LMFDB - Embedded newform 588.6.i.i.361.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [588,6,Mod(361,588)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(588, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 4]))

N = Newforms(chi, 6, names="a")

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

chi := DirichletCharacter("588.361");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 588 = 2^{2} \cdot 3 \cdot 7^{2} $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	588.i (of order $3$, degree $2$, not 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:	$94.3056860500$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{5569})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 1393x^{2} + 1392x + 1937664 $ x^4 - x^3 + 1393x^2 + 1392x + 1937664
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			361.1
Root			$18.9064 - 32.7469i$ of defining polynomial
Character	$\chi$	$=$	588.361
Dual form			588.6.i.i.373.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 7.79423i) q^{3} +(-38.8129 - 67.2259i) q^{5} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})$	\(q+(-4.50000 + 7.79423i) q^{3} +(-38.8129 - 67.2259i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(-238.690 + 413.423i) q^{11} +63.7544 q^{13} +698.632 q^{15} +(518.813 - 898.610i) q^{17} +(-333.509 - 577.654i) q^{19} +(-1625.81 - 2815.99i) q^{23} +(-1450.38 + 2512.13i) q^{25} +729.000 q^{27} +2300.97 q^{29} +(1858.53 - 3219.06i) q^{31} +(-2148.21 - 3720.81i) q^{33} +(-6122.93 - 10605.2i) q^{37} +(-286.895 + 496.916i) q^{39} +1829.65 q^{41} -20794.2 q^{43} +(-3143.84 + 5445.29i) q^{45} +(-2141.68 - 3709.51i) q^{47} +(4669.32 + 8087.49i) q^{51} +(-12859.2 + 22272.8i) q^{53} +37057.0 q^{55} +6003.16 q^{57} +(-1419.36 + 2458.40i) q^{59} +(8401.60 + 14552.0i) q^{61} +(-2474.49 - 4285.94i) q^{65} +(31267.5 - 54157.0i) q^{67} +29264.6 q^{69} +72301.0 q^{71} +(-27838.4 + 48217.6i) q^{73} +(-13053.4 - 22609.1i) q^{75} +(1994.60 + 3454.74i) q^{79} +(-3280.50 + 5681.99i) q^{81} +46092.2 q^{83} -80546.5 q^{85} +(-10354.4 + 17934.3i) q^{87} +(67692.3 + 117247. i) q^{89} +(16726.7 + 28971.6i) q^{93} +(-25888.9 + 44840.8i) q^{95} -142878. q^{97} +38667.8 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 18 q^{3} - 6 q^{5} - 162 q^{9}+O(q^{10}) $ 4 * q - 18 * q^3 - 6 * q^5 - 162 * q^9	$ 4 q - 18 q^{3} - 6 q^{5} - 162 q^{9} + 90 q^{11} - 1536 q^{13} + 108 q^{15} + 1926 q^{17} + 2248 q^{19} - 6354 q^{23} - 4906 q^{25} + 2916 q^{27} + 21144 q^{29} - 3312 q^{31} + 810 q^{33} - 2104 q^{37} + 6912 q^{39} - 2532 q^{41} - 11536 q^{43} - 486 q^{45} + 15612 q^{47} + 17334 q^{51} - 16512 q^{53} + 155392 q^{55} - 40464 q^{57} - 13140 q^{59} - 5796 q^{61} - 64524 q^{65} + 56116 q^{67} + 114372 q^{69} + 22044 q^{71} - 85384 q^{73} - 44154 q^{75} + 19620 q^{79} - 13122 q^{81} + 88848 q^{83} - 33832 q^{85} - 95148 q^{87} + 211218 q^{89} - 29808 q^{93} - 260568 q^{95} - 89728 q^{97} - 14580 q^{99}+O(q^{100}) $ 4 * q - 18 * q^3 - 6 * q^5 - 162 * q^9 + 90 * q^11 - 1536 * q^13 + 108 * q^15 + 1926 * q^17 + 2248 * q^19 - 6354 * q^23 - 4906 * q^25 + 2916 * q^27 + 21144 * q^29 - 3312 * q^31 + 810 * q^33 - 2104 * q^37 + 6912 * q^39 - 2532 * q^41 - 11536 * q^43 - 486 * q^45 + 15612 * q^47 + 17334 * q^51 - 16512 * q^53 + 155392 * q^55 - 40464 * q^57 - 13140 * q^59 - 5796 * q^61 - 64524 * q^65 + 56116 * q^67 + 114372 * q^69 + 22044 * q^71 - 85384 * q^73 - 44154 * q^75 + 19620 * q^79 - 13122 * q^81 + 88848 * q^83 - 33832 * q^85 - 95148 * q^87 + 211218 * q^89 - 29808 * q^93 - 260568 * q^95 - 89728 * q^97 - 14580 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/588\mathbb{Z}\right)^\times$.

$n$	$197$	$295$	$493$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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	+	7.79423i	−0.288675	+	0.500000i
$3$	−4.50000	+	7.79423i	−0.288675	+	0.500000i
$4$		0			0
$5$	−38.8129	−	67.2259i	−0.694306	−	1.20257i	−0.970414	−	0.241446i	$-0.922378\pi$
$5$	−38.8129	−	67.2259i	−0.694306	−	1.20257i	0.276109	−	0.961126i	$-0.410955\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	−238.690	+	413.423i	−0.594775	+	1.03018i	0.398804	+	0.917036i	$0.369425\pi$
$11$	−238.690	+	413.423i	−0.594775	+	1.03018i	−0.993579	+	0.113144i	$0.963908\pi$
$12$		0			0
$13$	63.7544			0.104629			0.0523145	−	0.998631i	$-0.483340\pi$
$13$	63.7544			0.104629			0.0523145	+	0.998631i	$0.483340\pi$
$14$		0			0
$15$	698.632			0.801715
$16$		0			0
$17$	518.813	−	898.610i	0.435400	−	0.754135i	−0.561928	−	0.827186i	$-0.689940\pi$
$17$	518.813	−	898.610i	0.435400	−	0.754135i	0.997328	+	0.0730511i	$0.0232736\pi$
$18$		0			0
$19$	−333.509	−	577.654i	−0.211945	−	0.367100i	0.740378	−	0.672191i	$-0.234646\pi$
$19$	−333.509	−	577.654i	−0.211945	−	0.367100i	−0.952323	+	0.305091i	$0.901313\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1625.81	−	2815.99i	−0.640842	−	1.10997i	−0.985245	−	0.171149i	$-0.945252\pi$
$23$	−1625.81	−	2815.99i	−0.640842	−	1.10997i	0.344403	−	0.938822i	$-0.388081\pi$
$24$		0			0
$25$	−1450.38	+	2512.13i	−0.464121	+	0.803881i
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	2300.97			0.508061			0.254031	−	0.967196i	$-0.418244\pi$
$29$	2300.97			0.508061			0.254031	+	0.967196i	$0.418244\pi$
$30$		0			0
$31$	1858.53	−	3219.06i	0.347348	−	0.601624i	−0.638430	−	0.769680i	$-0.720416\pi$
$31$	1858.53	−	3219.06i	0.347348	−	0.601624i	0.985777	+	0.168056i	$0.0537490\pi$
$32$		0			0
$33$	−2148.21	−	3720.81i	−0.343393	−	0.594775i
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−6122.93	−	10605.2i	−0.735284	−	1.27355i	−0.954599	−	0.297895i	$-0.903716\pi$
$37$	−6122.93	−	10605.2i	−0.735284	−	1.27355i	0.219315	−	0.975654i	$-0.429618\pi$
$38$		0			0
$39$	−286.895	+	496.916i	−0.0302038	+	0.0523145i
$40$		0			0
$41$	1829.65			0.169984			0.0849920	−	0.996382i	$-0.472914\pi$
$41$	1829.65			0.169984			0.0849920	+	0.996382i	$0.472914\pi$
$42$		0			0
$43$	−20794.2			−1.71503			−0.857513	−	0.514463i	$-0.827991\pi$
$43$	−20794.2			−1.71503			−0.857513	+	0.514463i	$0.827991\pi$
$44$		0			0
$45$	−3143.84	+	5445.29i	−0.231435	+	0.400858i
$46$		0			0
$47$	−2141.68	−	3709.51i	−0.141420	−	0.244947i	0.786612	−	0.617448i	$-0.211833\pi$
$47$	−2141.68	−	3709.51i	−0.141420	−	0.244947i	−0.928032	+	0.372502i	$0.878500\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$	4669.32	+	8087.49i	0.251378	+	0.435400i
$52$		0			0
$53$	−12859.2	+	22272.8i	−0.628818	+	1.08914i	0.358972	+	0.933348i	$0.383127\pi$
$53$	−12859.2	+	22272.8i	−0.628818	+	1.08914i	−0.987789	+	0.155796i	$0.950206\pi$
$54$		0			0
$55$	37057.0			1.65182
$56$		0			0
$57$	6003.16			0.244733
$58$		0			0
$59$	−1419.36	+	2458.40i	−0.0530837	+	0.0919437i	−0.891346	−	0.453323i	$-0.850238\pi$
$59$	−1419.36	+	2458.40i	−0.0530837	+	0.0919437i	0.838262	+	0.545267i	$0.183572\pi$
$60$		0			0
$61$	8401.60	+	14552.0i	0.289093	+	0.500723i	0.973593	−	0.228289i	$-0.0733131\pi$
$61$	8401.60	+	14552.0i	0.289093	+	0.500723i	−0.684501	+	0.729012i	$0.739980\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2474.49	−	4285.94i	−0.0726445	−	0.125824i
$66$		0			0
$67$	31267.5	−	54157.0i	0.850955	−	1.47390i	−0.0293918	−	0.999568i	$-0.509357\pi$
$67$	31267.5	−	54157.0i	0.850955	−	1.47390i	0.880347	−	0.474330i	$-0.157310\pi$
$68$		0			0
$69$	29264.6			0.739981
$70$		0			0
$71$	72301.0			1.70215			0.851077	−	0.525042i	$-0.175950\pi$
$71$	72301.0			1.70215			0.851077	+	0.525042i	$0.175950\pi$
$72$		0			0
$73$	−27838.4	+	48217.6i	−0.611417	+	1.05901i	0.379584	+	0.925157i	$0.376067\pi$
$73$	−27838.4	+	48217.6i	−0.611417	+	1.05901i	−0.991002	+	0.133849i	$0.957266\pi$
$74$		0			0
$75$	−13053.4	−	22609.1i	−0.267960	−	0.464121i
$76$		0			0
$77$		0			0
$78$		0			0
$79$	1994.60	+	3454.74i	0.0359573	+	0.0622799i	0.883444	−	0.468537i	$-0.155219\pi$
$79$	1994.60	+	3454.74i	0.0359573	+	0.0622799i	−0.847487	+	0.530817i	$0.821885\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	46092.2			0.734400			0.367200	−	0.930142i	$-0.380316\pi$
$83$	46092.2			0.734400			0.367200	+	0.930142i	$0.380316\pi$
$84$		0			0
$85$	−80546.5			−1.20920
$86$		0			0
$87$	−10354.4	+	17934.3i	−0.146665	+	0.254031i
$88$		0			0
$89$	67692.3	+	117247.i	0.905867	+	1.56901i	0.819749	+	0.572723i	$0.194113\pi$
$89$	67692.3	+	117247.i	0.905867	+	1.56901i	0.0861180	+	0.996285i	$0.472554\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$	16726.7	+	28971.6i	0.200541	+	0.347348i
$94$		0			0
$95$	−25888.9	+	44840.8i	−0.294309	+	0.509759i
$96$		0			0
$97$	−142878.			−1.54183			−0.770914	−	0.636939i	$-0.780200\pi$
$97$	−142878.			−1.54183			−0.770914	+	0.636939i	$0.780200\pi$
$98$		0			0
$99$	38667.8			0.396517
$100$		0			0
$101$	22233.5	−	38509.6i	0.216873	−	0.375635i	−0.736977	−	0.675917i	$-0.763748\pi$
$101$	22233.5	−	38509.6i	0.216873	−	0.375635i	0.953850	+	0.300283i	$0.0970810\pi$
$102$		0			0
$103$	101310.	+	175474.i	0.940931	+	1.62974i	0.763700	+	0.645572i	$0.223381\pi$
$103$	101310.	+	175474.i	0.940931	+	1.62974i	0.177232	+	0.984169i	$0.443286\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−49762.9	−	86192.0i	−0.420191	−	0.727792i	0.575767	−	0.817614i	$-0.304704\pi$
$107$	−49762.9	−	86192.0i	−0.420191	−	0.727792i	−0.995958	+	0.0898219i	$0.971370\pi$
$108$		0			0
$109$	−110465.	+	191331.i	−0.890551	+	1.54248i	−0.0513352	+	0.998681i	$0.516348\pi$
$109$	−110465.	+	191331.i	−0.890551	+	1.54248i	−0.839216	+	0.543798i	$0.816986\pi$
$110$		0			0
$111$	110213.			0.849033
$112$		0			0
$113$	29623.1			0.218240			0.109120	−	0.994029i	$-0.465197\pi$
$113$	29623.1			0.218240			0.109120	+	0.994029i	$0.465197\pi$
$114$		0			0
$115$	−126205.	+	218593.i	−0.889880	+	1.54132i
$116$		0			0
$117$	−2582.05	−	4472.25i	−0.0174382	−	0.0302038i
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−33420.4	−	57885.8i	−0.207514	−	0.359425i
$122$		0			0
$123$	−8233.42	+	14260.7i	−0.0490702	+	0.0849920i
$124$		0			0
$125$	−17407.2			−0.0996448
$126$		0			0
$127$	−264132.			−1.45315			−0.726577	−	0.687086i	$-0.758890\pi$
$127$	−264132.			−1.45315			−0.726577	+	0.687086i	$0.758890\pi$
$128$		0			0
$129$	93573.8	−	162075.i	0.495085	−	0.857513i
$130$		0			0
$131$	−38925.0	−	67420.2i	−0.198176	−	0.343251i	0.749761	−	0.661709i	$-0.230168\pi$
$131$	−38925.0	−	67420.2i	−0.198176	−	0.343251i	−0.947937	+	0.318458i	$0.896835\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$	−28294.6	−	49007.6i	−0.133619	−	0.231435i
$136$		0			0
$137$	−186201.	+	322510.i	−0.847581	+	1.46805i	0.0357792	+	0.999360i	$0.488609\pi$
$137$	−186201.	+	322510.i	−0.847581	+	1.46805i	−0.883361	+	0.468694i	$0.844725\pi$
$138$		0			0
$139$	274550.			1.20527			0.602636	−	0.798016i	$-0.294117\pi$
$139$	274550.			1.20527			0.602636	+	0.798016i	$0.294117\pi$
$140$		0			0
$141$	38550.3			0.163298
$142$		0			0
$143$	−15217.5	+	26357.6i	−0.0622307	+	0.107787i
$144$		0			0
$145$	−89307.3	−	154685.i	−0.352750	−	0.610981i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	156491.	+	271050.i	0.577462	+	1.00019i	0.995769	+	0.0918883i	$0.0292903\pi$
$149$	156491.	+	271050.i	0.577462	+	1.00019i	−0.418307	+	0.908306i	$0.637376\pi$
$150$		0			0
$151$	−216047.	+	374205.i	−0.771093	+	1.33557i	0.165872	+	0.986147i	$0.446956\pi$
$151$	−216047.	+	374205.i	−0.771093	+	1.33557i	−0.936965	+	0.349424i	$0.886377\pi$
$152$		0			0
$153$	−84047.7			−0.290267
$154$		0			0
$155$	−288539.			−0.964662
$156$		0			0
$157$	−42301.8	+	73268.9i	−0.136965	+	0.237231i	−0.926346	−	0.376673i	$-0.877068\pi$
$157$	−42301.8	+	73268.9i	−0.136965	+	0.237231i	0.789381	+	0.613903i	$0.210402\pi$
$158$		0			0
$159$	−115733.	−	200455.i	−0.363048	−	0.628818i
$160$		0			0
$161$		0			0
$162$		0			0
$163$	−153152.	−	265266.i	−0.451494	−	0.782011i	0.546985	−	0.837143i	$-0.315776\pi$
$163$	−153152.	−	265266.i	−0.451494	−	0.782011i	−0.998479	+	0.0551313i	$0.982442\pi$
$164$		0			0
$165$	−166756.	+	288831.i	−0.476840	+	0.825911i
$166$		0			0
$167$	606514.			1.68287			0.841433	−	0.540362i	$-0.181713\pi$
$167$	606514.			1.68287			0.841433	+	0.540362i	$0.181713\pi$
$168$		0			0
$169$	−367228.			−0.989053
$170$		0			0
$171$	−27014.2	+	46790.0i	−0.0706484	+	0.122367i
$172$		0			0
$173$	−144240.	−	249832.i	−0.366414	−	0.634648i	0.622588	−	0.782550i	$-0.286081\pi$
$173$	−144240.	−	249832.i	−0.366414	−	0.634648i	−0.989002	+	0.147902i	$0.952748\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$	−12774.2	−	22125.6i	−0.0306479	−	0.0530837i
$178$		0			0
$179$	−74429.1	+	128915.i	−0.173624	+	0.300726i	−0.939684	−	0.342043i	$-0.888881\pi$
$179$	−74429.1	+	128915.i	−0.173624	+	0.300726i	0.766060	+	0.642769i	$0.222214\pi$
$180$		0			0
$181$	−93377.8			−0.211859			−0.105930	−	0.994374i	$-0.533782\pi$
$181$	−93377.8			−0.211859			−0.105930	+	0.994374i	$0.533782\pi$
$182$		0			0
$183$	−151229.			−0.333816
$184$		0			0
$185$	−475297.	+	823238.i	−1.02102	+	1.76846i
$186$		0			0
$187$	247671.	+	428979.i	0.517930	+	0.897081i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−123457.	−	213835.i	−0.244869	−	0.424126i	0.717226	−	0.696841i	$-0.245412\pi$
$191$	−123457.	−	213835.i	−0.244869	−	0.424126i	−0.962095	+	0.272715i	$0.912078\pi$
$192$		0			0
$193$	−240718.	+	416937.i	−0.465175	+	0.805706i	−0.999209	−	0.0397564i	$-0.987342\pi$
$193$	−240718.	+	416937.i	−0.465175	+	0.805706i	0.534035	+	0.845463i	$0.320675\pi$
$194$		0			0
$195$	44540.8			0.0838826
$196$		0			0
$197$	−548236.			−1.00647			−0.503237	−	0.864149i	$-0.667858\pi$
$197$	−548236.			−1.00647			−0.503237	+	0.864149i	$0.667858\pi$
$198$		0			0
$199$	79089.6	−	136987.i	0.141575	−	0.245215i	−0.786515	−	0.617571i	$-0.788117\pi$
$199$	79089.6	−	136987.i	0.141575	−	0.245215i	0.928090	+	0.372356i	$0.121450\pi$
$200$		0			0
$201$	281408.	+	487413.i	0.491299	+	0.850955i
$202$		0			0
$203$		0			0
$204$		0			0
$205$	−71013.9	−	123000.i	−0.118021	−	0.204418i
$206$		0			0
$207$	−131691.	+	228095.i	−0.213614	+	0.369990i
$208$		0			0
$209$	318421.			0.504238
$210$		0			0
$211$	283510.			0.438392			0.219196	−	0.975681i	$-0.429657\pi$
$211$	283510.			0.438392			0.219196	+	0.975681i	$0.429657\pi$
$212$		0			0
$213$	−325355.	+	563531.i	−0.491369	+	0.851077i
$214$		0			0
$215$	807082.	+	1.39791e6i	1.19075	+	2.06244i
$216$		0			0
$217$		0			0
$218$		0			0
$219$	−250546.	−	433958.i	−0.353002	−	0.611417i
$220$		0			0
$221$	33076.6	−	57290.4i	0.0455554	−	0.0789043i
$222$		0			0
$223$	651135.			0.876817			0.438409	−	0.898776i	$-0.355542\pi$
$223$	651135.			0.876817			0.438409	+	0.898776i	$0.355542\pi$
$224$		0			0
$225$	234961.			0.309414
$226$		0			0
$227$	−189147.	+	327612.i	−0.243632	+	0.421983i	−0.961746	−	0.273942i	$-0.911672\pi$
$227$	−189147.	+	327612.i	−0.243632	+	0.421983i	0.718114	+	0.695925i	$0.245006\pi$
$228$		0			0
$229$	−11166.4	−	19340.8i	−0.0140710	−	0.0243717i	0.858904	−	0.512136i	$-0.171146\pi$
$229$	−11166.4	−	19340.8i	−0.0140710	−	0.0243717i	−0.872975	+	0.487765i	$0.837812\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	454470.	+	787165.i	0.548423	+	0.949896i	0.998383	+	0.0568472i	$0.0181048\pi$
$233$	454470.	+	787165.i	0.548423	+	0.949896i	−0.449960	+	0.893049i	$0.648562\pi$
$234$		0			0
$235$	−166250.	+	287953.i	−0.196377	+	0.340136i
$236$		0			0
$237$	−35902.7			−0.0415199
$238$		0			0
$239$	1.05363e6			1.19315			0.596573	−	0.802559i	$-0.296529\pi$
$239$	1.05363e6			1.19315			0.596573	+	0.802559i	$0.296529\pi$
$240$		0			0
$241$	526164.	−	911342.i	0.583550	−	1.01074i	−0.411504	−	0.911408i	$-0.634996\pi$
$241$	526164.	−	911342.i	0.583550	−	1.01074i	0.995054	−	0.0993307i	$-0.0316702\pi$
$242$		0			0
$243$	−29524.5	−	51137.9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$		0			0
$246$		0			0
$247$	−21262.7	−	36828.0i	−0.0221756	−	0.0384092i
$248$		0			0
$249$	−207415.	+	359253.i	−0.212003	+	0.367200i
$250$		0			0
$251$	−972876.			−0.974705			−0.487352	−	0.873205i	$-0.662037\pi$
$251$	−972876.			−0.974705			−0.487352	+	0.873205i	$0.662037\pi$
$252$		0			0
$253$	1.55226e6			1.52463
$254$		0			0
$255$	362459.	−	627798.i	0.349067	−	0.604601i
$256$		0			0
$257$	889738.	+	1.54107e6i	0.840290	+	1.45542i	0.889650	+	0.456644i	$0.150949\pi$
$257$	889738.	+	1.54107e6i	0.840290	+	1.45542i	−0.0493598	+	0.998781i	$0.515718\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$	−93189.3	−	161409.i	−0.0846769	−	0.146665i
$262$		0			0
$263$	−10979.8	+	19017.6i	−0.00978827	+	0.0169538i	−0.870878	−	0.491499i	$-0.836449\pi$
$263$	−10979.8	+	19017.6i	−0.00978827	+	0.0169538i	0.861090	+	0.508453i	$0.169782\pi$
$264$		0			0
$265$	1.99641e6			1.74637
$266$		0			0
$267$	−1.21846e6			−1.04601
$268$		0			0
$269$	−845555.	+	1.46454e6i	−0.712461	+	1.23402i	0.251470	+	0.967865i	$0.419086\pi$
$269$	−845555.	+	1.46454e6i	−0.712461	+	1.23402i	−0.963931	+	0.266153i	$0.914247\pi$
$270$		0			0
$271$	−233932.	−	405182.i	−0.193493	−	0.335140i	0.752912	−	0.658121i	$-0.228648\pi$
$271$	−233932.	−	405182.i	−0.193493	−	0.335140i	−0.946406	+	0.322981i	$0.895315\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−692381.	−	1.19924e6i	−0.552095	−	0.956256i
$276$		0			0
$277$	−453084.	+	784765.i	−0.354797	+	0.614526i	−0.987083	−	0.160209i	$-0.948783\pi$
$277$	−453084.	+	784765.i	−0.354797	+	0.614526i	0.632286	+	0.774735i	$0.282117\pi$
$278$		0			0
$279$	−301081.			−0.231565
$280$		0			0
$281$	−781388.			−0.590338			−0.295169	−	0.955445i	$-0.595376\pi$
$281$	−781388.			−0.590338			−0.295169	+	0.955445i	$0.595376\pi$
$282$		0			0
$283$	−749216.	+	1.29768e6i	−0.556085	+	0.963167i	0.441733	+	0.897146i	$0.354364\pi$
$283$	−749216.	+	1.29768e6i	−0.556085	+	0.963167i	−0.997818	+	0.0660209i	$0.978970\pi$
$284$		0			0
$285$	−233000.	−	403567.i	−0.169920	−	0.294309i
$286$		0			0
$287$		0			0
$288$		0			0
$289$	171595.	+	297211.i	0.120854	+	0.209325i
$290$		0			0
$291$	642951.	−	1.11362e6i	0.445087	−	0.770914i
$292$		0			0
$293$	1.56070e6			1.06206			0.531031	−	0.847352i	$-0.321805\pi$
$293$	1.56070e6			1.06206			0.531031	+	0.847352i	$0.321805\pi$
$294$		0			0
$295$	220357.			0.147425
$296$		0			0
$297$	−174005.	+	301386.i	−0.114464	+	0.198258i
$298$		0			0
$299$	−103653.	−	179532.i	−0.0670506	−	0.116135i
$300$		0			0
$301$		0			0
$302$		0			0
$303$	200102.	+	346587.i	0.125212	+	0.216873i
$304$		0			0
$305$	652180.	−	1.12961e6i	0.401438	−	0.695310i
$306$		0			0
$307$	889308.			0.538525			0.269263	−	0.963067i	$-0.413220\pi$
$307$	889308.			0.538525			0.269263	+	0.963067i	$0.413220\pi$
$308$		0			0
$309$	−1.82357e6			−1.08649
$310$		0			0
$311$	1.22413e6	−	2.12026e6i	0.717674	−	1.24305i	−0.244246	−	0.969713i	$-0.578540\pi$
$311$	1.22413e6	−	2.12026e6i	0.717674	−	1.24305i	0.961919	−	0.273334i	$-0.0881263\pi$
$312$		0			0
$313$	−1.36551e6	−	2.36514e6i	−0.787834	−	1.36457i	−0.927291	−	0.374341i	$-0.877869\pi$
$313$	−1.36551e6	−	2.36514e6i	−0.787834	−	1.36457i	0.139457	−	0.990228i	$-0.455464\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	124802.	+	216163.i	0.0697545	+	0.120818i	0.898793	−	0.438373i	$-0.144445\pi$
$317$	124802.	+	216163.i	0.0697545	+	0.120818i	−0.829039	+	0.559191i	$0.811112\pi$
$318$		0			0
$319$	−549219.	+	951275.i	−0.302182	+	0.523395i
$320$		0			0
$321$	895733.			0.485195
$322$		0			0
$323$	−692115.			−0.369124
$324$		0			0
$325$	−92467.9	+	160159.i	−0.0485605	+	0.0841092i
$326$		0			0
$327$	−994186.	−	1.72198e6i	−0.514160	−	0.890551i
$328$		0			0
$329$		0			0
$330$		0			0
$331$	−323308.	−	559986.i	−0.162199	−	0.280936i	0.773458	−	0.633847i	$-0.218525\pi$
$331$	−323308.	−	559986.i	−0.162199	−	0.280936i	−0.935657	+	0.352911i	$0.885192\pi$
$332$		0			0
$333$	−495957.	+	859023.i	−0.245095	+	0.424516i
$334$		0			0
$335$	−4.85433e6			−2.36329
$336$		0			0
$337$	−1.02782e6			−0.492994			−0.246497	−	0.969144i	$-0.579280\pi$
$337$	−1.02782e6			−0.492994			−0.246497	+	0.969144i	$0.579280\pi$
$338$		0			0
$339$	−133304.	+	230890.i	−0.0630006	+	0.109120i
$340$		0			0
$341$	887224.	+	1.53672e6i	0.413187	+	0.715662i
$342$		0			0
$343$		0			0
$344$		0			0
$345$	−1.13584e6	−	1.96734e6i	−0.513773	−	0.889880i
$346$		0			0
$347$	824579.	−	1.42821e6i	0.367628	−	0.636750i	−0.621566	−	0.783362i	$-0.713503\pi$
$347$	824579.	−	1.42821e6i	0.367628	−	0.636750i	0.989194	+	0.146611i	$0.0468367\pi$
$348$		0			0
$349$	−2.21201e6			−0.972128			−0.486064	−	0.873923i	$-0.661568\pi$
$349$	−2.21201e6			−0.972128			−0.486064	+	0.873923i	$0.661568\pi$
$350$		0			0
$351$	46477.0			0.0201358
$352$		0			0
$353$	550789.	−	953995.i	0.235260	−	0.407483i	−0.724088	−	0.689708i	$-0.757739\pi$
$353$	550789.	−	953995.i	0.235260	−	0.407483i	0.959348	+	0.282225i	$0.0910724\pi$
$354$		0			0
$355$	−2.80621e6	−	4.86050e6i	−1.18181	−	2.04696i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	649699.	+	1.12531e6i	0.266058	+	0.460826i	0.967840	−	0.251566i	$-0.0809455\pi$
$359$	649699.	+	1.12531e6i	0.266058	+	0.460826i	−0.701782	+	0.712391i	$0.747612\pi$
$360$		0			0
$361$	1.01559e6	−	1.75906e6i	0.410159	−	0.710416i
$362$		0			0
$363$	601567.			0.239617
$364$		0			0
$365$	4.32196e6			1.69804
$366$		0			0
$367$	778345.	−	1.34813e6i	0.301652	−	0.522477i	−0.674858	−	0.737948i	$-0.735795\pi$
$367$	778345.	−	1.34813e6i	0.301652	−	0.522477i	0.976510	+	0.215470i	$0.0691285\pi$
$368$		0			0
$369$	−74100.8	−	128346.i	−0.0283307	−	0.0490702i
$370$		0			0
$371$		0			0
$372$		0			0
$373$	1.56330e6	+	2.70772e6i	0.581796	+	1.00770i	0.995267	+	0.0971823i	$0.0309830\pi$
$373$	1.56330e6	+	2.70772e6i	0.581796	+	1.00770i	−0.413471	+	0.910517i	$0.635684\pi$
$374$		0			0
$375$	78332.5	−	135676.i	0.0287650	−	0.0498224i
$376$		0			0
$377$	146697.			0.0531579
$378$		0			0
$379$	−2.96497e6			−1.06029			−0.530143	−	0.847908i	$-0.677862\pi$
$379$	−2.96497e6			−1.06029			−0.530143	+	0.847908i	$0.677862\pi$
$380$		0			0
$381$	1.18859e6	−	2.05870e6i	0.419489	−	0.726577i
$382$		0			0
$383$	1.37387e6	+	2.37962e6i	0.478574	+	0.828915i	0.999698	−	0.0245659i	$-0.00782036\pi$
$383$	1.37387e6	+	2.37962e6i	0.478574	+	0.828915i	−0.521124	+	0.853481i	$0.674487\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$	842164.	+	1.45867e6i	0.285838	+	0.495085i
$388$		0			0
$389$	305671.	−	529438.i	0.102419	−	0.177395i	−0.810262	−	0.586068i	$-0.800675\pi$
$389$	305671.	−	529438.i	0.102419	−	0.177395i	0.912681	+	0.408673i	$0.134008\pi$
$390$		0			0
$391$	−3.37397e6			−1.11609
$392$		0			0
$393$	700651.			0.228834
$394$		0			0
$395$	154832.	−	268177.i	0.0499307	−	0.0864826i
$396$		0			0
$397$	1.08397e6	+	1.87749e6i	0.345176	+	0.597862i	0.985386	−	0.170338i	$-0.0544859\pi$
$397$	1.08397e6	+	1.87749e6i	0.345176	+	0.597862i	−0.640210	+	0.768200i	$0.721153\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	1.57322e6	+	2.72489e6i	0.488572	+	0.846231i	0.999914	−	0.0131465i	$-0.00418480\pi$
$401$	1.57322e6	+	2.72489e6i	0.488572	+	0.846231i	−0.511342	+	0.859377i	$0.670851\pi$
$402$		0			0
$403$	118489.	−	205229.i	0.0363426	−	0.0629473i
$404$		0			0
$405$	509302.			0.154290
$406$		0			0
$407$	5.84593e6			1.74931
$408$		0			0
$409$	2.79082e6	−	4.83385e6i	0.824943	−	1.42884i	−0.0770193	−	0.997030i	$-0.524540\pi$
$409$	2.79082e6	−	4.83385e6i	0.824943	−	1.42884i	0.901963	−	0.431814i	$-0.142126\pi$
$410$		0			0
$411$	−1.67581e6	−	2.90259e6i	−0.489351	−	0.847581i
$412$		0			0
$413$		0			0
$414$		0			0
$415$	−1.78897e6	−	3.09859e6i	−0.509898	−	0.883169i
$416$		0			0
$417$	−1.23548e6	+	2.13991e6i	−0.347932	+	0.602636i
$418$		0			0
$419$	−2.25054e6			−0.626257			−0.313128	−	0.949711i	$-0.601377\pi$
$419$	−2.25054e6			−0.626257			−0.313128	+	0.949711i	$0.601377\pi$
$420$		0			0
$421$	3.45914e6			0.951180			0.475590	−	0.879667i	$-0.342235\pi$
$421$	3.45914e6			0.951180			0.475590	+	0.879667i	$0.342235\pi$
$422$		0			0
$423$	−173476.	+	300470.i	−0.0471400	+	0.0816489i
$424$		0			0
$425$	1.50495e6	+	2.60665e6i	0.404156	+	0.700019i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	−136958.	−	237218.i	−0.0359289	−	0.0622307i
$430$		0			0
$431$	3.27963e6	−	5.68049e6i	0.850417	−	1.47297i	−0.0304157	−	0.999537i	$-0.509683\pi$
$431$	3.27963e6	−	5.68049e6i	0.850417	−	1.47297i	0.880833	−	0.473428i	$-0.156984\pi$
$432$		0			0
$433$	−5.05669e6			−1.29612			−0.648062	−	0.761587i	$-0.724420\pi$
$433$	−5.05669e6			−1.29612			−0.648062	+	0.761587i	$0.724420\pi$
$434$		0			0
$435$	1.60753e6			0.407320
$436$		0			0
$437$	−1.08445e6	+	1.87832e6i	−0.271647	+	0.470506i
$438$		0			0
$439$	2.11613e6	+	3.66524e6i	0.524059	+	0.907697i	0.999608	+	0.0280074i	$0.00891619\pi$
$439$	2.11613e6	+	3.66524e6i	0.524059	+	0.907697i	−0.475549	+	0.879689i	$0.657750\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.02763e6	+	5.24400e6i	0.732981	+	1.26956i	0.955603	+	0.294656i	$0.0952050\pi$
$443$	3.02763e6	+	5.24400e6i	0.732981	+	1.26956i	−0.222622	+	0.974905i	$0.571462\pi$
$444$		0			0
$445$	5.25467e6	−	9.10135e6i	1.25790	−	2.17874i
$446$		0			0
$447$	−2.81684e6			−0.666796
$448$		0			0
$449$	−299186.			−0.0700368			−0.0350184	−	0.999387i	$-0.511149\pi$
$449$	−299186.			−0.0700368			−0.0350184	+	0.999387i	$0.511149\pi$
$450$		0			0
$451$	−436719.	+	756420.i	−0.101102	+	0.175114i
$452$		0			0
$453$	−1.94443e6	−	3.36785e6i	−0.445191	−	0.771093i
$454$		0			0
$455$		0			0
$456$		0			0
$457$	−1.87589e6	−	3.24913e6i	−0.420161	−	0.727740i	0.575794	−	0.817595i	$-0.304693\pi$
$457$	−1.87589e6	−	3.24913e6i	−0.420161	−	0.727740i	−0.995955	+	0.0898545i	$0.971360\pi$
$458$		0			0
$459$	378215.	−	655087.i	0.0837928	−	0.145133i
$460$		0			0
$461$	−6.94525e6			−1.52207			−0.761036	−	0.648709i	$-0.775309\pi$
$461$	−6.94525e6			−1.52207			−0.761036	+	0.648709i	$0.775309\pi$
$462$		0			0
$463$	−9.13226e6			−1.97982			−0.989910	−	0.141697i	$-0.954744\pi$
$463$	−9.13226e6			−1.97982			−0.989910	+	0.141697i	$0.954744\pi$
$464$		0			0
$465$	1.29843e6	−	2.24894e6i	0.278474	−	0.482331i
$466$		0			0
$467$	−212068.	−	367313.i	−0.0449970	−	0.0779371i	0.842650	−	0.538462i	$-0.180994\pi$
$467$	−212068.	−	367313.i	−0.0449970	−	0.0779371i	−0.887647	+	0.460525i	$0.847661\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$	−380717.	−	659420.i	−0.0790769	−	0.136965i
$472$		0			0
$473$	4.96336e6	−	8.59680e6i	1.02005	−	1.76679i
$474$		0			0
$475$	1.93485e6			0.393472
$476$		0			0
$477$	2.08319e6			0.419212
$478$		0			0
$479$	−3.89303e6	+	6.74292e6i	−0.775262	+	1.34279i	0.159385	+	0.987216i	$0.449049\pi$
$479$	−3.89303e6	+	6.74292e6i	−0.775262	+	1.34279i	−0.934647	+	0.355577i	$0.884285\pi$
$480$		0			0
$481$	−390364.	−	676130.i	−0.0769320	−	0.133250i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	5.54550e6	+	9.60509e6i	1.07050	+	1.85416i
$486$		0			0
$487$	−1.16130e6	+	2.01143e6i	−0.221881	+	0.384310i	−0.955379	−	0.295382i	$-0.904553\pi$
$487$	−1.16130e6	+	2.01143e6i	−0.221881	+	0.384310i	0.733498	+	0.679692i	$0.237886\pi$
$488$		0			0
$489$	2.75673e6			0.521341
$490$		0			0
$491$	6.01036e6			1.12512			0.562558	−	0.826758i	$-0.309817\pi$
$491$	6.01036e6			1.12512			0.562558	+	0.826758i	$0.309817\pi$
$492$		0			0
$493$	1.19377e6	−	2.06768e6i	0.221210	−	0.383147i
$494$		0			0
$495$	−1.50081e6	−	2.59948e6i	−0.275304	−	0.476840i
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−1.68693e6	−	2.92184e6i	−0.303281	−	0.525298i	0.673596	−	0.739100i	$-0.264749\pi$
$499$	−1.68693e6	−	2.92184e6i	−0.303281	−	0.525298i	−0.976877	+	0.213802i	$0.931415\pi$
$500$		0			0
$501$	−2.72931e6	+	4.72731e6i	−0.485801	+	0.841433i
$502$		0			0
$503$	1.22068e6			0.215120			0.107560	−	0.994199i	$-0.465696\pi$
$503$	1.22068e6			0.215120			0.107560	+	0.994199i	$0.465696\pi$
$504$		0			0
$505$	−3.45179e6			−0.602304
$506$		0			0
$507$	1.65253e6	−	2.86226e6i	0.285515	−	0.494526i
$508$		0			0
$509$	784485.	+	1.35877e6i	0.134212	+	0.232461i	0.925296	−	0.379246i	$-0.123816\pi$
$509$	784485.	+	1.35877e6i	0.134212	+	0.232461i	−0.791084	+	0.611707i	$0.790483\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$	−243128.	−	421110.i	−0.0407888	−	0.0706484i
$514$		0			0
$515$	7.86424e6	−	1.36213e7i	1.30659	−	2.26308i
$516$		0			0
$517$	2.04480e6			0.336452
$518$		0			0
$519$	2.59633e6			0.423098
$520$		0			0
$521$	5.33893e6	−	9.24730e6i	0.861708	−	1.49252i	−0.00857106	−	0.999963i	$-0.502728\pi$
$521$	5.33893e6	−	9.24730e6i	0.861708	−	1.49252i	0.870279	−	0.492559i	$-0.163938\pi$
$522$		0			0
$523$	−6.05033e6	−	1.04795e7i	−0.967219	−	1.67527i	−0.703531	−	0.710665i	$-0.748394\pi$
$523$	−6.05033e6	−	1.04795e7i	−0.967219	−	1.67527i	−0.263689	−	0.964608i	$-0.584939\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−1.92845e6	−	3.34018e6i	−0.302470	−	0.523894i
$528$		0			0
$529$	−2.06836e6	+	3.58251e6i	−0.321357	+	0.556607i
$530$		0			0
$531$	229936.			0.0353892
$532$		0			0
$533$	116648.			0.0177852
$534$		0			0
$535$	−3.86289e6	+	6.69071e6i	−0.583482	+	1.01062i
$536$		0			0
$537$	−669861.	−	1.16023e6i	−0.100242	−	0.173624i
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.59903e6	+	2.76959e6i	0.234889	+	0.406839i	0.959240	−	0.282591i	$-0.0911941\pi$
$541$	1.59903e6	+	2.76959e6i	0.234889	+	0.406839i	−0.724352	+	0.689431i	$0.757861\pi$
$542$		0			0
$543$	420200.	−	727808.i	0.0611585	−	0.105930i
$544$		0			0
$545$	1.71499e7			2.47326
$546$		0			0
$547$	3.97811e6			0.568471			0.284235	−	0.958755i	$-0.408260\pi$
$547$	3.97811e6			0.568471			0.284235	+	0.958755i	$0.408260\pi$
$548$		0			0
$549$	680529.	−	1.17871e6i	0.0963643	−	0.166908i
$550$		0			0
$551$	−767394.	−	1.32917e6i	−0.107681	−	0.186509i
$552$		0			0
$553$		0			0
$554$		0			0
$555$	−4.27767e6	−	7.40915e6i	−0.589488	−	1.02102i
$556$		0			0
$557$	−5.70567e6	+	9.88251e6i	−0.779236	+	1.34968i	0.153147	+	0.988203i	$0.451059\pi$
$557$	−5.70567e6	+	9.88251e6i	−0.779236	+	1.34968i	−0.932383	+	0.361472i	$0.882274\pi$
$558$		0			0
$559$	−1.32572e6			−0.179441
$560$		0			0
$561$	−4.45808e6			−0.598054
$562$		0			0
$563$	−3.23486e6	+	5.60295e6i	−0.430115	+	0.744982i	−0.996883	−	0.0788962i	$-0.974860\pi$
$563$	−3.23486e6	+	5.60295e6i	−0.430115	+	0.744982i	0.566768	+	0.823878i	$0.308194\pi$
$564$		0			0
$565$	−1.14976e6	−	1.99144e6i	−0.151526	−	0.262450i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−358514.	−	620965.i	−0.0464222	−	0.0804056i	0.841881	−	0.539664i	$-0.181449\pi$
$569$	−358514.	−	620965.i	−0.0464222	−	0.0804056i	−0.888303	+	0.459258i	$0.848115\pi$
$570$		0			0
$571$	−3.00143e6	+	5.19863e6i	−0.385246	+	0.667265i	−0.991803	−	0.127774i	$-0.959217\pi$
$571$	−3.00143e6	+	5.19863e6i	−0.385246	+	0.667265i	0.606558	+	0.795040i	$0.292550\pi$
$572$		0			0
$573$	2.22223e6			0.282750
$574$		0			0
$575$	9.43217e6			1.18971
$576$		0			0
$577$	7.78560e6	−	1.34850e7i	0.973537	−	1.68621i	0.288855	−	0.957373i	$-0.406726\pi$
$577$	7.78560e6	−	1.34850e7i	0.973537	−	1.68621i	0.684682	−	0.728842i	$-0.259941\pi$
$578$		0			0
$579$	−2.16647e6	−	3.75243e6i	−0.268569	−	0.465175i
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−6.13873e6	−	1.06326e7i	−0.748010	−	1.29559i
$584$		0			0
$585$	−200434.	+	347161.i	−0.0242148	+	0.0419413i
$586$		0			0
$587$	−7.84621e6			−0.939863			−0.469931	−	0.882703i	$-0.655721\pi$
$587$	−7.84621e6			−0.939863			−0.469931	+	0.882703i	$0.655721\pi$
$588$		0			0
$589$	−2.47934e6			−0.294475
$590$		0			0
$591$	2.46706e6	−	4.27308e6i	0.290544	−	0.503237i
$592$		0			0
$593$	6.66240e6	+	1.15396e7i	0.778026	+	1.34758i	0.933078	+	0.359674i	$0.117112\pi$
$593$	6.66240e6	+	1.15396e7i	0.778026	+	1.34758i	−0.155052	+	0.987906i	$0.549555\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$	711807.	+	1.23289e6i	0.0817384	+	0.141575i
$598$		0			0
$599$	3.88958e6	−	6.73695e6i	0.442930	−	0.767178i	−0.554975	−	0.831867i	$-0.687272\pi$
$599$	3.88958e6	−	6.73695e6i	0.442930	−	0.767178i	0.997905	+	0.0646890i	$0.0206055\pi$
$600$		0			0
$601$	−8.62898e6			−0.974480			−0.487240	−	0.873268i	$-0.661996\pi$
$601$	−8.62898e6			−0.974480			−0.487240	+	0.873268i	$0.661996\pi$
$602$		0			0
$603$	−5.06534e6			−0.567304
$604$		0			0
$605$	−2.59428e6	+	4.49343e6i	−0.288157	+	0.499102i
$606$		0			0
$607$	−3.54676e6	−	6.14317e6i	−0.390715	−	0.676739i	0.601829	−	0.798625i	$-0.294439\pi$
$607$	−3.54676e6	−	6.14317e6i	−0.390715	−	0.676739i	−0.992544	+	0.121886i	$0.961106\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−136542.	−	236497.i	−0.0147966	−	0.0256285i
$612$		0			0
$613$	−2.26320e6	+	3.91998e6i	−0.243261	+	0.421340i	−0.961641	−	0.274311i	$-0.911550\pi$
$613$	−2.26320e6	+	3.91998e6i	−0.243261	+	0.421340i	0.718381	+	0.695650i	$0.244884\pi$
$614$		0			0
$615$	1.27825e6			0.136279
$616$		0			0
$617$	−8.38009e6			−0.886208			−0.443104	−	0.896470i	$-0.646123\pi$
$617$	−8.38009e6			−0.886208			−0.443104	+	0.896470i	$0.646123\pi$
$618$		0			0
$619$	−84171.0	+	145788.i	−0.00882949	+	0.0152931i	−0.870406	−	0.492334i	$-0.836144\pi$
$619$	−84171.0	+	145788.i	−0.00882949	+	0.0152931i	0.861577	+	0.507627i	$0.169477\pi$
$620$		0			0
$621$	−1.18522e6	−	2.05286e6i	−0.123330	−	0.213614i
$622$		0			0
$623$		0			0
$624$		0			0
$625$	5.20805e6	+	9.02061e6i	0.533305	+	0.923711i
$626$		0			0
$627$	−1.43289e6	+	2.48185e6i	−0.145561	+	0.252119i
$628$		0			0
$629$	−1.27066e7			−1.28057
$630$		0			0
$631$	1.66208e7			1.66180			0.830899	−	0.556423i	$-0.187826\pi$
$631$	1.66208e7			1.66180			0.830899	+	0.556423i	$0.187826\pi$
$632$		0			0
$633$	−1.27580e6	+	2.20974e6i	−0.126553	+	0.219196i
$634$		0			0
$635$	1.02517e7	+	1.77565e7i	1.00893	+	1.74752i
$636$		0			0
$637$		0			0
$638$		0			0
$639$	−2.92819e6	−	5.07178e6i	−0.283692	−	0.491369i
$640$		0			0
$641$	−512367.	+	887446.i	−0.0492534	+	0.0853094i	−0.889601	−	0.456738i	$-0.849018\pi$
$641$	−512367.	+	887446.i	−0.0492534	+	0.0853094i	0.840348	+	0.542048i	$0.182351\pi$
$642$		0			0
$643$	1.22962e7			1.17286			0.586428	−	0.810001i	$-0.300534\pi$
$643$	1.22962e7			1.17286			0.586428	+	0.810001i	$0.300534\pi$
$644$		0			0
$645$	−1.45275e7			−1.37496
$646$		0			0
$647$	−1.08269e6	+	1.87527e6i	−0.101682	+	0.176118i	−0.912378	−	0.409350i	$-0.865756\pi$
$647$	−1.08269e6	+	1.87527e6i	−0.101682	+	0.176118i	0.810696	+	0.585467i	$0.199089\pi$
$648$		0			0
$649$	−677573.	−	1.17359e6i	−0.0631458	−	0.109372i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	6.12507e6	+	1.06089e7i	0.562119	+	0.973619i	0.997311	+	0.0732814i	$0.0233471\pi$
$653$	6.12507e6	+	1.06089e7i	0.562119	+	0.973619i	−0.435192	+	0.900338i	$0.643320\pi$
$654$		0			0
$655$	−3.02158e6	+	5.23354e6i	−0.275189	+	0.476642i
$656$		0			0
$657$	4.50983e6			0.407612
$658$		0			0
$659$	1.18607e6			0.106389			0.0531944	−	0.998584i	$-0.483060\pi$
$659$	1.18607e6			0.106389			0.0531944	+	0.998584i	$0.483060\pi$
$660$		0			0
$661$	−7.26909e6	+	1.25904e7i	−0.647107	+	1.12082i	0.336703	+	0.941611i	$0.390688\pi$
$661$	−7.26909e6	+	1.25904e7i	−0.647107	+	1.12082i	−0.983811	+	0.179212i	$0.942645\pi$
$662$		0			0
$663$	297689.	+	515613.i	0.0263014	+	0.0455554i
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−3.74095e6	−	6.47951e6i	−0.325587	−	0.563933i
$668$		0			0
$669$	−2.93011e6	+	5.07510e6i	−0.253115	+	0.438409i
$670$		0			0
$671$	−8.02151e6			−0.687781
$672$		0			0
$673$	−5.99405e6			−0.510132			−0.255066	−	0.966924i	$-0.582097\pi$
$673$	−5.99405e6			−0.510132			−0.255066	+	0.966924i	$0.582097\pi$
$674$		0			0
$675$	−1.05732e6	+	1.83134e6i	−0.0893201	+	0.154707i
$676$		0			0
$677$	1.04194e7	+	1.80470e7i	0.873721	+	1.51333i	0.858119	+	0.513451i	$0.171633\pi$
$677$	1.04194e7	+	1.80470e7i	0.873721	+	1.51333i	0.0156016	+	0.999878i	$0.495034\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$	−1.70232e6	−	2.94851e6i	−0.140661	−	0.243632i
$682$		0			0
$683$	−2.27783e6	+	3.94531e6i	−0.186840	+	0.323616i	−0.944195	−	0.329387i	$-0.893158\pi$
$683$	−2.27783e6	+	3.94531e6i	−0.186840	+	0.323616i	0.757355	+	0.653003i	$0.226491\pi$
$684$		0			0
$685$	2.89080e7			2.35392
$686$		0			0
$687$	200995.			0.0162478
$688$		0			0
$689$	−819831.	+	1.41999e6i	−0.0657925	+	0.113956i
$690$		0			0
$691$	842712.	+	1.45962e6i	0.0671404	+	0.116291i	0.897641	−	0.440726i	$-0.145279\pi$
$691$	842712.	+	1.45962e6i	0.0671404	+	0.116291i	−0.830501	+	0.557017i	$0.811946\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.06561e7	−	1.84569e7i	−0.836827	−	1.44943i
$696$		0			0
$697$	949246.	−	1.64414e6i	0.0740111	−	0.128191i
$698$		0			0
$699$	−8.18046e6			−0.633264
$700$		0			0
$701$	2.45349e6			0.188577			0.0942887	−	0.995545i	$-0.469942\pi$
$701$	2.45349e6			0.188577			0.0942887	+	0.995545i	$0.469942\pi$
$702$		0			0
$703$	−4.08410e6	+	7.07387e6i	−0.311680	+	0.539845i
$704$		0			0
$705$	−1.49625e6	−	2.59158e6i	−0.113379	−	0.196377i
$706$		0			0
$707$		0			0
$708$		0			0
$709$	6.61569e6	+	1.14587e7i	0.494265	+	0.856092i	0.999978	−	0.00660968i	$-0.00210394\pi$
$709$	6.61569e6	+	1.14587e7i	0.494265	+	0.856092i	−0.505713	+	0.862702i	$0.668771\pi$
$710$		0			0
$711$	161562.	−	279834.i	0.0119858	−	0.0207600i
$712$		0			0
$713$	−1.20865e7			−0.890380
$714$		0			0
$715$	2.36255e6			0.172828
$716$		0			0
$717$	−4.74134e6	+	8.21224e6i	−0.344432	+	0.596573i
$718$		0			0
$719$	6.76685e6	+	1.17205e7i	0.488162	+	0.845522i	0.999907	−	0.0136156i	$-0.00433413\pi$
$719$	6.76685e6	+	1.17205e7i	0.488162	+	0.845522i	−0.511745	+	0.859137i	$0.671001\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$	4.73547e6	+	8.20208e6i	0.336913	+	0.583550i
$724$		0			0
$725$	−3.33728e6	+	5.78033e6i	−0.235802	+	0.408421i
$726$		0			0
$727$	−5.29416e6			−0.371502			−0.185751	−	0.982597i	$-0.559472\pi$
$727$	−5.29416e6			−0.371502			−0.185751	+	0.982597i	$0.559472\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−1.07883e7	+	1.86859e7i	−0.746722	+	1.29336i
$732$		0			0
$733$	−1.12507e7	−	1.94868e7i	−0.773426	−	1.33961i	−0.935675	−	0.352863i	$-0.885208\pi$
$733$	−1.12507e7	−	1.94868e7i	−0.773426	−	1.33961i	0.162249	−	0.986750i	$-0.448125\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	1.49265e7	+	2.58535e7i	1.01225	+	1.75327i
$738$		0			0
$739$	−8.03696e6	+	1.39204e7i	−0.541353	+	0.937652i	0.457473	+	0.889223i	$0.348755\pi$
$739$	−8.03696e6	+	1.39204e7i	−0.541353	+	0.937652i	−0.998827	+	0.0484282i	$0.984579\pi$
$740$		0			0
$741$	382728.			0.0256062
$742$		0			0
$743$	−2.31604e6			−0.153913			−0.0769563	−	0.997034i	$-0.524520\pi$
$743$	−2.31604e6			−0.153913			−0.0769563	+	0.997034i	$0.524520\pi$
$744$		0			0
$745$	1.21477e7	−	2.10405e7i	0.801871	−	1.38888i
$746$		0			0
$747$	−1.86674e6	−	3.23328e6i	−0.122400	−	0.212003i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	3.88632e6	+	6.73130e6i	0.251442	+	0.435511i	0.963923	−	0.266181i	$-0.0857618\pi$
$751$	3.88632e6	+	6.73130e6i	0.251442	+	0.435511i	−0.712481	+	0.701692i	$0.752428\pi$
$752$		0			0
$753$	4.37794e6	−	7.58282e6i	0.281373	−	0.487352i
$754$		0			0
$755$	3.35417e7			2.14150
$756$		0			0
$757$	−1.59716e7			−1.01300			−0.506498	−	0.862241i	$-0.669060\pi$
$757$	−1.59716e7			−1.01300			−0.506498	+	0.862241i	$0.669060\pi$
$758$		0			0
$759$	−6.98518e6	+	1.20987e7i	−0.440122	+	0.762313i
$760$		0			0
$761$	9.24701e6	+	1.60163e7i	0.578815	+	1.00254i	0.995616	+	0.0935391i	$0.0298180\pi$
$761$	9.24701e6	+	1.60163e7i	0.578815	+	1.00254i	−0.416801	+	0.908998i	$0.636849\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$	3.26213e6	+	5.65018e6i	0.201534	+	0.349067i
$766$		0			0
$767$	−90490.2	+	156734.i	−0.00555409	+	0.00961997i
$768$		0			0
$769$	2.33524e7			1.42402			0.712009	−	0.702170i	$-0.247786\pi$
$769$	2.33524e7			1.42402			0.712009	+	0.702170i	$0.247786\pi$
$770$		0			0
$771$	−1.60153e7			−0.970283
$772$		0			0
$773$	3.62631e6	−	6.28095e6i	0.218281	−	0.378074i	−0.736001	−	0.676980i	$-0.763288\pi$
$773$	3.62631e6	−	6.28095e6i	0.218281	−	0.378074i	0.954283	+	0.298906i	$0.0966217\pi$
$774$		0			0
$775$	5.39113e6	+	9.33771e6i	0.322423	+	0.558452i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−610204.	−	1.05690e6i	−0.0360273	−	0.0624011i
$780$		0			0
$781$	−1.72575e7	+	2.98909e7i	−1.01240	+	1.75352i
$782$		0			0
$783$	1.67741e6			0.0977764
$784$		0			0
$785$	6.56742e6			0.380383

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:	\( N \)	\(=\)	\( 588 = 2^{2} \cdot 3 \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	588.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 7.79423i) q^{3} +(-38.8129 - 67.2259i) q^{5} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)	\(q+(-4.50000 + 7.79423i) q^{3} +(-38.8129 - 67.2259i) q^{5} +(-40.5000 - 70.1481i) q^{9} +(-238.690 + 413.423i) q^{11} +63.7544 q^{13} +698.632 q^{15} +(518.813 - 898.610i) q^{17} +(-333.509 - 577.654i) q^{19} +(-1625.81 - 2815.99i) q^{23} +(-1450.38 + 2512.13i) q^{25} +729.000 q^{27} +2300.97 q^{29} +(1858.53 - 3219.06i) q^{31} +(-2148.21 - 3720.81i) q^{33} +(-6122.93 - 10605.2i) q^{37} +(-286.895 + 496.916i) q^{39} +1829.65 q^{41} -20794.2 q^{43} +(-3143.84 + 5445.29i) q^{45} +(-2141.68 - 3709.51i) q^{47} +(4669.32 + 8087.49i) q^{51} +(-12859.2 + 22272.8i) q^{53} +37057.0 q^{55} +6003.16 q^{57} +(-1419.36 + 2458.40i) q^{59} +(8401.60 + 14552.0i) q^{61} +(-2474.49 - 4285.94i) q^{65} +(31267.5 - 54157.0i) q^{67} +29264.6 q^{69} +72301.0 q^{71} +(-27838.4 + 48217.6i) q^{73} +(-13053.4 - 22609.1i) q^{75} +(1994.60 + 3454.74i) q^{79} +(-3280.50 + 5681.99i) q^{81} +46092.2 q^{83} -80546.5 q^{85} +(-10354.4 + 17934.3i) q^{87} +(67692.3 + 117247. i) q^{89} +(16726.7 + 28971.6i) q^{93} +(-25888.9 + 44840.8i) q^{95} -142878. q^{97} +38667.8 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{3} - 6 q^{5} - 162 q^{9}+O(q^{10}) \) 4 * q - 18 * q^3 - 6 * q^5 - 162 * q^9	\( 4 q - 18 q^{3} - 6 q^{5} - 162 q^{9} + 90 q^{11} - 1536 q^{13} + 108 q^{15} + 1926 q^{17} + 2248 q^{19} - 6354 q^{23} - 4906 q^{25} + 2916 q^{27} + 21144 q^{29} - 3312 q^{31} + 810 q^{33} - 2104 q^{37} + 6912 q^{39} - 2532 q^{41} - 11536 q^{43} - 486 q^{45} + 15612 q^{47} + 17334 q^{51} - 16512 q^{53} + 155392 q^{55} - 40464 q^{57} - 13140 q^{59} - 5796 q^{61} - 64524 q^{65} + 56116 q^{67} + 114372 q^{69} + 22044 q^{71} - 85384 q^{73} - 44154 q^{75} + 19620 q^{79} - 13122 q^{81} + 88848 q^{83} - 33832 q^{85} - 95148 q^{87} + 211218 q^{89} - 29808 q^{93} - 260568 q^{95} - 89728 q^{97} - 14580 q^{99}+O(q^{100}) \) 4 * q - 18 * q^3 - 6 * q^5 - 162 * q^9 + 90 * q^11 - 1536 * q^13 + 108 * q^15 + 1926 * q^17 + 2248 * q^19 - 6354 * q^23 - 4906 * q^25 + 2916 * q^27 + 21144 * q^29 - 3312 * q^31 + 810 * q^33 - 2104 * q^37 + 6912 * q^39 - 2532 * q^41 - 11536 * q^43 - 486 * q^45 + 15612 * q^47 + 17334 * q^51 - 16512 * q^53 + 155392 * q^55 - 40464 * q^57 - 13140 * q^59 - 5796 * q^61 - 64524 * q^65 + 56116 * q^67 + 114372 * q^69 + 22044 * q^71 - 85384 * q^73 - 44154 * q^75 + 19620 * q^79 - 13122 * q^81 + 88848 * q^83 - 33832 * q^85 - 95148 * q^87 + 211218 * q^89 - 29808 * q^93 - 260568 * q^95 - 89728 * q^97 - 14580 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more