LMFDB - Embedded newform 168.5.z.b.145.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [168,5,Mod(73,168)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(168, base_ring=CyclotomicField(6))

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

N = Newforms(chi, 5, names="a")

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

chi := DirichletCharacter("168.73");

S:= CuspForms(chi, 5);

N := Newforms(S);

Level:	$ N $	$=$	$ 168 = 2^{3} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 5 $
Character orbit:	$[\chi]$	$=$	168.z (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:	$17.3661537981$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} + 130 x^{14} + 6137 x^{12} + 133906 x^{10} + 1360384 x^{8} + 5425142 x^{6} + 5784425 x^{4} + \cdots + 117649 $ x^16 + 130x^14 + 6137x^12 + 133906x^10 + 1360384x^8 + 5425142x^6 + 5784425x^4 + 1717862*x^2 + 117649
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{36}\cdot 7^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.5
Root			$2.45155i$ of defining polynomial
Character	$\chi$	$=$	168.145
Dual form			168.5.z.b.73.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 - 2.59808i) q^{3} +(-1.04890 - 0.605581i) q^{5} +(43.9897 - 21.5849i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})$	\(q+(4.50000 - 2.59808i) q^{3} +(-1.04890 - 0.605581i) q^{5} +(43.9897 - 21.5849i) q^{7} +(13.5000 - 23.3827i) q^{9} +(-68.8822 - 119.308i) q^{11} +107.205i q^{13} -6.29338 q^{15} +(201.539 - 116.358i) q^{17} +(41.8553 + 24.1652i) q^{19} +(141.874 - 211.421i) q^{21} +(282.239 - 488.852i) q^{23} +(-311.767 - 539.995i) q^{25} -140.296i q^{27} -54.2202 q^{29} +(539.671 - 311.579i) q^{31} +(-619.940 - 357.923i) q^{33} +(-59.2120 - 3.99895i) q^{35} +(43.7715 - 75.8144i) q^{37} +(278.526 + 482.422i) q^{39} -1344.74i q^{41} +528.989 q^{43} +(-28.3202 + 16.3507i) q^{45} +(-88.0334 - 50.8261i) q^{47} +(1469.18 - 1899.03i) q^{49} +(604.616 - 1047.23i) q^{51} +(-1734.34 - 3003.96i) q^{53} +166.855i q^{55} +251.132 q^{57} +(1718.15 - 991.973i) q^{59} +(115.890 + 66.9093i) q^{61} +(89.1471 - 1319.99i) q^{63} +(64.9212 - 112.447i) q^{65} +(4032.57 + 6984.61i) q^{67} -2933.11i q^{69} +6569.08 q^{71} +(-4452.95 + 2570.91i) q^{73} +(-2805.90 - 1619.99i) q^{75} +(-5605.35 - 3761.48i) q^{77} +(-5470.09 + 9474.48i) q^{79} +(-364.500 - 631.333i) q^{81} +10621.5i q^{83} -281.858 q^{85} +(-243.991 + 140.868i) q^{87} +(-6947.10 - 4010.91i) q^{89} +(2314.01 + 4715.91i) q^{91} +(1619.01 - 2804.21i) q^{93} +(-29.2679 - 50.6935i) q^{95} +17996.3i q^{97} -3719.64 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9}+O(q^{10}) $ 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9	$ 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9} + 252 q^{11} - 72 q^{15} - 696 q^{17} + 156 q^{19} + 108 q^{21} - 672 q^{23} + 84 q^{25} + 1992 q^{29} + 2040 q^{31} + 2268 q^{33} + 2712 q^{35} + 2548 q^{37} - 396 q^{39} + 1304 q^{43} - 324 q^{45} - 744 q^{47} - 5608 q^{49} - 2088 q^{51} - 1164 q^{53} + 936 q^{57} + 8988 q^{59} + 816 q^{61} + 540 q^{63} + 8760 q^{65} + 3044 q^{67} - 4464 q^{71} - 15828 q^{73} + 756 q^{75} + 996 q^{77} - 11144 q^{79} - 5832 q^{81} - 15344 q^{85} + 8964 q^{87} + 22248 q^{89} + 1596 q^{91} + 6120 q^{93} + 3840 q^{95} + 13608 q^{99}+O(q^{100}) $ 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9 + 252 * q^11 - 72 * q^15 - 696 * q^17 + 156 * q^19 + 108 * q^21 - 672 * q^23 + 84 * q^25 + 1992 * q^29 + 2040 * q^31 + 2268 * q^33 + 2712 * q^35 + 2548 * q^37 - 396 * q^39 + 1304 * q^43 - 324 * q^45 - 744 * q^47 - 5608 * q^49 - 2088 * q^51 - 1164 * q^53 + 936 * q^57 + 8988 * q^59 + 816 * q^61 + 540 * q^63 + 8760 * q^65 + 3044 * q^67 - 4464 * q^71 - 15828 * q^73 + 756 * q^75 + 996 * q^77 - 11144 * q^79 - 5832 * q^81 - 15344 * q^85 + 8964 * q^87 + 22248 * q^89 + 1596 * q^91 + 6120 * q^93 + 3840 * q^95 + 13608 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/168\mathbb{Z}\right)^\times$.

$n$	$73$	$85$	$113$	$127$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$1$	$1$	$1$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	4.50000	−	2.59808i	0.500000	−	0.288675i
$3$	4.50000	−	2.59808i	0.500000	−	0.288675i
$4$		0			0
$5$	−1.04890	−	0.605581i	−0.0419559	−	0.0242232i	0.478875	−	0.877883i	$-0.341045\pi$
$5$	−1.04890	−	0.605581i	−0.0419559	−	0.0242232i	−0.520831	+	0.853660i	$0.674378\pi$
$6$		0			0
$7$	43.9897	−	21.5849i	0.897748	−	0.440509i
$7$	43.9897	−	21.5849i	0.897748	−	0.440509i
$8$		0			0
$9$	13.5000	−	23.3827i	0.166667	−	0.288675i
$10$		0			0
$11$	−68.8822	−	119.308i	−0.569275	−	0.986013i	−0.996638	−	0.0819327i	$-0.973891\pi$
$11$	−68.8822	−	119.308i	−0.569275	−	0.986013i	0.427363	−	0.904080i	$-0.359443\pi$
$12$		0			0
$13$			107.205i			0.634348i	0.948367	+	0.317174i	$0.102734\pi$
$13$			107.205i			0.634348i	−0.948367	+	0.317174i	$0.897266\pi$
$14$		0			0
$15$	−6.29338			−0.0279706
$16$		0			0
$17$	201.539	−	116.358i	0.697366	−	0.402624i	−0.109000	−	0.994042i	$-0.534765\pi$
$17$	201.539	−	116.358i	0.697366	−	0.402624i	0.806366	+	0.591417i	$0.201431\pi$
$18$		0			0
$19$	41.8553	+	24.1652i	0.115943	+	0.0669395i	0.556850	−	0.830613i	$-0.312010\pi$
$19$	41.8553	+	24.1652i	0.115943	+	0.0669395i	−0.440907	+	0.897553i	$0.645343\pi$
$20$		0			0
$21$	141.874	−	211.421i	0.321710	−	0.479412i
$22$		0			0
$23$	282.239	−	488.852i	0.533533	−	0.924106i	−0.465700	−	0.884943i	$-0.654197\pi$
$23$	282.239	−	488.852i	0.533533	−	0.924106i	0.999233	−	0.0391633i	$-0.0124692\pi$
$24$		0			0
$25$	−311.767	−	539.995i	−0.498826	−	0.863993i
$26$		0			0
$27$		−	140.296i		−	0.192450i
$28$		0			0
$29$	−54.2202			−0.0644711			−0.0322355	−	0.999480i	$-0.510263\pi$
$29$	−54.2202			−0.0644711			−0.0322355	+	0.999480i	$0.510263\pi$
$30$		0			0
$31$	539.671	−	311.579i	0.561572	−	0.324224i	−0.192204	−	0.981355i	$-0.561564\pi$
$31$	539.671	−	311.579i	0.561572	−	0.324224i	0.753776	+	0.657131i	$0.228230\pi$
$32$		0			0
$33$	−619.940	−	357.923i	−0.569275	−	0.328671i
$34$		0			0
$35$	−59.2120	−	3.99895i	−0.0483364	−	0.00326445i
$36$		0			0
$37$	43.7715	−	75.8144i	0.0319733	−	0.0553794i	−0.849596	−	0.527434i	$-0.823154\pi$
$37$	43.7715	−	75.8144i	0.0319733	−	0.0553794i	0.881569	+	0.472055i	$0.156488\pi$
$38$		0			0
$39$	278.526	+	482.422i	0.183121	+	0.317174i
$40$		0			0
$41$		−	1344.74i		−	0.799963i	−0.916523	−	0.399982i	$-0.869017\pi$
$41$		−	1344.74i		−	0.799963i	0.916523	−	0.399982i	$-0.130983\pi$
$42$		0			0
$43$	528.989			0.286094			0.143047	−	0.989716i	$-0.454310\pi$
$43$	528.989			0.286094			0.143047	+	0.989716i	$0.454310\pi$
$44$		0			0
$45$	−28.3202	+	16.3507i	−0.0139853	+	0.00807441i
$46$		0			0
$47$	−88.0334	−	50.8261i	−0.0398522	−	0.0230087i	0.479942	−	0.877300i	$-0.340658\pi$
$47$	−88.0334	−	50.8261i	−0.0398522	−	0.0230087i	−0.519794	+	0.854292i	$0.673991\pi$
$48$		0			0
$49$	1469.18	−	1899.03i	0.611904	−	0.790932i
$50$		0			0
$51$	604.616	−	1047.23i	0.232455	−	0.402624i
$52$		0			0
$53$	−1734.34	−	3003.96i	−0.617421	−	1.06941i	−0.989955	−	0.141386i	$-0.954844\pi$
$53$	−1734.34	−	3003.96i	−0.617421	−	1.06941i	0.372533	−	0.928019i	$-0.378489\pi$
$54$		0			0
$55$			166.855i			0.0551587i
$56$		0			0
$57$	251.132			0.0772951
$58$		0			0
$59$	1718.15	−	991.973i	0.493579	−	0.284968i	−0.232479	−	0.972601i	$-0.574684\pi$
$59$	1718.15	−	991.973i	0.493579	−	0.284968i	0.726058	+	0.687634i	$0.241350\pi$
$60$		0			0
$61$	115.890	+	66.9093i	0.0311450	+	0.0179815i	0.515492	−	0.856895i	$-0.327609\pi$
$61$	115.890	+	66.9093i	0.0311450	+	0.0179815i	−0.484347	+	0.874876i	$0.660943\pi$
$62$		0			0
$63$	89.1471	−	1319.99i	0.0224608	−	0.332576i
$64$		0			0
$65$	64.9212	−	112.447i	0.0153660	−	0.0266146i
$66$		0			0
$67$	4032.57	+	6984.61i	0.898322	+	1.55594i	0.829639	+	0.558301i	$0.188546\pi$
$67$	4032.57	+	6984.61i	0.898322	+	1.55594i	0.0686833	+	0.997639i	$0.478120\pi$
$68$		0			0
$69$		−	2933.11i		−	0.616071i
$70$		0			0
$71$	6569.08			1.30313			0.651565	−	0.758593i	$-0.274113\pi$
$71$	6569.08			1.30313			0.651565	+	0.758593i	$0.274113\pi$
$72$		0			0
$73$	−4452.95	+	2570.91i	−0.835607	+	0.482438i	−0.855769	−	0.517359i	$-0.826915\pi$
$73$	−4452.95	+	2570.91i	−0.835607	+	0.482438i	0.0201616	+	0.999797i	$0.493582\pi$
$74$		0			0
$75$	−2805.90	−	1619.99i	−0.498826	−	0.287998i
$76$		0			0
$77$	−5605.35	−	3761.48i	−0.945413	−	0.634421i
$78$		0			0
$79$	−5470.09	+	9474.48i	−0.876477	+	1.51810i	−0.0212956	+	0.999773i	$0.506779\pi$
$79$	−5470.09	+	9474.48i	−0.876477	+	1.51810i	−0.855181	+	0.518329i	$0.826554\pi$
$80$		0			0
$81$	−364.500	−	631.333i	−0.0555556	−	0.0962250i
$82$		0			0
$83$			10621.5i			1.54181i	0.636951	+	0.770904i	$0.280195\pi$
$83$			10621.5i			1.54181i	−0.636951	+	0.770904i	$0.719805\pi$
$84$		0			0
$85$	−281.858			−0.0390115
$86$		0			0
$87$	−243.991	+	140.868i	−0.0322355	+	0.0186112i
$88$		0			0
$89$	−6947.10	−	4010.91i	−0.877049	−	0.506364i	−0.00736451	−	0.999973i	$-0.502344\pi$
$89$	−6947.10	−	4010.91i	−0.877049	−	0.506364i	−0.869684	+	0.493609i	$0.835678\pi$
$90$		0			0
$91$	2314.01	+	4715.91i	0.279436	+	0.569485i
$92$		0			0
$93$	1619.01	−	2804.21i	0.187191	−	0.324224i
$94$		0			0
$95$	−29.2679	−	50.6935i	−0.00324298	−	0.00561701i
$96$		0			0
$97$			17996.3i			1.91266i	0.292281	+	0.956332i	$0.405586\pi$
$97$			17996.3i			1.91266i	−0.292281	+	0.956332i	$0.594414\pi$
$98$		0			0
$99$	−3719.64			−0.379517
$100$		0			0
$101$	4129.45	−	2384.14i	0.404808	−	0.233716i	−0.283748	−	0.958899i	$-0.591578\pi$
$101$	4129.45	−	2384.14i	0.404808	−	0.233716i	0.688557	+	0.725183i	$0.258245\pi$
$102$		0			0
$103$	3590.60	+	2073.04i	0.338449	+	0.195403i	0.659586	−	0.751629i	$-0.270732\pi$
$103$	3590.60	+	2073.04i	0.338449	+	0.195403i	−0.321137	+	0.947033i	$0.604065\pi$
$104$		0			0
$105$	−276.844	+	135.842i	−0.0251105	+	0.0123213i
$106$		0			0
$107$	−3637.79	+	6300.84i	−0.317739	+	0.550340i	−0.980016	−	0.198919i	$-0.936257\pi$
$107$	−3637.79	+	6300.84i	−0.317739	+	0.550340i	0.662277	+	0.749259i	$0.269590\pi$
$108$		0			0
$109$	6238.23	+	10804.9i	0.525060	+	0.909430i	0.999574	+	0.0291822i	$0.00929031\pi$
$109$	6238.23	+	10804.9i	0.525060	+	0.909430i	−0.474515	+	0.880248i	$0.657376\pi$
$110$		0			0
$111$		−	454.886i		−	0.0369196i
$112$		0			0
$113$	−19423.8			−1.52117			−0.760586	−	0.649237i	$-0.775088\pi$
$113$	−19423.8			−1.52117			−0.760586	+	0.649237i	$0.775088\pi$
$114$		0			0
$115$	−592.079	+	341.837i	−0.0447697	+	0.0258478i
$116$		0			0
$117$	2506.74	+	1447.27i	0.183121	+	0.105725i
$118$		0			0
$119$	6354.04	−	9468.77i	0.448700	−	0.668651i
$120$		0			0
$121$	−2169.03	+	3756.87i	−0.148148	+	0.256599i
$122$		0			0
$123$	−3493.73	−	6051.32i	−0.230929	−	0.399982i
$124$		0			0
$125$			1512.18i			0.0967792i
$126$		0			0
$127$	−4705.97			−0.291771			−0.145885	−	0.989302i	$-0.546603\pi$
$127$	−4705.97			−0.291771			−0.145885	+	0.989302i	$0.546603\pi$
$128$		0			0
$129$	2380.45	−	1374.35i	0.143047	−	0.0825883i
$130$		0			0
$131$	226.205	+	130.600i	0.0131814	+	0.00761026i	0.506576	−	0.862195i	$-0.330911\pi$
$131$	226.205	+	130.600i	0.0131814	+	0.00761026i	−0.493395	+	0.869805i	$0.664244\pi$
$132$		0			0
$133$	2362.80	+	159.574i	0.133575	+	0.00902111i
$134$		0			0
$135$	−84.9606	+	147.156i	−0.00466176	+	0.00807441i
$136$		0			0
$137$	5723.17	+	9912.82i	0.304927	+	0.528149i	0.977245	−	0.212114i	$-0.0680347\pi$
$137$	5723.17	+	9912.82i	0.304927	+	0.528149i	−0.672318	+	0.740262i	$0.734701\pi$
$138$		0			0
$139$			5916.90i			0.306242i	0.988207	+	0.153121i	$0.0489324\pi$
$139$			5916.90i			0.306242i	−0.988207	+	0.153121i	$0.951068\pi$
$140$		0			0
$141$	−528.200			−0.0265681
$142$		0			0
$143$	12790.4	−	7384.51i	0.625476	−	0.361119i
$144$		0			0
$145$	56.8714	+	32.8347i	0.00270494	+	0.00156170i
$146$		0			0
$147$	1677.50	−	12362.7i	0.0776298	−	0.572107i
$148$		0			0
$149$	3932.47	−	6811.24i	0.177130	−	0.306799i	−0.763766	−	0.645493i	$-0.776652\pi$
$149$	3932.47	−	6811.24i	0.177130	−	0.306799i	0.940896	+	0.338694i	$0.109985\pi$
$150$		0			0
$151$	5035.50	+	8721.74i	0.220845	+	0.382516i	0.955065	−	0.296397i	$-0.0957850\pi$
$151$	5035.50	+	8721.74i	0.220845	+	0.382516i	−0.734219	+	0.678912i	$0.762452\pi$
$152$		0			0
$153$		−	6283.36i		−	0.268416i
$154$		0			0
$155$	−754.745			−0.0314150
$156$		0			0
$157$	23270.5	−	13435.2i	0.944074	−	0.545061i	0.0528389	−	0.998603i	$-0.483173\pi$
$157$	23270.5	−	13435.2i	0.944074	−	0.545061i	0.891235	+	0.453542i	$0.149840\pi$
$158$		0			0
$159$	−15609.0	−	9011.88i	−0.617421	−	0.356468i
$160$		0			0
$161$	1863.76	−	27596.5i	0.0719016	−	1.06464i
$162$		0			0
$163$	−7649.12	+	13248.7i	−0.287896	+	0.498651i	−0.973307	−	0.229505i	$-0.926289\pi$
$163$	−7649.12	+	13248.7i	−0.287896	+	0.498651i	0.685411	+	0.728156i	$0.259623\pi$
$164$		0			0
$165$	433.502	+	750.848i	0.0159229	+	0.0275794i
$166$		0			0
$167$		−	54748.6i		−	1.96309i	−0.191237	−	0.981544i	$-0.561250\pi$
$167$		−	54748.6i		−	1.96309i	0.191237	−	0.981544i	$-0.438750\pi$
$168$		0			0
$169$	17068.1			0.597602
$170$		0			0
$171$	1130.09	−	652.459i	0.0386475	−	0.0223132i
$172$		0			0
$173$	−3790.01	−	2188.16i	−0.126633	−	0.0731117i	0.435345	−	0.900264i	$-0.356626\pi$
$173$	−3790.01	−	2188.16i	−0.126633	−	0.0731117i	−0.561978	+	0.827152i	$0.689960\pi$
$174$		0			0
$175$	−25370.3	−	17024.8i	−0.828417	−	0.555911i
$176$		0			0
$177$	5154.44	−	8927.76i	0.164526	−	0.284968i
$178$		0			0
$179$	14857.1	+	25733.3i	0.463690	+	0.803135i	0.999141	−	0.0414309i	$-0.0131917\pi$
$179$	14857.1	+	25733.3i	0.463690	+	0.803135i	−0.535451	+	0.844566i	$0.679858\pi$
$180$		0			0
$181$			20349.6i			0.621154i	0.950548	+	0.310577i	$0.100522\pi$
$181$			20349.6i			0.621154i	−0.950548	+	0.310577i	$0.899478\pi$
$182$		0			0
$183$	695.342			0.0207633
$184$		0			0
$185$	−91.8235	+	53.0143i	−0.00268294	+	0.00154899i
$186$		0			0
$187$	−27764.9	−	16030.1i	−0.793986	−	0.458408i
$188$		0			0
$189$	−3028.28	−	6171.58i	−0.0847759	−	0.172772i
$190$		0			0
$191$	3702.38	−	6412.71i	0.101488	−	0.175782i	−0.810810	−	0.585310i	$-0.800973\pi$
$191$	3702.38	−	6412.71i	0.101488	−	0.175782i	0.912298	+	0.409527i	$0.134306\pi$
$192$		0			0
$193$	13806.7	+	23913.9i	0.370659	+	0.642000i	0.989667	−	0.143384i	$-0.0457985\pi$
$193$	13806.7	+	23913.9i	0.370659	+	0.642000i	−0.619008	+	0.785385i	$0.712465\pi$
$194$		0			0
$195$		−	674.681i		−	0.0177431i
$196$		0			0
$197$	−47920.1			−1.23477			−0.617384	−	0.786662i	$-0.711808\pi$
$197$	−47920.1			−1.23477			−0.617384	+	0.786662i	$0.711808\pi$
$198$		0			0
$199$	56358.7	−	32538.7i	1.42316	−	0.821665i	0.426597	−	0.904442i	$-0.359712\pi$
$199$	56358.7	−	32538.7i	1.42316	−	0.821665i	0.996568	+	0.0827774i	$0.0263790\pi$
$200$		0			0
$201$	36293.1	+	20953.8i	0.898322	+	0.518646i
$202$		0			0
$203$	−2385.13	+	1170.34i	−0.0578788	+	0.0284001i
$204$		0			0
$205$	−814.348	+	1410.49i	−0.0193777	+	0.0335632i
$206$		0			0
$207$	−7620.45	−	13199.0i	−0.177844	−	0.308035i
$208$		0			0
$209$		−	6658.20i		−	0.152428i
$210$		0			0
$211$	−66870.6			−1.50200			−0.751000	−	0.660302i	$-0.770428\pi$
$211$	−66870.6			−1.50200			−0.751000	+	0.660302i	$0.770428\pi$
$212$		0			0
$213$	29560.9	−	17067.0i	0.651565	−	0.376181i
$214$		0			0
$215$	−554.854	−	320.345i	−0.0120033	−	0.00693013i
$216$		0			0
$217$	17014.5	−	25355.0i	0.361327	−	0.538449i
$218$		0			0
$219$	−13358.8	+	23138.2i	−0.278536	+	0.482438i
$220$		0			0
$221$	12474.2	+	21605.9i	0.255404	+	0.442373i
$222$		0			0
$223$		−	86596.2i		−	1.74136i	−0.491848	−	0.870681i	$-0.663678\pi$
$223$		−	86596.2i		−	1.74136i	0.491848	−	0.870681i	$-0.336322\pi$
$224$		0			0
$225$	−16835.4			−0.332551
$226$		0			0
$227$	41757.6	−	24108.8i	0.810371	−	0.467868i	−0.0367137	−	0.999326i	$-0.511689\pi$
$227$	41757.6	−	24108.8i	0.810371	−	0.467868i	0.847085	+	0.531458i	$0.178356\pi$
$228$		0			0
$229$	11529.2	+	6656.39i	0.219851	+	0.126931i	0.605881	−	0.795555i	$-0.292821\pi$
$229$	11529.2	+	6656.39i	0.219851	+	0.126931i	−0.386030	+	0.922486i	$0.626154\pi$
$230$		0			0
$231$	−34996.7	−	2363.54i	−0.655848	−	0.0442934i
$232$		0			0
$233$	1869.42	−	3237.93i	0.0344345	−	0.0596424i	−0.848295	−	0.529525i	$-0.822370\pi$
$233$	1869.42	−	3237.93i	0.0344345	−	0.0596424i	0.882729	+	0.469882i	$0.155704\pi$
$234$		0			0
$235$	61.5586	+	106.623i	0.00111469	+	0.00193070i
$236$		0			0
$237$			56846.9i			1.01207i
$238$		0			0
$239$	27748.8			0.485790			0.242895	−	0.970053i	$-0.421903\pi$
$239$	27748.8			0.485790			0.242895	+	0.970053i	$0.421903\pi$
$240$		0			0
$241$	68808.7	−	39726.7i	1.18470	−	0.683988i	0.227604	−	0.973754i	$-0.426911\pi$
$241$	68808.7	−	39726.7i	1.18470	−	0.683988i	0.957098	+	0.289766i	$0.0935773\pi$
$242$		0			0
$243$	−3280.50	−	1894.00i	−0.0555556	−	0.0320750i
$244$		0			0
$245$	−2691.04	+	1102.17i	−0.0448319	+	0.0183619i
$246$		0			0
$247$	−2590.62	+	4487.09i	−0.0424630	+	0.0735480i
$248$		0			0
$249$	27595.5	+	47796.8i	0.445082	+	0.770904i
$250$		0			0
$251$			84593.4i			1.34273i	0.741126	+	0.671366i	$0.234292\pi$
$251$			84593.4i			1.34273i	−0.741126	+	0.671366i	$0.765708\pi$
$252$		0			0
$253$	−77765.0			−1.21491
$254$		0			0
$255$	−1268.36	+	732.288i	−0.0195057	+	0.0112616i
$256$		0			0
$257$	4473.09	+	2582.54i	0.0677238	+	0.0391003i	0.533480	−	0.845813i	$-0.320884\pi$
$257$	4473.09	+	2582.54i	0.0677238	+	0.0391003i	−0.465756	+	0.884913i	$0.654217\pi$
$258$		0			0
$259$	289.044	−	4279.85i	0.00430889	−	0.0638013i
$260$		0			0
$261$	−731.972	+	1267.81i	−0.0107452	+	0.0186112i
$262$		0			0
$263$	53582.6	+	92807.8i	0.774662	+	1.34175i	0.934984	+	0.354689i	$0.115413\pi$
$263$	53582.6	+	92807.8i	0.774662	+	1.34175i	−0.160323	+	0.987065i	$0.551253\pi$
$264$		0			0
$265$			4201.12i			0.0598238i
$266$		0			0
$267$	−41682.6			−0.584699
$268$		0			0
$269$	−24548.5	+	14173.1i	−0.339251	+	0.195866i	−0.659941	−	0.751318i	$-0.729419\pi$
$269$	−24548.5	+	14173.1i	−0.339251	+	0.195866i	0.320690	+	0.947184i	$0.396085\pi$
$270$		0			0
$271$	−74874.3	−	43228.7i	−1.01952	−	0.588618i	−0.105553	−	0.994414i	$-0.533661\pi$
$271$	−74874.3	−	43228.7i	−1.01952	−	0.588618i	−0.913964	+	0.405795i	$0.866995\pi$
$272$		0			0
$273$	22665.3	+	15209.6i	0.304114	+	0.204076i
$274$		0			0
$275$	−42950.4	+	74392.2i	−0.567939	+	0.983699i
$276$		0			0
$277$	17573.9	+	30438.9i	0.229039	+	0.396707i	0.957524	−	0.288355i	$-0.0931084\pi$
$277$	17573.9	+	30438.9i	0.229039	+	0.396707i	−0.728485	+	0.685062i	$0.759775\pi$
$278$		0			0
$279$		−	16825.3i		−	0.216149i
$280$		0			0
$281$	−137829.			−1.74553			−0.872767	−	0.488136i	$-0.837677\pi$
$281$	−137829.			−1.74553			−0.872767	+	0.488136i	$0.837677\pi$
$282$		0			0
$283$	−17234.5	+	9950.36i	−0.215192	+	0.124241i	−0.603722	−	0.797195i	$-0.706316\pi$
$283$	−17234.5	+	9950.36i	−0.215192	+	0.124241i	0.388530	+	0.921436i	$0.372983\pi$
$284$		0			0
$285$	−263.411	−	152.081i	−0.00324298	−	0.00187234i
$286$		0			0
$287$	−29026.1	−	59154.6i	−0.352391	−	0.718166i
$288$		0			0
$289$	−14681.9	+	25429.8i	−0.175787	+	0.304472i
$290$		0			0
$291$	46755.7	+	80983.2i	0.552139	+	0.956332i
$292$		0			0
$293$			111152.i			1.29474i	0.762174	+	0.647372i	$0.224132\pi$
$293$			111152.i			1.29474i	−0.762174	+	0.647372i	$0.775868\pi$
$294$		0			0
$295$	−2402.88			−0.0276114
$296$		0			0
$297$	−16738.4	+	9663.91i	−0.189758	+	0.109557i
$298$		0			0
$299$	52407.3	+	30257.4i	0.586205	+	0.338446i
$300$		0			0
$301$	23270.0	−	11418.2i	0.256841	−	0.126027i
$302$		0			0
$303$	12388.3	−	21457.2i	0.134936	−	0.233716i
$304$		0			0
$305$	−81.0380	−	140.362i	−0.000871142	−	0.00150886i
$306$		0			0
$307$			3755.76i			0.0398494i	0.999801	+	0.0199247i	$0.00634265\pi$
$307$			3755.76i			0.0398494i	−0.999801	+	0.0199247i	$0.993657\pi$
$308$		0			0
$309$	21543.6			0.225632
$310$		0			0
$311$	64950.7	−	37499.3i	0.671527	−	0.387706i	−0.125128	−	0.992141i	$-0.539934\pi$
$311$	64950.7	−	37499.3i	0.671527	−	0.387706i	0.796655	+	0.604434i	$0.206601\pi$
$312$		0			0
$313$	−43181.4	−	24930.8i	−0.440766	−	0.254476i	0.263157	−	0.964753i	$-0.415236\pi$
$313$	−43181.4	−	24930.8i	−0.440766	−	0.254476i	−0.703922	+	0.710277i	$0.748570\pi$
$314$		0			0
$315$	−892.869	+	1330.55i	−0.00899842	+	0.0134094i
$316$		0			0
$317$	−15932.7	+	27596.2i	−0.158551	+	0.274619i	−0.934346	−	0.356366i	$-0.884016\pi$
$317$	−15932.7	+	27596.2i	−0.158551	+	0.274619i	0.775795	+	0.630985i	$0.217349\pi$
$318$		0			0
$319$	3734.81	+	6468.88i	0.0367018	+	0.0635693i
$320$		0			0
$321$			37805.1i			0.366893i
$322$		0			0
$323$	11247.3			0.107806
$324$		0			0
$325$	57890.2	−	33422.9i	0.548073	−	0.316430i
$326$		0			0
$327$	56144.1	+	32414.8i	0.525060	+	0.303143i
$328$		0			0
$329$	−4969.64	−	335.630i	−0.0459127	−	0.00310076i
$330$		0			0
$331$	−100781.	+	174559.i	−0.919866	+	1.59325i	−0.120250	+	0.992744i	$0.538370\pi$
$331$	−100781.	+	174559.i	−0.919866	+	1.59325i	−0.799616	+	0.600511i	$0.794964\pi$
$332$		0			0
$333$	−1181.83	−	2046.99i	−0.0106578	−	0.0184598i
$334$		0			0
$335$		−	9768.18i		−	0.0870410i
$336$		0			0
$337$	−104708.			−0.921981			−0.460991	−	0.887405i	$-0.652506\pi$
$337$	−104708.			−0.921981			−0.460991	+	0.887405i	$0.652506\pi$
$338$		0			0
$339$	−87407.3	+	50464.6i	−0.760586	+	0.439125i
$340$		0			0
$341$	−74347.5	−	42924.5i	−0.639378	−	0.369145i
$342$		0			0
$343$	23638.5	−	115250.i	0.200924	−	0.979607i
$344$		0			0
$345$	−1776.24	+	3076.53i	−0.0149232	+	0.0258478i
$346$		0			0
$347$	81631.5	+	141390.i	0.677952	+	1.17425i	0.975597	+	0.219570i	$0.0704655\pi$
$347$	81631.5	+	141390.i	0.677952	+	1.17425i	−0.297645	+	0.954677i	$0.596201\pi$
$348$		0			0
$349$		−	158645.i		−	1.30249i	−0.758867	−	0.651245i	$-0.774247\pi$
$349$		−	158645.i		−	1.30249i	0.758867	−	0.651245i	$-0.225753\pi$
$350$		0			0
$351$	15040.4			0.122080
$352$		0			0
$353$	96663.4	−	55808.6i	0.775733	−	0.447870i	−0.0591826	−	0.998247i	$-0.518849\pi$
$353$	96663.4	−	55808.6i	0.775733	−	0.447870i	0.834916	+	0.550377i	$0.185516\pi$
$354$		0			0
$355$	−6890.29	−	3978.11i	−0.0546740	−	0.0315660i
$356$		0			0
$357$	3992.58	−	59117.7i	0.0313269	−	0.463854i
$358$		0			0
$359$	−24169.9	+	41863.6i	−0.187537	+	0.324823i	−0.944428	−	0.328717i	$-0.893384\pi$
$359$	−24169.9	+	41863.6i	−0.187537	+	0.324823i	0.756892	+	0.653541i	$0.226717\pi$
$360$		0			0
$361$	−63992.6	−	110838.i	−0.491038	−	0.850503i
$362$		0			0
$363$			22541.2i			0.171066i
$364$		0			0
$365$	6227.58			0.0467448
$366$		0			0
$367$	116776.	−	67420.8i	0.867007	−	0.500567i	0.000654662	−	1.00000i	$-0.499792\pi$
$367$	116776.	−	67420.8i	0.867007	−	0.500567i	0.866353	+	0.499433i	$0.166458\pi$
$368$		0			0
$369$	−31443.6	−	18154.0i	−0.230929	−	0.133327i
$370$		0			0
$371$	−141133.	−	94707.6i	−1.02537	−	0.688077i
$372$		0			0
$373$	67477.1	−	116874.i	0.484997	−	0.840039i	−0.514855	−	0.857277i	$-0.672154\pi$
$373$	67477.1	−	116874.i	0.484997	−	0.840039i	0.999851	+	0.0172388i	$0.00548754\pi$
$374$		0			0
$375$	3928.75	+	6804.79i	0.0279378	+	0.0483896i
$376$		0			0
$377$		−	5812.67i		−	0.0408971i
$378$		0			0
$379$	268541.			1.86953			0.934764	−	0.355269i	$-0.115611\pi$
$379$	268541.			1.86953			0.934764	+	0.355269i	$0.115611\pi$
$380$		0			0
$381$	−21176.9	+	12226.5i	−0.145885	+	0.0842269i
$382$		0			0
$383$	215510.	+	124425.i	1.46916	+	0.848222i	0.999402	−	0.0345712i	$-0.0110065\pi$
$383$	215510.	+	124425.i	1.46916	+	0.848222i	0.469762	+	0.882793i	$0.344340\pi$
$384$		0			0
$385$	3601.55	+	7339.90i	0.0242979	+	0.0495186i
$386$		0			0
$387$	7141.34	−	12369.2i	0.0476824	−	0.0825883i
$388$		0			0
$389$	−20183.7	−	34959.1i	−0.133383	−	0.231026i	0.791595	−	0.611045i	$-0.209251\pi$
$389$	−20183.7	−	34959.1i	−0.133383	−	0.231026i	−0.924979	+	0.380019i	$0.875917\pi$
$390$		0			0
$391$		−	131364.i		−	0.859253i
$392$		0			0
$393$	1357.23			0.00878757
$394$		0			0
$395$	11475.1	−	6625.17i	0.0735467	−	0.0424622i
$396$		0			0
$397$	−255471.	−	147496.i	−1.62092	−	0.935837i	−0.986675	−	0.162702i	$-0.947979\pi$
$397$	−255471.	−	147496.i	−1.62092	−	0.935837i	−0.634241	−	0.773135i	$-0.718687\pi$
$398$		0			0
$399$	11047.2	−	5420.66i	0.0693915	−	0.0340492i
$400$		0			0
$401$	105410.	−	182576.i	0.655533	−	1.13542i	−0.326227	−	0.945292i	$-0.605777\pi$
$401$	105410.	−	182576.i	0.655533	−	1.13542i	0.981760	−	0.190125i	$-0.0608894\pi$
$402$		0			0
$403$	33402.8	+	57855.3i	0.205671	+	0.356232i
$404$		0			0
$405$			882.937i			0.00538294i
$406$		0			0
$407$	−12060.3			−0.0728064
$408$		0			0
$409$	192775.	−	111299.i	1.15240	−	0.665339i	0.202929	−	0.979193i	$-0.434954\pi$
$409$	192775.	−	111299.i	1.15240	−	0.665339i	0.949471	+	0.313855i	$0.101620\pi$
$410$		0			0
$411$	51508.5	+	29738.5i	0.304927	+	0.176050i
$412$		0			0
$413$	54169.1	−	80722.7i	0.317579	−	0.473255i
$414$		0			0
$415$	6432.19	−	11140.9i	0.0373476	−	0.0646879i
$416$		0			0
$417$	15372.6	+	26626.1i	0.0884044	+	0.153121i
$418$		0			0
$419$		−	10461.2i		−	0.0595875i	−0.999556	−	0.0297938i	$-0.990515\pi$
$419$		−	10461.2i		−	0.0595875i	0.999556	−	0.0297938i	$-0.00948505\pi$
$420$		0			0
$421$	−75620.7			−0.426655			−0.213327	−	0.976981i	$-0.568430\pi$
$421$	−75620.7			−0.426655			−0.213327	+	0.976981i	$0.568430\pi$
$422$		0			0
$423$	−2376.90	+	1372.31i	−0.0132841	+	0.00766955i
$424$		0			0
$425$	−125666.	−	72553.4i	−0.695729	−	0.401679i
$426$		0			0
$427$	6542.21	+	441.835i	0.0358814	+	0.00242328i
$428$		0			0
$429$	38371.1	−	66460.6i	0.208492	−	0.361119i
$430$		0			0
$431$	−114306.	−	197984.i	−0.615339	−	1.06580i	−0.990325	−	0.138768i	$-0.955686\pi$
$431$	−114306.	−	197984.i	−0.615339	−	1.06580i	0.374985	−	0.927031i	$-0.377648\pi$
$432$		0			0
$433$			312999.i			1.66943i	0.550685	+	0.834713i	$0.314367\pi$
$433$			312999.i			1.66943i	−0.550685	+	0.834713i	$0.685633\pi$
$434$		0			0
$435$	341.228			0.00180329
$436$		0			0
$437$	23626.4	−	13640.7i	0.123718	−	0.0714289i
$438$		0			0
$439$	−105531.	−	60928.5i	−0.547585	−	0.316149i	0.200562	−	0.979681i	$-0.435723\pi$
$439$	−105531.	−	60928.5i	−0.547585	−	0.316149i	−0.748148	+	0.663532i	$0.769056\pi$
$440$		0			0
$441$	−24570.4	−	59990.3i	−0.126338	−	0.308464i
$442$		0			0
$443$	32116.3	−	55627.0i	0.163651	−	0.283451i	−0.772525	−	0.634985i	$-0.781006\pi$
$443$	32116.3	−	55627.0i	0.163651	−	0.283451i	0.936175	+	0.351534i	$0.114340\pi$
$444$		0			0
$445$	4857.86	+	8414.06i	0.0245316	+	0.0424899i
$446$		0			0
$447$		−	40867.4i		−	0.204532i
$448$		0			0
$449$	−79630.3			−0.394989			−0.197495	−	0.980304i	$-0.563281\pi$
$449$	−79630.3			−0.394989			−0.197495	+	0.980304i	$0.563281\pi$
$450$		0			0
$451$	−160437.	+	92628.6i	−0.788774	+	0.455399i
$452$		0			0
$453$	45319.5	+	26165.2i	0.220845	+	0.127505i
$454$		0			0
$455$	428.707	−	6347.82i	0.00207080	−	0.0306621i
$456$		0			0
$457$	−76805.1	+	133030.i	−0.367754	+	0.636969i	−0.989214	−	0.146477i	$-0.953207\pi$
$457$	−76805.1	+	133030.i	−0.367754	+	0.636969i	0.621460	+	0.783446i	$0.286540\pi$
$458$		0			0
$459$	−16324.6	−	28275.1i	−0.0774851	−	0.134208i
$460$		0			0
$461$			247213.i			1.16324i	0.813460	+	0.581621i	$0.197581\pi$
$461$			247213.i			1.16324i	−0.813460	+	0.581621i	$0.802419\pi$
$462$		0			0
$463$	86869.1			0.405232			0.202616	−	0.979258i	$-0.435056\pi$
$463$	86869.1			0.405232			0.202616	+	0.979258i	$0.435056\pi$
$464$		0			0
$465$	−3396.35	+	1960.89i	−0.0157075	+	0.00906873i
$466$		0			0
$467$	−168666.	−	97379.5i	−0.773382	−	0.446513i	0.0606975	−	0.998156i	$-0.480667\pi$
$467$	−168666.	−	97379.5i	−0.773382	−	0.446513i	−0.834080	+	0.551644i	$0.814001\pi$
$468$		0			0
$469$	328154.	+	220208.i	1.49187	+	1.00112i
$470$		0			0
$471$	69811.4	−	120917.i	0.314691	−	0.545061i
$472$		0			0
$473$	−36437.9	−	63112.3i	−0.162866	−	0.282093i
$474$		0			0
$475$		−	30135.6i		−	0.133565i
$476$		0			0
$477$	−93654.2			−0.411614
$478$		0			0
$479$	−168631.	+	97359.4i	−0.734967	+	0.424333i	−0.820236	−	0.572025i	$-0.806158\pi$
$479$	−168631.	+	97359.4i	−0.734967	+	0.424333i	0.0852698	+	0.996358i	$0.472825\pi$
$480$		0			0
$481$	8127.67	+	4692.51i	0.0351298	+	0.0202822i
$482$		0			0
$483$	−63311.0	−	129027.i	−0.271384	−	0.553076i
$484$		0			0
$485$	10898.2	−	18876.2i	0.0463309	−	0.0802475i
$486$		0			0
$487$	112307.	+	194521.i	0.473530	+	0.820179i	0.999541	−	0.0302993i	$-0.00964604\pi$
$487$	112307.	+	194521.i	0.473530	+	0.820179i	−0.526010	+	0.850478i	$0.676313\pi$
$488$		0			0
$489$			79492.0i			0.332434i
$490$		0			0
$491$	214755.			0.890798			0.445399	−	0.895332i	$-0.353062\pi$
$491$	214755.			0.890798			0.445399	+	0.895332i	$0.353062\pi$
$492$		0			0
$493$	−10927.5	+	6308.98i	−0.0449599	+	0.0259576i
$494$		0			0
$495$	3901.52	+	2252.54i	0.0159229	+	0.00919312i
$496$		0			0
$497$	288972.	−	141793.i	1.16988	−	0.574040i
$498$		0			0
$499$	−113041.	+	195794.i	−0.453980	+	0.786317i	−0.998629	−	0.0523476i	$-0.983330\pi$
$499$	−113041.	+	195794.i	−0.453980	+	0.786317i	0.544649	+	0.838664i	$0.316663\pi$
$500$		0			0
$501$	−142241.	−	246368.i	−0.566695	−	0.981544i
$502$		0			0
$503$		−	251431.i		−	0.993763i	−0.867818	−	0.496882i	$-0.834478\pi$
$503$		−	251431.i		−	0.993763i	0.867818	−	0.496882i	$-0.165522\pi$
$504$		0			0
$505$	−5775.15			−0.0226454
$506$		0			0
$507$	76806.5	−	44344.3i	0.298801	−	0.172513i
$508$		0			0
$509$	313005.	+	180713.i	1.20813	+	0.697517i	0.962351	−	0.271808i	$-0.0876217\pi$
$509$	313005.	+	180713.i	1.20813	+	0.697517i	0.245783	+	0.969325i	$0.420955\pi$
$510$		0			0
$511$	−140391.	+	209210.i	−0.537647	+	0.801200i
$512$		0			0
$513$	3390.28	−	5872.14i	0.0128825	−	0.0223132i
$514$		0			0
$515$	−2510.78	−	4348.80i	−0.00946661	−	0.0163966i
$516$		0			0
$517$			14004.1i			0.0523930i
$518$		0			0
$519$	−22740.0			−0.0844221
$520$		0			0
$521$	−336468.	+	194260.i	−1.23956	+	0.715661i	−0.969005	−	0.247042i	$-0.920541\pi$
$521$	−336468.	+	194260.i	−1.23956	+	0.715661i	−0.270557	+	0.962704i	$0.587208\pi$
$522$		0			0
$523$	−20578.4	−	11880.9i	−0.0752329	−	0.0434357i	0.461912	−	0.886926i	$-0.347164\pi$
$523$	−20578.4	−	11880.9i	−0.0752329	−	0.0434357i	−0.537145	+	0.843490i	$0.680497\pi$
$524$		0			0
$525$	−158398.	−	10697.6i	−0.574686	−	0.0388120i
$526$		0			0
$527$	72509.7	−	125591.i	0.261081	−	0.452205i
$528$		0			0
$529$	−19397.0	−	33596.7i	−0.0693145	−	0.120056i
$530$		0			0
$531$		−	53566.5i		−	0.189979i
$532$		0			0
$533$	144163.			0.507455
$534$		0			0
$535$	7631.34	−	4405.96i	0.0266620	−	0.0153933i
$536$		0			0
$537$	133714.	+	77199.8i	0.463690	+	0.267712i
$538$		0			0
$539$	−327769.	−	44475.3i	−1.12821	−	0.153088i
$540$		0			0
$541$	−202643.	+	350987.i	−0.692367	+	1.19922i	0.278693	+	0.960380i	$0.410099\pi$
$541$	−202643.	+	350987.i	−0.692367	+	1.19922i	−0.971060	+	0.238835i	$0.923235\pi$
$542$		0			0
$543$	52869.8	+	91573.3i	0.179312	+	0.310577i
$544$		0			0
$545$		−	15111.0i		−	0.0508746i
$546$		0			0
$547$	367388.			1.22786			0.613932	−	0.789359i	$-0.289587\pi$
$547$	367388.			1.22786			0.613932	+	0.789359i	$0.289587\pi$
$548$		0			0
$549$	3129.04	−	1806.55i	0.0103817	−	0.00599385i
$550$		0			0
$551$	−2269.40	−	1310.24i	−0.00747495	−	0.00431566i
$552$		0			0
$553$	−36121.7	+	534851.i	−0.118118	+	1.74897i
$554$		0			0
$555$	−275.470	+	477.129i	−0.000894312	+	0.00154899i
$556$		0			0
$557$	140993.	+	244206.i	0.454450	+	0.787130i	0.998656	−	0.0518209i	$-0.0165025\pi$
$557$	140993.	+	244206.i	0.454450	+	0.787130i	−0.544206	+	0.838951i	$0.683169\pi$
$558$		0			0
$559$			56710.2i			0.181484i
$560$		0			0
$561$	−166589.			−0.529324
$562$		0			0
$563$	−491974.	+	284041.i	−1.55212	+	0.896118i	−0.554152	+	0.832415i	$0.686957\pi$
$563$	−491974.	+	284041.i	−1.55212	+	0.896118i	−0.997969	+	0.0637021i	$0.979709\pi$
$564$		0			0
$565$	20373.6	+	11762.7i	0.0638221	+	0.0368477i
$566$		0			0
$567$	−29661.5	−	19904.4i	−0.0922629	−	0.0619132i
$568$		0			0
$569$	119910.	−	207690.i	0.370365	−	0.641491i	−0.619257	−	0.785189i	$-0.712566\pi$
$569$	119910.	−	207690.i	0.370365	−	0.641491i	0.989622	+	0.143698i	$0.0458993\pi$
$570$		0			0
$571$	−28352.8	−	49108.4i	−0.0869608	−	0.150620i	0.819264	−	0.573416i	$-0.194382\pi$
$571$	−28352.8	−	49108.4i	−0.0869608	−	0.150620i	−0.906225	+	0.422796i	$0.861049\pi$
$572$		0			0
$573$		−	38476.3i		−	0.117188i
$574$		0			0
$575$	−351971.			−1.06456
$576$		0			0
$577$	236185.	−	136362.i	0.709416	−	0.409582i	−0.101429	−	0.994843i	$-0.532341\pi$
$577$	236185.	−	136362.i	0.709416	−	0.409582i	0.810845	+	0.585261i	$0.199008\pi$
$578$		0			0
$579$	124260.	+	71741.6i	0.370659	+	0.214000i
$580$		0			0
$581$	229265.	+	467237.i	0.679180	+	1.38416i
$582$		0			0
$583$	−238930.	+	413839.i	−0.702965	+	1.21757i
$584$		0			0
$585$	−1752.87	−	3036.07i	−0.00512199	−	0.00887155i
$586$		0			0
$587$		−	215131.i		−	0.624348i	−0.950025	−	0.312174i	$-0.898943\pi$
$587$		−	215131.i		−	0.624348i	0.950025	−	0.312174i	$-0.101057\pi$
$588$		0			0
$589$	30117.4			0.0868135
$590$		0			0
$591$	−215641.	+	124500.i	−0.617384	+	0.356447i
$592$		0			0
$593$	32238.1	+	18612.7i	0.0916770	+	0.0529297i	0.545138	−	0.838347i	$-0.316477\pi$
$593$	32238.1	+	18612.7i	0.0916770	+	0.0529297i	−0.453461	+	0.891276i	$0.649811\pi$
$594$		0			0
$595$	−12398.8	+	6083.88i	−0.0350225	+	0.0171849i
$596$		0			0
$597$	169076.	−	292849.i	0.474388	−	0.821665i
$598$		0			0
$599$	−181425.	−	314237.i	−0.505642	−	0.875798i	−0.999979	−	0.00652760i	$-0.997922\pi$
$599$	−181425.	−	314237.i	−0.505642	−	0.875798i	0.494336	−	0.869271i	$-0.335411\pi$
$600$		0			0
$601$			355030.i			0.982915i	0.870901	+	0.491458i	$0.163536\pi$
$601$			355030.i			0.982915i	−0.870901	+	0.491458i	$0.836464\pi$
$602$		0			0
$603$	217759.			0.598881
$604$		0			0
$605$	4550.17	−	2627.04i	0.0124313	−	0.00717722i
$606$		0			0
$607$	265069.	+	153038.i	0.719418	+	0.415356i	0.814539	−	0.580109i	$-0.196990\pi$
$607$	265069.	+	153038.i	0.719418	+	0.415356i	−0.0951201	+	0.995466i	$0.530324\pi$
$608$		0			0
$609$	−7692.45	+	11463.3i	−0.0207410	+	0.0309082i
$610$		0			0
$611$	5448.81	−	9437.61i	0.0145955	−	0.0252802i
$612$		0			0
$613$	−255797.	−	443054.i	−0.680730	−	1.17906i	−0.974759	−	0.223262i	$-0.928330\pi$
$613$	−255797.	−	443054.i	−0.680730	−	1.17906i	0.294029	−	0.955797i	$-0.405004\pi$
$614$		0			0
$615$			8462.95i			0.0223754i
$616$		0			0
$617$	115033.			0.302170			0.151085	−	0.988521i	$-0.451723\pi$
$617$	115033.			0.302170			0.151085	+	0.988521i	$0.451723\pi$
$618$		0			0
$619$	31220.8	−	18025.4i	0.0814823	−	0.0470438i	−0.458705	−	0.888588i	$-0.651687\pi$
$619$	31220.8	−	18025.4i	0.0814823	−	0.0470438i	0.540188	+	0.841545i	$0.318353\pi$
$620$		0			0
$621$	−68584.0	−	39597.0i	−0.177844	−	0.102678i
$622$		0			0
$623$	−392176.	−	26486.0i	−1.01043	−	0.0682402i
$624$		0			0
$625$	−193938.	+	335911.i	−0.496482	+	0.859932i
$626$		0			0
$627$	−17298.5	−	29961.9i	−0.0440021	−	0.0762140i
$628$		0			0
$629$		−	20372.7i		−	0.0514929i
$630$		0			0
$631$	147471.			0.370380			0.185190	−	0.982703i	$-0.440710\pi$
$631$	147471.			0.370380			0.185190	+	0.982703i	$0.440710\pi$
$632$		0			0
$633$	−300917.	+	173735.i	−0.751000	+	0.433590i
$634$		0			0
$635$	4936.07	+	2849.84i	0.0122415	+	0.00706763i
$636$		0			0
$637$	203585.	+	157504.i	0.501726	+	0.388161i
$638$		0			0
$639$	88682.6	−	153603.i	0.217188	−	0.376181i
$640$		0			0
$641$	88256.2	+	152864.i	0.214797	+	0.372040i	0.953210	−	0.302309i	$-0.0977575\pi$
$641$	88256.2	+	152864.i	0.214797	+	0.372040i	−0.738412	+	0.674349i	$0.764424\pi$
$642$		0			0
$643$		−	629700.i		−	1.52304i	−0.648141	−	0.761520i	$-0.724453\pi$
$643$		−	629700.i		−	1.52304i	0.648141	−	0.761520i	$-0.275547\pi$
$644$		0			0
$645$	−3329.13			−0.00800223
$646$		0			0
$647$	−154490.	+	89195.0i	−0.369056	+	0.213075i	−0.673046	−	0.739600i	$-0.735014\pi$
$647$	−154490.	+	89195.0i	−0.369056	+	0.213075i	0.303990	+	0.952675i	$0.401681\pi$
$648$		0			0
$649$	−236700.	−	136659.i	−0.561964	−	0.324450i
$650$		0			0
$651$	10691.1	−	158303.i	0.0252268	−	0.373530i
$652$		0			0
$653$	253669.	−	439368.i	0.594897	−	1.03039i	−0.398665	−	0.917097i	$-0.630526\pi$
$653$	253669.	−	439368.i	0.594897	−	1.03039i	0.993561	−	0.113295i	$-0.0361404\pi$
$654$		0			0
$655$	−158.177	−	273.971i	−0.000368690	−	0.000638590i
$656$		0			0
$657$			138829.i			0.321625i
$658$		0			0
$659$	362525.			0.834770			0.417385	−	0.908730i	$-0.362947\pi$
$659$	362525.			0.834770			0.417385	+	0.908730i	$0.362947\pi$
$660$		0			0
$661$	−271942.	+	157006.i	−0.622405	+	0.359346i	−0.777805	−	0.628506i	$-0.783667\pi$
$661$	−271942.	+	157006.i	−0.622405	+	0.359346i	0.155400	+	0.987852i	$0.450333\pi$
$662$		0			0
$663$	112268.	+	64817.8i	0.255404	+	0.147458i
$664$		0			0
$665$	−2381.70	−	1598.25i	−0.00538572	−	0.00361410i
$666$		0			0
$667$	−15303.0	+	26505.6i	−0.0343974	+	0.0595781i
$668$		0			0
$669$	−224984.	−	389683.i	−0.502688	−	0.870681i
$670$		0			0
$671$		−	18435.5i		−	0.0409458i
$672$		0			0
$673$	281603.			0.621738			0.310869	−	0.950453i	$-0.399380\pi$
$673$	281603.			0.621738			0.310869	+	0.950453i	$0.399380\pi$
$674$		0			0
$675$	−75759.3	+	43739.6i	−0.166275	+	0.0959992i
$676$		0			0
$677$	644409.	+	372050.i	1.40600	+	0.811753i	0.994999	−	0.0998830i	$-0.0318469\pi$
$677$	644409.	+	372050.i	1.40600	+	0.811753i	0.410998	+	0.911636i	$0.365180\pi$
$678$		0			0
$679$	388448.	+	791650.i	0.842545	+	1.71709i
$680$		0			0
$681$	125273.	−	216979.i	0.270124	−	0.467868i
$682$		0			0
$683$	−373570.	−	647042.i	−0.800812	−	1.38705i	−0.919082	−	0.394066i	$-0.871068\pi$
$683$	−373570.	−	647042.i	−0.800812	−	1.38705i	0.118270	−	0.992981i	$-0.462265\pi$
$684$		0			0
$685$		−	13863.4i		−	0.0295452i
$686$		0			0
$687$	69175.3			0.146567
$688$		0			0
$689$	322039.	−	185929.i	0.678375	−	0.391660i
$690$		0			0
$691$	−203574.	−	117534.i	−0.426350	−	0.246154i	0.271440	−	0.962455i	$-0.412500\pi$
$691$	−203574.	−	117534.i	−0.426350	−	0.246154i	−0.697791	+	0.716302i	$0.745833\pi$
$692$		0			0
$693$	−163626.	+	80288.2i	−0.340710	+	0.167180i
$694$		0			0
$695$	3583.16	−	6206.22i	0.00741817	−	0.0128486i
$696$		0			0
$697$	−156472.	−	271017.i	−0.322085	−	0.557867i
$698$		0			0
$699$		−	19427.6i		−	0.0397616i
$700$		0			0
$701$	−777252.			−1.58171			−0.790853	−	0.612006i	$-0.790363\pi$
$701$	−777252.			−1.58171			−0.790853	+	0.612006i	$0.790363\pi$
$702$		0			0
$703$	3664.13	−	2115.49i	0.00741414	−	0.00428056i
$704$		0			0
$705$	554.028	+	319.868i	0.00111469	+	0.000643565i
$706$		0			0
$707$	130192.	−	194011.i	0.260462	−	0.388140i
$708$		0			0
$709$	398232.	−	689757.i	0.792215	−	1.37216i	−0.132377	−	0.991199i	$-0.542261\pi$
$709$	398232.	−	689757.i	0.792215	−	1.37216i	0.924592	−	0.380958i	$-0.124406\pi$
$710$		0			0
$711$	147692.	+	255811.i	0.292159	+	0.506034i
$712$		0			0
$713$		−	351759.i		−	0.691936i
$714$		0			0
$715$	−17887.7			−0.0349898
$716$		0			0
$717$	124870.	−	72093.6i	0.242895	−	0.140236i
$718$		0			0
$719$	183054.	+	105686.i	0.354096	+	0.204437i	0.666488	−	0.745516i	$-0.267797\pi$
$719$	183054.	+	105686.i	0.354096	+	0.204437i	−0.312392	+	0.949953i	$0.601130\pi$
$720$		0			0
$721$	202696.	+	13689.3i	0.389919	+	0.0263336i
$722$		0			0
$723$	206426.	−	357540.i	0.394901	−	0.683988i
$724$		0			0
$725$	16904.0	+	29278.6i	0.0321599	+	0.0557025i
$726$		0			0
$727$		−	886461.i		−	1.67722i	−0.544730	−	0.838611i	$-0.683368\pi$
$727$		−	886461.i		−	1.67722i	0.544730	−	0.838611i	$-0.316632\pi$
$728$		0			0
$729$	−19683.0			−0.0370370
$730$		0			0
$731$	106612.	−	61552.3i	0.199513	−	0.115189i
$732$		0			0
$733$	−597812.	−	345147.i	−1.11264	−	0.642386i	−0.173132	−	0.984899i	$-0.555389\pi$
$733$	−597812.	−	345147.i	−1.11264	−	0.642386i	−0.939513	+	0.342513i	$0.888722\pi$
$734$		0			0
$735$	−9246.12	+	11951.3i	−0.0171153	+	0.0221228i
$736$		0			0
$737$	555545.	−	962231.i	1.02278	−	1.77151i
$738$		0			0
$739$	−243854.	−	422367.i	−0.446519	−	0.773394i	0.551637	−	0.834084i	$-0.314003\pi$
$739$	−243854.	−	422367.i	−0.446519	−	0.773394i	−0.998157	+	0.0606898i	$0.980670\pi$
$740$		0			0
$741$			26922.6i			0.0490320i
$742$		0			0
$743$	664586.			1.20385			0.601926	−	0.798552i	$-0.294400\pi$
$743$	664586.			1.20385			0.601926	+	0.798552i	$0.294400\pi$
$744$		0			0
$745$	−8249.51	+	4762.86i	−0.0148633	+	0.00858133i
$746$		0			0
$747$	248360.	+	143390.i	0.445082	+	0.256968i
$748$		0			0
$749$	−24022.1	+	355694.i	−0.0428201	+	0.634034i
$750$		0			0
$751$	−105892.	+	183410.i	−0.187752	+	0.325195i	−0.944500	−	0.328511i	$-0.893453\pi$
$751$	−105892.	+	183410.i	−0.187752	+	0.325195i	0.756749	+	0.653706i	$0.226787\pi$
$752$		0			0
$753$	219780.	+	380670.i	0.387613	+	0.671366i
$754$		0			0
$755$		−	12197.6i		−	0.0213984i
$756$		0			0
$757$	626993.			1.09414			0.547068	−	0.837088i	$-0.315744\pi$
$757$	626993.			1.09414			0.547068	+	0.837088i	$0.315744\pi$
$758$		0			0
$759$	−349942.	+	202039.i	−0.607454	+	0.350713i
$760$		0			0
$761$	−166918.	−	96370.0i	−0.288226	−	0.166407i	0.348916	−	0.937154i	$-0.386550\pi$
$761$	−166918.	−	96370.0i	−0.288226	−	0.166407i	−0.637142	+	0.770747i	$0.719883\pi$
$762$		0			0
$763$	507642.	+	340654.i	0.871983	+	0.585146i
$764$		0			0
$765$	−3805.08	+	6590.59i	−0.00650191	+	0.0112616i
$766$		0			0
$767$	106344.	+	184194.i	0.180769	+	0.313101i
$768$		0			0
$769$		−	627983.i		−	1.06193i	−0.847394	−	0.530964i	$-0.821830\pi$
$769$		−	627983.i		−	1.06193i	0.847394	−	0.530964i	$-0.178170\pi$
$770$		0			0
$771$	26838.5			0.0451492
$772$		0			0
$773$	−870327.	+	502483.i	−1.45654	+	0.840936i	−0.998839	−	0.0481689i	$-0.984661\pi$
$773$	−870327.	+	502483.i	−1.45654	+	0.840936i	−0.457704	+	0.889105i	$0.651328\pi$
$774$		0			0
$775$	−336503.	−	194280.i	−0.560254	−	0.323463i
$776$		0			0
$777$	−9818.69	−	20010.3i	−0.0162634	−	0.0331445i
$778$		0			0
$779$	32495.8	−	56284.4i	0.0535492	−	0.0927499i
$780$		0			0
$781$	−452493.	−	783741.i	−0.741839	−	1.28490i
$782$		0			0
$783$			7606.88i			0.0124075i
$784$		0			0
$785$	−32544.4			−0.0528126
$786$		0			0
$787$	765029.	−	441690.i	1.23517	−	0.713128i	0.267070	−	0.963677i	$-0.413944\pi$
$787$	765029.	−	441690.i	1.23517	−	0.713128i	0.968104	+	0.250549i	$0.0806111\pi$
$788$		0			0
$789$	482243.	+	278423.i	0.774662	+	0.447251i
$790$		0			0
$791$	−854449.	+	419262.i	−1.36563	+	0.670089i
$792$		0			0
$793$	−7173.01	+	12424.0i	−0.0114066	+	0.0197568i
$794$		0			0
$795$	10914.8	+	18905.1i	0.0172696	+	0.0299119i
$796$		0			0
$797$			989639.i			1.55797i	0.627041	+	0.778987i	$0.284266\pi$
$797$			989639.i			1.55797i	−0.627041	+	0.778987i	$0.715734\pi$
$798$		0			0
$799$	−23656.2			−0.0370554
$800$		0			0
$801$	−187572.	+	108295.i	−0.292350	+	0.168788i
$802$		0			0
$803$	613458.	+	354180.i	0.951380	+	0.549279i
$804$		0			0
$805$	−18666.8	+	27817.3i	−0.0288057	+	0.0429262i
$806$		0			0
$807$	−73645.6	+	127558.i	−0.113084	+	0.195866i
$808$		0			0
$809$	−417943.	−	723899.i	−0.638588	−	1.10607i	−0.985743	−	0.168259i	$-0.946186\pi$
$809$	−417943.	−	723899.i	−0.638588	−	1.10607i	0.347155	−	0.937808i	$-0.387148\pi$
$810$		0			0
$811$		−	449509.i		−	0.683435i	−0.939803	−	0.341717i	$-0.888991\pi$
$811$		−	449509.i		−	0.683435i	0.939803	−	0.341717i	$-0.111009\pi$
$812$		0			0
$813$	−449246.			−0.679678
$814$		0			0
$815$	16046.3	−	9264.32i	0.0241579	−	0.0139476i
$816$		0			0
$817$	22141.0	+	12783.1i	0.0331705	+	0.0191510i
$818$		0			0
$819$	141510.	+	9557.01i	0.210969	+	0.0142480i
$820$		0			0
$821$	−265386.	+	459662.i	−0.393724	+	0.681949i	−0.992937	−	0.118640i	$-0.962147\pi$
$821$	−265386.	+	459662.i	−0.393724	+	0.681949i	0.599214	+	0.800589i	$0.295480\pi$
$822$		0			0
$823$	81940.1	+	141924.i	0.120975	+	0.209535i	0.920153	−	0.391560i	$-0.128064\pi$
$823$	81940.1	+	141924.i	0.120975	+	0.209535i	−0.799177	+	0.601095i	$0.794731\pi$
$824$		0			0
$825$			446353.i			0.655799i
$826$		0			0
$827$	344293.			0.503405			0.251702	−	0.967805i	$-0.419010\pi$
$827$	344293.			0.503405			0.251702	+	0.967805i	$0.419010\pi$
$828$		0			0
$829$	−231695.	+	133769.i	−0.337138	+	0.194647i	−0.659006	−	0.752138i	$-0.729023\pi$
$829$	−231695.	+	133769.i	−0.337138	+	0.194647i	0.321868	+	0.946785i	$0.395689\pi$
$830$		0			0
$831$	158165.	+	91316.8i	0.229039	+	0.132236i
$832$		0			0
$833$	75129.3	−	553679.i	0.108273	−	0.797937i
$834$		0			0
$835$	−33154.7	+	57425.6i	−0.0475523	+	0.0823631i
$836$		0			0
$837$	−43713.3	−	75713.7i	−0.0623969	−	0.108075i
$838$		0			0
$839$			1.14814e6i			1.63106i	0.578712	+	0.815532i	$0.303555\pi$
$839$			1.14814e6i			1.63106i	−0.578712	+	0.815532i	$0.696445\pi$
$840$		0			0
$841$	−704341.			−0.995843
$842$		0			0
$843$	−620231.	+	358091.i	−0.872767	+	0.503893i
$844$		0			0
$845$	−17902.7	−	10336.1i	−0.0250729	−	0.0144759i
$846$		0			0
$847$	−14323.1	+	212082.i	−0.0199651	+	0.295622i
$848$		0			0
$849$	−51703.6	+	89553.2i	−0.0717307	+	0.124241i
$850$		0			0
$851$	−24708.0	−	42795.5i	−0.0341176	−	0.0590934i
$852$		0			0
$853$		−	589579.i		−	0.810296i	−0.914251	−	0.405148i	$-0.867220\pi$
$853$		−	589579.i		−	0.810296i	0.914251	−	0.405148i	$-0.132780\pi$
$854$		0			0
$855$	−1580.47			−0.00216199
$856$		0			0
$857$	−563906.	+	325572.i	−0.767795	+	0.443287i	−0.832088	−	0.554644i	$-0.812854\pi$
$857$	−563906.	+	325572.i	−0.767795	+	0.443287i	0.0642923	+	0.997931i	$0.479521\pi$
$858$		0			0
$859$	−825768.	−	476757.i	−1.11911	−	0.646117i	−0.177935	−	0.984042i	$-0.556942\pi$
$859$	−825768.	−	476757.i	−1.11911	−	0.646117i	−0.941173	+	0.337925i	$0.890275\pi$
$860$		0			0
$861$	−284305.	−	190784.i	−0.383512	−	0.257356i
$862$		0			0
$863$	−561635.	+	972780.i	−0.754106	+	1.30615i	0.191711	+	0.981451i	$0.438596\pi$
$863$	−561635.	+	972780.i	−0.754106	+	1.30615i	−0.945817	+	0.324699i	$0.894737\pi$
$864$		0			0
$865$	2650.22	+	4590.31i	0.00354200	+	0.00613493i
$866$		0			0
$867$			152579.i			0.202981i
$868$		0			0
$869$	1.50717e6			1.99582
$870$		0			0
$871$	−748784.	+	432311.i	−0.987008	+	0.569849i
$872$		0			0
$873$	420801.	+	242950.i	0.552139	+	0.318777i
$874$		0			0
$875$	32640.2	+	66520.1i	0.0426321	+	0.0868834i
$876$		0			0
$877$	−369967.	+	640802.i	−0.481021	+	0.833152i	−0.999763	−	0.0217785i	$-0.993067\pi$
$877$	−369967.	+	640802.i	−0.481021	+	0.833152i	0.518742	+	0.854931i	$0.326400\pi$
$878$		0			0
$879$	288783.	+	500186.i	0.373760	+	0.647372i
$880$		0			0
$881$		−	638892.i		−	0.823143i	−0.911377	−	0.411572i	$-0.864980\pi$
$881$		−	638892.i		−	0.823143i	0.911377	−	0.411572i	$-0.135020\pi$
$882$		0			0
$883$	1.26184e6			1.61839			0.809193	−	0.587543i	$-0.199905\pi$
$883$	1.26184e6			1.61839			0.809193	+	0.587543i	$0.199905\pi$
$884$		0			0
$885$	−10813.0	+	6242.86i	−0.0138057	+	0.00797072i
$886$		0			0
$887$	1.26501e6	+	730353.i	1.60785	+	0.928295i	0.989850	+	0.142119i	$0.0453917\pi$
$887$	1.26501e6	+	730353.i	1.60785	+	0.928295i	0.618004	+	0.786175i	$0.287942\pi$
$888$		0			0
$889$	−207014.	+	101578.i	−0.261937	+	0.128527i
$890$		0			0
$891$	−50215.2	+	86975.2i	−0.0632528	+	0.109557i
$892$		0			0
$893$	−2456.44	−	4254.68i	−0.00308038	−	0.00533537i
$894$		0			0
$895$		−	35988.7i		−	0.0449283i
$896$		0			0
$897$	314444.			0.390803
$898$		0			0
$899$	−29261.0	+	16893.9i	−0.0362051	+	0.0209031i
$900$		0			0
$901$	−699072.	−	403609.i	−0.861137	−	0.497178i
$902$		0			0
$903$	75049.8	−	111839.i	0.0920395	−	0.137157i
$904$		0			0
$905$	12323.3	−	21344.6i	0.0150463	−	0.0260610i
$906$		0			0
$907$	18504.7	+	32051.0i	0.0224940	+	0.0389608i	0.877053	−	0.480393i	$-0.159506\pi$
$907$	18504.7	+	32051.0i	0.0224940	+	0.0389608i	−0.854559	+	0.519354i	$0.826173\pi$
$908$		0			0
$909$		−	128743.i		−	0.155811i
$910$		0			0
$911$	−161359.			−0.194427			−0.0972134	−	0.995264i	$-0.530993\pi$
$911$	−161359.			−0.194427			−0.0972134	+	0.995264i	$0.530993\pi$
$912$		0			0
$913$	1.26723e6	−	731634.i	1.52024	−	0.877713i
$914$		0			0
$915$	−729.342	−	421.086i	−0.000871142	−	0.000502954i
$916$		0			0
$917$	12769.7	+	862.414i	0.0151859	+	0.00102560i
$918$		0			0
$919$	−405784.	+	702838.i	−0.480467	+	0.832194i	−0.999749	−	0.0224095i	$-0.992866\pi$
$919$	−405784.	+	702838.i	−0.480467	+	0.832194i	0.519282	+	0.854603i	$0.326200\pi$
$920$		0			0
$921$	9757.76	+	16900.9i	0.0115035	+	0.0199247i
$922$		0			0
$923$			704237.i			0.826639i
$924$		0			0
$925$	−54585.9			−0.0637965
$926$		0			0
$927$	96946.3	−	55972.0i	0.112816	−	0.0651345i
$928$		0			0
$929$	−168667.	−	97380.0i	−0.195434	−	0.112834i	0.399090	−	0.916912i	$-0.369326\pi$
$929$	−168667.	−	97380.0i	−0.195434	−	0.112834i	−0.594524	+	0.804078i	$0.702659\pi$
$930$		0			0
$931$	107383.	−	43981.3i	0.123890	−	0.0507421i
$932$		0			0
$933$	194852.	−	337494.i	0.223842	−	0.387706i
$934$		0			0
$935$	19415.0	+	33627.8i	0.0222082	+	0.0384658i
$936$		0			0
$937$			974675.i			1.11015i	0.831801	+	0.555074i	$0.187310\pi$
$937$			974675.i			1.11015i	−0.831801	+	0.555074i	$0.812690\pi$
$938$		0			0
$939$	−259088.			−0.293844
$940$		0			0
$941$	−1.50782e6	+	870541.i	−1.70283	+	0.983128i	−0.759953	+	0.649978i	$0.774778\pi$
$941$	−1.50782e6	+	870541.i	−1.70283	+	0.983128i	−0.942874	+	0.333150i	$0.891888\pi$
$942$		0			0
$943$	−657378.	−	379537.i	−0.739251	−	0.426807i
$944$		0			0
$945$	−561.037	+	8307.22i	−0.000628243	+	0.00930234i
$946$		0			0
$947$	566628.	−	981428.i	0.631827	−	1.09436i	−0.355351	−	0.934733i	$-0.615639\pi$
$947$	566628.	−	981428.i	0.631827	−	1.09436i	0.987178	−	0.159623i	$-0.0510279\pi$
$948$		0			0
$949$	−275614.	−	477378.i	−0.306034	−	0.530066i
$950$		0			0
$951$			165577.i			0.183079i
$952$		0			0
$953$	−1.63618e6			−1.80154			−0.900772	−	0.434293i	$-0.856998\pi$
$953$	−1.63618e6			−1.80154			−0.900772	+	0.434293i	$0.856998\pi$
$954$		0			0
$955$	−7766.83	+	4484.18i	−0.00851603	+	0.00491673i
$956$		0			0
$957$	33613.3	+	19406.6i	0.0367018	+	0.0211898i
$958$		0			0
$959$	465728.	+	312528.i	0.506401	+	0.339822i
$960$		0			0
$961$	−267598.	+	463493.i	−0.289758	+	0.501875i
$962$		0			0
$963$	98220.4	+	170123.i	0.105913	+	0.183447i
$964$		0			0
$965$		−	33444.2i		−	0.0359142i
$966$		0			0
$967$	464755.			0.497017			0.248508	−	0.968630i	$-0.420060\pi$
$967$	464755.			0.497017			0.248508	+	0.968630i	$0.420060\pi$
$968$		0			0
$969$	50612.8	−	29221.3i	0.0539030	−	0.0311209i
$970$		0			0
$971$	−1.15783e6	−	668475.i	−1.22803	−	0.709001i	−0.261409	−	0.965228i	$-0.584187\pi$
$971$	−1.15783e6	−	668475.i	−1.22803	−	0.709001i	−0.966617	+	0.256227i	$0.917521\pi$
$972$		0			0
$973$	127716.	+	260283.i	0.134902	+	0.274928i
$974$		0			0
$975$	173670.	−	300806.i	0.182691	−	0.316430i
$976$		0			0
$977$	886047.	+	1.53468e6i	0.928256	+	1.60779i	0.786239	+	0.617923i	$0.212026\pi$
$977$	886047.	+	1.53468e6i	0.928256	+	1.60779i	0.142017	+	0.989864i	$0.454641\pi$
$978$		0			0
$979$			1.10512e6i			1.15304i
$980$		0			0
$981$	336865.			0.350040
$982$		0			0
$983$	218434.	−	126113.i	0.226054	−	0.130512i	−0.382696	−	0.923874i	$-0.625004\pi$
$983$	218434.	−	126113.i	0.226054	−	0.130512i	0.608750	+	0.793362i	$0.291671\pi$
$984$		0			0
$985$	50263.3	+	29019.5i	0.0518058	+	0.0299101i
$986$		0			0
$987$	−23235.4	+	11401.2i	−0.0238515	+	0.0117035i
$988$		0			0
$989$	149301.	−	258597.i	0.152641	−	0.264382i
$990$		0			0
$991$	−214157.	−	370931.i	−0.218065	−	0.377699i	0.736151	−	0.676817i	$-0.236641\pi$
$991$	−214157.	−	370931.i	−0.218065	−	0.377699i	−0.954216	+	0.299117i	$0.903308\pi$
$992$		0			0
$993$			1.04735e6i			1.06217i
$994$		0			0
$995$	−78819.3			−0.0796135
$996$		0			0
$997$	1.38827e6	−	801516.i	1.39663	−	0.806347i	0.402596	−	0.915378i	$-0.368108\pi$
$997$	1.38827e6	−	801516.i	1.39663	−	0.806347i	0.994038	+	0.109031i	$0.0347747\pi$
$998$		0			0
$999$	−10636.5	−	6140.97i	−0.0106578	−	0.00615327i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.5.z.b.145.5	yes	16
3.2	odd	2		504.5.by.c.145.4		16
4.3	odd	2		336.5.bh.h.145.5		16
7.2	even	3		1176.5.f.a.97.12		16
7.3	odd	6	inner	168.5.z.b.73.5	✓	16
7.5	odd	6		1176.5.f.a.97.5		16
21.17	even	6		504.5.by.c.73.4		16
28.3	even	6		336.5.bh.h.241.5		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.5.z.b.73.5	✓	16	7.3	odd	6	inner
168.5.z.b.145.5	yes	16	1.1	even	1	trivial
336.5.bh.h.145.5		16	4.3	odd	2
336.5.bh.h.241.5		16	28.3	even	6
504.5.by.c.73.4		16	21.17	even	6
504.5.by.c.145.4		16	3.2	odd	2
1176.5.f.a.97.5		16	7.5	odd	6
1176.5.f.a.97.12		16	7.2	even	3

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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	168.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(4.50000 - 2.59808i) q^{3} +(-1.04890 - 0.605581i) q^{5} +(43.9897 - 21.5849i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})\)	\(q+(4.50000 - 2.59808i) q^{3} +(-1.04890 - 0.605581i) q^{5} +(43.9897 - 21.5849i) q^{7} +(13.5000 - 23.3827i) q^{9} +(-68.8822 - 119.308i) q^{11} +107.205i q^{13} -6.29338 q^{15} +(201.539 - 116.358i) q^{17} +(41.8553 + 24.1652i) q^{19} +(141.874 - 211.421i) q^{21} +(282.239 - 488.852i) q^{23} +(-311.767 - 539.995i) q^{25} -140.296i q^{27} -54.2202 q^{29} +(539.671 - 311.579i) q^{31} +(-619.940 - 357.923i) q^{33} +(-59.2120 - 3.99895i) q^{35} +(43.7715 - 75.8144i) q^{37} +(278.526 + 482.422i) q^{39} -1344.74i q^{41} +528.989 q^{43} +(-28.3202 + 16.3507i) q^{45} +(-88.0334 - 50.8261i) q^{47} +(1469.18 - 1899.03i) q^{49} +(604.616 - 1047.23i) q^{51} +(-1734.34 - 3003.96i) q^{53} +166.855i q^{55} +251.132 q^{57} +(1718.15 - 991.973i) q^{59} +(115.890 + 66.9093i) q^{61} +(89.1471 - 1319.99i) q^{63} +(64.9212 - 112.447i) q^{65} +(4032.57 + 6984.61i) q^{67} -2933.11i q^{69} +6569.08 q^{71} +(-4452.95 + 2570.91i) q^{73} +(-2805.90 - 1619.99i) q^{75} +(-5605.35 - 3761.48i) q^{77} +(-5470.09 + 9474.48i) q^{79} +(-364.500 - 631.333i) q^{81} +10621.5i q^{83} -281.858 q^{85} +(-243.991 + 140.868i) q^{87} +(-6947.10 - 4010.91i) q^{89} +(2314.01 + 4715.91i) q^{91} +(1619.01 - 2804.21i) q^{93} +(-29.2679 - 50.6935i) q^{95} +17996.3i q^{97} -3719.64 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9}+O(q^{10}) \) 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9	\( 16 q + 72 q^{3} - 12 q^{5} + 16 q^{7} + 216 q^{9} + 252 q^{11} - 72 q^{15} - 696 q^{17} + 156 q^{19} + 108 q^{21} - 672 q^{23} + 84 q^{25} + 1992 q^{29} + 2040 q^{31} + 2268 q^{33} + 2712 q^{35} + 2548 q^{37} - 396 q^{39} + 1304 q^{43} - 324 q^{45} - 744 q^{47} - 5608 q^{49} - 2088 q^{51} - 1164 q^{53} + 936 q^{57} + 8988 q^{59} + 816 q^{61} + 540 q^{63} + 8760 q^{65} + 3044 q^{67} - 4464 q^{71} - 15828 q^{73} + 756 q^{75} + 996 q^{77} - 11144 q^{79} - 5832 q^{81} - 15344 q^{85} + 8964 q^{87} + 22248 q^{89} + 1596 q^{91} + 6120 q^{93} + 3840 q^{95} + 13608 q^{99}+O(q^{100}) \) 16 * q + 72 * q^3 - 12 * q^5 + 16 * q^7 + 216 * q^9 + 252 * q^11 - 72 * q^15 - 696 * q^17 + 156 * q^19 + 108 * q^21 - 672 * q^23 + 84 * q^25 + 1992 * q^29 + 2040 * q^31 + 2268 * q^33 + 2712 * q^35 + 2548 * q^37 - 396 * q^39 + 1304 * q^43 - 324 * q^45 - 744 * q^47 - 5608 * q^49 - 2088 * q^51 - 1164 * q^53 + 936 * q^57 + 8988 * q^59 + 816 * q^61 + 540 * q^63 + 8760 * q^65 + 3044 * q^67 - 4464 * q^71 - 15828 * q^73 + 756 * q^75 + 996 * q^77 - 11144 * q^79 - 5832 * q^81 - 15344 * q^85 + 8964 * q^87 + 22248 * q^89 + 1596 * q^91 + 6120 * q^93 + 3840 * q^95 + 13608 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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