LMFDB - Embedded newform 336.6.q.i.193.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [336,6,Mod(193,336)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [8,0,-36,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	no
Analytic conductor:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 703x^{6} + 2770x^{5} + 427565x^{4} + 718170x^{3} + 42175732x^{2} - 40929504x + 3559792896 $ x^8 - 2x^7 + 703x^6 + 2770x^5 + 427565x^4 + 718170x^3 + 42175732x^2 - 40929504*x + 3559792896
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{3}\cdot 7 $
Twist minimal:	no (minimal twist has level 84)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			193.1
Root			$4.59067 - 7.95128i$ of defining polynomial
Character	$\chi$	$=$	336.193
Dual form			336.6.q.i.289.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.50000 + 7.79423i) q^{3} +(-39.3359 - 68.1317i) q^{5} +(100.606 - 81.7641i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(345.759 - 598.873i) q^{11} +818.732 q^{13} +708.046 q^{15} +(-554.638 + 960.661i) q^{17} +(-286.523 - 496.273i) q^{19} +(184.559 + 1152.09i) q^{21} +(1258.80 + 2180.30i) q^{23} +(-1532.12 + 2653.71i) q^{25} +729.000 q^{27} -3258.19 q^{29} +(5059.55 - 8763.40i) q^{31} +(3111.84 + 5389.86i) q^{33} +(-9528.17 - 3638.23i) q^{35} +(-2434.81 - 4217.21i) q^{37} +(-3684.30 + 6381.39i) q^{39} -13094.3 q^{41} +9303.64 q^{43} +(-3186.21 + 5518.67i) q^{45} +(6452.90 + 11176.8i) q^{47} +(3436.28 - 16452.0i) q^{49} +(-4991.74 - 8645.95i) q^{51} +(9770.12 - 16922.3i) q^{53} -54403.0 q^{55} +5157.42 q^{57} +(-12560.2 + 21755.0i) q^{59} +(15681.1 + 27160.5i) q^{61} +(-9810.15 - 3745.90i) q^{63} +(-32205.6 - 55781.7i) q^{65} +(27971.9 - 48448.8i) q^{67} -22658.4 q^{69} -20501.0 q^{71} +(33825.0 - 58586.6i) q^{73} +(-13789.1 - 23883.4i) q^{75} +(-14180.7 - 88521.1i) q^{77} +(7039.95 + 12193.5i) q^{79} +(-3280.50 + 5681.99i) q^{81} +77129.1 q^{83} +87268.6 q^{85} +(14661.9 - 25395.1i) q^{87} +(-160.396 - 277.815i) q^{89} +(82369.7 - 66942.9i) q^{91} +(45536.0 + 78870.6i) q^{93} +(-22541.3 + 39042.6i) q^{95} -112009. q^{97} -56013.0 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 36 q^{3} + 42 q^{7} - 324 q^{9} + 462 q^{11} - 1204 q^{13} + 228 q^{17} - 358 q^{19} + 1404 q^{21} + 2148 q^{23} - 5454 q^{25} + 5832 q^{27} - 11064 q^{29} - 830 q^{31} + 4158 q^{33} - 7692 q^{35}+ \cdots - 74844 q^{99}+O(q^{100}) $ 8 * q - 36 * q^3 + 42 * q^7 - 324 * q^9 + 462 * q^11 - 1204 * q^13 + 228 * q^17 - 358 * q^19 + 1404 * q^21 + 2148 * q^23 - 5454 * q^25 + 5832 * q^27 - 11064 * q^29 - 830 * q^31 + 4158 * q^33 - 7692 * q^35 - 3914 * q^37 + 5418 * q^39 - 16632 * q^41 + 29036 * q^43 - 41700 * q^47 + 41876 * q^49 + 2052 * q^51 + 22164 * q^53 - 7784 * q^55 + 6444 * q^57 - 32886 * q^59 + 83732 * q^61 - 16038 * q^63 - 93192 * q^65 + 80034 * q^67 - 38664 * q^69 - 89544 * q^71 - 22470 * q^73 - 49086 * q^75 + 138732 * q^77 + 75286 * q^79 - 26244 * q^81 + 34836 * q^83 + 278504 * q^85 + 49788 * q^87 + 28944 * q^89 + 12058 * q^91 - 7470 * q^93 + 144120 * q^95 - 433356 * q^97 - 74844 * q^99

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
$5$	−39.3359	−	68.1317i	−0.703662	−	1.21878i	−0.967172	−	0.254121i	$-0.918214\pi$
$5$	−39.3359	−	68.1317i	−0.703662	−	1.21878i	0.263511	−	0.964656i	$-0.415120\pi$
$6$		0			0
$7$	100.606	−	81.7641i	0.776033	−	0.630692i
$7$	100.606	−	81.7641i	0.776033	−	0.630692i
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	345.759	−	598.873i	0.861574	−	1.49229i	−0.00883597	−	0.999961i	$-0.502813\pi$
$11$	345.759	−	598.873i	0.861574	−	1.49229i	0.870410	−	0.492328i	$-0.163854\pi$
$12$		0			0
$13$	818.732			1.34364			0.671821	−	0.740714i	$-0.265512\pi$
$13$	818.732			1.34364			0.671821	+	0.740714i	$0.265512\pi$
$14$		0			0
$15$	708.046			0.812518
$16$		0			0
$17$	−554.638	+	960.661i	−0.465465	+	0.806209i	−0.999222	−	0.0394286i	$-0.987446\pi$
$17$	−554.638	+	960.661i	−0.465465	+	0.806209i	0.533757	+	0.845638i	$0.320780\pi$
$18$		0			0
$19$	−286.523	−	496.273i	−0.182086	−	0.315382i	0.760505	−	0.649332i	$-0.224951\pi$
$19$	−286.523	−	496.273i	−0.182086	−	0.315382i	−0.942591	+	0.333951i	$0.891618\pi$
$20$		0			0
$21$	184.559	+	1152.09i	0.0913245	+	0.570082i
$22$		0			0
$23$	1258.80	+	2180.30i	0.496177	+	0.859403i	0.999990	−	0.00440926i	$-0.00140352\pi$
$23$	1258.80	+	2180.30i	0.496177	+	0.859403i	−0.503814	+	0.863812i	$0.668070\pi$
$24$		0			0
$25$	−1532.12	+	2653.71i	−0.490279	+	0.849189i
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	−3258.19			−0.719419			−0.359710	−	0.933064i	$-0.617124\pi$
$29$	−3258.19			−0.719419			−0.359710	+	0.933064i	$0.617124\pi$
$30$		0			0
$31$	5059.55	−	8763.40i	0.945601	−	1.63783i	0.191056	−	0.981579i	$-0.438809\pi$
$31$	5059.55	−	8763.40i	0.945601	−	1.63783i	0.754544	−	0.656249i	$-0.227858\pi$
$32$		0			0
$33$	3111.84	+	5389.86i	0.497430	+	0.861574i
$34$		0			0
$35$	−9528.17	−	3638.23i	−1.31474	−	0.502018i
$36$		0			0
$37$	−2434.81	−	4217.21i	−0.292389	−	0.506432i	0.681985	−	0.731366i	$-0.261117\pi$
$37$	−2434.81	−	4217.21i	−0.292389	−	0.506432i	−0.974374	+	0.224934i	$0.927783\pi$
$38$		0			0
$39$	−3684.30	+	6381.39i	−0.387876	+	0.671821i
$40$		0			0
$41$	−13094.3			−1.21653			−0.608265	−	0.793734i	$-0.708134\pi$
$41$	−13094.3			−1.21653			−0.608265	+	0.793734i	$0.708134\pi$
$42$		0			0
$43$	9303.64			0.767329			0.383664	−	0.923473i	$-0.374662\pi$
$43$	9303.64			0.767329			0.383664	+	0.923473i	$0.374662\pi$
$44$		0			0
$45$	−3186.21	+	5518.67i	−0.234554	+	0.406259i
$46$		0			0
$47$	6452.90	+	11176.8i	0.426099	+	0.738025i	0.996522	−	0.0833256i	$-0.0265542\pi$
$47$	6452.90	+	11176.8i	0.426099	+	0.738025i	−0.570423	+	0.821351i	$0.693221\pi$
$48$		0			0
$49$	3436.28	−	16452.0i	0.204455	−	0.978876i
$50$		0			0
$51$	−4991.74	−	8645.95i	−0.268736	−	0.465465i
$52$		0			0
$53$	9770.12	−	16922.3i	0.477760	−	0.827505i	−0.521915	−	0.852998i	$-0.674782\pi$
$53$	9770.12	−	16922.3i	0.477760	−	0.827505i	0.999675	+	0.0254926i	$0.00811542\pi$
$54$		0			0
$55$	−54403.0			−2.42503
$56$		0			0
$57$	5157.42			0.210254
$58$		0			0
$59$	−12560.2	+	21755.0i	−0.469751	+	0.813632i	−0.999402	−	0.0345835i	$-0.988990\pi$
$59$	−12560.2	+	21755.0i	−0.469751	+	0.813632i	0.529651	+	0.848216i	$0.322323\pi$
$60$		0			0
$61$	15681.1	+	27160.5i	0.539575	+	0.934571i	0.998927	+	0.0463168i	$0.0147484\pi$
$61$	15681.1	+	27160.5i	0.539575	+	0.934571i	−0.459352	+	0.888254i	$0.651918\pi$
$62$		0			0
$63$	−9810.15	−	3745.90i	−0.311404	−	0.118906i
$64$		0			0
$65$	−32205.6	−	55781.7i	−0.945469	−	1.63760i
$66$		0			0
$67$	27971.9	−	48448.8i	0.761264	−	1.31855i	−0.180936	−	0.983495i	$-0.557913\pi$
$67$	27971.9	−	48448.8i	0.761264	−	1.31855i	0.942200	−	0.335052i	$-0.108754\pi$
$68$		0			0
$69$	−22658.4			−0.572935
$70$		0			0
$71$	−20501.0			−0.482647			−0.241323	−	0.970445i	$-0.577581\pi$
$71$	−20501.0			−0.482647			−0.241323	+	0.970445i	$0.577581\pi$
$72$		0			0
$73$	33825.0	−	58586.6i	0.742900	−	1.28674i	−0.208270	−	0.978071i	$-0.566783\pi$
$73$	33825.0	−	58586.6i	0.742900	−	1.28674i	0.951170	−	0.308669i	$-0.0998835\pi$
$74$		0			0
$75$	−13789.1	−	23883.4i	−0.283063	−	0.490279i
$76$		0			0
$77$	−14180.7	−	88521.1i	−0.272565	−	1.70145i
$78$		0			0
$79$	7039.95	+	12193.5i	0.126912	+	0.219818i	0.922479	−	0.386048i	$-0.126160\pi$
$79$	7039.95	+	12193.5i	0.126912	+	0.219818i	−0.795567	+	0.605866i	$0.792827\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	77129.1			1.22892			0.614459	−	0.788949i	$-0.289374\pi$
$83$	77129.1			1.22892			0.614459	+	0.788949i	$0.289374\pi$
$84$		0			0
$85$	87268.6			1.31012
$86$		0			0
$87$	14661.9	−	25395.1i	0.207678	−	0.359710i
$88$		0			0
$89$	−160.396	−	277.815i	−0.00214644	−	0.00371775i	0.864950	−	0.501858i	$-0.167350\pi$
$89$	−160.396	−	277.815i	−0.00214644	−	0.00371775i	−0.867097	+	0.498140i	$0.834017\pi$
$90$		0			0
$91$	82369.7	−	66942.9i	1.04271	−	0.847424i
$92$		0			0
$93$	45536.0	+	78870.6i	0.545943	+	0.945601i
$94$		0			0
$95$	−22541.3	+	39042.6i	−0.256253	+	0.443844i
$96$		0			0
$97$	−112009.			−1.20871			−0.604355	−	0.796715i	$-0.706569\pi$
$97$	−112009.			−1.20871			−0.604355	+	0.796715i	$0.706569\pi$
$98$		0			0
$99$	−56013.0			−0.574382
$100$		0			0
$101$	−33627.0	+	58243.7i	−0.328008	+	0.568127i	−0.982117	−	0.188274i	$-0.939711\pi$
$101$	−33627.0	+	58243.7i	−0.328008	+	0.568127i	0.654108	+	0.756401i	$0.273044\pi$
$102$		0			0
$103$	−58317.6	−	101009.i	−0.541635	−	0.938140i	−0.998810	−	0.0487630i	$-0.984472\pi$
$103$	−58317.6	−	101009.i	−0.541635	−	0.938140i	0.457175	−	0.889377i	$-0.348861\pi$
$104$		0			0
$105$	71233.9	−	57892.7i	0.630541	−	0.512449i
$106$		0			0
$107$	−80322.6	−	139123.i	−0.678232	−	1.17473i	−0.975513	−	0.219942i	$-0.929413\pi$
$107$	−80322.6	−	139123.i	−0.678232	−	1.17473i	0.297281	−	0.954790i	$-0.403920\pi$
$108$		0			0
$109$	−59668.3	+	103349.i	−0.481036	+	0.833179i	−0.999763	−	0.0217610i	$-0.993073\pi$
$109$	−59668.3	+	103349.i	−0.481036	+	0.833179i	0.518727	+	0.854940i	$0.326406\pi$
$110$		0			0
$111$	43826.6			0.337622
$112$		0			0
$113$	−110893.			−0.816972			−0.408486	−	0.912765i	$-0.633943\pi$
$113$	−110893.			−0.816972			−0.408486	+	0.912765i	$0.633943\pi$
$114$		0			0
$115$	99031.8	−	171528.i	0.698281	−	1.20946i
$116$		0			0
$117$	−33158.7	−	57432.5i	−0.223940	−	0.387876i
$118$		0			0
$119$	22747.4	+	141998.i	0.147253	+	0.919210i
$120$		0			0
$121$	−158574.	−	274658.i	−0.984618	−	1.70541i
$122$		0			0
$123$	58924.4	−	102060.i	0.351182	−	0.608265i
$124$		0			0
$125$	−4779.64			−0.0273603
$126$		0			0
$127$	−315184.			−1.73402			−0.867012	−	0.498287i	$-0.833962\pi$
$127$	−315184.			−1.73402			−0.867012	+	0.498287i	$0.833962\pi$
$128$		0			0
$129$	−41866.4	+	72514.7i	−0.221509	+	0.383664i
$130$		0			0
$131$	42832.3	+	74187.6i	0.218068	+	0.377705i	0.954217	−	0.299114i	$-0.0966911\pi$
$131$	42832.3	+	74187.6i	0.218068	+	0.377705i	−0.736149	+	0.676819i	$0.763358\pi$
$132$		0			0
$133$	−69403.3	−	26500.9i	−0.340213	−	0.129907i
$134$		0			0
$135$	−28675.9	−	49668.0i	−0.135420	−	0.234554i
$136$		0			0
$137$	16316.1	−	28260.3i	0.0742703	−	0.128640i	−0.826498	−	0.562939i	$-0.809671\pi$
$137$	16316.1	−	28260.3i	0.0742703	−	0.128640i	0.900769	+	0.434299i	$0.143004\pi$
$138$		0			0
$139$	−206590.			−0.906925			−0.453462	−	0.891275i	$-0.649811\pi$
$139$	−206590.			−0.906925			−0.453462	+	0.891275i	$0.649811\pi$
$140$		0			0
$141$	−116152.			−0.492017
$142$		0			0
$143$	283084.	−	490317.i	1.15765	−	2.00510i
$144$		0			0
$145$	128164.	+	221986.i	0.506228	+	0.876812i
$146$		0			0
$147$	112767.	+	100817.i	0.430417	+	0.384805i
$148$		0			0
$149$	−241855.	−	418906.i	−0.892463	−	1.54579i	−0.836913	−	0.547335i	$-0.815642\pi$
$149$	−241855.	−	418906.i	−0.892463	−	1.54579i	−0.0555495	−	0.998456i	$-0.517691\pi$
$150$		0			0
$151$	−83610.4	+	144818.i	−0.298413	+	0.516867i	−0.975773	−	0.218785i	$-0.929791\pi$
$151$	−83610.4	+	144818.i	−0.298413	+	0.516867i	0.677360	+	0.735652i	$0.263124\pi$
$152$		0			0
$153$	89851.3			0.310310
$154$		0			0
$155$	−796087.			−2.66153
$156$		0			0
$157$	7054.26	−	12218.3i	0.0228403	−	0.0395606i	−0.854379	−	0.519650i	$-0.826062\pi$
$157$	7054.26	−	12218.3i	0.0228403	−	0.0395606i	0.877220	+	0.480089i	$0.159396\pi$
$158$		0			0
$159$	87931.0	+	152301.i	0.275835	+	0.477760i
$160$		0			0
$161$	304913.	+	116428.i	0.927068	+	0.353991i
$162$		0			0
$163$	327188.	+	566706.i	0.964558	+	1.67066i	0.710797	+	0.703397i	$0.248335\pi$
$163$	327188.	+	566706.i	0.964558	+	1.67066i	0.253762	+	0.967267i	$0.418332\pi$
$164$		0			0
$165$	244814.	−	424030.i	0.700044	−	1.21251i
$166$		0			0
$167$	−182945.			−0.507610			−0.253805	−	0.967255i	$-0.581682\pi$
$167$	−182945.			−0.507610			−0.253805	+	0.967255i	$0.581682\pi$
$168$		0			0
$169$	299030.			0.805374
$170$		0			0
$171$	−23208.4	+	40198.1i	−0.0606952	+	0.105127i
$172$		0			0
$173$	146004.	+	252887.i	0.370894	+	0.642407i	0.989703	−	0.143134i	$-0.0457180\pi$
$173$	146004.	+	252887.i	0.370894	+	0.642407i	−0.618809	+	0.785541i	$0.712385\pi$
$174$		0			0
$175$	62837.2	+	392253.i	0.155103	+	0.968214i
$176$		0			0
$177$	−113042.	−	195795.i	−0.271211	−	0.469751i
$178$		0			0
$179$	97383.8	−	168674.i	0.227172	−	0.393473i	−0.729797	−	0.683664i	$-0.760385\pi$
$179$	97383.8	−	168674.i	0.227172	−	0.393473i	0.956969	+	0.290191i	$0.0937188\pi$
$180$		0			0
$181$	−256634.			−0.582261			−0.291130	−	0.956683i	$-0.594031\pi$
$181$	−256634.			−0.582261			−0.291130	+	0.956683i	$0.594031\pi$
$182$		0			0
$183$	−282260.			−0.623047
$184$		0			0
$185$	−191551.	+	331776.i	−0.411486	+	0.712714i
$186$		0			0
$187$	383542.	+	664315.i	0.802065	+	1.38922i
$188$		0			0
$189$	73342.0	−	59606.0i	0.149348	−	0.121377i
$190$		0			0
$191$	−146975.	−	254567.i	−0.291513	−	0.504916i	0.682654	−	0.730741i	$-0.260825\pi$
$191$	−146975.	−	254567.i	−0.291513	−	0.504916i	−0.974168	+	0.225825i	$0.927492\pi$
$192$		0			0
$193$	34565.0	−	59868.4i	0.0667950	−	0.115692i	−0.830694	−	0.556730i	$-0.812056\pi$
$193$	34565.0	−	59868.4i	0.0667950	−	0.115692i	0.897489	+	0.441037i	$0.145389\pi$
$194$		0			0
$195$	579700.			1.09173
$196$		0			0
$197$	331748.			0.609037			0.304518	−	0.952506i	$-0.401505\pi$
$197$	331748.			0.609037			0.304518	+	0.952506i	$0.401505\pi$
$198$		0			0
$199$	287214.	−	497470.i	0.514130	−	0.890500i	−0.485735	−	0.874106i	$-0.661448\pi$
$199$	287214.	−	497470.i	0.514130	−	0.890500i	0.999866	−	0.0163939i	$-0.00521858\pi$
$200$		0			0
$201$	251747.	+	436039.i	0.439516	+	0.761264i
$202$		0			0
$203$	−327795.	+	266403.i	−0.558293	+	0.453732i
$204$		0			0
$205$	515076.	+	892138.i	0.856025	+	1.48268i
$206$		0			0
$207$	101963.	−	176604.i	0.165392	−	0.286468i
$208$		0			0
$209$	−396272.			−0.627521
$210$		0			0
$211$	383999.			0.593777			0.296889	−	0.954912i	$-0.404051\pi$
$211$	383999.			0.593777			0.296889	+	0.954912i	$0.404051\pi$
$212$		0			0
$213$	92254.5	−	159790.i	0.139328	−	0.241323i
$214$		0			0
$215$	−365967.	−	633873.i	−0.539940	−	0.935203i
$216$		0			0
$217$	−207508.	−	1.29534e6i	−0.299148	−	1.86739i
$218$		0			0
$219$	304425.	+	527279.i	0.428913	+	0.742900i
$220$		0			0
$221$	−454100.	+	786524.i	−0.625418	+	1.08326i
$222$		0			0
$223$	347165.			0.467492			0.233746	−	0.972298i	$-0.424902\pi$
$223$	347165.			0.467492			0.233746	+	0.972298i	$0.424902\pi$
$224$		0			0
$225$	248204.			0.326853
$226$		0			0
$227$	180641.	−	312879.i	0.232676	−	0.403006i	−0.725919	−	0.687780i	$-0.758585\pi$
$227$	180641.	−	312879.i	0.232676	−	0.403006i	0.958595	+	0.284774i	$0.0919186\pi$
$228$		0			0
$229$	376492.	+	652102.i	0.474424	+	0.821726i	0.999571	−	0.0292852i	$-0.00932309\pi$
$229$	376492.	+	652102.i	0.474424	+	0.821726i	−0.525147	+	0.851011i	$0.675990\pi$
$230$		0			0
$231$	753767.	+	287818.i	0.929410	+	0.354885i
$232$		0			0
$233$	−357199.	−	618686.i	−0.431042	−	0.746587i	0.565921	−	0.824460i	$-0.308521\pi$
$233$	−357199.	−	618686.i	−0.431042	−	0.746587i	−0.996963	+	0.0778721i	$0.975187\pi$
$234$		0			0
$235$	507661.	−	879295.i	0.599659	−	1.03864i
$236$		0			0
$237$	−126719.			−0.146545
$238$		0			0
$239$	−343587.			−0.389083			−0.194541	−	0.980894i	$-0.562322\pi$
$239$	−343587.			−0.389083			−0.194541	+	0.980894i	$0.562322\pi$
$240$		0			0
$241$	38152.5	−	66082.0i	0.0423136	−	0.0732893i	−0.844093	−	0.536197i	$-0.819860\pi$
$241$	38152.5	−	66082.0i	0.0423136	−	0.0732893i	0.886407	+	0.462908i	$0.153194\pi$
$242$		0			0
$243$	−29524.5	−	51137.9i	−0.0320750	−	0.0555556i
$244$		0			0
$245$	−1.25607e6	+	413033.i	−1.33690	+	0.439612i
$246$		0			0
$247$	−234586.	−	406314.i	−0.244658	−	0.423760i
$248$		0			0
$249$	−347081.	+	601162.i	−0.354758	+	0.614459i
$250$		0			0
$251$	−196249.			−0.196618			−0.0983090	−	0.995156i	$-0.531343\pi$
$251$	−196249.			−0.196618			−0.0983090	+	0.995156i	$0.531343\pi$
$252$		0			0
$253$	1.74096e6			1.70997
$254$		0			0
$255$	−392709.	+	680192.i	−0.378199	+	0.655060i
$256$		0			0
$257$	854554.	+	1.48013e6i	0.807062	+	1.39787i	0.914890	+	0.403703i	$0.132277\pi$
$257$	854554.	+	1.48013e6i	0.807062	+	1.39787i	−0.107828	+	0.994170i	$0.534390\pi$
$258$		0			0
$259$	−589774.	−	225199.i	−0.546306	−	0.208601i
$260$		0			0
$261$	131957.	+	228556.i	0.119903	+	0.207678i
$262$		0			0
$263$	−395762.	+	685481.i	−0.352814	+	0.611091i	−0.986741	−	0.162301i	$-0.948108\pi$
$263$	−395762.	+	685481.i	−0.352814	+	0.611091i	0.633928	+	0.773392i	$0.281442\pi$
$264$		0			0
$265$	−1.53726e6			−1.34473
$266$		0			0
$267$	2887.13			0.00247850
$268$		0			0
$269$	−579502.	+	1.00373e6i	−0.488286	+	0.845736i	−0.999909	−	0.0134740i	$-0.995711\pi$
$269$	−579502.	+	1.00373e6i	−0.488286	+	0.845736i	0.511623	+	0.859210i	$0.329044\pi$
$270$		0			0
$271$	111567.	+	193240.i	0.0922811	+	0.159836i	0.908471	−	0.417949i	$-0.137251\pi$
$271$	111567.	+	193240.i	0.0922811	+	0.159836i	−0.816190	+	0.577784i	$0.803917\pi$
$272$		0			0
$273$	151105.	+	943251.i	0.122707	+	0.765986i
$274$		0			0
$275$	1.05949e6	+	1.83509e6i	0.844823	+	1.46328i
$276$		0			0
$277$	234523.	−	406205.i	0.183648	−	0.318087i	−0.759472	−	0.650540i	$-0.774543\pi$
$277$	234523.	−	406205.i	0.183648	−	0.318087i	0.943120	+	0.332452i	$0.107876\pi$
$278$		0			0
$279$	−819647.			−0.630400
$280$		0			0
$281$	−1.51447e6			−1.14418			−0.572090	−	0.820191i	$-0.693867\pi$
$281$	−1.51447e6			−1.14418			−0.572090	+	0.820191i	$0.693867\pi$
$282$		0			0
$283$	−535303.	+	927171.i	−0.397314	+	0.688167i	−0.993393	−	0.114758i	$-0.963391\pi$
$283$	−535303.	+	927171.i	−0.397314	+	0.688167i	0.596080	+	0.802925i	$0.296724\pi$
$284$		0			0
$285$	−202871.	−	351384.i	−0.147948	−	0.256253i
$286$		0			0
$287$	−1.31737e6	+	1.07064e6i	−0.944068	+	0.767256i
$288$		0			0
$289$	94682.6	+	163995.i	0.0666846	+	0.115501i
$290$		0			0
$291$	504039.	−	873021.i	0.348925	−	0.604355i
$292$		0			0
$293$	1.45450e6			0.989794			0.494897	−	0.868952i	$-0.335206\pi$
$293$	1.45450e6			0.989794			0.494897	+	0.868952i	$0.335206\pi$
$294$		0			0
$295$	1.97627e6			1.32218
$296$		0			0
$297$	252059.	−	436578.i	0.165810	−	0.287191i
$298$		0			0
$299$	1.03062e6	+	1.78508e6i	0.666684	+	1.15473i
$300$		0			0
$301$	936005.	−	760703.i	0.595473	−	0.483948i
$302$		0			0
$303$	−302643.	−	524193.i	−0.189376	−	0.328008i
$304$		0			0
$305$	1.23366e6	−	2.13676e6i	0.759356	−	1.31524i
$306$		0			0
$307$	2.23869e6			1.35565			0.677824	−	0.735224i	$-0.262923\pi$
$307$	2.23869e6			1.35565			0.677824	+	0.735224i	$0.262923\pi$
$308$		0			0
$309$	1.04972e6			0.625426
$310$		0			0
$311$	−750695.	+	1.30024e6i	−0.440111	+	0.762295i	−0.997697	−	0.0678240i	$-0.978394\pi$
$311$	−750695.	+	1.30024e6i	−0.440111	+	0.762295i	0.557586	+	0.830119i	$0.311728\pi$
$312$		0			0
$313$	890067.	+	1.54164e6i	0.513525	+	0.889452i	0.999877	+	0.0156888i	$0.00499410\pi$
$313$	890067.	+	1.54164e6i	0.513525	+	0.889452i	−0.486352	+	0.873763i	$0.661673\pi$
$314$		0			0
$315$	130676.	+	815731.i	0.0742028	+	0.463202i
$316$		0			0
$317$	−735500.	−	1.27392e6i	−0.411088	−	0.712025i	0.583921	−	0.811810i	$-0.301518\pi$
$317$	−735500.	−	1.27392e6i	−0.411088	−	0.712025i	−0.995009	+	0.0997854i	$0.968184\pi$
$318$		0			0
$319$	−1.12655e6	+	1.95124e6i	−0.619832	+	1.07358i
$320$		0			0
$321$	1.44581e6			0.783155
$322$		0			0
$323$	635666.			0.339018
$324$		0			0
$325$	−1.25440e6	+	2.17268e6i	−0.658760	+	1.14101i
$326$		0			0
$327$	−537015.	−	930137.i	−0.277726	−	0.481036i
$328$		0			0
$329$	1.56306e6	+	596837.i	0.796134	+	0.303995i
$330$		0			0
$331$	−914873.	−	1.58461e6i	−0.458977	−	0.794971i	0.539930	−	0.841710i	$-0.318451\pi$
$331$	−914873.	−	1.58461e6i	−0.458977	−	0.794971i	−0.998907	+	0.0467386i	$0.985117\pi$
$332$		0			0
$333$	−197220.	+	341594.i	−0.0974629	+	0.168811i
$334$		0			0
$335$	−4.40120e6			−2.14269
$336$		0			0
$337$	3.18627e6			1.52830			0.764148	−	0.645041i	$-0.223160\pi$
$337$	3.18627e6			1.52830			0.764148	+	0.645041i	$0.223160\pi$
$338$		0			0
$339$	499018.	−	864324.i	0.235840	−	0.408486i
$340$		0			0
$341$	−3.49878e6	−	6.06006e6i	−1.62941	−	2.82222i
$342$		0			0
$343$	−999468.	−	1.93614e6i	−0.458705	−	0.888589i
$344$		0			0
$345$	891286.	+	1.54375e6i	0.403153	+	0.698281i
$346$		0			0
$347$	1.23895e6	−	2.14592e6i	0.552370	−	0.956732i	−0.445733	−	0.895166i	$-0.647057\pi$
$347$	1.23895e6	−	2.14592e6i	0.552370	−	0.956732i	0.998103	−	0.0615664i	$-0.0196096\pi$
$348$		0			0
$349$	1.81720e6			0.798618			0.399309	−	0.916816i	$-0.369250\pi$
$349$	1.81720e6			0.798618			0.399309	+	0.916816i	$0.369250\pi$
$350$		0			0
$351$	596856.			0.258584
$352$		0			0
$353$	−1.57678e6	+	2.73106e6i	−0.673493	+	1.16652i	0.303413	+	0.952859i	$0.401874\pi$
$353$	−1.57678e6	+	2.73106e6i	−0.673493	+	1.16652i	−0.976907	+	0.213666i	$0.931460\pi$
$354$		0			0
$355$	806425.	+	1.39677e6i	0.339620	+	0.588239i
$356$		0			0
$357$	−1.20913e6	−	461692.i	−0.502113	−	0.191726i
$358$		0			0
$359$	1.35105e6	+	2.34008e6i	0.553266	+	0.958285i	0.998036	+	0.0626404i	$0.0199521\pi$
$359$	1.35105e6	+	2.34008e6i	0.553266	+	0.958285i	−0.444770	+	0.895645i	$0.646715\pi$
$360$		0			0
$361$	1.07386e6	−	1.85998e6i	0.433690	−	0.751173i
$362$		0			0
$363$	2.85433e6			1.13694
$364$		0			0
$365$	−5.32214e6			−2.09100
$366$		0			0
$367$	−1.80573e6	+	3.12762e6i	−0.699824	+	1.21213i	0.268703	+	0.963223i	$0.413405\pi$
$367$	−1.80573e6	+	3.12762e6i	−0.699824	+	1.21213i	−0.968527	+	0.248908i	$0.919928\pi$
$368$		0			0
$369$	530319.	+	918540.i	0.202755	+	0.351182i
$370$		0			0
$371$	−400703.	−	2.50134e6i	−0.151143	−	0.943491i
$372$		0			0
$373$	216759.	+	375438.i	0.0806688	+	0.139722i	0.903537	−	0.428509i	$-0.140961\pi$
$373$	216759.	+	375438.i	0.0806688	+	0.139722i	−0.822869	+	0.568232i	$0.807628\pi$
$374$		0			0
$375$	21508.4	−	37253.6i	0.00789823	−	0.0136801i
$376$		0			0
$377$	−2.66759e6			−0.966642
$378$		0			0
$379$	2.12163e6			0.758704			0.379352	−	0.925252i	$-0.376147\pi$
$379$	2.12163e6			0.758704			0.379352	+	0.925252i	$0.376147\pi$
$380$		0			0
$381$	1.41833e6	−	2.45662e6i	0.500570	−	0.867012i
$382$		0			0
$383$	−2.12339e6	−	3.67782e6i	−0.739662	−	1.28113i	−0.952647	−	0.304077i	$-0.901652\pi$
$383$	−2.12339e6	−	3.67782e6i	−0.739662	−	1.28113i	0.212985	−	0.977055i	$-0.431681\pi$
$384$		0			0
$385$	−5.47329e6	+	4.44821e6i	−1.88190	+	1.52944i
$386$		0			0
$387$	−376797.	−	652632.i	−0.127888	−	0.221509i
$388$		0			0
$389$	1.76088e6	−	3.04994e6i	0.590006	−	1.02192i	−0.404224	−	0.914660i	$-0.632459\pi$
$389$	1.76088e6	−	3.04994e6i	0.590006	−	1.02192i	0.994231	−	0.107261i	$-0.0342081\pi$
$390$		0			0
$391$	−2.79271e6			−0.923811
$392$		0			0
$393$	−770981.			−0.251804
$394$		0			0
$395$	553845.	−	959288.i	0.178606	−	0.309354i
$396$		0			0
$397$	367025.	+	635706.i	0.116874	+	0.202432i	0.918527	−	0.395357i	$-0.129379\pi$
$397$	367025.	+	635706.i	0.116874	+	0.202432i	−0.801653	+	0.597790i	$0.796046\pi$
$398$		0			0
$399$	518869.	−	421691.i	0.163164	−	0.132606i
$400$		0			0
$401$	111135.	+	192492.i	0.0345136	+	0.0597793i	0.882766	−	0.469812i	$-0.155678\pi$
$401$	111135.	+	192492.i	0.0345136	+	0.0597793i	−0.848253	+	0.529592i	$0.822345\pi$
$402$		0			0
$403$	4.14242e6	−	7.17488e6i	1.27055	−	2.20065i
$404$		0			0
$405$	516165.			0.156369
$406$		0			0
$407$	−3.36743e6			−1.00766
$408$		0			0
$409$	1.75379e6	−	3.03765e6i	0.518405	−	0.897904i	−0.481366	−	0.876520i	$-0.659859\pi$
$409$	1.75379e6	−	3.03765e6i	0.518405	−	0.897904i	0.999771	−	0.0213846i	$-0.00680745\pi$
$410$		0			0
$411$	146845.	+	254343.i	0.0428800	+	0.0742703i
$412$		0			0
$413$	515134.	+	3.21566e6i	0.148609	+	0.927674i
$414$		0			0
$415$	−3.03394e6	−	5.25494e6i	−0.864742	−	1.49778i
$416$		0			0
$417$	929653.	−	1.61021e6i	0.261807	−	0.453462i
$418$		0			0
$419$	1.97040e6			0.548302			0.274151	−	0.961687i	$-0.411603\pi$
$419$	1.97040e6			0.548302			0.274151	+	0.961687i	$0.411603\pi$
$420$		0			0
$421$	−7.19318e6			−1.97795			−0.988976	−	0.148079i	$-0.952691\pi$
$421$	−7.19318e6			−1.97795			−0.988976	+	0.148079i	$0.952691\pi$
$422$		0			0
$423$	522685.	−	905317.i	0.142033	−	0.246008i
$424$		0			0
$425$	−1.69955e6	−	2.94370e6i	−0.456416	−	0.790535i
$426$		0			0
$427$	3.79837e6	+	1.45036e6i	1.00815	+	0.384953i
$428$		0			0
$429$	2.54776e6	+	4.41285e6i	0.668367	+	1.15765i
$430$		0			0
$431$	262091.	−	453955.i	0.0679608	−	0.117712i	−0.830043	−	0.557700i	$-0.811684\pi$
$431$	262091.	−	453955.i	0.0679608	−	0.117712i	0.898004	+	0.439988i	$0.145017\pi$
$432$		0			0
$433$	1.49326e6			0.382750			0.191375	−	0.981517i	$-0.438705\pi$
$433$	1.49326e6			0.382750			0.191375	+	0.981517i	$0.438705\pi$
$434$		0			0
$435$	−2.30695e6			−0.584541
$436$		0			0
$437$	721349.	−	1.24941e6i	0.180693	−	0.312970i
$438$		0			0
$439$	1.32835e6	+	2.30077e6i	0.328966	+	0.569786i	0.982307	−	0.187278i	$-0.0599664\pi$
$439$	1.32835e6	+	2.30077e6i	0.328966	+	0.569786i	−0.653341	+	0.757064i	$0.726633\pi$
$440$		0			0
$441$	−1.29324e6	+	425256.i	−0.316653	+	0.104125i
$442$		0			0
$443$	−2.34399e6	−	4.05991e6i	−0.567474	−	0.982895i	−0.996815	−	0.0797518i	$-0.974587\pi$
$443$	−2.34399e6	−	4.05991e6i	−0.567474	−	0.982895i	0.429340	−	0.903143i	$-0.358746\pi$
$444$		0			0
$445$	−12618.7	+	21856.2i	−0.00302074	+	0.00523207i
$446$		0			0
$447$	4.35340e6			1.03053
$448$		0			0
$449$	−3.45899e6			−0.809716			−0.404858	−	0.914379i	$-0.632679\pi$
$449$	−3.45899e6			−0.809716			−0.404858	+	0.914379i	$0.632679\pi$
$450$		0			0
$451$	−4.52748e6	+	7.84182e6i	−1.04813	+	1.81541i
$452$		0			0
$453$	−752494.	−	1.30336e6i	−0.172289	−	0.298413i
$454$		0			0
$455$	−7.80102e6	−	2.97873e6i	−1.76654	−	0.674533i
$456$		0			0
$457$	−359140.	−	622048.i	−0.0804401	−	0.139326i	0.822999	−	0.568043i	$-0.192299\pi$
$457$	−359140.	−	622048.i	−0.0804401	−	0.139326i	−0.903439	+	0.428717i	$0.858966\pi$
$458$		0			0
$459$	−404331.	+	700322.i	−0.0895788	+	0.155155i
$460$		0			0
$461$	5.07998e6			1.11329			0.556647	−	0.830749i	$-0.312088\pi$
$461$	5.07998e6			1.11329			0.556647	+	0.830749i	$0.312088\pi$
$462$		0			0
$463$	−3.40607e6			−0.738416			−0.369208	−	0.929347i	$-0.620371\pi$
$463$	−3.40607e6			−0.738416			−0.369208	+	0.929347i	$0.620371\pi$
$464$		0			0
$465$	3.58239e6	−	6.20489e6i	0.768318	−	1.33077i
$466$		0			0
$467$	921342.	+	1.59581e6i	0.195492	+	0.338602i	0.947062	−	0.321052i	$-0.104036\pi$
$467$	921342.	+	1.59581e6i	0.195492	+	0.338602i	−0.751570	+	0.659654i	$0.770703\pi$
$468$		0			0
$469$	−1.14722e6	−	7.16135e6i	−0.240831	−	1.50336i
$470$		0			0
$471$	63488.3	+	109965.i	0.0131869	+	0.0228403i
$472$		0			0
$473$	3.21682e6	−	5.57170e6i	0.661110	−	1.14508i
$474$		0			0
$475$	1.75595e6			0.357091
$476$		0			0
$477$	−1.58276e6			−0.318507
$478$		0			0
$479$	4.55714e6	−	7.89319e6i	0.907514	−	1.57186i	0.0900071	−	0.995941i	$-0.471311\pi$
$479$	4.55714e6	−	7.89319e6i	0.907514	−	1.57186i	0.817507	−	0.575919i	$-0.195356\pi$
$480$		0			0
$481$	−1.99346e6	−	3.45277e6i	−0.392866	−	0.680464i
$482$		0			0
$483$	−2.27957e6	+	1.85264e6i	−0.444617	+	0.361346i
$484$		0			0
$485$	4.40596e6	+	7.63134e6i	0.850523	+	1.47315i
$486$		0			0
$487$	2.19220e6	−	3.79701e6i	0.418850	−	0.725469i	−0.576974	−	0.816762i	$-0.695767\pi$
$487$	2.19220e6	−	3.79701e6i	0.418850	−	0.725469i	0.995824	+	0.0912933i	$0.0291001\pi$
$488$		0			0
$489$	−5.88939e6			−1.11378
$490$		0			0
$491$	9.50592e6			1.77947			0.889735	−	0.456478i	$-0.150889\pi$
$491$	9.50592e6			1.77947			0.889735	+	0.456478i	$0.150889\pi$
$492$		0			0
$493$	1.80712e6	−	3.13002e6i	0.334864	−	0.580002i
$494$		0			0
$495$	2.20332e6	+	3.81627e6i	0.404171	+	0.700044i
$496$		0			0
$497$	−2.06253e6	+	1.67625e6i	−0.374550	+	0.304401i
$498$		0			0
$499$	1.68783e6	+	2.92341e6i	0.303444	+	0.525580i	0.976914	−	0.213635i	$-0.0685303\pi$
$499$	1.68783e6	+	2.92341e6i	0.303444	+	0.525580i	−0.673470	+	0.739215i	$0.735197\pi$
$500$		0			0
$501$	823255.	−	1.42592e6i	0.146535	−	0.253805i
$502$		0			0
$503$	−7.82645e6			−1.37926			−0.689628	−	0.724164i	$-0.742226\pi$
$503$	−7.82645e6			−1.37926			−0.689628	+	0.724164i	$0.742226\pi$
$504$		0			0
$505$	5.29099e6			0.923227
$506$		0			0
$507$	−1.34563e6	+	2.33070e6i	−0.232491	+	0.402687i
$508$		0			0
$509$	3.71219e6	+	6.42969e6i	0.635090	+	1.10001i	0.986496	+	0.163785i	$0.0523703\pi$
$509$	3.71219e6	+	6.42969e6i	0.635090	+	1.10001i	−0.351406	+	0.936223i	$0.614296\pi$
$510$		0			0
$511$	−1.38727e6	−	8.65985e6i	−0.235022	−	1.46709i
$512$		0			0
$513$	−208875.	−	361783.i	−0.0350424	−	0.0606952i
$514$		0			0
$515$	−4.58795e6	+	7.94656e6i	−0.762256	+	1.32027i
$516$		0			0
$517$	8.92461e6			1.46846
$518$		0			0
$519$	−2.62807e6			−0.428272
$520$		0			0
$521$	530710.	−	919217.i	0.0856570	−	0.148362i	−0.820014	−	0.572344i	$-0.806034\pi$
$521$	530710.	−	919217.i	0.0856570	−	0.148362i	0.905671	+	0.423981i	$0.139368\pi$
$522$		0			0
$523$	−956760.	−	1.65716e6i	−0.152950	−	0.264917i	0.779361	−	0.626575i	$-0.215544\pi$
$523$	−956760.	−	1.65716e6i	−0.152950	−	0.264917i	−0.932311	+	0.361659i	$0.882211\pi$
$524$		0			0
$525$	−3.34008e6	−	1.27537e6i	−0.528881	−	0.201948i
$526$		0			0
$527$	5.61243e6	+	9.72102e6i	0.880288	+	1.52470i
$528$		0			0
$529$	49029.1	−	84921.0i	0.00761755	−	0.0131940i
$530$		0			0
$531$	2.03476e6			0.313167
$532$		0			0
$533$	−1.07207e7			−1.63458
$534$		0			0
$535$	−6.31912e6	+	1.09450e7i	−0.954492	+	1.65323i
$536$		0			0
$537$	876454.	+	1.51806e6i	0.131158	+	0.227172i
$538$		0			0
$539$	−8.66451e6	−	7.74632e6i	−1.28461	−	1.14848i
$540$		0			0
$541$	−5.83140e6	−	1.01003e7i	−0.856603	−	1.48368i	−0.875150	−	0.483852i	$-0.839237\pi$
$541$	−5.83140e6	−	1.01003e7i	−0.856603	−	1.48368i	0.0185472	−	0.999828i	$-0.494096\pi$
$542$		0			0
$543$	1.15485e6	−	2.00026e6i	0.168084	−	0.291130i
$544$		0			0
$545$	9.38842e6			1.35395
$546$		0			0
$547$	6.36659e6			0.909785			0.454893	−	0.890546i	$-0.349678\pi$
$547$	6.36659e6			0.909785			0.454893	+	0.890546i	$0.349678\pi$
$548$		0			0
$549$	1.27017e6	−	2.20000e6i	0.179858	−	0.311524i
$550$		0			0
$551$	933548.	+	1.61695e6i	0.130996	+	0.226891i
$552$		0			0
$553$	1.70526e6	+	651134.i	0.237125	+	0.0905435i
$554$		0			0
$555$	−1.72396e6	−	2.98598e6i	−0.237571	−	0.411486i
$556$		0			0
$557$	−4.86221e6	+	8.42159e6i	−0.664042	+	1.15015i	0.315502	+	0.948925i	$0.397827\pi$
$557$	−4.86221e6	+	8.42159e6i	−0.664042	+	1.15015i	−0.979544	+	0.201230i	$0.935506\pi$
$558$		0			0
$559$	7.61719e6			1.03102
$560$		0			0
$561$	−6.90376e6			−0.926145
$562$		0			0
$563$	3.15133e6	−	5.45826e6i	0.419008	−	0.725744i	−0.576832	−	0.816863i	$-0.695711\pi$
$563$	3.15133e6	−	5.45826e6i	0.419008	−	0.725744i	0.995840	+	0.0911193i	$0.0290445\pi$
$564$		0			0
$565$	4.36207e6	+	7.55532e6i	0.574872	+	0.995708i
$566$		0			0
$567$	134544.	+	839872.i	0.0175754	+	0.109712i
$568$		0			0
$569$	−4.08244e6	−	7.07099e6i	−0.528614	−	0.915586i	−0.999443	−	0.0333620i	$-0.989379\pi$
$569$	−4.08244e6	−	7.07099e6i	−0.528614	−	0.915586i	0.470829	−	0.882224i	$-0.343955\pi$
$570$		0			0
$571$	−6.43857e6	+	1.11519e7i	−0.826416	+	1.43139i	0.0744158	+	0.997227i	$0.476291\pi$
$571$	−6.43857e6	+	1.11519e7i	−0.826416	+	1.43139i	−0.900832	+	0.434168i	$0.857043\pi$
$572$		0			0
$573$	2.64554e6			0.336611
$574$		0			0
$575$	−7.71453e6			−0.973061
$576$		0			0
$577$	−5.70887e6	+	9.88805e6i	−0.713856	+	1.23643i	0.249543	+	0.968364i	$0.419720\pi$
$577$	−5.70887e6	+	9.88805e6i	−0.713856	+	1.23643i	−0.963399	+	0.268071i	$0.913614\pi$
$578$		0			0
$579$	311085.	+	538816.i	0.0385641	+	0.0667950i
$580$		0			0
$581$	7.75968e6	−	6.30639e6i	0.953681	−	0.775069i
$582$		0			0
$583$	−6.75622e6	−	1.17021e7i	−0.823251	−	1.42591i
$584$		0			0
$585$	−2.60865e6	+	4.51831e6i	−0.315156	+	0.545867i
$586$		0			0
$587$	−2.12781e6			−0.254882			−0.127441	−	0.991846i	$-0.540676\pi$
$587$	−2.12781e6			−0.254882			−0.127441	+	0.991846i	$0.540676\pi$
$588$		0			0
$589$	−5.79871e6			−0.688721
$590$		0			0
$591$	−1.49287e6	+	2.58572e6i	−0.175814	+	0.304518i
$592$		0			0
$593$	4.28689e6	+	7.42511e6i	0.500617	+	0.867094i	1.00000		0.000712836i	$0.000226903\pi$
$593$	4.28689e6	+	7.42511e6i	0.500617	+	0.867094i	−0.499383	+	0.866382i	$0.666440\pi$
$594$		0			0
$595$	8.77978e6	−	7.13544e6i	1.01670	−	0.826282i
$596$		0			0
$597$	2.58493e6	+	4.47723e6i	0.296833	+	0.514130i
$598$		0			0
$599$	5.64372e6	−	9.77521e6i	0.642686	−	1.11316i	−0.342145	−	0.939647i	$-0.611154\pi$
$599$	5.64372e6	−	9.77521e6i	0.642686	−	1.11316i	0.984831	−	0.173517i	$-0.0555131\pi$
$600$		0			0
$601$	−1.58260e7			−1.78725			−0.893625	−	0.448814i	$-0.851847\pi$
$601$	−1.58260e7			−1.78725			−0.893625	+	0.448814i	$0.851847\pi$
$602$		0			0
$603$	−4.53145e6			−0.507509
$604$		0			0
$605$	−1.24753e7	+	2.16078e7i	−1.38568	+	2.40006i
$606$		0			0
$607$	4.87183e6	+	8.43826e6i	0.536686	+	0.929568i	0.999080	+	0.0428932i	$0.0136575\pi$
$607$	4.87183e6	+	8.43826e6i	0.536686	+	0.929568i	−0.462393	+	0.886675i	$0.653009\pi$
$608$		0			0
$609$	−601329.	−	3.75372e6i	−0.0657006	−	0.410128i
$610$		0			0
$611$	5.28320e6	+	9.15077e6i	0.572525	+	0.991642i
$612$		0			0
$613$	−3.07524e6	+	5.32647e6i	−0.330543	+	0.572517i	−0.982618	−	0.185637i	$-0.940565\pi$
$613$	−3.07524e6	+	5.32647e6i	−0.330543	+	0.572517i	0.652075	+	0.758154i	$0.273899\pi$
$614$		0			0
$615$	−9.27137e6			−0.988453
$616$		0			0
$617$	−3.36084e6			−0.355414			−0.177707	−	0.984083i	$-0.556868\pi$
$617$	−3.36084e6			−0.355414			−0.177707	+	0.984083i	$0.556868\pi$
$618$		0			0
$619$	3.44329e6	−	5.96396e6i	0.361200	−	0.625616i	−0.626959	−	0.779052i	$-0.715701\pi$
$619$	3.44329e6	−	5.96396e6i	0.361200	−	0.625616i	0.988159	+	0.153436i	$0.0490339\pi$
$620$		0			0
$621$	917663.	+	1.58944e6i	0.0954892	+	0.165392i
$622$		0			0
$623$	−38852.1	−	14835.3i	−0.00401046	−	0.00153135i
$624$		0			0
$625$	4.97590e6	+	8.61850e6i	0.509532	+	0.882535i
$626$		0			0
$627$	1.78323e6	−	3.08864e6i	0.181150	−	0.313760i
$628$		0			0
$629$	5.40175e6			0.544387
$630$		0			0
$631$	−3.59545e6			−0.359484			−0.179742	−	0.983714i	$-0.557526\pi$
$631$	−3.59545e6			−0.359484			−0.179742	+	0.983714i	$0.557526\pi$
$632$		0			0
$633$	−1.72799e6	+	2.99297e6i	−0.171409	+	0.296889i
$634$		0			0
$635$	1.23980e7	+	2.14740e7i	1.22017	+	2.11339i
$636$		0			0
$637$	2.81339e6	−	1.34698e7i	0.274715	−	1.31526i
$638$		0			0
$639$	830291.	+	1.43811e6i	0.0804411	+	0.139328i
$640$		0			0
$641$	9.91433e6	−	1.71721e7i	0.953056	−	1.65074i	0.214299	−	0.976768i	$-0.431253\pi$
$641$	9.91433e6	−	1.71721e7i	0.953056	−	1.65074i	0.738756	−	0.673973i	$-0.235413\pi$
$642$		0			0
$643$	8.10851e6			0.773417			0.386708	−	0.922202i	$-0.373612\pi$
$643$	8.10851e6			0.773417			0.386708	+	0.922202i	$0.373612\pi$
$644$		0			0
$645$	6.58740e6			0.623469
$646$		0			0
$647$	1.01373e6	−	1.75583e6i	0.0952053	−	0.164900i	−0.814489	−	0.580179i	$-0.802983\pi$
$647$	1.01373e6	−	1.75583e6i	0.0952053	−	0.164900i	0.909694	+	0.415279i	$0.136316\pi$
$648$		0			0
$649$	8.68564e6	+	1.50440e7i	0.809450	+	1.40201i
$650$		0			0
$651$	1.10300e7	+	4.21168e6i	1.02005	+	0.389496i
$652$		0			0
$653$	−757133.	−	1.31139e6i	−0.0694847	−	0.120351i	0.829190	−	0.558967i	$-0.188802\pi$
$653$	−757133.	−	1.31139e6i	−0.0694847	−	0.120351i	−0.898675	+	0.438616i	$0.855469\pi$
$654$		0			0
$655$	3.36969e6	−	5.83647e6i	0.306893	−	0.531554i
$656$		0			0
$657$	−5.47965e6			−0.495267
$658$		0			0
$659$	3.22358e6			0.289151			0.144576	−	0.989494i	$-0.453818\pi$
$659$	3.22358e6			0.289151			0.144576	+	0.989494i	$0.453818\pi$
$660$		0			0
$661$	7.19538e6	−	1.24628e7i	0.640546	−	1.10946i	−0.344765	−	0.938689i	$-0.612041\pi$
$661$	7.19538e6	−	1.24628e7i	0.640546	−	1.10946i	0.985311	−	0.170769i	$-0.0546253\pi$
$662$		0			0
$663$	−4.08690e6	−	7.07872e6i	−0.361085	−	0.625418i
$664$		0			0
$665$	924488.	+	5.77100e6i	0.0810676	+	0.506054i
$666$		0			0
$667$	−4.10141e6	−	7.10384e6i	−0.356959	−	0.618271i
$668$		0			0
$669$	−1.56224e6	+	2.70589e6i	−0.134953	+	0.233746i
$670$		0			0
$671$	2.16875e7			1.85953
$672$		0			0
$673$	7.33478e6			0.624237			0.312118	−	0.950043i	$-0.398961\pi$
$673$	7.33478e6			0.624237			0.312118	+	0.950043i	$0.398961\pi$
$674$		0			0
$675$	−1.11692e6	+	1.93456e6i	−0.0943543	+	0.163426i
$676$		0			0
$677$	1.52642e6	+	2.64384e6i	0.127998	+	0.221699i	0.922901	−	0.385038i	$-0.125812\pi$
$677$	1.52642e6	+	2.64384e6i	0.127998	+	0.221699i	−0.794903	+	0.606737i	$0.792478\pi$
$678$		0			0
$679$	−1.12688e7	+	9.15828e6i	−0.937999	+	0.762324i
$680$		0			0
$681$	1.62577e6	+	2.81591e6i	0.134335	+	0.232676i
$682$		0			0
$683$	7.98793e6	−	1.38355e7i	0.655213	−	1.13486i	−0.326627	−	0.945153i	$-0.605912\pi$
$683$	7.98793e6	−	1.38355e7i	0.655213	−	1.13486i	0.981840	−	0.189709i	$-0.0607546\pi$
$684$		0			0
$685$	−2.56724e6			−0.209045
$686$		0			0
$687$	−6.77685e6			−0.547817
$688$		0			0
$689$	7.99911e6	−	1.38549e7i	0.641939	−	1.11187i
$690$		0			0
$691$	−1.23267e6	−	2.13504e6i	−0.0982087	−	0.170103i	0.812735	−	0.582634i	$-0.197978\pi$
$691$	−1.23267e6	−	2.13504e6i	−0.0982087	−	0.170103i	−0.910943	+	0.412532i	$0.864645\pi$
$692$		0			0
$693$	−5.63527e6	+	4.57985e6i	−0.445740	+	0.362258i
$694$		0			0
$695$	8.12638e6	+	1.40753e7i	0.638168	+	1.10534i
$696$		0			0
$697$	7.26259e6	−	1.25792e7i	0.566252	−	0.980777i
$698$		0			0
$699$	6.42958e6			0.497725
$700$		0			0
$701$	1.74893e7			1.34424			0.672122	−	0.740440i	$-0.265383\pi$
$701$	1.74893e7			1.34424			0.672122	+	0.740440i	$0.265383\pi$
$702$		0			0
$703$	−1.39526e6	+	2.41666e6i	−0.106480	+	0.184428i
$704$		0			0
$705$	4.56895e6	+	7.91366e6i	0.346213	+	0.599659i
$706$		0			0
$707$	1.37915e6	+	8.60916e6i	0.103768	+	0.647757i
$708$		0			0
$709$	1.09216e7	+	1.89167e7i	0.815961	+	1.41329i	0.908636	+	0.417589i	$0.137125\pi$
$709$	1.09216e7	+	1.89167e7i	0.815961	+	1.41329i	−0.0926752	+	0.995696i	$0.529542\pi$
$710$		0			0
$711$	570236.	−	987677.i	0.0423039	−	0.0732725i
$712$		0			0
$713$	2.54758e7			1.87674
$714$		0			0
$715$	−4.45415e7			−3.25837
$716$		0			0
$717$	1.54614e6	−	2.67800e6i	0.112319	−	0.194541i
$718$		0			0
$719$	9.55868e6	+	1.65561e7i	0.689566	+	1.19436i	0.971978	+	0.235070i	$0.0755319\pi$
$719$	9.55868e6	+	1.65561e7i	0.689566	+	1.19436i	−0.282413	+	0.959293i	$0.591135\pi$
$720$		0			0
$721$	−1.41260e7	−	5.39387e6i	−1.01200	−	0.386423i
$722$		0			0
$723$	343372.	+	594738.i	0.0244298	+	0.0423136i
$724$		0			0
$725$	4.99195e6	−	8.64632e6i	0.352716	−	0.610923i
$726$		0			0
$727$	4.08900e6			0.286933			0.143467	−	0.989655i	$-0.454175\pi$
$727$	4.08900e6			0.286933			0.143467	+	0.989655i	$0.454175\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−5.16015e6	+	8.93764e6i	−0.357165	+	0.618628i
$732$		0			0
$733$	1.81634e6	+	3.14599e6i	0.124864	+	0.216271i	0.921680	−	0.387952i	$-0.126817\pi$
$733$	1.81634e6	+	3.14599e6i	0.124864	+	0.216271i	−0.796816	+	0.604222i	$0.793484\pi$
$734$		0			0
$735$	2.43304e6	−	1.16487e7i	0.166124	−	0.795355i
$736$		0			0
$737$	−1.93431e7	−	3.35032e7i	−1.31177	−	2.27205i
$738$		0			0
$739$	1.08978e6	−	1.88756e6i	0.0734056	−	0.127142i	−0.826986	−	0.562222i	$-0.809947\pi$
$739$	1.08978e6	−	1.88756e6i	0.0734056	−	0.127142i	0.900392	+	0.435080i	$0.143280\pi$
$740$		0			0
$741$	4.22254e6			0.282507
$742$		0			0
$743$	−1.21595e7			−0.808060			−0.404030	−	0.914746i	$-0.632391\pi$
$743$	−1.21595e7			−0.808060			−0.404030	+	0.914746i	$0.632391\pi$
$744$		0			0
$745$	−1.90272e7	+	3.29561e7i	−1.25598	+	2.17543i
$746$		0			0
$747$	−3.12373e6	−	5.41045e6i	−0.204820	−	0.354758i
$748$		0			0
$749$	−1.94562e7	−	7.42914e6i	−1.26722	−	0.483876i
$750$		0			0
$751$	1.16248e7	+	2.01347e7i	0.752115	+	1.30270i	0.946796	+	0.321835i	$0.104299\pi$
$751$	1.16248e7	+	2.01347e7i	0.752115	+	1.30270i	−0.194681	+	0.980867i	$0.562367\pi$
$752$		0			0
$753$	883121.	−	1.52961e6i	0.0567587	−	0.0983090i
$754$		0			0
$755$	1.31556e7			0.839928
$756$		0			0
$757$	−1.49939e7			−0.950985			−0.475492	−	0.879720i	$-0.657730\pi$
$757$	−1.49939e7			−0.950985			−0.475492	+	0.879720i	$0.657730\pi$
$758$		0			0
$759$	−7.83434e6	+	1.35695e7i	−0.493626	+	0.854985i
$760$		0			0
$761$	6.36596e6	+	1.10262e7i	0.398476	+	0.690181i	0.993538	−	0.113499i	$-0.0362058\pi$
$761$	6.36596e6	+	1.10262e7i	0.398476	+	0.690181i	−0.595062	+	0.803680i	$0.702873\pi$
$762$		0			0
$763$	2.44719e6	+	1.52762e7i	0.152179	+	0.949960i
$764$		0			0
$765$	−3.53438e6	−	6.12173e6i	−0.218353	−	0.378199i
$766$		0			0
$767$	−1.02835e7	+	1.78115e7i	−0.631177	+	1.09323i
$768$		0			0
$769$	2.34653e6			0.143090			0.0715451	−	0.997437i	$-0.477207\pi$
$769$	2.34653e6			0.143090			0.0715451	+	0.997437i	$0.477207\pi$
$770$		0			0
$771$	−1.53820e7			−0.931915
$772$		0			0
$773$	−1.35931e7	+	2.35440e7i	−0.818222	+	1.41720i	0.0887694	+	0.996052i	$0.471707\pi$
$773$	−1.35931e7	+	2.35440e7i	−0.818222	+	1.41720i	−0.906991	+	0.421150i	$0.861627\pi$
$774$		0			0
$775$	1.55037e7	+	2.68532e7i	0.927217	+	1.60599i
$776$		0			0
$777$	4.40923e6	−	3.58344e6i	0.262006	−	0.212935i
$778$		0			0
$779$	3.75182e6	+	6.49834e6i	0.221513	+	0.383671i
$780$		0			0
$781$	−7.08842e6	+	1.22775e7i	−0.415836	+	0.720249i
$782$		0			0
$783$	−2.37522e6			−0.138452
$784$		0			0
$785$	−1.10994e6			−0.0642874
$786$		0			0
$787$	2.15137e6	−	3.72628e6i	0.123816	−	0.214456i	−0.797453	−	0.603381i	$-0.793820\pi$
$787$	2.15137e6	−	3.72628e6i	0.123816	−	0.214456i	0.921270	+	0.388925i	$0.127153\pi$
$788$		0			0
$789$	−3.56186e6	−	6.16933e6i	−0.203697	−	0.352814i
$790$		0			0
$791$	−1.11565e7	+	9.06705e6i	−0.633998	+	0.515258i
$792$		0			0
$793$	1.28386e7	+	2.22371e7i	0.724995	+	1.25573i
$794$		0			0
$795$	6.91769e6	−	1.19818e7i	0.388189	−	0.672363i
$796$		0			0
$797$	1.92325e7			1.07248			0.536241	−	0.844065i	$-0.319844\pi$
$797$	1.92325e7			1.07248			0.536241	+	0.844065i	$0.319844\pi$
$798$		0			0
$799$	−1.43161e7			−0.793337
$800$		0			0
$801$	−12992.1	+	22503.0i	−0.000715481	+	0.00123925i
$802$		0			0
$803$	−2.33906e7	−	4.05137e7i	−1.28013	−	2.21724i
$804$		0			0
$805$	−4.06161e6	−	2.53541e7i	−0.220906	−	1.37898i
$806$		0			0
$807$	−5.21552e6	−	9.03354e6i	−0.281912	−	0.488286i
$808$		0			0
$809$	2.38183e6	−	4.12546e6i	0.127950	−	0.221616i	−0.794932	−	0.606698i	$-0.792494\pi$
$809$	2.38183e6	−	4.12546e6i	0.127950	−	0.221616i	0.922882	+	0.385082i	$0.125827\pi$
$810$		0			0
$811$	−1.05529e7			−0.563404			−0.281702	−	0.959502i	$-0.590899\pi$
$811$	−1.05529e7			−0.563404			−0.281702	+	0.959502i	$0.590899\pi$
$812$		0			0
$813$	−2.00821e6			−0.106557
$814$		0			0
$815$	2.57405e7	−	4.45838e7i	1.35745	−	2.35116i
$816$		0			0
$817$	−2.66571e6	−	4.61714e6i	−0.139720	−	0.242001i
$818$		0			0
$819$	−8.03188e6	−	3.06689e6i	−0.418415	−	0.159767i
$820$		0			0
$821$	7.55332e6	+	1.30827e7i	0.391093	+	0.677393i	0.992594	−	0.121479i	$-0.0387638\pi$
$821$	7.55332e6	+	1.30827e7i	0.391093	+	0.677393i	−0.601501	+	0.798872i	$0.705430\pi$
$822$		0			0
$823$	1.24076e7	−	2.14906e7i	0.638541	−	1.10599i	−0.347212	−	0.937787i	$-0.612872\pi$
$823$	1.24076e7	−	2.14906e7i	0.638541	−	1.10599i	0.985753	−	0.168199i	$-0.0537950\pi$
$824$		0			0
$825$	−1.90709e7			−0.975518
$826$		0			0
$827$	5.89505e6			0.299726			0.149863	−	0.988707i	$-0.452117\pi$
$827$	5.89505e6			0.299726			0.149863	+	0.988707i	$0.452117\pi$
$828$		0			0
$829$	4.58123e6	−	7.93493e6i	0.231524	−	0.401011i	−0.726733	−	0.686920i	$-0.758962\pi$
$829$	4.58123e6	−	7.93493e6i	0.231524	−	0.401011i	0.958257	+	0.285909i	$0.0922955\pi$
$830$		0			0
$831$	2.11071e6	+	3.65585e6i	0.106029	+	0.183648i
$832$		0			0
$833$	1.38989e7	+	1.24260e7i	0.694012	+	0.620466i
$834$		0			0
$835$	7.19632e6	+	1.24644e7i	0.357186	+	0.618664i
$836$		0			0
$837$	3.68841e6	−	6.38852e6i	0.181981	−	0.315200i
$838$		0			0
$839$	1.65127e7			0.809868			0.404934	−	0.914346i	$-0.367295\pi$
$839$	1.65127e7			0.809868			0.404934	+	0.914346i	$0.367295\pi$
$840$		0			0
$841$	−9.89532e6			−0.482436
$842$		0			0
$843$	6.81511e6	−	1.18041e7i	0.330296	−	0.572090i
$844$		0			0
$845$	−1.17626e7	−	2.03734e7i	−0.566710	−	0.981571i
$846$		0			0
$847$	−3.84107e7	−	1.46667e7i	−1.83968	−	0.702463i
$848$		0			0
$849$	−4.81772e6	−	8.34454e6i	−0.229389	−	0.397314i
$850$		0			0
$851$	6.12987e6	−	1.06172e7i	0.290153	−	0.502560i
$852$		0			0
$853$	−2.99081e7			−1.40740			−0.703699	−	0.710498i	$-0.748470\pi$
$853$	−2.99081e7			−1.40740			−0.703699	+	0.710498i	$0.748470\pi$
$854$		0			0
$855$	3.65169e6			0.170836
$856$		0			0
$857$	−6.63374e6	+	1.14900e7i	−0.308536	+	0.534400i	−0.978042	−	0.208406i	$-0.933172\pi$
$857$	−6.63374e6	+	1.14900e7i	−0.308536	+	0.534400i	0.669506	+	0.742807i	$0.266506\pi$
$858$		0			0
$859$	−1.10244e7	−	1.90949e7i	−0.509769	−	0.882945i	−0.999936	−	0.0113167i	$-0.996398\pi$
$859$	−1.10244e7	−	1.90949e7i	−0.509769	−	0.882945i	0.490167	−	0.871628i	$-0.336936\pi$
$860$		0			0
$861$	−2.41667e6	−	1.50858e7i	−0.111099	−	0.693521i
$862$		0			0
$863$	7.88661e6	+	1.36600e7i	0.360465	+	0.624344i	0.988037	−	0.154214i	$-0.0492847\pi$
$863$	7.88661e6	+	1.36600e7i	0.360465	+	0.624344i	−0.627572	+	0.778558i	$0.715951\pi$
$864$		0			0
$865$	1.14864e7	−	1.98950e7i	0.521968	−	0.904075i
$866$		0			0
$867$	−1.70429e6			−0.0770007
$868$		0			0
$869$	9.73652e6			0.437375
$870$		0			0
$871$	2.29015e7	−	3.96666e7i	1.02287	−	1.77166i
$872$		0			0
$873$	4.53635e6	+	7.85719e6i	0.201452	+	0.348925i
$874$		0			0
$875$	−480863.	+	390803.i	−0.0212325	+	0.0172559i
$876$		0			0
$877$	−1.07566e7	−	1.86309e7i	−0.472253	−	0.817967i	0.527243	−	0.849715i	$-0.323226\pi$
$877$	−1.07566e7	−	1.86309e7i	−0.472253	−	0.817967i	−0.999496	+	0.0317480i	$0.989893\pi$
$878$		0			0
$879$	−6.54525e6	+	1.13367e7i	−0.285729	+	0.494897i
$880$		0			0
$881$	−8.57354e6			−0.372152			−0.186076	−	0.982535i	$-0.559577\pi$
$881$	−8.57354e6			−0.372152			−0.186076	+	0.982535i	$0.559577\pi$
$882$		0			0
$883$	5.13016e6			0.221426			0.110713	−	0.993852i	$-0.464686\pi$
$883$	5.13016e6			0.221426			0.110713	+	0.993852i	$0.464686\pi$
$884$		0			0
$885$	−8.89322e6	+	1.54035e7i	−0.381681	+	0.661091i
$886$		0			0
$887$	−1.80668e6	−	3.12926e6i	−0.0771030	−	0.133546i	0.824896	−	0.565285i	$-0.191234\pi$
$887$	−1.80668e6	−	3.12926e6i	−0.0771030	−	0.133546i	−0.901999	+	0.431738i	$0.857900\pi$
$888$		0			0
$889$	−3.17095e7	+	2.57707e7i	−1.34566	+	1.09364i
$890$		0			0
$891$	2.26853e6	+	3.92921e6i	0.0957304	+	0.165810i
$892$		0			0
$893$	3.69781e6	−	6.40480e6i	0.155173	−	0.268768i
$894$		0			0
$895$	−1.53227e7			−0.639408
$896$		0			0
$897$	−1.85511e7			−0.769820
$898$		0			0
$899$	−1.64850e7	+	2.85529e7i	−0.680283	+	1.17828i
$900$		0			0
$901$	1.08377e7	+	1.87715e7i	0.444761	+	0.770349i
$902$		0			0
$903$	1.71707e6	+	1.07186e7i	0.0700759	+	0.437440i
$904$		0			0
$905$	1.00949e7	+	1.74849e7i	0.409715	+	0.709647i
$906$		0			0
$907$	1.34719e7	−	2.33340e7i	0.543765	−	0.941828i	−0.454919	−	0.890533i	$-0.650332\pi$
$907$	1.34719e7	−	2.33340e7i	0.543765	−	0.941828i	0.998684	−	0.0512953i	$-0.0163350\pi$
$908$		0			0
$909$	5.44757e6			0.218672
$910$		0			0
$911$	3.89588e7			1.55528			0.777641	−	0.628708i	$-0.216416\pi$
$911$	3.89588e7			1.55528			0.777641	+	0.628708i	$0.216416\pi$
$912$		0			0
$913$	2.66681e7	−	4.61905e7i	1.05880	−	1.83390i
$914$		0			0
$915$	1.11029e7	+	1.92308e7i	0.438415	+	0.759356i
$916$		0			0
$917$	1.03751e7	+	3.96161e6i	0.407444	+	0.155578i
$918$		0			0
$919$	1.73217e7	+	3.00021e7i	0.676553	+	1.17182i	0.976012	+	0.217716i	$0.0698605\pi$
$919$	1.73217e7	+	3.00021e7i	0.676553	+	1.17182i	−0.299459	+	0.954109i	$0.596806\pi$
$920$		0			0
$921$	−1.00741e7	+	1.74488e7i	−0.391342	+	0.677824i
$922$		0			0
$923$	−1.67848e7			−0.648504
$924$		0			0
$925$	1.49217e7			0.573409
$926$		0			0
$927$	−4.72373e6	+	8.18174e6i	−0.180545	+	0.312713i
$928$		0			0
$929$	1.76170e7	+	3.05136e7i	0.669721	+	1.15999i	0.977982	+	0.208689i	$0.0669195\pi$
$929$	1.76170e7	+	3.05136e7i	0.669721	+	1.15999i	−0.308261	+	0.951302i	$0.599747\pi$
$930$		0			0
$931$	−9.14923e6	+	3.00854e6i	−0.345948	+	0.113758i
$932$		0			0
$933$	−6.75626e6	−	1.17022e7i	−0.254098	−	0.440111i
$934$		0			0
$935$	3.01740e7	−	5.22628e7i	1.12876	−	1.95508i
$936$		0			0
$937$	7.21950e6			0.268632			0.134316	−	0.990939i	$-0.457116\pi$
$937$	7.21950e6			0.268632			0.134316	+	0.990939i	$0.457116\pi$
$938$		0			0
$939$	−1.60212e7			−0.592968
$940$		0			0
$941$	−1.86666e7	+	3.23315e7i	−0.687213	+	1.19029i	0.285523	+	0.958372i	$0.407833\pi$
$941$	−1.86666e7	+	3.23315e7i	−0.687213	+	1.19029i	−0.972736	+	0.231916i	$0.925501\pi$
$942$		0			0
$943$	−1.64831e7	−	2.85495e7i	−0.603614	−	1.04549i
$944$		0			0
$945$	−6.94603e6	−	2.65227e6i	−0.253021	−	0.0966135i
$946$		0			0
$947$	−2.08507e7	−	3.61145e7i	−0.755520	−	1.30860i	−0.945116	−	0.326736i	$-0.894051\pi$
$947$	−2.08507e7	−	3.61145e7i	−0.755520	−	1.30860i	0.189596	−	0.981862i	$-0.439282\pi$
$948$		0			0
$949$	2.76936e7	−	4.79667e7i	0.998191	−	1.72892i
$950$		0			0
$951$	1.32390e7			0.474683
$952$		0			0
$953$	5.07681e7			1.81075			0.905374	−	0.424614i	$-0.139590\pi$
$953$	5.07681e7			1.81075			0.905374	+	0.424614i	$0.139590\pi$
$954$		0			0
$955$	−1.15627e7	+	2.00273e7i	−0.410254	+	0.710580i
$956$		0			0
$957$	−1.01390e7	−	1.75612e7i	−0.357860	−	0.619832i
$958$		0			0
$959$	−669175.	−	4.17724e6i	−0.0234960	−	0.146671i
$960$		0			0
$961$	−3.68835e7	−	6.38841e7i	−1.28832	−	2.23144i
$962$		0			0
$963$	−6.50613e6	+	1.12689e7i	−0.226077	+	0.391577i
$964$		0			0
$965$	−5.43859e6			−0.188004
$966$		0			0
$967$	2.00129e7			0.688246			0.344123	−	0.938925i	$-0.388176\pi$
$967$	2.00129e7			0.688246			0.344123	+	0.938925i	$0.388176\pi$
$968$		0			0
$969$	−2.86050e6	+	4.95453e6i	−0.0978660	+	0.169509i
$970$		0			0
$971$	7.43621e6	+	1.28799e7i	0.253107	+	0.438393i	0.964380	−	0.264523i	$-0.0852143\pi$
$971$	7.43621e6	+	1.28799e7i	0.253107	+	0.438393i	−0.711273	+	0.702916i	$0.751881\pi$
$972$		0			0
$973$	−2.07842e7	+	1.68916e7i	−0.703804	+	0.571990i
$974$		0			0
$975$	−1.12896e7	−	1.95541e7i	−0.380335	−	0.658760i
$976$		0			0
$977$	2.43470e6	−	4.21703e6i	0.0816037	−	0.141342i	−0.822335	−	0.569003i	$-0.807329\pi$
$977$	2.43470e6	−	4.21703e6i	0.0816037	−	0.141342i	0.903939	+	0.427662i	$0.140662\pi$
$978$		0			0
$979$	−221834.			−0.00739727
$980$		0			0
$981$	9.66627e6			0.320691
$982$		0			0
$983$	−2.16555e7	+	3.75085e7i	−0.714800	+	1.23807i	0.248236	+	0.968700i	$0.420149\pi$
$983$	−2.16555e7	+	3.75085e7i	−0.714800	+	1.23807i	−0.963037	+	0.269371i	$0.913184\pi$
$984$		0			0
$985$	−1.30496e7	−	2.26026e7i	−0.428556	−	0.742280i
$986$		0			0
$987$	−1.16857e7	+	9.49708e6i	−0.381821	+	0.310311i
$988$		0			0
$989$	1.17114e7	+	2.02847e7i	0.380731	+	0.659445i
$990$		0			0
$991$	5.15180e6	−	8.92318e6i	0.166638	−	0.288626i	−0.770598	−	0.637322i	$-0.780042\pi$
$991$	5.15180e6	−	8.92318e6i	0.166638	−	0.288626i	0.937236	+	0.348696i	$0.113375\pi$
$992$		0			0
$993$	1.64677e7			0.529981
$994$		0			0
$995$	−4.51913e7			−1.44710
$996$		0			0
$997$	1.94218e7	−	3.36396e7i	0.618803	−	1.07180i	−0.370901	−	0.928672i	$-0.620951\pi$
$997$	1.94218e7	−	3.36396e7i	0.618803	−	1.07180i	0.989704	−	0.143126i	$-0.0457154\pi$
$998$		0			0
$999$	−1.77498e6	−	3.07435e6i	−0.0562703	−	0.0974629i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.q.i.193.1		8
4.3	odd	2		84.6.i.c.25.1	✓	8
7.2	even	3	inner	336.6.q.i.289.1		8
12.11	even	2		252.6.k.f.109.4		8
28.3	even	6		588.6.a.p.1.1		4
28.11	odd	6		588.6.a.n.1.4		4
28.19	even	6		588.6.i.o.373.4		8
28.23	odd	6		84.6.i.c.37.1	yes	8
28.27	even	2		588.6.i.o.361.4		8
84.23	even	6		252.6.k.f.37.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
84.6.i.c.25.1	✓	8	4.3	odd	2
84.6.i.c.37.1	yes	8	28.23	odd	6
252.6.k.f.37.4		8	84.23	even	6
252.6.k.f.109.4		8	12.11	even	2
336.6.q.i.193.1		8	1.1	even	1	trivial
336.6.q.i.289.1		8	7.2	even	3	inner
588.6.a.n.1.4		4	28.11	odd	6
588.6.a.p.1.1		4	28.3	even	6
588.6.i.o.361.4		8	28.27	even	2
588.6.i.o.373.4		8	28.19	even	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-4.50000 + 7.79423i) q^{3} +(-39.3359 - 68.1317i) q^{5} +(100.606 - 81.7641i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(345.759 - 598.873i) q^{11} +818.732 q^{13} +708.046 q^{15} +(-554.638 + 960.661i) q^{17} +(-286.523 - 496.273i) q^{19} +(184.559 + 1152.09i) q^{21} +(1258.80 + 2180.30i) q^{23} +(-1532.12 + 2653.71i) q^{25} +729.000 q^{27} -3258.19 q^{29} +(5059.55 - 8763.40i) q^{31} +(3111.84 + 5389.86i) q^{33} +(-9528.17 - 3638.23i) q^{35} +(-2434.81 - 4217.21i) q^{37} +(-3684.30 + 6381.39i) q^{39} -13094.3 q^{41} +9303.64 q^{43} +(-3186.21 + 5518.67i) q^{45} +(6452.90 + 11176.8i) q^{47} +(3436.28 - 16452.0i) q^{49} +(-4991.74 - 8645.95i) q^{51} +(9770.12 - 16922.3i) q^{53} -54403.0 q^{55} +5157.42 q^{57} +(-12560.2 + 21755.0i) q^{59} +(15681.1 + 27160.5i) q^{61} +(-9810.15 - 3745.90i) q^{63} +(-32205.6 - 55781.7i) q^{65} +(27971.9 - 48448.8i) q^{67} -22658.4 q^{69} -20501.0 q^{71} +(33825.0 - 58586.6i) q^{73} +(-13789.1 - 23883.4i) q^{75} +(-14180.7 - 88521.1i) q^{77} +(7039.95 + 12193.5i) q^{79} +(-3280.50 + 5681.99i) q^{81} +77129.1 q^{83} +87268.6 q^{85} +(14661.9 - 25395.1i) q^{87} +(-160.396 - 277.815i) q^{89} +(82369.7 - 66942.9i) q^{91} +(45536.0 + 78870.6i) q^{93} +(-22541.3 + 39042.6i) q^{95} -112009. q^{97} -56013.0 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 36 q^{3} + 42 q^{7} - 324 q^{9} + 462 q^{11} - 1204 q^{13} + 228 q^{17} - 358 q^{19} + 1404 q^{21} + 2148 q^{23} - 5454 q^{25} + 5832 q^{27} - 11064 q^{29} - 830 q^{31} + 4158 q^{33} - 7692 q^{35}+ \cdots - 74844 q^{99}+O(q^{100}) \) 8 * q - 36 * q^3 + 42 * q^7 - 324 * q^9 + 462 * q^11 - 1204 * q^13 + 228 * q^17 - 358 * q^19 + 1404 * q^21 + 2148 * q^23 - 5454 * q^25 + 5832 * q^27 - 11064 * q^29 - 830 * q^31 + 4158 * q^33 - 7692 * q^35 - 3914 * q^37 + 5418 * q^39 - 16632 * q^41 + 29036 * q^43 - 41700 * q^47 + 41876 * q^49 + 2052 * q^51 + 22164 * q^53 - 7784 * q^55 + 6444 * q^57 - 32886 * q^59 + 83732 * q^61 - 16038 * q^63 - 93192 * q^65 + 80034 * q^67 - 38664 * q^69 - 89544 * q^71 - 22470 * q^73 - 49086 * q^75 + 138732 * q^77 + 75286 * q^79 - 26244 * q^81 + 34836 * q^83 + 278504 * q^85 + 49788 * q^87 + 28944 * q^89 + 12058 * q^91 - 7470 * q^93 + 144120 * q^95 - 433356 * q^97 - 74844 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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