LMFDB - Embedded newform 560.6.a.e.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [560,6,Mod(1,560)]

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

lf = mfeigenbasis(mf)

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")

//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);

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

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$89.8149390953$
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)

$f(q)$

$=$

$q+9.00000 q^{3} +25.0000 q^{5} +49.0000 q^{7} -162.000 q^{9} +O(q^{10})$

$q+9.00000 q^{3} +25.0000 q^{5} +49.0000 q^{7} -162.000 q^{9} +187.000 q^{11} +627.000 q^{13} +225.000 q^{15} +1813.00 q^{17} -258.000 q^{19} +441.000 q^{21} -2970.00 q^{23} +625.000 q^{25} -3645.00 q^{27} +1299.00 q^{29} -1916.00 q^{31} +1683.00 q^{33} +1225.00 q^{35} +6578.00 q^{37} +5643.00 q^{39} +6676.00 q^{41} -3178.00 q^{43} -4050.00 q^{45} +22001.0 q^{47} +2401.00 q^{49} +16317.0 q^{51} +26168.0 q^{53} +4675.00 q^{55} -2322.00 q^{57} -3932.00 q^{59} -48740.0 q^{61} -7938.00 q^{63} +15675.0 q^{65} +44832.0 q^{67} -26730.0 q^{69} -63736.0 q^{71} +60470.0 q^{73} +5625.00 q^{75} +9163.00 q^{77} +43721.0 q^{79} +6561.00 q^{81} -97276.0 q^{83} +45325.0 q^{85} +11691.0 q^{87} +45560.0 q^{89} +30723.0 q^{91} -17244.0 q^{93} -6450.00 q^{95} -57295.0 q^{97} -30294.0 q^{99} +O(q^{100})$

Display 100 coefficients 10 coefficients

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
$3$	9.00000	0.577350	0.288675	−	0.957427i	$-0.406785\pi$
$3$	9.00000	0.577350	0.288675	+	0.957427i	$0.406785\pi$
$4$	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
$9$	−162.000	−0.666667
$10$	0	0
$11$	187.000	0.465972	0.232986	−	0.972480i	$-0.425150\pi$
$11$	187.000	0.465972	0.232986	+	0.972480i	$0.425150\pi$
$12$	0	0
$13$	627.000	1.02899	0.514493	−	0.857495i	$-0.327980\pi$
$13$	627.000	1.02899	0.514493	+	0.857495i	$0.327980\pi$
$14$	0	0
$15$	225.000	0.258199
$16$	0	0
$17$	1813.00	1.52151	0.760756	−	0.649038i	$-0.224828\pi$
$17$	1813.00	1.52151	0.760756	+	0.649038i	$0.224828\pi$
$18$	0	0
$19$	−258.000	−0.163959	−0.0819796	−	0.996634i	$-0.526124\pi$
$19$	−258.000	−0.163959	−0.0819796	+	0.996634i	$0.526124\pi$
$20$	0	0
$21$	441.000	0.218218
$22$	0	0
$23$	−2970.00	−1.17068	−0.585338	−	0.810789i	$-0.699038\pi$
$23$	−2970.00	−1.17068	−0.585338	+	0.810789i	$0.699038\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−3645.00	−0.962250
$28$	0	0
$29$	1299.00	0.286823	0.143412	−	0.989663i	$-0.454193\pi$
$29$	1299.00	0.286823	0.143412	+	0.989663i	$0.454193\pi$
$30$	0	0
$31$	−1916.00	−0.358089	−0.179045	−	0.983841i	$-0.557301\pi$
$31$	−1916.00	−0.358089	−0.179045	+	0.983841i	$0.557301\pi$
$32$	0	0
$33$	1683.00	0.269029
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	6578.00	0.789932	0.394966	−	0.918696i	$-0.370756\pi$
$37$	6578.00	0.789932	0.394966	+	0.918696i	$0.370756\pi$
$38$	0	0
$39$	5643.00	0.594085
$40$	0	0
$41$	6676.00	0.620236	0.310118	−	0.950698i	$-0.399632\pi$
$41$	6676.00	0.620236	0.310118	+	0.950698i	$0.399632\pi$
$42$	0	0
$43$	−3178.00	−0.262109	−0.131055	−	0.991375i	$-0.541836\pi$
$43$	−3178.00	−0.262109	−0.131055	+	0.991375i	$0.541836\pi$
$44$	0	0
$45$	−4050.00	−0.298142
$46$	0	0
$47$	22001.0	1.45277	0.726387	−	0.687286i	$-0.241198\pi$
$47$	22001.0	1.45277	0.726387	+	0.687286i	$0.241198\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	16317.0	0.878446
$52$	0	0
$53$	26168.0	1.27962	0.639810	−	0.768533i	$-0.279013\pi$
$53$	26168.0	1.27962	0.639810	+	0.768533i	$0.279013\pi$
$54$	0	0
$55$	4675.00	0.208389
$56$	0	0
$57$	−2322.00	−0.0946619
$58$	0	0
$59$	−3932.00	−0.147056	−0.0735281	−	0.997293i	$-0.523426\pi$
$59$	−3932.00	−0.147056	−0.0735281	+	0.997293i	$0.523426\pi$
$60$	0	0
$61$	−48740.0	−1.67711	−0.838554	−	0.544819i	$-0.816598\pi$
$61$	−48740.0	−1.67711	−0.838554	+	0.544819i	$0.816598\pi$
$62$	0	0
$63$	−7938.00	−0.251976
$64$	0	0
$65$	15675.0	0.460176
$66$	0	0
$67$	44832.0	1.22012	0.610058	−	0.792357i	$-0.291146\pi$
$67$	44832.0	1.22012	0.610058	+	0.792357i	$0.291146\pi$
$68$	0	0
$69$	−26730.0	−0.675890
$70$	0	0
$71$	−63736.0	−1.50051	−0.750255	−	0.661148i	$-0.770069\pi$
$71$	−63736.0	−1.50051	−0.750255	+	0.661148i	$0.770069\pi$
$72$	0	0
$73$	60470.0	1.32811	0.664053	−	0.747685i	$-0.268835\pi$
$73$	60470.0	1.32811	0.664053	+	0.747685i	$0.268835\pi$
$74$	0	0
$75$	5625.00	0.115470
$76$	0	0
$77$	9163.00	0.176121
$78$	0	0
$79$	43721.0	0.788174	0.394087	−	0.919073i	$-0.371061\pi$
$79$	43721.0	0.788174	0.394087	+	0.919073i	$0.371061\pi$
$80$	0	0
$81$	6561.00	0.111111
$82$	0	0
$83$	−97276.0	−1.54992	−0.774962	−	0.632008i	$-0.782231\pi$
$83$	−97276.0	−1.54992	−0.774962	+	0.632008i	$0.782231\pi$
$84$	0	0
$85$	45325.0	0.680441
$86$	0	0
$87$	11691.0	0.165597
$88$	0	0
$89$	45560.0	0.609689	0.304845	−	0.952402i	$-0.401395\pi$
$89$	45560.0	0.609689	0.304845	+	0.952402i	$0.401395\pi$
$90$	0	0
$91$	30723.0	0.388920
$92$	0	0
$93$	−17244.0	−0.206743
$94$	0	0
$95$	−6450.00	−0.0733248
$96$	0	0
$97$	−57295.0	−0.618283	−0.309142	−	0.951016i	$-0.600042\pi$
$97$	−57295.0	−0.618283	−0.309142	+	0.951016i	$0.600042\pi$
$98$	0	0
$99$	−30294.0	−0.310648
$100$	0	0
$101$	−44970.0	−0.438651	−0.219326	−	0.975652i	$-0.570386\pi$
$101$	−44970.0	−0.438651	−0.219326	+	0.975652i	$0.570386\pi$
$102$	0	0
$103$	101405.	0.941817	0.470908	−	0.882182i	$-0.343926\pi$
$103$	101405.	0.941817	0.470908	+	0.882182i	$0.343926\pi$
$104$	0	0
$105$	11025.0	0.0975900
$106$	0	0
$107$	166002.	1.40170	0.700848	−	0.713311i	$-0.252805\pi$
$107$	166002.	1.40170	0.700848	+	0.713311i	$0.252805\pi$
$108$	0	0
$109$	8289.00	0.0668245	0.0334123	−	0.999442i	$-0.489363\pi$
$109$	8289.00	0.0668245	0.0334123	+	0.999442i	$0.489363\pi$
$110$	0	0
$111$	59202.0	0.456067
$112$	0	0
$113$	263206.	1.93910	0.969549	−	0.244898i	$-0.0787545\pi$
$113$	263206.	1.93910	0.969549	+	0.244898i	$0.0787545\pi$
$114$	0	0
$115$	−74250.0	−0.523542
$116$	0	0
$117$	−101574.	−0.685990
$118$	0	0
$119$	88837.0	0.575078
$120$	0	0
$121$	−126082.	−0.782870
$122$	0	0
$123$	60084.0	0.358093
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	−30052.0	−0.165335	−0.0826674	−	0.996577i	$-0.526344\pi$
$127$	−30052.0	−0.165335	−0.0826674	+	0.996577i	$0.526344\pi$
$128$	0	0
$129$	−28602.0	−0.151329
$130$	0	0
$131$	120050.	0.611201	0.305600	−	0.952160i	$-0.401143\pi$
$131$	120050.	0.611201	0.305600	+	0.952160i	$0.401143\pi$
$132$	0	0
$133$	−12642.0	−0.0619707
$134$	0	0
$135$	−91125.0	−0.430331
$136$	0	0
$137$	31776.0	0.144643	0.0723216	−	0.997381i	$-0.476959\pi$
$137$	31776.0	0.144643	0.0723216	+	0.997381i	$0.476959\pi$
$138$	0	0
$139$	200162.	0.878708	0.439354	−	0.898314i	$-0.355207\pi$
$139$	200162.	0.878708	0.439354	+	0.898314i	$0.355207\pi$
$140$	0	0
$141$	198009.	0.838759
$142$	0	0
$143$	117249.	0.479478
$144$	0	0
$145$	32475.0	0.128271
$146$	0	0
$147$	21609.0	0.0824786
$148$	0	0
$149$	309642.	1.14260	0.571300	−	0.820741i	$-0.306439\pi$
$149$	309642.	1.14260	0.571300	+	0.820741i	$0.306439\pi$
$150$	0	0
$151$	208657.	0.744716	0.372358	−	0.928089i	$-0.378549\pi$
$151$	208657.	0.744716	0.372358	+	0.928089i	$0.378549\pi$
$152$	0	0
$153$	−293706.	−1.01434
$154$	0	0
$155$	−47900.0	−0.160142
$156$	0	0
$157$	36010.0	0.116593	0.0582967	−	0.998299i	$-0.481433\pi$
$157$	36010.0	0.116593	0.0582967	+	0.998299i	$0.481433\pi$
$158$	0	0
$159$	235512.	0.738789
$160$	0	0
$161$	−145530.	−0.442474
$162$	0	0
$163$	−175670.	−0.517879	−0.258940	−	0.965893i	$-0.583373\pi$
$163$	−175670.	−0.517879	−0.258940	+	0.965893i	$0.583373\pi$
$164$	0	0
$165$	42075.0	0.120313
$166$	0	0
$167$	157413.	0.436767	0.218383	−	0.975863i	$-0.429922\pi$
$167$	157413.	0.436767	0.218383	+	0.975863i	$0.429922\pi$
$168$	0	0
$169$	21836.0	0.0588107
$170$	0	0
$171$	41796.0	0.109306
$172$	0	0
$173$	−23471.0	−0.0596233	−0.0298117	−	0.999556i	$-0.509491\pi$
$173$	−23471.0	−0.0596233	−0.0298117	+	0.999556i	$0.509491\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	−35388.0	−0.0849030
$178$	0	0
$179$	−612228.	−1.42817	−0.714086	−	0.700058i	$-0.753158\pi$
$179$	−612228.	−1.42817	−0.714086	+	0.700058i	$0.753158\pi$
$180$	0	0
$181$	528832.	1.19983	0.599917	−	0.800062i	$-0.295200\pi$
$181$	528832.	1.19983	0.599917	+	0.800062i	$0.295200\pi$
$182$	0	0
$183$	−438660.	−0.968279
$184$	0	0
$185$	164450.	0.353268
$186$	0	0
$187$	339031.	0.708982
$188$	0	0
$189$	−178605.	−0.363696
$190$	0	0
$191$	−540369.	−1.07178	−0.535892	−	0.844287i	$-0.680024\pi$
$191$	−540369.	−1.07178	−0.535892	+	0.844287i	$0.680024\pi$
$192$	0	0
$193$	960320.	1.85576	0.927882	−	0.372874i	$-0.121628\pi$
$193$	960320.	1.85576	0.927882	+	0.372874i	$0.121628\pi$
$194$	0	0
$195$	141075.	0.265683
$196$	0	0
$197$	761944.	1.39881	0.699403	−	0.714728i	$-0.253449\pi$
$197$	761944.	1.39881	0.699403	+	0.714728i	$0.253449\pi$
$198$	0	0
$199$	−125084.	−0.223908	−0.111954	−	0.993713i	$-0.535711\pi$
$199$	−125084.	−0.223908	−0.111954	+	0.993713i	$0.535711\pi$
$200$	0	0
$201$	403488.	0.704434
$202$	0	0
$203$	63651.0	0.108409
$204$	0	0
$205$	166900.	0.277378
$206$	0	0
$207$	481140.	0.780451
$208$	0	0
$209$	−48246.0	−0.0764004
$210$	0	0
$211$	−627547.	−0.970376	−0.485188	−	0.874410i	$-0.661249\pi$
$211$	−627547.	−0.970376	−0.485188	+	0.874410i	$0.661249\pi$
$212$	0	0
$213$	−573624.	−0.866320
$214$	0	0
$215$	−79450.0	−0.117219
$216$	0	0
$217$	−93884.0	−0.135345
$218$	0	0
$219$	544230.	0.766782
$220$	0	0
$221$	1.13675e6	1.56561
$222$	0	0
$223$	1.22110e6	1.64433	0.822166	−	0.569248i	$-0.192766\pi$
$223$	1.22110e6	1.64433	0.822166	+	0.569248i	$0.192766\pi$
$224$	0	0
$225$	−101250.	−0.133333
$226$	0	0
$227$	−390547.	−0.503047	−0.251524	−	0.967851i	$-0.580932\pi$
$227$	−390547.	−0.503047	−0.251524	+	0.967851i	$0.580932\pi$
$228$	0	0
$229$	−712124.	−0.897360	−0.448680	−	0.893692i	$-0.648106\pi$
$229$	−712124.	−0.897360	−0.448680	+	0.893692i	$0.648106\pi$
$230$	0	0
$231$	82467.0	0.101683
$232$	0	0
$233$	561576.	0.677671	0.338835	−	0.940846i	$-0.389967\pi$
$233$	561576.	0.677671	0.338835	+	0.940846i	$0.389967\pi$
$234$	0	0
$235$	550025.	0.649700
$236$	0	0
$237$	393489.	0.455053
$238$	0	0
$239$	1.36084e6	1.54103	0.770515	−	0.637421i	$-0.219999\pi$
$239$	1.36084e6	1.54103	0.770515	+	0.637421i	$0.219999\pi$
$240$	0	0
$241$	530050.	0.587860	0.293930	−	0.955827i	$-0.405037\pi$
$241$	530050.	0.587860	0.293930	+	0.955827i	$0.405037\pi$
$242$	0	0
$243$	944784.	1.02640
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−161766.	−0.168712
$248$	0	0
$249$	−875484.	−0.894849
$250$	0	0
$251$	−990330.	−0.992192	−0.496096	−	0.868268i	$-0.665234\pi$
$251$	−990330.	−0.992192	−0.496096	+	0.868268i	$0.665234\pi$
$252$	0	0
$253$	−555390.	−0.545503
$254$	0	0
$255$	407925.	0.392853
$256$	0	0
$257$	−1.81643e6	−1.71548	−0.857740	−	0.514083i	$-0.828132\pi$
$257$	−1.81643e6	−1.71548	−0.857740	+	0.514083i	$0.828132\pi$
$258$	0	0
$259$	322322.	0.298566
$260$	0	0
$261$	−210438.	−0.191215
$262$	0	0
$263$	1.95847e6	1.74594	0.872968	−	0.487777i	$-0.162192\pi$
$263$	1.95847e6	1.74594	0.872968	+	0.487777i	$0.162192\pi$
$264$	0	0
$265$	654200.	0.572263
$266$	0	0
$267$	410040.	0.352004
$268$	0	0
$269$	218034.	0.183715	0.0918573	−	0.995772i	$-0.470720\pi$
$269$	218034.	0.183715	0.0918573	+	0.995772i	$0.470720\pi$
$270$	0	0
$271$	−1.26265e6	−1.04438	−0.522192	−	0.852828i	$-0.674886\pi$
$271$	−1.26265e6	−1.04438	−0.522192	+	0.852828i	$0.674886\pi$
$272$	0	0
$273$	276507.	0.224543
$274$	0	0
$275$	116875.	0.0931944
$276$	0	0
$277$	−1.10264e6	−0.863443	−0.431721	−	0.902007i	$-0.642094\pi$
$277$	−1.10264e6	−0.863443	−0.431721	+	0.902007i	$0.642094\pi$
$278$	0	0
$279$	310392.	0.238726
$280$	0	0
$281$	−998213.	−0.754149	−0.377075	−	0.926183i	$-0.623070\pi$
$281$	−998213.	−0.754149	−0.377075	+	0.926183i	$0.623070\pi$
$282$	0	0
$283$	386371.	0.286773	0.143387	−	0.989667i	$-0.454201\pi$
$283$	386371.	0.286773	0.143387	+	0.989667i	$0.454201\pi$
$284$	0	0
$285$	−58050.0	−0.0423341
$286$	0	0
$287$	327124.	0.234427
$288$	0	0
$289$	1.86711e6	1.31500
$290$	0	0
$291$	−515655.	−0.356966
$292$	0	0
$293$	−783571.	−0.533224	−0.266612	−	0.963804i	$-0.585904\pi$
$293$	−783571.	−0.533224	−0.266612	+	0.963804i	$0.585904\pi$
$294$	0	0
$295$	−98300.0	−0.0657656
$296$	0	0
$297$	−681615.	−0.448382
$298$	0	0
$299$	−1.86219e6	−1.20461
$300$	0	0
$301$	−155722.	−0.0990681
$302$	0	0
$303$	−404730.	−0.253255
$304$	0	0
$305$	−1.21850e6	−0.750025
$306$	0	0
$307$	−2.81773e6	−1.70629	−0.853147	−	0.521670i	$-0.825309\pi$
$307$	−2.81773e6	−1.70629	−0.853147	+	0.521670i	$0.825309\pi$
$308$	0	0
$309$	912645.	0.543758
$310$	0	0
$311$	847398.	0.496806	0.248403	−	0.968657i	$-0.420094\pi$
$311$	847398.	0.496806	0.248403	+	0.968657i	$0.420094\pi$
$312$	0	0
$313$	364955.	0.210561	0.105281	−	0.994443i	$-0.466426\pi$
$313$	364955.	0.210561	0.105281	+	0.994443i	$0.466426\pi$
$314$	0	0
$315$	−198450.	−0.112687
$316$	0	0
$317$	1.93744e6	1.08288	0.541439	−	0.840740i	$-0.317880\pi$
$317$	1.93744e6	1.08288	0.541439	+	0.840740i	$0.317880\pi$
$318$	0	0
$319$	242913.	0.133652
$320$	0	0
$321$	1.49402e6	0.809270
$322$	0	0
$323$	−467754.	−0.249466
$324$	0	0
$325$	391875.	0.205797
$326$	0	0
$327$	74601.0	0.0385812
$328$	0	0
$329$	1.07805e6	0.549097
$330$	0	0
$331$	61460.0	0.0308335	0.0154167	−	0.999881i	$-0.495093\pi$
$331$	61460.0	0.0308335	0.0154167	+	0.999881i	$0.495093\pi$
$332$	0	0
$333$	−1.06564e6	−0.526621
$334$	0	0
$335$	1.12080e6	0.545652
$336$	0	0
$337$	−3.74116e6	−1.79445	−0.897225	−	0.441574i	$-0.854420\pi$
$337$	−3.74116e6	−1.79445	−0.897225	+	0.441574i	$0.854420\pi$
$338$	0	0
$339$	2.36885e6	1.11954
$340$	0	0
$341$	−358292.	−0.166860
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	−668250.	−0.302267
$346$	0	0
$347$	−211334.	−0.0942206	−0.0471103	−	0.998890i	$-0.515001\pi$
$347$	−211334.	−0.0942206	−0.0471103	+	0.998890i	$0.515001\pi$
$348$	0	0
$349$	3.39558e6	1.49228	0.746140	−	0.665789i	$-0.231905\pi$
$349$	3.39558e6	1.49228	0.746140	+	0.665789i	$0.231905\pi$
$350$	0	0
$351$	−2.28542e6	−0.990142
$352$	0	0
$353$	−3.88094e6	−1.65768	−0.828838	−	0.559489i	$-0.810997\pi$
$353$	−3.88094e6	−1.65768	−0.828838	+	0.559489i	$0.810997\pi$
$354$	0	0
$355$	−1.59340e6	−0.671049
$356$	0	0
$357$	799533.	0.332021
$358$	0	0
$359$	−3.24210e6	−1.32767	−0.663836	−	0.747878i	$-0.731073\pi$
$359$	−3.24210e6	−1.32767	−0.663836	+	0.747878i	$0.731073\pi$
$360$	0	0
$361$	−2.40954e6	−0.973117
$362$	0	0
$363$	−1.13474e6	−0.451990
$364$	0	0
$365$	1.51175e6	0.593947
$366$	0	0
$367$	−1.44430e6	−0.559749	−0.279874	−	0.960037i	$-0.590293\pi$
$367$	−1.44430e6	−0.559749	−0.279874	+	0.960037i	$0.590293\pi$
$368$	0	0
$369$	−1.08151e6	−0.413490
$370$	0	0
$371$	1.28223e6	0.483651
$372$	0	0
$373$	3.43542e6	1.27852	0.639260	−	0.768991i	$-0.279241\pi$
$373$	3.43542e6	1.27852	0.639260	+	0.768991i	$0.279241\pi$
$374$	0	0
$375$	140625.	0.0516398
$376$	0	0
$377$	814473.	0.295137
$378$	0	0
$379$	1.68635e6	0.603044	0.301522	−	0.953459i	$-0.402505\pi$
$379$	1.68635e6	0.603044	0.301522	+	0.953459i	$0.402505\pi$
$380$	0	0
$381$	−270468.	−0.0954560
$382$	0	0
$383$	−2.64354e6	−0.920850	−0.460425	−	0.887699i	$-0.652303\pi$
$383$	−2.64354e6	−0.920850	−0.460425	+	0.887699i	$0.652303\pi$
$384$	0	0
$385$	229075.	0.0787637
$386$	0	0
$387$	514836.	0.174740
$388$	0	0
$389$	−452099.	−0.151481	−0.0757407	−	0.997128i	$-0.524132\pi$
$389$	−452099.	−0.151481	−0.0757407	+	0.997128i	$0.524132\pi$
$390$	0	0
$391$	−5.38461e6	−1.78120
$392$	0	0
$393$	1.08045e6	0.352877
$394$	0	0
$395$	1.09302e6	0.352482
$396$	0	0
$397$	−1.95530e6	−0.622641	−0.311321	−	0.950305i	$-0.600771\pi$
$397$	−1.95530e6	−0.622641	−0.311321	+	0.950305i	$0.600771\pi$
$398$	0	0
$399$	−113778.	−0.0357788
$400$	0	0
$401$	−4.76737e6	−1.48053	−0.740266	−	0.672314i	$-0.765300\pi$
$401$	−4.76737e6	−1.48053	−0.740266	+	0.672314i	$0.765300\pi$
$402$	0	0
$403$	−1.20133e6	−0.368469
$404$	0	0
$405$	164025.	0.0496904
$406$	0	0
$407$	1.23009e6	0.368086
$408$	0	0
$409$	−4.13199e6	−1.22138	−0.610690	−	0.791870i	$-0.709108\pi$
$409$	−4.13199e6	−1.22138	−0.610690	+	0.791870i	$0.709108\pi$
$410$	0	0
$411$	285984.	0.0835097
$412$	0	0
$413$	−192668.	−0.0555820
$414$	0	0
$415$	−2.43190e6	−0.693147
$416$	0	0
$417$	1.80146e6	0.507322
$418$	0	0
$419$	−190512.	−0.0530136	−0.0265068	−	0.999649i	$-0.508438\pi$
$419$	−190512.	−0.0530136	−0.0265068	+	0.999649i	$0.508438\pi$
$420$	0	0
$421$	−5.19186e6	−1.42764	−0.713818	−	0.700332i	$-0.753035\pi$
$421$	−5.19186e6	−1.42764	−0.713818	+	0.700332i	$0.753035\pi$
$422$	0	0
$423$	−3.56416e6	−0.968515
$424$	0	0
$425$	1.13312e6	0.304302
$426$	0	0
$427$	−2.38826e6	−0.633887
$428$	0	0
$429$	1.05524e6	0.276827
$430$	0	0
$431$	4.21781e6	1.09369	0.546845	−	0.837234i	$-0.315829\pi$
$431$	4.21781e6	1.09369	0.546845	+	0.837234i	$0.315829\pi$
$432$	0	0
$433$	−4.86027e6	−1.24578	−0.622890	−	0.782310i	$-0.714041\pi$
$433$	−4.86027e6	−1.24578	−0.622890	+	0.782310i	$0.714041\pi$
$434$	0	0
$435$	292275.	0.0740574
$436$	0	0
$437$	766260.	0.191943
$438$	0	0
$439$	2.03113e6	0.503011	0.251505	−	0.967856i	$-0.419074\pi$
$439$	2.03113e6	0.503011	0.251505	+	0.967856i	$0.419074\pi$
$440$	0	0
$441$	−388962.	−0.0952381
$442$	0	0
$443$	−2.84199e6	−0.688038	−0.344019	−	0.938963i	$-0.611789\pi$
$443$	−2.84199e6	−0.688038	−0.344019	+	0.938963i	$0.611789\pi$
$444$	0	0
$445$	1.13900e6	0.272661
$446$	0	0
$447$	2.78678e6	0.659680
$448$	0	0
$449$	−4.59682e6	−1.07607	−0.538037	−	0.842921i	$-0.680834\pi$
$449$	−4.59682e6	−1.07607	−0.538037	+	0.842921i	$0.680834\pi$
$450$	0	0
$451$	1.24841e6	0.289012
$452$	0	0
$453$	1.87791e6	0.429962
$454$	0	0
$455$	768075.	0.173930
$456$	0	0
$457$	4.93367e6	1.10504	0.552522	−	0.833498i	$-0.313665\pi$
$457$	4.93367e6	1.10504	0.552522	+	0.833498i	$0.313665\pi$
$458$	0	0
$459$	−6.60838e6	−1.46408
$460$	0	0
$461$	−4.75667e6	−1.04244	−0.521220	−	0.853422i	$-0.674523\pi$
$461$	−4.75667e6	−1.04244	−0.521220	+	0.853422i	$0.674523\pi$
$462$	0	0
$463$	−4.08619e6	−0.885862	−0.442931	−	0.896556i	$-0.646061\pi$
$463$	−4.08619e6	−0.885862	−0.442931	+	0.896556i	$0.646061\pi$
$464$	0	0
$465$	−431100.	−0.0924582
$466$	0	0
$467$	4.15932e6	0.882531	0.441266	−	0.897377i	$-0.354530\pi$
$467$	4.15932e6	0.882531	0.441266	+	0.897377i	$0.354530\pi$
$468$	0	0
$469$	2.19677e6	0.461160
$470$	0	0
$471$	324090.	0.0673152
$472$	0	0
$473$	−594286.	−0.122136
$474$	0	0
$475$	−161250.	−0.0327918
$476$	0	0
$477$	−4.23922e6	−0.853080
$478$	0	0
$479$	3.36040e6	0.669195	0.334597	−	0.942361i	$-0.391400\pi$
$479$	3.36040e6	0.669195	0.334597	+	0.942361i	$0.391400\pi$
$480$	0	0
$481$	4.12441e6	0.812828
$482$	0	0
$483$	−1.30977e6	−0.255463
$484$	0	0
$485$	−1.43238e6	−0.276505
$486$	0	0
$487$	−7.05243e6	−1.34746	−0.673730	−	0.738977i	$-0.735309\pi$
$487$	−7.05243e6	−1.34746	−0.673730	+	0.738977i	$0.735309\pi$
$488$	0	0
$489$	−1.58103e6	−0.298998
$490$	0	0
$491$	83937.0	0.0157127	0.00785633	−	0.999969i	$-0.497499\pi$
$491$	83937.0	0.0157127	0.00785633	+	0.999969i	$0.497499\pi$
$492$	0	0
$493$	2.35509e6	0.436405
$494$	0	0
$495$	−757350.	−0.138926
$496$	0	0
$497$	−3.12306e6	−0.567140
$498$	0	0
$499$	7.10526e6	1.27741	0.638703	−	0.769454i	$-0.279471\pi$
$499$	7.10526e6	1.27741	0.638703	+	0.769454i	$0.279471\pi$
$500$	0	0
$501$	1.41672e6	0.252167
$502$	0	0
$503$	−2.89147e6	−0.509564	−0.254782	−	0.966999i	$-0.582004\pi$
$503$	−2.89147e6	−0.509564	−0.254782	+	0.966999i	$0.582004\pi$
$504$	0	0
$505$	−1.12425e6	−0.196171
$506$	0	0
$507$	196524.	0.0339544
$508$	0	0
$509$	1.03548e6	0.177153	0.0885764	−	0.996069i	$-0.471768\pi$
$509$	1.03548e6	0.177153	0.0885764	+	0.996069i	$0.471768\pi$
$510$	0	0
$511$	2.96303e6	0.501977
$512$	0	0
$513$	940410.	0.157770
$514$	0	0
$515$	2.53513e6	0.421193
$516$	0	0
$517$	4.11419e6	0.676952
$518$	0	0
$519$	−211239.	−0.0344236
$520$	0	0
$521$	−7.49715e6	−1.21005	−0.605023	−	0.796208i	$-0.706836\pi$
$521$	−7.49715e6	−1.21005	−0.605023	+	0.796208i	$0.706836\pi$
$522$	0	0
$523$	−3.53223e6	−0.564670	−0.282335	−	0.959316i	$-0.591109\pi$
$523$	−3.53223e6	−0.564670	−0.282335	+	0.959316i	$0.591109\pi$
$524$	0	0
$525$	275625.	0.0436436
$526$	0	0
$527$	−3.47371e6	−0.544837
$528$	0	0
$529$	2.38456e6	0.370483
$530$	0	0
$531$	636984.	0.0980375
$532$	0	0
$533$	4.18585e6	0.638213
$534$	0	0
$535$	4.15005e6	0.626858
$536$	0	0
$537$	−5.51005e6	−0.824556
$538$	0	0
$539$	448987.	0.0665674
$540$	0	0
$541$	−4.99188e6	−0.733281	−0.366641	−	0.930363i	$-0.619492\pi$
$541$	−4.99188e6	−0.733281	−0.366641	+	0.930363i	$0.619492\pi$
$542$	0	0
$543$	4.75949e6	0.692725
$544$	0	0
$545$	207225.	0.0298848
$546$	0	0
$547$	−5.12634e6	−0.732553	−0.366277	−	0.930506i	$-0.619368\pi$
$547$	−5.12634e6	−0.732553	−0.366277	+	0.930506i	$0.619368\pi$
$548$	0	0
$549$	7.89588e6	1.11807
$550$	0	0
$551$	−335142.	−0.0470273
$552$	0	0
$553$	2.14233e6	0.297902
$554$	0	0
$555$	1.48005e6	0.203959
$556$	0	0
$557$	8.86866e6	1.21121	0.605606	−	0.795765i	$-0.292931\pi$
$557$	8.86866e6	1.21121	0.605606	+	0.795765i	$0.292931\pi$
$558$	0	0
$559$	−1.99261e6	−0.269707
$560$	0	0
$561$	3.05128e6	0.409331
$562$	0	0
$563$	−9.07277e6	−1.20634	−0.603169	−	0.797613i	$-0.706096\pi$
$563$	−9.07277e6	−1.20634	−0.603169	+	0.797613i	$0.706096\pi$
$564$	0	0
$565$	6.58015e6	0.867191
$566$	0	0
$567$	321489.	0.0419961
$568$	0	0
$569$	2.08310e6	0.269730	0.134865	−	0.990864i	$-0.456940\pi$
$569$	2.08310e6	0.269730	0.134865	+	0.990864i	$0.456940\pi$
$570$	0	0
$571$	5.46368e6	0.701286	0.350643	−	0.936509i	$-0.385963\pi$
$571$	5.46368e6	0.701286	0.350643	+	0.936509i	$0.385963\pi$
$572$	0	0
$573$	−4.86332e6	−0.618794
$574$	0	0
$575$	−1.85625e6	−0.234135
$576$	0	0
$577$	7.66246e6	0.958140	0.479070	−	0.877777i	$-0.340974\pi$
$577$	7.66246e6	0.958140	0.479070	+	0.877777i	$0.340974\pi$
$578$	0	0
$579$	8.64288e6	1.07143
$580$	0	0
$581$	−4.76652e6	−0.585816
$582$	0	0
$583$	4.89342e6	0.596267
$584$	0	0
$585$	−2.53935e6	−0.306784
$586$	0	0
$587$	1.57465e7	1.88620	0.943100	−	0.332510i	$-0.107896\pi$
$587$	1.57465e7	1.88620	0.943100	+	0.332510i	$0.107896\pi$
$588$	0	0
$589$	494328.	0.0587120
$590$	0	0
$591$	6.85750e6	0.807601
$592$	0	0
$593$	−1.62409e7	−1.89658	−0.948292	−	0.317398i	$-0.897191\pi$
$593$	−1.62409e7	−1.89658	−0.948292	+	0.317398i	$0.897191\pi$
$594$	0	0
$595$	2.22093e6	0.257183
$596$	0	0
$597$	−1.12576e6	−0.129273
$598$	0	0
$599$	1.90793e6	0.217268	0.108634	−	0.994082i	$-0.465352\pi$
$599$	1.90793e6	0.217268	0.108634	+	0.994082i	$0.465352\pi$
$600$	0	0
$601$	3.52970e6	0.398613	0.199306	−	0.979937i	$-0.436131\pi$
$601$	3.52970e6	0.398613	0.199306	+	0.979937i	$0.436131\pi$
$602$	0	0
$603$	−7.26278e6	−0.813411
$604$	0	0
$605$	−3.15205e6	−0.350110
$606$	0	0
$607$	−3.37799e6	−0.372123	−0.186061	−	0.982538i	$-0.559572\pi$
$607$	−3.37799e6	−0.372123	−0.186061	+	0.982538i	$0.559572\pi$
$608$	0	0
$609$	572859.	0.0625899
$610$	0	0
$611$	1.37946e7	1.49488
$612$	0	0
$613$	1.20412e6	0.129425	0.0647127	−	0.997904i	$-0.479387\pi$
$613$	1.20412e6	0.129425	0.0647127	+	0.997904i	$0.479387\pi$
$614$	0	0
$615$	1.50210e6	0.160144
$616$	0	0
$617$	−5.47330e6	−0.578810	−0.289405	−	0.957207i	$-0.593457\pi$
$617$	−5.47330e6	−0.578810	−0.289405	+	0.957207i	$0.593457\pi$
$618$	0	0
$619$	−3.22662e6	−0.338471	−0.169236	−	0.985576i	$-0.554130\pi$
$619$	−3.22662e6	−0.338471	−0.169236	+	0.985576i	$0.554130\pi$
$620$	0	0
$621$	1.08256e7	1.12648
$622$	0	0
$623$	2.23244e6	0.230441
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−434214.	−0.0441098
$628$	0	0
$629$	1.19259e7	1.20189
$630$	0	0
$631$	−1.36282e7	−1.36259	−0.681297	−	0.732007i	$-0.738584\pi$
$631$	−1.36282e7	−1.36259	−0.681297	+	0.732007i	$0.738584\pi$
$632$	0	0
$633$	−5.64792e6	−0.560247
$634$	0	0
$635$	−751300.	−0.0739399
$636$	0	0
$637$	1.50543e6	0.146998
$638$	0	0
$639$	1.03252e7	1.00034
$640$	0	0
$641$	−1.92472e7	−1.85021	−0.925106	−	0.379710i	$-0.876024\pi$
$641$	−1.92472e7	−1.85021	−0.925106	+	0.379710i	$0.876024\pi$
$642$	0	0
$643$	1.28399e7	1.22472	0.612358	−	0.790580i	$-0.290221\pi$
$643$	1.28399e7	1.22472	0.612358	+	0.790580i	$0.290221\pi$
$644$	0	0
$645$	−715050.	−0.0676764
$646$	0	0
$647$	−2.00233e7	−1.88050	−0.940251	−	0.340481i	$-0.889410\pi$
$647$	−2.00233e7	−1.88050	−0.940251	+	0.340481i	$0.889410\pi$
$648$	0	0
$649$	−735284.	−0.0685241
$650$	0	0
$651$	−844956.	−0.0781415
$652$	0	0
$653$	−7.23655e6	−0.664124	−0.332062	−	0.943258i	$-0.607744\pi$
$653$	−7.23655e6	−0.664124	−0.332062	+	0.943258i	$0.607744\pi$
$654$	0	0
$655$	3.00125e6	0.273337
$656$	0	0
$657$	−9.79614e6	−0.885404
$658$	0	0
$659$	−1.42474e7	−1.27798	−0.638989	−	0.769216i	$-0.720647\pi$
$659$	−1.42474e7	−1.27798	−0.638989	+	0.769216i	$0.720647\pi$
$660$	0	0
$661$	1.49265e7	1.32878	0.664391	−	0.747385i	$-0.268691\pi$
$661$	1.49265e7	1.32878	0.664391	+	0.747385i	$0.268691\pi$
$662$	0	0
$663$	1.02308e7	0.903908
$664$	0	0
$665$	−316050.	−0.0277142
$666$	0	0
$667$	−3.85803e6	−0.335777
$668$	0	0
$669$	1.09899e7	0.949355
$670$	0	0
$671$	−9.11438e6	−0.781485
$672$	0	0
$673$	−1.55062e7	−1.31967	−0.659837	−	0.751409i	$-0.729375\pi$
$673$	−1.55062e7	−1.31967	−0.659837	+	0.751409i	$0.729375\pi$
$674$	0	0
$675$	−2.27812e6	−0.192450
$676$	0	0
$677$	−7.80065e6	−0.654122	−0.327061	−	0.945003i	$-0.606058\pi$
$677$	−7.80065e6	−0.654122	−0.327061	+	0.945003i	$0.606058\pi$
$678$	0	0
$679$	−2.80745e6	−0.233689
$680$	0	0
$681$	−3.51492e6	−0.290434
$682$	0	0
$683$	−1.58547e7	−1.30049	−0.650243	−	0.759727i	$-0.725333\pi$
$683$	−1.58547e7	−1.30049	−0.650243	+	0.759727i	$0.725333\pi$
$684$	0	0
$685$	794400.	0.0646864
$686$	0	0
$687$	−6.40912e6	−0.518091
$688$	0	0
$689$	1.64073e7	1.31671
$690$	0	0
$691$	−2.03656e7	−1.62257	−0.811284	−	0.584652i	$-0.801231\pi$
$691$	−2.03656e7	−1.62257	−0.811284	+	0.584652i	$0.801231\pi$
$692$	0	0
$693$	−1.48441e6	−0.117414
$694$	0	0
$695$	5.00405e6	0.392970
$696$	0	0
$697$	1.21036e7	0.943696
$698$	0	0
$699$	5.05418e6	0.391253
$700$	0	0
$701$	2.48036e7	1.90643	0.953213	−	0.302300i	$-0.0977543\pi$
$701$	2.48036e7	1.90643	0.953213	+	0.302300i	$0.0977543\pi$
$702$	0	0
$703$	−1.69712e6	−0.129517
$704$	0	0
$705$	4.95022e6	0.375104
$706$	0	0
$707$	−2.20353e6	−0.165795
$708$	0	0
$709$	1.81917e7	1.35912	0.679560	−	0.733620i	$-0.262171\pi$
$709$	1.81917e7	1.35912	0.679560	+	0.733620i	$0.262171\pi$
$710$	0	0
$711$	−7.08280e6	−0.525450
$712$	0	0
$713$	5.69052e6	0.419207
$714$	0	0
$715$	2.93122e6	0.214429
$716$	0	0
$717$	1.22475e7	0.889715
$718$	0	0
$719$	1.66202e7	1.19899	0.599493	−	0.800380i	$-0.295369\pi$
$719$	1.66202e7	1.19899	0.599493	+	0.800380i	$0.295369\pi$
$720$	0	0
$721$	4.96884e6	0.355973
$722$	0	0
$723$	4.77045e6	0.339401
$724$	0	0
$725$	811875.	0.0573646
$726$	0	0
$727$	1.57591e7	1.10585	0.552925	−	0.833231i	$-0.313511\pi$
$727$	1.57591e7	1.10585	0.552925	+	0.833231i	$0.313511\pi$
$728$	0	0
$729$	6.90873e6	0.481481
$730$	0	0
$731$	−5.76171e6	−0.398803
$732$	0	0
$733$	−2.15238e6	−0.147965	−0.0739827	−	0.997260i	$-0.523571\pi$
$733$	−2.15238e6	−0.147965	−0.0739827	+	0.997260i	$0.523571\pi$
$734$	0	0
$735$	540225.	0.0368856
$736$	0	0
$737$	8.38358e6	0.568540
$738$	0	0
$739$	2.27267e7	1.53083	0.765413	−	0.643540i	$-0.222535\pi$
$739$	2.27267e7	1.53083	0.765413	+	0.643540i	$0.222535\pi$
$740$	0	0
$741$	−1.45589e6	−0.0974057
$742$	0	0
$743$	−1.21153e7	−0.805123	−0.402561	−	0.915393i	$-0.631880\pi$
$743$	−1.21153e7	−0.805123	−0.402561	+	0.915393i	$0.631880\pi$
$744$	0	0
$745$	7.74105e6	0.510986
$746$	0	0
$747$	1.57587e7	1.03328
$748$	0	0
$749$	8.13410e6	0.529791
$750$	0	0
$751$	2.07590e7	1.34310	0.671549	−	0.740961i	$-0.265629\pi$
$751$	2.07590e7	1.34310	0.671549	+	0.740961i	$0.265629\pi$
$752$	0	0
$753$	−8.91297e6	−0.572842
$754$	0	0
$755$	5.21642e6	0.333047
$756$	0	0
$757$	−1.86222e7	−1.18111	−0.590556	−	0.806997i	$-0.701091\pi$
$757$	−1.86222e7	−1.18111	−0.590556	+	0.806997i	$0.701091\pi$
$758$	0	0
$759$	−4.99851e6	−0.314946
$760$	0	0
$761$	2.65336e7	1.66087	0.830434	−	0.557117i	$-0.188093\pi$
$761$	2.65336e7	1.66087	0.830434	+	0.557117i	$0.188093\pi$
$762$	0	0
$763$	406161.	0.0252573
$764$	0	0
$765$	−7.34265e6	−0.453627
$766$	0	0
$767$	−2.46536e6	−0.151319
$768$	0	0
$769$	−2.01595e7	−1.22931	−0.614657	−	0.788794i	$-0.710706\pi$
$769$	−2.01595e7	−1.22931	−0.614657	+	0.788794i	$0.710706\pi$
$770$	0	0
$771$	−1.63479e7	−0.990433
$772$	0	0
$773$	−5.86488e6	−0.353029	−0.176514	−	0.984298i	$-0.556482\pi$
$773$	−5.86488e6	−0.353029	−0.176514	+	0.984298i	$0.556482\pi$
$774$	0	0
$775$	−1.19750e6	−0.0716178
$776$	0	0
$777$	2.90090e6	0.172377
$778$	0	0
$779$	−1.72241e6	−0.101693
$780$	0	0
$781$	−1.19186e7	−0.699196
$782$	0	0
$783$	−4.73486e6	−0.275996
$784$	0	0
$785$	900250.	0.0521422
$786$	0	0
$787$	1.63347e6	0.0940100	0.0470050	−	0.998895i	$-0.485032\pi$
$787$	1.63347e6	0.0940100	0.0470050	+	0.998895i	$0.485032\pi$
$788$	0	0
$789$	1.76263e7	1.00802
$790$	0	0
$791$	1.28971e7	0.732910
$792$	0	0
$793$	−3.05600e7	−1.72572
$794$	0	0
$795$	5.88780e6	0.330396
$796$	0	0
$797$	2.07673e7	1.15807	0.579034	−	0.815303i	$-0.303430\pi$
$797$	2.07673e7	1.15807	0.579034	+	0.815303i	$0.303430\pi$
$798$	0	0
$799$	3.98878e7	2.21041
$800$	0	0
$801$	−7.38072e6	−0.406460
$802$	0	0
$803$	1.13079e7	0.618860
$804$	0	0
$805$	−3.63825e6	−0.197880
$806$	0	0
$807$	1.96231e6	0.106068
$808$	0	0
$809$	3.53936e6	0.190131	0.0950656	−	0.995471i	$-0.469694\pi$
$809$	3.53936e6	0.190131	0.0950656	+	0.995471i	$0.469694\pi$
$810$	0	0
$811$	2.11480e7	1.12906	0.564530	−	0.825412i	$-0.309057\pi$
$811$	2.11480e7	1.12906	0.564530	+	0.825412i	$0.309057\pi$
$812$	0	0
$813$	−1.13639e7	−0.602976
$814$	0	0
$815$	−4.39175e6	−0.231603
$816$	0	0
$817$	819924.	0.0429753
$818$	0	0
$819$	−4.97713e6	−0.259280
$820$	0	0
$821$	265389.	0.0137412	0.00687061	−	0.999976i	$-0.497813\pi$
$821$	265389.	0.0137412	0.00687061	+	0.999976i	$0.497813\pi$
$822$	0	0
$823$	−3.09261e7	−1.59157	−0.795785	−	0.605579i	$-0.792942\pi$
$823$	−3.09261e7	−1.59157	−0.795785	+	0.605579i	$0.792942\pi$
$824$	0	0
$825$	1.05188e6	0.0538058
$826$	0	0
$827$	2.84152e7	1.44473	0.722367	−	0.691510i	$-0.243054\pi$
$827$	2.84152e7	1.44473	0.722367	+	0.691510i	$0.243054\pi$
$828$	0	0
$829$	−3.33547e7	−1.68566	−0.842832	−	0.538177i	$-0.819113\pi$
$829$	−3.33547e7	−1.68566	−0.842832	+	0.538177i	$0.819113\pi$
$830$	0	0
$831$	−9.92374e6	−0.498509
$832$	0	0
$833$	4.35301e6	0.217359
$834$	0	0
$835$	3.93532e6	0.195328
$836$	0	0
$837$	6.98382e6	0.344572
$838$	0	0
$839$	5.66205e6	0.277695	0.138848	−	0.990314i	$-0.455660\pi$
$839$	5.66205e6	0.277695	0.138848	+	0.990314i	$0.455660\pi$
$840$	0	0
$841$	−1.88237e7	−0.917732
$842$	0	0
$843$	−8.98392e6	−0.435408
$844$	0	0
$845$	545900.	0.0263009
$846$	0	0
$847$	−6.17802e6	−0.295897
$848$	0	0
$849$	3.47734e6	0.165569
$850$	0	0
$851$	−1.95367e7	−0.924754
$852$	0	0
$853$	2.19983e7	1.03518	0.517592	−	0.855628i	$-0.326829\pi$
$853$	2.19983e7	1.03518	0.517592	+	0.855628i	$0.326829\pi$
$854$	0	0
$855$	1.04490e6	0.0488832
$856$	0	0
$857$	−2.17568e7	−1.01191	−0.505956	−	0.862559i	$-0.668860\pi$
$857$	−2.17568e7	−1.01191	−0.505956	+	0.862559i	$0.668860\pi$
$858$	0	0
$859$	4.09384e7	1.89299	0.946494	−	0.322721i	$-0.104598\pi$
$859$	4.09384e7	1.89299	0.946494	+	0.322721i	$0.104598\pi$
$860$	0	0
$861$	2.94412e6	0.135347
$862$	0	0
$863$	−5.65597e6	−0.258512	−0.129256	−	0.991611i	$-0.541259\pi$
$863$	−5.65597e6	−0.258512	−0.129256	+	0.991611i	$0.541259\pi$
$864$	0	0
$865$	−586775.	−0.0266644
$866$	0	0
$867$	1.68040e7	0.759216
$868$	0	0
$869$	8.17583e6	0.367267
$870$	0	0
$871$	2.81097e7	1.25548
$872$	0	0
$873$	9.28179e6	0.412189
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	2.61067e7	1.14618	0.573089	−	0.819493i	$-0.305745\pi$
$877$	2.61067e7	1.14618	0.573089	+	0.819493i	$0.305745\pi$
$878$	0	0
$879$	−7.05214e6	−0.307857
$880$	0	0
$881$	1.44294e6	0.0626339	0.0313170	−	0.999510i	$-0.490030\pi$
$881$	1.44294e6	0.0626339	0.0313170	+	0.999510i	$0.490030\pi$
$882$	0	0
$883$	1.52432e7	0.657921	0.328960	−	0.944344i	$-0.393302\pi$
$883$	1.52432e7	0.657921	0.328960	+	0.944344i	$0.393302\pi$
$884$	0	0
$885$	−884700.	−0.0379698
$886$	0	0
$887$	3.31500e7	1.41473	0.707366	−	0.706847i	$-0.249883\pi$
$887$	3.31500e7	1.41473	0.707366	+	0.706847i	$0.249883\pi$
$888$	0	0
$889$	−1.47255e6	−0.0624907
$890$	0	0
$891$	1.22691e6	0.0517747
$892$	0	0
$893$	−5.67626e6	−0.238195
$894$	0	0
$895$	−1.53057e7	−0.638698
$896$	0	0
$897$	−1.67597e7	−0.695481
$898$	0	0
$899$	−2.48888e6	−0.102708
$900$	0	0
$901$	4.74426e7	1.94696
$902$	0	0
$903$	−1.40150e6	−0.0571970
$904$	0	0
$905$	1.32208e7	0.536582
$906$	0	0
$907$	−1.16963e7	−0.472096	−0.236048	−	0.971741i	$-0.575852\pi$
$907$	−1.16963e7	−0.472096	−0.236048	+	0.971741i	$0.575852\pi$
$908$	0	0
$909$	7.28514e6	0.292434
$910$	0	0
$911$	−2.89321e7	−1.15501	−0.577503	−	0.816389i	$-0.695973\pi$
$911$	−2.89321e7	−1.15501	−0.577503	+	0.816389i	$0.695973\pi$
$912$	0	0
$913$	−1.81906e7	−0.722221
$914$	0	0
$915$	−1.09665e7	−0.433027
$916$	0	0
$917$	5.88245e6	0.231012
$918$	0	0
$919$	4.57838e7	1.78823	0.894115	−	0.447838i	$-0.147806\pi$
$919$	4.57838e7	1.78823	0.894115	+	0.447838i	$0.147806\pi$
$920$	0	0
$921$	−2.53596e7	−0.985129
$922$	0	0
$923$	−3.99625e7	−1.54400
$924$	0	0
$925$	4.11125e6	0.157986
$926$	0	0
$927$	−1.64276e7	−0.627878
$928$	0	0
$929$	2.46947e7	0.938782	0.469391	−	0.882990i	$-0.344474\pi$
$929$	2.46947e7	0.938782	0.469391	+	0.882990i	$0.344474\pi$
$930$	0	0
$931$	−619458.	−0.0234227
$932$	0	0
$933$	7.62658e6	0.286831
$934$	0	0
$935$	8.47578e6	0.317067
$936$	0	0
$937$	−1.98926e7	−0.740187	−0.370094	−	0.928994i	$-0.620674\pi$
$937$	−1.98926e7	−0.740187	−0.370094	+	0.928994i	$0.620674\pi$
$938$	0	0
$939$	3.28460e6	0.121568
$940$	0	0
$941$	−3.73454e7	−1.37488	−0.687438	−	0.726243i	$-0.741265\pi$
$941$	−3.73454e7	−1.37488	−0.687438	+	0.726243i	$0.741265\pi$
$942$	0	0
$943$	−1.98277e7	−0.726095
$944$	0	0
$945$	−4.46512e6	−0.162650
$946$	0	0
$947$	−5.10396e7	−1.84941	−0.924703	−	0.380689i	$-0.875687\pi$
$947$	−5.10396e7	−1.84941	−0.924703	+	0.380689i	$0.875687\pi$
$948$	0	0
$949$	3.79147e7	1.36660
$950$	0	0
$951$	1.74369e7	0.625200
$952$	0	0
$953$	254832.	0.00908912	0.00454456	−	0.999990i	$-0.498553\pi$
$953$	254832.	0.00908912	0.00454456	+	0.999990i	$0.498553\pi$
$954$	0	0
$955$	−1.35092e7	−0.479316
$956$	0	0
$957$	2.18622e6	0.0771638
$958$	0	0
$959$	1.55702e6	0.0546700
$960$	0	0
$961$	−2.49581e7	−0.871772
$962$	0	0
$963$	−2.68923e7	−0.934464
$964$	0	0
$965$	2.40080e7	0.829923
$966$	0	0
$967$	−1.17012e7	−0.402405	−0.201203	−	0.979550i	$-0.564485\pi$
$967$	−1.17012e7	−0.402405	−0.201203	+	0.979550i	$0.564485\pi$
$968$	0	0
$969$	−4.20979e6	−0.144029
$970$	0	0
$971$	−3.59080e7	−1.22220	−0.611101	−	0.791553i	$-0.709273\pi$
$971$	−3.59080e7	−1.22220	−0.611101	+	0.791553i	$0.709273\pi$
$972$	0	0
$973$	9.80794e6	0.332120
$974$	0	0
$975$	3.52688e6	0.118817
$976$	0	0
$977$	−5.50592e7	−1.84541	−0.922706	−	0.385504i	$-0.874028\pi$
$977$	−5.50592e7	−1.84541	−0.922706	+	0.385504i	$0.874028\pi$
$978$	0	0
$979$	8.51972e6	0.284098
$980$	0	0
$981$	−1.34282e6	−0.0445497
$982$	0	0
$983$	1.81317e7	0.598488	0.299244	−	0.954177i	$-0.403266\pi$
$983$	1.81317e7	0.598488	0.299244	+	0.954177i	$0.403266\pi$
$984$	0	0
$985$	1.90486e7	0.625565
$986$	0	0
$987$	9.70244e6	0.317021
$988$	0	0
$989$	9.43866e6	0.306845
$990$	0	0
$991$	2.02908e7	0.656318	0.328159	−	0.944623i	$-0.393572\pi$
$991$	2.02908e7	0.656318	0.328159	+	0.944623i	$0.393572\pi$
$992$	0	0
$993$	553140.	0.0178017
$994$	0	0
$995$	−3.12710e6	−0.100135
$996$	0	0
$997$	4.75390e7	1.51465	0.757325	−	0.653038i	$-0.226506\pi$
$997$	4.75390e7	1.51465	0.757325	+	0.653038i	$0.226506\pi$
$998$	0	0
$999$	−2.39768e7	−0.760112

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
70.6.a.b.1.1	✓	1	4.3	odd	2
350.6.a.l.1.1		1	20.19	odd	2
350.6.c.g.99.1		2	20.7	even	4
350.6.c.g.99.2		2	20.3	even	4
490.6.a.g.1.1		1	28.27	even	2
560.6.a.e.1.1		1	1.1	even	1	trivial
630.6.a.i.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 $\F_{q}$

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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