LMFDB - Embedded newform 280.6.a.b.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [280,6,Mod(1,280)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(280, 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("280.1");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 280 = 2^{3} \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	280.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:	$44.9074695476$
Analytic rank:	$1$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	280.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q+4.00000 q^{3} +25.0000 q^{5} -49.0000 q^{7} -227.000 q^{9} +O(q^{10})$

$q+4.00000 q^{3} +25.0000 q^{5} -49.0000 q^{7} -227.000 q^{9} +124.000 q^{11} +766.000 q^{13} +100.000 q^{15} -1102.00 q^{17} -764.000 q^{19} -196.000 q^{21} +168.000 q^{23} +625.000 q^{25} -1880.00 q^{27} -6866.00 q^{29} -4096.00 q^{31} +496.000 q^{33} -1225.00 q^{35} -4682.00 q^{37} +3064.00 q^{39} +13130.0 q^{41} +18220.0 q^{43} -5675.00 q^{45} -7104.00 q^{47} +2401.00 q^{49} -4408.00 q^{51} -20026.0 q^{53} +3100.00 q^{55} -3056.00 q^{57} -38964.0 q^{59} -56274.0 q^{61} +11123.0 q^{63} +19150.0 q^{65} -24060.0 q^{67} +672.000 q^{69} -31896.0 q^{71} -23670.0 q^{73} +2500.00 q^{75} -6076.00 q^{77} +37744.0 q^{79} +47641.0 q^{81} -68204.0 q^{83} -27550.0 q^{85} -27464.0 q^{87} -19078.0 q^{89} -37534.0 q^{91} -16384.0 q^{93} -19100.0 q^{95} -115646. q^{97} -28148.0 q^{99} +O(q^{100})$

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
$3$	4.00000	0.256600	0.128300	−	0.991735i	$-0.459048\pi$
$3$	4.00000	0.256600	0.128300	+	0.991735i	$0.459048\pi$
$4$	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
$9$	−227.000	−0.934156
$10$	0	0
$11$	124.000	0.308987	0.154493	−	0.987994i	$-0.450625\pi$
$11$	124.000	0.308987	0.154493	+	0.987994i	$0.450625\pi$
$12$	0	0
$13$	766.000	1.25710	0.628551	−	0.777769i	$-0.283648\pi$
$13$	766.000	1.25710	0.628551	+	0.777769i	$0.283648\pi$
$14$	0	0
$15$	100.000	0.114755
$16$	0	0
$17$	−1102.00	−0.924824	−0.462412	−	0.886665i	$-0.653016\pi$
$17$	−1102.00	−0.924824	−0.462412	+	0.886665i	$0.653016\pi$
$18$	0	0
$19$	−764.000	−0.485522	−0.242761	−	0.970086i	$-0.578053\pi$
$19$	−764.000	−0.485522	−0.242761	+	0.970086i	$0.578053\pi$
$20$	0	0
$21$	−196.000	−0.0969857
$22$	0	0
$23$	168.000	0.0662201	0.0331100	−	0.999452i	$-0.489459\pi$
$23$	168.000	0.0662201	0.0331100	+	0.999452i	$0.489459\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−1880.00	−0.496305
$28$	0	0
$29$	−6866.00	−1.51603	−0.758017	−	0.652235i	$-0.773832\pi$
$29$	−6866.00	−1.51603	−0.758017	+	0.652235i	$0.773832\pi$
$30$	0	0
$31$	−4096.00	−0.765519	−0.382759	−	0.923848i	$-0.625026\pi$
$31$	−4096.00	−0.765519	−0.382759	+	0.923848i	$0.625026\pi$
$32$	0	0
$33$	496.000	0.0792861
$34$	0	0
$35$	−1225.00	−0.169031
$36$	0	0
$37$	−4682.00	−0.562247	−0.281123	−	0.959672i	$-0.590707\pi$
$37$	−4682.00	−0.562247	−0.281123	+	0.959672i	$0.590707\pi$
$38$	0	0
$39$	3064.00	0.322572
$40$	0	0
$41$	13130.0	1.21985	0.609923	−	0.792461i	$-0.291200\pi$
$41$	13130.0	1.21985	0.609923	+	0.792461i	$0.291200\pi$
$42$	0	0
$43$	18220.0	1.50272	0.751359	−	0.659894i	$-0.229399\pi$
$43$	18220.0	1.50272	0.751359	+	0.659894i	$0.229399\pi$
$44$	0	0
$45$	−5675.00	−0.417767
$46$	0	0
$47$	−7104.00	−0.469092	−0.234546	−	0.972105i	$-0.575360\pi$
$47$	−7104.00	−0.469092	−0.234546	+	0.972105i	$0.575360\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	−4408.00	−0.237310
$52$	0	0
$53$	−20026.0	−0.979275	−0.489637	−	0.871926i	$-0.662871\pi$
$53$	−20026.0	−0.979275	−0.489637	+	0.871926i	$0.662871\pi$
$54$	0	0
$55$	3100.00	0.138183
$56$	0	0
$57$	−3056.00	−0.124585
$58$	0	0
$59$	−38964.0	−1.45725	−0.728624	−	0.684914i	$-0.759840\pi$
$59$	−38964.0	−1.45725	−0.728624	+	0.684914i	$0.759840\pi$
$60$	0	0
$61$	−56274.0	−1.93635	−0.968174	−	0.250280i	$-0.919477\pi$
$61$	−56274.0	−1.93635	−0.968174	+	0.250280i	$0.919477\pi$
$62$	0	0
$63$	11123.0	0.353078
$64$	0	0
$65$	19150.0	0.562193
$66$	0	0
$67$	−24060.0	−0.654800	−0.327400	−	0.944886i	$-0.606172\pi$
$67$	−24060.0	−0.654800	−0.327400	+	0.944886i	$0.606172\pi$
$68$	0	0
$69$	672.000	0.0169921
$70$	0	0
$71$	−31896.0	−0.750914	−0.375457	−	0.926840i	$-0.622514\pi$
$71$	−31896.0	−0.750914	−0.375457	+	0.926840i	$0.622514\pi$
$72$	0	0
$73$	−23670.0	−0.519866	−0.259933	−	0.965627i	$-0.583700\pi$
$73$	−23670.0	−0.519866	−0.259933	+	0.965627i	$0.583700\pi$
$74$	0	0
$75$	2500.00	0.0513200
$76$	0	0
$77$	−6076.00	−0.116786
$78$	0	0
$79$	37744.0	0.680425	0.340212	−	0.940349i	$-0.389501\pi$
$79$	37744.0	0.680425	0.340212	+	0.940349i	$0.389501\pi$
$80$	0	0
$81$	47641.0	0.806805
$82$	0	0
$83$	−68204.0	−1.08671	−0.543356	−	0.839502i	$-0.682847\pi$
$83$	−68204.0	−1.08671	−0.543356	+	0.839502i	$0.682847\pi$
$84$	0	0
$85$	−27550.0	−0.413594
$86$	0	0
$87$	−27464.0	−0.389014
$88$	0	0
$89$	−19078.0	−0.255304	−0.127652	−	0.991819i	$-0.540744\pi$
$89$	−19078.0	−0.255304	−0.127652	+	0.991819i	$0.540744\pi$
$90$	0	0
$91$	−37534.0	−0.475140
$92$	0	0
$93$	−16384.0	−0.196432
$94$	0	0
$95$	−19100.0	−0.217132
$96$	0	0
$97$	−115646.	−1.24796	−0.623981	−	0.781440i	$-0.714486\pi$
$97$	−115646.	−1.24796	−0.623981	+	0.781440i	$0.714486\pi$
$98$	0	0
$99$	−28148.0	−0.288642
$100$	0	0
$101$	100790.	0.983137	0.491568	−	0.870839i	$-0.336424\pi$
$101$	100790.	0.983137	0.491568	+	0.870839i	$0.336424\pi$
$102$	0	0
$103$	13304.0	0.123563	0.0617816	−	0.998090i	$-0.480322\pi$
$103$	13304.0	0.123563	0.0617816	+	0.998090i	$0.480322\pi$
$104$	0	0
$105$	−4900.00	−0.0433733
$106$	0	0
$107$	128204.	1.08254	0.541268	−	0.840850i	$-0.317945\pi$
$107$	128204.	1.08254	0.541268	+	0.840850i	$0.317945\pi$
$108$	0	0
$109$	−176450.	−1.42251	−0.711255	−	0.702934i	$-0.751873\pi$
$109$	−176450.	−1.42251	−0.711255	+	0.702934i	$0.751873\pi$
$110$	0	0
$111$	−18728.0	−0.144273
$112$	0	0
$113$	−166734.	−1.22837	−0.614183	−	0.789163i	$-0.710514\pi$
$113$	−166734.	−1.22837	−0.614183	+	0.789163i	$0.710514\pi$
$114$	0	0
$115$	4200.00	0.0296145
$116$	0	0
$117$	−173882.	−1.17433
$118$	0	0
$119$	53998.0	0.349551
$120$	0	0
$121$	−145675.	−0.904527
$122$	0	0
$123$	52520.0	0.313013
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	228112.	1.25499	0.627493	−	0.778622i	$-0.284081\pi$
$127$	228112.	1.25499	0.627493	+	0.778622i	$0.284081\pi$
$128$	0	0
$129$	72880.0	0.385597
$130$	0	0
$131$	131796.	0.671002	0.335501	−	0.942040i	$-0.391094\pi$
$131$	131796.	0.671002	0.335501	+	0.942040i	$0.391094\pi$
$132$	0	0
$133$	37436.0	0.183510
$134$	0	0
$135$	−47000.0	−0.221954
$136$	0	0
$137$	−187862.	−0.855141	−0.427570	−	0.903982i	$-0.640630\pi$
$137$	−187862.	−0.855141	−0.427570	+	0.903982i	$0.640630\pi$
$138$	0	0
$139$	−106404.	−0.467112	−0.233556	−	0.972343i	$-0.575036\pi$
$139$	−106404.	−0.467112	−0.233556	+	0.972343i	$0.575036\pi$
$140$	0	0
$141$	−28416.0	−0.120369
$142$	0	0
$143$	94984.0	0.388428
$144$	0	0
$145$	−171650.	−0.677991
$146$	0	0
$147$	9604.00	0.0366572
$148$	0	0
$149$	−177466.	−0.654862	−0.327431	−	0.944875i	$-0.606183\pi$
$149$	−177466.	−0.654862	−0.327431	+	0.944875i	$0.606183\pi$
$150$	0	0
$151$	117048.	0.417755	0.208877	−	0.977942i	$-0.433019\pi$
$151$	117048.	0.417755	0.208877	+	0.977942i	$0.433019\pi$
$152$	0	0
$153$	250154.	0.863931
$154$	0	0
$155$	−102400.	−0.342350
$156$	0	0
$157$	−31986.0	−0.103564	−0.0517822	−	0.998658i	$-0.516490\pi$
$157$	−31986.0	−0.103564	−0.0517822	+	0.998658i	$0.516490\pi$
$158$	0	0
$159$	−80104.0	−0.251282
$160$	0	0
$161$	−8232.00	−0.0250288
$162$	0	0
$163$	37060.0	0.109254	0.0546269	−	0.998507i	$-0.482603\pi$
$163$	37060.0	0.109254	0.0546269	+	0.998507i	$0.482603\pi$
$164$	0	0
$165$	12400.0	0.0354578
$166$	0	0
$167$	185880.	0.515753	0.257876	−	0.966178i	$-0.416977\pi$
$167$	185880.	0.515753	0.257876	+	0.966178i	$0.416977\pi$
$168$	0	0
$169$	215463.	0.580305
$170$	0	0
$171$	173428.	0.453554
$172$	0	0
$173$	464990.	1.18121	0.590607	−	0.806960i	$-0.298888\pi$
$173$	464990.	1.18121	0.590607	+	0.806960i	$0.298888\pi$
$174$	0	0
$175$	−30625.0	−0.0755929
$176$	0	0
$177$	−155856.	−0.373930
$178$	0	0
$179$	835300.	1.94854	0.974271	−	0.225378i	$-0.0723618\pi$
$179$	835300.	1.94854	0.974271	+	0.225378i	$0.0723618\pi$
$180$	0	0
$181$	−498682.	−1.13143	−0.565714	−	0.824601i	$-0.691400\pi$
$181$	−498682.	−1.13143	−0.565714	+	0.824601i	$0.691400\pi$
$182$	0	0
$183$	−225096.	−0.496867
$184$	0	0
$185$	−117050.	−0.251444
$186$	0	0
$187$	−136648.	−0.285759
$188$	0	0
$189$	92120.0	0.187586
$190$	0	0
$191$	295168.	0.585445	0.292722	−	0.956197i	$-0.405439\pi$
$191$	295168.	0.585445	0.292722	+	0.956197i	$0.405439\pi$
$192$	0	0
$193$	77506.0	0.149776	0.0748880	−	0.997192i	$-0.476140\pi$
$193$	77506.0	0.149776	0.0748880	+	0.997192i	$0.476140\pi$
$194$	0	0
$195$	76600.0	0.144259
$196$	0	0
$197$	310486.	0.570002	0.285001	−	0.958527i	$-0.408006\pi$
$197$	310486.	0.570002	0.285001	+	0.958527i	$0.408006\pi$
$198$	0	0
$199$	639080.	1.14399	0.571995	−	0.820257i	$-0.306170\pi$
$199$	639080.	1.14399	0.571995	+	0.820257i	$0.306170\pi$
$200$	0	0
$201$	−96240.0	−0.168022
$202$	0	0
$203$	336434.	0.573007
$204$	0	0
$205$	328250.	0.545532
$206$	0	0
$207$	−38136.0	−0.0618599
$208$	0	0
$209$	−94736.0	−0.150020
$210$	0	0
$211$	985636.	1.52409	0.762045	−	0.647524i	$-0.224196\pi$
$211$	985636.	1.52409	0.762045	+	0.647524i	$0.224196\pi$
$212$	0	0
$213$	−127584.	−0.192685
$214$	0	0
$215$	455500.	0.672036
$216$	0	0
$217$	200704.	0.289339
$218$	0	0
$219$	−94680.0	−0.133398
$220$	0	0
$221$	−844132.	−1.16260
$222$	0	0
$223$	1.26920e6	1.70910	0.854551	−	0.519368i	$-0.173833\pi$
$223$	1.26920e6	1.70910	0.854551	+	0.519368i	$0.173833\pi$
$224$	0	0
$225$	−141875.	−0.186831
$226$	0	0
$227$	−1.01843e6	−1.31179	−0.655897	−	0.754850i	$-0.727709\pi$
$227$	−1.01843e6	−1.31179	−0.655897	+	0.754850i	$0.727709\pi$
$228$	0	0
$229$	442358.	0.557423	0.278712	−	0.960375i	$-0.410093\pi$
$229$	442358.	0.557423	0.278712	+	0.960375i	$0.410093\pi$
$230$	0	0
$231$	−24304.0	−0.0299673
$232$	0	0
$233$	396810.	0.478843	0.239421	−	0.970916i	$-0.423042\pi$
$233$	396810.	0.478843	0.239421	+	0.970916i	$0.423042\pi$
$234$	0	0
$235$	−177600.	−0.209784
$236$	0	0
$237$	150976.	0.174597
$238$	0	0
$239$	649872.	0.735924	0.367962	−	0.929841i	$-0.380056\pi$
$239$	649872.	0.735924	0.367962	+	0.929841i	$0.380056\pi$
$240$	0	0
$241$	716082.	0.794182	0.397091	−	0.917779i	$-0.370020\pi$
$241$	716082.	0.794182	0.397091	+	0.917779i	$0.370020\pi$
$242$	0	0
$243$	647404.	0.703331
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−585224.	−0.610351
$248$	0	0
$249$	−272816.	−0.278851
$250$	0	0
$251$	−411860.	−0.412634	−0.206317	−	0.978485i	$-0.566148\pi$
$251$	−411860.	−0.412634	−0.206317	+	0.978485i	$0.566148\pi$
$252$	0	0
$253$	20832.0	0.0204611
$254$	0	0
$255$	−110200.	−0.106128
$256$	0	0
$257$	−1.51856e6	−1.43416	−0.717082	−	0.696989i	$-0.754523\pi$
$257$	−1.51856e6	−1.43416	−0.717082	+	0.696989i	$0.754523\pi$
$258$	0	0
$259$	229418.	0.212509
$260$	0	0
$261$	1.55858e6	1.41621
$262$	0	0
$263$	−1.77546e6	−1.58279	−0.791394	−	0.611307i	$-0.790644\pi$
$263$	−1.77546e6	−1.58279	−0.791394	+	0.611307i	$0.790644\pi$
$264$	0	0
$265$	−500650.	−0.437945
$266$	0	0
$267$	−76312.0	−0.0655111
$268$	0	0
$269$	550014.	0.463439	0.231720	−	0.972783i	$-0.425565\pi$
$269$	550014.	0.463439	0.231720	+	0.972783i	$0.425565\pi$
$270$	0	0
$271$	−64144.0	−0.0530558	−0.0265279	−	0.999648i	$-0.508445\pi$
$271$	−64144.0	−0.0530558	−0.0265279	+	0.999648i	$0.508445\pi$
$272$	0	0
$273$	−150136.	−0.121921
$274$	0	0
$275$	77500.0	0.0617974
$276$	0	0
$277$	112998.	0.0884853	0.0442427	−	0.999021i	$-0.485913\pi$
$277$	112998.	0.0884853	0.0442427	+	0.999021i	$0.485913\pi$
$278$	0	0
$279$	929792.	0.715114
$280$	0	0
$281$	606330.	0.458082	0.229041	−	0.973417i	$-0.426441\pi$
$281$	606330.	0.458082	0.229041	+	0.973417i	$0.426441\pi$
$282$	0	0
$283$	−565444.	−0.419685	−0.209843	−	0.977735i	$-0.567295\pi$
$283$	−565444.	−0.419685	−0.209843	+	0.977735i	$0.567295\pi$
$284$	0	0
$285$	−76400.0	−0.0557162
$286$	0	0
$287$	−643370.	−0.461059
$288$	0	0
$289$	−205453.	−0.144700
$290$	0	0
$291$	−462584.	−0.320227
$292$	0	0
$293$	−726474.	−0.494369	−0.247184	−	0.968968i	$-0.579505\pi$
$293$	−726474.	−0.494369	−0.247184	+	0.968968i	$0.579505\pi$
$294$	0	0
$295$	−974100.	−0.651701
$296$	0	0
$297$	−233120.	−0.153352
$298$	0	0
$299$	128688.	0.0832454
$300$	0	0
$301$	−892780.	−0.567974
$302$	0	0
$303$	403160.	0.252273
$304$	0	0
$305$	−1.40685e6	−0.865961
$306$	0	0
$307$	1.69346e6	1.02548	0.512742	−	0.858543i	$-0.328630\pi$
$307$	1.69346e6	1.02548	0.512742	+	0.858543i	$0.328630\pi$
$308$	0	0
$309$	53216.0	0.0317063
$310$	0	0
$311$	2.52684e6	1.48142	0.740708	−	0.671828i	$-0.234490\pi$
$311$	2.52684e6	1.48142	0.740708	+	0.671828i	$0.234490\pi$
$312$	0	0
$313$	1.00966e6	0.582523	0.291262	−	0.956643i	$-0.405925\pi$
$313$	1.00966e6	0.582523	0.291262	+	0.956643i	$0.405925\pi$
$314$	0	0
$315$	278075.	0.157901
$316$	0	0
$317$	−1.14792e6	−0.641600	−0.320800	−	0.947147i	$-0.603952\pi$
$317$	−1.14792e6	−0.641600	−0.320800	+	0.947147i	$0.603952\pi$
$318$	0	0
$319$	−851384.	−0.468434
$320$	0	0
$321$	512816.	0.277779
$322$	0	0
$323$	841928.	0.449023
$324$	0	0
$325$	478750.	0.251420
$326$	0	0
$327$	−705800.	−0.365016
$328$	0	0
$329$	348096.	0.177300
$330$	0	0
$331$	−3.02349e6	−1.51684	−0.758418	−	0.651768i	$-0.774028\pi$
$331$	−3.02349e6	−1.51684	−0.758418	+	0.651768i	$0.774028\pi$
$332$	0	0
$333$	1.06281e6	0.525227
$334$	0	0
$335$	−601500.	−0.292835
$336$	0	0
$337$	−763054.	−0.366000	−0.183000	−	0.983113i	$-0.558581\pi$
$337$	−763054.	−0.366000	−0.183000	+	0.983113i	$0.558581\pi$
$338$	0	0
$339$	−666936.	−0.315199
$340$	0	0
$341$	−507904.	−0.236535
$342$	0	0
$343$	−117649.	−0.0539949
$344$	0	0
$345$	16800.0	0.00759909
$346$	0	0
$347$	−3.93575e6	−1.75470	−0.877351	−	0.479848i	$-0.840692\pi$
$347$	−3.93575e6	−1.75470	−0.877351	+	0.479848i	$0.840692\pi$
$348$	0	0
$349$	4.28424e6	1.88283	0.941413	−	0.337257i	$-0.109499\pi$
$349$	4.28424e6	1.88283	0.941413	+	0.337257i	$0.109499\pi$
$350$	0	0
$351$	−1.44008e6	−0.623906
$352$	0	0
$353$	603906.	0.257948	0.128974	−	0.991648i	$-0.458832\pi$
$353$	603906.	0.257948	0.128974	+	0.991648i	$0.458832\pi$
$354$	0	0
$355$	−797400.	−0.335819
$356$	0	0
$357$	215992.	0.0896948
$358$	0	0
$359$	2.78858e6	1.14195	0.570976	−	0.820967i	$-0.306565\pi$
$359$	2.78858e6	1.14195	0.570976	+	0.820967i	$0.306565\pi$
$360$	0	0
$361$	−1.89240e6	−0.764268
$362$	0	0
$363$	−582700.	−0.232102
$364$	0	0
$365$	−591750.	−0.232491
$366$	0	0
$367$	−3.04883e6	−1.18159	−0.590797	−	0.806820i	$-0.701187\pi$
$367$	−3.04883e6	−1.18159	−0.590797	+	0.806820i	$0.701187\pi$
$368$	0	0
$369$	−2.98051e6	−1.13953
$370$	0	0
$371$	981274.	0.370131
$372$	0	0
$373$	−1.57465e6	−0.586019	−0.293010	−	0.956109i	$-0.594657\pi$
$373$	−1.57465e6	−0.586019	−0.293010	+	0.956109i	$0.594657\pi$
$374$	0	0
$375$	62500.0	0.0229510
$376$	0	0
$377$	−5.25936e6	−1.90581
$378$	0	0
$379$	4.00929e6	1.43374	0.716869	−	0.697208i	$-0.245575\pi$
$379$	4.00929e6	1.43374	0.716869	+	0.697208i	$0.245575\pi$
$380$	0	0
$381$	912448.	0.322030
$382$	0	0
$383$	−3.44590e6	−1.20035	−0.600173	−	0.799870i	$-0.704901\pi$
$383$	−3.44590e6	−1.20035	−0.600173	+	0.799870i	$0.704901\pi$
$384$	0	0
$385$	−151900.	−0.0522283
$386$	0	0
$387$	−4.13594e6	−1.40377
$388$	0	0
$389$	1.09125e6	0.365638	0.182819	−	0.983147i	$-0.441478\pi$
$389$	1.09125e6	0.365638	0.182819	+	0.983147i	$0.441478\pi$
$390$	0	0
$391$	−185136.	−0.0612419
$392$	0	0
$393$	527184.	0.172179
$394$	0	0
$395$	943600.	0.304295
$396$	0	0
$397$	68990.0	0.0219690	0.0109845	−	0.999940i	$-0.496503\pi$
$397$	68990.0	0.0219690	0.0109845	+	0.999940i	$0.496503\pi$
$398$	0	0
$399$	149744.	0.0470888
$400$	0	0
$401$	249938.	0.0776196	0.0388098	−	0.999247i	$-0.487643\pi$
$401$	249938.	0.0776196	0.0388098	+	0.999247i	$0.487643\pi$
$402$	0	0
$403$	−3.13754e6	−0.962335
$404$	0	0
$405$	1.19102e6	0.360814
$406$	0	0
$407$	−580568.	−0.173727
$408$	0	0
$409$	1.21836e6	0.360137	0.180069	−	0.983654i	$-0.442368\pi$
$409$	1.21836e6	0.360137	0.180069	+	0.983654i	$0.442368\pi$
$410$	0	0
$411$	−751448.	−0.219429
$412$	0	0
$413$	1.90924e6	0.550788
$414$	0	0
$415$	−1.70510e6	−0.485993
$416$	0	0
$417$	−425616.	−0.119861
$418$	0	0
$419$	−1.21502e6	−0.338102	−0.169051	−	0.985607i	$-0.554070\pi$
$419$	−1.21502e6	−0.338102	−0.169051	+	0.985607i	$0.554070\pi$
$420$	0	0
$421$	882646.	0.242706	0.121353	−	0.992609i	$-0.461277\pi$
$421$	882646.	0.242706	0.121353	+	0.992609i	$0.461277\pi$
$422$	0	0
$423$	1.61261e6	0.438206
$424$	0	0
$425$	−688750.	−0.184965
$426$	0	0
$427$	2.75743e6	0.731870
$428$	0	0
$429$	379936.	0.0996706
$430$	0	0
$431$	−6.95086e6	−1.80238	−0.901188	−	0.433427i	$-0.857304\pi$
$431$	−6.95086e6	−1.80238	−0.901188	+	0.433427i	$0.857304\pi$
$432$	0	0
$433$	7.09723e6	1.81915	0.909577	−	0.415536i	$-0.136406\pi$
$433$	7.09723e6	1.81915	0.909577	+	0.415536i	$0.136406\pi$
$434$	0	0
$435$	−686600.	−0.173973
$436$	0	0
$437$	−128352.	−0.0321513
$438$	0	0
$439$	4.40444e6	1.09076	0.545380	−	0.838189i	$-0.316385\pi$
$439$	4.40444e6	1.09076	0.545380	+	0.838189i	$0.316385\pi$
$440$	0	0
$441$	−545027.	−0.133451
$442$	0	0
$443$	−2.59066e6	−0.627193	−0.313596	−	0.949556i	$-0.601534\pi$
$443$	−2.59066e6	−0.627193	−0.313596	+	0.949556i	$0.601534\pi$
$444$	0	0
$445$	−476950.	−0.114175
$446$	0	0
$447$	−709864.	−0.168038
$448$	0	0
$449$	1.61888e6	0.378965	0.189483	−	0.981884i	$-0.439319\pi$
$449$	1.61888e6	0.378965	0.189483	+	0.981884i	$0.439319\pi$
$450$	0	0
$451$	1.62812e6	0.376916
$452$	0	0
$453$	468192.	0.107196
$454$	0	0
$455$	−938350.	−0.212489
$456$	0	0
$457$	−7.32412e6	−1.64046	−0.820228	−	0.572036i	$-0.806154\pi$
$457$	−7.32412e6	−1.64046	−0.820228	+	0.572036i	$0.806154\pi$
$458$	0	0
$459$	2.07176e6	0.458995
$460$	0	0
$461$	5.86086e6	1.28443	0.642213	−	0.766526i	$-0.278016\pi$
$461$	5.86086e6	1.28443	0.642213	+	0.766526i	$0.278016\pi$
$462$	0	0
$463$	−1.15974e6	−0.251426	−0.125713	−	0.992067i	$-0.540122\pi$
$463$	−1.15974e6	−0.251426	−0.125713	+	0.992067i	$0.540122\pi$
$464$	0	0
$465$	−409600.	−0.0878471
$466$	0	0
$467$	−6.37889e6	−1.35348	−0.676742	−	0.736220i	$-0.736609\pi$
$467$	−6.37889e6	−1.35348	−0.676742	+	0.736220i	$0.736609\pi$
$468$	0	0
$469$	1.17894e6	0.247491
$470$	0	0
$471$	−127944.	−0.0265747
$472$	0	0
$473$	2.25928e6	0.464320
$474$	0	0
$475$	−477500.	−0.0971045
$476$	0	0
$477$	4.54590e6	0.914796
$478$	0	0
$479$	−2.52893e6	−0.503614	−0.251807	−	0.967777i	$-0.581025\pi$
$479$	−2.52893e6	−0.503614	−0.251807	+	0.967777i	$0.581025\pi$
$480$	0	0
$481$	−3.58641e6	−0.706802
$482$	0	0
$483$	−32928.0	−0.00642240
$484$	0	0
$485$	−2.89115e6	−0.558105
$486$	0	0
$487$	−991272.	−0.189396	−0.0946979	−	0.995506i	$-0.530189\pi$
$487$	−991272.	−0.189396	−0.0946979	+	0.995506i	$0.530189\pi$
$488$	0	0
$489$	148240.	0.0280345
$490$	0	0
$491$	−6.58506e6	−1.23270	−0.616348	−	0.787474i	$-0.711389\pi$
$491$	−6.58506e6	−1.23270	−0.616348	+	0.787474i	$0.711389\pi$
$492$	0	0
$493$	7.56633e6	1.40206
$494$	0	0
$495$	−703700.	−0.129085
$496$	0	0
$497$	1.56290e6	0.283819
$498$	0	0
$499$	6.39831e6	1.15031	0.575154	−	0.818045i	$-0.304942\pi$
$499$	6.39831e6	1.15031	0.575154	+	0.818045i	$0.304942\pi$
$500$	0	0
$501$	743520.	0.132342
$502$	0	0
$503$	1.91546e6	0.337562	0.168781	−	0.985654i	$-0.446017\pi$
$503$	1.91546e6	0.337562	0.168781	+	0.985654i	$0.446017\pi$
$504$	0	0
$505$	2.51975e6	0.439672
$506$	0	0
$507$	861852.	0.148906
$508$	0	0
$509$	−2.72437e6	−0.466092	−0.233046	−	0.972466i	$-0.574869\pi$
$509$	−2.72437e6	−0.466092	−0.233046	+	0.972466i	$0.574869\pi$
$510$	0	0
$511$	1.15983e6	0.196491
$512$	0	0
$513$	1.43632e6	0.240967
$514$	0	0
$515$	332600.	0.0552591
$516$	0	0
$517$	−880896.	−0.144943
$518$	0	0
$519$	1.85996e6	0.303099
$520$	0	0
$521$	4.28260e6	0.691215	0.345608	−	0.938379i	$-0.387673\pi$
$521$	4.28260e6	0.691215	0.345608	+	0.938379i	$0.387673\pi$
$522$	0	0
$523$	3.61035e6	0.577158	0.288579	−	0.957456i	$-0.406817\pi$
$523$	3.61035e6	0.577158	0.288579	+	0.957456i	$0.406817\pi$
$524$	0	0
$525$	−122500.	−0.0193971
$526$	0	0
$527$	4.51379e6	0.707970
$528$	0	0
$529$	−6.40812e6	−0.995615
$530$	0	0
$531$	8.84483e6	1.36130
$532$	0	0
$533$	1.00576e7	1.53347
$534$	0	0
$535$	3.20510e6	0.484125
$536$	0	0
$537$	3.34120e6	0.499996
$538$	0	0
$539$	297724.	0.0441410
$540$	0	0
$541$	1.17755e7	1.72977	0.864883	−	0.501973i	$-0.167392\pi$
$541$	1.17755e7	1.72977	0.864883	+	0.501973i	$0.167392\pi$
$542$	0	0
$543$	−1.99473e6	−0.290325
$544$	0	0
$545$	−4.41125e6	−0.636166
$546$	0	0
$547$	−1.07968e7	−1.54286	−0.771428	−	0.636317i	$-0.780457\pi$
$547$	−1.07968e7	−1.54286	−0.771428	+	0.636317i	$0.780457\pi$
$548$	0	0
$549$	1.27742e7	1.80885
$550$	0	0
$551$	5.24562e6	0.736068
$552$	0	0
$553$	−1.84946e6	−0.257176
$554$	0	0
$555$	−468200.	−0.0645207
$556$	0	0
$557$	−1.11383e7	−1.52118	−0.760589	−	0.649234i	$-0.775090\pi$
$557$	−1.11383e7	−1.52118	−0.760589	+	0.649234i	$0.775090\pi$
$558$	0	0
$559$	1.39565e7	1.88907
$560$	0	0
$561$	−546592.	−0.0733257
$562$	0	0
$563$	4.49836e6	0.598112	0.299056	−	0.954235i	$-0.403328\pi$
$563$	4.49836e6	0.598112	0.299056	+	0.954235i	$0.403328\pi$
$564$	0	0
$565$	−4.16835e6	−0.549342
$566$	0	0
$567$	−2.33441e6	−0.304943
$568$	0	0
$569$	−5.14929e6	−0.666755	−0.333378	−	0.942793i	$-0.608188\pi$
$569$	−5.14929e6	−0.666755	−0.333378	+	0.942793i	$0.608188\pi$
$570$	0	0
$571$	1.95544e6	0.250988	0.125494	−	0.992094i	$-0.459948\pi$
$571$	1.95544e6	0.250988	0.125494	+	0.992094i	$0.459948\pi$
$572$	0	0
$573$	1.18067e6	0.150225
$574$	0	0
$575$	105000.	0.0132440
$576$	0	0
$577$	2.50179e6	0.312833	0.156416	−	0.987691i	$-0.450006\pi$
$577$	2.50179e6	0.312833	0.156416	+	0.987691i	$0.450006\pi$
$578$	0	0
$579$	310024.	0.0384325
$580$	0	0
$581$	3.34200e6	0.410739
$582$	0	0
$583$	−2.48322e6	−0.302583
$584$	0	0
$585$	−4.34705e6	−0.525176
$586$	0	0
$587$	1.31352e7	1.57341	0.786703	−	0.617332i	$-0.211786\pi$
$587$	1.31352e7	1.57341	0.786703	+	0.617332i	$0.211786\pi$
$588$	0	0
$589$	3.12934e6	0.371676
$590$	0	0
$591$	1.24194e6	0.146263
$592$	0	0
$593$	−1.61671e7	−1.88797	−0.943983	−	0.329994i	$-0.892953\pi$
$593$	−1.61671e7	−1.88797	−0.943983	+	0.329994i	$0.892953\pi$
$594$	0	0
$595$	1.34995e6	0.156324
$596$	0	0
$597$	2.55632e6	0.293548
$598$	0	0
$599$	−1.39864e7	−1.59272	−0.796359	−	0.604824i	$-0.793244\pi$
$599$	−1.39864e7	−1.59272	−0.796359	+	0.604824i	$0.793244\pi$
$600$	0	0
$601$	5.95609e6	0.672628	0.336314	−	0.941750i	$-0.390820\pi$
$601$	5.95609e6	0.672628	0.336314	+	0.941750i	$0.390820\pi$
$602$	0	0
$603$	5.46162e6	0.611686
$604$	0	0
$605$	−3.64188e6	−0.404517
$606$	0	0
$607$	−1.44142e7	−1.58789	−0.793943	−	0.607992i	$-0.791975\pi$
$607$	−1.44142e7	−1.58789	−0.793943	+	0.607992i	$0.791975\pi$
$608$	0	0
$609$	1.34574e6	0.147034
$610$	0	0
$611$	−5.44166e6	−0.589697
$612$	0	0
$613$	−1.08923e6	−0.117076	−0.0585379	−	0.998285i	$-0.518644\pi$
$613$	−1.08923e6	−0.117076	−0.0585379	+	0.998285i	$0.518644\pi$
$614$	0	0
$615$	1.31300e6	0.139984
$616$	0	0
$617$	−983542.	−0.104011	−0.0520056	−	0.998647i	$-0.516561\pi$
$617$	−983542.	−0.104011	−0.0520056	+	0.998647i	$0.516561\pi$
$618$	0	0
$619$	1.30483e7	1.36875	0.684377	−	0.729128i	$-0.260074\pi$
$619$	1.30483e7	1.36875	0.684377	+	0.729128i	$0.260074\pi$
$620$	0	0
$621$	−315840.	−0.0328653
$622$	0	0
$623$	934822.	0.0964959
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	−378944.	−0.0384952
$628$	0	0
$629$	5.15956e6	0.519980
$630$	0	0
$631$	8.59951e6	0.859806	0.429903	−	0.902875i	$-0.358548\pi$
$631$	8.59951e6	0.859806	0.429903	+	0.902875i	$0.358548\pi$
$632$	0	0
$633$	3.94254e6	0.391082
$634$	0	0
$635$	5.70280e6	0.561247
$636$	0	0
$637$	1.83917e6	0.179586
$638$	0	0
$639$	7.24039e6	0.701471
$640$	0	0
$641$	−7.40288e6	−0.711632	−0.355816	−	0.934556i	$-0.615797\pi$
$641$	−7.40288e6	−0.711632	−0.355816	+	0.934556i	$0.615797\pi$
$642$	0	0
$643$	−7.68038e6	−0.732580	−0.366290	−	0.930501i	$-0.619372\pi$
$643$	−7.68038e6	−0.732580	−0.366290	+	0.930501i	$0.619372\pi$
$644$	0	0
$645$	1.82200e6	0.172444
$646$	0	0
$647$	−2.00321e6	−0.188133	−0.0940666	−	0.995566i	$-0.529987\pi$
$647$	−2.00321e6	−0.188133	−0.0940666	+	0.995566i	$0.529987\pi$
$648$	0	0
$649$	−4.83154e6	−0.450270
$650$	0	0
$651$	802816.	0.0742444
$652$	0	0
$653$	3.64089e6	0.334137	0.167069	−	0.985945i	$-0.446570\pi$
$653$	3.64089e6	0.334137	0.167069	+	0.985945i	$0.446570\pi$
$654$	0	0
$655$	3.29490e6	0.300081
$656$	0	0
$657$	5.37309e6	0.485636
$658$	0	0
$659$	3.02720e6	0.271536	0.135768	−	0.990741i	$-0.456650\pi$
$659$	3.02720e6	0.271536	0.135768	+	0.990741i	$0.456650\pi$
$660$	0	0
$661$	−5.76758e6	−0.513440	−0.256720	−	0.966486i	$-0.582642\pi$
$661$	−5.76758e6	−0.513440	−0.256720	+	0.966486i	$0.582642\pi$
$662$	0	0
$663$	−3.37653e6	−0.298323
$664$	0	0
$665$	935900.	0.0820683
$666$	0	0
$667$	−1.15349e6	−0.100392
$668$	0	0
$669$	5.07680e6	0.438556
$670$	0	0
$671$	−6.97798e6	−0.598306
$672$	0	0
$673$	1.43872e7	1.22444	0.612220	−	0.790687i	$-0.290277\pi$
$673$	1.43872e7	1.22444	0.612220	+	0.790687i	$0.290277\pi$
$674$	0	0
$675$	−1.17500e6	−0.0992610
$676$	0	0
$677$	−4.13262e6	−0.346540	−0.173270	−	0.984874i	$-0.555433\pi$
$677$	−4.13262e6	−0.346540	−0.173270	+	0.984874i	$0.555433\pi$
$678$	0	0
$679$	5.66665e6	0.471685
$680$	0	0
$681$	−4.07371e6	−0.336607
$682$	0	0
$683$	−5.15356e6	−0.422722	−0.211361	−	0.977408i	$-0.567790\pi$
$683$	−5.15356e6	−0.422722	−0.211361	+	0.977408i	$0.567790\pi$
$684$	0	0
$685$	−4.69655e6	−0.382431
$686$	0	0
$687$	1.76943e6	0.143035
$688$	0	0
$689$	−1.53399e7	−1.23105
$690$	0	0
$691$	−1.01299e6	−0.0807066	−0.0403533	−	0.999185i	$-0.512848\pi$
$691$	−1.01299e6	−0.0807066	−0.0403533	+	0.999185i	$0.512848\pi$
$692$	0	0
$693$	1.37925e6	0.109096
$694$	0	0
$695$	−2.66010e6	−0.208899
$696$	0	0
$697$	−1.44693e7	−1.12814
$698$	0	0
$699$	1.58724e6	0.122871
$700$	0	0
$701$	7.19041e6	0.552661	0.276331	−	0.961063i	$-0.410882\pi$
$701$	7.19041e6	0.552661	0.276331	+	0.961063i	$0.410882\pi$
$702$	0	0
$703$	3.57705e6	0.272984
$704$	0	0
$705$	−710400.	−0.0538307
$706$	0	0
$707$	−4.93871e6	−0.371591
$708$	0	0
$709$	−2.41861e7	−1.80697	−0.903484	−	0.428621i	$-0.859000\pi$
$709$	−2.41861e7	−1.80697	−0.903484	+	0.428621i	$0.859000\pi$
$710$	0	0
$711$	−8.56789e6	−0.635623
$712$	0	0
$713$	−688128.	−0.0506927
$714$	0	0
$715$	2.37460e6	0.173710
$716$	0	0
$717$	2.59949e6	0.188838
$718$	0	0
$719$	1.88305e7	1.35844	0.679218	−	0.733937i	$-0.262319\pi$
$719$	1.88305e7	1.35844	0.679218	+	0.733937i	$0.262319\pi$
$720$	0	0
$721$	−651896.	−0.0467025
$722$	0	0
$723$	2.86433e6	0.203787
$724$	0	0
$725$	−4.29125e6	−0.303207
$726$	0	0
$727$	1.42513e7	1.00005	0.500023	−	0.866012i	$-0.333325\pi$
$727$	1.42513e7	1.00005	0.500023	+	0.866012i	$0.333325\pi$
$728$	0	0
$729$	−8.98715e6	−0.626330
$730$	0	0
$731$	−2.00784e7	−1.38975
$732$	0	0
$733$	−945586.	−0.0650042	−0.0325021	−	0.999472i	$-0.510348\pi$
$733$	−945586.	−0.0650042	−0.0325021	+	0.999472i	$0.510348\pi$
$734$	0	0
$735$	240100.	0.0163936
$736$	0	0
$737$	−2.98344e6	−0.202325
$738$	0	0
$739$	−2.01534e6	−0.135749	−0.0678746	−	0.997694i	$-0.521622\pi$
$739$	−2.01534e6	−0.135749	−0.0678746	+	0.997694i	$0.521622\pi$
$740$	0	0
$741$	−2.34090e6	−0.156616
$742$	0	0
$743$	−1.22830e7	−0.816264	−0.408132	−	0.912923i	$-0.633820\pi$
$743$	−1.22830e7	−0.816264	−0.408132	+	0.912923i	$0.633820\pi$
$744$	0	0
$745$	−4.43665e6	−0.292863
$746$	0	0
$747$	1.54823e7	1.01516
$748$	0	0
$749$	−6.28200e6	−0.409160
$750$	0	0
$751$	6.43979e6	0.416651	0.208325	−	0.978060i	$-0.433199\pi$
$751$	6.43979e6	0.416651	0.208325	+	0.978060i	$0.433199\pi$
$752$	0	0
$753$	−1.64744e6	−0.105882
$754$	0	0
$755$	2.92620e6	0.186826
$756$	0	0
$757$	1.08870e7	0.690507	0.345253	−	0.938510i	$-0.387793\pi$
$757$	1.08870e7	0.690507	0.345253	+	0.938510i	$0.387793\pi$
$758$	0	0
$759$	83328.0	0.00525033
$760$	0	0
$761$	−2.57412e7	−1.61126	−0.805632	−	0.592417i	$-0.798174\pi$
$761$	−2.57412e7	−1.61126	−0.805632	+	0.592417i	$0.798174\pi$
$762$	0	0
$763$	8.64605e6	0.537658
$764$	0	0
$765$	6.25385e6	0.386362
$766$	0	0
$767$	−2.98464e7	−1.83191
$768$	0	0
$769$	−2.38746e7	−1.45586	−0.727930	−	0.685652i	$-0.759517\pi$
$769$	−2.38746e7	−1.45586	−0.727930	+	0.685652i	$0.759517\pi$
$770$	0	0
$771$	−6.07423e6	−0.368006
$772$	0	0
$773$	1.70534e7	1.02651	0.513254	−	0.858237i	$-0.328440\pi$
$773$	1.70534e7	1.02651	0.513254	+	0.858237i	$0.328440\pi$
$774$	0	0
$775$	−2.56000e6	−0.153104
$776$	0	0
$777$	917672.	0.0545299
$778$	0	0
$779$	−1.00313e7	−0.592263
$780$	0	0
$781$	−3.95510e6	−0.232023
$782$	0	0
$783$	1.29081e7	0.752415
$784$	0	0
$785$	−799650.	−0.0463154
$786$	0	0
$787$	2.15326e7	1.23925	0.619626	−	0.784897i	$-0.287284\pi$
$787$	2.15326e7	1.23925	0.619626	+	0.784897i	$0.287284\pi$
$788$	0	0
$789$	−7.10186e6	−0.406143
$790$	0	0
$791$	8.16997e6	0.464279
$792$	0	0
$793$	−4.31059e7	−2.43419
$794$	0	0
$795$	−2.00260e6	−0.112377
$796$	0	0
$797$	−7.38450e6	−0.411789	−0.205895	−	0.978574i	$-0.566010\pi$
$797$	−7.38450e6	−0.411789	−0.205895	+	0.978574i	$0.566010\pi$
$798$	0	0
$799$	7.82861e6	0.433828
$800$	0	0
$801$	4.33071e6	0.238494
$802$	0	0
$803$	−2.93508e6	−0.160632
$804$	0	0
$805$	−205800.	−0.0111932
$806$	0	0
$807$	2.20006e6	0.118919
$808$	0	0
$809$	−2.80415e7	−1.50637	−0.753183	−	0.657811i	$-0.771482\pi$
$809$	−2.80415e7	−1.50637	−0.753183	+	0.657811i	$0.771482\pi$
$810$	0	0
$811$	1.07165e7	0.572138	0.286069	−	0.958209i	$-0.407651\pi$
$811$	1.07165e7	0.572138	0.286069	+	0.958209i	$0.407651\pi$
$812$	0	0
$813$	−256576.	−0.0136141
$814$	0	0
$815$	926500.	0.0488598
$816$	0	0
$817$	−1.39201e7	−0.729603
$818$	0	0
$819$	8.52022e6	0.443855
$820$	0	0
$821$	−2.28481e7	−1.18302	−0.591510	−	0.806298i	$-0.701468\pi$
$821$	−2.28481e7	−1.18302	−0.591510	+	0.806298i	$0.701468\pi$
$822$	0	0
$823$	−1.36644e7	−0.703222	−0.351611	−	0.936146i	$-0.614366\pi$
$823$	−1.36644e7	−0.703222	−0.351611	+	0.936146i	$0.614366\pi$
$824$	0	0
$825$	310000.	0.0158572
$826$	0	0
$827$	−1.59497e7	−0.810938	−0.405469	−	0.914109i	$-0.632892\pi$
$827$	−1.59497e7	−0.810938	−0.405469	+	0.914109i	$0.632892\pi$
$828$	0	0
$829$	−615250.	−0.0310932	−0.0155466	−	0.999879i	$-0.504949\pi$
$829$	−615250.	−0.0310932	−0.0155466	+	0.999879i	$0.504949\pi$
$830$	0	0
$831$	451992.	0.0227053
$832$	0	0
$833$	−2.64590e6	−0.132118
$834$	0	0
$835$	4.64700e6	0.230652
$836$	0	0
$837$	7.70048e6	0.379930
$838$	0	0
$839$	−5.07702e6	−0.249002	−0.124501	−	0.992219i	$-0.539733\pi$
$839$	−5.07702e6	−0.249002	−0.124501	+	0.992219i	$0.539733\pi$
$840$	0	0
$841$	2.66308e7	1.29836
$842$	0	0
$843$	2.42532e6	0.117544
$844$	0	0
$845$	5.38658e6	0.259520
$846$	0	0
$847$	7.13808e6	0.341879
$848$	0	0
$849$	−2.26178e6	−0.107691
$850$	0	0
$851$	−786576.	−0.0372320
$852$	0	0
$853$	2.28325e7	1.07444	0.537218	−	0.843443i	$-0.319475\pi$
$853$	2.28325e7	1.07444	0.537218	+	0.843443i	$0.319475\pi$
$854$	0	0
$855$	4.33570e6	0.202835
$856$	0	0
$857$	1.03857e7	0.483040	0.241520	−	0.970396i	$-0.422354\pi$
$857$	1.03857e7	0.483040	0.241520	+	0.970396i	$0.422354\pi$
$858$	0	0
$859$	−9.66607e6	−0.446958	−0.223479	−	0.974709i	$-0.571741\pi$
$859$	−9.66607e6	−0.446958	−0.223479	+	0.974709i	$0.571741\pi$
$860$	0	0
$861$	−2.57348e6	−0.118308
$862$	0	0
$863$	−2.47160e7	−1.12967	−0.564835	−	0.825204i	$-0.691060\pi$
$863$	−2.47160e7	−1.12967	−0.564835	+	0.825204i	$0.691060\pi$
$864$	0	0
$865$	1.16248e7	0.528255
$866$	0	0
$867$	−821812.	−0.0371300
$868$	0	0
$869$	4.68026e6	0.210242
$870$	0	0
$871$	−1.84300e7	−0.823150
$872$	0	0
$873$	2.62516e7	1.16579
$874$	0	0
$875$	−765625.	−0.0338062
$876$	0	0
$877$	3.87672e7	1.70202	0.851012	−	0.525147i	$-0.175990\pi$
$877$	3.87672e7	1.70202	0.851012	+	0.525147i	$0.175990\pi$
$878$	0	0
$879$	−2.90590e6	−0.126855
$880$	0	0
$881$	1.21677e7	0.528162	0.264081	−	0.964500i	$-0.414931\pi$
$881$	1.21677e7	0.528162	0.264081	+	0.964500i	$0.414931\pi$
$882$	0	0
$883$	−4.54978e7	−1.96376	−0.981881	−	0.189497i	$-0.939314\pi$
$883$	−4.54978e7	−1.96376	−0.981881	+	0.189497i	$0.939314\pi$
$884$	0	0
$885$	−3.89640e6	−0.167227
$886$	0	0
$887$	1.30952e7	0.558860	0.279430	−	0.960166i	$-0.409854\pi$
$887$	1.30952e7	0.558860	0.279430	+	0.960166i	$0.409854\pi$
$888$	0	0
$889$	−1.11775e7	−0.474340
$890$	0	0
$891$	5.90748e6	0.249292
$892$	0	0
$893$	5.42746e6	0.227755
$894$	0	0
$895$	2.08825e7	0.871415
$896$	0	0
$897$	514752.	0.0213608
$898$	0	0
$899$	2.81231e7	1.16055
$900$	0	0
$901$	2.20687e7	0.905657
$902$	0	0
$903$	−3.57112e6	−0.145742
$904$	0	0
$905$	−1.24670e7	−0.505990
$906$	0	0
$907$	9.98552e6	0.403044	0.201522	−	0.979484i	$-0.435411\pi$
$907$	9.98552e6	0.403044	0.201522	+	0.979484i	$0.435411\pi$
$908$	0	0
$909$	−2.28793e7	−0.918404
$910$	0	0
$911$	−1.58347e7	−0.632140	−0.316070	−	0.948736i	$-0.602363\pi$
$911$	−1.58347e7	−0.632140	−0.316070	+	0.948736i	$0.602363\pi$
$912$	0	0
$913$	−8.45730e6	−0.335780
$914$	0	0
$915$	−5.62740e6	−0.222206
$916$	0	0
$917$	−6.45800e6	−0.253615
$918$	0	0
$919$	4.40462e7	1.72036	0.860180	−	0.509991i	$-0.170351\pi$
$919$	4.40462e7	1.72036	0.860180	+	0.509991i	$0.170351\pi$
$920$	0	0
$921$	6.77384e6	0.263139
$922$	0	0
$923$	−2.44323e7	−0.943976
$924$	0	0
$925$	−2.92625e6	−0.112449
$926$	0	0
$927$	−3.02001e6	−0.115427
$928$	0	0
$929$	−1.33648e7	−0.508068	−0.254034	−	0.967195i	$-0.581758\pi$
$929$	−1.33648e7	−0.508068	−0.254034	+	0.967195i	$0.581758\pi$
$930$	0	0
$931$	−1.83436e6	−0.0693604
$932$	0	0
$933$	1.01074e7	0.380131
$934$	0	0
$935$	−3.41620e6	−0.127795
$936$	0	0
$937$	−2.72180e7	−1.01276	−0.506381	−	0.862310i	$-0.669017\pi$
$937$	−2.72180e7	−1.01276	−0.506381	+	0.862310i	$0.669017\pi$
$938$	0	0
$939$	4.03863e6	0.149476
$940$	0	0
$941$	3.09420e7	1.13913	0.569566	−	0.821946i	$-0.307111\pi$
$941$	3.09420e7	1.13913	0.569566	+	0.821946i	$0.307111\pi$
$942$	0	0
$943$	2.20584e6	0.0807783
$944$	0	0
$945$	2.30300e6	0.0838908
$946$	0	0
$947$	6.39058e6	0.231561	0.115780	−	0.993275i	$-0.463063\pi$
$947$	6.39058e6	0.231561	0.115780	+	0.993275i	$0.463063\pi$
$948$	0	0
$949$	−1.81312e7	−0.653524
$950$	0	0
$951$	−4.59169e6	−0.164635
$952$	0	0
$953$	−3.72447e7	−1.32841	−0.664205	−	0.747550i	$-0.731230\pi$
$953$	−3.72447e7	−1.32841	−0.664205	+	0.747550i	$0.731230\pi$
$954$	0	0
$955$	7.37920e6	0.261819
$956$	0	0
$957$	−3.40554e6	−0.120200
$958$	0	0
$959$	9.20524e6	0.323213
$960$	0	0
$961$	−1.18519e7	−0.413981
$962$	0	0
$963$	−2.91023e7	−1.01126
$964$	0	0
$965$	1.93765e6	0.0669818
$966$	0	0
$967$	4.34627e7	1.49469	0.747344	−	0.664437i	$-0.231329\pi$
$967$	4.34627e7	1.49469	0.747344	+	0.664437i	$0.231329\pi$
$968$	0	0
$969$	3.36771e6	0.115219
$970$	0	0
$971$	−4.88474e7	−1.66262	−0.831311	−	0.555807i	$-0.812409\pi$
$971$	−4.88474e7	−1.66262	−0.831311	+	0.555807i	$0.812409\pi$
$972$	0	0
$973$	5.21380e6	0.176552
$974$	0	0
$975$	1.91500e6	0.0645145
$976$	0	0
$977$	1.46440e7	0.490823	0.245411	−	0.969419i	$-0.421077\pi$
$977$	1.46440e7	0.490823	0.245411	+	0.969419i	$0.421077\pi$
$978$	0	0
$979$	−2.36567e6	−0.0788856
$980$	0	0
$981$	4.00542e7	1.32885
$982$	0	0
$983$	−4.58487e7	−1.51336	−0.756681	−	0.653784i	$-0.773180\pi$
$983$	−4.58487e7	−1.51336	−0.756681	+	0.653784i	$0.773180\pi$
$984$	0	0
$985$	7.76215e6	0.254913
$986$	0	0
$987$	1.39238e6	0.0454953
$988$	0	0
$989$	3.06096e6	0.0995100
$990$	0	0
$991$	1.86037e7	0.601748	0.300874	−	0.953664i	$-0.402722\pi$
$991$	1.86037e7	0.601748	0.300874	+	0.953664i	$0.402722\pi$
$992$	0	0
$993$	−1.20940e7	−0.389220
$994$	0	0
$995$	1.59770e7	0.511608
$996$	0	0
$997$	3.26338e7	1.03975	0.519876	−	0.854242i	$-0.325978\pi$
$997$	3.26338e7	1.03975	0.519876	+	0.854242i	$0.325978\pi$
$998$	0	0
$999$	8.80216e6	0.279046

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.b.1.1	✓	1
4.3	odd	2		560.6.a.b.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.b.1.1	✓	1	1.1	even	1	trivial
560.6.a.b.1.1		1	4.3	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 280 = 2^{3} \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	280.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more