LMFDB - Embedded newform 187.2.d.a.67.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [187,2,Mod(67,187)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(187, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

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

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

chi := DirichletCharacter("187.67");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 187 = 11 \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	187.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$1.49320251780$
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} + 21x^{14} + 172x^{12} + 700x^{10} + 1492x^{8} + 1620x^{6} + 840x^{4} + 196x^{2} + 16 $ x^16 + 21x^14 + 172x^12 + 700x^10 + 1492x^8 + 1620x^6 + 840x^4 + 196*x^2 + 16
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			67.1
Root			$-2.61227i$ of defining polynomial
Character	$\chi$	$=$	187.67
Dual form			187.2.d.a.67.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.61227 q^{2} -1.70629i q^{3} +4.82394 q^{4} +0.846648i q^{5} +4.45730i q^{6} +3.51966i q^{7} -7.37688 q^{8} +0.0885594 q^{9} +O(q^{10})$	\(q-2.61227 q^{2} -1.70629i q^{3} +4.82394 q^{4} +0.846648i q^{5} +4.45730i q^{6} +3.51966i q^{7} -7.37688 q^{8} +0.0885594 q^{9} -2.21167i q^{10} +1.00000i q^{11} -8.23106i q^{12} +0.0408112 q^{13} -9.19429i q^{14} +1.44463 q^{15} +9.62251 q^{16} +(0.793539 + 4.04602i) q^{17} -0.231341 q^{18} +6.25620 q^{19} +4.08418i q^{20} +6.00558 q^{21} -2.61227i q^{22} +2.40934i q^{23} +12.5871i q^{24} +4.28319 q^{25} -0.106610 q^{26} -5.26999i q^{27} +16.9786i q^{28} -9.14108i q^{29} -3.77376 q^{30} +2.81641i q^{31} -10.3828 q^{32} +1.70629 q^{33} +(-2.07294 - 10.5693i) q^{34} -2.97992 q^{35} +0.427205 q^{36} +1.51824i q^{37} -16.3429 q^{38} -0.0696360i q^{39} -6.24562i q^{40} +2.95083i q^{41} -15.6882 q^{42} -8.15145 q^{43} +4.82394i q^{44} +0.0749787i q^{45} -6.29384i q^{46} -0.595765 q^{47} -16.4188i q^{48} -5.38801 q^{49} -11.1888 q^{50} +(6.90370 - 1.35401i) q^{51} +0.196871 q^{52} +13.0420 q^{53} +13.7666i q^{54} -0.846648 q^{55} -25.9641i q^{56} -10.6749i q^{57} +23.8789i q^{58} -6.48733 q^{59} +6.96881 q^{60} +6.01792i q^{61} -7.35721i q^{62} +0.311699i q^{63} +7.87761 q^{64} +0.0345528i q^{65} -4.45730 q^{66} -11.2311 q^{67} +(3.82798 + 19.5178i) q^{68} +4.11104 q^{69} +7.78433 q^{70} +16.6629i q^{71} -0.653292 q^{72} -6.52734i q^{73} -3.96605i q^{74} -7.30838i q^{75} +30.1795 q^{76} -3.51966 q^{77} +0.181908i q^{78} -3.93839i q^{79} +8.14688i q^{80} -8.72648 q^{81} -7.70836i q^{82} +5.74662 q^{83} +28.9705 q^{84} +(-3.42556 + 0.671849i) q^{85} +21.2938 q^{86} -15.5974 q^{87} -7.37688i q^{88} +13.0491 q^{89} -0.195864i q^{90} +0.143642i q^{91} +11.6225i q^{92} +4.80562 q^{93} +1.55630 q^{94} +5.29680i q^{95} +17.7161i q^{96} -11.6580i q^{97} +14.0749 q^{98} +0.0885594i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 2 q^{2} + 10 q^{4} - 6 q^{8} - 20 q^{9}+O(q^{10}) $ 16 * q - 2 * q^2 + 10 * q^4 - 6 * q^8 - 20 * q^9	$ 16 q - 2 q^{2} + 10 q^{4} - 6 q^{8} - 20 q^{9} - 16 q^{13} + 4 q^{15} + 6 q^{16} + 4 q^{17} - 10 q^{18} + 20 q^{19} + 12 q^{21} + 4 q^{25} - 12 q^{26} + 28 q^{30} - 34 q^{32} + 4 q^{33} - 6 q^{34} + 12 q^{35} - 18 q^{36} + 8 q^{43} + 14 q^{47} - 42 q^{49} - 34 q^{50} - 18 q^{51} - 44 q^{52} + 26 q^{53} + 8 q^{55} - 30 q^{59} + 72 q^{60} - 10 q^{64} - 8 q^{66} + 10 q^{67} + 22 q^{68} + 4 q^{69} - 8 q^{70} - 46 q^{72} + 36 q^{76} - 10 q^{77} - 8 q^{81} - 8 q^{83} + 92 q^{84} - 2 q^{85} + 56 q^{86} + 8 q^{87} + 10 q^{89} - 20 q^{93} + 8 q^{94} + 38 q^{98}+O(q^{100}) $ 16 * q - 2 * q^2 + 10 * q^4 - 6 * q^8 - 20 * q^9 - 16 * q^13 + 4 * q^15 + 6 * q^16 + 4 * q^17 - 10 * q^18 + 20 * q^19 + 12 * q^21 + 4 * q^25 - 12 * q^26 + 28 * q^30 - 34 * q^32 + 4 * q^33 - 6 * q^34 + 12 * q^35 - 18 * q^36 + 8 * q^43 + 14 * q^47 - 42 * q^49 - 34 * q^50 - 18 * q^51 - 44 * q^52 + 26 * q^53 + 8 * q^55 - 30 * q^59 + 72 * q^60 - 10 * q^64 - 8 * q^66 + 10 * q^67 + 22 * q^68 + 4 * q^69 - 8 * q^70 - 46 * q^72 + 36 * q^76 - 10 * q^77 - 8 * q^81 - 8 * q^83 + 92 * q^84 - 2 * q^85 + 56 * q^86 + 8 * q^87 + 10 * q^89 - 20 * q^93 + 8 * q^94 + 38 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/187\mathbb{Z}\right)^\times$.

$n$	$35$	$122$
$\chi(n)$	$1$	$-1$

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 $

$2$	−2.61227			−1.84715			−0.923576	−	0.383416i	$-0.874748\pi$
$2$	−2.61227			−1.84715			−0.923576	+	0.383416i	$0.874748\pi$
$3$		−	1.70629i		−	0.985130i	−0.870276	−	0.492565i	$-0.836059\pi$
$3$		−	1.70629i		−	0.985130i	0.870276	−	0.492565i	$-0.163941\pi$
$4$	4.82394			2.41197
$5$			0.846648i			0.378633i	0.981916	+	0.189316i	$0.0606272\pi$
$5$			0.846648i			0.378633i	−0.981916	+	0.189316i	$0.939373\pi$
$6$			4.45730i			1.81968i
$7$			3.51966i			1.33031i	0.746707	+	0.665153i	$0.231634\pi$
$7$			3.51966i			1.33031i	−0.746707	+	0.665153i	$0.768366\pi$
$8$	−7.37688			−2.60812
$9$	0.0885594			0.0295198
$10$		−	2.21167i		−	0.699392i
$11$			1.00000i			0.301511i
$11$			1.00000i			0.301511i
$12$		−	8.23106i		−	2.37610i
$13$	0.0408112			0.0113190			0.00565950	−	0.999984i	$-0.498199\pi$
$13$	0.0408112			0.0113190			0.00565950	+	0.999984i	$0.498199\pi$
$14$		−	9.19429i		−	2.45728i
$15$	1.44463			0.373002
$16$	9.62251			2.40563
$17$	0.793539	+	4.04602i	0.192462	+	0.981305i
$17$	0.793539	+	4.04602i	0.192462	+	0.981305i
$18$	−0.231341			−0.0545276
$19$	6.25620			1.43527			0.717635	−	0.696419i	$-0.245225\pi$
$19$	6.25620			1.43527			0.717635	+	0.696419i	$0.245225\pi$
$20$			4.08418i			0.913250i
$21$	6.00558			1.31052
$22$		−	2.61227i		−	0.556937i
$23$			2.40934i			0.502382i	0.967938	+	0.251191i	$0.0808222\pi$
$23$			2.40934i			0.502382i	−0.967938	+	0.251191i	$0.919178\pi$
$24$			12.5871i			2.56934i
$25$	4.28319			0.856637
$26$	−0.106610			−0.0209079
$27$		−	5.26999i		−	1.01421i
$28$			16.9786i			3.20866i
$29$		−	9.14108i		−	1.69746i	−0.528830	−	0.848728i	$-0.677369\pi$
$29$		−	9.14108i		−	1.69746i	0.528830	−	0.848728i	$-0.322631\pi$
$30$	−3.77376			−0.688992
$31$			2.81641i			0.505842i	0.967487	+	0.252921i	$0.0813913\pi$
$31$			2.81641i			0.505842i	−0.967487	+	0.252921i	$0.918609\pi$
$32$	−10.3828			−1.83544
$33$	1.70629			0.297028
$34$	−2.07294	−	10.5693i	−0.355506	−	1.81262i
$35$	−2.97992			−0.503698
$36$	0.427205			0.0712009
$37$			1.51824i			0.249597i	0.992182	+	0.124799i	$0.0398284\pi$
$37$			1.51824i			0.249597i	−0.992182	+	0.124799i	$0.960172\pi$
$38$	−16.3429			−2.65116
$39$		−	0.0696360i		−	0.0111507i
$40$		−	6.24562i		−	0.987520i
$41$			2.95083i			0.460842i	0.973091	+	0.230421i	$0.0740104\pi$
$41$			2.95083i			0.460842i	−0.973091	+	0.230421i	$0.925990\pi$
$42$	−15.6882			−2.42074
$43$	−8.15145			−1.24308			−0.621542	−	0.783381i	$-0.713494\pi$
$43$	−8.15145			−1.24308			−0.621542	+	0.783381i	$0.713494\pi$
$44$			4.82394i			0.727236i
$45$			0.0749787i			0.0111772i
$46$		−	6.29384i		−	0.927976i
$47$	−0.595765			−0.0869012			−0.0434506	−	0.999056i	$-0.513835\pi$
$47$	−0.595765			−0.0869012			−0.0434506	+	0.999056i	$0.513835\pi$
$48$		−	16.4188i		−	2.36985i
$49$	−5.38801			−0.769716
$50$	−11.1888			−1.58234
$51$	6.90370	−	1.35401i	0.966712	−	0.189600i
$52$	0.196871			0.0273011
$53$	13.0420			1.79146			0.895730	−	0.444598i	$-0.146653\pi$
$53$	13.0420			1.79146			0.895730	+	0.444598i	$0.146653\pi$
$54$			13.7666i			1.87340i
$55$	−0.846648			−0.114162
$56$		−	25.9641i		−	3.46960i
$57$		−	10.6749i		−	1.41393i
$58$			23.8789i			3.13546i
$59$	−6.48733			−0.844578			−0.422289	−	0.906461i	$-0.638773\pi$
$59$	−6.48733			−0.844578			−0.422289	+	0.906461i	$0.638773\pi$
$60$	6.96881			0.899670
$61$			6.01792i			0.770515i	0.922809	+	0.385258i	$0.125887\pi$
$61$			6.01792i			0.770515i	−0.922809	+	0.385258i	$0.874113\pi$
$62$		−	7.35721i		−	0.934367i
$63$			0.311699i			0.0392704i
$64$	7.87761			0.984702
$65$			0.0345528i			0.00428574i
$66$	−4.45730			−0.548655
$67$	−11.2311			−1.37209			−0.686046	−	0.727558i	$-0.740655\pi$
$67$	−11.2311			−1.37209			−0.686046	+	0.727558i	$0.740655\pi$
$68$	3.82798	+	19.5178i	0.464211	+	2.36688i
$69$	4.11104			0.494911
$70$	7.78433			0.930406
$71$			16.6629i			1.97752i	0.149509	+	0.988760i	$0.452231\pi$
$71$			16.6629i			1.97752i	−0.149509	+	0.988760i	$0.547769\pi$
$72$	−0.653292			−0.0769913
$73$		−	6.52734i		−	0.763968i	−0.924169	−	0.381984i	$-0.875241\pi$
$73$		−	6.52734i		−	0.763968i	0.924169	−	0.381984i	$-0.124759\pi$
$74$		−	3.96605i		−	0.461044i
$75$		−	7.30838i		−	0.843899i
$76$	30.1795			3.46183
$77$	−3.51966			−0.401103
$78$			0.181908i			0.0205970i
$79$		−	3.93839i		−	0.443104i	−0.975149	−	0.221552i	$-0.928888\pi$
$79$		−	3.93839i		−	0.443104i	0.975149	−	0.221552i	$-0.0711122\pi$
$80$			8.14688i			0.910849i
$81$	−8.72648			−0.969609
$82$		−	7.70836i		−	0.851246i
$83$	5.74662			0.630774			0.315387	−	0.948963i	$-0.397866\pi$
$83$	5.74662			0.630774			0.315387	+	0.948963i	$0.397866\pi$
$84$	28.9705			3.16095
$85$	−3.42556	+	0.671849i	−0.371554	+	0.0728722i
$86$	21.2938			2.29617
$87$	−15.5974			−1.67221
$88$		−	7.37688i		−	0.786378i
$89$	13.0491			1.38320			0.691600	−	0.722280i	$-0.256906\pi$
$89$	13.0491			1.38320			0.691600	+	0.722280i	$0.256906\pi$
$90$		−	0.195864i		−	0.0206459i
$91$			0.143642i			0.0150577i
$92$			11.6225i			1.21173i
$93$	4.80562			0.498320
$94$	1.55630			0.160520
$95$			5.29680i			0.543440i
$96$			17.7161i			1.80814i
$97$		−	11.6580i		−	1.18369i	−0.806050	−	0.591847i	$-0.798399\pi$
$97$		−	11.6580i		−	1.18369i	0.806050	−	0.591847i	$-0.201601\pi$
$98$	14.0749			1.42178
$99$			0.0885594i			0.00890056i
$100$	20.6618			2.06618
$101$	−14.2555			−1.41848			−0.709239	−	0.704968i	$-0.750961\pi$
$101$	−14.2555			−1.41848			−0.709239	+	0.704968i	$0.750961\pi$
$102$	−18.0343	+	3.53704i	−1.78566	+	0.350219i
$103$	−7.03176			−0.692860			−0.346430	−	0.938076i	$-0.612606\pi$
$103$	−7.03176			−0.692860			−0.346430	+	0.938076i	$0.612606\pi$
$104$	−0.301060			−0.0295213
$105$			5.08461i			0.496207i
$106$	−34.0693			−3.30910
$107$			9.42118i			0.910780i	0.890292	+	0.455390i	$0.150500\pi$
$107$			9.42118i			0.910780i	−0.890292	+	0.455390i	$0.849500\pi$
$108$		−	25.4221i		−	2.44624i
$109$		−	5.29704i		−	0.507365i	−0.967288	−	0.253682i	$-0.918358\pi$
$109$		−	5.29704i		−	0.507365i	0.967288	−	0.253682i	$-0.0816418\pi$
$110$	2.21167			0.210875
$111$	2.59056			0.245885
$112$			33.8680i			3.20022i
$113$		−	14.7703i		−	1.38948i	−0.719263	−	0.694738i	$-0.755520\pi$
$113$		−	14.7703i		−	1.38948i	0.719263	−	0.694738i	$-0.244480\pi$
$114$			27.8857i			2.61174i
$115$	−2.03986			−0.190218
$116$		−	44.0960i		−	4.09421i
$117$	0.00361422			0.000334135
$118$	16.9466			1.56006
$119$	−14.2406	+	2.79299i	−1.30544	+	0.256033i
$120$	−10.6569			−0.972835
$121$	−1.00000			−0.0909091
$122$		−	15.7204i		−	1.42326i
$123$	5.03498			0.453989
$124$			13.5862i			1.22008i
$125$			7.85959i			0.702984i
$126$		−	0.814242i		−	0.0725384i
$127$	−4.20909			−0.373496			−0.186748	−	0.982408i	$-0.559795\pi$
$127$	−4.20909			−0.373496			−0.186748	+	0.982408i	$0.559795\pi$
$128$	0.187159			0.0165426
$129$			13.9088i			1.22460i
$130$		−	0.0902610i		−	0.00791641i
$131$		−	14.8820i		−	1.30025i	−0.759829	−	0.650123i	$-0.774717\pi$
$131$		−	14.8820i		−	1.30025i	0.759829	−	0.650123i	$-0.225283\pi$
$132$	8.23106			0.716422
$133$			22.0197i			1.90935i
$134$	29.3385			2.53446
$135$	4.46183			0.384013
$136$	−5.85385	−	29.8470i	−0.501963	−	2.55936i
$137$	−2.41761			−0.206550			−0.103275	−	0.994653i	$-0.532932\pi$
$137$	−2.41761			−0.206550			−0.103275	+	0.994653i	$0.532932\pi$
$138$	−10.7391			−0.914176
$139$		−	13.4959i		−	1.14471i	−0.820006	−	0.572354i	$-0.806030\pi$
$139$		−	13.4959i		−	1.14471i	0.820006	−	0.572354i	$-0.193970\pi$
$140$	−14.3749			−1.21490
$141$			1.01655i			0.0856089i
$142$		−	43.5279i		−	3.65278i
$143$			0.0408112i			0.00341281i
$144$	0.852164			0.0710136
$145$	7.73928			0.642712
$146$			17.0512i			1.41116i
$147$			9.19354i			0.758270i
$148$			7.32389i			0.602020i
$149$	14.9197			1.22227			0.611135	−	0.791526i	$-0.290713\pi$
$149$	14.9197			1.22227			0.611135	+	0.791526i	$0.290713\pi$
$150$			19.0914i			1.55881i
$151$	−8.25660			−0.671913			−0.335956	−	0.941878i	$-0.609059\pi$
$151$	−8.25660			−0.671913			−0.335956	+	0.941878i	$0.609059\pi$
$152$	−46.1512			−3.74336
$153$	0.0702754	+	0.358313i	0.00568143	+	0.0289679i
$154$	9.19429			0.740897
$155$	−2.38451			−0.191528
$156$		−	0.335920i		−	0.0268951i
$157$	13.9006			1.10939			0.554695	−	0.832054i	$-0.312835\pi$
$157$	13.9006			1.10939			0.554695	+	0.832054i	$0.312835\pi$
$158$			10.2881i			0.818480i
$159$		−	22.2535i		−	1.76482i
$160$		−	8.79057i		−	0.694956i
$161$	−8.48006			−0.668322
$162$	22.7959			1.79101
$163$		−	17.3415i		−	1.35829i	−0.734003	−	0.679146i	$-0.762350\pi$
$163$		−	17.3415i		−	1.35829i	0.734003	−	0.679146i	$-0.237650\pi$
$164$			14.2346i			1.11154i
$165$			1.44463i			0.112464i
$166$	−15.0117			−1.16513
$167$		−	2.43840i		−	0.188689i	−0.995540	−	0.0943445i	$-0.969924\pi$
$167$		−	2.43840i		−	0.188689i	0.995540	−	0.0943445i	$-0.0300755\pi$
$168$	−44.3024			−3.41801
$169$	−12.9983			−0.999872
$170$	8.94847	−	1.75505i	0.686317	−	0.134606i
$171$	0.554045			0.0423689
$172$	−39.3221			−2.99828
$173$		−	12.5043i		−	0.950682i	−0.879802	−	0.475341i	$-0.842325\pi$
$173$		−	12.5043i		−	0.950682i	0.879802	−	0.475341i	$-0.157675\pi$
$174$	40.7445			3.08883
$175$			15.0754i			1.13959i
$176$			9.62251i			0.725324i
$177$			11.0693i			0.832019i
$178$	−34.0877			−2.55498
$179$	−16.2191			−1.21227			−0.606135	−	0.795362i	$-0.707281\pi$
$179$	−16.2191			−1.21227			−0.606135	+	0.795362i	$0.707281\pi$
$180$			0.361693i			0.0269590i
$181$			20.4054i			1.51672i	0.651834	+	0.758362i	$0.274000\pi$
$181$			20.4054i			1.51672i	−0.651834	+	0.758362i	$0.726000\pi$
$182$		−	0.375230i		−	0.0278139i
$183$	10.2683			0.759057
$184$		−	17.7734i		−	1.31027i
$185$	−1.28542			−0.0945056
$186$	−12.5536			−0.920472
$187$	−4.04602	+	0.793539i	−0.295874	+	0.0580293i
$188$	−2.87393			−0.209603
$189$	18.5486			1.34921
$190$		−	13.8367i		−	1.00382i
$191$	−0.314466			−0.0227540			−0.0113770	−	0.999935i	$-0.503621\pi$
$191$	−0.314466			−0.0227540			−0.0113770	+	0.999935i	$0.503621\pi$
$192$		−	13.4415i		−	0.970059i
$193$			6.52903i			0.469970i	0.971999	+	0.234985i	$0.0755041\pi$
$193$			6.52903i			0.469970i	−0.971999	+	0.234985i	$0.924496\pi$
$194$			30.4539i			2.18646i
$195$	0.0589572			0.00422201
$196$	−25.9915			−1.85653
$197$		−	3.27780i		−	0.233534i	−0.993159	−	0.116767i	$-0.962747\pi$
$197$		−	3.27780i		−	0.233534i	0.993159	−	0.116767i	$-0.0372530\pi$
$198$		−	0.231341i		−	0.0164407i
$199$			9.26984i			0.657121i	0.944483	+	0.328561i	$0.106564\pi$
$199$			9.26984i			0.657121i	−0.944483	+	0.328561i	$0.893436\pi$
$200$	−31.5966			−2.23421
$201$			19.1635i			1.35169i
$202$	37.2393			2.62014
$203$	32.1735			2.25814
$204$	33.3030	−	6.53167i	2.33168	−	0.457308i
$205$	−2.49832			−0.174490
$206$	18.3688			1.27982
$207$			0.213370i			0.0148302i
$208$	0.392706			0.0272293
$209$			6.25620i			0.432750i
$210$		−	13.2824i		−	0.916570i
$211$		−	18.3608i		−	1.26401i	−0.774964	−	0.632006i	$-0.782232\pi$
$211$		−	18.3608i		−	1.26401i	0.774964	−	0.632006i	$-0.217768\pi$
$212$	62.9140			4.32095
$213$	28.4318			1.94811
$214$		−	24.6106i		−	1.68235i
$215$		−	6.90141i		−	0.470672i
$216$			38.8761i			2.64518i
$217$	−9.91281			−0.672925
$218$			13.8373i			0.937180i
$219$	−11.1376			−0.752607
$220$	−4.08418			−0.275355
$221$	0.0323853	+	0.165123i	0.00217847	+	0.0111074i
$222$	−6.76724			−0.454188
$223$	−6.75345			−0.452245			−0.226122	−	0.974099i	$-0.572605\pi$
$223$	−6.75345			−0.452245			−0.226122	+	0.974099i	$0.572605\pi$
$224$		−	36.5439i		−	2.44169i
$225$	0.379317			0.0252878
$226$			38.5840i			2.56657i
$227$		−	11.8382i		−	0.785726i	−0.919597	−	0.392863i	$-0.871485\pi$
$227$		−	11.8382i		−	0.785726i	0.919597	−	0.392863i	$-0.128515\pi$
$228$		−	51.4951i		−	3.41035i
$229$	5.10721			0.337494			0.168747	−	0.985659i	$-0.446028\pi$
$229$	5.10721			0.337494			0.168747	+	0.985659i	$0.446028\pi$
$230$	5.32867			0.351362
$231$			6.00558i			0.395138i
$232$			67.4326i			4.42717i
$233$		−	9.99997i		−	0.655120i	−0.944830	−	0.327560i	$-0.893774\pi$
$233$		−	9.99997i		−	0.655120i	0.944830	−	0.327560i	$-0.106226\pi$
$234$	−0.00944130			−0.000617197
$235$		−	0.504403i		−	0.0329036i
$236$	−31.2945			−2.03710
$237$	−6.72005			−0.436514
$238$	37.2003	−	7.29603i	2.41134	−	0.472932i
$239$	−10.8383			−0.701071			−0.350536	−	0.936549i	$-0.614000\pi$
$239$	−10.8383			−0.701071			−0.350536	+	0.936549i	$0.614000\pi$
$240$	13.9010			0.897304
$241$		−	16.7967i		−	1.08197i	−0.841033	−	0.540984i	$-0.818052\pi$
$241$		−	16.7967i		−	1.08197i	0.841033	−	0.540984i	$-0.181948\pi$
$242$	2.61227			0.167923
$243$		−	0.920033i		−	0.0590201i
$244$			29.0301i			1.85846i
$245$		−	4.56175i		−	0.291440i
$246$	−13.1527			−0.838587
$247$	0.255323			0.0162458
$248$		−	20.7763i		−	1.31930i
$249$		−	9.80543i		−	0.621394i
$250$		−	20.5314i		−	1.29852i
$251$	2.28525			0.144244			0.0721220	−	0.997396i	$-0.477023\pi$
$251$	2.28525			0.144244			0.0721220	+	0.997396i	$0.477023\pi$
$252$			1.50362i			0.0947190i
$253$	−2.40934			−0.151474
$254$	10.9953			0.689904
$255$	1.14637	+	5.84501i	0.0717886	+	0.366029i
$256$	−16.2441			−1.01526
$257$	11.0311			0.688101			0.344050	−	0.938951i	$-0.388201\pi$
$257$	11.0311			0.688101			0.344050	+	0.938951i	$0.388201\pi$
$258$		−	36.3334i		−	2.26202i
$259$	−5.34369			−0.332041
$260$			0.166680i			0.0103371i
$261$		−	0.809529i		−	0.0501086i
$262$			38.8757i			2.40175i
$263$	16.1149			0.993685			0.496842	−	0.867841i	$-0.334493\pi$
$263$	16.1149			0.993685			0.496842	+	0.867841i	$0.334493\pi$
$264$	−12.5871			−0.774684
$265$			11.0420i			0.678306i
$266$		−	57.5213i		−	3.52686i
$267$		−	22.2656i		−	1.36263i
$268$	−54.1779			−3.30944
$269$		−	7.16171i		−	0.436657i	−0.975875	−	0.218328i	$-0.929940\pi$
$269$		−	7.16171i		−	0.436657i	0.975875	−	0.218328i	$-0.0700604\pi$
$270$	−11.6555			−0.709331
$271$	15.9431			0.968474			0.484237	−	0.874937i	$-0.339097\pi$
$271$	15.9431			0.968474			0.484237	+	0.874937i	$0.339097\pi$
$272$	7.63584	+	38.9329i	0.462991	+	2.36065i
$273$	0.245095			0.0148338
$274$	6.31544			0.381530
$275$			4.28319i			0.258286i
$276$	19.8314			1.19371
$277$			23.6002i			1.41800i	0.705210	+	0.708999i	$0.250853\pi$
$277$			23.6002i			1.41800i	−0.705210	+	0.708999i	$0.749147\pi$
$278$			35.2549i			2.11445i
$279$			0.249420i			0.0149324i
$280$	21.9825			1.31370
$281$	−25.0551			−1.49466			−0.747330	−	0.664453i	$-0.768665\pi$
$281$	−25.0551			−1.49466			−0.747330	+	0.664453i	$0.768665\pi$
$282$		−	2.65550i		−	0.158133i
$283$		−	4.94168i		−	0.293752i	−0.989155	−	0.146876i	$-0.953078\pi$
$283$		−	4.94168i		−	0.293752i	0.989155	−	0.146876i	$-0.0469219\pi$
$284$			80.3807i			4.76972i
$285$	9.03790			0.535359
$286$		−	0.106610i		−	0.00630397i
$287$	−10.3859			−0.613062
$288$	−0.919494			−0.0541817
$289$	−15.7406	+	6.42136i	−0.925917	+	0.377727i
$290$	−20.2171			−1.18719
$291$	−19.8920			−1.16609
$292$		−	31.4875i		−	1.84267i
$293$	−30.3808			−1.77487			−0.887433	−	0.460937i	$-0.847514\pi$
$293$	−30.3808			−1.77487			−0.887433	+	0.460937i	$0.847514\pi$
$294$		−	24.0160i		−	1.40064i
$295$		−	5.49249i		−	0.319785i
$296$		−	11.1999i		−	0.650979i
$297$	5.26999			0.305796
$298$	−38.9743			−2.25772
$299$			0.0983281i			0.00568646i
$300$		−	35.2552i		−	2.03546i
$301$		−	28.6903i		−	1.65368i
$302$	21.5685			1.24112
$303$			24.3241i			1.39738i
$304$	60.2003			3.45272
$305$	−5.09506			−0.291742
$306$	−0.183578	−	0.936010i	−0.0104945	−	0.0535082i
$307$	27.8806			1.59123			0.795614	−	0.605804i	$-0.207148\pi$
$307$	27.8806			1.59123			0.795614	+	0.605804i	$0.207148\pi$
$308$	−16.9786			−0.967447
$309$			11.9983i			0.682557i
$310$	6.22897			0.353782
$311$		−	27.8391i		−	1.57861i	−0.614003	−	0.789304i	$-0.710442\pi$
$311$		−	27.8391i		−	1.57861i	0.614003	−	0.789304i	$-0.289558\pi$
$312$			0.513696i			0.0290823i
$313$		−	14.9467i		−	0.844839i	−0.906400	−	0.422420i	$-0.861181\pi$
$313$		−	14.9467i		−	0.844839i	0.906400	−	0.422420i	$-0.138819\pi$
$314$	−36.3121			−2.04921
$315$	−0.263900			−0.0148691
$316$		−	18.9986i		−	1.06875i
$317$		−	8.68582i		−	0.487844i	−0.969795	−	0.243922i	$-0.921566\pi$
$317$		−	8.68582i		−	0.487844i	0.969795	−	0.243922i	$-0.0784341\pi$
$318$			58.1322i			3.25989i
$319$	9.14108			0.511802
$320$			6.66957i			0.372840i
$321$	16.0753			0.897236
$322$	22.1522			1.23449
$323$	4.96454	+	25.3127i	0.276234	+	1.40844i
$324$	−42.0960			−2.33867
$325$	0.174802			0.00969627
$326$			45.3007i			2.50897i
$327$	−9.03832			−0.499820
$328$		−	21.7679i		−	1.20193i
$329$		−	2.09689i		−	0.115605i
$330$		−	3.77376i		−	0.207739i
$331$	−8.37957			−0.460583			−0.230292	−	0.973122i	$-0.573968\pi$
$331$	−8.37957			−0.460583			−0.230292	+	0.973122i	$0.573968\pi$
$332$	27.7214			1.52141
$333$			0.134454i			0.00736806i
$334$			6.36975i			0.348537i
$335$		−	9.50876i		−	0.519519i
$336$	57.7887			3.15263
$337$			6.25552i			0.340760i	0.985378	+	0.170380i	$0.0544995\pi$
$337$			6.25552i			0.340760i	−0.985378	+	0.170380i	$0.945501\pi$
$338$	33.9551			1.84692
$339$	−25.2025			−1.36881
$340$	−16.5247	+	3.24096i	−0.896177	+	0.175766i
$341$	−2.81641			−0.152517
$342$	−1.44731			−0.0782618
$343$			5.67364i			0.306348i
$344$	60.1323			3.24212
$345$			3.48061i			0.187390i
$346$			32.6645i			1.75605i
$347$			16.9283i			0.908760i	0.890808	+	0.454380i	$0.150139\pi$
$347$			16.9283i			0.908760i	−0.890808	+	0.454380i	$0.849861\pi$
$348$	−75.2408			−4.03333
$349$	21.5285			1.15239			0.576197	−	0.817311i	$-0.304536\pi$
$349$	21.5285			1.15239			0.576197	+	0.817311i	$0.304536\pi$
$350$		−	39.3809i		−	2.10500i
$351$		−	0.215075i		−	0.0114798i
$352$		−	10.3828i		−	0.553405i
$353$	−6.07733			−0.323464			−0.161732	−	0.986835i	$-0.551708\pi$
$353$	−6.07733			−0.323464			−0.161732	+	0.986835i	$0.551708\pi$
$354$		−	28.9159i		−	1.53687i
$355$	−14.1076			−0.748754
$356$	62.9480			3.33624
$357$	4.76566	+	24.2987i	0.252226	+	1.28602i
$358$	42.3685			2.23925
$359$	−2.74997			−0.145138			−0.0725689	−	0.997363i	$-0.523120\pi$
$359$	−2.74997			−0.145138			−0.0725689	+	0.997363i	$0.523120\pi$
$360$		−	0.553109i		−	0.0291514i
$361$	20.1400			1.06000
$362$		−	53.3044i		−	2.80162i
$363$			1.70629i			0.0895572i
$364$			0.692919i			0.0363188i
$365$	5.52636			0.289263
$366$	−26.8236			−1.40209
$367$		−	11.5377i		−	0.602263i	−0.953583	−	0.301131i	$-0.902636\pi$
$367$		−	11.5377i		−	0.602263i	0.953583	−	0.301131i	$-0.0973643\pi$
$368$			23.1839i			1.20854i
$369$			0.261324i			0.0136040i
$370$	3.35785			0.174566
$371$			45.9035i			2.38319i
$372$	23.1820			1.20193
$373$	9.91490			0.513374			0.256687	−	0.966495i	$-0.417369\pi$
$373$	9.91490			0.513374			0.256687	+	0.966495i	$0.417369\pi$
$374$	10.5693	−	2.07294i	0.546525	−	0.107189i
$375$	13.4108			0.692530
$376$	4.39488			0.226649
$377$		−	0.373059i		−	0.0192135i
$378$	−48.4539			−2.49220
$379$			11.6088i			0.596306i	0.954518	+	0.298153i	$0.0963706\pi$
$379$			11.6088i			0.596306i	−0.954518	+	0.298153i	$0.903629\pi$
$380$			25.5514i			1.31076i
$381$			7.18194i			0.367942i
$382$	0.821470			0.0420301
$383$	3.61128			0.184528			0.0922638	−	0.995735i	$-0.470590\pi$
$383$	3.61128			0.184528			0.0922638	+	0.995735i	$0.470590\pi$
$384$		−	0.319348i		−	0.0162966i
$385$		−	2.97992i		−	0.151871i
$386$		−	17.0556i		−	0.868106i
$387$	−0.721888			−0.0366956
$388$		−	56.2376i		−	2.85503i
$389$	−11.8292			−0.599765			−0.299883	−	0.953976i	$-0.596948\pi$
$389$	−11.8292			−0.599765			−0.299883	+	0.953976i	$0.596948\pi$
$390$	−0.154012			−0.00779869
$391$	−9.74824	+	1.91191i	−0.492990	+	0.0966892i
$392$	39.7467			2.00751
$393$	−25.3931			−1.28091
$394$			8.56249i			0.431372i
$395$	3.33443			0.167774
$396$			0.427205i			0.0214679i
$397$		−	27.0276i		−	1.35648i	−0.734842	−	0.678238i	$-0.762744\pi$
$397$		−	27.0276i		−	1.35648i	0.734842	−	0.678238i	$-0.237256\pi$
$398$		−	24.2153i		−	1.21380i
$399$	37.5721			1.88096
$400$	41.2150			2.06075
$401$		−	6.84011i		−	0.341579i	−0.985308	−	0.170789i	$-0.945368\pi$
$401$		−	6.84011i		−	0.341579i	0.985308	−	0.170789i	$-0.0546318\pi$
$402$		−	50.0602i		−	2.49677i
$403$			0.114941i			0.00572562i
$404$	−68.7678			−3.42133
$405$		−	7.38826i		−	0.367126i
$406$	−84.0458			−4.17112
$407$	−1.51824			−0.0752563
$408$	−50.9278	+	9.98838i	−2.52130	+	0.494499i
$409$	−7.26303			−0.359134			−0.179567	−	0.983746i	$-0.557470\pi$
$409$	−7.26303			−0.359134			−0.179567	+	0.983746i	$0.557470\pi$
$410$	6.52627			0.322309
$411$			4.12515i			0.203479i
$412$	−33.9208			−1.67116
$413$		−	22.8332i		−	1.12355i
$414$		−	0.557379i		−	0.0273937i
$415$			4.86537i			0.238832i
$416$	−0.423734			−0.0207753
$417$	−23.0280			−1.12769
$418$		−	16.3429i		−	0.799355i
$419$			13.5510i			0.662009i	0.943629	+	0.331004i	$0.107387\pi$
$419$			13.5510i			0.662009i	−0.943629	+	0.331004i	$0.892613\pi$
$420$			24.5279i			1.19684i
$421$	7.08574			0.345338			0.172669	−	0.984980i	$-0.444761\pi$
$421$	7.08574			0.345338			0.172669	+	0.984980i	$0.444761\pi$
$422$			47.9634i			2.33482i
$423$	−0.0527606			−0.00256531
$424$	−96.2095			−4.67235
$425$	3.39888	+	17.3299i	0.164870	+	0.840622i
$426$	−74.2714			−3.59846
$427$	−21.1810			−1.02502
$428$			45.4472i			2.19677i
$429$	0.0696360			0.00336206
$430$			18.0283i			0.869403i
$431$			9.04004i			0.435443i	0.976011	+	0.217722i	$0.0698625\pi$
$431$			9.04004i			0.435443i	−0.976011	+	0.217722i	$0.930137\pi$
$432$		−	50.7105i		−	2.43981i
$433$	7.24759			0.348297			0.174148	−	0.984719i	$-0.444283\pi$
$433$	7.24759			0.348297			0.174148	+	0.984719i	$0.444283\pi$
$434$	25.8949			1.24299
$435$		−	13.2055i		−	0.633155i
$436$		−	25.5526i		−	1.22375i
$437$			15.0733i			0.721054i
$438$	29.0943			1.39018
$439$			11.3563i			0.542008i	0.962578	+	0.271004i	$0.0873557\pi$
$439$			11.3563i			0.542008i	−0.962578	+	0.271004i	$0.912644\pi$
$440$	6.24562			0.297748
$441$	−0.477160			−0.0227219
$442$	−0.0845991	−	0.431346i	−0.00402397	−	0.0205170i
$443$	−0.811975			−0.0385781			−0.0192890	−	0.999814i	$-0.506140\pi$
$443$	−0.811975			−0.0385781			−0.0192890	+	0.999814i	$0.506140\pi$
$444$	12.4967			0.593068
$445$			11.0480i			0.523725i
$446$	17.6418			0.835365
$447$		−	25.4574i		−	1.20409i
$448$			27.7265i			1.30996i
$449$		−	6.03970i		−	0.285031i	−0.989793	−	0.142516i	$-0.954481\pi$
$449$		−	6.03970i		−	0.285031i	0.989793	−	0.142516i	$-0.0455191\pi$
$450$	−0.990876			−0.0467104
$451$	−2.95083			−0.138949
$452$		−	71.2511i		−	3.35137i
$453$			14.0882i			0.661921i
$454$			30.9244i			1.45135i
$455$	−0.121614			−0.00570135
$456$			78.7476i			3.68769i
$457$	4.79885			0.224481			0.112240	−	0.993681i	$-0.464197\pi$
$457$	4.79885			0.224481			0.112240	+	0.993681i	$0.464197\pi$
$458$	−13.3414			−0.623403
$459$	21.3225	−	4.18195i	0.995249	−	0.195197i
$460$	−9.84017			−0.458801
$461$	15.9345			0.742144			0.371072	−	0.928604i	$-0.378990\pi$
$461$	15.9345			0.742144			0.371072	+	0.928604i	$0.378990\pi$
$462$		−	15.6882i		−	0.729880i
$463$	−10.5912			−0.492215			−0.246107	−	0.969243i	$-0.579152\pi$
$463$	−10.5912			−0.492215			−0.246107	+	0.969243i	$0.579152\pi$
$464$		−	87.9601i		−	4.08344i
$465$			4.06867i			0.188680i
$466$			26.1226i			1.21011i
$467$	15.3667			0.711087			0.355544	−	0.934660i	$-0.384296\pi$
$467$	15.3667			0.711087			0.355544	+	0.934660i	$0.384296\pi$
$468$	0.0174348			0.000805922
$469$		−	39.5295i		−	1.82530i
$470$			1.31764i			0.0607780i
$471$		−	23.7186i		−	1.09289i
$472$	47.8563			2.20276
$473$		−	8.15145i		−	0.374804i
$474$	17.5546			0.806308
$475$	26.7965			1.22951
$476$	−68.6959	+	13.4732i	−3.14867	+	0.617544i
$477$	1.15500			0.0528836
$478$	28.3125			1.29499
$479$		−	11.7210i		−	0.535546i	−0.963482	−	0.267773i	$-0.913712\pi$
$479$		−	11.7210i		−	0.535546i	0.963482	−	0.267773i	$-0.0862878\pi$
$480$	−14.9993			−0.684622
$481$			0.0619612i			0.00282519i
$482$			43.8774i			1.99856i
$483$			14.4695i			0.658384i
$484$	−4.82394			−0.219270
$485$	9.87025			0.448185
$486$			2.40337i			0.109019i
$487$			17.3358i			0.785560i	0.919633	+	0.392780i	$0.128487\pi$
$487$			17.3358i			0.785560i	−0.919633	+	0.392780i	$0.871513\pi$
$488$		−	44.3934i		−	2.00960i
$489$	−29.5897			−1.33809
$490$			11.9165i			0.538333i
$491$	34.0963			1.53874			0.769371	−	0.638802i	$-0.220570\pi$
$491$	34.0963			1.53874			0.769371	+	0.638802i	$0.220570\pi$
$492$	24.2885			1.09501
$493$	36.9850	−	7.25380i	1.66572	−	0.326695i
$494$	−0.666972			−0.0300085
$495$	−0.0749787			−0.00337004
$496$			27.1009i			1.21687i
$497$	−58.6477			−2.63071
$498$			25.6144i			1.14781i
$499$			12.5100i			0.560023i	0.959997	+	0.280012i	$0.0903383\pi$
$499$			12.5100i			0.560023i	−0.959997	+	0.280012i	$0.909662\pi$
$500$			37.9142i			1.69557i
$501$	−4.16063			−0.185883
$502$	−5.96969			−0.266440
$503$			28.9561i			1.29109i	0.763723	+	0.645544i	$0.223369\pi$
$503$			28.9561i			1.29109i	−0.763723	+	0.645544i	$0.776631\pi$
$504$		−	2.29937i		−	0.102422i
$505$		−	12.0694i		−	0.537082i
$506$	6.29384			0.279795
$507$			22.1790i			0.985003i
$508$	−20.3044			−0.900861
$509$	−16.9674			−0.752069			−0.376034	−	0.926606i	$-0.622713\pi$
$509$	−16.9674			−0.752069			−0.376034	+	0.926606i	$0.622713\pi$
$510$	−2.99463	−	15.2687i	−0.132604	−	0.676111i
$511$	22.9740			1.01631
$512$	42.0597			1.85879
$513$		−	32.9701i		−	1.45567i
$514$	−28.8162			−1.27103
$515$		−	5.95343i		−	0.262339i
$516$			67.0951i			2.95370i
$517$		−	0.595765i		−	0.0262017i
$518$	13.9591			0.613329
$519$	−21.3360			−0.936545
$520$		−	0.254892i		−	0.0111777i
$521$		−	32.3726i		−	1.41827i	−0.705073	−	0.709135i	$-0.749086\pi$
$521$		−	32.3726i		−	1.41827i	0.705073	−	0.709135i	$-0.250914\pi$
$522$			2.11471i			0.0925581i
$523$	35.4044			1.54813			0.774064	−	0.633107i	$-0.218221\pi$
$523$	35.4044			1.54813			0.774064	+	0.633107i	$0.218221\pi$
$524$		−	71.7898i		−	3.13615i
$525$	25.7230			1.12264
$526$	−42.0963			−1.83549
$527$	−11.3953	+	2.23493i	−0.496385	+	0.0973551i
$528$	16.4188			0.714538
$529$	17.1951			0.747612
$530$		−	28.8447i		−	1.25293i
$531$	−0.574514			−0.0249318
$532$			106.222i			4.60529i
$533$			0.120427i			0.00521627i
$534$			58.1637i			2.51699i
$535$	−7.97642			−0.344851
$536$	82.8502			3.57858
$537$			27.6745i			1.19424i
$538$			18.7083i			0.806572i
$539$		−	5.38801i		−	0.232078i
$540$	21.5236			0.926228
$541$		−	28.2751i		−	1.21564i	−0.794074	−	0.607821i	$-0.792044\pi$
$541$		−	28.2751i		−	1.21564i	0.794074	−	0.607821i	$-0.207956\pi$
$542$	−41.6476			−1.78892
$543$	34.8177			1.49417
$544$	−8.23915	−	42.0090i	−0.353251	−	1.80112i
$545$	4.48473			0.192105
$546$	−0.640253			−0.0274003
$547$		−	18.2643i		−	0.780924i	−0.920619	−	0.390462i	$-0.872315\pi$
$547$		−	18.2643i		−	0.780924i	0.920619	−	0.390462i	$-0.127685\pi$
$548$	−11.6624			−0.498193
$549$			0.532943i			0.0227455i
$550$		−	11.1888i		−	0.477093i
$551$		−	57.1884i		−	2.43631i
$552$	−30.3267			−1.29079
$553$	13.8618			0.589464
$554$		−	61.6500i		−	2.61926i
$555$			2.19330i			0.0931003i
$556$		−	65.1035i		−	2.76100i
$557$	−19.9786			−0.846519			−0.423259	−	0.906009i	$-0.639114\pi$
$557$	−19.9786			−0.846519			−0.423259	+	0.906009i	$0.639114\pi$
$558$		−	0.651551i		−	0.0275823i
$559$	−0.332671			−0.0140705
$560$	−28.6743			−1.21171
$561$	1.35401	+	6.90370i	0.0571664	+	0.291475i
$562$	65.4506			2.76087
$563$	8.96391			0.377784			0.188892	−	0.981998i	$-0.439510\pi$
$563$	8.96391			0.377784			0.188892	+	0.981998i	$0.439510\pi$
$564$			4.90377i			0.206486i
$565$	12.5053			0.526101
$566$			12.9090i			0.542605i
$567$		−	30.7142i		−	1.28988i
$568$		−	122.920i		−	5.15761i
$569$	−13.0865			−0.548615			−0.274308	−	0.961642i	$-0.588449\pi$
$569$	−13.0865			−0.548615			−0.274308	+	0.961642i	$0.588449\pi$
$570$	−23.6094			−0.988889
$571$			38.9603i			1.63044i	0.579153	+	0.815219i	$0.303383\pi$
$571$			38.9603i			1.63044i	−0.579153	+	0.815219i	$0.696617\pi$
$572$			0.196871i			0.00823158i
$573$			0.536572i			0.0224156i
$574$	27.1308			1.13242
$575$			10.3196i			0.430359i
$576$	0.697637			0.0290682
$577$	−27.3839			−1.14000			−0.570002	−	0.821643i	$-0.693058\pi$
$577$	−27.3839			−1.14000			−0.570002	+	0.821643i	$0.693058\pi$
$578$	41.1186	−	16.7743i	1.71031	−	0.697719i
$579$	11.1404			0.462981
$580$	37.3338			1.55020
$581$			20.2262i			0.839123i
$582$	51.9633			2.15395
$583$			13.0420i			0.540146i
$584$			48.1514i			1.99252i
$585$			0.00305997i			0.000126514i
$586$	79.3628			3.27845
$587$	−17.9676			−0.741604			−0.370802	−	0.928712i	$-0.620917\pi$
$587$	−17.9676			−0.741604			−0.370802	+	0.928712i	$0.620917\pi$
$588$			44.3491i			1.82892i
$589$			17.6200i			0.726020i
$590$			14.3478i			0.590691i
$591$	−5.59289			−0.230061
$592$			14.6093i			0.600437i
$593$	−18.1593			−0.745711			−0.372856	−	0.927889i	$-0.621621\pi$
$593$	−18.1593			−0.745711			−0.372856	+	0.927889i	$0.621621\pi$
$594$	−13.7666			−0.564851
$595$	−2.36468	−	12.0568i	−0.0969424	−	0.494281i
$596$	71.9718			2.94808
$597$	15.8171			0.647350
$598$		−	0.256859i		−	0.0105038i
$599$	−2.72410			−0.111304			−0.0556518	−	0.998450i	$-0.517724\pi$
$599$	−2.72410			−0.111304			−0.0556518	+	0.998450i	$0.517724\pi$
$600$			53.9130i			2.20099i
$601$			30.2236i			1.23285i	0.787415	+	0.616423i	$0.211419\pi$
$601$			30.2236i			1.23285i	−0.787415	+	0.616423i	$0.788581\pi$
$602$			74.9468i			3.05461i
$603$	−0.994616			−0.0405039
$604$	−39.8293			−1.62063
$605$		−	0.846648i		−	0.0344212i
$606$		−	63.5411i		−	2.58118i
$607$		−	2.04841i		−	0.0831426i	−0.999136	−	0.0415713i	$-0.986764\pi$
$607$		−	2.04841i		−	0.0831426i	0.999136	−	0.0415713i	$-0.0132364\pi$
$608$	−64.9568			−2.63435
$609$		−	54.8975i		−	2.22456i
$610$	13.3097			0.538892
$611$	−0.0243139			−0.000983634
$612$	0.339004	+	1.72848i	0.0137034	+	0.0698697i
$613$	18.8003			0.759338			0.379669	−	0.925122i	$-0.376038\pi$
$613$	18.8003			0.759338			0.379669	+	0.925122i	$0.376038\pi$
$614$	−72.8315			−2.93924
$615$			4.26286i			0.171895i
$616$	25.9641			1.04612
$617$			37.1443i			1.49537i	0.664051	+	0.747687i	$0.268836\pi$
$617$			37.1443i			1.49537i	−0.664051	+	0.747687i	$0.731164\pi$
$618$		−	31.3426i		−	1.26079i
$619$			24.3473i			0.978601i	0.872115	+	0.489300i	$0.162748\pi$
$619$			24.3473i			0.978601i	−0.872115	+	0.489300i	$0.837252\pi$
$620$	−11.5027			−0.461960
$621$	12.6972			0.509521
$622$			72.7230i			2.91593i
$623$			45.9284i			1.84008i
$624$		−	0.670072i		−	0.0268244i
$625$	14.7616			0.590465
$626$			39.0448i			1.56055i
$627$	10.6749			0.426315
$628$	67.0558			2.67582
$629$	−6.14283	+	1.20478i	−0.244931	+	0.0480378i
$630$	0.689376			0.0274654
$631$	−24.9568			−0.993516			−0.496758	−	0.867889i	$-0.665476\pi$
$631$	−24.9568			−0.993516			−0.496758	+	0.867889i	$0.665476\pi$
$632$			29.0530i			1.15567i
$633$	−31.3290			−1.24521
$634$			22.6897i			0.901122i
$635$		−	3.56362i		−	0.141418i
$636$		−	107.350i		−	4.25669i
$637$	−0.219891			−0.00871242
$638$	−23.8789			−0.945376
$639$			1.47566i			0.0583760i
$640$			0.158458i			0.00626358i
$641$		−	10.7558i		−	0.424828i	−0.977180	−	0.212414i	$-0.931867\pi$
$641$		−	10.7558i		−	0.424828i	0.977180	−	0.212414i	$-0.0681325\pi$
$642$	−41.9930			−1.65733
$643$		−	15.1132i		−	0.596007i	−0.954565	−	0.298004i	$-0.903679\pi$
$643$		−	15.1132i		−	0.596007i	0.954565	−	0.298004i	$-0.0963207\pi$
$644$	−40.9073			−1.61197
$645$	−11.7758			−0.463673
$646$	−12.9687	−	66.1236i	−0.510247	−	2.60160i
$647$	22.5859			0.887945			0.443972	−	0.896040i	$-0.353569\pi$
$647$	22.5859			0.887945			0.443972	+	0.896040i	$0.353569\pi$
$648$	64.3742			2.52886
$649$		−	6.48733i		−	0.254650i
$650$	−0.456630			−0.0179105
$651$			16.9142i			0.662918i
$652$		−	83.6544i		−	3.27616i
$653$		−	0.772124i		−	0.0302155i	−0.999886	−	0.0151078i	$-0.995191\pi$
$653$		−	0.772124i		−	0.0302155i	0.999886	−	0.0151078i	$-0.00480913\pi$
$654$	23.6105			0.923243
$655$	12.5998			0.492315
$656$			28.3944i			1.10861i
$657$		−	0.578058i		−	0.0225522i
$658$			5.47763i			0.213540i
$659$	−27.8591			−1.08524			−0.542618	−	0.839980i	$-0.682567\pi$
$659$	−27.8591			−1.08524			−0.542618	+	0.839980i	$0.682567\pi$
$660$			6.96881i			0.271261i
$661$	−34.1963			−1.33008			−0.665042	−	0.746806i	$-0.731586\pi$
$661$	−34.1963			−1.33008			−0.665042	+	0.746806i	$0.731586\pi$
$662$	21.8897			0.850767
$663$	0.281749	−	0.0552589i	0.0109422	−	0.00214608i
$664$	−42.3922			−1.64513
$665$	−18.6429			−0.722942
$666$		−	0.351231i		−	0.0136099i
$667$	22.0240			0.852771
$668$		−	11.7627i		−	0.455112i
$669$			11.5234i			0.445520i
$670$			24.8394i			0.959630i
$671$	−6.01792			−0.232319
$672$	−62.3547			−2.40538
$673$			47.8499i			1.84448i	0.386622	+	0.922238i	$0.373642\pi$
$673$			47.8499i			1.84448i	−0.386622	+	0.922238i	$0.626358\pi$
$674$		−	16.3411i		−	0.629435i
$675$		−	22.5724i		−	0.868810i
$676$	−62.7032			−2.41166
$677$			44.8342i			1.72312i	0.507658	+	0.861559i	$0.330512\pi$
$677$			44.8342i			1.72312i	−0.507658	+	0.861559i	$0.669488\pi$
$678$	65.8357			2.52841
$679$	41.0323			1.57468
$680$	25.2699	−	4.95615i	0.969058	−	0.190060i
$681$	−20.1994			−0.774042
$682$	7.35721			0.281722
$683$			6.09522i			0.233227i	0.993177	+	0.116614i	$0.0372039\pi$
$683$			6.09522i			0.233227i	−0.993177	+	0.116614i	$0.962796\pi$
$684$	2.67268			0.102192
$685$		−	2.04686i		−	0.0782067i
$686$		−	14.8211i		−	0.565871i
$687$		−	8.71441i		−	0.332475i
$688$	−78.4374			−2.99040
$689$	0.532261			0.0202775
$690$		−	9.09228i		−	0.346137i
$691$			16.3208i			0.620873i	0.950594	+	0.310437i	$0.100475\pi$
$691$			16.3208i			0.620873i	−0.950594	+	0.310437i	$0.899525\pi$
$692$		−	60.3198i		−	2.29302i
$693$	−0.311699			−0.0118405
$694$		−	44.2213i		−	1.67862i
$695$	11.4263			0.433424
$696$	115.060			4.36134
$697$	−11.9391	+	2.34160i	−0.452227	+	0.0886944i
$698$	−56.2382			−2.12865
$699$	−17.0629			−0.645378
$700$			72.7226i			2.74866i
$701$	7.96878			0.300977			0.150488	−	0.988612i	$-0.451915\pi$
$701$	7.96878			0.300977			0.150488	+	0.988612i	$0.451915\pi$
$702$			0.561833i			0.0212050i
$703$			9.49840i			0.358239i
$704$			7.87761i			0.296899i
$705$	−0.860660			−0.0324143
$706$	15.8756			0.597486
$707$		−	50.1746i		−	1.88701i
$708$			53.3976i			2.00680i
$709$			7.11520i			0.267217i	0.991034	+	0.133609i	$0.0426565\pi$
$709$			7.11520i			0.267217i	−0.991034	+	0.133609i	$0.957344\pi$
$710$	36.8528			1.38306
$711$		−	0.348782i		−	0.0130803i
$712$	−96.2616			−3.60755
$713$	−6.78568			−0.254126
$714$	−12.4492	−	63.4747i	−0.465899	−	2.37548i
$715$	−0.0345528			−0.00129220
$716$	−78.2397			−2.92396
$717$			18.4933i			0.690646i
$718$	7.18366			0.268092
$719$		−	29.1625i		−	1.08758i	−0.839222	−	0.543789i	$-0.816989\pi$
$719$		−	29.1625i		−	1.08758i	0.839222	−	0.543789i	$-0.183011\pi$
$720$			0.721483i			0.0268881i
$721$		−	24.7494i		−	0.921716i
$722$	−52.6110			−1.95798
$723$	−28.6601			−1.06588
$724$			98.4346i			3.65829i
$725$		−	39.1529i		−	1.45410i
$726$		−	4.45730i		−	0.165426i
$727$	−14.2813			−0.529663			−0.264832	−	0.964295i	$-0.585316\pi$
$727$	−14.2813			−0.529663			−0.264832	+	0.964295i	$0.585316\pi$
$728$		−	1.05963i		−	0.0392724i
$729$	−27.7493			−1.02775
$730$	−14.4363			−0.534313
$731$	−6.46850	−	32.9810i	−0.239246	−	1.21984i
$732$	49.5338			1.83082
$733$	6.51116			0.240495			0.120248	−	0.992744i	$-0.461631\pi$
$733$	6.51116			0.240495			0.120248	+	0.992744i	$0.461631\pi$
$734$			30.1395i			1.11247i
$735$	−7.78370			−0.287106
$736$		−	25.0157i		−	0.922090i
$737$		−	11.2311i		−	0.413701i
$738$		−	0.682648i		−	0.0251286i
$739$	−35.1873			−1.29439			−0.647193	−	0.762326i	$-0.724057\pi$
$739$	−35.1873			−1.29439			−0.647193	+	0.762326i	$0.724057\pi$
$740$	−6.20076			−0.227945
$741$		−	0.435656i		−	0.0160042i
$742$		−	119.912i		−	4.40212i
$743$		−	14.7090i		−	0.539620i	−0.962914	−	0.269810i	$-0.913039\pi$
$743$		−	14.7090i		−	0.539620i	0.962914	−	0.269810i	$-0.0869609\pi$
$744$	−35.4505			−1.29968
$745$			12.6317i			0.462791i
$746$	−25.9004			−0.948280
$747$	0.508918			0.0186203
$748$	−19.5178	+	3.82798i	−0.713640	+	0.139965i
$749$	−33.1594			−1.21162
$750$	−35.0325			−1.27921
$751$		−	16.4817i		−	0.601426i	−0.953715	−	0.300713i	$-0.902775\pi$
$751$		−	16.4817i		−	0.601426i	0.953715	−	0.300713i	$-0.0972246\pi$
$752$	−5.73275			−0.209052
$753$		−	3.89932i		−	0.142099i
$754$			0.974529i			0.0354902i
$755$		−	6.99044i		−	0.254408i
$756$	89.4772			3.25426
$757$	−41.4419			−1.50623			−0.753115	−	0.657889i	$-0.771450\pi$
$757$	−41.4419			−1.50623			−0.753115	+	0.657889i	$0.771450\pi$
$758$		−	30.3254i		−	1.10147i
$759$			4.11104i			0.149221i
$760$		−	39.0739i		−	1.41736i
$761$	27.9819			1.01434			0.507171	−	0.861845i	$-0.330691\pi$
$761$	27.9819			1.01434			0.507171	+	0.861845i	$0.330691\pi$
$762$		−	18.7611i		−	0.679645i
$763$	18.6438			0.674951
$764$	−1.51697			−0.0548819
$765$	−0.303366	+	0.0594986i	−0.0109682	+	0.00215117i
$766$	−9.43361			−0.340850
$767$	−0.264756			−0.00955978
$768$			27.7173i			1.00016i
$769$	21.2233			0.765332			0.382666	−	0.923887i	$-0.375006\pi$
$769$	21.2233			0.765332			0.382666	+	0.923887i	$0.375006\pi$
$770$			7.78433i			0.280528i
$771$		−	18.8223i		−	0.677868i
$772$			31.4956i			1.13355i
$773$	−5.53916			−0.199230			−0.0996148	−	0.995026i	$-0.531761\pi$
$773$	−5.53916			−0.199230			−0.0996148	+	0.995026i	$0.531761\pi$
$774$	1.88576			0.0677824
$775$			12.0632i			0.433323i
$776$			85.9999i			3.08722i
$777$			9.11791i			0.327103i
$778$	30.9011			1.10786
$779$			18.4610i			0.661433i
$780$	0.284406			0.0101834
$781$	−16.6629			−0.596245
$782$	25.4650	−	4.99441i	0.910627	−	0.178600i
$783$	−48.1734			−1.72158
$784$	−51.8462			−1.85165
$785$			11.7689i			0.420052i
$786$	66.3334			2.36604
$787$		−	42.8906i		−	1.52889i	−0.644692	−	0.764443i	$-0.723014\pi$
$787$		−	42.8906i		−	1.52889i	0.644692	−	0.764443i	$-0.276986\pi$
$788$		−	15.8119i		−	0.563276i
$789$		−	27.4967i		−	0.978908i
$790$	−8.71043			−0.309903
$791$	51.9865			1.84843
$792$		−	0.653292i		−	0.0232137i
$793$			0.245598i			0.00872146i
$794$			70.6033i			2.50562i
$795$	18.8409			0.668219
$796$			44.7171i			1.58496i
$797$	−16.9556			−0.600599			−0.300299	−	0.953845i	$-0.597087\pi$
$797$	−16.9556			−0.600599			−0.300299	+	0.953845i	$0.597087\pi$
$798$	−98.1483			−3.47441
$799$	−0.472763	−	2.41048i	−0.0167251	−	0.0852765i
$800$	−44.4714			−1.57230
$801$	1.15562			0.0408318
$802$			17.8682i			0.630948i
$803$	6.52734			0.230345
$804$			92.4435i			3.26023i
$805$		−	7.17963i		−	0.253049i
$806$		−	0.300257i		−	0.0105761i
$807$	−12.2200			−0.430164
$808$	105.161			3.69956
$809$		−	2.71776i		−	0.0955514i	−0.998858	−	0.0477757i	$-0.984787\pi$
$809$		−	2.71776i		−	0.0955514i	0.998858	−	0.0477757i	$-0.0152133\pi$
$810$			19.3001i			0.678137i
$811$			30.0848i			1.05642i	0.849114	+	0.528210i	$0.177137\pi$
$811$			30.0848i			1.05642i	−0.849114	+	0.528210i	$0.822863\pi$
$812$	155.203			5.44656
$813$		−	27.2036i		−	0.954072i
$814$	3.96605			0.139010
$815$	14.6822			0.514294
$816$	66.4309	−	13.0290i	2.32555	−	0.456106i
$817$	−50.9971			−1.78416
$818$	18.9730			0.663375
$819$			0.0127208i			0.000444502i
$820$	−12.0517			−0.420864
$821$			2.06536i			0.0720816i	0.999350	+	0.0360408i	$0.0114746\pi$
$821$			2.06536i			0.0720816i	−0.999350	+	0.0360408i	$0.988525\pi$
$822$		−	10.7760i		−	0.375856i
$823$			20.9681i			0.730904i	0.930830	+	0.365452i	$0.119085\pi$
$823$			20.9681i			0.730904i	−0.930830	+	0.365452i	$0.880915\pi$
$824$	51.8725			1.80706
$825$	7.30838			0.254445
$826$			59.6464i			2.07536i
$827$			39.3285i			1.36759i	0.729676	+	0.683793i	$0.239671\pi$
$827$			39.3285i			1.36759i	−0.729676	+	0.683793i	$0.760329\pi$
$828$			1.02928i			0.0357700i
$829$	23.4969			0.816082			0.408041	−	0.912964i	$-0.366212\pi$
$829$	23.4969			0.816082			0.408041	+	0.912964i	$0.366212\pi$
$830$		−	12.7096i		−	0.441158i
$831$	40.2689			1.39691
$832$	0.321495			0.0111458
$833$	−4.27560	−	21.8000i	−0.148141	−	0.755326i
$834$	60.1553			2.08301
$835$	2.06447			0.0714438
$836$			30.1795i			1.04378i
$837$	14.8425			0.513030
$838$		−	35.3988i		−	1.22283i
$839$		−	6.51649i		−	0.224974i	−0.993653	−	0.112487i	$-0.964118\pi$
$839$		−	6.51649i		−	0.224974i	0.993653	−	0.112487i	$-0.0358817\pi$
$840$		−	37.5086i		−	1.29417i
$841$	−54.5593			−1.88136
$842$	−18.5099			−0.637892
$843$			42.7513i			1.47243i
$844$		−	88.5715i		−	3.04876i
$845$		−	11.0050i		−	0.378584i
$846$	0.137825			0.00473851
$847$		−	3.51966i		−	0.120937i
$848$	125.497			4.30959
$849$	−8.43196			−0.289384
$850$	−8.87877	−	45.2702i	−0.304539	−	1.55276i
$851$	−3.65795			−0.125393
$852$	137.153			4.69879
$853$		−	34.2075i		−	1.17124i	−0.810585	−	0.585621i	$-0.800851\pi$
$853$		−	34.2075i		−	1.17124i	0.810585	−	0.585621i	$-0.199149\pi$
$854$	55.3305			1.89337
$855$			0.469082i			0.0160422i
$856$		−	69.4989i		−	2.37542i
$857$		−	17.7043i		−	0.604767i	−0.953186	−	0.302384i	$-0.902218\pi$
$857$		−	17.7043i		−	0.604767i	0.953186	−	0.302384i	$-0.0977824\pi$
$858$	−0.181908			−0.00621023
$859$	28.4323			0.970098			0.485049	−	0.874487i	$-0.338802\pi$
$859$	28.4323			0.970098			0.485049	+	0.874487i	$0.338802\pi$
$860$		−	33.2920i		−	1.13525i
$861$			17.7214i			0.603945i
$862$		−	23.6150i		−	0.804330i
$863$	−21.9955			−0.748736			−0.374368	−	0.927280i	$-0.622140\pi$
$863$	−21.9955			−0.748736			−0.374368	+	0.927280i	$0.622140\pi$
$864$			54.7172i			1.86152i
$865$	10.5867			0.359959
$866$	−18.9326			−0.643357
$867$	10.9567	+	26.8581i	0.372110	+	0.912148i
$868$	−47.8188			−1.62307
$869$	3.93839			0.133601
$870$			34.4963i			1.16953i
$871$	−0.458353			−0.0155307
$872$			39.0757i			1.32327i
$873$		−	1.03243i		−	0.0349424i
$874$		−	39.3755i		−	1.33190i
$875$	−27.6631			−0.935184
$876$	−53.7269			−1.81527
$877$			39.6572i			1.33913i	0.742754	+	0.669564i	$0.233519\pi$
$877$			39.6572i			1.33913i	−0.742754	+	0.669564i	$0.766481\pi$
$878$		−	29.6658i		−	1.00117i
$879$			51.8386i			1.74847i
$880$	−8.14688			−0.274631
$881$		−	20.8558i		−	0.702649i	−0.936254	−	0.351324i	$-0.885731\pi$
$881$		−	20.8558i		−	0.702649i	0.936254	−	0.351324i	$-0.114269\pi$
$882$	1.24647			0.0419708
$883$	0.0129833			0.000436923			0.000218462	−	1.00000i	$-0.499930\pi$
$883$	0.0129833			0.000436923			0.000218462		1.00000i	$0.499930\pi$
$884$	0.156225	+	0.796544i	0.00525441	+	0.0267907i
$885$	−9.37180			−0.315030
$886$	2.12109			0.0712596
$887$			42.2133i			1.41738i	0.705518	+	0.708692i	$0.250714\pi$
$887$			42.2133i			1.41738i	−0.705518	+	0.708692i	$0.749286\pi$
$888$	−19.1103			−0.641299
$889$		−	14.8146i		−	0.496864i
$890$		−	28.8603i		−	0.967399i
$891$		−	8.72648i		−	0.292348i
$892$	−32.5782			−1.09080
$893$	−3.72722			−0.124727
$894$			66.5016i			2.22415i
$895$		−	13.7318i		−	0.459005i
$896$			0.658735i			0.0220068i
$897$	0.167777			0.00560190
$898$			15.7773i			0.526496i
$899$	25.7450			0.858644
$900$	1.82980			0.0609933
$901$	10.3494	+	52.7684i	0.344787	+	1.75797i
$902$	7.70836			0.256660
$903$	−48.9542			−1.62909
$904$			108.959i			3.62392i
$905$	−17.2762			−0.574281
$906$		−	36.8021i		−	1.22267i
$907$		−	50.3120i		−	1.67058i	−0.549808	−	0.835291i	$-0.685299\pi$
$907$		−	50.3120i		−	1.67058i	0.549808	−	0.835291i	$-0.314701\pi$
$908$		−	57.1065i		−	1.89515i
$909$	−1.26246			−0.0418732
$910$	0.317688			0.0105313
$911$		−	37.8857i		−	1.25521i	−0.778533	−	0.627604i	$-0.784036\pi$
$911$		−	37.8857i		−	1.25521i	0.778533	−	0.627604i	$-0.215964\pi$
$912$		−	102.719i		−	3.40138i
$913$			5.74662i			0.190185i
$914$	−12.5359			−0.414650
$915$			8.69367i			0.287404i
$916$	24.6369			0.814026
$917$	52.3796			1.72973
$918$	−55.7001	+	10.9244i	−1.83838	+	0.360558i
$919$	−36.0579			−1.18944			−0.594721	−	0.803933i	$-0.702737\pi$
$919$	−36.0579			−1.18944			−0.594721	+	0.803933i	$0.702737\pi$
$920$	15.0478			0.496112
$921$		−	47.5725i		−	1.56757i
$922$	−41.6252			−1.37085
$923$			0.680033i			0.0223835i
$924$			28.9705i			0.953061i
$925$			6.50290i			0.213814i
$926$	27.6670			0.909195
$927$	−0.622729			−0.0204531
$928$			94.9099i			3.11557i
$929$			38.4271i			1.26075i	0.776289	+	0.630377i	$0.217100\pi$
$929$			38.4271i			1.26075i	−0.776289	+	0.630377i	$0.782900\pi$
$930$		−	10.6285i		−	0.348521i
$931$	−33.7085			−1.10475
$932$		−	48.2393i		−	1.58013i
$933$	−47.5016			−1.55513
$934$	−40.1420			−1.31349
$935$	−0.671849	−	3.42556i	−0.0219718	−	0.112028i
$936$	−0.0266617			−0.000871464
$937$	−4.31889			−0.141092			−0.0705461	−	0.997509i	$-0.522474\pi$
$937$	−4.31889			−0.141092			−0.0705461	+	0.997509i	$0.522474\pi$
$938$			103.262i			3.37161i
$939$	−25.5035			−0.832276
$940$		−	2.43321i		−	0.0793625i
$941$			24.5986i			0.801893i	0.916101	+	0.400946i	$0.131319\pi$
$941$			24.5986i			0.801893i	−0.916101	+	0.400946i	$0.868681\pi$
$942$			61.9592i			2.01874i
$943$	−7.10955			−0.231519
$944$	−62.4244			−2.03174
$945$			15.7041i			0.510855i
$946$			21.2938i			0.692320i
$947$		−	8.46266i		−	0.274999i	−0.990502	−	0.137500i	$-0.956093\pi$
$947$		−	8.46266i		−	0.274999i	0.990502	−	0.137500i	$-0.0439066\pi$
$948$	−32.4171			−1.05286
$949$		−	0.266389i		−	0.00864735i
$950$	−69.9995			−2.27108
$951$	−14.8206			−0.480590
$952$	105.051	−	20.6036i	3.40474	−	0.667765i
$953$	23.1966			0.751413			0.375707	−	0.926739i	$-0.377400\pi$
$953$	23.1966			0.751413			0.375707	+	0.926739i	$0.377400\pi$
$954$	−3.01716			−0.0976840
$955$		−	0.266243i		−	0.00861541i
$956$	−52.2833			−1.69096
$957$		−	15.5974i		−	0.504191i
$958$			30.6184i			0.989235i
$959$		−	8.50916i		−	0.274775i
$960$	11.3802			0.367296
$961$	23.0678			0.744124
$962$		−	0.161859i		−	0.00521855i
$963$			0.834334i			0.0268860i
$964$		−	81.0261i		−	2.60967i
$965$	−5.52779			−0.177946
$966$		−	37.7981i		−	1.21613i
$967$	−0.647242			−0.0208139			−0.0104069	−	0.999946i	$-0.503313\pi$
$967$	−0.647242			−0.0208139			−0.0104069	+	0.999946i	$0.503313\pi$
$968$	7.37688			0.237102
$969$	43.1909	−	8.47096i	1.38749	−	0.272127i
$970$	−25.7837			−0.827866
$971$	58.4544			1.87589			0.937945	−	0.346783i	$-0.112726\pi$
$971$	58.4544			1.87589			0.937945	+	0.346783i	$0.112726\pi$
$972$		−	4.43818i		−	0.142355i
$973$	47.5010			1.52281
$974$		−	45.2857i		−	1.45105i
$975$		−	0.298264i		−	0.00955209i
$976$			57.9074i			1.85357i
$977$	−18.6427			−0.596432			−0.298216	−	0.954498i	$-0.596392\pi$
$977$	−18.6427			−0.596432			−0.298216	+	0.954498i	$0.596392\pi$
$978$	77.2963			2.47166
$979$			13.0491i			0.417051i
$980$		−	22.0056i		−	0.702944i
$981$		−	0.469103i		−	0.0149773i
$982$	−89.0685			−2.84229
$983$		−	12.6846i		−	0.404576i	−0.979326	−	0.202288i	$-0.935162\pi$
$983$		−	12.6846i		−	0.404576i	0.979326	−	0.202288i	$-0.0648377\pi$
$984$	−37.1425			−1.18406
$985$	2.77515			0.0884235
$986$	−96.6147	+	18.9489i	−3.07684	+	0.603455i
$987$	−3.57791			−0.113886
$988$	1.23166			0.0391844
$989$		−	19.6396i		−	0.624503i
$990$	0.195864			0.00622498
$991$		−	21.4050i		−	0.679954i	−0.940434	−	0.339977i	$-0.889581\pi$
$991$		−	21.4050i		−	0.679954i	0.940434	−	0.339977i	$-0.110419\pi$
$992$		−	29.2422i		−	0.928440i
$993$			14.2980i			0.453734i
$994$	153.203			4.85932
$995$	−7.84829			−0.248808
$996$		−	47.3008i		−	1.49878i
$997$		−	28.3739i		−	0.898612i	−0.893378	−	0.449306i	$-0.851671\pi$
$997$		−	28.3739i		−	0.898612i	0.893378	−	0.449306i	$-0.148329\pi$
$998$		−	32.6794i		−	1.03445i
$999$	8.00111			0.253144

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	187.2.d.a.67.1	✓	16
3.2	odd	2		1683.2.g.b.1189.15		16
4.3	odd	2		2992.2.b.g.1937.12		16
17.4	even	4		3179.2.a.bb.1.8		8
17.13	even	4		3179.2.a.bc.1.8		8
17.16	even	2	inner	187.2.d.a.67.2	yes	16
51.50	odd	2		1683.2.g.b.1189.16		16
68.67	odd	2		2992.2.b.g.1937.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
187.2.d.a.67.1	✓	16	1.1	even	1	trivial
187.2.d.a.67.2	yes	16	17.16	even	2	inner
1683.2.g.b.1189.15		16	3.2	odd	2
1683.2.g.b.1189.16		16	51.50	odd	2
2992.2.b.g.1937.5		16	68.67	odd	2
2992.2.b.g.1937.12		16	4.3	odd	2
3179.2.a.bb.1.8		8	17.4	even	4
3179.2.a.bc.1.8		8	17.13	even	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 187 = 11 \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	187.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.61227 q^{2} -1.70629i q^{3} +4.82394 q^{4} +0.846648i q^{5} +4.45730i q^{6} +3.51966i q^{7} -7.37688 q^{8} +0.0885594 q^{9} +O(q^{10})\)	\(q-2.61227 q^{2} -1.70629i q^{3} +4.82394 q^{4} +0.846648i q^{5} +4.45730i q^{6} +3.51966i q^{7} -7.37688 q^{8} +0.0885594 q^{9} -2.21167i q^{10} +1.00000i q^{11} -8.23106i q^{12} +0.0408112 q^{13} -9.19429i q^{14} +1.44463 q^{15} +9.62251 q^{16} +(0.793539 + 4.04602i) q^{17} -0.231341 q^{18} +6.25620 q^{19} +4.08418i q^{20} +6.00558 q^{21} -2.61227i q^{22} +2.40934i q^{23} +12.5871i q^{24} +4.28319 q^{25} -0.106610 q^{26} -5.26999i q^{27} +16.9786i q^{28} -9.14108i q^{29} -3.77376 q^{30} +2.81641i q^{31} -10.3828 q^{32} +1.70629 q^{33} +(-2.07294 - 10.5693i) q^{34} -2.97992 q^{35} +0.427205 q^{36} +1.51824i q^{37} -16.3429 q^{38} -0.0696360i q^{39} -6.24562i q^{40} +2.95083i q^{41} -15.6882 q^{42} -8.15145 q^{43} +4.82394i q^{44} +0.0749787i q^{45} -6.29384i q^{46} -0.595765 q^{47} -16.4188i q^{48} -5.38801 q^{49} -11.1888 q^{50} +(6.90370 - 1.35401i) q^{51} +0.196871 q^{52} +13.0420 q^{53} +13.7666i q^{54} -0.846648 q^{55} -25.9641i q^{56} -10.6749i q^{57} +23.8789i q^{58} -6.48733 q^{59} +6.96881 q^{60} +6.01792i q^{61} -7.35721i q^{62} +0.311699i q^{63} +7.87761 q^{64} +0.0345528i q^{65} -4.45730 q^{66} -11.2311 q^{67} +(3.82798 + 19.5178i) q^{68} +4.11104 q^{69} +7.78433 q^{70} +16.6629i q^{71} -0.653292 q^{72} -6.52734i q^{73} -3.96605i q^{74} -7.30838i q^{75} +30.1795 q^{76} -3.51966 q^{77} +0.181908i q^{78} -3.93839i q^{79} +8.14688i q^{80} -8.72648 q^{81} -7.70836i q^{82} +5.74662 q^{83} +28.9705 q^{84} +(-3.42556 + 0.671849i) q^{85} +21.2938 q^{86} -15.5974 q^{87} -7.37688i q^{88} +13.0491 q^{89} -0.195864i q^{90} +0.143642i q^{91} +11.6225i q^{92} +4.80562 q^{93} +1.55630 q^{94} +5.29680i q^{95} +17.7161i q^{96} -11.6580i q^{97} +14.0749 q^{98} +0.0885594i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 2 q^{2} + 10 q^{4} - 6 q^{8} - 20 q^{9}+O(q^{10}) \) 16 * q - 2 * q^2 + 10 * q^4 - 6 * q^8 - 20 * q^9	\( 16 q - 2 q^{2} + 10 q^{4} - 6 q^{8} - 20 q^{9} - 16 q^{13} + 4 q^{15} + 6 q^{16} + 4 q^{17} - 10 q^{18} + 20 q^{19} + 12 q^{21} + 4 q^{25} - 12 q^{26} + 28 q^{30} - 34 q^{32} + 4 q^{33} - 6 q^{34} + 12 q^{35} - 18 q^{36} + 8 q^{43} + 14 q^{47} - 42 q^{49} - 34 q^{50} - 18 q^{51} - 44 q^{52} + 26 q^{53} + 8 q^{55} - 30 q^{59} + 72 q^{60} - 10 q^{64} - 8 q^{66} + 10 q^{67} + 22 q^{68} + 4 q^{69} - 8 q^{70} - 46 q^{72} + 36 q^{76} - 10 q^{77} - 8 q^{81} - 8 q^{83} + 92 q^{84} - 2 q^{85} + 56 q^{86} + 8 q^{87} + 10 q^{89} - 20 q^{93} + 8 q^{94} + 38 q^{98}+O(q^{100}) \) 16 * q - 2 * q^2 + 10 * q^4 - 6 * q^8 - 20 * q^9 - 16 * q^13 + 4 * q^15 + 6 * q^16 + 4 * q^17 - 10 * q^18 + 20 * q^19 + 12 * q^21 + 4 * q^25 - 12 * q^26 + 28 * q^30 - 34 * q^32 + 4 * q^33 - 6 * q^34 + 12 * q^35 - 18 * q^36 + 8 * q^43 + 14 * q^47 - 42 * q^49 - 34 * q^50 - 18 * q^51 - 44 * q^52 + 26 * q^53 + 8 * q^55 - 30 * q^59 + 72 * q^60 - 10 * q^64 - 8 * q^66 + 10 * q^67 + 22 * q^68 + 4 * q^69 - 8 * q^70 - 46 * q^72 + 36 * q^76 - 10 * q^77 - 8 * q^81 - 8 * q^83 + 92 * q^84 - 2 * q^85 + 56 * q^86 + 8 * q^87 + 10 * q^89 - 20 * q^93 + 8 * q^94 + 38 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more