LMFDB - Embedded newform 273.2.c.c.64.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [273,2,Mod(64,273)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(273, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 1]))

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

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

chi := DirichletCharacter("273.64");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 273 = 3 \cdot 7 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	273.c (of order $2$, degree $1$, minimal)

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:	no
Analytic conductor:	$2.17991597518$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 15x^{6} + 67x^{4} + 77x^{2} + 4 $ x^8 + 15x^6 + 67x^4 + 77*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			64.5
Root			$0.233455i$ of defining polynomial
Character	$\chi$	$=$	273.64
Dual form			273.2.c.c.64.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.233455i q^{2} -1.00000 q^{3} +1.94550 q^{4} +2.94550i q^{5} -0.233455i q^{6} -1.00000i q^{7} +0.921097i q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+0.233455i q^{2} -1.00000 q^{3} +1.94550 q^{4} +2.94550i q^{5} -0.233455i q^{6} -1.00000i q^{7} +0.921097i q^{8} +1.00000 q^{9} -0.687642 q^{10} +3.62146i q^{11} -1.94550 q^{12} +(-3.38801 - 1.23346i) q^{13} +0.233455 q^{14} -2.94550i q^{15} +3.67596 q^{16} +1.53309 q^{17} +0.233455i q^{18} +4.10005i q^{19} +5.73046i q^{20} +1.00000i q^{21} -0.845448 q^{22} +7.03387 q^{23} -0.921097i q^{24} -3.67596 q^{25} +(0.287956 - 0.790947i) q^{26} -1.00000 q^{27} -1.94550i q^{28} -3.79095 q^{29} +0.687642 q^{30} +1.79095i q^{31} +2.70037i q^{32} -3.62146i q^{33} +0.357908i q^{34} +2.94550 q^{35} +1.94550 q^{36} -7.24292i q^{37} -0.957177 q^{38} +(3.38801 + 1.23346i) q^{39} -2.71309 q^{40} -9.15455i q^{41} -0.233455 q^{42} +5.79095 q^{43} +7.04555i q^{44} +2.94550i q^{45} +1.64209i q^{46} -7.25460i q^{47} -3.67596 q^{48} -1.00000 q^{49} -0.858172i q^{50} -1.53309 q^{51} +(-6.59136 - 2.39969i) q^{52} -13.3430 q^{53} -0.233455i q^{54} -10.6670 q^{55} +0.921097 q^{56} -4.10005i q^{57} -0.885016i q^{58} +4.84545i q^{59} -5.73046i q^{60} +2.77601 q^{61} -0.418106 q^{62} -1.00000i q^{63} +6.72151 q^{64} +(3.63314 - 9.97937i) q^{65} +0.845448 q^{66} -14.8248i q^{67} +2.98262 q^{68} -7.03387 q^{69} +0.687642i q^{70} -3.62146i q^{71} +0.921097i q^{72} -14.5670i q^{73} +1.69090 q^{74} +3.67596 q^{75} +7.97664i q^{76} +3.62146 q^{77} +(-0.287956 + 0.790947i) q^{78} +6.56696 q^{79} +10.8275i q^{80} +1.00000 q^{81} +2.13718 q^{82} +7.25460i q^{83} +1.94550i q^{84} +4.51571i q^{85} +1.35193i q^{86} +3.79095 q^{87} -3.33572 q^{88} +0.636396i q^{89} -0.687642 q^{90} +(-1.23346 + 3.38801i) q^{91} +13.6844 q^{92} -1.79095i q^{93} +1.69362 q^{94} -12.0767 q^{95} -2.70037i q^{96} +11.3240i q^{97} -0.233455i q^{98} +3.62146i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9}+O(q^{10}) $ 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9	$ 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9} + 4 q^{10} + 14 q^{12} - 6 q^{13} - 2 q^{14} + 34 q^{16} + 20 q^{17} - 24 q^{22} - 6 q^{23} - 34 q^{25} + 28 q^{26} - 8 q^{27} - 18 q^{29} - 4 q^{30} - 6 q^{35} - 14 q^{36} + 36 q^{38} + 6 q^{39} - 8 q^{40} + 2 q^{42} + 34 q^{43} - 34 q^{48} - 8 q^{49} - 20 q^{51} + 18 q^{52} - 10 q^{53} + 16 q^{55} - 6 q^{56} - 20 q^{61} - 28 q^{62} - 18 q^{64} - 10 q^{65} + 24 q^{66} - 24 q^{68} + 6 q^{69} + 48 q^{74} + 34 q^{75} + 4 q^{77} - 28 q^{78} - 2 q^{79} + 8 q^{81} - 48 q^{82} + 18 q^{87} - 8 q^{88} + 4 q^{90} - 6 q^{91} + 56 q^{92} - 72 q^{94} - 78 q^{95}+O(q^{100}) $ 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9 + 4 * q^10 + 14 * q^12 - 6 * q^13 - 2 * q^14 + 34 * q^16 + 20 * q^17 - 24 * q^22 - 6 * q^23 - 34 * q^25 + 28 * q^26 - 8 * q^27 - 18 * q^29 - 4 * q^30 - 6 * q^35 - 14 * q^36 + 36 * q^38 + 6 * q^39 - 8 * q^40 + 2 * q^42 + 34 * q^43 - 34 * q^48 - 8 * q^49 - 20 * q^51 + 18 * q^52 - 10 * q^53 + 16 * q^55 - 6 * q^56 - 20 * q^61 - 28 * q^62 - 18 * q^64 - 10 * q^65 + 24 * q^66 - 24 * q^68 + 6 * q^69 + 48 * q^74 + 34 * q^75 + 4 * q^77 - 28 * q^78 - 2 * q^79 + 8 * q^81 - 48 * q^82 + 18 * q^87 - 8 * q^88 + 4 * q^90 - 6 * q^91 + 56 * q^92 - 72 * q^94 - 78 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/273\mathbb{Z}\right)^\times$.

$n$	$92$	$106$	$157$
$\chi(n)$	$1$	$-1$	$1$

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 $

$2$			0.233455i			0.165078i	0.996588	+	0.0825388i	$0.0263028\pi$
$2$			0.233455i			0.165078i	−0.996588	+	0.0825388i	$0.973697\pi$
$3$	−1.00000			−0.577350
$3$	−1.00000			−0.577350
$4$	1.94550			0.972749
$5$			2.94550i			1.31727i	0.752464	+	0.658634i	$0.228865\pi$
$5$			2.94550i			1.31727i	−0.752464	+	0.658634i	$0.771135\pi$
$6$		−	0.233455i		−	0.0953076i
$7$		−	1.00000i		−	0.377964i
$7$		−	1.00000i		−	0.377964i
$8$			0.921097i			0.325657i
$9$	1.00000			0.333333
$10$	−0.687642			−0.217451
$11$			3.62146i			1.09191i	0.837814	+	0.545956i	$0.183833\pi$
$11$			3.62146i			1.09191i	−0.837814	+	0.545956i	$0.816167\pi$
$12$	−1.94550			−0.561617
$13$	−3.38801	−	1.23346i	−0.939664	−	0.342099i
$13$	−3.38801	−	1.23346i	−0.939664	−	0.342099i
$14$	0.233455			0.0623935
$15$		−	2.94550i		−	0.760525i
$16$	3.67596			0.918991
$17$	1.53309			0.371829			0.185914	−	0.982566i	$-0.440475\pi$
$17$	1.53309			0.371829			0.185914	+	0.982566i	$0.440475\pi$
$18$			0.233455i			0.0550259i
$19$			4.10005i			0.940616i	0.882502	+	0.470308i	$0.155857\pi$
$19$			4.10005i			0.940616i	−0.882502	+	0.470308i	$0.844143\pi$
$20$			5.73046i			1.28137i
$21$			1.00000i			0.218218i
$22$	−0.845448			−0.180250
$23$	7.03387			1.46666			0.733332	−	0.679871i	$-0.237964\pi$
$23$	7.03387			1.46666			0.733332	+	0.679871i	$0.237964\pi$
$24$		−	0.921097i		−	0.188018i
$25$	−3.67596			−0.735193
$26$	0.287956	−	0.790947i	0.0564729	−	0.155118i
$27$	−1.00000			−0.192450
$28$		−	1.94550i		−	0.367665i
$29$	−3.79095			−0.703961			−0.351981	−	0.936007i	$-0.614492\pi$
$29$	−3.79095			−0.703961			−0.351981	+	0.936007i	$0.614492\pi$
$30$	0.687642			0.125546
$31$			1.79095i			0.321664i	0.986982	+	0.160832i	$0.0514177\pi$
$31$			1.79095i			0.321664i	−0.986982	+	0.160832i	$0.948582\pi$
$32$			2.70037i			0.477362i
$33$		−	3.62146i		−	0.630416i
$34$			0.357908i			0.0613806i
$35$	2.94550			0.497880
$36$	1.94550			0.324250
$37$		−	7.24292i		−	1.19073i	−0.803456	−	0.595365i	$-0.797008\pi$
$37$		−	7.24292i		−	1.19073i	0.803456	−	0.595365i	$-0.202992\pi$
$38$	−0.957177			−0.155275
$39$	3.38801	+	1.23346i	0.542515	+	0.197511i
$40$	−2.71309			−0.428977
$41$		−	9.15455i		−	1.42970i	−0.699277	−	0.714850i	$-0.746495\pi$
$41$		−	9.15455i		−	1.42970i	0.699277	−	0.714850i	$-0.253505\pi$
$42$	−0.233455			−0.0360229
$43$	5.79095			0.883111			0.441556	−	0.897234i	$-0.354427\pi$
$43$	5.79095			0.883111			0.441556	+	0.897234i	$0.354427\pi$
$44$			7.04555i			1.06216i
$45$			2.94550i			0.439089i
$46$			1.64209i			0.242113i
$47$		−	7.25460i		−	1.05819i	−0.848562	−	0.529096i	$-0.822531\pi$
$47$		−	7.25460i		−	1.05819i	0.848562	−	0.529096i	$-0.177469\pi$
$48$	−3.67596			−0.530580
$49$	−1.00000			−0.142857
$50$		−	0.858172i		−	0.121364i
$51$	−1.53309			−0.214676
$52$	−6.59136	−	2.39969i	−0.914058	−	0.332776i
$53$	−13.3430			−1.83280			−0.916399	−	0.400266i	$-0.868918\pi$
$53$	−13.3430			−1.83280			−0.916399	+	0.400266i	$0.868918\pi$
$54$		−	0.233455i		−	0.0317692i
$55$	−10.6670			−1.43834
$56$	0.921097			0.123087
$57$		−	4.10005i		−	0.543065i
$58$		−	0.885016i		−	0.116208i
$59$			4.84545i			0.630824i	0.948955	+	0.315412i	$0.102143\pi$
$59$			4.84545i			0.630824i	−0.948955	+	0.315412i	$0.897857\pi$
$60$		−	5.73046i		−	0.739800i
$61$	2.77601			0.355432			0.177716	−	0.984082i	$-0.443129\pi$
$61$	2.77601			0.355432			0.177716	+	0.984082i	$0.443129\pi$
$62$	−0.418106			−0.0530995
$63$		−	1.00000i		−	0.125988i
$64$	6.72151			0.840189
$65$	3.63314	−	9.97937i	0.450636	−	1.23779i
$66$	0.845448			0.104068
$67$		−	14.8248i		−	1.81114i	−0.424197	−	0.905570i	$-0.639444\pi$
$67$		−	14.8248i		−	1.81114i	0.424197	−	0.905570i	$-0.360556\pi$
$68$	2.98262			0.361696
$69$	−7.03387			−0.846778
$70$			0.687642i			0.0821889i
$71$		−	3.62146i		−	0.429788i	−0.976637	−	0.214894i	$-0.931059\pi$
$71$		−	3.62146i		−	0.429788i	0.976637	−	0.214894i	$-0.0689407\pi$
$72$			0.921097i			0.108552i
$73$		−	14.5670i		−	1.70493i	−0.522781	−	0.852467i	$-0.675106\pi$
$73$		−	14.5670i		−	1.70493i	0.522781	−	0.852467i	$-0.324894\pi$
$74$	1.69090			0.196563
$75$	3.67596			0.424464
$76$			7.97664i			0.914984i
$77$	3.62146			0.412704
$78$	−0.287956	+	0.790947i	−0.0326046	+	0.0895571i
$79$	6.56696			0.738841			0.369420	−	0.929262i	$-0.379556\pi$
$79$	6.56696			0.738841			0.369420	+	0.929262i	$0.379556\pi$
$80$			10.8275i			1.21056i
$81$	1.00000			0.111111
$82$	2.13718			0.236012
$83$			7.25460i			0.796296i	0.917321	+	0.398148i	$0.130347\pi$
$83$			7.25460i			0.796296i	−0.917321	+	0.398148i	$0.869653\pi$
$84$			1.94550i			0.212271i
$85$			4.51571i			0.489798i
$86$			1.35193i			0.145782i
$87$	3.79095			0.406432
$88$	−3.33572			−0.355588
$89$			0.636396i			0.0674578i	0.999431	+	0.0337289i	$0.0107383\pi$
$89$			0.636396i			0.0674578i	−0.999431	+	0.0337289i	$0.989262\pi$
$90$	−0.687642			−0.0724838
$91$	−1.23346	+	3.38801i	−0.129301	+	0.355160i
$92$	13.6844			1.42670
$93$		−	1.79095i		−	0.185713i
$94$	1.69362			0.174684
$95$	−12.0767			−1.23904
$96$		−	2.70037i		−	0.275605i
$97$			11.3240i			1.14978i	0.818230	+	0.574891i	$0.194956\pi$
$97$			11.3240i			1.14978i	−0.818230	+	0.574891i	$0.805044\pi$
$98$		−	0.233455i		−	0.0235825i
$99$			3.62146i			0.363971i
$100$	−7.15158			−0.715158
$101$	−6.04880			−0.601879			−0.300939	−	0.953643i	$-0.597300\pi$
$101$	−6.04880			−0.601879			−0.300939	+	0.953643i	$0.597300\pi$
$102$		−	0.357908i		−	0.0354381i
$103$	18.6670			1.83932			0.919658	−	0.392721i	$-0.128466\pi$
$103$	18.6670			1.83932			0.919658	+	0.392721i	$0.128466\pi$
$104$	1.13613	−	3.12068i	0.111407	−	0.306008i
$105$	−2.94550			−0.287451
$106$		−	3.11498i		−	0.302554i
$107$	−4.82482			−0.466433			−0.233216	−	0.972425i	$-0.574925\pi$
$107$	−4.82482			−0.466433			−0.233216	+	0.972425i	$0.574925\pi$
$108$	−1.94550			−0.187206
$109$		−	4.95718i		−	0.474811i	−0.971411	−	0.237406i	$-0.923703\pi$
$109$		−	4.95718i		−	0.474811i	0.971411	−	0.237406i	$-0.0762971\pi$
$110$		−	2.49027i		−	0.237438i
$111$			7.24292i			0.687468i
$112$		−	3.67596i		−	0.347346i
$113$	3.79095			0.356622			0.178311	−	0.983974i	$-0.442937\pi$
$113$	3.79095			0.356622			0.178311	+	0.983974i	$0.442937\pi$
$114$	0.957177			0.0896479
$115$			20.7183i			1.93199i
$116$	−7.37528			−0.684778
$117$	−3.38801	−	1.23346i	−0.313221	−	0.114033i
$118$	−1.13119			−0.104135
$119$		−	1.53309i		−	0.140538i
$120$	2.71309			0.247670
$121$	−2.11498			−0.192271
$122$			0.648074i			0.0586739i
$123$			9.15455i			0.825438i
$124$			3.48429i			0.312898i
$125$			3.89995i			0.348822i
$126$	0.233455			0.0207978
$127$	−10.4903			−0.930861			−0.465430	−	0.885085i	$-0.654100\pi$
$127$	−10.4903			−0.930861			−0.465430	+	0.885085i	$0.654100\pi$
$128$			6.96990i			0.616058i
$129$	−5.79095			−0.509864
$130$	2.32973	+	0.848175i	0.204331	+	0.0743899i
$131$	5.35193			0.467600			0.233800	−	0.972285i	$-0.424884\pi$
$131$	5.35193			0.467600			0.233800	+	0.972285i	$0.424884\pi$
$132$		−	7.04555i		−	0.613236i
$133$	4.10005			0.355519
$134$	3.46093			0.298979
$135$		−	2.94550i		−	0.253508i
$136$			1.41212i			0.121089i
$137$			11.6936i			0.999054i	0.866298	+	0.499527i	$0.166493\pi$
$137$			11.6936i			0.999054i	−0.866298	+	0.499527i	$0.833507\pi$
$138$		−	1.64209i		−	0.139784i
$139$	−14.6670			−1.24404			−0.622020	−	0.783002i	$-0.713688\pi$
$139$	−14.6670			−1.24404			−0.622020	+	0.783002i	$0.713688\pi$
$140$	5.73046			0.484313
$141$			7.25460i			0.610948i
$142$	0.845448			0.0709485
$143$	4.46691	−	12.2695i	0.373542	−	1.02603i
$144$	3.67596			0.306330
$145$		−	11.1662i		−	0.927305i
$146$	3.40073			0.281446
$147$	1.00000			0.0824786
$148$		−	14.0911i		−	1.15828i
$149$		−	12.6274i		−	1.03448i	−0.855840	−	0.517240i	$-0.826959\pi$
$149$		−	12.6274i		−	1.03448i	0.855840	−	0.517240i	$-0.173041\pi$
$150$			0.858172i			0.0700695i
$151$			18.4858i			1.50436i	0.658959	+	0.752178i	$0.270997\pi$
$151$			18.4858i			1.50436i	−0.658959	+	0.752178i	$0.729003\pi$
$152$	−3.77654			−0.306318
$153$	1.53309			0.123943
$154$			0.845448i			0.0681282i
$155$	−5.27523			−0.423717
$156$	6.59136	+	2.39969i	0.527731	+	0.192129i
$157$	−1.06618			−0.0850904			−0.0425452	−	0.999095i	$-0.513547\pi$
$157$	−1.06618			−0.0850904			−0.0425452	+	0.999095i	$0.513547\pi$
$158$			1.53309i			0.121966i
$159$	13.3430			1.05817
$160$	−7.95392			−0.628813
$161$		−	7.03387i		−	0.554347i
$162$			0.233455i			0.0183420i
$163$			23.4430i			1.83620i	0.396350	+	0.918100i	$0.370277\pi$
$163$			23.4430i			1.83620i	−0.396350	+	0.918100i	$0.629723\pi$
$164$		−	17.8102i		−	1.39074i
$165$	10.6670			0.830426
$166$	−1.69362			−0.131451
$167$		−	7.67271i		−	0.593732i	−0.954919	−	0.296866i	$-0.904059\pi$
$167$		−	7.67271i		−	0.593732i	0.954919	−	0.296866i	$-0.0959415\pi$
$168$	−0.921097			−0.0710641
$169$	9.95718	+	8.35791i	0.765937	+	0.642916i
$170$	−1.05422			−0.0808547
$171$			4.10005i			0.313539i
$172$	11.2663			0.859046
$173$	−2.66701			−0.202769			−0.101385	−	0.994847i	$-0.532327\pi$
$173$	−2.66701			−0.202769			−0.101385	+	0.994847i	$0.532327\pi$
$174$			0.885016i			0.0670929i
$175$			3.67596i			0.277877i
$176$			13.3124i			1.00346i
$177$		−	4.84545i		−	0.364206i
$178$	−0.148570			−0.0111358
$179$	−11.2340			−0.839666			−0.419833	−	0.907601i	$-0.637911\pi$
$179$	−11.2340			−0.839666			−0.419833	+	0.907601i	$0.637911\pi$
$180$			5.73046i			0.427124i
$181$	9.58189			0.712217			0.356108	−	0.934445i	$-0.384103\pi$
$181$	9.58189			0.712217			0.356108	+	0.934445i	$0.384103\pi$
$182$	−0.790947	−	0.287956i	−0.0586289	−	0.0213447i
$183$	−2.77601			−0.205209
$184$			6.47887i			0.477629i
$185$	21.3340			1.56851
$186$	0.418106			0.0306570
$187$			5.55203i			0.406004i
$188$		−	14.1138i		−	1.02936i
$189$			1.00000i			0.0727393i
$190$		−	2.81936i		−	0.204538i
$191$	−7.89100			−0.570973			−0.285486	−	0.958383i	$-0.592155\pi$
$191$	−7.89100			−0.570973			−0.285486	+	0.958383i	$0.592155\pi$
$192$	−6.72151			−0.485083
$193$			5.89100i			0.424043i	0.977265	+	0.212022i	$0.0680047\pi$
$193$			5.89100i			0.424043i	−0.977265	+	0.212022i	$0.931995\pi$
$194$	−2.64365			−0.189803
$195$	−3.63314	+	9.97937i	−0.260175	+	0.714637i
$196$	−1.94550			−0.138964
$197$		−	19.1546i		−	1.36471i	−0.731023	−	0.682353i	$-0.760957\pi$
$197$		−	19.1546i		−	1.36471i	0.731023	−	0.682353i	$-0.239043\pi$
$198$	−0.845448			−0.0600834
$199$	8.20010			0.581290			0.290645	−	0.956831i	$-0.406130\pi$
$199$	8.20010			0.581290			0.290645	+	0.956831i	$0.406130\pi$
$200$		−	3.38592i		−	0.239420i
$201$			14.8248i			1.04566i
$202$		−	1.41212i		−	0.0993567i
$203$			3.79095i			0.266072i
$204$	−2.98262			−0.208825
$205$	26.9647			1.88330
$206$			4.35791i			0.303630i
$207$	7.03387			0.488888
$208$	−12.4542	−	4.53413i	−0.863542	−	0.314386i
$209$	−14.8482			−1.02707
$210$		−	0.687642i		−	0.0474518i
$211$	−11.7009			−0.805522			−0.402761	−	0.915305i	$-0.631949\pi$
$211$	−11.7009			−0.805522			−0.402761	+	0.915305i	$0.631949\pi$
$212$	−25.9587			−1.78285
$213$			3.62146i			0.248138i
$214$		−	1.12638i		−	0.0769976i
$215$			17.0572i			1.16329i
$216$		−	0.921097i		−	0.0626727i
$217$	1.79095			0.121577
$218$	1.15728			0.0783808
$219$			14.5670i			0.984344i
$220$	−20.7527			−1.39914
$221$	−5.19412	−	1.89100i	−0.349394	−	0.127202i
$222$	−1.69090			−0.113486
$223$			5.45198i			0.365091i	0.983197	+	0.182546i	$0.0584337\pi$
$223$			5.45198i			0.365091i	−0.983197	+	0.182546i	$0.941566\pi$
$224$	2.70037			0.180426
$225$	−3.67596			−0.245064
$226$			0.885016i			0.0588704i
$227$		−	27.1133i		−	1.79957i	−0.436331	−	0.899786i	$-0.643722\pi$
$227$		−	27.1133i		−	1.79957i	0.436331	−	0.899786i	$-0.356278\pi$
$228$		−	7.97664i		−	0.528266i
$229$		−	17.4918i		−	1.15589i	−0.816075	−	0.577946i	$-0.803854\pi$
$229$		−	17.4918i		−	1.15589i	0.816075	−	0.577946i	$-0.196146\pi$
$230$	−4.83678			−0.318928
$231$	−3.62146			−0.238275
$232$		−	3.49183i		−	0.229250i
$233$	13.3430			0.874127			0.437064	−	0.899431i	$-0.356018\pi$
$233$	13.3430			0.874127			0.437064	+	0.899431i	$0.356018\pi$
$234$	0.287956	−	0.790947i	0.0188243	−	0.0517058i
$235$	21.3684			1.39392
$236$			9.42681i			0.613633i
$237$	−6.56696			−0.426570
$238$	0.357908			0.0231997
$239$		−	12.7554i		−	0.825077i	−0.910940	−	0.412539i	$-0.864642\pi$
$239$		−	12.7554i		−	0.825077i	0.910940	−	0.412539i	$-0.135358\pi$
$240$		−	10.8275i		−	0.698915i
$241$		−	9.94875i		−	0.640856i	−0.947273	−	0.320428i	$-0.896173\pi$
$241$		−	9.94875i		−	0.640856i	0.947273	−	0.320428i	$-0.103827\pi$
$242$		−	0.493754i		−	0.0317397i
$243$	−1.00000			−0.0641500
$244$	5.40073			0.345746
$245$		−	2.94550i		−	0.188181i
$246$	−2.13718			−0.136261
$247$	5.05723	−	13.8910i	0.321784	−	0.883863i
$248$	−1.64964			−0.104752
$249$		−	7.25460i		−	0.459742i
$250$	−0.910463			−0.0575827
$251$	24.4002			1.54013			0.770064	−	0.637967i	$-0.220224\pi$
$251$	24.4002			1.54013			0.770064	+	0.637967i	$0.220224\pi$
$252$		−	1.94550i		−	0.122555i
$253$			25.4729i			1.60147i
$254$		−	2.44901i		−	0.153664i
$255$		−	4.51571i		−	0.282785i
$256$	11.8159			0.738492
$257$	−28.2190			−1.76026			−0.880128	−	0.474737i	$-0.842543\pi$
$257$	−28.2190			−1.76026			−0.880128	+	0.474737i	$0.842543\pi$
$258$		−	1.35193i		−	0.0841672i
$259$	−7.24292			−0.450053
$260$	7.06827	−	19.4149i	0.438355	−	1.20406i
$261$	−3.79095			−0.234654
$262$			1.24943i			0.0771903i
$263$	−11.4520			−0.706159			−0.353080	−	0.935593i	$-0.614866\pi$
$263$	−11.4520			−0.706159			−0.353080	+	0.935593i	$0.614866\pi$
$264$	3.33572			0.205299
$265$		−	39.3017i		−	2.41428i
$266$			0.957177i			0.0586883i
$267$		−	0.636396i		−	0.0389468i
$268$		−	28.8417i		−	1.76179i
$269$	19.8009			1.20728			0.603642	−	0.797255i	$-0.293716\pi$
$269$	19.8009			1.20728			0.603642	+	0.797255i	$0.293716\pi$
$270$	0.687642			0.0418485
$271$			12.5391i			0.761694i	0.924638	+	0.380847i	$0.124368\pi$
$271$			12.5391i			0.761694i	−0.924638	+	0.380847i	$0.875632\pi$
$272$	5.63558			0.341707
$273$	1.23346	−	3.38801i	0.0746521	−	0.205051i
$274$	−2.72994			−0.164921
$275$		−	13.3124i		−	0.802765i
$276$	−13.6844			−0.823703
$277$	−2.43902			−0.146547			−0.0732733	−	0.997312i	$-0.523345\pi$
$277$	−2.43902			−0.146547			−0.0732733	+	0.997312i	$0.523345\pi$
$278$		−	3.42409i		−	0.205363i
$279$			1.79095i			0.107221i
$280$			2.71309i			0.162138i
$281$			25.9794i			1.54980i	0.632084	+	0.774900i	$0.282200\pi$
$281$			25.9794i			1.54980i	−0.632084	+	0.774900i	$0.717800\pi$
$282$	−1.69362			−0.100854
$283$	−7.78199			−0.462592			−0.231296	−	0.972883i	$-0.574297\pi$
$283$	−7.78199			−0.462592			−0.231296	+	0.972883i	$0.574297\pi$
$284$		−	7.04555i		−	0.418076i
$285$	12.0767			0.715362
$286$	2.86438	+	1.04282i	0.169375	+	0.0616634i
$287$	−9.15455			−0.540376
$288$			2.70037i			0.159121i
$289$	−14.6496			−0.861743
$290$	2.60681			0.153077
$291$		−	11.3240i		−	0.663827i
$292$		−	28.3400i		−	1.65847i
$293$			7.14560i			0.417450i	0.977974	+	0.208725i	$0.0669314\pi$
$293$			7.14560i			0.417450i	−0.977974	+	0.208725i	$0.933069\pi$
$294$			0.233455i			0.0136154i
$295$	−14.2723			−0.830963
$296$	6.67143			0.387769
$297$		−	3.62146i		−	0.210139i
$298$	2.94794			0.170770
$299$	−23.8308	−	8.67596i	−1.37817	−	0.501744i
$300$	7.15158			0.412897
$301$		−	5.79095i		−	0.333785i
$302$	−4.31561			−0.248336
$303$	6.04880			0.347495
$304$			15.0716i			0.864417i
$305$			8.17674i			0.468199i
$306$			0.357908i			0.0204602i
$307$			10.4092i			0.594082i	0.954865	+	0.297041i	$0.0959998\pi$
$307$			10.4092i			0.594082i	−0.954865	+	0.297041i	$0.904000\pi$
$308$	7.04555			0.401457
$309$	−18.6670			−1.06193
$310$		−	1.23153i		−	0.0699462i
$311$	4.51571			0.256063			0.128031	−	0.991770i	$-0.459134\pi$
$311$	4.51571			0.256063			0.128031	+	0.991770i	$0.459134\pi$
$312$	−1.13613	+	3.12068i	−0.0643208	+	0.176674i
$313$	−21.6242			−1.22227			−0.611136	−	0.791526i	$-0.709287\pi$
$313$	−21.6242			−1.22227			−0.611136	+	0.791526i	$0.709287\pi$
$314$		−	0.248905i		−	0.0140465i
$315$	2.94550			0.165960
$316$	12.7760			0.718707
$317$			16.8275i			0.945129i	0.881296	+	0.472565i	$0.156672\pi$
$317$			16.8275i			0.945129i	−0.881296	+	0.472565i	$0.843328\pi$
$318$			3.11498i			0.174680i
$319$		−	13.7288i		−	0.768664i
$320$			19.7982i			1.10675i
$321$	4.82482			0.269295
$322$	1.64209			0.0915102
$323$			6.28575i			0.349748i
$324$	1.94550			0.108083
$325$	12.4542	+	4.53413i	0.690834	+	0.251509i
$326$	−5.47289			−0.303115
$327$			4.95718i			0.274133i
$328$	8.43223			0.465592
$329$	−7.25460			−0.399959
$330$			2.49027i			0.137085i
$331$			36.1355i			1.98619i	0.117331	+	0.993093i	$0.462566\pi$
$331$			36.1355i			1.98619i	−0.117331	+	0.993093i	$0.537434\pi$
$332$			14.1138i			0.774596i
$333$		−	7.24292i		−	0.396910i
$334$	1.79123			0.0980119
$335$	43.6665			2.38575
$336$			3.67596i			0.200540i
$337$	−33.4595			−1.82266			−0.911328	−	0.411681i	$-0.864942\pi$
$337$	−33.4595			−1.82266			−0.911328	+	0.411681i	$0.864942\pi$
$338$	−1.95120	+	2.32455i	−0.106131	+	0.126439i
$339$	−3.79095			−0.205896
$340$			8.78532i			0.476451i
$341$	−6.48585			−0.351228
$342$	−0.957177			−0.0517582
$343$			1.00000i			0.0539949i
$344$			5.33402i			0.287591i
$345$		−	20.7183i		−	1.11543i
$346$		−	0.622627i		−	0.0334726i
$347$	−8.72721			−0.468501			−0.234251	−	0.972176i	$-0.575264\pi$
$347$	−8.72721			−0.468501			−0.234251	+	0.972176i	$0.575264\pi$
$348$	7.37528			0.395357
$349$			0.852706i			0.0456443i	0.999740	+	0.0228222i	$0.00726515\pi$
$349$			0.852706i			0.0456443i	−0.999740	+	0.0228222i	$0.992735\pi$
$350$	−0.858172			−0.0458712
$351$	3.38801	+	1.23346i	0.180838	+	0.0658370i
$352$	−9.77927			−0.521237
$353$			7.46365i			0.397250i	0.980076	+	0.198625i	$0.0636476\pi$
$353$			7.46365i			0.397250i	−0.980076	+	0.198625i	$0.936352\pi$
$354$	1.13119			0.0601223
$355$	10.6670			0.566146
$356$			1.23811i			0.0656195i
$357$			1.53309i			0.0811397i
$358$		−	2.62263i		−	0.138610i
$359$			12.4019i			0.654547i	0.944930	+	0.327274i	$0.106130\pi$
$359$			12.4019i			0.654547i	−0.944930	+	0.327274i	$0.893870\pi$
$360$	−2.71309			−0.142992
$361$	2.18959			0.115241
$362$			2.23694i			0.117571i
$363$	2.11498			0.111008
$364$	−2.39969	+	6.59136i	−0.125778	+	0.345481i
$365$	42.9070			2.24585
$366$		−	0.648074i		−	0.0338754i
$367$	−9.31508			−0.486243			−0.243122	−	0.969996i	$-0.578171\pi$
$367$	−9.31508			−0.486243			−0.243122	+	0.969996i	$0.578171\pi$
$368$	25.8562			1.34785
$369$		−	9.15455i		−	0.476567i
$370$			4.98053i			0.258926i
$371$			13.3430i			0.692733i
$372$		−	3.48429i		−	0.180652i
$373$	−3.73319			−0.193297			−0.0966486	−	0.995319i	$-0.530812\pi$
$373$	−3.73319			−0.193297			−0.0966486	+	0.995319i	$0.530812\pi$
$374$	−1.29615			−0.0670222
$375$		−	3.89995i		−	0.201393i
$376$	6.68219			0.344608
$377$	12.8438	+	4.67596i	0.661487	+	0.240824i
$378$	−0.233455			−0.0120076
$379$		−	12.0000i		−	0.616399i	−0.951322	−	0.308199i	$-0.900274\pi$
$379$		−	12.0000i		−	0.616399i	0.951322	−	0.308199i	$-0.0997264\pi$
$380$	−23.4952			−1.20528
$381$	10.4903			0.537433
$382$		−	1.84219i		−	0.0942548i
$383$			8.80419i			0.449873i	0.974373	+	0.224936i	$0.0722175\pi$
$383$			8.80419i			0.449873i	−0.974373	+	0.224936i	$0.927783\pi$
$384$		−	6.96990i		−	0.355681i
$385$			10.6670i			0.543641i
$386$	−1.37528			−0.0700001
$387$	5.79095			0.294370
$388$			22.0309i			1.11845i
$389$	−26.4978			−1.34349			−0.671746	−	0.740781i	$-0.734455\pi$
$389$	−26.4978			−1.34349			−0.671746	+	0.740781i	$0.734455\pi$
$390$	−2.32973	−	0.848175i	−0.117971	−	0.0429490i
$391$	10.7836			0.545348
$392$		−	0.921097i		−	0.0465224i
$393$	−5.35193			−0.269969
$394$	4.47173			0.225282
$395$			19.3430i			0.973251i
$396$			7.04555i			0.354052i
$397$		−	16.0399i		−	0.805017i	−0.915416	−	0.402509i	$-0.868138\pi$
$397$		−	16.0399i		−	0.805017i	0.915416	−	0.402509i	$-0.131862\pi$
$398$			1.91435i			0.0959579i
$399$	−4.10005			−0.205259
$400$	−13.5127			−0.675635
$401$			12.2885i			0.613657i	0.951765	+	0.306829i	$0.0992678\pi$
$401$			12.2885i			0.613657i	−0.951765	+	0.306829i	$0.900732\pi$
$402$	−3.46093			−0.172615
$403$	2.20905	−	6.06774i	0.110041	−	0.302256i
$404$	−11.7679			−0.585477
$405$			2.94550i			0.146363i
$406$	−0.885016			−0.0439226
$407$	26.2300			1.30017
$408$		−	1.41212i		−	0.0699105i
$409$		−	24.8114i		−	1.22685i	−0.789754	−	0.613423i	$-0.789792\pi$
$409$		−	24.8114i		−	1.22685i	0.789754	−	0.613423i	$-0.210208\pi$
$410$			6.29505i			0.310890i
$411$		−	11.6936i		−	0.576804i
$412$	36.3166			1.78919
$413$	4.84545			0.238429
$414$			1.64209i			0.0807044i
$415$	−21.3684			−1.04893
$416$	3.33078	−	9.14885i	0.163305	−	0.448560i
$417$	14.6670			0.718247
$418$		−	3.46638i		−	0.169546i
$419$	21.6496			1.05765			0.528827	−	0.848730i	$-0.322632\pi$
$419$	21.6496			1.05765			0.528827	+	0.848730i	$0.322632\pi$
$420$	−5.73046			−0.279618
$421$			22.4679i			1.09502i	0.836799	+	0.547510i	$0.184424\pi$
$421$			22.4679i			1.09502i	−0.836799	+	0.547510i	$0.815576\pi$
$422$		−	2.73163i		−	0.132974i
$423$		−	7.25460i		−	0.352731i
$424$		−	12.2902i		−	0.596863i
$425$	−5.63558			−0.273366
$426$	−0.845448			−0.0409621
$427$		−	2.77601i		−	0.134341i
$428$	−9.38668			−0.453722
$429$	−4.46691	+	12.2695i	−0.215664	+	0.592379i
$430$	−3.98210			−0.192034
$431$		−	28.7787i		−	1.38622i	−0.720830	−	0.693112i	$-0.756239\pi$
$431$		−	28.7787i		−	1.38622i	0.720830	−	0.693112i	$-0.243761\pi$
$432$	−3.67596			−0.176860
$433$	−29.5217			−1.41872			−0.709361	−	0.704845i	$-0.751016\pi$
$433$	−29.5217			−1.41872			−0.709361	+	0.704845i	$0.751016\pi$
$434$			0.418106i			0.0200697i
$435$			11.1662i			0.535380i
$436$		−	9.64418i		−	0.461873i
$437$			28.8392i			1.37957i
$438$	−3.40073			−0.162493
$439$	18.6670			0.890928			0.445464	−	0.895300i	$-0.353039\pi$
$439$	18.6670			0.890928			0.445464	+	0.895300i	$0.353039\pi$
$440$		−	9.82535i		−	0.468405i
$441$	−1.00000			−0.0476190
$442$	0.441463	−	1.21259i	0.0209982	−	0.0576772i
$443$	15.2340			0.723788			0.361894	−	0.932219i	$-0.382130\pi$
$443$	15.2340			0.723788			0.361894	+	0.932219i	$0.382130\pi$
$444$			14.0911i			0.668734i
$445$	−1.87450			−0.0888599
$446$	−1.27279			−0.0602684
$447$			12.6274i			0.597258i
$448$		−	6.72151i		−	0.317562i
$449$		−	11.2755i		−	0.532125i	−0.963956	−	0.266062i	$-0.914277\pi$
$449$		−	11.2755i		−	0.532125i	0.963956	−	0.266062i	$-0.0857227\pi$
$450$		−	0.858172i		−	0.0404546i
$451$	33.1529			1.56111
$452$	7.37528			0.346904
$453$		−	18.4858i		−	0.868541i
$454$	6.32973			0.297069
$455$	−9.97937	−	3.63314i	−0.467840	−	0.170324i
$456$	3.77654			0.176853
$457$			24.0234i			1.12377i	0.827217	+	0.561883i	$0.189923\pi$
$457$			24.0234i			1.12377i	−0.827217	+	0.561883i	$0.810077\pi$
$458$	4.08356			0.190812
$459$	−1.53309			−0.0715585
$460$			40.3073i			1.87934i
$461$		−	1.91163i		−	0.0890334i	−0.999009	−	0.0445167i	$-0.985825\pi$
$461$		−	1.91163i		−	0.0890334i	0.999009	−	0.0445167i	$-0.0141748\pi$
$462$		−	0.845448i		−	0.0393338i
$463$		−	15.2250i		−	0.707567i	−0.935327	−	0.353783i	$-0.884895\pi$
$463$		−	15.2250i		−	0.707567i	0.935327	−	0.353783i	$-0.115105\pi$
$464$	−13.9354			−0.646934
$465$	5.27523			0.244633
$466$			3.11498i			0.144299i
$467$	31.0483			1.43674			0.718371	−	0.695660i	$-0.244888\pi$
$467$	31.0483			1.43674			0.718371	+	0.695660i	$0.244888\pi$
$468$	−6.59136	−	2.39969i	−0.304686	−	0.110925i
$469$	−14.8248			−0.684546
$470$			4.98857i			0.230105i
$471$	1.06618			0.0491270
$472$	−4.46313			−0.205432
$473$			20.9717i			0.964279i
$474$		−	1.53309i		−	0.0704172i
$475$		−	15.0716i		−	0.691534i
$476$		−	2.98262i		−	0.136708i
$477$	−13.3430			−0.610933
$478$	2.97781			0.136202
$479$		−	39.6314i		−	1.81081i	−0.424552	−	0.905403i	$-0.639568\pi$
$479$		−	39.6314i		−	1.81081i	0.424552	−	0.905403i	$-0.360432\pi$
$480$	7.95392			0.363045
$481$	−8.93382	+	24.5391i	−0.407347	+	1.11889i
$482$	2.32259			0.105791
$483$			7.03387i			0.320052i
$484$	−4.11470			−0.187032
$485$	−33.3549			−1.51457
$486$		−	0.233455i		−	0.0105897i
$487$		−	5.91435i		−	0.268005i	−0.990981	−	0.134002i	$-0.957217\pi$
$487$		−	5.91435i		−	0.268005i	0.990981	−	0.134002i	$-0.0427830\pi$
$488$			2.55698i			0.115749i
$489$		−	23.4430i		−	1.06013i
$490$	0.687642			0.0310645
$491$	−40.6068			−1.83256			−0.916280	−	0.400539i	$-0.868823\pi$
$491$	−40.6068			−1.83256			−0.916280	+	0.400539i	$0.868823\pi$
$492$			17.8102i			0.802944i
$493$	−5.81186			−0.261753
$494$	3.24292	+	1.18064i	0.145906	+	0.0531193i
$495$	−10.6670			−0.479446
$496$			6.58346i			0.295606i
$497$	−3.62146			−0.162445
$498$	1.69362			0.0758931
$499$		−	27.9821i		−	1.25265i	−0.779562	−	0.626325i	$-0.784558\pi$
$499$		−	27.9821i		−	1.25265i	0.779562	−	0.626325i	$-0.215442\pi$
$500$			7.58735i			0.339316i
$501$			7.67271i			0.342791i
$502$			5.69635i			0.254241i
$503$	−26.9458			−1.20145			−0.600727	−	0.799455i	$-0.705122\pi$
$503$	−26.9458			−1.20145			−0.600727	+	0.799455i	$0.705122\pi$
$504$	0.921097			0.0410289
$505$		−	17.8167i		−	0.792835i
$506$	−5.94677			−0.264366
$507$	−9.95718	−	8.35791i	−0.442214	−	0.371188i
$508$	−20.4088			−0.905494
$509$		−	0.836496i		−	0.0370770i	−0.999828	−	0.0185385i	$-0.994099\pi$
$509$		−	0.836496i		−	0.0370770i	0.999828	−	0.0185385i	$-0.00590133\pi$
$510$	1.05422			0.0466815
$511$	−14.5670			−0.644404
$512$			16.6983i			0.737966i
$513$		−	4.10005i		−	0.181022i
$514$		−	6.58788i		−	0.290579i
$515$			54.9837i			2.42287i
$516$	−11.2663			−0.495970
$517$	26.2723			1.15545
$518$		−	1.69090i		−	0.0742937i
$519$	2.66701			0.117069
$520$	9.19196	+	3.34647i	0.403094	+	0.146753i
$521$	22.3047			0.977186			0.488593	−	0.872512i	$-0.337510\pi$
$521$	22.3047			0.977186			0.488593	+	0.872512i	$0.337510\pi$
$522$		−	0.885016i		−	0.0387361i
$523$	28.0010			1.22440			0.612200	−	0.790703i	$-0.290285\pi$
$523$	28.0010			1.22440			0.612200	+	0.790703i	$0.290285\pi$
$524$	10.4122			0.454858
$525$		−	3.67596i		−	0.160432i
$526$		−	2.67352i		−	0.116571i
$527$			2.74568i			0.119604i
$528$		−	13.3124i		−	0.579346i
$529$	26.4753			1.15110
$530$	9.17518			0.398544
$531$			4.84545i			0.210275i
$532$	7.97664			0.345831
$533$	−11.2917	+	31.0157i	−0.489099	+	1.34344i
$534$	0.148570			0.00642924
$535$		−	14.2115i		−	0.614416i
$536$	13.6551			0.589810
$537$	11.2340			0.484782
$538$			4.62263i			0.199296i
$539$		−	3.62146i		−	0.155987i
$540$		−	5.73046i		−	0.246600i
$541$		−	5.86764i		−	0.252270i	−0.992013	−	0.126135i	$-0.959743\pi$
$541$		−	5.86764i		−	0.252270i	0.992013	−	0.126135i	$-0.0402572\pi$
$542$	−2.92731			−0.125739
$543$	−9.58189			−0.411198
$544$			4.13990i			0.177497i
$545$	14.6014			0.625454
$546$	0.790947	+	0.287956i	0.0338494	+	0.0123234i
$547$	26.5068			1.13335			0.566674	−	0.823942i	$-0.308230\pi$
$547$	26.5068			1.13335			0.566674	+	0.823942i	$0.308230\pi$
$548$			22.7499i			0.971829i
$549$	2.77601			0.118477
$550$	3.10784			0.132519
$551$		−	15.5431i		−	0.662157i
$552$		−	6.47887i		−	0.275759i
$553$		−	6.56696i		−	0.279256i
$554$		−	0.569402i		−	0.0241916i
$555$	−21.3340			−0.905579
$556$	−28.5347			−1.21014
$557$		−	32.5862i		−	1.38072i	−0.723466	−	0.690360i	$-0.757452\pi$
$557$		−	32.5862i		−	1.38072i	0.723466	−	0.690360i	$-0.242548\pi$
$558$	−0.418106			−0.0176998
$559$	−19.6198	−	7.14287i	−0.829828	−	0.302111i
$560$	10.8275			0.457547
$561$		−	5.55203i		−	0.234407i
$562$	−6.06501			−0.255837
$563$	−9.71425			−0.409407			−0.204703	−	0.978824i	$-0.565623\pi$
$563$	−9.71425			−0.409407			−0.204703	+	0.978824i	$0.565623\pi$
$564$			14.1138i			0.594299i
$565$			11.1662i			0.469767i
$566$		−	1.81675i		−	0.0763635i
$567$		−	1.00000i		−	0.0419961i
$568$	3.33572			0.139964
$569$	3.79095			0.158925			0.0794624	−	0.996838i	$-0.474680\pi$
$569$	3.79095			0.158925			0.0794624	+	0.996838i	$0.474680\pi$
$570$			2.81936i			0.118090i
$571$	−3.76550			−0.157581			−0.0787906	−	0.996891i	$-0.525106\pi$
$571$	−3.76550			−0.157581			−0.0787906	+	0.996891i	$0.525106\pi$
$572$	8.69037	−	23.8704i	0.363363	−	0.998070i
$573$	7.89100			0.329651
$574$		−	2.13718i		−	0.0892040i
$575$	−25.8562			−1.07828
$576$	6.72151			0.280063
$577$		−	10.7228i		−	0.446396i	−0.974773	−	0.223198i	$-0.928350\pi$
$577$		−	10.7228i		−	0.446396i	0.974773	−	0.223198i	$-0.0716496\pi$
$578$		−	3.42003i		−	0.142255i
$579$		−	5.89100i		−	0.244822i
$580$		−	21.7239i		−	0.902035i
$581$	7.25460			0.300972
$582$	2.64365			0.109583
$583$		−	48.3211i		−	2.00125i
$584$	13.4176			0.555223
$585$	3.63314	−	9.97937i	0.150212	−	0.412596i
$586$	−1.66818			−0.0689117
$587$		−	34.0794i		−	1.40661i	−0.710889	−	0.703304i	$-0.751707\pi$
$587$		−	34.0794i		−	1.40661i	0.710889	−	0.703304i	$-0.248293\pi$
$588$	1.94550			0.0802310
$589$	−7.34297			−0.302562
$590$		−	3.33193i		−	0.137173i
$591$			19.1546i			0.787913i
$592$		−	26.6247i		−	1.09427i
$593$			38.9600i			1.59990i	0.600069	+	0.799948i	$0.295140\pi$
$593$			38.9600i			1.59990i	−0.600069	+	0.799948i	$0.704860\pi$
$594$	0.845448			0.0346892
$595$	4.51571			0.185126
$596$		−	24.5667i		−	1.00629i
$597$	−8.20010			−0.335608
$598$	2.02545	−	5.56342i	0.0828267	−	0.227505i
$599$	−4.90151			−0.200270			−0.100135	−	0.994974i	$-0.531928\pi$
$599$	−4.90151			−0.200270			−0.100135	+	0.994974i	$0.531928\pi$
$600$			3.38592i			0.138229i
$601$	9.73970			0.397291			0.198645	−	0.980071i	$-0.436346\pi$
$601$	9.73970			0.397291			0.198645	+	0.980071i	$0.436346\pi$
$602$	1.35193			0.0551004
$603$		−	14.8248i		−	0.603713i
$604$			35.9642i			1.46336i
$605$		−	6.22968i		−	0.253273i
$606$			1.41212i			0.0573636i
$607$	−17.8308			−0.723730			−0.361865	−	0.932231i	$-0.617860\pi$
$607$	−17.8308			−0.723730			−0.361865	+	0.932231i	$0.617860\pi$
$608$	−11.0716			−0.449014
$609$		−	3.79095i		−	0.153617i
$610$	−1.90890			−0.0772892
$611$	−8.94823	+	24.5786i	−0.362006	+	0.994345i
$612$	2.98262			0.120565
$613$		−	7.83777i		−	0.316565i	−0.987394	−	0.158282i	$-0.949404\pi$
$613$		−	7.83777i		−	0.316565i	0.987394	−	0.158282i	$-0.0505956\pi$
$614$	−2.43007			−0.0980696
$615$	−26.9647			−1.08732
$616$			3.33572i			0.134400i
$617$		−	40.0471i		−	1.61224i	−0.591755	−	0.806118i	$-0.701565\pi$
$617$		−	40.0471i		−	1.61224i	0.591755	−	0.806118i	$-0.298435\pi$
$618$		−	4.35791i		−	0.175301i
$619$			9.07269i			0.364662i	0.983237	+	0.182331i	$0.0583643\pi$
$619$			9.07269i			0.364662i	−0.983237	+	0.182331i	$0.941636\pi$
$620$	−10.2630			−0.412170
$621$	−7.03387			−0.282259
$622$			1.05422i			0.0422702i
$623$	0.636396			0.0254967
$624$	12.4542	+	4.53413i	0.498566	+	0.181511i
$625$	−29.8671			−1.19468
$626$		−	5.04828i		−	0.201770i
$627$	14.8482			0.592979
$628$	−2.07425			−0.0827717
$629$		−	11.1041i		−	0.442748i
$630$			0.687642i			0.0273963i
$631$		−	9.11056i		−	0.362686i	−0.983420	−	0.181343i	$-0.941956\pi$
$631$		−	9.11056i		−	0.362686i	0.983420	−	0.181343i	$-0.0580444\pi$
$632$			6.04880i			0.240609i
$633$	11.7009			0.465068
$634$	−3.92847			−0.156020
$635$		−	30.8991i		−	1.22619i
$636$	25.9587			1.02933
$637$	3.38801	+	1.23346i	0.134238	+	0.0488713i
$638$	3.20505			0.126889
$639$		−	3.62146i		−	0.143263i
$640$	−20.5298			−0.811513
$641$	22.8392			0.902095			0.451048	−	0.892500i	$-0.351050\pi$
$641$	22.8392			0.902095			0.451048	+	0.892500i	$0.351050\pi$
$642$			1.12638i			0.0444546i
$643$			15.6844i			0.618532i	0.950976	+	0.309266i	$0.100083\pi$
$643$			15.6844i			0.618532i	−0.950976	+	0.309266i	$0.899917\pi$
$644$		−	13.6844i		−	0.539240i
$645$		−	17.0572i		−	0.671628i
$646$	−1.46744			−0.0577356
$647$	−47.6198			−1.87213			−0.936063	−	0.351832i	$-0.885559\pi$
$647$	−47.6198			−1.87213			−0.936063	+	0.351832i	$0.885559\pi$
$648$			0.921097i			0.0361841i
$649$	−17.5476			−0.688804
$650$	−1.05852	+	2.90749i	−0.0415184	+	0.114041i
$651$	−1.79095			−0.0701928
$652$			45.6084i			1.78616i
$653$	42.4002			1.65925			0.829624	−	0.558322i	$-0.188555\pi$
$653$	42.4002			1.65925			0.829624	+	0.558322i	$0.188555\pi$
$654$	−1.15728			−0.0452532
$655$			15.7641i			0.615954i
$656$		−	33.6518i		−	1.31388i
$657$		−	14.5670i		−	0.568311i
$658$		−	1.69362i		−	0.0660243i
$659$	15.7497			0.613521			0.306760	−	0.951787i	$-0.400755\pi$
$659$	15.7497			0.613521			0.306760	+	0.951787i	$0.400755\pi$
$660$	20.7527			0.807796
$661$		−	17.3684i		−	0.675553i	−0.941226	−	0.337777i	$-0.890325\pi$
$661$		−	17.3684i		−	0.675553i	0.941226	−	0.337777i	$-0.109675\pi$
$662$	−8.43601			−0.327875
$663$	5.19412	+	1.89100i	0.201723	+	0.0734403i
$664$	−6.68219			−0.259319
$665$			12.0767i			0.468314i
$666$	1.69090			0.0655209
$667$	−26.6650			−1.03247
$668$		−	14.9272i		−	0.577552i
$669$		−	5.45198i		−	0.210786i
$670$			10.1942i			0.393835i
$671$			10.0532i			0.388100i
$672$	−2.70037			−0.104169
$673$	−35.6754			−1.37519			−0.687593	−	0.726096i	$-0.741333\pi$
$673$	−35.6754			−1.37519			−0.687593	+	0.726096i	$0.741333\pi$
$674$		−	7.81129i		−	0.300880i
$675$	3.67596			0.141488
$676$	19.3717	+	16.2603i	0.745064	+	0.625396i
$677$	−30.7228			−1.18077			−0.590386	−	0.807121i	$-0.701025\pi$
$677$	−30.7228			−1.18077			−0.590386	+	0.807121i	$0.701025\pi$
$678$		−	0.885016i		−	0.0339888i
$679$	11.3240			0.434577
$680$	−4.15941			−0.159506
$681$			27.1133i			1.03898i
$682$		−	1.51415i		−	0.0579799i
$683$			12.2018i			0.466889i	0.972370	+	0.233444i	$0.0749997\pi$
$683$			12.2018i			0.466889i	−0.972370	+	0.233444i	$0.925000\pi$
$684$			7.97664i			0.304995i
$685$	−34.4436			−1.31602
$686$	−0.233455			−0.00891335
$687$			17.4918i			0.667355i
$688$	21.2873			0.811571
$689$	45.2061	+	16.4580i	1.72221	+	0.626998i
$690$	4.83678			0.184133
$691$			19.7198i			0.750177i	0.926989	+	0.375089i	$0.122388\pi$
$691$			19.7198i			0.750177i	−0.926989	+	0.375089i	$0.877612\pi$
$692$	−5.18867			−0.197243
$693$	3.62146			0.137568
$694$		−	2.03741i		−	0.0773391i
$695$		−	43.2017i		−	1.63873i
$696$			3.49183i			0.132357i
$697$		−	14.0348i		−	0.531604i
$698$	−0.199069			−0.00753486
$699$	−13.3430			−0.504678
$700$			7.15158i			0.270304i
$701$	−38.4211			−1.45115			−0.725573	−	0.688145i	$-0.758425\pi$
$701$	−38.4211			−1.45115			−0.725573	+	0.688145i	$0.758425\pi$
$702$	−0.287956	+	0.790947i	−0.0108682	+	0.0298524i
$703$	29.6963			1.12002
$704$			24.3417i			0.917412i
$705$	−21.3684			−0.804781
$706$	−1.74243			−0.0655771
$707$			6.04880i			0.227489i
$708$		−	9.42681i		−	0.354281i
$709$		−	29.1573i		−	1.09502i	−0.836798	−	0.547512i	$-0.815575\pi$
$709$		−	29.1573i		−	1.09502i	0.836798	−	0.547512i	$-0.184425\pi$
$710$			2.49027i			0.0934581i
$711$	6.56696			0.246280
$712$	−0.586182			−0.0219681
$713$			12.5973i			0.471772i
$714$	−0.357908			−0.0133944
$715$	36.1399	+	13.1573i	1.35156	+	0.492054i
$716$	−21.8557			−0.816785
$717$			12.7554i			0.476358i
$718$	−2.89528			−0.108051
$719$	−32.5157			−1.21263			−0.606316	−	0.795224i	$-0.707353\pi$
$719$	−32.5157			−1.21263			−0.606316	+	0.795224i	$0.707353\pi$
$720$			10.8275i			0.403519i
$721$		−	18.6670i		−	0.695196i
$722$			0.511170i			0.0190238i
$723$			9.94875i			0.369998i
$724$	18.6416			0.692808
$725$	13.9354			0.517547
$726$			0.493754i			0.0183249i
$727$	19.9821			0.741095			0.370547	−	0.928814i	$-0.379170\pi$
$727$	19.9821			0.741095			0.370547	+	0.928814i	$0.379170\pi$
$728$	−3.12068	−	1.13613i	−0.115660	−	0.0421078i
$729$	1.00000			0.0370370
$730$			10.0168i			0.370740i
$731$	8.87804			0.328366
$732$	−5.40073			−0.199617
$733$		−	26.1956i		−	0.967555i	−0.875191	−	0.483778i	$-0.839264\pi$
$733$		−	26.1956i		−	0.967555i	0.875191	−	0.483778i	$-0.160736\pi$
$734$		−	2.17465i		−	0.0802679i
$735$			2.94550i			0.108646i
$736$			18.9940i			0.700129i
$737$	53.6875			1.97760
$738$	2.13718			0.0786705
$739$			14.6247i			0.537979i	0.963143	+	0.268989i	$0.0866897\pi$
$739$			14.6247i			0.537979i	−0.963143	+	0.268989i	$0.913310\pi$
$740$	41.5053			1.52577
$741$	−5.05723	+	13.8910i	−0.185782	+	0.510299i
$742$	−3.11498			−0.114355
$743$			15.4268i			0.565955i	0.959127	+	0.282977i	$0.0913222\pi$
$743$			15.4268i			0.565955i	−0.959127	+	0.282977i	$0.908678\pi$
$744$	1.64964			0.0604786
$745$	37.1941			1.36269
$746$		−	0.871532i		−	0.0319091i
$747$			7.25460i			0.265432i
$748$			10.8015i			0.394941i
$749$			4.82482i			0.176295i
$750$	0.910463			0.0332454
$751$	2.62716			0.0958664			0.0479332	−	0.998851i	$-0.484737\pi$
$751$	2.62716			0.0958664			0.0479332	+	0.998851i	$0.484737\pi$
$752$		−	26.6676i		−	0.972469i
$753$	−24.4002			−0.889193
$754$	−1.09163	+	2.99844i	−0.0397547	+	0.109197i
$755$	−54.4500			−1.98164
$756$			1.94550i			0.0707571i
$757$	35.9454			1.30646			0.653228	−	0.757161i	$-0.273414\pi$
$757$	35.9454			1.30646			0.653228	+	0.757161i	$0.273414\pi$
$758$	2.80146			0.101754
$759$		−	25.4729i		−	0.924607i
$760$		−	11.1238i		−	0.403503i
$761$		−	24.1028i		−	0.873725i	−0.899528	−	0.436862i	$-0.856090\pi$
$761$		−	24.1028i		−	0.873725i	0.899528	−	0.436862i	$-0.143910\pi$
$762$			2.44901i			0.0887181i
$763$	−4.95718			−0.179462
$764$	−15.3519			−0.555413
$765$			4.51571i			0.163266i
$766$	−2.05538			−0.0742640
$767$	5.97664	−	16.4164i	0.215804	−	0.592762i
$768$	−11.8159			−0.426368
$769$			51.8831i			1.87095i	0.353391	+	0.935476i	$0.385029\pi$
$769$			51.8831i			1.87095i	−0.353391	+	0.935476i	$0.614971\pi$
$770$	−2.49027			−0.0897430
$771$	28.2190			1.01628
$772$			11.4609i			0.412488i
$773$			30.8833i			1.11080i	0.831585	+	0.555398i	$0.187434\pi$
$773$			30.8833i			1.11080i	−0.831585	+	0.555398i	$0.812566\pi$
$774$			1.35193i			0.0485940i
$775$		−	6.58346i		−	0.236485i
$776$	−10.4305			−0.374434
$777$	7.24292			0.259838
$778$		−	6.18605i		−	0.221781i
$779$	37.5341			1.34480
$780$	−7.06827	+	19.4149i	−0.253085	+	0.695163i
$781$	13.1150			0.469291
$782$			2.51748i			0.0900247i
$783$	3.79095			0.135477
$784$	−3.67596			&mi

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:	\( N \)	\(=\)	\( 273 = 3 \cdot 7 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	273.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.233455i q^{2} -1.00000 q^{3} +1.94550 q^{4} +2.94550i q^{5} -0.233455i q^{6} -1.00000i q^{7} +0.921097i q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+0.233455i q^{2} -1.00000 q^{3} +1.94550 q^{4} +2.94550i q^{5} -0.233455i q^{6} -1.00000i q^{7} +0.921097i q^{8} +1.00000 q^{9} -0.687642 q^{10} +3.62146i q^{11} -1.94550 q^{12} +(-3.38801 - 1.23346i) q^{13} +0.233455 q^{14} -2.94550i q^{15} +3.67596 q^{16} +1.53309 q^{17} +0.233455i q^{18} +4.10005i q^{19} +5.73046i q^{20} +1.00000i q^{21} -0.845448 q^{22} +7.03387 q^{23} -0.921097i q^{24} -3.67596 q^{25} +(0.287956 - 0.790947i) q^{26} -1.00000 q^{27} -1.94550i q^{28} -3.79095 q^{29} +0.687642 q^{30} +1.79095i q^{31} +2.70037i q^{32} -3.62146i q^{33} +0.357908i q^{34} +2.94550 q^{35} +1.94550 q^{36} -7.24292i q^{37} -0.957177 q^{38} +(3.38801 + 1.23346i) q^{39} -2.71309 q^{40} -9.15455i q^{41} -0.233455 q^{42} +5.79095 q^{43} +7.04555i q^{44} +2.94550i q^{45} +1.64209i q^{46} -7.25460i q^{47} -3.67596 q^{48} -1.00000 q^{49} -0.858172i q^{50} -1.53309 q^{51} +(-6.59136 - 2.39969i) q^{52} -13.3430 q^{53} -0.233455i q^{54} -10.6670 q^{55} +0.921097 q^{56} -4.10005i q^{57} -0.885016i q^{58} +4.84545i q^{59} -5.73046i q^{60} +2.77601 q^{61} -0.418106 q^{62} -1.00000i q^{63} +6.72151 q^{64} +(3.63314 - 9.97937i) q^{65} +0.845448 q^{66} -14.8248i q^{67} +2.98262 q^{68} -7.03387 q^{69} +0.687642i q^{70} -3.62146i q^{71} +0.921097i q^{72} -14.5670i q^{73} +1.69090 q^{74} +3.67596 q^{75} +7.97664i q^{76} +3.62146 q^{77} +(-0.287956 + 0.790947i) q^{78} +6.56696 q^{79} +10.8275i q^{80} +1.00000 q^{81} +2.13718 q^{82} +7.25460i q^{83} +1.94550i q^{84} +4.51571i q^{85} +1.35193i q^{86} +3.79095 q^{87} -3.33572 q^{88} +0.636396i q^{89} -0.687642 q^{90} +(-1.23346 + 3.38801i) q^{91} +13.6844 q^{92} -1.79095i q^{93} +1.69362 q^{94} -12.0767 q^{95} -2.70037i q^{96} +11.3240i q^{97} -0.233455i q^{98} +3.62146i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9}+O(q^{10}) \) 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9	\( 8 q - 8 q^{3} - 14 q^{4} + 8 q^{9} + 4 q^{10} + 14 q^{12} - 6 q^{13} - 2 q^{14} + 34 q^{16} + 20 q^{17} - 24 q^{22} - 6 q^{23} - 34 q^{25} + 28 q^{26} - 8 q^{27} - 18 q^{29} - 4 q^{30} - 6 q^{35} - 14 q^{36} + 36 q^{38} + 6 q^{39} - 8 q^{40} + 2 q^{42} + 34 q^{43} - 34 q^{48} - 8 q^{49} - 20 q^{51} + 18 q^{52} - 10 q^{53} + 16 q^{55} - 6 q^{56} - 20 q^{61} - 28 q^{62} - 18 q^{64} - 10 q^{65} + 24 q^{66} - 24 q^{68} + 6 q^{69} + 48 q^{74} + 34 q^{75} + 4 q^{77} - 28 q^{78} - 2 q^{79} + 8 q^{81} - 48 q^{82} + 18 q^{87} - 8 q^{88} + 4 q^{90} - 6 q^{91} + 56 q^{92} - 72 q^{94} - 78 q^{95}+O(q^{100}) \) 8 * q - 8 * q^3 - 14 * q^4 + 8 * q^9 + 4 * q^10 + 14 * q^12 - 6 * q^13 - 2 * q^14 + 34 * q^16 + 20 * q^17 - 24 * q^22 - 6 * q^23 - 34 * q^25 + 28 * q^26 - 8 * q^27 - 18 * q^29 - 4 * q^30 - 6 * q^35 - 14 * q^36 + 36 * q^38 + 6 * q^39 - 8 * q^40 + 2 * q^42 + 34 * q^43 - 34 * q^48 - 8 * q^49 - 20 * q^51 + 18 * q^52 - 10 * q^53 + 16 * q^55 - 6 * q^56 - 20 * q^61 - 28 * q^62 - 18 * q^64 - 10 * q^65 + 24 * q^66 - 24 * q^68 + 6 * q^69 + 48 * q^74 + 34 * q^75 + 4 * q^77 - 28 * q^78 - 2 * q^79 + 8 * q^81 - 48 * q^82 + 18 * q^87 - 8 * q^88 + 4 * q^90 - 6 * q^91 + 56 * q^92 - 72 * q^94 - 78 * q^95

Introduction

L-functions

Modular forms

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