LMFDB - Embedded newform 585.8.a.a.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [585,8,Mod(1,585)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(585, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 8, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("585.1");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 585 = 3^{2} \cdot 5 \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	585.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:	$182.745239908$
Analytic rank:	$2$
Dimension:	$1$
Coefficient field:	$\mathbb{Q}$
Coefficient ring:	$\mathbb{Z}$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 195)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Character	$\chi$	$=$	585.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$

$=$

$q-22.0000 q^{2} +356.000 q^{4} +125.000 q^{5} -1366.00 q^{7} -5016.00 q^{8} +O(q^{10})$

\(q-22.0000 q^{2} +356.000 q^{4} +125.000 q^{5} -1366.00 q^{7} -5016.00 q^{8} -2750.00 q^{10} -4204.00 q^{11} -2197.00 q^{13} +30052.0 q^{14} +64784.0 q^{16} -22300.0 q^{17} +36138.0 q^{19} +44500.0 q^{20} +92488.0 q^{22} -49830.0 q^{23} +15625.0 q^{25} +48334.0 q^{26} -486296. q^{28} -194704. q^{29} -146992. q^{31} -783200. q^{32} +490600. q^{34} -170750. q^{35} +314494. q^{37} -795036. q^{38} -627000. q^{40} -227838. q^{41} -1.02828e6 q^{43} -1.49662e6 q^{44} +1.09626e6 q^{46} -612488. q^{47} +1.04241e6 q^{49} -343750. q^{50} -782132. q^{52} -890902. q^{53} -525500. q^{55} +6.85186e6 q^{56} +4.28349e6 q^{58} +2.16930e6 q^{59} -2.55795e6 q^{61} +3.23382e6 q^{62} +8.93805e6 q^{64} -274625. q^{65} +748436. q^{67} -7.93880e6 q^{68} +3.75650e6 q^{70} -1.95418e6 q^{71} -3.18433e6 q^{73} -6.91887e6 q^{74} +1.28651e7 q^{76} +5.74266e6 q^{77} +2.81321e6 q^{79} +8.09800e6 q^{80} +5.01244e6 q^{82} -3.94232e6 q^{83} -2.78750e6 q^{85} +2.26221e7 q^{86} +2.10873e7 q^{88} -1.54285e6 q^{89} +3.00110e6 q^{91} -1.77395e7 q^{92} +1.34747e7 q^{94} +4.51725e6 q^{95} -4.72480e6 q^{97} -2.29331e7 q^{98} +O(q^{100})\)

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 $

$2$	−22.0000	−1.94454	−0.972272	−	0.233854i	$-0.924866\pi$
$2$	−22.0000	−1.94454	−0.972272	+	0.233854i	$0.924866\pi$
$3$	0	0
$3$	0	0
$4$	356.000	2.78125
$5$	125.000	0.447214
$5$	125.000	0.447214
$6$	0	0
$7$	−1366.00	−1.50525	−0.752623	−	0.658452i	$-0.771212\pi$
$7$	−1366.00	−1.50525	−0.752623	+	0.658452i	$0.771212\pi$
$8$	−5016.00	−3.46372
$9$	0	0
$10$	−2750.00	−0.869626
$11$	−4204.00	−0.952332	−0.476166	−	0.879355i	$-0.657974\pi$
$11$	−4204.00	−0.952332	−0.476166	+	0.879355i	$0.657974\pi$
$12$	0	0
$13$	−2197.00	−0.277350
$13$	−2197.00	−0.277350
$14$	30052.0	2.92702
$15$	0	0
$16$	64784.0	3.95410
$17$	−22300.0	−1.10086	−0.550432	−	0.834880i	$-0.685537\pi$
$17$	−22300.0	−1.10086	−0.550432	+	0.834880i	$0.685537\pi$
$18$	0	0
$19$	36138.0	1.20872	0.604361	−	0.796711i	$-0.293428\pi$
$19$	36138.0	1.20872	0.604361	+	0.796711i	$0.293428\pi$
$20$	44500.0	1.24381
$21$	0	0
$22$	92488.0	1.85185
$23$	−49830.0	−0.853972	−0.426986	−	0.904258i	$-0.640425\pi$
$23$	−49830.0	−0.853972	−0.426986	+	0.904258i	$0.640425\pi$
$24$	0	0
$25$	15625.0	0.200000
$26$	48334.0	0.539319
$27$	0	0
$28$	−486296.	−4.18647
$29$	−194704.	−1.48246	−0.741228	−	0.671253i	$-0.765756\pi$
$29$	−194704.	−1.48246	−0.741228	+	0.671253i	$0.765756\pi$
$30$	0	0
$31$	−146992.	−0.886192	−0.443096	−	0.896474i	$-0.646120\pi$
$31$	−146992.	−0.886192	−0.443096	+	0.896474i	$0.646120\pi$
$32$	−783200.	−4.22520
$33$	0	0
$34$	490600.	2.14068
$35$	−170750.	−0.673167
$36$	0	0
$37$	314494.	1.02072	0.510360	−	0.859961i	$-0.329512\pi$
$37$	314494.	1.02072	0.510360	+	0.859961i	$0.329512\pi$
$38$	−795036.	−2.35041
$39$	0	0
$40$	−627000.	−1.54902
$41$	−227838.	−0.516277	−0.258138	−	0.966108i	$-0.583109\pi$
$41$	−227838.	−0.516277	−0.258138	+	0.966108i	$0.583109\pi$
$42$	0	0
$43$	−1.02828e6	−1.97229	−0.986143	−	0.165895i	$-0.946949\pi$
$43$	−1.02828e6	−1.97229	−0.986143	+	0.165895i	$0.946949\pi$
$44$	−1.49662e6	−2.64867
$45$	0	0
$46$	1.09626e6	1.66059
$47$	−612488.	−0.860508	−0.430254	−	0.902708i	$-0.641576\pi$
$47$	−612488.	−0.860508	−0.430254	+	0.902708i	$0.641576\pi$
$48$	0	0
$49$	1.04241e6	1.26577
$50$	−343750.	−0.388909
$51$	0	0
$52$	−782132.	−0.771380
$53$	−890902.	−0.821986	−0.410993	−	0.911639i	$-0.634818\pi$
$53$	−890902.	−0.821986	−0.410993	+	0.911639i	$0.634818\pi$
$54$	0	0
$55$	−525500.	−0.425896
$56$	6.85186e6	5.21375
$57$	0	0
$58$	4.28349e6	2.88270
$59$	2.16930e6	1.37511	0.687555	−	0.726132i	$-0.258684\pi$
$59$	2.16930e6	1.37511	0.687555	+	0.726132i	$0.258684\pi$
$60$	0	0
$61$	−2.55795e6	−1.44291	−0.721453	−	0.692463i	$-0.756526\pi$
$61$	−2.55795e6	−1.44291	−0.721453	+	0.692463i	$0.756526\pi$
$62$	3.23382e6	1.72324
$63$	0	0
$64$	8.93805e6	4.26199
$65$	−274625.	−0.124035
$66$	0	0
$67$	748436.	0.304013	0.152007	−	0.988379i	$-0.451426\pi$
$67$	748436.	0.304013	0.152007	+	0.988379i	$0.451426\pi$
$68$	−7.93880e6	−3.06178
$69$	0	0
$70$	3.75650e6	1.30900
$71$	−1.95418e6	−0.647977	−0.323988	−	0.946061i	$-0.605024\pi$
$71$	−1.95418e6	−0.647977	−0.323988	+	0.946061i	$0.605024\pi$
$72$	0	0
$73$	−3.18433e6	−0.958050	−0.479025	−	0.877801i	$-0.659010\pi$
$73$	−3.18433e6	−0.958050	−0.479025	+	0.877801i	$0.659010\pi$
$74$	−6.91887e6	−1.98483
$75$	0	0
$76$	1.28651e7	3.36176
$77$	5.74266e6	1.43349
$78$	0	0
$79$	2.81321e6	0.641959	0.320979	−	0.947086i	$-0.395988\pi$
$79$	2.81321e6	0.641959	0.320979	+	0.947086i	$0.395988\pi$
$80$	8.09800e6	1.76833
$81$	0	0
$82$	5.01244e6	1.00392
$83$	−3.94232e6	−0.756795	−0.378397	−	0.925643i	$-0.623525\pi$
$83$	−3.94232e6	−0.756795	−0.378397	+	0.925643i	$0.623525\pi$
$84$	0	0
$85$	−2.78750e6	−0.492321
$86$	2.26221e7	3.83520
$87$	0	0
$88$	2.10873e7	3.29861
$89$	−1.54285e6	−0.231984	−0.115992	−	0.993250i	$-0.537005\pi$
$89$	−1.54285e6	−0.231984	−0.115992	+	0.993250i	$0.537005\pi$
$90$	0	0
$91$	3.00110e6	0.417480
$92$	−1.77395e7	−2.37511
$93$	0	0
$94$	1.34747e7	1.67330
$95$	4.51725e6	0.540557
$96$	0	0
$97$	−4.72480e6	−0.525633	−0.262817	−	0.964846i	$-0.584651\pi$
$97$	−4.72480e6	−0.525633	−0.262817	+	0.964846i	$0.584651\pi$
$98$	−2.29331e7	−2.46134
$99$	0	0
$100$	5.56250e6	0.556250
$101$	−1.78948e7	−1.72824	−0.864118	−	0.503289i	$-0.832123\pi$
$101$	−1.78948e7	−1.72824	−0.864118	+	0.503289i	$0.832123\pi$
$102$	0	0
$103$	−3.95441e6	−0.356576	−0.178288	−	0.983978i	$-0.557056\pi$
$103$	−3.95441e6	−0.356576	−0.178288	+	0.983978i	$0.557056\pi$
$104$	1.10202e7	0.960663
$105$	0	0
$106$	1.95998e7	1.59839
$107$	−5.02284e6	−0.396375	−0.198187	−	0.980164i	$-0.563505\pi$
$107$	−5.02284e6	−0.396375	−0.198187	+	0.980164i	$0.563505\pi$
$108$	0	0
$109$	74680.0	0.00552346	0.00276173	−	0.999996i	$-0.499121\pi$
$109$	74680.0	0.00552346	0.00276173	+	0.999996i	$0.499121\pi$
$110$	1.15610e7	0.828173
$111$	0	0
$112$	−8.84949e7	−5.95190
$113$	2.25526e7	1.47035	0.735176	−	0.677876i	$-0.237099\pi$
$113$	2.25526e7	1.47035	0.735176	+	0.677876i	$0.237099\pi$
$114$	0	0
$115$	−6.22875e6	−0.381908
$116$	−6.93146e7	−4.12308
$117$	0	0
$118$	−4.77246e7	−2.67396
$119$	3.04618e7	1.65707
$120$	0	0
$121$	−1.81356e6	−0.0930640
$122$	5.62750e7	2.80580
$123$	0	0
$124$	−5.23292e7	−2.46472
$125$	1.95312e6	0.0894427
$126$	0	0
$127$	−1.19597e7	−0.518093	−0.259047	−	0.965865i	$-0.583408\pi$
$127$	−1.19597e7	−0.518093	−0.259047	+	0.965865i	$0.583408\pi$
$128$	−9.63875e7	−4.06243
$129$	0	0
$130$	6.04175e6	0.241191
$131$	−2.83610e7	−1.10223	−0.551115	−	0.834429i	$-0.685798\pi$
$131$	−2.83610e7	−1.10223	−0.551115	+	0.834429i	$0.685798\pi$
$132$	0	0
$133$	−4.93645e7	−1.81942
$134$	−1.64656e7	−0.591168
$135$	0	0
$136$	1.11857e8	3.81308
$137$	−1.48900e7	−0.494735	−0.247367	−	0.968922i	$-0.579565\pi$
$137$	−1.48900e7	−0.494735	−0.247367	+	0.968922i	$0.579565\pi$
$138$	0	0
$139$	1.04927e7	0.331386	0.165693	−	0.986177i	$-0.447014\pi$
$139$	1.04927e7	0.331386	0.165693	+	0.986177i	$0.447014\pi$
$140$	−6.07870e7	−1.87224
$141$	0	0
$142$	4.29919e7	1.26002
$143$	9.23619e6	0.264129
$144$	0	0
$145$	−2.43380e7	−0.662975
$146$	7.00553e7	1.86297
$147$	0	0
$148$	1.11960e8	2.83888
$149$	−2.79470e7	−0.692122	−0.346061	−	0.938212i	$-0.612481\pi$
$149$	−2.79470e7	−0.692122	−0.346061	+	0.938212i	$0.612481\pi$
$150$	0	0
$151$	5.60904e7	1.32577	0.662886	−	0.748720i	$-0.269331\pi$
$151$	5.60904e7	1.32577	0.662886	+	0.748720i	$0.269331\pi$
$152$	−1.81268e8	−4.18667
$153$	0	0
$154$	−1.26339e8	−2.78749
$155$	−1.83740e7	−0.396317
$156$	0	0
$157$	−7.26436e7	−1.49813	−0.749063	−	0.662498i	$-0.769496\pi$
$157$	−7.26436e7	−1.49813	−0.749063	+	0.662498i	$0.769496\pi$
$158$	−6.18906e7	−1.24832
$159$	0	0
$160$	−9.79000e7	−1.88957
$161$	6.80678e7	1.28544
$162$	0	0
$163$	−9.20053e7	−1.66401	−0.832006	−	0.554767i	$-0.812807\pi$
$163$	−9.20053e7	−1.66401	−0.832006	+	0.554767i	$0.812807\pi$
$164$	−8.11103e7	−1.43590
$165$	0	0
$166$	8.67310e7	1.47162
$167$	8.91443e7	1.48111	0.740553	−	0.671998i	$-0.234564\pi$
$167$	8.91443e7	1.48111	0.740553	+	0.671998i	$0.234564\pi$
$168$	0	0
$169$	4.82681e6	0.0769231
$170$	6.13250e7	0.957340
$171$	0	0
$172$	−3.66066e8	−5.48542
$173$	−5.50650e7	−0.808564	−0.404282	−	0.914634i	$-0.632478\pi$
$173$	−5.50650e7	−0.808564	−0.404282	+	0.914634i	$0.632478\pi$
$174$	0	0
$175$	−2.13438e7	−0.301049
$176$	−2.72352e8	−3.76562
$177$	0	0
$178$	3.39427e7	0.451104
$179$	−8.95216e7	−1.16665	−0.583327	−	0.812237i	$-0.698249\pi$
$179$	−8.95216e7	−1.16665	−0.583327	+	0.812237i	$0.698249\pi$
$180$	0	0
$181$	6.22521e7	0.780331	0.390166	−	0.920745i	$-0.372418\pi$
$181$	6.22521e7	0.780331	0.390166	+	0.920745i	$0.372418\pi$
$182$	−6.60242e7	−0.811808
$183$	0	0
$184$	2.49947e8	2.95792
$185$	3.93118e7	0.456480
$186$	0	0
$187$	9.37492e7	1.04839
$188$	−2.18046e8	−2.39329
$189$	0	0
$190$	−9.93795e7	−1.05114
$191$	1.22650e8	1.27365	0.636824	−	0.771009i	$-0.280248\pi$
$191$	1.22650e8	1.27365	0.636824	+	0.771009i	$0.280248\pi$
$192$	0	0
$193$	1.09034e7	0.109172	0.0545858	−	0.998509i	$-0.482616\pi$
$193$	1.09034e7	0.109172	0.0545858	+	0.998509i	$0.482616\pi$
$194$	1.03946e8	1.02212
$195$	0	0
$196$	3.71099e8	3.52041
$197$	1.74166e8	1.62305	0.811526	−	0.584316i	$-0.198637\pi$
$197$	1.74166e8	1.62305	0.811526	+	0.584316i	$0.198637\pi$
$198$	0	0
$199$	−2.09993e8	−1.88894	−0.944471	−	0.328596i	$-0.893425\pi$
$199$	−2.09993e8	−1.88894	−0.944471	+	0.328596i	$0.893425\pi$
$200$	−7.83750e7	−0.692744
$201$	0	0
$202$	3.93687e8	3.36063
$203$	2.65966e8	2.23146
$204$	0	0
$205$	−2.84798e7	−0.230886
$206$	8.69971e7	0.693377
$207$	0	0
$208$	−1.42330e8	−1.09667
$209$	−1.51924e8	−1.15110
$210$	0	0
$211$	−2.60205e8	−1.90690	−0.953449	−	0.301555i	$-0.902494\pi$
$211$	−2.60205e8	−1.90690	−0.953449	+	0.301555i	$0.902494\pi$
$212$	−3.17161e8	−2.28615
$213$	0	0
$214$	1.10502e8	0.770768
$215$	−1.28534e8	−0.882034
$216$	0	0
$217$	2.00791e8	1.33394
$218$	−1.64296e6	−0.0107406
$219$	0	0
$220$	−1.87078e8	−1.18452
$221$	4.89931e7	0.305325
$222$	0	0
$223$	9.82511e7	0.593295	0.296647	−	0.954987i	$-0.404131\pi$
$223$	9.82511e7	0.593295	0.296647	+	0.954987i	$0.404131\pi$
$224$	1.06985e9	6.35997
$225$	0	0
$226$	−4.96157e8	−2.85917
$227$	1.06826e8	0.606160	0.303080	−	0.952965i	$-0.401985\pi$
$227$	1.06826e8	0.606160	0.303080	+	0.952965i	$0.401985\pi$
$228$	0	0
$229$	−1.06260e8	−0.584716	−0.292358	−	0.956309i	$-0.594440\pi$
$229$	−1.06260e8	−0.584716	−0.292358	+	0.956309i	$0.594440\pi$
$230$	1.37032e8	0.742636
$231$	0	0
$232$	9.76635e8	5.13481
$233$	−9.00162e7	−0.466203	−0.233101	−	0.972452i	$-0.574887\pi$
$233$	−9.00162e7	−0.466203	−0.233101	+	0.972452i	$0.574887\pi$
$234$	0	0
$235$	−7.65610e7	−0.384831
$236$	7.72271e8	3.82453
$237$	0	0
$238$	−6.70160e8	−3.22225
$239$	−1.50288e8	−0.712083	−0.356042	−	0.934470i	$-0.615874\pi$
$239$	−1.50288e8	−0.712083	−0.356042	+	0.934470i	$0.615874\pi$
$240$	0	0
$241$	3.78027e8	1.73965	0.869827	−	0.493358i	$-0.164231\pi$
$241$	3.78027e8	1.73965	0.869827	+	0.493358i	$0.164231\pi$
$242$	3.98982e7	0.180967
$243$	0	0
$244$	−9.10632e8	−4.01308
$245$	1.30302e8	0.566068
$246$	0	0
$247$	−7.93952e7	−0.335239
$248$	7.37312e8	3.06952
$249$	0	0
$250$	−4.29688e7	−0.173925
$251$	1.45162e8	0.579423	0.289712	−	0.957114i	$-0.406441\pi$
$251$	1.45162e8	0.579423	0.289712	+	0.957114i	$0.406441\pi$
$252$	0	0
$253$	2.09485e8	0.813264
$254$	2.63114e8	1.00746
$255$	0	0
$256$	9.76454e8	3.63757
$257$	−2.04235e7	−0.0750522	−0.0375261	−	0.999296i	$-0.511948\pi$
$257$	−2.04235e7	−0.0750522	−0.0375261	+	0.999296i	$0.511948\pi$
$258$	0	0
$259$	−4.29599e8	−1.53643
$260$	−9.77665e7	−0.344972
$261$	0	0
$262$	6.23943e8	2.14334
$263$	−5.55857e8	−1.88416	−0.942081	−	0.335385i	$-0.891133\pi$
$263$	−5.55857e8	−1.88416	−0.942081	+	0.335385i	$0.891133\pi$
$264$	0	0
$265$	−1.11363e8	−0.367603
$266$	1.08602e9	3.53795
$267$	0	0
$268$	2.66443e8	0.845538
$269$	4.30843e8	1.34954	0.674771	−	0.738027i	$-0.264242\pi$
$269$	4.30843e8	1.34954	0.674771	+	0.738027i	$0.264242\pi$
$270$	0	0
$271$	−2.37258e8	−0.724149	−0.362075	−	0.932149i	$-0.617931\pi$
$271$	−2.37258e8	−0.724149	−0.362075	+	0.932149i	$0.617931\pi$
$272$	−1.44468e9	−4.35293
$273$	0	0
$274$	3.27580e8	0.962033
$275$	−6.56875e7	−0.190466
$276$	0	0
$277$	−1.69266e8	−0.478510	−0.239255	−	0.970957i	$-0.576903\pi$
$277$	−1.69266e8	−0.478510	−0.239255	+	0.970957i	$0.576903\pi$
$278$	−2.30839e8	−0.644395
$279$	0	0
$280$	8.56482e8	2.33166
$281$	−4.97892e8	−1.33864	−0.669319	−	0.742976i	$-0.733414\pi$
$281$	−4.97892e8	−1.33864	−0.669319	+	0.742976i	$0.733414\pi$
$282$	0	0
$283$	4.40831e8	1.15616	0.578082	−	0.815978i	$-0.303801\pi$
$283$	4.40831e8	1.15616	0.578082	+	0.815978i	$0.303801\pi$
$284$	−6.95687e8	−1.80219
$285$	0	0
$286$	−2.03196e8	−0.513611
$287$	3.11227e8	0.777124
$288$	0	0
$289$	8.69513e7	0.211901
$290$	5.35436e8	1.28918
$291$	0	0
$292$	−1.13362e9	−2.66458
$293$	−2.05174e8	−0.476525	−0.238262	−	0.971201i	$-0.576578\pi$
$293$	−2.05174e8	−0.476525	−0.238262	+	0.971201i	$0.576578\pi$
$294$	0	0
$295$	2.71162e8	0.614968
$296$	−1.57750e9	−3.53548
$297$	0	0
$298$	6.14833e8	1.34586
$299$	1.09477e8	0.236849
$300$	0	0
$301$	1.40463e9	2.96878
$302$	−1.23399e9	−2.57802
$303$	0	0
$304$	2.34116e9	4.77941
$305$	−3.19744e8	−0.645288
$306$	0	0
$307$	−6.52312e8	−1.28668	−0.643340	−	0.765580i	$-0.722452\pi$
$307$	−6.52312e8	−1.28668	−0.643340	+	0.765580i	$0.722452\pi$
$308$	2.04439e9	3.98691
$309$	0	0
$310$	4.04228e8	0.770656
$311$	2.49011e7	0.0469415	0.0234707	−	0.999725i	$-0.492528\pi$
$311$	2.49011e7	0.0469415	0.0234707	+	0.999725i	$0.492528\pi$
$312$	0	0
$313$	−3.97227e8	−0.732207	−0.366104	−	0.930574i	$-0.619308\pi$
$313$	−3.97227e8	−0.732207	−0.366104	+	0.930574i	$0.619308\pi$
$314$	1.59816e9	2.91317
$315$	0	0
$316$	1.00150e9	1.78545
$317$	−3.75738e8	−0.662487	−0.331244	−	0.943545i	$-0.607468\pi$
$317$	−3.75738e8	−0.662487	−0.331244	+	0.943545i	$0.607468\pi$
$318$	0	0
$319$	8.18536e8	1.41179
$320$	1.11726e9	1.90602
$321$	0	0
$322$	−1.49749e9	−2.49959
$323$	−8.05877e8	−1.33064
$324$	0	0
$325$	−3.43281e7	−0.0554700
$326$	2.02412e9	3.23574
$327$	0	0
$328$	1.14284e9	1.78824
$329$	8.36659e8	1.29528
$330$	0	0
$331$	−1.00405e8	−0.152180	−0.0760900	−	0.997101i	$-0.524244\pi$
$331$	−1.00405e8	−0.152180	−0.0760900	+	0.997101i	$0.524244\pi$
$332$	−1.40346e9	−2.10484
$333$	0	0
$334$	−1.96117e9	−2.88007
$335$	9.35545e7	0.135959
$336$	0	0
$337$	−1.16187e9	−1.65369	−0.826844	−	0.562431i	$-0.809866\pi$
$337$	−1.16187e9	−1.65369	−0.826844	+	0.562431i	$0.809866\pi$
$338$	−1.06190e8	−0.149580
$339$	0	0
$340$	−9.92350e8	−1.36927
$341$	6.17954e8	0.843949
$342$	0	0
$343$	−2.98976e8	−0.400044
$344$	5.15783e9	6.83145
$345$	0	0
$346$	1.21143e9	1.57229
$347$	1.16859e9	1.50144	0.750720	−	0.660621i	$-0.229707\pi$
$347$	1.16859e9	1.50144	0.750720	+	0.660621i	$0.229707\pi$
$348$	0	0
$349$	3.82782e8	0.482017	0.241009	−	0.970523i	$-0.422522\pi$
$349$	3.82782e8	0.482017	0.241009	+	0.970523i	$0.422522\pi$
$350$	4.69562e8	0.585403
$351$	0	0
$352$	3.29257e9	4.02380
$353$	−7.35893e8	−0.890437	−0.445219	−	0.895422i	$-0.646874\pi$
$353$	−7.35893e8	−0.890437	−0.445219	+	0.895422i	$0.646874\pi$
$354$	0	0
$355$	−2.44272e8	−0.289784
$356$	−5.49255e8	−0.645206
$357$	0	0
$358$	1.96948e9	2.26861
$359$	−1.10649e9	−1.26216	−0.631082	−	0.775716i	$-0.717389\pi$
$359$	−1.10649e9	−1.26216	−0.631082	+	0.775716i	$0.717389\pi$
$360$	0	0
$361$	4.12083e8	0.461009
$362$	−1.36955e9	−1.51739
$363$	0	0
$364$	1.06839e9	1.16112
$365$	−3.98042e8	−0.428453
$366$	0	0
$367$	4.09778e8	0.432730	0.216365	−	0.976313i	$-0.430580\pi$
$367$	4.09778e8	0.432730	0.216365	+	0.976313i	$0.430580\pi$
$368$	−3.22819e9	−3.37669
$369$	0	0
$370$	−8.64858e8	−0.887644
$371$	1.21697e9	1.23729
$372$	0	0
$373$	1.15025e9	1.14766	0.573828	−	0.818976i	$-0.305458\pi$
$373$	1.15025e9	1.14766	0.573828	+	0.818976i	$0.305458\pi$
$374$	−2.06248e9	−2.03864
$375$	0	0
$376$	3.07224e9	2.98056
$377$	4.27765e8	0.411159
$378$	0	0
$379$	−754370.	−0.000711782 0	−0.000355891	−	1.00000i	$-0.500113\pi$
$379$	−754370.	−0.000711782 0	−0.000355891		1.00000i	$0.500113\pi$
$380$	1.60814e9	1.50342
$381$	0	0
$382$	−2.69829e9	−2.47666
$383$	1.80512e9	1.64176	0.820879	−	0.571102i	$-0.193484\pi$
$383$	1.80512e9	1.64176	0.820879	+	0.571102i	$0.193484\pi$
$384$	0	0
$385$	7.17833e8	0.641078
$386$	−2.39874e8	−0.212289
$387$	0	0
$388$	−1.68203e9	−1.46192
$389$	1.23790e9	1.06626	0.533129	−	0.846034i	$-0.321016\pi$
$389$	1.23790e9	1.06626	0.533129	+	0.846034i	$0.321016\pi$
$390$	0	0
$391$	1.11121e9	0.940107
$392$	−5.22874e9	−4.38426
$393$	0	0
$394$	−3.83166e9	−3.15610
$395$	3.51651e8	0.287093
$396$	0	0
$397$	−1.51011e9	−1.21127	−0.605635	−	0.795742i	$-0.707081\pi$
$397$	−1.51011e9	−1.21127	−0.605635	+	0.795742i	$0.707081\pi$
$398$	4.61984e9	3.67313
$399$	0	0
$400$	1.01225e9	0.790820
$401$	−9.74555e8	−0.754746	−0.377373	−	0.926061i	$-0.623173\pi$
$401$	−9.74555e8	−0.754746	−0.377373	+	0.926061i	$0.623173\pi$
$402$	0	0
$403$	3.22941e8	0.245785
$404$	−6.37056e9	−4.80666
$405$	0	0
$406$	−5.85124e9	−4.33917
$407$	−1.32213e9	−0.972064
$408$	0	0
$409$	1.73413e7	0.0125328	0.00626642	−	0.999980i	$-0.498005\pi$
$409$	1.73413e7	0.0125328	0.00626642	+	0.999980i	$0.498005\pi$
$410$	6.26554e8	0.448968
$411$	0	0
$412$	−1.40777e9	−0.991726
$413$	−2.96326e9	−2.06988
$414$	0	0
$415$	−4.92790e8	−0.338449
$416$	1.72069e9	1.17186
$417$	0	0
$418$	3.34233e9	2.23837
$419$	−3.08156e8	−0.204655	−0.102327	−	0.994751i	$-0.532629\pi$
$419$	−3.08156e8	−0.204655	−0.102327	+	0.994751i	$0.532629\pi$
$420$	0	0
$421$	−7.65338e8	−0.499880	−0.249940	−	0.968261i	$-0.580411\pi$
$421$	−7.65338e8	−0.499880	−0.249940	+	0.968261i	$0.580411\pi$
$422$	5.72451e9	3.70805
$423$	0	0
$424$	4.46876e9	2.84713
$425$	−3.48438e8	−0.220173
$426$	0	0
$427$	3.49417e9	2.17193
$428$	−1.78813e9	−1.10242
$429$	0	0
$430$	2.82776e9	1.71515
$431$	1.90108e9	1.14375	0.571873	−	0.820342i	$-0.306217\pi$
$431$	1.90108e9	1.14375	0.571873	+	0.820342i	$0.306217\pi$
$432$	0	0
$433$	8.59670e8	0.508890	0.254445	−	0.967087i	$-0.418107\pi$
$433$	8.59670e8	0.508890	0.254445	+	0.967087i	$0.418107\pi$
$434$	−4.41740e9	−2.59390
$435$	0	0
$436$	2.65861e7	0.0153621
$437$	−1.80076e9	−1.03221
$438$	0	0
$439$	2.47359e9	1.39541	0.697704	−	0.716386i	$-0.254205\pi$
$439$	2.47359e9	1.39541	0.697704	+	0.716386i	$0.254205\pi$
$440$	2.63591e9	1.47518
$441$	0	0
$442$	−1.07785e9	−0.593717
$443$	1.96001e8	0.107114	0.0535570	−	0.998565i	$-0.482944\pi$
$443$	1.96001e8	0.107114	0.0535570	+	0.998565i	$0.482944\pi$
$444$	0	0
$445$	−1.92856e8	−0.103747
$446$	−2.16152e9	−1.15369
$447$	0	0
$448$	−1.22094e10	−6.41535
$449$	−2.10979e9	−1.09996	−0.549980	−	0.835178i	$-0.685365\pi$
$449$	−2.10979e9	−1.09996	−0.549980	+	0.835178i	$0.685365\pi$
$450$	0	0
$451$	9.57831e8	0.491667
$452$	8.02872e9	4.08942
$453$	0	0
$454$	−2.35018e9	−1.17871
$455$	3.75138e8	0.186703
$456$	0	0
$457$	2.40495e8	0.117869	0.0589344	−	0.998262i	$-0.481230\pi$
$457$	2.40495e8	0.117869	0.0589344	+	0.998262i	$0.481230\pi$
$458$	2.33772e9	1.13701
$459$	0	0
$460$	−2.21744e9	−1.06218
$461$	−5.19250e8	−0.246844	−0.123422	−	0.992354i	$-0.539387\pi$
$461$	−5.19250e8	−0.246844	−0.123422	+	0.992354i	$0.539387\pi$
$462$	0	0
$463$	2.99539e9	1.40256	0.701279	−	0.712887i	$-0.252613\pi$
$463$	2.99539e9	1.40256	0.701279	+	0.712887i	$0.252613\pi$
$464$	−1.26137e10	−5.86178
$465$	0	0
$466$	1.98036e9	0.906552
$467$	−2.18358e9	−0.992112	−0.496056	−	0.868291i	$-0.665219\pi$
$467$	−2.18358e9	−0.992112	−0.496056	+	0.868291i	$0.665219\pi$
$468$	0	0
$469$	−1.02236e9	−0.457615
$470$	1.68434e9	0.748321
$471$	0	0
$472$	−1.08812e10	−4.76300
$473$	4.32287e9	1.87827
$474$	0	0
$475$	5.64656e8	0.241744
$476$	1.08444e10	4.60873
$477$	0	0
$478$	3.30633e9	1.38468
$479$	9.89639e8	0.411436	0.205718	−	0.978611i	$-0.434047\pi$
$479$	9.89639e8	0.411436	0.205718	+	0.978611i	$0.434047\pi$
$480$	0	0
$481$	−6.90943e8	−0.283097
$482$	−8.31658e9	−3.38283
$483$	0	0
$484$	−6.45626e8	−0.258834
$485$	−5.90600e8	−0.235070
$486$	0	0
$487$	3.65074e9	1.43228	0.716142	−	0.697955i	$-0.245906\pi$
$487$	3.65074e9	1.43228	0.716142	+	0.697955i	$0.245906\pi$
$488$	1.28307e10	4.99782
$489$	0	0
$490$	−2.86664e9	−1.10074
$491$	−1.89207e8	−0.0721360	−0.0360680	−	0.999349i	$-0.511483\pi$
$491$	−1.89207e8	−0.0721360	−0.0360680	+	0.999349i	$0.511483\pi$
$492$	0	0
$493$	4.34190e9	1.63198
$494$	1.74669e9	0.651887
$495$	0	0
$496$	−9.52273e9	−3.50409
$497$	2.66940e9	0.975365
$498$	0	0
$499$	3.67755e9	1.32497	0.662486	−	0.749074i	$-0.269501\pi$
$499$	3.67755e9	1.32497	0.662486	+	0.749074i	$0.269501\pi$
$500$	6.95312e8	0.248763
$501$	0	0
$502$	−3.19357e9	−1.12671
$503$	1.36625e9	0.478676	0.239338	−	0.970936i	$-0.423070\pi$
$503$	1.36625e9	0.478676	0.239338	+	0.970936i	$0.423070\pi$
$504$	0	0
$505$	−2.23686e9	−0.772891
$506$	−4.60868e9	−1.58143
$507$	0	0
$508$	−4.25766e9	−1.44095
$509$	3.12483e8	0.105030	0.0525151	−	0.998620i	$-0.483276\pi$
$509$	3.12483e8	0.105030	0.0525151	+	0.998620i	$0.483276\pi$
$510$	0	0
$511$	4.34980e9	1.44210
$512$	−9.14439e9	−3.01099
$513$	0	0
$514$	4.49316e8	0.145942
$515$	−4.94302e8	−0.159465
$516$	0	0
$517$	2.57490e9	0.819489
$518$	9.45117e9	2.98766
$519$	0	0
$520$	1.37752e9	0.429621
$521$	2.84217e9	0.880477	0.440238	−	0.897881i	$-0.354894\pi$
$521$	2.84217e9	0.880477	0.440238	+	0.897881i	$0.354894\pi$
$522$	0	0
$523$	−6.74440e7	−0.0206152	−0.0103076	−	0.999947i	$-0.503281\pi$
$523$	−6.74440e7	−0.0206152	−0.0103076	+	0.999947i	$0.503281\pi$
$524$	−1.00965e10	−3.06558
$525$	0	0
$526$	1.22289e10	3.66384
$527$	3.27792e9	0.975577
$528$	0	0
$529$	−9.21797e8	−0.270732
$530$	2.44998e9	0.714821
$531$	0	0
$532$	−1.75738e10	−5.06027
$533$	5.00560e8	0.143189
$534$	0	0
$535$	−6.27854e8	−0.177264
$536$	−3.75415e9	−1.05302
$537$	0	0
$538$	−9.47855e9	−2.62424
$539$	−4.38230e9	−1.20543
$540$	0	0
$541$	3.87005e9	1.05081	0.525407	−	0.850851i	$-0.323913\pi$
$541$	3.87005e9	1.05081	0.525407	+	0.850851i	$0.323913\pi$
$542$	5.21967e9	1.40814
$543$	0	0
$544$	1.74654e10	4.65138
$545$	9.33500e6	0.00247017
$546$	0	0
$547$	−6.39425e9	−1.67045	−0.835226	−	0.549906i	$-0.814663\pi$
$547$	−6.39425e9	−1.67045	−0.835226	+	0.549906i	$0.814663\pi$
$548$	−5.30084e9	−1.37598
$549$	0	0
$550$	1.44512e9	0.370370
$551$	−7.03621e9	−1.79188
$552$	0	0
$553$	−3.84284e9	−0.966306
$554$	3.72386e9	0.930484
$555$	0	0
$556$	3.73539e9	0.921668
$557$	−2.02546e9	−0.496626	−0.248313	−	0.968680i	$-0.579876\pi$
$557$	−2.02546e9	−0.496626	−0.248313	+	0.968680i	$0.579876\pi$
$558$	0	0
$559$	2.25912e9	0.547014
$560$	−1.10619e10	−2.66177
$561$	0	0
$562$	1.09536e10	2.60304
$563$	6.49741e9	1.53448	0.767239	−	0.641361i	$-0.221630\pi$
$563$	6.49741e9	1.53448	0.767239	+	0.641361i	$0.221630\pi$
$564$	0	0
$565$	2.81907e9	0.657562
$566$	−9.69828e9	−2.24821
$567$	0	0
$568$	9.80215e9	2.24441
$569$	1.68488e9	0.383421	0.191711	−	0.981452i	$-0.438597\pi$
$569$	1.68488e9	0.383421	0.191711	+	0.981452i	$0.438597\pi$
$570$	0	0
$571$	−1.60376e9	−0.360507	−0.180253	−	0.983620i	$-0.557692\pi$
$571$	−1.60376e9	−0.360507	−0.180253	+	0.983620i	$0.557692\pi$
$572$	3.28808e9	0.734610
$573$	0	0
$574$	−6.84699e9	−1.51115
$575$	−7.78594e8	−0.170794
$576$	0	0
$577$	3.36601e8	0.0729457	0.0364728	−	0.999335i	$-0.488388\pi$
$577$	3.36601e8	0.0729457	0.0364728	+	0.999335i	$0.488388\pi$
$578$	−1.91293e9	−0.412051
$579$	0	0
$580$	−8.66433e9	−1.84390
$581$	5.38520e9	1.13916
$582$	0	0
$583$	3.74535e9	0.782803
$584$	1.59726e10	3.31842
$585$	0	0
$586$	4.51383e9	0.926623
$587$	7.14305e9	1.45764	0.728821	−	0.684705i	$-0.240069\pi$
$587$	7.14305e9	1.45764	0.728821	+	0.684705i	$0.240069\pi$
$588$	0	0
$589$	−5.31200e9	−1.07116
$590$	−5.96558e9	−1.19583
$591$	0	0
$592$	2.03742e10	4.03603
$593$	−7.44580e9	−1.46629	−0.733145	−	0.680072i	$-0.761948\pi$
$593$	−7.44580e9	−1.46629	−0.733145	+	0.680072i	$0.761948\pi$
$594$	0	0
$595$	3.80772e9	0.741065
$596$	−9.94911e9	−1.92496
$597$	0	0
$598$	−2.40848e9	−0.460563
$599$	−1.26681e8	−0.0240834	−0.0120417	−	0.999927i	$-0.503833\pi$
$599$	−1.26681e8	−0.0240834	−0.0120417	+	0.999927i	$0.503833\pi$
$600$	0	0
$601$	−3.11735e9	−0.585768	−0.292884	−	0.956148i	$-0.594615\pi$
$601$	−3.11735e9	−0.585768	−0.292884	+	0.956148i	$0.594615\pi$
$602$	−3.09018e10	−5.77292
$603$	0	0
$604$	1.99682e10	3.68730
$605$	−2.26694e8	−0.0416195
$606$	0	0
$607$	−1.39422e9	−0.253030	−0.126515	−	0.991965i	$-0.540379\pi$
$607$	−1.39422e9	−0.253030	−0.126515	+	0.991965i	$0.540379\pi$
$608$	−2.83033e10	−5.10710
$609$	0	0
$610$	7.03437e9	1.25479
$611$	1.34564e9	0.238662
$612$	0	0
$613$	−3.00660e9	−0.527186	−0.263593	−	0.964634i	$-0.584908\pi$
$613$	−3.00660e9	−0.527186	−0.263593	+	0.964634i	$0.584908\pi$
$614$	1.43509e10	2.50201
$615$	0	0
$616$	−2.88052e10	−4.96522
$617$	4.05590e8	0.0695168	0.0347584	−	0.999396i	$-0.488934\pi$
$617$	4.05590e8	0.0695168	0.0347584	+	0.999396i	$0.488934\pi$
$618$	0	0
$619$	−9.32589e9	−1.58042	−0.790211	−	0.612835i	$-0.790029\pi$
$619$	−9.32589e9	−1.58042	−0.790211	+	0.612835i	$0.790029\pi$
$620$	−6.54114e9	−1.10226
$621$	0	0
$622$	−5.47823e8	−0.0912797
$623$	2.10753e9	0.349194
$624$	0	0
$625$	2.44141e8	0.0400000
$626$	8.73900e9	1.42381
$627$	0	0
$628$	−2.58611e10	−4.16667
$629$	−7.01322e9	−1.12367
$630$	0	0
$631$	−1.04559e10	−1.65675	−0.828374	−	0.560175i	$-0.810734\pi$
$631$	−1.04559e10	−1.65675	−0.828374	+	0.560175i	$0.810734\pi$
$632$	−1.41111e10	−2.22356
$633$	0	0
$634$	8.26623e9	1.28824
$635$	−1.49497e9	−0.231698
$636$	0	0
$637$	−2.29018e9	−0.351060
$638$	−1.80078e10	−2.74529
$639$	0	0
$640$	−1.20484e10	−1.81677
$641$	−4.09010e9	−0.613381	−0.306691	−	0.951809i	$-0.599222\pi$
$641$	−4.09010e9	−0.613381	−0.306691	+	0.951809i	$0.599222\pi$
$642$	0	0
$643$	−1.12738e10	−1.67237	−0.836184	−	0.548449i	$-0.815219\pi$
$643$	−1.12738e10	−1.67237	−0.836184	+	0.548449i	$0.815219\pi$
$644$	2.42321e10	3.57512
$645$	0	0
$646$	1.77293e10	2.58749
$647$	6.99463e9	1.01531	0.507656	−	0.861560i	$-0.330512\pi$
$647$	6.99463e9	1.01531	0.507656	+	0.861560i	$0.330512\pi$
$648$	0	0
$649$	−9.11974e9	−1.30956
$650$	7.55219e8	0.107864
$651$	0	0
$652$	−3.27539e10	−4.62803
$653$	−6.88780e9	−0.968021	−0.484010	−	0.875062i	$-0.660820\pi$
$653$	−6.88780e9	−0.968021	−0.484010	+	0.875062i	$0.660820\pi$
$654$	0	0
$655$	−3.54513e9	−0.492932
$656$	−1.47603e10	−2.04141
$657$	0	0
$658$	−1.84065e10	−2.51872
$659$	5.96148e9	0.811437	0.405719	−	0.913998i	$-0.367021\pi$
$659$	5.96148e9	0.811437	0.405719	+	0.913998i	$0.367021\pi$
$660$	0	0
$661$	−1.53630e8	−0.0206905	−0.0103453	−	0.999946i	$-0.503293\pi$
$661$	−1.53630e8	−0.0206905	−0.0103453	+	0.999946i	$0.503293\pi$
$662$	2.20891e9	0.295921
$663$	0	0
$664$	1.97747e10	2.62132
$665$	−6.17056e9	−0.813671
$666$	0	0
$667$	9.70210e9	1.26598
$668$	3.17354e10	4.11932
$669$	0	0
$670$	−2.05820e9	−0.264378
$671$	1.07536e10	1.37413
$672$	0	0
$673$	6.19881e9	0.783891	0.391946	−	0.919988i	$-0.371802\pi$
$673$	6.19881e9	0.783891	0.391946	+	0.919988i	$0.371802\pi$
$674$	2.55612e10	3.21567
$675$	0	0
$676$	1.71834e9	0.213942
$677$	−1.36357e9	−0.168896	−0.0844478	−	0.996428i	$-0.526913\pi$
$677$	−1.36357e9	−0.168896	−0.0844478	+	0.996428i	$0.526913\pi$
$678$	0	0
$679$	6.45408e9	0.791207
$680$	1.39821e10	1.70526
$681$	0	0
$682$	−1.35950e10	−1.64110
$683$	1.29270e10	1.55247	0.776237	−	0.630442i	$-0.217126\pi$
$683$	1.29270e10	1.55247	0.776237	+	0.630442i	$0.217126\pi$
$684$	0	0
$685$	−1.86125e9	−0.221252
$686$	6.57748e9	0.777903
$687$	0	0
$688$	−6.66158e10	−7.79862
$689$	1.95731e9	0.227978
$690$	0	0
$691$	1.08207e10	1.24762	0.623810	−	0.781576i	$-0.285584\pi$
$691$	1.08207e10	1.24762	0.623810	+	0.781576i	$0.285584\pi$
$692$	−1.96031e10	−2.24882
$693$	0	0
$694$	−2.57089e10	−2.91961
$695$	1.31158e9	0.148200
$696$	0	0
$697$	5.08079e9	0.568351
$698$	−8.42120e9	−0.937303
$699$	0	0
$700$	−7.59838e9	−0.837293
$701$	6.24458e9	0.684684	0.342342	−	0.939575i	$-0.388780\pi$
$701$	6.24458e9	0.684684	0.342342	+	0.939575i	$0.388780\pi$
$702$	0	0
$703$	1.13652e10	1.23377
$704$	−3.75756e10	−4.05883
$705$	0	0
$706$	1.61897e10	1.73149
$707$	2.44444e10	2.60142
$708$	0	0
$709$	−2.39202e9	−0.252059	−0.126030	−	0.992026i	$-0.540223\pi$
$709$	−2.39202e9	−0.252059	−0.126030	+	0.992026i	$0.540223\pi$
$710$	5.37398e9	0.563498
$711$	0	0
$712$	7.73894e9	0.803528
$713$	7.32461e9	0.756783
$714$	0	0
$715$	1.15452e9	0.118122
$716$	−3.18697e10	−3.24476
$717$	0	0
$718$	2.43427e10	2.45433
$719$	9.82846e9	0.986130	0.493065	−	0.869992i	$-0.335876\pi$
$719$	9.82846e9	0.986130	0.493065	+	0.869992i	$0.335876\pi$
$720$	0	0
$721$	5.40173e9	0.536734
$722$	−9.06583e9	−0.896453
$723$	0	0
$724$	2.21617e10	2.17030
$725$	−3.04225e9	−0.296491
$726$	0	0
$727$	8.27780e9	0.798996	0.399498	−	0.916734i	$-0.369185\pi$
$727$	8.27780e9	0.798996	0.399498	+	0.916734i	$0.369185\pi$
$728$	−1.50535e10	−1.44603
$729$	0	0
$730$	8.75691e9	0.833146
$731$	2.29306e10	2.17122
$732$	0	0
$733$	6.51630e9	0.611135	0.305567	−	0.952170i	$-0.401154\pi$
$733$	6.51630e9	0.611135	0.305567	+	0.952170i	$0.401154\pi$
$734$	−9.01512e9	−0.841463
$735$	0	0
$736$	3.90269e10	3.60821
$737$	−3.14642e9	−0.289522
$738$	0	0
$739$	1.16715e10	1.06383	0.531915	−	0.846798i	$-0.321473\pi$
$739$	1.16715e10	1.06383	0.531915	+	0.846798i	$0.321473\pi$
$740$	1.39950e10	1.26958
$741$	0	0
$742$	−2.67734e10	−2.40597
$743$	1.99992e10	1.78876	0.894378	−	0.447313i	$-0.147619\pi$
$743$	1.99992e10	1.78876	0.894378	+	0.447313i	$0.147619\pi$
$744$	0	0
$745$	−3.49337e9	−0.309526
$746$	−2.53055e10	−2.23167
$747$	0	0
$748$	3.33747e10	2.91583
$749$	6.86119e9	0.596641
$750$	0	0
$751$	−1.75756e8	−0.0151416	−0.00757078	−	0.999971i	$-0.502410\pi$
$751$	−1.75756e8	−0.0151416	−0.00757078	+	0.999971i	$0.502410\pi$
$752$	−3.96794e10	−3.40254
$753$	0	0
$754$	−9.41082e9	−0.799517
$755$	7.01130e9	0.592903
$756$	0	0
$757$	2.42700e8	0.0203345	0.0101673	−	0.999948i	$-0.496764\pi$
$757$	2.42700e8	0.0203345	0.0101673	+	0.999948i	$0.496764\pi$
$758$	1.65961e7	0.00138409
$759$	0	0
$760$	−2.26585e10	−1.87234
$761$	5.06318e9	0.416463	0.208232	−	0.978080i	$-0.433229\pi$
$761$	5.06318e9	0.416463	0.208232	+	0.978080i	$0.433229\pi$
$762$	0	0
$763$	−1.02013e8	−0.00831417
$764$	4.36633e10	3.54233
$765$	0	0
$766$	−3.97125e10	−3.19247
$767$	−4.76595e9	−0.381387
$768$	0	0
$769$	9.60207e9	0.761417	0.380709	−	0.924695i	$-0.375680\pi$
$769$	9.60207e9	0.761417	0.380709	+	0.924695i	$0.375680\pi$
$770$	−1.57923e10	−1.24660
$771$	0	0
$772$	3.88160e9	0.303634
$773$	1.12869e8	0.00878915	0.00439457	−	0.999990i	$-0.498601\pi$
$773$	1.12869e8	0.00878915	0.00439457	+	0.999990i	$0.498601\pi$
$774$	0	0
$775$	−2.29675e9	−0.177238
$776$	2.36996e10	1.82065
$777$	0	0
$778$	−2.72338e10	−2.07338
$779$	−8.23361e9	−0.624035
$780$	0	0
$781$	8.21536e9	0.617089
$782$	−2.44466e10	−1.82808
$783$	0	0
$784$	6.75317e10	5.00497
$785$	−9.08045e9	−0.669983
$786$	0	0
$787$	−1.20413e10	−0.880563	−0.440281	−	0.897860i	$-0.645121\pi$
$787$	−1.20413e10	−0.880563	−0.440281	+	0.897860i	$0.645121\pi$
$788$	6.20032e10	4.51411
$789$	0	0
$790$	−7.73632e9	−0.558264
$791$	−3.08068e10	−2.21324
$792$	0	0
$793$	5.61982e9	0.400190
$794$	3.32224e10	2.35537
$795$	0	0
$796$	−7.47574e10	−5.25362
$797$	1.30995e9	0.0916536	0.0458268	−	0.998949i	$-0.485408\pi$
$797$	1.30995e9	0.0916536	0.0458268	+	0.998949i	$0.485408\pi$
$798$	0	0
$799$	1.36585e10	0.947303
$800$	−1.22375e10	−0.845041
$801$	0	0
$802$	2.14402e10	1.46764
$803$	1.33869e10	0.912382
$804$	0	0
$805$	8.50847e9	0.574865
$806$	−7.10471e9	−0.477940
$807$	0	0
$808$	8.97605e10	5.98612
$809$	1.47002e10	0.976118	0.488059	−	0.872811i	$-0.337705\pi$
$809$	1.47002e10	0.976118	0.488059	+	0.872811i	$0.337705\pi$
$810$	0	0
$811$	−1.45417e10	−0.957286	−0.478643	−	0.878010i	$-0.658871\pi$
$811$	−1.45417e10	−0.957286	−0.478643	+	0.878010i	$0.658871\pi$
$812$	9.46838e10	6.20625
$813$	0	0
$814$	2.90869e10	1.89022
$815$	−1.15007e10	−0.744169
$816$	0	0
$817$	−3.71598e10	−2.38395
$818$	−3.81508e8	−0.0243707
$819$	0	0
$820$	−1.01388e10	−0.642152
$821$	1.26875e10	0.800154	0.400077	−	0.916482i	$-0.368983\pi$
$821$	1.26875e10	0.800154	0.400077	+	0.916482i	$0.368983\pi$
$822$	0	0
$823$	−2.68452e8	−0.0167868	−0.00839339	−	0.999965i	$-0.502672\pi$
$823$	−2.68452e8	−0.0167868	−0.00839339	+	0.999965i	$0.502672\pi$
$824$	1.98353e10	1.23508
$825$	0	0
$826$	6.51918e10	4.02497
$827$	2.96268e10	1.82144	0.910720	−	0.413024i	$-0.135528\pi$
$827$	2.96268e10	1.82144	0.910720	+	0.413024i	$0.135528\pi$
$828$	0	0
$829$	1.00623e10	0.613415	0.306708	−	0.951804i	$-0.400773\pi$
$829$	1.00623e10	0.613415	0.306708	+	0.951804i	$0.400773\pi$
$830$	1.08414e10	0.658129
$831$	0	0
$832$	−1.96369e10	−1.18206
$833$	−2.32458e10	−1.39344
$834$	0	0
$835$	1.11430e10	0.662371
$836$	−5.40850e10	−3.20151
$837$	0	0
$838$	6.77944e9	0.397960
$839$	2.87983e10	1.68345	0.841724	−	0.539908i	$-0.181541\pi$
$839$	2.87983e10	1.68345	0.841724	+	0.539908i	$0.181541\pi$
$840$	0	0
$841$	2.06598e10	1.19768
$842$	1.68374e10	0.972039
$843$	0	0
$844$	−9.26330e10	−5.30356
$845$	6.03351e8	0.0344010
$846$	0	0
$847$	2.47732e9	0.140084
$848$	−5.77162e10	−3.25022
$849$	0	0
$850$	7.66563e9	0.428136
$851$	−1.56712e10	−0.871665
$852$	0	0
$853$	−1.22110e10	−0.673640	−0.336820	−	0.941569i	$-0.609351\pi$
$853$	−1.22110e10	−0.673640	−0.336820	+	0.941569i	$0.609351\pi$
$854$	−7.68716e10	−4.22341
$855$	0	0
$856$	2.51945e10	1.37293
$857$	2.86754e10	1.55624	0.778121	−	0.628114i	$-0.216173\pi$
$857$	2.86754e10	1.55624	0.778121	+	0.628114i	$0.216173\pi$
$858$	0	0
$859$	7.29011e9	0.392426	0.196213	−	0.980561i	$-0.437136\pi$
$859$	7.29011e9	0.392426	0.196213	+	0.980561i	$0.437136\pi$
$860$	−4.57583e10	−2.45316
$861$	0	0
$862$	−4.18237e10	−2.22406
$863$	−1.06825e10	−0.565763	−0.282881	−	0.959155i	$-0.591290\pi$
$863$	−1.06825e10	−0.565763	−0.282881	+	0.959155i	$0.591290\pi$
$864$	0	0
$865$	−6.88312e9	−0.361601
$866$	−1.89127e10	−0.989559
$867$	0	0
$868$	7.14816e10	3.71001
$869$	−1.18267e10	−0.611358
$870$	0	0
$871$	−1.64431e9	−0.0843182
$872$	−3.74595e8	−0.0191317
$873$	0	0
$874$	3.96166e10	2.00719
$875$	−2.66797e9	−0.134633
$876$	0	0
$877$	1.96948e10	0.985946	0.492973	−	0.870045i	$-0.335910\pi$
$877$	1.96948e10	0.985946	0.492973	+	0.870045i	$0.335910\pi$
$878$	−5.44189e10	−2.71343
$879$	0	0
$880$	−3.40440e10	−1.68404
$881$	−9.84626e9	−0.485127	−0.242564	−	0.970135i	$-0.577988\pi$
$881$	−9.84626e9	−0.485127	−0.242564	+	0.970135i	$0.577988\pi$
$882$	0	0
$883$	−2.86588e10	−1.40086	−0.700432	−	0.713719i	$-0.747009\pi$
$883$	−2.86588e10	−1.40086	−0.700432	+	0.713719i	$0.747009\pi$
$884$	1.74415e10	0.849184
$885$	0	0
$886$	−4.31203e9	−0.208288
$887$	7.83167e9	0.376809	0.188405	−	0.982091i	$-0.439668\pi$
$887$	7.83167e9	0.376809	0.188405	+	0.982091i	$0.439668\pi$
$888$	0	0
$889$	1.63370e10	0.779858
$890$	4.24284e9	0.201740
$891$	0	0
$892$	3.49774e10	1.65010
$893$	−2.21341e10	−1.04012
$894$	0	0
$895$	−1.11902e10	−0.521744
$896$	1.31665e11	6.11495
$897$	0	0
$898$	4.64154e10	2.13892
$899$	2.86199e10	1.31374
$900$	0	0
$901$	1.98671e10	0.904895
$902$	−2.10723e10	−0.956068
$903$	0	0
$904$	−1.13124e11	−5.09289
$905$	7.78151e9	0.348975
$906$	0	0
$907$	−2.40982e9	−0.107241	−0.0536204	−	0.998561i	$-0.517076\pi$
$907$	−2.40982e9	−0.107241	−0.0536204	+	0.998561i	$0.517076\pi$
$908$	3.80302e10	1.68588
$909$	0	0
$910$	−8.25303e9	−0.363052
$911$	−3.75328e9	−0.164474	−0.0822368	−	0.996613i	$-0.526206\pi$
$911$	−3.75328e9	−0.164474	−0.0822368	+	0.996613i	$0.526206\pi$
$912$	0	0
$913$	1.65735e10	0.720720
$914$	−5.29089e9	−0.229201
$915$	0	0
$916$	−3.78285e10	−1.62624
$917$	3.87412e10	1.65913
$918$	0	0
$919$	−1.97797e10	−0.840649	−0.420325	−	0.907374i	$-0.638084\pi$
$919$	−1.97797e10	−0.840649	−0.420325	+	0.907374i	$0.638084\pi$
$920$	3.12434e10	1.32282
$921$	0	0
$922$	1.14235e10	0.480000
$923$	4.29332e9	0.179716
$924$	0	0
$925$	4.91397e9	0.204144
$926$	−6.58987e10	−2.72733
$927$	0	0
$928$	1.52492e11	6.26368
$929$	−1.69708e10	−0.694460	−0.347230	−	0.937780i	$-0.612878\pi$
$929$	−1.69708e10	−0.694460	−0.347230	+	0.937780i	$0.612878\pi$
$930$	0	0
$931$	3.76707e10	1.52996
$932$	−3.20458e10	−1.29663
$933$	0	0
$934$	4.80388e10	1.92920
$935$	1.17186e10	0.468853
$936$	0	0
$937$	−3.23352e10	−1.28406	−0.642031	−	0.766678i	$-0.721908\pi$
$937$	−3.23352e10	−1.28406	−0.642031	+	0.766678i	$0.721908\pi$
$938$	2.24920e10	0.889853
$939$	0	0
$940$	−2.72557e10	−1.07031
$941$	5.51297e9	0.215686	0.107843	−	0.994168i	$-0.465606\pi$
$941$	5.51297e9	0.215686	0.107843	+	0.994168i	$0.465606\pi$
$942$	0	0
$943$	1.13532e10	0.440886
$944$	1.40536e11	5.43733
$945$	0	0
$946$	−9.51032e10	−3.65238
$947$	−1.80849e10	−0.691977	−0.345988	−	0.938239i	$-0.612456\pi$
$947$	−1.80849e10	−0.691977	−0.345988	+	0.938239i	$0.612456\pi$
$948$	0	0
$949$	6.99598e9	0.265715
$950$	−1.24224e10	−0.470083
$951$	0	0
$952$	−1.52796e11	−5.73963
$953$	1.71947e10	0.643531	0.321766	−	0.946819i	$-0.395724\pi$
$953$	1.71947e10	0.643531	0.321766	+	0.946819i	$0.395724\pi$
$954$	0	0
$955$	1.53312e10	0.569593
$956$	−5.35024e10	−1.98048
$957$	0	0
$958$	−2.17721e10	−0.800056
$959$	2.03397e10	0.744697
$960$	0	0
$961$	−5.90597e9	−0.214664
$962$	1.52008e10	0.550494
$963$	0	0
$964$	1.34577e11	4.83841
$965$	1.36292e9	0.0488230
$966$	0	0
$967$	−8.82434e9	−0.313826	−0.156913	−	0.987612i	$-0.550154\pi$
$967$	−8.82434e9	−0.313826	−0.156913	+	0.987612i	$0.550154\pi$
$968$	9.09679e9	0.322348
$969$	0	0
$970$	1.29932e10	0.457104
$971$	−1.96655e10	−0.689347	−0.344674	−	0.938723i	$-0.612010\pi$
$971$	−1.96655e10	−0.689347	−0.344674	+	0.938723i	$0.612010\pi$
$972$	0	0
$973$	−1.43330e10	−0.498818
$974$	−8.03162e10	−2.78514
$975$	0	0
$976$	−1.65714e11	−5.70540
$977$	−1.10816e10	−0.380164	−0.190082	−	0.981768i	$-0.560875\pi$
$977$	−1.10816e10	−0.380164	−0.190082	+	0.981768i	$0.560875\pi$
$978$	0	0
$979$	6.48614e9	0.220926
$980$	4.63874e10	1.57438
$981$	0	0
$982$	4.16255e9	0.140272
$983$	−1.86653e9	−0.0626755	−0.0313377	−	0.999509i	$-0.509977\pi$
$983$	−1.86653e9	−0.0626755	−0.0313377	+	0.999509i	$0.509977\pi$
$984$	0	0
$985$	2.17708e10	0.725851
$986$	−9.55218e10	−3.17346
$987$	0	0
$988$	−2.82647e10	−0.932384
$989$	5.12390e10	1.68428
$990$	0	0
$991$	−4.96346e10	−1.62004	−0.810022	−	0.586399i	$-0.800545\pi$
$991$	−4.96346e10	−1.62004	−0.810022	+	0.586399i	$0.800545\pi$
$992$	1.15124e11	3.74434
$993$	0	0
$994$	−5.87269e10	−1.89664
$995$	−2.62491e10	−0.844760
$996$	0	0
$997$	2.92052e10	0.933311	0.466656	−	0.884439i	$-0.345459\pi$
$997$	2.92052e10	0.933311	0.466656	+	0.884439i	$0.345459\pi$
$998$	−8.09061e10	−2.57647
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	585.8.a.a.1.1		1
3.2	odd	2		195.8.a.a.1.1	✓	1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
195.8.a.a.1.1	✓	1	3.2	odd	2
585.8.a.a.1.1		1	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 585 = 3^{2} \cdot 5 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	585.a (trivial)

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

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