LMFDB - Embedded newform 1323.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1323,4,Mod(1,1323)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1323, 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("1323.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$1323 = 3^{3} \cdot 7^{2}$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	1323.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:	$78.0595269376$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 189)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1323.1

$q$ -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^{2} +1.00000 q^{4} +6.00000 q^{5} -21.0000 q^{8} +18.0000 q^{10} -57.0000 q^{11} -62.0000 q^{13} -71.0000 q^{16} +12.0000 q^{17} +124.000 q^{19} +6.00000 q^{20} -171.000 q^{22} +156.000 q^{23} -89.0000 q^{25} -186.000 q^{26} +261.000 q^{29} +109.000 q^{31} -45.0000 q^{32} +36.0000 q^{34} +368.000 q^{37} +372.000 q^{38} -126.000 q^{40} -54.0000 q^{41} +152.000 q^{43} -57.0000 q^{44} +468.000 q^{46} +78.0000 q^{47} -267.000 q^{50} -62.0000 q^{52} +222.000 q^{53} -342.000 q^{55} +783.000 q^{58} -285.000 q^{59} +712.000 q^{61} +327.000 q^{62} +433.000 q^{64} -372.000 q^{65} +170.000 q^{67} +12.0000 q^{68} +396.000 q^{71} +475.000 q^{73} +1104.00 q^{74} +124.000 q^{76} -163.000 q^{79} -426.000 q^{80} -162.000 q^{82} -27.0000 q^{83} +72.0000 q^{85} +456.000 q^{86} +1197.00 q^{88} -642.000 q^{89} +156.000 q^{92} +234.000 q^{94} +744.000 q^{95} -1835.00 q^{97} +O(q^{100})

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	3.00000	1.06066	0.530330	−	0.847791i	$-0.322068\pi$
$2$	3.00000	1.06066	0.530330	+	0.847791i	$0.322068\pi$
$3$	0	0
$3$	0	0
$4$	1.00000	0.125000
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	−21.0000	−0.928078
$9$	0	0
$10$	18.0000	0.569210
$11$	−57.0000	−1.56238	−0.781188	−	0.624295i	$-0.785386\pi$
$11$	−57.0000	−1.56238	−0.781188	+	0.624295i	$0.785386\pi$
$12$	0	0
$13$	−62.0000	−1.32275	−0.661373	−	0.750057i	$-0.730026\pi$
$13$	−62.0000	−1.32275	−0.661373	+	0.750057i	$0.730026\pi$
$14$	0	0
$15$	0	0
$16$	−71.0000	−1.10938
$17$	12.0000	0.171202	0.0856008	−	0.996330i	$-0.472719\pi$
$17$	12.0000	0.171202	0.0856008	+	0.996330i	$0.472719\pi$
$18$	0	0
$19$	124.000	1.49724	0.748620	−	0.663000i	$-0.230717\pi$
$19$	124.000	1.49724	0.748620	+	0.663000i	$0.230717\pi$
$20$	6.00000	0.0670820
$21$	0	0
$22$	−171.000	−1.65715
$23$	156.000	1.41427	0.707136	−	0.707078i	$-0.249987\pi$
$23$	156.000	1.41427	0.707136	+	0.707078i	$0.249987\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	−186.000	−1.40298
$27$	0	0
$28$	0	0
$29$	261.000	1.67126	0.835629	−	0.549294i	$-0.185103\pi$
$29$	261.000	1.67126	0.835629	+	0.549294i	$0.185103\pi$
$30$	0	0
$31$	109.000	0.631515	0.315758	−	0.948840i	$-0.397741\pi$
$31$	109.000	0.631515	0.315758	+	0.948840i	$0.397741\pi$
$32$	−45.0000	−0.248592
$33$	0	0
$34$	36.0000	0.181587
$35$	0	0
$36$	0	0
$37$	368.000	1.63510	0.817552	−	0.575855i	$-0.195331\pi$
$37$	368.000	1.63510	0.817552	+	0.575855i	$0.195331\pi$
$38$	372.000	1.58806
$39$	0	0
$40$	−126.000	−0.498059
$41$	−54.0000	−0.205692	−0.102846	−	0.994697i	$-0.532795\pi$
$41$	−54.0000	−0.205692	−0.102846	+	0.994697i	$0.532795\pi$
$42$	0	0
$43$	152.000	0.539065	0.269532	−	0.962991i	$-0.413131\pi$
$43$	152.000	0.539065	0.269532	+	0.962991i	$0.413131\pi$
$44$	−57.0000	−0.195297
$45$	0	0
$46$	468.000	1.50006
$47$	78.0000	0.242074	0.121037	−	0.992648i	$-0.461378\pi$
$47$	78.0000	0.242074	0.121037	+	0.992648i	$0.461378\pi$
$48$	0	0
$49$	0	0
$50$	−267.000	−0.755190
$51$	0	0
$52$	−62.0000	−0.165343
$53$	222.000	0.575359	0.287680	−	0.957727i	$-0.407116\pi$
$53$	222.000	0.575359	0.287680	+	0.957727i	$0.407116\pi$
$54$	0	0
$55$	−342.000	−0.838459
$56$	0	0
$57$	0	0
$58$	783.000	1.77264
$59$	−285.000	−0.628879	−0.314439	−	0.949278i	$-0.601816\pi$
$59$	−285.000	−0.628879	−0.314439	+	0.949278i	$0.601816\pi$
$60$	0	0
$61$	712.000	1.49446	0.747232	−	0.664564i	$-0.231383\pi$
$61$	712.000	1.49446	0.747232	+	0.664564i	$0.231383\pi$
$62$	327.000	0.669823
$63$	0	0
$64$	433.000	0.845703
$65$	−372.000	−0.709860
$66$	0	0
$67$	170.000	0.309982	0.154991	−	0.987916i	$-0.450465\pi$
$67$	170.000	0.309982	0.154991	+	0.987916i	$0.450465\pi$
$68$	12.0000	0.0214002
$69$	0	0
$70$	0	0
$71$	396.000	0.661923	0.330962	−	0.943644i	$-0.392627\pi$
$71$	396.000	0.661923	0.330962	+	0.943644i	$0.392627\pi$
$72$	0	0
$73$	475.000	0.761569	0.380785	−	0.924664i	$-0.375654\pi$
$73$	475.000	0.761569	0.380785	+	0.924664i	$0.375654\pi$
$74$	1104.00	1.73429
$75$	0	0
$76$	124.000	0.187155
$77$	0	0
$78$	0	0
$79$	−163.000	−0.232138	−0.116069	−	0.993241i	$-0.537029\pi$
$79$	−163.000	−0.232138	−0.116069	+	0.993241i	$0.537029\pi$
$80$	−426.000	−0.595353
$81$	0	0
$82$	−162.000	−0.218170
$83$	−27.0000	−0.0357064	−0.0178532	−	0.999841i	$-0.505683\pi$
$83$	−27.0000	−0.0357064	−0.0178532	+	0.999841i	$0.505683\pi$
$84$	0	0
$85$	72.0000	0.0918764
$86$	456.000	0.571764
$87$	0	0
$88$	1197.00	1.45001
$89$	−642.000	−0.764628	−0.382314	−	0.924033i	$-0.624873\pi$
$89$	−642.000	−0.764628	−0.382314	+	0.924033i	$0.624873\pi$
$90$	0	0
$91$	0	0
$92$	156.000	0.176784
$93$	0	0
$94$	234.000	0.256758
$95$	744.000	0.803503
$96$	0	0
$97$	−1835.00	−1.92078	−0.960392	−	0.278653i	$-0.910112\pi$
$97$	−1835.00	−1.92078	−0.960392	+	0.278653i	$0.910112\pi$
$98$	0	0
$99$	0	0
$100$	−89.0000	−0.0890000
$101$	−1257.00	−1.23838	−0.619189	−	0.785242i	$-0.712539\pi$
$101$	−1257.00	−1.23838	−0.619189	+	0.785242i	$0.712539\pi$
$102$	0	0
$103$	−872.000	−0.834182	−0.417091	−	0.908865i	$-0.636950\pi$
$103$	−872.000	−0.834182	−0.417091	+	0.908865i	$0.636950\pi$
$104$	1302.00	1.22761
$105$	0	0
$106$	666.000	0.610261
$107$	−636.000	−0.574621	−0.287310	−	0.957838i	$-0.592761\pi$
$107$	−636.000	−0.574621	−0.287310	+	0.957838i	$0.592761\pi$
$108$	0	0
$109$	−64.0000	−0.0562393	−0.0281197	−	0.999605i	$-0.508952\pi$
$109$	−64.0000	−0.0562393	−0.0281197	+	0.999605i	$0.508952\pi$
$110$	−1026.00	−0.889321
$111$	0	0
$112$	0	0
$113$	900.000	0.749247	0.374623	−	0.927177i	$-0.377772\pi$
$113$	900.000	0.749247	0.374623	+	0.927177i	$0.377772\pi$
$114$	0	0
$115$	936.000	0.758978
$116$	261.000	0.208907
$117$	0	0
$118$	−855.000	−0.667027
$119$	0	0
$120$	0	0
$121$	1918.00	1.44102
$122$	2136.00	1.58512
$123$	0	0
$124$	109.000	0.0789394
$125$	−1284.00	−0.918756
$126$	0	0
$127$	−1936.00	−1.35269	−0.676347	−	0.736583i	$-0.736438\pi$
$127$	−1936.00	−1.35269	−0.676347	+	0.736583i	$0.736438\pi$
$128$	1659.00	1.14560
$129$	0	0
$130$	−1116.00	−0.752921
$131$	1113.00	0.742315	0.371157	−	0.928570i	$-0.378961\pi$
$131$	1113.00	0.742315	0.371157	+	0.928570i	$0.378961\pi$
$132$	0	0
$133$	0	0
$134$	510.000	0.328786
$135$	0	0
$136$	−252.000	−0.158888
$137$	1968.00	1.22728	0.613641	−	0.789585i	$-0.289704\pi$
$137$	1968.00	1.22728	0.613641	+	0.789585i	$0.289704\pi$
$138$	0	0
$139$	496.000	0.302663	0.151332	−	0.988483i	$-0.451644\pi$
$139$	496.000	0.302663	0.151332	+	0.988483i	$0.451644\pi$
$140$	0	0
$141$	0	0
$142$	1188.00	0.702076
$143$	3534.00	2.06663
$144$	0	0
$145$	1566.00	0.896891
$146$	1425.00	0.807766
$147$	0	0
$148$	368.000	0.204388
$149$	−141.000	−0.0775246	−0.0387623	−	0.999248i	$-0.512342\pi$
$149$	−141.000	−0.0775246	−0.0387623	+	0.999248i	$0.512342\pi$
$150$	0	0
$151$	71.0000	0.0382642	0.0191321	−	0.999817i	$-0.493910\pi$
$151$	71.0000	0.0382642	0.0191321	+	0.999817i	$0.493910\pi$
$152$	−2604.00	−1.38955
$153$	0	0
$154$	0	0
$155$	654.000	0.338907
$156$	0	0
$157$	−614.000	−0.312118	−0.156059	−	0.987748i	$-0.549879\pi$
$157$	−614.000	−0.312118	−0.156059	+	0.987748i	$0.549879\pi$
$158$	−489.000	−0.246220
$159$	0	0
$160$	−270.000	−0.133409
$161$	0	0
$162$	0	0
$163$	818.000	0.393072	0.196536	−	0.980497i	$-0.437031\pi$
$163$	818.000	0.393072	0.196536	+	0.980497i	$0.437031\pi$
$164$	−54.0000	−0.0257115
$165$	0	0
$166$	−81.0000	−0.0378724
$167$	3474.00	1.60974	0.804869	−	0.593453i	$-0.202236\pi$
$167$	3474.00	1.60974	0.804869	+	0.593453i	$0.202236\pi$
$168$	0	0
$169$	1647.00	0.749659
$170$	216.000	0.0974497
$171$	0	0
$172$	152.000	0.0673831
$173$	−3891.00	−1.70998	−0.854992	−	0.518641i	$-0.826438\pi$
$173$	−3891.00	−1.70998	−0.854992	+	0.518641i	$0.826438\pi$
$174$	0	0
$175$	0	0
$176$	4047.00	1.73326
$177$	0	0
$178$	−1926.00	−0.811010
$179$	2499.00	1.04349	0.521743	−	0.853103i	$-0.325282\pi$
$179$	2499.00	1.04349	0.521743	+	0.853103i	$0.325282\pi$
$180$	0	0
$181$	1888.00	0.775326	0.387663	−	0.921801i	$-0.373282\pi$
$181$	1888.00	0.775326	0.387663	+	0.921801i	$0.373282\pi$
$182$	0	0
$183$	0	0
$184$	−3276.00	−1.31255
$185$	2208.00	0.877489
$186$	0	0
$187$	−684.000	−0.267481
$188$	78.0000	0.0302592
$189$	0	0
$190$	2232.00	0.852244
$191$	570.000	0.215936	0.107968	−	0.994154i	$-0.465566\pi$
$191$	570.000	0.215936	0.107968	+	0.994154i	$0.465566\pi$
$192$	0	0
$193$	2141.00	0.798511	0.399255	−	0.916840i	$-0.369269\pi$
$193$	2141.00	0.798511	0.399255	+	0.916840i	$0.369269\pi$
$194$	−5505.00	−2.03730
$195$	0	0
$196$	0	0
$197$	−531.000	−0.192042	−0.0960208	−	0.995379i	$-0.530612\pi$
$197$	−531.000	−0.192042	−0.0960208	+	0.995379i	$0.530612\pi$
$198$	0	0
$199$	91.0000	0.0324162	0.0162081	−	0.999869i	$-0.494841\pi$
$199$	91.0000	0.0324162	0.0162081	+	0.999869i	$0.494841\pi$
$200$	1869.00	0.660791
$201$	0	0
$202$	−3771.00	−1.31350
$203$	0	0
$204$	0	0
$205$	−324.000	−0.110386
$206$	−2616.00	−0.884783
$207$	0	0
$208$	4402.00	1.46742
$209$	−7068.00	−2.33925
$210$	0	0
$211$	1610.00	0.525294	0.262647	−	0.964892i	$-0.415405\pi$
$211$	1610.00	0.525294	0.262647	+	0.964892i	$0.415405\pi$
$212$	222.000	0.0719199
$213$	0	0
$214$	−1908.00	−0.609478
$215$	912.000	0.289292
$216$	0	0
$217$	0	0
$218$	−192.000	−0.0596508
$219$	0	0
$220$	−342.000	−0.104807
$221$	−744.000	−0.226456
$222$	0	0
$223$	4273.00	1.28314	0.641572	−	0.767063i	$-0.278282\pi$
$223$	4273.00	1.28314	0.641572	+	0.767063i	$0.278282\pi$
$224$	0	0
$225$	0	0
$226$	2700.00	0.794696
$227$	−15.0000	−0.00438584	−0.00219292	−	0.999998i	$-0.500698\pi$
$227$	−15.0000	−0.00438584	−0.00219292	+	0.999998i	$0.500698\pi$
$228$	0	0
$229$	−1988.00	−0.573671	−0.286836	−	0.957980i	$-0.592603\pi$
$229$	−1988.00	−0.573671	−0.286836	+	0.957980i	$0.592603\pi$
$230$	2808.00	0.805018
$231$	0	0
$232$	−5481.00	−1.55106
$233$	5070.00	1.42552	0.712761	−	0.701407i	$-0.247444\pi$
$233$	5070.00	1.42552	0.712761	+	0.701407i	$0.247444\pi$
$234$	0	0
$235$	468.000	0.129910
$236$	−285.000	−0.0786098
$237$	0	0
$238$	0	0
$239$	1872.00	0.506651	0.253326	−	0.967381i	$-0.418476\pi$
$239$	1872.00	0.506651	0.253326	+	0.967381i	$0.418476\pi$
$240$	0	0
$241$	5185.00	1.38587	0.692936	−	0.720999i	$-0.256317\pi$
$241$	5185.00	1.38587	0.692936	+	0.720999i	$0.256317\pi$
$242$	5754.00	1.52843
$243$	0	0
$244$	712.000	0.186808
$245$	0	0
$246$	0	0
$247$	−7688.00	−1.98047
$248$	−2289.00	−0.586095
$249$	0	0
$250$	−3852.00	−0.974487
$251$	7479.00	1.88076	0.940379	−	0.340128i	$-0.110470\pi$
$251$	7479.00	1.88076	0.940379	+	0.340128i	$0.110470\pi$
$252$	0	0
$253$	−8892.00	−2.20963
$254$	−5808.00	−1.43475
$255$	0	0
$256$	1513.00	0.369385
$257$	6288.00	1.52620	0.763102	−	0.646278i	$-0.223675\pi$
$257$	6288.00	1.52620	0.763102	+	0.646278i	$0.223675\pi$
$258$	0	0
$259$	0	0
$260$	−372.000	−0.0887325
$261$	0	0
$262$	3339.00	0.787344
$263$	7764.00	1.82034	0.910169	−	0.414238i	$-0.135952\pi$
$263$	7764.00	1.82034	0.910169	+	0.414238i	$0.135952\pi$
$264$	0	0
$265$	1332.00	0.308770
$266$	0	0
$267$	0	0
$268$	170.000	0.0387478
$269$	1299.00	0.294429	0.147215	−	0.989105i	$-0.452969\pi$
$269$	1299.00	0.294429	0.147215	+	0.989105i	$0.452969\pi$
$270$	0	0
$271$	−6128.00	−1.37361	−0.686807	−	0.726840i	$-0.740988\pi$
$271$	−6128.00	−1.37361	−0.686807	+	0.726840i	$0.740988\pi$
$272$	−852.000	−0.189927
$273$	0	0
$274$	5904.00	1.30173
$275$	5073.00	1.11241
$276$	0	0
$277$	−3646.00	−0.790855	−0.395428	−	0.918497i	$-0.629404\pi$
$277$	−3646.00	−0.790855	−0.395428	+	0.918497i	$0.629404\pi$
$278$	1488.00	0.321023
$279$	0	0
$280$	0	0
$281$	4950.00	1.05086	0.525431	−	0.850836i	$-0.323904\pi$
$281$	4950.00	1.05086	0.525431	+	0.850836i	$0.323904\pi$
$282$	0	0
$283$	−4904.00	−1.03008	−0.515040	−	0.857166i	$-0.672223\pi$
$283$	−4904.00	−1.03008	−0.515040	+	0.857166i	$0.672223\pi$
$284$	396.000	0.0827404
$285$	0	0
$286$	10602.0	2.19199
$287$	0	0
$288$	0	0
$289$	−4769.00	−0.970690
$290$	4698.00	0.951297
$291$	0	0
$292$	475.000	0.0951961
$293$	7623.00	1.51993	0.759967	−	0.649962i	$-0.225215\pi$
$293$	7623.00	1.51993	0.759967	+	0.649962i	$0.225215\pi$
$294$	0	0
$295$	−1710.00	−0.337492
$296$	−7728.00	−1.51750
$297$	0	0
$298$	−423.000	−0.0822273
$299$	−9672.00	−1.87072
$300$	0	0
$301$	0	0
$302$	213.000	0.0405853
$303$	0	0
$304$	−8804.00	−1.66100
$305$	4272.00	0.802013
$306$	0	0
$307$	226.000	0.0420147	0.0210073	−	0.999779i	$-0.493313\pi$
$307$	226.000	0.0420147	0.0210073	+	0.999779i	$0.493313\pi$
$308$	0	0
$309$	0	0
$310$	1962.00	0.359465
$311$	−3426.00	−0.624664	−0.312332	−	0.949973i	$-0.601110\pi$
$311$	−3426.00	−0.624664	−0.312332	+	0.949973i	$0.601110\pi$
$312$	0	0
$313$	−3242.00	−0.585459	−0.292730	−	0.956195i	$-0.594564\pi$
$313$	−3242.00	−0.585459	−0.292730	+	0.956195i	$0.594564\pi$
$314$	−1842.00	−0.331051
$315$	0	0
$316$	−163.000	−0.0290173
$317$	−5091.00	−0.902016	−0.451008	−	0.892520i	$-0.648935\pi$
$317$	−5091.00	−0.902016	−0.451008	+	0.892520i	$0.648935\pi$
$318$	0	0
$319$	−14877.0	−2.61114
$320$	2598.00	0.453852
$321$	0	0
$322$	0	0
$323$	1488.00	0.256330
$324$	0	0
$325$	5518.00	0.941796
$326$	2454.00	0.416916
$327$	0	0
$328$	1134.00	0.190898
$329$	0	0
$330$	0	0
$331$	−6640.00	−1.10262	−0.551310	−	0.834300i	$-0.685872\pi$
$331$	−6640.00	−1.10262	−0.551310	+	0.834300i	$0.685872\pi$
$332$	−27.0000	−0.00446331
$333$	0	0
$334$	10422.0	1.70738
$335$	1020.00	0.166354
$336$	0	0
$337$	2981.00	0.481856	0.240928	−	0.970543i	$-0.422548\pi$
$337$	2981.00	0.481856	0.240928	+	0.970543i	$0.422548\pi$
$338$	4941.00	0.795133
$339$	0	0
$340$	72.0000	0.0114846
$341$	−6213.00	−0.986665
$342$	0	0
$343$	0	0
$344$	−3192.00	−0.500294
$345$	0	0
$346$	−11673.0	−1.81371
$347$	−9561.00	−1.47914	−0.739570	−	0.673080i	$-0.764971\pi$
$347$	−9561.00	−1.47914	−0.739570	+	0.673080i	$0.764971\pi$
$348$	0	0
$349$	−13004.0	−1.99452	−0.997261	−	0.0739631i	$-0.976435\pi$
$349$	−13004.0	−1.99452	−0.997261	+	0.0739631i	$0.976435\pi$
$350$	0	0
$351$	0	0
$352$	2565.00	0.388395
$353$	642.000	0.0967995	0.0483997	−	0.998828i	$-0.484588\pi$
$353$	642.000	0.0967995	0.0483997	+	0.998828i	$0.484588\pi$
$354$	0	0
$355$	2376.00	0.355225
$356$	−642.000	−0.0955785
$357$	0	0
$358$	7497.00	1.10678
$359$	3576.00	0.525722	0.262861	−	0.964834i	$-0.415334\pi$
$359$	3576.00	0.525722	0.262861	+	0.964834i	$0.415334\pi$
$360$	0	0
$361$	8517.00	1.24173
$362$	5664.00	0.822357
$363$	0	0
$364$	0	0
$365$	2850.00	0.408701
$366$	0	0
$367$	−2972.00	−0.422717	−0.211358	−	0.977409i	$-0.567789\pi$
$367$	−2972.00	−0.422717	−0.211358	+	0.977409i	$0.567789\pi$
$368$	−11076.0	−1.56896
$369$	0	0
$370$	6624.00	0.930717
$371$	0	0
$372$	0	0
$373$	−7108.00	−0.986698	−0.493349	−	0.869832i	$-0.664227\pi$
$373$	−7108.00	−0.986698	−0.493349	+	0.869832i	$0.664227\pi$
$374$	−2052.00	−0.283707
$375$	0	0
$376$	−1638.00	−0.224663
$377$	−16182.0	−2.21065
$378$	0	0
$379$	−5878.00	−0.796656	−0.398328	−	0.917243i	$-0.630409\pi$
$379$	−5878.00	−0.796656	−0.398328	+	0.917243i	$0.630409\pi$
$380$	744.000	0.100438
$381$	0	0
$382$	1710.00	0.229035
$383$	−4332.00	−0.577950	−0.288975	−	0.957337i	$-0.593314\pi$
$383$	−4332.00	−0.577950	−0.288975	+	0.957337i	$0.593314\pi$
$384$	0	0
$385$	0	0
$386$	6423.00	0.846948
$387$	0	0
$388$	−1835.00	−0.240098
$389$	−1173.00	−0.152888	−0.0764440	−	0.997074i	$-0.524357\pi$
$389$	−1173.00	−0.152888	−0.0764440	+	0.997074i	$0.524357\pi$
$390$	0	0
$391$	1872.00	0.242126
$392$	0	0
$393$	0	0
$394$	−1593.00	−0.203691
$395$	−978.000	−0.124579
$396$	0	0
$397$	8866.00	1.12084	0.560418	−	0.828210i	$-0.310641\pi$
$397$	8866.00	1.12084	0.560418	+	0.828210i	$0.310641\pi$
$398$	273.000	0.0343825
$399$	0	0
$400$	6319.00	0.789875
$401$	8778.00	1.09315	0.546574	−	0.837411i	$-0.315932\pi$
$401$	8778.00	1.09315	0.546574	+	0.837411i	$0.315932\pi$
$402$	0	0
$403$	−6758.00	−0.835335
$404$	−1257.00	−0.154797
$405$	0	0
$406$	0	0
$407$	−20976.0	−2.55465
$408$	0	0
$409$	5482.00	0.662757	0.331378	−	0.943498i	$-0.392486\pi$
$409$	5482.00	0.662757	0.331378	+	0.943498i	$0.392486\pi$
$410$	−972.000	−0.117082
$411$	0	0
$412$	−872.000	−0.104273
$413$	0	0
$414$	0	0
$415$	−162.000	−0.0191621
$416$	2790.00	0.328825
$417$	0	0
$418$	−21204.0	−2.48115
$419$	−3780.00	−0.440728	−0.220364	−	0.975418i	$-0.570725\pi$
$419$	−3780.00	−0.440728	−0.220364	+	0.975418i	$0.570725\pi$
$420$	0	0
$421$	1262.00	0.146095	0.0730476	−	0.997328i	$-0.476727\pi$
$421$	1262.00	0.146095	0.0730476	+	0.997328i	$0.476727\pi$
$422$	4830.00	0.557158
$423$	0	0
$424$	−4662.00	−0.533978
$425$	−1068.00	−0.121896
$426$	0	0
$427$	0	0
$428$	−636.000	−0.0718276
$429$	0	0
$430$	2736.00	0.306841
$431$	5010.00	0.559915	0.279957	−	0.960012i	$-0.409680\pi$
$431$	5010.00	0.559915	0.279957	+	0.960012i	$0.409680\pi$
$432$	0	0
$433$	8263.00	0.917077	0.458539	−	0.888674i	$-0.348373\pi$
$433$	8263.00	0.917077	0.458539	+	0.888674i	$0.348373\pi$
$434$	0	0
$435$	0	0
$436$	−64.0000	−0.00702992
$437$	19344.0	2.11750
$438$	0	0
$439$	−3755.00	−0.408238	−0.204119	−	0.978946i	$-0.565433\pi$
$439$	−3755.00	−0.408238	−0.204119	+	0.978946i	$0.565433\pi$
$440$	7182.00	0.778155
$441$	0	0
$442$	−2232.00	−0.240193
$443$	11721.0	1.25707	0.628534	−	0.777782i	$-0.283655\pi$
$443$	11721.0	1.25707	0.628534	+	0.777782i	$0.283655\pi$
$444$	0	0
$445$	−3852.00	−0.410342
$446$	12819.0	1.36098
$447$	0	0
$448$	0	0
$449$	7398.00	0.777580	0.388790	−	0.921327i	$-0.372893\pi$
$449$	7398.00	0.777580	0.388790	+	0.921327i	$0.372893\pi$
$450$	0	0
$451$	3078.00	0.321369
$452$	900.000	0.0936558
$453$	0	0
$454$	−45.0000	−0.00465188
$455$	0	0
$456$	0	0
$457$	6182.00	0.632783	0.316391	−	0.948629i	$-0.397529\pi$
$457$	6182.00	0.632783	0.316391	+	0.948629i	$0.397529\pi$
$458$	−5964.00	−0.608470
$459$	0	0
$460$	936.000	0.0948722
$461$	−15291.0	−1.54484	−0.772422	−	0.635110i	$-0.780955\pi$
$461$	−15291.0	−1.54484	−0.772422	+	0.635110i	$0.780955\pi$
$462$	0	0
$463$	−18349.0	−1.84179	−0.920897	−	0.389807i	$-0.872542\pi$
$463$	−18349.0	−1.84179	−0.920897	+	0.389807i	$0.872542\pi$
$464$	−18531.0	−1.85405
$465$	0	0
$466$	15210.0	1.51199
$467$	5883.00	0.582940	0.291470	−	0.956580i	$-0.405856\pi$
$467$	5883.00	0.582940	0.291470	+	0.956580i	$0.405856\pi$
$468$	0	0
$469$	0	0
$470$	1404.00	0.137791
$471$	0	0
$472$	5985.00	0.583648
$473$	−8664.00	−0.842222
$474$	0	0
$475$	−11036.0	−1.06603
$476$	0	0
$477$	0	0
$478$	5616.00	0.537385
$479$	−11742.0	−1.12005	−0.560027	−	0.828474i	$-0.689209\pi$
$479$	−11742.0	−1.12005	−0.560027	+	0.828474i	$0.689209\pi$
$480$	0	0
$481$	−22816.0	−2.16283
$482$	15555.0	1.46994
$483$	0	0
$484$	1918.00	0.180128
$485$	−11010.0	−1.03080
$486$	0	0
$487$	6923.00	0.644171	0.322085	−	0.946711i	$-0.395616\pi$
$487$	6923.00	0.644171	0.322085	+	0.946711i	$0.395616\pi$
$488$	−14952.0	−1.38698
$489$	0	0
$490$	0	0
$491$	−8172.00	−0.751114	−0.375557	−	0.926799i	$-0.622549\pi$
$491$	−8172.00	−0.751114	−0.375557	+	0.926799i	$0.622549\pi$
$492$	0	0
$493$	3132.00	0.286122
$494$	−23064.0	−2.10060
$495$	0	0
$496$	−7739.00	−0.700587
$497$	0	0
$498$	0	0
$499$	9944.00	0.892093	0.446047	−	0.895010i	$-0.352832\pi$
$499$	9944.00	0.892093	0.446047	+	0.895010i	$0.352832\pi$
$500$	−1284.00	−0.114844
$501$	0	0
$502$	22437.0	1.99485
$503$	6606.00	0.585580	0.292790	−	0.956177i	$-0.405416\pi$
$503$	6606.00	0.585580	0.292790	+	0.956177i	$0.405416\pi$
$504$	0	0
$505$	−7542.00	−0.664583
$506$	−26676.0	−2.34366
$507$	0	0
$508$	−1936.00	−0.169087
$509$	20103.0	1.75059	0.875295	−	0.483590i	$-0.160668\pi$
$509$	20103.0	1.75059	0.875295	+	0.483590i	$0.160668\pi$
$510$	0	0
$511$	0	0
$512$	−8733.00	−0.753804
$513$	0	0
$514$	18864.0	1.61878
$515$	−5232.00	−0.447669
$516$	0	0
$517$	−4446.00	−0.378211
$518$	0	0
$519$	0	0
$520$	7812.00	0.658806
$521$	−4056.00	−0.341068	−0.170534	−	0.985352i	$-0.554549\pi$
$521$	−4056.00	−0.341068	−0.170534	+	0.985352i	$0.554549\pi$
$522$	0	0
$523$	20494.0	1.71346	0.856730	−	0.515764i	$-0.172492\pi$
$523$	20494.0	1.71346	0.856730	+	0.515764i	$0.172492\pi$
$524$	1113.00	0.0927894
$525$	0	0
$526$	23292.0	1.93076
$527$	1308.00	0.108116
$528$	0	0
$529$	12169.0	1.00016
$530$	3996.00	0.327500
$531$	0	0
$532$	0	0
$533$	3348.00	0.272079
$534$	0	0
$535$	−3816.00	−0.308374
$536$	−3570.00	−0.287688
$537$	0	0
$538$	3897.00	0.312289
$539$	0	0
$540$	0	0
$541$	7490.00	0.595232	0.297616	−	0.954686i	$-0.403809\pi$
$541$	7490.00	0.595232	0.297616	+	0.954686i	$0.403809\pi$
$542$	−18384.0	−1.45694
$543$	0	0
$544$	−540.000	−0.0425594
$545$	−384.000	−0.0301812
$546$	0	0
$547$	6326.00	0.494479	0.247240	−	0.968954i	$-0.420476\pi$
$547$	6326.00	0.494479	0.247240	+	0.968954i	$0.420476\pi$
$548$	1968.00	0.153410
$549$	0	0
$550$	15219.0	1.17989
$551$	32364.0	2.50227
$552$	0	0
$553$	0	0
$554$	−10938.0	−0.838829
$555$	0	0
$556$	496.000	0.0378329
$557$	10239.0	0.778888	0.389444	−	0.921050i	$-0.372667\pi$
$557$	10239.0	0.778888	0.389444	+	0.921050i	$0.372667\pi$
$558$	0	0
$559$	−9424.00	−0.713046
$560$	0	0
$561$	0	0
$562$	14850.0	1.11461
$563$	−9132.00	−0.683602	−0.341801	−	0.939772i	$-0.611037\pi$
$563$	−9132.00	−0.683602	−0.341801	+	0.939772i	$0.611037\pi$
$564$	0	0
$565$	5400.00	0.402088
$566$	−14712.0	−1.09256
$567$	0	0
$568$	−8316.00	−0.614316
$569$	21324.0	1.57109	0.785544	−	0.618806i	$-0.212383\pi$
$569$	21324.0	1.57109	0.785544	+	0.618806i	$0.212383\pi$
$570$	0	0
$571$	17840.0	1.30750	0.653748	−	0.756712i	$-0.273195\pi$
$571$	17840.0	1.30750	0.653748	+	0.756712i	$0.273195\pi$
$572$	3534.00	0.258329
$573$	0	0
$574$	0	0
$575$	−13884.0	−1.00696
$576$	0	0
$577$	1189.00	0.0857863	0.0428932	−	0.999080i	$-0.486342\pi$
$577$	1189.00	0.0857863	0.0428932	+	0.999080i	$0.486342\pi$
$578$	−14307.0	−1.02957
$579$	0	0
$580$	1566.00	0.112111
$581$	0	0
$582$	0	0
$583$	−12654.0	−0.898928
$584$	−9975.00	−0.706795
$585$	0	0
$586$	22869.0	1.61213
$587$	−20124.0	−1.41500	−0.707501	−	0.706712i	$-0.750178\pi$
$587$	−20124.0	−1.41500	−0.707501	+	0.706712i	$0.750178\pi$
$588$	0	0
$589$	13516.0	0.945530
$590$	−5130.00	−0.357964
$591$	0	0
$592$	−26128.0	−1.81394
$593$	7260.00	0.502753	0.251376	−	0.967889i	$-0.419117\pi$
$593$	7260.00	0.502753	0.251376	+	0.967889i	$0.419117\pi$
$594$	0	0
$595$	0	0
$596$	−141.000	−0.00969058
$597$	0	0
$598$	−29016.0	−1.98420
$599$	19122.0	1.30435	0.652173	−	0.758070i	$-0.273857\pi$
$599$	19122.0	1.30435	0.652173	+	0.758070i	$0.273857\pi$
$600$	0	0
$601$	−5318.00	−0.360941	−0.180471	−	0.983580i	$-0.557762\pi$
$601$	−5318.00	−0.360941	−0.180471	+	0.983580i	$0.557762\pi$
$602$	0	0
$603$	0	0
$604$	71.0000	0.00478303
$605$	11508.0	0.773333
$606$	0	0
$607$	−7175.00	−0.479776	−0.239888	−	0.970801i	$-0.577111\pi$
$607$	−7175.00	−0.479776	−0.239888	+	0.970801i	$0.577111\pi$
$608$	−5580.00	−0.372202
$609$	0	0
$610$	12816.0	0.850663
$611$	−4836.00	−0.320202
$612$	0	0
$613$	4346.00	0.286351	0.143176	−	0.989697i	$-0.454269\pi$
$613$	4346.00	0.286351	0.143176	+	0.989697i	$0.454269\pi$
$614$	678.000	0.0445633
$615$	0	0
$616$	0	0
$617$	10926.0	0.712908	0.356454	−	0.934313i	$-0.383986\pi$
$617$	10926.0	0.712908	0.356454	+	0.934313i	$0.383986\pi$
$618$	0	0
$619$	388.000	0.0251939	0.0125970	−	0.999921i	$-0.495990\pi$
$619$	388.000	0.0251939	0.0125970	+	0.999921i	$0.495990\pi$
$620$	654.000	0.0423633
$621$	0	0
$622$	−10278.0	−0.662557
$623$	0	0
$624$	0	0
$625$	3421.00	0.218944
$626$	−9726.00	−0.620973
$627$	0	0
$628$	−614.000	−0.0390148
$629$	4416.00	0.279932
$630$	0	0
$631$	−16579.0	−1.04596	−0.522979	−	0.852346i	$-0.675179\pi$
$631$	−16579.0	−1.04596	−0.522979	+	0.852346i	$0.675179\pi$
$632$	3423.00	0.215442
$633$	0	0
$634$	−15273.0	−0.956732
$635$	−11616.0	−0.725932
$636$	0	0
$637$	0	0
$638$	−44631.0	−2.76953
$639$	0	0
$640$	9954.00	0.614791
$641$	−19038.0	−1.17310	−0.586549	−	0.809914i	$-0.699514\pi$
$641$	−19038.0	−1.17310	−0.586549	+	0.809914i	$0.699514\pi$
$642$	0	0
$643$	−20438.0	−1.25349	−0.626747	−	0.779223i	$-0.715614\pi$
$643$	−20438.0	−1.25349	−0.626747	+	0.779223i	$0.715614\pi$
$644$	0	0
$645$	0	0
$646$	4464.00	0.271879
$647$	9678.00	0.588070	0.294035	−	0.955795i	$-0.405002\pi$
$647$	9678.00	0.588070	0.294035	+	0.955795i	$0.405002\pi$
$648$	0	0
$649$	16245.0	0.982545
$650$	16554.0	0.998925
$651$	0	0
$652$	818.000	0.0491340
$653$	−3786.00	−0.226888	−0.113444	−	0.993544i	$-0.536188\pi$
$653$	−3786.00	−0.226888	−0.113444	+	0.993544i	$0.536188\pi$
$654$	0	0
$655$	6678.00	0.398368
$656$	3834.00	0.228190
$657$	0	0
$658$	0	0
$659$	3051.00	0.180349	0.0901746	−	0.995926i	$-0.471257\pi$
$659$	3051.00	0.180349	0.0901746	+	0.995926i	$0.471257\pi$
$660$	0	0
$661$	15910.0	0.936199	0.468099	−	0.883676i	$-0.344939\pi$
$661$	15910.0	0.936199	0.468099	+	0.883676i	$0.344939\pi$
$662$	−19920.0	−1.16951
$663$	0	0
$664$	567.000	0.0331384
$665$	0	0
$666$	0	0
$667$	40716.0	2.36361
$668$	3474.00	0.201217
$669$	0	0
$670$	3060.00	0.176445
$671$	−40584.0	−2.33491
$672$	0	0
$673$	−13534.0	−0.775182	−0.387591	−	0.921831i	$-0.626693\pi$
$673$	−13534.0	−0.775182	−0.387591	+	0.921831i	$0.626693\pi$
$674$	8943.00	0.511085
$675$	0	0
$676$	1647.00	0.0937073
$677$	28221.0	1.60210	0.801050	−	0.598598i	$-0.204275\pi$
$677$	28221.0	1.60210	0.801050	+	0.598598i	$0.204275\pi$
$678$	0	0
$679$	0	0
$680$	−1512.00	−0.0852685
$681$	0	0
$682$	−18639.0	−1.04652
$683$	−8841.00	−0.495302	−0.247651	−	0.968849i	$-0.579659\pi$
$683$	−8841.00	−0.495302	−0.247651	+	0.968849i	$0.579659\pi$
$684$	0	0
$685$	11808.0	0.658628
$686$	0	0
$687$	0	0
$688$	−10792.0	−0.598025
$689$	−13764.0	−0.761055
$690$	0	0
$691$	−27728.0	−1.52652	−0.763258	−	0.646094i	$-0.776402\pi$
$691$	−27728.0	−1.52652	−0.763258	+	0.646094i	$0.776402\pi$
$692$	−3891.00	−0.213748
$693$	0	0
$694$	−28683.0	−1.56886
$695$	2976.00	0.162426
$696$	0	0
$697$	−648.000	−0.0352148
$698$	−39012.0	−2.11551
$699$	0	0
$700$	0	0
$701$	21798.0	1.17446	0.587232	−	0.809419i	$-0.300218\pi$
$701$	21798.0	1.17446	0.587232	+	0.809419i	$0.300218\pi$
$702$	0	0
$703$	45632.0	2.44814
$704$	−24681.0	−1.32131
$705$	0	0
$706$	1926.00	0.102671
$707$	0	0
$708$	0	0
$709$	−6904.00	−0.365705	−0.182853	−	0.983140i	$-0.558533\pi$
$709$	−6904.00	−0.365705	−0.182853	+	0.983140i	$0.558533\pi$
$710$	7128.00	0.376773
$711$	0	0
$712$	13482.0	0.709634
$713$	17004.0	0.893134
$714$	0	0
$715$	21204.0	1.10907
$716$	2499.00	0.130436
$717$	0	0
$718$	10728.0	0.557612
$719$	15270.0	0.792037	0.396019	−	0.918242i	$-0.370392\pi$
$719$	15270.0	0.792037	0.396019	+	0.918242i	$0.370392\pi$
$720$	0	0
$721$	0	0
$722$	25551.0	1.31705
$723$	0	0
$724$	1888.00	0.0969157
$725$	−23229.0	−1.18994
$726$	0	0
$727$	3664.00	0.186919	0.0934596	−	0.995623i	$-0.470207\pi$
$727$	3664.00	0.186919	0.0934596	+	0.995623i	$0.470207\pi$
$728$	0	0
$729$	0	0
$730$	8550.00	0.433493
$731$	1824.00	0.0922888
$732$	0	0
$733$	−10124.0	−0.510148	−0.255074	−	0.966922i	$-0.582100\pi$
$733$	−10124.0	−0.510148	−0.255074	+	0.966922i	$0.582100\pi$
$734$	−8916.00	−0.448359
$735$	0	0
$736$	−7020.00	−0.351577
$737$	−9690.00	−0.484309
$738$	0	0
$739$	10034.0	0.499468	0.249734	−	0.968315i	$-0.419657\pi$
$739$	10034.0	0.499468	0.249734	+	0.968315i	$0.419657\pi$
$740$	2208.00	0.109686
$741$	0	0
$742$	0	0
$743$	−20898.0	−1.03186	−0.515931	−	0.856630i	$-0.672554\pi$
$743$	−20898.0	−1.03186	−0.515931	+	0.856630i	$0.672554\pi$
$744$	0	0
$745$	−846.000	−0.0416041
$746$	−21324.0	−1.04655
$747$	0	0
$748$	−684.000	−0.0334352
$749$	0	0
$750$	0	0
$751$	−9268.00	−0.450325	−0.225163	−	0.974321i	$-0.572291\pi$
$751$	−9268.00	−0.450325	−0.225163	+	0.974321i	$0.572291\pi$
$752$	−5538.00	−0.268551
$753$	0	0
$754$	−48546.0	−2.34475
$755$	426.000	0.0205347
$756$	0	0
$757$	12086.0	0.580282	0.290141	−	0.956984i	$-0.406298\pi$
$757$	12086.0	0.580282	0.290141	+	0.956984i	$0.406298\pi$
$758$	−17634.0	−0.844981
$759$	0	0
$760$	−15624.0	−0.745713
$761$	2688.00	0.128042	0.0640210	−	0.997949i	$-0.479608\pi$
$761$	2688.00	0.128042	0.0640210	+	0.997949i	$0.479608\pi$
$762$	0	0
$763$	0	0
$764$	570.000	0.0269920
$765$	0	0
$766$	−12996.0	−0.613009
$767$	17670.0	0.831847
$768$	0	0
$769$	9871.00	0.462883	0.231442	−	0.972849i	$-0.425656\pi$
$769$	9871.00	0.462883	0.231442	+	0.972849i	$0.425656\pi$
$770$	0	0
$771$	0	0
$772$	2141.00	0.0998138
$773$	24546.0	1.14212	0.571060	−	0.820909i	$-0.306532\pi$
$773$	24546.0	1.14212	0.571060	+	0.820909i	$0.306532\pi$
$774$	0	0
$775$	−9701.00	−0.449639
$776$	38535.0	1.78264
$777$	0	0
$778$	−3519.00	−0.162162
$779$	−6696.00	−0.307971
$780$	0	0
$781$	−22572.0	−1.03417
$782$	5616.00	0.256813
$783$	0	0
$784$	0	0
$785$	−3684.00	−0.167500
$786$	0	0
$787$	−30194.0	−1.36760	−0.683799	−	0.729670i	$-0.739674\pi$
$787$	−30194.0	−1.36760	−0.683799	+	0.729670i	$0.739674\pi$
$788$	−531.000	−0.0240052
$789$	0	0
$790$	−2934.00	−0.132135
$791$	0	0
$792$	0	0
$793$	−44144.0	−1.97680
$794$	26598.0	1.18883
$795$	0	0
$796$	91.0000	0.00405202
$797$	−26451.0	−1.17559	−0.587793	−	0.809011i	$-0.700003\pi$
$797$	−26451.0	−1.17559	−0.587793	+	0.809011i	$0.700003\pi$
$798$	0	0
$799$	936.000	0.0414434
$800$	4005.00	0.176998
$801$	0	0
$802$	26334.0	1.15946
$803$	−27075.0	−1.18986
$804$	0	0
$805$	0	0
$806$	−20274.0	−0.886006
$807$	0	0
$808$	26397.0	1.14931
$809$	−1650.00	−0.0717069	−0.0358535	−	0.999357i	$-0.511415\pi$
$809$	−1650.00	−0.0717069	−0.0358535	+	0.999357i	$0.511415\pi$
$810$	0	0
$811$	−1190.00	−0.0515247	−0.0257624	−	0.999668i	$-0.508201\pi$
$811$	−1190.00	−0.0515247	−0.0257624	+	0.999668i	$0.508201\pi$
$812$	0	0
$813$	0	0
$814$	−62928.0	−2.70961
$815$	4908.00	0.210944
$816$	0	0
$817$	18848.0	0.807109
$818$	16446.0	0.702960
$819$	0	0
$820$	−324.000	−0.0137983
$821$	26805.0	1.13947	0.569733	−	0.821830i	$-0.307047\pi$
$821$	26805.0	1.13947	0.569733	+	0.821830i	$0.307047\pi$
$822$	0	0
$823$	−23707.0	−1.00410	−0.502050	−	0.864839i	$-0.667421\pi$
$823$	−23707.0	−1.00410	−0.502050	+	0.864839i	$0.667421\pi$
$824$	18312.0	0.774185
$825$	0	0
$826$	0	0
$827$	30897.0	1.29915	0.649573	−	0.760299i	$-0.274948\pi$
$827$	30897.0	1.29915	0.649573	+	0.760299i	$0.274948\pi$
$828$	0	0
$829$	−47126.0	−1.97437	−0.987186	−	0.159577i	$-0.948987\pi$
$829$	−47126.0	−1.97437	−0.987186	+	0.159577i	$0.948987\pi$
$830$	−486.000	−0.0203245
$831$	0	0
$832$	−26846.0	−1.11865
$833$	0	0
$834$	0	0
$835$	20844.0	0.863876
$836$	−7068.00	−0.292407
$837$	0	0
$838$	−11340.0	−0.467463
$839$	2916.00	0.119990	0.0599949	−	0.998199i	$-0.480892\pi$
$839$	2916.00	0.119990	0.0599949	+	0.998199i	$0.480892\pi$
$840$	0	0
$841$	43732.0	1.79310
$842$	3786.00	0.154957
$843$	0	0
$844$	1610.00	0.0656617
$845$	9882.00	0.402309
$846$	0	0
$847$	0	0
$848$	−15762.0	−0.638289
$849$	0	0
$850$	−3204.00	−0.129290
$851$	57408.0	2.31248
$852$	0	0
$853$	25198.0	1.01145	0.505723	−	0.862696i	$-0.331226\pi$
$853$	25198.0	1.01145	0.505723	+	0.862696i	$0.331226\pi$
$854$	0	0
$855$	0	0
$856$	13356.0	0.533293
$857$	−42090.0	−1.67767	−0.838837	−	0.544382i	$-0.816764\pi$
$857$	−42090.0	−1.67767	−0.838837	+	0.544382i	$0.816764\pi$
$858$	0	0
$859$	25984.0	1.03209	0.516043	−	0.856562i	$-0.327404\pi$
$859$	25984.0	1.03209	0.516043	+	0.856562i	$0.327404\pi$
$860$	912.000	0.0361616
$861$	0	0
$862$	15030.0	0.593879
$863$	24942.0	0.983819	0.491909	−	0.870646i	$-0.336299\pi$
$863$	24942.0	0.983819	0.491909	+	0.870646i	$0.336299\pi$
$864$	0	0
$865$	−23346.0	−0.917674
$866$	24789.0	0.972707
$867$	0	0
$868$	0	0
$869$	9291.00	0.362688
$870$	0	0
$871$	−10540.0	−0.410028
$872$	1344.00	0.0521945
$873$	0	0
$874$	58032.0	2.24595
$875$	0	0
$876$	0	0
$877$	−9190.00	−0.353847	−0.176924	−	0.984225i	$-0.556615\pi$
$877$	−9190.00	−0.353847	−0.176924	+	0.984225i	$0.556615\pi$
$878$	−11265.0	−0.433002
$879$	0	0
$880$	24282.0	0.930166
$881$	−30870.0	−1.18052	−0.590259	−	0.807214i	$-0.700974\pi$
$881$	−30870.0	−1.18052	−0.590259	+	0.807214i	$0.700974\pi$
$882$	0	0
$883$	27002.0	1.02909	0.514547	−	0.857462i	$-0.327960\pi$
$883$	27002.0	1.02909	0.514547	+	0.857462i	$0.327960\pi$
$884$	−744.000	−0.0283070
$885$	0	0
$886$	35163.0	1.33332
$887$	−10866.0	−0.411324	−0.205662	−	0.978623i	$-0.565935\pi$
$887$	−10866.0	−0.411324	−0.205662	+	0.978623i	$0.565935\pi$
$888$	0	0
$889$	0	0
$890$	−11556.0	−0.435234
$891$	0	0
$892$	4273.00	0.160393
$893$	9672.00	0.362442
$894$	0	0
$895$	14994.0	0.559993
$896$	0	0
$897$	0	0
$898$	22194.0	0.824748
$899$	28449.0	1.05543
$900$	0	0
$901$	2664.00	0.0985025
$902$	9234.00	0.340863
$903$	0	0
$904$	−18900.0	−0.695359
$905$	11328.0	0.416083
$906$	0	0
$907$	25190.0	0.922183	0.461092	−	0.887353i	$-0.347458\pi$
$907$	25190.0	0.922183	0.461092	+	0.887353i	$0.347458\pi$
$908$	−15.0000	−0.000548230 0
$909$	0	0
$910$	0	0
$911$	−3852.00	−0.140091	−0.0700453	−	0.997544i	$-0.522314\pi$
$911$	−3852.00	−0.140091	−0.0700453	+	0.997544i	$0.522314\pi$
$912$	0	0
$913$	1539.00	0.0557869
$914$	18546.0	0.671168
$915$	0	0
$916$	−1988.00	−0.0717089
$917$	0	0
$918$	0	0
$919$	9557.00	0.343043	0.171521	−	0.985180i	$-0.445132\pi$
$919$	9557.00	0.343043	0.171521	+	0.985180i	$0.445132\pi$
$920$	−19656.0	−0.704390
$921$	0	0
$922$	−45873.0	−1.63855
$923$	−24552.0	−0.875557
$924$	0	0
$925$	−32752.0	−1.16419
$926$	−55047.0	−1.95352
$927$	0	0
$928$	−11745.0	−0.415462
$929$	15180.0	0.536103	0.268051	−	0.963405i	$-0.413620\pi$
$929$	15180.0	0.536103	0.268051	+	0.963405i	$0.413620\pi$
$930$	0	0
$931$	0	0
$932$	5070.00	0.178190
$933$	0	0
$934$	17649.0	0.618301
$935$	−4104.00	−0.143546
$936$	0	0
$937$	−494.000	−0.0172233	−0.00861167	−	0.999963i	$-0.502741\pi$
$937$	−494.000	−0.0172233	−0.00861167	+	0.999963i	$0.502741\pi$
$938$	0	0
$939$	0	0
$940$	468.000	0.0162388
$941$	−20679.0	−0.716383	−0.358191	−	0.933648i	$-0.616606\pi$
$941$	−20679.0	−0.716383	−0.358191	+	0.933648i	$0.616606\pi$
$942$	0	0
$943$	−8424.00	−0.290905
$944$	20235.0	0.697662
$945$	0	0
$946$	−25992.0	−0.893312
$947$	2667.00	0.0915162	0.0457581	−	0.998953i	$-0.485430\pi$
$947$	2667.00	0.0915162	0.0457581	+	0.998953i	$0.485430\pi$
$948$	0	0
$949$	−29450.0	−1.00736
$950$	−33108.0	−1.13070
$951$	0	0
$952$	0	0
$953$	−19602.0	−0.666287	−0.333143	−	0.942876i	$-0.608109\pi$
$953$	−19602.0	−0.666287	−0.333143	+	0.942876i	$0.608109\pi$
$954$	0	0
$955$	3420.00	0.115883
$956$	1872.00	0.0633314
$957$	0	0
$958$	−35226.0	−1.18800
$959$	0	0
$960$	0	0
$961$	−17910.0	−0.601188
$962$	−68448.0	−2.29403
$963$	0	0
$964$	5185.00	0.173234
$965$	12846.0	0.428526
$966$	0	0
$967$	−3160.00	−0.105087	−0.0525433	−	0.998619i	$-0.516733\pi$
$967$	−3160.00	−0.105087	−0.0525433	+	0.998619i	$0.516733\pi$
$968$	−40278.0	−1.33738
$969$	0	0
$970$	−33030.0	−1.09333
$971$	−45012.0	−1.48765	−0.743823	−	0.668377i	$-0.766989\pi$
$971$	−45012.0	−1.48765	−0.743823	+	0.668377i	$0.766989\pi$
$972$	0	0
$973$	0	0
$974$	20769.0	0.683246
$975$	0	0
$976$	−50552.0	−1.65792
$977$	32658.0	1.06942	0.534709	−	0.845036i	$-0.320421\pi$
$977$	32658.0	1.06942	0.534709	+	0.845036i	$0.320421\pi$
$978$	0	0
$979$	36594.0	1.19464
$980$	0	0
$981$	0	0
$982$	−24516.0	−0.796677
$983$	20460.0	0.663858	0.331929	−	0.943304i	$-0.392301\pi$
$983$	20460.0	0.663858	0.331929	+	0.943304i	$0.392301\pi$
$984$	0	0
$985$	−3186.00	−0.103060
$986$	9396.00	0.303478
$987$	0	0
$988$	−7688.00	−0.247559
$989$	23712.0	0.762384
$990$	0	0
$991$	−42904.0	−1.37527	−0.687634	−	0.726058i	$-0.741351\pi$
$991$	−42904.0	−1.37527	−0.687634	+	0.726058i	$0.741351\pi$
$992$	−4905.00	−0.156990
$993$	0	0
$994$	0	0
$995$	546.000	0.0173963
$996$	0	0
$997$	−19430.0	−0.617206	−0.308603	−	0.951191i	$-0.599861\pi$
$997$	−19430.0	−0.617206	−0.308603	+	0.951191i	$0.599861\pi$
$998$	29832.0	0.946208
$999$	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	1323.4.a.m.1.1		1
3.2	odd	2		1323.4.a.b.1.1		1
7.3	odd	6		189.4.e.a.163.1	yes	2
7.5	odd	6		189.4.e.a.109.1	✓	2
7.6	odd	2		1323.4.a.l.1.1		1
21.5	even	6		189.4.e.d.109.1	yes	2
21.17	even	6		189.4.e.d.163.1	yes	2
21.20	even	2		1323.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
189.4.e.a.109.1	✓	2	7.5	odd	6
189.4.e.a.163.1	yes	2	7.3	odd	6
189.4.e.d.109.1	yes	2	21.5	even	6
189.4.e.d.163.1	yes	2	21.17	even	6
1323.4.a.b.1.1		1	3.2	odd	2
1323.4.a.c.1.1		1	21.20	even	2
1323.4.a.l.1.1		1	7.6	odd	2
1323.4.a.m.1.1		1	1.1	even	1	trivial

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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

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Embedding invariants

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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	1323.4.a.m.1.1

Level	$1323$
Weight	$4$
Character	1323.1
Self dual	yes
Analytic conductor	$78.060$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$