LMFDB - Embedded newform 8019.2.a.l.1.19

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			1.19
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)

$f(q)$	$=$	$q-0.933308 q^{2} -1.12894 q^{4} +1.97519 q^{5} +5.04438 q^{7} +2.92026 q^{8} +O(q^{10})$	\(q-0.933308 q^{2} -1.12894 q^{4} +1.97519 q^{5} +5.04438 q^{7} +2.92026 q^{8} -1.84346 q^{10} +1.00000 q^{11} +3.71225 q^{13} -4.70796 q^{14} -0.467634 q^{16} +4.82677 q^{17} -2.33194 q^{19} -2.22987 q^{20} -0.933308 q^{22} -3.35323 q^{23} -1.09861 q^{25} -3.46467 q^{26} -5.69478 q^{28} -5.55516 q^{29} +4.45726 q^{31} -5.40408 q^{32} -4.50486 q^{34} +9.96362 q^{35} +5.21762 q^{37} +2.17642 q^{38} +5.76808 q^{40} +10.0626 q^{41} -9.29780 q^{43} -1.12894 q^{44} +3.12960 q^{46} +0.325411 q^{47} +18.4457 q^{49} +1.02535 q^{50} -4.19089 q^{52} -2.55924 q^{53} +1.97519 q^{55} +14.7309 q^{56} +5.18468 q^{58} -3.78761 q^{59} +13.6134 q^{61} -4.16000 q^{62} +5.97894 q^{64} +7.33241 q^{65} -9.81707 q^{67} -5.44911 q^{68} -9.29913 q^{70} +6.04703 q^{71} +10.0675 q^{73} -4.86965 q^{74} +2.63261 q^{76} +5.04438 q^{77} +5.12790 q^{79} -0.923667 q^{80} -9.39150 q^{82} +7.23420 q^{83} +9.53380 q^{85} +8.67771 q^{86} +2.92026 q^{88} +4.73283 q^{89} +18.7260 q^{91} +3.78558 q^{92} -0.303709 q^{94} -4.60603 q^{95} -0.553084 q^{97} -17.2156 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−0.933308	−0.659949	−0.329974	−	0.943990i	$-0.607040\pi$
$2$	−0.933308	−0.659949	−0.329974	+	0.943990i	$0.607040\pi$
$3$	0	0
$3$	0	0
$4$	−1.12894	−0.564468
$5$	1.97519	0.883333	0.441667	−	0.897179i	$-0.354387\pi$
$5$	1.97519	0.883333	0.441667	+	0.897179i	$0.354387\pi$
$6$	0	0
$7$	5.04438	1.90660	0.953298	−	0.302033i	$-0.0976651\pi$
$7$	5.04438	1.90660	0.953298	+	0.302033i	$0.0976651\pi$
$8$	2.92026	1.03247
$9$	0	0
$10$	−1.84346	−0.582954
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	3.71225	1.02959	0.514796	−	0.857312i	$-0.327868\pi$
$13$	3.71225	1.02959	0.514796	+	0.857312i	$0.327868\pi$
$14$	−4.70796	−1.25826
$15$	0	0
$16$	−0.467634	−0.116908
$17$	4.82677	1.17066	0.585332	−	0.810794i	$-0.300964\pi$
$17$	4.82677	1.17066	0.585332	+	0.810794i	$0.300964\pi$
$18$	0	0
$19$	−2.33194	−0.534983	−0.267492	−	0.963560i	$-0.586195\pi$
$19$	−2.33194	−0.534983	−0.267492	+	0.963560i	$0.586195\pi$
$20$	−2.22987	−0.498613
$21$	0	0
$22$	−0.933308	−0.198982
$23$	−3.35323	−0.699198	−0.349599	−	0.936899i	$-0.613682\pi$
$23$	−3.35323	−0.699198	−0.349599	+	0.936899i	$0.613682\pi$
$24$	0	0
$25$	−1.09861	−0.219723
$26$	−3.46467	−0.679478
$27$	0	0
$28$	−5.69478	−1.07621
$29$	−5.55516	−1.03157	−0.515784	−	0.856719i	$-0.672499\pi$
$29$	−5.55516	−1.03157	−0.515784	+	0.856719i	$0.672499\pi$
$30$	0	0
$31$	4.45726	0.800548	0.400274	−	0.916396i	$-0.368915\pi$
$31$	4.45726	0.800548	0.400274	+	0.916396i	$0.368915\pi$
$32$	−5.40408	−0.955315
$33$	0	0
$34$	−4.50486	−0.772578
$35$	9.96362	1.68416
$36$	0	0
$37$	5.21762	0.857772	0.428886	−	0.903359i	$-0.358906\pi$
$37$	5.21762	0.857772	0.428886	+	0.903359i	$0.358906\pi$
$38$	2.17642	0.353061
$39$	0	0
$40$	5.76808	0.912013
$41$	10.0626	1.57151	0.785756	−	0.618537i	$-0.212274\pi$
$41$	10.0626	1.57151	0.785756	+	0.618537i	$0.212274\pi$
$42$	0	0
$43$	−9.29780	−1.41790	−0.708950	−	0.705258i	$-0.750831\pi$
$43$	−9.29780	−1.41790	−0.708950	+	0.705258i	$0.750831\pi$
$44$	−1.12894	−0.170193
$45$	0	0
$46$	3.12960	0.461435
$47$	0.325411	0.0474661	0.0237330	−	0.999718i	$-0.492445\pi$
$47$	0.325411	0.0474661	0.0237330	+	0.999718i	$0.492445\pi$
$48$	0	0
$49$	18.4457	2.63511
$50$	1.02535	0.145006
$51$	0	0
$52$	−4.19089	−0.581172
$53$	−2.55924	−0.351538	−0.175769	−	0.984431i	$-0.556241\pi$
$53$	−2.55924	−0.351538	−0.175769	+	0.984431i	$0.556241\pi$
$54$	0	0
$55$	1.97519	0.266335
$56$	14.7309	1.96850
$57$	0	0
$58$	5.18468	0.680782
$59$	−3.78761	−0.493105	−0.246552	−	0.969129i	$-0.579298\pi$
$59$	−3.78761	−0.493105	−0.246552	+	0.969129i	$0.579298\pi$
$60$	0	0
$61$	13.6134	1.74302	0.871511	−	0.490377i	$-0.163141\pi$
$61$	13.6134	1.74302	0.871511	+	0.490377i	$0.163141\pi$
$62$	−4.16000	−0.528321
$63$	0	0
$64$	5.97894	0.747367
$65$	7.33241	0.909473
$66$	0	0
$67$	−9.81707	−1.19935	−0.599673	−	0.800245i	$-0.704703\pi$
$67$	−9.81707	−1.19935	−0.599673	+	0.800245i	$0.704703\pi$
$68$	−5.44911	−0.660802
$69$	0	0
$70$	−9.29913	−1.11146
$71$	6.04703	0.717651	0.358825	−	0.933405i	$-0.383177\pi$
$71$	6.04703	0.717651	0.358825	+	0.933405i	$0.383177\pi$
$72$	0	0
$73$	10.0675	1.17831	0.589153	−	0.808021i	$-0.299461\pi$
$73$	10.0675	1.17831	0.589153	+	0.808021i	$0.299461\pi$
$74$	−4.86965	−0.566085
$75$	0	0
$76$	2.63261	0.301981
$77$	5.04438	0.574860
$78$	0	0
$79$	5.12790	0.576934	0.288467	−	0.957490i	$-0.406854\pi$
$79$	5.12790	0.576934	0.288467	+	0.957490i	$0.406854\pi$
$80$	−0.923667	−0.103269
$81$	0	0
$82$	−9.39150	−1.03712
$83$	7.23420	0.794057	0.397028	−	0.917806i	$-0.370041\pi$
$83$	7.23420	0.794057	0.397028	+	0.917806i	$0.370041\pi$
$84$	0	0
$85$	9.53380	1.03409
$86$	8.67771	0.935742
$87$	0	0
$88$	2.92026	0.311301
$89$	4.73283	0.501678	0.250839	−	0.968029i	$-0.419293\pi$
$89$	4.73283	0.501678	0.250839	+	0.968029i	$0.419293\pi$
$90$	0	0
$91$	18.7260	1.96302
$92$	3.78558	0.394675
$93$	0	0
$94$	−0.303709	−0.0313252
$95$	−4.60603	−0.472568
$96$	0	0
$97$	−0.553084	−0.0561571	−0.0280786	−	0.999606i	$-0.508939\pi$
$97$	−0.553084	−0.0561571	−0.0280786	+	0.999606i	$0.508939\pi$
$98$	−17.2156	−1.73903
$99$	0	0
$100$	1.24026	0.124026
$101$	0.324534	0.0322924	0.0161462	−	0.999870i	$-0.494860\pi$
$101$	0.324534	0.0322924	0.0161462	+	0.999870i	$0.494860\pi$
$102$	0	0
$103$	−18.4739	−1.82029	−0.910143	−	0.414294i	$-0.864029\pi$
$103$	−18.4739	−1.82029	−0.910143	+	0.414294i	$0.864029\pi$
$104$	10.8407	1.06302
$105$	0	0
$106$	2.38856	0.231997
$107$	7.15345	0.691550	0.345775	−	0.938317i	$-0.387616\pi$
$107$	7.15345	0.691550	0.345775	+	0.938317i	$0.387616\pi$
$108$	0	0
$109$	5.33622	0.511118	0.255559	−	0.966794i	$-0.417741\pi$
$109$	5.33622	0.511118	0.255559	+	0.966794i	$0.417741\pi$
$110$	−1.84346	−0.175767
$111$	0	0
$112$	−2.35892	−0.222897
$113$	−18.3048	−1.72197	−0.860984	−	0.508632i	$-0.830151\pi$
$113$	−18.3048	−1.72197	−0.860984	+	0.508632i	$0.830151\pi$
$114$	0	0
$115$	−6.62328	−0.617624
$116$	6.27142	0.582287
$117$	0	0
$118$	3.53501	0.325424
$119$	24.3480	2.23198
$120$	0	0
$121$	1.00000	0.0909091
$122$	−12.7055	−1.15030
$123$	0	0
$124$	−5.03196	−0.451884
$125$	−12.0459	−1.07742
$126$	0	0
$127$	7.05110	0.625684	0.312842	−	0.949805i	$-0.398719\pi$
$127$	7.05110	0.625684	0.312842	+	0.949805i	$0.398719\pi$
$128$	5.22796	0.462091
$129$	0	0
$130$	−6.84340	−0.600206
$131$	−15.0817	−1.31770	−0.658849	−	0.752275i	$-0.728956\pi$
$131$	−15.0817	−1.31770	−0.658849	+	0.752275i	$0.728956\pi$
$132$	0	0
$133$	−11.7632	−1.02000
$134$	9.16236	0.791507
$135$	0	0
$136$	14.0954	1.20867
$137$	2.36548	0.202097	0.101048	−	0.994882i	$-0.467780\pi$
$137$	2.36548	0.202097	0.101048	+	0.994882i	$0.467780\pi$
$138$	0	0
$139$	13.8508	1.17481	0.587403	−	0.809295i	$-0.300150\pi$
$139$	13.8508	1.17481	0.587403	+	0.809295i	$0.300150\pi$
$140$	−11.2483	−0.950653
$141$	0	0
$142$	−5.64375	−0.473613
$143$	3.71225	0.310434
$144$	0	0
$145$	−10.9725	−0.911218
$146$	−9.39604	−0.777622
$147$	0	0
$148$	−5.89036	−0.484185
$149$	−7.51992	−0.616056	−0.308028	−	0.951377i	$-0.599669\pi$
$149$	−7.51992	−0.616056	−0.308028	+	0.951377i	$0.599669\pi$
$150$	0	0
$151$	−0.162368	−0.0132133	−0.00660667	−	0.999978i	$-0.502103\pi$
$151$	−0.162368	−0.0132133	−0.00660667	+	0.999978i	$0.502103\pi$
$152$	−6.80987	−0.552353
$153$	0	0
$154$	−4.70796	−0.379378
$155$	8.80395	0.707151
$156$	0	0
$157$	1.65343	0.131958	0.0659789	−	0.997821i	$-0.478983\pi$
$157$	1.65343	0.131958	0.0659789	+	0.997821i	$0.478983\pi$
$158$	−4.78591	−0.380747
$159$	0	0
$160$	−10.6741	−0.843861
$161$	−16.9150	−1.33309
$162$	0	0
$163$	−0.913204	−0.0715276	−0.0357638	−	0.999360i	$-0.511386\pi$
$163$	−0.913204	−0.0715276	−0.0357638	+	0.999360i	$0.511386\pi$
$164$	−11.3600	−0.887068
$165$	0	0
$166$	−6.75174	−0.524037
$167$	5.21455	0.403514	0.201757	−	0.979436i	$-0.435335\pi$
$167$	5.21455	0.403514	0.201757	+	0.979436i	$0.435335\pi$
$168$	0	0
$169$	0.780795	0.0600611
$170$	−8.89797	−0.682443
$171$	0	0
$172$	10.4966	0.800359
$173$	−4.24044	−0.322394	−0.161197	−	0.986922i	$-0.551536\pi$
$173$	−4.24044	−0.322394	−0.161197	+	0.986922i	$0.551536\pi$
$174$	0	0
$175$	−5.54182	−0.418922
$176$	−0.467634	−0.0352492
$177$	0	0
$178$	−4.41719	−0.331082
$179$	−9.17356	−0.685664	−0.342832	−	0.939397i	$-0.611386\pi$
$179$	−9.17356	−0.685664	−0.342832	+	0.939397i	$0.611386\pi$
$180$	0	0
$181$	10.0298	0.745509	0.372754	−	0.927930i	$-0.378413\pi$
$181$	10.0298	0.745509	0.372754	+	0.927930i	$0.378413\pi$
$182$	−17.4771	−1.29549
$183$	0	0
$184$	−9.79232	−0.721900
$185$	10.3058	0.757698
$186$	0	0
$187$	4.82677	0.352968
$188$	−0.367368	−0.0267931
$189$	0	0
$190$	4.29884	0.311871
$191$	−21.8081	−1.57798	−0.788989	−	0.614407i	$-0.789395\pi$
$191$	−21.8081	−1.57798	−0.788989	+	0.614407i	$0.789395\pi$
$192$	0	0
$193$	−4.48905	−0.323129	−0.161564	−	0.986862i	$-0.551654\pi$
$193$	−4.48905	−0.323129	−0.161564	+	0.986862i	$0.551654\pi$
$194$	0.516198	0.0370608
$195$	0	0
$196$	−20.8240	−1.48743
$197$	26.0553	1.85636	0.928182	−	0.372127i	$-0.121371\pi$
$197$	26.0553	1.85636	0.928182	+	0.372127i	$0.121371\pi$
$198$	0	0
$199$	3.49249	0.247576	0.123788	−	0.992309i	$-0.460496\pi$
$199$	3.49249	0.247576	0.123788	+	0.992309i	$0.460496\pi$
$200$	−3.20824	−0.226857
$201$	0	0
$202$	−0.302891	−0.0213113
$203$	−28.0223	−1.96678
$204$	0	0
$205$	19.8755	1.38817
$206$	17.2418	1.20130
$207$	0	0
$208$	−1.73597	−0.120368
$209$	−2.33194	−0.161303
$210$	0	0
$211$	2.35037	0.161806	0.0809031	−	0.996722i	$-0.474220\pi$
$211$	2.35037	0.161806	0.0809031	+	0.996722i	$0.474220\pi$
$212$	2.88921	0.198432
$213$	0	0
$214$	−6.67638	−0.456388
$215$	−18.3649	−1.25248
$216$	0	0
$217$	22.4841	1.52632
$218$	−4.98034	−0.337311
$219$	0	0
$220$	−2.22987	−0.150337
$221$	17.9182	1.20531
$222$	0	0
$223$	−9.64473	−0.645859	−0.322930	−	0.946423i	$-0.604668\pi$
$223$	−9.64473	−0.645859	−0.322930	+	0.946423i	$0.604668\pi$
$224$	−27.2602	−1.82140
$225$	0	0
$226$	17.0840	1.13641
$227$	0.683630	0.0453741	0.0226871	−	0.999743i	$-0.492778\pi$
$227$	0.683630	0.0453741	0.0226871	+	0.999743i	$0.492778\pi$
$228$	0	0
$229$	9.73487	0.643298	0.321649	−	0.946859i	$-0.395763\pi$
$229$	9.73487	0.643298	0.321649	+	0.946859i	$0.395763\pi$
$230$	6.18157	0.407600
$231$	0	0
$232$	−16.2225	−1.06506
$233$	−13.6883	−0.896753	−0.448377	−	0.893845i	$-0.647998\pi$
$233$	−13.6883	−0.896753	−0.448377	+	0.893845i	$0.647998\pi$
$234$	0	0
$235$	0.642749	0.0419283
$236$	4.27597	0.278342
$237$	0	0
$238$	−22.7242	−1.47299
$239$	−24.9662	−1.61493	−0.807466	−	0.589914i	$-0.799162\pi$
$239$	−24.9662	−1.61493	−0.807466	+	0.589914i	$0.799162\pi$
$240$	0	0
$241$	27.4302	1.76693	0.883466	−	0.468494i	$-0.155203\pi$
$241$	27.4302	1.76693	0.883466	+	0.468494i	$0.155203\pi$
$242$	−0.933308	−0.0599953
$243$	0	0
$244$	−15.3687	−0.983879
$245$	36.4339	2.32768
$246$	0	0
$247$	−8.65673	−0.550815
$248$	13.0164	0.826541
$249$	0	0
$250$	11.2426	0.711043
$251$	−12.6220	−0.796696	−0.398348	−	0.917234i	$-0.630416\pi$
$251$	−12.6220	−0.796696	−0.398348	+	0.917234i	$0.630416\pi$
$252$	0	0
$253$	−3.35323	−0.210816
$254$	−6.58085	−0.412920
$255$	0	0
$256$	−16.8372	−1.05232
$257$	−6.78084	−0.422977	−0.211488	−	0.977380i	$-0.567831\pi$
$257$	−6.78084	−0.422977	−0.211488	+	0.977380i	$0.567831\pi$
$258$	0	0
$259$	26.3197	1.63542
$260$	−8.27781	−0.513368
$261$	0	0
$262$	14.0759	0.869613
$263$	−13.5665	−0.836549	−0.418275	−	0.908321i	$-0.637365\pi$
$263$	−13.5665	−0.836549	−0.418275	+	0.908321i	$0.637365\pi$
$264$	0	0
$265$	−5.05499	−0.310525
$266$	10.9787	0.673145
$267$	0	0
$268$	11.0828	0.676992
$269$	17.9997	1.09746	0.548732	−	0.835999i	$-0.315111\pi$
$269$	17.9997	1.09746	0.548732	+	0.835999i	$0.315111\pi$
$270$	0	0
$271$	7.29430	0.443097	0.221549	−	0.975149i	$-0.428889\pi$
$271$	7.29430	0.443097	0.221549	+	0.975149i	$0.428889\pi$
$272$	−2.25716	−0.136860
$273$	0	0
$274$	−2.20772	−0.133373
$275$	−1.09861	−0.0662489
$276$	0	0
$277$	23.9929	1.44159	0.720795	−	0.693148i	$-0.243777\pi$
$277$	23.9929	1.44159	0.720795	+	0.693148i	$0.243777\pi$
$278$	−12.9270	−0.775312
$279$	0	0
$280$	29.0964	1.73884
$281$	−11.2921	−0.673633	−0.336816	−	0.941570i	$-0.609350\pi$
$281$	−11.2921	−0.673633	−0.336816	+	0.941570i	$0.609350\pi$
$282$	0	0
$283$	5.20005	0.309111	0.154555	−	0.987984i	$-0.450605\pi$
$283$	5.20005	0.309111	0.154555	+	0.987984i	$0.450605\pi$
$284$	−6.82671	−0.405091
$285$	0	0
$286$	−3.46467	−0.204870
$287$	50.7595	2.99624
$288$	0	0
$289$	6.29770	0.370453
$290$	10.2407	0.601357
$291$	0	0
$292$	−11.3655	−0.665116
$293$	−4.81180	−0.281108	−0.140554	−	0.990073i	$-0.544888\pi$
$293$	−4.81180	−0.281108	−0.140554	+	0.990073i	$0.544888\pi$
$294$	0	0
$295$	−7.48126	−0.435576
$296$	15.2368	0.885622
$297$	0	0
$298$	7.01840	0.406565
$299$	−12.4480	−0.719889
$300$	0	0
$301$	−46.9016	−2.70336
$302$	0.151540	0.00872012
$303$	0	0
$304$	1.09049	0.0625441
$305$	26.8892	1.53967
$306$	0	0
$307$	12.4814	0.712354	0.356177	−	0.934419i	$-0.384080\pi$
$307$	12.4814	0.712354	0.356177	+	0.934419i	$0.384080\pi$
$308$	−5.69478	−0.324490
$309$	0	0
$310$	−8.21680	−0.466683
$311$	−22.5168	−1.27681	−0.638406	−	0.769700i	$-0.720406\pi$
$311$	−22.5168	−1.27681	−0.638406	+	0.769700i	$0.720406\pi$
$312$	0	0
$313$	18.5868	1.05059	0.525293	−	0.850921i	$-0.323956\pi$
$313$	18.5868	1.05059	0.525293	+	0.850921i	$0.323956\pi$
$314$	−1.54316	−0.0870853
$315$	0	0
$316$	−5.78907	−0.325661
$317$	6.68582	0.375513	0.187757	−	0.982216i	$-0.439878\pi$
$317$	6.68582	0.375513	0.187757	+	0.982216i	$0.439878\pi$
$318$	0	0
$319$	−5.55516	−0.311029
$320$	11.8096	0.660174
$321$	0	0
$322$	15.7869	0.879769
$323$	−11.2557	−0.626285
$324$	0	0
$325$	−4.07833	−0.226225
$326$	0.852301	0.0472046
$327$	0	0
$328$	29.3854	1.62254
$329$	1.64150	0.0904986
$330$	0	0
$331$	−7.97359	−0.438268	−0.219134	−	0.975695i	$-0.570323\pi$
$331$	−7.97359	−0.438268	−0.219134	+	0.975695i	$0.570323\pi$
$332$	−8.16694	−0.448219
$333$	0	0
$334$	−4.86678	−0.266298
$335$	−19.3906	−1.05942
$336$	0	0
$337$	4.16890	0.227094	0.113547	−	0.993533i	$-0.463779\pi$
$337$	4.16890	0.227094	0.113547	+	0.993533i	$0.463779\pi$
$338$	−0.728722	−0.0396373
$339$	0	0
$340$	−10.7630	−0.583708
$341$	4.45726	0.241374
$342$	0	0
$343$	57.7366	3.11748
$344$	−27.1520	−1.46394
$345$	0	0
$346$	3.95764	0.212764
$347$	15.6505	0.840162	0.420081	−	0.907487i	$-0.362002\pi$
$347$	15.6505	0.840162	0.420081	+	0.907487i	$0.362002\pi$
$348$	0	0
$349$	−19.4273	−1.03992	−0.519961	−	0.854190i	$-0.674053\pi$
$349$	−19.4273	−1.03992	−0.519961	+	0.854190i	$0.674053\pi$
$350$	5.17223	0.276467
$351$	0	0
$352$	−5.40408	−0.288038
$353$	6.60191	0.351384	0.175692	−	0.984445i	$-0.443784\pi$
$353$	6.60191	0.351384	0.175692	+	0.984445i	$0.443784\pi$
$354$	0	0
$355$	11.9441	0.633925
$356$	−5.34305	−0.283181
$357$	0	0
$358$	8.56176	0.452503
$359$	−7.32379	−0.386535	−0.193267	−	0.981146i	$-0.561909\pi$
$359$	−7.32379	−0.386535	−0.193267	+	0.981146i	$0.561909\pi$
$360$	0	0
$361$	−13.5621	−0.713793
$362$	−9.36089	−0.491998
$363$	0	0
$364$	−21.1404	−1.10806
$365$	19.8852	1.04084
$366$	0	0
$367$	−25.8165	−1.34761	−0.673805	−	0.738909i	$-0.735341\pi$
$367$	−25.8165	−1.34761	−0.673805	+	0.738909i	$0.735341\pi$
$368$	1.56809	0.0817421
$369$	0	0
$370$	−9.61850	−0.500042
$371$	−12.9098	−0.670241
$372$	0	0
$373$	−7.09447	−0.367338	−0.183669	−	0.982988i	$-0.558797\pi$
$373$	−7.09447	−0.367338	−0.183669	+	0.982988i	$0.558797\pi$
$374$	−4.50486	−0.232941
$375$	0	0
$376$	0.950285	0.0490072
$377$	−20.6222	−1.06209
$378$	0	0
$379$	10.6368	0.546374	0.273187	−	0.961961i	$-0.411922\pi$
$379$	10.6368	0.546374	0.273187	+	0.961961i	$0.411922\pi$
$380$	5.19991	0.266750
$381$	0	0
$382$	20.3537	1.04138
$383$	−27.2686	−1.39336	−0.696680	−	0.717382i	$-0.745340\pi$
$383$	−27.2686	−1.39336	−0.696680	+	0.717382i	$0.745340\pi$
$384$	0	0
$385$	9.96362	0.507793
$386$	4.18966	0.213248
$387$	0	0
$388$	0.624396	0.0316989
$389$	−16.5592	−0.839585	−0.419793	−	0.907620i	$-0.637897\pi$
$389$	−16.5592	−0.839585	−0.419793	+	0.907620i	$0.637897\pi$
$390$	0	0
$391$	−16.1853	−0.818525
$392$	53.8664	2.72066
$393$	0	0
$394$	−24.3176	−1.22510
$395$	10.1286	0.509625
$396$	0	0
$397$	−16.0870	−0.807381	−0.403691	−	0.914896i	$-0.632273\pi$
$397$	−16.0870	−0.807381	−0.403691	+	0.914896i	$0.632273\pi$
$398$	−3.25957	−0.163388
$399$	0	0
$400$	0.513749	0.0256875
$401$	−29.7923	−1.48776	−0.743879	−	0.668314i	$-0.767016\pi$
$401$	−29.7923	−1.48776	−0.743879	+	0.668314i	$0.767016\pi$
$402$	0	0
$403$	16.5465	0.824238
$404$	−0.366378	−0.0182280
$405$	0	0
$406$	26.1535	1.29798
$407$	5.21762	0.258628
$408$	0	0
$409$	28.4589	1.40720	0.703600	−	0.710596i	$-0.251575\pi$
$409$	28.4589	1.40720	0.703600	+	0.710596i	$0.251575\pi$
$410$	−18.5500	−0.916120
$411$	0	0
$412$	20.8558	1.02749
$413$	−19.1061	−0.940152
$414$	0	0
$415$	14.2889	0.701416
$416$	−20.0613	−0.983585
$417$	0	0
$418$	2.17642	0.106452
$419$	2.48607	0.121453	0.0607263	−	0.998154i	$-0.480658\pi$
$419$	2.48607	0.121453	0.0607263	+	0.998154i	$0.480658\pi$
$420$	0	0
$421$	34.8776	1.69983	0.849914	−	0.526921i	$-0.176654\pi$
$421$	34.8776	1.69983	0.849914	+	0.526921i	$0.176654\pi$
$422$	−2.19362	−0.106784
$423$	0	0
$424$	−7.47365	−0.362952
$425$	−5.30275	−0.257221
$426$	0	0
$427$	68.6713	3.32324
$428$	−8.07579	−0.390358
$429$	0	0
$430$	17.1402	0.826572
$431$	29.8719	1.43888	0.719439	−	0.694556i	$-0.244399\pi$
$431$	29.8719	1.43888	0.719439	+	0.694556i	$0.244399\pi$
$432$	0	0
$433$	−0.976527	−0.0469289	−0.0234645	−	0.999725i	$-0.507470\pi$
$433$	−0.976527	−0.0469289	−0.0234645	+	0.999725i	$0.507470\pi$
$434$	−20.9846	−1.00729
$435$	0	0
$436$	−6.02425	−0.288509
$437$	7.81953	0.374059
$438$	0	0
$439$	−14.6860	−0.700925	−0.350462	−	0.936577i	$-0.613976\pi$
$439$	−14.6860	−0.700925	−0.350462	+	0.936577i	$0.613976\pi$
$440$	5.76808	0.274982
$441$	0	0
$442$	−16.7232	−0.795440
$443$	25.6221	1.21734	0.608672	−	0.793422i	$-0.291703\pi$
$443$	25.6221	1.21734	0.608672	+	0.793422i	$0.291703\pi$
$444$	0	0
$445$	9.34824	0.443149
$446$	9.00151	0.426234
$447$	0	0
$448$	30.1600	1.42493
$449$	−27.0326	−1.27575	−0.637873	−	0.770142i	$-0.720185\pi$
$449$	−27.0326	−1.27575	−0.637873	+	0.770142i	$0.720185\pi$
$450$	0	0
$451$	10.0626	0.473829
$452$	20.6649	0.971996
$453$	0	0
$454$	−0.638038	−0.0299446
$455$	36.9874	1.73400
$456$	0	0
$457$	−27.9150	−1.30581	−0.652903	−	0.757441i	$-0.726449\pi$
$457$	−27.9150	−1.30581	−0.652903	+	0.757441i	$0.726449\pi$
$458$	−9.08563	−0.424544
$459$	0	0
$460$	7.47726	0.348629
$461$	−34.1042	−1.58839	−0.794196	−	0.607662i	$-0.792108\pi$
$461$	−34.1042	−1.58839	−0.794196	+	0.607662i	$0.792108\pi$
$462$	0	0
$463$	−21.7200	−1.00941	−0.504707	−	0.863291i	$-0.668399\pi$
$463$	−21.7200	−1.00941	−0.504707	+	0.863291i	$0.668399\pi$
$464$	2.59778	0.120599
$465$	0	0
$466$	12.7755	0.591811
$467$	−28.4314	−1.31565	−0.657824	−	0.753172i	$-0.728523\pi$
$467$	−28.4314	−1.31565	−0.657824	+	0.753172i	$0.728523\pi$
$468$	0	0
$469$	−49.5210	−2.28667
$470$	−0.599883	−0.0276706
$471$	0	0
$472$	−11.0608	−0.509115
$473$	−9.29780	−0.427513
$474$	0	0
$475$	2.56190	0.117548
$476$	−27.4874	−1.25988
$477$	0	0
$478$	23.3012	1.06577
$479$	2.59173	0.118419	0.0592097	−	0.998246i	$-0.481142\pi$
$479$	2.59173	0.118419	0.0592097	+	0.998246i	$0.481142\pi$
$480$	0	0
$481$	19.3691	0.883156
$482$	−25.6008	−1.16609
$483$	0	0
$484$	−1.12894	−0.0513152
$485$	−1.09245	−0.0496054
$486$	0	0
$487$	22.5441	1.02157	0.510785	−	0.859708i	$-0.329355\pi$
$487$	22.5441	1.02157	0.510785	+	0.859708i	$0.329355\pi$
$488$	39.7548	1.79961
$489$	0	0
$490$	−34.0041	−1.53615
$491$	26.9579	1.21659	0.608297	−	0.793710i	$-0.291853\pi$
$491$	26.9579	1.21659	0.608297	+	0.793710i	$0.291853\pi$
$492$	0	0
$493$	−26.8135	−1.20762
$494$	8.07940	0.363509
$495$	0	0
$496$	−2.08437	−0.0935908
$497$	30.5035	1.36827
$498$	0	0
$499$	39.8326	1.78315	0.891575	−	0.452872i	$-0.149601\pi$
$499$	39.8326	1.78315	0.891575	+	0.452872i	$0.149601\pi$
$500$	13.5991	0.608170
$501$	0	0
$502$	11.7803	0.525778
$503$	30.9243	1.37885	0.689424	−	0.724358i	$-0.257864\pi$
$503$	30.9243	1.37885	0.689424	+	0.724358i	$0.257864\pi$
$504$	0	0
$505$	0.641018	0.0285249
$506$	3.12960	0.139128
$507$	0	0
$508$	−7.96024	−0.353179
$509$	−8.66460	−0.384052	−0.192026	−	0.981390i	$-0.561506\pi$
$509$	−8.66460	−0.384052	−0.192026	+	0.981390i	$0.561506\pi$
$510$	0	0
$511$	50.7840	2.24655
$512$	5.25836	0.232389
$513$	0	0
$514$	6.32861	0.279143
$515$	−36.4895	−1.60792
$516$	0	0
$517$	0.325411	0.0143116
$518$	−24.5644	−1.07930
$519$	0	0
$520$	21.4125	0.939002
$521$	−21.4666	−0.940467	−0.470234	−	0.882542i	$-0.655830\pi$
$521$	−21.4666	−0.940467	−0.470234	+	0.882542i	$0.655830\pi$
$522$	0	0
$523$	14.5974	0.638298	0.319149	−	0.947705i	$-0.396603\pi$
$523$	14.5974	0.638298	0.319149	+	0.947705i	$0.396603\pi$
$524$	17.0263	0.743798
$525$	0	0
$526$	12.6618	0.552080
$527$	21.5142	0.937172
$528$	0	0
$529$	−11.7558	−0.511123
$530$	4.71786	0.204931
$531$	0	0
$532$	13.2799	0.575755
$533$	37.3548	1.61802
$534$	0	0
$535$	14.1294	0.610869
$536$	−28.6684	−1.23829
$537$	0	0
$538$	−16.7993	−0.724269
$539$	18.4457	0.794514
$540$	0	0
$541$	−1.44041	−0.0619280	−0.0309640	−	0.999520i	$-0.509858\pi$
$541$	−1.44041	−0.0619280	−0.0309640	+	0.999520i	$0.509858\pi$
$542$	−6.80783	−0.292421
$543$	0	0
$544$	−26.0842	−1.11835
$545$	10.5401	0.451487
$546$	0	0
$547$	−11.2858	−0.482547	−0.241274	−	0.970457i	$-0.577565\pi$
$547$	−11.2858	−0.482547	−0.241274	+	0.970457i	$0.577565\pi$
$548$	−2.67047	−0.114077
$549$	0	0
$550$	1.02535	0.0437209
$551$	12.9543	0.551871
$552$	0	0
$553$	25.8671	1.09998
$554$	−22.3927	−0.951376
$555$	0	0
$556$	−15.6366	−0.663140
$557$	18.6443	0.789984	0.394992	−	0.918685i	$-0.370747\pi$
$557$	18.6443	0.789984	0.394992	+	0.918685i	$0.370747\pi$
$558$	0	0
$559$	−34.5157	−1.45986
$560$	−4.65932	−0.196892
$561$	0	0
$562$	10.5390	0.444563
$563$	−14.4314	−0.608209	−0.304105	−	0.952639i	$-0.598357\pi$
$563$	−14.4314	−0.608209	−0.304105	+	0.952639i	$0.598357\pi$
$564$	0	0
$565$	−36.1555	−1.52107
$566$	−4.85325	−0.203997
$567$	0	0
$568$	17.6589	0.740952
$569$	10.3633	0.434452	0.217226	−	0.976121i	$-0.430299\pi$
$569$	10.3633	0.434452	0.217226	+	0.976121i	$0.430299\pi$
$570$	0	0
$571$	9.66990	0.404673	0.202336	−	0.979316i	$-0.435147\pi$
$571$	9.66990	0.404673	0.202336	+	0.979316i	$0.435147\pi$
$572$	−4.19089	−0.175230
$573$	0	0
$574$	−47.3742	−1.97736
$575$	3.68391	0.153630
$576$	0	0
$577$	−35.1061	−1.46149	−0.730743	−	0.682653i	$-0.760826\pi$
$577$	−35.1061	−1.46149	−0.730743	+	0.682653i	$0.760826\pi$
$578$	−5.87769	−0.244480
$579$	0	0
$580$	12.3873	0.514353
$581$	36.4920	1.51394
$582$	0	0
$583$	−2.55924	−0.105993
$584$	29.3996	1.21656
$585$	0	0
$586$	4.49089	0.185517
$587$	19.3964	0.800574	0.400287	−	0.916390i	$-0.368910\pi$
$587$	19.3964	0.800574	0.400287	+	0.916390i	$0.368910\pi$
$588$	0	0
$589$	−10.3941	−0.428280
$590$	6.98232	0.287458
$591$	0	0
$592$	−2.43994	−0.100281
$593$	0.316042	0.0129783	0.00648914	−	0.999979i	$-0.497934\pi$
$593$	0.316042	0.0129783	0.00648914	+	0.999979i	$0.497934\pi$
$594$	0	0
$595$	48.0921	1.97158
$596$	8.48950	0.347744
$597$	0	0
$598$	11.6179	0.475090
$599$	−2.19220	−0.0895707	−0.0447854	−	0.998997i	$-0.514260\pi$
$599$	−2.19220	−0.0895707	−0.0447854	+	0.998997i	$0.514260\pi$
$600$	0	0
$601$	−24.2863	−0.990657	−0.495329	−	0.868706i	$-0.664952\pi$
$601$	−24.2863	−0.990657	−0.495329	+	0.868706i	$0.664952\pi$
$602$	43.7737	1.78408
$603$	0	0
$604$	0.183303	0.00745850
$605$	1.97519	0.0803030
$606$	0	0
$607$	15.7943	0.641073	0.320536	−	0.947236i	$-0.396137\pi$
$607$	15.7943	0.641073	0.320536	+	0.947236i	$0.396137\pi$
$608$	12.6020	0.511077
$609$	0	0
$610$	−25.0959	−1.01610
$611$	1.20801	0.0488707
$612$	0	0
$613$	42.4197	1.71332	0.856658	−	0.515884i	$-0.172537\pi$
$613$	42.4197	1.71332	0.856658	+	0.515884i	$0.172537\pi$
$614$	−11.6490	−0.470117
$615$	0	0
$616$	14.7309	0.593525
$617$	−23.4043	−0.942222	−0.471111	−	0.882074i	$-0.656147\pi$
$617$	−23.4043	−0.942222	−0.471111	+	0.882074i	$0.656147\pi$
$618$	0	0
$619$	−14.9103	−0.599297	−0.299648	−	0.954050i	$-0.596869\pi$
$619$	−14.9103	−0.599297	−0.299648	+	0.954050i	$0.596869\pi$
$620$	−9.93909	−0.399164
$621$	0	0
$622$	21.0151	0.842630
$623$	23.8742	0.956498
$624$	0	0
$625$	−18.3000	−0.731999
$626$	−17.3472	−0.693333
$627$	0	0
$628$	−1.86661	−0.0744859
$629$	25.1843	1.00416
$630$	0	0
$631$	45.4694	1.81011	0.905053	−	0.425298i	$-0.139831\pi$
$631$	45.4694	1.81011	0.905053	+	0.425298i	$0.139831\pi$
$632$	14.9748	0.595666
$633$	0	0
$634$	−6.23993	−0.247819
$635$	13.9273	0.552688
$636$	0	0
$637$	68.4752	2.71309
$638$	5.18468	0.205263
$639$	0	0
$640$	10.3262	0.408180
$641$	8.85876	0.349900	0.174950	−	0.984577i	$-0.444024\pi$
$641$	8.85876	0.349900	0.174950	+	0.984577i	$0.444024\pi$
$642$	0	0
$643$	4.41396	0.174069	0.0870347	−	0.996205i	$-0.472261\pi$
$643$	4.41396	0.174069	0.0870347	+	0.996205i	$0.472261\pi$
$644$	19.0959	0.752485
$645$	0	0
$646$	10.5051	0.413316
$647$	−38.4336	−1.51098	−0.755491	−	0.655159i	$-0.772602\pi$
$647$	−38.4336	−1.51098	−0.755491	+	0.655159i	$0.772602\pi$
$648$	0	0
$649$	−3.78761	−0.148677
$650$	3.80634	0.149297
$651$	0	0
$652$	1.03095	0.0403750
$653$	37.1731	1.45469	0.727347	−	0.686270i	$-0.240753\pi$
$653$	37.1731	1.45469	0.727347	+	0.686270i	$0.240753\pi$
$654$	0	0
$655$	−29.7893	−1.16397
$656$	−4.70561	−0.183723
$657$	0	0
$658$	−1.53202	−0.0597244
$659$	−9.03745	−0.352049	−0.176024	−	0.984386i	$-0.556324\pi$
$659$	−9.03745	−0.352049	−0.176024	+	0.984386i	$0.556324\pi$
$660$	0	0
$661$	48.7779	1.89724	0.948621	−	0.316415i	$-0.102479\pi$
$661$	48.7779	1.89724	0.948621	+	0.316415i	$0.102479\pi$
$662$	7.44182	0.289235
$663$	0	0
$664$	21.1258	0.819838
$665$	−23.2345	−0.900996
$666$	0	0
$667$	18.6278	0.721270
$668$	−5.88689	−0.227771
$669$	0	0
$670$	18.0974	0.699164
$671$	13.6134	0.525541
$672$	0	0
$673$	−28.1173	−1.08384	−0.541920	−	0.840430i	$-0.682303\pi$
$673$	−28.1173	−1.08384	−0.541920	+	0.840430i	$0.682303\pi$
$674$	−3.89087	−0.149871
$675$	0	0
$676$	−0.881467	−0.0339026
$677$	16.8699	0.648361	0.324181	−	0.945995i	$-0.394911\pi$
$677$	16.8699	0.648361	0.324181	+	0.945995i	$0.394911\pi$
$678$	0	0
$679$	−2.78996	−0.107069
$680$	27.8412	1.06766
$681$	0	0
$682$	−4.16000	−0.159295
$683$	37.7780	1.44553	0.722767	−	0.691091i	$-0.242870\pi$
$683$	37.7780	1.44553	0.722767	+	0.691091i	$0.242870\pi$
$684$	0	0
$685$	4.67228	0.178519
$686$	−53.8861	−2.05738
$687$	0	0
$688$	4.34797	0.165765
$689$	−9.50053	−0.361941
$690$	0	0
$691$	14.2682	0.542790	0.271395	−	0.962468i	$-0.412515\pi$
$691$	14.2682	0.542790	0.271395	+	0.962468i	$0.412515\pi$
$692$	4.78718	0.181981
$693$	0	0
$694$	−14.6067	−0.554464
$695$	27.3579	1.03774
$696$	0	0
$697$	48.5698	1.83971
$698$	18.1317	0.686295
$699$	0	0
$700$	6.25636	0.236468
$701$	42.4821	1.60453	0.802263	−	0.596970i	$-0.203629\pi$
$701$	42.4821	1.60453	0.802263	+	0.596970i	$0.203629\pi$
$702$	0	0
$703$	−12.1672	−0.458893
$704$	5.97894	0.225340
$705$	0	0
$706$	−6.16161	−0.231895
$707$	1.63707	0.0615685
$708$	0	0
$709$	28.1025	1.05541	0.527706	−	0.849427i	$-0.323052\pi$
$709$	28.1025	1.05541	0.527706	+	0.849427i	$0.323052\pi$
$710$	−11.1475	−0.418358
$711$	0	0
$712$	13.8211	0.517967
$713$	−14.9462	−0.559741
$714$	0	0
$715$	7.33241	0.274216
$716$	10.3564	0.387035
$717$	0	0
$718$	6.83536	0.255093
$719$	25.0424	0.933924	0.466962	−	0.884277i	$-0.345348\pi$
$719$	25.0424	0.933924	0.466962	+	0.884277i	$0.345348\pi$
$720$	0	0
$721$	−93.1893	−3.47055
$722$	12.6576	0.471067
$723$	0	0
$724$	−11.3230	−0.420816
$725$	6.10298	0.226659
$726$	0	0
$727$	−22.2031	−0.823467	−0.411734	−	0.911304i	$-0.635077\pi$
$727$	−22.2031	−0.823467	−0.411734	+	0.911304i	$0.635077\pi$
$728$	54.6848	2.02675
$729$	0	0
$730$	−18.5590	−0.686899
$731$	−44.8783	−1.65988
$732$	0	0
$733$	25.0090	0.923728	0.461864	−	0.886951i	$-0.347181\pi$
$733$	25.0090	0.923728	0.461864	+	0.886951i	$0.347181\pi$
$734$	24.0948	0.889354
$735$	0	0
$736$	18.1211	0.667954
$737$	−9.81707	−0.361617
$738$	0	0
$739$	19.0355	0.700231	0.350116	−	0.936707i	$-0.386142\pi$
$739$	19.0355	0.700231	0.350116	+	0.936707i	$0.386142\pi$
$740$	−11.6346	−0.427696
$741$	0	0
$742$	12.0488	0.442325
$743$	23.5630	0.864442	0.432221	−	0.901768i	$-0.357730\pi$
$743$	23.5630	0.864442	0.432221	+	0.901768i	$0.357730\pi$
$744$	0	0
$745$	−14.8533	−0.544182
$746$	6.62133	0.242424
$747$	0	0
$748$	−5.44911	−0.199239
$749$	36.0847	1.31851
$750$	0	0
$751$	−33.2359	−1.21279	−0.606397	−	0.795162i	$-0.707386\pi$
$751$	−33.2359	−1.21279	−0.606397	+	0.795162i	$0.707386\pi$
$752$	−0.152173	−0.00554918
$753$	0	0
$754$	19.2468	0.700928
$755$	−0.320708	−0.0116718
$756$	0	0
$757$	−10.5580	−0.383736	−0.191868	−	0.981421i	$-0.561455\pi$
$757$	−10.5580	−0.383736	−0.191868	+	0.981421i	$0.561455\pi$
$758$	−9.92738	−0.360579
$759$	0	0
$760$	−13.4508	−0.487912
$761$	−47.8352	−1.73402	−0.867012	−	0.498288i	$-0.833962\pi$
$761$	−47.8352	−1.73402	−0.867012	+	0.498288i	$0.833962\pi$
$762$	0	0
$763$	26.9179	0.974494
$764$	24.6199	0.890718
$765$	0	0
$766$	25.4500	0.919546
$767$	−14.0606	−0.507697
$768$	0	0
$769$	−39.2064	−1.41382	−0.706909	−	0.707305i	$-0.749911\pi$
$769$	−39.2064	−1.41382	−0.706909	+	0.707305i	$0.749911\pi$
$770$	−9.29913	−0.335117
$771$	0	0
$772$	5.06784	0.182396
$773$	−8.80583	−0.316724	−0.158362	−	0.987381i	$-0.550621\pi$
$773$	−8.80583	−0.316724	−0.158362	+	0.987381i	$0.550621\pi$
$774$	0	0
$775$	−4.89681	−0.175899
$776$	−1.61515	−0.0579805
$777$	0	0
$778$	15.4548	0.554083
$779$	−23.4653	−0.840732
$780$	0	0
$781$	6.04703	0.216380
$782$	15.1059	0.540185
$783$	0	0
$784$	−8.62585	−0.308066
$785$	3.26583	0.116563
$786$	0	0
$787$	−6.04164	−0.215361	−0.107681	−	0.994186i	$-0.534342\pi$
$787$	−6.04164	−0.215361	−0.107681	+	0.994186i	$0.534342\pi$
$788$	−29.4148	−1.04786
$789$	0	0
$790$	−9.45310	−0.336326
$791$	−92.3362	−3.28310
$792$	0	0
$793$	50.5365	1.79460
$794$	15.0141	0.532830
$795$	0	0
$796$	−3.94280	−0.139749
$797$	−1.17712	−0.0416957	−0.0208479	−	0.999783i	$-0.506637\pi$
$797$	−1.17712	−0.0416957	−0.0208479	+	0.999783i	$0.506637\pi$
$798$	0	0
$799$	1.57068	0.0555668
$800$	5.93699	0.209904
$801$	0	0
$802$	27.8054	0.981844
$803$	10.0675	0.355273
$804$	0	0
$805$	−33.4103	−1.17756
$806$	−15.4430	−0.543955
$807$	0	0
$808$	0.947725	0.0333408
$809$	14.8940	0.523644	0.261822	−	0.965116i	$-0.415677\pi$
$809$	14.8940	0.523644	0.261822	+	0.965116i	$0.415677\pi$
$810$	0	0
$811$	−3.18288	−0.111766	−0.0558831	−	0.998437i	$-0.517797\pi$
$811$	−3.18288	−0.111766	−0.0558831	+	0.998437i	$0.517797\pi$
$812$	31.6354	1.11019
$813$	0	0
$814$	−4.86965	−0.170681
$815$	−1.80375	−0.0631827
$816$	0	0
$817$	21.6819	0.758553
$818$	−26.5609	−0.928680
$819$	0	0
$820$	−22.4382	−0.783576
$821$	27.2844	0.952231	0.476116	−	0.879383i	$-0.342044\pi$
$821$	27.2844	0.952231	0.476116	+	0.879383i	$0.342044\pi$
$822$	0	0
$823$	−15.2379	−0.531159	−0.265579	−	0.964089i	$-0.585563\pi$
$823$	−15.2379	−0.531159	−0.265579	+	0.964089i	$0.585563\pi$
$824$	−53.9486	−1.87939
$825$	0	0
$826$	17.8319	0.620452
$827$	−51.9803	−1.80753	−0.903766	−	0.428026i	$-0.859209\pi$
$827$	−51.9803	−1.80753	−0.903766	+	0.428026i	$0.859209\pi$
$828$	0	0
$829$	−36.9400	−1.28298	−0.641490	−	0.767131i	$-0.721684\pi$
$829$	−36.9400	−1.28298	−0.641490	+	0.767131i	$0.721684\pi$
$830$	−13.3360	−0.462899
$831$	0	0
$832$	22.1953	0.769484
$833$	89.0333	3.08482
$834$	0	0
$835$	10.2997	0.356437
$836$	2.63261	0.0910506
$837$	0	0
$838$	−2.32027	−0.0801525
$839$	10.3756	0.358206	0.179103	−	0.983830i	$-0.442680\pi$
$839$	10.3756	0.358206	0.179103	+	0.983830i	$0.442680\pi$
$840$	0	0
$841$	1.85984	0.0641324
$842$	−32.5515	−1.12180
$843$	0	0
$844$	−2.65342	−0.0913343
$845$	1.54222	0.0530540
$846$	0	0
$847$	5.04438	0.173327
$848$	1.19679	0.0410978
$849$	0	0
$850$	4.94911	0.169753
$851$	−17.4959	−0.599752
$852$	0	0
$853$	−42.2887	−1.44794	−0.723969	−	0.689832i	$-0.757684\pi$
$853$	−42.2887	−1.44794	−0.723969	+	0.689832i	$0.757684\pi$
$854$	−64.0915	−2.19317
$855$	0	0
$856$	20.8900	0.714004
$857$	−28.4199	−0.970804	−0.485402	−	0.874291i	$-0.661327\pi$
$857$	−28.4199	−0.970804	−0.485402	+	0.874291i	$0.661327\pi$
$858$	0	0
$859$	−24.5275	−0.836867	−0.418434	−	0.908247i	$-0.637421\pi$
$859$	−24.5275	−0.836867	−0.418434	+	0.908247i	$0.637421\pi$
$860$	20.7328	0.706984
$861$	0	0
$862$	−27.8797	−0.949585
$863$	12.7120	0.432720	0.216360	−	0.976314i	$-0.430581\pi$
$863$	12.7120	0.432720	0.216360	+	0.976314i	$0.430581\pi$
$864$	0	0
$865$	−8.37568	−0.284782
$866$	0.911401	0.0309707
$867$	0	0
$868$	−25.3831	−0.861559
$869$	5.12790	0.173952
$870$	0	0
$871$	−36.4434	−1.23484
$872$	15.5832	0.527713
$873$	0	0
$874$	−7.29803	−0.246860
$875$	−60.7642	−2.05421
$876$	0	0
$877$	19.7930	0.668362	0.334181	−	0.942509i	$-0.391540\pi$
$877$	19.7930	0.668362	0.334181	+	0.942509i	$0.391540\pi$
$878$	13.7066	0.462574
$879$	0	0
$880$	−0.923667	−0.0311368
$881$	−6.59893	−0.222324	−0.111162	−	0.993802i	$-0.535457\pi$
$881$	−6.59893	−0.222324	−0.111162	+	0.993802i	$0.535457\pi$
$882$	0	0
$883$	49.5752	1.66834	0.834169	−	0.551509i	$-0.185948\pi$
$883$	49.5752	1.66834	0.834169	+	0.551509i	$0.185948\pi$
$884$	−20.2285	−0.680357
$885$	0	0
$886$	−23.9133	−0.803385
$887$	−10.5478	−0.354162	−0.177081	−	0.984196i	$-0.556665\pi$
$887$	−10.5478	−0.354162	−0.177081	+	0.984196i	$0.556665\pi$
$888$	0	0
$889$	35.5684	1.19293
$890$	−8.72479	−0.292456
$891$	0	0
$892$	10.8883	0.364567
$893$	−0.758838	−0.0253935
$894$	0	0
$895$	−18.1195	−0.605670
$896$	26.3718	0.881020
$897$	0	0
$898$	25.2297	0.841926
$899$	−24.7608	−0.825820
$900$	0	0
$901$	−12.3529	−0.411533
$902$	−9.39150	−0.312703
$903$	0	0
$904$	−53.4547	−1.77788
$905$	19.8108	0.658533
$906$	0	0
$907$	−39.4323	−1.30933	−0.654663	−	0.755920i	$-0.727190\pi$
$907$	−39.4323	−1.30933	−0.654663	+	0.755920i	$0.727190\pi$
$908$	−0.771774	−0.0256122
$909$	0	0
$910$	−34.5207	−1.14435
$911$	34.9264	1.15716	0.578582	−	0.815624i	$-0.303606\pi$
$911$	34.9264	1.15716	0.578582	+	0.815624i	$0.303606\pi$
$912$	0	0
$913$	7.23420	0.239417
$914$	26.0533	0.861765
$915$	0	0
$916$	−10.9900	−0.363121
$917$	−76.0780	−2.51232
$918$	0	0
$919$	−32.9888	−1.08820	−0.544100	−	0.839021i	$-0.683129\pi$
$919$	−32.9888	−1.08820	−0.544100	+	0.839021i	$0.683129\pi$
$920$	−19.3417	−0.637678
$921$	0	0
$922$	31.8298	1.04826
$923$	22.4481	0.738888
$924$	0	0
$925$	−5.73215	−0.188472
$926$	20.2714	0.666161
$927$	0	0
$928$	30.0205	0.985472
$929$	37.0549	1.21573	0.607867	−	0.794039i	$-0.292026\pi$
$929$	37.0549	1.21573	0.607867	+	0.794039i	$0.292026\pi$
$930$	0	0
$931$	−43.0143	−1.40974
$932$	15.4533	0.506188
$933$	0	0
$934$	26.5353	0.868260
$935$	9.53380	0.311789
$936$	0	0
$937$	−3.00751	−0.0982511	−0.0491255	−	0.998793i	$-0.515643\pi$
$937$	−3.00751	−0.0982511	−0.0491255	+	0.998793i	$0.515643\pi$
$938$	46.2184	1.50908
$939$	0	0
$940$	−0.725622	−0.0236672
$941$	−22.9417	−0.747877	−0.373939	−	0.927453i	$-0.621993\pi$
$941$	−22.9417	−0.747877	−0.373939	+	0.927453i	$0.621993\pi$
$942$	0	0
$943$	−33.7422	−1.09880
$944$	1.77122	0.0576481
$945$	0	0
$946$	8.67771	0.282137
$947$	−17.1833	−0.558383	−0.279191	−	0.960235i	$-0.590066\pi$
$947$	−17.1833	−0.558383	−0.279191	+	0.960235i	$0.590066\pi$
$948$	0	0
$949$	37.3729	1.21318
$950$	−2.39104	−0.0775756
$951$	0	0
$952$	71.1026	2.30445
$953$	−24.7412	−0.801446	−0.400723	−	0.916199i	$-0.631241\pi$
$953$	−24.7412	−0.801446	−0.400723	+	0.916199i	$0.631241\pi$
$954$	0	0
$955$	−43.0752	−1.39388
$956$	28.1853	0.911577
$957$	0	0
$958$	−2.41889	−0.0781507
$959$	11.9324	0.385316
$960$	0	0
$961$	−11.1328	−0.359123
$962$	−18.0774	−0.582837
$963$	0	0
$964$	−30.9669	−0.997377
$965$	−8.86673	−0.285430
$966$	0	0
$967$	−27.9460	−0.898684	−0.449342	−	0.893360i	$-0.648342\pi$
$967$	−27.9460	−0.898684	−0.449342	+	0.893360i	$0.648342\pi$
$968$	2.92026	0.0938608
$969$	0	0
$970$	1.01959	0.0327371
$971$	−13.2398	−0.424886	−0.212443	−	0.977173i	$-0.568142\pi$
$971$	−13.2398	−0.424886	−0.212443	+	0.977173i	$0.568142\pi$
$972$	0	0
$973$	69.8684	2.23988
$974$	−21.0406	−0.674184
$975$	0	0
$976$	−6.36610	−0.203774
$977$	43.0393	1.37695	0.688475	−	0.725260i	$-0.258280\pi$
$977$	43.0393	1.37695	0.688475	+	0.725260i	$0.258280\pi$
$978$	0	0
$979$	4.73283	0.151262
$980$	−41.1315	−1.31390
$981$	0	0
$982$	−25.1601	−0.802890
$983$	20.2817	0.646886	0.323443	−	0.946248i	$-0.395160\pi$
$983$	20.2817	0.646886	0.323443	+	0.946248i	$0.395160\pi$
$984$	0	0
$985$	51.4643	1.63979
$986$	25.0253	0.796966
$987$	0	0
$988$	9.77289	0.310917
$989$	31.1777	0.991393
$990$	0	0
$991$	−7.90004	−0.250953	−0.125476	−	0.992097i	$-0.540046\pi$
$991$	−7.90004	−0.250953	−0.125476	+	0.992097i	$0.540046\pi$
$992$	−24.0874	−0.764775
$993$	0	0
$994$	−28.4692	−0.902988
$995$	6.89834	0.218692
$996$	0	0
$997$	6.62859	0.209929	0.104965	−	0.994476i	$-0.466527\pi$
$997$	6.62859	0.209929	0.104965	+	0.994476i	$0.466527\pi$
$998$	−37.1761	−1.17679
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8019.2.a.l.1.19		51
3.2	odd	2		8019.2.a.k.1.33		51
27.2	odd	18		891.2.j.c.793.7		102
27.13	even	9		297.2.j.c.34.11	✓	102
27.14	odd	18		891.2.j.c.100.7		102
27.25	even	9		297.2.j.c.166.11	yes	102

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.j.c.34.11	✓	102	27.13	even	9
297.2.j.c.166.11	yes	102	27.25	even	9
891.2.j.c.100.7		102	27.14	odd	18
891.2.j.c.793.7		102	27.2	odd	18
8019.2.a.k.1.33		51	3.2	odd	2
8019.2.a.l.1.19		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)

\(f(q)\)	\(=\)	\(q-0.933308 q^{2} -1.12894 q^{4} +1.97519 q^{5} +5.04438 q^{7} +2.92026 q^{8} +O(q^{10})\)	\(q-0.933308 q^{2} -1.12894 q^{4} +1.97519 q^{5} +5.04438 q^{7} +2.92026 q^{8} -1.84346 q^{10} +1.00000 q^{11} +3.71225 q^{13} -4.70796 q^{14} -0.467634 q^{16} +4.82677 q^{17} -2.33194 q^{19} -2.22987 q^{20} -0.933308 q^{22} -3.35323 q^{23} -1.09861 q^{25} -3.46467 q^{26} -5.69478 q^{28} -5.55516 q^{29} +4.45726 q^{31} -5.40408 q^{32} -4.50486 q^{34} +9.96362 q^{35} +5.21762 q^{37} +2.17642 q^{38} +5.76808 q^{40} +10.0626 q^{41} -9.29780 q^{43} -1.12894 q^{44} +3.12960 q^{46} +0.325411 q^{47} +18.4457 q^{49} +1.02535 q^{50} -4.19089 q^{52} -2.55924 q^{53} +1.97519 q^{55} +14.7309 q^{56} +5.18468 q^{58} -3.78761 q^{59} +13.6134 q^{61} -4.16000 q^{62} +5.97894 q^{64} +7.33241 q^{65} -9.81707 q^{67} -5.44911 q^{68} -9.29913 q^{70} +6.04703 q^{71} +10.0675 q^{73} -4.86965 q^{74} +2.63261 q^{76} +5.04438 q^{77} +5.12790 q^{79} -0.923667 q^{80} -9.39150 q^{82} +7.23420 q^{83} +9.53380 q^{85} +8.67771 q^{86} +2.92026 q^{88} +4.73283 q^{89} +18.7260 q^{91} +3.78558 q^{92} -0.303709 q^{94} -4.60603 q^{95} -0.553084 q^{97} -17.2156 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 51 q + 60 q^{4} + 6 q^{5} + 12 q^{7}+O(q^{10}) \) 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

Fields

Representations

Groups

Database

Properties

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