LMFDB - Embedded newform 2040.2.cf.d.1441.6

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} + 56 x^{18} + 1252 x^{16} + 14246 x^{14} + 87676 x^{12} + 294068 x^{10} + 556337 x^{8} + \cdots + 16384 $ x^20 + 56x^18 + 1252x^16 + 14246x^14 + 87676x^12 + 294068x^10 + 556337x^8 + 606400x^6 + 376064x^4 + 122880*x^2 + 16384
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1441.6
Root			$1.24830i$ of defining polynomial
Character	$\chi$	$=$	2040.1441
Dual form			2040.2.cf.d.361.6

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.707107 + 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{5} +(-2.60629 + 2.60629i) q^{7} +1.00000i q^{9} +(0.984240 - 0.984240i) q^{11} -3.11298 q^{13} +1.00000i q^{15} +(-3.85878 - 1.45252i) q^{17} -6.44375i q^{19} -3.68584 q^{21} +(-5.42787 + 5.42787i) q^{23} +1.00000i q^{25} +(-0.707107 + 0.707107i) q^{27} +(1.84322 + 1.84322i) q^{29} +(-5.06877 - 5.06877i) q^{31} +1.39193 q^{33} -3.68584 q^{35} +(-0.390138 - 0.390138i) q^{37} +(-2.20121 - 2.20121i) q^{39} +(2.22853 - 2.22853i) q^{41} +8.16481i q^{43} +(-0.707107 + 0.707107i) q^{45} -7.24229 q^{47} -6.58545i q^{49} +(-1.70148 - 3.75566i) q^{51} -8.36985i q^{53} +1.39193 q^{55} +(4.55642 - 4.55642i) q^{57} +4.72663i q^{59} +(7.75320 - 7.75320i) q^{61} +(-2.60629 - 2.60629i) q^{63} +(-2.20121 - 2.20121i) q^{65} +13.0434 q^{67} -7.67616 q^{69} +(-10.3569 - 10.3569i) q^{71} +(-2.07683 - 2.07683i) q^{73} +(-0.707107 + 0.707107i) q^{75} +5.13042i q^{77} +(-10.1798 + 10.1798i) q^{79} -1.00000 q^{81} +3.73346i q^{83} +(-1.70148 - 3.75566i) q^{85} +2.60671i q^{87} +9.27359 q^{89} +(8.11332 - 8.11332i) q^{91} -7.16833i q^{93} +(4.55642 - 4.55642i) q^{95} +(-10.0783 - 10.0783i) q^{97} +(0.984240 + 0.984240i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 4 q^{7} - 4 q^{11} + 8 q^{13} + 4 q^{17} - 4 q^{23} + 12 q^{29} + 8 q^{31} - 8 q^{33} - 12 q^{37} + 4 q^{39} - 12 q^{41} - 48 q^{47} + 16 q^{51} - 8 q^{55} + 8 q^{57} + 24 q^{61} - 4 q^{63} + 4 q^{65}+ \cdots - 4 q^{99}+O(q^{100}) $ 20 * q - 4 * q^7 - 4 * q^11 + 8 * q^13 + 4 * q^17 - 4 * q^23 + 12 * q^29 + 8 * q^31 - 8 * q^33 - 12 * q^37 + 4 * q^39 - 12 * q^41 - 48 * q^47 + 16 * q^51 - 8 * q^55 + 8 * q^57 + 24 * q^61 - 4 * q^63 + 4 * q^65 + 56 * q^67 - 24 * q^69 + 16 * q^71 + 4 * q^73 + 12 * q^79 - 20 * q^81 + 16 * q^85 - 44 * q^91 + 8 * q^95 - 40 * 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
$7$	−2.60629	+	2.60629i	−0.985083	+	0.985083i	−0.999890	−	0.0148070i	$-0.995287\pi$
$7$	−2.60629	+	2.60629i	−0.985083	+	0.985083i	0.0148070	+	0.999890i	$0.495287\pi$
$8$		0			0
$9$			1.00000i			0.333333i
$10$		0			0
$11$	0.984240	−	0.984240i	0.296759	−	0.296759i	−0.542984	−	0.839743i	$-0.682705\pi$
$11$	0.984240	−	0.984240i	0.296759	−	0.296759i	0.839743	+	0.542984i	$0.182705\pi$
$12$		0			0
$13$	−3.11298			−0.863386			−0.431693	−	0.902021i	$-0.642084\pi$
$13$	−3.11298			−0.863386			−0.431693	+	0.902021i	$0.642084\pi$
$14$		0			0
$15$			1.00000i			0.258199i
$16$		0			0
$17$	−3.85878	−	1.45252i	−0.935892	−	0.352288i
$17$	−3.85878	−	1.45252i	−0.935892	−	0.352288i
$18$		0			0
$19$		−	6.44375i		−	1.47830i	−0.673542	−	0.739149i	$-0.735228\pi$
$19$		−	6.44375i		−	1.47830i	0.673542	−	0.739149i	$-0.264772\pi$
$20$		0			0
$21$	−3.68584			−0.804317
$22$		0			0
$23$	−5.42787	+	5.42787i	−1.13179	+	1.13179i	−0.141909	+	0.989880i	$0.545324\pi$
$23$	−5.42787	+	5.42787i	−1.13179	+	1.13179i	−0.989880	+	0.141909i	$0.954676\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$	−0.707107	+	0.707107i	−0.136083	+	0.136083i
$28$		0			0
$29$	1.84322	+	1.84322i	0.342278	+	0.342278i	0.857223	−	0.514945i	$-0.172188\pi$
$29$	1.84322	+	1.84322i	0.342278	+	0.342278i	−0.514945	+	0.857223i	$0.672188\pi$
$30$		0			0
$31$	−5.06877	−	5.06877i	−0.910378	−	0.910378i	0.0859235	−	0.996302i	$-0.472616\pi$
$31$	−5.06877	−	5.06877i	−0.910378	−	0.910378i	−0.996302	+	0.0859235i	$0.972616\pi$
$32$		0			0
$33$	1.39193			0.242303
$34$		0			0
$35$	−3.68584			−0.623021
$36$		0			0
$37$	−0.390138	−	0.390138i	−0.0641384	−	0.0641384i	0.674310	−	0.738448i	$-0.264441\pi$
$37$	−0.390138	−	0.390138i	−0.0641384	−	0.0641384i	−0.738448	+	0.674310i	$0.764441\pi$
$38$		0			0
$39$	−2.20121	−	2.20121i	−0.352476	−	0.352476i
$40$		0			0
$41$	2.22853	−	2.22853i	0.348037	−	0.348037i	−0.511341	−	0.859378i	$-0.670851\pi$
$41$	2.22853	−	2.22853i	0.348037	−	0.348037i	0.859378	+	0.511341i	$0.170851\pi$
$42$		0			0
$43$			8.16481i			1.24512i	0.782572	+	0.622561i	$0.213908\pi$
$43$			8.16481i			1.24512i	−0.782572	+	0.622561i	$0.786092\pi$
$44$		0			0
$45$	−0.707107	+	0.707107i	−0.105409	+	0.105409i
$46$		0			0
$47$	−7.24229			−1.05640			−0.528198	−	0.849121i	$-0.677132\pi$
$47$	−7.24229			−1.05640			−0.528198	+	0.849121i	$0.677132\pi$
$48$		0			0
$49$		−	6.58545i		−	0.940778i
$50$		0			0
$51$	−1.70148	−	3.75566i	−0.238255	−	0.525897i
$52$		0			0
$53$		−	8.36985i		−	1.14969i	−0.818263	−	0.574844i	$-0.805063\pi$
$53$		−	8.36985i		−	1.14969i	0.818263	−	0.574844i	$-0.194937\pi$
$54$		0			0
$55$	1.39193			0.187687
$56$		0			0
$57$	4.55642	−	4.55642i	0.603512	−	0.603512i
$58$		0			0
$59$			4.72663i			0.615354i	0.951491	+	0.307677i	$0.0995517\pi$
$59$			4.72663i			0.615354i	−0.951491	+	0.307677i	$0.900448\pi$
$60$		0			0
$61$	7.75320	−	7.75320i	0.992696	−	0.992696i	−0.00727757	−	0.999974i	$-0.502317\pi$
$61$	7.75320	−	7.75320i	0.992696	−	0.992696i	0.999974	+	0.00727757i	$0.00231654\pi$
$62$		0			0
$63$	−2.60629	−	2.60629i	−0.328361	−	0.328361i
$64$		0			0
$65$	−2.20121	−	2.20121i	−0.273027	−	0.273027i
$66$		0			0
$67$	13.0434			1.59351			0.796753	−	0.604306i	$-0.206549\pi$
$67$	13.0434			1.59351			0.796753	+	0.604306i	$0.206549\pi$
$68$		0			0
$69$	−7.67616			−0.924101
$70$		0			0
$71$	−10.3569	−	10.3569i	−1.22913	−	1.22913i	−0.964292	−	0.264842i	$-0.914680\pi$
$71$	−10.3569	−	10.3569i	−1.22913	−	1.22913i	−0.264842	−	0.964292i	$-0.585320\pi$
$72$		0			0
$73$	−2.07683	−	2.07683i	−0.243075	−	0.243075i	0.575046	−	0.818121i	$-0.304984\pi$
$73$	−2.07683	−	2.07683i	−0.243075	−	0.243075i	−0.818121	+	0.575046i	$0.804984\pi$
$74$		0			0
$75$	−0.707107	+	0.707107i	−0.0816497	+	0.0816497i
$76$		0			0
$77$			5.13042i			0.584666i
$78$		0			0
$79$	−10.1798	+	10.1798i	−1.14531	+	1.14531i	−0.157850	+	0.987463i	$0.550456\pi$
$79$	−10.1798	+	10.1798i	−1.14531	+	1.14531i	−0.987463	+	0.157850i	$0.949544\pi$
$80$		0			0
$81$	−1.00000			−0.111111
$82$		0			0
$83$			3.73346i			0.409801i	0.978783	+	0.204901i	$0.0656871\pi$
$83$			3.73346i			0.409801i	−0.978783	+	0.204901i	$0.934313\pi$
$84$		0			0
$85$	−1.70148	−	3.75566i	−0.184552	−	0.407358i
$86$		0			0
$87$			2.60671i			0.279468i
$88$		0			0
$89$	9.27359			0.982998			0.491499	−	0.870878i	$-0.336449\pi$
$89$	9.27359			0.982998			0.491499	+	0.870878i	$0.336449\pi$
$90$		0			0
$91$	8.11332	−	8.11332i	0.850507	−	0.850507i
$92$		0			0
$93$		−	7.16833i		−	0.743321i
$94$		0			0
$95$	4.55642	−	4.55642i	0.467479	−	0.467479i
$96$		0			0
$97$	−10.0783	−	10.0783i	−1.02329	−	1.02329i	−0.999722	−	0.0235726i	$-0.992496\pi$
$97$	−10.0783	−	10.0783i	−1.02329	−	1.02329i	−0.0235726	−	0.999722i	$-0.507504\pi$
$98$		0			0
$99$	0.984240	+	0.984240i	0.0989198	+	0.0989198i
$100$		0			0
$101$	4.53448			0.451197			0.225599	−	0.974220i	$-0.427566\pi$
$101$	4.53448			0.451197			0.225599	+	0.974220i	$0.427566\pi$
$102$		0			0
$103$	−7.50109			−0.739104			−0.369552	−	0.929210i	$-0.620489\pi$
$103$	−7.50109			−0.739104			−0.369552	+	0.929210i	$0.620489\pi$
$104$		0			0
$105$	−2.60629	−	2.60629i	−0.254347	−	0.254347i
$106$		0			0
$107$	−10.7566	−	10.7566i	−1.03988	−	1.03988i	−0.999171	−	0.0407064i	$-0.987039\pi$
$107$	−10.7566	−	10.7566i	−1.03988	−	1.03988i	−0.0407064	−	0.999171i	$-0.512961\pi$
$108$		0			0
$109$	−10.2067	+	10.2067i	−0.977627	+	0.977627i	−0.999755	−	0.0221277i	$-0.992956\pi$
$109$	−10.2067	+	10.2067i	−0.977627	+	0.977627i	0.0221277	+	0.999755i	$0.492956\pi$
$110$		0			0
$111$		−	0.551739i		−	0.0523687i
$112$		0			0
$113$	−6.87012	+	6.87012i	−0.646286	+	0.646286i	−0.952093	−	0.305807i	$-0.901074\pi$
$113$	−6.87012	+	6.87012i	−0.646286	+	0.646286i	0.305807	+	0.952093i	$0.401074\pi$
$114$		0			0
$115$	−7.67616			−0.715806
$116$		0			0
$117$		−	3.11298i		−	0.287795i
$118$		0			0
$119$	13.8428	−	6.27141i	1.26896	−	0.574899i
$120$		0			0
$121$			9.06254i			0.823868i
$122$		0			0
$123$	3.15161			0.284171
$124$		0			0
$125$	−0.707107	+	0.707107i	−0.0632456	+	0.0632456i
$126$		0			0
$127$			21.3097i			1.89093i	0.325718	+	0.945467i	$0.394394\pi$
$127$			21.3097i			1.89093i	−0.325718	+	0.945467i	$0.605606\pi$
$128$		0			0
$129$	−5.77339	+	5.77339i	−0.508319	+	0.508319i
$130$		0			0
$131$	−11.9425	−	11.9425i	−1.04342	−	1.04342i	−0.999013	−	0.0444111i	$-0.985859\pi$
$131$	−11.9425	−	11.9425i	−1.04342	−	1.04342i	−0.0444111	−	0.999013i	$-0.514141\pi$
$132$		0			0
$133$	16.7943	+	16.7943i	1.45625	+	1.45625i
$134$		0			0
$135$	−1.00000			−0.0860663
$136$		0			0
$137$	19.2409			1.64386			0.821932	−	0.569586i	$-0.192897\pi$
$137$	19.2409			1.64386			0.821932	+	0.569586i	$0.192897\pi$
$138$		0			0
$139$	8.23532	+	8.23532i	0.698511	+	0.698511i	0.964089	−	0.265578i	$-0.0855629\pi$
$139$	8.23532	+	8.23532i	0.698511	+	0.698511i	−0.265578	+	0.964089i	$0.585563\pi$
$140$		0			0
$141$	−5.12107	−	5.12107i	−0.431272	−	0.431272i
$142$		0			0
$143$	−3.06392	+	3.06392i	−0.256218	+	0.256218i
$144$		0			0
$145$			2.60671i			0.216475i
$146$		0			0
$147$	4.65662	−	4.65662i	0.384071	−	0.384071i
$148$		0			0
$149$	−1.77251			−0.145209			−0.0726047	−	0.997361i	$-0.523131\pi$
$149$	−1.77251			−0.145209			−0.0726047	+	0.997361i	$0.523131\pi$
$150$		0			0
$151$			18.0554i			1.46933i	0.678432	+	0.734664i	$0.262660\pi$
$151$			18.0554i			1.46933i	−0.678432	+	0.734664i	$0.737340\pi$
$152$		0			0
$153$	1.45252	−	3.85878i	0.117429	−	0.311964i
$154$		0			0
$155$		−	7.16833i		−	0.575774i
$156$		0			0
$157$	−22.6892			−1.81079			−0.905396	−	0.424569i	$-0.860426\pi$
$157$	−22.6892			−1.81079			−0.905396	+	0.424569i	$0.860426\pi$
$158$		0			0
$159$	5.91838	−	5.91838i	0.469358	−	0.469358i
$160$		0			0
$161$		−	28.2931i		−	2.22981i
$162$		0			0
$163$	2.99444	−	2.99444i	0.234543	−	0.234543i	−0.580043	−	0.814586i	$-0.696964\pi$
$163$	2.99444	−	2.99444i	0.234543	−	0.234543i	0.814586	+	0.580043i	$0.196964\pi$
$164$		0			0
$165$	0.984240	+	0.984240i	0.0766230	+	0.0766230i
$166$		0			0
$167$	−3.53740	−	3.53740i	−0.273732	−	0.273732i	0.556868	−	0.830601i	$-0.312003\pi$
$167$	−3.53740	−	3.53740i	−0.273732	−	0.273732i	−0.830601	+	0.556868i	$0.812003\pi$
$168$		0			0
$169$	−3.30934			−0.254565
$170$		0			0
$171$	6.44375			0.492766
$172$		0			0
$173$	1.02328	+	1.02328i	0.0777988	+	0.0777988i	0.744935	−	0.667137i	$-0.232480\pi$
$173$	1.02328	+	1.02328i	0.0777988	+	0.0777988i	−0.667137	+	0.744935i	$0.732480\pi$
$174$		0			0
$175$	−2.60629	−	2.60629i	−0.197017	−	0.197017i
$176$		0			0
$177$	−3.34223	+	3.34223i	−0.251217	+	0.251217i
$178$		0			0
$179$			5.35529i			0.400273i	0.979768	+	0.200137i	$0.0641386\pi$
$179$			5.35529i			0.400273i	−0.979768	+	0.200137i	$0.935861\pi$
$180$		0			0
$181$	2.41206	−	2.41206i	0.179287	−	0.179287i	−0.611758	−	0.791045i	$-0.709537\pi$
$181$	2.41206	−	2.41206i	0.179287	−	0.179287i	0.791045	+	0.611758i	$0.209537\pi$
$182$		0			0
$183$	10.9647			0.810533
$184$		0			0
$185$		−	0.551739i		−	0.0405647i
$186$		0			0
$187$	−5.22759	+	2.36834i	−0.382279	+	0.173190i
$188$		0			0
$189$		−	3.68584i		−	0.268106i
$190$		0			0
$191$	11.3707			0.822756			0.411378	−	0.911465i	$-0.365048\pi$
$191$	11.3707			0.822756			0.411378	+	0.911465i	$0.365048\pi$
$192$		0			0
$193$	−0.162023	+	0.162023i	−0.0116627	+	0.0116627i	−0.712914	−	0.701251i	$-0.752625\pi$
$193$	−0.162023	+	0.162023i	−0.0116627	+	0.0116627i	0.701251	+	0.712914i	$0.252625\pi$
$194$		0			0
$195$		−	3.11298i		−	0.222925i
$196$		0			0
$197$	5.21844	−	5.21844i	0.371798	−	0.371798i	−0.496333	−	0.868132i	$-0.665321\pi$
$197$	5.21844	−	5.21844i	0.371798	−	0.371798i	0.868132	+	0.496333i	$0.165321\pi$
$198$		0			0
$199$	15.2500	+	15.2500i	1.08105	+	1.08105i	0.996412	+	0.0846337i	$0.0269720\pi$
$199$	15.2500	+	15.2500i	1.08105	+	1.08105i	0.0846337	+	0.996412i	$0.473028\pi$
$200$		0			0
$201$	9.22308	+	9.22308i	0.650546	+	0.650546i
$202$		0			0
$203$	−9.60792			−0.674344
$204$		0			0
$205$	3.15161			0.220118
$206$		0			0
$207$	−5.42787	−	5.42787i	−0.377263	−	0.377263i
$208$		0			0
$209$	−6.34220	−	6.34220i	−0.438699	−	0.438699i
$210$		0			0
$211$	−19.3832	+	19.3832i	−1.33439	+	1.33439i	−0.432998	+	0.901395i	$0.642544\pi$
$211$	−19.3832	+	19.3832i	−1.33439	+	1.33439i	−0.901395	+	0.432998i	$0.857456\pi$
$212$		0			0
$213$		−	14.6468i		−	1.00358i
$214$		0			0
$215$	−5.77339	+	5.77339i	−0.393742	+	0.393742i
$216$		0			0
$217$	26.4213			1.79360
$218$		0			0
$219$		−	2.93709i		−	0.198470i
$220$		0			0
$221$	12.0123	+	4.52166i	0.808036	+	0.304160i
$222$		0			0
$223$		−	16.6332i		−	1.11384i	−0.830567	−	0.556919i	$-0.811983\pi$
$223$		−	16.6332i		−	1.11384i	0.830567	−	0.556919i	$-0.188017\pi$
$224$		0			0
$225$	−1.00000			−0.0666667
$226$		0			0
$227$	−17.7041	+	17.7041i	−1.17506	+	1.17506i	−0.194075	+	0.980987i	$0.562170\pi$
$227$	−17.7041	+	17.7041i	−1.17506	+	1.17506i	−0.980987	+	0.194075i	$0.937830\pi$
$228$		0			0
$229$		−	1.14016i		−	0.0753436i	−0.999290	−	0.0376718i	$-0.988006\pi$
$229$		−	1.14016i		−	0.0753436i	0.999290	−	0.0376718i	$-0.0119941\pi$
$230$		0			0
$231$	−3.62775	+	3.62775i	−0.238689	+	0.238689i
$232$		0			0
$233$	−18.4627	−	18.4627i	−1.20953	−	1.20953i	−0.971179	−	0.238353i	$-0.923393\pi$
$233$	−18.4627	−	18.4627i	−1.20953	−	1.20953i	−0.238353	−	0.971179i	$-0.576607\pi$
$234$		0			0
$235$	−5.12107	−	5.12107i	−0.334062	−	0.334062i
$236$		0			0
$237$	−14.3964			−0.935144
$238$		0			0
$239$	4.55032			0.294336			0.147168	−	0.989112i	$-0.452984\pi$
$239$	4.55032			0.294336			0.147168	+	0.989112i	$0.452984\pi$
$240$		0			0
$241$	3.08685	+	3.08685i	0.198842	+	0.198842i	0.799503	−	0.600662i	$-0.205096\pi$
$241$	3.08685	+	3.08685i	0.198842	+	0.198842i	−0.600662	+	0.799503i	$0.705096\pi$
$242$		0			0
$243$	−0.707107	−	0.707107i	−0.0453609	−	0.0453609i
$244$		0			0
$245$	4.65662	−	4.65662i	0.297500	−	0.297500i
$246$		0			0
$247$			20.0593i			1.27634i
$248$		0			0
$249$	−2.63996	+	2.63996i	−0.167301	+	0.167301i
$250$		0			0
$251$	−29.1745			−1.84148			−0.920738	−	0.390181i	$-0.872412\pi$
$251$	−29.1745			−1.84148			−0.920738	+	0.390181i	$0.872412\pi$
$252$		0			0
$253$			10.6846i			0.671738i
$254$		0			0
$255$	1.45252	−	3.85878i	0.0909603	−	0.241646i
$256$		0			0
$257$		−	3.17986i		−	0.198354i	−0.995070	−	0.0991772i	$-0.968379\pi$
$257$		−	3.17986i		−	0.198354i	0.995070	−	0.0991772i	$-0.0316211\pi$
$258$		0			0
$259$	2.03362			0.126363
$260$		0			0
$261$	−1.84322	+	1.84322i	−0.114093	+	0.114093i
$262$		0			0
$263$			23.5499i			1.45215i	0.687617	+	0.726074i	$0.258657\pi$
$263$			23.5499i			1.45215i	−0.687617	+	0.726074i	$0.741343\pi$
$264$		0			0
$265$	5.91838	−	5.91838i	0.363563	−	0.363563i
$266$		0			0
$267$	6.55741	+	6.55741i	0.401307	+	0.401307i
$268$		0			0
$269$	1.53837	+	1.53837i	0.0937961	+	0.0937961i	0.752448	−	0.658652i	$-0.228873\pi$
$269$	1.53837	+	1.53837i	0.0937961	+	0.0937961i	−0.658652	+	0.752448i	$0.728873\pi$
$270$		0			0
$271$	−14.4182			−0.875844			−0.437922	−	0.899013i	$-0.644285\pi$
$271$	−14.4182			−0.875844			−0.437922	+	0.899013i	$0.644285\pi$
$272$		0			0
$273$	11.4740			0.694436
$274$		0			0
$275$	0.984240	+	0.984240i	0.0593519	+	0.0593519i
$276$		0			0
$277$	−0.508820	−	0.508820i	−0.0305720	−	0.0305720i	0.691656	−	0.722228i	$-0.256882\pi$
$277$	−0.508820	−	0.508820i	−0.0305720	−	0.0305720i	−0.722228	+	0.691656i	$0.756882\pi$
$278$		0			0
$279$	5.06877	−	5.06877i	0.303459	−	0.303459i
$280$		0			0
$281$		−	25.9728i		−	1.54940i	−0.632326	−	0.774702i	$-0.717900\pi$
$281$		−	25.9728i		−	1.54940i	0.632326	−	0.774702i	$-0.282100\pi$
$282$		0			0
$283$	16.0895	−	16.0895i	0.956424	−	0.956424i	−0.0426654	−	0.999089i	$-0.513585\pi$
$283$	16.0895	−	16.0895i	0.956424	−	0.956424i	0.999089	+	0.0426654i	$0.0135850\pi$
$284$		0			0
$285$	6.44375			0.381695
$286$		0			0
$287$			11.6164i			0.685692i
$288$		0			0
$289$	12.7804	+	11.2099i	0.751787	+	0.659406i
$290$		0			0
$291$		−	14.2528i		−	0.835517i
$292$		0			0
$293$	15.0783			0.880882			0.440441	−	0.897782i	$-0.354822\pi$
$293$	15.0783			0.880882			0.440441	+	0.897782i	$0.354822\pi$
$294$		0			0
$295$	−3.34223	+	3.34223i	−0.194592	+	0.194592i
$296$		0			0
$297$			1.39193i			0.0807677i
$298$		0			0
$299$	16.8969	−	16.8969i	0.977170	−	0.977170i
$300$		0			0
$301$	−21.2798	−	21.2798i	−1.22655	−	1.22655i
$302$		0			0
$303$	3.20636	+	3.20636i	0.184201	+	0.184201i
$304$		0			0
$305$	10.9647			0.627836
$306$		0			0
$307$	14.7150			0.839829			0.419915	−	0.907564i	$-0.362060\pi$
$307$	14.7150			0.839829			0.419915	+	0.907564i	$0.362060\pi$
$308$		0			0
$309$	−5.30407	−	5.30407i	−0.301738	−	0.301738i
$310$		0			0
$311$	−7.47281	−	7.47281i	−0.423744	−	0.423744i	0.462746	−	0.886491i	$-0.346864\pi$
$311$	−7.47281	−	7.47281i	−0.423744	−	0.423744i	−0.886491	+	0.462746i	$0.846864\pi$
$312$		0			0
$313$	0.490796	−	0.490796i	0.0277414	−	0.0277414i	−0.693100	−	0.720841i	$-0.743756\pi$
$313$	0.490796	−	0.490796i	0.0277414	−	0.0277414i	0.720841	+	0.693100i	$0.243756\pi$
$314$		0			0
$315$		−	3.68584i		−	0.207674i
$316$		0			0
$317$	−0.871532	+	0.871532i	−0.0489501	+	0.0489501i	−0.731158	−	0.682208i	$-0.761020\pi$
$317$	−0.871532	+	0.871532i	−0.0489501	+	0.0489501i	0.682208	+	0.731158i	$0.261020\pi$
$318$		0			0
$319$	3.62834			0.203148
$320$		0			0
$321$		−	15.2121i		−	0.849056i
$322$		0			0
$323$	−9.35967	+	24.8650i	−0.520786	+	1.38353i
$324$		0			0
$325$		−	3.11298i		−	0.172677i
$326$		0			0
$327$	−14.4345			−0.798229
$328$		0			0
$329$	18.8755	−	18.8755i	1.04064	−	1.04064i
$330$		0			0
$331$		−	0.385234i		−	0.0211744i	−0.999944	−	0.0105872i	$-0.996630\pi$
$331$		−	0.385234i		−	0.0211744i	0.999944	−	0.0105872i	$-0.00337007\pi$
$332$		0			0
$333$	0.390138	−	0.390138i	0.0213795	−	0.0213795i
$334$		0			0
$335$	9.22308	+	9.22308i	0.503911	+	0.503911i
$336$		0			0
$337$	21.7632	+	21.7632i	1.18552	+	1.18552i	0.978294	+	0.207224i	$0.0664428\pi$
$337$	21.7632	+	21.7632i	1.18552	+	1.18552i	0.207224	+	0.978294i	$0.433557\pi$
$338$		0			0
$339$	−9.71581			−0.527691
$340$		0			0
$341$	−9.97777			−0.540327
$342$		0			0
$343$	−1.08044	−	1.08044i	−0.0583381	−	0.0583381i
$344$		0			0
$345$	−5.42787	−	5.42787i	−0.292227	−	0.292227i
$346$		0			0
$347$	−1.11918	+	1.11918i	−0.0600809	+	0.0600809i	−0.736509	−	0.676428i	$-0.763527\pi$
$347$	−1.11918	+	1.11918i	−0.0600809	+	0.0600809i	0.676428	+	0.736509i	$0.263527\pi$
$348$		0			0
$349$			25.0150i			1.33902i	0.742801	+	0.669512i	$0.233497\pi$
$349$			25.0150i			1.33902i	−0.742801	+	0.669512i	$0.766503\pi$
$350$		0			0
$351$	2.20121	−	2.20121i	0.117492	−	0.117492i
$352$		0			0
$353$	−9.53899			−0.507709			−0.253855	−	0.967242i	$-0.581698\pi$
$353$	−9.53899			−0.507709			−0.253855	+	0.967242i	$0.581698\pi$
$354$		0			0
$355$		−	14.6468i		−	0.777373i
$356$		0			0
$357$	14.2229	+	5.35376i	0.752754	+	0.283351i
$358$		0			0
$359$			15.8414i			0.836079i	0.908429	+	0.418040i	$0.137283\pi$
$359$			15.8414i			0.836079i	−0.908429	+	0.418040i	$0.862717\pi$
$360$		0			0
$361$	−22.5219			−1.18536
$362$		0			0
$363$	−6.40819	+	6.40819i	−0.336343	+	0.336343i
$364$		0			0
$365$		−	2.93709i		−	0.153734i
$366$		0			0
$367$	−13.6825	+	13.6825i	−0.714221	+	0.714221i	−0.967415	−	0.253195i	$-0.918519\pi$
$367$	−13.6825	+	13.6825i	−0.714221	+	0.714221i	0.253195	+	0.967415i	$0.418519\pi$
$368$		0			0
$369$	2.22853	+	2.22853i	0.116012	+	0.116012i
$370$		0			0
$371$	21.8142	+	21.8142i	1.13254	+	1.13254i
$372$		0			0
$373$	20.7592			1.07487			0.537436	−	0.843304i	$-0.319393\pi$
$373$	20.7592			1.07487			0.537436	+	0.843304i	$0.319393\pi$
$374$		0			0
$375$	−1.00000			−0.0516398
$376$		0			0
$377$	−5.73791	−	5.73791i	−0.295518	−	0.295518i
$378$		0			0
$379$	0.437405	+	0.437405i	0.0224680	+	0.0224680i	0.718252	−	0.695784i	$-0.244943\pi$
$379$	0.437405	+	0.437405i	0.0224680	+	0.0224680i	−0.695784	+	0.718252i	$0.744943\pi$
$380$		0			0
$381$	−15.0683	+	15.0683i	−0.771970	+	0.771970i
$382$		0			0
$383$		−	12.4097i		−	0.634107i	−0.948408	−	0.317054i	$-0.897306\pi$
$383$		−	12.4097i		−	0.634107i	0.948408	−	0.317054i	$-0.102694\pi$
$384$		0			0
$385$	−3.62775	+	3.62775i	−0.184888	+	0.184888i
$386$		0			0
$387$	−8.16481			−0.415040
$388$		0			0
$389$			14.5525i			0.737841i	0.929461	+	0.368921i	$0.120273\pi$
$389$			14.5525i			0.737841i	−0.929461	+	0.368921i	$0.879727\pi$
$390$		0			0
$391$	28.8290	−	13.0609i	1.45795	−	0.660517i
$392$		0			0
$393$		−	16.8893i		−	0.851952i
$394$		0			0
$395$	−14.3964			−0.724360
$396$		0			0
$397$	7.68172	−	7.68172i	0.385534	−	0.385534i	−0.487557	−	0.873091i	$-0.662112\pi$
$397$	7.68172	−	7.68172i	0.385534	−	0.385534i	0.873091	+	0.487557i	$0.162112\pi$
$398$		0			0
$399$			23.7507i			1.18902i
$400$		0			0
$401$	3.99065	−	3.99065i	0.199284	−	0.199284i	−0.600409	−	0.799693i	$-0.704996\pi$
$401$	3.99065	−	3.99065i	0.199284	−	0.199284i	0.799693	+	0.600409i	$0.204996\pi$
$402$		0			0
$403$	15.7790	+	15.7790i	0.786008	+	0.786008i
$404$		0			0
$405$	−0.707107	−	0.707107i	−0.0351364	−	0.0351364i
$406$		0			0
$407$	−0.767979			−0.0380673
$408$		0			0
$409$	16.4058			0.811215			0.405607	−	0.914047i	$-0.367060\pi$
$409$	16.4058			0.811215			0.405607	+	0.914047i	$0.367060\pi$
$410$		0			0
$411$	13.6054	+	13.6054i	0.671104	+	0.671104i
$412$		0			0
$413$	−12.3189	−	12.3189i	−0.606175	−	0.606175i
$414$		0			0
$415$	−2.63996	+	2.63996i	−0.129590	+	0.129590i
$416$		0			0
$417$			11.6465i			0.570332i
$418$		0			0
$419$	−2.79134	+	2.79134i	−0.136366	+	0.136366i	−0.771995	−	0.635629i	$-0.780741\pi$
$419$	−2.79134	+	2.79134i	−0.136366	+	0.136366i	0.635629	+	0.771995i	$0.280741\pi$
$420$		0			0
$421$	15.9717			0.778412			0.389206	−	0.921151i	$-0.372749\pi$
$421$	15.9717			0.778412			0.389206	+	0.921151i	$0.372749\pi$
$422$		0			0
$423$		−	7.24229i		−	0.352132i
$424$		0			0
$425$	1.45252	−	3.85878i	0.0704575	−	0.187178i
$426$		0			0
$427$			40.4141i			1.95578i
$428$		0			0
$429$	−4.33304			−0.209201
$430$		0			0
$431$	6.75116	−	6.75116i	0.325192	−	0.325192i	−0.525563	−	0.850755i	$-0.676145\pi$
$431$	6.75116	−	6.75116i	0.325192	−	0.325192i	0.850755	+	0.525563i	$0.176145\pi$
$432$		0			0
$433$			24.0298i			1.15480i	0.816461	+	0.577400i	$0.195933\pi$
$433$			24.0298i			1.15480i	−0.816461	+	0.577400i	$0.804067\pi$
$434$		0			0
$435$	−1.84322	+	1.84322i	−0.0883757	+	0.0883757i
$436$		0			0
$437$	34.9758	+	34.9758i	1.67312	+	1.67312i
$438$		0			0
$439$	16.4415	+	16.4415i	0.784710	+	0.784710i	0.980622	−	0.195912i	$-0.0627667\pi$
$439$	16.4415	+	16.4415i	0.784710	+	0.784710i	−0.195912	+	0.980622i	$0.562767\pi$
$440$		0			0
$441$	6.58545			0.313593
$442$		0			0
$443$	10.8518			0.515584			0.257792	−	0.966200i	$-0.417005\pi$
$443$	10.8518			0.515584			0.257792	+	0.966200i	$0.417005\pi$
$444$		0			0
$445$	6.55741	+	6.55741i	0.310851	+	0.310851i
$446$		0			0
$447$	−1.25335	−	1.25335i	−0.0592815	−	0.0592815i
$448$		0			0
$449$	−19.4339	+	19.4339i	−0.917141	+	0.917141i	−0.996821	−	0.0796795i	$-0.974610\pi$
$449$	−19.4339	+	19.4339i	−0.917141	+	0.917141i	0.0796795	+	0.996821i	$0.474610\pi$
$450$		0			0
$451$		−	4.38681i		−	0.206567i
$452$		0			0
$453$	−12.7671	+	12.7671i	−0.599850	+	0.599850i
$454$		0			0
$455$	11.4740			0.537908
$456$		0			0
$457$			11.8665i			0.555093i	0.960712	+	0.277546i	$0.0895212\pi$
$457$			11.8665i			0.555093i	−0.960712	+	0.277546i	$0.910479\pi$
$458$		0			0
$459$	3.75566	−	1.70148i	0.175299	−	0.0794185i
$460$		0			0
$461$			4.71164i			0.219443i	0.993962	+	0.109722i	$0.0349959\pi$
$461$			4.71164i			0.219443i	−0.993962	+	0.109722i	$0.965004\pi$
$462$		0			0
$463$	24.2724			1.12803			0.564017	−	0.825763i	$-0.309255\pi$
$463$	24.2724			1.12803			0.564017	+	0.825763i	$0.309255\pi$
$464$		0			0
$465$	5.06877	−	5.06877i	0.235059	−	0.235059i
$466$		0			0
$467$			8.98717i			0.415876i	0.978142	+	0.207938i	$0.0666753\pi$
$467$			8.98717i			0.415876i	−0.978142	+	0.207938i	$0.933325\pi$
$468$		0			0
$469$	−33.9948	+	33.9948i	−1.56974	+	1.56974i
$470$		0			0
$471$	−16.0437	−	16.0437i	−0.739253	−	0.739253i
$472$		0			0
$473$	8.03613	+	8.03613i	0.369502	+	0.369502i
$474$		0			0
$475$	6.44375			0.295660
$476$		0			0
$477$	8.36985			0.383229
$478$		0			0
$479$	−27.4728	−	27.4728i	−1.25526	−	1.25526i	−0.953328	−	0.301935i	$-0.902367\pi$
$479$	−27.4728	−	27.4728i	−1.25526	−	1.25526i	−0.301935	−	0.953328i	$-0.597633\pi$
$480$		0			0
$481$	1.21449	+	1.21449i	0.0553761	+	0.0553761i
$482$		0			0
$483$	20.0063	−	20.0063i	0.910317	−	0.910317i
$484$		0			0
$485$		−	14.2528i		−	0.647188i
$486$		0			0
$487$	4.03323	−	4.03323i	0.182763	−	0.182763i	−0.609795	−	0.792559i	$-0.708748\pi$
$487$	4.03323	−	4.03323i	0.182763	−	0.182763i	0.792559	+	0.609795i	$0.208748\pi$
$488$		0			0
$489$	4.23478			0.191503
$490$		0			0
$491$		−	5.56937i		−	0.251342i	−0.992072	−	0.125671i	$-0.959892\pi$
$491$		−	5.56937i		−	0.251342i	0.992072	−	0.125671i	$-0.0401084\pi$
$492$		0			0
$493$	−4.43527	−	9.78990i	−0.199755	−	0.440915i
$494$		0			0
$495$			1.39193i			0.0625624i
$496$		0			0
$497$	53.9859			2.42160
$498$		0			0
$499$	−13.8153	+	13.8153i	−0.618456	+	0.618456i	−0.945135	−	0.326679i	$-0.894070\pi$
$499$	−13.8153	+	13.8153i	−0.618456	+	0.618456i	0.326679	+	0.945135i	$0.394070\pi$
$500$		0			0
$501$		−	5.00264i		−	0.223501i
$502$		0			0
$503$	9.87055	−	9.87055i	0.440106	−	0.440106i	−0.451941	−	0.892048i	$-0.649268\pi$
$503$	9.87055	−	9.87055i	0.440106	−	0.440106i	0.892048	+	0.451941i	$0.149268\pi$
$504$		0			0
$505$	3.20636	+	3.20636i	0.142681	+	0.142681i
$506$		0			0
$507$	−2.34006	−	2.34006i	−0.103926	−	0.103926i
$508$		0			0
$509$	−7.51858			−0.333255			−0.166628	−	0.986020i	$-0.553288\pi$
$509$	−7.51858			−0.333255			−0.166628	+	0.986020i	$0.553288\pi$
$510$		0			0
$511$	10.8256			0.478899
$512$		0			0
$513$	4.55642	+	4.55642i	0.201171	+	0.201171i
$514$		0			0
$515$	−5.30407	−	5.30407i	−0.233725	−	0.233725i
$516$		0			0
$517$	−7.12815	+	7.12815i	−0.313496	+	0.313496i
$518$		0			0
$519$			1.44714i			0.0635225i
$520$		0			0
$521$	15.7494	−	15.7494i	0.689995	−	0.689995i	−0.272236	−	0.962231i	$-0.587763\pi$
$521$	15.7494	−	15.7494i	0.689995	−	0.689995i	0.962231	+	0.272236i	$0.0877630\pi$
$522$		0			0
$523$	10.5246			0.460209			0.230104	−	0.973166i	$-0.426093\pi$
$523$	10.5246			0.460209			0.230104	+	0.973166i	$0.426093\pi$
$524$		0			0
$525$		−	3.68584i		−	0.160863i
$526$		0			0
$527$	12.1968	+	26.9218i	0.531301	+	1.17273i
$528$		0			0
$529$		−	35.9235i		−	1.56189i
$530$		0			0
$531$	−4.72663			−0.205118
$532$		0			0
$533$	−6.93736	+	6.93736i	−0.300491	+	0.300491i
$534$		0			0
$535$		−	15.2121i		−	0.657676i
$536$		0			0
$537$	−3.78676	+	3.78676i	−0.163411	+	0.163411i
$538$		0			0
$539$	−6.48166	−	6.48166i	−0.279185	−	0.279185i
$540$		0			0
$541$	2.75141	+	2.75141i	0.118292	+	0.118292i	0.763775	−	0.645483i	$-0.223344\pi$
$541$	2.75141	+	2.75141i	0.118292	+	0.118292i	−0.645483	+	0.763775i	$0.723344\pi$
$542$		0			0
$543$	3.41116			0.146387
$544$		0			0
$545$	−14.4345			−0.618306
$546$		0			0
$547$	−8.07559	−	8.07559i	−0.345287	−	0.345287i	0.513063	−	0.858351i	$-0.328511\pi$
$547$	−8.07559	−	8.07559i	−0.345287	−	0.345287i	−0.858351	+	0.513063i	$0.828511\pi$
$548$		0			0
$549$	7.75320	+	7.75320i	0.330899	+	0.330899i
$550$		0			0
$551$	11.8773	−	11.8773i	0.505988	−	0.505988i
$552$		0			0
$553$		−	53.0628i		−	2.25646i
$554$		0			0
$555$	0.390138	−	0.390138i	0.0165605	−	0.0165605i
$556$		0			0
$557$	4.10126			0.173776			0.0868880	−	0.996218i	$-0.472308\pi$
$557$	4.10126			0.173776			0.0868880	+	0.996218i	$0.472308\pi$
$558$		0			0
$559$		−	25.4169i		−	1.07502i
$560$		0			0
$561$	−5.37113	−	2.02180i	−0.226769	−	0.0853604i
$562$		0			0
$563$			22.8644i			0.963621i	0.876275	+	0.481810i	$0.160021\pi$
$563$			22.8644i			0.963621i	−0.876275	+	0.481810i	$0.839979\pi$
$564$		0			0
$565$	−9.71581			−0.408747
$566$		0			0
$567$	2.60629	−	2.60629i	0.109454	−	0.109454i
$568$		0			0
$569$		−	9.54376i		−	0.400095i	−0.979786	−	0.200048i	$-0.935890\pi$
$569$		−	9.54376i		−	0.400095i	0.979786	−	0.200048i	$-0.0641097\pi$
$570$		0			0
$571$	28.1691	−	28.1691i	1.17884	−	1.17884i	0.198802	−	0.980040i	$-0.436295\pi$
$571$	28.1691	−	28.1691i	1.17884	−	1.17884i	0.980040	−	0.198802i	$-0.0637050\pi$
$572$		0			0
$573$	8.04031	+	8.04031i	0.335889	+	0.335889i
$574$		0			0
$575$	−5.42787	−	5.42787i	−0.226358	−	0.226358i
$576$		0			0
$577$	4.15731			0.173071			0.0865356	−	0.996249i	$-0.472420\pi$
$577$	4.15731			0.173071			0.0865356	+	0.996249i	$0.472420\pi$
$578$		0			0
$579$	−0.229135			−0.00952254
$580$		0			0
$581$	−9.73048	−	9.73048i	−0.403688	−	0.403688i
$582$		0			0
$583$	−8.23794	−	8.23794i	−0.341181	−	0.341181i
$584$		0			0
$585$	2.20121	−	2.20121i	0.0910089	−	0.0910089i
$586$		0			0
$587$		−	43.9526i		−	1.81412i	−0.421005	−	0.907058i	$-0.638323\pi$
$587$		−	43.9526i		−	1.81412i	0.421005	−	0.907058i	$-0.361677\pi$
$588$		0			0
$589$	−32.6619	+	32.6619i	−1.34581	+	1.34581i
$590$		0			0
$591$	7.37999			0.303572
$592$		0			0
$593$			35.9267i			1.47533i	0.675165	+	0.737666i	$0.264072\pi$
$593$			35.9267i			1.47533i	−0.675165	+	0.737666i	$0.735928\pi$
$594$		0			0
$595$	14.2229	+	5.35376i	0.583081	+	0.219483i
$596$		0			0
$597$			21.5668i			0.882670i
$598$		0			0
$599$	35.1075			1.43445			0.717226	−	0.696841i	$-0.245412\pi$
$599$	35.1075			1.43445			0.717226	+	0.696841i	$0.245412\pi$
$600$		0			0
$601$	−3.65146	+	3.65146i	−0.148946	+	0.148946i	−0.777647	−	0.628701i	$-0.783587\pi$
$601$	−3.65146	+	3.65146i	−0.148946	+	0.148946i	0.628701	+	0.777647i	$0.283587\pi$
$602$		0			0
$603$			13.0434i			0.531168i
$604$		0			0
$605$	−6.40819	+	6.40819i	−0.260530	+	0.260530i
$606$		0			0
$607$	8.89399	+	8.89399i	0.360996	+	0.360996i	0.864179	−	0.503184i	$-0.167838\pi$
$607$	8.89399	+	8.89399i	0.360996	+	0.360996i	−0.503184	+	0.864179i	$0.667838\pi$
$608$		0			0
$609$	−6.79383	−	6.79383i	−0.275300	−	0.275300i
$610$		0			0
$611$	22.5451			0.912078
$612$		0			0
$613$	−33.1417			−1.33858			−0.669291	−	0.743000i	$-0.733402\pi$
$613$	−33.1417			−1.33858			−0.669291	+	0.743000i	$0.733402\pi$
$614$		0			0
$615$	2.22853	+	2.22853i	0.0898629	+	0.0898629i
$616$		0			0
$617$	−1.83786	−	1.83786i	−0.0739894	−	0.0739894i	0.669144	−	0.743133i	$-0.266661\pi$
$617$	−1.83786	−	1.83786i	−0.0739894	−	0.0739894i	−0.743133	+	0.669144i	$0.766661\pi$
$618$		0			0
$619$	4.14663	−	4.14663i	0.166667	−	0.166667i	−0.618846	−	0.785513i	$-0.712399\pi$
$619$	4.14663	−	4.14663i	0.166667	−	0.166667i	0.785513	+	0.618846i	$0.212399\pi$
$620$		0			0
$621$		−	7.67616i		−	0.308034i
$622$		0			0
$623$	−24.1696	+	24.1696i	−0.968335	+	0.968335i
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		−	8.96922i		−	0.358196i
$628$		0			0
$629$	0.938775	+	2.07214i	0.0374314	+	0.0826217i
$630$		0			0
$631$		−	12.6801i		−	0.504786i	−0.967625	−	0.252393i	$-0.918782\pi$
$631$		−	12.6801i		−	0.504786i	0.967625	−	0.252393i	$-0.0812176\pi$
$632$		0			0
$633$	−27.4120			−1.08953
$634$		0			0
$635$	−15.0683	+	15.0683i	−0.597966	+	0.597966i
$636$		0			0
$637$			20.5004i			0.812255i
$638$		0			0
$639$	10.3569	−	10.3569i	0.409711	−	0.409711i
$640$		0			0
$641$	−22.4667	−	22.4667i	−0.887380	−	0.887380i	0.106891	−	0.994271i	$-0.465910\pi$
$641$	−22.4667	−	22.4667i	−0.887380	−	0.887380i	−0.994271	+	0.106891i	$0.965910\pi$
$642$		0			0
$643$	9.09155	+	9.09155i	0.358536	+	0.358536i	0.863273	−	0.504737i	$-0.168411\pi$
$643$	9.09155	+	9.09155i	0.358536	+	0.358536i	−0.504737	+	0.863273i	$0.668411\pi$
$644$		0			0
$645$	−8.16481			−0.321489
$646$		0			0
$647$	18.2921			0.719135			0.359567	−	0.933119i	$-0.382924\pi$
$647$	18.2921			0.719135			0.359567	+	0.933119i	$0.382924\pi$
$648$		0			0
$649$	4.65213	+	4.65213i	0.182612	+	0.182612i
$650$		0			0
$651$	18.6827	+	18.6827i	0.732233	+	0.732233i
$652$		0			0
$653$	1.01717	−	1.01717i	0.0398048	−	0.0398048i	−0.686924	−	0.726729i	$-0.741040\pi$
$653$	1.01717	−	1.01717i	0.0398048	−	0.0398048i	0.726729	+	0.686924i	$0.241040\pi$
$654$		0			0
$655$		−	16.8893i		−	0.659920i
$656$		0			0
$657$	2.07683	−	2.07683i	0.0810250	−	0.0810250i
$658$		0			0
$659$	−30.6681			−1.19466			−0.597329	−	0.801996i	$-0.703771\pi$
$659$	−30.6681			−1.19466			−0.597329	+	0.801996i	$0.703771\pi$
$660$		0			0
$661$		−	6.42933i		−	0.250072i	−0.992152	−	0.125036i	$-0.960095\pi$
$661$		−	6.42933i		−	0.250072i	0.992152	−	0.125036i	$-0.0399046\pi$
$662$		0			0
$663$	5.29669	+	11.6913i	0.205706	+	0.454052i
$664$		0			0
$665$			23.7507i			0.921011i
$666$		0			0
$667$	−20.0095			−0.774772
$668$		0			0
$669$	11.7614	−	11.7614i	0.454723	−	0.454723i
$670$		0			0
$671$		−	15.2620i		−	0.589184i
$672$		0			0
$673$	14.9520	−	14.9520i	0.576356	−	0.576356i	−0.357542	−	0.933897i	$-0.616385\pi$
$673$	14.9520	−	14.9520i	0.576356	−	0.576356i	0.933897	+	0.357542i	$0.116385\pi$
$674$		0			0
$675$	−0.707107	−	0.707107i	−0.0272166	−	0.0272166i
$676$		0			0
$677$	10.1772	+	10.1772i	0.391141	+	0.391141i	0.875094	−	0.483953i	$-0.160799\pi$
$677$	10.1772	+	10.1772i	0.391141	+	0.391141i	−0.483953	+	0.875094i	$0.660799\pi$
$678$		0			0
$679$	52.5338			2.01606
$680$		0			0
$681$	−25.0374			−0.959434
$682$		0			0
$683$	−26.9866	−	26.9866i	−1.03261	−	1.03261i	−0.999450	−	0.0331645i	$-0.989441\pi$
$683$	−26.9866	−	26.9866i	−1.03261	−	1.03261i	−0.0331645	−	0.999450i	$-0.510559\pi$
$684$		0			0
$685$	13.6054	+	13.6054i	0.519835	+	0.519835i
$686$		0			0
$687$	0.806212	−	0.806212i	0.0307589	−	0.0307589i
$688$		0			0
$689$			26.0552i			0.992624i
$690$		0			0
$691$	−24.4464	+	24.4464i	−0.929985	+	0.929985i	−0.997704	−	0.0677193i	$-0.978428\pi$
$691$	−24.4464	+	24.4464i	−0.929985	+	0.929985i	0.0677193	+	0.997704i	$0.478428\pi$
$692$		0			0
$693$	−5.13042			−0.194889
$694$		0			0
$695$			11.6465i			0.441777i
$696$		0			0
$697$	−11.8364	+	5.36242i	−0.448335	+	0.203116i
$698$		0			0
$699$		−	26.1102i		−	0.987578i
$700$		0			0
$701$	−32.7092			−1.23541			−0.617705	−	0.786410i	$-0.711937\pi$
$701$	−32.7092			−1.23541			−0.617705	+	0.786410i	$0.711937\pi$
$702$		0			0
$703$	−2.51395	+	2.51395i	−0.0948156	+	0.0948156i
$704$		0			0
$705$		−	7.24229i		−	0.272760i
$706$		0			0
$707$	−11.8181	+	11.8181i	−0.444467	+	0.444467i
$708$		0			0
$709$	−8.26375	−	8.26375i	−0.310352	−	0.310352i	0.534694	−	0.845046i	$-0.320427\pi$
$709$	−8.26375	−	8.26375i	−0.310352	−	0.310352i	−0.845046	+	0.534694i	$0.820427\pi$
$710$		0			0
$711$	−10.1798	−	10.1798i	−0.381771	−	0.381771i
$712$		0			0
$713$	55.0252			2.06071
$714$		0			0
$715$	−4.33304			−0.162046
$716$		0			0
$717$	3.21756	+	3.21756i	0.120162	+	0.120162i
$718$		0			0
$719$	25.9650	+	25.9650i	0.968333	+	0.968333i	0.999514	−	0.0311810i	$-0.00992684\pi$
$719$	25.9650	+	25.9650i	0.968333	+	0.968333i	−0.0311810	+	0.999514i	$0.509927\pi$
$720$		0			0
$721$	19.5500	−	19.5500i	0.728079	−	0.728079i
$722$		0			0
$723$			4.36547i			0.162353i
$724$		0			0
$725$	−1.84322	+	1.84322i	−0.0684555	+	0.0684555i
$726$		0			0
$727$	−14.2255			−0.527593			−0.263796	−	0.964578i	$-0.584975\pi$
$727$	−14.2255			−0.527593			−0.263796	+	0.964578i	$0.584975\pi$
$728$		0			0
$729$		−	1.00000i		−	0.0370370i
$730$		0			0
$731$	11.8595	−	31.5062i	0.438641	−	1.16530i
$732$		0			0
$733$			38.5540i			1.42402i	0.702167	+	0.712012i	$0.252216\pi$
$733$			38.5540i			1.42402i	−0.702167	+	0.712012i	$0.747784\pi$
$734$		0			0
$735$	6.58545			0.242908
$736$		0			0
$737$	12.8378	−	12.8378i	0.472888	−	0.472888i
$738$		0			0
$739$		−	36.4229i		−	1.33984i	−0.742433	−	0.669920i	$-0.766328\pi$
$739$		−	36.4229i		−	1.33984i	0.742433	−	0.669920i	$-0.233672\pi$
$740$		0			0
$741$	−14.1841	+	14.1841i	−0.521064	+	0.521064i
$742$		0			0
$743$	−9.71602	−	9.71602i	−0.356446	−	0.356446i	0.506055	−	0.862501i	$-0.331103\pi$
$743$	−9.71602	−	9.71602i	−0.356446	−	0.356446i	−0.862501	+	0.506055i	$0.831103\pi$
$744$		0			0
$745$	−1.25335	−	1.25335i	−0.0459193	−	0.0459193i
$746$		0			0
$747$	−3.73346			−0.136600
$748$		0			0
$749$	56.0694			2.04873
$750$		0			0
$751$	−7.23612	−	7.23612i	−0.264050	−	0.264050i	0.562647	−	0.826697i	$-0.309783\pi$
$751$	−7.23612	−	7.23612i	−0.264050	−	0.264050i	−0.826697	+	0.562647i	$0.809783\pi$
$752$		0			0
$753$	−20.6295	−	20.6295i	−0.751779	−	0.751779i
$754$		0			0
$755$	−12.7671	+	12.7671i	−0.464642	+	0.464642i
$756$		0			0
$757$			14.7376i			0.535646i	0.963468	+	0.267823i	$0.0863043\pi$
$757$			14.7376i			0.535646i	−0.963468	+	0.267823i	$0.913696\pi$
$758$		0			0
$759$	−7.55519	+	7.55519i	−0.274236	+	0.274236i
$760$		0			0
$761$	51.0059			1.84896			0.924481	−	0.381227i	$-0.124498\pi$
$761$	51.0059			1.84896			0.924481	+	0.381227i	$0.124498\pi$
$762$		0			0
$763$		−	53.2033i		−	1.92609i
$764$		0			0
$765$	3.75566	−	1.70148i	0.135786	−	0.0615173i
$766$		0			0
$767$		−	14.7139i		−	0.531288i
$768$		0			0
$769$	−19.6050			−0.706975			−0.353488	−	0.935439i	$-0.615004\pi$
$769$	−19.6050			−0.706975			−0.353488	+	0.935439i	$0.615004\pi$
$770$		0			0
$771$	2.24850	−	2.24850i	0.0809778	−	0.0809778i
$772$		0			0
$773$			36.5059i			1.31303i	0.754315	+	0.656513i	$0.227969\pi$
$773$			36.5059i			1.31303i	−0.754315	+	0.656513i	$0.772031\pi$
$774$		0			0
$775$	5.06877	−	5.06877i	0.182076	−	0.182076i
$776$		0			0
$777$	1.43799	+	1.43799i	0.0515876	+	0.0515876i
$778$		0			0
$779$	−14.3601	−	14.3601i	−0.514503	−	0.514503i
$780$		0			0
$781$	−20.3873			−0.729514
$782$		0			0
$783$	−2.60671			−0.0931561
$784$		0			0
$785$	−16.0437	−	16.0437i	−0.572623	−	0.572623i
$786$		0			0
$787$	5.10153	+	5.10153i	0.181850	+	0.181850i	0.792161	−	0.610312i	$-0.208956\pi$
$787$	5.10153	+	5.10153i	0.181850	+	0.181850i	−0.610312	+	0.792161i	$0.708956\pi$
$788$		0			0
$789$	−16.6523	+	16.6523i	−0.592837	+	0.592837i
$790$		0			0
$791$		−	35.8110i		−	1.27329i
$792$		0			0
$793$	−24.1356	+	24.1356i	−0.857080	+	0.857080i
$794$		0			0
$795$	8.36985			0.296848
$796$		0			0
$797$		−	18.4209i		−	0.652503i	−0.945283	−	0.326251i	$-0.894214\pi$
$797$		−	18.4209i		−	0.652503i	0.945283	−	0.326251i	$-0.105786\pi$
$798$		0			0
$799$	27.9464	+	10.5196i	0.988673	+	0.372155i
$800$		0			0
$801$			9.27359i			0.327666i
$802$		0			0
$803$	−4.08821			−0.144270
$804$		0			0
$805$	20.0063	−	20.0063i	0.705128	−	0.705128i
$806$		0			0
$807$			2.17558i			0.0765842i
$808$		0			0
$809$	−37.1750	+	37.1750i	−1.30700	+	1.30700i	−0.383435	+	0.923568i	$0.625259\pi$
$809$	−37.1750	+	37.1750i	−1.30700	+	1.30700i	−0.923568	+	0.383435i	$0.874741\pi$
$810$		0			0
$811$	6.19841	+	6.19841i	0.217656	+	0.217656i	0.807510	−	0.589854i	$-0.200815\pi$
$811$	6.19841	+	6.19841i	0.217656	+	0.217656i	−0.589854	+	0.807510i	$0.700815\pi$
$812$		0			0
$813$	−10.1952	−	10.1952i	−0.357562	−	0.357562i
$814$		0			0
$815$	4.23478			0.148338
$816$		0			0
$817$	52.6120			1.84066
$818$		0			0
$819$	8.11332	+	8.11332i	0.283502	+	0.283502i
$820$		0			0
$821$	13.9232	+	13.9232i	0.485923	+	0.485923i	0.907017	−	0.421094i	$-0.138354\pi$
$821$	13.9232	+	13.9232i	0.485923	+	0.485923i	−0.421094	+	0.907017i	$0.638354\pi$
$822$		0			0
$823$	−30.8697	+	30.8697i	−1.07605	+	1.07605i	−0.0791918	+	0.996859i	$0.525234\pi$
$823$	−30.8697	+	30.8697i	−1.07605	+	1.07605i	−0.996859	+	0.0791918i	$0.974766\pi$
$824$		0			0
$825$			1.39193i			0.0484606i
$826$		0			0
$827$	−29.5016	+	29.5016i	−1.02587	+	1.02587i	−0.0262139	+	0.999656i	$0.508345\pi$
$827$	−29.5016	+	29.5016i	−1.02587	+	1.02587i	−0.999656	+	0.0262139i	$0.991655\pi$
$828$		0			0
$829$	−13.2016			−0.458512			−0.229256	−	0.973366i	$-0.573629\pi$
$829$	−13.2016			−0.458512			−0.229256	+	0.973366i	$0.573629\pi$
$830$		0			0
$831$		−	0.719579i		−	0.0249619i
$832$		0			0
$833$	−9.56549	+	25.4118i	−0.331425	+	0.880467i
$834$		0			0
$835$		−	5.00264i		−	0.173124i
$836$		0			0
$837$	7.16833			0.247774
$838$		0			0
$839$	18.9990	−	18.9990i	0.655918	−	0.655918i	−0.298493	−	0.954412i	$-0.596484\pi$
$839$	18.9990	−	18.9990i	0.655918	−	0.655918i	0.954412	+	0.298493i	$0.0964841\pi$
$840$		0			0
$841$		−	22.2051i		−	0.765692i
$842$		0			0
$843$	18.3655	−	18.3655i	0.632542	−	0.632542i
$844$		0			0
$845$	−2.34006	−	2.34006i	−0.0805005	−	0.0805005i
$846$		0			0
$847$	−23.6196	−	23.6196i	−0.811578	−	0.811578i
$848$		0			0
$849$	22.7540			0.780917
$850$		0			0
$851$	4.23524			0.145182
$852$		0			0
$853$	20.4559	+	20.4559i	0.700395	+	0.700395i	0.964495	−	0.264100i	$-0.0850749\pi$
$853$	20.4559	+	20.4559i	0.700395	+	0.700395i	−0.264100	+	0.964495i	$0.585075\pi$
$854$		0			0
$855$	4.55642	+	4.55642i	0.155826	+	0.155826i
$856$		0			0
$857$	22.4495	−	22.4495i	0.766859	−	0.766859i	−0.210693	−	0.977552i	$-0.567572\pi$
$857$	22.4495	−	22.4495i	0.766859	−	0.766859i	0.977552	+	0.210693i	$0.0675722\pi$
$858$		0			0
$859$		−	8.69843i		−	0.296787i	−0.988928	−	0.148393i	$-0.952590\pi$
$859$		−	8.69843i		−	0.296787i	0.988928	−	0.148393i	$-0.0474101\pi$
$860$		0			0
$861$	−8.21400	+	8.21400i	−0.279932	+	0.279932i
$862$		0			0
$863$	−34.4058			−1.17119			−0.585593	−	0.810605i	$-0.699138\pi$
$863$	−34.4058			−1.17119			−0.585593	+	0.810605i	$0.699138\pi$
$864$		0			0
$865$			1.44714i			0.0492043i
$866$		0			0
$867$	1.11049	+	16.9637i	0.0377143	+	0.576117i
$868$		0			0
$869$			20.0387i			0.679765i
$870$		0			0
$871$	−40.6039			−1.37581
$872$		0			0
$873$	10.0783	−	10.0783i	0.341098	−	0.341098i
$874$		0			0
$875$		−	3.68584i		−	0.124604i
$876$		0			0
$877$	2.31022	−	2.31022i	0.0780105	−	0.0780105i	−0.667025	−	0.745035i	$-0.732433\pi$
$877$	2.31022	−	2.31022i	0.0780105	−	0.0780105i	0.745035	+	0.667025i	$0.232433\pi$
$878$		0			0
$879$	10.6619	+	10.6619i	0.359619	+	0.359619i
$880$		0			0
$881$	−26.2289	−	26.2289i	−0.883674	−	0.883674i	0.110231	−	0.993906i	$-0.464841\pi$
$881$	−26.2289	−	26.2289i	−0.883674	−	0.883674i	−0.993906	+	0.110231i	$0.964841\pi$
$882$		0			0
$883$	−28.1516			−0.947377			−0.473688	−	0.880693i	$-0.657078\pi$
$883$	−28.1516			−0.947377			−0.473688	+	0.880693i	$0.657078\pi$
$884$		0			0
$885$	−4.72663			−0.158884
$886$		0			0
$887$	−40.8071	−	40.8071i	−1.37017	−	1.37017i	−0.860182	−	0.509987i	$-0.829650\pi$
$887$	−40.8071	−	40.8071i	−1.37017	−	1.37017i	−0.509987	−	0.860182i	$-0.670350\pi$
$888$		0			0
$889$	−55.5393	−	55.5393i	−1.86273	−	1.86273i
$890$		0			0
$891$	−0.984240	+	0.984240i	−0.0329733	+	0.0329733i
$892$		0			0
$893$			46.6675i			1.56167i
$894$		0			0
$895$	−3.78676	+	3.78676i	−0.126578	+	0.126578i
$896$		0			0
$897$	23.8958			0.797856
$898$		0			0
$899$		−	18.6857i		−	0.623204i
$900$		0			0
$901$	−12.1574	+	32.2974i	−0.405021	+	1.07598i
$902$		0			0
$903$		−	30.0942i		−	1.00147i
$904$		0			0
$905$	3.41116			0.113391
$906$		0			0
$907$	1.54996	−	1.54996i	0.0514656	−	0.0514656i	−0.680906	−	0.732371i	$-0.738414\pi$
$907$	1.54996	−	1.54996i	0.0514656	−	0.0514656i	0.732371	+	0.680906i	$0.238414\pi$
$908$		0			0
$909$			4.53448i			0.150399i
$910$		0			0
$911$	30.4027	−	30.4027i	1.00728	−	1.00728i	0.00731159	−	0.999973i	$-0.497673\pi$
$911$	30.4027	−	30.4027i	1.00728	−	1.00728i	0.999973	−	0.00731159i	$-0.00232737\pi$
$912$		0			0
$913$	3.67462	+	3.67462i	0.121612	+	0.121612i
$914$		0			0
$915$	7.75320	+	7.75320i	0.256313	+	0.256313i
$916$		0			0
$917$	62.2513			2.05572
$918$		0			0
$919$	0.696653			0.0229805			0.0114902	−	0.999934i	$-0.496342\pi$
$919$	0.696653			0.0229805			0.0114902	+	0.999934i	$0.496342\pi$
$920$		0			0
$921$	10.4051	+	10.4051i	0.342859	+	0.342859i
$922$		0			0
$923$	32.2407	+	32.2407i	1.06122	+	1.06122i
$924$		0			0
$925$	0.390138	−	0.390138i	0.0128277	−	0.0128277i
$926$		0			0
$927$		−	7.50109i		−	0.246368i
$928$		0			0
$929$	1.90744	−	1.90744i	0.0625811	−	0.0625811i	−0.675124	−	0.737705i	$-0.735910\pi$
$929$	1.90744	−	1.90744i	0.0625811	−	0.0625811i	0.737705	+	0.675124i	$0.235910\pi$
$930$		0			0
$931$	−42.4350			−1.39075
$932$		0			0
$933$		−	10.5682i		−	0.345986i
$934$		0			0
$935$	−5.37113	−	2.02180i	−0.175655	−	0.0661199i
$936$		0			0
$937$			28.8392i			0.942136i	0.882097	+	0.471068i	$0.156131\pi$
$937$			28.8392i			0.942136i	−0.882097	+	0.471068i	$0.843869\pi$
$938$		0			0
$939$	0.694091			0.0226508
$940$		0			0
$941$	11.7247	−	11.7247i	0.382214	−	0.382214i	−0.489685	−	0.871899i	$-0.662888\pi$
$941$	11.7247	−	11.7247i	0.382214	−	0.382214i	0.871899	+	0.489685i	$0.162888\pi$
$942$		0			0
$943$			24.1923i			0.787809i
$944$		0			0
$945$	2.60629	−	2.60629i	0.0847825	−	0.0847825i
$946$		0			0
$947$	−11.0521	−	11.0521i	−0.359145	−	0.359145i	0.504353	−	0.863498i	$-0.331731\pi$
$947$	−11.0521	−	11.0521i	−0.359145	−	0.359145i	−0.863498	+	0.504353i	$0.831731\pi$
$948$		0			0
$949$	6.46515	+	6.46515i	0.209868	+	0.209868i
$950$		0			0
$951$	−1.23253			−0.0399676
$952$		0			0
$953$	−15.4158			−0.499365			−0.249683	−	0.968328i	$-0.580326\pi$
$953$	−15.4158			−0.499365			−0.249683	+	0.968328i	$0.580326\pi$
$954$		0			0
$955$	8.04031	+	8.04031i	0.260178	+	0.260178i
$956$		0			0
$957$	2.56563	+	2.56563i	0.0829349	+	0.0829349i
$958$		0			0
$959$	−50.1473	+	50.1473i	−1.61934	+	1.61934i
$960$		0			0
$961$			20.3849i			0.657577i
$962$		0			0
$963$	10.7566	−	10.7566i	0.346626	−	0.346626i
$964$		0			0
$965$	−0.229135			−0.00737613
$966$		0			0
$967$		−	10.6770i		−	0.343349i	−0.985154	−	0.171674i	$-0.945082\pi$
$967$		−	10.6770i		−	0.343349i	0.985154	−	0.171674i	$-0.0549177\pi$
$968$		0			0
$969$	−24.2005	+	10.9639i	−0.777432	+	0.352212i
$970$		0			0
$971$			7.37080i			0.236540i	0.992981	+	0.118270i	$0.0377349\pi$
$971$			7.37080i			0.236540i	−0.992981	+	0.118270i	$0.962265\pi$
$972$		0			0
$973$	−42.9272			−1.37618
$974$		0			0
$975$	2.20121	−	2.20121i	0.0704952	−	0.0704952i
$976$		0			0
$977$		−	30.2732i		−	0.968526i	−0.874922	−	0.484263i	$-0.839088\pi$
$977$		−	30.2732i		−	0.968526i	0.874922	−	0.484263i	$-0.160912\pi$
$978$		0			0
$979$	9.12743	−	9.12743i	0.291714	−	0.291714i
$980$		0			0
$981$	−10.2067	−	10.2067i	−0.325876	−	0.325876i
$982$		0			0
$983$	14.1772	+	14.1772i	0.452183	+	0.452183i	0.896078	−	0.443896i	$-0.146404\pi$
$983$	14.1772	+	14.1772i	0.452183	+	0.452183i	−0.443896	+	0.896078i	$0.646404\pi$
$984$		0			0
$985$	7.37999			0.235146
$986$		0			0
$987$	26.6940			0.849678
$988$		0			0
$989$	−44.3175	−	44.3175i	−1.40921	−	1.40921i
$990$		0			0
$991$	−18.1132	−	18.1132i	−0.575386	−	0.575386i	0.358243	−	0.933629i	$-0.383376\pi$
$991$	−18.1132	−	18.1132i	−0.575386	−	0.575386i	−0.933629	+	0.358243i	$0.883376\pi$
$992$		0			0
$993$	0.272402	−	0.272402i	0.00864441	−	0.00864441i
$994$		0			0
$995$			21.5668i			0.683713i
$996$		0			0
$997$	−2.91628	+	2.91628i	−0.0923595	+	0.0923595i	−0.751777	−	0.659417i	$-0.770803\pi$
$997$	−2.91628	+	2.91628i	−0.0923595	+	0.0923595i	0.659417	+	0.751777i	$0.270803\pi$
$998$		0			0
$999$	0.551739			0.0174562

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2040.2.cf.d.1441.6	yes	20
17.4	even	4	inner	2040.2.cf.d.361.6	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2040.2.cf.d.361.6	✓	20	17.4	even	4	inner
2040.2.cf.d.1441.6	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)

\(f(q)\)	\(=\)	\(q+(0.707107 + 0.707107i) q^{3} +(0.707107 + 0.707107i) q^{5} +(-2.60629 + 2.60629i) q^{7} +1.00000i q^{9} +(0.984240 - 0.984240i) q^{11} -3.11298 q^{13} +1.00000i q^{15} +(-3.85878 - 1.45252i) q^{17} -6.44375i q^{19} -3.68584 q^{21} +(-5.42787 + 5.42787i) q^{23} +1.00000i q^{25} +(-0.707107 + 0.707107i) q^{27} +(1.84322 + 1.84322i) q^{29} +(-5.06877 - 5.06877i) q^{31} +1.39193 q^{33} -3.68584 q^{35} +(-0.390138 - 0.390138i) q^{37} +(-2.20121 - 2.20121i) q^{39} +(2.22853 - 2.22853i) q^{41} +8.16481i q^{43} +(-0.707107 + 0.707107i) q^{45} -7.24229 q^{47} -6.58545i q^{49} +(-1.70148 - 3.75566i) q^{51} -8.36985i q^{53} +1.39193 q^{55} +(4.55642 - 4.55642i) q^{57} +4.72663i q^{59} +(7.75320 - 7.75320i) q^{61} +(-2.60629 - 2.60629i) q^{63} +(-2.20121 - 2.20121i) q^{65} +13.0434 q^{67} -7.67616 q^{69} +(-10.3569 - 10.3569i) q^{71} +(-2.07683 - 2.07683i) q^{73} +(-0.707107 + 0.707107i) q^{75} +5.13042i q^{77} +(-10.1798 + 10.1798i) q^{79} -1.00000 q^{81} +3.73346i q^{83} +(-1.70148 - 3.75566i) q^{85} +2.60671i q^{87} +9.27359 q^{89} +(8.11332 - 8.11332i) q^{91} -7.16833i q^{93} +(4.55642 - 4.55642i) q^{95} +(-10.0783 - 10.0783i) q^{97} +(0.984240 + 0.984240i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 4 q^{7} - 4 q^{11} + 8 q^{13} + 4 q^{17} - 4 q^{23} + 12 q^{29} + 8 q^{31} - 8 q^{33} - 12 q^{37} + 4 q^{39} - 12 q^{41} - 48 q^{47} + 16 q^{51} - 8 q^{55} + 8 q^{57} + 24 q^{61} - 4 q^{63} + 4 q^{65}+ \cdots - 4 q^{99}+O(q^{100}) \) 20 * q - 4 * q^7 - 4 * q^11 + 8 * q^13 + 4 * q^17 - 4 * q^23 + 12 * q^29 + 8 * q^31 - 8 * q^33 - 12 * q^37 + 4 * q^39 - 12 * q^41 - 48 * q^47 + 16 * q^51 - 8 * q^55 + 8 * q^57 + 24 * q^61 - 4 * q^63 + 4 * q^65 + 56 * q^67 - 24 * q^69 + 16 * q^71 + 4 * q^73 + 12 * q^79 - 20 * q^81 + 16 * q^85 - 44 * q^91 + 8 * q^95 - 40 * q^97 - 4 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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