LMFDB - Embedded newform 1638.4.a.j.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1638,4,Mod(1,1638)] 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("1638.1"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1638, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	1638.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,2,0,4,0,0,7,8,0,0,-39] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$96.6451285894$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 182)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	1638.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+2.00000 q^{2} +4.00000 q^{4} +7.00000 q^{7} +8.00000 q^{8} -39.0000 q^{11} +13.0000 q^{13} +14.0000 q^{14} +16.0000 q^{16} -24.0000 q^{17} +38.0000 q^{19} -78.0000 q^{22} -39.0000 q^{23} -125.000 q^{25} +26.0000 q^{26} +28.0000 q^{28} +96.0000 q^{29} +227.000 q^{31} +32.0000 q^{32} -48.0000 q^{34} +425.000 q^{37} +76.0000 q^{38} +105.000 q^{41} +344.000 q^{43} -156.000 q^{44} -78.0000 q^{46} -99.0000 q^{47} +49.0000 q^{49} -250.000 q^{50} +52.0000 q^{52} +540.000 q^{53} +56.0000 q^{56} +192.000 q^{58} -114.000 q^{59} -565.000 q^{61} +454.000 q^{62} +64.0000 q^{64} -385.000 q^{67} -96.0000 q^{68} +156.000 q^{71} -673.000 q^{73} +850.000 q^{74} +152.000 q^{76} -273.000 q^{77} +749.000 q^{79} +210.000 q^{82} +1044.00 q^{83} +688.000 q^{86} -312.000 q^{88} +690.000 q^{89} +91.0000 q^{91} -156.000 q^{92} -198.000 q^{94} +317.000 q^{97} +98.0000 q^{98} +O(q^{100})$

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 $

$2$	2.00000	0.707107
$2$	2.00000	0.707107
$3$	0	0
$3$	0	0
$4$	4.00000	0.500000
$5$	0	0		−	1.00000i	$-0.5\pi$
$5$	0	0			1.00000i	$0.5\pi$
$6$	0	0
$7$	7.00000	0.377964
$7$	7.00000	0.377964
$8$	8.00000	0.353553
$9$	0	0
$10$	0	0
$11$	−39.0000	−1.06899	−0.534497	−	0.845170i	$-0.679499\pi$
$11$	−39.0000	−1.06899	−0.534497	+	0.845170i	$0.679499\pi$
$12$	0	0
$13$	13.0000	0.277350
$13$	13.0000	0.277350
$14$	14.0000	0.267261
$15$	0	0
$16$	16.0000	0.250000
$17$	−24.0000	−0.342403	−0.171202	−	0.985236i	$-0.554765\pi$
$17$	−24.0000	−0.342403	−0.171202	+	0.985236i	$0.554765\pi$
$18$	0	0
$19$	38.0000	0.458831	0.229416	−	0.973329i	$-0.426318\pi$
$19$	38.0000	0.458831	0.229416	+	0.973329i	$0.426318\pi$
$20$	0	0
$21$	0	0
$22$	−78.0000	−0.755893
$23$	−39.0000	−0.353568	−0.176784	−	0.984250i	$-0.556569\pi$
$23$	−39.0000	−0.353568	−0.176784	+	0.984250i	$0.556569\pi$
$24$	0	0
$25$	−125.000	−1.00000
$26$	26.0000	0.196116
$27$	0	0
$28$	28.0000	0.188982
$29$	96.0000	0.614716	0.307358	−	0.951594i	$-0.400555\pi$
$29$	96.0000	0.614716	0.307358	+	0.951594i	$0.400555\pi$
$30$	0	0
$31$	227.000	1.31517	0.657587	−	0.753378i	$-0.271577\pi$
$31$	227.000	1.31517	0.657587	+	0.753378i	$0.271577\pi$
$32$	32.0000	0.176777
$33$	0	0
$34$	−48.0000	−0.242116
$35$	0	0
$36$	0	0
$37$	425.000	1.88837	0.944183	−	0.329420i	$-0.106853\pi$
$37$	425.000	1.88837	0.944183	+	0.329420i	$0.106853\pi$
$38$	76.0000	0.324443
$39$	0	0
$40$	0	0
$41$	105.000	0.399957	0.199979	−	0.979800i	$-0.435913\pi$
$41$	105.000	0.399957	0.199979	+	0.979800i	$0.435913\pi$
$42$	0	0
$43$	344.000	1.21999	0.609994	−	0.792406i	$-0.291172\pi$
$43$	344.000	1.21999	0.609994	+	0.792406i	$0.291172\pi$
$44$	−156.000	−0.534497
$45$	0	0
$46$	−78.0000	−0.250010
$47$	−99.0000	−0.307248	−0.153624	−	0.988129i	$-0.549094\pi$
$47$	−99.0000	−0.307248	−0.153624	+	0.988129i	$0.549094\pi$
$48$	0	0
$49$	49.0000	0.142857
$50$	−250.000	−0.707107
$51$	0	0
$52$	52.0000	0.138675
$53$	540.000	1.39952	0.699761	−	0.714377i	$-0.253290\pi$
$53$	540.000	1.39952	0.699761	+	0.714377i	$0.253290\pi$
$54$	0	0
$55$	0	0
$56$	56.0000	0.133631
$57$	0	0
$58$	192.000	0.434670
$59$	−114.000	−0.251551	−0.125776	−	0.992059i	$-0.540142\pi$
$59$	−114.000	−0.251551	−0.125776	+	0.992059i	$0.540142\pi$
$60$	0	0
$61$	−565.000	−1.18592	−0.592958	−	0.805234i	$-0.702040\pi$
$61$	−565.000	−1.18592	−0.592958	+	0.805234i	$0.702040\pi$
$62$	454.000	0.929969
$63$	0	0
$64$	64.0000	0.125000
$65$	0	0
$66$	0	0
$67$	−385.000	−0.702018	−0.351009	−	0.936372i	$-0.614161\pi$
$67$	−385.000	−0.702018	−0.351009	+	0.936372i	$0.614161\pi$
$68$	−96.0000	−0.171202
$69$	0	0
$70$	0	0
$71$	156.000	0.260758	0.130379	−	0.991464i	$-0.458381\pi$
$71$	156.000	0.260758	0.130379	+	0.991464i	$0.458381\pi$
$72$	0	0
$73$	−673.000	−1.07902	−0.539512	−	0.841978i	$-0.681391\pi$
$73$	−673.000	−1.07902	−0.539512	+	0.841978i	$0.681391\pi$
$74$	850.000	1.33528
$75$	0	0
$76$	152.000	0.229416
$77$	−273.000	−0.404042
$78$	0	0
$79$	749.000	1.06670	0.533349	−	0.845896i	$-0.320933\pi$
$79$	749.000	1.06670	0.533349	+	0.845896i	$0.320933\pi$
$80$	0	0
$81$	0	0
$82$	210.000	0.282812
$83$	1044.00	1.38065	0.690325	−	0.723500i	$-0.257468\pi$
$83$	1044.00	1.38065	0.690325	+	0.723500i	$0.257468\pi$
$84$	0	0
$85$	0	0
$86$	688.000	0.862662
$87$	0	0
$88$	−312.000	−0.377947
$89$	690.000	0.821796	0.410898	−	0.911681i	$-0.365215\pi$
$89$	690.000	0.821796	0.410898	+	0.911681i	$0.365215\pi$
$90$	0	0
$91$	91.0000	0.104828
$92$	−156.000	−0.176784
$93$	0	0
$94$	−198.000	−0.217257
$95$	0	0
$96$	0	0
$97$	317.000	0.331819	0.165910	−	0.986141i	$-0.446944\pi$
$97$	317.000	0.331819	0.165910	+	0.986141i	$0.446944\pi$
$98$	98.0000	0.101015
$99$	0	0
$100$	−500.000	−0.500000
$101$	663.000	0.653178	0.326589	−	0.945166i	$-0.394101\pi$
$101$	663.000	0.653178	0.326589	+	0.945166i	$0.394101\pi$
$102$	0	0
$103$	−646.000	−0.617983	−0.308992	−	0.951065i	$-0.599992\pi$
$103$	−646.000	−0.617983	−0.308992	+	0.951065i	$0.599992\pi$
$104$	104.000	0.0980581
$105$	0	0
$106$	1080.00	0.989612
$107$	744.000	0.672198	0.336099	−	0.941827i	$-0.390892\pi$
$107$	744.000	0.672198	0.336099	+	0.941827i	$0.390892\pi$
$108$	0	0
$109$	218.000	0.191565	0.0957826	−	0.995402i	$-0.469465\pi$
$109$	218.000	0.191565	0.0957826	+	0.995402i	$0.469465\pi$
$110$	0	0
$111$	0	0
$112$	112.000	0.0944911
$113$	1623.00	1.35114	0.675571	−	0.737295i	$-0.263897\pi$
$113$	1623.00	1.35114	0.675571	+	0.737295i	$0.263897\pi$
$114$	0	0
$115$	0	0
$116$	384.000	0.307358
$117$	0	0
$118$	−228.000	−0.177874
$119$	−168.000	−0.129416
$120$	0	0
$121$	190.000	0.142750
$122$	−1130.00	−0.838569
$123$	0	0
$124$	908.000	0.657587
$125$	0	0
$126$	0	0
$127$	659.000	0.460447	0.230224	−	0.973138i	$-0.426054\pi$
$127$	659.000	0.460447	0.230224	+	0.973138i	$0.426054\pi$
$128$	128.000	0.0883883
$129$	0	0
$130$	0	0
$131$	216.000	0.144061	0.0720306	−	0.997402i	$-0.477052\pi$
$131$	216.000	0.144061	0.0720306	+	0.997402i	$0.477052\pi$
$132$	0	0
$133$	266.000	0.173422
$134$	−770.000	−0.496402
$135$	0	0
$136$	−192.000	−0.121058
$137$	−1842.00	−1.14871	−0.574353	−	0.818608i	$-0.694746\pi$
$137$	−1842.00	−1.14871	−0.574353	+	0.818608i	$0.694746\pi$
$138$	0	0
$139$	−628.000	−0.383211	−0.191605	−	0.981472i	$-0.561369\pi$
$139$	−628.000	−0.383211	−0.191605	+	0.981472i	$0.561369\pi$
$140$	0	0
$141$	0	0
$142$	312.000	0.184384
$143$	−507.000	−0.296486
$144$	0	0
$145$	0	0
$146$	−1346.00	−0.762985
$147$	0	0
$148$	1700.00	0.944183
$149$	−321.000	−0.176492	−0.0882461	−	0.996099i	$-0.528126\pi$
$149$	−321.000	−0.176492	−0.0882461	+	0.996099i	$0.528126\pi$
$150$	0	0
$151$	−1600.00	−0.862292	−0.431146	−	0.902282i	$-0.641891\pi$
$151$	−1600.00	−0.862292	−0.431146	+	0.902282i	$0.641891\pi$
$152$	304.000	0.162221
$153$	0	0
$154$	−546.000	−0.285701
$155$	0	0
$156$	0	0
$157$	1127.00	0.572894	0.286447	−	0.958096i	$-0.407526\pi$
$157$	1127.00	0.572894	0.286447	+	0.958096i	$0.407526\pi$
$158$	1498.00	0.754269
$159$	0	0
$160$	0	0
$161$	−273.000	−0.133636
$162$	0	0
$163$	−1204.00	−0.578556	−0.289278	−	0.957245i	$-0.593415\pi$
$163$	−1204.00	−0.578556	−0.289278	+	0.957245i	$0.593415\pi$
$164$	420.000	0.199979
$165$	0	0
$166$	2088.00	0.976266
$167$	600.000	0.278020	0.139010	−	0.990291i	$-0.455608\pi$
$167$	600.000	0.278020	0.139010	+	0.990291i	$0.455608\pi$
$168$	0	0
$169$	169.000	0.0769231
$170$	0	0
$171$	0	0
$172$	1376.00	0.609994
$173$	2154.00	0.946622	0.473311	−	0.880895i	$-0.343059\pi$
$173$	2154.00	0.946622	0.473311	+	0.880895i	$0.343059\pi$
$174$	0	0
$175$	−875.000	−0.377964
$176$	−624.000	−0.267249
$177$	0	0
$178$	1380.00	0.581098
$179$	2850.00	1.19005	0.595025	−	0.803707i	$-0.297142\pi$
$179$	2850.00	1.19005	0.595025	+	0.803707i	$0.297142\pi$
$180$	0	0
$181$	4205.00	1.72682	0.863412	−	0.504499i	$-0.168323\pi$
$181$	4205.00	1.72682	0.863412	+	0.504499i	$0.168323\pi$
$182$	182.000	0.0741249
$183$	0	0
$184$	−312.000	−0.125005
$185$	0	0
$186$	0	0
$187$	936.000	0.366027
$188$	−396.000	−0.153624
$189$	0	0
$190$	0	0
$191$	4152.00	1.57292	0.786461	−	0.617640i	$-0.211911\pi$
$191$	4152.00	1.57292	0.786461	+	0.617640i	$0.211911\pi$
$192$	0	0
$193$	−3148.00	−1.17408	−0.587041	−	0.809557i	$-0.699707\pi$
$193$	−3148.00	−1.17408	−0.587041	+	0.809557i	$0.699707\pi$
$194$	634.000	0.234632
$195$	0	0
$196$	196.000	0.0714286
$197$	−1173.00	−0.424227	−0.212114	−	0.977245i	$-0.568035\pi$
$197$	−1173.00	−0.424227	−0.212114	+	0.977245i	$0.568035\pi$
$198$	0	0
$199$	3512.00	1.25105	0.625525	−	0.780204i	$-0.284885\pi$
$199$	3512.00	1.25105	0.625525	+	0.780204i	$0.284885\pi$
$200$	−1000.00	−0.353553
$201$	0	0
$202$	1326.00	0.461867
$203$	672.000	0.232341
$204$	0	0
$205$	0	0
$206$	−1292.00	−0.436980
$207$	0	0
$208$	208.000	0.0693375
$209$	−1482.00	−0.490488
$210$	0	0
$211$	−3418.00	−1.11519	−0.557594	−	0.830114i	$-0.688276\pi$
$211$	−3418.00	−1.11519	−0.557594	+	0.830114i	$0.688276\pi$
$212$	2160.00	0.699761
$213$	0	0
$214$	1488.00	0.475316
$215$	0	0
$216$	0	0
$217$	1589.00	0.497089
$218$	436.000	0.135457
$219$	0	0
$220$	0	0
$221$	−312.000	−0.0949656
$222$	0	0
$223$	4241.00	1.27354	0.636768	−	0.771056i	$-0.280271\pi$
$223$	4241.00	1.27354	0.636768	+	0.771056i	$0.280271\pi$
$224$	224.000	0.0668153
$225$	0	0
$226$	3246.00	0.955401
$227$	−888.000	−0.259642	−0.129821	−	0.991537i	$-0.541440\pi$
$227$	−888.000	−0.259642	−0.129821	+	0.991537i	$0.541440\pi$
$228$	0	0
$229$	4700.00	1.35627	0.678133	−	0.734940i	$-0.262789\pi$
$229$	4700.00	1.35627	0.678133	+	0.734940i	$0.262789\pi$
$230$	0	0
$231$	0	0
$232$	768.000	0.217335
$233$	−6363.00	−1.78907	−0.894536	−	0.446995i	$-0.852494\pi$
$233$	−6363.00	−1.78907	−0.894536	+	0.446995i	$0.852494\pi$
$234$	0	0
$235$	0	0
$236$	−456.000	−0.125776
$237$	0	0
$238$	−336.000	−0.0915111
$239$	3078.00	0.833051	0.416526	−	0.909124i	$-0.363248\pi$
$239$	3078.00	0.833051	0.416526	+	0.909124i	$0.363248\pi$
$240$	0	0
$241$	3674.00	0.982005	0.491002	−	0.871158i	$-0.336631\pi$
$241$	3674.00	0.982005	0.491002	+	0.871158i	$0.336631\pi$
$242$	380.000	0.100939
$243$	0	0
$244$	−2260.00	−0.592958
$245$	0	0
$246$	0	0
$247$	494.000	0.127257
$248$	1816.00	0.464984
$249$	0	0
$250$	0	0
$251$	345.000	0.0867578	0.0433789	−	0.999059i	$-0.486188\pi$
$251$	345.000	0.0867578	0.0433789	+	0.999059i	$0.486188\pi$
$252$	0	0
$253$	1521.00	0.377962
$254$	1318.00	0.325585
$255$	0	0
$256$	256.000	0.0625000
$257$	−6888.00	−1.67184	−0.835918	−	0.548855i	$-0.815064\pi$
$257$	−6888.00	−1.67184	−0.835918	+	0.548855i	$0.815064\pi$
$258$	0	0
$259$	2975.00	0.713736
$260$	0	0
$261$	0	0
$262$	432.000	0.101867
$263$	1248.00	0.292604	0.146302	−	0.989240i	$-0.453263\pi$
$263$	1248.00	0.292604	0.146302	+	0.989240i	$0.453263\pi$
$264$	0	0
$265$	0	0
$266$	532.000	0.122628
$267$	0	0
$268$	−1540.00	−0.351009
$269$	2253.00	0.510661	0.255331	−	0.966854i	$-0.417816\pi$
$269$	2253.00	0.510661	0.255331	+	0.966854i	$0.417816\pi$
$270$	0	0
$271$	1397.00	0.313143	0.156571	−	0.987667i	$-0.449956\pi$
$271$	1397.00	0.313143	0.156571	+	0.987667i	$0.449956\pi$
$272$	−384.000	−0.0856008
$273$	0	0
$274$	−3684.00	−0.812258
$275$	4875.00	1.06899
$276$	0	0
$277$	−2302.00	−0.499328	−0.249664	−	0.968333i	$-0.580320\pi$
$277$	−2302.00	−0.499328	−0.249664	+	0.968333i	$0.580320\pi$
$278$	−1256.00	−0.270971
$279$	0	0
$280$	0	0
$281$	−8532.00	−1.81130	−0.905652	−	0.424022i	$-0.860618\pi$
$281$	−8532.00	−1.81130	−0.905652	+	0.424022i	$0.860618\pi$
$282$	0	0
$283$	−3769.00	−0.791674	−0.395837	−	0.918321i	$-0.629546\pi$
$283$	−3769.00	−0.791674	−0.395837	+	0.918321i	$0.629546\pi$
$284$	624.000	0.130379
$285$	0	0
$286$	−1014.00	−0.209647
$287$	735.000	0.151170
$288$	0	0
$289$	−4337.00	−0.882760
$290$	0	0
$291$	0	0
$292$	−2692.00	−0.539512
$293$	6150.00	1.22623	0.613117	−	0.789992i	$-0.289915\pi$
$293$	6150.00	1.22623	0.613117	+	0.789992i	$0.289915\pi$
$294$	0	0
$295$	0	0
$296$	3400.00	0.667638
$297$	0	0
$298$	−642.000	−0.124799
$299$	−507.000	−0.0980621
$300$	0	0
$301$	2408.00	0.461112
$302$	−3200.00	−0.609733
$303$	0	0
$304$	608.000	0.114708
$305$	0	0
$306$	0	0
$307$	4592.00	0.853678	0.426839	−	0.904328i	$-0.359627\pi$
$307$	4592.00	0.853678	0.426839	+	0.904328i	$0.359627\pi$
$308$	−1092.00	−0.202021
$309$	0	0
$310$	0	0
$311$	−6498.00	−1.18478	−0.592392	−	0.805650i	$-0.701816\pi$
$311$	−6498.00	−1.18478	−0.592392	+	0.805650i	$0.701816\pi$
$312$	0	0
$313$	5762.00	1.04054	0.520268	−	0.854003i	$-0.325832\pi$
$313$	5762.00	1.04054	0.520268	+	0.854003i	$0.325832\pi$
$314$	2254.00	0.405097
$315$	0	0
$316$	2996.00	0.533349
$317$	−2565.00	−0.454463	−0.227231	−	0.973841i	$-0.572967\pi$
$317$	−2565.00	−0.454463	−0.227231	+	0.973841i	$0.572967\pi$
$318$	0	0
$319$	−3744.00	−0.657128
$320$	0	0
$321$	0	0
$322$	−546.000	−0.0944950
$323$	−912.000	−0.157105
$324$	0	0
$325$	−1625.00	−0.277350
$326$	−2408.00	−0.409101
$327$	0	0
$328$	840.000	0.141406
$329$	−693.000	−0.116129
$330$	0	0
$331$	−745.000	−0.123713	−0.0618563	−	0.998085i	$-0.519702\pi$
$331$	−745.000	−0.123713	−0.0618563	+	0.998085i	$0.519702\pi$
$332$	4176.00	0.690325
$333$	0	0
$334$	1200.00	0.196590
$335$	0	0
$336$	0	0
$337$	8345.00	1.34891	0.674453	−	0.738318i	$-0.264380\pi$
$337$	8345.00	1.34891	0.674453	+	0.738318i	$0.264380\pi$
$338$	338.000	0.0543928
$339$	0	0
$340$	0	0
$341$	−8853.00	−1.40591
$342$	0	0
$343$	343.000	0.0539949
$344$	2752.00	0.431331
$345$	0	0
$346$	4308.00	0.669363
$347$	1776.00	0.274757	0.137378	−	0.990519i	$-0.456132\pi$
$347$	1776.00	0.274757	0.137378	+	0.990519i	$0.456132\pi$
$348$	0	0
$349$	−8602.00	−1.31935	−0.659677	−	0.751549i	$-0.729307\pi$
$349$	−8602.00	−1.31935	−0.659677	+	0.751549i	$0.729307\pi$
$350$	−1750.00	−0.267261
$351$	0	0
$352$	−1248.00	−0.188973
$353$	6477.00	0.976589	0.488295	−	0.872679i	$-0.337619\pi$
$353$	6477.00	0.976589	0.488295	+	0.872679i	$0.337619\pi$
$354$	0	0
$355$	0	0
$356$	2760.00	0.410898
$357$	0	0
$358$	5700.00	0.841493
$359$	−7920.00	−1.16435	−0.582175	−	0.813064i	$-0.697798\pi$
$359$	−7920.00	−1.16435	−0.582175	+	0.813064i	$0.697798\pi$
$360$	0	0
$361$	−5415.00	−0.789474
$362$	8410.00	1.22105
$363$	0	0
$364$	364.000	0.0524142
$365$	0	0
$366$	0	0
$367$	3404.00	0.484162	0.242081	−	0.970256i	$-0.422170\pi$
$367$	3404.00	0.484162	0.242081	+	0.970256i	$0.422170\pi$
$368$	−624.000	−0.0883920
$369$	0	0
$370$	0	0
$371$	3780.00	0.528970
$372$	0	0
$373$	10604.0	1.47200	0.735998	−	0.676984i	$-0.236713\pi$
$373$	10604.0	1.47200	0.735998	+	0.676984i	$0.236713\pi$
$374$	1872.00	0.258820
$375$	0	0
$376$	−792.000	−0.108628
$377$	1248.00	0.170491
$378$	0	0
$379$	−11680.0	−1.58301	−0.791506	−	0.611162i	$-0.790702\pi$
$379$	−11680.0	−1.58301	−0.791506	+	0.611162i	$0.790702\pi$
$380$	0	0
$381$	0	0
$382$	8304.00	1.11222
$383$	−8133.00	−1.08506	−0.542529	−	0.840037i	$-0.682533\pi$
$383$	−8133.00	−1.08506	−0.542529	+	0.840037i	$0.682533\pi$
$384$	0	0
$385$	0	0
$386$	−6296.00	−0.830202
$387$	0	0
$388$	1268.00	0.165910
$389$	−2556.00	−0.333147	−0.166574	−	0.986029i	$-0.553270\pi$
$389$	−2556.00	−0.333147	−0.166574	+	0.986029i	$0.553270\pi$
$390$	0	0
$391$	936.000	0.121063
$392$	392.000	0.0505076
$393$	0	0
$394$	−2346.00	−0.299974
$395$	0	0
$396$	0	0
$397$	12620.0	1.59541	0.797707	−	0.603045i	$-0.206046\pi$
$397$	12620.0	1.59541	0.797707	+	0.603045i	$0.206046\pi$
$398$	7024.00	0.884626
$399$	0	0
$400$	−2000.00	−0.250000
$401$	−2064.00	−0.257036	−0.128518	−	0.991707i	$-0.541022\pi$
$401$	−2064.00	−0.257036	−0.128518	+	0.991707i	$0.541022\pi$
$402$	0	0
$403$	2951.00	0.364764
$404$	2652.00	0.326589
$405$	0	0
$406$	1344.00	0.164290
$407$	−16575.0	−2.01865
$408$	0	0
$409$	−5974.00	−0.722238	−0.361119	−	0.932520i	$-0.617605\pi$
$409$	−5974.00	−0.722238	−0.361119	+	0.932520i	$0.617605\pi$
$410$	0	0
$411$	0	0
$412$	−2584.00	−0.308992
$413$	−798.000	−0.0950775
$414$	0	0
$415$	0	0
$416$	416.000	0.0490290
$417$	0	0
$418$	−2964.00	−0.346828
$419$	−12459.0	−1.45265	−0.726327	−	0.687349i	$-0.758774\pi$
$419$	−12459.0	−1.45265	−0.726327	+	0.687349i	$0.758774\pi$
$420$	0	0
$421$	−2221.00	−0.257114	−0.128557	−	0.991702i	$-0.541034\pi$
$421$	−2221.00	−0.257114	−0.128557	+	0.991702i	$0.541034\pi$
$422$	−6836.00	−0.788558
$423$	0	0
$424$	4320.00	0.494806
$425$	3000.00	0.342403
$426$	0	0
$427$	−3955.00	−0.448234
$428$	2976.00	0.336099
$429$	0	0
$430$	0	0
$431$	15162.0	1.69450	0.847248	−	0.531197i	$-0.178258\pi$
$431$	15162.0	1.69450	0.847248	+	0.531197i	$0.178258\pi$
$432$	0	0
$433$	−10978.0	−1.21840	−0.609202	−	0.793015i	$-0.708510\pi$
$433$	−10978.0	−1.21840	−0.609202	+	0.793015i	$0.708510\pi$
$434$	3178.00	0.351495
$435$	0	0
$436$	872.000	0.0957826
$437$	−1482.00	−0.162228
$438$	0	0
$439$	−3274.00	−0.355944	−0.177972	−	0.984036i	$-0.556954\pi$
$439$	−3274.00	−0.355944	−0.177972	+	0.984036i	$0.556954\pi$
$440$	0	0
$441$	0	0
$442$	−624.000	−0.0671508
$443$	−3888.00	−0.416985	−0.208493	−	0.978024i	$-0.566856\pi$
$443$	−3888.00	−0.416985	−0.208493	+	0.978024i	$0.566856\pi$
$444$	0	0
$445$	0	0
$446$	8482.00	0.900525
$447$	0	0
$448$	448.000	0.0472456
$449$	11262.0	1.18371	0.591856	−	0.806044i	$-0.298395\pi$
$449$	11262.0	1.18371	0.591856	+	0.806044i	$0.298395\pi$
$450$	0	0
$451$	−4095.00	−0.427552
$452$	6492.00	0.675571
$453$	0	0
$454$	−1776.00	−0.183594
$455$	0	0
$456$	0	0
$457$	−9718.00	−0.994724	−0.497362	−	0.867543i	$-0.665698\pi$
$457$	−9718.00	−0.994724	−0.497362	+	0.867543i	$0.665698\pi$
$458$	9400.00	0.959024
$459$	0	0
$460$	0	0
$461$	13656.0	1.37966	0.689830	−	0.723971i	$-0.257685\pi$
$461$	13656.0	1.37966	0.689830	+	0.723971i	$0.257685\pi$
$462$	0	0
$463$	12008.0	1.20531	0.602656	−	0.798001i	$-0.294109\pi$
$463$	12008.0	1.20531	0.602656	+	0.798001i	$0.294109\pi$
$464$	1536.00	0.153679
$465$	0	0
$466$	−12726.0	−1.26507
$467$	−2268.00	−0.224733	−0.112367	−	0.993667i	$-0.535843\pi$
$467$	−2268.00	−0.224733	−0.112367	+	0.993667i	$0.535843\pi$
$468$	0	0
$469$	−2695.00	−0.265338
$470$	0	0
$471$	0	0
$472$	−912.000	−0.0889369
$473$	−13416.0	−1.30416
$474$	0	0
$475$	−4750.00	−0.458831
$476$	−672.000	−0.0647081
$477$	0	0
$478$	6156.00	0.589056
$479$	10536.0	1.00501	0.502507	−	0.864573i	$-0.332411\pi$
$479$	10536.0	1.00501	0.502507	+	0.864573i	$0.332411\pi$
$480$	0	0
$481$	5525.00	0.523739
$482$	7348.00	0.694382
$483$	0	0
$484$	760.000	0.0713749
$485$	0	0
$486$	0	0
$487$	12566.0	1.16924	0.584620	−	0.811307i	$-0.301244\pi$
$487$	12566.0	1.16924	0.584620	+	0.811307i	$0.301244\pi$
$488$	−4520.00	−0.419284
$489$	0	0
$490$	0	0
$491$	12444.0	1.14377	0.571884	−	0.820335i	$-0.306213\pi$
$491$	12444.0	1.14377	0.571884	+	0.820335i	$0.306213\pi$
$492$	0	0
$493$	−2304.00	−0.210481
$494$	988.000	0.0899843
$495$	0	0
$496$	3632.00	0.328794
$497$	1092.00	0.0985571
$498$	0	0
$499$	−6091.00	−0.546434	−0.273217	−	0.961952i	$-0.588088\pi$
$499$	−6091.00	−0.546434	−0.273217	+	0.961952i	$0.588088\pi$
$500$	0	0
$501$	0	0
$502$	690.000	0.0613470
$503$	−9204.00	−0.815877	−0.407938	−	0.913009i	$-0.633752\pi$
$503$	−9204.00	−0.815877	−0.407938	+	0.913009i	$0.633752\pi$
$504$	0	0
$505$	0	0
$506$	3042.00	0.267260
$507$	0	0
$508$	2636.00	0.230224
$509$	−21522.0	−1.87416	−0.937078	−	0.349119i	$-0.886481\pi$
$509$	−21522.0	−1.87416	−0.937078	+	0.349119i	$0.886481\pi$
$510$	0	0
$511$	−4711.00	−0.407832
$512$	512.000	0.0441942
$513$	0	0
$514$	−13776.0	−1.18217
$515$	0	0
$516$	0	0
$517$	3861.00	0.328446
$518$	5950.00	0.504687
$519$	0	0
$520$	0	0
$521$	12474.0	1.04894	0.524468	−	0.851430i	$-0.324264\pi$
$521$	12474.0	1.04894	0.524468	+	0.851430i	$0.324264\pi$
$522$	0	0
$523$	16607.0	1.38848	0.694238	−	0.719745i	$-0.255741\pi$
$523$	16607.0	1.38848	0.694238	+	0.719745i	$0.255741\pi$
$524$	864.000	0.0720306
$525$	0	0
$526$	2496.00	0.206903
$527$	−5448.00	−0.450320
$528$	0	0
$529$	−10646.0	−0.874990
$530$	0	0
$531$	0	0
$532$	1064.00	0.0867110
$533$	1365.00	0.110928
$534$	0	0
$535$	0	0
$536$	−3080.00	−0.248201
$537$	0	0
$538$	4506.00	0.361092
$539$	−1911.00	−0.152714
$540$	0	0
$541$	−19690.0	−1.56477	−0.782384	−	0.622797i	$-0.785996\pi$
$541$	−19690.0	−1.56477	−0.782384	+	0.622797i	$0.785996\pi$
$542$	2794.00	0.221425
$543$	0	0
$544$	−768.000	−0.0605289
$545$	0	0
$546$	0	0
$547$	−19960.0	−1.56020	−0.780099	−	0.625656i	$-0.784831\pi$
$547$	−19960.0	−1.56020	−0.780099	+	0.625656i	$0.784831\pi$
$548$	−7368.00	−0.574353
$549$	0	0
$550$	9750.00	0.755893
$551$	3648.00	0.282051
$552$	0	0
$553$	5243.00	0.403174
$554$	−4604.00	−0.353078
$555$	0	0
$556$	−2512.00	−0.191605
$557$	−19137.0	−1.45576	−0.727882	−	0.685702i	$-0.759495\pi$
$557$	−19137.0	−1.45576	−0.727882	+	0.685702i	$0.759495\pi$
$558$	0	0
$559$	4472.00	0.338364
$560$	0	0
$561$	0	0
$562$	−17064.0	−1.28079
$563$	−6711.00	−0.502371	−0.251186	−	0.967939i	$-0.580820\pi$
$563$	−6711.00	−0.502371	−0.251186	+	0.967939i	$0.580820\pi$
$564$	0	0
$565$	0	0
$566$	−7538.00	−0.559798
$567$	0	0
$568$	1248.00	0.0921918
$569$	5595.00	0.412222	0.206111	−	0.978529i	$-0.433919\pi$
$569$	5595.00	0.412222	0.206111	+	0.978529i	$0.433919\pi$
$570$	0	0
$571$	−21274.0	−1.55918	−0.779588	−	0.626293i	$-0.784571\pi$
$571$	−21274.0	−1.55918	−0.779588	+	0.626293i	$0.784571\pi$
$572$	−2028.00	−0.148243
$573$	0	0
$574$	1470.00	0.106893
$575$	4875.00	0.353568
$576$	0	0
$577$	−15694.0	−1.13232	−0.566161	−	0.824295i	$-0.691572\pi$
$577$	−15694.0	−1.13232	−0.566161	+	0.824295i	$0.691572\pi$
$578$	−8674.00	−0.624206
$579$	0	0
$580$	0	0
$581$	7308.00	0.521836
$582$	0	0
$583$	−21060.0	−1.49608
$584$	−5384.00	−0.381492
$585$	0	0
$586$	12300.0	0.867079
$587$	−21054.0	−1.48039	−0.740197	−	0.672390i	$-0.765268\pi$
$587$	−21054.0	−1.48039	−0.740197	+	0.672390i	$0.765268\pi$
$588$	0	0
$589$	8626.00	0.603443
$590$	0	0
$591$	0	0
$592$	6800.00	0.472092
$593$	17910.0	1.24026	0.620131	−	0.784498i	$-0.287079\pi$
$593$	17910.0	1.24026	0.620131	+	0.784498i	$0.287079\pi$
$594$	0	0
$595$	0	0
$596$	−1284.00	−0.0882461
$597$	0	0
$598$	−1014.00	−0.0693404
$599$	−3213.00	−0.219165	−0.109582	−	0.993978i	$-0.534951\pi$
$599$	−3213.00	−0.219165	−0.109582	+	0.993978i	$0.534951\pi$
$600$	0	0
$601$	15158.0	1.02880	0.514399	−	0.857551i	$-0.328015\pi$
$601$	15158.0	1.02880	0.514399	+	0.857551i	$0.328015\pi$
$602$	4816.00	0.326056
$603$	0	0
$604$	−6400.00	−0.431146
$605$	0	0
$606$	0	0
$607$	−26206.0	−1.75234	−0.876169	−	0.482005i	$-0.839909\pi$
$607$	−26206.0	−1.75234	−0.876169	+	0.482005i	$0.839909\pi$
$608$	1216.00	0.0811107
$609$	0	0
$610$	0	0
$611$	−1287.00	−0.0852151
$612$	0	0
$613$	−6145.00	−0.404885	−0.202442	−	0.979294i	$-0.564888\pi$
$613$	−6145.00	−0.404885	−0.202442	+	0.979294i	$0.564888\pi$
$614$	9184.00	0.603642
$615$	0	0
$616$	−2184.00	−0.142850
$617$	−9474.00	−0.618167	−0.309083	−	0.951035i	$-0.600022\pi$
$617$	−9474.00	−0.618167	−0.309083	+	0.951035i	$0.600022\pi$
$618$	0	0
$619$	−14326.0	−0.930227	−0.465114	−	0.885251i	$-0.653987\pi$
$619$	−14326.0	−0.930227	−0.465114	+	0.885251i	$0.653987\pi$
$620$	0	0
$621$	0	0
$622$	−12996.0	−0.837769
$623$	4830.00	0.310610
$624$	0	0
$625$	15625.0	1.00000
$626$	11524.0	0.735769
$627$	0	0
$628$	4508.00	0.286447
$629$	−10200.0	−0.646583
$630$	0	0
$631$	−15478.0	−0.976497	−0.488248	−	0.872705i	$-0.662364\pi$
$631$	−15478.0	−0.976497	−0.488248	+	0.872705i	$0.662364\pi$
$632$	5992.00	0.377134
$633$	0	0
$634$	−5130.00	−0.321354
$635$	0	0
$636$	0	0
$637$	637.000	0.0396214
$638$	−7488.00	−0.464659
$639$	0	0
$640$	0	0
$641$	−17439.0	−1.07457	−0.537285	−	0.843401i	$-0.680550\pi$
$641$	−17439.0	−1.07457	−0.537285	+	0.843401i	$0.680550\pi$
$642$	0	0
$643$	30296.0	1.85810	0.929049	−	0.369956i	$-0.120627\pi$
$643$	30296.0	1.85810	0.929049	+	0.369956i	$0.120627\pi$
$644$	−1092.00	−0.0668181
$645$	0	0
$646$	−1824.00	−0.111090
$647$	17124.0	1.04052	0.520258	−	0.854009i	$-0.325836\pi$
$647$	17124.0	1.04052	0.520258	+	0.854009i	$0.325836\pi$
$648$	0	0
$649$	4446.00	0.268907
$650$	−3250.00	−0.196116
$651$	0	0
$652$	−4816.00	−0.289278
$653$	−27120.0	−1.62525	−0.812625	−	0.582788i	$-0.801962\pi$
$653$	−27120.0	−1.62525	−0.812625	+	0.582788i	$0.801962\pi$
$654$	0	0
$655$	0	0
$656$	1680.00	0.0999893
$657$	0	0
$658$	−1386.00	−0.0821154
$659$	138.000	0.00815739	0.00407869	−	0.999992i	$-0.498702\pi$
$659$	138.000	0.00815739	0.00407869	+	0.999992i	$0.498702\pi$
$660$	0	0
$661$	29720.0	1.74883	0.874413	−	0.485182i	$-0.161247\pi$
$661$	29720.0	1.74883	0.874413	+	0.485182i	$0.161247\pi$
$662$	−1490.00	−0.0874781
$663$	0	0
$664$	8352.00	0.488133
$665$	0	0
$666$	0	0
$667$	−3744.00	−0.217344
$668$	2400.00	0.139010
$669$	0	0
$670$	0	0
$671$	22035.0	1.26774
$672$	0	0
$673$	−32929.0	−1.88606	−0.943031	−	0.332705i	$-0.892039\pi$
$673$	−32929.0	−1.88606	−0.943031	+	0.332705i	$0.892039\pi$
$674$	16690.0	0.953820
$675$	0	0
$676$	676.000	0.0384615
$677$	−33021.0	−1.87459	−0.937297	−	0.348532i	$-0.886680\pi$
$677$	−33021.0	−1.87459	−0.937297	+	0.348532i	$0.886680\pi$
$678$	0	0
$679$	2219.00	0.125416
$680$	0	0
$681$	0	0
$682$	−17706.0	−0.994132
$683$	−18519.0	−1.03750	−0.518748	−	0.854927i	$-0.673602\pi$
$683$	−18519.0	−1.03750	−0.518748	+	0.854927i	$0.673602\pi$
$684$	0	0
$685$	0	0
$686$	686.000	0.0381802
$687$	0	0
$688$	5504.00	0.304997
$689$	7020.00	0.388158
$690$	0	0
$691$	15248.0	0.839452	0.419726	−	0.907651i	$-0.362126\pi$
$691$	15248.0	0.839452	0.419726	+	0.907651i	$0.362126\pi$
$692$	8616.00	0.473311
$693$	0	0
$694$	3552.00	0.194283
$695$	0	0
$696$	0	0
$697$	−2520.00	−0.136947
$698$	−17204.0	−0.932924
$699$	0	0
$700$	−3500.00	−0.188982
$701$	7740.00	0.417027	0.208513	−	0.978020i	$-0.433138\pi$
$701$	7740.00	0.417027	0.208513	+	0.978020i	$0.433138\pi$
$702$	0	0
$703$	16150.0	0.866442
$704$	−2496.00	−0.133624
$705$	0	0
$706$	12954.0	0.690553
$707$	4641.00	0.246878
$708$	0	0
$709$	29747.0	1.57570	0.787851	−	0.615867i	$-0.211194\pi$
$709$	29747.0	1.57570	0.787851	+	0.615867i	$0.211194\pi$
$710$	0	0
$711$	0	0
$712$	5520.00	0.290549
$713$	−8853.00	−0.465003
$714$	0	0
$715$	0	0
$716$	11400.0	0.595025
$717$	0	0
$718$	−15840.0	−0.823320
$719$	−10266.0	−0.532486	−0.266243	−	0.963906i	$-0.585782\pi$
$719$	−10266.0	−0.532486	−0.266243	+	0.963906i	$0.585782\pi$
$720$	0	0
$721$	−4522.00	−0.233576
$722$	−10830.0	−0.558242
$723$	0	0
$724$	16820.0	0.863412
$725$	−12000.0	−0.614716
$726$	0	0
$727$	−8026.00	−0.409447	−0.204723	−	0.978820i	$-0.565629\pi$
$727$	−8026.00	−0.409447	−0.204723	+	0.978820i	$0.565629\pi$
$728$	728.000	0.0370625
$729$	0	0
$730$	0	0
$731$	−8256.00	−0.417728
$732$	0	0
$733$	13268.0	0.668574	0.334287	−	0.942471i	$-0.391505\pi$
$733$	13268.0	0.668574	0.334287	+	0.942471i	$0.391505\pi$
$734$	6808.00	0.342354
$735$	0	0
$736$	−1248.00	−0.0625026
$737$	15015.0	0.750454
$738$	0	0
$739$	−8080.00	−0.402202	−0.201101	−	0.979570i	$-0.564452\pi$
$739$	−8080.00	−0.402202	−0.201101	+	0.979570i	$0.564452\pi$
$740$	0	0
$741$	0	0
$742$	7560.00	0.374038
$743$	27096.0	1.33789	0.668947	−	0.743310i	$-0.266745\pi$
$743$	27096.0	1.33789	0.668947	+	0.743310i	$0.266745\pi$
$744$	0	0
$745$	0	0
$746$	21208.0	1.04086
$747$	0	0
$748$	3744.00	0.183014
$749$	5208.00	0.254067
$750$	0	0
$751$	25067.0	1.21799	0.608993	−	0.793175i	$-0.291574\pi$
$751$	25067.0	1.21799	0.608993	+	0.793175i	$0.291574\pi$
$752$	−1584.00	−0.0768119
$753$	0	0
$754$	2496.00	0.120556
$755$	0	0
$756$	0	0
$757$	−6442.00	−0.309298	−0.154649	−	0.987969i	$-0.549425\pi$
$757$	−6442.00	−0.309298	−0.154649	+	0.987969i	$0.549425\pi$
$758$	−23360.0	−1.11936
$759$	0	0
$760$	0	0
$761$	−26511.0	−1.26284	−0.631421	−	0.775440i	$-0.717528\pi$
$761$	−26511.0	−1.26284	−0.631421	+	0.775440i	$0.717528\pi$
$762$	0	0
$763$	1526.00	0.0724049
$764$	16608.0	0.786461
$765$	0	0
$766$	−16266.0	−0.767251
$767$	−1482.00	−0.0697678
$768$	0	0
$769$	−17665.0	−0.828370	−0.414185	−	0.910193i	$-0.635933\pi$
$769$	−17665.0	−0.828370	−0.414185	+	0.910193i	$0.635933\pi$
$770$	0	0
$771$	0	0
$772$	−12592.0	−0.587041
$773$	−36258.0	−1.68708	−0.843538	−	0.537070i	$-0.819531\pi$
$773$	−36258.0	−1.68708	−0.843538	+	0.537070i	$0.819531\pi$
$774$	0	0
$775$	−28375.0	−1.31517
$776$	2536.00	0.117316
$777$	0	0
$778$	−5112.00	−0.235571
$779$	3990.00	0.183513
$780$	0	0
$781$	−6084.00	−0.278749
$782$	1872.00	0.0856043
$783$	0	0
$784$	784.000	0.0357143
$785$	0	0
$786$	0	0
$787$	21296.0	0.964575	0.482287	−	0.876013i	$-0.339806\pi$
$787$	21296.0	0.964575	0.482287	+	0.876013i	$0.339806\pi$
$788$	−4692.00	−0.212114
$789$	0	0
$790$	0	0
$791$	11361.0	0.510684
$792$	0	0
$793$	−7345.00	−0.328914
$794$	25240.0	1.12813
$795$	0	0
$796$	14048.0	0.625525
$797$	28947.0	1.28652	0.643259	−	0.765648i	$-0.277582\pi$
$797$	28947.0	1.28652	0.643259	+	0.765648i	$0.277582\pi$
$798$	0	0
$799$	2376.00	0.105203
$800$	−4000.00	−0.176777
$801$	0	0
$802$	−4128.00	−0.181752
$803$	26247.0	1.15347
$804$	0	0
$805$	0	0
$806$	5902.00	0.257927
$807$	0	0
$808$	5304.00	0.230933
$809$	20418.0	0.887341	0.443670	−	0.896190i	$-0.353676\pi$
$809$	20418.0	0.887341	0.443670	+	0.896190i	$0.353676\pi$
$810$	0	0
$811$	−14524.0	−0.628861	−0.314431	−	0.949280i	$-0.601814\pi$
$811$	−14524.0	−0.628861	−0.314431	+	0.949280i	$0.601814\pi$
$812$	2688.00	0.116170
$813$	0	0
$814$	−33150.0	−1.42740
$815$	0	0
$816$	0	0
$817$	13072.0	0.559769
$818$	−11948.0	−0.510699
$819$	0	0
$820$	0	0
$821$	7710.00	0.327748	0.163874	−	0.986481i	$-0.447601\pi$
$821$	7710.00	0.327748	0.163874	+	0.986481i	$0.447601\pi$
$822$	0	0
$823$	2531.00	0.107199	0.0535997	−	0.998563i	$-0.482931\pi$
$823$	2531.00	0.107199	0.0535997	+	0.998563i	$0.482931\pi$
$824$	−5168.00	−0.218490
$825$	0	0
$826$	−1596.00	−0.0672300
$827$	−24516.0	−1.03084	−0.515420	−	0.856938i	$-0.672364\pi$
$827$	−24516.0	−1.03084	−0.515420	+	0.856938i	$0.672364\pi$
$828$	0	0
$829$	−15586.0	−0.652985	−0.326492	−	0.945200i	$-0.605867\pi$
$829$	−15586.0	−0.652985	−0.326492	+	0.945200i	$0.605867\pi$
$830$	0	0
$831$	0	0
$832$	832.000	0.0346688
$833$	−1176.00	−0.0489147
$834$	0	0
$835$	0	0
$836$	−5928.00	−0.245244
$837$	0	0
$838$	−24918.0	−1.02718
$839$	24915.0	1.02522	0.512611	−	0.858621i	$-0.328678\pi$
$839$	24915.0	1.02522	0.512611	+	0.858621i	$0.328678\pi$
$840$	0	0
$841$	−15173.0	−0.622125
$842$	−4442.00	−0.181807
$843$	0	0
$844$	−13672.0	−0.557594
$845$	0	0
$846$	0	0
$847$	1330.00	0.0539544
$848$	8640.00	0.349881
$849$	0	0
$850$	6000.00	0.242116
$851$	−16575.0	−0.667666
$852$	0	0
$853$	−20500.0	−0.822868	−0.411434	−	0.911439i	$-0.634972\pi$
$853$	−20500.0	−0.822868	−0.411434	+	0.911439i	$0.634972\pi$
$854$	−7910.00	−0.316949
$855$	0	0
$856$	5952.00	0.237658
$857$	−26694.0	−1.06400	−0.532001	−	0.846744i	$-0.678560\pi$
$857$	−26694.0	−1.06400	−0.532001	+	0.846744i	$0.678560\pi$
$858$	0	0
$859$	20801.0	0.826218	0.413109	−	0.910682i	$-0.364443\pi$
$859$	20801.0	0.826218	0.413109	+	0.910682i	$0.364443\pi$
$860$	0	0
$861$	0	0
$862$	30324.0	1.19819
$863$	37404.0	1.47537	0.737687	−	0.675143i	$-0.235918\pi$
$863$	37404.0	1.47537	0.737687	+	0.675143i	$0.235918\pi$
$864$	0	0
$865$	0	0
$866$	−21956.0	−0.861542
$867$	0	0
$868$	6356.00	0.248545
$869$	−29211.0	−1.14029
$870$	0	0
$871$	−5005.00	−0.194705
$872$	1744.00	0.0677285
$873$	0	0
$874$	−2964.00	−0.114713
$875$	0	0
$876$	0	0
$877$	−20581.0	−0.792441	−0.396221	−	0.918155i	$-0.629678\pi$
$877$	−20581.0	−0.792441	−0.396221	+	0.918155i	$0.629678\pi$
$878$	−6548.00	−0.251691
$879$	0	0
$880$	0	0
$881$	34314.0	1.31222	0.656111	−	0.754664i	$-0.272200\pi$
$881$	34314.0	1.31222	0.656111	+	0.754664i	$0.272200\pi$
$882$	0	0
$883$	−12058.0	−0.459552	−0.229776	−	0.973244i	$-0.573799\pi$
$883$	−12058.0	−0.459552	−0.229776	+	0.973244i	$0.573799\pi$
$884$	−1248.00	−0.0474828
$885$	0	0
$886$	−7776.00	−0.294853
$887$	−20406.0	−0.772454	−0.386227	−	0.922404i	$-0.626222\pi$
$887$	−20406.0	−0.772454	−0.386227	+	0.922404i	$0.626222\pi$
$888$	0	0
$889$	4613.00	0.174033
$890$	0	0
$891$	0	0
$892$	16964.0	0.636768
$893$	−3762.00	−0.140975
$894$	0	0
$895$	0	0
$896$	896.000	0.0334077
$897$	0	0
$898$	22524.0	0.837011
$899$	21792.0	0.808458
$900$	0	0
$901$	−12960.0	−0.479201
$902$	−8190.00	−0.302325
$903$	0	0
$904$	12984.0	0.477701
$905$	0	0
$906$	0	0
$907$	−2914.00	−0.106679	−0.0533395	−	0.998576i	$-0.516987\pi$
$907$	−2914.00	−0.106679	−0.0533395	+	0.998576i	$0.516987\pi$
$908$	−3552.00	−0.129821
$909$	0	0
$910$	0	0
$911$	1044.00	0.0379685	0.0189842	−	0.999820i	$-0.493957\pi$
$911$	1044.00	0.0379685	0.0189842	+	0.999820i	$0.493957\pi$
$912$	0	0
$913$	−40716.0	−1.47591
$914$	−19436.0	−0.703376
$915$	0	0
$916$	18800.0	0.678133
$917$	1512.00	0.0544500
$918$	0	0
$919$	20693.0	0.742763	0.371381	−	0.928480i	$-0.378884\pi$
$919$	20693.0	0.742763	0.371381	+	0.928480i	$0.378884\pi$
$920$	0	0
$921$	0	0
$922$	27312.0	0.975567
$923$	2028.00	0.0723212
$924$	0	0
$925$	−53125.0	−1.88837
$926$	24016.0	0.852284
$927$	0	0
$928$	3072.00	0.108667
$929$	52653.0	1.85951	0.929757	−	0.368173i	$-0.120017\pi$
$929$	52653.0	1.85951	0.929757	+	0.368173i	$0.120017\pi$
$930$	0	0
$931$	1862.00	0.0655474
$932$	−25452.0	−0.894536
$933$	0	0
$934$	−4536.00	−0.158911
$935$	0	0
$936$	0	0
$937$	34868.0	1.21568	0.607838	−	0.794061i	$-0.292037\pi$
$937$	34868.0	1.21568	0.607838	+	0.794061i	$0.292037\pi$
$938$	−5390.00	−0.187622
$939$	0	0
$940$	0	0
$941$	−25542.0	−0.884852	−0.442426	−	0.896805i	$-0.645882\pi$
$941$	−25542.0	−0.884852	−0.442426	+	0.896805i	$0.645882\pi$
$942$	0	0
$943$	−4095.00	−0.141412
$944$	−1824.00	−0.0628879
$945$	0	0
$946$	−26832.0	−0.922181
$947$	672.000	0.0230592	0.0115296	−	0.999934i	$-0.496330\pi$
$947$	672.000	0.0230592	0.0115296	+	0.999934i	$0.496330\pi$
$948$	0	0
$949$	−8749.00	−0.299267
$950$	−9500.00	−0.324443
$951$	0	0
$952$	−1344.00	−0.0457556
$953$	−52278.0	−1.77697	−0.888484	−	0.458908i	$-0.848241\pi$
$953$	−52278.0	−1.77697	−0.888484	+	0.458908i	$0.848241\pi$
$954$	0	0
$955$	0	0
$956$	12312.0	0.416526
$957$	0	0
$958$	21072.0	0.710653
$959$	−12894.0	−0.434170
$960$	0	0
$961$	21738.0	0.729683
$962$	11050.0	0.370339
$963$	0	0
$964$	14696.0	0.491002
$965$	0	0
$966$	0	0
$967$	758.000	0.0252075	0.0126037	−	0.999921i	$-0.495988\pi$
$967$	758.000	0.0252075	0.0126037	+	0.999921i	$0.495988\pi$
$968$	1520.00	0.0504697
$969$	0	0
$970$	0	0
$971$	−27285.0	−0.901769	−0.450884	−	0.892582i	$-0.648891\pi$
$971$	−27285.0	−0.901769	−0.450884	+	0.892582i	$0.648891\pi$
$972$	0	0
$973$	−4396.00	−0.144840
$974$	25132.0	0.826777
$975$	0	0
$976$	−9040.00	−0.296479
$977$	−18786.0	−0.615166	−0.307583	−	0.951521i	$-0.599520\pi$
$977$	−18786.0	−0.615166	−0.307583	+	0.951521i	$0.599520\pi$
$978$	0	0
$979$	−26910.0	−0.878496
$980$	0	0
$981$	0	0
$982$	24888.0	0.808766
$983$	−37152.0	−1.20546	−0.602729	−	0.797946i	$-0.705920\pi$
$983$	−37152.0	−1.20546	−0.602729	+	0.797946i	$0.705920\pi$
$984$	0	0
$985$	0	0
$986$	−4608.00	−0.148832
$987$	0	0
$988$	1976.00	0.0636285
$989$	−13416.0	−0.431349
$990$	0	0
$991$	−17143.0	−0.549511	−0.274755	−	0.961514i	$-0.588597\pi$
$991$	−17143.0	−0.549511	−0.274755	+	0.961514i	$0.588597\pi$
$992$	7264.00	0.232492
$993$	0	0
$994$	2184.00	0.0696904
$995$	0	0
$996$	0	0
$997$	−14137.0	−0.449070	−0.224535	−	0.974466i	$-0.572086\pi$
$997$	−14137.0	−0.449070	−0.224535	+	0.974466i	$0.572086\pi$
$998$	−12182.0	−0.386387
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1638.4.a.j.1.1		1
3.2	odd	2		182.4.a.a.1.1	✓	1
12.11	even	2		1456.4.a.b.1.1		1
21.20	even	2		1274.4.a.a.1.1		1
39.38	odd	2		2366.4.a.g.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
182.4.a.a.1.1	✓	1	3.2	odd	2
1274.4.a.a.1.1		1	21.20	even	2
1456.4.a.b.1.1		1	12.11	even	2
1638.4.a.j.1.1		1	1.1	even	1	trivial
2366.4.a.g.1.1		1	39.38	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:	\( N \)	\(=\)	\( 1638 = 2 \cdot 3^{2} \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	1638.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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