LMFDB - Embedded newform 1368.2.g.b.685.14

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1368,2,Mod(685,1368)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1368, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 1, 0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1368.685"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.g (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [16,0,0,2,0,0,-8] 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:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 2 x^{15} + 3 x^{14} - 4 x^{13} + 4 x^{12} + 4 x^{11} - 10 x^{10} + 24 x^{9} - 40 x^{8} + \cdots + 256 $ x^16 - 2x^15 + 3x^14 - 4x^13 + 4x^12 + 4x^11 - 10x^10 + 24x^9 - 40x^8 + 48x^7 - 40x^6 + 32x^5 + 64x^4 - 128x^3 + 192x^2 - 256*x + 256
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			685.14
Root			$0.340606 + 1.37258i$ of defining polynomial
Character	$\chi$	$=$	1368.685
Dual form			1368.2.g.b.685.13

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.37258 + 0.340606i) q^{2} +(1.76798 + 0.935021i) q^{4} +2.13486i q^{5} +3.29464 q^{7} +(2.10822 + 1.88558i) q^{8} +(-0.727145 + 2.93027i) q^{10} -3.71210i q^{11} -2.32843i q^{13} +(4.52218 + 1.12218i) q^{14} +(2.25147 + 3.30619i) q^{16} +6.48822 q^{17} +1.00000i q^{19} +(-1.99614 + 3.77437i) q^{20} +(1.26436 - 5.09517i) q^{22} -7.32651 q^{23} +0.442384 q^{25} +(0.793079 - 3.19597i) q^{26} +(5.82485 + 3.08056i) q^{28} -2.59857i q^{29} -1.34204 q^{31} +(1.96423 + 5.30489i) q^{32} +(8.90563 + 2.20993i) q^{34} +7.03360i q^{35} +3.72986i q^{37} +(-0.340606 + 1.37258i) q^{38} +(-4.02544 + 4.50075i) q^{40} -6.52385 q^{41} +1.97202i q^{43} +(3.47089 - 6.56290i) q^{44} +(-10.0563 - 2.49545i) q^{46} -5.45991 q^{47} +3.85468 q^{49} +(0.607210 + 0.150679i) q^{50} +(2.17714 - 4.11661i) q^{52} +4.98640i q^{53} +7.92480 q^{55} +(6.94584 + 6.21231i) q^{56} +(0.885089 - 3.56676i) q^{58} +9.67136i q^{59} -8.15570i q^{61} +(-1.84206 - 0.457105i) q^{62} +(0.889191 + 7.95043i) q^{64} +4.97088 q^{65} +0.524986i q^{67} +(11.4710 + 6.06662i) q^{68} +(-2.39569 + 9.65420i) q^{70} -7.17489 q^{71} -6.33130 q^{73} +(-1.27041 + 5.11955i) q^{74} +(-0.935021 + 1.76798i) q^{76} -12.2300i q^{77} -8.75644 q^{79} +(-7.05824 + 4.80657i) q^{80} +(-8.95453 - 2.22206i) q^{82} -7.74008i q^{83} +13.8514i q^{85} +(-0.671680 + 2.70676i) q^{86} +(6.99945 - 7.82592i) q^{88} +1.04368 q^{89} -7.67136i q^{91} +(-12.9531 - 6.85044i) q^{92} +(-7.49419 - 1.85968i) q^{94} -2.13486 q^{95} -0.117594 q^{97} +(5.29088 + 1.31293i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 2 q^{4} - 8 q^{7} + 12 q^{8} - 8 q^{10} - 4 q^{14} + 2 q^{16} + 8 q^{17} - 8 q^{20} + 20 q^{22} - 24 q^{25} + 10 q^{26} - 14 q^{28} + 16 q^{31} + 20 q^{32} - 2 q^{38} + 28 q^{40} - 16 q^{41} + 28 q^{44}+ \cdots + 48 q^{98}+O(q^{100}) $ 16 * q + 2 * q^4 - 8 * q^7 + 12 * q^8 - 8 * q^10 - 4 * q^14 + 2 * q^16 + 8 * q^17 - 8 * q^20 + 20 * q^22 - 24 * q^25 + 10 * q^26 - 14 * q^28 + 16 * q^31 + 20 * q^32 - 2 * q^38 + 28 * q^40 - 16 * q^41 + 28 * q^44 - 48 * q^46 - 24 * q^47 + 24 * q^49 - 12 * q^50 + 8 * q^52 + 16 * q^55 + 48 * q^56 + 38 * q^58 + 16 * q^62 + 14 * q^64 - 16 * q^65 + 26 * q^68 - 32 * q^70 - 48 * q^71 + 20 * q^74 - 4 * q^76 - 48 * q^79 - 4 * q^80 - 12 * q^82 - 48 * q^86 + 40 * q^88 + 16 * q^89 - 62 * q^92 - 36 * q^94 - 16 * q^95 + 32 * q^97 + 48 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1368\mathbb{Z}\right)^\times$.

$n$	$343$	$685$	$1009$	$1217$
$\chi(n)$	$1$	$-1$	$1$	$1$

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 $

$2$	1.37258	+	0.340606i	0.970564	+	0.240845i
$2$	1.37258	+	0.340606i	0.970564	+	0.240845i
$3$		0			0
$3$		0			0
$4$	1.76798	+	0.935021i	0.883988	+	0.467510i
$5$			2.13486i			0.954737i	0.878703	+	0.477369i	$0.158409\pi$
$5$			2.13486i			0.954737i	−0.878703	+	0.477369i	$0.841591\pi$
$6$		0			0
$7$	3.29464			1.24526			0.622629	−	0.782517i	$-0.286064\pi$
$7$	3.29464			1.24526			0.622629	+	0.782517i	$0.286064\pi$
$8$	2.10822	+	1.88558i	0.745369	+	0.666653i
$9$		0			0
$10$	−0.727145	+	2.93027i	−0.229944	+	0.926633i
$11$		−	3.71210i		−	1.11924i	−0.828749	−	0.559620i	$-0.810947\pi$
$11$		−	3.71210i		−	1.11924i	0.828749	−	0.559620i	$-0.189053\pi$
$12$		0			0
$13$		−	2.32843i		−	0.645792i	−0.946435	−	0.322896i	$-0.895344\pi$
$13$		−	2.32843i		−	0.645792i	0.946435	−	0.322896i	$-0.104656\pi$
$14$	4.52218	+	1.12218i	1.20860	+	0.299914i
$15$		0			0
$16$	2.25147	+	3.30619i	0.562868	+	0.826547i
$17$	6.48822			1.57362			0.786812	−	0.617192i	$-0.211730\pi$
$17$	6.48822			1.57362			0.786812	+	0.617192i	$0.211730\pi$
$18$		0			0
$19$			1.00000i			0.229416i
$19$			1.00000i			0.229416i
$20$	−1.99614	+	3.77437i	−0.446350	+	0.843976i
$21$		0			0
$22$	1.26436	−	5.09517i	0.269563	−	1.08629i
$23$	−7.32651			−1.52768			−0.763842	−	0.645404i	$-0.776689\pi$
$23$	−7.32651			−1.52768			−0.763842	+	0.645404i	$0.776689\pi$
$24$		0			0
$25$	0.442384			0.0884769
$26$	0.793079	−	3.19597i	0.155536	−	0.626782i
$27$		0			0
$28$	5.82485	+	3.08056i	1.10079	+	0.582171i
$29$		−	2.59857i		−	0.482542i	−0.970458	−	0.241271i	$-0.922436\pi$
$29$		−	2.59857i		−	0.482542i	0.970458	−	0.241271i	$-0.0775643\pi$
$30$		0			0
$31$	−1.34204			−0.241037			−0.120518	−	0.992711i	$-0.538456\pi$
$31$	−1.34204			−0.241037			−0.120518	+	0.992711i	$0.538456\pi$
$32$	1.96423	+	5.30489i	0.347230	+	0.937780i
$33$		0			0
$34$	8.90563	+	2.20993i	1.52730	+	0.378999i
$35$			7.03360i			1.18889i
$36$		0			0
$37$			3.72986i			0.613186i	0.951841	+	0.306593i	$0.0991890\pi$
$37$			3.72986i			0.613186i	−0.951841	+	0.306593i	$0.900811\pi$
$38$	−0.340606	+	1.37258i	−0.0552536	+	0.222663i
$39$		0			0
$40$	−4.02544	+	4.50075i	−0.636478	+	0.711631i
$41$	−6.52385			−1.01885			−0.509427	−	0.860514i	$-0.670143\pi$
$41$	−6.52385			−1.01885			−0.509427	+	0.860514i	$0.670143\pi$
$42$		0			0
$43$			1.97202i			0.300729i	0.988631	+	0.150365i	$0.0480448\pi$
$43$			1.97202i			0.300729i	−0.988631	+	0.150365i	$0.951955\pi$
$44$	3.47089	−	6.56290i	0.523256	−	0.989394i
$45$		0			0
$46$	−10.0563	−	2.49545i	−1.48271	−	0.367935i
$47$	−5.45991			−0.796410			−0.398205	−	0.917297i	$-0.630367\pi$
$47$	−5.45991			−0.796410			−0.398205	+	0.917297i	$0.630367\pi$
$48$		0			0
$49$	3.85468			0.550669
$50$	0.607210	+	0.150679i	0.0858724	+	0.0213092i
$51$		0			0
$52$	2.17714	−	4.11661i	0.301914	−	0.570872i
$53$			4.98640i			0.684935i	0.939530	+	0.342467i	$0.111263\pi$
$53$			4.98640i			0.684935i	−0.939530	+	0.342467i	$0.888737\pi$
$54$		0			0
$55$	7.92480			1.06858
$56$	6.94584	+	6.21231i	0.928177	+	0.830155i
$57$		0			0
$58$	0.885089	−	3.56676i	0.116218	−	0.468338i
$59$			9.67136i			1.25910i	0.776958	+	0.629552i	$0.216762\pi$
$59$			9.67136i			1.25910i	−0.776958	+	0.629552i	$0.783238\pi$
$60$		0			0
$61$		−	8.15570i		−	1.04423i	−0.852875	−	0.522115i	$-0.825143\pi$
$61$		−	8.15570i		−	1.04423i	0.852875	−	0.522115i	$-0.174857\pi$
$62$	−1.84206	−	0.457105i	−0.233941	−	0.0580525i
$63$		0			0
$64$	0.889191	+	7.95043i	0.111149	+	0.993804i
$65$	4.97088			0.616561
$66$		0			0
$67$			0.524986i			0.0641372i	0.999486	+	0.0320686i	$0.0102095\pi$
$67$			0.524986i			0.0641372i	−0.999486	+	0.0320686i	$0.989790\pi$
$68$	11.4710	+	6.06662i	1.39106	+	0.735686i
$69$		0			0
$70$	−2.39569	+	9.65420i	−0.286339	+	1.15390i
$71$	−7.17489			−0.851503			−0.425751	−	0.904840i	$-0.639990\pi$
$71$	−7.17489			−0.851503			−0.425751	+	0.904840i	$0.639990\pi$
$72$		0			0
$73$	−6.33130			−0.741022			−0.370511	−	0.928828i	$-0.620817\pi$
$73$	−6.33130			−0.741022			−0.370511	+	0.928828i	$0.620817\pi$
$74$	−1.27041	+	5.11955i	−0.147683	+	0.595136i
$75$		0			0
$76$	−0.935021	+	1.76798i	−0.107254	+	0.202801i
$77$		−	12.2300i		−	1.39374i
$78$		0			0
$79$	−8.75644			−0.985176			−0.492588	−	0.870263i	$-0.663949\pi$
$79$	−8.75644			−0.985176			−0.492588	+	0.870263i	$0.663949\pi$
$80$	−7.05824	+	4.80657i	−0.789135	+	0.537391i
$81$		0			0
$82$	−8.95453	−	2.22206i	−0.988862	−	0.245386i
$83$		−	7.74008i		−	0.849585i	−0.905291	−	0.424792i	$-0.860347\pi$
$83$		−	7.74008i		−	0.849585i	0.905291	−	0.424792i	$-0.139653\pi$
$84$		0			0
$85$			13.8514i			1.50240i
$86$	−0.671680	+	2.70676i	−0.0724291	+	0.291877i
$87$		0			0
$88$	6.99945	−	7.82592i	0.746144	−	0.834246i
$89$	1.04368			0.110630			0.0553148	−	0.998469i	$-0.482384\pi$
$89$	1.04368			0.110630			0.0553148	+	0.998469i	$0.482384\pi$
$90$		0			0
$91$		−	7.67136i		−	0.804178i
$92$	−12.9531	−	6.85044i	−1.35045	−	0.714208i
$93$		0			0
$94$	−7.49419	−	1.85968i	−0.772966	−	0.191811i
$95$	−2.13486			−0.219032
$96$		0			0
$97$	−0.117594			−0.0119398			−0.00596992	−	0.999982i	$-0.501900\pi$
$97$	−0.117594			−0.0119398			−0.00596992	+	0.999982i	$0.501900\pi$
$98$	5.29088	+	1.31293i	0.534459	+	0.132626i
$99$		0			0
$100$	0.782125	+	0.413639i	0.0782125	+	0.0413639i
$101$		−	10.6142i		−	1.05615i	−0.849197	−	0.528077i	$-0.822913\pi$
$101$		−	10.6142i		−	1.05615i	0.849197	−	0.528077i	$-0.177087\pi$
$102$		0			0
$103$	12.4190			1.22368			0.611840	−	0.790982i	$-0.290430\pi$
$103$	12.4190			1.22368			0.611840	+	0.790982i	$0.290430\pi$
$104$	4.39045	−	4.90885i	0.430519	−	0.481353i
$105$		0			0
$106$	−1.69840	+	6.84425i	−0.164963	+	0.664773i
$107$		−	16.5439i		−	1.59936i	−0.600430	−	0.799678i	$-0.705004\pi$
$107$		−	16.5439i		−	1.59936i	0.600430	−	0.799678i	$-0.294996\pi$
$108$		0			0
$109$			17.4437i			1.67080i	0.549641	+	0.835401i	$0.314765\pi$
$109$			17.4437i			1.67080i	−0.549641	+	0.835401i	$0.685235\pi$
$110$	10.8775	+	2.69924i	1.03712	+	0.257362i
$111$		0			0
$112$	7.41780	+	10.8927i	0.700916	+	1.02926i
$113$	14.3193			1.34705			0.673523	−	0.739166i	$-0.264780\pi$
$113$	14.3193			1.34705			0.673523	+	0.739166i	$0.264780\pi$
$114$		0			0
$115$		−	15.6411i		−	1.45854i
$116$	2.42972	−	4.59421i	0.225594	−	0.426561i
$117$		0			0
$118$	−3.29413	+	13.2748i	−0.303249	+	1.22204i
$119$	21.3764			1.95957
$120$		0			0
$121$	−2.77968			−0.252698
$122$	2.77788	−	11.1944i	0.251498	−	1.01349i
$123$		0			0
$124$	−2.37269	−	1.25483i	−0.213073	−	0.112687i
$125$			11.6187i			1.03921i
$126$		0			0
$127$	2.63985			0.234248			0.117124	−	0.993117i	$-0.462632\pi$
$127$	2.63985			0.234248			0.117124	+	0.993117i	$0.462632\pi$
$128$	−1.48748	+	11.2155i	−0.131475	+	0.991319i
$129$		0			0
$130$	6.82295	+	1.69311i	0.598412	+	0.148496i
$131$		−	15.3670i		−	1.34263i	−0.741174	−	0.671313i	$-0.765731\pi$
$131$		−	15.3670i		−	1.34263i	0.741174	−	0.671313i	$-0.234269\pi$
$132$		0			0
$133$			3.29464i			0.285682i
$134$	−0.178813	+	0.720587i	−0.0154471	+	0.0622492i
$135$		0			0
$136$	13.6786	+	12.2340i	1.17293	+	1.04906i
$137$	4.91853			0.420218			0.210109	−	0.977678i	$-0.432618\pi$
$137$	4.91853			0.420218			0.210109	+	0.977678i	$0.432618\pi$
$138$		0			0
$139$		−	9.80377i		−	0.831545i	−0.909469	−	0.415773i	$-0.863511\pi$
$139$		−	9.80377i		−	0.831545i	0.909469	−	0.415773i	$-0.136489\pi$
$140$	−6.57656	+	12.4352i	−0.555821	+	1.05097i
$141$		0			0
$142$	−9.84814	−	2.44381i	−0.826438	−	0.205080i
$143$	−8.64338			−0.722796
$144$		0			0
$145$	5.54758			0.460701
$146$	−8.69024	−	2.15648i	−0.719209	−	0.178471i
$147$		0			0
$148$	−3.48750	+	6.59431i	−0.286671	+	0.542049i
$149$		−	0.724147i		−	0.0593244i	−0.999560	−	0.0296622i	$-0.990557\pi$
$149$		−	0.724147i		−	0.0593244i	0.999560	−	0.0296622i	$-0.00944316\pi$
$150$		0			0
$151$	−15.7252			−1.27970			−0.639850	−	0.768500i	$-0.721003\pi$
$151$	−15.7252			−1.27970			−0.639850	+	0.768500i	$0.721003\pi$
$152$	−1.88558	+	2.10822i	−0.152941	+	0.170999i
$153$		0			0
$154$	4.16563	−	16.7868i	0.335676	−	1.35272i
$155$		−	2.86505i		−	0.230127i
$156$		0			0
$157$		−	0.141127i		−	0.0112632i	−0.999984	−	0.00563160i	$-0.998207\pi$
$157$		−	0.141127i		−	0.0112632i	0.999984	−	0.00563160i	$-0.00179260\pi$
$158$	−12.0189	−	2.98249i	−0.956176	−	0.237275i
$159$		0			0
$160$	−11.3252	+	4.19334i	−0.895334	+	0.331513i
$161$	−24.1383			−1.90236
$162$		0			0
$163$		−	8.41859i		−	0.659395i	−0.944087	−	0.329697i	$-0.893053\pi$
$163$		−	8.41859i		−	0.659395i	0.944087	−	0.329697i	$-0.106947\pi$
$164$	−11.5340	−	6.09993i	−0.900654	−	0.476325i
$165$		0			0
$166$	2.63632	−	10.6239i	0.204618	−	0.824576i
$167$	−5.16538			−0.399709			−0.199854	−	0.979826i	$-0.564047\pi$
$167$	−5.16538			−0.399709			−0.199854	+	0.979826i	$0.564047\pi$
$168$		0			0
$169$	7.57839			0.582953
$170$	−4.71788	+	19.0123i	−0.361845	+	1.45817i
$171$		0			0
$172$	−1.84388	+	3.48647i	−0.140594	+	0.265841i
$173$		−	3.13988i		−	0.238721i	−0.992851	−	0.119360i	$-0.961916\pi$
$173$		−	3.13988i		−	0.238721i	0.992851	−	0.119360i	$-0.0380844\pi$
$174$		0			0
$175$	1.45750			0.110177
$176$	12.2729	−	8.35769i	0.925104	−	0.629984i
$177$		0			0
$178$	1.43254	+	0.355483i	0.107373	+	0.0266446i
$179$		−	18.1898i		−	1.35957i	−0.733410	−	0.679786i	$-0.762073\pi$
$179$		−	18.1898i		−	1.35957i	0.733410	−	0.679786i	$-0.237927\pi$
$180$		0			0
$181$			14.0798i			1.04654i	0.852166	+	0.523271i	$0.175288\pi$
$181$			14.0798i			1.04654i	−0.852166	+	0.523271i	$0.824712\pi$
$182$	2.61291	−	10.5296i	0.193682	−	0.780505i
$183$		0			0
$184$	−15.4459	−	13.8147i	−1.13869	−	1.01843i
$185$	−7.96273			−0.585431
$186$		0			0
$187$		−	24.0849i		−	1.76126i
$188$	−9.65299	−	5.10513i	−0.704016	−	0.372330i
$189$		0			0
$190$	−2.93027	−	0.727145i	−0.212584	−	0.0527527i
$191$	−4.87409			−0.352677			−0.176338	−	0.984330i	$-0.556425\pi$
$191$	−4.87409			−0.352677			−0.176338	+	0.984330i	$0.556425\pi$
$192$		0			0
$193$	−9.99845			−0.719704			−0.359852	−	0.933009i	$-0.617173\pi$
$193$	−9.99845			−0.719704			−0.359852	+	0.933009i	$0.617173\pi$
$194$	−0.161407	−	0.0400531i	−0.0115884	−	0.00287565i
$195$		0			0
$196$	6.81498	+	3.60421i	0.486785	+	0.257444i
$197$			19.2209i			1.36943i	0.728810	+	0.684716i	$0.240074\pi$
$197$			19.2209i			1.36943i	−0.728810	+	0.684716i	$0.759926\pi$
$198$		0			0
$199$	−11.7323			−0.831683			−0.415842	−	0.909437i	$-0.636513\pi$
$199$	−11.7323			−0.831683			−0.415842	+	0.909437i	$0.636513\pi$
$200$	0.932644	+	0.834150i	0.0659479	+	0.0589833i
$201$		0			0
$202$	3.61526	−	14.5689i	0.254369	−	1.02506i
$203$		−	8.56137i		−	0.600890i
$204$		0			0
$205$		−	13.9275i		−	0.972737i
$206$	17.0461	+	4.22998i	1.18766	+	0.294717i
$207$		0			0
$208$	7.69824	−	5.24240i	0.533777	−	0.363495i
$209$	3.71210			0.256771
$210$		0			0
$211$		−	0.399383i		−	0.0274947i	−0.999906	−	0.0137473i	$-0.995624\pi$
$211$		−	0.399383i		−	0.0274947i	0.999906	−	0.0137473i	$-0.00437605\pi$
$212$	−4.66239	+	8.81583i	−0.320214	+	0.605474i
$213$		0			0
$214$	5.63494	−	22.7078i	0.385196	−	1.55228i
$215$	−4.20997			−0.287118
$216$		0			0
$217$	−4.42153			−0.300153
$218$	−5.94142	+	23.9429i	−0.402404	+	1.62162i
$219$		0			0
$220$	14.0109	+	7.40986i	0.944611	+	0.499572i
$221$		−	15.1074i		−	1.01623i
$222$		0			0
$223$	−19.2548			−1.28939			−0.644697	−	0.764438i	$-0.723017\pi$
$223$	−19.2548			−1.28939			−0.644697	+	0.764438i	$0.723017\pi$
$224$	6.47143	+	17.4777i	0.432391	+	1.16778i
$225$		0			0
$226$	19.6545	+	4.87724i	1.30739	+	0.324429i
$227$			20.0414i			1.33019i	0.746758	+	0.665096i	$0.231609\pi$
$227$			20.0414i			1.33019i	−0.746758	+	0.665096i	$0.768391\pi$
$228$		0			0
$229$		−	18.2013i		−	1.20277i	−0.798958	−	0.601387i	$-0.794615\pi$
$229$		−	18.2013i		−	1.20277i	0.798958	−	0.601387i	$-0.205385\pi$
$230$	5.32744	−	21.4687i	0.351281	−	1.41560i
$231$		0			0
$232$	4.89981	−	5.47836i	0.321688	−	0.359672i
$233$	−14.2376			−0.932739			−0.466369	−	0.884590i	$-0.654438\pi$
$233$	−14.2376			−0.932739			−0.466369	+	0.884590i	$0.654438\pi$
$234$		0			0
$235$		−	11.6561i		−	0.760362i
$236$	−9.04293	+	17.0987i	−0.588645	+	1.11303i
$237$		0			0
$238$	29.3409	+	7.28093i	1.90189	+	0.471952i
$239$	−8.31368			−0.537767			−0.268884	−	0.963173i	$-0.586655\pi$
$239$	−8.31368			−0.537767			−0.268884	+	0.963173i	$0.586655\pi$
$240$		0			0
$241$	−26.1776			−1.68625			−0.843125	−	0.537718i	$-0.819287\pi$
$241$	−26.1776			−1.68625			−0.843125	+	0.537718i	$0.819287\pi$
$242$	−3.81534	−	0.946775i	−0.245260	−	0.0608610i
$243$		0			0
$244$	7.62575	−	14.4191i	0.488189	−	0.923087i
$245$			8.22920i			0.525744i
$246$		0			0
$247$	2.32843			0.148155
$248$	−2.82931	−	2.53051i	−0.179661	−	0.160688i
$249$		0			0
$250$	−3.95740	+	15.9477i	−0.250288	+	1.00862i
$251$		−	29.4740i		−	1.86038i	−0.367075	−	0.930191i	$-0.619641\pi$
$251$		−	29.4740i		−	1.86038i	0.367075	−	0.930191i	$-0.380359\pi$
$252$		0			0
$253$			27.1967i			1.70984i
$254$	3.62341	+	0.899147i	0.227353	+	0.0564175i
$255$		0			0
$256$	−5.86175	+	14.8876i	−0.366360	+	0.930473i
$257$	−1.28195			−0.0799657			−0.0399828	−	0.999200i	$-0.512730\pi$
$257$	−1.28195			−0.0799657			−0.0399828	+	0.999200i	$0.512730\pi$
$258$		0			0
$259$			12.2886i			0.763575i
$260$	8.78838	+	4.64787i	0.545032	+	0.288249i
$261$		0			0
$262$	5.23411	−	21.0926i	0.323364	−	1.30310i
$263$	29.2736			1.80509			0.902544	−	0.430597i	$-0.141697\pi$
$263$	29.2736			1.80509			0.902544	+	0.430597i	$0.141697\pi$
$264$		0			0
$265$	−10.6453			−0.653933
$266$	−1.12218	+	4.52218i	−0.0688050	+	0.277272i
$267$		0			0
$268$	−0.490873	+	0.928161i	−0.0299848	+	0.0566965i
$269$		−	14.1850i		−	0.864875i	−0.901664	−	0.432437i	$-0.857654\pi$
$269$		−	14.1850i		−	0.864875i	0.901664	−	0.432437i	$-0.142346\pi$
$270$		0			0
$271$	26.3185			1.59874			0.799369	−	0.600840i	$-0.205167\pi$
$271$	26.3185			1.59874			0.799369	+	0.600840i	$0.205167\pi$
$272$	14.6080	+	21.4513i	0.885743	+	1.30067i
$273$		0			0
$274$	6.75110	+	1.67528i	0.407849	+	0.101207i
$275$		−	1.64217i		−	0.0990269i
$276$		0			0
$277$		−	3.58863i		−	0.215620i	−0.994172	−	0.107810i	$-0.965616\pi$
$277$		−	3.58863i		−	0.215620i	0.994172	−	0.107810i	$-0.0343838\pi$
$278$	3.33922	−	13.4565i	0.200273	−	0.807068i
$279$		0			0
$280$	−13.2624	+	14.8284i	−0.792580	+	0.886165i
$281$	4.42597			0.264031			0.132016	−	0.991248i	$-0.457855\pi$
$281$	4.42597			0.264031			0.132016	+	0.991248i	$0.457855\pi$
$282$		0			0
$283$		−	0.493151i		−	0.0293148i	−0.999893	−	0.0146574i	$-0.995334\pi$
$283$		−	0.493151i		−	0.0293148i	0.999893	−	0.0146574i	$-0.00466576\pi$
$284$	−12.6850	−	6.70867i	−0.752718	−	0.398086i
$285$		0			0
$286$	−11.8638	−	2.94399i	−0.701519	−	0.174082i
$287$	−21.4938			−1.26874
$288$		0			0
$289$	25.0970			1.47630
$290$	7.61452	+	1.88954i	0.447140	+	0.110957i
$291$		0			0
$292$	−11.1936	−	5.91989i	−0.655054	−	0.346436i
$293$			12.7228i			0.743275i	0.928378	+	0.371637i	$0.121204\pi$
$293$			12.7228i			0.743275i	−0.928378	+	0.371637i	$0.878796\pi$
$294$		0			0
$295$	−20.6470			−1.20211
$296$	−7.03295	+	7.86338i	−0.408782	+	0.457050i
$297$		0			0
$298$	0.246649	−	0.993952i	0.0142880	−	0.0575781i
$299$			17.0593i			0.986565i
$300$		0			0
$301$			6.49709i			0.374486i
$302$	−21.5842	−	5.35610i	−1.24203	−	0.308209i
$303$		0			0
$304$	−3.30619	+	2.25147i	−0.189623	+	0.129131i
$305$	17.4113			0.996966
$306$		0			0
$307$			10.2166i			0.583092i	0.956557	+	0.291546i	$0.0941697\pi$
$307$			10.2166i			0.583092i	−0.956557	+	0.291546i	$0.905830\pi$
$308$	11.4354	−	21.6224i	0.651590	−	1.23205i
$309$		0			0
$310$	0.975855	−	3.93253i	0.0554248	−	0.223353i
$311$	−0.184934			−0.0104867			−0.00524333	−	0.999986i	$-0.501669\pi$
$311$	−0.184934			−0.0104867			−0.00524333	+	0.999986i	$0.501669\pi$
$312$		0			0
$313$	−19.0733			−1.07809			−0.539044	−	0.842278i	$-0.681214\pi$
$313$	−19.0733			−1.07809			−0.539044	+	0.842278i	$0.681214\pi$
$314$	0.0480689	−	0.193709i	0.00271268	−	0.0109317i
$315$		0			0
$316$	−15.4812	−	8.18745i	−0.870883	−	0.460580i
$317$			0.911635i			0.0512025i	0.999672	+	0.0256013i	$0.00815002\pi$
$317$			0.911635i			0.0512025i	−0.999672	+	0.0256013i	$0.991850\pi$
$318$		0			0
$319$	−9.64615			−0.540081
$320$	−16.9730	+	1.89830i	−0.948821	+	0.106118i
$321$		0			0
$322$	−33.1318	−	8.22164i	−1.84636	−	0.458174i
$323$			6.48822i			0.361014i
$324$		0			0
$325$		−	1.03006i		−	0.0571376i
$326$	2.86742	−	11.5552i	0.158812	−	0.639984i
$327$		0			0
$328$	−13.7537	−	12.3012i	−0.759422	−	0.679221i
$329$	−17.9885			−0.991736
$330$		0			0
$331$		−	2.45230i		−	0.134791i	−0.997726	−	0.0673953i	$-0.978531\pi$
$331$		−	2.45230i		−	0.134791i	0.997726	−	0.0673953i	$-0.0214689\pi$
$332$	7.23714	−	13.6843i	0.397190	−	0.751022i
$333$		0			0
$334$	−7.08992	−	1.75936i	−0.387943	−	0.0962678i
$335$	−1.12077			−0.0612342
$336$		0			0
$337$	35.1467			1.91456			0.957282	−	0.289156i	$-0.0933746\pi$
$337$	35.1467			1.91456			0.957282	+	0.289156i	$0.0933746\pi$
$338$	10.4020	+	2.58125i	0.565793	+	0.140401i
$339$		0			0
$340$	−12.9514	+	24.4890i	−0.702387	+	1.32810i
$341$			4.98177i			0.269778i
$342$		0			0
$343$	−10.3627			−0.559533
$344$	−3.71839	+	4.15744i	−0.200482	+	0.224154i
$345$		0			0
$346$	1.06946	−	4.30975i	0.0574946	−	0.231694i
$347$			5.30643i			0.284864i	0.989805	+	0.142432i	$0.0454922\pi$
$347$			5.30643i			0.284864i	−0.989805	+	0.142432i	$0.954508\pi$
$348$		0			0
$349$			26.8913i			1.43946i	0.694255	+	0.719729i	$0.255734\pi$
$349$			26.8913i			1.43946i	−0.694255	+	0.719729i	$0.744266\pi$
$350$	2.00054	+	0.496433i	0.106933	+	0.0265355i
$351$		0			0
$352$	19.6923	−	7.29140i	1.04960	−	0.388633i
$353$	3.21383			0.171055			0.0855275	−	0.996336i	$-0.472742\pi$
$353$	3.21383			0.171055			0.0855275	+	0.996336i	$0.472742\pi$
$354$		0			0
$355$		−	15.3174i		−	0.812961i
$356$	1.84520	+	0.975861i	0.0977952	+	0.0517205i
$357$		0			0
$358$	6.19557	−	24.9671i	0.327446	−	1.31955i
$359$	15.6827			0.827699			0.413850	−	0.910345i	$-0.364184\pi$
$359$	15.6827			0.827699			0.413850	+	0.910345i	$0.364184\pi$
$360$		0			0
$361$	−1.00000			−0.0526316
$362$	−4.79566	+	19.3257i	−0.252054	+	1.01574i
$363$		0			0
$364$	7.17289	−	13.5628i	0.375961	−	0.710883i
$365$		−	13.5164i		−	0.707481i
$366$		0			0
$367$	14.3509			0.749111			0.374555	−	0.927205i	$-0.377795\pi$
$367$	14.3509			0.749111			0.374555	+	0.927205i	$0.377795\pi$
$368$	−16.4954	−	24.2228i	−0.859884	−	1.26270i
$369$		0			0
$370$	−10.9295	−	2.71215i	−0.568198	−	0.140998i
$371$			16.4284i			0.852921i
$372$		0			0
$373$			9.08701i			0.470508i	0.971934	+	0.235254i	$0.0755921\pi$
$373$			9.08701i			0.470508i	−0.971934	+	0.235254i	$0.924408\pi$
$374$	8.20347	−	33.0586i	0.424191	−	1.70942i
$375$		0			0
$376$	−11.5107	−	10.2951i	−0.593619	−	0.530929i
$377$	−6.05060			−0.311622
$378$		0			0
$379$			18.2006i			0.934900i	0.884020	+	0.467450i	$0.154827\pi$
$379$			18.2006i			0.934900i	−0.884020	+	0.467450i	$0.845173\pi$
$380$	−3.77437	−	1.99614i	−0.193621	−	0.102400i
$381$		0			0
$382$	−6.69010	−	1.66015i	−0.342295	−	0.0849404i
$383$	22.4154			1.14537			0.572686	−	0.819775i	$-0.305901\pi$
$383$	22.4154			1.14537			0.572686	+	0.819775i	$0.305901\pi$
$384$		0			0
$385$	26.1094			1.33066
$386$	−13.7237	−	3.40553i	−0.698518	−	0.173337i
$387$		0			0
$388$	−0.207903	−	0.109953i	−0.0105547	−	0.00558200i
$389$			34.9213i			1.77058i	0.465038	+	0.885291i	$0.346041\pi$
$389$			34.9213i			1.77058i	−0.465038	+	0.885291i	$0.653959\pi$
$390$		0			0
$391$	−47.5360			−2.40400
$392$	8.12652	+	7.26831i	0.410451	+	0.367105i
$393$		0			0
$394$	−6.54676	+	26.3823i	−0.329821	+	1.32912i
$395$		−	18.6937i		−	0.940584i
$396$		0			0
$397$		−	31.3443i		−	1.57313i	−0.617510	−	0.786563i	$-0.711858\pi$
$397$		−	31.3443i		−	1.57313i	0.617510	−	0.786563i	$-0.288142\pi$
$398$	−16.1036	−	3.99610i	−0.807201	−	0.200307i
$399$		0			0
$400$	0.996016	+	1.46261i	0.0498008	+	0.0731303i
$401$	−1.64321			−0.0820579			−0.0410290	−	0.999158i	$-0.513064\pi$
$401$	−1.64321			−0.0820579			−0.0410290	+	0.999158i	$0.513064\pi$
$402$		0			0
$403$			3.12484i			0.155659i
$404$	9.92451	−	18.7657i	0.493763	−	0.933627i
$405$		0			0
$406$	2.91605	−	11.7512i	0.144721	−	0.583202i
$407$	13.8456			0.686302
$408$		0			0
$409$	−2.46645			−0.121958			−0.0609791	−	0.998139i	$-0.519422\pi$
$409$	−2.46645			−0.121958			−0.0609791	+	0.998139i	$0.519422\pi$
$410$	4.74378	−	19.1166i	0.234279	−	0.944104i
$411$		0			0
$412$	21.9565	+	11.6120i	1.08172	+	0.572083i
$413$			31.8637i			1.56791i
$414$		0			0
$415$	16.5240			0.811130
$416$	12.3521	−	4.57357i	0.605611	−	0.224238i
$417$		0			0
$418$	5.09517	+	1.26436i	0.249213	+	0.0618420i
$419$			4.25063i			0.207657i	0.994595	+	0.103828i	$0.0331093\pi$
$419$			4.25063i			0.207657i	−0.994595	+	0.103828i	$0.966891\pi$
$420$		0			0
$421$		−	31.4900i		−	1.53473i	−0.641212	−	0.767364i	$-0.721568\pi$
$421$		−	31.4900i		−	1.53473i	0.641212	−	0.767364i	$-0.278432\pi$
$422$	0.136032	−	0.548187i	0.00662195	−	0.0266853i
$423$		0			0
$424$	−9.40224	+	10.5124i	−0.456613	+	0.510529i
$425$	2.87029			0.139229
$426$		0			0
$427$		−	26.8701i		−	1.30034i
$428$	15.4689	−	29.2491i	0.747715	−	1.41381i
$429$		0			0
$430$	−5.77854	−	1.43394i	−0.278666	−	0.0691508i
$431$	21.0564			1.01425			0.507125	−	0.861872i	$-0.330708\pi$
$431$	21.0564			1.01425			0.507125	+	0.861872i	$0.330708\pi$
$432$		0			0
$433$	−7.53125			−0.361929			−0.180964	−	0.983490i	$-0.557922\pi$
$433$	−7.53125			−0.361929			−0.180964	+	0.983490i	$0.557922\pi$
$434$	−6.06892	−	1.50600i	−0.291318	−	0.0722903i
$435$		0			0
$436$	−16.3102	+	30.8400i	−0.781117	+	1.47697i
$437$		−	7.32651i		−	0.350475i
$438$		0			0
$439$	0.823995			0.0393271			0.0196636	−	0.999807i	$-0.493740\pi$
$439$	0.823995			0.0393271			0.0196636	+	0.999807i	$0.493740\pi$
$440$	16.7072	+	14.9428i	0.796486	+	0.712372i
$441$		0			0
$442$	5.14567	−	20.7362i	0.244755	−	0.986319i
$443$		−	8.58526i		−	0.407898i	−0.978982	−	0.203949i	$-0.934622\pi$
$443$		−	8.58526i		−	0.407898i	0.978982	−	0.203949i	$-0.0653777\pi$
$444$		0			0
$445$			2.22810i			0.105622i
$446$	−26.4288	−	6.55829i	−1.25144	−	0.310544i
$447$		0			0
$448$	2.92957	+	26.1938i	0.138409	+	1.23754i
$449$	−2.49451			−0.117723			−0.0588615	−	0.998266i	$-0.518747\pi$
$449$	−2.49451			−0.117723			−0.0588615	+	0.998266i	$0.518747\pi$
$450$		0			0
$451$			24.2172i			1.14034i
$452$	25.3162	+	13.3888i	1.19077	+	0.629759i
$453$		0			0
$454$	−6.82621	+	27.5085i	−0.320370	+	1.29104i
$455$	16.3773			0.767778
$456$		0			0
$457$	14.5004			0.678301			0.339150	−	0.940732i	$-0.389860\pi$
$457$	14.5004			0.678301			0.339150	+	0.940732i	$0.389860\pi$
$458$	6.19947	−	24.9828i	0.289682	−	1.16737i
$459$		0			0
$460$	14.6247	−	27.6530i	0.681881	−	1.28933i
$461$		−	23.1930i		−	1.08020i	−0.841600	−	0.540102i	$-0.818386\pi$
$461$		−	23.1930i		−	1.08020i	0.841600	−	0.540102i	$-0.181614\pi$
$462$		0			0
$463$	26.0637			1.21128			0.605642	−	0.795737i	$-0.292916\pi$
$463$	26.0637			1.21128			0.605642	+	0.795737i	$0.292916\pi$
$464$	8.59136	−	5.85061i	0.398844	−	0.271608i
$465$		0			0
$466$	−19.5424	−	4.84943i	−0.905282	−	0.224645i
$467$			12.1953i			0.564332i	0.959366	+	0.282166i	$0.0910529\pi$
$467$			12.1953i			0.564332i	−0.959366	+	0.282166i	$0.908947\pi$
$468$		0			0
$469$			1.72964i			0.0798674i
$470$	3.97015	−	15.9990i	0.183129	−	0.737980i
$471$		0			0
$472$	−18.2361	+	20.3894i	−0.839385	+	0.938497i
$473$	7.32031			0.336588
$474$		0			0
$475$			0.442384i			0.0202980i
$476$	37.7929	+	19.9874i	1.73224	+	0.916120i
$477$		0			0
$478$	−11.4112	−	2.83169i	−0.521937	−	0.129518i
$479$	−18.1883			−0.831045			−0.415523	−	0.909583i	$-0.636401\pi$
$479$	−18.1883			−0.831045			−0.415523	+	0.909583i	$0.636401\pi$
$480$		0			0
$481$	8.68475			0.395990
$482$	−35.9310	−	8.91626i	−1.63661	−	0.406125i
$483$		0			0
$484$	−4.91440	−	2.59906i	−0.223382	−	0.118139i
$485$		−	0.251046i		−	0.0113994i
$486$		0			0
$487$	−2.13693			−0.0968334			−0.0484167	−	0.998827i	$-0.515418\pi$
$487$	−2.13693			−0.0968334			−0.0484167	+	0.998827i	$0.515418\pi$
$488$	15.3782	−	17.1940i	0.696139	−	0.778337i
$489$		0			0
$490$	−2.80291	+	11.2953i	−0.126623	+	0.510268i
$491$		−	2.32773i		−	0.105049i	−0.998620	−	0.0525244i	$-0.983273\pi$
$491$		−	2.32773i		−	0.105049i	0.998620	−	0.0525244i	$-0.0167267\pi$
$492$		0			0
$493$		−	16.8601i		−	0.759341i
$494$	3.19597	+	0.793079i	0.143794	+	0.0356823i
$495$		0			0
$496$	−3.02156	−	4.43702i	−0.135672	−	0.199228i
$497$	−23.6387			−1.06034
$498$		0			0
$499$		−	16.9382i		−	0.758260i	−0.925343	−	0.379130i	$-0.876223\pi$
$499$		−	16.9382i		−	0.758260i	0.925343	−	0.379130i	$-0.123777\pi$
$500$	−10.8637	+	20.5416i	−0.485841	+	0.918648i
$501$		0			0
$502$	10.0390	−	40.4555i	0.448063	−	1.80562i
$503$	−8.22413			−0.366696			−0.183348	−	0.983048i	$-0.558693\pi$
$503$	−8.22413			−0.366696			−0.183348	+	0.983048i	$0.558693\pi$
$504$		0			0
$505$	22.6598			1.00835
$506$	−9.26337	+	37.3298i	−0.411807	+	1.65951i
$507$		0			0
$508$	4.66718	+	2.46831i	0.207073	+	0.109514i
$509$			19.8145i			0.878261i	0.898423	+	0.439131i	$0.144713\pi$
$509$			19.8145i			0.878261i	−0.898423	+	0.439131i	$0.855287\pi$
$510$		0			0
$511$	−20.8594			−0.922764
$512$	−13.1165	+	18.4379i	−0.579675	+	0.814848i
$513$		0			0
$514$	−1.75958	−	0.436639i	−0.0776118	−	0.0192593i
$515$			26.5128i			1.16829i
$516$		0			0
$517$			20.2677i			0.891374i
$518$	−4.18556	+	16.8671i	−0.183903	+	0.741098i
$519$		0			0
$520$	10.4797	+	9.37297i	0.459565	+	0.411032i
$521$	36.0978			1.58147			0.790737	−	0.612156i	$-0.209698\pi$
$521$	36.0978			1.58147			0.790737	+	0.612156i	$0.209698\pi$
$522$		0			0
$523$		−	4.19475i		−	0.183424i	−0.995786	−	0.0917119i	$-0.970766\pi$
$523$		−	4.19475i		−	0.183424i	0.995786	−	0.0917119i	$-0.0292339\pi$
$524$	14.3685	−	27.1686i	0.627691	−	1.18686i
$525$		0			0
$526$	40.1805	+	9.97077i	1.75195	+	0.434746i
$527$	−8.70743			−0.379301
$528$		0			0
$529$	30.6778			1.33382
$530$	−14.6115	−	3.62584i	−0.634683	−	0.157496i
$531$		0			0
$532$	−3.08056	+	5.82485i	−0.133559	+	0.252539i
$533$			15.1903i			0.657967i
$534$		0			0
$535$	35.3188			1.52696
$536$	−0.989901	+	1.10679i	−0.0427572	+	0.0478059i
$537$		0			0
$538$	4.83150	−	19.4701i	0.208301	−	0.839416i
$539$		−	14.3090i		−	0.616331i
$540$		0			0
$541$			6.12211i			0.263210i	0.991302	+	0.131605i	$0.0420131\pi$
$541$			6.12211i			0.263210i	−0.991302	+	0.131605i	$0.957987\pi$
$542$	36.1244	+	8.96426i	1.55168	+	0.385048i
$543$		0			0
$544$	12.7443	+	34.4193i	0.546409	+	1.47571i
$545$	−37.2398			−1.59518
$546$		0			0
$547$			4.88879i			0.209029i	0.994523	+	0.104515i	$0.0333289\pi$
$547$			4.88879i			0.209029i	−0.994523	+	0.104515i	$0.966671\pi$
$548$	8.69584	+	4.59893i	0.371468	+	0.196457i
$549$		0			0
$550$	0.559335	−	2.25402i	0.0238501	−	0.0961119i
$551$	2.59857			0.110703
$552$		0			0
$553$	−28.8493			−1.22680
$554$	1.22231	−	4.92570i	0.0519310	−	0.209273i
$555$		0			0
$556$	9.16673	−	17.3328i	0.388756	−	0.735076i
$557$			28.5987i			1.21176i	0.795554	+	0.605882i	$0.207180\pi$
$557$			28.5987i			1.21176i	−0.795554	+	0.605882i	$0.792820\pi$
$558$		0			0
$559$	4.59171			0.194209
$560$	−23.2544	+	15.8359i	−0.982677	+	0.669191i
$561$		0			0
$562$	6.07501	+	1.50751i	0.256259	+	0.0635905i
$563$			9.83345i			0.414431i	0.978295	+	0.207215i	$0.0664401\pi$
$563$			9.83345i			0.414431i	−0.978295	+	0.207215i	$0.933560\pi$
$564$		0			0
$565$			30.5697i			1.28608i
$566$	0.167970	−	0.676891i	0.00706032	−	0.0284519i
$567$		0			0
$568$	−15.1263	−	13.5288i	−0.634683	−	0.567656i
$569$	10.3543			0.434075			0.217037	−	0.976163i	$-0.430361\pi$
$569$	10.3543			0.434075			0.217037	+	0.976163i	$0.430361\pi$
$570$		0			0
$571$			36.7988i			1.53998i	0.638055	+	0.769991i	$0.279739\pi$
$571$			36.7988i			1.53998i	−0.638055	+	0.769991i	$0.720261\pi$
$572$	−15.2813	−	8.08174i	−0.638942	−	0.337915i
$573$		0			0
$574$	−29.5020	−	7.32090i	−1.23139	−	0.305569i
$575$	−3.24113			−0.135165
$576$		0			0
$577$	−8.22871			−0.342566			−0.171283	−	0.985222i	$-0.554791\pi$
$577$	−8.22871			−0.342566			−0.171283	+	0.985222i	$0.554791\pi$
$578$	34.4478	+	8.54820i	1.43284	+	0.355558i
$579$		0			0
$580$	9.80798	+	5.18710i	0.407254	+	0.215383i
$581$		−	25.5008i		−	1.05795i
$582$		0			0
$583$	18.5100			0.766606
$584$	−13.3478	−	11.9382i	−0.552335	−	0.494004i
$585$		0			0
$586$	−4.33347	+	17.4631i	−0.179014	+	0.721395i
$587$		−	7.18047i		−	0.296370i	−0.988960	−	0.148185i	$-0.952657\pi$
$587$		−	7.18047i		−	0.296370i	0.988960	−	0.148185i	$-0.0473431\pi$
$588$		0			0
$589$		−	1.34204i		−	0.0552976i
$590$	−28.3397	−	7.03249i	−1.16673	−	0.289523i
$591$		0			0
$592$	−12.3316	+	8.39768i	−0.506827	+	0.345143i
$593$	24.7687			1.01713			0.508564	−	0.861024i	$-0.330177\pi$
$593$	24.7687			1.01713			0.508564	+	0.861024i	$0.330177\pi$
$594$		0			0
$595$			45.6355i			1.87087i
$596$	0.677092	−	1.28027i	0.0277348	−	0.0524420i
$597$		0			0
$598$	−5.81050	+	23.4153i	−0.237609	+	0.957524i
$599$	−41.9307			−1.71324			−0.856621	−	0.515946i	$-0.827440\pi$
$599$	−41.9307			−1.71324			−0.856621	+	0.515946i	$0.827440\pi$
$600$		0			0
$601$	2.10027			0.0856718			0.0428359	−	0.999082i	$-0.486361\pi$
$601$	2.10027			0.0856718			0.0428359	+	0.999082i	$0.486361\pi$
$602$	−2.21295	+	8.91780i	−0.0901930	+	0.363462i
$603$		0			0
$604$	−27.8018	−	14.7034i	−1.13124	−	0.598273i
$605$		−	5.93422i		−	0.241260i
$606$		0			0
$607$	−11.2877			−0.458152			−0.229076	−	0.973409i	$-0.573570\pi$
$607$	−11.2877			−0.458152			−0.229076	+	0.973409i	$0.573570\pi$
$608$	−5.30489	+	1.96423i	−0.215142	+	0.0796599i
$609$		0			0
$610$	23.8984	+	5.93038i	0.967619	+	0.240114i
$611$			12.7130i			0.514315i
$612$		0			0
$613$			31.4983i			1.27220i	0.771605	+	0.636102i	$0.219454\pi$
$613$			31.4983i			1.27220i	−0.771605	+	0.636102i	$0.780546\pi$
$614$	−3.47984	+	14.0231i	−0.140435	+	0.565928i
$615$		0			0
$616$	23.0607	−	25.7836i	0.929142	−	1.03885i
$617$	11.4563			0.461214			0.230607	−	0.973047i	$-0.425929\pi$
$617$	11.4563			0.461214			0.230607	+	0.973047i	$0.425929\pi$
$618$		0			0
$619$		−	37.6816i		−	1.51455i	−0.653095	−	0.757276i	$-0.726530\pi$
$619$		−	37.6816i		−	1.51455i	0.653095	−	0.757276i	$-0.273470\pi$
$620$	2.67889	−	5.06534i	0.107587	−	0.203429i
$621$		0			0
$622$	−0.253838	−	0.0629898i	−0.0101780	−	0.00252566i
$623$	3.43855			0.137763
$624$		0			0
$625$	−22.5924			−0.903695
$626$	−26.1797	−	6.49649i	−1.04635	−	0.259652i
$627$		0			0
$628$	0.131957	−	0.249510i	0.00526566	−	0.00995653i
$629$			24.2002i			0.964925i
$630$		0			0
$631$	6.99670			0.278534			0.139267	−	0.990255i	$-0.455525\pi$
$631$	6.99670			0.278534			0.139267	+	0.990255i	$0.455525\pi$
$632$	−18.4605	−	16.5109i	−0.734319	−	0.656770i
$633$		0			0
$634$	−0.310508	+	1.25130i	−0.0123319	+	0.0496953i
$635$			5.63569i			0.223646i
$636$		0			0
$637$		−	8.97538i		−	0.355617i
$638$	−13.2402	−	3.28554i	−0.524183	−	0.130076i
$639$		0			0
$640$	−23.9435	−	3.17555i	−0.946450	−	0.125525i
$641$	6.20749			0.245181			0.122591	−	0.992457i	$-0.460880\pi$
$641$	6.20749			0.245181			0.122591	+	0.992457i	$0.460880\pi$
$642$		0			0
$643$		−	29.4008i		−	1.15945i	−0.814811	−	0.579726i	$-0.803159\pi$
$643$		−	29.4008i		−	1.15945i	0.814811	−	0.579726i	$-0.196841\pi$
$644$	−42.6758	−	22.5698i	−1.68166	−	0.889374i
$645$		0			0
$646$	−2.20993	+	8.90563i	−0.0869484	+	0.350387i
$647$	−5.62285			−0.221057			−0.110529	−	0.993873i	$-0.535254\pi$
$647$	−5.62285			−0.221057			−0.110529	+	0.993873i	$0.535254\pi$
$648$		0			0
$649$	35.9011			1.40924
$650$	0.350846	−	1.41385i	0.0137613	−	0.0554557i
$651$		0			0
$652$	7.87156	−	14.8839i	0.308274	−	0.582897i
$653$		−	34.9447i		−	1.36749i	−0.729721	−	0.683745i	$-0.760350\pi$
$653$		−	34.9447i		−	1.36749i	0.729721	−	0.683745i	$-0.239650\pi$
$654$		0			0
$655$	32.8064			1.28185
$656$	−14.6883	−	21.5691i	−0.573480	−	0.842130i
$657$		0			0
$658$	−24.6907	−	6.12698i	−0.962543	−	0.238855i
$659$			37.8768i			1.47547i	0.675090	+	0.737736i	$0.264105\pi$
$659$			37.8768i			1.47547i	−0.675090	+	0.737736i	$0.735895\pi$
$660$		0			0
$661$			6.77264i			0.263425i	0.991288	+	0.131713i	$0.0420476\pi$
$661$			6.77264i			0.263425i	−0.991288	+	0.131713i	$0.957952\pi$
$662$	0.835268	−	3.36599i	0.0324636	−	0.130823i
$663$		0			0
$664$	14.5945	−	16.3178i	0.566378	−	0.633254i
$665$	−7.03360			−0.272751
$666$		0			0
$667$			19.0385i			0.737172i
$668$	−9.13226	−	4.82974i	−0.353338	−	0.186868i
$669$		0			0
$670$	−1.53835	−	0.381741i	−0.0594317	−	0.0147479i
$671$	−30.2748			−1.16874
$672$		0			0
$673$	24.6355			0.949630			0.474815	−	0.880086i	$-0.342515\pi$
$673$	24.6355			0.949630			0.474815	+	0.880086i	$0.342515\pi$
$674$	48.2418	+	11.9712i	1.85821	+	0.461113i
$675$		0			0
$676$	13.3984	+	7.08596i	0.515323	+	0.272537i
$677$		−	28.2956i		−	1.08749i	−0.839251	−	0.543744i	$-0.817006\pi$
$677$		−	28.2956i		−	1.08749i	0.839251	−	0.543744i	$-0.182994\pi$
$678$		0			0
$679$	−0.387430			−0.0148682
$680$	−26.1179	+	29.2019i	−1.00158	+	1.11984i
$681$		0			0
$682$	−1.69682	+	6.83790i	−0.0649746	+	0.261837i
$683$			34.4333i			1.31755i	0.752339	+	0.658777i	$0.228926\pi$
$683$			34.4333i			1.31755i	−0.752339	+	0.658777i	$0.771074\pi$
$684$		0			0
$685$			10.5004i			0.401198i
$686$	−14.2237	−	3.52960i	−0.543063	−	0.134761i
$687$		0			0
$688$	−6.51985	+	4.43994i	−0.248567	+	0.169271i
$689$	11.6105			0.442325
$690$		0			0
$691$		−	2.19928i		−	0.0836646i	−0.999125	−	0.0418323i	$-0.986680\pi$
$691$		−	2.19928i		−	0.0836646i	0.999125	−	0.0418323i	$-0.0133195\pi$
$692$	2.93585	−	5.55123i	0.111604	−	0.211026i
$693$		0			0
$694$	−1.80740	+	7.28352i	−0.0686080	+	0.276479i
$695$	20.9297			0.793907
$696$		0			0
$697$	−42.3282			−1.60329
$698$	−9.15934	+	36.9106i	−0.346686	+	1.39709i
$699$		0			0
$700$	2.57682	+	1.36279i	0.0973947	+	0.0515087i
$701$			36.0608i			1.36200i	0.732285	+	0.680999i	$0.238454\pi$
$701$			36.0608i			1.36200i	−0.732285	+	0.680999i	$0.761546\pi$
$702$		0			0
$703$	−3.72986			−0.140675
$704$	29.5128	−	3.30076i	1.11230	−	0.124402i
$705$		0			0
$706$	4.41126	+	1.09465i	0.166020	+	0.0411977i
$707$		−	34.9701i		−	1.31518i
$708$		0			0
$709$			10.7299i			0.402970i	0.979492	+	0.201485i	$0.0645768\pi$
$709$			10.7299i			0.402970i	−0.979492	+	0.201485i	$0.935423\pi$
$710$	5.21719	−	21.0244i	0.195798	−	0.789031i
$711$		0			0
$712$	2.20030	+	1.96794i	0.0824599	+	0.0737515i
$713$	9.83244			0.368228
$714$		0			0
$715$		−	18.4524i		−	0.690080i
$716$	17.0079	−	32.1592i	0.635614	−	1.20185i
$717$		0			0
$718$	21.5258	+	5.34161i	0.803335	+	0.199347i
$719$	−29.1037			−1.08539			−0.542693	−	0.839931i	$-0.682595\pi$
$719$	−29.1037			−1.08539			−0.542693	+	0.839931i	$0.682595\pi$
$720$		0			0
$721$	40.9162			1.52380
$722$	−1.37258	−	0.340606i	−0.0510823	−	0.0126760i
$723$		0			0
$724$	−13.1649	+	24.8927i	−0.489269	+	0.925130i
$725$		−	1.14957i		−	0.0426938i
$726$		0			0
$727$	−40.5850			−1.50521			−0.752607	−	0.658470i	$-0.771204\pi$
$727$	−40.5850			−1.50521			−0.752607	+	0.658470i	$0.771204\pi$
$728$	14.4650	−	16.1729i	0.536107	−	0.599409i
$729$		0			0
$730$	4.60377	−	18.5524i	0.170393	−	0.686656i
$731$			12.7949i			0.473235i
$732$		0			0
$733$		−	18.6206i		−	0.687766i	−0.939012	−	0.343883i	$-0.888258\pi$
$733$		−	18.6206i		−	0.687766i	0.939012	−	0.343883i	$-0.111742\pi$
$734$	19.6978	+	4.88800i	0.727059	+	0.180419i
$735$		0			0
$736$	−14.3909	−	38.8663i	−0.530457	−	1.43263i
$737$	1.94880			0.0717849
$738$		0			0
$739$		−	7.89939i		−	0.290584i	−0.989389	−	0.145292i	$-0.953588\pi$
$739$		−	7.89939i		−	0.290584i	0.989389	−	0.145292i	$-0.0464121\pi$
$740$	−14.0779	−	7.44532i	−0.517514	−	0.273695i
$741$		0			0
$742$	−5.59562	+	22.5494i	−0.205422	+	0.827814i
$743$	14.4064			0.528521			0.264261	−	0.964451i	$-0.414872\pi$
$743$	14.4064			0.528521			0.264261	+	0.964451i	$0.414872\pi$
$744$		0			0
$745$	1.54595			0.0566392
$746$	−3.09509	+	12.4727i	−0.113319	+	0.456658i
$747$		0			0
$748$	22.5199	−	42.5815i	0.823409	−	1.55694i
$749$		−	54.5061i		−	1.99161i
$750$		0			0
$751$	14.0914			0.514204			0.257102	−	0.966384i	$-0.417232\pi$
$751$	14.0914			0.514204			0.257102	+	0.966384i	$0.417232\pi$
$752$	−12.2928	−	18.0515i	−0.448273	−	0.658270i
$753$		0			0
$754$	−8.30496	−	2.06087i	−0.302449	−	0.0750525i
$755$		−	33.5711i		−	1.22178i
$756$		0			0
$757$		−	50.2311i		−	1.82568i	−0.408318	−	0.912840i	$-0.633884\pi$
$757$		−	50.2311i		−	1.82568i	0.408318	−	0.912840i	$-0.366116\pi$
$758$	−6.19922	+	24.9818i	−0.225166	+	0.907380i
$759$		0			0
$760$	−4.50075	−	4.02544i	−0.163259	−	0.146018i
$761$	−23.7483			−0.860875			−0.430437	−	0.902620i	$-0.641641\pi$
$761$	−23.7483			−0.860875			−0.430437	+	0.902620i	$0.641641\pi$
$762$		0			0
$763$			57.4707i			2.08058i
$764$	−8.61727	−	4.55738i	−0.311762	−	0.164880i
$765$		0			0
$766$	30.7670	+	7.63481i	1.11166	+	0.275857i
$767$	22.5191			0.813119
$768$		0			0
$769$	−47.4035			−1.70942			−0.854708	−	0.519109i	$-0.826264\pi$
$769$	−47.4035			−1.70942			−0.854708	+	0.519109i	$0.826264\pi$
$770$	35.8374	+	8.89302i	1.29149	+	0.320482i
$771$		0			0
$772$	−17.6770	−	9.34876i	−0.636209	−	0.336469i
$773$			11.8508i			0.426242i	0.977026	+	0.213121i	$0.0683628\pi$
$773$			11.8508i			0.426242i	−0.977026	+	0.213121i	$0.931637\pi$
$774$		0			0
$775$	−0.593696			−0.0213262
$776$	−0.247914	−	0.221732i	−0.00889958	−	0.00795972i
$777$		0			0
$778$	−11.8944	+	47.9325i	−0.426435	+	1.71846i
$779$		−	6.52385i		−	0.233741i
$780$		0			0
$781$			26.6339i			0.953036i
$782$	−65.2472	−	16.1911i	−2.33324	−	0.578991i
$783$		0			0
$784$	8.67871	+	12.7443i	0.309954	+	0.455154i
$785$	0.301287			0.0107534
$786$		0			0
$787$		−	26.3903i		−	0.940713i	−0.882477	−	0.470356i	$-0.844125\pi$
$787$		−	26.3903i		−	0.940713i	0.882477	−	0.470356i	$-0.155875\pi$
$788$	−17.9719	+	33.9821i	−0.640224	+	1.21056i
$789$		0			0
$790$	6.36720	−	25.6587i	0.226535	−	0.912897i
$791$	47.1770			1.67742
$792$		0			0
$793$	−18.9900			−0.674356
$794$	10.6761	−	43.0227i	0.378879	−	1.52682i
$795$		0			0
$796$	−20.7425	−	10.9700i	−0.735197	−	0.388821i
$797$			14.1893i			0.502610i	0.967908	+	0.251305i	$0.0808597\pi$
$797$			14.1893i			0.502610i	−0.967908	+	0.251305i	$0.919140\pi$
$798$		0			0
$799$	−35.4251			−1.25325
$800$	0.868943	+	2.34680i	0.0307218	+	0.0829719i
$801$		0			0
$802$	−2.25544	−	0.559687i	−0.0796424	−	0.0197632i
$803$			23.5024i			0.829382i
$804$		0			0
$805$		−	51.5317i		−	1.81625i
$806$	−1.06434	+	4.28911i	−0.0374898	+	0.151077i
$807$		0			0
$808$	20.0139	−	22.3771i	0.704087	−	0.787224i
$809$	29.9998			1.05474			0.527368	−	0.849637i	$-0.323179\pi$
$809$	29.9998			1.05474			0.527368	+	0.849637i	$0.323179\pi$
$810$		0			0
$811$			39.8833i			1.40049i	0.713901	+	0.700246i	$0.246927\pi$
$811$			39.8833i			1.40049i	−0.713901	+	0.700246i	$0.753073\pi$
$812$	8.00506	−	15.1363i	0.280922	−	0.531179i
$813$		0			0
$814$	19.0043	+	4.71590i	0.666100	+	0.165292i
$815$	17.9725			0.629549
$816$		0			0
$817$	−1.97202			−0.0689921
$818$	−3.38541	−	0.840089i	−0.118368	−	0.0293730i
$819$		0			0
$820$	13.0225	−	24.6234i	0.454765	−	0.859888i
$821$			56.4395i			1.96975i	0.173256	+	0.984877i	$0.444571\pi$
$821$			56.4395i			1.96975i	−0.173256	+	0.984877i	$0.555429\pi$
$822$		0			0
$823$	−41.7360			−1.45482			−0.727412	−	0.686201i	$-0.759277\pi$
$823$	−41.7360			−1.45482			−0.727412	+	0.686201i	$0.759277\pi$
$824$	26.1820	+	23.4170i	0.912093	+	0.815769i
$825$		0			0
$826$	−10.8530	+	43.7356i	−0.377623	+	1.52176i
$827$			17.4248i			0.605921i	0.953003	+	0.302960i	$0.0979750\pi$
$827$			17.4248i			0.605921i	−0.953003	+	0.302960i	$0.902025\pi$
$828$		0			0
$829$			52.2723i			1.81549i	0.419519	+	0.907747i	$0.362199\pi$
$829$			52.2723i			1.81549i	−0.419519	+	0.907747i	$0.637801\pi$
$830$	22.6805	+	5.62817i	0.787253	+	0.195356i
$831$		0			0
$832$	18.5121	−	2.07042i	0.641790	−	0.0717790i
$833$	25.0100			0.866547
$834$		0			0
$835$		−	11.0273i		−	0.381617i
$836$	6.56290	+	3.47089i	0.226983	+	0.120043i
$837$		0			0
$838$	−1.44779	+	5.83434i	−0.0500130	+	0.201544i
$839$	−40.8429			−1.41005			−0.705027	−	0.709180i	$-0.749065\pi$
$839$	−40.8429			−1.41005			−0.705027	+	0.709180i	$0.749065\pi$
$840$		0			0
$841$	22.2474			0.767153
$842$	10.7257	−	43.2226i	0.369631	−	1.48955i
$843$		0			0
$844$	0.373432	−	0.706100i	0.0128541	−	0.0243050i
$845$			16.1788i			0.556567i
$846$		0			0
$847$	−9.15805			−0.314674
$848$	−16.4860	+	11.2267i	−0.566131	+	0.385528i
$849$		0			0
$850$	3.93971	+	0.977638i	0.135131	+	0.0335327i
$851$		−	27.3269i		−	0.936754i
$852$		0			0
$853$			42.8153i			1.46597i	0.680245	+	0.732984i	$0.261873\pi$
$853$			42.8153i			1.46597i	−0.680245	+	0.732984i	$0.738127\pi$
$854$	9.15213	−	36.8815i	0.313180	−	1.26206i
$855$		0			0
$856$	31.1947	−	34.8781i	1.06621	−	1.19211i
$857$	−23.2947			−0.795732			−0.397866	−	0.917444i	$-0.630249\pi$
$857$	−23.2947			−0.795732			−0.397866	+	0.917444i	$0.630249\pi$
$858$		0			0
$859$		−	17.5879i		−	0.600091i	−0.953925	−	0.300045i	$-0.902998\pi$
$859$		−	17.5879i		−	0.600091i	0.953925	−	0.300045i	$-0.0970018\pi$
$860$	−7.44312	−	3.93641i	−0.253808	−	0.134230i
$861$		0			0
$862$	28.9017	+	7.17193i	0.984394	+	0.244277i
$863$	11.8886			0.404692			0.202346	−	0.979314i	$-0.435143\pi$
$863$	11.8886			0.404692			0.202346	+	0.979314i	$0.435143\pi$
$864$		0			0
$865$	6.70319			0.227915
$866$	−10.3373	−	2.56519i	−0.351275	−	0.0871687i
$867$		0			0
$868$	−7.81716	−	4.13422i	−0.265332	−	0.140325i
$869$			32.5048i			1.10265i
$870$		0			0
$871$	1.22239			0.0414193
$872$	−32.8914	+	36.7751i	−1.11384	+	1.24536i
$873$		0			0
$874$	2.49545	−	10.0563i	0.0844100	−	0.340158i
$875$			38.2795i			1.29408i
$876$		0			0
$877$		−	28.6698i		−	0.968111i	−0.875037	−	0.484056i	$-0.839163\pi$
$877$		−	28.6698i		−	0.968111i	0.875037	−	0.484056i	$-0.160837\pi$
$878$	1.13100	+	0.280658i	0.0381695	+	0.00947173i
$879$		0			0
$880$	17.8425	+	26.2009i	0.601469	+	0.883231i
$881$	44.2297			1.49014			0.745069	−	0.666988i	$-0.232417\pi$
$881$	44.2297			1.49014			0.745069	+	0.666988i	$0.232417\pi$
$882$		0			0
$883$			26.5869i			0.894719i	0.894354	+	0.447359i	$0.147636\pi$
$883$			26.5869i			0.894719i	−0.894354	+	0.447359i	$0.852364\pi$
$884$	14.1257	−	26.7095i	0.475100	−	0.898338i
$885$		0			0
$886$	2.92419	−	11.7840i	0.0982401	−	0.395891i
$887$	−16.4124			−0.551074			−0.275537	−	0.961290i	$-0.588856\pi$
$887$	−16.4124			−0.551074			−0.275537	+	0.961290i	$0.588856\pi$
$888$		0			0
$889$	8.69735			0.291700
$890$	−0.758906	+	3.05826i	−0.0254386	+	0.102513i
$891$		0			0
$892$	−34.0420	−	18.0036i	−1.13981	−	0.602806i
$893$		−	5.45991i		−	0.182709i
$894$		0			0
$895$	38.8327			1.29803
$896$	−4.90070	+	36.9511i	−0.163721	+	1.23445i
$897$		0			0
$898$	−3.42392	−	0.849644i	−0.114258	−	0.0283530i
$899$			3.48737i			0.116310i
$900$		0			0
$901$			32.3529i			1.07783i
$902$	−8.24851	+	33.2401i	−0.274645	+	1.10677i
$903$		0			0
$904$	30.1883	+	27.0002i	1.00405	+	0.898012i
$905$	−30.0583			−0.999172
$906$		0			0
$907$			47.3380i			1.57183i	0.618333	+	0.785916i	$0.287808\pi$
$907$			47.3380i			1.57183i	−0.618333	+	0.785916i	$0.712192\pi$
$908$	−18.7391	+	35.4326i	−0.621879	+	1.17587i
$909$		0			0
$910$	22.4792	+	5.57820i	0.745178	+	0.184915i
$911$	−5.30896			−0.175894			−0.0879469	−	0.996125i	$-0.528031\pi$
$911$	−5.30896			−0.175894			−0.0879469	+	0.996125i	$0.528031\pi$
$912$		0			0
$913$	−28.7320			−0.950889
$914$	19.9030	+	4.93893i	0.658334	+	0.163365i
$915$		0			0
$916$	17.0186	−	32.1794i	0.562310	−	1.06324i
$917$		−	50.6290i		−	1.67192i
$918$		0			0
$919$	49.2513			1.62465			0.812326	−	0.583204i	$-0.198201\pi$
$919$	49.2513			1.62465			0.812326	+	0.583204i	$0.198201\pi$
$920$	29.4924	−	32.9748i	0.972337	−	1.08715i
$921$		0			0
$922$	7.89966	−	31.8343i	0.260162	−	1.04841i
$923$			16.7063i			0.549893i
$924$		0			0
$925$			1.65003i			0.0542528i
$926$	35.7747	+	8.87746i	1.17563	+	0.291732i
$927$		0			0
$928$	13.7851	−	5.10418i	0.452519	−	0.167553i
$929$	29.2990			0.961270			0.480635	−	0.876921i	$-0.340406\pi$
$929$	29.2990			0.961270			0.480635	+	0.876921i	$0.340406\pi$
$930$		0			0
$931$			3.85468i			0.126332i
$932$	−25.1718	−	13.3125i	−0.824529	−	0.436065i
$933$		0			0
$934$	−4.15380	+	16.7391i	−0.135916	+	0.547720i
$935$	51.4179			1.68154
$936$		0			0
$937$	−6.35345			−0.207558			−0.103779	−	0.994600i	$-0.533093\pi$
$937$	−6.35345			−0.207558			−0.103779	+	0.994600i	$0.533093\pi$
$938$	−0.589126	+	2.37408i	−0.0192357	+	0.0775164i
$939$		0			0
$940$	10.8987	−	20.6077i	0.355477	−	0.672151i
$941$			2.07523i			0.0676505i	0.999428	+	0.0338253i	$0.0107690\pi$
$941$			2.07523i			0.0676505i	−0.999428	+	0.0338253i	$0.989231\pi$
$942$		0			0
$943$	47.7970			1.55649
$944$	−31.9753	+	21.7748i	−1.04071	+	0.708709i
$945$		0			0
$946$	10.0477	+	2.49334i	0.326680	+	0.0810656i
$947$			13.3618i			0.434201i	0.976149	+	0.217101i	$0.0696600\pi$
$947$			13.3618i			0.434201i	−0.976149	+	0.217101i	$0.930340\pi$
$948$		0			0
$949$			14.7420i			0.478546i
$950$	−0.150679	+	0.607210i	−0.00488867	+	0.0197005i
$951$		0			0
$952$	45.0661	+	40.3068i	1.46060	+	1.30635i
$953$	52.6796			1.70646			0.853231	−	0.521534i	$-0.174640\pi$
$953$	52.6796			1.70646			0.853231	+	0.521534i	$0.174640\pi$
$954$		0			0
$955$		−	10.4055i		−	0.336714i
$956$	−14.6984	−	7.77347i	−0.475380	−	0.251412i
$957$		0			0
$958$	−24.9650	−	6.19505i	−0.806582	−	0.200153i
$959$	16.2048			0.523281
$960$		0			0
$961$	−29.1989			−0.941901
$962$	11.9205	+	2.95808i	0.384334	+	0.0953722i
$963$		0			0
$964$	−46.2814	−	24.4766i	−1.49062	−	0.788339i
$965$		−	21.3453i		−	0.687128i
$966$		0			0
$967$	−7.17519			−0.230739			−0.115369	−	0.993323i	$-0.536805\pi$
$967$	−7.17519			−0.230739			−0.115369	+	0.993323i	$0.536805\pi$
$968$	−5.86018	−	5.24130i	−0.188353	−	0.168462i
$969$		0			0
$970$	0.0855077	−	0.344582i	0.00274549	−	0.0110638i
$971$			14.6991i			0.471718i	0.971787	+	0.235859i	$0.0757903\pi$
$971$			14.6991i			0.471718i	−0.971787	+	0.235859i	$0.924210\pi$
$972$		0			0
$973$		−	32.3000i		−	1.03549i
$974$	−2.93311	−	0.727851i	−0.0939830	−	0.0233218i
$975$		0			0
$976$	26.9643	−	18.3623i	0.863106	−	0.587764i
$977$	−19.5470			−0.625365			−0.312682	−	0.949858i	$-0.601228\pi$
$977$	−19.5470			−0.625365			−0.312682	+	0.949858i	$0.601228\pi$
$978$		0			0
$979$		−	3.87424i		−	0.123821i
$980$	−7.69447	+	14.5490i	−0.245791	+	0.464751i
$981$		0			0
$982$	0.792838	−	3.19500i	0.0253005	−	0.101957i
$983$	−24.0339			−0.766562			−0.383281	−	0.923632i	$-0.625206\pi$
$983$	−24.0339			−0.766562			−0.383281	+	0.923632i	$0.625206\pi$
$984$		0			0
$985$	−41.0339			−1.30745
$986$	5.74265	−	23.1419i	0.182883	−	0.736988i
$987$		0			0
$988$	4.11661	+	2.17714i	0.130967	+	0.0692639i
$989$		−	14.4480i		−	0.459419i
$990$		0			0
$991$	50.5419			1.60552			0.802758	−	0.596305i	$-0.203365\pi$
$991$	50.5419			1.60552			0.802758	+	0.596305i	$0.203365\pi$
$992$	−2.63606	−	7.11935i	−0.0836951	−	0.226039i
$993$		0			0
$994$	−32.4461	−	8.05149i	−1.02913	−	0.255378i
$995$		−	25.0469i		−	0.794039i
$996$		0			0
$997$			21.9225i			0.694293i	0.937811	+	0.347147i	$0.112849\pi$
$997$			21.9225i			0.694293i	−0.937811	+	0.347147i	$0.887151\pi$
$998$	5.76927	−	23.2492i	0.182623	−	0.735939i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.g.b.685.14		16
3.2	odd	2		152.2.c.b.77.3	✓	16
4.3	odd	2		5472.2.g.b.2737.13		16
8.3	odd	2		5472.2.g.b.2737.4		16
8.5	even	2	inner	1368.2.g.b.685.13		16
12.11	even	2		608.2.c.b.305.15		16
24.5	odd	2		152.2.c.b.77.4	yes	16
24.11	even	2		608.2.c.b.305.2		16
48.5	odd	4		4864.2.a.bq.1.1		8
48.11	even	4		4864.2.a.bp.1.8		8
48.29	odd	4		4864.2.a.bo.1.8		8
48.35	even	4		4864.2.a.bn.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.c.b.77.3	✓	16	3.2	odd	2
152.2.c.b.77.4	yes	16	24.5	odd	2
608.2.c.b.305.2		16	24.11	even	2
608.2.c.b.305.15		16	12.11	even	2
1368.2.g.b.685.13		16	8.5	even	2	inner
1368.2.g.b.685.14		16	1.1	even	1	trivial
4864.2.a.bn.1.1		8	48.35	even	4
4864.2.a.bo.1.8		8	48.29	odd	4
4864.2.a.bp.1.8		8	48.11	even	4
4864.2.a.bq.1.1		8	48.5	odd	4
5472.2.g.b.2737.4		16	8.3	odd	2
5472.2.g.b.2737.13		16	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1368 = 2^{3} \cdot 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1368.g (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(1.37258 + 0.340606i) q^{2} +(1.76798 + 0.935021i) q^{4} +2.13486i q^{5} +3.29464 q^{7} +(2.10822 + 1.88558i) q^{8} +(-0.727145 + 2.93027i) q^{10} -3.71210i q^{11} -2.32843i q^{13} +(4.52218 + 1.12218i) q^{14} +(2.25147 + 3.30619i) q^{16} +6.48822 q^{17} +1.00000i q^{19} +(-1.99614 + 3.77437i) q^{20} +(1.26436 - 5.09517i) q^{22} -7.32651 q^{23} +0.442384 q^{25} +(0.793079 - 3.19597i) q^{26} +(5.82485 + 3.08056i) q^{28} -2.59857i q^{29} -1.34204 q^{31} +(1.96423 + 5.30489i) q^{32} +(8.90563 + 2.20993i) q^{34} +7.03360i q^{35} +3.72986i q^{37} +(-0.340606 + 1.37258i) q^{38} +(-4.02544 + 4.50075i) q^{40} -6.52385 q^{41} +1.97202i q^{43} +(3.47089 - 6.56290i) q^{44} +(-10.0563 - 2.49545i) q^{46} -5.45991 q^{47} +3.85468 q^{49} +(0.607210 + 0.150679i) q^{50} +(2.17714 - 4.11661i) q^{52} +4.98640i q^{53} +7.92480 q^{55} +(6.94584 + 6.21231i) q^{56} +(0.885089 - 3.56676i) q^{58} +9.67136i q^{59} -8.15570i q^{61} +(-1.84206 - 0.457105i) q^{62} +(0.889191 + 7.95043i) q^{64} +4.97088 q^{65} +0.524986i q^{67} +(11.4710 + 6.06662i) q^{68} +(-2.39569 + 9.65420i) q^{70} -7.17489 q^{71} -6.33130 q^{73} +(-1.27041 + 5.11955i) q^{74} +(-0.935021 + 1.76798i) q^{76} -12.2300i q^{77} -8.75644 q^{79} +(-7.05824 + 4.80657i) q^{80} +(-8.95453 - 2.22206i) q^{82} -7.74008i q^{83} +13.8514i q^{85} +(-0.671680 + 2.70676i) q^{86} +(6.99945 - 7.82592i) q^{88} +1.04368 q^{89} -7.67136i q^{91} +(-12.9531 - 6.85044i) q^{92} +(-7.49419 - 1.85968i) q^{94} -2.13486 q^{95} -0.117594 q^{97} +(5.29088 + 1.31293i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 2 q^{4} - 8 q^{7} + 12 q^{8} - 8 q^{10} - 4 q^{14} + 2 q^{16} + 8 q^{17} - 8 q^{20} + 20 q^{22} - 24 q^{25} + 10 q^{26} - 14 q^{28} + 16 q^{31} + 20 q^{32} - 2 q^{38} + 28 q^{40} - 16 q^{41} + 28 q^{44}+ \cdots + 48 q^{98}+O(q^{100}) \) 16 * q + 2 * q^4 - 8 * q^7 + 12 * q^8 - 8 * q^10 - 4 * q^14 + 2 * q^16 + 8 * q^17 - 8 * q^20 + 20 * q^22 - 24 * q^25 + 10 * q^26 - 14 * q^28 + 16 * q^31 + 20 * q^32 - 2 * q^38 + 28 * q^40 - 16 * q^41 + 28 * q^44 - 48 * q^46 - 24 * q^47 + 24 * q^49 - 12 * q^50 + 8 * q^52 + 16 * q^55 + 48 * q^56 + 38 * q^58 + 16 * q^62 + 14 * q^64 - 16 * q^65 + 26 * q^68 - 32 * q^70 - 48 * q^71 + 20 * q^74 - 4 * q^76 - 48 * q^79 - 4 * q^80 - 12 * q^82 - 48 * q^86 + 40 * q^88 + 16 * q^89 - 62 * q^92 - 36 * q^94 - 16 * q^95 + 32 * q^97 + 48 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more