LMFDB - Embedded newform 384.4.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [384,4,Mod(1,384)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(384, 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("384.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 384 = 2^{7} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	384.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:	$22.6567334422$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	384.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+3.00000 q^{3} -8.00000 q^{5} +10.0000 q^{7} +9.00000 q^{9} +O(q^{10})$

$q+3.00000 q^{3} -8.00000 q^{5} +10.0000 q^{7} +9.00000 q^{9} -68.0000 q^{11} +46.0000 q^{13} -24.0000 q^{15} -74.0000 q^{17} -16.0000 q^{19} +30.0000 q^{21} +20.0000 q^{23} -61.0000 q^{25} +27.0000 q^{27} -228.000 q^{29} +162.000 q^{31} -204.000 q^{33} -80.0000 q^{35} -262.000 q^{37} +138.000 q^{39} +30.0000 q^{41} -264.000 q^{43} -72.0000 q^{45} -124.000 q^{47} -243.000 q^{49} -222.000 q^{51} +204.000 q^{53} +544.000 q^{55} -48.0000 q^{57} -340.000 q^{59} -950.000 q^{61} +90.0000 q^{63} -368.000 q^{65} +436.000 q^{67} +60.0000 q^{69} +780.000 q^{71} +518.000 q^{73} -183.000 q^{75} -680.000 q^{77} +1010.00 q^{79} +81.0000 q^{81} -852.000 q^{83} +592.000 q^{85} -684.000 q^{87} -686.000 q^{89} +460.000 q^{91} +486.000 q^{93} +128.000 q^{95} -806.000 q^{97} -612.000 q^{99} +O(q^{100})$

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 $

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	−8.00000	−0.715542	−0.357771	−	0.933809i	$-0.616463\pi$
$5$	−8.00000	−0.715542	−0.357771	+	0.933809i	$0.616463\pi$
$6$	0	0
$7$	10.0000	0.539949	0.269975	−	0.962867i	$-0.412985\pi$
$7$	10.0000	0.539949	0.269975	+	0.962867i	$0.412985\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−68.0000	−1.86389	−0.931944	−	0.362602i	$-0.881889\pi$
$11$	−68.0000	−1.86389	−0.931944	+	0.362602i	$0.881889\pi$
$12$	0	0
$13$	46.0000	0.981393	0.490696	−	0.871331i	$-0.336742\pi$
$13$	46.0000	0.981393	0.490696	+	0.871331i	$0.336742\pi$
$14$	0	0
$15$	−24.0000	−0.413118
$16$	0	0
$17$	−74.0000	−1.05574	−0.527872	−	0.849324i	$-0.677010\pi$
$17$	−74.0000	−1.05574	−0.527872	+	0.849324i	$0.677010\pi$
$18$	0	0
$19$	−16.0000	−0.193192	−0.0965961	−	0.995324i	$-0.530796\pi$
$19$	−16.0000	−0.193192	−0.0965961	+	0.995324i	$0.530796\pi$
$20$	0	0
$21$	30.0000	0.311740
$22$	0	0
$23$	20.0000	0.181317	0.0906584	−	0.995882i	$-0.471103\pi$
$23$	20.0000	0.181317	0.0906584	+	0.995882i	$0.471103\pi$
$24$	0	0
$25$	−61.0000	−0.488000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−228.000	−1.45995	−0.729975	−	0.683474i	$-0.760468\pi$
$29$	−228.000	−1.45995	−0.729975	+	0.683474i	$0.760468\pi$
$30$	0	0
$31$	162.000	0.938583	0.469291	−	0.883043i	$-0.344509\pi$
$31$	162.000	0.938583	0.469291	+	0.883043i	$0.344509\pi$
$32$	0	0
$33$	−204.000	−1.07612
$34$	0	0
$35$	−80.0000	−0.386356
$36$	0	0
$37$	−262.000	−1.16412	−0.582061	−	0.813145i	$-0.697754\pi$
$37$	−262.000	−1.16412	−0.582061	+	0.813145i	$0.697754\pi$
$38$	0	0
$39$	138.000	0.566607
$40$	0	0
$41$	30.0000	0.114273	0.0571367	−	0.998366i	$-0.481803\pi$
$41$	30.0000	0.114273	0.0571367	+	0.998366i	$0.481803\pi$
$42$	0	0
$43$	−264.000	−0.936270	−0.468135	−	0.883657i	$-0.655074\pi$
$43$	−264.000	−0.936270	−0.468135	+	0.883657i	$0.655074\pi$
$44$	0	0
$45$	−72.0000	−0.238514
$46$	0	0
$47$	−124.000	−0.384835	−0.192418	−	0.981313i	$-0.561633\pi$
$47$	−124.000	−0.384835	−0.192418	+	0.981313i	$0.561633\pi$
$48$	0	0
$49$	−243.000	−0.708455
$50$	0	0
$51$	−222.000	−0.609534
$52$	0	0
$53$	204.000	0.528709	0.264354	−	0.964426i	$-0.414841\pi$
$53$	204.000	0.528709	0.264354	+	0.964426i	$0.414841\pi$
$54$	0	0
$55$	544.000	1.33369
$56$	0	0
$57$	−48.0000	−0.111540
$58$	0	0
$59$	−340.000	−0.750241	−0.375121	−	0.926976i	$-0.622399\pi$
$59$	−340.000	−0.750241	−0.375121	+	0.926976i	$0.622399\pi$
$60$	0	0
$61$	−950.000	−1.99402	−0.997008	−	0.0772921i	$-0.975373\pi$
$61$	−950.000	−1.99402	−0.997008	+	0.0772921i	$0.975373\pi$
$62$	0	0
$63$	90.0000	0.179983
$64$	0	0
$65$	−368.000	−0.702227
$66$	0	0
$67$	436.000	0.795013	0.397507	−	0.917599i	$-0.369876\pi$
$67$	436.000	0.795013	0.397507	+	0.917599i	$0.369876\pi$
$68$	0	0
$69$	60.0000	0.104683
$70$	0	0
$71$	780.000	1.30379	0.651894	−	0.758310i	$-0.273975\pi$
$71$	780.000	1.30379	0.651894	+	0.758310i	$0.273975\pi$
$72$	0	0
$73$	518.000	0.830511	0.415256	−	0.909705i	$-0.363692\pi$
$73$	518.000	0.830511	0.415256	+	0.909705i	$0.363692\pi$
$74$	0	0
$75$	−183.000	−0.281747
$76$	0	0
$77$	−680.000	−1.00641
$78$	0	0
$79$	1010.00	1.43840	0.719202	−	0.694801i	$-0.244508\pi$
$79$	1010.00	1.43840	0.719202	+	0.694801i	$0.244508\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−852.000	−1.12674	−0.563368	−	0.826206i	$-0.690495\pi$
$83$	−852.000	−1.12674	−0.563368	+	0.826206i	$0.690495\pi$
$84$	0	0
$85$	592.000	0.755428
$86$	0	0
$87$	−684.000	−0.842902
$88$	0	0
$89$	−686.000	−0.817032	−0.408516	−	0.912751i	$-0.633954\pi$
$89$	−686.000	−0.817032	−0.408516	+	0.912751i	$0.633954\pi$
$90$	0	0
$91$	460.000	0.529902
$92$	0	0
$93$	486.000	0.541891
$94$	0	0
$95$	128.000	0.138237
$96$	0	0
$97$	−806.000	−0.843679	−0.421840	−	0.906670i	$-0.638615\pi$
$97$	−806.000	−0.843679	−0.421840	+	0.906670i	$0.638615\pi$
$98$	0	0
$99$	−612.000	−0.621296
$100$	0	0
$101$	636.000	0.626578	0.313289	−	0.949658i	$-0.398569\pi$
$101$	636.000	0.626578	0.313289	+	0.949658i	$0.398569\pi$
$102$	0	0
$103$	−1586.00	−1.51722	−0.758608	−	0.651547i	$-0.774120\pi$
$103$	−1586.00	−1.51722	−0.758608	+	0.651547i	$0.774120\pi$
$104$	0	0
$105$	−240.000	−0.223063
$106$	0	0
$107$	−92.0000	−0.0831213	−0.0415606	−	0.999136i	$-0.513233\pi$
$107$	−92.0000	−0.0831213	−0.0415606	+	0.999136i	$0.513233\pi$
$108$	0	0
$109$	−2010.00	−1.76627	−0.883133	−	0.469122i	$-0.844570\pi$
$109$	−2010.00	−1.76627	−0.883133	+	0.469122i	$0.844570\pi$
$110$	0	0
$111$	−786.000	−0.672106
$112$	0	0
$113$	−2062.00	−1.71661	−0.858304	−	0.513142i	$-0.828481\pi$
$113$	−2062.00	−1.71661	−0.858304	+	0.513142i	$0.828481\pi$
$114$	0	0
$115$	−160.000	−0.129740
$116$	0	0
$117$	414.000	0.327131
$118$	0	0
$119$	−740.000	−0.570048
$120$	0	0
$121$	3293.00	2.47408
$122$	0	0
$123$	90.0000	0.0659758
$124$	0	0
$125$	1488.00	1.06473
$126$	0	0
$127$	1046.00	0.730846	0.365423	−	0.930841i	$-0.380924\pi$
$127$	1046.00	0.730846	0.365423	+	0.930841i	$0.380924\pi$
$128$	0	0
$129$	−792.000	−0.540556
$130$	0	0
$131$	1884.00	1.25653	0.628267	−	0.777998i	$-0.283765\pi$
$131$	1884.00	1.25653	0.628267	+	0.777998i	$0.283765\pi$
$132$	0	0
$133$	−160.000	−0.104314
$134$	0	0
$135$	−216.000	−0.137706
$136$	0	0
$137$	2374.00	1.48047	0.740235	−	0.672348i	$-0.234714\pi$
$137$	2374.00	1.48047	0.740235	+	0.672348i	$0.234714\pi$
$138$	0	0
$139$	2156.00	1.31561	0.657804	−	0.753189i	$-0.271485\pi$
$139$	2156.00	1.31561	0.657804	+	0.753189i	$0.271485\pi$
$140$	0	0
$141$	−372.000	−0.222185
$142$	0	0
$143$	−3128.00	−1.82921
$144$	0	0
$145$	1824.00	1.04465
$146$	0	0
$147$	−729.000	−0.409027
$148$	0	0
$149$	2952.00	1.62307	0.811534	−	0.584305i	$-0.198633\pi$
$149$	2952.00	1.62307	0.811534	+	0.584305i	$0.198633\pi$
$150$	0	0
$151$	3410.00	1.83776	0.918880	−	0.394536i	$-0.129095\pi$
$151$	3410.00	1.83776	0.918880	+	0.394536i	$0.129095\pi$
$152$	0	0
$153$	−666.000	−0.351914
$154$	0	0
$155$	−1296.00	−0.671595
$156$	0	0
$157$	666.000	0.338552	0.169276	−	0.985569i	$-0.445857\pi$
$157$	666.000	0.338552	0.169276	+	0.985569i	$0.445857\pi$
$158$	0	0
$159$	612.000	0.305250
$160$	0	0
$161$	200.000	0.0979019
$162$	0	0
$163$	704.000	0.338292	0.169146	−	0.985591i	$-0.445899\pi$
$163$	704.000	0.338292	0.169146	+	0.985591i	$0.445899\pi$
$164$	0	0
$165$	1632.00	0.770006
$166$	0	0
$167$	−1648.00	−0.763629	−0.381815	−	0.924239i	$-0.624701\pi$
$167$	−1648.00	−0.763629	−0.381815	+	0.924239i	$0.624701\pi$
$168$	0	0
$169$	−81.0000	−0.0368685
$170$	0	0
$171$	−144.000	−0.0643974
$172$	0	0
$173$	360.000	0.158210	0.0791049	−	0.996866i	$-0.474794\pi$
$173$	360.000	0.158210	0.0791049	+	0.996866i	$0.474794\pi$
$174$	0	0
$175$	−610.000	−0.263495
$176$	0	0
$177$	−1020.00	−0.433152
$178$	0	0
$179$	1956.00	0.816750	0.408375	−	0.912814i	$-0.366096\pi$
$179$	1956.00	0.816750	0.408375	+	0.912814i	$0.366096\pi$
$180$	0	0
$181$	−218.000	−0.0895238	−0.0447619	−	0.998998i	$-0.514253\pi$
$181$	−218.000	−0.0895238	−0.0447619	+	0.998998i	$0.514253\pi$
$182$	0	0
$183$	−2850.00	−1.15125
$184$	0	0
$185$	2096.00	0.832978
$186$	0	0
$187$	5032.00	1.96779
$188$	0	0
$189$	270.000	0.103913
$190$	0	0
$191$	−680.000	−0.257608	−0.128804	−	0.991670i	$-0.541114\pi$
$191$	−680.000	−0.257608	−0.128804	+	0.991670i	$0.541114\pi$
$192$	0	0
$193$	4510.00	1.68206	0.841028	−	0.540991i	$-0.181951\pi$
$193$	4510.00	1.68206	0.841028	+	0.540991i	$0.181951\pi$
$194$	0	0
$195$	−1104.00	−0.405431
$196$	0	0
$197$	−796.000	−0.287881	−0.143941	−	0.989586i	$-0.545977\pi$
$197$	−796.000	−0.287881	−0.143941	+	0.989586i	$0.545977\pi$
$198$	0	0
$199$	−986.000	−0.351235	−0.175617	−	0.984459i	$-0.556192\pi$
$199$	−986.000	−0.351235	−0.175617	+	0.984459i	$0.556192\pi$
$200$	0	0
$201$	1308.00	0.459001
$202$	0	0
$203$	−2280.00	−0.788299
$204$	0	0
$205$	−240.000	−0.0817674
$206$	0	0
$207$	180.000	0.0604390
$208$	0	0
$209$	1088.00	0.360089
$210$	0	0
$211$	4196.00	1.36903	0.684513	−	0.729001i	$-0.260015\pi$
$211$	4196.00	1.36903	0.684513	+	0.729001i	$0.260015\pi$
$212$	0	0
$213$	2340.00	0.752743
$214$	0	0
$215$	2112.00	0.669940
$216$	0	0
$217$	1620.00	0.506787
$218$	0	0
$219$	1554.00	0.479496
$220$	0	0
$221$	−3404.00	−1.03610
$222$	0	0
$223$	2502.00	0.751329	0.375664	−	0.926756i	$-0.377415\pi$
$223$	2502.00	0.751329	0.375664	+	0.926756i	$0.377415\pi$
$224$	0	0
$225$	−549.000	−0.162667
$226$	0	0
$227$	−324.000	−0.0947341	−0.0473670	−	0.998878i	$-0.515083\pi$
$227$	−324.000	−0.0947341	−0.0473670	+	0.998878i	$0.515083\pi$
$228$	0	0
$229$	−2194.00	−0.633116	−0.316558	−	0.948573i	$-0.602527\pi$
$229$	−2194.00	−0.633116	−0.316558	+	0.948573i	$0.602527\pi$
$230$	0	0
$231$	−2040.00	−0.581048
$232$	0	0
$233$	−4230.00	−1.18934	−0.594671	−	0.803969i	$-0.702718\pi$
$233$	−4230.00	−1.18934	−0.594671	+	0.803969i	$0.702718\pi$
$234$	0	0
$235$	992.000	0.275366
$236$	0	0
$237$	3030.00	0.830463
$238$	0	0
$239$	−6672.00	−1.80576	−0.902878	−	0.429896i	$-0.858550\pi$
$239$	−6672.00	−1.80576	−0.902878	+	0.429896i	$0.858550\pi$
$240$	0	0
$241$	−1490.00	−0.398255	−0.199127	−	0.979974i	$-0.563811\pi$
$241$	−1490.00	−0.398255	−0.199127	+	0.979974i	$0.563811\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	1944.00	0.506929
$246$	0	0
$247$	−736.000	−0.189597
$248$	0	0
$249$	−2556.00	−0.650522
$250$	0	0
$251$	−7380.00	−1.85586	−0.927931	−	0.372751i	$-0.878414\pi$
$251$	−7380.00	−1.85586	−0.927931	+	0.372751i	$0.878414\pi$
$252$	0	0
$253$	−1360.00	−0.337954
$254$	0	0
$255$	1776.00	0.436147
$256$	0	0
$257$	−1998.00	−0.484949	−0.242474	−	0.970158i	$-0.577959\pi$
$257$	−1998.00	−0.484949	−0.242474	+	0.970158i	$0.577959\pi$
$258$	0	0
$259$	−2620.00	−0.628567
$260$	0	0
$261$	−2052.00	−0.486650
$262$	0	0
$263$	3768.00	0.883440	0.441720	−	0.897153i	$-0.354368\pi$
$263$	3768.00	0.883440	0.441720	+	0.897153i	$0.354368\pi$
$264$	0	0
$265$	−1632.00	−0.378313
$266$	0	0
$267$	−2058.00	−0.471714
$268$	0	0
$269$	7964.00	1.80511	0.902553	−	0.430578i	$-0.141690\pi$
$269$	7964.00	1.80511	0.902553	+	0.430578i	$0.141690\pi$
$270$	0	0
$271$	−2330.00	−0.522278	−0.261139	−	0.965301i	$-0.584098\pi$
$271$	−2330.00	−0.522278	−0.261139	+	0.965301i	$0.584098\pi$
$272$	0	0
$273$	1380.00	0.305939
$274$	0	0
$275$	4148.00	0.909577
$276$	0	0
$277$	−2154.00	−0.467225	−0.233612	−	0.972330i	$-0.575055\pi$
$277$	−2154.00	−0.467225	−0.233612	+	0.972330i	$0.575055\pi$
$278$	0	0
$279$	1458.00	0.312861
$280$	0	0
$281$	−5598.00	−1.18843	−0.594215	−	0.804306i	$-0.702537\pi$
$281$	−5598.00	−1.18843	−0.594215	+	0.804306i	$0.702537\pi$
$282$	0	0
$283$	−1884.00	−0.395732	−0.197866	−	0.980229i	$-0.563401\pi$
$283$	−1884.00	−0.395732	−0.197866	+	0.980229i	$0.563401\pi$
$284$	0	0
$285$	384.000	0.0798112
$286$	0	0
$287$	300.000	0.0617019
$288$	0	0
$289$	563.000	0.114594
$290$	0	0
$291$	−2418.00	−0.487099
$292$	0	0
$293$	−2788.00	−0.555893	−0.277947	−	0.960597i	$-0.589654\pi$
$293$	−2788.00	−0.555893	−0.277947	+	0.960597i	$0.589654\pi$
$294$	0	0
$295$	2720.00	0.536829
$296$	0	0
$297$	−1836.00	−0.358705
$298$	0	0
$299$	920.000	0.177943
$300$	0	0
$301$	−2640.00	−0.505538
$302$	0	0
$303$	1908.00	0.361755
$304$	0	0
$305$	7600.00	1.42680
$306$	0	0
$307$	−9180.00	−1.70661	−0.853307	−	0.521409i	$-0.825406\pi$
$307$	−9180.00	−1.70661	−0.853307	+	0.521409i	$0.825406\pi$
$308$	0	0
$309$	−4758.00	−0.875965
$310$	0	0
$311$	8032.00	1.46448	0.732239	−	0.681047i	$-0.238475\pi$
$311$	8032.00	1.46448	0.732239	+	0.681047i	$0.238475\pi$
$312$	0	0
$313$	214.000	0.0386454	0.0193227	−	0.999813i	$-0.493849\pi$
$313$	214.000	0.0386454	0.0193227	+	0.999813i	$0.493849\pi$
$314$	0	0
$315$	−720.000	−0.128785
$316$	0	0
$317$	1428.00	0.253011	0.126505	−	0.991966i	$-0.459624\pi$
$317$	1428.00	0.253011	0.126505	+	0.991966i	$0.459624\pi$
$318$	0	0
$319$	15504.0	2.72118
$320$	0	0
$321$	−276.000	−0.0479901
$322$	0	0
$323$	1184.00	0.203961
$324$	0	0
$325$	−2806.00	−0.478920
$326$	0	0
$327$	−6030.00	−1.01975
$328$	0	0
$329$	−1240.00	−0.207792
$330$	0	0
$331$	2708.00	0.449683	0.224842	−	0.974395i	$-0.427814\pi$
$331$	2708.00	0.449683	0.224842	+	0.974395i	$0.427814\pi$
$332$	0	0
$333$	−2358.00	−0.388041
$334$	0	0
$335$	−3488.00	−0.568865
$336$	0	0
$337$	−6338.00	−1.02449	−0.512245	−	0.858840i	$-0.671186\pi$
$337$	−6338.00	−1.02449	−0.512245	+	0.858840i	$0.671186\pi$
$338$	0	0
$339$	−6186.00	−0.991084
$340$	0	0
$341$	−11016.0	−1.74941
$342$	0	0
$343$	−5860.00	−0.922479
$344$	0	0
$345$	−480.000	−0.0749053
$346$	0	0
$347$	−1980.00	−0.306317	−0.153158	−	0.988202i	$-0.548944\pi$
$347$	−1980.00	−0.306317	−0.153158	+	0.988202i	$0.548944\pi$
$348$	0	0
$349$	7418.00	1.13775	0.568877	−	0.822422i	$-0.307378\pi$
$349$	7418.00	1.13775	0.568877	+	0.822422i	$0.307378\pi$
$350$	0	0
$351$	1242.00	0.188869
$352$	0	0
$353$	−3438.00	−0.518375	−0.259187	−	0.965827i	$-0.583455\pi$
$353$	−3438.00	−0.518375	−0.259187	+	0.965827i	$0.583455\pi$
$354$	0	0
$355$	−6240.00	−0.932915
$356$	0	0
$357$	−2220.00	−0.329117
$358$	0	0
$359$	−2372.00	−0.348717	−0.174358	−	0.984682i	$-0.555785\pi$
$359$	−2372.00	−0.348717	−0.174358	+	0.984682i	$0.555785\pi$
$360$	0	0
$361$	−6603.00	−0.962677
$362$	0	0
$363$	9879.00	1.42841
$364$	0	0
$365$	−4144.00	−0.594265
$366$	0	0
$367$	−6046.00	−0.859942	−0.429971	−	0.902843i	$-0.641476\pi$
$367$	−6046.00	−0.859942	−0.429971	+	0.902843i	$0.641476\pi$
$368$	0	0
$369$	270.000	0.0380912
$370$	0	0
$371$	2040.00	0.285476
$372$	0	0
$373$	−9238.00	−1.28237	−0.641187	−	0.767385i	$-0.721558\pi$
$373$	−9238.00	−1.28237	−0.641187	+	0.767385i	$0.721558\pi$
$374$	0	0
$375$	4464.00	0.614720
$376$	0	0
$377$	−10488.0	−1.43278
$378$	0	0
$379$	−2936.00	−0.397921	−0.198961	−	0.980007i	$-0.563757\pi$
$379$	−2936.00	−0.397921	−0.198961	+	0.980007i	$0.563757\pi$
$380$	0	0
$381$	3138.00	0.421954
$382$	0	0
$383$	12600.0	1.68102	0.840509	−	0.541798i	$-0.182256\pi$
$383$	12600.0	1.68102	0.840509	+	0.541798i	$0.182256\pi$
$384$	0	0
$385$	5440.00	0.720125
$386$	0	0
$387$	−2376.00	−0.312090
$388$	0	0
$389$	−3112.00	−0.405616	−0.202808	−	0.979219i	$-0.565007\pi$
$389$	−3112.00	−0.405616	−0.202808	+	0.979219i	$0.565007\pi$
$390$	0	0
$391$	−1480.00	−0.191424
$392$	0	0
$393$	5652.00	0.725460
$394$	0	0
$395$	−8080.00	−1.02924
$396$	0	0
$397$	2130.00	0.269274	0.134637	−	0.990895i	$-0.457013\pi$
$397$	2130.00	0.269274	0.134637	+	0.990895i	$0.457013\pi$
$398$	0	0
$399$	−480.000	−0.0602257
$400$	0	0
$401$	−1554.00	−0.193524	−0.0967619	−	0.995308i	$-0.530849\pi$
$401$	−1554.00	−0.193524	−0.0967619	+	0.995308i	$0.530849\pi$
$402$	0	0
$403$	7452.00	0.921118
$404$	0	0
$405$	−648.000	−0.0795046
$406$	0	0
$407$	17816.0	2.16979
$408$	0	0
$409$	−8942.00	−1.08106	−0.540530	−	0.841325i	$-0.681776\pi$
$409$	−8942.00	−1.08106	−0.540530	+	0.841325i	$0.681776\pi$
$410$	0	0
$411$	7122.00	0.854750
$412$	0	0
$413$	−3400.00	−0.405092
$414$	0	0
$415$	6816.00	0.806227
$416$	0	0
$417$	6468.00	0.759567
$418$	0	0
$419$	6908.00	0.805436	0.402718	−	0.915324i	$-0.368065\pi$
$419$	6908.00	0.805436	0.402718	+	0.915324i	$0.368065\pi$
$420$	0	0
$421$	7862.00	0.910144	0.455072	−	0.890455i	$-0.349614\pi$
$421$	7862.00	0.910144	0.455072	+	0.890455i	$0.349614\pi$
$422$	0	0
$423$	−1116.00	−0.128278
$424$	0	0
$425$	4514.00	0.515203
$426$	0	0
$427$	−9500.00	−1.07667
$428$	0	0
$429$	−9384.00	−1.05609
$430$	0	0
$431$	−12100.0	−1.35229	−0.676144	−	0.736769i	$-0.736350\pi$
$431$	−12100.0	−1.35229	−0.676144	+	0.736769i	$0.736350\pi$
$432$	0	0
$433$	12882.0	1.42972	0.714861	−	0.699267i	$-0.246490\pi$
$433$	12882.0	1.42972	0.714861	+	0.699267i	$0.246490\pi$
$434$	0	0
$435$	5472.00	0.603132
$436$	0	0
$437$	−320.000	−0.0350290
$438$	0	0
$439$	−10818.0	−1.17612	−0.588058	−	0.808819i	$-0.700107\pi$
$439$	−10818.0	−1.17612	−0.588058	+	0.808819i	$0.700107\pi$
$440$	0	0
$441$	−2187.00	−0.236152
$442$	0	0
$443$	852.000	0.0913764	0.0456882	−	0.998956i	$-0.485452\pi$
$443$	852.000	0.0913764	0.0456882	+	0.998956i	$0.485452\pi$
$444$	0	0
$445$	5488.00	0.584621
$446$	0	0
$447$	8856.00	0.937079
$448$	0	0
$449$	5222.00	0.548867	0.274434	−	0.961606i	$-0.411510\pi$
$449$	5222.00	0.548867	0.274434	+	0.961606i	$0.411510\pi$
$450$	0	0
$451$	−2040.00	−0.212993
$452$	0	0
$453$	10230.0	1.06103
$454$	0	0
$455$	−3680.00	−0.379167
$456$	0	0
$457$	−11598.0	−1.18716	−0.593579	−	0.804775i	$-0.702286\pi$
$457$	−11598.0	−1.18716	−0.593579	+	0.804775i	$0.702286\pi$
$458$	0	0
$459$	−1998.00	−0.203178
$460$	0	0
$461$	4088.00	0.413009	0.206504	−	0.978446i	$-0.433791\pi$
$461$	4088.00	0.413009	0.206504	+	0.978446i	$0.433791\pi$
$462$	0	0
$463$	−15394.0	−1.54518	−0.772592	−	0.634903i	$-0.781040\pi$
$463$	−15394.0	−1.54518	−0.772592	+	0.634903i	$0.781040\pi$
$464$	0	0
$465$	−3888.00	−0.387746
$466$	0	0
$467$	−3092.00	−0.306383	−0.153191	−	0.988197i	$-0.548955\pi$
$467$	−3092.00	−0.306383	−0.153191	+	0.988197i	$0.548955\pi$
$468$	0	0
$469$	4360.00	0.429267
$470$	0	0
$471$	1998.00	0.195463
$472$	0	0
$473$	17952.0	1.74510
$474$	0	0
$475$	976.000	0.0942778
$476$	0	0
$477$	1836.00	0.176236
$478$	0	0
$479$	−9612.00	−0.916876	−0.458438	−	0.888726i	$-0.651591\pi$
$479$	−9612.00	−0.916876	−0.458438	+	0.888726i	$0.651591\pi$
$480$	0	0
$481$	−12052.0	−1.14246
$482$	0	0
$483$	600.000	0.0565237
$484$	0	0
$485$	6448.00	0.603688
$486$	0	0
$487$	−13606.0	−1.26601	−0.633005	−	0.774148i	$-0.718179\pi$
$487$	−13606.0	−1.26601	−0.633005	+	0.774148i	$0.718179\pi$
$488$	0	0
$489$	2112.00	0.195313
$490$	0	0
$491$	−2020.00	−0.185665	−0.0928323	−	0.995682i	$-0.529592\pi$
$491$	−2020.00	−0.185665	−0.0928323	+	0.995682i	$0.529592\pi$
$492$	0	0
$493$	16872.0	1.54133
$494$	0	0
$495$	4896.00	0.444563
$496$	0	0
$497$	7800.00	0.703980
$498$	0	0
$499$	−19348.0	−1.73574	−0.867871	−	0.496789i	$-0.834512\pi$
$499$	−19348.0	−1.73574	−0.867871	+	0.496789i	$0.834512\pi$
$500$	0	0
$501$	−4944.00	−0.440881
$502$	0	0
$503$	−596.000	−0.0528317	−0.0264158	−	0.999651i	$-0.508409\pi$
$503$	−596.000	−0.0528317	−0.0264158	+	0.999651i	$0.508409\pi$
$504$	0	0
$505$	−5088.00	−0.448343
$506$	0	0
$507$	−243.000	−0.0212860
$508$	0	0
$509$	−4300.00	−0.374448	−0.187224	−	0.982317i	$-0.559949\pi$
$509$	−4300.00	−0.374448	−0.187224	+	0.982317i	$0.559949\pi$
$510$	0	0
$511$	5180.00	0.448434
$512$	0	0
$513$	−432.000	−0.0371799
$514$	0	0
$515$	12688.0	1.08563
$516$	0	0
$517$	8432.00	0.717290
$518$	0	0
$519$	1080.00	0.0913425
$520$	0	0
$521$	9158.00	0.770095	0.385047	−	0.922897i	$-0.374185\pi$
$521$	9158.00	0.770095	0.385047	+	0.922897i	$0.374185\pi$
$522$	0	0
$523$	10040.0	0.839424	0.419712	−	0.907657i	$-0.362131\pi$
$523$	10040.0	0.839424	0.419712	+	0.907657i	$0.362131\pi$
$524$	0	0
$525$	−1830.00	−0.152129
$526$	0	0
$527$	−11988.0	−0.990902
$528$	0	0
$529$	−11767.0	−0.967124
$530$	0	0
$531$	−3060.00	−0.250080
$532$	0	0
$533$	1380.00	0.112147
$534$	0	0
$535$	736.000	0.0594767
$536$	0	0
$537$	5868.00	0.471551
$538$	0	0
$539$	16524.0	1.32048
$540$	0	0
$541$	−1906.00	−0.151470	−0.0757351	−	0.997128i	$-0.524130\pi$
$541$	−1906.00	−0.151470	−0.0757351	+	0.997128i	$0.524130\pi$
$542$	0	0
$543$	−654.000	−0.0516866
$544$	0	0
$545$	16080.0	1.26384
$546$	0	0
$547$	10264.0	0.802298	0.401149	−	0.916013i	$-0.368611\pi$
$547$	10264.0	0.802298	0.401149	+	0.916013i	$0.368611\pi$
$548$	0	0
$549$	−8550.00	−0.664672
$550$	0	0
$551$	3648.00	0.282051
$552$	0	0
$553$	10100.0	0.776665
$554$	0	0
$555$	6288.00	0.480920
$556$	0	0
$557$	15856.0	1.20618	0.603088	−	0.797674i	$-0.293937\pi$
$557$	15856.0	1.20618	0.603088	+	0.797674i	$0.293937\pi$
$558$	0	0
$559$	−12144.0	−0.918849
$560$	0	0
$561$	15096.0	1.13610
$562$	0	0
$563$	19764.0	1.47949	0.739745	−	0.672887i	$-0.234946\pi$
$563$	19764.0	1.47949	0.739745	+	0.672887i	$0.234946\pi$
$564$	0	0
$565$	16496.0	1.22830
$566$	0	0
$567$	810.000	0.0599944
$568$	0	0
$569$	6302.00	0.464312	0.232156	−	0.972679i	$-0.425422\pi$
$569$	6302.00	0.464312	0.232156	+	0.972679i	$0.425422\pi$
$570$	0	0
$571$	764.000	0.0559937	0.0279969	−	0.999608i	$-0.491087\pi$
$571$	764.000	0.0559937	0.0279969	+	0.999608i	$0.491087\pi$
$572$	0	0
$573$	−2040.00	−0.148730
$574$	0	0
$575$	−1220.00	−0.0884826
$576$	0	0
$577$	−10618.0	−0.766089	−0.383044	−	0.923730i	$-0.625124\pi$
$577$	−10618.0	−0.766089	−0.383044	+	0.923730i	$0.625124\pi$
$578$	0	0
$579$	13530.0	0.971136
$580$	0	0
$581$	−8520.00	−0.608381
$582$	0	0
$583$	−13872.0	−0.985454
$584$	0	0
$585$	−3312.00	−0.234076
$586$	0	0
$587$	−4036.00	−0.283788	−0.141894	−	0.989882i	$-0.545319\pi$
$587$	−4036.00	−0.283788	−0.141894	+	0.989882i	$0.545319\pi$
$588$	0	0
$589$	−2592.00	−0.181327
$590$	0	0
$591$	−2388.00	−0.166208
$592$	0	0
$593$	−24750.0	−1.71393	−0.856965	−	0.515374i	$-0.827653\pi$
$593$	−24750.0	−1.71393	−0.856965	+	0.515374i	$0.827653\pi$
$594$	0	0
$595$	5920.00	0.407893
$596$	0	0
$597$	−2958.00	−0.202785
$598$	0	0
$599$	4908.00	0.334784	0.167392	−	0.985890i	$-0.446465\pi$
$599$	4908.00	0.334784	0.167392	+	0.985890i	$0.446465\pi$
$600$	0	0
$601$	22062.0	1.49738	0.748692	−	0.662918i	$-0.230682\pi$
$601$	22062.0	1.49738	0.748692	+	0.662918i	$0.230682\pi$
$602$	0	0
$603$	3924.00	0.265004
$604$	0	0
$605$	−26344.0	−1.77031
$606$	0	0
$607$	−12986.0	−0.868345	−0.434173	−	0.900830i	$-0.642959\pi$
$607$	−12986.0	−0.868345	−0.434173	+	0.900830i	$0.642959\pi$
$608$	0	0
$609$	−6840.00	−0.455124
$610$	0	0
$611$	−5704.00	−0.377675
$612$	0	0
$613$	24986.0	1.64629	0.823144	−	0.567832i	$-0.192218\pi$
$613$	24986.0	1.64629	0.823144	+	0.567832i	$0.192218\pi$
$614$	0	0
$615$	−720.000	−0.0472085
$616$	0	0
$617$	−3230.00	−0.210753	−0.105377	−	0.994432i	$-0.533605\pi$
$617$	−3230.00	−0.210753	−0.105377	+	0.994432i	$0.533605\pi$
$618$	0	0
$619$	−15500.0	−1.00646	−0.503229	−	0.864153i	$-0.667855\pi$
$619$	−15500.0	−1.00646	−0.503229	+	0.864153i	$0.667855\pi$
$620$	0	0
$621$	540.000	0.0348945
$622$	0	0
$623$	−6860.00	−0.441156
$624$	0	0
$625$	−4279.00	−0.273856
$626$	0	0
$627$	3264.00	0.207897
$628$	0	0
$629$	19388.0	1.22901
$630$	0	0
$631$	16874.0	1.06457	0.532285	−	0.846565i	$-0.321334\pi$
$631$	16874.0	1.06457	0.532285	+	0.846565i	$0.321334\pi$
$632$	0	0
$633$	12588.0	0.790408
$634$	0	0
$635$	−8368.00	−0.522951
$636$	0	0
$637$	−11178.0	−0.695272
$638$	0	0
$639$	7020.00	0.434596
$640$	0	0
$641$	−978.000	−0.0602631	−0.0301316	−	0.999546i	$-0.509593\pi$
$641$	−978.000	−0.0602631	−0.0301316	+	0.999546i	$0.509593\pi$
$642$	0	0
$643$	−1992.00	−0.122172	−0.0610862	−	0.998132i	$-0.519456\pi$
$643$	−1992.00	−0.122172	−0.0610862	+	0.998132i	$0.519456\pi$
$644$	0	0
$645$	6336.00	0.386790
$646$	0	0
$647$	3588.00	0.218020	0.109010	−	0.994041i	$-0.465232\pi$
$647$	3588.00	0.218020	0.109010	+	0.994041i	$0.465232\pi$
$648$	0	0
$649$	23120.0	1.39837
$650$	0	0
$651$	4860.00	0.292594
$652$	0	0
$653$	7536.00	0.451618	0.225809	−	0.974172i	$-0.427497\pi$
$653$	7536.00	0.451618	0.225809	+	0.974172i	$0.427497\pi$
$654$	0	0
$655$	−15072.0	−0.899102
$656$	0	0
$657$	4662.00	0.276837
$658$	0	0
$659$	3684.00	0.217767	0.108883	−	0.994055i	$-0.465272\pi$
$659$	3684.00	0.217767	0.108883	+	0.994055i	$0.465272\pi$
$660$	0	0
$661$	4418.00	0.259970	0.129985	−	0.991516i	$-0.458507\pi$
$661$	4418.00	0.259970	0.129985	+	0.991516i	$0.458507\pi$
$662$	0	0
$663$	−10212.0	−0.598192
$664$	0	0
$665$	1280.00	0.0746410
$666$	0	0
$667$	−4560.00	−0.264714
$668$	0	0
$669$	7506.00	0.433780
$670$	0	0
$671$	64600.0	3.71662
$672$	0	0
$673$	−158.000	−0.00904971	−0.00452485	−	0.999990i	$-0.501440\pi$
$673$	−158.000	−0.00904971	−0.00452485	+	0.999990i	$0.501440\pi$
$674$	0	0
$675$	−1647.00	−0.0939156
$676$	0	0
$677$	−10736.0	−0.609480	−0.304740	−	0.952436i	$-0.598570\pi$
$677$	−10736.0	−0.609480	−0.304740	+	0.952436i	$0.598570\pi$
$678$	0	0
$679$	−8060.00	−0.455544
$680$	0	0
$681$	−972.000	−0.0546947
$682$	0	0
$683$	6132.00	0.343535	0.171768	−	0.985138i	$-0.445052\pi$
$683$	6132.00	0.343535	0.171768	+	0.985138i	$0.445052\pi$
$684$	0	0
$685$	−18992.0	−1.05934
$686$	0	0
$687$	−6582.00	−0.365530
$688$	0	0
$689$	9384.00	0.518871
$690$	0	0
$691$	−28320.0	−1.55911	−0.779554	−	0.626335i	$-0.784554\pi$
$691$	−28320.0	−1.55911	−0.779554	+	0.626335i	$0.784554\pi$
$692$	0	0
$693$	−6120.00	−0.335468
$694$	0	0
$695$	−17248.0	−0.941373
$696$	0	0
$697$	−2220.00	−0.120643
$698$	0	0
$699$	−12690.0	−0.686666
$700$	0	0
$701$	−23420.0	−1.26186	−0.630928	−	0.775841i	$-0.717326\pi$
$701$	−23420.0	−1.26186	−0.630928	+	0.775841i	$0.717326\pi$
$702$	0	0
$703$	4192.00	0.224899
$704$	0	0
$705$	2976.00	0.158982
$706$	0	0
$707$	6360.00	0.338320
$708$	0	0
$709$	−13634.0	−0.722194	−0.361097	−	0.932528i	$-0.617598\pi$
$709$	−13634.0	−0.722194	−0.361097	+	0.932528i	$0.617598\pi$
$710$	0	0
$711$	9090.00	0.479468
$712$	0	0
$713$	3240.00	0.170181
$714$	0	0
$715$	25024.0	1.30887
$716$	0	0
$717$	−20016.0	−1.04255
$718$	0	0
$719$	4044.00	0.209758	0.104879	−	0.994485i	$-0.466555\pi$
$719$	4044.00	0.209758	0.104879	+	0.994485i	$0.466555\pi$
$720$	0	0
$721$	−15860.0	−0.819220
$722$	0	0
$723$	−4470.00	−0.229932
$724$	0	0
$725$	13908.0	0.712455
$726$	0	0
$727$	25842.0	1.31833	0.659166	−	0.751998i	$-0.270910\pi$
$727$	25842.0	1.31833	0.659166	+	0.751998i	$0.270910\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	19536.0	0.988461
$732$	0	0
$733$	33414.0	1.68373	0.841865	−	0.539688i	$-0.181458\pi$
$733$	33414.0	1.68373	0.841865	+	0.539688i	$0.181458\pi$
$734$	0	0
$735$	5832.00	0.292676
$736$	0	0
$737$	−29648.0	−1.48182
$738$	0	0
$739$	−21708.0	−1.08057	−0.540285	−	0.841482i	$-0.681684\pi$
$739$	−21708.0	−1.08057	−0.540285	+	0.841482i	$0.681684\pi$
$740$	0	0
$741$	−2208.00	−0.109464
$742$	0	0
$743$	−12488.0	−0.616609	−0.308304	−	0.951288i	$-0.599762\pi$
$743$	−12488.0	−0.616609	−0.308304	+	0.951288i	$0.599762\pi$
$744$	0	0
$745$	−23616.0	−1.16137
$746$	0	0
$747$	−7668.00	−0.375579
$748$	0	0
$749$	−920.000	−0.0448813
$750$	0	0
$751$	13522.0	0.657024	0.328512	−	0.944500i	$-0.393453\pi$
$751$	13522.0	0.657024	0.328512	+	0.944500i	$0.393453\pi$
$752$	0	0
$753$	−22140.0	−1.07148
$754$	0	0
$755$	−27280.0	−1.31499
$756$	0	0
$757$	−10178.0	−0.488673	−0.244337	−	0.969690i	$-0.578570\pi$
$757$	−10178.0	−0.488673	−0.244337	+	0.969690i	$0.578570\pi$
$758$	0	0
$759$	−4080.00	−0.195118
$760$	0	0
$761$	−482.000	−0.0229599	−0.0114800	−	0.999934i	$-0.503654\pi$
$761$	−482.000	−0.0229599	−0.0114800	+	0.999934i	$0.503654\pi$
$762$	0	0
$763$	−20100.0	−0.953694
$764$	0	0
$765$	5328.00	0.251809
$766$	0	0
$767$	−15640.0	−0.736281
$768$	0	0
$769$	−6706.00	−0.314466	−0.157233	−	0.987562i	$-0.550257\pi$
$769$	−6706.00	−0.314466	−0.157233	+	0.987562i	$0.550257\pi$
$770$	0	0
$771$	−5994.00	−0.279985
$772$	0	0
$773$	40788.0	1.89786	0.948928	−	0.315493i	$-0.102170\pi$
$773$	40788.0	1.89786	0.948928	+	0.315493i	$0.102170\pi$
$774$	0	0
$775$	−9882.00	−0.458028
$776$	0	0
$777$	−7860.00	−0.362903
$778$	0	0
$779$	−480.000	−0.0220767
$780$	0	0
$781$	−53040.0	−2.43012
$782$	0	0
$783$	−6156.00	−0.280967
$784$	0	0
$785$	−5328.00	−0.242248
$786$	0	0
$787$	29720.0	1.34613	0.673065	−	0.739584i	$-0.264978\pi$
$787$	29720.0	1.34613	0.673065	+	0.739584i	$0.264978\pi$
$788$	0	0
$789$	11304.0	0.510055
$790$	0	0
$791$	−20620.0	−0.926881
$792$	0	0
$793$	−43700.0	−1.95691
$794$	0	0
$795$	−4896.00	−0.218419
$796$	0	0
$797$	−20736.0	−0.921589	−0.460795	−	0.887507i	$-0.652436\pi$
$797$	−20736.0	−0.921589	−0.460795	+	0.887507i	$0.652436\pi$
$798$	0	0
$799$	9176.00	0.406287
$800$	0	0
$801$	−6174.00	−0.272344
$802$	0	0
$803$	−35224.0	−1.54798
$804$	0	0
$805$	−1600.00	−0.0700529
$806$	0	0
$807$	23892.0	1.04218
$808$	0	0
$809$	−9834.00	−0.427373	−0.213687	−	0.976902i	$-0.568547\pi$
$809$	−9834.00	−0.427373	−0.213687	+	0.976902i	$0.568547\pi$
$810$	0	0
$811$	−4176.00	−0.180813	−0.0904064	−	0.995905i	$-0.528817\pi$
$811$	−4176.00	−0.180813	−0.0904064	+	0.995905i	$0.528817\pi$
$812$	0	0
$813$	−6990.00	−0.301538
$814$	0	0
$815$	−5632.00	−0.242062
$816$	0	0
$817$	4224.00	0.180880
$818$	0	0
$819$	4140.00	0.176634
$820$	0	0
$821$	−1996.00	−0.0848488	−0.0424244	−	0.999100i	$-0.513508\pi$
$821$	−1996.00	−0.0848488	−0.0424244	+	0.999100i	$0.513508\pi$
$822$	0	0
$823$	14386.0	0.609313	0.304656	−	0.952462i	$-0.401458\pi$
$823$	14386.0	0.609313	0.304656	+	0.952462i	$0.401458\pi$
$824$	0	0
$825$	12444.0	0.525145
$826$	0	0
$827$	−33836.0	−1.42272	−0.711362	−	0.702826i	$-0.751921\pi$
$827$	−33836.0	−1.42272	−0.711362	+	0.702826i	$0.751921\pi$
$828$	0	0
$829$	16358.0	0.685328	0.342664	−	0.939458i	$-0.388671\pi$
$829$	16358.0	0.685328	0.342664	+	0.939458i	$0.388671\pi$
$830$	0	0
$831$	−6462.00	−0.269752
$832$	0	0
$833$	17982.0	0.747946
$834$	0	0
$835$	13184.0	0.546409
$836$	0	0
$837$	4374.00	0.180630
$838$	0	0
$839$	26244.0	1.07991	0.539954	−	0.841694i	$-0.318441\pi$
$839$	26244.0	1.07991	0.539954	+	0.841694i	$0.318441\pi$
$840$	0	0
$841$	27595.0	1.13145
$842$	0	0
$843$	−16794.0	−0.686140
$844$	0	0
$845$	648.000	0.0263809
$846$	0	0
$847$	32930.0	1.33588
$848$	0	0
$849$	−5652.00	−0.228476
$850$	0	0
$851$	−5240.00	−0.211075
$852$	0	0
$853$	−39854.0	−1.59974	−0.799868	−	0.600176i	$-0.795097\pi$
$853$	−39854.0	−1.59974	−0.799868	+	0.600176i	$0.795097\pi$
$854$	0	0
$855$	1152.00	0.0460790
$856$	0	0
$857$	−2706.00	−0.107859	−0.0539295	−	0.998545i	$-0.517175\pi$
$857$	−2706.00	−0.107859	−0.0539295	+	0.998545i	$0.517175\pi$
$858$	0	0
$859$	17728.0	0.704158	0.352079	−	0.935970i	$-0.385475\pi$
$859$	17728.0	0.704158	0.352079	+	0.935970i	$0.385475\pi$
$860$	0	0
$861$	900.000	0.0356236
$862$	0	0
$863$	−20056.0	−0.791094	−0.395547	−	0.918446i	$-0.629445\pi$
$863$	−20056.0	−0.791094	−0.395547	+	0.918446i	$0.629445\pi$
$864$	0	0
$865$	−2880.00	−0.113206
$866$	0	0
$867$	1689.00	0.0661608
$868$	0	0
$869$	−68680.0	−2.68102
$870$	0	0
$871$	20056.0	0.780220
$872$	0	0
$873$	−7254.00	−0.281226
$874$	0	0
$875$	14880.0	0.574898
$876$	0	0
$877$	−26534.0	−1.02165	−0.510826	−	0.859684i	$-0.670661\pi$
$877$	−26534.0	−1.02165	−0.510826	+	0.859684i	$0.670661\pi$
$878$	0	0
$879$	−8364.00	−0.320945
$880$	0	0
$881$	−45838.0	−1.75292	−0.876459	−	0.481476i	$-0.840101\pi$
$881$	−45838.0	−1.75292	−0.876459	+	0.481476i	$0.840101\pi$
$882$	0	0
$883$	23200.0	0.884193	0.442096	−	0.896968i	$-0.354235\pi$
$883$	23200.0	0.884193	0.442096	+	0.896968i	$0.354235\pi$
$884$	0	0
$885$	8160.00	0.309938
$886$	0	0
$887$	25272.0	0.956652	0.478326	−	0.878182i	$-0.341244\pi$
$887$	25272.0	0.956652	0.478326	+	0.878182i	$0.341244\pi$
$888$	0	0
$889$	10460.0	0.394620
$890$	0	0
$891$	−5508.00	−0.207099
$892$	0	0
$893$	1984.00	0.0743472
$894$	0	0
$895$	−15648.0	−0.584419
$896$	0	0
$897$	2760.00	0.102735
$898$	0	0
$899$	−36936.0	−1.37028
$900$	0	0
$901$	−15096.0	−0.558181
$902$	0	0
$903$	−7920.00	−0.291873
$904$	0	0
$905$	1744.00	0.0640580
$906$	0	0
$907$	−42448.0	−1.55398	−0.776992	−	0.629511i	$-0.783255\pi$
$907$	−42448.0	−1.55398	−0.776992	+	0.629511i	$0.783255\pi$
$908$	0	0
$909$	5724.00	0.208859
$910$	0	0
$911$	49776.0	1.81027	0.905133	−	0.425128i	$-0.139771\pi$
$911$	49776.0	1.81027	0.905133	+	0.425128i	$0.139771\pi$
$912$	0	0
$913$	57936.0	2.10011
$914$	0	0
$915$	22800.0	0.823765
$916$	0	0
$917$	18840.0	0.678464
$918$	0	0
$919$	3042.00	0.109191	0.0545954	−	0.998509i	$-0.482613\pi$
$919$	3042.00	0.109191	0.0545954	+	0.998509i	$0.482613\pi$
$920$	0	0
$921$	−27540.0	−0.985314
$922$	0	0
$923$	35880.0	1.27953
$924$	0	0
$925$	15982.0	0.568092
$926$	0	0
$927$	−14274.0	−0.505739
$928$	0	0
$929$	−11490.0	−0.405785	−0.202893	−	0.979201i	$-0.565034\pi$
$929$	−11490.0	−0.405785	−0.202893	+	0.979201i	$0.565034\pi$
$930$	0	0
$931$	3888.00	0.136868
$932$	0	0
$933$	24096.0	0.845517
$934$	0	0
$935$	−40256.0	−1.40803
$936$	0	0
$937$	19882.0	0.693187	0.346594	−	0.938015i	$-0.387338\pi$
$937$	19882.0	0.693187	0.346594	+	0.938015i	$0.387338\pi$
$938$	0	0
$939$	642.000	0.0223119
$940$	0	0
$941$	41336.0	1.43200	0.716002	−	0.698099i	$-0.245970\pi$
$941$	41336.0	1.43200	0.716002	+	0.698099i	$0.245970\pi$
$942$	0	0
$943$	600.000	0.0207197
$944$	0	0
$945$	−2160.00	−0.0743543
$946$	0	0
$947$	−35036.0	−1.20224	−0.601118	−	0.799160i	$-0.705278\pi$
$947$	−35036.0	−1.20224	−0.601118	+	0.799160i	$0.705278\pi$
$948$	0	0
$949$	23828.0	0.815058
$950$	0	0
$951$	4284.00	0.146076
$952$	0	0
$953$	9814.00	0.333585	0.166793	−	0.985992i	$-0.446659\pi$
$953$	9814.00	0.333585	0.166793	+	0.985992i	$0.446659\pi$
$954$	0	0
$955$	5440.00	0.184329
$956$	0	0
$957$	46512.0	1.57108
$958$	0	0
$959$	23740.0	0.799379
$960$	0	0
$961$	−3547.00	−0.119063
$962$	0	0
$963$	−828.000	−0.0277071
$964$	0	0
$965$	−36080.0	−1.20358
$966$	0	0
$967$	26006.0	0.864836	0.432418	−	0.901673i	$-0.357660\pi$
$967$	26006.0	0.864836	0.432418	+	0.901673i	$0.357660\pi$
$968$	0	0
$969$	3552.00	0.117757
$970$	0	0
$971$	−17404.0	−0.575202	−0.287601	−	0.957750i	$-0.592858\pi$
$971$	−17404.0	−0.575202	−0.287601	+	0.957750i	$0.592858\pi$
$972$	0	0
$973$	21560.0	0.710362
$974$	0	0
$975$	−8418.00	−0.276504
$976$	0	0
$977$	44526.0	1.45805	0.729024	−	0.684488i	$-0.239974\pi$
$977$	44526.0	1.45805	0.729024	+	0.684488i	$0.239974\pi$
$978$	0	0
$979$	46648.0	1.52286
$980$	0	0
$981$	−18090.0	−0.588756
$982$	0	0
$983$	−47480.0	−1.54057	−0.770283	−	0.637702i	$-0.779885\pi$
$983$	−47480.0	−1.54057	−0.770283	+	0.637702i	$0.779885\pi$
$984$	0	0
$985$	6368.00	0.205991
$986$	0	0
$987$	−3720.00	−0.119968
$988$	0	0
$989$	−5280.00	−0.169762
$990$	0	0
$991$	18866.0	0.604741	0.302370	−	0.953190i	$-0.402222\pi$
$991$	18866.0	0.604741	0.302370	+	0.953190i	$0.402222\pi$
$992$	0	0
$993$	8124.00	0.259625
$994$	0	0
$995$	7888.00	0.251323
$996$	0	0
$997$	−17550.0	−0.557487	−0.278743	−	0.960366i	$-0.589918\pi$
$997$	−17550.0	−0.557487	−0.278743	+	0.960366i	$0.589918\pi$
$998$	0	0
$999$	−7074.00	−0.224035

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	384.4.a.e.1.1	yes	1
3.2	odd	2		1152.4.a.l.1.1		1
4.3	odd	2		384.4.a.a.1.1	✓	1
8.3	odd	2		384.4.a.h.1.1	yes	1
8.5	even	2		384.4.a.d.1.1	yes	1
12.11	even	2		1152.4.a.k.1.1		1
16.3	odd	4		768.4.d.l.385.1		2
16.5	even	4		768.4.d.e.385.1		2
16.11	odd	4		768.4.d.l.385.2		2
16.13	even	4		768.4.d.e.385.2		2
24.5	odd	2		1152.4.a.b.1.1		1
24.11	even	2		1152.4.a.a.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.a.a.1.1	✓	1	4.3	odd	2
384.4.a.d.1.1	yes	1	8.5	even	2
384.4.a.e.1.1	yes	1	1.1	even	1	trivial
384.4.a.h.1.1	yes	1	8.3	odd	2
768.4.d.e.385.1		2	16.5	even	4
768.4.d.e.385.2		2	16.13	even	4
768.4.d.l.385.1		2	16.3	odd	4
768.4.d.l.385.2		2	16.11	odd	4
1152.4.a.a.1.1		1	24.11	even	2
1152.4.a.b.1.1		1	24.5	odd	2
1152.4.a.k.1.1		1	12.11	even	2
1152.4.a.l.1.1		1	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 384 = 2^{7} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	384.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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