LMFDB - Embedded newform 8019.2.a.l.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8019,2,Mod(1,8019)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8019, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("8019.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8019 = 3^{6} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8019.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:	$64.0320373809$
Analytic rank:	$0$
Dimension:	$51$
Twist minimal:	no (minimal twist has level 297)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Character	$\chi$	$=$	8019.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.79315 q^{2} +1.21539 q^{4} +3.19791 q^{5} -3.70301 q^{7} +1.40692 q^{8} +O(q^{10})$	\(q-1.79315 q^{2} +1.21539 q^{4} +3.19791 q^{5} -3.70301 q^{7} +1.40692 q^{8} -5.73435 q^{10} +1.00000 q^{11} +4.82389 q^{13} +6.64005 q^{14} -4.95361 q^{16} +5.48525 q^{17} -4.18293 q^{19} +3.88672 q^{20} -1.79315 q^{22} +8.64429 q^{23} +5.22666 q^{25} -8.64996 q^{26} -4.50060 q^{28} -4.03535 q^{29} -1.82854 q^{31} +6.06872 q^{32} -9.83589 q^{34} -11.8419 q^{35} +7.55622 q^{37} +7.50062 q^{38} +4.49921 q^{40} +5.69173 q^{41} +1.26606 q^{43} +1.21539 q^{44} -15.5005 q^{46} +7.48060 q^{47} +6.71225 q^{49} -9.37219 q^{50} +5.86291 q^{52} +2.85046 q^{53} +3.19791 q^{55} -5.20984 q^{56} +7.23600 q^{58} -6.76596 q^{59} -8.15773 q^{61} +3.27886 q^{62} -0.974926 q^{64} +15.4264 q^{65} -4.62250 q^{67} +6.66673 q^{68} +21.2343 q^{70} +0.919786 q^{71} +13.2592 q^{73} -13.5494 q^{74} -5.08390 q^{76} -3.70301 q^{77} -3.72774 q^{79} -15.8412 q^{80} -10.2061 q^{82} +11.8648 q^{83} +17.5414 q^{85} -2.27024 q^{86} +1.40692 q^{88} -5.92634 q^{89} -17.8629 q^{91} +10.5062 q^{92} -13.4139 q^{94} -13.3766 q^{95} +11.8580 q^{97} -12.0361 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) $ 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7	$ 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7} + 18 q^{10} + 51 q^{11} + 30 q^{13} + 12 q^{14} + 78 q^{16} + 30 q^{19} + 18 q^{20} + 3 q^{23} + 75 q^{25} + 9 q^{26} + 36 q^{28} + 42 q^{31} - 15 q^{32} + 42 q^{34} - 9 q^{35} + 48 q^{37} - 3 q^{38} + 54 q^{40} + 18 q^{43} + 60 q^{44} + 42 q^{46} + 30 q^{47} + 99 q^{49} - 30 q^{50} + 60 q^{52} + 18 q^{53} + 6 q^{55} + 21 q^{56} + 30 q^{58} + 24 q^{59} + 99 q^{61} + 114 q^{64} - 15 q^{65} + 39 q^{67} - 39 q^{68} + 48 q^{70} + 30 q^{71} + 69 q^{73} + 90 q^{76} + 12 q^{77} + 48 q^{79} + 42 q^{80} + 42 q^{82} - 21 q^{83} + 84 q^{85} + 24 q^{86} + 15 q^{89} + 69 q^{91} - 66 q^{92} + 66 q^{94} - 12 q^{95} + 72 q^{97} - 57 q^{98}+O(q^{100}) $ 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7 + 18 * q^10 + 51 * q^11 + 30 * q^13 + 12 * q^14 + 78 * q^16 + 30 * q^19 + 18 * q^20 + 3 * q^23 + 75 * q^25 + 9 * q^26 + 36 * q^28 + 42 * q^31 - 15 * q^32 + 42 * q^34 - 9 * q^35 + 48 * q^37 - 3 * q^38 + 54 * q^40 + 18 * q^43 + 60 * q^44 + 42 * q^46 + 30 * q^47 + 99 * q^49 - 30 * q^50 + 60 * q^52 + 18 * q^53 + 6 * q^55 + 21 * q^56 + 30 * q^58 + 24 * q^59 + 99 * q^61 + 114 * q^64 - 15 * q^65 + 39 * q^67 - 39 * q^68 + 48 * q^70 + 30 * q^71 + 69 * q^73 + 90 * q^76 + 12 * q^77 + 48 * q^79 + 42 * q^80 + 42 * q^82 - 21 * q^83 + 84 * q^85 + 24 * q^86 + 15 * q^89 + 69 * q^91 - 66 * q^92 + 66 * q^94 - 12 * q^95 + 72 * q^97 - 57 * q^98

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 $

$2$	−1.79315	−1.26795	−0.633975	−	0.773354i	$-0.718578\pi$
$2$	−1.79315	−1.26795	−0.633975	+	0.773354i	$0.718578\pi$
$3$	0	0
$3$	0	0
$4$	1.21539	0.607696
$5$	3.19791	1.43015	0.715075	−	0.699047i	$-0.246392\pi$
$5$	3.19791	1.43015	0.715075	+	0.699047i	$0.246392\pi$
$6$	0	0
$7$	−3.70301	−1.39960	−0.699802	−	0.714337i	$-0.746729\pi$
$7$	−3.70301	−1.39960	−0.699802	+	0.714337i	$0.746729\pi$
$8$	1.40692	0.497422
$9$	0	0
$10$	−5.73435	−1.81336
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	4.82389	1.33791	0.668953	−	0.743305i	$-0.266743\pi$
$13$	4.82389	1.33791	0.668953	+	0.743305i	$0.266743\pi$
$14$	6.64005	1.77463
$15$	0	0
$16$	−4.95361	−1.23840
$17$	5.48525	1.33037	0.665184	−	0.746679i	$-0.268353\pi$
$17$	5.48525	1.33037	0.665184	+	0.746679i	$0.268353\pi$
$18$	0	0
$19$	−4.18293	−0.959629	−0.479815	−	0.877370i	$-0.659296\pi$
$19$	−4.18293	−0.959629	−0.479815	+	0.877370i	$0.659296\pi$
$20$	3.88672	0.869097
$21$	0	0
$22$	−1.79315	−0.382301
$23$	8.64429	1.80246	0.901230	−	0.433341i	$-0.142665\pi$
$23$	8.64429	1.80246	0.901230	+	0.433341i	$0.142665\pi$
$24$	0	0
$25$	5.22666	1.04533
$26$	−8.64996	−1.69640
$27$	0	0
$28$	−4.50060	−0.850534
$29$	−4.03535	−0.749346	−0.374673	−	0.927157i	$-0.622245\pi$
$29$	−4.03535	−0.749346	−0.374673	+	0.927157i	$0.622245\pi$
$30$	0	0
$31$	−1.82854	−0.328416	−0.164208	−	0.986426i	$-0.552507\pi$
$31$	−1.82854	−0.328416	−0.164208	+	0.986426i	$0.552507\pi$
$32$	6.06872	1.07281
$33$	0	0
$34$	−9.83589	−1.68684
$35$	−11.8419	−2.00165
$36$	0	0
$37$	7.55622	1.24223	0.621117	−	0.783718i	$-0.286679\pi$
$37$	7.55622	1.24223	0.621117	+	0.783718i	$0.286679\pi$
$38$	7.50062	1.21676
$39$	0	0
$40$	4.49921	0.711388
$41$	5.69173	0.888899	0.444450	−	0.895804i	$-0.353399\pi$
$41$	5.69173	0.888899	0.444450	+	0.895804i	$0.353399\pi$
$42$	0	0
$43$	1.26606	0.193072	0.0965362	−	0.995329i	$-0.469224\pi$
$43$	1.26606	0.193072	0.0965362	+	0.995329i	$0.469224\pi$
$44$	1.21539	0.183227
$45$	0	0
$46$	−15.5005	−2.28543
$47$	7.48060	1.09116	0.545579	−	0.838059i	$-0.316310\pi$
$47$	7.48060	1.09116	0.545579	+	0.838059i	$0.316310\pi$
$48$	0	0
$49$	6.71225	0.958893
$50$	−9.37219	−1.32543
$51$	0	0
$52$	5.86291	0.813040
$53$	2.85046	0.391541	0.195771	−	0.980650i	$-0.437279\pi$
$53$	2.85046	0.391541	0.195771	+	0.980650i	$0.437279\pi$
$54$	0	0
$55$	3.19791	0.431207
$56$	−5.20984	−0.696194
$57$	0	0
$58$	7.23600	0.950133
$59$	−6.76596	−0.880853	−0.440427	−	0.897789i	$-0.645173\pi$
$59$	−6.76596	−0.880853	−0.440427	+	0.897789i	$0.645173\pi$
$60$	0	0
$61$	−8.15773	−1.04449	−0.522245	−	0.852796i	$-0.674905\pi$
$61$	−8.15773	−1.04449	−0.522245	+	0.852796i	$0.674905\pi$
$62$	3.27886	0.416415
$63$	0	0
$64$	−0.974926	−0.121866
$65$	15.4264	1.91341
$66$	0	0
$67$	−4.62250	−0.564728	−0.282364	−	0.959307i	$-0.591119\pi$
$67$	−4.62250	−0.564728	−0.282364	+	0.959307i	$0.591119\pi$
$68$	6.66673	0.808460
$69$	0	0
$70$	21.2343	2.53799
$71$	0.919786	0.109158	0.0545792	−	0.998509i	$-0.482618\pi$
$71$	0.919786	0.109158	0.0545792	+	0.998509i	$0.482618\pi$
$72$	0	0
$73$	13.2592	1.55187	0.775937	−	0.630811i	$-0.217278\pi$
$73$	13.2592	1.55187	0.775937	+	0.630811i	$0.217278\pi$
$74$	−13.5494	−1.57509
$75$	0	0
$76$	−5.08390	−0.583163
$77$	−3.70301	−0.421997
$78$	0	0
$79$	−3.72774	−0.419404	−0.209702	−	0.977765i	$-0.567249\pi$
$79$	−3.72774	−0.419404	−0.209702	+	0.977765i	$0.567249\pi$
$80$	−15.8412	−1.77110
$81$	0	0
$82$	−10.2061	−1.12708
$83$	11.8648	1.30233	0.651167	−	0.758935i	$-0.274280\pi$
$83$	11.8648	1.30233	0.651167	+	0.758935i	$0.274280\pi$
$84$	0	0
$85$	17.5414	1.90263
$86$	−2.27024	−0.244806
$87$	0	0
$88$	1.40692	0.149978
$89$	−5.92634	−0.628191	−0.314096	−	0.949391i	$-0.601701\pi$
$89$	−5.92634	−0.628191	−0.314096	+	0.949391i	$0.601701\pi$
$90$	0	0
$91$	−17.8629	−1.87254
$92$	10.5062	1.09535
$93$	0	0
$94$	−13.4139	−1.38353
$95$	−13.3766	−1.37242
$96$	0	0
$97$	11.8580	1.20400	0.602000	−	0.798496i	$-0.294371\pi$
$97$	11.8580	1.20400	0.602000	+	0.798496i	$0.294371\pi$
$98$	−12.0361	−1.21583
$99$	0	0
$100$	6.35244	0.635244
$101$	−17.5494	−1.74623	−0.873117	−	0.487512i	$-0.837905\pi$
$101$	−17.5494	−1.74623	−0.873117	+	0.487512i	$0.837905\pi$
$102$	0	0
$103$	1.06166	0.104608	0.0523041	−	0.998631i	$-0.483344\pi$
$103$	1.06166	0.104608	0.0523041	+	0.998631i	$0.483344\pi$
$104$	6.78683	0.665503
$105$	0	0
$106$	−5.11131	−0.496454
$107$	−9.62989	−0.930957	−0.465478	−	0.885059i	$-0.654118\pi$
$107$	−9.62989	−0.930957	−0.465478	+	0.885059i	$0.654118\pi$
$108$	0	0
$109$	2.64027	0.252892	0.126446	−	0.991974i	$-0.459643\pi$
$109$	2.64027	0.252892	0.126446	+	0.991974i	$0.459643\pi$
$110$	−5.73435	−0.546748
$111$	0	0
$112$	18.3432	1.73327
$113$	13.8928	1.30693	0.653465	−	0.756957i	$-0.273315\pi$
$113$	13.8928	1.30693	0.653465	+	0.756957i	$0.273315\pi$
$114$	0	0
$115$	27.6437	2.57779
$116$	−4.90453	−0.455375
$117$	0	0
$118$	12.1324	1.11688
$119$	−20.3119	−1.86199
$120$	0	0
$121$	1.00000	0.0909091
$122$	14.6280	1.32436
$123$	0	0
$124$	−2.22240	−0.199577
$125$	0.724837	0.0648314
$126$	0	0
$127$	0.495926	0.0440063	0.0220032	−	0.999758i	$-0.492996\pi$
$127$	0.495926	0.0440063	0.0220032	+	0.999758i	$0.492996\pi$
$128$	−10.3893	−0.918289
$129$	0	0
$130$	−27.6618	−2.42610
$131$	−8.56767	−0.748560	−0.374280	−	0.927316i	$-0.622110\pi$
$131$	−8.56767	−0.748560	−0.374280	+	0.927316i	$0.622110\pi$
$132$	0	0
$133$	15.4894	1.34310
$134$	8.28885	0.716047
$135$	0	0
$136$	7.71732	0.661755
$137$	−5.31872	−0.454409	−0.227204	−	0.973847i	$-0.572959\pi$
$137$	−5.31872	−0.454409	−0.227204	+	0.973847i	$0.572959\pi$
$138$	0	0
$139$	−4.07813	−0.345902	−0.172951	−	0.984930i	$-0.555330\pi$
$139$	−4.07813	−0.345902	−0.172951	+	0.984930i	$0.555330\pi$
$140$	−14.3925	−1.21639
$141$	0	0
$142$	−1.64932	−0.138407
$143$	4.82389	0.403394
$144$	0	0
$145$	−12.9047	−1.07168
$146$	−23.7758	−1.96770
$147$	0	0
$148$	9.18376	0.754901
$149$	11.0681	0.906736	0.453368	−	0.891323i	$-0.350222\pi$
$149$	11.0681	0.906736	0.453368	+	0.891323i	$0.350222\pi$
$150$	0	0
$151$	−1.45991	−0.118806	−0.0594029	−	0.998234i	$-0.518920\pi$
$151$	−1.45991	−0.118806	−0.0594029	+	0.998234i	$0.518920\pi$
$152$	−5.88505	−0.477341
$153$	0	0
$154$	6.64005	0.535070
$155$	−5.84753	−0.469685
$156$	0	0
$157$	−8.29775	−0.662232	−0.331116	−	0.943590i	$-0.607425\pi$
$157$	−8.29775	−0.662232	−0.331116	+	0.943590i	$0.607425\pi$
$158$	6.68441	0.531783
$159$	0	0
$160$	19.4073	1.53428
$161$	−32.0099	−2.52273
$162$	0	0
$163$	−5.55818	−0.435350	−0.217675	−	0.976021i	$-0.569847\pi$
$163$	−5.55818	−0.435350	−0.217675	+	0.976021i	$0.569847\pi$
$164$	6.91768	0.540180
$165$	0	0
$166$	−21.2754	−1.65129
$167$	−11.1085	−0.859601	−0.429800	−	0.902924i	$-0.641416\pi$
$167$	−11.1085	−0.859601	−0.429800	+	0.902924i	$0.641416\pi$
$168$	0	0
$169$	10.2699	0.789991
$170$	−31.4543	−2.41244
$171$	0	0
$172$	1.53876	0.117329
$173$	−2.00089	−0.152125	−0.0760625	−	0.997103i	$-0.524235\pi$
$173$	−2.00089	−0.152125	−0.0760625	+	0.997103i	$0.524235\pi$
$174$	0	0
$175$	−19.3544	−1.46305
$176$	−4.95361	−0.373392
$177$	0	0
$178$	10.6268	0.796515
$179$	1.44830	0.108251	0.0541255	−	0.998534i	$-0.482763\pi$
$179$	1.44830	0.108251	0.0541255	+	0.998534i	$0.482763\pi$
$180$	0	0
$181$	23.7790	1.76748	0.883740	−	0.467978i	$-0.155018\pi$
$181$	23.7790	1.76748	0.883740	+	0.467978i	$0.155018\pi$
$182$	32.0309	2.37428
$183$	0	0
$184$	12.1618	0.896583
$185$	24.1641	1.77658
$186$	0	0
$187$	5.48525	0.401121
$188$	9.09186	0.663092
$189$	0	0
$190$	23.9864	1.74015
$191$	13.7163	0.992474	0.496237	−	0.868187i	$-0.334715\pi$
$191$	13.7163	0.992474	0.496237	+	0.868187i	$0.334715\pi$
$192$	0	0
$193$	3.84107	0.276486	0.138243	−	0.990398i	$-0.455854\pi$
$193$	3.84107	0.276486	0.138243	+	0.990398i	$0.455854\pi$
$194$	−21.2632	−1.52661
$195$	0	0
$196$	8.15802	0.582715
$197$	1.92519	0.137164	0.0685820	−	0.997645i	$-0.478153\pi$
$197$	1.92519	0.137164	0.0685820	+	0.997645i	$0.478153\pi$
$198$	0	0
$199$	24.5507	1.74036	0.870178	−	0.492738i	$-0.164004\pi$
$199$	24.5507	1.74036	0.870178	+	0.492738i	$0.164004\pi$
$200$	7.35350	0.519971
$201$	0	0
$202$	31.4688	2.21414
$203$	14.9429	1.04879
$204$	0	0
$205$	18.2017	1.27126
$206$	−1.90371	−0.132638
$207$	0	0
$208$	−23.8956	−1.65686
$209$	−4.18293	−0.289339
$210$	0	0
$211$	−9.67792	−0.666256	−0.333128	−	0.942882i	$-0.608104\pi$
$211$	−9.67792	−0.666256	−0.333128	+	0.942882i	$0.608104\pi$
$212$	3.46443	0.237938
$213$	0	0
$214$	17.2679	1.18041
$215$	4.04876	0.276123
$216$	0	0
$217$	6.77111	0.459653
$218$	−4.73440	−0.320654
$219$	0	0
$220$	3.88672	0.262043
$221$	26.4602	1.77991
$222$	0	0
$223$	−2.40637	−0.161142	−0.0805711	−	0.996749i	$-0.525674\pi$
$223$	−2.40637	−0.161142	−0.0805711	+	0.996749i	$0.525674\pi$
$224$	−22.4725	−1.50151
$225$	0	0
$226$	−24.9120	−1.65712
$227$	19.9151	1.32181	0.660904	−	0.750470i	$-0.270173\pi$
$227$	19.9151	1.32181	0.660904	+	0.750470i	$0.270173\pi$
$228$	0	0
$229$	−24.7830	−1.63771	−0.818854	−	0.574002i	$-0.805390\pi$
$229$	−24.7830	−1.63771	−0.818854	+	0.574002i	$0.805390\pi$
$230$	−49.5694	−3.26851
$231$	0	0
$232$	−5.67742	−0.372741
$233$	−7.43800	−0.487280	−0.243640	−	0.969866i	$-0.578342\pi$
$233$	−7.43800	−0.487280	−0.243640	+	0.969866i	$0.578342\pi$
$234$	0	0
$235$	23.9223	1.56052
$236$	−8.22329	−0.535291
$237$	0	0
$238$	36.4223	2.36091
$239$	−8.90183	−0.575811	−0.287906	−	0.957659i	$-0.592959\pi$
$239$	−8.90183	−0.575811	−0.287906	+	0.957659i	$0.592959\pi$
$240$	0	0
$241$	1.07752	0.0694094	0.0347047	−	0.999398i	$-0.488951\pi$
$241$	1.07752	0.0694094	0.0347047	+	0.999398i	$0.488951\pi$
$242$	−1.79315	−0.115268
$243$	0	0
$244$	−9.91483	−0.634732
$245$	21.4652	1.37136
$246$	0	0
$247$	−20.1780	−1.28389
$248$	−2.57262	−0.163361
$249$	0	0
$250$	−1.29974	−0.0822029
$251$	13.9016	0.877458	0.438729	−	0.898619i	$-0.355429\pi$
$251$	13.9016	0.877458	0.438729	+	0.898619i	$0.355429\pi$
$252$	0	0
$253$	8.64429	0.543462
$254$	−0.889270	−0.0557978
$255$	0	0
$256$	20.5794	1.28621
$257$	−11.7470	−0.732760	−0.366380	−	0.930465i	$-0.619403\pi$
$257$	−11.7470	−0.732760	−0.366380	+	0.930465i	$0.619403\pi$
$258$	0	0
$259$	−27.9807	−1.73864
$260$	18.7491	1.16277
$261$	0	0
$262$	15.3631	0.949137
$263$	−8.10819	−0.499972	−0.249986	−	0.968249i	$-0.580426\pi$
$263$	−8.10819	−0.499972	−0.249986	+	0.968249i	$0.580426\pi$
$264$	0	0
$265$	9.11554	0.559963
$266$	−27.7749	−1.70299
$267$	0	0
$268$	−5.61815	−0.343183
$269$	4.81314	0.293463	0.146731	−	0.989176i	$-0.453125\pi$
$269$	4.81314	0.293463	0.146731	+	0.989176i	$0.453125\pi$
$270$	0	0
$271$	−10.0565	−0.610889	−0.305445	−	0.952210i	$-0.598805\pi$
$271$	−10.0565	−0.610889	−0.305445	+	0.952210i	$0.598805\pi$
$272$	−27.1718	−1.64753
$273$	0	0
$274$	9.53727	0.576167
$275$	5.22666	0.315179
$276$	0	0
$277$	15.2011	0.913348	0.456674	−	0.889634i	$-0.349041\pi$
$277$	15.2011	0.913348	0.456674	+	0.889634i	$0.349041\pi$
$278$	7.31270	0.438587
$279$	0	0
$280$	−16.6606	−0.995662
$281$	−0.225239	−0.0134366	−0.00671831	−	0.999977i	$-0.502139\pi$
$281$	−0.225239	−0.0134366	−0.00671831	+	0.999977i	$0.502139\pi$
$282$	0	0
$283$	−21.8642	−1.29969	−0.649846	−	0.760066i	$-0.725167\pi$
$283$	−21.8642	−1.29969	−0.649846	+	0.760066i	$0.725167\pi$
$284$	1.11790	0.0663352
$285$	0	0
$286$	−8.64996	−0.511483
$287$	−21.0765	−1.24411
$288$	0	0
$289$	13.0880	0.769881
$290$	23.1401	1.35883
$291$	0	0
$292$	16.1151	0.943067
$293$	25.4121	1.48459	0.742296	−	0.670072i	$-0.233737\pi$
$293$	25.4121	1.48459	0.742296	+	0.670072i	$0.233737\pi$
$294$	0	0
$295$	−21.6370	−1.25975
$296$	10.6310	0.617914
$297$	0	0
$298$	−19.8468	−1.14970
$299$	41.6991	2.41152
$300$	0	0
$301$	−4.68823	−0.270225
$302$	2.61784	0.150640
$303$	0	0
$304$	20.7206	1.18841
$305$	−26.0877	−1.49378
$306$	0	0
$307$	−26.4230	−1.50804	−0.754021	−	0.656850i	$-0.771888\pi$
$307$	−26.4230	−1.50804	−0.754021	+	0.656850i	$0.771888\pi$
$308$	−4.50060	−0.256446
$309$	0	0
$310$	10.4855	0.595537
$311$	12.0563	0.683648	0.341824	−	0.939764i	$-0.388955\pi$
$311$	12.0563	0.683648	0.341824	+	0.939764i	$0.388955\pi$
$312$	0	0
$313$	28.8195	1.62898	0.814489	−	0.580180i	$-0.197018\pi$
$313$	28.8195	1.62898	0.814489	+	0.580180i	$0.197018\pi$
$314$	14.8791	0.839677
$315$	0	0
$316$	−4.53067	−0.254870
$317$	−7.26561	−0.408077	−0.204039	−	0.978963i	$-0.565407\pi$
$317$	−7.26561	−0.408077	−0.204039	+	0.978963i	$0.565407\pi$
$318$	0	0
$319$	−4.03535	−0.225936
$320$	−3.11773	−0.174286
$321$	0	0
$322$	57.3985	3.19870
$323$	−22.9444	−1.27666
$324$	0	0
$325$	25.2128	1.39856
$326$	9.96666	0.552002
$327$	0	0
$328$	8.00782	0.442158
$329$	−27.7007	−1.52719
$330$	0	0
$331$	15.5863	0.856701	0.428351	−	0.903613i	$-0.359095\pi$
$331$	15.5863	0.856701	0.428351	+	0.903613i	$0.359095\pi$
$332$	14.4204	0.791423
$333$	0	0
$334$	19.9192	1.08993
$335$	−14.7824	−0.807647
$336$	0	0
$337$	−10.9156	−0.594612	−0.297306	−	0.954782i	$-0.596088\pi$
$337$	−10.9156	−0.594612	−0.297306	+	0.954782i	$0.596088\pi$
$338$	−18.4155	−1.00167
$339$	0	0
$340$	21.3196	1.15622
$341$	−1.82854	−0.0990212
$342$	0	0
$343$	1.06553	0.0575331
$344$	1.78125	0.0960385
$345$	0	0
$346$	3.58790	0.192887
$347$	22.0306	1.18267	0.591333	−	0.806428i	$-0.298602\pi$
$347$	22.0306	1.18267	0.591333	+	0.806428i	$0.298602\pi$
$348$	0	0
$349$	1.95136	0.104454	0.0522271	−	0.998635i	$-0.483368\pi$
$349$	1.95136	0.104454	0.0522271	+	0.998635i	$0.483368\pi$
$350$	34.7053	1.85508
$351$	0	0
$352$	6.06872	0.323464
$353$	−27.1444	−1.44475	−0.722374	−	0.691502i	$-0.756949\pi$
$353$	−27.1444	−1.44475	−0.722374	+	0.691502i	$0.756949\pi$
$354$	0	0
$355$	2.94140	0.156113
$356$	−7.20283	−0.381749
$357$	0	0
$358$	−2.59702	−0.137257
$359$	−3.89422	−0.205529	−0.102765	−	0.994706i	$-0.532769\pi$
$359$	−3.89422	−0.205529	−0.102765	+	0.994706i	$0.532769\pi$
$360$	0	0
$361$	−1.50311	−0.0791113
$362$	−42.6394	−2.24108
$363$	0	0
$364$	−21.7104	−1.13793
$365$	42.4018	2.21941
$366$	0	0
$367$	−16.7234	−0.872956	−0.436478	−	0.899715i	$-0.643774\pi$
$367$	−16.7234	−0.872956	−0.436478	+	0.899715i	$0.643774\pi$
$368$	−42.8204	−2.23217
$369$	0	0
$370$	−43.3300	−2.25262
$371$	−10.5553	−0.548003
$372$	0	0
$373$	−9.75893	−0.505298	−0.252649	−	0.967558i	$-0.581302\pi$
$373$	−9.75893	−0.505298	−0.252649	+	0.967558i	$0.581302\pi$
$374$	−9.83589	−0.508602
$375$	0	0
$376$	10.5246	0.542766
$377$	−19.4661	−1.00255
$378$	0	0
$379$	−1.43173	−0.0735429	−0.0367714	−	0.999324i	$-0.511707\pi$
$379$	−1.43173	−0.0735429	−0.0367714	+	0.999324i	$0.511707\pi$
$380$	−16.2579	−0.834011
$381$	0	0
$382$	−24.5953	−1.25841
$383$	−0.106884	−0.00546152	−0.00273076	−	0.999996i	$-0.500869\pi$
$383$	−0.106884	−0.00546152	−0.00273076	+	0.999996i	$0.500869\pi$
$384$	0	0
$385$	−11.8419	−0.603519
$386$	−6.88762	−0.350571
$387$	0	0
$388$	14.4121	0.731666
$389$	36.2430	1.83759	0.918796	−	0.394734i	$-0.129163\pi$
$389$	36.2430	1.83759	0.918796	+	0.394734i	$0.129163\pi$
$390$	0	0
$391$	47.4161	2.39794
$392$	9.44361	0.476974
$393$	0	0
$394$	−3.45215	−0.173917
$395$	−11.9210	−0.599811
$396$	0	0
$397$	4.53142	0.227425	0.113713	−	0.993514i	$-0.463726\pi$
$397$	4.53142	0.227425	0.113713	+	0.993514i	$0.463726\pi$
$398$	−44.0232	−2.20668
$399$	0	0
$400$	−25.8908	−1.29454
$401$	−25.8843	−1.29260	−0.646300	−	0.763083i	$-0.723685\pi$
$401$	−25.8843	−1.29260	−0.646300	+	0.763083i	$0.723685\pi$
$402$	0	0
$403$	−8.82069	−0.439390
$404$	−21.3294	−1.06118
$405$	0	0
$406$	−26.7949	−1.32981
$407$	7.55622	0.374548
$408$	0	0
$409$	−4.78503	−0.236604	−0.118302	−	0.992978i	$-0.537745\pi$
$409$	−4.78503	−0.236604	−0.118302	+	0.992978i	$0.537745\pi$
$410$	−32.6384	−1.61189
$411$	0	0
$412$	1.29033	0.0635699
$413$	25.0544	1.23285
$414$	0	0
$415$	37.9427	1.86253
$416$	29.2748	1.43532
$417$	0	0
$418$	7.50062	0.366867
$419$	−18.0658	−0.882572	−0.441286	−	0.897366i	$-0.645478\pi$
$419$	−18.0658	−0.882572	−0.441286	+	0.897366i	$0.645478\pi$
$420$	0	0
$421$	−2.53408	−0.123503	−0.0617517	−	0.998092i	$-0.519669\pi$
$421$	−2.53408	−0.123503	−0.0617517	+	0.998092i	$0.519669\pi$
$422$	17.3540	0.844778
$423$	0	0
$424$	4.01038	0.194761
$425$	28.6695	1.39068
$426$	0	0
$427$	30.2081	1.46187
$428$	−11.7041	−0.565739
$429$	0	0
$430$	−7.26003	−0.350110
$431$	−23.9012	−1.15128	−0.575639	−	0.817704i	$-0.695247\pi$
$431$	−23.9012	−1.15128	−0.575639	+	0.817704i	$0.695247\pi$
$432$	0	0
$433$	−7.79860	−0.374777	−0.187388	−	0.982286i	$-0.560002\pi$
$433$	−7.79860	−0.374777	−0.187388	+	0.982286i	$0.560002\pi$
$434$	−12.1416	−0.582817
$435$	0	0
$436$	3.20896	0.153681
$437$	−36.1585	−1.72969
$438$	0	0
$439$	11.5915	0.553230	0.276615	−	0.960981i	$-0.410787\pi$
$439$	11.5915	0.553230	0.276615	+	0.960981i	$0.410787\pi$
$440$	4.49921	0.214492
$441$	0	0
$442$	−47.4472	−2.25683
$443$	21.3300	1.01342	0.506709	−	0.862117i	$-0.330862\pi$
$443$	21.3300	1.01342	0.506709	+	0.862117i	$0.330862\pi$
$444$	0	0
$445$	−18.9519	−0.898408
$446$	4.31498	0.204320
$447$	0	0
$448$	3.61016	0.170564
$449$	−5.51518	−0.260277	−0.130139	−	0.991496i	$-0.541542\pi$
$449$	−5.51518	−0.260277	−0.130139	+	0.991496i	$0.541542\pi$
$450$	0	0
$451$	5.69173	0.268013
$452$	16.8853	0.794215
$453$	0	0
$454$	−35.7107	−1.67599
$455$	−57.1240	−2.67801
$456$	0	0
$457$	−27.6712	−1.29440	−0.647202	−	0.762318i	$-0.724061\pi$
$457$	−27.6712	−1.29440	−0.647202	+	0.762318i	$0.724061\pi$
$458$	44.4397	2.07653
$459$	0	0
$460$	33.5979	1.56651
$461$	−9.58640	−0.446483	−0.223242	−	0.974763i	$-0.571664\pi$
$461$	−9.58640	−0.446483	−0.223242	+	0.974763i	$0.571664\pi$
$462$	0	0
$463$	−4.38498	−0.203788	−0.101894	−	0.994795i	$-0.532490\pi$
$463$	−4.38498	−0.203788	−0.101894	+	0.994795i	$0.532490\pi$
$464$	19.9895	0.927991
$465$	0	0
$466$	13.3375	0.617846
$467$	2.66661	0.123396	0.0616981	−	0.998095i	$-0.480348\pi$
$467$	2.66661	0.123396	0.0616981	+	0.998095i	$0.480348\pi$
$468$	0	0
$469$	17.1172	0.790397
$470$	−42.8964	−1.97866
$471$	0	0
$472$	−9.51918	−0.438156
$473$	1.26606	0.0582135
$474$	0	0
$475$	−21.8627	−1.00313
$476$	−24.6869	−1.13152
$477$	0	0
$478$	15.9623	0.730100
$479$	−21.4978	−0.982259	−0.491130	−	0.871086i	$-0.663416\pi$
$479$	−21.4978	−0.982259	−0.491130	+	0.871086i	$0.663416\pi$
$480$	0	0
$481$	36.4503	1.66199
$482$	−1.93216	−0.0880076
$483$	0	0
$484$	1.21539	0.0552451
$485$	37.9209	1.72190
$486$	0	0
$487$	23.1203	1.04768	0.523841	−	0.851816i	$-0.324499\pi$
$487$	23.1203	1.04768	0.523841	+	0.851816i	$0.324499\pi$
$488$	−11.4773	−0.519552
$489$	0	0
$490$	−38.4904	−1.73882
$491$	8.10982	0.365991	0.182996	−	0.983114i	$-0.441421\pi$
$491$	8.10982	0.365991	0.182996	+	0.983114i	$0.441421\pi$
$492$	0	0
$493$	−22.1349	−0.996907
$494$	36.1822	1.62791
$495$	0	0
$496$	9.05789	0.406711
$497$	−3.40597	−0.152779
$498$	0	0
$499$	−35.6731	−1.59695	−0.798474	−	0.602029i	$-0.794359\pi$
$499$	−35.6731	−1.59695	−0.798474	+	0.602029i	$0.794359\pi$
$500$	0.880961	0.0393978
$501$	0	0
$502$	−24.9276	−1.11257
$503$	−22.4422	−1.00065	−0.500325	−	0.865838i	$-0.666786\pi$
$503$	−22.4422	−1.00065	−0.500325	+	0.865838i	$0.666786\pi$
$504$	0	0
$505$	−56.1216	−2.49738
$506$	−15.5005	−0.689083
$507$	0	0
$508$	0.602744	0.0267425
$509$	5.75483	0.255078	0.127539	−	0.991834i	$-0.459292\pi$
$509$	5.75483	0.255078	0.127539	+	0.991834i	$0.459292\pi$
$510$	0	0
$511$	−49.0990	−2.17201
$512$	−16.1234	−0.712560
$513$	0	0
$514$	21.0642	0.929103
$515$	3.39509	0.149605
$516$	0	0
$517$	7.48060	0.328996
$518$	50.1737	2.20450
$519$	0	0
$520$	21.7037	0.951770
$521$	−10.6027	−0.464511	−0.232256	−	0.972655i	$-0.574611\pi$
$521$	−10.6027	−0.464511	−0.232256	+	0.972655i	$0.574611\pi$
$522$	0	0
$523$	24.2238	1.05923	0.529617	−	0.848237i	$-0.322336\pi$
$523$	24.2238	1.05923	0.529617	+	0.848237i	$0.322336\pi$
$524$	−10.4131	−0.454897
$525$	0	0
$526$	14.5392	0.633940
$527$	−10.0300	−0.436915
$528$	0	0
$529$	51.7238	2.24886
$530$	−16.3455	−0.710005
$531$	0	0
$532$	18.8257	0.816197
$533$	27.4563	1.18926
$534$	0	0
$535$	−30.7956	−1.33141
$536$	−6.50350	−0.280908
$537$	0	0
$538$	−8.63070	−0.372096
$539$	6.71225	0.289117
$540$	0	0
$541$	10.5688	0.454386	0.227193	−	0.973850i	$-0.427045\pi$
$541$	10.5688	0.454386	0.227193	+	0.973850i	$0.427045\pi$
$542$	18.0328	0.774577
$543$	0	0
$544$	33.2885	1.42723
$545$	8.44335	0.361674
$546$	0	0
$547$	33.0336	1.41242	0.706208	−	0.708004i	$-0.250404\pi$
$547$	33.0336	1.41242	0.706208	+	0.708004i	$0.250404\pi$
$548$	−6.46433	−0.276142
$549$	0	0
$550$	−9.37219	−0.399632
$551$	16.8796	0.719095
$552$	0	0
$553$	13.8039	0.587000
$554$	−27.2579	−1.15808
$555$	0	0
$556$	−4.95652	−0.210203
$557$	−22.3794	−0.948247	−0.474124	−	0.880458i	$-0.657235\pi$
$557$	−22.3794	−0.948247	−0.474124	+	0.880458i	$0.657235\pi$
$558$	0	0
$559$	6.10734	0.258313
$560$	58.6601	2.47884
$561$	0	0
$562$	0.403887	0.0170370
$563$	37.1527	1.56580	0.782901	−	0.622147i	$-0.213739\pi$
$563$	37.1527	1.56580	0.782901	+	0.622147i	$0.213739\pi$
$564$	0	0
$565$	44.4281	1.86911
$566$	39.2058	1.64794
$567$	0	0
$568$	1.29407	0.0542978
$569$	−25.3073	−1.06094	−0.530468	−	0.847705i	$-0.677984\pi$
$569$	−25.3073	−1.06094	−0.530468	+	0.847705i	$0.677984\pi$
$570$	0	0
$571$	14.6130	0.611535	0.305767	−	0.952106i	$-0.401087\pi$
$571$	14.6130	0.611535	0.305767	+	0.952106i	$0.401087\pi$
$572$	5.86291	0.245141
$573$	0	0
$574$	37.7934	1.57747
$575$	45.1808	1.88417
$576$	0	0
$577$	38.2873	1.59392	0.796961	−	0.604031i	$-0.206440\pi$
$577$	38.2873	1.59392	0.796961	+	0.604031i	$0.206440\pi$
$578$	−23.4687	−0.976170
$579$	0	0
$580$	−15.6843	−0.651254
$581$	−43.9355	−1.82275
$582$	0	0
$583$	2.85046	0.118054
$584$	18.6547	0.771936
$585$	0	0
$586$	−45.5678	−1.88239
$587$	0.531608	0.0219418	0.0109709	−	0.999940i	$-0.496508\pi$
$587$	0.531608	0.0219418	0.0109709	+	0.999940i	$0.496508\pi$
$588$	0	0
$589$	7.64867	0.315158
$590$	38.7984	1.59730
$591$	0	0
$592$	−37.4305	−1.53838
$593$	41.0711	1.68659	0.843294	−	0.537453i	$-0.180614\pi$
$593$	41.0711	1.68659	0.843294	+	0.537453i	$0.180614\pi$
$594$	0	0
$595$	−64.9558	−2.66293
$596$	13.4521	0.551020
$597$	0	0
$598$	−74.7728	−3.05769
$599$	25.3755	1.03681	0.518407	−	0.855134i	$-0.326525\pi$
$599$	25.3755	1.03681	0.518407	+	0.855134i	$0.326525\pi$
$600$	0	0
$601$	−38.0912	−1.55377	−0.776887	−	0.629640i	$-0.783202\pi$
$601$	−38.0912	−1.55377	−0.776887	+	0.629640i	$0.783202\pi$
$602$	8.40671	0.342632
$603$	0	0
$604$	−1.77436	−0.0721977
$605$	3.19791	0.130014
$606$	0	0
$607$	−24.2023	−0.982342	−0.491171	−	0.871063i	$-0.663431\pi$
$607$	−24.2023	−0.982342	−0.491171	+	0.871063i	$0.663431\pi$
$608$	−25.3850	−1.02950
$609$	0	0
$610$	46.7792	1.89403
$611$	36.0856	1.45987
$612$	0	0
$613$	−18.1913	−0.734741	−0.367371	−	0.930075i	$-0.619742\pi$
$613$	−18.1913	−0.734741	−0.367371	+	0.930075i	$0.619742\pi$
$614$	47.3805	1.91212
$615$	0	0
$616$	−5.20984	−0.209910
$617$	43.8569	1.76561	0.882806	−	0.469738i	$-0.155652\pi$
$617$	43.8569	1.76561	0.882806	+	0.469738i	$0.155652\pi$
$618$	0	0
$619$	28.4240	1.14246	0.571228	−	0.820791i	$-0.306467\pi$
$619$	28.4240	1.14246	0.571228	+	0.820791i	$0.306467\pi$
$620$	−7.10704	−0.285426
$621$	0	0
$622$	−21.6187	−0.866831
$623$	21.9453	0.879219
$624$	0	0
$625$	−23.8153	−0.952613
$626$	−51.6778	−2.06546
$627$	0	0
$628$	−10.0850	−0.402436
$629$	41.4477	1.65263
$630$	0	0
$631$	−35.0274	−1.39442	−0.697209	−	0.716867i	$-0.745575\pi$
$631$	−35.0274	−1.39442	−0.697209	+	0.716867i	$0.745575\pi$
$632$	−5.24464	−0.208621
$633$	0	0
$634$	13.0283	0.517421
$635$	1.58593	0.0629357
$636$	0	0
$637$	32.3792	1.28291
$638$	7.23600	0.286476
$639$	0	0
$640$	−33.2240	−1.31329
$641$	21.1456	0.835200	0.417600	−	0.908631i	$-0.362871\pi$
$641$	21.1456	0.835200	0.417600	+	0.908631i	$0.362871\pi$
$642$	0	0
$643$	−36.7993	−1.45122	−0.725611	−	0.688105i	$-0.758443\pi$
$643$	−36.7993	−1.45122	−0.725611	+	0.688105i	$0.758443\pi$
$644$	−38.9045	−1.53305
$645$	0	0
$646$	41.1428	1.61874
$647$	−5.33180	−0.209615	−0.104807	−	0.994493i	$-0.533423\pi$
$647$	−5.33180	−0.209615	−0.104807	+	0.994493i	$0.533423\pi$
$648$	0	0
$649$	−6.76596	−0.265587
$650$	−45.2104	−1.77330
$651$	0	0
$652$	−6.75537	−0.264561
$653$	39.7088	1.55392	0.776962	−	0.629547i	$-0.216760\pi$
$653$	39.7088	1.55392	0.776962	+	0.629547i	$0.216760\pi$
$654$	0	0
$655$	−27.3987	−1.07055
$656$	−28.1946	−1.10081
$657$	0	0
$658$	49.6716	1.93640
$659$	16.5765	0.645730	0.322865	−	0.946445i	$-0.395354\pi$
$659$	16.5765	0.645730	0.322865	+	0.946445i	$0.395354\pi$
$660$	0	0
$661$	10.5001	0.408407	0.204204	−	0.978928i	$-0.434540\pi$
$661$	10.5001	0.408407	0.204204	+	0.978928i	$0.434540\pi$
$662$	−27.9486	−1.08625
$663$	0	0
$664$	16.6929	0.647809
$665$	49.5338	1.92084
$666$	0	0
$667$	−34.8828	−1.35067
$668$	−13.5012	−0.522376
$669$	0	0
$670$	26.5070	1.02406
$671$	−8.15773	−0.314925
$672$	0	0
$673$	21.9300	0.845340	0.422670	−	0.906284i	$-0.361093\pi$
$673$	21.9300	0.845340	0.422670	+	0.906284i	$0.361093\pi$
$674$	19.5734	0.753938
$675$	0	0
$676$	12.4819	0.480074
$677$	3.57109	0.137248	0.0686241	−	0.997643i	$-0.478139\pi$
$677$	3.57109	0.137248	0.0686241	+	0.997643i	$0.478139\pi$
$678$	0	0
$679$	−43.9103	−1.68512
$680$	24.6793	0.946409
$681$	0	0
$682$	3.27886	0.125554
$683$	−6.25131	−0.239200	−0.119600	−	0.992822i	$-0.538161\pi$
$683$	−6.25131	−0.239200	−0.119600	+	0.992822i	$0.538161\pi$
$684$	0	0
$685$	−17.0088	−0.649873
$686$	−1.91065	−0.0729491
$687$	0	0
$688$	−6.27157	−0.239101
$689$	13.7503	0.523845
$690$	0	0
$691$	−10.4515	−0.397594	−0.198797	−	0.980041i	$-0.563704\pi$
$691$	−10.4515	−0.397594	−0.198797	+	0.980041i	$0.563704\pi$
$692$	−2.43187	−0.0924458
$693$	0	0
$694$	−39.5042	−1.49956
$695$	−13.0415	−0.494693
$696$	0	0
$697$	31.2206	1.18256
$698$	−3.49909	−0.132443
$699$	0	0
$700$	−23.5231	−0.889090
$701$	−13.7160	−0.518047	−0.259024	−	0.965871i	$-0.583401\pi$
$701$	−13.7160	−0.518047	−0.259024	+	0.965871i	$0.583401\pi$
$702$	0	0
$703$	−31.6071	−1.19208
$704$	−0.974926	−0.0367439
$705$	0	0
$706$	48.6739	1.83187
$707$	64.9856	2.44404
$708$	0	0
$709$	1.98721	0.0746312	0.0373156	−	0.999304i	$-0.488119\pi$
$709$	1.98721	0.0746312	0.0373156	+	0.999304i	$0.488119\pi$
$710$	−5.27437	−0.197944
$711$	0	0
$712$	−8.33790	−0.312476
$713$	−15.8065	−0.591957
$714$	0	0
$715$	15.4264	0.576914
$716$	1.76025	0.0657837
$717$	0	0
$718$	6.98293	0.260601
$719$	19.7038	0.734829	0.367414	−	0.930057i	$-0.380243\pi$
$719$	19.7038	0.734829	0.367414	+	0.930057i	$0.380243\pi$
$720$	0	0
$721$	−3.93132	−0.146410
$722$	2.69531	0.100309
$723$	0	0
$724$	28.9008	1.07409
$725$	−21.0914	−0.783315
$726$	0	0
$727$	−10.5565	−0.391520	−0.195760	−	0.980652i	$-0.562717\pi$
$727$	−10.5565	−0.391520	−0.195760	+	0.980652i	$0.562717\pi$
$728$	−25.1317	−0.931442
$729$	0	0
$730$	−76.0329	−2.81410
$731$	6.94466	0.256858
$732$	0	0
$733$	33.3988	1.23361	0.616806	−	0.787116i	$-0.288426\pi$
$733$	33.3988	1.23361	0.616806	+	0.787116i	$0.288426\pi$
$734$	29.9876	1.10686
$735$	0	0
$736$	52.4598	1.93369
$737$	−4.62250	−0.170272
$738$	0	0
$739$	−6.13230	−0.225580	−0.112790	−	0.993619i	$-0.535979\pi$
$739$	−6.13230	−0.225580	−0.112790	+	0.993619i	$0.535979\pi$
$740$	29.3689	1.07962
$741$	0	0
$742$	18.9272	0.694840
$743$	39.0558	1.43282	0.716408	−	0.697681i	$-0.245785\pi$
$743$	39.0558	1.43282	0.716408	+	0.697681i	$0.245785\pi$
$744$	0	0
$745$	35.3949	1.29677
$746$	17.4992	0.640693
$747$	0	0
$748$	6.66673	0.243760
$749$	35.6596	1.30297
$750$	0	0
$751$	39.6418	1.44655	0.723276	−	0.690559i	$-0.242636\pi$
$751$	39.6418	1.44655	0.723276	+	0.690559i	$0.242636\pi$
$752$	−37.0560	−1.35129
$753$	0	0
$754$	34.9056	1.27119
$755$	−4.66866	−0.169910
$756$	0	0
$757$	31.6638	1.15084	0.575421	−	0.817857i	$-0.304838\pi$
$757$	31.6638	1.15084	0.575421	+	0.817857i	$0.304838\pi$
$758$	2.56730	0.0932487
$759$	0	0
$760$	−18.8199	−0.682669
$761$	12.4397	0.450940	0.225470	−	0.974250i	$-0.427608\pi$
$761$	12.4397	0.450940	0.225470	+	0.974250i	$0.427608\pi$
$762$	0	0
$763$	−9.77693	−0.353949
$764$	16.6706	0.603122
$765$	0	0
$766$	0.191659	0.00692494
$767$	−32.6382	−1.17850
$768$	0	0
$769$	51.0285	1.84013	0.920067	−	0.391761i	$-0.128134\pi$
$769$	51.0285	1.84013	0.920067	+	0.391761i	$0.128134\pi$
$770$	21.2343	0.765232
$771$	0	0
$772$	4.66841	0.168020
$773$	49.2403	1.77105	0.885525	−	0.464592i	$-0.153799\pi$
$773$	49.2403	1.77105	0.885525	+	0.464592i	$0.153799\pi$
$774$	0	0
$775$	−9.55718	−0.343304
$776$	16.6833	0.598896
$777$	0	0
$778$	−64.9891	−2.32997
$779$	−23.8081	−0.853014
$780$	0	0
$781$	0.919786	0.0329125
$782$	−85.0243	−3.04046
$783$	0	0
$784$	−33.2499	−1.18750
$785$	−26.5355	−0.947092
$786$	0	0
$787$	45.9304	1.63724	0.818621	−	0.574334i	$-0.194739\pi$
$787$	45.9304	1.63724	0.818621	+	0.574334i	$0.194739\pi$
$788$	2.33986	0.0833540
$789$	0	0
$790$	21.3762	0.760530
$791$	−51.4453	−1.82918
$792$	0	0
$793$	−39.3519	−1.39743
$794$	−8.12552	−0.288364
$795$	0	0
$796$	29.8388	1.05761
$797$	−12.5023	−0.442854	−0.221427	−	0.975177i	$-0.571071\pi$
$797$	−12.5023	−0.442854	−0.221427	+	0.975177i	$0.571071\pi$
$798$	0	0
$799$	41.0330	1.45164
$800$	31.7191	1.12144
$801$	0	0
$802$	46.4145	1.63895
$803$	13.2592	0.467908
$804$	0	0
$805$	−102.365	−3.60789
$806$	15.8168	0.557124
$807$	0	0
$808$	−24.6907	−0.868614
$809$	−21.9152	−0.770497	−0.385249	−	0.922813i	$-0.625884\pi$
$809$	−21.9152	−0.770497	−0.385249	+	0.922813i	$0.625884\pi$
$810$	0	0
$811$	49.8717	1.75123	0.875616	−	0.483008i	$-0.160456\pi$
$811$	49.8717	1.75123	0.875616	+	0.483008i	$0.160456\pi$
$812$	18.1615	0.637344
$813$	0	0
$814$	−13.5494	−0.474908
$815$	−17.7746	−0.622617
$816$	0	0
$817$	−5.29584	−0.185278
$818$	8.58028	0.300003
$819$	0	0
$820$	22.1222	0.772540
$821$	14.9001	0.520016	0.260008	−	0.965606i	$-0.416275\pi$
$821$	14.9001	0.520016	0.260008	+	0.965606i	$0.416275\pi$
$822$	0	0
$823$	−1.54227	−0.0537601	−0.0268800	−	0.999639i	$-0.508557\pi$
$823$	−1.54227	−0.0537601	−0.0268800	+	0.999639i	$0.508557\pi$
$824$	1.49367	0.0520344
$825$	0	0
$826$	−44.9263	−1.56319
$827$	−0.945513	−0.0328787	−0.0164394	−	0.999865i	$-0.505233\pi$
$827$	−0.945513	−0.0328787	−0.0164394	+	0.999865i	$0.505233\pi$
$828$	0	0
$829$	40.2839	1.39912	0.699559	−	0.714575i	$-0.253380\pi$
$829$	40.2839	1.39912	0.699559	+	0.714575i	$0.253380\pi$
$830$	−68.0370	−2.36160
$831$	0	0
$832$	−4.70293	−0.163045
$833$	36.8184	1.27568
$834$	0	0
$835$	−35.5240	−1.22936
$836$	−5.08390	−0.175830
$837$	0	0
$838$	32.3947	1.11906
$839$	34.7679	1.20032	0.600161	−	0.799879i	$-0.295103\pi$
$839$	34.7679	1.20032	0.600161	+	0.799879i	$0.295103\pi$
$840$	0	0
$841$	−12.7159	−0.438480
$842$	4.54399	0.156596
$843$	0	0
$844$	−11.7625	−0.404881
$845$	32.8422	1.12981
$846$	0	0
$847$	−3.70301	−0.127237
$848$	−14.1201	−0.484885
$849$	0	0
$850$	−51.4088	−1.76331
$851$	65.3182	2.23908
$852$	0	0
$853$	−31.5242	−1.07937	−0.539683	−	0.841868i	$-0.681456\pi$
$853$	−31.5242	−1.07937	−0.539683	+	0.841868i	$0.681456\pi$
$854$	−54.1677	−1.85358
$855$	0	0
$856$	−13.5485	−0.463078
$857$	−19.9742	−0.682306	−0.341153	−	0.940008i	$-0.610817\pi$
$857$	−19.9742	−0.682306	−0.341153	+	0.940008i	$0.610817\pi$
$858$	0	0
$859$	39.4390	1.34564	0.672820	−	0.739806i	$-0.265083\pi$
$859$	39.4390	1.34564	0.672820	+	0.739806i	$0.265083\pi$
$860$	4.92082	0.167799
$861$	0	0
$862$	42.8584	1.45976
$863$	22.5802	0.768638	0.384319	−	0.923200i	$-0.374436\pi$
$863$	22.5802	0.768638	0.384319	+	0.923200i	$0.374436\pi$
$864$	0	0
$865$	−6.39869	−0.217562
$866$	13.9841	0.475198
$867$	0	0
$868$	8.22955	0.279329
$869$	−3.72774	−0.126455
$870$	0	0
$871$	−22.2984	−0.755553
$872$	3.71465	0.125794
$873$	0	0
$874$	64.8376	2.19316
$875$	−2.68408	−0.0907383
$876$	0	0
$877$	44.3242	1.49672	0.748360	−	0.663293i	$-0.230842\pi$
$877$	44.3242	1.49672	0.748360	+	0.663293i	$0.230842\pi$
$878$	−20.7852	−0.701468
$879$	0	0
$880$	−15.8412	−0.534007
$881$	54.9134	1.85008	0.925041	−	0.379868i	$-0.124031\pi$
$881$	54.9134	1.85008	0.925041	+	0.379868i	$0.124031\pi$
$882$	0	0
$883$	6.27138	0.211049	0.105524	−	0.994417i	$-0.466348\pi$
$883$	6.27138	0.211049	0.105524	+	0.994417i	$0.466348\pi$
$884$	32.1595	1.08164
$885$	0	0
$886$	−38.2479	−1.28496
$887$	−19.6718	−0.660513	−0.330257	−	0.943891i	$-0.607135\pi$
$887$	−19.6718	−0.660513	−0.330257	+	0.943891i	$0.607135\pi$
$888$	0	0
$889$	−1.83642	−0.0615914
$890$	33.9837	1.13914
$891$	0	0
$892$	−2.92468	−0.0979255
$893$	−31.2908	−1.04711
$894$	0	0
$895$	4.63154	0.154815
$896$	38.4715	1.28524
$897$	0	0
$898$	9.88955	0.330018
$899$	7.37882	0.246097
$900$	0	0
$901$	15.6355	0.520894
$902$	−10.2061	−0.339827
$903$	0	0
$904$	19.5461	0.650095
$905$	76.0432	2.52776
$906$	0	0
$907$	−12.9218	−0.429061	−0.214531	−	0.976717i	$-0.568822\pi$
$907$	−12.9218	−0.429061	−0.214531	+	0.976717i	$0.568822\pi$
$908$	24.2046	0.803258
$909$	0	0
$910$	102.432	3.39559
$911$	−44.3624	−1.46979	−0.734897	−	0.678179i	$-0.762769\pi$
$911$	−44.3624	−1.46979	−0.734897	+	0.678179i	$0.762769\pi$
$912$	0	0
$913$	11.8648	0.392668
$914$	49.6186	1.64124
$915$	0	0
$916$	−30.1211	−0.995229
$917$	31.7261	1.04769
$918$	0	0
$919$	−43.4630	−1.43371	−0.716856	−	0.697221i	$-0.754420\pi$
$919$	−43.4630	−1.43371	−0.716856	+	0.697221i	$0.754420\pi$
$920$	38.8925	1.28225
$921$	0	0
$922$	17.1899	0.566118
$923$	4.43694	0.146044
$924$	0	0
$925$	39.4938	1.29855
$926$	7.86294	0.258392
$927$	0	0
$928$	−24.4894	−0.803905
$929$	46.8174	1.53603	0.768015	−	0.640432i	$-0.221245\pi$
$929$	46.8174	1.53603	0.768015	+	0.640432i	$0.221245\pi$
$930$	0	0
$931$	−28.0769	−0.920182
$932$	−9.04009	−0.296118
$933$	0	0
$934$	−4.78164	−0.156460
$935$	17.5414	0.573664
$936$	0	0
$937$	7.52447	0.245814	0.122907	−	0.992418i	$-0.460778\pi$
$937$	7.52447	0.245814	0.122907	+	0.992418i	$0.460778\pi$
$938$	−30.6936	−1.00218
$939$	0	0
$940$	29.0750	0.948322
$941$	−36.9920	−1.20591	−0.602953	−	0.797777i	$-0.706009\pi$
$941$	−36.9920	−1.20591	−0.602953	+	0.797777i	$0.706009\pi$
$942$	0	0
$943$	49.2010	1.60221
$944$	33.5159	1.09085
$945$	0	0
$946$	−2.27024	−0.0738118
$947$	−43.9680	−1.42877	−0.714384	−	0.699754i	$-0.753293\pi$
$947$	−43.9680	−1.42877	−0.714384	+	0.699754i	$0.753293\pi$
$948$	0	0
$949$	63.9610	2.07626
$950$	39.2032	1.27192
$951$	0	0
$952$	−28.5773	−0.926195
$953$	47.3152	1.53269	0.766345	−	0.642429i	$-0.222073\pi$
$953$	47.3152	1.53269	0.766345	+	0.642429i	$0.222073\pi$
$954$	0	0
$955$	43.8634	1.41939
$956$	−10.8192	−0.349918
$957$	0	0
$958$	38.5488	1.24546
$959$	19.6952	0.635993
$960$	0	0
$961$	−27.6564	−0.892143
$962$	−65.3610	−2.10732
$963$	0	0
$964$	1.30961	0.0421798
$965$	12.2834	0.395417
$966$	0	0
$967$	58.2223	1.87230	0.936151	−	0.351598i	$-0.114361\pi$
$967$	58.2223	1.87230	0.936151	+	0.351598i	$0.114361\pi$
$968$	1.40692	0.0452202
$969$	0	0
$970$	−67.9980	−2.18328
$971$	−4.90150	−0.157297	−0.0786484	−	0.996902i	$-0.525060\pi$
$971$	−4.90150	−0.157297	−0.0786484	+	0.996902i	$0.525060\pi$
$972$	0	0
$973$	15.1013	0.484126
$974$	−41.4582	−1.32841
$975$	0	0
$976$	40.4102	1.29350
$977$	−44.8956	−1.43634	−0.718169	−	0.695869i	$-0.755020\pi$
$977$	−44.8956	−1.43634	−0.718169	+	0.695869i	$0.755020\pi$
$978$	0	0
$979$	−5.92634	−0.189407
$980$	26.0886	0.833371
$981$	0	0
$982$	−14.5421	−0.464058
$983$	−37.4391	−1.19412	−0.597061	−	0.802196i	$-0.703665\pi$
$983$	−37.4391	−1.19412	−0.597061	+	0.802196i	$0.703665\pi$
$984$	0	0
$985$	6.15659	0.196165
$986$	39.6913	1.26403
$987$	0	0
$988$	−24.5241	−0.780217
$989$	10.9442	0.348005
$990$	0	0
$991$	41.1230	1.30631	0.653157	−	0.757223i	$-0.273444\pi$
$991$	41.1230	1.30631	0.653157	+	0.757223i	$0.273444\pi$
$992$	−11.0969	−0.352328
$993$	0	0
$994$	6.10742	0.193716
$995$	78.5112	2.48897
$996$	0	0
$997$	−12.0044	−0.380184	−0.190092	−	0.981766i	$-0.560879\pi$
$997$	−12.0044	−0.380184	−0.190092	+	0.981766i	$0.560879\pi$
$998$	63.9673	2.02485
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.l.1.12		51
3.2	odd	2		8019.2.a.k.1.40		51
27.5	odd	18		891.2.j.c.397.14		102
27.11	odd	18		891.2.j.c.496.14		102
27.16	even	9		297.2.j.c.67.4	✓	102
27.22	even	9		297.2.j.c.133.4	yes	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.67.4	✓	102	27.16	even	9
297.2.j.c.133.4	yes	102	27.22	even	9
891.2.j.c.397.14		102	27.5	odd	18
891.2.j.c.496.14		102	27.11	odd	18
8019.2.a.k.1.40		51	3.2	odd	2
8019.2.a.l.1.12		51	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:	\( N \)	\(=\)	\( 8019 = 3^{6} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8019.a (trivial)

\(f(q)\)	\(=\)	\(q-1.79315 q^{2} +1.21539 q^{4} +3.19791 q^{5} -3.70301 q^{7} +1.40692 q^{8} +O(q^{10})\)	\(q-1.79315 q^{2} +1.21539 q^{4} +3.19791 q^{5} -3.70301 q^{7} +1.40692 q^{8} -5.73435 q^{10} +1.00000 q^{11} +4.82389 q^{13} +6.64005 q^{14} -4.95361 q^{16} +5.48525 q^{17} -4.18293 q^{19} +3.88672 q^{20} -1.79315 q^{22} +8.64429 q^{23} +5.22666 q^{25} -8.64996 q^{26} -4.50060 q^{28} -4.03535 q^{29} -1.82854 q^{31} +6.06872 q^{32} -9.83589 q^{34} -11.8419 q^{35} +7.55622 q^{37} +7.50062 q^{38} +4.49921 q^{40} +5.69173 q^{41} +1.26606 q^{43} +1.21539 q^{44} -15.5005 q^{46} +7.48060 q^{47} +6.71225 q^{49} -9.37219 q^{50} +5.86291 q^{52} +2.85046 q^{53} +3.19791 q^{55} -5.20984 q^{56} +7.23600 q^{58} -6.76596 q^{59} -8.15773 q^{61} +3.27886 q^{62} -0.974926 q^{64} +15.4264 q^{65} -4.62250 q^{67} +6.66673 q^{68} +21.2343 q^{70} +0.919786 q^{71} +13.2592 q^{73} -13.5494 q^{74} -5.08390 q^{76} -3.70301 q^{77} -3.72774 q^{79} -15.8412 q^{80} -10.2061 q^{82} +11.8648 q^{83} +17.5414 q^{85} -2.27024 q^{86} +1.40692 q^{88} -5.92634 q^{89} -17.8629 q^{91} +10.5062 q^{92} -13.4139 q^{94} -13.3766 q^{95} +11.8580 q^{97} -12.0361 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) \) 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7} + 18 q^{10} + 51 q^{11} + 30 q^{13} + 12 q^{14} + 78 q^{16} + 30 q^{19} + 18 q^{20} + 3 q^{23} + 75 q^{25} + 9 q^{26} + 36 q^{28} + 42 q^{31} - 15 q^{32} + 42 q^{34} - 9 q^{35} + 48 q^{37} - 3 q^{38} + 54 q^{40} + 18 q^{43} + 60 q^{44} + 42 q^{46} + 30 q^{47} + 99 q^{49} - 30 q^{50} + 60 q^{52} + 18 q^{53} + 6 q^{55} + 21 q^{56} + 30 q^{58} + 24 q^{59} + 99 q^{61} + 114 q^{64} - 15 q^{65} + 39 q^{67} - 39 q^{68} + 48 q^{70} + 30 q^{71} + 69 q^{73} + 90 q^{76} + 12 q^{77} + 48 q^{79} + 42 q^{80} + 42 q^{82} - 21 q^{83} + 84 q^{85} + 24 q^{86} + 15 q^{89} + 69 q^{91} - 66 q^{92} + 66 q^{94} - 12 q^{95} + 72 q^{97} - 57 q^{98}+O(q^{100}) \) 51 * q + 60 * q^4 + 6 * q^5 + 12 * q^7 + 18 * q^10 + 51 * q^11 + 30 * q^13 + 12 * q^14 + 78 * q^16 + 30 * q^19 + 18 * q^20 + 3 * q^23 + 75 * q^25 + 9 * q^26 + 36 * q^28 + 42 * q^31 - 15 * q^32 + 42 * q^34 - 9 * q^35 + 48 * q^37 - 3 * q^38 + 54 * q^40 + 18 * q^43 + 60 * q^44 + 42 * q^46 + 30 * q^47 + 99 * q^49 - 30 * q^50 + 60 * q^52 + 18 * q^53 + 6 * q^55 + 21 * q^56 + 30 * q^58 + 24 * q^59 + 99 * q^61 + 114 * q^64 - 15 * q^65 + 39 * q^67 - 39 * q^68 + 48 * q^70 + 30 * q^71 + 69 * q^73 + 90 * q^76 + 12 * q^77 + 48 * q^79 + 42 * q^80 + 42 * q^82 - 21 * q^83 + 84 * q^85 + 24 * q^86 + 15 * q^89 + 69 * q^91 - 66 * q^92 + 66 * q^94 - 12 * q^95 + 72 * q^97 - 57 * q^98

Introduction

L-functions

Modular forms

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Database

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