LMFDB - Embedded newform 1386.2.r.a.89.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1386,2,Mod(89,1386)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1386, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 5, 0]))

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

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

chi := DirichletCharacter("1386.89");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1386.r (of order $6$, degree $2$, 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:	$11.0672657201$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			89.2
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	1386.89
Dual form			1386.2.r.a.1277.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.741181 + 1.28376i) q^{5} +(-1.48356 + 2.19067i) q^{7} -1.00000i q^{8} +O(q^{10})$	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.741181 + 1.28376i) q^{5} +(-1.48356 + 2.19067i) q^{7} -1.00000i q^{8} +(1.28376 - 0.741181i) q^{10} +(-0.866025 + 0.500000i) q^{11} -2.44949i q^{13} +(2.38014 - 1.15539i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.182163 + 0.315515i) q^{17} +(7.06350 + 4.07812i) q^{19} -1.48236 q^{20} +1.00000 q^{22} +(-5.18034 - 2.99087i) q^{23} +(1.40130 + 2.42713i) q^{25} +(-1.22474 + 2.12132i) q^{26} +(-2.63896 - 0.189469i) q^{28} +0.332449i q^{29} +(0.752011 - 0.434174i) q^{31} +(0.866025 - 0.500000i) q^{32} -0.364326i q^{34} +(-1.71271 - 3.52823i) q^{35} +(-4.64394 + 8.04354i) q^{37} +(-4.07812 - 7.06350i) q^{38} +(1.28376 + 0.741181i) q^{40} -9.38891 q^{41} -2.94406 q^{43} +(-0.866025 - 0.500000i) q^{44} +(2.99087 + 5.18034i) q^{46} +(0.637756 - 1.10463i) q^{47} +(-2.59808 - 6.50000i) q^{49} -2.80260i q^{50} +(2.12132 - 1.22474i) q^{52} +(-4.65722 + 2.68885i) q^{53} -1.48236i q^{55} +(2.19067 + 1.48356i) q^{56} +(0.166225 - 0.287909i) q^{58} +(-4.43916 - 7.68885i) q^{59} +(-7.28945 - 4.20857i) q^{61} -0.868348 q^{62} -1.00000 q^{64} +(3.14456 + 1.81552i) q^{65} +(-5.08044 - 8.79958i) q^{67} +(-0.182163 + 0.315515i) q^{68} +(-0.280861 + 3.91189i) q^{70} -2.07911i q^{71} +(-7.86798 + 4.54258i) q^{73} +(8.04354 - 4.64394i) q^{74} +8.15623i q^{76} +(0.189469 - 2.63896i) q^{77} +(3.00913 - 5.21197i) q^{79} +(-0.741181 - 1.28376i) q^{80} +(8.13103 + 4.69445i) q^{82} -15.5699 q^{83} -0.540062 q^{85} +(2.54963 + 1.47203i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-0.459822 + 0.796435i) q^{89} +(5.36603 + 3.63397i) q^{91} -5.98174i q^{92} +(-1.10463 + 0.637756i) q^{94} +(-10.4707 + 6.04524i) q^{95} -1.79920i q^{97} +(-1.00000 + 6.92820i) q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{4} - 8 q^{5}+O(q^{10}) $ 8 * q + 4 * q^4 - 8 * q^5	$ 8 q + 4 q^{4} - 8 q^{5} - 4 q^{16} + 4 q^{17} + 24 q^{19} - 16 q^{20} + 8 q^{22} - 24 q^{23} - 4 q^{25} + 12 q^{31} - 16 q^{35} + 12 q^{37} + 8 q^{38} - 16 q^{41} - 32 q^{43} + 8 q^{46} - 48 q^{53} - 4 q^{58} - 16 q^{59} - 24 q^{62} - 8 q^{64} + 12 q^{65} - 24 q^{67} - 4 q^{68} - 20 q^{70} - 24 q^{73} + 12 q^{74} + 40 q^{79} - 8 q^{80} + 12 q^{82} - 72 q^{83} - 32 q^{85} + 24 q^{86} + 4 q^{88} + 16 q^{89} + 36 q^{91} - 24 q^{95} - 8 q^{98}+O(q^{100}) $ 8 * q + 4 * q^4 - 8 * q^5 - 4 * q^16 + 4 * q^17 + 24 * q^19 - 16 * q^20 + 8 * q^22 - 24 * q^23 - 4 * q^25 + 12 * q^31 - 16 * q^35 + 12 * q^37 + 8 * q^38 - 16 * q^41 - 32 * q^43 + 8 * q^46 - 48 * q^53 - 4 * q^58 - 16 * q^59 - 24 * q^62 - 8 * q^64 + 12 * q^65 - 24 * q^67 - 4 * q^68 - 20 * q^70 - 24 * q^73 + 12 * q^74 + 40 * q^79 - 8 * q^80 + 12 * q^82 - 72 * q^83 - 32 * q^85 + 24 * q^86 + 4 * q^88 + 16 * q^89 + 36 * q^91 - 24 * q^95 - 8 * q^98

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1386\mathbb{Z}\right)^\times$.

$n$	$155$	$199$	$1135$
$\chi(n)$	$-1$	$e\left(\frac{5}{6}\right)$	$1$

Coefficient data

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

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

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

$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$2$	−0.866025	−	0.500000i	−0.612372	−	0.353553i
$3$		0			0
$3$		0			0
$4$	0.500000	+	0.866025i	0.250000	+	0.433013i
$5$	−0.741181	+	1.28376i	−0.331466	+	0.574116i	−0.982800	−	0.184675i	$-0.940877\pi$
$5$	−0.741181	+	1.28376i	−0.331466	+	0.574116i	0.651333	+	0.758792i	$0.274210\pi$
$6$		0			0
$7$	−1.48356	+	2.19067i	−0.560734	+	0.827996i
$7$	−1.48356	+	2.19067i	−0.560734	+	0.827996i
$8$		−	1.00000i		−	0.353553i
$9$		0			0
$10$	1.28376	−	0.741181i	0.405962	−	0.234382i
$11$	−0.866025	+	0.500000i	−0.261116	+	0.150756i
$11$	−0.866025	+	0.500000i	−0.261116	+	0.150756i
$12$		0			0
$13$		−	2.44949i		−	0.679366i	−0.940540	−	0.339683i	$-0.889680\pi$
$13$		−	2.44949i		−	0.679366i	0.940540	−	0.339683i	$-0.110320\pi$
$14$	2.38014	−	1.15539i	0.636119	−	0.308792i
$15$		0			0
$16$	−0.500000	+	0.866025i	−0.125000	+	0.216506i
$17$	0.182163	+	0.315515i	0.0441810	+	0.0765237i	0.887270	−	0.461250i	$-0.152599\pi$
$17$	0.182163	+	0.315515i	0.0441810	+	0.0765237i	−0.843089	+	0.537774i	$0.819266\pi$
$18$		0			0
$19$	7.06350	+	4.07812i	1.62048	+	0.935584i	0.986792	+	0.161995i	$0.0517928\pi$
$19$	7.06350	+	4.07812i	1.62048	+	0.935584i	0.633687	+	0.773589i	$0.281541\pi$
$20$	−1.48236			−0.331466
$21$		0			0
$22$	1.00000			0.213201
$23$	−5.18034	−	2.99087i	−1.08018	−	0.623639i	−0.149232	−	0.988802i	$-0.547680\pi$
$23$	−5.18034	−	2.99087i	−1.08018	−	0.623639i	−0.930944	+	0.365163i	$0.881013\pi$
$24$		0			0
$25$	1.40130	+	2.42713i	0.280260	+	0.485425i
$26$	−1.22474	+	2.12132i	−0.240192	+	0.416025i
$27$		0			0
$28$	−2.63896	−	0.189469i	−0.498716	−	0.0358062i
$29$			0.332449i			0.0617343i	0.999523	+	0.0308671i	$0.00982687\pi$
$29$			0.332449i			0.0617343i	−0.999523	+	0.0308671i	$0.990173\pi$
$30$		0			0
$31$	0.752011	−	0.434174i	0.135065	−	0.0779799i	−0.430945	−	0.902378i	$-0.641820\pi$
$31$	0.752011	−	0.434174i	0.135065	−	0.0779799i	0.566010	+	0.824398i	$0.308486\pi$
$32$	0.866025	−	0.500000i	0.153093	−	0.0883883i
$33$		0			0
$34$		−	0.364326i		−	0.0624813i
$35$	−1.71271	−	3.52823i	−0.289501	−	0.596379i
$36$		0			0
$37$	−4.64394	+	8.04354i	−0.763459	+	1.32235i	0.177598	+	0.984103i	$0.443167\pi$
$37$	−4.64394	+	8.04354i	−0.763459	+	1.32235i	−0.941057	+	0.338247i	$0.890166\pi$
$38$	−4.07812	−	7.06350i	−0.661558	−	1.14585i
$39$		0			0
$40$	1.28376	+	0.741181i	0.202981	+	0.117191i
$41$	−9.38891			−1.46630			−0.733150	−	0.680067i	$-0.761951\pi$
$41$	−9.38891			−1.46630			−0.733150	+	0.680067i	$0.761951\pi$
$42$		0			0
$43$	−2.94406			−0.448965			−0.224482	−	0.974478i	$-0.572069\pi$
$43$	−2.94406			−0.448965			−0.224482	+	0.974478i	$0.572069\pi$
$44$	−0.866025	−	0.500000i	−0.130558	−	0.0753778i
$45$		0			0
$46$	2.99087	+	5.18034i	0.440980	+	0.763799i
$47$	0.637756	−	1.10463i	0.0930263	−	0.161126i	−0.815757	−	0.578395i	$-0.803679\pi$
$47$	0.637756	−	1.10463i	0.0930263	−	0.161126i	0.908783	+	0.417269i	$0.137013\pi$
$48$		0			0
$49$	−2.59808	−	6.50000i	−0.371154	−	0.928571i
$50$		−	2.80260i		−	0.396348i
$51$		0			0
$52$	2.12132	−	1.22474i	0.294174	−	0.169842i
$53$	−4.65722	+	2.68885i	−0.639718	+	0.369341i	−0.784506	−	0.620121i	$-0.787083\pi$
$53$	−4.65722	+	2.68885i	−0.639718	+	0.369341i	0.144788	+	0.989463i	$0.453750\pi$
$54$		0			0
$55$		−	1.48236i		−	0.199882i
$56$	2.19067	+	1.48356i	0.292741	+	0.198250i
$57$		0			0
$58$	0.166225	−	0.287909i	0.0218264	−	0.0378044i
$59$	−4.43916	−	7.68885i	−0.577929	−	1.00100i	−0.995717	−	0.0924578i	$-0.970528\pi$
$59$	−4.43916	−	7.68885i	−0.577929	−	1.00100i	0.417788	−	0.908545i	$-0.362806\pi$
$60$		0			0
$61$	−7.28945	−	4.20857i	−0.933319	−	0.538852i	−0.0454590	−	0.998966i	$-0.514475\pi$
$61$	−7.28945	−	4.20857i	−0.933319	−	0.538852i	−0.887860	+	0.460114i	$0.847808\pi$
$62$	−0.868348			−0.110280
$63$		0			0
$64$	−1.00000			−0.125000
$65$	3.14456	+	1.81552i	0.390035	+	0.225187i
$66$		0			0
$67$	−5.08044	−	8.79958i	−0.620674	−	1.07504i	−0.989360	−	0.145485i	$-0.953526\pi$
$67$	−5.08044	−	8.79958i	−0.620674	−	1.07504i	0.368686	−	0.929554i	$-0.379808\pi$
$68$	−0.182163	+	0.315515i	−0.0220905	+	0.0382618i
$69$		0			0
$70$	−0.280861	+	3.91189i	−0.0335693	+	0.467560i
$71$		−	2.07911i		−	0.246745i	−0.992360	−	0.123373i	$-0.960629\pi$
$71$		−	2.07911i		−	0.246745i	0.992360	−	0.123373i	$-0.0393710\pi$
$72$		0			0
$73$	−7.86798	+	4.54258i	−0.920878	+	0.531669i	−0.883915	−	0.467648i	$-0.845102\pi$
$73$	−7.86798	+	4.54258i	−0.920878	+	0.531669i	−0.0369628	+	0.999317i	$0.511768\pi$
$74$	8.04354	−	4.64394i	0.935043	−	0.539847i
$75$		0			0
$76$			8.15623i			0.935584i
$77$	0.189469	−	2.63896i	0.0215920	−	0.300737i
$78$		0			0
$79$	3.00913	−	5.21197i	0.338554	−	0.586392i	−0.645607	−	0.763670i	$-0.723396\pi$
$79$	3.00913	−	5.21197i	0.338554	−	0.586392i	0.984161	+	0.177278i	$0.0567290\pi$
$80$	−0.741181	−	1.28376i	−0.0828665	−	0.143529i
$81$		0			0
$82$	8.13103	+	4.69445i	0.897922	+	0.518416i
$83$	−15.5699			−1.70902			−0.854511	−	0.519433i	$-0.826143\pi$
$83$	−15.5699			−1.70902			−0.854511	+	0.519433i	$0.826143\pi$
$84$		0			0
$85$	−0.540062			−0.0585780
$86$	2.54963	+	1.47203i	0.274934	+	0.158733i
$87$		0			0
$88$	0.500000	+	0.866025i	0.0533002	+	0.0923186i
$89$	−0.459822	+	0.796435i	−0.0487410	+	0.0844219i	−0.889367	−	0.457195i	$-0.848854\pi$
$89$	−0.459822	+	0.796435i	−0.0487410	+	0.0844219i	0.840626	+	0.541617i	$0.182188\pi$
$90$		0			0
$91$	5.36603	+	3.63397i	0.562512	+	0.380944i
$92$		−	5.98174i		−	0.623639i
$93$		0			0
$94$	−1.10463	+	0.637756i	−0.113934	+	0.0657796i
$95$	−10.4707	+	6.04524i	−1.07427	+	0.620229i
$96$		0			0
$97$		−	1.79920i		−	0.182681i	−0.995820	−	0.0913407i	$-0.970885\pi$
$97$		−	1.79920i		−	0.182681i	0.995820	−	0.0913407i	$-0.0291152\pi$
$98$	−1.00000	+	6.92820i	−0.101015	+	0.699854i
$99$		0			0
$100$	−1.40130	+	2.42713i	−0.140130	+	0.242713i
$101$	5.31079	+	9.19856i	0.528443	+	0.915291i	0.999450	+	0.0331610i	$0.0105574\pi$
$101$	5.31079	+	9.19856i	0.528443	+	0.915291i	−0.471007	+	0.882130i	$0.656109\pi$
$102$		0			0
$103$	2.50133	+	1.44414i	0.246463	+	0.142295i	0.618144	−	0.786065i	$-0.287885\pi$
$103$	2.50133	+	1.44414i	0.246463	+	0.142295i	−0.371681	+	0.928361i	$0.621218\pi$
$104$	−2.44949			−0.240192
$105$		0			0
$106$	5.37769			0.522328
$107$	4.29850	+	2.48174i	0.415552	+	0.239919i	0.693172	−	0.720772i	$-0.256213\pi$
$107$	4.29850	+	2.48174i	0.415552	+	0.239919i	−0.277621	+	0.960691i	$0.589546\pi$
$108$		0			0
$109$	2.23351	+	3.86855i	0.213932	+	0.370540i	0.952942	−	0.303154i	$-0.0980397\pi$
$109$	2.23351	+	3.86855i	0.213932	+	0.370540i	−0.739010	+	0.673694i	$0.764706\pi$
$110$	−0.741181	+	1.28376i	−0.0706688	+	0.122402i
$111$		0			0
$112$	−1.15539	−	2.38014i	−0.109175	−	0.224902i
$113$		−	8.35827i		−	0.786280i	−0.919479	−	0.393140i	$-0.871389\pi$
$113$		−	8.35827i		−	0.786280i	0.919479	−	0.393140i	$-0.128611\pi$
$114$		0			0
$115$	7.67914	−	4.43355i	0.716083	−	0.413431i
$116$	−0.287909	+	0.166225i	−0.0267317	+	0.0154336i
$117$		0			0
$118$			8.87832i			0.817315i
$119$	−0.961440	−	0.0690283i	−0.0881350	−	0.00632781i
$120$		0			0
$121$	0.500000	−	0.866025i	0.0454545	−	0.0787296i
$122$	4.20857	+	7.28945i	0.381026	+	0.659956i
$123$		0			0
$124$	0.752011	+	0.434174i	0.0675326	+	0.0389900i
$125$	−11.5663			−1.03452
$126$		0			0
$127$	−18.1879			−1.61391			−0.806956	−	0.590612i	$-0.798886\pi$
$127$	−18.1879			−1.61391			−0.806956	+	0.590612i	$0.798886\pi$
$128$	0.866025	+	0.500000i	0.0765466	+	0.0441942i
$129$		0			0
$130$	−1.81552	−	3.14456i	−0.159231	−	0.275797i
$131$	−9.24264	+	16.0087i	−0.807533	+	1.39869i	0.107034	+	0.994255i	$0.465865\pi$
$131$	−9.24264	+	16.0087i	−0.807533	+	1.39869i	−0.914567	+	0.404433i	$0.867469\pi$
$132$		0			0
$133$	−19.4130	+	9.42367i	−1.68332	+	0.817135i
$134$			10.1609i			0.877766i
$135$		0			0
$136$	0.315515	−	0.182163i	0.0270552	−	0.0156203i
$137$	13.8248	−	7.98174i	1.18113	−	0.681926i	0.224855	−	0.974392i	$-0.427809\pi$
$137$	13.8248	−	7.98174i	1.18113	−	0.681926i	0.956276	+	0.292466i	$0.0944759\pi$
$138$		0			0
$139$		−	9.59111i		−	0.813507i	−0.913538	−	0.406754i	$-0.866661\pi$
$139$		−	9.59111i		−	0.813507i	0.913538	−	0.406754i	$-0.133339\pi$
$140$	2.19918	−	3.24737i	0.185865	−	0.274453i
$141$		0			0
$142$	−1.03956	+	1.80056i	−0.0872376	+	0.151100i
$143$	1.22474	+	2.12132i	0.102418	+	0.177394i
$144$		0			0
$145$	−0.426786	−	0.246405i	−0.0354426	−	0.0204628i
$146$	9.08516			0.751894
$147$		0			0
$148$	−9.28788			−0.763459
$149$	17.4212	+	10.0582i	1.42720	+	0.823996i	0.996899	−	0.0786878i	$-0.0250730\pi$
$149$	17.4212	+	10.0582i	1.42720	+	0.823996i	0.430304	+	0.902684i	$0.358406\pi$
$150$		0			0
$151$	2.54466	+	4.40749i	0.207082	+	0.358676i	0.950794	−	0.309824i	$-0.100270\pi$
$151$	2.54466	+	4.40749i	0.207082	+	0.358676i	−0.743712	+	0.668500i	$0.766937\pi$
$152$	4.07812	−	7.06350i	0.330779	−	0.572926i
$153$		0			0
$154$	−1.48356	+	2.19067i	−0.119549	+	0.176529i
$155$			1.28721i			0.103391i
$156$		0			0
$157$	−13.7026	+	7.91119i	−1.09358	+	0.631381i	−0.934529	−	0.355888i	$-0.884178\pi$
$157$	−13.7026	+	7.91119i	−1.09358	+	0.631381i	−0.159056	+	0.987270i	$0.550845\pi$
$158$	−5.21197	+	3.00913i	−0.414642	+	0.239394i
$159$		0			0
$160$			1.48236i			0.117191i
$161$	14.2374	−	6.91127i	1.12206	−	0.544684i
$162$		0			0
$163$	8.89595	−	15.4082i	0.696785	−	1.20687i	−0.272790	−	0.962073i	$-0.587947\pi$
$163$	8.89595	−	15.4082i	0.696785	−	1.20687i	0.969575	−	0.244793i	$-0.0787201\pi$
$164$	−4.69445	−	8.13103i	−0.366575	−	0.634927i
$165$		0			0
$166$	13.4840	+	7.78497i	1.04656	+	0.604230i
$167$	−12.7627			−0.987606			−0.493803	−	0.869574i	$-0.664394\pi$
$167$	−12.7627			−0.987606			−0.493803	+	0.869574i	$0.664394\pi$
$168$		0			0
$169$	7.00000			0.538462
$170$	0.467708	+	0.270031i	0.0358715	+	0.0207104i
$171$		0			0
$172$	−1.47203	−	2.54963i	−0.112241	−	0.194407i
$173$	−0.636379	+	1.10224i	−0.0483830	+	0.0838018i	−0.889203	−	0.457514i	$-0.848740\pi$
$173$	−0.636379	+	1.10224i	−0.0483830	+	0.0838018i	0.840820	+	0.541315i	$0.182073\pi$
$174$		0			0
$175$	−7.39595	−	0.531006i	−0.559082	−	0.0401402i
$176$		−	1.00000i		−	0.0753778i
$177$		0			0
$178$	0.796435	−	0.459822i	0.0596953	−	0.0344651i
$179$	7.10671	−	4.10306i	0.531180	−	0.306677i	−0.210317	−	0.977633i	$-0.567449\pi$
$179$	7.10671	−	4.10306i	0.531180	−	0.306677i	0.741497	+	0.670956i	$0.234116\pi$
$180$		0			0
$181$		−	3.53125i		−	0.262476i	−0.991351	−	0.131238i	$-0.958105\pi$
$181$		−	3.53125i		−	0.262476i	0.991351	−	0.131238i	$-0.0418952\pi$
$182$	−2.83013	−	5.83013i	−0.209783	−	0.432158i
$183$		0			0
$184$	−2.99087	+	5.18034i	−0.220490	+	0.381900i
$185$	−6.88400	−	11.9234i	−0.506122	−	0.876629i
$186$		0			0
$187$	−0.315515	−	0.182163i	−0.0230728	−	0.0133211i
$188$	1.27551			0.0930263
$189$		0			0
$190$	12.0905			0.877136
$191$	3.03516	+	1.75235i	0.219616	+	0.126796i	0.605773	−	0.795638i	$-0.292864\pi$
$191$	3.03516	+	1.75235i	0.219616	+	0.126796i	−0.386156	+	0.922433i	$0.626197\pi$
$192$		0			0
$193$	10.2639	+	17.7777i	0.738814	+	1.27966i	0.953030	+	0.302877i	$0.0979473\pi$
$193$	10.2639	+	17.7777i	0.738814	+	1.27966i	−0.214215	+	0.976786i	$0.568719\pi$
$194$	−0.899602	+	1.55816i	−0.0645876	+	0.111869i
$195$		0			0
$196$	4.33013	−	5.50000i	0.309295	−	0.392857i
$197$			25.2772i			1.80093i	0.434934	+	0.900463i	$0.356772\pi$
$197$			25.2772i			1.80093i	−0.434934	+	0.900463i	$0.643228\pi$
$198$		0			0
$199$	−18.4746	+	10.6663i	−1.30963	+	0.756114i	−0.982034	−	0.188704i	$-0.939571\pi$
$199$	−18.4746	+	10.6663i	−1.30963	+	0.756114i	−0.327595	+	0.944818i	$0.606238\pi$
$200$	2.42713	−	1.40130i	0.171624	−	0.0990870i
$201$		0			0
$202$		−	10.6216i		−	0.747332i
$203$	−0.728287	−	0.493210i	−0.0511157	−	0.0346165i
$204$		0			0
$205$	6.95888	−	12.0531i	0.486029	−	0.841827i
$206$	−1.44414	−	2.50133i	−0.100618	−	0.174276i
$207$		0			0
$208$	2.12132	+	1.22474i	0.147087	+	0.0849208i
$209$	−8.15623			−0.564178
$210$		0			0
$211$	−27.8630			−1.91817			−0.959083	−	0.283125i	$-0.908629\pi$
$211$	−27.8630			−1.91817			−0.959083	+	0.283125i	$0.908629\pi$
$212$	−4.65722	−	2.68885i	−0.319859	−	0.184671i
$213$		0			0
$214$	−2.48174	−	4.29850i	−0.169648	−	0.293839i
$215$	2.18208	−	3.77947i	0.148817	−	0.257758i
$216$		0			0
$217$	−0.164525	+	2.29153i	−0.0111687	+	0.155559i
$218$		−	4.46702i		−	0.302545i
$219$		0			0
$220$	1.28376	−	0.741181i	0.0865513	−	0.0499704i
$221$	0.772851	−	0.446206i	0.0519876	−	0.0300151i
$222$		0			0
$223$		−	5.97418i		−	0.400060i	−0.979790	−	0.200030i	$-0.935896\pi$
$223$		−	5.97418i		−	0.400060i	0.979790	−	0.200030i	$-0.0641040\pi$
$224$	−0.189469	+	2.63896i	−0.0126594	+	0.176323i
$225$		0			0
$226$	−4.17914	+	7.23848i	−0.277992	+	0.481496i
$227$	1.77721	+	3.07822i	0.117958	+	0.204309i	0.918958	−	0.394355i	$-0.129032\pi$
$227$	1.77721	+	3.07822i	0.117958	+	0.204309i	−0.801000	+	0.598664i	$0.795699\pi$
$228$		0			0
$229$	8.68046	+	5.01167i	0.573621	+	0.331180i	0.758594	−	0.651563i	$-0.225886\pi$
$229$	8.68046	+	5.01167i	0.573621	+	0.331180i	−0.184973	+	0.982744i	$0.559220\pi$
$230$	−8.86710			−0.584679
$231$		0			0
$232$	0.332449			0.0218264
$233$	15.1427	+	8.74264i	0.992031	+	0.572749i	0.905881	−	0.423533i	$-0.139210\pi$
$233$	15.1427	+	8.74264i	0.992031	+	0.572749i	0.0861503	+	0.996282i	$0.472543\pi$
$234$		0			0
$235$	0.945386	+	1.63746i	0.0616702	+	0.106816i
$236$	4.43916	−	7.68885i	0.288965	−	0.500501i
$237$		0			0
$238$	0.798117	+	0.540500i	0.0517343	+	0.0350354i
$239$			13.9742i			0.903914i	0.892040	+	0.451957i	$0.149274\pi$
$239$			13.9742i			0.903914i	−0.892040	+	0.451957i	$0.850726\pi$
$240$		0			0
$241$	−0.247649	+	0.142980i	−0.0159525	+	0.00921018i	−0.507955	−	0.861384i	$-0.669598\pi$
$241$	−0.247649	+	0.142980i	−0.0159525	+	0.00921018i	0.492003	+	0.870594i	$0.336265\pi$
$242$	−0.866025	+	0.500000i	−0.0556702	+	0.0321412i
$243$		0			0
$244$		−	8.41713i		−	0.538852i
$245$	10.2701	+	1.48236i	0.656133	+	0.0947046i
$246$		0			0
$247$	9.98930	−	17.3020i	0.635604	−	1.10090i
$248$	−0.434174	−	0.752011i	−0.0275701	−	0.0477527i
$249$		0			0
$250$	10.0167	+	5.78314i	0.633511	+	0.365758i
$251$	4.29150			0.270877			0.135438	−	0.990786i	$-0.456756\pi$
$251$	4.29150			0.270877			0.135438	+	0.990786i	$0.456756\pi$
$252$		0			0
$253$	5.98174			0.376069
$254$	15.7511	+	9.09393i	0.988315	+	0.570604i
$255$		0			0
$256$	−0.500000	−	0.866025i	−0.0312500	−	0.0541266i
$257$	−6.88865	+	11.9315i	−0.429702	+	0.744266i	−0.996847	−	0.0793525i	$-0.974715\pi$
$257$	−6.88865	+	11.9315i	−0.429702	+	0.744266i	0.567145	+	0.823618i	$0.308048\pi$
$258$		0			0
$259$	−10.7312	−	22.1065i	−0.666803	−	1.37363i
$260$			3.63103i			0.225187i
$261$		0			0
$262$	16.0087	−	9.24264i	0.989022	−	0.571012i
$263$	−4.92134	+	2.84134i	−0.303463	+	0.175204i	−0.643998	−	0.765028i	$-0.722725\pi$
$263$	−4.92134	+	2.84134i	−0.303463	+	0.175204i	0.340535	+	0.940232i	$0.389392\pi$
$264$		0			0
$265$		−	7.97169i		−	0.489697i
$266$	21.5240	+	1.54535i	1.31972	+	0.0947515i
$267$		0			0
$268$	5.08044	−	8.79958i	0.310337	−	0.537520i
$269$	−6.83328	−	11.8356i	−0.416632	−	0.721628i	0.578966	−	0.815352i	$-0.303456\pi$
$269$	−6.83328	−	11.8356i	−0.416632	−	0.721628i	−0.995598	+	0.0937234i	$0.970123\pi$
$270$		0			0
$271$	15.5431	+	8.97381i	0.944176	+	0.545120i	0.891267	−	0.453479i	$-0.149817\pi$
$271$	15.5431	+	8.97381i	0.944176	+	0.545120i	0.0529090	+	0.998599i	$0.483151\pi$
$272$	−0.364326			−0.0220905
$273$		0			0
$274$	−15.9635			−0.964389
$275$	−2.42713	−	1.40130i	−0.146361	−	0.0845017i
$276$		0			0
$277$	−8.33814	−	14.4421i	−0.500990	−	0.867740i	−0.999999	−	0.00114365i	$-0.999636\pi$
$277$	−8.33814	−	14.4421i	−0.500990	−	0.867740i	0.499009	−	0.866597i	$-0.333697\pi$
$278$	−4.79555	+	8.30614i	−0.287618	+	0.498169i
$279$		0			0
$280$	−3.52823	+	1.71271i	−0.210852	+	0.102354i
$281$			13.1377i			0.783730i	0.920023	+	0.391865i	$0.128170\pi$
$281$			13.1377i			0.783730i	−0.920023	+	0.391865i	$0.871830\pi$
$282$		0			0
$283$	25.2985	−	14.6061i	1.50384	−	0.868242i	0.503849	−	0.863792i	$-0.331917\pi$
$283$	25.2985	−	14.6061i	1.50384	−	0.868242i	0.999990	−	0.00444975i	$-0.00141641\pi$
$284$	1.80056	−	1.03956i	0.106844	−	0.0616863i
$285$		0			0
$286$		−	2.44949i		−	0.144841i
$287$	13.9290	−	20.5680i	0.822205	−	1.21409i
$288$		0			0
$289$	8.43363	−	14.6075i	0.496096	−	0.859264i
$290$	0.246405	+	0.426786i	0.0144694	+	0.0250617i
$291$		0			0
$292$	−7.86798	−	4.54258i	−0.460439	−	0.265835i
$293$	25.6875			1.50068			0.750339	−	0.661053i	$-0.229890\pi$
$293$	25.6875			1.50068			0.750339	+	0.661053i	$0.229890\pi$
$294$		0			0
$295$	13.1609			0.766256
$296$	8.04354	+	4.64394i	0.467521	+	0.269924i
$297$		0			0
$298$	−10.0582	−	17.4212i	−0.582653	−	1.00919i
$299$	−7.32611	+	12.6892i	−0.423680	+	0.733835i
$300$		0			0
$301$	4.36770	−	6.44946i	0.251750	−	0.371741i
$302$		−	5.08933i		−	0.292858i
$303$		0			0
$304$	−7.06350	+	4.07812i	−0.405120	+	0.233896i
$305$	10.8056	−	6.23862i	0.618727	−	0.357222i
$306$		0			0
$307$			0.928932i			0.0530170i	0.999649	+	0.0265085i	$0.00843890\pi$
$307$			0.928932i			0.0530170i	−0.999649	+	0.0265085i	$0.991561\pi$
$308$	2.38014	−	1.15539i	0.135621	−	0.0658347i
$309$		0			0
$310$	0.643603	−	1.11475i	0.0365542	−	0.0633137i
$311$	−4.63676	−	8.03111i	−0.262927	−	0.455402i	0.704092	−	0.710109i	$-0.251354\pi$
$311$	−4.63676	−	8.03111i	−0.262927	−	0.455402i	−0.967018	+	0.254707i	$0.918021\pi$
$312$		0			0
$313$	−0.884368	−	0.510590i	−0.0499874	−	0.0288602i	0.474798	−	0.880095i	$-0.342521\pi$
$313$	−0.884368	−	0.510590i	−0.0499874	−	0.0288602i	−0.524785	+	0.851235i	$0.675854\pi$
$314$	15.8224			0.892908
$315$		0			0
$316$	6.01826			0.338554
$317$	−9.92460	−	5.72997i	−0.557421	−	0.321827i	0.194689	−	0.980865i	$-0.437630\pi$
$317$	−9.92460	−	5.72997i	−0.557421	−	0.321827i	−0.752110	+	0.659038i	$0.770964\pi$
$318$		0			0
$319$	−0.166225	−	0.287909i	−0.00930679	−	0.0161198i
$320$	0.741181	−	1.28376i	0.0414333	−	0.0717645i
$321$		0			0
$322$	−15.7856	−	1.13335i	−0.879695	−	0.0631593i
$323$			2.97152i			0.165340i
$324$		0			0
$325$	5.94522	−	3.43247i	0.329781	−	0.190399i
$326$	−15.4082	+	8.89595i	−0.853384	+	0.492701i
$327$		0			0
$328$			9.38891i			0.518416i
$329$	1.47372	+	3.03590i	0.0812488	+	0.167374i
$330$		0			0
$331$	10.9670	−	18.9954i	0.602802	−	1.04408i	−0.389593	−	0.920987i	$-0.627384\pi$
$331$	10.9670	−	18.9954i	0.602802	−	1.04408i	0.992395	−	0.123096i	$-0.0392824\pi$
$332$	−7.78497	−	13.4840i	−0.427255	−	0.740028i
$333$		0			0
$334$	11.0528	+	6.38134i	0.604783	+	0.349171i
$335$	15.0621			0.822930
$336$		0			0
$337$	−10.7768			−0.587049			−0.293524	−	0.955952i	$-0.594828\pi$
$337$	−10.7768			−0.587049			−0.293524	+	0.955952i	$0.594828\pi$
$338$	−6.06218	−	3.50000i	−0.329739	−	0.190375i
$339$		0			0
$340$	−0.270031	−	0.467708i	−0.0146445	−	0.0253650i
$341$	−0.434174	+	0.752011i	−0.0235118	+	0.0407237i
$342$		0			0
$343$	18.0938	+	3.95164i	0.976972	+	0.213368i
$344$			2.94406i			0.158733i
$345$		0			0
$346$	1.10224	−	0.636379i	0.0592568	−	0.0342119i
$347$	27.5550	−	15.9089i	1.47923	−	0.854033i	0.479504	−	0.877540i	$-0.340817\pi$
$347$	27.5550	−	15.9089i	1.47923	−	0.854033i	0.999724	+	0.0235069i	$0.00748318\pi$
$348$		0			0
$349$		−	20.3088i		−	1.08711i	−0.839375	−	0.543553i	$-0.817079\pi$
$349$		−	20.3088i		−	1.08711i	0.839375	−	0.543553i	$-0.182921\pi$
$350$	6.13958	+	4.15784i	0.328174	+	0.222246i
$351$		0			0
$352$	−0.500000	+	0.866025i	−0.0266501	+	0.0461593i
$353$	6.06314	+	10.5017i	0.322708	+	0.558947i	0.981046	−	0.193776i	$-0.0620734\pi$
$353$	6.06314	+	10.5017i	0.322708	+	0.558947i	−0.658338	+	0.752723i	$0.728740\pi$
$354$		0			0
$355$	2.66909	+	1.54100i	0.141660	+	0.0817877i
$356$	−0.919644			−0.0487410
$357$		0			0
$358$	−8.20612			−0.433707
$359$	11.6763	+	6.74131i	0.616251	+	0.355793i	0.775408	−	0.631460i	$-0.217544\pi$
$359$	11.6763	+	6.74131i	0.616251	+	0.355793i	−0.159157	+	0.987253i	$0.550877\pi$
$360$		0			0
$361$	23.7621	+	41.1571i	1.25063	+	2.16616i
$362$	−1.76563	+	3.05816i	−0.0927993	+	0.160733i
$363$		0			0
$364$	−0.464102	+	6.46410i	−0.0243255	+	0.338811i
$365$		−	13.4675i		−	0.704921i
$366$		0			0
$367$	−27.0615	+	15.6240i	−1.41260	+	0.815565i	−0.995633	−	0.0933561i	$-0.970240\pi$
$367$	−27.0615	+	15.6240i	−1.41260	+	0.815565i	−0.416968	+	0.908921i	$0.636907\pi$
$368$	5.18034	−	2.99087i	0.270044	−	0.155910i
$369$		0			0
$370$			13.7680i			0.715765i
$371$	1.01890	−	14.1915i	0.0528989	−	0.736786i
$372$		0			0
$373$	−0.444639	+	0.770138i	−0.0230226	+	0.0398762i	−0.877307	−	0.479929i	$-0.840662\pi$
$373$	−0.444639	+	0.770138i	−0.0230226	+	0.0398762i	0.854285	+	0.519806i	$0.173996\pi$
$374$	0.182163	+	0.315515i	0.00941941	+	0.0163149i
$375$		0			0
$376$	−1.10463	−	0.637756i	−0.0569668	−	0.0328898i
$377$	0.814331			0.0419402
$378$		0			0
$379$	20.9635			1.07682			0.538411	−	0.842683i	$-0.319025\pi$
$379$	20.9635			1.07682			0.538411	+	0.842683i	$0.319025\pi$
$380$	−10.4707	−	6.04524i	−0.537134	−	0.310114i
$381$		0			0
$382$	−1.75235	−	3.03516i	−0.0896581	−	0.155292i
$383$	0.824147	−	1.42747i	0.0421120	−	0.0729401i	−0.844201	−	0.536026i	$-0.819925\pi$
$383$	0.824147	−	1.42747i	0.0421120	−	0.0729401i	0.886313	+	0.463086i	$0.153258\pi$
$384$		0			0
$385$	3.24737	+	2.19918i	0.165501	+	0.112081i
$386$		−	20.5279i		−	1.04484i
$387$		0			0
$388$	1.55816	−	0.899602i	0.0791034	−	0.0456704i
$389$	−7.89537	+	4.55840i	−0.400311	+	0.231120i	−0.686618	−	0.727018i	$-0.740906\pi$
$389$	−7.89537	+	4.55840i	−0.400311	+	0.231120i	0.286307	+	0.958138i	$0.407572\pi$
$390$		0			0
$391$		−	2.17930i		−	0.110212i
$392$	−6.50000	+	2.59808i	−0.328300	+	0.131223i
$393$		0			0
$394$	12.6386	−	21.8907i	0.636723	−	1.10284i
$395$	4.46062	+	7.72602i	0.224438	+	0.388738i
$396$		0			0
$397$	6.49063	+	3.74737i	0.325755	+	0.188075i	0.653955	−	0.756533i	$-0.273109\pi$
$397$	6.49063	+	3.74737i	0.325755	+	0.188075i	−0.328200	+	0.944608i	$0.606442\pi$
$398$	21.3326			1.06931
$399$		0			0
$400$	−2.80260			−0.140130
$401$	5.45101	+	3.14714i	0.272211	+	0.157161i	0.629892	−	0.776683i	$-0.283099\pi$
$401$	5.45101	+	3.14714i	0.272211	+	0.157161i	−0.357681	+	0.933844i	$0.616433\pi$
$402$		0			0
$403$	−1.06350	−	1.84204i	−0.0529769	−	0.0917587i
$404$	−5.31079	+	9.19856i	−0.264222	+	0.457645i
$405$		0			0
$406$	0.384110	+	0.791275i	0.0190631	+	0.0392703i
$407$		−	9.28788i		−	0.460383i
$408$		0			0
$409$	−11.7467	+	6.78194i	−0.580835	+	0.335345i	−0.761465	−	0.648206i	$-0.775520\pi$
$409$	−11.7467	+	6.78194i	−0.580835	+	0.335345i	0.180630	+	0.983551i	$0.442186\pi$
$410$	−12.0531	+	6.95888i	−0.595262	+	0.343674i
$411$		0			0
$412$			2.88828i			0.142295i
$413$	23.4295	+	1.68216i	1.15289	+	0.0827738i
$414$		0			0
$415$	11.5401	−	19.9881i	0.566483	−	0.981177i
$416$	−1.22474	−	2.12132i	−0.0600481	−	0.104006i
$417$		0			0
$418$	7.06350	+	4.07812i	0.345487	+	0.199467i
$419$	−20.4037			−0.996785			−0.498393	−	0.866951i	$-0.666076\pi$
$419$	−20.4037			−0.996785			−0.498393	+	0.866951i	$0.666076\pi$
$420$		0			0
$421$	−19.0673			−0.929284			−0.464642	−	0.885499i	$-0.653817\pi$
$421$	−19.0673			−0.929284			−0.464642	+	0.885499i	$0.653817\pi$
$422$	24.1300	+	13.9315i	1.17463	+	0.678174i
$423$		0			0
$424$	2.68885	+	4.65722i	0.130582	+	0.226175i
$425$	−0.510530	+	0.884264i	−0.0247643	+	0.0428931i
$426$		0			0
$427$	20.0339	−	9.72511i	0.969511	−	0.470631i
$428$			4.96348i			0.239919i
$429$		0			0
$430$	−3.77947	+	2.18208i	−0.182262	+	0.105229i
$431$	−24.1846	+	13.9630i	−1.16493	+	0.672574i	−0.952481	−	0.304598i	$-0.901478\pi$
$431$	−24.1846	+	13.9630i	−1.16493	+	0.672574i	−0.212451	+	0.977172i	$0.568145\pi$
$432$		0			0
$433$			24.8145i			1.19251i	0.802796	+	0.596254i	$0.203345\pi$
$433$			24.8145i			1.19251i	−0.802796	+	0.596254i	$0.796655\pi$
$434$	1.28825	−	1.90226i	0.0618379	−	0.0913116i
$435$		0			0
$436$	−2.23351	+	3.86855i	−0.106966	+	0.185270i
$437$	−24.3942	−	42.2520i	−1.16693	−	2.02119i
$438$		0			0
$439$	−12.6859	−	7.32420i	−0.605464	−	0.349565i	0.165724	−	0.986172i	$-0.447004\pi$
$439$	−12.6859	−	7.32420i	−0.605464	−	0.349565i	−0.771188	+	0.636607i	$0.780337\pi$
$440$	−1.48236			−0.0706688
$441$		0			0
$442$	−0.892412			−0.0424477
$443$	27.7172	+	16.0026i	1.31689	+	0.760304i	0.983226	−	0.182389i	$-0.0583831\pi$
$443$	27.7172	+	16.0026i	1.31689	+	0.760304i	0.333659	+	0.942694i	$0.391716\pi$
$444$		0			0
$445$	−0.681622	−	1.18060i	−0.0323120	−	0.0559660i
$446$	−2.98709	+	5.17379i	−0.141443	+	0.244986i
$447$		0			0
$448$	1.48356	−	2.19067i	0.0700918	−	0.103499i
$449$		−	38.8938i		−	1.83551i	−0.397146	−	0.917755i	$-0.630000\pi$
$449$		−	38.8938i		−	1.83551i	0.397146	−	0.917755i	$-0.370000\pi$
$450$		0			0
$451$	8.13103	−	4.69445i	0.382875	−	0.221053i
$452$	7.23848	−	4.17914i	0.340469	−	0.196570i
$453$		0			0
$454$		−	3.55443i		−	0.166817i
$455$	−8.64236	+	4.19527i	−0.405160	+	0.196677i
$456$		0			0
$457$	−11.8239	+	20.4796i	−0.553099	+	0.957995i	0.444950	+	0.895555i	$0.353221\pi$
$457$	−11.8239	+	20.4796i	−0.553099	+	0.957995i	−0.998049	+	0.0624396i	$0.980112\pi$
$458$	−5.01167	−	8.68046i	−0.234180	−	0.405611i
$459$		0			0
$460$	7.67914	+	4.43355i	0.358042	+	0.206715i
$461$	7.97181			0.371284			0.185642	−	0.982617i	$-0.440564\pi$
$461$	7.97181			0.371284			0.185642	+	0.982617i	$0.440564\pi$
$462$		0			0
$463$	−33.7108			−1.56667			−0.783337	−	0.621597i	$-0.786484\pi$
$463$	−33.7108			−1.56667			−0.783337	+	0.621597i	$0.786484\pi$
$464$	−0.287909	−	0.166225i	−0.0133659	−	0.00771678i
$465$		0			0
$466$	−8.74264	−	15.1427i	−0.404995	−	0.701472i
$467$	9.33814	−	16.1741i	0.432117	−	0.748449i	−0.564938	−	0.825133i	$-0.691100\pi$
$467$	9.33814	−	16.1741i	0.432117	−	0.748449i	0.997055	+	0.0766840i	$0.0244333\pi$
$468$		0			0
$469$	26.8141	+	1.92517i	1.23816	+	0.0888960i
$470$		−	1.89077i		−	0.0872148i
$471$		0			0
$472$	−7.68885	+	4.43916i	−0.353908	+	0.204329i
$473$	2.54963	−	1.47203i	0.117232	−	0.0676840i
$474$		0			0
$475$			22.8587i			1.04883i
$476$	−0.420940	−	0.867145i	−0.0192937	−	0.0397455i
$477$		0			0
$478$	6.98709	−	12.1020i	0.319582	−	0.553532i
$479$	8.06866	+	13.9753i	0.368667	+	0.638549i	0.989357	−	0.145506i	$-0.0464811\pi$
$479$	8.06866	+	13.9753i	0.368667	+	0.638549i	−0.620691	+	0.784056i	$0.713148\pi$
$480$		0			0
$481$	19.7026	+	11.3753i	0.898360	+	0.518669i
$482$	0.285961			0.0130252
$483$		0			0
$484$	1.00000			0.0454545
$485$	2.30975	+	1.33354i	0.104880	+	0.0605527i
$486$		0			0
$487$	−12.7790	−	22.1339i	−0.579072	−	1.00298i	−0.995586	−	0.0938523i	$-0.970082\pi$
$487$	−12.7790	−	22.1339i	−0.579072	−	1.00298i	0.416515	−	0.909129i	$-0.363251\pi$
$488$	−4.20857	+	7.28945i	−0.190513	+	0.329978i
$489$		0			0
$490$	−8.15299	−	6.41882i	−0.368315	−	0.289973i
$491$		−	21.5909i		−	0.974384i	−0.873295	−	0.487192i	$-0.838021\pi$
$491$		−	21.5909i		−	0.974384i	0.873295	−	0.487192i	$-0.161979\pi$
$492$		0			0
$493$	−0.104893	+	0.0605598i	−0.00472413	+	0.00272748i
$494$	−17.3020	+	9.98930i	−0.778453	+	0.449440i
$495$		0			0
$496$			0.868348i			0.0389900i
$497$	4.55465	+	3.08450i	0.204304	+	0.138359i
$498$		0			0
$499$	0.470154	−	0.814331i	0.0210470	−	0.0364544i	−0.855310	−	0.518116i	$-0.826633\pi$
$499$	0.470154	−	0.814331i	0.0210470	−	0.0364544i	0.876357	+	0.481662i	$0.159967\pi$
$500$	−5.78314	−	10.0167i	−0.258630	−	0.447960i
$501$		0			0
$502$	−3.71655	−	2.14575i	−0.165878	−	0.0957695i
$503$	−37.4660			−1.67053			−0.835264	−	0.549849i	$-0.814685\pi$
$503$	−37.4660			−1.67053			−0.835264	+	0.549849i	$0.814685\pi$
$504$		0			0
$505$	−15.7450			−0.700644
$506$	−5.18034	−	2.99087i	−0.230294	−	0.132960i
$507$		0			0
$508$	−9.09393	−	15.7511i	−0.403478	−	0.698844i
$509$	−8.52849	+	14.7718i	−0.378018	+	0.654747i	−0.990774	−	0.135525i	$-0.956728\pi$
$509$	−8.52849	+	14.7718i	−0.378018	+	0.654747i	0.612755	+	0.790273i	$0.290061\pi$
$510$		0			0
$511$	1.72135	−	23.9754i	0.0761482	−	1.06061i
$512$			1.00000i			0.0441942i
$513$		0			0
$514$	11.9315	−	6.88865i	0.526275	−	0.303845i
$515$	−3.70787	+	2.14074i	−0.163388	+	0.0943323i
$516$		0			0
$517$			1.27551i			0.0560970i
$518$	−1.75976	+	24.5103i	−0.0773196	+	1.07692i
$519$		0			0
$520$	1.81552	−	3.14456i	0.0796156	−	0.137898i
$521$	14.2022	+	24.5989i	0.622210	+	1.07770i	0.989073	+	0.147424i	$0.0470983\pi$
$521$	14.2022	+	24.5989i	0.622210	+	1.07770i	−0.366863	+	0.930275i	$0.619568\pi$
$522$		0			0
$523$	13.1571	+	7.59627i	0.575321	+	0.332162i	0.759272	−	0.650774i	$-0.225555\pi$
$523$	13.1571	+	7.59627i	0.575321	+	0.332162i	−0.183951	+	0.982935i	$0.558889\pi$
$524$	−18.4853			−0.807533
$525$		0			0
$526$	5.68268			0.247777
$527$	0.273977	+	0.158181i	0.0119346	+	0.00689045i
$528$		0			0
$529$	6.39060	+	11.0689i	0.277852	+	0.481254i
$530$	−3.98584	+	6.90368i	−0.173134	+	0.299877i
$531$		0			0
$532$	−17.8676	−	12.1003i	−0.774660	−	0.524614i
$533$			22.9980i			0.996155i
$534$		0			0
$535$	−6.37193	+	3.67884i	−0.275483	+	0.159050i
$536$	−8.79958	+	5.08044i	−0.380084	+	0.219442i
$537$		0			0
$538$			13.6666i			0.589207i
$539$	5.50000	+	4.33013i	0.236902	+	0.186512i
$540$		0			0
$541$	−15.8538	+	27.4597i	−0.681610	+	1.18058i	0.292879	+	0.956149i	$0.405387\pi$
$541$	−15.8538	+	27.4597i	−0.681610	+	1.18058i	−0.974489	+	0.224434i	$0.927947\pi$
$542$	−8.97381	−	15.5431i	−0.385458	−	0.667633i
$543$		0			0
$544$	0.315515	+	0.182163i	0.0135276	+	0.00781016i
$545$	−6.62174			−0.283644
$546$		0			0
$547$	−6.10562			−0.261057			−0.130529	−	0.991445i	$-0.541667\pi$
$547$	−6.10562			−0.261057			−0.130529	+	0.991445i	$0.541667\pi$
$548$	13.8248	+	7.98174i	0.590565	+	0.340963i
$549$		0			0
$550$	1.40130	+	2.42713i	0.0597517	+	0.103493i
$551$	−1.35577	+	2.34826i	−0.0577576	+	0.100039i
$552$		0			0
$553$	6.95346	+	14.3243i	0.295691	+	0.609131i
$554$			16.6763i			0.708507i
$555$		0			0
$556$	8.30614	−	4.79555i	0.352259	−	0.203377i
$557$	−4.29153	+	2.47772i	−0.181838	+	0.104984i	−0.588156	−	0.808748i	$-0.700146\pi$
$557$	−4.29153	+	2.47772i	−0.181838	+	0.104984i	0.406318	+	0.913732i	$0.366813\pi$
$558$		0			0
$559$			7.21144i			0.305012i
$560$	3.91189	+	0.280861i	0.165308	+	0.0118686i
$561$		0			0
$562$	6.56885	−	11.3776i	0.277090	−	0.479935i
$563$	−13.7773	−	23.8630i	−0.580644	−	1.00570i	−0.995403	−	0.0957734i	$-0.969468\pi$
$563$	−13.7773	−	23.8630i	−0.580644	−	1.00570i	0.414759	−	0.909931i	$-0.363866\pi$
$564$		0			0
$565$	10.7300	+	6.19499i	0.451416	+	0.260625i
$566$	−29.2122			−1.22788
$567$		0			0
$568$	−2.07911			−0.0872376
$569$	11.0129	+	6.35827i	0.461683	+	0.266553i	0.712752	−	0.701417i	$-0.247449\pi$
$569$	11.0129	+	6.35827i	0.461683	+	0.266553i	−0.251069	+	0.967969i	$0.580782\pi$
$570$		0			0
$571$	20.4985	+	35.5045i	0.857837	+	1.48582i	0.873988	+	0.485948i	$0.161525\pi$
$571$	20.4985	+	35.5045i	0.857837	+	1.48582i	−0.0161511	+	0.999870i	$0.505141\pi$
$572$	−1.22474	+	2.12132i	−0.0512092	+	0.0886969i
$573$		0			0
$574$	−22.3469	+	10.8479i	−0.932742	+	0.452782i
$575$		−	16.7644i		−	0.699126i
$576$		0			0
$577$	−22.1999	+	12.8171i	−0.924193	+	0.533583i	−0.884970	−	0.465647i	$-0.845822\pi$
$577$	−22.1999	+	12.8171i	−0.924193	+	0.533583i	−0.0392228	+	0.999230i	$0.512488\pi$
$578$	−14.6075	+	8.43363i	−0.607591	+	0.350793i
$579$		0			0
$580$		−	0.492810i		−	0.0204628i
$581$	23.0990	−	34.1086i	0.958307	−	1.41506i
$582$		0			0
$583$	2.68885	−	4.65722i	0.111361	−	0.192882i
$584$	4.54258	+	7.86798i	0.187973	+	0.325579i
$585$		0			0
$586$	−22.2460	−	12.8437i	−0.918974	−	0.530570i
$587$	−12.7060			−0.524433			−0.262216	−	0.965009i	$-0.584453\pi$
$587$	−12.7060			−0.524433			−0.262216	+	0.965009i	$0.584453\pi$
$588$		0			0
$589$	7.08244			0.291827
$590$	−11.3977	−	6.58044i	−0.469234	−	0.270912i
$591$		0			0
$592$	−4.64394	−	8.04354i	−0.190865	−	0.330588i
$593$	−13.4689	+	23.3288i	−0.553102	+	0.958000i	0.444947	+	0.895557i	$0.353223\pi$
$593$	−13.4689	+	23.3288i	−0.553102	+	0.958000i	−0.998049	+	0.0624434i	$0.980111\pi$
$594$		0			0
$595$	0.801217	−	1.18310i	0.0328467	−	0.0485023i
$596$			20.1163i			0.823996i
$597$		0			0
$598$	12.6892	−	7.32611i	0.518899	−	0.299587i
$599$	8.83280	−	5.09962i	0.360898	−	0.208365i	−0.308576	−	0.951200i	$-0.599853\pi$
$599$	8.83280	−	5.09962i	0.360898	−	0.208365i	0.669475	+	0.742835i	$0.266519\pi$
$600$		0			0
$601$			5.89774i			0.240574i	0.992739	+	0.120287i	$0.0383814\pi$
$601$			5.89774i			0.240574i	−0.992739	+	0.120287i	$0.961619\pi$
$602$	−7.00727	+	3.40155i	−0.285595	+	0.138637i
$603$		0			0
$604$	−2.54466	+	4.40749i	−0.103541	+	0.179338i
$605$	0.741181	+	1.28376i	0.0301333	+	0.0521924i
$606$		0			0
$607$	−23.4361	−	13.5308i	−0.951243	−	0.549200i	−0.0577759	−	0.998330i	$-0.518401\pi$
$607$	−23.4361	−	13.5308i	−0.951243	−	0.549200i	−0.893467	+	0.449129i	$0.851734\pi$
$608$	8.15623			0.330779
$609$		0			0
$610$	−12.4772			−0.505189
$611$	−2.70577	−	1.56218i	−0.109464	−	0.0631990i
$612$		0			0
$613$	16.5001	+	28.5790i	0.666433	+	1.15430i	0.978895	+	0.204365i	$0.0655130\pi$
$613$	16.5001	+	28.5790i	0.666433	+	1.15430i	−0.312462	+	0.949930i	$0.601154\pi$
$614$	0.464466	−	0.804479i	0.0187443	−	0.0324661i
$615$		0			0
$616$	−2.63896	−	0.189469i	−0.106327	−	0.00763391i
$617$			26.6032i			1.07101i	0.844533	+	0.535503i	$0.179878\pi$
$617$			26.6032i			1.07101i	−0.844533	+	0.535503i	$0.820122\pi$
$618$		0			0
$619$	18.5141	−	10.6891i	0.744143	−	0.429631i	−0.0794305	−	0.996840i	$-0.525310\pi$
$619$	18.5141	−	10.6891i	0.744143	−	0.429631i	0.823574	+	0.567209i	$0.191977\pi$
$620$	−1.11475	+	0.643603i	−0.0447695	+	0.0258477i
$621$		0			0
$622$			9.27352i			0.371834i
$623$	−1.06255	−	2.18888i	−0.0425702	−	0.0876956i
$624$		0			0
$625$	1.56620	−	2.71274i	0.0626480	−	0.108510i
$626$	0.510590	+	0.884368i	0.0204073	+	0.0353464i
$627$		0			0
$628$	−13.7026	−	7.91119i	−0.546792	−	0.315691i
$629$	−3.38381			−0.134921
$630$		0			0
$631$	34.2068			1.36175			0.680876	−	0.732399i	$-0.261599\pi$
$631$	34.2068			1.36175			0.680876	+	0.732399i	$0.261599\pi$
$632$	−5.21197	−	3.00913i	−0.207321	−	0.119697i
$633$		0			0
$634$	5.72997	+	9.92460i	0.227566	+	0.394156i
$635$	13.4805	−	23.3489i	0.534957	−	0.926573i
$636$		0			0
$637$	−15.9217	+	6.36396i	−0.630840	+	0.252149i
$638$			0.332449i			0.0131618i
$639$		0			0
$640$	−1.28376	+	0.741181i	−0.0507452	+	0.0292977i
$641$	6.68319	−	3.85854i	0.263970	−	0.152403i	−0.362174	−	0.932110i	$-0.617965\pi$
$641$	6.68319	−	3.85854i	0.263970	−	0.152403i	0.626144	+	0.779707i	$0.284632\pi$
$642$		0			0
$643$		−	31.0411i		−	1.22414i	−0.790803	−	0.612071i	$-0.790337\pi$
$643$		−	31.0411i		−	1.22414i	0.790803	−	0.612071i	$-0.209663\pi$
$644$	13.1040	+	8.87429i	0.516371	+	0.349696i
$645$		0			0
$646$	1.48576	−	2.57341i	0.0584565	−	0.101250i
$647$	−19.7719	−	34.2460i	−0.777315	−	1.34635i	−0.933484	−	0.358618i	$-0.883248\pi$
$647$	−19.7719	−	34.2460i	−0.777315	−	1.34635i	0.156170	−	0.987730i	$-0.450085\pi$
$648$		0			0
$649$	7.68885	+	4.43916i	0.301814	+	0.174252i
$650$	−6.86495			−0.269265
$651$		0			0
$652$	17.7919			0.696785
$653$	−5.85305	−	3.37926i	−0.229048	−	0.132241i	0.381085	−	0.924540i	$-0.375550\pi$
$653$	−5.85305	−	3.37926i	−0.229048	−	0.132241i	−0.610133	+	0.792299i	$0.708884\pi$
$654$		0			0
$655$	−13.7009	−	23.7307i	−0.535340	−	0.927236i
$656$	4.69445	−	8.13103i	0.183288	−	0.317463i
$657$		0			0
$658$	0.241670	−	3.36603i	0.00942127	−	0.131221i
$659$			36.2611i			1.41253i	0.707947	+	0.706266i	$0.249622\pi$
$659$			36.2611i			1.41253i	−0.707947	+	0.706266i	$0.750378\pi$
$660$		0			0
$661$	−26.4271	+	15.2577i	−1.02790	+	0.593456i	−0.916381	−	0.400307i	$-0.868904\pi$
$661$	−26.4271	+	15.2577i	−1.02790	+	0.593456i	−0.111515	+	0.993763i	$0.535570\pi$
$662$	−18.9954	+	10.9670i	−0.738279	+	0.426245i
$663$		0			0
$664$			15.5699i			0.604230i
$665$	2.29077	−	31.9063i	0.0888322	−	1.23727i
$666$		0			0
$667$	0.994312	−	1.72220i	0.0384999	−	0.0666838i
$668$	−6.38134	−	11.0528i	−0.246902	−	0.427646i
$669$		0			0
$670$	−13.0442	−	7.53105i	−0.503940	−	0.290950i
$671$	8.41713			0.324940
$672$		0			0
$673$	−18.4702			−0.711972			−0.355986	−	0.934491i	$-0.615855\pi$
$673$	−18.4702			−0.711972			−0.355986	+	0.934491i	$0.615855\pi$
$674$	9.33296	+	5.38839i	0.359492	+	0.207553i
$675$		0			0
$676$	3.50000	+	6.06218i	0.134615	+	0.233161i
$677$	−18.9418	+	32.8082i	−0.727993	+	1.26092i	0.229737	+	0.973253i	$0.426213\pi$
$677$	−18.9418	+	32.8082i	−0.727993	+	1.26092i	−0.957730	+	0.287669i	$0.907120\pi$
$678$		0			0
$679$	3.94146	+	2.66923i	0.151259	+	0.102436i
$680$			0.540062i			0.0207104i
$681$		0			0
$682$	0.752011	−	0.434174i	0.0287960	−	0.0166254i
$683$	−3.63276	+	2.09737i	−0.139004	+	0.0802537i	−0.567889	−	0.823105i	$-0.692240\pi$
$683$	−3.63276	+	2.09737i	−0.139004	+	0.0802537i	0.428885	+	0.903359i	$0.358906\pi$
$684$		0			0
$685$			23.6637i			0.904142i
$686$	−13.6938	−	12.4691i	−0.522834	−	0.476073i
$687$		0			0
$688$	1.47203	−	2.54963i	0.0561206	−	0.0972037i
$689$	6.58630	+	11.4078i	0.250918	+	0.434603i
$690$		0			0
$691$	25.8937	+	14.9498i	0.985044	+	0.568716i	0.903789	−	0.427978i	$-0.140774\pi$
$691$	25.8937	+	14.9498i	0.985044	+	0.568716i	0.0812550	+	0.996693i	$0.474107\pi$
$692$	−1.27276			−0.0483830
$693$		0			0
$694$	−31.8177			−1.20778
$695$	12.3127	+	7.10875i	0.467048	+	0.269650i
$696$		0			0
$697$	−1.71031	−	2.96234i	−0.0647826	−	0.112207i
$698$	−10.1544	+	17.5879i	−0.384350	+	0.665713i
$699$		0			0
$700$	−3.23811	−	6.67059i	−0.122389	−	0.252124i
$701$		−	6.63368i		−	0.250551i	−0.992122	−	0.125275i	$-0.960019\pi$
$701$		−	6.63368i		−	0.250551i	0.992122	−	0.125275i	$-0.0399814\pi$
$702$		0			0
$703$	−65.6050	+	37.8771i	−2.47434	+	1.42856i
$704$	0.866025	−	0.500000i	0.0326396	−	0.0188445i
$705$		0			0
$706$		−	12.1263i		−	0.456379i
$707$	−28.0299	−	2.01246i	−1.05417	−	0.0756862i
$708$		0			0
$709$	−9.77143	+	16.9246i	−0.366974	+	0.635617i	−0.989091	−	0.147307i	$-0.952939\pi$
$709$	−9.77143	+	16.9246i	−0.366974	+	0.635617i	0.622117	+	0.782924i	$0.286273\pi$
$710$	−1.54100	−	2.66909i	−0.0578326	−	0.100169i
$711$		0			0
$712$	0.796435	+	0.459822i	0.0298477	+	0.0172326i
$713$	−5.19423			−0.194525
$714$		0			0
$715$	−3.63103			−0.135793
$716$	7.10671	+	4.10306i	0.265590	+	0.153339i
$717$		0			0
$718$	−6.74131	−	11.6763i	−0.251584	−	0.435756i
$719$	−18.5505	+	32.1304i	−0.691816	+	1.19826i	0.279427	+	0.960167i	$0.409856\pi$
$719$	−18.5505	+	32.1304i	−0.691816	+	1.19826i	−0.971242	+	0.238093i	$0.923478\pi$
$720$		0			0
$721$	−6.87452	+	3.33711i	−0.256020	+	0.124280i
$722$		−	47.5241i		−	1.76866i
$723$		0			0
$724$	3.05816	−	1.76563i	0.113655	−	0.0656190i
$725$	−0.806896	+	0.465861i	−0.0299674	+	0.0173017i
$726$		0			0
$727$		−	6.35238i		−	0.235597i	−0.993038	−	0.117798i	$-0.962416\pi$
$727$		−	6.35238i		−	0.235597i	0.993038	−	0.117798i	$-0.0375837\pi$
$728$	3.63397	−	5.36603i	0.134684	−	0.198878i
$729$		0			0
$730$	−6.73375	+	11.6632i	−0.249227	+	0.431674i
$731$	−0.536298	−	0.928895i	−0.0198357	−	0.0343564i
$732$		0			0
$733$	−11.6975	−	6.75355i	−0.432057	−	0.249448i	0.268166	−	0.963373i	$-0.413583\pi$
$733$	−11.6975	−	6.75355i	−0.432057	−	0.249448i	−0.700223	+	0.713925i	$0.746916\pi$
$734$	31.2480			1.15338
$735$		0			0
$736$	−5.98174			−0.220490
$737$	8.79958	+	5.08044i	0.324137	+	0.187140i
$738$		0			0
$739$	21.1056	+	36.5560i	0.776383	+	1.34473i	0.934014	+	0.357236i	$0.116281\pi$
$739$	21.1056	+	36.5560i	0.776383	+	1.34473i	−0.157632	+	0.987498i	$0.550386\pi$
$740$	6.88400	−	11.9234i	0.253061	−	0.438314i
$741$		0			0
$742$	−7.97815	+	11.7808i	−0.292887	+	0.432485i
$743$		−	44.2670i		−	1.62400i	−0.583657	−	0.812000i	$-0.698379\pi$
$743$		−	44.2670i		−	1.62400i	0.583657	−	0.812000i	$-0.301621\pi$
$744$		0			0
$745$	−25.8246	+	14.9098i	−0.946139	+	0.546254i
$746$	0.770138	−	0.444639i	0.0281968	−	0.0162794i
$747$		0			0
$748$		−	0.364326i		−	0.0133211i
$749$	−11.8138	+	5.73478i	−0.431666	+	0.209544i
$750$		0			0
$751$	9.76492	−	16.9133i	0.356327	−	0.617177i	−0.631017	−	0.775769i	$-0.717362\pi$
$751$	9.76492	−	16.9133i	0.356327	−	0.617177i	0.987344	+	0.158592i	$0.0506955\pi$
$752$	0.637756	+	1.10463i	0.0232566	+	0.0402816i
$753$		0			0
$754$	−0.705231	−	0.407165i	−0.0256830	−	0.0148281i
$755$	−7.54423			−0.274563
$756$		0			0
$757$	13.7687			0.500433			0.250217	−	0.968190i	$-0.419498\pi$
$757$	13.7687			0.500433			0.250217	+	0.968190i	$0.419498\pi$
$758$	−18.1549	−	10.4817i	−0.659416	−	0.380714i
$759$		0			0
$760$	6.04524	+	10.4707i	0.219284	+	0.379811i
$761$	5.03177	−	8.71528i	0.182402	−	0.315929i	−0.760296	−	0.649576i	$-0.774946\pi$
$761$	5.03177	−	8.71528i	0.182402	−	0.315929i	0.942698	+	0.333648i	$0.108280\pi$
$762$		0			0
$763$	−11.7883	−	0.846361i	−0.426764	−	0.0306403i
$764$			3.50470i			0.126796i
$765$		0			0
$766$	−1.42747	+	0.824147i	−0.0515764	+	0.0297777i
$767$	−18.8338	+	10.8737i	−0.680047	+	0.392626i
$768$		0			0
$769$		−	0.355784i		−	0.0128299i	−0.999979	−	0.00641495i	$-0.997958\pi$
$769$		−	0.355784i		−	0.0128299i	0.999979	−	0.00641495i	$-0.00204196\pi$
$770$	−1.71271	−	3.52823i	−0.0617219	−	0.127149i
$771$		0			0
$772$	−10.2639	+	17.7777i	−0.369407	+	0.639832i
$773$	−24.7902	−	42.9379i	−0.891642	−	1.54437i	−0.837906	−	0.545815i	$-0.816220\pi$
$773$	−24.7902	−	42.9379i	−0.891642	−	1.54437i	−0.0537365	−	0.998555i	$-0.517113\pi$
$774$		0			0
$775$	2.10759	+	1.21682i	0.0757068	+	0.0437093i
$776$	−1.79920			−0.0645876
$777$		0			0
$778$	9.11679			0.326853
$779$	−66.3186	−	38.2890i	−2.37611	−	1.37185i
$780$		0			0
$781$	1.03956	+	1.80056i	0.0371982	+	0.0644292i
$782$	−1.08965	+	1.88733i	−0.0389658	+	0.0674908i
$783$		0			0
$784$	6.92820	+	1.00000i	0.247436	+	0.0357143i
$785$		−	23.4545i		−	0.837126i
$786$		0			0
$787$	−25.9077	+	14.9578i	−0.923511	+	0.533189i	−0.884754	−	0.466059i	$-0.845673\pi$
$787$	−25.9077	+	14.9578i	−0.923511	+	0.533189i	−0.0387576	+	0.999249i	$0.512340\pi$
$788$	−21.8907	+	12.6386i	−0.779823	+	0.450231i
$789$		0			0
$790$		−	8.92124i		−	0.317403i
$791$	18.3102	+	12.4000i	0.651037	+	0.440894i
$792$		0			0
$793$	−10.3088	+	17.8554i	−0.366078	+	0.634065i
$794$	−3.74737	−	6.49063i	−0.132989	−	0.230344i
$795$		0			0
$796$	−18.4746	−	10.6663i	−0.654814	−	0.378057i
$797$	32.6828			1.15768			0.578841	−	0.815440i	$-0.303505\pi$
$797$	32.6828			1.15768			0.578841	+	0.815440i	$0.303505\pi$
$798$		0			0
$799$	0.464702			0.0164400
$800$	2.42713	+	1.40130i	0.0858118	+	0.0495435i
$801$		0			0
$802$	−3.14714	−	5.45101i	−0.111130	−	0.192482i
$803$	4.54258	−	7.86798i	0.160304	−	0.277655i
$804$		0			0
$805$	−1.68004	+	23.3999i	−0.0592136	+	0.824739i
$806$			2.12701i			0.0749207i
$807$		0			0
$808$	9.19856	−	5.31079i	0.323604	−	0.186833i
$809$	−26.2413	+	15.1505i	−0.922597	+	0.532661i	−0.884463	−	0.466611i	$-0.845475\pi$
$809$	−26.2413	+	15.1505i	−0.922597	+	0.532661i	−0.0381342	+	0.999273i	$0.512141\pi$
$810$		0			0
$811$		−	8.42099i		−	0.295701i	−0.989010	−	0.147851i	$-0.952765\pi$
$811$		−	8.42099i		−	0.295701i	0.989010	−	0.147851i	$-0.0472355\pi$
$812$	0.0629887	−	0.877319i	0.00221047	−	0.0307879i
$813$		0			0
$814$	−4.64394	+	8.04354i	−0.162770	+	0.281926i
$815$	13.1870	+	22.8406i	0.461921	+	0.800071i
$816$		0			0
$817$	−20.7954	−	12.0062i	−0.727538	−	0.420044i
$818$	13.5639			0.474250
$819$		0			0
$820$	13.9178			0.486029
$821$	13.9888	+	8.07642i	0.488211	+	0.281869i	0.723832	−	0.689976i	$-0.242379\pi$
$821$	13.9888	+	8.07642i	0.488211	+	0.281869i	−0.235621	+	0.971845i	$0.575712\pi$
$822$		0			0
$823$	7.82086	+	13.5461i	0.272618	+	0.472189i	0.969531	−	0.244967i	$-0.0787771\pi$
$823$	7.82086	+	13.5461i	0.272618	+	0.472189i	−0.696913	+	0.717155i	$0.745444\pi$
$824$	1.44414	−	2.50133i	0.0503091	−	0.0871378i
$825$		0			0
$826$	−19.4495	−	13.1715i	−0.676733	−	0.458297i
$827$		−	0.555670i		−	0.0193225i	−0.999953	−	0.00966127i	$-0.996925\pi$
$827$		−	0.555670i		−	0.0193225i	0.999953	−	0.00966127i	$-0.00307532\pi$
$828$		0			0
$829$	14.9532	−	8.63325i	0.519347	−	0.299845i	−0.217321	−	0.976100i	$-0.569732\pi$
$829$	14.9532	−	8.63325i	0.519347	−	0.299845i	0.736667	+	0.676255i	$0.236398\pi$
$830$	−19.9881	+	11.5401i	−0.693797	+	0.400564i
$831$		0			0
$832$			2.44949i			0.0849208i
$833$	1.57758	−	2.00379i	0.0546598	−	0.0694272i
$834$		0			0
$835$	9.45946	−	16.3843i	0.327358	−	0.567001i
$836$	−4.07812	−	7.06350i	−0.141045	−	0.244296i
$837$		0			0
$838$	17.6701	+	10.2018i	0.610404	+	0.352417i
$839$	−17.0727			−0.589417			−0.294708	−	0.955587i	$-0.595222\pi$
$839$	−17.0727			−0.589417			−0.294708	+	0.955587i	$0.595222\pi$
$840$		0			0
$841$	28.8895			0.996189
$842$	16.5128	+	9.53366i	0.569068	+	0.328552i
$843$		0			0
$844$	−13.9315	−	24.1300i	−0.479542	−	0.830590i
$845$	−5.18827	+	8.98634i	−0.178482	+	0.309140i
$846$		0			0
$847$	1.15539	+	2.38014i	0.0396998	+	0.0817826i
$848$		−	5.37769i		−	0.184671i
$849$		0			0
$850$	0.884264	−	0.510530i	0.0303300	−	0.0175110i
$851$	48.1144	−	27.7789i	1.64934	−	0.952247i
$852$		0			0
$853$			6.24047i			0.213670i	0.994277	+	0.106835i	$0.0340716\pi$
$853$			6.24047i			0.213670i	−0.994277	+	0.106835i	$0.965928\pi$
$854$	−22.2125	−	1.59478i	−0.760095	−	0.0545724i
$855$		0			0
$856$	2.48174	−	4.29850i	0.0848241	−	0.146920i
$857$	−2.02423	−	3.50607i	−0.0691464	−	0.119765i	0.829379	−	0.558686i	$-0.188694\pi$
$857$	−2.02423	−	3.50607i	−0.0691464	−	0.119765i	−0.898526	+	0.438921i	$0.855361\pi$
$858$		0			0
$859$	7.41776	+	4.28264i	0.253091	+	0.146122i	0.621179	−	0.783669i	$-0.286654\pi$
$859$	7.41776	+	4.28264i	0.253091	+	0.146122i	−0.368088	+	0.929791i	$0.619987\pi$
$860$	4.36416			0.148817
$861$		0			0
$862$	27.9260			0.951163
$863$	13.8196	+	7.97878i	0.470426	+	0.271601i	0.716418	−	0.697671i	$-0.245780\pi$
$863$	13.8196	+	7.97878i	0.470426	+	0.271601i	−0.245992	+	0.969272i	$0.579114\pi$
$864$		0			0
$865$	−0.943343	−	1.63392i	−0.0320746	−	0.0555549i
$866$	12.4072	−	21.4900i	0.421615	−	0.730259i
$867$		0			0
$868$	−2.06679	+	1.00328i	−0.0701514	+	0.0340537i
$869$			6.01826i			0.204155i
$870$		0			0
$871$	−21.5545	+	12.4445i	−0.730345	+	0.421665i
$872$	3.86855	−	2.23351i	0.131006	−	0.0756362i
$873$		0			0
$874$			48.7885i			1.65029i
$875$	17.1593	−	25.3379i	0.580091	−	0.856578i
$876$		0			0
$877$	19.2806	−	33.3950i	0.651061	−	1.12767i	−0.331805	−	0.943348i	$-0.607658\pi$
$877$	19.2806	−	33.3950i	0.651061	−	1.12767i	0.982866	−	0.184322i	$-0.0590090\pi$
$878$	7.32420	+	12.6859i	0.247180	+	0.428128i
$879$		0			0
$880$	1.28376	+	0.741181i	0.0432756	+	0.0249852i
$881$	29.5113			0.994260			0.497130	−	0.867676i	$-0.334387\pi$
$881$	29.5113			0.994260			0.497130	+	0.867676i	$0.334387\pi$
$882$		0			0
$883$	58.8660			1.98100			0.990499	−	0.137521i	$-0.0439134\pi$
$883$	58.8660			1.98100			0.990499	+	0.137521i	$0.0439134\pi$
$884$	0.772851	+	0.446206i	0.0259938	+	0.0150075i
$885$		0			0
$886$	−16.0026	−	27.7172i	−0.537616	−	0.931179i
$887$	−21.9177	+	37.9626i	−0.735925	+	1.27466i	0.218391	+	0.975861i	$0.429919\pi$
$887$	−21.9177	+	37.9626i	−0.735925	+	1.27466i	−0.954316	+	0.298799i	$0.903414\pi$
$888$		0			0
$889$	26.9829	−	39.8436i	0.904976	−	1.33631i
$890$			1.36324i			0.0456961i
$891$		0			0
$892$	5.17379	−	2.98709i	0.173231	−	0.100015i
$893$	9.00959	−	5.20169i	0.301494	−	0.174068i
$894$		0			0
$895$			12.1644i			0.406612i
$896$	−2.38014	+	1.15539i	−0.0795149	+	0.0385990i
$897$		0			0
$898$	−19.4469	+	33.6830i	−0.648951	+	1.12402i
$899$	0.144341	+	0.250005i	0.00481403	+	0.00833815i
$900$		0			0
$901$	−1.69674	−	0.979615i	−0.0565267	−	0.0326357i
$902$	−9.38891			−0.312616
$903$		0			0
$904$	−8.35827			−0.277992
$905$	4.53329	+	2.61730i	0.150692	+	0.0870019i
$906$		0			0
$907$	8.01005	+	13.8738i	0.265969	+	0.460672i	0.967817	−	0.251655i	$-0.0809747\pi$
$907$	8.01005	+	13.8738i	0.265969	+	0.460672i	−0.701848	+	0.712327i	$0.747641\pi$
$908$	−1.77721	+	3.07822i	−0.0589789	+	0.102154i
$909$		0			0
$910$	9.58214	+	0.687967i	0.317645	+	0.0228059i
$911$		−	40.5903i		−	1.34482i	−0.740181	−	0.672408i	$-0.765260\pi$
$911$		−	40.5903i		−	1.34482i	0.740181	−	0.672408i	$-0.234740\pi$
$912$		0			0
$913$	13.4840	−	7.78497i	0.446254	−	0.257645i
$914$	20.4796	−	11.8239i	0.677405	−	0.391100i
$915$		0			0
$916$			10.0233i			0.331180i
$917$	−21.3578	−	43.9975i	−0.705296	−	1.45293i
$918$		0			0
$919$	0.338544	−	0.586375i	0.0111675	−	0.0193427i	−0.860388	−	0.509640i	$-0.829779\pi$
$919$	0.338544	−	0.586375i	0.0111675	−	0.0193427i	0.871555	+	0.490297i	$0.163112\pi$
$920$	−4.43355	−	7.67914i	−0.146170	−	0.253174i
$921$		0			0
$922$	−6.90379	−	3.98590i	−0.227364	−	0.131269i
$923$	−5.09276			−0.167630
$924$		0			0
$925$	−26.0303			−0.855870
$926$	29.1944	+	16.8554i	0.959388	+	0.553903i
$927$		0			0
$928$	0.166225	+	0.287909i	0.00545659	+	0.00945109i
$929$	1.13578	−	1.96723i	0.0372638	−	0.0645427i	−0.846792	−	0.531924i	$-0.821469\pi$
$929$	1.13578	−	1.96723i	0.0372638	−	0.0645427i	0.884056	+	0.467381i	$0.154802\pi$
$930$		0			0
$931$	8.15623	−	56.5080i	0.267310	−	1.85198i
$932$			17.4853i			0.572749i
$933$		0			0
$934$	−16.1741	+	9.33814i	−0.529234	+	0.305553i
$935$	0.467708	−	0.270031i	0.0152957	−	0.00883096i
$936$		0			0
$937$		−	10.2711i		−	0.335543i	−0.985826	−	0.167772i	$-0.946343\pi$
$937$		−	10.2711i		−	0.335543i	0.985826	−	0.167772i	$-0.0536571\pi$
$938$	−22.2591	−	15.0743i	−0.726786	−	0.492194i
$939$		0			0
$940$	−0.945386	+	1.63746i	−0.0308351	+	0.0534079i
$941$	26.8997	+	46.5916i	0.876904	+	1.51884i	0.854721	+	0.519088i	$0.173728\pi$
$941$	26.8997	+	46.5916i	0.876904	+	1.51884i	0.0221830	+	0.999754i	$0.492938\pi$
$942$		0			0
$943$	48.6377	+	28.0810i	1.58386	+	0.914443i
$944$	8.87832			0.288965
$945$		0			0
$946$	−2.94406			−0.0957196
$947$	−19.6731	−	11.3583i	−0.639290	−	0.369094i	0.145051	−	0.989424i	$-0.453665\pi$
$947$	−19.6731	−	11.3583i	−0.639290	−	0.369094i	−0.784341	+	0.620330i	$0.786999\pi$
$948$		0			0
$949$	11.1270	+	19.2725i	0.361198	+	0.625613i
$950$	11.4293	−	19.7962i	0.370817	−	0.642274i
$951$		0			0
$952$	−0.0690283	+	0.961440i	−0.00223722	+	0.0311604i
$953$			12.6569i			0.409996i	0.978762	+	0.204998i	$0.0657187\pi$
$953$			12.6569i			0.409996i	−0.978762	+	0.204998i	$0.934281\pi$
$954$		0			0
$955$	−4.49921	+	2.59762i	−0.145591	+	0.0840569i
$956$	−12.1020	+	6.98709i	−0.391406	+	0.225979i
$957$		0			0
$958$		−	16.1373i		−	0.521373i
$959$	−3.02458	+	42.1270i	−0.0976688	+	1.36035i
$960$		0			0
$961$	−15.1230	+	26.1938i	−0.487838	+	0.844961i
$962$	−11.3753	−	19.7026i	−0.366754	−	0.635237i
$963$		0			0
$964$	−0.247649	−	0.142980i	−0.00797625	−	0.00460509i
$965$	−30.4297			−0.979568
$966$		0			0
$967$	−17.4083			−0.559813			−0.279906	−	0.960027i	$-0.590303\pi$
$967$	−17.4083			−0.559813			−0.279906	+	0.960027i	$0.590303\pi$
$968$	−0.866025	−	0.500000i	−0.0278351	−	0.0160706i
$969$		0			0
$970$	−1.33354	−	2.30975i	−0.0428172	−	0.0741616i
$971$	−24.8581	+	43.0555i	−0.797734	+	1.38172i	0.123355	+	0.992363i	$0.460635\pi$
$971$	−24.8581	+	43.0555i	−0.797734	+	1.38172i	−0.921089	+	0.389353i	$0.872699\pi$
$972$		0			0
$973$	21.0110	+	14.2290i	0.673581	+	0.456162i
$974$			25.5580i			0.818931i
$975$		0			0
$976$	7.28945	−	4.20857i	0.233330	−	0.134713i
$977$	−27.9697	+	16.1483i	−0.894829	+	0.516630i	−0.875519	−	0.483184i	$-0.839480\pi$
$977$	−27.9697	+	16.1483i	−0.894829	+	0.516630i	−0.0193103	+	0.999814i	$0.506147\pi$
$978$		0			0
$979$		−	0.919644i		−	0.0293919i
$980$	3.85129	+	9.63535i	0.123025	+	0.307790i
$981$		0			0
$982$	−10.7954	+	18.6983i	−0.344497	+	0.596686i
$983$	11.3865	+	19.7219i	0.363172	+	0.629032i	0.988481	−	0.151345i	$-0.0483605\pi$
$983$	11.3865	+	19.7219i	0.363172	+	0.629032i	−0.625309	+	0.780377i	$0.715027\pi$
$984$		0			0
$985$	−32.4499	−	18.7350i	−1.03394	−	0.596946i
$986$	0.121120			0.00385724
$987$		0			0
$988$	19.9786			0.635604
$989$	15.2512	+	8.80530i	0.484961	+	0.279992i
$990$		0			0
$991$	22.3719	+	38.7492i	0.710666	+	1.23091i	0.964608	+	0.263690i	$0.0849395\pi$
$991$	22.3719	+	38.7492i	0.710666	+	1.23091i	−0.253942	+	0.967219i	$0.581727\pi$
$992$	0.434174	−	0.752011i	0.0137850	−	0.0238764i
$993$		0			0
$994$	−2.40219	−	4.94858i	−0.0761930	−	0.156959i
$995$		−	31.6227i		−	1.00251i
$996$		0			0
$997$	26.5845	−	15.3486i	0.841940	−	0.486095i	−0.0159829	−	0.999872i	$-0.505088\pi$
$997$	26.5845	−	15.3486i	0.841940	−	0.486095i	0.857923	+	0.513778i	$0.171754\pi$
$998$	−0.814331	+	0.470154i	−0.0257772	+	0.0148825i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1386.2.r.a.89.2	✓	8
3.2	odd	2		1386.2.r.c.89.3	yes	8
7.3	odd	6		1386.2.r.c.1277.3	yes	8
21.17	even	6	inner	1386.2.r.a.1277.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1386.2.r.a.89.2	✓	8	1.1	even	1	trivial
1386.2.r.a.1277.2	yes	8	21.17	even	6	inner
1386.2.r.c.89.3	yes	8	3.2	odd	2
1386.2.r.c.1277.3	yes	8	7.3	odd	6

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:	\( N \)	\(=\)	\( 1386 = 2 \cdot 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1386.r (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.741181 + 1.28376i) q^{5} +(-1.48356 + 2.19067i) q^{7} -1.00000i q^{8} +O(q^{10})\)	\(q+(-0.866025 - 0.500000i) q^{2} +(0.500000 + 0.866025i) q^{4} +(-0.741181 + 1.28376i) q^{5} +(-1.48356 + 2.19067i) q^{7} -1.00000i q^{8} +(1.28376 - 0.741181i) q^{10} +(-0.866025 + 0.500000i) q^{11} -2.44949i q^{13} +(2.38014 - 1.15539i) q^{14} +(-0.500000 + 0.866025i) q^{16} +(0.182163 + 0.315515i) q^{17} +(7.06350 + 4.07812i) q^{19} -1.48236 q^{20} +1.00000 q^{22} +(-5.18034 - 2.99087i) q^{23} +(1.40130 + 2.42713i) q^{25} +(-1.22474 + 2.12132i) q^{26} +(-2.63896 - 0.189469i) q^{28} +0.332449i q^{29} +(0.752011 - 0.434174i) q^{31} +(0.866025 - 0.500000i) q^{32} -0.364326i q^{34} +(-1.71271 - 3.52823i) q^{35} +(-4.64394 + 8.04354i) q^{37} +(-4.07812 - 7.06350i) q^{38} +(1.28376 + 0.741181i) q^{40} -9.38891 q^{41} -2.94406 q^{43} +(-0.866025 - 0.500000i) q^{44} +(2.99087 + 5.18034i) q^{46} +(0.637756 - 1.10463i) q^{47} +(-2.59808 - 6.50000i) q^{49} -2.80260i q^{50} +(2.12132 - 1.22474i) q^{52} +(-4.65722 + 2.68885i) q^{53} -1.48236i q^{55} +(2.19067 + 1.48356i) q^{56} +(0.166225 - 0.287909i) q^{58} +(-4.43916 - 7.68885i) q^{59} +(-7.28945 - 4.20857i) q^{61} -0.868348 q^{62} -1.00000 q^{64} +(3.14456 + 1.81552i) q^{65} +(-5.08044 - 8.79958i) q^{67} +(-0.182163 + 0.315515i) q^{68} +(-0.280861 + 3.91189i) q^{70} -2.07911i q^{71} +(-7.86798 + 4.54258i) q^{73} +(8.04354 - 4.64394i) q^{74} +8.15623i q^{76} +(0.189469 - 2.63896i) q^{77} +(3.00913 - 5.21197i) q^{79} +(-0.741181 - 1.28376i) q^{80} +(8.13103 + 4.69445i) q^{82} -15.5699 q^{83} -0.540062 q^{85} +(2.54963 + 1.47203i) q^{86} +(0.500000 + 0.866025i) q^{88} +(-0.459822 + 0.796435i) q^{89} +(5.36603 + 3.63397i) q^{91} -5.98174i q^{92} +(-1.10463 + 0.637756i) q^{94} +(-10.4707 + 6.04524i) q^{95} -1.79920i q^{97} +(-1.00000 + 6.92820i) q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{4} - 8 q^{5}+O(q^{10}) \) 8 * q + 4 * q^4 - 8 * q^5	\( 8 q + 4 q^{4} - 8 q^{5} - 4 q^{16} + 4 q^{17} + 24 q^{19} - 16 q^{20} + 8 q^{22} - 24 q^{23} - 4 q^{25} + 12 q^{31} - 16 q^{35} + 12 q^{37} + 8 q^{38} - 16 q^{41} - 32 q^{43} + 8 q^{46} - 48 q^{53} - 4 q^{58} - 16 q^{59} - 24 q^{62} - 8 q^{64} + 12 q^{65} - 24 q^{67} - 4 q^{68} - 20 q^{70} - 24 q^{73} + 12 q^{74} + 40 q^{79} - 8 q^{80} + 12 q^{82} - 72 q^{83} - 32 q^{85} + 24 q^{86} + 4 q^{88} + 16 q^{89} + 36 q^{91} - 24 q^{95} - 8 q^{98}+O(q^{100}) \) 8 * q + 4 * q^4 - 8 * q^5 - 4 * q^16 + 4 * q^17 + 24 * q^19 - 16 * q^20 + 8 * q^22 - 24 * q^23 - 4 * q^25 + 12 * q^31 - 16 * q^35 + 12 * q^37 + 8 * q^38 - 16 * q^41 - 32 * q^43 + 8 * q^46 - 48 * q^53 - 4 * q^58 - 16 * q^59 - 24 * q^62 - 8 * q^64 + 12 * q^65 - 24 * q^67 - 4 * q^68 - 20 * q^70 - 24 * q^73 + 12 * q^74 + 40 * q^79 - 8 * q^80 + 12 * q^82 - 72 * q^83 - 32 * q^85 + 24 * q^86 + 4 * q^88 + 16 * q^89 + 36 * q^91 - 24 * q^95 - 8 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more