LMFDB - Embedded newform 560.6.a.h.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [560,6,Mod(1,560)] 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("560.1"); S:= CuspForms(chi, 6); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(560, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 6, names="a")

Level:	$ N $	$=$	$ 560 = 2^{4} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	560.a (trivial)

Newform invariants

comment:select newform

sage:traces = [1,0,17,0,25,0,49,0,46] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(9)] == traces)

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

Self dual:	yes
Analytic conductor:	$89.8149390953$
Analytic rank:	$0$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 70)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	560.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$

$=$

$q+17.0000 q^{3} +25.0000 q^{5} +49.0000 q^{7} +46.0000 q^{9} +715.000 q^{11} +331.000 q^{13} +425.000 q^{15} -1699.00 q^{17} +1718.00 q^{19} +833.000 q^{21} +3950.00 q^{23} +625.000 q^{25} -3349.00 q^{27} +4579.00 q^{29} -6756.00 q^{31} +12155.0 q^{33} +1225.00 q^{35} -16518.0 q^{37} +5627.00 q^{39} +18876.0 q^{41} -2258.00 q^{43} +1150.00 q^{45} +537.000 q^{47} +2401.00 q^{49} -28883.0 q^{51} -10984.0 q^{53} +17875.0 q^{55} +29206.0 q^{57} +25956.0 q^{59} +39188.0 q^{61} +2254.00 q^{63} +8275.00 q^{65} -4416.00 q^{67} +67150.0 q^{69} +31880.0 q^{71} -5018.00 q^{73} +10625.0 q^{75} +35035.0 q^{77} +27977.0 q^{79} -68111.0 q^{81} -37644.0 q^{83} -42475.0 q^{85} +77843.0 q^{87} -17216.0 q^{89} +16219.0 q^{91} -114852. q^{93} +42950.0 q^{95} -63175.0 q^{97} +32890.0 q^{99} +O(q^{100})$

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
$3$	17.0000	1.09055	0.545275	−	0.838257i	$-0.316425\pi$
$3$	17.0000	1.09055	0.545275	+	0.838257i	$0.316425\pi$
$4$	0	0
$5$	25.0000	0.447214
$5$	25.0000	0.447214
$6$	0	0
$7$	49.0000	0.377964
$7$	49.0000	0.377964
$8$	0	0
$9$	46.0000	0.189300
$10$	0	0
$11$	715.000	1.78166	0.890829	−	0.454339i	$-0.150124\pi$
$11$	715.000	1.78166	0.890829	+	0.454339i	$0.150124\pi$
$12$	0	0
$13$	331.000	0.543212	0.271606	−	0.962408i	$-0.412445\pi$
$13$	331.000	0.543212	0.271606	+	0.962408i	$0.412445\pi$
$14$	0	0
$15$	425.000	0.487709
$16$	0	0
$17$	−1699.00	−1.42584	−0.712920	−	0.701245i	$-0.752628\pi$
$17$	−1699.00	−1.42584	−0.712920	+	0.701245i	$0.752628\pi$
$18$	0	0
$19$	1718.00	1.09179	0.545895	−	0.837854i	$-0.316190\pi$
$19$	1718.00	1.09179	0.545895	+	0.837854i	$0.316190\pi$
$20$	0	0
$21$	833.000	0.412189
$22$	0	0
$23$	3950.00	1.55696	0.778480	−	0.627669i	$-0.215991\pi$
$23$	3950.00	1.55696	0.778480	+	0.627669i	$0.215991\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−3349.00	−0.884109
$28$	0	0
$29$	4579.00	1.01106	0.505529	−	0.862810i	$-0.331298\pi$
$29$	4579.00	1.01106	0.505529	+	0.862810i	$0.331298\pi$
$30$	0	0
$31$	−6756.00	−1.26266	−0.631329	−	0.775516i	$-0.717490\pi$
$31$	−6756.00	−1.26266	−0.631329	+	0.775516i	$0.717490\pi$
$32$	0	0
$33$	12155.0	1.94299
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	−16518.0	−1.98360	−0.991798	−	0.127816i	$-0.959203\pi$
$37$	−16518.0	−1.98360	−0.991798	+	0.127816i	$0.959203\pi$
$38$	0	0
$39$	5627.00	0.592400
$40$	0	0
$41$	18876.0	1.75368	0.876840	−	0.480782i	$-0.159647\pi$
$41$	18876.0	1.75368	0.876840	+	0.480782i	$0.159647\pi$
$42$	0	0
$43$	−2258.00	−0.186231	−0.0931157	−	0.995655i	$-0.529683\pi$
$43$	−2258.00	−0.186231	−0.0931157	+	0.995655i	$0.529683\pi$
$44$	0	0
$45$	1150.00	0.0846577
$46$	0	0
$47$	537.000	0.0354593	0.0177296	−	0.999843i	$-0.494356\pi$
$47$	537.000	0.0354593	0.0177296	+	0.999843i	$0.494356\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−28883.0	−1.55495
$52$	0	0
$53$	−10984.0	−0.537119	−0.268560	−	0.963263i	$-0.586548\pi$
$53$	−10984.0	−0.537119	−0.268560	+	0.963263i	$0.586548\pi$
$54$	0	0
$55$	17875.0	0.796782
$56$	0	0
$57$	29206.0	1.19065
$58$	0	0
$59$	25956.0	0.970751	0.485375	−	0.874306i	$-0.338683\pi$
$59$	25956.0	0.970751	0.485375	+	0.874306i	$0.338683\pi$
$60$	0	0
$61$	39188.0	1.34843	0.674215	−	0.738535i	$-0.264482\pi$
$61$	39188.0	1.34843	0.674215	+	0.738535i	$0.264482\pi$
$62$	0	0
$63$	2254.00	0.0715488
$64$	0	0
$65$	8275.00	0.242932
$66$	0	0
$67$	−4416.00	−0.120183	−0.0600914	−	0.998193i	$-0.519139\pi$
$67$	−4416.00	−0.120183	−0.0600914	+	0.998193i	$0.519139\pi$
$68$	0	0
$69$	67150.0	1.69794
$70$	0	0
$71$	31880.0	0.750538	0.375269	−	0.926916i	$-0.377550\pi$
$71$	31880.0	0.750538	0.375269	+	0.926916i	$0.377550\pi$
$72$	0	0
$73$	−5018.00	−0.110211	−0.0551053	−	0.998481i	$-0.517549\pi$
$73$	−5018.00	−0.110211	−0.0551053	+	0.998481i	$0.517549\pi$
$74$	0	0
$75$	10625.0	0.218110
$76$	0	0
$77$	35035.0	0.673403
$78$	0	0
$79$	27977.0	0.504352	0.252176	−	0.967681i	$-0.418854\pi$
$79$	27977.0	0.504352	0.252176	+	0.967681i	$0.418854\pi$
$80$	0	0
$81$	−68111.0	−1.15347
$82$	0	0
$83$	−37644.0	−0.599792	−0.299896	−	0.953972i	$-0.596952\pi$
$83$	−37644.0	−0.599792	−0.299896	+	0.953972i	$0.596952\pi$
$84$	0	0
$85$	−42475.0	−0.637655
$86$	0	0
$87$	77843.0	1.10261
$88$	0	0
$89$	−17216.0	−0.230387	−0.115193	−	0.993343i	$-0.536749\pi$
$89$	−17216.0	−0.230387	−0.115193	+	0.993343i	$0.536749\pi$
$90$	0	0
$91$	16219.0	0.205315
$92$	0	0
$93$	−114852.	−1.37699
$94$	0	0
$95$	42950.0	0.488263
$96$	0	0
$97$	−63175.0	−0.681736	−0.340868	−	0.940111i	$-0.610721\pi$
$97$	−63175.0	−0.681736	−0.340868	+	0.940111i	$0.610721\pi$
$98$	0	0
$99$	32890.0	0.337269
$100$	0	0
$101$	−29250.0	−0.285314	−0.142657	−	0.989772i	$-0.545565\pi$
$101$	−29250.0	−0.285314	−0.142657	+	0.989772i	$0.545565\pi$
$102$	0	0
$103$	149189.	1.38562	0.692809	−	0.721121i	$-0.256373\pi$
$103$	149189.	1.38562	0.692809	+	0.721121i	$0.256373\pi$
$104$	0	0
$105$	20825.0	0.184337
$106$	0	0
$107$	−83742.0	−0.707105	−0.353552	−	0.935415i	$-0.615026\pi$
$107$	−83742.0	−0.707105	−0.353552	+	0.935415i	$0.615026\pi$
$108$	0	0
$109$	105377.	0.849532	0.424766	−	0.905303i	$-0.360356\pi$
$109$	105377.	0.849532	0.424766	+	0.905303i	$0.360356\pi$
$110$	0	0
$111$	−280806.	−2.16321
$112$	0	0
$113$	−122754.	−0.904356	−0.452178	−	0.891928i	$-0.649353\pi$
$113$	−122754.	−0.904356	−0.452178	+	0.891928i	$0.649353\pi$
$114$	0	0
$115$	98750.0	0.696294
$116$	0	0
$117$	15226.0	0.102830
$118$	0	0
$119$	−83251.0	−0.538917
$120$	0	0
$121$	350174.	2.17431
$122$	0	0
$123$	320892.	1.91248
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	219196.	1.20593	0.602967	−	0.797766i	$-0.293985\pi$
$127$	219196.	1.20593	0.602967	+	0.797766i	$0.293985\pi$
$128$	0	0
$129$	−38386.0	−0.203095
$130$	0	0
$131$	96682.0	0.492229	0.246115	−	0.969241i	$-0.420846\pi$
$131$	96682.0	0.492229	0.246115	+	0.969241i	$0.420846\pi$
$132$	0	0
$133$	84182.0	0.412658
$134$	0	0
$135$	−83725.0	−0.395385
$136$	0	0
$137$	187288.	0.852528	0.426264	−	0.904599i	$-0.359830\pi$
$137$	187288.	0.852528	0.426264	+	0.904599i	$0.359830\pi$
$138$	0	0
$139$	−176894.	−0.776562	−0.388281	−	0.921541i	$-0.626931\pi$
$139$	−176894.	−0.776562	−0.388281	+	0.921541i	$0.626931\pi$
$140$	0	0
$141$	9129.00	0.0386701
$142$	0	0
$143$	236665.	0.967819
$144$	0	0
$145$	114475.	0.452158
$146$	0	0
$147$	40817.0	0.155793
$148$	0	0
$149$	−199078.	−0.734611	−0.367306	−	0.930100i	$-0.619720\pi$
$149$	−199078.	−0.734611	−0.367306	+	0.930100i	$0.619720\pi$
$150$	0	0
$151$	−471583.	−1.68312	−0.841561	−	0.540162i	$-0.818363\pi$
$151$	−471583.	−1.68312	−0.841561	+	0.540162i	$0.818363\pi$
$152$	0	0
$153$	−78154.0	−0.269912
$154$	0	0
$155$	−168900.	−0.564677
$156$	0	0
$157$	−72054.0	−0.233297	−0.116648	−	0.993173i	$-0.537215\pi$
$157$	−72054.0	−0.233297	−0.116648	+	0.993173i	$0.537215\pi$
$158$	0	0
$159$	−186728.	−0.585756
$160$	0	0
$161$	193550.	0.588476
$162$	0	0
$163$	−385334.	−1.13597	−0.567987	−	0.823038i	$-0.692278\pi$
$163$	−385334.	−1.13597	−0.567987	+	0.823038i	$0.692278\pi$
$164$	0	0
$165$	303875.	0.868931
$166$	0	0
$167$	542957.	1.50652	0.753259	−	0.657724i	$-0.228481\pi$
$167$	542957.	1.50652	0.753259	+	0.657724i	$0.228481\pi$
$168$	0	0
$169$	−261732.	−0.704920
$170$	0	0
$171$	79028.0	0.206676
$172$	0	0
$173$	370953.	0.942331	0.471166	−	0.882045i	$-0.343833\pi$
$173$	370953.	0.942331	0.471166	+	0.882045i	$0.343833\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	441252.	1.05865
$178$	0	0
$179$	754172.	1.75929	0.879646	−	0.475629i	$-0.157780\pi$
$179$	754172.	1.75929	0.879646	+	0.475629i	$0.157780\pi$
$180$	0	0
$181$	303840.	0.689364	0.344682	−	0.938720i	$-0.387987\pi$
$181$	303840.	0.689364	0.344682	+	0.938720i	$0.387987\pi$
$182$	0	0
$183$	666196.	1.47053
$184$	0	0
$185$	−412950.	−0.887091
$186$	0	0
$187$	−1.21478e6	−2.54036
$188$	0	0
$189$	−164101.	−0.334162
$190$	0	0
$191$	186271.	0.369455	0.184728	−	0.982790i	$-0.440860\pi$
$191$	186271.	0.369455	0.184728	+	0.982790i	$0.440860\pi$
$192$	0	0
$193$	92504.0	0.178759	0.0893794	−	0.995998i	$-0.471512\pi$
$193$	92504.0	0.178759	0.0893794	+	0.995998i	$0.471512\pi$
$194$	0	0
$195$	140675.	0.264930
$196$	0	0
$197$	−736368.	−1.35185	−0.675926	−	0.736969i	$-0.736256\pi$
$197$	−736368.	−1.35185	−0.675926	+	0.736969i	$0.736256\pi$
$198$	0	0
$199$	481620.	0.862128	0.431064	−	0.902321i	$-0.358138\pi$
$199$	481620.	0.862128	0.431064	+	0.902321i	$0.358138\pi$
$200$	0	0
$201$	−75072.0	−0.131065
$202$	0	0
$203$	224371.	0.382144
$204$	0	0
$205$	471900.	0.784269
$206$	0	0
$207$	181700.	0.294733
$208$	0	0
$209$	1.22837e6	1.94520
$210$	0	0
$211$	−189531.	−0.293072	−0.146536	−	0.989205i	$-0.546812\pi$
$211$	−189531.	−0.293072	−0.146536	+	0.989205i	$0.546812\pi$
$212$	0	0
$213$	541960.	0.818499
$214$	0	0
$215$	−56450.0	−0.0832852
$216$	0	0
$217$	−331044.	−0.477240
$218$	0	0
$219$	−85306.0	−0.120190
$220$	0	0
$221$	−562369.	−0.774534
$222$	0	0
$223$	22597.0	0.0304291	0.0152145	−	0.999884i	$-0.495157\pi$
$223$	22597.0	0.0304291	0.0152145	+	0.999884i	$0.495157\pi$
$224$	0	0
$225$	28750.0	0.0378601
$226$	0	0
$227$	998117.	1.28563	0.642816	−	0.766020i	$-0.277766\pi$
$227$	998117.	1.28563	0.642816	+	0.766020i	$0.277766\pi$
$228$	0	0
$229$	−854644.	−1.07695	−0.538476	−	0.842641i	$-0.681000\pi$
$229$	−854644.	−1.07695	−0.538476	+	0.842641i	$0.681000\pi$
$230$	0	0
$231$	595595.	0.734380
$232$	0	0
$233$	1.25818e6	1.51829	0.759144	−	0.650922i	$-0.225618\pi$
$233$	1.25818e6	1.51829	0.759144	+	0.650922i	$0.225618\pi$
$234$	0	0
$235$	13425.0	0.0158579
$236$	0	0
$237$	475609.	0.550021
$238$	0	0
$239$	706581.	0.800142	0.400071	−	0.916484i	$-0.368985\pi$
$239$	706581.	0.800142	0.400071	+	0.916484i	$0.368985\pi$
$240$	0	0
$241$	616330.	0.683551	0.341775	−	0.939782i	$-0.388972\pi$
$241$	616330.	0.683551	0.341775	+	0.939782i	$0.388972\pi$
$242$	0	0
$243$	−344080.	−0.373804
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	568658.	0.593074
$248$	0	0
$249$	−639948.	−0.654103
$250$	0	0
$251$	−190842.	−0.191201	−0.0956004	−	0.995420i	$-0.530477\pi$
$251$	−190842.	−0.191201	−0.0956004	+	0.995420i	$0.530477\pi$
$252$	0	0
$253$	2.82425e6	2.77397
$254$	0	0
$255$	−722075.	−0.695395
$256$	0	0
$257$	−1.13094e6	−1.06809	−0.534045	−	0.845456i	$-0.679329\pi$
$257$	−1.13094e6	−1.06809	−0.534045	+	0.845456i	$0.679329\pi$
$258$	0	0
$259$	−809382.	−0.749729
$260$	0	0
$261$	210634.	0.191394
$262$	0	0
$263$	1.67377e6	1.49213	0.746065	−	0.665874i	$-0.231941\pi$
$263$	1.67377e6	1.49213	0.746065	+	0.665874i	$0.231941\pi$
$264$	0	0
$265$	−274600.	−0.240207
$266$	0	0
$267$	−292672.	−0.251248
$268$	0	0
$269$	−630942.	−0.531629	−0.265815	−	0.964024i	$-0.585641\pi$
$269$	−630942.	−0.531629	−0.265815	+	0.964024i	$0.585641\pi$
$270$	0	0
$271$	372476.	0.308088	0.154044	−	0.988064i	$-0.450770\pi$
$271$	372476.	0.308088	0.154044	+	0.988064i	$0.450770\pi$
$272$	0	0
$273$	275723.	0.223906
$274$	0	0
$275$	446875.	0.356332
$276$	0	0
$277$	867010.	0.678930	0.339465	−	0.940619i	$-0.389754\pi$
$277$	867010.	0.678930	0.339465	+	0.940619i	$0.389754\pi$
$278$	0	0
$279$	−310776.	−0.239021
$280$	0	0
$281$	−1.94498e6	−1.46943	−0.734716	−	0.678375i	$-0.762685\pi$
$281$	−1.94498e6	−1.46943	−0.734716	+	0.678375i	$0.762685\pi$
$282$	0	0
$283$	−1.18501e6	−0.879543	−0.439771	−	0.898110i	$-0.644941\pi$
$283$	−1.18501e6	−0.879543	−0.439771	+	0.898110i	$0.644941\pi$
$284$	0	0
$285$	730150.	0.532476
$286$	0	0
$287$	924924.	0.662829
$288$	0	0
$289$	1.46674e6	1.03302
$290$	0	0
$291$	−1.07398e6	−0.743467
$292$	0	0
$293$	33669.0	0.0229119	0.0114560	−	0.999934i	$-0.496353\pi$
$293$	33669.0	0.0229119	0.0114560	+	0.999934i	$0.496353\pi$
$294$	0	0
$295$	648900.	0.434133
$296$	0	0
$297$	−2.39454e6	−1.57518
$298$	0	0
$299$	1.30745e6	0.845760
$300$	0	0
$301$	−110642.	−0.0703888
$302$	0	0
$303$	−497250.	−0.311149
$304$	0	0
$305$	979700.	0.603036
$306$	0	0
$307$	27043.0	0.0163760	0.00818802	−	0.999966i	$-0.497394\pi$
$307$	27043.0	0.0163760	0.00818802	+	0.999966i	$0.497394\pi$
$308$	0	0
$309$	2.53621e6	1.51109
$310$	0	0
$311$	−2.14919e6	−1.26001	−0.630004	−	0.776592i	$-0.716947\pi$
$311$	−2.14919e6	−1.26001	−0.630004	+	0.776592i	$0.716947\pi$
$312$	0	0
$313$	−2.67052e6	−1.54076	−0.770381	−	0.637583i	$-0.779934\pi$
$313$	−2.67052e6	−1.54076	−0.770381	+	0.637583i	$0.779934\pi$
$314$	0	0
$315$	56350.0	0.0319976
$316$	0	0
$317$	−250514.	−0.140018	−0.0700090	−	0.997546i	$-0.522303\pi$
$317$	−250514.	−0.140018	−0.0700090	+	0.997546i	$0.522303\pi$
$318$	0	0
$319$	3.27398e6	1.80136
$320$	0	0
$321$	−1.42361e6	−0.771134
$322$	0	0
$323$	−2.91888e6	−1.55672
$324$	0	0
$325$	206875.	0.108642
$326$	0	0
$327$	1.79141e6	0.926457
$328$	0	0
$329$	26313.0	0.0134023
$330$	0	0
$331$	−1.05899e6	−0.531277	−0.265639	−	0.964073i	$-0.585583\pi$
$331$	−1.05899e6	−0.531277	−0.265639	+	0.964073i	$0.585583\pi$
$332$	0	0
$333$	−759828.	−0.375495
$334$	0	0
$335$	−110400.	−0.0537473
$336$	0	0
$337$	−2.85025e6	−1.36712	−0.683562	−	0.729893i	$-0.739570\pi$
$337$	−2.85025e6	−1.36712	−0.683562	+	0.729893i	$0.739570\pi$
$338$	0	0
$339$	−2.08682e6	−0.986246
$340$	0	0
$341$	−4.83054e6	−2.24962
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	1.67875e6	0.759344
$346$	0	0
$347$	−1.89141e6	−0.843259	−0.421630	−	0.906768i	$-0.638542\pi$
$347$	−1.89141e6	−0.843259	−0.421630	+	0.906768i	$0.638542\pi$
$348$	0	0
$349$	−1.04232e6	−0.458075	−0.229038	−	0.973418i	$-0.573558\pi$
$349$	−1.04232e6	−0.458075	−0.229038	+	0.973418i	$0.573558\pi$
$350$	0	0
$351$	−1.10852e6	−0.480259
$352$	0	0
$353$	−2.30309e6	−0.983725	−0.491862	−	0.870673i	$-0.663684\pi$
$353$	−2.30309e6	−0.983725	−0.491862	+	0.870673i	$0.663684\pi$
$354$	0	0
$355$	797000.	0.335651
$356$	0	0
$357$	−1.41527e6	−0.587716
$358$	0	0
$359$	1.67594e6	0.686315	0.343157	−	0.939278i	$-0.388504\pi$
$359$	1.67594e6	0.686315	0.343157	+	0.939278i	$0.388504\pi$
$360$	0	0
$361$	475425.	0.192006
$362$	0	0
$363$	5.95296e6	2.37119
$364$	0	0
$365$	−125450.	−0.0492877
$366$	0	0
$367$	−94663.0	−0.0366872	−0.0183436	−	0.999832i	$-0.505839\pi$
$367$	−94663.0	−0.0366872	−0.0183436	+	0.999832i	$0.505839\pi$
$368$	0	0
$369$	868296.	0.331972
$370$	0	0
$371$	−538216.	−0.203012
$372$	0	0
$373$	−953536.	−0.354867	−0.177433	−	0.984133i	$-0.556779\pi$
$373$	−953536.	−0.354867	−0.177433	+	0.984133i	$0.556779\pi$
$374$	0	0
$375$	265625.	0.0975418
$376$	0	0
$377$	1.51565e6	0.549219
$378$	0	0
$379$	−3.88824e6	−1.39045	−0.695225	−	0.718792i	$-0.744695\pi$
$379$	−3.88824e6	−1.39045	−0.695225	+	0.718792i	$0.744695\pi$
$380$	0	0
$381$	3.72633e6	1.31513
$382$	0	0
$383$	−2.93636e6	−1.02285	−0.511425	−	0.859328i	$-0.670882\pi$
$383$	−2.93636e6	−1.02285	−0.511425	+	0.859328i	$0.670882\pi$
$384$	0	0
$385$	875875.	0.301155
$386$	0	0
$387$	−103868.	−0.0352537
$388$	0	0
$389$	1.70377e6	0.570871	0.285435	−	0.958398i	$-0.407862\pi$
$389$	1.70377e6	0.570871	0.285435	+	0.958398i	$0.407862\pi$
$390$	0	0
$391$	−6.71105e6	−2.21998
$392$	0	0
$393$	1.64359e6	0.536801
$394$	0	0
$395$	699425.	0.225553
$396$	0	0
$397$	1.19110e6	0.379292	0.189646	−	0.981853i	$-0.439266\pi$
$397$	1.19110e6	0.379292	0.189646	+	0.981853i	$0.439266\pi$
$398$	0	0
$399$	1.43109e6	0.450024
$400$	0	0
$401$	−3.38330e6	−1.05070	−0.525351	−	0.850885i	$-0.676066\pi$
$401$	−3.38330e6	−1.05070	−0.525351	+	0.850885i	$0.676066\pi$
$402$	0	0
$403$	−2.23624e6	−0.685891
$404$	0	0
$405$	−1.70278e6	−0.515846
$406$	0	0
$407$	−1.18104e7	−3.53409
$408$	0	0
$409$	−1.33185e6	−0.393682	−0.196841	−	0.980435i	$-0.563068\pi$
$409$	−1.33185e6	−0.393682	−0.196841	+	0.980435i	$0.563068\pi$
$410$	0	0
$411$	3.18390e6	0.929725
$412$	0	0
$413$	1.27184e6	0.366909
$414$	0	0
$415$	−941100.	−0.268235
$416$	0	0
$417$	−3.00720e6	−0.846880
$418$	0	0
$419$	−5.82786e6	−1.62171	−0.810856	−	0.585246i	$-0.800998\pi$
$419$	−5.82786e6	−1.62171	−0.810856	+	0.585246i	$0.800998\pi$
$420$	0	0
$421$	2.47430e6	0.680374	0.340187	−	0.940358i	$-0.389510\pi$
$421$	2.47430e6	0.680374	0.340187	+	0.940358i	$0.389510\pi$
$422$	0	0
$423$	24702.0	0.00671245
$424$	0	0
$425$	−1.06188e6	−0.285168
$426$	0	0
$427$	1.92021e6	0.509659
$428$	0	0
$429$	4.02331e6	1.05546
$430$	0	0
$431$	−4.61851e6	−1.19759	−0.598796	−	0.800902i	$-0.704354\pi$
$431$	−4.61851e6	−1.19759	−0.598796	+	0.800902i	$0.704354\pi$
$432$	0	0
$433$	58606.0	0.0150218	0.00751091	−	0.999972i	$-0.497609\pi$
$433$	58606.0	0.0150218	0.00751091	+	0.999972i	$0.497609\pi$
$434$	0	0
$435$	1.94608e6	0.493102
$436$	0	0
$437$	6.78610e6	1.69987
$438$	0	0
$439$	−7.04298e6	−1.74419	−0.872097	−	0.489332i	$-0.837241\pi$
$439$	−7.04298e6	−1.74419	−0.872097	+	0.489332i	$0.837241\pi$
$440$	0	0
$441$	110446.	0.0270429
$442$	0	0
$443$	−1.46894e6	−0.355627	−0.177813	−	0.984064i	$-0.556902\pi$
$443$	−1.46894e6	−0.355627	−0.177813	+	0.984064i	$0.556902\pi$
$444$	0	0
$445$	−430400.	−0.103032
$446$	0	0
$447$	−3.38433e6	−0.801131
$448$	0	0
$449$	−7.48414e6	−1.75197	−0.875983	−	0.482341i	$-0.839787\pi$
$449$	−7.48414e6	−1.75197	−0.875983	+	0.482341i	$0.839787\pi$
$450$	0	0
$451$	1.34963e7	3.12446
$452$	0	0
$453$	−8.01691e6	−1.83553
$454$	0	0
$455$	405475.	0.0918196
$456$	0	0
$457$	170320.	0.0381483	0.0190741	−	0.999818i	$-0.493928\pi$
$457$	170320.	0.0381483	0.0190741	+	0.999818i	$0.493928\pi$
$458$	0	0
$459$	5.68995e6	1.26060
$460$	0	0
$461$	−4.28685e6	−0.939476	−0.469738	−	0.882806i	$-0.655652\pi$
$461$	−4.28685e6	−0.939476	−0.469738	+	0.882806i	$0.655652\pi$
$462$	0	0
$463$	−3.38317e6	−0.733452	−0.366726	−	0.930329i	$-0.619521\pi$
$463$	−3.38317e6	−0.733452	−0.366726	+	0.930329i	$0.619521\pi$
$464$	0	0
$465$	−2.87130e6	−0.615809
$466$	0	0
$467$	−5.18029e6	−1.09916	−0.549581	−	0.835440i	$-0.685213\pi$
$467$	−5.18029e6	−1.09916	−0.549581	+	0.835440i	$0.685213\pi$
$468$	0	0
$469$	−216384.	−0.0454248
$470$	0	0
$471$	−1.22492e6	−0.254422
$472$	0	0
$473$	−1.61447e6	−0.331801
$474$	0	0
$475$	1.07375e6	0.218358
$476$	0	0
$477$	−505264.	−0.101677
$478$	0	0
$479$	8.76779e6	1.74603	0.873014	−	0.487695i	$-0.162162\pi$
$479$	8.76779e6	1.74603	0.873014	+	0.487695i	$0.162162\pi$
$480$	0	0
$481$	−5.46746e6	−1.07751
$482$	0	0
$483$	3.29035e6	0.641762
$484$	0	0
$485$	−1.57938e6	−0.304881
$486$	0	0
$487$	−270154.	−0.0516166	−0.0258083	−	0.999667i	$-0.508216\pi$
$487$	−270154.	−0.0516166	−0.0258083	+	0.999667i	$0.508216\pi$
$488$	0	0
$489$	−6.55068e6	−1.23884
$490$	0	0
$491$	−4.85550e6	−0.908930	−0.454465	−	0.890765i	$-0.650170\pi$
$491$	−4.85550e6	−0.908930	−0.454465	+	0.890765i	$0.650170\pi$
$492$	0	0
$493$	−7.77972e6	−1.44161
$494$	0	0
$495$	822250.	0.150831
$496$	0	0
$497$	1.56212e6	0.283677
$498$	0	0
$499$	2.98576e6	0.536789	0.268394	−	0.963309i	$-0.413507\pi$
$499$	2.98576e6	0.536789	0.268394	+	0.963309i	$0.413507\pi$
$500$	0	0
$501$	9.23027e6	1.64293
$502$	0	0
$503$	8.28783e6	1.46057	0.730283	−	0.683145i	$-0.239388\pi$
$503$	8.28783e6	1.46057	0.730283	+	0.683145i	$0.239388\pi$
$504$	0	0
$505$	−731250.	−0.127596
$506$	0	0
$507$	−4.44944e6	−0.768751
$508$	0	0
$509$	6.24307e6	1.06808	0.534040	−	0.845459i	$-0.320673\pi$
$509$	6.24307e6	1.06808	0.534040	+	0.845459i	$0.320673\pi$
$510$	0	0
$511$	−245882.	−0.0416557
$512$	0	0
$513$	−5.75358e6	−0.965261
$514$	0	0
$515$	3.72972e6	0.619668
$516$	0	0
$517$	383955.	0.0631763
$518$	0	0
$519$	6.30620e6	1.02766
$520$	0	0
$521$	7.49509e6	1.20971	0.604856	−	0.796335i	$-0.293230\pi$
$521$	7.49509e6	1.20971	0.604856	+	0.796335i	$0.293230\pi$
$522$	0	0
$523$	−3.80957e6	−0.609007	−0.304503	−	0.952511i	$-0.598490\pi$
$523$	−3.80957e6	−0.609007	−0.304503	+	0.952511i	$0.598490\pi$
$524$	0	0
$525$	520625.	0.0824379
$526$	0	0
$527$	1.14784e7	1.80035
$528$	0	0
$529$	9.16616e6	1.42413
$530$	0	0
$531$	1.19398e6	0.183764
$532$	0	0
$533$	6.24796e6	0.952621
$534$	0	0
$535$	−2.09355e6	−0.316227
$536$	0	0
$537$	1.28209e7	1.91860
$538$	0	0
$539$	1.71672e6	0.254523
$540$	0	0
$541$	7.67156e6	1.12691	0.563457	−	0.826145i	$-0.309471\pi$
$541$	7.67156e6	1.12691	0.563457	+	0.826145i	$0.309471\pi$
$542$	0	0
$543$	5.16528e6	0.751786
$544$	0	0
$545$	2.63442e6	0.379922
$546$	0	0
$547$	9.53845e6	1.36304	0.681522	−	0.731798i	$-0.261319\pi$
$547$	9.53845e6	1.36304	0.681522	+	0.731798i	$0.261319\pi$
$548$	0	0
$549$	1.80265e6	0.255258
$550$	0	0
$551$	7.86672e6	1.10386
$552$	0	0
$553$	1.37087e6	0.190627
$554$	0	0
$555$	−7.02015e6	−0.967417
$556$	0	0
$557$	−7.45022e6	−1.01749	−0.508746	−	0.860916i	$-0.669891\pi$
$557$	−7.45022e6	−1.01749	−0.508746	+	0.860916i	$0.669891\pi$
$558$	0	0
$559$	−747398.	−0.101163
$560$	0	0
$561$	−2.06513e7	−2.77039
$562$	0	0
$563$	−3.36698e6	−0.447682	−0.223841	−	0.974626i	$-0.571860\pi$
$563$	−3.36698e6	−0.447682	−0.223841	+	0.974626i	$0.571860\pi$
$564$	0	0
$565$	−3.06885e6	−0.404440
$566$	0	0
$567$	−3.33744e6	−0.435969
$568$	0	0
$569$	−4.05501e6	−0.525063	−0.262532	−	0.964923i	$-0.584557\pi$
$569$	−4.05501e6	−0.525063	−0.262532	+	0.964923i	$0.584557\pi$
$570$	0	0
$571$	−7.31585e6	−0.939020	−0.469510	−	0.882927i	$-0.655569\pi$
$571$	−7.31585e6	−0.939020	−0.469510	+	0.882927i	$0.655569\pi$
$572$	0	0
$573$	3.16661e6	0.402910
$574$	0	0
$575$	2.46875e6	0.311392
$576$	0	0
$577$	−9.76895e6	−1.22154	−0.610771	−	0.791807i	$-0.709140\pi$
$577$	−9.76895e6	−1.22154	−0.610771	+	0.791807i	$0.709140\pi$
$578$	0	0
$579$	1.57257e6	0.194945
$580$	0	0
$581$	−1.84456e6	−0.226700
$582$	0	0
$583$	−7.85356e6	−0.956963
$584$	0	0
$585$	380650.	0.0459871
$586$	0	0
$587$	−3.75689e6	−0.450021	−0.225011	−	0.974356i	$-0.572242\pi$
$587$	−3.75689e6	−0.450021	−0.225011	+	0.974356i	$0.572242\pi$
$588$	0	0
$589$	−1.16068e7	−1.37856
$590$	0	0
$591$	−1.25183e7	−1.47426
$592$	0	0
$593$	2.89048e6	0.337546	0.168773	−	0.985655i	$-0.446020\pi$
$593$	2.89048e6	0.337546	0.168773	+	0.985655i	$0.446020\pi$
$594$	0	0
$595$	−2.08128e6	−0.241011
$596$	0	0
$597$	8.18754e6	0.940194
$598$	0	0
$599$	−1.32233e7	−1.50582	−0.752910	−	0.658124i	$-0.771350\pi$
$599$	−1.32233e7	−1.50582	−0.752910	+	0.658124i	$0.771350\pi$
$600$	0	0
$601$	3.47399e6	0.392321	0.196161	−	0.980572i	$-0.437153\pi$
$601$	3.47399e6	0.392321	0.196161	+	0.980572i	$0.437153\pi$
$602$	0	0
$603$	−203136.	−0.0227506
$604$	0	0
$605$	8.75435e6	0.972379
$606$	0	0
$607$	−6.45088e6	−0.710636	−0.355318	−	0.934746i	$-0.615627\pi$
$607$	−6.45088e6	−0.710636	−0.355318	+	0.934746i	$0.615627\pi$
$608$	0	0
$609$	3.81431e6	0.416747
$610$	0	0
$611$	177747.	0.0192619
$612$	0	0
$613$	8.43820e6	0.906982	0.453491	−	0.891261i	$-0.350178\pi$
$613$	8.43820e6	0.906982	0.453491	+	0.891261i	$0.350178\pi$
$614$	0	0
$615$	8.02230e6	0.855285
$616$	0	0
$617$	9.45501e6	0.999882	0.499941	−	0.866059i	$-0.333355\pi$
$617$	9.45501e6	0.999882	0.499941	+	0.866059i	$0.333355\pi$
$618$	0	0
$619$	−1.43145e6	−0.150158	−0.0750790	−	0.997178i	$-0.523921\pi$
$619$	−1.43145e6	−0.150158	−0.0750790	+	0.997178i	$0.523921\pi$
$620$	0	0
$621$	−1.32286e7	−1.37652
$622$	0	0
$623$	−843584.	−0.0870780
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	2.08823e7	2.12134
$628$	0	0
$629$	2.80641e7	2.82829
$630$	0	0
$631$	1.01813e7	1.01795	0.508977	−	0.860780i	$-0.330024\pi$
$631$	1.01813e7	1.01795	0.508977	+	0.860780i	$0.330024\pi$
$632$	0	0
$633$	−3.22203e6	−0.319610
$634$	0	0
$635$	5.47990e6	0.539310
$636$	0	0
$637$	794731.	0.0776018
$638$	0	0
$639$	1.46648e6	0.142077
$640$	0	0
$641$	−1.76908e7	−1.70060	−0.850300	−	0.526298i	$-0.823580\pi$
$641$	−1.76908e7	−1.70060	−0.850300	+	0.526298i	$0.823580\pi$
$642$	0	0
$643$	−1.82748e7	−1.74311	−0.871556	−	0.490296i	$-0.836889\pi$
$643$	−1.82748e7	−1.74311	−0.871556	+	0.490296i	$0.836889\pi$
$644$	0	0
$645$	−959650.	−0.0908267
$646$	0	0
$647$	−1.52897e6	−0.143594	−0.0717972	−	0.997419i	$-0.522873\pi$
$647$	−1.52897e6	−0.143594	−0.0717972	+	0.997419i	$0.522873\pi$
$648$	0	0
$649$	1.85585e7	1.72955
$650$	0	0
$651$	−5.62775e6	−0.520454
$652$	0	0
$653$	−9.10088e6	−0.835219	−0.417610	−	0.908627i	$-0.637132\pi$
$653$	−9.10088e6	−0.835219	−0.417610	+	0.908627i	$0.637132\pi$
$654$	0	0
$655$	2.41705e6	0.220132
$656$	0	0
$657$	−230828.	−0.0208629
$658$	0	0
$659$	430119.	0.0385811	0.0192906	−	0.999814i	$-0.493859\pi$
$659$	430119.	0.0385811	0.0192906	+	0.999814i	$0.493859\pi$
$660$	0	0
$661$	7.65248e6	0.681238	0.340619	−	0.940202i	$-0.389363\pi$
$661$	7.65248e6	0.681238	0.340619	+	0.940202i	$0.389363\pi$
$662$	0	0
$663$	−9.56027e6	−0.844669
$664$	0	0
$665$	2.10455e6	0.184546
$666$	0	0
$667$	1.80870e7	1.57418
$668$	0	0
$669$	384149.	0.0331844
$670$	0	0
$671$	2.80194e7	2.40244
$672$	0	0
$673$	−2.18404e7	−1.85876	−0.929378	−	0.369128i	$-0.879656\pi$
$673$	−2.18404e7	−1.85876	−0.929378	+	0.369128i	$0.879656\pi$
$674$	0	0
$675$	−2.09312e6	−0.176822
$676$	0	0
$677$	1.39504e7	1.16981	0.584905	−	0.811102i	$-0.301132\pi$
$677$	1.39504e7	1.16981	0.584905	+	0.811102i	$0.301132\pi$
$678$	0	0
$679$	−3.09558e6	−0.257672
$680$	0	0
$681$	1.69680e7	1.40205
$682$	0	0
$683$	−2.29121e7	−1.87937	−0.939686	−	0.342040i	$-0.888882\pi$
$683$	−2.29121e7	−1.87937	−0.939686	+	0.342040i	$0.888882\pi$
$684$	0	0
$685$	4.68220e6	0.381262
$686$	0	0
$687$	−1.45289e7	−1.17447
$688$	0	0
$689$	−3.63570e6	−0.291770
$690$	0	0
$691$	1.69127e7	1.34747	0.673734	−	0.738974i	$-0.264690\pi$
$691$	1.69127e7	1.34747	0.673734	+	0.738974i	$0.264690\pi$
$692$	0	0
$693$	1.61161e6	0.127476
$694$	0	0
$695$	−4.42235e6	−0.347289
$696$	0	0
$697$	−3.20703e7	−2.50047
$698$	0	0
$699$	2.13891e7	1.65577
$700$	0	0
$701$	−1.90087e7	−1.46102	−0.730510	−	0.682902i	$-0.760718\pi$
$701$	−1.90087e7	−1.46102	−0.730510	+	0.682902i	$0.760718\pi$
$702$	0	0
$703$	−2.83779e7	−2.16567
$704$	0	0
$705$	228225.	0.0172938
$706$	0	0
$707$	−1.43325e6	−0.107838
$708$	0	0
$709$	1.66079e7	1.24079	0.620396	−	0.784289i	$-0.286972\pi$
$709$	1.66079e7	1.24079	0.620396	+	0.784289i	$0.286972\pi$
$710$	0	0
$711$	1.28694e6	0.0954740
$712$	0	0
$713$	−2.66862e7	−1.96591
$714$	0	0
$715$	5.91662e6	0.432822
$716$	0	0
$717$	1.20119e7	0.872596
$718$	0	0
$719$	−5.93610e6	−0.428232	−0.214116	−	0.976808i	$-0.568687\pi$
$719$	−5.93610e6	−0.428232	−0.214116	+	0.976808i	$0.568687\pi$
$720$	0	0
$721$	7.31026e6	0.523715
$722$	0	0
$723$	1.04776e7	0.745447
$724$	0	0
$725$	2.86188e6	0.202211
$726$	0	0
$727$	−1.73276e7	−1.21591	−0.607957	−	0.793970i	$-0.708011\pi$
$727$	−1.73276e7	−1.21591	−0.607957	+	0.793970i	$0.708011\pi$
$728$	0	0
$729$	1.07016e7	0.745814
$730$	0	0
$731$	3.83634e6	0.265536
$732$	0	0
$733$	1.39829e7	0.961255	0.480627	−	0.876925i	$-0.340409\pi$
$733$	1.39829e7	0.961255	0.480627	+	0.876925i	$0.340409\pi$
$734$	0	0
$735$	1.02042e6	0.0696727
$736$	0	0
$737$	−3.15744e6	−0.214125
$738$	0	0
$739$	1.14263e7	0.769649	0.384824	−	0.922990i	$-0.374262\pi$
$739$	1.14263e7	0.769649	0.384824	+	0.922990i	$0.374262\pi$
$740$	0	0
$741$	9.66719e6	0.646777
$742$	0	0
$743$	−1.23126e7	−0.818236	−0.409118	−	0.912481i	$-0.634164\pi$
$743$	−1.23126e7	−0.818236	−0.409118	+	0.912481i	$0.634164\pi$
$744$	0	0
$745$	−4.97695e6	−0.328528
$746$	0	0
$747$	−1.73162e6	−0.113541
$748$	0	0
$749$	−4.10336e6	−0.267261
$750$	0	0
$751$	1.43093e7	0.925806	0.462903	−	0.886409i	$-0.346808\pi$
$751$	1.43093e7	0.925806	0.462903	+	0.886409i	$0.346808\pi$
$752$	0	0
$753$	−3.24431e6	−0.208514
$754$	0	0
$755$	−1.17896e7	−0.752715
$756$	0	0
$757$	−5.34505e6	−0.339010	−0.169505	−	0.985529i	$-0.554217\pi$
$757$	−5.34505e6	−0.339010	−0.169505	+	0.985529i	$0.554217\pi$
$758$	0	0
$759$	4.80122e7	3.02515
$760$	0	0
$761$	6.22568e6	0.389695	0.194848	−	0.980834i	$-0.437579\pi$
$761$	6.22568e6	0.389695	0.194848	+	0.980834i	$0.437579\pi$
$762$	0	0
$763$	5.16347e6	0.321093
$764$	0	0
$765$	−1.95385e6	−0.120708
$766$	0	0
$767$	8.59144e6	0.527324
$768$	0	0
$769$	1.57888e7	0.962793	0.481397	−	0.876503i	$-0.340130\pi$
$769$	1.57888e7	0.962793	0.481397	+	0.876503i	$0.340130\pi$
$770$	0	0
$771$	−1.92260e7	−1.16481
$772$	0	0
$773$	−2.50453e7	−1.50757	−0.753785	−	0.657121i	$-0.771774\pi$
$773$	−2.50453e7	−1.50757	−0.753785	+	0.657121i	$0.771774\pi$
$774$	0	0
$775$	−4.22250e6	−0.252531
$776$	0	0
$777$	−1.37595e7	−0.817617
$778$	0	0
$779$	3.24290e7	1.91465
$780$	0	0
$781$	2.27942e7	1.33720
$782$	0	0
$783$	−1.53351e7	−0.893884
$784$	0	0
$785$	−1.80135e6	−0.104334
$786$	0	0
$787$	1.28020e6	0.0736784	0.0368392	−	0.999321i	$-0.488271\pi$
$787$	1.28020e6	0.0736784	0.0368392	+	0.999321i	$0.488271\pi$
$788$	0	0
$789$	2.84541e7	1.62724
$790$	0	0
$791$	−6.01495e6	−0.341815
$792$	0	0
$793$	1.29712e7	0.732484
$794$	0	0
$795$	−4.66820e6	−0.261958
$796$	0	0
$797$	−1.13798e7	−0.634584	−0.317292	−	0.948328i	$-0.602774\pi$
$797$	−1.13798e7	−0.634584	−0.317292	+	0.948328i	$0.602774\pi$
$798$	0	0
$799$	−912363.	−0.0505593
$800$	0	0
$801$	−791936.	−0.0436123
$802$	0	0
$803$	−3.58787e6	−0.196358
$804$	0	0
$805$	4.83875e6	0.263174
$806$	0	0
$807$	−1.07260e7	−0.579768
$808$	0	0
$809$	−1.70542e7	−0.916138	−0.458069	−	0.888917i	$-0.651459\pi$
$809$	−1.70542e7	−0.916138	−0.458069	+	0.888917i	$0.651459\pi$
$810$	0	0
$811$	2.21494e7	1.18252	0.591262	−	0.806480i	$-0.298630\pi$
$811$	2.21494e7	1.18252	0.591262	+	0.806480i	$0.298630\pi$
$812$	0	0
$813$	6.33209e6	0.335986
$814$	0	0
$815$	−9.63335e6	−0.508023
$816$	0	0
$817$	−3.87924e6	−0.203326
$818$	0	0
$819$	746074.	0.0388662
$820$	0	0
$821$	−1.01068e7	−0.523307	−0.261654	−	0.965162i	$-0.584268\pi$
$821$	−1.01068e7	−0.523307	−0.261654	+	0.965162i	$0.584268\pi$
$822$	0	0
$823$	1.83993e7	0.946895	0.473447	−	0.880822i	$-0.343009\pi$
$823$	1.83993e7	0.946895	0.473447	+	0.880822i	$0.343009\pi$
$824$	0	0
$825$	7.59688e6	0.388598
$826$	0	0
$827$	2.48056e7	1.26121	0.630604	−	0.776105i	$-0.282807\pi$
$827$	2.48056e7	1.26121	0.630604	+	0.776105i	$0.282807\pi$
$828$	0	0
$829$	−1.19708e6	−0.0604976	−0.0302488	−	0.999542i	$-0.509630\pi$
$829$	−1.19708e6	−0.0604976	−0.0302488	+	0.999542i	$0.509630\pi$
$830$	0	0
$831$	1.47392e7	0.740407
$832$	0	0
$833$	−4.07930e6	−0.203692
$834$	0	0
$835$	1.35739e7	0.673735
$836$	0	0
$837$	2.26258e7	1.11633
$838$	0	0
$839$	3.17171e7	1.55557	0.777783	−	0.628533i	$-0.216344\pi$
$839$	3.17171e7	1.55557	0.777783	+	0.628533i	$0.216344\pi$
$840$	0	0
$841$	456092.	0.0222363
$842$	0	0
$843$	−3.30647e7	−1.60249
$844$	0	0
$845$	−6.54330e6	−0.315250
$846$	0	0
$847$	1.71585e7	0.821810
$848$	0	0
$849$	−2.01452e7	−0.959186
$850$	0	0
$851$	−6.52461e7	−3.08838
$852$	0	0
$853$	3.18237e7	1.49754	0.748769	−	0.662831i	$-0.230645\pi$
$853$	3.18237e7	1.49754	0.748769	+	0.662831i	$0.230645\pi$
$854$	0	0
$855$	1.97570e6	0.0924285
$856$	0	0
$857$	2.27853e7	1.05975	0.529874	−	0.848076i	$-0.322239\pi$
$857$	2.27853e7	1.05975	0.529874	+	0.848076i	$0.322239\pi$
$858$	0	0
$859$	1.85966e7	0.859907	0.429953	−	0.902851i	$-0.358530\pi$
$859$	1.85966e7	0.859907	0.429953	+	0.902851i	$0.358530\pi$
$860$	0	0
$861$	1.57237e7	0.722848
$862$	0	0
$863$	2.77046e7	1.26627	0.633133	−	0.774043i	$-0.281769\pi$
$863$	2.77046e7	1.26627	0.633133	+	0.774043i	$0.281769\pi$
$864$	0	0
$865$	9.27382e6	0.421423
$866$	0	0
$867$	2.49346e7	1.12656
$868$	0	0
$869$	2.00036e7	0.898582
$870$	0	0
$871$	−1.46170e6	−0.0652847
$872$	0	0
$873$	−2.90605e6	−0.129053
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	−2.41150e7	−1.05874	−0.529370	−	0.848391i	$-0.677572\pi$
$877$	−2.41150e7	−1.05874	−0.529370	+	0.848391i	$0.677572\pi$
$878$	0	0
$879$	572373.	0.0249866
$880$	0	0
$881$	−1.26207e7	−0.547827	−0.273914	−	0.961754i	$-0.588318\pi$
$881$	−1.26207e7	−0.547827	−0.273914	+	0.961754i	$0.588318\pi$
$882$	0	0
$883$	6.01876e6	0.259780	0.129890	−	0.991528i	$-0.458538\pi$
$883$	6.01876e6	0.259780	0.129890	+	0.991528i	$0.458538\pi$
$884$	0	0
$885$	1.10313e7	0.473444
$886$	0	0
$887$	−2.36901e7	−1.01102	−0.505509	−	0.862821i	$-0.668695\pi$
$887$	−2.36901e7	−1.01102	−0.505509	+	0.862821i	$0.668695\pi$
$888$	0	0
$889$	1.07406e7	0.455800
$890$	0	0
$891$	−4.86994e7	−2.05508
$892$	0	0
$893$	922566.	0.0387141
$894$	0	0
$895$	1.88543e7	0.786779
$896$	0	0
$897$	2.22266e7	0.922344
$898$	0	0
$899$	−3.09357e7	−1.27662
$900$	0	0
$901$	1.86618e7	0.765847
$902$	0	0
$903$	−1.88091e6	−0.0767626
$904$	0	0
$905$	7.59600e6	0.308293
$906$	0	0
$907$	1.10583e7	0.446346	0.223173	−	0.974779i	$-0.428359\pi$
$907$	1.10583e7	0.446346	0.223173	+	0.974779i	$0.428359\pi$
$908$	0	0
$909$	−1.34550e6	−0.0540100
$910$	0	0
$911$	−3.07573e6	−0.122787	−0.0613934	−	0.998114i	$-0.519554\pi$
$911$	−3.07573e6	−0.122787	−0.0613934	+	0.998114i	$0.519554\pi$
$912$	0	0
$913$	−2.69155e7	−1.06862
$914$	0	0
$915$	1.66549e7	0.657642
$916$	0	0
$917$	4.73742e6	0.186045
$918$	0	0
$919$	1.89018e7	0.738270	0.369135	−	0.929376i	$-0.379654\pi$
$919$	1.89018e7	0.738270	0.369135	+	0.929376i	$0.379654\pi$
$920$	0	0
$921$	459731.	0.0178589
$922$	0	0
$923$	1.05523e7	0.407701
$924$	0	0
$925$	−1.03238e7	−0.396719
$926$	0	0
$927$	6.86269e6	0.262298
$928$	0	0
$929$	−1.81458e7	−0.689821	−0.344911	−	0.938636i	$-0.612091\pi$
$929$	−1.81458e7	−0.689821	−0.344911	+	0.938636i	$0.612091\pi$
$930$	0	0
$931$	4.12492e6	0.155970
$932$	0	0
$933$	−3.65362e7	−1.37410
$934$	0	0
$935$	−3.03696e7	−1.13608
$936$	0	0
$937$	−2.17350e7	−0.808744	−0.404372	−	0.914595i	$-0.632510\pi$
$937$	−2.17350e7	−0.808744	−0.404372	+	0.914595i	$0.632510\pi$
$938$	0	0
$939$	−4.53989e7	−1.68028
$940$	0	0
$941$	1.86808e7	0.687735	0.343868	−	0.939018i	$-0.388263\pi$
$941$	1.86808e7	0.687735	0.343868	+	0.939018i	$0.388263\pi$
$942$	0	0
$943$	7.45602e7	2.73041
$944$	0	0
$945$	−4.10252e6	−0.149442
$946$	0	0
$947$	2.24778e6	0.0814476	0.0407238	−	0.999170i	$-0.487034\pi$
$947$	2.24778e6	0.0814476	0.0407238	+	0.999170i	$0.487034\pi$
$948$	0	0
$949$	−1.66096e6	−0.0598678
$950$	0	0
$951$	−4.25874e6	−0.152697
$952$	0	0
$953$	−3.73293e7	−1.33143	−0.665714	−	0.746207i	$-0.731873\pi$
$953$	−3.73293e7	−1.33143	−0.665714	+	0.746207i	$0.731873\pi$
$954$	0	0
$955$	4.65678e6	0.165225
$956$	0	0
$957$	5.56577e7	1.96447
$958$	0	0
$959$	9.17711e6	0.322225
$960$	0	0
$961$	1.70144e7	0.594303
$962$	0	0
$963$	−3.85213e6	−0.133855
$964$	0	0
$965$	2.31260e6	0.0799433
$966$	0	0
$967$	−2.61870e7	−0.900573	−0.450287	−	0.892884i	$-0.648678\pi$
$967$	−2.61870e7	−0.900573	−0.450287	+	0.892884i	$0.648678\pi$
$968$	0	0
$969$	−4.96210e7	−1.69768
$970$	0	0
$971$	3.91957e7	1.33410	0.667052	−	0.745011i	$-0.267556\pi$
$971$	3.91957e7	1.33410	0.667052	+	0.745011i	$0.267556\pi$
$972$	0	0
$973$	−8.66781e6	−0.293513
$974$	0	0
$975$	3.51688e6	0.118480
$976$	0	0
$977$	3.03935e6	0.101870	0.0509348	−	0.998702i	$-0.483780\pi$
$977$	3.03935e6	0.101870	0.0509348	+	0.998702i	$0.483780\pi$
$978$	0	0
$979$	−1.23094e7	−0.410470
$980$	0	0
$981$	4.84734e6	0.160817
$982$	0	0
$983$	1.59937e7	0.527915	0.263957	−	0.964534i	$-0.414972\pi$
$983$	1.59937e7	0.527915	0.263957	+	0.964534i	$0.414972\pi$
$984$	0	0
$985$	−1.84092e7	−0.604567
$986$	0	0
$987$	447321.	0.0146159
$988$	0	0
$989$	−8.91910e6	−0.289955
$990$	0	0
$991$	−3.63186e6	−0.117475	−0.0587375	−	0.998273i	$-0.518707\pi$
$991$	−3.63186e6	−0.117475	−0.0587375	+	0.998273i	$0.518707\pi$
$992$	0	0
$993$	−1.80028e7	−0.579384
$994$	0	0
$995$	1.20405e7	0.385555
$996$	0	0
$997$	−4.33287e7	−1.38051	−0.690253	−	0.723568i	$-0.742501\pi$
$997$	−4.33287e7	−1.38051	−0.690253	+	0.723568i	$0.742501\pi$
$998$	0	0
$999$	5.53188e7	1.75371

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	560.6.a.h.1.1		1
4.3	odd	2		70.6.a.e.1.1	✓	1
12.11	even	2		630.6.a.b.1.1		1
20.3	even	4		350.6.c.a.99.1		2
20.7	even	4		350.6.c.a.99.2		2
20.19	odd	2		350.6.a.e.1.1		1
28.27	even	2		490.6.a.m.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.e.1.1	✓	1	4.3	odd	2
350.6.a.e.1.1		1	20.19	odd	2
350.6.c.a.99.1		2	20.3	even	4
350.6.c.a.99.2		2	20.7	even	4
490.6.a.m.1.1		1	28.27	even	2
560.6.a.h.1.1		1	1.1	even	1	trivial
630.6.a.b.1.1		1	12.11	even	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:	\( N \)	\(=\)	\( 560 = 2^{4} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	560.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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