LMFDB - Embedded newform 209.2.a.b.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [209,2,Mod(1,209)] 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("209.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(209, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 209 = 11 \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	209.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$1.66887340224$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{8})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 2 $ x^2 - 2
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			1.1
Root			$-1.41421$ of defining polynomial
Character	$\chi$	$=$	209.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421 q^{2} +0.414214 q^{3} -1.00000 q^{5} -0.585786 q^{6} -0.585786 q^{7} +2.82843 q^{8} -2.82843 q^{9} +1.41421 q^{10} -1.00000 q^{11} -6.24264 q^{13} +0.828427 q^{14} -0.414214 q^{15} -4.00000 q^{16} +0.585786 q^{17} +4.00000 q^{18} -1.00000 q^{19} -0.242641 q^{21} +1.41421 q^{22} -3.00000 q^{23} +1.17157 q^{24} -4.00000 q^{25} +8.82843 q^{26} -2.41421 q^{27} +2.24264 q^{29} +0.585786 q^{30} -3.58579 q^{31} -0.414214 q^{33} -0.828427 q^{34} +0.585786 q^{35} -4.07107 q^{37} +1.41421 q^{38} -2.58579 q^{39} -2.82843 q^{40} +9.65685 q^{41} +0.343146 q^{42} +11.6569 q^{43} +2.82843 q^{45} +4.24264 q^{46} +3.17157 q^{47} -1.65685 q^{48} -6.65685 q^{49} +5.65685 q^{50} +0.242641 q^{51} +12.4853 q^{53} +3.41421 q^{54} +1.00000 q^{55} -1.65685 q^{56} -0.414214 q^{57} -3.17157 q^{58} -4.41421 q^{59} +3.07107 q^{61} +5.07107 q^{62} +1.65685 q^{63} +8.00000 q^{64} +6.24264 q^{65} +0.585786 q^{66} -7.58579 q^{67} -1.24264 q^{69} -0.828427 q^{70} -9.58579 q^{71} -8.00000 q^{72} +12.4853 q^{73} +5.75736 q^{74} -1.65685 q^{75} +0.585786 q^{77} +3.65685 q^{78} -17.4142 q^{79} +4.00000 q^{80} +7.48528 q^{81} -13.6569 q^{82} +0.585786 q^{83} -0.585786 q^{85} -16.4853 q^{86} +0.928932 q^{87} -2.82843 q^{88} -14.8995 q^{89} -4.00000 q^{90} +3.65685 q^{91} -1.48528 q^{93} -4.48528 q^{94} +1.00000 q^{95} -0.414214 q^{97} +9.41421 q^{98} +2.82843 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{3} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 2 q^{11} - 4 q^{13} - 4 q^{14} + 2 q^{15} - 8 q^{16} + 4 q^{17} + 8 q^{18} - 2 q^{19} + 8 q^{21} - 6 q^{23} + 8 q^{24} - 8 q^{25} + 12 q^{26} - 2 q^{27} - 4 q^{29}+ \cdots + 16 q^{98}+O(q^{100}) $ 2 * q - 2 * q^3 - 2 * q^5 - 4 * q^6 - 4 * q^7 - 2 * q^11 - 4 * q^13 - 4 * q^14 + 2 * q^15 - 8 * q^16 + 4 * q^17 + 8 * q^18 - 2 * q^19 + 8 * q^21 - 6 * q^23 + 8 * q^24 - 8 * q^25 + 12 * q^26 - 2 * q^27 - 4 * q^29 + 4 * q^30 - 10 * q^31 + 2 * q^33 + 4 * q^34 + 4 * q^35 + 6 * q^37 - 8 * q^39 + 8 * q^41 + 12 * q^42 + 12 * q^43 + 12 * q^47 + 8 * q^48 - 2 * q^49 - 8 * q^51 + 8 * q^53 + 4 * q^54 + 2 * q^55 + 8 * q^56 + 2 * q^57 - 12 * q^58 - 6 * q^59 - 8 * q^61 - 4 * q^62 - 8 * q^63 + 16 * q^64 + 4 * q^65 + 4 * q^66 - 18 * q^67 + 6 * q^69 + 4 * q^70 - 22 * q^71 - 16 * q^72 + 8 * q^73 + 20 * q^74 + 8 * q^75 + 4 * q^77 - 4 * q^78 - 32 * q^79 + 8 * q^80 - 2 * q^81 - 16 * q^82 + 4 * q^83 - 4 * q^85 - 16 * q^86 + 16 * q^87 - 10 * q^89 - 8 * q^90 - 4 * q^91 + 14 * q^93 + 8 * q^94 + 2 * q^95 + 2 * q^97 + 16 * 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 $

$2$	−1.41421	−1.00000	−0.500000	−	0.866025i	$-0.666667\pi$
$2$	−1.41421	−1.00000	−0.500000	+	0.866025i	$0.666667\pi$
$3$	0.414214	0.239146	0.119573	−	0.992825i	$-0.461847\pi$
$3$	0.414214	0.239146	0.119573	+	0.992825i	$0.461847\pi$
$4$	0	0
$5$	−1.00000	−0.447214	−0.223607	−	0.974679i	$-0.571783\pi$
$5$	−1.00000	−0.447214	−0.223607	+	0.974679i	$0.571783\pi$
$6$	−0.585786	−0.239146
$7$	−0.585786	−0.221406	−0.110703	−	0.993854i	$-0.535310\pi$
$7$	−0.585786	−0.221406	−0.110703	+	0.993854i	$0.535310\pi$
$8$	2.82843	1.00000
$9$	−2.82843	−0.942809
$10$	1.41421	0.447214
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−6.24264	−1.73140	−0.865699	−	0.500566i	$-0.833125\pi$
$13$	−6.24264	−1.73140	−0.865699	+	0.500566i	$0.833125\pi$
$14$	0.828427	0.221406
$15$	−0.414214	−0.106949
$16$	−4.00000	−1.00000
$17$	0.585786	0.142074	0.0710370	−	0.997474i	$-0.477369\pi$
$17$	0.585786	0.142074	0.0710370	+	0.997474i	$0.477369\pi$
$18$	4.00000	0.942809
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	0	0
$21$	−0.242641	−0.0529485
$22$	1.41421	0.301511
$23$	−3.00000	−0.625543	−0.312772	−	0.949828i	$-0.601257\pi$
$23$	−3.00000	−0.625543	−0.312772	+	0.949828i	$0.601257\pi$
$24$	1.17157	0.239146
$25$	−4.00000	−0.800000
$26$	8.82843	1.73140
$27$	−2.41421	−0.464616
$28$	0	0
$29$	2.24264	0.416448	0.208224	−	0.978081i	$-0.433232\pi$
$29$	2.24264	0.416448	0.208224	+	0.978081i	$0.433232\pi$
$30$	0.585786	0.106949
$31$	−3.58579	−0.644026	−0.322013	−	0.946735i	$-0.604360\pi$
$31$	−3.58579	−0.644026	−0.322013	+	0.946735i	$0.604360\pi$
$32$	0	0
$33$	−0.414214	−0.0721053
$34$	−0.828427	−0.142074
$35$	0.585786	0.0990160
$36$	0	0
$37$	−4.07107	−0.669279	−0.334640	−	0.942346i	$-0.608615\pi$
$37$	−4.07107	−0.669279	−0.334640	+	0.942346i	$0.608615\pi$
$38$	1.41421	0.229416
$39$	−2.58579	−0.414057
$40$	−2.82843	−0.447214
$41$	9.65685	1.50815	0.754074	−	0.656790i	$-0.228086\pi$
$41$	9.65685	1.50815	0.754074	+	0.656790i	$0.228086\pi$
$42$	0.343146	0.0529485
$43$	11.6569	1.77765	0.888827	−	0.458243i	$-0.151521\pi$
$43$	11.6569	1.77765	0.888827	+	0.458243i	$0.151521\pi$
$44$	0	0
$45$	2.82843	0.421637
$46$	4.24264	0.625543
$47$	3.17157	0.462621	0.231311	−	0.972880i	$-0.425699\pi$
$47$	3.17157	0.462621	0.231311	+	0.972880i	$0.425699\pi$
$48$	−1.65685	−0.239146
$49$	−6.65685	−0.950979
$50$	5.65685	0.800000
$51$	0.242641	0.0339765
$52$	0	0
$53$	12.4853	1.71499	0.857493	−	0.514496i	$-0.172021\pi$
$53$	12.4853	1.71499	0.857493	+	0.514496i	$0.172021\pi$
$54$	3.41421	0.464616
$55$	1.00000	0.134840
$56$	−1.65685	−0.221406
$57$	−0.414214	−0.0548639
$58$	−3.17157	−0.416448
$59$	−4.41421	−0.574682	−0.287341	−	0.957828i	$-0.592771\pi$
$59$	−4.41421	−0.574682	−0.287341	+	0.957828i	$0.592771\pi$
$60$	0	0
$61$	3.07107	0.393210	0.196605	−	0.980483i	$-0.437008\pi$
$61$	3.07107	0.393210	0.196605	+	0.980483i	$0.437008\pi$
$62$	5.07107	0.644026
$63$	1.65685	0.208744
$64$	8.00000	1.00000
$65$	6.24264	0.774304
$66$	0.585786	0.0721053
$67$	−7.58579	−0.926751	−0.463376	−	0.886162i	$-0.653362\pi$
$67$	−7.58579	−0.926751	−0.463376	+	0.886162i	$0.653362\pi$
$68$	0	0
$69$	−1.24264	−0.149596
$70$	−0.828427	−0.0990160
$71$	−9.58579	−1.13762	−0.568812	−	0.822468i	$-0.692597\pi$
$71$	−9.58579	−1.13762	−0.568812	+	0.822468i	$0.692597\pi$
$72$	−8.00000	−0.942809
$73$	12.4853	1.46129	0.730646	−	0.682757i	$-0.239219\pi$
$73$	12.4853	1.46129	0.730646	+	0.682757i	$0.239219\pi$
$74$	5.75736	0.669279
$75$	−1.65685	−0.191317
$76$	0	0
$77$	0.585786	0.0667566
$78$	3.65685	0.414057
$79$	−17.4142	−1.95925	−0.979626	−	0.200830i	$-0.935636\pi$
$79$	−17.4142	−1.95925	−0.979626	+	0.200830i	$0.935636\pi$
$80$	4.00000	0.447214
$81$	7.48528	0.831698
$82$	−13.6569	−1.50815
$83$	0.585786	0.0642984	0.0321492	−	0.999483i	$-0.489765\pi$
$83$	0.585786	0.0642984	0.0321492	+	0.999483i	$0.489765\pi$
$84$	0	0
$85$	−0.585786	−0.0635375
$86$	−16.4853	−1.77765
$87$	0.928932	0.0995920
$88$	−2.82843	−0.301511
$89$	−14.8995	−1.57934	−0.789672	−	0.613530i	$-0.789749\pi$
$89$	−14.8995	−1.57934	−0.789672	+	0.613530i	$0.789749\pi$
$90$	−4.00000	−0.421637
$91$	3.65685	0.383342
$92$	0	0
$93$	−1.48528	−0.154017
$94$	−4.48528	−0.462621
$95$	1.00000	0.102598
$96$	0	0
$97$	−0.414214	−0.0420570	−0.0210285	−	0.999779i	$-0.506694\pi$
$97$	−0.414214	−0.0420570	−0.0210285	+	0.999779i	$0.506694\pi$
$98$	9.41421	0.950979
$99$	2.82843	0.284268
$100$	0	0
$101$	−2.24264	−0.223151	−0.111576	−	0.993756i	$-0.535590\pi$
$101$	−2.24264	−0.223151	−0.111576	+	0.993756i	$0.535590\pi$
$102$	−0.343146	−0.0339765
$103$	2.34315	0.230877	0.115439	−	0.993315i	$-0.463173\pi$
$103$	2.34315	0.230877	0.115439	+	0.993315i	$0.463173\pi$
$104$	−17.6569	−1.73140
$105$	0.242641	0.0236793
$106$	−17.6569	−1.71499
$107$	4.34315	0.419868	0.209934	−	0.977716i	$-0.432675\pi$
$107$	4.34315	0.419868	0.209934	+	0.977716i	$0.432675\pi$
$108$	0	0
$109$	−11.6569	−1.11652	−0.558262	−	0.829665i	$-0.688532\pi$
$109$	−11.6569	−1.11652	−0.558262	+	0.829665i	$0.688532\pi$
$110$	−1.41421	−0.134840
$111$	−1.68629	−0.160056
$112$	2.34315	0.221406
$113$	−6.41421	−0.603398	−0.301699	−	0.953403i	$-0.597554\pi$
$113$	−6.41421	−0.603398	−0.301699	+	0.953403i	$0.597554\pi$
$114$	0.585786	0.0548639
$115$	3.00000	0.279751
$116$	0	0
$117$	17.6569	1.63238
$118$	6.24264	0.574682
$119$	−0.343146	−0.0314561
$120$	−1.17157	−0.106949
$121$	1.00000	0.0909091
$122$	−4.34315	−0.393210
$123$	4.00000	0.360668
$124$	0	0
$125$	9.00000	0.804984
$126$	−2.34315	−0.208744
$127$	−16.5858	−1.47175	−0.735875	−	0.677117i	$-0.763229\pi$
$127$	−16.5858	−1.47175	−0.735875	+	0.677117i	$0.763229\pi$
$128$	−11.3137	−1.00000
$129$	4.82843	0.425119
$130$	−8.82843	−0.774304
$131$	−2.48528	−0.217140	−0.108570	−	0.994089i	$-0.534627\pi$
$131$	−2.48528	−0.217140	−0.108570	+	0.994089i	$0.534627\pi$
$132$	0	0
$133$	0.585786	0.0507941
$134$	10.7279	0.926751
$135$	2.41421	0.207782
$136$	1.65685	0.142074
$137$	8.65685	0.739605	0.369802	−	0.929110i	$-0.379425\pi$
$137$	8.65685	0.739605	0.369802	+	0.929110i	$0.379425\pi$
$138$	1.75736	0.149596
$139$	−16.3848	−1.38974	−0.694869	−	0.719136i	$-0.744538\pi$
$139$	−16.3848	−1.38974	−0.694869	+	0.719136i	$0.744538\pi$
$140$	0	0
$141$	1.31371	0.110634
$142$	13.5563	1.13762
$143$	6.24264	0.522036
$144$	11.3137	0.942809
$145$	−2.24264	−0.186241
$146$	−17.6569	−1.46129
$147$	−2.75736	−0.227423
$148$	0	0
$149$	−1.31371	−0.107623	−0.0538116	−	0.998551i	$-0.517137\pi$
$149$	−1.31371	−0.107623	−0.0538116	+	0.998551i	$0.517137\pi$
$150$	2.34315	0.191317
$151$	6.48528	0.527765	0.263882	−	0.964555i	$-0.414997\pi$
$151$	6.48528	0.527765	0.263882	+	0.964555i	$0.414997\pi$
$152$	−2.82843	−0.229416
$153$	−1.65685	−0.133949
$154$	−0.828427	−0.0667566
$155$	3.58579	0.288017
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	24.6274	1.95925
$159$	5.17157	0.410132
$160$	0	0
$161$	1.75736	0.138499
$162$	−10.5858	−0.831698
$163$	8.14214	0.637741	0.318871	−	0.947798i	$-0.396696\pi$
$163$	8.14214	0.637741	0.318871	+	0.947798i	$0.396696\pi$
$164$	0	0
$165$	0.414214	0.0322465
$166$	−0.828427	−0.0642984
$167$	6.72792	0.520622	0.260311	−	0.965525i	$-0.416175\pi$
$167$	6.72792	0.520622	0.260311	+	0.965525i	$0.416175\pi$
$168$	−0.686292	−0.0529485
$169$	25.9706	1.99774
$170$	0.828427	0.0635375
$171$	2.82843	0.216295
$172$	0	0
$173$	−17.8995	−1.36087	−0.680437	−	0.732807i	$-0.738210\pi$
$173$	−17.8995	−1.36087	−0.680437	+	0.732807i	$0.738210\pi$
$174$	−1.31371	−0.0995920
$175$	2.34315	0.177125
$176$	4.00000	0.301511
$177$	−1.82843	−0.137433
$178$	21.0711	1.57934
$179$	1.92893	0.144175	0.0720876	−	0.997398i	$-0.477034\pi$
$179$	1.92893	0.144175	0.0720876	+	0.997398i	$0.477034\pi$
$180$	0	0
$181$	17.3848	1.29220	0.646100	−	0.763253i	$-0.276399\pi$
$181$	17.3848	1.29220	0.646100	+	0.763253i	$0.276399\pi$
$182$	−5.17157	−0.383342
$183$	1.27208	0.0940347
$184$	−8.48528	−0.625543
$185$	4.07107	0.299311
$186$	2.10051	0.154017
$187$	−0.585786	−0.0428369
$188$	0	0
$189$	1.41421	0.102869
$190$	−1.41421	−0.102598
$191$	−14.3137	−1.03570	−0.517852	−	0.855470i	$-0.673268\pi$
$191$	−14.3137	−1.03570	−0.517852	+	0.855470i	$0.673268\pi$
$192$	3.31371	0.239146
$193$	6.82843	0.491521	0.245760	−	0.969331i	$-0.420962\pi$
$193$	6.82843	0.491521	0.245760	+	0.969331i	$0.420962\pi$
$194$	0.585786	0.0420570
$195$	2.58579	0.185172
$196$	0	0
$197$	−3.89949	−0.277828	−0.138914	−	0.990304i	$-0.544361\pi$
$197$	−3.89949	−0.277828	−0.138914	+	0.990304i	$0.544361\pi$
$198$	−4.00000	−0.284268
$199$	−16.1421	−1.14429	−0.572143	−	0.820154i	$-0.693888\pi$
$199$	−16.1421	−1.14429	−0.572143	+	0.820154i	$0.693888\pi$
$200$	−11.3137	−0.800000
$201$	−3.14214	−0.221629
$202$	3.17157	0.223151
$203$	−1.31371	−0.0922043
$204$	0	0
$205$	−9.65685	−0.674464
$206$	−3.31371	−0.230877
$207$	8.48528	0.589768
$208$	24.9706	1.73140
$209$	1.00000	0.0691714
$210$	−0.343146	−0.0236793
$211$	−15.4142	−1.06116	−0.530579	−	0.847635i	$-0.678026\pi$
$211$	−15.4142	−1.06116	−0.530579	+	0.847635i	$0.678026\pi$
$212$	0	0
$213$	−3.97056	−0.272058
$214$	−6.14214	−0.419868
$215$	−11.6569	−0.794991
$216$	−6.82843	−0.464616
$217$	2.10051	0.142592
$218$	16.4853	1.11652
$219$	5.17157	0.349463
$220$	0	0
$221$	−3.65685	−0.245987
$222$	2.38478	0.160056
$223$	−13.5858	−0.909772	−0.454886	−	0.890550i	$-0.650320\pi$
$223$	−13.5858	−0.909772	−0.454886	+	0.890550i	$0.650320\pi$
$224$	0	0
$225$	11.3137	0.754247
$226$	9.07107	0.603398
$227$	19.0711	1.26579	0.632896	−	0.774237i	$-0.281866\pi$
$227$	19.0711	1.26579	0.632896	+	0.774237i	$0.281866\pi$
$228$	0	0
$229$	8.31371	0.549385	0.274693	−	0.961532i	$-0.411424\pi$
$229$	8.31371	0.549385	0.274693	+	0.961532i	$0.411424\pi$
$230$	−4.24264	−0.279751
$231$	0.242641	0.0159646
$232$	6.34315	0.416448
$233$	8.24264	0.539993	0.269997	−	0.962861i	$-0.412977\pi$
$233$	8.24264	0.539993	0.269997	+	0.962861i	$0.412977\pi$
$234$	−24.9706	−1.63238
$235$	−3.17157	−0.206891
$236$	0	0
$237$	−7.21320	−0.468548
$238$	0.485281	0.0314561
$239$	6.00000	0.388108	0.194054	−	0.980991i	$-0.437836\pi$
$239$	6.00000	0.388108	0.194054	+	0.980991i	$0.437836\pi$
$240$	1.65685	0.106949
$241$	16.9706	1.09317	0.546585	−	0.837404i	$-0.315928\pi$
$241$	16.9706	1.09317	0.546585	+	0.837404i	$0.315928\pi$
$242$	−1.41421	−0.0909091
$243$	10.3431	0.663513
$244$	0	0
$245$	6.65685	0.425291
$246$	−5.65685	−0.360668
$247$	6.24264	0.397210
$248$	−10.1421	−0.644026
$249$	0.242641	0.0153767
$250$	−12.7279	−0.804984
$251$	−22.6569	−1.43009	−0.715044	−	0.699079i	$-0.753593\pi$
$251$	−22.6569	−1.43009	−0.715044	+	0.699079i	$0.753593\pi$
$252$	0	0
$253$	3.00000	0.188608
$254$	23.4558	1.47175
$255$	−0.242641	−0.0151947
$256$	0	0
$257$	−24.8284	−1.54875	−0.774377	−	0.632724i	$-0.781937\pi$
$257$	−24.8284	−1.54875	−0.774377	+	0.632724i	$0.781937\pi$
$258$	−6.82843	−0.425119
$259$	2.38478	0.148183
$260$	0	0
$261$	−6.34315	−0.392631
$262$	3.51472	0.217140
$263$	−26.4853	−1.63315	−0.816576	−	0.577238i	$-0.804131\pi$
$263$	−26.4853	−1.63315	−0.816576	+	0.577238i	$0.804131\pi$
$264$	−1.17157	−0.0721053
$265$	−12.4853	−0.766965
$266$	−0.828427	−0.0507941
$267$	−6.17157	−0.377694
$268$	0	0
$269$	−2.00000	−0.121942	−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000	−0.121942	−0.0609711	+	0.998140i	$0.519420\pi$
$270$	−3.41421	−0.207782
$271$	−22.1421	−1.34504	−0.672519	−	0.740079i	$-0.734788\pi$
$271$	−22.1421	−1.34504	−0.672519	+	0.740079i	$0.734788\pi$
$272$	−2.34315	−0.142074
$273$	1.51472	0.0916749
$274$	−12.2426	−0.739605
$275$	4.00000	0.241209
$276$	0	0
$277$	14.4853	0.870336	0.435168	−	0.900349i	$-0.356689\pi$
$277$	14.4853	0.870336	0.435168	+	0.900349i	$0.356689\pi$
$278$	23.1716	1.38974
$279$	10.1421	0.607194
$280$	1.65685	0.0990160
$281$	12.3431	0.736330	0.368165	−	0.929760i	$-0.379986\pi$
$281$	12.3431	0.736330	0.368165	+	0.929760i	$0.379986\pi$
$282$	−1.85786	−0.110634
$283$	8.72792	0.518821	0.259411	−	0.965767i	$-0.416472\pi$
$283$	8.72792	0.518821	0.259411	+	0.965767i	$0.416472\pi$
$284$	0	0
$285$	0.414214	0.0245359
$286$	−8.82843	−0.522036
$287$	−5.65685	−0.333914
$288$	0	0
$289$	−16.6569	−0.979815
$290$	3.17157	0.186241
$291$	−0.171573	−0.0100578
$292$	0	0
$293$	27.6985	1.61816	0.809081	−	0.587697i	$-0.199965\pi$
$293$	27.6985	1.61816	0.809081	+	0.587697i	$0.199965\pi$
$294$	3.89949	0.227423
$295$	4.41421	0.257005
$296$	−11.5147	−0.669279
$297$	2.41421	0.140087
$298$	1.85786	0.107623
$299$	18.7279	1.08306
$300$	0	0
$301$	−6.82843	−0.393584
$302$	−9.17157	−0.527765
$303$	−0.928932	−0.0533658
$304$	4.00000	0.229416
$305$	−3.07107	−0.175849
$306$	2.34315	0.133949
$307$	4.58579	0.261725	0.130862	−	0.991401i	$-0.458225\pi$
$307$	4.58579	0.261725	0.130862	+	0.991401i	$0.458225\pi$
$308$	0	0
$309$	0.970563	0.0552134
$310$	−5.07107	−0.288017
$311$	−0.343146	−0.0194580	−0.00972901	−	0.999953i	$-0.503097\pi$
$311$	−0.343146	−0.0194580	−0.00972901	+	0.999953i	$0.503097\pi$
$312$	−7.31371	−0.414057
$313$	19.9706	1.12880	0.564401	−	0.825500i	$-0.309107\pi$
$313$	19.9706	1.12880	0.564401	+	0.825500i	$0.309107\pi$
$314$	−7.07107	−0.399043
$315$	−1.65685	−0.0933532
$316$	0	0
$317$	−25.5858	−1.43704	−0.718520	−	0.695506i	$-0.755180\pi$
$317$	−25.5858	−1.43704	−0.718520	+	0.695506i	$0.755180\pi$
$318$	−7.31371	−0.410132
$319$	−2.24264	−0.125564
$320$	−8.00000	−0.447214
$321$	1.79899	0.100410
$322$	−2.48528	−0.138499
$323$	−0.585786	−0.0325940
$324$	0	0
$325$	24.9706	1.38512
$326$	−11.5147	−0.637741
$327$	−4.82843	−0.267013
$328$	27.3137	1.50815
$329$	−1.85786	−0.102427
$330$	−0.585786	−0.0322465
$331$	−8.21320	−0.451438	−0.225719	−	0.974192i	$-0.572473\pi$
$331$	−8.21320	−0.451438	−0.225719	+	0.974192i	$0.572473\pi$
$332$	0	0
$333$	11.5147	0.631003
$334$	−9.51472	−0.520622
$335$	7.58579	0.414456
$336$	0.970563	0.0529485
$337$	−8.72792	−0.475440	−0.237720	−	0.971334i	$-0.576400\pi$
$337$	−8.72792	−0.475440	−0.237720	+	0.971334i	$0.576400\pi$
$338$	−36.7279	−1.99774
$339$	−2.65685	−0.144301
$340$	0	0
$341$	3.58579	0.194181
$342$	−4.00000	−0.216295
$343$	8.00000	0.431959
$344$	32.9706	1.77765
$345$	1.24264	0.0669015
$346$	25.3137	1.36087
$347$	−25.6985	−1.37957	−0.689783	−	0.724016i	$-0.742294\pi$
$347$	−25.6985	−1.37957	−0.689783	+	0.724016i	$0.742294\pi$
$348$	0	0
$349$	−19.2132	−1.02846	−0.514230	−	0.857653i	$-0.671922\pi$
$349$	−19.2132	−1.02846	−0.514230	+	0.857653i	$0.671922\pi$
$350$	−3.31371	−0.177125
$351$	15.0711	0.804434
$352$	0	0
$353$	−1.68629	−0.0897522	−0.0448761	−	0.998993i	$-0.514289\pi$
$353$	−1.68629	−0.0897522	−0.0448761	+	0.998993i	$0.514289\pi$
$354$	2.58579	0.137433
$355$	9.58579	0.508761
$356$	0	0
$357$	−0.142136	−0.00752261
$358$	−2.72792	−0.144175
$359$	−0.485281	−0.0256122	−0.0128061	−	0.999918i	$-0.504076\pi$
$359$	−0.485281	−0.0256122	−0.0128061	+	0.999918i	$0.504076\pi$
$360$	8.00000	0.421637
$361$	1.00000	0.0526316
$362$	−24.5858	−1.29220
$363$	0.414214	0.0217406
$364$	0	0
$365$	−12.4853	−0.653509
$366$	−1.79899	−0.0940347
$367$	−4.17157	−0.217754	−0.108877	−	0.994055i	$-0.534726\pi$
$367$	−4.17157	−0.217754	−0.108877	+	0.994055i	$0.534726\pi$
$368$	12.0000	0.625543
$369$	−27.3137	−1.42189
$370$	−5.75736	−0.299311
$371$	−7.31371	−0.379709
$372$	0	0
$373$	−25.3137	−1.31069	−0.655347	−	0.755328i	$-0.727478\pi$
$373$	−25.3137	−1.31069	−0.655347	+	0.755328i	$0.727478\pi$
$374$	0.828427	0.0428369
$375$	3.72792	0.192509
$376$	8.97056	0.462621
$377$	−14.0000	−0.721037
$378$	−2.00000	−0.102869
$379$	28.6985	1.47414	0.737071	−	0.675815i	$-0.236208\pi$
$379$	28.6985	1.47414	0.737071	+	0.675815i	$0.236208\pi$
$380$	0	0
$381$	−6.87006	−0.351964
$382$	20.2426	1.03570
$383$	−31.5269	−1.61095	−0.805475	−	0.592630i	$-0.798090\pi$
$383$	−31.5269	−1.61095	−0.805475	+	0.592630i	$0.798090\pi$
$384$	−4.68629	−0.239146
$385$	−0.585786	−0.0298544
$386$	−9.65685	−0.491521
$387$	−32.9706	−1.67599
$388$	0	0
$389$	−5.68629	−0.288306	−0.144153	−	0.989555i	$-0.546046\pi$
$389$	−5.68629	−0.288306	−0.144153	+	0.989555i	$0.546046\pi$
$390$	−3.65685	−0.185172
$391$	−1.75736	−0.0888735
$392$	−18.8284	−0.950979
$393$	−1.02944	−0.0519282
$394$	5.51472	0.277828
$395$	17.4142	0.876204
$396$	0	0
$397$	26.0000	1.30490	0.652451	−	0.757831i	$-0.273741\pi$
$397$	26.0000	1.30490	0.652451	+	0.757831i	$0.273741\pi$
$398$	22.8284	1.14429
$399$	0.242641	0.0121472
$400$	16.0000	0.800000
$401$	−1.02944	−0.0514076	−0.0257038	−	0.999670i	$-0.508183\pi$
$401$	−1.02944	−0.0514076	−0.0257038	+	0.999670i	$0.508183\pi$
$402$	4.44365	0.221629
$403$	22.3848	1.11507
$404$	0	0
$405$	−7.48528	−0.371947
$406$	1.85786	0.0922043
$407$	4.07107	0.201795
$408$	0.686292	0.0339765
$409$	−24.7279	−1.22272	−0.611359	−	0.791354i	$-0.709377\pi$
$409$	−24.7279	−1.22272	−0.611359	+	0.791354i	$0.709377\pi$
$410$	13.6569	0.674464
$411$	3.58579	0.176874
$412$	0	0
$413$	2.58579	0.127238
$414$	−12.0000	−0.589768
$415$	−0.585786	−0.0287551
$416$	0	0
$417$	−6.78680	−0.332351
$418$	−1.41421	−0.0691714
$419$	23.4558	1.14589	0.572946	−	0.819593i	$-0.305800\pi$
$419$	23.4558	1.14589	0.572946	+	0.819593i	$0.305800\pi$
$420$	0	0
$421$	0.142136	0.00692727	0.00346363	−	0.999994i	$-0.498897\pi$
$421$	0.142136	0.00692727	0.00346363	+	0.999994i	$0.498897\pi$
$422$	21.7990	1.06116
$423$	−8.97056	−0.436164
$424$	35.3137	1.71499
$425$	−2.34315	−0.113659
$426$	5.61522	0.272058
$427$	−1.79899	−0.0870592
$428$	0	0
$429$	2.58579	0.124843
$430$	16.4853	0.794991
$431$	13.5147	0.650981	0.325491	−	0.945545i	$-0.394471\pi$
$431$	13.5147	0.650981	0.325491	+	0.945545i	$0.394471\pi$
$432$	9.65685	0.464616
$433$	−25.3848	−1.21991	−0.609957	−	0.792434i	$-0.708813\pi$
$433$	−25.3848	−1.21991	−0.609957	+	0.792434i	$0.708813\pi$
$434$	−2.97056	−0.142592
$435$	−0.928932	−0.0445389
$436$	0	0
$437$	3.00000	0.143509
$438$	−7.31371	−0.349463
$439$	−5.75736	−0.274784	−0.137392	−	0.990517i	$-0.543872\pi$
$439$	−5.75736	−0.274784	−0.137392	+	0.990517i	$0.543872\pi$
$440$	2.82843	0.134840
$441$	18.8284	0.896592
$442$	5.17157	0.245987
$443$	21.9706	1.04385	0.521926	−	0.852990i	$-0.325214\pi$
$443$	21.9706	1.04385	0.521926	+	0.852990i	$0.325214\pi$
$444$	0	0
$445$	14.8995	0.706304
$446$	19.2132	0.909772
$447$	−0.544156	−0.0257377
$448$	−4.68629	−0.221406
$449$	23.3848	1.10360	0.551798	−	0.833978i	$-0.313942\pi$
$449$	23.3848	1.10360	0.551798	+	0.833978i	$0.313942\pi$
$450$	−16.0000	−0.754247
$451$	−9.65685	−0.454724
$452$	0	0
$453$	2.68629	0.126213
$454$	−26.9706	−1.26579
$455$	−3.65685	−0.171436
$456$	−1.17157	−0.0548639
$457$	31.0711	1.45344	0.726722	−	0.686932i	$-0.241043\pi$
$457$	31.0711	1.45344	0.726722	+	0.686932i	$0.241043\pi$
$458$	−11.7574	−0.549385
$459$	−1.41421	−0.0660098
$460$	0	0
$461$	−20.9706	−0.976696	−0.488348	−	0.872649i	$-0.662400\pi$
$461$	−20.9706	−0.976696	−0.488348	+	0.872649i	$0.662400\pi$
$462$	−0.343146	−0.0159646
$463$	−38.4558	−1.78719	−0.893597	−	0.448870i	$-0.851827\pi$
$463$	−38.4558	−1.78719	−0.893597	+	0.448870i	$0.851827\pi$
$464$	−8.97056	−0.416448
$465$	1.48528	0.0688783
$466$	−11.6569	−0.539993
$467$	32.3137	1.49530	0.747650	−	0.664093i	$-0.231182\pi$
$467$	32.3137	1.49530	0.747650	+	0.664093i	$0.231182\pi$
$468$	0	0
$469$	4.44365	0.205189
$470$	4.48528	0.206891
$471$	2.07107	0.0954298
$472$	−12.4853	−0.574682
$473$	−11.6569	−0.535983
$474$	10.2010	0.468548
$475$	4.00000	0.183533
$476$	0	0
$477$	−35.3137	−1.61690
$478$	−8.48528	−0.388108
$479$	24.5269	1.12066	0.560332	−	0.828268i	$-0.310674\pi$
$479$	24.5269	1.12066	0.560332	+	0.828268i	$0.310674\pi$
$480$	0	0
$481$	25.4142	1.15879
$482$	−24.0000	−1.09317
$483$	0.727922	0.0331216
$484$	0	0
$485$	0.414214	0.0188085
$486$	−14.6274	−0.663513
$487$	−18.5563	−0.840868	−0.420434	−	0.907323i	$-0.638122\pi$
$487$	−18.5563	−0.840868	−0.420434	+	0.907323i	$0.638122\pi$
$488$	8.68629	0.393210
$489$	3.37258	0.152513
$490$	−9.41421	−0.425291
$491$	36.2426	1.63561	0.817804	−	0.575497i	$-0.195191\pi$
$491$	36.2426	1.63561	0.817804	+	0.575497i	$0.195191\pi$
$492$	0	0
$493$	1.31371	0.0591665
$494$	−8.82843	−0.397210
$495$	−2.82843	−0.127128
$496$	14.3431	0.644026
$497$	5.61522	0.251877
$498$	−0.343146	−0.0153767
$499$	12.6274	0.565281	0.282640	−	0.959226i	$-0.408790\pi$
$499$	12.6274	0.565281	0.282640	+	0.959226i	$0.408790\pi$
$500$	0	0
$501$	2.78680	0.124505
$502$	32.0416	1.43009
$503$	0.142136	0.00633751	0.00316876	−	0.999995i	$-0.498991\pi$
$503$	0.142136	0.00633751	0.00316876	+	0.999995i	$0.498991\pi$
$504$	4.68629	0.208744
$505$	2.24264	0.0997962
$506$	−4.24264	−0.188608
$507$	10.7574	0.477751
$508$	0	0
$509$	34.2132	1.51647	0.758237	−	0.651979i	$-0.226061\pi$
$509$	34.2132	1.51647	0.758237	+	0.651979i	$0.226061\pi$
$510$	0.343146	0.0151947
$511$	−7.31371	−0.323539
$512$	22.6274	1.00000
$513$	2.41421	0.106590
$514$	35.1127	1.54875
$515$	−2.34315	−0.103251
$516$	0	0
$517$	−3.17157	−0.139486
$518$	−3.37258	−0.148183
$519$	−7.41421	−0.325448
$520$	17.6569	0.774304
$521$	−30.5563	−1.33870	−0.669349	−	0.742948i	$-0.733427\pi$
$521$	−30.5563	−1.33870	−0.669349	+	0.742948i	$0.733427\pi$
$522$	8.97056	0.392631
$523$	13.6569	0.597173	0.298586	−	0.954383i	$-0.403485\pi$
$523$	13.6569	0.597173	0.298586	+	0.954383i	$0.403485\pi$
$524$	0	0
$525$	0.970563	0.0423588
$526$	37.4558	1.63315
$527$	−2.10051	−0.0914994
$528$	1.65685	0.0721053
$529$	−14.0000	−0.608696
$530$	17.6569	0.766965
$531$	12.4853	0.541815
$532$	0	0
$533$	−60.2843	−2.61120
$534$	8.72792	0.377694
$535$	−4.34315	−0.187771
$536$	−21.4558	−0.926751
$537$	0.798990	0.0344790
$538$	2.82843	0.121942
$539$	6.65685	0.286731
$540$	0	0
$541$	−33.2132	−1.42795	−0.713974	−	0.700173i	$-0.753106\pi$
$541$	−33.2132	−1.42795	−0.713974	+	0.700173i	$0.753106\pi$
$542$	31.3137	1.34504
$543$	7.20101	0.309025
$544$	0	0
$545$	11.6569	0.499325
$546$	−2.14214	−0.0916749
$547$	10.7279	0.458693	0.229346	−	0.973345i	$-0.426341\pi$
$547$	10.7279	0.458693	0.229346	+	0.973345i	$0.426341\pi$
$548$	0	0
$549$	−8.68629	−0.370722
$550$	−5.65685	−0.241209
$551$	−2.24264	−0.0955397
$552$	−3.51472	−0.149596
$553$	10.2010	0.433791
$554$	−20.4853	−0.870336
$555$	1.68629	0.0715791
$556$	0	0
$557$	39.9411	1.69236	0.846180	−	0.532897i	$-0.178897\pi$
$557$	39.9411	1.69236	0.846180	+	0.532897i	$0.178897\pi$
$558$	−14.3431	−0.607194
$559$	−72.7696	−3.07782
$560$	−2.34315	−0.0990160
$561$	−0.242641	−0.0102443
$562$	−17.4558	−0.736330
$563$	19.2132	0.809740	0.404870	−	0.914374i	$-0.367317\pi$
$563$	19.2132	0.809740	0.404870	+	0.914374i	$0.367317\pi$
$564$	0	0
$565$	6.41421	0.269848
$566$	−12.3431	−0.518821
$567$	−4.38478	−0.184143
$568$	−27.1127	−1.13762
$569$	−17.7574	−0.744427	−0.372214	−	0.928147i	$-0.621401\pi$
$569$	−17.7574	−0.744427	−0.372214	+	0.928147i	$0.621401\pi$
$570$	−0.585786	−0.0245359
$571$	−25.6985	−1.07545	−0.537724	−	0.843121i	$-0.680716\pi$
$571$	−25.6985	−1.07545	−0.537724	+	0.843121i	$0.680716\pi$
$572$	0	0
$573$	−5.92893	−0.247685
$574$	8.00000	0.333914
$575$	12.0000	0.500435
$576$	−22.6274	−0.942809
$577$	−3.97056	−0.165297	−0.0826483	−	0.996579i	$-0.526338\pi$
$577$	−3.97056	−0.165297	−0.0826483	+	0.996579i	$0.526338\pi$
$578$	23.5563	0.979815
$579$	2.82843	0.117545
$580$	0	0
$581$	−0.343146	−0.0142361
$582$	0.242641	0.0100578
$583$	−12.4853	−0.517088
$584$	35.3137	1.46129
$585$	−17.6569	−0.730021
$586$	−39.1716	−1.61816
$587$	−3.65685	−0.150935	−0.0754673	−	0.997148i	$-0.524045\pi$
$587$	−3.65685	−0.150935	−0.0754673	+	0.997148i	$0.524045\pi$
$588$	0	0
$589$	3.58579	0.147750
$590$	−6.24264	−0.257005
$591$	−1.61522	−0.0664414
$592$	16.2843	0.669279
$593$	3.02944	0.124404	0.0622020	−	0.998064i	$-0.480188\pi$
$593$	3.02944	0.124404	0.0622020	+	0.998064i	$0.480188\pi$
$594$	−3.41421	−0.140087
$595$	0.343146	0.0140676
$596$	0	0
$597$	−6.68629	−0.273652
$598$	−26.4853	−1.08306
$599$	−9.31371	−0.380548	−0.190274	−	0.981731i	$-0.560938\pi$
$599$	−9.31371	−0.380548	−0.190274	+	0.981731i	$0.560938\pi$
$600$	−4.68629	−0.191317
$601$	−19.5563	−0.797720	−0.398860	−	0.917012i	$-0.630594\pi$
$601$	−19.5563	−0.797720	−0.398860	+	0.917012i	$0.630594\pi$
$602$	9.65685	0.393584
$603$	21.4558	0.873750
$604$	0	0
$605$	−1.00000	−0.0406558
$606$	1.31371	0.0533658
$607$	−21.8995	−0.888873	−0.444437	−	0.895810i	$-0.646596\pi$
$607$	−21.8995	−0.888873	−0.444437	+	0.895810i	$0.646596\pi$
$608$	0	0
$609$	−0.544156	−0.0220503
$610$	4.34315	0.175849
$611$	−19.7990	−0.800981
$612$	0	0
$613$	−10.5858	−0.427556	−0.213778	−	0.976882i	$-0.568577\pi$
$613$	−10.5858	−0.427556	−0.213778	+	0.976882i	$0.568577\pi$
$614$	−6.48528	−0.261725
$615$	−4.00000	−0.161296
$616$	1.65685	0.0667566
$617$	17.1716	0.691301	0.345651	−	0.938363i	$-0.387658\pi$
$617$	17.1716	0.691301	0.345651	+	0.938363i	$0.387658\pi$
$618$	−1.37258	−0.0552134
$619$	6.51472	0.261849	0.130924	−	0.991392i	$-0.458206\pi$
$619$	6.51472	0.261849	0.130924	+	0.991392i	$0.458206\pi$
$620$	0	0
$621$	7.24264	0.290637
$622$	0.485281	0.0194580
$623$	8.72792	0.349677
$624$	10.3431	0.414057
$625$	11.0000	0.440000
$626$	−28.2426	−1.12880
$627$	0.414214	0.0165421
$628$	0	0
$629$	−2.38478	−0.0950873
$630$	2.34315	0.0933532
$631$	−41.9706	−1.67082	−0.835411	−	0.549626i	$-0.814770\pi$
$631$	−41.9706	−1.67082	−0.835411	+	0.549626i	$0.814770\pi$
$632$	−49.2548	−1.95925
$633$	−6.38478	−0.253772
$634$	36.1838	1.43704
$635$	16.5858	0.658187
$636$	0	0
$637$	41.5563	1.64652
$638$	3.17157	0.125564
$639$	27.1127	1.07256
$640$	11.3137	0.447214
$641$	−18.8995	−0.746485	−0.373243	−	0.927734i	$-0.621754\pi$
$641$	−18.8995	−0.746485	−0.373243	+	0.927734i	$0.621754\pi$
$642$	−2.54416	−0.100410
$643$	13.3431	0.526202	0.263101	−	0.964768i	$-0.415255\pi$
$643$	13.3431	0.526202	0.263101	+	0.964768i	$0.415255\pi$
$644$	0	0
$645$	−4.82843	−0.190119
$646$	0.828427	0.0325940
$647$	36.9411	1.45231	0.726153	−	0.687533i	$-0.241307\pi$
$647$	36.9411	1.45231	0.726153	+	0.687533i	$0.241307\pi$
$648$	21.1716	0.831698
$649$	4.41421	0.173273
$650$	−35.3137	−1.38512
$651$	0.870058	0.0341002
$652$	0	0
$653$	−25.4853	−0.997316	−0.498658	−	0.866799i	$-0.666174\pi$
$653$	−25.4853	−0.997316	−0.498658	+	0.866799i	$0.666174\pi$
$654$	6.82843	0.267013
$655$	2.48528	0.0971080
$656$	−38.6274	−1.50815
$657$	−35.3137	−1.37772
$658$	2.62742	0.102427
$659$	−17.7990	−0.693350	−0.346675	−	0.937985i	$-0.612689\pi$
$659$	−17.7990	−0.693350	−0.346675	+	0.937985i	$0.612689\pi$
$660$	0	0
$661$	3.87006	0.150528	0.0752639	−	0.997164i	$-0.476020\pi$
$661$	3.87006	0.150528	0.0752639	+	0.997164i	$0.476020\pi$
$662$	11.6152	0.451438
$663$	−1.51472	−0.0588268
$664$	1.65685	0.0642984
$665$	−0.585786	−0.0227158
$666$	−16.2843	−0.631003
$667$	−6.72792	−0.260506
$668$	0	0
$669$	−5.62742	−0.217569
$670$	−10.7279	−0.414456
$671$	−3.07107	−0.118557
$672$	0	0
$673$	−32.1421	−1.23899	−0.619494	−	0.785001i	$-0.712662\pi$
$673$	−32.1421	−1.23899	−0.619494	+	0.785001i	$0.712662\pi$
$674$	12.3431	0.475440
$675$	9.65685	0.371692
$676$	0	0
$677$	−1.31371	−0.0504899	−0.0252450	−	0.999681i	$-0.508037\pi$
$677$	−1.31371	−0.0504899	−0.0252450	+	0.999681i	$0.508037\pi$
$678$	3.75736	0.144301
$679$	0.242641	0.00931169
$680$	−1.65685	−0.0635375
$681$	7.89949	0.302709
$682$	−5.07107	−0.194181
$683$	−39.1127	−1.49661	−0.748303	−	0.663357i	$-0.769131\pi$
$683$	−39.1127	−1.49661	−0.748303	+	0.663357i	$0.769131\pi$
$684$	0	0
$685$	−8.65685	−0.330761
$686$	−11.3137	−0.431959
$687$	3.44365	0.131383
$688$	−46.6274	−1.77765
$689$	−77.9411	−2.96932
$690$	−1.75736	−0.0669015
$691$	−7.97056	−0.303214	−0.151607	−	0.988441i	$-0.548445\pi$
$691$	−7.97056	−0.303214	−0.151607	+	0.988441i	$0.548445\pi$
$692$	0	0
$693$	−1.65685	−0.0629387
$694$	36.3431	1.37957
$695$	16.3848	0.621510
$696$	2.62742	0.0995920
$697$	5.65685	0.214269
$698$	27.1716	1.02846
$699$	3.41421	0.129137
$700$	0	0
$701$	21.3137	0.805008	0.402504	−	0.915418i	$-0.368140\pi$
$701$	21.3137	0.805008	0.402504	+	0.915418i	$0.368140\pi$
$702$	−21.3137	−0.804434
$703$	4.07107	0.153543
$704$	−8.00000	−0.301511
$705$	−1.31371	−0.0494771
$706$	2.38478	0.0897522
$707$	1.31371	0.0494071
$708$	0	0
$709$	31.2843	1.17491	0.587453	−	0.809258i	$-0.300131\pi$
$709$	31.2843	1.17491	0.587453	+	0.809258i	$0.300131\pi$
$710$	−13.5563	−0.508761
$711$	49.2548	1.84720
$712$	−42.1421	−1.57934
$713$	10.7574	0.402866
$714$	0.201010	0.00752261
$715$	−6.24264	−0.233462
$716$	0	0
$717$	2.48528	0.0928145
$718$	0.686292	0.0256122
$719$	43.9706	1.63983	0.819913	−	0.572489i	$-0.194022\pi$
$719$	43.9706	1.63983	0.819913	+	0.572489i	$0.194022\pi$
$720$	−11.3137	−0.421637
$721$	−1.37258	−0.0511177
$722$	−1.41421	−0.0526316
$723$	7.02944	0.261428
$724$	0	0
$725$	−8.97056	−0.333158
$726$	−0.585786	−0.0217406
$727$	44.1127	1.63605	0.818025	−	0.575183i	$-0.195069\pi$
$727$	44.1127	1.63605	0.818025	+	0.575183i	$0.195069\pi$
$728$	10.3431	0.383342
$729$	−18.1716	−0.673021
$730$	17.6569	0.653509
$731$	6.82843	0.252559
$732$	0	0
$733$	10.5858	0.390995	0.195497	−	0.980704i	$-0.437368\pi$
$733$	10.5858	0.390995	0.195497	+	0.980704i	$0.437368\pi$
$734$	5.89949	0.217754
$735$	2.75736	0.101707
$736$	0	0
$737$	7.58579	0.279426
$738$	38.6274	1.42189
$739$	−24.5858	−0.904403	−0.452201	−	0.891916i	$-0.649361\pi$
$739$	−24.5858	−0.904403	−0.452201	+	0.891916i	$0.649361\pi$
$740$	0	0
$741$	2.58579	0.0949912
$742$	10.3431	0.379709
$743$	−21.0711	−0.773023	−0.386511	−	0.922285i	$-0.626320\pi$
$743$	−21.0711	−0.773023	−0.386511	+	0.922285i	$0.626320\pi$
$744$	−4.20101	−0.154017
$745$	1.31371	0.0481306
$746$	35.7990	1.31069
$747$	−1.65685	−0.0606211
$748$	0	0
$749$	−2.54416	−0.0929614
$750$	−5.27208	−0.192509
$751$	34.5563	1.26098	0.630490	−	0.776198i	$-0.282854\pi$
$751$	34.5563	1.26098	0.630490	+	0.776198i	$0.282854\pi$
$752$	−12.6863	−0.462621
$753$	−9.38478	−0.342000
$754$	19.7990	0.721037
$755$	−6.48528	−0.236024
$756$	0	0
$757$	−25.9411	−0.942846	−0.471423	−	0.881907i	$-0.656260\pi$
$757$	−25.9411	−0.942846	−0.471423	+	0.881907i	$0.656260\pi$
$758$	−40.5858	−1.47414
$759$	1.24264	0.0451050
$760$	2.82843	0.102598
$761$	34.1421	1.23765	0.618826	−	0.785528i	$-0.287609\pi$
$761$	34.1421	1.23765	0.618826	+	0.785528i	$0.287609\pi$
$762$	9.71573	0.351964
$763$	6.82843	0.247206
$764$	0	0
$765$	1.65685	0.0599037
$766$	44.5858	1.61095
$767$	27.5563	0.995002
$768$	0	0
$769$	30.1421	1.08695	0.543477	−	0.839424i	$-0.317108\pi$
$769$	30.1421	1.08695	0.543477	+	0.839424i	$0.317108\pi$
$770$	0.828427	0.0298544
$771$	−10.2843	−0.370379
$772$	0	0
$773$	8.34315	0.300082	0.150041	−	0.988680i	$-0.452059\pi$
$773$	8.34315	0.300082	0.150041	+	0.988680i	$0.452059\pi$
$774$	46.6274	1.67599
$775$	14.3431	0.515221
$776$	−1.17157	−0.0420570
$777$	0.987807	0.0354374
$778$	8.04163	0.288306
$779$	−9.65685	−0.345993
$780$	0	0
$781$	9.58579	0.343006
$782$	2.48528	0.0888735
$783$	−5.41421	−0.193488
$784$	26.6274	0.950979
$785$	−5.00000	−0.178458
$786$	1.45584	0.0519282
$787$	36.0000	1.28326	0.641631	−	0.767014i	$-0.278258\pi$
$787$	36.0000	1.28326	0.641631	+	0.767014i	$0.278258\pi$
$788$	0	0
$789$	−10.9706	−0.390562
$790$	−24.6274	−0.876204
$791$	3.75736	0.133596
$792$	8.00000	0.284268
$793$	−19.1716	−0.680803
$794$	−36.7696	−1.30490
$795$	−5.17157	−0.183417
$796$	0	0
$797$	−4.75736	−0.168514	−0.0842572	−	0.996444i	$-0.526852\pi$
$797$	−4.75736	−0.168514	−0.0842572	+	0.996444i	$0.526852\pi$
$798$	−0.343146	−0.0121472
$799$	1.85786	0.0657265
$800$	0	0
$801$	42.1421	1.48902
$802$	1.45584	0.0514076
$803$	−12.4853	−0.440596
$804$	0	0
$805$	−1.75736	−0.0619388
$806$	−31.6569	−1.11507
$807$	−0.828427	−0.0291620
$808$	−6.34315	−0.223151
$809$	52.4853	1.84528	0.922642	−	0.385657i	$-0.126025\pi$
$809$	52.4853	1.84528	0.922642	+	0.385657i	$0.126025\pi$
$810$	10.5858	0.371947
$811$	52.2426	1.83449	0.917244	−	0.398327i	$-0.130409\pi$
$811$	52.2426	1.83449	0.917244	+	0.398327i	$0.130409\pi$
$812$	0	0
$813$	−9.17157	−0.321661
$814$	−5.75736	−0.201795
$815$	−8.14214	−0.285207
$816$	−0.970563	−0.0339765
$817$	−11.6569	−0.407822
$818$	34.9706	1.22272
$819$	−10.3431	−0.361419
$820$	0	0
$821$	26.5269	0.925796	0.462898	−	0.886412i	$-0.346810\pi$
$821$	26.5269	0.925796	0.462898	+	0.886412i	$0.346810\pi$
$822$	−5.07107	−0.176874
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	6.62742	0.230877
$825$	1.65685	0.0576843
$826$	−3.65685	−0.127238
$827$	4.10051	0.142589	0.0712943	−	0.997455i	$-0.477287\pi$
$827$	4.10051	0.142589	0.0712943	+	0.997455i	$0.477287\pi$
$828$	0	0
$829$	11.0416	0.383492	0.191746	−	0.981445i	$-0.438585\pi$
$829$	11.0416	0.383492	0.191746	+	0.981445i	$0.438585\pi$
$830$	0.828427	0.0287551
$831$	6.00000	0.208138
$832$	−49.9411	−1.73140
$833$	−3.89949	−0.135109
$834$	9.59798	0.332351
$835$	−6.72792	−0.232829
$836$	0	0
$837$	8.65685	0.299225
$838$	−33.1716	−1.14589
$839$	40.5563	1.40016	0.700080	−	0.714064i	$-0.253148\pi$
$839$	40.5563	1.40016	0.700080	+	0.714064i	$0.253148\pi$
$840$	0.686292	0.0236793
$841$	−23.9706	−0.826571
$842$	−0.201010	−0.00692727
$843$	5.11270	0.176091
$844$	0	0
$845$	−25.9706	−0.893415
$846$	12.6863	0.436164
$847$	−0.585786	−0.0201279
$848$	−49.9411	−1.71499
$849$	3.61522	0.124074
$850$	3.31371	0.113659
$851$	12.2132	0.418663
$852$	0	0
$853$	−14.0000	−0.479351	−0.239675	−	0.970853i	$-0.577041\pi$
$853$	−14.0000	−0.479351	−0.239675	+	0.970853i	$0.577041\pi$
$854$	2.54416	0.0870592
$855$	−2.82843	−0.0967302
$856$	12.2843	0.419868
$857$	47.6985	1.62935	0.814675	−	0.579918i	$-0.196916\pi$
$857$	47.6985	1.62935	0.814675	+	0.579918i	$0.196916\pi$
$858$	−3.65685	−0.124843
$859$	−29.2843	−0.999166	−0.499583	−	0.866266i	$-0.666513\pi$
$859$	−29.2843	−0.999166	−0.499583	+	0.866266i	$0.666513\pi$
$860$	0	0
$861$	−2.34315	−0.0798542
$862$	−19.1127	−0.650981
$863$	47.5980	1.62025	0.810127	−	0.586254i	$-0.199398\pi$
$863$	47.5980	1.62025	0.810127	+	0.586254i	$0.199398\pi$
$864$	0	0
$865$	17.8995	0.608601
$866$	35.8995	1.21991
$867$	−6.89949	−0.234319
$868$	0	0
$869$	17.4142	0.590737
$870$	1.31371	0.0445389
$871$	47.3553	1.60457
$872$	−32.9706	−1.11652
$873$	1.17157	0.0396517
$874$	−4.24264	−0.143509
$875$	−5.27208	−0.178229
$876$	0	0
$877$	−9.85786	−0.332876	−0.166438	−	0.986052i	$-0.553227\pi$
$877$	−9.85786	−0.332876	−0.166438	+	0.986052i	$0.553227\pi$
$878$	8.14214	0.274784
$879$	11.4731	0.386978
$880$	−4.00000	−0.134840
$881$	21.7696	0.733435	0.366717	−	0.930332i	$-0.380482\pi$
$881$	21.7696	0.733435	0.366717	+	0.930332i	$0.380482\pi$
$882$	−26.6274	−0.896592
$883$	33.5147	1.12786	0.563930	−	0.825823i	$-0.309289\pi$
$883$	33.5147	1.12786	0.563930	+	0.825823i	$0.309289\pi$
$884$	0	0
$885$	1.82843	0.0614619
$886$	−31.0711	−1.04385
$887$	−51.2548	−1.72097	−0.860484	−	0.509477i	$-0.829839\pi$
$887$	−51.2548	−1.72097	−0.860484	+	0.509477i	$0.829839\pi$
$888$	−4.76955	−0.160056
$889$	9.71573	0.325855
$890$	−21.0711	−0.706304
$891$	−7.48528	−0.250766
$892$	0	0
$893$	−3.17157	−0.106133
$894$	0.769553	0.0257377
$895$	−1.92893	−0.0644771
$896$	6.62742	0.221406
$897$	7.75736	0.259011
$898$	−33.0711	−1.10360
$899$	−8.04163	−0.268203
$900$	0	0
$901$	7.31371	0.243655
$902$	13.6569	0.454724
$903$	−2.82843	−0.0941242
$904$	−18.1421	−0.603398
$905$	−17.3848	−0.577890
$906$	−3.79899	−0.126213
$907$	−50.1421	−1.66494	−0.832471	−	0.554068i	$-0.813075\pi$
$907$	−50.1421	−1.66494	−0.832471	+	0.554068i	$0.813075\pi$
$908$	0	0
$909$	6.34315	0.210389
$910$	5.17157	0.171436
$911$	−46.4264	−1.53818	−0.769088	−	0.639143i	$-0.779289\pi$
$911$	−46.4264	−1.53818	−0.769088	+	0.639143i	$0.779289\pi$
$912$	1.65685	0.0548639
$913$	−0.585786	−0.0193867
$914$	−43.9411	−1.45344
$915$	−1.27208	−0.0420536
$916$	0	0
$917$	1.45584	0.0480762
$918$	2.00000	0.0660098
$919$	−19.6152	−0.647047	−0.323523	−	0.946220i	$-0.604867\pi$
$919$	−19.6152	−0.647047	−0.323523	+	0.946220i	$0.604867\pi$
$920$	8.48528	0.279751
$921$	1.89949	0.0625905
$922$	29.6569	0.976696
$923$	59.8406	1.96968
$924$	0	0
$925$	16.2843	0.535424
$926$	54.3848	1.78719
$927$	−6.62742	−0.217673
$928$	0	0
$929$	−56.2843	−1.84663	−0.923314	−	0.384047i	$-0.874530\pi$
$929$	−56.2843	−1.84663	−0.923314	+	0.384047i	$0.874530\pi$
$930$	−2.10051	−0.0688783
$931$	6.65685	0.218170
$932$	0	0
$933$	−0.142136	−0.00465331
$934$	−45.6985	−1.49530
$935$	0.585786	0.0191573
$936$	49.9411	1.63238
$937$	23.5563	0.769552	0.384776	−	0.923010i	$-0.374279\pi$
$937$	23.5563	0.769552	0.384776	+	0.923010i	$0.374279\pi$
$938$	−6.28427	−0.205189
$939$	8.27208	0.269949
$940$	0	0
$941$	19.7574	0.644072	0.322036	−	0.946728i	$-0.395633\pi$
$941$	19.7574	0.644072	0.322036	+	0.946728i	$0.395633\pi$
$942$	−2.92893	−0.0954298
$943$	−28.9706	−0.943411
$944$	17.6569	0.574682
$945$	−1.41421	−0.0460044
$946$	16.4853	0.535983
$947$	5.97056	0.194017	0.0970086	−	0.995284i	$-0.469073\pi$
$947$	5.97056	0.194017	0.0970086	+	0.995284i	$0.469073\pi$
$948$	0	0
$949$	−77.9411	−2.53008
$950$	−5.65685	−0.183533
$951$	−10.5980	−0.343663
$952$	−0.970563	−0.0314561
$953$	25.8995	0.838967	0.419483	−	0.907763i	$-0.362211\pi$
$953$	25.8995	0.838967	0.419483	+	0.907763i	$0.362211\pi$
$954$	49.9411	1.61690
$955$	14.3137	0.463181
$956$	0	0
$957$	−0.928932	−0.0300281
$958$	−34.6863	−1.12066
$959$	−5.07107	−0.163753
$960$	−3.31371	−0.106949
$961$	−18.1421	−0.585230
$962$	−35.9411	−1.15879
$963$	−12.2843	−0.395855
$964$	0	0
$965$	−6.82843	−0.219815
$966$	−1.02944	−0.0331216
$967$	−27.5563	−0.886152	−0.443076	−	0.896484i	$-0.646113\pi$
$967$	−27.5563	−0.886152	−0.443076	+	0.896484i	$0.646113\pi$
$968$	2.82843	0.0909091
$969$	−0.242641	−0.00779474
$970$	−0.585786	−0.0188085
$971$	25.7279	0.825648	0.412824	−	0.910811i	$-0.364542\pi$
$971$	25.7279	0.825648	0.412824	+	0.910811i	$0.364542\pi$
$972$	0	0
$973$	9.59798	0.307697
$974$	26.2426	0.840868
$975$	10.3431	0.331246
$976$	−12.2843	−0.393210
$977$	−44.8406	−1.43458	−0.717289	−	0.696776i	$-0.754617\pi$
$977$	−44.8406	−1.43458	−0.717289	+	0.696776i	$0.754617\pi$
$978$	−4.76955	−0.152513
$979$	14.8995	0.476190
$980$	0	0
$981$	32.9706	1.05267
$982$	−51.2548	−1.63561
$983$	13.1005	0.417841	0.208921	−	0.977933i	$-0.433005\pi$
$983$	13.1005	0.417841	0.208921	+	0.977933i	$0.433005\pi$
$984$	11.3137	0.360668
$985$	3.89949	0.124248
$986$	−1.85786	−0.0591665
$987$	−0.769553	−0.0244951
$988$	0	0
$989$	−34.9706	−1.11200
$990$	4.00000	0.127128
$991$	−8.68629	−0.275929	−0.137965	−	0.990437i	$-0.544056\pi$
$991$	−8.68629	−0.275929	−0.137965	+	0.990437i	$0.544056\pi$
$992$	0	0
$993$	−3.40202	−0.107960
$994$	−7.94113	−0.251877
$995$	16.1421	0.511740
$996$	0	0
$997$	−37.5563	−1.18942	−0.594711	−	0.803940i	$-0.702733\pi$
$997$	−37.5563	−1.18942	−0.594711	+	0.803940i	$0.702733\pi$
$998$	−17.8579	−0.565281
$999$	9.82843	0.310958

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	209.2.a.b.1.1	✓	2
3.2	odd	2		1881.2.a.d.1.2		2
4.3	odd	2		3344.2.a.n.1.1		2
5.4	even	2		5225.2.a.f.1.2		2
11.10	odd	2		2299.2.a.f.1.2		2
19.18	odd	2		3971.2.a.d.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
209.2.a.b.1.1	✓	2	1.1	even	1	trivial
1881.2.a.d.1.2		2	3.2	odd	2
2299.2.a.f.1.2		2	11.10	odd	2
3344.2.a.n.1.1		2	4.3	odd	2
3971.2.a.d.1.2		2	19.18	odd	2
5225.2.a.f.1.2		2	5.4	even	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:	\( N \)	\(=\)	\( 209 = 11 \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	209.a (trivial)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} +0.414214 q^{3} -1.00000 q^{5} -0.585786 q^{6} -0.585786 q^{7} +2.82843 q^{8} -2.82843 q^{9} +1.41421 q^{10} -1.00000 q^{11} -6.24264 q^{13} +0.828427 q^{14} -0.414214 q^{15} -4.00000 q^{16} +0.585786 q^{17} +4.00000 q^{18} -1.00000 q^{19} -0.242641 q^{21} +1.41421 q^{22} -3.00000 q^{23} +1.17157 q^{24} -4.00000 q^{25} +8.82843 q^{26} -2.41421 q^{27} +2.24264 q^{29} +0.585786 q^{30} -3.58579 q^{31} -0.414214 q^{33} -0.828427 q^{34} +0.585786 q^{35} -4.07107 q^{37} +1.41421 q^{38} -2.58579 q^{39} -2.82843 q^{40} +9.65685 q^{41} +0.343146 q^{42} +11.6569 q^{43} +2.82843 q^{45} +4.24264 q^{46} +3.17157 q^{47} -1.65685 q^{48} -6.65685 q^{49} +5.65685 q^{50} +0.242641 q^{51} +12.4853 q^{53} +3.41421 q^{54} +1.00000 q^{55} -1.65685 q^{56} -0.414214 q^{57} -3.17157 q^{58} -4.41421 q^{59} +3.07107 q^{61} +5.07107 q^{62} +1.65685 q^{63} +8.00000 q^{64} +6.24264 q^{65} +0.585786 q^{66} -7.58579 q^{67} -1.24264 q^{69} -0.828427 q^{70} -9.58579 q^{71} -8.00000 q^{72} +12.4853 q^{73} +5.75736 q^{74} -1.65685 q^{75} +0.585786 q^{77} +3.65685 q^{78} -17.4142 q^{79} +4.00000 q^{80} +7.48528 q^{81} -13.6569 q^{82} +0.585786 q^{83} -0.585786 q^{85} -16.4853 q^{86} +0.928932 q^{87} -2.82843 q^{88} -14.8995 q^{89} -4.00000 q^{90} +3.65685 q^{91} -1.48528 q^{93} -4.48528 q^{94} +1.00000 q^{95} -0.414214 q^{97} +9.41421 q^{98} +2.82843 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{3} - 2 q^{5} - 4 q^{6} - 4 q^{7} - 2 q^{11} - 4 q^{13} - 4 q^{14} + 2 q^{15} - 8 q^{16} + 4 q^{17} + 8 q^{18} - 2 q^{19} + 8 q^{21} - 6 q^{23} + 8 q^{24} - 8 q^{25} + 12 q^{26} - 2 q^{27} - 4 q^{29}+ \cdots + 16 q^{98}+O(q^{100}) \) 2 * q - 2 * q^3 - 2 * q^5 - 4 * q^6 - 4 * q^7 - 2 * q^11 - 4 * q^13 - 4 * q^14 + 2 * q^15 - 8 * q^16 + 4 * q^17 + 8 * q^18 - 2 * q^19 + 8 * q^21 - 6 * q^23 + 8 * q^24 - 8 * q^25 + 12 * q^26 - 2 * q^27 - 4 * q^29 + 4 * q^30 - 10 * q^31 + 2 * q^33 + 4 * q^34 + 4 * q^35 + 6 * q^37 - 8 * q^39 + 8 * q^41 + 12 * q^42 + 12 * q^43 + 12 * q^47 + 8 * q^48 - 2 * q^49 - 8 * q^51 + 8 * q^53 + 4 * q^54 + 2 * q^55 + 8 * q^56 + 2 * q^57 - 12 * q^58 - 6 * q^59 - 8 * q^61 - 4 * q^62 - 8 * q^63 + 16 * q^64 + 4 * q^65 + 4 * q^66 - 18 * q^67 + 6 * q^69 + 4 * q^70 - 22 * q^71 - 16 * q^72 + 8 * q^73 + 20 * q^74 + 8 * q^75 + 4 * q^77 - 4 * q^78 - 32 * q^79 + 8 * q^80 - 2 * q^81 - 16 * q^82 + 4 * q^83 - 4 * q^85 - 16 * q^86 + 16 * q^87 - 10 * q^89 - 8 * q^90 - 4 * q^91 + 14 * q^93 + 8 * q^94 + 2 * q^95 + 2 * q^97 + 16 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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