LMFDB - Embedded newform 168.5.z.a.145.1

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} - 6 x^{15} + 99 x^{14} - 1810 x^{13} + 14212 x^{12} - 199882 x^{11} + 1800935 x^{10} + \cdots + 41390114348800 $ x^16 - 6x^15 + 99x^14 - 1810x^13 + 14212x^12 - 199882x^11 + 1800935x^10 - 9073270x^9 + 62016647x^8 - 13497916x^7 - 872612906x^6 + 8208467240x^5 + 11144877252x^4 - 185824445600x^3 + 3750935164160x^2 - 13970712492800*x + 41390114348800
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{36}\cdot 7^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			145.1
Root			$-0.154925 - 7.21452i$ of defining polynomial
Character	$\chi$	$=$	168.145
Dual form			168.5.z.a.73.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 2.59808i) q^{3} +(-40.6814 - 23.4874i) q^{5} +(48.9386 - 2.45242i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})$	\(q+(-4.50000 + 2.59808i) q^{3} +(-40.6814 - 23.4874i) q^{5} +(48.9386 - 2.45242i) q^{7} +(13.5000 - 23.3827i) q^{9} +(-69.3346 - 120.091i) q^{11} +65.6883i q^{13} +244.088 q^{15} +(-229.572 + 132.544i) q^{17} +(385.133 + 222.356i) q^{19} +(-213.852 + 138.182i) q^{21} +(-212.083 + 367.338i) q^{23} +(790.818 + 1369.74i) q^{25} +140.296i q^{27} +580.814 q^{29} +(-1457.39 + 841.424i) q^{31} +(624.011 + 360.273i) q^{33} +(-2048.49 - 1049.67i) q^{35} +(762.580 - 1320.83i) q^{37} +(-170.663 - 295.598i) q^{39} +2590.55i q^{41} -1593.54 q^{43} +(-1098.40 + 634.160i) q^{45} +(2487.57 + 1436.20i) q^{47} +(2388.97 - 240.036i) q^{49} +(688.717 - 1192.89i) q^{51} +(1122.91 + 1944.94i) q^{53} +6513.96i q^{55} -2310.80 q^{57} +(416.680 - 240.570i) q^{59} +(4006.19 + 2312.97i) q^{61} +(603.327 - 1177.42i) q^{63} +(1542.85 - 2672.29i) q^{65} +(-2863.69 - 4960.06i) q^{67} -2204.03i q^{69} +3649.07 q^{71} +(-4734.10 + 2733.24i) q^{73} +(-7117.36 - 4109.21i) q^{75} +(-3687.65 - 5707.05i) q^{77} +(479.526 - 830.564i) q^{79} +(-364.500 - 631.333i) q^{81} -7572.28i q^{83} +12452.4 q^{85} +(-2613.67 + 1509.00i) q^{87} +(-3494.50 - 2017.55i) q^{89} +(161.095 + 3214.69i) q^{91} +(4372.17 - 7572.81i) q^{93} +(-10445.2 - 18091.5i) q^{95} +3801.42i q^{97} -3744.07 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9}+O(q^{10}) $ 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9	$ 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9} - 108 q^{11} - 72 q^{15} - 168 q^{17} + 948 q^{19} - 252 q^{21} - 768 q^{23} + 1908 q^{25} - 1608 q^{29} - 3216 q^{31} + 972 q^{33} + 696 q^{35} - 1820 q^{37} - 1188 q^{39} + 2888 q^{43} + 324 q^{45} + 744 q^{47} - 3784 q^{49} + 504 q^{51} + 4476 q^{53} - 5688 q^{57} - 4668 q^{59} + 17760 q^{61} + 1188 q^{63} + 8760 q^{65} + 1580 q^{67} + 48 q^{71} + 588 q^{73} - 17172 q^{75} - 17508 q^{77} - 3824 q^{79} - 5832 q^{81} + 11440 q^{85} + 7236 q^{87} - 360 q^{89} + 25860 q^{91} + 9648 q^{93} - 21792 q^{95} - 5832 q^{99}+O(q^{100}) $ 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9 - 108 * q^11 - 72 * q^15 - 168 * q^17 + 948 * q^19 - 252 * q^21 - 768 * q^23 + 1908 * q^25 - 1608 * q^29 - 3216 * q^31 + 972 * q^33 + 696 * q^35 - 1820 * q^37 - 1188 * q^39 + 2888 * q^43 + 324 * q^45 + 744 * q^47 - 3784 * q^49 + 504 * q^51 + 4476 * q^53 - 5688 * q^57 - 4668 * q^59 + 17760 * q^61 + 1188 * q^63 + 8760 * q^65 + 1580 * q^67 + 48 * q^71 + 588 * q^73 - 17172 * q^75 - 17508 * q^77 - 3824 * q^79 - 5832 * q^81 + 11440 * q^85 + 7236 * q^87 - 360 * q^89 + 25860 * q^91 + 9648 * q^93 - 21792 * q^95 - 5832 * 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
$5$	−40.6814	−	23.4874i	−1.62726	−	0.939497i	−0.984907	−	0.173085i	$-0.944627\pi$
$5$	−40.6814	−	23.4874i	−1.62726	−	0.939497i	−0.642349	−	0.766412i	$-0.722040\pi$
$6$		0			0
$7$	48.9386	−	2.45242i	0.998747	−	0.0500493i
$7$	48.9386	−	2.45242i	0.998747	−	0.0500493i
$8$		0			0
$9$	13.5000	−	23.3827i	0.166667	−	0.288675i
$10$		0			0
$11$	−69.3346	−	120.091i	−0.573013	−	0.992488i	−0.996254	−	0.0864706i	$-0.972441\pi$
$11$	−69.3346	−	120.091i	−0.573013	−	0.992488i	0.423241	−	0.906017i	$-0.360892\pi$
$12$		0			0
$13$			65.6883i			0.388688i	0.980933	+	0.194344i	$0.0622579\pi$
$13$			65.6883i			0.388688i	−0.980933	+	0.194344i	$0.937742\pi$
$14$		0			0
$15$	244.088			1.08484
$16$		0			0
$17$	−229.572	+	132.544i	−0.794368	+	0.458629i	−0.841498	−	0.540260i	$-0.818326\pi$
$17$	−229.572	+	132.544i	−0.794368	+	0.458629i	0.0471297	+	0.998889i	$0.484993\pi$
$18$		0			0
$19$	385.133	+	222.356i	1.06685	+	0.615946i	0.927319	−	0.374272i	$-0.122107\pi$
$19$	385.133	+	222.356i	1.06685	+	0.615946i	0.139530	+	0.990218i	$0.455441\pi$
$20$		0			0
$21$	−213.852	+	138.182i	−0.484925	+	0.313338i
$22$		0			0
$23$	−212.083	+	367.338i	−0.400912	+	0.694401i	−0.993836	−	0.110858i	$-0.964640\pi$
$23$	−212.083	+	367.338i	−0.400912	+	0.694401i	0.592924	+	0.805259i	$0.297974\pi$
$24$		0			0
$25$	790.818	+	1369.74i	1.26531	+	2.19158i
$26$		0			0
$27$			140.296i			0.192450i
$28$		0			0
$29$	580.814			0.690624			0.345312	−	0.938488i	$-0.387773\pi$
$29$	580.814			0.690624			0.345312	+	0.938488i	$0.387773\pi$
$30$		0			0
$31$	−1457.39	+	841.424i	−1.51653	+	0.875571i	−0.516722	+	0.856153i	$0.672848\pi$
$31$	−1457.39	+	841.424i	−1.51653	+	0.875571i	−0.999811	+	0.0194175i	$0.993819\pi$
$32$		0			0
$33$	624.011	+	360.273i	0.573013	+	0.330829i
$34$		0			0
$35$	−2048.49	−	1049.67i	−1.67224	−	0.856876i
$36$		0			0
$37$	762.580	−	1320.83i	0.557034	−	0.964812i	−0.440708	−	0.897651i	$-0.645273\pi$
$37$	762.580	−	1320.83i	0.557034	−	0.964812i	0.997742	−	0.0671610i	$-0.0213941\pi$
$38$		0			0
$39$	−170.663	−	295.598i	−0.112205	−	0.194344i
$40$		0			0
$41$			2590.55i			1.54108i	0.637392	+	0.770539i	$0.280013\pi$
$41$			2590.55i			1.54108i	−0.637392	+	0.770539i	$0.719987\pi$
$42$		0			0
$43$	−1593.54			−0.861840			−0.430920	−	0.902390i	$-0.641811\pi$
$43$	−1593.54			−0.861840			−0.430920	+	0.902390i	$0.641811\pi$
$44$		0			0
$45$	−1098.40	+	634.160i	−0.542419	+	0.313166i
$46$		0			0
$47$	2487.57	+	1436.20i	1.12611	+	0.650159i	0.942953	−	0.332925i	$-0.108036\pi$
$47$	2487.57	+	1436.20i	1.12611	+	0.650159i	0.183155	+	0.983084i	$0.441369\pi$
$48$		0			0
$49$	2388.97	−	240.036i	0.994990	−	0.0999732i
$50$		0			0
$51$	688.717	−	1192.89i	0.264789	−	0.458629i
$52$		0			0
$53$	1122.91	+	1944.94i	0.399755	+	0.692395i	0.993695	−	0.112113i	$-0.0357620\pi$
$53$	1122.91	+	1944.94i	0.399755	+	0.692395i	−0.593941	+	0.804509i	$0.702429\pi$
$54$		0			0
$55$			6513.96i			2.15338i
$56$		0			0
$57$	−2310.80			−0.711233
$58$		0			0
$59$	416.680	−	240.570i	0.119701	−	0.0691095i	−0.438954	−	0.898510i	$-0.644651\pi$
$59$	416.680	−	240.570i	0.119701	−	0.0691095i	0.558655	+	0.829400i	$0.311318\pi$
$60$		0			0
$61$	4006.19	+	2312.97i	1.07664	+	0.621600i	0.929989	−	0.367588i	$-0.119816\pi$
$61$	4006.19	+	2312.97i	1.07664	+	0.621600i	0.146654	+	0.989188i	$0.453150\pi$
$62$		0			0
$63$	603.327	−	1177.42i	0.152010	−	0.296655i
$64$		0			0
$65$	1542.85	−	2672.29i	0.365171	−	0.632496i
$66$		0			0
$67$	−2863.69	−	4960.06i	−0.637935	−	1.10494i	−0.985885	−	0.167423i	$-0.946455\pi$
$67$	−2863.69	−	4960.06i	−0.637935	−	1.10494i	0.347950	−	0.937513i	$-0.386878\pi$
$68$		0			0
$69$		−	2204.03i		−	0.462934i
$70$		0			0
$71$	3649.07			0.723878			0.361939	−	0.932202i	$-0.382115\pi$
$71$	3649.07			0.723878			0.361939	+	0.932202i	$0.382115\pi$
$72$		0			0
$73$	−4734.10	+	2733.24i	−0.888366	+	0.512898i	−0.873408	−	0.486990i	$-0.838095\pi$
$73$	−4734.10	+	2733.24i	−0.888366	+	0.512898i	−0.0149583	+	0.999888i	$0.504762\pi$
$74$		0			0
$75$	−7117.36	−	4109.21i	−1.26531	−	0.730526i
$76$		0			0
$77$	−3687.65	−	5707.05i	−0.621968	−	0.962565i
$78$		0			0
$79$	479.526	−	830.564i	0.0768348	−	0.133082i	−0.825048	−	0.565063i	$-0.808852\pi$
$79$	479.526	−	830.564i	0.0768348	−	0.133082i	0.901883	+	0.431981i	$0.142185\pi$
$80$		0			0
$81$	−364.500	−	631.333i	−0.0555556	−	0.0962250i
$82$		0			0
$83$		−	7572.28i		−	1.09918i	−0.835433	−	0.549592i	$-0.814783\pi$
$83$		−	7572.28i		−	1.09918i	0.835433	−	0.549592i	$-0.185217\pi$
$84$		0			0
$85$	12452.4			1.72352
$86$		0			0
$87$	−2613.67	+	1509.00i	−0.345312	+	0.199366i
$88$		0			0
$89$	−3494.50	−	2017.55i	−0.441169	−	0.254709i	0.262925	−	0.964816i	$-0.415313\pi$
$89$	−3494.50	−	2017.55i	−0.441169	−	0.254709i	−0.704093	+	0.710108i	$0.748646\pi$
$90$		0			0
$91$	161.095	+	3214.69i	0.0194536	+	0.388201i
$92$		0			0
$93$	4372.17	−	7572.81i	0.505511	−	0.875571i
$94$		0			0
$95$	−10445.2	−	18091.5i	−1.15736	−	2.00460i
$96$		0			0
$97$			3801.42i			0.404019i	0.979384	+	0.202010i	$0.0647473\pi$
$97$			3801.42i			0.404019i	−0.979384	+	0.202010i	$0.935253\pi$
$98$		0			0
$99$	−3744.07			−0.382009
$100$		0			0
$101$	−13079.8	+	7551.64i	−1.28221	+	0.740284i	−0.977252	−	0.212082i	$-0.931976\pi$
$101$	−13079.8	+	7551.64i	−1.28221	+	0.740284i	−0.304957	+	0.952366i	$0.598642\pi$
$102$		0			0
$103$	2312.77	+	1335.28i	0.218001	+	0.125863i	0.605024	−	0.796207i	$-0.293163\pi$
$103$	2312.77	+	1335.28i	0.218001	+	0.125863i	−0.387023	+	0.922070i	$0.626497\pi$
$104$		0			0
$105$	11945.3	−	598.607i	1.08348	−	0.0542954i
$106$		0			0
$107$	−4238.44	+	7341.19i	−0.370202	+	0.641208i	−0.989596	−	0.143872i	$-0.954045\pi$
$107$	−4238.44	+	7341.19i	−0.370202	+	0.641208i	0.619395	+	0.785080i	$0.287378\pi$
$108$		0			0
$109$	7061.17	+	12230.3i	0.594325	+	1.02940i	0.993642	+	0.112588i	$0.0359139\pi$
$109$	7061.17	+	12230.3i	0.594325	+	1.02940i	−0.399317	+	0.916813i	$0.630753\pi$
$110$		0			0
$111$			7924.96i			0.643208i
$112$		0			0
$113$	12740.3			0.997754			0.498877	−	0.866673i	$-0.333746\pi$
$113$	12740.3			0.997754			0.498877	+	0.866673i	$0.333746\pi$
$114$		0			0
$115$	17255.6	−	9962.55i	1.30477	−	0.753312i
$116$		0			0
$117$	1535.97	+	886.793i	0.112205	+	0.0647814i
$118$		0			0
$119$	−10909.9	+	7049.51i	−0.770419	+	0.497812i
$120$		0			0
$121$	−2294.06	+	3973.44i	−0.156688	+	0.271391i
$122$		0			0
$123$	−6730.46	−	11657.5i	−0.444871	−	0.770539i
$124$		0			0
$125$		−	44937.8i		−	2.87602i
$126$		0			0
$127$	−2932.80			−0.181834			−0.0909170	−	0.995858i	$-0.528980\pi$
$127$	−2932.80			−0.181834			−0.0909170	+	0.995858i	$0.528980\pi$
$128$		0			0
$129$	7170.94	−	4140.14i	0.430920	−	0.248792i
$130$		0			0
$131$	−17683.4	−	10209.5i	−1.03044	−	0.594925i	−0.113330	−	0.993557i	$-0.536152\pi$
$131$	−17683.4	−	10209.5i	−1.03044	−	0.594925i	−0.917111	+	0.398632i	$0.869485\pi$
$132$		0			0
$133$	19393.2	+	9937.30i	1.09634	+	0.561779i
$134$		0			0
$135$	3295.19	−	5707.44i	0.180806	−	0.313166i
$136$		0			0
$137$	−3177.42	−	5503.45i	−0.169291	−	0.293220i	0.768880	−	0.639393i	$-0.220814\pi$
$137$	−3177.42	−	5503.45i	−0.169291	−	0.293220i	−0.938171	+	0.346173i	$0.887481\pi$
$138$		0			0
$139$		−	294.000i		−	0.0152166i	−0.999971	−	0.00760830i	$-0.997578\pi$
$139$		−	294.000i		−	0.0152166i	0.999971	−	0.00760830i	$-0.00242182\pi$
$140$		0			0
$141$	−14925.4			−0.750739
$142$		0			0
$143$	7888.58	−	4554.47i	0.385768	−	0.222723i
$144$		0			0
$145$	−23628.3	−	13641.8i	−1.12382	−	0.648839i
$146$		0			0
$147$	−10126.7	+	7286.89i	−0.468635	+	0.337216i
$148$		0			0
$149$	−16664.0	+	28862.9i	−0.750598	+	1.30007i	0.196936	+	0.980416i	$0.436901\pi$
$149$	−16664.0	+	28862.9i	−0.750598	+	1.30007i	−0.947533	+	0.319657i	$0.896432\pi$
$150$		0			0
$151$	−4033.78	−	6986.71i	−0.176912	−	0.306421i	0.763909	−	0.645324i	$-0.223278\pi$
$151$	−4033.78	−	6986.71i	−0.176912	−	0.306421i	−0.940821	+	0.338903i	$0.889944\pi$
$152$		0			0
$153$			7157.36i			0.305753i
$154$		0			0
$155$	79051.5			3.29038
$156$		0			0
$157$	−18560.0	+	10715.6i	−0.752971	+	0.434728i	−0.826766	−	0.562546i	$-0.809822\pi$
$157$	−18560.0	+	10715.6i	−0.752971	+	0.434728i	0.0737955	+	0.997273i	$0.476489\pi$
$158$		0			0
$159$	−10106.2	−	5834.81i	−0.399755	−	0.230798i
$160$		0			0
$161$	−9478.16	+	18497.1i	−0.365656	+	0.713596i
$162$		0			0
$163$	−14264.5	+	24706.8i	−0.536883	+	0.929909i	0.462186	+	0.886783i	$0.347065\pi$
$163$	−14264.5	+	24706.8i	−0.536883	+	0.929909i	−0.999070	+	0.0431264i	$0.986268\pi$
$164$		0			0
$165$	−16923.8	−	29312.8i	−0.621626	−	1.07669i
$166$		0			0
$167$			15947.1i			0.571808i	0.958258	+	0.285904i	$0.0922938\pi$
$167$			15947.1i			0.571808i	−0.958258	+	0.285904i	$0.907706\pi$
$168$		0			0
$169$	24246.0			0.848921
$170$		0			0
$171$	10398.6	−	6003.62i	0.355616	−	0.205315i
$172$		0			0
$173$	21622.5	+	12483.8i	0.722460	+	0.417113i	0.815658	−	0.578535i	$-0.196375\pi$
$173$	21622.5	+	12483.8i	0.722460	+	0.417113i	−0.0931972	+	0.995648i	$0.529709\pi$
$174$		0			0
$175$	42060.7	+	65093.6i	1.37341	+	2.12550i
$176$		0			0
$177$	−1250.04	+	2165.13i	−0.0399004	+	0.0691095i
$178$		0			0
$179$	19159.0	+	33184.4i	0.597954	+	1.03569i	0.993123	+	0.117078i	$0.0373527\pi$
$179$	19159.0	+	33184.4i	0.597954	+	1.03569i	−0.395169	+	0.918608i	$0.629314\pi$
$180$		0			0
$181$			54508.3i			1.66382i	0.554914	+	0.831908i	$0.312751\pi$
$181$			54508.3i			1.66382i	−0.554914	+	0.831908i	$0.687249\pi$
$182$		0			0
$183$	−24037.1			−0.717762
$184$		0			0
$185$	−62045.6	+	35822.1i	−1.81287	+	1.04666i
$186$		0			0
$187$	31834.6	+	18379.7i	0.910367	+	0.525601i
$188$		0			0
$189$	344.065	+	6865.89i	0.00963200	+	0.192209i
$190$		0			0
$191$	8226.95	−	14249.5i	0.225513	−	0.390601i	−0.730960	−	0.682420i	$-0.760927\pi$
$191$	8226.95	−	14249.5i	0.225513	−	0.390601i	0.956473	+	0.291820i	$0.0942607\pi$
$192$		0			0
$193$	−17451.0	−	30226.1i	−0.468497	−	0.811460i	0.530855	−	0.847463i	$-0.321871\pi$
$193$	−17451.0	−	30226.1i	−0.468497	−	0.811460i	−0.999352	+	0.0360023i	$0.988538\pi$
$194$		0			0
$195$			16033.8i			0.421664i
$196$		0			0
$197$	−39922.2			−1.02868			−0.514341	−	0.857585i	$-0.671964\pi$
$197$	−39922.2			−1.02868			−0.514341	+	0.857585i	$0.671964\pi$
$198$		0			0
$199$	−38419.8	+	22181.7i	−0.970172	+	0.560129i	−0.899289	−	0.437356i	$-0.855915\pi$
$199$	−38419.8	+	22181.7i	−0.970172	+	0.560129i	−0.0708834	+	0.997485i	$0.522582\pi$
$200$		0			0
$201$	25773.2	+	14880.2i	0.637935	+	0.368312i
$202$		0			0
$203$	28424.2	−	1424.40i	0.689758	−	0.0345653i
$204$		0			0
$205$	60845.4	−	105387.i	1.44784	−	2.50773i
$206$		0			0
$207$	5726.23	+	9918.13i	0.133637	+	0.231467i
$208$		0			0
$209$		−	61667.9i		−	1.41178i
$210$		0			0
$211$	−55733.4			−1.25185			−0.625923	−	0.779885i	$-0.715278\pi$
$211$	−55733.4			−1.25185			−0.625923	+	0.779885i	$0.715278\pi$
$212$		0			0
$213$	−16420.8	+	9480.55i	−0.361939	+	0.208965i
$214$		0			0
$215$	64827.5	+	37428.2i	1.40243	+	0.809696i
$216$		0			0
$217$	−69259.0	+	44752.2i	−1.47081	+	0.950375i
$218$		0			0
$219$	14202.3	−	24599.1i	0.296122	−	0.512898i
$220$		0			0
$221$	−8706.58	−	15080.2i	−0.178264	−	0.308762i
$222$		0			0
$223$		−	7505.09i		−	0.150920i	−0.997149	−	0.0754599i	$-0.975958\pi$
$223$		−	7505.09i		−	0.150920i	0.997149	−	0.0754599i	$-0.0240425\pi$
$224$		0			0
$225$	42704.2			0.843539
$226$		0			0
$227$	−51943.4	+	29989.5i	−1.00804	+	0.581993i	−0.910618	−	0.413250i	$-0.864394\pi$
$227$	−51943.4	+	29989.5i	−1.00804	+	0.581993i	−0.0974237	+	0.995243i	$0.531060\pi$
$228$		0			0
$229$	24401.5	+	14088.2i	0.465313	+	0.268648i	0.714276	−	0.699865i	$-0.246756\pi$
$229$	24401.5	+	14088.2i	0.465313	+	0.268648i	−0.248963	+	0.968513i	$0.580090\pi$
$230$		0			0
$231$	31421.8	+	16100.9i	0.588853	+	0.301736i
$232$		0			0
$233$	−18550.9	+	32131.1i	−0.341706	+	0.591852i	−0.984750	−	0.173977i	$-0.944338\pi$
$233$	−18550.9	+	32131.1i	−0.341706	+	0.591852i	0.643044	+	0.765830i	$0.277671\pi$
$234$		0			0
$235$	−67465.3	−	116853.i	−1.22164	−	2.11595i
$236$		0			0
$237$			4983.38i			0.0887212i
$238$		0			0
$239$	34583.8			0.605448			0.302724	−	0.953078i	$-0.402104\pi$
$239$	34583.8			0.605448			0.302724	+	0.953078i	$0.402104\pi$
$240$		0			0
$241$	55889.0	−	32267.5i	0.962260	−	0.555561i	0.0653919	−	0.997860i	$-0.479170\pi$
$241$	55889.0	−	32267.5i	0.962260	−	0.555561i	0.896868	+	0.442299i	$0.145837\pi$
$242$		0			0
$243$	3280.50	+	1894.00i	0.0555556	+	0.0320750i
$244$		0			0
$245$	−102825.	−	46345.8i	−1.71303	−	0.772108i
$246$		0			0
$247$	−14606.2	+	25298.7i	−0.239411	+	0.414672i
$248$		0			0
$249$	19673.4	+	34075.3i	0.317307	+	0.549592i
$250$		0			0
$251$		−	109204.i		−	1.73337i	−0.498856	−	0.866685i	$-0.666246\pi$
$251$		−	109204.i		−	1.73337i	0.498856	−	0.866685i	$-0.333754\pi$
$252$		0			0
$253$	58818.6			0.918912
$254$		0			0
$255$	−56036.0	+	32352.4i	−0.861761	+	0.497538i
$256$		0			0
$257$	25933.1	+	14972.5i	0.392635	+	0.226688i	0.683301	−	0.730137i	$-0.260544\pi$
$257$	25933.1	+	14972.5i	0.392635	+	0.226688i	−0.290666	+	0.956824i	$0.593877\pi$
$258$		0			0
$259$	34080.4	−	66509.6i	0.508048	−	0.991482i
$260$		0			0
$261$	7841.00	−	13581.0i	0.115104	−	0.199366i
$262$		0			0
$263$	33999.4	+	58888.6i	0.491540	+	0.851373i	0.999953	−	0.00974099i	$-0.00310070\pi$
$263$	33999.4	+	58888.6i	0.491540	+	0.851373i	−0.508412	+	0.861114i	$0.669767\pi$
$264$		0			0
$265$		−	105497.i		−	1.50227i
$266$		0			0
$267$	20967.0			0.294112
$268$		0			0
$269$	−83890.6	+	48434.3i	−1.15934	+	0.669342i	−0.951145	−	0.308746i	$-0.900091\pi$
$269$	−83890.6	+	48434.3i	−1.15934	+	0.669342i	−0.208191	+	0.978088i	$0.566757\pi$
$270$		0			0
$271$	6201.24	+	3580.29i	0.0844384	+	0.0487505i	0.541625	−	0.840620i	$-0.317809\pi$
$271$	6201.24	+	3580.29i	0.0844384	+	0.0487505i	−0.457186	+	0.889371i	$0.651143\pi$
$272$		0			0
$273$	−9076.95	−	14047.6i	−0.121791	−	0.188485i
$274$		0			0
$275$	109662.	−	189940.i	1.45008	−	2.51161i
$276$		0			0
$277$	−17556.7	−	30409.1i	−0.228815	−	0.396319i	0.728642	−	0.684894i	$-0.240152\pi$
$277$	−17556.7	−	30409.1i	−0.228815	−	0.396319i	−0.957457	+	0.288576i	$0.906818\pi$
$278$		0			0
$279$			45436.9i			0.583714i
$280$		0			0
$281$	59670.9			0.755701			0.377851	−	0.925867i	$-0.376663\pi$
$281$	59670.9			0.755701			0.377851	+	0.925867i	$0.376663\pi$
$282$		0			0
$283$	−78242.8	+	45173.5i	−0.976948	+	0.564041i	−0.901347	−	0.433097i	$-0.857421\pi$
$283$	−78242.8	+	45173.5i	−0.976948	+	0.564041i	−0.0756005	+	0.997138i	$0.524087\pi$
$284$		0			0
$285$	94006.4	+	54274.6i	1.15736	+	0.668201i
$286$		0			0
$287$	6353.12	+	126778.i	0.0771300	+	1.53915i
$288$		0			0
$289$	−6624.82	+	11474.5i	−0.0793192	+	0.137385i
$290$		0			0
$291$	−9876.38	−	17106.4i	−0.116630	−	0.202010i
$292$		0			0
$293$			134578.i			1.56761i	0.621007	+	0.783805i	$0.286724\pi$
$293$			134578.i			1.56761i	−0.621007	+	0.783805i	$0.713276\pi$
$294$		0			0
$295$	−22601.5			−0.259713
$296$		0			0
$297$	16848.3	−	9727.37i	0.191004	−	0.110276i
$298$		0			0
$299$	−24129.8	−	13931.4i	−0.269905	−	0.155830i
$300$		0			0
$301$	−77985.7	+	3908.03i	−0.860760	+	0.0431345i
$302$		0			0
$303$	39239.5	−	67964.7i	0.427403	−	0.740284i
$304$		0			0
$305$	−108652.	−	188190.i	−1.16798	−	2.02300i
$306$		0			0
$307$		−	121419.i		−	1.28827i	−0.764910	−	0.644137i	$-0.777217\pi$
$307$		−	121419.i		−	1.28827i	0.764910	−	0.644137i	$-0.222783\pi$
$308$		0			0
$309$	−13876.6			−0.145334
$310$		0			0
$311$	68474.8	−	39533.9i	0.707962	−	0.408742i	−0.102344	−	0.994749i	$-0.532634\pi$
$311$	68474.8	−	39533.9i	0.707962	−	0.408742i	0.810306	+	0.586007i	$0.199301\pi$
$312$		0			0
$313$	−113889.	−	65754.1i	−1.16251	−	0.671173i	−0.210602	−	0.977572i	$-0.567542\pi$
$313$	−113889.	−	65754.1i	−1.16251	−	0.671173i	−0.951903	+	0.306399i	$0.900876\pi$
$314$		0			0
$315$	−52198.8	+	33728.6i	−0.526065	+	0.339921i
$316$		0			0
$317$	21490.4	−	37222.5i	0.213858	−	0.370413i	−0.739061	−	0.673639i	$-0.764730\pi$
$317$	21490.4	−	37222.5i	0.213858	−	0.370413i	0.952919	+	0.303226i	$0.0980636\pi$
$318$		0			0
$319$	−40270.5	−	69750.6i	−0.395736	−	0.685435i
$320$		0			0
$321$		−	44047.2i		−	0.427472i
$322$		0			0
$323$	−117888.			−1.12996
$324$		0			0
$325$	−89975.7	+	51947.5i	−0.851841	+	0.491811i
$326$		0			0
$327$	−63550.5	−	36690.9i	−0.594325	−	0.343133i
$328$		0			0
$329$	125261.	+	64185.1i	1.15724	+	0.592983i
$330$		0			0
$331$	41286.5	−	71510.3i	0.376836	−	0.652699i	−0.613764	−	0.789489i	$-0.710345\pi$
$331$	41286.5	−	71510.3i	0.376836	−	0.652699i	0.990600	+	0.136791i	$0.0436787\pi$
$332$		0			0
$333$	−20589.7	−	35662.3i	−0.185678	−	0.321604i
$334$		0			0
$335$			269043.i			2.39735i
$336$		0			0
$337$	119721.			1.05417			0.527084	−	0.849813i	$-0.323285\pi$
$337$	119721.			1.05417			0.527084	+	0.849813i	$0.323285\pi$
$338$		0			0
$339$	−57331.5	+	33100.3i	−0.498877	+	0.288027i
$340$		0			0
$341$	202095.	+	116679.i	1.73799	+	1.00343i
$342$		0			0
$343$	116324.	−	17605.8i	0.988740	−	0.149647i
$344$		0			0
$345$	−51766.9	+	89662.9i	−0.434925	+	0.753312i
$346$		0			0
$347$	−90181.4	−	156199.i	−0.748959	−	1.29724i	−0.948322	−	0.317310i	$-0.897221\pi$
$347$	−90181.4	−	156199.i	−0.748959	−	1.29724i	0.199363	−	0.979926i	$-0.436113\pi$
$348$		0			0
$349$		−	167994.i		−	1.37925i	−0.724166	−	0.689625i	$-0.757775\pi$
$349$		−	167994.i		−	1.37925i	0.724166	−	0.689625i	$-0.242225\pi$
$350$		0			0
$351$	−9215.82			−0.0748031
$352$		0			0
$353$	69258.6	−	39986.5i	0.555807	−	0.320896i	−0.195654	−	0.980673i	$-0.562683\pi$
$353$	69258.6	−	39986.5i	0.555807	−	0.320896i	0.751461	+	0.659778i	$0.229349\pi$
$354$		0			0
$355$	−148449.	−	85707.2i	−1.17793	−	0.680081i
$356$		0			0
$357$	30779.4	−	60067.6i	0.241504	−	0.471307i
$358$		0			0
$359$	−1186.90	+	2055.77i	−0.00920925	+	0.0159509i	−0.870593	−	0.492003i	$-0.836265\pi$
$359$	−1186.90	+	2055.77i	−0.00920925	+	0.0159509i	0.861384	+	0.507954i	$0.169598\pi$
$360$		0			0
$361$	33724.2	+	58412.1i	0.258778	+	0.448217i
$362$		0			0
$363$		−	23840.6i		−	0.180927i
$364$		0			0
$365$	256787.			1.92747
$366$		0			0
$367$	−144688.	+	83535.5i	−1.07424	+	0.620211i	−0.929336	−	0.369236i	$-0.879620\pi$
$367$	−144688.	+	83535.5i	−1.07424	+	0.620211i	−0.144900	+	0.989446i	$0.546286\pi$
$368$		0			0
$369$	60574.1	+	34972.5i	0.444871	+	0.256846i
$370$		0			0
$371$	59723.5	+	92428.7i	0.433908	+	0.671520i
$372$		0			0
$373$	−78324.5	+	135662.i	−0.562963	+	0.975080i	0.434273	+	0.900781i	$0.357005\pi$
$373$	−78324.5	+	135662.i	−0.562963	+	0.975080i	−0.997236	+	0.0742991i	$0.976328\pi$
$374$		0			0
$375$	116752.	+	202220.i	0.830235	+	1.43801i
$376$		0			0
$377$			38152.7i			0.268437i
$378$		0			0
$379$	89789.6			0.625098			0.312549	−	0.949902i	$-0.398817\pi$
$379$	89789.6			0.625098			0.312549	+	0.949902i	$0.398817\pi$
$380$		0			0
$381$	13197.6	−	7619.64i	0.0909170	−	0.0524910i
$382$		0			0
$383$	−19546.7	−	11285.3i	−0.133253	−	0.0769335i	0.431892	−	0.901925i	$-0.357846\pi$
$383$	−19546.7	−	11285.3i	−0.133253	−	0.0769335i	−0.565144	+	0.824992i	$0.691180\pi$
$384$		0			0
$385$	15974.9	+	318784.i	0.107775	+	2.15068i
$386$		0			0
$387$	−21512.8	+	37261.3i	−0.143640	+	0.248792i
$388$		0			0
$389$	49537.7	+	85801.8i	0.327368	+	0.567019i	0.981989	−	0.188939i	$-0.0605049\pi$
$389$	49537.7	+	85801.8i	0.327368	+	0.567019i	−0.654620	+	0.755958i	$0.727172\pi$
$390$		0			0
$391$		−	112441.i		−	0.735480i
$392$		0			0
$393$	106100.			0.686960
$394$		0			0
$395$	−39015.6	+	22525.7i	−0.250060	+	0.144372i
$396$		0			0
$397$	−131155.	−	75722.3i	−0.832153	−	0.480444i	0.0224361	−	0.999748i	$-0.492858\pi$
$397$	−131155.	−	75722.3i	−0.832153	−	0.480444i	−0.854589	+	0.519304i	$0.826191\pi$
$398$		0			0
$399$	−113087.	+	5667.03i	−0.710342	+	0.0355967i
$400$		0			0
$401$	−70987.5	+	122954.i	−0.441462	+	0.764634i	−0.997798	−	0.0663231i	$-0.978873\pi$
$401$	−70987.5	+	122954.i	−0.441462	+	0.764634i	0.556337	+	0.830957i	$0.312207\pi$
$402$		0			0
$403$	−55271.7	−	95733.4i	−0.340324	−	0.589459i
$404$		0			0
$405$			34244.7i			0.208777i
$406$		0			0
$407$	−211493.			−1.27675
$408$		0			0
$409$	272578.	−	157373.i	1.62946	−	0.940770i	0.645208	−	0.764007i	$-0.276771\pi$
$409$	272578.	−	157373.i	1.62946	−	0.940770i	0.984253	−	0.176764i	$-0.0565628\pi$
$410$		0			0
$411$	28596.8	+	16510.4i	0.169291	+	0.0977401i
$412$		0			0
$413$	19801.7	−	12795.0i	0.116092	−	0.0750139i
$414$		0			0
$415$	−177853.	+	308051.i	−1.03268	+	1.78865i
$416$		0			0
$417$	763.834	+	1323.00i	0.00439265	+	0.00760830i
$418$		0			0
$419$			277371.i			1.57991i	0.613165	+	0.789955i	$0.289896\pi$
$419$			277371.i			1.57991i	−0.613165	+	0.789955i	$0.710104\pi$
$420$		0			0
$421$	142610.			0.804608			0.402304	−	0.915506i	$-0.368209\pi$
$421$	142610.			0.804608			0.402304	+	0.915506i	$0.368209\pi$
$422$		0			0
$423$	67164.5	−	38777.4i	0.375370	−	0.216720i
$424$		0			0
$425$	−363100.	−	209636.i	−2.01024	−	1.16061i
$426$		0			0
$427$	201730.	+	103369.i	1.10640	+	0.566936i
$428$		0			0
$429$	−23665.7	+	40990.3i	−0.128589	+	0.222723i
$430$		0			0
$431$	−90347.8	−	156487.i	−0.486366	−	0.842410i	0.513511	−	0.858083i	$-0.328344\pi$
$431$	−90347.8	−	156487.i	−0.486366	−	0.842410i	−0.999877	+	0.0156724i	$0.995011\pi$
$432$		0			0
$433$		−	6338.24i		−	0.0338059i	−0.999857	−	0.0169030i	$-0.994619\pi$
$433$		−	6338.24i		−	0.0338059i	0.999857	−	0.0169030i	$-0.00538064\pi$
$434$		0			0
$435$	141770.			0.749214
$436$		0			0
$437$	−163360.	+	94315.9i	−0.855426	+	0.493881i
$438$		0			0
$439$	39718.8	+	22931.7i	0.206095	+	0.118989i	0.599495	−	0.800378i	$-0.295368\pi$
$439$	39718.8	+	22931.7i	0.206095	+	0.118989i	−0.393400	+	0.919367i	$0.628701\pi$
$440$		0			0
$441$	26638.4	−	59101.0i	0.136972	−	0.303891i
$442$		0			0
$443$	78324.8	−	135663.i	0.399109	−	0.691278i	−0.594507	−	0.804091i	$-0.702653\pi$
$443$	78324.8	−	135663.i	0.399109	−	0.691278i	0.993616	+	0.112813i	$0.0359860\pi$
$444$		0			0
$445$	94774.0	+	164153.i	0.478596	+	0.828953i
$446$		0			0
$447$		−	173178.i		−	0.866716i
$448$		0			0
$449$	−288769.			−1.43238			−0.716190	−	0.697906i	$-0.754116\pi$
$449$	−288769.			−1.43238			−0.716190	+	0.697906i	$0.754116\pi$
$450$		0			0
$451$	311102.	−	179615.i	1.52950	−	0.883058i
$452$		0			0
$453$	36304.0	+	20960.1i	0.176912	+	0.102140i
$454$		0			0
$455$	68951.3	−	134562.i	0.333058	−	0.649979i
$456$		0			0
$457$	−203187.	+	351930.i	−0.972888	+	1.68509i	−0.286152	+	0.958184i	$0.592376\pi$
$457$	−203187.	+	351930.i	−0.972888	+	1.68509i	−0.686736	+	0.726907i	$0.740957\pi$
$458$		0			0
$459$	−18595.4	−	32208.1i	−0.0882632	−	0.152876i
$460$		0			0
$461$		−	155423.i		−	0.731328i	−0.930747	−	0.365664i	$-0.880842\pi$
$461$		−	155423.i		−	0.731328i	0.930747	−	0.365664i	$-0.119158\pi$
$462$		0			0
$463$	22511.1			0.105011			0.0525055	−	0.998621i	$-0.483279\pi$
$463$	22511.1			0.105011			0.0525055	+	0.998621i	$0.483279\pi$
$464$		0			0
$465$	−355732.	+	205382.i	−1.64519	+	0.949852i
$466$		0			0
$467$	−78632.7	−	45398.6i	−0.360553	−	0.208166i	0.308770	−	0.951137i	$-0.400083\pi$
$467$	−78632.7	−	45398.6i	−0.360553	−	0.208166i	−0.669323	+	0.742971i	$0.733416\pi$
$468$		0			0
$469$	−152309.	−	235715.i	−0.692437	−	1.07162i
$470$		0			0
$471$	55679.9	−	96440.5i	0.250990	−	0.434728i
$472$		0			0
$473$	110488.	+	191370.i	0.493846	+	0.855366i
$474$		0			0
$475$			703374.i			3.11744i
$476$		0			0
$477$	60637.2			0.266503
$478$		0			0
$479$	47778.2	−	27584.8i	0.208238	−	0.120226i	−0.392254	−	0.919857i	$-0.628305\pi$
$479$	47778.2	−	27584.8i	0.208238	−	0.120226i	0.600492	+	0.799631i	$0.294971\pi$
$480$		0			0
$481$	86762.9	+	50092.6i	0.375011	+	0.216513i
$482$		0			0
$483$	−5405.20	−	107862.i	−0.0231695	−	0.462354i
$484$		0			0
$485$	89285.5	−	154647.i	0.379575	−	0.657443i
$486$		0			0
$487$	−39283.5	−	68041.1i	−0.165635	−	0.286889i	0.771245	−	0.636538i	$-0.219634\pi$
$487$	−39283.5	−	68041.1i	−0.165635	−	0.286889i	−0.936881	+	0.349649i	$0.886301\pi$
$488$		0			0
$489$		−	148241.i		−	0.619940i
$490$		0			0
$491$	1639.55			0.00680081			0.00340040	−	0.999994i	$-0.498918\pi$
$491$	1639.55			0.00680081			0.00340040	+	0.999994i	$0.498918\pi$
$492$		0			0
$493$	−133339.	+	76983.3i	−0.548610	+	0.316740i
$494$		0			0
$495$	152314.	+	87938.5i	0.621626	+	0.358896i
$496$		0			0
$497$	178580.	−	8949.03i	0.722970	−	0.0362296i
$498$		0			0
$499$	−31092.7	+	53854.2i	−0.124870	+	0.216281i	−0.921682	−	0.387946i	$-0.873185\pi$
$499$	−31092.7	+	53854.2i	−0.124870	+	0.216281i	0.796812	+	0.604227i	$0.206518\pi$
$500$		0			0
$501$	−41431.9	−	71762.1i	−0.165067	−	0.285904i
$502$		0			0
$503$			192285.i			0.759994i	0.924988	+	0.379997i	$0.124075\pi$
$503$			192285.i			0.759994i	−0.924988	+	0.379997i	$0.875925\pi$
$504$		0			0
$505$	709474.			2.78198
$506$		0			0
$507$	−109107.	+	62993.1i	−0.424461	+	0.245062i
$508$		0			0
$509$	385164.	+	222374.i	1.48665	+	0.858320i	0.999884	−	0.0152104i	$-0.00484179\pi$
$509$	385164.	+	222374.i	1.48665	+	0.858320i	0.486770	+	0.873530i	$0.338175\pi$
$510$		0			0
$511$	−224977.	+	145371.i	−0.861582	+	0.556718i
$512$		0			0
$513$	−31195.7	+	54032.6i	−0.118539	+	0.205315i
$514$		0			0
$515$	−62724.5	−	108642.i	−0.236496	−	0.409622i
$516$		0			0
$517$		−	398314.i		−	1.49020i
$518$		0			0
$519$	−129735.			−0.481640
$520$		0			0
$521$	−80812.6	+	46657.2i	−0.297717	+	0.171887i	−0.641417	−	0.767193i	$-0.721653\pi$
$521$	−80812.6	+	46657.2i	−0.297717	+	0.171887i	0.343700	+	0.939080i	$0.388320\pi$
$522$		0			0
$523$	−57289.6	−	33076.2i	−0.209446	−	0.120924i	0.391608	−	0.920132i	$-0.371919\pi$
$523$	−57289.6	−	33076.2i	−0.209446	−	0.120924i	−0.601054	+	0.799208i	$0.705252\pi$
$524$		0			0
$525$	−358391.	−	183644.i	−1.30028	−	0.666283i
$526$		0			0
$527$	223051.	−	386335.i	0.803124	−	1.39105i
$528$		0			0
$529$	49962.4	+	86537.4i	0.178538	+	0.309238i
$530$		0			0
$531$		−	12990.8i		−	0.0460730i
$532$		0			0
$533$	−170169.			−0.598999
$534$		0			0
$535$	344851.	−	199100.i	1.20483	−	0.695607i
$536$		0			0
$537$	−172431.	−	99553.3i	−0.597954	−	0.345229i
$538$		0			0
$539$	−194464.	−	270251.i	−0.669364	−	0.930229i
$540$		0			0
$541$	145960.	−	252810.i	0.498700	−	0.863773i	−0.501299	−	0.865274i	$-0.667144\pi$
$541$	145960.	−	252810.i	0.498700	−	0.863773i	0.999999	+	0.00150098i	$0.000477776\pi$
$542$		0			0
$543$	−141617.	−	245287.i	−0.480302	−	0.831908i
$544$		0			0
$545$		−	663395.i		−	2.23346i
$546$		0			0
$547$	−439985.			−1.47050			−0.735248	−	0.677799i	$-0.762934\pi$
$547$	−439985.			−1.47050			−0.735248	+	0.677799i	$0.762934\pi$
$548$		0			0
$549$	108167.	−	62450.3i	0.358881	−	0.207200i
$550$		0			0
$551$	223691.	+	129148.i	0.736791	+	0.425387i
$552$		0			0
$553$	21430.4	−	41822.6i	0.0700779	−	0.136761i
$554$		0			0
$555$	186137.	−	322399.i	0.604292	−	1.04666i
$556$		0			0
$557$	184356.	+	319315.i	0.594221	+	1.02922i	0.993656	+	0.112459i	$0.0358727\pi$
$557$	184356.	+	319315.i	0.594221	+	1.02922i	−0.399436	+	0.916761i	$0.630794\pi$
$558$		0			0
$559$		−	104677.i		−	0.334987i
$560$		0			0
$561$	−191008.			−0.606911
$562$		0			0
$563$	199387.	−	115116.i	0.629042	−	0.363178i	−0.151339	−	0.988482i	$-0.548359\pi$
$563$	199387.	−	115116.i	0.629042	−	0.363178i	0.780381	+	0.625304i	$0.215025\pi$
$564$		0			0
$565$	−518294.	−	299237.i	−1.62360	−	0.937387i
$566$		0			0
$567$	−19386.4	−	30002.6i	−0.0603019	−	0.0933239i
$568$		0			0
$569$	200192.	−	346743.i	0.618334	−	1.07099i	−0.371456	−	0.928451i	$-0.621141\pi$
$569$	200192.	−	346743.i	0.618334	−	1.07099i	0.989790	−	0.142535i	$-0.0455254\pi$
$570$		0			0
$571$	46434.9	+	80427.6i	0.142420	+	0.246680i	0.928408	−	0.371563i	$-0.121178\pi$
$571$	46434.9	+	80427.6i	0.142420	+	0.246680i	−0.785987	+	0.618243i	$0.787845\pi$
$572$		0			0
$573$			85497.0i			0.260400i
$574$		0			0
$575$	−670875.			−2.02911
$576$		0			0
$577$	−218515.	+	126159.i	−0.656340	+	0.378938i	−0.790881	−	0.611970i	$-0.790377\pi$
$577$	−218515.	+	126159.i	−0.656340	+	0.378938i	0.134541	+	0.990908i	$0.457044\pi$
$578$		0			0
$579$	157059.	+	90678.3i	0.468497	+	0.270487i
$580$		0			0
$581$	−18570.4	−	370577.i	−0.0550134	−	1.09781i
$582$		0			0
$583$	155713.	−	269703.i	0.458129	−	0.793503i
$584$		0			0
$585$	−41656.9	−	72151.9i	−0.121724	−	0.210832i
$586$		0			0
$587$			543282.i			1.57670i	0.615228	+	0.788350i	$0.289064\pi$
$587$			543282.i			1.57670i	−0.615228	+	0.788350i	$0.710936\pi$
$588$		0			0
$589$	−748384.			−2.15722
$590$		0			0
$591$	179650.	−	103721.i	0.514341	−	0.296955i
$592$		0			0
$593$	−347546.	−	200656.i	−0.988332	−	0.570614i	−0.0835564	−	0.996503i	$-0.526628\pi$
$593$	−347546.	−	200656.i	−0.988332	−	0.570614i	−0.904775	+	0.425890i	$0.859961\pi$
$594$		0			0
$595$	609405.	−	30538.6i	1.72136	−	0.0862611i
$596$		0			0
$597$	115259.	−	199635.i	0.323391	−	0.560129i
$598$		0			0
$599$	41205.2	+	71369.5i	0.114841	+	0.198911i	0.917716	−	0.397236i	$-0.130031\pi$
$599$	41205.2	+	71369.5i	0.114841	+	0.198911i	−0.802875	+	0.596148i	$0.796697\pi$
$600$		0			0
$601$		−	122915.i		−	0.340295i	−0.985419	−	0.170148i	$-0.945576\pi$
$601$		−	122915.i		−	0.340295i	0.985419	−	0.170148i	$-0.0544245\pi$
$602$		0			0
$603$	−154639.			−0.425290
$604$		0			0
$605$	186652.	−	107763.i	0.509942	−	0.294415i
$606$		0			0
$607$	607801.	+	350914.i	1.64962	+	0.952408i	0.977222	+	0.212218i	$0.0680686\pi$
$607$	607801.	+	350914.i	1.64962	+	0.952408i	0.672397	+	0.740191i	$0.265265\pi$
$608$		0			0
$609$	−124208.	+	80258.1i	−0.334901	+	0.216399i
$610$		0			0
$611$	−94341.7	+	163405.i	−0.252709	+	0.437705i
$612$		0			0
$613$	185221.	+	320811.i	0.492911	+	0.853746i	0.999967	−	0.00816675i	$-0.00259959\pi$
$613$	185221.	+	320811.i	0.492911	+	0.853746i	−0.507056	+	0.861913i	$0.669266\pi$
$614$		0			0
$615$			632324.i			1.67182i
$616$		0			0
$617$	100287.			0.263435			0.131717	−	0.991287i	$-0.457951\pi$
$617$	100287.			0.263435			0.131717	+	0.991287i	$0.457951\pi$
$618$		0			0
$619$	367978.	−	212452.i	0.960375	−	0.554473i	0.0640864	−	0.997944i	$-0.479587\pi$
$619$	367978.	−	212452.i	0.960375	−	0.554473i	0.896288	+	0.443472i	$0.146253\pi$
$620$		0			0
$621$	−51536.1	−	29754.4i	−0.133637	−	0.0771556i
$622$		0			0
$623$	−175964.	−	90166.0i	−0.453364	−	0.232309i
$624$		0			0
$625$	−561212.	+	972047.i	−1.43670	+	2.48844i
$626$		0			0
$627$	160218.	+	277506.i	0.407546	+	0.705890i
$628$		0			0
$629$			404301.i			1.02189i
$630$		0			0
$631$	516897.			1.29821			0.649106	−	0.760698i	$-0.275143\pi$
$631$	516897.			1.29821			0.649106	+	0.760698i	$0.275143\pi$
$632$		0			0
$633$	250800.	−	144800.i	0.625923	−	0.361377i
$634$		0			0
$635$	119310.	+	68883.9i	0.295891	+	0.170832i
$636$		0			0
$637$	15767.5	+	156928.i	0.0388584	+	0.386741i
$638$		0			0
$639$	49262.4	−	85325.0i	0.120646	−	0.208965i
$640$		0			0
$641$	87217.2	+	151065.i	0.212269	+	0.367660i	0.952424	−	0.304776i	$-0.0985815\pi$
$641$	87217.2	+	151065.i	0.212269	+	0.367660i	−0.740155	+	0.672436i	$0.765248\pi$
$642$		0			0
$643$		−	170938.i		−	0.413444i	−0.978400	−	0.206722i	$-0.933720\pi$
$643$		−	170938.i		−	0.413444i	0.978400	−	0.206722i	$-0.0662796\pi$
$644$		0			0
$645$	−388965.			−0.934956
$646$		0			0
$647$	−547638.	+	316179.i	−1.30823	+	0.755308i	−0.981801	−	0.189913i	$-0.939180\pi$
$647$	−547638.	+	316179.i	−1.30823	+	0.755308i	−0.326431	+	0.945221i	$0.605846\pi$
$648$		0			0
$649$	−57780.6	−	33359.7i	−0.137181	−	0.0792013i
$650$		0			0
$651$	195396.	−	381325.i	0.461056	−	0.899774i
$652$		0			0
$653$	−44862.4	+	77704.0i	−0.105210	+	0.182229i	−0.913824	−	0.406111i	$-0.866885\pi$
$653$	−44862.4	+	77704.0i	−0.105210	+	0.182229i	0.808614	+	0.588339i	$0.200218\pi$
$654$		0			0
$655$	479590.	+	830674.i	1.11786	+	1.93619i
$656$		0			0
$657$			147595.i			0.341932i
$658$		0			0
$659$	−157986.			−0.363788			−0.181894	−	0.983318i	$-0.558223\pi$
$659$	−157986.			−0.363788			−0.181894	+	0.983318i	$0.558223\pi$
$660$		0			0
$661$	362182.	−	209106.i	0.828941	−	0.478590i	−0.0245487	−	0.999699i	$-0.507815\pi$
$661$	362182.	−	209106.i	0.828941	−	0.478590i	0.853490	+	0.521109i	$0.174482\pi$
$662$		0			0
$663$	78359.2	+	45240.7i	0.178264	+	0.102921i
$664$		0			0
$665$	−555539.	−	859759.i	−1.25624	−	1.94417i
$666$		0			0
$667$	−123181.	+	213355.i	−0.276880	+	0.479570i
$668$		0			0
$669$	19498.8	+	33772.9i	0.0435668	+	0.0754599i
$670$		0			0
$671$		−	641476.i		−	1.42474i
$672$		0			0
$673$	−47167.4			−0.104139			−0.0520693	−	0.998643i	$-0.516582\pi$
$673$	−47167.4			−0.104139			−0.0520693	+	0.998643i	$0.516582\pi$
$674$		0			0
$675$	−192169.	+	110949.i	−0.421769	+	0.243509i
$676$		0			0
$677$	480196.	+	277241.i	1.04771	+	0.604896i	0.922008	−	0.387172i	$-0.126548\pi$
$677$	480196.	+	277241.i	1.04771	+	0.604896i	0.125703	+	0.992068i	$0.459881\pi$
$678$		0			0
$679$	9322.67	+	186036.i	0.0202209	+	0.403513i
$680$		0			0
$681$	155830.	−	269906.i	0.336014	−	0.581993i
$682$		0			0
$683$	12255.6	+	21227.4i	0.0262721	+	0.0455046i	0.878863	−	0.477075i	$-0.158303\pi$
$683$	12255.6	+	21227.4i	0.0262721	+	0.0455046i	−0.852590	+	0.522580i	$0.824970\pi$
$684$		0			0
$685$			298517.i			0.636193i
$686$		0			0
$687$	−146409.			−0.310208
$688$		0			0
$689$	−127760.	+	73762.1i	−0.269126	+	0.155380i
$690$		0			0
$691$	−715670.	−	413192.i	−1.49884	−	0.865358i	−0.498846	−	0.866691i	$-0.666243\pi$
$691$	−715670.	−	413192.i	−1.49884	−	0.865358i	−0.999999	+	0.00133250i	$0.999576\pi$
$692$		0			0
$693$	−183229.	+	9182.01i	−0.381530	+	0.0191193i
$694$		0			0
$695$	−6905.30	+	11960.3i	−0.0142959	+	0.0247613i
$696$		0			0
$697$	−343362.	−	594720.i	−0.706783	−	1.22418i
$698$		0			0
$699$		−	192786.i		−	0.394568i
$700$		0			0
$701$	624144.			1.27013			0.635066	−	0.772458i	$-0.280973\pi$
$701$	624144.			1.27013			0.635066	+	0.772458i	$0.280973\pi$
$702$		0			0
$703$	587389.	−	339129.i	1.18854	−	0.686206i
$704$		0			0
$705$	607188.	+	350560.i	1.22164	+	0.705317i
$706$		0			0
$707$	−621588.	+	401644.i	−1.24355	+	0.803530i
$708$		0			0
$709$	−463012.	+	801960.i	−0.921084	+	1.59536i	−0.123344	+	0.992364i	$0.539362\pi$
$709$	−463012.	+	801960.i	−0.921084	+	1.59536i	−0.797741	+	0.603001i	$0.793972\pi$
$710$		0			0
$711$	−12947.2	−	22425.2i	−0.0256116	−	0.0443606i
$712$		0			0
$713$		−	713805.i		−	1.40411i
$714$		0			0
$715$	−427891.			−0.836992
$716$		0			0
$717$	−155627.	+	89851.3i	−0.302724	+	0.174778i
$718$		0			0
$719$	−572626.	−	330606.i	−1.10768	−	0.639518i	−0.169451	−	0.985539i	$-0.554199\pi$
$719$	−572626.	−	330606.i	−1.10768	−	0.639518i	−0.938227	+	0.346020i	$0.887533\pi$
$720$		0			0
$721$	116458.	+	59674.8i	0.224027	+	0.114794i
$722$		0			0
$723$	−167667.	+	290408.i	−0.320753	+	0.555561i
$724$		0			0
$725$	459318.	+	795563.i	0.873852	+	1.51356i
$726$		0			0
$727$		−	474354.i		−	0.897499i	−0.893658	−	0.448749i	$-0.851870\pi$
$727$		−	474354.i		−	0.897499i	0.893658	−	0.448749i	$-0.148130\pi$
$728$		0			0
$729$	−19683.0			−0.0370370
$730$		0			0
$731$	365834.	−	211214.i	0.684619	−	0.395265i
$732$		0			0
$733$	481346.	+	277905.i	0.895879	+	0.517236i	0.875861	−	0.482564i	$-0.160294\pi$
$733$	481346.	+	277905.i	0.895879	+	0.517236i	0.0200182	+	0.999800i	$0.493628\pi$
$734$		0			0
$735$	583120.	−	58589.9i	1.07940	−	0.108455i
$736$		0			0
$737$	−397105.	+	687807.i	−0.731090	+	1.26629i
$738$		0			0
$739$	198723.	+	344199.i	0.363882	+	0.630262i	0.988596	−	0.150592i	$-0.0481179\pi$
$739$	198723.	+	344199.i	0.363882	+	0.630262i	−0.624714	+	0.780853i	$0.714785\pi$
$740$		0			0
$741$		−	151792.i		−	0.276448i
$742$		0			0
$743$	−197129.			−0.357085			−0.178543	−	0.983932i	$-0.557138\pi$
$743$	−197129.			−0.357085			−0.178543	+	0.983932i	$0.557138\pi$
$744$		0			0
$745$	1.35583e6	−	782790.i	2.44283	−	1.41037i
$746$		0			0
$747$	−177060.	−	102226.i	−0.317307	−	0.183197i
$748$		0			0
$749$	−189420.	+	369662.i	−0.337646	+	0.658933i
$750$		0			0
$751$	−125618.	+	217577.i	−0.222727	+	0.385775i	−0.955635	−	0.294553i	$-0.904829\pi$
$751$	−125618.	+	217577.i	−0.222727	+	0.385775i	0.732908	+	0.680328i	$0.238163\pi$
$752$		0			0
$753$	283720.	+	491418.i	0.500381	+	0.866685i
$754$		0			0
$755$			378972.i			0.664834i
$756$		0			0
$757$	−512610.			−0.894531			−0.447265	−	0.894401i	$-0.647602\pi$
$757$	−512610.			−0.894531			−0.447265	+	0.894401i	$0.647602\pi$
$758$		0			0
$759$	−264684.	+	152815.i	−0.459456	+	0.265267i
$760$		0			0
$761$	922933.	+	532856.i	1.59368	+	0.920111i	0.992668	+	0.120870i	$0.0385685\pi$
$761$	922933.	+	532856.i	1.59368	+	0.920111i	0.601011	+	0.799241i	$0.294765\pi$
$762$		0			0
$763$	375558.	+	581217.i	0.645101	+	0.998365i
$764$		0			0
$765$	168108.	−	291172.i	0.287254	−	0.497538i
$766$		0			0
$767$	15802.7	+	27371.0i	0.0268621	+	0.0465265i
$768$		0			0
$769$			626032.i			1.05863i	0.848426	+	0.529314i	$0.177551\pi$
$769$			626032.i			1.05863i	−0.848426	+	0.529314i	$0.822449\pi$
$770$		0			0
$771$	−155599.			−0.261756
$772$		0			0
$773$	84028.3	−	48513.8i	0.140626	−	0.0811906i	−0.428036	−	0.903762i	$-0.640794\pi$
$773$	84028.3	−	48513.8i	0.140626	−	0.0811906i	0.568662	+	0.822571i	$0.307461\pi$
$774$		0			0
$775$	−2.30506e6	−	1.33083e6i	−3.83776	−	2.21573i
$776$		0			0
$777$	19435.3	+	387836.i	0.0321921	+	0.642402i
$778$		0			0
$779$	−576026.	+	997707.i	−0.949221	+	1.64410i
$780$		0			0
$781$	−253006.	−	438220.i	−0.414791	−	0.718439i
$782$		0			0
$783$			81486.0i			0.132911i
$784$		0			0
$785$	1.00673e6			1.63370
$786$		0			0
$787$	−123458.	+	71278.2i	−0.199328	+	0.115082i	−0.596342	−	0.802731i	$-0.703380\pi$
$787$	−123458.	+	71278.2i	−0.199328	+	0.115082i	0.397014	+	0.917813i	$0.370046\pi$
$788$		0			0
$789$	−305994.	−	176666.i	−0.491540	−	0.283791i
$790$		0			0
$791$	623493.	−	31244.6i	0.996504	−	0.0499369i
$792$		0			0
$793$	−151935.	+	263160.i	−0.241609	+	0.418479i
$794$		0			0
$795$	274089.	+	474737.i	0.433669	+	0.751136i
$796$		0			0
$797$		−	118334.i		−	0.186291i	−0.995652	−	0.0931456i	$-0.970308\pi$
$797$		−	118334.i		−	0.186291i	0.995652	−	0.0931456i	$-0.0296922\pi$
$798$		0			0
$799$	−761438.			−1.19273
$800$		0			0
$801$	−94351.4	+	54473.8i	−0.147056	+	0.0849030i
$802$		0			0
$803$	656474.	+	379015.i	1.01809	+	0.587795i
$804$		0			0
$805$	820034.	−	529871.i	1.26544	−	0.817671i
$806$		0			0
$807$	251672.	−	435909.i	0.386445	−	0.669342i
$808$		0			0
$809$	215958.	+	374051.i	0.329969	+	0.571523i	0.982505	−	0.186235i	$-0.0596285\pi$
$809$	215958.	+	374051.i	0.329969	+	0.571523i	−0.652537	+	0.757757i	$0.726295\pi$
$810$		0			0
$811$			607886.i			0.924231i	0.886820	+	0.462116i	$0.152910\pi$
$811$			607886.i			0.924231i	−0.886820	+	0.462116i	$0.847090\pi$
$812$		0			0
$813$	−37207.4			−0.0562922
$814$		0			0
$815$	1.16060e6	−	670070.i	1.74729	−	1.00880i
$816$		0			0
$817$	−613725.	−	354334.i	−0.919454	−	0.530847i
$818$		0			0
$819$	77343.0	+	39631.5i	0.115306	+	0.0590844i
$820$		0			0
$821$	−358359.	+	620696.i	−0.531657	+	0.920857i	0.467660	+	0.883908i	$0.345097\pi$
$821$	−358359.	+	620696.i	−0.531657	+	0.920857i	−0.999317	+	0.0369488i	$0.988236\pi$
$822$		0			0
$823$	−437212.	−	757273.i	−0.645494	−	1.11803i	−0.984187	−	0.177132i	$-0.943318\pi$
$823$	−437212.	−	757273.i	−0.645494	−	1.11803i	0.338693	−	0.940897i	$-0.390015\pi$
$824$		0			0
$825$			1.13964e6i			1.67440i
$826$		0			0
$827$	446731.			0.653184			0.326592	−	0.945165i	$-0.394100\pi$
$827$	446731.			0.653184			0.326592	+	0.945165i	$0.394100\pi$
$828$		0			0
$829$	800235.	−	462016.i	1.16442	−	0.672277i	0.212059	−	0.977257i	$-0.431983\pi$
$829$	800235.	−	462016.i	1.16442	−	0.672277i	0.952359	+	0.304980i	$0.0986499\pi$
$830$		0			0
$831$	158010.	+	91227.4i	0.228815	+	0.132106i
$832$		0			0
$833$	−516627.	+	371749.i	−0.744538	+	0.535747i
$834$		0			0
$835$	374557.	−	648752.i	0.537211	−	0.930477i
$836$		0			0
$837$	−118048.	−	204466.i	−0.168504	−	0.291857i
$838$		0			0
$839$		−	230940.i		−	0.328077i	−0.986454	−	0.164039i	$-0.947548\pi$
$839$		−	230940.i		−	0.328077i	0.986454	−	0.164039i	$-0.0524522\pi$
$840$		0			0
$841$	−369936.			−0.523039
$842$		0			0
$843$	−268519.	+	155030.i	−0.377851	+	0.218152i
$844$		0			0
$845$	−986363.	−	569477.i	−1.38141	−	0.797559i
$846$		0			0
$847$	−102524.	+	200080.i	−0.142908	+	0.278893i
$848$		0			0
$849$	234728.	−	406561.i	0.325649	−	0.564041i
$850$		0			0
$851$	323460.	+	560249.i	0.446644	+	0.773610i
$852$		0			0
$853$		−	990422.i		−	1.36120i	−0.732655	−	0.680600i	$-0.761719\pi$
$853$		−	990422.i		−	1.36120i	0.732655	−	0.680600i	$-0.238281\pi$
$854$		0			0
$855$	−564038.			−0.771572
$856$		0			0
$857$	−124244.	+	71732.1i	−0.169166	+	0.0976680i	−0.582192	−	0.813051i	$-0.697805\pi$
$857$	−124244.	+	71732.1i	−0.169166	+	0.0976680i	0.413027	+	0.910719i	$0.364472\pi$
$858$		0			0
$859$	−680116.	−	392665.i	−0.921715	−	0.532153i	−0.0375335	−	0.999295i	$-0.511950\pi$
$859$	−680116.	−	392665.i	−0.921715	−	0.532153i	−0.884182	+	0.467143i	$0.845283\pi$
$860$		0			0
$861$	−357968.	−	553995.i	−0.482879	−	0.747308i
$862$		0			0
$863$	−108331.	+	187634.i	−0.145455	+	0.251936i	−0.929543	−	0.368715i	$-0.879798\pi$
$863$	−108331.	+	187634.i	−0.145455	+	0.251936i	0.784087	+	0.620650i	$0.213131\pi$
$864$		0			0
$865$	−586423.	−	1.01571e6i	−0.783752	−	1.35750i
$866$		0			0
$867$		−	68847.1i		−	0.0915899i
$868$		0			0
$869$	−132991.			−0.176109
$870$		0			0
$871$	325818.	−	188111.i	0.429476	−	0.247958i
$872$		0			0
$873$	88887.4	+	51319.2i	0.116630	+	0.0673366i
$874$		0			0
$875$	−110206.	−	2.19919e6i	−0.143943	−	2.87241i
$876$		0			0
$877$	−350970.	+	607898.i	−0.456322	+	0.790372i	−0.998763	−	0.0497214i	$-0.984167\pi$
$877$	−350970.	+	607898.i	−0.456322	+	0.790372i	0.542442	+	0.840094i	$0.317500\pi$
$878$		0			0
$879$	−349643.	−	605600.i	−0.452530	−	0.783805i
$880$		0			0
$881$		−	288948.i		−	0.372278i	−0.982523	−	0.186139i	$-0.940403\pi$
$881$		−	288948.i		−	0.372278i	0.982523	−	0.186139i	$-0.0595975\pi$
$882$		0			0
$883$	−555854.			−0.712918			−0.356459	−	0.934311i	$-0.616016\pi$
$883$	−555854.			−0.712918			−0.356459	+	0.934311i	$0.616016\pi$
$884$		0			0
$885$	101707.	−	58720.4i	0.129856	−	0.0749726i
$886$		0			0
$887$	826643.	+	477262.i	1.05068	+	0.606611i	0.922840	−	0.385184i	$-0.125862\pi$
$887$	826643.	+	477262.i	1.05068	+	0.606611i	0.127841	+	0.991795i	$0.459195\pi$
$888$		0			0
$889$	−143527.	+	7192.45i	−0.181606	+	0.00910067i
$890$		0			0
$891$	−50544.9	+	87546.3i	−0.0636681	+	0.110276i
$892$		0			0
$893$	638697.	+	1.10626e6i	0.800926	+	1.38724i
$894$		0			0
$895$		−	1.79999e6i		−	2.24710i
$896$		0			0
$897$	144779.			0.179937
$898$		0			0
$899$	−846472.	+	488711.i	−1.04735	+	0.604690i
$900$		0			0
$901$	−515579.	−	297670.i	−0.635105	−	0.366678i
$902$		0			0
$903$	340782.	−	220199.i	0.417928	−	0.270047i
$904$		0			0
$905$	1.28026e6	−	2.21747e6i	1.56315	−	2.70745i
$906$		0			0
$907$	488687.	+	846431.i	0.594041	+	1.02891i	0.993681	+	0.112237i	$0.0358017\pi$
$907$	488687.	+	846431.i	0.594041	+	1.02891i	−0.399640	+	0.916672i	$0.630865\pi$
$908$		0			0
$909$			407788.i			0.493523i
$910$		0			0
$911$	−711654.			−0.857496			−0.428748	−	0.903424i	$-0.641045\pi$
$911$	−711654.			−0.857496			−0.428748	+	0.903424i	$0.641045\pi$
$912$		0			0
$913$	−909363.	+	525021.i	−1.09093	+	0.629847i
$914$		0			0
$915$	977864.	+	564570.i	1.16798	+	0.674335i
$916$		0			0
$917$	−890438.	−	456272.i	−1.05892	−	0.542607i
$918$		0			0
$919$	−5284.19	+	9152.49i	−0.00625673	+	0.0108370i	−0.869137	−	0.494572i	$-0.835325\pi$
$919$	−5284.19	+	9152.49i	−0.00625673	+	0.0108370i	0.862880	+	0.505409i	$0.168658\pi$
$920$		0			0
$921$	315455.	+	546383.i	0.371893	+	0.644137i
$922$		0			0
$923$			239701.i			0.281363i
$924$		0			0
$925$	2.41225e6			2.81928
$926$		0			0
$927$	62444.8	−	36052.5i	0.0726670	−	0.0419543i
$928$		0			0
$929$	−99475.1	−	57432.0i	−0.115261	−	0.0665461i	0.441261	−	0.897379i	$-0.354531\pi$
$929$	−99475.1	−	57432.0i	−0.115261	−	0.0665461i	−0.556522	+	0.830833i	$0.687865\pi$
$930$		0			0
$931$	973444.	+	438758.i	1.12308	+	0.506204i
$932$		0			0
$933$	−205424.	+	355805.i	−0.235987	+	0.408742i
$934$		0			0
$935$	−863385.	−	1.49543e6i	−0.987600	−	1.71057i
$936$		0			0
$937$			307042.i			0.349719i	0.984593	+	0.174859i	$0.0559471\pi$
$937$			307042.i			0.349719i	−0.984593	+	0.174859i	$0.944053\pi$
$938$		0			0
$939$	683337.			0.775003
$940$		0			0
$941$	747779.	−	431730.i	0.844489	−	0.487566i	−0.0142988	−	0.999898i	$-0.504552\pi$
$941$	747779.	−	431730.i	0.844489	−	0.487566i	0.858787	+	0.512332i	$0.171218\pi$
$942$		0			0
$943$	−951609.	−	549412.i	−1.07013	−	0.617838i
$944$		0			0
$945$	147265.	−	287395.i	0.164906	−	0.321822i
$946$		0			0
$947$	−216580.	+	375127.i	−0.241500	+	0.418291i	−0.961142	−	0.276055i	$-0.910973\pi$
$947$	−216580.	+	375127.i	−0.241500	+	0.418291i	0.719642	+	0.694346i	$0.244306\pi$
$948$		0			0
$949$	−179542.	−	310975.i	−0.199358	−	0.345298i
$950$		0			0
$951$			223335.i			0.246942i
$952$		0			0
$953$	−1.54744e6			−1.70384			−0.851920	−	0.523672i	$-0.824562\pi$
$953$	−1.54744e6			−1.70384			−0.851920	+	0.523672i	$0.824562\pi$
$954$		0			0
$955$	−669368.	+	386460.i	−0.733936	+	0.423738i
$956$		0			0
$957$	362435.	+	209252.i	0.395736	+	0.228478i
$958$		0			0
$959$	−168995.	−	261539.i	−0.183754	−	0.284380i
$960$		0			0
$961$	954227.	−	1.65277e6i	1.03325	−	1.78964i
$962$		0			0
$963$	114438.	+	198212.i	0.123401	+	0.213736i
$964$		0			0
$965$			1.63952e6i			1.76061i
$966$		0			0
$967$	415384.			0.444219			0.222109	−	0.975022i	$-0.428706\pi$
$967$	415384.			0.444219			0.222109	+	0.975022i	$0.428706\pi$
$968$		0			0
$969$	530495.	−	306281.i	0.564981	−	0.326192i
$970$		0			0
$971$	−367889.	−	212401.i	−0.390192	−	0.225278i	0.292051	−	0.956403i	$-0.405662\pi$
$971$	−367889.	−	212401.i	−0.390192	−	0.225278i	−0.682244	+	0.731125i	$0.738996\pi$
$972$		0			0
$973$	−721.010	−	14387.9i	−0.000761581	−	0.0151975i
$974$		0			0
$975$	269927.	−	467527.i	0.283947	−	0.491811i
$976$		0			0
$977$	−588913.	−	1.02003e6i	−0.616967	−	1.06862i	−0.990036	−	0.140815i	$-0.955028\pi$
$977$	−588913.	−	1.02003e6i	−0.616967	−	1.06862i	0.373069	−	0.927804i	$-0.378306\pi$
$978$		0			0
$979$			559544.i			0.583806i
$980$		0			0
$981$	381303.			0.396216
$982$		0			0
$983$	−343117.	+	198099.i	−0.355088	+	0.205010i	−0.666924	−	0.745126i	$-0.732389\pi$
$983$	−343117.	+	198099.i	−0.355088	+	0.205010i	0.311836	+	0.950136i	$0.399056\pi$
$984$		0			0
$985$	1.62409e6	+	937668.i	1.67393	+	0.966444i
$986$		0			0
$987$	−730430.	+	36603.4i	−0.749798	+	0.0375740i
$988$		0			0
$989$	337963.	−	585369.i	0.345522	−	0.598462i
$990$		0			0
$991$	391846.	+	678697.i	0.398996	+	0.691081i	0.993602	−	0.112936i	$-0.0360255\pi$
$991$	391846.	+	678697.i	0.398996	+	0.691081i	−0.594607	+	0.804017i	$0.702692\pi$
$992$		0			0
$993$			429062.i			0.435133i
$994$		0			0
$995$	2.08396e6			2.10496
$996$		0			0
$997$	514705.	−	297165.i	0.517807	−	0.298956i	−0.218230	−	0.975897i	$-0.570028\pi$
$997$	514705.	−	297165.i	0.517807	−	0.298956i	0.736037	+	0.676941i	$0.236695\pi$
$998$		0			0
$999$	185307.	+	106987.i	0.185678	+	0.107201i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	168.5.z.a.145.1	yes	16
3.2	odd	2		504.5.by.a.145.8		16
4.3	odd	2		336.5.bh.i.145.1		16
7.2	even	3		1176.5.f.b.97.8		16
7.3	odd	6	inner	168.5.z.a.73.1	✓	16
7.5	odd	6		1176.5.f.b.97.9		16
21.17	even	6		504.5.by.a.73.8		16
28.3	even	6		336.5.bh.i.241.1		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.5.z.a.73.1	✓	16	7.3	odd	6	inner
168.5.z.a.145.1	yes	16	1.1	even	1	trivial
336.5.bh.i.145.1		16	4.3	odd	2
336.5.bh.i.241.1		16	28.3	even	6
504.5.by.a.73.8		16	21.17	even	6
504.5.by.a.145.8		16	3.2	odd	2
1176.5.f.b.97.8		16	7.2	even	3
1176.5.f.b.97.9		16	7.5	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:	\( N \)	\(=\)	\( 168 = 2^{3} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 5 \)
Character orbit:	\([\chi]\)	\(=\)	168.z (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 2.59808i) q^{3} +(-40.6814 - 23.4874i) q^{5} +(48.9386 - 2.45242i) q^{7} +(13.5000 - 23.3827i) q^{9} +O(q^{10})\)	\(q+(-4.50000 + 2.59808i) q^{3} +(-40.6814 - 23.4874i) q^{5} +(48.9386 - 2.45242i) q^{7} +(13.5000 - 23.3827i) q^{9} +(-69.3346 - 120.091i) q^{11} +65.6883i q^{13} +244.088 q^{15} +(-229.572 + 132.544i) q^{17} +(385.133 + 222.356i) q^{19} +(-213.852 + 138.182i) q^{21} +(-212.083 + 367.338i) q^{23} +(790.818 + 1369.74i) q^{25} +140.296i q^{27} +580.814 q^{29} +(-1457.39 + 841.424i) q^{31} +(624.011 + 360.273i) q^{33} +(-2048.49 - 1049.67i) q^{35} +(762.580 - 1320.83i) q^{37} +(-170.663 - 295.598i) q^{39} +2590.55i q^{41} -1593.54 q^{43} +(-1098.40 + 634.160i) q^{45} +(2487.57 + 1436.20i) q^{47} +(2388.97 - 240.036i) q^{49} +(688.717 - 1192.89i) q^{51} +(1122.91 + 1944.94i) q^{53} +6513.96i q^{55} -2310.80 q^{57} +(416.680 - 240.570i) q^{59} +(4006.19 + 2312.97i) q^{61} +(603.327 - 1177.42i) q^{63} +(1542.85 - 2672.29i) q^{65} +(-2863.69 - 4960.06i) q^{67} -2204.03i q^{69} +3649.07 q^{71} +(-4734.10 + 2733.24i) q^{73} +(-7117.36 - 4109.21i) q^{75} +(-3687.65 - 5707.05i) q^{77} +(479.526 - 830.564i) q^{79} +(-364.500 - 631.333i) q^{81} -7572.28i q^{83} +12452.4 q^{85} +(-2613.67 + 1509.00i) q^{87} +(-3494.50 - 2017.55i) q^{89} +(161.095 + 3214.69i) q^{91} +(4372.17 - 7572.81i) q^{93} +(-10445.2 - 18091.5i) q^{95} +3801.42i q^{97} -3744.07 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9}+O(q^{10}) \) 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9	\( 16 q - 72 q^{3} + 12 q^{5} + 40 q^{7} + 216 q^{9} - 108 q^{11} - 72 q^{15} - 168 q^{17} + 948 q^{19} - 252 q^{21} - 768 q^{23} + 1908 q^{25} - 1608 q^{29} - 3216 q^{31} + 972 q^{33} + 696 q^{35} - 1820 q^{37} - 1188 q^{39} + 2888 q^{43} + 324 q^{45} + 744 q^{47} - 3784 q^{49} + 504 q^{51} + 4476 q^{53} - 5688 q^{57} - 4668 q^{59} + 17760 q^{61} + 1188 q^{63} + 8760 q^{65} + 1580 q^{67} + 48 q^{71} + 588 q^{73} - 17172 q^{75} - 17508 q^{77} - 3824 q^{79} - 5832 q^{81} + 11440 q^{85} + 7236 q^{87} - 360 q^{89} + 25860 q^{91} + 9648 q^{93} - 21792 q^{95} - 5832 q^{99}+O(q^{100}) \) 16 * q - 72 * q^3 + 12 * q^5 + 40 * q^7 + 216 * q^9 - 108 * q^11 - 72 * q^15 - 168 * q^17 + 948 * q^19 - 252 * q^21 - 768 * q^23 + 1908 * q^25 - 1608 * q^29 - 3216 * q^31 + 972 * q^33 + 696 * q^35 - 1820 * q^37 - 1188 * q^39 + 2888 * q^43 + 324 * q^45 + 744 * q^47 - 3784 * q^49 + 504 * q^51 + 4476 * q^53 - 5688 * q^57 - 4668 * q^59 + 17760 * q^61 + 1188 * q^63 + 8760 * q^65 + 1580 * q^67 + 48 * q^71 + 588 * q^73 - 17172 * q^75 - 17508 * q^77 - 3824 * q^79 - 5832 * q^81 + 11440 * q^85 + 7236 * q^87 - 360 * q^89 + 25860 * q^91 + 9648 * q^93 - 21792 * q^95 - 5832 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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