LMFDB - Embedded newform 336.6.q.e.289.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

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

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

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

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

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

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

chi := DirichletCharacter("336.193");

S:= CuspForms(chi, 6);

N := Newforms(S);

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

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$53.8889634572$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-83})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 20x^{2} - 21x + 441 $ x^4 - x^3 - 20x^2 - 21x + 441
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 21)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			289.2
Root			$4.19493 - 1.84460i$ of defining polynomial
Character	$\chi$	$=$	336.289
Dual form			336.6.q.e.193.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-4.50000 - 7.79423i) q^{3} +(35.8645 - 62.1192i) q^{5} +(87.5000 - 95.6596i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-280.305 - 485.503i) q^{11} +533.509 q^{13} -645.562 q^{15} +(-502.850 - 870.962i) q^{17} +(684.263 - 1185.18i) q^{19} +(-1139.34 - 251.527i) q^{21} +(1614.04 - 2795.60i) q^{23} +(-1010.03 - 1749.42i) q^{25} +729.000 q^{27} -753.456 q^{29} +(4103.21 + 7106.97i) q^{31} +(-2522.75 + 4369.52i) q^{33} +(-2804.15 - 8866.21i) q^{35} +(1404.33 - 2432.37i) q^{37} +(-2400.79 - 4158.29i) q^{39} +245.827 q^{41} +17504.5 q^{43} +(2905.03 + 5031.65i) q^{45} +(-8172.74 + 14155.6i) q^{47} +(-1494.50 - 16740.4i) q^{49} +(-4525.65 + 7838.66i) q^{51} +(14820.8 + 25670.4i) q^{53} -40212.0 q^{55} -12316.7 q^{57} +(-5178.05 - 8968.65i) q^{59} +(-477.089 + 826.343i) q^{61} +(3166.58 + 10012.2i) q^{63} +(19134.0 - 33141.1i) q^{65} +(-9907.60 - 17160.5i) q^{67} -29052.7 q^{69} -62125.4 q^{71} +(-13554.8 - 23477.6i) q^{73} +(-9090.27 + 15744.8i) q^{75} +(-70969.7 - 15667.6i) q^{77} +(22343.7 - 38700.4i) q^{79} +(-3280.50 - 5681.99i) q^{81} -15606.6 q^{83} -72137.9 q^{85} +(3390.55 + 5872.61i) q^{87} +(-6817.66 + 11808.5i) q^{89} +(46682.0 - 51035.2i) q^{91} +(36928.9 - 63962.7i) q^{93} +(-49081.6 - 85011.8i) q^{95} -12919.5 q^{97} +45409.4 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 18 q^{3} + 33 q^{5} + 350 q^{7} - 162 q^{9} - 1137 q^{11} + 1850 q^{13} - 594 q^{15} + 324 q^{17} + 2311 q^{19} - 1575 q^{21} + 1596 q^{23} - 395 q^{25} + 2916 q^{27} - 4434 q^{29} + 4294 q^{31} - 10233 q^{33}+ \cdots + 184194 q^{99}+O(q^{100}) $ 4 * q - 18 * q^3 + 33 * q^5 + 350 * q^7 - 162 * q^9 - 1137 * q^11 + 1850 * q^13 - 594 * q^15 + 324 * q^17 + 2311 * q^19 - 1575 * q^21 + 1596 * q^23 - 395 * q^25 + 2916 * q^27 - 4434 * q^29 + 4294 * q^31 - 10233 * q^33 - 15414 * q^35 + 19109 * q^37 - 8325 * q^39 - 25716 * q^41 + 5542 * q^43 + 2673 * q^45 - 23160 * q^47 - 5978 * q^49 + 2916 * q^51 + 31653 * q^53 - 35778 * q^55 - 41598 * q^57 + 41097 * q^59 - 42052 * q^61 - 14175 * q^63 + 23106 * q^65 + 30763 * q^67 - 28728 * q^69 - 204192 * q^71 + 28577 * q^73 - 3555 * q^75 - 96873 * q^77 - 18464 * q^79 - 13122 * q^81 - 122358 * q^83 - 247272 * q^85 + 19953 * q^87 - 29322 * q^89 + 161875 * q^91 + 38646 * q^93 - 61662 * q^95 - 19582 * q^97 + 184194 * q^99

Character values

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

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

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
$5$	35.8645	−	62.1192i	0.641564	−	1.11122i	−0.343519	−	0.939146i	$-0.611619\pi$
$5$	35.8645	−	62.1192i	0.641564	−	1.11122i	0.985084	−	0.172076i	$-0.0550476\pi$
$6$		0			0
$7$	87.5000	−	95.6596i	0.674937	−	0.737876i
$7$	87.5000	−	95.6596i	0.674937	−	0.737876i
$8$		0			0
$9$	−40.5000	+	70.1481i	−0.166667	+	0.288675i
$10$		0			0
$11$	−280.305	−	485.503i	−0.698472	−	1.20979i	−0.968996	−	0.247076i	$-0.920530\pi$
$11$	−280.305	−	485.503i	−0.698472	−	1.20979i	0.270524	−	0.962713i	$-0.412803\pi$
$12$		0			0
$13$	533.509			0.875555			0.437777	−	0.899083i	$-0.355766\pi$
$13$	533.509			0.875555			0.437777	+	0.899083i	$0.355766\pi$
$14$		0			0
$15$	−645.562			−0.740815
$16$		0			0
$17$	−502.850	−	870.962i	−0.422004	−	0.730932i	0.574132	−	0.818763i	$-0.305340\pi$
$17$	−502.850	−	870.962i	−0.422004	−	0.730932i	−0.996135	+	0.0878311i	$0.972006\pi$
$18$		0			0
$19$	684.263	−	1185.18i	0.434850	−	0.753182i	−0.562434	−	0.826842i	$-0.690135\pi$
$19$	684.263	−	1185.18i	0.434850	−	0.753182i	0.997283	+	0.0736606i	$0.0234682\pi$
$20$		0			0
$21$	−1139.34	−	251.527i	−0.563775	−	0.124462i
$22$		0			0
$23$	1614.04	−	2795.60i	0.636201	−	1.10193i	−0.350058	−	0.936728i	$-0.613838\pi$
$23$	1614.04	−	2795.60i	0.636201	−	1.10193i	0.986259	−	0.165205i	$-0.0528286\pi$
$24$		0			0
$25$	−1010.03	−	1749.42i	−0.323209	−	0.559815i
$26$		0			0
$27$	729.000			0.192450
$28$		0			0
$29$	−753.456			−0.166365			−0.0831827	−	0.996534i	$-0.526508\pi$
$29$	−753.456			−0.166365			−0.0831827	+	0.996534i	$0.526508\pi$
$30$		0			0
$31$	4103.21	+	7106.97i	0.766866	+	1.32825i	0.939254	+	0.343222i	$0.111518\pi$
$31$	4103.21	+	7106.97i	0.766866	+	1.32825i	−0.172389	+	0.985029i	$0.555148\pi$
$32$		0			0
$33$	−2522.75	+	4369.52i	−0.403263	+	0.698472i
$34$		0			0
$35$	−2804.15	−	8866.21i	−0.386929	−	1.22340i
$36$		0			0
$37$	1404.33	−	2432.37i	0.168642	−	0.292096i	−0.769301	−	0.638887i	$-0.779395\pi$
$37$	1404.33	−	2432.37i	0.168642	−	0.292096i	0.937943	+	0.346791i	$0.112729\pi$
$38$		0			0
$39$	−2400.79	−	4158.29i	−0.252751	−	0.437777i
$40$		0			0
$41$	245.827			0.0228387			0.0114193	−	0.999935i	$-0.496365\pi$
$41$	245.827			0.0228387			0.0114193	+	0.999935i	$0.496365\pi$
$42$		0			0
$43$	17504.5			1.44371			0.721853	−	0.692047i	$-0.243291\pi$
$43$	17504.5			1.44371			0.721853	+	0.692047i	$0.243291\pi$
$44$		0			0
$45$	2905.03	+	5031.65i	0.213855	+	0.370407i
$46$		0			0
$47$	−8172.74	+	14155.6i	−0.539663	+	0.934725i	0.459258	+	0.888303i	$0.348115\pi$
$47$	−8172.74	+	14155.6i	−0.539663	+	0.934725i	−0.998922	+	0.0464219i	$0.985218\pi$
$48$		0			0
$49$	−1494.50	−	16740.4i	−0.0889213	−	0.996039i
$50$		0			0
$51$	−4525.65	+	7838.66i	−0.243644	+	0.422004i
$52$		0			0
$53$	14820.8	+	25670.4i	0.724741	+	1.25529i	0.959081	+	0.283133i	$0.0913739\pi$
$53$	14820.8	+	25670.4i	0.724741	+	1.25529i	−0.234340	+	0.972155i	$0.575293\pi$
$54$		0			0
$55$	−40212.0			−1.79246
$56$		0			0
$57$	−12316.7			−0.502121
$58$		0			0
$59$	−5178.05	−	8968.65i	−0.193659	−	0.335426i	0.752801	−	0.658248i	$-0.228702\pi$
$59$	−5178.05	−	8968.65i	−0.193659	−	0.335426i	−0.946460	+	0.322821i	$0.895369\pi$
$60$		0			0
$61$	−477.089	+	826.343i	−0.0164163	+	0.0284339i	−0.874117	−	0.485716i	$-0.838559\pi$
$61$	−477.089	+	826.343i	−0.0164163	+	0.0284339i	0.857701	+	0.514150i	$0.171892\pi$
$62$		0			0
$63$	3166.58	+	10012.2i	0.100517	+	0.317817i
$64$		0			0
$65$	19134.0	−	33141.1i	0.561725	−	0.972935i
$66$		0			0
$67$	−9907.60	−	17160.5i	−0.269638	−	0.467027i	0.699130	−	0.714994i	$-0.253571\pi$
$67$	−9907.60	−	17160.5i	−0.269638	−	0.467027i	−0.968768	+	0.247967i	$0.920237\pi$
$68$		0			0
$69$	−29052.7			−0.734622
$70$		0			0
$71$	−62125.4			−1.46259			−0.731296	−	0.682060i	$-0.761084\pi$
$71$	−62125.4			−1.46259			−0.731296	+	0.682060i	$0.761084\pi$
$72$		0			0
$73$	−13554.8	−	23477.6i	−0.297705	−	0.515641i	0.677905	−	0.735149i	$-0.262888\pi$
$73$	−13554.8	−	23477.6i	−0.297705	−	0.515641i	−0.975611	+	0.219509i	$0.929555\pi$
$74$		0			0
$75$	−9090.27	+	15744.8i	−0.186605	+	0.323209i
$76$		0			0
$77$	−70969.7	−	15667.6i	−1.36410	−	0.301145i
$78$		0			0
$79$	22343.7	−	38700.4i	0.402798	−	0.697666i	−0.591265	−	0.806477i	$-0.701371\pi$
$79$	22343.7	−	38700.4i	0.402798	−	0.697666i	0.994062	+	0.108812i	$0.0347045\pi$
$80$		0			0
$81$	−3280.50	−	5681.99i	−0.0555556	−	0.0962250i
$82$		0			0
$83$	−15606.6			−0.248665			−0.124332	−	0.992241i	$-0.539679\pi$
$83$	−15606.6			−0.248665			−0.124332	+	0.992241i	$0.539679\pi$
$84$		0			0
$85$	−72137.9			−1.08297
$86$		0			0
$87$	3390.55	+	5872.61i	0.0480255	+	0.0831827i
$88$		0			0
$89$	−6817.66	+	11808.5i	−0.0912347	+	0.158023i	−0.908031	−	0.418903i	$-0.862415\pi$
$89$	−6817.66	+	11808.5i	−0.0912347	+	0.158023i	0.816796	+	0.576926i	$0.195748\pi$
$90$		0			0
$91$	46682.0	−	51035.2i	0.590944	−	0.646050i
$92$		0			0
$93$	36928.9	−	63962.7i	0.442750	−	0.766866i
$94$		0			0
$95$	−49081.6	−	85011.8i	−0.557968	−	0.966429i
$96$		0			0
$97$	−12919.5			−0.139417			−0.0697086	−	0.997567i	$-0.522207\pi$
$97$	−12919.5			−0.139417			−0.0697086	+	0.997567i	$0.522207\pi$
$98$		0			0
$99$	45409.4			0.465648
$100$		0			0
$101$	−12571.5	−	21774.4i	−0.122626	−	0.212394i	0.798177	−	0.602424i	$-0.205798\pi$
$101$	−12571.5	−	21774.4i	−0.122626	−	0.212394i	−0.920802	+	0.390029i	$0.872465\pi$
$102$		0			0
$103$	−80376.6	+	139216.i	−0.746511	+	1.29300i	0.202974	+	0.979184i	$0.434939\pi$
$103$	−80376.6	+	139216.i	−0.746511	+	1.29300i	−0.949485	+	0.313811i	$0.898394\pi$
$104$		0			0
$105$	−56486.6	+	61754.1i	−0.500003	+	0.546629i
$106$		0			0
$107$	−47187.8	+	81731.6i	−0.398446	+	0.690129i	−0.993534	−	0.113531i	$-0.963784\pi$
$107$	−47187.8	+	81731.6i	−0.398446	+	0.690129i	0.595088	+	0.803661i	$0.297117\pi$
$108$		0			0
$109$	41696.7	+	72220.9i	0.336152	+	0.582233i	0.983705	−	0.179788i	$-0.0575410\pi$
$109$	41696.7	+	72220.9i	0.336152	+	0.582233i	−0.647553	+	0.762020i	$0.724208\pi$
$110$		0			0
$111$	−25278.0			−0.194731
$112$		0			0
$113$	−179254.			−1.32060			−0.660301	−	0.751001i	$-0.729571\pi$
$113$	−179254.			−1.32060			−0.660301	+	0.751001i	$0.729571\pi$
$114$		0			0
$115$	−115774.	−	200526.i	−0.816328	−	1.41392i
$116$		0			0
$117$	−21607.1	+	37424.6i	−0.145926	+	0.252751i
$118$		0			0
$119$	−127315.	−	28106.8i	−0.824163	−	0.181946i
$120$		0			0
$121$	−76616.4	+	132703.i	−0.475727	+	0.823984i
$122$		0			0
$123$	−1106.22	−	1916.04i	−0.00659295	−	0.0114193i
$124$		0			0
$125$	79256.4			0.453690
$126$		0			0
$127$	143674.			0.790440			0.395220	−	0.918586i	$-0.370668\pi$
$127$	143674.			0.790440			0.395220	+	0.918586i	$0.370668\pi$
$128$		0			0
$129$	−78770.2	−	136434.i	−0.416762	−	0.721853i
$130$		0			0
$131$	26144.9	−	45284.3i	0.133110	−	0.230553i	−0.791764	−	0.610827i	$-0.790837\pi$
$131$	26144.9	−	45284.3i	0.133110	−	0.230553i	0.924874	+	0.380274i	$0.124170\pi$
$132$		0			0
$133$	−53500.6	−	169159.i	−0.262259	−	0.829215i
$134$		0			0
$135$	26145.2	−	45284.9i	0.123469	−	0.213855i
$136$		0			0
$137$	−4705.05	−	8149.39i	−0.0214172	−	0.0370957i	0.855118	−	0.518433i	$-0.173484\pi$
$137$	−4705.05	−	8149.39i	−0.0214172	−	0.0370957i	−0.876535	+	0.481337i	$0.840151\pi$
$138$		0			0
$139$	−183094.			−0.803781			−0.401890	−	0.915688i	$-0.631647\pi$
$139$	−183094.			−0.803781			−0.401890	+	0.915688i	$0.631647\pi$
$140$		0			0
$141$	147109.			0.623150
$142$		0			0
$143$	−149545.	−	259020.i	−0.611551	−	1.05924i
$144$		0			0
$145$	−27022.3	+	46804.1i	−0.106734	+	0.184869i
$146$		0			0
$147$	−123753.	+	86980.4i	−0.472350	+	0.331992i
$148$		0			0
$149$	83500.8	−	144628.i	0.308123	−	0.533685i	−0.669828	−	0.742516i	$-0.733632\pi$
$149$	83500.8	−	144628.i	0.308123	−	0.533685i	0.977952	+	0.208830i	$0.0669657\pi$
$150$		0			0
$151$	188132.	+	325855.i	0.671461	+	1.16300i	0.977490	+	0.210982i	$0.0676662\pi$
$151$	188132.	+	325855.i	0.671461	+	1.16300i	−0.306029	+	0.952022i	$0.599000\pi$
$152$		0			0
$153$	81461.7			0.281336
$154$		0			0
$155$	588639.			1.96797
$156$		0			0
$157$	−19537.5	−	33840.0i	−0.0632587	−	0.109567i	0.832662	−	0.553782i	$-0.186816\pi$
$157$	−19537.5	−	33840.0i	−0.0632587	−	0.109567i	−0.895920	+	0.444215i	$0.853483\pi$
$158$		0			0
$159$	133387.	−	231034.i	0.418429	−	0.724741i
$160$		0			0
$161$	−126197.	−	399013.i	−0.383694	−	1.21317i
$162$		0			0
$163$	−238959.	+	413890.i	−0.704458	+	1.22016i	0.262428	+	0.964951i	$0.415477\pi$
$163$	−238959.	+	413890.i	−0.704458	+	1.22016i	−0.966887	+	0.255206i	$0.917857\pi$
$164$		0			0
$165$	180954.	+	313422.i	0.517439	+	0.896230i
$166$		0			0
$167$	−39793.4			−0.110413			−0.0552064	−	0.998475i	$-0.517582\pi$
$167$	−39793.4			−0.110413			−0.0552064	+	0.998475i	$0.517582\pi$
$168$		0			0
$169$	−86661.4			−0.233404
$170$		0			0
$171$	55425.3	+	95999.5i	0.144950	+	0.251061i
$172$		0			0
$173$	−24169.1	+	41862.1i	−0.0613968	+	0.106342i	−0.895090	−	0.445886i	$-0.852889\pi$
$173$	−24169.1	+	41862.1i	−0.0613968	+	0.106342i	0.833693	+	0.552228i	$0.186222\pi$
$174$		0			0
$175$	−255727.	−	56455.5i	−0.631220	−	0.139351i
$176$		0			0
$177$	−46602.5	+	80717.9i	−0.111809	+	0.193659i
$178$		0			0
$179$	−71455.3	−	123764.i	−0.166687	−	0.288711i	0.770566	−	0.637360i	$-0.219974\pi$
$179$	−71455.3	−	123764.i	−0.166687	−	0.288711i	−0.937253	+	0.348650i	$0.886640\pi$
$180$		0			0
$181$	−77245.3			−0.175257			−0.0876285	−	0.996153i	$-0.527929\pi$
$181$	−77245.3			−0.175257			−0.0876285	+	0.996153i	$0.527929\pi$
$182$		0			0
$183$	8587.61			0.0189559
$184$		0			0
$185$	−100731.	−	174472.i	−0.216389	−	0.374797i
$186$		0			0
$187$	−281903.	+	488270.i	−0.589516	+	1.02107i
$188$		0			0
$189$	63787.5	−	69735.8i	0.129892	−	0.142004i
$190$		0			0
$191$	136027.	−	235606.i	0.269800	−	0.467307i	−0.699010	−	0.715112i	$-0.746376\pi$
$191$	136027.	−	235606.i	0.269800	−	0.467307i	0.968810	+	0.247805i	$0.0797092\pi$
$192$		0			0
$193$	8016.89	+	13885.7i	0.0154922	+	0.0268333i	0.873668	−	0.486523i	$-0.161735\pi$
$193$	8016.89	+	13885.7i	0.0154922	+	0.0268333i	−0.858175	+	0.513357i	$0.828402\pi$
$194$		0			0
$195$	−344413.			−0.648624
$196$		0			0
$197$	1.03228e6			1.89510			0.947552	−	0.319603i	$-0.103549\pi$
$197$	1.03228e6			1.89510			0.947552	+	0.319603i	$0.103549\pi$
$198$		0			0
$199$	−440868.	−	763606.i	−0.789180	−	1.36690i	−0.926470	−	0.376369i	$-0.877173\pi$
$199$	−440868.	−	763606.i	−0.789180	−	1.36690i	0.137290	−	0.990531i	$-0.456161\pi$
$200$		0			0
$201$	−89168.4	+	154444.i	−0.155676	+	0.269638i
$202$		0			0
$203$	−65927.4	+	72075.3i	−0.112286	+	0.122757i
$204$		0			0
$205$	8816.49	−	15270.6i	0.0146525	−	0.0253788i
$206$		0			0
$207$	130737.	+	226443.i	0.212067	+	0.367311i
$208$		0			0
$209$	−767210.			−1.21492
$210$		0			0
$211$	372813.			0.576480			0.288240	−	0.957558i	$-0.406930\pi$
$211$	372813.			0.576480			0.288240	+	0.957558i	$0.406930\pi$
$212$		0			0
$213$	279564.	+	484219.i	0.422214	+	0.731296i
$214$		0			0
$215$	627791.	−	1.08737e6i	0.926230	−	1.60428i
$216$		0			0
$217$	1.03888e6	+	229348.i	1.49767	+	0.330633i
$218$		0			0
$219$	−121993.	+	211299.i	−0.171880	+	0.297705i
$220$		0			0
$221$	−268275.	−	464666.i	−0.369487	−	0.639971i
$222$		0			0
$223$	1.08205e6			1.45708			0.728541	−	0.685002i	$-0.240199\pi$
$223$	1.08205e6			1.45708			0.728541	+	0.685002i	$0.240199\pi$
$224$		0			0
$225$	163625.			0.215473
$226$		0			0
$227$	276524.	+	478954.i	0.356179	+	0.616921i	0.987319	−	0.158748i	$-0.0507459\pi$
$227$	276524.	+	478954.i	0.356179	+	0.616921i	−0.631140	+	0.775669i	$0.717413\pi$
$228$		0			0
$229$	−261512.	+	452952.i	−0.329536	+	0.570773i	−0.982420	−	0.186685i	$-0.940226\pi$
$229$	−261512.	+	452952.i	−0.329536	+	0.570773i	0.652884	+	0.757458i	$0.273559\pi$
$230$		0			0
$231$	197246.	+	623658.i	0.243209	+	0.768983i
$232$		0			0
$233$	182090.	−	315390.i	0.219734	−	0.380590i	−0.734993	−	0.678075i	$-0.762814\pi$
$233$	182090.	−	315390.i	0.219734	−	0.380590i	0.954727	+	0.297485i	$0.0961477\pi$
$234$		0			0
$235$	586223.	+	1.01537e6i	0.692458	+	1.19937i
$236$		0			0
$237$	−402186.			−0.465111
$238$		0			0
$239$	−371841.			−0.421078			−0.210539	−	0.977585i	$-0.567522\pi$
$239$	−371841.			−0.421078			−0.210539	+	0.977585i	$0.567522\pi$
$240$		0			0
$241$	855737.	+	1.48218e6i	0.949069	+	1.64384i	0.747391	+	0.664384i	$0.231306\pi$
$241$	855737.	+	1.48218e6i	0.949069	+	1.64384i	0.201678	+	0.979452i	$0.435360\pi$
$242$		0			0
$243$	−29524.5	+	51137.9i	−0.0320750	+	0.0555556i
$244$		0			0
$245$	−1.09350e6	−	507550.i	−1.16387	−	0.540212i
$246$		0			0
$247$	365060.	−	632303.i	0.380735	−	0.659452i
$248$		0			0
$249$	70229.9	+	121642.i	0.0717833	+	0.124332i
$250$		0			0
$251$	−58134.1			−0.0582434			−0.0291217	−	0.999576i	$-0.509271\pi$
$251$	−58134.1			−0.0582434			−0.0291217	+	0.999576i	$0.509271\pi$
$252$		0			0
$253$	−1.80969e6			−1.77748
$254$		0			0
$255$	324621.	+	562260.i	0.312627	+	0.541485i
$256$		0			0
$257$	155920.	−	270061.i	0.147254	−	0.255052i	−0.782957	−	0.622075i	$-0.786290\pi$
$257$	155920.	−	270061.i	0.147254	−	0.255052i	0.930212	+	0.367023i	$0.119623\pi$
$258$		0			0
$259$	−109801.	−	347171.i	−0.101708	−	0.321583i
$260$		0			0
$261$	30515.0	−	52853.5i	0.0277276	−	0.0480255i
$262$		0			0
$263$	431983.	+	748216.i	0.385103	+	0.667018i	0.991783	−	0.127928i	$-0.0408326\pi$
$263$	431983.	+	748216.i	0.385103	+	0.667018i	−0.606681	+	0.794946i	$0.707499\pi$
$264$		0			0
$265$	2.12617e6			1.85987
$266$		0			0
$267$	122718.			0.105349
$268$		0			0
$269$	560346.	+	970548.i	0.472145	+	0.817779i	0.999492	−	0.0318707i	$-0.0101465\pi$
$269$	560346.	+	970548.i	0.472145	+	0.817779i	−0.527347	+	0.849650i	$0.676813\pi$
$270$		0			0
$271$	570058.	−	987369.i	0.471515	−	0.816688i	−0.527954	−	0.849273i	$-0.677041\pi$
$271$	570058.	−	987369.i	0.471515	−	0.816688i	0.999469	+	0.0325848i	$0.0103739\pi$
$272$		0			0
$273$	−607849.	−	134192.i	−0.493616	−	0.108973i
$274$		0			0
$275$	−566233.	+	980744.i	−0.451506	+	0.782031i
$276$		0			0
$277$	994005.	+	1.72167e6i	0.778375	+	1.34819i	0.932878	+	0.360193i	$0.117289\pi$
$277$	994005.	+	1.72167e6i	0.778375	+	1.34819i	−0.154502	+	0.987992i	$0.549377\pi$
$278$		0			0
$279$	−664720.			−0.511244
$280$		0			0
$281$	532321.			0.402168			0.201084	−	0.979574i	$-0.435554\pi$
$281$	532321.			0.402168			0.201084	+	0.979574i	$0.435554\pi$
$282$		0			0
$283$	1.31237e6	+	2.27308e6i	0.974067	+	1.68713i	0.682980	+	0.730437i	$0.260684\pi$
$283$	1.31237e6	+	2.27308e6i	0.974067	+	1.68713i	0.291087	+	0.956697i	$0.405983\pi$
$284$		0			0
$285$	−441734.	+	765106.i	−0.322143	+	0.557968i
$286$		0			0
$287$	21509.9	−	23515.7i	0.0154146	−	0.0168521i
$288$		0			0
$289$	204212.	−	353705.i	0.143826	−	0.249113i
$290$		0			0
$291$	58137.7	+	100697.i	0.0402463	+	0.0697086i
$292$		0			0
$293$	609962.			0.415082			0.207541	−	0.978226i	$-0.433454\pi$
$293$	609962.			0.415082			0.207541	+	0.978226i	$0.433454\pi$
$294$		0			0
$295$	−742834.			−0.496978
$296$		0			0
$297$	−204342.	−	353931.i	−0.134421	−	0.232824i
$298$		0			0
$299$	861104.	−	1.49148e6i	0.557029	−	0.964802i
$300$		0			0
$301$	1.53164e6	−	1.67447e6i	0.974409	−	1.06528i
$302$		0			0
$303$	−113143.	+	195970.i	−0.0707981	+	0.122626i
$304$		0			0
$305$	34221.2	+	59272.8i	0.0210642	+	0.0364843i
$306$		0			0
$307$	1.34843e6			0.816551			0.408275	−	0.912859i	$-0.366130\pi$
$307$	1.34843e6			0.816551			0.408275	+	0.912859i	$0.366130\pi$
$308$		0			0
$309$	1.44678e6			0.861997
$310$		0			0
$311$	−1.66530e6	−	2.88439e6i	−0.976320	−	1.69104i	−0.675507	−	0.737353i	$-0.736075\pi$
$311$	−1.66530e6	−	2.88439e6i	−0.976320	−	1.69104i	−0.300813	−	0.953683i	$-0.597258\pi$
$312$		0			0
$313$	1.41835e6	−	2.45666e6i	0.818320	−	1.41737i	−0.0885991	−	0.996067i	$-0.528239\pi$
$313$	1.41835e6	−	2.45666e6i	0.818320	−	1.41737i	0.906919	−	0.421305i	$-0.138428\pi$
$314$		0			0
$315$	735516.	+	162376.i	0.417653	+	0.0922032i
$316$		0			0
$317$	544195.	−	942574.i	0.304163	−	0.526826i	−0.672912	−	0.739723i	$-0.734957\pi$
$317$	544195.	−	942574.i	0.304163	−	0.526826i	0.977075	+	0.212897i	$0.0682899\pi$
$318$		0			0
$319$	211198.	+	365805.i	0.116202	+	0.201267i
$320$		0			0
$321$	849380.			0.460086
$322$		0			0
$323$	−1.37633e6			−0.734033
$324$		0			0
$325$	−538860.	−	933332.i	−0.282988	−	0.490149i
$326$		0			0
$327$	375271.	−	649988.i	0.194078	−	0.336152i
$328$		0			0
$329$	639004.	+	2.02042e6i	0.325472	+	1.02908i
$330$		0			0
$331$	652773.	−	1.13064e6i	0.327485	−	0.567221i	−0.654527	−	0.756039i	$-0.727132\pi$
$331$	652773.	−	1.13064e6i	0.327485	−	0.567221i	0.982012	+	0.188817i	$0.0604654\pi$
$332$		0			0
$333$	113751.	+	197022.i	0.0562140	+	0.0973654i
$334$		0			0
$335$	−1.42133e6			−0.691961
$336$		0			0
$337$	−3.17016e6			−1.52057			−0.760285	−	0.649590i	$-0.774941\pi$
$337$	−3.17016e6			−1.52057			−0.760285	+	0.649590i	$0.774941\pi$
$338$		0			0
$339$	806641.	+	1.39714e6i	0.381225	+	0.660301i
$340$		0			0
$341$	2.30030e6	−	3.98424e6i	1.07127	−	1.85549i
$342$		0			0
$343$	−1.73215e6	−	1.32182e6i	−0.794969	−	0.606650i
$344$		0			0
$345$	−1.04196e6	+	1.80473e6i	−0.471307	+	0.816328i
$346$		0			0
$347$	−857958.	−	1.48603e6i	−0.382510	−	0.662526i	0.608911	−	0.793239i	$-0.291607\pi$
$347$	−857958.	−	1.48603e6i	−0.382510	−	0.662526i	−0.991420	+	0.130713i	$0.958273\pi$
$348$		0			0
$349$	−2.95822e6			−1.30007			−0.650034	−	0.759905i	$-0.725245\pi$
$349$	−2.95822e6			−1.30007			−0.650034	+	0.759905i	$0.725245\pi$
$350$		0			0
$351$	388928.			0.168501
$352$		0			0
$353$	−1.88490e6	−	3.26474e6i	−0.805103	−	1.39448i	−0.916222	−	0.400672i	$-0.868777\pi$
$353$	−1.88490e6	−	3.26474e6i	−0.805103	−	1.39448i	0.111119	−	0.993807i	$-0.464557\pi$
$354$		0			0
$355$	−2.22810e6	+	3.85918e6i	−0.938347	+	1.62526i
$356$		0			0
$357$	353848.	+	1.11880e6i	0.146942	+	0.464605i
$358$		0			0
$359$	964935.	−	1.67132e6i	0.395150	−	0.684420i	−0.597970	−	0.801518i	$-0.704026\pi$
$359$	964935.	−	1.67132e6i	0.395150	−	0.684420i	0.993120	+	0.117099i	$0.0373593\pi$
$360$		0			0
$361$	301617.	+	522416.i	0.121811	+	0.210984i
$362$		0			0
$363$	1.37909e6			0.549323
$364$		0			0
$365$	−1.94455e6			−0.763988
$366$		0			0
$367$	1.18371e6	+	2.05025e6i	0.458754	+	0.794586i	0.998895	−	0.0469885i	$-0.0149624\pi$
$367$	1.18371e6	+	2.05025e6i	0.458754	+	0.794586i	−0.540141	+	0.841575i	$0.681629\pi$
$368$		0			0
$369$	−9956.01	+	17244.3i	−0.00380644	+	0.00659295i
$370$		0			0
$371$	3.75244e6	+	828409.i	1.41540	+	0.312471i
$372$		0			0
$373$	−1.76914e6	+	3.06425e6i	−0.658402	+	1.14039i	0.322627	+	0.946526i	$0.395434\pi$
$373$	−1.76914e6	+	3.06425e6i	−0.658402	+	1.14039i	−0.981029	+	0.193860i	$0.937899\pi$
$374$		0			0
$375$	−356654.	−	617742.i	−0.130969	−	0.226845i
$376$		0			0
$377$	−401975.			−0.145662
$378$		0			0
$379$	−1.79847e6			−0.643139			−0.321569	−	0.946886i	$-0.604210\pi$
$379$	−1.79847e6			−0.643139			−0.321569	+	0.946886i	$0.604210\pi$
$380$		0			0
$381$	−646533.	−	1.11983e6i	−0.228180	−	0.395220i
$382$		0			0
$383$	1.30407e6	−	2.25872e6i	0.454261	−	0.786802i	−0.544385	−	0.838836i	$-0.683237\pi$
$383$	1.30407e6	−	2.25872e6i	0.454261	−	0.786802i	0.998645	+	0.0520333i	$0.0165702\pi$
$384$		0			0
$385$	−3.51855e6	+	3.84667e6i	−1.20980	+	1.32261i
$386$		0			0
$387$	−708932.	+	1.22791e6i	−0.240618	+	0.416762i
$388$		0			0
$389$	1.41498e6	+	2.45081e6i	0.474106	+	0.821175i	0.999560	−	0.0296466i	$-0.00943818\pi$
$389$	1.41498e6	+	2.45081e6i	0.474106	+	0.821175i	−0.525455	+	0.850821i	$0.676105\pi$
$390$		0			0
$391$	−3.24648e6			−1.07392
$392$		0			0
$393$	−470609.			−0.153702
$394$		0			0
$395$	−1.60269e6	−	2.77594e6i	−0.516841	−	0.895195i
$396$		0			0
$397$	1.21531e6	−	2.10498e6i	0.387000	−	0.670303i	−0.605045	−	0.796191i	$-0.706845\pi$
$397$	1.21531e6	−	2.10498e6i	0.387000	−	0.670303i	0.992044	+	0.125888i	$0.0401781\pi$
$398$		0			0
$399$	−1.07771e6	+	1.17821e6i	−0.338900	+	0.370503i
$400$		0			0
$401$	1.21592e6	−	2.10604e6i	0.377611	−	0.654041i	−0.613103	−	0.790003i	$-0.710079\pi$
$401$	1.21592e6	−	2.10604e6i	0.377611	−	0.654041i	0.990714	+	0.135962i	$0.0434124\pi$
$402$		0			0
$403$	2.18910e6	+	3.79163e6i	0.671433	+	1.16296i
$404$		0			0
$405$	−470614.			−0.142570
$406$		0			0
$407$	−1.57457e6			−0.471167
$408$		0			0
$409$	2.38733e6	+	4.13497e6i	0.705674	+	1.22226i	0.966448	+	0.256863i	$0.0826889\pi$
$409$	2.38733e6	+	4.13497e6i	0.705674	+	1.22226i	−0.260774	+	0.965400i	$0.583978\pi$
$410$		0			0
$411$	−42345.5	+	73344.5i	−0.0123652	+	0.0214172i
$412$		0			0
$413$	−1.31102e6	−	289427.i	−0.378210	−	0.0834956i
$414$		0			0
$415$	−559725.	+	969472.i	−0.159534	+	0.276322i
$416$		0			0
$417$	823924.	+	1.42708e6i	0.232032	+	0.401890i
$418$		0			0
$419$	457181.			0.127219			0.0636097	−	0.997975i	$-0.479739\pi$
$419$	457181.			0.127219			0.0636097	+	0.997975i	$0.479739\pi$
$420$		0			0
$421$	−1.82396e6			−0.501545			−0.250773	−	0.968046i	$-0.580685\pi$
$421$	−1.82396e6			−0.501545			−0.250773	+	0.968046i	$0.580685\pi$
$422$		0			0
$423$	−661992.	−	1.14660e6i	−0.179888	−	0.311575i
$424$		0			0
$425$	−1.01579e6	+	1.75939e6i	−0.272791	+	0.472488i
$426$		0			0
$427$	37302.3	+	117943.i	0.00990069	+	0.0313042i
$428$		0			0
$429$	−1.34591e6	+	2.33118e6i	−0.353079	+	0.611551i
$430$		0			0
$431$	−1.69124e6	−	2.92932e6i	−0.438544	−	0.759580i	0.559034	−	0.829145i	$-0.311172\pi$
$431$	−1.69124e6	−	2.92932e6i	−0.438544	−	0.759580i	−0.997577	+	0.0695651i	$0.977839\pi$
$432$		0			0
$433$	−285266.			−0.0731190			−0.0365595	−	0.999331i	$-0.511640\pi$
$433$	−285266.			−0.0731190			−0.0365595	+	0.999331i	$0.511640\pi$
$434$		0			0
$435$	486402.			0.123246
$436$		0			0
$437$	−2.20886e6	−	3.82585e6i	−0.553304	−	0.958351i
$438$		0			0
$439$	2.17610e6	−	3.76911e6i	0.538911	−	0.933421i	−0.460052	−	0.887892i	$-0.652169\pi$
$439$	2.17610e6	−	3.76911e6i	0.538911	−	0.933421i	0.998963	−	0.0455294i	$-0.0144975\pi$
$440$		0			0
$441$	1.23484e6	+	573151.i	0.302352	+	0.140337i
$442$		0			0
$443$	−2.55280e6	+	4.42158e6i	−0.618027	+	1.07045i	0.371819	+	0.928305i	$0.378734\pi$
$443$	−2.55280e6	+	4.42158e6i	−0.618027	+	1.07045i	−0.989845	+	0.142148i	$0.954599\pi$
$444$		0			0
$445$	489024.	+	847015.i	0.117066	+	0.202764i
$446$		0			0
$447$	−1.50301e6			−0.355790
$448$		0			0
$449$	3.04163e6			0.712016			0.356008	−	0.934483i	$-0.384138\pi$
$449$	3.04163e6			0.712016			0.356008	+	0.934483i	$0.384138\pi$
$450$		0			0
$451$	−68906.7	−	119350.i	−0.0159522	−	0.0276300i
$452$		0			0
$453$	1.69319e6	−	2.93269e6i	0.387668	−	0.671461i
$454$		0			0
$455$	−1.49604e6	−	4.73020e6i	−0.338777	−	1.07115i
$456$		0			0
$457$	872162.	−	1.51063e6i	0.195347	−	0.338351i	−0.751667	−	0.659542i	$-0.770750\pi$
$457$	872162.	−	1.51063e6i	0.195347	−	0.338351i	0.947014	+	0.321192i	$0.104083\pi$
$458$		0			0
$459$	−366578.	−	634931.i	−0.0812147	−	0.140668i
$460$		0			0
$461$	−6.85701e6			−1.50273			−0.751367	−	0.659884i	$-0.770605\pi$
$461$	−6.85701e6			−1.50273			−0.751367	+	0.659884i	$0.770605\pi$
$462$		0			0
$463$	−5.13844e6			−1.11398			−0.556992	−	0.830518i	$-0.688045\pi$
$463$	−5.13844e6			−1.11398			−0.556992	+	0.830518i	$0.688045\pi$
$464$		0			0
$465$	−2.64887e6	−	4.58798e6i	−0.568105	−	0.983987i
$466$		0			0
$467$	−2.29085e6	+	3.96788e6i	−0.486077	+	0.841910i	−0.999872	−	0.0160029i	$-0.994906\pi$
$467$	−2.29085e6	+	3.96788e6i	−0.486077	+	0.841910i	0.513795	+	0.857913i	$0.328239\pi$
$468$		0			0
$469$	−2.50848e6	−	553784.i	−0.526597	−	0.116254i
$470$		0			0
$471$	−175838.	+	304560.i	−0.0365224	+	0.0632587i
$472$		0			0
$473$	−4.90660e6	−	8.49848e6i	−1.00839	−	1.74658i
$474$		0			0
$475$	−2.76450e6			−0.562190
$476$		0			0
$477$	−2.40097e6			−0.483161
$478$		0			0
$479$	−144261.	−	249868.i	−0.0287284	−	0.0497590i	0.851304	−	0.524673i	$-0.175812\pi$
$479$	−144261.	−	249868.i	−0.0287284	−	0.0497590i	−0.880032	+	0.474914i	$0.842479\pi$
$480$		0			0
$481$	749223.	−	1.29769e6i	0.147655	−	0.255746i
$482$		0			0
$483$	−2.54211e6	+	2.77917e6i	−0.495823	+	0.542060i
$484$		0			0
$485$	−463352.	+	802549.i	−0.0894451	+	0.154923i
$486$		0			0
$487$	−3.90842e6	−	6.76959e6i	−0.746757	−	1.29342i	−0.949369	−	0.314162i	$-0.898277\pi$
$487$	−3.90842e6	−	6.76959e6i	−0.746757	−	1.29342i	0.202613	−	0.979259i	$-0.435057\pi$
$488$		0			0
$489$	4.30127e6			0.813438
$490$		0			0
$491$	−3.14467e6			−0.588669			−0.294335	−	0.955702i	$-0.595098\pi$
$491$	−3.14467e6			−0.588669			−0.294335	+	0.955702i	$0.595098\pi$
$492$		0			0
$493$	378875.	+	656232.i	0.0702068	+	0.121602i
$494$		0			0
$495$	1.62859e6	−	2.82080e6i	0.298743	−	0.517439i
$496$		0			0
$497$	−5.43597e6	+	5.94289e6i	−0.987157	+	1.07921i
$498$		0			0
$499$	3.36281e6	−	5.82456e6i	0.604577	−	1.04716i	−0.387541	−	0.921852i	$-0.626676\pi$
$499$	3.36281e6	−	5.82456e6i	0.604577	−	1.04716i	0.992118	−	0.125305i	$-0.0399911\pi$
$500$		0			0
$501$	179070.	+	310159.i	0.0318734	+	0.0552064i
$502$		0			0
$503$	9.45056e6			1.66547			0.832737	−	0.553669i	$-0.186773\pi$
$503$	9.45056e6			1.66547			0.832737	+	0.553669i	$0.186773\pi$
$504$		0			0
$505$	−1.80348e6			−0.314690
$506$		0			0
$507$	389976.	+	675458.i	0.0673780	+	0.116702i
$508$		0			0
$509$	4.41851e6	−	7.65308e6i	0.755930	−	1.30931i	−0.188981	−	0.981981i	$-0.560519\pi$
$509$	4.41851e6	−	7.65308e6i	0.755930	−	1.30931i	0.944911	−	0.327328i	$-0.106148\pi$
$510$		0			0
$511$	−3.43191e6	−	757645.i	−0.581411	−	0.128355i
$512$		0			0
$513$	498828.	−	863995.i	0.0836869	−	0.144950i
$514$		0			0
$515$	5.76534e6	+	9.98585e6i	0.957870	+	1.65908i
$516$		0			0
$517$	9.16344e6			1.50776
$518$		0			0
$519$	435044.			0.0708949
$520$		0			0
$521$	−347492.	−	601874.i	−0.0560855	−	0.0971430i	0.836619	−	0.547784i	$-0.184529\pi$
$521$	−347492.	−	601874.i	−0.0560855	−	0.0971430i	−0.892705	+	0.450641i	$0.851195\pi$
$522$		0			0
$523$	1.79105e6	−	3.10219e6i	0.286321	−	0.495923i	−0.686607	−	0.727028i	$-0.740901\pi$
$523$	1.79105e6	−	3.10219e6i	0.286321	−	0.495923i	0.972929	+	0.231105i	$0.0742342\pi$
$524$		0			0
$525$	710743.	+	2.24724e6i	0.112542	+	0.355837i
$526$		0			0
$527$	4.12660e6	−	7.14748e6i	0.647240	−	1.12105i
$528$		0			0
$529$	−1.99208e6	−	3.45038e6i	−0.309504	−	0.536077i
$530$		0			0
$531$	838845.			0.129106
$532$		0			0
$533$	131151.			0.0199965
$534$		0			0
$535$	3.38473e6	+	5.86253e6i	0.511258	+	0.885525i
$536$		0			0
$537$	−643098.	+	1.11388e6i	−0.0962368	+	0.166687i
$538$		0			0
$539$	−7.70860e6	+	5.41801e6i	−1.14289	+	0.803282i
$540$		0			0
$541$	2.58423e6	−	4.47602e6i	0.379610	−	0.657504i	−0.611395	−	0.791325i	$-0.709391\pi$
$541$	2.58423e6	−	4.47602e6i	0.379610	−	0.657504i	0.991005	+	0.133821i	$0.0427248\pi$
$542$		0			0
$543$	347604.	+	602068.i	0.0505924	+	0.0876285i
$544$		0			0
$545$	5.98174e6			0.862653
$546$		0			0
$547$	−8.47489e6			−1.21106			−0.605530	−	0.795822i	$-0.707039\pi$
$547$	−8.47489e6			−1.21106			−0.605530	+	0.795822i	$0.707039\pi$
$548$		0			0
$549$	−38644.2	−	66933.8i	−0.00547210	−	0.00947795i
$550$		0			0
$551$	−515562.	+	892980.i	−0.0723439	+	0.125303i
$552$		0			0
$553$	−1.74699e6	−	5.52367e6i	−0.242928	−	0.768095i
$554$		0			0
$555$	−906583.	+	1.57025e6i	−0.124932	+	0.216389i
$556$		0			0
$557$	5.21063e6	+	9.02508e6i	0.711627	+	1.23257i	0.964246	+	0.265009i	$0.0853748\pi$
$557$	5.21063e6	+	9.02508e6i	0.711627	+	1.23257i	−0.252619	+	0.967566i	$0.581292\pi$
$558$		0			0
$559$	9.33880e6			1.26404
$560$		0			0
$561$	5.07425e6			0.680714
$562$		0			0
$563$	−3.62039e6	−	6.27069e6i	−0.481375	−	0.833767i	0.518396	−	0.855141i	$-0.326529\pi$
$563$	−3.62039e6	−	6.27069e6i	−0.481375	−	0.833767i	−0.999772	+	0.0213739i	$0.993196\pi$
$564$		0			0
$565$	−6.42885e6	+	1.11351e7i	−0.847251	+	1.46748i
$566$		0			0
$567$	−830581.	−	183363.i	−0.108499	−	0.0239527i
$568$		0			0
$569$	−458268.	+	793743.i	−0.0593388	+	0.102778i	−0.894169	−	0.447730i	$-0.852233\pi$
$569$	−458268.	+	793743.i	−0.0593388	+	0.102778i	0.834830	+	0.550508i	$0.185566\pi$
$570$		0			0
$571$	5.03538e6	+	8.72154e6i	0.646312	+	1.11944i	0.983997	+	0.178186i	$0.0570228\pi$
$571$	5.03538e6	+	8.72154e6i	0.646312	+	1.11944i	−0.337685	+	0.941259i	$0.609644\pi$
$572$		0			0
$573$	−2.44849e6			−0.311538
$574$		0			0
$575$	−6.52091e6			−0.822505
$576$		0			0
$577$	−5.84987e6	−	1.01323e7i	−0.731488	−	1.26697i	−0.956247	−	0.292560i	$-0.905493\pi$
$577$	−5.84987e6	−	1.01323e7i	−0.731488	−	1.26697i	0.224760	−	0.974414i	$-0.427840\pi$
$578$		0			0
$579$	72152.0	−	124971.i	0.00894442	−	0.0154922i
$580$		0			0
$581$	−1.36558e6	+	1.49292e6i	−0.167833	+	0.183484i
$582$		0			0
$583$	8.30871e6	−	1.43911e7i	1.01242	−	1.75357i
$584$		0			0
$585$	1.54986e6	+	2.68443e6i	0.187242	+	0.324312i
$586$		0			0
$587$	6.92367e6			0.829357			0.414678	−	0.909968i	$-0.363894\pi$
$587$	6.92367e6			0.829357			0.414678	+	0.909968i	$0.363894\pi$
$588$		0			0
$589$	1.12307e7			1.33389
$590$		0			0
$591$	−4.64527e6	−	8.04584e6i	−0.547069	−	0.947552i
$592$		0			0
$593$	−7.88850e6	+	1.36633e7i	−0.921208	+	1.59558i	−0.123660	+	0.992325i	$0.539463\pi$
$593$	−7.88850e6	+	1.36633e7i	−0.921208	+	1.59558i	−0.797548	+	0.603255i	$0.793870\pi$
$594$		0			0
$595$	−6.31207e6	+	6.90068e6i	−0.730936	+	0.799097i
$596$		0			0
$597$	−3.96781e6	+	6.87245e6i	−0.455633	+	0.789180i
$598$		0			0
$599$	7.13541e6	+	1.23589e7i	0.812553	+	1.40738i	0.911072	+	0.412248i	$0.135256\pi$
$599$	7.13541e6	+	1.23589e7i	0.812553	+	1.40738i	−0.0985183	+	0.995135i	$0.531410\pi$
$600$		0			0
$601$	7.63222e6			0.861916			0.430958	−	0.902372i	$-0.358176\pi$
$601$	7.63222e6			0.861916			0.430958	+	0.902372i	$0.358176\pi$
$602$		0			0
$603$	1.60503e6			0.179759
$604$		0			0
$605$	5.49562e6	+	9.51869e6i	0.610419	+	1.05728i
$606$		0			0
$607$	−1.78017e6	+	3.08335e6i	−0.196106	+	0.339665i	−0.947262	−	0.320459i	$-0.896163\pi$
$607$	−1.78017e6	+	3.08335e6i	−0.196106	+	0.339665i	0.751157	+	0.660124i	$0.229496\pi$
$608$		0			0
$609$	858444.	+	189515.i	0.0937927	+	0.0207061i
$610$		0			0
$611$	−4.36023e6	+	7.55214e6i	−0.472505	+	0.818402i
$612$		0			0
$613$	6.86420e6	+	1.18891e7i	0.737800	+	1.27791i	0.953484	+	0.301445i	$0.0974688\pi$
$613$	6.86420e6	+	1.18891e7i	0.737800	+	1.27791i	−0.215683	+	0.976463i	$0.569198\pi$
$614$		0			0
$615$	−158697.			−0.0169192
$616$		0			0
$617$	−6.51173e6			−0.688626			−0.344313	−	0.938855i	$-0.611888\pi$
$617$	−6.51173e6			−0.688626			−0.344313	+	0.938855i	$0.611888\pi$
$618$		0			0
$619$	4.42555e6	+	7.66527e6i	0.464238	+	0.804083i	0.999167	−	0.0408136i	$-0.0129950\pi$
$619$	4.42555e6	+	7.66527e6i	0.464238	+	0.804083i	−0.534929	+	0.844897i	$0.679662\pi$
$620$		0			0
$621$	1.17663e6	−	2.03799e6i	0.122437	−	0.212067i
$622$		0			0
$623$	533054.	+	1.68542e6i	0.0550238	+	0.173976i
$624$		0			0
$625$	5.99884e6	−	1.03903e7i	0.614281	−	1.06397i
$626$		0			0
$627$	3.45244e6	+	5.97981e6i	0.350718	+	0.607461i
$628$		0			0
$629$	−2.82467e6			−0.284670
$630$		0			0
$631$	6.89663e6			0.689546			0.344773	−	0.938686i	$-0.387956\pi$
$631$	6.89663e6			0.689546			0.344773	+	0.938686i	$0.387956\pi$
$632$		0			0
$633$	−1.67766e6	−	2.90579e6i	−0.166416	−	0.288240i
$634$		0			0
$635$	5.15280e6	−	8.92492e6i	0.507118	−	0.878355i
$636$		0			0
$637$	−797329.	−	8.93116e6i	−0.0778554	−	0.872086i
$638$		0			0
$639$	2.51608e6	−	4.35797e6i	0.243765	−	0.422214i
$640$		0			0
$641$	8.33473e6	+	1.44362e7i	0.801210	+	1.38774i	0.918820	+	0.394677i	$0.129144\pi$
$641$	8.33473e6	+	1.44362e7i	0.801210	+	1.38774i	−0.117610	+	0.993060i	$0.537523\pi$
$642$		0			0
$643$	1.28697e7			1.22756			0.613779	−	0.789478i	$-0.289649\pi$
$643$	1.28697e7			1.22756			0.613779	+	0.789478i	$0.289649\pi$
$644$		0			0
$645$	−1.13002e7			−1.06952
$646$		0			0
$647$	−5.73656e6	−	9.93601e6i	−0.538754	−	0.933150i	−0.998971	−	0.0453435i	$-0.985562\pi$
$647$	−5.73656e6	−	9.93601e6i	−0.538754	−	0.933150i	0.460217	−	0.887806i	$-0.347772\pi$
$648$		0			0
$649$	−2.90287e6	+	5.02792e6i	−0.270530	+	0.468572i
$650$		0			0
$651$	−2.88737e6	−	9.12934e6i	−0.267023	−	0.844280i
$652$		0			0
$653$	6.59005e6	−	1.14143e7i	0.604791	−	1.04753i	−0.387293	−	0.921957i	$-0.626590\pi$
$653$	6.59005e6	−	1.14143e7i	0.604791	−	1.04753i	0.992084	−	0.125573i	$-0.0400768\pi$
$654$		0			0
$655$	−1.87535e6	−	3.24820e6i	−0.170797	−	0.295829i
$656$		0			0
$657$	2.19588e6			0.198470
$658$		0			0
$659$	1.13068e7			1.01421			0.507104	−	0.861885i	$-0.330716\pi$
$659$	1.13068e7			1.01421			0.507104	+	0.861885i	$0.330716\pi$
$660$		0			0
$661$	−1.10160e6	−	1.90802e6i	−0.0980661	−	0.169856i	0.812818	−	0.582518i	$-0.197932\pi$
$661$	−1.10160e6	−	1.90802e6i	−0.0980661	−	0.169856i	−0.910884	+	0.412662i	$0.864599\pi$
$662$		0			0
$663$	−2.41447e6	+	4.18199e6i	−0.213324	+	0.369487i
$664$		0			0
$665$	−1.24268e7	−	2.74341e6i	−1.08970	−	0.240567i
$666$		0			0
$667$	−1.21611e6	+	2.10636e6i	−0.105842	+	0.183323i
$668$		0			0
$669$	−4.86921e6	−	8.43372e6i	−0.420623	−	0.728541i
$670$		0			0
$671$	534922.			0.0458653
$672$		0			0
$673$	−1.89787e7			−1.61521			−0.807606	−	0.589723i	$-0.799237\pi$
$673$	−1.89787e7			−1.61521			−0.807606	+	0.589723i	$0.799237\pi$
$674$		0			0
$675$	−736312.	−	1.27533e6i	−0.0622017	−	0.107736i
$676$		0			0
$677$	9.82377e6	−	1.70153e7i	0.823771	−	1.42681i	−0.0790842	−	0.996868i	$-0.525200\pi$
$677$	9.82377e6	−	1.70153e7i	0.823771	−	1.42681i	0.902855	−	0.429945i	$-0.141467\pi$
$678$		0			0
$679$	−1.13046e6	+	1.23587e6i	−0.0940977	+	0.102873i
$680$		0			0
$681$	2.48872e6	−	4.31059e6i	0.205640	−	0.356179i
$682$		0			0
$683$	8.13523e6	+	1.40906e7i	0.667295	+	1.15579i	0.978658	+	0.205498i	$0.0658813\pi$
$683$	8.13523e6	+	1.40906e7i	0.667295	+	1.15579i	−0.311363	+	0.950291i	$0.600785\pi$
$684$		0			0
$685$	−674978.			−0.0549621
$686$		0			0
$687$	4.70722e6			0.380515
$688$		0			0
$689$	7.90704e6	+	1.36954e7i	0.634550	+	1.09907i
$690$		0			0
$691$	9.72335e6	−	1.68413e7i	0.774677	−	1.34178i	−0.160299	−	0.987069i	$-0.551246\pi$
$691$	9.72335e6	−	1.68413e7i	0.774677	−	1.34178i	0.934976	−	0.354712i	$-0.115421\pi$
$692$		0			0
$693$	3.97332e6	−	4.34384e6i	0.314283	−	0.343591i
$694$		0			0
$695$	−6.56659e6	+	1.13737e7i	−0.515677	+	0.893179i
$696$		0			0
$697$	−123614.	−	214106.i	−0.00963800	−	0.0166935i
$698$		0			0
$699$	−3.27763e6			−0.253727
$700$		0			0
$701$	1.49625e7			1.15003			0.575014	−	0.818144i	$-0.304997\pi$
$701$	1.49625e7			1.15003			0.575014	+	0.818144i	$0.304997\pi$
$702$		0			0
$703$	−1.92187e6	−	3.32877e6i	−0.146668	−	0.254036i
$704$		0			0
$705$	5.27601e6	−	9.13831e6i	0.399791	−	0.692458i
$706$		0			0
$707$	−3.18293e6	−	702680.i	−0.239485	−	0.0528700i
$708$		0			0
$709$	−793193.	+	1.37385e6i	−0.0592603	+	0.102642i	−0.894134	−	0.447800i	$-0.852208\pi$
$709$	−793193.	+	1.37385e6i	−0.0592603	+	0.102642i	0.834873	+	0.550442i	$0.185541\pi$
$710$		0			0
$711$	1.80984e6	+	3.13473e6i	0.134266	+	0.232555i
$712$		0			0
$713$	2.64910e7			1.95152
$714$		0			0
$715$	−2.14535e7			−1.56940
$716$		0			0
$717$	1.67329e6	+	2.89822e6i	0.121555	+	0.210539i
$718$		0			0
$719$	−9.08376e6	+	1.57335e7i	−0.655305	+	1.13502i	0.326512	+	0.945193i	$0.394127\pi$
$719$	−9.08376e6	+	1.57335e7i	−0.655305	+	1.13502i	−0.981817	+	0.189829i	$0.939207\pi$
$720$		0			0
$721$	6.28442e6	+	1.98702e7i	0.450222	+	1.42352i
$722$		0			0
$723$	7.70164e6	−	1.33396e7i	0.547945	−	0.949069i
$724$		0			0
$725$	761013.	+	1.31811e6i	0.0537709	+	0.0931339i
$726$		0			0
$727$	−1.26903e7			−0.890506			−0.445253	−	0.895405i	$-0.646886\pi$
$727$	−1.26903e7			−0.890506			−0.445253	+	0.895405i	$0.646886\pi$
$728$		0			0
$729$	531441.			0.0370370
$730$		0			0
$731$	−8.80214e6	−	1.52458e7i	−0.609249	−	1.05525i
$732$		0			0
$733$	1.55701e6	−	2.69681e6i	0.107036	−	0.185392i	−0.807532	−	0.589824i	$-0.799197\pi$
$733$	1.55701e6	−	2.69681e6i	0.107036	−	0.185392i	0.914568	+	0.404432i	$0.132531\pi$
$734$		0			0
$735$	964792.	+	1.08070e7i	0.0658742	+	0.737880i
$736$		0			0
$737$	−5.55430e6	+	9.62033e6i	−0.376670	+	0.652411i
$738$		0			0
$739$	4.54950e6	+	7.87996e6i	0.306445	+	0.530778i	0.977582	−	0.210555i	$-0.0675272\pi$
$739$	4.54950e6	+	7.87996e6i	0.306445	+	0.530778i	−0.671137	+	0.741333i	$0.734194\pi$
$740$		0			0
$741$	−6.57109e6			−0.439634
$742$		0			0
$743$	8.79218e6			0.584285			0.292142	−	0.956375i	$-0.405632\pi$
$743$	8.79218e6			0.584285			0.292142	+	0.956375i	$0.405632\pi$
$744$		0			0
$745$	−5.98943e6	−	1.03740e7i	−0.395362	−	0.684787i
$746$		0			0
$747$	632069.	−	1.09478e6i	0.0414441	−	0.0717833i
$748$		0			0
$749$	3.68948e6	+	1.16655e7i	0.240304	+	0.759798i
$750$		0			0
$751$	−1.40459e7	+	2.43282e7i	−0.908759	+	1.57402i	−0.0929698	+	0.995669i	$0.529636\pi$
$751$	−1.40459e7	+	2.43282e7i	−0.908759	+	1.57402i	−0.815790	+	0.578349i	$0.803697\pi$
$752$		0			0
$753$	261603.	+	453110.i	0.0168134	+	0.0291217i
$754$		0			0
$755$	2.69891e7			1.72314
$756$		0			0
$757$	−4.18815e6			−0.265634			−0.132817	−	0.991141i	$-0.542402\pi$
$757$	−4.18815e6			−0.265634			−0.132817	+	0.991141i	$0.542402\pi$
$758$		0			0
$759$	8.14362e6	+	1.41052e7i	0.513113	+	0.888738i
$760$		0			0
$761$	196591.	−	340506.i	0.0123056	−	0.0213139i	−0.859807	−	0.510619i	$-0.829416\pi$
$761$	196591.	−	340506.i	0.0123056	−	0.0213139i	0.872113	+	0.489305i	$0.162750\pi$
$762$		0			0
$763$	1.05571e7	+	2.33063e6i	0.656497	+	0.144932i
$764$		0			0
$765$	2.92159e6	−	5.06034e6i	0.180495	−	0.312627i
$766$		0			0
$767$	−2.76254e6	−	4.78486e6i	−0.169559	−	0.293684i
$768$		0			0
$769$	1.57517e7			0.960530			0.480265	−	0.877124i	$-0.340541\pi$
$769$	1.57517e7			0.960530			0.480265	+	0.877124i	$0.340541\pi$
$770$		0			0
$771$	−2.80655e6			−0.170035
$772$		0			0
$773$	−1.44236e6	−	2.49824e6i	−0.0868210	−	0.150378i	0.819345	−	0.573301i	$-0.194337\pi$
$773$	−1.44236e6	−	2.49824e6i	−0.0868210	−	0.150378i	−0.906166	+	0.422923i	$0.861004\pi$
$774$		0			0
$775$	8.28873e6	−	1.43565e7i	0.495717	−	0.858606i
$776$		0			0
$777$	−2.21182e6	+	2.41808e6i	−0.131431	+	0.143687i
$778$		0			0
$779$	168211.	−	291349.i	0.00993139	−	0.0172017i
$780$		0			0
$781$	1.74141e7	+	3.01620e7i	1.02158	+	1.76943i
$782$		0			0
$783$	−549269.			−0.0320170
$784$		0			0
$785$	−2.80281e6			−0.162338
$786$		0			0
$787$	−1.48933e7	−	2.57960e7i	−0.857145	−	1.48462i	−0.874641	−	0.484771i	$-0.838903\pi$
$787$	−1.48933e7	−	2.57960e7i	−0.857145	−	1.48462i	0.0174963	−	0.999847i	$-0.494430\pi$
$788$		0			0
$789$	3.88784e6	−	6.73394e6i	0.222339	−	0.385103i
$790$		0			0
$791$	−1.56847e7	+	1.71473e7i	−0.891322	+	0.974440i
$792$		0			0
$793$	−254531.	+	440861.i	−0.0143734	+	0.0248954i
$794$		0			0
$795$	−9.56776e6	−	1.65718e7i	−0.536899	−	0.929936i
$796$		0			0
$797$	8.11321e6			0.452425			0.226213	−	0.974078i	$-0.427366\pi$
$797$	8.11321e6			0.452425			0.226213	+	0.974078i	$0.427366\pi$
$798$		0			0
$799$	1.64387e7			0.910960
$800$		0			0
$801$	−552230.	−	956491.i	−0.0304116	−	0.0526744i
$802$		0			0
$803$	−7.59897e6	+	1.31618e7i	−0.415878	+	0.720321i
$804$		0			0
$805$	−2.93124e7	−	6.47115e6i	−1.59427	−	0.351959i
$806$		0			0
$807$	5.04311e6	−	8.73493e6i	0.272593	−	0.472145i
$808$		0			0
$809$	−55798.5	−	96645.8i	−0.00299744	−	0.00519173i	0.864523	−	0.502594i	$-0.167621\pi$
$809$	−55798.5	−	96645.8i	−0.00299744	−	0.00519173i	−0.867520	+	0.497402i	$0.834287\pi$
$810$		0			0
$811$	209250.			0.0111716			0.00558578	−	0.999984i	$-0.498222\pi$
$811$	209250.			0.0111716			0.00558578	+	0.999984i	$0.498222\pi$
$812$		0			0
$813$	−1.02610e7			−0.544459
$814$		0			0
$815$	1.71403e7	+	2.96879e7i	0.903910	+	1.56562i
$816$		0			0
$817$	1.19777e7	−	2.07460e7i	0.627795	−	1.08737i
$818$		0			0
$819$	1.68940e6	+	5.34158e6i	0.0880081	+	0.278266i
$820$		0			0
$821$	−3.42307e6	+	5.92893e6i	−0.177238	+	0.306986i	−0.940934	−	0.338591i	$-0.890050\pi$
$821$	−3.42307e6	+	5.92893e6i	−0.177238	+	0.306986i	0.763695	+	0.645577i	$0.223383\pi$
$822$		0			0
$823$	−2.80605e6	−	4.86022e6i	−0.144409	−	0.250125i	0.784743	−	0.619821i	$-0.212795\pi$
$823$	−2.80605e6	−	4.86022e6i	−0.144409	−	0.250125i	−0.929152	+	0.369697i	$0.879462\pi$
$824$		0			0
$825$	1.01922e7			0.521354
$826$		0			0
$827$	2.15868e7			1.09755			0.548776	−	0.835969i	$-0.315094\pi$
$827$	2.15868e7			1.09755			0.548776	+	0.835969i	$0.315094\pi$
$828$		0			0
$829$	−9.62135e6	−	1.66647e7i	−0.486239	−	0.842190i	0.513636	−	0.858008i	$-0.328298\pi$
$829$	−9.62135e6	−	1.66647e7i	−0.486239	−	0.842190i	−0.999875	+	0.0158177i	$0.994965\pi$
$830$		0			0
$831$	8.94604e6	−	1.54950e7i	0.449395	−	0.778375i
$832$		0			0
$833$	−1.38288e7	+	9.71958e6i	−0.690511	+	0.485327i
$834$		0			0
$835$	−1.42717e6	+	2.47193e6i	−0.0708369	+	0.122693i
$836$		0			0
$837$	2.99124e6	+	5.18098e6i	0.147583	+	0.255622i
$838$		0			0
$839$	−2.47678e7			−1.21474			−0.607369	−	0.794420i	$-0.707775\pi$
$839$	−2.47678e7			−1.21474			−0.607369	+	0.794420i	$0.707775\pi$
$840$		0			0
$841$	−1.99435e7			−0.972323
$842$		0			0
$843$	−2.39544e6	−	4.14903e6i	−0.116096	−	0.201084i
$844$		0			0
$845$	−3.10807e6	+	5.38333e6i	−0.149744	+	0.259364i
$846$		0			0
$847$	5.99042e6	+	1.89406e7i	0.286912	+	0.907165i
$848$		0			0
$849$	1.18113e7	−	2.04578e7i	0.562378	−	0.974067i
$850$		0			0
$851$	−4.53329e6	−	7.85190e6i	−0.214580	−	0.371664i
$852$		0			0
$853$	−2.91267e7			−1.37062			−0.685312	−	0.728250i	$-0.740334\pi$
$853$	−2.91267e7			−1.37062			−0.685312	+	0.728250i	$0.740334\pi$
$854$		0			0
$855$	7.95121e6			0.371979
$856$		0			0
$857$	1.61293e6	+	2.79367e6i	0.0750175	+	0.129934i	0.901094	−	0.433624i	$-0.142765\pi$
$857$	1.61293e6	+	2.79367e6i	0.0750175	+	0.129934i	−0.826076	+	0.563558i	$0.809432\pi$
$858$		0			0
$859$	1.32335e7	−	2.29212e7i	0.611918	−	1.05987i	−0.378999	−	0.925397i	$-0.623732\pi$
$859$	1.32335e7	−	2.29212e7i	0.611918	−	1.05987i	0.990917	−	0.134476i	$-0.0429350\pi$
$860$		0			0
$861$	−280082.	−	61832.2i	−0.0128759	−	0.00284254i
$862$		0			0
$863$	7.92552e6	−	1.37274e7i	0.362243	−	0.627424i	−0.626086	−	0.779754i	$-0.715344\pi$
$863$	7.92552e6	−	1.37274e7i	0.362243	−	0.627424i	0.988330	+	0.152330i	$0.0486776\pi$
$864$		0			0
$865$	1.73363e6	+	3.00273e6i	0.0787799	+	0.136451i
$866$		0			0
$867$	−3.67582e6			−0.166076
$868$		0			0
$869$	−2.50522e7			−1.12537
$870$		0			0
$871$	−5.28579e6	−	9.15526e6i	−0.236083	−	0.408908i
$872$		0			0
$873$	523240.	−	906277.i	0.0232362	−	0.0402463i
$874$		0			0
$875$	6.93493e6	−	7.58163e6i	0.306212	−	0.334767i
$876$		0			0
$877$	7.45659e6	−	1.29152e7i	0.327372	−	0.567024i	−0.654618	−	0.755960i	$-0.727170\pi$
$877$	7.45659e6	−	1.29152e7i	0.327372	−	0.567024i	0.981989	+	0.188936i	$0.0605037\pi$
$878$		0			0
$879$	−2.74483e6	−	4.75419e6i	−0.119824	−	0.207541i
$880$		0			0
$881$	−3.70679e7			−1.60901			−0.804504	−	0.593947i	$-0.797569\pi$
$881$	−3.70679e7			−1.60901			−0.804504	+	0.593947i	$0.797569\pi$
$882$		0			0
$883$	−1.50440e7			−0.649325			−0.324663	−	0.945830i	$-0.605251\pi$
$883$	−1.50440e7			−0.649325			−0.324663	+	0.945830i	$0.605251\pi$
$884$		0			0
$885$	3.34275e6	+	5.78982e6i	0.143465	+	0.248489i
$886$		0			0
$887$	8.90504e6	−	1.54240e7i	0.380038	−	0.658245i	−0.611030	−	0.791608i	$-0.709244\pi$
$887$	8.90504e6	−	1.54240e7i	0.380038	−	0.658245i	0.991067	+	0.133363i	$0.0425777\pi$
$888$		0			0
$889$	1.25715e7	−	1.37438e7i	0.533497	−	0.583247i
$890$		0			0
$891$	−1.83908e6	+	3.18538e6i	−0.0776080	+	0.134421i
$892$		0			0
$893$	1.11846e7	+	1.93723e7i	0.469345	+	0.812929i
$894$		0			0
$895$	−1.02508e7			−0.427762
$896$		0			0
$897$	−1.54999e7			−0.643202
$898$		0			0
$899$	−3.09159e6	−	5.35479e6i	−0.127580	−	0.220975i
$900$		0			0
$901$	1.49053e7	−	2.58168e7i	0.611687	−	1.05947i
$902$		0			0
$903$	−1.99436e7	−	4.40285e6i	−0.813925	−	0.179686i
$904$		0			0
$905$	−2.77037e6	+	4.79842e6i	−0.112439	+	0.194750i
$906$		0			0
$907$	−584680.	−	1.01270e6i	−0.0235993	−	0.0408753i	0.853985	−	0.520298i	$-0.174179\pi$
$907$	−584680.	−	1.01270e6i	−0.0235993	−	0.0408753i	−0.877584	+	0.479423i	$0.840846\pi$
$908$		0			0
$909$	2.03658e6			0.0817506
$910$		0			0
$911$	−2.74389e7			−1.09539			−0.547697	−	0.836677i	$-0.684495\pi$
$911$	−2.74389e7			−1.09539			−0.547697	+	0.836677i	$0.684495\pi$
$912$		0			0
$913$	4.37462e6	+	7.57707e6i	0.173685	+	0.300832i
$914$		0			0
$915$	307991.	−	533455.i	0.0121614	−	0.0210642i
$916$		0			0
$917$	−2.04420e6	−	6.46339e6i	−0.0802786	−	0.253827i
$918$		0			0
$919$	5.51091e6	−	9.54518e6i	0.215246	−	0.372817i	−0.738103	−	0.674688i	$-0.764278\pi$
$919$	5.51091e6	−	9.54518e6i	0.215246	−	0.372817i	0.953349	+	0.301872i	$0.0976114\pi$
$920$		0			0
$921$	−6.06794e6	−	1.05100e7i	−0.235718	−	0.408275i
$922$		0			0
$923$	−3.31444e7			−1.28058
$924$		0			0
$925$	−5.67367e6			−0.218027
$926$		0			0
$927$	−6.51050e6	−	1.12765e7i	−0.248837	−	0.430998i
$928$		0			0
$929$	−2.12526e6	+	3.68105e6i	−0.0807927	+	0.139937i	−0.903591	−	0.428397i	$-0.859079\pi$
$929$	−2.12526e6	+	3.68105e6i	−0.0807927	+	0.139937i	0.822798	+	0.568334i	$0.192412\pi$
$930$		0			0
$931$	−2.08630e7	−	9.68361e6i	−0.788866	−	0.366153i
$932$		0			0
$933$	−1.49877e7	+	2.59595e7i	−0.563679	+	0.976320i
$934$		0			0
$935$	2.02206e7	+	3.50232e7i	0.756425	+	1.31017i
$936$		0			0
$937$	3.75738e7			1.39809			0.699046	−	0.715077i	$-0.253608\pi$
$937$	3.75738e7			1.39809			0.699046	+	0.715077i	$0.253608\pi$
$938$		0			0
$939$	−2.55303e7			−0.944915
$940$		0			0
$941$	−3.43521e6	−	5.94997e6i	−0.126468	−	0.219049i	0.795838	−	0.605510i	$-0.207031\pi$
$941$	−3.43521e6	−	5.94997e6i	−0.126468	−	0.219049i	−0.922306	+	0.386461i	$0.873697\pi$
$942$		0			0
$943$	396775.	−	687235.i	0.0145300	−	0.0251667i
$944$		0			0
$945$	−2.04422e6	−	6.46347e6i	−0.0744644	−	0.235443i
$946$		0			0
$947$	−1.15462e7	+	1.99985e7i	−0.418372	+	0.724641i	−0.995776	−	0.0918172i	$-0.970732\pi$
$947$	−1.15462e7	+	1.99985e7i	−0.418372	+	0.724641i	0.577404	+	0.816459i	$0.304066\pi$
$948$		0			0
$949$	−7.23161e6	−	1.25255e7i	−0.260657	−	0.451471i
$950$		0			0
$951$	−9.79551e6			−0.351217
$952$		0			0
$953$	−2.95399e7			−1.05360			−0.526802	−	0.849988i	$-0.676609\pi$
$953$	−2.95399e7			−1.05360			−0.526802	+	0.849988i	$0.676609\pi$
$954$		0			0
$955$	−9.75709e6	−	1.68998e7i	−0.346188	−	0.599615i
$956$		0			0
$957$	1.90078e6	−	3.29224e6i	0.0670890	−	0.116202i
$958$		0			0
$959$	−1.19126e6	−	262988.i	−0.0418273	−	0.00923400i
$960$		0			0
$961$	−1.93581e7	+	3.35292e7i	−0.676166	+	1.17115i
$962$		0			0
$963$	−3.82221e6	−	6.62026e6i	−0.132815	−	0.230043i
$964$		0			0
$965$	1.15009e6			0.0397569
$966$		0			0
$967$	−1.49151e7			−0.512931			−0.256466	−	0.966553i	$-0.582558\pi$
$967$	−1.49151e7			−0.512931			−0.256466	+	0.966553i	$0.582558\pi$
$968$		0			0
$969$	6.19347e6	+	1.07274e7i	0.211897	+	0.367016i
$970$		0			0
$971$	7.74150e6	−	1.34087e7i	0.263498	−	0.456391i	−0.703671	−	0.710526i	$-0.748457\pi$
$971$	7.74150e6	−	1.34087e7i	0.263498	−	0.456391i	0.967169	+	0.254134i	$0.0817905\pi$
$972$		0			0
$973$	−1.60207e7	+	1.75147e7i	−0.542501	+	0.593091i
$974$		0			0
$975$	−4.84974e6	+	8.39999e6i	−0.163383	+	0.282988i
$976$		0			0
$977$	1.41714e7	+	2.45455e7i	0.474980	+	0.822690i	0.999589	−	0.0286531i	$-0.00912183\pi$
$977$	1.41714e7	+	2.45455e7i	0.474980	+	0.822690i	−0.524609	+	0.851343i	$0.675788\pi$
$978$		0			0
$979$	7.64410e6			0.254900
$980$		0			0
$981$	−6.75487e6			−0.224101
$982$		0			0
$983$	1.28497e7	+	2.22563e7i	0.424139	+	0.734630i	0.996340	−	0.0854834i	$-0.0272434\pi$
$983$	1.28497e7	+	2.22563e7i	0.424139	+	0.734630i	−0.572201	+	0.820114i	$0.693910\pi$
$984$		0			0
$985$	3.70223e7	−	6.41245e7i	1.21583	−	2.10588i
$986$		0			0
$987$	1.28721e7	−	1.40724e7i	0.420587	−	0.459807i
$988$		0			0
$989$	2.82530e7	−	4.89355e7i	0.918487	−	1.59087i
$990$		0			0
$991$	2.23202e7	+	3.86598e7i	0.721962	+	1.25048i	0.960212	+	0.279271i	$0.0900929\pi$
$991$	2.23202e7	+	3.86598e7i	0.721962	+	1.25048i	−0.238250	+	0.971204i	$0.576574\pi$
$992$		0			0
$993$	−1.17499e7			−0.378148
$994$		0			0
$995$	−6.32461e7			−2.02524
$996$		0			0
$997$	−1.15277e7	−	1.99665e7i	−0.367286	−	0.636158i	0.621854	−	0.783133i	$-0.286380\pi$
$997$	−1.15277e7	−	1.99665e7i	−0.367286	−	0.636158i	−0.989140	+	0.146975i	$0.953046\pi$
$998$		0			0
$999$	1.02376e6	−	1.77320e6i	0.0324551	−	0.0562140i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.6.q.e.289.2		4
4.3	odd	2		21.6.e.b.16.2	yes	4
7.4	even	3	inner	336.6.q.e.193.2		4
12.11	even	2		63.6.e.c.37.1		4
28.3	even	6		147.6.e.l.67.2		4
28.11	odd	6		21.6.e.b.4.2	✓	4
28.19	even	6		147.6.a.k.1.1		2
28.23	odd	6		147.6.a.i.1.1		2
28.27	even	2		147.6.e.l.79.2		4
84.11	even	6		63.6.e.c.46.1		4
84.23	even	6		441.6.a.t.1.2		2
84.47	odd	6		441.6.a.s.1.2		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
21.6.e.b.4.2	✓	4	28.11	odd	6
21.6.e.b.16.2	yes	4	4.3	odd	2
63.6.e.c.37.1		4	12.11	even	2
63.6.e.c.46.1		4	84.11	even	6
147.6.a.i.1.1		2	28.23	odd	6
147.6.a.k.1.1		2	28.19	even	6
147.6.e.l.67.2		4	28.3	even	6
147.6.e.l.79.2		4	28.27	even	2
336.6.q.e.193.2		4	7.4	even	3	inner
336.6.q.e.289.2		4	1.1	even	1	trivial
441.6.a.s.1.2		2	84.47	odd	6
441.6.a.t.1.2		2	84.23	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)

\(f(q)\)	\(=\)	\(q+(-4.50000 - 7.79423i) q^{3} +(35.8645 - 62.1192i) q^{5} +(87.5000 - 95.6596i) q^{7} +(-40.5000 + 70.1481i) q^{9} +(-280.305 - 485.503i) q^{11} +533.509 q^{13} -645.562 q^{15} +(-502.850 - 870.962i) q^{17} +(684.263 - 1185.18i) q^{19} +(-1139.34 - 251.527i) q^{21} +(1614.04 - 2795.60i) q^{23} +(-1010.03 - 1749.42i) q^{25} +729.000 q^{27} -753.456 q^{29} +(4103.21 + 7106.97i) q^{31} +(-2522.75 + 4369.52i) q^{33} +(-2804.15 - 8866.21i) q^{35} +(1404.33 - 2432.37i) q^{37} +(-2400.79 - 4158.29i) q^{39} +245.827 q^{41} +17504.5 q^{43} +(2905.03 + 5031.65i) q^{45} +(-8172.74 + 14155.6i) q^{47} +(-1494.50 - 16740.4i) q^{49} +(-4525.65 + 7838.66i) q^{51} +(14820.8 + 25670.4i) q^{53} -40212.0 q^{55} -12316.7 q^{57} +(-5178.05 - 8968.65i) q^{59} +(-477.089 + 826.343i) q^{61} +(3166.58 + 10012.2i) q^{63} +(19134.0 - 33141.1i) q^{65} +(-9907.60 - 17160.5i) q^{67} -29052.7 q^{69} -62125.4 q^{71} +(-13554.8 - 23477.6i) q^{73} +(-9090.27 + 15744.8i) q^{75} +(-70969.7 - 15667.6i) q^{77} +(22343.7 - 38700.4i) q^{79} +(-3280.50 - 5681.99i) q^{81} -15606.6 q^{83} -72137.9 q^{85} +(3390.55 + 5872.61i) q^{87} +(-6817.66 + 11808.5i) q^{89} +(46682.0 - 51035.2i) q^{91} +(36928.9 - 63962.7i) q^{93} +(-49081.6 - 85011.8i) q^{95} -12919.5 q^{97} +45409.4 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 18 q^{3} + 33 q^{5} + 350 q^{7} - 162 q^{9} - 1137 q^{11} + 1850 q^{13} - 594 q^{15} + 324 q^{17} + 2311 q^{19} - 1575 q^{21} + 1596 q^{23} - 395 q^{25} + 2916 q^{27} - 4434 q^{29} + 4294 q^{31} - 10233 q^{33}+ \cdots + 184194 q^{99}+O(q^{100}) \) 4 * q - 18 * q^3 + 33 * q^5 + 350 * q^7 - 162 * q^9 - 1137 * q^11 + 1850 * q^13 - 594 * q^15 + 324 * q^17 + 2311 * q^19 - 1575 * q^21 + 1596 * q^23 - 395 * q^25 + 2916 * q^27 - 4434 * q^29 + 4294 * q^31 - 10233 * q^33 - 15414 * q^35 + 19109 * q^37 - 8325 * q^39 - 25716 * q^41 + 5542 * q^43 + 2673 * q^45 - 23160 * q^47 - 5978 * q^49 + 2916 * q^51 + 31653 * q^53 - 35778 * q^55 - 41598 * q^57 + 41097 * q^59 - 42052 * q^61 - 14175 * q^63 + 23106 * q^65 + 30763 * q^67 - 28728 * q^69 - 204192 * q^71 + 28577 * q^73 - 3555 * q^75 - 96873 * q^77 - 18464 * q^79 - 13122 * q^81 - 122358 * q^83 - 247272 * q^85 + 19953 * q^87 - 29322 * q^89 + 161875 * q^91 + 38646 * q^93 - 61662 * q^95 - 19582 * q^97 + 184194 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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