LMFDB - Embedded newform 1472.4.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1472,4,Mod(1,1472)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1472.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1472, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1472 = 2^{6} \cdot 23 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1472.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,-8,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	yes
Analytic conductor:	$86.8508115285$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 184)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1472.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q-8.00000 q^{3} +4.00000 q^{5} -4.00000 q^{7} +37.0000 q^{9} -26.0000 q^{11} -70.0000 q^{13} -32.0000 q^{15} +94.0000 q^{17} -54.0000 q^{19} +32.0000 q^{21} -23.0000 q^{23} -109.000 q^{25} -80.0000 q^{27} +86.0000 q^{29} -144.000 q^{31} +208.000 q^{33} -16.0000 q^{35} +172.000 q^{37} +560.000 q^{39} -42.0000 q^{41} -386.000 q^{43} +148.000 q^{45} -80.0000 q^{47} -327.000 q^{49} -752.000 q^{51} +108.000 q^{53} -104.000 q^{55} +432.000 q^{57} -164.000 q^{59} +400.000 q^{61} -148.000 q^{63} -280.000 q^{65} -398.000 q^{67} +184.000 q^{69} -320.000 q^{71} -810.000 q^{73} +872.000 q^{75} +104.000 q^{77} -204.000 q^{79} -359.000 q^{81} -102.000 q^{83} +376.000 q^{85} -688.000 q^{87} +1018.00 q^{89} +280.000 q^{91} +1152.00 q^{93} -216.000 q^{95} -1370.00 q^{97} -962.000 q^{99} +O(q^{100})$

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
$3$	−8.00000	−1.53960	−0.769800	−	0.638285i	$-0.779644\pi$
$3$	−8.00000	−1.53960	−0.769800	+	0.638285i	$0.779644\pi$
$4$	0	0
$5$	4.00000	0.357771	0.178885	−	0.983870i	$-0.442751\pi$
$5$	4.00000	0.357771	0.178885	+	0.983870i	$0.442751\pi$
$6$	0	0
$7$	−4.00000	−0.215980	−0.107990	−	0.994152i	$-0.534441\pi$
$7$	−4.00000	−0.215980	−0.107990	+	0.994152i	$0.534441\pi$
$8$	0	0
$9$	37.0000	1.37037
$10$	0	0
$11$	−26.0000	−0.712663	−0.356332	−	0.934360i	$-0.615973\pi$
$11$	−26.0000	−0.712663	−0.356332	+	0.934360i	$0.615973\pi$
$12$	0	0
$13$	−70.0000	−1.49342	−0.746712	−	0.665148i	$-0.768369\pi$
$13$	−70.0000	−1.49342	−0.746712	+	0.665148i	$0.768369\pi$
$14$	0	0
$15$	−32.0000	−0.550824
$16$	0	0
$17$	94.0000	1.34108	0.670540	−	0.741874i	$-0.266063\pi$
$17$	94.0000	1.34108	0.670540	+	0.741874i	$0.266063\pi$
$18$	0	0
$19$	−54.0000	−0.652024	−0.326012	−	0.945366i	$-0.605705\pi$
$19$	−54.0000	−0.652024	−0.326012	+	0.945366i	$0.605705\pi$
$20$	0	0
$21$	32.0000	0.332522
$22$	0	0
$23$	−23.0000	−0.208514
$23$	−23.0000	−0.208514
$24$	0	0
$25$	−109.000	−0.872000
$26$	0	0
$27$	−80.0000	−0.570222
$28$	0	0
$29$	86.0000	0.550683	0.275341	−	0.961347i	$-0.411209\pi$
$29$	86.0000	0.550683	0.275341	+	0.961347i	$0.411209\pi$
$30$	0	0
$31$	−144.000	−0.834296	−0.417148	−	0.908839i	$-0.636970\pi$
$31$	−144.000	−0.834296	−0.417148	+	0.908839i	$0.636970\pi$
$32$	0	0
$33$	208.000	1.09722
$34$	0	0
$35$	−16.0000	−0.0772712
$36$	0	0
$37$	172.000	0.764233	0.382117	−	0.924114i	$-0.375195\pi$
$37$	172.000	0.764233	0.382117	+	0.924114i	$0.375195\pi$
$38$	0	0
$39$	560.000	2.29928
$40$	0	0
$41$	−42.0000	−0.159983	−0.0799914	−	0.996796i	$-0.525489\pi$
$41$	−42.0000	−0.159983	−0.0799914	+	0.996796i	$0.525489\pi$
$42$	0	0
$43$	−386.000	−1.36894	−0.684470	−	0.729041i	$-0.739966\pi$
$43$	−386.000	−1.36894	−0.684470	+	0.729041i	$0.739966\pi$
$44$	0	0
$45$	148.000	0.490279
$46$	0	0
$47$	−80.0000	−0.248281	−0.124140	−	0.992265i	$-0.539617\pi$
$47$	−80.0000	−0.248281	−0.124140	+	0.992265i	$0.539617\pi$
$48$	0	0
$49$	−327.000	−0.953353
$50$	0	0
$51$	−752.000	−2.06473
$52$	0	0
$53$	108.000	0.279905	0.139952	−	0.990158i	$-0.455305\pi$
$53$	108.000	0.279905	0.139952	+	0.990158i	$0.455305\pi$
$54$	0	0
$55$	−104.000	−0.254970
$56$	0	0
$57$	432.000	1.00386
$58$	0	0
$59$	−164.000	−0.361881	−0.180941	−	0.983494i	$-0.557914\pi$
$59$	−164.000	−0.361881	−0.180941	+	0.983494i	$0.557914\pi$
$60$	0	0
$61$	400.000	0.839586	0.419793	−	0.907620i	$-0.362103\pi$
$61$	400.000	0.839586	0.419793	+	0.907620i	$0.362103\pi$
$62$	0	0
$63$	−148.000	−0.295972
$64$	0	0
$65$	−280.000	−0.534303
$66$	0	0
$67$	−398.000	−0.725723	−0.362861	−	0.931843i	$-0.618200\pi$
$67$	−398.000	−0.725723	−0.362861	+	0.931843i	$0.618200\pi$
$68$	0	0
$69$	184.000	0.321029
$70$	0	0
$71$	−320.000	−0.534888	−0.267444	−	0.963573i	$-0.586179\pi$
$71$	−320.000	−0.534888	−0.267444	+	0.963573i	$0.586179\pi$
$72$	0	0
$73$	−810.000	−1.29868	−0.649338	−	0.760500i	$-0.724954\pi$
$73$	−810.000	−1.29868	−0.649338	+	0.760500i	$0.724954\pi$
$74$	0	0
$75$	872.000	1.34253
$76$	0	0
$77$	104.000	0.153921
$78$	0	0
$79$	−204.000	−0.290529	−0.145265	−	0.989393i	$-0.546403\pi$
$79$	−204.000	−0.290529	−0.145265	+	0.989393i	$0.546403\pi$
$80$	0	0
$81$	−359.000	−0.492455
$82$	0	0
$83$	−102.000	−0.134891	−0.0674455	−	0.997723i	$-0.521485\pi$
$83$	−102.000	−0.134891	−0.0674455	+	0.997723i	$0.521485\pi$
$84$	0	0
$85$	376.000	0.479799
$86$	0	0
$87$	−688.000	−0.847832
$88$	0	0
$89$	1018.00	1.21245	0.606224	−	0.795294i	$-0.292684\pi$
$89$	1018.00	1.21245	0.606224	+	0.795294i	$0.292684\pi$
$90$	0	0
$91$	280.000	0.322549
$92$	0	0
$93$	1152.00	1.28448
$94$	0	0
$95$	−216.000	−0.233275
$96$	0	0
$97$	−1370.00	−1.43405	−0.717023	−	0.697050i	$-0.754496\pi$
$97$	−1370.00	−1.43405	−0.717023	+	0.697050i	$0.754496\pi$
$98$	0	0
$99$	−962.000	−0.976613
$100$	0	0
$101$	1330.00	1.31030	0.655148	−	0.755500i	$-0.272606\pi$
$101$	1330.00	1.31030	0.655148	+	0.755500i	$0.272606\pi$
$102$	0	0
$103$	−1160.00	−1.10969	−0.554846	−	0.831953i	$-0.687223\pi$
$103$	−1160.00	−1.10969	−0.554846	+	0.831953i	$0.687223\pi$
$104$	0	0
$105$	128.000	0.118967
$106$	0	0
$107$	402.000	0.363204	0.181602	−	0.983372i	$-0.441872\pi$
$107$	402.000	0.363204	0.181602	+	0.983372i	$0.441872\pi$
$108$	0	0
$109$	−2040.00	−1.79263	−0.896315	−	0.443419i	$-0.853765\pi$
$109$	−2040.00	−1.79263	−0.896315	+	0.443419i	$0.853765\pi$
$110$	0	0
$111$	−1376.00	−1.17661
$112$	0	0
$113$	1362.00	1.13386	0.566930	−	0.823766i	$-0.308131\pi$
$113$	1362.00	1.13386	0.566930	+	0.823766i	$0.308131\pi$
$114$	0	0
$115$	−92.0000	−0.0746004
$116$	0	0
$117$	−2590.00	−2.04654
$118$	0	0
$119$	−376.000	−0.289646
$120$	0	0
$121$	−655.000	−0.492111
$122$	0	0
$123$	336.000	0.246310
$124$	0	0
$125$	−936.000	−0.669747
$126$	0	0
$127$	1664.00	1.16265	0.581323	−	0.813673i	$-0.302535\pi$
$127$	1664.00	1.16265	0.581323	+	0.813673i	$0.302535\pi$
$128$	0	0
$129$	3088.00	2.10762
$130$	0	0
$131$	−2484.00	−1.65670	−0.828351	−	0.560209i	$-0.810721\pi$
$131$	−2484.00	−1.65670	−0.828351	+	0.560209i	$0.810721\pi$
$132$	0	0
$133$	216.000	0.140824
$134$	0	0
$135$	−320.000	−0.204009
$136$	0	0
$137$	1222.00	0.762062	0.381031	−	0.924562i	$-0.375569\pi$
$137$	1222.00	0.762062	0.381031	+	0.924562i	$0.375569\pi$
$138$	0	0
$139$	1900.00	1.15939	0.579697	−	0.814832i	$-0.303171\pi$
$139$	1900.00	1.15939	0.579697	+	0.814832i	$0.303171\pi$
$140$	0	0
$141$	640.000	0.382253
$142$	0	0
$143$	1820.00	1.06431
$144$	0	0
$145$	344.000	0.197018
$146$	0	0
$147$	2616.00	1.46778
$148$	0	0
$149$	3372.00	1.85399	0.926997	−	0.375070i	$-0.122381\pi$
$149$	3372.00	1.85399	0.926997	+	0.375070i	$0.122381\pi$
$150$	0	0
$151$	56.0000	0.0301802	0.0150901	−	0.999886i	$-0.495196\pi$
$151$	56.0000	0.0301802	0.0150901	+	0.999886i	$0.495196\pi$
$152$	0	0
$153$	3478.00	1.83778
$154$	0	0
$155$	−576.000	−0.298487
$156$	0	0
$157$	1116.00	0.567303	0.283651	−	0.958928i	$-0.408454\pi$
$157$	1116.00	0.567303	0.283651	+	0.958928i	$0.408454\pi$
$158$	0	0
$159$	−864.000	−0.430941
$160$	0	0
$161$	92.0000	0.0450349
$162$	0	0
$163$	−160.000	−0.0768845	−0.0384422	−	0.999261i	$-0.512240\pi$
$163$	−160.000	−0.0768845	−0.0384422	+	0.999261i	$0.512240\pi$
$164$	0	0
$165$	832.000	0.392552
$166$	0	0
$167$	4176.00	1.93502	0.967511	−	0.252830i	$-0.0813613\pi$
$167$	4176.00	1.93502	0.967511	+	0.252830i	$0.0813613\pi$
$168$	0	0
$169$	2703.00	1.23031
$170$	0	0
$171$	−1998.00	−0.893514
$172$	0	0
$173$	1214.00	0.533519	0.266759	−	0.963763i	$-0.414047\pi$
$173$	1214.00	0.533519	0.266759	+	0.963763i	$0.414047\pi$
$174$	0	0
$175$	436.000	0.188334
$176$	0	0
$177$	1312.00	0.557152
$178$	0	0
$179$	2216.00	0.925316	0.462658	−	0.886537i	$-0.346896\pi$
$179$	2216.00	0.925316	0.462658	+	0.886537i	$0.346896\pi$
$180$	0	0
$181$	2592.00	1.06443	0.532215	−	0.846609i	$-0.321360\pi$
$181$	2592.00	1.06443	0.532215	+	0.846609i	$0.321360\pi$
$182$	0	0
$183$	−3200.00	−1.29263
$184$	0	0
$185$	688.000	0.273420
$186$	0	0
$187$	−2444.00	−0.955738
$188$	0	0
$189$	320.000	0.123156
$190$	0	0
$191$	−1068.00	−0.404596	−0.202298	−	0.979324i	$-0.564841\pi$
$191$	−1068.00	−0.404596	−0.202298	+	0.979324i	$0.564841\pi$
$192$	0	0
$193$	−2918.00	−1.08830	−0.544151	−	0.838987i	$-0.683148\pi$
$193$	−2918.00	−1.08830	−0.544151	+	0.838987i	$0.683148\pi$
$194$	0	0
$195$	2240.00	0.822614
$196$	0	0
$197$	−1578.00	−0.570700	−0.285350	−	0.958423i	$-0.592110\pi$
$197$	−1578.00	−0.570700	−0.285350	+	0.958423i	$0.592110\pi$
$198$	0	0
$199$	3484.00	1.24108	0.620538	−	0.784176i	$-0.286914\pi$
$199$	3484.00	1.24108	0.620538	+	0.784176i	$0.286914\pi$
$200$	0	0
$201$	3184.00	1.11732
$202$	0	0
$203$	−344.000	−0.118936
$204$	0	0
$205$	−168.000	−0.0572372
$206$	0	0
$207$	−851.000	−0.285742
$208$	0	0
$209$	1404.00	0.464673
$210$	0	0
$211$	4876.00	1.59089	0.795445	−	0.606026i	$-0.207237\pi$
$211$	4876.00	1.59089	0.795445	+	0.606026i	$0.207237\pi$
$212$	0	0
$213$	2560.00	0.823513
$214$	0	0
$215$	−1544.00	−0.489767
$216$	0	0
$217$	576.000	0.180191
$218$	0	0
$219$	6480.00	1.99944
$220$	0	0
$221$	−6580.00	−2.00280
$222$	0	0
$223$	5888.00	1.76811	0.884057	−	0.467378i	$-0.154801\pi$
$223$	5888.00	1.76811	0.884057	+	0.467378i	$0.154801\pi$
$224$	0	0
$225$	−4033.00	−1.19496
$226$	0	0
$227$	−4498.00	−1.31517	−0.657583	−	0.753382i	$-0.728421\pi$
$227$	−4498.00	−1.31517	−0.657583	+	0.753382i	$0.728421\pi$
$228$	0	0
$229$	−352.000	−0.101576	−0.0507878	−	0.998709i	$-0.516173\pi$
$229$	−352.000	−0.101576	−0.0507878	+	0.998709i	$0.516173\pi$
$230$	0	0
$231$	−832.000	−0.236977
$232$	0	0
$233$	4362.00	1.22646	0.613228	−	0.789906i	$-0.289871\pi$
$233$	4362.00	1.22646	0.613228	+	0.789906i	$0.289871\pi$
$234$	0	0
$235$	−320.000	−0.0888277
$236$	0	0
$237$	1632.00	0.447299
$238$	0	0
$239$	2936.00	0.794619	0.397310	−	0.917685i	$-0.369944\pi$
$239$	2936.00	0.794619	0.397310	+	0.917685i	$0.369944\pi$
$240$	0	0
$241$	−7306.00	−1.95278	−0.976392	−	0.216007i	$-0.930697\pi$
$241$	−7306.00	−1.95278	−0.976392	+	0.216007i	$0.930697\pi$
$242$	0	0
$243$	5032.00	1.32841
$244$	0	0
$245$	−1308.00	−0.341082
$246$	0	0
$247$	3780.00	0.973748
$248$	0	0
$249$	816.000	0.207678
$250$	0	0
$251$	3930.00	0.988284	0.494142	−	0.869381i	$-0.335482\pi$
$251$	3930.00	0.988284	0.494142	+	0.869381i	$0.335482\pi$
$252$	0	0
$253$	598.000	0.148601
$254$	0	0
$255$	−3008.00	−0.738699
$256$	0	0
$257$	−138.000	−0.0334950	−0.0167475	−	0.999860i	$-0.505331\pi$
$257$	−138.000	−0.0334950	−0.0167475	+	0.999860i	$0.505331\pi$
$258$	0	0
$259$	−688.000	−0.165059
$260$	0	0
$261$	3182.00	0.754639
$262$	0	0
$263$	3664.00	0.859057	0.429528	−	0.903053i	$-0.358680\pi$
$263$	3664.00	0.859057	0.429528	+	0.903053i	$0.358680\pi$
$264$	0	0
$265$	432.000	0.100142
$266$	0	0
$267$	−8144.00	−1.86668
$268$	0	0
$269$	−3354.00	−0.760212	−0.380106	−	0.924943i	$-0.624112\pi$
$269$	−3354.00	−0.760212	−0.380106	+	0.924943i	$0.624112\pi$
$270$	0	0
$271$	−7688.00	−1.72329	−0.861647	−	0.507508i	$-0.830567\pi$
$271$	−7688.00	−1.72329	−0.861647	+	0.507508i	$0.830567\pi$
$272$	0	0
$273$	−2240.00	−0.496597
$274$	0	0
$275$	2834.00	0.621442
$276$	0	0
$277$	7606.00	1.64982	0.824910	−	0.565264i	$-0.191226\pi$
$277$	7606.00	1.64982	0.824910	+	0.565264i	$0.191226\pi$
$278$	0	0
$279$	−5328.00	−1.14329
$280$	0	0
$281$	4794.00	1.01774	0.508872	−	0.860842i	$-0.330063\pi$
$281$	4794.00	1.01774	0.508872	+	0.860842i	$0.330063\pi$
$282$	0	0
$283$	−1898.00	−0.398673	−0.199336	−	0.979931i	$-0.563879\pi$
$283$	−1898.00	−0.398673	−0.199336	+	0.979931i	$0.563879\pi$
$284$	0	0
$285$	1728.00	0.359150
$286$	0	0
$287$	168.000	0.0345531
$288$	0	0
$289$	3923.00	0.798494
$290$	0	0
$291$	10960.0	2.20786
$292$	0	0
$293$	5380.00	1.07271	0.536353	−	0.843994i	$-0.319802\pi$
$293$	5380.00	1.07271	0.536353	+	0.843994i	$0.319802\pi$
$294$	0	0
$295$	−656.000	−0.129470
$296$	0	0
$297$	2080.00	0.406377
$298$	0	0
$299$	1610.00	0.311400
$300$	0	0
$301$	1544.00	0.295663
$302$	0	0
$303$	−10640.0	−2.01733
$304$	0	0
$305$	1600.00	0.300379
$306$	0	0
$307$	3112.00	0.578538	0.289269	−	0.957248i	$-0.406588\pi$
$307$	3112.00	0.578538	0.289269	+	0.957248i	$0.406588\pi$
$308$	0	0
$309$	9280.00	1.70848
$310$	0	0
$311$	9504.00	1.73287	0.866435	−	0.499290i	$-0.166406\pi$
$311$	9504.00	1.73287	0.866435	+	0.499290i	$0.166406\pi$
$312$	0	0
$313$	−1930.00	−0.348531	−0.174265	−	0.984699i	$-0.555755\pi$
$313$	−1930.00	−0.348531	−0.174265	+	0.984699i	$0.555755\pi$
$314$	0	0
$315$	−592.000	−0.105890
$316$	0	0
$317$	−1866.00	−0.330615	−0.165308	−	0.986242i	$-0.552862\pi$
$317$	−1866.00	−0.330615	−0.165308	+	0.986242i	$0.552862\pi$
$318$	0	0
$319$	−2236.00	−0.392451
$320$	0	0
$321$	−3216.00	−0.559189
$322$	0	0
$323$	−5076.00	−0.874415
$324$	0	0
$325$	7630.00	1.30227
$326$	0	0
$327$	16320.0	2.75993
$328$	0	0
$329$	320.000	0.0536236
$330$	0	0
$331$	−6772.00	−1.12454	−0.562270	−	0.826954i	$-0.690072\pi$
$331$	−6772.00	−1.12454	−0.562270	+	0.826954i	$0.690072\pi$
$332$	0	0
$333$	6364.00	1.04728
$334$	0	0
$335$	−1592.00	−0.259643
$336$	0	0
$337$	8522.00	1.37752	0.688758	−	0.724991i	$-0.258156\pi$
$337$	8522.00	1.37752	0.688758	+	0.724991i	$0.258156\pi$
$338$	0	0
$339$	−10896.0	−1.74569
$340$	0	0
$341$	3744.00	0.594572
$342$	0	0
$343$	2680.00	0.421885
$344$	0	0
$345$	736.000	0.114855
$346$	0	0
$347$	−11584.0	−1.79211	−0.896054	−	0.443944i	$-0.853579\pi$
$347$	−11584.0	−1.79211	−0.896054	+	0.443944i	$0.853579\pi$
$348$	0	0
$349$	446.000	0.0684064	0.0342032	−	0.999415i	$-0.489111\pi$
$349$	446.000	0.0684064	0.0342032	+	0.999415i	$0.489111\pi$
$350$	0	0
$351$	5600.00	0.851584
$352$	0	0
$353$	−2046.00	−0.308492	−0.154246	−	0.988032i	$-0.549295\pi$
$353$	−2046.00	−0.308492	−0.154246	+	0.988032i	$0.549295\pi$
$354$	0	0
$355$	−1280.00	−0.191367
$356$	0	0
$357$	3008.00	0.445939
$358$	0	0
$359$	−2412.00	−0.354597	−0.177299	−	0.984157i	$-0.556736\pi$
$359$	−2412.00	−0.354597	−0.177299	+	0.984157i	$0.556736\pi$
$360$	0	0
$361$	−3943.00	−0.574865
$362$	0	0
$363$	5240.00	0.757655
$364$	0	0
$365$	−3240.00	−0.464628
$366$	0	0
$367$	9804.00	1.39445	0.697227	−	0.716850i	$-0.254417\pi$
$367$	9804.00	1.39445	0.697227	+	0.716850i	$0.254417\pi$
$368$	0	0
$369$	−1554.00	−0.219236
$370$	0	0
$371$	−432.000	−0.0604537
$372$	0	0
$373$	3116.00	0.432548	0.216274	−	0.976333i	$-0.430610\pi$
$373$	3116.00	0.432548	0.216274	+	0.976333i	$0.430610\pi$
$374$	0	0
$375$	7488.00	1.03114
$376$	0	0
$377$	−6020.00	−0.822403
$378$	0	0
$379$	−8558.00	−1.15988	−0.579941	−	0.814659i	$-0.696924\pi$
$379$	−8558.00	−1.15988	−0.579941	+	0.814659i	$0.696924\pi$
$380$	0	0
$381$	−13312.0	−1.79001
$382$	0	0
$383$	−10584.0	−1.41206	−0.706028	−	0.708184i	$-0.749515\pi$
$383$	−10584.0	−1.41206	−0.706028	+	0.708184i	$0.749515\pi$
$384$	0	0
$385$	416.000	0.0550684
$386$	0	0
$387$	−14282.0	−1.87596
$388$	0	0
$389$	−9560.00	−1.24604	−0.623022	−	0.782204i	$-0.714095\pi$
$389$	−9560.00	−1.24604	−0.623022	+	0.782204i	$0.714095\pi$
$390$	0	0
$391$	−2162.00	−0.279634
$392$	0	0
$393$	19872.0	2.55066
$394$	0	0
$395$	−816.000	−0.103943
$396$	0	0
$397$	5770.00	0.729441	0.364720	−	0.931117i	$-0.381165\pi$
$397$	5770.00	0.729441	0.364720	+	0.931117i	$0.381165\pi$
$398$	0	0
$399$	−1728.00	−0.216813
$400$	0	0
$401$	2862.00	0.356413	0.178206	−	0.983993i	$-0.442971\pi$
$401$	2862.00	0.356413	0.178206	+	0.983993i	$0.442971\pi$
$402$	0	0
$403$	10080.0	1.24596
$404$	0	0
$405$	−1436.00	−0.176186
$406$	0	0
$407$	−4472.00	−0.544641
$408$	0	0
$409$	11378.0	1.37556	0.687782	−	0.725917i	$-0.258584\pi$
$409$	11378.0	1.37556	0.687782	+	0.725917i	$0.258584\pi$
$410$	0	0
$411$	−9776.00	−1.17327
$412$	0	0
$413$	656.000	0.0781590
$414$	0	0
$415$	−408.000	−0.0482601
$416$	0	0
$417$	−15200.0	−1.78501
$418$	0	0
$419$	1754.00	0.204507	0.102254	−	0.994758i	$-0.467395\pi$
$419$	1754.00	0.204507	0.102254	+	0.994758i	$0.467395\pi$
$420$	0	0
$421$	−12148.0	−1.40631	−0.703156	−	0.711036i	$-0.748226\pi$
$421$	−12148.0	−1.40631	−0.703156	+	0.711036i	$0.748226\pi$
$422$	0	0
$423$	−2960.00	−0.340237
$424$	0	0
$425$	−10246.0	−1.16942
$426$	0	0
$427$	−1600.00	−0.181334
$428$	0	0
$429$	−14560.0	−1.63861
$430$	0	0
$431$	4884.00	0.545833	0.272916	−	0.962038i	$-0.412012\pi$
$431$	4884.00	0.545833	0.272916	+	0.962038i	$0.412012\pi$
$432$	0	0
$433$	2278.00	0.252826	0.126413	−	0.991978i	$-0.459654\pi$
$433$	2278.00	0.252826	0.126413	+	0.991978i	$0.459654\pi$
$434$	0	0
$435$	−2752.00	−0.303329
$436$	0	0
$437$	1242.00	0.135956
$438$	0	0
$439$	−11480.0	−1.24809	−0.624044	−	0.781389i	$-0.714511\pi$
$439$	−11480.0	−1.24809	−0.624044	+	0.781389i	$0.714511\pi$
$440$	0	0
$441$	−12099.0	−1.30645
$442$	0	0
$443$	11840.0	1.26983	0.634916	−	0.772581i	$-0.281035\pi$
$443$	11840.0	1.26983	0.634916	+	0.772581i	$0.281035\pi$
$444$	0	0
$445$	4072.00	0.433778
$446$	0	0
$447$	−26976.0	−2.85441
$448$	0	0
$449$	−2290.00	−0.240694	−0.120347	−	0.992732i	$-0.538401\pi$
$449$	−2290.00	−0.240694	−0.120347	+	0.992732i	$0.538401\pi$
$450$	0	0
$451$	1092.00	0.114014
$452$	0	0
$453$	−448.000	−0.0464655
$454$	0	0
$455$	1120.00	0.115399
$456$	0	0
$457$	−19370.0	−1.98269	−0.991346	−	0.131274i	$-0.958093\pi$
$457$	−19370.0	−1.98269	−0.991346	+	0.131274i	$0.958093\pi$
$458$	0	0
$459$	−7520.00	−0.764714
$460$	0	0
$461$	−18458.0	−1.86480	−0.932402	−	0.361423i	$-0.882291\pi$
$461$	−18458.0	−1.86480	−0.932402	+	0.361423i	$0.882291\pi$
$462$	0	0
$463$	−7496.00	−0.752416	−0.376208	−	0.926535i	$-0.622772\pi$
$463$	−7496.00	−0.752416	−0.376208	+	0.926535i	$0.622772\pi$
$464$	0	0
$465$	4608.00	0.459550
$466$	0	0
$467$	12074.0	1.19640	0.598199	−	0.801347i	$-0.295883\pi$
$467$	12074.0	1.19640	0.598199	+	0.801347i	$0.295883\pi$
$468$	0	0
$469$	1592.00	0.156741
$470$	0	0
$471$	−8928.00	−0.873419
$472$	0	0
$473$	10036.0	0.975594
$474$	0	0
$475$	5886.00	0.568565
$476$	0	0
$477$	3996.00	0.383573
$478$	0	0
$479$	9912.00	0.945492	0.472746	−	0.881199i	$-0.343263\pi$
$479$	9912.00	0.945492	0.472746	+	0.881199i	$0.343263\pi$
$480$	0	0
$481$	−12040.0	−1.14132
$482$	0	0
$483$	−736.000	−0.0693357
$484$	0	0
$485$	−5480.00	−0.513060
$486$	0	0
$487$	5272.00	0.490549	0.245274	−	0.969454i	$-0.421122\pi$
$487$	5272.00	0.490549	0.245274	+	0.969454i	$0.421122\pi$
$488$	0	0
$489$	1280.00	0.118371
$490$	0	0
$491$	17936.0	1.64855	0.824277	−	0.566186i	$-0.191582\pi$
$491$	17936.0	1.64855	0.824277	+	0.566186i	$0.191582\pi$
$492$	0	0
$493$	8084.00	0.738509
$494$	0	0
$495$	−3848.00	−0.349404
$496$	0	0
$497$	1280.00	0.115525
$498$	0	0
$499$	−1304.00	−0.116984	−0.0584920	−	0.998288i	$-0.518629\pi$
$499$	−1304.00	−0.116984	−0.0584920	+	0.998288i	$0.518629\pi$
$500$	0	0
$501$	−33408.0	−2.97916
$502$	0	0
$503$	−9364.00	−0.830060	−0.415030	−	0.909808i	$-0.636229\pi$
$503$	−9364.00	−0.830060	−0.415030	+	0.909808i	$0.636229\pi$
$504$	0	0
$505$	5320.00	0.468786
$506$	0	0
$507$	−21624.0	−1.89419
$508$	0	0
$509$	9798.00	0.853219	0.426610	−	0.904436i	$-0.359708\pi$
$509$	9798.00	0.853219	0.426610	+	0.904436i	$0.359708\pi$
$510$	0	0
$511$	3240.00	0.280488
$512$	0	0
$513$	4320.00	0.371799
$514$	0	0
$515$	−4640.00	−0.397015
$516$	0	0
$517$	2080.00	0.176941
$518$	0	0
$519$	−9712.00	−0.821406
$520$	0	0
$521$	21510.0	1.80877	0.904386	−	0.426715i	$-0.140329\pi$
$521$	21510.0	1.80877	0.904386	+	0.426715i	$0.140329\pi$
$522$	0	0
$523$	3498.00	0.292461	0.146230	−	0.989251i	$-0.453286\pi$
$523$	3498.00	0.292461	0.146230	+	0.989251i	$0.453286\pi$
$524$	0	0
$525$	−3488.00	−0.289960
$526$	0	0
$527$	−13536.0	−1.11886
$528$	0	0
$529$	529.000	0.0434783
$530$	0	0
$531$	−6068.00	−0.495911
$532$	0	0
$533$	2940.00	0.238922
$534$	0	0
$535$	1608.00	0.129944
$536$	0	0
$537$	−17728.0	−1.42462
$538$	0	0
$539$	8502.00	0.679419
$540$	0	0
$541$	−10722.0	−0.852079	−0.426040	−	0.904705i	$-0.640092\pi$
$541$	−10722.0	−0.852079	−0.426040	+	0.904705i	$0.640092\pi$
$542$	0	0
$543$	−20736.0	−1.63880
$544$	0	0
$545$	−8160.00	−0.641350
$546$	0	0
$547$	5628.00	0.439919	0.219960	−	0.975509i	$-0.429407\pi$
$547$	5628.00	0.439919	0.219960	+	0.975509i	$0.429407\pi$
$548$	0	0
$549$	14800.0	1.15054
$550$	0	0
$551$	−4644.00	−0.359058
$552$	0	0
$553$	816.000	0.0627484
$554$	0	0
$555$	−5504.00	−0.420958
$556$	0	0
$557$	4440.00	0.337754	0.168877	−	0.985637i	$-0.445986\pi$
$557$	4440.00	0.337754	0.168877	+	0.985637i	$0.445986\pi$
$558$	0	0
$559$	27020.0	2.04441
$560$	0	0
$561$	19552.0	1.47145
$562$	0	0
$563$	7926.00	0.593323	0.296662	−	0.954983i	$-0.404127\pi$
$563$	7926.00	0.593323	0.296662	+	0.954983i	$0.404127\pi$
$564$	0	0
$565$	5448.00	0.405662
$566$	0	0
$567$	1436.00	0.106360
$568$	0	0
$569$	−18706.0	−1.37820	−0.689100	−	0.724666i	$-0.741994\pi$
$569$	−18706.0	−1.37820	−0.689100	+	0.724666i	$0.741994\pi$
$570$	0	0
$571$	−16774.0	−1.22937	−0.614685	−	0.788773i	$-0.710717\pi$
$571$	−16774.0	−1.22937	−0.614685	+	0.788773i	$0.710717\pi$
$572$	0	0
$573$	8544.00	0.622916
$574$	0	0
$575$	2507.00	0.181825
$576$	0	0
$577$	2238.00	0.161472	0.0807358	−	0.996736i	$-0.474273\pi$
$577$	2238.00	0.161472	0.0807358	+	0.996736i	$0.474273\pi$
$578$	0	0
$579$	23344.0	1.67555
$580$	0	0
$581$	408.000	0.0291337
$582$	0	0
$583$	−2808.00	−0.199478
$584$	0	0
$585$	−10360.0	−0.732194
$586$	0	0
$587$	23524.0	1.65407	0.827035	−	0.562150i	$-0.190026\pi$
$587$	23524.0	1.65407	0.827035	+	0.562150i	$0.190026\pi$
$588$	0	0
$589$	7776.00	0.543980
$590$	0	0
$591$	12624.0	0.878650
$592$	0	0
$593$	−1274.00	−0.0882241	−0.0441121	−	0.999027i	$-0.514046\pi$
$593$	−1274.00	−0.0882241	−0.0441121	+	0.999027i	$0.514046\pi$
$594$	0	0
$595$	−1504.00	−0.103627
$596$	0	0
$597$	−27872.0	−1.91076
$598$	0	0
$599$	400.000	0.0272847	0.0136424	−	0.999907i	$-0.495657\pi$
$599$	400.000	0.0272847	0.0136424	+	0.999907i	$0.495657\pi$
$600$	0	0
$601$	−8746.00	−0.593605	−0.296803	−	0.954939i	$-0.595920\pi$
$601$	−8746.00	−0.593605	−0.296803	+	0.954939i	$0.595920\pi$
$602$	0	0
$603$	−14726.0	−0.994509
$604$	0	0
$605$	−2620.00	−0.176063
$606$	0	0
$607$	27056.0	1.80917	0.904587	−	0.426288i	$-0.140179\pi$
$607$	27056.0	1.80917	0.904587	+	0.426288i	$0.140179\pi$
$608$	0	0
$609$	2752.00	0.183114
$610$	0	0
$611$	5600.00	0.370788
$612$	0	0
$613$	−328.000	−0.0216114	−0.0108057	−	0.999942i	$-0.503440\pi$
$613$	−328.000	−0.0216114	−0.0108057	+	0.999942i	$0.503440\pi$
$614$	0	0
$615$	1344.00	0.0881225
$616$	0	0
$617$	29778.0	1.94298	0.971489	−	0.237086i	$-0.0761922\pi$
$617$	29778.0	1.94298	0.971489	+	0.237086i	$0.0761922\pi$
$618$	0	0
$619$	10646.0	0.691274	0.345637	−	0.938368i	$-0.387663\pi$
$619$	10646.0	0.691274	0.345637	+	0.938368i	$0.387663\pi$
$620$	0	0
$621$	1840.00	0.118900
$622$	0	0
$623$	−4072.00	−0.261864
$624$	0	0
$625$	9881.00	0.632384
$626$	0	0
$627$	−11232.0	−0.715411
$628$	0	0
$629$	16168.0	1.02490
$630$	0	0
$631$	8056.00	0.508248	0.254124	−	0.967172i	$-0.418213\pi$
$631$	8056.00	0.508248	0.254124	+	0.967172i	$0.418213\pi$
$632$	0	0
$633$	−39008.0	−2.44933
$634$	0	0
$635$	6656.00	0.415961
$636$	0	0
$637$	22890.0	1.42376
$638$	0	0
$639$	−11840.0	−0.732994
$640$	0	0
$641$	−25458.0	−1.56869	−0.784345	−	0.620325i	$-0.787001\pi$
$641$	−25458.0	−1.56869	−0.784345	+	0.620325i	$0.787001\pi$
$642$	0	0
$643$	3806.00	0.233428	0.116714	−	0.993166i	$-0.462764\pi$
$643$	3806.00	0.233428	0.116714	+	0.993166i	$0.462764\pi$
$644$	0	0
$645$	12352.0	0.754046
$646$	0	0
$647$	2168.00	0.131736	0.0658678	−	0.997828i	$-0.479018\pi$
$647$	2168.00	0.131736	0.0658678	+	0.997828i	$0.479018\pi$
$648$	0	0
$649$	4264.00	0.257899
$650$	0	0
$651$	−4608.00	−0.277422
$652$	0	0
$653$	−6462.00	−0.387255	−0.193628	−	0.981075i	$-0.562025\pi$
$653$	−6462.00	−0.387255	−0.193628	+	0.981075i	$0.562025\pi$
$654$	0	0
$655$	−9936.00	−0.592720
$656$	0	0
$657$	−29970.0	−1.77967
$658$	0	0
$659$	29354.0	1.73516	0.867579	−	0.497299i	$-0.165675\pi$
$659$	29354.0	1.73516	0.867579	+	0.497299i	$0.165675\pi$
$660$	0	0
$661$	−17152.0	−1.00928	−0.504641	−	0.863329i	$-0.668375\pi$
$661$	−17152.0	−1.00928	−0.504641	+	0.863329i	$0.668375\pi$
$662$	0	0
$663$	52640.0	3.08351
$664$	0	0
$665$	864.000	0.0503827
$666$	0	0
$667$	−1978.00	−0.114825
$668$	0	0
$669$	−47104.0	−2.72219
$670$	0	0
$671$	−10400.0	−0.598342
$672$	0	0
$673$	766.000	0.0438739	0.0219369	−	0.999759i	$-0.493017\pi$
$673$	766.000	0.0438739	0.0219369	+	0.999759i	$0.493017\pi$
$674$	0	0
$675$	8720.00	0.497234
$676$	0	0
$677$	−3788.00	−0.215044	−0.107522	−	0.994203i	$-0.534292\pi$
$677$	−3788.00	−0.215044	−0.107522	+	0.994203i	$0.534292\pi$
$678$	0	0
$679$	5480.00	0.309725
$680$	0	0
$681$	35984.0	2.02483
$682$	0	0
$683$	412.000	0.0230816	0.0115408	−	0.999933i	$-0.496326\pi$
$683$	412.000	0.0230816	0.0115408	+	0.999933i	$0.496326\pi$
$684$	0	0
$685$	4888.00	0.272644
$686$	0	0
$687$	2816.00	0.156386
$688$	0	0
$689$	−7560.00	−0.418016
$690$	0	0
$691$	−28292.0	−1.55757	−0.778783	−	0.627293i	$-0.784163\pi$
$691$	−28292.0	−1.55757	−0.778783	+	0.627293i	$0.784163\pi$
$692$	0	0
$693$	3848.00	0.210928
$694$	0	0
$695$	7600.00	0.414798
$696$	0	0
$697$	−3948.00	−0.214550
$698$	0	0
$699$	−34896.0	−1.88825
$700$	0	0
$701$	−18648.0	−1.00474	−0.502372	−	0.864652i	$-0.667539\pi$
$701$	−18648.0	−1.00474	−0.502372	+	0.864652i	$0.667539\pi$
$702$	0	0
$703$	−9288.00	−0.498298
$704$	0	0
$705$	2560.00	0.136759
$706$	0	0
$707$	−5320.00	−0.282997
$708$	0	0
$709$	−18844.0	−0.998168	−0.499084	−	0.866554i	$-0.666330\pi$
$709$	−18844.0	−0.998168	−0.499084	+	0.866554i	$0.666330\pi$
$710$	0	0
$711$	−7548.00	−0.398132
$712$	0	0
$713$	3312.00	0.173963
$714$	0	0
$715$	7280.00	0.380778
$716$	0	0
$717$	−23488.0	−1.22340
$718$	0	0
$719$	31744.0	1.64652	0.823262	−	0.567661i	$-0.192152\pi$
$719$	31744.0	1.64652	0.823262	+	0.567661i	$0.192152\pi$
$720$	0	0
$721$	4640.00	0.239671
$722$	0	0
$723$	58448.0	3.00651
$724$	0	0
$725$	−9374.00	−0.480195
$726$	0	0
$727$	3900.00	0.198959	0.0994794	−	0.995040i	$-0.468282\pi$
$727$	3900.00	0.198959	0.0994794	+	0.995040i	$0.468282\pi$
$728$	0	0
$729$	−30563.0	−1.55276
$730$	0	0
$731$	−36284.0	−1.83586
$732$	0	0
$733$	−7948.00	−0.400499	−0.200250	−	0.979745i	$-0.564175\pi$
$733$	−7948.00	−0.400499	−0.200250	+	0.979745i	$0.564175\pi$
$734$	0	0
$735$	10464.0	0.525130
$736$	0	0
$737$	10348.0	0.517196
$738$	0	0
$739$	17368.0	0.864536	0.432268	−	0.901745i	$-0.357713\pi$
$739$	17368.0	0.864536	0.432268	+	0.901745i	$0.357713\pi$
$740$	0	0
$741$	−30240.0	−1.49918
$742$	0	0
$743$	25704.0	1.26916	0.634582	−	0.772856i	$-0.281172\pi$
$743$	25704.0	1.26916	0.634582	+	0.772856i	$0.281172\pi$
$744$	0	0
$745$	13488.0	0.663305
$746$	0	0
$747$	−3774.00	−0.184851
$748$	0	0
$749$	−1608.00	−0.0784446
$750$	0	0
$751$	6672.00	0.324187	0.162094	−	0.986775i	$-0.448175\pi$
$751$	6672.00	0.324187	0.162094	+	0.986775i	$0.448175\pi$
$752$	0	0
$753$	−31440.0	−1.52156
$754$	0	0
$755$	224.000	0.0107976
$756$	0	0
$757$	6648.00	0.319188	0.159594	−	0.987183i	$-0.448981\pi$
$757$	6648.00	0.319188	0.159594	+	0.987183i	$0.448981\pi$
$758$	0	0
$759$	−4784.00	−0.228786
$760$	0	0
$761$	−3470.00	−0.165292	−0.0826461	−	0.996579i	$-0.526337\pi$
$761$	−3470.00	−0.165292	−0.0826461	+	0.996579i	$0.526337\pi$
$762$	0	0
$763$	8160.00	0.387171
$764$	0	0
$765$	13912.0	0.657503
$766$	0	0
$767$	11480.0	0.540442
$768$	0	0
$769$	9302.00	0.436201	0.218101	−	0.975926i	$-0.430014\pi$
$769$	9302.00	0.436201	0.218101	+	0.975926i	$0.430014\pi$
$770$	0	0
$771$	1104.00	0.0515689
$772$	0	0
$773$	4176.00	0.194308	0.0971541	−	0.995269i	$-0.469026\pi$
$773$	4176.00	0.194308	0.0971541	+	0.995269i	$0.469026\pi$
$774$	0	0
$775$	15696.0	0.727506
$776$	0	0
$777$	5504.00	0.254125
$778$	0	0
$779$	2268.00	0.104313
$780$	0	0
$781$	8320.00	0.381195
$782$	0	0
$783$	−6880.00	−0.314012
$784$	0	0
$785$	4464.00	0.202964
$786$	0	0
$787$	−6090.00	−0.275839	−0.137919	−	0.990443i	$-0.544041\pi$
$787$	−6090.00	−0.275839	−0.137919	+	0.990443i	$0.544041\pi$
$788$	0	0
$789$	−29312.0	−1.32260
$790$	0	0
$791$	−5448.00	−0.244891
$792$	0	0
$793$	−28000.0	−1.25386
$794$	0	0
$795$	−3456.00	−0.154178
$796$	0	0
$797$	29824.0	1.32550	0.662748	−	0.748843i	$-0.269390\pi$
$797$	29824.0	1.32550	0.662748	+	0.748843i	$0.269390\pi$
$798$	0	0
$799$	−7520.00	−0.332964
$800$	0	0
$801$	37666.0	1.66150
$802$	0	0
$803$	21060.0	0.925518
$804$	0	0
$805$	368.000	0.0161122
$806$	0	0
$807$	26832.0	1.17042
$808$	0	0
$809$	−29038.0	−1.26196	−0.630978	−	0.775801i	$-0.717346\pi$
$809$	−29038.0	−1.26196	−0.630978	+	0.775801i	$0.717346\pi$
$810$	0	0
$811$	−28012.0	−1.21287	−0.606433	−	0.795135i	$-0.707400\pi$
$811$	−28012.0	−1.21287	−0.606433	+	0.795135i	$0.707400\pi$
$812$	0	0
$813$	61504.0	2.65319
$814$	0	0
$815$	−640.000	−0.0275070
$816$	0	0
$817$	20844.0	0.892582
$818$	0	0
$819$	10360.0	0.442012
$820$	0	0
$821$	−4950.00	−0.210422	−0.105211	−	0.994450i	$-0.533552\pi$
$821$	−4950.00	−0.210422	−0.105211	+	0.994450i	$0.533552\pi$
$822$	0	0
$823$	2680.00	0.113510	0.0567551	−	0.998388i	$-0.481925\pi$
$823$	2680.00	0.113510	0.0567551	+	0.998388i	$0.481925\pi$
$824$	0	0
$825$	−22672.0	−0.956773
$826$	0	0
$827$	−606.000	−0.0254809	−0.0127404	−	0.999919i	$-0.504056\pi$
$827$	−606.000	−0.0254809	−0.0127404	+	0.999919i	$0.504056\pi$
$828$	0	0
$829$	−17642.0	−0.739122	−0.369561	−	0.929207i	$-0.620492\pi$
$829$	−17642.0	−0.739122	−0.369561	+	0.929207i	$0.620492\pi$
$830$	0	0
$831$	−60848.0	−2.54006
$832$	0	0
$833$	−30738.0	−1.27852
$834$	0	0
$835$	16704.0	0.692294
$836$	0	0
$837$	11520.0	0.475734
$838$	0	0
$839$	−24264.0	−0.998434	−0.499217	−	0.866477i	$-0.666379\pi$
$839$	−24264.0	−0.998434	−0.499217	+	0.866477i	$0.666379\pi$
$840$	0	0
$841$	−16993.0	−0.696749
$842$	0	0
$843$	−38352.0	−1.56692
$844$	0	0
$845$	10812.0	0.440171
$846$	0	0
$847$	2620.00	0.106286
$848$	0	0
$849$	15184.0	0.613797
$850$	0	0
$851$	−3956.00	−0.159354
$852$	0	0
$853$	−12998.0	−0.521739	−0.260869	−	0.965374i	$-0.584009\pi$
$853$	−12998.0	−0.521739	−0.260869	+	0.965374i	$0.584009\pi$
$854$	0	0
$855$	−7992.00	−0.319673
$856$	0	0
$857$	−44518.0	−1.77445	−0.887226	−	0.461334i	$-0.847371\pi$
$857$	−44518.0	−1.77445	−0.887226	+	0.461334i	$0.847371\pi$
$858$	0	0
$859$	−8000.00	−0.317761	−0.158880	−	0.987298i	$-0.550788\pi$
$859$	−8000.00	−0.317761	−0.158880	+	0.987298i	$0.550788\pi$
$860$	0	0
$861$	−1344.00	−0.0531979
$862$	0	0
$863$	−1848.00	−0.0728930	−0.0364465	−	0.999336i	$-0.511604\pi$
$863$	−1848.00	−0.0728930	−0.0364465	+	0.999336i	$0.511604\pi$
$864$	0	0
$865$	4856.00	0.190877
$866$	0	0
$867$	−31384.0	−1.22936
$868$	0	0
$869$	5304.00	0.207049
$870$	0	0
$871$	27860.0	1.08381
$872$	0	0
$873$	−50690.0	−1.96517
$874$	0	0
$875$	3744.00	0.144652
$876$	0	0
$877$	902.000	0.0347302	0.0173651	−	0.999849i	$-0.494472\pi$
$877$	902.000	0.0347302	0.0173651	+	0.999849i	$0.494472\pi$
$878$	0	0
$879$	−43040.0	−1.65154
$880$	0	0
$881$	786.000	0.0300579	0.0150290	−	0.999887i	$-0.495216\pi$
$881$	786.000	0.0300579	0.0150290	+	0.999887i	$0.495216\pi$
$882$	0	0
$883$	8316.00	0.316937	0.158469	−	0.987364i	$-0.449344\pi$
$883$	8316.00	0.316937	0.158469	+	0.987364i	$0.449344\pi$
$884$	0	0
$885$	5248.00	0.199333
$886$	0	0
$887$	8280.00	0.313433	0.156717	−	0.987644i	$-0.449909\pi$
$887$	8280.00	0.313433	0.156717	+	0.987644i	$0.449909\pi$
$888$	0	0
$889$	−6656.00	−0.251108
$890$	0	0
$891$	9334.00	0.350955
$892$	0	0
$893$	4320.00	0.161885
$894$	0	0
$895$	8864.00	0.331051
$896$	0	0
$897$	−12880.0	−0.479432
$898$	0	0
$899$	−12384.0	−0.459432
$900$	0	0
$901$	10152.0	0.375374
$902$	0	0
$903$	−12352.0	−0.455204
$904$	0	0
$905$	10368.0	0.380822
$906$	0	0
$907$	4318.00	0.158078	0.0790391	−	0.996872i	$-0.474815\pi$
$907$	4318.00	0.158078	0.0790391	+	0.996872i	$0.474815\pi$
$908$	0	0
$909$	49210.0	1.79559
$910$	0	0
$911$	−30732.0	−1.11767	−0.558835	−	0.829279i	$-0.688751\pi$
$911$	−30732.0	−1.11767	−0.558835	+	0.829279i	$0.688751\pi$
$912$	0	0
$913$	2652.00	0.0961319
$914$	0	0
$915$	−12800.0	−0.462464
$916$	0	0
$917$	9936.00	0.357814
$918$	0	0
$919$	53692.0	1.92724	0.963621	−	0.267272i	$-0.0861222\pi$
$919$	53692.0	1.92724	0.963621	+	0.267272i	$0.0861222\pi$
$920$	0	0
$921$	−24896.0	−0.890718
$922$	0	0
$923$	22400.0	0.798814
$924$	0	0
$925$	−18748.0	−0.666411
$926$	0	0
$927$	−42920.0	−1.52069
$928$	0	0
$929$	12014.0	0.424291	0.212146	−	0.977238i	$-0.431955\pi$
$929$	12014.0	0.424291	0.212146	+	0.977238i	$0.431955\pi$
$930$	0	0
$931$	17658.0	0.621609
$932$	0	0
$933$	−76032.0	−2.66793
$934$	0	0
$935$	−9776.00	−0.341935
$936$	0	0
$937$	39514.0	1.37766	0.688829	−	0.724924i	$-0.258125\pi$
$937$	39514.0	1.37766	0.688829	+	0.724924i	$0.258125\pi$
$938$	0	0
$939$	15440.0	0.536598
$940$	0	0
$941$	−35484.0	−1.22927	−0.614636	−	0.788811i	$-0.710697\pi$
$941$	−35484.0	−1.22927	−0.614636	+	0.788811i	$0.710697\pi$
$942$	0	0
$943$	966.000	0.0333587
$944$	0	0
$945$	1280.00	0.0440618
$946$	0	0
$947$	21128.0	0.724992	0.362496	−	0.931985i	$-0.381925\pi$
$947$	21128.0	0.724992	0.362496	+	0.931985i	$0.381925\pi$
$948$	0	0
$949$	56700.0	1.93947
$950$	0	0
$951$	14928.0	0.509015
$952$	0	0
$953$	−6962.00	−0.236644	−0.118322	−	0.992975i	$-0.537751\pi$
$953$	−6962.00	−0.236644	−0.118322	+	0.992975i	$0.537751\pi$
$954$	0	0
$955$	−4272.00	−0.144753
$956$	0	0
$957$	17888.0	0.604218
$958$	0	0
$959$	−4888.00	−0.164590
$960$	0	0
$961$	−9055.00	−0.303951
$962$	0	0
$963$	14874.0	0.497724
$964$	0	0
$965$	−11672.0	−0.389363
$966$	0	0
$967$	33616.0	1.11791	0.558954	−	0.829198i	$-0.311203\pi$
$967$	33616.0	1.11791	0.558954	+	0.829198i	$0.311203\pi$
$968$	0	0
$969$	40608.0	1.34625
$970$	0	0
$971$	−31602.0	−1.04445	−0.522223	−	0.852809i	$-0.674897\pi$
$971$	−31602.0	−1.04445	−0.522223	+	0.852809i	$0.674897\pi$
$972$	0	0
$973$	−7600.00	−0.250406
$974$	0	0
$975$	−61040.0	−2.00497
$976$	0	0
$977$	−19482.0	−0.637957	−0.318979	−	0.947762i	$-0.603340\pi$
$977$	−19482.0	−0.637957	−0.318979	+	0.947762i	$0.603340\pi$
$978$	0	0
$979$	−26468.0	−0.864066
$980$	0	0
$981$	−75480.0	−2.45657
$982$	0	0
$983$	42716.0	1.38599	0.692995	−	0.720942i	$-0.256291\pi$
$983$	42716.0	1.38599	0.692995	+	0.720942i	$0.256291\pi$
$984$	0	0
$985$	−6312.00	−0.204180
$986$	0	0
$987$	−2560.00	−0.0825590
$988$	0	0
$989$	8878.00	0.285444
$990$	0	0
$991$	26328.0	0.843932	0.421966	−	0.906612i	$-0.361340\pi$
$991$	26328.0	0.843932	0.421966	+	0.906612i	$0.361340\pi$
$992$	0	0
$993$	54176.0	1.73134
$994$	0	0
$995$	13936.0	0.444021
$996$	0	0
$997$	47614.0	1.51249	0.756244	−	0.654290i	$-0.227032\pi$
$997$	47614.0	1.51249	0.756244	+	0.654290i	$0.227032\pi$
$998$	0	0
$999$	−13760.0	−0.435783

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1472.4.a.b.1.1		1
4.3	odd	2		1472.4.a.i.1.1		1
8.3	odd	2		368.4.a.a.1.1		1
8.5	even	2		184.4.a.b.1.1	✓	1
24.5	odd	2		1656.4.a.c.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
184.4.a.b.1.1	✓	1	8.5	even	2
368.4.a.a.1.1		1	8.3	odd	2
1472.4.a.b.1.1		1	1.1	even	1	trivial
1472.4.a.i.1.1		1	4.3	odd	2
1656.4.a.c.1.1		1	24.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1472 = 2^{6} \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1472.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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