LMFDB - Embedded newform 8019.2.a.l.1.9

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			1.9
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)

$f(q)$	$=$	$q-1.98291 q^{2} +1.93194 q^{4} +0.594867 q^{5} -1.24839 q^{7} +0.134957 q^{8} +O(q^{10})$	\(q-1.98291 q^{2} +1.93194 q^{4} +0.594867 q^{5} -1.24839 q^{7} +0.134957 q^{8} -1.17957 q^{10} +1.00000 q^{11} +5.57436 q^{13} +2.47546 q^{14} -4.13149 q^{16} +6.43369 q^{17} +5.69526 q^{19} +1.14925 q^{20} -1.98291 q^{22} -9.13802 q^{23} -4.64613 q^{25} -11.0535 q^{26} -2.41182 q^{28} +2.60888 q^{29} -3.60529 q^{31} +7.92246 q^{32} -12.7574 q^{34} -0.742629 q^{35} +7.88375 q^{37} -11.2932 q^{38} +0.0802814 q^{40} -2.17028 q^{41} +5.71494 q^{43} +1.93194 q^{44} +18.1199 q^{46} -0.711651 q^{47} -5.44151 q^{49} +9.21287 q^{50} +10.7693 q^{52} +7.07446 q^{53} +0.594867 q^{55} -0.168479 q^{56} -5.17319 q^{58} -2.32477 q^{59} +14.0099 q^{61} +7.14897 q^{62} -7.44657 q^{64} +3.31600 q^{65} +11.9152 q^{67} +12.4295 q^{68} +1.47257 q^{70} +1.19258 q^{71} +8.74213 q^{73} -15.6328 q^{74} +11.0029 q^{76} -1.24839 q^{77} +3.98017 q^{79} -2.45769 q^{80} +4.30348 q^{82} -1.78700 q^{83} +3.82719 q^{85} -11.3322 q^{86} +0.134957 q^{88} +6.84126 q^{89} -6.95900 q^{91} -17.6541 q^{92} +1.41114 q^{94} +3.38792 q^{95} +4.93852 q^{97} +10.7900 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−1.98291	−1.40213	−0.701065	−	0.713097i	$-0.747292\pi$
$2$	−1.98291	−1.40213	−0.701065	+	0.713097i	$0.747292\pi$
$3$	0	0
$3$	0	0
$4$	1.93194	0.965970
$5$	0.594867	0.266033	0.133016	−	0.991114i	$-0.457534\pi$
$5$	0.594867	0.266033	0.133016	+	0.991114i	$0.457534\pi$
$6$	0	0
$7$	−1.24839	−0.471849	−0.235924	−	0.971771i	$-0.575812\pi$
$7$	−1.24839	−0.471849	−0.235924	+	0.971771i	$0.575812\pi$
$8$	0.134957	0.0477144
$9$	0	0
$10$	−1.17957	−0.373013
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	5.57436	1.54605	0.773024	−	0.634377i	$-0.218743\pi$
$13$	5.57436	1.54605	0.773024	+	0.634377i	$0.218743\pi$
$14$	2.47546	0.661594
$15$	0	0
$16$	−4.13149	−1.03287
$17$	6.43369	1.56040	0.780199	−	0.625531i	$-0.215118\pi$
$17$	6.43369	1.56040	0.780199	+	0.625531i	$0.215118\pi$
$18$	0	0
$19$	5.69526	1.30658	0.653291	−	0.757107i	$-0.273388\pi$
$19$	5.69526	1.30658	0.653291	+	0.757107i	$0.273388\pi$
$20$	1.14925	0.256980
$21$	0	0
$22$	−1.98291	−0.422758
$23$	−9.13802	−1.90541	−0.952704	−	0.303899i	$-0.901711\pi$
$23$	−9.13802	−1.90541	−0.952704	+	0.303899i	$0.901711\pi$
$24$	0	0
$25$	−4.64613	−0.929227
$26$	−11.0535	−2.16776
$27$	0	0
$28$	−2.41182	−0.455792
$29$	2.60888	0.484457	0.242229	−	0.970219i	$-0.422122\pi$
$29$	2.60888	0.484457	0.242229	+	0.970219i	$0.422122\pi$
$30$	0	0
$31$	−3.60529	−0.647529	−0.323765	−	0.946138i	$-0.604949\pi$
$31$	−3.60529	−0.647529	−0.323765	+	0.946138i	$0.604949\pi$
$32$	7.92246	1.40051
$33$	0	0
$34$	−12.7574	−2.18788
$35$	−0.742629	−0.125527
$36$	0	0
$37$	7.88375	1.29608	0.648040	−	0.761606i	$-0.275589\pi$
$37$	7.88375	1.29608	0.648040	+	0.761606i	$0.275589\pi$
$38$	−11.2932	−1.83200
$39$	0	0
$40$	0.0802814	0.0126936
$41$	−2.17028	−0.338942	−0.169471	−	0.985535i	$-0.554206\pi$
$41$	−2.17028	−0.338942	−0.169471	+	0.985535i	$0.554206\pi$
$42$	0	0
$43$	5.71494	0.871521	0.435760	−	0.900063i	$-0.356479\pi$
$43$	5.71494	0.871521	0.435760	+	0.900063i	$0.356479\pi$
$44$	1.93194	0.291251
$45$	0	0
$46$	18.1199	2.67163
$47$	−0.711651	−0.103805	−0.0519024	−	0.998652i	$-0.516528\pi$
$47$	−0.711651	−0.103805	−0.0519024	+	0.998652i	$0.516528\pi$
$48$	0	0
$49$	−5.44151	−0.777359
$50$	9.21287	1.30290
$51$	0	0
$52$	10.7693	1.49344
$53$	7.07446	0.971751	0.485876	−	0.874028i	$-0.338501\pi$
$53$	7.07446	0.971751	0.485876	+	0.874028i	$0.338501\pi$
$54$	0	0
$55$	0.594867	0.0802119
$56$	−0.168479	−0.0225140
$57$	0	0
$58$	−5.17319	−0.679273
$59$	−2.32477	−0.302660	−0.151330	−	0.988483i	$-0.548356\pi$
$59$	−2.32477	−0.302660	−0.151330	+	0.988483i	$0.548356\pi$
$60$	0	0
$61$	14.0099	1.79379	0.896894	−	0.442246i	$-0.145818\pi$
$61$	14.0099	1.79379	0.896894	+	0.442246i	$0.145818\pi$
$62$	7.14897	0.907920
$63$	0	0
$64$	−7.44657	−0.930821
$65$	3.31600	0.411299
$66$	0	0
$67$	11.9152	1.45567	0.727835	−	0.685752i	$-0.240527\pi$
$67$	11.9152	1.45567	0.727835	+	0.685752i	$0.240527\pi$
$68$	12.4295	1.50730
$69$	0	0
$70$	1.47257	0.176006
$71$	1.19258	0.141534	0.0707669	−	0.997493i	$-0.477455\pi$
$71$	1.19258	0.141534	0.0707669	+	0.997493i	$0.477455\pi$
$72$	0	0
$73$	8.74213	1.02319	0.511594	−	0.859227i	$-0.329055\pi$
$73$	8.74213	1.02319	0.511594	+	0.859227i	$0.329055\pi$
$74$	−15.6328	−1.81727
$75$	0	0
$76$	11.0029	1.26212
$77$	−1.24839	−0.142268
$78$	0	0
$79$	3.98017	0.447804	0.223902	−	0.974612i	$-0.428120\pi$
$79$	3.98017	0.447804	0.223902	+	0.974612i	$0.428120\pi$
$80$	−2.45769	−0.274778
$81$	0	0
$82$	4.30348	0.475240
$83$	−1.78700	−0.196149	−0.0980744	−	0.995179i	$-0.531268\pi$
$83$	−1.78700	−0.196149	−0.0980744	+	0.995179i	$0.531268\pi$
$84$	0	0
$85$	3.82719	0.415117
$86$	−11.3322	−1.22199
$87$	0	0
$88$	0.134957	0.0143864
$89$	6.84126	0.725172	0.362586	−	0.931950i	$-0.381894\pi$
$89$	6.84126	0.725172	0.362586	+	0.931950i	$0.381894\pi$
$90$	0	0
$91$	−6.95900	−0.729501
$92$	−17.6541	−1.84057
$93$	0	0
$94$	1.41114	0.145548
$95$	3.38792	0.347594
$96$	0	0
$97$	4.93852	0.501431	0.250716	−	0.968061i	$-0.419334\pi$
$97$	4.93852	0.501431	0.250716	+	0.968061i	$0.419334\pi$
$98$	10.7900	1.08996
$99$	0	0
$100$	−8.97605	−0.897605
$101$	−2.44844	−0.243629	−0.121814	−	0.992553i	$-0.538871\pi$
$101$	−2.44844	−0.243629	−0.121814	+	0.992553i	$0.538871\pi$
$102$	0	0
$103$	8.42683	0.830320	0.415160	−	0.909748i	$-0.363726\pi$
$103$	8.42683	0.830320	0.415160	+	0.909748i	$0.363726\pi$
$104$	0.752297	0.0737688
$105$	0	0
$106$	−14.0280	−1.36252
$107$	−11.4829	−1.11009	−0.555046	−	0.831820i	$-0.687299\pi$
$107$	−11.4829	−1.11009	−0.555046	+	0.831820i	$0.687299\pi$
$108$	0	0
$109$	−14.0761	−1.34825	−0.674123	−	0.738619i	$-0.735478\pi$
$109$	−14.0761	−1.34825	−0.674123	+	0.738619i	$0.735478\pi$
$110$	−1.17957	−0.112468
$111$	0	0
$112$	5.15773	0.487359
$113$	−14.0724	−1.32382	−0.661909	−	0.749585i	$-0.730253\pi$
$113$	−14.0724	−1.32382	−0.661909	+	0.749585i	$0.730253\pi$
$114$	0	0
$115$	−5.43591	−0.506901
$116$	5.04021	0.467971
$117$	0	0
$118$	4.60982	0.424368
$119$	−8.03178	−0.736272
$120$	0	0
$121$	1.00000	0.0909091
$122$	−27.7805	−2.51512
$123$	0	0
$124$	−6.96520	−0.625494
$125$	−5.73817	−0.513237
$126$	0	0
$127$	6.69300	0.593907	0.296954	−	0.954892i	$-0.404029\pi$
$127$	6.69300	0.593907	0.296954	+	0.954892i	$0.404029\pi$
$128$	−1.07903	−0.0953736
$129$	0	0
$130$	−6.57534	−0.576695
$131$	16.4880	1.44057	0.720283	−	0.693681i	$-0.244012\pi$
$131$	16.4880	1.44057	0.720283	+	0.693681i	$0.244012\pi$
$132$	0	0
$133$	−7.10993	−0.616509
$134$	−23.6268	−2.04104
$135$	0	0
$136$	0.868270	0.0744535
$137$	−16.4149	−1.40242	−0.701210	−	0.712955i	$-0.747357\pi$
$137$	−16.4149	−1.40242	−0.701210	+	0.712955i	$0.747357\pi$
$138$	0	0
$139$	−11.0455	−0.936871	−0.468436	−	0.883498i	$-0.655182\pi$
$139$	−11.0455	−0.936871	−0.468436	+	0.883498i	$0.655182\pi$
$140$	−1.43471	−0.121256
$141$	0	0
$142$	−2.36479	−0.198449
$143$	5.57436	0.466151
$144$	0	0
$145$	1.55194	0.128882
$146$	−17.3349	−1.43464
$147$	0	0
$148$	15.2309	1.25198
$149$	1.23552	0.101217	0.0506087	−	0.998719i	$-0.483884\pi$
$149$	1.23552	0.101217	0.0506087	+	0.998719i	$0.483884\pi$
$150$	0	0
$151$	−3.75905	−0.305907	−0.152954	−	0.988233i	$-0.548879\pi$
$151$	−3.75905	−0.305907	−0.152954	+	0.988233i	$0.548879\pi$
$152$	0.768614	0.0623429
$153$	0	0
$154$	2.47546	0.199478
$155$	−2.14467	−0.172264
$156$	0	0
$157$	21.6451	1.72746	0.863732	−	0.503951i	$-0.168121\pi$
$157$	21.6451	1.72746	0.863732	+	0.503951i	$0.168121\pi$
$158$	−7.89232	−0.627879
$159$	0	0
$160$	4.71281	0.372581
$161$	11.4079	0.899065
$162$	0	0
$163$	−1.99715	−0.156429	−0.0782143	−	0.996937i	$-0.524922\pi$
$163$	−1.99715	−0.156429	−0.0782143	+	0.996937i	$0.524922\pi$
$164$	−4.19286	−0.327407
$165$	0	0
$166$	3.54346	0.275026
$167$	−11.7227	−0.907131	−0.453565	−	0.891223i	$-0.649848\pi$
$167$	−11.7227	−0.907131	−0.453565	+	0.891223i	$0.649848\pi$
$168$	0	0
$169$	18.0734	1.39026
$170$	−7.58898	−0.582048
$171$	0	0
$172$	11.0409	0.841863
$173$	0.690170	0.0524727	0.0262363	−	0.999656i	$-0.491648\pi$
$173$	0.690170	0.0524727	0.0262363	+	0.999656i	$0.491648\pi$
$174$	0	0
$175$	5.80021	0.438454
$176$	−4.13149	−0.311423
$177$	0	0
$178$	−13.5656	−1.01679
$179$	−2.20803	−0.165036	−0.0825180	−	0.996590i	$-0.526296\pi$
$179$	−2.20803	−0.165036	−0.0825180	+	0.996590i	$0.526296\pi$
$180$	0	0
$181$	−5.13288	−0.381524	−0.190762	−	0.981636i	$-0.561096\pi$
$181$	−5.13288	−0.381524	−0.190762	+	0.981636i	$0.561096\pi$
$182$	13.7991	1.02286
$183$	0	0
$184$	−1.23324	−0.0909155
$185$	4.68978	0.344800
$186$	0	0
$187$	6.43369	0.470478
$188$	−1.37487	−0.100272
$189$	0	0
$190$	−6.71796	−0.487372
$191$	−16.2003	−1.17222	−0.586108	−	0.810233i	$-0.699341\pi$
$191$	−16.2003	−1.17222	−0.586108	+	0.810233i	$0.699341\pi$
$192$	0	0
$193$	−4.18408	−0.301177	−0.150588	−	0.988597i	$-0.548117\pi$
$193$	−4.18408	−0.301177	−0.150588	+	0.988597i	$0.548117\pi$
$194$	−9.79266	−0.703072
$195$	0	0
$196$	−10.5127	−0.750905
$197$	−8.93397	−0.636519	−0.318259	−	0.948004i	$-0.603098\pi$
$197$	−8.93397	−0.636519	−0.318259	+	0.948004i	$0.603098\pi$
$198$	0	0
$199$	−16.3429	−1.15852	−0.579260	−	0.815143i	$-0.696659\pi$
$199$	−16.3429	−1.15852	−0.579260	+	0.815143i	$0.696659\pi$
$200$	−0.627027	−0.0443375
$201$	0	0
$202$	4.85504	0.341599
$203$	−3.25692	−0.228591
$204$	0	0
$205$	−1.29103	−0.0901695
$206$	−16.7097	−1.16422
$207$	0	0
$208$	−23.0304	−1.59687
$209$	5.69526	0.393949
$210$	0	0
$211$	10.4820	0.721614	0.360807	−	0.932641i	$-0.382501\pi$
$211$	10.4820	0.721614	0.360807	+	0.932641i	$0.382501\pi$
$212$	13.6674	0.938683
$213$	0	0
$214$	22.7695	1.55649
$215$	3.39963	0.231853
$216$	0	0
$217$	4.50082	0.305536
$218$	27.9117	1.89042
$219$	0	0
$220$	1.14925	0.0774823
$221$	35.8637	2.41245
$222$	0	0
$223$	8.78584	0.588343	0.294172	−	0.955753i	$-0.404956\pi$
$223$	8.78584	0.588343	0.294172	+	0.955753i	$0.404956\pi$
$224$	−9.89036	−0.660828
$225$	0	0
$226$	27.9043	1.85616
$227$	−24.7854	−1.64507	−0.822534	−	0.568716i	$-0.807440\pi$
$227$	−24.7854	−1.64507	−0.822534	+	0.568716i	$0.807440\pi$
$228$	0	0
$229$	−3.96533	−0.262037	−0.131018	−	0.991380i	$-0.541825\pi$
$229$	−3.96533	−0.262037	−0.131018	+	0.991380i	$0.541825\pi$
$230$	10.7789	0.710741
$231$	0	0
$232$	0.352087	0.0231156
$233$	14.0587	0.921014	0.460507	−	0.887656i	$-0.347668\pi$
$233$	14.0587	0.921014	0.460507	+	0.887656i	$0.347668\pi$
$234$	0	0
$235$	−0.423338	−0.0276155
$236$	−4.49132	−0.292360
$237$	0	0
$238$	15.9263	1.03235
$239$	10.5509	0.682482	0.341241	−	0.939976i	$-0.389153\pi$
$239$	10.5509	0.682482	0.341241	+	0.939976i	$0.389153\pi$
$240$	0	0
$241$	10.8998	0.702115	0.351058	−	0.936354i	$-0.385822\pi$
$241$	10.8998	0.702115	0.351058	+	0.936354i	$0.385822\pi$
$242$	−1.98291	−0.127466
$243$	0	0
$244$	27.0663	1.73275
$245$	−3.23698	−0.206803
$246$	0	0
$247$	31.7474	2.02004
$248$	−0.486558	−0.0308965
$249$	0	0
$250$	11.3783	0.719626
$251$	−27.9276	−1.76277	−0.881386	−	0.472396i	$-0.843389\pi$
$251$	−27.9276	−1.76277	−0.881386	+	0.472396i	$0.843389\pi$
$252$	0	0
$253$	−9.13802	−0.574502
$254$	−13.2716	−0.832736
$255$	0	0
$256$	17.0328	1.06455
$257$	9.73742	0.607404	0.303702	−	0.952767i	$-0.401777\pi$
$257$	9.73742	0.607404	0.303702	+	0.952767i	$0.401777\pi$
$258$	0	0
$259$	−9.84203	−0.611554
$260$	6.40632	0.397303
$261$	0	0
$262$	−32.6943	−2.01986
$263$	−19.1008	−1.17781	−0.588903	−	0.808203i	$-0.700440\pi$
$263$	−19.1008	−1.17781	−0.588903	+	0.808203i	$0.700440\pi$
$264$	0	0
$265$	4.20836	0.258518
$266$	14.0984	0.864427
$267$	0	0
$268$	23.0194	1.40613
$269$	29.3520	1.78962	0.894812	−	0.446443i	$-0.147309\pi$
$269$	29.3520	1.78962	0.894812	+	0.446443i	$0.147309\pi$
$270$	0	0
$271$	4.62696	0.281068	0.140534	−	0.990076i	$-0.455118\pi$
$271$	4.62696	0.281068	0.140534	+	0.990076i	$0.455118\pi$
$272$	−26.5807	−1.61169
$273$	0	0
$274$	32.5493	1.96638
$275$	−4.64613	−0.280172
$276$	0	0
$277$	−9.69788	−0.582689	−0.291345	−	0.956618i	$-0.594103\pi$
$277$	−9.69788	−0.582689	−0.291345	+	0.956618i	$0.594103\pi$
$278$	21.9024	1.31362
$279$	0	0
$280$	−0.100223	−0.00598946
$281$	29.7691	1.77588	0.887939	−	0.459961i	$-0.152137\pi$
$281$	29.7691	1.77588	0.887939	+	0.459961i	$0.152137\pi$
$282$	0	0
$283$	7.61235	0.452507	0.226254	−	0.974068i	$-0.427352\pi$
$283$	7.61235	0.452507	0.226254	+	0.974068i	$0.427352\pi$
$284$	2.30400	0.136717
$285$	0	0
$286$	−11.0535	−0.653605
$287$	2.70937	0.159929
$288$	0	0
$289$	24.3923	1.43484
$290$	−3.07736	−0.180709
$291$	0	0
$292$	16.8893	0.988370
$293$	4.81805	0.281473	0.140737	−	0.990047i	$-0.455053\pi$
$293$	4.81805	0.281473	0.140737	+	0.990047i	$0.455053\pi$
$294$	0	0
$295$	−1.38293	−0.0805174
$296$	1.06397	0.0618418
$297$	0	0
$298$	−2.44992	−0.141920
$299$	−50.9386	−2.94585
$300$	0	0
$301$	−7.13451	−0.411226
$302$	7.45387	0.428922
$303$	0	0
$304$	−23.5299	−1.34953
$305$	8.33405	0.477206
$306$	0	0
$307$	3.01933	0.172322	0.0861611	−	0.996281i	$-0.472540\pi$
$307$	3.01933	0.172322	0.0861611	+	0.996281i	$0.472540\pi$
$308$	−2.41182	−0.137426
$309$	0	0
$310$	4.25269	0.241536
$311$	−5.14190	−0.291570	−0.145785	−	0.989316i	$-0.546571\pi$
$311$	−5.14190	−0.291570	−0.145785	+	0.989316i	$0.546571\pi$
$312$	0	0
$313$	13.0431	0.737242	0.368621	−	0.929580i	$-0.379830\pi$
$313$	13.0431	0.737242	0.368621	+	0.929580i	$0.379830\pi$
$314$	−42.9203	−2.42213
$315$	0	0
$316$	7.68944	0.432565
$317$	−3.51382	−0.197356	−0.0986778	−	0.995119i	$-0.531461\pi$
$317$	−3.51382	−0.197356	−0.0986778	+	0.995119i	$0.531461\pi$
$318$	0	0
$319$	2.60888	0.146069
$320$	−4.42972	−0.247629
$321$	0	0
$322$	−22.6208	−1.26061
$323$	36.6415	2.03879
$324$	0	0
$325$	−25.8992	−1.43663
$326$	3.96016	0.219333
$327$	0	0
$328$	−0.292895	−0.0161724
$329$	0.888421	0.0489802
$330$	0	0
$331$	−18.6654	−1.02594	−0.512972	−	0.858405i	$-0.671456\pi$
$331$	−18.6654	−1.02594	−0.512972	+	0.858405i	$0.671456\pi$
$332$	−3.45238	−0.189474
$333$	0	0
$334$	23.2451	1.27192
$335$	7.08795	0.387256
$336$	0	0
$337$	19.5844	1.06683	0.533415	−	0.845853i	$-0.320908\pi$
$337$	19.5844	1.06683	0.533415	+	0.845853i	$0.320908\pi$
$338$	−35.8380	−1.94933
$339$	0	0
$340$	7.39390	0.400990
$341$	−3.60529	−0.195237
$342$	0	0
$343$	15.5319	0.838645
$344$	0.771271	0.0415841
$345$	0	0
$346$	−1.36855	−0.0735735
$347$	−0.135314	−0.00726404	−0.00363202	−	0.999993i	$-0.501156\pi$
$347$	−0.135314	−0.00726404	−0.00363202	+	0.999993i	$0.501156\pi$
$348$	0	0
$349$	13.0131	0.696578	0.348289	−	0.937387i	$-0.386763\pi$
$349$	13.0131	0.696578	0.348289	+	0.937387i	$0.386763\pi$
$350$	−11.5013	−0.614770
$351$	0	0
$352$	7.92246	0.422269
$353$	11.6510	0.620123	0.310061	−	0.950717i	$-0.399650\pi$
$353$	11.6510	0.620123	0.310061	+	0.950717i	$0.399650\pi$
$354$	0	0
$355$	0.709430	0.0376526
$356$	13.2169	0.700494
$357$	0	0
$358$	4.37833	0.231402
$359$	8.81094	0.465024	0.232512	−	0.972594i	$-0.425306\pi$
$359$	8.81094	0.465024	0.232512	+	0.972594i	$0.425306\pi$
$360$	0	0
$361$	13.4360	0.707158
$362$	10.1780	0.534946
$363$	0	0
$364$	−13.4444	−0.704676
$365$	5.20041	0.272202
$366$	0	0
$367$	−23.7508	−1.23978	−0.619890	−	0.784689i	$-0.712823\pi$
$367$	−23.7508	−1.23978	−0.619890	+	0.784689i	$0.712823\pi$
$368$	37.7536	1.96804
$369$	0	0
$370$	−9.29943	−0.483454
$371$	−8.83171	−0.458520
$372$	0	0
$373$	−34.2810	−1.77500	−0.887501	−	0.460806i	$-0.847561\pi$
$373$	−34.2810	−1.77500	−0.887501	+	0.460806i	$0.847561\pi$
$374$	−12.7574	−0.659671
$375$	0	0
$376$	−0.0960421	−0.00495299
$377$	14.5428	0.748994
$378$	0	0
$379$	−17.6242	−0.905295	−0.452647	−	0.891690i	$-0.649520\pi$
$379$	−17.6242	−0.905295	−0.452647	+	0.891690i	$0.649520\pi$
$380$	6.54527	0.335765
$381$	0	0
$382$	32.1239	1.64360
$383$	20.4485	1.04487	0.522436	−	0.852679i	$-0.325024\pi$
$383$	20.4485	1.04487	0.522436	+	0.852679i	$0.325024\pi$
$384$	0	0
$385$	−0.742629	−0.0378479
$386$	8.29667	0.422289
$387$	0	0
$388$	9.54093	0.484367
$389$	−23.5354	−1.19329	−0.596646	−	0.802505i	$-0.703500\pi$
$389$	−23.5354	−1.19329	−0.596646	+	0.802505i	$0.703500\pi$
$390$	0	0
$391$	−58.7911	−2.97320
$392$	−0.734369	−0.0370912
$393$	0	0
$394$	17.7153	0.892482
$395$	2.36767	0.119130
$396$	0	0
$397$	−24.4452	−1.22687	−0.613436	−	0.789745i	$-0.710213\pi$
$397$	−24.4452	−1.22687	−0.613436	+	0.789745i	$0.710213\pi$
$398$	32.4066	1.62440
$399$	0	0
$400$	19.1954	0.959772
$401$	−23.6966	−1.18335	−0.591676	−	0.806176i	$-0.701533\pi$
$401$	−23.6966	−1.18335	−0.591676	+	0.806176i	$0.701533\pi$
$402$	0	0
$403$	−20.0972	−1.00111
$404$	−4.73023	−0.235338
$405$	0	0
$406$	6.45818	0.320514
$407$	7.88375	0.390783
$408$	0	0
$409$	−31.9971	−1.58216	−0.791078	−	0.611715i	$-0.790480\pi$
$409$	−31.9971	−1.58216	−0.791078	+	0.611715i	$0.790480\pi$
$410$	2.56000	0.126429
$411$	0	0
$412$	16.2801	0.802064
$413$	2.90223	0.142810
$414$	0	0
$415$	−1.06303	−0.0521820
$416$	44.1626	2.16525
$417$	0	0
$418$	−11.2932	−0.552369
$419$	22.6536	1.10670	0.553351	−	0.832948i	$-0.313349\pi$
$419$	22.6536	1.10670	0.553351	+	0.832948i	$0.313349\pi$
$420$	0	0
$421$	10.2685	0.500455	0.250227	−	0.968187i	$-0.419495\pi$
$421$	10.2685	0.500455	0.250227	+	0.968187i	$0.419495\pi$
$422$	−20.7850	−1.01180
$423$	0	0
$424$	0.954746	0.0463666
$425$	−29.8918	−1.44996
$426$	0	0
$427$	−17.4899	−0.846397
$428$	−22.1842	−1.07232
$429$	0	0
$430$	−6.74117	−0.325088
$431$	1.01829	0.0490493	0.0245246	−	0.999699i	$-0.492193\pi$
$431$	1.01829	0.0490493	0.0245246	+	0.999699i	$0.492193\pi$
$432$	0	0
$433$	−18.1760	−0.873481	−0.436740	−	0.899588i	$-0.643867\pi$
$433$	−18.1760	−0.873481	−0.436740	+	0.899588i	$0.643867\pi$
$434$	−8.92474	−0.428401
$435$	0	0
$436$	−27.1942	−1.30237
$437$	−52.0434	−2.48957
$438$	0	0
$439$	13.2984	0.634696	0.317348	−	0.948309i	$-0.397208\pi$
$439$	13.2984	0.634696	0.317348	+	0.948309i	$0.397208\pi$
$440$	0.0802814	0.00382726
$441$	0	0
$442$	−71.1145	−3.38257
$443$	16.5205	0.784912	0.392456	−	0.919771i	$-0.371626\pi$
$443$	16.5205	0.784912	0.392456	+	0.919771i	$0.371626\pi$
$444$	0	0
$445$	4.06964	0.192919
$446$	−17.4216	−0.824934
$447$	0	0
$448$	9.29626	0.439207
$449$	−8.93629	−0.421730	−0.210865	−	0.977515i	$-0.567628\pi$
$449$	−8.93629	−0.421730	−0.210865	+	0.977515i	$0.567628\pi$
$450$	0	0
$451$	−2.17028	−0.102195
$452$	−27.1870	−1.27877
$453$	0	0
$454$	49.1473	2.30660
$455$	−4.13968	−0.194071
$456$	0	0
$457$	25.0746	1.17294	0.586471	−	0.809970i	$-0.300517\pi$
$457$	25.0746	1.17294	0.586471	+	0.809970i	$0.300517\pi$
$458$	7.86290	0.367409
$459$	0	0
$460$	−10.5018	−0.489651
$461$	30.7120	1.43040	0.715201	−	0.698919i	$-0.246335\pi$
$461$	30.7120	1.43040	0.715201	+	0.698919i	$0.246335\pi$
$462$	0	0
$463$	34.0716	1.58344	0.791721	−	0.610883i	$-0.209185\pi$
$463$	34.0716	1.58344	0.791721	+	0.610883i	$0.209185\pi$
$464$	−10.7786	−0.500382
$465$	0	0
$466$	−27.8771	−1.29138
$467$	−26.6921	−1.23516	−0.617582	−	0.786506i	$-0.711888\pi$
$467$	−26.6921	−1.23516	−0.617582	+	0.786506i	$0.711888\pi$
$468$	0	0
$469$	−14.8748	−0.686857
$470$	0.839441	0.0387205
$471$	0	0
$472$	−0.313744	−0.0144412
$473$	5.71494	0.262773
$474$	0	0
$475$	−26.4609	−1.21411
$476$	−15.5169	−0.711217
$477$	0	0
$478$	−20.9215	−0.956928
$479$	19.6602	0.898298	0.449149	−	0.893457i	$-0.351727\pi$
$479$	19.6602	0.898298	0.449149	+	0.893457i	$0.351727\pi$
$480$	0	0
$481$	43.9468	2.00380
$482$	−21.6133	−0.984458
$483$	0	0
$484$	1.93194	0.0878155
$485$	2.93777	0.133397
$486$	0	0
$487$	10.2757	0.465638	0.232819	−	0.972520i	$-0.425205\pi$
$487$	10.2757	0.465638	0.232819	+	0.972520i	$0.425205\pi$
$488$	1.89074	0.0855896
$489$	0	0
$490$	6.41864	0.289965
$491$	22.5795	1.01900	0.509499	−	0.860471i	$-0.329831\pi$
$491$	22.5795	1.01900	0.509499	+	0.860471i	$0.329831\pi$
$492$	0	0
$493$	16.7847	0.755946
$494$	−62.9523	−2.83236
$495$	0	0
$496$	14.8952	0.668815
$497$	−1.48882	−0.0667826
$498$	0	0
$499$	7.34717	0.328905	0.164452	−	0.986385i	$-0.447414\pi$
$499$	7.34717	0.328905	0.164452	+	0.986385i	$0.447414\pi$
$500$	−11.0858	−0.495772
$501$	0	0
$502$	55.3779	2.47164
$503$	20.5595	0.916703	0.458351	−	0.888771i	$-0.348440\pi$
$503$	20.5595	0.916703	0.458351	+	0.888771i	$0.348440\pi$
$504$	0	0
$505$	−1.45650	−0.0648132
$506$	18.1199	0.805527
$507$	0	0
$508$	12.9305	0.573697
$509$	−5.37046	−0.238042	−0.119021	−	0.992892i	$-0.537976\pi$
$509$	−5.37046	−0.238042	−0.119021	+	0.992892i	$0.537976\pi$
$510$	0	0
$511$	−10.9136	−0.482790
$512$	−31.6164	−1.39726
$513$	0	0
$514$	−19.3084	−0.851659
$515$	5.01284	0.220892
$516$	0	0
$517$	−0.711651	−0.0312984
$518$	19.5159	0.857479
$519$	0	0
$520$	0.447517	0.0196249
$521$	4.19741	0.183892	0.0919460	−	0.995764i	$-0.470691\pi$
$521$	4.19741	0.183892	0.0919460	+	0.995764i	$0.470691\pi$
$522$	0	0
$523$	−3.88046	−0.169681	−0.0848404	−	0.996395i	$-0.527038\pi$
$523$	−3.88046	−0.169681	−0.0848404	+	0.996395i	$0.527038\pi$
$524$	31.8539	1.39154
$525$	0	0
$526$	37.8752	1.65144
$527$	−23.1953	−1.01040
$528$	0	0
$529$	60.5034	2.63058
$530$	−8.34481	−0.362475
$531$	0	0
$532$	−13.7360	−0.595530
$533$	−12.0979	−0.524020
$534$	0	0
$535$	−6.83079	−0.295321
$536$	1.60803	0.0694565
$537$	0	0
$538$	−58.2025	−2.50929
$539$	−5.44151	−0.234382
$540$	0	0
$541$	30.8963	1.32834	0.664168	−	0.747583i	$-0.268786\pi$
$541$	30.8963	1.32834	0.664168	+	0.747583i	$0.268786\pi$
$542$	−9.17485	−0.394094
$543$	0	0
$544$	50.9706	2.18535
$545$	−8.37342	−0.358678
$546$	0	0
$547$	−10.3226	−0.441362	−0.220681	−	0.975346i	$-0.570828\pi$
$547$	−10.3226	−0.441362	−0.220681	+	0.975346i	$0.570828\pi$
$548$	−31.7126	−1.35470
$549$	0	0
$550$	9.21287	0.392838
$551$	14.8583	0.632984
$552$	0	0
$553$	−4.96882	−0.211296
$554$	19.2300	0.817006
$555$	0	0
$556$	−21.3393	−0.904990
$557$	−1.71110	−0.0725018	−0.0362509	−	0.999343i	$-0.511542\pi$
$557$	−1.71110	−0.0725018	−0.0362509	+	0.999343i	$0.511542\pi$
$558$	0	0
$559$	31.8571	1.34741
$560$	3.06816	0.129654
$561$	0	0
$562$	−59.0296	−2.49001
$563$	−24.2136	−1.02048	−0.510241	−	0.860031i	$-0.670444\pi$
$563$	−24.2136	−1.02048	−0.510241	+	0.860031i	$0.670444\pi$
$564$	0	0
$565$	−8.37119	−0.352179
$566$	−15.0946	−0.634474
$567$	0	0
$568$	0.160947	0.00675321
$569$	−11.0133	−0.461700	−0.230850	−	0.972989i	$-0.574151\pi$
$569$	−11.0133	−0.461700	−0.230850	+	0.972989i	$0.574151\pi$
$570$	0	0
$571$	−21.6779	−0.907193	−0.453596	−	0.891207i	$-0.649859\pi$
$571$	−21.6779	−0.907193	−0.453596	+	0.891207i	$0.649859\pi$
$572$	10.7693	0.450288
$573$	0	0
$574$	−5.37245	−0.224242
$575$	42.4564	1.77056
$576$	0	0
$577$	−34.7494	−1.44663	−0.723317	−	0.690516i	$-0.757384\pi$
$577$	−34.7494	−1.44663	−0.723317	+	0.690516i	$0.757384\pi$
$578$	−48.3678	−2.01184
$579$	0	0
$580$	2.99825	0.124496
$581$	2.23088	0.0925526
$582$	0	0
$583$	7.07446	0.292994
$584$	1.17981	0.0488209
$585$	0	0
$586$	−9.55376	−0.394662
$587$	9.57384	0.395155	0.197577	−	0.980287i	$-0.436693\pi$
$587$	9.57384	0.395155	0.197577	+	0.980287i	$0.436693\pi$
$588$	0	0
$589$	−20.5331	−0.846050
$590$	2.74223	0.112896
$591$	0	0
$592$	−32.5716	−1.33869
$593$	−12.1826	−0.500281	−0.250141	−	0.968210i	$-0.580477\pi$
$593$	−12.1826	−0.500281	−0.250141	+	0.968210i	$0.580477\pi$
$594$	0	0
$595$	−4.77784	−0.195872
$596$	2.38694	0.0977729
$597$	0	0
$598$	101.007	4.13047
$599$	15.6022	0.637488	0.318744	−	0.947841i	$-0.396739\pi$
$599$	15.6022	0.637488	0.318744	+	0.947841i	$0.396739\pi$
$600$	0	0
$601$	19.6560	0.801784	0.400892	−	0.916125i	$-0.368700\pi$
$601$	19.6560	0.801784	0.400892	+	0.916125i	$0.368700\pi$
$602$	14.1471	0.576593
$603$	0	0
$604$	−7.26227	−0.295497
$605$	0.594867	0.0241848
$606$	0	0
$607$	13.7220	0.556959	0.278479	−	0.960442i	$-0.410170\pi$
$607$	13.7220	0.556959	0.278479	+	0.960442i	$0.410170\pi$
$608$	45.1205	1.82988
$609$	0	0
$610$	−16.5257	−0.669105
$611$	−3.96699	−0.160487
$612$	0	0
$613$	−27.1817	−1.09786	−0.548929	−	0.835869i	$-0.684964\pi$
$613$	−27.1817	−1.09786	−0.548929	+	0.835869i	$0.684964\pi$
$614$	−5.98706	−0.241618
$615$	0	0
$616$	−0.168479	−0.00678823
$617$	2.87522	0.115752	0.0578759	−	0.998324i	$-0.481567\pi$
$617$	2.87522	0.115752	0.0578759	+	0.998324i	$0.481567\pi$
$618$	0	0
$619$	37.7754	1.51832	0.759161	−	0.650903i	$-0.225609\pi$
$619$	37.7754	1.51832	0.759161	+	0.650903i	$0.225609\pi$
$620$	−4.14337	−0.166402
$621$	0	0
$622$	10.1959	0.408820
$623$	−8.54059	−0.342172
$624$	0	0
$625$	19.8172	0.792689
$626$	−25.8634	−1.03371
$627$	0	0
$628$	41.8170	1.66868
$629$	50.7216	2.02240
$630$	0	0
$631$	−37.3497	−1.48687	−0.743435	−	0.668808i	$-0.766805\pi$
$631$	−37.3497	−1.48687	−0.743435	+	0.668808i	$0.766805\pi$
$632$	0.537151	0.0213667
$633$	0	0
$634$	6.96759	0.276718
$635$	3.98144	0.157999
$636$	0	0
$637$	−30.3329	−1.20183
$638$	−5.17319	−0.204808
$639$	0	0
$640$	−0.641879	−0.0253725
$641$	39.0718	1.54324	0.771621	−	0.636082i	$-0.219446\pi$
$641$	39.0718	1.54324	0.771621	+	0.636082i	$0.219446\pi$
$642$	0	0
$643$	−7.62560	−0.300724	−0.150362	−	0.988631i	$-0.548044\pi$
$643$	−7.62560	−0.300724	−0.150362	+	0.988631i	$0.548044\pi$
$644$	22.0393	0.868470
$645$	0	0
$646$	−72.6569	−2.85865
$647$	0.837635	0.0329308	0.0164654	−	0.999864i	$-0.494759\pi$
$647$	0.837635	0.0329308	0.0164654	+	0.999864i	$0.494759\pi$
$648$	0	0
$649$	−2.32477	−0.0912553
$650$	51.3558	2.01434
$651$	0	0
$652$	−3.85837	−0.151105
$653$	37.3531	1.46174	0.730869	−	0.682517i	$-0.239115\pi$
$653$	37.3531	1.46174	0.730869	+	0.682517i	$0.239115\pi$
$654$	0	0
$655$	9.80818	0.383237
$656$	8.96651	0.350083
$657$	0	0
$658$	−1.76166	−0.0686767
$659$	45.0815	1.75613	0.878063	−	0.478546i	$-0.158836\pi$
$659$	45.0815	1.75613	0.878063	+	0.478546i	$0.158836\pi$
$660$	0	0
$661$	32.1438	1.25025	0.625124	−	0.780526i	$-0.285049\pi$
$661$	32.1438	1.25025	0.625124	+	0.780526i	$0.285049\pi$
$662$	37.0119	1.43851
$663$	0	0
$664$	−0.241168	−0.00935913
$665$	−4.22947	−0.164012
$666$	0	0
$667$	−23.8400	−0.923089
$668$	−22.6476	−0.876261
$669$	0	0
$670$	−14.0548	−0.542983
$671$	14.0099	0.540847
$672$	0	0
$673$	9.37561	0.361403	0.180702	−	0.983538i	$-0.442163\pi$
$673$	9.37561	0.361403	0.180702	+	0.983538i	$0.442163\pi$
$674$	−38.8342	−1.49584
$675$	0	0
$676$	34.9168	1.34295
$677$	10.0639	0.386785	0.193393	−	0.981121i	$-0.438051\pi$
$677$	10.0639	0.386785	0.193393	+	0.981121i	$0.438051\pi$
$678$	0	0
$679$	−6.16523	−0.236600
$680$	0.516505	0.0198071
$681$	0	0
$682$	7.14897	0.273748
$683$	−4.40902	−0.168706	−0.0843532	−	0.996436i	$-0.526882\pi$
$683$	−4.40902	−0.168706	−0.0843532	+	0.996436i	$0.526882\pi$
$684$	0	0
$685$	−9.76469	−0.373090
$686$	−30.7984	−1.17589
$687$	0	0
$688$	−23.6112	−0.900169
$689$	39.4355	1.50237
$690$	0	0
$691$	11.7064	0.445333	0.222667	−	0.974895i	$-0.428524\pi$
$691$	11.7064	0.445333	0.222667	+	0.974895i	$0.428524\pi$
$692$	1.33337	0.0506870
$693$	0	0
$694$	0.268316	0.0101851
$695$	−6.57063	−0.249238
$696$	0	0
$697$	−13.9629	−0.528884
$698$	−25.8039	−0.976693
$699$	0	0
$700$	11.2057	0.423534
$701$	13.3602	0.504608	0.252304	−	0.967648i	$-0.418812\pi$
$701$	13.3602	0.504608	0.252304	+	0.967648i	$0.418812\pi$
$702$	0	0
$703$	44.9000	1.69344
$704$	−7.44657	−0.280653
$705$	0	0
$706$	−23.1030	−0.869493
$707$	3.05662	0.114956
$708$	0	0
$709$	−12.9680	−0.487025	−0.243513	−	0.969898i	$-0.578300\pi$
$709$	−12.9680	−0.487025	−0.243513	+	0.969898i	$0.578300\pi$
$710$	−1.40674	−0.0527939
$711$	0	0
$712$	0.923274	0.0346012
$713$	32.9452	1.23381
$714$	0	0
$715$	3.31600	0.124011
$716$	−4.26578	−0.159420
$717$	0	0
$718$	−17.4713	−0.652024
$719$	−10.8533	−0.404760	−0.202380	−	0.979307i	$-0.564868\pi$
$719$	−10.8533	−0.404760	−0.202380	+	0.979307i	$0.564868\pi$
$720$	0	0
$721$	−10.5200	−0.391785
$722$	−26.6424	−0.991528
$723$	0	0
$724$	−9.91642	−0.368541
$725$	−12.1212	−0.450171
$726$	0	0
$727$	3.40864	0.126419	0.0632097	−	0.998000i	$-0.479866\pi$
$727$	3.40864	0.126419	0.0632097	+	0.998000i	$0.479866\pi$
$728$	−0.939164	−0.0348077
$729$	0	0
$730$	−10.3119	−0.381662
$731$	36.7682	1.35992
$732$	0	0
$733$	3.53306	0.130497	0.0652484	−	0.997869i	$-0.479216\pi$
$733$	3.53306	0.130497	0.0652484	+	0.997869i	$0.479216\pi$
$734$	47.0957	1.73833
$735$	0	0
$736$	−72.3956	−2.66854
$737$	11.9152	0.438901
$738$	0	0
$739$	−29.6120	−1.08930	−0.544648	−	0.838665i	$-0.683337\pi$
$739$	−29.6120	−1.08930	−0.544648	+	0.838665i	$0.683337\pi$
$740$	9.06038	0.333066
$741$	0	0
$742$	17.5125	0.642905
$743$	12.6382	0.463651	0.231825	−	0.972757i	$-0.425530\pi$
$743$	12.6382	0.463651	0.231825	+	0.972757i	$0.425530\pi$
$744$	0	0
$745$	0.734968	0.0269271
$746$	67.9762	2.48878
$747$	0	0
$748$	12.4295	0.454467
$749$	14.3352	0.523796
$750$	0	0
$751$	42.2466	1.54160	0.770799	−	0.637078i	$-0.219857\pi$
$751$	42.2466	1.54160	0.770799	+	0.637078i	$0.219857\pi$
$752$	2.94018	0.107217
$753$	0	0
$754$	−28.8372	−1.05019
$755$	−2.23614	−0.0813814
$756$	0	0
$757$	29.3409	1.06641	0.533207	−	0.845985i	$-0.320987\pi$
$757$	29.3409	1.06641	0.533207	+	0.845985i	$0.320987\pi$
$758$	34.9473	1.26934
$759$	0	0
$760$	0.457223	0.0165852
$761$	−43.2228	−1.56683	−0.783413	−	0.621501i	$-0.786523\pi$
$761$	−43.2228	−1.56683	−0.783413	+	0.621501i	$0.786523\pi$
$762$	0	0
$763$	17.5725	0.636169
$764$	−31.2981	−1.13233
$765$	0	0
$766$	−40.5476	−1.46505
$767$	−12.9591	−0.467926
$768$	0	0
$769$	41.3541	1.49127	0.745633	−	0.666357i	$-0.232147\pi$
$769$	41.3541	1.49127	0.745633	+	0.666357i	$0.232147\pi$
$770$	1.47257	0.0530677
$771$	0	0
$772$	−8.08340	−0.290928
$773$	8.38159	0.301465	0.150732	−	0.988575i	$-0.451837\pi$
$773$	8.38159	0.301465	0.150732	+	0.988575i	$0.451837\pi$
$774$	0	0
$775$	16.7507	0.601701
$776$	0.666487	0.0239255
$777$	0	0
$778$	46.6686	1.67315
$779$	−12.3603	−0.442855
$780$	0	0
$781$	1.19258	0.0426740
$782$	116.578	4.16881
$783$	0	0
$784$	22.4815	0.802912
$785$	12.8759	0.459562
$786$	0	0
$787$	−36.6487	−1.30638	−0.653192	−	0.757192i	$-0.726570\pi$
$787$	−36.6487	−1.30638	−0.653192	+	0.757192i	$0.726570\pi$
$788$	−17.2599	−0.614858
$789$	0	0
$790$	−4.69488	−0.167036
$791$	17.5679	0.624642
$792$	0	0
$793$	78.0963	2.77328
$794$	48.4727	1.72023
$795$	0	0
$796$	−31.5736	−1.11910
$797$	23.0271	0.815663	0.407831	−	0.913057i	$-0.366285\pi$
$797$	23.0271	0.815663	0.407831	+	0.913057i	$0.366285\pi$
$798$	0	0
$799$	−4.57854	−0.161977
$800$	−36.8088	−1.30139
$801$	0	0
$802$	46.9882	1.65921
$803$	8.74213	0.308503
$804$	0	0
$805$	6.78616	0.239181
$806$	39.8509	1.40369
$807$	0	0
$808$	−0.330433	−0.0116246
$809$	31.8739	1.12063	0.560314	−	0.828280i	$-0.310680\pi$
$809$	31.8739	1.12063	0.560314	+	0.828280i	$0.310680\pi$
$810$	0	0
$811$	−33.6519	−1.18168	−0.590838	−	0.806790i	$-0.701203\pi$
$811$	−33.6519	−1.18168	−0.590838	+	0.806790i	$0.701203\pi$
$812$	−6.29217	−0.220812
$813$	0	0
$814$	−15.6328	−0.547929
$815$	−1.18804	−0.0416151
$816$	0	0
$817$	32.5481	1.13871
$818$	63.4475	2.21839
$819$	0	0
$820$	−2.49420	−0.0871011
$821$	25.8678	0.902793	0.451397	−	0.892323i	$-0.350926\pi$
$821$	25.8678	0.902793	0.451397	+	0.892323i	$0.350926\pi$
$822$	0	0
$823$	−49.3280	−1.71947	−0.859733	−	0.510744i	$-0.829370\pi$
$823$	−49.3280	−1.71947	−0.859733	+	0.510744i	$0.829370\pi$
$824$	1.13726	0.0396182
$825$	0	0
$826$	−5.75488	−0.200238
$827$	29.8291	1.03726	0.518630	−	0.854999i	$-0.326442\pi$
$827$	29.8291	1.03726	0.518630	+	0.854999i	$0.326442\pi$
$828$	0	0
$829$	−26.3449	−0.914995	−0.457498	−	0.889211i	$-0.651254\pi$
$829$	−26.3449	−0.914995	−0.457498	+	0.889211i	$0.651254\pi$
$830$	2.10789	0.0731659
$831$	0	0
$832$	−41.5098	−1.43909
$833$	−35.0090	−1.21299
$834$	0	0
$835$	−6.97346	−0.241326
$836$	11.0029	0.380543
$837$	0	0
$838$	−44.9201	−1.55174
$839$	37.0520	1.27918	0.639589	−	0.768717i	$-0.279105\pi$
$839$	37.0520	1.27918	0.639589	+	0.768717i	$0.279105\pi$
$840$	0	0
$841$	−22.1937	−0.765301
$842$	−20.3615	−0.701703
$843$	0	0
$844$	20.2507	0.697057
$845$	10.7513	0.369856
$846$	0	0
$847$	−1.24839	−0.0428953
$848$	−29.2280	−1.00369
$849$	0	0
$850$	59.2727	2.03304
$851$	−72.0418	−2.46956
$852$	0	0
$853$	−20.6609	−0.707416	−0.353708	−	0.935356i	$-0.615079\pi$
$853$	−20.6609	−0.707416	−0.353708	+	0.935356i	$0.615079\pi$
$854$	34.6810	1.18676
$855$	0	0
$856$	−1.54969	−0.0529674
$857$	−23.3980	−0.799260	−0.399630	−	0.916676i	$-0.630861\pi$
$857$	−23.3980	−0.799260	−0.399630	+	0.916676i	$0.630861\pi$
$858$	0	0
$859$	8.03073	0.274005	0.137003	−	0.990571i	$-0.456253\pi$
$859$	8.03073	0.274005	0.137003	+	0.990571i	$0.456253\pi$
$860$	6.56789	0.223963
$861$	0	0
$862$	−2.01918	−0.0687735
$863$	−41.1668	−1.40133	−0.700667	−	0.713489i	$-0.747114\pi$
$863$	−41.1668	−1.40133	−0.700667	+	0.713489i	$0.747114\pi$
$864$	0	0
$865$	0.410560	0.0139594
$866$	36.0413	1.22473
$867$	0	0
$868$	8.69532	0.295138
$869$	3.98017	0.135018
$870$	0	0
$871$	66.4194	2.25054
$872$	−1.89967	−0.0643308
$873$	0	0
$874$	103.197	3.49071
$875$	7.16350	0.242170
$876$	0	0
$877$	23.4484	0.791797	0.395898	−	0.918294i	$-0.370433\pi$
$877$	23.4484	0.791797	0.395898	+	0.918294i	$0.370433\pi$
$878$	−26.3695	−0.889926
$879$	0	0
$880$	−2.45769	−0.0828486
$881$	9.12961	0.307584	0.153792	−	0.988103i	$-0.450851\pi$
$881$	9.12961	0.307584	0.153792	+	0.988103i	$0.450851\pi$
$882$	0	0
$883$	44.3941	1.49398	0.746990	−	0.664835i	$-0.231498\pi$
$883$	44.3941	1.49398	0.746990	+	0.664835i	$0.231498\pi$
$884$	69.2864	2.33035
$885$	0	0
$886$	−32.7587	−1.10055
$887$	−1.16604	−0.0391518	−0.0195759	−	0.999808i	$-0.506232\pi$
$887$	−1.16604	−0.0391518	−0.0195759	+	0.999808i	$0.506232\pi$
$888$	0	0
$889$	−8.35550	−0.280234
$890$	−8.06974	−0.270498
$891$	0	0
$892$	16.9737	0.568322
$893$	−4.05304	−0.135630
$894$	0	0
$895$	−1.31349	−0.0439050
$896$	1.34705	0.0450019
$897$	0	0
$898$	17.7199	0.591320
$899$	−9.40578	−0.313700
$900$	0	0
$901$	45.5148	1.51632
$902$	4.30348	0.143290
$903$	0	0
$904$	−1.89916	−0.0631652
$905$	−3.05338	−0.101498
$906$	0	0
$907$	13.4121	0.445340	0.222670	−	0.974894i	$-0.428523\pi$
$907$	13.4121	0.445340	0.222670	+	0.974894i	$0.428523\pi$
$908$	−47.8840	−1.58909
$909$	0	0
$910$	8.20862	0.272113
$911$	−15.5787	−0.516145	−0.258073	−	0.966126i	$-0.583087\pi$
$911$	−15.5787	−0.516145	−0.258073	+	0.966126i	$0.583087\pi$
$912$	0	0
$913$	−1.78700	−0.0591411
$914$	−49.7208	−1.64462
$915$	0	0
$916$	−7.66078	−0.253119
$917$	−20.5836	−0.679729
$918$	0	0
$919$	−14.7124	−0.485316	−0.242658	−	0.970112i	$-0.578019\pi$
$919$	−14.7124	−0.485316	−0.242658	+	0.970112i	$0.578019\pi$
$920$	−0.733613	−0.0241865
$921$	0	0
$922$	−60.8992	−2.00561
$923$	6.64789	0.218818
$924$	0	0
$925$	−36.6290	−1.20435
$926$	−67.5610	−2.22019
$927$	0	0
$928$	20.6688	0.678486
$929$	−19.4271	−0.637382	−0.318691	−	0.947859i	$-0.603243\pi$
$929$	−19.4271	−0.637382	−0.318691	+	0.947859i	$0.603243\pi$
$930$	0	0
$931$	−30.9908	−1.01568
$932$	27.1605	0.889672
$933$	0	0
$934$	52.9282	1.73186
$935$	3.82719	0.125162
$936$	0	0
$937$	27.3590	0.893779	0.446889	−	0.894589i	$-0.352532\pi$
$937$	27.3590	0.893779	0.446889	+	0.894589i	$0.352532\pi$
$938$	29.4955	0.963063
$939$	0	0
$940$	−0.817863	−0.0266757
$941$	−0.136616	−0.00445356	−0.00222678	−	0.999998i	$-0.500709\pi$
$941$	−0.136616	−0.00445356	−0.00222678	+	0.999998i	$0.500709\pi$
$942$	0	0
$943$	19.8321	0.645822
$944$	9.60477	0.312609
$945$	0	0
$946$	−11.3322	−0.368443
$947$	33.7242	1.09589	0.547945	−	0.836514i	$-0.315410\pi$
$947$	33.7242	1.09589	0.547945	+	0.836514i	$0.315410\pi$
$948$	0	0
$949$	48.7317	1.58190
$950$	52.4697	1.70234
$951$	0	0
$952$	−1.08394	−0.0351308
$953$	4.05681	0.131413	0.0657065	−	0.997839i	$-0.479070\pi$
$953$	4.05681	0.131413	0.0657065	+	0.997839i	$0.479070\pi$
$954$	0	0
$955$	−9.63706	−0.311848
$956$	20.3837	0.659257
$957$	0	0
$958$	−38.9845	−1.25953
$959$	20.4923	0.661730
$960$	0	0
$961$	−18.0019	−0.580706
$962$	−87.1427	−2.80959
$963$	0	0
$964$	21.0577	0.678223
$965$	−2.48897	−0.0801229
$966$	0	0
$967$	41.7505	1.34261	0.671303	−	0.741183i	$-0.265735\pi$
$967$	41.7505	1.34261	0.671303	+	0.741183i	$0.265735\pi$
$968$	0.134957	0.00433768
$969$	0	0
$970$	−5.82533	−0.187040
$971$	0.681779	0.0218793	0.0109397	−	0.999940i	$-0.496518\pi$
$971$	0.681779	0.0218793	0.0109397	+	0.999940i	$0.496518\pi$
$972$	0	0
$973$	13.7892	0.442062
$974$	−20.3759	−0.652886
$975$	0	0
$976$	−57.8819	−1.85275
$977$	−13.8474	−0.443018	−0.221509	−	0.975158i	$-0.571098\pi$
$977$	−13.8474	−0.443018	−0.221509	+	0.975158i	$0.571098\pi$
$978$	0	0
$979$	6.84126	0.218648
$980$	−6.25364	−0.199765
$981$	0	0
$982$	−44.7732	−1.42877
$983$	14.5944	0.465488	0.232744	−	0.972538i	$-0.425230\pi$
$983$	14.5944	0.465488	0.232744	+	0.972538i	$0.425230\pi$
$984$	0	0
$985$	−5.31452	−0.169335
$986$	−33.2827	−1.05994
$987$	0	0
$988$	61.3341	1.95130
$989$	−52.2233	−1.66060
$990$	0	0
$991$	−15.4979	−0.492306	−0.246153	−	0.969231i	$-0.579167\pi$
$991$	−15.4979	−0.492306	−0.246153	+	0.969231i	$0.579167\pi$
$992$	−28.5628	−0.906869
$993$	0	0
$994$	2.95219	0.0936379
$995$	−9.72188	−0.308204
$996$	0	0
$997$	4.92781	0.156065	0.0780326	−	0.996951i	$-0.475136\pi$
$997$	4.92781	0.156065	0.0780326	+	0.996951i	$0.475136\pi$
$998$	−14.5688	−0.461167
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.l.1.9		51
3.2	odd	2		8019.2.a.k.1.43		51
27.4	even	9		297.2.j.c.232.3	✓	102
27.7	even	9		297.2.j.c.265.3	yes	102
27.20	odd	18		891.2.j.c.199.15		102
27.23	odd	18		891.2.j.c.694.15		102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.232.3	✓	102	27.4	even	9
297.2.j.c.265.3	yes	102	27.7	even	9
891.2.j.c.199.15		102	27.20	odd	18
891.2.j.c.694.15		102	27.23	odd	18
8019.2.a.k.1.43		51	3.2	odd	2
8019.2.a.l.1.9		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)

\(f(q)\)	\(=\)	\(q-1.98291 q^{2} +1.93194 q^{4} +0.594867 q^{5} -1.24839 q^{7} +0.134957 q^{8} +O(q^{10})\)	\(q-1.98291 q^{2} +1.93194 q^{4} +0.594867 q^{5} -1.24839 q^{7} +0.134957 q^{8} -1.17957 q^{10} +1.00000 q^{11} +5.57436 q^{13} +2.47546 q^{14} -4.13149 q^{16} +6.43369 q^{17} +5.69526 q^{19} +1.14925 q^{20} -1.98291 q^{22} -9.13802 q^{23} -4.64613 q^{25} -11.0535 q^{26} -2.41182 q^{28} +2.60888 q^{29} -3.60529 q^{31} +7.92246 q^{32} -12.7574 q^{34} -0.742629 q^{35} +7.88375 q^{37} -11.2932 q^{38} +0.0802814 q^{40} -2.17028 q^{41} +5.71494 q^{43} +1.93194 q^{44} +18.1199 q^{46} -0.711651 q^{47} -5.44151 q^{49} +9.21287 q^{50} +10.7693 q^{52} +7.07446 q^{53} +0.594867 q^{55} -0.168479 q^{56} -5.17319 q^{58} -2.32477 q^{59} +14.0099 q^{61} +7.14897 q^{62} -7.44657 q^{64} +3.31600 q^{65} +11.9152 q^{67} +12.4295 q^{68} +1.47257 q^{70} +1.19258 q^{71} +8.74213 q^{73} -15.6328 q^{74} +11.0029 q^{76} -1.24839 q^{77} +3.98017 q^{79} -2.45769 q^{80} +4.30348 q^{82} -1.78700 q^{83} +3.82719 q^{85} -11.3322 q^{86} +0.134957 q^{88} +6.84126 q^{89} -6.95900 q^{91} -17.6541 q^{92} +1.41114 q^{94} +3.38792 q^{95} +4.93852 q^{97} +10.7900 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) \) 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

Varieties

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Representations

Groups

Database

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