095
$837$	0	0
$838$	−26964.0	−1.11152
$839$	31164.0	1.28236	0.641180	−	0.767390i	$-0.278445\pi$
$839$	31164.0	1.28236	0.641180	+	0.767390i	$0.278445\pi$
$840$	0	0
$841$	−53.0000	−0.00217311
$842$	3074.00	0.125816
$843$	0	0
$844$	−1228.00	−0.0500823
$845$	0	0
$846$	0	0
$847$	8056.00	0.326809
$848$	2832.00	0.114683
$849$	0	0
$850$	0	0
$851$	4662.00	0.187792
$852$	0	0
$853$	−49160.0	−1.97328	−0.986639	−	0.162921i	$-0.947908\pi$
$853$	−49160.0	−1.97328	−0.986639	+	0.162921i	$0.947908\pi$
$854$	−14512.0	−0.581488
$855$	0	0
$856$	−7584.00	−0.302822
$857$	2349.00	0.0936293	0.0468147	−	0.998904i	$-0.485093\pi$
$857$	2349.00	0.0936293	0.0468147	+	0.998904i	$0.485093\pi$
$858$	0	0
$859$	−28195.0	−1.11991	−0.559954	−	0.828524i	$-0.689181\pi$
$859$	−28195.0	−1.11991	−0.559954	+	0.828524i	$0.689181\pi$
$860$	0	0
$861$	0	0
$862$	20736.0	0.819340
$863$	23997.0	0.946544	0.473272	−	0.880916i	$-0.343073\pi$
$863$	23997.0	0.946544	0.473272	+	0.880916i	$0.343073\pi$
$864$	0	0
$865$	0	0
$866$	−26336.0	−1.03341
$867$	0	0
$868$	2720.00	0.106363
$869$	−20214.0	−0.789083
$870$	0	0
$871$	−2576.00	−0.100212
$872$	−4744.00	−0.184234
$873$	0	0
$874$	2898.00	0.112158
$875$	0	0
$876$	0	0
$877$	−46286.0	−1.78217	−0.891087	−	0.453832i	$-0.850057\pi$
$877$	−46286.0	−1.78217	−0.891087	+	0.453832i	$0.850057\pi$
$878$	−14638.0	−0.562652
$879$	0	0
$880$	0	0
$881$	39636.0	1.51574	0.757872	−	0.652403i	$-0.226239\pi$
$881$	39636.0	1.51574	0.757872	+	0.652403i	$0.226239\pi$
$882$	0	0
$883$	16744.0	0.638143	0.319072	−	0.947731i	$-0.396629\pi$
$883$	16744.0	0.638143	0.319072	+	0.947731i	$0.396629\pi$
$884$	480.000	0.0182626
$885$	0	0
$886$	−8238.00	−0.312371
$887$	−1251.00	−0.0473557	−0.0236778	−	0.999720i	$-0.507538\pi$
$887$	−1251.00	−0.0473557	−0.0236778	+	0.999720i	$0.507538\pi$
$888$	0	0
$889$	2608.00	0.0983909
$890$	0	0
$891$	0	0
$892$	−20936.0	−0.785862
$893$	−6624.00	−0.248224
$894$	0	0
$895$	0	0
$896$	1024.00	0.0381802
$897$	0	0
$898$	10776.0	0.400445
$899$	−13260.0	−0.491931
$900$	0	0
$901$	−2655.00	−0.0981697
$902$	−8856.00	−0.326910
$903$	0	0
$904$	−8496.00	−0.312580
$905$	0	0
$906$	0	0
$907$	36988.0	1.35410	0.677049	−	0.735938i	$-0.263259\pi$
$907$	36988.0	1.35410	0.677049	+	0.735938i	$0.263259\pi$
$908$	−6036.00	−0.220608
$909$	0	0
$910$	0	0
$911$	16404.0	0.596585	0.298292	−	0.954475i	$-0.403583\pi$
$911$	16404.0	0.596585	0.298292	+	0.954475i	$0.403583\pi$
$912$	0	0
$913$	13878.0	0.503061
$914$	−5504.00	−0.199186
$915$	0	0
$916$	4844.00	0.174727
$917$	7920.00	0.285214
$918$	0	0
$919$	−664.000	−0.0238339	−0.0119169	−	0.999929i	$-0.503793\pi$
$919$	−664.000	−0.0238339	−0.0119169	+	0.999929i	$0.503793\pi$
$920$	0	0
$921$	0	0
$922$	8628.00	0.308187
$923$	2160.00	0.0770285
$924$	0	0
$925$	0	0
$926$	−11588.0	−0.411237
$927$	0	0
$928$	−4992.00	−0.176585
$929$	−39642.0	−1.40001	−0.700006	−	0.714137i	$-0.746820\pi$
$929$	−39642.0	−1.40001	−0.700006	+	0.714137i	$0.746820\pi$
$930$	0	0
$931$	−6417.00	−0.225895
$932$	−24984.0	−0.878088
$933$	0	0
$934$	12618.0	0.442049
$935$	0	0
$936$	0	0
$937$	36028.0	1.25612	0.628059	−	0.778165i	$-0.283849\pi$
$937$	36028.0	1.25612	0.628059	+	0.778165i	$0.283849\pi$
$938$	5152.00	0.179338
$939$	0	0
$940$	0	0
$941$	23058.0	0.798798	0.399399	−	0.916777i	$-0.369219\pi$
$941$	23058.0	0.798798	0.399399	+	0.916777i	$0.369219\pi$
$942$	0	0
$943$	−15498.0	−0.535190
$944$	12672.0	0.436905
$945$	0	0
$946$	−6840.00	−0.235082
$947$	−19953.0	−0.684673	−0.342337	−	0.939577i	$-0.611218\pi$
$947$	−19953.0	−0.684673	−0.342337	+	0.939577i	$0.611218\pi$
$948$	0	0
$949$	2032.00	0.0695063
$950$	0	0
$951$	0	0
$952$	−960.000	−0.0326825
$953$	−25638.0	−0.871455	−0.435727	−	0.900079i	$-0.643509\pi$
$953$	−25638.0	−0.871455	−0.435727	+	0.900079i	$0.643509\pi$
$954$	0	0
$955$	0	0
$956$	18600.0	0.629254
$957$	0	0
$958$	−29652.0	−1.00001
$959$	−1176.00	−0.0395986
$960$	0	0
$961$	−22566.0	−0.757477
$962$	−1184.00	−0.0396816
$963$	0	0
$964$	−12580.0	−0.420306
$965$	0	0
$966$	0	0
$967$	27034.0	0.899023	0.449511	−	0.893275i	$-0.351598\pi$
$967$	27034.0	0.899023	0.449511	+	0.893275i	$0.351598\pi$
$968$	8056.00	0.267489
$969$	0	0
$970$	0	0
$971$	14802.0	0.489206	0.244603	−	0.969623i	$-0.421342\pi$
$971$	14802.0	0.489206	0.244603	+	0.969623i	$0.421342\pi$
$972$	0	0
$973$	−12832.0	−0.422790
$974$	13516.0	0.444641
$975$	0	0
$976$	−14512.0	−0.475940
$977$	9186.00	0.300805	0.150402	−	0.988625i	$-0.451943\pi$
$977$	9186.00	0.300805	0.150402	+	0.988625i	$0.451943\pi$
$978$	0	0
$979$	−3564.00	−0.116349
$980$	0	0
$981$	0	0
$982$	29148.0	0.947200
$983$	−31647.0	−1.02684	−0.513419	−	0.858138i	$-0.671621\pi$
$983$	−31647.0	−1.02684	−0.513419	+	0.858138i	$0.671621\pi$
$984$	0	0
$985$	0	0
$986$	4680.00	0.151158
$987$	0	0
$988$	−736.000	−0.0236997
$989$	−11970.0	−0.384857
$990$	0	0
$991$	−48823.0	−1.56500	−0.782499	−	0.622651i	$-0.786055\pi$
$991$	−48823.0	−1.56500	−0.782499	+	0.622651i	$0.786055\pi$
$992$	2720.00	0.0870565
$993$	0	0
$994$	−4320.00	−0.137849
$995$	0	0
$996$	0	0
$997$	13066.0	0.415050	0.207525	−	0.978230i	$-0.433459\pi$
$997$	13066.0	0.415050	0.207525	+	0.978230i	$0.433459\pi$
$998$	−25222.0	−0.799988
$999$	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	1350.4.a.f.1.1		1
3.2	odd	2		1350.4.a.t.1.1		1
5.2	odd	4		1350.4.c.n.649.1		2
5.3	odd	4		1350.4.c.n.649.2		2
5.4	even	2		270.4.a.l.1.1	yes	1
15.2	even	4		1350.4.c.g.649.2		2
15.8	even	4		1350.4.c.g.649.1		2
15.14	odd	2		270.4.a.b.1.1	✓	1
20.19	odd	2		2160.4.a.m.1.1		1
45.4	even	6		810.4.e.b.541.1		2
45.14	odd	6		810.4.e.v.541.1		2
45.29	odd	6		810.4.e.v.271.1		2
45.34	even	6		810.4.e.b.271.1		2
60.59	even	2		2160.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
270.4.a.b.1.1	✓	1	15.14	odd	2
270.4.a.l.1.1	yes	1	5.4	even	2
810.4.e.b.271.1		2	45.34	even	6
810.4.e.b.541.1		2	45.4	even	6
810.4.e.v.271.1		2	45.29	odd	6
810.4.e.v.541.1		2	45.14	odd	6
1350.4.a.f.1.1		1	1.1	even	1	trivial
1350.4.a.t.1.1		1	3.2	odd	2
1350.4.c.g.649.1		2	15.8	even	4
1350.4.c.g.649.2		2	15.2	even	4
1350.4.c.n.649.1		2	5.2	odd	4
1350.4.c.n.649.2		2	5.3	odd	4
2160.4.a.c.1.1		1	60.59	even	2
2160.4.a.m.1.1		1	20.19	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 $\F_{q}$
Belyi maps

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

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	1350.4.a.f.1.1

Level	$1350$
Weight	$4$
Character	1350.1
Self dual	yes
Analytic conductor	$79.653$
Analytic rank	$1$
Dimension	$1$
CM	no
Inner twists	$1$