LMFDB - Embedded newform 1386.4.a.h.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:	$1$
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)

$f(q)$

$=$

$q-2.00000 q^{2} +4.00000 q^{4} +17.0000 q^{5} +7.00000 q^{7} -8.00000 q^{8} +O(q^{10})$

\(q-2.00000 q^{2} +4.00000 q^{4} +17.0000 q^{5} +7.00000 q^{7} -8.00000 q^{8} -34.0000 q^{10} +11.0000 q^{11} -21.0000 q^{13} -14.0000 q^{14} +16.0000 q^{16} +104.000 q^{17} -161.000 q^{19} +68.0000 q^{20} -22.0000 q^{22} -194.000 q^{23} +164.000 q^{25} +42.0000 q^{26} +28.0000 q^{28} -9.00000 q^{29} -180.000 q^{31} -32.0000 q^{32} -208.000 q^{34} +119.000 q^{35} -363.000 q^{37} +322.000 q^{38} -136.000 q^{40} +108.000 q^{41} -386.000 q^{43} +44.0000 q^{44} +388.000 q^{46} -333.000 q^{47} +49.0000 q^{49} -328.000 q^{50} -84.0000 q^{52} +122.000 q^{53} +187.000 q^{55} -56.0000 q^{56} +18.0000 q^{58} -537.000 q^{59} -950.000 q^{61} +360.000 q^{62} +64.0000 q^{64} -357.000 q^{65} -83.0000 q^{67} +416.000 q^{68} -238.000 q^{70} -180.000 q^{71} +177.000 q^{73} +726.000 q^{74} -644.000 q^{76} +77.0000 q^{77} -220.000 q^{79} +272.000 q^{80} -216.000 q^{82} -1112.00 q^{83} +1768.00 q^{85} +772.000 q^{86} -88.0000 q^{88} +394.000 q^{89} -147.000 q^{91} -776.000 q^{92} +666.000 q^{94} -2737.00 q^{95} +826.000 q^{97} -98.0000 q^{98} +O(q^{100})\)

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
$5$	17.0000	1.52053	0.760263	−	0.649615i	$-0.225070\pi$
$5$	17.0000	1.52053	0.760263	+	0.649615i	$0.225070\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	−8.00000	−0.353553
$9$	0	0
$10$	−34.0000	−1.07517
$11$	11.0000	0.301511
$11$	11.0000	0.301511
$12$	0	0
$13$	−21.0000	−0.448027	−0.224014	−	0.974586i	$-0.571916\pi$
$13$	−21.0000	−0.448027	−0.224014	+	0.974586i	$0.571916\pi$
$14$	−14.0000	−0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	104.000	1.48375	0.741874	−	0.670540i	$-0.233937\pi$
$17$	104.000	1.48375	0.741874	+	0.670540i	$0.233937\pi$
$18$	0	0
$19$	−161.000	−1.94400	−0.971998	−	0.234988i	$-0.924495\pi$
$19$	−161.000	−1.94400	−0.971998	+	0.234988i	$0.924495\pi$
$20$	68.0000	0.760263
$21$	0	0
$22$	−22.0000	−0.213201
$23$	−194.000	−1.75877	−0.879387	−	0.476108i	$-0.842047\pi$
$23$	−194.000	−1.75877	−0.879387	+	0.476108i	$0.842047\pi$
$24$	0	0
$25$	164.000	1.31200
$26$	42.0000	0.316803
$27$	0	0
$28$	28.0000	0.188982
$29$	−9.00000	−0.0576296	−0.0288148	−	0.999585i	$-0.509173\pi$
$29$	−9.00000	−0.0576296	−0.0288148	+	0.999585i	$0.509173\pi$
$30$	0	0
$31$	−180.000	−1.04287	−0.521435	−	0.853291i	$-0.674603\pi$
$31$	−180.000	−1.04287	−0.521435	+	0.853291i	$0.674603\pi$
$32$	−32.0000	−0.176777
$33$	0	0
$34$	−208.000	−1.04917
$35$	119.000	0.574705
$36$	0	0
$37$	−363.000	−1.61289	−0.806444	−	0.591311i	$-0.798611\pi$
$37$	−363.000	−1.61289	−0.806444	+	0.591311i	$0.798611\pi$
$38$	322.000	1.37461
$39$	0	0
$40$	−136.000	−0.537587
$41$	108.000	0.411385	0.205692	−	0.978617i	$-0.434055\pi$
$41$	108.000	0.411385	0.205692	+	0.978617i	$0.434055\pi$
$42$	0	0
$43$	−386.000	−1.36894	−0.684470	−	0.729041i	$-0.739966\pi$
$43$	−386.000	−1.36894	−0.684470	+	0.729041i	$0.739966\pi$
$44$	44.0000	0.150756
$45$	0	0
$46$	388.000	1.24364
$47$	−333.000	−1.03347	−0.516734	−	0.856146i	$-0.672853\pi$
$47$	−333.000	−1.03347	−0.516734	+	0.856146i	$0.672853\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−328.000	−0.927724
$51$	0	0
$52$	−84.0000	−0.224014
$53$	122.000	0.316188	0.158094	−	0.987424i	$-0.449465\pi$
$53$	122.000	0.316188	0.158094	+	0.987424i	$0.449465\pi$
$54$	0	0
$55$	187.000	0.458456
$56$	−56.0000	−0.133631
$57$	0	0
$58$	18.0000	0.0407503
$59$	−537.000	−1.18494	−0.592470	−	0.805593i	$-0.701847\pi$
$59$	−537.000	−1.18494	−0.592470	+	0.805593i	$0.701847\pi$
$60$	0	0
$61$	−950.000	−1.99402	−0.997008	−	0.0772921i	$-0.975373\pi$
$61$	−950.000	−1.99402	−0.997008	+	0.0772921i	$0.975373\pi$
$62$	360.000	0.737420
$63$	0	0
$64$	64.0000	0.125000
$65$	−357.000	−0.681237
$66$	0	0
$67$	−83.0000	−0.151344	−0.0756721	−	0.997133i	$-0.524110\pi$
$67$	−83.0000	−0.151344	−0.0756721	+	0.997133i	$0.524110\pi$
$68$	416.000	0.741874
$69$	0	0
$70$	−238.000	−0.406378
$71$	−180.000	−0.300874	−0.150437	−	0.988620i	$-0.548068\pi$
$71$	−180.000	−0.300874	−0.150437	+	0.988620i	$0.548068\pi$
$72$	0	0
$73$	177.000	0.283785	0.141892	−	0.989882i	$-0.454681\pi$
$73$	177.000	0.283785	0.141892	+	0.989882i	$0.454681\pi$
$74$	726.000	1.14048
$75$	0	0
$76$	−644.000	−0.971998
$77$	77.0000	0.113961
$78$	0	0
$79$	−220.000	−0.313316	−0.156658	−	0.987653i	$-0.550072\pi$
$79$	−220.000	−0.313316	−0.156658	+	0.987653i	$0.550072\pi$
$80$	272.000	0.380132
$81$	0	0
$82$	−216.000	−0.290893
$83$	−1112.00	−1.47058	−0.735288	−	0.677754i	$-0.762953\pi$
$83$	−1112.00	−1.47058	−0.735288	+	0.677754i	$0.762953\pi$
$84$	0	0
$85$	1768.00	2.25608
$86$	772.000	0.967987
$87$	0	0
$88$	−88.0000	−0.106600
$89$	394.000	0.469257	0.234629	−	0.972085i	$-0.424613\pi$
$89$	394.000	0.469257	0.234629	+	0.972085i	$0.424613\pi$
$90$	0	0
$91$	−147.000	−0.169338
$92$	−776.000	−0.879387
$93$	0	0
$94$	666.000	0.730773
$95$	−2737.00	−2.95590
$96$	0	0
$97$	826.000	0.864614	0.432307	−	0.901726i	$-0.357700\pi$
$97$	826.000	0.864614	0.432307	+	0.901726i	$0.357700\pi$
$98$	−98.0000	−0.101015
$99$	0	0
$100$	656.000	0.656000
$101$	1688.00	1.66299	0.831496	−	0.555530i	$-0.187485\pi$
$101$	1688.00	1.66299	0.831496	+	0.555530i	$0.187485\pi$
$102$	0	0
$103$	1338.00	1.27997	0.639986	−	0.768387i	$-0.278940\pi$
$103$	1338.00	1.27997	0.639986	+	0.768387i	$0.278940\pi$
$104$	168.000	0.158401
$105$	0	0
$106$	−244.000	−0.223579
$107$	535.000	0.483368	0.241684	−	0.970355i	$-0.422300\pi$
$107$	535.000	0.483368	0.241684	+	0.970355i	$0.422300\pi$
$108$	0	0
$109$	216.000	0.189808	0.0949039	−	0.995486i	$-0.469746\pi$
$109$	216.000	0.189808	0.0949039	+	0.995486i	$0.469746\pi$
$110$	−374.000	−0.324177
$111$	0	0
$112$	112.000	0.0944911
$113$	−502.000	−0.417913	−0.208957	−	0.977925i	$-0.567007\pi$
$113$	−502.000	−0.417913	−0.208957	+	0.977925i	$0.567007\pi$
$114$	0	0
$115$	−3298.00	−2.67426
$116$	−36.0000	−0.0288148
$117$	0	0
$118$	1074.00	0.837879
$119$	728.000	0.560804
$120$	0	0
$121$	121.000	0.0909091
$122$	1900.00	1.40998
$123$	0	0
$124$	−720.000	−0.521435
$125$	663.000	0.474404
$126$	0	0
$127$	1752.00	1.22413	0.612066	−	0.790806i	$-0.290339\pi$
$127$	1752.00	1.22413	0.612066	+	0.790806i	$0.290339\pi$
$128$	−128.000	−0.0883883
$129$	0	0
$130$	714.000	0.481707
$131$	112.000	0.0746984	0.0373492	−	0.999302i	$-0.488109\pi$
$131$	112.000	0.0746984	0.0373492	+	0.999302i	$0.488109\pi$
$132$	0	0
$133$	−1127.00	−0.734762
$134$	166.000	0.107017
$135$	0	0
$136$	−832.000	−0.524584
$137$	1724.00	1.07512	0.537559	−	0.843226i	$-0.319346\pi$
$137$	1724.00	1.07512	0.537559	+	0.843226i	$0.319346\pi$
$138$	0	0
$139$	2516.00	1.53528	0.767641	−	0.640880i	$-0.221430\pi$
$139$	2516.00	1.53528	0.767641	+	0.640880i	$0.221430\pi$
$140$	476.000	0.287352
$141$	0	0
$142$	360.000	0.212750
$143$	−231.000	−0.135085
$144$	0	0
$145$	−153.000	−0.0876273
$146$	−354.000	−0.200666
$147$	0	0
$148$	−1452.00	−0.806444
$149$	251.000	0.138005	0.0690024	−	0.997616i	$-0.478018\pi$
$149$	251.000	0.138005	0.0690024	+	0.997616i	$0.478018\pi$
$150$	0	0
$151$	3040.00	1.63836	0.819178	−	0.573540i	$-0.194430\pi$
$151$	3040.00	1.63836	0.819178	+	0.573540i	$0.194430\pi$
$152$	1288.00	0.687307
$153$	0	0
$154$	−154.000	−0.0805823
$155$	−3060.00	−1.58571
$156$	0	0
$157$	−496.000	−0.252134	−0.126067	−	0.992022i	$-0.540236\pi$
$157$	−496.000	−0.252134	−0.126067	+	0.992022i	$0.540236\pi$
$158$	440.000	0.221548
$159$	0	0
$160$	−544.000	−0.268794
$161$	−1358.00	−0.664754
$162$	0	0
$163$	−1259.00	−0.604985	−0.302492	−	0.953152i	$-0.597819\pi$
$163$	−1259.00	−0.604985	−0.302492	+	0.953152i	$0.597819\pi$
$164$	432.000	0.205692
$165$	0	0
$166$	2224.00	1.03985
$167$	2130.00	0.986972	0.493486	−	0.869754i	$-0.335722\pi$
$167$	2130.00	0.986972	0.493486	+	0.869754i	$0.335722\pi$
$168$	0	0
$169$	−1756.00	−0.799272
$170$	−3536.00	−1.59529
$171$	0	0
$172$	−1544.00	−0.684470
$173$	−988.000	−0.434198	−0.217099	−	0.976150i	$-0.569659\pi$
$173$	−988.000	−0.434198	−0.217099	+	0.976150i	$0.569659\pi$
$174$	0	0
$175$	1148.00	0.495889
$176$	176.000	0.0753778
$177$	0	0
$178$	−788.000	−0.331815
$179$	−4276.00	−1.78549	−0.892746	−	0.450559i	$-0.851225\pi$
$179$	−4276.00	−1.78549	−0.892746	+	0.450559i	$0.851225\pi$
$180$	0	0
$181$	370.000	0.151944	0.0759721	−	0.997110i	$-0.475794\pi$
$181$	370.000	0.151944	0.0759721	+	0.997110i	$0.475794\pi$
$182$	294.000	0.119740
$183$	0	0
$184$	1552.00	0.621820
$185$	−6171.00	−2.45244
$186$	0	0
$187$	1144.00	0.447367
$188$	−1332.00	−0.516734
$189$	0	0
$190$	5474.00	2.09014
$191$	3384.00	1.28198	0.640989	−	0.767550i	$-0.278525\pi$
$191$	3384.00	1.28198	0.640989	+	0.767550i	$0.278525\pi$
$192$	0	0
$193$	−2894.00	−1.07935	−0.539675	−	0.841873i	$-0.681453\pi$
$193$	−2894.00	−1.07935	−0.539675	+	0.841873i	$0.681453\pi$
$194$	−1652.00	−0.611375
$195$	0	0
$196$	196.000	0.0714286
$197$	−2654.00	−0.959846	−0.479923	−	0.877311i	$-0.659335\pi$
$197$	−2654.00	−0.959846	−0.479923	+	0.877311i	$0.659335\pi$
$198$	0	0
$199$	−4114.00	−1.46550	−0.732748	−	0.680500i	$-0.761763\pi$
$199$	−4114.00	−1.46550	−0.732748	+	0.680500i	$0.761763\pi$
$200$	−1312.00	−0.463862
$201$	0	0
$202$	−3376.00	−1.17591
$203$	−63.0000	−0.0217819
$204$	0	0
$205$	1836.00	0.625521
$206$	−2676.00	−0.905076
$207$	0	0
$208$	−336.000	−0.112007
$209$	−1771.00	−0.586137
$210$	0	0
$211$	470.000	0.153347	0.0766733	−	0.997056i	$-0.475570\pi$
$211$	470.000	0.153347	0.0766733	+	0.997056i	$0.475570\pi$
$212$	488.000	0.158094
$213$	0	0
$214$	−1070.00	−0.341793
$215$	−6562.00	−2.08151
$216$	0	0
$217$	−1260.00	−0.394168
$218$	−432.000	−0.134214
$219$	0	0
$220$	748.000	0.229228
$221$	−2184.00	−0.664759
$222$	0	0
$223$	2958.00	0.888262	0.444131	−	0.895962i	$-0.353513\pi$
$223$	2958.00	0.888262	0.444131	+	0.895962i	$0.353513\pi$
$224$	−224.000	−0.0668153
$225$	0	0
$226$	1004.00	0.295509
$227$	−4878.00	−1.42627	−0.713137	−	0.701025i	$-0.752726\pi$
$227$	−4878.00	−1.42627	−0.713137	+	0.701025i	$0.752726\pi$
$228$	0	0
$229$	−6538.00	−1.88665	−0.943326	−	0.331868i	$-0.892321\pi$
$229$	−6538.00	−1.88665	−0.943326	+	0.331868i	$0.892321\pi$
$230$	6596.00	1.89099
$231$	0	0
$232$	72.0000	0.0203751
$233$	3262.00	0.917170	0.458585	−	0.888650i	$-0.348356\pi$
$233$	3262.00	0.917170	0.458585	+	0.888650i	$0.348356\pi$
$234$	0	0
$235$	−5661.00	−1.57142
$236$	−2148.00	−0.592470
$237$	0	0
$238$	−1456.00	−0.396548
$239$	2609.00	0.706118	0.353059	−	0.935601i	$-0.385142\pi$
$239$	2609.00	0.706118	0.353059	+	0.935601i	$0.385142\pi$
$240$	0	0
$241$	6559.00	1.75312	0.876561	−	0.481291i	$-0.159832\pi$
$241$	6559.00	1.75312	0.876561	+	0.481291i	$0.159832\pi$
$242$	−242.000	−0.0642824
$243$	0	0
$244$	−3800.00	−0.997008
$245$	833.000	0.217218
$246$	0	0
$247$	3381.00	0.870963
$248$	1440.00	0.368710
$249$	0	0
$250$	−1326.00	−0.335454
$251$	2705.00	0.680231	0.340116	−	0.940384i	$-0.389534\pi$
$251$	2705.00	0.680231	0.340116	+	0.940384i	$0.389534\pi$
$252$	0	0
$253$	−2134.00	−0.530290
$254$	−3504.00	−0.865593
$255$	0	0
$256$	256.000	0.0625000
$257$	−2995.00	−0.726938	−0.363469	−	0.931606i	$-0.618408\pi$
$257$	−2995.00	−0.726938	−0.363469	+	0.931606i	$0.618408\pi$
$258$	0	0
$259$	−2541.00	−0.609614
$260$	−1428.00	−0.340618
$261$	0	0
$262$	−224.000	−0.0528197
$263$	39.0000	0.00914389	0.00457194	−	0.999990i	$-0.498545\pi$
$263$	39.0000	0.00914389	0.00457194	+	0.999990i	$0.498545\pi$
$264$	0	0
$265$	2074.00	0.480773
$266$	2254.00	0.519555
$267$	0	0
$268$	−332.000	−0.0756721
$269$	−2494.00	−0.565286	−0.282643	−	0.959225i	$-0.591211\pi$
$269$	−2494.00	−0.565286	−0.282643	+	0.959225i	$0.591211\pi$
$270$	0	0
$271$	−1681.00	−0.376803	−0.188401	−	0.982092i	$-0.560331\pi$
$271$	−1681.00	−0.376803	−0.188401	+	0.982092i	$0.560331\pi$
$272$	1664.00	0.370937
$273$	0	0
$274$	−3448.00	−0.760224
$275$	1804.00	0.395583
$276$	0	0
$277$	272.000	0.0589996	0.0294998	−	0.999565i	$-0.490609\pi$
$277$	272.000	0.0589996	0.0294998	+	0.999565i	$0.490609\pi$
$278$	−5032.00	−1.08561
$279$	0	0
$280$	−952.000	−0.203189
$281$	−1813.00	−0.384892	−0.192446	−	0.981308i	$-0.561642\pi$
$281$	−1813.00	−0.384892	−0.192446	+	0.981308i	$0.561642\pi$
$282$	0	0
$283$	−2513.00	−0.527853	−0.263926	−	0.964543i	$-0.585018\pi$
$283$	−2513.00	−0.527853	−0.263926	+	0.964543i	$0.585018\pi$
$284$	−720.000	−0.150437
$285$	0	0
$286$	462.000	0.0955197
$287$	756.000	0.155489
$288$	0	0
$289$	5903.00	1.20151
$290$	306.000	0.0619619
$291$	0	0
$292$	708.000	0.141892
$293$	7648.00	1.52492	0.762459	−	0.647037i	$-0.223992\pi$
$293$	7648.00	1.52492	0.762459	+	0.647037i	$0.223992\pi$
$294$	0	0
$295$	−9129.00	−1.80173
$296$	2904.00	0.570242
$297$	0	0
$298$	−502.000	−0.0975842
$299$	4074.00	0.787978
$300$	0	0
$301$	−2702.00	−0.517411
$302$	−6080.00	−1.15849
$303$	0	0
$304$	−2576.00	−0.485999
$305$	−16150.0	−3.03196
$306$	0	0
$307$	−5444.00	−1.01207	−0.506035	−	0.862513i	$-0.668889\pi$
$307$	−5444.00	−1.01207	−0.506035	+	0.862513i	$0.668889\pi$
$308$	308.000	0.0569803
$309$	0	0
$310$	6120.00	1.12127
$311$	3400.00	0.619924	0.309962	−	0.950749i	$-0.399684\pi$
$311$	3400.00	0.619924	0.309962	+	0.950749i	$0.399684\pi$
$312$	0	0
$313$	−7348.00	−1.32694	−0.663472	−	0.748201i	$-0.730918\pi$
$313$	−7348.00	−1.32694	−0.663472	+	0.748201i	$0.730918\pi$
$314$	992.000	0.178286
$315$	0	0
$316$	−880.000	−0.156658
$317$	486.000	0.0861088	0.0430544	−	0.999073i	$-0.486291\pi$
$317$	486.000	0.0861088	0.0430544	+	0.999073i	$0.486291\pi$
$318$	0	0
$319$	−99.0000	−0.0173760
$320$	1088.00	0.190066
$321$	0	0
$322$	2716.00	0.470052
$323$	−16744.0	−2.88440
$324$	0	0
$325$	−3444.00	−0.587812
$326$	2518.00	0.427789
$327$	0	0
$328$	−864.000	−0.145446
$329$	−2331.00	−0.390615
$330$	0	0
$331$	−4468.00	−0.741944	−0.370972	−	0.928644i	$-0.620975\pi$
$331$	−4468.00	−0.741944	−0.370972	+	0.928644i	$0.620975\pi$
$332$	−4448.00	−0.735288
$333$	0	0
$334$	−4260.00	−0.697895
$335$	−1411.00	−0.230123
$336$	0	0
$337$	450.000	0.0727391	0.0363695	−	0.999338i	$-0.488421\pi$
$337$	450.000	0.0727391	0.0363695	+	0.999338i	$0.488421\pi$
$338$	3512.00	0.565170
$339$	0	0
$340$	7072.00	1.12804
$341$	−1980.00	−0.314437
$342$	0	0
$343$	343.000	0.0539949
$344$	3088.00	0.483994
$345$	0	0
$346$	1976.00	0.307024
$347$	11288.0	1.74632	0.873158	−	0.487437i	$-0.162068\pi$
$347$	11288.0	1.74632	0.873158	+	0.487437i	$0.162068\pi$
$348$	0	0
$349$	−11121.0	−1.70571	−0.852856	−	0.522146i	$-0.825132\pi$
$349$	−11121.0	−1.70571	−0.852856	+	0.522146i	$0.825132\pi$
$350$	−2296.00	−0.350647
$351$	0	0
$352$	−352.000	−0.0533002
$353$	−8853.00	−1.33484	−0.667419	−	0.744683i	$-0.732601\pi$
$353$	−8853.00	−1.33484	−0.667419	+	0.744683i	$0.732601\pi$
$354$	0	0
$355$	−3060.00	−0.457487
$356$	1576.00	0.234629
$357$	0	0
$358$	8552.00	1.26253
$359$	5400.00	0.793875	0.396937	−	0.917846i	$-0.370073\pi$
$359$	5400.00	0.793875	0.396937	+	0.917846i	$0.370073\pi$
$360$	0	0
$361$	19062.0	2.77912
$362$	−740.000	−0.107441
$363$	0	0
$364$	−588.000	−0.0846692
$365$	3009.00	0.431502
$366$	0	0
$367$	8570.00	1.21894	0.609469	−	0.792810i	$-0.291383\pi$
$367$	8570.00	1.21894	0.609469	+	0.792810i	$0.291383\pi$
$368$	−3104.00	−0.439693
$369$	0	0
$370$	12342.0	1.73414
$371$	854.000	0.119508
$372$	0	0
$373$	12244.0	1.69965	0.849826	−	0.527063i	$-0.176707\pi$
$373$	12244.0	1.69965	0.849826	+	0.527063i	$0.176707\pi$
$374$	−2288.00	−0.316336
$375$	0	0
$376$	2664.00	0.365386
$377$	189.000	0.0258196
$378$	0	0
$379$	323.000	0.0437768	0.0218884	−	0.999760i	$-0.493032\pi$
$379$	323.000	0.0437768	0.0218884	+	0.999760i	$0.493032\pi$
$380$	−10948.0	−1.47795
$381$	0	0
$382$	−6768.00	−0.906495
$383$	6920.00	0.923226	0.461613	−	0.887081i	$-0.347271\pi$
$383$	6920.00	0.923226	0.461613	+	0.887081i	$0.347271\pi$
$384$	0	0
$385$	1309.00	0.173280
$386$	5788.00	0.763216
$387$	0	0
$388$	3304.00	0.432307
$389$	5388.00	0.702268	0.351134	−	0.936325i	$-0.385796\pi$
$389$	5388.00	0.702268	0.351134	+	0.936325i	$0.385796\pi$
$390$	0	0
$391$	−20176.0	−2.60958
$392$	−392.000	−0.0505076
$393$	0	0
$394$	5308.00	0.678714
$395$	−3740.00	−0.476405
$396$	0	0
$397$	−12392.0	−1.56659	−0.783296	−	0.621650i	$-0.786463\pi$
$397$	−12392.0	−1.56659	−0.783296	+	0.621650i	$0.786463\pi$
$398$	8228.00	1.03626
$399$	0	0
$400$	2624.00	0.328000
$401$	3826.00	0.476462	0.238231	−	0.971209i	$-0.423432\pi$
$401$	3826.00	0.476462	0.238231	+	0.971209i	$0.423432\pi$
$402$	0	0
$403$	3780.00	0.467234
$404$	6752.00	0.831496
$405$	0	0
$406$	126.000	0.0154022
$407$	−3993.00	−0.486304
$408$	0	0
$409$	−7094.00	−0.857642	−0.428821	−	0.903389i	$-0.641071\pi$
$409$	−7094.00	−0.857642	−0.428821	+	0.903389i	$0.641071\pi$
$410$	−3672.00	−0.442310
$411$	0	0
$412$	5352.00	0.639986
$413$	−3759.00	−0.447865
$414$	0	0
$415$	−18904.0	−2.23605
$416$	672.000	0.0792007
$417$	0	0
$418$	3542.00	0.414461
$419$	−8283.00	−0.965754	−0.482877	−	0.875688i	$-0.660408\pi$
$419$	−8283.00	−0.965754	−0.482877	+	0.875688i	$0.660408\pi$
$420$	0	0
$421$	7043.00	0.815332	0.407666	−	0.913131i	$-0.366343\pi$
$421$	7043.00	0.815332	0.407666	+	0.913131i	$0.366343\pi$
$422$	−940.000	−0.108432
$423$	0	0
$424$	−976.000	−0.111790
$425$	17056.0	1.94668
$426$	0	0
$427$	−6650.00	−0.753668
$428$	2140.00	0.241684
$429$	0	0
$430$	13124.0	1.47185
$431$	16623.0	1.85778	0.928888	−	0.370360i	$-0.120766\pi$
$431$	16623.0	1.85778	0.928888	+	0.370360i	$0.120766\pi$
$432$	0	0
$433$	6896.00	0.765359	0.382680	−	0.923881i	$-0.375001\pi$
$433$	6896.00	0.765359	0.382680	+	0.923881i	$0.375001\pi$
$434$	2520.00	0.278719
$435$	0	0
$436$	864.000	0.0949039
$437$	31234.0	3.41905
$438$	0	0
$439$	−15809.0	−1.71873	−0.859365	−	0.511363i	$-0.829141\pi$
$439$	−15809.0	−1.71873	−0.859365	+	0.511363i	$0.829141\pi$
$440$	−1496.00	−0.162089
$441$	0	0
$442$	4368.00	0.470056
$443$	628.000	0.0673526	0.0336763	−	0.999433i	$-0.489278\pi$
$443$	628.000	0.0673526	0.0336763	+	0.999433i	$0.489278\pi$
$444$	0	0
$445$	6698.00	0.713518
$446$	−5916.00	−0.628096
$447$	0	0
$448$	448.000	0.0472456
$449$	−6616.00	−0.695386	−0.347693	−	0.937608i	$-0.613035\pi$
$449$	−6616.00	−0.695386	−0.347693	+	0.937608i	$0.613035\pi$
$450$	0	0
$451$	1188.00	0.124037
$452$	−2008.00	−0.208957
$453$	0	0
$454$	9756.00	1.00853
$455$	−2499.00	−0.257483
$456$	0	0
$457$	−7238.00	−0.740874	−0.370437	−	0.928858i	$-0.620792\pi$
$457$	−7238.00	−0.740874	−0.370437	+	0.928858i	$0.620792\pi$
$458$	13076.0	1.33406
$459$	0	0
$460$	−13192.0	−1.33713
$461$	7400.00	0.747619	0.373810	−	0.927506i	$-0.378051\pi$
$461$	7400.00	0.747619	0.373810	+	0.927506i	$0.378051\pi$
$462$	0	0
$463$	−18447.0	−1.85163	−0.925815	−	0.377977i	$-0.876620\pi$
$463$	−18447.0	−1.85163	−0.925815	+	0.377977i	$0.876620\pi$
$464$	−144.000	−0.0144074
$465$	0	0
$466$	−6524.00	−0.648537
$467$	−12393.0	−1.22801	−0.614004	−	0.789303i	$-0.710442\pi$
$467$	−12393.0	−1.22801	−0.614004	+	0.789303i	$0.710442\pi$
$468$	0	0
$469$	−581.000	−0.0572027
$470$	11322.0	1.11116
$471$	0	0
$472$	4296.00	0.418939
$473$	−4246.00	−0.412751
$474$	0	0
$475$	−26404.0	−2.55052
$476$	2912.00	0.280402
$477$	0	0
$478$	−5218.00	−0.499301
$479$	−2628.00	−0.250681	−0.125341	−	0.992114i	$-0.540002\pi$
$479$	−2628.00	−0.250681	−0.125341	+	0.992114i	$0.540002\pi$
$480$	0	0
$481$	7623.00	0.722617
$482$	−13118.0	−1.23964
$483$	0	0
$484$	484.000	0.0454545
$485$	14042.0	1.31467
$486$	0	0
$487$	−6712.00	−0.624537	−0.312269	−	0.949994i	$-0.601089\pi$
$487$	−6712.00	−0.624537	−0.312269	+	0.949994i	$0.601089\pi$
$488$	7600.00	0.704991
$489$	0	0
$490$	−1666.00	−0.153596
$491$	5423.00	0.498445	0.249223	−	0.968446i	$-0.419825\pi$
$491$	5423.00	0.498445	0.249223	+	0.968446i	$0.419825\pi$
$492$	0	0
$493$	−936.000	−0.0855077
$494$	−6762.00	−0.615864
$495$	0	0
$496$	−2880.00	−0.260717
$497$	−1260.00	−0.113720
$498$	0	0
$499$	−709.000	−0.0636056	−0.0318028	−	0.999494i	$-0.510125\pi$
$499$	−709.000	−0.0636056	−0.0318028	+	0.999494i	$0.510125\pi$
$500$	2652.00	0.237202
$501$	0	0
$502$	−5410.00	−0.480996
$503$	−3118.00	−0.276391	−0.138196	−	0.990405i	$-0.544130\pi$
$503$	−3118.00	−0.276391	−0.138196	+	0.990405i	$0.544130\pi$
$504$	0	0
$505$	28696.0	2.52862
$506$	4268.00	0.374972
$507$	0	0
$508$	7008.00	0.612066
$509$	−6214.00	−0.541121	−0.270561	−	0.962703i	$-0.587209\pi$
$509$	−6214.00	−0.541121	−0.270561	+	0.962703i	$0.587209\pi$
$510$	0	0
$511$	1239.00	0.107261
$512$	−512.000	−0.0441942
$513$	0	0
$514$	5990.00	0.514023
$515$	22746.0	1.94623
$516$	0	0
$517$	−3663.00	−0.311603
$518$	5082.00	0.431062
$519$	0	0
$520$	2856.00	0.240854
$521$	4377.00	0.368061	0.184031	−	0.982921i	$-0.441085\pi$
$521$	4377.00	0.368061	0.184031	+	0.982921i	$0.441085\pi$
$522$	0	0
$523$	701.000	0.0586092	0.0293046	−	0.999571i	$-0.490671\pi$
$523$	701.000	0.0586092	0.0293046	+	0.999571i	$0.490671\pi$
$524$	448.000	0.0373492
$525$	0	0
$526$	−78.0000	−0.00646571
$527$	−18720.0	−1.54735
$528$	0	0
$529$	25469.0	2.09329
$530$	−4148.00	−0.339958
$531$	0	0
$532$	−4508.00	−0.367381
$533$	−2268.00	−0.184311
$534$	0	0
$535$	9095.00	0.734974
$536$	664.000	0.0535083
$537$	0	0
$538$	4988.00	0.399717
$539$	539.000	0.0430730
$540$	0	0
$541$	−6668.00	−0.529907	−0.264954	−	0.964261i	$-0.585357\pi$
$541$	−6668.00	−0.529907	−0.264954	+	0.964261i	$0.585357\pi$
$542$	3362.00	0.266440
$543$	0	0
$544$	−3328.00	−0.262292
$545$	3672.00	0.288608
$546$	0	0
$547$	10628.0	0.830750	0.415375	−	0.909650i	$-0.363650\pi$
$547$	10628.0	0.830750	0.415375	+	0.909650i	$0.363650\pi$
$548$	6896.00	0.537559
$549$	0	0
$550$	−3608.00	−0.279719
$551$	1449.00	0.112032
$552$	0	0
$553$	−1540.00	−0.118422
$554$	−544.000	−0.0417190
$555$	0	0
$556$	10064.0	0.767641
$557$	1191.00	0.0906002	0.0453001	−	0.998973i	$-0.485576\pi$
$557$	1191.00	0.0906002	0.0453001	+	0.998973i	$0.485576\pi$
$558$	0	0
$559$	8106.00	0.613322
$560$	1904.00	0.143676
$561$	0	0
$562$	3626.00	0.272159
$563$	550.000	0.0411718	0.0205859	−	0.999788i	$-0.493447\pi$
$563$	550.000	0.0411718	0.0205859	+	0.999788i	$0.493447\pi$
$564$	0	0
$565$	−8534.00	−0.635448
$566$	5026.00	0.373248
$567$	0	0
$568$	1440.00	0.106375
$569$	8074.00	0.594868	0.297434	−	0.954742i	$-0.403869\pi$
$569$	8074.00	0.594868	0.297434	+	0.954742i	$0.403869\pi$
$570$	0	0
$571$	1264.00	0.0926388	0.0463194	−	0.998927i	$-0.485251\pi$
$571$	1264.00	0.0926388	0.0463194	+	0.998927i	$0.485251\pi$
$572$	−924.000	−0.0675426
$573$	0	0
$574$	−1512.00	−0.109947
$575$	−31816.0	−2.30751
$576$	0	0
$577$	12764.0	0.920922	0.460461	−	0.887680i	$-0.347684\pi$
$577$	12764.0	0.920922	0.460461	+	0.887680i	$0.347684\pi$
$578$	−11806.0	−0.849593
$579$	0	0
$580$	−612.000	−0.0438136
$581$	−7784.00	−0.555826
$582$	0	0
$583$	1342.00	0.0953344
$584$	−1416.00	−0.100333
$585$	0	0
$586$	−15296.0	−1.07828
$587$	1821.00	0.128042	0.0640211	−	0.997949i	$-0.479608\pi$
$587$	1821.00	0.128042	0.0640211	+	0.997949i	$0.479608\pi$
$588$	0	0
$589$	28980.0	2.02733
$590$	18258.0	1.27402
$591$	0	0
$592$	−5808.00	−0.403222
$593$	16284.0	1.12766	0.563831	−	0.825890i	$-0.309327\pi$
$593$	16284.0	1.12766	0.563831	+	0.825890i	$0.309327\pi$
$594$	0	0
$595$	12376.0	0.852717
$596$	1004.00	0.0690024
$597$	0	0
$598$	−8148.00	−0.557185
$599$	−14508.0	−0.989617	−0.494809	−	0.869002i	$-0.664762\pi$
$599$	−14508.0	−0.989617	−0.494809	+	0.869002i	$0.664762\pi$
$600$	0	0
$601$	−10607.0	−0.719914	−0.359957	−	0.932969i	$-0.617209\pi$
$601$	−10607.0	−0.719914	−0.359957	+	0.932969i	$0.617209\pi$
$602$	5404.00	0.365865
$603$	0	0
$604$	12160.0	0.819178
$605$	2057.00	0.138230
$606$	0	0
$607$	26707.0	1.78584	0.892919	−	0.450217i	$-0.148653\pi$
$607$	26707.0	1.78584	0.892919	+	0.450217i	$0.148653\pi$
$608$	5152.00	0.343653
$609$	0	0
$610$	32300.0	2.14392
$611$	6993.00	0.463022
$612$	0	0
$613$	−10848.0	−0.714758	−0.357379	−	0.933959i	$-0.616330\pi$
$613$	−10848.0	−0.714758	−0.357379	+	0.933959i	$0.616330\pi$
$614$	10888.0	0.715642
$615$	0	0
$616$	−616.000	−0.0402911
$617$	−2974.00	−0.194050	−0.0970249	−	0.995282i	$-0.530933\pi$
$617$	−2974.00	−0.194050	−0.0970249	+	0.995282i	$0.530933\pi$
$618$	0	0
$619$	6776.00	0.439985	0.219992	−	0.975502i	$-0.429397\pi$
$619$	6776.00	0.439985	0.219992	+	0.975502i	$0.429397\pi$
$620$	−12240.0	−0.792855
$621$	0	0
$622$	−6800.00	−0.438352
$623$	2758.00	0.177363
$624$	0	0
$625$	−9229.00	−0.590656
$626$	14696.0	0.938291
$627$	0	0
$628$	−1984.00	−0.126067
$629$	−37752.0	−2.39312
$630$	0	0
$631$	−29600.0	−1.86744	−0.933722	−	0.357998i	$-0.883459\pi$
$631$	−29600.0	−1.86744	−0.933722	+	0.357998i	$0.883459\pi$
$632$	1760.00	0.110774
$633$	0	0
$634$	−972.000	−0.0608881
$635$	29784.0	1.86133
$636$	0	0
$637$	−1029.00	−0.0640039
$638$	198.000	0.0122867
$639$	0	0
$640$	−2176.00	−0.134397
$641$	−7200.00	−0.443655	−0.221828	−	0.975086i	$-0.571202\pi$
$641$	−7200.00	−0.443655	−0.221828	+	0.975086i	$0.571202\pi$
$642$	0	0
$643$	−25246.0	−1.54837	−0.774187	−	0.632956i	$-0.781841\pi$
$643$	−25246.0	−1.54837	−0.774187	+	0.632956i	$0.781841\pi$
$644$	−5432.00	−0.332377
$645$	0	0
$646$	33488.0	2.03958
$647$	28181.0	1.71238	0.856190	−	0.516662i	$-0.172825\pi$
$647$	28181.0	1.71238	0.856190	+	0.516662i	$0.172825\pi$
$648$	0	0
$649$	−5907.00	−0.357273
$650$	6888.00	0.415646
$651$	0	0
$652$	−5036.00	−0.302492
$653$	2664.00	0.159648	0.0798242	−	0.996809i	$-0.474564\pi$
$653$	2664.00	0.159648	0.0798242	+	0.996809i	$0.474564\pi$
$654$	0	0
$655$	1904.00	0.113581
$656$	1728.00	0.102846
$657$	0	0
$658$	4662.00	0.276206
$659$	−2205.00	−0.130341	−0.0651704	−	0.997874i	$-0.520759\pi$
$659$	−2205.00	−0.130341	−0.0651704	+	0.997874i	$0.520759\pi$
$660$	0	0
$661$	−10770.0	−0.633743	−0.316872	−	0.948468i	$-0.602632\pi$
$661$	−10770.0	−0.633743	−0.316872	+	0.948468i	$0.602632\pi$
$662$	8936.00	0.524634
$663$	0	0
$664$	8896.00	0.519927
$665$	−19159.0	−1.11722
$666$	0	0
$667$	1746.00	0.101357
$668$	8520.00	0.493486
$669$	0	0
$670$	2822.00	0.162721
$671$	−10450.0	−0.601219
$672$	0	0
$673$	−274.000	−0.0156938	−0.00784690	−	0.999969i	$-0.502498\pi$
$673$	−274.000	−0.0156938	−0.00784690	+	0.999969i	$0.502498\pi$
$674$	−900.000	−0.0514343
$675$	0	0
$676$	−7024.00	−0.399636
$677$	−3290.00	−0.186772	−0.0933862	−	0.995630i	$-0.529769\pi$
$677$	−3290.00	−0.186772	−0.0933862	+	0.995630i	$0.529769\pi$
$678$	0	0
$679$	5782.00	0.326794
$680$	−14144.0	−0.797644
$681$	0	0
$682$	3960.00	0.222341
$683$	−24780.0	−1.38826	−0.694129	−	0.719851i	$-0.744210\pi$
$683$	−24780.0	−1.38826	−0.694129	+	0.719851i	$0.744210\pi$
$684$	0	0
$685$	29308.0	1.63475
$686$	−686.000	−0.0381802
$687$	0	0
$688$	−6176.00	−0.342235
$689$	−2562.00	−0.141661
$690$	0	0
$691$	4014.00	0.220984	0.110492	−	0.993877i	$-0.464757\pi$
$691$	4014.00	0.220984	0.110492	+	0.993877i	$0.464757\pi$
$692$	−3952.00	−0.217099
$693$	0	0
$694$	−22576.0	−1.23483
$695$	42772.0	2.33444
$696$	0	0
$697$	11232.0	0.610391
$698$	22242.0	1.20612
$699$	0	0
$700$	4592.00	0.247945
$701$	−27378.0	−1.47511	−0.737555	−	0.675287i	$-0.764020\pi$
$701$	−27378.0	−1.47511	−0.737555	+	0.675287i	$0.764020\pi$
$702$	0	0
$703$	58443.0	3.13545
$704$	704.000	0.0376889
$705$	0	0
$706$	17706.0	0.943873
$707$	11816.0	0.628552
$708$	0	0
$709$	−11289.0	−0.597979	−0.298990	−	0.954256i	$-0.596650\pi$
$709$	−11289.0	−0.597979	−0.298990	+	0.954256i	$0.596650\pi$
$710$	6120.00	0.323492
$711$	0	0
$712$	−3152.00	−0.165908
$713$	34920.0	1.83417
$714$	0	0
$715$	−3927.00	−0.205401
$716$	−17104.0	−0.892746
$717$	0	0
$718$	−10800.0	−0.561354
$719$	21573.0	1.11897	0.559483	−	0.828842i	$-0.311000\pi$
$719$	21573.0	1.11897	0.559483	+	0.828842i	$0.311000\pi$
$720$	0	0
$721$	9366.00	0.483784
$722$	−38124.0	−1.96514
$723$	0	0
$724$	1480.00	0.0759721
$725$	−1476.00	−0.0756100
$726$	0	0
$727$	31590.0	1.61157	0.805783	−	0.592211i	$-0.201745\pi$
$727$	31590.0	1.61157	0.805783	+	0.592211i	$0.201745\pi$
$728$	1176.00	0.0598701
$729$	0	0
$730$	−6018.00	−0.305118
$731$	−40144.0	−2.03116
$732$	0	0
$733$	−27814.0	−1.40155	−0.700773	−	0.713384i	$-0.747162\pi$
$733$	−27814.0	−1.40155	−0.700773	+	0.713384i	$0.747162\pi$
$734$	−17140.0	−0.861920
$735$	0	0
$736$	6208.00	0.310910
$737$	−913.000	−0.0456320
$738$	0	0
$739$	−4942.00	−0.246001	−0.123000	−	0.992407i	$-0.539252\pi$
$739$	−4942.00	−0.246001	−0.123000	+	0.992407i	$0.539252\pi$
$740$	−24684.0	−1.22622
$741$	0	0
$742$	−1708.00	−0.0845049
$743$	−17897.0	−0.883684	−0.441842	−	0.897093i	$-0.645675\pi$
$743$	−17897.0	−0.883684	−0.441842	+	0.897093i	$0.645675\pi$
$744$	0	0
$745$	4267.00	0.209840
$746$	−24488.0	−1.20184
$747$	0	0
$748$	4576.00	0.223683
$749$	3745.00	0.182696
$750$	0	0
$751$	−2443.00	−0.118704	−0.0593518	−	0.998237i	$-0.518903\pi$
$751$	−2443.00	−0.118704	−0.0593518	+	0.998237i	$0.518903\pi$
$752$	−5328.00	−0.258367
$753$	0	0
$754$	−378.000	−0.0182572
$755$	51680.0	2.49116
$756$	0	0
$757$	16589.0	0.796483	0.398241	−	0.917281i	$-0.369621\pi$
$757$	16589.0	0.796483	0.398241	+	0.917281i	$0.369621\pi$
$758$	−646.000	−0.0309549
$759$	0	0
$760$	21896.0	1.04507
$761$	−17040.0	−0.811695	−0.405847	−	0.913941i	$-0.633024\pi$
$761$	−17040.0	−0.811695	−0.405847	+	0.913941i	$0.633024\pi$
$762$	0	0
$763$	1512.00	0.0717406
$764$	13536.0	0.640989
$765$	0	0
$766$	−13840.0	−0.652819
$767$	11277.0	0.530885
$768$	0	0
$769$	23527.0	1.10326	0.551629	−	0.834090i	$-0.314006\pi$
$769$	23527.0	1.10326	0.551629	+	0.834090i	$0.314006\pi$
$770$	−2618.00	−0.122527
$771$	0	0
$772$	−11576.0	−0.539675
$773$	−16287.0	−0.757830	−0.378915	−	0.925431i	$-0.623703\pi$
$773$	−16287.0	−0.757830	−0.378915	+	0.925431i	$0.623703\pi$
$774$	0	0
$775$	−29520.0	−1.36824
$776$	−6608.00	−0.305687
$777$	0	0
$778$	−10776.0	−0.496579
$779$	−17388.0	−0.799730
$780$	0	0
$781$	−1980.00	−0.0907170
$782$	40352.0	1.84525
$783$	0	0
$784$	784.000	0.0357143
$785$	−8432.00	−0.383377
$786$	0	0
$787$	31753.0	1.43821	0.719106	−	0.694901i	$-0.244552\pi$
$787$	31753.0	1.43821	0.719106	+	0.694901i	$0.244552\pi$
$788$	−10616.0	−0.479923
$789$	0	0
$790$	7480.00	0.336869
$791$	−3514.00	−0.157956
$792$	0	0
$793$	19950.0	0.893374
$794$	24784.0	1.10775
$795$	0	0
$796$	−16456.0	−0.732748
$797$	12879.0	0.572393	0.286197	−	0.958171i	$-0.407609\pi$
$797$	12879.0	0.572393	0.286197	+	0.958171i	$0.407609\pi$
$798$	0	0
$799$	−34632.0	−1.53341
$800$	−5248.00	−0.231931
$801$	0	0
$802$	−7652.00	−0.336910
$803$	1947.00	0.0855643
$804$	0	0
$805$	−23086.0	−1.01078
$806$	−7560.00	−0.330384
$807$	0	0
$808$	−13504.0	−0.587957
$809$	−7601.00	−0.330330	−0.165165	−	0.986266i	$-0.552816\pi$
$809$	−7601.00	−0.330330	−0.165165	+	0.986266i	$0.552816\pi$
$810$	0	0
$811$	−28363.0	−1.22806	−0.614032	−	0.789281i	$-0.710453\pi$
$811$	−28363.0	−1.22806	−0.614032	+	0.789281i	$0.710453\pi$
$812$	−252.000	−0.0108910
$813$	0	0
$814$	7986.00	0.343869
$815$	−21403.0	−0.919895
$816$	0	0
$817$	62146.0	2.66122
$818$	14188.0	0.606445
$819$	0	0
$820$	7344.00	0.312760
$821$	13369.0	0.568309	0.284154	−	0.958779i	$-0.408287\pi$
$821$	13369.0	0.568309	0.284154	+	0.958779i	$0.408287\pi$
$822$	0	0
$823$	5669.00	0.240108	0.120054	−	0.992767i	$-0.461693\pi$
$823$	5669.00	0.240108	0.120054	+	0.992767i	$0.461693\pi$
$824$	−10704.0	−0.452538
$825$	0	0
$826$	7518.00	0.316688
$827$	20171.0	0.848143	0.424072	−	0.905629i	$-0.360600\pi$
$827$	20171.0	0.848143	0.424072	+	0.905629i	$0.360600\pi$
$828$	0	0
$829$	41152.0	1.72409	0.862043	−	0.506834i	$-0.169184\pi$
$829$	41152.0	1.72409	0.862043	+	0.506834i	$0.169184\pi$
$830$	37808.0	1.58113
$831$	0	0
$832$	−1344.00	−0.0560034
$833$	5096.00	0.211964
$834$	0	0
$835$	36210.0	1.50072
$836$	−7084.00	−0.293068
$837$	0	0
$838$	16566.0	0.682891
$839$	17091.0	0.703274	0.351637	−	0.936136i	$-0.385625\pi$
$839$	17091.0	0.703274	0.351637	+	0.936136i	$0.385625\pi$
$840$	0	0
$841$	−24308.0	−0.996679
$842$	−14086.0	−0.576527
$843$	0	0
$844$	1880.00	0.0766733
$845$	−29852.0	−1.21531
$846$	0	0
$847$	847.000	0.0343604
$848$	1952.00	0.0790471
$849$	0	0
$850$	−34112.0	−1.37651
$851$	70422.0	2.83670
$852$	0	0
$853$	37314.0	1.49778	0.748890	−	0.662694i	$-0.230587\pi$
$853$	37314.0	1.49778	0.748890	+	0.662694i	$0.230587\pi$
$854$	13300.0	0.532923
$855$	0	0
$856$	−4280.00	−0.170896
$857$	14594.0	0.581705	0.290853	−	0.956768i	$-0.406061\pi$
$857$	14594.0	0.581705	0.290853	+	0.956768i	$0.406061\pi$
$858$	0	0
$859$	19024.0	0.755635	0.377818	−	0.925880i	$-0.376675\pi$
$859$	19024.0	0.755635	0.377818	+	0.925880i	$0.376675\pi$
$860$	−26248.0	−1.04076
$861$	0	0
$862$	−33246.0	−1.31365
$863$	2414.00	0.0952184	0.0476092	−	0.998866i	$-0.484840\pi$
$863$	2414.00	0.0952184	0.0476092	+	0.998866i	$0.484840\pi$
$864$	0	0
$865$	−16796.0	−0.660209
$866$	−13792.0	−0.541191
$867$	0	0
$868$	−5040.00	−0.197084
$869$	−2420.00	−0.0944682
$870$	0	0
$871$	1743.00	0.0678063
$872$	−1728.00	−0.0671072
$873$	0	0
$874$	−62468.0	−2.41763
$875$	4641.00	0.179308
$876$	0	0
$877$	9756.00	0.375640	0.187820	−	0.982203i	$-0.439858\pi$
$877$	9756.00	0.375640	0.187820	+	0.982203i	$0.439858\pi$
$878$	31618.0	1.21533
$879$	0	0
$880$	2992.00	0.114614
$881$	−31283.0	−1.19631	−0.598156	−	0.801380i	$-0.704100\pi$
$881$	−31283.0	−1.19631	−0.598156	+	0.801380i	$0.704100\pi$
$882$	0	0
$883$	11479.0	0.437485	0.218742	−	0.975783i	$-0.429805\pi$
$883$	11479.0	0.437485	0.218742	+	0.975783i	$0.429805\pi$
$884$	−8736.00	−0.332379
$885$	0	0
$886$	−1256.00	−0.0476254
$887$	−33672.0	−1.27463	−0.637314	−	0.770604i	$-0.719955\pi$
$887$	−33672.0	−1.27463	−0.637314	+	0.770604i	$0.719955\pi$
$888$	0	0
$889$	12264.0	0.462679
$890$	−13396.0	−0.504534
$891$	0	0
$892$	11832.0	0.444131
$893$	53613.0	2.00906
$894$	0	0
$895$	−72692.0	−2.71489
$896$	−896.000	−0.0334077
$897$	0	0
$898$	13232.0	0.491712
$899$	1620.00	0.0601001
$900$	0	0
$901$	12688.0	0.469144
$902$	−2376.00	−0.0877075
$903$	0	0
$904$	4016.00	0.147755
$905$	6290.00	0.231035
$906$	0	0
$907$	39004.0	1.42790	0.713951	−	0.700196i	$-0.246904\pi$
$907$	39004.0	1.42790	0.713951	+	0.700196i	$0.246904\pi$
$908$	−19512.0	−0.713137
$909$	0	0
$910$	4998.00	0.182068
$911$	5850.00	0.212754	0.106377	−	0.994326i	$-0.466075\pi$
$911$	5850.00	0.212754	0.106377	+	0.994326i	$0.466075\pi$
$912$	0	0
$913$	−12232.0	−0.443396
$914$	14476.0	0.523877
$915$	0	0
$916$	−26152.0	−0.943326
$917$	784.000	0.0282333
$918$	0	0
$919$	−49054.0	−1.76076	−0.880382	−	0.474265i	$-0.842714\pi$
$919$	−49054.0	−1.76076	−0.880382	+	0.474265i	$0.842714\pi$
$920$	26384.0	0.945494
$921$	0	0
$922$	−14800.0	−0.528646
$923$	3780.00	0.134800
$924$	0	0
$925$	−59532.0	−2.11611
$926$	36894.0	1.30930
$927$	0	0
$928$	288.000	0.0101876
$929$	17799.0	0.628597	0.314298	−	0.949324i	$-0.398231\pi$
$929$	17799.0	0.628597	0.314298	+	0.949324i	$0.398231\pi$
$930$	0	0
$931$	−7889.00	−0.277714
$932$	13048.0	0.458585
$933$	0	0
$934$	24786.0	0.868333
$935$	19448.0	0.680233
$936$	0	0
$937$	−12938.0	−0.451084	−0.225542	−	0.974233i	$-0.572415\pi$
$937$	−12938.0	−0.451084	−0.225542	+	0.974233i	$0.572415\pi$
$938$	1162.00	0.0404484
$939$	0	0
$940$	−22644.0	−0.785708
$941$	16200.0	0.561217	0.280608	−	0.959822i	$-0.409464\pi$
$941$	16200.0	0.561217	0.280608	+	0.959822i	$0.409464\pi$
$942$	0	0
$943$	−20952.0	−0.723532
$944$	−8592.00	−0.296235
$945$	0	0
$946$	8492.00	0.291859
$947$	28302.0	0.971163	0.485582	−	0.874191i	$-0.338608\pi$
$947$	28302.0	0.971163	0.485582	+	0.874191i	$0.338608\pi$
$948$	0	0
$949$	−3717.00	−0.127143
$950$	52808.0	1.80349
$951$	0	0
$952$	−5824.00	−0.198274
$953$	17787.0	0.604593	0.302297	−	0.953214i	$-0.402247\pi$
$953$	17787.0	0.604593	0.302297	+	0.953214i	$0.402247\pi$
$954$	0	0
$955$	57528.0	1.94928
$956$	10436.0	0.353059
$957$	0	0
$958$	5256.00	0.177259
$959$	12068.0	0.406357
$960$	0	0
$961$	2609.00	0.0875768
$962$	−15246.0	−0.510968
$963$	0	0
$964$	26236.0	0.876561
$965$	−49198.0	−1.64118
$966$	0	0
$967$	−11990.0	−0.398731	−0.199365	−	0.979925i	$-0.563888\pi$
$967$	−11990.0	−0.398731	−0.199365	+	0.979925i	$0.563888\pi$
$968$	−968.000	−0.0321412
$969$	0	0
$970$	−28084.0	−0.929611
$971$	−44763.0	−1.47942	−0.739708	−	0.672928i	$-0.765036\pi$
$971$	−44763.0	−1.47942	−0.739708	+	0.672928i	$0.765036\pi$
$972$	0	0
$973$	17612.0	0.580282
$974$	13424.0	0.441615
$975$	0	0
$976$	−15200.0	−0.498504
$977$	−49474.0	−1.62008	−0.810038	−	0.586378i	$-0.800553\pi$
$977$	−49474.0	−1.62008	−0.810038	+	0.586378i	$0.800553\pi$
$978$	0	0
$979$	4334.00	0.141486
$980$	3332.00	0.108609
$981$	0	0
$982$	−10846.0	−0.352454
$983$	36276.0	1.17703	0.588517	−	0.808485i	$-0.299712\pi$
$983$	36276.0	1.17703	0.588517	+	0.808485i	$0.299712\pi$
$984$	0	0
$985$	−45118.0	−1.45947
$986$	1872.00	0.0604631
$987$	0	0
$988$	13524.0	0.435482
$989$	74884.0	2.40766
$990$	0	0
$991$	−47035.0	−1.50769	−0.753843	−	0.657055i	$-0.771802\pi$
$991$	−47035.0	−1.50769	−0.753843	+	0.657055i	$0.771802\pi$
$992$	5760.00	0.184355
$993$	0	0
$994$	2520.00	0.0804120
$995$	−69938.0	−2.22833
$996$	0	0
$997$	−20450.0	−0.649607	−0.324803	−	0.945782i	$-0.605298\pi$
$997$	−20450.0	−0.649607	−0.324803	+	0.945782i	$0.605298\pi$
$998$	1418.00	0.0449760
$999$	0	0

Twists

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

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

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