LMFDB - Embedded newform 336.6.q.m.289.4

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:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 1094 x^{8} - 12883 x^{7} + 1063781 x^{6} - 7555708 x^{5} + 199315216 x^{4} + \cdots + 37456183296 $ x^10 - x^9 + 1094x^8 - 12883x^7 + 1063781x^6 - 7555708x^5 + 199315216x^4 + 540268032x^3 + 20356706304x^2 - 26680098816x + 37456183296
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{12}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 168)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.4
Root			$12.6192 + 21.8570i$ of defining polynomial
Character	$\chi$	$=$	336.289
Dual form			336.6.q.m.193.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(4.50000 + 7.79423i) q^{3} +(10.3382 - 17.9063i) q^{5} +(-74.2549 + 106.270i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})$	\(q+(4.50000 + 7.79423i) q^{3} +(10.3382 - 17.9063i) q^{5} +(-74.2549 + 106.270i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-187.535 - 324.820i) q^{11} +973.265 q^{13} +186.088 q^{15} +(-851.164 - 1474.26i) q^{17} +(361.677 - 626.443i) q^{19} +(-1162.44 - 100.547i) q^{21} +(-227.469 + 393.988i) q^{23} +(1348.74 + 2336.09i) q^{25} -729.000 q^{27} +3993.24 q^{29} +(4462.08 + 7728.56i) q^{31} +(1687.82 - 2923.38i) q^{33} +(1135.23 + 2428.27i) q^{35} +(5696.75 - 9867.06i) q^{37} +(4379.69 + 7585.85i) q^{39} -8799.48 q^{41} +6466.17 q^{43} +(837.397 + 1450.41i) q^{45} +(-6583.14 + 11402.3i) q^{47} +(-5779.42 - 15782.1i) q^{49} +(7660.47 - 13268.3i) q^{51} +(419.928 + 727.337i) q^{53} -7755.12 q^{55} +6510.19 q^{57} +(18514.0 + 32067.1i) q^{59} +(21786.9 - 37735.9i) q^{61} +(-4447.28 - 9512.75i) q^{63} +(10061.8 - 17427.6i) q^{65} +(17287.4 + 29942.6i) q^{67} -4094.45 q^{69} +4363.10 q^{71} +(35587.3 + 61639.0i) q^{73} +(-12138.7 + 21024.8i) q^{75} +(48443.9 + 4190.23i) q^{77} +(6320.92 - 10948.2i) q^{79} +(-3280.50 - 5681.99i) q^{81} +45853.6 q^{83} -35198.1 q^{85} +(17969.6 + 31124.2i) q^{87} +(-68605.8 + 118829. i) q^{89} +(-72269.7 + 103428. i) q^{91} +(-40158.8 + 69557.0i) q^{93} +(-7478.20 - 12952.6i) q^{95} +178723. q^{97} +30380.7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 45 q^{3} - 75 q^{5} + 113 q^{7} - 405 q^{9}+O(q^{10}) $ 10 * q + 45 * q^3 - 75 * q^5 + 113 * q^7 - 405 * q^9	$ 10 q + 45 q^{3} - 75 q^{5} + 113 q^{7} - 405 q^{9} - 91 q^{11} + 580 q^{13} - 1350 q^{15} - 1128 q^{17} - 282 q^{19} + 279 q^{21} + 1808 q^{23} - 202 q^{25} - 7290 q^{27} + 5026 q^{29} + 5069 q^{31} + 819 q^{33} + 884 q^{35} - 5010 q^{37} + 2610 q^{39} + 12872 q^{41} - 17328 q^{43} - 6075 q^{45} + 50 q^{47} + 29135 q^{49} + 10152 q^{51} + 1167 q^{53} - 97410 q^{55} - 5076 q^{57} + 42797 q^{59} - 26546 q^{61} - 6642 q^{63} - 2216 q^{65} + 13440 q^{67} + 32544 q^{69} + 39356 q^{71} - 27768 q^{73} + 1818 q^{75} + 125797 q^{77} + 123369 q^{79} - 32805 q^{81} - 334250 q^{83} - 324936 q^{85} + 22617 q^{87} - 59350 q^{89} - 113850 q^{91} - 45621 q^{93} - 41864 q^{95} + 525282 q^{97} + 14742 q^{99}+O(q^{100}) $ 10 * q + 45 * q^3 - 75 * q^5 + 113 * q^7 - 405 * q^9 - 91 * q^11 + 580 * q^13 - 1350 * q^15 - 1128 * q^17 - 282 * q^19 + 279 * q^21 + 1808 * q^23 - 202 * q^25 - 7290 * q^27 + 5026 * q^29 + 5069 * q^31 + 819 * q^33 + 884 * q^35 - 5010 * q^37 + 2610 * q^39 + 12872 * q^41 - 17328 * q^43 - 6075 * q^45 + 50 * q^47 + 29135 * q^49 + 10152 * q^51 + 1167 * q^53 - 97410 * q^55 - 5076 * q^57 + 42797 * q^59 - 26546 * q^61 - 6642 * q^63 - 2216 * q^65 + 13440 * q^67 + 32544 * q^69 + 39356 * q^71 - 27768 * q^73 + 1818 * q^75 + 125797 * q^77 + 123369 * q^79 - 32805 * q^81 - 334250 * q^83 - 324936 * q^85 + 22617 * q^87 - 59350 * q^89 - 113850 * q^91 - 45621 * q^93 - 41864 * q^95 + 525282 * q^97 + 14742 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	4.50000	+	7.79423i	0.288675	+	0.500000i
$3$	4.50000	+	7.79423i	0.288675	+	0.500000i
$4$		0			0
$5$	10.3382	−	17.9063i	0.184936	−	0.320318i	−0.758619	−	0.651534i	$-0.774126\pi$
$5$	10.3382	−	17.9063i	0.184936	−	0.320318i	0.943555	+	0.331216i	$0.107459\pi$
$6$		0			0
$7$	−74.2549	+	106.270i	−0.572770	+	0.819716i
$7$	−74.2549	+	106.270i	−0.572770	+	0.819716i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	−187.535	−	324.820i	−0.467305	−	0.809397i	0.531997	−	0.846746i	$-0.321442\pi$
$11$	−187.535	−	324.820i	−0.467305	−	0.809397i	−0.999302	+	0.0373497i	$0.988108\pi$
$12$		0			0
$13$	973.265			1.59725			0.798625	−	0.601829i	$-0.205561\pi$
$13$	973.265			1.59725			0.798625	+	0.601829i	$0.205561\pi$
$14$		0			0
$15$	186.088			0.213546
$16$		0			0
$17$	−851.164	−	1474.26i	−0.714317	−	1.23723i	−0.963223	−	0.268705i	$-0.913404\pi$
$17$	−851.164	−	1474.26i	−0.714317	−	1.23723i	0.248906	−	0.968528i	$-0.419929\pi$
$18$		0			0
$19$	361.677	−	626.443i	0.229846	−	0.398105i	−0.727916	−	0.685666i	$-0.759511\pi$
$19$	361.677	−	626.443i	0.229846	−	0.398105i	0.957762	+	0.287561i	$0.0928444\pi$
$20$		0			0
$21$	−1162.44	−	100.547i	−0.575203	−	0.0497531i
$22$		0			0
$23$	−227.469	+	393.988i	−0.0896609	+	0.155297i	−0.907368	−	0.420338i	$-0.861912\pi$
$23$	−227.469	+	393.988i	−0.0896609	+	0.155297i	0.817707	+	0.575635i	$0.195245\pi$
$24$		0			0
$25$	1348.74	+	2336.09i	0.431597	+	0.747549i
$26$		0			0
$27$	−729.000			−0.192450
$28$		0			0
$29$	3993.24			0.881719			0.440860	−	0.897576i	$-0.354674\pi$
$29$	3993.24			0.881719			0.440860	+	0.897576i	$0.354674\pi$
$30$		0			0
$31$	4462.08	+	7728.56i	0.833938	+	1.44442i	0.894892	+	0.446282i	$0.147252\pi$
$31$	4462.08	+	7728.56i	0.833938	+	1.44442i	−0.0609548	+	0.998141i	$0.519415\pi$
$32$		0			0
$33$	1687.82	−	2923.38i	0.269799	−	0.467305i
$34$		0			0
$35$	1135.23	+	2428.27i	0.156645	+	0.335064i
$36$		0			0
$37$	5696.75	−	9867.06i	0.684105	−	1.18491i	−0.289612	−	0.957144i	$-0.593526\pi$
$37$	5696.75	−	9867.06i	0.684105	−	1.18491i	0.973717	−	0.227761i	$-0.0731405\pi$
$38$		0			0
$39$	4379.69	+	7585.85i	0.461086	+	0.798625i
$40$		0			0
$41$	−8799.48			−0.817518			−0.408759	−	0.912642i	$-0.634038\pi$
$41$	−8799.48			−0.817518			−0.408759	+	0.912642i	$0.634038\pi$
$42$		0			0
$43$	6466.17			0.533305			0.266653	−	0.963793i	$-0.414082\pi$
$43$	6466.17			0.533305			0.266653	+	0.963793i	$0.414082\pi$
$44$		0			0
$45$	837.397	+	1450.41i	0.0616453	+	0.106773i
$46$		0			0
$47$	−6583.14	+	11402.3i	−0.434699	+	0.752920i	−0.997271	−	0.0738278i	$-0.976478\pi$
$47$	−6583.14	+	11402.3i	−0.434699	+	0.752920i	0.562572	+	0.826748i	$0.309812\pi$
$48$		0			0
$49$	−5779.42	−	15782.1i	−0.343870	−	0.939017i
$50$		0			0
$51$	7660.47	−	13268.3i	0.412411	−	0.714317i
$52$		0			0
$53$	419.928	+	727.337i	0.0205346	+	0.0355669i	0.876110	−	0.482111i	$-0.160130\pi$
$53$	419.928	+	727.337i	0.0205346	+	0.0355669i	−0.855576	+	0.517678i	$0.826797\pi$
$54$		0			0
$55$	−7755.12			−0.345686
$56$		0			0
$57$	6510.19			0.265403
$58$		0			0
$59$	18514.0	+	32067.1i	0.692419	+	1.19931i	0.971043	+	0.238905i	$0.0767884\pi$
$59$	18514.0	+	32067.1i	0.692419	+	1.19931i	−0.278624	+	0.960400i	$0.589878\pi$
$60$		0			0
$61$	21786.9	−	37735.9i	0.749670	−	1.29847i	−0.198311	−	0.980139i	$-0.563546\pi$
$61$	21786.9	−	37735.9i	0.749670	−	1.29847i	0.947981	−	0.318327i	$-0.103121\pi$
$62$		0			0
$63$	−4447.28	−	9512.75i	−0.141170	−	0.301964i
$64$		0			0
$65$	10061.8	−	17427.6i	0.295389	−	0.511628i
$66$		0			0
$67$	17287.4	+	29942.6i	0.470481	+	0.814898i	0.999430	−	0.0337560i	$-0.0107469\pi$
$67$	17287.4	+	29942.6i	0.470481	+	0.814898i	−0.528949	+	0.848654i	$0.677414\pi$
$68$		0			0
$69$	−4094.45			−0.103532
$70$		0			0
$71$	4363.10			0.102719			0.0513593	−	0.998680i	$-0.483645\pi$
$71$	4363.10			0.102719			0.0513593	+	0.998680i	$0.483645\pi$
$72$		0			0
$73$	35587.3	+	61639.0i	0.781606	+	1.35378i	0.931006	+	0.365005i	$0.118933\pi$
$73$	35587.3	+	61639.0i	0.781606	+	1.35378i	−0.149399	+	0.988777i	$0.547734\pi$
$74$		0			0
$75$	−12138.7	+	21024.8i	−0.249183	+	0.431597i
$76$		0			0
$77$	48443.9	+	4190.23i	0.931134	+	0.0805399i
$78$		0			0
$79$	6320.92	−	10948.2i	0.113950	−	0.197366i	−0.803410	−	0.595426i	$-0.796983\pi$
$79$	6320.92	−	10948.2i	0.113950	−	0.197366i	0.917359	+	0.398060i	$0.130316\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	45853.6			0.730598			0.365299	−	0.930890i	$-0.380967\pi$
$83$	45853.6			0.730598			0.365299	+	0.930890i	$0.380967\pi$
$84$		0			0
$85$	−35198.1			−0.528411
$86$		0			0
$87$	17969.6	+	31124.2i	0.254530	+	0.440860i
$88$		0			0
$89$	−68605.8	+	118829.i	−0.918091	+	1.59018i	−0.115780	+	0.993275i	$0.536937\pi$
$89$	−68605.8	+	118829.i	−0.918091	+	1.59018i	−0.802312	+	0.596906i	$0.796397\pi$
$90$		0			0
$91$	−72269.7	+	103428.i	−0.914856	+	1.30929i
$92$		0			0
$93$	−40158.8	+	69557.0i	−0.481474	+	0.833938i
$94$		0			0
$95$	−7478.20	−	12952.6i	−0.0850136	−	0.147248i
$96$		0			0
$97$	178723.			1.92864			0.964320	−	0.264739i	$-0.0852857\pi$
$97$	178723.			1.92864			0.964320	+	0.264739i	$0.0852857\pi$
$98$		0			0
$99$	30380.7			0.311537
$100$		0			0
$101$	−56134.5	−	97227.7i	−0.547553	−	0.948389i	−0.998441	−	0.0558092i	$-0.982226\pi$
$101$	−56134.5	−	97227.7i	−0.547553	−	0.948389i	0.450889	−	0.892580i	$-0.351107\pi$
$102$		0			0
$103$	−10533.9	+	18245.3i	−0.0978355	+	0.169456i	−0.910788	−	0.412873i	$-0.864525\pi$
$103$	−10533.9	+	18245.3i	−0.0978355	+	0.169456i	0.812953	+	0.582329i	$0.197859\pi$
$104$		0			0
$105$	−13818.0	+	19775.5i	−0.122312	+	0.175047i
$106$		0			0
$107$	87578.6	−	151691.i	0.739500	−	1.28085i	−0.213220	−	0.977004i	$-0.568395\pi$
$107$	87578.6	−	151691.i	0.739500	−	1.28085i	0.952720	−	0.303848i	$-0.0982715\pi$
$108$		0			0
$109$	69865.2	+	121010.i	0.563242	+	0.975563i	0.997211	+	0.0746354i	$0.0237793\pi$
$109$	69865.2	+	121010.i	0.563242	+	0.975563i	−0.433969	+	0.900928i	$0.642887\pi$
$110$		0			0
$111$	102542.			0.789937
$112$		0			0
$113$	133307.			0.982103			0.491051	−	0.871131i	$-0.336613\pi$
$113$	133307.			0.982103			0.491051	+	0.871131i	$0.336613\pi$
$114$		0			0
$115$	4703.26	+	8146.29i	0.0331631	+	0.0574401i
$116$		0			0
$117$	−39417.2	+	68272.7i	−0.266208	+	0.461086i
$118$		0			0
$119$	219872.	+	19018.2i	1.42332	+	0.123112i
$120$		0			0
$121$	10186.7	−	17643.9i	0.0632515	−	0.109555i
$122$		0			0
$123$	−39597.6	−	68585.1i	−0.235997	−	0.408759i
$124$		0			0
$125$	120388.			0.689143
$126$		0			0
$127$	32431.7			0.178427			0.0892134	−	0.996013i	$-0.471565\pi$
$127$	32431.7			0.178427			0.0892134	+	0.996013i	$0.471565\pi$
$128$		0			0
$129$	29097.8	+	50398.8i	0.153952	+	0.266653i
$130$		0			0
$131$	65982.6	−	114285.i	0.335932	−	0.581851i	−0.647731	−	0.761869i	$-0.724282\pi$
$131$	65982.6	−	114285.i	0.335932	−	0.581851i	0.983663	+	0.180018i	$0.0576155\pi$
$132$		0			0
$133$	39715.5	+	84951.7i	0.194684	+	0.416431i
$134$		0			0
$135$	−7536.57	+	13053.7i	−0.0355909	+	0.0616453i
$136$		0			0
$137$	68676.0	+	118950.i	0.312610	+	0.541457i	0.978927	−	0.204212i	$-0.0654632\pi$
$137$	68676.0	+	118950.i	0.312610	+	0.541457i	−0.666316	+	0.745669i	$0.732130\pi$
$138$		0			0
$139$	−29016.9			−0.127384			−0.0636919	−	0.997970i	$-0.520288\pi$
$139$	−29016.9			−0.127384			−0.0636919	+	0.997970i	$0.520288\pi$
$140$		0			0
$141$	−118497.			−0.501947
$142$		0			0
$143$	−182521.	−	316136.i	−0.746403	−	1.29281i
$144$		0			0
$145$	41283.0	−	71504.3i	0.163062	−	0.282431i
$146$		0			0
$147$	97001.6	−	116065.i	0.370242	−	0.443006i
$148$		0			0
$149$	−159143.	+	275643.i	−0.587247	+	1.01714i	0.407344	+	0.913275i	$0.366455\pi$
$149$	−159143.	+	275643.i	−0.587247	+	1.01714i	−0.994591	+	0.103867i	$0.966878\pi$
$150$		0			0
$151$	10509.8	+	18203.4i	0.0375103	+	0.0649697i	0.884171	−	0.467163i	$-0.154724\pi$
$151$	10509.8	+	18203.4i	0.0375103	+	0.0649697i	−0.846661	+	0.532133i	$0.821391\pi$
$152$		0			0
$153$	137889.			0.476211
$154$		0			0
$155$	184520.			0.616900
$156$		0			0
$157$	−51950.5	−	89980.9i	−0.168206	−	0.291341i	0.769583	−	0.638546i	$-0.220464\pi$
$157$	−51950.5	−	89980.9i	−0.168206	−	0.291341i	−0.937789	+	0.347206i	$0.887131\pi$
$158$		0			0
$159$	−3779.35	+	6546.03i	−0.0118556	+	0.0205346i
$160$		0			0
$161$	−24978.3	−	53428.6i	−0.0759447	−	0.162446i
$162$		0			0
$163$	−73528.2	+	127355.i	−0.216763	+	0.375444i	−0.953816	−	0.300390i	$-0.902883\pi$
$163$	−73528.2	+	127355.i	−0.216763	+	0.375444i	0.737054	+	0.675834i	$0.236217\pi$
$164$		0			0
$165$	−34898.1	−	60445.2i	−0.0997910	−	0.172843i
$166$		0			0
$167$	−308950.			−0.857230			−0.428615	−	0.903487i	$-0.640998\pi$
$167$	−308950.			−0.857230			−0.428615	+	0.903487i	$0.640998\pi$
$168$		0			0
$169$	575952.			1.55121
$170$		0			0
$171$	29295.8	+	50741.9i	0.0766153	+	0.132702i
$172$		0			0
$173$	94403.5	−	163512.i	0.239813	−	0.415368i	−0.720847	−	0.693094i	$-0.756247\pi$
$173$	94403.5	−	163512.i	0.239813	−	0.415368i	0.960661	+	0.277725i	$0.0895805\pi$
$174$		0			0
$175$	−348406.	−	30135.9i	−0.859984	−	0.0743857i
$176$		0			0
$177$	−166626.	+	288604.i	−0.399768	+	0.692419i
$178$		0			0
$179$	289784.	+	501921.i	0.675993	+	1.17085i	0.976178	+	0.216973i	$0.0696182\pi$
$179$	289784.	+	501921.i	0.675993	+	1.17085i	−0.300185	+	0.953881i	$0.597048\pi$
$180$		0			0
$181$	489449.			1.11048			0.555241	−	0.831690i	$-0.312626\pi$
$181$	489449.			1.11048			0.555241	+	0.831690i	$0.312626\pi$
$182$		0			0
$183$	392163.			0.865644
$184$		0			0
$185$	−117789.	−	204016.i	−0.253031	−	0.438263i
$186$		0			0
$187$	−319246.	+	552950.i	−0.667608	+	1.15633i
$188$		0			0
$189$	54131.8	−	77470.5i	0.110230	−	0.157754i
$190$		0			0
$191$	453802.	−	786008.i	0.900084	−	1.55899i	0.0727012	−	0.997354i	$-0.476838\pi$
$191$	453802.	−	786008.i	0.900084	−	1.55899i	0.827383	−	0.561638i	$-0.189829\pi$
$192$		0			0
$193$	−228930.	−	396519.i	−0.442395	−	0.766250i	0.555472	−	0.831535i	$-0.312538\pi$
$193$	−228930.	−	396519.i	−0.442395	−	0.766250i	−0.997867	+	0.0652853i	$0.979204\pi$
$194$		0			0
$195$	181113.			0.341086
$196$		0			0
$197$	−796515.			−1.46227			−0.731136	−	0.682232i	$-0.761010\pi$
$197$	−796515.			−1.46227			−0.731136	+	0.682232i	$0.761010\pi$
$198$		0			0
$199$	237147.	+	410751.i	0.424508	+	0.735269i	0.996374	−	0.0850776i	$-0.0271138\pi$
$199$	237147.	+	410751.i	0.424508	+	0.735269i	−0.571867	+	0.820347i	$0.693780\pi$
$200$		0			0
$201$	−155587.	+	269484.i	−0.271633	+	0.470481i
$202$		0			0
$203$	−296518.	+	424360.i	−0.505022	+	0.722760i
$204$		0			0
$205$	−90971.0	+	157566.i	−0.151188	+	0.261866i
$206$		0			0
$207$	−18425.0	−	31913.1i	−0.0298870	−	0.0517658i
$208$		0			0
$209$	−271309.			−0.429633
$210$		0			0
$211$	−858152.			−1.32696			−0.663481	−	0.748193i	$-0.730922\pi$
$211$	−858152.			−1.32696			−0.663481	+	0.748193i	$0.730922\pi$
$212$		0			0
$213$	19633.9	+	34007.0i	0.0296523	+	0.0513593i
$214$		0			0
$215$	66848.8	−	115785.i	0.0986273	−	0.170828i
$216$		0			0
$217$	−1.15264e6	−	99699.6i	−1.66167	−	0.143729i
$218$		0			0
$219$	−320286.	+	554751.i	−0.451261	+	0.781606i
$220$		0			0
$221$	−828408.	−	1.43484e6i	−1.14094	−	1.97617i
$222$		0			0
$223$	−611765.			−0.823802			−0.411901	−	0.911229i	$-0.635135\pi$
$223$	−611765.			−0.823802			−0.411901	+	0.911229i	$0.635135\pi$
$224$		0			0
$225$	−218496.			−0.287732
$226$		0			0
$227$	−338148.	−	585690.i	−0.435555	−	0.754403i	0.561786	−	0.827283i	$-0.310114\pi$
$227$	−338148.	−	585690.i	−0.435555	−	0.754403i	−0.997341	+	0.0728798i	$0.976781\pi$
$228$		0			0
$229$	360941.	−	625169.i	0.454829	−	0.787786i	−0.543850	−	0.839183i	$-0.683034\pi$
$229$	360941.	−	625169.i	0.454829	−	0.787786i	0.998678	+	0.0513964i	$0.0163672\pi$
$230$		0			0
$231$	185338.	+	396439.i	0.228525	+	0.488817i
$232$		0			0
$233$	794358.	−	1.37587e6i	0.958576	−	1.66030i	0.232613	−	0.972569i	$-0.425272\pi$
$233$	794358.	−	1.37587e6i	0.958576	−	1.66030i	0.725963	−	0.687734i	$-0.241394\pi$
$234$		0			0
$235$	136116.	+	235760.i	0.160783	+	0.278484i
$236$		0			0
$237$	113777.			0.131578
$238$		0			0
$239$	−1.08599e6			−1.22979			−0.614893	−	0.788610i	$-0.710801\pi$
$239$	−1.08599e6			−1.22979			−0.614893	+	0.788610i	$0.710801\pi$
$240$		0			0
$241$	134356.	+	232712.i	0.149010	+	0.258093i	0.930862	−	0.365372i	$-0.119058\pi$
$241$	134356.	+	232712.i	0.149010	+	0.258093i	−0.781852	+	0.623464i	$0.785725\pi$
$242$		0			0
$243$	29524.5	−	51137.9i	0.0320750	−	0.0555556i
$244$		0			0
$245$	−342348.	−	59670.3i	−0.364378	−	0.0635102i
$246$		0			0
$247$	352008.	−	609695.i	0.367121	−	0.635873i
$248$		0			0
$249$	206341.	+	357394.i	0.210906	+	0.365299i
$250$		0			0
$251$	57808.1			0.0579168			0.0289584	−	0.999581i	$-0.490781\pi$
$251$	57808.1			0.0579168			0.0289584	+	0.999581i	$0.490781\pi$
$252$		0			0
$253$	170634.			0.167596
$254$		0			0
$255$	−158391.	−	274342.i	−0.152539	−	0.264206i
$256$		0			0
$257$	60568.5	−	104908.i	0.0572024	−	0.0990774i	−0.836006	−	0.548720i	$-0.815115\pi$
$257$	60568.5	−	104908.i	0.0572024	−	0.0990774i	0.893209	+	0.449643i	$0.148449\pi$
$258$		0			0
$259$	625556.	+	1.33807e6i	0.579452	+	1.23945i
$260$		0			0
$261$	−161726.	+	280118.i	−0.146953	+	0.254530i
$262$		0			0
$263$	−651374.	−	1.12821e6i	−0.580685	−	1.00578i	−0.995398	−	0.0958243i	$-0.969451\pi$
$263$	−651374.	−	1.12821e6i	−0.580685	−	1.00578i	0.414713	−	0.909952i	$-0.363882\pi$
$264$		0			0
$265$	17365.3			0.0151903
$266$		0			0
$267$	−1.23490e6			−1.06012
$268$		0			0
$269$	145752.	+	252450.i	0.122810	+	0.212714i	0.920875	−	0.389858i	$-0.127476\pi$
$269$	145752.	+	252450.i	0.122810	+	0.212714i	−0.798065	+	0.602572i	$0.794143\pi$
$270$		0			0
$271$	−162596.	+	281624.i	−0.134489	+	0.232941i	−0.925402	−	0.378987i	$-0.876272\pi$
$271$	−162596.	+	281624.i	−0.134489	+	0.232941i	0.790913	+	0.611928i	$0.209606\pi$
$272$		0			0
$273$	−1.13136e6	−	97858.7i	−0.918742	−	0.0794681i
$274$		0			0
$275$	505873.	−	876197.i	0.403376	−	0.698667i
$276$		0			0
$277$	−423711.	−	733890.i	−0.331796	−	0.574687i	0.651068	−	0.759019i	$-0.274321\pi$
$277$	−423711.	−	733890.i	−0.331796	−	0.574687i	−0.982864	+	0.184332i	$0.940988\pi$
$278$		0			0
$279$	−722858.			−0.555958
$280$		0			0
$281$	−362414.			−0.273803			−0.136902	−	0.990585i	$-0.543714\pi$
$281$	−362414.			−0.273803			−0.136902	+	0.990585i	$0.543714\pi$
$282$		0			0
$283$	−407692.	−	706144.i	−0.302598	−	0.524115i	0.674125	−	0.738617i	$-0.264521\pi$
$283$	−407692.	−	706144.i	−0.302598	−	0.524115i	−0.976724	+	0.214501i	$0.931187\pi$
$284$		0			0
$285$	67303.8	−	116574.i	0.0490826	−	0.0850136i
$286$		0			0
$287$	653404.	−	935116.i	0.468249	−	0.670133i
$288$		0			0
$289$	−739030.	+	1.28004e6i	−0.520496	+	0.901526i
$290$		0			0
$291$	804254.	+	1.39301e6i	0.556751	+	0.964320i
$292$		0			0
$293$	1.60046e6			1.08912			0.544562	−	0.838721i	$-0.316696\pi$
$293$	1.60046e6			1.08912			0.544562	+	0.838721i	$0.316696\pi$
$294$		0			0
$295$	765606.			0.512213
$296$		0			0
$297$	136713.	+	236794.i	0.0899329	+	0.155768i
$298$		0			0
$299$	−221388.	+	383455.i	−0.143211	+	0.248049i
$300$		0			0
$301$	−480145.	+	687157.i	−0.305461	+	0.437159i
$302$		0			0
$303$	505210.	−	875050.i	0.316130	−	0.547553i
$304$		0			0
$305$	−450475.	−	780246.i	−0.277282	−	0.480266i
$306$		0			0
$307$	71714.6			0.0434272			0.0217136	−	0.999764i	$-0.493088\pi$
$307$	71714.6			0.0434272			0.0217136	+	0.999764i	$0.493088\pi$
$308$		0			0
$309$	−189610.			−0.112971
$310$		0			0
$311$	−1.29986e6	−	2.25142e6i	−0.762069	−	1.31994i	−0.941782	−	0.336223i	$-0.890850\pi$
$311$	−1.29986e6	−	2.25142e6i	−0.762069	−	1.31994i	0.179713	−	0.983719i	$-0.442483\pi$
$312$		0			0
$313$	−470841.	+	815520.i	−0.271652	+	0.470515i	−0.969285	−	0.245940i	$-0.920903\pi$
$313$	−470841.	+	815520.i	−0.271652	+	0.470515i	0.697633	+	0.716455i	$0.254237\pi$
$314$		0			0
$315$	−216316.	−	18710.6i	−0.122832	−	0.0106245i
$316$		0			0
$317$	−754769.	+	1.30730e6i	−0.421858	+	0.730679i	−0.996121	−	0.0879916i	$-0.971955\pi$
$317$	−754769.	+	1.30730e6i	−0.421858	+	0.730679i	0.574264	+	0.818670i	$0.305288\pi$
$318$		0			0
$319$	−748872.	−	1.29708e6i	−0.412032	−	0.713661i
$320$		0			0
$321$	1.57641e6			0.853902
$322$		0			0
$323$	−1.23139e6			−0.656731
$324$		0			0
$325$	1.31268e6	+	2.27363e6i	0.689369	+	1.19402i
$326$		0			0
$327$	−628787.	+	1.08909e6i	−0.325188	+	0.563242i
$328$		0			0
$329$	−722890.	−	1.54627e6i	−0.368199	−	0.787580i
$330$		0			0
$331$	961864.	−	1.66600e6i	0.482551	−	0.835804i	−0.517248	−	0.855836i	$-0.673043\pi$
$331$	961864.	−	1.66600e6i	0.482551	−	0.835804i	0.999799	+	0.0200319i	$0.00637679\pi$
$332$		0			0
$333$	461437.	+	799232.i	0.228035	+	0.394968i
$334$		0			0
$335$	714884.			0.348036
$336$		0			0
$337$	−2.74797e6			−1.31806			−0.659032	−	0.752115i	$-0.729034\pi$
$337$	−2.74797e6			−1.31806			−0.659032	+	0.752115i	$0.729034\pi$
$338$		0			0
$339$	599882.	+	1.03903e6i	0.283509	+	0.491051i
$340$		0			0
$341$	1.67359e6	−	2.89875e6i	0.779407	−	1.34997i
$342$		0			0
$343$	2.10630e6	+	557719.i	0.966686	+	0.255965i
$344$		0			0
$345$	−42329.4	+	73316.6i	−0.0191467	+	0.0331631i
$346$		0			0
$347$	1.00410e6	+	1.73916e6i	0.447666	+	0.775380i	0.998234	−	0.0594101i	$-0.0189220\pi$
$347$	1.00410e6	+	1.73916e6i	0.447666	+	0.775380i	−0.550568	+	0.834791i	$0.685589\pi$
$348$		0			0
$349$	−905291.			−0.397855			−0.198927	−	0.980014i	$-0.563746\pi$
$349$	−905291.			−0.397855			−0.198927	+	0.980014i	$0.563746\pi$
$350$		0			0
$351$	−709510.			−0.307391
$352$		0			0
$353$	−488270.	−	845708.i	−0.208556	−	0.361230i	0.742704	−	0.669620i	$-0.233543\pi$
$353$	−488270.	−	845708.i	−0.208556	−	0.361230i	−0.951260	+	0.308390i	$0.900210\pi$
$354$		0			0
$355$	45106.7	−	78127.1i	0.0189964	−	0.0329027i
$356$		0			0
$357$	841191.	+	1.79931e6i	0.349321	+	0.747199i
$358$		0			0
$359$	1.53777e6	−	2.66350e6i	0.629732	−	1.09073i	−0.357873	−	0.933770i	$-0.616498\pi$
$359$	1.53777e6	−	2.66350e6i	0.629732	−	1.09073i	0.987605	−	0.156958i	$-0.0501687\pi$
$360$		0			0
$361$	976429.	+	1.69122e6i	0.394342	+	0.683020i
$362$		0			0
$363$	183361.			0.0730365
$364$		0			0
$365$	1.47164e6			0.578188
$366$		0			0
$367$	−2.08093e6	−	3.60427e6i	−0.806476	−	1.39686i	−0.915290	−	0.402795i	$-0.868039\pi$
$367$	−2.08093e6	−	3.60427e6i	−0.806476	−	1.39686i	0.108814	−	0.994062i	$-0.465295\pi$
$368$		0			0
$369$	356379.	−	617266.i	0.136253	−	0.235997i
$370$		0			0
$371$	−108475.	−	9382.76i	−0.0409163	−	0.00353912i
$372$		0			0
$373$	−1.40054e6	+	2.42581e6i	−0.521224	+	0.902787i	0.478471	+	0.878103i	$0.341191\pi$
$373$	−1.40054e6	+	2.42581e6i	−0.521224	+	0.902787i	−0.999695	+	0.0246838i	$0.992142\pi$
$374$		0			0
$375$	541748.	+	938334.i	0.198939	+	0.344572i
$376$		0			0
$377$	3.88648e6			1.40833
$378$		0			0
$379$	−42978.0			−0.0153691			−0.00768454	−	0.999970i	$-0.502446\pi$
$379$	−42978.0			−0.0153691			−0.00768454	+	0.999970i	$0.502446\pi$
$380$		0			0
$381$	145943.	+	252780.i	0.0515074	+	0.0892134i
$382$		0			0
$383$	1.86229e6	−	3.22557e6i	0.648708	−	1.12360i	−0.334723	−	0.942316i	$-0.608643\pi$
$383$	1.86229e6	−	3.22557e6i	0.648708	−	1.12360i	0.983432	−	0.181279i	$-0.0580238\pi$
$384$		0			0
$385$	575856.	−	824133.i	0.197999	−	0.283365i
$386$		0			0
$387$	−261880.	+	453589.i	−0.0888842	+	0.153952i
$388$		0			0
$389$	−56867.1	−	98496.7i	−0.0190540	−	0.0330026i	0.856341	−	0.516411i	$-0.172732\pi$
$389$	−56867.1	−	98496.7i	−0.0190540	−	0.0330026i	−0.875395	+	0.483408i	$0.839399\pi$
$390$		0			0
$391$	774454.			0.256185
$392$		0			0
$393$	1.18769e6			0.387901
$394$		0			0
$395$	−130694.	−	226369.i	−0.0421467	−	0.0730003i
$396$		0			0
$397$	−1.06152e6	+	1.83860e6i	−0.338026	+	0.585478i	−0.984061	−	0.177829i	$-0.943092\pi$
$397$	−1.06152e6	+	1.83860e6i	−0.338026	+	0.585478i	0.646035	+	0.763307i	$0.276426\pi$
$398$		0			0
$399$	−483413.	+	691835.i	−0.152015	+	0.217555i
$400$		0			0
$401$	−1.88822e6	+	3.27049e6i	−0.586396	+	1.01567i	0.408303	+	0.912846i	$0.366121\pi$
$401$	−1.88822e6	+	3.27049e6i	−0.586396	+	1.01567i	−0.994700	+	0.102822i	$0.967213\pi$
$402$		0			0
$403$	4.34279e6	+	7.52194e6i	1.33201	+	2.30710i
$404$		0			0
$405$	−135658.			−0.0410969
$406$		0			0
$407$	−4.27336e6			−1.27874
$408$		0			0
$409$	−80869.1	−	140069.i	−0.0239042	−	0.0414033i	0.853826	−	0.520559i	$-0.174276\pi$
$409$	−80869.1	−	140069.i	−0.0239042	−	0.0414033i	−0.877730	+	0.479155i	$0.840943\pi$
$410$		0			0
$411$	−618084.	+	1.07055e6i	−0.180486	+	0.312610i
$412$		0			0
$413$	−4.78251e6	−	413671.i	−1.37969	−	0.119338i
$414$		0			0
$415$	474046.	−	821071.i	0.135114	−	0.234024i
$416$		0			0
$417$	−130576.	−	226165.i	−0.0367726	−	0.0636919i
$418$		0			0
$419$	−3.95038e6			−1.09927			−0.549635	−	0.835405i	$-0.685233\pi$
$419$	−3.95038e6			−1.09927			−0.549635	+	0.835405i	$0.685233\pi$
$420$		0			0
$421$	2.73532e6			0.752149			0.376074	−	0.926590i	$-0.377274\pi$
$421$	2.73532e6			0.752149			0.376074	+	0.926590i	$0.377274\pi$
$422$		0			0
$423$	−533234.	−	923589.i	−0.144900	−	0.250973i
$424$		0			0
$425$	2.29600e6	−	3.97679e6i	0.616594	−	1.06797i
$426$		0			0
$427$	2.39240e6	+	5.11736e6i	0.634986	+	1.35824i
$428$		0			0
$429$	1.64269e6	−	2.84523e6i	0.430936	−	0.746403i
$430$		0			0
$431$	1.82401e6	+	3.15928e6i	0.472971	+	0.819210i	0.999521	−	0.0309338i	$-0.00984809\pi$
$431$	1.82401e6	+	3.15928e6i	0.472971	+	0.819210i	−0.526550	+	0.850144i	$0.676515\pi$
$432$		0			0
$433$	955063.			0.244800			0.122400	−	0.992481i	$-0.460941\pi$
$433$	955063.			0.244800			0.122400	+	0.992481i	$0.460941\pi$
$434$		0			0
$435$	743095.			0.188287
$436$		0			0
$437$	164541.	+	284993.i	0.0412164	+	0.0713889i
$438$		0			0
$439$	1.96411e6	−	3.40194e6i	0.486412	−	0.842491i	−0.513466	−	0.858110i	$-0.671639\pi$
$439$	1.96411e6	−	3.40194e6i	0.486412	−	0.842491i	0.999878	+	0.0156192i	$0.00497193\pi$
$440$		0			0
$441$	1.34115e6	+	233758.i	0.328383	+	0.0572362i
$442$		0			0
$443$	−3.23554e6	+	5.60413e6i	−0.783318	+	1.35675i	0.146681	+	0.989184i	$0.453141\pi$
$443$	−3.23554e6	+	5.60413e6i	−0.783318	+	1.35675i	−0.929999	+	0.367562i	$0.880192\pi$
$444$		0			0
$445$	1.41853e6	+	2.45696e6i	0.339576	+	0.588163i
$446$		0			0
$447$	−2.86457e6			−0.678095
$448$		0			0
$449$	1.43743e6			0.336490			0.168245	−	0.985745i	$-0.446190\pi$
$449$	1.43743e6			0.336490			0.168245	+	0.985745i	$0.446190\pi$
$450$		0			0
$451$	1.65021e6	+	2.85825e6i	0.382030	+	0.661696i
$452$		0			0
$453$	−94587.8	+	163831.i	−0.0216566	+	0.0375103i
$454$		0			0
$455$	1.10488e6	+	2.36335e6i	0.250200	+	0.535180i
$456$		0			0
$457$	−3.15202e6	+	5.45945e6i	−0.705989	+	1.22281i	0.260344	+	0.965516i	$0.416164\pi$
$457$	−3.15202e6	+	5.45945e6i	−0.705989	+	1.22281i	−0.966333	+	0.257293i	$0.917169\pi$
$458$		0			0
$459$	620498.	+	1.07473e6i	0.137470	+	0.238106i
$460$		0			0
$461$	7.38105e6			1.61758			0.808790	−	0.588098i	$-0.200123\pi$
$461$	7.38105e6			1.61758			0.808790	+	0.588098i	$0.200123\pi$
$462$		0			0
$463$	4.32714e6			0.938098			0.469049	−	0.883172i	$-0.344597\pi$
$463$	4.32714e6			0.938098			0.469049	+	0.883172i	$0.344597\pi$
$464$		0			0
$465$	830341.	+	1.43819e6i	0.178084	+	0.308450i
$466$		0			0
$467$	1.30544e6	−	2.26109e6i	0.276991	−	0.479762i	−0.693645	−	0.720317i	$-0.743996\pi$
$467$	1.30544e6	−	2.26109e6i	0.276991	−	0.479762i	0.970635	+	0.240555i	$0.0773295\pi$
$468$		0			0
$469$	−4.46566e6	−	386265.i	−0.937463	−	0.0810873i
$470$		0			0
$471$	467555.	−	809829.i	0.0971136	−	0.168206i
$472$		0			0
$473$	−1.21263e6	−	2.10034e6i	−0.249216	−	0.431656i
$474$		0			0
$475$	1.95124e6			0.396804
$476$		0			0
$477$	−68028.4			−0.0136897
$478$		0			0
$479$	3.94726e6	+	6.83686e6i	0.786063	+	1.36150i	0.928362	+	0.371677i	$0.121217\pi$
$479$	3.94726e6	+	6.83686e6i	0.786063	+	1.36150i	−0.142299	+	0.989824i	$0.545450\pi$
$480$		0			0
$481$	5.54445e6	−	9.60327e6i	1.09269	−	1.89259i
$482$		0			0
$483$	304033.	−	435115.i	0.0592997	−	0.0848665i
$484$		0			0
$485$	1.84768e6	−	3.20028e6i	0.356675	−	0.617779i
$486$		0			0
$487$	88470.7	+	153236.i	0.0169035	+	0.0292778i	0.874353	−	0.485290i	$-0.161286\pi$
$487$	88470.7	+	153236.i	0.0169035	+	0.0292778i	−0.857450	+	0.514567i	$0.827953\pi$
$488$		0			0
$489$	−1.32351e6			−0.250296
$490$		0			0
$491$	7.54095e6			1.41163			0.705817	−	0.708394i	$-0.250580\pi$
$491$	7.54095e6			1.41163			0.705817	+	0.708394i	$0.250580\pi$
$492$		0			0
$493$	−3.39890e6	−	5.88707e6i	−0.629827	−	1.09089i
$494$		0			0
$495$	314082.	−	544007.i	0.0576144	−	0.0997910i
$496$		0			0
$497$	−323981.	+	463664.i	−0.0588341	+	0.0842001i
$498$		0			0
$499$	2.94214e6	−	5.09594e6i	0.528947	−	0.916163i	−0.470483	−	0.882409i	$-0.655920\pi$
$499$	2.94214e6	−	5.09594e6i	0.528947	−	0.916163i	0.999430	−	0.0337540i	$-0.0107463\pi$
$500$		0			0
$501$	−1.39028e6	−	2.40803e6i	−0.247461	−	0.428615i
$502$		0			0
$503$	−3.01834e6			−0.531923			−0.265961	−	0.963984i	$-0.585689\pi$
$503$	−3.01834e6			−0.531923			−0.265961	+	0.963984i	$0.585689\pi$
$504$		0			0
$505$	−2.32132e6			−0.405049
$506$		0			0
$507$	2.59178e6	+	4.48910e6i	0.447795	+	0.775603i
$508$		0			0
$509$	963791.	−	1.66934e6i	0.164888	−	0.285594i	−0.771728	−	0.635953i	$-0.780607\pi$
$509$	963791.	−	1.66934e6i	0.164888	−	0.285594i	0.936615	+	0.350359i	$0.113940\pi$
$510$		0			0
$511$	−9.19288e6	−	795153.i	−1.55740	−	0.134710i
$512$		0			0
$513$	−263663.	+	456677.i	−0.0442339	+	0.0766153i
$514$		0			0
$515$	217804.	+	377247.i	0.0361866	+	0.0626770i
$516$		0			0
$517$	4.93828e6			0.812548
$518$		0			0
$519$	1.69926e6			0.276912
$520$		0			0
$521$	1.67758e6	+	2.90565e6i	0.270763	+	0.468975i	0.969057	−	0.246836i	$-0.0793908\pi$
$521$	1.67758e6	+	2.90565e6i	0.270763	+	0.468975i	−0.698295	+	0.715810i	$0.746057\pi$
$522$		0			0
$523$	−5.40910e6	+	9.36883e6i	−0.864710	+	1.49772i	0.00262503	+	0.999997i	$0.499164\pi$
$523$	−5.40910e6	+	9.36883e6i	−0.864710	+	1.49772i	−0.867335	+	0.497725i	$0.834169\pi$
$524$		0			0
$525$	−1.33294e6	−	2.85117e6i	−0.211063	−	0.451465i
$526$		0			0
$527$	7.59593e6	−	1.31565e7i	1.19139	−	2.06355i
$528$		0			0
$529$	3.11469e6	+	5.39480e6i	0.483922	+	0.838177i
$530$		0			0
$531$	−2.99926e6			−0.461613
$532$		0			0
$533$	−8.56422e6			−1.30578
$534$		0			0
$535$	−1.81082e6	−	3.13642e6i	−0.273520	−	0.473751i
$536$		0			0
$537$	−2.60806e6	+	4.51729e6i	−0.390284	+	0.675993i
$538$		0			0
$539$	−4.04249e6	+	4.83696e6i	−0.599345	+	0.717135i
$540$		0			0
$541$	1.27674e6	−	2.21138e6i	0.187547	−	0.324841i	−0.756885	−	0.653548i	$-0.773280\pi$
$541$	1.27674e6	−	2.21138e6i	0.187547	−	0.324841i	0.944432	+	0.328707i	$0.106613\pi$
$542$		0			0
$543$	2.20252e6	+	3.81488e6i	0.320568	+	0.555241i
$544$		0			0
$545$	2.88913e6			0.416654
$546$		0			0
$547$	−1.22840e7			−1.75538			−0.877689	−	0.479230i	$-0.840916\pi$
$547$	−1.22840e7			−1.75538			−0.877689	+	0.479230i	$0.840916\pi$
$548$		0			0
$549$	1.76474e6	+	3.05661e6i	0.249890	+	0.432822i
$550$		0			0
$551$	1.44426e6	−	2.50154e6i	0.202660	−	0.351017i
$552$		0			0
$553$	694096.	+	1.48468e6i	0.0965177	+	0.206452i
$554$		0			0
$555$	1.06010e6	−	1.83614e6i	0.146088	−	0.253031i
$556$		0			0
$557$	2.65819e6	+	4.60412e6i	0.363035	+	0.628795i	0.988459	−	0.151490i	$-0.0484072\pi$
$557$	2.65819e6	+	4.60412e6i	0.363035	+	0.628795i	−0.625424	+	0.780285i	$0.715074\pi$
$558$		0			0
$559$	6.29330e6			0.851822
$560$		0			0
$561$	−5.74643e6			−0.770887
$562$		0			0
$563$	−3.34814e6	−	5.79915e6i	−0.445177	−	0.771069i	0.552888	−	0.833256i	$-0.313526\pi$
$563$	−3.34814e6	−	5.79915e6i	−0.445177	−	0.771069i	−0.998065	+	0.0621870i	$0.980192\pi$
$564$		0			0
$565$	1.37816e6	−	2.38704e6i	0.181626	−	0.314586i
$566$		0			0
$567$	847416.	+	73298.6i	0.110698	+	0.00957498i
$568$		0			0
$569$	2.20160e6	−	3.81328e6i	0.285074	−	0.493762i	−0.687553	−	0.726134i	$-0.741315\pi$
$569$	2.20160e6	−	3.81328e6i	0.285074	−	0.493762i	0.972627	+	0.232372i	$0.0746486\pi$
$570$		0			0
$571$	3.37418e6	+	5.84425e6i	0.433090	+	0.750134i	0.997138	−	0.0756086i	$-0.0240900\pi$
$571$	3.37418e6	+	5.84425e6i	0.433090	+	0.750134i	−0.564048	+	0.825742i	$0.690757\pi$
$572$		0			0
$573$	8.16844e6			1.03933
$574$		0			0
$575$	−1.22719e6			−0.154790
$576$		0			0
$577$	738944.	+	1.27989e6i	0.0924000	+	0.160042i	0.908521	−	0.417840i	$-0.137213\pi$
$577$	738944.	+	1.27989e6i	0.0924000	+	0.160042i	−0.816121	+	0.577882i	$0.803879\pi$
$578$		0			0
$579$	2.06037e6	−	3.56867e6i	0.255417	−	0.442395i
$580$		0			0
$581$	−3.40486e6	+	4.87285e6i	−0.418465	+	0.598883i
$582$		0			0
$583$	157502.	−	272802.i	0.0191918	−	0.0332412i
$584$		0			0
$585$	815009.	+	1.41164e6i	0.0984629	+	0.170543i
$586$		0			0
$587$	6.64183e6			0.795596			0.397798	−	0.917473i	$-0.369775\pi$
$587$	6.64183e6			0.795596			0.397798	+	0.917473i	$0.369775\pi$
$588$		0			0
$589$	6.45534e6			0.766709
$590$		0			0
$591$	−3.58432e6	−	6.20822e6i	−0.422122	−	0.731136i
$592$		0			0
$593$	−231052.	+	400194.i	−0.0269820	+	0.0467341i	−0.879201	−	0.476451i	$-0.841923\pi$
$593$	−231052.	+	400194.i	−0.0269820	+	0.0467341i	0.852219	+	0.523185i	$0.175256\pi$
$594$		0			0
$595$	2.61363e6	−	3.74049e6i	0.302658	−	0.433147i
$596$		0			0
$597$	−2.13433e6	+	3.69676e6i	−0.245090	+	0.424508i
$598$		0			0
$599$	646081.	+	1.11905e6i	0.0735733	+	0.127433i	0.900465	−	0.434929i	$-0.143226\pi$
$599$	646081.	+	1.11905e6i	0.0735733	+	0.127433i	−0.826892	+	0.562361i	$0.809893\pi$
$600$		0			0
$601$	−6.71853e6			−0.758732			−0.379366	−	0.925247i	$-0.623858\pi$
$601$	−6.71853e6			−0.758732			−0.379366	+	0.925247i	$0.623858\pi$
$602$		0			0
$603$	−2.80056e6			−0.313654
$604$		0			0
$605$	−210625.	−	364814.i	−0.0233949	−	0.0405212i
$606$		0			0
$607$	7.64797e6	−	1.32467e7i	0.842509	−	1.45927i	−0.0452574	−	0.998975i	$-0.514411\pi$
$607$	7.64797e6	−	1.32467e7i	0.842509	−	1.45927i	0.887767	−	0.460294i	$-0.152256\pi$
$608$		0			0
$609$	−4.64188e6	−	401507.i	−0.507167	−	0.0438682i
$610$		0			0
$611$	−6.40714e6	+	1.10975e7i	−0.694322	+	1.20260i
$612$		0			0
$613$	5.24003e6	+	9.07600e6i	0.563226	+	0.975536i	0.997212	+	0.0746164i	$0.0237732\pi$
$613$	5.24003e6	+	9.07600e6i	0.563226	+	0.975536i	−0.433986	+	0.900919i	$0.642893\pi$
$614$		0			0
$615$	−1.63748e6			−0.174577
$616$		0			0
$617$	−2.35934e6			−0.249504			−0.124752	−	0.992188i	$-0.539814\pi$
$617$	−2.35934e6			−0.249504			−0.124752	+	0.992188i	$0.539814\pi$
$618$		0			0
$619$	−5.18252e6	−	8.97639e6i	−0.543644	−	0.941618i	−0.998691	−	0.0511510i	$-0.983711\pi$
$619$	−5.18252e6	−	8.97639e6i	−0.543644	−	0.941618i	0.455047	−	0.890467i	$-0.349622\pi$
$620$		0			0
$621$	165825.	−	287218.i	0.0172553	−	0.0298870i
$622$		0			0
$623$	−7.53356e6	−	1.61143e7i	−0.777642	−	1.66338i
$624$		0			0
$625$	−2.97022e6	+	5.14456e6i	−0.304150	+	0.526803i
$626$		0			0
$627$	−1.22089e6	−	2.11464e6i	−0.124024	−	0.214817i
$628$		0			0
$629$	−1.93955e7			−1.95467
$630$		0			0
$631$	−1.56167e7			−1.56141			−0.780704	−	0.624901i	$-0.785139\pi$
$631$	−1.56167e7			−1.56141			−0.780704	+	0.624901i	$0.785139\pi$
$632$		0			0
$633$	−3.86169e6	−	6.68864e6i	−0.383061	−	0.663481i
$634$		0			0
$635$	335286.	−	580733.i	0.0329975	−	0.0571534i
$636$		0			0
$637$	−5.62491e6	−	1.53601e7i	−0.549246	−	1.49984i
$638$		0			0
$639$	−176705.	+	306063.i	−0.0171198	+	0.0296523i
$640$		0			0
$641$	6.55025e6	+	1.13454e7i	0.629670	+	1.09062i	0.987618	+	0.156878i	$0.0501431\pi$
$641$	6.55025e6	+	1.13454e7i	0.629670	+	1.09062i	−0.357948	+	0.933741i	$0.616524\pi$
$642$		0			0
$643$	−5.02084e6			−0.478905			−0.239452	−	0.970908i	$-0.576968\pi$
$643$	−5.02084e6			−0.478905			−0.239452	+	0.970908i	$0.576968\pi$
$644$		0			0
$645$	1.20328e6			0.113885
$646$		0			0
$647$	3.08395e6	+	5.34156e6i	0.289632	+	0.501658i	0.973722	−	0.227740i	$-0.0731336\pi$
$647$	3.08395e6	+	5.34156e6i	0.289632	+	0.501658i	−0.684090	+	0.729398i	$0.739800\pi$
$648$		0			0
$649$	6.94403e6	−	1.20274e7i	0.647142	−	1.12088i
$650$		0			0
$651$	−4.40981e6	−	9.43260e6i	−0.407819	−	0.872326i
$652$		0			0
$653$	−7.81312e6	+	1.35327e7i	−0.717037	+	1.24194i	0.245132	+	0.969490i	$0.421169\pi$
$653$	−7.81312e6	+	1.35327e7i	−0.717037	+	1.24194i	−0.962169	+	0.272454i	$0.912165\pi$
$654$		0			0
$655$	−1.36429e6	−	2.36301e6i	−0.124252	−	0.215210i
$656$		0			0
$657$	−5.76514e6			−0.521071
$658$		0			0
$659$	−2.59701e6			−0.232949			−0.116474	−	0.993194i	$-0.537159\pi$
$659$	−2.59701e6			−0.232949			−0.116474	+	0.993194i	$0.537159\pi$
$660$		0			0
$661$	−7.96178e6	−	1.37902e7i	−0.708772	−	1.22763i	−0.965313	−	0.261096i	$-0.915916\pi$
$661$	−7.96178e6	−	1.37902e7i	−0.708772	−	1.22763i	0.256541	−	0.966533i	$-0.417417\pi$
$662$		0			0
$663$	7.45567e6	−	1.29136e7i	0.658723	−	1.14094i
$664$		0			0
$665$	1.93176e6	+	167091.i	0.169395	+	0.0146521i
$666$		0			0
$667$	−908339.	+	1.57329e6i	−0.0790558	+	0.136929i
$668$		0			0
$669$	−2.75294e6	−	4.76824e6i	−0.237811	−	0.411901i
$670$		0			0
$671$	−1.63432e7			−1.40130
$672$		0			0
$673$	−856750.			−0.0729150			−0.0364575	−	0.999335i	$-0.511607\pi$
$673$	−856750.			−0.0729150			−0.0364575	+	0.999335i	$0.511607\pi$
$674$		0			0
$675$	−983233.	−	1.70301e6i	−0.0830610	−	0.143866i
$676$		0			0
$677$	−9.65260e6	+	1.67188e7i	−0.809418	+	1.40195i	0.103850	+	0.994593i	$0.466884\pi$
$677$	−9.65260e6	+	1.67188e7i	−0.809418	+	1.40195i	−0.913268	+	0.407360i	$0.866450\pi$
$678$		0			0
$679$	−1.32711e7	+	1.89928e7i	−1.10467	+	1.58094i
$680$		0			0
$681$	3.04334e6	−	5.27121e6i	0.251468	−	0.435555i
$682$		0			0
$683$	7.99184e6	+	1.38423e7i	0.655534	+	1.13542i	0.981760	+	0.190126i	$0.0608896\pi$
$683$	7.99184e6	+	1.38423e7i	0.655534	+	1.13542i	−0.326226	+	0.945292i	$0.605777\pi$
$684$		0			0
$685$	2.83995e6			0.231252
$686$		0			0
$687$	6.49694e6			0.525191
$688$		0			0
$689$	408701.	+	707892.i	0.0327988	+	0.0568092i
$690$		0			0
$691$	3.78089e6	−	6.54870e6i	0.301231	−	0.521747i	−0.675184	−	0.737649i	$-0.735936\pi$
$691$	3.78089e6	−	6.54870e6i	0.301231	−	0.521747i	0.976415	+	0.215902i	$0.0692692\pi$
$692$		0			0
$693$	−2.25591e6	+	3.22854e6i	−0.178439	+	0.255372i
$694$		0			0
$695$	−299984.	+	519587.i	−0.0235579	+	0.0408034i
$696$		0			0
$697$	7.48979e6	+	1.29727e7i	0.583966	+	1.01146i
$698$		0			0
$699$	1.42985e7			1.10687
$700$		0			0
$701$	−2.26517e7			−1.74103			−0.870513	−	0.492146i	$-0.836213\pi$
$701$	−2.26517e7			−1.74103			−0.870513	+	0.492146i	$0.836213\pi$
$702$		0			0
$703$	−4.12077e6	−	7.13738e6i	−0.314478	−	0.544692i
$704$		0			0
$705$	−1.22504e6	+	2.12184e6i	−0.0928280	+	0.160783i
$706$		0			0
$707$	1.45006e7	+	1.25425e6i	1.09103	+	0.0943706i
$708$		0			0
$709$	5.00961e6	−	8.67689e6i	0.374272	−	0.648259i	−0.615945	−	0.787789i	$-0.711226\pi$
$709$	5.00961e6	−	8.67689e6i	0.374272	−	0.648259i	0.990218	+	0.139530i	$0.0445591\pi$
$710$		0			0
$711$	511995.	+	886801.i	0.0379832	+	0.0657888i
$712$		0			0
$713$	−4.05995e6			−0.299087
$714$		0			0
$715$	−7.54779e6			−0.552147
$716$		0			0
$717$	−4.88694e6	−	8.46443e6i	−0.355009	−	0.614893i
$718$		0			0
$719$	−2.42584e6	+	4.20168e6i	−0.175001	+	0.303110i	−0.940162	−	0.340729i	$-0.889326\pi$
$719$	−2.42584e6	+	4.20168e6i	−0.175001	+	0.303110i	0.765161	+	0.643839i	$0.222659\pi$
$720$		0			0
$721$	−1.15672e6	−	2.47423e6i	−0.0828687	−	0.177257i
$722$		0			0
$723$	−1.20921e6	+	2.09441e6i	−0.0860309	+	0.149010i
$724$		0			0
$725$	5.38585e6	+	9.32856e6i	0.380548	+	0.659128i
$726$		0			0
$727$	1.38566e7			0.972345			0.486173	−	0.873863i	$-0.338393\pi$
$727$	1.38566e7			0.972345			0.486173	+	0.873863i	$0.338393\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−5.50377e6	−	9.53281e6i	−0.380949	−	0.659823i
$732$		0			0
$733$	7.32643e6	−	1.26897e7i	0.503654	−	0.872355i	−0.496337	−	0.868130i	$-0.665322\pi$
$733$	7.32643e6	−	1.26897e7i	0.503654	−	0.872355i	0.999991	−	0.00422470i	$-0.00134477\pi$
$734$		0			0
$735$	−1.07548e6	−	2.93686e6i	−0.0734319	−	0.200523i
$736$		0			0
$737$	6.48398e6	−	1.12306e7i	0.439717	−	0.761612i
$738$		0			0
$739$	−1.05349e7	−	1.82469e7i	−0.709607	−	1.22908i	−0.965003	−	0.262239i	$-0.915539\pi$
$739$	−1.05349e7	−	1.82469e7i	−0.709607	−	1.22908i	0.255395	−	0.966837i	$-0.417794\pi$
$740$		0			0
$741$	6.33614e6			0.423915
$742$		0			0
$743$	−1.48722e7			−0.988330			−0.494165	−	0.869368i	$-0.664526\pi$
$743$	−1.48722e7			−0.988330			−0.494165	+	0.869368i	$0.664526\pi$
$744$		0			0
$745$	3.29051e6	+	5.69933e6i	0.217206	+	0.376212i
$746$		0			0
$747$	−1.85707e6	+	3.21654e6i	−0.121766	+	0.210906i
$748$		0			0
$749$	9.61694e6	+	2.05707e7i	0.626372	+	1.33981i
$750$		0			0
$751$	−8.56624e6	+	1.48372e7i	−0.554230	+	0.959955i	0.443733	+	0.896159i	$0.353654\pi$
$751$	−8.56624e6	+	1.48372e7i	−0.554230	+	0.959955i	−0.997963	+	0.0637958i	$0.979679\pi$
$752$		0			0
$753$	260136.	+	450570.i	0.0167191	+	0.0289584i
$754$		0			0
$755$	434609.			0.0277480
$756$		0			0
$757$	−5.11399e6			−0.324354			−0.162177	−	0.986762i	$-0.551852\pi$
$757$	−5.11399e6			−0.324354			−0.162177	+	0.986762i	$0.551852\pi$
$758$		0			0
$759$	767852.	+	1.32996e6i	0.0483808	+	0.0837981i
$760$		0			0
$761$	1.14345e7	−	1.98052e7i	0.715742	−	1.23970i	−0.246931	−	0.969033i	$-0.579422\pi$
$761$	1.14345e7	−	1.98052e7i	0.715742	−	1.23970i	0.962673	−	0.270668i	$-0.0872447\pi$
$762$		0			0
$763$	−1.80475e7	−	1.56105e6i	−1.12229	−	0.0970745i
$764$		0			0
$765$	1.42552e6	−	2.46908e6i	0.0880685	−	0.152539i
$766$		0			0
$767$	1.80190e7	+	3.12098e7i	1.10597	+	1.91559i
$768$		0			0
$769$	−2.70939e7			−1.65218			−0.826088	−	0.563541i	$-0.809439\pi$
$769$	−2.70939e7			−1.65218			−0.826088	+	0.563541i	$0.809439\pi$
$770$		0			0
$771$	1.09023e6			0.0660516
$772$		0			0
$773$	−8.04517e6	−	1.39346e7i	−0.484269	−	0.838778i	0.515568	−	0.856849i	$-0.327581\pi$
$773$	−8.04517e6	−	1.39346e7i	−0.484269	−	0.838778i	−0.999837	+	0.0180706i	$0.994248\pi$
$774$		0			0
$775$	−1.20364e7	+	2.08477e7i	−0.719851	+	1.24682i
$776$		0			0
$777$	−7.61421e6	+	1.08970e7i	−0.452452	+	0.647524i
$778$		0			0
$779$	−3.18257e6	+	5.51237e6i	−0.187903	+	0.325458i
$780$		0			0
$781$	−818234.	−	1.41722e6i	−0.0480010	−	0.0831401i
$782$		0			0
$783$	−2.91107e6			−0.169687
$784$		0			0
$785$	−2.14831e6			−0.124429
$786$		0			0
$787$	−1.94608e6	−	3.37070e6i	−0.112001	−	0.193992i	0.804576	−	0.593850i	$-0.202393\pi$
$787$	−1.94608e6	−	3.37070e6i	−0.112001	−	0.193992i	−0.916577	+	0.399858i	$0.869059\pi$
$788$		0			0
$789$	5.86236e6	−	1.01539e7i	0.335259	−	0.580685i
$790$		0			0
$791$	−9.89870e6	+	1.41665e7i	−0.562519	+	0.805046i
$792$		0			0
$793$	2.12044e7	−	3.67271e7i	1.19741	−	2.07397i
$794$		0			0
$795$	78143.7	+	135349.i	0.00438506	+	0.00759515i
$796$		0			0
$797$	1.62260e7			0.904828			0.452414	−	0.891808i	$-0.350563\pi$
$797$	1.62260e7			0.904828			0.452414	+	0.891808i	$0.350563\pi$
$798$		0			0
$799$	2.24133e7			1.24205
$800$		0			0
$801$	−5.55707e6	−	9.62513e6i	−0.306030	−	0.530060i
$802$		0			0
$803$	1.33477e7	−	2.31190e7i	0.730497	−	1.26526i
$804$		0			0
$805$	−1.21494e6	−	105088.i	−0.0660794	−	0.00571564i
$806$		0			0
$807$	−1.31177e6	+	2.27205e6i	−0.0709045	+	0.122810i
$808$		0			0
$809$	4.87336e6	+	8.44091e6i	0.261793	+	0.453438i	0.966718	−	0.255843i	$-0.0823531\pi$
$809$	4.87336e6	+	8.44091e6i	0.261793	+	0.453438i	−0.704926	+	0.709281i	$0.749020\pi$
$810$		0			0
$811$	3.24349e7			1.73165			0.865827	−	0.500344i	$-0.166793\pi$
$811$	3.24349e7			1.73165			0.865827	+	0.500344i	$0.166793\pi$
$812$		0			0
$813$	−2.92672e6			−0.155294
$814$		0			0
$815$	1.52030e6	+	2.63324e6i	0.0801745	+	0.138866i
$816$		0			0
$817$	2.33867e6	−	4.05069e6i	0.122578	−	0.212312i
$818$		0			0
$819$	−4.32838e6	−	9.25843e6i	−0.225484	−	0.482311i
$820$		0			0
$821$	1.01754e7	−	1.76243e7i	0.526859	−	0.912546i	−0.472651	−	0.881250i	$-0.656703\pi$
$821$	1.01754e7	−	1.76243e7i	0.526859	−	0.912546i	0.999510	−	0.0312968i	$-0.00996371\pi$
$822$		0			0
$823$	−9.66202e6	−	1.67351e7i	−0.497243	−	0.861249i	0.502752	−	0.864430i	$-0.332321\pi$
$823$	−9.66202e6	−	1.67351e7i	−0.497243	−	0.861249i	−0.999995	+	0.00318111i	$0.998987\pi$
$824$		0			0
$825$	9.10571e6			0.465778
$826$		0			0
$827$	−4.48192e6			−0.227877			−0.113939	−	0.993488i	$-0.536347\pi$
$827$	−4.48192e6			−0.227877			−0.113939	+	0.993488i	$0.536347\pi$
$828$		0			0
$829$	1.17952e7	+	2.04298e7i	0.596099	+	1.03247i	0.993391	+	0.114781i	$0.0366166\pi$
$829$	1.17952e7	+	2.04298e7i	0.596099	+	1.03247i	−0.397292	+	0.917692i	$0.630050\pi$
$830$		0			0
$831$	3.81340e6	−	6.60501e6i	0.191562	−	0.331796i
$832$		0			0
$833$	−1.83476e7	+	2.19535e7i	−0.916151	+	1.09620i
$834$		0			0
$835$	−3.19400e6	+	5.53217e6i	−0.158533	+	0.274587i
$836$		0			0
$837$	−3.25286e6	−	5.63412e6i	−0.160491	−	0.277979i
$838$		0			0
$839$	3.70111e7			1.81521			0.907607	−	0.419822i	$-0.137907\pi$
$839$	3.70111e7			1.81521			0.907607	+	0.419822i	$0.137907\pi$
$840$		0			0
$841$	−4.56519e6			−0.222571
$842$		0			0
$843$	−1.63086e6	−	2.82473e6i	−0.0790402	−	0.136902i
$844$		0			0
$845$	5.95432e6	−	1.03132e7i	0.286874	−	0.496880i
$846$		0			0
$847$	1.11860e6	+	2.39268e6i	0.0535753	+	0.114598i
$848$		0			0
$849$	3.66923e6	−	6.35529e6i	0.174705	−	0.302598i
$850$		0			0
$851$	2.59167e6	+	4.48891e6i	0.122675	+	0.212479i
$852$		0			0
$853$	−1.59180e7			−0.749059			−0.374530	−	0.927215i	$-0.622196\pi$
$853$	−1.59180e7			−0.749059			−0.374530	+	0.927215i	$0.622196\pi$
$854$		0			0
$855$	1.21147e6			0.0566757
$856$		0			0
$857$	−5.61104e6	−	9.71861e6i	−0.260970	−	0.452014i	0.705530	−	0.708680i	$-0.250709\pi$
$857$	−5.61104e6	−	9.71861e6i	−0.260970	−	0.452014i	−0.966500	+	0.256666i	$0.917376\pi$
$858$		0			0
$859$	3.00567e6	−	5.20598e6i	0.138982	−	0.240724i	−0.788129	−	0.615509i	$-0.788950\pi$
$859$	3.00567e6	−	5.20598e6i	0.138982	−	0.240724i	0.927112	+	0.374785i	$0.122284\pi$
$860$		0			0
$861$	1.02288e7	+	884759.i	0.470238	+	0.0406740i
$862$		0			0
$863$	−1.29004e7	+	2.23442e7i	−0.589626	+	1.02126i	0.404656	+	0.914469i	$0.367391\pi$
$863$	−1.29004e7	+	2.23442e7i	−0.589626	+	1.02126i	−0.994281	+	0.106792i	$0.965942\pi$
$864$		0			0
$865$	−1.95193e6	−	3.38084e6i	−0.0887001	−	0.153633i
$866$		0			0
$867$	−1.33025e7			−0.601017
$868$		0			0
$869$	−4.74158e6			−0.212997
$870$		0			0
$871$	1.68252e7	+	2.91421e7i	0.751476	+	1.30160i
$872$		0			0
$873$	−7.23828e6	+	1.25371e7i	−0.321440	+	0.556751i
$874$		0			0
$875$	−8.93943e6	+	1.27936e7i	−0.394720	+	0.564902i
$876$		0			0
$877$	2.45108e6	−	4.24539e6i	0.107611	−	0.186388i	−0.807191	−	0.590291i	$-0.799013\pi$
$877$	2.45108e6	−	4.24539e6i	0.107611	−	0.186388i	0.914802	+	0.403902i	$0.132346\pi$
$878$		0			0
$879$	7.20209e6	+	1.24744e7i	0.314403	+	0.544562i
$880$		0			0
$881$	2.57453e7			1.11753			0.558763	−	0.829327i	$-0.311276\pi$
$881$	2.57453e7			1.11753			0.558763	+	0.829327i	$0.311276\pi$
$882$		0			0
$883$	1.74696e7			0.754018			0.377009	−	0.926210i	$-0.376953\pi$
$883$	1.74696e7			0.754018			0.377009	+	0.926210i	$0.376953\pi$
$884$		0			0
$885$	3.44523e6	+	5.96731e6i	0.147863	+	0.256106i
$886$		0			0
$887$	1.54474e7	−	2.67557e7i	0.659244	−	1.14184i	−0.321567	−	0.946887i	$-0.604210\pi$
$887$	1.54474e7	−	2.67557e7i	0.659244	−	1.14184i	0.980812	−	0.194958i	$-0.0624570\pi$
$888$		0			0
$889$	−2.40821e6	+	3.44650e6i	−0.102197	+	0.146259i
$890$		0			0
$891$	−1.23042e6	+	2.13115e6i	−0.0519228	+	0.0899329i
$892$		0			0
$893$	4.76194e6	+	8.24793e6i	0.199828	+	0.346112i
$894$		0			0
$895$	1.19834e7			0.500061
$896$		0			0
$897$	−3.98498e6			−0.165366
$898$		0			0
$899$	1.78182e7	+	3.08620e7i	0.735299	+	1.27358i
$900$		0			0
$901$	714855.	−	1.23817e6i	0.0293363	−	0.0508120i
$902$		0			0
$903$	−7.51651e6	−	650152.i	−0.306759	−	0.0265336i
$904$		0			0
$905$	5.06004e6	−	8.76425e6i	0.205368	−	0.355708i
$906$		0			0
$907$	−2.13421e6	−	3.69656e6i	−0.0861427	−	0.149204i	0.819735	−	0.572743i	$-0.194121\pi$
$907$	−2.13421e6	−	3.69656e6i	−0.0861427	−	0.149204i	−0.905878	+	0.423540i	$0.860787\pi$
$908$		0			0
$909$	9.09378e6			0.365035
$910$		0			0
$911$	−2.91542e7			−1.16387			−0.581936	−	0.813234i	$-0.697705\pi$
$911$	−2.91542e7			−1.16387			−0.581936	+	0.813234i	$0.697705\pi$
$912$		0			0
$913$	−8.59917e6	−	1.48942e7i	−0.341412	−	0.591344i
$914$		0			0
$915$	4.05428e6	−	7.02221e6i	0.160089	−	0.277282i
$916$		0			0
$917$	7.24551e6	+	1.54982e7i	0.284541	+	0.608636i
$918$		0			0
$919$	1.75388e7	−	3.03780e7i	0.685031	−	1.18651i	−0.288396	−	0.957511i	$-0.593122\pi$
$919$	1.75388e7	−	3.03780e7i	0.685031	−	1.18651i	0.973427	−	0.228998i	$-0.0735448\pi$
$920$		0			0
$921$	322716.	+	558960.i	0.0125363	+	0.0217136i
$922$		0			0
$923$	4.24645e6			0.164067
$924$		0			0
$925$	3.07338e7			1.18103
$926$		0			0
$927$	−853246.	−	1.47787e6i	−0.0326118	−	0.0564853i
$928$		0			0
$929$	−1.95588e7	+	3.38769e7i	−0.743538	+	1.28785i	0.207336	+	0.978270i	$0.433521\pi$
$929$	−1.95588e7	+	3.38769e7i	−0.743538	+	1.28785i	−0.950875	+	0.309576i	$0.899813\pi$
$930$		0			0
$931$	−1.19768e7	−	2.08753e6i	−0.452865	−	0.0789331i
$932$		0			0
$933$	1.16987e7	−	2.02628e7i	0.439981	−	0.762069i
$934$		0			0
$935$	6.60088e6	+	1.14331e7i	0.246929	+	0.427694i
$936$		0			0
$937$	−3.62115e7			−1.34740			−0.673701	−	0.739004i	$-0.735297\pi$
$937$	−3.62115e7			−1.34740			−0.673701	+	0.739004i	$0.735297\pi$
$938$		0			0
$939$	−8.47513e6			−0.313677
$940$		0			0
$941$	2.27277e7	+	3.93655e7i	0.836722	+	1.44925i	0.892620	+	0.450809i	$0.148864\pi$
$941$	2.27277e7	+	3.93655e7i	0.836722	+	1.44925i	−0.0558980	+	0.998436i	$0.517802\pi$
$942$		0			0
$943$	2.00161e6	−	3.46689e6i	0.0732994	−	0.126958i
$944$		0			0
$945$	−827586.	−	1.77021e6i	−0.0301463	−	0.0644830i
$946$		0			0
$947$	−1.54509e7	+	2.67618e7i	−0.559861	+	0.969708i	0.437646	+	0.899147i	$0.355812\pi$
$947$	−1.54509e7	+	2.67618e7i	−0.559861	+	0.969708i	−0.997507	+	0.0705606i	$0.977521\pi$
$948$		0			0
$949$	3.46359e7	+	5.99911e7i	1.24842	+	2.16233i
$950$		0			0
$951$	−1.35858e7			−0.487119
$952$		0			0
$953$	1.98293e7			0.707254			0.353627	−	0.935386i	$-0.384948\pi$
$953$	1.98293e7			0.707254			0.353627	+	0.935386i	$0.384948\pi$
$954$		0			0
$955$	−9.38302e6	−	1.62519e7i	−0.332916	−	0.576627i
$956$		0			0
$957$	6.73985e6	−	1.16738e7i	0.237887	−	0.412032i
$958$		0			0
$959$	−1.77403e7	−	1.53448e6i	−0.622895	−	0.0538783i
$960$		0			0
$961$	−2.55058e7	+	4.41774e7i	−0.890904	+	1.54309i
$962$		0			0
$963$	7.09386e6	+	1.22869e7i	0.246500	+	0.426951i
$964$		0			0
$965$	−9.46693e6			−0.327259
$966$		0			0
$967$	−7.62077e6			−0.262079			−0.131040	−	0.991377i	$-0.541832\pi$
$967$	−7.62077e6			−0.262079			−0.131040	+	0.991377i	$0.541832\pi$
$968$		0			0
$969$	−5.54124e6	−	9.59770e6i	−0.189582	−	0.328366i
$970$		0			0
$971$	1.66605e7	−	2.88568e7i	0.567073	−	0.982200i	−0.429780	−	0.902933i	$-0.641409\pi$
$971$	1.66605e7	−	2.88568e7i	0.567073	−	0.982200i	0.996853	−	0.0792661i	$-0.0252577\pi$
$972$		0			0
$973$	2.15465e6	−	3.08362e6i	0.0729616	−	0.104419i
$974$		0			0
$975$	−1.18142e7	+	2.04627e7i	−0.398007	+	0.689369i
$976$		0			0
$977$	9.65232e6	+	1.67183e7i	0.323516	+	0.560346i	0.981211	−	0.192938i	$-0.0618018\pi$
$977$	9.65232e6	+	1.67183e7i	0.323516	+	0.560346i	−0.657695	+	0.753284i	$0.728468\pi$
$978$		0			0
$979$	5.14640e7			1.71612
$980$		0			0
$981$	−1.13182e7			−0.375494
$982$		0			0
$983$	1.85314e7	+	3.20974e7i	0.611681	+	1.05946i	0.990957	+	0.134180i	$0.0428399\pi$
$983$	1.85314e7	+	3.20974e7i	0.611681	+	1.05946i	−0.379276	+	0.925284i	$0.623827\pi$
$984$		0			0
$985$	−8.23455e6	+	1.42627e7i	−0.270427	+	0.468393i
$986$		0			0
$987$	8.79895e6	−	1.25926e7i	0.287500	−	0.411454i
$988$		0			0
$989$	−1.47086e6	+	2.54760e6i	−0.0478167	+	0.0828209i
$990$		0			0
$991$	−2.82451e6	−	4.89219e6i	−0.0913605	−	0.158241i	0.816723	−	0.577029i	$-0.195788\pi$
$991$	−2.82451e6	−	4.89219e6i	−0.0913605	−	0.158241i	−0.908084	+	0.418788i	$0.862455\pi$
$992$		0			0
$993$	1.73136e7			0.557202
$994$		0			0
$995$	9.80674e6			0.314027
$996$		0			0
$997$	−3.98926e6	−	6.90960e6i	−0.127103	−	0.220148i	0.795450	−	0.606019i	$-0.207234\pi$
$997$	−3.98926e6	−	6.90960e6i	−0.127103	−	0.220148i	−0.922553	+	0.385871i	$0.873901\pi$
$998$		0			0
$999$	−4.15293e6	+	7.19309e6i	−0.131656	+	0.228035i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.q.m.289.4		10
4.3	odd	2		168.6.q.b.121.4	yes	10
7.4	even	3	inner	336.6.q.m.193.4		10
12.11	even	2		504.6.s.e.289.2		10
28.11	odd	6		168.6.q.b.25.4	✓	10
84.11	even	6		504.6.s.e.361.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
168.6.q.b.25.4	✓	10	28.11	odd	6
168.6.q.b.121.4	yes	10	4.3	odd	2
336.6.q.m.193.4		10	7.4	even	3	inner
336.6.q.m.289.4		10	1.1	even	1	trivial
504.6.s.e.289.2		10	12.11	even	2
504.6.s.e.361.2		10	84.11	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 $\F_{q}$

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

\(f(q)\)	\(=\)	\(q+(4.50000 + 7.79423i) q^{3} +(10.3382 - 17.9063i) q^{5} +(-74.2549 + 106.270i) q^{7} +(-40.5000 + 70.1481i) q^{9} +O(q^{10})\)	\(q+(4.50000 + 7.79423i) q^{3} +(10.3382 - 17.9063i) q^{5} +(-74.2549 + 106.270i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-187.535 - 324.820i) q^{11} +973.265 q^{13} +186.088 q^{15} +(-851.164 - 1474.26i) q^{17} +(361.677 - 626.443i) q^{19} +(-1162.44 - 100.547i) q^{21} +(-227.469 + 393.988i) q^{23} +(1348.74 + 2336.09i) q^{25} -729.000 q^{27} +3993.24 q^{29} +(4462.08 + 7728.56i) q^{31} +(1687.82 - 2923.38i) q^{33} +(1135.23 + 2428.27i) q^{35} +(5696.75 - 9867.06i) q^{37} +(4379.69 + 7585.85i) q^{39} -8799.48 q^{41} +6466.17 q^{43} +(837.397 + 1450.41i) q^{45} +(-6583.14 + 11402.3i) q^{47} +(-5779.42 - 15782.1i) q^{49} +(7660.47 - 13268.3i) q^{51} +(419.928 + 727.337i) q^{53} -7755.12 q^{55} +6510.19 q^{57} +(18514.0 + 32067.1i) q^{59} +(21786.9 - 37735.9i) q^{61} +(-4447.28 - 9512.75i) q^{63} +(10061.8 - 17427.6i) q^{65} +(17287.4 + 29942.6i) q^{67} -4094.45 q^{69} +4363.10 q^{71} +(35587.3 + 61639.0i) q^{73} +(-12138.7 + 21024.8i) q^{75} +(48443.9 + 4190.23i) q^{77} +(6320.92 - 10948.2i) q^{79} +(-3280.50 - 5681.99i) q^{81} +45853.6 q^{83} -35198.1 q^{85} +(17969.6 + 31124.2i) q^{87} +(-68605.8 + 118829. i) q^{89} +(-72269.7 + 103428. i) q^{91} +(-40158.8 + 69557.0i) q^{93} +(-7478.20 - 12952.6i) q^{95} +178723. q^{97} +30380.7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 45 q^{3} - 75 q^{5} + 113 q^{7} - 405 q^{9}+O(q^{10}) \) 10 * q + 45 * q^3 - 75 * q^5 + 113 * q^7 - 405 * q^9	\( 10 q + 45 q^{3} - 75 q^{5} + 113 q^{7} - 405 q^{9} - 91 q^{11} + 580 q^{13} - 1350 q^{15} - 1128 q^{17} - 282 q^{19} + 279 q^{21} + 1808 q^{23} - 202 q^{25} - 7290 q^{27} + 5026 q^{29} + 5069 q^{31} + 819 q^{33} + 884 q^{35} - 5010 q^{37} + 2610 q^{39} + 12872 q^{41} - 17328 q^{43} - 6075 q^{45} + 50 q^{47} + 29135 q^{49} + 10152 q^{51} + 1167 q^{53} - 97410 q^{55} - 5076 q^{57} + 42797 q^{59} - 26546 q^{61} - 6642 q^{63} - 2216 q^{65} + 13440 q^{67} + 32544 q^{69} + 39356 q^{71} - 27768 q^{73} + 1818 q^{75} + 125797 q^{77} + 123369 q^{79} - 32805 q^{81} - 334250 q^{83} - 324936 q^{85} + 22617 q^{87} - 59350 q^{89} - 113850 q^{91} - 45621 q^{93} - 41864 q^{95} + 525282 q^{97} + 14742 q^{99}+O(q^{100}) \) 10 * q + 45 * q^3 - 75 * q^5 + 113 * q^7 - 405 * q^9 - 91 * q^11 + 580 * q^13 - 1350 * q^15 - 1128 * q^17 - 282 * q^19 + 279 * q^21 + 1808 * q^23 - 202 * q^25 - 7290 * q^27 + 5026 * q^29 + 5069 * q^31 + 819 * q^33 + 884 * q^35 - 5010 * q^37 + 2610 * q^39 + 12872 * q^41 - 17328 * q^43 - 6075 * q^45 + 50 * q^47 + 29135 * q^49 + 10152 * q^51 + 1167 * q^53 - 97410 * q^55 - 5076 * q^57 + 42797 * q^59 - 26546 * q^61 - 6642 * q^63 - 2216 * q^65 + 13440 * q^67 + 32544 * q^69 + 39356 * q^71 - 27768 * q^73 + 1818 * q^75 + 125797 * q^77 + 123369 * q^79 - 32805 * q^81 - 334250 * q^83 - 324936 * q^85 + 22617 * q^87 - 59350 * q^89 - 113850 * q^91 - 45621 * q^93 - 41864 * q^95 + 525282 * q^97 + 14742 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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