LMFDB - Embedded newform 336.6.q.k.193.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("336.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$8$
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} - x^{7} + 118x^{6} + 555x^{5} + 12174x^{4} + 28215x^{3} + 199593x^{2} - 283824x + 1679616 $ x^8 - x^7 + 118x^6 + 555x^5 + 12174x^4 + 28215x^3 + 199593x^2 - 283824x + 1679616
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{6}\cdot 3\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			193.1
Root			$-2.82121 + 4.88647i$ of defining polynomial
Character	$\chi$	$=$	336.193
Dual form			336.6.q.k.289.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 - 7.79423i) q^{3} +(-30.1755 - 52.2655i) q^{5} +(84.0301 + 98.7215i) q^{7} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 - 7.79423i) q^{3} +(-30.1755 - 52.2655i) q^{5} +(84.0301 + 98.7215i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(3.84136 - 6.65343i) q^{11} -42.5391 q^{13} -543.159 q^{15} +(-68.9458 + 119.418i) q^{17} +(-542.790 - 940.140i) q^{19} +(1147.59 - 210.703i) q^{21} +(-1328.93 - 2301.78i) q^{23} +(-258.622 + 447.946i) q^{25} -729.000 q^{27} -2635.67 q^{29} +(1286.72 - 2228.66i) q^{31} +(-34.5722 - 59.8808i) q^{33} +(2624.08 - 7370.85i) q^{35} +(2872.65 + 4975.58i) q^{37} +(-191.426 + 331.560i) q^{39} -18194.0 q^{41} -1725.55 q^{43} +(-2444.22 + 4233.51i) q^{45} +(-7384.51 - 12790.3i) q^{47} +(-2684.87 + 16591.2i) q^{49} +(620.512 + 1074.76i) q^{51} +(-9397.31 + 16276.6i) q^{53} -463.660 q^{55} -9770.22 q^{57} +(-3977.27 + 6888.84i) q^{59} +(25481.1 + 44134.6i) q^{61} +(3521.90 - 9892.77i) q^{63} +(1283.64 + 2223.33i) q^{65} +(-11682.8 + 20235.2i) q^{67} -23920.8 q^{69} -60027.2 q^{71} +(-10029.5 + 17371.6i) q^{73} +(2327.59 + 4031.51i) q^{75} +(979.626 - 179.864i) q^{77} +(-23591.6 - 40861.9i) q^{79} +(-3280.50 + 5681.99i) q^{81} +42207.0 q^{83} +8321.90 q^{85} +(-11860.5 + 20543.0i) q^{87} +(-17189.2 - 29772.6i) q^{89} +(-3574.57 - 4199.53i) q^{91} +(-11580.4 - 20057.9i) q^{93} +(-32757.9 + 56738.4i) q^{95} -85606.4 q^{97} -622.300 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 36 q^{3} + 64 q^{5} - 42 q^{7} - 324 q^{9}+O(q^{10}) $ 8 * q + 36 * q^3 + 64 * q^5 - 42 * q^7 - 324 * q^9	$ 8 q + 36 q^{3} + 64 q^{5} - 42 q^{7} - 324 q^{9} - 70 q^{11} + 1356 q^{13} + 1152 q^{15} + 944 q^{17} - 606 q^{19} + 1512 q^{21} - 648 q^{23} - 2582 q^{25} - 5832 q^{27} - 8720 q^{29} - 4354 q^{31} + 630 q^{33} - 5824 q^{35} + 19302 q^{37} + 6102 q^{39} + 1832 q^{41} - 16788 q^{43} + 5184 q^{45} - 5104 q^{47} - 15484 q^{49} - 8496 q^{51} - 13244 q^{53} - 35744 q^{55} - 10908 q^{57} - 30742 q^{59} - 2428 q^{61} + 17010 q^{63} + 126004 q^{65} + 8258 q^{67} - 11664 q^{69} - 29664 q^{71} - 12758 q^{73} + 23238 q^{75} - 91672 q^{77} - 21382 q^{79} - 26244 q^{81} + 110500 q^{83} + 275960 q^{85} - 39240 q^{87} - 49072 q^{89} + 35658 q^{91} + 39186 q^{93} + 25292 q^{95} - 135228 q^{97} + 11340 q^{99}+O(q^{100}) $ 8 * q + 36 * q^3 + 64 * q^5 - 42 * q^7 - 324 * q^9 - 70 * q^11 + 1356 * q^13 + 1152 * q^15 + 944 * q^17 - 606 * q^19 + 1512 * q^21 - 648 * q^23 - 2582 * q^25 - 5832 * q^27 - 8720 * q^29 - 4354 * q^31 + 630 * q^33 - 5824 * q^35 + 19302 * q^37 + 6102 * q^39 + 1832 * q^41 - 16788 * q^43 + 5184 * q^45 - 5104 * q^47 - 15484 * q^49 - 8496 * q^51 - 13244 * q^53 - 35744 * q^55 - 10908 * q^57 - 30742 * q^59 - 2428 * q^61 + 17010 * q^63 + 126004 * q^65 + 8258 * q^67 - 11664 * q^69 - 29664 * q^71 - 12758 * q^73 + 23238 * q^75 - 91672 * q^77 - 21382 * q^79 - 26244 * q^81 + 110500 * q^83 + 275960 * q^85 - 39240 * q^87 - 49072 * q^89 + 35658 * q^91 + 39186 * q^93 + 25292 * q^95 - 135228 * q^97 + 11340 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	4.50000	−	7.79423i	0.288675	−	0.500000i
$3$	4.50000	−	7.79423i	0.288675	−	0.500000i
$4$		0			0
$5$	−30.1755	−	52.2655i	−0.539796	−	0.934954i	−0.998915	−	0.0465788i	$-0.985168\pi$
$5$	−30.1755	−	52.2655i	−0.539796	−	0.934954i	0.459119	−	0.888375i	$-0.348165\pi$
$6$		0			0
$7$	84.0301	+	98.7215i	0.648172	+	0.761494i
$7$	84.0301	+	98.7215i	0.648172	+	0.761494i
$8$		0			0
$9$	−40.5000	−	70.1481i	−0.166667	−	0.288675i
$10$		0			0
$11$	3.84136	−	6.65343i	0.00957201	−	0.0165792i	−0.861200	−	0.508267i	$-0.830286\pi$
$11$	3.84136	−	6.65343i	0.00957201	−	0.0165792i	0.870772	+	0.491687i	$0.163620\pi$
$12$		0			0
$13$	−42.5391			−0.0698120			−0.0349060	−	0.999391i	$-0.511113\pi$
$13$	−42.5391			−0.0698120			−0.0349060	+	0.999391i	$0.511113\pi$
$14$		0			0
$15$	−543.159			−0.623302
$16$		0			0
$17$	−68.9458	+	119.418i	−0.0578610	+	0.100218i	−0.893505	−	0.449053i	$-0.851761\pi$
$17$	−68.9458	+	119.418i	−0.0578610	+	0.100218i	0.835644	+	0.549271i	$0.185095\pi$
$18$		0			0
$19$	−542.790	−	940.140i	−0.344943	−	0.597460i	0.640400	−	0.768042i	$-0.278769\pi$
$19$	−542.790	−	940.140i	−0.344943	−	0.597460i	−0.985343	+	0.170582i	$0.945435\pi$
$20$		0			0
$21$	1147.59	−	210.703i	0.567858	−	0.104261i
$22$		0			0
$23$	−1328.93	−	2301.78i	−0.523822	−	0.907286i	−0.999615	−	0.0277290i	$-0.991172\pi$
$23$	−1328.93	−	2301.78i	−0.523822	−	0.907286i	0.475794	−	0.879557i	$-0.342161\pi$
$24$		0			0
$25$	−258.622	+	447.946i	−0.0827589	+	0.143343i
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	−2635.67			−0.581963			−0.290982	−	0.956729i	$-0.593982\pi$
$29$	−2635.67			−0.581963			−0.290982	+	0.956729i	$0.593982\pi$
$30$		0			0
$31$	1286.72	−	2228.66i	0.240480	−	0.416523i	−0.720371	−	0.693589i	$-0.756029\pi$
$31$	1286.72	−	2228.66i	0.240480	−	0.416523i	0.960851	+	0.277065i	$0.0893619\pi$
$32$		0			0
$33$	−34.5722	−	59.8808i	−0.00552640	−	0.00957201i
$34$		0			0
$35$	2624.08	−	7370.85i	0.362082	−	1.01706i
$36$		0			0
$37$	2872.65	+	4975.58i	0.344968	+	0.597502i	0.985348	−	0.170556i	$-0.0545564\pi$
$37$	2872.65	+	4975.58i	0.344968	+	0.597502i	−0.640380	+	0.768058i	$0.721223\pi$
$38$		0			0
$39$	−191.426	+	331.560i	−0.0201530	+	0.0349060i
$40$		0			0
$41$	−18194.0			−1.69032			−0.845161	−	0.534511i	$-0.820496\pi$
$41$	−18194.0			−1.69032			−0.845161	+	0.534511i	$0.820496\pi$
$42$		0			0
$43$	−1725.55			−0.142317			−0.0711585	−	0.997465i	$-0.522670\pi$
$43$	−1725.55			−0.142317			−0.0711585	+	0.997465i	$0.522670\pi$
$44$		0			0
$45$	−2444.22	+	4233.51i	−0.179932	+	0.311651i
$46$		0			0
$47$	−7384.51	−	12790.3i	−0.487615	−	0.844574i	0.512284	−	0.858816i	$-0.328800\pi$
$47$	−7384.51	−	12790.3i	−0.487615	−	0.844574i	−0.999899	+	0.0142427i	$0.995466\pi$
$48$		0			0
$49$	−2684.87	+	16591.2i	−0.159747	+	0.987158i
$50$		0			0
$51$	620.512	+	1074.76i	0.0334060	+	0.0578610i
$52$		0			0
$53$	−9397.31	+	16276.6i	−0.459530	+	0.795929i	−0.998936	−	0.0461167i	$-0.985315\pi$
$53$	−9397.31	+	16276.6i	−0.459530	+	0.795929i	0.539406	+	0.842046i	$0.318649\pi$
$54$		0			0
$55$	−463.660			−0.0206677
$56$		0			0
$57$	−9770.22			−0.398306
$58$		0			0
$59$	−3977.27	+	6888.84i	−0.148750	+	0.257642i	−0.930766	−	0.365617i	$-0.880858\pi$
$59$	−3977.27	+	6888.84i	−0.148750	+	0.257642i	0.782016	+	0.623258i	$0.214191\pi$
$60$		0			0
$61$	25481.1	+	44134.6i	0.876787	+	1.51864i	0.854847	+	0.518881i	$0.173651\pi$
$61$	25481.1	+	44134.6i	0.876787	+	1.51864i	0.0219404	+	0.999759i	$0.493016\pi$
$62$		0			0
$63$	3521.90	−	9892.77i	0.111796	−	0.314027i
$64$		0			0
$65$	1283.64	+	2223.33i	0.0376842	+	0.0652710i
$66$		0			0
$67$	−11682.8	+	20235.2i	−0.317951	+	0.550707i	−0.980060	−	0.198700i	$-0.936328\pi$
$67$	−11682.8	+	20235.2i	−0.317951	+	0.550707i	0.662109	+	0.749407i	$0.269661\pi$
$68$		0			0
$69$	−23920.8			−0.604857
$70$		0			0
$71$	−60027.2			−1.41320			−0.706598	−	0.707615i	$-0.749771\pi$
$71$	−60027.2			−1.41320			−0.706598	+	0.707615i	$0.749771\pi$
$72$		0			0
$73$	−10029.5	+	17371.6i	−0.220278	+	0.381532i	−0.954892	−	0.296952i	$-0.904030\pi$
$73$	−10029.5	+	17371.6i	−0.220278	+	0.381532i	0.734614	+	0.678485i	$0.237363\pi$
$74$		0			0
$75$	2327.59	+	4031.51i	0.0477809	+	0.0827589i
$76$		0			0
$77$	979.626	−	179.864i	0.0188293	−	0.00345714i
$78$		0			0
$79$	−23591.6	−	40861.9i	−0.425295	−	0.736632i	0.571153	−	0.820843i	$-0.306496\pi$
$79$	−23591.6	−	40861.9i	−0.425295	−	0.736632i	−0.996448	+	0.0842114i	$0.973163\pi$
$80$		0			0
$81$	−3280.50	+	5681.99i	−0.0555556	+	0.0962250i
$82$		0			0
$83$	42207.0			0.672496			0.336248	−	0.941773i	$-0.390842\pi$
$83$	42207.0			0.672496			0.336248	+	0.941773i	$0.390842\pi$
$84$		0			0
$85$	8321.90			0.124932
$86$		0			0
$87$	−11860.5	+	20543.0i	−0.167998	+	0.290982i
$88$		0			0
$89$	−17189.2	−	29772.6i	−0.230029	−	0.398421i	0.727788	−	0.685803i	$-0.240549\pi$
$89$	−17189.2	−	29772.6i	−0.230029	−	0.398421i	−0.957816	+	0.287381i	$0.907215\pi$
$90$		0			0
$91$	−3574.57	−	4199.53i	−0.0452502	−	0.0531615i
$92$		0			0
$93$	−11580.4	−	20057.9i	−0.138841	−	0.240480i
$94$		0			0
$95$	−32757.9	+	56738.4i	−0.372398	+	0.645012i
$96$		0			0
$97$	−85606.4			−0.923797			−0.461899	−	0.886933i	$-0.652832\pi$
$97$	−85606.4			−0.923797			−0.461899	+	0.886933i	$0.652832\pi$
$98$		0			0
$99$	−622.300			−0.00638134
$100$		0			0
$101$	36100.5	−	62528.0i	0.352136	−	0.609917i	−0.634488	−	0.772933i	$-0.718789\pi$
$101$	36100.5	−	62528.0i	0.352136	−	0.609917i	0.986623	+	0.163016i	$0.0521222\pi$
$102$		0			0
$103$	−17672.1	−	30609.1i	−0.164133	−	0.284287i	0.772214	−	0.635363i	$-0.219149\pi$
$103$	−17672.1	−	30609.1i	−0.164133	−	0.284287i	−0.936347	+	0.351076i	$0.885816\pi$
$104$		0			0
$105$	−45641.7	−	53621.5i	−0.404007	−	0.474641i
$106$		0			0
$107$	24080.8	+	41709.1i	0.203335	+	0.352186i	0.949601	−	0.313462i	$-0.101489\pi$
$107$	24080.8	+	41709.1i	0.203335	+	0.352186i	−0.746266	+	0.665648i	$0.768155\pi$
$108$		0			0
$109$	8178.25	−	14165.1i	0.0659316	−	0.114197i	−0.831175	−	0.556011i	$-0.812331\pi$
$109$	8178.25	−	14165.1i	0.0659316	−	0.114197i	0.897107	+	0.441814i	$0.145665\pi$
$110$		0			0
$111$	51707.8			0.398335
$112$		0			0
$113$	−146029.			−1.07583			−0.537913	−	0.843000i	$-0.680787\pi$
$113$	−146029.			−1.07583			−0.537913	+	0.843000i	$0.680787\pi$
$114$		0			0
$115$	−80202.4	+	138915.i	−0.565514	+	0.979498i
$116$		0			0
$117$	1722.83	+	2984.04i	0.0116353	+	0.0201530i
$118$		0			0
$119$	−17582.6	+	3228.25i	−0.113819	+	0.0208977i
$120$		0			0
$121$	80496.0	+	139423.i	0.499817	+	0.865708i
$122$		0			0
$123$	−81873.2	+	141809.i	−0.487954	+	0.845161i
$124$		0			0
$125$	−157381.			−0.900900
$126$		0			0
$127$	−348590.			−1.91781			−0.958905	−	0.283727i	$-0.908429\pi$
$127$	−348590.			−1.91781			−0.958905	+	0.283727i	$0.908429\pi$
$128$		0			0
$129$	−7764.98	+	13449.3i	−0.0410834	+	0.0711585i
$130$		0			0
$131$	149743.	+	259363.i	0.762376	+	1.32047i	0.941623	+	0.336669i	$0.109300\pi$
$131$	149743.	+	259363.i	0.762376	+	1.32047i	−0.179247	+	0.983804i	$0.557366\pi$
$132$		0			0
$133$	47201.3	−	132585.i	0.231380	−	0.649929i
$134$		0			0
$135$	21997.9	+	38101.5i	0.103884	+	0.179932i
$136$		0			0
$137$	−49324.5	+	85432.5i	−0.224523	+	0.388885i	−0.956176	−	0.292792i	$-0.905416\pi$
$137$	−49324.5	+	85432.5i	−0.224523	+	0.388885i	0.731653	+	0.681677i	$0.238749\pi$
$138$		0			0
$139$	−297028.			−1.30395			−0.651974	−	0.758241i	$-0.726059\pi$
$139$	−297028.			−1.30395			−0.651974	+	0.758241i	$0.726059\pi$
$140$		0			0
$141$	−132921.			−0.563049
$142$		0			0
$143$	−163.408	+	283.031i	−0.000668241	+	0.00115743i
$144$		0			0
$145$	79532.6	+	137754.i	0.314141	+	0.544109i
$146$		0			0
$147$	117233.	+	95586.7i	0.447464	+	0.364842i
$148$		0			0
$149$	−65087.1	−	112734.i	−0.240176	−	0.415997i	0.720588	−	0.693363i	$-0.243872\pi$
$149$	−65087.1	−	112734.i	−0.240176	−	0.415997i	−0.960764	+	0.277366i	$0.910538\pi$
$150$		0			0
$151$	−82198.7	+	142372.i	−0.293375	+	0.508140i	−0.974606	−	0.223929i	$-0.928112\pi$
$151$	−82198.7	+	142372.i	−0.293375	+	0.508140i	0.681231	+	0.732069i	$0.261445\pi$
$152$		0			0
$153$	11169.2			0.0385740
$154$		0			0
$155$	−155309.			−0.519240
$156$		0			0
$157$	41102.3	−	71191.2i	0.133081	−	0.230503i	−0.791782	−	0.610804i	$-0.790846\pi$
$157$	41102.3	−	71191.2i	0.133081	−	0.230503i	0.924863	+	0.380301i	$0.124180\pi$
$158$		0			0
$159$	84575.8	+	146490.i	0.265310	+	0.459530i
$160$		0			0
$161$	115565.	−	324613.i	0.351367	−	0.986964i
$162$		0			0
$163$	−216939.	−	375749.i	−0.639541	−	1.10772i	−0.985534	−	0.169480i	$-0.945791\pi$
$163$	−216939.	−	375749.i	−0.639541	−	1.10772i	0.345992	−	0.938237i	$-0.387542\pi$
$164$		0			0
$165$	−2086.47	+	3613.87i	−0.00596626	+	0.0103339i
$166$		0			0
$167$	353734.			0.981488			0.490744	−	0.871304i	$-0.336725\pi$
$167$	353734.			0.981488			0.490744	+	0.871304i	$0.336725\pi$
$168$		0			0
$169$	−369483.			−0.995126
$170$		0			0
$171$	−43966.0	+	76151.3i	−0.114981	+	0.199153i
$172$		0			0
$173$	−201313.	−	348685.i	−0.511396	−	0.885763i	−0.999913	−	0.0132089i	$-0.995795\pi$
$173$	−201313.	−	348685.i	−0.511396	−	0.885763i	0.488517	−	0.872554i	$-0.337538\pi$
$174$		0			0
$175$	−65953.9	+	12109.4i	−0.162797	+	0.0298902i
$176$		0			0
$177$	35795.5	+	61999.6i	0.0858806	+	0.148750i
$178$		0			0
$179$	206755.	−	358111.i	0.482308	−	0.835382i	−0.517486	−	0.855692i	$-0.673132\pi$
$179$	206755.	−	358111.i	0.482308	−	0.835382i	0.999794	+	0.0203099i	$0.00646529\pi$
$180$		0			0
$181$	552811.			1.25424			0.627119	−	0.778923i	$-0.284234\pi$
$181$	552811.			1.25424			0.627119	+	0.778923i	$0.284234\pi$
$182$		0			0
$183$	458660.			1.01243
$184$		0			0
$185$	173367.	−	300281.i	0.372425	−	0.645058i
$186$		0			0
$187$	529.691	+	917.452i	0.00110769	+	0.00191858i
$188$		0			0
$189$	−61258.0	−	71968.0i	−0.124741	−	0.146550i
$190$		0			0
$191$	−395164.	−	684444.i	−0.783779	−	1.35755i	−0.929726	−	0.368253i	$-0.879956\pi$
$191$	−395164.	−	684444.i	−0.783779	−	1.35755i	0.145947	−	0.989292i	$-0.453377\pi$
$192$		0			0
$193$	−235888.	+	408569.i	−0.455839	+	0.789537i	−0.998736	−	0.0502625i	$-0.983994\pi$
$193$	−235888.	+	408569.i	−0.455839	+	0.789537i	0.542897	+	0.839800i	$0.317328\pi$
$194$		0			0
$195$	23105.5			0.0435140
$196$		0			0
$197$	188443.			0.345951			0.172976	−	0.984926i	$-0.444662\pi$
$197$	188443.			0.345951			0.172976	+	0.984926i	$0.444662\pi$
$198$		0			0
$199$	533185.	−	923504.i	0.954433	−	1.65313i	0.218773	−	0.975776i	$-0.429794\pi$
$199$	533185.	−	923504.i	0.954433	−	1.65313i	0.735660	−	0.677351i	$-0.236872\pi$
$200$		0			0
$201$	105145.	+	182117.i	0.183569	+	0.317951i
$202$		0			0
$203$	−221476.	−	260197.i	−0.377212	−	0.443162i
$204$		0			0
$205$	549014.	+	950921.i	0.912429	+	1.58037i
$206$		0			0
$207$	−107644.	+	186444.i	−0.174607	+	0.302429i
$208$		0			0
$209$	−8340.20			−0.0132072
$210$		0			0
$211$	69397.7			0.107310			0.0536548	−	0.998560i	$-0.482913\pi$
$211$	69397.7			0.107310			0.0536548	+	0.998560i	$0.482913\pi$
$212$		0			0
$213$	−270122.	+	467866.i	−0.407954	+	0.706598i
$214$		0			0
$215$	52069.4	+	90186.8i	0.0768221	+	0.133060i
$216$		0			0
$217$	328139.	−	60247.9i	0.473052	−	0.0868545i
$218$		0			0
$219$	90265.2	+	156344.i	0.127177	+	0.220278i
$220$		0			0
$221$	2932.89	−	5079.92i	0.00403939	−	0.00699643i
$222$		0			0
$223$	1.15636e6			1.55715			0.778574	−	0.627553i	$-0.215943\pi$
$223$	1.15636e6			1.55715			0.778574	+	0.627553i	$0.215943\pi$
$224$		0			0
$225$	41896.7			0.0551726
$226$		0			0
$227$	309983.	−	536906.i	0.399275	−	0.691565i	−0.594361	−	0.804198i	$-0.702595\pi$
$227$	309983.	−	536906.i	0.399275	−	0.691565i	0.993637	+	0.112633i	$0.0359284\pi$
$228$		0			0
$229$	324585.	+	562198.i	0.409016	+	0.708436i	0.994780	−	0.102045i	$-0.0325387\pi$
$229$	324585.	+	562198.i	0.409016	+	0.708436i	−0.585764	+	0.810482i	$0.699205\pi$
$230$		0			0
$231$	3006.42	−	8444.82i	0.00370697	−	0.0104126i
$232$		0			0
$233$	133434.	+	231115.i	0.161019	+	0.278893i	0.935234	−	0.354029i	$-0.115189\pi$
$233$	133434.	+	231115.i	0.161019	+	0.278893i	−0.774215	+	0.632922i	$0.781855\pi$
$234$		0			0
$235$	−445662.	+	771910.i	−0.526425	+	0.911794i
$236$		0			0
$237$	−424649.			−0.491088
$238$		0			0
$239$	504518.			0.571323			0.285661	−	0.958331i	$-0.407787\pi$
$239$	504518.			0.571323			0.285661	+	0.958331i	$0.407787\pi$
$240$		0			0
$241$	−452336.	+	783469.i	−0.501670	+	0.868918i	0.498328	+	0.866989i	$0.333948\pi$
$241$	−452336.	+	783469.i	−0.501670	+	0.868918i	−0.999998	+	0.00192971i	$0.999386\pi$
$242$		0			0
$243$	29524.5	+	51137.9i	0.0320750	+	0.0555556i
$244$		0			0
$245$	948163.	−	360320.i	1.00918	−	0.383507i
$246$		0			0
$247$	23089.8	+	39992.7i	0.0240812	+	0.0417099i
$248$		0			0
$249$	189932.	−	328971.i	0.194133	−	0.336248i
$250$		0			0
$251$	1.43747e6			1.44018			0.720089	−	0.693882i	$-0.244101\pi$
$251$	1.43747e6			1.44018			0.720089	+	0.693882i	$0.244101\pi$
$252$		0			0
$253$	−20419.6			−0.0200561
$254$		0			0
$255$	37448.5	−	64862.8i	0.0360649	−	0.0624662i
$256$		0			0
$257$	−550635.	−	953728.i	−0.520033	−	0.900724i	−0.999729	−	0.0232892i	$-0.992586\pi$
$257$	−550635.	−	953728.i	−0.520033	−	0.900724i	0.479695	−	0.877435i	$-0.340747\pi$
$258$		0			0
$259$	−249807.	+	701691.i	−0.231396	+	0.649975i
$260$		0			0
$261$	106745.	+	184887.i	0.0969939	+	0.167998i
$262$		0			0
$263$	630499.	−	1.09206e6i	0.562076	−	0.973544i	−0.435239	−	0.900315i	$-0.643336\pi$
$263$	630499.	−	1.09206e6i	0.562076	−	0.973544i	0.997315	−	0.0732293i	$-0.0233305\pi$
$264$		0			0
$265$	1.13427e6			0.992209
$266$		0			0
$267$	−309406.			−0.265614
$268$		0			0
$269$	−241521.	+	418327.i	−0.203505	+	0.352481i	−0.949655	−	0.313297i	$-0.898567\pi$
$269$	−241521.	+	418327.i	−0.203505	+	0.352481i	0.746150	+	0.665777i	$0.231900\pi$
$270$		0			0
$271$	−259591.	−	449624.i	−0.214717	−	0.371900i	0.738468	−	0.674288i	$-0.235549\pi$
$271$	−259591.	−	449624.i	−0.214717	−	0.371900i	−0.953185	+	0.302388i	$0.902216\pi$
$272$		0			0
$273$	−48817.6	+	8963.13i	−0.0396433	+	0.00727869i
$274$		0			0
$275$	1986.92	+	3441.44i	0.00158434	+	0.00274415i
$276$		0			0
$277$	1.18853e6	−	2.05860e6i	0.930704	−	1.61203i	0.148582	−	0.988900i	$-0.452529\pi$
$277$	1.18853e6	−	2.05860e6i	0.930704	−	1.61203i	0.782122	−	0.623126i	$-0.214138\pi$
$278$		0			0
$279$	−208448.			−0.160320
$280$		0			0
$281$	940528.			0.710568			0.355284	−	0.934758i	$-0.384384\pi$
$281$	940528.			0.710568			0.355284	+	0.934758i	$0.384384\pi$
$282$		0			0
$283$	−324746.	+	562477.i	−0.241034	+	0.417483i	−0.961009	−	0.276517i	$-0.910820\pi$
$283$	−324746.	+	562477.i	−0.241034	+	0.417483i	0.719975	+	0.694000i	$0.244153\pi$
$284$		0			0
$285$	294821.	+	510646.i	0.215004	+	0.372398i
$286$		0			0
$287$	−1.52885e6	−	1.79614e6i	−1.09562	−	1.28717i
$288$		0			0
$289$	700421.	+	1.21317e6i	0.493304	+	0.854428i
$290$		0			0
$291$	−385229.	+	667236.i	−0.266677	+	0.461899i
$292$		0			0
$293$	−793396.			−0.539909			−0.269955	−	0.962873i	$-0.587009\pi$
$293$	−793396.			−0.539909			−0.269955	+	0.962873i	$0.587009\pi$
$294$		0			0
$295$	480065.			0.321177
$296$		0			0
$297$	−2800.35	+	4850.35i	−0.00184213	+	0.00319067i
$298$		0			0
$299$	56531.6	+	97915.7i	0.0365691	+	0.0633395i
$300$		0			0
$301$	−144998.	−	170349.i	−0.0922458	−	0.108374i
$302$		0			0
$303$	−324905.	−	562752.i	−0.203306	−	0.352136i
$304$		0			0
$305$	1.53781e6	−	2.66357e6i	0.946572	−	1.63951i
$306$		0			0
$307$	2.08282e6			1.26126			0.630632	−	0.776082i	$-0.282796\pi$
$307$	2.08282e6			1.26126			0.630632	+	0.776082i	$0.282796\pi$
$308$		0			0
$309$	−318099.			−0.189525
$310$		0			0
$311$	598523.	−	1.03667e6i	0.350897	−	0.607772i	−0.635510	−	0.772093i	$-0.719210\pi$
$311$	598523.	−	1.03667e6i	0.350897	−	0.607772i	0.986407	+	0.164321i	$0.0525433\pi$
$312$		0			0
$313$	−1.29207e6	−	2.23793e6i	−0.745460	−	1.29117i	−0.949980	−	0.312311i	$-0.898897\pi$
$313$	−1.29207e6	−	2.23793e6i	−0.745460	−	1.29117i	0.204520	−	0.978862i	$-0.434437\pi$
$314$		0			0
$315$	−623326.	+	114445.i	−0.353947	+	0.0649863i
$316$		0			0
$317$	−645757.	−	1.11848e6i	−0.360928	−	0.625146i	0.627185	−	0.778870i	$-0.284207\pi$
$317$	−645757.	−	1.11848e6i	−0.360928	−	0.625146i	−0.988114	+	0.153724i	$0.950873\pi$
$318$		0			0
$319$	−10124.5	+	17536.2i	−0.00557056	+	0.00964849i
$320$		0			0
$321$	433454.			0.234791
$322$		0			0
$323$	149692.			0.0798350
$324$		0			0
$325$	11001.5	−	19055.2i	0.00577756	−	0.0100070i
$326$		0			0
$327$	−73604.2	−	127486.i	−0.0380657	−	0.0659316i
$328$		0			0
$329$	642161.	−	1.80378e6i	0.327080	−	0.918744i
$330$		0			0
$331$	1.55407e6	+	2.69172e6i	0.779650	+	1.35039i	0.932143	+	0.362089i	$0.117937\pi$
$331$	1.55407e6	+	2.69172e6i	0.779650	+	1.35039i	−0.152493	+	0.988305i	$0.548730\pi$
$332$		0			0
$333$	232685.	−	403022.i	0.114989	−	0.199167i
$334$		0			0
$335$	1.41014e6			0.686515
$336$		0			0
$337$	−3.45367e6			−1.65656			−0.828278	−	0.560318i	$-0.810679\pi$
$337$	−3.45367e6			−1.65656			−0.828278	+	0.560318i	$0.810679\pi$
$338$		0			0
$339$	−657129.	+	1.13818e6i	−0.310564	+	0.537913i
$340$		0			0
$341$	−9885.48	−	17122.1i	−0.00460375	−	0.00797393i
$342$		0			0
$343$	−1.86351e6	+	1.12910e6i	−0.855259	+	0.518201i
$344$		0			0
$345$	721822.	+	1.25023e6i	0.326499	+	0.565514i
$346$		0			0
$347$	−1.52325e6	+	2.63834e6i	−0.679119	+	1.17627i	0.296127	+	0.955149i	$0.404305\pi$
$347$	−1.52325e6	+	2.63834e6i	−0.679119	+	1.17627i	−0.975246	+	0.221121i	$0.929028\pi$
$348$		0			0
$349$	−3.94475e6			−1.73363			−0.866814	−	0.498631i	$-0.833836\pi$
$349$	−3.94475e6			−1.73363			−0.866814	+	0.498631i	$0.833836\pi$
$350$		0			0
$351$	31011.0			0.0134353
$352$		0			0
$353$	−163086.	+	282473.i	−0.0696594	+	0.120654i	−0.898751	−	0.438459i	$-0.855525\pi$
$353$	−163086.	+	282473.i	−0.0696594	+	0.120654i	0.829092	+	0.559112i	$0.188858\pi$
$354$		0			0
$355$	1.81135e6	+	3.13735e6i	0.762837	+	1.32127i
$356$		0			0
$357$	−53960.1	+	151570.i	−0.0224079	+	0.0629423i
$358$		0			0
$359$	−1.83256e6	−	3.17408e6i	−0.750450	−	1.29982i	−0.947605	−	0.319445i	$-0.896504\pi$
$359$	−1.83256e6	−	3.17408e6i	−0.750450	−	1.29982i	0.197155	−	0.980372i	$-0.436830\pi$
$360$		0			0
$361$	648807.	−	1.12377e6i	0.262028	−	0.453846i
$362$		0			0
$363$	1.44893e6			0.577139
$364$		0			0
$365$	1.21058e6			0.475620
$366$		0			0
$367$	419401.	−	726423.i	0.162541	−	0.281530i	−0.773238	−	0.634116i	$-0.781364\pi$
$367$	419401.	−	726423.i	0.162541	−	0.281530i	0.935779	+	0.352586i	$0.114698\pi$
$368$		0			0
$369$	736859.	+	1.27628e6i	0.281720	+	0.487954i
$370$		0			0
$371$	−2.39651e6	+	440010.i	−0.903950	+	0.165969i
$372$		0			0
$373$	883385.	+	1.53007e6i	0.328759	+	0.569427i	0.982266	−	0.187493i	$-0.0600362\pi$
$373$	883385.	+	1.53007e6i	0.328759	+	0.569427i	−0.653507	+	0.756921i	$0.726703\pi$
$374$		0			0
$375$	−708213.	+	1.22666e6i	−0.260067	+	0.450450i
$376$		0			0
$377$	112119.			0.0406280
$378$		0			0
$379$	−1.02568e6			−0.366786			−0.183393	−	0.983040i	$-0.558708\pi$
$379$	−1.02568e6			−0.366786			−0.183393	+	0.983040i	$0.558708\pi$
$380$		0			0
$381$	−1.56866e6	+	2.71699e6i	−0.553624	+	0.958905i
$382$		0			0
$383$	−981203.	−	1.69949e6i	−0.341792	−	0.592001i	0.642974	−	0.765888i	$-0.277701\pi$
$383$	−981203.	−	1.69949e6i	−0.341792	−	0.592001i	−0.984766	+	0.173887i	$0.944367\pi$
$384$		0			0
$385$	−38961.4	−	45773.2i	−0.0133962	−	0.0157383i
$386$		0			0
$387$	69884.8	+	121044.i	0.0237195	+	0.0410834i
$388$		0			0
$389$	1.81474e6	−	3.14322e6i	0.608052	−	1.05318i	−0.383509	−	0.923537i	$-0.625285\pi$
$389$	1.81474e6	−	3.14322e6i	0.608052	−	1.05318i	0.991561	−	0.129640i	$-0.0413821\pi$
$390$		0			0
$391$	366498.			0.121235
$392$		0			0
$393$	2.69538e6			0.880316
$394$		0			0
$395$	−1.42378e6	+	2.46606e6i	−0.459145	+	0.795262i
$396$		0			0
$397$	−1.49711e6	−	2.59307e6i	−0.476735	−	0.825728i	0.522910	−	0.852388i	$-0.324846\pi$
$397$	−1.49711e6	−	2.59307e6i	−0.476735	−	0.825728i	−0.999645	+	0.0266595i	$0.991513\pi$
$398$		0			0
$399$	−820993.	−	964531.i	−0.258171	−	0.303308i
$400$		0			0
$401$	2.13003e6	+	3.68931e6i	0.661491	+	1.14574i	0.980224	+	0.197892i	$0.0634094\pi$
$401$	2.13003e6	+	3.68931e6i	0.661491	+	1.14574i	−0.318733	+	0.947845i	$0.603257\pi$
$402$		0			0
$403$	−54735.8	+	94805.2i	−0.0167884	+	0.0290783i
$404$		0			0
$405$	395963.			0.119955
$406$		0			0
$407$	44139.6			0.0132081
$408$		0			0
$409$	−1.44228e6	+	2.49811e6i	−0.426326	+	0.738419i	−0.996543	−	0.0830751i	$-0.973526\pi$
$409$	−1.44228e6	+	2.49811e6i	−0.426326	+	0.738419i	0.570217	+	0.821494i	$0.306859\pi$
$410$		0			0
$411$	443920.	+	768893.i	0.129628	+	0.224523i
$412$		0			0
$413$	−1.01429e6	+	186228.i	−0.292608	+	0.0537241i
$414$		0			0
$415$	−1.27362e6	−	2.20597e6i	−0.363010	−	0.628753i
$416$		0			0
$417$	−1.33663e6	+	2.31510e6i	−0.376417	+	0.651974i
$418$		0			0
$419$	−232038.			−0.0645690			−0.0322845	−	0.999479i	$-0.510278\pi$
$419$	−232038.			−0.0645690			−0.0322845	+	0.999479i	$0.510278\pi$
$420$		0			0
$421$	−3.39524e6			−0.933609			−0.466805	−	0.884360i	$-0.654595\pi$
$421$	−3.39524e6			−0.933609			−0.466805	+	0.884360i	$0.654595\pi$
$422$		0			0
$423$	−598145.	+	1.03602e6i	−0.162538	+	0.281525i
$424$		0			0
$425$	−35661.7	−	61768.0i	−0.00957702	−	0.0165879i
$426$		0			0
$427$	−2.21585e6	+	6.22417e6i	−0.588127	+	1.65201i
$428$		0			0
$429$	1470.67	+	2547.28i	0.000385809	+	0.000668241i
$430$		0			0
$431$	2.43893e6	−	4.22435e6i	0.632421	−	1.09539i	−0.354634	−	0.935005i	$-0.615395\pi$
$431$	2.43893e6	−	4.22435e6i	0.632421	−	1.09539i	0.987055	−	0.160380i	$-0.0512720\pi$
$432$		0			0
$433$	931802.			0.238838			0.119419	−	0.992844i	$-0.461897\pi$
$433$	931802.			0.238838			0.119419	+	0.992844i	$0.461897\pi$
$434$		0			0
$435$	1.43159e6			0.362739
$436$		0			0
$437$	−1.44266e6	+	2.49877e6i	−0.361378	+	0.625925i
$438$		0			0
$439$	741706.	+	1.28467e6i	0.183684	+	0.318149i	0.943132	−	0.332418i	$-0.107865\pi$
$439$	741706.	+	1.28467e6i	0.183684	+	0.318149i	−0.759449	+	0.650567i	$0.774531\pi$
$440$		0			0
$441$	1.27258e6	−	483604.i	0.311592	−	0.118411i
$442$		0			0
$443$	−2.26750e6	−	3.92743e6i	−0.548957	−	0.950821i	−0.998346	−	0.0574849i	$-0.981692\pi$
$443$	−2.26750e6	−	3.92743e6i	−0.548957	−	0.950821i	0.449390	−	0.893336i	$-0.351641\pi$
$444$		0			0
$445$	−1.03739e6	+	1.79681e6i	−0.248337	+	0.430132i
$446$		0			0
$447$	−1.17157e6			−0.277331
$448$		0			0
$449$	−5.80527e6			−1.35896			−0.679479	−	0.733695i	$-0.737794\pi$
$449$	−5.80527e6			−1.35896			−0.679479	+	0.733695i	$0.737794\pi$
$450$		0			0
$451$	−69889.8	+	121053.i	−0.0161798	+	0.0280242i
$452$		0			0
$453$	739789.	+	1.28135e6i	0.169380	+	0.293375i
$454$		0			0
$455$	−111626.	+	313549.i	−0.0252776	+	0.0710031i
$456$		0			0
$457$	1.97166e6	+	3.41502e6i	0.441613	+	0.764897i	0.997809	−	0.0661543i	$-0.0210730\pi$
$457$	1.97166e6	+	3.41502e6i	0.441613	+	0.764897i	−0.556196	+	0.831051i	$0.687740\pi$
$458$		0			0
$459$	50261.5	−	87055.5i	0.0111353	−	0.0192870i
$460$		0			0
$461$	−3.03473e6			−0.665070			−0.332535	−	0.943091i	$-0.607904\pi$
$461$	−3.03473e6			−0.665070			−0.332535	+	0.943091i	$0.607904\pi$
$462$		0			0
$463$	−1.47871e6			−0.320575			−0.160288	−	0.987070i	$-0.551242\pi$
$463$	−1.47871e6			−0.320575			−0.160288	+	0.987070i	$0.551242\pi$
$464$		0			0
$465$	−698892.	+	1.21052e6i	−0.149892	+	0.259620i
$466$		0			0
$467$	511990.	+	886792.i	0.108635	+	0.188161i	0.915217	−	0.402960i	$-0.132019\pi$
$467$	511990.	+	886792.i	0.108635	+	0.188161i	−0.806583	+	0.591121i	$0.798685\pi$
$468$		0			0
$469$	−2.97936e6	+	547024.i	−0.625447	+	0.114835i
$470$		0			0
$471$	−369920.	−	640721.i	−0.0768345	−	0.133081i
$472$		0			0
$473$	−6628.46	+	11480.8i	−0.00136226	+	0.00235950i
$474$		0			0
$475$	561509.			0.114189
$476$		0			0
$477$	1.52236e6			0.306353
$478$		0			0
$479$	3.17762e6	−	5.50380e6i	0.632796	−	1.09603i	−0.354182	−	0.935176i	$-0.615241\pi$
$479$	3.17762e6	−	5.50380e6i	0.632796	−	1.09603i	0.986978	−	0.160858i	$-0.0514260\pi$
$480$		0			0
$481$	−122200.	−	211657.i	−0.0240829	−	0.0417128i
$482$		0			0
$483$	−2.01007e6	−	2.36150e6i	−0.392051	−	0.460595i
$484$		0			0
$485$	2.58321e6	+	4.47426e6i	0.498662	+	0.863708i
$486$		0			0
$487$	4.00131e6	−	6.93047e6i	0.764504	−	1.32416i	−0.176005	−	0.984389i	$-0.556317\pi$
$487$	4.00131e6	−	6.93047e6i	0.764504	−	1.32416i	0.940508	−	0.339770i	$-0.110349\pi$
$488$		0			0
$489$	−3.90490e6			−0.738478
$490$		0			0
$491$	−2.89285e6			−0.541530			−0.270765	−	0.962646i	$-0.587277\pi$
$491$	−2.89285e6			−0.541530			−0.270765	+	0.962646i	$0.587277\pi$
$492$		0			0
$493$	181718.	−	314745.i	0.0336730	−	0.0583233i
$494$		0			0
$495$	18778.2	+	32524.8i	0.00344462	+	0.00596626i
$496$		0			0
$497$	−5.04409e6	−	5.92598e6i	−0.915993	−	1.07614i
$498$		0			0
$499$	−2.83363e6	−	4.90799e6i	−0.509438	−	0.882373i	−0.999940	−	0.0109328i	$-0.996520\pi$
$499$	−2.83363e6	−	4.90799e6i	−0.509438	−	0.882373i	0.490502	−	0.871440i	$-0.336813\pi$
$500$		0			0
$501$	1.59180e6	−	2.75708e6i	0.283331	−	0.490744i
$502$		0			0
$503$	458347.			0.0807746			0.0403873	−	0.999184i	$-0.487141\pi$
$503$	458347.			0.0807746			0.0403873	+	0.999184i	$0.487141\pi$
$504$		0			0
$505$	−4.35741e6			−0.760326
$506$		0			0
$507$	−1.66268e6	+	2.87984e6i	−0.287268	+	0.497563i
$508$		0			0
$509$	4.46450e6	+	7.73274e6i	0.763798	+	1.32294i	0.940880	+	0.338740i	$0.110001\pi$
$509$	4.46450e6	+	7.73274e6i	0.763798	+	1.32294i	−0.177082	+	0.984196i	$0.556666\pi$
$510$		0			0
$511$	−2.55772e6	+	469609.i	−0.433313	+	0.0795581i
$512$		0			0
$513$	395694.	+	685362.i	0.0663844	+	0.114981i
$514$		0			0
$515$	−1.06653e6	+	1.84729e6i	−0.177197	+	0.306914i
$516$		0			0
$517$	−113466.			−0.0186698
$518$		0			0
$519$	−3.62364e6			−0.590509
$520$		0			0
$521$	−2.51245e6	+	4.35170e6i	−0.405512	+	0.702367i	−0.994381	−	0.105861i	$-0.966240\pi$
$521$	−2.51245e6	+	4.35170e6i	−0.405512	+	0.702367i	0.588869	+	0.808229i	$0.299573\pi$
$522$		0			0
$523$	3.06664e6	+	5.31158e6i	0.490240	+	0.849121i	0.999937	−	0.0112329i	$-0.00357563\pi$
$523$	3.06664e6	+	5.31158e6i	0.490240	+	0.849121i	−0.509696	+	0.860354i	$0.670242\pi$
$524$		0			0
$525$	−202409.	+	568552.i	−0.0320502	+	0.0900268i
$526$		0			0
$527$	177427.	+	307313.i	0.0278288	+	0.0482009i
$528$		0			0
$529$	−313955.	+	543786.i	−0.0487785	+	0.0844868i
$530$		0			0
$531$	644319.			0.0991663
$532$		0			0
$533$	773959.			0.118005
$534$		0			0
$535$	1.45330e6	−	2.51719e6i	0.219518	−	0.380217i
$536$		0			0
$537$	−1.86080e6	−	3.22300e6i	−0.278461	−	0.482308i
$538$		0			0
$539$	100075.	+	81596.2i	0.0148372	+	0.0120976i
$540$		0			0
$541$	1.24987e6	+	2.16484e6i	0.183599	+	0.318004i	0.943104	−	0.332499i	$-0.107892\pi$
$541$	1.24987e6	+	2.16484e6i	0.183599	+	0.318004i	−0.759504	+	0.650502i	$0.774558\pi$
$542$		0			0
$543$	2.48765e6	−	4.30873e6i	0.362068	−	0.627119i
$544$		0			0
$545$	−987131.			−0.142358
$546$		0			0
$547$	−7.07863e6			−1.01153			−0.505767	−	0.862670i	$-0.668791\pi$
$547$	−7.07863e6			−1.01153			−0.505767	+	0.862670i	$0.668791\pi$
$548$		0			0
$549$	2.06397e6	−	3.57490e6i	0.292262	−	0.506213i
$550$		0			0
$551$	1.43061e6	+	2.47790e6i	0.200744	+	0.347700i
$552$		0			0
$553$	2.05154e6	−	5.76263e6i	0.285277	−	0.801323i
$554$		0			0
$555$	−1.56031e6	−	2.70253e6i	−0.215019	−	0.372425i
$556$		0			0
$557$	5.86600e6	−	1.01602e7i	0.801132	−	1.38760i	−0.117739	−	0.993045i	$-0.537565\pi$
$557$	5.86600e6	−	1.01602e7i	0.801132	−	1.38760i	0.918871	−	0.394557i	$-0.129102\pi$
$558$		0			0
$559$	73403.5			0.00993544
$560$		0			0
$561$	9534.44			0.00127905
$562$		0			0
$563$	−4.77645e6	+	8.27305e6i	−0.635088	+	1.10001i	0.351408	+	0.936222i	$0.385703\pi$
$563$	−4.77645e6	+	8.27305e6i	−0.635088	+	1.10001i	−0.986496	+	0.163783i	$0.947630\pi$
$564$		0			0
$565$	4.40649e6	+	7.63226e6i	0.580726	+	1.00585i
$566$		0			0
$567$	−836596.	+	153603.i	−0.109284	+	0.0200651i
$568$		0			0
$569$	−3.92055e6	−	6.79058e6i	−0.507652	−	0.879279i	−0.999961	−	0.00885809i	$-0.997180\pi$
$569$	−3.92055e6	−	6.79058e6i	−0.507652	−	0.879279i	0.492309	−	0.870420i	$-0.336153\pi$
$570$		0			0
$571$	2.22895e6	−	3.86066e6i	0.286095	−	0.495531i	−0.686779	−	0.726866i	$-0.740976\pi$
$571$	2.22895e6	−	3.86066e6i	0.286095	−	0.495531i	0.972874	+	0.231335i	$0.0743093\pi$
$572$		0			0
$573$	−7.11295e6			−0.905030
$574$		0			0
$575$	1.37476e6			0.173404
$576$		0			0
$577$	2.72772e6	−	4.72456e6i	0.341084	−	0.590774i	−0.643551	−	0.765404i	$-0.722539\pi$
$577$	2.72772e6	−	4.72456e6i	0.341084	−	0.590774i	0.984634	+	0.174629i	$0.0558728\pi$
$578$		0			0
$579$	2.12299e6	+	3.67712e6i	0.263179	+	0.455839i
$580$		0			0
$581$	3.54666e6	+	4.16674e6i	0.435893	+	0.512102i
$582$		0			0
$583$	72196.8	+	125049.i	0.00879725	+	0.0152373i
$584$		0			0
$585$	103975.	−	180090.i	0.0125614	−	0.0217570i
$586$		0			0
$587$	5.52379e6			0.661670			0.330835	−	0.943689i	$-0.392670\pi$
$587$	5.52379e6			0.661670			0.330835	+	0.943689i	$0.392670\pi$
$588$		0			0
$589$	−2.79367e6			−0.331808
$590$		0			0
$591$	847995.	−	1.46877e6i	0.0998676	−	0.172976i
$592$		0			0
$593$	4.32543e6	+	7.49187e6i	0.505118	+	0.874890i	0.999982	+	0.00591957i	$0.00188427\pi$
$593$	4.32543e6	+	7.49187e6i	0.505118	+	0.874890i	−0.494865	+	0.868970i	$0.664782\pi$
$594$		0			0
$595$	699290.	+	821550.i	0.0809776	+	0.0951353i
$596$		0			0
$597$	−4.79867e6	−	8.31154e6i	−0.551042	−	0.954433i
$598$		0			0
$599$	−6.10796e6	+	1.05793e7i	−0.695551	+	1.20473i	0.274444	+	0.961603i	$0.411506\pi$
$599$	−6.10796e6	+	1.05793e7i	−0.695551	+	1.20473i	−0.969995	+	0.243127i	$0.921827\pi$
$600$		0			0
$601$	1.61479e7			1.82360			0.911801	−	0.410632i	$-0.134692\pi$
$601$	1.61479e7			1.82360			0.911801	+	0.410632i	$0.134692\pi$
$602$		0			0
$603$	1.89262e6			0.211967
$604$		0			0
$605$	4.85801e6	−	8.41433e6i	0.539598	−	0.934611i
$606$		0			0
$607$	5.52286e6	+	9.56587e6i	0.608404	+	1.05379i	0.991504	+	0.130080i	$0.0415233\pi$
$607$	5.52286e6	+	9.56587e6i	0.608404	+	1.05379i	−0.383100	+	0.923707i	$0.625143\pi$
$608$		0			0
$609$	−3.02468e6	+	555344.i	−0.330473	+	0.0606763i
$610$		0			0
$611$	314130.	+	544090.i	0.0340414	+	0.0589614i
$612$		0			0
$613$	5.17071e6	−	8.95593e6i	0.555775	−	0.962631i	−0.442068	−	0.896982i	$-0.645755\pi$
$613$	5.17071e6	−	8.95593e6i	0.555775	−	0.962631i	0.997843	−	0.0656490i	$-0.0209118\pi$
$614$		0			0
$615$	9.88226e6			1.05358
$616$		0			0
$617$	−1.11422e7			−1.17831			−0.589154	−	0.808021i	$-0.700539\pi$
$617$	−1.11422e7			−1.17831			−0.589154	+	0.808021i	$0.700539\pi$
$618$		0			0
$619$	−570310.	+	987806.i	−0.0598252	+	0.103620i	−0.894387	−	0.447294i	$-0.852388\pi$
$619$	−570310.	+	987806.i	−0.0598252	+	0.103620i	0.834562	+	0.550915i	$0.185721\pi$
$620$		0			0
$621$	968792.	+	1.67800e6i	0.100810	+	0.174607i
$622$		0			0
$623$	1.49479e6	−	4.19875e6i	0.154297	−	0.433411i
$624$		0			0
$625$	5.55723e6	+	9.62541e6i	0.569061	+	0.985642i
$626$		0			0
$627$	−37530.9	+	65005.5i	−0.00381259	+	0.00660360i
$628$		0			0
$629$	−792230.			−0.0798407
$630$		0			0
$631$	−7.26422e6			−0.726299			−0.363150	−	0.931731i	$-0.618299\pi$
$631$	−7.26422e6			−0.726299			−0.363150	+	0.931731i	$0.618299\pi$
$632$		0			0
$633$	312290.	−	540901.i	0.0309776	−	0.0536548i
$634$		0			0
$635$	1.05189e7	+	1.82192e7i	1.03523	+	1.79306i
$636$		0			0
$637$	114212.	−	705774.i	0.0111523	−	0.0689155i
$638$		0			0
$639$	2.43110e6	+	4.21079e6i	0.235533	+	0.407954i
$640$		0			0
$641$	5.32494e6	−	9.22307e6i	0.511882	−	0.886605i	−0.488023	−	0.872831i	$-0.662282\pi$
$641$	5.32494e6	−	9.22307e6i	0.511882	−	0.886605i	0.999905	−	0.0137746i	$-0.00438474\pi$
$642$		0			0
$643$	−1.95575e6			−0.186546			−0.0932730	−	0.995641i	$-0.529733\pi$
$643$	−1.95575e6			−0.186546			−0.0932730	+	0.995641i	$0.529733\pi$
$644$		0			0
$645$	937249.			0.0887065
$646$		0			0
$647$	645496.	−	1.11803e6i	0.0606224	−	0.105001i	−0.834121	−	0.551581i	$-0.814025\pi$
$647$	645496.	−	1.11803e6i	0.0606224	−	0.105001i	0.894744	+	0.446580i	$0.147358\pi$
$648$		0			0
$649$	30556.3	+	52925.0i	0.00284766	+	0.00493230i
$650$		0			0
$651$	1.00704e6	−	2.82871e6i	0.0931312	−	0.261599i
$652$		0			0
$653$	−5.31437e6	−	9.20475e6i	−0.487718	−	0.844752i	0.512183	−	0.858877i	$-0.328837\pi$
$653$	−5.31437e6	−	9.20475e6i	−0.487718	−	0.844752i	−0.999900	+	0.0141249i	$0.995504\pi$
$654$		0			0
$655$	9.03716e6	−	1.56528e7i	0.823054	−	1.42557i
$656$		0			0
$657$	1.62477e6			0.146852
$658$		0			0
$659$	−3.25503e6			−0.291972			−0.145986	−	0.989287i	$-0.546635\pi$
$659$	−3.25503e6			−0.291972			−0.145986	+	0.989287i	$0.546635\pi$
$660$		0			0
$661$	3.40454e6	−	5.89684e6i	0.303078	−	0.524947i	−0.673753	−	0.738956i	$-0.735319\pi$
$661$	3.40454e6	−	5.89684e6i	0.303078	−	0.524947i	0.976832	+	0.214009i	$0.0686522\pi$
$662$		0			0
$663$	−26396.1	−	45719.3i	−0.00233214	−	0.00403939i
$664$		0			0
$665$	−8.35395e6	+	1.53382e6i	−0.732551	+	0.134500i
$666$		0			0
$667$	3.50263e6	+	6.06673e6i	0.304845	+	0.528007i
$668$		0			0
$669$	5.20361e6	−	9.01291e6i	0.449510	−	0.778574i
$670$		0			0
$671$	391529.			0.0335705
$672$		0			0
$673$	1.20192e7			1.02291			0.511456	−	0.859310i	$-0.329107\pi$
$673$	1.20192e7			1.02291			0.511456	+	0.859310i	$0.329107\pi$
$674$		0			0
$675$	188535.	−	326552.i	0.0159270	−	0.0275863i
$676$		0			0
$677$	1.15506e7	+	2.00063e7i	0.968576	+	1.67762i	0.699684	+	0.714452i	$0.253324\pi$
$677$	1.15506e7	+	2.00063e7i	0.968576	+	1.67762i	0.268892	+	0.963170i	$0.413343\pi$
$678$		0			0
$679$	−7.19352e6	−	8.45119e6i	−0.598779	−	0.703466i
$680$		0			0
$681$	−2.78984e6	−	4.83215e6i	−0.230522	−	0.399275i
$682$		0			0
$683$	7.06846e6	−	1.22429e7i	0.579793	−	1.00423i	−0.415709	−	0.909497i	$-0.636467\pi$
$683$	7.06846e6	−	1.22429e7i	0.579793	−	1.00423i	0.995503	−	0.0947337i	$-0.0302000\pi$
$684$		0			0
$685$	5.95356e6			0.484787
$686$		0			0
$687$	5.84254e6			0.472291
$688$		0			0
$689$	399753.	−	692393.i	0.0320807	−	0.0555654i
$690$		0			0
$691$	−5.72809e6	−	9.92134e6i	−0.456367	−	0.790452i	0.542398	−	0.840122i	$-0.317516\pi$
$691$	−5.72809e6	−	9.92134e6i	−0.456367	−	0.790452i	−0.998766	+	0.0496699i	$0.984183\pi$
$692$		0			0
$693$	−52291.9	−	61434.4i	−0.00413620	−	0.00485935i
$694$		0			0
$695$	8.96297e6	+	1.55243e7i	0.703866	+	1.21913i
$696$		0			0
$697$	1.25440e6	−	2.17269e6i	0.0978037	−	0.169401i
$698$		0			0
$699$	2.40181e6			0.185929
$700$		0			0
$701$	6.39944e6			0.491866			0.245933	−	0.969287i	$-0.420906\pi$
$701$	6.39944e6			0.491866			0.245933	+	0.969287i	$0.420906\pi$
$702$		0			0
$703$	3.11850e6	−	5.40139e6i	0.237989	−	0.412209i
$704$		0			0
$705$	4.01096e6	+	6.94719e6i	0.303931	+	0.526425i
$706$		0			0
$707$	9.20639e6	−	1.69033e6i	0.692693	−	0.127182i
$708$		0			0
$709$	−3.57396e6	−	6.19028e6i	−0.267014	−	0.462482i	0.701075	−	0.713087i	$-0.252704\pi$
$709$	−3.57396e6	−	6.19028e6i	−0.267014	−	0.462482i	−0.968089	+	0.250606i	$0.919370\pi$
$710$		0			0
$711$	−1.91092e6	+	3.30981e6i	−0.141765	+	0.245544i
$712$		0			0
$713$	−6.83984e6			−0.503874
$714$		0			0
$715$	19723.7			0.00144285
$716$		0			0
$717$	2.27033e6	−	3.93233e6i	0.164927	−	0.285661i
$718$		0			0
$719$	641705.	+	1.11147e6i	0.0462928	+	0.0801814i	0.888243	−	0.459373i	$-0.151926\pi$
$719$	641705.	+	1.11147e6i	0.0462928	+	0.0801814i	−0.841951	+	0.539555i	$0.818593\pi$
$720$		0			0
$721$	1.53678e6	−	4.31670e6i	0.110096	−	0.309253i
$722$		0			0
$723$	4.07102e6	+	7.05122e6i	0.289639	+	0.501670i
$724$		0			0
$725$	681640.	−	1.18064e6i	0.0481626	−	0.0834201i
$726$		0			0
$727$	−1.39582e7			−0.979477			−0.489739	−	0.871869i	$-0.662908\pi$
$727$	−1.39582e7			−0.979477			−0.489739	+	0.871869i	$0.662908\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	118970.	−	206061.i	0.00823460	−	0.0142627i
$732$		0			0
$733$	−1.71410e6	−	2.96891e6i	−0.117836	−	0.204097i	0.801074	−	0.598565i	$-0.204262\pi$
$733$	−1.71410e6	−	2.96891e6i	−0.117836	−	0.204097i	−0.918910	+	0.394468i	$0.870929\pi$
$734$		0			0
$735$	1.45831e6	−	9.01164e6i	0.0995708	−	0.615298i
$736$		0			0
$737$	89755.7	+	155461.i	0.00608686	+	0.0105428i
$738$		0			0
$739$	1.19542e7	−	2.07052e7i	0.805208	−	1.39466i	−0.110942	−	0.993827i	$-0.535387\pi$
$739$	1.19542e7	−	2.07052e7i	0.805208	−	1.39466i	0.916150	−	0.400835i	$-0.131280\pi$
$740$		0			0
$741$	415617.			0.0278066
$742$		0			0
$743$	−4.30858e6			−0.286327			−0.143163	−	0.989699i	$-0.545727\pi$
$743$	−4.30858e6			−0.286327			−0.143163	+	0.989699i	$0.545727\pi$
$744$		0			0
$745$	−3.92807e6	+	6.80362e6i	−0.259292	+	0.449107i
$746$		0			0
$747$	−1.70939e6	−	2.96074e6i	−0.112083	−	0.194133i
$748$		0			0
$749$	−2.09408e6	+	5.88212e6i	−0.136392	+	0.383115i
$750$		0			0
$751$	−2.97785e6	−	5.15779e6i	−0.192665	−	0.333706i	0.753467	−	0.657485i	$-0.228380\pi$
$751$	−2.97785e6	−	5.15779e6i	−0.192665	−	0.333706i	−0.946133	+	0.323779i	$0.895046\pi$
$752$		0			0
$753$	6.46864e6	−	1.12040e7i	0.415743	−	0.720089i
$754$		0			0
$755$	9.92155e6			0.633450
$756$		0			0
$757$	−1.58463e7			−1.00505			−0.502526	−	0.864562i	$-0.667596\pi$
$757$	−1.58463e7			−1.00505			−0.502526	+	0.864562i	$0.667596\pi$
$758$		0			0
$759$	−91888.3	+	159155.i	−0.00578970	+	0.0100281i
$760$		0			0
$761$	3.40913e6	+	5.90479e6i	0.213394	+	0.369609i	0.952775	−	0.303679i	$-0.0982149\pi$
$761$	3.40913e6	+	5.90479e6i	0.213394	+	0.369609i	−0.739381	+	0.673288i	$0.764882\pi$
$762$		0			0
$763$	2.08562e6	−	382930.i	0.129695	−	0.0238126i
$764$		0			0
$765$	−337037.	−	583765.i	−0.0208221	−	0.0360649i
$766$		0			0
$767$	169190.	−	293045.i	0.0103845	−	0.0179865i
$768$		0			0
$769$	2.93983e7			1.79270			0.896349	−	0.443349i	$-0.146210\pi$
$769$	2.93983e7			1.79270			0.896349	+	0.443349i	$0.146210\pi$
$770$		0			0
$771$	−9.91144e6			−0.600483
$772$		0			0
$773$	−1.02235e7	+	1.77077e7i	−0.615393	+	1.06589i	0.374923	+	0.927056i	$0.377669\pi$
$773$	−1.02235e7	+	1.77077e7i	−0.615393	+	1.06589i	−0.990316	+	0.138835i	$0.955664\pi$
$774$		0			0
$775$	665545.	+	1.15276e6i	0.0398037	+	0.0689420i
$776$		0			0
$777$	4.34501e6	+	5.10467e6i	0.258189	+	0.303330i
$778$		0			0
$779$	9.87555e6	+	1.71049e7i	0.583066	+	1.00990i
$780$		0			0
$781$	−230586.	+	399387.i	−0.0135271	+	0.0234297i
$782$		0			0
$783$	1.92140e6			0.111999
$784$		0			0
$785$	−4.96113e6			−0.287347
$786$		0			0
$787$	1.22719e6	−	2.12556e6i	0.0706279	−	0.122331i	−0.828549	−	0.559917i	$-0.810833\pi$
$787$	1.22719e6	−	2.12556e6i	0.0706279	−	0.122331i	0.899177	+	0.437586i	$0.144166\pi$
$788$		0			0
$789$	−5.67449e6	−	9.82851e6i	−0.324515	−	0.562076i
$790$		0			0
$791$	−1.22708e7	−	1.44162e7i	−0.697320	−	0.819235i
$792$		0			0
$793$	−1.08395e6	−	1.87745e6i	−0.0612103	−	0.106019i
$794$		0			0
$795$	5.10423e6	−	8.84079e6i	0.286426	−	0.496104i
$796$		0			0
$797$	6.18924e6			0.345137			0.172569	−	0.984998i	$-0.444793\pi$
$797$	6.18924e6			0.345137			0.172569	+	0.984998i	$0.444793\pi$
$798$		0			0
$799$	2.03652e6			0.112855
$800$		0			0
$801$	−1.39233e6	+	2.41158e6i	−0.0766762	+	0.132807i
$802$		0			0
$803$	77053.6	+	133461.i	0.00421700	+	0.00730406i
$804$		0			0
$805$	−2.04533e7	+	3.75532e6i	−1.11243	+	0.204248i
$806$		0			0
$807$	2.17369e6	+	3.76494e6i	0.117494	+	0.203505i
$808$		0			0
$809$	−1.24448e7	+	2.15550e7i	−0.668524	+	1.15792i	0.309794	+	0.950804i	$0.399740\pi$
$809$	−1.24448e7	+	2.15550e7i	−0.668524	+	1.15792i	−0.978317	+	0.207113i	$0.933593\pi$
$810$		0			0
$811$	−3.04458e7			−1.62546			−0.812728	−	0.582643i	$-0.802019\pi$
$811$	−3.04458e7			−1.62546			−0.812728	+	0.582643i	$0.802019\pi$
$812$		0			0
$813$	−4.67263e6			−0.247933
$814$		0			0
$815$	−1.30925e7	+	2.26768e7i	−0.690443	+	1.19588i
$816$		0			0
$817$	936612.	+	1.62226e6i	0.0490913	+	0.0850287i
$818$		0			0
$819$	−149819.	+	420830.i	−0.00780470	+	0.0219228i
$820$		0			0
$821$	8.66575e6	+	1.50095e7i	0.448692	+	0.777157i	0.998301	−	0.0582649i	$-0.0185568\pi$
$821$	8.66575e6	+	1.50095e7i	0.448692	+	0.777157i	−0.549609	+	0.835422i	$0.685223\pi$
$822$		0			0
$823$	1.06372e7	−	1.84242e7i	0.547430	−	0.948177i	−0.451019	−	0.892514i	$-0.648939\pi$
$823$	1.06372e7	−	1.84242e7i	0.547430	−	0.948177i	0.998450	−	0.0556628i	$-0.0177272\pi$
$824$		0			0
$825$	35764.5			0.00182944
$826$		0			0
$827$	−1.30833e7			−0.665202			−0.332601	−	0.943068i	$-0.607926\pi$
$827$	−1.30833e7			−0.665202			−0.332601	+	0.943068i	$0.607926\pi$
$828$		0			0
$829$	−1.07609e7	+	1.86385e7i	−0.543830	+	0.941941i	0.454849	+	0.890568i	$0.349693\pi$
$829$	−1.07609e7	+	1.86385e7i	−0.543830	+	0.941941i	−0.998680	+	0.0513730i	$0.983640\pi$
$830$		0			0
$831$	−1.06968e7	−	1.85274e7i	−0.537342	−	0.930704i
$832$		0			0
$833$	−1.79617e6	−	1.46451e6i	−0.0896880	−	0.0731275i
$834$		0			0
$835$	−1.06741e7	−	1.84881e7i	−0.529803	−	0.917646i
$836$		0			0
$837$	−938016.	+	1.62469e6i	−0.0462804	+	0.0801599i
$838$		0			0
$839$	−1.15427e7			−0.566112			−0.283056	−	0.959103i	$-0.591348\pi$
$839$	−1.15427e7			−0.566112			−0.283056	+	0.959103i	$0.591348\pi$
$840$		0			0
$841$	−1.35644e7			−0.661319
$842$		0			0
$843$	4.23237e6	−	7.33069e6i	0.205123	−	0.355284i
$844$		0			0
$845$	1.11493e7	+	1.93112e7i	0.537165	+	0.930397i
$846$		0			0
$847$	−6.99997e6	+	1.96624e7i	−0.335265	+	0.941735i
$848$		0			0
$849$	2.92272e6	+	5.06229e6i	0.139161	+	0.241034i
$850$		0			0
$851$	7.63513e6	−	1.32244e7i	0.361404	−	0.625969i
$852$		0			0
$853$	2.91491e7			1.37168			0.685840	−	0.727752i	$-0.259435\pi$
$853$	2.91491e7			1.37168			0.685840	+	0.727752i	$0.259435\pi$
$854$		0			0
$855$	5.30678e6			0.248265
$856$		0			0
$857$	−4.51794e6	+	7.82531e6i	−0.210130	+	0.363956i	−0.951755	−	0.306859i	$-0.900722\pi$
$857$	−4.51794e6	+	7.82531e6i	−0.210130	+	0.363956i	0.741625	+	0.670815i	$0.234055\pi$
$858$		0			0
$859$	2.05181e6	+	3.55383e6i	0.0948754	+	0.164329i	0.909557	−	0.415580i	$-0.136421\pi$
$859$	2.05181e6	+	3.55383e6i	0.0948754	+	0.164329i	−0.814681	+	0.579909i	$0.803088\pi$
$860$		0			0
$861$	−2.08794e7	+	3.83355e6i	−0.959863	+	0.176235i
$862$		0			0
$863$	7.02861e6	+	1.21739e7i	0.321249	+	0.556420i	0.980746	−	0.195287i	$-0.0625639\pi$
$863$	7.02861e6	+	1.21739e7i	0.321249	+	0.556420i	−0.659497	+	0.751707i	$0.729231\pi$
$864$		0			0
$865$	−1.21495e7	+	2.10435e7i	−0.552098	+	0.956262i
$866$		0			0
$867$	1.26076e7			0.569619
$868$		0			0
$869$	−362495.			−0.0162837
$870$		0			0
$871$	496976.	−	860788.i	0.0221968	−	0.0384460i
$872$		0			0
$873$	3.46706e6	+	6.00512e6i	0.153966	+	0.266677i
$874$		0			0
$875$	−1.32247e7	−	1.55369e7i	−0.583938	−	0.686030i
$876$		0			0
$877$	−1.81039e7	−	3.13569e7i	−0.794828	−	1.37668i	−0.922948	−	0.384924i	$-0.874228\pi$
$877$	−1.81039e7	−	3.13569e7i	−0.794828	−	1.37668i	0.128121	−	0.991759i	$-0.459106\pi$
$878$		0			0
$879$	−3.57028e6	+	6.18391e6i	−0.155858	+	0.269955i
$880$		0			0
$881$	1.92414e7			0.835213			0.417607	−	0.908628i	$-0.362869\pi$
$881$	1.92414e7			0.835213			0.417607	+	0.908628i	$0.362869\pi$
$882$		0			0
$883$	1.97804e7			0.853756			0.426878	−	0.904309i	$-0.359613\pi$
$883$	1.97804e7			0.853756			0.426878	+	0.904309i	$0.359613\pi$
$884$		0			0
$885$	2.16029e6	−	3.74174e6i	0.0927159	−	0.160589i
$886$		0			0
$887$	1.90057e7	+	3.29189e7i	0.811101	+	1.40487i	0.912094	+	0.409981i	$0.134465\pi$
$887$	1.90057e7	+	3.29189e7i	0.811101	+	1.40487i	−0.100993	+	0.994887i	$0.532202\pi$
$888$		0			0
$889$	−2.92921e7	−	3.44133e7i	−1.24307	−	1.46040i
$890$		0			0
$891$	25203.1	+	43653.1i	0.00106356	+	0.00184213i
$892$		0			0
$893$	−8.01647e6	+	1.38849e7i	−0.336399	+	0.582660i
$894$		0			0
$895$	−2.49558e7			−1.04139
$896$		0			0
$897$	1.01757e6			0.0422263
$898$		0			0
$899$	−3.39136e6	+	5.87400e6i	−0.139950	+	0.242401i
$900$		0			0
$901$	−1.29581e6	−	2.24441e6i	−0.0531777	−	0.0921064i
$902$		0			0
$903$	−1.98023e6	+	363579.i	−0.0808159	+	0.0148382i
$904$		0			0
$905$	−1.66813e7	−	2.88929e7i	−0.677033	−	1.17266i
$906$		0			0
$907$	1.32603e7	−	2.29675e7i	0.535222	−	0.927032i	−0.463930	−	0.885872i	$-0.653561\pi$
$907$	1.32603e7	−	2.29675e7i	0.535222	−	0.927032i	0.999153	−	0.0411606i	$-0.0131055\pi$
$908$		0			0
$909$	−5.84829e6			−0.234757
$910$		0			0
$911$	−3.66343e7			−1.46249			−0.731244	−	0.682116i	$-0.761060\pi$
$911$	−3.66343e7			−1.46249			−0.731244	+	0.682116i	$0.761060\pi$
$912$		0			0
$913$	162132.	−	280821.i	0.00643714	−	0.0111494i
$914$		0			0
$915$	−1.38403e7	−	2.39721e7i	−0.546504	−	0.946572i
$916$		0			0
$917$	−1.30218e7	+	3.65772e7i	−0.511383	+	1.43644i
$918$		0			0
$919$	1.77545e7	+	3.07516e7i	0.693456	+	1.20110i	0.970699	+	0.240300i	$0.0772460\pi$
$919$	1.77545e7	+	3.07516e7i	0.693456	+	1.20110i	−0.277243	+	0.960800i	$0.589421\pi$
$920$		0			0
$921$	9.37269e6	−	1.62340e7i	0.364095	−	0.630632i
$922$		0			0
$923$	2.55350e6			0.0986580
$924$		0			0
$925$	−2.97172e6			−0.114197
$926$		0			0
$927$	−1.43144e6	+	2.47933e6i	−0.0547110	+	0.0947623i
$928$		0			0
$929$	−1.17173e7	−	2.02950e7i	−0.445441	−	0.771526i	0.552642	−	0.833419i	$-0.313620\pi$
$929$	−1.17173e7	−	2.02950e7i	−0.445441	−	0.771526i	−0.998083	+	0.0618929i	$0.980286\pi$
$930$		0			0
$931$	1.70553e7	−	6.48136e6i	0.644891	−	0.245071i
$932$		0			0
$933$	−5.38671e6	−	9.33005e6i	−0.202591	−	0.350897i
$934$		0			0
$935$	31967.4	−	55369.1i	0.00119585	−	0.00207128i
$936$		0			0
$937$	3.37868e6			0.125718			0.0628592	−	0.998022i	$-0.479978\pi$
$937$	3.37868e6			0.125718			0.0628592	+	0.998022i	$0.479978\pi$
$938$		0			0
$939$	−2.32572e7			−0.860783
$940$		0			0
$941$	1.38835e7	−	2.40470e7i	0.511123	−	0.885292i	−0.488793	−	0.872400i	$-0.662563\pi$
$941$	1.38835e7	−	2.40470e7i	0.511123	−	0.885292i	0.999917	−	0.0128922i	$-0.00410382\pi$
$942$		0			0
$943$	2.41787e7	+	4.18787e7i	0.885428	+	1.53361i
$944$		0			0
$945$	−1.91295e6	+	5.37335e6i	−0.0696826	+	0.195734i
$946$		0			0
$947$	1.08752e7	+	1.88363e7i	0.394058	+	0.682529i	0.992981	−	0.118278i	$-0.0377374\pi$
$947$	1.08752e7	+	1.88363e7i	0.394058	+	0.682529i	−0.598922	+	0.800807i	$0.704404\pi$
$948$		0			0
$949$	426645.	−	738971.i	0.0153780	−	0.0266355i
$950$		0			0
$951$	−1.16236e7			−0.416764
$952$		0			0
$953$	−2.00654e7			−0.715676			−0.357838	−	0.933784i	$-0.616486\pi$
$953$	−2.00654e7			−0.715676			−0.357838	+	0.933784i	$0.616486\pi$
$954$		0			0
$955$	−2.38485e7	+	4.13069e7i	−0.846161	+	1.46559i
$956$		0			0
$957$	91120.9	+	157826.i	0.00321616	+	0.00557056i
$958$		0			0
$959$	−1.25788e7	+	2.30952e6i	−0.441664	+	0.0810914i
$960$		0			0
$961$	1.10033e7	+	1.90583e7i	0.384339	+	0.665695i
$962$		0			0
$963$	1.95054e6	−	3.37844e6i	0.0677782	−	0.117395i
$964$		0			0
$965$	2.84721e7			0.984241
$966$		0			0
$967$	3.68366e7			1.26681			0.633407	−	0.773819i	$-0.281656\pi$
$967$	3.68366e7			1.26681			0.633407	+	0.773819i	$0.281656\pi$
$968$		0			0
$969$	673616.	−	1.16674e6i	0.0230464	−	0.0399175i
$970$		0			0
$971$	−2.34625e7	−	4.06383e7i	−0.798595	−	1.38321i	−0.920531	−	0.390669i	$-0.872244\pi$
$971$	−2.34625e7	−	4.06383e7i	−0.798595	−	1.38321i	0.121937	−	0.992538i	$-0.461090\pi$
$972$		0			0
$973$	−2.49593e7	−	2.93230e7i	−0.845182	−	0.992949i
$974$		0			0
$975$	−99013.8	−	171497.i	−0.00333568	−	0.00577756i
$976$		0			0
$977$	1.18284e7	−	2.04873e7i	0.396449	−	0.686671i	−0.596836	−	0.802364i	$-0.703576\pi$
$977$	1.18284e7	−	2.04873e7i	0.396449	−	0.686671i	0.993285	+	0.115693i	$0.0369089\pi$
$978$		0			0
$979$	−264120.			−0.00880734
$980$		0			0
$981$	−1.32488e6			−0.0439544
$982$		0			0
$983$	−1.17046e7	+	2.02729e7i	−0.386342	+	0.669165i	−0.991954	−	0.126595i	$-0.959595\pi$
$983$	−1.17046e7	+	2.02729e7i	−0.386342	+	0.669165i	0.605612	+	0.795760i	$0.292928\pi$
$984$		0			0
$985$	−5.68637e6	−	9.84908e6i	−0.186743	−	0.323449i
$986$		0			0
$987$	−1.11694e7	−	1.31222e7i	−0.364952	−	0.428759i
$988$		0			0
$989$	2.29314e6	+	3.97184e6i	0.0745487	+	0.129122i
$990$		0			0
$991$	−2.03205e6	+	3.51961e6i	−0.0657280	+	0.113844i	−0.897017	−	0.441997i	$-0.854270\pi$
$991$	−2.03205e6	+	3.51961e6i	−0.0657280	+	0.113844i	0.831289	+	0.555841i	$0.187604\pi$
$992$		0			0
$993$	2.79732e7			0.900263
$994$		0			0
$995$	−6.43565e7			−2.06080
$996$		0			0
$997$	−6.48780e6	+	1.12372e7i	−0.206709	+	0.358031i	−0.950676	−	0.310186i	$-0.899609\pi$
$997$	−6.48780e6	+	1.12372e7i	−0.206709	+	0.358031i	0.743967	+	0.668217i	$0.232942\pi$
$998$		0			0
$999$	−2.09416e6	−	3.62720e6i	−0.0663891	−	0.114989i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.q.k.193.1		8
4.3	odd	2		168.6.q.a.25.1	✓	8
7.2	even	3	inner	336.6.q.k.289.1		8
12.11	even	2		504.6.s.a.361.4		8
28.23	odd	6		168.6.q.a.121.1	yes	8
84.23	even	6		504.6.s.a.289.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.q.a.25.1	✓	8	4.3	odd	2
168.6.q.a.121.1	yes	8	28.23	odd	6
336.6.q.k.193.1		8	1.1	even	1	trivial
336.6.q.k.289.1		8	7.2	even	3	inner
504.6.s.a.289.4		8	84.23	even	6
504.6.s.a.361.4		8	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(4.50000 - 7.79423i) q^{3} +(-30.1755 - 52.2655i) q^{5} +(84.0301 + 98.7215i) q^{7} +(-40.5000 - 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 - 7.79423i) q^{3} +(-30.1755 - 52.2655i) q^{5} +(84.0301 + 98.7215i) q^{7} +(-40.5000 - 70.1481i) q^{9} +(3.84136 - 6.65343i) q^{11} -42.5391 q^{13} -543.159 q^{15} +(-68.9458 + 119.418i) q^{17} +(-542.790 - 940.140i) q^{19} +(1147.59 - 210.703i) q^{21} +(-1328.93 - 2301.78i) q^{23} +(-258.622 + 447.946i) q^{25} -729.000 q^{27} -2635.67 q^{29} +(1286.72 - 2228.66i) q^{31} +(-34.5722 - 59.8808i) q^{33} +(2624.08 - 7370.85i) q^{35} +(2872.65 + 4975.58i) q^{37} +(-191.426 + 331.560i) q^{39} -18194.0 q^{41} -1725.55 q^{43} +(-2444.22 + 4233.51i) q^{45} +(-7384.51 - 12790.3i) q^{47} +(-2684.87 + 16591.2i) q^{49} +(620.512 + 1074.76i) q^{51} +(-9397.31 + 16276.6i) q^{53} -463.660 q^{55} -9770.22 q^{57} +(-3977.27 + 6888.84i) q^{59} +(25481.1 + 44134.6i) q^{61} +(3521.90 - 9892.77i) q^{63} +(1283.64 + 2223.33i) q^{65} +(-11682.8 + 20235.2i) q^{67} -23920.8 q^{69} -60027.2 q^{71} +(-10029.5 + 17371.6i) q^{73} +(2327.59 + 4031.51i) q^{75} +(979.626 - 179.864i) q^{77} +(-23591.6 - 40861.9i) q^{79} +(-3280.50 + 5681.99i) q^{81} +42207.0 q^{83} +8321.90 q^{85} +(-11860.5 + 20543.0i) q^{87} +(-17189.2 - 29772.6i) q^{89} +(-3574.57 - 4199.53i) q^{91} +(-11580.4 - 20057.9i) q^{93} +(-32757.9 + 56738.4i) q^{95} -85606.4 q^{97} -622.300 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 36 q^{3} + 64 q^{5} - 42 q^{7} - 324 q^{9}+O(q^{10}) \) 8 * q + 36 * q^3 + 64 * q^5 - 42 * q^7 - 324 * q^9	\( 8 q + 36 q^{3} + 64 q^{5} - 42 q^{7} - 324 q^{9} - 70 q^{11} + 1356 q^{13} + 1152 q^{15} + 944 q^{17} - 606 q^{19} + 1512 q^{21} - 648 q^{23} - 2582 q^{25} - 5832 q^{27} - 8720 q^{29} - 4354 q^{31} + 630 q^{33} - 5824 q^{35} + 19302 q^{37} + 6102 q^{39} + 1832 q^{41} - 16788 q^{43} + 5184 q^{45} - 5104 q^{47} - 15484 q^{49} - 8496 q^{51} - 13244 q^{53} - 35744 q^{55} - 10908 q^{57} - 30742 q^{59} - 2428 q^{61} + 17010 q^{63} + 126004 q^{65} + 8258 q^{67} - 11664 q^{69} - 29664 q^{71} - 12758 q^{73} + 23238 q^{75} - 91672 q^{77} - 21382 q^{79} - 26244 q^{81} + 110500 q^{83} + 275960 q^{85} - 39240 q^{87} - 49072 q^{89} + 35658 q^{91} + 39186 q^{93} + 25292 q^{95} - 135228 q^{97} + 11340 q^{99}+O(q^{100}) \) 8 * q + 36 * q^3 + 64 * q^5 - 42 * q^7 - 324 * q^9 - 70 * q^11 + 1356 * q^13 + 1152 * q^15 + 944 * q^17 - 606 * q^19 + 1512 * q^21 - 648 * q^23 - 2582 * q^25 - 5832 * q^27 - 8720 * q^29 - 4354 * q^31 + 630 * q^33 - 5824 * q^35 + 19302 * q^37 + 6102 * q^39 + 1832 * q^41 - 16788 * q^43 + 5184 * q^45 - 5104 * q^47 - 15484 * q^49 - 8496 * q^51 - 13244 * q^53 - 35744 * q^55 - 10908 * q^57 - 30742 * q^59 - 2428 * q^61 + 17010 * q^63 + 126004 * q^65 + 8258 * q^67 - 11664 * q^69 - 29664 * q^71 - 12758 * q^73 + 23238 * q^75 - 91672 * q^77 - 21382 * q^79 - 26244 * q^81 + 110500 * q^83 + 275960 * q^85 - 39240 * q^87 - 49072 * q^89 + 35658 * q^91 + 39186 * q^93 + 25292 * q^95 - 135228 * q^97 + 11340 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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