LMFDB - Embedded newform 1386.4.a.e.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)

$f(q)$

$=$

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

\(q-2.00000 q^{2} +4.00000 q^{4} +4.00000 q^{5} -7.00000 q^{7} -8.00000 q^{8} -8.00000 q^{10} +11.0000 q^{11} +62.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} +120.000 q^{17} +118.000 q^{19} +16.0000 q^{20} -22.0000 q^{22} +188.000 q^{23} -109.000 q^{25} -124.000 q^{26} -28.0000 q^{28} -62.0000 q^{29} -322.000 q^{31} -32.0000 q^{32} -240.000 q^{34} -28.0000 q^{35} -198.000 q^{37} -236.000 q^{38} -32.0000 q^{40} -48.0000 q^{41} +32.0000 q^{43} +44.0000 q^{44} -376.000 q^{46} +326.000 q^{47} +49.0000 q^{49} +218.000 q^{50} +248.000 q^{52} +482.000 q^{53} +44.0000 q^{55} +56.0000 q^{56} +124.000 q^{58} -400.000 q^{59} +70.0000 q^{61} +644.000 q^{62} +64.0000 q^{64} +248.000 q^{65} -124.000 q^{67} +480.000 q^{68} +56.0000 q^{70} +712.000 q^{71} +304.000 q^{73} +396.000 q^{74} +472.000 q^{76} -77.0000 q^{77} -1016.00 q^{79} +64.0000 q^{80} +96.0000 q^{82} -430.000 q^{83} +480.000 q^{85} -64.0000 q^{86} -88.0000 q^{88} -442.000 q^{89} -434.000 q^{91} +752.000 q^{92} -652.000 q^{94} +472.000 q^{95} -966.000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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
$5$	4.00000	0.357771	0.178885	−	0.983870i	$-0.442751\pi$
$5$	4.00000	0.357771	0.178885	+	0.983870i	$0.442751\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−8.00000	−0.252982
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	62.0000	1.32275	0.661373	−	0.750057i	$-0.269974\pi$
$13$	62.0000	1.32275	0.661373	+	0.750057i	$0.269974\pi$
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	120.000	1.71202	0.856008	−	0.516962i	$-0.172937\pi$
$17$	120.000	1.71202	0.856008	+	0.516962i	$0.172937\pi$
$18$	0	0
$19$	118.000	1.42479	0.712396	−	0.701777i	$-0.247610\pi$
$19$	118.000	1.42479	0.712396	+	0.701777i	$0.247610\pi$
$20$	16.0000	0.178885
$21$	0	0
$22$	−22.0000	−0.213201
$23$	188.000	1.70438	0.852189	−	0.523234i	$-0.175274\pi$
$23$	188.000	1.70438	0.852189	+	0.523234i	$0.175274\pi$
$24$	0	0
$25$	−109.000	−0.872000
$26$	−124.000	−0.935323
$27$	0	0
$28$	−28.0000	−0.188982
$29$	−62.0000	−0.397004	−0.198502	−	0.980101i	$-0.563608\pi$
$29$	−62.0000	−0.397004	−0.198502	+	0.980101i	$0.563608\pi$
$30$	0	0
$31$	−322.000	−1.86558	−0.932789	−	0.360423i	$-0.882632\pi$
$31$	−322.000	−1.86558	−0.932789	+	0.360423i	$0.882632\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−240.000	−1.21058
$35$	−28.0000	−0.135225
$36$	0	0
$37$	−198.000	−0.879757	−0.439878	−	0.898057i	$-0.644978\pi$
$37$	−198.000	−0.879757	−0.439878	+	0.898057i	$0.644978\pi$
$38$	−236.000	−1.00748
$39$	0	0
$40$	−32.0000	−0.126491
$41$	−48.0000	−0.182838	−0.0914188	−	0.995813i	$-0.529140\pi$
$41$	−48.0000	−0.182838	−0.0914188	+	0.995813i	$0.529140\pi$
$42$	0	0
$43$	32.0000	0.113487	0.0567437	−	0.998389i	$-0.481928\pi$
$43$	32.0000	0.113487	0.0567437	+	0.998389i	$0.481928\pi$
$44$	44.0000	0.150756
$45$	0	0
$46$	−376.000	−1.20518
$47$	326.000	1.01174	0.505872	−	0.862608i	$-0.331171\pi$
$47$	326.000	1.01174	0.505872	+	0.862608i	$0.331171\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	218.000	0.616597
$51$	0	0
$52$	248.000	0.661373
$53$	482.000	1.24920	0.624602	−	0.780943i	$-0.285261\pi$
$53$	482.000	1.24920	0.624602	+	0.780943i	$0.285261\pi$
$54$	0	0
$55$	44.0000	0.107872
$56$	56.0000	0.133631
$57$	0	0
$58$	124.000	0.280724
$59$	−400.000	−0.882637	−0.441318	−	0.897351i	$-0.645489\pi$
$59$	−400.000	−0.882637	−0.441318	+	0.897351i	$0.645489\pi$
$60$	0	0
$61$	70.0000	0.146928	0.0734638	−	0.997298i	$-0.476595\pi$
$61$	70.0000	0.146928	0.0734638	+	0.997298i	$0.476595\pi$
$62$	644.000	1.31916
$63$	0	0
$64$	64.0000	0.125000
$65$	248.000	0.473240
$66$	0	0
$67$	−124.000	−0.226105	−0.113052	−	0.993589i	$-0.536063\pi$
$67$	−124.000	−0.226105	−0.113052	+	0.993589i	$0.536063\pi$
$68$	480.000	0.856008
$69$	0	0
$70$	56.0000	0.0956183
$71$	712.000	1.19012	0.595062	−	0.803680i	$-0.297127\pi$
$71$	712.000	1.19012	0.595062	+	0.803680i	$0.297127\pi$
$72$	0	0
$73$	304.000	0.487404	0.243702	−	0.969850i	$-0.421638\pi$
$73$	304.000	0.487404	0.243702	+	0.969850i	$0.421638\pi$
$74$	396.000	0.622082
$75$	0	0
$76$	472.000	0.712396
$77$	−77.0000	−0.113961
$78$	0	0
$79$	−1016.00	−1.44695	−0.723474	−	0.690351i	$-0.757456\pi$
$79$	−1016.00	−1.44695	−0.723474	+	0.690351i	$0.757456\pi$
$80$	64.0000	0.0894427
$81$	0	0
$82$	96.0000	0.129286
$83$	−430.000	−0.568658	−0.284329	−	0.958727i	$-0.591771\pi$
$83$	−430.000	−0.568658	−0.284329	+	0.958727i	$0.591771\pi$
$84$	0	0
$85$	480.000	0.612510
$86$	−64.0000	−0.0802476
$87$	0	0
$88$	−88.0000	−0.106600
$89$	−442.000	−0.526426	−0.263213	−	0.964738i	$-0.584782\pi$
$89$	−442.000	−0.526426	−0.263213	+	0.964738i	$0.584782\pi$
$90$	0	0
$91$	−434.000	−0.499951
$92$	752.000	0.852189
$93$	0	0
$94$	−652.000	−0.715411
$95$	472.000	0.509749
$96$	0	0
$97$	−966.000	−1.01116	−0.505580	−	0.862780i	$-0.668721\pi$
$97$	−966.000	−1.01116	−0.505580	+	0.862780i	$0.668721\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	−436.000	−0.436000
$101$	−546.000	−0.537911	−0.268956	−	0.963153i	$-0.586678\pi$
$101$	−546.000	−0.537911	−0.268956	+	0.963153i	$0.586678\pi$
$102$	0	0
$103$	98.0000	0.0937498	0.0468749	−	0.998901i	$-0.485074\pi$
$103$	98.0000	0.0937498	0.0468749	+	0.998901i	$0.485074\pi$
$104$	−496.000	−0.467662
$105$	0	0
$106$	−964.000	−0.883320
$107$	2064.00	1.86481	0.932404	−	0.361418i	$-0.117707\pi$
$107$	2064.00	1.86481	0.932404	+	0.361418i	$0.117707\pi$
$108$	0	0
$109$	1306.00	1.14763	0.573817	−	0.818984i	$-0.305462\pi$
$109$	1306.00	1.14763	0.573817	+	0.818984i	$0.305462\pi$
$110$	−88.0000	−0.0762770
$111$	0	0
$112$	−112.000	−0.0944911
$113$	−478.000	−0.397933	−0.198967	−	0.980006i	$-0.563759\pi$
$113$	−478.000	−0.397933	−0.198967	+	0.980006i	$0.563759\pi$
$114$	0	0
$115$	752.000	0.609777
$116$	−248.000	−0.198502
$117$	0	0
$118$	800.000	0.624118
$119$	−840.000	−0.647081
$120$	0	0
$121$	121.000	0.0909091
$122$	−140.000	−0.103893
$123$	0	0
$124$	−1288.00	−0.932789
$125$	−936.000	−0.669747
$126$	0	0
$127$	824.000	0.575734	0.287867	−	0.957670i	$-0.407054\pi$
$127$	824.000	0.575734	0.287867	+	0.957670i	$0.407054\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	−496.000	−0.334631
$131$	22.0000	0.0146729	0.00733645	−	0.999973i	$-0.497665\pi$
$131$	22.0000	0.0146729	0.00733645	+	0.999973i	$0.497665\pi$
$132$	0	0
$133$	−826.000	−0.538521
$134$	248.000	0.159880
$135$	0	0
$136$	−960.000	−0.605289
$137$	2106.00	1.31334	0.656671	−	0.754178i	$-0.271964\pi$
$137$	2106.00	1.31334	0.656671	+	0.754178i	$0.271964\pi$
$138$	0	0
$139$	578.000	0.352700	0.176350	−	0.984328i	$-0.443571\pi$
$139$	578.000	0.352700	0.176350	+	0.984328i	$0.443571\pi$
$140$	−112.000	−0.0676123
$141$	0	0
$142$	−1424.00	−0.841545
$143$	682.000	0.398823
$144$	0	0
$145$	−248.000	−0.142036
$146$	−608.000	−0.344647
$147$	0	0
$148$	−792.000	−0.439878
$149$	−946.000	−0.520130	−0.260065	−	0.965591i	$-0.583744\pi$
$149$	−946.000	−0.520130	−0.260065	+	0.965591i	$0.583744\pi$
$150$	0	0
$151$	2368.00	1.27619	0.638096	−	0.769956i	$-0.279722\pi$
$151$	2368.00	1.27619	0.638096	+	0.769956i	$0.279722\pi$
$152$	−944.000	−0.503740
$153$	0	0
$154$	154.000	0.0805823
$155$	−1288.00	−0.667449
$156$	0	0
$157$	208.000	0.105734	0.0528669	−	0.998602i	$-0.483164\pi$
$157$	208.000	0.105734	0.0528669	+	0.998602i	$0.483164\pi$
$158$	2032.00	1.02315
$159$	0	0
$160$	−128.000	−0.0632456
$161$	−1316.00	−0.644195
$162$	0	0
$163$	3852.00	1.85099	0.925497	−	0.378756i	$-0.123648\pi$
$163$	3852.00	1.85099	0.925497	+	0.378756i	$0.123648\pi$
$164$	−192.000	−0.0914188
$165$	0	0
$166$	860.000	0.402102
$167$	−28.0000	−0.0129743	−0.00648714	−	0.999979i	$-0.502065\pi$
$167$	−28.0000	−0.0129743	−0.00648714	+	0.999979i	$0.502065\pi$
$168$	0	0
$169$	1647.00	0.749659
$170$	−960.000	−0.433110
$171$	0	0
$172$	128.000	0.0567437
$173$	−3646.00	−1.60231	−0.801157	−	0.598455i	$-0.795782\pi$
$173$	−3646.00	−1.60231	−0.801157	+	0.598455i	$0.795782\pi$
$174$	0	0
$175$	763.000	0.329585
$176$	176.000	0.0753778
$177$	0	0
$178$	884.000	0.372239
$179$	−948.000	−0.395848	−0.197924	−	0.980217i	$-0.563420\pi$
$179$	−948.000	−0.395848	−0.197924	+	0.980217i	$0.563420\pi$
$180$	0	0
$181$	2228.00	0.914950	0.457475	−	0.889223i	$-0.348754\pi$
$181$	2228.00	0.914950	0.457475	+	0.889223i	$0.348754\pi$
$182$	868.000	0.353519
$183$	0	0
$184$	−1504.00	−0.602589
$185$	−792.000	−0.314751
$186$	0	0
$187$	1320.00	0.516192
$188$	1304.00	0.505872
$189$	0	0
$190$	−944.000	−0.360447
$191$	3412.00	1.29258	0.646292	−	0.763090i	$-0.276319\pi$
$191$	3412.00	1.29258	0.646292	+	0.763090i	$0.276319\pi$
$192$	0	0
$193$	−3894.00	−1.45231	−0.726156	−	0.687530i	$-0.758695\pi$
$193$	−3894.00	−1.45231	−0.726156	+	0.687530i	$0.758695\pi$
$194$	1932.00	0.714998
$195$	0	0
$196$	196.000	0.0714286
$197$	1122.00	0.405783	0.202891	−	0.979201i	$-0.434966\pi$
$197$	1122.00	0.405783	0.202891	+	0.979201i	$0.434966\pi$
$198$	0	0
$199$	−1406.00	−0.500848	−0.250424	−	0.968136i	$-0.580570\pi$
$199$	−1406.00	−0.500848	−0.250424	+	0.968136i	$0.580570\pi$
$200$	872.000	0.308299
$201$	0	0
$202$	1092.00	0.380361
$203$	434.000	0.150053
$204$	0	0
$205$	−192.000	−0.0654140
$206$	−196.000	−0.0662911
$207$	0	0
$208$	992.000	0.330687
$209$	1298.00	0.429591
$210$	0	0
$211$	−4288.00	−1.39904	−0.699522	−	0.714612i	$-0.746604\pi$
$211$	−4288.00	−1.39904	−0.699522	+	0.714612i	$0.746604\pi$
$212$	1928.00	0.624602
$213$	0	0
$214$	−4128.00	−1.31862
$215$	128.000	0.0406025
$216$	0	0
$217$	2254.00	0.705122
$218$	−2612.00	−0.811500
$219$	0	0
$220$	176.000	0.0539360
$221$	7440.00	2.26456
$222$	0	0
$223$	4102.00	1.23179	0.615897	−	0.787826i	$-0.288794\pi$
$223$	4102.00	1.23179	0.615897	+	0.787826i	$0.288794\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	956.000	0.281381
$227$	6670.00	1.95024	0.975118	−	0.221688i	$-0.0711565\pi$
$227$	6670.00	1.95024	0.975118	+	0.221688i	$0.0711565\pi$
$228$	0	0
$229$	−1708.00	−0.492873	−0.246436	−	0.969159i	$-0.579260\pi$
$229$	−1708.00	−0.492873	−0.246436	+	0.969159i	$0.579260\pi$
$230$	−1504.00	−0.431177
$231$	0	0
$232$	496.000	0.140362
$233$	−3198.00	−0.899176	−0.449588	−	0.893236i	$-0.648429\pi$
$233$	−3198.00	−0.899176	−0.449588	+	0.893236i	$0.648429\pi$
$234$	0	0
$235$	1304.00	0.361973
$236$	−1600.00	−0.441318
$237$	0	0
$238$	1680.00	0.457556
$239$	−3760.00	−1.01763	−0.508816	−	0.860875i	$-0.669917\pi$
$239$	−3760.00	−1.01763	−0.508816	+	0.860875i	$0.669917\pi$
$240$	0	0
$241$	6848.00	1.83037	0.915184	−	0.403037i	$-0.132046\pi$
$241$	6848.00	1.83037	0.915184	+	0.403037i	$0.132046\pi$
$242$	−242.000	−0.0642824
$243$	0	0
$244$	280.000	0.0734638
$245$	196.000	0.0511101
$246$	0	0
$247$	7316.00	1.88464
$248$	2576.00	0.659581
$249$	0	0
$250$	1872.00	0.473583
$251$	6800.00	1.71001	0.855004	−	0.518621i	$-0.173554\pi$
$251$	6800.00	1.71001	0.855004	+	0.518621i	$0.173554\pi$
$252$	0	0
$253$	2068.00	0.513890
$254$	−1648.00	−0.407105
$255$	0	0
$256$	256.000	0.0625000
$257$	−1466.00	−0.355823	−0.177912	−	0.984046i	$-0.556934\pi$
$257$	−1466.00	−0.355823	−0.177912	+	0.984046i	$0.556934\pi$
$258$	0	0
$259$	1386.00	0.332517
$260$	992.000	0.236620
$261$	0	0
$262$	−44.0000	−0.0103753
$263$	−576.000	−0.135048	−0.0675241	−	0.997718i	$-0.521510\pi$
$263$	−576.000	−0.135048	−0.0675241	+	0.997718i	$0.521510\pi$
$264$	0	0
$265$	1928.00	0.446929
$266$	1652.00	0.380792
$267$	0	0
$268$	−496.000	−0.113052
$269$	1420.00	0.321855	0.160927	−	0.986966i	$-0.448552\pi$
$269$	1420.00	0.321855	0.160927	+	0.986966i	$0.448552\pi$
$270$	0	0
$271$	−2644.00	−0.592663	−0.296331	−	0.955085i	$-0.595763\pi$
$271$	−2644.00	−0.592663	−0.296331	+	0.955085i	$0.595763\pi$
$272$	1920.00	0.428004
$273$	0	0
$274$	−4212.00	−0.928672
$275$	−1199.00	−0.262918
$276$	0	0
$277$	−78.0000	−0.0169190	−0.00845951	−	0.999964i	$-0.502693\pi$
$277$	−78.0000	−0.0169190	−0.00845951	+	0.999964i	$0.502693\pi$
$278$	−1156.00	−0.249397
$279$	0	0
$280$	224.000	0.0478091
$281$	330.000	0.0700575	0.0350287	−	0.999386i	$-0.488848\pi$
$281$	330.000	0.0700575	0.0350287	+	0.999386i	$0.488848\pi$
$282$	0	0
$283$	−6078.00	−1.27668	−0.638339	−	0.769756i	$-0.720378\pi$
$283$	−6078.00	−1.27668	−0.638339	+	0.769756i	$0.720378\pi$
$284$	2848.00	0.595062
$285$	0	0
$286$	−1364.00	−0.282011
$287$	336.000	0.0691061
$288$	0	0
$289$	9487.00	1.93100
$290$	496.000	0.100435
$291$	0	0
$292$	1216.00	0.243702
$293$	−5382.00	−1.07311	−0.536553	−	0.843867i	$-0.680274\pi$
$293$	−5382.00	−1.07311	−0.536553	+	0.843867i	$0.680274\pi$
$294$	0	0
$295$	−1600.00	−0.315782
$296$	1584.00	0.311041
$297$	0	0
$298$	1892.00	0.367787
$299$	11656.0	2.25446
$300$	0	0
$301$	−224.000	−0.0428942
$302$	−4736.00	−0.902405
$303$	0	0
$304$	1888.00	0.356198
$305$	280.000	0.0525664
$306$	0	0
$307$	10594.0	1.96948	0.984742	−	0.174021i	$-0.0556762\pi$
$307$	10594.0	1.96948	0.984742	+	0.174021i	$0.0556762\pi$
$308$	−308.000	−0.0569803
$309$	0	0
$310$	2576.00	0.471958
$311$	1458.00	0.265838	0.132919	−	0.991127i	$-0.457565\pi$
$311$	1458.00	0.265838	0.132919	+	0.991127i	$0.457565\pi$
$312$	0	0
$313$	7662.00	1.38365	0.691824	−	0.722066i	$-0.256807\pi$
$313$	7662.00	1.38365	0.691824	+	0.722066i	$0.256807\pi$
$314$	−416.000	−0.0747651
$315$	0	0
$316$	−4064.00	−0.723474
$317$	−2262.00	−0.400778	−0.200389	−	0.979716i	$-0.564221\pi$
$317$	−2262.00	−0.400778	−0.200389	+	0.979716i	$0.564221\pi$
$318$	0	0
$319$	−682.000	−0.119701
$320$	256.000	0.0447214
$321$	0	0
$322$	2632.00	0.455514
$323$	14160.0	2.43927
$324$	0	0
$325$	−6758.00	−1.15344
$326$	−7704.00	−1.30885
$327$	0	0
$328$	384.000	0.0646428
$329$	−2282.00	−0.382403
$330$	0	0
$331$	−8780.00	−1.45798	−0.728992	−	0.684523i	$-0.760011\pi$
$331$	−8780.00	−1.45798	−0.728992	+	0.684523i	$0.760011\pi$
$332$	−1720.00	−0.284329
$333$	0	0
$334$	56.0000	0.00917420
$335$	−496.000	−0.0808937
$336$	0	0
$337$	2174.00	0.351410	0.175705	−	0.984443i	$-0.443779\pi$
$337$	2174.00	0.351410	0.175705	+	0.984443i	$0.443779\pi$
$338$	−3294.00	−0.530089
$339$	0	0
$340$	1920.00	0.306255
$341$	−3542.00	−0.562493
$342$	0	0
$343$	−343.000	−0.0539949
$344$	−256.000	−0.0401238
$345$	0	0
$346$	7292.00	1.13301
$347$	−7656.00	−1.18443	−0.592213	−	0.805782i	$-0.701745\pi$
$347$	−7656.00	−1.18443	−0.592213	+	0.805782i	$0.701745\pi$
$348$	0	0
$349$	−3046.00	−0.467188	−0.233594	−	0.972334i	$-0.575049\pi$
$349$	−3046.00	−0.467188	−0.233594	+	0.972334i	$0.575049\pi$
$350$	−1526.00	−0.233052
$351$	0	0
$352$	−352.000	−0.0533002
$353$	10390.0	1.56658	0.783292	−	0.621654i	$-0.213539\pi$
$353$	10390.0	1.56658	0.783292	+	0.621654i	$0.213539\pi$
$354$	0	0
$355$	2848.00	0.425792
$356$	−1768.00	−0.263213
$357$	0	0
$358$	1896.00	0.279907
$359$	1848.00	0.271682	0.135841	−	0.990731i	$-0.456626\pi$
$359$	1848.00	0.271682	0.135841	+	0.990731i	$0.456626\pi$
$360$	0	0
$361$	7065.00	1.03003
$362$	−4456.00	−0.646967
$363$	0	0
$364$	−1736.00	−0.249976
$365$	1216.00	0.174379
$366$	0	0
$367$	−7314.00	−1.04029	−0.520147	−	0.854077i	$-0.674123\pi$
$367$	−7314.00	−1.04029	−0.520147	+	0.854077i	$0.674123\pi$
$368$	3008.00	0.426095
$369$	0	0
$370$	1584.00	0.222563
$371$	−3374.00	−0.472155
$372$	0	0
$373$	−6034.00	−0.837610	−0.418805	−	0.908076i	$-0.637551\pi$
$373$	−6034.00	−0.837610	−0.418805	+	0.908076i	$0.637551\pi$
$374$	−2640.00	−0.365003
$375$	0	0
$376$	−2608.00	−0.357706
$377$	−3844.00	−0.525135
$378$	0	0
$379$	−3220.00	−0.436412	−0.218206	−	0.975903i	$-0.570021\pi$
$379$	−3220.00	−0.436412	−0.218206	+	0.975903i	$0.570021\pi$
$380$	1888.00	0.254875
$381$	0	0
$382$	−6824.00	−0.913995
$383$	5074.00	0.676943	0.338472	−	0.940977i	$-0.390090\pi$
$383$	5074.00	0.676943	0.338472	+	0.940977i	$0.390090\pi$
$384$	0	0
$385$	−308.000	−0.0407718
$386$	7788.00	1.02694
$387$	0	0
$388$	−3864.00	−0.505580
$389$	11166.0	1.45537	0.727685	−	0.685912i	$-0.240596\pi$
$389$	11166.0	1.45537	0.727685	+	0.685912i	$0.240596\pi$
$390$	0	0
$391$	22560.0	2.91792
$392$	−392.000	−0.0505076
$393$	0	0
$394$	−2244.00	−0.286932
$395$	−4064.00	−0.517676
$396$	0	0
$397$	−4608.00	−0.582541	−0.291271	−	0.956641i	$-0.594078\pi$
$397$	−4608.00	−0.582541	−0.291271	+	0.956641i	$0.594078\pi$
$398$	2812.00	0.354153
$399$	0	0
$400$	−1744.00	−0.218000
$401$	2498.00	0.311083	0.155541	−	0.987829i	$-0.450288\pi$
$401$	2498.00	0.311083	0.155541	+	0.987829i	$0.450288\pi$
$402$	0	0
$403$	−19964.0	−2.46769
$404$	−2184.00	−0.268956
$405$	0	0
$406$	−868.000	−0.106104
$407$	−2178.00	−0.265257
$408$	0	0
$409$	8876.00	1.07308	0.536540	−	0.843875i	$-0.319731\pi$
$409$	8876.00	1.07308	0.536540	+	0.843875i	$0.319731\pi$
$410$	384.000	0.0462547
$411$	0	0
$412$	392.000	0.0468749
$413$	2800.00	0.333605
$414$	0	0
$415$	−1720.00	−0.203449
$416$	−1984.00	−0.233831
$417$	0	0
$418$	−2596.00	−0.303767
$419$	−9252.00	−1.07873	−0.539367	−	0.842071i	$-0.681337\pi$
$419$	−9252.00	−1.07873	−0.539367	+	0.842071i	$0.681337\pi$
$420$	0	0
$421$	4750.00	0.549883	0.274942	−	0.961461i	$-0.411341\pi$
$421$	4750.00	0.549883	0.274942	+	0.961461i	$0.411341\pi$
$422$	8576.00	0.989273
$423$	0	0
$424$	−3856.00	−0.441660
$425$	−13080.0	−1.49288
$426$	0	0
$427$	−490.000	−0.0555334
$428$	8256.00	0.932404
$429$	0	0
$430$	−256.000	−0.0287103
$431$	8008.00	0.894969	0.447485	−	0.894292i	$-0.352320\pi$
$431$	8008.00	0.894969	0.447485	+	0.894292i	$0.352320\pi$
$432$	0	0
$433$	−11734.0	−1.30231	−0.651155	−	0.758945i	$-0.725715\pi$
$433$	−11734.0	−1.30231	−0.651155	+	0.758945i	$0.725715\pi$
$434$	−4508.00	−0.498597
$435$	0	0
$436$	5224.00	0.573817
$437$	22184.0	2.42839
$438$	0	0
$439$	−8212.00	−0.892796	−0.446398	−	0.894835i	$-0.647293\pi$
$439$	−8212.00	−0.892796	−0.446398	+	0.894835i	$0.647293\pi$
$440$	−352.000	−0.0381385
$441$	0	0
$442$	−14880.0	−1.60129
$443$	5012.00	0.537533	0.268767	−	0.963205i	$-0.413384\pi$
$443$	5012.00	0.537533	0.268767	+	0.963205i	$0.413384\pi$
$444$	0	0
$445$	−1768.00	−0.188340
$446$	−8204.00	−0.871010
$447$	0	0
$448$	−448.000	−0.0472456
$449$	13602.0	1.42966	0.714831	−	0.699297i	$-0.246504\pi$
$449$	13602.0	1.42966	0.714831	+	0.699297i	$0.246504\pi$
$450$	0	0
$451$	−528.000	−0.0551276
$452$	−1912.00	−0.198967
$453$	0	0
$454$	−13340.0	−1.37902
$455$	−1736.00	−0.178868
$456$	0	0
$457$	5962.00	0.610264	0.305132	−	0.952310i	$-0.401299\pi$
$457$	5962.00	0.610264	0.305132	+	0.952310i	$0.401299\pi$
$458$	3416.00	0.348514
$459$	0	0
$460$	3008.00	0.304889
$461$	−9506.00	−0.960387	−0.480194	−	0.877162i	$-0.659434\pi$
$461$	−9506.00	−0.960387	−0.480194	+	0.877162i	$0.659434\pi$
$462$	0	0
$463$	−9768.00	−0.980470	−0.490235	−	0.871590i	$-0.663089\pi$
$463$	−9768.00	−0.980470	−0.490235	+	0.871590i	$0.663089\pi$
$464$	−992.000	−0.0992510
$465$	0	0
$466$	6396.00	0.635813
$467$	2068.00	0.204916	0.102458	−	0.994737i	$-0.467329\pi$
$467$	2068.00	0.204916	0.102458	+	0.994737i	$0.467329\pi$
$468$	0	0
$469$	868.000	0.0854595
$470$	−2608.00	−0.255953
$471$	0	0
$472$	3200.00	0.312059
$473$	352.000	0.0342177
$474$	0	0
$475$	−12862.0	−1.24242
$476$	−3360.00	−0.323541
$477$	0	0
$478$	7520.00	0.719575
$479$	10920.0	1.04164	0.520822	−	0.853665i	$-0.325626\pi$
$479$	10920.0	1.04164	0.520822	+	0.853665i	$0.325626\pi$
$480$	0	0
$481$	−12276.0	−1.16370
$482$	−13696.0	−1.29426
$483$	0	0
$484$	484.000	0.0454545
$485$	−3864.00	−0.361763
$486$	0	0
$487$	−2724.00	−0.253462	−0.126731	−	0.991937i	$-0.540449\pi$
$487$	−2724.00	−0.253462	−0.126731	+	0.991937i	$0.540449\pi$
$488$	−560.000	−0.0519467
$489$	0	0
$490$	−392.000	−0.0361403
$491$	12460.0	1.14524	0.572619	−	0.819822i	$-0.305927\pi$
$491$	12460.0	1.14524	0.572619	+	0.819822i	$0.305927\pi$
$492$	0	0
$493$	−7440.00	−0.679677
$494$	−14632.0	−1.33264
$495$	0	0
$496$	−5152.00	−0.466394
$497$	−4984.00	−0.449825
$498$	0	0
$499$	22028.0	1.97617	0.988085	−	0.153910i	$-0.0491866\pi$
$499$	22028.0	1.97617	0.988085	+	0.153910i	$0.0491866\pi$
$500$	−3744.00	−0.334874
$501$	0	0
$502$	−13600.0	−1.20916
$503$	20724.0	1.83705	0.918526	−	0.395360i	$-0.129380\pi$
$503$	20724.0	1.83705	0.918526	+	0.395360i	$0.129380\pi$
$504$	0	0
$505$	−2184.00	−0.192449
$506$	−4136.00	−0.363375
$507$	0	0
$508$	3296.00	0.287867
$509$	−3740.00	−0.325683	−0.162841	−	0.986652i	$-0.552066\pi$
$509$	−3740.00	−0.325683	−0.162841	+	0.986652i	$0.552066\pi$
$510$	0	0
$511$	−2128.00	−0.184221
$512$	−512.000	−0.0441942
$513$	0	0
$514$	2932.00	0.251605
$515$	392.000	0.0335409
$516$	0	0
$517$	3586.00	0.305052
$518$	−2772.00	−0.235125
$519$	0	0
$520$	−1984.00	−0.167316
$521$	−15090.0	−1.26892	−0.634458	−	0.772958i	$-0.718777\pi$
$521$	−15090.0	−1.26892	−0.634458	+	0.772958i	$0.718777\pi$
$522$	0	0
$523$	−10694.0	−0.894103	−0.447052	−	0.894508i	$-0.647526\pi$
$523$	−10694.0	−0.894103	−0.447052	+	0.894508i	$0.647526\pi$
$524$	88.0000	0.00733645
$525$	0	0
$526$	1152.00	0.0954935
$527$	−38640.0	−3.19390
$528$	0	0
$529$	23177.0	1.90491
$530$	−3856.00	−0.316026
$531$	0	0
$532$	−3304.00	−0.269260
$533$	−2976.00	−0.241848
$534$	0	0
$535$	8256.00	0.667174
$536$	992.000	0.0799401
$537$	0	0
$538$	−2840.00	−0.227586
$539$	539.000	0.0430730
$540$	0	0
$541$	10438.0	0.829510	0.414755	−	0.909933i	$-0.363867\pi$
$541$	10438.0	0.829510	0.414755	+	0.909933i	$0.363867\pi$
$542$	5288.00	0.419076
$543$	0	0
$544$	−3840.00	−0.302645
$545$	5224.00	0.410590
$546$	0	0
$547$	−5852.00	−0.457429	−0.228714	−	0.973494i	$-0.573452\pi$
$547$	−5852.00	−0.457429	−0.228714	+	0.973494i	$0.573452\pi$
$548$	8424.00	0.656671
$549$	0	0
$550$	2398.00	0.185911
$551$	−7316.00	−0.565648
$552$	0	0
$553$	7112.00	0.546895
$554$	156.000	0.0119635
$555$	0	0
$556$	2312.00	0.176350
$557$	−9990.00	−0.759946	−0.379973	−	0.924998i	$-0.624067\pi$
$557$	−9990.00	−0.759946	−0.379973	+	0.924998i	$0.624067\pi$
$558$	0	0
$559$	1984.00	0.150115
$560$	−448.000	−0.0338062
$561$	0	0
$562$	−660.000	−0.0495381
$563$	−3258.00	−0.243887	−0.121943	−	0.992537i	$-0.538913\pi$
$563$	−3258.00	−0.243887	−0.121943	+	0.992537i	$0.538913\pi$
$564$	0	0
$565$	−1912.00	−0.142369
$566$	12156.0	0.902747
$567$	0	0
$568$	−5696.00	−0.420773
$569$	−10478.0	−0.771987	−0.385994	−	0.922501i	$-0.626141\pi$
$569$	−10478.0	−0.771987	−0.385994	+	0.922501i	$0.626141\pi$
$570$	0	0
$571$	−8116.00	−0.594823	−0.297412	−	0.954749i	$-0.596123\pi$
$571$	−8116.00	−0.594823	−0.297412	+	0.954749i	$0.596123\pi$
$572$	2728.00	0.199412
$573$	0	0
$574$	−672.000	−0.0488654
$575$	−20492.0	−1.48622
$576$	0	0
$577$	5910.00	0.426406	0.213203	−	0.977008i	$-0.431610\pi$
$577$	5910.00	0.426406	0.213203	+	0.977008i	$0.431610\pi$
$578$	−18974.0	−1.36542
$579$	0	0
$580$	−992.000	−0.0710182
$581$	3010.00	0.214933
$582$	0	0
$583$	5302.00	0.376649
$584$	−2432.00	−0.172323
$585$	0	0
$586$	10764.0	0.758800
$587$	−12888.0	−0.906209	−0.453105	−	0.891457i	$-0.649684\pi$
$587$	−12888.0	−0.906209	−0.453105	+	0.891457i	$0.649684\pi$
$588$	0	0
$589$	−37996.0	−2.65806
$590$	3200.00	0.223291
$591$	0	0
$592$	−3168.00	−0.219939
$593$	−16440.0	−1.13847	−0.569233	−	0.822177i	$-0.692760\pi$
$593$	−16440.0	−1.13847	−0.569233	+	0.822177i	$0.692760\pi$
$594$	0	0
$595$	−3360.00	−0.231507
$596$	−3784.00	−0.260065
$597$	0	0
$598$	−23312.0	−1.59414
$599$	18840.0	1.28511	0.642556	−	0.766239i	$-0.277874\pi$
$599$	18840.0	1.28511	0.642556	+	0.766239i	$0.277874\pi$
$600$	0	0
$601$	−16068.0	−1.09056	−0.545281	−	0.838254i	$-0.683577\pi$
$601$	−16068.0	−1.09056	−0.545281	+	0.838254i	$0.683577\pi$
$602$	448.000	0.0303308
$603$	0	0
$604$	9472.00	0.638096
$605$	484.000	0.0325246
$606$	0	0
$607$	13272.0	0.887469	0.443735	−	0.896158i	$-0.353653\pi$
$607$	13272.0	0.887469	0.443735	+	0.896158i	$0.353653\pi$
$608$	−3776.00	−0.251870
$609$	0	0
$610$	−560.000	−0.0371701
$611$	20212.0	1.33828
$612$	0	0
$613$	25022.0	1.64866	0.824330	−	0.566109i	$-0.191552\pi$
$613$	25022.0	1.64866	0.824330	+	0.566109i	$0.191552\pi$
$614$	−21188.0	−1.39264
$615$	0	0
$616$	616.000	0.0402911
$617$	−7202.00	−0.469922	−0.234961	−	0.972005i	$-0.575496\pi$
$617$	−7202.00	−0.469922	−0.234961	+	0.972005i	$0.575496\pi$
$618$	0	0
$619$	16228.0	1.05373	0.526865	−	0.849949i	$-0.323367\pi$
$619$	16228.0	1.05373	0.526865	+	0.849949i	$0.323367\pi$
$620$	−5152.00	−0.333725
$621$	0	0
$622$	−2916.00	−0.187976
$623$	3094.00	0.198970
$624$	0	0
$625$	9881.00	0.632384
$626$	−15324.0	−0.978387
$627$	0	0
$628$	832.000	0.0528669
$629$	−23760.0	−1.50616
$630$	0	0
$631$	−25348.0	−1.59919	−0.799594	−	0.600541i	$-0.794952\pi$
$631$	−25348.0	−1.59919	−0.799594	+	0.600541i	$0.794952\pi$
$632$	8128.00	0.511574
$633$	0	0
$634$	4524.00	0.283393
$635$	3296.00	0.205981
$636$	0	0
$637$	3038.00	0.188964
$638$	1364.00	0.0846415
$639$	0	0
$640$	−512.000	−0.0316228
$641$	−28938.0	−1.78312	−0.891562	−	0.452899i	$-0.850390\pi$
$641$	−28938.0	−1.78312	−0.891562	+	0.452899i	$0.850390\pi$
$642$	0	0
$643$	20776.0	1.27422	0.637112	−	0.770772i	$-0.280129\pi$
$643$	20776.0	1.27422	0.637112	+	0.770772i	$0.280129\pi$
$644$	−5264.00	−0.322097
$645$	0	0
$646$	−28320.0	−1.72482
$647$	30114.0	1.82984	0.914918	−	0.403640i	$-0.132255\pi$
$647$	30114.0	1.82984	0.914918	+	0.403640i	$0.132255\pi$
$648$	0	0
$649$	−4400.00	−0.266125
$650$	13516.0	0.815602
$651$	0	0
$652$	15408.0	0.925497
$653$	−9850.00	−0.590291	−0.295146	−	0.955452i	$-0.595368\pi$
$653$	−9850.00	−0.590291	−0.295146	+	0.955452i	$0.595368\pi$
$654$	0	0
$655$	88.0000	0.00524953
$656$	−768.000	−0.0457094
$657$	0	0
$658$	4564.00	0.270400
$659$	−32360.0	−1.91285	−0.956424	−	0.291982i	$-0.905685\pi$
$659$	−32360.0	−1.91285	−0.956424	+	0.291982i	$0.905685\pi$
$660$	0	0
$661$	29836.0	1.75565	0.877826	−	0.478980i	$-0.158993\pi$
$661$	29836.0	1.75565	0.877826	+	0.478980i	$0.158993\pi$
$662$	17560.0	1.03095
$663$	0	0
$664$	3440.00	0.201051
$665$	−3304.00	−0.192667
$666$	0	0
$667$	−11656.0	−0.676645
$668$	−112.000	−0.00648714
$669$	0	0
$670$	992.000	0.0572005
$671$	770.000	0.0443003
$672$	0	0
$673$	−24082.0	−1.37934	−0.689668	−	0.724126i	$-0.742243\pi$
$673$	−24082.0	−1.37934	−0.689668	+	0.724126i	$0.742243\pi$
$674$	−4348.00	−0.248485
$675$	0	0
$676$	6588.00	0.374829
$677$	3690.00	0.209480	0.104740	−	0.994500i	$-0.466599\pi$
$677$	3690.00	0.209480	0.104740	+	0.994500i	$0.466599\pi$
$678$	0	0
$679$	6762.00	0.382182
$680$	−3840.00	−0.216555
$681$	0	0
$682$	7084.00	0.397742
$683$	7596.00	0.425553	0.212777	−	0.977101i	$-0.431749\pi$
$683$	7596.00	0.425553	0.212777	+	0.977101i	$0.431749\pi$
$684$	0	0
$685$	8424.00	0.469875
$686$	686.000	0.0381802
$687$	0	0
$688$	512.000	0.0283718
$689$	29884.0	1.65238
$690$	0	0
$691$	12960.0	0.713490	0.356745	−	0.934202i	$-0.383886\pi$
$691$	12960.0	0.713490	0.356745	+	0.934202i	$0.383886\pi$
$692$	−14584.0	−0.801157
$693$	0	0
$694$	15312.0	0.837515
$695$	2312.00	0.126186
$696$	0	0
$697$	−5760.00	−0.313021
$698$	6092.00	0.330352
$699$	0	0
$700$	3052.00	0.164793
$701$	−11034.0	−0.594506	−0.297253	−	0.954799i	$-0.596070\pi$
$701$	−11034.0	−0.594506	−0.297253	+	0.954799i	$0.596070\pi$
$702$	0	0
$703$	−23364.0	−1.25347
$704$	704.000	0.0376889
$705$	0	0
$706$	−20780.0	−1.10774
$707$	3822.00	0.203311
$708$	0	0
$709$	−694.000	−0.0367612	−0.0183806	−	0.999831i	$-0.505851\pi$
$709$	−694.000	−0.0367612	−0.0183806	+	0.999831i	$0.505851\pi$
$710$	−5696.00	−0.301080
$711$	0	0
$712$	3536.00	0.186120
$713$	−60536.0	−3.17965
$714$	0	0
$715$	2728.00	0.142687
$716$	−3792.00	−0.197924
$717$	0	0
$718$	−3696.00	−0.192108
$719$	3762.00	0.195131	0.0975653	−	0.995229i	$-0.468895\pi$
$719$	3762.00	0.195131	0.0975653	+	0.995229i	$0.468895\pi$
$720$	0	0
$721$	−686.000	−0.0354341
$722$	−14130.0	−0.728344
$723$	0	0
$724$	8912.00	0.457475
$725$	6758.00	0.346187
$726$	0	0
$727$	−1274.00	−0.0649932	−0.0324966	−	0.999472i	$-0.510346\pi$
$727$	−1274.00	−0.0649932	−0.0324966	+	0.999472i	$0.510346\pi$
$728$	3472.00	0.176759
$729$	0	0
$730$	−2432.00	−0.123305
$731$	3840.00	0.194292
$732$	0	0
$733$	−21026.0	−1.05950	−0.529750	−	0.848154i	$-0.677714\pi$
$733$	−21026.0	−1.05950	−0.529750	+	0.848154i	$0.677714\pi$
$734$	14628.0	0.735599
$735$	0	0
$736$	−6016.00	−0.301294
$737$	−1364.00	−0.0681731
$738$	0	0
$739$	12692.0	0.631776	0.315888	−	0.948796i	$-0.397698\pi$
$739$	12692.0	0.631776	0.315888	+	0.948796i	$0.397698\pi$
$740$	−3168.00	−0.157376
$741$	0	0
$742$	6748.00	0.333864
$743$	−2152.00	−0.106257	−0.0531287	−	0.998588i	$-0.516919\pi$
$743$	−2152.00	−0.106257	−0.0531287	+	0.998588i	$0.516919\pi$
$744$	0	0
$745$	−3784.00	−0.186087
$746$	12068.0	0.592280
$747$	0	0
$748$	5280.00	0.258096
$749$	−14448.0	−0.704831
$750$	0	0
$751$	18740.0	0.910562	0.455281	−	0.890348i	$-0.349539\pi$
$751$	18740.0	0.910562	0.455281	+	0.890348i	$0.349539\pi$
$752$	5216.00	0.252936
$753$	0	0
$754$	7688.00	0.371327
$755$	9472.00	0.456585
$756$	0	0
$757$	−36894.0	−1.77138	−0.885690	−	0.464276i	$-0.846314\pi$
$757$	−36894.0	−1.77138	−0.885690	+	0.464276i	$0.846314\pi$
$758$	6440.00	0.308590
$759$	0	0
$760$	−3776.00	−0.180224
$761$	18732.0	0.892292	0.446146	−	0.894960i	$-0.352796\pi$
$761$	18732.0	0.892292	0.446146	+	0.894960i	$0.352796\pi$
$762$	0	0
$763$	−9142.00	−0.433765
$764$	13648.0	0.646292
$765$	0	0
$766$	−10148.0	−0.478671
$767$	−24800.0	−1.16750
$768$	0	0
$769$	−16292.0	−0.763985	−0.381993	−	0.924165i	$-0.624762\pi$
$769$	−16292.0	−0.763985	−0.381993	+	0.924165i	$0.624762\pi$
$770$	616.000	0.0288300
$771$	0	0
$772$	−15576.0	−0.726156
$773$	−23336.0	−1.08582	−0.542909	−	0.839791i	$-0.682677\pi$
$773$	−23336.0	−1.08582	−0.542909	+	0.839791i	$0.682677\pi$
$774$	0	0
$775$	35098.0	1.62678
$776$	7728.00	0.357499
$777$	0	0
$778$	−22332.0	−1.02910
$779$	−5664.00	−0.260506
$780$	0	0
$781$	7832.00	0.358836
$782$	−45120.0	−2.06328
$783$	0	0
$784$	784.000	0.0357143
$785$	832.000	0.0378285
$786$	0	0
$787$	9110.00	0.412626	0.206313	−	0.978486i	$-0.433854\pi$
$787$	9110.00	0.412626	0.206313	+	0.978486i	$0.433854\pi$
$788$	4488.00	0.202891
$789$	0	0
$790$	8128.00	0.366052
$791$	3346.00	0.150405
$792$	0	0
$793$	4340.00	0.194348
$794$	9216.00	0.411919
$795$	0	0
$796$	−5624.00	−0.250424
$797$	−20144.0	−0.895279	−0.447639	−	0.894214i	$-0.647735\pi$
$797$	−20144.0	−0.895279	−0.447639	+	0.894214i	$0.647735\pi$
$798$	0	0
$799$	39120.0	1.73212
$800$	3488.00	0.154149
$801$	0	0
$802$	−4996.00	−0.219969
$803$	3344.00	0.146958
$804$	0	0
$805$	−5264.00	−0.230474
$806$	39928.0	1.74492
$807$	0	0
$808$	4368.00	0.190180
$809$	−10946.0	−0.475699	−0.237850	−	0.971302i	$-0.576443\pi$
$809$	−10946.0	−0.475699	−0.237850	+	0.971302i	$0.576443\pi$
$810$	0	0
$811$	13874.0	0.600718	0.300359	−	0.953826i	$-0.402894\pi$
$811$	13874.0	0.600718	0.300359	+	0.953826i	$0.402894\pi$
$812$	1736.00	0.0750267
$813$	0	0
$814$	4356.00	0.187565
$815$	15408.0	0.662232
$816$	0	0
$817$	3776.00	0.161696
$818$	−17752.0	−0.758783
$819$	0	0
$820$	−768.000	−0.0327070
$821$	−7570.00	−0.321796	−0.160898	−	0.986971i	$-0.551439\pi$
$821$	−7570.00	−0.321796	−0.160898	+	0.986971i	$0.551439\pi$
$822$	0	0
$823$	14372.0	0.608720	0.304360	−	0.952557i	$-0.401557\pi$
$823$	14372.0	0.608720	0.304360	+	0.952557i	$0.401557\pi$
$824$	−784.000	−0.0331456
$825$	0	0
$826$	−5600.00	−0.235895
$827$	−25036.0	−1.05270	−0.526352	−	0.850266i	$-0.676441\pi$
$827$	−25036.0	−1.05270	−0.526352	+	0.850266i	$0.676441\pi$
$828$	0	0
$829$	−38684.0	−1.62069	−0.810344	−	0.585954i	$-0.800720\pi$
$829$	−38684.0	−1.62069	−0.810344	+	0.585954i	$0.800720\pi$
$830$	3440.00	0.143860
$831$	0	0
$832$	3968.00	0.165343
$833$	5880.00	0.244574
$834$	0	0
$835$	−112.000	−0.00464182
$836$	5192.00	0.214796
$837$	0	0
$838$	18504.0	0.762781
$839$	−31110.0	−1.28014	−0.640069	−	0.768317i	$-0.721094\pi$
$839$	−31110.0	−1.28014	−0.640069	+	0.768317i	$0.721094\pi$
$840$	0	0
$841$	−20545.0	−0.842388
$842$	−9500.00	−0.388826
$843$	0	0
$844$	−17152.0	−0.699522
$845$	6588.00	0.268206
$846$	0	0
$847$	−847.000	−0.0343604
$848$	7712.00	0.312301
$849$	0	0
$850$	26160.0	1.05562
$851$	−37224.0	−1.49944
$852$	0	0
$853$	−25922.0	−1.04051	−0.520253	−	0.854012i	$-0.674163\pi$
$853$	−25922.0	−1.04051	−0.520253	+	0.854012i	$0.674163\pi$
$854$	980.000	0.0392680
$855$	0	0
$856$	−16512.0	−0.659309
$857$	14076.0	0.561058	0.280529	−	0.959845i	$-0.409490\pi$
$857$	14076.0	0.561058	0.280529	+	0.959845i	$0.409490\pi$
$858$	0	0
$859$	−23920.0	−0.950105	−0.475052	−	0.879958i	$-0.657571\pi$
$859$	−23920.0	−0.950105	−0.475052	+	0.879958i	$0.657571\pi$
$860$	512.000	0.0203012
$861$	0	0
$862$	−16016.0	−0.632839
$863$	34316.0	1.35357	0.676785	−	0.736181i	$-0.263373\pi$
$863$	34316.0	1.35357	0.676785	+	0.736181i	$0.263373\pi$
$864$	0	0
$865$	−14584.0	−0.573261
$866$	23468.0	0.920872
$867$	0	0
$868$	9016.00	0.352561
$869$	−11176.0	−0.436271
$870$	0	0
$871$	−7688.00	−0.299079
$872$	−10448.0	−0.405750
$873$	0	0
$874$	−44368.0	−1.71713
$875$	6552.00	0.253141
$876$	0	0
$877$	2954.00	0.113739	0.0568697	−	0.998382i	$-0.481888\pi$
$877$	2954.00	0.113739	0.0568697	+	0.998382i	$0.481888\pi$
$878$	16424.0	0.631302
$879$	0	0
$880$	704.000	0.0269680
$881$	35994.0	1.37647	0.688234	−	0.725489i	$-0.258386\pi$
$881$	35994.0	1.37647	0.688234	+	0.725489i	$0.258386\pi$
$882$	0	0
$883$	−36844.0	−1.40419	−0.702095	−	0.712084i	$-0.747752\pi$
$883$	−36844.0	−1.40419	−0.702095	+	0.712084i	$0.747752\pi$
$884$	29760.0	1.13228
$885$	0	0
$886$	−10024.0	−0.380094
$887$	−35000.0	−1.32490	−0.662449	−	0.749107i	$-0.730483\pi$
$887$	−35000.0	−1.32490	−0.662449	+	0.749107i	$0.730483\pi$
$888$	0	0
$889$	−5768.00	−0.217607
$890$	3536.00	0.133176
$891$	0	0
$892$	16408.0	0.615897
$893$	38468.0	1.44153
$894$	0	0
$895$	−3792.00	−0.141623
$896$	896.000	0.0334077
$897$	0	0
$898$	−27204.0	−1.01092
$899$	19964.0	0.740641
$900$	0	0
$901$	57840.0	2.13866
$902$	1056.00	0.0389811
$903$	0	0
$904$	3824.00	0.140691
$905$	8912.00	0.327342
$906$	0	0
$907$	−5460.00	−0.199886	−0.0999428	−	0.994993i	$-0.531866\pi$
$907$	−5460.00	−0.199886	−0.0999428	+	0.994993i	$0.531866\pi$
$908$	26680.0	0.975118
$909$	0	0
$910$	3472.00	0.126479
$911$	−27116.0	−0.986162	−0.493081	−	0.869984i	$-0.664129\pi$
$911$	−27116.0	−0.986162	−0.493081	+	0.869984i	$0.664129\pi$
$912$	0	0
$913$	−4730.00	−0.171457
$914$	−11924.0	−0.431522
$915$	0	0
$916$	−6832.00	−0.246436
$917$	−154.000	−0.00554583
$918$	0	0
$919$	50056.0	1.79673	0.898365	−	0.439249i	$-0.144756\pi$
$919$	50056.0	1.79673	0.898365	+	0.439249i	$0.144756\pi$
$920$	−6016.00	−0.215589
$921$	0	0
$922$	19012.0	0.679096
$923$	44144.0	1.57423
$924$	0	0
$925$	21582.0	0.767148
$926$	19536.0	0.693297
$927$	0	0
$928$	1984.00	0.0701810
$929$	28294.0	0.999242	0.499621	−	0.866244i	$-0.333473\pi$
$929$	28294.0	0.999242	0.499621	+	0.866244i	$0.333473\pi$
$930$	0	0
$931$	5782.00	0.203542
$932$	−12792.0	−0.449588
$933$	0	0
$934$	−4136.00	−0.144897
$935$	5280.00	0.184679
$936$	0	0
$937$	15368.0	0.535806	0.267903	−	0.963446i	$-0.413669\pi$
$937$	15368.0	0.535806	0.267903	+	0.963446i	$0.413669\pi$
$938$	−1736.00	−0.0604290
$939$	0	0
$940$	5216.00	0.180986
$941$	2394.00	0.0829354	0.0414677	−	0.999140i	$-0.486797\pi$
$941$	2394.00	0.0829354	0.0414677	+	0.999140i	$0.486797\pi$
$942$	0	0
$943$	−9024.00	−0.311624
$944$	−6400.00	−0.220659
$945$	0	0
$946$	−704.000	−0.0241956
$947$	−20772.0	−0.712776	−0.356388	−	0.934338i	$-0.615992\pi$
$947$	−20772.0	−0.712776	−0.356388	+	0.934338i	$0.615992\pi$
$948$	0	0
$949$	18848.0	0.644712
$950$	25724.0	0.878523
$951$	0	0
$952$	6720.00	0.228778
$953$	49790.0	1.69240	0.846200	−	0.532866i	$-0.178885\pi$
$953$	49790.0	1.69240	0.846200	+	0.532866i	$0.178885\pi$
$954$	0	0
$955$	13648.0	0.462449
$956$	−15040.0	−0.508816
$957$	0	0
$958$	−21840.0	−0.736554
$959$	−14742.0	−0.496396
$960$	0	0
$961$	73893.0	2.48038
$962$	24552.0	0.822857
$963$	0	0
$964$	27392.0	0.915184
$965$	−15576.0	−0.519595
$966$	0	0
$967$	24456.0	0.813291	0.406645	−	0.913586i	$-0.366699\pi$
$967$	24456.0	0.813291	0.406645	+	0.913586i	$0.366699\pi$
$968$	−968.000	−0.0321412
$969$	0	0
$970$	7728.00	0.255805
$971$	−32820.0	−1.08470	−0.542350	−	0.840153i	$-0.682465\pi$
$971$	−32820.0	−1.08470	−0.542350	+	0.840153i	$0.682465\pi$
$972$	0	0
$973$	−4046.00	−0.133308
$974$	5448.00	0.179225
$975$	0	0
$976$	1120.00	0.0367319
$977$	−21650.0	−0.708951	−0.354475	−	0.935065i	$-0.615341\pi$
$977$	−21650.0	−0.708951	−0.354475	+	0.935065i	$0.615341\pi$
$978$	0	0
$979$	−4862.00	−0.158723
$980$	784.000	0.0255551
$981$	0	0
$982$	−24920.0	−0.809806
$983$	36502.0	1.18437	0.592184	−	0.805803i	$-0.298266\pi$
$983$	36502.0	1.18437	0.592184	+	0.805803i	$0.298266\pi$
$984$	0	0
$985$	4488.00	0.145177
$986$	14880.0	0.480604
$987$	0	0
$988$	29264.0	0.942320
$989$	6016.00	0.193425
$990$	0	0
$991$	−39016.0	−1.25064	−0.625320	−	0.780368i	$-0.715032\pi$
$991$	−39016.0	−1.25064	−0.625320	+	0.780368i	$0.715032\pi$
$992$	10304.0	0.329791
$993$	0	0
$994$	9968.00	0.318074
$995$	−5624.00	−0.179189
$996$	0	0
$997$	−12558.0	−0.398913	−0.199456	−	0.979907i	$-0.563918\pi$
$997$	−12558.0	−0.398913	−0.199456	+	0.979907i	$0.563918\pi$
$998$	−44056.0	−1.39736
$999$	0	0

Twists

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

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