LMFDB - Embedded newform 280.6.a.j.1.3

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:	$0$
Dimension:	$5$
Coefficient field:	$\mathbb{Q}[x]/(x^{5} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 1151x^{3} - 5642x^{2} + 193596x + 1258056 $ x^5 - 1151x^3 - 5642x^2 + 193596*x + 1258056
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-7.03843$ of defining polynomial
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)

$f(q)$	$=$	$q+10.0384 q^{3} +25.0000 q^{5} +49.0000 q^{7} -142.230 q^{9} +O(q^{10})$	$q+10.0384 q^{3} +25.0000 q^{5} +49.0000 q^{7} -142.230 q^{9} +422.924 q^{11} -731.815 q^{13} +250.961 q^{15} +1498.36 q^{17} +1370.58 q^{19} +491.883 q^{21} +2789.75 q^{23} +625.000 q^{25} -3867.10 q^{27} -6216.76 q^{29} +4080.91 q^{31} +4245.49 q^{33} +1225.00 q^{35} +14739.6 q^{37} -7346.27 q^{39} +9008.85 q^{41} +2356.52 q^{43} -3555.75 q^{45} -6761.78 q^{47} +2401.00 q^{49} +15041.2 q^{51} -25405.2 q^{53} +10573.1 q^{55} +13758.5 q^{57} +16243.0 q^{59} +14602.7 q^{61} -6969.27 q^{63} -18295.4 q^{65} +49347.9 q^{67} +28004.7 q^{69} -36061.5 q^{71} -2178.41 q^{73} +6274.02 q^{75} +20723.3 q^{77} +108080. q^{79} -4257.76 q^{81} +28981.6 q^{83} +37459.0 q^{85} -62406.5 q^{87} +53791.5 q^{89} -35858.9 q^{91} +40965.9 q^{93} +34264.5 q^{95} -95770.6 q^{97} -60152.4 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9}+O(q^{10}) $ 5 * q + 15 * q^3 + 125 * q^5 + 245 * q^7 + 1132 * q^9	$ 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9} + 281 q^{11} + 909 q^{13} + 375 q^{15} + 1495 q^{17} - 422 q^{19} + 735 q^{21} - 62 q^{23} + 3125 q^{25} - 3363 q^{27} - 2047 q^{29} + 1636 q^{31} + 19181 q^{33} + 6125 q^{35} - 10358 q^{37} + 15685 q^{39} + 6424 q^{41} + 28306 q^{43} + 28300 q^{45} + 20955 q^{47} + 12005 q^{49} - 23577 q^{51} + 43748 q^{53} + 7025 q^{55} + 13690 q^{57} + 45788 q^{59} + 50432 q^{61} + 55468 q^{63} + 22725 q^{65} + 40712 q^{67} + 35050 q^{69} - 3096 q^{71} + 135438 q^{73} + 9375 q^{75} + 13769 q^{77} + 13191 q^{79} + 381101 q^{81} + 35108 q^{83} + 37375 q^{85} + 297289 q^{87} + 213772 q^{89} + 44541 q^{91} + 134244 q^{93} - 10550 q^{95} + 10659 q^{97} + 39462 q^{99}+O(q^{100}) $ 5 * q + 15 * q^3 + 125 * q^5 + 245 * q^7 + 1132 * q^9 + 281 * q^11 + 909 * q^13 + 375 * q^15 + 1495 * q^17 - 422 * q^19 + 735 * q^21 - 62 * q^23 + 3125 * q^25 - 3363 * q^27 - 2047 * q^29 + 1636 * q^31 + 19181 * q^33 + 6125 * q^35 - 10358 * q^37 + 15685 * q^39 + 6424 * q^41 + 28306 * q^43 + 28300 * q^45 + 20955 * q^47 + 12005 * q^49 - 23577 * q^51 + 43748 * q^53 + 7025 * q^55 + 13690 * q^57 + 45788 * q^59 + 50432 * q^61 + 55468 * q^63 + 22725 * q^65 + 40712 * q^67 + 35050 * q^69 - 3096 * q^71 + 135438 * q^73 + 9375 * q^75 + 13769 * q^77 + 13191 * q^79 + 381101 * q^81 + 35108 * q^83 + 37375 * q^85 + 297289 * q^87 + 213772 * q^89 + 44541 * q^91 + 134244 * q^93 - 10550 * q^95 + 10659 * q^97 + 39462 * q^99

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
$3$	10.0384	0.643965	0.321983	−	0.946746i	$-0.395651\pi$
$3$	10.0384	0.643965	0.321983	+	0.946746i	$0.395651\pi$
$4$	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
$9$	−142.230	−0.585308
$10$	0	0
$11$	422.924	1.05385	0.526927	−	0.849911i	$-0.323344\pi$
$11$	422.924	1.05385	0.526927	+	0.849911i	$0.323344\pi$
$12$	0	0
$13$	−731.815	−1.20100	−0.600500	−	0.799625i	$-0.705032\pi$
$13$	−731.815	−1.20100	−0.600500	+	0.799625i	$0.705032\pi$
$14$	0	0
$15$	250.961	0.287990
$16$	0	0
$17$	1498.36	1.25746	0.628729	−	0.777625i	$-0.283575\pi$
$17$	1498.36	1.25746	0.628729	+	0.777625i	$0.283575\pi$
$18$	0	0
$19$	1370.58	0.871004	0.435502	−	0.900188i	$-0.356571\pi$
$19$	1370.58	0.871004	0.435502	+	0.900188i	$0.356571\pi$
$20$	0	0
$21$	491.883	0.243396
$22$	0	0
$23$	2789.75	1.09963	0.549813	−	0.835288i	$-0.314699\pi$
$23$	2789.75	1.09963	0.549813	+	0.835288i	$0.314699\pi$
$24$	0	0
$25$	625.000	0.200000
$26$	0	0
$27$	−3867.10	−1.02088
$28$	0	0
$29$	−6216.76	−1.37268	−0.686340	−	0.727281i	$-0.740784\pi$
$29$	−6216.76	−1.37268	−0.686340	+	0.727281i	$0.740784\pi$
$30$	0	0
$31$	4080.91	0.762698	0.381349	−	0.924431i	$-0.375460\pi$
$31$	4080.91	0.762698	0.381349	+	0.924431i	$0.375460\pi$
$32$	0	0
$33$	4245.49	0.678645
$34$	0	0
$35$	1225.00	0.169031
$36$	0	0
$37$	14739.6	1.77004	0.885018	−	0.465557i	$-0.154146\pi$
$37$	14739.6	1.77004	0.885018	+	0.465557i	$0.154146\pi$
$38$	0	0
$39$	−7346.27	−0.773402
$40$	0	0
$41$	9008.85	0.836969	0.418485	−	0.908224i	$-0.362561\pi$
$41$	9008.85	0.836969	0.418485	+	0.908224i	$0.362561\pi$
$42$	0	0
$43$	2356.52	0.194357	0.0971784	−	0.995267i	$-0.469018\pi$
$43$	2356.52	0.194357	0.0971784	+	0.995267i	$0.469018\pi$
$44$	0	0
$45$	−3555.75	−0.261758
$46$	0	0
$47$	−6761.78	−0.446495	−0.223247	−	0.974762i	$-0.571666\pi$
$47$	−6761.78	−0.446495	−0.223247	+	0.974762i	$0.571666\pi$
$48$	0	0
$49$	2401.00	0.142857
$50$	0	0
$51$	15041.2	0.809760
$52$	0	0
$53$	−25405.2	−1.24232	−0.621158	−	0.783685i	$-0.713338\pi$
$53$	−25405.2	−1.24232	−0.621158	+	0.783685i	$0.713338\pi$
$54$	0	0
$55$	10573.1	0.471298
$56$	0	0
$57$	13758.5	0.560897
$58$	0	0
$59$	16243.0	0.607487	0.303743	−	0.952754i	$-0.401763\pi$
$59$	16243.0	0.607487	0.303743	+	0.952754i	$0.401763\pi$
$60$	0	0
$61$	14602.7	0.502469	0.251234	−	0.967926i	$-0.419164\pi$
$61$	14602.7	0.502469	0.251234	+	0.967926i	$0.419164\pi$
$62$	0	0
$63$	−6969.27	−0.221226
$64$	0	0
$65$	−18295.4	−0.537103
$66$	0	0
$67$	49347.9	1.34302	0.671509	−	0.740996i	$-0.265646\pi$
$67$	49347.9	1.34302	0.671509	+	0.740996i	$0.265646\pi$
$68$	0	0
$69$	28004.7	0.708121
$70$	0	0
$71$	−36061.5	−0.848981	−0.424490	−	0.905432i	$-0.639547\pi$
$71$	−36061.5	−0.848981	−0.424490	+	0.905432i	$0.639547\pi$
$72$	0	0
$73$	−2178.41	−0.0478446	−0.0239223	−	0.999714i	$-0.507615\pi$
$73$	−2178.41	−0.0478446	−0.0239223	+	0.999714i	$0.507615\pi$
$74$	0	0
$75$	6274.02	0.128793
$76$	0	0
$77$	20723.3	0.398319
$78$	0	0
$79$	108080.	1.94841	0.974203	−	0.225673i	$-0.0724581\pi$
$79$	108080.	1.94841	0.974203	+	0.225673i	$0.0724581\pi$
$80$	0	0
$81$	−4257.76	−0.0721056
$82$	0	0
$83$	28981.6	0.461771	0.230886	−	0.972981i	$-0.425838\pi$
$83$	28981.6	0.461771	0.230886	+	0.972981i	$0.425838\pi$
$84$	0	0
$85$	37459.0	0.562352
$86$	0	0
$87$	−62406.5	−0.883959
$88$	0	0
$89$	53791.5	0.719845	0.359922	−	0.932982i	$-0.382803\pi$
$89$	53791.5	0.719845	0.359922	+	0.932982i	$0.382803\pi$
$90$	0	0
$91$	−35858.9	−0.453935
$92$	0	0
$93$	40965.9	0.491151
$94$	0	0
$95$	34264.5	0.389525
$96$	0	0
$97$	−95770.6	−1.03348	−0.516741	−	0.856142i	$-0.672855\pi$
$97$	−95770.6	−1.03348	−0.516741	+	0.856142i	$0.672855\pi$
$98$	0	0
$99$	−60152.4	−0.616829
$100$	0	0
$101$	103047.	1.00515	0.502577	−	0.864532i	$-0.332385\pi$
$101$	103047.	1.00515	0.502577	+	0.864532i	$0.332385\pi$
$102$	0	0
$103$	−110405.	−1.02541	−0.512704	−	0.858565i	$-0.671356\pi$
$103$	−110405.	−1.02541	−0.512704	+	0.858565i	$0.671356\pi$
$104$	0	0
$105$	12297.1	0.108850
$106$	0	0
$107$	−82760.3	−0.698816	−0.349408	−	0.936971i	$-0.613617\pi$
$107$	−82760.3	−0.698816	−0.349408	+	0.936971i	$0.613617\pi$
$108$	0	0
$109$	96804.8	0.780424	0.390212	−	0.920725i	$-0.372402\pi$
$109$	96804.8	0.780424	0.390212	+	0.920725i	$0.372402\pi$
$110$	0	0
$111$	147963.	1.13984
$112$	0	0
$113$	218042.	1.60636	0.803182	−	0.595733i	$-0.203138\pi$
$113$	218042.	1.60636	0.803182	+	0.595733i	$0.203138\pi$
$114$	0	0
$115$	69743.6	0.491768
$116$	0	0
$117$	104086.	0.702955
$118$	0	0
$119$	73419.6	0.475274
$120$	0	0
$121$	17813.4	0.110607
$122$	0	0
$123$	90434.7	0.538979
$124$	0	0
$125$	15625.0	0.0894427
$126$	0	0
$127$	234551.	1.29041	0.645205	−	0.764010i	$-0.276772\pi$
$127$	234551.	1.29041	0.645205	+	0.764010i	$0.276772\pi$
$128$	0	0
$129$	23655.8	0.125159
$130$	0	0
$131$	−81880.4	−0.416871	−0.208435	−	0.978036i	$-0.566837\pi$
$131$	−81880.4	−0.416871	−0.208435	+	0.978036i	$0.566837\pi$
$132$	0	0
$133$	67158.4	0.329209
$134$	0	0
$135$	−96677.6	−0.456553
$136$	0	0
$137$	136807.	0.622739	0.311369	−	0.950289i	$-0.399212\pi$
$137$	136807.	0.622739	0.311369	+	0.950289i	$0.399212\pi$
$138$	0	0
$139$	−287262.	−1.26107	−0.630537	−	0.776159i	$-0.717165\pi$
$139$	−287262.	−1.26107	−0.630537	+	0.776159i	$0.717165\pi$
$140$	0	0
$141$	−67877.6	−0.287527
$142$	0	0
$143$	−309502.	−1.26568
$144$	0	0
$145$	−155419.	−0.613881
$146$	0	0
$147$	24102.3	0.0919951
$148$	0	0
$149$	309740.	1.14296	0.571480	−	0.820616i	$-0.306369\pi$
$149$	309740.	1.14296	0.571480	+	0.820616i	$0.306369\pi$
$150$	0	0
$151$	−486149.	−1.73511	−0.867556	−	0.497340i	$-0.834310\pi$
$151$	−486149.	−1.73511	−0.867556	+	0.497340i	$0.834310\pi$
$152$	0	0
$153$	−213111.	−0.736001
$154$	0	0
$155$	102023.	0.341089
$156$	0	0
$157$	507730.	1.64393	0.821966	−	0.569537i	$-0.192877\pi$
$157$	507730.	1.64393	0.821966	+	0.569537i	$0.192877\pi$
$158$	0	0
$159$	−255028.	−0.800009
$160$	0	0
$161$	136698.	0.415620
$162$	0	0
$163$	−210698.	−0.621143	−0.310571	−	0.950550i	$-0.600520\pi$
$163$	−210698.	−0.621143	−0.310571	+	0.950550i	$0.600520\pi$
$164$	0	0
$165$	106137.	0.303499
$166$	0	0
$167$	−454974.	−1.26240	−0.631198	−	0.775622i	$-0.717437\pi$
$167$	−454974.	−1.26240	−0.631198	+	0.775622i	$0.717437\pi$
$168$	0	0
$169$	164260.	0.442399
$170$	0	0
$171$	−194937.	−0.509806
$172$	0	0
$173$	−206691.	−0.525057	−0.262529	−	0.964924i	$-0.584556\pi$
$173$	−206691.	−0.525057	−0.262529	+	0.964924i	$0.584556\pi$
$174$	0	0
$175$	30625.0	0.0755929
$176$	0	0
$177$	163054.	0.391200
$178$	0	0
$179$	−235999.	−0.550526	−0.275263	−	0.961369i	$-0.588765\pi$
$179$	−235999.	−0.550526	−0.275263	+	0.961369i	$0.588765\pi$
$180$	0	0
$181$	−365358.	−0.828939	−0.414470	−	0.910063i	$-0.636033\pi$
$181$	−365358.	−0.828939	−0.414470	+	0.910063i	$0.636033\pi$
$182$	0	0
$183$	146588.	0.323573
$184$	0	0
$185$	368490.	0.791584
$186$	0	0
$187$	633691.	1.32518
$188$	0	0
$189$	−189488.	−0.385858
$190$	0	0
$191$	143765.	0.285148	0.142574	−	0.989784i	$-0.454462\pi$
$191$	143765.	0.285148	0.142574	+	0.989784i	$0.454462\pi$
$192$	0	0
$193$	−615006.	−1.18846	−0.594232	−	0.804294i	$-0.702544\pi$
$193$	−615006.	−1.18846	−0.594232	+	0.804294i	$0.702544\pi$
$194$	0	0
$195$	−183657.	−0.345876
$196$	0	0
$197$	−915222.	−1.68020	−0.840099	−	0.542432i	$-0.817503\pi$
$197$	−915222.	−1.68020	−0.840099	+	0.542432i	$0.817503\pi$
$198$	0	0
$199$	−582204.	−1.04218	−0.521089	−	0.853502i	$-0.674474\pi$
$199$	−582204.	−1.04218	−0.521089	+	0.853502i	$0.674474\pi$
$200$	0	0
$201$	495376.	0.864858
$202$	0	0
$203$	−304621.	−0.518824
$204$	0	0
$205$	225221.	0.374304
$206$	0	0
$207$	−396785.	−0.643620
$208$	0	0
$209$	579651.	0.917911
$210$	0	0
$211$	−862136.	−1.33312	−0.666561	−	0.745451i	$-0.732234\pi$
$211$	−862136.	−1.33312	−0.666561	+	0.745451i	$0.732234\pi$
$212$	0	0
$213$	−362001.	−0.546714
$214$	0	0
$215$	58913.0	0.0869190
$216$	0	0
$217$	199964.	0.288273
$218$	0	0
$219$	−21867.8	−0.0308103
$220$	0	0
$221$	−1.09652e6	−1.51021
$222$	0	0
$223$	−684947.	−0.922349	−0.461174	−	0.887310i	$-0.652572\pi$
$223$	−684947.	−0.922349	−0.461174	+	0.887310i	$0.652572\pi$
$224$	0	0
$225$	−88893.7	−0.117062
$226$	0	0
$227$	−160804.	−0.207125	−0.103563	−	0.994623i	$-0.533024\pi$
$227$	−160804.	−0.207125	−0.103563	+	0.994623i	$0.533024\pi$
$228$	0	0
$229$	−1.47603e6	−1.85997	−0.929987	−	0.367592i	$-0.880182\pi$
$229$	−1.47603e6	−1.85997	−0.929987	+	0.367592i	$0.880182\pi$
$230$	0	0
$231$	208029.	0.256504
$232$	0	0
$233$	−1.33729e6	−1.61375	−0.806874	−	0.590724i	$-0.798842\pi$
$233$	−1.33729e6	−1.61375	−0.806874	+	0.590724i	$0.798842\pi$
$234$	0	0
$235$	−169044.	−0.199679
$236$	0	0
$237$	1.08496e6	1.25471
$238$	0	0
$239$	−31064.0	−0.0351773	−0.0175886	−	0.999845i	$-0.505599\pi$
$239$	−31064.0	−0.0351773	−0.0175886	+	0.999845i	$0.505599\pi$
$240$	0	0
$241$	924703.	1.02556	0.512779	−	0.858521i	$-0.328616\pi$
$241$	924703.	1.02556	0.512779	+	0.858521i	$0.328616\pi$
$242$	0	0
$243$	896965.	0.974450
$244$	0	0
$245$	60025.0	0.0638877
$246$	0	0
$247$	−1.00301e6	−1.04608
$248$	0	0
$249$	290930.	0.297365
$250$	0	0
$251$	561062.	0.562117	0.281058	−	0.959691i	$-0.409315\pi$
$251$	561062.	0.562117	0.281058	+	0.959691i	$0.409315\pi$
$252$	0	0
$253$	1.17985e6	1.15884
$254$	0	0
$255$	376029.	0.362135
$256$	0	0
$257$	316039.	0.298475	0.149238	−	0.988801i	$-0.452318\pi$
$257$	316039.	0.298475	0.149238	+	0.988801i	$0.452318\pi$
$258$	0	0
$259$	722241.	0.669010
$260$	0	0
$261$	884210.	0.803441
$262$	0	0
$263$	−262038.	−0.233601	−0.116800	−	0.993155i	$-0.537264\pi$
$263$	−262038.	−0.233601	−0.116800	+	0.993155i	$0.537264\pi$
$264$	0	0
$265$	−635129.	−0.555581
$266$	0	0
$267$	539982.	0.463555
$268$	0	0
$269$	−1.21153e6	−1.02083	−0.510413	−	0.859929i	$-0.670507\pi$
$269$	−1.21153e6	−1.02083	−0.510413	+	0.859929i	$0.670507\pi$
$270$	0	0
$271$	152730.	0.126328	0.0631641	−	0.998003i	$-0.479881\pi$
$271$	152730.	0.126328	0.0631641	+	0.998003i	$0.479881\pi$
$272$	0	0
$273$	−359967.	−0.292319
$274$	0	0
$275$	264327.	0.210771
$276$	0	0
$277$	645375.	0.505374	0.252687	−	0.967548i	$-0.418686\pi$
$277$	645375.	0.505374	0.252687	+	0.967548i	$0.418686\pi$
$278$	0	0
$279$	−580427.	−0.446414
$280$	0	0
$281$	871336.	0.658294	0.329147	−	0.944279i	$-0.393239\pi$
$281$	871336.	0.658294	0.329147	+	0.944279i	$0.393239\pi$
$282$	0	0
$283$	1.22474e6	0.909033	0.454516	−	0.890738i	$-0.349812\pi$
$283$	1.22474e6	0.909033	0.454516	+	0.890738i	$0.349812\pi$
$284$	0	0
$285$	343962.	0.250841
$286$	0	0
$287$	441433.	0.316345
$288$	0	0
$289$	825222.	0.581200
$290$	0	0
$291$	−961386.	−0.665526
$292$	0	0
$293$	180700.	0.122967	0.0614835	−	0.998108i	$-0.480417\pi$
$293$	180700.	0.122967	0.0614835	+	0.998108i	$0.480417\pi$
$294$	0	0
$295$	406075.	0.271676
$296$	0	0
$297$	−1.63549e6	−1.07586
$298$	0	0
$299$	−2.04158e6	−1.32065
$300$	0	0
$301$	115469.	0.0734600
$302$	0	0
$303$	1.03443e6	0.647284
$304$	0	0
$305$	365068.	0.224711
$306$	0	0
$307$	2.56849e6	1.55536	0.777682	−	0.628658i	$-0.216395\pi$
$307$	2.56849e6	1.55536	0.777682	+	0.628658i	$0.216395\pi$
$308$	0	0
$309$	−1.10830e6	−0.660328
$310$	0	0
$311$	−2.83718e6	−1.66336	−0.831680	−	0.555255i	$-0.812621\pi$
$311$	−2.83718e6	−1.66336	−0.831680	+	0.555255i	$0.812621\pi$
$312$	0	0
$313$	−1.30062e6	−0.750395	−0.375197	−	0.926945i	$-0.622425\pi$
$313$	−1.30062e6	−0.750395	−0.375197	+	0.926945i	$0.622425\pi$
$314$	0	0
$315$	−174232.	−0.0989352
$316$	0	0
$317$	2.37328e6	1.32648	0.663241	−	0.748406i	$-0.269180\pi$
$317$	2.37328e6	1.32648	0.663241	+	0.748406i	$0.269180\pi$
$318$	0	0
$319$	−2.62922e6	−1.44660
$320$	0	0
$321$	−830783.	−0.450013
$322$	0	0
$323$	2.05362e6	1.09525
$324$	0	0
$325$	−457384.	−0.240200
$326$	0	0
$327$	971768.	0.502566
$328$	0	0
$329$	−331327.	−0.168759
$330$	0	0
$331$	−462474.	−0.232016	−0.116008	−	0.993248i	$-0.537010\pi$
$331$	−462474.	−0.232016	−0.116008	+	0.993248i	$0.537010\pi$
$332$	0	0
$333$	−2.09642e6	−1.03602
$334$	0	0
$335$	1.23370e6	0.600616
$336$	0	0
$337$	3.41385e6	1.63746	0.818729	−	0.574180i	$-0.194679\pi$
$337$	3.41385e6	1.63746	0.818729	+	0.574180i	$0.194679\pi$
$338$	0	0
$339$	2.18880e6	1.03444
$340$	0	0
$341$	1.72591e6	0.803772
$342$	0	0
$343$	117649.	0.0539949
$344$	0	0
$345$	700116.	0.316681
$346$	0	0
$347$	1.06822e6	0.476252	0.238126	−	0.971234i	$-0.423467\pi$
$347$	1.06822e6	0.476252	0.238126	+	0.971234i	$0.423467\pi$
$348$	0	0
$349$	−3.36926e6	−1.48071	−0.740356	−	0.672215i	$-0.765343\pi$
$349$	−3.36926e6	−1.48071	−0.740356	+	0.672215i	$0.765343\pi$
$350$	0	0
$351$	2.83000e6	1.22608
$352$	0	0
$353$	3.66070e6	1.56361	0.781803	−	0.623525i	$-0.214300\pi$
$353$	3.66070e6	1.56361	0.781803	+	0.623525i	$0.214300\pi$
$354$	0	0
$355$	−901537.	−0.379676
$356$	0	0
$357$	737017.	0.306060
$358$	0	0
$359$	−3.67332e6	−1.50426	−0.752129	−	0.659016i	$-0.770973\pi$
$359$	−3.67332e6	−1.50426	−0.752129	+	0.659016i	$0.770973\pi$
$360$	0	0
$361$	−597611.	−0.241352
$362$	0	0
$363$	178819.	0.0712273
$364$	0	0
$365$	−54460.3	−0.0213967
$366$	0	0
$367$	1.86123e6	0.721330	0.360665	−	0.932695i	$-0.382550\pi$
$367$	1.86123e6	0.721330	0.360665	+	0.932695i	$0.382550\pi$
$368$	0	0
$369$	−1.28133e6	−0.489885
$370$	0	0
$371$	−1.24485e6	−0.469551
$372$	0	0
$373$	−3.59890e6	−1.33936	−0.669681	−	0.742648i	$-0.733569\pi$
$373$	−3.59890e6	−1.33936	−0.669681	+	0.742648i	$0.733569\pi$
$374$	0	0
$375$	156850.	0.0575980
$376$	0	0
$377$	4.54952e6	1.64859
$378$	0	0
$379$	−2.20715e6	−0.789283	−0.394642	−	0.918835i	$-0.629131\pi$
$379$	−2.20715e6	−0.789283	−0.394642	+	0.918835i	$0.629131\pi$
$380$	0	0
$381$	2.35452e6	0.830979
$382$	0	0
$383$	2.51980e6	0.877746	0.438873	−	0.898549i	$-0.355378\pi$
$383$	2.51980e6	0.877746	0.438873	+	0.898549i	$0.355378\pi$
$384$	0	0
$385$	518081.	0.178134
$386$	0	0
$387$	−335168.	−0.113759
$388$	0	0
$389$	−131011.	−0.0438970	−0.0219485	−	0.999759i	$-0.506987\pi$
$389$	−131011.	−0.0438970	−0.0219485	+	0.999759i	$0.506987\pi$
$390$	0	0
$391$	4.18004e6	1.38273
$392$	0	0
$393$	−821950.	−0.268450
$394$	0	0
$395$	2.70201e6	0.871354
$396$	0	0
$397$	−2.86220e6	−0.911431	−0.455716	−	0.890125i	$-0.650617\pi$
$397$	−2.86220e6	−0.911431	−0.455716	+	0.890125i	$0.650617\pi$
$398$	0	0
$399$	674165.	0.211999
$400$	0	0
$401$	−2.19911e6	−0.682946	−0.341473	−	0.939892i	$-0.610926\pi$
$401$	−2.19911e6	−0.682946	−0.341473	+	0.939892i	$0.610926\pi$
$402$	0	0
$403$	−2.98647e6	−0.916000
$404$	0	0
$405$	−106444.	−0.0322466
$406$	0	0
$407$	6.23373e6	1.86536
$408$	0	0
$409$	4.04568e6	1.19587	0.597934	−	0.801545i	$-0.295989\pi$
$409$	4.04568e6	1.19587	0.597934	+	0.801545i	$0.295989\pi$
$410$	0	0
$411$	1.37332e6	0.401022
$412$	0	0
$413$	795908.	0.229608
$414$	0	0
$415$	724540.	0.206510
$416$	0	0
$417$	−2.88365e6	−0.812088
$418$	0	0
$419$	4.10057e6	1.14106	0.570531	−	0.821276i	$-0.306737\pi$
$419$	4.10057e6	1.14106	0.570531	+	0.821276i	$0.306737\pi$
$420$	0	0
$421$	−6.28257e6	−1.72755	−0.863777	−	0.503873i	$-0.831908\pi$
$421$	−6.28257e6	−1.72755	−0.863777	+	0.503873i	$0.831908\pi$
$422$	0	0
$423$	961728.	0.261337
$424$	0	0
$425$	936474.	0.251492
$426$	0	0
$427$	715533.	0.189915
$428$	0	0
$429$	−3.10691e6	−0.815053
$430$	0	0
$431$	2.76782e6	0.717703	0.358852	−	0.933395i	$-0.383168\pi$
$431$	2.76782e6	0.717703	0.358852	+	0.933395i	$0.383168\pi$
$432$	0	0
$433$	1.52060e6	0.389759	0.194879	−	0.980827i	$-0.437568\pi$
$433$	1.52060e6	0.389759	0.194879	+	0.980827i	$0.437568\pi$
$434$	0	0
$435$	−1.56016e6	−0.395318
$436$	0	0
$437$	3.82357e6	0.957779
$438$	0	0
$439$	1.69329e6	0.419343	0.209671	−	0.977772i	$-0.432761\pi$
$439$	1.69329e6	0.419343	0.209671	+	0.977772i	$0.432761\pi$
$440$	0	0
$441$	−341494.	−0.0836155
$442$	0	0
$443$	−363356.	−0.0879677	−0.0439839	−	0.999032i	$-0.514005\pi$
$443$	−363356.	−0.0879677	−0.0439839	+	0.999032i	$0.514005\pi$
$444$	0	0
$445$	1.34479e6	0.321924
$446$	0	0
$447$	3.10930e6	0.736027
$448$	0	0
$449$	487162.	0.114040	0.0570200	−	0.998373i	$-0.481840\pi$
$449$	487162.	0.114040	0.0570200	+	0.998373i	$0.481840\pi$
$450$	0	0
$451$	3.81005e6	0.882043
$452$	0	0
$453$	−4.88018e6	−1.11735
$454$	0	0
$455$	−896473.	−0.203006
$456$	0	0
$457$	5.38947e6	1.20713	0.603567	−	0.797312i	$-0.293746\pi$
$457$	5.38947e6	1.20713	0.603567	+	0.797312i	$0.293746\pi$
$458$	0	0
$459$	−5.79431e6	−1.28372
$460$	0	0
$461$	1.69670e6	0.371837	0.185919	−	0.982565i	$-0.440474\pi$
$461$	1.69670e6	0.371837	0.185919	+	0.982565i	$0.440474\pi$
$462$	0	0
$463$	−5.84875e6	−1.26797	−0.633987	−	0.773344i	$-0.718583\pi$
$463$	−5.84875e6	−1.26797	−0.633987	+	0.773344i	$0.718583\pi$
$464$	0	0
$465$	1.02415e6	0.219649
$466$	0	0
$467$	−1.01745e6	−0.215883	−0.107942	−	0.994157i	$-0.534426\pi$
$467$	−1.01745e6	−0.215883	−0.107942	+	0.994157i	$0.534426\pi$
$468$	0	0
$469$	2.41805e6	0.507613
$470$	0	0
$471$	5.09681e6	1.05864
$472$	0	0
$473$	996628.	0.204824
$474$	0	0
$475$	856612.	0.174201
$476$	0	0
$477$	3.61337e6	0.727138
$478$	0	0
$479$	−6.29901e6	−1.25439	−0.627196	−	0.778861i	$-0.715798\pi$
$479$	−6.29901e6	−1.25439	−0.627196	+	0.778861i	$0.715798\pi$
$480$	0	0
$481$	−1.07867e7	−2.12581
$482$	0	0
$483$	1.37223e6	0.267645
$484$	0	0
$485$	−2.39426e6	−0.462187
$486$	0	0
$487$	−79494.0	−0.0151884	−0.00759420	−	0.999971i	$-0.502417\pi$
$487$	−79494.0	−0.0151884	−0.00759420	+	0.999971i	$0.502417\pi$
$488$	0	0
$489$	−2.11508e6	−0.399994
$490$	0	0
$491$	−2.10200e6	−0.393485	−0.196743	−	0.980455i	$-0.563036\pi$
$491$	−2.10200e6	−0.393485	−0.196743	+	0.980455i	$0.563036\pi$
$492$	0	0
$493$	−9.31494e6	−1.72609
$494$	0	0
$495$	−1.50381e6	−0.275854
$496$	0	0
$497$	−1.76701e6	−0.320885
$498$	0	0
$499$	−7.02043e6	−1.26215	−0.631077	−	0.775720i	$-0.717387\pi$
$499$	−7.02043e6	−1.26215	−0.631077	+	0.775720i	$0.717387\pi$
$500$	0	0
$501$	−4.56723e6	−0.812940
$502$	0	0
$503$	9.36878e6	1.65106	0.825530	−	0.564358i	$-0.190876\pi$
$503$	9.36878e6	1.65106	0.825530	+	0.564358i	$0.190876\pi$
$504$	0	0
$505$	2.57618e6	0.449518
$506$	0	0
$507$	1.64891e6	0.284890
$508$	0	0
$509$	−6.06864e6	−1.03824	−0.519119	−	0.854702i	$-0.673740\pi$
$509$	−6.06864e6	−1.03824	−0.519119	+	0.854702i	$0.673740\pi$
$510$	0	0
$511$	−106742.	−0.0180836
$512$	0	0
$513$	−5.30017e6	−0.889194
$514$	0	0
$515$	−2.76013e6	−0.458577
$516$	0	0
$517$	−2.85972e6	−0.470540
$518$	0	0
$519$	−2.07485e6	−0.338119
$520$	0	0
$521$	9.15802e6	1.47811	0.739056	−	0.673644i	$-0.235272\pi$
$521$	9.15802e6	1.47811	0.739056	+	0.673644i	$0.235272\pi$
$522$	0	0
$523$	7.72844e6	1.23549	0.617743	−	0.786380i	$-0.288047\pi$
$523$	7.72844e6	1.23549	0.617743	+	0.786380i	$0.288047\pi$
$524$	0	0
$525$	307427.	0.0486792
$526$	0	0
$527$	6.11466e6	0.959061
$528$	0	0
$529$	1.34634e6	0.209177
$530$	0	0
$531$	−2.31024e6	−0.355567
$532$	0	0
$533$	−6.59281e6	−1.00520
$534$	0	0
$535$	−2.06901e6	−0.312520
$536$	0	0
$537$	−2.36906e6	−0.354520
$538$	0	0
$539$	1.01544e6	0.150551
$540$	0	0
$541$	−7.74918e6	−1.13832	−0.569158	−	0.822228i	$-0.692731\pi$
$541$	−7.74918e6	−1.13832	−0.569158	+	0.822228i	$0.692731\pi$
$542$	0	0
$543$	−3.66762e6	−0.533808
$544$	0	0
$545$	2.42012e6	0.349016
$546$	0	0
$547$	−4.29607e6	−0.613907	−0.306954	−	0.951725i	$-0.599310\pi$
$547$	−4.29607e6	−0.613907	−0.306954	+	0.951725i	$0.599310\pi$
$548$	0	0
$549$	−2.07694e6	−0.294099
$550$	0	0
$551$	−8.52057e6	−1.19561
$552$	0	0
$553$	5.29594e6	0.736428
$554$	0	0
$555$	3.69907e6	0.509753
$556$	0	0
$557$	−1.16671e7	−1.59341	−0.796703	−	0.604371i	$-0.793424\pi$
$557$	−1.16671e7	−1.59341	−0.796703	+	0.604371i	$0.793424\pi$
$558$	0	0
$559$	−1.72454e6	−0.233422
$560$	0	0
$561$	6.36126e6	0.853368
$562$	0	0
$563$	7.33119e6	0.974773	0.487386	−	0.873186i	$-0.337950\pi$
$563$	7.33119e6	0.974773	0.487386	+	0.873186i	$0.337950\pi$
$564$	0	0
$565$	5.45105e6	0.718388
$566$	0	0
$567$	−208630.	−0.0272533
$568$	0	0
$569$	−2.23824e6	−0.289818	−0.144909	−	0.989445i	$-0.546289\pi$
$569$	−2.23824e6	−0.289818	−0.144909	+	0.989445i	$0.546289\pi$
$570$	0	0
$571$	−1.04904e7	−1.34648	−0.673240	−	0.739424i	$-0.735098\pi$
$571$	−1.04904e7	−1.34648	−0.673240	+	0.739424i	$0.735098\pi$
$572$	0	0
$573$	1.44317e6	0.183625
$574$	0	0
$575$	1.74359e6	0.219925
$576$	0	0
$577$	4.40235e6	0.550484	0.275242	−	0.961375i	$-0.411242\pi$
$577$	4.40235e6	0.550484	0.275242	+	0.961375i	$0.411242\pi$
$578$	0	0
$579$	−6.17369e6	−0.765329
$580$	0	0
$581$	1.42010e6	0.174533
$582$	0	0
$583$	−1.07444e7	−1.30922
$584$	0	0
$585$	2.60215e6	0.314371
$586$	0	0
$587$	1.01305e7	1.21349	0.606747	−	0.794895i	$-0.292474\pi$
$587$	1.01305e7	1.21349	0.606747	+	0.794895i	$0.292474\pi$
$588$	0	0
$589$	5.59321e6	0.664313
$590$	0	0
$591$	−9.18739e6	−1.08199
$592$	0	0
$593$	−1.52764e7	−1.78396	−0.891980	−	0.452074i	$-0.850684\pi$
$593$	−1.52764e7	−1.78396	−0.891980	+	0.452074i	$0.850684\pi$
$594$	0	0
$595$	1.83549e6	0.212549
$596$	0	0
$597$	−5.84441e6	−0.671127
$598$	0	0
$599$	−1.61379e7	−1.83772	−0.918862	−	0.394580i	$-0.870890\pi$
$599$	−1.61379e7	−1.83772	−0.918862	+	0.394580i	$0.870890\pi$
$600$	0	0
$601$	4.52964e6	0.511538	0.255769	−	0.966738i	$-0.417671\pi$
$601$	4.52964e6	0.511538	0.255769	+	0.966738i	$0.417671\pi$
$602$	0	0
$603$	−7.01876e6	−0.786080
$604$	0	0
$605$	445335.	0.0494651
$606$	0	0
$607$	−2.28753e6	−0.251997	−0.125999	−	0.992030i	$-0.540213\pi$
$607$	−2.28753e6	−0.251997	−0.125999	+	0.992030i	$0.540213\pi$
$608$	0	0
$609$	−3.05792e6	−0.334105
$610$	0	0
$611$	4.94837e6	0.536240
$612$	0	0
$613$	−3.62618e6	−0.389761	−0.194880	−	0.980827i	$-0.562432\pi$
$613$	−3.62618e6	−0.389761	−0.194880	+	0.980827i	$0.562432\pi$
$614$	0	0
$615$	2.26087e6	0.241039
$616$	0	0
$617$	7.84457e6	0.829576	0.414788	−	0.909918i	$-0.363856\pi$
$617$	7.84457e6	0.829576	0.414788	+	0.909918i	$0.363856\pi$
$618$	0	0
$619$	−6.84380e6	−0.717911	−0.358955	−	0.933355i	$-0.616867\pi$
$619$	−6.84380e6	−0.717911	−0.358955	+	0.933355i	$0.616867\pi$
$620$	0	0
$621$	−1.07882e7	−1.12259
$622$	0	0
$623$	2.63578e6	0.272076
$624$	0	0
$625$	390625.	0.0400000
$626$	0	0
$627$	5.81878e6	0.591103
$628$	0	0
$629$	2.20852e7	2.22575
$630$	0	0
$631$	1.57191e7	1.57165	0.785825	−	0.618449i	$-0.212239\pi$
$631$	1.57191e7	1.57165	0.785825	+	0.618449i	$0.212239\pi$
$632$	0	0
$633$	−8.65450e6	−0.858485
$634$	0	0
$635$	5.86377e6	0.577089
$636$	0	0
$637$	−1.75709e6	−0.171571
$638$	0	0
$639$	5.12903e6	0.496916
$640$	0	0
$641$	−7.46226e6	−0.717340	−0.358670	−	0.933464i	$-0.616770\pi$
$641$	−7.46226e6	−0.717340	−0.358670	+	0.933464i	$0.616770\pi$
$642$	0	0
$643$	−1.86336e7	−1.77734	−0.888668	−	0.458551i	$-0.848369\pi$
$643$	−1.86336e7	−1.77734	−0.888668	+	0.458551i	$0.848369\pi$
$644$	0	0
$645$	591394.	0.0559729
$646$	0	0
$647$	1.78598e7	1.67732	0.838661	−	0.544654i	$-0.183339\pi$
$647$	1.78598e7	1.67732	0.838661	+	0.544654i	$0.183339\pi$
$648$	0	0
$649$	6.86956e6	0.640202
$650$	0	0
$651$	2.00733e6	0.185638
$652$	0	0
$653$	−549913.	−0.0504674	−0.0252337	−	0.999682i	$-0.508033\pi$
$653$	−549913.	−0.0504674	−0.0252337	+	0.999682i	$0.508033\pi$
$654$	0	0
$655$	−2.04701e6	−0.186430
$656$	0	0
$657$	309835.	0.0280038
$658$	0	0
$659$	8.42882e6	0.756055	0.378027	−	0.925794i	$-0.376603\pi$
$659$	8.42882e6	0.756055	0.378027	+	0.925794i	$0.376603\pi$
$660$	0	0
$661$	7.35056e6	0.654360	0.327180	−	0.944962i	$-0.393902\pi$
$661$	7.35056e6	0.654360	0.327180	+	0.944962i	$0.393902\pi$
$662$	0	0
$663$	−1.10073e7	−0.972521
$664$	0	0
$665$	1.67896e6	0.147227
$666$	0	0
$667$	−1.73432e7	−1.50943
$668$	0	0
$669$	−6.87580e6	−0.593961
$670$	0	0
$671$	6.17583e6	0.529529
$672$	0	0
$673$	1.31577e7	1.11981	0.559903	−	0.828558i	$-0.310838\pi$
$673$	1.31577e7	1.11981	0.559903	+	0.828558i	$0.310838\pi$
$674$	0	0
$675$	−2.41694e6	−0.204177
$676$	0	0
$677$	−739294.	−0.0619934	−0.0309967	−	0.999519i	$-0.509868\pi$
$677$	−739294.	−0.0619934	−0.0309967	+	0.999519i	$0.509868\pi$
$678$	0	0
$679$	−4.69276e6	−0.390619
$680$	0	0
$681$	−1.61422e6	−0.133382
$682$	0	0
$683$	2.02676e6	0.166246	0.0831228	−	0.996539i	$-0.473511\pi$
$683$	2.02676e6	0.166246	0.0831228	+	0.996539i	$0.473511\pi$
$684$	0	0
$685$	3.42017e6	0.278497
$686$	0	0
$687$	−1.48170e7	−1.19776
$688$	0	0
$689$	1.85919e7	1.49202
$690$	0	0
$691$	−2.86420e6	−0.228196	−0.114098	−	0.993470i	$-0.536398\pi$
$691$	−2.86420e6	−0.228196	−0.114098	+	0.993470i	$0.536398\pi$
$692$	0	0
$693$	−2.94747e6	−0.233140
$694$	0	0
$695$	−7.18154e6	−0.563969
$696$	0	0
$697$	1.34985e7	1.05245
$698$	0	0
$699$	−1.34243e7	−1.03920
$700$	0	0
$701$	6.71236e6	0.515917	0.257959	−	0.966156i	$-0.416950\pi$
$701$	6.71236e6	0.515917	0.257959	+	0.966156i	$0.416950\pi$
$702$	0	0
$703$	2.02018e7	1.54171
$704$	0	0
$705$	−1.69694e6	−0.128586
$706$	0	0
$707$	5.04931e6	0.379912
$708$	0	0
$709$	6.24637e6	0.466672	0.233336	−	0.972396i	$-0.425036\pi$
$709$	6.24637e6	0.466672	0.233336	+	0.972396i	$0.425036\pi$
$710$	0	0
$711$	−1.53723e7	−1.14042
$712$	0	0
$713$	1.13847e7	0.838682
$714$	0	0
$715$	−7.73754e6	−0.566028
$716$	0	0
$717$	−311833.	−0.0226530
$718$	0	0
$719$	1.33936e7	0.966215	0.483107	−	0.875561i	$-0.339508\pi$
$719$	1.33936e7	0.966215	0.483107	+	0.875561i	$0.339508\pi$
$720$	0	0
$721$	−5.40986e6	−0.387568
$722$	0	0
$723$	9.28257e6	0.660423
$724$	0	0
$725$	−3.88548e6	−0.274536
$726$	0	0
$727$	1.81735e7	1.27527	0.637635	−	0.770338i	$-0.279913\pi$
$727$	1.81735e7	1.27527	0.637635	+	0.770338i	$0.279913\pi$
$728$	0	0
$729$	1.00388e7	0.699618
$730$	0	0
$731$	3.53091e6	0.244396
$732$	0	0
$733$	9.72669e6	0.668660	0.334330	−	0.942456i	$-0.391490\pi$
$733$	9.72669e6	0.668660	0.334330	+	0.942456i	$0.391490\pi$
$734$	0	0
$735$	602557.	0.0411414
$736$	0	0
$737$	2.08704e7	1.41535
$738$	0	0
$739$	−1.17199e7	−0.789431	−0.394716	−	0.918803i	$-0.629157\pi$
$739$	−1.17199e7	−0.789431	−0.394716	+	0.918803i	$0.629157\pi$
$740$	0	0
$741$	−1.00686e7	−0.673637
$742$	0	0
$743$	−1.66089e7	−1.10374	−0.551872	−	0.833929i	$-0.686086\pi$
$743$	−1.66089e7	−1.10374	−0.551872	+	0.833929i	$0.686086\pi$
$744$	0	0
$745$	7.74349e6	0.511147
$746$	0	0
$747$	−4.12205e6	−0.270279
$748$	0	0
$749$	−4.05525e6	−0.264127
$750$	0	0
$751$	4.81282e6	0.311386	0.155693	−	0.987805i	$-0.450239\pi$
$751$	4.81282e6	0.311386	0.155693	+	0.987805i	$0.450239\pi$
$752$	0	0
$753$	5.63218e6	0.361984
$754$	0	0
$755$	−1.21537e7	−0.775965
$756$	0	0
$757$	1.44840e7	0.918645	0.459323	−	0.888270i	$-0.348092\pi$
$757$	1.44840e7	0.918645	0.459323	+	0.888270i	$0.348092\pi$
$758$	0	0
$759$	1.18438e7	0.746256
$760$	0	0
$761$	6.97379e6	0.436523	0.218262	−	0.975890i	$-0.429961\pi$
$761$	6.97379e6	0.436523	0.218262	+	0.975890i	$0.429961\pi$
$762$	0	0
$763$	4.74343e6	0.294972
$764$	0	0
$765$	−5.32779e6	−0.329150
$766$	0	0
$767$	−1.18869e7	−0.729591
$768$	0	0
$769$	1.57839e7	0.962494	0.481247	−	0.876585i	$-0.340184\pi$
$769$	1.57839e7	0.962494	0.481247	+	0.876585i	$0.340184\pi$
$770$	0	0
$771$	3.17254e6	0.192208
$772$	0	0
$773$	2.86689e7	1.72569	0.862843	−	0.505472i	$-0.168682\pi$
$773$	2.86689e7	1.72569	0.862843	+	0.505472i	$0.168682\pi$
$774$	0	0
$775$	2.55057e6	0.152540
$776$	0	0
$777$	7.25017e6	0.430820
$778$	0	0
$779$	1.23473e7	0.729004
$780$	0	0
$781$	−1.52513e7	−0.894701
$782$	0	0
$783$	2.40409e7	1.40135
$784$	0	0
$785$	1.26933e7	0.735188
$786$	0	0
$787$	−2.82453e7	−1.62558	−0.812791	−	0.582556i	$-0.802053\pi$
$787$	−2.82453e7	−1.62558	−0.812791	+	0.582556i	$0.802053\pi$
$788$	0	0
$789$	−2.63045e6	−0.150431
$790$	0	0
$791$	1.06841e7	0.607149
$792$	0	0
$793$	−1.06865e7	−0.603465
$794$	0	0
$795$	−6.37570e6	−0.357775
$796$	0	0
$797$	−2.95525e7	−1.64797	−0.823983	−	0.566615i	$-0.808253\pi$
$797$	−2.95525e7	−1.64797	−0.823983	+	0.566615i	$0.808253\pi$
$798$	0	0
$799$	−1.01316e7	−0.561448
$800$	0	0
$801$	−7.65077e6	−0.421331
$802$	0	0
$803$	−921302.	−0.0504212
$804$	0	0
$805$	3.41744e6	0.185871
$806$	0	0
$807$	−1.21618e7	−0.657377
$808$	0	0
$809$	−1.18511e7	−0.636631	−0.318316	−	0.947985i	$-0.603117\pi$
$809$	−1.18511e7	−0.636631	−0.318316	+	0.947985i	$0.603117\pi$
$810$	0	0
$811$	3.41901e7	1.82536	0.912680	−	0.408676i	$-0.134009\pi$
$811$	3.41901e7	1.82536	0.912680	+	0.408676i	$0.134009\pi$
$812$	0	0
$813$	1.53317e6	0.0813510
$814$	0	0
$815$	−5.26745e6	−0.277783
$816$	0	0
$817$	3.22980e6	0.169286
$818$	0	0
$819$	5.10021e6	0.265692
$820$	0	0
$821$	1.92970e7	0.999152	0.499576	−	0.866270i	$-0.333489\pi$
$821$	1.92970e7	0.999152	0.499576	+	0.866270i	$0.333489\pi$
$822$	0	0
$823$	−1.68887e7	−0.869156	−0.434578	−	0.900634i	$-0.643103\pi$
$823$	−1.68887e7	−0.869156	−0.434578	+	0.900634i	$0.643103\pi$
$824$	0	0
$825$	2.65343e6	0.135729
$826$	0	0
$827$	−2.80486e6	−0.142609	−0.0713046	−	0.997455i	$-0.522716\pi$
$827$	−2.80486e6	−0.142609	−0.0713046	+	0.997455i	$0.522716\pi$
$828$	0	0
$829$	−7.98746e6	−0.403666	−0.201833	−	0.979420i	$-0.564690\pi$
$829$	−7.98746e6	−0.403666	−0.201833	+	0.979420i	$0.564690\pi$
$830$	0	0
$831$	6.47856e6	0.325444
$832$	0	0
$833$	3.59756e6	0.179637
$834$	0	0
$835$	−1.13744e7	−0.564561
$836$	0	0
$837$	−1.57813e7	−0.778626
$838$	0	0
$839$	−1.93272e7	−0.947901	−0.473950	−	0.880552i	$-0.657172\pi$
$839$	−1.93272e7	−0.947901	−0.473950	+	0.880552i	$0.657172\pi$
$840$	0	0
$841$	1.81370e7	0.884251
$842$	0	0
$843$	8.74684e6	0.423918
$844$	0	0
$845$	4.10650e6	0.197847
$846$	0	0
$847$	872857.	0.0418056
$848$	0	0
$849$	1.22945e7	0.585386
$850$	0	0
$851$	4.11198e7	1.94638
$852$	0	0
$853$	−3.94339e7	−1.85566	−0.927828	−	0.373009i	$-0.878326\pi$
$853$	−3.94339e7	−1.85566	−0.927828	+	0.373009i	$0.878326\pi$
$854$	0	0
$855$	−4.87344e6	−0.227992
$856$	0	0
$857$	6.72300e6	0.312688	0.156344	−	0.987703i	$-0.450029\pi$
$857$	6.72300e6	0.312688	0.156344	+	0.987703i	$0.450029\pi$
$858$	0	0
$859$	−5.35827e6	−0.247766	−0.123883	−	0.992297i	$-0.539535\pi$
$859$	−5.35827e6	−0.247766	−0.123883	+	0.992297i	$0.539535\pi$
$860$	0	0
$861$	4.43130e6	0.203715
$862$	0	0
$863$	5.33529e6	0.243855	0.121927	−	0.992539i	$-0.461092\pi$
$863$	5.33529e6	0.243855	0.121927	+	0.992539i	$0.461092\pi$
$864$	0	0
$865$	−5.16728e6	−0.234813
$866$	0	0
$867$	8.28393e6	0.374273
$868$	0	0
$869$	4.57098e7	2.05333
$870$	0	0
$871$	−3.61136e7	−1.61296
$872$	0	0
$873$	1.36214e7	0.604905
$874$	0	0
$875$	765625.	0.0338062
$876$	0	0
$877$	−1.59912e7	−0.702073	−0.351036	−	0.936362i	$-0.614171\pi$
$877$	−1.59912e7	−0.702073	−0.351036	+	0.936362i	$0.614171\pi$
$878$	0	0
$879$	1.81394e6	0.0791865
$880$	0	0
$881$	4.36775e6	0.189591	0.0947956	−	0.995497i	$-0.469780\pi$
$881$	4.36775e6	0.189591	0.0947956	+	0.995497i	$0.469780\pi$
$882$	0	0
$883$	2.91052e7	1.25623	0.628114	−	0.778122i	$-0.283827\pi$
$883$	2.91052e7	1.25623	0.628114	+	0.778122i	$0.283827\pi$
$884$	0	0
$885$	4.07636e6	0.174950
$886$	0	0
$887$	3.99268e7	1.70395	0.851973	−	0.523586i	$-0.175406\pi$
$887$	3.99268e7	1.70395	0.851973	+	0.523586i	$0.175406\pi$
$888$	0	0
$889$	1.14930e7	0.487729
$890$	0	0
$891$	−1.80071e6	−0.0759887
$892$	0	0
$893$	−9.26756e6	−0.388899
$894$	0	0
$895$	−5.89998e6	−0.246203
$896$	0	0
$897$	−2.04942e7	−0.850453
$898$	0	0
$899$	−2.53700e7	−1.04694
$900$	0	0
$901$	−3.80660e7	−1.56216
$902$	0	0
$903$	1.15913e6	0.0473057
$904$	0	0
$905$	−9.13396e6	−0.370713
$906$	0	0
$907$	−4.59920e7	−1.85637	−0.928183	−	0.372123i	$-0.878630\pi$
$907$	−4.59920e7	−1.85637	−0.928183	+	0.372123i	$0.878630\pi$
$908$	0	0
$909$	−1.46564e7	−0.588325
$910$	0	0
$911$	−2.02730e7	−0.809322	−0.404661	−	0.914467i	$-0.632610\pi$
$911$	−2.02730e7	−0.809322	−0.404661	+	0.914467i	$0.632610\pi$
$912$	0	0
$913$	1.22570e7	0.486639
$914$	0	0
$915$	3.66471e6	0.144706
$916$	0	0
$917$	−4.01214e6	−0.157562
$918$	0	0
$919$	−1.94257e7	−0.758730	−0.379365	−	0.925247i	$-0.623858\pi$
$919$	−1.94257e7	−0.758730	−0.379365	+	0.925247i	$0.623858\pi$
$920$	0	0
$921$	2.57836e7	1.00160
$922$	0	0
$923$	2.63903e7	1.01963
$924$	0	0
$925$	9.21226e6	0.354007
$926$	0	0
$927$	1.57029e7	0.600181
$928$	0	0
$929$	4.59308e7	1.74608	0.873040	−	0.487648i	$-0.162145\pi$
$929$	4.59308e7	1.74608	0.873040	+	0.487648i	$0.162145\pi$
$930$	0	0
$931$	3.29076e6	0.124429
$932$	0	0
$933$	−2.84809e7	−1.07115
$934$	0	0
$935$	1.58423e7	0.592637
$936$	0	0
$937$	−7.92867e6	−0.295020	−0.147510	−	0.989061i	$-0.547126\pi$
$937$	−7.92867e6	−0.295020	−0.147510	+	0.989061i	$0.547126\pi$
$938$	0	0
$939$	−1.30562e7	−0.483228
$940$	0	0
$941$	−4.31133e7	−1.58722	−0.793610	−	0.608426i	$-0.791801\pi$
$941$	−4.31133e7	−1.58722	−0.793610	+	0.608426i	$0.791801\pi$
$942$	0	0
$943$	2.51324e7	0.920353
$944$	0	0
$945$	−4.73720e6	−0.172561
$946$	0	0
$947$	4.33497e7	1.57077	0.785383	−	0.619011i	$-0.212466\pi$
$947$	4.33497e7	1.57077	0.785383	+	0.619011i	$0.212466\pi$
$948$	0	0
$949$	1.59419e6	0.0574613
$950$	0	0
$951$	2.38240e7	0.854209
$952$	0	0
$953$	3.08760e6	0.110126	0.0550628	−	0.998483i	$-0.482464\pi$
$953$	3.08760e6	0.110126	0.0550628	+	0.998483i	$0.482464\pi$
$954$	0	0
$955$	3.59412e6	0.127522
$956$	0	0
$957$	−2.63932e7	−0.931563
$958$	0	0
$959$	6.70353e6	0.235373
$960$	0	0
$961$	−1.19753e7	−0.418292
$962$	0	0
$963$	1.17710e7	0.409023
$964$	0	0
$965$	−1.53751e7	−0.531497
$966$	0	0
$967$	−5.37145e6	−0.184725	−0.0923623	−	0.995725i	$-0.529442\pi$
$967$	−5.37145e6	−0.184725	−0.0923623	+	0.995725i	$0.529442\pi$
$968$	0	0
$969$	2.06151e7	0.705304
$970$	0	0
$971$	−588590.	−0.0200339	−0.0100169	−	0.999950i	$-0.503189\pi$
$971$	−588590.	−0.0200339	−0.0100169	+	0.999950i	$0.503189\pi$
$972$	0	0
$973$	−1.40758e7	−0.476641
$974$	0	0
$975$	−4.59142e6	−0.154680
$976$	0	0
$977$	−8.96258e6	−0.300398	−0.150199	−	0.988656i	$-0.547991\pi$
$977$	−8.96258e6	−0.300398	−0.150199	+	0.988656i	$0.547991\pi$
$978$	0	0
$979$	2.27497e7	0.758611
$980$	0	0
$981$	−1.37685e7	−0.456789
$982$	0	0
$983$	1.47149e7	0.485705	0.242853	−	0.970063i	$-0.421917\pi$
$983$	1.47149e7	0.485705	0.242853	+	0.970063i	$0.421917\pi$
$984$	0	0
$985$	−2.28805e7	−0.751408
$986$	0	0
$987$	−3.32600e6	−0.108675
$988$	0	0
$989$	6.57409e6	0.213720
$990$	0	0
$991$	−3.67288e7	−1.18802	−0.594008	−	0.804459i	$-0.702455\pi$
$991$	−3.67288e7	−1.18802	−0.594008	+	0.804459i	$0.702455\pi$
$992$	0	0
$993$	−4.64251e6	−0.149410
$994$	0	0
$995$	−1.45551e7	−0.466077
$996$	0	0
$997$	2.22158e6	0.0707821	0.0353910	−	0.999374i	$-0.488732\pi$
$997$	2.22158e6	0.0707821	0.0353910	+	0.999374i	$0.488732\pi$
$998$	0	0
$999$	−5.69996e7	−1.80700

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	280.6.a.j.1.3	✓	5
4.3	odd	2		560.6.a.y.1.3		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
280.6.a.j.1.3	✓	5	1.1	even	1	trivial
560.6.a.y.1.3		5	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)

\(f(q)\)	\(=\)	\(q+10.0384 q^{3} +25.0000 q^{5} +49.0000 q^{7} -142.230 q^{9} +O(q^{10})\)	\(q+10.0384 q^{3} +25.0000 q^{5} +49.0000 q^{7} -142.230 q^{9} +422.924 q^{11} -731.815 q^{13} +250.961 q^{15} +1498.36 q^{17} +1370.58 q^{19} +491.883 q^{21} +2789.75 q^{23} +625.000 q^{25} -3867.10 q^{27} -6216.76 q^{29} +4080.91 q^{31} +4245.49 q^{33} +1225.00 q^{35} +14739.6 q^{37} -7346.27 q^{39} +9008.85 q^{41} +2356.52 q^{43} -3555.75 q^{45} -6761.78 q^{47} +2401.00 q^{49} +15041.2 q^{51} -25405.2 q^{53} +10573.1 q^{55} +13758.5 q^{57} +16243.0 q^{59} +14602.7 q^{61} -6969.27 q^{63} -18295.4 q^{65} +49347.9 q^{67} +28004.7 q^{69} -36061.5 q^{71} -2178.41 q^{73} +6274.02 q^{75} +20723.3 q^{77} +108080. q^{79} -4257.76 q^{81} +28981.6 q^{83} +37459.0 q^{85} -62406.5 q^{87} +53791.5 q^{89} -35858.9 q^{91} +40965.9 q^{93} +34264.5 q^{95} -95770.6 q^{97} -60152.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9}+O(q^{10}) \) 5 * q + 15 * q^3 + 125 * q^5 + 245 * q^7 + 1132 * q^9	\( 5 q + 15 q^{3} + 125 q^{5} + 245 q^{7} + 1132 q^{9} + 281 q^{11} + 909 q^{13} + 375 q^{15} + 1495 q^{17} - 422 q^{19} + 735 q^{21} - 62 q^{23} + 3125 q^{25} - 3363 q^{27} - 2047 q^{29} + 1636 q^{31} + 19181 q^{33} + 6125 q^{35} - 10358 q^{37} + 15685 q^{39} + 6424 q^{41} + 28306 q^{43} + 28300 q^{45} + 20955 q^{47} + 12005 q^{49} - 23577 q^{51} + 43748 q^{53} + 7025 q^{55} + 13690 q^{57} + 45788 q^{59} + 50432 q^{61} + 55468 q^{63} + 22725 q^{65} + 40712 q^{67} + 35050 q^{69} - 3096 q^{71} + 135438 q^{73} + 9375 q^{75} + 13769 q^{77} + 13191 q^{79} + 381101 q^{81} + 35108 q^{83} + 37375 q^{85} + 297289 q^{87} + 213772 q^{89} + 44541 q^{91} + 134244 q^{93} - 10550 q^{95} + 10659 q^{97} + 39462 q^{99}+O(q^{100}) \) 5 * q + 15 * q^3 + 125 * q^5 + 245 * q^7 + 1132 * q^9 + 281 * q^11 + 909 * q^13 + 375 * q^15 + 1495 * q^17 - 422 * q^19 + 735 * q^21 - 62 * q^23 + 3125 * q^25 - 3363 * q^27 - 2047 * q^29 + 1636 * q^31 + 19181 * q^33 + 6125 * q^35 - 10358 * q^37 + 15685 * q^39 + 6424 * q^41 + 28306 * q^43 + 28300 * q^45 + 20955 * q^47 + 12005 * q^49 - 23577 * q^51 + 43748 * q^53 + 7025 * q^55 + 13690 * q^57 + 45788 * q^59 + 50432 * q^61 + 55468 * q^63 + 22725 * q^65 + 40712 * q^67 + 35050 * q^69 - 3096 * q^71 + 135438 * q^73 + 9375 * q^75 + 13769 * q^77 + 13191 * q^79 + 381101 * q^81 + 35108 * q^83 + 37375 * q^85 + 297289 * q^87 + 213772 * q^89 + 44541 * q^91 + 134244 * q^93 - 10550 * q^95 + 10659 * q^97 + 39462 * q^99

Introduction

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