LMFDB - Embedded newform 2299.2.a.p.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2299,2,Mod(1,2299)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2299.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2299, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2299 = 11^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2299.a (trivial)

Newform invariants

comment:select newform

sage:traces = [7,-2,-1,6,-1,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(6)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$18.3576074247$
Analytic rank:	$1$
Dimension:	$7$
Coefficient field:	$\mathbb{Q}[x]/(x^{7} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{7} - 2x^{6} - 8x^{5} + 14x^{4} + 17x^{3} - 18x^{2} - 16x - 1 $ x^7 - 2x^6 - 8x^5 + 14x^4 + 17x^3 - 18x^2 - 16x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$-1.09476$ of defining polynomial
Character	$\chi$	$=$	2299.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.09476 q^{2} -0.248246 q^{3} -0.801502 q^{4} +1.63186 q^{5} -0.271770 q^{6} +4.48973 q^{7} -3.06697 q^{8} -2.93837 q^{9} +1.78650 q^{10} +0.198970 q^{12} -5.57331 q^{13} +4.91517 q^{14} -0.405103 q^{15} -1.75459 q^{16} -5.40998 q^{17} -3.21681 q^{18} +1.00000 q^{19} -1.30794 q^{20} -1.11456 q^{21} -5.02103 q^{23} +0.761363 q^{24} -2.33703 q^{25} -6.10144 q^{26} +1.47418 q^{27} -3.59853 q^{28} -0.639772 q^{29} -0.443491 q^{30} -10.9928 q^{31} +4.21309 q^{32} -5.92262 q^{34} +7.32662 q^{35} +2.35511 q^{36} +2.01891 q^{37} +1.09476 q^{38} +1.38355 q^{39} -5.00487 q^{40} +9.64435 q^{41} -1.22017 q^{42} -7.62145 q^{43} -4.79502 q^{45} -5.49681 q^{46} -0.859862 q^{47} +0.435570 q^{48} +13.1577 q^{49} -2.55848 q^{50} +1.34301 q^{51} +4.46702 q^{52} +1.31006 q^{53} +1.61387 q^{54} -13.7699 q^{56} -0.248246 q^{57} -0.700397 q^{58} +5.59646 q^{59} +0.324691 q^{60} +5.11170 q^{61} -12.0344 q^{62} -13.1925 q^{63} +8.12150 q^{64} -9.09488 q^{65} +5.13847 q^{67} +4.33611 q^{68} +1.24645 q^{69} +8.02088 q^{70} +0.467163 q^{71} +9.01191 q^{72} -6.37081 q^{73} +2.21022 q^{74} +0.580158 q^{75} -0.801502 q^{76} +1.51466 q^{78} -12.2372 q^{79} -2.86325 q^{80} +8.44916 q^{81} +10.5582 q^{82} -8.18208 q^{83} +0.893320 q^{84} -8.82833 q^{85} -8.34366 q^{86} +0.158821 q^{87} -13.7128 q^{89} -5.24939 q^{90} -25.0227 q^{91} +4.02436 q^{92} +2.72891 q^{93} -0.941342 q^{94} +1.63186 q^{95} -1.04588 q^{96} +14.6114 q^{97} +14.4045 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 7 q - 2 q^{2} - q^{3} + 6 q^{4} - q^{5} - 2 q^{7} - 6 q^{8} + 4 q^{9} + 3 q^{10} - 9 q^{12} - 12 q^{13} + 3 q^{14} + 5 q^{15} - 12 q^{17} + 3 q^{18} + 7 q^{19} - 17 q^{20} - 9 q^{21} - 9 q^{23} - 27 q^{24}+ \cdots - 9 q^{98}+O(q^{100}) $ 7 * q - 2 * q^2 - q^3 + 6 * q^4 - q^5 - 2 * q^7 - 6 * q^8 + 4 * q^9 + 3 * q^10 - 9 * q^12 - 12 * q^13 + 3 * q^14 + 5 * q^15 - 12 * q^17 + 3 * q^18 + 7 * q^19 - 17 * q^20 - 9 * q^21 - 9 * q^23 - 27 * q^24 + 4 * q^25 + 18 * q^26 + 23 * q^27 + 10 * q^28 - 18 * q^29 - 4 * q^30 - 5 * q^31 + 16 * q^32 - 6 * q^34 - 20 * q^35 - 2 * q^36 + 4 * q^37 - 2 * q^38 - 21 * q^39 - 17 * q^40 - 25 * q^41 - 19 * q^42 + 10 * q^43 - 16 * q^45 - 29 * q^46 - 14 * q^47 + 30 * q^48 + 25 * q^49 - 13 * q^50 - q^51 + 3 * q^52 - 15 * q^53 + 28 * q^54 - 3 * q^56 - q^57 + 6 * q^58 - 9 * q^59 + 29 * q^60 - 3 * q^61 - 41 * q^62 - 52 * q^63 - 20 * q^64 + 11 * q^65 - 15 * q^67 - 9 * q^68 - 11 * q^69 + 42 * q^70 + 13 * q^71 - 2 * q^72 + 28 * q^73 - 22 * q^74 - 22 * q^75 + 6 * q^76 - 38 * q^78 - 18 * q^79 + 32 * q^80 + 23 * q^81 + 8 * q^82 + 10 * q^83 + 19 * q^84 - 14 * q^85 - 30 * q^86 + 22 * q^87 + 2 * q^89 - 30 * q^90 - 19 * q^91 + 4 * q^92 + 17 * q^93 - 9 * q^94 - q^95 + q^96 + 29 * q^97 - 9 * q^98

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 $

$2$	1.09476	0.774112	0.387056	−	0.922056i	$-0.373492\pi$
$2$	1.09476	0.774112	0.387056	+	0.922056i	$0.373492\pi$
$3$	−0.248246	−0.143325	−0.0716625	−	0.997429i	$-0.522830\pi$
$3$	−0.248246	−0.143325	−0.0716625	+	0.997429i	$0.522830\pi$
$4$	−0.801502	−0.400751
$5$	1.63186	0.729791	0.364895	−	0.931049i	$-0.381105\pi$
$5$	1.63186	0.729791	0.364895	+	0.931049i	$0.381105\pi$
$6$	−0.271770	−0.110950
$7$	4.48973	1.69696	0.848479	−	0.529229i	$-0.177519\pi$
$7$	4.48973	1.69696	0.848479	+	0.529229i	$0.177519\pi$
$8$	−3.06697	−1.08434
$9$	−2.93837	−0.979458
$10$	1.78650	0.564940
$11$	0	0
$11$	0	0
$12$	0.198970	0.0574376
$13$	−5.57331	−1.54576	−0.772880	−	0.634553i	$-0.781184\pi$
$13$	−5.57331	−1.54576	−0.772880	+	0.634553i	$0.781184\pi$
$14$	4.91517	1.31364
$15$	−0.405103	−0.104597
$16$	−1.75459	−0.438648
$17$	−5.40998	−1.31211	−0.656056	−	0.754712i	$-0.727776\pi$
$17$	−5.40998	−1.31211	−0.656056	+	0.754712i	$0.727776\pi$
$18$	−3.21681	−0.758210
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	−1.30794	−0.292464
$21$	−1.11456	−0.243216
$22$	0	0
$23$	−5.02103	−1.04696	−0.523478	−	0.852039i	$-0.675366\pi$
$23$	−5.02103	−1.04696	−0.523478	+	0.852039i	$0.675366\pi$
$24$	0.761363	0.155413
$25$	−2.33703	−0.467405
$26$	−6.10144	−1.19659
$27$	1.47418	0.283706
$28$	−3.59853	−0.680058
$29$	−0.639772	−0.118803	−0.0594014	−	0.998234i	$-0.518919\pi$
$29$	−0.639772	−0.118803	−0.0594014	+	0.998234i	$0.518919\pi$
$30$	−0.443491	−0.0809699
$31$	−10.9928	−1.97436	−0.987178	−	0.159622i	$-0.948972\pi$
$31$	−10.9928	−1.97436	−0.987178	+	0.159622i	$0.948972\pi$
$32$	4.21309	0.744776
$33$	0	0
$34$	−5.92262	−1.01572
$35$	7.32662	1.23842
$36$	2.35511	0.392519
$37$	2.01891	0.331907	0.165953	−	0.986134i	$-0.446930\pi$
$37$	2.01891	0.331907	0.165953	+	0.986134i	$0.446930\pi$
$38$	1.09476	0.177593
$39$	1.38355	0.221546
$40$	−5.00487	−0.791340
$41$	9.64435	1.50619	0.753097	−	0.657909i	$-0.228559\pi$
$41$	9.64435	1.50619	0.753097	+	0.657909i	$0.228559\pi$
$42$	−1.22017	−0.188277
$43$	−7.62145	−1.16226	−0.581130	−	0.813810i	$-0.697389\pi$
$43$	−7.62145	−1.16226	−0.581130	+	0.813810i	$0.697389\pi$
$44$	0	0
$45$	−4.79502	−0.714799
$46$	−5.49681	−0.810461
$47$	−0.859862	−0.125424	−0.0627119	−	0.998032i	$-0.519975\pi$
$47$	−0.859862	−0.125424	−0.0627119	+	0.998032i	$0.519975\pi$
$48$	0.435570	0.0628691
$49$	13.1577	1.87967
$50$	−2.55848	−0.361824
$51$	1.34301	0.188058
$52$	4.46702	0.619465
$53$	1.31006	0.179951	0.0899755	−	0.995944i	$-0.471321\pi$
$53$	1.31006	0.179951	0.0899755	+	0.995944i	$0.471321\pi$
$54$	1.61387	0.219620
$55$	0	0
$56$	−13.7699	−1.84008
$57$	−0.248246	−0.0328810
$58$	−0.700397	−0.0919666
$59$	5.59646	0.728597	0.364299	−	0.931282i	$-0.381309\pi$
$59$	5.59646	0.728597	0.364299	+	0.931282i	$0.381309\pi$
$60$	0.324691	0.0419175
$61$	5.11170	0.654486	0.327243	−	0.944940i	$-0.393880\pi$
$61$	5.11170	0.654486	0.327243	+	0.944940i	$0.393880\pi$
$62$	−12.0344	−1.52837
$63$	−13.1925	−1.66210
$64$	8.12150	1.01519
$65$	−9.09488	−1.12808
$66$	0	0
$67$	5.13847	0.627764	0.313882	−	0.949462i	$-0.398370\pi$
$67$	5.13847	0.627764	0.313882	+	0.949462i	$0.398370\pi$
$68$	4.33611	0.525830
$69$	1.24645	0.150055
$70$	8.02088	0.958679
$71$	0.467163	0.0554420	0.0277210	−	0.999616i	$-0.491175\pi$
$71$	0.467163	0.0554420	0.0277210	+	0.999616i	$0.491175\pi$
$72$	9.01191	1.06206
$73$	−6.37081	−0.745647	−0.372824	−	0.927902i	$-0.621610\pi$
$73$	−6.37081	−0.745647	−0.372824	+	0.927902i	$0.621610\pi$
$74$	2.21022	0.256933
$75$	0.580158	0.0669908
$76$	−0.801502	−0.0919386
$77$	0	0
$78$	1.51466	0.171501
$79$	−12.2372	−1.37679	−0.688394	−	0.725337i	$-0.741684\pi$
$79$	−12.2372	−1.37679	−0.688394	+	0.725337i	$0.741684\pi$
$80$	−2.86325	−0.320121
$81$	8.44916	0.938796
$82$	10.5582	1.16596
$83$	−8.18208	−0.898100	−0.449050	−	0.893507i	$-0.648237\pi$
$83$	−8.18208	−0.898100	−0.449050	+	0.893507i	$0.648237\pi$
$84$	0.893320	0.0974692
$85$	−8.82833	−0.957567
$86$	−8.34366	−0.899720
$87$	0.158821	0.0170274
$88$	0	0
$89$	−13.7128	−1.45355	−0.726776	−	0.686875i	$-0.758982\pi$
$89$	−13.7128	−1.45355	−0.726776	+	0.686875i	$0.758982\pi$
$90$	−5.24939	−0.553335
$91$	−25.0227	−2.62309
$92$	4.02436	0.419569
$93$	2.72891	0.282975
$94$	−0.941342	−0.0970921
$95$	1.63186	0.167426
$96$	−1.04588	−0.106745
$97$	14.6114	1.48357	0.741783	−	0.670640i	$-0.233980\pi$
$97$	14.6114	1.48357	0.741783	+	0.670640i	$0.233980\pi$
$98$	14.4045	1.45507
$99$	0	0
$100$	1.87313	0.187313
$101$	−17.5938	−1.75065	−0.875325	−	0.483535i	$-0.839353\pi$
$101$	−17.5938	−1.75065	−0.875325	+	0.483535i	$0.839353\pi$
$102$	1.47027	0.145578
$103$	−6.21422	−0.612305	−0.306153	−	0.951982i	$-0.599042\pi$
$103$	−6.21422	−0.612305	−0.306153	+	0.951982i	$0.599042\pi$
$104$	17.0932	1.67613
$105$	−1.81880	−0.177497
$106$	1.43420	0.139302
$107$	4.60291	0.444980	0.222490	−	0.974935i	$-0.428582\pi$
$107$	4.60291	0.444980	0.222490	+	0.974935i	$0.428582\pi$
$108$	−1.18156	−0.113695
$109$	−10.3589	−0.992205	−0.496103	−	0.868264i	$-0.665236\pi$
$109$	−10.3589	−0.992205	−0.496103	+	0.868264i	$0.665236\pi$
$110$	0	0
$111$	−0.501187	−0.0475706
$112$	−7.87763	−0.744367
$113$	7.36127	0.692490	0.346245	−	0.938144i	$-0.387457\pi$
$113$	7.36127	0.692490	0.346245	+	0.938144i	$0.387457\pi$
$114$	−0.271770	−0.0254536
$115$	−8.19362	−0.764059
$116$	0.512779	0.0476103
$117$	16.3765	1.51401
$118$	6.12678	0.564016
$119$	−24.2893	−2.22660
$120$	1.24244	0.113419
$121$	0	0
$122$	5.59608	0.506645
$123$	−2.39417	−0.215875
$124$	8.81071	0.791225
$125$	−11.9730	−1.07090
$126$	−14.4426	−1.28665
$127$	−3.13201	−0.277921	−0.138961	−	0.990298i	$-0.544376\pi$
$127$	−3.13201	−0.277921	−0.138961	+	0.990298i	$0.544376\pi$
$128$	0.464910	0.0410926
$129$	1.89200	0.166581
$130$	−9.95670	−0.873261
$131$	20.8979	1.82586	0.912931	−	0.408114i	$-0.133813\pi$
$131$	20.8979	1.82586	0.912931	+	0.408114i	$0.133813\pi$
$132$	0	0
$133$	4.48973	0.389309
$134$	5.62538	0.485959
$135$	2.40566	0.207046
$136$	16.5922	1.42277
$137$	7.52119	0.642578	0.321289	−	0.946981i	$-0.395884\pi$
$137$	7.52119	0.642578	0.321289	+	0.946981i	$0.395884\pi$
$138$	1.36456	0.116159
$139$	6.82918	0.579244	0.289622	−	0.957141i	$-0.406470\pi$
$139$	6.82918	0.579244	0.289622	+	0.957141i	$0.406470\pi$
$140$	−5.87230	−0.496300
$141$	0.213457	0.0179764
$142$	0.511431	0.0429183
$143$	0	0
$144$	5.15564	0.429637
$145$	−1.04402	−0.0867012
$146$	−6.97451	−0.577214
$147$	−3.26634	−0.269403
$148$	−1.61816	−0.133012
$149$	−21.5299	−1.76380	−0.881900	−	0.471436i	$-0.843736\pi$
$149$	−21.5299	−1.76380	−0.881900	+	0.471436i	$0.843736\pi$
$150$	0.635133	0.0518584
$151$	−15.9754	−1.30006	−0.650032	−	0.759907i	$-0.725244\pi$
$151$	−15.9754	−1.30006	−0.650032	+	0.759907i	$0.725244\pi$
$152$	−3.06697	−0.248764
$153$	15.8965	1.28516
$154$	0	0
$155$	−17.9387	−1.44087
$156$	−1.10892	−0.0887847
$157$	20.4576	1.63269	0.816347	−	0.577562i	$-0.195996\pi$
$157$	20.4576	1.63269	0.816347	+	0.577562i	$0.195996\pi$
$158$	−13.3967	−1.06579
$159$	−0.325218	−0.0257915
$160$	6.87518	0.543530
$161$	−22.5430	−1.77664
$162$	9.24980	0.726733
$163$	4.91421	0.384910	0.192455	−	0.981306i	$-0.438355\pi$
$163$	4.91421	0.384910	0.192455	+	0.981306i	$0.438355\pi$
$164$	−7.72997	−0.603609
$165$	0	0
$166$	−8.95741	−0.695230
$167$	−3.07085	−0.237629	−0.118815	−	0.992916i	$-0.537909\pi$
$167$	−3.07085	−0.237629	−0.118815	+	0.992916i	$0.537909\pi$
$168$	3.41832	0.263729
$169$	18.0618	1.38937
$170$	−9.66490	−0.741264
$171$	−2.93837	−0.224703
$172$	6.10861	0.465777
$173$	−4.27240	−0.324824	−0.162412	−	0.986723i	$-0.551927\pi$
$173$	−4.27240	−0.324824	−0.162412	+	0.986723i	$0.551927\pi$
$174$	0.173871	0.0131811
$175$	−10.4926	−0.793167
$176$	0	0
$177$	−1.38930	−0.104426
$178$	−15.0122	−1.12521
$179$	−9.67103	−0.722847	−0.361424	−	0.932402i	$-0.617709\pi$
$179$	−9.67103	−0.722847	−0.361424	+	0.932402i	$0.617709\pi$
$180$	3.84322	0.286457
$181$	6.56442	0.487929	0.243965	−	0.969784i	$-0.421552\pi$
$181$	6.56442	0.487929	0.243965	+	0.969784i	$0.421552\pi$
$182$	−27.3938	−2.03056
$183$	−1.26896	−0.0938042
$184$	15.3993	1.13525
$185$	3.29458	0.242223
$186$	2.98750	0.219054
$187$	0	0
$188$	0.689182	0.0502637
$189$	6.61866	0.481437
$190$	1.78650	0.129606
$191$	8.85771	0.640921	0.320461	−	0.947262i	$-0.396162\pi$
$191$	8.85771	0.640921	0.320461	+	0.947262i	$0.396162\pi$
$192$	−2.01613	−0.145502
$193$	20.5836	1.48164	0.740818	−	0.671705i	$-0.234438\pi$
$193$	20.5836	1.48164	0.740818	+	0.671705i	$0.234438\pi$
$194$	15.9960	1.14845
$195$	2.25777	0.161682
$196$	−10.5459	−0.753278
$197$	−11.2619	−0.802374	−0.401187	−	0.915996i	$-0.631402\pi$
$197$	−11.2619	−0.802374	−0.401187	+	0.915996i	$0.631402\pi$
$198$	0	0
$199$	9.75531	0.691536	0.345768	−	0.938320i	$-0.387618\pi$
$199$	9.75531	0.691536	0.345768	+	0.938320i	$0.387618\pi$
$200$	7.16759	0.506825
$201$	−1.27560	−0.0899742
$202$	−19.2610	−1.35520
$203$	−2.87240	−0.201603
$204$	−1.07642	−0.0753646
$205$	15.7382	1.09921
$206$	−6.80308	−0.473993
$207$	14.7537	1.02545
$208$	9.77888	0.678044
$209$	0	0
$210$	−1.99115	−0.137403
$211$	5.77759	0.397745	0.198873	−	0.980025i	$-0.436272\pi$
$211$	5.77759	0.397745	0.198873	+	0.980025i	$0.436272\pi$
$212$	−1.05002	−0.0721155
$213$	−0.115971	−0.00794622
$214$	5.03908	0.344464
$215$	−12.4372	−0.848207
$216$	−4.52126	−0.307633
$217$	−49.3545	−3.35040
$218$	−11.3405	−0.768078
$219$	1.58153	0.106870
$220$	0	0
$221$	30.1515	2.02821
$222$	−0.548679	−0.0368249
$223$	−8.83978	−0.591956	−0.295978	−	0.955195i	$-0.595645\pi$
$223$	−8.83978	−0.591956	−0.295978	+	0.955195i	$0.595645\pi$
$224$	18.9156	1.26385
$225$	6.86706	0.457804
$226$	8.05882	0.536065
$227$	−10.7967	−0.716601	−0.358300	−	0.933606i	$-0.616644\pi$
$227$	−10.7967	−0.716601	−0.358300	+	0.933606i	$0.616644\pi$
$228$	0.198970	0.0131771
$229$	−12.0337	−0.795206	−0.397603	−	0.917557i	$-0.630158\pi$
$229$	−12.0337	−0.795206	−0.397603	+	0.917557i	$0.630158\pi$
$230$	−8.97004	−0.591467
$231$	0	0
$232$	1.96216	0.128822
$233$	−13.4819	−0.883226	−0.441613	−	0.897206i	$-0.645594\pi$
$233$	−13.4819	−0.883226	−0.441613	+	0.897206i	$0.645594\pi$
$234$	17.9283	1.17201
$235$	−1.40318	−0.0915332
$236$	−4.48557	−0.291986
$237$	3.03783	0.197328
$238$	−26.5910	−1.72364
$239$	−18.1842	−1.17624	−0.588119	−	0.808774i	$-0.700131\pi$
$239$	−18.1842	−1.17624	−0.588119	+	0.808774i	$0.700131\pi$
$240$	0.710790	0.0458813
$241$	1.48941	0.0959415	0.0479708	−	0.998849i	$-0.484725\pi$
$241$	1.48941	0.0959415	0.0479708	+	0.998849i	$0.484725\pi$
$242$	0	0
$243$	−6.52001	−0.418259
$244$	−4.09704	−0.262286
$245$	21.4715	1.37176
$246$	−2.62104	−0.167112
$247$	−5.57331	−0.354622
$248$	33.7144	2.14087
$249$	2.03117	0.128720
$250$	−13.1076	−0.828995
$251$	18.6760	1.17882	0.589409	−	0.807835i	$-0.299361\pi$
$251$	18.6760	1.17882	0.589409	+	0.807835i	$0.299361\pi$
$252$	10.5738	0.666088
$253$	0	0
$254$	−3.42880	−0.215142
$255$	2.19160	0.137243
$256$	−15.7340	−0.983377
$257$	−12.4889	−0.779034	−0.389517	−	0.921019i	$-0.627358\pi$
$257$	−12.4889	−0.779034	−0.389517	+	0.921019i	$0.627358\pi$
$258$	2.07128	0.128952
$259$	9.06436	0.563232
$260$	7.28957	0.452080
$261$	1.87989	0.116362
$262$	22.8782	1.41342
$263$	6.20578	0.382665	0.191332	−	0.981525i	$-0.438719\pi$
$263$	6.20578	0.382665	0.191332	+	0.981525i	$0.438719\pi$
$264$	0	0
$265$	2.13784	0.131327
$266$	4.91517	0.301369
$267$	3.40414	0.208330
$268$	−4.11849	−0.251577
$269$	20.4957	1.24964	0.624822	−	0.780768i	$-0.285172\pi$
$269$	20.4957	1.24964	0.624822	+	0.780768i	$0.285172\pi$
$270$	2.63361	0.160277
$271$	−28.2020	−1.71315	−0.856574	−	0.516024i	$-0.827412\pi$
$271$	−28.2020	−1.71315	−0.856574	+	0.516024i	$0.827412\pi$
$272$	9.49229	0.575555
$273$	6.21178	0.375954
$274$	8.23389	0.497427
$275$	0	0
$276$	−0.999032	−0.0601347
$277$	12.8309	0.770937	0.385468	−	0.922721i	$-0.374040\pi$
$277$	12.8309	0.770937	0.385468	+	0.922721i	$0.374040\pi$
$278$	7.47631	0.448399
$279$	32.3008	1.93380
$280$	−22.4705	−1.34287
$281$	6.08165	0.362801	0.181400	−	0.983409i	$-0.441937\pi$
$281$	6.08165	0.362801	0.181400	+	0.983409i	$0.441937\pi$
$282$	0.233685	0.0139157
$283$	0.387478	0.0230332	0.0115166	−	0.999934i	$-0.496334\pi$
$283$	0.387478	0.0230332	0.0115166	+	0.999934i	$0.496334\pi$
$284$	−0.374432	−0.0222184
$285$	−0.405103	−0.0239963
$286$	0	0
$287$	43.3005	2.55595
$288$	−12.3796	−0.729476
$289$	12.2678	0.721638
$290$	−1.14295	−0.0671164
$291$	−3.62723	−0.212632
$292$	5.10622	0.298819
$293$	−12.4415	−0.726839	−0.363419	−	0.931626i	$-0.618391\pi$
$293$	−12.4415	−0.726839	−0.363419	+	0.931626i	$0.618391\pi$
$294$	−3.57586	−0.208548
$295$	9.13265	0.531723
$296$	−6.19194	−0.359899
$297$	0	0
$298$	−23.5701	−1.36538
$299$	27.9838	1.61834
$300$	−0.464998	−0.0268466
$301$	−34.2183	−1.97231
$302$	−17.4893	−1.00639
$303$	4.36760	0.250912
$304$	−1.75459	−0.100633
$305$	8.34159	0.477638
$306$	17.4029	0.994856
$307$	−13.8906	−0.792779	−0.396389	−	0.918082i	$-0.629737\pi$
$307$	−13.8906	−0.792779	−0.396389	+	0.918082i	$0.629737\pi$
$308$	0	0
$309$	1.54266	0.0877586
$310$	−19.6385	−1.11539
$311$	−34.7334	−1.96955	−0.984775	−	0.173834i	$-0.944384\pi$
$311$	−34.7334	−1.96955	−0.984775	+	0.173834i	$0.944384\pi$
$312$	−4.24332	−0.240231
$313$	2.06279	0.116596	0.0582980	−	0.998299i	$-0.481433\pi$
$313$	2.06279	0.116596	0.0582980	+	0.998299i	$0.481433\pi$
$314$	22.3961	1.26389
$315$	−21.5283	−1.21298
$316$	9.80811	0.551749
$317$	−0.891517	−0.0500726	−0.0250363	−	0.999687i	$-0.507970\pi$
$317$	−0.891517	−0.0500726	−0.0250363	+	0.999687i	$0.507970\pi$
$318$	−0.356035	−0.0199655
$319$	0	0
$320$	13.2532	0.740874
$321$	−1.14265	−0.0637767
$322$	−24.6792	−1.37532
$323$	−5.40998	−0.301019
$324$	−6.77202	−0.376223
$325$	13.0250	0.722496
$326$	5.37987	0.297964
$327$	2.57156	0.142208
$328$	−29.5789	−1.63322
$329$	−3.86055	−0.212839
$330$	0	0
$331$	8.08006	0.444120	0.222060	−	0.975033i	$-0.428722\pi$
$331$	8.08006	0.444120	0.222060	+	0.975033i	$0.428722\pi$
$332$	6.55795	0.359914
$333$	−5.93232	−0.325089
$334$	−3.36184	−0.183952
$335$	8.38527	0.458136
$336$	1.95559	0.106686
$337$	21.4963	1.17098	0.585490	−	0.810680i	$-0.300902\pi$
$337$	21.4963	1.17098	0.585490	+	0.810680i	$0.300902\pi$
$338$	19.7734	1.07553
$339$	−1.82741	−0.0992511
$340$	7.07593	0.383746
$341$	0	0
$342$	−3.21681	−0.173945
$343$	27.6463	1.49276
$344$	23.3748	1.26028
$345$	2.03403	0.109509
$346$	−4.67725	−0.251450
$347$	−8.94541	−0.480215	−0.240107	−	0.970746i	$-0.577183\pi$
$347$	−8.94541	−0.480215	−0.240107	+	0.970746i	$0.577183\pi$
$348$	−0.127295	−0.00682375
$349$	28.4646	1.52368	0.761838	−	0.647768i	$-0.224297\pi$
$349$	28.4646	1.52368	0.761838	+	0.647768i	$0.224297\pi$
$350$	−11.4869	−0.614000
$351$	−8.21606	−0.438541
$352$	0	0
$353$	−1.90094	−0.101177	−0.0505884	−	0.998720i	$-0.516110\pi$
$353$	−1.90094	−0.101177	−0.0505884	+	0.998720i	$0.516110\pi$
$354$	−1.52095	−0.0808375
$355$	0.762345	0.0404611
$356$	10.9908	0.582512
$357$	6.02973	0.319127
$358$	−10.5875	−0.559564
$359$	28.0108	1.47835	0.739176	−	0.673512i	$-0.235215\pi$
$359$	28.0108	1.47835	0.739176	+	0.673512i	$0.235215\pi$
$360$	14.7062	0.775084
$361$	1.00000	0.0526316
$362$	7.18646	0.377712
$363$	0	0
$364$	20.0557	1.05121
$365$	−10.3963	−0.544166
$366$	−1.38921	−0.0726149
$367$	10.6227	0.554502	0.277251	−	0.960798i	$-0.410577\pi$
$367$	10.6227	0.554502	0.277251	+	0.960798i	$0.410577\pi$
$368$	8.80984	0.459245
$369$	−28.3387	−1.47525
$370$	3.60678	0.187507
$371$	5.88183	0.305369
$372$	−2.18722	−0.113402
$373$	21.0351	1.08916	0.544578	−	0.838710i	$-0.316690\pi$
$373$	21.0351	1.08916	0.544578	+	0.838710i	$0.316690\pi$
$374$	0	0
$375$	2.97225	0.153487
$376$	2.63717	0.136002
$377$	3.56565	0.183640
$378$	7.24584	0.372686
$379$	21.0358	1.08053	0.540267	−	0.841493i	$-0.318323\pi$
$379$	21.0358	1.08053	0.540267	+	0.841493i	$0.318323\pi$
$380$	−1.30794	−0.0670959
$381$	0.777510	0.0398330
$382$	9.69706	0.496145
$383$	30.8255	1.57511	0.787554	−	0.616245i	$-0.211347\pi$
$383$	30.8255	1.57511	0.787554	+	0.616245i	$0.211347\pi$
$384$	−0.115412	−0.00588960
$385$	0	0
$386$	22.5340	1.14695
$387$	22.3947	1.13839
$388$	−11.7111	−0.594541
$389$	−9.92456	−0.503195	−0.251598	−	0.967832i	$-0.580956\pi$
$389$	−9.92456	−0.503195	−0.251598	+	0.967832i	$0.580956\pi$
$390$	2.47171	0.125160
$391$	27.1636	1.37372
$392$	−40.3542	−2.03819
$393$	−5.18783	−0.261692
$394$	−12.3290	−0.621127
$395$	−19.9694	−1.00477
$396$	0	0
$397$	−17.7195	−0.889314	−0.444657	−	0.895701i	$-0.646674\pi$
$397$	−17.7195	−0.889314	−0.444657	+	0.895701i	$0.646674\pi$
$398$	10.6797	0.535326
$399$	−1.11456	−0.0557977
$400$	4.10052	0.205026
$401$	5.07648	0.253508	0.126754	−	0.991934i	$-0.459544\pi$
$401$	5.07648	0.253508	0.126754	+	0.991934i	$0.459544\pi$
$402$	−1.39648	−0.0696501
$403$	61.2661	3.05188
$404$	14.1015	0.701575
$405$	13.7879	0.685125
$406$	−3.14459	−0.156063
$407$	0	0
$408$	−4.11896	−0.203919
$409$	−17.3841	−0.859587	−0.429793	−	0.902927i	$-0.641414\pi$
$409$	−17.3841	−0.859587	−0.429793	+	0.902927i	$0.641414\pi$
$410$	17.2296	0.850909
$411$	−1.86711	−0.0920975
$412$	4.98071	0.245382
$413$	25.1266	1.23640
$414$	16.1517	0.793813
$415$	−13.3520	−0.655425
$416$	−23.4809	−1.15124
$417$	−1.69532	−0.0830201
$418$	0	0
$419$	12.5295	0.612106	0.306053	−	0.952015i	$-0.400992\pi$
$419$	12.5295	0.612106	0.306053	+	0.952015i	$0.400992\pi$
$420$	1.45778	0.0711322
$421$	−1.93325	−0.0942210	−0.0471105	−	0.998890i	$-0.515001\pi$
$421$	−1.93325	−0.0942210	−0.0471105	+	0.998890i	$0.515001\pi$
$422$	6.32507	0.307899
$423$	2.52660	0.122847
$424$	−4.01792	−0.195128
$425$	12.6433	0.613288
$426$	−0.126961	−0.00615126
$427$	22.9501	1.11064
$428$	−3.68924	−0.178326
$429$	0	0
$430$	−13.6157	−0.656607
$431$	8.24025	0.396919	0.198460	−	0.980109i	$-0.436406\pi$
$431$	8.24025	0.396919	0.198460	+	0.980109i	$0.436406\pi$
$432$	−2.58658	−0.124447
$433$	11.0612	0.531568	0.265784	−	0.964033i	$-0.414369\pi$
$433$	11.0612	0.531568	0.265784	+	0.964033i	$0.414369\pi$
$434$	−54.0313	−2.59358
$435$	0.259174	0.0124264
$436$	8.30270	0.397627
$437$	−5.02103	−0.240188
$438$	1.73139	0.0827292
$439$	6.85793	0.327311	0.163656	−	0.986518i	$-0.447671\pi$
$439$	6.85793	0.327311	0.163656	+	0.986518i	$0.447671\pi$
$440$	0	0
$441$	−38.6621	−1.84105
$442$	33.0086	1.57006
$443$	−0.201359	−0.00956684	−0.00478342	−	0.999989i	$-0.501523\pi$
$443$	−0.201359	−0.00956684	−0.00478342	+	0.999989i	$0.501523\pi$
$444$	0.401702	0.0190639
$445$	−22.3774	−1.06079
$446$	−9.67743	−0.458240
$447$	5.34472	0.252797
$448$	36.4633	1.72273
$449$	17.3565	0.819104	0.409552	−	0.912287i	$-0.365685\pi$
$449$	17.3565	0.819104	0.409552	+	0.912287i	$0.365685\pi$
$450$	7.51778	0.354391
$451$	0	0
$452$	−5.90007	−0.277516
$453$	3.96584	0.186332
$454$	−11.8198	−0.554729
$455$	−40.8335	−1.91431
$456$	0.761363	0.0356541
$457$	−25.9186	−1.21242	−0.606209	−	0.795305i	$-0.707311\pi$
$457$	−25.9186	−1.21242	−0.606209	+	0.795305i	$0.707311\pi$
$458$	−13.1740	−0.615579
$459$	−7.97527	−0.372254
$460$	6.56720	0.306197
$461$	−17.2823	−0.804916	−0.402458	−	0.915438i	$-0.631844\pi$
$461$	−17.2823	−0.804916	−0.402458	+	0.915438i	$0.631844\pi$
$462$	0	0
$463$	−6.95313	−0.323140	−0.161570	−	0.986861i	$-0.551656\pi$
$463$	−6.95313	−0.323140	−0.161570	+	0.986861i	$0.551656\pi$
$464$	1.12254	0.0521125
$465$	4.45320	0.206512
$466$	−14.7594	−0.683716
$467$	−14.9411	−0.691390	−0.345695	−	0.938347i	$-0.612357\pi$
$467$	−14.9411	−0.691390	−0.345695	+	0.938347i	$0.612357\pi$
$468$	−13.1258	−0.606740
$469$	23.0703	1.06529
$470$	−1.53614	−0.0708569
$471$	−5.07852	−0.234006
$472$	−17.1642	−0.790045
$473$	0	0
$474$	3.32569	0.152754
$475$	−2.33703	−0.107230
$476$	19.4679	0.892312
$477$	−3.84945	−0.176254
$478$	−19.9073	−0.910540
$479$	−34.8266	−1.59127	−0.795633	−	0.605778i	$-0.792862\pi$
$479$	−34.8266	−1.59127	−0.795633	+	0.605778i	$0.792862\pi$
$480$	−1.70674	−0.0779015
$481$	−11.2520	−0.513048
$482$	1.63055	0.0742694
$483$	5.59622	0.254637
$484$	0	0
$485$	23.8438	1.08269
$486$	−7.13784	−0.323779
$487$	20.0445	0.908303	0.454151	−	0.890925i	$-0.349943\pi$
$487$	20.0445	0.908303	0.454151	+	0.890925i	$0.349943\pi$
$488$	−15.6774	−0.709684
$489$	−1.21993	−0.0551672
$490$	23.5061	1.06190
$491$	23.1239	1.04357	0.521784	−	0.853078i	$-0.325267\pi$
$491$	23.1239	1.04357	0.521784	+	0.853078i	$0.325267\pi$
$492$	1.91893	0.0865122
$493$	3.46115	0.155882
$494$	−6.10144	−0.274517
$495$	0	0
$496$	19.2878	0.866047
$497$	2.09743	0.0940827
$498$	2.22364	0.0996438
$499$	−18.6912	−0.836735	−0.418367	−	0.908278i	$-0.637398\pi$
$499$	−18.6912	−0.836735	−0.418367	+	0.908278i	$0.637398\pi$
$500$	9.59640	0.429164
$501$	0.762326	0.0340582
$502$	20.4457	0.912537
$503$	−9.49186	−0.423221	−0.211611	−	0.977354i	$-0.567871\pi$
$503$	−9.49186	−0.423221	−0.211611	+	0.977354i	$0.567871\pi$
$504$	40.4610	1.80228
$505$	−28.7107	−1.27761
$506$	0	0
$507$	−4.48378	−0.199132
$508$	2.51032	0.111377
$509$	−3.10660	−0.137698	−0.0688488	−	0.997627i	$-0.521933\pi$
$509$	−3.10660	−0.137698	−0.0688488	+	0.997627i	$0.521933\pi$
$510$	2.39927	0.106242
$511$	−28.6032	−1.26533
$512$	−18.1548	−0.802336
$513$	1.47418	0.0650866
$514$	−13.6723	−0.603060
$515$	−10.1407	−0.446855
$516$	−1.51644	−0.0667575
$517$	0	0
$518$	9.92330	0.436005
$519$	1.06061	0.0465554
$520$	27.8937	1.22322
$521$	−0.816498	−0.0357714	−0.0178857	−	0.999840i	$-0.505694\pi$
$521$	−0.816498	−0.0357714	−0.0178857	+	0.999840i	$0.505694\pi$
$522$	2.05803	0.0900774
$523$	−6.80565	−0.297590	−0.148795	−	0.988868i	$-0.547539\pi$
$523$	−6.80565	−0.297590	−0.148795	+	0.988868i	$0.547539\pi$
$524$	−16.7497	−0.731716
$525$	2.60475	0.113681
$526$	6.79383	0.296225
$527$	59.4705	2.59058
$528$	0	0
$529$	2.21070	0.0961174
$530$	2.34042	0.101661
$531$	−16.4445	−0.713630
$532$	−3.59853	−0.156016
$533$	−53.7510	−2.32821
$534$	3.72672	0.161271
$535$	7.51131	0.324742
$536$	−15.7595	−0.680708
$537$	2.40080	0.103602
$538$	22.4378	0.967363
$539$	0	0
$540$	−1.92814	−0.0829738
$541$	−17.7329	−0.762396	−0.381198	−	0.924494i	$-0.624488\pi$
$541$	−17.7329	−0.762396	−0.381198	+	0.924494i	$0.624488\pi$
$542$	−30.8744	−1.32617
$543$	−1.62959	−0.0699325
$544$	−22.7927	−0.977229
$545$	−16.9043	−0.724102
$546$	6.80040	0.291030
$547$	38.8630	1.66166	0.830831	−	0.556526i	$-0.187866\pi$
$547$	38.8630	1.66166	0.830831	+	0.556526i	$0.187866\pi$
$548$	−6.02825	−0.257514
$549$	−15.0201	−0.641042
$550$	0	0
$551$	−0.639772	−0.0272552
$552$	−3.82283	−0.162710
$553$	−54.9415	−2.33635
$554$	14.0468	0.596791
$555$	−0.817868	−0.0347166
$556$	−5.47361	−0.232133
$557$	−34.3279	−1.45452	−0.727259	−	0.686363i	$-0.759206\pi$
$557$	−34.3279	−1.45452	−0.727259	+	0.686363i	$0.759206\pi$
$558$	35.3616	1.49698
$559$	42.4768	1.79658
$560$	−12.8552	−0.543232
$561$	0	0
$562$	6.65794	0.280848
$563$	25.4704	1.07345	0.536725	−	0.843757i	$-0.319661\pi$
$563$	25.4704	1.07345	0.536725	+	0.843757i	$0.319661\pi$
$564$	−0.171087	−0.00720405
$565$	12.0126	0.505373
$566$	0.424196	0.0178303
$567$	37.9345	1.59310
$568$	−1.43277	−0.0601178
$569$	−26.1038	−1.09433	−0.547163	−	0.837026i	$-0.684292\pi$
$569$	−26.1038	−1.09433	−0.547163	+	0.837026i	$0.684292\pi$
$570$	−0.443491	−0.0185758
$571$	−10.8101	−0.452387	−0.226194	−	0.974082i	$-0.572628\pi$
$571$	−10.8101	−0.452387	−0.226194	+	0.974082i	$0.572628\pi$
$572$	0	0
$573$	−2.19889	−0.0918600
$574$	47.4036	1.97859
$575$	11.7343	0.489353
$576$	−23.8640	−0.994333
$577$	−6.08671	−0.253393	−0.126697	−	0.991942i	$-0.540437\pi$
$577$	−6.08671	−0.253393	−0.126697	+	0.991942i	$0.540437\pi$
$578$	13.4303	0.558628
$579$	−5.10979	−0.212356
$580$	0.836784	0.0347456
$581$	−36.7353	−1.52404
$582$	−3.97094	−0.164601
$583$	0	0
$584$	19.5391	0.808533
$585$	26.7242	1.10491
$586$	−13.6204	−0.562655
$587$	−0.981023	−0.0404911	−0.0202456	−	0.999795i	$-0.506445\pi$
$587$	−0.981023	−0.0404911	−0.0202456	+	0.999795i	$0.506445\pi$
$588$	2.61798	0.107964
$589$	−10.9928	−0.452948
$590$	9.99805	0.411613
$591$	2.79571	0.115000
$592$	−3.54236	−0.145590
$593$	−13.3473	−0.548108	−0.274054	−	0.961714i	$-0.588365\pi$
$593$	−13.3473	−0.548108	−0.274054	+	0.961714i	$0.588365\pi$
$594$	0	0
$595$	−39.6368	−1.62495
$596$	17.2563	0.706845
$597$	−2.42172	−0.0991143
$598$	30.6355	1.25278
$599$	37.2790	1.52318	0.761589	−	0.648060i	$-0.224420\pi$
$599$	37.2790	1.52318	0.761589	+	0.648060i	$0.224420\pi$
$600$	−1.77933	−0.0726407
$601$	−0.551555	−0.0224984	−0.0112492	−	0.999937i	$-0.503581\pi$
$601$	−0.551555	−0.0224984	−0.0112492	+	0.999937i	$0.503581\pi$
$602$	−37.4608	−1.52679
$603$	−15.0987	−0.614868
$604$	12.8044	0.521002
$605$	0	0
$606$	4.78147	0.194234
$607$	−27.5738	−1.11919	−0.559594	−	0.828767i	$-0.689043\pi$
$607$	−27.5738	−1.11919	−0.559594	+	0.828767i	$0.689043\pi$
$608$	4.21309	0.170863
$609$	0.713063	0.0288948
$610$	9.13203	0.369745
$611$	4.79228	0.193875
$612$	−12.7411	−0.515029
$613$	29.9724	1.21057	0.605287	−	0.796008i	$-0.293059\pi$
$613$	29.9724	1.21057	0.605287	+	0.796008i	$0.293059\pi$
$614$	−15.2069	−0.613699
$615$	−3.90696	−0.157544
$616$	0	0
$617$	−7.62612	−0.307016	−0.153508	−	0.988147i	$-0.549057\pi$
$617$	−7.62612	−0.307016	−0.153508	+	0.988147i	$0.549057\pi$
$618$	1.68884	0.0679350
$619$	28.8420	1.15926	0.579628	−	0.814881i	$-0.303198\pi$
$619$	28.8420	1.15926	0.579628	+	0.814881i	$0.303198\pi$
$620$	14.3779	0.577429
$621$	−7.40189	−0.297027
$622$	−38.0247	−1.52465
$623$	−61.5666	−2.46662
$624$	−2.42757	−0.0971806
$625$	−7.85317	−0.314127
$626$	2.25826	0.0902583
$627$	0	0
$628$	−16.3968	−0.654304
$629$	−10.9223	−0.435499
$630$	−23.5684	−0.938986
$631$	−4.50477	−0.179332	−0.0896660	−	0.995972i	$-0.528580\pi$
$631$	−4.50477	−0.179332	−0.0896660	+	0.995972i	$0.528580\pi$
$632$	37.5310	1.49290
$633$	−1.43426	−0.0570068
$634$	−0.975996	−0.0387618
$635$	−5.11101	−0.202824
$636$	0.260663	0.0103360
$637$	−73.3318	−2.90551
$638$	0	0
$639$	−1.37270	−0.0543031
$640$	0.758669	0.0299890
$641$	34.2229	1.35172	0.675861	−	0.737029i	$-0.263772\pi$
$641$	34.2229	1.35172	0.675861	+	0.737029i	$0.263772\pi$
$642$	−1.25093	−0.0493703
$643$	−2.05393	−0.0809993	−0.0404996	−	0.999180i	$-0.512895\pi$
$643$	−2.05393	−0.0809993	−0.0404996	+	0.999180i	$0.512895\pi$
$644$	18.0683	0.711991
$645$	3.08748	0.121569
$646$	−5.92262	−0.233022
$647$	−18.1609	−0.713979	−0.356990	−	0.934108i	$-0.616197\pi$
$647$	−18.1609	−0.713979	−0.356990	+	0.934108i	$0.616197\pi$
$648$	−25.9133	−1.01797
$649$	0	0
$650$	14.2592	0.559293
$651$	12.2521	0.480196
$652$	−3.93875	−0.154253
$653$	−45.5267	−1.78160	−0.890800	−	0.454396i	$-0.849855\pi$
$653$	−45.5267	−1.78160	−0.890800	+	0.454396i	$0.849855\pi$
$654$	2.81524	0.110085
$655$	34.1026	1.33250
$656$	−16.9219	−0.660688
$657$	18.7198	0.730330
$658$	−4.22637	−0.164761
$659$	−44.4071	−1.72985	−0.864927	−	0.501897i	$-0.832636\pi$
$659$	−44.4071	−1.72985	−0.864927	+	0.501897i	$0.832636\pi$
$660$	0	0
$661$	−6.94517	−0.270136	−0.135068	−	0.990836i	$-0.543125\pi$
$661$	−6.94517	−0.270136	−0.135068	+	0.990836i	$0.543125\pi$
$662$	8.84572	0.343799
$663$	−7.48499	−0.290693
$664$	25.0942	0.973844
$665$	7.32662	0.284114
$666$	−6.49446	−0.251655
$667$	3.21231	0.124381
$668$	2.46129	0.0952303
$669$	2.19444	0.0848420
$670$	9.17985	0.354649
$671$	0	0
$672$	−4.69573	−0.181142
$673$	18.8199	0.725454	0.362727	−	0.931895i	$-0.381846\pi$
$673$	18.8199	0.725454	0.362727	+	0.931895i	$0.381846\pi$
$674$	23.5333	0.906469
$675$	−3.44519	−0.132606
$676$	−14.4766	−0.556792
$677$	2.28065	0.0876523	0.0438262	−	0.999039i	$-0.486045\pi$
$677$	2.28065	0.0876523	0.0438262	+	0.999039i	$0.486045\pi$
$678$	−2.00057	−0.0768314
$679$	65.6014	2.51755
$680$	27.0762	1.03833
$681$	2.68023	0.102707
$682$	0	0
$683$	7.91546	0.302877	0.151438	−	0.988467i	$-0.451610\pi$
$683$	7.91546	0.302877	0.151438	+	0.988467i	$0.451610\pi$
$684$	2.35511	0.0900500
$685$	12.2735	0.468948
$686$	30.2660	1.15556
$687$	2.98731	0.113973
$688$	13.3725	0.509823
$689$	−7.30139	−0.278161
$690$	2.22678	0.0847720
$691$	−37.3359	−1.42033	−0.710163	−	0.704037i	$-0.751379\pi$
$691$	−37.3359	−1.42033	−0.710163	+	0.704037i	$0.751379\pi$
$692$	3.42433	0.130174
$693$	0	0
$694$	−9.79307	−0.371740
$695$	11.1443	0.422727
$696$	−0.487099	−0.0184635
$697$	−52.1757	−1.97630
$698$	31.1619	1.17950
$699$	3.34682	0.126588
$700$	8.40985	0.317863
$701$	28.2380	1.06653	0.533267	−	0.845947i	$-0.320964\pi$
$701$	28.2380	1.06653	0.533267	+	0.845947i	$0.320964\pi$
$702$	−8.99461	−0.339480
$703$	2.01891	0.0761447
$704$	0	0
$705$	0.348333	0.0131190
$706$	−2.08107	−0.0783221
$707$	−78.9915	−2.97078
$708$	1.11353	0.0418489
$709$	−45.5805	−1.71181	−0.855905	−	0.517133i	$-0.826999\pi$
$709$	−45.5805	−1.71181	−0.855905	+	0.517133i	$0.826999\pi$
$710$	0.834584	0.0313214
$711$	35.9574	1.34851
$712$	42.0567	1.57614
$713$	55.1949	2.06706
$714$	6.60110	0.247040
$715$	0	0
$716$	7.75135	0.289682
$717$	4.51416	0.168584
$718$	30.6651	1.14441
$719$	−18.2212	−0.679536	−0.339768	−	0.940509i	$-0.610349\pi$
$719$	−18.2212	−0.679536	−0.339768	+	0.940509i	$0.610349\pi$
$720$	8.41330	0.313545
$721$	−27.9002	−1.03906
$722$	1.09476	0.0407427
$723$	−0.369741	−0.0137508
$724$	−5.26140	−0.195538
$725$	1.49516	0.0555290
$726$	0	0
$727$	44.7081	1.65813	0.829066	−	0.559151i	$-0.188873\pi$
$727$	44.7081	1.65813	0.829066	+	0.559151i	$0.188873\pi$
$728$	76.7438	2.84431
$729$	−23.7289	−0.878849
$730$	−11.3814	−0.421246
$731$	41.2319	1.52502
$732$	1.01707	0.0375921
$733$	14.9819	0.553369	0.276685	−	0.960961i	$-0.410764\pi$
$733$	14.9819	0.553369	0.276685	+	0.960961i	$0.410764\pi$
$734$	11.6293	0.429246
$735$	−5.33022	−0.196608
$736$	−21.1540	−0.779747
$737$	0	0
$738$	−31.0241	−1.14201
$739$	−14.2504	−0.524209	−0.262104	−	0.965040i	$-0.584416\pi$
$739$	−14.2504	−0.524209	−0.262104	+	0.965040i	$0.584416\pi$
$740$	−2.64062	−0.0970710
$741$	1.38355	0.0508261
$742$	6.43918	0.236390
$743$	9.33615	0.342510	0.171255	−	0.985227i	$-0.445218\pi$
$743$	9.33615	0.342510	0.171255	+	0.985227i	$0.445218\pi$
$744$	−8.36948	−0.306840
$745$	−35.1339	−1.28721
$746$	23.0284	0.843128
$747$	24.0420	0.879651
$748$	0	0
$749$	20.6658	0.755113
$750$	3.25390	0.118816
$751$	−44.2037	−1.61302	−0.806509	−	0.591222i	$-0.798646\pi$
$751$	−44.2037	−1.61302	−0.806509	+	0.591222i	$0.798646\pi$
$752$	1.50871	0.0550169
$753$	−4.63624	−0.168954
$754$	3.90353	0.142158
$755$	−26.0697	−0.948774
$756$	−5.30487	−0.192936
$757$	−34.3770	−1.24945	−0.624727	−	0.780843i	$-0.714790\pi$
$757$	−34.3770	−1.24945	−0.624727	+	0.780843i	$0.714790\pi$
$758$	23.0291	0.836455
$759$	0	0
$760$	−5.00487	−0.181546
$761$	−35.6083	−1.29080	−0.645400	−	0.763845i	$-0.723309\pi$
$761$	−35.6083	−1.29080	−0.645400	+	0.763845i	$0.723309\pi$
$762$	0.851186	0.0308352
$763$	−46.5088	−1.68373
$764$	−7.09947	−0.256850
$765$	25.9409	0.937897
$766$	33.7465	1.21931
$767$	−31.1908	−1.12624
$768$	3.90591	0.140942
$769$	−18.9211	−0.682312	−0.341156	−	0.940007i	$-0.610819\pi$
$769$	−18.9211	−0.682312	−0.341156	+	0.940007i	$0.610819\pi$
$770$	0	0
$771$	3.10031	0.111655
$772$	−16.4978	−0.593767
$773$	−1.93704	−0.0696703	−0.0348352	−	0.999393i	$-0.511091\pi$
$773$	−1.93704	−0.0696703	−0.0348352	+	0.999393i	$0.511091\pi$
$774$	24.5168	0.881238
$775$	25.6903	0.922825
$776$	−44.8128	−1.60869
$777$	−2.25019	−0.0807252
$778$	−10.8650	−0.389529
$779$	9.64435	0.345545
$780$	−1.80961	−0.0647943
$781$	0	0
$782$	29.7376	1.06342
$783$	−0.943138	−0.0337050
$784$	−23.0863	−0.824511
$785$	33.3840	1.19153
$786$	−5.67943	−0.202579
$787$	−27.5288	−0.981295	−0.490648	−	0.871358i	$-0.663240\pi$
$787$	−27.5288	−0.981295	−0.490648	+	0.871358i	$0.663240\pi$
$788$	9.02640	0.321552
$789$	−1.54056	−0.0548454
$790$	−21.8616	−0.777802
$791$	33.0501	1.17513
$792$	0	0
$793$	−28.4891	−1.01168
$794$	−19.3985	−0.688429
$795$	−0.530711	−0.0188224
$796$	−7.81890	−0.277134
$797$	−33.0802	−1.17176	−0.585881	−	0.810397i	$-0.699251\pi$
$797$	−33.0802	−1.17176	−0.585881	+	0.810397i	$0.699251\pi$
$798$	−1.22017	−0.0431936
$799$	4.65184	0.164570
$800$	−9.84610	−0.348112
$801$	40.2933	1.42369
$802$	5.55753	0.196243
$803$	0	0
$804$	1.02240	0.0360572
$805$	−36.7871	−1.29658
$806$	67.0716	2.36250
$807$	−5.08797	−0.179105
$808$	53.9597	1.89830
$809$	39.4195	1.38591	0.692957	−	0.720979i	$-0.256308\pi$
$809$	39.4195	1.38591	0.692957	+	0.720979i	$0.256308\pi$
$810$	15.0944	0.530363
$811$	43.7164	1.53509	0.767544	−	0.640996i	$-0.221478\pi$
$811$	43.7164	1.53509	0.767544	+	0.640996i	$0.221478\pi$
$812$	2.30224	0.0807927
$813$	7.00103	0.245537
$814$	0	0
$815$	8.01931	0.280904
$816$	−2.35642	−0.0824913
$817$	−7.62145	−0.266641
$818$	−19.0314	−0.665416
$819$	73.5260	2.56921
$820$	−12.6142	−0.440508
$821$	−51.8046	−1.80799	−0.903996	−	0.427541i	$-0.859380\pi$
$821$	−51.8046	−1.80799	−0.903996	+	0.427541i	$0.859380\pi$
$822$	−2.04403	−0.0712938
$823$	−7.98486	−0.278335	−0.139167	−	0.990269i	$-0.544443\pi$
$823$	−7.98486	−0.278335	−0.139167	+	0.990269i	$0.544443\pi$
$824$	19.0588	0.663946
$825$	0	0
$826$	27.5076	0.957111
$827$	−39.0227	−1.35695	−0.678477	−	0.734622i	$-0.737360\pi$
$827$	−39.0227	−1.35695	−0.678477	+	0.734622i	$0.737360\pi$
$828$	−11.8251	−0.410950
$829$	22.6572	0.786917	0.393459	−	0.919342i	$-0.371278\pi$
$829$	22.6572	0.786917	0.393459	+	0.919342i	$0.371278\pi$
$830$	−14.6173	−0.507372
$831$	−3.18523	−0.110494
$832$	−45.2637	−1.56923
$833$	−71.1827	−2.46633
$834$	−1.85597	−0.0642668
$835$	−5.01120	−0.173420
$836$	0	0
$837$	−16.2053	−0.560136
$838$	13.7168	0.473838
$839$	30.3795	1.04882	0.524409	−	0.851467i	$-0.324286\pi$
$839$	30.3795	1.04882	0.524409	+	0.851467i	$0.324286\pi$
$840$	5.57822	0.192467
$841$	−28.5907	−0.985886
$842$	−2.11645	−0.0729376
$843$	−1.50975	−0.0519984
$844$	−4.63075	−0.159397
$845$	29.4744	1.01395
$846$	2.76602	0.0950976
$847$	0	0
$848$	−2.29862	−0.0789350
$849$	−0.0961900	−0.00330123
$850$	13.8413	0.474753
$851$	−10.1370	−0.347492
$852$	0.0929512	0.00318446
$853$	29.7746	1.01946	0.509731	−	0.860334i	$-0.329745\pi$
$853$	29.7746	1.01946	0.509731	+	0.860334i	$0.329745\pi$
$854$	25.1249	0.859756
$855$	−4.79502	−0.163986
$856$	−14.1170	−0.482509
$857$	27.0696	0.924679	0.462339	−	0.886703i	$-0.347010\pi$
$857$	27.0696	0.924679	0.462339	+	0.886703i	$0.347010\pi$
$858$	0	0
$859$	19.1748	0.654237	0.327119	−	0.944983i	$-0.393922\pi$
$859$	19.1748	0.654237	0.327119	+	0.944983i	$0.393922\pi$
$860$	9.96841	0.339920
$861$	−10.7492	−0.366331
$862$	9.02110	0.307260
$863$	13.3654	0.454964	0.227482	−	0.973782i	$-0.426951\pi$
$863$	13.3654	0.454964	0.227482	+	0.973782i	$0.426951\pi$
$864$	6.21084	0.211297
$865$	−6.97196	−0.237054
$866$	12.1094	0.411493
$867$	−3.04544	−0.103429
$868$	39.5577	1.34268
$869$	0	0
$870$	0.283733	0.00961945
$871$	−28.6383	−0.970371
$872$	31.7705	1.07589
$873$	−42.9338	−1.45309
$874$	−5.49681	−0.185933
$875$	−53.7556	−1.81727
$876$	−1.26760	−0.0428282
$877$	−36.3535	−1.22757	−0.613785	−	0.789474i	$-0.710354\pi$
$877$	−36.3535	−1.22757	−0.613785	+	0.789474i	$0.710354\pi$
$878$	7.50778	0.253375
$879$	3.08855	0.104174
$880$	0	0
$881$	13.3651	0.450283	0.225141	−	0.974326i	$-0.427716\pi$
$881$	13.3651	0.450283	0.225141	+	0.974326i	$0.427716\pi$
$882$	−42.3257	−1.42518
$883$	−5.98933	−0.201557	−0.100779	−	0.994909i	$-0.532133\pi$
$883$	−5.98933	−0.201557	−0.100779	+	0.994909i	$0.532133\pi$
$884$	−24.1665	−0.812807
$885$	−2.26714	−0.0762092
$886$	−0.220439	−0.00740581
$887$	8.56387	0.287547	0.143773	−	0.989611i	$-0.454076\pi$
$887$	8.56387	0.287547	0.143773	+	0.989611i	$0.454076\pi$
$888$	1.53713	0.0515825
$889$	−14.0619	−0.471621
$890$	−24.4978	−0.821169
$891$	0	0
$892$	7.08510	0.237227
$893$	−0.859862	−0.0287742
$894$	5.85118	0.195693
$895$	−15.7818	−0.527527
$896$	2.08732	0.0697325
$897$	−6.94686	−0.231949
$898$	19.0012	0.634078
$899$	7.03286	0.234559
$900$	−5.50396	−0.183465
$901$	−7.08741	−0.236116
$902$	0	0
$903$	8.49455	0.282681
$904$	−22.5768	−0.750893
$905$	10.7122	0.356086
$906$	4.34164	0.144241
$907$	−25.2554	−0.838593	−0.419296	−	0.907849i	$-0.637723\pi$
$907$	−25.2554	−0.838593	−0.419296	+	0.907849i	$0.637723\pi$
$908$	8.65356	0.287179
$909$	51.6972	1.71469
$910$	−44.7029	−1.48189
$911$	34.6945	1.14948	0.574739	−	0.818337i	$-0.305103\pi$
$911$	34.6945	1.14948	0.574739	+	0.818337i	$0.305103\pi$
$912$	0.435570	0.0144232
$913$	0	0
$914$	−28.3746	−0.938548
$915$	−2.07077	−0.0684574
$916$	9.64500	0.318680
$917$	93.8261	3.09841
$918$	−8.73100	−0.288166
$919$	−2.29770	−0.0757941	−0.0378970	−	0.999282i	$-0.512066\pi$
$919$	−2.29770	−0.0757941	−0.0378970	+	0.999282i	$0.512066\pi$
$920$	25.1296	0.828498
$921$	3.44829	0.113625
$922$	−18.9199	−0.623095
$923$	−2.60364	−0.0857000
$924$	0	0
$925$	−4.71825	−0.155135
$926$	−7.61201	−0.250146
$927$	18.2597	0.599727
$928$	−2.69542	−0.0884814
$929$	−49.5517	−1.62574	−0.812869	−	0.582447i	$-0.802095\pi$
$929$	−49.5517	−1.62574	−0.812869	+	0.582447i	$0.802095\pi$
$930$	4.87518	0.159864
$931$	13.1577	0.431225
$932$	10.8057	0.353954
$933$	8.62243	0.282286
$934$	−16.3569	−0.535213
$935$	0	0
$936$	−50.2262	−1.64169
$937$	14.4425	0.471817	0.235908	−	0.971775i	$-0.424193\pi$
$937$	14.4425	0.471817	0.235908	+	0.971775i	$0.424193\pi$
$938$	25.2564	0.824652
$939$	−0.512080	−0.0167111
$940$	1.12465	0.0366820
$941$	−0.846479	−0.0275944	−0.0137972	−	0.999905i	$-0.504392\pi$
$941$	−0.846479	−0.0275944	−0.0137972	+	0.999905i	$0.504392\pi$
$942$	−5.55976	−0.181147
$943$	−48.4245	−1.57692
$944$	−9.81949	−0.319597
$945$	10.8007	0.351348
$946$	0	0
$947$	−29.3178	−0.952702	−0.476351	−	0.879255i	$-0.658041\pi$
$947$	−29.3178	−0.952702	−0.476351	+	0.879255i	$0.658041\pi$
$948$	−2.43483	−0.0790795
$949$	35.5065	1.15259
$950$	−2.55848	−0.0830081
$951$	0.221316	0.00717665
$952$	74.4946	2.41439
$953$	−46.9852	−1.52200	−0.761000	−	0.648752i	$-0.775291\pi$
$953$	−46.9852	−1.52200	−0.761000	+	0.648752i	$0.775291\pi$
$954$	−4.21422	−0.136441
$955$	14.4546	0.467738
$956$	14.5747	0.471379
$957$	0	0
$958$	−38.1267	−1.23182
$959$	33.7681	1.09043
$960$	−3.29005	−0.106186
$961$	89.8406	2.89808
$962$	−12.3183	−0.397157
$963$	−13.5251	−0.435839
$964$	−1.19377	−0.0384487
$965$	33.5895	1.08128
$966$	6.12652	0.197117
$967$	27.6316	0.888573	0.444287	−	0.895885i	$-0.353457\pi$
$967$	27.6316	0.888573	0.444287	+	0.895885i	$0.353457\pi$
$968$	0	0
$969$	1.34301	0.0431435
$970$	26.1033	0.838125
$971$	2.40771	0.0772671	0.0386336	−	0.999253i	$-0.487699\pi$
$971$	2.40771	0.0772671	0.0386336	+	0.999253i	$0.487699\pi$
$972$	5.22580	0.167618
$973$	30.6612	0.982952
$974$	21.9439	0.703128
$975$	−3.23340	−0.103552
$976$	−8.96894	−0.287089
$977$	−33.3546	−1.06711	−0.533554	−	0.845766i	$-0.679144\pi$
$977$	−33.3546	−1.06711	−0.533554	+	0.845766i	$0.679144\pi$
$978$	−1.33553	−0.0427056
$979$	0	0
$980$	−17.2095	−0.549736
$981$	30.4384	0.971823
$982$	25.3151	0.807838
$983$	−42.1197	−1.34341	−0.671705	−	0.740819i	$-0.734438\pi$
$983$	−42.1197	−1.34341	−0.671705	+	0.740819i	$0.734438\pi$
$984$	7.34285	0.234082
$985$	−18.3778	−0.585565
$986$	3.78913	0.120670
$987$	0.958366	0.0305051
$988$	4.46702	0.142115
$989$	38.2675	1.21684
$990$	0	0
$991$	17.5474	0.557412	0.278706	−	0.960376i	$-0.410094\pi$
$991$	17.5474	0.557412	0.278706	+	0.960376i	$0.410094\pi$
$992$	−46.3134	−1.47045
$993$	−2.00584	−0.0636535
$994$	2.29618	0.0728306
$995$	15.9193	0.504676
$996$	−1.62799	−0.0515847
$997$	−22.8361	−0.723226	−0.361613	−	0.932328i	$-0.617774\pi$
$997$	−22.8361	−0.723226	−0.361613	+	0.932328i	$0.617774\pi$
$998$	−20.4624	−0.647726
$999$	2.97623	0.0941639

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2299.2.a.p.1.6	✓	7
11.10	odd	2		2299.2.a.r.1.2	yes	7

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2299.2.a.p.1.6	✓	7	1.1	even	1	trivial
2299.2.a.r.1.2	yes	7	11.10	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 2299 = 11^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2299.a (trivial)

\(f(q)\)	\(=\)	\(q+1.09476 q^{2} -0.248246 q^{3} -0.801502 q^{4} +1.63186 q^{5} -0.271770 q^{6} +4.48973 q^{7} -3.06697 q^{8} -2.93837 q^{9} +1.78650 q^{10} +0.198970 q^{12} -5.57331 q^{13} +4.91517 q^{14} -0.405103 q^{15} -1.75459 q^{16} -5.40998 q^{17} -3.21681 q^{18} +1.00000 q^{19} -1.30794 q^{20} -1.11456 q^{21} -5.02103 q^{23} +0.761363 q^{24} -2.33703 q^{25} -6.10144 q^{26} +1.47418 q^{27} -3.59853 q^{28} -0.639772 q^{29} -0.443491 q^{30} -10.9928 q^{31} +4.21309 q^{32} -5.92262 q^{34} +7.32662 q^{35} +2.35511 q^{36} +2.01891 q^{37} +1.09476 q^{38} +1.38355 q^{39} -5.00487 q^{40} +9.64435 q^{41} -1.22017 q^{42} -7.62145 q^{43} -4.79502 q^{45} -5.49681 q^{46} -0.859862 q^{47} +0.435570 q^{48} +13.1577 q^{49} -2.55848 q^{50} +1.34301 q^{51} +4.46702 q^{52} +1.31006 q^{53} +1.61387 q^{54} -13.7699 q^{56} -0.248246 q^{57} -0.700397 q^{58} +5.59646 q^{59} +0.324691 q^{60} +5.11170 q^{61} -12.0344 q^{62} -13.1925 q^{63} +8.12150 q^{64} -9.09488 q^{65} +5.13847 q^{67} +4.33611 q^{68} +1.24645 q^{69} +8.02088 q^{70} +0.467163 q^{71} +9.01191 q^{72} -6.37081 q^{73} +2.21022 q^{74} +0.580158 q^{75} -0.801502 q^{76} +1.51466 q^{78} -12.2372 q^{79} -2.86325 q^{80} +8.44916 q^{81} +10.5582 q^{82} -8.18208 q^{83} +0.893320 q^{84} -8.82833 q^{85} -8.34366 q^{86} +0.158821 q^{87} -13.7128 q^{89} -5.24939 q^{90} -25.0227 q^{91} +4.02436 q^{92} +2.72891 q^{93} -0.941342 q^{94} +1.63186 q^{95} -1.04588 q^{96} +14.6114 q^{97} +14.4045 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 7 q - 2 q^{2} - q^{3} + 6 q^{4} - q^{5} - 2 q^{7} - 6 q^{8} + 4 q^{9} + 3 q^{10} - 9 q^{12} - 12 q^{13} + 3 q^{14} + 5 q^{15} - 12 q^{17} + 3 q^{18} + 7 q^{19} - 17 q^{20} - 9 q^{21} - 9 q^{23} - 27 q^{24}+ \cdots - 9 q^{98}+O(q^{100}) \) 7 * q - 2 * q^2 - q^3 + 6 * q^4 - q^5 - 2 * q^7 - 6 * q^8 + 4 * q^9 + 3 * q^10 - 9 * q^12 - 12 * q^13 + 3 * q^14 + 5 * q^15 - 12 * q^17 + 3 * q^18 + 7 * q^19 - 17 * q^20 - 9 * q^21 - 9 * q^23 - 27 * q^24 + 4 * q^25 + 18 * q^26 + 23 * q^27 + 10 * q^28 - 18 * q^29 - 4 * q^30 - 5 * q^31 + 16 * q^32 - 6 * q^34 - 20 * q^35 - 2 * q^36 + 4 * q^37 - 2 * q^38 - 21 * q^39 - 17 * q^40 - 25 * q^41 - 19 * q^42 + 10 * q^43 - 16 * q^45 - 29 * q^46 - 14 * q^47 + 30 * q^48 + 25 * q^49 - 13 * q^50 - q^51 + 3 * q^52 - 15 * q^53 + 28 * q^54 - 3 * q^56 - q^57 + 6 * q^58 - 9 * q^59 + 29 * q^60 - 3 * q^61 - 41 * q^62 - 52 * q^63 - 20 * q^64 + 11 * q^65 - 15 * q^67 - 9 * q^68 - 11 * q^69 + 42 * q^70 + 13 * q^71 - 2 * q^72 + 28 * q^73 - 22 * q^74 - 22 * q^75 + 6 * q^76 - 38 * q^78 - 18 * q^79 + 32 * q^80 + 23 * q^81 + 8 * q^82 + 10 * q^83 + 19 * q^84 - 14 * q^85 - 30 * q^86 + 22 * q^87 + 2 * q^89 - 30 * q^90 - 19 * q^91 + 4 * q^92 + 17 * q^93 - 9 * q^94 - q^95 + q^96 + 29 * q^97 - 9 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more