LMFDB - Embedded newform 48.4.a.c.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [48,4,Mod(1,48)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(48, 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("48.1");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$N$	$=$	$48 = 2^{4} \cdot 3$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	48.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:	$2.83209168028$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 6)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	48.1

$q$ -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} +6.00000 q^{5} +16.0000 q^{7} +9.00000 q^{9} -12.0000 q^{11} +38.0000 q^{13} +18.0000 q^{15} -126.000 q^{17} -20.0000 q^{19} +48.0000 q^{21} -168.000 q^{23} -89.0000 q^{25} +27.0000 q^{27} +30.0000 q^{29} +88.0000 q^{31} -36.0000 q^{33} +96.0000 q^{35} +254.000 q^{37} +114.000 q^{39} +42.0000 q^{41} +52.0000 q^{43} +54.0000 q^{45} +96.0000 q^{47} -87.0000 q^{49} -378.000 q^{51} +198.000 q^{53} -72.0000 q^{55} -60.0000 q^{57} +660.000 q^{59} -538.000 q^{61} +144.000 q^{63} +228.000 q^{65} -884.000 q^{67} -504.000 q^{69} -792.000 q^{71} +218.000 q^{73} -267.000 q^{75} -192.000 q^{77} +520.000 q^{79} +81.0000 q^{81} +492.000 q^{83} -756.000 q^{85} +90.0000 q^{87} +810.000 q^{89} +608.000 q^{91} +264.000 q^{93} -120.000 q^{95} +1154.00 q^{97} -108.000 q^{99} +O(q^{100})

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$	0	0
$2$	0	0
$3$	3.00000	0.577350
$3$	3.00000	0.577350
$4$	0	0
$5$	6.00000	0.536656	0.268328	−	0.963328i	$-0.413529\pi$
$5$	6.00000	0.536656	0.268328	+	0.963328i	$0.413529\pi$
$6$	0	0
$7$	16.0000	0.863919	0.431959	−	0.901893i	$-0.357822\pi$
$7$	16.0000	0.863919	0.431959	+	0.901893i	$0.357822\pi$
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	−12.0000	−0.328921	−0.164461	−	0.986384i	$-0.552588\pi$
$11$	−12.0000	−0.328921	−0.164461	+	0.986384i	$0.552588\pi$
$12$	0	0
$13$	38.0000	0.810716	0.405358	−	0.914158i	$-0.367147\pi$
$13$	38.0000	0.810716	0.405358	+	0.914158i	$0.367147\pi$
$14$	0	0
$15$	18.0000	0.309839
$16$	0	0
$17$	−126.000	−1.79762	−0.898808	−	0.438342i	$-0.855566\pi$
$17$	−126.000	−1.79762	−0.898808	+	0.438342i	$0.855566\pi$
$18$	0	0
$19$	−20.0000	−0.241490	−0.120745	−	0.992684i	$-0.538528\pi$
$19$	−20.0000	−0.241490	−0.120745	+	0.992684i	$0.538528\pi$
$20$	0	0
$21$	48.0000	0.498784
$22$	0	0
$23$	−168.000	−1.52306	−0.761531	−	0.648129i	$-0.775552\pi$
$23$	−168.000	−1.52306	−0.761531	+	0.648129i	$0.775552\pi$
$24$	0	0
$25$	−89.0000	−0.712000
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	30.0000	0.192099	0.0960493	−	0.995377i	$-0.469379\pi$
$29$	30.0000	0.192099	0.0960493	+	0.995377i	$0.469379\pi$
$30$	0	0
$31$	88.0000	0.509847	0.254924	−	0.966961i	$-0.417950\pi$
$31$	88.0000	0.509847	0.254924	+	0.966961i	$0.417950\pi$
$32$	0	0
$33$	−36.0000	−0.189903
$34$	0	0
$35$	96.0000	0.463627
$36$	0	0
$37$	254.000	1.12858	0.564288	−	0.825578i	$-0.309151\pi$
$37$	254.000	1.12858	0.564288	+	0.825578i	$0.309151\pi$
$38$	0	0
$39$	114.000	0.468067
$40$	0	0
$41$	42.0000	0.159983	0.0799914	−	0.996796i	$-0.474511\pi$
$41$	42.0000	0.159983	0.0799914	+	0.996796i	$0.474511\pi$
$42$	0	0
$43$	52.0000	0.184417	0.0922084	−	0.995740i	$-0.470607\pi$
$43$	52.0000	0.184417	0.0922084	+	0.995740i	$0.470607\pi$
$44$	0	0
$45$	54.0000	0.178885
$46$	0	0
$47$	96.0000	0.297937	0.148969	−	0.988842i	$-0.452405\pi$
$47$	96.0000	0.297937	0.148969	+	0.988842i	$0.452405\pi$
$48$	0	0
$49$	−87.0000	−0.253644
$50$	0	0
$51$	−378.000	−1.03785
$52$	0	0
$53$	198.000	0.513158	0.256579	−	0.966523i	$-0.417405\pi$
$53$	198.000	0.513158	0.256579	+	0.966523i	$0.417405\pi$
$54$	0	0
$55$	−72.0000	−0.176518
$56$	0	0
$57$	−60.0000	−0.139424
$58$	0	0
$59$	660.000	1.45635	0.728175	−	0.685391i	$-0.240369\pi$
$59$	660.000	1.45635	0.728175	+	0.685391i	$0.240369\pi$
$60$	0	0
$61$	−538.000	−1.12924	−0.564622	−	0.825350i	$-0.690978\pi$
$61$	−538.000	−1.12924	−0.564622	+	0.825350i	$0.690978\pi$
$62$	0	0
$63$	144.000	0.287973
$64$	0	0
$65$	228.000	0.435076
$66$	0	0
$67$	−884.000	−1.61191	−0.805954	−	0.591979i	$-0.798347\pi$
$67$	−884.000	−1.61191	−0.805954	+	0.591979i	$0.798347\pi$
$68$	0	0
$69$	−504.000	−0.879340
$70$	0	0
$71$	−792.000	−1.32385	−0.661923	−	0.749572i	$-0.730260\pi$
$71$	−792.000	−1.32385	−0.661923	+	0.749572i	$0.730260\pi$
$72$	0	0
$73$	218.000	0.349520	0.174760	−	0.984611i	$-0.444085\pi$
$73$	218.000	0.349520	0.174760	+	0.984611i	$0.444085\pi$
$74$	0	0
$75$	−267.000	−0.411073
$76$	0	0
$77$	−192.000	−0.284161
$78$	0	0
$79$	520.000	0.740564	0.370282	−	0.928919i	$-0.379261\pi$
$79$	520.000	0.740564	0.370282	+	0.928919i	$0.379261\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	492.000	0.650651	0.325325	−	0.945602i	$-0.394526\pi$
$83$	492.000	0.650651	0.325325	+	0.945602i	$0.394526\pi$
$84$	0	0
$85$	−756.000	−0.964703
$86$	0	0
$87$	90.0000	0.110908
$88$	0	0
$89$	810.000	0.964717	0.482359	−	0.875974i	$-0.339780\pi$
$89$	810.000	0.964717	0.482359	+	0.875974i	$0.339780\pi$
$90$	0	0
$91$	608.000	0.700393
$92$	0	0
$93$	264.000	0.294360
$94$	0	0
$95$	−120.000	−0.129597
$96$	0	0
$97$	1154.00	1.20795	0.603974	−	0.797004i	$-0.293583\pi$
$97$	1154.00	1.20795	0.603974	+	0.797004i	$0.293583\pi$
$98$	0	0
$99$	−108.000	−0.109640
$100$	0	0
$101$	−618.000	−0.608845	−0.304422	−	0.952537i	$-0.598463\pi$
$101$	−618.000	−0.608845	−0.304422	+	0.952537i	$0.598463\pi$
$102$	0	0
$103$	−128.000	−0.122449	−0.0612243	−	0.998124i	$-0.519501\pi$
$103$	−128.000	−0.122449	−0.0612243	+	0.998124i	$0.519501\pi$
$104$	0	0
$105$	288.000	0.267675
$106$	0	0
$107$	1476.00	1.33355	0.666777	−	0.745257i	$-0.267673\pi$
$107$	1476.00	1.33355	0.666777	+	0.745257i	$0.267673\pi$
$108$	0	0
$109$	1190.00	1.04570	0.522850	−	0.852425i	$-0.324869\pi$
$109$	1190.00	1.04570	0.522850	+	0.852425i	$0.324869\pi$
$110$	0	0
$111$	762.000	0.651584
$112$	0	0
$113$	−462.000	−0.384613	−0.192307	−	0.981335i	$-0.561597\pi$
$113$	−462.000	−0.384613	−0.192307	+	0.981335i	$0.561597\pi$
$114$	0	0
$115$	−1008.00	−0.817361
$116$	0	0
$117$	342.000	0.270239
$118$	0	0
$119$	−2016.00	−1.55300
$120$	0	0
$121$	−1187.00	−0.891811
$122$	0	0
$123$	126.000	0.0923662
$124$	0	0
$125$	−1284.00	−0.918756
$126$	0	0
$127$	2536.00	1.77192	0.885959	−	0.463763i	$-0.153501\pi$
$127$	2536.00	1.77192	0.885959	+	0.463763i	$0.153501\pi$
$128$	0	0
$129$	156.000	0.106473
$130$	0	0
$131$	−2292.00	−1.52865	−0.764324	−	0.644832i	$-0.776927\pi$
$131$	−2292.00	−1.52865	−0.764324	+	0.644832i	$0.776927\pi$
$132$	0	0
$133$	−320.000	−0.208628
$134$	0	0
$135$	162.000	0.103280
$136$	0	0
$137$	−726.000	−0.452747	−0.226374	−	0.974041i	$-0.572687\pi$
$137$	−726.000	−0.452747	−0.226374	+	0.974041i	$0.572687\pi$
$138$	0	0
$139$	−380.000	−0.231879	−0.115939	−	0.993256i	$-0.536988\pi$
$139$	−380.000	−0.231879	−0.115939	+	0.993256i	$0.536988\pi$
$140$	0	0
$141$	288.000	0.172014
$142$	0	0
$143$	−456.000	−0.266662
$144$	0	0
$145$	180.000	0.103091
$146$	0	0
$147$	−261.000	−0.146442
$148$	0	0
$149$	1590.00	0.874214	0.437107	−	0.899410i	$-0.356003\pi$
$149$	1590.00	0.874214	0.437107	+	0.899410i	$0.356003\pi$
$150$	0	0
$151$	−2432.00	−1.31068	−0.655342	−	0.755332i	$-0.727476\pi$
$151$	−2432.00	−1.31068	−0.655342	+	0.755332i	$0.727476\pi$
$152$	0	0
$153$	−1134.00	−0.599206
$154$	0	0
$155$	528.000	0.273613
$156$	0	0
$157$	614.000	0.312118	0.156059	−	0.987748i	$-0.450121\pi$
$157$	614.000	0.312118	0.156059	+	0.987748i	$0.450121\pi$
$158$	0	0
$159$	594.000	0.296272
$160$	0	0
$161$	−2688.00	−1.31580
$162$	0	0
$163$	1852.00	0.889938	0.444969	−	0.895546i	$-0.353215\pi$
$163$	1852.00	0.889938	0.444969	+	0.895546i	$0.353215\pi$
$164$	0	0
$165$	−216.000	−0.101913
$166$	0	0
$167$	2136.00	0.989752	0.494876	−	0.868964i	$-0.335213\pi$
$167$	2136.00	0.989752	0.494876	+	0.868964i	$0.335213\pi$
$168$	0	0
$169$	−753.000	−0.342740
$170$	0	0
$171$	−180.000	−0.0804967
$172$	0	0
$173$	1758.00	0.772591	0.386296	−	0.922375i	$-0.373754\pi$
$173$	1758.00	0.772591	0.386296	+	0.922375i	$0.373754\pi$
$174$	0	0
$175$	−1424.00	−0.615110
$176$	0	0
$177$	1980.00	0.840824
$178$	0	0
$179$	540.000	0.225483	0.112742	−	0.993624i	$-0.464037\pi$
$179$	540.000	0.225483	0.112742	+	0.993624i	$0.464037\pi$
$180$	0	0
$181$	1982.00	0.813928	0.406964	−	0.913444i	$-0.366588\pi$
$181$	1982.00	0.813928	0.406964	+	0.913444i	$0.366588\pi$
$182$	0	0
$183$	−1614.00	−0.651969
$184$	0	0
$185$	1524.00	0.605658
$186$	0	0
$187$	1512.00	0.591275
$188$	0	0
$189$	432.000	0.166261
$190$	0	0
$191$	2688.00	1.01831	0.509154	−	0.860675i	$-0.329958\pi$
$191$	2688.00	1.01831	0.509154	+	0.860675i	$0.329958\pi$
$192$	0	0
$193$	−2302.00	−0.858557	−0.429279	−	0.903172i	$-0.641232\pi$
$193$	−2302.00	−0.858557	−0.429279	+	0.903172i	$0.641232\pi$
$194$	0	0
$195$	684.000	0.251191
$196$	0	0
$197$	4374.00	1.58190	0.790951	−	0.611880i	$-0.209586\pi$
$197$	4374.00	1.58190	0.790951	+	0.611880i	$0.209586\pi$
$198$	0	0
$199$	1600.00	0.569955	0.284977	−	0.958534i	$-0.408014\pi$
$199$	1600.00	0.569955	0.284977	+	0.958534i	$0.408014\pi$
$200$	0	0
$201$	−2652.00	−0.930635
$202$	0	0
$203$	480.000	0.165958
$204$	0	0
$205$	252.000	0.0858558
$206$	0	0
$207$	−1512.00	−0.507687
$208$	0	0
$209$	240.000	0.0794313
$210$	0	0
$211$	−3332.00	−1.08713	−0.543565	−	0.839367i	$-0.682926\pi$
$211$	−3332.00	−1.08713	−0.543565	+	0.839367i	$0.682926\pi$
$212$	0	0
$213$	−2376.00	−0.764323
$214$	0	0
$215$	312.000	0.0989685
$216$	0	0
$217$	1408.00	0.440467
$218$	0	0
$219$	654.000	0.201796
$220$	0	0
$221$	−4788.00	−1.45736
$222$	0	0
$223$	−2648.00	−0.795171	−0.397586	−	0.917565i	$-0.630152\pi$
$223$	−2648.00	−0.795171	−0.397586	+	0.917565i	$0.630152\pi$
$224$	0	0
$225$	−801.000	−0.237333
$226$	0	0
$227$	−2244.00	−0.656121	−0.328061	−	0.944657i	$-0.606395\pi$
$227$	−2244.00	−0.656121	−0.328061	+	0.944657i	$0.606395\pi$
$228$	0	0
$229$	−5650.00	−1.63040	−0.815202	−	0.579177i	$-0.803374\pi$
$229$	−5650.00	−1.63040	−0.815202	+	0.579177i	$0.803374\pi$
$230$	0	0
$231$	−576.000	−0.164061
$232$	0	0
$233$	4698.00	1.32093	0.660464	−	0.750858i	$-0.270360\pi$
$233$	4698.00	1.32093	0.660464	+	0.750858i	$0.270360\pi$
$234$	0	0
$235$	576.000	0.159890
$236$	0	0
$237$	1560.00	0.427565
$238$	0	0
$239$	1200.00	0.324776	0.162388	−	0.986727i	$-0.448080\pi$
$239$	1200.00	0.324776	0.162388	+	0.986727i	$0.448080\pi$
$240$	0	0
$241$	−718.000	−0.191911	−0.0959553	−	0.995386i	$-0.530591\pi$
$241$	−718.000	−0.191911	−0.0959553	+	0.995386i	$0.530591\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	−522.000	−0.136120
$246$	0	0
$247$	−760.000	−0.195780
$248$	0	0
$249$	1476.00	0.375653
$250$	0	0
$251$	−6012.00	−1.51185	−0.755924	−	0.654659i	$-0.772812\pi$
$251$	−6012.00	−1.51185	−0.755924	+	0.654659i	$0.772812\pi$
$252$	0	0
$253$	2016.00	0.500968
$254$	0	0
$255$	−2268.00	−0.556971
$256$	0	0
$257$	−2046.00	−0.496599	−0.248300	−	0.968683i	$-0.579872\pi$
$257$	−2046.00	−0.496599	−0.248300	+	0.968683i	$0.579872\pi$
$258$	0	0
$259$	4064.00	0.974999
$260$	0	0
$261$	270.000	0.0640329
$262$	0	0
$263$	6072.00	1.42363	0.711817	−	0.702365i	$-0.247873\pi$
$263$	6072.00	1.42363	0.711817	+	0.702365i	$0.247873\pi$
$264$	0	0
$265$	1188.00	0.275390
$266$	0	0
$267$	2430.00	0.556980
$268$	0	0
$269$	−6930.00	−1.57074	−0.785371	−	0.619025i	$-0.787528\pi$
$269$	−6930.00	−1.57074	−0.785371	+	0.619025i	$0.787528\pi$
$270$	0	0
$271$	−1352.00	−0.303056	−0.151528	−	0.988453i	$-0.548419\pi$
$271$	−1352.00	−0.303056	−0.151528	+	0.988453i	$0.548419\pi$
$272$	0	0
$273$	1824.00	0.404372
$274$	0	0
$275$	1068.00	0.234192
$276$	0	0
$277$	−1186.00	−0.257256	−0.128628	−	0.991693i	$-0.541057\pi$
$277$	−1186.00	−0.257256	−0.128628	+	0.991693i	$0.541057\pi$
$278$	0	0
$279$	792.000	0.169949
$280$	0	0
$281$	2442.00	0.518425	0.259213	−	0.965820i	$-0.416537\pi$
$281$	2442.00	0.518425	0.259213	+	0.965820i	$0.416537\pi$
$282$	0	0
$283$	−2828.00	−0.594018	−0.297009	−	0.954875i	$-0.595989\pi$
$283$	−2828.00	−0.594018	−0.297009	+	0.954875i	$0.595989\pi$
$284$	0	0
$285$	−360.000	−0.0748230
$286$	0	0
$287$	672.000	0.138212
$288$	0	0
$289$	10963.0	2.23143
$290$	0	0
$291$	3462.00	0.697409
$292$	0	0
$293$	4758.00	0.948687	0.474344	−	0.880340i	$-0.342685\pi$
$293$	4758.00	0.948687	0.474344	+	0.880340i	$0.342685\pi$
$294$	0	0
$295$	3960.00	0.781560
$296$	0	0
$297$	−324.000	−0.0633010
$298$	0	0
$299$	−6384.00	−1.23477
$300$	0	0
$301$	832.000	0.159321
$302$	0	0
$303$	−1854.00	−0.351517
$304$	0	0
$305$	−3228.00	−0.606016
$306$	0	0
$307$	8476.00	1.57574	0.787868	−	0.615844i	$-0.211185\pi$
$307$	8476.00	1.57574	0.787868	+	0.615844i	$0.211185\pi$
$308$	0	0
$309$	−384.000	−0.0706958
$310$	0	0
$311$	−4632.00	−0.844555	−0.422278	−	0.906467i	$-0.638769\pi$
$311$	−4632.00	−0.844555	−0.422278	+	0.906467i	$0.638769\pi$
$312$	0	0
$313$	−4822.00	−0.870785	−0.435392	−	0.900241i	$-0.643390\pi$
$313$	−4822.00	−0.870785	−0.435392	+	0.900241i	$0.643390\pi$
$314$	0	0
$315$	864.000	0.154542
$316$	0	0
$317$	−3426.00	−0.607014	−0.303507	−	0.952829i	$-0.598158\pi$
$317$	−3426.00	−0.607014	−0.303507	+	0.952829i	$0.598158\pi$
$318$	0	0
$319$	−360.000	−0.0631854
$320$	0	0
$321$	4428.00	0.769928
$322$	0	0
$323$	2520.00	0.434107
$324$	0	0
$325$	−3382.00	−0.577230
$326$	0	0
$327$	3570.00	0.603735
$328$	0	0
$329$	1536.00	0.257393
$330$	0	0
$331$	2788.00	0.462968	0.231484	−	0.972839i	$-0.425642\pi$
$331$	2788.00	0.462968	0.231484	+	0.972839i	$0.425642\pi$
$332$	0	0
$333$	2286.00	0.376192
$334$	0	0
$335$	−5304.00	−0.865040
$336$	0	0
$337$	434.000	0.0701528	0.0350764	−	0.999385i	$-0.488833\pi$
$337$	434.000	0.0701528	0.0350764	+	0.999385i	$0.488833\pi$
$338$	0	0
$339$	−1386.00	−0.222057
$340$	0	0
$341$	−1056.00	−0.167700
$342$	0	0
$343$	−6880.00	−1.08305
$344$	0	0
$345$	−3024.00	−0.471903
$346$	0	0
$347$	−6684.00	−1.03405	−0.517026	−	0.855970i	$-0.672961\pi$
$347$	−6684.00	−1.03405	−0.517026	+	0.855970i	$0.672961\pi$
$348$	0	0
$349$	2630.00	0.403383	0.201692	−	0.979449i	$-0.435356\pi$
$349$	2630.00	0.403383	0.201692	+	0.979449i	$0.435356\pi$
$350$	0	0
$351$	1026.00	0.156022
$352$	0	0
$353$	−7422.00	−1.11907	−0.559537	−	0.828805i	$-0.689021\pi$
$353$	−7422.00	−1.11907	−0.559537	+	0.828805i	$0.689021\pi$
$354$	0	0
$355$	−4752.00	−0.710451
$356$	0	0
$357$	−6048.00	−0.896622
$358$	0	0
$359$	10440.0	1.53482	0.767412	−	0.641154i	$-0.221544\pi$
$359$	10440.0	1.53482	0.767412	+	0.641154i	$0.221544\pi$
$360$	0	0
$361$	−6459.00	−0.941682
$362$	0	0
$363$	−3561.00	−0.514887
$364$	0	0
$365$	1308.00	0.187572
$366$	0	0
$367$	−10424.0	−1.48264	−0.741319	−	0.671153i	$-0.765800\pi$
$367$	−10424.0	−1.48264	−0.741319	+	0.671153i	$0.765800\pi$
$368$	0	0
$369$	378.000	0.0533276
$370$	0	0
$371$	3168.00	0.443327
$372$	0	0
$373$	3278.00	0.455036	0.227518	−	0.973774i	$-0.426939\pi$
$373$	3278.00	0.455036	0.227518	+	0.973774i	$0.426939\pi$
$374$	0	0
$375$	−3852.00	−0.530444
$376$	0	0
$377$	1140.00	0.155737
$378$	0	0
$379$	−6140.00	−0.832165	−0.416083	−	0.909327i	$-0.636597\pi$
$379$	−6140.00	−0.832165	−0.416083	+	0.909327i	$0.636597\pi$
$380$	0	0
$381$	7608.00	1.02302
$382$	0	0
$383$	3072.00	0.409848	0.204924	−	0.978778i	$-0.434305\pi$
$383$	3072.00	0.409848	0.204924	+	0.978778i	$0.434305\pi$
$384$	0	0
$385$	−1152.00	−0.152497
$386$	0	0
$387$	468.000	0.0614723
$388$	0	0
$389$	6150.00	0.801587	0.400794	−	0.916168i	$-0.368734\pi$
$389$	6150.00	0.801587	0.400794	+	0.916168i	$0.368734\pi$
$390$	0	0
$391$	21168.0	2.73788
$392$	0	0
$393$	−6876.00	−0.882566
$394$	0	0
$395$	3120.00	0.397428
$396$	0	0
$397$	−106.000	−0.0134005	−0.00670024	−	0.999978i	$-0.502133\pi$
$397$	−106.000	−0.0134005	−0.00670024	+	0.999978i	$0.502133\pi$
$398$	0	0
$399$	−960.000	−0.120451
$400$	0	0
$401$	−1758.00	−0.218929	−0.109464	−	0.993991i	$-0.534914\pi$
$401$	−1758.00	−0.218929	−0.109464	+	0.993991i	$0.534914\pi$
$402$	0	0
$403$	3344.00	0.413341
$404$	0	0
$405$	486.000	0.0596285
$406$	0	0
$407$	−3048.00	−0.371213
$408$	0	0
$409$	−3670.00	−0.443691	−0.221846	−	0.975082i	$-0.571208\pi$
$409$	−3670.00	−0.443691	−0.221846	+	0.975082i	$0.571208\pi$
$410$	0	0
$411$	−2178.00	−0.261394
$412$	0	0
$413$	10560.0	1.25817
$414$	0	0
$415$	2952.00	0.349176
$416$	0	0
$417$	−1140.00	−0.133875
$418$	0	0
$419$	9660.00	1.12631	0.563153	−	0.826353i	$-0.309588\pi$
$419$	9660.00	1.12631	0.563153	+	0.826353i	$0.309588\pi$
$420$	0	0
$421$	8462.00	0.979602	0.489801	−	0.871834i	$-0.337069\pi$
$421$	8462.00	0.979602	0.489801	+	0.871834i	$0.337069\pi$
$422$	0	0
$423$	864.000	0.0993123
$424$	0	0
$425$	11214.0	1.27990
$426$	0	0
$427$	−8608.00	−0.975575
$428$	0	0
$429$	−1368.00	−0.153957
$430$	0	0
$431$	−9792.00	−1.09435	−0.547174	−	0.837019i	$-0.684296\pi$
$431$	−9792.00	−1.09435	−0.547174	+	0.837019i	$0.684296\pi$
$432$	0	0
$433$	−7342.00	−0.814859	−0.407430	−	0.913237i	$-0.633575\pi$
$433$	−7342.00	−0.814859	−0.407430	+	0.913237i	$0.633575\pi$
$434$	0	0
$435$	540.000	0.0595196
$436$	0	0
$437$	3360.00	0.367805
$438$	0	0
$439$	−10640.0	−1.15676	−0.578382	−	0.815766i	$-0.696316\pi$
$439$	−10640.0	−1.15676	−0.578382	+	0.815766i	$0.696316\pi$
$440$	0	0
$441$	−783.000	−0.0845481
$442$	0	0
$443$	17412.0	1.86742	0.933712	−	0.358024i	$-0.116549\pi$
$443$	17412.0	1.86742	0.933712	+	0.358024i	$0.116549\pi$
$444$	0	0
$445$	4860.00	0.517722
$446$	0	0
$447$	4770.00	0.504728
$448$	0	0
$449$	−1710.00	−0.179732	−0.0898662	−	0.995954i	$-0.528644\pi$
$449$	−1710.00	−0.179732	−0.0898662	+	0.995954i	$0.528644\pi$
$450$	0	0
$451$	−504.000	−0.0526218
$452$	0	0
$453$	−7296.00	−0.756724
$454$	0	0
$455$	3648.00	0.375870
$456$	0	0
$457$	−646.000	−0.0661239	−0.0330619	−	0.999453i	$-0.510526\pi$
$457$	−646.000	−0.0661239	−0.0330619	+	0.999453i	$0.510526\pi$
$458$	0	0
$459$	−3402.00	−0.345952
$460$	0	0
$461$	−6018.00	−0.607996	−0.303998	−	0.952673i	$-0.598322\pi$
$461$	−6018.00	−0.607996	−0.303998	+	0.952673i	$0.598322\pi$
$462$	0	0
$463$	6712.00	0.673722	0.336861	−	0.941554i	$-0.390635\pi$
$463$	6712.00	0.673722	0.336861	+	0.941554i	$0.390635\pi$
$464$	0	0
$465$	1584.00	0.157970
$466$	0	0
$467$	−5364.00	−0.531512	−0.265756	−	0.964040i	$-0.585622\pi$
$467$	−5364.00	−0.531512	−0.265756	+	0.964040i	$0.585622\pi$
$468$	0	0
$469$	−14144.0	−1.39256
$470$	0	0
$471$	1842.00	0.180201
$472$	0	0
$473$	−624.000	−0.0606587
$474$	0	0
$475$	1780.00	0.171941
$476$	0	0
$477$	1782.00	0.171053
$478$	0	0
$479$	−9840.00	−0.938624	−0.469312	−	0.883032i	$-0.655498\pi$
$479$	−9840.00	−0.938624	−0.469312	+	0.883032i	$0.655498\pi$
$480$	0	0
$481$	9652.00	0.914955
$482$	0	0
$483$	−8064.00	−0.759678
$484$	0	0
$485$	6924.00	0.648253
$486$	0	0
$487$	−1424.00	−0.132500	−0.0662501	−	0.997803i	$-0.521104\pi$
$487$	−1424.00	−0.132500	−0.0662501	+	0.997803i	$0.521104\pi$
$488$	0	0
$489$	5556.00	0.513806
$490$	0	0
$491$	4548.00	0.418021	0.209011	−	0.977913i	$-0.432976\pi$
$491$	4548.00	0.418021	0.209011	+	0.977913i	$0.432976\pi$
$492$	0	0
$493$	−3780.00	−0.345320
$494$	0	0
$495$	−648.000	−0.0588393
$496$	0	0
$497$	−12672.0	−1.14370
$498$	0	0
$499$	−6500.00	−0.583126	−0.291563	−	0.956552i	$-0.594175\pi$
$499$	−6500.00	−0.583126	−0.291563	+	0.956552i	$0.594175\pi$
$500$	0	0
$501$	6408.00	0.571434
$502$	0	0
$503$	−12168.0	−1.07862	−0.539308	−	0.842108i	$-0.681314\pi$
$503$	−12168.0	−1.07862	−0.539308	+	0.842108i	$0.681314\pi$
$504$	0	0
$505$	−3708.00	−0.326740
$506$	0	0
$507$	−2259.00	−0.197881
$508$	0	0
$509$	−21090.0	−1.83654	−0.918269	−	0.395957i	$-0.870413\pi$
$509$	−21090.0	−1.83654	−0.918269	+	0.395957i	$0.870413\pi$
$510$	0	0
$511$	3488.00	0.301957
$512$	0	0
$513$	−540.000	−0.0464748
$514$	0	0
$515$	−768.000	−0.0657129
$516$	0	0
$517$	−1152.00	−0.0979979
$518$	0	0
$519$	5274.00	0.446056
$520$	0	0
$521$	−5238.00	−0.440462	−0.220231	−	0.975448i	$-0.570681\pi$
$521$	−5238.00	−0.440462	−0.220231	+	0.975448i	$0.570681\pi$
$522$	0	0
$523$	−8588.00	−0.718025	−0.359012	−	0.933333i	$-0.616886\pi$
$523$	−8588.00	−0.718025	−0.359012	+	0.933333i	$0.616886\pi$
$524$	0	0
$525$	−4272.00	−0.355134
$526$	0	0
$527$	−11088.0	−0.916510
$528$	0	0
$529$	16057.0	1.31972
$530$	0	0
$531$	5940.00	0.485450
$532$	0	0
$533$	1596.00	0.129701
$534$	0	0
$535$	8856.00	0.715660
$536$	0	0
$537$	1620.00	0.130183
$538$	0	0
$539$	1044.00	0.0834291
$540$	0	0
$541$	3062.00	0.243338	0.121669	−	0.992571i	$-0.461175\pi$
$541$	3062.00	0.243338	0.121669	+	0.992571i	$0.461175\pi$
$542$	0	0
$543$	5946.00	0.469921
$544$	0	0
$545$	7140.00	0.561182
$546$	0	0
$547$	8476.00	0.662537	0.331268	−	0.943537i	$-0.392523\pi$
$547$	8476.00	0.662537	0.331268	+	0.943537i	$0.392523\pi$
$548$	0	0
$549$	−4842.00	−0.376414
$550$	0	0
$551$	−600.000	−0.0463899
$552$	0	0
$553$	8320.00	0.639787
$554$	0	0
$555$	4572.00	0.349677
$556$	0	0
$557$	−12546.0	−0.954383	−0.477191	−	0.878799i	$-0.658345\pi$
$557$	−12546.0	−0.954383	−0.477191	+	0.878799i	$0.658345\pi$
$558$	0	0
$559$	1976.00	0.149510
$560$	0	0
$561$	4536.00	0.341373
$562$	0	0
$563$	12.0000	0.000898294 0	0.000449147	−	1.00000i	$-0.499857\pi$
$563$	12.0000	0.000898294 0	0.000449147		1.00000i	$0.499857\pi$
$564$	0	0
$565$	−2772.00	−0.206405
$566$	0	0
$567$	1296.00	0.0959910
$568$	0	0
$569$	19290.0	1.42123	0.710614	−	0.703582i	$-0.248417\pi$
$569$	19290.0	1.42123	0.710614	+	0.703582i	$0.248417\pi$
$570$	0	0
$571$	12148.0	0.890329	0.445165	−	0.895449i	$-0.353145\pi$
$571$	12148.0	0.890329	0.445165	+	0.895449i	$0.353145\pi$
$572$	0	0
$573$	8064.00	0.587920
$574$	0	0
$575$	14952.0	1.08442
$576$	0	0
$577$	−10366.0	−0.747907	−0.373953	−	0.927447i	$-0.621998\pi$
$577$	−10366.0	−0.747907	−0.373953	+	0.927447i	$0.621998\pi$
$578$	0	0
$579$	−6906.00	−0.495688
$580$	0	0
$581$	7872.00	0.562109
$582$	0	0
$583$	−2376.00	−0.168789
$584$	0	0
$585$	2052.00	0.145025
$586$	0	0
$587$	−7644.00	−0.537482	−0.268741	−	0.963213i	$-0.586607\pi$
$587$	−7644.00	−0.537482	−0.268741	+	0.963213i	$0.586607\pi$
$588$	0	0
$589$	−1760.00	−0.123123
$590$	0	0
$591$	13122.0	0.913311
$592$	0	0
$593$	8658.00	0.599564	0.299782	−	0.954008i	$-0.403086\pi$
$593$	8658.00	0.599564	0.299782	+	0.954008i	$0.403086\pi$
$594$	0	0
$595$	−12096.0	−0.833425
$596$	0	0
$597$	4800.00	0.329064
$598$	0	0
$599$	−25800.0	−1.75987	−0.879933	−	0.475098i	$-0.842413\pi$
$599$	−25800.0	−1.75987	−0.879933	+	0.475098i	$0.842413\pi$
$600$	0	0
$601$	16202.0	1.09966	0.549828	−	0.835278i	$-0.314693\pi$
$601$	16202.0	1.09966	0.549828	+	0.835278i	$0.314693\pi$
$602$	0	0
$603$	−7956.00	−0.537302
$604$	0	0
$605$	−7122.00	−0.478596
$606$	0	0
$607$	24136.0	1.61392	0.806960	−	0.590605i	$-0.201111\pi$
$607$	24136.0	1.61392	0.806960	+	0.590605i	$0.201111\pi$
$608$	0	0
$609$	1440.00	0.0958157
$610$	0	0
$611$	3648.00	0.241542
$612$	0	0
$613$	−4642.00	−0.305854	−0.152927	−	0.988237i	$-0.548870\pi$
$613$	−4642.00	−0.305854	−0.152927	+	0.988237i	$0.548870\pi$
$614$	0	0
$615$	756.000	0.0495689
$616$	0	0
$617$	−6726.00	−0.438863	−0.219432	−	0.975628i	$-0.570420\pi$
$617$	−6726.00	−0.438863	−0.219432	+	0.975628i	$0.570420\pi$
$618$	0	0
$619$	21220.0	1.37787	0.688937	−	0.724821i	$-0.258078\pi$
$619$	21220.0	1.37787	0.688937	+	0.724821i	$0.258078\pi$
$620$	0	0
$621$	−4536.00	−0.293113
$622$	0	0
$623$	12960.0	0.833437
$624$	0	0
$625$	3421.00	0.218944
$626$	0	0
$627$	720.000	0.0458597
$628$	0	0
$629$	−32004.0	−2.02875
$630$	0	0
$631$	−29792.0	−1.87956	−0.939779	−	0.341783i	$-0.888969\pi$
$631$	−29792.0	−1.87956	−0.939779	+	0.341783i	$0.888969\pi$
$632$	0	0
$633$	−9996.00	−0.627655
$634$	0	0
$635$	15216.0	0.950911
$636$	0	0
$637$	−3306.00	−0.205633
$638$	0	0
$639$	−7128.00	−0.441282
$640$	0	0
$641$	−10158.0	−0.625923	−0.312962	−	0.949766i	$-0.601321\pi$
$641$	−10158.0	−0.625923	−0.312962	+	0.949766i	$0.601321\pi$
$642$	0	0
$643$	−29828.0	−1.82940	−0.914698	−	0.404138i	$-0.867571\pi$
$643$	−29828.0	−1.82940	−0.914698	+	0.404138i	$0.867571\pi$
$644$	0	0
$645$	936.000	0.0571395
$646$	0	0
$647$	−1944.00	−0.118124	−0.0590622	−	0.998254i	$-0.518811\pi$
$647$	−1944.00	−0.118124	−0.0590622	+	0.998254i	$0.518811\pi$
$648$	0	0
$649$	−7920.00	−0.479025
$650$	0	0
$651$	4224.00	0.254304
$652$	0	0
$653$	26718.0	1.60116	0.800579	−	0.599227i	$-0.204525\pi$
$653$	26718.0	1.60116	0.800579	+	0.599227i	$0.204525\pi$
$654$	0	0
$655$	−13752.0	−0.820359
$656$	0	0
$657$	1962.00	0.116507
$658$	0	0
$659$	−4260.00	−0.251815	−0.125907	−	0.992042i	$-0.540184\pi$
$659$	−4260.00	−0.251815	−0.125907	+	0.992042i	$0.540184\pi$
$660$	0	0
$661$	22862.0	1.34528	0.672639	−	0.739971i	$-0.265161\pi$
$661$	22862.0	1.34528	0.672639	+	0.739971i	$0.265161\pi$
$662$	0	0
$663$	−14364.0	−0.841405
$664$	0	0
$665$	−1920.00	−0.111962
$666$	0	0
$667$	−5040.00	−0.292578
$668$	0	0
$669$	−7944.00	−0.459092
$670$	0	0
$671$	6456.00	0.371432
$672$	0	0
$673$	−32542.0	−1.86390	−0.931948	−	0.362592i	$-0.881892\pi$
$673$	−32542.0	−1.86390	−0.931948	+	0.362592i	$0.881892\pi$
$674$	0	0
$675$	−2403.00	−0.137024
$676$	0	0
$677$	14214.0	0.806925	0.403463	−	0.914996i	$-0.367807\pi$
$677$	14214.0	0.806925	0.403463	+	0.914996i	$0.367807\pi$
$678$	0	0
$679$	18464.0	1.04357
$680$	0	0
$681$	−6732.00	−0.378812
$682$	0	0
$683$	7092.00	0.397317	0.198659	−	0.980069i	$-0.436341\pi$
$683$	7092.00	0.397317	0.198659	+	0.980069i	$0.436341\pi$
$684$	0	0
$685$	−4356.00	−0.242970
$686$	0	0
$687$	−16950.0	−0.941314
$688$	0	0
$689$	7524.00	0.416026
$690$	0	0
$691$	13228.0	0.728244	0.364122	−	0.931351i	$-0.381369\pi$
$691$	13228.0	0.728244	0.364122	+	0.931351i	$0.381369\pi$
$692$	0	0
$693$	−1728.00	−0.0947205
$694$	0	0
$695$	−2280.00	−0.124439
$696$	0	0
$697$	−5292.00	−0.287588
$698$	0	0
$699$	14094.0	0.762638
$700$	0	0
$701$	28062.0	1.51196	0.755982	−	0.654592i	$-0.227160\pi$
$701$	28062.0	1.51196	0.755982	+	0.654592i	$0.227160\pi$
$702$	0	0
$703$	−5080.00	−0.272540
$704$	0	0
$705$	1728.00	0.0923124
$706$	0	0
$707$	−9888.00	−0.525992
$708$	0	0
$709$	−27250.0	−1.44343	−0.721717	−	0.692188i	$-0.756647\pi$
$709$	−27250.0	−1.44343	−0.721717	+	0.692188i	$0.756647\pi$
$710$	0	0
$711$	4680.00	0.246855
$712$	0	0
$713$	−14784.0	−0.776529
$714$	0	0
$715$	−2736.00	−0.143106
$716$	0	0
$717$	3600.00	0.187510
$718$	0	0
$719$	14400.0	0.746912	0.373456	−	0.927648i	$-0.378173\pi$
$719$	14400.0	0.746912	0.373456	+	0.927648i	$0.378173\pi$
$720$	0	0
$721$	−2048.00	−0.105786
$722$	0	0
$723$	−2154.00	−0.110800
$724$	0	0
$725$	−2670.00	−0.136774
$726$	0	0
$727$	−17984.0	−0.917455	−0.458727	−	0.888577i	$-0.651695\pi$
$727$	−17984.0	−0.917455	−0.458727	+	0.888577i	$0.651695\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−6552.00	−0.331511
$732$	0	0
$733$	16598.0	0.836373	0.418186	−	0.908361i	$-0.362666\pi$
$733$	16598.0	0.836373	0.418186	+	0.908361i	$0.362666\pi$
$734$	0	0
$735$	−1566.00	−0.0785888
$736$	0	0
$737$	10608.0	0.530191
$738$	0	0
$739$	−1460.00	−0.0726752	−0.0363376	−	0.999340i	$-0.511569\pi$
$739$	−1460.00	−0.0726752	−0.0363376	+	0.999340i	$0.511569\pi$
$740$	0	0
$741$	−2280.00	−0.113034
$742$	0	0
$743$	30072.0	1.48484	0.742419	−	0.669936i	$-0.233678\pi$
$743$	30072.0	1.48484	0.742419	+	0.669936i	$0.233678\pi$
$744$	0	0
$745$	9540.00	0.469152
$746$	0	0
$747$	4428.00	0.216884
$748$	0	0
$749$	23616.0	1.15208
$750$	0	0
$751$	18088.0	0.878882	0.439441	−	0.898271i	$-0.355177\pi$
$751$	18088.0	0.878882	0.439441	+	0.898271i	$0.355177\pi$
$752$	0	0
$753$	−18036.0	−0.872866
$754$	0	0
$755$	−14592.0	−0.703387
$756$	0	0
$757$	24734.0	1.18755	0.593773	−	0.804633i	$-0.297638\pi$
$757$	24734.0	1.18755	0.593773	+	0.804633i	$0.297638\pi$
$758$	0	0
$759$	6048.00	0.289234
$760$	0	0
$761$	−22278.0	−1.06120	−0.530602	−	0.847621i	$-0.678034\pi$
$761$	−22278.0	−1.06120	−0.530602	+	0.847621i	$0.678034\pi$
$762$	0	0
$763$	19040.0	0.903400
$764$	0	0
$765$	−6804.00	−0.321568
$766$	0	0
$767$	25080.0	1.18069
$768$	0	0
$769$	16130.0	0.756388	0.378194	−	0.925726i	$-0.376545\pi$
$769$	16130.0	0.756388	0.378194	+	0.925726i	$0.376545\pi$
$770$	0	0
$771$	−6138.00	−0.286712
$772$	0	0
$773$	29718.0	1.38277	0.691386	−	0.722486i	$-0.257001\pi$
$773$	29718.0	1.38277	0.691386	+	0.722486i	$0.257001\pi$
$774$	0	0
$775$	−7832.00	−0.363011
$776$	0	0
$777$	12192.0	0.562916
$778$	0	0
$779$	−840.000	−0.0386343
$780$	0	0
$781$	9504.00	0.435442
$782$	0	0
$783$	810.000	0.0369694
$784$	0	0
$785$	3684.00	0.167500
$786$	0	0
$787$	−9524.00	−0.431377	−0.215689	−	0.976462i	$-0.569200\pi$
$787$	−9524.00	−0.431377	−0.215689	+	0.976462i	$0.569200\pi$
$788$	0	0
$789$	18216.0	0.821935
$790$	0	0
$791$	−7392.00	−0.332275
$792$	0	0
$793$	−20444.0	−0.915495
$794$	0	0
$795$	3564.00	0.158996
$796$	0	0
$797$	−33906.0	−1.50692	−0.753458	−	0.657496i	$-0.771616\pi$
$797$	−33906.0	−1.50692	−0.753458	+	0.657496i	$0.771616\pi$
$798$	0	0
$799$	−12096.0	−0.535577
$800$	0	0
$801$	7290.00	0.321572
$802$	0	0
$803$	−2616.00	−0.114965
$804$	0	0
$805$	−16128.0	−0.706133
$806$	0	0
$807$	−20790.0	−0.906868
$808$	0	0
$809$	−630.000	−0.0273790	−0.0136895	−	0.999906i	$-0.504358\pi$
$809$	−630.000	−0.0273790	−0.0136895	+	0.999906i	$0.504358\pi$
$810$	0	0
$811$	20788.0	0.900081	0.450040	−	0.893008i	$-0.351410\pi$
$811$	20788.0	0.900081	0.450040	+	0.893008i	$0.351410\pi$
$812$	0	0
$813$	−4056.00	−0.174969
$814$	0	0
$815$	11112.0	0.477591
$816$	0	0
$817$	−1040.00	−0.0445349
$818$	0	0
$819$	5472.00	0.233464
$820$	0	0
$821$	−43098.0	−1.83207	−0.916036	−	0.401097i	$-0.868629\pi$
$821$	−43098.0	−1.83207	−0.916036	+	0.401097i	$0.868629\pi$
$822$	0	0
$823$	14272.0	0.604484	0.302242	−	0.953231i	$-0.402265\pi$
$823$	14272.0	0.604484	0.302242	+	0.953231i	$0.402265\pi$
$824$	0	0
$825$	3204.00	0.135211
$826$	0	0
$827$	−13644.0	−0.573698	−0.286849	−	0.957976i	$-0.592608\pi$
$827$	−13644.0	−0.573698	−0.286849	+	0.957976i	$0.592608\pi$
$828$	0	0
$829$	−2410.00	−0.100968	−0.0504842	−	0.998725i	$-0.516076\pi$
$829$	−2410.00	−0.100968	−0.0504842	+	0.998725i	$0.516076\pi$
$830$	0	0
$831$	−3558.00	−0.148527
$832$	0	0
$833$	10962.0	0.455955
$834$	0	0
$835$	12816.0	0.531157
$836$	0	0
$837$	2376.00	0.0981202
$838$	0	0
$839$	−23160.0	−0.953006	−0.476503	−	0.879173i	$-0.658096\pi$
$839$	−23160.0	−0.953006	−0.476503	+	0.879173i	$0.658096\pi$
$840$	0	0
$841$	−23489.0	−0.963098
$842$	0	0
$843$	7326.00	0.299313
$844$	0	0
$845$	−4518.00	−0.183934
$846$	0	0
$847$	−18992.0	−0.770452
$848$	0	0
$849$	−8484.00	−0.342957
$850$	0	0
$851$	−42672.0	−1.71889
$852$	0	0
$853$	32078.0	1.28761	0.643804	−	0.765190i	$-0.277355\pi$
$853$	32078.0	1.28761	0.643804	+	0.765190i	$0.277355\pi$
$854$	0	0
$855$	−1080.00	−0.0431991
$856$	0	0
$857$	−14406.0	−0.574212	−0.287106	−	0.957899i	$-0.592693\pi$
$857$	−14406.0	−0.574212	−0.287106	+	0.957899i	$0.592693\pi$
$858$	0	0
$859$	−30620.0	−1.21623	−0.608115	−	0.793849i	$-0.708074\pi$
$859$	−30620.0	−1.21623	−0.608115	+	0.793849i	$0.708074\pi$
$860$	0	0
$861$	2016.00	0.0797969
$862$	0	0
$863$	−17568.0	−0.692957	−0.346478	−	0.938058i	$-0.612623\pi$
$863$	−17568.0	−0.692957	−0.346478	+	0.938058i	$0.612623\pi$
$864$	0	0
$865$	10548.0	0.414616
$866$	0	0
$867$	32889.0	1.28831
$868$	0	0
$869$	−6240.00	−0.243587
$870$	0	0
$871$	−33592.0	−1.30680
$872$	0	0
$873$	10386.0	0.402649
$874$	0	0
$875$	−20544.0	−0.793730
$876$	0	0
$877$	−21706.0	−0.835758	−0.417879	−	0.908503i	$-0.637226\pi$
$877$	−21706.0	−0.835758	−0.417879	+	0.908503i	$0.637226\pi$
$878$	0	0
$879$	14274.0	0.547725
$880$	0	0
$881$	−14958.0	−0.572018	−0.286009	−	0.958227i	$-0.592329\pi$
$881$	−14958.0	−0.572018	−0.286009	+	0.958227i	$0.592329\pi$
$882$	0	0
$883$	32812.0	1.25052	0.625261	−	0.780415i	$-0.284992\pi$
$883$	32812.0	1.25052	0.625261	+	0.780415i	$0.284992\pi$
$884$	0	0
$885$	11880.0	0.451234
$886$	0	0
$887$	38856.0	1.47086	0.735432	−	0.677598i	$-0.236979\pi$
$887$	38856.0	1.47086	0.735432	+	0.677598i	$0.236979\pi$
$888$	0	0
$889$	40576.0	1.53079
$890$	0	0
$891$	−972.000	−0.0365468
$892$	0	0
$893$	−1920.00	−0.0719489
$894$	0	0
$895$	3240.00	0.121007
$896$	0	0
$897$	−19152.0	−0.712895
$898$	0	0
$899$	2640.00	0.0979410
$900$	0	0
$901$	−24948.0	−0.922462
$902$	0	0
$903$	2496.00	0.0919841
$904$	0	0
$905$	11892.0	0.436799
$906$	0	0
$907$	28276.0	1.03516	0.517579	−	0.855635i	$-0.326833\pi$
$907$	28276.0	1.03516	0.517579	+	0.855635i	$0.326833\pi$
$908$	0	0
$909$	−5562.00	−0.202948
$910$	0	0
$911$	−8112.00	−0.295019	−0.147510	−	0.989061i	$-0.547126\pi$
$911$	−8112.00	−0.295019	−0.147510	+	0.989061i	$0.547126\pi$
$912$	0	0
$913$	−5904.00	−0.214013
$914$	0	0
$915$	−9684.00	−0.349883
$916$	0	0
$917$	−36672.0	−1.32063
$918$	0	0
$919$	26080.0	0.936126	0.468063	−	0.883695i	$-0.344952\pi$
$919$	26080.0	0.936126	0.468063	+	0.883695i	$0.344952\pi$
$920$	0	0
$921$	25428.0	0.909751
$922$	0	0
$923$	−30096.0	−1.07326
$924$	0	0
$925$	−22606.0	−0.803547
$926$	0	0
$927$	−1152.00	−0.0408162
$928$	0	0
$929$	49170.0	1.73651	0.868254	−	0.496120i	$-0.165243\pi$
$929$	49170.0	1.73651	0.868254	+	0.496120i	$0.165243\pi$
$930$	0	0
$931$	1740.00	0.0612526
$932$	0	0
$933$	−13896.0	−0.487604
$934$	0	0
$935$	9072.00	0.317311
$936$	0	0
$937$	48314.0	1.68447	0.842236	−	0.539110i	$-0.181239\pi$
$937$	48314.0	1.68447	0.842236	+	0.539110i	$0.181239\pi$
$938$	0	0
$939$	−14466.0	−0.502748
$940$	0	0
$941$	34782.0	1.20495	0.602477	−	0.798137i	$-0.294181\pi$
$941$	34782.0	1.20495	0.602477	+	0.798137i	$0.294181\pi$
$942$	0	0
$943$	−7056.00	−0.243664
$944$	0	0
$945$	2592.00	0.0892251
$946$	0	0
$947$	25116.0	0.861838	0.430919	−	0.902391i	$-0.358190\pi$
$947$	25116.0	0.861838	0.430919	+	0.902391i	$0.358190\pi$
$948$	0	0
$949$	8284.00	0.283361
$950$	0	0
$951$	−10278.0	−0.350460
$952$	0	0
$953$	−15462.0	−0.525565	−0.262782	−	0.964855i	$-0.584640\pi$
$953$	−15462.0	−0.525565	−0.262782	+	0.964855i	$0.584640\pi$
$954$	0	0
$955$	16128.0	0.546481
$956$	0	0
$957$	−1080.00	−0.0364801
$958$	0	0
$959$	−11616.0	−0.391137
$960$	0	0
$961$	−22047.0	−0.740056
$962$	0	0
$963$	13284.0	0.444518
$964$	0	0
$965$	−13812.0	−0.460750
$966$	0	0
$967$	736.000	0.0244759	0.0122379	−	0.999925i	$-0.496104\pi$
$967$	736.000	0.0244759	0.0122379	+	0.999925i	$0.496104\pi$
$968$	0	0
$969$	7560.00	0.250632
$970$	0	0
$971$	29268.0	0.967307	0.483653	−	0.875260i	$-0.339310\pi$
$971$	29268.0	0.967307	0.483653	+	0.875260i	$0.339310\pi$
$972$	0	0
$973$	−6080.00	−0.200325
$974$	0	0
$975$	−10146.0	−0.333264
$976$	0	0
$977$	16674.0	0.546007	0.273003	−	0.962013i	$-0.411983\pi$
$977$	16674.0	0.546007	0.273003	+	0.962013i	$0.411983\pi$
$978$	0	0
$979$	−9720.00	−0.317316
$980$	0	0
$981$	10710.0	0.348567
$982$	0	0
$983$	31272.0	1.01467	0.507336	−	0.861749i	$-0.330630\pi$
$983$	31272.0	1.01467	0.507336	+	0.861749i	$0.330630\pi$
$984$	0	0
$985$	26244.0	0.848937
$986$	0	0
$987$	4608.00	0.148606
$988$	0	0
$989$	−8736.00	−0.280878
$990$	0	0
$991$	15928.0	0.510565	0.255282	−	0.966867i	$-0.417832\pi$
$991$	15928.0	0.510565	0.255282	+	0.966867i	$0.417832\pi$
$992$	0	0
$993$	8364.00	0.267295
$994$	0	0
$995$	9600.00	0.305870
$996$	0	0
$997$	42014.0	1.33460	0.667300	−	0.744789i	$-0.267450\pi$
$997$	42014.0	1.33460	0.667300	+	0.744789i	$0.267450\pi$
$998$	0	0
$999$	6858.00	0.217195

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	48.4.a.c.1.1		1
3.2	odd	2		144.4.a.c.1.1		1
4.3	odd	2		6.4.a.a.1.1	✓	1
5.2	odd	4		1200.4.f.j.49.1		2
5.3	odd	4		1200.4.f.j.49.2		2
5.4	even	2		1200.4.a.b.1.1		1
7.6	odd	2		2352.4.a.e.1.1		1
8.3	odd	2		192.4.a.i.1.1		1
8.5	even	2		192.4.a.c.1.1		1
12.11	even	2		18.4.a.a.1.1		1
16.3	odd	4		768.4.d.n.385.1		2
16.5	even	4		768.4.d.c.385.1		2
16.11	odd	4		768.4.d.n.385.2		2
16.13	even	4		768.4.d.c.385.2		2
20.3	even	4		150.4.c.d.49.2		2
20.7	even	4		150.4.c.d.49.1		2
20.19	odd	2		150.4.a.i.1.1		1
24.5	odd	2		576.4.a.r.1.1		1
24.11	even	2		576.4.a.q.1.1		1
28.3	even	6		294.4.e.g.79.1		2
28.11	odd	6		294.4.e.h.79.1		2
28.19	even	6		294.4.e.g.67.1		2
28.23	odd	6		294.4.e.h.67.1		2
28.27	even	2		294.4.a.e.1.1		1
36.7	odd	6		162.4.c.f.109.1		2
36.11	even	6		162.4.c.c.109.1		2
36.23	even	6		162.4.c.c.55.1		2
36.31	odd	6		162.4.c.f.55.1		2
44.43	even	2		726.4.a.f.1.1		1
52.31	even	4		1014.4.b.d.337.2		2
52.47	even	4		1014.4.b.d.337.1		2
52.51	odd	2		1014.4.a.g.1.1		1
60.23	odd	4		450.4.c.e.199.1		2
60.47	odd	4		450.4.c.e.199.2		2
60.59	even	2		450.4.a.h.1.1		1
68.67	odd	2		1734.4.a.d.1.1		1
76.75	even	2		2166.4.a.i.1.1		1
84.11	even	6		882.4.g.i.667.1		2
84.23	even	6		882.4.g.i.361.1		2
84.47	odd	6		882.4.g.f.361.1		2
84.59	odd	6		882.4.g.f.667.1		2
84.83	odd	2		882.4.a.n.1.1		1
132.131	odd	2		2178.4.a.e.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
6.4.a.a.1.1	✓	1	4.3	odd	2
18.4.a.a.1.1		1	12.11	even	2
48.4.a.c.1.1		1	1.1	even	1	trivial
144.4.a.c.1.1		1	3.2	odd	2
150.4.a.i.1.1		1	20.19	odd	2
150.4.c.d.49.1		2	20.7	even	4
150.4.c.d.49.2		2	20.3	even	4
162.4.c.c.55.1		2	36.23	even	6
162.4.c.c.109.1		2	36.11	even	6
162.4.c.f.55.1		2	36.31	odd	6
162.4.c.f.109.1		2	36.7	odd	6
192.4.a.c.1.1		1	8.5	even	2
192.4.a.i.1.1		1	8.3	odd	2
294.4.a.e.1.1		1	28.27	even	2
294.4.e.g.67.1		2	28.19	even	6
294.4.e.g.79.1		2	28.3	even	6
294.4.e.h.67.1		2	28.23	odd	6
294.4.e.h.79.1		2	28.11	odd	6
450.4.a.h.1.1		1	60.59	even	2
450.4.c.e.199.1		2	60.23	odd	4
450.4.c.e.199.2		2	60.47	odd	4
576.4.a.q.1.1		1	24.11	even	2
576.4.a.r.1.1		1	24.5	odd	2
726.4.a.f.1.1		1	44.43	even	2
768.4.d.c.385.1		2	16.5	even	4
768.4.d.c.385.2		2	16.13	even	4
768.4.d.n.385.1		2	16.3	odd	4
768.4.d.n.385.2		2	16.11	odd	4
882.4.a.n.1.1		1	84.83	odd	2
882.4.g.f.361.1		2	84.47	odd	6
882.4.g.f.667.1		2	84.59	odd	6
882.4.g.i.361.1		2	84.23	even	6
882.4.g.i.667.1		2	84.11	even	6
1014.4.a.g.1.1		1	52.51	odd	2
1014.4.b.d.337.1		2	52.47	even	4
1014.4.b.d.337.2		2	52.31	even	4
1200.4.a.b.1.1		1	5.4	even	2
1200.4.f.j.49.1		2	5.2	odd	4
1200.4.f.j.49.2		2	5.3	odd	4
1734.4.a.d.1.1		1	68.67	odd	2
2166.4.a.i.1.1		1	76.75	even	2
2178.4.a.e.1.1		1	132.131	odd	2
2352.4.a.e.1.1		1	7.6	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 $\F_{q}$

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more

Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	48.4.a.c.1.1

Level	$48$
Weight	$4$
Character	48.1
Self dual	yes
Analytic conductor	$2.832$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$