LMFDB - Embedded newform 336.7.bh.c.241.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [336,7,Mod(145,336)]

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, 5]))

N = Newforms(chi, 7, names="a")

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

chi := DirichletCharacter("336.145");

S:= CuspForms(chi, 7);

N := Newforms(S);

Level:	$ N $	$=$	$ 336 = 2^{4} \cdot 3 \cdot 7 $
Weight:	$ k $	$=$	$ 7 $
Character orbit:	$[\chi]$	$=$	336.bh (of order $6$, 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:	$77.2981720963$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} + 33x^{6} + 2x^{5} + 701x^{4} - 28x^{3} + 6468x^{2} + 5488x + 38416 $ x^8 - 2x^7 + 33x^6 + 2x^5 + 701x^4 - 28x^3 + 6468x^2 + 5488*x + 38416
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6}\cdot 3^{4}\cdot 7^{2} $
Twist minimal:	no (minimal twist has level 42)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			241.4
Root			$-1.97725 + 3.42469i$ of defining polynomial
Character	$\chi$	$=$	336.241
Dual form			336.7.bh.c.145.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-13.5000 - 7.79423i) q^{3} +(182.133 - 105.155i) q^{5} +(-186.289 + 288.003i) q^{7} +(121.500 + 210.444i) q^{9} +O(q^{10})$	\(q+(-13.5000 - 7.79423i) q^{3} +(182.133 - 105.155i) q^{5} +(-186.289 + 288.003i) q^{7} +(121.500 + 210.444i) q^{9} +(-485.969 + 841.723i) q^{11} +3559.78i q^{13} -3278.39 q^{15} +(-2916.87 - 1684.05i) q^{17} +(1621.56 - 936.209i) q^{19} +(4759.66 - 2436.06i) q^{21} +(-9942.27 - 17220.5i) q^{23} +(14302.4 - 24772.6i) q^{25} -3788.00i q^{27} -18697.0 q^{29} +(17628.0 + 10177.5i) q^{31} +(13121.2 - 7575.51i) q^{33} +(-3644.56 + 72043.9i) q^{35} +(-44846.5 - 77676.4i) q^{37} +(27745.8 - 48057.1i) q^{39} +10931.8i q^{41} -3887.60 q^{43} +(44258.3 + 25552.5i) q^{45} +(67906.5 - 39205.8i) q^{47} +(-48242.0 - 107303. i) q^{49} +(26251.8 + 45469.4i) q^{51} +(-53856.2 + 93281.7i) q^{53} +204407. i q^{55} -29188.1 q^{57} +(-212952. - 122948. i) q^{59} +(274561. - 158518. i) q^{61} +(-83242.6 - 4211.08i) q^{63} +(374327. + 648354. i) q^{65} +(242122. - 419367. i) q^{67} +309969. i q^{69} +69288.4 q^{71} +(-141741. - 81834.1i) q^{73} +(-386166. + 222953. i) q^{75} +(-151888. - 296764. i) q^{77} +(-91467.9 - 158427. i) q^{79} +(-29524.5 + 51137.9i) q^{81} +364891. i q^{83} -708343. q^{85} +(252410. + 145729. i) q^{87} +(-981810. + 566848. i) q^{89} +(-1.02523e6 - 663148. i) q^{91} +(-158652. - 274793. i) q^{93} +(196893. - 341029. i) q^{95} -1.38167e6i q^{97} -236181. q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 108 q^{3} + 210 q^{5} + 608 q^{7} + 972 q^{9}+O(q^{10}) $ 8 * q - 108 * q^3 + 210 * q^5 + 608 * q^7 + 972 * q^9	$ 8 q - 108 q^{3} + 210 q^{5} + 608 q^{7} + 972 q^{9} - 2058 q^{11} - 3780 q^{15} - 11244 q^{17} - 21834 q^{19} - 4482 q^{21} - 15504 q^{23} - 6550 q^{25} + 35316 q^{29} + 51060 q^{31} + 55566 q^{33} - 71460 q^{35} + 20282 q^{37} + 101682 q^{39} - 387812 q^{43} + 51030 q^{45} + 55212 q^{47} - 277780 q^{49} + 101196 q^{51} - 336174 q^{53} + 393012 q^{57} + 560454 q^{59} + 850728 q^{61} - 26730 q^{63} + 826380 q^{65} + 947882 q^{67} - 147192 q^{71} - 533034 q^{73} + 176850 q^{75} - 1848102 q^{77} + 6260 q^{79} - 236196 q^{81} + 560040 q^{85} - 476766 q^{87} + 413460 q^{89} - 256074 q^{91} - 459540 q^{93} + 170880 q^{95} - 1000188 q^{99}+O(q^{100}) $ 8 * q - 108 * q^3 + 210 * q^5 + 608 * q^7 + 972 * q^9 - 2058 * q^11 - 3780 * q^15 - 11244 * q^17 - 21834 * q^19 - 4482 * q^21 - 15504 * q^23 - 6550 * q^25 + 35316 * q^29 + 51060 * q^31 + 55566 * q^33 - 71460 * q^35 + 20282 * q^37 + 101682 * q^39 - 387812 * q^43 + 51030 * q^45 + 55212 * q^47 - 277780 * q^49 + 101196 * q^51 - 336174 * q^53 + 393012 * q^57 + 560454 * q^59 + 850728 * q^61 - 26730 * q^63 + 826380 * q^65 + 947882 * q^67 - 147192 * q^71 - 533034 * q^73 + 176850 * q^75 - 1848102 * q^77 + 6260 * q^79 - 236196 * q^81 + 560040 * q^85 - 476766 * q^87 + 413460 * q^89 - 256074 * q^91 - 459540 * q^93 + 170880 * q^95 - 1000188 * q^99

Display 100 coefficients 10 coefficients

Character values

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

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

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−13.5000	−	7.79423i	−0.500000	−	0.288675i
$3$	−13.5000	−	7.79423i	−0.500000	−	0.288675i
$4$		0			0
$5$	182.133	−	105.155i	1.45706	−	0.841236i	0.458198	−	0.888850i	$-0.348495\pi$
$5$	182.133	−	105.155i	1.45706	−	0.841236i	0.998866	+	0.0476139i	$0.0151617\pi$
$6$		0			0
$7$	−186.289	+	288.003i	−0.543116	+	0.839658i
$7$	−186.289	+	288.003i	−0.543116	+	0.839658i
$8$		0			0
$9$	121.500	+	210.444i	0.166667	+	0.288675i
$10$		0			0
$11$	−485.969	+	841.723i	−0.365116	+	0.632399i	−0.988795	−	0.149281i	$-0.952304\pi$
$11$	−485.969	+	841.723i	−0.365116	+	0.632399i	0.623679	+	0.781681i	$0.285637\pi$
$12$		0			0
$13$			3559.78i			1.62029i	0.586227	+	0.810147i	$0.300613\pi$
$13$			3559.78i			1.62029i	−0.586227	+	0.810147i	$0.699387\pi$
$14$		0			0
$15$	−3278.39			−0.971376
$16$		0			0
$17$	−2916.87	−	1684.05i	−0.593704	−	0.342775i	0.172857	−	0.984947i	$-0.444700\pi$
$17$	−2916.87	−	1684.05i	−0.593704	−	0.342775i	−0.766561	+	0.642172i	$0.778034\pi$
$18$		0			0
$19$	1621.56	−	936.209i	0.236414	−	0.136494i	−0.377114	−	0.926167i	$-0.623083\pi$
$19$	1621.56	−	936.209i	0.236414	−	0.136494i	0.613527	+	0.789673i	$0.289750\pi$
$20$		0			0
$21$	4759.66	−	2436.06i	0.513946	−	0.263045i
$22$		0			0
$23$	−9942.27	−	17220.5i	−0.817150	−	1.41535i	−0.907774	−	0.419460i	$-0.862219\pi$
$23$	−9942.27	−	17220.5i	−0.817150	−	1.41535i	0.0906234	−	0.995885i	$-0.471114\pi$
$24$		0			0
$25$	14302.4	−	24772.6i	0.915357	−	1.58544i
$26$		0			0
$27$		−	3788.00i		−	0.192450i
$28$		0			0
$29$	−18697.0			−0.766616			−0.383308	−	0.923621i	$-0.625215\pi$
$29$	−18697.0			−0.766616			−0.383308	+	0.923621i	$0.625215\pi$
$30$		0			0
$31$	17628.0	+	10177.5i	0.591721	+	0.341630i	0.765778	−	0.643105i	$-0.222354\pi$
$31$	17628.0	+	10177.5i	0.591721	+	0.341630i	−0.174057	+	0.984736i	$0.555688\pi$
$32$		0			0
$33$	13121.2	−	7575.51i	0.365116	−	0.210800i
$34$		0			0
$35$	−3644.56	+	72043.9i	−0.0850042	+	1.68032i
$36$		0			0
$37$	−44846.5	−	77676.4i	−0.885367	−	1.53350i	−0.845292	−	0.534304i	$-0.820574\pi$
$37$	−44846.5	−	77676.4i	−0.885367	−	1.53350i	−0.0400751	−	0.999197i	$-0.512760\pi$
$38$		0			0
$39$	27745.8	−	48057.1i	0.467738	−	0.810147i
$40$		0			0
$41$			10931.8i			0.158613i	0.996850	+	0.0793067i	$0.0252706\pi$
$41$			10931.8i			0.158613i	−0.996850	+	0.0793067i	$0.974729\pi$
$42$		0			0
$43$	−3887.60			−0.0488963			−0.0244482	−	0.999701i	$-0.507783\pi$
$43$	−3887.60			−0.0488963			−0.0244482	+	0.999701i	$0.507783\pi$
$44$		0			0
$45$	44258.3	+	25552.5i	0.485688	+	0.280412i
$46$		0			0
$47$	67906.5	−	39205.8i	0.654060	−	0.377622i	−0.135950	−	0.990716i	$-0.543409\pi$
$47$	67906.5	−	39205.8i	0.654060	−	0.377622i	0.790010	+	0.613094i	$0.210075\pi$
$48$		0			0
$49$	−48242.0	−	107303.i	−0.410050	−	0.912063i
$50$		0			0
$51$	26251.8	+	45469.4i	0.197901	+	0.342775i
$52$		0			0
$53$	−53856.2	+	93281.7i	−0.361750	+	0.626569i	−0.988249	−	0.152853i	$-0.951154\pi$
$53$	−53856.2	+	93281.7i	−0.361750	+	0.626569i	0.626499	+	0.779422i	$0.284487\pi$
$54$		0			0
$55$			204407.i			1.22859i
$56$		0			0
$57$	−29188.1			−0.157609
$58$		0			0
$59$	−212952.	−	122948.i	−1.03688	−	0.598640i	−0.117929	−	0.993022i	$-0.537625\pi$
$59$	−212952.	−	122948.i	−1.03688	−	0.598640i	−0.918947	+	0.394382i	$0.870959\pi$
$60$		0			0
$61$	274561.	−	158518.i	1.20962	−	0.698376i	0.246946	−	0.969029i	$-0.420573\pi$
$61$	274561.	−	158518.i	1.20962	−	0.698376i	0.962677	+	0.270653i	$0.0872397\pi$
$62$		0			0
$63$	−83242.6	−	4211.08i	−0.332908	−	0.0168411i
$64$		0			0
$65$	374327.	+	648354.i	1.36305	+	2.36087i
$66$		0			0
$67$	242122.	−	419367.i	0.805025	−	1.39434i	−0.111250	−	0.993793i	$-0.535485\pi$
$67$	242122.	−	419367.i	0.805025	−	1.39434i	0.916274	−	0.400551i	$-0.131181\pi$
$68$		0			0
$69$			309969.i			0.943564i
$70$		0			0
$71$	69288.4			0.193591			0.0967956	−	0.995304i	$-0.469141\pi$
$71$	69288.4			0.193591			0.0967956	+	0.995304i	$0.469141\pi$
$72$		0			0
$73$	−141741.	−	81834.1i	−0.364356	−	0.210361i	0.306634	−	0.951828i	$-0.400797\pi$
$73$	−141741.	−	81834.1i	−0.364356	−	0.210361i	−0.670990	+	0.741466i	$0.734131\pi$
$74$		0			0
$75$	−386166.	+	222953.i	−0.915357	+	0.528481i
$76$		0			0
$77$	−151888.	−	296764.i	−0.332699	−	0.650039i
$78$		0			0
$79$	−91467.9	−	158427.i	−0.185519	−	0.321328i	0.758233	−	0.651984i	$-0.226063\pi$
$79$	−91467.9	−	158427.i	−0.185519	−	0.321328i	−0.943751	+	0.330657i	$0.892730\pi$
$80$		0			0
$81$	−29524.5	+	51137.9i	−0.0555556	+	0.0962250i
$82$		0			0
$83$			364891.i			0.638160i	0.947728	+	0.319080i	$0.103374\pi$
$83$			364891.i			0.638160i	−0.947728	+	0.319080i	$0.896626\pi$
$84$		0			0
$85$	−708343.			−1.15342
$86$		0			0
$87$	252410.	+	145729.i	0.383308	+	0.221303i
$88$		0			0
$89$	−981810.	+	566848.i	−1.39270	+	0.804075i	−0.993613	−	0.112838i	$-0.964006\pi$
$89$	−981810.	+	566848.i	−1.39270	+	0.804075i	−0.399086	+	0.916913i	$0.630673\pi$
$90$		0			0
$91$	−1.02523e6	−	663148.i	−1.36049	−	0.880007i
$92$		0			0
$93$	−158652.	−	274793.i	−0.197240	−	0.341630i
$94$		0			0
$95$	196893.	−	341029.i	0.229647	−	0.397760i
$96$		0			0
$97$		−	1.38167e6i		−	1.51387i	−0.653488	−	0.756937i	$-0.726695\pi$
$97$		−	1.38167e6i		−	1.51387i	0.653488	−	0.756937i	$-0.273305\pi$
$98$		0			0
$99$	−236181.			−0.243411
$100$		0			0
$101$	−1.45252e6	−	838613.i	−1.40980	−	0.813950i	−0.414433	−	0.910080i	$-0.636020\pi$
$101$	−1.45252e6	−	838613.i	−1.40980	−	0.813950i	−0.995369	+	0.0961301i	$0.969354\pi$
$102$		0			0
$103$	1.06488e6	−	614811.i	0.974520	−	0.562639i	0.0739088	−	0.997265i	$-0.476453\pi$
$103$	1.06488e6	−	614811.i	0.974520	−	0.562639i	0.900611	+	0.434626i	$0.143119\pi$
$104$		0			0
$105$	610728.	−	944186.i	0.527570	−	0.815623i
$106$		0			0
$107$	−884286.	−	1.53163e6i	−0.721841	−	1.25026i	−0.960261	−	0.279103i	$-0.909963\pi$
$107$	−884286.	−	1.53163e6i	−0.721841	−	1.25026i	0.238420	−	0.971162i	$-0.423370\pi$
$108$		0			0
$109$	637301.	−	1.10384e6i	0.492113	−	0.852365i	−0.507846	−	0.861448i	$-0.669558\pi$
$109$	637301.	−	1.10384e6i	0.492113	−	0.852365i	0.999959	+	0.00908330i	$0.00289134\pi$
$110$		0			0
$111$			1.39818e6i			1.02233i
$112$		0			0
$113$	−1.13727e6			−0.788188			−0.394094	−	0.919070i	$-0.628942\pi$
$113$	−1.13727e6			−0.788188			−0.394094	+	0.919070i	$0.628942\pi$
$114$		0			0
$115$	−3.62163e6	−	2.09095e6i	−2.38128	−	1.37483i
$116$		0			0
$117$	−749136.	+	432514.i	−0.467738	+	0.270049i
$118$		0			0
$119$	1.02839e6	−	526345.i	0.610264	−	0.312341i
$120$		0			0
$121$	413448.	+	716113.i	0.233381	+	0.404227i
$122$		0			0
$123$	85204.9	−	147579.i	0.0457877	−	0.0793067i
$124$		0			0
$125$		−	2.72979e6i		−	1.39765i
$126$		0			0
$127$	−696850.			−0.340195			−0.170097	−	0.985427i	$-0.554408\pi$
$127$	−696850.			−0.340195			−0.170097	+	0.985427i	$0.554408\pi$
$128$		0			0
$129$	52482.6	+	30300.9i	0.0244482	+	0.0141152i
$130$		0			0
$131$	−2.17981e6	+	1.25851e6i	−0.969626	+	0.559814i	−0.899122	−	0.437697i	$-0.855794\pi$
$131$	−2.17981e6	+	1.25851e6i	−0.969626	+	0.559814i	−0.0705041	+	0.997511i	$0.522461\pi$
$132$		0			0
$133$	−32448.1	+	641419.i	−0.0137922	+	0.272638i
$134$		0			0
$135$	−398325.	−	689919.i	−0.161896	−	0.280412i
$136$		0			0
$137$	−301363.	+	521976.i	−0.117200	+	0.202997i	−0.918657	−	0.395056i	$-0.870725\pi$
$137$	−301363.	+	521976.i	−0.117200	+	0.202997i	0.801457	+	0.598052i	$0.204059\pi$
$138$		0			0
$139$			3.66866e6i			1.36604i	0.730400	+	0.683020i	$0.239334\pi$
$139$			3.66866e6i			1.36604i	−0.730400	+	0.683020i	$0.760666\pi$
$140$		0			0
$141$	−1.22232e6			−0.436040
$142$		0			0
$143$	−2.99635e6	−	1.72995e6i	−1.02467	−	0.591595i
$144$		0			0
$145$	−3.40534e6	+	1.96607e6i	−1.11701	+	0.644905i
$146$		0			0
$147$	−185080.	+	1.82460e6i	−0.0582649	+	0.574403i
$148$		0			0
$149$	−94655.9	−	163949.i	−0.0286147	−	0.0495621i	0.851363	−	0.524576i	$-0.175776\pi$
$149$	−94655.9	−	163949.i	−0.0286147	−	0.0495621i	−0.879978	+	0.475014i	$0.842443\pi$
$150$		0			0
$151$	7645.03	−	13241.6i	0.00222049	−	0.00384600i	−0.864913	−	0.501922i	$-0.832627\pi$
$151$	7645.03	−	13241.6i	0.00222049	−	0.00384600i	0.867134	+	0.498076i	$0.165960\pi$
$152$		0			0
$153$		−	818450.i		−	0.228517i
$154$		0			0
$155$	4.28084e6			1.14957
$156$		0			0
$157$	2.54625e6	+	1.47008e6i	0.657964	+	0.379876i	0.791501	−	0.611168i	$-0.209300\pi$
$157$	2.54625e6	+	1.47008e6i	0.657964	+	0.379876i	−0.133537	+	0.991044i	$0.542633\pi$
$158$		0			0
$159$	1.45412e6	−	839535.i	0.361750	−	0.208856i
$160$		0			0
$161$	6.81168e6	+	344590.i	1.63221	+	0.0825704i
$162$		0			0
$163$	458195.	+	793618.i	0.105801	+	0.183252i	0.914065	−	0.405568i	$-0.132926\pi$
$163$	458195.	+	793618.i	0.105801	+	0.183252i	−0.808264	+	0.588820i	$0.799593\pi$
$164$		0			0
$165$	1.59320e6	−	2.75950e6i	0.354665	−	0.614297i
$166$		0			0
$167$		−	3.49297e6i		−	0.749972i	−0.927031	−	0.374986i	$-0.877648\pi$
$167$		−	3.49297e6i		−	0.749972i	0.927031	−	0.374986i	$-0.122352\pi$
$168$		0			0
$169$	−7.84526e6			−1.62535
$170$		0			0
$171$	394039.	+	227499.i	0.0788046	+	0.0454978i
$172$		0			0
$173$	595541.	−	343836.i	0.115020	−	0.0664069i	−0.441386	−	0.897317i	$-0.645513\pi$
$173$	595541.	−	343836.i	0.115020	−	0.0664069i	0.556406	+	0.830910i	$0.312180\pi$
$174$		0			0
$175$	4.47018e6	+	8.73399e6i	0.834086	+	1.62967i
$176$		0			0
$177$	1.91657e6	+	3.31960e6i	0.345625	+	0.598640i
$178$		0			0
$179$	−39555.6	+	68512.3i	−0.00689681	+	0.0119456i	−0.869453	−	0.494015i	$-0.835529\pi$
$179$	−39555.6	+	68512.3i	−0.00689681	+	0.0119456i	0.862556	+	0.505961i	$0.168862\pi$
$180$		0			0
$181$			4.28637e6i			0.722860i	0.932399	+	0.361430i	$0.117711\pi$
$181$			4.28637e6i			0.722860i	−0.932399	+	0.361430i	$0.882289\pi$
$182$		0			0
$183$	−4.94211e6			−0.806415
$184$		0			0
$185$	−1.63361e7	−	9.43163e6i	−2.58007	−	1.48961i
$186$		0			0
$187$	2.83501e6	−	1.63680e6i	0.433541	−	0.250305i
$188$		0			0
$189$	1.09095e6	+	705661.i	0.161592	+	0.104523i
$190$		0			0
$191$	−4.72334e6	−	8.18106e6i	−0.677874	−	1.17411i	−0.975620	−	0.219467i	$-0.929568\pi$
$191$	−4.72334e6	−	8.18106e6i	−0.677874	−	1.17411i	0.297746	−	0.954645i	$-0.403765\pi$
$192$		0			0
$193$	4.68418e6	−	8.11323e6i	0.651570	−	1.12855i	−0.331171	−	0.943571i	$-0.607444\pi$
$193$	4.68418e6	−	8.11323e6i	0.651570	−	1.12855i	0.982742	−	0.184982i	$-0.0592228\pi$
$194$		0			0
$195$		−	1.16704e7i		−	1.57391i
$196$		0			0
$197$	3.86436e6			0.505450			0.252725	−	0.967538i	$-0.418673\pi$
$197$	3.86436e6			0.505450			0.252725	+	0.967538i	$0.418673\pi$
$198$		0			0
$199$	−8.61379e6	−	4.97318e6i	−1.09304	−	0.631066i	−0.158654	−	0.987334i	$-0.550716\pi$
$199$	−8.61379e6	−	4.97318e6i	−1.09304	−	0.631066i	−0.934384	+	0.356268i	$0.884049\pi$
$200$		0			0
$201$	−6.53729e6	+	3.77430e6i	−0.805025	+	0.464781i
$202$		0			0
$203$	3.48304e6	−	5.38478e6i	0.416361	−	0.643695i
$204$		0			0
$205$	1.14953e6	+	1.99104e6i	0.133431	+	0.231110i
$206$		0			0
$207$	2.41597e6	−	4.18458e6i	0.272383	−	0.471782i
$208$		0			0
$209$			1.81988e6i			0.199344i
$210$		0			0
$211$	−4.42876e6			−0.471449			−0.235724	−	0.971820i	$-0.575746\pi$
$211$	−4.42876e6			−0.471449			−0.235724	+	0.971820i	$0.575746\pi$
$212$		0			0
$213$	−935394.	−	540050.i	−0.0967956	−	0.0558850i
$214$		0			0
$215$	−708061.	+	408799.i	−0.0712451	+	0.0411334i
$216$		0			0
$217$	−6.21504e6	+	3.18094e6i	−0.608225	+	0.311298i
$218$		0			0
$219$	1.27567e6	+	2.20952e6i	0.121452	+	0.210361i
$220$		0			0
$221$	5.99487e6	−	1.03834e7i	0.555396	−	0.961974i
$222$		0			0
$223$			1.12907e7i			1.01814i	0.860726	+	0.509068i	$0.170010\pi$
$223$			1.12907e7i			1.01814i	−0.860726	+	0.509068i	$0.829990\pi$
$224$		0			0
$225$	6.95099e6			0.610238
$226$		0			0
$227$	−7.74694e6	−	4.47270e6i	−0.662297	−	0.382377i	0.130855	−	0.991402i	$-0.458228\pi$
$227$	−7.74694e6	−	4.47270e6i	−0.662297	−	0.382377i	−0.793151	+	0.609024i	$0.791561\pi$
$228$		0			0
$229$	5.93107e6	−	3.42431e6i	0.493886	−	0.285145i	−0.232299	−	0.972644i	$-0.574625\pi$
$229$	5.93107e6	−	3.42431e6i	0.493886	−	0.285145i	0.726185	+	0.687499i	$0.241291\pi$
$230$		0			0
$231$	−262560.	+	5.19016e6i	−0.0213006	+	0.421061i
$232$		0			0
$233$	1.00855e7	+	1.74686e7i	0.797313	+	1.38099i	0.921360	+	0.388710i	$0.127079\pi$
$233$	1.00855e7	+	1.74686e7i	0.797313	+	1.38099i	−0.124048	+	0.992276i	$0.539588\pi$
$234$		0			0
$235$	8.24534e6	−	1.42814e7i	0.635339	−	1.10044i
$236$		0			0
$237$			2.85169e6i			0.214218i
$238$		0			0
$239$	−2.24773e7			−1.64645			−0.823227	−	0.567712i	$-0.807829\pi$
$239$	−2.24773e7			−1.64645			−0.823227	+	0.567712i	$0.807829\pi$
$240$		0			0
$241$	−4.19282e6	−	2.42073e6i	−0.299540	−	0.172940i	0.342696	−	0.939446i	$-0.388660\pi$
$241$	−4.19282e6	−	2.42073e6i	−0.299540	−	0.172940i	−0.642236	+	0.766507i	$0.721993\pi$
$242$		0			0
$243$	797162.	−	460241.i	0.0555556	−	0.0320750i
$244$		0			0
$245$	−2.00699e7	−	1.44706e7i	−1.36473	−	0.983985i
$246$		0			0
$247$	3.33270e6	+	5.77241e6i	0.221160	+	0.383060i
$248$		0			0
$249$	2.84405e6	−	4.92603e6i	0.184221	−	0.319080i
$250$		0			0
$251$			2.19206e7i			1.38622i	0.720834	+	0.693108i	$0.243759\pi$
$251$			2.19206e7i			1.38622i	−0.720834	+	0.693108i	$0.756241\pi$
$252$		0			0
$253$	1.93265e7			1.19342
$254$		0			0
$255$	9.56264e6	+	5.52099e6i	0.576709	+	0.332963i
$256$		0			0
$257$	748568.	−	432186.i	0.0440993	−	0.0254608i	−0.477788	−	0.878475i	$-0.658561\pi$
$257$	748568.	−	432186.i	0.0440993	−	0.0254608i	0.521888	+	0.853014i	$0.325228\pi$
$258$		0			0
$259$	3.07254e7	+	1.55434e6i	1.76847	+	0.0894635i
$260$		0			0
$261$	−2.27169e6	−	3.93467e6i	−0.127769	−	0.221303i
$262$		0			0
$263$	−1.17280e7	+	2.03134e7i	−0.644696	+	1.11665i	0.339675	+	0.940543i	$0.389683\pi$
$263$	−1.17280e7	+	2.03134e7i	−0.644696	+	1.11665i	−0.984372	+	0.176104i	$0.943651\pi$
$264$		0			0
$265$			2.26529e7i			1.21727i
$266$		0			0
$267$	1.76726e7			0.928466
$268$		0			0
$269$	−1.21794e7	−	7.03179e6i	−0.625705	−	0.361251i	0.153382	−	0.988167i	$-0.450984\pi$
$269$	−1.21794e7	−	7.03179e6i	−0.625705	−	0.361251i	−0.779087	+	0.626916i	$0.784317\pi$
$270$		0			0
$271$	1.08536e6	−	626633.i	0.0545338	−	0.0314851i	−0.472485	−	0.881339i	$-0.656643\pi$
$271$	1.08536e6	−	626633.i	0.0545338	−	0.0314851i	0.527019	+	0.849853i	$0.323310\pi$
$272$		0			0
$273$	8.67184e6	+	1.69434e7i	0.426210	+	0.832744i
$274$		0			0
$275$	1.39011e7	+	2.40774e7i	0.668423	+	1.15774i
$276$		0			0
$277$	−1.62307e7	+	2.81125e7i	−0.763658	+	1.32270i	0.177295	+	0.984158i	$0.443265\pi$
$277$	−1.62307e7	+	2.81125e7i	−0.763658	+	1.32270i	−0.940953	+	0.338537i	$0.890068\pi$
$278$		0			0
$279$			4.94627e6i			0.227753i
$280$		0			0
$281$	1.81052e7			0.815989			0.407995	−	0.912984i	$-0.366228\pi$
$281$	1.81052e7			0.815989			0.407995	+	0.912984i	$0.366228\pi$
$282$		0			0
$283$	1.18049e7	+	6.81558e6i	0.520840	+	0.300707i	0.737278	−	0.675589i	$-0.236111\pi$
$283$	1.18049e7	+	6.81558e6i	0.520840	+	0.300707i	−0.216439	+	0.976296i	$0.569444\pi$
$284$		0			0
$285$	−5.31612e6	+	3.06926e6i	−0.229647	+	0.132587i
$286$		0			0
$287$	−3.14838e6	−	2.03647e6i	−0.133181	−	0.0861454i
$288$		0			0
$289$	−6.39671e6	−	1.10794e7i	−0.265011	−	0.459012i
$290$		0			0
$291$	−1.07691e7	+	1.86526e7i	−0.437018	+	0.756937i
$292$		0			0
$293$			5.71340e6i			0.227139i	0.993530	+	0.113570i	$0.0362285\pi$
$293$			5.71340e6i			0.227139i	−0.993530	+	0.113570i	$0.963772\pi$
$294$		0			0
$295$	−5.17142e7			−2.01439
$296$		0			0
$297$	3.18844e6	+	1.84085e6i	0.121705	+	0.0702666i
$298$		0			0
$299$	6.13013e7	−	3.53923e7i	2.29328	−	1.32402i
$300$		0			0
$301$	724217.	−	1.11964e6i	0.0265564	−	0.0410562i
$302$		0			0
$303$	1.30727e7	+	2.26426e7i	0.469934	+	0.813950i
$304$		0			0
$305$	3.33378e7	−	5.77427e7i	1.17500	−	2.03516i
$306$		0			0
$307$			7.04957e6i			0.243639i	0.992552	+	0.121820i	$0.0388730\pi$
$307$			7.04957e6i			0.243639i	−0.992552	+	0.121820i	$0.961127\pi$
$308$		0			0
$309$	−1.91679e7			−0.649680
$310$		0			0
$311$	4.50891e6	+	2.60322e6i	0.149896	+	0.0865426i	0.573072	−	0.819505i	$-0.305751\pi$
$311$	4.50891e6	+	2.60322e6i	0.149896	+	0.0865426i	−0.423176	+	0.906047i	$0.639085\pi$
$312$		0			0
$313$	2.26504e7	−	1.30772e7i	0.738656	−	0.426463i	−0.0829244	−	0.996556i	$-0.526426\pi$
$313$	2.26504e7	−	1.30772e7i	0.738656	−	0.426463i	0.821580	+	0.570093i	$0.193093\pi$
$314$		0			0
$315$	−1.56040e7	+	7.98636e6i	−0.499235	+	0.255515i
$316$		0			0
$317$	3.29139e6	+	5.70085e6i	0.103324	+	0.178963i	0.913052	−	0.407843i	$-0.133719\pi$
$317$	3.29139e6	+	5.70085e6i	0.103324	+	0.178963i	−0.809728	+	0.586805i	$0.800385\pi$
$318$		0			0
$319$	9.08617e6	−	1.57377e7i	0.279904	−	0.484808i
$320$		0			0
$321$			2.75693e7i			0.833510i
$322$		0			0
$323$	−6.30650e6			−0.187146
$324$		0			0
$325$	8.81850e7	+	5.09136e7i	2.56889	+	1.48315i
$326$		0			0
$327$	−1.72071e7	+	9.93453e6i	−0.492113	+	0.284122i
$328$		0			0
$329$	−1.35884e6	+	2.68609e7i	−0.0381575	+	0.754279i
$330$		0			0
$331$	−7.38720e6	−	1.27950e7i	−0.203702	−	0.352823i	0.746016	−	0.665928i	$-0.231964\pi$
$331$	−7.38720e6	−	1.27950e7i	−0.203702	−	0.352823i	−0.949718	+	0.313105i	$0.898631\pi$
$332$		0			0
$333$	1.08977e7	−	1.88754e7i	0.295122	−	0.511167i
$334$		0			0
$335$		−	1.01841e8i		−	2.70886i
$336$		0			0
$337$	−6.86189e6			−0.179289			−0.0896445	−	0.995974i	$-0.528573\pi$
$337$	−6.86189e6			−0.179289			−0.0896445	+	0.995974i	$0.528573\pi$
$338$		0			0
$339$	1.53532e7	+	8.86417e6i	0.394094	+	0.227530i
$340$		0			0
$341$	−1.71333e7	+	9.89191e6i	−0.432093	+	0.249469i
$342$		0			0
$343$	3.98906e7	+	6.09559e6i	0.988525	+	0.151054i
$344$		0			0
$345$	3.25947e7	+	5.64556e7i	0.793760	+	1.37483i
$346$		0			0
$347$	8.62186e6	−	1.49335e7i	0.206354	−	0.357415i	−0.744209	−	0.667946i	$-0.767174\pi$
$347$	8.62186e6	−	1.49335e7i	0.206354	−	0.357415i	0.950563	+	0.310531i	$0.100507\pi$
$348$		0			0
$349$		−	6.43913e7i		−	1.51478i	−0.652960	−	0.757392i	$-0.726473\pi$
$349$		−	6.43913e7i		−	1.51478i	0.652960	−	0.757392i	$-0.273527\pi$
$350$		0			0
$351$	1.34844e7			0.311826
$352$		0			0
$353$	5.92558e6	+	3.42114e6i	0.134712	+	0.0777761i	0.565842	−	0.824514i	$-0.308551\pi$
$353$	5.92558e6	+	3.42114e6i	0.134712	+	0.0777761i	−0.431129	+	0.902290i	$0.641885\pi$
$354$		0			0
$355$	1.26197e7	−	7.28599e6i	0.282075	−	0.162856i
$356$		0			0
$357$	−1.79857e7	−	909862.i	−0.395297	−	0.0199973i
$358$		0			0
$359$	1.61260e7	+	2.79311e7i	0.348533	+	0.603677i	0.985989	−	0.166810i	$-0.0533467\pi$
$359$	1.61260e7	+	2.79311e7i	0.348533	+	0.603677i	−0.637456	+	0.770487i	$0.720013\pi$
$360$		0			0
$361$	−2.17700e7	+	3.77067e7i	−0.462739	+	0.801488i
$362$		0			0
$363$		−	1.28900e7i		−	0.269485i
$364$		0			0
$365$	−3.44209e7			−0.707854
$366$		0			0
$367$	−1.25873e6	−	726727.i	−0.0254644	−	0.0147019i	0.487214	−	0.873283i	$-0.338013\pi$
$367$	−1.25873e6	−	726727.i	−0.0254644	−	0.0147019i	−0.512678	+	0.858581i	$0.671347\pi$
$368$		0			0
$369$	−2.30053e6	+	1.32821e6i	−0.0457877	+	0.0264356i
$370$		0			0
$371$	−1.68326e7	−	3.28881e7i	−0.329631	−	0.644046i
$372$		0			0
$373$	1.58502e7	+	2.74533e7i	0.305427	+	0.529015i	0.977356	−	0.211600i	$-0.0678675\pi$
$373$	1.58502e7	+	2.74533e7i	0.305427	+	0.529015i	−0.671929	+	0.740615i	$0.734534\pi$
$374$		0			0
$375$	−2.12766e7	+	3.68522e7i	−0.403468	+	0.698826i
$376$		0			0
$377$		−	6.65573e7i		−	1.24214i
$378$		0			0
$379$	6.98752e7			1.28353			0.641764	−	0.766902i	$-0.278203\pi$
$379$	6.98752e7			1.28353			0.641764	+	0.766902i	$0.278203\pi$
$380$		0			0
$381$	9.40747e6	+	5.43140e6i	0.170097	+	0.0982058i
$382$		0			0
$383$	−6.53554e7	+	3.77329e7i	−1.16328	+	0.671621i	−0.952088	−	0.305823i	$-0.901068\pi$
$383$	−6.53554e7	+	3.77329e7i	−1.16328	+	0.671621i	−0.211193	+	0.977444i	$0.567735\pi$
$384$		0			0
$385$	−5.88699e7	−	3.80788e7i	−1.03160	−	0.667270i
$386$		0			0
$387$	−472344.	−	818123.i	−0.00814939	−	0.0141152i
$388$		0			0
$389$	−4.30868e7	+	7.46285e7i	−0.731973	+	1.26781i	0.224066	+	0.974574i	$0.428067\pi$
$389$	−4.30868e7	+	7.46285e7i	−0.731973	+	1.26781i	−0.956039	+	0.293240i	$0.905266\pi$
$390$		0			0
$391$			6.69732e7i			1.12039i
$392$		0			0
$393$	3.92366e7			0.646418
$394$		0			0
$395$	−3.33186e7	−	1.92365e7i	−0.540625	−	0.312130i
$396$		0			0
$397$	6.71952e7	−	3.87951e7i	1.07391	−	0.620020i	0.144660	−	0.989481i	$-0.453791\pi$
$397$	6.71952e7	−	3.87951e7i	1.07391	−	0.620020i	0.929246	+	0.369461i	$0.120458\pi$
$398$		0			0
$399$	5.43742e6	−	8.40625e6i	0.0856000	−	0.132338i
$400$		0			0
$401$	1.84833e6	+	3.20140e6i	0.0286647	+	0.0496486i	0.880002	−	0.474970i	$-0.157541\pi$
$401$	1.84833e6	+	3.20140e6i	0.0286647	+	0.0496486i	−0.851337	+	0.524619i	$0.824208\pi$
$402$		0			0
$403$	−3.62297e7	+	6.27517e7i	−0.553541	+	0.958761i
$404$		0			0
$405$			1.24185e7i			0.186941i
$406$		0			0
$407$	8.71761e7			1.29305
$408$		0			0
$409$	−5.00824e7	−	2.89151e7i	−0.732007	−	0.422624i	0.0871493	−	0.996195i	$-0.472224\pi$
$409$	−5.00824e7	−	2.89151e7i	−0.732007	−	0.422624i	−0.819156	+	0.573571i	$0.805558\pi$
$410$		0			0
$411$	8.13680e6	−	4.69779e6i	0.117200	−	0.0676656i
$412$		0			0
$413$	7.50800e7	−	3.84270e7i	1.06580	−	0.545489i
$414$		0			0
$415$	3.83700e7	+	6.64587e7i	0.536843	+	0.929839i
$416$		0			0
$417$	2.85944e7	−	4.95269e7i	0.394342	−	0.683020i
$418$		0			0
$419$		−	8.02169e7i		−	1.09050i	−0.838275	−	0.545248i	$-0.816435\pi$
$419$		−	8.02169e7i		−	1.09050i	0.838275	−	0.545248i	$-0.183565\pi$
$420$		0			0
$421$	−1.93511e6			−0.0259334			−0.0129667	−	0.999916i	$-0.504128\pi$
$421$	−1.93511e6			−0.0259334			−0.0129667	+	0.999916i	$0.504128\pi$
$422$		0			0
$423$	1.65013e7	+	9.52702e6i	0.218020	+	0.125874i
$424$		0			0
$425$	−8.34367e7	+	4.81722e7i	−1.08690	+	0.627523i
$426$		0			0
$427$	−5.49409e6	+	1.08605e8i	−0.0705687	+	1.39497i
$428$		0			0
$429$	2.69672e7	+	4.67085e7i	0.341557	+	0.591595i
$430$		0			0
$431$	5.67563e6	−	9.83048e6i	0.0708896	−	0.122784i	−0.828402	−	0.560134i	$-0.810750\pi$
$431$	5.67563e6	−	9.83048e6i	0.0708896	−	0.122784i	0.899291	+	0.437350i	$0.144083\pi$
$432$		0			0
$433$		−	9.23707e7i		−	1.13781i	−0.822402	−	0.568906i	$-0.807367\pi$
$433$		−	9.23707e7i		−	1.13781i	0.822402	−	0.568906i	$-0.192633\pi$
$434$		0			0
$435$	6.12961e7			0.744672
$436$		0			0
$437$	−3.22440e7	−	1.86161e7i	−0.386371	−	0.223071i
$438$		0			0
$439$	5.24472e7	−	3.02804e7i	0.619910	−	0.357905i	−0.156924	−	0.987611i	$-0.550158\pi$
$439$	5.24472e7	−	3.02804e7i	0.619910	−	0.357905i	0.776834	+	0.629705i	$0.216824\pi$
$440$		0			0
$441$	1.67200e7	−	2.31896e7i	0.194948	−	0.270382i
$442$		0			0
$443$	−6.26388e7	−	1.08494e8i	−0.720498	−	1.24794i	−0.960801	−	0.277241i	$-0.910580\pi$
$443$	−6.26388e7	−	1.08494e8i	−0.720498	−	1.24794i	0.240303	−	0.970698i	$-0.422753\pi$
$444$		0			0
$445$	−1.19213e8	+	2.06484e8i	−1.35283	+	2.34318i
$446$		0			0
$447$			2.95108e6i			0.0330414i
$448$		0			0
$449$	4.26832e6			0.0471539			0.0235770	−	0.999722i	$-0.492495\pi$
$449$	4.26832e6			0.0471539			0.0235770	+	0.999722i	$0.492495\pi$
$450$		0			0
$451$	−9.20154e6	−	5.31251e6i	−0.100307	−	0.0579122i
$452$		0			0
$453$	−206416.	+	119174.i	−0.00222049	+	0.00128200i
$454$		0			0
$455$	−2.56461e8	−	1.29738e7i	−2.72262	−	0.137732i
$456$		0			0
$457$	4.34066e7	+	7.51825e7i	0.454786	+	0.787713i	0.998676	−	0.0514437i	$-0.0163823\pi$
$457$	4.34066e7	+	7.51825e7i	0.454786	+	0.787713i	−0.543890	+	0.839157i	$0.683049\pi$
$458$		0			0
$459$	−6.37919e6	+	1.10491e7i	−0.0659671	+	0.114258i
$460$		0			0
$461$		−	1.65906e8i		−	1.69339i	−0.532075	−	0.846697i	$-0.678588\pi$
$461$		−	1.65906e8i		−	1.69339i	0.532075	−	0.846697i	$-0.321412\pi$
$462$		0			0
$463$	−1.41475e8			−1.42540			−0.712699	−	0.701470i	$-0.752527\pi$
$463$	−1.41475e8			−1.42540			−0.712699	+	0.701470i	$0.752527\pi$
$464$		0			0
$465$	−5.77914e7	−	3.33659e7i	−0.574783	−	0.331851i
$466$		0			0
$467$	4.53153e7	−	2.61628e7i	0.444932	−	0.256882i	−0.260755	−	0.965405i	$-0.583972\pi$
$467$	4.53153e7	−	2.61628e7i	0.444932	−	0.256882i	0.705688	+	0.708523i	$0.250638\pi$
$468$		0			0
$469$	7.56742e7	+	1.47855e8i	0.733550	+	1.43324i
$470$		0			0
$471$	−2.29162e7	−	3.96921e7i	−0.219321	−	0.379876i
$472$		0			0
$473$	1.88926e6	−	3.27229e6i	0.0178528	−	0.0309220i
$474$		0			0
$475$		−	5.35603e7i		−	0.499761i
$476$		0			0
$477$	−2.61741e7			−0.241166
$478$		0			0
$479$	1.29684e8	+	7.48728e7i	1.17999	+	0.681268i	0.956012	−	0.293327i	$-0.0947623\pi$
$479$	1.29684e8	+	7.48728e7i	1.17999	+	0.681268i	0.223978	+	0.974594i	$0.428096\pi$
$480$		0			0
$481$	2.76511e8	−	1.59644e8i	2.48472	−	1.43455i
$482$		0			0
$483$	−8.92719e7	−	5.77438e7i	−0.792271	−	0.512465i
$484$		0			0
$485$	−1.45289e8	−	2.51648e8i	−1.27353	−	2.20581i
$486$		0			0
$487$	−9.37604e7	+	1.62398e8i	−0.811769	+	1.40603i	0.0998558	+	0.995002i	$0.468162\pi$
$487$	−9.37604e7	+	1.62398e8i	−0.811769	+	1.40603i	−0.911625	+	0.411023i	$0.865171\pi$
$488$		0			0
$489$		−	1.42851e7i		−	0.122168i
$490$		0			0
$491$	−1.41921e8			−1.19895			−0.599476	−	0.800393i	$-0.704624\pi$
$491$	−1.41921e8			−1.19895			−0.599476	+	0.800393i	$0.704624\pi$
$492$		0			0
$493$	5.45366e7	+	3.14867e7i	0.455143	+	0.262777i
$494$		0			0
$495$	−4.30164e7	+	2.48355e7i	−0.354665	+	0.204766i
$496$		0			0
$497$	−1.29077e7	+	1.99553e7i	−0.105143	+	0.162550i
$498$		0			0
$499$	−5.33587e7	−	9.24200e7i	−0.429441	−	0.743814i	0.567382	−	0.823454i	$-0.307956\pi$
$499$	−5.33587e7	−	9.24200e7i	−0.429441	−	0.743814i	−0.996824	+	0.0796405i	$0.974623\pi$
$500$		0			0
$501$	−2.72250e7	+	4.71550e7i	−0.216498	+	0.374986i
$502$		0			0
$503$		−	2.73080e6i		−	0.0214578i	−0.999942	−	0.0107289i	$-0.996585\pi$
$503$		−	2.73080e6i		−	0.0214578i	0.999942	−	0.0107289i	$-0.00341518\pi$
$504$		0			0
$505$	−3.52736e8			−2.73890
$506$		0			0
$507$	1.05911e8	+	6.11477e7i	0.812675	+	0.469198i
$508$		0			0
$509$	−1.39488e8	+	8.05333e7i	−1.05775	+	0.610692i	−0.924809	−	0.380432i	$-0.875775\pi$
$509$	−1.39488e8	+	8.05333e7i	−1.05775	+	0.610692i	−0.132941	+	0.991124i	$0.542442\pi$
$510$		0			0
$511$	4.99731e7	−	2.55769e7i	0.374519	−	0.191684i
$512$		0			0
$513$	−3.54636e6	−	6.14247e6i	−0.0262682	−	0.0454978i
$514$		0			0
$515$	1.29300e8	−	2.23955e8i	0.946625	−	1.63960i
$516$		0			0
$517$			7.62113e7i			0.551503i
$518$		0			0
$519$	−1.07197e7			−0.0766800
$520$		0			0
$521$	−3.57813e7	−	2.06583e7i	−0.253013	−	0.146077i	0.368130	−	0.929774i	$-0.379998\pi$
$521$	−3.57813e7	−	2.06583e7i	−0.253013	−	0.146077i	−0.621143	+	0.783697i	$0.713331\pi$
$522$		0			0
$523$	5.39683e7	−	3.11586e7i	0.377254	−	0.217807i	−0.299369	−	0.954137i	$-0.596776\pi$
$523$	5.39683e7	−	3.11586e7i	0.377254	−	0.217807i	0.676623	+	0.736330i	$0.263443\pi$
$524$		0			0
$525$	7.72734e6	−	1.52751e8i	0.0534014	−	1.05561i
$526$		0			0
$527$	−3.42789e7	−	5.93728e7i	−0.234205	−	0.405654i
$528$		0			0
$529$	−1.23679e8	+	2.14219e8i	−0.835469	+	1.44707i
$530$		0			0
$531$		−	5.97528e7i		−	0.399093i
$532$		0			0
$533$	−3.89148e7			−0.257000
$534$		0			0
$535$	−3.22115e8	−	1.85973e8i	−2.10354	−	1.21448i
$536$		0			0
$537$	1.06800e6	−	616610.i	0.00689681	−	0.00398188i
$538$		0			0
$539$	1.13764e8	+	1.15397e7i	0.726504	+	0.0736934i
$540$		0			0
$541$	−1.54969e8	−	2.68415e8i	−0.978709	−	1.69517i	−0.667108	−	0.744961i	$-0.732468\pi$
$541$	−1.54969e8	−	2.68415e8i	−0.978709	−	1.69517i	−0.311602	−	0.950213i	$-0.600865\pi$
$542$		0			0
$543$	3.34090e7	−	5.78660e7i	0.208672	−	0.361430i
$544$		0			0
$545$		−	2.68060e8i		−	1.65593i
$546$		0			0
$547$	1.89819e8			1.15979			0.579893	−	0.814693i	$-0.303095\pi$
$547$	1.89819e8			1.15979			0.579893	+	0.814693i	$0.303095\pi$
$548$		0			0
$549$	6.67184e7	+	3.85199e7i	0.403208	+	0.232792i
$550$		0			0
$551$	−3.03183e7	+	1.75043e7i	−0.181239	+	0.104638i
$552$		0			0
$553$	6.26668e7	+	3.17019e6i	0.370563	+	0.0187461i
$554$		0			0
$555$	1.47025e8	+	2.54654e8i	0.860024	+	1.48961i
$556$		0			0
$557$	1.40106e8	−	2.42671e8i	0.810758	−	1.40427i	−0.101575	−	0.994828i	$-0.532388\pi$
$557$	1.40106e8	−	2.42671e8i	0.810758	−	1.40427i	0.912334	−	0.409447i	$-0.134278\pi$
$558$		0			0
$559$		−	1.38390e7i		−	0.0792264i
$560$		0			0
$561$	−5.10303e7			−0.289028
$562$		0			0
$563$	6.08041e7	+	3.51053e7i	0.340728	+	0.196719i	0.660594	−	0.750743i	$-0.270305\pi$
$563$	6.08041e7	+	3.51053e7i	0.340728	+	0.196719i	−0.319866	+	0.947463i	$0.603638\pi$
$564$		0			0
$565$	−2.07135e8	+	1.19590e8i	−1.14844	+	0.663052i
$566$		0			0
$567$	−9.22777e6	−	1.80296e7i	−0.0506230	−	0.0989090i
$568$		0			0
$569$	1.10988e8	+	1.92237e8i	0.602476	+	1.04352i	0.992445	+	0.122691i	$0.0391526\pi$
$569$	1.10988e8	+	1.92237e8i	0.602476	+	1.04352i	−0.389968	+	0.920828i	$0.627514\pi$
$570$		0			0
$571$	9.94189e7	−	1.72199e8i	0.534024	−	0.924956i	−0.465186	−	0.885213i	$-0.654013\pi$
$571$	9.94189e7	−	1.72199e8i	0.534024	−	0.924956i	0.999210	−	0.0397433i	$-0.0126540\pi$
$572$		0			0
$573$			1.47259e8i			0.782741i
$574$		0			0
$575$	−5.68795e8			−2.99194
$576$		0			0
$577$	1.93148e8	+	1.11514e8i	1.00546	+	0.580501i	0.909859	−	0.414919i	$-0.136190\pi$
$577$	1.93148e8	+	1.11514e8i	1.00546	+	0.580501i	0.0955992	+	0.995420i	$0.469523\pi$
$578$		0			0
$579$	−1.26473e8	+	7.30191e7i	−0.651570	+	0.376184i
$580$		0			0
$581$	−1.05090e8	−	6.79752e7i	−0.535836	−	0.346595i
$582$		0			0
$583$	−5.23449e7	−	9.06641e7i	−0.264161	−	0.457541i
$584$		0			0
$585$	−9.09616e7	+	1.57550e8i	−0.454350	+	0.786957i
$586$		0			0
$587$		−	7.09053e7i		−	0.350561i	−0.984518	−	0.175281i	$-0.943917\pi$
$587$		−	7.09053e7i		−	0.350561i	0.984518	−	0.175281i	$-0.0560833\pi$
$588$		0			0
$589$	3.81131e7			0.186521
$590$		0			0
$591$	−5.21688e7	−	3.01197e7i	−0.252725	−	0.145911i
$592$		0			0
$593$	2.20611e8	−	1.27370e8i	1.05795	−	0.610805i	0.133083	−	0.991105i	$-0.457512\pi$
$593$	2.20611e8	−	1.27370e8i	1.05795	−	0.610805i	0.924863	+	0.380300i	$0.124179\pi$
$594$		0			0
$595$	1.31956e8	−	2.04005e8i	0.626440	−	0.968477i
$596$		0			0
$597$	7.75241e7	+	1.34276e8i	0.364346	+	0.631066i
$598$		0			0
$599$	−9.52522e7	+	1.64982e8i	−0.443195	+	0.767636i	−0.997924	−	0.0643949i	$-0.979488\pi$
$599$	−9.52522e7	+	1.64982e8i	−0.443195	+	0.767636i	0.554730	+	0.832031i	$0.312822\pi$
$600$		0			0
$601$			8.79125e7i			0.404974i	0.979285	+	0.202487i	$0.0649024\pi$
$601$			8.79125e7i			0.404974i	−0.979285	+	0.202487i	$0.935098\pi$
$602$		0			0
$603$	1.17671e8			0.536683
$604$		0			0
$605$	1.50605e8	+	8.69519e7i	0.680101	+	0.392657i
$606$		0			0
$607$	−2.49954e8	+	1.44311e8i	−1.11762	+	0.645259i	−0.940792	−	0.338983i	$-0.889917\pi$
$607$	−2.49954e8	+	1.44311e8i	−1.11762	+	0.645259i	−0.176828	+	0.984242i	$0.556584\pi$
$608$		0			0
$609$	−8.89913e7	+	4.55470e7i	−0.394000	+	0.201654i
$610$		0			0
$611$	1.39564e8	+	2.41733e8i	0.611858	+	1.05977i
$612$		0			0
$613$	1.82049e8	−	3.15319e8i	0.790329	−	1.36889i	−0.135435	−	0.990786i	$-0.543243\pi$
$613$	1.82049e8	−	3.15319e8i	0.790329	−	1.36889i	0.925763	−	0.378103i	$-0.123424\pi$
$614$		0			0
$615$		−	3.58387e7i		−	0.154073i
$616$		0			0
$617$	−2.33610e8			−0.994571			−0.497286	−	0.867587i	$-0.665670\pi$
$617$	−2.33610e8			−0.994571			−0.497286	+	0.867587i	$0.665670\pi$
$618$		0			0
$619$	3.44277e8	+	1.98769e8i	1.45157	+	0.838062i	0.998570	−	0.0534515i	$-0.0170222\pi$
$619$	3.44277e8	+	1.98769e8i	1.45157	+	0.838062i	0.452995	+	0.891513i	$0.350356\pi$
$620$		0			0
$621$	−6.52312e7	+	3.76613e7i	−0.272383	+	0.157261i
$622$		0			0
$623$	1.96464e7	−	3.88361e8i	0.0812493	−	1.60610i
$624$		0			0
$625$	−6.35740e7	−	1.10113e8i	−0.260399	−	0.451025i
$626$		0			0
$627$	1.41845e7	−	2.45683e7i	0.0575456	−	0.0996719i
$628$		0			0
$629$			3.02096e8i			1.21393i
$630$		0			0
$631$	1.47291e8			0.586258			0.293129	−	0.956073i	$-0.405304\pi$
$631$	1.47291e8			0.586258			0.293129	+	0.956073i	$0.405304\pi$
$632$		0			0
$633$	5.97882e7	+	3.45187e7i	0.235724	+	0.136096i
$634$		0			0
$635$	−1.26919e8	+	7.32769e7i	−0.495686	+	0.286184i
$636$		0			0
$637$	3.81977e8	−	1.71731e8i	1.47781	−	0.664401i
$638$		0			0
$639$	8.41855e6	+	1.45813e7i	0.0322652	+	0.0558850i
$640$		0			0
$641$	398057.	−	689455.i	0.00151137	−	0.00261777i	−0.865269	−	0.501308i	$-0.832852\pi$
$641$	398057.	−	689455.i	0.00151137	−	0.00261777i	0.866780	+	0.498691i	$0.166186\pi$
$642$		0			0
$643$		−	3.43701e8i		−	1.29285i	−0.762978	−	0.646425i	$-0.776263\pi$
$643$		−	3.43701e8i		−	1.29285i	0.762978	−	0.646425i	$-0.223737\pi$
$644$		0			0
$645$	1.27451e7			0.0474967
$646$		0			0
$647$	3.65961e7	+	2.11288e7i	0.135121	+	0.0780120i	0.566037	−	0.824380i	$-0.308476\pi$
$647$	3.65961e7	+	2.11288e7i	0.135121	+	0.0780120i	−0.430916	+	0.902392i	$0.641809\pi$
$648$		0			0
$649$	2.06977e8	−	1.19498e8i	0.757159	−	0.437146i
$650$		0			0
$651$	1.08696e8	+	5.49871e6i	0.393977	+	0.0199305i
$652$		0			0
$653$	1.10631e8	+	1.91618e8i	0.397316	+	0.688171i	0.993394	−	0.114756i	$-0.0366085\pi$
$653$	1.10631e8	+	1.91618e8i	0.397316	+	0.688171i	−0.596078	+	0.802926i	$0.703275\pi$
$654$		0			0
$655$	−2.64677e8	+	4.58433e8i	−0.941872	+	1.63137i
$656$		0			0
$657$		−	3.97714e7i		−	0.140241i
$658$		0			0
$659$	1.16774e8			0.408027			0.204014	−	0.978968i	$-0.434601\pi$
$659$	1.16774e8			0.408027			0.204014	+	0.978968i	$0.434601\pi$
$660$		0			0
$661$	−4.61650e8	−	2.66534e8i	−1.59848	−	0.922886i	−0.991780	−	0.127959i	$-0.959158\pi$
$661$	−4.61650e8	−	2.66534e8i	−1.59848	−	0.922886i	−0.606705	−	0.794927i	$-0.707509\pi$
$662$		0			0
$663$	−1.61861e8	+	9.34507e7i	−0.555396	+	0.320658i
$664$		0			0
$665$	6.15383e7	+	1.20236e8i	0.209257	+	0.408854i
$666$		0			0
$667$	1.85891e8	+	3.21972e8i	0.626441	+	1.08503i
$668$		0			0
$669$	8.80022e7	−	1.52424e8i	0.293911	−	0.509068i
$670$		0			0
$671$			3.08140e8i			1.01995i
$672$		0			0
$673$	7.89690e7			0.259067			0.129533	−	0.991575i	$-0.458652\pi$
$673$	7.89690e7			0.259067			0.129533	+	0.991575i	$0.458652\pi$
$674$		0			0
$675$	−9.38384e7	−	5.41776e7i	−0.305119	−	0.176160i
$676$		0			0
$677$	−7.85243e7	+	4.53360e7i	−0.253068	+	0.146109i	−0.621168	−	0.783677i	$-0.713342\pi$
$677$	−7.85243e7	+	4.53360e7i	−0.253068	+	0.146109i	0.368100	+	0.929786i	$0.380008\pi$
$678$		0			0
$679$	3.97925e8	+	2.57390e8i	1.27114	+	0.822209i
$680$		0			0
$681$	6.97224e7	+	1.20763e8i	0.220766	+	0.382377i
$682$		0			0
$683$	1.21588e8	−	2.10596e8i	0.381617	−	0.660981i	−0.609676	−	0.792651i	$-0.708701\pi$
$683$	1.21588e8	−	2.10596e8i	0.381617	−	0.660981i	0.991294	+	0.131670i	$0.0420339\pi$
$684$		0			0
$685$			1.26759e8i			0.394372i
$686$		0			0
$687$	−1.06759e8			−0.329257
$688$		0			0
$689$	−3.32063e8	−	1.91717e8i	−1.01523	−	0.586141i
$690$		0			0
$691$	1.35131e8	−	7.80178e7i	0.409563	−	0.236461i	−0.281039	−	0.959696i	$-0.590679\pi$
$691$	1.35131e8	−	7.80178e7i	0.409563	−	0.236461i	0.690602	+	0.723235i	$0.257346\pi$
$692$		0			0
$693$	4.39979e7	−	6.80208e7i	0.132200	−	0.204382i
$694$		0			0
$695$	3.85776e8	+	6.68184e8i	1.14916	+	1.99041i
$696$		0			0
$697$	1.84097e7	−	3.18866e7i	0.0543687	−	0.0941693i
$698$		0			0
$699$		−	3.14434e8i		−	0.920657i
$700$		0			0
$701$	1.55020e8			0.450023			0.225012	−	0.974356i	$-0.427758\pi$
$701$	1.55020e8			0.450023			0.225012	+	0.974356i	$0.427758\pi$
$702$		0			0
$703$	−1.45443e8	−	8.39714e7i	−0.418626	−	0.241694i
$704$		0			0
$705$	−2.22624e8	+	1.28532e8i	−0.635339	+	0.366813i
$706$		0			0
$707$	5.12111e8	−	2.62105e8i	1.44913	−	0.741682i
$708$		0			0
$709$	−3.22565e8	−	5.58698e8i	−0.905061	−	1.56761i	−0.820835	−	0.571165i	$-0.806492\pi$
$709$	−3.22565e8	−	5.58698e8i	−0.905061	−	1.56761i	−0.0842258	−	0.996447i	$-0.526842\pi$
$710$		0			0
$711$	2.22267e7	−	3.84978e7i	0.0618395	−	0.107109i
$712$		0			0
$713$		−	4.04750e8i		−	1.11665i
$714$		0			0
$715$	−7.27647e8			−1.99068
$716$		0			0
$717$	3.03443e8	+	1.75193e8i	0.823227	+	0.475291i
$718$		0			0
$719$	−4.75380e8	+	2.74461e8i	−1.27895	+	0.738404i	−0.976656	−	0.214810i	$-0.931087\pi$
$719$	−4.75380e8	+	2.74461e8i	−1.27895	+	0.738404i	−0.302297	+	0.953214i	$0.597753\pi$
$720$		0			0
$721$	−2.13088e7	+	4.21222e8i	−0.0568529	+	1.12384i
$722$		0			0
$723$	3.77354e7	+	6.53596e7i	0.0998468	+	0.172940i
$724$		0			0
$725$	−2.67413e8	+	4.63173e8i	−0.701727	+	1.21543i
$726$		0			0
$727$		−	3.37493e8i		−	0.878338i	−0.898404	−	0.439169i	$-0.855273\pi$
$727$		−	3.37493e8i		−	0.878338i	0.898404	−	0.439169i	$-0.144727\pi$
$728$		0			0
$729$	−1.43489e7			−0.0370370
$730$		0			0
$731$	1.13396e7	+	6.54693e6i	0.0290299	+	0.0167604i
$732$		0			0
$733$	−4.10304e8	+	2.36889e8i	−1.04182	+	0.601497i	−0.920349	−	0.391098i	$-0.872095\pi$
$733$	−4.10304e8	+	2.36889e8i	−1.04182	+	0.601497i	−0.121474	+	0.992595i	$0.538762\pi$
$734$		0			0
$735$	1.58156e8	+	3.51782e8i	0.398313	+	0.885956i
$736$		0			0
$737$	2.35327e8	+	4.07599e8i	0.587855	+	1.01819i
$738$		0			0
$739$	−2.55575e7	+	4.42670e7i	−0.0633265	+	0.109685i	−0.895950	−	0.444154i	$-0.853504\pi$
$739$	−2.55575e7	+	4.42670e7i	−0.0633265	+	0.109685i	0.832624	+	0.553839i	$0.186838\pi$
$740$		0			0
$741$		−	1.03903e8i		−	0.255373i
$742$		0			0
$743$	−3.07500e8			−0.749684			−0.374842	−	0.927089i	$-0.622303\pi$
$743$	−3.07500e8			−0.749684			−0.374842	+	0.927089i	$0.622303\pi$
$744$		0			0
$745$	−3.44799e7	−	1.99070e7i	−0.0833868	−	0.0481434i
$746$		0			0
$747$	−7.67893e7	+	4.43343e7i	−0.184221	+	0.106360i
$748$		0			0
$749$	6.05845e8	+	3.06485e7i	1.44184	+	0.0729397i
$750$		0			0
$751$	−2.90979e8	−	5.03991e8i	−0.686977	−	1.18988i	−0.972811	−	0.231599i	$-0.925604\pi$
$751$	−2.90979e8	−	5.03991e8i	−0.686977	−	1.18988i	0.285835	−	0.958279i	$-0.407729\pi$
$752$		0			0
$753$	1.70854e8	−	2.95928e8i	0.400166	−	0.693108i
$754$		0			0
$755$		−	3.21564e6i		−	0.00747182i
$756$		0			0
$757$	4.40250e7			0.101487			0.0507436	−	0.998712i	$-0.483841\pi$
$757$	4.40250e7			0.101487			0.0507436	+	0.998712i	$0.483841\pi$
$758$		0			0
$759$	−2.60908e8	−	1.50636e8i	−0.596709	−	0.344510i
$760$		0			0
$761$	−2.09687e8	+	1.21063e8i	−0.475792	+	0.274698i	−0.718661	−	0.695361i	$-0.755245\pi$
$761$	−2.09687e8	+	1.21063e8i	−0.475792	+	0.274698i	0.242869	+	0.970059i	$0.421911\pi$
$762$		0			0
$763$	1.99186e8	+	3.89177e8i	0.448420	+	0.876139i
$764$		0			0
$765$	−8.60637e7	−	1.49067e8i	−0.192236	−	0.332963i
$766$		0			0
$767$	4.37669e8	−	7.58065e8i	0.969973	−	1.68004i
$768$		0			0
$769$			2.23515e8i			0.491505i	0.969333	+	0.245753i	$0.0790351\pi$
$769$			2.23515e8i			0.491505i	−0.969333	+	0.245753i	$0.920965\pi$
$770$		0			0
$771$	−1.34742e7			−0.0293995
$772$		0			0
$773$	1.95041e8	+	1.12607e8i	0.422266	+	0.243796i	0.696047	−	0.717997i	$-0.254941\pi$
$773$	1.95041e8	+	1.12607e8i	0.422266	+	0.243796i	−0.273780	+	0.961792i	$0.588274\pi$
$774$		0			0
$775$	5.04246e8	−	2.91126e8i	1.08327	−	0.625427i
$776$		0			0
$777$	−4.02678e8	−	2.60464e8i	−0.858411	−	0.555246i
$778$		0			0
$779$	1.02344e7	+	1.77266e7i	0.0216497	+	0.0374984i
$780$		0			0
$781$	−3.36721e7	+	5.83217e7i	−0.0706832	+	0.122427i
$782$		0			0
$783$			7.08241e7i			0.147535i
$784$		0			0
$785$	6.18341e8			1.27826
$786$		0			0
$787$	4.82030e8	+	2.78300e8i	0.988895	+	0.570939i	0.904944	−	0.425531i	$-0.139913\pi$
$787$	4.82030e8	+	2.78300e8i	0.988895	+	0.570939i	0.0839513	+	0.996470i	$0.473246\pi$
$788$		0			0
$789$	3.16655e8	−	1.82821e8i	0.644696	−	0.372216i
$790$		0			0
$791$	2.11861e8	−	3.27538e8i	0.428078	−	0.661808i
$792$		0			0
$793$	5.64290e8	+	9.77379e8i	1.13157	+	1.95994i
$794$		0			0
$795$	1.76562e8	−	3.05814e8i	0.351395	−	0.608634i
$796$		0			0
$797$			1.25601e6i			0.00248095i	0.999999	+	0.00124048i	$0.000394856\pi$
$797$			1.25601e6i			0.00248095i	−0.999999	+	0.00124048i	$0.999605\pi$
$798$		0			0
$799$	−2.64099e8			−0.517757
$800$		0			0
$801$	−2.38580e8	−	1.37744e8i	−0.464233	−	0.268025i
$802$		0			0
$803$	1.37763e8	−	7.95377e7i	0.266065	−	0.153612i
$804$		0			0
$805$	1.27687e9	−	6.53518e8i	2.44770	−	1.25277i
$806$		0			0
$807$	1.09615e8	+	1.89858e8i	0.208568	+	0.361251i
$808$		0			0
$809$	−8.24366e7	+	1.42784e8i	−0.155695	+	0.269672i	−0.933312	−	0.359067i	$-0.883095\pi$
$809$	−8.24366e7	+	1.42784e8i	−0.155695	+	0.269672i	0.777617	+	0.628738i	$0.216428\pi$
$810$		0			0
$811$		−	1.67902e8i		−	0.314771i	−0.987537	−	0.157385i	$-0.949694\pi$
$811$		−	1.67902e8i		−	0.314771i	0.987537	−	0.157385i	$-0.0503064\pi$
$812$		0			0
$813$	−1.95365e7			−0.0363559
$814$		0			0
$815$	1.66905e8	+	9.63626e7i	0.308316	+	0.178006i
$816$		0			0
$817$	−6.30399e6	+	3.63961e6i	−0.0115598	+	0.00667403i
$818$		0			0
$819$	1.49905e7	−	2.96326e8i	0.0272876	−	0.539408i
$820$		0			0
$821$	7.86482e7	+	1.36223e8i	0.142121	+	0.246161i	0.928295	−	0.371844i	$-0.121274\pi$
$821$	7.86482e7	+	1.36223e8i	0.142121	+	0.246161i	−0.786174	+	0.618005i	$0.787941\pi$
$822$		0			0
$823$	−2.44122e8	+	4.22832e8i	−0.437933	+	0.758522i	−0.997530	−	0.0702428i	$-0.977623\pi$
$823$	−2.44122e8	+	4.22832e8i	−0.437933	+	0.758522i	0.559597	+	0.828765i	$0.310956\pi$
$824$		0			0
$825$		−	4.33393e8i		−	0.771828i
$826$		0			0
$827$	3.44930e8			0.609839			0.304919	−	0.952378i	$-0.401370\pi$
$827$	3.44930e8			0.609839			0.304919	+	0.952378i	$0.401370\pi$
$828$		0			0
$829$	1.72654e8	+	9.96817e7i	0.303049	+	0.174965i	0.643812	−	0.765184i	$-0.277352\pi$
$829$	1.72654e8	+	9.96817e7i	0.303049	+	0.174965i	−0.340763	+	0.940149i	$0.610685\pi$
$830$		0			0
$831$	4.38230e8	−	2.53012e8i	0.763658	−	0.440898i
$832$		0			0
$833$	−3.99891e7	+	3.94231e8i	−0.0691842	+	0.682050i
$834$		0			0
$835$	−3.67301e8	−	6.36184e8i	−0.630903	−	1.09276i
$836$		0			0
$837$	3.85523e7	−	6.67746e7i	0.0657467	−	0.113877i
$838$		0			0
$839$			6.44765e8i			1.09173i	0.837873	+	0.545865i	$0.183799\pi$
$839$			6.44765e8i			1.09173i	−0.837873	+	0.545865i	$0.816201\pi$
$840$		0			0
$841$	−2.45245e8			−0.412300
$842$		0			0
$843$	−2.44420e8	−	1.41116e8i	−0.407995	−	0.235556i
$844$		0			0
$845$	−1.42888e9	+	8.24964e8i	−2.36824	+	1.36730i
$846$		0			0
$847$	−2.83263e8	−	1.43297e7i	−0.466165	−	0.0235824i
$848$		0			0
$849$	−1.06244e8	−	1.84021e8i	−0.173613	−	0.300707i
$850$		0			0
$851$	−8.91752e8	+	1.54456e9i	−1.44696	+	2.50620i
$852$		0			0
$853$			6.39556e8i			1.03046i	0.857052	+	0.515231i	$0.172294\pi$
$853$			6.39556e8i			1.03046i	−0.857052	+	0.515231i	$0.827706\pi$
$854$		0			0
$855$	9.56901e7			0.153098
$856$		0			0
$857$	7.38721e7	+	4.26501e7i	0.117365	+	0.0677606i	0.557533	−	0.830155i	$-0.311748\pi$
$857$	7.38721e7	+	4.26501e7i	0.117365	+	0.0677606i	−0.440168	+	0.897915i	$0.645081\pi$
$858$		0			0
$859$	−5.01049e8	+	2.89281e8i	−0.790499	+	0.456395i	−0.840138	−	0.542373i	$-0.817526\pi$
$859$	−5.01049e8	+	2.89281e8i	−0.790499	+	0.456395i	0.0496394	+	0.998767i	$0.484193\pi$
$860$		0			0
$861$	2.66305e7	+	5.20316e7i	0.0417224	+	0.0815187i
$862$		0			0
$863$	2.74923e8	+	4.76180e8i	0.427738	+	0.740864i	0.996672	−	0.0815193i	$-0.0259772\pi$
$863$	2.74923e8	+	4.76180e8i	0.427738	+	0.740864i	−0.568934	+	0.822383i	$0.692644\pi$
$864$		0			0
$865$	7.23118e7	−	1.25248e8i	0.111728	−	0.193518i
$866$		0			0
$867$			1.99430e8i			0.306008i
$868$		0			0
$869$	1.77802e8			0.270943
$870$		0			0
$871$	1.49286e9	+	8.61901e8i	2.25925	+	1.30438i
$872$		0			0
$873$	2.90765e8	−	1.67873e8i	0.437018	−	0.252312i
$874$		0			0
$875$	7.86187e8	+	5.08529e8i	1.17355	+	0.759087i
$876$		0			0
$877$	5.43435e8	+	9.41257e8i	0.805655	+	1.39543i	0.915848	+	0.401525i	$0.131520\pi$
$877$	5.43435e8	+	9.41257e8i	0.805655	+	1.39543i	−0.110194	+	0.993910i	$0.535147\pi$
$878$		0			0
$879$	4.45316e7	−	7.71309e7i	0.0655694	−	0.113570i
$880$		0			0
$881$		−	1.50334e8i		−	0.219852i	−0.993940	−	0.109926i	$-0.964939\pi$
$881$		−	1.50334e8i		−	0.219852i	0.993940	−	0.109926i	$-0.0350614\pi$
$882$		0			0
$883$	6.63235e8			0.963352			0.481676	−	0.876349i	$-0.340028\pi$
$883$	6.63235e8			0.963352			0.481676	+	0.876349i	$0.340028\pi$
$884$		0			0
$885$	6.98142e8	+	4.03072e8i	1.00720	+	0.581505i
$886$		0			0
$887$	3.83371e8	−	2.21339e8i	0.549349	−	0.317167i	−0.199510	−	0.979896i	$-0.563935\pi$
$887$	3.83371e8	−	2.21339e8i	0.549349	−	0.317167i	0.748859	+	0.662729i	$0.230602\pi$
$888$		0			0
$889$	1.29815e8	−	2.00694e8i	0.184765	−	0.285647i
$890$		0			0
$891$	−2.86960e7	−	4.97029e7i	−0.0405684	−	0.0702666i
$892$		0			0
$893$	7.34097e7	−	1.27149e8i	0.103086	−	0.178550i
$894$		0			0
$895$			1.66378e7i			0.0232074i
$896$		0			0
$897$	−1.10342e9			−1.52885
$898$		0			0
$899$	−3.29590e8	−	1.90289e8i	−0.453623	−	0.261899i
$900$		0			0
$901$	3.14183e8	−	1.81393e8i	0.429544	−	0.247998i
$902$		0			0
$903$	−1.85037e7	+	9.47042e6i	−0.0251301	+	0.0128619i
$904$		0			0
$905$	4.50731e8	+	7.80690e8i	0.608096	+	1.05325i
$906$		0			0
$907$	−2.05909e8	+	3.56644e8i	−0.275964	+	0.477984i	−0.970378	−	0.241592i	$-0.922330\pi$
$907$	−2.05909e8	+	3.56644e8i	−0.275964	+	0.477984i	0.694414	+	0.719576i	$0.255664\pi$
$908$		0			0
$909$		−	4.07566e8i		−	0.542633i
$910$		0			0
$911$	8.39816e8			1.11078			0.555391	−	0.831589i	$-0.312568\pi$
$911$	8.39816e8			1.11078			0.555391	+	0.831589i	$0.312568\pi$
$912$		0			0
$913$	−3.07138e8	−	1.77326e8i	−0.403572	−	0.233002i
$914$		0			0
$915$	−9.00120e8	+	5.19685e8i	−1.17500	+	0.678386i
$916$		0			0
$917$	4.36189e7	−	8.62237e8i	0.0565674	−	1.11820i
$918$		0			0
$919$	2.39254e8	+	4.14400e8i	0.308257	+	0.533917i	0.977981	−	0.208693i	$-0.0669210\pi$
$919$	2.39254e8	+	4.14400e8i	0.308257	+	0.533917i	−0.669724	+	0.742610i	$0.733588\pi$
$920$		0			0
$921$	5.49460e7	−	9.51692e7i	0.0703326	−	0.121820i
$922$		0			0
$923$			2.46652e8i			0.313675i
$924$		0			0
$925$	−2.56566e9			−3.24171
$926$		0			0
$927$	2.58767e8	+	1.49399e8i	0.324840	+	0.187546i
$928$		0			0
$929$	7.68390e8	−	4.43630e8i	0.958373	−	0.553317i	0.0627011	−	0.998032i	$-0.480029\pi$
$929$	7.68390e8	−	4.43630e8i	0.958373	−	0.553317i	0.895672	+	0.444715i	$0.146695\pi$
$930$		0			0
$931$	−1.78686e8	−	1.28834e8i	−0.221432	−	0.159655i
$932$		0			0
$933$	−4.05802e7	−	7.02869e7i	−0.0499654	−	0.0865426i
$934$		0			0
$935$	3.44233e8	−	5.96229e8i	0.421132	−	0.729421i
$936$		0			0
$937$		−	1.19816e9i		−	1.45646i	−0.685334	−	0.728229i	$-0.740344\pi$
$937$		−	1.19816e9i		−	1.45646i	0.685334	−	0.728229i	$-0.259656\pi$
$938$		0			0
$939$	−4.07707e8			−0.492437
$940$		0			0
$941$	9.75385e8	+	5.63139e8i	1.17060	+	0.675844i	0.953820	−	0.300378i	$-0.0971127\pi$
$941$	9.75385e8	+	5.63139e8i	1.17060	+	0.675844i	0.216776	+	0.976221i	$0.430446\pi$
$942$		0			0
$943$	1.88251e8	−	1.08687e8i	0.224493	−	0.129611i
$944$		0			0
$945$	2.72902e8	+	1.38056e7i	0.323378	+	0.0163591i
$946$		0			0
$947$	−3.53545e8	−	6.12359e8i	−0.416289	−	0.721034i	0.579274	−	0.815133i	$-0.303336\pi$
$947$	−3.53545e8	−	6.12359e8i	−0.416289	−	0.721034i	−0.995563	+	0.0940990i	$0.970003\pi$
$948$		0			0
$949$	2.91312e8	−	5.04567e8i	0.340847	−	0.590364i
$950$		0			0
$951$		−	1.02615e8i		−	0.119308i
$952$		0			0
$953$	2.87519e8			0.332191			0.166095	−	0.986110i	$-0.446884\pi$
$953$	2.87519e8			0.332191			0.166095	+	0.986110i	$0.446884\pi$
$954$		0			0
$955$	−1.72055e9	−	9.93361e8i	−1.97541	−	1.14050i
$956$		0			0
$957$	−2.45327e8	+	1.41639e8i	−0.279904	+	0.161603i
$958$		0			0
$959$	−9.41899e7	−	1.84032e8i	−0.106794	−	0.208659i
$960$		0			0
$961$	−2.36589e8	−	4.09784e8i	−0.266578	−	0.461726i
$962$		0			0
$963$	2.14881e8	−	3.72186e8i	0.240614	−	0.416755i
$964$		0			0
$965$		−	1.97025e9i		−	2.19250i
$966$		0			0
$967$	−1.48175e9			−1.63869			−0.819343	−	0.573304i	$-0.805661\pi$
$967$	−1.48175e9			−1.63869			−0.819343	+	0.573304i	$0.805661\pi$
$968$		0			0
$969$	8.51378e7	+	4.91543e7i	0.0935731	+	0.0540245i
$970$		0			0
$971$	1.50641e9	−	8.69728e8i	1.64546	−	0.950005i	0.666610	−	0.745406i	$-0.267745\pi$
$971$	1.50641e9	−	8.69728e8i	1.64546	−	0.950005i	0.978846	−	0.204598i	$-0.0655888\pi$
$972$		0			0
$973$	−1.05658e9	−	6.83430e8i	−1.14701	−	0.741918i
$974$		0			0
$975$	−7.93665e8	−	1.37467e9i	−0.856295	−	1.48315i
$976$		0			0
$977$	−5.55082e8	+	9.61429e8i	−0.595214	+	1.03094i	0.398303	+	0.917254i	$0.369599\pi$
$977$	−5.55082e8	+	9.61429e8i	−0.595214	+	1.03094i	−0.993517	+	0.113687i	$0.963734\pi$
$978$		0			0
$979$		−	1.10188e9i		−	1.17432i
$980$		0			0
$981$	3.09728e8			0.328075
$982$		0			0
$983$	−9.45759e8	−	5.46034e8i	−0.995681	−	0.574857i	−0.0887133	−	0.996057i	$-0.528275\pi$
$983$	−9.45759e8	−	5.46034e8i	−0.995681	−	0.574857i	−0.906968	+	0.421201i	$0.861609\pi$
$984$		0			0
$985$	7.03827e8	−	4.06355e8i	0.736473	−	0.425203i
$986$		0			0
$987$	2.27704e8	−	3.52031e8i	0.236820	−	0.366125i
$988$		0			0
$989$	3.86516e7	+	6.69465e7i	0.0399557	+	0.0692052i
$990$		0			0
$991$	−1.15604e8	+	2.00232e8i	−0.118782	+	0.205737i	−0.919285	−	0.393591i	$-0.871232\pi$
$991$	−1.15604e8	+	2.00232e8i	−0.118782	+	0.205737i	0.800503	+	0.599329i	$0.204566\pi$
$992$		0			0
$993$			2.30310e8i			0.235215i
$994$		0			0
$995$	−2.09181e9			−2.12350
$996$		0			0
$997$	5.60304e8	+	3.23492e8i	0.565377	+	0.326421i	0.755301	−	0.655378i	$-0.227491\pi$
$997$	5.60304e8	+	3.23492e8i	0.565377	+	0.326421i	−0.189924	+	0.981799i	$0.560824\pi$
$998$		0			0
$999$	−2.94238e8	+	1.69878e8i	−0.295122	+	0.170389i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	336.7.bh.c.241.4		8
4.3	odd	2		42.7.g.b.31.4	yes	8
7.5	odd	6	inner	336.7.bh.c.145.4		8
12.11	even	2		126.7.n.b.73.1		8
28.3	even	6		294.7.c.a.97.3		8
28.11	odd	6		294.7.c.a.97.2		8
28.19	even	6		42.7.g.b.19.4	✓	8
28.23	odd	6		294.7.g.b.19.3		8
28.27	even	2		294.7.g.b.31.3		8
84.47	odd	6		126.7.n.b.19.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
42.7.g.b.19.4	✓	8	28.19	even	6
42.7.g.b.31.4	yes	8	4.3	odd	2
126.7.n.b.19.1		8	84.47	odd	6
126.7.n.b.73.1		8	12.11	even	2
294.7.c.a.97.2		8	28.11	odd	6
294.7.c.a.97.3		8	28.3	even	6
294.7.g.b.19.3		8	28.23	odd	6
294.7.g.b.31.3		8	28.27	even	2
336.7.bh.c.145.4		8	7.5	odd	6	inner
336.7.bh.c.241.4		8	1.1	even	1	trivial

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:	\( k \)	\(=\)	\( 7 \)
Character orbit:	\([\chi]\)	\(=\)	336.bh (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-13.5000 - 7.79423i) q^{3} +(182.133 - 105.155i) q^{5} +(-186.289 + 288.003i) q^{7} +(121.500 + 210.444i) q^{9} +O(q^{10})\)	\(q+(-13.5000 - 7.79423i) q^{3} +(182.133 - 105.155i) q^{5} +(-186.289 + 288.003i) q^{7} +(121.500 + 210.444i) q^{9} +(-485.969 + 841.723i) q^{11} +3559.78i q^{13} -3278.39 q^{15} +(-2916.87 - 1684.05i) q^{17} +(1621.56 - 936.209i) q^{19} +(4759.66 - 2436.06i) q^{21} +(-9942.27 - 17220.5i) q^{23} +(14302.4 - 24772.6i) q^{25} -3788.00i q^{27} -18697.0 q^{29} +(17628.0 + 10177.5i) q^{31} +(13121.2 - 7575.51i) q^{33} +(-3644.56 + 72043.9i) q^{35} +(-44846.5 - 77676.4i) q^{37} +(27745.8 - 48057.1i) q^{39} +10931.8i q^{41} -3887.60 q^{43} +(44258.3 + 25552.5i) q^{45} +(67906.5 - 39205.8i) q^{47} +(-48242.0 - 107303. i) q^{49} +(26251.8 + 45469.4i) q^{51} +(-53856.2 + 93281.7i) q^{53} +204407. i q^{55} -29188.1 q^{57} +(-212952. - 122948. i) q^{59} +(274561. - 158518. i) q^{61} +(-83242.6 - 4211.08i) q^{63} +(374327. + 648354. i) q^{65} +(242122. - 419367. i) q^{67} +309969. i q^{69} +69288.4 q^{71} +(-141741. - 81834.1i) q^{73} +(-386166. + 222953. i) q^{75} +(-151888. - 296764. i) q^{77} +(-91467.9 - 158427. i) q^{79} +(-29524.5 + 51137.9i) q^{81} +364891. i q^{83} -708343. q^{85} +(252410. + 145729. i) q^{87} +(-981810. + 566848. i) q^{89} +(-1.02523e6 - 663148. i) q^{91} +(-158652. - 274793. i) q^{93} +(196893. - 341029. i) q^{95} -1.38167e6i q^{97} -236181. q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 108 q^{3} + 210 q^{5} + 608 q^{7} + 972 q^{9}+O(q^{10}) \) 8 * q - 108 * q^3 + 210 * q^5 + 608 * q^7 + 972 * q^9	\( 8 q - 108 q^{3} + 210 q^{5} + 608 q^{7} + 972 q^{9} - 2058 q^{11} - 3780 q^{15} - 11244 q^{17} - 21834 q^{19} - 4482 q^{21} - 15504 q^{23} - 6550 q^{25} + 35316 q^{29} + 51060 q^{31} + 55566 q^{33} - 71460 q^{35} + 20282 q^{37} + 101682 q^{39} - 387812 q^{43} + 51030 q^{45} + 55212 q^{47} - 277780 q^{49} + 101196 q^{51} - 336174 q^{53} + 393012 q^{57} + 560454 q^{59} + 850728 q^{61} - 26730 q^{63} + 826380 q^{65} + 947882 q^{67} - 147192 q^{71} - 533034 q^{73} + 176850 q^{75} - 1848102 q^{77} + 6260 q^{79} - 236196 q^{81} + 560040 q^{85} - 476766 q^{87} + 413460 q^{89} - 256074 q^{91} - 459540 q^{93} + 170880 q^{95} - 1000188 q^{99}+O(q^{100}) \) 8 * q - 108 * q^3 + 210 * q^5 + 608 * q^7 + 972 * q^9 - 2058 * q^11 - 3780 * q^15 - 11244 * q^17 - 21834 * q^19 - 4482 * q^21 - 15504 * q^23 - 6550 * q^25 + 35316 * q^29 + 51060 * q^31 + 55566 * q^33 - 71460 * q^35 + 20282 * q^37 + 101682 * q^39 - 387812 * q^43 + 51030 * q^45 + 55212 * q^47 - 277780 * q^49 + 101196 * q^51 - 336174 * q^53 + 393012 * q^57 + 560454 * q^59 + 850728 * q^61 - 26730 * q^63 + 826380 * q^65 + 947882 * q^67 - 147192 * q^71 - 533034 * q^73 + 176850 * q^75 - 1848102 * q^77 + 6260 * q^79 - 236196 * q^81 + 560040 * q^85 - 476766 * q^87 + 413460 * q^89 - 256074 * q^91 - 459540 * q^93 + 170880 * q^95 - 1000188 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more