LMFDB - Embedded newform 1386.4.a.m.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,4,Mod(1,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 4, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1386.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1386.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:	$81.7766472680$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 462)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1386.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +14.0000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +O(q^{10})$

\(q+2.00000 q^{2} +4.00000 q^{4} +14.0000 q^{5} -7.00000 q^{7} +8.00000 q^{8} +28.0000 q^{10} -11.0000 q^{11} +38.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} -54.0000 q^{17} +40.0000 q^{19} +56.0000 q^{20} -22.0000 q^{22} -8.00000 q^{23} +71.0000 q^{25} +76.0000 q^{26} -28.0000 q^{28} +170.000 q^{29} +92.0000 q^{31} +32.0000 q^{32} -108.000 q^{34} -98.0000 q^{35} +294.000 q^{37} +80.0000 q^{38} +112.000 q^{40} +258.000 q^{41} -52.0000 q^{43} -44.0000 q^{44} -16.0000 q^{46} +76.0000 q^{47} +49.0000 q^{49} +142.000 q^{50} +152.000 q^{52} +322.000 q^{53} -154.000 q^{55} -56.0000 q^{56} +340.000 q^{58} -260.000 q^{59} +22.0000 q^{61} +184.000 q^{62} +64.0000 q^{64} +532.000 q^{65} -436.000 q^{67} -216.000 q^{68} -196.000 q^{70} +368.000 q^{71} -2.00000 q^{73} +588.000 q^{74} +160.000 q^{76} +77.0000 q^{77} -200.000 q^{79} +224.000 q^{80} +516.000 q^{82} +952.000 q^{83} -756.000 q^{85} -104.000 q^{86} -88.0000 q^{88} +70.0000 q^{89} -266.000 q^{91} -32.0000 q^{92} +152.000 q^{94} +560.000 q^{95} -1086.00 q^{97} +98.0000 q^{98} +O(q^{100})\)

Display 100 coefficients 10 coefficients

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
$5$	14.0000	1.25220	0.626099	−	0.779744i	$-0.284651\pi$
$5$	14.0000	1.25220	0.626099	+	0.779744i	$0.284651\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	28.0000	0.885438
$11$	−11.0000	−0.301511
$11$	−11.0000	−0.301511
$12$	0	0
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−54.0000	−0.770407	−0.385204	−	0.922832i	$-0.625869\pi$
$17$	−54.0000	−0.770407	−0.385204	+	0.922832i	$0.625869\pi$
$18$	0	0
$19$	40.0000	0.482980	0.241490	−	0.970403i	$-0.422364\pi$
$19$	40.0000	0.482980	0.241490	+	0.970403i	$0.422364\pi$
$20$	56.0000	0.626099
$21$	0	0
$22$	−22.0000	−0.213201
$23$	−8.00000	−0.0725268	−0.0362634	−	0.999342i	$-0.511546\pi$
$23$	−8.00000	−0.0725268	−0.0362634	+	0.999342i	$0.511546\pi$
$24$	0	0
$25$	71.0000	0.568000
$26$	76.0000	0.573263
$27$	0	0
$28$	−28.0000	−0.188982
$29$	170.000	1.08856	0.544279	−	0.838904i	$-0.316803\pi$
$29$	170.000	1.08856	0.544279	+	0.838904i	$0.316803\pi$
$30$	0	0
$31$	92.0000	0.533022	0.266511	−	0.963832i	$-0.414129\pi$
$31$	92.0000	0.533022	0.266511	+	0.963832i	$0.414129\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−108.000	−0.544760
$35$	−98.0000	−0.473286
$36$	0	0
$37$	294.000	1.30631	0.653153	−	0.757226i	$-0.273446\pi$
$37$	294.000	1.30631	0.653153	+	0.757226i	$0.273446\pi$
$38$	80.0000	0.341519
$39$	0	0
$40$	112.000	0.442719
$41$	258.000	0.982752	0.491376	−	0.870948i	$-0.336494\pi$
$41$	258.000	0.982752	0.491376	+	0.870948i	$0.336494\pi$
$42$	0	0
$43$	−52.0000	−0.184417	−0.0922084	−	0.995740i	$-0.529393\pi$
$43$	−52.0000	−0.184417	−0.0922084	+	0.995740i	$0.529393\pi$
$44$	−44.0000	−0.150756
$45$	0	0
$46$	−16.0000	−0.0512842
$47$	76.0000	0.235867	0.117933	−	0.993022i	$-0.462373\pi$
$47$	76.0000	0.235867	0.117933	+	0.993022i	$0.462373\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	142.000	0.401637
$51$	0	0
$52$	152.000	0.405358
$53$	322.000	0.834530	0.417265	−	0.908785i	$-0.362989\pi$
$53$	322.000	0.834530	0.417265	+	0.908785i	$0.362989\pi$
$54$	0	0
$55$	−154.000	−0.377552
$56$	−56.0000	−0.133631
$57$	0	0
$58$	340.000	0.769727
$59$	−260.000	−0.573714	−0.286857	−	0.957973i	$-0.592610\pi$
$59$	−260.000	−0.573714	−0.286857	+	0.957973i	$0.592610\pi$
$60$	0	0
$61$	22.0000	0.0461772	0.0230886	−	0.999733i	$-0.492650\pi$
$61$	22.0000	0.0461772	0.0230886	+	0.999733i	$0.492650\pi$
$62$	184.000	0.376904
$63$	0	0
$64$	64.0000	0.125000
$65$	532.000	1.01518
$66$	0	0
$67$	−436.000	−0.795013	−0.397507	−	0.917599i	$-0.630124\pi$
$67$	−436.000	−0.795013	−0.397507	+	0.917599i	$0.630124\pi$
$68$	−216.000	−0.385204
$69$	0	0
$70$	−196.000	−0.334664
$71$	368.000	0.615121	0.307560	−	0.951529i	$-0.400487\pi$
$71$	368.000	0.615121	0.307560	+	0.951529i	$0.400487\pi$
$72$	0	0
$73$	−2.00000	−0.00320661	−0.00160330	−	0.999999i	$-0.500510\pi$
$73$	−2.00000	−0.00320661	−0.00160330	+	0.999999i	$0.500510\pi$
$74$	588.000	0.923697
$75$	0	0
$76$	160.000	0.241490
$77$	77.0000	0.113961
$78$	0	0
$79$	−200.000	−0.284832	−0.142416	−	0.989807i	$-0.545487\pi$
$79$	−200.000	−0.284832	−0.142416	+	0.989807i	$0.545487\pi$
$80$	224.000	0.313050
$81$	0	0
$82$	516.000	0.694911
$83$	952.000	1.25898	0.629491	−	0.777007i	$-0.283263\pi$
$83$	952.000	1.25898	0.629491	+	0.777007i	$0.283263\pi$
$84$	0	0
$85$	−756.000	−0.964703
$86$	−104.000	−0.130402
$87$	0	0
$88$	−88.0000	−0.106600
$89$	70.0000	0.0833706	0.0416853	−	0.999131i	$-0.486727\pi$
$89$	70.0000	0.0833706	0.0416853	+	0.999131i	$0.486727\pi$
$90$	0	0
$91$	−266.000	−0.306422
$92$	−32.0000	−0.0362634
$93$	0	0
$94$	152.000	0.166783
$95$	560.000	0.604787
$96$	0	0
$97$	−1086.00	−1.13677	−0.568385	−	0.822763i	$-0.692431\pi$
$97$	−1086.00	−1.13677	−0.568385	+	0.822763i	$0.692431\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	284.000	0.284000
$101$	−102.000	−0.100489	−0.0502445	−	0.998737i	$-0.516000\pi$
$101$	−102.000	−0.100489	−0.0502445	+	0.998737i	$0.516000\pi$
$102$	0	0
$103$	188.000	0.179847	0.0899233	−	0.995949i	$-0.471338\pi$
$103$	188.000	0.179847	0.0899233	+	0.995949i	$0.471338\pi$
$104$	304.000	0.286631
$105$	0	0
$106$	644.000	0.590102
$107$	1716.00	1.55039	0.775196	−	0.631721i	$-0.217651\pi$
$107$	1716.00	1.55039	0.775196	+	0.631721i	$0.217651\pi$
$108$	0	0
$109$	310.000	0.272409	0.136205	−	0.990681i	$-0.456510\pi$
$109$	310.000	0.272409	0.136205	+	0.990681i	$0.456510\pi$
$110$	−308.000	−0.266970
$111$	0	0
$112$	−112.000	−0.0944911
$113$	1822.00	1.51681	0.758404	−	0.651785i	$-0.225979\pi$
$113$	1822.00	1.51681	0.758404	+	0.651785i	$0.225979\pi$
$114$	0	0
$115$	−112.000	−0.0908179
$116$	680.000	0.544279
$117$	0	0
$118$	−520.000	−0.405677
$119$	378.000	0.291187
$120$	0	0
$121$	121.000	0.0909091
$122$	44.0000	0.0326522
$123$	0	0
$124$	368.000	0.266511
$125$	−756.000	−0.540950
$126$	0	0
$127$	−1576.00	−1.10116	−0.550580	−	0.834782i	$-0.685594\pi$
$127$	−1576.00	−1.10116	−0.550580	+	0.834782i	$0.685594\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	1064.00	0.717838
$131$	−1072.00	−0.714970	−0.357485	−	0.933919i	$-0.616366\pi$
$131$	−1072.00	−0.714970	−0.357485	+	0.933919i	$0.616366\pi$
$132$	0	0
$133$	−280.000	−0.182549
$134$	−872.000	−0.562159
$135$	0	0
$136$	−432.000	−0.272380
$137$	246.000	0.153410	0.0767051	−	0.997054i	$-0.475560\pi$
$137$	246.000	0.153410	0.0767051	+	0.997054i	$0.475560\pi$
$138$	0	0
$139$	1640.00	1.00074	0.500370	−	0.865811i	$-0.333197\pi$
$139$	1640.00	1.00074	0.500370	+	0.865811i	$0.333197\pi$
$140$	−392.000	−0.236643
$141$	0	0
$142$	736.000	0.434956
$143$	−418.000	−0.244440
$144$	0	0
$145$	2380.00	1.36309
$146$	−4.00000	−0.00226741
$147$	0	0
$148$	1176.00	0.653153
$149$	850.000	0.467347	0.233674	−	0.972315i	$-0.424925\pi$
$149$	850.000	0.467347	0.233674	+	0.972315i	$0.424925\pi$
$150$	0	0
$151$	−1448.00	−0.780375	−0.390187	−	0.920735i	$-0.627590\pi$
$151$	−1448.00	−0.780375	−0.390187	+	0.920735i	$0.627590\pi$
$152$	320.000	0.170759
$153$	0	0
$154$	154.000	0.0805823
$155$	1288.00	0.667449
$156$	0	0
$157$	−1886.00	−0.958721	−0.479360	−	0.877618i	$-0.659131\pi$
$157$	−1886.00	−0.958721	−0.479360	+	0.877618i	$0.659131\pi$
$158$	−400.000	−0.201407
$159$	0	0
$160$	448.000	0.221359
$161$	56.0000	0.0274125
$162$	0	0
$163$	228.000	0.109560	0.0547802	−	0.998498i	$-0.482554\pi$
$163$	228.000	0.109560	0.0547802	+	0.998498i	$0.482554\pi$
$164$	1032.00	0.491376
$165$	0	0
$166$	1904.00	0.890235
$167$	−1664.00	−0.771043	−0.385522	−	0.922699i	$-0.625978\pi$
$167$	−1664.00	−0.771043	−0.385522	+	0.922699i	$0.625978\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	−1512.00	−0.682148
$171$	0	0
$172$	−208.000	−0.0922084
$173$	−2438.00	−1.07143	−0.535716	−	0.844398i	$-0.679958\pi$
$173$	−2438.00	−1.07143	−0.535716	+	0.844398i	$0.679958\pi$
$174$	0	0
$175$	−497.000	−0.214684
$176$	−176.000	−0.0753778
$177$	0	0
$178$	140.000	0.0589519
$179$	−1620.00	−0.676450	−0.338225	−	0.941065i	$-0.609826\pi$
$179$	−1620.00	−0.676450	−0.338225	+	0.941065i	$0.609826\pi$
$180$	0	0
$181$	3602.00	1.47920	0.739598	−	0.673049i	$-0.235016\pi$
$181$	3602.00	1.47920	0.739598	+	0.673049i	$0.235016\pi$
$182$	−532.000	−0.216673
$183$	0	0
$184$	−64.0000	−0.0256421
$185$	4116.00	1.63575
$186$	0	0
$187$	594.000	0.232287
$188$	304.000	0.117933
$189$	0	0
$190$	1120.00	0.427649
$191$	−3472.00	−1.31531	−0.657657	−	0.753317i	$-0.728453\pi$
$191$	−3472.00	−1.31531	−0.657657	+	0.753317i	$0.728453\pi$
$192$	0	0
$193$	−222.000	−0.0827975	−0.0413987	−	0.999143i	$-0.513181\pi$
$193$	−222.000	−0.0827975	−0.0413987	+	0.999143i	$0.513181\pi$
$194$	−2172.00	−0.803817
$195$	0	0
$196$	196.000	0.0714286
$197$	3906.00	1.41264	0.706322	−	0.707890i	$-0.250353\pi$
$197$	3906.00	1.41264	0.706322	+	0.707890i	$0.250353\pi$
$198$	0	0
$199$	1900.00	0.676821	0.338411	−	0.940999i	$-0.390111\pi$
$199$	1900.00	0.676821	0.338411	+	0.940999i	$0.390111\pi$
$200$	568.000	0.200818
$201$	0	0
$202$	−204.000	−0.0710564
$203$	−1190.00	−0.411437
$204$	0	0
$205$	3612.00	1.23060
$206$	376.000	0.127171
$207$	0	0
$208$	608.000	0.202679
$209$	−440.000	−0.145624
$210$	0	0
$211$	1172.00	0.382388	0.191194	−	0.981552i	$-0.438764\pi$
$211$	1172.00	0.382388	0.191194	+	0.981552i	$0.438764\pi$
$212$	1288.00	0.417265
$213$	0	0
$214$	3432.00	1.09629
$215$	−728.000	−0.230926
$216$	0	0
$217$	−644.000	−0.201463
$218$	620.000	0.192622
$219$	0	0
$220$	−616.000	−0.188776
$221$	−2052.00	−0.624581
$222$	0	0
$223$	−2372.00	−0.712291	−0.356145	−	0.934431i	$-0.615909\pi$
$223$	−2372.00	−0.712291	−0.356145	+	0.934431i	$0.615909\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	3644.00	1.07255
$227$	−1024.00	−0.299406	−0.149703	−	0.988731i	$-0.547832\pi$
$227$	−1024.00	−0.299406	−0.149703	+	0.988731i	$0.547832\pi$
$228$	0	0
$229$	290.000	0.0836845	0.0418422	−	0.999124i	$-0.486677\pi$
$229$	290.000	0.0836845	0.0418422	+	0.999124i	$0.486677\pi$
$230$	−224.000	−0.0642179
$231$	0	0
$232$	1360.00	0.384864
$233$	5022.00	1.41203	0.706013	−	0.708199i	$-0.250492\pi$
$233$	5022.00	1.41203	0.706013	+	0.708199i	$0.250492\pi$
$234$	0	0
$235$	1064.00	0.295352
$236$	−1040.00	−0.286857
$237$	0	0
$238$	756.000	0.205900
$239$	−200.000	−0.0541294	−0.0270647	−	0.999634i	$-0.508616\pi$
$239$	−200.000	−0.0541294	−0.0270647	+	0.999634i	$0.508616\pi$
$240$	0	0
$241$	−1498.00	−0.400393	−0.200196	−	0.979756i	$-0.564158\pi$
$241$	−1498.00	−0.400393	−0.200196	+	0.979756i	$0.564158\pi$
$242$	242.000	0.0642824
$243$	0	0
$244$	88.0000	0.0230886
$245$	686.000	0.178885
$246$	0	0
$247$	1520.00	0.391560
$248$	736.000	0.188452
$249$	0	0
$250$	−1512.00	−0.382509
$251$	7468.00	1.87799	0.938996	−	0.343928i	$-0.111758\pi$
$251$	7468.00	1.87799	0.938996	+	0.343928i	$0.111758\pi$
$252$	0	0
$253$	88.0000	0.0218676
$254$	−3152.00	−0.778638
$255$	0	0
$256$	256.000	0.0625000
$257$	−2194.00	−0.532521	−0.266261	−	0.963901i	$-0.585788\pi$
$257$	−2194.00	−0.532521	−0.266261	+	0.963901i	$0.585788\pi$
$258$	0	0
$259$	−2058.00	−0.493737
$260$	2128.00	0.507588
$261$	0	0
$262$	−2144.00	−0.505560
$263$	−3288.00	−0.770900	−0.385450	−	0.922729i	$-0.625954\pi$
$263$	−3288.00	−0.770900	−0.385450	+	0.922729i	$0.625954\pi$
$264$	0	0
$265$	4508.00	1.04500
$266$	−560.000	−0.129082
$267$	0	0
$268$	−1744.00	−0.397507
$269$	6710.00	1.52088	0.760439	−	0.649410i	$-0.224984\pi$
$269$	6710.00	1.52088	0.760439	+	0.649410i	$0.224984\pi$
$270$	0	0
$271$	4952.00	1.11001	0.555005	−	0.831847i	$-0.312716\pi$
$271$	4952.00	1.11001	0.555005	+	0.831847i	$0.312716\pi$
$272$	−864.000	−0.192602
$273$	0	0
$274$	492.000	0.108477
$275$	−781.000	−0.171258
$276$	0	0
$277$	−3666.00	−0.795193	−0.397597	−	0.917560i	$-0.630156\pi$
$277$	−3666.00	−0.795193	−0.397597	+	0.917560i	$0.630156\pi$
$278$	3280.00	0.707631
$279$	0	0
$280$	−784.000	−0.167332
$281$	6798.00	1.44318	0.721592	−	0.692319i	$-0.243411\pi$
$281$	6798.00	1.44318	0.721592	+	0.692319i	$0.243411\pi$
$282$	0	0
$283$	−1992.00	−0.418417	−0.209209	−	0.977871i	$-0.567089\pi$
$283$	−1992.00	−0.418417	−0.209209	+	0.977871i	$0.567089\pi$
$284$	1472.00	0.307560
$285$	0	0
$286$	−836.000	−0.172845
$287$	−1806.00	−0.371445
$288$	0	0
$289$	−1997.00	−0.406473
$290$	4760.00	0.963851
$291$	0	0
$292$	−8.00000	−0.00160330
$293$	4002.00	0.797950	0.398975	−	0.916962i	$-0.369366\pi$
$293$	4002.00	0.797950	0.398975	+	0.916962i	$0.369366\pi$
$294$	0	0
$295$	−3640.00	−0.718403
$296$	2352.00	0.461849
$297$	0	0
$298$	1700.00	0.330464
$299$	−304.000	−0.0587986
$300$	0	0
$301$	364.000	0.0697030
$302$	−2896.00	−0.551808
$303$	0	0
$304$	640.000	0.120745
$305$	308.000	0.0578230
$306$	0	0
$307$	5344.00	0.993479	0.496740	−	0.867900i	$-0.334530\pi$
$307$	5344.00	0.993479	0.496740	+	0.867900i	$0.334530\pi$
$308$	308.000	0.0569803
$309$	0	0
$310$	2576.00	0.471958
$311$	2268.00	0.413526	0.206763	−	0.978391i	$-0.433707\pi$
$311$	2268.00	0.413526	0.206763	+	0.978391i	$0.433707\pi$
$312$	0	0
$313$	−7422.00	−1.34031	−0.670154	−	0.742222i	$-0.733772\pi$
$313$	−7422.00	−1.34031	−0.670154	+	0.742222i	$0.733772\pi$
$314$	−3772.00	−0.677918
$315$	0	0
$316$	−800.000	−0.142416
$317$	7626.00	1.35116	0.675582	−	0.737285i	$-0.263893\pi$
$317$	7626.00	1.35116	0.675582	+	0.737285i	$0.263893\pi$
$318$	0	0
$319$	−1870.00	−0.328213
$320$	896.000	0.156525
$321$	0	0
$322$	112.000	0.0193836
$323$	−2160.00	−0.372092
$324$	0	0
$325$	2698.00	0.460487
$326$	456.000	0.0774709
$327$	0	0
$328$	2064.00	0.347455
$329$	−532.000	−0.0891493
$330$	0	0
$331$	1492.00	0.247758	0.123879	−	0.992297i	$-0.460467\pi$
$331$	1492.00	0.247758	0.123879	+	0.992297i	$0.460467\pi$
$332$	3808.00	0.629491
$333$	0	0
$334$	−3328.00	−0.545210
$335$	−6104.00	−0.995514
$336$	0	0
$337$	74.0000	0.0119615	0.00598077	−	0.999982i	$-0.498096\pi$
$337$	74.0000	0.0119615	0.00598077	+	0.999982i	$0.498096\pi$
$338$	−1506.00	−0.242354
$339$	0	0
$340$	−3024.00	−0.482351
$341$	−1012.00	−0.160712
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−416.000	−0.0652012
$345$	0	0
$346$	−4876.00	−0.757617
$347$	−11724.0	−1.81377	−0.906884	−	0.421381i	$-0.861546\pi$
$347$	−11724.0	−1.81377	−0.906884	+	0.421381i	$0.861546\pi$
$348$	0	0
$349$	6350.00	0.973948	0.486974	−	0.873417i	$-0.338101\pi$
$349$	6350.00	0.973948	0.486974	+	0.873417i	$0.338101\pi$
$350$	−994.000	−0.151804
$351$	0	0
$352$	−352.000	−0.0533002
$353$	5822.00	0.877829	0.438915	−	0.898529i	$-0.355363\pi$
$353$	5822.00	0.877829	0.438915	+	0.898529i	$0.355363\pi$
$354$	0	0
$355$	5152.00	0.770253
$356$	280.000	0.0416853
$357$	0	0
$358$	−3240.00	−0.478322
$359$	−13320.0	−1.95822	−0.979112	−	0.203320i	$-0.934827\pi$
$359$	−13320.0	−1.95822	−0.979112	+	0.203320i	$0.934827\pi$
$360$	0	0
$361$	−5259.00	−0.766730
$362$	7204.00	1.04595
$363$	0	0
$364$	−1064.00	−0.153211
$365$	−28.0000	−0.00401531
$366$	0	0
$367$	−11076.0	−1.57537	−0.787687	−	0.616075i	$-0.788722\pi$
$367$	−11076.0	−1.57537	−0.787687	+	0.616075i	$0.788722\pi$
$368$	−128.000	−0.0181317
$369$	0	0
$370$	8232.00	1.15665
$371$	−2254.00	−0.315423
$372$	0	0
$373$	−3922.00	−0.544433	−0.272216	−	0.962236i	$-0.587757\pi$
$373$	−3922.00	−0.544433	−0.272216	+	0.962236i	$0.587757\pi$
$374$	1188.00	0.164251
$375$	0	0
$376$	608.000	0.0833915
$377$	6460.00	0.882512
$378$	0	0
$379$	−11620.0	−1.57488	−0.787440	−	0.616392i	$-0.788594\pi$
$379$	−11620.0	−1.57488	−0.787440	+	0.616392i	$0.788594\pi$
$380$	2240.00	0.302394
$381$	0	0
$382$	−6944.00	−0.930068
$383$	−3628.00	−0.484026	−0.242013	−	0.970273i	$-0.577808\pi$
$383$	−3628.00	−0.484026	−0.242013	+	0.970273i	$0.577808\pi$
$384$	0	0
$385$	1078.00	0.142701
$386$	−444.000	−0.0585466
$387$	0	0
$388$	−4344.00	−0.568385
$389$	−9750.00	−1.27081	−0.635404	−	0.772180i	$-0.719167\pi$
$389$	−9750.00	−1.27081	−0.635404	+	0.772180i	$0.719167\pi$
$390$	0	0
$391$	432.000	0.0558751
$392$	392.000	0.0505076
$393$	0	0
$394$	7812.00	0.998891
$395$	−2800.00	−0.356667
$396$	0	0
$397$	−6606.00	−0.835128	−0.417564	−	0.908648i	$-0.637116\pi$
$397$	−6606.00	−0.835128	−0.417564	+	0.908648i	$0.637116\pi$
$398$	3800.00	0.478585
$399$	0	0
$400$	1136.00	0.142000
$401$	−2.00000	−0.000249065 0	−0.000124533	−	1.00000i	$-0.500040\pi$
$401$	−2.00000	−0.000249065 0	−0.000124533		1.00000i	$0.500040\pi$
$402$	0	0
$403$	3496.00	0.432129
$404$	−408.000	−0.0502445
$405$	0	0
$406$	−2380.00	−0.290930
$407$	−3234.00	−0.393866
$408$	0	0
$409$	−13930.0	−1.68409	−0.842047	−	0.539405i	$-0.818649\pi$
$409$	−13930.0	−1.68409	−0.842047	+	0.539405i	$0.818649\pi$
$410$	7224.00	0.870166
$411$	0	0
$412$	752.000	0.0899233
$413$	1820.00	0.216843
$414$	0	0
$415$	13328.0	1.57650
$416$	1216.00	0.143316
$417$	0	0
$418$	−880.000	−0.102972
$419$	7740.00	0.902443	0.451222	−	0.892412i	$-0.350988\pi$
$419$	7740.00	0.902443	0.451222	+	0.892412i	$0.350988\pi$
$420$	0	0
$421$	−5618.00	−0.650367	−0.325184	−	0.945651i	$-0.605426\pi$
$421$	−5618.00	−0.650367	−0.325184	+	0.945651i	$0.605426\pi$
$422$	2344.00	0.270389
$423$	0	0
$424$	2576.00	0.295051
$425$	−3834.00	−0.437591
$426$	0	0
$427$	−154.000	−0.0174534
$428$	6864.00	0.775196
$429$	0	0
$430$	−1456.00	−0.163290
$431$	9008.00	1.00673	0.503364	−	0.864074i	$-0.332095\pi$
$431$	9008.00	1.00673	0.503364	+	0.864074i	$0.332095\pi$
$432$	0	0
$433$	−10702.0	−1.18777	−0.593886	−	0.804549i	$-0.702407\pi$
$433$	−10702.0	−1.18777	−0.593886	+	0.804549i	$0.702407\pi$
$434$	−1288.00	−0.142456
$435$	0	0
$436$	1240.00	0.136205
$437$	−320.000	−0.0350290
$438$	0	0
$439$	−4960.00	−0.539243	−0.269622	−	0.962966i	$-0.586899\pi$
$439$	−4960.00	−0.539243	−0.269622	+	0.962966i	$0.586899\pi$
$440$	−1232.00	−0.133485
$441$	0	0
$442$	−4104.00	−0.441646
$443$	−188.000	−0.0201629	−0.0100814	−	0.999949i	$-0.503209\pi$
$443$	−188.000	−0.0201629	−0.0100814	+	0.999949i	$0.503209\pi$
$444$	0	0
$445$	980.000	0.104397
$446$	−4744.00	−0.503666
$447$	0	0
$448$	−448.000	−0.0472456
$449$	3150.00	0.331086	0.165543	−	0.986203i	$-0.447062\pi$
$449$	3150.00	0.331086	0.165543	+	0.986203i	$0.447062\pi$
$450$	0	0
$451$	−2838.00	−0.296311
$452$	7288.00	0.758404
$453$	0	0
$454$	−2048.00	−0.211712
$455$	−3724.00	−0.383701
$456$	0	0
$457$	−12806.0	−1.31081	−0.655404	−	0.755278i	$-0.727502\pi$
$457$	−12806.0	−1.31081	−0.655404	+	0.755278i	$0.727502\pi$
$458$	580.000	0.0591738
$459$	0	0
$460$	−448.000	−0.0454089
$461$	1778.00	0.179631	0.0898153	−	0.995958i	$-0.471372\pi$
$461$	1778.00	0.179631	0.0898153	+	0.995958i	$0.471372\pi$
$462$	0	0
$463$	−15672.0	−1.57309	−0.786544	−	0.617534i	$-0.788132\pi$
$463$	−15672.0	−1.57309	−0.786544	+	0.617534i	$0.788132\pi$
$464$	2720.00	0.272140
$465$	0	0
$466$	10044.0	0.998453
$467$	−4.00000	−0.000396355 0	−0.000198178	−	1.00000i	$-0.500063\pi$
$467$	−4.00000	−0.000396355 0	−0.000198178		1.00000i	$0.500063\pi$
$468$	0	0
$469$	3052.00	0.300487
$470$	2128.00	0.208845
$471$	0	0
$472$	−2080.00	−0.202838
$473$	572.000	0.0556038
$474$	0	0
$475$	2840.00	0.274333
$476$	1512.00	0.145593
$477$	0	0
$478$	−400.000	−0.0382753
$479$	−14640.0	−1.39649	−0.698245	−	0.715859i	$-0.746035\pi$
$479$	−14640.0	−1.39649	−0.698245	+	0.715859i	$0.746035\pi$
$480$	0	0
$481$	11172.0	1.05904
$482$	−2996.00	−0.283120
$483$	0	0
$484$	484.000	0.0454545
$485$	−15204.0	−1.42346
$486$	0	0
$487$	−19896.0	−1.85128	−0.925640	−	0.378404i	$-0.876473\pi$
$487$	−19896.0	−1.85128	−0.925640	+	0.378404i	$0.876473\pi$
$488$	176.000	0.0163261
$489$	0	0
$490$	1372.00	0.126491
$491$	−7972.00	−0.732732	−0.366366	−	0.930471i	$-0.619398\pi$
$491$	−7972.00	−0.732732	−0.366366	+	0.930471i	$0.619398\pi$
$492$	0	0
$493$	−9180.00	−0.838634
$494$	3040.00	0.276875
$495$	0	0
$496$	1472.00	0.133256
$497$	−2576.00	−0.232494
$498$	0	0
$499$	−2260.00	−0.202748	−0.101374	−	0.994848i	$-0.532324\pi$
$499$	−2260.00	−0.202748	−0.101374	+	0.994848i	$0.532324\pi$
$500$	−3024.00	−0.270475
$501$	0	0
$502$	14936.0	1.32794
$503$	−6288.00	−0.557392	−0.278696	−	0.960379i	$-0.589902\pi$
$503$	−6288.00	−0.557392	−0.278696	+	0.960379i	$0.589902\pi$
$504$	0	0
$505$	−1428.00	−0.125832
$506$	176.000	0.0154628
$507$	0	0
$508$	−6304.00	−0.550580
$509$	−12010.0	−1.04584	−0.522921	−	0.852381i	$-0.675158\pi$
$509$	−12010.0	−1.04584	−0.522921	+	0.852381i	$0.675158\pi$
$510$	0	0
$511$	14.0000	0.00121198
$512$	512.000	0.0441942
$513$	0	0
$514$	−4388.00	−0.376549
$515$	2632.00	0.225203
$516$	0	0
$517$	−836.000	−0.0711165
$518$	−4116.00	−0.349125
$519$	0	0
$520$	4256.00	0.358919
$521$	−10842.0	−0.911702	−0.455851	−	0.890056i	$-0.650665\pi$
$521$	−10842.0	−0.911702	−0.455851	+	0.890056i	$0.650665\pi$
$522$	0	0
$523$	18808.0	1.57250	0.786249	−	0.617910i	$-0.212020\pi$
$523$	18808.0	1.57250	0.786249	+	0.617910i	$0.212020\pi$
$524$	−4288.00	−0.357485
$525$	0	0
$526$	−6576.00	−0.545109
$527$	−4968.00	−0.410644
$528$	0	0
$529$	−12103.0	−0.994740
$530$	9016.00	0.738925
$531$	0	0
$532$	−1120.00	−0.0912747
$533$	9804.00	0.796732
$534$	0	0
$535$	24024.0	1.94140
$536$	−3488.00	−0.281080
$537$	0	0
$538$	13420.0	1.07542
$539$	−539.000	−0.0430730
$540$	0	0
$541$	15622.0	1.24148	0.620741	−	0.784015i	$-0.286832\pi$
$541$	15622.0	1.24148	0.620741	+	0.784015i	$0.286832\pi$
$542$	9904.00	0.784895
$543$	0	0
$544$	−1728.00	−0.136190
$545$	4340.00	0.341110
$546$	0	0
$547$	2284.00	0.178532	0.0892658	−	0.996008i	$-0.471548\pi$
$547$	2284.00	0.178532	0.0892658	+	0.996008i	$0.471548\pi$
$548$	984.000	0.0767051
$549$	0	0
$550$	−1562.00	−0.121098
$551$	6800.00	0.525753
$552$	0	0
$553$	1400.00	0.107657
$554$	−7332.00	−0.562287
$555$	0	0
$556$	6560.00	0.500370
$557$	−16854.0	−1.28209	−0.641047	−	0.767501i	$-0.721500\pi$
$557$	−16854.0	−1.28209	−0.641047	+	0.767501i	$0.721500\pi$
$558$	0	0
$559$	−1976.00	−0.149510
$560$	−1568.00	−0.118322
$561$	0	0
$562$	13596.0	1.02049
$563$	−528.000	−0.0395250	−0.0197625	−	0.999805i	$-0.506291\pi$
$563$	−528.000	−0.0395250	−0.0197625	+	0.999805i	$0.506291\pi$
$564$	0	0
$565$	25508.0	1.89934
$566$	−3984.00	−0.295866
$567$	0	0
$568$	2944.00	0.217478
$569$	−8050.00	−0.593099	−0.296550	−	0.955017i	$-0.595836\pi$
$569$	−8050.00	−0.593099	−0.296550	+	0.955017i	$0.595836\pi$
$570$	0	0
$571$	−11308.0	−0.828765	−0.414383	−	0.910103i	$-0.636002\pi$
$571$	−11308.0	−0.828765	−0.414383	+	0.910103i	$0.636002\pi$
$572$	−1672.00	−0.122220
$573$	0	0
$574$	−3612.00	−0.262652
$575$	−568.000	−0.0411952
$576$	0	0
$577$	5274.00	0.380519	0.190260	−	0.981734i	$-0.439067\pi$
$577$	5274.00	0.380519	0.190260	+	0.981734i	$0.439067\pi$
$578$	−3994.00	−0.287420
$579$	0	0
$580$	9520.00	0.681546
$581$	−6664.00	−0.475851
$582$	0	0
$583$	−3542.00	−0.251620
$584$	−16.0000	−0.00113371
$585$	0	0
$586$	8004.00	0.564236
$587$	−11604.0	−0.815926	−0.407963	−	0.912999i	$-0.633761\pi$
$587$	−11604.0	−0.815926	−0.407963	+	0.912999i	$0.633761\pi$
$588$	0	0
$589$	3680.00	0.257439
$590$	−7280.00	−0.507988
$591$	0	0
$592$	4704.00	0.326576
$593$	13002.0	0.900385	0.450192	−	0.892932i	$-0.351355\pi$
$593$	13002.0	0.900385	0.450192	+	0.892932i	$0.351355\pi$
$594$	0	0
$595$	5292.00	0.364623
$596$	3400.00	0.233674
$597$	0	0
$598$	−608.000	−0.0415769
$599$	−16320.0	−1.11322	−0.556609	−	0.830775i	$-0.687898\pi$
$599$	−16320.0	−1.11322	−0.556609	+	0.830775i	$0.687898\pi$
$600$	0	0
$601$	−9258.00	−0.628356	−0.314178	−	0.949364i	$-0.601729\pi$
$601$	−9258.00	−0.628356	−0.314178	+	0.949364i	$0.601729\pi$
$602$	728.000	0.0492875
$603$	0	0
$604$	−5792.00	−0.390187
$605$	1694.00	0.113836
$606$	0	0
$607$	25824.0	1.72679	0.863397	−	0.504525i	$-0.168332\pi$
$607$	25824.0	1.72679	0.863397	+	0.504525i	$0.168332\pi$
$608$	1280.00	0.0853797
$609$	0	0
$610$	616.000	0.0408871
$611$	2888.00	0.191221
$612$	0	0
$613$	15518.0	1.02246	0.511228	−	0.859445i	$-0.329191\pi$
$613$	15518.0	1.02246	0.511228	+	0.859445i	$0.329191\pi$
$614$	10688.0	0.702496
$615$	0	0
$616$	616.000	0.0402911
$617$	14486.0	0.945194	0.472597	−	0.881279i	$-0.343317\pi$
$617$	14486.0	0.945194	0.472597	+	0.881279i	$0.343317\pi$
$618$	0	0
$619$	12460.0	0.809062	0.404531	−	0.914524i	$-0.367435\pi$
$619$	12460.0	0.809062	0.404531	+	0.914524i	$0.367435\pi$
$620$	5152.00	0.333725
$621$	0	0
$622$	4536.00	0.292407
$623$	−490.000	−0.0315111
$624$	0	0
$625$	−19459.0	−1.24538
$626$	−14844.0	−0.947741
$627$	0	0
$628$	−7544.00	−0.479360
$629$	−15876.0	−1.00639
$630$	0	0
$631$	−16648.0	−1.05031	−0.525156	−	0.851006i	$-0.675993\pi$
$631$	−16648.0	−1.05031	−0.525156	+	0.851006i	$0.675993\pi$
$632$	−1600.00	−0.100703
$633$	0	0
$634$	15252.0	0.955417
$635$	−22064.0	−1.37887
$636$	0	0
$637$	1862.00	0.115817
$638$	−3740.00	−0.232082
$639$	0	0
$640$	1792.00	0.110680
$641$	10638.0	0.655500	0.327750	−	0.944764i	$-0.393710\pi$
$641$	10638.0	0.655500	0.327750	+	0.944764i	$0.393710\pi$
$642$	0	0
$643$	4588.00	0.281389	0.140694	−	0.990053i	$-0.455066\pi$
$643$	4588.00	0.281389	0.140694	+	0.990053i	$0.455066\pi$
$644$	224.000	0.0137063
$645$	0	0
$646$	−4320.00	−0.263109
$647$	8796.00	0.534477	0.267238	−	0.963630i	$-0.413889\pi$
$647$	8796.00	0.534477	0.267238	+	0.963630i	$0.413889\pi$
$648$	0	0
$649$	2860.00	0.172981
$650$	5396.00	0.325613
$651$	0	0
$652$	912.000	0.0547802
$653$	−18878.0	−1.13132	−0.565661	−	0.824638i	$-0.691379\pi$
$653$	−18878.0	−1.13132	−0.565661	+	0.824638i	$0.691379\pi$
$654$	0	0
$655$	−15008.0	−0.895284
$656$	4128.00	0.245688
$657$	0	0
$658$	−1064.00	−0.0630381
$659$	20780.0	1.22834	0.614168	−	0.789175i	$-0.289492\pi$
$659$	20780.0	1.22834	0.614168	+	0.789175i	$0.289492\pi$
$660$	0	0
$661$	10402.0	0.612089	0.306045	−	0.952017i	$-0.400994\pi$
$661$	10402.0	0.612089	0.306045	+	0.952017i	$0.400994\pi$
$662$	2984.00	0.175191
$663$	0	0
$664$	7616.00	0.445118
$665$	−3920.00	−0.228588
$666$	0	0
$667$	−1360.00	−0.0789496
$668$	−6656.00	−0.385522
$669$	0	0
$670$	−12208.0	−0.703935
$671$	−242.000	−0.0139230
$672$	0	0
$673$	23018.0	1.31839	0.659197	−	0.751971i	$-0.270896\pi$
$673$	23018.0	1.31839	0.659197	+	0.751971i	$0.270896\pi$
$674$	148.000	0.00845808
$675$	0	0
$676$	−3012.00	−0.171370
$677$	16866.0	0.957479	0.478739	−	0.877957i	$-0.341094\pi$
$677$	16866.0	0.957479	0.478739	+	0.877957i	$0.341094\pi$
$678$	0	0
$679$	7602.00	0.429658
$680$	−6048.00	−0.341074
$681$	0	0
$682$	−2024.00	−0.113641
$683$	−4668.00	−0.261517	−0.130758	−	0.991414i	$-0.541741\pi$
$683$	−4668.00	−0.261517	−0.130758	+	0.991414i	$0.541741\pi$
$684$	0	0
$685$	3444.00	0.192100
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−832.000	−0.0461042
$689$	12236.0	0.676567
$690$	0	0
$691$	−33108.0	−1.82270	−0.911351	−	0.411629i	$-0.864960\pi$
$691$	−33108.0	−1.82270	−0.911351	+	0.411629i	$0.864960\pi$
$692$	−9752.00	−0.535716
$693$	0	0
$694$	−23448.0	−1.28253
$695$	22960.0	1.25313
$696$	0	0
$697$	−13932.0	−0.757119
$698$	12700.0	0.688685
$699$	0	0
$700$	−1988.00	−0.107342
$701$	−7302.00	−0.393428	−0.196714	−	0.980461i	$-0.563027\pi$
$701$	−7302.00	−0.393428	−0.196714	+	0.980461i	$0.563027\pi$
$702$	0	0
$703$	11760.0	0.630920
$704$	−704.000	−0.0376889
$705$	0	0
$706$	11644.0	0.620719
$707$	714.000	0.0379812
$708$	0	0
$709$	−10570.0	−0.559894	−0.279947	−	0.960015i	$-0.590317\pi$
$709$	−10570.0	−0.559894	−0.279947	+	0.960015i	$0.590317\pi$
$710$	10304.0	0.544651
$711$	0	0
$712$	560.000	0.0294760
$713$	−736.000	−0.0386584
$714$	0	0
$715$	−5852.00	−0.306087
$716$	−6480.00	−0.338225
$717$	0	0
$718$	−26640.0	−1.38467
$719$	−20220.0	−1.04879	−0.524394	−	0.851476i	$-0.675708\pi$
$719$	−20220.0	−1.04879	−0.524394	+	0.851476i	$0.675708\pi$
$720$	0	0
$721$	−1316.00	−0.0679756
$722$	−10518.0	−0.542160
$723$	0	0
$724$	14408.0	0.739598
$725$	12070.0	0.618301
$726$	0	0
$727$	−29996.0	−1.53025	−0.765124	−	0.643883i	$-0.777322\pi$
$727$	−29996.0	−1.53025	−0.765124	+	0.643883i	$0.777322\pi$
$728$	−2128.00	−0.108336
$729$	0	0
$730$	−56.0000	−0.00283925
$731$	2808.00	0.142076
$732$	0	0
$733$	−2282.00	−0.114990	−0.0574949	−	0.998346i	$-0.518311\pi$
$733$	−2282.00	−0.114990	−0.0574949	+	0.998346i	$0.518311\pi$
$734$	−22152.0	−1.11396
$735$	0	0
$736$	−256.000	−0.0128210
$737$	4796.00	0.239705
$738$	0	0
$739$	3260.00	0.162275	0.0811374	−	0.996703i	$-0.474145\pi$
$739$	3260.00	0.162275	0.0811374	+	0.996703i	$0.474145\pi$
$740$	16464.0	0.817877
$741$	0	0
$742$	−4508.00	−0.223038
$743$	14512.0	0.716546	0.358273	−	0.933617i	$-0.383366\pi$
$743$	14512.0	0.716546	0.358273	+	0.933617i	$0.383366\pi$
$744$	0	0
$745$	11900.0	0.585211
$746$	−7844.00	−0.384972
$747$	0	0
$748$	2376.00	0.116143
$749$	−12012.0	−0.585993
$750$	0	0
$751$	−20128.0	−0.978004	−0.489002	−	0.872283i	$-0.662639\pi$
$751$	−20128.0	−0.978004	−0.489002	+	0.872283i	$0.662639\pi$
$752$	1216.00	0.0589667
$753$	0	0
$754$	12920.0	0.624030
$755$	−20272.0	−0.977184
$756$	0	0
$757$	−3066.00	−0.147207	−0.0736035	−	0.997288i	$-0.523450\pi$
$757$	−3066.00	−0.147207	−0.0736035	+	0.997288i	$0.523450\pi$
$758$	−23240.0	−1.11361
$759$	0	0
$760$	4480.00	0.213825
$761$	−26982.0	−1.28528	−0.642639	−	0.766169i	$-0.722161\pi$
$761$	−26982.0	−1.28528	−0.642639	+	0.766169i	$0.722161\pi$
$762$	0	0
$763$	−2170.00	−0.102961
$764$	−13888.0	−0.657657
$765$	0	0
$766$	−7256.00	−0.342258
$767$	−9880.00	−0.465119
$768$	0	0
$769$	3550.00	0.166471	0.0832355	−	0.996530i	$-0.473475\pi$
$769$	3550.00	0.166471	0.0832355	+	0.996530i	$0.473475\pi$
$770$	2156.00	0.100905
$771$	0	0
$772$	−888.000	−0.0413987
$773$	33062.0	1.53837	0.769183	−	0.639028i	$-0.220663\pi$
$773$	33062.0	1.53837	0.769183	+	0.639028i	$0.220663\pi$
$774$	0	0
$775$	6532.00	0.302757
$776$	−8688.00	−0.401909
$777$	0	0
$778$	−19500.0	−0.898598
$779$	10320.0	0.474650
$780$	0	0
$781$	−4048.00	−0.185466
$782$	864.000	0.0395097
$783$	0	0
$784$	784.000	0.0357143
$785$	−26404.0	−1.20051
$786$	0	0
$787$	18824.0	0.852609	0.426304	−	0.904580i	$-0.359815\pi$
$787$	18824.0	0.852609	0.426304	+	0.904580i	$0.359815\pi$
$788$	15624.0	0.706322
$789$	0	0
$790$	−5600.00	−0.252201
$791$	−12754.0	−0.573300
$792$	0	0
$793$	836.000	0.0374366
$794$	−13212.0	−0.590524
$795$	0	0
$796$	7600.00	0.338411
$797$	−41314.0	−1.83616	−0.918078	−	0.396399i	$-0.870260\pi$
$797$	−41314.0	−1.83616	−0.918078	+	0.396399i	$0.870260\pi$
$798$	0	0
$799$	−4104.00	−0.181713
$800$	2272.00	0.100409
$801$	0	0
$802$	−4.00000	−0.000176116 0
$803$	22.0000	0.000966828 0
$804$	0	0
$805$	784.000	0.0343259
$806$	6992.00	0.305562
$807$	0	0
$808$	−816.000	−0.0355282
$809$	−16090.0	−0.699251	−0.349626	−	0.936889i	$-0.613691\pi$
$809$	−16090.0	−0.699251	−0.349626	+	0.936889i	$0.613691\pi$
$810$	0	0
$811$	4472.00	0.193629	0.0968145	−	0.995302i	$-0.469135\pi$
$811$	4472.00	0.193629	0.0968145	+	0.995302i	$0.469135\pi$
$812$	−4760.00	−0.205718
$813$	0	0
$814$	−6468.00	−0.278505
$815$	3192.00	0.137191
$816$	0	0
$817$	−2080.00	−0.0890698
$818$	−27860.0	−1.19083
$819$	0	0
$820$	14448.0	0.615300
$821$	44338.0	1.88478	0.942392	−	0.334512i	$-0.108571\pi$
$821$	44338.0	1.88478	0.942392	+	0.334512i	$0.108571\pi$
$822$	0	0
$823$	−19432.0	−0.823034	−0.411517	−	0.911402i	$-0.635001\pi$
$823$	−19432.0	−0.823034	−0.411517	+	0.911402i	$0.635001\pi$
$824$	1504.00	0.0635853
$825$	0	0
$826$	3640.00	0.153331
$827$	−15764.0	−0.662839	−0.331420	−	0.943483i	$-0.607528\pi$
$827$	−15764.0	−0.662839	−0.331420	+	0.943483i	$0.607528\pi$
$828$	0	0
$829$	16570.0	0.694210	0.347105	−	0.937826i	$-0.387165\pi$
$829$	16570.0	0.694210	0.347105	+	0.937826i	$0.387165\pi$
$830$	26656.0	1.11475
$831$	0	0
$832$	2432.00	0.101339
$833$	−2646.00	−0.110058
$834$	0	0
$835$	−23296.0	−0.965499
$836$	−1760.00	−0.0728120
$837$	0	0
$838$	15480.0	0.638124
$839$	5580.00	0.229610	0.114805	−	0.993388i	$-0.463376\pi$
$839$	5580.00	0.229610	0.114805	+	0.993388i	$0.463376\pi$
$840$	0	0
$841$	4511.00	0.184960
$842$	−11236.0	−0.459879
$843$	0	0
$844$	4688.00	0.191194
$845$	−10542.0	−0.429178
$846$	0	0
$847$	−847.000	−0.0343604
$848$	5152.00	0.208633
$849$	0	0
$850$	−7668.00	−0.309424
$851$	−2352.00	−0.0947421
$852$	0	0
$853$	−17522.0	−0.703332	−0.351666	−	0.936126i	$-0.614385\pi$
$853$	−17522.0	−0.703332	−0.351666	+	0.936126i	$0.614385\pi$
$854$	−308.000	−0.0123414
$855$	0	0
$856$	13728.0	0.548146
$857$	37146.0	1.48061	0.740305	−	0.672271i	$-0.234681\pi$
$857$	37146.0	1.48061	0.740305	+	0.672271i	$0.234681\pi$
$858$	0	0
$859$	46820.0	1.85969	0.929847	−	0.367945i	$-0.119939\pi$
$859$	46820.0	1.85969	0.929847	+	0.367945i	$0.119939\pi$
$860$	−2912.00	−0.115463
$861$	0	0
$862$	18016.0	0.711865
$863$	−25808.0	−1.01798	−0.508989	−	0.860773i	$-0.669981\pi$
$863$	−25808.0	−1.01798	−0.508989	+	0.860773i	$0.669981\pi$
$864$	0	0
$865$	−34132.0	−1.34164
$866$	−21404.0	−0.839882
$867$	0	0
$868$	−2576.00	−0.100732
$869$	2200.00	0.0858802
$870$	0	0
$871$	−16568.0	−0.644530
$872$	2480.00	0.0963112
$873$	0	0
$874$	−640.000	−0.0247692
$875$	5292.00	0.204460
$876$	0	0
$877$	−43546.0	−1.67667	−0.838337	−	0.545152i	$-0.816472\pi$
$877$	−43546.0	−1.67667	−0.838337	+	0.545152i	$0.816472\pi$
$878$	−9920.00	−0.381303
$879$	0	0
$880$	−2464.00	−0.0943880
$881$	10278.0	0.393047	0.196524	−	0.980499i	$-0.437035\pi$
$881$	10278.0	0.393047	0.196524	+	0.980499i	$0.437035\pi$
$882$	0	0
$883$	20708.0	0.789218	0.394609	−	0.918849i	$-0.370880\pi$
$883$	20708.0	0.789218	0.394609	+	0.918849i	$0.370880\pi$
$884$	−8208.00	−0.312291
$885$	0	0
$886$	−376.000	−0.0142573
$887$	30296.0	1.14683	0.573416	−	0.819264i	$-0.305618\pi$
$887$	30296.0	1.14683	0.573416	+	0.819264i	$0.305618\pi$
$888$	0	0
$889$	11032.0	0.416200
$890$	1960.00	0.0738195
$891$	0	0
$892$	−9488.00	−0.356145
$893$	3040.00	0.113919
$894$	0	0
$895$	−22680.0	−0.847049
$896$	−896.000	−0.0334077
$897$	0	0
$898$	6300.00	0.234113
$899$	15640.0	0.580226
$900$	0	0
$901$	−17388.0	−0.642928
$902$	−5676.00	−0.209523
$903$	0	0
$904$	14576.0	0.536273
$905$	50428.0	1.85225
$906$	0	0
$907$	29604.0	1.08378	0.541888	−	0.840451i	$-0.317710\pi$
$907$	29604.0	1.08378	0.541888	+	0.840451i	$0.317710\pi$
$908$	−4096.00	−0.149703
$909$	0	0
$910$	−7448.00	−0.271317
$911$	−12112.0	−0.440492	−0.220246	−	0.975444i	$-0.570686\pi$
$911$	−12112.0	−0.440492	−0.220246	+	0.975444i	$0.570686\pi$
$912$	0	0
$913$	−10472.0	−0.379598
$914$	−25612.0	−0.926881
$915$	0	0
$916$	1160.00	0.0418422
$917$	7504.00	0.270233
$918$	0	0
$919$	−33320.0	−1.19600	−0.598001	−	0.801496i	$-0.704038\pi$
$919$	−33320.0	−1.19600	−0.598001	+	0.801496i	$0.704038\pi$
$920$	−896.000	−0.0321090
$921$	0	0
$922$	3556.00	0.127018
$923$	13984.0	0.498688
$924$	0	0
$925$	20874.0	0.741982
$926$	−31344.0	−1.11234
$927$	0	0
$928$	5440.00	0.192432
$929$	45950.0	1.62279	0.811394	−	0.584499i	$-0.198709\pi$
$929$	45950.0	1.62279	0.811394	+	0.584499i	$0.198709\pi$
$930$	0	0
$931$	1960.00	0.0689972
$932$	20088.0	0.706013
$933$	0	0
$934$	−8.00000	−0.000280266 0
$935$	8316.00	0.290869
$936$	0	0
$937$	11054.0	0.385399	0.192699	−	0.981258i	$-0.438276\pi$
$937$	11054.0	0.385399	0.192699	+	0.981258i	$0.438276\pi$
$938$	6104.00	0.212476
$939$	0	0
$940$	4256.00	0.147676
$941$	55818.0	1.93370	0.966852	−	0.255339i	$-0.0821870\pi$
$941$	55818.0	1.93370	0.966852	+	0.255339i	$0.0821870\pi$
$942$	0	0
$943$	−2064.00	−0.0712758
$944$	−4160.00	−0.143428
$945$	0	0
$946$	1144.00	0.0393178
$947$	33636.0	1.15420	0.577098	−	0.816675i	$-0.304185\pi$
$947$	33636.0	1.15420	0.577098	+	0.816675i	$0.304185\pi$
$948$	0	0
$949$	−76.0000	−0.00259965
$950$	5680.00	0.193983
$951$	0	0
$952$	3024.00	0.102950
$953$	−31418.0	−1.06792	−0.533961	−	0.845509i	$-0.679297\pi$
$953$	−31418.0	−1.06792	−0.533961	+	0.845509i	$0.679297\pi$
$954$	0	0
$955$	−48608.0	−1.64703
$956$	−800.000	−0.0270647
$957$	0	0
$958$	−29280.0	−0.987467
$959$	−1722.00	−0.0579836
$960$	0	0
$961$	−21327.0	−0.715887
$962$	22344.0	0.748856
$963$	0	0
$964$	−5992.00	−0.200196
$965$	−3108.00	−0.103679
$966$	0	0
$967$	−13176.0	−0.438171	−0.219086	−	0.975706i	$-0.570307\pi$
$967$	−13176.0	−0.438171	−0.219086	+	0.975706i	$0.570307\pi$
$968$	968.000	0.0321412
$969$	0	0
$970$	−30408.0	−1.00654
$971$	−27132.0	−0.896712	−0.448356	−	0.893855i	$-0.647990\pi$
$971$	−27132.0	−0.896712	−0.448356	+	0.893855i	$0.647990\pi$
$972$	0	0
$973$	−11480.0	−0.378245
$974$	−39792.0	−1.30905
$975$	0	0
$976$	352.000	0.0115443
$977$	−27154.0	−0.889185	−0.444592	−	0.895733i	$-0.646651\pi$
$977$	−27154.0	−0.889185	−0.444592	+	0.895733i	$0.646651\pi$
$978$	0	0
$979$	−770.000	−0.0251372
$980$	2744.00	0.0894427
$981$	0	0
$982$	−15944.0	−0.518120
$983$	−35068.0	−1.13784	−0.568919	−	0.822393i	$-0.692638\pi$
$983$	−35068.0	−1.13784	−0.568919	+	0.822393i	$0.692638\pi$
$984$	0	0
$985$	54684.0	1.76891
$986$	−18360.0	−0.593004
$987$	0	0
$988$	6080.00	0.195780
$989$	416.000	0.0133752
$990$	0	0
$991$	26072.0	0.835726	0.417863	−	0.908510i	$-0.362779\pi$
$991$	26072.0	0.835726	0.417863	+	0.908510i	$0.362779\pi$
$992$	2944.00	0.0942259
$993$	0	0
$994$	−5152.00	−0.164398
$995$	26600.0	0.847514
$996$	0	0
$997$	−19866.0	−0.631056	−0.315528	−	0.948916i	$-0.602182\pi$
$997$	−19866.0	−0.631056	−0.315528	+	0.948916i	$0.602182\pi$
$998$	−4520.00	−0.143365
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.4.a.m.1.1		1
3.2	odd	2		462.4.a.b.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
462.4.a.b.1.1	✓	1	3.2	odd	2
1386.4.a.m.1.1		1	1.1	even	1	trivial

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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1386.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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