LMFDB - Embedded newform 2040.2.cf.e.1441.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2040,2,Mod(361,2040)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2040, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2040.361"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2040.cf (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [20,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$16.2894820123$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 8 x^{19} + 34 x^{18} - 92 x^{17} + 162 x^{16} - 344 x^{15} + 2614 x^{14} + 5936 x^{13} + \cdots + 7688 $ x^20 - 8x^19 + 34x^18 - 92x^17 + 162x^16 - 344x^15 + 2614x^14 + 5936x^13 + 8057x^12 - 8592x^11 - 39536x^10 - 20244x^9 + 125120x^8 + 306832x^7 + 493180x^6 + 543664x^5 + 555536x^4 + 473520x^3 + 253744x^2 + 69440*x + 7688
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1441.4
Root			$1.72040 + 4.15341i$ of defining polynomial
Character	$\chi$	$=$	2040.1441
Dual form			2040.2.cf.e.361.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.707107 - 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{5} +(1.99447 - 1.99447i) q^{7} +1.00000i q^{9} +(-0.973731 + 0.973731i) q^{11} -5.30344 q^{13} -1.00000i q^{15} +(4.04255 - 0.811061i) q^{17} +7.91334i q^{19} -2.82061 q^{21} +(-6.39296 + 6.39296i) q^{23} +1.00000i q^{25} +(0.707107 - 0.707107i) q^{27} +(2.61466 + 2.61466i) q^{29} +(5.62185 + 5.62185i) q^{31} +1.37706 q^{33} +2.82061 q^{35} +(-6.31432 - 6.31432i) q^{37} +(3.75010 + 3.75010i) q^{39} +(7.51005 - 7.51005i) q^{41} +3.41059i q^{43} +(-0.707107 + 0.707107i) q^{45} +0.558773 q^{47} -0.955847i q^{49} +(-3.43202 - 2.28501i) q^{51} +9.29257i q^{53} -1.37706 q^{55} +(5.59558 - 5.59558i) q^{57} -6.38214i q^{59} +(-0.253278 + 0.253278i) q^{61} +(1.99447 + 1.99447i) q^{63} +(-3.75010 - 3.75010i) q^{65} -11.3627 q^{67} +9.04101 q^{69} +(3.96077 + 3.96077i) q^{71} +(2.81184 + 2.81184i) q^{73} +(0.707107 - 0.707107i) q^{75} +3.88416i q^{77} +(-6.24848 + 6.24848i) q^{79} -1.00000 q^{81} +0.373918i q^{83} +(3.43202 + 2.28501i) q^{85} -3.69768i q^{87} -6.65909 q^{89} +(-10.5776 + 10.5776i) q^{91} -7.95049i q^{93} +(-5.59558 + 5.59558i) q^{95} +(13.6102 + 13.6102i) q^{97} +(-0.973731 - 0.973731i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q + 4 q^{7} + 4 q^{11} - 16 q^{13} + 4 q^{17} - 8 q^{21} + 4 q^{23} + 12 q^{29} + 16 q^{31} - 8 q^{33} + 8 q^{35} - 20 q^{37} - 4 q^{39} - 4 q^{41} + 8 q^{51} + 8 q^{55} - 8 q^{57} + 4 q^{63} + 4 q^{65}+ \cdots + 4 q^{99}+O(q^{100}) $ 20 * q + 4 * q^7 + 4 * q^11 - 16 * q^13 + 4 * q^17 - 8 * q^21 + 4 * q^23 + 12 * q^29 + 16 * q^31 - 8 * q^33 + 8 * q^35 - 20 * q^37 - 4 * q^39 - 4 * q^41 + 8 * q^51 + 8 * q^55 - 8 * q^57 + 4 * q^63 + 4 * q^65 - 16 * q^67 + 24 * q^69 - 8 * q^71 + 12 * q^73 - 28 * q^79 - 20 * q^81 - 8 * q^85 - 32 * q^89 + 12 * q^91 + 8 * q^95 + 8 * q^97 + 4 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2040\mathbb{Z}\right)^\times$.

$n$	$241$	$511$	$817$	$1021$	$1361$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$3$	−0.707107	−	0.707107i	−0.408248	−	0.408248i
$4$		0			0
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$5$	0.707107	+	0.707107i	0.316228	+	0.316228i
$6$		0			0
$7$	1.99447	−	1.99447i	0.753840	−	0.753840i	−0.221354	−	0.975194i	$-0.571047\pi$
$7$	1.99447	−	1.99447i	0.753840	−	0.753840i	0.975194	+	0.221354i	$0.0710475\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	−0.973731	+	0.973731i	−0.293591	+	0.293591i	−0.838497	−	0.544906i	$-0.816565\pi$
$11$	−0.973731	+	0.973731i	−0.293591	+	0.293591i	0.544906	+	0.838497i	$0.316565\pi$
$12$		0			0
$13$	−5.30344			−1.47091			−0.735454	−	0.677574i	$-0.763031\pi$
$13$	−5.30344			−1.47091			−0.735454	+	0.677574i	$0.763031\pi$
$14$		0			0
$15$		−	1.00000i		−	0.258199i
$16$		0			0
$17$	4.04255	−	0.811061i	0.980461	−	0.196711i
$17$	4.04255	−	0.811061i	0.980461	−	0.196711i
$18$		0			0
$19$			7.91334i			1.81544i	0.419571	+	0.907722i	$0.362180\pi$
$19$			7.91334i			1.81544i	−0.419571	+	0.907722i	$0.637820\pi$
$20$		0			0
$21$	−2.82061			−0.615508
$22$		0			0
$23$	−6.39296	+	6.39296i	−1.33302	+	1.33302i	−0.430373	+	0.902651i	$0.641618\pi$
$23$	−6.39296	+	6.39296i	−1.33302	+	1.33302i	−0.902651	+	0.430373i	$0.858382\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$	0.707107	−	0.707107i	0.136083	−	0.136083i
$28$		0			0
$29$	2.61466	+	2.61466i	0.485529	+	0.485529i	0.906892	−	0.421363i	$-0.138448\pi$
$29$	2.61466	+	2.61466i	0.485529	+	0.485529i	−0.421363	+	0.906892i	$0.638448\pi$
$30$		0			0
$31$	5.62185	+	5.62185i	1.00971	+	1.00971i	0.999952	+	0.00976085i	$0.00310702\pi$
$31$	5.62185	+	5.62185i	1.00971	+	1.00971i	0.00976085	+	0.999952i	$0.496893\pi$
$32$		0			0
$33$	1.37706			0.239716
$34$		0			0
$35$	2.82061			0.476770
$36$		0			0
$37$	−6.31432	−	6.31432i	−1.03807	−	1.03807i	−0.999246	−	0.0388212i	$-0.987640\pi$
$37$	−6.31432	−	6.31432i	−1.03807	−	1.03807i	−0.0388212	−	0.999246i	$-0.512360\pi$
$38$		0			0
$39$	3.75010	+	3.75010i	0.600496	+	0.600496i
$40$		0			0
$41$	7.51005	−	7.51005i	1.17287	−	1.17287i	0.191352	−	0.981522i	$-0.438713\pi$
$41$	7.51005	−	7.51005i	1.17287	−	1.17287i	0.981522	−	0.191352i	$-0.0612871\pi$
$42$		0			0
$43$			3.41059i			0.520110i	0.965594	+	0.260055i	$0.0837406\pi$
$43$			3.41059i			0.520110i	−0.965594	+	0.260055i	$0.916259\pi$
$44$		0			0
$45$	−0.707107	+	0.707107i	−0.105409	+	0.105409i
$46$		0			0
$47$	0.558773			0.0815054			0.0407527	−	0.999169i	$-0.487024\pi$
$47$	0.558773			0.0815054			0.0407527	+	0.999169i	$0.487024\pi$
$48$		0			0
$49$		−	0.955847i		−	0.136550i
$50$		0			0
$51$	−3.43202	−	2.28501i	−0.480579	−	0.319965i
$52$		0			0
$53$			9.29257i			1.27643i	0.769857	+	0.638216i	$0.220327\pi$
$53$			9.29257i			1.27643i	−0.769857	+	0.638216i	$0.779673\pi$
$54$		0			0
$55$	−1.37706			−0.185683
$56$		0			0
$57$	5.59558	−	5.59558i	0.741152	−	0.741152i
$58$		0			0
$59$		−	6.38214i		−	0.830884i	−0.909620	−	0.415442i	$-0.863627\pi$
$59$		−	6.38214i		−	0.830884i	0.909620	−	0.415442i	$-0.136373\pi$
$60$		0			0
$61$	−0.253278	+	0.253278i	−0.0324290	+	0.0324290i	−0.723135	−	0.690706i	$-0.757300\pi$
$61$	−0.253278	+	0.253278i	−0.0324290	+	0.0324290i	0.690706	+	0.723135i	$0.257300\pi$
$62$		0			0
$63$	1.99447	+	1.99447i	0.251280	+	0.251280i
$64$		0			0
$65$	−3.75010	−	3.75010i	−0.465142	−	0.465142i
$66$		0			0
$67$	−11.3627			−1.38818			−0.694089	−	0.719889i	$-0.744192\pi$
$67$	−11.3627			−1.38818			−0.694089	+	0.719889i	$0.744192\pi$
$68$		0			0
$69$	9.04101			1.08841
$70$		0			0
$71$	3.96077	+	3.96077i	0.470057	+	0.470057i	0.901933	−	0.431876i	$-0.142148\pi$
$71$	3.96077	+	3.96077i	0.470057	+	0.470057i	−0.431876	+	0.901933i	$0.642148\pi$
$72$		0			0
$73$	2.81184	+	2.81184i	0.329102	+	0.329102i	0.852245	−	0.523143i	$-0.175241\pi$
$73$	2.81184	+	2.81184i	0.329102	+	0.329102i	−0.523143	+	0.852245i	$0.675241\pi$
$74$		0			0
$75$	0.707107	−	0.707107i	0.0816497	−	0.0816497i
$76$		0			0
$77$			3.88416i			0.442641i
$78$		0			0
$79$	−6.24848	+	6.24848i	−0.703009	+	0.703009i	−0.965055	−	0.262047i	$-0.915603\pi$
$79$	−6.24848	+	6.24848i	−0.703009	+	0.703009i	0.262047	+	0.965055i	$0.415603\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$			0.373918i			0.0410429i	0.999789	+	0.0205214i	$0.00653263\pi$
$83$			0.373918i			0.0410429i	−0.999789	+	0.0205214i	$0.993467\pi$
$84$		0			0
$85$	3.43202	+	2.28501i	0.372255	+	0.247844i
$86$		0			0
$87$		−	3.69768i		−	0.396433i
$88$		0			0
$89$	−6.65909			−0.705863			−0.352931	−	0.935649i	$-0.614815\pi$
$89$	−6.65909			−0.705863			−0.352931	+	0.935649i	$0.614815\pi$
$90$		0			0
$91$	−10.5776	+	10.5776i	−1.10883	+	1.10883i
$92$		0			0
$93$		−	7.95049i		−	0.824427i
$94$		0			0
$95$	−5.59558	+	5.59558i	−0.574094	+	0.574094i
$96$		0			0
$97$	13.6102	+	13.6102i	1.38191	+	1.38191i	0.841223	+	0.540688i	$0.181836\pi$
$97$	13.6102	+	13.6102i	1.38191	+	1.38191i	0.540688	+	0.841223i	$0.318164\pi$
$98$		0			0
$99$	−0.973731	−	0.973731i	−0.0978636	−	0.0978636i
$100$		0			0
$101$	14.1034			1.40334			0.701669	−	0.712503i	$-0.252439\pi$
$101$	14.1034			1.40334			0.701669	+	0.712503i	$0.252439\pi$
$102$		0			0
$103$	2.96694			0.292341			0.146171	−	0.989259i	$-0.453305\pi$
$103$	2.96694			0.292341			0.146171	+	0.989259i	$0.453305\pi$
$104$		0			0
$105$	−1.99447	−	1.99447i	−0.194641	−	0.194641i
$106$		0			0
$107$	1.47679	+	1.47679i	0.142766	+	0.142766i	0.774878	−	0.632111i	$-0.217811\pi$
$107$	1.47679	+	1.47679i	0.142766	+	0.142766i	−0.632111	+	0.774878i	$0.717811\pi$
$108$		0			0
$109$	1.81143	−	1.81143i	0.173504	−	0.173504i	−0.615013	−	0.788517i	$-0.710849\pi$
$109$	1.81143	−	1.81143i	0.173504	−	0.173504i	0.788517	+	0.615013i	$0.210849\pi$
$110$		0			0
$111$			8.92979i			0.847578i
$112$		0			0
$113$	9.09420	−	9.09420i	0.855510	−	0.855510i	−0.135295	−	0.990805i	$-0.543198\pi$
$113$	9.09420	−	9.09420i	0.855510	−	0.855510i	0.990805	+	0.135295i	$0.0431983\pi$
$114$		0			0
$115$	−9.04101			−0.843079
$116$		0			0
$117$		−	5.30344i		−	0.490303i
$118$		0			0
$119$	6.44511	−	9.68039i	0.590822	−	0.887400i
$120$		0			0
$121$			9.10370i			0.827609i
$122$		0			0
$123$	−10.6208			−0.957647
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$		−	7.02190i		−	0.623093i	−0.950231	−	0.311547i	$-0.899153\pi$
$127$		−	7.02190i		−	0.623093i	0.950231	−	0.311547i	$-0.100847\pi$
$128$		0			0
$129$	2.41165	−	2.41165i	0.212334	−	0.212334i
$130$		0			0
$131$	−1.01645	−	1.01645i	−0.0888080	−	0.0888080i	0.661307	−	0.750115i	$-0.270002\pi$
$131$	−1.01645	−	1.01645i	−0.0888080	−	0.0888080i	−0.750115	+	0.661307i	$0.770002\pi$
$132$		0			0
$133$	15.7829	+	15.7829i	1.36855	+	1.36855i
$134$		0			0
$135$	1.00000			0.0860663
$136$		0			0
$137$	7.84170			0.669962			0.334981	−	0.942225i	$-0.391270\pi$
$137$	7.84170			0.669962			0.334981	+	0.942225i	$0.391270\pi$
$138$		0			0
$139$	8.51929	+	8.51929i	0.722597	+	0.722597i	0.969133	−	0.246537i	$-0.0792926\pi$
$139$	8.51929	+	8.51929i	0.722597	+	0.722597i	−0.246537	+	0.969133i	$0.579293\pi$
$140$		0			0
$141$	−0.395112	−	0.395112i	−0.0332744	−	0.0332744i
$142$		0			0
$143$	5.16412	−	5.16412i	0.431845	−	0.431845i
$144$		0			0
$145$			3.69768i			0.307076i
$146$		0			0
$147$	−0.675886	+	0.675886i	−0.0557462	+	0.0557462i
$148$		0			0
$149$	−9.20202			−0.753859			−0.376929	−	0.926242i	$-0.623020\pi$
$149$	−9.20202			−0.753859			−0.376929	+	0.926242i	$0.623020\pi$
$150$		0			0
$151$			16.5906i			1.35012i	0.737763	+	0.675060i	$0.235882\pi$
$151$			16.5906i			1.35012i	−0.737763	+	0.675060i	$0.764118\pi$
$152$		0			0
$153$	0.811061	+	4.04255i	0.0655704	+	0.326820i
$154$		0			0
$155$			7.95049i			0.638599i
$156$		0			0
$157$	−2.12713			−0.169763			−0.0848816	−	0.996391i	$-0.527051\pi$
$157$	−2.12713			−0.169763			−0.0848816	+	0.996391i	$0.527051\pi$
$158$		0			0
$159$	6.57084	−	6.57084i	0.521101	−	0.521101i
$160$		0			0
$161$			25.5012i			2.00977i
$162$		0			0
$163$	6.53718	−	6.53718i	0.512031	−	0.512031i	−0.403117	−	0.915148i	$-0.632073\pi$
$163$	6.53718	−	6.53718i	0.512031	−	0.512031i	0.915148	+	0.403117i	$0.132073\pi$
$164$		0			0
$165$	0.973731	+	0.973731i	0.0758048	+	0.0758048i
$166$		0			0
$167$	−8.00033	−	8.00033i	−0.619084	−	0.619084i	0.326212	−	0.945297i	$-0.394228\pi$
$167$	−8.00033	−	8.00033i	−0.619084	−	0.619084i	−0.945297	+	0.326212i	$0.894228\pi$
$168$		0			0
$169$	15.1264			1.16357
$170$		0			0
$171$	−7.91334			−0.605148
$172$		0			0
$173$	−1.59040	−	1.59040i	−0.120916	−	0.120916i	0.644060	−	0.764975i	$-0.277249\pi$
$173$	−1.59040	−	1.59040i	−0.120916	−	0.120916i	−0.764975	+	0.644060i	$0.777249\pi$
$174$		0			0
$175$	1.99447	+	1.99447i	0.150768	+	0.150768i
$176$		0			0
$177$	−4.51286	+	4.51286i	−0.339207	+	0.339207i
$178$		0			0
$179$			19.0227i			1.42182i	0.703281	+	0.710912i	$0.251718\pi$
$179$			19.0227i			1.42182i	−0.703281	+	0.710912i	$0.748282\pi$
$180$		0			0
$181$	−8.79621	+	8.79621i	−0.653817	+	0.653817i	−0.953910	−	0.300093i	$-0.902982\pi$
$181$	−8.79621	+	8.79621i	−0.653817	+	0.653817i	0.300093	+	0.953910i	$0.402982\pi$
$182$		0			0
$183$	0.358190			0.0264781
$184$		0			0
$185$		−	8.92979i		−	0.656531i
$186$		0			0
$187$	−3.14660	+	4.72611i	−0.230102	+	0.345607i
$188$		0			0
$189$		−	2.82061i		−	0.205169i
$190$		0			0
$191$	16.0650			1.16243			0.581213	−	0.813752i	$-0.302578\pi$
$191$	16.0650			1.16243			0.581213	+	0.813752i	$0.302578\pi$
$192$		0			0
$193$	−6.21299	+	6.21299i	−0.447220	+	0.447220i	−0.894429	−	0.447209i	$-0.852418\pi$
$193$	−6.21299	+	6.21299i	−0.447220	+	0.447220i	0.447209	+	0.894429i	$0.352418\pi$
$194$		0			0
$195$			5.30344i			0.379787i
$196$		0			0
$197$	3.22826	−	3.22826i	0.230004	−	0.230004i	−0.582690	−	0.812694i	$-0.698000\pi$
$197$	3.22826	−	3.22826i	0.230004	−	0.230004i	0.812694	+	0.582690i	$0.198000\pi$
$198$		0			0
$199$	8.82355	+	8.82355i	0.625485	+	0.625485i	0.946929	−	0.321444i	$-0.104168\pi$
$199$	8.82355	+	8.82355i	0.625485	+	0.625485i	−0.321444	+	0.946929i	$0.604168\pi$
$200$		0			0
$201$	8.03466	+	8.03466i	0.566721	+	0.566721i
$202$		0			0
$203$	10.4297			0.732023
$204$		0			0
$205$	10.6208			0.741790
$206$		0			0
$207$	−6.39296	−	6.39296i	−0.444341	−	0.444341i
$208$		0			0
$209$	−7.70546	−	7.70546i	−0.532998	−	0.532998i
$210$		0			0
$211$	0.508728	−	0.508728i	0.0350223	−	0.0350223i	−0.689379	−	0.724401i	$-0.742116\pi$
$211$	0.508728	−	0.508728i	0.0350223	−	0.0350223i	0.724401	+	0.689379i	$0.242116\pi$
$212$		0			0
$213$		−	5.60137i		−	0.383800i
$214$		0			0
$215$	−2.41165	+	2.41165i	−0.164473	+	0.164473i
$216$		0			0
$217$	22.4252			1.52232
$218$		0			0
$219$		−	3.97655i		−	0.268710i
$220$		0			0
$221$	−21.4394	+	4.30141i	−1.44217	+	0.289344i
$222$		0			0
$223$			7.90359i			0.529264i	0.964350	+	0.264632i	$0.0852504\pi$
$223$			7.90359i			0.529264i	−0.964350	+	0.264632i	$0.914750\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−13.8064	+	13.8064i	−0.916365	+	0.916365i	−0.996763	−	0.0803974i	$-0.974381\pi$
$227$	−13.8064	+	13.8064i	−0.916365	+	0.916365i	0.0803974	+	0.996763i	$0.474381\pi$
$228$		0			0
$229$		−	14.6319i		−	0.966900i	−0.875372	−	0.483450i	$-0.839383\pi$
$229$		−	14.6319i		−	0.966900i	0.875372	−	0.483450i	$-0.160617\pi$
$230$		0			0
$231$	2.74652	−	2.74652i	0.180707	−	0.180707i
$232$		0			0
$233$	−19.1321	−	19.1321i	−1.25339	−	1.25339i	−0.954192	−	0.299195i	$-0.903282\pi$
$233$	−19.1321	−	19.1321i	−1.25339	−	1.25339i	−0.299195	−	0.954192i	$-0.596718\pi$
$234$		0			0
$235$	0.395112	+	0.395112i	0.0257743	+	0.0257743i
$236$		0			0
$237$	8.83668			0.574004
$238$		0			0
$239$	−19.7735			−1.27904			−0.639521	−	0.768774i	$-0.720867\pi$
$239$	−19.7735			−1.27904			−0.639521	+	0.768774i	$0.720867\pi$
$240$		0			0
$241$	−17.5625	−	17.5625i	−1.13130	−	1.13130i	−0.989961	−	0.141338i	$-0.954860\pi$
$241$	−17.5625	−	17.5625i	−1.13130	−	1.13130i	−0.141338	−	0.989961i	$-0.545140\pi$
$242$		0			0
$243$	0.707107	+	0.707107i	0.0453609	+	0.0453609i
$244$		0			0
$245$	0.675886	−	0.675886i	0.0431808	−	0.0431808i
$246$		0			0
$247$		−	41.9679i		−	2.67035i
$248$		0			0
$249$	0.264400	−	0.264400i	0.0167557	−	0.0167557i
$250$		0			0
$251$	11.2063			0.707333			0.353666	−	0.935372i	$-0.384935\pi$
$251$	11.2063			0.707333			0.353666	+	0.935372i	$0.384935\pi$
$252$		0			0
$253$		−	12.4500i		−	0.782728i
$254$		0			0
$255$	−0.811061	−	4.04255i	−0.0507906	−	0.253154i
$256$		0			0
$257$		−	19.4660i		−	1.21426i	−0.794604	−	0.607128i	$-0.792321\pi$
$257$		−	19.4660i		−	1.21426i	0.794604	−	0.607128i	$-0.207679\pi$
$258$		0			0
$259$	−25.1875			−1.56507
$260$		0			0
$261$	−2.61466	+	2.61466i	−0.161843	+	0.161843i
$262$		0			0
$263$			8.50318i			0.524329i	0.965023	+	0.262164i	$0.0844362\pi$
$263$			8.50318i			0.524329i	−0.965023	+	0.262164i	$0.915564\pi$
$264$		0			0
$265$	−6.57084	+	6.57084i	−0.403643	+	0.403643i
$266$		0			0
$267$	4.70869	+	4.70869i	0.288167	+	0.288167i
$268$		0			0
$269$	4.83042	+	4.83042i	0.294516	+	0.294516i	0.838861	−	0.544345i	$-0.183222\pi$
$269$	4.83042	+	4.83042i	0.294516	+	0.294516i	−0.544345	+	0.838861i	$0.683222\pi$
$270$		0			0
$271$	−22.5790			−1.37158			−0.685788	−	0.727802i	$-0.740542\pi$
$271$	−22.5790			−1.37158			−0.685788	+	0.727802i	$0.740542\pi$
$272$		0			0
$273$	14.9589			0.905356
$274$		0			0
$275$	−0.973731	−	0.973731i	−0.0587182	−	0.0587182i
$276$		0			0
$277$	8.27912	+	8.27912i	0.497444	+	0.497444i	0.910641	−	0.413197i	$-0.135588\pi$
$277$	8.27912	+	8.27912i	0.497444	+	0.497444i	−0.413197	+	0.910641i	$0.635588\pi$
$278$		0			0
$279$	−5.62185	+	5.62185i	−0.336571	+	0.336571i
$280$		0			0
$281$		−	15.7552i		−	0.939877i	−0.882699	−	0.469939i	$-0.844276\pi$
$281$		−	15.7552i		−	0.939877i	0.882699	−	0.469939i	$-0.155724\pi$
$282$		0			0
$283$	9.68630	−	9.68630i	0.575790	−	0.575790i	−0.357950	−	0.933741i	$-0.616524\pi$
$283$	9.68630	−	9.68630i	0.575790	−	0.575790i	0.933741	+	0.357950i	$0.116524\pi$
$284$		0			0
$285$	7.91334			0.468746
$286$		0			0
$287$		−	29.9572i		−	1.76832i
$288$		0			0
$289$	15.6844	−	6.55750i	0.922609	−	0.385735i
$290$		0			0
$291$		−	19.2478i		−	1.12833i
$292$		0			0
$293$	−8.27088			−0.483190			−0.241595	−	0.970377i	$-0.577671\pi$
$293$	−8.27088			−0.483190			−0.241595	+	0.970377i	$0.577671\pi$
$294$		0			0
$295$	4.51286	−	4.51286i	0.262749	−	0.262749i
$296$		0			0
$297$			1.37706i			0.0799053i
$298$		0			0
$299$	33.9047	−	33.9047i	1.96076	−	1.96076i
$300$		0			0
$301$	6.80232	+	6.80232i	0.392079	+	0.392079i
$302$		0			0
$303$	−9.97259	−	9.97259i	−0.572910	−	0.572910i
$304$		0			0
$305$	−0.358190			−0.0205099
$306$		0			0
$307$	−27.3011			−1.55816			−0.779079	−	0.626925i	$-0.784313\pi$
$307$	−27.3011			−1.55816			−0.779079	+	0.626925i	$0.784313\pi$
$308$		0			0
$309$	−2.09794	−	2.09794i	−0.119348	−	0.119348i
$310$		0			0
$311$	−4.24922	−	4.24922i	−0.240951	−	0.240951i	0.576292	−	0.817244i	$-0.304499\pi$
$311$	−4.24922	−	4.24922i	−0.240951	−	0.240951i	−0.817244	+	0.576292i	$0.804499\pi$
$312$		0			0
$313$	−16.3492	+	16.3492i	−0.924111	+	0.924111i	−0.997317	−	0.0732054i	$-0.976677\pi$
$313$	−16.3492	+	16.3492i	−0.924111	+	0.924111i	0.0732054	+	0.997317i	$0.476677\pi$
$314$		0			0
$315$			2.82061i			0.158923i
$316$		0			0
$317$	18.1828	−	18.1828i	1.02125	−	1.02125i	0.0214783	−	0.999769i	$-0.493163\pi$
$317$	18.1828	−	18.1828i	1.02125	−	1.02125i	0.999769	−	0.0214783i	$-0.00683729\pi$
$318$		0			0
$319$	−5.09194			−0.285094
$320$		0			0
$321$		−	2.08849i		−	0.116568i
$322$		0			0
$323$	6.41820	+	31.9900i	0.357118	+	1.77997i
$324$		0			0
$325$		−	5.30344i		−	0.294182i
$326$		0			0
$327$	−2.56176			−0.141665
$328$		0			0
$329$	1.11446	−	1.11446i	0.0614420	−	0.0614420i
$330$		0			0
$331$			18.5030i			1.01702i	0.861057	+	0.508508i	$0.169803\pi$
$331$			18.5030i			1.01702i	−0.861057	+	0.508508i	$0.830197\pi$
$332$		0			0
$333$	6.31432	−	6.31432i	0.346022	−	0.346022i
$334$		0			0
$335$	−8.03466	−	8.03466i	−0.438980	−	0.438980i
$336$		0			0
$337$	13.8123	+	13.8123i	0.752403	+	0.752403i	0.974927	−	0.222524i	$-0.0714295\pi$
$337$	13.8123	+	13.8123i	0.752403	+	0.752403i	−0.222524	+	0.974927i	$0.571430\pi$
$338$		0			0
$339$	−12.8611			−0.698521
$340$		0			0
$341$	−10.9483			−0.592885
$342$		0			0
$343$	12.0549	+	12.0549i	0.650903	+	0.650903i
$344$		0			0
$345$	6.39296	+	6.39296i	0.344185	+	0.344185i
$346$		0			0
$347$	18.8927	−	18.8927i	1.01421	−	1.01421i	0.0143162	−	0.999898i	$-0.495443\pi$
$347$	18.8927	−	18.8927i	1.01421	−	1.01421i	0.999898	−	0.0143162i	$-0.00455714\pi$
$348$		0			0
$349$			20.6784i			1.10689i	0.832887	+	0.553444i	$0.186687\pi$
$349$			20.6784i			1.10689i	−0.832887	+	0.553444i	$0.813313\pi$
$350$		0			0
$351$	−3.75010	+	3.75010i	−0.200165	+	0.200165i
$352$		0			0
$353$	19.0870			1.01590			0.507949	−	0.861387i	$-0.330404\pi$
$353$	19.0870			1.01590			0.507949	+	0.861387i	$0.330404\pi$
$354$		0			0
$355$			5.60137i			0.297290i
$356$		0			0
$357$	−11.4025	+	2.28769i	−0.603482	+	0.121077i
$358$		0			0
$359$			6.43264i			0.339502i	0.985487	+	0.169751i	$0.0542963\pi$
$359$			6.43264i			0.339502i	−0.985487	+	0.169751i	$0.945704\pi$
$360$		0			0
$361$	−43.6209			−2.29584
$362$		0			0
$363$	6.43729	−	6.43729i	0.337870	−	0.337870i
$364$		0			0
$365$			3.97655i			0.208142i
$366$		0			0
$367$	−5.54698	+	5.54698i	−0.289550	+	0.289550i	−0.836902	−	0.547352i	$-0.815636\pi$
$367$	−5.54698	+	5.54698i	−0.289550	+	0.289550i	0.547352	+	0.836902i	$0.315636\pi$
$368$		0			0
$369$	7.51005	+	7.51005i	0.390958	+	0.390958i
$370$		0			0
$371$	18.5338	+	18.5338i	0.962226	+	0.962226i
$372$		0			0
$373$	−32.0731			−1.66068			−0.830341	−	0.557256i	$-0.811854\pi$
$373$	−32.0731			−1.66068			−0.830341	+	0.557256i	$0.811854\pi$
$374$		0			0
$375$	1.00000			0.0516398
$376$		0			0
$377$	−13.8667	−	13.8667i	−0.714169	−	0.714169i
$378$		0			0
$379$	13.7164	+	13.7164i	0.704567	+	0.704567i	0.965387	−	0.260821i	$-0.0839931\pi$
$379$	13.7164	+	13.7164i	0.704567	+	0.704567i	−0.260821	+	0.965387i	$0.583993\pi$
$380$		0			0
$381$	−4.96523	+	4.96523i	−0.254377	+	0.254377i
$382$		0			0
$383$		−	17.8366i		−	0.911410i	−0.890131	−	0.455705i	$-0.849387\pi$
$383$		−	17.8366i		−	0.911410i	0.890131	−	0.455705i	$-0.150613\pi$
$384$		0			0
$385$	−2.74652	+	2.74652i	−0.139975	+	0.139975i
$386$		0			0
$387$	−3.41059			−0.173370
$388$		0			0
$389$		−	31.0499i		−	1.57429i	−0.616766	−	0.787147i	$-0.711557\pi$
$389$		−	31.0499i		−	1.57429i	0.616766	−	0.787147i	$-0.288443\pi$
$390$		0			0
$391$	−20.6588	+	31.0289i	−1.04476	+	1.56920i
$392$		0			0
$393$			1.43748i			0.0725114i
$394$		0			0
$395$	−8.83668			−0.444622
$396$		0			0
$397$	−12.0033	+	12.0033i	−0.602428	+	0.602428i	−0.940956	−	0.338528i	$-0.890071\pi$
$397$	−12.0033	+	12.0033i	−0.602428	+	0.602428i	0.338528	+	0.940956i	$0.390071\pi$
$398$		0			0
$399$		−	22.3205i		−	1.11742i
$400$		0			0
$401$	−6.81399	+	6.81399i	−0.340275	+	0.340275i	−0.856471	−	0.516196i	$-0.827348\pi$
$401$	−6.81399	+	6.81399i	−0.340275	+	0.340275i	0.516196	+	0.856471i	$0.327348\pi$
$402$		0			0
$403$	−29.8151	−	29.8151i	−1.48520	−	1.48520i
$404$		0			0
$405$	−0.707107	−	0.707107i	−0.0351364	−	0.0351364i
$406$		0			0
$407$	12.2969			0.609534
$408$		0			0
$409$	11.0129			0.544551			0.272276	−	0.962219i	$-0.412224\pi$
$409$	11.0129			0.544551			0.272276	+	0.962219i	$0.412224\pi$
$410$		0			0
$411$	−5.54492	−	5.54492i	−0.273511	−	0.273511i
$412$		0			0
$413$	−12.7290	−	12.7290i	−0.626354	−	0.626354i
$414$		0			0
$415$	−0.264400	+	0.264400i	−0.0129789	+	0.0129789i
$416$		0			0
$417$		−	12.0481i		−	0.589998i
$418$		0			0
$419$	−25.2640	+	25.2640i	−1.23423	+	1.23423i	−0.271902	+	0.962325i	$0.587653\pi$
$419$	−25.2640	+	25.2640i	−1.23423	+	1.23423i	−0.962325	+	0.271902i	$0.912347\pi$
$420$		0			0
$421$	4.24585			0.206930			0.103465	−	0.994633i	$-0.467007\pi$
$421$	4.24585			0.206930			0.103465	+	0.994633i	$0.467007\pi$
$422$		0			0
$423$			0.558773i			0.0271685i
$424$		0			0
$425$	0.811061	+	4.04255i	0.0393422	+	0.196092i
$426$		0			0
$427$			1.01031i			0.0488925i
$428$		0			0
$429$	−7.30317			−0.352600
$430$		0			0
$431$	−11.7542	+	11.7542i	−0.566182	+	0.566182i	−0.931057	−	0.364875i	$-0.881112\pi$
$431$	−11.7542	+	11.7542i	−0.566182	+	0.566182i	0.364875	+	0.931057i	$0.381112\pi$
$432$		0			0
$433$		−	35.5584i		−	1.70883i	−0.519593	−	0.854414i	$-0.673916\pi$
$433$		−	35.5584i		−	1.70883i	0.519593	−	0.854414i	$-0.326084\pi$
$434$		0			0
$435$	2.61466	−	2.61466i	0.125363	−	0.125363i
$436$		0			0
$437$	−50.5897	−	50.5897i	−2.42003	−	2.42003i
$438$		0			0
$439$	−28.4156	−	28.4156i	−1.35620	−	1.35620i	−0.878549	−	0.477652i	$-0.841488\pi$
$439$	−28.4156	−	28.4156i	−1.35620	−	1.35620i	−0.477652	−	0.878549i	$-0.658512\pi$
$440$		0			0
$441$	0.955847			0.0455165
$442$		0			0
$443$	14.0331			0.666731			0.333365	−	0.942798i	$-0.391816\pi$
$443$	14.0331			0.666731			0.333365	+	0.942798i	$0.391816\pi$
$444$		0			0
$445$	−4.70869	−	4.70869i	−0.223213	−	0.223213i
$446$		0			0
$447$	6.50681	+	6.50681i	0.307762	+	0.307762i
$448$		0			0
$449$	8.76010	−	8.76010i	0.413415	−	0.413415i	−0.469512	−	0.882926i	$-0.655570\pi$
$449$	8.76010	−	8.76010i	0.413415	−	0.413415i	0.882926	+	0.469512i	$0.155570\pi$
$450$		0			0
$451$			14.6255i			0.688690i
$452$		0			0
$453$	11.7313	−	11.7313i	0.551184	−	0.551184i
$454$		0			0
$455$	−14.9589			−0.701286
$456$		0			0
$457$		−	27.3654i		−	1.28010i	−0.768334	−	0.640049i	$-0.778914\pi$
$457$		−	27.3654i		−	1.28010i	0.768334	−	0.640049i	$-0.221086\pi$
$458$		0			0
$459$	2.28501	−	3.43202i	0.106655	−	0.160193i
$460$		0			0
$461$			16.9348i			0.788731i	0.918954	+	0.394365i	$0.129036\pi$
$461$			16.9348i			0.788731i	−0.918954	+	0.394365i	$0.870964\pi$
$462$		0			0
$463$	23.2955			1.08263			0.541317	−	0.840819i	$-0.317926\pi$
$463$	23.2955			1.08263			0.541317	+	0.840819i	$0.317926\pi$
$464$		0			0
$465$	5.62185	−	5.62185i	0.260707	−	0.260707i
$466$		0			0
$467$		−	26.1680i		−	1.21091i	−0.795879	−	0.605456i	$-0.792991\pi$
$467$		−	26.1680i		−	1.21091i	0.795879	−	0.605456i	$-0.207009\pi$
$468$		0			0
$469$	−22.6626	+	22.6626i	−1.04646	+	1.04646i
$470$		0			0
$471$	1.50411	+	1.50411i	0.0693055	+	0.0693055i
$472$		0			0
$473$	−3.32099	−	3.32099i	−0.152699	−	0.152699i
$474$		0			0
$475$	−7.91334			−0.363089
$476$		0			0
$477$	−9.29257			−0.425477
$478$		0			0
$479$	−9.96509	−	9.96509i	−0.455317	−	0.455317i	0.441798	−	0.897115i	$-0.354341\pi$
$479$	−9.96509	−	9.96509i	−0.455317	−	0.455317i	−0.897115	+	0.441798i	$0.854341\pi$
$480$		0			0
$481$	33.4876	+	33.4876i	1.52690	+	1.52690i
$482$		0			0
$483$	18.0321	−	18.0321i	0.820487	−	0.820487i
$484$		0			0
$485$			19.2478i			0.873997i
$486$		0			0
$487$	−22.3785	+	22.3785i	−1.01407	+	1.01407i	−0.0141677	+	0.999900i	$0.504510\pi$
$487$	−22.3785	+	22.3785i	−1.01407	+	1.01407i	−0.999900	+	0.0141677i	$0.995490\pi$
$488$		0			0
$489$	−9.24496			−0.418072
$490$		0			0
$491$		−	39.7145i		−	1.79229i	−0.443759	−	0.896146i	$-0.646355\pi$
$491$		−	39.7145i		−	1.79229i	0.443759	−	0.896146i	$-0.353645\pi$
$492$		0			0
$493$	12.6905	+	8.44922i	0.571552	+	0.380534i
$494$		0			0
$495$		−	1.37706i		−	0.0618944i
$496$		0			0
$497$	15.7993			0.708695
$498$		0			0
$499$	19.3831	−	19.3831i	0.867706	−	0.867706i	−0.124512	−	0.992218i	$-0.539736\pi$
$499$	19.3831	−	19.3831i	0.867706	−	0.867706i	0.992218	+	0.124512i	$0.0397365\pi$
$500$		0			0
$501$			11.3142i			0.505480i
$502$		0			0
$503$	9.43380	−	9.43380i	0.420632	−	0.420632i	−0.464789	−	0.885421i	$-0.653870\pi$
$503$	9.43380	−	9.43380i	0.420632	−	0.420632i	0.885421	+	0.464789i	$0.153870\pi$
$504$		0			0
$505$	9.97259	+	9.97259i	0.443774	+	0.443774i
$506$		0			0
$507$	−10.6960	−	10.6960i	−0.475026	−	0.475026i
$508$		0			0
$509$	0.824026			0.0365243			0.0182622	−	0.999833i	$-0.494187\pi$
$509$	0.824026			0.0365243			0.0182622	+	0.999833i	$0.494187\pi$
$510$		0			0
$511$	11.2163			0.496180
$512$		0			0
$513$	5.59558	+	5.59558i	0.247051	+	0.247051i
$514$		0			0
$515$	2.09794	+	2.09794i	0.0924465	+	0.0924465i
$516$		0			0
$517$	−0.544094	+	0.544094i	−0.0239292	+	0.0239292i
$518$		0			0
$519$			2.24916i			0.0987272i
$520$		0			0
$521$	10.7266	−	10.7266i	0.469940	−	0.469940i	−0.431955	−	0.901895i	$-0.642176\pi$
$521$	10.7266	−	10.7266i	0.469940	−	0.469940i	0.901895	+	0.431955i	$0.142176\pi$
$522$		0			0
$523$	−12.4696			−0.545256			−0.272628	−	0.962119i	$-0.587893\pi$
$523$	−12.4696			−0.545256			−0.272628	+	0.962119i	$0.587893\pi$
$524$		0			0
$525$		−	2.82061i		−	0.123102i
$526$		0			0
$527$	27.2862	+	18.1669i	1.18861	+	0.791363i
$528$		0			0
$529$		−	58.7399i		−	2.55391i
$530$		0			0
$531$	6.38214			0.276961
$532$		0			0
$533$	−39.8291	+	39.8291i	−1.72519	+	1.72519i
$534$		0			0
$535$			2.08849i			0.0902934i
$536$		0			0
$537$	13.4511	−	13.4511i	0.580457	−	0.580457i
$538$		0			0
$539$	0.930738	+	0.930738i	0.0400897	+	0.0400897i
$540$		0			0
$541$	−6.01645	−	6.01645i	−0.258668	−	0.258668i	0.565844	−	0.824512i	$-0.308550\pi$
$541$	−6.01645	−	6.01645i	−0.258668	−	0.258668i	−0.824512	+	0.565844i	$0.808550\pi$
$542$		0			0
$543$	12.4397			0.533839
$544$		0			0
$545$	2.56176			0.109734
$546$		0			0
$547$	−0.586915	−	0.586915i	−0.0250947	−	0.0250947i	0.694448	−	0.719543i	$-0.255649\pi$
$547$	−0.586915	−	0.586915i	−0.0250947	−	0.0250947i	−0.719543	+	0.694448i	$0.755649\pi$
$548$		0			0
$549$	−0.253278	−	0.253278i	−0.0108097	−	0.0108097i
$550$		0			0
$551$	−20.6907	+	20.6907i	−0.881452	+	0.881452i
$552$		0			0
$553$			24.9248i			1.05991i
$554$		0			0
$555$	−6.31432	+	6.31432i	−0.268028	+	0.268028i
$556$		0			0
$557$	17.5926			0.745421			0.372710	−	0.927948i	$-0.378428\pi$
$557$	17.5926			0.745421			0.372710	+	0.927948i	$0.378428\pi$
$558$		0			0
$559$		−	18.0878i		−	0.765034i
$560$		0			0
$561$	5.56684	−	1.11688i	0.235032	−	0.0471548i
$562$		0			0
$563$			19.3293i			0.814633i	0.913287	+	0.407317i	$0.133535\pi$
$563$			19.3293i			0.814633i	−0.913287	+	0.407317i	$0.866465\pi$
$564$		0			0
$565$	12.8611			0.541072
$566$		0			0
$567$	−1.99447	+	1.99447i	−0.0837600	+	0.0837600i
$568$		0			0
$569$			0.408193i			0.0171123i	0.999963	+	0.00855617i	$0.00272355\pi$
$569$			0.408193i			0.0171123i	−0.999963	+	0.00855617i	$0.997276\pi$
$570$		0			0
$571$	13.7975	−	13.7975i	0.577407	−	0.577407i	−0.356781	−	0.934188i	$-0.616126\pi$
$571$	13.7975	−	13.7975i	0.577407	−	0.577407i	0.934188	+	0.356781i	$0.116126\pi$
$572$		0			0
$573$	−11.3597	−	11.3597i	−0.474558	−	0.474558i
$574$		0			0
$575$	−6.39296	−	6.39296i	−0.266605	−	0.266605i
$576$		0			0
$577$	24.5084			1.02030			0.510149	−	0.860086i	$-0.329590\pi$
$577$	24.5084			1.02030			0.510149	+	0.860086i	$0.329590\pi$
$578$		0			0
$579$	8.78649			0.365154
$580$		0			0
$581$	0.745770	+	0.745770i	0.0309398	+	0.0309398i
$582$		0			0
$583$	−9.04846	−	9.04846i	−0.374749	−	0.374749i
$584$		0			0
$585$	3.75010	−	3.75010i	0.155047	−	0.155047i
$586$		0			0
$587$		−	10.7151i		−	0.442261i	−0.975244	−	0.221130i	$-0.929025\pi$
$587$		−	10.7151i		−	0.442261i	0.975244	−	0.221130i	$-0.0709747\pi$
$588$		0			0
$589$	−44.4876	+	44.4876i	−1.83308	+	1.83308i
$590$		0			0
$591$	−4.56546			−0.187798
$592$		0			0
$593$		−	15.0740i		−	0.619015i	−0.950897	−	0.309508i	$-0.899836\pi$
$593$		−	15.0740i		−	0.619015i	0.950897	−	0.309508i	$-0.100164\pi$
$594$		0			0
$595$	11.4025	−	2.28769i	0.467455	−	0.0937860i
$596$		0			0
$597$		−	12.4784i		−	0.510706i
$598$		0			0
$599$	33.3521			1.36273			0.681365	−	0.731944i	$-0.261387\pi$
$599$	33.3521			1.36273			0.681365	+	0.731944i	$0.261387\pi$
$600$		0			0
$601$	17.1773	−	17.1773i	0.700676	−	0.700676i	−0.263879	−	0.964556i	$-0.585002\pi$
$601$	17.1773	−	17.1773i	0.700676	−	0.700676i	0.964556	+	0.263879i	$0.0850020\pi$
$602$		0			0
$603$		−	11.3627i		−	0.462726i
$604$		0			0
$605$	−6.43729	+	6.43729i	−0.261713	+	0.261713i
$606$		0			0
$607$	−1.72524	−	1.72524i	−0.0700253	−	0.0700253i	0.671227	−	0.741252i	$-0.265768\pi$
$607$	−1.72524	−	1.72524i	−0.0700253	−	0.0700253i	−0.741252	+	0.671227i	$0.765768\pi$
$608$		0			0
$609$	−7.37493	−	7.37493i	−0.298847	−	0.298847i
$610$		0			0
$611$	−2.96342			−0.119887
$612$		0			0
$613$	24.6312			0.994845			0.497423	−	0.867508i	$-0.334280\pi$
$613$	24.6312			0.994845			0.497423	+	0.867508i	$0.334280\pi$
$614$		0			0
$615$	−7.51005	−	7.51005i	−0.302835	−	0.302835i
$616$		0			0
$617$	32.3258	+	32.3258i	1.30139	+	1.30139i	0.927458	+	0.373928i	$0.121989\pi$
$617$	32.3258	+	32.3258i	1.30139	+	1.30139i	0.373928	+	0.927458i	$0.378011\pi$
$618$		0			0
$619$	−11.2002	+	11.2002i	−0.450175	+	0.450175i	−0.895413	−	0.445237i	$-0.853119\pi$
$619$	−11.2002	+	11.2002i	−0.450175	+	0.450175i	0.445237	+	0.895413i	$0.353119\pi$
$620$		0			0
$621$			9.04101i			0.362803i
$622$		0			0
$623$	−13.2814	+	13.2814i	−0.532107	+	0.532107i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$			10.8972i			0.435191i
$628$		0			0
$629$	−30.6472	−	20.4046i	−1.22198	−	0.813586i
$630$		0			0
$631$		−	8.97590i		−	0.357325i	−0.983910	−	0.178663i	$-0.942823\pi$
$631$		−	8.97590i		−	0.357325i	0.983910	−	0.178663i	$-0.0571770\pi$
$632$		0			0
$633$	−0.719450			−0.0285956
$634$		0			0
$635$	4.96523	−	4.96523i	0.197039	−	0.197039i
$636$		0			0
$637$			5.06928i			0.200852i
$638$		0			0
$639$	−3.96077	+	3.96077i	−0.156686	+	0.156686i
$640$		0			0
$641$	11.4869	+	11.4869i	0.453703	+	0.453703i	0.896582	−	0.442878i	$-0.146043\pi$
$641$	11.4869	+	11.4869i	0.453703	+	0.453703i	−0.442878	+	0.896582i	$0.646043\pi$
$642$		0			0
$643$	−3.99287	−	3.99287i	−0.157464	−	0.157464i	0.623978	−	0.781442i	$-0.285515\pi$
$643$	−3.99287	−	3.99287i	−0.157464	−	0.157464i	−0.781442	+	0.623978i	$0.785515\pi$
$644$		0			0
$645$	3.41059			0.134292
$646$		0			0
$647$	21.9021			0.861059			0.430529	−	0.902577i	$-0.358327\pi$
$647$	21.9021			0.861059			0.430529	+	0.902577i	$0.358327\pi$
$648$		0			0
$649$	6.21449	+	6.21449i	0.243940	+	0.243940i
$650$		0			0
$651$	−15.8570	−	15.8570i	−0.621486	−	0.621486i
$652$		0			0
$653$	−16.0555	+	16.0555i	−0.628302	+	0.628302i	−0.947641	−	0.319339i	$-0.896539\pi$
$653$	−16.0555	+	16.0555i	−0.628302	+	0.628302i	0.319339	+	0.947641i	$0.396539\pi$
$654$		0			0
$655$		−	1.43748i		−	0.0561671i
$656$		0			0
$657$	−2.81184	+	2.81184i	−0.109701	+	0.109701i
$658$		0			0
$659$	40.9096			1.59361			0.796805	−	0.604236i	$-0.206522\pi$
$659$	40.9096			1.59361			0.796805	+	0.604236i	$0.206522\pi$
$660$		0			0
$661$		−	4.04244i		−	0.157233i	−0.996905	−	0.0786164i	$-0.974950\pi$
$661$		−	4.04244i		−	0.157233i	0.996905	−	0.0786164i	$-0.0250502\pi$
$662$		0			0
$663$	18.2015	+	12.1184i	0.706887	+	0.470639i
$664$		0			0
$665$			22.3205i			0.865550i
$666$		0			0
$667$	−33.4308			−1.29444
$668$		0			0
$669$	5.58868	−	5.58868i	0.216071	−	0.216071i
$670$		0			0
$671$		−	0.493250i		−	0.0190417i
$672$		0			0
$673$	12.5535	−	12.5535i	0.483902	−	0.483902i	−0.422474	−	0.906375i	$-0.638838\pi$
$673$	12.5535	−	12.5535i	0.483902	−	0.483902i	0.906375	+	0.422474i	$0.138838\pi$
$674$		0			0
$675$	0.707107	+	0.707107i	0.0272166	+	0.0272166i
$676$		0			0
$677$	26.0901	+	26.0901i	1.00273	+	1.00273i	0.999996	+	0.00272902i	$0.000868674\pi$
$677$	26.0901	+	26.0901i	1.00273	+	1.00273i	0.00272902	+	0.999996i	$0.499131\pi$
$678$		0			0
$679$	54.2905			2.08348
$680$		0			0
$681$	19.5253			0.748209
$682$		0			0
$683$	−17.2264	−	17.2264i	−0.659149	−	0.659149i	0.296030	−	0.955179i	$-0.404337\pi$
$683$	−17.2264	−	17.2264i	−0.659149	−	0.659149i	−0.955179	+	0.296030i	$0.904337\pi$
$684$		0			0
$685$	5.54492	+	5.54492i	0.211860	+	0.211860i
$686$		0			0
$687$	−10.3463	+	10.3463i	−0.394735	+	0.394735i
$688$		0			0
$689$		−	49.2825i		−	1.87751i
$690$		0			0
$691$	−32.2960	+	32.2960i	−1.22860	+	1.22860i	−0.264105	+	0.964494i	$0.585076\pi$
$691$	−32.2960	+	32.2960i	−1.22860	+	1.22860i	−0.964494	+	0.264105i	$0.914924\pi$
$692$		0			0
$693$	−3.88416			−0.147547
$694$		0			0
$695$			12.0481i			0.457010i
$696$		0			0
$697$	24.2686	−	36.4508i	0.919240	−	1.38067i
$698$		0			0
$699$			27.0569i			1.02339i
$700$		0			0
$701$	8.77977			0.331607			0.165804	−	0.986159i	$-0.446978\pi$
$701$	8.77977			0.331607			0.165804	+	0.986159i	$0.446978\pi$
$702$		0			0
$703$	49.9673	−	49.9673i	1.88455	−	1.88455i
$704$		0			0
$705$		−	0.558773i		−	0.0210446i
$706$		0			0
$707$	28.1288	−	28.1288i	1.05789	−	1.05789i
$708$		0			0
$709$	−35.0224	−	35.0224i	−1.31529	−	1.31529i	−0.917456	−	0.397837i	$-0.869761\pi$
$709$	−35.0224	−	35.0224i	−1.31529	−	1.31529i	−0.397837	−	0.917456i	$-0.630239\pi$
$710$		0			0
$711$	−6.24848	−	6.24848i	−0.234336	−	0.234336i
$712$		0			0
$713$	−71.8805			−2.69194
$714$		0			0
$715$	7.30317			0.273123
$716$		0			0
$717$	13.9820	+	13.9820i	0.522166	+	0.522166i
$718$		0			0
$719$	−3.15250	−	3.15250i	−0.117568	−	0.117568i	0.645875	−	0.763443i	$-0.276493\pi$
$719$	−3.15250	−	3.15250i	−0.117568	−	0.117568i	−0.763443	+	0.645875i	$0.776493\pi$
$720$		0			0
$721$	5.91748	−	5.91748i	0.220379	−	0.220379i
$722$		0			0
$723$			24.8371i			0.923702i
$724$		0			0
$725$	−2.61466	+	2.61466i	−0.0971059	+	0.0971059i
$726$		0			0
$727$	−15.6067			−0.578821			−0.289411	−	0.957205i	$-0.593459\pi$
$727$	−15.6067			−0.578821			−0.289411	+	0.957205i	$0.593459\pi$
$728$		0			0
$729$		−	1.00000i		−	0.0370370i
$730$		0			0
$731$	2.76619	+	13.7875i	0.102311	+	0.509947i
$732$		0			0
$733$			4.94469i			0.182636i	0.995822	+	0.0913182i	$0.0291080\pi$
$733$			4.94469i			0.182636i	−0.995822	+	0.0913182i	$0.970892\pi$
$734$		0			0
$735$	−0.955847			−0.0352570
$736$		0			0
$737$	11.0642	−	11.0642i	0.407556	−	0.407556i
$738$		0			0
$739$		−	13.3654i		−	0.491653i	−0.969314	−	0.245826i	$-0.920941\pi$
$739$		−	13.3654i		−	0.491653i	0.969314	−	0.245826i	$-0.0790593\pi$
$740$		0			0
$741$	−29.6758	+	29.6758i	−1.09017	+	1.09017i
$742$		0			0
$743$	−20.9973	−	20.9973i	−0.770317	−	0.770317i	0.207845	−	0.978162i	$-0.433355\pi$
$743$	−20.9973	−	20.9973i	−0.770317	−	0.770317i	−0.978162	+	0.207845i	$0.933355\pi$
$744$		0			0
$745$	−6.50681	−	6.50681i	−0.238391	−	0.238391i
$746$		0			0
$747$	−0.373918			−0.0136810
$748$		0			0
$749$	5.89082			0.215246
$750$		0			0
$751$	−12.7780	−	12.7780i	−0.466276	−	0.466276i	0.434430	−	0.900706i	$-0.356950\pi$
$751$	−12.7780	−	12.7780i	−0.466276	−	0.466276i	−0.900706	+	0.434430i	$0.856950\pi$
$752$		0			0
$753$	−7.92402	−	7.92402i	−0.288767	−	0.288767i
$754$		0			0
$755$	−11.7313	+	11.7313i	−0.426945	+	0.426945i
$756$		0			0
$757$		−	20.8097i		−	0.756342i	−0.925736	−	0.378171i	$-0.876553\pi$
$757$		−	20.8097i		−	0.756342i	0.925736	−	0.378171i	$-0.123447\pi$
$758$		0			0
$759$	−8.80351	+	8.80351i	−0.319547	+	0.319547i
$760$		0			0
$761$	−25.3503			−0.918948			−0.459474	−	0.888191i	$-0.651962\pi$
$761$	−25.3503			−0.918948			−0.459474	+	0.888191i	$0.651962\pi$
$762$		0			0
$763$		−	7.22572i		−	0.261588i
$764$		0			0
$765$	−2.28501	+	3.43202i	−0.0826145	+	0.124085i
$766$		0			0
$767$			33.8473i			1.22215i
$768$		0			0
$769$	27.4442			0.989664			0.494832	−	0.868989i	$-0.335230\pi$
$769$	27.4442			0.989664			0.494832	+	0.868989i	$0.335230\pi$
$770$		0			0
$771$	−13.7645	+	13.7645i	−0.495718	+	0.495718i
$772$		0			0
$773$		−	8.59641i		−	0.309192i	−0.987978	−	0.154596i	$-0.950592\pi$
$773$		−	8.59641i		−	0.309192i	0.987978	−	0.154596i	$-0.0494075\pi$
$774$		0			0
$775$	−5.62185	+	5.62185i	−0.201943	+	0.201943i
$776$		0			0
$777$	17.8102	+	17.8102i	0.638939	+	0.638939i
$778$		0			0
$779$	59.4296	+	59.4296i	2.12929	+	2.12929i
$780$		0			0
$781$	−7.71345			−0.276009
$782$		0			0
$783$	3.69768			0.132144
$784$		0			0
$785$	−1.50411	−	1.50411i	−0.0536838	−	0.0536838i
$786$		0			0
$787$	−2.48348	−	2.48348i	−0.0885263	−	0.0885263i	0.661457	−	0.749983i	$-0.269938\pi$
$787$	−2.48348	−	2.48348i	−0.0885263	−	0.0885263i	−0.749983	+	0.661457i	$0.769938\pi$
$788$		0			0
$789$	6.01266	−	6.01266i	0.214056	−	0.214056i
$790$		0			0
$791$		−	36.2763i		−	1.28984i
$792$		0			0
$793$	1.34325	−	1.34325i	0.0477000	−	0.0477000i
$794$		0			0
$795$	9.29257			0.329573
$796$		0			0
$797$			49.2122i			1.74319i	0.490231	+	0.871593i	$0.336913\pi$
$797$			49.2122i			1.74319i	−0.490231	+	0.871593i	$0.663087\pi$
$798$		0			0
$799$	2.25887	−	0.453199i	0.0799129	−	0.0160330i
$800$		0			0
$801$		−	6.65909i		−	0.235288i
$802$		0			0
$803$	−5.47596			−0.193242
$804$		0			0
$805$	−18.0321	+	18.0321i	−0.635546	+	0.635546i
$806$		0			0
$807$		−	6.83124i		−	0.240471i
$808$		0			0
$809$	−4.17134	+	4.17134i	−0.146656	+	0.146656i	−0.776623	−	0.629966i	$-0.783069\pi$
$809$	−4.17134	+	4.17134i	−0.146656	+	0.146656i	0.629966	+	0.776623i	$0.283069\pi$
$810$		0			0
$811$	27.1111	+	27.1111i	0.951998	+	0.951998i	0.998900	−	0.0469014i	$-0.0149347\pi$
$811$	27.1111	+	27.1111i	0.951998	+	0.951998i	−0.0469014	+	0.998900i	$0.514935\pi$
$812$		0			0
$813$	15.9657	+	15.9657i	0.559943	+	0.559943i
$814$		0			0
$815$	9.24496			0.323837
$816$		0			0
$817$	−26.9891			−0.944230
$818$		0			0
$819$	−10.5776	−	10.5776i	−0.369610	−	0.369610i
$820$		0			0
$821$	−34.4268	−	34.4268i	−1.20150	−	1.20150i	−0.973709	−	0.227793i	$-0.926849\pi$
$821$	−34.4268	−	34.4268i	−1.20150	−	1.20150i	−0.227793	−	0.973709i	$-0.573151\pi$
$822$		0			0
$823$	−35.3270	+	35.3270i	−1.23142	+	1.23142i	−0.268005	+	0.963418i	$0.586364\pi$
$823$	−35.3270	+	35.3270i	−1.23142	+	1.23142i	−0.963418	+	0.268005i	$0.913636\pi$
$824$		0			0
$825$			1.37706i			0.0479432i
$826$		0			0
$827$	0.101760	−	0.101760i	0.00353854	−	0.00353854i	−0.705335	−	0.708874i	$-0.749204\pi$
$827$	0.101760	−	0.101760i	0.00353854	−	0.00353854i	0.708874	+	0.705335i	$0.249204\pi$
$828$		0			0
$829$	−31.0155			−1.07721			−0.538606	−	0.842558i	$-0.681049\pi$
$829$	−31.0155			−1.07721			−0.538606	+	0.842558i	$0.681049\pi$
$830$		0			0
$831$		−	11.7084i		−	0.406161i
$832$		0			0
$833$	−0.775250	−	3.86406i	−0.0268608	−	0.133882i
$834$		0			0
$835$		−	11.3142i		−	0.391543i
$836$		0			0
$837$	7.95049			0.274809
$838$		0			0
$839$	28.9112	−	28.9112i	0.998127	−	0.998127i	−0.00187119	−	0.999998i	$-0.500596\pi$
$839$	28.9112	−	28.9112i	0.998127	−	0.998127i	0.999998	+	0.00187119i	$0.000595619\pi$
$840$		0			0
$841$		−	15.3272i		−	0.528522i
$842$		0			0
$843$	−11.1406	+	11.1406i	−0.383703	+	0.383703i
$844$		0			0
$845$	10.6960	+	10.6960i	0.367954	+	0.367954i
$846$		0			0
$847$	18.1571	+	18.1571i	0.623885	+	0.623885i
$848$		0			0
$849$	−13.6985			−0.470131
$850$		0			0
$851$	80.7344			2.76754
$852$		0			0
$853$	−6.53196	−	6.53196i	−0.223650	−	0.223650i	0.586383	−	0.810034i	$-0.300551\pi$
$853$	−6.53196	−	6.53196i	−0.223650	−	0.223650i	−0.810034	+	0.586383i	$0.800551\pi$
$854$		0			0
$855$	−5.59558	−	5.59558i	−0.191365	−	0.191365i
$856$		0			0
$857$	−24.3626	+	24.3626i	−0.832211	+	0.832211i	−0.987819	−	0.155608i	$-0.950266\pi$
$857$	−24.3626	+	24.3626i	−0.832211	+	0.832211i	0.155608	+	0.987819i	$0.450266\pi$
$858$		0			0
$859$		−	14.5832i		−	0.497571i	−0.968559	−	0.248785i	$-0.919969\pi$
$859$		−	14.5832i		−	0.497571i	0.968559	−	0.248785i	$-0.0800314\pi$
$860$		0			0
$861$	−21.1829	+	21.1829i	−0.721913	+	0.721913i
$862$		0			0
$863$	−9.62593			−0.327671			−0.163835	−	0.986488i	$-0.552387\pi$
$863$	−9.62593			−0.327671			−0.163835	+	0.986488i	$0.552387\pi$
$864$		0			0
$865$		−	2.24916i		−	0.0764738i
$866$		0			0
$867$	−15.7274	−	6.45367i	−0.534130	−	0.219178i
$868$		0			0
$869$		−	12.1687i		−	0.412794i
$870$		0			0
$871$	60.2615			2.04188
$872$		0			0
$873$	−13.6102	+	13.6102i	−0.460637	+	0.460637i
$874$		0			0
$875$			2.82061i			0.0953541i
$876$		0			0
$877$	23.5693	−	23.5693i	0.795878	−	0.795878i	−0.186565	−	0.982443i	$-0.559735\pi$
$877$	23.5693	−	23.5693i	0.795878	−	0.795878i	0.982443	+	0.186565i	$0.0597355\pi$
$878$		0			0
$879$	5.84840	+	5.84840i	0.197262	+	0.197262i
$880$		0			0
$881$	38.5206	+	38.5206i	1.29779	+	1.29779i	0.929847	+	0.367946i	$0.119939\pi$
$881$	38.5206	+	38.5206i	1.29779	+	1.29779i	0.367946	+	0.929847i	$0.380061\pi$
$882$		0			0
$883$	3.93822			0.132532			0.0662658	−	0.997802i	$-0.478891\pi$
$883$	3.93822			0.132532			0.0662658	+	0.997802i	$0.478891\pi$
$884$		0			0
$885$	−6.38214			−0.214533
$886$		0			0
$887$	18.3742	+	18.3742i	0.616945	+	0.616945i	0.944747	−	0.327801i	$-0.106308\pi$
$887$	18.3742	+	18.3742i	0.616945	+	0.616945i	−0.327801	+	0.944747i	$0.606308\pi$
$888$		0			0
$889$	−14.0050	−	14.0050i	−0.469713	−	0.469713i
$890$		0			0
$891$	0.973731	−	0.973731i	0.0326212	−	0.0326212i
$892$		0			0
$893$			4.42176i			0.147969i
$894$		0			0
$895$	−13.4511	+	13.4511i	−0.449620	+	0.449620i
$896$		0			0
$897$	−47.9484			−1.60095
$898$		0			0
$899$			29.3984i			0.980491i
$900$		0			0
$901$	7.53683	+	37.5656i	0.251088	+	1.25149i
$902$		0			0
$903$		−	9.61994i		−	0.320131i
$904$		0			0
$905$	−12.4397			−0.413510
$906$		0			0
$907$	4.64445	−	4.64445i	0.154216	−	0.154216i	−0.625782	−	0.779998i	$-0.715220\pi$
$907$	4.64445	−	4.64445i	0.154216	−	0.154216i	0.779998	+	0.625782i	$0.215220\pi$
$908$		0			0
$909$			14.1034i			0.467779i
$910$		0			0
$911$	−15.2547	+	15.2547i	−0.505412	+	0.505412i	−0.913115	−	0.407703i	$-0.866330\pi$
$911$	−15.2547	+	15.2547i	−0.505412	+	0.505412i	0.407703	+	0.913115i	$0.366330\pi$
$912$		0			0
$913$	−0.364096	−	0.364096i	−0.0120498	−	0.0120498i
$914$		0			0
$915$	0.253278	+	0.253278i	0.00837312	+	0.00837312i
$916$		0			0
$917$	−4.05458			−0.133894
$918$		0			0
$919$	55.4164			1.82802			0.914009	−	0.405693i	$-0.132970\pi$
$919$	55.4164			1.82802			0.914009	+	0.405693i	$0.132970\pi$
$920$		0			0
$921$	19.3048	+	19.3048i	0.636116	+	0.636116i
$922$		0			0
$923$	−21.0057	−	21.0057i	−0.691411	−	0.691411i
$924$		0			0
$925$	6.31432	−	6.31432i	0.207613	−	0.207613i
$926$		0			0
$927$			2.96694i			0.0974471i
$928$		0			0
$929$	−39.4595	+	39.4595i	−1.29463	+	1.29463i	−0.362732	+	0.931894i	$0.618156\pi$
$929$	−39.4595	+	39.4595i	−1.29463	+	1.29463i	−0.931894	+	0.362732i	$0.881844\pi$
$930$		0			0
$931$	7.56395			0.247898
$932$		0			0
$933$			6.00931i			0.196736i
$934$		0			0
$935$	−5.56684	+	1.11688i	−0.182055	+	0.0365259i
$936$		0			0
$937$			55.3322i			1.80763i	0.427928	+	0.903813i	$0.359244\pi$
$937$			55.3322i			1.80763i	−0.427928	+	0.903813i	$0.640756\pi$
$938$		0			0
$939$	23.1213			0.754534
$940$		0			0
$941$	−17.0994	+	17.0994i	−0.557426	+	0.557426i	−0.928574	−	0.371148i	$-0.878964\pi$
$941$	−17.0994	+	17.0994i	−0.557426	+	0.557426i	0.371148	+	0.928574i	$0.378964\pi$
$942$		0			0
$943$			96.0229i			3.12694i
$944$		0			0
$945$	1.99447	−	1.99447i	0.0648802	−	0.0648802i
$946$		0			0
$947$	−0.756856	−	0.756856i	−0.0245945	−	0.0245945i	0.694703	−	0.719297i	$-0.255536\pi$
$947$	−0.756856	−	0.756856i	−0.0245945	−	0.0245945i	−0.719297	+	0.694703i	$0.755536\pi$
$948$		0			0
$949$	−14.9124	−	14.9124i	−0.484078	−	0.484078i
$950$		0			0
$951$	−25.7144			−0.833845
$952$		0			0
$953$	−34.1782			−1.10714			−0.553570	−	0.832803i	$-0.686735\pi$
$953$	−34.1782			−1.10714			−0.553570	+	0.832803i	$0.686735\pi$
$954$		0			0
$955$	11.3597	+	11.3597i	0.367591	+	0.367591i
$956$		0			0
$957$	3.60055	+	3.60055i	0.116389	+	0.116389i
$958$		0			0
$959$	15.6401	−	15.6401i	0.505044	−	0.505044i
$960$		0			0
$961$			32.2103i			1.03904i
$962$		0			0
$963$	−1.47679	+	1.47679i	−0.0475888	+	0.0475888i
$964$		0			0
$965$	−8.78649			−0.282847
$966$		0			0
$967$		−	53.6202i		−	1.72431i	−0.506644	−	0.862155i	$-0.669114\pi$
$967$		−	53.6202i		−	1.72431i	0.506644	−	0.862155i	$-0.330886\pi$
$968$		0			0
$969$	18.0820	−	27.1587i	0.580878	−	0.872464i
$970$		0			0
$971$		−	41.2678i		−	1.32435i	−0.749351	−	0.662173i	$-0.769635\pi$
$971$		−	41.2678i		−	1.32435i	0.749351	−	0.662173i	$-0.230365\pi$
$972$		0			0
$973$	33.9830			1.08944
$974$		0			0
$975$	−3.75010	+	3.75010i	−0.120099	+	0.120099i
$976$		0			0
$977$		−	4.35594i		−	0.139359i	−0.997569	−	0.0696795i	$-0.977802\pi$
$977$		−	4.35594i		−	0.139359i	0.997569	−	0.0696795i	$-0.0221977\pi$
$978$		0			0
$979$	6.48416	−	6.48416i	0.207235	−	0.207235i
$980$		0			0
$981$	1.81143	+	1.81143i	0.0578347	+	0.0578347i
$982$		0			0
$983$	27.4424	+	27.4424i	0.875275	+	0.875275i	0.993041	−	0.117766i	$-0.0375733\pi$
$983$	27.4424	+	27.4424i	0.875275	+	0.875275i	−0.117766	+	0.993041i	$0.537573\pi$
$984$		0			0
$985$	4.56546			0.145467
$986$		0			0
$987$	−1.57608			−0.0501672
$988$		0			0
$989$	−21.8037	−	21.8037i	−0.693319	−	0.693319i
$990$		0			0
$991$	−32.7439	−	32.7439i	−1.04014	−	1.04014i	−0.999160	−	0.0409834i	$-0.986951\pi$
$991$	−32.7439	−	32.7439i	−1.04014	−	1.04014i	−0.0409834	−	0.999160i	$-0.513049\pi$
$992$		0			0
$993$	13.0836	−	13.0836i	0.415195	−	0.415195i
$994$		0			0
$995$			12.4784i			0.395591i
$996$		0			0
$997$	−15.6319	+	15.6319i	−0.495066	+	0.495066i	−0.909898	−	0.414832i	$-0.863840\pi$
$997$	−15.6319	+	15.6319i	−0.495066	+	0.495066i	0.414832	+	0.909898i	$0.363840\pi$
$998$		0			0
$999$	−8.92979			−0.282526

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.2.cf.e.1441.4	yes	20
17.4	even	4	inner	2040.2.cf.e.361.4	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.cf.e.361.4	✓	20	17.4	even	4	inner
2040.2.cf.e.1441.4	yes	20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2040 = 2^{3} \cdot 3 \cdot 5 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2040.cf (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{5} +(1.99447 - 1.99447i) q^{7} +1.00000i q^{9} +(-0.973731 + 0.973731i) q^{11} -5.30344 q^{13} -1.00000i q^{15} +(4.04255 - 0.811061i) q^{17} +7.91334i q^{19} -2.82061 q^{21} +(-6.39296 + 6.39296i) q^{23} +1.00000i q^{25} +(0.707107 - 0.707107i) q^{27} +(2.61466 + 2.61466i) q^{29} +(5.62185 + 5.62185i) q^{31} +1.37706 q^{33} +2.82061 q^{35} +(-6.31432 - 6.31432i) q^{37} +(3.75010 + 3.75010i) q^{39} +(7.51005 - 7.51005i) q^{41} +3.41059i q^{43} +(-0.707107 + 0.707107i) q^{45} +0.558773 q^{47} -0.955847i q^{49} +(-3.43202 - 2.28501i) q^{51} +9.29257i q^{53} -1.37706 q^{55} +(5.59558 - 5.59558i) q^{57} -6.38214i q^{59} +(-0.253278 + 0.253278i) q^{61} +(1.99447 + 1.99447i) q^{63} +(-3.75010 - 3.75010i) q^{65} -11.3627 q^{67} +9.04101 q^{69} +(3.96077 + 3.96077i) q^{71} +(2.81184 + 2.81184i) q^{73} +(0.707107 - 0.707107i) q^{75} +3.88416i q^{77} +(-6.24848 + 6.24848i) q^{79} -1.00000 q^{81} +0.373918i q^{83} +(3.43202 + 2.28501i) q^{85} -3.69768i q^{87} -6.65909 q^{89} +(-10.5776 + 10.5776i) q^{91} -7.95049i q^{93} +(-5.59558 + 5.59558i) q^{95} +(13.6102 + 13.6102i) q^{97} +(-0.973731 - 0.973731i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q + 4 q^{7} + 4 q^{11} - 16 q^{13} + 4 q^{17} - 8 q^{21} + 4 q^{23} + 12 q^{29} + 16 q^{31} - 8 q^{33} + 8 q^{35} - 20 q^{37} - 4 q^{39} - 4 q^{41} + 8 q^{51} + 8 q^{55} - 8 q^{57} + 4 q^{63} + 4 q^{65}+ \cdots + 4 q^{99}+O(q^{100}) \) 20 * q + 4 * q^7 + 4 * q^11 - 16 * q^13 + 4 * q^17 - 8 * q^21 + 4 * q^23 + 12 * q^29 + 16 * q^31 - 8 * q^33 + 8 * q^35 - 20 * q^37 - 4 * q^39 - 4 * q^41 + 8 * q^51 + 8 * q^55 - 8 * q^57 + 4 * q^63 + 4 * q^65 - 16 * q^67 + 24 * q^69 - 8 * q^71 + 12 * q^73 - 28 * q^79 - 20 * q^81 - 8 * q^85 - 32 * q^89 + 12 * q^91 + 8 * q^95 + 8 * q^97 + 4 * q^99

Introduction

L-functions

Modular forms

Varieties

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Groups

Database

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