LMFDB - Embedded newform 1368.2.e.e.379.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1368,2,Mod(379,1368)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1368, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 1, 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("1368.379");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1368 = 2^{3} \cdot 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1368.e (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:	$10.9235349965$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	12.0.319794774016000000.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 2x^{10} + 2x^{8} + 8x^{4} - 32x^{2} + 64 $ x^12 - 2x^10 + 2x^8 + 8x^4 - 32x^2 + 64
Coefficient ring:	$\Z[a_1, \ldots, a_{29}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 152)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			379.2
Root			$1.37364 - 0.336338i$ of defining polynomial
Character	$\chi$	$=$	1368.379
Dual form			1368.2.e.e.379.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.37364 + 0.336338i) q^{2} +(1.77375 - 0.924013i) q^{4} +3.04222i q^{5} +2.23607i q^{7} +(-2.12571 + 1.86584i) q^{8} +O(q^{10})$	\(q+(-1.37364 + 0.336338i) q^{2} +(1.77375 - 0.924013i) q^{4} +3.04222i q^{5} +2.23607i q^{7} +(-2.12571 + 1.86584i) q^{8} +(-1.02321 - 4.17890i) q^{10} +2.29240 q^{11} -3.09769 q^{13} +(-0.752075 - 3.07154i) q^{14} +(2.29240 - 3.27794i) q^{16} -5.58480 q^{17} +(-4.29240 - 0.758478i) q^{19} +(2.81105 + 5.39615i) q^{20} +(-3.14893 + 0.771022i) q^{22} +5.51401i q^{23} -4.25511 q^{25} +(4.25511 - 1.04187i) q^{26} +(2.06615 + 3.96623i) q^{28} -1.85457 q^{29} -4.25142 q^{31} +(-2.04643 + 5.27372i) q^{32} +(7.67149 - 1.87838i) q^{34} -6.80261 q^{35} +1.24312 q^{37} +(6.15130 - 0.401826i) q^{38} +(-5.67629 - 6.46688i) q^{40} -8.33272i q^{41} +4.87720 q^{43} +(4.06615 - 2.11821i) q^{44} +(-1.85457 - 7.57424i) q^{46} -9.36238i q^{47} +2.00000 q^{49} +(5.84497 - 1.43115i) q^{50} +(-5.49455 + 2.86231i) q^{52} +11.4420 q^{53} +6.97399i q^{55} +(-4.17214 - 4.75323i) q^{56} +(2.54751 - 0.623763i) q^{58} -2.49721i q^{59} +2.26613i q^{61} +(5.83991 - 1.42992i) q^{62} +(1.03730 - 7.93247i) q^{64} -9.42387i q^{65} +4.49015i q^{67} +(-9.90606 + 5.16043i) q^{68} +(9.34431 - 2.28798i) q^{70} -14.6982 q^{71} +1.51021 q^{73} +(-1.70760 + 0.418110i) q^{74} +(-8.31450 + 2.62088i) q^{76} +5.12597i q^{77} -11.5314 q^{79} +(9.97222 + 6.97399i) q^{80} +(2.80261 + 11.4461i) q^{82} -2.00000 q^{83} -16.9902i q^{85} +(-6.69951 + 1.64039i) q^{86} +(-4.87298 + 4.27725i) q^{88} +5.29881i q^{89} -6.92665i q^{91} +(5.09501 + 9.78049i) q^{92} +(3.14893 + 12.8605i) q^{94} +(2.30746 - 13.0584i) q^{95} +1.17375i q^{97} +(-2.74727 + 0.672676i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 4 q^{4}+O(q^{10}) $ 12 * q + 4 * q^4	$ 12 q + 4 q^{4} - 12 q^{17} - 24 q^{19} - 4 q^{20} - 44 q^{25} + 44 q^{26} - 20 q^{28} - 40 q^{35} - 4 q^{38} - 24 q^{43} + 4 q^{44} + 24 q^{49} - 4 q^{58} + 8 q^{62} - 8 q^{64} - 12 q^{68} + 4 q^{73} - 48 q^{74} - 12 q^{76} - 32 q^{80} - 8 q^{82} - 24 q^{83} - 8 q^{92}+O(q^{100}) $ 12 * q + 4 * q^4 - 12 * q^17 - 24 * q^19 - 4 * q^20 - 44 * q^25 + 44 * q^26 - 20 * q^28 - 40 * q^35 - 4 * q^38 - 24 * q^43 + 4 * q^44 + 24 * q^49 - 4 * q^58 + 8 * q^62 - 8 * q^64 - 12 * q^68 + 4 * q^73 - 48 * q^74 - 12 * q^76 - 32 * q^80 - 8 * q^82 - 24 * q^83 - 8 * q^92

Display 100 coefficients 10 coefficients

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 $

$2$	−1.37364	+	0.336338i	−0.971308	+	0.237827i
$2$	−1.37364	+	0.336338i	−0.971308	+	0.237827i
$3$		0			0
$3$		0			0
$4$	1.77375	−	0.924013i	0.886877	−	0.462006i
$5$			3.04222i			1.36052i	0.732970	+	0.680261i	$0.238134\pi$
$5$			3.04222i			1.36052i	−0.732970	+	0.680261i	$0.761866\pi$
$6$		0			0
$7$			2.23607i			0.845154i	0.906327	+	0.422577i	$0.138874\pi$
$7$			2.23607i			0.845154i	−0.906327	+	0.422577i	$0.861126\pi$
$8$	−2.12571	+	1.86584i	−0.751552	+	0.659673i
$9$		0			0
$10$	−1.02321	−	4.17890i	−0.323569	−	1.32149i
$11$	2.29240			0.691185			0.345593	−	0.938385i	$-0.387678\pi$
$11$	2.29240			0.691185			0.345593	+	0.938385i	$0.387678\pi$
$12$		0			0
$13$	−3.09769			−0.859146			−0.429573	−	0.903032i	$-0.641336\pi$
$13$	−3.09769			−0.859146			−0.429573	+	0.903032i	$0.641336\pi$
$14$	−0.752075	−	3.07154i	−0.201000	−	0.820905i
$15$		0			0
$16$	2.29240	−	3.27794i	0.573100	−	0.819485i
$17$	−5.58480			−1.35451			−0.677257	−	0.735747i	$-0.736831\pi$
$17$	−5.58480			−1.35451			−0.677257	+	0.735747i	$0.736831\pi$
$18$		0			0
$19$	−4.29240	−	0.758478i	−0.984744	−	0.174007i
$19$	−4.29240	−	0.758478i	−0.984744	−	0.174007i
$20$	2.81105	+	5.39615i	0.628570	+	1.20662i
$21$		0			0
$22$	−3.14893	+	0.771022i	−0.671353	+	0.164382i
$23$			5.51401i			1.14975i	0.818241	+	0.574875i	$0.194949\pi$
$23$			5.51401i			1.14975i	−0.818241	+	0.574875i	$0.805051\pi$
$24$		0			0
$25$	−4.25511			−0.851021
$26$	4.25511	−	1.04187i	0.834495	−	0.204328i
$27$		0			0
$28$	2.06615	+	3.96623i	0.390467	+	0.749548i
$29$	−1.85457			−0.344385			−0.172193	−	0.985063i	$-0.555085\pi$
$29$	−1.85457			−0.344385			−0.172193	+	0.985063i	$0.555085\pi$
$30$		0			0
$31$	−4.25142			−0.763578			−0.381789	−	0.924250i	$-0.624692\pi$
$31$	−4.25142			−0.763578			−0.381789	+	0.924250i	$0.624692\pi$
$32$	−2.04643	+	5.27372i	−0.361761	+	0.932271i
$33$		0			0
$34$	7.67149	−	1.87838i	1.31565	−	0.322140i
$35$	−6.80261			−1.14985
$36$		0			0
$37$	1.24312			0.204368			0.102184	−	0.994766i	$-0.467417\pi$
$37$	1.24312			0.204368			0.102184	+	0.994766i	$0.467417\pi$
$38$	6.15130	−	0.401826i	0.997873	−	0.0651847i
$39$		0			0
$40$	−5.67629	−	6.46688i	−0.897500	−	1.02250i
$41$		−	8.33272i		−	1.30135i	−0.759355	−	0.650676i	$-0.774486\pi$
$41$		−	8.33272i		−	1.30135i	0.759355	−	0.650676i	$-0.225514\pi$
$42$		0			0
$43$	4.87720			0.743767			0.371883	−	0.928279i	$-0.378712\pi$
$43$	4.87720			0.743767			0.371883	+	0.928279i	$0.378712\pi$
$44$	4.06615	−	2.11821i	0.612996	−	0.319332i
$45$		0			0
$46$	−1.85457	−	7.57424i	−0.273442	−	1.11676i
$47$		−	9.36238i		−	1.36564i	−0.730585	−	0.682822i	$-0.760753\pi$
$47$		−	9.36238i		−	1.36564i	0.730585	−	0.682822i	$-0.239247\pi$
$48$		0			0
$49$	2.00000			0.285714
$50$	5.84497	−	1.43115i	0.826603	−	0.202396i
$51$		0			0
$52$	−5.49455	+	2.86231i	−0.761956	+	0.396931i
$53$	11.4420			1.57168			0.785838	−	0.618432i	$-0.212232\pi$
$53$	11.4420			1.57168			0.785838	+	0.618432i	$0.212232\pi$
$54$		0			0
$55$			6.97399i			0.940373i
$56$	−4.17214	−	4.75323i	−0.557526	−	0.635178i
$57$		0			0
$58$	2.54751	−	0.623763i	0.334504	−	0.0819041i
$59$		−	2.49721i		−	0.325110i	−0.986700	−	0.162555i	$-0.948027\pi$
$59$		−	2.49721i		−	0.325110i	0.986700	−	0.162555i	$-0.0519734\pi$
$60$		0			0
$61$			2.26613i			0.290149i	0.989421	+	0.145074i	$0.0463422\pi$
$61$			2.26613i			0.290149i	−0.989421	+	0.145074i	$0.953658\pi$
$62$	5.83991	−	1.42992i	0.741669	−	0.181599i
$63$		0			0
$64$	1.03730	−	7.93247i	0.129662	−	0.991558i
$65$		−	9.42387i		−	1.16889i
$66$		0			0
$67$			4.49015i			0.548560i	0.961650	+	0.274280i	$0.0884394\pi$
$67$			4.49015i			0.548560i	−0.961650	+	0.274280i	$0.911561\pi$
$68$	−9.90606	+	5.16043i	−1.20129	+	0.625794i
$69$		0			0
$70$	9.34431	−	2.28798i	1.11686	−	0.273466i
$71$	−14.6982			−1.74436			−0.872180	−	0.489186i	$-0.837294\pi$
$71$	−14.6982			−1.74436			−0.872180	+	0.489186i	$0.837294\pi$
$72$		0			0
$73$	1.51021			0.176757			0.0883784	−	0.996087i	$-0.471832\pi$
$73$	1.51021			0.176757			0.0883784	+	0.996087i	$0.471832\pi$
$74$	−1.70760	+	0.418110i	−0.198504	+	0.0486043i
$75$		0			0
$76$	−8.31450	+	2.62088i	−0.953739	+	0.300636i
$77$			5.12597i			0.584158i
$78$		0			0
$79$	−11.5314			−1.29738			−0.648690	−	0.761053i	$-0.724683\pi$
$79$	−11.5314			−1.29738			−0.648690	+	0.761053i	$0.724683\pi$
$80$	9.97222	+	6.97399i	1.11493	+	0.779716i
$81$		0			0
$82$	2.80261	+	11.4461i	0.309497	+	1.26401i
$83$	−2.00000			−0.219529			−0.109764	−	0.993958i	$-0.535010\pi$
$83$	−2.00000			−0.219529			−0.109764	+	0.993958i	$0.535010\pi$
$84$		0			0
$85$		−	16.9902i		−	1.84285i
$86$	−6.69951	+	1.64039i	−0.722426	+	0.176888i
$87$		0			0
$88$	−4.87298	+	4.27725i	−0.519462	+	0.455956i
$89$			5.29881i			0.561673i	0.959756	+	0.280836i	$0.0906118\pi$
$89$			5.29881i			0.561673i	−0.959756	+	0.280836i	$0.909388\pi$
$90$		0			0
$91$		−	6.92665i		−	0.726111i
$92$	5.09501	+	9.78049i	0.531192	+	1.01969i
$93$		0			0
$94$	3.14893	+	12.8605i	0.324787	+	1.32646i
$95$	2.30746	−	13.0584i	0.236740	−	1.33977i
$96$		0			0
$97$			1.17375i			0.119176i	0.998223	+	0.0595881i	$0.0189787\pi$
$97$			1.17375i			0.119176i	−0.998223	+	0.0595881i	$0.981021\pi$
$98$	−2.74727	+	0.672676i	−0.277516	+	0.0679506i
$99$		0			0
$100$	−7.54751	+	3.93177i	−0.754751	+	0.393177i
$101$		−	12.6403i		−	1.25776i	−0.777503	−	0.628880i	$-0.783514\pi$
$101$		−	12.6403i		−	1.25776i	0.777503	−	0.628880i	$-0.216486\pi$
$102$		0			0
$103$	−3.70914			−0.365473			−0.182736	−	0.983162i	$-0.558495\pi$
$103$	−3.70914			−0.365473			−0.182736	+	0.983162i	$0.558495\pi$
$104$	6.58480	−	5.77980i	0.645693	−	0.566756i
$105$		0			0
$106$	−15.7171	+	3.84837i	−1.52658	+	0.373787i
$107$			11.1343i			1.07639i	0.842819	+	0.538197i	$0.180895\pi$
$107$			11.1343i			1.07639i	−0.842819	+	0.538197i	$0.819105\pi$
$108$		0			0
$109$	11.4420			1.09594			0.547971	−	0.836497i	$-0.315400\pi$
$109$	11.4420			1.09594			0.547971	+	0.836497i	$0.315400\pi$
$110$	−2.34562	−	9.57973i	−0.223646	−	0.913391i
$111$		0			0
$112$	7.32970	+	5.12597i	0.692591	+	0.484358i
$113$			17.5348i			1.64954i	0.565471	+	0.824768i	$0.308694\pi$
$113$			17.5348i			1.64954i	−0.565471	+	0.824768i	$0.691306\pi$
$114$		0			0
$115$	−16.7748			−1.56426
$116$	−3.28955	+	1.71365i	−0.305427	+	0.159108i
$117$		0			0
$118$	0.839908	+	3.43026i	0.0773198	+	0.315781i
$119$		−	12.4880i		−	1.14477i
$120$		0			0
$121$	−5.74489			−0.522263
$122$	−0.762188	−	3.11284i	−0.0690052	−	0.281824i
$123$		0			0
$124$	−7.54097	+	3.92837i	−0.677200	+	0.352778i
$125$			2.26613i			0.202689i
$126$		0			0
$127$	−2.84974			−0.252873			−0.126437	−	0.991975i	$-0.540354\pi$
$127$	−2.84974			−0.252873			−0.126437	+	0.991975i	$0.540354\pi$
$128$	1.24312	+	11.2452i	0.109878	+	0.993945i
$129$		0			0
$130$	3.16961	+	12.9450i	0.277993	+	1.13535i
$131$	−7.97222			−0.696536			−0.348268	−	0.937395i	$-0.613230\pi$
$131$	−7.97222			−0.696536			−0.348268	+	0.937395i	$0.613230\pi$
$132$		0			0
$133$	1.69601	−	9.59810i	0.147063	−	0.832261i
$134$	−1.51021	−	6.16784i	−0.130462	−	0.532820i
$135$		0			0
$136$	11.8717	−	10.4203i	1.01799	−	0.893537i
$137$	−9.19003			−0.785157			−0.392578	−	0.919719i	$-0.628417\pi$
$137$	−9.19003			−0.785157			−0.392578	+	0.919719i	$0.628417\pi$
$138$		0			0
$139$	−3.89762			−0.330592			−0.165296	−	0.986244i	$-0.552858\pi$
$139$	−3.89762			−0.330592			−0.165296	+	0.986244i	$0.552858\pi$
$140$	−12.0662	+	6.28570i	−1.01978	+	0.531238i
$141$		0			0
$142$	20.1900	−	4.94358i	1.69431	−	0.414856i
$143$	−7.10116			−0.593829
$144$		0			0
$145$		−	5.64202i		−	0.468544i
$146$	−2.07448	+	0.507941i	−0.171685	+	0.0420375i
$147$		0			0
$148$	2.20499	−	1.14866i	0.181249	−	0.0944193i
$149$		−	10.4343i		−	0.854813i	−0.904060	−	0.427406i	$-0.859427\pi$
$149$		−	10.4343i		−	0.854813i	0.904060	−	0.427406i	$-0.140573\pi$
$150$		0			0
$151$	−10.9891			−0.894279			−0.447140	−	0.894464i	$-0.647557\pi$
$151$	−10.9891			−0.894279			−0.447140	+	0.894464i	$0.647557\pi$
$152$	10.5396	−	6.39662i	0.854875	−	0.518835i
$153$		0			0
$154$	−1.72406	−	7.04121i	−0.138929	−	0.567397i
$155$		−	12.9338i		−	1.03886i
$156$		0			0
$157$		−	5.77980i		−	0.461278i	−0.973039	−	0.230639i	$-0.925918\pi$
$157$		−	5.77980i		−	0.461278i	0.973039	−	0.230639i	$-0.0740816\pi$
$158$	15.8399	−	3.87844i	1.26016	−	0.308552i
$159$		0			0
$160$	−16.0438	−	6.22569i	−1.26838	−	0.492184i
$161$	−12.3297			−0.971716
$162$		0			0
$163$	6.65940			0.521604			0.260802	−	0.965392i	$-0.416013\pi$
$163$	6.65940			0.521604			0.260802	+	0.965392i	$0.416013\pi$
$164$	−7.69954	−	14.7802i	−0.601233	−	1.15414i
$165$		0			0
$166$	2.74727	−	0.672676i	0.213230	−	0.0522098i
$167$	−1.94397			−0.150429			−0.0752143	−	0.997167i	$-0.523964\pi$
$167$	−1.94397			−0.150429			−0.0752143	+	0.997167i	$0.523964\pi$
$168$		0			0
$169$	−3.40429			−0.261869
$170$	5.71445	+	23.3384i	0.438279	+	1.78997i
$171$		0			0
$172$	8.65096	−	4.50660i	0.659629	−	0.343625i
$173$	−20.5765			−1.56440			−0.782201	−	0.623026i	$-0.785903\pi$
$173$	−20.5765			−1.56440			−0.782201	+	0.623026i	$0.785903\pi$
$174$		0			0
$175$		−	9.51470i		−	0.719244i
$176$	5.25511	−	7.51436i	0.396118	−	0.566416i
$177$		0			0
$178$	−1.78219	−	7.27864i	−0.133581	−	0.545557i
$179$			8.72610i			0.652220i	0.945332	+	0.326110i	$0.105738\pi$
$179$			8.72610i			0.652220i	−0.945332	+	0.326110i	$0.894262\pi$
$180$		0			0
$181$	8.50284			0.632011			0.316006	−	0.948757i	$-0.397658\pi$
$181$	8.50284			0.632011			0.316006	+	0.948757i	$0.397658\pi$
$182$	2.32970	+	9.51470i	0.172689	+	0.705277i
$183$		0			0
$184$	−10.2882	−	11.7212i	−0.758460	−	0.864098i
$185$			3.78185i			0.278047i
$186$		0			0
$187$	−12.8026			−0.936220
$188$	−8.65096	−	16.6066i	−0.630936	−	1.21116i
$189$		0			0
$190$	1.22244	+	18.7136i	0.0886853	+	1.35763i
$191$			11.2405i			0.813332i	0.913577	+	0.406666i	$0.133309\pi$
$191$			11.2405i			0.813332i	−0.913577	+	0.406666i	$0.866691\pi$
$192$		0			0
$193$			7.76773i			0.559134i	0.960126	+	0.279567i	$0.0901909\pi$
$193$			7.76773i			0.559134i	−0.960126	+	0.279567i	$0.909809\pi$
$194$	−0.394777	−	1.61230i	−0.0283433	−	0.115757i
$195$		0			0
$196$	3.54751	−	1.84803i	0.253393	−	0.132002i
$197$			3.63592i			0.259048i	0.991576	+	0.129524i	$0.0413450\pi$
$197$			3.63592i			0.259048i	−0.991576	+	0.129524i	$0.958655\pi$
$198$		0			0
$199$		−	5.46068i		−	0.387097i	−0.981091	−	0.193549i	$-0.938000\pi$
$199$		−	5.46068i		−	0.387097i	0.981091	−	0.193549i	$-0.0619997\pi$
$200$	9.04512	−	7.93934i	0.639587	−	0.561396i
$201$		0			0
$202$	4.25142	+	17.3632i	0.299129	+	1.22167i
$203$		−	4.14695i		−	0.291059i
$204$		0			0
$205$	25.3500			1.77052
$206$	5.09501	−	1.24753i	0.354986	−	0.0869193i
$207$		0			0
$208$	−7.10116	+	10.1541i	−0.492377	+	0.704057i
$209$	−9.83991	−	1.73874i	−0.680641	−	0.120271i
$210$		0			0
$211$		−	15.3420i		−	1.05619i	−0.849187	−	0.528093i	$-0.822907\pi$
$211$		−	15.3420i		−	1.05619i	0.849187	−	0.528093i	$-0.177093\pi$
$212$	20.2952	−	10.5725i	1.39388	−	0.726124i
$213$		0			0
$214$	−3.74489	−	15.2945i	−0.255996	−	1.04551i
$215$			14.8375i			1.01191i
$216$		0			0
$217$		−	9.50647i		−	0.645341i
$218$	−15.7171	+	3.84837i	−1.06450	+	0.260645i
$219$		0			0
$220$	6.44406	+	12.3701i	0.434458	+	0.833995i
$221$	17.3000			1.16372
$222$		0			0
$223$	−10.4468			−0.699570			−0.349785	−	0.936830i	$-0.613745\pi$
$223$	−10.4468			−0.699570			−0.349785	+	0.936830i	$0.613745\pi$
$224$	−11.7924	−	4.57596i	−0.787913	−	0.305744i
$225$		0			0
$226$	−5.89762	−	24.0864i	−0.392304	−	1.60221i
$227$			0.193492i			0.0128425i	0.999979	+	0.00642124i	$0.00204396\pi$
$227$			0.193492i			0.0128425i	−0.999979	+	0.00642124i	$0.997956\pi$
$228$		0			0
$229$			19.3185i			1.27660i	0.769788	+	0.638300i	$0.220362\pi$
$229$			19.3185i			1.27660i	−0.769788	+	0.638300i	$0.779638\pi$
$230$	23.0425	−	5.64202i	1.51938	−	0.372023i
$231$		0			0
$232$	3.94228	−	3.46033i	0.258824	−	0.227182i
$233$	12.4247			0.813970			0.406985	−	0.913435i	$-0.366580\pi$
$233$	12.4247			0.813970			0.406985	+	0.913435i	$0.366580\pi$
$234$		0			0
$235$	28.4824			1.85799
$236$	−2.30746	−	4.42944i	−0.150203	−	0.288332i
$237$		0			0
$238$	4.20019	+	17.1540i	0.272258	+	1.11193i
$239$		−	0.152323i		−	0.00985293i	−0.999988	−	0.00492647i	$-0.998432\pi$
$239$		−	0.152323i		−	0.00985293i	0.999988	−	0.00492647i	$-0.00156815\pi$
$240$		0			0
$241$		−	11.9754i		−	0.771403i	−0.922624	−	0.385701i	$-0.873959\pi$
$241$		−	11.9754i		−	0.771403i	0.922624	−	0.385701i	$-0.126041\pi$
$242$	7.89140	−	1.93223i	0.507278	−	0.124208i
$243$		0			0
$244$	2.09394	+	4.01956i	0.134051	+	0.257326i
$245$			6.08444i			0.388721i
$246$		0			0
$247$	13.2965	+	2.34953i	0.846039	+	0.149497i
$248$	9.03730	−	7.93247i	0.573869	−	0.503712i
$249$		0			0
$250$	−0.762188	−	3.11284i	−0.0482050	−	0.196874i
$251$	−7.82303			−0.493785			−0.246893	−	0.969043i	$-0.579410\pi$
$251$	−7.82303			−0.493785			−0.246893	+	0.969043i	$0.579410\pi$
$252$		0			0
$253$			12.6403i			0.794690i
$254$	3.91450	−	0.958475i	0.245618	−	0.0601401i
$255$		0			0
$256$	−5.48979	−	15.0287i	−0.343112	−	0.939295i
$257$			13.4097i			0.836477i	0.908337	+	0.418239i	$0.137352\pi$
$257$			13.4097i			0.836477i	−0.908337	+	0.418239i	$0.862648\pi$
$258$		0			0
$259$			2.77971i			0.172723i
$260$	−8.70777	−	16.7156i	−0.540033	−	1.03666i
$261$		0			0
$262$	10.9509	−	2.68136i	0.676551	−	0.165655i
$263$		−	4.11416i		−	0.253690i	−0.991923	−	0.126845i	$-0.959515\pi$
$263$		−	4.11416i		−	0.253690i	0.991923	−	0.126845i	$-0.0404851\pi$
$264$		0			0
$265$			34.8090i			2.13830i
$266$	0.898510	+	13.7547i	0.0550912	+	0.843357i
$267$		0			0
$268$	4.14896	+	7.96443i	0.253438	+	0.486505i
$269$	26.4345			1.61174			0.805871	−	0.592091i	$-0.201698\pi$
$269$	26.4345			1.61174			0.805871	+	0.592091i	$0.201698\pi$
$270$		0			0
$271$			19.2952i			1.17210i	0.810275	+	0.586050i	$0.199318\pi$
$271$			19.2952i			1.17210i	−0.810275	+	0.586050i	$0.800682\pi$
$272$	−12.8026	+	18.3067i	−0.776272	+	1.11000i
$273$		0			0
$274$	12.6238	−	3.09096i	0.762629	−	0.186732i
$275$	−9.75441			−0.588213
$276$		0			0
$277$		−	26.7106i		−	1.60488i	−0.596731	−	0.802441i	$-0.703534\pi$
$277$		−	26.7106i		−	1.60488i	0.596731	−	0.802441i	$-0.296466\pi$
$278$	5.35392	−	1.31092i	0.321107	−	0.0786238i
$279$		0			0
$280$	14.4604	−	12.6926i	0.864173	−	0.758526i
$281$			23.4424i			1.39846i	0.714899	+	0.699228i	$0.246473\pi$
$281$			23.4424i			1.39846i	−0.714899	+	0.699228i	$0.753527\pi$
$282$		0			0
$283$	−11.5366			−0.685780			−0.342890	−	0.939376i	$-0.611406\pi$
$283$	−11.5366			−0.685780			−0.342890	+	0.939376i	$0.611406\pi$
$284$	−26.0710	+	13.5814i	−1.54703	+	0.805905i
$285$		0			0
$286$	9.75441	−	2.38839i	0.576790	−	0.141228i
$287$	18.6325			1.09984
$288$		0			0
$289$	14.1900			0.834707
$290$	1.89762	+	7.75008i	0.111432	+	0.455100i
$291$		0			0
$292$	2.67874	−	1.39545i	0.156761	−	0.0816627i
$293$	11.7793			0.688156			0.344078	−	0.938941i	$-0.388192\pi$
$293$	11.7793			0.688156			0.344078	+	0.938941i	$0.388192\pi$
$294$		0			0
$295$	7.59707			0.442319
$296$	−2.64252	+	2.31947i	−0.153593	+	0.134816i
$297$		0			0
$298$	3.50946	+	14.3330i	0.203298	+	0.830286i
$299$		−	17.0807i		−	0.987803i
$300$		0			0
$301$			10.9058i			0.628598i
$302$	15.0950	−	3.69605i	0.868620	−	0.212684i
$303$		0			0
$304$	−12.3262	+	12.3315i	−0.706953	+	0.707260i
$305$	−6.89408			−0.394754
$306$		0			0
$307$		−	3.16669i		−	0.180733i	−0.995909	−	0.0903663i	$-0.971196\pi$
$307$		−	3.16669i		−	0.180733i	0.995909	−	0.0903663i	$-0.0288038\pi$
$308$	4.73646	+	9.09220i	0.269885	+	0.518076i
$309$		0			0
$310$	4.35012	+	17.7663i	0.247070	+	1.00906i
$311$			9.73483i			0.552012i	0.961156	+	0.276006i	$0.0890109\pi$
$311$			9.73483i			0.552012i	−0.961156	+	0.276006i	$0.910989\pi$
$312$		0			0
$313$	−21.0299			−1.18868			−0.594341	−	0.804213i	$-0.702587\pi$
$313$	−21.0299			−1.18868			−0.594341	+	0.804213i	$0.702587\pi$
$314$	1.94397	+	7.93934i	0.109704	+	0.448043i
$315$		0			0
$316$	−20.4538	+	10.6551i	−1.15062	+	0.599398i
$317$	−10.1989			−0.572825			−0.286412	−	0.958106i	$-0.592463\pi$
$317$	−10.1989			−0.572825			−0.286412	+	0.958106i	$0.592463\pi$
$318$		0			0
$319$	−4.25142			−0.238034
$320$	24.1323	+	3.15568i	1.34904	+	0.176408i
$321$		0			0
$322$	16.9365	−	4.14695i	0.943835	−	0.231100i
$323$	23.9722	+	4.23595i	1.33385	+	0.235695i
$324$		0			0
$325$	13.1810			0.731151
$326$	−9.14759	+	2.23981i	−0.506638	+	0.124052i
$327$		0			0
$328$	15.5475	+	17.7130i	0.858467	+	0.978034i
$329$	20.9349			1.15418
$330$		0			0
$331$			31.4100i			1.72645i	0.504819	+	0.863225i	$0.331559\pi$
$331$			31.4100i			1.72645i	−0.504819	+	0.863225i	$0.668441\pi$
$332$	−3.54751	+	1.84803i	−0.194695	+	0.101424i
$333$		0			0
$334$	2.67030	−	0.653830i	0.146112	−	0.0357760i
$335$	−13.6600			−0.746328
$336$		0			0
$337$			23.0554i			1.25591i	0.778251	+	0.627954i	$0.216107\pi$
$337$			23.0554i			1.25591i	−0.778251	+	0.627954i	$0.783893\pi$
$338$	4.67626	−	1.14499i	0.254355	−	0.0622794i
$339$		0			0
$340$	−15.6992	−	30.1364i	−0.851407	−	1.63438i
$341$	−9.74597			−0.527774
$342$		0			0
$343$			20.1246i			1.08663i
$344$	−10.3675	+	9.10007i	−0.558980	+	0.490643i
$345$		0			0
$346$	28.2646	−	6.92066i	1.51952	−	0.372057i
$347$	−29.3319			−1.57462			−0.787308	−	0.616560i	$-0.788526\pi$
$347$	−29.3319			−1.57462			−0.787308	+	0.616560i	$0.788526\pi$
$348$		0			0
$349$			11.9865i			0.641622i	0.947143	+	0.320811i	$0.103955\pi$
$349$			11.9865i			0.641622i	−0.947143	+	0.320811i	$0.896045\pi$
$350$	3.20016	+	13.0697i	0.171056	+	0.698607i
$351$		0			0
$352$	−4.69124	+	12.0895i	−0.250044	+	0.644372i
$353$	12.5197			0.666358			0.333179	−	0.942864i	$-0.391879\pi$
$353$	12.5197			0.666358			0.333179	+	0.942864i	$0.391879\pi$
$354$		0			0
$355$		−	44.7153i		−	2.37324i
$356$	4.89617	+	9.39878i	0.259496	+	0.498134i
$357$		0			0
$358$	−2.93492	−	11.9865i	−0.155115	−	0.633506i
$359$			16.0173i			0.845358i	0.906279	+	0.422679i	$0.138910\pi$
$359$			16.0173i			0.845358i	−0.906279	+	0.422679i	$0.861090\pi$
$360$		0			0
$361$	17.8494	+	6.51138i	0.939443	+	0.342704i
$362$	−11.6798	+	2.85983i	−0.613878	+	0.150309i
$363$		0			0
$364$	−6.40032	−	12.2862i	−0.335468	−	0.643971i
$365$			4.59439i			0.240481i
$366$		0			0
$367$		−	28.0338i		−	1.46335i	−0.681652	−	0.731677i	$-0.738738\pi$
$367$		−	28.0338i		−	1.46335i	0.681652	−	0.731677i	$-0.261262\pi$
$368$	18.0746	+	12.6403i	0.942203	+	0.658922i
$369$		0			0
$370$	−1.27198	−	5.19489i	−0.0661272	−	0.270070i
$371$			25.5850i			1.32831i
$372$		0			0
$373$	13.7697			0.712966			0.356483	−	0.934302i	$-0.383976\pi$
$373$	13.7697			0.712966			0.356483	+	0.934302i	$0.383976\pi$
$374$	17.5861	−	4.30601i	0.909357	−	0.222658i
$375$		0			0
$376$	17.4687	+	19.9017i	0.900879	+	1.02635i
$377$	5.74489			0.295877
$378$		0			0
$379$			29.8817i			1.53492i	0.641097	+	0.767460i	$0.278480\pi$
$379$			29.8817i			1.53492i	−0.641097	+	0.767460i	$0.721520\pi$
$380$	−7.97330	−	25.2946i	−0.409021	−	1.29758i
$381$		0			0
$382$	−3.78060	−	15.4403i	−0.193432	−	0.789996i
$383$	−8.72800			−0.445980			−0.222990	−	0.974821i	$-0.571582\pi$
$383$	−8.72800			−0.445980			−0.222990	+	0.974821i	$0.571582\pi$
$384$		0			0
$385$	−15.5943			−0.794760
$386$	−2.61259	−	10.6700i	−0.132977	−	0.543091i
$387$		0			0
$388$	1.08456	+	2.08194i	0.0550602	+	0.105695i
$389$		−	20.6261i		−	1.04579i	−0.852398	−	0.522893i	$-0.824853\pi$
$389$		−	20.6261i		−	1.04579i	0.852398	−	0.522893i	$-0.175147\pi$
$390$		0			0
$391$		−	30.7947i		−	1.55735i
$392$	−4.25142	+	3.73168i	−0.214729	+	0.188478i
$393$		0			0
$394$	−1.22290	−	4.99443i	−0.0616087	−	0.251616i
$395$		−	35.0810i		−	1.76512i
$396$		0			0
$397$			23.3793i			1.17337i	0.809814	+	0.586686i	$0.199568\pi$
$397$			23.3793i			1.17337i	−0.809814	+	0.586686i	$0.800432\pi$
$398$	1.83663	+	7.50099i	0.0920622	+	0.375990i
$399$		0			0
$400$	−9.75441	+	13.9480i	−0.487720	+	0.697399i
$401$			3.17309i			0.158457i	0.996856	+	0.0792283i	$0.0252456\pi$
$401$			3.17309i			0.158457i	−0.996856	+	0.0792283i	$0.974754\pi$
$402$		0			0
$403$	13.1696			0.656025
$404$	−11.6798	−	22.4208i	−0.581093	−	1.11548i
$405$		0			0
$406$	1.39478	+	5.69640i	0.0692216	+	0.282707i
$407$	2.84974			0.141256
$408$		0			0
$409$		−	8.55450i		−	0.422993i	−0.977379	−	0.211496i	$-0.932166\pi$
$409$		−	8.55450i		−	0.422993i	0.977379	−	0.211496i	$-0.0678337\pi$
$410$	−34.8216	+	8.52616i	−1.71972	+	0.421077i
$411$		0			0
$412$	−6.57910	+	3.42729i	−0.324129	+	0.168851i
$413$	5.58394			0.274768
$414$		0			0
$415$		−	6.08444i		−	0.298673i
$416$	6.33921	−	16.3364i	0.310805	−	0.800957i
$417$		0			0
$418$	14.1013	−	0.921146i	0.689715	−	0.0450547i
$419$	35.5497			1.73671			0.868357	−	0.495939i	$-0.165176\pi$
$419$	35.5497			1.73671			0.868357	+	0.495939i	$0.165176\pi$
$420$		0			0
$421$	28.6265			1.39517			0.697584	−	0.716503i	$-0.254258\pi$
$421$	28.6265			1.39517			0.697584	+	0.716503i	$0.254258\pi$
$422$	5.16009	+	21.0743i	0.251189	+	1.02588i
$423$		0			0
$424$	−24.3223	+	21.3489i	−1.18120	+	1.03679i
$425$	23.7639			1.15272
$426$		0			0
$427$	−5.06723			−0.245221
$428$	10.2882	+	19.7495i	0.497301	+	0.954630i
$429$		0			0
$430$	−4.99043	−	20.3814i	−0.240660	−	0.982877i
$431$	−31.5656			−1.52046			−0.760230	−	0.649654i	$-0.774914\pi$
$431$	−31.5656			−1.52046			−0.760230	+	0.649654i	$0.774914\pi$
$432$		0			0
$433$			18.4868i			0.888418i	0.895923	+	0.444209i	$0.146515\pi$
$433$			18.4868i			0.888418i	−0.895923	+	0.444209i	$0.853485\pi$
$434$	3.19739	+	13.0584i	0.153480	+	0.626825i
$435$		0			0
$436$	20.2952	−	10.5725i	0.971966	−	0.506332i
$437$	4.18225	−	23.6683i	0.200064	−	1.13221i
$438$		0			0
$439$	−10.6256			−0.507132			−0.253566	−	0.967318i	$-0.581604\pi$
$439$	−10.6256			−0.507132			−0.253566	+	0.967318i	$0.581604\pi$
$440$	−13.0123	−	14.8247i	−0.620339	−	0.706739i
$441$		0			0
$442$	−23.7639	+	5.81865i	−1.13033	+	0.276765i
$443$	28.7470			1.36581			0.682907	−	0.730506i	$-0.260716\pi$
$443$	28.7470			1.36581			0.682907	+	0.730506i	$0.260716\pi$
$444$		0			0
$445$	−16.1201			−0.764168
$446$	14.3501	−	3.51366i	0.679498	−	0.166377i
$447$		0			0
$448$	17.7375	+	2.31947i	0.838020	+	0.109584i
$449$		−	28.2150i		−	1.33155i	−0.746153	−	0.665775i	$-0.768101\pi$
$449$		−	28.2150i		−	1.33155i	0.746153	−	0.665775i	$-0.231899\pi$
$450$		0			0
$451$		−	19.1019i		−	0.899475i
$452$	16.2024	+	31.1024i	0.762096	+	1.46293i
$453$		0			0
$454$	−0.0650786	−	0.265787i	−0.00305429	−	0.0124740i
$455$	21.0724			0.987890
$456$		0			0
$457$	23.1154			1.08129			0.540647	−	0.841249i	$-0.318179\pi$
$457$	23.1154			1.08129			0.540647	+	0.841249i	$0.318179\pi$
$458$	−6.49753	−	26.5365i	−0.303610	−	1.23997i
$459$		0			0
$460$	−29.7544	+	15.5002i	−1.38731	+	0.722698i
$461$		−	1.90136i		−	0.0885550i	−0.999019	−	0.0442775i	$-0.985901\pi$
$461$		−	1.90136i		−	0.0885550i	0.999019	−	0.0442775i	$-0.0140986\pi$
$462$		0			0
$463$			20.5572i			0.955374i	0.878530	+	0.477687i	$0.158525\pi$
$463$			20.5572i			0.955374i	−0.878530	+	0.477687i	$0.841475\pi$
$464$	−4.25142	+	6.07918i	−0.197367	+	0.282219i
$465$		0			0
$466$	−17.0670	+	4.17890i	−0.790615	+	0.193584i
$467$	−15.1228			−0.699800			−0.349900	−	0.936787i	$-0.613784\pi$
$467$	−15.1228			−0.699800			−0.349900	+	0.936787i	$0.613784\pi$
$468$		0			0
$469$	−10.0403			−0.463617
$470$	−39.1245	+	9.57973i	−1.80468	+	0.441880i
$471$		0			0
$472$	4.65940	+	5.30836i	0.214466	+	0.244337i
$473$	11.1805			0.514080
$474$		0			0
$475$	18.2646	+	3.22740i	0.838038	+	0.148083i
$476$	−11.5391	−	22.1506i	−0.528892	−	1.01527i
$477$		0			0
$478$	0.0512319	+	0.209236i	0.00234329	+	0.00957023i
$479$			17.4171i			0.795808i	0.917427	+	0.397904i	$0.130262\pi$
$479$			17.4171i			0.795808i	−0.917427	+	0.397904i	$0.869738\pi$
$480$		0			0
$481$	−3.85081			−0.175582
$482$	4.02778	+	16.4498i	0.183460	+	0.749270i
$483$		0			0
$484$	−10.1900	+	5.30836i	−0.463183	+	0.241289i
$485$	−3.57080			−0.162142
$486$		0			0
$487$	−42.3759			−1.92023			−0.960117	−	0.279597i	$-0.909799\pi$
$487$	−42.3759			−1.92023			−0.960117	+	0.279597i	$0.909799\pi$
$488$	−4.22824	−	4.81715i	−0.191403	−	0.218062i
$489$		0			0
$490$	−2.04643	−	8.35781i	−0.0924483	−	0.377567i
$491$	6.65940			0.300534			0.150267	−	0.988645i	$-0.451987\pi$
$491$	6.65940			0.300534			0.150267	+	0.988645i	$0.451987\pi$
$492$		0			0
$493$	10.3574			0.466475
$494$	−19.0549	+	1.24473i	−0.857319	+	0.0560032i
$495$		0			0
$496$	−9.74597	+	13.9359i	−0.437607	+	0.625741i
$497$		−	32.8662i		−	1.47425i
$498$		0			0
$499$	−19.5028			−0.873067			−0.436534	−	0.899688i	$-0.643794\pi$
$499$	−19.5028			−0.873067			−0.436534	+	0.899688i	$0.643794\pi$
$500$	2.09394	+	4.01956i	0.0936437	+	0.179760i
$501$		0			0
$502$	10.7460	−	2.63118i	0.479617	−	0.117435i
$503$			18.8238i			0.839310i	0.907684	+	0.419655i	$0.137849\pi$
$503$			18.8238i			0.839310i	−0.907684	+	0.419655i	$0.862151\pi$
$504$		0			0
$505$	38.4546			1.71121
$506$	−4.25142	−	17.3632i	−0.188999	−	0.771889i
$507$		0			0
$508$	−5.05473	+	2.63319i	−0.224267	+	0.116829i
$509$	19.8091			0.878021			0.439011	−	0.898482i	$-0.355329\pi$
$509$	19.8091			0.878021			0.439011	+	0.898482i	$0.355329\pi$
$510$		0			0
$511$			3.37693i			0.149387i
$512$	12.5957	+	18.7976i	0.556657	+	0.830743i
$513$		0			0
$514$	−4.51021	−	18.4201i	−0.198937	−	0.812477i
$515$		−	11.2840i		−	0.497234i
$516$		0			0
$517$		−	21.4623i		−	0.943913i
$518$	−0.934921	−	3.81831i	−0.0410781	−	0.167767i
$519$		0			0
$520$	17.5834	+	20.0324i	0.771084	+	0.878480i
$521$		−	29.7708i		−	1.30428i	−0.758097	−	0.652141i	$-0.773871\pi$
$521$		−	29.7708i		−	1.30428i	0.758097	−	0.652141i	$-0.226129\pi$
$522$		0			0
$523$		−	2.05365i		−	0.0898000i	−0.998991	−	0.0449000i	$-0.985703\pi$
$523$		−	2.05365i		−	0.0898000i	0.998991	−	0.0449000i	$-0.0142969\pi$
$524$	−14.1407	+	7.36643i	−0.617741	+	0.321804i
$525$		0			0
$526$	1.38375	+	5.65136i	0.0603343	+	0.246411i
$527$	23.7434			1.03428
$528$		0			0
$529$	−7.40429			−0.321926
$530$	−11.7076	−	47.8149i	−0.508546	−	2.07695i
$531$		0			0
$532$	−5.86047	−	18.5918i	−0.254083	−	0.806057i
$533$			25.8122i			1.11805i
$534$		0			0
$535$	−33.8730			−1.46446
$536$	−8.37790	−	9.54477i	−0.361870	−	0.412271i
$537$		0			0
$538$	−36.3114	+	8.89094i	−1.56550	+	0.383316i
$539$	4.58480			0.197481
$540$		0			0
$541$			24.9916i			1.07447i	0.843432	+	0.537236i	$0.180532\pi$
$541$			24.9916i			1.07447i	−0.843432	+	0.537236i	$0.819468\pi$
$542$	−6.48971	−	26.5046i	−0.278757	−	1.13847i
$543$		0			0
$544$	11.4289	−	29.4527i	0.490010	−	1.26277i
$545$			34.8090i			1.49105i
$546$		0			0
$547$			26.0893i			1.11550i	0.830010	+	0.557749i	$0.188335\pi$
$547$			26.0893i			1.11550i	−0.830010	+	0.557749i	$0.811665\pi$
$548$	−16.3008	+	8.49170i	−0.696337	+	0.362747i
$549$		0			0
$550$	13.3990	−	3.28078i	0.571336	−	0.139893i
$551$	7.96057	+	1.40665i	0.339131	+	0.0599254i
$552$		0			0
$553$		−	25.7849i		−	1.09649i
$554$	8.98378	+	36.6906i	0.381684	+	1.55883i
$555$		0			0
$556$	−6.91342	+	3.60145i	−0.293195	+	0.152736i
$557$			10.9058i			0.462092i	0.972943	+	0.231046i	$0.0742148\pi$
$557$			10.9058i			0.462092i	−0.972943	+	0.231046i	$0.925785\pi$
$558$		0			0
$559$	−15.1081			−0.639004
$560$	−15.5943	+	22.2986i	−0.658980	+	0.942286i
$561$		0			0
$562$	−7.88457	−	32.2013i	−0.332590	−	1.35833i
$563$			27.7390i			1.16906i	0.811372	+	0.584531i	$0.198721\pi$
$563$			27.7390i			1.16906i	−0.811372	+	0.584531i	$0.801279\pi$
$564$		0			0
$565$	−53.3448			−2.24423
$566$	15.8471	−	3.88020i	0.666103	−	0.163097i
$567$		0			0
$568$	31.2442	−	27.4245i	1.31098	−	1.15071i
$569$		−	45.2675i		−	1.89771i	−0.315712	−	0.948855i	$-0.602243\pi$
$569$		−	45.2675i		−	1.89771i	0.315712	−	0.948855i	$-0.397757\pi$
$570$		0			0
$571$	−7.80997			−0.326837			−0.163419	−	0.986557i	$-0.552252\pi$
$571$	−7.80997			−0.326837			−0.163419	+	0.986557i	$0.552252\pi$
$572$	−12.5957	+	6.56156i	−0.526653	+	0.274353i
$573$		0			0
$574$	−25.5943	+	6.26683i	−1.06829	+	0.261572i
$575$		−	23.4627i		−	0.978462i
$576$		0			0
$577$	−12.1696			−0.506627			−0.253314	−	0.967384i	$-0.581520\pi$
$577$	−12.1696			−0.506627			−0.253314	+	0.967384i	$0.581520\pi$
$578$	−19.4919	+	4.77265i	−0.810758	+	0.198516i
$579$		0			0
$580$	−5.21329	−	10.0075i	−0.216470	−	0.415541i
$581$		−	4.47214i		−	0.185535i
$582$		0			0
$583$	26.2296			1.08632
$584$	−3.21027	+	2.81781i	−0.132842	+	0.116602i
$585$		0			0
$586$	−16.1805	+	3.96184i	−0.668411	+	0.163662i
$587$	6.84345			0.282459			0.141230	−	0.989977i	$-0.454894\pi$
$587$	6.84345			0.282459			0.141230	+	0.989977i	$0.454894\pi$
$588$		0			0
$589$	18.2488	+	3.22461i	0.751929	+	0.132868i
$590$	−10.4356	+	2.55519i	−0.429628	+	0.105195i
$591$		0			0
$592$	2.84974	−	4.07488i	0.117123	−	0.167477i
$593$	−17.0950			−0.702008			−0.351004	−	0.936374i	$-0.614160\pi$
$593$	−17.0950			−0.702008			−0.351004	+	0.936374i	$0.614160\pi$
$594$		0			0
$595$	37.9912			1.55749
$596$	−9.64144	−	18.5079i	−0.394929	−	0.758114i
$597$		0			0
$598$	5.74489	+	23.4627i	0.234926	+	0.959461i
$599$	1.76518			0.0721232			0.0360616	−	0.999350i	$-0.488519\pi$
$599$	1.76518			0.0721232			0.0360616	+	0.999350i	$0.488519\pi$
$600$		0			0
$601$		−	48.7011i		−	1.98656i	−0.115731	−	0.993281i	$-0.536921\pi$
$601$		−	48.7011i		−	1.98656i	0.115731	−	0.993281i	$-0.463079\pi$
$602$	−3.66802	−	14.9805i	−0.149497	−	0.610562i
$603$		0			0
$604$	−19.4919	+	10.1541i	−0.793116	+	0.413163i
$605$		−	17.4772i		−	0.710551i
$606$		0			0
$607$	45.9062			1.86328			0.931638	−	0.363387i	$-0.118380\pi$
$607$	45.9062			1.86328			0.931638	+	0.363387i	$0.118380\pi$
$608$	12.7841	−	21.0848i	0.518464	−	0.855100i
$609$		0			0
$610$	9.46996	−	2.31874i	0.383427	−	0.0938831i
$611$			29.0018i			1.17329i
$612$		0			0
$613$		−	20.9308i		−	0.845386i	−0.906273	−	0.422693i	$-0.861085\pi$
$613$		−	20.9308i		−	0.845386i	0.906273	−	0.422693i	$-0.138915\pi$
$614$	1.06508	+	4.34988i	0.0429831	+	0.175547i
$615$		0			0
$616$	−9.56422	−	10.8963i	−0.385353	−	0.439025i
$617$	−32.8048			−1.32067			−0.660335	−	0.750971i	$-0.729586\pi$
$617$	−32.8048			−1.32067			−0.660335	+	0.750971i	$0.729586\pi$
$618$		0			0
$619$	22.0190			0.885020			0.442510	−	0.896764i	$-0.354088\pi$
$619$	22.0190			0.885020			0.442510	+	0.896764i	$0.354088\pi$
$620$	−11.9510	−	22.9413i	−0.479962	−	0.921345i
$621$		0			0
$622$	−3.27419	−	13.3721i	−0.131283	−	0.536173i
$623$	−11.8485			−0.474700
$624$		0			0
$625$	−28.1696			−1.12678
$626$	28.8875	−	7.07317i	1.15458	−	0.282701i
$627$		0			0
$628$	−5.34060	−	10.2519i	−0.213113	−	0.409097i
$629$	−6.94260			−0.276819
$630$		0			0
$631$			9.72716i			0.387232i	0.981077	+	0.193616i	$0.0620216\pi$
$631$			9.72716i			0.387232i	−0.981077	+	0.193616i	$0.937978\pi$
$632$	24.5124	−	21.5157i	0.975049	−	0.855847i
$633$		0			0
$634$	14.0095	−	3.43026i	0.556389	−	0.136233i
$635$		−	8.66953i		−	0.344040i
$636$		0			0
$637$	−6.19539			−0.245470
$638$	5.83991	−	1.42992i	0.231204	−	0.0566109i
$639$		0			0
$640$	−34.2104	+	3.78185i	−1.35228	+	0.149491i
$641$			15.1485i			0.598329i	0.954202	+	0.299165i	$0.0967080\pi$
$641$			15.1485i			0.598329i	−0.954202	+	0.299165i	$0.903292\pi$
$642$		0			0
$643$	−1.46201			−0.0576560			−0.0288280	−	0.999584i	$-0.509178\pi$
$643$	−1.46201			−0.0576560			−0.0288280	+	0.999584i	$0.509178\pi$
$644$	−21.8698	+	11.3928i	−0.861793	+	0.448939i
$645$		0			0
$646$	−34.3538	+	2.24412i	−1.35163	+	0.0882936i
$647$		−	7.90920i		−	0.310943i	−0.987840	−	0.155471i	$-0.950310\pi$
$647$		−	7.90920i		−	0.310943i	0.987840	−	0.155471i	$-0.0496897\pi$
$648$		0			0
$649$		−	5.72462i		−	0.224711i
$650$	−18.1059	+	4.43328i	−0.710173	+	0.173887i
$651$		0			0
$652$	11.8121	−	6.15337i	0.462599	−	0.240984i
$653$			32.5505i			1.27380i	0.770947	+	0.636900i	$0.219783\pi$
$653$			32.5505i			1.27380i	−0.770947	+	0.636900i	$0.780217\pi$
$654$		0			0
$655$		−	24.2532i		−	0.947653i
$656$	−27.3142	−	19.1019i	−1.06644	−	0.745805i
$657$		0			0
$658$	−28.7570	+	7.04121i	−1.12106	+	0.274495i
$659$		−	12.8731i		−	0.501463i	−0.968057	−	0.250731i	$-0.919329\pi$
$659$		−	12.8731i		−	0.501463i	0.968057	−	0.250731i	$-0.0806711\pi$
$660$		0			0
$661$	2.01313			0.0783019			0.0391509	−	0.999233i	$-0.487535\pi$
$661$	2.01313			0.0783019			0.0391509	+	0.999233i	$0.487535\pi$
$662$	−10.5644	−	43.1459i	−0.410596	−	1.67691i
$663$		0			0
$664$	4.25142	−	3.73168i	0.164987	−	0.144817i
$665$	29.1995	+	5.15963i	1.13231	+	0.200082i
$666$		0			0
$667$		−	10.2261i		−	0.395957i
$668$	−3.44812	+	1.79625i	−0.133412	+	0.0694990i
$669$		0			0
$670$	18.7639	−	4.59439i	0.724914	−	0.177497i
$671$			5.19489i			0.200547i
$672$		0			0
$673$		−	21.9254i		−	0.845163i	−0.906325	−	0.422582i	$-0.861124\pi$
$673$		−	21.9254i		−	0.845163i	0.906325	−	0.422582i	$-0.138876\pi$
$674$	−7.75441	−	31.6697i	−0.298689	−	1.21987i
$675$		0			0
$676$	−6.03837	+	3.14561i	−0.232245	+	0.120985i
$677$	−42.2401			−1.62342			−0.811710	−	0.584061i	$-0.801463\pi$
$677$	−42.2401			−1.62342			−0.811710	+	0.584061i	$0.801463\pi$
$678$		0			0
$679$	−2.62458			−0.100722
$680$	31.7010	+	36.1163i	1.21568	+	1.38500i
$681$		0			0
$682$	13.3874	−	3.27794i	0.512631	−	0.125519i
$683$			6.89837i			0.263959i	0.991252	+	0.131979i	$0.0421332\pi$
$683$			6.89837i			0.263959i	−0.991252	+	0.131979i	$0.957867\pi$
$684$		0			0
$685$		−	27.9581i		−	1.06822i
$686$	−6.76867	−	27.6439i	−0.258429	−	1.05545i
$687$		0			0
$688$	11.1805	−	15.9872i	0.426253	−	0.609506i
$689$	−35.4437			−1.35030
$690$		0			0
$691$	−35.4620			−1.34904			−0.674519	−	0.738257i	$-0.735649\pi$
$691$	−35.4620			−1.34904			−0.674519	+	0.738257i	$0.735649\pi$
$692$	−36.4976	+	19.0129i	−1.38743	+	0.722764i
$693$		0			0
$694$	40.2913	−	9.86542i	1.52944	−	0.374486i
$695$		−	11.8574i		−	0.449778i
$696$		0			0
$697$			46.5366i			1.76270i
$698$	−4.03151	−	16.4651i	−0.152595	−	0.623213i
$699$		0			0
$700$	−8.79171	−	16.8767i	−0.332295	−	0.637881i
$701$		−	10.3586i		−	0.391239i	−0.980680	−	0.195619i	$-0.937328\pi$
$701$		−	10.3586i		−	0.391239i	0.980680	−	0.195619i	$-0.0626717\pi$
$702$		0			0
$703$	−5.33598	−	0.942881i	−0.201250	−	0.0355614i
$704$	2.37790	−	18.1844i	0.0896205	−	0.685350i
$705$		0			0
$706$	−17.1975	+	4.21086i	−0.647238	+	0.158478i
$707$	28.2646			1.06300
$708$		0			0
$709$		−	5.24822i		−	0.197101i	−0.995132	−	0.0985506i	$-0.968579\pi$
$709$		−	5.24822i		−	0.197101i	0.995132	−	0.0985506i	$-0.0314206\pi$
$710$	15.0394	+	61.4225i	0.564420	+	2.30515i
$711$		0			0
$712$	−9.88672	−	11.2637i	−0.370520	−	0.422126i
$713$		−	23.4424i		−	0.877924i
$714$		0			0
$715$		−	21.6033i		−	0.807917i
$716$	8.06303	+	15.4780i	0.301330	+	0.578438i
$717$		0			0
$718$	−5.38721	−	22.0019i	−0.201049	−	0.821103i
$719$			4.31981i			0.161102i	0.996750	+	0.0805509i	$0.0256680\pi$
$719$			4.31981i			0.161102i	−0.996750	+	0.0805509i	$0.974332\pi$
$720$		0			0
$721$		−	8.29390i		−	0.308881i
$722$	−26.7086	−	2.94083i	−0.993993	−	0.109446i
$723$		0			0
$724$	15.0819	−	7.85674i	0.560516	−	0.291993i
$725$	7.89140			0.293079
$726$		0			0
$727$		−	7.37763i		−	0.273621i	−0.990597	−	0.136811i	$-0.956315\pi$
$727$		−	7.37763i		−	0.273621i	0.990597	−	0.136811i	$-0.0436852\pi$
$728$	12.9240	+	14.7241i	0.478996	+	0.545710i
$729$		0			0
$730$	−1.54527	−	6.31102i	−0.0571930	−	0.233581i
$731$	−27.2382			−1.00744
$732$		0			0
$733$			6.72268i			0.248308i	0.992263	+	0.124154i	$0.0396216\pi$
$733$			6.72268i			0.248308i	−0.992263	+	0.124154i	$0.960378\pi$
$734$	9.42884	+	38.5083i	0.348025	+	1.42137i
$735$		0			0
$736$	−29.0793	−	11.2840i	−1.07188	−	0.415935i
$737$			10.2932i			0.379156i
$738$		0			0
$739$	36.1622			1.33025			0.665125	−	0.746732i	$-0.268378\pi$
$739$	36.1622			1.33025			0.665125	+	0.746732i	$0.268378\pi$
$740$	3.49448	+	6.70807i	0.128460	+	0.246594i
$741$		0			0
$742$	−8.60522	−	35.1445i	−0.315908	−	1.29020i
$743$	−11.6697			−0.428120			−0.214060	−	0.976821i	$-0.568669\pi$
$743$	−11.6697			−0.428120			−0.214060	+	0.976821i	$0.568669\pi$
$744$		0			0
$745$	31.7435			1.16299
$746$	−18.9145	+	4.63126i	−0.692509	+	0.169563i
$747$		0			0
$748$	−22.7087	+	11.8298i	−0.830311	+	0.432539i
$749$	−24.8971			−0.909720
$750$		0			0
$751$	8.36451			0.305225			0.152613	−	0.988286i	$-0.451231\pi$
$751$	8.36451			0.305225			0.152613	+	0.988286i	$0.451231\pi$
$752$	−30.6893	−	21.4623i	−1.11912	−	0.782651i
$753$		0			0
$754$	−7.89140	+	1.93223i	−0.287388	+	0.0703676i
$755$		−	33.4312i		−	1.21669i
$756$		0			0
$757$			44.5991i			1.62098i	0.585751	+	0.810491i	$0.300800\pi$
$757$			44.5991i			1.62098i	−0.585751	+	0.810491i	$0.699200\pi$
$758$	−10.0504	−	41.0466i	−0.365045	−	1.49088i
$759$		0			0
$760$	19.4599	+	32.0638i	0.705886	+	1.16308i
$761$	40.6243			1.47263			0.736314	−	0.676640i	$-0.236565\pi$
$761$	40.6243			1.47263			0.736314	+	0.676640i	$0.236565\pi$
$762$		0			0
$763$			25.5850i			0.926240i
$764$	10.3863	+	19.9378i	0.375765	+	0.721325i
$765$		0			0
$766$	11.9891	−	2.93556i	0.433184	−	0.106066i
$767$			7.73560i			0.279316i
$768$		0			0
$769$	13.7748			0.496733			0.248367	−	0.968666i	$-0.420106\pi$
$769$	13.7748			0.496733			0.248367	+	0.968666i	$0.420106\pi$
$770$	21.4209	−	5.24496i	0.771956	−	0.189015i
$771$		0			0
$772$	7.17748	+	13.7780i	0.258323	+	0.495883i
$773$	22.7684			0.818923			0.409462	−	0.912327i	$-0.365717\pi$
$773$	22.7684			0.818923			0.409462	+	0.912327i	$0.365717\pi$
$774$		0			0
$775$	18.0902			0.649821
$776$	−2.19003	−	2.49505i	−0.0786174	−	0.0895672i
$777$		0			0
$778$	6.93735	+	28.3328i	0.248716	+	1.01578i
$779$	−6.32018	+	35.7674i	−0.226444	+	1.28150i
$780$		0			0
$781$	−33.6943			−1.20568
$782$	10.3574	+	42.3007i	0.370380	+	1.51267i
$783$		0			0
$784$	4.58480	−	6.55588i	0.163743	−	0.234139i
$785$	17.5834			0.627579
$786$		0			0
$787$		−	6.09611i		−	0.217303i	−0.994080	−	0.108651i	$-0.965347\pi$
$787$		−	6.09611i		−	0.217303i	0.994080	−	0.108651i	$-0.0346532\pi$
$788$	3.35963	+	6.44922i	0.119682	+	0.229744i
$789$		0			0
$790$	11.7991	+	48.1885i	0.419792	+	1.71447i
$791$	−39.2090			−1.39411
$792$		0			0
$793$		−	7.01979i		−	0.249280i
$794$	−7.86335	−	32.1146i	−0.279060	−	1.13971i
$795$		0			0
$796$	−5.04573	−	9.68590i	−0.178841	−	0.343307i
$797$	17.2798			0.612081			0.306041	−	0.952018i	$-0.400996\pi$
$797$	17.2798			0.612081			0.306041	+	0.952018i	$0.400996\pi$
$798$		0			0
$799$			52.2871i			1.84978i
$800$	8.70777	−	22.4402i	0.307866	−	0.793382i
$801$		0			0
$802$	−1.06723	−	4.35867i	−0.0376852	−	0.153910i
$803$	3.46201			0.122172
$804$		0			0
$805$		−	37.5097i		−	1.32204i
$806$	−18.0902	+	4.42944i	−0.637202	+	0.156020i
$807$		0			0
$808$	23.5848	+	26.8697i	0.829710	+	0.945272i
$809$	39.3801			1.38453			0.692264	−	0.721644i	$-0.256613\pi$
$809$	39.3801			1.38453			0.692264	+	0.721644i	$0.256613\pi$
$810$		0			0
$811$			20.9452i			0.735484i	0.929928	+	0.367742i	$0.119869\pi$
$811$			20.9452i			0.735484i	−0.929928	+	0.367742i	$0.880131\pi$
$812$	−3.83183	−	7.35566i	−0.134471	−	0.258133i
$813$		0			0
$814$	−3.91450	+	0.958475i	−0.137203	+	0.0335945i
$815$			20.2594i			0.709654i
$816$		0			0
$817$	−20.9349	−	3.69925i	−0.732420	−	0.129420i
$818$	2.87720	+	11.7508i	0.100599	+	0.410856i
$819$		0			0
$820$	44.9646	−	23.4237i	1.57023	−	0.817991i
$821$			52.2181i			1.82243i	0.411936	+	0.911213i	$0.364853\pi$
$821$			52.2181i			1.82243i	−0.411936	+	0.911213i	$0.635147\pi$
$822$		0			0
$823$			39.5789i			1.37963i	0.723983	+	0.689817i	$0.242309\pi$
$823$			39.5789i			1.37963i	−0.723983	+	0.689817i	$0.757691\pi$
$824$	7.88457	−	6.92066i	0.274672	−	0.241093i
$825$		0			0
$826$	−7.67030	+	1.87809i	−0.266884	+	0.0653472i
$827$			19.5172i			0.678680i	0.940664	+	0.339340i	$0.110204\pi$
$827$			19.5172i			0.678680i	−0.940664	+	0.339340i	$0.889796\pi$
$828$		0			0
$829$	25.0759			0.870921			0.435461	−	0.900208i	$-0.356586\pi$
$829$	25.0759			0.870921			0.435461	+	0.900208i	$0.356586\pi$
$830$	2.04643	+	8.35781i	0.0710326	+	0.290104i
$831$		0			0
$832$	−3.21323	+	24.5724i	−0.111399	+	0.851893i
$833$	−11.1696			−0.387004
$834$		0			0
$835$		−	5.91397i		−	0.204661i
$836$	−19.0602	+	6.00811i	−0.659210	+	0.207795i
$837$		0			0
$838$	−48.8323	+	11.9567i	−1.68688	+	0.413038i
$839$	10.0833			0.348115			0.174057	−	0.984736i	$-0.444312\pi$
$839$	10.0833			0.348115			0.174057	+	0.984736i	$0.444312\pi$
$840$		0			0
$841$	−25.5606			−0.881399
$842$	−39.3223	+	9.62817i	−1.35514	+	0.331809i
$843$		0			0
$844$	−14.1762	−	27.2129i	−0.487964	−	0.936706i
$845$		−	10.3566i		−	0.356278i
$846$		0			0
$847$		−	12.8460i		−	0.441393i
$848$	26.2296	−	37.5061i	0.900728	−	1.28797i
$849$		0			0
$850$	−32.6430	+	7.99271i	−1.11965	+	0.274148i
$851$			6.85459i			0.234972i
$852$		0			0
$853$			38.4525i			1.31659i	0.752760	+	0.658295i	$0.228722\pi$
$853$			38.4525i			1.31659i	−0.752760	+	0.658295i	$0.771278\pi$
$854$	6.96053	−	1.70430i	0.238185	−	0.0583201i
$855$		0			0
$856$	−20.7748	−	23.6683i	−0.710069	−	0.808967i
$857$			42.6333i			1.45633i	0.685404	+	0.728163i	$0.259626\pi$
$857$			42.6333i			1.45633i	−0.685404	+	0.728163i	$0.740374\pi$
$858$		0			0
$859$	−28.5570			−0.974353			−0.487176	−	0.873304i	$-0.661973\pi$
$859$	−28.5570			−0.974353			−0.487176	+	0.873304i	$0.661973\pi$
$860$	13.7101	+	26.3181i	0.467509	+	0.897440i
$861$		0			0
$862$	43.3596	−	10.6167i	1.47683	−	0.361607i
$863$	−17.0521			−0.580459			−0.290229	−	0.956957i	$-0.593732\pi$
$863$	−17.0521			−0.580459			−0.290229	+	0.956957i	$0.593732\pi$
$864$		0			0
$865$		−	62.5982i		−	2.12840i
$866$	−6.21781	−	25.3941i	−0.211290	−	0.862927i
$867$		0			0
$868$	−8.78410	−	16.8621i	−0.298152	−	0.572338i
$869$	−26.4345			−0.896730
$870$		0			0
$871$		−	13.9091i		−	0.471293i
$872$	−24.3223	+	21.3489i	−0.823658	+	0.722964i
$873$		0			0
$874$	2.21567	+	33.9183i	0.0749462	+	1.14731i
$875$	−5.06723			−0.171304
$876$		0			0
$877$	−10.5160			−0.355099			−0.177550	−	0.984112i	$-0.556817\pi$
$877$	−10.5160			−0.355099			−0.177550	+	0.984112i	$0.556817\pi$
$878$	14.5957	−	3.57379i	0.492581	−	0.120610i
$879$		0			0
$880$	22.8603	+	15.9872i	0.770622	+	0.538928i
$881$	−37.5943			−1.26658			−0.633292	−	0.773913i	$-0.718297\pi$
$881$	−37.5943			−1.26658			−0.633292	+	0.773913i	$0.718297\pi$
$882$		0			0
$883$	−13.2720			−0.446638			−0.223319	−	0.974745i	$-0.571689\pi$
$883$	−13.2720			−0.446638			−0.223319	+	0.974745i	$0.571689\pi$
$884$	30.6860	−	15.9854i	1.03208	−	0.537648i
$885$		0			0
$886$	−39.4880	+	9.66873i	−1.32662	+	0.324827i
$887$	27.6777			0.929325			0.464662	−	0.885488i	$-0.346176\pi$
$887$	27.6777			0.929325			0.464662	+	0.885488i	$0.346176\pi$
$888$		0			0
$889$		−	6.37220i		−	0.213717i
$890$	22.1432	−	5.42182i	0.742242	−	0.181740i
$891$		0			0
$892$	−18.5301	+	9.65298i	−0.620432	+	0.323206i
$893$	−7.10116	+	40.1871i	−0.237631	+	1.34481i
$894$		0			0
$895$	−26.5467			−0.887359
$896$	−25.1450	+	2.77971i	−0.840037	+	0.0928635i
$897$		0			0
$898$	9.48979	+	38.7572i	0.316678	+	1.29334i
$899$	7.88457			0.262965
$900$		0			0
$901$	−63.9012			−2.12886
$902$	6.42471	+	26.2391i	0.213919	+	0.873667i
$903$		0			0
$904$	−32.7171	−	37.2739i	−1.08815	−	1.23971i
$905$			25.8675i			0.859866i
$906$		0			0
$907$		−	44.3324i		−	1.47203i	−0.676963	−	0.736017i	$-0.736704\pi$
$907$		−	44.3324i		−	1.47203i	0.676963	−	0.736017i	$-0.263296\pi$
$908$	0.178789	+	0.343206i	0.00593331	+	0.0113897i
$909$		0			0
$910$	−28.9458	+	7.08746i	−0.959545	+	0.234947i
$911$	40.9742			1.35754			0.678768	−	0.734353i	$-0.262514\pi$
$911$	40.9742			1.35754			0.678768	+	0.734353i	$0.262514\pi$
$912$		0			0
$913$	−4.58480			−0.151735
$914$	−31.7522	+	7.77460i	−1.05027	+	0.257161i
$915$		0			0
$916$	17.8505	+	34.2662i	0.589797	+	1.13219i
$917$		−	17.8264i		−	0.588680i
$918$		0			0
$919$			37.2439i			1.22856i	0.789087	+	0.614281i	$0.210554\pi$
$919$			37.2439i			1.22856i	−0.789087	+	0.614281i	$0.789446\pi$
$920$	35.6584	−	31.2991i	1.17562	−	1.03190i
$921$		0			0
$922$	0.639499	+	2.61177i	0.0210608	+	0.0860141i
$923$	45.5306			1.49866
$924$		0			0
$925$	−5.28962			−0.173922
$926$	−6.91417	−	28.2381i	−0.227214	−	0.927962i
$927$		0			0
$928$	3.79525	−	9.78049i	0.124585	−	0.321060i
$929$	3.86033			0.126653			0.0633266	−	0.997993i	$-0.479829\pi$
$929$	3.86033			0.126653			0.0633266	+	0.997993i	$0.479829\pi$
$930$		0			0
$931$	−8.58480	−	1.51696i	−0.281356	−	0.0497162i
$932$	22.0384	−	11.4806i	0.721891	−	0.376059i
$933$		0			0
$934$	20.7732	−	5.08637i	0.679721	−	0.166431i
$935$		−	38.9484i		−	1.27375i
$936$		0			0
$937$	20.6798			0.675580			0.337790	−	0.941221i	$-0.390321\pi$
$937$	20.6798			0.675580			0.337790	+	0.941221i	$0.390321\pi$
$938$	13.7917	−	3.37693i	0.450315	−	0.110261i
$939$		0			0
$940$	50.5208	−	26.3181i	1.64781	−	0.858402i
$941$	−9.29308			−0.302946			−0.151473	−	0.988461i	$-0.548402\pi$
$941$	−9.29308			−0.302946			−0.151473	+	0.988461i	$0.548402\pi$
$942$		0			0
$943$	45.9467			1.49623
$944$	−8.18572	−	5.72462i	−0.266422	−	0.186320i
$945$		0			0
$946$	−15.3580	+	3.76043i	−0.499330	+	0.122262i
$947$	−37.3934			−1.21512			−0.607561	−	0.794273i	$-0.707852\pi$
$947$	−37.3934			−1.21512			−0.607561	+	0.794273i	$0.707852\pi$
$948$		0			0
$949$	−4.67817			−0.151860
$950$	−26.1744	+	1.70981i	−0.849211	+	0.0554736i
$951$		0			0
$952$	23.3006	+	26.5459i	0.755176	+	0.860357i
$953$		−	6.08558i		−	0.197131i	−0.995131	−	0.0985656i	$-0.968575\pi$
$953$		−	6.08558i		−	0.197131i	0.995131	−	0.0985656i	$-0.0314254\pi$
$954$		0			0
$955$	−34.1960			−1.10656
$956$	−0.140748	−	0.270183i	−0.00455212	−	0.00873833i
$957$		0			0
$958$	−5.85804	−	23.9248i	−0.189265	−	0.772974i
$959$		−	20.5495i		−	0.663579i
$960$		0			0
$961$	−12.9254			−0.416949
$962$	5.28962	−	1.29518i	0.170544	−	0.0417581i
$963$		0			0
$964$	−11.0654	−	21.2414i	−0.356393	−	0.684139i
$965$	−23.6312			−0.760714
$966$		0			0
$967$		−	18.0088i		−	0.579124i	−0.957159	−	0.289562i	$-0.906490\pi$
$967$		−	18.0088i		−	0.579124i	0.957159	−	0.289562i	$-0.0935097\pi$
$968$	12.2120	−	10.7190i	0.392508	−	0.344523i
$969$		0			0
$970$	4.90499	−	1.20100i	0.157490	−	0.0385617i
$971$		−	55.6999i		−	1.78749i	−0.448572	−	0.893747i	$-0.648067\pi$
$971$		−	55.6999i		−	1.78749i	0.448572	−	0.893747i	$-0.351933\pi$
$972$		0			0
$973$		−	8.71535i		−	0.279401i
$974$	58.2091	−	14.2526i	1.86514	−	0.456684i
$975$		0			0
$976$	7.42826	+	5.19489i	0.237773	+	0.166284i
$977$		−	33.1269i		−	1.05982i	−0.848053	−	0.529911i	$-0.822225\pi$
$977$		−	33.1269i		−	1.05982i	0.848053	−	0.529911i	$-0.177775\pi$
$978$		0			0
$979$			12.1470i			0.388220i
$980$	5.62210	+	10.7923i	0.179591	+	0.344747i
$981$		0			0
$982$	−9.14759	+	2.23981i	−0.291911	+	0.0714752i
$983$	26.2760			0.838073			0.419037	−	0.907969i	$-0.362368\pi$
$983$	26.2760			0.838073			0.419037	+	0.907969i	$0.362368\pi$
$984$		0			0
$985$	−11.0613			−0.352441
$986$	−14.2273	+	3.48359i	−0.453090	+	0.110940i
$987$		0			0
$988$	25.7558	−	8.11869i	0.819401	−	0.258290i
$989$			26.8929i			0.855146i
$990$		0			0
$991$	47.6714			1.51433			0.757166	−	0.653222i	$-0.226583\pi$
$991$	47.6714			1.51433			0.757166	+	0.653222i	$0.226583\pi$
$992$	8.70024	−	22.4208i	0.276233	−	0.711862i
$993$		0			0
$994$	11.0542	+	45.1463i	0.350617	+	1.43195i
$995$	16.6126			0.526654
$996$		0			0
$997$			2.73758i			0.0866999i	0.999060	+	0.0433499i	$0.0138030\pi$
$997$			2.73758i			0.0866999i	−0.999060	+	0.0433499i	$0.986197\pi$
$998$	26.7898	−	6.55955i	0.848017	−	0.207639i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1368.2.e.e.379.2		12
3.2	odd	2		152.2.b.c.75.11	yes	12
4.3	odd	2		5472.2.e.e.5167.9		12
8.3	odd	2	inner	1368.2.e.e.379.12		12
8.5	even	2		5472.2.e.e.5167.4		12
12.11	even	2		608.2.b.c.303.1		12
19.18	odd	2	inner	1368.2.e.e.379.11		12
24.5	odd	2		608.2.b.c.303.2		12
24.11	even	2		152.2.b.c.75.1	✓	12
57.56	even	2		152.2.b.c.75.2	yes	12
76.75	even	2		5472.2.e.e.5167.10		12
152.37	odd	2		5472.2.e.e.5167.3		12
152.75	even	2	inner	1368.2.e.e.379.1		12
228.227	odd	2		608.2.b.c.303.11		12
456.227	odd	2		152.2.b.c.75.12	yes	12
456.341	even	2		608.2.b.c.303.12		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.b.c.75.1	✓	12	24.11	even	2
152.2.b.c.75.2	yes	12	57.56	even	2
152.2.b.c.75.11	yes	12	3.2	odd	2
152.2.b.c.75.12	yes	12	456.227	odd	2
608.2.b.c.303.1		12	12.11	even	2
608.2.b.c.303.2		12	24.5	odd	2
608.2.b.c.303.11		12	228.227	odd	2
608.2.b.c.303.12		12	456.341	even	2
1368.2.e.e.379.1		12	152.75	even	2	inner
1368.2.e.e.379.2		12	1.1	even	1	trivial
1368.2.e.e.379.11		12	19.18	odd	2	inner
1368.2.e.e.379.12		12	8.3	odd	2	inner
5472.2.e.e.5167.3		12	152.37	odd	2
5472.2.e.e.5167.4		12	8.5	even	2
5472.2.e.e.5167.9		12	4.3	odd	2
5472.2.e.e.5167.10		12	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-1.37364 + 0.336338i) q^{2} +(1.77375 - 0.924013i) q^{4} +3.04222i q^{5} +2.23607i q^{7} +(-2.12571 + 1.86584i) q^{8} +O(q^{10})\)	\(q+(-1.37364 + 0.336338i) q^{2} +(1.77375 - 0.924013i) q^{4} +3.04222i q^{5} +2.23607i q^{7} +(-2.12571 + 1.86584i) q^{8} +(-1.02321 - 4.17890i) q^{10} +2.29240 q^{11} -3.09769 q^{13} +(-0.752075 - 3.07154i) q^{14} +(2.29240 - 3.27794i) q^{16} -5.58480 q^{17} +(-4.29240 - 0.758478i) q^{19} +(2.81105 + 5.39615i) q^{20} +(-3.14893 + 0.771022i) q^{22} +5.51401i q^{23} -4.25511 q^{25} +(4.25511 - 1.04187i) q^{26} +(2.06615 + 3.96623i) q^{28} -1.85457 q^{29} -4.25142 q^{31} +(-2.04643 + 5.27372i) q^{32} +(7.67149 - 1.87838i) q^{34} -6.80261 q^{35} +1.24312 q^{37} +(6.15130 - 0.401826i) q^{38} +(-5.67629 - 6.46688i) q^{40} -8.33272i q^{41} +4.87720 q^{43} +(4.06615 - 2.11821i) q^{44} +(-1.85457 - 7.57424i) q^{46} -9.36238i q^{47} +2.00000 q^{49} +(5.84497 - 1.43115i) q^{50} +(-5.49455 + 2.86231i) q^{52} +11.4420 q^{53} +6.97399i q^{55} +(-4.17214 - 4.75323i) q^{56} +(2.54751 - 0.623763i) q^{58} -2.49721i q^{59} +2.26613i q^{61} +(5.83991 - 1.42992i) q^{62} +(1.03730 - 7.93247i) q^{64} -9.42387i q^{65} +4.49015i q^{67} +(-9.90606 + 5.16043i) q^{68} +(9.34431 - 2.28798i) q^{70} -14.6982 q^{71} +1.51021 q^{73} +(-1.70760 + 0.418110i) q^{74} +(-8.31450 + 2.62088i) q^{76} +5.12597i q^{77} -11.5314 q^{79} +(9.97222 + 6.97399i) q^{80} +(2.80261 + 11.4461i) q^{82} -2.00000 q^{83} -16.9902i q^{85} +(-6.69951 + 1.64039i) q^{86} +(-4.87298 + 4.27725i) q^{88} +5.29881i q^{89} -6.92665i q^{91} +(5.09501 + 9.78049i) q^{92} +(3.14893 + 12.8605i) q^{94} +(2.30746 - 13.0584i) q^{95} +1.17375i q^{97} +(-2.74727 + 0.672676i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 4 q^{4}+O(q^{10}) \) 12 * q + 4 * q^4	\( 12 q + 4 q^{4} - 12 q^{17} - 24 q^{19} - 4 q^{20} - 44 q^{25} + 44 q^{26} - 20 q^{28} - 40 q^{35} - 4 q^{38} - 24 q^{43} + 4 q^{44} + 24 q^{49} - 4 q^{58} + 8 q^{62} - 8 q^{64} - 12 q^{68} + 4 q^{73} - 48 q^{74} - 12 q^{76} - 32 q^{80} - 8 q^{82} - 24 q^{83} - 8 q^{92}+O(q^{100}) \) 12 * q + 4 * q^4 - 12 * q^17 - 24 * q^19 - 4 * q^20 - 44 * q^25 + 44 * q^26 - 20 * q^28 - 40 * q^35 - 4 * q^38 - 24 * q^43 + 4 * q^44 + 24 * q^49 - 4 * q^58 + 8 * q^62 - 8 * q^64 - 12 * q^68 + 4 * q^73 - 48 * q^74 - 12 * q^76 - 32 * q^80 - 8 * q^82 - 24 * q^83 - 8 * q^92

Introduction

L-functions

Modular forms

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Database

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