LMFDB - Embedded newform 336.6.bl.e.31.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,6,Mod(31,336)]

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([3, 0, 0, 1]))

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

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

chi := DirichletCharacter("336.31");

S:= CuspForms(chi, 6);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 6 $
Character orbit:	$[\chi]$	$=$	336.bl (of order $6$, degree $2$, minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} - x^{13} + 434 x^{12} - 1773 x^{11} + 161210 x^{10} - 411219 x^{9} + 13011143 x^{8} + \cdots + 122943744 $ x^14 - x^13 + 434x^12 - 1773x^11 + 161210x^10 - 411219x^9 + 13011143x^8 + 40309634x^7 + 721029148x^6 + 394325472x^5 + 1128825600x^4 - 1034041536x^3 + 1221323328x^2 - 407417472x + 122943744
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{8}\cdot 7^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			31.4
Root			$8.48408 + 14.6949i$ of defining polynomial
Character	$\chi$	$=$	336.31
Dual form			336.6.bl.e.271.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.50000 + 7.79423i) q^{3} +(5.98712 - 3.45667i) q^{5} +(77.5174 - 103.914i) q^{7} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})$	\(q+(-4.50000 + 7.79423i) q^{3} +(5.98712 - 3.45667i) q^{5} +(77.5174 - 103.914i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(-220.062 - 127.053i) q^{11} -516.718i q^{13} +62.2200i q^{15} +(5.26414 + 3.03925i) q^{17} +(229.635 + 397.739i) q^{19} +(461.098 + 1071.80i) q^{21} +(-1844.37 + 1064.85i) q^{23} +(-1538.60 + 2664.94i) q^{25} +729.000 q^{27} -2453.67 q^{29} +(3228.41 - 5591.77i) q^{31} +(1980.56 - 1143.47i) q^{33} +(104.911 - 890.096i) q^{35} +(4076.46 + 7060.63i) q^{37} +(4027.42 + 2325.23i) q^{39} +20008.3i q^{41} +8725.33i q^{43} +(-484.957 - 279.990i) q^{45} +(-13069.3 - 22636.7i) q^{47} +(-4789.10 - 16110.2i) q^{49} +(-47.3772 + 27.3533i) q^{51} +(-233.700 + 404.780i) q^{53} -1756.72 q^{55} -4133.42 q^{57} +(-16282.0 + 28201.3i) q^{59} +(-17778.3 + 10264.3i) q^{61} +(-10428.8 - 1229.19i) q^{63} +(-1786.12 - 3093.65i) q^{65} +(-41169.7 - 23769.3i) q^{67} -19167.3i q^{69} -52443.1i q^{71} +(12930.4 + 7465.35i) q^{73} +(-13847.4 - 23984.4i) q^{75} +(-30261.1 + 13018.6i) q^{77} +(-36986.7 + 21354.3i) q^{79} +(-3280.50 + 5681.99i) q^{81} -67116.1 q^{83} +42.0227 q^{85} +(11041.5 - 19124.5i) q^{87} +(-23673.5 + 13667.9i) q^{89} +(-53694.0 - 40054.6i) q^{91} +(29055.7 + 50325.9i) q^{93} +(2749.70 + 1587.54i) q^{95} +62780.5i q^{97} +20582.5i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 63 q^{3} - 33 q^{5} + 13 q^{7} - 567 q^{9}+O(q^{10}) $ 14 * q - 63 * q^3 - 33 * q^5 + 13 * q^7 - 567 * q^9	$ 14 q - 63 q^{3} - 33 q^{5} + 13 q^{7} - 567 q^{9} + 549 q^{11} - 870 q^{17} + 146 q^{19} - 1773 q^{21} + 6396 q^{23} + 5134 q^{25} + 10206 q^{27} + 16458 q^{29} - 3137 q^{31} - 4941 q^{33} - 20910 q^{35} + 2162 q^{37} + 2376 q^{39} + 2673 q^{45} - 720 q^{47} + 23477 q^{49} + 7830 q^{51} - 44835 q^{53} - 72666 q^{55} - 2628 q^{57} - 44841 q^{59} + 93060 q^{61} + 14904 q^{63} - 4884 q^{65} + 68484 q^{67} - 163632 q^{73} + 46206 q^{75} + 99681 q^{77} + 134205 q^{79} - 45927 q^{81} + 188634 q^{83} + 276972 q^{85} - 74061 q^{87} - 49302 q^{89} + 110976 q^{91} - 28233 q^{93} + 60222 q^{95}+O(q^{100}) $ 14 * q - 63 * q^3 - 33 * q^5 + 13 * q^7 - 567 * q^9 + 549 * q^11 - 870 * q^17 + 146 * q^19 - 1773 * q^21 + 6396 * q^23 + 5134 * q^25 + 10206 * q^27 + 16458 * q^29 - 3137 * q^31 - 4941 * q^33 - 20910 * q^35 + 2162 * q^37 + 2376 * q^39 + 2673 * q^45 - 720 * q^47 + 23477 * q^49 + 7830 * q^51 - 44835 * q^53 - 72666 * q^55 - 2628 * q^57 - 44841 * q^59 + 93060 * q^61 + 14904 * q^63 - 4884 * q^65 + 68484 * q^67 - 163632 * q^73 + 46206 * q^75 + 99681 * q^77 + 134205 * q^79 - 45927 * q^81 + 188634 * q^83 + 276972 * q^85 - 74061 * q^87 - 49302 * q^89 + 110976 * q^91 - 28233 * q^93 + 60222 * q^95

Display 100 coefficients 10 coefficients

Character values

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

$n$	$85$	$113$	$127$	$241$
$\chi(n)$	$1$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$5$	5.98712	−	3.45667i	0.107101	−	0.0618347i	−0.445493	−	0.895286i	$-0.646971\pi$
$5$	5.98712	−	3.45667i	0.107101	−	0.0618347i	0.552594	+	0.833451i	$0.313638\pi$
$6$		0			0
$7$	77.5174	−	103.914i	0.597935	−	0.801544i
$7$	77.5174	−	103.914i	0.597935	−	0.801544i
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	−220.062	−	127.053i	−0.548356	−	0.316594i	0.200102	−	0.979775i	$-0.435872\pi$
$11$	−220.062	−	127.053i	−0.548356	−	0.316594i	−0.748459	+	0.663181i	$0.769206\pi$
$12$		0			0
$13$		−	516.718i		−	0.847998i	−0.905663	−	0.423999i	$-0.860626\pi$
$13$		−	516.718i		−	0.847998i	0.905663	−	0.423999i	$-0.139374\pi$
$14$		0			0
$15$			62.2200i			0.0714006i
$16$		0			0
$17$	5.26414	+	3.03925i	0.00441779	+	0.00255061i	0.502207	−	0.864747i	$-0.332521\pi$
$17$	5.26414	+	3.03925i	0.00441779	+	0.00255061i	−0.497789	+	0.867298i	$0.665855\pi$
$18$		0			0
$19$	229.635	+	397.739i	0.145933	+	0.252763i	0.929721	−	0.368266i	$-0.120048\pi$
$19$	229.635	+	397.739i	0.145933	+	0.252763i	−0.783788	+	0.621029i	$0.786715\pi$
$20$		0			0
$21$	461.098	+	1071.80i	0.228163	+	0.530354i
$22$		0			0
$23$	−1844.37	+	1064.85i	−0.726991	+	0.419729i	−0.817320	−	0.576183i	$-0.804541\pi$
$23$	−1844.37	+	1064.85i	−0.726991	+	0.419729i	0.0903293	+	0.995912i	$0.471208\pi$
$24$		0			0
$25$	−1538.60	+	2664.94i	−0.492353	+	0.852780i
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	−2453.67			−0.541778			−0.270889	−	0.962611i	$-0.587318\pi$
$29$	−2453.67			−0.541778			−0.270889	+	0.962611i	$0.587318\pi$
$30$		0			0
$31$	3228.41	−	5591.77i	0.603371	−	1.04507i	−0.388935	−	0.921265i	$-0.627157\pi$
$31$	3228.41	−	5591.77i	0.603371	−	1.04507i	0.992307	−	0.123805i	$-0.0395096\pi$
$32$		0			0
$33$	1980.56	−	1143.47i	0.316594	−	0.182785i
$34$		0			0
$35$	104.911	−	890.096i	0.0144761	−	0.122819i
$36$		0			0
$37$	4076.46	+	7060.63i	0.489529	+	0.847889i	0.999927	−	0.0120488i	$-0.00383533\pi$
$37$	4076.46	+	7060.63i	0.489529	+	0.847889i	−0.510398	+	0.859938i	$0.670502\pi$
$38$		0			0
$39$	4027.42	+	2325.23i	0.423999	+	0.244796i
$40$		0			0
$41$			20008.3i			1.85888i	0.368977	+	0.929439i	$0.379708\pi$
$41$			20008.3i			1.85888i	−0.368977	+	0.929439i	$0.620292\pi$
$42$		0			0
$43$			8725.33i			0.719633i	0.933023	+	0.359816i	$0.117161\pi$
$43$			8725.33i			0.719633i	−0.933023	+	0.359816i	$0.882839\pi$
$44$		0			0
$45$	−484.957	−	279.990i	−0.0357003	−	0.0206116i
$46$		0			0
$47$	−13069.3	−	22636.7i	−0.862992	−	1.49475i	−0.869027	−	0.494765i	$-0.835254\pi$
$47$	−13069.3	−	22636.7i	−0.862992	−	1.49475i	0.00603452	−	0.999982i	$-0.498079\pi$
$48$		0			0
$49$	−4789.10	−	16110.2i	−0.284947	−	0.958543i
$50$		0			0
$51$	−47.3772	+	27.3533i	−0.00255061	+	0.00147260i
$52$		0			0
$53$	−233.700	+	404.780i	−0.0114280	+	0.0197938i	−0.871683	−	0.490070i	$-0.836971\pi$
$53$	−233.700	+	404.780i	−0.0114280	+	0.0197938i	0.860255	+	0.509864i	$0.170304\pi$
$54$		0			0
$55$	−1756.72			−0.0783059
$56$		0			0
$57$	−4133.42			−0.168509
$58$		0			0
$59$	−16282.0	+	28201.3i	−0.608945	+	1.05472i	0.382470	+	0.923968i	$0.375074\pi$
$59$	−16282.0	+	28201.3i	−0.608945	+	1.05472i	−0.991415	+	0.130756i	$0.958260\pi$
$60$		0			0
$61$	−17778.3	+	10264.3i	−0.611738	+	0.353187i	−0.773645	−	0.633619i	$-0.781569\pi$
$61$	−17778.3	+	10264.3i	−0.611738	+	0.353187i	0.161908	+	0.986806i	$0.448235\pi$
$62$		0			0
$63$	−10428.8	−	1229.19i	−0.331042	−	0.0390183i
$64$		0			0
$65$	−1786.12	−	3093.65i	−0.0524357	−	0.0908214i
$66$		0			0
$67$	−41169.7	−	23769.3i	−1.12044	−	0.646889i	−0.178930	−	0.983862i	$-0.557264\pi$
$67$	−41169.7	−	23769.3i	−1.12044	−	0.646889i	−0.941514	+	0.336973i	$0.890597\pi$
$68$		0			0
$69$		−	19167.3i		−	0.484661i
$70$		0			0
$71$		−	52443.1i		−	1.23465i	−0.786709	−	0.617324i	$-0.788217\pi$
$71$		−	52443.1i		−	1.23465i	0.786709	−	0.617324i	$-0.211783\pi$
$72$		0			0
$73$	12930.4	+	7465.35i	0.283990	+	0.163962i	0.635229	−	0.772324i	$-0.280906\pi$
$73$	12930.4	+	7465.35i	0.283990	+	0.163962i	−0.351238	+	0.936286i	$0.614239\pi$
$74$		0			0
$75$	−13847.4	−	23984.4i	−0.284260	−	0.492353i
$76$		0			0
$77$	−30261.1	+	13018.6i	−0.581646	+	0.250229i
$78$		0			0
$79$	−36986.7	+	21354.3i	−0.666772	+	0.384961i	−0.794852	−	0.606803i	$-0.792452\pi$
$79$	−36986.7	+	21354.3i	−0.666772	+	0.384961i	0.128080	+	0.991764i	$0.459118\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	−67116.1			−1.06938			−0.534690	−	0.845049i	$-0.679571\pi$
$83$	−67116.1			−1.06938			−0.534690	+	0.845049i	$0.679571\pi$
$84$		0			0
$85$	42.0227			0.000630865
$86$		0			0
$87$	11041.5	−	19124.5i	0.156398	−	0.270889i
$88$		0			0
$89$	−23673.5	+	13667.9i	−0.316802	+	0.182906i	−0.649966	−	0.759963i	$-0.725217\pi$
$89$	−23673.5	+	13667.9i	−0.316802	+	0.182906i	0.333164	+	0.942869i	$0.391884\pi$
$90$		0			0
$91$	−53694.0	−	40054.6i	−0.679708	−	0.507048i
$92$		0			0
$93$	29055.7	+	50325.9i	0.348357	+	0.603371i
$94$		0			0
$95$	2749.70	+	1587.54i	0.0312591	+	0.0180474i
$96$		0			0
$97$			62780.5i			0.677478i	0.940880	+	0.338739i	$0.110000\pi$
$97$			62780.5i			0.677478i	−0.940880	+	0.338739i	$0.890000\pi$
$98$		0			0
$99$			20582.5i			0.211062i
$100$		0			0
$101$	−10843.5	−	6260.51i	−0.105771	−	0.0610669i	0.446181	−	0.894943i	$-0.352784\pi$
$101$	−10843.5	−	6260.51i	−0.105771	−	0.0610669i	−0.551952	+	0.833876i	$0.686117\pi$
$102$		0			0
$103$	−77188.2	−	133694.i	−0.716899	−	1.24170i	−0.962223	−	0.272264i	$-0.912228\pi$
$103$	−77188.2	−	133694.i	−0.716899	−	1.24170i	0.245324	−	0.969441i	$-0.421106\pi$
$104$		0			0
$105$	6465.51	+	4823.13i	0.0572307	+	0.0426929i
$106$		0			0
$107$	−21031.2	+	12142.3i	−0.177584	+	0.102528i	−0.586157	−	0.810197i	$-0.699360\pi$
$107$	−21031.2	+	12142.3i	−0.177584	+	0.102528i	0.408573	+	0.912726i	$0.366027\pi$
$108$		0			0
$109$	−56808.5	+	98395.3i	−0.457981	+	0.793246i	−0.998854	−	0.0478580i	$-0.984761\pi$
$109$	−56808.5	+	98395.3i	−0.457981	+	0.793246i	0.540873	+	0.841104i	$0.318094\pi$
$110$		0			0
$111$	−73376.2			−0.565260
$112$		0			0
$113$	−61774.1			−0.455103			−0.227552	−	0.973766i	$-0.573072\pi$
$113$	−61774.1			−0.455103			−0.227552	+	0.973766i	$0.573072\pi$
$114$		0			0
$115$	−7361.66	+	12750.8i	−0.0519076	+	0.0899066i
$116$		0			0
$117$	−36246.8	+	20927.1i	−0.244796	+	0.141333i
$118$		0			0
$119$	723.882	−	311.421i	0.00468598	−	0.00201595i
$120$		0			0
$121$	−48240.7	−	83555.4i	−0.299537	−	0.518813i
$122$		0			0
$123$	−155949.	−	90037.4i	−0.929439	−	0.536612i
$124$		0			0
$125$			42877.9i			0.245447i
$126$		0			0
$127$		−	92785.1i		−	0.510468i	−0.966879	−	0.255234i	$-0.917847\pi$
$127$		−	92785.1i		−	0.510468i	0.966879	−	0.255234i	$-0.0821526\pi$
$128$		0			0
$129$	−68007.2	−	39264.0i	−0.359816	−	0.207740i
$130$		0			0
$131$	−14247.6	−	24677.6i	−0.0725379	−	0.125639i	0.827475	−	0.561502i	$-0.189776\pi$
$131$	−14247.6	−	24677.6i	−0.0725379	−	0.125639i	−0.900013	+	0.435863i	$0.856443\pi$
$132$		0			0
$133$	59131.2	+	6969.51i	0.289859	+	0.0341644i
$134$		0			0
$135$	4364.61	−	2519.91i	0.0206116	−	0.0119001i
$136$		0			0
$137$	−128177.	+	222010.i	−0.583458	+	1.01058i	0.411607	+	0.911361i	$0.364968\pi$
$137$	−128177.	+	222010.i	−0.583458	+	1.01058i	−0.995066	+	0.0992182i	$0.968366\pi$
$138$		0			0
$139$	−172900.			−0.759030			−0.379515	−	0.925185i	$-0.623909\pi$
$139$	−172900.			−0.759030			−0.379515	+	0.925185i	$0.623909\pi$
$140$		0			0
$141$	235247.			0.996498
$142$		0			0
$143$	−65650.4	+	113710.i	−0.268471	+	0.465005i
$144$		0			0
$145$	−14690.4	+	8481.53i	−0.0580249	+	0.0335007i
$146$		0			0
$147$	147118.	+	35168.7i	0.561529	+	0.134234i
$148$		0			0
$149$	−10328.4	−	17889.3i	−0.0381124	−	0.0660126i	0.846340	−	0.532643i	$-0.178801\pi$
$149$	−10328.4	−	17889.3i	−0.0381124	−	0.0660126i	−0.884452	+	0.466630i	$0.845468\pi$
$150$		0			0
$151$	−181611.	−	104853.i	−0.648185	−	0.374230i	0.139575	−	0.990211i	$-0.455426\pi$
$151$	−181611.	−	104853.i	−0.648185	−	0.374230i	−0.787761	+	0.615982i	$0.788760\pi$
$152$		0			0
$153$		−	492.359i		−	0.00170041i
$154$		0			0
$155$		−	44638.2i		−	0.149237i
$156$		0			0
$157$	166527.	+	96144.3i	0.539182	+	0.311297i	0.744747	−	0.667347i	$-0.232570\pi$
$157$	166527.	+	96144.3i	0.539182	+	0.311297i	−0.205565	+	0.978643i	$0.565903\pi$
$158$		0			0
$159$	−2103.30	−	3643.02i	−0.00659793	−	0.0114280i
$160$		0			0
$161$	−32318.7	+	274200.i	−0.0982627	+	0.833686i
$162$		0			0
$163$	214863.	−	124051.i	0.633421	−	0.365706i	−0.148655	−	0.988889i	$-0.547494\pi$
$163$	214863.	−	124051.i	0.633421	−	0.365706i	0.782076	+	0.623184i	$0.214161\pi$
$164$		0			0
$165$	7905.22	−	13692.2i	0.0226050	−	0.0391530i
$166$		0			0
$167$	−61234.3			−0.169904			−0.0849520	−	0.996385i	$-0.527074\pi$
$167$	−61234.3			−0.169904			−0.0849520	+	0.996385i	$0.527074\pi$
$168$		0			0
$169$	104296.			0.280899
$170$		0			0
$171$	18600.4	−	32216.8i	0.0486443	−	0.0842544i
$172$		0			0
$173$	−486604.	+	280941.i	−1.23612	+	0.713673i	−0.968299	−	0.249796i	$-0.919637\pi$
$173$	−486604.	+	280941.i	−1.23612	+	0.713673i	−0.267820	+	0.963469i	$0.586303\pi$
$174$		0			0
$175$	157655.	+	366461.i	0.389146	+	0.904550i
$176$		0			0
$177$	−146538.	−	253812.i	−0.351575	−	0.608945i
$178$		0			0
$179$	228944.	+	132181.i	0.534069	+	0.308345i	0.742672	−	0.669656i	$-0.233558\pi$
$179$	228944.	+	132181.i	0.534069	+	0.308345i	−0.208603	+	0.978000i	$0.566892\pi$
$180$		0			0
$181$			221949.i			0.503565i	0.967784	+	0.251783i	$0.0810168\pi$
$181$			221949.i			0.503565i	−0.967784	+	0.251783i	$0.918983\pi$
$182$		0			0
$183$		−	184757.i		−	0.407825i
$184$		0			0
$185$	48812.5	+	28181.9i	0.104858	+	0.0605398i
$186$		0			0
$187$	−772.290	−	1337.65i	−0.00161501	−	0.00279729i
$188$		0			0
$189$	56510.2	−	75753.1i	0.115073	−	0.154257i
$190$		0			0
$191$	363167.	−	209675.i	0.720316	−	0.415875i	−0.0945529	−	0.995520i	$-0.530142\pi$
$191$	363167.	−	209675.i	0.720316	−	0.415875i	0.814869	+	0.579645i	$0.196809\pi$
$192$		0			0
$193$	378256.	−	655158.i	0.730957	−	1.26606i	−0.225517	−	0.974239i	$-0.572407\pi$
$193$	378256.	−	655158.i	0.730957	−	1.26606i	0.956474	−	0.291816i	$-0.0942595\pi$
$194$		0			0
$195$	32150.2			0.0605476
$196$		0			0
$197$	−458778.			−0.842242			−0.421121	−	0.907005i	$-0.638363\pi$
$197$	−458778.			−0.842242			−0.421121	+	0.907005i	$0.638363\pi$
$198$		0			0
$199$	132688.	−	229823.i	0.237520	−	0.411396i	−0.722482	−	0.691389i	$-0.756999\pi$
$199$	132688.	−	229823.i	0.237520	−	0.411396i	0.960002	+	0.279993i	$0.0903323\pi$
$200$		0			0
$201$	370527.	−	213924.i	0.646889	−	0.373482i
$202$		0			0
$203$	−190202.	+	254970.i	−0.323948	+	0.434259i
$204$		0			0
$205$	69162.0	+	119792.i	0.114943	+	0.199087i
$206$		0			0
$207$	149394.	+	86252.8i	0.242330	+	0.139910i
$208$		0			0
$209$		−	116703.i		−	0.184806i
$210$		0			0
$211$			394916.i			0.610659i	0.952247	+	0.305330i	$0.0987666\pi$
$211$			394916.i			0.610659i	−0.952247	+	0.305330i	$0.901233\pi$
$212$		0			0
$213$	408754.	+	235994.i	0.617324	+	0.356412i
$214$		0			0
$215$	30160.6	+	52239.6i	0.0444983	+	0.0770733i
$216$		0			0
$217$	−330803.	−	768936.i	−0.476893	−	1.10851i
$218$		0			0
$219$	−116373.	+	67188.1i	−0.163962	+	0.0946635i
$220$		0			0
$221$	1570.44	−	2720.07i	0.00216291	−	0.00374628i
$222$		0			0
$223$	429057.			0.577767			0.288884	−	0.957364i	$-0.406716\pi$
$223$	429057.			0.577767			0.288884	+	0.957364i	$0.406716\pi$
$224$		0			0
$225$	249254.			0.328235
$226$		0			0
$227$	−533314.	+	923727.i	−0.686939	+	1.18981i	0.285884	+	0.958264i	$0.407713\pi$
$227$	−533314.	+	923727.i	−0.686939	+	1.18981i	−0.972823	+	0.231549i	$0.925621\pi$
$228$		0			0
$229$	1.27378e6	−	735415.i	1.60511	−	0.926710i	0.614666	−	0.788787i	$-0.289291\pi$
$229$	1.27378e6	−	735415.i	1.60511	−	0.926710i	0.990443	−	0.137923i	$-0.0440426\pi$
$230$		0			0
$231$	34705.0	−	294446.i	0.0427919	−	0.363058i
$232$		0			0
$233$	−588039.	−	1.01851e6i	−0.709605	−	1.22907i	−0.965004	−	0.262236i	$-0.915540\pi$
$233$	−588039.	−	1.01851e6i	−0.709605	−	1.22907i	0.255399	−	0.966836i	$-0.417793\pi$
$234$		0			0
$235$	−156495.	−	90352.3i	−0.184854	−	0.106726i
$236$		0			0
$237$		−	384377.i		−	0.444515i
$238$		0			0
$239$			1.05050e6i			1.18960i	0.803872	+	0.594802i	$0.202770\pi$
$239$			1.05050e6i			1.18960i	−0.803872	+	0.594802i	$0.797230\pi$
$240$		0			0
$241$	−140862.	−	81326.9i	−0.156226	−	0.0901969i	0.419849	−	0.907594i	$-0.362083\pi$
$241$	−140862.	−	81326.9i	−0.156226	−	0.0901969i	−0.576075	+	0.817397i	$0.695416\pi$
$242$		0			0
$243$	−29524.5	−	51137.9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	−84360.6	−	79899.6i	−0.0897893	−	0.0850412i
$246$		0			0
$247$	205519.	−	118656.i	0.214343	−	0.123751i
$248$		0			0
$249$	302023.	−	523118.i	0.308703	−	0.534690i
$250$		0			0
$251$	1.70118e6			1.70438			0.852191	−	0.523231i	$-0.175274\pi$
$251$	1.70118e6			1.70438			0.852191	+	0.523231i	$0.175274\pi$
$252$		0			0
$253$	541168.			0.531534
$254$		0			0
$255$	−189.102	+	327.535i	−0.000182115	+	0.000315433i
$256$		0			0
$257$	−1.12687e6	+	650600.i	−1.06425	+	0.614442i	−0.926604	−	0.376040i	$-0.877286\pi$
$257$	−1.12687e6	+	650600.i	−1.06425	+	0.614442i	−0.137642	+	0.990482i	$0.543952\pi$
$258$		0			0
$259$	1.04969e6	+	123722.i	0.972328	+	0.114604i
$260$		0			0
$261$	99373.7	+	172120.i	0.0902964	+	0.156398i
$262$		0			0
$263$	1.36195e6	+	786324.i	1.21415	+	0.700990i	0.963661	−	0.267129i	$-0.0860750\pi$
$263$	1.36195e6	+	786324.i	1.21415	+	0.700990i	0.250490	+	0.968119i	$0.419408\pi$
$264$		0			0
$265$			3231.29i			0.00282658i
$266$		0			0
$267$		−	246022.i		−	0.211201i
$268$		0			0
$269$	−196636.	−	113528.i	−0.165684	−	0.0956580i	0.414865	−	0.909883i	$-0.363829\pi$
$269$	−196636.	−	113528.i	−0.165684	−	0.0956580i	−0.580549	+	0.814225i	$0.697162\pi$
$270$		0			0
$271$	−476479.	−	825285.i	−0.394113	−	0.682623i	0.598875	−	0.800843i	$-0.295615\pi$
$271$	−476479.	−	825285.i	−0.394113	−	0.682623i	−0.992987	+	0.118220i	$0.962281\pi$
$272$		0			0
$273$	553818.	−	238258.i	0.449739	−	0.193482i
$274$		0			0
$275$	677175.	−	390967.i	0.539970	−	0.311752i
$276$		0			0
$277$	4215.42	−	7301.33i	0.00330097	−	0.00571745i	−0.864370	−	0.502856i	$-0.832283\pi$
$277$	4215.42	−	7301.33i	0.00330097	−	0.00571745i	0.867671	+	0.497139i	$0.165616\pi$
$278$		0			0
$279$	−523003.			−0.402247
$280$		0			0
$281$	688929.			0.520486			0.260243	−	0.965543i	$-0.416197\pi$
$281$	688929.			0.520486			0.260243	+	0.965543i	$0.416197\pi$
$282$		0			0
$283$	−261313.	+	452607.i	−0.193952	+	0.335935i	−0.946557	−	0.322538i	$-0.895464\pi$
$283$	−261313.	+	452607.i	−0.193952	+	0.335935i	0.752604	+	0.658473i	$0.228797\pi$
$284$		0			0
$285$	−24747.3	+	14287.9i	−0.0180474	+	0.0104197i
$286$		0			0
$287$	2.07914e6	+	1.55099e6i	1.48997	+	1.11149i
$288$		0			0
$289$	−709910.	−	1.22960e6i	−0.499987	−	0.866003i
$290$		0			0
$291$	−489325.	−	282512.i	−0.338739	−	0.195571i
$292$		0			0
$293$			2.48856e6i			1.69348i	0.532011	+	0.846738i	$0.321437\pi$
$293$			2.48856e6i			1.69348i	−0.532011	+	0.846738i	$0.678563\pi$
$294$		0			0
$295$			225126.i			0.150616i
$296$		0			0
$297$	−160425.	−	92621.4i	−0.105531	−	0.0609285i
$298$		0			0
$299$	550227.	+	953020.i	0.355929	+	0.616487i
$300$		0			0
$301$	906681.	+	676365.i	0.576817	+	0.430294i
$302$		0			0
$303$	97591.6	−	56344.5i	0.0610669	−	0.0352570i
$304$		0			0
$305$	−70960.5	+	122907.i	−0.0436784	+	0.0756533i
$306$		0			0
$307$	444669.			0.269272			0.134636	−	0.990895i	$-0.457014\pi$
$307$	444669.			0.269272			0.134636	+	0.990895i	$0.457014\pi$
$308$		0			0
$309$	1.38939e6			0.827803
$310$		0			0
$311$	133147.	−	230618.i	0.0780606	−	0.135205i	−0.824353	−	0.566077i	$-0.808461\pi$
$311$	133147.	−	230618.i	0.0780606	−	0.135205i	0.902413	+	0.430872i	$0.141794\pi$
$312$		0			0
$313$	385023.	−	222293.i	0.222140	−	0.128252i	−0.384801	−	0.923000i	$-0.625730\pi$
$313$	385023.	−	222293.i	0.222140	−	0.128252i	0.606941	+	0.794747i	$0.292397\pi$
$314$		0			0
$315$	−66687.4	+	28689.5i	−0.0378676	+	0.0162910i
$316$		0			0
$317$	289251.	+	500997.i	0.161669	+	0.280018i	0.935467	−	0.353413i	$-0.114979\pi$
$317$	289251.	+	500997.i	0.161669	+	0.280018i	−0.773799	+	0.633432i	$0.781646\pi$
$318$		0			0
$319$	539959.	+	311746.i	0.297088	+	0.171524i
$320$		0			0
$321$		−	218562.i		−	0.118389i
$322$		0			0
$323$			2791.67i			0.00148887i
$324$		0			0
$325$	1.37702e6	+	795024.i	0.723156	+	0.417515i
$326$		0			0
$327$	−511277.	−	885557.i	−0.264415	−	0.457981i
$328$		0			0
$329$	−3.36535e6	−	396659.i	−1.71412	−	0.202035i
$330$		0			0
$331$	644352.	−	372017.i	0.323261	−	0.186635i	−0.329584	−	0.944126i	$-0.606909\pi$
$331$	644352.	−	372017.i	0.323261	−	0.186635i	0.652845	+	0.757491i	$0.273575\pi$
$332$		0			0
$333$	330193.	−	571911.i	0.163176	−	0.282630i
$334$		0			0
$335$	−328650.			−0.160001
$336$		0			0
$337$	1.24847e6			0.598831			0.299416	−	0.954123i	$-0.403208\pi$
$337$	1.24847e6			0.598831			0.299416	+	0.954123i	$0.403208\pi$
$338$		0			0
$339$	277983.	−	481481.i	0.131377	−	0.227552i
$340$		0			0
$341$	−1.42090e6	+	820357.i	−0.661725	+	0.382047i
$342$		0			0
$343$	−2.04531e6	−	751171.i	−0.938695	−	0.344750i
$344$		0			0
$345$	−66254.9	−	114757.i	−0.0299689	−	0.0519076i
$346$		0			0
$347$	−345693.	−	199586.i	−0.154123	−	0.0889829i	0.420955	−	0.907081i	$-0.361695\pi$
$347$	−345693.	−	199586.i	−0.154123	−	0.0889829i	−0.575078	+	0.818099i	$0.695028\pi$
$348$		0			0
$349$			118183.i			0.0519386i	0.999663	+	0.0259693i	$0.00826722\pi$
$349$			118183.i			0.0519386i	−0.999663	+	0.0259693i	$0.991733\pi$
$350$		0			0
$351$		−	376687.i		−	0.163197i
$352$		0			0
$353$	−3.24291e6	−	1.87229e6i	−1.38515	−	0.799719i	−0.392389	−	0.919799i	$-0.628351\pi$
$353$	−3.24291e6	−	1.87229e6i	−1.38515	−	0.799719i	−0.992764	+	0.120080i	$0.961685\pi$
$354$		0			0
$355$	−181278.	−	313983.i	−0.0763441	−	0.132232i
$356$		0			0
$357$	−830.184	+	7043.50i	−0.000344750	+	0.00292495i
$358$		0			0
$359$	−2.55241e6	+	1.47364e6i	−1.04524	+	0.603468i	−0.921312	−	0.388824i	$-0.872882\pi$
$359$	−2.55241e6	+	1.47364e6i	−1.04524	+	0.603468i	−0.123925	+	0.992292i	$0.539548\pi$
$360$		0			0
$361$	1.13259e6	−	1.96170e6i	0.457407	−	0.792252i
$362$		0			0
$363$	868333.			0.345875
$364$		0			0
$365$	103221.			0.0405542
$366$		0			0
$367$	−1.44334e6	+	2.49994e6i	−0.559376	+	0.968867i	0.438173	+	0.898891i	$0.355626\pi$
$367$	−1.44334e6	+	2.49994e6i	−0.559376	+	0.968867i	−0.997549	+	0.0699765i	$0.977708\pi$
$368$		0			0
$369$	1.40354e6	−	810336.i	0.536612	−	0.309813i
$370$		0			0
$371$	23946.4	+	55662.1i	0.00903243	+	0.0209954i
$372$		0			0
$373$	−184972.	−	320381.i	−0.0688389	−	0.119232i	0.829552	−	0.558430i	$-0.188596\pi$
$373$	−184972.	−	320381.i	−0.0688389	−	0.119232i	−0.898390	+	0.439198i	$0.855263\pi$
$374$		0			0
$375$	−334200.	−	192951.i	−0.122724	−	0.0708546i
$376$		0			0
$377$			1.26786e6i			0.459427i
$378$		0			0
$379$		−	5.05403e6i		−	1.80734i	−0.428230	−	0.903670i	$-0.640863\pi$
$379$		−	5.05403e6i		−	1.80734i	0.428230	−	0.903670i	$-0.359137\pi$
$380$		0			0
$381$	723188.	+	417533.i	0.255234	+	0.147360i
$382$		0			0
$383$	543918.	+	942093.i	0.189468	+	0.328169i	0.945073	−	0.326859i	$-0.105990\pi$
$383$	543918.	+	942093.i	0.189468	+	0.328169i	−0.755605	+	0.655028i	$0.772657\pi$
$384$		0			0
$385$	−136176.	+	182547.i	−0.0468219	+	0.0627657i
$386$		0			0
$387$	612065.	−	353376.i	0.207740	−	0.119939i
$388$		0			0
$389$	−2.39348e6	+	4.14563e6i	−0.801966	+	1.38905i	0.116355	+	0.993208i	$0.462879\pi$
$389$	−2.39348e6	+	4.14563e6i	−0.801966	+	1.38905i	−0.918321	+	0.395837i	$0.870454\pi$
$390$		0			0
$391$	−12945.4			−0.00428226
$392$		0			0
$393$	256458.			0.0837595
$394$		0			0
$395$	−147629.	+	255701.i	−0.0476079	+	0.0824593i
$396$		0			0
$397$	−2.46880e6	+	1.42536e6i	−0.786159	+	0.453889i	−0.838609	−	0.544735i	$-0.816630\pi$
$397$	−2.46880e6	+	1.42536e6i	−0.786159	+	0.453889i	0.0524497	+	0.998624i	$0.483297\pi$
$398$		0			0
$399$	−320412.	+	429519.i	−0.100757	+	0.135067i
$400$		0			0
$401$	−694963.	−	1.20371e6i	−0.215824	−	0.373819i	0.737703	−	0.675126i	$-0.235911\pi$
$401$	−694963.	−	1.20371e6i	−0.215824	−	0.373819i	−0.953527	+	0.301307i	$0.902577\pi$
$402$		0			0
$403$	−2.88937e6	−	1.66818e6i	−0.886217	−	0.511658i
$404$		0			0
$405$			45358.4i			0.0137410i
$406$		0			0
$407$		−	2.07170e6i		−	0.619927i
$408$		0			0
$409$	−1.50530e6	−	869084.i	−0.444953	−	0.256894i	0.260743	−	0.965408i	$-0.416032\pi$
$409$	−1.50530e6	−	869084.i	−0.444953	−	0.256894i	−0.705696	+	0.708515i	$0.749366\pi$
$410$		0			0
$411$	−1.15360e6	−	1.99809e6i	−0.336860	−	0.583458i
$412$		0			0
$413$	1.66836e6	+	3.87801e6i	0.481298	+	1.11875i
$414$		0			0
$415$	−401832.	+	231998.i	−0.114531	+	0.0661248i
$416$		0			0
$417$	778052.	−	1.34763e6i	0.219113	−	0.379515i
$418$		0			0
$419$	−3.71914e6			−1.03492			−0.517461	−	0.855707i	$-0.673123\pi$
$419$	−3.71914e6			−1.03492			−0.517461	+	0.855707i	$0.673123\pi$
$420$		0			0
$421$	6.86213e6			1.88692			0.943461	−	0.331483i	$-0.107549\pi$
$421$	6.86213e6			1.88692			0.943461	+	0.331483i	$0.107549\pi$
$422$		0			0
$423$	−1.05861e6	+	1.83357e6i	−0.287664	+	0.498249i
$424$		0			0
$425$	−16198.8	+	9352.40i	−0.00435022	+	0.00251160i
$426$		0			0
$427$	−311526.	+	2.64307e6i	−0.0826846	+	0.701518i
$428$		0			0
$429$	−590854.	−	1.02339e6i	−0.155002	−	0.268471i
$430$		0			0
$431$	−3.94696e6	−	2.27878e6i	−1.02346	−	0.590893i	−0.108353	−	0.994112i	$-0.534558\pi$
$431$	−3.94696e6	−	2.27878e6i	−1.02346	−	0.590893i	−0.915103	+	0.403220i	$0.867891\pi$
$432$		0			0
$433$			3.04050e6i			0.779338i	0.920955	+	0.389669i	$0.127411\pi$
$433$			3.04050e6i			0.779338i	−0.920955	+	0.389669i	$0.872589\pi$
$434$		0			0
$435$		−	152667.i		−	0.0386833i
$436$		0			0
$437$	−847063.	−	489052.i	−0.212184	−	0.122504i
$438$		0			0
$439$	−791316.	−	1.37060e6i	−0.195970	−	0.339429i	0.751248	−	0.660020i	$-0.229452\pi$
$439$	−791316.	−	1.37060e6i	−0.195970	−	0.339429i	−0.947218	+	0.320590i	$0.896119\pi$
$440$		0			0
$441$	−936143.	+	988411.i	−0.229217	+	0.242014i
$442$		0			0
$443$	−1.88414e6	+	1.08781e6i	−0.456146	+	0.263356i	−0.710422	−	0.703776i	$-0.751496\pi$
$443$	−1.88414e6	+	1.08781e6i	−0.456146	+	0.263356i	0.254277	+	0.967132i	$0.418163\pi$
$444$		0			0
$445$	−94490.8	+	163663.i	−0.0226198	+	0.0391787i
$446$		0			0
$447$	185911.			0.0440084
$448$		0			0
$449$	−5.64422e6			−1.32126			−0.660630	−	0.750712i	$-0.729711\pi$
$449$	−5.64422e6			−1.32126			−0.660630	+	0.750712i	$0.729711\pi$
$450$		0			0
$451$	2.54211e6	−	4.40306e6i	0.588509	−	1.01933i
$452$		0			0
$453$	1.63450e6	−	943677.i	0.374230	−	0.216062i
$454$		0			0
$455$	−459928.	−	54209.6i	−0.104151	−	0.0122757i
$456$		0			0
$457$	−3.38915e6	−	5.87018e6i	−0.759102	−	1.31480i	−0.943309	−	0.331916i	$-0.892305\pi$
$457$	−3.38915e6	−	5.87018e6i	−0.759102	−	1.31480i	0.184207	−	0.982887i	$-0.441028\pi$
$458$		0			0
$459$	3837.56	+	2215.61i	0.000850204	+	0.000490865i
$460$		0			0
$461$			8.49455e6i			1.86161i	0.365520	+	0.930803i	$0.380891\pi$
$461$			8.49455e6i			1.86161i	−0.365520	+	0.930803i	$0.619109\pi$
$462$		0			0
$463$		−	2.11828e6i		−	0.459231i	−0.973281	−	0.229615i	$-0.926253\pi$
$463$		−	2.11828e6i		−	0.459231i	0.973281	−	0.229615i	$-0.0737468\pi$
$464$		0			0
$465$	347920.	+	200872.i	0.0746186	+	0.0430811i
$466$		0			0
$467$	−1.69837e6	−	2.94167e6i	−0.360363	−	0.624168i	0.627657	−	0.778490i	$-0.284014\pi$
$467$	−1.69837e6	−	2.94167e6i	−0.360363	−	0.624168i	−0.988021	+	0.154322i	$0.950681\pi$
$468$		0			0
$469$	−5.66133e6	+	2.43556e6i	−1.18846	+	0.511288i
$470$		0			0
$471$	−1.49874e6	+	865299.i	−0.311297	+	0.179727i
$472$		0			0
$473$	1.10858e6	−	1.92011e6i	0.227831	−	0.394615i
$474$		0			0
$475$	−1.41327e6			−0.287402
$476$		0			0
$477$	37859.4			0.00761864
$478$		0			0
$479$	309067.	−	535320.i	0.0615480	−	0.106604i	−0.833610	−	0.552354i	$-0.813730\pi$
$479$	309067.	−	535320.i	0.0615480	−	0.106604i	0.895158	+	0.445750i	$0.147063\pi$
$480$		0			0
$481$	3.64835e6	−	2.10638e6i	0.719009	−	0.415120i
$482$		0			0
$483$	−1.99174e6	−	1.48580e6i	−0.388477	−	0.289796i
$484$		0			0
$485$	217011.	+	375874.i	0.0418917	+	0.0725585i
$486$		0			0
$487$	−6.18166e6	−	3.56899e6i	−1.18109	−	0.681903i	−0.224823	−	0.974400i	$-0.572181\pi$
$487$	−6.18166e6	−	3.56899e6i	−1.18109	−	0.681903i	−0.956266	+	0.292497i	$0.905514\pi$
$488$		0			0
$489$			2.23292e6i			0.422280i
$490$		0			0
$491$			3.67363e6i			0.687689i	0.939027	+	0.343844i	$0.111729\pi$
$491$			3.67363e6i			0.687689i	−0.939027	+	0.343844i	$0.888271\pi$
$492$		0			0
$493$	−12916.5	−	7457.33i	−0.00239346	−	0.00138187i
$494$		0			0
$495$	71147.0	+	123230.i	0.0130510	+	0.0226050i
$496$		0			0
$497$	−5.44956e6	−	4.06526e6i	−0.989624	−	0.738239i
$498$		0			0
$499$	−8.38426e6	+	4.84066e6i	−1.50735	+	0.870268i	−0.507384	+	0.861720i	$0.669388\pi$
$499$	−8.38426e6	+	4.84066e6i	−1.50735	+	0.870268i	−0.999963	+	0.00854775i	$0.997279\pi$
$500$		0			0
$501$	275554.	−	477274.i	0.0490470	−	0.0849520i
$502$		0			0
$503$	1.00906e7			1.77826			0.889131	−	0.457652i	$-0.151309\pi$
$503$	1.00906e7			1.77826			0.889131	+	0.457652i	$0.151309\pi$
$504$		0			0
$505$	−86561.9			−0.0151042
$506$		0			0
$507$	−469331.	+	812904.i	−0.0810884	+	0.140449i
$508$		0			0
$509$	2.24312e6	−	1.29507e6i	0.383759	−	0.221563i	−0.295693	−	0.955283i	$-0.595551\pi$
$509$	2.24312e6	−	1.29507e6i	0.383759	−	0.221563i	0.679452	+	0.733720i	$0.262217\pi$
$510$		0			0
$511$	1.77808e6	−	764947.i	0.301231	−	0.129592i
$512$		0			0
$513$	167404.	+	289952.i	0.0280848	+	0.0486443i
$514$		0			0
$515$	−924270.	−	533628.i	−0.153561	−	0.0886585i
$516$		0			0
$517$			6.64195e6i			1.09287i
$518$		0			0
$519$		−	5.05693e6i		−	0.824079i
$520$		0			0
$521$	112854.	+	65156.5i	0.0182148	+	0.0105163i	0.509080	−	0.860719i	$-0.329986\pi$
$521$	112854.	+	65156.5i	0.0182148	+	0.0105163i	−0.490865	+	0.871236i	$0.663319\pi$
$522$		0			0
$523$	2.12360e6	+	3.67818e6i	0.339483	+	0.588001i	0.984335	−	0.176306i	$-0.0564147\pi$
$523$	2.12360e6	+	3.67818e6i	0.339483	+	0.588001i	−0.644853	+	0.764307i	$0.723081\pi$
$524$		0			0
$525$	−3.56573e6	−	420276.i	−0.564612	−	0.0665482i
$526$		0			0
$527$	33989.6	−	19623.9i	0.00533113	−	0.00307793i
$528$		0			0
$529$	−950364.	+	1.64608e6i	−0.147656	+	0.255748i
$530$		0			0
$531$	2.63769e6			0.405963
$532$		0			0
$533$	1.03386e7			1.57633
$534$		0			0
$535$	−83944.1	+	145395.i	−0.0126796	+	0.0219617i
$536$		0			0
$537$	−2.06050e6	+	1.18963e6i	−0.308345	+	0.178023i
$538$		0			0
$539$	−992952.	+	4.15372e6i	−0.147216	+	0.615836i
$540$		0			0
$541$	3.18948e6	+	5.52434e6i	0.468518	+	0.811497i	0.999353	−	0.0359783i	$-0.0114547\pi$
$541$	3.18948e6	+	5.52434e6i	0.468518	+	0.811497i	−0.530834	+	0.847476i	$0.678121\pi$
$542$		0			0
$543$	−1.72992e6	−	998769.i	−0.251783	−	0.145367i
$544$		0			0
$545$			785472.i			0.113276i
$546$		0			0
$547$		−	3.87435e6i		−	0.553644i	−0.960921	−	0.276822i	$-0.910719\pi$
$547$		−	3.87435e6i		−	0.553644i	0.960921	−	0.276822i	$-0.0892812\pi$
$548$		0			0
$549$	1.44004e6	+	831408.i	0.203913	+	0.117729i
$550$		0			0
$551$	−563448.	−	975920.i	−0.0790633	−	0.136942i
$552$		0			0
$553$	−648111.	+	5.49875e6i	−0.0901232	+	0.764629i
$554$		0			0
$555$	−439312.	+	253637.i	−0.0605398	+	0.0349527i
$556$		0			0
$557$	3.75684e6	−	6.50703e6i	0.513079	−	0.888679i	−0.486806	−	0.873510i	$-0.661838\pi$
$557$	3.75684e6	−	6.50703e6i	0.513079	−	0.888679i	0.999885	−	0.0151691i	$-0.00482867\pi$
$558$		0			0
$559$	4.50854e6			0.610247
$560$		0			0
$561$	13901.2			0.00186486
$562$		0			0
$563$	2.96146e6	−	5.12941e6i	0.393763	−	0.682018i	−0.599179	−	0.800615i	$-0.704506\pi$
$563$	2.96146e6	−	5.12941e6i	0.393763	−	0.682018i	0.992943	+	0.118597i	$0.0378396\pi$
$564$		0			0
$565$	−369849.	+	213532.i	−0.0487420	+	0.0281412i
$566$		0			0
$567$	336141.	+	781342.i	0.0439100	+	0.102067i
$568$		0			0
$569$	−2.53715e6	−	4.39446e6i	−0.328522	−	0.569017i	0.653697	−	0.756757i	$-0.273217\pi$
$569$	−2.53715e6	−	4.39446e6i	−0.328522	−	0.569017i	−0.982219	+	0.187740i	$0.939884\pi$
$570$		0			0
$571$	1.00559e7	+	5.80579e6i	1.29072	+	0.745197i	0.978782	−	0.204906i	$-0.0656888\pi$
$571$	1.00559e7	+	5.80579e6i	1.29072	+	0.745197i	0.311937	+	0.950103i	$0.399022\pi$
$572$		0			0
$573$			3.77414e6i			0.480211i
$574$		0			0
$575$		−	6.55352e6i		−	0.826618i
$576$		0			0
$577$	−6.49447e6	−	3.74959e6i	−0.812090	−	0.468861i	0.0355908	−	0.999366i	$-0.488669\pi$
$577$	−6.49447e6	−	3.74959e6i	−0.812090	−	0.468861i	−0.847681	+	0.530506i	$0.822002\pi$
$578$		0			0
$579$	3.40430e6	+	5.89642e6i	0.422018	+	0.730957i
$580$		0			0
$581$	−5.20267e6	+	6.97428e6i	−0.639420	+	0.857155i
$582$		0			0
$583$	102857.	−	59384.4i	0.0125332	−	0.00723604i
$584$		0			0
$585$	−144676.	+	250586.i	−0.0174786	+	0.0302738i
$586$		0			0
$587$	3.60829e6			0.432221			0.216111	−	0.976369i	$-0.430663\pi$
$587$	3.60829e6			0.432221			0.216111	+	0.976369i	$0.430663\pi$
$588$		0			0
$589$	2.96542e6			0.352207
$590$		0			0
$591$	2.06450e6	−	3.57582e6i	0.243134	−	0.421121i
$592$		0			0
$593$	1.13287e6	−	654060.i	0.132294	−	0.0763802i	−0.432392	−	0.901686i	$-0.642330\pi$
$593$	1.13287e6	−	654060.i	0.132294	−	0.0763802i	0.564687	+	0.825305i	$0.308997\pi$
$594$		0			0
$595$	3257.49	−	4366.73i	0.000377217	−	0.000505667i
$596$		0			0
$597$	1.19419e6	+	2.06840e6i	0.137132	+	0.237520i
$598$		0			0
$599$	1.04418e7	+	6.02856e6i	1.18907	+	0.686510i	0.958095	−	0.286450i	$-0.0924752\pi$
$599$	1.04418e7	+	6.02856e6i	1.18907	+	0.686510i	0.230975	+	0.972960i	$0.425809\pi$
$600$		0			0
$601$		−	1.27539e6i		−	0.144031i	−0.997404	−	0.0720156i	$-0.977057\pi$
$601$		−	1.27539e6i		−	0.144031i	0.997404	−	0.0720156i	$-0.0229431\pi$
$602$		0			0
$603$			3.85063e6i			0.431259i
$604$		0			0
$605$	−577646.	−	333504.i	−0.0641613	−	0.0370436i
$606$		0			0
$607$	1.96489e6	+	3.40329e6i	0.216454	+	0.374910i	0.953721	−	0.300691i	$-0.0972174\pi$
$607$	1.96489e6	+	3.40329e6i	0.216454	+	0.374910i	−0.737267	+	0.675601i	$0.763884\pi$
$608$		0			0
$609$	−1.13138e6	−	2.62985e6i	−0.123614	−	0.287334i
$610$		0			0
$611$	−1.16968e7	+	6.75313e6i	−1.26754	+	0.731816i
$612$		0			0
$613$	4.78881e6	−	8.29446e6i	0.514726	−	0.891532i	−0.485128	−	0.874443i	$-0.661227\pi$
$613$	4.78881e6	−	8.29446e6i	0.514726	−	0.891532i	0.999854	−	0.0170885i	$-0.00543970\pi$
$614$		0			0
$615$	−1.24492e6			−0.132725
$616$		0			0
$617$	1.87060e7			1.97819			0.989094	−	0.147289i	$-0.0470547\pi$
$617$	1.87060e7			1.97819			0.989094	+	0.147289i	$0.0470547\pi$
$618$		0			0
$619$	−5.19180e6	+	8.99247e6i	−0.544618	+	0.943305i	0.454013	+	0.890995i	$0.349992\pi$
$619$	−5.19180e6	+	8.99247e6i	−0.544618	+	0.943305i	−0.998631	+	0.0523104i	$0.983341\pi$
$620$		0			0
$621$	−1.34455e6	+	776275.i	−0.139910	+	0.0807768i
$622$		0			0
$623$	−414827.	+	3.51950e6i	−0.0428201	+	0.363296i
$624$		0			0
$625$	−4.65992e6	−	8.07122e6i	−0.477176	−	0.826493i
$626$		0			0
$627$	909608.	+	525162.i	0.0924029	+	0.0533488i
$628$		0			0
$629$			49557.5i			0.00499439i
$630$		0			0
$631$		−	283563.i		−	0.0283515i	−0.999900	−	0.0141758i	$-0.995488\pi$
$631$		−	283563.i		−	0.0283515i	0.999900	−	0.0141758i	$-0.00451244\pi$
$632$		0			0
$633$	−3.07807e6	−	1.77712e6i	−0.305330	−	0.176282i
$634$		0			0
$635$	−320727.	−	555516.i	−0.0315647	−	0.0546716i
$636$		0			0
$637$	−8.32445e6	+	2.47461e6i	−0.812843	+	0.241634i
$638$		0			0
$639$	−3.67878e6	+	2.12395e6i	−0.356412	+	0.205775i
$640$		0			0
$641$	8.36211e6	−	1.44836e7i	0.803842	−	1.39230i	−0.113227	−	0.993569i	$-0.536119\pi$
$641$	8.36211e6	−	1.44836e7i	0.803842	−	1.39230i	0.917070	−	0.398727i	$-0.130548\pi$
$642$		0			0
$643$	1.43367e7			1.36749			0.683743	−	0.729723i	$-0.260351\pi$
$643$	1.43367e7			1.36749			0.683743	+	0.729723i	$0.260351\pi$
$644$		0			0
$645$	−542890.			−0.0513822
$646$		0			0
$647$	284135.	−	492136.i	0.0266848	−	0.0462194i	−0.852375	−	0.522932i	$-0.824838\pi$
$647$	284135.	−	492136.i	0.0266848	−	0.0462194i	0.879059	+	0.476712i	$0.158172\pi$
$648$		0			0
$649$	7.16610e6	−	4.13735e6i	0.667838	−	0.385576i
$650$		0			0
$651$	7.48188e6	+	881854.i	0.691923	+	0.0815538i
$652$		0			0
$653$	−9.29093e6	−	1.60924e7i	−0.852661	−	1.47685i	−0.878798	−	0.477194i	$-0.841654\pi$
$653$	−9.29093e6	−	1.60924e7i	−0.852661	−	1.47685i	0.0261372	−	0.999658i	$-0.491679\pi$
$654$		0			0
$655$	−170605.	−	98498.7i	−0.0155377	−	0.00897072i
$656$		0			0
$657$		−	1.20939e6i		−	0.109308i
$658$		0			0
$659$		−	1.33728e7i		−	1.19953i	−0.800178	−	0.599763i	$-0.795262\pi$
$659$		−	1.33728e7i		−	1.19953i	0.800178	−	0.599763i	$-0.204738\pi$
$660$		0			0
$661$	−3.42838e6	−	1.97938e6i	−0.305201	−	0.176208i	0.339576	−	0.940579i	$-0.389716\pi$
$661$	−3.42838e6	−	1.97938e6i	−0.305201	−	0.176208i	−0.644777	+	0.764371i	$0.723050\pi$
$662$		0			0
$663$	14133.9	+	24480.7i	0.00124876	+	0.00216291i
$664$		0			0
$665$	378117.	−	162669.i	0.0331567	−	0.0142643i
$666$		0			0
$667$	4.52549e6	−	2.61279e6i	0.393868	−	0.227400i
$668$		0			0
$669$	−1.93076e6	+	3.34417e6i	−0.166787	+	0.288884i
$670$		0			0
$671$	5.21643e6			0.447267
$672$		0			0
$673$	5.90220e6			0.502315			0.251158	−	0.967946i	$-0.419189\pi$
$673$	5.90220e6			0.502315			0.251158	+	0.967946i	$0.419189\pi$
$674$		0			0
$675$	−1.12164e6	+	1.94274e6i	−0.0947534	+	0.164118i
$676$		0			0
$677$	1.48927e7	−	8.59829e6i	1.24882	−	0.721009i	0.277949	−	0.960596i	$-0.410345\pi$
$677$	1.48927e7	−	8.59829e6i	1.24882	−	0.721009i	0.970875	+	0.239587i	$0.0770121\pi$
$678$		0			0
$679$	6.52375e6	+	4.86658e6i	0.543029	+	0.405088i
$680$		0			0
$681$	−4.79983e6	−	8.31354e6i	−0.396605	−	0.686939i
$682$		0			0
$683$	−8.54357e6	−	4.93263e6i	−0.700789	−	0.404601i	0.106852	−	0.994275i	$-0.465923\pi$
$683$	−8.54357e6	−	4.93263e6i	−0.700789	−	0.404601i	−0.807641	+	0.589674i	$0.799256\pi$
$684$		0			0
$685$			1.77227e6i			0.144312i
$686$		0			0
$687$			1.32375e7i			1.07007i
$688$		0			0
$689$	209157.	+	120757.i	0.0167851	+	0.00969089i
$690$		0			0
$691$	2.25864e6	+	3.91208e6i	0.179950	+	0.311683i	0.941863	−	0.335997i	$-0.109073\pi$
$691$	2.25864e6	+	3.91208e6i	0.179950	+	0.311683i	−0.761913	+	0.647679i	$0.775740\pi$
$692$		0			0
$693$	2.13881e6	+	1.59551e6i	0.169176	+	0.126202i
$694$		0			0
$695$	−1.03518e6	+	597659.i	−0.0812928	+	0.0469344i
$696$		0			0
$697$	−60810.3	+	105326.i	−0.00474127	+	0.00821213i
$698$		0			0
$699$	1.05847e7			0.819381
$700$		0			0
$701$	1.90322e7			1.46283			0.731416	−	0.681932i	$-0.238860\pi$
$701$	1.90322e7			1.46283			0.731416	+	0.681932i	$0.238860\pi$
$702$		0			0
$703$	−1.87219e6	+	3.24273e6i	−0.142877	+	0.247470i
$704$		0			0
$705$	1.40845e6	−	813170.i	0.106726	−	0.0616182i
$706$		0			0
$707$	−1.49111e6	+	641491.i	−0.112192	+	0.0482661i
$708$		0			0
$709$	−7.96300e6	−	1.37923e7i	−0.594924	−	1.03044i	−0.993558	−	0.113328i	$-0.963849\pi$
$709$	−7.96300e6	−	1.37923e7i	−0.594924	−	1.03044i	0.398634	−	0.917110i	$-0.369484\pi$
$710$		0			0
$711$	2.99592e6	+	1.72969e6i	0.222257	+	0.128320i
$712$		0			0
$713$			1.37511e7i			1.01301i
$714$		0			0
$715$			907726.i			0.0664033i
$716$		0			0
$717$	−8.18786e6	−	4.72726e6i	−0.594802	−	0.343409i
$718$		0			0
$719$	1.19051e6	+	2.06202e6i	0.0858834	+	0.148754i	0.905767	−	0.423775i	$-0.139295\pi$
$719$	1.19051e6	+	2.06202e6i	0.0858834	+	0.148754i	−0.819884	+	0.572530i	$0.805962\pi$
$720$		0			0
$721$	−1.98760e7	−	2.34270e6i	−1.42394	−	0.167833i
$722$		0			0
$723$	1.26776e6	−	731942.i	0.0901969	−	0.0520752i
$724$		0			0
$725$	3.77523e6	−	6.53889e6i	0.266746	−	0.462018i
$726$		0			0
$727$	1.51883e7			1.06579			0.532895	−	0.846181i	$-0.321104\pi$
$727$	1.51883e7			1.06579			0.532895	+	0.846181i	$0.321104\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−26518.5	+	45931.3i	−0.00183550	+	0.00317918i
$732$		0			0
$733$	−3.31964e6	+	1.91660e6i	−0.228208	+	0.131756i	−0.609745	−	0.792598i	$-0.708728\pi$
$733$	−3.31964e6	+	1.91660e6i	−0.228208	+	0.131756i	0.381537	+	0.924354i	$0.375395\pi$
$734$		0			0
$735$	1.00238e6	−	297978.i	0.0684406	−	0.0203454i
$736$		0			0
$737$	6.03992e6	+	1.04614e7i	0.409602	+	0.709452i
$738$		0			0
$739$	−137384.	−	79318.8i	−0.00925391	−	0.00534275i	0.495366	−	0.868684i	$-0.335034\pi$
$739$	−137384.	−	79318.8i	−0.00925391	−	0.00534275i	−0.504620	+	0.863342i	$0.668367\pi$
$740$		0			0
$741$			2.13581e6i			0.142895i
$742$		0			0
$743$		−	2.10879e7i		−	1.40140i	−0.713458	−	0.700698i	$-0.752872\pi$
$743$		−	2.10879e7i		−	1.40140i	0.713458	−	0.700698i	$-0.247128\pi$
$744$		0			0
$745$	−123675.	−	71403.5i	−0.00816375	−	0.00471334i
$746$		0			0
$747$	2.71820e6	+	4.70807e6i	0.178230	+	0.308703i
$748$		0			0
$749$	−368526.	+	3.12667e6i	−0.0240029	+	0.203647i
$750$		0			0
$751$	−1.19067e7	+	6.87433e6i	−0.770355	+	0.444765i	−0.833001	−	0.553271i	$-0.813379\pi$
$751$	−1.19067e7	+	6.87433e6i	−0.770355	+	0.444765i	0.0626460	+	0.998036i	$0.480046\pi$
$752$		0			0
$753$	−7.65532e6	+	1.32594e7i	−0.492013	+	0.852191i
$754$		0			0
$755$	−1.44977e6			−0.0925616
$756$		0			0
$757$	2.69473e7			1.70913			0.854566	−	0.519343i	$-0.173823\pi$
$757$	2.69473e7			1.70913			0.854566	+	0.519343i	$0.173823\pi$
$758$		0			0
$759$	−2.43526e6	+	4.21799e6i	−0.153441	+	0.265767i
$760$		0			0
$761$	1.44676e7	−	8.35289e6i	0.905599	−	0.522848i	0.0265864	−	0.999647i	$-0.491536\pi$
$761$	1.44676e7	−	8.35289e6i	0.905599	−	0.522848i	0.879012	+	0.476799i	$0.158203\pi$
$762$		0			0
$763$	5.82096e6	+	1.35305e7i	0.361979	+	0.841402i
$764$		0			0
$765$	−1701.92	−	2947.81i	−0.000105144	−	0.000182115i
$766$		0			0
$767$	1.45721e7	+	8.41321e6i	0.894404	+	0.516385i
$768$		0			0
$769$			2.53462e7i			1.54560i	0.634651	+	0.772799i	$0.281144\pi$
$769$			2.53462e7i			1.54560i	−0.634651	+	0.772799i	$0.718856\pi$
$770$		0			0
$771$		−	1.17108e7i		−	0.709497i
$772$		0			0
$773$	−1.29309e7	−	7.46565e6i	−0.778358	−	0.449385i	0.0574900	−	0.998346i	$-0.481690\pi$
$773$	−1.29309e7	−	7.46565e6i	−0.778358	−	0.449385i	−0.835848	+	0.548961i	$0.815024\pi$
$774$		0			0
$775$	9.93448e6	+	1.72070e7i	0.594143	+	1.02909i
$776$		0			0
$777$	−5.68794e6	+	7.62479e6i	−0.337989	+	0.453081i
$778$		0			0
$779$	−7.95808e6	+	4.59460e6i	−0.469856	+	0.271271i
$780$		0			0
$781$	−6.66304e6	+	1.15407e7i	−0.390881	+	0.677027i
$782$		0			0
$783$	−1.78873e6			−0.104265
$784$		0			0
$785$	1.32936e6			0.0769958
$786$		0			0
$787$	1.04205e7	−	1.80489e7i	0.599726	−	1.03876i	−0.393135	−	0.919481i	$-0.628610\pi$
$787$	1.04205e7	−	1.80489e7i	0.599726	−	1.03876i	0.992861	−	0.119275i	$-0.0380570\pi$
$788$		0			0
$789$	−1.22576e7	+	7.07691e6i	−0.700990	+	0.404717i
$790$		0			0
$791$	−4.78857e6	+	6.41917e6i	−0.272122	+	0.364786i
$792$		0			0
$793$	5.30375e6	+	9.18636e6i	0.299502	+	0.518753i
$794$		0			0
$795$	−25185.4	−	14540.8i	−0.00141329	−	0.000815963i
$796$		0			0
$797$		−	6.52948e6i		−	0.364110i	−0.983288	−	0.182055i	$-0.941725\pi$
$797$		−	6.52948e6i		−	0.364110i	0.983288	−	0.182055i	$-0.0582750\pi$
$798$		0			0
$799$		−	158883.i		−	0.00880463i
$800$		0			0
$801$	1.91755e6	+	1.10710e6i	0.105601	+	0.0609685i
$802$		0			0
$803$	−1.89699e6	−	3.28568e6i	−0.103819	−	0.179819i
$804$		0			0
$805$	754322.	+	1.75338e6i	0.0410267	+	0.0953646i
$806$		0			0
$807$	1.76972e6	−	1.02175e6i	0.0956580	−	0.0552281i
$808$		0			0
$809$	1.65469e7	−	2.86600e7i	0.888883	−	1.53959i	0.0476853	−	0.998862i	$-0.484816\pi$
$809$	1.65469e7	−	2.86600e7i	0.888883	−	1.53959i	0.841198	−	0.540728i	$-0.181851\pi$
$810$		0			0
$811$	8.65278e6			0.461959			0.230979	−	0.972959i	$-0.425807\pi$
$811$	8.65278e6			0.461959			0.230979	+	0.972959i	$0.425807\pi$
$812$		0			0
$813$	8.57662e6			0.455082
$814$		0			0
$815$	857606.	−	1.48542e6i	0.0452266	−	0.0783348i
$816$		0			0
$817$	−3.47040e6	+	2.00364e6i	−0.181897	+	0.105018i
$818$		0			0
$819$	−635146.	+	5.38875e6i	−0.0330875	+	0.280723i
$820$		0			0
$821$	−8.58238e6	−	1.48651e7i	−0.444375	−	0.769681i	0.553633	−	0.832761i	$-0.313241\pi$
$821$	−8.58238e6	−	1.48651e7i	−0.444375	−	0.769681i	−0.998008	+	0.0630799i	$0.979908\pi$
$822$		0			0
$823$	−2.69281e7	−	1.55470e7i	−1.38582	−	0.800104i	−0.392979	−	0.919547i	$-0.628556\pi$
$823$	−2.69281e7	−	1.55470e7i	−1.38582	−	0.800104i	−0.992841	+	0.119444i	$0.961889\pi$
$824$		0			0
$825$			7.03741e6i			0.359980i
$826$		0			0
$827$		−	1.58173e7i		−	0.804208i	−0.915594	−	0.402104i	$-0.868279\pi$
$827$		−	1.58173e7i		−	0.804208i	0.915594	−	0.402104i	$-0.131721\pi$
$828$		0			0
$829$	−1.16685e7	−	6.73682e6i	−0.589697	−	0.340462i	0.175281	−	0.984519i	$-0.443917\pi$
$829$	−1.16685e7	−	6.73682e6i	−0.589697	−	0.340462i	−0.764978	+	0.644057i	$0.777250\pi$
$830$		0			0
$831$	37938.8	+	65711.9i	0.00190582	+	0.00330097i
$832$		0			0
$833$	23752.6	−	99361.8i	0.00118604	−	0.00496143i
$834$		0			0
$835$	−366617.	+	211666.i	−0.0181969	+	0.0105060i
$836$		0			0
$837$	2.35351e6	−	4.07640e6i	0.116119	−	0.201124i
$838$		0			0
$839$	−8.68023e6			−0.425722			−0.212861	−	0.977082i	$-0.568278\pi$
$839$	−8.68023e6			−0.425722			−0.212861	+	0.977082i	$0.568278\pi$
$840$		0			0
$841$	−1.44906e7			−0.706476
$842$		0			0
$843$	−3.10018e6	+	5.36967e6i	−0.150251	+	0.260243i
$844$		0			0
$845$	624431.	−	360515.i	0.0300845	−	0.0173693i
$846$		0			0
$847$	−1.24220e7	−	1.46413e6i	−0.594955	−	0.0701246i
$848$		0			0
$849$	−2.35182e6	−	4.07346e6i	−0.111978	−	0.193952i
$850$		0			0
$851$	−1.50370e7	−	8.68162e6i	−0.711767	−	0.410939i
$852$		0			0
$853$			3.25522e7i			1.53182i	0.642949	+	0.765909i	$0.277711\pi$
$853$			3.25522e7i			1.53182i	−0.642949	+	0.765909i	$0.722289\pi$
$854$		0			0
$855$		−	257181.i		−	0.0120316i
$856$		0			0
$857$	2.38083e7	+	1.37457e7i	1.10733	+	0.639317i	0.938136	−	0.346267i	$-0.112551\pi$
$857$	2.38083e7	+	1.37457e7i	1.10733	+	0.639317i	0.169192	+	0.985583i	$0.445884\pi$
$858$		0			0
$859$	−1.32093e7	−	2.28791e7i	−0.610795	−	1.05793i	−0.991107	−	0.133070i	$-0.957516\pi$
$859$	−1.32093e7	−	2.28791e7i	−0.610795	−	1.05793i	0.380311	−	0.924859i	$-0.375817\pi$
$860$		0			0
$861$	−2.14449e7	+	9.22580e6i	−0.985862	+	0.424127i
$862$		0			0
$863$	3.01761e7	−	1.74222e7i	1.37923	−	0.796297i	0.387160	−	0.922013i	$-0.373456\pi$
$863$	3.01761e7	−	1.74222e7i	1.37923	−	0.796297i	0.992066	+	0.125716i	$0.0401228\pi$
$864$		0			0
$865$	−1.94224e6	+	3.36405e6i	−0.0882596	+	0.152870i
$866$		0			0
$867$	1.27784e7			0.577335
$868$		0			0
$869$	1.08525e7			0.487505
$870$		0			0
$871$	−1.22820e7	+	2.12731e7i	−0.548561	+	0.950136i
$872$		0			0
$873$	4.40393e6	−	2.54261e6i	0.195571	−	0.112913i
$874$		0			0
$875$	4.45560e6	+	3.32378e6i	0.196737	+	0.146762i
$876$		0			0
$877$	−1.75277e7	−	3.03588e7i	−0.769530	−	1.33287i	−0.937818	−	0.347127i	$-0.887157\pi$
$877$	−1.75277e7	−	3.03588e7i	−0.769530	−	1.33287i	0.168288	−	0.985738i	$-0.446176\pi$
$878$		0			0
$879$	−1.93964e7	−	1.11985e7i	−0.846738	−	0.488864i
$880$		0			0
$881$			3.38281e7i			1.46838i	0.678946	+	0.734188i	$0.262437\pi$
$881$			3.38281e7i			1.46838i	−0.678946	+	0.734188i	$0.737563\pi$
$882$		0			0
$883$			1.62781e7i			0.702588i	0.936265	+	0.351294i	$0.114258\pi$
$883$			1.62781e7i			0.702588i	−0.936265	+	0.351294i	$0.885742\pi$
$884$		0			0
$885$	−1.75468e6	−	1.01307e6i	−0.0753079	−	0.0434790i
$886$		0			0
$887$	−1.59326e7	−	2.75960e7i	−0.679950	−	1.17771i	−0.974995	−	0.222225i	$-0.928668\pi$
$887$	−1.59326e7	−	2.75960e7i	−0.679950	−	1.17771i	0.295045	−	0.955483i	$-0.404665\pi$
$888$		0			0
$889$	−9.64164e6	−	7.19246e6i	−0.409163	−	0.305227i
$890$		0			0
$891$	1.44383e6	−	833593.i	0.0609285	−	0.0351771i
$892$		0			0
$893$	6.00232e6	−	1.03963e7i	0.251878	−	0.436265i
$894$		0			0
$895$	1.82762e6			0.0762656
$896$		0			0
$897$	−9.90408e6			−0.410992
$898$		0			0
$899$	−7.92146e6	+	1.37204e7i	−0.326893	+	0.566196i
$900$		0			0
$901$	−2460.45	+	1420.54i	−0.000100973	+	5.82965e-5i
$902$		0			0
$903$	−9.35181e6	+	4.02324e6i	−0.381660	+	0.164194i
$904$		0			0
$905$	767202.	+	1.32883e6i	0.0311378	+	0.0539323i
$906$		0			0
$907$	−6.52918e6	−	3.76962e6i	−0.263536	−	0.152153i	0.362410	−	0.932019i	$-0.381954\pi$
$907$	−6.52918e6	−	3.76962e6i	−0.263536	−	0.152153i	−0.625947	+	0.779866i	$0.715287\pi$
$908$		0			0
$909$			1.01420e6i			0.0407113i
$910$		0			0
$911$		−	6.36943e6i		−	0.254276i	−0.991885	−	0.127138i	$-0.959421\pi$
$911$		−	6.36943e6i		−	0.254276i	0.991885	−	0.127138i	$-0.0405790\pi$
$912$		0			0
$913$	1.47697e7	+	8.52729e6i	0.586401	+	0.338559i
$914$		0			0
$915$	−638644.	−	1.10616e6i	−0.0252178	−	0.0436784i
$916$		0			0
$917$	−3.66878e6	−	432422.i	−0.144078	−	0.0169818i
$918$		0			0
$919$	−1.84757e7	+	1.06669e7i	−0.721625	+	0.416630i	−0.815350	−	0.578968i	$-0.803456\pi$
$919$	−1.84757e7	+	1.06669e7i	−0.721625	+	0.416630i	0.0937255	+	0.995598i	$0.470122\pi$
$920$		0			0
$921$	−2.00101e6	+	3.46585e6i	−0.0777321	+	0.134636i
$922$		0			0
$923$	−2.70983e7			−1.04698
$924$		0			0
$925$	−2.50882e7			−0.964085
$926$		0			0
$927$	−6.25224e6	+	1.08292e7i	−0.238966	+	0.413902i
$928$		0			0
$929$	−1.56736e7	+	9.04915e6i	−0.595839	+	0.344008i	−0.767403	−	0.641165i	$-0.778451\pi$
$929$	−1.56736e7	+	9.04915e6i	−0.595839	+	0.344008i	0.171564	+	0.985173i	$0.445118\pi$
$930$		0			0
$931$	5.30792e6	−	5.60428e6i	0.200701	−	0.211907i
$932$		0			0
$933$	1.19833e6	+	2.07556e6i	0.0450683	+	0.0780606i
$934$		0			0
$935$	−9247.59	−	5339.10i	−0.000345939	−	0.000199728i
$936$		0			0
$937$			4.52982e7i			1.68551i	0.538297	+	0.842755i	$0.319068\pi$
$937$			4.52982e7i			1.68551i	−0.538297	+	0.842755i	$0.680932\pi$
$938$		0			0
$939$			4.00128e6i			0.148093i
$940$		0			0
$941$	−4.58805e6	−	2.64891e6i	−0.168909	−	0.0975199i	0.413162	−	0.910658i	$-0.364424\pi$
$941$	−4.58805e6	−	2.64891e6i	−0.168909	−	0.0975199i	−0.582072	+	0.813138i	$0.697758\pi$
$942$		0			0
$943$	−2.13058e7	−	3.69028e7i	−0.780224	−	1.35139i
$944$		0			0
$945$	76480.4	−	648880.i	0.00278593	−	0.0236366i
$946$		0			0
$947$	−3.56706e7	+	2.05945e7i	−1.29252	+	0.746235i	−0.979100	−	0.203381i	$-0.934807\pi$
$947$	−3.56706e7	+	2.05945e7i	−1.29252	+	0.746235i	−0.313417	+	0.949616i	$0.601474\pi$
$948$		0			0
$949$	3.85748e6	−	6.68135e6i	0.139039	−	0.240823i
$950$		0			0
$951$	−5.20651e6			−0.186679
$952$		0			0
$953$	3.99749e7			1.42579			0.712894	−	0.701272i	$-0.247384\pi$
$953$	3.99749e7			1.42579			0.712894	+	0.701272i	$0.247384\pi$
$954$		0			0
$955$	1.44955e6	−	2.51070e6i	0.0514310	−	0.0890811i
$956$		0			0
$957$	−4.85963e6	+	2.80571e6i	−0.171524	+	0.0990292i
$958$		0			0
$959$	1.31339e7	+	3.05290e7i	0.461154	+	1.07193i
$960$		0			0
$961$	−6.53070e6	−	1.13115e7i	−0.228114	−	0.395104i
$962$		0			0
$963$	1.70352e6	+	983530.i	0.0591947	+	0.0341760i
$964$		0			0
$965$		−	5.23001e6i		−	0.180794i
$966$		0			0
$967$		−	4.74154e7i		−	1.63062i	−0.579024	−	0.815311i	$-0.696566\pi$
$967$		−	4.74154e7i		−	1.63062i	0.579024	−	0.815311i	$-0.303434\pi$
$968$		0			0
$969$	−21758.9	−	12562.5i	−0.000744436	−	0.000429800i
$970$		0			0
$971$	−2.03781e7	−	3.52959e7i	−0.693611	−	1.20137i	−0.970647	−	0.240509i	$-0.922686\pi$
$971$	−2.03781e7	−	3.52959e7i	−0.693611	−	1.20137i	0.277036	−	0.960860i	$-0.410648\pi$
$972$		0			0
$973$	−1.34028e7	+	1.79667e7i	−0.453851	+	0.608397i
$974$		0			0
$975$	−1.23932e7	+	7.15521e6i	−0.417515	+	0.241052i
$976$		0			0
$977$	−1.39288e7	+	2.41254e7i	−0.466850	+	0.808608i	−0.999283	−	0.0378641i	$-0.987945\pi$
$977$	−1.39288e7	+	2.41254e7i	−0.466850	+	0.808608i	0.532433	+	0.846472i	$0.321278\pi$
$978$		0			0
$979$	6.94618e6			0.231627
$980$		0			0
$981$	9.20298e6			0.305321
$982$		0			0
$983$	2.20200e7	−	3.81398e7i	0.726832	−	1.25891i	−0.231384	−	0.972862i	$-0.574325\pi$
$983$	2.20200e7	−	3.81398e7i	0.726832	−	1.25891i	0.958216	−	0.286047i	$-0.0923413\pi$
$984$		0			0
$985$	−2.74676e6	+	1.58584e6i	−0.0902048	+	0.0520798i
$986$		0			0
$987$	1.82357e7	−	2.44454e7i	0.595841	−	0.798737i
$988$		0			0
$989$	−9.29116e6	−	1.60928e7i	−0.302050	−	0.523167i
$990$		0			0
$991$	1.70732e7	+	9.85721e6i	0.552243	+	0.318838i	0.750026	−	0.661408i	$-0.230041\pi$
$991$	1.70732e7	+	9.85721e6i	0.552243	+	0.318838i	−0.197783	+	0.980246i	$0.563374\pi$
$992$		0			0
$993$			6.69631e6i			0.215507i
$994$		0			0
$995$		−	1.83463e6i		−	0.0587478i
$996$		0			0
$997$	−1.15452e7	−	6.66560e6i	−0.367842	−	0.212374i	0.304673	−	0.952457i	$-0.401453\pi$
$997$	−1.15452e7	−	6.66560e6i	−0.367842	−	0.212374i	−0.672515	+	0.740083i	$0.734786\pi$
$998$		0			0
$999$	2.97174e6	+	5.14720e6i	0.0942099	+	0.163176i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.bl.e.31.4	✓	14
4.3	odd	2		336.6.bl.g.31.4	yes	14
7.5	odd	6		336.6.bl.g.271.4	yes	14
28.19	even	6	inner	336.6.bl.e.271.4	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
336.6.bl.e.31.4	✓	14	1.1	even	1	trivial
336.6.bl.e.271.4	yes	14	28.19	even	6	inner
336.6.bl.g.31.4	yes	14	4.3	odd	2
336.6.bl.g.271.4	yes	14	7.5	odd	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 336 = 2^{4} \cdot 3 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 6 \)
Character orbit:	\([\chi]\)	\(=\)	336.bl (of order \(6\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(-4.50000 + 7.79423i) q^{3} +(5.98712 - 3.45667i) q^{5} +(77.5174 - 103.914i) q^{7} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)	\(q+(-4.50000 + 7.79423i) q^{3} +(5.98712 - 3.45667i) q^{5} +(77.5174 - 103.914i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(-220.062 - 127.053i) q^{11} -516.718i q^{13} +62.2200i q^{15} +(5.26414 + 3.03925i) q^{17} +(229.635 + 397.739i) q^{19} +(461.098 + 1071.80i) q^{21} +(-1844.37 + 1064.85i) q^{23} +(-1538.60 + 2664.94i) q^{25} +729.000 q^{27} -2453.67 q^{29} +(3228.41 - 5591.77i) q^{31} +(1980.56 - 1143.47i) q^{33} +(104.911 - 890.096i) q^{35} +(4076.46 + 7060.63i) q^{37} +(4027.42 + 2325.23i) q^{39} +20008.3i q^{41} +8725.33i q^{43} +(-484.957 - 279.990i) q^{45} +(-13069.3 - 22636.7i) q^{47} +(-4789.10 - 16110.2i) q^{49} +(-47.3772 + 27.3533i) q^{51} +(-233.700 + 404.780i) q^{53} -1756.72 q^{55} -4133.42 q^{57} +(-16282.0 + 28201.3i) q^{59} +(-17778.3 + 10264.3i) q^{61} +(-10428.8 - 1229.19i) q^{63} +(-1786.12 - 3093.65i) q^{65} +(-41169.7 - 23769.3i) q^{67} -19167.3i q^{69} -52443.1i q^{71} +(12930.4 + 7465.35i) q^{73} +(-13847.4 - 23984.4i) q^{75} +(-30261.1 + 13018.6i) q^{77} +(-36986.7 + 21354.3i) q^{79} +(-3280.50 + 5681.99i) q^{81} -67116.1 q^{83} +42.0227 q^{85} +(11041.5 - 19124.5i) q^{87} +(-23673.5 + 13667.9i) q^{89} +(-53694.0 - 40054.6i) q^{91} +(29055.7 + 50325.9i) q^{93} +(2749.70 + 1587.54i) q^{95} +62780.5i q^{97} +20582.5i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 63 q^{3} - 33 q^{5} + 13 q^{7} - 567 q^{9}+O(q^{10}) \) 14 * q - 63 * q^3 - 33 * q^5 + 13 * q^7 - 567 * q^9	\( 14 q - 63 q^{3} - 33 q^{5} + 13 q^{7} - 567 q^{9} + 549 q^{11} - 870 q^{17} + 146 q^{19} - 1773 q^{21} + 6396 q^{23} + 5134 q^{25} + 10206 q^{27} + 16458 q^{29} - 3137 q^{31} - 4941 q^{33} - 20910 q^{35} + 2162 q^{37} + 2376 q^{39} + 2673 q^{45} - 720 q^{47} + 23477 q^{49} + 7830 q^{51} - 44835 q^{53} - 72666 q^{55} - 2628 q^{57} - 44841 q^{59} + 93060 q^{61} + 14904 q^{63} - 4884 q^{65} + 68484 q^{67} - 163632 q^{73} + 46206 q^{75} + 99681 q^{77} + 134205 q^{79} - 45927 q^{81} + 188634 q^{83} + 276972 q^{85} - 74061 q^{87} - 49302 q^{89} + 110976 q^{91} - 28233 q^{93} + 60222 q^{95}+O(q^{100}) \) 14 * q - 63 * q^3 - 33 * q^5 + 13 * q^7 - 567 * q^9 + 549 * q^11 - 870 * q^17 + 146 * q^19 - 1773 * q^21 + 6396 * q^23 + 5134 * q^25 + 10206 * q^27 + 16458 * q^29 - 3137 * q^31 - 4941 * q^33 - 20910 * q^35 + 2162 * q^37 + 2376 * q^39 + 2673 * q^45 - 720 * q^47 + 23477 * q^49 + 7830 * q^51 - 44835 * q^53 - 72666 * q^55 - 2628 * q^57 - 44841 * q^59 + 93060 * q^61 + 14904 * q^63 - 4884 * q^65 + 68484 * q^67 - 163632 * q^73 + 46206 * q^75 + 99681 * q^77 + 134205 * q^79 - 45927 * q^81 + 188634 * q^83 + 276972 * q^85 - 74061 * q^87 - 49302 * q^89 + 110976 * q^91 - 28233 * q^93 + 60222 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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