LMFDB - Embedded newform 336.4.a.l.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,4,Mod(1,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(336, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 4, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("336.1"); S:= CuspForms(chi, 4); N := Newforms(S);

Level:	$N$	$=$	$336 = 2^{4} \cdot 3 \cdot 7$
Weight:	$k$	$=$	$4$
Character orbit:	$[\chi]$	$=$	336.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,3,0,18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$19.8246417619$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$1$
Twist minimal:	no (minimal twist has level 42)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	336.1

$q$ -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} +18.0000 q^{5} -7.00000 q^{7} +9.00000 q^{9} +72.0000 q^{11} -34.0000 q^{13} +54.0000 q^{15} +6.00000 q^{17} -92.0000 q^{19} -21.0000 q^{21} +180.000 q^{23} +199.000 q^{25} +27.0000 q^{27} -114.000 q^{29} -56.0000 q^{31} +216.000 q^{33} -126.000 q^{35} -34.0000 q^{37} -102.000 q^{39} +6.00000 q^{41} -164.000 q^{43} +162.000 q^{45} -168.000 q^{47} +49.0000 q^{49} +18.0000 q^{51} +654.000 q^{53} +1296.00 q^{55} -276.000 q^{57} +492.000 q^{59} -250.000 q^{61} -63.0000 q^{63} -612.000 q^{65} +124.000 q^{67} +540.000 q^{69} -36.0000 q^{71} +1010.00 q^{73} +597.000 q^{75} -504.000 q^{77} -56.0000 q^{79} +81.0000 q^{81} -228.000 q^{83} +108.000 q^{85} -342.000 q^{87} +390.000 q^{89} +238.000 q^{91} -168.000 q^{93} -1656.00 q^{95} -70.0000 q^{97} +648.000 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
$5$	18.0000	1.60997	0.804984	−	0.593296i	$-0.202174\pi$
$5$	18.0000	1.60997	0.804984	+	0.593296i	$0.202174\pi$
$6$	0	0
$7$	−7.00000	−0.377964
$7$	−7.00000	−0.377964
$8$	0	0
$9$	9.00000	0.333333
$10$	0	0
$11$	72.0000	1.97353	0.986764	−	0.162160i	$-0.0518462\pi$
$11$	72.0000	1.97353	0.986764	+	0.162160i	$0.0518462\pi$
$12$	0	0
$13$	−34.0000	−0.725377	−0.362689	−	0.931910i	$-0.618141\pi$
$13$	−34.0000	−0.725377	−0.362689	+	0.931910i	$0.618141\pi$
$14$	0	0
$15$	54.0000	0.929516
$16$	0	0
$17$	6.00000	0.0856008	0.0428004	−	0.999084i	$-0.486372\pi$
$17$	6.00000	0.0856008	0.0428004	+	0.999084i	$0.486372\pi$
$18$	0	0
$19$	−92.0000	−1.11086	−0.555428	−	0.831565i	$-0.687445\pi$
$19$	−92.0000	−1.11086	−0.555428	+	0.831565i	$0.687445\pi$
$20$	0	0
$21$	−21.0000	−0.218218
$22$	0	0
$23$	180.000	1.63185	0.815926	−	0.578156i	$-0.196228\pi$
$23$	180.000	1.63185	0.815926	+	0.578156i	$0.196228\pi$
$24$	0	0
$25$	199.000	1.59200
$26$	0	0
$27$	27.0000	0.192450
$28$	0	0
$29$	−114.000	−0.729975	−0.364987	−	0.931012i	$-0.618927\pi$
$29$	−114.000	−0.729975	−0.364987	+	0.931012i	$0.618927\pi$
$30$	0	0
$31$	−56.0000	−0.324448	−0.162224	−	0.986754i	$-0.551867\pi$
$31$	−56.0000	−0.324448	−0.162224	+	0.986754i	$0.551867\pi$
$32$	0	0
$33$	216.000	1.13942
$34$	0	0
$35$	−126.000	−0.608511
$36$	0	0
$37$	−34.0000	−0.151069	−0.0755347	−	0.997143i	$-0.524066\pi$
$37$	−34.0000	−0.151069	−0.0755347	+	0.997143i	$0.524066\pi$
$38$	0	0
$39$	−102.000	−0.418797
$40$	0	0
$41$	6.00000	0.0228547	0.0114273	−	0.999935i	$-0.496362\pi$
$41$	6.00000	0.0228547	0.0114273	+	0.999935i	$0.496362\pi$
$42$	0	0
$43$	−164.000	−0.581622	−0.290811	−	0.956780i	$-0.593925\pi$
$43$	−164.000	−0.581622	−0.290811	+	0.956780i	$0.593925\pi$
$44$	0	0
$45$	162.000	0.536656
$46$	0	0
$47$	−168.000	−0.521390	−0.260695	−	0.965421i	$-0.583952\pi$
$47$	−168.000	−0.521390	−0.260695	+	0.965421i	$0.583952\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	0	0
$51$	18.0000	0.0494217
$52$	0	0
$53$	654.000	1.69498	0.847489	−	0.530813i	$-0.178113\pi$
$53$	654.000	1.69498	0.847489	+	0.530813i	$0.178113\pi$
$54$	0	0
$55$	1296.00	3.17732
$56$	0	0
$57$	−276.000	−0.641353
$58$	0	0
$59$	492.000	1.08564	0.542822	−	0.839848i	$-0.317356\pi$
$59$	492.000	1.08564	0.542822	+	0.839848i	$0.317356\pi$
$60$	0	0
$61$	−250.000	−0.524741	−0.262371	−	0.964967i	$-0.584504\pi$
$61$	−250.000	−0.524741	−0.262371	+	0.964967i	$0.584504\pi$
$62$	0	0
$63$	−63.0000	−0.125988
$64$	0	0
$65$	−612.000	−1.16783
$66$	0	0
$67$	124.000	0.226105	0.113052	−	0.993589i	$-0.463937\pi$
$67$	124.000	0.226105	0.113052	+	0.993589i	$0.463937\pi$
$68$	0	0
$69$	540.000	0.942150
$70$	0	0
$71$	−36.0000	−0.0601748	−0.0300874	−	0.999547i	$-0.509579\pi$
$71$	−36.0000	−0.0601748	−0.0300874	+	0.999547i	$0.509579\pi$
$72$	0	0
$73$	1010.00	1.61934	0.809668	−	0.586888i	$-0.199647\pi$
$73$	1010.00	1.61934	0.809668	+	0.586888i	$0.199647\pi$
$74$	0	0
$75$	597.000	0.919142
$76$	0	0
$77$	−504.000	−0.745924
$78$	0	0
$79$	−56.0000	−0.0797531	−0.0398765	−	0.999205i	$-0.512696\pi$
$79$	−56.0000	−0.0797531	−0.0398765	+	0.999205i	$0.512696\pi$
$80$	0	0
$81$	81.0000	0.111111
$82$	0	0
$83$	−228.000	−0.301521	−0.150761	−	0.988570i	$-0.548172\pi$
$83$	−228.000	−0.301521	−0.150761	+	0.988570i	$0.548172\pi$
$84$	0	0
$85$	108.000	0.137815
$86$	0	0
$87$	−342.000	−0.421451
$88$	0	0
$89$	390.000	0.464493	0.232247	−	0.972657i	$-0.425392\pi$
$89$	390.000	0.464493	0.232247	+	0.972657i	$0.425392\pi$
$90$	0	0
$91$	238.000	0.274167
$92$	0	0
$93$	−168.000	−0.187320
$94$	0	0
$95$	−1656.00	−1.78844
$96$	0	0
$97$	−70.0000	−0.0732724	−0.0366362	−	0.999329i	$-0.511664\pi$
$97$	−70.0000	−0.0732724	−0.0366362	+	0.999329i	$0.511664\pi$
$98$	0	0
$99$	648.000	0.657843
$100$	0	0
$101$	−1350.00	−1.33000	−0.665000	−	0.746843i	$-0.731569\pi$
$101$	−1350.00	−1.33000	−0.665000	+	0.746843i	$0.731569\pi$
$102$	0	0
$103$	−2000.00	−1.91326	−0.956630	−	0.291305i	$-0.905911\pi$
$103$	−2000.00	−1.91326	−0.956630	+	0.291305i	$0.905911\pi$
$104$	0	0
$105$	−378.000	−0.351324
$106$	0	0
$107$	−696.000	−0.628830	−0.314415	−	0.949286i	$-0.601808\pi$
$107$	−696.000	−0.628830	−0.314415	+	0.949286i	$0.601808\pi$
$108$	0	0
$109$	−1114.00	−0.978916	−0.489458	−	0.872027i	$-0.662805\pi$
$109$	−1114.00	−0.978916	−0.489458	+	0.872027i	$0.662805\pi$
$110$	0	0
$111$	−102.000	−0.0872199
$112$	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
$115$	3240.00	2.62723
$116$	0	0
$117$	−306.000	−0.241792
$118$	0	0
$119$	−42.0000	−0.0323541
$120$	0	0
$121$	3853.00	2.89482
$122$	0	0
$123$	18.0000	0.0131952
$124$	0	0
$125$	1332.00	0.953102
$126$	0	0
$127$	−1064.00	−0.743423	−0.371712	−	0.928348i	$-0.621229\pi$
$127$	−1064.00	−0.743423	−0.371712	+	0.928348i	$0.621229\pi$
$128$	0	0
$129$	−492.000	−0.335800
$130$	0	0
$131$	−180.000	−0.120051	−0.0600255	−	0.998197i	$-0.519118\pi$
$131$	−180.000	−0.120051	−0.0600255	+	0.998197i	$0.519118\pi$
$132$	0	0
$133$	644.000	0.419864
$134$	0	0
$135$	486.000	0.309839
$136$	0	0
$137$	−2718.00	−1.69500	−0.847498	−	0.530799i	$-0.821892\pi$
$137$	−2718.00	−1.69500	−0.847498	+	0.530799i	$0.821892\pi$
$138$	0	0
$139$	1348.00	0.822560	0.411280	−	0.911509i	$-0.365082\pi$
$139$	1348.00	0.822560	0.411280	+	0.911509i	$0.365082\pi$
$140$	0	0
$141$	−504.000	−0.301025
$142$	0	0
$143$	−2448.00	−1.43155
$144$	0	0
$145$	−2052.00	−1.17524
$146$	0	0
$147$	147.000	0.0824786
$148$	0	0
$149$	558.000	0.306800	0.153400	−	0.988164i	$-0.450978\pi$
$149$	558.000	0.306800	0.153400	+	0.988164i	$0.450978\pi$
$150$	0	0
$151$	−1928.00	−1.03906	−0.519531	−	0.854451i	$-0.673893\pi$
$151$	−1928.00	−1.03906	−0.519531	+	0.854451i	$0.673893\pi$
$152$	0	0
$153$	54.0000	0.0285336
$154$	0	0
$155$	−1008.00	−0.522352
$156$	0	0
$157$	−2410.00	−1.22509	−0.612544	−	0.790436i	$-0.709854\pi$
$157$	−2410.00	−1.22509	−0.612544	+	0.790436i	$0.709854\pi$
$158$	0	0
$159$	1962.00	0.978596
$160$	0	0
$161$	−1260.00	−0.616782
$162$	0	0
$163$	−740.000	−0.355591	−0.177795	−	0.984067i	$-0.556896\pi$
$163$	−740.000	−0.355591	−0.177795	+	0.984067i	$0.556896\pi$
$164$	0	0
$165$	3888.00	1.83443
$166$	0	0
$167$	−3984.00	−1.84605	−0.923027	−	0.384734i	$-0.874293\pi$
$167$	−3984.00	−1.84605	−0.923027	+	0.384734i	$0.874293\pi$
$168$	0	0
$169$	−1041.00	−0.473828
$170$	0	0
$171$	−828.000	−0.370285
$172$	0	0
$173$	−1038.00	−0.456172	−0.228086	−	0.973641i	$-0.573247\pi$
$173$	−1038.00	−0.456172	−0.228086	+	0.973641i	$0.573247\pi$
$174$	0	0
$175$	−1393.00	−0.601719
$176$	0	0
$177$	1476.00	0.626796
$178$	0	0
$179$	2568.00	1.07230	0.536149	−	0.844123i	$-0.319879\pi$
$179$	2568.00	1.07230	0.536149	+	0.844123i	$0.319879\pi$
$180$	0	0
$181$	−2698.00	−1.10796	−0.553980	−	0.832530i	$-0.686892\pi$
$181$	−2698.00	−1.10796	−0.553980	+	0.832530i	$0.686892\pi$
$182$	0	0
$183$	−750.000	−0.302960
$184$	0	0
$185$	−612.000	−0.243217
$186$	0	0
$187$	432.000	0.168936
$188$	0	0
$189$	−189.000	−0.0727393
$190$	0	0
$191$	4116.00	1.55928	0.779642	−	0.626225i	$-0.215401\pi$
$191$	4116.00	1.55928	0.779642	+	0.626225i	$0.215401\pi$
$192$	0	0
$193$	−3310.00	−1.23450	−0.617251	−	0.786766i	$-0.711754\pi$
$193$	−3310.00	−1.23450	−0.617251	+	0.786766i	$0.711754\pi$
$194$	0	0
$195$	−1836.00	−0.674250
$196$	0	0
$197$	1278.00	0.462202	0.231101	−	0.972930i	$-0.425767\pi$
$197$	1278.00	0.462202	0.231101	+	0.972930i	$0.425767\pi$
$198$	0	0
$199$	−2936.00	−1.04587	−0.522933	−	0.852374i	$-0.675162\pi$
$199$	−2936.00	−1.04587	−0.522933	+	0.852374i	$0.675162\pi$
$200$	0	0
$201$	372.000	0.130542
$202$	0	0
$203$	798.000	0.275905
$204$	0	0
$205$	108.000	0.0367954
$206$	0	0
$207$	1620.00	0.543951
$208$	0	0
$209$	−6624.00	−2.19230
$210$	0	0
$211$	3508.00	1.14455	0.572276	−	0.820061i	$-0.306060\pi$
$211$	3508.00	1.14455	0.572276	+	0.820061i	$0.306060\pi$
$212$	0	0
$213$	−108.000	−0.0347420
$214$	0	0
$215$	−2952.00	−0.936394
$216$	0	0
$217$	392.000	0.122630
$218$	0	0
$219$	3030.00	0.934924
$220$	0	0
$221$	−204.000	−0.0620929
$222$	0	0
$223$	1888.00	0.566950	0.283475	−	0.958980i	$-0.408513\pi$
$223$	1888.00	0.566950	0.283475	+	0.958980i	$0.408513\pi$
$224$	0	0
$225$	1791.00	0.530667
$226$	0	0
$227$	−3564.00	−1.04207	−0.521037	−	0.853534i	$-0.674455\pi$
$227$	−3564.00	−1.04207	−0.521037	+	0.853534i	$0.674455\pi$
$228$	0	0
$229$	1334.00	0.384948	0.192474	−	0.981302i	$-0.438349\pi$
$229$	1334.00	0.384948	0.192474	+	0.981302i	$0.438349\pi$
$230$	0	0
$231$	−1512.00	−0.430659
$232$	0	0
$233$	2658.00	0.747345	0.373672	−	0.927561i	$-0.378098\pi$
$233$	2658.00	0.747345	0.373672	+	0.927561i	$0.378098\pi$
$234$	0	0
$235$	−3024.00	−0.839421
$236$	0	0
$237$	−168.000	−0.0460455
$238$	0	0
$239$	588.000	0.159140	0.0795702	−	0.996829i	$-0.474645\pi$
$239$	588.000	0.159140	0.0795702	+	0.996829i	$0.474645\pi$
$240$	0	0
$241$	5690.00	1.52085	0.760426	−	0.649425i	$-0.224990\pi$
$241$	5690.00	1.52085	0.760426	+	0.649425i	$0.224990\pi$
$242$	0	0
$243$	243.000	0.0641500
$244$	0	0
$245$	882.000	0.229996
$246$	0	0
$247$	3128.00	0.805789
$248$	0	0
$249$	−684.000	−0.174083
$250$	0	0
$251$	−180.000	−0.0452649	−0.0226325	−	0.999744i	$-0.507205\pi$
$251$	−180.000	−0.0452649	−0.0226325	+	0.999744i	$0.507205\pi$
$252$	0	0
$253$	12960.0	3.22051
$254$	0	0
$255$	324.000	0.0795673
$256$	0	0
$257$	5310.00	1.28883	0.644414	−	0.764677i	$-0.277101\pi$
$257$	5310.00	1.28883	0.644414	+	0.764677i	$0.277101\pi$
$258$	0	0
$259$	238.000	0.0570988
$260$	0	0
$261$	−1026.00	−0.243325
$262$	0	0
$263$	−828.000	−0.194132	−0.0970659	−	0.995278i	$-0.530946\pi$
$263$	−828.000	−0.194132	−0.0970659	+	0.995278i	$0.530946\pi$
$264$	0	0
$265$	11772.0	2.72886
$266$	0	0
$267$	1170.00	0.268175
$268$	0	0
$269$	−4134.00	−0.937005	−0.468503	−	0.883462i	$-0.655206\pi$
$269$	−4134.00	−0.937005	−0.468503	+	0.883462i	$0.655206\pi$
$270$	0	0
$271$	2968.00	0.665288	0.332644	−	0.943052i	$-0.392059\pi$
$271$	2968.00	0.665288	0.332644	+	0.943052i	$0.392059\pi$
$272$	0	0
$273$	714.000	0.158290
$274$	0	0
$275$	14328.0	3.14186
$276$	0	0
$277$	−4786.00	−1.03813	−0.519067	−	0.854734i	$-0.673720\pi$
$277$	−4786.00	−1.03813	−0.519067	+	0.854734i	$0.673720\pi$
$278$	0	0
$279$	−504.000	−0.108149
$280$	0	0
$281$	−4398.00	−0.933675	−0.466838	−	0.884343i	$-0.654607\pi$
$281$	−4398.00	−0.933675	−0.466838	+	0.884343i	$0.654607\pi$
$282$	0	0
$283$	−4772.00	−1.00235	−0.501177	−	0.865345i	$-0.667099\pi$
$283$	−4772.00	−1.00235	−0.501177	+	0.865345i	$0.667099\pi$
$284$	0	0
$285$	−4968.00	−1.03256
$286$	0	0
$287$	−42.0000	−0.00863826
$288$	0	0
$289$	−4877.00	−0.992673
$290$	0	0
$291$	−210.000	−0.0423038
$292$	0	0
$293$	6522.00	1.30041	0.650204	−	0.759760i	$-0.274684\pi$
$293$	6522.00	1.30041	0.650204	+	0.759760i	$0.274684\pi$
$294$	0	0
$295$	8856.00	1.74785
$296$	0	0
$297$	1944.00	0.379806
$298$	0	0
$299$	−6120.00	−1.18371
$300$	0	0
$301$	1148.00	0.219833
$302$	0	0
$303$	−4050.00	−0.767876
$304$	0	0
$305$	−4500.00	−0.844817
$306$	0	0
$307$	6244.00	1.16079	0.580397	−	0.814333i	$-0.302897\pi$
$307$	6244.00	1.16079	0.580397	+	0.814333i	$0.302897\pi$
$308$	0	0
$309$	−6000.00	−1.10462
$310$	0	0
$311$	528.000	0.0962705	0.0481353	−	0.998841i	$-0.484672\pi$
$311$	528.000	0.0962705	0.0481353	+	0.998841i	$0.484672\pi$
$312$	0	0
$313$	−5830.00	−1.05281	−0.526407	−	0.850232i	$-0.676461\pi$
$313$	−5830.00	−1.05281	−0.526407	+	0.850232i	$0.676461\pi$
$314$	0	0
$315$	−1134.00	−0.202837
$316$	0	0
$317$	5046.00	0.894043	0.447021	−	0.894523i	$-0.352485\pi$
$317$	5046.00	0.894043	0.447021	+	0.894523i	$0.352485\pi$
$318$	0	0
$319$	−8208.00	−1.44063
$320$	0	0
$321$	−2088.00	−0.363055
$322$	0	0
$323$	−552.000	−0.0950901
$324$	0	0
$325$	−6766.00	−1.15480
$326$	0	0
$327$	−3342.00	−0.565177
$328$	0	0
$329$	1176.00	0.197067
$330$	0	0
$331$	5020.00	0.833608	0.416804	−	0.908996i	$-0.363150\pi$
$331$	5020.00	0.833608	0.416804	+	0.908996i	$0.363150\pi$
$332$	0	0
$333$	−306.000	−0.0503564
$334$	0	0
$335$	2232.00	0.364021
$336$	0	0
$337$	−7486.00	−1.21005	−0.605027	−	0.796205i	$-0.706838\pi$
$337$	−7486.00	−1.21005	−0.605027	+	0.796205i	$0.706838\pi$
$338$	0	0
$339$	−1386.00	−0.222057
$340$	0	0
$341$	−4032.00	−0.640308
$342$	0	0
$343$	−343.000	−0.0539949
$344$	0	0
$345$	9720.00	1.51683
$346$	0	0
$347$	10032.0	1.55201	0.776003	−	0.630729i	$-0.217244\pi$
$347$	10032.0	1.55201	0.776003	+	0.630729i	$0.217244\pi$
$348$	0	0
$349$	5942.00	0.911370	0.455685	−	0.890141i	$-0.349394\pi$
$349$	5942.00	0.911370	0.455685	+	0.890141i	$0.349394\pi$
$350$	0	0
$351$	−918.000	−0.139599
$352$	0	0
$353$	−90.0000	−0.0135700	−0.00678501	−	0.999977i	$-0.502160\pi$
$353$	−90.0000	−0.0135700	−0.00678501	+	0.999977i	$0.502160\pi$
$354$	0	0
$355$	−648.000	−0.0968796
$356$	0	0
$357$	−126.000	−0.0186796
$358$	0	0
$359$	−10596.0	−1.55776	−0.778880	−	0.627174i	$-0.784212\pi$
$359$	−10596.0	−1.55776	−0.778880	+	0.627174i	$0.784212\pi$
$360$	0	0
$361$	1605.00	0.233999
$362$	0	0
$363$	11559.0	1.67132
$364$	0	0
$365$	18180.0	2.60708
$366$	0	0
$367$	−4016.00	−0.571208	−0.285604	−	0.958348i	$-0.592194\pi$
$367$	−4016.00	−0.571208	−0.285604	+	0.958348i	$0.592194\pi$
$368$	0	0
$369$	54.0000	0.00761823
$370$	0	0
$371$	−4578.00	−0.640641
$372$	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
$375$	3996.00	0.550273
$376$	0	0
$377$	3876.00	0.529507
$378$	0	0
$379$	−4628.00	−0.627241	−0.313621	−	0.949548i	$-0.601542\pi$
$379$	−4628.00	−0.627241	−0.313621	+	0.949548i	$0.601542\pi$
$380$	0	0
$381$	−3192.00	−0.429216
$382$	0	0
$383$	2880.00	0.384233	0.192116	−	0.981372i	$-0.438465\pi$
$383$	2880.00	0.384233	0.192116	+	0.981372i	$0.438465\pi$
$384$	0	0
$385$	−9072.00	−1.20091
$386$	0	0
$387$	−1476.00	−0.193874
$388$	0	0
$389$	7974.00	1.03933	0.519663	−	0.854371i	$-0.326057\pi$
$389$	7974.00	1.03933	0.519663	+	0.854371i	$0.326057\pi$
$390$	0	0
$391$	1080.00	0.139688
$392$	0	0
$393$	−540.000	−0.0693114
$394$	0	0
$395$	−1008.00	−0.128400
$396$	0	0
$397$	−12346.0	−1.56078	−0.780388	−	0.625296i	$-0.784978\pi$
$397$	−12346.0	−1.56078	−0.780388	+	0.625296i	$0.784978\pi$
$398$	0	0
$399$	1932.00	0.242408
$400$	0	0
$401$	9738.00	1.21270	0.606350	−	0.795198i	$-0.292633\pi$
$401$	9738.00	1.21270	0.606350	+	0.795198i	$0.292633\pi$
$402$	0	0
$403$	1904.00	0.235347
$404$	0	0
$405$	1458.00	0.178885
$406$	0	0
$407$	−2448.00	−0.298140
$408$	0	0
$409$	−430.000	−0.0519857	−0.0259928	−	0.999662i	$-0.508275\pi$
$409$	−430.000	−0.0519857	−0.0259928	+	0.999662i	$0.508275\pi$
$410$	0	0
$411$	−8154.00	−0.978606
$412$	0	0
$413$	−3444.00	−0.410335
$414$	0	0
$415$	−4104.00	−0.485440
$416$	0	0
$417$	4044.00	0.474905
$418$	0	0
$419$	1812.00	0.211270	0.105635	−	0.994405i	$-0.466313\pi$
$419$	1812.00	0.211270	0.105635	+	0.994405i	$0.466313\pi$
$420$	0	0
$421$	−10690.0	−1.23753	−0.618763	−	0.785577i	$-0.712366\pi$
$421$	−10690.0	−1.23753	−0.618763	+	0.785577i	$0.712366\pi$
$422$	0	0
$423$	−1512.00	−0.173797
$424$	0	0
$425$	1194.00	0.136276
$426$	0	0
$427$	1750.00	0.198334
$428$	0	0
$429$	−7344.00	−0.826507
$430$	0	0
$431$	4116.00	0.460002	0.230001	−	0.973190i	$-0.426127\pi$
$431$	4116.00	0.460002	0.230001	+	0.973190i	$0.426127\pi$
$432$	0	0
$433$	9938.00	1.10298	0.551489	−	0.834182i	$-0.314060\pi$
$433$	9938.00	1.10298	0.551489	+	0.834182i	$0.314060\pi$
$434$	0	0
$435$	−6156.00	−0.678523
$436$	0	0
$437$	−16560.0	−1.81275
$438$	0	0
$439$	−1784.00	−0.193954	−0.0969769	−	0.995287i	$-0.530917\pi$
$439$	−1784.00	−0.193954	−0.0969769	+	0.995287i	$0.530917\pi$
$440$	0	0
$441$	441.000	0.0476190
$442$	0	0
$443$	11712.0	1.25610	0.628052	−	0.778172i	$-0.283853\pi$
$443$	11712.0	1.25610	0.628052	+	0.778172i	$0.283853\pi$
$444$	0	0
$445$	7020.00	0.747820
$446$	0	0
$447$	1674.00	0.177131
$448$	0	0
$449$	7650.00	0.804066	0.402033	−	0.915625i	$-0.368304\pi$
$449$	7650.00	0.804066	0.402033	+	0.915625i	$0.368304\pi$
$450$	0	0
$451$	432.000	0.0451044
$452$	0	0
$453$	−5784.00	−0.599903
$454$	0	0
$455$	4284.00	0.441400
$456$	0	0
$457$	3674.00	0.376067	0.188033	−	0.982163i	$-0.439789\pi$
$457$	3674.00	0.376067	0.188033	+	0.982163i	$0.439789\pi$
$458$	0	0
$459$	162.000	0.0164739
$460$	0	0
$461$	−3102.00	−0.313394	−0.156697	−	0.987647i	$-0.550085\pi$
$461$	−3102.00	−0.313394	−0.156697	+	0.987647i	$0.550085\pi$
$462$	0	0
$463$	−8984.00	−0.901775	−0.450888	−	0.892581i	$-0.648892\pi$
$463$	−8984.00	−0.901775	−0.450888	+	0.892581i	$0.648892\pi$
$464$	0	0
$465$	−3024.00	−0.301580
$466$	0	0
$467$	−3612.00	−0.357909	−0.178954	−	0.983857i	$-0.557271\pi$
$467$	−3612.00	−0.357909	−0.178954	+	0.983857i	$0.557271\pi$
$468$	0	0
$469$	−868.000	−0.0854595
$470$	0	0
$471$	−7230.00	−0.707305
$472$	0	0
$473$	−11808.0	−1.14785
$474$	0	0
$475$	−18308.0	−1.76848
$476$	0	0
$477$	5886.00	0.564993
$478$	0	0
$479$	9288.00	0.885970	0.442985	−	0.896529i	$-0.353920\pi$
$479$	9288.00	0.885970	0.442985	+	0.896529i	$0.353920\pi$
$480$	0	0
$481$	1156.00	0.109582
$482$	0	0
$483$	−3780.00	−0.356099
$484$	0	0
$485$	−1260.00	−0.117966
$486$	0	0
$487$	5848.00	0.544144	0.272072	−	0.962277i	$-0.412291\pi$
$487$	5848.00	0.544144	0.272072	+	0.962277i	$0.412291\pi$
$488$	0	0
$489$	−2220.00	−0.205300
$490$	0	0
$491$	5952.00	0.547067	0.273534	−	0.961862i	$-0.411808\pi$
$491$	5952.00	0.547067	0.273534	+	0.961862i	$0.411808\pi$
$492$	0	0
$493$	−684.000	−0.0624864
$494$	0	0
$495$	11664.0	1.05911
$496$	0	0
$497$	252.000	0.0227440
$498$	0	0
$499$	−10748.0	−0.964222	−0.482111	−	0.876110i	$-0.660130\pi$
$499$	−10748.0	−0.964222	−0.482111	+	0.876110i	$0.660130\pi$
$500$	0	0
$501$	−11952.0	−1.06582
$502$	0	0
$503$	16488.0	1.46156	0.730779	−	0.682614i	$-0.239157\pi$
$503$	16488.0	1.46156	0.730779	+	0.682614i	$0.239157\pi$
$504$	0	0
$505$	−24300.0	−2.14126
$506$	0	0
$507$	−3123.00	−0.273565
$508$	0	0
$509$	14058.0	1.22418	0.612092	−	0.790786i	$-0.290328\pi$
$509$	14058.0	1.22418	0.612092	+	0.790786i	$0.290328\pi$
$510$	0	0
$511$	−7070.00	−0.612052
$512$	0	0
$513$	−2484.00	−0.213784
$514$	0	0
$515$	−36000.0	−3.08029
$516$	0	0
$517$	−12096.0	−1.02898
$518$	0	0
$519$	−3114.00	−0.263371
$520$	0	0
$521$	−14466.0	−1.21644	−0.608222	−	0.793767i	$-0.708117\pi$
$521$	−14466.0	−1.21644	−0.608222	+	0.793767i	$0.708117\pi$
$522$	0	0
$523$	−18524.0	−1.54875	−0.774377	−	0.632725i	$-0.781936\pi$
$523$	−18524.0	−1.54875	−0.774377	+	0.632725i	$0.781936\pi$
$524$	0	0
$525$	−4179.00	−0.347403
$526$	0	0
$527$	−336.000	−0.0277730
$528$	0	0
$529$	20233.0	1.66294
$530$	0	0
$531$	4428.00	0.361881
$532$	0	0
$533$	−204.000	−0.0165783
$534$	0	0
$535$	−12528.0	−1.01240
$536$	0	0
$537$	7704.00	0.619092
$538$	0	0
$539$	3528.00	0.281933
$540$	0	0
$541$	4358.00	0.346331	0.173165	−	0.984893i	$-0.444600\pi$
$541$	4358.00	0.346331	0.173165	+	0.984893i	$0.444600\pi$
$542$	0	0
$543$	−8094.00	−0.639681
$544$	0	0
$545$	−20052.0	−1.57602
$546$	0	0
$547$	2140.00	0.167276	0.0836378	−	0.996496i	$-0.473346\pi$
$547$	2140.00	0.167276	0.0836378	+	0.996496i	$0.473346\pi$
$548$	0	0
$549$	−2250.00	−0.174914
$550$	0	0
$551$	10488.0	0.810896
$552$	0	0
$553$	392.000	0.0301438
$554$	0	0
$555$	−1836.00	−0.140421
$556$	0	0
$557$	2022.00	0.153815	0.0769074	−	0.997038i	$-0.475495\pi$
$557$	2022.00	0.153815	0.0769074	+	0.997038i	$0.475495\pi$
$558$	0	0
$559$	5576.00	0.421896
$560$	0	0
$561$	1296.00	0.0975350
$562$	0	0
$563$	−7356.00	−0.550654	−0.275327	−	0.961351i	$-0.588786\pi$
$563$	−7356.00	−0.550654	−0.275327	+	0.961351i	$0.588786\pi$
$564$	0	0
$565$	−8316.00	−0.619215
$566$	0	0
$567$	−567.000	−0.0419961
$568$	0	0
$569$	11202.0	0.825329	0.412665	−	0.910883i	$-0.364598\pi$
$569$	11202.0	0.825329	0.412665	+	0.910883i	$0.364598\pi$
$570$	0	0
$571$	10564.0	0.774238	0.387119	−	0.922030i	$-0.373470\pi$
$571$	10564.0	0.774238	0.387119	+	0.922030i	$0.373470\pi$
$572$	0	0
$573$	12348.0	0.900253
$574$	0	0
$575$	35820.0	2.59791
$576$	0	0
$577$	−18574.0	−1.34011	−0.670057	−	0.742310i	$-0.733730\pi$
$577$	−18574.0	−1.34011	−0.670057	+	0.742310i	$0.733730\pi$
$578$	0	0
$579$	−9930.00	−0.712740
$580$	0	0
$581$	1596.00	0.113964
$582$	0	0
$583$	47088.0	3.34509
$584$	0	0
$585$	−5508.00	−0.389278
$586$	0	0
$587$	−13188.0	−0.927303	−0.463652	−	0.886018i	$-0.653461\pi$
$587$	−13188.0	−0.927303	−0.463652	+	0.886018i	$0.653461\pi$
$588$	0	0
$589$	5152.00	0.360415
$590$	0	0
$591$	3834.00	0.266852
$592$	0	0
$593$	−22506.0	−1.55853	−0.779267	−	0.626692i	$-0.784408\pi$
$593$	−22506.0	−1.55853	−0.779267	+	0.626692i	$0.784408\pi$
$594$	0	0
$595$	−756.000	−0.0520890
$596$	0	0
$597$	−8808.00	−0.603832
$598$	0	0
$599$	−10596.0	−0.722773	−0.361386	−	0.932416i	$-0.617696\pi$
$599$	−10596.0	−0.722773	−0.361386	+	0.932416i	$0.617696\pi$
$600$	0	0
$601$	14618.0	0.992148	0.496074	−	0.868280i	$-0.334775\pi$
$601$	14618.0	0.992148	0.496074	+	0.868280i	$0.334775\pi$
$602$	0	0
$603$	1116.00	0.0753682
$604$	0	0
$605$	69354.0	4.66056
$606$	0	0
$607$	−5168.00	−0.345573	−0.172786	−	0.984959i	$-0.555277\pi$
$607$	−5168.00	−0.345573	−0.172786	+	0.984959i	$0.555277\pi$
$608$	0	0
$609$	2394.00	0.159294
$610$	0	0
$611$	5712.00	0.378204
$612$	0	0
$613$	5726.00	0.377277	0.188639	−	0.982047i	$-0.439593\pi$
$613$	5726.00	0.377277	0.188639	+	0.982047i	$0.439593\pi$
$614$	0	0
$615$	324.000	0.0212438
$616$	0	0
$617$	−7806.00	−0.509332	−0.254666	−	0.967029i	$-0.581965\pi$
$617$	−7806.00	−0.509332	−0.254666	+	0.967029i	$0.581965\pi$
$618$	0	0
$619$	18052.0	1.17217	0.586083	−	0.810251i	$-0.300669\pi$
$619$	18052.0	1.17217	0.586083	+	0.810251i	$0.300669\pi$
$620$	0	0
$621$	4860.00	0.314050
$622$	0	0
$623$	−2730.00	−0.175562
$624$	0	0
$625$	−899.000	−0.0575360
$626$	0	0
$627$	−19872.0	−1.26573
$628$	0	0
$629$	−204.000	−0.0129317
$630$	0	0
$631$	6208.00	0.391659	0.195829	−	0.980638i	$-0.437260\pi$
$631$	6208.00	0.391659	0.195829	+	0.980638i	$0.437260\pi$
$632$	0	0
$633$	10524.0	0.660808
$634$	0	0
$635$	−19152.0	−1.19689
$636$	0	0
$637$	−1666.00	−0.103625
$638$	0	0
$639$	−324.000	−0.0200583
$640$	0	0
$641$	−21510.0	−1.32542	−0.662710	−	0.748876i	$-0.730594\pi$
$641$	−21510.0	−1.32542	−0.662710	+	0.748876i	$0.730594\pi$
$642$	0	0
$643$	11140.0	0.683233	0.341616	−	0.939839i	$-0.389026\pi$
$643$	11140.0	0.683233	0.341616	+	0.939839i	$0.389026\pi$
$644$	0	0
$645$	−8856.00	−0.540627
$646$	0	0
$647$	−9312.00	−0.565831	−0.282915	−	0.959145i	$-0.591302\pi$
$647$	−9312.00	−0.565831	−0.282915	+	0.959145i	$0.591302\pi$
$648$	0	0
$649$	35424.0	2.14255
$650$	0	0
$651$	1176.00	0.0708004
$652$	0	0
$653$	4878.00	0.292329	0.146165	−	0.989260i	$-0.453307\pi$
$653$	4878.00	0.292329	0.146165	+	0.989260i	$0.453307\pi$
$654$	0	0
$655$	−3240.00	−0.193278
$656$	0	0
$657$	9090.00	0.539779
$658$	0	0
$659$	9744.00	0.575982	0.287991	−	0.957633i	$-0.407013\pi$
$659$	9744.00	0.575982	0.287991	+	0.957633i	$0.407013\pi$
$660$	0	0
$661$	2990.00	0.175942	0.0879709	−	0.996123i	$-0.471962\pi$
$661$	2990.00	0.175942	0.0879709	+	0.996123i	$0.471962\pi$
$662$	0	0
$663$	−612.000	−0.0358493
$664$	0	0
$665$	11592.0	0.675968
$666$	0	0
$667$	−20520.0	−1.19121
$668$	0	0
$669$	5664.00	0.327329
$670$	0	0
$671$	−18000.0	−1.03559
$672$	0	0
$673$	33266.0	1.90536	0.952682	−	0.303969i	$-0.0983118\pi$
$673$	33266.0	1.90536	0.952682	+	0.303969i	$0.0983118\pi$
$674$	0	0
$675$	5373.00	0.306381
$676$	0	0
$677$	5370.00	0.304854	0.152427	−	0.988315i	$-0.451291\pi$
$677$	5370.00	0.304854	0.152427	+	0.988315i	$0.451291\pi$
$678$	0	0
$679$	490.000	0.0276944
$680$	0	0
$681$	−10692.0	−0.601642
$682$	0	0
$683$	−384.000	−0.0215130	−0.0107565	−	0.999942i	$-0.503424\pi$
$683$	−384.000	−0.0215130	−0.0107565	+	0.999942i	$0.503424\pi$
$684$	0	0
$685$	−48924.0	−2.72889
$686$	0	0
$687$	4002.00	0.222250
$688$	0	0
$689$	−22236.0	−1.22950
$690$	0	0
$691$	14524.0	0.799593	0.399797	−	0.916604i	$-0.369081\pi$
$691$	14524.0	0.799593	0.399797	+	0.916604i	$0.369081\pi$
$692$	0	0
$693$	−4536.00	−0.248641
$694$	0	0
$695$	24264.0	1.32430
$696$	0	0
$697$	36.0000	0.00195638
$698$	0	0
$699$	7974.00	0.431480
$700$	0	0
$701$	24750.0	1.33352	0.666758	−	0.745274i	$-0.267682\pi$
$701$	24750.0	1.33352	0.666758	+	0.745274i	$0.267682\pi$
$702$	0	0
$703$	3128.00	0.167816
$704$	0	0
$705$	−9072.00	−0.484640
$706$	0	0
$707$	9450.00	0.502693
$708$	0	0
$709$	−1042.00	−0.0551948	−0.0275974	−	0.999619i	$-0.508786\pi$
$709$	−1042.00	−0.0551948	−0.0275974	+	0.999619i	$0.508786\pi$
$710$	0	0
$711$	−504.000	−0.0265844
$712$	0	0
$713$	−10080.0	−0.529452
$714$	0	0
$715$	−44064.0	−2.30476
$716$	0	0
$717$	1764.00	0.0918798
$718$	0	0
$719$	36960.0	1.91707	0.958536	−	0.284970i	$-0.0919836\pi$
$719$	36960.0	1.91707	0.958536	+	0.284970i	$0.0919836\pi$
$720$	0	0
$721$	14000.0	0.723145
$722$	0	0
$723$	17070.0	0.878064
$724$	0	0
$725$	−22686.0	−1.16212
$726$	0	0
$727$	16288.0	0.830933	0.415467	−	0.909608i	$-0.363618\pi$
$727$	16288.0	0.830933	0.415467	+	0.909608i	$0.363618\pi$
$728$	0	0
$729$	729.000	0.0370370
$730$	0	0
$731$	−984.000	−0.0497874
$732$	0	0
$733$	−7810.00	−0.393546	−0.196773	−	0.980449i	$-0.563046\pi$
$733$	−7810.00	−0.393546	−0.196773	+	0.980449i	$0.563046\pi$
$734$	0	0
$735$	2646.00	0.132788
$736$	0	0
$737$	8928.00	0.446224
$738$	0	0
$739$	36700.0	1.82684	0.913418	−	0.407024i	$-0.133433\pi$
$739$	36700.0	1.82684	0.913418	+	0.407024i	$0.133433\pi$
$740$	0	0
$741$	9384.00	0.465222
$742$	0	0
$743$	−29508.0	−1.45699	−0.728495	−	0.685051i	$-0.759780\pi$
$743$	−29508.0	−1.45699	−0.728495	+	0.685051i	$0.759780\pi$
$744$	0	0
$745$	10044.0	0.493938
$746$	0	0
$747$	−2052.00	−0.100507
$748$	0	0
$749$	4872.00	0.237676
$750$	0	0
$751$	15136.0	0.735447	0.367723	−	0.929935i	$-0.380137\pi$
$751$	15136.0	0.735447	0.367723	+	0.929935i	$0.380137\pi$
$752$	0	0
$753$	−540.000	−0.0261337
$754$	0	0
$755$	−34704.0	−1.67286
$756$	0	0
$757$	3422.00	0.164299	0.0821497	−	0.996620i	$-0.473821\pi$
$757$	3422.00	0.164299	0.0821497	+	0.996620i	$0.473821\pi$
$758$	0	0
$759$	38880.0	1.85936
$760$	0	0
$761$	31446.0	1.49792	0.748960	−	0.662616i	$-0.230554\pi$
$761$	31446.0	1.49792	0.748960	+	0.662616i	$0.230554\pi$
$762$	0	0
$763$	7798.00	0.369995
$764$	0	0
$765$	972.000	0.0459382
$766$	0	0
$767$	−16728.0	−0.787501
$768$	0	0
$769$	−18718.0	−0.877748	−0.438874	−	0.898549i	$-0.644623\pi$
$769$	−18718.0	−0.877748	−0.438874	+	0.898549i	$0.644623\pi$
$770$	0	0
$771$	15930.0	0.744105
$772$	0	0
$773$	−1686.00	−0.0784492	−0.0392246	−	0.999230i	$-0.512489\pi$
$773$	−1686.00	−0.0784492	−0.0392246	+	0.999230i	$0.512489\pi$
$774$	0	0
$775$	−11144.0	−0.516522
$776$	0	0
$777$	714.000	0.0329660
$778$	0	0
$779$	−552.000	−0.0253883
$780$	0	0
$781$	−2592.00	−0.118757
$782$	0	0
$783$	−3078.00	−0.140484
$784$	0	0
$785$	−43380.0	−1.97235
$786$	0	0
$787$	−5492.00	−0.248753	−0.124377	−	0.992235i	$-0.539693\pi$
$787$	−5492.00	−0.248753	−0.124377	+	0.992235i	$0.539693\pi$
$788$	0	0
$789$	−2484.00	−0.112082
$790$	0	0
$791$	3234.00	0.145370
$792$	0	0
$793$	8500.00	0.380635
$794$	0	0
$795$	35316.0	1.57551
$796$	0	0
$797$	−17310.0	−0.769325	−0.384662	−	0.923057i	$-0.625682\pi$
$797$	−17310.0	−0.769325	−0.384662	+	0.923057i	$0.625682\pi$
$798$	0	0
$799$	−1008.00	−0.0446314
$800$	0	0
$801$	3510.00	0.154831
$802$	0	0
$803$	72720.0	3.19581
$804$	0	0
$805$	−22680.0	−0.993000
$806$	0	0
$807$	−12402.0	−0.540980
$808$	0	0
$809$	35754.0	1.55382	0.776912	−	0.629609i	$-0.216785\pi$
$809$	35754.0	1.55382	0.776912	+	0.629609i	$0.216785\pi$
$810$	0	0
$811$	−33644.0	−1.45672	−0.728360	−	0.685194i	$-0.759717\pi$
$811$	−33644.0	−1.45672	−0.728360	+	0.685194i	$0.759717\pi$
$812$	0	0
$813$	8904.00	0.384104
$814$	0	0
$815$	−13320.0	−0.572490
$816$	0	0
$817$	15088.0	0.646098
$818$	0	0
$819$	2142.00	0.0913889
$820$	0	0
$821$	28734.0	1.22147	0.610733	−	0.791837i	$-0.290875\pi$
$821$	28734.0	1.22147	0.610733	+	0.791837i	$0.290875\pi$
$822$	0	0
$823$	28672.0	1.21439	0.607195	−	0.794553i	$-0.292295\pi$
$823$	28672.0	1.21439	0.607195	+	0.794553i	$0.292295\pi$
$824$	0	0
$825$	42984.0	1.81395
$826$	0	0
$827$	15912.0	0.669062	0.334531	−	0.942385i	$-0.391422\pi$
$827$	15912.0	0.669062	0.334531	+	0.942385i	$0.391422\pi$
$828$	0	0
$829$	17534.0	0.734597	0.367299	−	0.930103i	$-0.380283\pi$
$829$	17534.0	0.734597	0.367299	+	0.930103i	$0.380283\pi$
$830$	0	0
$831$	−14358.0	−0.599366
$832$	0	0
$833$	294.000	0.0122287
$834$	0	0
$835$	−71712.0	−2.97209
$836$	0	0
$837$	−1512.00	−0.0624401
$838$	0	0
$839$	−40656.0	−1.67295	−0.836473	−	0.548009i	$-0.815386\pi$
$839$	−40656.0	−1.67295	−0.836473	+	0.548009i	$0.815386\pi$
$840$	0	0
$841$	−11393.0	−0.467137
$842$	0	0
$843$	−13194.0	−0.539058
$844$	0	0
$845$	−18738.0	−0.762848
$846$	0	0
$847$	−26971.0	−1.09414
$848$	0	0
$849$	−14316.0	−0.578709
$850$	0	0
$851$	−6120.00	−0.246523
$852$	0	0
$853$	23870.0	0.958140	0.479070	−	0.877777i	$-0.340974\pi$
$853$	23870.0	0.958140	0.479070	+	0.877777i	$0.340974\pi$
$854$	0	0
$855$	−14904.0	−0.596147
$856$	0	0
$857$	−29610.0	−1.18023	−0.590116	−	0.807319i	$-0.700918\pi$
$857$	−29610.0	−1.18023	−0.590116	+	0.807319i	$0.700918\pi$
$858$	0	0
$859$	45484.0	1.80663	0.903314	−	0.428979i	$-0.141127\pi$
$859$	45484.0	1.80663	0.903314	+	0.428979i	$0.141127\pi$
$860$	0	0
$861$	−126.000	−0.00498730
$862$	0	0
$863$	−46164.0	−1.82090	−0.910452	−	0.413614i	$-0.864266\pi$
$863$	−46164.0	−1.82090	−0.910452	+	0.413614i	$0.864266\pi$
$864$	0	0
$865$	−18684.0	−0.734422
$866$	0	0
$867$	−14631.0	−0.573120
$868$	0	0
$869$	−4032.00	−0.157395
$870$	0	0
$871$	−4216.00	−0.164011
$872$	0	0
$873$	−630.000	−0.0244241
$874$	0	0
$875$	−9324.00	−0.360239
$876$	0	0
$877$	−2986.00	−0.114972	−0.0574858	−	0.998346i	$-0.518308\pi$
$877$	−2986.00	−0.114972	−0.0574858	+	0.998346i	$0.518308\pi$
$878$	0	0
$879$	19566.0	0.750790
$880$	0	0
$881$	6534.00	0.249871	0.124935	−	0.992165i	$-0.460128\pi$
$881$	6534.00	0.249871	0.124935	+	0.992165i	$0.460128\pi$
$882$	0	0
$883$	−29756.0	−1.13405	−0.567027	−	0.823699i	$-0.691906\pi$
$883$	−29756.0	−1.13405	−0.567027	+	0.823699i	$0.691906\pi$
$884$	0	0
$885$	26568.0	1.00912
$886$	0	0
$887$	−29952.0	−1.13381	−0.566905	−	0.823783i	$-0.691859\pi$
$887$	−29952.0	−1.13381	−0.566905	+	0.823783i	$0.691859\pi$
$888$	0	0
$889$	7448.00	0.280988
$890$	0	0
$891$	5832.00	0.219281
$892$	0	0
$893$	15456.0	0.579188
$894$	0	0
$895$	46224.0	1.72637
$896$	0	0
$897$	−18360.0	−0.683414
$898$	0	0
$899$	6384.00	0.236839
$900$	0	0
$901$	3924.00	0.145091
$902$	0	0
$903$	3444.00	0.126920
$904$	0	0
$905$	−48564.0	−1.78378
$906$	0	0
$907$	36268.0	1.32774	0.663869	−	0.747848i	$-0.268913\pi$
$907$	36268.0	1.32774	0.663869	+	0.747848i	$0.268913\pi$
$908$	0	0
$909$	−12150.0	−0.443333
$910$	0	0
$911$	23604.0	0.858436	0.429218	−	0.903201i	$-0.358789\pi$
$911$	23604.0	0.858436	0.429218	+	0.903201i	$0.358789\pi$
$912$	0	0
$913$	−16416.0	−0.595061
$914$	0	0
$915$	−13500.0	−0.487755
$916$	0	0
$917$	1260.00	0.0453750
$918$	0	0
$919$	−34184.0	−1.22701	−0.613507	−	0.789689i	$-0.710242\pi$
$919$	−34184.0	−1.22701	−0.613507	+	0.789689i	$0.710242\pi$
$920$	0	0
$921$	18732.0	0.670185
$922$	0	0
$923$	1224.00	0.0436495
$924$	0	0
$925$	−6766.00	−0.240502
$926$	0	0
$927$	−18000.0	−0.637754
$928$	0	0
$929$	−53922.0	−1.90433	−0.952165	−	0.305583i	$-0.901149\pi$
$929$	−53922.0	−1.90433	−0.952165	+	0.305583i	$0.901149\pi$
$930$	0	0
$931$	−4508.00	−0.158694
$932$	0	0
$933$	1584.00	0.0555818
$934$	0	0
$935$	7776.00	0.271981
$936$	0	0
$937$	40538.0	1.41336	0.706680	−	0.707533i	$-0.250192\pi$
$937$	40538.0	1.41336	0.706680	+	0.707533i	$0.250192\pi$
$938$	0	0
$939$	−17490.0	−0.607843
$940$	0	0
$941$	−3606.00	−0.124923	−0.0624613	−	0.998047i	$-0.519895\pi$
$941$	−3606.00	−0.124923	−0.0624613	+	0.998047i	$0.519895\pi$
$942$	0	0
$943$	1080.00	0.0372955
$944$	0	0
$945$	−3402.00	−0.117108
$946$	0	0
$947$	−14064.0	−0.482596	−0.241298	−	0.970451i	$-0.577573\pi$
$947$	−14064.0	−0.482596	−0.241298	+	0.970451i	$0.577573\pi$
$948$	0	0
$949$	−34340.0	−1.17463
$950$	0	0
$951$	15138.0	0.516176
$952$	0	0
$953$	33066.0	1.12394	0.561969	−	0.827158i	$-0.310044\pi$
$953$	33066.0	1.12394	0.561969	+	0.827158i	$0.310044\pi$
$954$	0	0
$955$	74088.0	2.51040
$956$	0	0
$957$	−24624.0	−0.831746
$958$	0	0
$959$	19026.0	0.640648
$960$	0	0
$961$	−26655.0	−0.894733
$962$	0	0
$963$	−6264.00	−0.209610
$964$	0	0
$965$	−59580.0	−1.98751
$966$	0	0
$967$	26368.0	0.876875	0.438437	−	0.898762i	$-0.355532\pi$
$967$	26368.0	0.876875	0.438437	+	0.898762i	$0.355532\pi$
$968$	0	0
$969$	−1656.00	−0.0549003
$970$	0	0
$971$	−55884.0	−1.84696	−0.923482	−	0.383641i	$-0.874670\pi$
$971$	−55884.0	−1.84696	−0.923482	+	0.383641i	$0.874670\pi$
$972$	0	0
$973$	−9436.00	−0.310899
$974$	0	0
$975$	−20298.0	−0.666724
$976$	0	0
$977$	−51126.0	−1.67417	−0.837086	−	0.547072i	$-0.815743\pi$
$977$	−51126.0	−1.67417	−0.837086	+	0.547072i	$0.815743\pi$
$978$	0	0
$979$	28080.0	0.916691
$980$	0	0
$981$	−10026.0	−0.326305
$982$	0	0
$983$	14184.0	0.460223	0.230112	−	0.973164i	$-0.426091\pi$
$983$	14184.0	0.460223	0.230112	+	0.973164i	$0.426091\pi$
$984$	0	0
$985$	23004.0	0.744130
$986$	0	0
$987$	3528.00	0.113777
$988$	0	0
$989$	−29520.0	−0.949122
$990$	0	0
$991$	−51680.0	−1.65658	−0.828289	−	0.560301i	$-0.810686\pi$
$991$	−51680.0	−1.65658	−0.828289	+	0.560301i	$0.810686\pi$
$992$	0	0
$993$	15060.0	0.481284
$994$	0	0
$995$	−52848.0	−1.68381
$996$	0	0
$997$	52094.0	1.65480	0.827399	−	0.561615i	$-0.189820\pi$
$997$	52094.0	1.65480	0.827399	+	0.561615i	$0.189820\pi$
$998$	0	0
$999$	−918.000	−0.0290733

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	336.4.a.l.1.1		1
3.2	odd	2		1008.4.a.b.1.1		1
4.3	odd	2		42.4.a.a.1.1	✓	1
7.6	odd	2		2352.4.a.a.1.1		1
8.3	odd	2		1344.4.a.o.1.1		1
8.5	even	2		1344.4.a.a.1.1		1
12.11	even	2		126.4.a.a.1.1		1
20.3	even	4		1050.4.g.a.799.1		2
20.7	even	4		1050.4.g.a.799.2		2
20.19	odd	2		1050.4.a.g.1.1		1
28.3	even	6		294.4.e.b.79.1		2
28.11	odd	6		294.4.e.c.79.1		2
28.19	even	6		294.4.e.b.67.1		2
28.23	odd	6		294.4.e.c.67.1		2
28.27	even	2		294.4.a.i.1.1		1
84.11	even	6		882.4.g.w.667.1		2
84.23	even	6		882.4.g.w.361.1		2
84.47	odd	6		882.4.g.o.361.1		2
84.59	odd	6		882.4.g.o.667.1		2
84.83	odd	2		882.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.4.a.a.1.1	✓	1	4.3	odd	2
126.4.a.a.1.1		1	12.11	even	2
294.4.a.i.1.1		1	28.27	even	2
294.4.e.b.67.1		2	28.19	even	6
294.4.e.b.79.1		2	28.3	even	6
294.4.e.c.67.1		2	28.23	odd	6
294.4.e.c.79.1		2	28.11	odd	6
336.4.a.l.1.1		1	1.1	even	1	trivial
882.4.a.g.1.1		1	84.83	odd	2
882.4.g.o.361.1		2	84.47	odd	6
882.4.g.o.667.1		2	84.59	odd	6
882.4.g.w.361.1		2	84.23	even	6
882.4.g.w.667.1		2	84.11	even	6
1008.4.a.b.1.1		1	3.2	odd	2
1050.4.a.g.1.1		1	20.19	odd	2
1050.4.g.a.799.1		2	20.3	even	4
1050.4.g.a.799.2		2	20.7	even	4
1344.4.a.a.1.1		1	8.5	even	2
1344.4.a.o.1.1		1	8.3	odd	2
2352.4.a.a.1.1		1	7.6	odd	2

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Classical	Maass
Hilbert	Bianchi

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

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Fields

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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	336.4.a.l.1.1

Level	$336$
Weight	$4$
Character	336.1
Self dual	yes
Analytic conductor	$19.825$
Analytic rank	$0$
Dimension	$1$
CM	no
Inner twists	$1$