LMFDB - Embedded newform 336.6.q.f.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(193,336)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(336, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 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("336.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.q (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:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{9601})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 2401x^{2} + 2400x + 5760000 $ x^4 - x^3 + 2401x^2 + 2400x + 5760000
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Root			$24.7462 - 42.8616i$ of defining polynomial
Character	$\chi$	$=$	336.289
Dual form			336.6.q.f.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 - 7.79423i) q^{3} +(37.7462 - 65.3783i) q^{5} +(-99.4847 + 83.1252i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(-4.50000 - 7.79423i) q^{3} +(37.7462 - 65.3783i) q^{5} +(-99.4847 + 83.1252i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-74.7309 - 129.438i) q^{11} +349.416 q^{13} -679.431 q^{15} +(574.923 + 995.797i) q^{17} +(-1397.60 + 2420.72i) q^{19} +(1095.58 + 401.343i) q^{21} +(906.985 - 1570.94i) q^{23} +(-1287.05 - 2229.23i) q^{25} +729.000 q^{27} -759.033 q^{29} +(4515.87 + 7821.72i) q^{31} +(-672.578 + 1164.94i) q^{33} +(1679.42 + 9641.80i) q^{35} +(-3897.45 + 6750.57i) q^{37} +(-1572.37 - 2723.43i) q^{39} +7640.49 q^{41} -12188.8 q^{43} +(3057.44 + 5295.64i) q^{45} +(12299.4 - 21303.2i) q^{47} +(2987.41 - 16539.4i) q^{49} +(5174.31 - 8962.17i) q^{51} +(-6798.11 - 11774.7i) q^{53} -11283.2 q^{55} +25156.8 q^{57} +(-13179.4 - 22827.4i) q^{59} +(-17660.9 + 30589.5i) q^{61} +(-1801.94 - 10345.2i) q^{63} +(13189.1 - 22844.2i) q^{65} +(27186.0 + 47087.5i) q^{67} -16325.7 q^{69} +70145.7 q^{71} +(22234.4 + 38511.1i) q^{73} +(-11583.4 + 20063.1i) q^{75} +(18194.1 + 6665.05i) q^{77} +(30806.2 - 53357.9i) q^{79} +(-3280.50 - 5681.99i) q^{81} +87142.0 q^{83} +86804.6 q^{85} +(3415.65 + 5916.08i) q^{87} +(-49284.7 + 85363.6i) q^{89} +(-34761.5 + 29045.3i) q^{91} +(40642.9 - 70395.5i) q^{93} +(105508. + 182745. i) q^{95} +32342.3 q^{97} +12106.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 18 q^{3} + 53 q^{5} - 6 q^{7} - 162 q^{9}+O(q^{10}) $ 4 * q - 18 * q^3 + 53 * q^5 - 6 * q^7 - 162 * q^9	$ 4 q - 18 q^{3} + 53 q^{5} - 6 q^{7} - 162 q^{9} + 191 q^{11} - 758 q^{13} - 954 q^{15} + 340 q^{17} - 1769 q^{19} - 27 q^{21} + 3236 q^{23} + 45 q^{25} + 2916 q^{27} + 8918 q^{29} + 1994 q^{31} + 1719 q^{33} + 4562 q^{35} - 20587 q^{37} + 3411 q^{39} + 17628 q^{41} - 31706 q^{43} + 4293 q^{45} + 33912 q^{47} + 9598 q^{49} + 3060 q^{51} - 49239 q^{53} - 37882 q^{55} + 31842 q^{57} - 56735 q^{59} - 67508 q^{61} + 729 q^{63} + 42762 q^{65} + 75723 q^{67} - 58248 q^{69} + 17984 q^{71} + 3201 q^{73} + 405 q^{75} + 120299 q^{77} + 26612 q^{79} - 13122 q^{81} + 1898 q^{83} + 210040 q^{85} - 40131 q^{87} - 176562 q^{89} - 210085 q^{91} + 17946 q^{93} + 234098 q^{95} - 258846 q^{97} - 30942 q^{99}+O(q^{100}) $ 4 * q - 18 * q^3 + 53 * q^5 - 6 * q^7 - 162 * q^9 + 191 * q^11 - 758 * q^13 - 954 * q^15 + 340 * q^17 - 1769 * q^19 - 27 * q^21 + 3236 * q^23 + 45 * q^25 + 2916 * q^27 + 8918 * q^29 + 1994 * q^31 + 1719 * q^33 + 4562 * q^35 - 20587 * q^37 + 3411 * q^39 + 17628 * q^41 - 31706 * q^43 + 4293 * q^45 + 33912 * q^47 + 9598 * q^49 + 3060 * q^51 - 49239 * q^53 - 37882 * q^55 + 31842 * q^57 - 56735 * q^59 - 67508 * q^61 + 729 * q^63 + 42762 * q^65 + 75723 * q^67 - 58248 * q^69 + 17984 * q^71 + 3201 * q^73 + 405 * q^75 + 120299 * q^77 + 26612 * q^79 - 13122 * q^81 + 1898 * q^83 + 210040 * q^85 - 40131 * q^87 - 176562 * q^89 - 210085 * q^91 + 17946 * q^93 + 234098 * q^95 - 258846 * q^97 - 30942 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	37.7462	−	65.3783i	0.675224	−	1.16952i	−0.301179	−	0.953568i	$-0.597380\pi$
$5$	37.7462	−	65.3783i	0.675224	−	1.16952i	0.976403	−	0.215955i	$-0.0692864\pi$
$6$		0			0
$7$	−99.4847	+	83.1252i	−0.767381	+	0.641191i
$7$	−99.4847	+	83.1252i	−0.767381	+	0.641191i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	−74.7309	−	129.438i	−0.186217	−	0.322537i	0.757769	−	0.652523i	$-0.226289\pi$
$11$	−74.7309	−	129.438i	−0.186217	−	0.322537i	−0.943986	+	0.329986i	$0.892956\pi$
$12$		0			0
$13$	349.416			0.573435			0.286717	−	0.958015i	$-0.407436\pi$
$13$	349.416			0.573435			0.286717	+	0.958015i	$0.407436\pi$
$14$		0			0
$15$	−679.431			−0.779682
$16$		0			0
$17$	574.923	+	995.797i	0.482489	+	0.835696i	0.999798	−	0.0201029i	$-0.00639938\pi$
$17$	574.923	+	995.797i	0.482489	+	0.835696i	−0.517309	+	0.855799i	$0.673066\pi$
$18$		0			0
$19$	−1397.60	+	2420.72i	−0.888176	+	1.53837i	−0.0461468	+	0.998935i	$0.514694\pi$
$19$	−1397.60	+	2420.72i	−0.888176	+	1.53837i	−0.842029	+	0.539432i	$0.818639\pi$
$20$		0			0
$21$	1095.58	+	401.343i	0.542119	+	0.198595i
$22$		0			0
$23$	906.985	−	1570.94i	0.357504	−	0.619214i	−0.630040	−	0.776563i	$-0.716961\pi$
$23$	906.985	−	1570.94i	0.357504	−	0.619214i	0.987543	+	0.157349i	$0.0502947\pi$
$24$		0			0
$25$	−1287.05	−	2229.23i	−0.411855	−	0.713354i
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	−759.033			−0.167597			−0.0837984	−	0.996483i	$-0.526705\pi$
$29$	−759.033			−0.167597			−0.0837984	+	0.996483i	$0.526705\pi$
$30$		0			0
$31$	4515.87	+	7821.72i	0.843990	+	1.46183i	0.886495	+	0.462738i	$0.153133\pi$
$31$	4515.87	+	7821.72i	0.843990	+	1.46183i	−0.0425050	+	0.999096i	$0.513534\pi$
$32$		0			0
$33$	−672.578	+	1164.94i	−0.107512	+	0.186217i
$34$		0			0
$35$	1679.42	+	9641.80i	0.231733	+	1.33042i
$36$		0			0
$37$	−3897.45	+	6750.57i	−0.468032	+	0.810655i	−0.999333	−	0.0365280i	$-0.988370\pi$
$37$	−3897.45	+	6750.57i	−0.468032	+	0.810655i	0.531300	+	0.847183i	$0.321704\pi$
$38$		0			0
$39$	−1572.37	−	2723.43i	−0.165536	−	0.286717i
$40$		0			0
$41$	7640.49			0.709842			0.354921	−	0.934896i	$-0.384508\pi$
$41$	7640.49			0.709842			0.354921	+	0.934896i	$0.384508\pi$
$42$		0			0
$43$	−12188.8			−1.00529			−0.502645	−	0.864493i	$-0.667640\pi$
$43$	−12188.8			−1.00529			−0.502645	+	0.864493i	$0.667640\pi$
$44$		0			0
$45$	3057.44	+	5295.64i	0.225075	+	0.389841i
$46$		0			0
$47$	12299.4	−	21303.2i	0.812156	−	1.40670i	−0.0991964	−	0.995068i	$-0.531627\pi$
$47$	12299.4	−	21303.2i	0.812156	−	1.40670i	0.911352	−	0.411627i	$-0.135039\pi$
$48$		0			0
$49$	2987.41	−	16539.4i	0.177748	−	0.984076i
$50$		0			0
$51$	5174.31	−	8962.17i	0.278565	−	0.482489i
$52$		0			0
$53$	−6798.11	−	11774.7i	−0.332429	−	0.575783i	0.650559	−	0.759456i	$-0.274535\pi$
$53$	−6798.11	−	11774.7i	−0.332429	−	0.575783i	−0.982988	+	0.183672i	$0.941201\pi$
$54$		0			0
$55$	−11283.2			−0.502952
$56$		0			0
$57$	25156.8			1.02558
$58$		0			0
$59$	−13179.4	−	22827.4i	−0.492908	−	0.853742i	0.507059	−	0.861912i	$-0.330733\pi$
$59$	−13179.4	−	22827.4i	−0.492908	−	0.853742i	−0.999967	+	0.00816991i	$0.997399\pi$
$60$		0			0
$61$	−17660.9	+	30589.5i	−0.607698	+	1.05256i	0.383921	+	0.923366i	$0.374573\pi$
$61$	−17660.9	+	30589.5i	−0.607698	+	1.05256i	−0.991619	+	0.129198i	$0.958760\pi$
$62$		0			0
$63$	−1801.94	−	10345.2i	−0.0571991	−	0.328389i
$64$		0			0
$65$	13189.1	−	22844.2i	0.387197	−	0.670645i
$66$		0			0
$67$	27186.0	+	47087.5i	0.739874	+	1.28150i	0.952552	+	0.304377i	$0.0984482\pi$
$67$	27186.0	+	47087.5i	0.739874	+	1.28150i	−0.212678	+	0.977122i	$0.568219\pi$
$68$		0			0
$69$	−16325.7			−0.412810
$70$		0			0
$71$	70145.7			1.65141			0.825706	−	0.564101i	$-0.190777\pi$
$71$	70145.7			1.65141			0.825706	+	0.564101i	$0.190777\pi$
$72$		0			0
$73$	22234.4	+	38511.1i	0.488335	+	0.845822i	0.999910	−	0.0134170i	$-0.00427090\pi$
$73$	22234.4	+	38511.1i	0.488335	+	0.845822i	−0.511574	+	0.859239i	$0.670938\pi$
$74$		0			0
$75$	−11583.4	+	20063.1i	−0.237785	+	0.411855i
$76$		0			0
$77$	18194.1	+	6665.05i	0.349707	+	0.128108i
$78$		0			0
$79$	30806.2	−	53357.9i	0.555355	−	0.961903i	−0.442521	−	0.896758i	$-0.645916\pi$
$79$	30806.2	−	53357.9i	0.555355	−	0.961903i	0.997876	−	0.0651450i	$-0.0207510\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	87142.0			1.38846			0.694228	−	0.719755i	$-0.255746\pi$
$83$	87142.0			1.38846			0.694228	+	0.719755i	$0.255746\pi$
$84$		0			0
$85$	86804.6			1.30315
$86$		0			0
$87$	3415.65	+	5916.08i	0.0483810	+	0.0837984i
$88$		0			0
$89$	−49284.7	+	85363.6i	−0.659534	+	1.14235i	0.321203	+	0.947010i	$0.395913\pi$
$89$	−49284.7	+	85363.6i	−0.659534	+	1.14235i	−0.980736	+	0.195336i	$0.937420\pi$
$90$		0			0
$91$	−34761.5	+	29045.3i	−0.440043	+	0.367681i
$92$		0			0
$93$	40642.9	−	70395.5i	0.487278	−	0.843990i
$94$		0			0
$95$	105508.	+	182745.i	1.19944	+	2.07748i
$96$		0			0
$97$	32342.3			0.349013			0.174507	−	0.984656i	$-0.444167\pi$
$97$	32342.3			0.349013			0.174507	+	0.984656i	$0.444167\pi$
$98$		0			0
$99$	12106.4			0.124144
$100$		0			0
$101$	−15673.2	−	27146.8i	−0.152881	−	0.264798i	0.779404	−	0.626521i	$-0.215522\pi$
$101$	−15673.2	−	27146.8i	−0.152881	−	0.264798i	−0.932286	+	0.361723i	$0.882189\pi$
$102$		0			0
$103$	49666.5	−	86024.8i	0.461286	−	0.798970i	−0.537740	−	0.843111i	$-0.680722\pi$
$103$	49666.5	−	86024.8i	0.461286	−	0.798970i	0.999025	+	0.0441406i	$0.0140549\pi$
$104$		0			0
$105$	67593.0	−	56477.8i	0.598313	−	0.499925i
$106$		0			0
$107$	−72634.0	+	125806.i	−0.613310	+	1.06228i	0.377368	+	0.926063i	$0.376829\pi$
$107$	−72634.0	+	125806.i	−0.613310	+	1.06228i	−0.990678	+	0.136221i	$0.956504\pi$
$108$		0			0
$109$	90425.4	+	156621.i	0.728994	+	1.26265i	0.957309	+	0.289067i	$0.0933452\pi$
$109$	90425.4	+	156621.i	0.728994	+	1.26265i	−0.228315	+	0.973587i	$0.573321\pi$
$110$		0			0
$111$	70154.0			0.540437
$112$		0			0
$113$	197832.			1.45748			0.728738	−	0.684793i	$-0.240107\pi$
$113$	197832.			1.45748			0.728738	+	0.684793i	$0.240107\pi$
$114$		0			0
$115$	−68470.4	−	118594.i	−0.482790	−	0.836217i
$116$		0			0
$117$	−14151.3	+	24510.8i	−0.0955725	+	0.165536i
$118$		0			0
$119$	−139972.	−	51275.9i	−0.906094	−	0.331930i
$120$		0			0
$121$	69356.1	−	120128.i	0.430647	−	0.745902i
$122$		0			0
$123$	−34382.2	−	59551.8i	−0.204914	−	0.354921i
$124$		0			0
$125$	41589.2			0.238070
$126$		0			0
$127$	33517.2			0.184399			0.0921996	−	0.995741i	$-0.470610\pi$
$127$	33517.2			0.184399			0.0921996	+	0.995741i	$0.470610\pi$
$128$		0			0
$129$	54849.8	+	95002.6i	0.290202	+	0.502645i
$130$		0			0
$131$	−5404.45	+	9360.79i	−0.0275153	+	0.0476578i	−0.879455	−	0.475982i	$-0.842093\pi$
$131$	−5404.45	+	9360.79i	−0.0275153	+	0.0476578i	0.851940	+	0.523640i	$0.175426\pi$
$132$		0			0
$133$	−62182.5	−	357000.i	−0.304817	−	1.75000i
$134$		0			0
$135$	27517.0	−	47660.8i	0.129947	−	0.225075i
$136$		0			0
$137$	9466.01	+	16395.6i	0.0430889	+	0.0746322i	0.886766	−	0.462220i	$-0.152947\pi$
$137$	9466.01	+	16395.6i	0.0430889	+	0.0746322i	−0.843677	+	0.536852i	$0.819613\pi$
$138$		0			0
$139$	168897.			0.741457			0.370729	−	0.928741i	$-0.379108\pi$
$139$	168897.			0.741457			0.370729	+	0.928741i	$0.379108\pi$
$140$		0			0
$141$	−221389.			−0.937797
$142$		0			0
$143$	−26112.1	−	45227.6i	−0.106783	−	0.184954i
$144$		0			0
$145$	−28650.6	+	49624.3i	−0.113165	+	0.196008i
$146$		0			0
$147$	−142355.	+	51142.6i	−0.543349	+	0.195204i
$148$		0			0
$149$	−132001.	+	228633.i	−0.487093	+	0.843670i	−0.999890	−	0.0148402i	$-0.995276\pi$
$149$	−132001.	+	228633.i	−0.487093	+	0.843670i	0.512797	+	0.858510i	$0.328609\pi$
$150$		0			0
$151$	139587.	+	241772.i	0.498200	+	0.862908i	0.999998	−	0.00207707i	$-0.000661153\pi$
$151$	139587.	+	241772.i	0.498200	+	0.862908i	−0.501798	+	0.864985i	$0.667328\pi$
$152$		0			0
$153$	−93137.6			−0.321660
$154$		0			0
$155$	681828.			2.27953
$156$		0			0
$157$	−94301.2	−	163334.i	−0.305329	−	0.528845i	0.672006	−	0.740546i	$-0.265433\pi$
$157$	−94301.2	−	163334.i	−0.305329	−	0.528845i	−0.977334	+	0.211701i	$0.932100\pi$
$158$		0			0
$159$	−61183.0	+	105972.i	−0.191928	+	0.332429i
$160$		0			0
$161$	40353.9	+	231678.i	0.122693	+	0.704402i
$162$		0			0
$163$	44858.7	−	77697.5i	0.132244	−	0.229054i	−0.792297	−	0.610136i	$-0.791115\pi$
$163$	44858.7	−	77697.5i	0.132244	−	0.229054i	0.924541	+	0.381081i	$0.124448\pi$
$164$		0			0
$165$	50774.5	+	87944.0i	0.145190	+	0.251476i
$166$		0			0
$167$	−529411.			−1.46893			−0.734467	−	0.678645i	$-0.762568\pi$
$167$	−529411.			−1.46893			−0.734467	+	0.678645i	$0.762568\pi$
$168$		0			0
$169$	−249202.			−0.671172
$170$		0			0
$171$	−113206.	−	196078.i	−0.296059	−	0.512789i
$172$		0			0
$173$	−16919.2	+	29304.8i	−0.0429797	+	0.0744430i	−0.886715	−	0.462316i	$-0.847018\pi$
$173$	−16919.2	+	29304.8i	−0.0429797	+	0.0744430i	0.843735	+	0.536759i	$0.180352\pi$
$174$		0			0
$175$	313347.	+	114788.i	0.773446	+	0.283337i
$176$		0			0
$177$	−118615.	+	205447.i	−0.284581	+	0.492908i
$178$		0			0
$179$	−123872.	−	214552.i	−0.288962	−	0.500496i	0.684601	−	0.728918i	$-0.259977\pi$
$179$	−123872.	−	214552.i	−0.288962	−	0.500496i	−0.973562	+	0.228422i	$0.926643\pi$
$180$		0			0
$181$	369470.			0.838268			0.419134	−	0.907924i	$-0.362334\pi$
$181$	369470.			0.838268			0.419134	+	0.907924i	$0.362334\pi$
$182$		0			0
$183$	317896.			0.701709
$184$		0			0
$185$	294227.	+	509617.i	0.632053	+	1.09475i
$186$		0			0
$187$	85929.1	−	148833.i	0.179695	−	0.311241i
$188$		0			0
$189$	−72524.3	+	60598.3i	−0.147683	+	0.123397i
$190$		0			0
$191$	−242612.	+	420217.i	−0.481204	+	0.833470i	−0.999767	−	0.0215693i	$-0.993134\pi$
$191$	−242612.	+	420217.i	−0.481204	+	0.833470i	0.518563	+	0.855039i	$0.326467\pi$
$192$		0			0
$193$	165409.	+	286496.i	0.319643	+	0.553637i	0.980413	−	0.196950i	$-0.0631038\pi$
$193$	165409.	+	286496.i	0.319643	+	0.553637i	−0.660771	+	0.750588i	$0.729770\pi$
$194$		0			0
$195$	−237404.			−0.447097
$196$		0			0
$197$	−161963.			−0.297337			−0.148669	−	0.988887i	$-0.547499\pi$
$197$	−161963.			−0.297337			−0.148669	+	0.988887i	$0.547499\pi$
$198$		0			0
$199$	27698.1	+	47974.5i	0.0495813	+	0.0858773i	0.889751	−	0.456446i	$-0.150878\pi$
$199$	27698.1	+	47974.5i	0.0495813	+	0.0858773i	−0.840170	+	0.542324i	$0.817545\pi$
$200$		0			0
$201$	244674.	−	423787.i	0.427166	−	0.739874i
$202$		0			0
$203$	75512.2	−	63094.8i	0.128611	−	0.107462i
$204$		0			0
$205$	288399.	−	499522.i	0.479303	−	0.830176i
$206$		0			0
$207$	73465.8	+	127246.i	0.119168	+	0.206405i
$208$		0			0
$209$	417776.			0.661572
$210$		0			0
$211$	−481748.			−0.744926			−0.372463	−	0.928047i	$-0.621487\pi$
$211$	−481748.			−0.744926			−0.372463	+	0.928047i	$0.621487\pi$
$212$		0			0
$213$	−315656.	−	546732.i	−0.476722	−	0.825706i
$214$		0			0
$215$	−460082.	+	796885.i	−0.678795	+	1.17571i
$216$		0			0
$217$	−1.09944e6	−	402759.i	−1.58498	−	0.580625i
$218$		0			0
$219$	200110.	−	346600.i	0.281941	−	0.488335i
$220$		0			0
$221$	200887.	+	347947.i	0.276676	+	0.479217i
$222$		0			0
$223$	−638779.			−0.860178			−0.430089	−	0.902787i	$-0.641518\pi$
$223$	−638779.			−0.860178			−0.430089	+	0.902787i	$0.641518\pi$
$224$		0			0
$225$	208502.			0.274570
$226$		0			0
$227$	−262841.	−	455254.i	−0.338554	−	0.586394i	0.645607	−	0.763670i	$-0.276605\pi$
$227$	−262841.	−	455254.i	−0.338554	−	0.586394i	−0.984161	+	0.177277i	$0.943271\pi$
$228$		0			0
$229$	−466343.	+	807730.i	−0.587647	+	1.01784i	0.406892	+	0.913476i	$0.366612\pi$
$229$	−466343.	+	807730.i	−0.587647	+	1.01784i	−0.994540	+	0.104359i	$0.966721\pi$
$230$		0			0
$231$	−29924.6	−	171802.i	−0.0368976	−	0.211835i
$232$		0			0
$233$	353896.	−	612967.i	0.427058	−	0.739685i	−0.569553	−	0.821955i	$-0.692884\pi$
$233$	353896.	−	612967.i	0.427058	−	0.739685i	0.996610	+	0.0822696i	$0.0262168\pi$
$234$		0			0
$235$	−928511.	−	1.60823e6i	−1.09677	−	1.89967i
$236$		0			0
$237$	−554512.			−0.641269
$238$		0			0
$239$	500614.			0.566902			0.283451	−	0.958987i	$-0.408521\pi$
$239$	500614.			0.566902			0.283451	+	0.958987i	$0.408521\pi$
$240$		0			0
$241$	−604109.	−	1.04635e6i	−0.669997	−	1.16047i	−0.977904	−	0.209052i	$-0.932962\pi$
$241$	−604109.	−	1.04635e6i	−0.669997	−	1.16047i	0.307907	−	0.951416i	$-0.400371\pi$
$242$		0			0
$243$	−29524.5	+	51137.9i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	−968552.	−	819609.i	−1.03088	−	0.872352i
$246$		0			0
$247$	−488344.	+	845836.i	−0.509311	+	0.882153i
$248$		0			0
$249$	−392139.	−	679204.i	−0.400813	−	0.694228i
$250$		0			0
$251$	−97826.7			−0.0980106			−0.0490053	−	0.998799i	$-0.515605\pi$
$251$	−97826.7			−0.0980106			−0.0490053	+	0.998799i	$0.515605\pi$
$252$		0			0
$253$	−271119.			−0.266292
$254$		0			0
$255$	−390621.	−	676575.i	−0.376188	−	0.651577i
$256$		0			0
$257$	553956.	−	959481.i	0.523170	−	0.906157i	−0.476466	−	0.879193i	$-0.658083\pi$
$257$	553956.	−	959481.i	0.523170	−	0.906157i	0.999636	−	0.0269643i	$-0.00858403\pi$
$258$		0			0
$259$	−173406.	−	995555.i	−0.160626	−	0.922180i
$260$		0			0
$261$	30740.8	−	53244.7i	0.0279328	−	0.0483810i
$262$		0			0
$263$	753427.	+	1.30497e6i	0.671663	+	1.16336i	0.977432	+	0.211249i	$0.0677532\pi$
$263$	753427.	+	1.30497e6i	0.671663	+	1.16336i	−0.305769	+	0.952106i	$0.598913\pi$
$264$		0			0
$265$	−1.02641e6			−0.897856
$266$		0			0
$267$	887125.			0.761564
$268$		0			0
$269$	1.06751e6	+	1.84898e6i	0.899481	+	1.55795i	0.828159	+	0.560493i	$0.189388\pi$
$269$	1.06751e6	+	1.84898e6i	0.899481	+	1.55795i	0.0713221	+	0.997453i	$0.477278\pi$
$270$		0			0
$271$	614299.	−	1.06400e6i	0.508108	−	0.880070i	−0.491847	−	0.870681i	$-0.663678\pi$
$271$	614299.	−	1.06400e6i	0.508108	−	0.880070i	0.999956	−	0.00938828i	$-0.00298843\pi$
$272$		0			0
$273$	382812.	+	140236.i	0.310870	+	0.113881i
$274$		0			0
$275$	−192364.	+	333185.i	−0.153388	+	0.265677i
$276$		0			0
$277$	923412.	+	1.59940e6i	0.723097	+	1.25244i	0.959753	+	0.280847i	$0.0906152\pi$
$277$	923412.	+	1.59940e6i	0.723097	+	1.25244i	−0.236656	+	0.971593i	$0.576051\pi$
$278$		0			0
$279$	−731571.			−0.562660
$280$		0			0
$281$	−2.28326e6			−1.72500			−0.862500	−	0.506056i	$-0.831103\pi$
$281$	−2.28326e6			−1.72500			−0.862500	+	0.506056i	$0.831103\pi$
$282$		0			0
$283$	668463.	+	1.15781e6i	0.496148	+	0.859354i	0.999990	−	0.00444218i	$-0.00141400\pi$
$283$	668463.	+	1.15781e6i	0.496148	+	0.859354i	−0.503842	+	0.863796i	$0.668081\pi$
$284$		0			0
$285$	949573.	−	1.64471e6i	0.692495	−	1.19944i
$286$		0			0
$287$	−760112.	+	635118.i	−0.544720	+	0.455145i
$288$		0			0
$289$	48854.5	−	84618.5i	0.0344081	−	0.0595965i
$290$		0			0
$291$	−145541.	−	252084.i	−0.100751	−	0.174507i
$292$		0			0
$293$	−2.23033e6			−1.51775			−0.758875	−	0.651236i	$-0.774251\pi$
$293$	−2.23033e6			−1.51775			−0.758875	+	0.651236i	$0.774251\pi$
$294$		0			0
$295$	−1.98989e6			−1.33129
$296$		0			0
$297$	−54478.8	−	94360.1i	−0.0358374	−	0.0620722i
$298$		0			0
$299$	316915.	−	548913.i	0.205005	−	0.355079i
$300$		0			0
$301$	1.21260e6	−	1.01320e6i	0.771440	−	0.644583i
$302$		0			0
$303$	−141059.	+	244321.i	−0.0882661	+	0.152881i
$304$		0			0
$305$	1.33326e6	+	2.30928e6i	0.820664	+	1.42143i
$306$		0			0
$307$	−1.77782e6			−1.07657			−0.538284	−	0.842763i	$-0.680927\pi$
$307$	−1.77782e6			−1.07657			−0.538284	+	0.842763i	$0.680927\pi$
$308$		0			0
$309$	−893996.			−0.532647
$310$		0			0
$311$	701358.	+	1.21479e6i	0.411187	+	0.712196i	0.995020	−	0.0996776i	$-0.0317811\pi$
$311$	701358.	+	1.21479e6i	0.411187	+	0.712196i	−0.583833	+	0.811874i	$0.698448\pi$
$312$		0			0
$313$	−609472.	+	1.05564e6i	−0.351636	+	0.609051i	−0.986536	−	0.163543i	$-0.947708\pi$
$313$	−609472.	+	1.05564e6i	−0.351636	+	0.609051i	0.634900	+	0.772594i	$0.281041\pi$
$314$		0			0
$315$	−744370.	−	272685.i	−0.422681	−	0.154841i
$316$		0			0
$317$	−1.34001e6	+	2.32096e6i	−0.748961	+	1.29724i	0.199361	+	0.979926i	$0.436113\pi$
$317$	−1.34001e6	+	2.32096e6i	−0.748961	+	1.29724i	−0.948321	+	0.317312i	$0.897220\pi$
$318$		0			0
$319$	56723.2	+	98247.5i	0.0312093	+	0.0540561i
$320$		0			0
$321$	1.30741e6			0.708190
$322$		0			0
$323$	−3.21405e6			−1.71414
$324$		0			0
$325$	−449715.	−	778929.i	−0.236172	−	0.409062i
$326$		0			0
$327$	813828.	−	1.40959e6i	0.420885	−	0.728994i
$328$		0			0
$329$	547229.	+	3.14173e6i	0.278727	+	1.60022i
$330$		0			0
$331$	71280.1	−	123461.i	0.0357601	−	0.0619383i	−0.847591	−	0.530649i	$-0.821948\pi$
$331$	71280.1	−	123461.i	0.0357601	−	0.0619383i	0.883352	+	0.468711i	$0.155281\pi$
$332$		0			0
$333$	−315693.	−	546796.i	−0.156011	−	0.270218i
$334$		0			0
$335$	4.10466e6			1.99832
$336$		0			0
$337$	−1.21206e6			−0.581367			−0.290683	−	0.956819i	$-0.593883\pi$
$337$	−1.21206e6			−0.581367			−0.290683	+	0.956819i	$0.593883\pi$
$338$		0			0
$339$	−890246.	−	1.54195e6i	−0.420737	−	0.728738i
$340$		0			0
$341$	674950.	−	1.16905e6i	0.314330	−	0.544435i
$342$		0			0
$343$	1.07764e6	+	1.89374e6i	0.494580	+	0.869132i
$344$		0			0
$345$	−616234.	+	1.06735e6i	−0.278739	+	0.482790i
$346$		0			0
$347$	1.73426e6	+	3.00383e6i	0.773199	+	1.33922i	0.935801	+	0.352529i	$0.114678\pi$
$347$	1.73426e6	+	3.00383e6i	0.773199	+	1.33922i	−0.162602	+	0.986692i	$0.551989\pi$
$348$		0			0
$349$	1.01692e6			0.446911			0.223456	−	0.974714i	$-0.428266\pi$
$349$	1.01692e6			0.446911			0.223456	+	0.974714i	$0.428266\pi$
$350$		0			0
$351$	254724.			0.110358
$352$		0			0
$353$	−781927.	−	1.35434e6i	−0.333987	−	0.578483i	0.649303	−	0.760530i	$-0.275061\pi$
$353$	−781927.	−	1.35434e6i	−0.333987	−	0.578483i	−0.983290	+	0.182048i	$0.941728\pi$
$354$		0			0
$355$	2.64773e6	−	4.58601e6i	1.11507	−	1.93136i
$356$		0			0
$357$	230217.	+	1.32171e6i	0.0956021	+	0.548867i
$358$		0			0
$359$	24776.6	−	42914.3i	0.0101462	−	0.0175738i	−0.860908	−	0.508761i	$-0.830104\pi$
$359$	24776.6	−	42914.3i	0.0101462	−	0.0175738i	0.871054	+	0.491187i	$0.163437\pi$
$360$		0			0
$361$	−2.66853e6	−	4.62202e6i	−1.07771	−	1.86666i
$362$		0			0
$363$	−1.24841e6			−0.497268
$364$		0			0
$365$	3.35705e6			1.31894
$366$		0			0
$367$	1.77416e6	+	3.07293e6i	0.687585	+	1.19093i	0.972617	+	0.232414i	$0.0746626\pi$
$367$	1.77416e6	+	3.07293e6i	0.687585	+	1.19093i	−0.285032	+	0.958518i	$0.592004\pi$
$368$		0			0
$369$	−309440.	+	535966.i	−0.118307	+	0.204914i
$370$		0			0
$371$	1.65508e6	+	606306.i	0.624287	+	0.228695i
$372$		0			0
$373$	1.12787e6	−	1.95352e6i	0.419745	−	0.727020i	−0.576169	−	0.817331i	$-0.695453\pi$
$373$	1.12787e6	−	1.95352e6i	0.419745	−	0.727020i	0.995914	+	0.0903112i	$0.0287862\pi$
$374$		0			0
$375$	−187151.	−	324155.i	−0.0687250	−	0.119035i
$376$		0			0
$377$	−265218.			−0.0961059
$378$		0			0
$379$	−4.39503e6			−1.57168			−0.785840	−	0.618430i	$-0.787769\pi$
$379$	−4.39503e6			−1.57168			−0.785840	+	0.618430i	$0.787769\pi$
$380$		0			0
$381$	−150828.	−	261241.i	−0.0532315	−	0.0921996i
$382$		0			0
$383$	−613904.	+	1.06331e6i	−0.213847	+	0.370394i	−0.952915	−	0.303237i	$-0.901933\pi$
$383$	−613904.	+	1.06331e6i	−0.213847	+	0.370394i	0.739068	+	0.673631i	$0.235266\pi$
$384$		0			0
$385$	1.12251e6	−	937919.i	0.385956	−	0.322488i
$386$		0			0
$387$	493648.	−	855023.i	0.167548	−	0.290202i
$388$		0			0
$389$	−1.02712e6	−	1.77903e6i	−0.344150	−	0.596086i	0.641049	−	0.767500i	$-0.278500\pi$
$389$	−1.02712e6	−	1.77903e6i	−0.344150	−	0.596086i	−0.985199	+	0.171415i	$0.945166\pi$
$390$		0			0
$391$	2.08579e6			0.689967
$392$		0			0
$393$	97280.2			0.0317719
$394$		0			0
$395$	−2.32563e6	−	4.02812e6i	−0.749978	−	1.29900i
$396$		0			0
$397$	1.90184e6	−	3.29408e6i	0.605615	−	1.04896i	−0.386339	−	0.922357i	$-0.626260\pi$
$397$	1.90184e6	−	3.29408e6i	0.605615	−	1.04896i	0.991954	−	0.126599i	$-0.0404062\pi$
$398$		0			0
$399$	−2.50272e6	+	2.09116e6i	−0.787009	+	0.657591i
$400$		0			0
$401$	−592622.	+	1.02645e6i	−0.184042	+	0.318770i	−0.943253	−	0.332074i	$-0.892252\pi$
$401$	−592622.	+	1.02645e6i	−0.184042	+	0.318770i	0.759211	+	0.650844i	$0.225585\pi$
$402$		0			0
$403$	1.57792e6	+	2.73303e6i	0.483974	+	0.838267i
$404$		0			0
$405$	−495305.			−0.150050
$406$		0			0
$407$	1.16504e6			0.348621
$408$		0			0
$409$	2.15197e6	+	3.72731e6i	0.636102	+	1.10176i	0.986280	+	0.165079i	$0.0527878\pi$
$409$	2.15197e6	+	3.72731e6i	0.636102	+	1.10176i	−0.350178	+	0.936683i	$0.613879\pi$
$410$		0			0
$411$	85194.1	−	147560.i	0.0248774	−	0.0430889i
$412$		0			0
$413$	3.20868e6	+	1.17544e6i	0.925660	+	0.339097i
$414$		0			0
$415$	3.28928e6	−	5.69719e6i	0.937519	−	1.62383i
$416$		0			0
$417$	−760038.	−	1.31643e6i	−0.214040	−	0.370729i
$418$		0			0
$419$	113725.			0.0316461			0.0158230	−	0.999875i	$-0.494963\pi$
$419$	113725.			0.0316461			0.0158230	+	0.999875i	$0.494963\pi$
$420$		0			0
$421$	443417.			0.121929			0.0609645	−	0.998140i	$-0.480582\pi$
$421$	443417.			0.121929			0.0609645	+	0.998140i	$0.480582\pi$
$422$		0			0
$423$	996252.	+	1.72556e6i	0.270719	+	0.468898i
$424$		0			0
$425$	1.47991e6	−	2.56327e6i	0.397431	−	0.688371i
$426$		0			0
$427$	−785774.	−	4.51125e6i	−0.208559	−	1.19737i
$428$		0			0
$429$	−235009.	+	407048.i	−0.0616512	+	0.106783i
$430$		0			0
$431$	2.31655e6	+	4.01239e6i	0.600688	+	1.04042i	0.992717	+	0.120469i	$0.0384399\pi$
$431$	2.31655e6	+	4.01239e6i	0.600688	+	1.04042i	−0.392029	+	0.919953i	$0.628227\pi$
$432$		0			0
$433$	6.57955e6			1.68646			0.843231	−	0.537552i	$-0.180651\pi$
$433$	6.57955e6			1.68646			0.843231	+	0.537552i	$0.180651\pi$
$434$		0			0
$435$	515711.			0.130672
$436$		0			0
$437$	2.53520e6	+	4.39110e6i	0.635052	+	1.09994i
$438$		0			0
$439$	−910348.	+	1.57677e6i	−0.225448	+	0.390487i	−0.956454	−	0.291884i	$-0.905718\pi$
$439$	−910348.	+	1.57677e6i	−0.225448	+	0.390487i	0.731006	+	0.682371i	$0.239051\pi$
$440$		0			0
$441$	1.03921e6	+	879405.i	0.254454	+	0.215324i
$442$		0			0
$443$	−1.03495e6	+	1.79259e6i	−0.250559	+	0.433982i	−0.963680	−	0.267060i	$-0.913948\pi$
$443$	−1.03495e6	+	1.79259e6i	−0.250559	+	0.433982i	0.713121	+	0.701041i	$0.247281\pi$
$444$		0			0
$445$	3.72062e6	+	6.44430e6i	0.890666	+	1.54268i
$446$		0			0
$447$	2.37602e6			0.562447
$448$		0			0
$449$	5.72581e6			1.34036			0.670180	−	0.742199i	$-0.266217\pi$
$449$	5.72581e6			1.34036			0.670180	+	0.742199i	$0.266217\pi$
$450$		0			0
$451$	−570981.	−	988968.i	−0.132184	−	0.228950i
$452$		0			0
$453$	1.25629e6	−	2.17595e6i	0.287636	−	0.498200i
$454$		0			0
$455$	586814.	+	3.36900e6i	0.132884	+	0.762908i
$456$		0			0
$457$	−159338.	+	275982.i	−0.0356886	+	0.0618144i	−0.883318	−	0.468774i	$-0.844696\pi$
$457$	−159338.	+	275982.i	−0.0356886	+	0.0618144i	0.847629	+	0.530589i	$0.178029\pi$
$458$		0			0
$459$	419119.	+	725936.i	0.0928551	+	0.160830i
$460$		0			0
$461$	−3.42470e6			−0.750535			−0.375267	−	0.926917i	$-0.622449\pi$
$461$	−3.42470e6			−0.750535			−0.375267	+	0.926917i	$0.622449\pi$
$462$		0			0
$463$	−3.82945e6			−0.830201			−0.415101	−	0.909775i	$-0.636254\pi$
$463$	−3.82945e6			−0.830201			−0.415101	+	0.909775i	$0.636254\pi$
$464$		0			0
$465$	−3.06822e6	−	5.31432e6i	−0.658044	−	1.13977i
$466$		0			0
$467$	1.42231e6	−	2.46351e6i	0.301788	−	0.522712i	−0.674753	−	0.738044i	$-0.735750\pi$
$467$	1.42231e6	−	2.46351e6i	0.301788	−	0.522712i	0.976541	+	0.215332i	$0.0690832\pi$
$468$		0			0
$469$	−6.61874e6	−	2.42464e6i	−1.38945	−	0.508998i
$470$		0			0
$471$	−848710.	+	1.47001e6i	−0.176282	+	0.305329i
$472$		0			0
$473$	910882.	+	1.57769e6i	0.187202	+	0.324243i
$474$		0			0
$475$	7.19511e6			1.46320
$476$		0			0
$477$	1.10129e6			0.221619
$478$		0			0
$479$	651827.	+	1.12900e6i	0.129806	+	0.224830i	0.923601	−	0.383355i	$-0.125231\pi$
$479$	651827.	+	1.12900e6i	0.129806	+	0.224830i	−0.793796	+	0.608185i	$0.791898\pi$
$480$		0			0
$481$	−1.36183e6	+	2.35876e6i	−0.268386	+	0.464858i
$482$		0			0
$483$	1.62416e6	−	1.35708e6i	0.316782	−	0.264690i
$484$		0			0
$485$	1.22080e6	−	2.11449e6i	0.235662	−	0.408179i
$486$		0			0
$487$	3.11812e6	+	5.40074e6i	0.595759	+	1.03188i	0.993439	+	0.114360i	$0.0364818\pi$
$487$	3.11812e6	+	5.40074e6i	0.595759	+	1.03188i	−0.397681	+	0.917524i	$0.630185\pi$
$488$		0			0
$489$	−807456.			−0.152703
$490$		0			0
$491$	3.93928e6			0.737417			0.368709	−	0.929545i	$-0.379800\pi$
$491$	3.93928e6			0.737417			0.368709	+	0.929545i	$0.379800\pi$
$492$		0			0
$493$	−436386.	−	755843.i	−0.0808637	−	0.140060i
$494$		0			0
$495$	456970.	−	791496.i	0.0838253	−	0.145190i
$496$		0			0
$497$	−6.97843e6	+	5.83088e6i	−1.26726	+	1.05887i
$498$		0			0
$499$	3.98785e6	−	6.90717e6i	0.716948	−	1.24179i	−0.245255	−	0.969459i	$-0.578872\pi$
$499$	3.98785e6	−	6.90717e6i	0.716948	−	1.24179i	0.962203	−	0.272333i	$-0.0877951\pi$
$500$		0			0
$501$	2.38235e6	+	4.12635e6i	0.424045	+	0.734467i
$502$		0			0
$503$	2.70777e6			0.477190			0.238595	−	0.971119i	$-0.423313\pi$
$503$	2.70777e6			0.477190			0.238595	+	0.971119i	$0.423313\pi$
$504$		0			0
$505$	−2.36641e6			−0.412917
$506$		0			0
$507$	1.12141e6	+	1.94233e6i	0.193751	+	0.335586i
$508$		0			0
$509$	−1.95025e6	+	3.37793e6i	−0.333653	+	0.577904i	−0.983225	−	0.182396i	$-0.941615\pi$
$509$	−1.95025e6	+	3.37793e6i	−0.333653	+	0.577904i	0.649572	+	0.760300i	$0.274948\pi$
$510$		0			0
$511$	−5.41323e6	−	1.98303e6i	−0.917073	−	0.335951i
$512$		0			0
$513$	−1.01885e6	+	1.76470e6i	−0.170930	+	0.296059i
$514$		0			0
$515$	−3.74944e6	−	6.49422e6i	−0.622943	−	1.07897i
$516$		0			0
$517$	−3.67658e6			−0.604947
$518$		0			0
$519$	304545.			0.0496287
$520$		0			0
$521$	−3.50032e6	−	6.06273e6i	−0.564954	−	0.978529i	−0.997054	−	0.0767034i	$-0.975561\pi$
$521$	−3.50032e6	−	6.06273e6i	−0.564954	−	0.978529i	0.432100	−	0.901826i	$-0.357773\pi$
$522$		0			0
$523$	−1.05604e6	+	1.82911e6i	−0.168821	+	0.292406i	−0.938005	−	0.346620i	$-0.887329\pi$
$523$	−1.05604e6	+	1.82911e6i	−0.168821	+	0.292406i	0.769185	+	0.639026i	$0.220663\pi$
$524$		0			0
$525$	−515374.	−	2.95884e6i	−0.0816064	−	0.468515i
$526$		0			0
$527$	−5.19256e6	+	8.99378e6i	−0.814433	+	1.41064i
$528$		0			0
$529$	1.57293e6	+	2.72439e6i	0.244382	+	0.423283i
$530$		0			0
$531$	2.13506e6			0.328605
$532$		0			0
$533$	2.66971e6			0.407048
$534$		0			0
$535$	5.48331e6	+	9.49737e6i	0.828244	+	1.43456i
$536$		0			0
$537$	−1.11485e6	+	1.93097e6i	−0.166832	+	0.288962i
$538$		0			0
$539$	−2.36407e6	+	849318.i	−0.350500	+	0.125921i
$540$		0			0
$541$	−2.85283e6	+	4.94125e6i	−0.419067	+	0.725845i	−0.995846	−	0.0910551i	$-0.970976\pi$
$541$	−2.85283e6	+	4.94125e6i	−0.419067	+	0.725845i	0.576779	+	0.816900i	$0.304309\pi$
$542$		0			0
$543$	−1.66262e6	−	2.87974e6i	−0.241987	−	0.419134i
$544$		0			0
$545$	1.36528e7			1.96894
$546$		0			0
$547$	−2.28054e6			−0.325888			−0.162944	−	0.986635i	$-0.552099\pi$
$547$	−2.28054e6			−0.325888			−0.162944	+	0.986635i	$0.552099\pi$
$548$		0			0
$549$	−1.43053e6	−	2.47775e6i	−0.202566	−	0.350855i
$550$		0			0
$551$	1.06083e6	−	1.83740e6i	0.148855	−	0.257825i
$552$		0			0
$553$	1.37064e6	+	7.86907e6i	0.190595	+	1.09424i
$554$		0			0
$555$	2.64805e6	−	4.58655e6i	0.364916	−	0.632053i
$556$		0			0
$557$	3.36970e6	+	5.83650e6i	0.460208	+	0.797103i	0.998971	−	0.0453541i	$-0.0144416\pi$
$557$	3.36970e6	+	5.83650e6i	0.460208	+	0.797103i	−0.538763	+	0.842457i	$0.681108\pi$
$558$		0			0
$559$	−4.25897e6			−0.576468
$560$		0			0
$561$	−1.54672e6			−0.207494
$562$		0			0
$563$	−4.83410e6	−	8.37291e6i	−0.642754	−	1.11328i	−0.984815	−	0.173605i	$-0.944458\pi$
$563$	−4.83410e6	−	8.37291e6i	−0.642754	−	1.11328i	0.342061	−	0.939678i	$-0.388875\pi$
$564$		0			0
$565$	7.46742e6	−	1.29339e7i	0.984123	−	1.70455i
$566$		0			0
$567$	798676.	+	292579.i	0.104331	+	0.0382196i
$568$		0			0
$569$	6.97934e6	−	1.20886e7i	0.903719	−	1.56529i	0.0810922	−	0.996707i	$-0.474159\pi$
$569$	6.97934e6	−	1.20886e7i	0.903719	−	1.56529i	0.822627	−	0.568581i	$-0.192507\pi$
$570$		0			0
$571$	−1.17920e6	−	2.04244e6i	−0.151355	−	0.262155i	0.780371	−	0.625317i	$-0.215030\pi$
$571$	−1.17920e6	−	2.04244e6i	−0.151355	−	0.262155i	−0.931726	+	0.363162i	$0.881697\pi$
$572$		0			0
$573$	4.36702e6			0.555647
$574$		0			0
$575$	−4.66933e6			−0.588959
$576$		0			0
$577$	−2.50549e6	−	4.33963e6i	−0.313295	−	0.542642i	0.665779	−	0.746149i	$-0.268099\pi$
$577$	−2.50549e6	−	4.33963e6i	−0.313295	−	0.542642i	−0.979074	+	0.203507i	$0.934766\pi$
$578$		0			0
$579$	1.48868e6	−	2.57846e6i	0.184546	−	0.319643i
$580$		0			0
$581$	−8.66929e6	+	7.24369e6i	−1.06547	+	0.890266i
$582$		0			0
$583$	−1.01606e6	+	1.75986e6i	−0.123807	+	0.214441i
$584$		0			0
$585$	1.06832e6	+	1.85038e6i	0.129066	+	0.223548i
$586$		0			0
$587$	−3.95106e6			−0.473280			−0.236640	−	0.971597i	$-0.576046\pi$
$587$	−3.95106e6			−0.473280			−0.236640	+	0.971597i	$0.576046\pi$
$588$		0			0
$589$	−2.52455e7			−2.99845
$590$		0			0
$591$	728832.	+	1.26237e6i	0.0858339	+	0.148669i
$592$		0			0
$593$	1.26840e6	−	2.19694e6i	0.148122	−	0.256555i	−0.782411	−	0.622762i	$-0.786010\pi$
$593$	1.26840e6	−	2.19694e6i	0.148122	−	0.256555i	0.930533	+	0.366207i	$0.119344\pi$
$594$		0			0
$595$	−8.63573e6	+	7.21565e6i	−1.00002	+	0.835571i
$596$		0			0
$597$	249283.	−	431771.i	0.0286258	−	0.0495813i
$598$		0			0
$599$	3.58857e6	+	6.21558e6i	0.408652	+	0.707807i	0.994739	−	0.102442i	$-0.0326655\pi$
$599$	3.58857e6	+	6.21558e6i	0.408652	+	0.707807i	−0.586087	+	0.810248i	$0.699332\pi$
$600$		0			0
$601$	1.12527e7			1.27078			0.635392	−	0.772190i	$-0.280838\pi$
$601$	1.12527e7			1.27078			0.635392	+	0.772190i	$0.280838\pi$
$602$		0			0
$603$	−4.40413e6			−0.493249
$604$		0			0
$605$	−5.23585e6	−	9.06877e6i	−0.581566	−	1.00730i
$606$		0			0
$607$	−7.01851e6	+	1.21564e7i	−0.773167	+	1.33916i	0.162652	+	0.986684i	$0.447995\pi$
$607$	−7.01851e6	+	1.21564e7i	−0.773167	+	1.33916i	−0.935819	+	0.352481i	$0.885338\pi$
$608$		0			0
$609$	−831580.	−	304633.i	−0.0908575	−	0.0332838i
$610$		0			0
$611$	4.29761e6	−	7.44367e6i	0.465719	−	0.806648i
$612$		0			0
$613$	3.81436e6	+	6.60667e6i	0.409988	+	0.710119i	0.994888	−	0.100986i	$-0.0321997\pi$
$613$	3.81436e6	+	6.60667e6i	0.409988	+	0.710119i	−0.584900	+	0.811105i	$0.698866\pi$
$614$		0			0
$615$	−5.19119e6			−0.553451
$616$		0			0
$617$	−4.41080e6			−0.466450			−0.233225	−	0.972423i	$-0.574928\pi$
$617$	−4.41080e6			−0.466450			−0.233225	+	0.972423i	$0.574928\pi$
$618$		0			0
$619$	−4.52178e6	−	7.83195e6i	−0.474332	−	0.821567i	0.525236	−	0.850957i	$-0.323977\pi$
$619$	−4.52178e6	−	7.83195e6i	−0.474332	−	0.821567i	−0.999568	+	0.0293895i	$0.990644\pi$
$620$		0			0
$621$	661192.	−	1.14522e6i	0.0688016	−	0.119168i
$622$		0			0
$623$	−2.19279e6	−	1.25892e7i	−0.226348	−	1.29950i
$624$		0			0
$625$	5.59185e6	−	9.68538e6i	0.572606	−	0.991782i
$626$		0			0
$627$	−1.87999e6	−	3.25624e6i	−0.190980	−	0.330786i
$628$		0			0
$629$	−8.96293e6			−0.903282
$630$		0			0
$631$	7.33039e6			0.732916			0.366458	−	0.930435i	$-0.380570\pi$
$631$	7.33039e6			0.732916			0.366458	+	0.930435i	$0.380570\pi$
$632$		0			0
$633$	2.16786e6	+	3.75485e6i	0.215042	+	0.372463i
$634$		0			0
$635$	1.26515e6	−	2.19130e6i	0.124511	−	0.215659i
$636$		0			0
$637$	1.04385e6	−	5.77912e6i	0.101927	−	0.564304i
$638$		0			0
$639$	−2.84090e6	+	4.92059e6i	−0.275235	+	0.476722i
$640$		0			0
$641$	−1.84412e6	−	3.19411e6i	−0.177273	−	0.307047i	0.763672	−	0.645604i	$-0.223394\pi$
$641$	−1.84412e6	−	3.19411e6i	−0.177273	−	0.307047i	−0.940946	+	0.338558i	$0.890061\pi$
$642$		0			0
$643$	1.17584e7			1.12155			0.560776	−	0.827968i	$-0.310503\pi$
$643$	1.17584e7			1.12155			0.560776	+	0.827968i	$0.310503\pi$
$644$		0			0
$645$	8.28147e6			0.783806
$646$		0			0
$647$	−4.97025e6	−	8.60873e6i	−0.466786	−	0.808497i	0.532494	−	0.846434i	$-0.321255\pi$
$647$	−4.97025e6	−	8.60873e6i	−0.466786	−	0.808497i	−0.999280	+	0.0379368i	$0.987921\pi$
$648$		0			0
$649$	−1.96982e6	+	3.41182e6i	−0.183575	+	0.317962i
$650$		0			0
$651$	1.80830e6	+	1.03817e7i	0.167231	+	0.960101i
$652$		0			0
$653$	513100.	−	888714.i	0.0470889	−	0.0815604i	−0.841520	−	0.540226i	$-0.818339\pi$
$653$	513100.	−	888714.i	0.0470889	−	0.0815604i	0.888609	+	0.458665i	$0.151672\pi$
$654$		0			0
$655$	407995.	+	706668.i	0.0371579	+	0.0643594i
$656$		0			0
$657$	−3.60197e6			−0.325557
$658$		0			0
$659$	−1.00207e7			−0.898846			−0.449423	−	0.893319i	$-0.648370\pi$
$659$	−1.00207e7			−0.898846			−0.449423	+	0.893319i	$0.648370\pi$
$660$		0			0
$661$	−1.31413e6	−	2.27614e6i	−0.116986	−	0.202626i	0.801586	−	0.597880i	$-0.203990\pi$
$661$	−1.31413e6	−	2.27614e6i	−0.116986	−	0.202626i	−0.918572	+	0.395254i	$0.870657\pi$
$662$		0			0
$663$	1.80799e6	−	3.13152e6i	0.159739	−	0.276676i
$664$		0			0
$665$	−2.56872e7	−	9.40999e6i	−2.25249	−	0.825154i
$666$		0			0
$667$	−688431.	+	1.19240e6i	−0.0599165	+	0.103778i
$668$		0			0
$669$	2.87450e6	+	4.97879e6i	0.248312	+	0.430089i
$670$		0			0
$671$	5.27925e6			0.452654
$672$		0			0
$673$	−1.50220e7			−1.27847			−0.639233	−	0.769013i	$-0.720748\pi$
$673$	−1.50220e7			−1.27847			−0.639233	+	0.769013i	$0.720748\pi$
$674$		0			0
$675$	−938257.	−	1.62511e6i	−0.0792616	−	0.137285i
$676$		0			0
$677$	3.01655e6	−	5.22482e6i	0.252953	−	0.438127i	−0.711385	−	0.702803i	$-0.751932\pi$
$677$	3.01655e6	−	5.22482e6i	0.252953	−	0.438127i	0.964337	+	0.264676i	$0.0852650\pi$
$678$		0			0
$679$	−3.21757e6	+	2.68846e6i	−0.267826	+	0.223784i
$680$		0			0
$681$	−2.36557e6	+	4.09729e6i	−0.195465	+	0.338554i
$682$		0			0
$683$	5.53601e6	+	9.58865e6i	0.454093	+	0.786513i	0.998636	−	0.0522205i	$-0.0166299\pi$
$683$	5.53601e6	+	9.58865e6i	0.454093	+	0.786513i	−0.544542	+	0.838734i	$0.683297\pi$
$684$		0			0
$685$	1.42922e6			0.116379
$686$		0			0
$687$	8.39417e6			0.678557
$688$		0			0
$689$	−2.37537e6	−	4.11426e6i	−0.190626	−	0.330174i
$690$		0			0
$691$	−5.27739e6	+	9.14071e6i	−0.420460	+	0.728257i	−0.995984	−	0.0895269i	$-0.971464\pi$
$691$	−5.27739e6	+	9.14071e6i	−0.420460	+	0.728257i	0.575525	+	0.817784i	$0.304798\pi$
$692$		0			0
$693$	−1.20440e6	+	1.00635e6i	−0.0952661	+	0.0796003i
$694$		0			0
$695$	6.37523e6	−	1.10422e7i	0.500650	−	0.867151i
$696$		0			0
$697$	4.39270e6	+	7.60838e6i	0.342491	+	0.593212i
$698$		0			0
$699$	−6.37014e6			−0.493124
$700$		0			0
$701$	−7.20675e6			−0.553917			−0.276958	−	0.960882i	$-0.589326\pi$
$701$	−7.20675e6			−0.553917			−0.276958	+	0.960882i	$0.589326\pi$
$702$		0			0
$703$	−1.08941e7	−	1.88692e7i	−0.831390	−	1.44001i
$704$		0			0
$705$	−8.35660e6	+	1.44741e7i	−0.633223	+	1.09677i
$706$		0			0
$707$	3.81583e6	+	1.39785e6i	0.287104	+	0.105175i
$708$		0			0
$709$	−1.34687e6	+	2.33284e6i	−0.100626	+	0.174289i	−0.911943	−	0.410318i	$-0.865418\pi$
$709$	−1.34687e6	+	2.33284e6i	−0.100626	+	0.174289i	0.811317	+	0.584607i	$0.198751\pi$
$710$		0			0
$711$	2.49530e6	+	4.32199e6i	0.185118	+	0.320634i
$712$		0			0
$713$	1.63833e7			1.20692
$714$		0			0
$715$	−3.94253e6			−0.288410
$716$		0			0
$717$	−2.25276e6	−	3.90190e6i	−0.163651	−	0.283451i
$718$		0			0
$719$	3.46275e6	−	5.99766e6i	0.249804	−	0.432673i	−0.713667	−	0.700485i	$-0.752967\pi$
$719$	3.46275e6	−	5.99766e6i	0.249804	−	0.432673i	0.963471	+	0.267812i	$0.0863005\pi$
$720$		0			0
$721$	2.20978e6	+	1.26867e7i	0.158311	+	0.908887i
$722$		0			0
$723$	−5.43698e6	+	9.41713e6i	−0.386823	+	0.669997i
$724$		0			0
$725$	976911.	+	1.69206e6i	0.0690256	+	0.119556i
$726$		0			0
$727$	−4.11366e6			−0.288664			−0.144332	−	0.989529i	$-0.546103\pi$
$727$	−4.11366e6			−0.288664			−0.144332	+	0.989529i	$0.546103\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−7.00765e6	−	1.21376e7i	−0.485041	−	0.840116i
$732$		0			0
$733$	−3.96019e6	+	6.85925e6i	−0.272243	+	0.471538i	−0.969436	−	0.245345i	$-0.921099\pi$
$733$	−3.96019e6	+	6.85925e6i	−0.272243	+	0.471538i	0.697193	+	0.716883i	$0.254432\pi$
$734$		0			0
$735$	−2.02974e6	+	1.12374e7i	−0.138587	+	0.767266i
$736$		0			0
$737$	4.06326e6	−	7.03777e6i	0.275554	−	0.477273i
$738$		0			0
$739$	8.02050e6	+	1.38919e7i	0.540245	+	0.935731i	0.998890	+	0.0471115i	$0.0150016\pi$
$739$	8.02050e6	+	1.38919e7i	0.540245	+	0.935731i	−0.458645	+	0.888620i	$0.651665\pi$
$740$		0			0
$741$	8.79019e6			0.588102
$742$		0			0
$743$	−1.53453e7			−1.01977			−0.509887	−	0.860241i	$-0.670313\pi$
$743$	−1.53453e7			−1.01977			−0.509887	+	0.860241i	$0.670313\pi$
$744$		0			0
$745$	9.96507e6	+	1.72600e7i	0.657794	+	1.13933i
$746$		0			0
$747$	−3.52925e6	+	6.11284e6i	−0.231409	+	0.400813i
$748$		0			0
$749$	−3.23166e6	−	1.85535e7i	−0.210485	−	1.20843i
$750$		0			0
$751$	1.12488e7	−	1.94835e7i	0.727790	−	1.26057i	−0.230025	−	0.973185i	$-0.573881\pi$
$751$	1.12488e7	−	1.94835e7i	0.727790	−	1.26057i	0.957815	−	0.287385i	$-0.0927860\pi$
$752$		0			0
$753$	440220.	+	762483.i	0.0282932	+	0.0490053i
$754$		0			0
$755$	2.10756e7			1.34559
$756$		0			0
$757$	2.30349e7			1.46099			0.730494	−	0.682919i	$-0.239290\pi$
$757$	2.30349e7			1.46099			0.730494	+	0.682919i	$0.239290\pi$
$758$		0			0
$759$	1.22004e6	+	2.11316e6i	0.0768720	+	0.133146i
$760$		0			0
$761$	−2.70368e6	+	4.68290e6i	−0.169236	+	0.293126i	−0.938152	−	0.346225i	$-0.887463\pi$
$761$	−2.70368e6	+	4.68290e6i	−0.169236	+	0.293126i	0.768915	+	0.639351i	$0.220797\pi$
$762$		0			0
$763$	−2.20151e7	−	8.06480e6i	−1.36902	−	0.501513i
$764$		0			0
$765$	−3.51559e6	+	6.08918e6i	−0.217192	+	0.376188i
$766$		0			0
$767$	−4.60509e6	−	7.97626e6i	−0.282651	−	0.489565i
$768$		0			0
$769$	−7.93100e6			−0.483629			−0.241814	−	0.970323i	$-0.577742\pi$
$769$	−7.93100e6			−0.483629			−0.241814	+	0.970323i	$0.577742\pi$
$770$		0			0
$771$	−9.97122e6			−0.604105
$772$		0			0
$773$	−134790.	−	233462.i	−0.00811349	−	0.0140530i	0.861940	−	0.507010i	$-0.169249\pi$
$773$	−134790.	−	233462.i	−0.00811349	−	0.0140530i	−0.870054	+	0.492957i	$0.835916\pi$
$774$		0			0
$775$	1.16243e7	−	2.01338e7i	0.695203	−	1.20413i
$776$		0			0
$777$	−6.97925e6	+	5.83157e6i	−0.414721	+	0.346523i
$778$		0			0
$779$	−1.06784e7	+	1.84955e7i	−0.630465	+	1.09200i
$780$		0			0
$781$	−5.24205e6	−	9.07950e6i	−0.307520	−	0.532641i
$782$

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:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 - 7.79423i) q^{3} +(37.7462 - 65.3783i) q^{5} +(-99.4847 + 83.1252i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(-4.50000 - 7.79423i) q^{3} +(37.7462 - 65.3783i) q^{5} +(-99.4847 + 83.1252i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-74.7309 - 129.438i) q^{11} +349.416 q^{13} -679.431 q^{15} +(574.923 + 995.797i) q^{17} +(-1397.60 + 2420.72i) q^{19} +(1095.58 + 401.343i) q^{21} +(906.985 - 1570.94i) q^{23} +(-1287.05 - 2229.23i) q^{25} +729.000 q^{27} -759.033 q^{29} +(4515.87 + 7821.72i) q^{31} +(-672.578 + 1164.94i) q^{33} +(1679.42 + 9641.80i) q^{35} +(-3897.45 + 6750.57i) q^{37} +(-1572.37 - 2723.43i) q^{39} +7640.49 q^{41} -12188.8 q^{43} +(3057.44 + 5295.64i) q^{45} +(12299.4 - 21303.2i) q^{47} +(2987.41 - 16539.4i) q^{49} +(5174.31 - 8962.17i) q^{51} +(-6798.11 - 11774.7i) q^{53} -11283.2 q^{55} +25156.8 q^{57} +(-13179.4 - 22827.4i) q^{59} +(-17660.9 + 30589.5i) q^{61} +(-1801.94 - 10345.2i) q^{63} +(13189.1 - 22844.2i) q^{65} +(27186.0 + 47087.5i) q^{67} -16325.7 q^{69} +70145.7 q^{71} +(22234.4 + 38511.1i) q^{73} +(-11583.4 + 20063.1i) q^{75} +(18194.1 + 6665.05i) q^{77} +(30806.2 - 53357.9i) q^{79} +(-3280.50 - 5681.99i) q^{81} +87142.0 q^{83} +86804.6 q^{85} +(3415.65 + 5916.08i) q^{87} +(-49284.7 + 85363.6i) q^{89} +(-34761.5 + 29045.3i) q^{91} +(40642.9 - 70395.5i) q^{93} +(105508. + 182745. i) q^{95} +32342.3 q^{97} +12106.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{3} + 53 q^{5} - 6 q^{7} - 162 q^{9}+O(q^{10}) \) 4 * q - 18 * q^3 + 53 * q^5 - 6 * q^7 - 162 * q^9	\( 4 q - 18 q^{3} + 53 q^{5} - 6 q^{7} - 162 q^{9} + 191 q^{11} - 758 q^{13} - 954 q^{15} + 340 q^{17} - 1769 q^{19} - 27 q^{21} + 3236 q^{23} + 45 q^{25} + 2916 q^{27} + 8918 q^{29} + 1994 q^{31} + 1719 q^{33} + 4562 q^{35} - 20587 q^{37} + 3411 q^{39} + 17628 q^{41} - 31706 q^{43} + 4293 q^{45} + 33912 q^{47} + 9598 q^{49} + 3060 q^{51} - 49239 q^{53} - 37882 q^{55} + 31842 q^{57} - 56735 q^{59} - 67508 q^{61} + 729 q^{63} + 42762 q^{65} + 75723 q^{67} - 58248 q^{69} + 17984 q^{71} + 3201 q^{73} + 405 q^{75} + 120299 q^{77} + 26612 q^{79} - 13122 q^{81} + 1898 q^{83} + 210040 q^{85} - 40131 q^{87} - 176562 q^{89} - 210085 q^{91} + 17946 q^{93} + 234098 q^{95} - 258846 q^{97} - 30942 q^{99}+O(q^{100}) \) 4 * q - 18 * q^3 + 53 * q^5 - 6 * q^7 - 162 * q^9 + 191 * q^11 - 758 * q^13 - 954 * q^15 + 340 * q^17 - 1769 * q^19 - 27 * q^21 + 3236 * q^23 + 45 * q^25 + 2916 * q^27 + 8918 * q^29 + 1994 * q^31 + 1719 * q^33 + 4562 * q^35 - 20587 * q^37 + 3411 * q^39 + 17628 * q^41 - 31706 * q^43 + 4293 * q^45 + 33912 * q^47 + 9598 * q^49 + 3060 * q^51 - 49239 * q^53 - 37882 * q^55 + 31842 * q^57 - 56735 * q^59 - 67508 * q^61 + 729 * q^63 + 42762 * q^65 + 75723 * q^67 - 58248 * q^69 + 17984 * q^71 + 3201 * q^73 + 405 * q^75 + 120299 * q^77 + 26612 * q^79 - 13122 * q^81 + 1898 * q^83 + 210040 * q^85 - 40131 * q^87 - 176562 * q^89 - 210085 * q^91 + 17946 * q^93 + 234098 * q^95 - 258846 * q^97 - 30942 * q^99

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