LMFDB - Embedded newform 1666.2.a.p.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1666,2,Mod(1,1666)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1666, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1666.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1666 = 2 \cdot 7^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1666.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$13.3030769767$
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:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$1.41421$ of defining polynomial
Character	$\chi$	$=$	1666.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -0.585786 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})$	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -0.585786 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +0.585786 q^{10} -4.24264 q^{11} +1.41421 q^{12} +4.82843 q^{13} -0.828427 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -6.82843 q^{19} -0.585786 q^{20} +4.24264 q^{22} -2.82843 q^{23} -1.41421 q^{24} -4.65685 q^{25} -4.82843 q^{26} -5.65685 q^{27} +9.07107 q^{29} +0.828427 q^{30} +4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -1.00000 q^{34} -1.00000 q^{36} +1.75736 q^{37} +6.82843 q^{38} +6.82843 q^{39} +0.585786 q^{40} -6.00000 q^{41} -8.48528 q^{43} -4.24264 q^{44} +0.585786 q^{45} +2.82843 q^{46} -12.8284 q^{47} +1.41421 q^{48} +4.65685 q^{50} +1.41421 q^{51} +4.82843 q^{52} -3.17157 q^{53} +5.65685 q^{54} +2.48528 q^{55} -9.65685 q^{57} -9.07107 q^{58} -11.3137 q^{59} -0.828427 q^{60} +2.24264 q^{61} -4.00000 q^{62} +1.00000 q^{64} -2.82843 q^{65} +6.00000 q^{66} -1.17157 q^{67} +1.00000 q^{68} -4.00000 q^{69} -6.82843 q^{71} +1.00000 q^{72} +10.4853 q^{73} -1.75736 q^{74} -6.58579 q^{75} -6.82843 q^{76} -6.82843 q^{78} +4.48528 q^{79} -0.585786 q^{80} -5.00000 q^{81} +6.00000 q^{82} -5.65685 q^{83} -0.585786 q^{85} +8.48528 q^{86} +12.8284 q^{87} +4.24264 q^{88} +1.65685 q^{89} -0.585786 q^{90} -2.82843 q^{92} +5.65685 q^{93} +12.8284 q^{94} +4.00000 q^{95} -1.41421 q^{96} -14.9706 q^{97} +4.24264 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} - 2 q^{9}+O(q^{10}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 - 2 * q^9	$ 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} - 2 q^{9} + 4 q^{10} + 4 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 8 q^{19} - 4 q^{20} + 2 q^{25} - 4 q^{26} + 4 q^{29} - 4 q^{30} + 8 q^{31} - 2 q^{32} - 12 q^{33} - 2 q^{34} - 2 q^{36} + 12 q^{37} + 8 q^{38} + 8 q^{39} + 4 q^{40} - 12 q^{41} + 4 q^{45} - 20 q^{47} - 2 q^{50} + 4 q^{52} - 12 q^{53} - 12 q^{55} - 8 q^{57} - 4 q^{58} + 4 q^{60} - 4 q^{61} - 8 q^{62} + 2 q^{64} + 12 q^{66} - 8 q^{67} + 2 q^{68} - 8 q^{69} - 8 q^{71} + 2 q^{72} + 4 q^{73} - 12 q^{74} - 16 q^{75} - 8 q^{76} - 8 q^{78} - 8 q^{79} - 4 q^{80} - 10 q^{81} + 12 q^{82} - 4 q^{85} + 20 q^{87} - 8 q^{89} - 4 q^{90} + 20 q^{94} + 8 q^{95} + 4 q^{97}+O(q^{100}) $ 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 - 2 * q^9 + 4 * q^10 + 4 * q^13 + 4 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 - 8 * q^19 - 4 * q^20 + 2 * q^25 - 4 * q^26 + 4 * q^29 - 4 * q^30 + 8 * q^31 - 2 * q^32 - 12 * q^33 - 2 * q^34 - 2 * q^36 + 12 * q^37 + 8 * q^38 + 8 * q^39 + 4 * q^40 - 12 * q^41 + 4 * q^45 - 20 * q^47 - 2 * q^50 + 4 * q^52 - 12 * q^53 - 12 * q^55 - 8 * q^57 - 4 * q^58 + 4 * q^60 - 4 * q^61 - 8 * q^62 + 2 * q^64 + 12 * q^66 - 8 * q^67 + 2 * q^68 - 8 * q^69 - 8 * q^71 + 2 * q^72 + 4 * q^73 - 12 * q^74 - 16 * q^75 - 8 * q^76 - 8 * q^78 - 8 * q^79 - 4 * q^80 - 10 * q^81 + 12 * q^82 - 4 * q^85 + 20 * q^87 - 8 * q^89 - 4 * q^90 + 20 * q^94 + 8 * q^95 + 4 * q^97

Display 100 coefficients 10 coefficients

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	−1.00000	−0.707107
$2$	−1.00000	−0.707107
$3$	1.41421	0.816497	0.408248	−	0.912871i	$-0.366140\pi$
$3$	1.41421	0.816497	0.408248	+	0.912871i	$0.366140\pi$
$4$	1.00000	0.500000
$5$	−0.585786	−0.261972	−0.130986	−	0.991384i	$-0.541814\pi$
$5$	−0.585786	−0.261972	−0.130986	+	0.991384i	$0.541814\pi$
$6$	−1.41421	−0.577350
$7$	0	0
$7$	0	0
$8$	−1.00000	−0.353553
$9$	−1.00000	−0.333333
$10$	0.585786	0.185242
$11$	−4.24264	−1.27920	−0.639602	−	0.768706i	$-0.720901\pi$
$11$	−4.24264	−1.27920	−0.639602	+	0.768706i	$0.720901\pi$
$12$	1.41421	0.408248
$13$	4.82843	1.33916	0.669582	−	0.742738i	$-0.266473\pi$
$13$	4.82843	1.33916	0.669582	+	0.742738i	$0.266473\pi$
$14$	0	0
$15$	−0.828427	−0.213899
$16$	1.00000	0.250000
$17$	1.00000	0.242536
$17$	1.00000	0.242536
$18$	1.00000	0.235702
$19$	−6.82843	−1.56655	−0.783274	−	0.621676i	$-0.786452\pi$
$19$	−6.82843	−1.56655	−0.783274	+	0.621676i	$0.786452\pi$
$20$	−0.585786	−0.130986
$21$	0	0
$22$	4.24264	0.904534
$23$	−2.82843	−0.589768	−0.294884	−	0.955533i	$-0.595281\pi$
$23$	−2.82843	−0.589768	−0.294884	+	0.955533i	$0.595281\pi$
$24$	−1.41421	−0.288675
$25$	−4.65685	−0.931371
$26$	−4.82843	−0.946932
$27$	−5.65685	−1.08866
$28$	0	0
$29$	9.07107	1.68446	0.842228	−	0.539122i	$-0.181244\pi$
$29$	9.07107	1.68446	0.842228	+	0.539122i	$0.181244\pi$
$30$	0.828427	0.151249
$31$	4.00000	0.718421	0.359211	−	0.933257i	$-0.383046\pi$
$31$	4.00000	0.718421	0.359211	+	0.933257i	$0.383046\pi$
$32$	−1.00000	−0.176777
$33$	−6.00000	−1.04447
$34$	−1.00000	−0.171499
$35$	0	0
$36$	−1.00000	−0.166667
$37$	1.75736	0.288908	0.144454	−	0.989512i	$-0.453857\pi$
$37$	1.75736	0.288908	0.144454	+	0.989512i	$0.453857\pi$
$38$	6.82843	1.10772
$39$	6.82843	1.09342
$40$	0.585786	0.0926210
$41$	−6.00000	−0.937043	−0.468521	−	0.883452i	$-0.655213\pi$
$41$	−6.00000	−0.937043	−0.468521	+	0.883452i	$0.655213\pi$
$42$	0	0
$43$	−8.48528	−1.29399	−0.646997	−	0.762493i	$-0.723975\pi$
$43$	−8.48528	−1.29399	−0.646997	+	0.762493i	$0.723975\pi$
$44$	−4.24264	−0.639602
$45$	0.585786	0.0873239
$46$	2.82843	0.417029
$47$	−12.8284	−1.87122	−0.935609	−	0.353037i	$-0.885149\pi$
$47$	−12.8284	−1.87122	−0.935609	+	0.353037i	$0.885149\pi$
$48$	1.41421	0.204124
$49$	0	0
$50$	4.65685	0.658579
$51$	1.41421	0.198030
$52$	4.82843	0.669582
$53$	−3.17157	−0.435649	−0.217825	−	0.975988i	$-0.569896\pi$
$53$	−3.17157	−0.435649	−0.217825	+	0.975988i	$0.569896\pi$
$54$	5.65685	0.769800
$55$	2.48528	0.335115
$56$	0	0
$57$	−9.65685	−1.27908
$58$	−9.07107	−1.19109
$59$	−11.3137	−1.47292	−0.736460	−	0.676481i	$-0.763504\pi$
$59$	−11.3137	−1.47292	−0.736460	+	0.676481i	$0.763504\pi$
$60$	−0.828427	−0.106949
$61$	2.24264	0.287141	0.143570	−	0.989640i	$-0.454142\pi$
$61$	2.24264	0.287141	0.143570	+	0.989640i	$0.454142\pi$
$62$	−4.00000	−0.508001
$63$	0	0
$64$	1.00000	0.125000
$65$	−2.82843	−0.350823
$66$	6.00000	0.738549
$67$	−1.17157	−0.143130	−0.0715652	−	0.997436i	$-0.522799\pi$
$67$	−1.17157	−0.143130	−0.0715652	+	0.997436i	$0.522799\pi$
$68$	1.00000	0.121268
$69$	−4.00000	−0.481543
$70$	0	0
$71$	−6.82843	−0.810385	−0.405193	−	0.914231i	$-0.632796\pi$
$71$	−6.82843	−0.810385	−0.405193	+	0.914231i	$0.632796\pi$
$72$	1.00000	0.117851
$73$	10.4853	1.22721	0.613605	−	0.789613i	$-0.289719\pi$
$73$	10.4853	1.22721	0.613605	+	0.789613i	$0.289719\pi$
$74$	−1.75736	−0.204289
$75$	−6.58579	−0.760461
$76$	−6.82843	−0.783274
$77$	0	0
$78$	−6.82843	−0.773167
$79$	4.48528	0.504634	0.252317	−	0.967645i	$-0.418807\pi$
$79$	4.48528	0.504634	0.252317	+	0.967645i	$0.418807\pi$
$80$	−0.585786	−0.0654929
$81$	−5.00000	−0.555556
$82$	6.00000	0.662589
$83$	−5.65685	−0.620920	−0.310460	−	0.950586i	$-0.600483\pi$
$83$	−5.65685	−0.620920	−0.310460	+	0.950586i	$0.600483\pi$
$84$	0	0
$85$	−0.585786	−0.0635375
$86$	8.48528	0.914991
$87$	12.8284	1.37535
$88$	4.24264	0.452267
$89$	1.65685	0.175626	0.0878131	−	0.996137i	$-0.472012\pi$
$89$	1.65685	0.175626	0.0878131	+	0.996137i	$0.472012\pi$
$90$	−0.585786	−0.0617473
$91$	0	0
$92$	−2.82843	−0.294884
$93$	5.65685	0.586588
$94$	12.8284	1.32315
$95$	4.00000	0.410391
$96$	−1.41421	−0.144338
$97$	−14.9706	−1.52003	−0.760015	−	0.649905i	$-0.774809\pi$
$97$	−14.9706	−1.52003	−0.760015	+	0.649905i	$0.774809\pi$
$98$	0	0
$99$	4.24264	0.426401
$100$	−4.65685	−0.465685
$101$	−3.17157	−0.315583	−0.157792	−	0.987472i	$-0.550437\pi$
$101$	−3.17157	−0.315583	−0.157792	+	0.987472i	$0.550437\pi$
$102$	−1.41421	−0.140028
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	−4.82843	−0.473466
$105$	0	0
$106$	3.17157	0.308050
$107$	4.24264	0.410152	0.205076	−	0.978746i	$-0.434256\pi$
$107$	4.24264	0.410152	0.205076	+	0.978746i	$0.434256\pi$
$108$	−5.65685	−0.544331
$109$	7.89949	0.756634	0.378317	−	0.925676i	$-0.376503\pi$
$109$	7.89949	0.756634	0.378317	+	0.925676i	$0.376503\pi$
$110$	−2.48528	−0.236962
$111$	2.48528	0.235892
$112$	0	0
$113$	9.31371	0.876160	0.438080	−	0.898936i	$-0.355659\pi$
$113$	9.31371	0.876160	0.438080	+	0.898936i	$0.355659\pi$
$114$	9.65685	0.904447
$115$	1.65685	0.154502
$116$	9.07107	0.842228
$117$	−4.82843	−0.446388
$118$	11.3137	1.04151
$119$	0	0
$120$	0.828427	0.0756247
$121$	7.00000	0.636364
$122$	−2.24264	−0.203039
$123$	−8.48528	−0.765092
$124$	4.00000	0.359211
$125$	5.65685	0.505964
$126$	0	0
$127$	16.8284	1.49328	0.746641	−	0.665228i	$-0.231665\pi$
$127$	16.8284	1.49328	0.746641	+	0.665228i	$0.231665\pi$
$128$	−1.00000	−0.0883883
$129$	−12.0000	−1.05654
$130$	2.82843	0.248069
$131$	−14.5858	−1.27437	−0.637183	−	0.770713i	$-0.719900\pi$
$131$	−14.5858	−1.27437	−0.637183	+	0.770713i	$0.719900\pi$
$132$	−6.00000	−0.522233
$133$	0	0
$134$	1.17157	0.101208
$135$	3.31371	0.285199
$136$	−1.00000	−0.0857493
$137$	−1.65685	−0.141555	−0.0707773	−	0.997492i	$-0.522548\pi$
$137$	−1.65685	−0.141555	−0.0707773	+	0.997492i	$0.522548\pi$
$138$	4.00000	0.340503
$139$	−0.928932	−0.0787910	−0.0393955	−	0.999224i	$-0.512543\pi$
$139$	−0.928932	−0.0787910	−0.0393955	+	0.999224i	$0.512543\pi$
$140$	0	0
$141$	−18.1421	−1.52784
$142$	6.82843	0.573029
$143$	−20.4853	−1.71307
$144$	−1.00000	−0.0833333
$145$	−5.31371	−0.441279
$146$	−10.4853	−0.867768
$147$	0	0
$148$	1.75736	0.144454
$149$	−17.3137	−1.41839	−0.709197	−	0.705010i	$-0.750942\pi$
$149$	−17.3137	−1.41839	−0.709197	+	0.705010i	$0.750942\pi$
$150$	6.58579	0.537727
$151$	14.4853	1.17880	0.589398	−	0.807843i	$-0.299365\pi$
$151$	14.4853	1.17880	0.589398	+	0.807843i	$0.299365\pi$
$152$	6.82843	0.553859
$153$	−1.00000	−0.0808452
$154$	0	0
$155$	−2.34315	−0.188206
$156$	6.82843	0.546712
$157$	−14.9706	−1.19478	−0.597390	−	0.801950i	$-0.703796\pi$
$157$	−14.9706	−1.19478	−0.597390	+	0.801950i	$0.703796\pi$
$158$	−4.48528	−0.356830
$159$	−4.48528	−0.355706
$160$	0.585786	0.0463105
$161$	0	0
$162$	5.00000	0.392837
$163$	4.24264	0.332309	0.166155	−	0.986100i	$-0.446865\pi$
$163$	4.24264	0.332309	0.166155	+	0.986100i	$0.446865\pi$
$164$	−6.00000	−0.468521
$165$	3.51472	0.273620
$166$	5.65685	0.439057
$167$	−12.0000	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$167$	−12.0000	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$168$	0	0
$169$	10.3137	0.793362
$170$	0.585786	0.0449278
$171$	6.82843	0.522183
$172$	−8.48528	−0.646997
$173$	−4.10051	−0.311756	−0.155878	−	0.987776i	$-0.549821\pi$
$173$	−4.10051	−0.311756	−0.155878	+	0.987776i	$0.549821\pi$
$174$	−12.8284	−0.972521
$175$	0	0
$176$	−4.24264	−0.319801
$177$	−16.0000	−1.20263
$178$	−1.65685	−0.124186
$179$	21.6569	1.61871	0.809355	−	0.587320i	$-0.199817\pi$
$179$	21.6569	1.61871	0.809355	+	0.587320i	$0.199817\pi$
$180$	0.585786	0.0436619
$181$	18.2426	1.35596	0.677982	−	0.735078i	$-0.262855\pi$
$181$	18.2426	1.35596	0.677982	+	0.735078i	$0.262855\pi$
$182$	0	0
$183$	3.17157	0.234449
$184$	2.82843	0.208514
$185$	−1.02944	−0.0756857
$186$	−5.65685	−0.414781
$187$	−4.24264	−0.310253
$188$	−12.8284	−0.935609
$189$	0	0
$190$	−4.00000	−0.290191
$191$	11.3137	0.818631	0.409316	−	0.912393i	$-0.365768\pi$
$191$	11.3137	0.818631	0.409316	+	0.912393i	$0.365768\pi$
$192$	1.41421	0.102062
$193$	−24.8284	−1.78719	−0.893595	−	0.448875i	$-0.851825\pi$
$193$	−24.8284	−1.78719	−0.893595	+	0.448875i	$0.851825\pi$
$194$	14.9706	1.07482
$195$	−4.00000	−0.286446
$196$	0	0
$197$	−22.2426	−1.58472	−0.792361	−	0.610052i	$-0.791148\pi$
$197$	−22.2426	−1.58472	−0.792361	+	0.610052i	$0.791148\pi$
$198$	−4.24264	−0.301511
$199$	−7.51472	−0.532704	−0.266352	−	0.963876i	$-0.585818\pi$
$199$	−7.51472	−0.532704	−0.266352	+	0.963876i	$0.585818\pi$
$200$	4.65685	0.329289
$201$	−1.65685	−0.116865
$202$	3.17157	0.223151
$203$	0	0
$204$	1.41421	0.0990148
$205$	3.51472	0.245479
$206$	0	0
$207$	2.82843	0.196589
$208$	4.82843	0.334791
$209$	28.9706	2.00394
$210$	0	0
$211$	−7.55635	−0.520201	−0.260100	−	0.965582i	$-0.583756\pi$
$211$	−7.55635	−0.520201	−0.260100	+	0.965582i	$0.583756\pi$
$212$	−3.17157	−0.217825
$213$	−9.65685	−0.661677
$214$	−4.24264	−0.290021
$215$	4.97056	0.338990
$216$	5.65685	0.384900
$217$	0	0
$218$	−7.89949	−0.535021
$219$	14.8284	1.00201
$220$	2.48528	0.167558
$221$	4.82843	0.324795
$222$	−2.48528	−0.166801
$223$	−21.7990	−1.45977	−0.729884	−	0.683571i	$-0.760426\pi$
$223$	−21.7990	−1.45977	−0.729884	+	0.683571i	$0.760426\pi$
$224$	0	0
$225$	4.65685	0.310457
$226$	−9.31371	−0.619539
$227$	21.2132	1.40797	0.703985	−	0.710215i	$-0.251402\pi$
$227$	21.2132	1.40797	0.703985	+	0.710215i	$0.251402\pi$
$228$	−9.65685	−0.639541
$229$	26.0000	1.71813	0.859064	−	0.511868i	$-0.171046\pi$
$229$	26.0000	1.71813	0.859064	+	0.511868i	$0.171046\pi$
$230$	−1.65685	−0.109250
$231$	0	0
$232$	−9.07107	−0.595545
$233$	−11.6569	−0.763666	−0.381833	−	0.924231i	$-0.624707\pi$
$233$	−11.6569	−0.763666	−0.381833	+	0.924231i	$0.624707\pi$
$234$	4.82843	0.315644
$235$	7.51472	0.490206
$236$	−11.3137	−0.736460
$237$	6.34315	0.412032
$238$	0	0
$239$	13.6569	0.883388	0.441694	−	0.897166i	$-0.354378\pi$
$239$	13.6569	0.883388	0.441694	+	0.897166i	$0.354378\pi$
$240$	−0.828427	−0.0534747
$241$	−25.3137	−1.63060	−0.815300	−	0.579039i	$-0.803428\pi$
$241$	−25.3137	−1.63060	−0.815300	+	0.579039i	$0.803428\pi$
$242$	−7.00000	−0.449977
$243$	9.89949	0.635053
$244$	2.24264	0.143570
$245$	0	0
$246$	8.48528	0.541002
$247$	−32.9706	−2.09787
$248$	−4.00000	−0.254000
$249$	−8.00000	−0.506979
$250$	−5.65685	−0.357771
$251$	−5.65685	−0.357057	−0.178529	−	0.983935i	$-0.557134\pi$
$251$	−5.65685	−0.357057	−0.178529	+	0.983935i	$0.557134\pi$
$252$	0	0
$253$	12.0000	0.754434
$254$	−16.8284	−1.05591
$255$	−0.828427	−0.0518781
$256$	1.00000	0.0625000
$257$	−16.6274	−1.03719	−0.518595	−	0.855020i	$-0.673545\pi$
$257$	−16.6274	−1.03719	−0.518595	+	0.855020i	$0.673545\pi$
$258$	12.0000	0.747087
$259$	0	0
$260$	−2.82843	−0.175412
$261$	−9.07107	−0.561485
$262$	14.5858	0.901113
$263$	19.4558	1.19970	0.599849	−	0.800113i	$-0.295227\pi$
$263$	19.4558	1.19970	0.599849	+	0.800113i	$0.295227\pi$
$264$	6.00000	0.369274
$265$	1.85786	0.114128
$266$	0	0
$267$	2.34315	0.143398
$268$	−1.17157	−0.0715652
$269$	−18.7279	−1.14186	−0.570931	−	0.820998i	$-0.693418\pi$
$269$	−18.7279	−1.14186	−0.570931	+	0.820998i	$0.693418\pi$
$270$	−3.31371	−0.201666
$271$	9.79899	0.595246	0.297623	−	0.954683i	$-0.403806\pi$
$271$	9.79899	0.595246	0.297623	+	0.954683i	$0.403806\pi$
$272$	1.00000	0.0606339
$273$	0	0
$274$	1.65685	0.100094
$275$	19.7574	1.19141
$276$	−4.00000	−0.240772
$277$	2.24264	0.134747	0.0673736	−	0.997728i	$-0.478538\pi$
$277$	2.24264	0.134747	0.0673736	+	0.997728i	$0.478538\pi$
$278$	0.928932	0.0557137
$279$	−4.00000	−0.239474
$280$	0	0
$281$	22.6274	1.34984	0.674919	−	0.737892i	$-0.264178\pi$
$281$	22.6274	1.34984	0.674919	+	0.737892i	$0.264178\pi$
$282$	18.1421	1.08035
$283$	−12.7279	−0.756596	−0.378298	−	0.925684i	$-0.623491\pi$
$283$	−12.7279	−0.756596	−0.378298	+	0.925684i	$0.623491\pi$
$284$	−6.82843	−0.405193
$285$	5.65685	0.335083
$286$	20.4853	1.21132
$287$	0	0
$288$	1.00000	0.0589256
$289$	1.00000	0.0588235
$290$	5.31371	0.312032
$291$	−21.1716	−1.24110
$292$	10.4853	0.613605
$293$	29.3137	1.71253	0.856263	−	0.516541i	$-0.172781\pi$
$293$	29.3137	1.71253	0.856263	+	0.516541i	$0.172781\pi$
$294$	0	0
$295$	6.62742	0.385863
$296$	−1.75736	−0.102144
$297$	24.0000	1.39262
$298$	17.3137	1.00296
$299$	−13.6569	−0.789796
$300$	−6.58579	−0.380231
$301$	0	0
$302$	−14.4853	−0.833534
$303$	−4.48528	−0.257673
$304$	−6.82843	−0.391637
$305$	−1.31371	−0.0752227
$306$	1.00000	0.0571662
$307$	24.4853	1.39745	0.698724	−	0.715391i	$-0.253751\pi$
$307$	24.4853	1.39745	0.698724	+	0.715391i	$0.253751\pi$
$308$	0	0
$309$	0	0
$310$	2.34315	0.133082
$311$	−25.6569	−1.45487	−0.727433	−	0.686178i	$-0.759287\pi$
$311$	−25.6569	−1.45487	−0.727433	+	0.686178i	$0.759287\pi$
$312$	−6.82843	−0.386584
$313$	22.0000	1.24351	0.621757	−	0.783210i	$-0.286419\pi$
$313$	22.0000	1.24351	0.621757	+	0.783210i	$0.286419\pi$
$314$	14.9706	0.844838
$315$	0	0
$316$	4.48528	0.252317
$317$	10.9289	0.613830	0.306915	−	0.951737i	$-0.400703\pi$
$317$	10.9289	0.613830	0.306915	+	0.951737i	$0.400703\pi$
$318$	4.48528	0.251522
$319$	−38.4853	−2.15476
$320$	−0.585786	−0.0327465
$321$	6.00000	0.334887
$322$	0	0
$323$	−6.82843	−0.379944
$324$	−5.00000	−0.277778
$325$	−22.4853	−1.24726
$326$	−4.24264	−0.234978
$327$	11.1716	0.617789
$328$	6.00000	0.331295
$329$	0	0
$330$	−3.51472	−0.193479
$331$	−27.7990	−1.52797	−0.763985	−	0.645234i	$-0.776760\pi$
$331$	−27.7990	−1.52797	−0.763985	+	0.645234i	$0.776760\pi$
$332$	−5.65685	−0.310460
$333$	−1.75736	−0.0963027
$334$	12.0000	0.656611
$335$	0.686292	0.0374961
$336$	0	0
$337$	6.00000	0.326841	0.163420	−	0.986557i	$-0.447747\pi$
$337$	6.00000	0.326841	0.163420	+	0.986557i	$0.447747\pi$
$338$	−10.3137	−0.560992
$339$	13.1716	0.715382
$340$	−0.585786	−0.0317687
$341$	−16.9706	−0.919007
$342$	−6.82843	−0.369239
$343$	0	0
$344$	8.48528	0.457496
$345$	2.34315	0.126151
$346$	4.10051	0.220445
$347$	−7.55635	−0.405646	−0.202823	−	0.979215i	$-0.565012\pi$
$347$	−7.55635	−0.405646	−0.202823	+	0.979215i	$0.565012\pi$
$348$	12.8284	0.687676
$349$	−4.82843	−0.258460	−0.129230	−	0.991615i	$-0.541251\pi$
$349$	−4.82843	−0.258460	−0.129230	+	0.991615i	$0.541251\pi$
$350$	0	0
$351$	−27.3137	−1.45790
$352$	4.24264	0.226134
$353$	1.31371	0.0699216	0.0349608	−	0.999389i	$-0.488869\pi$
$353$	1.31371	0.0699216	0.0349608	+	0.999389i	$0.488869\pi$
$354$	16.0000	0.850390
$355$	4.00000	0.212298
$356$	1.65685	0.0878131
$357$	0	0
$358$	−21.6569	−1.14460
$359$	16.9706	0.895672	0.447836	−	0.894116i	$-0.352195\pi$
$359$	16.9706	0.895672	0.447836	+	0.894116i	$0.352195\pi$
$360$	−0.585786	−0.0308737
$361$	27.6274	1.45407
$362$	−18.2426	−0.958812
$363$	9.89949	0.519589
$364$	0	0
$365$	−6.14214	−0.321494
$366$	−3.17157	−0.165781
$367$	22.1421	1.15581	0.577905	−	0.816104i	$-0.303870\pi$
$367$	22.1421	1.15581	0.577905	+	0.816104i	$0.303870\pi$
$368$	−2.82843	−0.147442
$369$	6.00000	0.312348
$370$	1.02944	0.0535179
$371$	0	0
$372$	5.65685	0.293294
$373$	−12.8284	−0.664231	−0.332115	−	0.943239i	$-0.607762\pi$
$373$	−12.8284	−0.664231	−0.332115	+	0.943239i	$0.607762\pi$
$374$	4.24264	0.219382
$375$	8.00000	0.413118
$376$	12.8284	0.661576
$377$	43.7990	2.25576
$378$	0	0
$379$	5.21320	0.267784	0.133892	−	0.990996i	$-0.457252\pi$
$379$	5.21320	0.267784	0.133892	+	0.990996i	$0.457252\pi$
$380$	4.00000	0.205196
$381$	23.7990	1.21926
$382$	−11.3137	−0.578860
$383$	2.34315	0.119729	0.0598646	−	0.998207i	$-0.480933\pi$
$383$	2.34315	0.119729	0.0598646	+	0.998207i	$0.480933\pi$
$384$	−1.41421	−0.0721688
$385$	0	0
$386$	24.8284	1.26373
$387$	8.48528	0.431331
$388$	−14.9706	−0.760015
$389$	4.34315	0.220206	0.110103	−	0.993920i	$-0.464882\pi$
$389$	4.34315	0.220206	0.110103	+	0.993920i	$0.464882\pi$
$390$	4.00000	0.202548
$391$	−2.82843	−0.143040
$392$	0	0
$393$	−20.6274	−1.04052
$394$	22.2426	1.12057
$395$	−2.62742	−0.132200
$396$	4.24264	0.213201
$397$	37.5563	1.88490	0.942450	−	0.334348i	$-0.108516\pi$
$397$	37.5563	1.88490	0.942450	+	0.334348i	$0.108516\pi$
$398$	7.51472	0.376679
$399$	0	0
$400$	−4.65685	−0.232843
$401$	−17.7990	−0.888839	−0.444420	−	0.895819i	$-0.646590\pi$
$401$	−17.7990	−0.888839	−0.444420	+	0.895819i	$0.646590\pi$
$402$	1.65685	0.0826364
$403$	19.3137	0.962084
$404$	−3.17157	−0.157792
$405$	2.92893	0.145540
$406$	0	0
$407$	−7.45584	−0.369572
$408$	−1.41421	−0.0700140
$409$	−2.00000	−0.0988936	−0.0494468	−	0.998777i	$-0.515746\pi$
$409$	−2.00000	−0.0988936	−0.0494468	+	0.998777i	$0.515746\pi$
$410$	−3.51472	−0.173580
$411$	−2.34315	−0.115579
$412$	0	0
$413$	0	0
$414$	−2.82843	−0.139010
$415$	3.31371	0.162664
$416$	−4.82843	−0.236733
$417$	−1.31371	−0.0643326
$418$	−28.9706	−1.41700
$419$	−1.41421	−0.0690889	−0.0345444	−	0.999403i	$-0.510998\pi$
$419$	−1.41421	−0.0690889	−0.0345444	+	0.999403i	$0.510998\pi$
$420$	0	0
$421$	14.4853	0.705969	0.352985	−	0.935629i	$-0.385167\pi$
$421$	14.4853	0.705969	0.352985	+	0.935629i	$0.385167\pi$
$422$	7.55635	0.367837
$423$	12.8284	0.623739
$424$	3.17157	0.154025
$425$	−4.65685	−0.225891
$426$	9.65685	0.467876
$427$	0	0
$428$	4.24264	0.205076
$429$	−28.9706	−1.39871
$430$	−4.97056	−0.239702
$431$	−38.8284	−1.87030	−0.935150	−	0.354253i	$-0.884735\pi$
$431$	−38.8284	−1.87030	−0.935150	+	0.354253i	$0.884735\pi$
$432$	−5.65685	−0.272166
$433$	12.0000	0.576683	0.288342	−	0.957528i	$-0.406896\pi$
$433$	12.0000	0.576683	0.288342	+	0.957528i	$0.406896\pi$
$434$	0	0
$435$	−7.51472	−0.360303
$436$	7.89949	0.378317
$437$	19.3137	0.923900
$438$	−14.8284	−0.708530
$439$	15.5147	0.740477	0.370239	−	0.928937i	$-0.379276\pi$
$439$	15.5147	0.740477	0.370239	+	0.928937i	$0.379276\pi$
$440$	−2.48528	−0.118481
$441$	0	0
$442$	−4.82843	−0.229665
$443$	22.8284	1.08461	0.542306	−	0.840181i	$-0.317551\pi$
$443$	22.8284	1.08461	0.542306	+	0.840181i	$0.317551\pi$
$444$	2.48528	0.117946
$445$	−0.970563	−0.0460091
$446$	21.7990	1.03221
$447$	−24.4853	−1.15811
$448$	0	0
$449$	41.7990	1.97262	0.986308	−	0.164913i	$-0.0527343\pi$
$449$	41.7990	1.97262	0.986308	+	0.164913i	$0.0527343\pi$
$450$	−4.65685	−0.219526
$451$	25.4558	1.19867
$452$	9.31371	0.438080
$453$	20.4853	0.962482
$454$	−21.2132	−0.995585
$455$	0	0
$456$	9.65685	0.452224
$457$	39.9411	1.86837	0.934184	−	0.356793i	$-0.116130\pi$
$457$	39.9411	1.86837	0.934184	+	0.356793i	$0.116130\pi$
$458$	−26.0000	−1.21490
$459$	−5.65685	−0.264039
$460$	1.65685	0.0772512
$461$	−19.4558	−0.906149	−0.453074	−	0.891473i	$-0.649673\pi$
$461$	−19.4558	−0.906149	−0.453074	+	0.891473i	$0.649673\pi$
$462$	0	0
$463$	−0.828427	−0.0385003	−0.0192501	−	0.999815i	$-0.506128\pi$
$463$	−0.828427	−0.0385003	−0.0192501	+	0.999815i	$0.506128\pi$
$464$	9.07107	0.421114
$465$	−3.31371	−0.153670
$466$	11.6569	0.539993
$467$	−13.1716	−0.609508	−0.304754	−	0.952431i	$-0.598574\pi$
$467$	−13.1716	−0.609508	−0.304754	+	0.952431i	$0.598574\pi$
$468$	−4.82843	−0.223194
$469$	0	0
$470$	−7.51472	−0.346628
$471$	−21.1716	−0.975535
$472$	11.3137	0.520756
$473$	36.0000	1.65528
$474$	−6.34315	−0.291350
$475$	31.7990	1.45904
$476$	0	0
$477$	3.17157	0.145216
$478$	−13.6569	−0.624650
$479$	−5.17157	−0.236295	−0.118148	−	0.992996i	$-0.537696\pi$
$479$	−5.17157	−0.236295	−0.118148	+	0.992996i	$0.537696\pi$
$480$	0.828427	0.0378124
$481$	8.48528	0.386896
$482$	25.3137	1.15301
$483$	0	0
$484$	7.00000	0.318182
$485$	8.76955	0.398205
$486$	−9.89949	−0.449050
$487$	6.14214	0.278327	0.139163	−	0.990269i	$-0.455559\pi$
$487$	6.14214	0.278327	0.139163	+	0.990269i	$0.455559\pi$
$488$	−2.24264	−0.101520
$489$	6.00000	0.271329
$490$	0	0
$491$	21.6569	0.977360	0.488680	−	0.872463i	$-0.337479\pi$
$491$	21.6569	0.977360	0.488680	+	0.872463i	$0.337479\pi$
$492$	−8.48528	−0.382546
$493$	9.07107	0.408540
$494$	32.9706	1.48342
$495$	−2.48528	−0.111705
$496$	4.00000	0.179605
$497$	0	0
$498$	8.00000	0.358489
$499$	23.5563	1.05453	0.527264	−	0.849702i	$-0.323218\pi$
$499$	23.5563	1.05453	0.527264	+	0.849702i	$0.323218\pi$
$500$	5.65685	0.252982
$501$	−16.9706	−0.758189
$502$	5.65685	0.252478
$503$	−3.51472	−0.156714	−0.0783568	−	0.996925i	$-0.524967\pi$
$503$	−3.51472	−0.156714	−0.0783568	+	0.996925i	$0.524967\pi$
$504$	0	0
$505$	1.85786	0.0826739
$506$	−12.0000	−0.533465
$507$	14.5858	0.647778
$508$	16.8284	0.746641
$509$	16.6274	0.736997	0.368499	−	0.929628i	$-0.379872\pi$
$509$	16.6274	0.736997	0.368499	+	0.929628i	$0.379872\pi$
$510$	0.828427	0.0366834
$511$	0	0
$512$	−1.00000	−0.0441942
$513$	38.6274	1.70544
$514$	16.6274	0.733404
$515$	0	0
$516$	−12.0000	−0.528271
$517$	54.4264	2.39367
$518$	0	0
$519$	−5.79899	−0.254547
$520$	2.82843	0.124035
$521$	−5.31371	−0.232798	−0.116399	−	0.993203i	$-0.537135\pi$
$521$	−5.31371	−0.232798	−0.116399	+	0.993203i	$0.537135\pi$
$522$	9.07107	0.397030
$523$	22.3431	0.976998	0.488499	−	0.872565i	$-0.337545\pi$
$523$	22.3431	0.976998	0.488499	+	0.872565i	$0.337545\pi$
$524$	−14.5858	−0.637183
$525$	0	0
$526$	−19.4558	−0.848315
$527$	4.00000	0.174243
$528$	−6.00000	−0.261116
$529$	−15.0000	−0.652174
$530$	−1.85786	−0.0807005
$531$	11.3137	0.490973
$532$	0	0
$533$	−28.9706	−1.25485
$534$	−2.34315	−0.101398
$535$	−2.48528	−0.107448
$536$	1.17157	0.0506042
$537$	30.6274	1.32167
$538$	18.7279	0.807418
$539$	0	0
$540$	3.31371	0.142599
$541$	−1.75736	−0.0755548	−0.0377774	−	0.999286i	$-0.512028\pi$
$541$	−1.75736	−0.0755548	−0.0377774	+	0.999286i	$0.512028\pi$
$542$	−9.79899	−0.420903
$543$	25.7990	1.10714
$544$	−1.00000	−0.0428746
$545$	−4.62742	−0.198217
$546$	0	0
$547$	−36.2426	−1.54962	−0.774812	−	0.632192i	$-0.782155\pi$
$547$	−36.2426	−1.54962	−0.774812	+	0.632192i	$0.782155\pi$
$548$	−1.65685	−0.0707773
$549$	−2.24264	−0.0957136
$550$	−19.7574	−0.842457
$551$	−61.9411	−2.63878
$552$	4.00000	0.170251
$553$	0	0
$554$	−2.24264	−0.0952807
$555$	−1.45584	−0.0617971
$556$	−0.928932	−0.0393955
$557$	−36.8284	−1.56047	−0.780235	−	0.625486i	$-0.784901\pi$
$557$	−36.8284	−1.56047	−0.780235	+	0.625486i	$0.784901\pi$
$558$	4.00000	0.169334
$559$	−40.9706	−1.73287
$560$	0	0
$561$	−6.00000	−0.253320
$562$	−22.6274	−0.954480
$563$	12.0000	0.505740	0.252870	−	0.967500i	$-0.418626\pi$
$563$	12.0000	0.505740	0.252870	+	0.967500i	$0.418626\pi$
$564$	−18.1421	−0.763922
$565$	−5.45584	−0.229529
$566$	12.7279	0.534994
$567$	0	0
$568$	6.82843	0.286514
$569$	22.2843	0.934205	0.467103	−	0.884203i	$-0.345298\pi$
$569$	22.2843	0.934205	0.467103	+	0.884203i	$0.345298\pi$
$570$	−5.65685	−0.236940
$571$	6.10051	0.255298	0.127649	−	0.991819i	$-0.459257\pi$
$571$	6.10051	0.255298	0.127649	+	0.991819i	$0.459257\pi$
$572$	−20.4853	−0.856533
$573$	16.0000	0.668410
$574$	0	0
$575$	13.1716	0.549293
$576$	−1.00000	−0.0416667
$577$	−2.97056	−0.123666	−0.0618331	−	0.998087i	$-0.519695\pi$
$577$	−2.97056	−0.123666	−0.0618331	+	0.998087i	$0.519695\pi$
$578$	−1.00000	−0.0415945
$579$	−35.1127	−1.45923
$580$	−5.31371	−0.220640
$581$	0	0
$582$	21.1716	0.877590
$583$	13.4558	0.557284
$584$	−10.4853	−0.433884
$585$	2.82843	0.116941
$586$	−29.3137	−1.21094
$587$	−25.4558	−1.05068	−0.525338	−	0.850894i	$-0.676061\pi$
$587$	−25.4558	−1.05068	−0.525338	+	0.850894i	$0.676061\pi$
$588$	0	0
$589$	−27.3137	−1.12544
$590$	−6.62742	−0.272846
$591$	−31.4558	−1.29392
$592$	1.75736	0.0722270
$593$	1.65685	0.0680388	0.0340194	−	0.999421i	$-0.489169\pi$
$593$	1.65685	0.0680388	0.0340194	+	0.999421i	$0.489169\pi$
$594$	−24.0000	−0.984732
$595$	0	0
$596$	−17.3137	−0.709197
$597$	−10.6274	−0.434951
$598$	13.6569	0.558470
$599$	0.828427	0.0338486	0.0169243	−	0.999857i	$-0.494613\pi$
$599$	0.828427	0.0338486	0.0169243	+	0.999857i	$0.494613\pi$
$600$	6.58579	0.268864
$601$	−40.1421	−1.63743	−0.818716	−	0.574199i	$-0.805314\pi$
$601$	−40.1421	−1.63743	−0.818716	+	0.574199i	$0.805314\pi$
$602$	0	0
$603$	1.17157	0.0477101
$604$	14.4853	0.589398
$605$	−4.10051	−0.166709
$606$	4.48528	0.182202
$607$	−20.0000	−0.811775	−0.405887	−	0.913923i	$-0.633038\pi$
$607$	−20.0000	−0.811775	−0.405887	+	0.913923i	$0.633038\pi$
$608$	6.82843	0.276929
$609$	0	0
$610$	1.31371	0.0531905
$611$	−61.9411	−2.50587
$612$	−1.00000	−0.0404226
$613$	38.9706	1.57401	0.787003	−	0.616949i	$-0.211632\pi$
$613$	38.9706	1.57401	0.787003	+	0.616949i	$0.211632\pi$
$614$	−24.4853	−0.988146
$615$	4.97056	0.200432
$616$	0	0
$617$	−24.3431	−0.980018	−0.490009	−	0.871717i	$-0.663007\pi$
$617$	−24.3431	−0.980018	−0.490009	+	0.871717i	$0.663007\pi$
$618$	0	0
$619$	−11.7574	−0.472568	−0.236284	−	0.971684i	$-0.575930\pi$
$619$	−11.7574	−0.472568	−0.236284	+	0.971684i	$0.575930\pi$
$620$	−2.34315	−0.0941030
$621$	16.0000	0.642058
$622$	25.6569	1.02875
$623$	0	0
$624$	6.82843	0.273356
$625$	19.9706	0.798823
$626$	−22.0000	−0.879297
$627$	40.9706	1.63621
$628$	−14.9706	−0.597390
$629$	1.75736	0.0700705
$630$	0	0
$631$	−24.0000	−0.955425	−0.477712	−	0.878516i	$-0.658534\pi$
$631$	−24.0000	−0.955425	−0.477712	+	0.878516i	$0.658534\pi$
$632$	−4.48528	−0.178415
$633$	−10.6863	−0.424742
$634$	−10.9289	−0.434043
$635$	−9.85786	−0.391197
$636$	−4.48528	−0.177853
$637$	0	0
$638$	38.4853	1.52365
$639$	6.82843	0.270128
$640$	0.585786	0.0231552
$641$	3.85786	0.152376	0.0761882	−	0.997093i	$-0.475725\pi$
$641$	3.85786	0.152376	0.0761882	+	0.997093i	$0.475725\pi$
$642$	−6.00000	−0.236801
$643$	−0.443651	−0.0174959	−0.00874794	−	0.999962i	$-0.502785\pi$
$643$	−0.443651	−0.0174959	−0.00874794	+	0.999962i	$0.502785\pi$
$644$	0	0
$645$	7.02944	0.276784
$646$	6.82843	0.268661
$647$	15.1716	0.596456	0.298228	−	0.954495i	$-0.403604\pi$
$647$	15.1716	0.596456	0.298228	+	0.954495i	$0.403604\pi$
$648$	5.00000	0.196419
$649$	48.0000	1.88416
$650$	22.4853	0.881945
$651$	0	0
$652$	4.24264	0.166155
$653$	−9.07107	−0.354978	−0.177489	−	0.984123i	$-0.556797\pi$
$653$	−9.07107	−0.354978	−0.177489	+	0.984123i	$0.556797\pi$
$654$	−11.1716	−0.436843
$655$	8.54416	0.333848
$656$	−6.00000	−0.234261
$657$	−10.4853	−0.409070
$658$	0	0
$659$	−8.68629	−0.338370	−0.169185	−	0.985584i	$-0.554114\pi$
$659$	−8.68629	−0.338370	−0.169185	+	0.985584i	$0.554114\pi$
$660$	3.51472	0.136810
$661$	2.48528	0.0966662	0.0483331	−	0.998831i	$-0.484609\pi$
$661$	2.48528	0.0966662	0.0483331	+	0.998831i	$0.484609\pi$
$662$	27.7990	1.08044
$663$	6.82843	0.265194
$664$	5.65685	0.219529
$665$	0	0
$666$	1.75736	0.0680963
$667$	−25.6569	−0.993437
$668$	−12.0000	−0.464294
$669$	−30.8284	−1.19190
$670$	−0.686292	−0.0265138
$671$	−9.51472	−0.367312
$672$	0	0
$673$	−23.1716	−0.893198	−0.446599	−	0.894734i	$-0.647365\pi$
$673$	−23.1716	−0.893198	−0.446599	+	0.894734i	$0.647365\pi$
$674$	−6.00000	−0.231111
$675$	26.3431	1.01395
$676$	10.3137	0.396681
$677$	3.21320	0.123493	0.0617467	−	0.998092i	$-0.480333\pi$
$677$	3.21320	0.123493	0.0617467	+	0.998092i	$0.480333\pi$
$678$	−13.1716	−0.505851
$679$	0	0
$680$	0.585786	0.0224639
$681$	30.0000	1.14960
$682$	16.9706	0.649836
$683$	−0.443651	−0.0169758	−0.00848791	−	0.999964i	$-0.502702\pi$
$683$	−0.443651	−0.0169758	−0.00848791	+	0.999964i	$0.502702\pi$
$684$	6.82843	0.261091
$685$	0.970563	0.0370833
$686$	0	0
$687$	36.7696	1.40285
$688$	−8.48528	−0.323498
$689$	−15.3137	−0.583406
$690$	−2.34315	−0.0892020
$691$	13.2132	0.502654	0.251327	−	0.967902i	$-0.419133\pi$
$691$	13.2132	0.502654	0.251327	+	0.967902i	$0.419133\pi$
$692$	−4.10051	−0.155878
$693$	0	0
$694$	7.55635	0.286835
$695$	0.544156	0.0206410
$696$	−12.8284	−0.486260
$697$	−6.00000	−0.227266
$698$	4.82843	0.182759
$699$	−16.4853	−0.623531
$700$	0	0
$701$	−29.1127	−1.09957	−0.549786	−	0.835306i	$-0.685291\pi$
$701$	−29.1127	−1.09957	−0.549786	+	0.835306i	$0.685291\pi$
$702$	27.3137	1.03089
$703$	−12.0000	−0.452589
$704$	−4.24264	−0.159901
$705$	10.6274	0.400252
$706$	−1.31371	−0.0494421
$707$	0	0
$708$	−16.0000	−0.601317
$709$	−36.5858	−1.37401	−0.687004	−	0.726654i	$-0.741075\pi$
$709$	−36.5858	−1.37401	−0.687004	+	0.726654i	$0.741075\pi$
$710$	−4.00000	−0.150117
$711$	−4.48528	−0.168211
$712$	−1.65685	−0.0620932
$713$	−11.3137	−0.423702
$714$	0	0
$715$	12.0000	0.448775
$716$	21.6569	0.809355
$717$	19.3137	0.721284
$718$	−16.9706	−0.633336
$719$	48.7696	1.81880	0.909399	−	0.415925i	$-0.136542\pi$
$719$	48.7696	1.81880	0.909399	+	0.415925i	$0.136542\pi$
$720$	0.585786	0.0218310
$721$	0	0
$722$	−27.6274	−1.02819
$723$	−35.7990	−1.33138
$724$	18.2426	0.677982
$725$	−42.2426	−1.56885
$726$	−9.89949	−0.367405
$727$	−25.5147	−0.946289	−0.473144	−	0.880985i	$-0.656881\pi$
$727$	−25.5147	−0.946289	−0.473144	+	0.880985i	$0.656881\pi$
$728$	0	0
$729$	29.0000	1.07407
$730$	6.14214	0.227331
$731$	−8.48528	−0.313839
$732$	3.17157	0.117225
$733$	−19.4558	−0.718618	−0.359309	−	0.933219i	$-0.616988\pi$
$733$	−19.4558	−0.718618	−0.359309	+	0.933219i	$0.616988\pi$
$734$	−22.1421	−0.817281
$735$	0	0
$736$	2.82843	0.104257
$737$	4.97056	0.183093
$738$	−6.00000	−0.220863
$739$	−30.6274	−1.12665	−0.563324	−	0.826236i	$-0.690478\pi$
$739$	−30.6274	−1.12665	−0.563324	+	0.826236i	$0.690478\pi$
$740$	−1.02944	−0.0378429
$741$	−46.6274	−1.71290
$742$	0	0
$743$	−0.686292	−0.0251776	−0.0125888	−	0.999921i	$-0.504007\pi$
$743$	−0.686292	−0.0251776	−0.0125888	+	0.999921i	$0.504007\pi$
$744$	−5.65685	−0.207390
$745$	10.1421	0.371579
$746$	12.8284	0.469682
$747$	5.65685	0.206973
$748$	−4.24264	−0.155126
$749$	0	0
$750$	−8.00000	−0.292119
$751$	3.51472	0.128254	0.0641270	−	0.997942i	$-0.479574\pi$
$751$	3.51472	0.128254	0.0641270	+	0.997942i	$0.479574\pi$
$752$	−12.8284	−0.467805
$753$	−8.00000	−0.291536
$754$	−43.7990	−1.59507
$755$	−8.48528	−0.308811
$756$	0	0
$757$	51.9411	1.88783	0.943916	−	0.330185i	$-0.107111\pi$
$757$	51.9411	1.88783	0.943916	+	0.330185i	$0.107111\pi$
$758$	−5.21320	−0.189352
$759$	16.9706	0.615992
$760$	−4.00000	−0.145095
$761$	−52.6274	−1.90774	−0.953871	−	0.300216i	$-0.902941\pi$
$761$	−52.6274	−1.90774	−0.953871	+	0.300216i	$0.902941\pi$
$762$	−23.7990	−0.862146
$763$	0	0
$764$	11.3137	0.409316
$765$	0.585786	0.0211792
$766$	−2.34315	−0.0846613
$767$	−54.6274	−1.97248
$768$	1.41421	0.0510310
$769$	−33.3137	−1.20132	−0.600662	−	0.799503i	$-0.705096\pi$
$769$	−33.3137	−1.20132	−0.600662	+	0.799503i	$0.705096\pi$
$770$	0	0
$771$	−23.5147	−0.846862
$772$	−24.8284	−0.893595
$773$	29.5980	1.06457	0.532283	−	0.846567i	$-0.321334\pi$
$773$	29.5980	1.06457	0.532283	+	0.846567i	$0.321334\pi$
$774$	−8.48528	−0.304997
$775$	−18.6274	−0.669117
$776$	14.9706	0.537412
$777$	0	0
$778$	−4.34315	−0.155709
$779$	40.9706	1.46792
$780$	−4.00000	−0.143223
$781$	28.9706	1.03665
$782$	2.82843	0.101144
$783$	−51.3137	−1.83380
$784$	0	0
$785$	8.76955	0.312999
$786$	20.6274	0.735756
$787$	19.2721	0.686975	0.343488	−	0.939157i	$-0.388392\pi$
$787$	19.2721	0.686975	0.343488	+	0.939157i	$0.388392\pi$
$788$	−22.2426	−0.792361
$789$	27.5147	0.979550
$790$	2.62742	0.0934793
$791$	0	0
$792$	−4.24264	−0.150756
$793$	10.8284	0.384529
$794$	−37.5563	−1.33282
$795$	2.62742	0.0931849
$796$	−7.51472	−0.266352
$797$	2.48528	0.0880332	0.0440166	−	0.999031i	$-0.485985\pi$
$797$	2.48528	0.0880332	0.0440166	+	0.999031i	$0.485985\pi$
$798$	0	0
$799$	−12.8284	−0.453837
$800$	4.65685	0.164645
$801$	−1.65685	−0.0585421
$802$	17.7990	0.628504
$803$	−44.4853	−1.56985
$804$	−1.65685	−0.0584327
$805$	0	0
$806$	−19.3137	−0.680296
$807$	−26.4853	−0.932326
$808$	3.17157	0.111576
$809$	−14.4853	−0.509275	−0.254638	−	0.967037i	$-0.581956\pi$
$809$	−14.4853	−0.509275	−0.254638	+	0.967037i	$0.581956\pi$
$810$	−2.92893	−0.102912
$811$	−44.7279	−1.57061	−0.785305	−	0.619109i	$-0.787494\pi$
$811$	−44.7279	−1.57061	−0.785305	+	0.619109i	$0.787494\pi$
$812$	0	0
$813$	13.8579	0.486017
$814$	7.45584	0.261327
$815$	−2.48528	−0.0870556
$816$	1.41421	0.0495074
$817$	57.9411	2.02710
$818$	2.00000	0.0699284
$819$	0	0
$820$	3.51472	0.122739
$821$	−20.8701	−0.728370	−0.364185	−	0.931327i	$-0.618652\pi$
$821$	−20.8701	−0.728370	−0.364185	+	0.931327i	$0.618652\pi$
$822$	2.34315	0.0817266
$823$	−32.4853	−1.13237	−0.566183	−	0.824280i	$-0.691580\pi$
$823$	−32.4853	−1.13237	−0.566183	+	0.824280i	$0.691580\pi$
$824$	0	0
$825$	27.9411	0.972785
$826$	0	0
$827$	14.1005	0.490323	0.245161	−	0.969482i	$-0.421159\pi$
$827$	14.1005	0.490323	0.245161	+	0.969482i	$0.421159\pi$
$828$	2.82843	0.0982946
$829$	−19.6569	−0.682711	−0.341355	−	0.939934i	$-0.610886\pi$
$829$	−19.6569	−0.682711	−0.341355	+	0.939934i	$0.610886\pi$
$830$	−3.31371	−0.115021
$831$	3.17157	0.110021
$832$	4.82843	0.167396
$833$	0	0
$834$	1.31371	0.0454900
$835$	7.02944	0.243264
$836$	28.9706	1.00197
$837$	−22.6274	−0.782118
$838$	1.41421	0.0488532
$839$	−18.3431	−0.633276	−0.316638	−	0.948547i	$-0.602554\pi$
$839$	−18.3431	−0.633276	−0.316638	+	0.948547i	$0.602554\pi$
$840$	0	0
$841$	53.2843	1.83739
$842$	−14.4853	−0.499196
$843$	32.0000	1.10214
$844$	−7.55635	−0.260100
$845$	−6.04163	−0.207838
$846$	−12.8284	−0.441050
$847$	0	0
$848$	−3.17157	−0.108912
$849$	−18.0000	−0.617758
$850$	4.65685	0.159729
$851$	−4.97056	−0.170389
$852$	−9.65685	−0.330838
$853$	−48.5858	−1.66355	−0.831773	−	0.555116i	$-0.812674\pi$
$853$	−48.5858	−1.66355	−0.831773	+	0.555116i	$0.812674\pi$
$854$	0	0
$855$	−4.00000	−0.136797
$856$	−4.24264	−0.145010
$857$	−25.5980	−0.874410	−0.437205	−	0.899362i	$-0.644032\pi$
$857$	−25.5980	−0.874410	−0.437205	+	0.899362i	$0.644032\pi$
$858$	28.9706	0.989039
$859$	−44.9706	−1.53438	−0.767188	−	0.641422i	$-0.778345\pi$
$859$	−44.9706	−1.53438	−0.767188	+	0.641422i	$0.778345\pi$
$860$	4.97056	0.169495
$861$	0	0
$862$	38.8284	1.32250
$863$	−21.7990	−0.742046	−0.371023	−	0.928624i	$-0.620993\pi$
$863$	−21.7990	−0.742046	−0.371023	+	0.928624i	$0.620993\pi$
$864$	5.65685	0.192450
$865$	2.40202	0.0816711
$866$	−12.0000	−0.407777
$867$	1.41421	0.0480292
$868$	0	0
$869$	−19.0294	−0.645529
$870$	7.51472	0.254773
$871$	−5.65685	−0.191675
$872$	−7.89949	−0.267511
$873$	14.9706	0.506677
$874$	−19.3137	−0.653296
$875$	0	0
$876$	14.8284	0.501006
$877$	44.3848	1.49877	0.749384	−	0.662136i	$-0.230350\pi$
$877$	44.3848	1.49877	0.749384	+	0.662136i	$0.230350\pi$
$878$	−15.5147	−0.523596
$879$	41.4558	1.39827
$880$	2.48528	0.0837788
$881$	49.1127	1.65465	0.827324	−	0.561724i	$-0.189862\pi$
$881$	49.1127	1.65465	0.827324	+	0.561724i	$0.189862\pi$
$882$	0	0
$883$	45.4558	1.52971	0.764855	−	0.644202i	$-0.222810\pi$
$883$	45.4558	1.52971	0.764855	+	0.644202i	$0.222810\pi$
$884$	4.82843	0.162398
$885$	9.37258	0.315056
$886$	−22.8284	−0.766936
$887$	5.17157	0.173644	0.0868222	−	0.996224i	$-0.472329\pi$
$887$	5.17157	0.173644	0.0868222	+	0.996224i	$0.472329\pi$
$888$	−2.48528	−0.0834006
$889$	0	0
$890$	0.970563	0.0325333
$891$	21.2132	0.710669
$892$	−21.7990	−0.729884
$893$	87.5980	2.93135
$894$	24.4853	0.818910
$895$	−12.6863	−0.424056
$896$	0	0
$897$	−19.3137	−0.644866
$898$	−41.7990	−1.39485
$899$	36.2843	1.21015
$900$	4.65685	0.155228
$901$	−3.17157	−0.105660
$902$	−25.4558	−0.847587
$903$	0	0
$904$	−9.31371	−0.309769
$905$	−10.6863	−0.355224
$906$	−20.4853	−0.680578
$907$	−38.5858	−1.28122	−0.640610	−	0.767866i	$-0.721318\pi$
$907$	−38.5858	−1.28122	−0.640610	+	0.767866i	$0.721318\pi$
$908$	21.2132	0.703985
$909$	3.17157	0.105194
$910$	0	0
$911$	18.3431	0.607736	0.303868	−	0.952714i	$-0.401722\pi$
$911$	18.3431	0.607736	0.303868	+	0.952714i	$0.401722\pi$
$912$	−9.65685	−0.319770
$913$	24.0000	0.794284
$914$	−39.9411	−1.32114
$915$	−1.85786	−0.0614191
$916$	26.0000	0.859064
$917$	0	0
$918$	5.65685	0.186704
$919$	24.9706	0.823703	0.411851	−	0.911251i	$-0.364882\pi$
$919$	24.9706	0.823703	0.411851	+	0.911251i	$0.364882\pi$
$920$	−1.65685	−0.0546249
$921$	34.6274	1.14101
$922$	19.4558	0.640744
$923$	−32.9706	−1.08524
$924$	0	0
$925$	−8.18377	−0.269081
$926$	0.828427	0.0272238
$927$	0	0
$928$	−9.07107	−0.297772
$929$	−37.5980	−1.23355	−0.616775	−	0.787140i	$-0.711561\pi$
$929$	−37.5980	−1.23355	−0.616775	+	0.787140i	$0.711561\pi$
$930$	3.31371	0.108661
$931$	0	0
$932$	−11.6569	−0.381833
$933$	−36.2843	−1.18789
$934$	13.1716	0.430987
$935$	2.48528	0.0812774
$936$	4.82843	0.157822
$937$	22.9706	0.750416	0.375208	−	0.926941i	$-0.377571\pi$
$937$	22.9706	0.750416	0.375208	+	0.926941i	$0.377571\pi$
$938$	0	0
$939$	31.1127	1.01532
$940$	7.51472	0.245103
$941$	−26.4437	−0.862038	−0.431019	−	0.902343i	$-0.641846\pi$
$941$	−26.4437	−0.862038	−0.431019	+	0.902343i	$0.641846\pi$
$942$	21.1716	0.689807
$943$	16.9706	0.552638
$944$	−11.3137	−0.368230
$945$	0	0
$946$	−36.0000	−1.17046
$947$	−31.5563	−1.02544	−0.512722	−	0.858555i	$-0.671363\pi$
$947$	−31.5563	−1.02544	−0.512722	+	0.858555i	$0.671363\pi$
$948$	6.34315	0.206016
$949$	50.6274	1.64344
$950$	−31.7990	−1.03170
$951$	15.4558	0.501190
$952$	0	0
$953$	−33.5980	−1.08835	−0.544173	−	0.838973i	$-0.683156\pi$
$953$	−33.5980	−1.08835	−0.544173	+	0.838973i	$0.683156\pi$
$954$	−3.17157	−0.102683
$955$	−6.62742	−0.214458
$956$	13.6569	0.441694
$957$	−54.4264	−1.75936
$958$	5.17157	0.167086
$959$	0	0
$960$	−0.828427	−0.0267374
$961$	−15.0000	−0.483871
$962$	−8.48528	−0.273576
$963$	−4.24264	−0.136717
$964$	−25.3137	−0.815300
$965$	14.5442	0.468193
$966$	0	0
$967$	10.3431	0.332613	0.166307	−	0.986074i	$-0.446816\pi$
$967$	10.3431	0.332613	0.166307	+	0.986074i	$0.446816\pi$
$968$	−7.00000	−0.224989
$969$	−9.65685	−0.310223
$970$	−8.76955	−0.281573
$971$	2.62742	0.0843178	0.0421589	−	0.999111i	$-0.486576\pi$
$971$	2.62742	0.0843178	0.0421589	+	0.999111i	$0.486576\pi$
$972$	9.89949	0.317526
$973$	0	0
$974$	−6.14214	−0.196807
$975$	−31.7990	−1.01838
$976$	2.24264	0.0717852
$977$	42.2843	1.35279	0.676397	−	0.736537i	$-0.263540\pi$
$977$	42.2843	1.35279	0.676397	+	0.736537i	$0.263540\pi$
$978$	−6.00000	−0.191859
$979$	−7.02944	−0.224662
$980$	0	0
$981$	−7.89949	−0.252211
$982$	−21.6569	−0.691098
$983$	25.4558	0.811915	0.405958	−	0.913892i	$-0.366938\pi$
$983$	25.4558	0.811915	0.405958	+	0.913892i	$0.366938\pi$
$984$	8.48528	0.270501
$985$	13.0294	0.415152
$986$	−9.07107	−0.288882
$987$	0	0
$988$	−32.9706	−1.04893
$989$	24.0000	0.763156
$990$	2.48528	0.0789874
$991$	−46.4264	−1.47478	−0.737392	−	0.675465i	$-0.763943\pi$
$991$	−46.4264	−1.47478	−0.737392	+	0.675465i	$0.763943\pi$
$992$	−4.00000	−0.127000
$993$	−39.3137	−1.24758
$994$	0	0
$995$	4.40202	0.139553
$996$	−8.00000	−0.253490
$997$	35.4142	1.12158	0.560790	−	0.827958i	$-0.310498\pi$
$997$	35.4142	1.12158	0.560790	+	0.827958i	$0.310498\pi$
$998$	−23.5563	−0.745663
$999$	−9.94113	−0.314523

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1666.2.a.p.1.2	✓	2
7.6	odd	2		1666.2.a.q.1.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1666.2.a.p.1.2	✓	2	1.1	even	1	trivial
1666.2.a.q.1.1	yes	2	7.6	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1666 = 2 \cdot 7^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1666.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -0.585786 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +O(q^{10})\)	\(q-1.00000 q^{2} +1.41421 q^{3} +1.00000 q^{4} -0.585786 q^{5} -1.41421 q^{6} -1.00000 q^{8} -1.00000 q^{9} +0.585786 q^{10} -4.24264 q^{11} +1.41421 q^{12} +4.82843 q^{13} -0.828427 q^{15} +1.00000 q^{16} +1.00000 q^{17} +1.00000 q^{18} -6.82843 q^{19} -0.585786 q^{20} +4.24264 q^{22} -2.82843 q^{23} -1.41421 q^{24} -4.65685 q^{25} -4.82843 q^{26} -5.65685 q^{27} +9.07107 q^{29} +0.828427 q^{30} +4.00000 q^{31} -1.00000 q^{32} -6.00000 q^{33} -1.00000 q^{34} -1.00000 q^{36} +1.75736 q^{37} +6.82843 q^{38} +6.82843 q^{39} +0.585786 q^{40} -6.00000 q^{41} -8.48528 q^{43} -4.24264 q^{44} +0.585786 q^{45} +2.82843 q^{46} -12.8284 q^{47} +1.41421 q^{48} +4.65685 q^{50} +1.41421 q^{51} +4.82843 q^{52} -3.17157 q^{53} +5.65685 q^{54} +2.48528 q^{55} -9.65685 q^{57} -9.07107 q^{58} -11.3137 q^{59} -0.828427 q^{60} +2.24264 q^{61} -4.00000 q^{62} +1.00000 q^{64} -2.82843 q^{65} +6.00000 q^{66} -1.17157 q^{67} +1.00000 q^{68} -4.00000 q^{69} -6.82843 q^{71} +1.00000 q^{72} +10.4853 q^{73} -1.75736 q^{74} -6.58579 q^{75} -6.82843 q^{76} -6.82843 q^{78} +4.48528 q^{79} -0.585786 q^{80} -5.00000 q^{81} +6.00000 q^{82} -5.65685 q^{83} -0.585786 q^{85} +8.48528 q^{86} +12.8284 q^{87} +4.24264 q^{88} +1.65685 q^{89} -0.585786 q^{90} -2.82843 q^{92} +5.65685 q^{93} +12.8284 q^{94} +4.00000 q^{95} -1.41421 q^{96} -14.9706 q^{97} +4.24264 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} - 2 q^{9}+O(q^{10}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 - 2 * q^9	\( 2 q - 2 q^{2} + 2 q^{4} - 4 q^{5} - 2 q^{8} - 2 q^{9} + 4 q^{10} + 4 q^{13} + 4 q^{15} + 2 q^{16} + 2 q^{17} + 2 q^{18} - 8 q^{19} - 4 q^{20} + 2 q^{25} - 4 q^{26} + 4 q^{29} - 4 q^{30} + 8 q^{31} - 2 q^{32} - 12 q^{33} - 2 q^{34} - 2 q^{36} + 12 q^{37} + 8 q^{38} + 8 q^{39} + 4 q^{40} - 12 q^{41} + 4 q^{45} - 20 q^{47} - 2 q^{50} + 4 q^{52} - 12 q^{53} - 12 q^{55} - 8 q^{57} - 4 q^{58} + 4 q^{60} - 4 q^{61} - 8 q^{62} + 2 q^{64} + 12 q^{66} - 8 q^{67} + 2 q^{68} - 8 q^{69} - 8 q^{71} + 2 q^{72} + 4 q^{73} - 12 q^{74} - 16 q^{75} - 8 q^{76} - 8 q^{78} - 8 q^{79} - 4 q^{80} - 10 q^{81} + 12 q^{82} - 4 q^{85} + 20 q^{87} - 8 q^{89} - 4 q^{90} + 20 q^{94} + 8 q^{95} + 4 q^{97}+O(q^{100}) \) 2 * q - 2 * q^2 + 2 * q^4 - 4 * q^5 - 2 * q^8 - 2 * q^9 + 4 * q^10 + 4 * q^13 + 4 * q^15 + 2 * q^16 + 2 * q^17 + 2 * q^18 - 8 * q^19 - 4 * q^20 + 2 * q^25 - 4 * q^26 + 4 * q^29 - 4 * q^30 + 8 * q^31 - 2 * q^32 - 12 * q^33 - 2 * q^34 - 2 * q^36 + 12 * q^37 + 8 * q^38 + 8 * q^39 + 4 * q^40 - 12 * q^41 + 4 * q^45 - 20 * q^47 - 2 * q^50 + 4 * q^52 - 12 * q^53 - 12 * q^55 - 8 * q^57 - 4 * q^58 + 4 * q^60 - 4 * q^61 - 8 * q^62 + 2 * q^64 + 12 * q^66 - 8 * q^67 + 2 * q^68 - 8 * q^69 - 8 * q^71 + 2 * q^72 + 4 * q^73 - 12 * q^74 - 16 * q^75 - 8 * q^76 - 8 * q^78 - 8 * q^79 - 4 * q^80 - 10 * q^81 + 12 * q^82 - 4 * q^85 + 20 * q^87 - 8 * q^89 - 4 * q^90 + 20 * q^94 + 8 * q^95 + 4 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more