LMFDB - Embedded newform 336.6.q.f.193.1

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			193.1
Root			$-24.2462 - 41.9956i$ of defining polynomial
Character	$\chi$	$=$	336.193
Dual form			336.6.q.f.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 7.79423i) q^{3} +(-11.2462 - 19.4789i) q^{5} +(96.4847 + 86.5893i) q^{7} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})$	\(q+(-4.50000 + 7.79423i) q^{3} +(-11.2462 - 19.4789i) q^{5} +(96.4847 + 86.5893i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(170.231 - 294.849i) q^{11} -728.416 q^{13} +202.431 q^{15} +(-404.923 + 701.348i) q^{17} +(513.101 + 888.717i) q^{19} +(-1109.08 + 362.372i) q^{21} +(711.015 + 1231.51i) q^{23} +(1309.55 - 2268.20i) q^{25} +729.000 q^{27} +5218.03 q^{29} +(-3518.87 + 6094.87i) q^{31} +(1532.08 + 2653.64i) q^{33} +(601.584 - 2853.22i) q^{35} +(-6396.05 - 11078.3i) q^{37} +(3277.87 - 5677.44i) q^{39} +1173.51 q^{41} -3664.17 q^{43} +(-910.940 + 1577.79i) q^{45} +(4656.60 + 8065.46i) q^{47} +(1811.59 + 16709.1i) q^{49} +(-3644.31 - 6312.13i) q^{51} +(-17821.4 + 30867.6i) q^{53} -7657.78 q^{55} -9235.81 q^{57} +(-15188.1 + 26306.5i) q^{59} +(-16093.1 - 27874.1i) q^{61} +(2166.44 - 10275.1i) q^{63} +(8191.89 + 14188.8i) q^{65} +(10675.5 - 18490.6i) q^{67} -12798.3 q^{69} -61153.7 q^{71} +(-20633.9 + 35739.0i) q^{73} +(11785.9 + 20413.8i) q^{75} +(41955.4 - 13708.2i) q^{77} +(-17500.2 - 30311.3i) q^{79} +(-3280.50 + 5681.99i) q^{81} -86193.0 q^{83} +18215.4 q^{85} +(-23481.1 + 40670.5i) q^{87} +(-38996.3 - 67543.6i) q^{89} +(-70281.0 - 63073.0i) q^{91} +(-31669.9 - 54853.8i) q^{93} +(11540.8 - 19989.3i) q^{95} -161765. q^{97} -27577.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/336\mathbb{Z}\right)^\times$.

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	−11.2462	−	19.4789i	−0.201178	−	0.348450i	0.747730	−	0.664002i	$-0.231144\pi$
$5$	−11.2462	−	19.4789i	−0.201178	−	0.348450i	−0.948908	+	0.315552i	$0.897810\pi$
$6$		0			0
$7$	96.4847	+	86.5893i	0.744241	+	0.667912i
$7$	96.4847	+	86.5893i	0.744241	+	0.667912i
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	170.231	−	294.849i	0.424186	−	0.734712i	−0.572158	−	0.820144i	$-0.693894\pi$
$11$	170.231	−	294.849i	0.424186	−	0.734712i	0.996344	+	0.0854314i	$0.0272268\pi$
$12$		0			0
$13$	−728.416			−1.19542			−0.597711	−	0.801712i	$-0.703923\pi$
$13$	−728.416			−1.19542			−0.597711	+	0.801712i	$0.703923\pi$
$14$		0			0
$15$	202.431			0.232300
$16$		0			0
$17$	−404.923	+	701.348i	−0.339821	+	0.588588i	−0.984399	−	0.175951i	$-0.943700\pi$
$17$	−404.923	+	701.348i	−0.339821	+	0.588588i	0.644578	+	0.764539i	$0.277033\pi$
$18$		0			0
$19$	513.101	+	888.717i	0.326076	+	0.564780i	0.981730	−	0.190282i	$-0.0609402\pi$
$19$	513.101	+	888.717i	0.326076	+	0.564780i	−0.655654	+	0.755062i	$0.727607\pi$
$20$		0			0
$21$	−1109.08	+	362.372i	−0.548800	+	0.179311i
$22$		0			0
$23$	711.015	+	1231.51i	0.280259	+	0.485423i	0.971448	−	0.237251i	$-0.0762464\pi$
$23$	711.015	+	1231.51i	0.280259	+	0.485423i	−0.691190	+	0.722674i	$0.742913\pi$
$24$		0			0
$25$	1309.55	−	2268.20i	0.419055	−	0.725825i
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	5218.03			1.15216			0.576079	−	0.817394i	$-0.304582\pi$
$29$	5218.03			1.15216			0.576079	+	0.817394i	$0.304582\pi$
$30$		0			0
$31$	−3518.87	+	6094.87i	−0.657657	+	1.13909i	0.323564	+	0.946206i	$0.395119\pi$
$31$	−3518.87	+	6094.87i	−0.657657	+	1.13909i	−0.981221	+	0.192889i	$0.938214\pi$
$32$		0			0
$33$	1532.08	+	2653.64i	0.244904	+	0.424186i
$34$		0			0
$35$	601.584	−	2853.22i	0.0830092	−	0.393699i
$36$		0			0
$37$	−6396.05	−	11078.3i	−0.768082	−	1.33036i	−0.938601	−	0.345004i	$-0.887878\pi$
$37$	−6396.05	−	11078.3i	−0.768082	−	1.33036i	0.170519	−	0.985354i	$-0.445456\pi$
$38$		0			0
$39$	3277.87	−	5677.44i	0.345088	−	0.597711i
$40$		0			0
$41$	1173.51			0.109025			0.0545124	−	0.998513i	$-0.482640\pi$
$41$	1173.51			0.109025			0.0545124	+	0.998513i	$0.482640\pi$
$42$		0			0
$43$	−3664.17			−0.302207			−0.151103	−	0.988518i	$-0.548283\pi$
$43$	−3664.17			−0.302207			−0.151103	+	0.988518i	$0.548283\pi$
$44$		0			0
$45$	−910.940	+	1577.79i	−0.0670592	+	0.116150i
$46$		0			0
$47$	4656.60	+	8065.46i	0.307485	+	0.532580i	0.977812	−	0.209487i	$-0.0671793\pi$
$47$	4656.60	+	8065.46i	0.307485	+	0.532580i	−0.670326	+	0.742066i	$0.733846\pi$
$48$		0			0
$49$	1811.59	+	16709.1i	0.107788	+	0.994174i
$50$		0			0
$51$	−3644.31	−	6312.13i	−0.196196	−	0.339821i
$52$		0			0
$53$	−17821.4	+	30867.6i	−0.871469	+	1.50943i	−0.0109916	+	0.999940i	$0.503499\pi$
$53$	−17821.4	+	30867.6i	−0.871469	+	1.50943i	−0.860477	+	0.509489i	$0.829835\pi$
$54$		0			0
$55$	−7657.78			−0.341347
$56$		0			0
$57$	−9235.81			−0.376520
$58$		0			0
$59$	−15188.1	+	26306.5i	−0.568033	+	0.983861i	0.428728	+	0.903434i	$0.358962\pi$
$59$	−15188.1	+	26306.5i	−0.568033	+	0.983861i	−0.996760	+	0.0804276i	$0.974371\pi$
$60$		0			0
$61$	−16093.1	−	27874.1i	−0.553753	−	0.959128i	−0.997999	−	0.0632231i	$-0.979862\pi$
$61$	−16093.1	−	27874.1i	−0.553753	−	0.959128i	0.444247	−	0.895904i	$-0.353471\pi$
$62$		0			0
$63$	2166.44	−	10275.1i	0.0687694	−	0.326162i
$64$		0			0
$65$	8191.89	+	14188.8i	0.240492	+	0.416544i
$66$		0			0
$67$	10675.5	−	18490.6i	0.290538	−	0.503226i	−0.683399	−	0.730045i	$-0.739499\pi$
$67$	10675.5	−	18490.6i	0.290538	−	0.503226i	0.973937	+	0.226819i	$0.0728324\pi$
$68$		0			0
$69$	−12798.3			−0.323615
$70$		0			0
$71$	−61153.7			−1.43972			−0.719859	−	0.694121i	$-0.755793\pi$
$71$	−61153.7			−1.43972			−0.719859	+	0.694121i	$0.755793\pi$
$72$		0			0
$73$	−20633.9	+	35739.0i	−0.453184	+	0.784937i	−0.998582	−	0.0532401i	$-0.983045\pi$
$73$	−20633.9	+	35739.0i	−0.453184	+	0.784937i	0.545398	+	0.838177i	$0.316378\pi$
$74$		0			0
$75$	11785.9	+	20413.8i	0.241942	+	0.419055i
$76$		0			0
$77$	41955.4	−	13708.2i	0.806419	−	0.263484i
$78$		0			0
$79$	−17500.2	−	30311.3i	−0.315483	−	0.546433i	0.664057	−	0.747682i	$-0.268833\pi$
$79$	−17500.2	−	30311.3i	−0.315483	−	0.546433i	−0.979540	+	0.201249i	$0.935500\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−86193.0			−1.37334			−0.686668	−	0.726972i	$-0.740927\pi$
$83$	−86193.0			−1.37334			−0.686668	+	0.726972i	$0.740927\pi$
$84$		0			0
$85$	18215.4			0.273458
$86$		0			0
$87$	−23481.1	+	40670.5i	−0.332599	+	0.576079i
$88$		0			0
$89$	−38996.3	−	67543.6i	−0.521853	−	0.903876i	−0.999677	−	0.0254206i	$-0.991908\pi$
$89$	−38996.3	−	67543.6i	−0.521853	−	0.903876i	0.477824	−	0.878456i	$-0.341426\pi$
$90$		0			0
$91$	−70281.0	−	63073.0i	−0.889681	−	0.798436i
$92$		0			0
$93$	−31669.9	−	54853.8i	−0.379698	−	0.657657i
$94$		0			0
$95$	11540.8	−	19989.3i	0.131198	−	0.227242i
$96$		0			0
$97$	−161765.			−1.74565			−0.872823	−	0.488037i	$-0.837713\pi$
$97$	−161765.			−1.74565			−0.872823	+	0.488037i	$0.837713\pi$
$98$		0			0
$99$	−27577.4			−0.282791
$100$		0			0
$101$	32927.2	−	57031.6i	0.321182	−	0.556304i	−0.659550	−	0.751661i	$-0.729253\pi$
$101$	32927.2	−	57031.6i	0.321182	−	0.556304i	0.980732	+	0.195357i	$0.0625865\pi$
$102$		0			0
$103$	65099.0	+	112755.i	0.604619	+	1.04723i	0.992112	+	0.125358i	$0.0400078\pi$
$103$	65099.0	+	112755.i	0.604619	+	1.04723i	−0.387493	+	0.921873i	$0.626659\pi$
$104$		0			0
$105$	19531.5	+	17528.4i	0.172887	+	0.155156i
$106$		0			0
$107$	1295.48	+	2243.83i	0.0109388	+	0.0189466i	0.871443	−	0.490497i	$-0.163185\pi$
$107$	1295.48	+	2243.83i	0.0109388	+	0.0189466i	−0.860504	+	0.509443i	$0.829851\pi$
$108$		0			0
$109$	−55326.9	+	95828.9i	−0.446036	+	0.772557i	−0.998124	−	0.0612291i	$-0.980498\pi$
$109$	−55326.9	+	95828.9i	−0.446036	+	0.772557i	0.552088	+	0.833786i	$0.313831\pi$
$110$		0			0
$111$	115129.			0.886905
$112$		0			0
$113$	−193910.			−1.42858			−0.714291	−	0.699849i	$-0.753251\pi$
$113$	−193910.			−1.42858			−0.714291	+	0.699849i	$0.753251\pi$
$114$		0			0
$115$	15992.4	−	27699.7i	0.112764	−	0.195312i
$116$		0			0
$117$	29500.8	+	51097.0i	0.199237	+	0.345088i
$118$		0			0
$119$	−99798.1	+	32607.3i	−0.646033	+	0.211080i
$120$		0			0
$121$	22568.4	+	39089.6i	0.140132	+	0.242716i
$122$		0			0
$123$	−5280.77	+	9146.57i	−0.0314728	+	0.0545124i
$124$		0			0
$125$	−129198.			−0.739573
$126$		0			0
$127$	−27429.2			−0.150905			−0.0754526	−	0.997149i	$-0.524040\pi$
$127$	−27429.2			−0.150905			−0.0754526	+	0.997149i	$0.524040\pi$
$128$		0			0
$129$	16488.7	−	28559.3i	0.0872395	−	0.151103i
$130$		0			0
$131$	75188.0	+	130229.i	0.382798	+	0.663026i	0.991461	−	0.130403i	$-0.0416271\pi$
$131$	75188.0	+	130229.i	0.382798	+	0.663026i	−0.608663	+	0.793429i	$0.708294\pi$
$132$		0			0
$133$	−27447.0	+	130177.i	−0.134544	+	0.638122i
$134$		0			0
$135$	−8198.46	−	14200.1i	−0.0387167	−	0.0670592i
$136$		0			0
$137$	−48051.0	+	83226.8i	−0.218726	+	0.378845i	−0.954419	−	0.298471i	$-0.903524\pi$
$137$	−48051.0	+	83226.8i	−0.218726	+	0.378845i	0.735693	+	0.677316i	$0.236857\pi$
$138$		0			0
$139$	−100854.			−0.442749			−0.221375	−	0.975189i	$-0.571054\pi$
$139$	−100854.			−0.442749			−0.221375	+	0.975189i	$0.571054\pi$
$140$		0			0
$141$	−83818.7			−0.355053
$142$		0			0
$143$	−123999.	+	214772.i	−0.507081	+	0.878291i
$144$		0			0
$145$	−58682.9	−	101642.i	−0.231788	−	0.401469i
$146$		0			0
$147$	−138387.	−	61070.9i	−0.528203	−	0.233099i
$148$		0			0
$149$	180472.	+	312587.i	0.665954	+	1.15347i	0.979026	+	0.203737i	$0.0653087\pi$
$149$	180472.	+	312587.i	0.665954	+	1.15347i	−0.313071	+	0.949730i	$0.601358\pi$
$150$		0			0
$151$	−217028.	+	375903.i	−0.774592	+	1.34163i	0.160431	+	0.987047i	$0.448712\pi$
$151$	−217028.	+	375903.i	−0.774592	+	1.34163i	−0.935023	+	0.354586i	$0.884622\pi$
$152$		0			0
$153$	65597.6			0.226548
$154$		0			0
$155$	158295.			0.529223
$156$		0			0
$157$	255798.	−	443055.i	0.828225	−	1.43453i	−0.0712046	−	0.997462i	$-0.522684\pi$
$157$	255798.	−	443055.i	0.828225	−	1.43453i	0.899430	−	0.437066i	$-0.143982\pi$
$158$		0			0
$159$	−160393.	−	277808.i	−0.503143	−	0.871469i
$160$		0			0
$161$	−38033.9	+	180389.i	−0.115639	+	0.548459i
$162$		0			0
$163$	−125635.	−	217606.i	−0.370374	−	0.641507i	0.619249	−	0.785195i	$-0.287437\pi$
$163$	−125635.	−	217606.i	−0.370374	−	0.641507i	−0.989623	+	0.143688i	$0.954104\pi$
$164$		0			0
$165$	34460.0	−	59686.5i	0.0985384	−	0.170674i
$166$		0			0
$167$	−419277.			−1.16335			−0.581674	−	0.813422i	$-0.697602\pi$
$167$	−419277.			−1.16335			−0.581674	+	0.813422i	$0.697602\pi$
$168$		0			0
$169$	159297.			0.429032
$170$		0			0
$171$	41561.2	−	71986.0i	0.108692	−	0.188260i
$172$		0			0
$173$	230688.	+	399564.i	0.586017	+	1.01501i	0.994748	+	0.102356i	$0.0326381\pi$
$173$	230688.	+	399564.i	0.586017	+	1.01501i	−0.408731	+	0.912655i	$0.634029\pi$
$174$		0			0
$175$	322753.	−	105454.i	0.796665	−	0.260296i
$176$		0			0
$177$	−136693.	−	236759.i	−0.327954	−	0.568033i
$178$		0			0
$179$	370853.	−	642336.i	0.865105	−	1.49841i	−0.00183697	−	0.999998i	$-0.500585\pi$
$179$	370853.	−	642336.i	0.865105	−	1.49841i	0.866942	−	0.498408i	$-0.166082\pi$
$180$		0			0
$181$	301371.			0.683762			0.341881	−	0.939743i	$-0.388936\pi$
$181$	301371.			0.683762			0.341881	+	0.939743i	$0.388936\pi$
$182$		0			0
$183$	289676.			0.639418
$184$		0			0
$185$	−143862.	+	249177.i	−0.309042	+	0.535277i
$186$		0			0
$187$	137861.	+	238782.i	0.288295	+	0.499342i
$188$		0			0
$189$	70337.3	+	63123.6i	0.143229	+	0.128540i
$190$		0			0
$191$	131591.	+	227923.i	0.261002	+	0.452069i	0.966508	−	0.256635i	$-0.0826138\pi$
$191$	131591.	+	227923.i	0.261002	+	0.452069i	−0.705507	+	0.708703i	$0.749280\pi$
$192$		0			0
$193$	415563.	−	719777.i	0.803053	−	1.39093i	−0.114545	−	0.993418i	$-0.536541\pi$
$193$	415563.	−	719777.i	0.803053	−	1.39093i	0.917598	−	0.397510i	$-0.130126\pi$
$194$		0			0
$195$	−147454.			−0.277696
$196$		0			0
$197$	−1.05421e6			−1.93536			−0.967681	−	0.252178i	$-0.918853\pi$
$197$	−1.05421e6			−1.93536			−0.967681	+	0.252178i	$0.918853\pi$
$198$		0			0
$199$	349284.	−	604977.i	0.625239	−	1.08295i	−0.363256	−	0.931689i	$-0.618335\pi$
$199$	349284.	−	604977.i	0.625239	−	1.08295i	0.988495	−	0.151256i	$-0.0483317\pi$
$200$		0			0
$201$	96079.9	+	166415.i	0.167742	+	0.290538i
$202$		0			0
$203$	503460.	+	451826.i	0.857482	+	0.769539i
$204$		0			0
$205$	−13197.4	−	22858.6i	−0.0219334	−	0.0379897i
$206$		0			0
$207$	57592.2	−	99752.7i	0.0934196	−	0.161808i
$208$		0			0
$209$	349382.			0.553268
$210$		0			0
$211$	99693.6			0.154156			0.0770781	−	0.997025i	$-0.475441\pi$
$211$	99693.6			0.154156			0.0770781	+	0.997025i	$0.475441\pi$
$212$		0			0
$213$	275192.	−	476646.i	0.415611	−	0.719859i
$214$		0			0
$215$	41207.8	+	71374.1i	0.0607972	+	0.105304i
$216$		0			0
$217$	−867267.	+	283365.i	−1.25027	+	0.408504i
$218$		0			0
$219$	−185705.	−	321651.i	−0.261646	−	0.453184i
$220$		0			0
$221$	294953.	−	510873.i	0.406230	−	0.703610i
$222$		0			0
$223$	−526194.			−0.708572			−0.354286	−	0.935137i	$-0.615276\pi$
$223$	−526194.			−0.708572			−0.354286	+	0.935137i	$0.615276\pi$
$224$		0			0
$225$	−212147.			−0.279370
$226$		0			0
$227$	−490802.	+	850095.i	−0.632182	+	1.09497i	0.354923	+	0.934896i	$0.384507\pi$
$227$	−490802.	+	850095.i	−0.632182	+	1.09497i	−0.987105	+	0.160076i	$0.948826\pi$
$228$		0			0
$229$	42540.5	+	73682.3i	0.0536061	+	0.0928484i	0.891583	−	0.452857i	$-0.149595\pi$
$229$	42540.5	+	73682.3i	0.0536061	+	0.0928484i	−0.837977	+	0.545705i	$0.816262\pi$
$230$		0			0
$231$	−81954.4	+	388697.i	−0.101051	+	0.479271i
$232$		0			0
$233$	−56561.4	−	97967.3i	−0.0682544	−	0.118220i	0.829879	−	0.557944i	$-0.188410\pi$
$233$	−56561.4	−	97967.3i	−0.0682544	−	0.118220i	−0.898133	+	0.439724i	$0.855076\pi$
$234$		0			0
$235$	104738.	−	181411.i	0.123718	−	0.214286i
$236$		0			0
$237$	315004.			0.364288
$238$		0			0
$239$	895100.			1.01362			0.506812	−	0.862057i	$-0.330824\pi$
$239$	895100.			1.01362			0.506812	+	0.862057i	$0.330824\pi$
$240$		0			0
$241$	−263857.	+	457014.i	−0.292635	+	0.506859i	−0.974432	−	0.224683i	$-0.927865\pi$
$241$	−263857.	+	457014.i	−0.292635	+	0.506859i	0.681797	+	0.731542i	$0.261199\pi$
$242$		0			0
$243$	−29524.5	−	51137.9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	305102.	−	223201.i	0.324735	−	0.237564i
$246$		0			0
$247$	−373751.	−	647355.i	−0.389798	−	0.675150i
$248$		0			0
$249$	387868.	−	671808.i	0.396448	−	0.686668i
$250$		0			0
$251$	1.19293e6			1.19517			0.597584	−	0.801806i	$-0.296127\pi$
$251$	1.19293e6			1.19517			0.597584	+	0.801806i	$0.296127\pi$
$252$		0			0
$253$	484147.			0.475528
$254$		0			0
$255$	−81969.1	+	141975.i	−0.0789405	+	0.136729i
$256$		0			0
$257$	597462.	+	1.03483e6i	0.564257	+	0.977322i	0.997118	+	0.0758617i	$0.0241707\pi$
$257$	597462.	+	1.03483e6i	0.564257	+	0.977322i	−0.432861	+	0.901461i	$0.642496\pi$
$258$		0			0
$259$	342140.	−	1.62272e6i	0.316923	−	1.50312i
$260$		0			0
$261$	−211330.	−	366035.i	−0.192026	−	0.332599i
$262$		0			0
$263$	−1.06115e6	+	1.83797e6i	−0.945993	+	1.63851i	−0.192244	+	0.981347i	$0.561576\pi$
$263$	−1.06115e6	+	1.83797e6i	−0.945993	+	1.63851i	−0.753750	+	0.657161i	$0.771757\pi$
$264$		0			0
$265$	801690.			0.701280
$266$		0			0
$267$	701933.			0.602584
$268$		0			0
$269$	−77488.3	+	134214.i	−0.0652913	+	0.113088i	−0.896823	−	0.442389i	$-0.854131\pi$
$269$	−77488.3	+	134214.i	−0.0652913	+	0.113088i	0.831532	+	0.555477i	$0.187464\pi$
$270$		0			0
$271$	979439.	+	1.69644e6i	0.810129	+	1.40318i	0.912773	+	0.408466i	$0.133936\pi$
$271$	979439.	+	1.69644e6i	0.810129	+	1.40318i	−0.102645	+	0.994718i	$0.532730\pi$
$272$		0			0
$273$	807870.	−	263957.i	0.656047	−	0.214352i
$274$		0			0
$275$	−445851.	−	772236.i	−0.355515	−	0.615770i
$276$		0			0
$277$	874763.	−	1.51513e6i	0.685001	−	1.18646i	−0.288436	−	0.957499i	$-0.593135\pi$
$277$	874763.	−	1.51513e6i	0.685001	−	1.18646i	0.973437	−	0.228957i	$-0.0735315\pi$
$278$		0			0
$279$	570057.			0.438438
$280$		0			0
$281$	1.40665e6			1.06272			0.531361	−	0.847145i	$-0.321681\pi$
$281$	1.40665e6			1.06272			0.531361	+	0.847145i	$0.321681\pi$
$282$		0			0
$283$	978536.	−	1.69487e6i	0.726291	−	1.25797i	−0.232150	−	0.972680i	$-0.574576\pi$
$283$	978536.	−	1.69487e6i	0.726291	−	1.25797i	0.958441	−	0.285292i	$-0.0920907\pi$
$284$		0			0
$285$	103868.	+	179904.i	0.0757474	+	0.131198i
$286$		0			0
$287$	113225.	+	101613.i	0.0811407	+	0.0728189i
$288$		0			0
$289$	382002.	+	661648.i	0.269043	+	0.465996i
$290$		0			0
$291$	727944.	−	1.26084e6i	0.503925	−	0.872823i
$292$		0			0
$293$	−1.26998e6			−0.864229			−0.432114	−	0.901819i	$-0.642232\pi$
$293$	−1.26998e6			−0.864229			−0.432114	+	0.901819i	$0.642232\pi$
$294$		0			0
$295$	683232.			0.457102
$296$		0			0
$297$	124098.	−	214945.i	0.0816347	−	0.141395i
$298$		0			0
$299$	−517915.	−	897055.i	−0.335027	−	0.580285i
$300$		0			0
$301$	−353536.	−	317277.i	−0.224914	−	0.201847i
$302$		0			0
$303$	296345.	+	513284.i	0.185435	+	0.321182i
$304$		0			0
$305$	−361972.	+	626954.i	−0.222805	+	0.385910i
$306$		0			0
$307$	41854.4			0.0253451			0.0126726	−	0.999920i	$-0.495966\pi$
$307$	41854.4			0.0253451			0.0126726	+	0.999920i	$0.495966\pi$
$308$		0			0
$309$	−1.17178e6			−0.698154
$310$		0			0
$311$	−1.14595e6	+	1.98484e6i	−0.671837	+	1.16366i	0.305546	+	0.952177i	$0.401161\pi$
$311$	−1.14595e6	+	1.98484e6i	−0.671837	+	1.16366i	−0.977383	+	0.211478i	$0.932172\pi$
$312$		0			0
$313$	−1.13575e6	−	1.96717e6i	−0.655271	−	1.13496i	−0.981826	−	0.189785i	$-0.939221\pi$
$313$	−1.13575e6	−	1.96717e6i	−0.655271	−	1.13496i	0.326555	−	0.945178i	$-0.394112\pi$
$314$		0			0
$315$	−224512.	+	73355.4i	−0.127486	+	0.0416539i
$316$		0			0
$317$	992372.	+	1.71884e6i	0.554659	+	0.960698i	0.997930	+	0.0643100i	$0.0204847\pi$
$317$	992372.	+	1.71884e6i	0.554659	+	0.960698i	−0.443271	+	0.896388i	$0.646182\pi$
$318$		0			0
$319$	888270.	−	1.53853e6i	0.488729	−	0.846504i
$320$		0			0
$321$	−23318.6			−0.0126310
$322$		0			0
$323$	−831066.			−0.443230
$324$		0			0
$325$	−953895.	+	1.65219e6i	−0.500947	+	0.867666i
$326$		0			0
$327$	−497942.	−	862460.i	−0.257519	−	0.446036i
$328$		0			0
$329$	−249092.	+	1.18141e6i	−0.126873	+	0.601740i
$330$		0			0
$331$	319230.	+	552923.i	0.160153	+	0.277393i	0.934923	−	0.354850i	$-0.115468\pi$
$331$	319230.	+	552923.i	0.160153	+	0.277393i	−0.774771	+	0.632242i	$0.782135\pi$
$332$		0			0
$333$	−518080.	+	897342.i	−0.256027	+	0.443453i
$334$		0			0
$335$	−480236.			−0.233799
$336$		0			0
$337$	2.72026e6			1.30478			0.652388	−	0.757886i	$-0.273767\pi$
$337$	2.72026e6			1.30478			0.652388	+	0.757886i	$0.273767\pi$
$338$		0			0
$339$	872597.	−	1.51138e6i	0.412396	−	0.714291i
$340$		0			0
$341$	1.19804e6	+	2.07507e6i	0.557938	+	0.966377i
$342$		0			0
$343$	−1.27204e6	+	1.76903e6i	−0.583800	+	0.811897i
$344$		0			0
$345$	143932.	+	249297.i	0.0651041	+	0.112764i
$346$		0			0
$347$	1.31038e6	−	2.26965e6i	0.584217	−	1.01189i	−0.410755	−	0.911746i	$-0.634735\pi$
$347$	1.31038e6	−	2.26965e6i	0.584217	−	1.01189i	0.994973	−	0.100148i	$-0.0319317\pi$
$348$		0			0
$349$	575592.			0.252960			0.126480	−	0.991969i	$-0.459632\pi$
$349$	575592.			0.252960			0.126480	+	0.991969i	$0.459632\pi$
$350$		0			0
$351$	−531015.			−0.230059
$352$		0			0
$353$	1.29486e6	−	2.24276e6i	0.553077	−	0.957957i	−0.444974	−	0.895544i	$-0.646787\pi$
$353$	1.29486e6	−	2.24276e6i	0.553077	−	0.957957i	0.998050	−	0.0624135i	$-0.0198797\pi$
$354$		0			0
$355$	687746.	+	1.19121e6i	0.289639	+	0.501669i
$356$		0			0
$357$	194943.	−	924582.i	0.0809537	−	0.383950i
$358$		0			0
$359$	1.43389e6	+	2.48358e6i	0.587193	+	1.01705i	0.994598	+	0.103801i	$0.0331005\pi$
$359$	1.43389e6	+	2.48358e6i	0.587193	+	1.01705i	−0.407405	+	0.913248i	$0.633566\pi$
$360$		0			0
$361$	711505.	−	1.23236e6i	0.287349	−	0.497703i
$362$		0			0
$363$	−406231.			−0.161811
$364$		0			0
$365$	928210.			0.364682
$366$		0			0
$367$	−977255.	+	1.69265e6i	−0.378741	+	0.655999i	−0.990879	−	0.134752i	$-0.956976\pi$
$367$	−977255.	+	1.69265e6i	−0.378741	+	0.655999i	0.612138	+	0.790751i	$0.290310\pi$
$368$		0			0
$369$	−47527.0	−	82319.1i	−0.0181708	−	0.0314728i
$370$		0			0
$371$	−4.39229e6	+	1.43510e6i	−1.65675	+	0.541314i
$372$		0			0
$373$	−1.10555e6	−	1.91486e6i	−0.411439	−	0.712633i	0.583608	−	0.812035i	$-0.301640\pi$
$373$	−1.10555e6	−	1.91486e6i	−0.411439	−	0.712633i	−0.995047	+	0.0994020i	$0.968307\pi$
$374$		0			0
$375$	581392.	−	1.00700e6i	0.213496	−	0.369787i
$376$		0			0
$377$	−3.80090e6			−1.37731
$378$		0			0
$379$	−3.81232e6			−1.36330			−0.681649	−	0.731679i	$-0.738737\pi$
$379$	−3.81232e6			−1.36330			−0.681649	+	0.731679i	$0.738737\pi$
$380$		0			0
$381$	123432.	−	213790.i	0.0435626	−	0.0754526i
$382$		0			0
$383$	1.90303e6	+	3.29614e6i	0.662901	+	1.14818i	0.979850	+	0.199735i	$0.0640082\pi$
$383$	1.90303e6	+	3.29614e6i	0.662901	+	1.14818i	−0.316949	+	0.948442i	$0.602658\pi$
$384$		0			0
$385$	−738859.	−	663082.i	−0.254044	−	0.227990i
$386$		0			0
$387$	148399.	+	257034.i	0.0503678	+	0.0872395i
$388$		0			0
$389$	−1.53762e6	+	2.66324e6i	−0.515200	+	0.892352i	0.484645	+	0.874711i	$0.338949\pi$
$389$	−1.53762e6	+	2.66324e6i	−0.515200	+	0.892352i	−0.999844	+	0.0176410i	$0.994384\pi$
$390$		0			0
$391$	−1.15163e6			−0.380952
$392$		0			0
$393$	−1.35338e6			−0.442017
$394$		0			0
$395$	−393621.	+	681772.i	−0.126936	+	0.219860i
$396$		0			0
$397$	−1.09566e6	−	1.89774e6i	−0.348900	−	0.604312i	0.637154	−	0.770736i	$-0.280111\pi$
$397$	−1.09566e6	−	1.89774e6i	−0.348900	−	0.604312i	−0.986054	+	0.166424i	$0.946778\pi$
$398$		0			0
$399$	−891115.	−	799722.i	−0.280221	−	0.251482i
$400$		0			0
$401$	−888340.	−	1.53865e6i	−0.275879	−	0.477836i	0.694478	−	0.719514i	$-0.255635\pi$
$401$	−888340.	−	1.53865e6i	−0.275879	−	0.477836i	−0.970356	+	0.241678i	$0.922302\pi$
$402$		0			0
$403$	2.56320e6	−	4.43960e6i	0.786177	−	1.36170i
$404$		0			0
$405$	147572.			0.0447061
$406$		0			0
$407$	−4.35522e6			−1.30324
$408$		0			0
$409$	−1.18059e6	+	2.04484e6i	−0.348973	+	0.604438i	−0.986067	−	0.166347i	$-0.946803\pi$
$409$	−1.18059e6	+	2.04484e6i	−0.348973	+	0.604438i	0.637095	+	0.770786i	$0.280136\pi$
$410$		0			0
$411$	−432459.	−	749041.i	−0.126282	−	0.218726i
$412$		0			0
$413$	−3.74328e6	+	1.22305e6i	−1.07989	+	0.352834i
$414$		0			0
$415$	969341.	+	1.67895e6i	0.276284	+	0.478539i
$416$		0			0
$417$	453845.	−	786082.i	0.127811	−	0.221375i
$418$		0			0
$419$	3.23493e6			0.900181			0.450090	−	0.892983i	$-0.351392\pi$
$419$	3.23493e6			0.900181			0.450090	+	0.892983i	$0.351392\pi$
$420$		0			0
$421$	−2.85759e6			−0.785769			−0.392884	−	0.919588i	$-0.628523\pi$
$421$	−2.85759e6			−0.785769			−0.392884	+	0.919588i	$0.628523\pi$
$422$		0			0
$423$	377184.	−	653302.i	0.102495	−	0.177527i
$424$		0			0
$425$	1.06053e6	+	1.83690e6i	0.284808	+	0.493301i
$426$		0			0
$427$	860859.	−	4.08292e6i	0.228487	−	1.08368i
$428$		0			0
$429$	−1.11599e6	−	1.93295e6i	−0.292764	−	0.507081i
$430$		0			0
$431$	−44193.1	+	76544.7i	−0.0114594	+	0.0198482i	−0.871698	−	0.490043i	$-0.836981\pi$
$431$	−44193.1	+	76544.7i	−0.0114594	+	0.0198482i	0.860239	+	0.509891i	$0.170314\pi$
$432$		0			0
$433$	−3.09418e6			−0.793097			−0.396549	−	0.918014i	$-0.629792\pi$
$433$	−3.09418e6			−0.793097			−0.396549	+	0.918014i	$0.629792\pi$
$434$		0			0
$435$	1.05629e6			0.267646
$436$		0			0
$437$	−729645.	+	1.26378e6i	−0.182771	+	0.316569i
$438$		0			0
$439$	242589.	+	420177.i	0.0600773	+	0.104057i	0.894500	−	0.447068i	$-0.147532\pi$
$439$	242589.	+	420177.i	0.0600773	+	0.104057i	−0.834422	+	0.551125i	$0.814199\pi$
$440$		0			0
$441$	1.09874e6	−	803797.i	0.269029	−	0.196811i
$442$		0			0
$443$	637598.	+	1.10435e6i	0.154361	+	0.267361i	0.932826	−	0.360327i	$-0.117335\pi$
$443$	637598.	+	1.10435e6i	0.154361	+	0.267361i	−0.778465	+	0.627688i	$0.784001\pi$
$444$		0			0
$445$	−877118.	+	1.51921e6i	−0.209970	+	0.363679i
$446$		0			0
$447$	−3.24850e6			−0.768978
$448$		0			0
$449$	−3.79144e6			−0.887541			−0.443770	−	0.896141i	$-0.646359\pi$
$449$	−3.79144e6			−0.887541			−0.443770	+	0.896141i	$0.646359\pi$
$450$		0			0
$451$	199767.	−	346006.i	0.0462468	−	0.0801019i
$452$		0			0
$453$	−1.95325e6	−	3.38313e6i	−0.447211	−	0.774592i
$454$		0			0
$455$	−438203.	+	2.07833e6i	−0.0992310	+	0.470637i
$456$		0			0
$457$	−851404.	−	1.47467e6i	−0.190698	−	0.330298i	0.754784	−	0.655973i	$-0.227742\pi$
$457$	−851404.	−	1.47467e6i	−0.190698	−	0.330298i	−0.945482	+	0.325675i	$0.894408\pi$
$458$		0			0
$459$	−295189.	+	511283.i	−0.0653986	+	0.113274i
$460$		0			0
$461$	4.55537e6			0.998323			0.499161	−	0.866509i	$-0.333642\pi$
$461$	4.55537e6			0.998323			0.499161	+	0.866509i	$0.333642\pi$
$462$		0			0
$463$	−5.82647e6			−1.26314			−0.631572	−	0.775317i	$-0.717590\pi$
$463$	−5.82647e6			−1.26314			−0.631572	+	0.775317i	$0.717590\pi$
$464$		0			0
$465$	−712329.	+	1.23379e6i	−0.152774	+	0.264612i
$466$		0			0
$467$	−1.77131e6	−	3.06799e6i	−0.375839	−	0.650972i	0.614614	−	0.788828i	$-0.289312\pi$
$467$	−1.77131e6	−	3.06799e6i	−0.375839	−	0.650972i	−0.990452	+	0.137857i	$0.955979\pi$
$468$		0			0
$469$	2.63111e6	−	859670.i	0.552341	−	0.180468i
$470$		0			0
$471$	2.30218e6	+	3.98750e6i	0.478176	+	0.828225i
$472$		0			0
$473$	−623754.	+	1.08037e6i	−0.128192	+	0.222035i
$474$		0			0
$475$	2.68772e6			0.546575
$476$		0			0
$477$	2.88707e6			0.580979
$478$		0			0
$479$	−2.11771e6	+	3.66798e6i	−0.421723	+	0.730446i	−0.996108	−	0.0881390i	$-0.971908\pi$
$479$	−2.11771e6	+	3.66798e6i	−0.421723	+	0.730446i	0.574385	+	0.818585i	$0.305241\pi$
$480$		0			0
$481$	4.65899e6	+	8.06960e6i	0.918182	+	1.59034i
$482$		0			0
$483$	−1.23484e6	−	1.10819e6i	−0.240847	−	0.216146i
$484$		0			0
$485$	1.81924e6	+	3.15102e6i	0.351185	+	0.608270i
$486$		0			0
$487$	−2.82740e6	+	4.89719e6i	−0.540212	+	0.935675i	0.458679	+	0.888602i	$0.348323\pi$
$487$	−2.82740e6	+	4.89719e6i	−0.540212	+	0.935675i	−0.998891	+	0.0470729i	$0.985011\pi$
$488$		0			0
$489$	2.26142e6			0.427671
$490$		0			0
$491$	−8.33183e6			−1.55968			−0.779842	−	0.625976i	$-0.784701\pi$
$491$	−8.33183e6			−1.55968			−0.779842	+	0.625976i	$0.784701\pi$
$492$		0			0
$493$	−2.11290e6	+	3.65966e6i	−0.391528	+	0.678146i
$494$		0			0
$495$	310140.	+	537179.i	0.0568912	+	0.0985384i
$496$		0			0
$497$	−5.90040e6	−	5.29526e6i	−1.07150	−	0.961604i
$498$		0			0
$499$	−3.61884e6	−	6.26802e6i	−0.650607	−	1.12688i	−0.982976	−	0.183734i	$-0.941181\pi$
$499$	−3.61884e6	−	6.26802e6i	−0.650607	−	1.12688i	0.332369	−	0.943149i	$-0.392152\pi$
$500$		0			0
$501$	1.88674e6	−	3.26794e6i	0.335829	−	0.581674i
$502$		0			0
$503$	6.16761e6			1.08692			0.543459	−	0.839436i	$-0.317114\pi$
$503$	6.16761e6			1.08692			0.543459	+	0.839436i	$0.317114\pi$
$504$		0			0
$505$	−1.48122e6			−0.258459
$506$		0			0
$507$	−716835.	+	1.24159e6i	−0.123851	+	0.214516i
$508$		0			0
$509$	−917439.	−	1.58905e6i	−0.156958	−	0.271859i	0.776812	−	0.629732i	$-0.216835\pi$
$509$	−917439.	−	1.58905e6i	−0.156958	−	0.271859i	−0.933770	+	0.357873i	$0.883502\pi$
$510$		0			0
$511$	−5.08547e6	+	1.66159e6i	−0.861546	+	0.281495i
$512$		0			0
$513$	374050.	+	647874.i	0.0627533	+	0.108692i
$514$		0			0
$515$	1.46423e6	−	2.53612e6i	0.243272	−	0.421359i
$516$		0			0
$517$	3.17079e6			0.521724
$518$		0			0
$519$	−4.15239e6			−0.676674
$520$		0			0
$521$	−2.65520e6	+	4.59894e6i	−0.428551	+	0.742273i	−0.996745	−	0.0806224i	$-0.974309\pi$
$521$	−2.65520e6	+	4.59894e6i	−0.428551	+	0.742273i	0.568193	+	0.822895i	$0.307643\pi$
$522$		0			0
$523$	−1.49418e6	−	2.58799e6i	−0.238862	−	0.413722i	0.721526	−	0.692388i	$-0.243441\pi$
$523$	−1.49418e6	−	2.58799e6i	−0.238862	−	0.413722i	−0.960388	+	0.278666i	$0.910108\pi$
$524$		0			0
$525$	−630457.	+	2.99016e6i	−0.0998291	+	0.473473i
$526$		0			0
$527$	−2.84975e6	−	4.93591e6i	−0.446972	−	0.774178i
$528$		0			0
$529$	2.20709e6	−	3.82279e6i	0.342910	−	0.593937i
$530$		0			0
$531$	2.46047e6			0.378688
$532$		0			0
$533$	−854800.			−0.130331
$534$		0			0
$535$	29138.3	−	50469.0i	0.00440128	−	0.00762325i
$536$		0			0
$537$	3.33768e6	+	5.78102e6i	0.499469	+	0.865105i
$538$		0			0
$539$	5.23504e6	+	2.31026e6i	0.776154	+	0.342522i
$540$		0			0
$541$	−1.08416e6	−	1.87782e6i	−0.159258	−	0.275843i	0.775343	−	0.631540i	$-0.217577\pi$
$541$	−1.08416e6	−	1.87782e6i	−0.159258	−	0.275843i	−0.934601	+	0.355697i	$0.884243\pi$
$542$		0			0
$543$	−1.35617e6	+	2.34895e6i	−0.197385	+	0.341881i
$544$		0			0
$545$	2.48886e6			0.358930
$546$		0			0
$547$	9.13272e6			1.30506			0.652532	−	0.757761i	$-0.273707\pi$
$547$	9.13272e6			1.30506			0.652532	+	0.757761i	$0.273707\pi$
$548$		0			0
$549$	−1.30354e6	+	2.25780e6i	−0.184584	+	0.319709i
$550$		0			0
$551$	2.67738e6	+	4.63735e6i	0.375691	+	0.650715i
$552$		0			0
$553$	936128.	−	4.43991e6i	0.130173	−	0.617392i
$554$		0			0
$555$	−1.29476e6	−	2.24259e6i	−0.178426	−	0.309042i
$556$		0			0
$557$	4.61014e6	−	7.98500e6i	0.629617	−	1.09053i	−0.358012	−	0.933717i	$-0.616545\pi$
$557$	4.61014e6	−	7.98500e6i	0.629617	−	1.09053i	0.987629	−	0.156811i	$-0.0501214\pi$
$558$		0			0
$559$	2.66904e6			0.361264
$560$		0			0
$561$	−2.48150e6			−0.332894
$562$		0			0
$563$	4.24638e6	−	7.35495e6i	0.564610	−	0.977933i	−0.432476	−	0.901645i	$-0.642360\pi$
$563$	4.24638e6	−	7.35495e6i	0.564610	−	0.977933i	0.997086	−	0.0762872i	$-0.0243066\pi$
$564$		0			0
$565$	2.18075e6	+	3.77717e6i	0.287399	+	0.497789i
$566$		0			0
$567$	−808518.	+	264169.i	−0.105617	+	0.0345084i
$568$		0			0
$569$	2.35456e6	+	4.07822e6i	0.304880	+	0.528068i	0.977235	−	0.212162i	$-0.0680503\pi$
$569$	2.35456e6	+	4.07822e6i	0.304880	+	0.528068i	−0.672355	+	0.740229i	$0.734717\pi$
$570$		0			0
$571$	−864035.	+	1.49655e6i	−0.110902	+	0.192089i	−0.916134	−	0.400871i	$-0.868707\pi$
$571$	−864035.	+	1.49655e6i	−0.110902	+	0.192089i	0.805232	+	0.592960i	$0.202041\pi$
$572$		0			0
$573$	−2.36864e6			−0.301379
$574$		0			0
$575$	3.72443e6			0.469776
$576$		0			0
$577$	−1.19249e6	+	2.06546e6i	−0.149113	+	0.258272i	−0.930900	−	0.365274i	$-0.880975\pi$
$577$	−1.19249e6	+	2.06546e6i	−0.149113	+	0.258272i	0.781787	+	0.623546i	$0.214309\pi$
$578$		0			0
$579$	3.74007e6	+	6.47799e6i	0.463643	+	0.803053i
$580$		0			0
$581$	−8.31630e6	−	7.46339e6i	−1.02209	−	0.917267i
$582$		0			0
$583$	6.06750e6	+	1.05092e7i	0.739330	+	1.28056i
$584$		0			0
$585$	663543.	−	1.14929e6i	0.0801640	−	0.138848i
$586$		0			0
$587$	3.44904e6			0.413145			0.206573	−	0.978431i	$-0.433769\pi$
$587$	3.44904e6			0.413145			0.206573	+	0.978431i	$0.433769\pi$
$588$		0			0
$589$	−7.22214e6			−0.857784
$590$		0			0
$591$	4.74395e6	−	8.21676e6i	0.558691	−	0.967681i
$592$		0			0
$593$	−567341.	−	982664.i	−0.0662533	−	0.114754i	0.830996	−	0.556278i	$-0.187771\pi$
$593$	−567341.	−	982664.i	−0.0662533	−	0.114754i	−0.897249	+	0.441524i	$0.854438\pi$
$594$		0			0
$595$	1.75750e6	+	1.57725e6i	0.203518	+	0.182646i
$596$		0			0
$597$	3.14355e6	+	5.44480e6i	0.360982	+	0.625239i
$598$		0			0
$599$	3.80933e6	−	6.59795e6i	0.433792	−	0.751349i	−0.563404	−	0.826181i	$-0.690509\pi$
$599$	3.80933e6	−	6.59795e6i	0.433792	−	0.751349i	0.997196	+	0.0748320i	$0.0238420\pi$
$600$		0			0
$601$	9.78414e6			1.10493			0.552467	−	0.833535i	$-0.313686\pi$
$601$	9.78414e6			1.10493			0.552467	+	0.833535i	$0.313686\pi$
$602$		0			0
$603$	−1.72944e6			−0.193692
$604$		0			0
$605$	507616.	−	879217.i	0.0563829	−	0.0976580i
$606$		0			0
$607$	926673.	+	1.60504e6i	0.102083	+	0.176813i	0.912543	−	0.408981i	$-0.134116\pi$
$607$	926673.	+	1.60504e6i	0.102083	+	0.176813i	−0.810460	+	0.585795i	$0.800783\pi$
$608$		0			0
$609$	−5.78720e6	+	1.89087e6i	−0.632304	+	0.206594i
$610$		0			0
$611$	−3.39194e6	−	5.87501e6i	−0.367574	−	0.636657i
$612$		0			0
$613$	3.56048e6	−	6.16694e6i	0.382699	−	0.662855i	−0.608748	−	0.793364i	$-0.708328\pi$
$613$	3.56048e6	−	6.16694e6i	0.382699	−	0.662855i	0.991447	+	0.130509i	$0.0416612\pi$
$614$		0			0
$615$	237554.			0.0253265
$616$		0			0
$617$	−2.75212e6			−0.291041			−0.145521	−	0.989355i	$-0.546486\pi$
$617$	−2.75212e6			−0.291041			−0.145521	+	0.989355i	$0.546486\pi$
$618$		0			0
$619$	−6.51660e6	+	1.12871e7i	−0.683588	+	1.18401i	0.290291	+	0.956939i	$0.406248\pi$
$619$	−6.51660e6	+	1.12871e7i	−0.683588	+	1.18401i	−0.973878	+	0.227070i	$0.927085\pi$
$620$		0			0
$621$	518330.	+	897774.i	0.0539358	+	0.0934196i
$622$		0			0
$623$	2.08600e6	−	9.89358e6i	0.215325	−	1.02125i
$624$		0			0
$625$	−2.63935e6	−	4.57149e6i	−0.270269	−	0.468120i
$626$		0			0
$627$	−1.57222e6	+	2.72317e6i	−0.159715	+	0.276634i
$628$		0			0
$629$	1.03597e7			1.04404
$630$		0			0
$631$	4.40820e6			0.440745			0.220373	−	0.975416i	$-0.429273\pi$
$631$	4.40820e6			0.440745			0.220373	+	0.975416i	$0.429273\pi$
$632$		0			0
$633$	−448621.	+	777035.i	−0.0445011	+	0.0770781i
$634$		0			0
$635$	308474.	+	534293.i	0.0303588	+	0.0525829i
$636$		0			0
$637$	−1.31959e6	−	1.21712e7i	−0.128852	−	1.18846i
$638$		0			0
$639$	2.47673e6	+	4.28982e6i	0.239953	+	0.415611i
$640$		0			0
$641$	3.99195e6	−	6.91426e6i	0.383742	−	0.664661i	−0.607851	−	0.794051i	$-0.707968\pi$
$641$	3.99195e6	−	6.91426e6i	0.383742	−	0.664661i	0.991594	+	0.129389i	$0.0413017\pi$
$642$		0			0
$643$	−1.52102e6			−0.145080			−0.0725398	−	0.997366i	$-0.523110\pi$
$643$	−1.52102e6			−0.145080			−0.0725398	+	0.997366i	$0.523110\pi$
$644$		0			0
$645$	−741741.			−0.0702026
$646$		0			0
$647$	−6.26247e6	+	1.08469e7i	−0.588146	+	1.01870i	0.406329	+	0.913727i	$0.366809\pi$
$647$	−6.26247e6	+	1.08469e7i	−0.588146	+	1.01870i	−0.994475	+	0.104972i	$0.966525\pi$
$648$		0			0
$649$	5.17096e6	+	8.95637e6i	0.481903	+	0.834681i
$650$		0			0
$651$	1.69409e6	−	8.03482e6i	0.156670	−	0.743060i
$652$		0			0
$653$	8.53702e6	+	1.47865e7i	0.783471	+	1.35701i	0.929908	+	0.367792i	$0.119886\pi$
$653$	8.53702e6	+	1.47865e7i	0.783471	+	1.35701i	−0.146436	+	0.989220i	$0.546780\pi$
$654$		0			0
$655$	1.69115e6	−	2.92916e6i	0.154021	−	0.266772i
$656$		0			0
$657$	3.34269e6			0.302122
$658$		0			0
$659$	2.12585e6			0.190686			0.0953432	−	0.995444i	$-0.469605\pi$
$659$	2.12585e6			0.190686			0.0953432	+	0.995444i	$0.469605\pi$
$660$		0			0
$661$	1.30005e6	−	2.25176e6i	0.115733	−	0.200456i	−0.802339	−	0.596868i	$-0.796412\pi$
$661$	1.30005e6	−	2.25176e6i	0.115733	−	0.200456i	0.918073	+	0.396412i	$0.129745\pi$
$662$		0			0
$663$	2.65457e6	+	4.59786e6i	0.234537	+	0.406230i
$664$		0			0
$665$	2.84438e6	−	929350.i	0.249421	−	0.0814940i
$666$		0			0
$667$	3.71010e6	+	6.42608e6i	0.322902	+	0.559283i
$668$		0			0
$669$	2.36787e6	−	4.10128e6i	0.204547	−	0.354286i
$670$		0			0
$671$	−1.09582e7			−0.939577
$672$		0			0
$673$	−1.44746e7			−1.23188			−0.615942	−	0.787792i	$-0.711224\pi$
$673$	−1.44746e7			−1.23188			−0.615942	+	0.787792i	$0.711224\pi$
$674$		0			0
$675$	954660.	−	1.65352e6i	0.0806472	−	0.139685i
$676$		0			0
$677$	1.64423e6	+	2.84788e6i	0.137876	+	0.238809i	0.926693	−	0.375820i	$-0.122639\pi$
$677$	1.64423e6	+	2.84788e6i	0.137876	+	0.238809i	−0.788816	+	0.614629i	$0.789306\pi$
$678$		0			0
$679$	−1.56079e7	−	1.40071e7i	−1.29918	−	1.16594i
$680$		0			0
$681$	−4.41722e6	−	7.65085e6i	−0.364990	−	0.632182i
$682$		0			0
$683$	4.40943e6	−	7.63736e6i	0.361685	−	0.626458i	−0.626553	−	0.779379i	$-0.715535\pi$
$683$	4.40943e6	−	7.63736e6i	0.361685	−	0.626458i	0.988238	+	0.152921i	$0.0488681\pi$
$684$		0			0
$685$	2.16156e6			0.176011
$686$		0			0
$687$	−765729.			−0.0618989
$688$		0			0
$689$	1.29814e7	−	2.24844e7i	1.04177	−	1.80440i
$690$		0			0
$691$	−4.29808e6	−	7.44450e6i	−0.342436	−	0.593117i	0.642448	−	0.766329i	$-0.277919\pi$
$691$	−4.29808e6	−	7.44450e6i	−0.342436	−	0.593117i	−0.984885	+	0.173212i	$0.944585\pi$
$692$		0			0
$693$	−2.66080e6	−	2.38791e6i	−0.210464	−	0.188879i
$694$		0			0
$695$	1.13423e6	+	1.96454e6i	0.0890713	+	0.154276i
$696$		0			0
$697$	−475180.	+	823035.i	−0.0370490	+	0.0641707i
$698$		0			0
$699$	1.01811e6			0.0788134
$700$		0			0
$701$	2.06437e7			1.58669			0.793347	−	0.608770i	$-0.208337\pi$
$701$	2.06437e7			1.58669			0.793347	+	0.608770i	$0.208337\pi$
$702$		0			0
$703$	6.56364e6	−	1.13686e7i	0.500906	−	0.867595i
$704$		0			0
$705$	942640.	+	1.63270e6i	0.0714288	+	0.123718i
$706$		0			0
$707$	8.11530e6	−	2.65153e6i	0.610599	−	0.199503i
$708$		0			0
$709$	213441.	+	369691.i	0.0159464	+	0.0276200i	0.873888	−	0.486126i	$-0.161591\pi$
$709$	213441.	+	369691.i	0.0159464	+	0.0276200i	−0.857942	+	0.513746i	$0.828257\pi$
$710$		0			0
$711$	−1.41752e6	+	2.45521e6i	−0.105161	+	0.182144i
$712$		0			0
$713$	−1.00079e7			−0.737257
$714$		0			0
$715$	5.57805e6			0.408054
$716$		0			0
$717$	−4.02795e6	+	6.97662e6i	−0.292608	+	0.506812i
$718$		0			0
$719$	1.90362e6	+	3.29716e6i	0.137328	+	0.237858i	0.926484	−	0.376334i	$-0.122815\pi$
$719$	1.90362e6	+	3.29716e6i	0.137328	+	0.237858i	−0.789157	+	0.614192i	$0.789482\pi$
$720$		0			0
$721$	−3.48230e6	+	1.65160e7i	−0.249476	+	1.18322i
$722$		0			0
$723$	−2.37472e6	−	4.11313e6i	−0.168953	−	0.292635i
$724$		0			0
$725$	6.83326e6	−	1.18356e7i	0.482817	−	0.836264i
$726$		0			0
$727$	2.22044e7			1.55813			0.779065	−	0.626943i	$-0.215694\pi$
$727$	2.22044e7			1.55813			0.779065	+	0.626943i	$0.215694\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	1.48371e6	−	2.56986e6i	0.102696	−	0.177875i
$732$		0			0
$733$	1.45286e7	+	2.51643e7i	0.998766	+	1.72991i	0.542401	+	0.840120i	$0.317515\pi$
$733$	1.45286e7	+	2.51643e7i	0.998766	+	1.72991i	0.456365	+	0.889793i	$0.349151\pi$
$734$		0			0
$735$	366723.	+	3.38244e6i	0.0250391	+	0.230947i
$736$		0			0
$737$	−3.63461e6	−	6.29533e6i	−0.246484	−	0.426923i
$738$		0			0
$739$	1.06761e7	−	1.84916e7i	0.719122	−	1.24556i	−0.242226	−	0.970220i	$-0.577877\pi$
$739$	1.06761e7	−	1.84916e7i	0.719122	−	1.24556i	0.961348	−	0.275337i	$-0.0887893\pi$
$740$		0			0
$741$	6.72751e6			0.450100
$742$		0			0
$743$	−3.91874e6			−0.260420			−0.130210	−	0.991486i	$-0.541565\pi$
$743$	−3.91874e6			−0.260420			−0.130210	+	0.991486i	$0.541565\pi$
$744$		0			0
$745$	4.05924e6	−	7.03081e6i	0.267950	−	0.464103i
$746$		0			0
$747$	3.49081e6	+	6.04627e6i	0.228889	+	0.396448i
$748$		0			0
$749$	−69298.0	+	328669.i	−0.00451353	+	0.0214069i
$750$		0			0
$751$	2.83456e6	+	4.90960e6i	0.183394	+	0.317648i	0.943034	−	0.332696i	$-0.107958\pi$
$751$	2.83456e6	+	4.90960e6i	0.183394	+	0.317648i	−0.759640	+	0.650344i	$0.774625\pi$
$752$		0			0
$753$	−5.36817e6	+	9.29794e6i	−0.345015	+	0.597584i
$754$		0			0
$755$	9.76293e6			0.623323
$756$		0			0
$757$	−1.91706e7			−1.21590			−0.607949	−	0.793976i	$-0.708007\pi$
$757$	−1.91706e7			−1.21590			−0.607949	+	0.793976i	$0.708007\pi$
$758$		0			0
$759$	−2.17866e6	+	3.77355e6i	−0.137273	+	0.237764i
$760$		0			0
$761$	−7.31680e6	−	1.26731e7i	−0.457993	−	0.793268i	0.540861	−	0.841112i	$-0.318098\pi$
$761$	−7.31680e6	−	1.26731e7i	−0.457993	−	0.793268i	−0.998855	+	0.0478438i	$0.984765\pi$
$762$		0			0
$763$	−1.36360e7	+	4.45531e6i	−0.847958	+	0.277056i
$764$		0			0
$765$	−737722.	−	1.27777e6i	−0.0455763	−	0.0789405i
$766$		0			0
$767$	1.10632e7	−	1.91621e7i	0.679038	−	1.17613i
$768$		0			0
$769$	1.57337e7			0.959432			0.479716	−	0.877424i	$-0.340740\pi$
$769$	1.57337e7			0.959432			0.479716	+	0.877424i	$0.340740\pi$
$770$		0			0
$771$	−1.07543e7			−0.651548
$772$		0			0
$773$	−2.02452e6	+	3.50658e6i	−0.121864	+	0.211074i	−0.920503	−	0.390736i	$-0.872220\pi$
$773$	−2.02452e6	+	3.50658e6i	−0.121864	+	0.211074i	0.798639	+	0.601810i	$0.205554\pi$
$774$		0			0
$775$	9.21626e6	+	1.59630e7i	0.551189	+	0.954687i
$776$		0			0
$777$	1.11082e7	+	9.96894e6i	0.660071	+	0.592374i
$778$		0			0
$779$	602126.	+	1.04291e6i	0.0355504	+	0.0615750i
$780$		0			0
$781$	−1.04103e7	+	1.80311e7i	−0.610708	+	1.05778i
$782$		0

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)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 7.79423i) q^{3} +(-11.2462 - 19.4789i) q^{5} +(96.4847 + 86.5893i) q^{7} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)	\(q+(-4.50000 + 7.79423i) q^{3} +(-11.2462 - 19.4789i) q^{5} +(96.4847 + 86.5893i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(170.231 - 294.849i) q^{11} -728.416 q^{13} +202.431 q^{15} +(-404.923 + 701.348i) q^{17} +(513.101 + 888.717i) q^{19} +(-1109.08 + 362.372i) q^{21} +(711.015 + 1231.51i) q^{23} +(1309.55 - 2268.20i) q^{25} +729.000 q^{27} +5218.03 q^{29} +(-3518.87 + 6094.87i) q^{31} +(1532.08 + 2653.64i) q^{33} +(601.584 - 2853.22i) q^{35} +(-6396.05 - 11078.3i) q^{37} +(3277.87 - 5677.44i) q^{39} +1173.51 q^{41} -3664.17 q^{43} +(-910.940 + 1577.79i) q^{45} +(4656.60 + 8065.46i) q^{47} +(1811.59 + 16709.1i) q^{49} +(-3644.31 - 6312.13i) q^{51} +(-17821.4 + 30867.6i) q^{53} -7657.78 q^{55} -9235.81 q^{57} +(-15188.1 + 26306.5i) q^{59} +(-16093.1 - 27874.1i) q^{61} +(2166.44 - 10275.1i) q^{63} +(8191.89 + 14188.8i) q^{65} +(10675.5 - 18490.6i) q^{67} -12798.3 q^{69} -61153.7 q^{71} +(-20633.9 + 35739.0i) q^{73} +(11785.9 + 20413.8i) q^{75} +(41955.4 - 13708.2i) q^{77} +(-17500.2 - 30311.3i) q^{79} +(-3280.50 + 5681.99i) q^{81} -86193.0 q^{83} +18215.4 q^{85} +(-23481.1 + 40670.5i) q^{87} +(-38996.3 - 67543.6i) q^{89} +(-70281.0 - 63073.0i) q^{91} +(-31669.9 - 54853.8i) q^{93} +(11540.8 - 19989.3i) q^{95} -161765. q^{97} -27577.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{3} + 53 q^{5} - 6 q^{7} - 162 q^{9}+O(q^{10}) \) 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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