LMFDB - Embedded newform 112.8.i.d.81.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [112,8,Mod(65,112)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(112, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2]))

N = Newforms(chi, 8, names="a")

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

chi := DirichletCharacter("112.65");

S:= CuspForms(chi, 8);

N := Newforms(S);

Level:	$ N $	$=$	$ 112 = 2^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	112.i (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:	$34.9871228542$
Analytic rank:	$0$
Dimension:	$10$
Relative dimension:	$5$ over $\Q(\zeta_{3})$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - x^{9} + 342 x^{8} + 2165 x^{7} + 113605 x^{6} + 319380 x^{5} + 1438128 x^{4} + 1705752 x^{3} + \cdots + 23619600 $ x^10 - x^9 + 342x^8 + 2165x^7 + 113605x^6 + 319380x^5 + 1438128x^4 + 1705752x^3 + 7544016x^2 + 8573040x + 23619600
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{4}\cdot 7^{5} $
Twist minimal:	no (minimal twist has level 28)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			81.1
Root			$-1.38371 - 2.39666i$ of defining polynomial
Character	$\chi$	$=$	112.81
Dual form			112.8.i.d.65.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-37.9099 + 65.6620i) q^{3} +(274.614 + 475.646i) q^{5} +(-907.484 + 4.07000i) q^{7} +(-1780.83 - 3084.49i) q^{9} +(-2225.01 + 3853.83i) q^{11} -3918.83 q^{13} -41642.4 q^{15} +(-3001.55 + 5198.83i) q^{17} +(1058.25 + 1832.95i) q^{19} +(34135.4 - 59741.4i) q^{21} +(32443.3 + 56193.4i) q^{23} +(-111763. + 193580. i) q^{25} +104226. q^{27} +118135. q^{29} +(42724.4 - 74000.8i) q^{31} +(-168700. - 292197. i) q^{33} +(-251144. - 430523. i) q^{35} +(13176.1 + 22821.6i) q^{37} +(148563. - 257318. i) q^{39} -459420. q^{41} +511775. q^{43} +(978081. - 1.69409e6i) q^{45} +(-76217.4 - 132012. i) q^{47} +(823510. - 7386.92i) q^{49} +(-227577. - 394175. i) q^{51} +(30522.5 - 52866.5i) q^{53} -2.44408e6 q^{55} -160473. q^{57} +(1.03829e6 - 1.79837e6i) q^{59} +(475849. + 824194. i) q^{61} +(1.62863e6 + 2.79187e6i) q^{63} +(-1.07617e6 - 1.86397e6i) q^{65} +(-558691. + 967681. i) q^{67} -4.91969e6 q^{69} +4.62273e6 q^{71} +(-1.25326e6 + 2.17070e6i) q^{73} +(-8.47389e6 - 1.46772e7i) q^{75} +(2.00348e6 - 3.50635e6i) q^{77} +(1.81828e6 + 3.14935e6i) q^{79} +(-56543.1 + 97935.5i) q^{81} -194336. q^{83} -3.29707e6 q^{85} +(-4.47850e6 + 7.75699e6i) q^{87} +(-2.85298e6 - 4.94150e6i) q^{89} +(3.55627e6 - 15949.7i) q^{91} +(3.23936e6 + 5.61073e6i) q^{93} +(-581222. + 1.00671e6i) q^{95} +1.01861e7 q^{97} +1.58495e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 27 q^{3} + 249 q^{5} - 332 q^{7} - 5702 q^{9} - 6399 q^{11} - 26988 q^{13} - 19294 q^{15} + 3609 q^{17} + 12403 q^{19} + 16099 q^{21} + 13959 q^{23} - 162364 q^{25} - 161550 q^{27} + 26148 q^{29}+ \cdots + 119277812 q^{99}+O(q^{100}) $ 10 * q - 27 * q^3 + 249 * q^5 - 332 * q^7 - 5702 * q^9 - 6399 * q^11 - 26988 * q^13 - 19294 * q^15 + 3609 * q^17 + 12403 * q^19 + 16099 * q^21 + 13959 * q^23 - 162364 * q^25 - 161550 * q^27 + 26148 * q^29 + 20181 * q^31 + 264239 * q^33 - 791715 * q^35 - 54763 * q^37 - 4750 * q^39 - 1824468 * q^41 + 1938424 * q^43 + 1936430 * q^45 - 1217253 * q^47 + 1722058 * q^49 + 1353871 * q^51 - 2349159 * q^53 - 3123494 * q^55 - 9823194 * q^57 - 2188611 * q^59 + 6107445 * q^61 - 10275990 * q^63 - 3022014 * q^65 + 3171581 * q^67 - 11255938 * q^69 + 18938976 * q^71 + 8034853 * q^73 - 29173588 * q^75 + 24100191 * q^77 + 7589559 * q^79 - 9846485 * q^81 + 13452408 * q^83 - 48717670 * q^85 - 30033142 * q^87 + 29864757 * q^89 - 34267512 * q^91 - 20217029 * q^93 + 25812009 * q^95 + 285580 * q^97 + 119277812 * q^99

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/112\mathbb{Z}\right)^\times$.

$n$	$15$	$17$	$85$
$\chi(n)$	$1$	$e\left(\frac{2}{3}\right)$	$1$

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
$3$	−37.9099	+	65.6620i	−0.810641	+	1.40407i	0.101774	+	0.994808i	$0.467548\pi$
$3$	−37.9099	+	65.6620i	−0.810641	+	1.40407i	−0.912416	+	0.409265i	$0.865785\pi$
$4$		0			0
$5$	274.614	+	475.646i	0.982489	+	1.70172i	0.652601	+	0.757702i	$0.273678\pi$
$5$	274.614	+	475.646i	0.982489	+	1.70172i	0.329888	+	0.944020i	$0.392989\pi$
$6$		0			0
$7$	−907.484	+	4.07000i	−0.999990	+	0.00448489i
$7$	−907.484	+	4.07000i	−0.999990	+	0.00448489i
$8$		0			0
$9$	−1780.83	−	3084.49i	−0.814279	−	1.41037i
$10$		0			0
$11$	−2225.01	+	3853.83i	−0.504032	+	0.873009i	0.495957	+	0.868347i	$0.334817\pi$
$11$	−2225.01	+	3853.83i	−0.504032	+	0.873009i	−0.999989	+	0.00466169i	$0.998516\pi$
$12$		0			0
$13$	−3918.83			−0.494715			−0.247357	−	0.968924i	$-0.579562\pi$
$13$	−3918.83			−0.494715			−0.247357	+	0.968924i	$0.579562\pi$
$14$		0			0
$15$	−41642.4			−3.18579
$16$		0			0
$17$	−3001.55	+	5198.83i	−0.148175	+	0.256646i	−0.930553	−	0.366158i	$-0.880673\pi$
$17$	−3001.55	+	5198.83i	−0.148175	+	0.256646i	0.782378	+	0.622804i	$0.214006\pi$
$18$		0			0
$19$	1058.25	+	1832.95i	0.0353958	+	0.0613073i	0.883181	−	0.469033i	$-0.155397\pi$
$19$	1058.25	+	1832.95i	0.0353958	+	0.0613073i	−0.847785	+	0.530340i	$0.822064\pi$
$20$		0			0
$21$	34135.4	−	59741.4i	0.804336	−	1.40769i
$22$		0			0
$23$	32443.3	+	56193.4i	0.556003	+	0.963026i	0.997825	+	0.0659227i	$0.0209991\pi$
$23$	32443.3	+	56193.4i	0.556003	+	0.963026i	−0.441822	+	0.897103i	$0.645668\pi$
$24$		0			0
$25$	−111763.	+	193580.i	−1.43057	+	2.47782i
$26$		0			0
$27$	104226.			1.01907
$28$		0			0
$29$	118135.			0.899470			0.449735	−	0.893162i	$-0.351518\pi$
$29$	118135.			0.899470			0.449735	+	0.893162i	$0.351518\pi$
$30$		0			0
$31$	42724.4	−	74000.8i	0.257579	−	0.446139i	−0.708014	−	0.706198i	$-0.750409\pi$
$31$	42724.4	−	74000.8i	0.257579	−	0.446139i	0.965593	+	0.260059i	$0.0837420\pi$
$32$		0			0
$33$	−168700.	−	292197.i	−0.817178	−	1.41539i
$34$		0			0
$35$	−251144.	−	430523.i	−0.990112	−	1.69730i
$36$		0			0
$37$	13176.1	+	22821.6i	0.0427641	+	0.0740696i	0.886615	−	0.462508i	$-0.153050\pi$
$37$	13176.1	+	22821.6i	0.0427641	+	0.0740696i	−0.843851	+	0.536577i	$0.819717\pi$
$38$		0			0
$39$	148563.	−	257318.i	0.401036	−	0.694615i
$40$		0			0
$41$	−459420.			−1.04104			−0.520519	−	0.853850i	$-0.674262\pi$
$41$	−459420.			−1.04104			−0.520519	+	0.853850i	$0.674262\pi$
$42$		0			0
$43$	511775.			0.981611			0.490806	−	0.871269i	$-0.336703\pi$
$43$	511775.			0.981611			0.490806	+	0.871269i	$0.336703\pi$
$44$		0			0
$45$	978081.	−	1.69409e6i	1.60004	−	2.77135i
$46$		0			0
$47$	−76217.4	−	132012.i	−0.107081	−	0.185470i	0.807506	−	0.589860i	$-0.200817\pi$
$47$	−76217.4	−	132012.i	−0.107081	−	0.185470i	−0.914586	+	0.404390i	$0.867484\pi$
$48$		0			0
$49$	823510.	−	7386.92i	0.999960	−	0.00896969i
$50$		0			0
$51$	−227577.	−	394175.i	−0.240233	−	0.416096i
$52$		0			0
$53$	30522.5	−	52866.5i	0.0281614	−	0.0487770i	−0.851601	−	0.524190i	$-0.824368\pi$
$53$	30522.5	−	52866.5i	0.0281614	−	0.0487770i	0.879763	+	0.475413i	$0.157701\pi$
$54$		0			0
$55$	−2.44408e6			−1.98082
$56$		0			0
$57$	−160473.			−0.114773
$58$		0			0
$59$	1.03829e6	−	1.79837e6i	0.658167	−	1.13998i	−0.322923	−	0.946425i	$-0.604666\pi$
$59$	1.03829e6	−	1.79837e6i	0.658167	−	1.13998i	0.981090	−	0.193553i	$-0.0620011\pi$
$60$		0			0
$61$	475849.	+	824194.i	0.268420	+	0.464917i	0.968454	−	0.249193i	$-0.0801652\pi$
$61$	475849.	+	824194.i	0.268420	+	0.464917i	−0.700034	+	0.714109i	$0.746832\pi$
$62$		0			0
$63$	1.62863e6	+	2.79187e6i	0.820596	+	1.40671i
$64$		0			0
$65$	−1.07617e6	−	1.86397e6i	−0.486052	−	0.841866i
$66$		0			0
$67$	−558691.	+	967681.i	−0.226939	+	0.393071i	−0.956900	−	0.290419i	$-0.906205\pi$
$67$	−558691.	+	967681.i	−0.226939	+	0.393071i	0.729960	+	0.683490i	$0.239539\pi$
$68$		0			0
$69$	−4.91969e6			−1.80288
$70$		0			0
$71$	4.62273e6			1.53283			0.766415	−	0.642346i	$-0.222039\pi$
$71$	4.62273e6			1.53283			0.766415	+	0.642346i	$0.222039\pi$
$72$		0			0
$73$	−1.25326e6	+	2.17070e6i	−0.377059	+	0.653086i	−0.990633	−	0.136552i	$-0.956398\pi$
$73$	−1.25326e6	+	2.17070e6i	−0.377059	+	0.653086i	0.613574	+	0.789637i	$0.289731\pi$
$74$		0			0
$75$	−8.47389e6	−	1.46772e7i	−2.31936	−	4.01725i
$76$		0			0
$77$	2.00348e6	−	3.50635e6i	0.500111	−	0.875260i
$78$		0			0
$79$	1.81828e6	+	3.14935e6i	0.414921	+	0.718664i	0.995420	−	0.0955966i	$-0.0304759\pi$
$79$	1.81828e6	+	3.14935e6i	0.414921	+	0.718664i	−0.580499	+	0.814261i	$0.697143\pi$
$80$		0			0
$81$	−56543.1	+	97935.5i	−0.0118218	+	0.0204759i
$82$		0			0
$83$	−194336.			−0.0373060			−0.0186530	−	0.999826i	$-0.505938\pi$
$83$	−194336.			−0.0373060			−0.0186530	+	0.999826i	$0.505938\pi$
$84$		0			0
$85$	−3.29707e6			−0.582320
$86$		0			0
$87$	−4.47850e6	+	7.75699e6i	−0.729147	+	1.26292i
$88$		0			0
$89$	−2.85298e6	−	4.94150e6i	−0.428977	−	0.743009i	0.567806	−	0.823162i	$-0.307792\pi$
$89$	−2.85298e6	−	4.94150e6i	−0.428977	−	0.743009i	−0.996783	+	0.0801532i	$0.974459\pi$
$90$		0			0
$91$	3.55627e6	−	15949.7i	0.494710	−	0.00221874i
$92$		0			0
$93$	3.23936e6	+	5.61073e6i	0.417608	+	0.723318i
$94$		0			0
$95$	−581222.	+	1.00671e6i	−0.0695520	+	0.120468i
$96$		0			0
$97$	1.01861e7			1.13320			0.566601	−	0.823992i	$-0.308258\pi$
$97$	1.01861e7			1.13320			0.566601	+	0.823992i	$0.308258\pi$
$98$		0			0
$99$	1.58495e7			1.64169
$100$		0			0
$101$	7.53754e6	−	1.30554e7i	0.727956	−	1.26086i	−0.229790	−	0.973240i	$-0.573804\pi$
$101$	7.53754e6	−	1.30554e7i	0.727956	−	1.26086i	0.957746	−	0.287616i	$-0.0928627\pi$
$102$		0			0
$103$	−1.82282e6	−	3.15722e6i	−0.164367	−	0.284692i	0.772063	−	0.635546i	$-0.219225\pi$
$103$	−1.82282e6	−	3.15722e6i	−0.164367	−	0.284692i	−0.936430	+	0.350854i	$0.885891\pi$
$104$		0			0
$105$	3.77898e7	−	169485.i	3.18575	−	0.0142879i
$106$		0			0
$107$	4.56669e6	+	7.90974e6i	0.360378	+	0.624193i	0.988023	−	0.154307i	$-0.0493144\pi$
$107$	4.56669e6	+	7.90974e6i	0.360378	+	0.624193i	−0.627645	+	0.778500i	$0.715981\pi$
$108$		0			0
$109$	−1.20607e7	+	2.08898e7i	−0.892033	+	1.54505i	−0.0545997	+	0.998508i	$0.517388\pi$
$109$	−1.20607e7	+	2.08898e7i	−0.892033	+	1.54505i	−0.837434	+	0.546539i	$0.815945\pi$
$110$		0			0
$111$	−1.99801e6			−0.138665
$112$		0			0
$113$	−1.51746e7			−0.989337			−0.494668	−	0.869082i	$-0.664710\pi$
$113$	−1.51746e7			−0.989337			−0.494668	+	0.869082i	$0.664710\pi$
$114$		0			0
$115$	−1.78188e7	+	3.08630e7i	−1.09253	+	1.89232i
$116$		0			0
$117$	6.97876e6	+	1.20876e7i	0.402836	+	0.697732i
$118$		0			0
$119$	2.70269e6	−	4.73007e6i	0.147022	−	0.257308i
$120$		0			0
$121$	−157767.	−	273260.i	−0.00809593	−	0.0140226i
$122$		0			0
$123$	1.74166e7	−	3.01664e7i	0.843909	−	1.46169i
$124$		0			0
$125$	−7.98587e7			−3.65710
$126$		0			0
$127$	−3.12338e7			−1.35305			−0.676523	−	0.736422i	$-0.736514\pi$
$127$	−3.12338e7			−1.35305			−0.676523	+	0.736422i	$0.736514\pi$
$128$		0			0
$129$	−1.94014e7	+	3.36042e7i	−0.795735	+	1.37825i
$130$		0			0
$131$	4.64512e6	+	8.04559e6i	0.180529	+	0.312686i	0.942061	−	0.335442i	$-0.108886\pi$
$131$	4.64512e6	+	8.04559e6i	0.180529	+	0.312686i	−0.761532	+	0.648128i	$0.775552\pi$
$132$		0			0
$133$	−967806.	−	1.65906e6i	−0.0356704	−	0.0611479i
$134$		0			0
$135$	2.86220e7	+	4.95748e7i	1.00123	+	1.73417i
$136$		0			0
$137$	4.48735e6	−	7.77232e6i	0.149097	−	0.258243i	−0.781797	−	0.623533i	$-0.785697\pi$
$137$	4.48735e6	−	7.77232e6i	0.149097	−	0.258243i	0.930894	+	0.365290i	$0.119030\pi$
$138$		0			0
$139$	−4.36201e7			−1.37764			−0.688819	−	0.724934i	$-0.741870\pi$
$139$	−4.36201e7			−1.37764			−0.688819	+	0.724934i	$0.741870\pi$
$140$		0			0
$141$	1.15576e7			0.347217
$142$		0			0
$143$	8.71944e6	−	1.51025e7i	0.249352	−	0.431890i
$144$		0			0
$145$	3.24416e7	+	5.61905e7i	0.883719	+	1.53065i
$146$		0			0
$147$	−3.07342e7	+	5.43533e7i	−0.798015	+	1.41129i
$148$		0			0
$149$	2.62443e7	+	4.54565e7i	0.649955	+	1.12575i	0.983133	+	0.182891i	$0.0585455\pi$
$149$	2.62443e7	+	4.54565e7i	0.649955	+	1.12575i	−0.333178	+	0.942864i	$0.608121\pi$
$150$		0			0
$151$	2.09853e7	−	3.63477e7i	0.496017	−	0.859127i	−0.503972	−	0.863720i	$-0.668129\pi$
$151$	2.09853e7	−	3.63477e7i	0.496017	−	0.859127i	0.999989	+	0.00459273i	$0.00146192\pi$
$152$		0			0
$153$	2.13810e7			0.482622
$154$		0			0
$155$	4.69309e7			1.01227
$156$		0			0
$157$	1.76464e7	−	3.05645e7i	0.363922	−	0.630331i	−0.624681	−	0.780880i	$-0.714771\pi$
$157$	1.76464e7	−	3.05645e7i	0.363922	−	0.630331i	0.988603	+	0.150549i	$0.0481042\pi$
$158$		0			0
$159$	2.31421e6	+	4.00833e6i	0.0456576	+	0.0790813i
$160$		0			0
$161$	−2.96704e7	−	5.08625e7i	−0.560317	−	0.960522i
$162$		0			0
$163$	2.54191e7	+	4.40271e7i	0.459730	+	0.796276i	0.998946	−	0.0458910i	$-0.0146127\pi$
$163$	2.54191e7	+	4.40271e7i	0.459730	+	0.796276i	−0.539216	+	0.842167i	$0.681279\pi$
$164$		0			0
$165$	9.26549e7	−	1.60483e8i	1.60574	−	2.78122i
$166$		0			0
$167$	−6.77967e7			−1.12642			−0.563211	−	0.826313i	$-0.690434\pi$
$167$	−6.77967e7			−1.12642			−0.563211	+	0.826313i	$0.690434\pi$
$168$		0			0
$169$	−4.73913e7			−0.755257
$170$		0			0
$171$	3.76913e6	−	6.52832e6i	0.0576441	−	0.0998425i
$172$		0			0
$173$	−2.54193e7	−	4.40275e7i	−0.373252	−	0.646492i	0.616812	−	0.787111i	$-0.288424\pi$
$173$	−2.54193e7	−	4.40275e7i	−0.373252	−	0.646492i	−0.990064	+	0.140619i	$0.955091\pi$
$174$		0			0
$175$	1.00636e8	−	1.76125e8i	1.41944	−	2.48421i
$176$		0			0
$177$	7.87229e7	+	1.36352e8i	1.06707	+	1.84823i
$178$		0			0
$179$	−2.33223e7	+	4.03954e7i	−0.303939	+	0.526437i	−0.977024	−	0.213127i	$-0.931635\pi$
$179$	−2.33223e7	+	4.03954e7i	−0.303939	+	0.526437i	0.673086	+	0.739565i	$0.264969\pi$
$180$		0			0
$181$	5.38874e7			0.675480			0.337740	−	0.941239i	$-0.390338\pi$
$181$	5.38874e7			0.675480			0.337740	+	0.941239i	$0.390338\pi$
$182$		0			0
$183$	−7.21576e7			−0.870369
$184$		0			0
$185$	−7.23666e6	+	1.25343e7i	−0.0840306	+	0.145545i
$186$		0			0
$187$	−1.33570e7	−	2.31349e7i	−0.149369	−	0.258715i
$188$		0			0
$189$	−9.45837e7	+	424202.i	−1.01906	+	0.00457042i
$190$		0			0
$191$	−4.24528e7	−	7.35305e7i	−0.440849	−	0.763573i	0.556904	−	0.830577i	$-0.311989\pi$
$191$	−4.24528e7	−	7.35305e7i	−0.440849	−	0.763573i	−0.997753	+	0.0670042i	$0.978656\pi$
$192$		0			0
$193$	3.14432e7	−	5.44613e7i	0.314830	−	0.545302i	−0.664571	−	0.747225i	$-0.731386\pi$
$193$	3.14432e7	−	5.44613e7i	0.314830	−	0.545302i	0.979401	+	0.201923i	$0.0647190\pi$
$194$		0			0
$195$	1.63190e8			1.57605
$196$		0			0
$197$	−1.65100e8			−1.53856			−0.769281	−	0.638910i	$-0.779385\pi$
$197$	−1.65100e8			−1.53856			−0.769281	+	0.638910i	$0.779385\pi$
$198$		0			0
$199$	2.85712e6	−	4.94867e6i	0.0257005	−	0.0445146i	−0.852889	−	0.522092i	$-0.825152\pi$
$199$	2.85712e6	−	4.94867e6i	0.0257005	−	0.0445146i	0.878590	+	0.477578i	$0.158485\pi$
$200$		0			0
$201$	−4.23599e7	−	7.33695e7i	−0.367933	−	0.637279i
$202$		0			0
$203$	−1.07206e8	+	480811.i	−0.899461	+	0.00403402i
$204$		0			0
$205$	−1.26163e8	−	2.18521e8i	−1.02281	−	1.77156i
$206$		0			0
$207$	1.15552e8	−	2.00142e8i	0.905483	−	1.56834i
$208$		0			0
$209$	−9.41849e6			−0.0713624
$210$		0			0
$211$	−1.18425e8			−0.867870			−0.433935	−	0.900944i	$-0.642875\pi$
$211$	−1.18425e8			−0.867870			−0.433935	+	0.900944i	$0.642875\pi$
$212$		0			0
$213$	−1.75247e8	+	3.03537e8i	−1.24258	+	2.15220i
$214$		0			0
$215$	1.40541e8	+	2.43424e8i	0.964423	+	1.67043i
$216$		0			0
$217$	−3.84705e7	+	6.73284e7i	−0.255575	+	0.447290i
$218$		0			0
$219$	−9.50217e7	−	1.64582e8i	−0.611320	−	1.05884i
$220$		0			0
$221$	1.17626e7	−	2.03733e7i	0.0733041	−	0.126967i
$222$		0			0
$223$	−2.90507e8			−1.75424			−0.877121	−	0.480270i	$-0.840539\pi$
$223$	−2.90507e8			−1.75424			−0.877121	+	0.480270i	$0.840539\pi$
$224$		0			0
$225$	7.96125e8			4.65954
$226$		0			0
$227$	3.42802e7	−	5.93750e7i	0.194515	−	0.336909i	−0.752227	−	0.658905i	$-0.771020\pi$
$227$	3.42802e7	−	5.93750e7i	0.194515	−	0.336909i	0.946741	+	0.321995i	$0.104353\pi$
$228$		0			0
$229$	−1.27632e8	−	2.21064e8i	−0.702318	−	1.21645i	−0.967651	−	0.252294i	$-0.918815\pi$
$229$	−1.27632e8	−	2.21064e8i	−0.702318	−	1.21645i	0.265332	−	0.964157i	$-0.414518\pi$
$230$		0			0
$231$	1.54282e8	+	2.64478e8i	0.823518	+	1.41171i
$232$		0			0
$233$	−2.98377e7	−	5.16805e7i	−0.154533	−	0.267658i	0.778356	−	0.627823i	$-0.216054\pi$
$233$	−2.98377e7	−	5.16805e7i	−0.154533	−	0.267658i	−0.932889	+	0.360165i	$0.882721\pi$
$234$		0			0
$235$	4.18608e7	−	7.25050e7i	0.210412	−	0.364444i
$236$		0			0
$237$	−2.75723e8			−1.34541
$238$		0			0
$239$	−3.11142e8			−1.47423			−0.737117	−	0.675765i	$-0.763813\pi$
$239$	−3.11142e8			−1.47423			−0.737117	+	0.675765i	$0.763813\pi$
$240$		0			0
$241$	1.80211e8	−	3.12135e8i	0.829319	−	1.43642i	−0.0692534	−	0.997599i	$-0.522062\pi$
$241$	1.80211e8	−	3.12135e8i	0.829319	−	1.43642i	0.898573	−	0.438824i	$-0.144605\pi$
$242$		0			0
$243$	1.09684e8	+	1.89979e8i	0.490369	+	0.849344i
$244$		0			0
$245$	2.29661e8	+	3.89670e8i	0.997714	+	1.69284i
$246$		0			0
$247$	−4.14711e6	−	7.18300e6i	−0.0175108	−	0.0303296i
$248$		0			0
$249$	7.36725e6	−	1.27605e7i	0.0302418	−	0.0523804i
$250$		0			0
$251$	1.56577e8			0.624985			0.312492	−	0.949920i	$-0.398836\pi$
$251$	1.56577e8			0.624985			0.312492	+	0.949920i	$0.398836\pi$
$252$		0			0
$253$	−2.88747e8			−1.12097
$254$		0			0
$255$	1.24992e8	−	2.16492e8i	0.472053	−	0.817619i
$256$		0			0
$257$	1.70225e8	+	2.94838e8i	0.625544	+	1.08347i	0.988435	+	0.151642i	$0.0484561\pi$
$257$	1.70225e8	+	2.94838e8i	0.625544	+	1.08347i	−0.362892	+	0.931831i	$0.618211\pi$
$258$		0			0
$259$	−1.20499e7	−	2.06566e7i	−0.0430959	−	0.0738771i
$260$		0			0
$261$	−2.10379e8	−	3.64386e8i	−0.732419	−	1.26859i
$262$		0			0
$263$	−1.95033e8	+	3.37808e8i	−0.661095	+	1.14505i	0.319233	+	0.947676i	$0.396575\pi$
$263$	−1.95033e8	+	3.37808e8i	−0.661095	+	1.14505i	−0.980328	+	0.197374i	$0.936759\pi$
$264$		0			0
$265$	3.35276e7			0.110673
$266$		0			0
$267$	4.32625e8			1.39098
$268$		0			0
$269$	1.80155e8	−	3.12038e8i	0.564306	−	0.977406i	−0.432808	−	0.901486i	$-0.642477\pi$
$269$	1.80155e8	−	3.12038e8i	0.564306	−	0.977406i	0.997114	−	0.0759200i	$-0.0241894\pi$
$270$		0			0
$271$	1.24337e8	+	2.15358e8i	0.379496	+	0.657307i	0.990989	−	0.133943i	$-0.0427640\pi$
$271$	1.24337e8	+	2.15358e8i	0.379496	+	0.657307i	−0.611493	+	0.791250i	$0.709431\pi$
$272$		0			0
$273$	−1.33771e8	+	2.34117e8i	−0.397917	+	0.696407i
$274$		0			0
$275$	−4.97349e8	−	8.61435e8i	−1.44211	−	2.49780i
$276$		0			0
$277$	−1.93090e8	+	3.34442e8i	−0.545860	+	0.945458i	0.452692	+	0.891667i	$0.350464\pi$
$277$	−1.93090e8	+	3.34442e8i	−0.545860	+	0.945458i	−0.998552	+	0.0537907i	$0.982870\pi$
$278$		0			0
$279$	−3.04339e8			−0.838964
$280$		0			0
$281$	−1.38669e8			−0.372826			−0.186413	−	0.982471i	$-0.559686\pi$
$281$	−1.38669e8			−0.372826			−0.186413	+	0.982471i	$0.559686\pi$
$282$		0			0
$283$	2.13509e8	−	3.69809e8i	0.559969	−	0.969894i	−0.437529	−	0.899204i	$-0.644146\pi$
$283$	2.13509e8	−	3.69809e8i	0.559969	−	0.969894i	0.997498	−	0.0706904i	$-0.0225202\pi$
$284$		0			0
$285$	−4.40682e7	−	7.63283e7i	−0.112763	−	0.195312i
$286$		0			0
$287$	4.16916e8	−	1.86984e6i	1.04103	−	0.00466894i
$288$		0			0
$289$	1.87151e8	+	3.24155e8i	0.456089	+	0.789969i
$290$		0			0
$291$	−3.86155e8	+	6.68840e8i	−0.918620	+	1.59110i
$292$		0			0
$293$	5.59769e8			1.30009			0.650043	−	0.759898i	$-0.274751\pi$
$293$	5.59769e8			1.30009			0.650043	+	0.759898i	$0.274751\pi$
$294$		0			0
$295$	1.14051e9			2.58657
$296$		0			0
$297$	−2.31905e8	+	4.01671e8i	−0.513644	+	0.889657i
$298$		0			0
$299$	−1.27140e8	−	2.20212e8i	−0.275063	−	0.476423i
$300$		0			0
$301$	−4.64427e8	+	2.08293e6i	−0.981601	+	0.00440242i
$302$		0			0
$303$	5.71496e8	+	9.89860e8i	1.18022	+	2.04420i
$304$		0			0
$305$	−2.61350e8	+	4.52671e8i	−0.527439	+	0.913552i
$306$		0			0
$307$	1.50479e8			0.296819			0.148409	−	0.988926i	$-0.452585\pi$
$307$	1.50479e8			0.296819			0.148409	+	0.988926i	$0.452585\pi$
$308$		0			0
$309$	2.76413e8			0.532971
$310$		0			0
$311$	−4.31084e8	+	7.46659e8i	−0.812644	+	1.40754i	0.0983630	+	0.995151i	$0.468639\pi$
$311$	−4.31084e8	+	7.46659e8i	−0.812644	+	1.40754i	−0.911007	+	0.412390i	$0.864694\pi$
$312$		0			0
$313$	4.79502e8	+	8.30522e8i	0.883864	+	1.53090i	0.847010	+	0.531577i	$0.178400\pi$
$313$	4.79502e8	+	8.30522e8i	0.883864	+	1.53090i	0.0368540	+	0.999321i	$0.488266\pi$
$314$		0			0
$315$	−8.80698e8	+	1.54134e9i	−1.58760	+	2.77850i
$316$		0			0
$317$	7.64346e7	+	1.32389e8i	0.134767	+	0.233423i	0.925508	−	0.378727i	$-0.123638\pi$
$317$	7.64346e7	+	1.32389e8i	0.134767	+	0.233423i	−0.790742	+	0.612150i	$0.790305\pi$
$318$		0			0
$319$	−2.62852e8	+	4.55274e8i	−0.453361	+	0.785245i
$320$		0			0
$321$	−6.92492e8			−1.16855
$322$		0			0
$323$	−1.27056e7			−0.0209790
$324$		0			0
$325$	4.37982e8	−	7.58606e8i	0.707724	−	1.22581i
$326$		0			0
$327$	−9.14444e8	−	1.58386e9i	−1.44624	−	2.50496i
$328$		0			0
$329$	6.97034e7	+	1.19489e8i	0.107912	+	0.184987i
$330$		0			0
$331$	−3.64647e8	−	6.31587e8i	−0.552681	−	0.957271i	−0.998080	−	0.0619390i	$-0.980272\pi$
$331$	−3.64647e8	−	6.31587e8i	−0.552681	−	0.957271i	0.445399	−	0.895332i	$-0.353062\pi$
$332$		0			0
$333$	4.69286e7	−	8.12827e7i	0.0696438	−	0.120627i
$334$		0			0
$335$	−6.13698e8			−0.891862
$336$		0			0
$337$	9.99520e8			1.42261			0.711306	−	0.702882i	$-0.248104\pi$
$337$	9.99520e8			1.42261			0.711306	+	0.702882i	$0.248104\pi$
$338$		0			0
$339$	5.75270e8	−	9.96397e8i	0.801997	−	1.38910i
$340$		0			0
$341$	1.90125e8	+	3.29305e8i	0.259656	+	0.449737i
$342$		0			0
$343$	−7.47292e8	+	1.00552e7i	−0.999909	+	0.0134543i
$344$		0			0
$345$	−1.35102e9	−	2.34003e9i	−1.77131	−	3.06799i
$346$		0			0
$347$	2.89931e8	−	5.02175e8i	0.372513	−	0.645211i	−0.617439	−	0.786619i	$-0.711830\pi$
$347$	2.89931e8	−	5.02175e8i	0.372513	−	0.645211i	0.989951	+	0.141408i	$0.0451630\pi$
$348$		0			0
$349$	−1.18267e9			−1.48928			−0.744638	−	0.667469i	$-0.767378\pi$
$349$	−1.18267e9			−1.48928			−0.744638	+	0.667469i	$0.767378\pi$
$350$		0			0
$351$	−4.08445e8			−0.504149
$352$		0			0
$353$	−2.27190e7	+	3.93505e7i	−0.0274903	+	0.0476145i	−0.879443	−	0.476004i	$-0.842085\pi$
$353$	−2.27190e7	+	3.93505e7i	−0.0274903	+	0.0476145i	0.851953	+	0.523618i	$0.175418\pi$
$354$		0			0
$355$	1.26947e9	+	2.19878e9i	1.50599	+	2.60845i
$356$		0			0
$357$	2.08127e8	+	3.56781e8i	0.242097	+	0.415014i
$358$		0			0
$359$	4.45017e7	+	7.70792e7i	0.0507628	+	0.0879238i	0.890290	−	0.455393i	$-0.150501\pi$
$359$	4.45017e7	+	7.70792e7i	0.0507628	+	0.0879238i	−0.839527	+	0.543317i	$0.817168\pi$
$360$		0			0
$361$	4.44696e8	−	7.70236e8i	0.497494	−	0.861685i
$362$		0			0
$363$	2.39237e7			0.0262516
$364$		0			0
$365$	−1.37665e9			−1.48183
$366$		0			0
$367$	−2.39219e8	+	4.14340e8i	−0.252619	+	0.437548i	−0.964246	−	0.265009i	$-0.914625\pi$
$367$	−2.39219e8	+	4.14340e8i	−0.252619	+	0.437548i	0.711627	+	0.702557i	$0.247958\pi$
$368$		0			0
$369$	8.18149e8	+	1.41708e9i	0.847696	+	1.46825i
$370$		0			0
$371$	−2.74835e7	+	4.80997e7i	−0.0279424	+	0.0489028i
$372$		0			0
$373$	5.82022e8	+	1.00809e9i	0.580709	+	1.00582i	0.995395	+	0.0958535i	$0.0305580\pi$
$373$	5.82022e8	+	1.00809e9i	0.580709	+	1.00582i	−0.414686	+	0.909965i	$0.636109\pi$
$374$		0			0
$375$	3.02744e9	−	5.24368e9i	2.96460	−	5.13484i
$376$		0			0
$377$	−4.62952e8			−0.444981
$378$		0			0
$379$	1.78772e9			1.68679			0.843395	−	0.537294i	$-0.180553\pi$
$379$	1.78772e9			1.68679			0.843395	+	0.537294i	$0.180553\pi$
$380$		0			0
$381$	1.18407e9	−	2.05088e9i	1.09683	−	1.89977i
$382$		0			0
$383$	1.98793e8	+	3.44319e8i	0.180803	+	0.313159i	0.942154	−	0.335180i	$-0.108797\pi$
$383$	1.98793e8	+	3.44319e8i	0.180803	+	0.313159i	−0.761352	+	0.648339i	$0.775464\pi$
$384$		0			0
$385$	2.21796e9	−	9.94741e6i	1.98080	−	0.00888377i
$386$		0			0
$387$	−9.11384e8	−	1.57856e9i	−0.799305	−	1.38444i
$388$		0			0
$389$	2.46267e8	−	4.26547e8i	0.212121	−	0.367404i	−0.740257	−	0.672324i	$-0.765296\pi$
$389$	2.46267e8	−	4.26547e8i	0.212121	−	0.367404i	0.952378	+	0.304920i	$0.0986297\pi$
$390$		0			0
$391$	−3.89520e8			−0.329542
$392$		0			0
$393$	−7.04386e8			−0.585378
$394$		0			0
$395$	−9.98650e8	+	1.72971e9i	−0.815311	+	1.41216i
$396$		0			0
$397$	2.85231e7	+	4.94034e7i	0.0228786	+	0.0396269i	0.877238	−	0.480056i	$-0.159384\pi$
$397$	2.85231e7	+	4.94034e7i	0.0228786	+	0.0396269i	−0.854359	+	0.519683i	$0.826050\pi$
$398$		0			0
$399$	1.45627e8	−	653126.i	0.114772	−	0.000514745i
$400$		0			0
$401$	−9.15297e8	−	1.58534e9i	−0.708854	−	1.22777i	−0.965283	−	0.261207i	$-0.915879\pi$
$401$	−9.15297e8	−	1.58534e9i	−0.708854	−	1.22777i	0.256429	−	0.966563i	$-0.417454\pi$
$402$		0			0
$403$	−1.67430e8	+	2.89997e8i	−0.127428	+	0.220712i
$404$		0			0
$405$	−6.21101e7			−0.0464590
$406$		0			0
$407$	−1.17268e8			−0.0862179
$408$		0			0
$409$	−1.03052e9	+	1.78492e9i	−0.744777	+	1.28999i	0.205522	+	0.978653i	$0.434111\pi$
$409$	−1.03052e9	+	1.78492e9i	−0.744777	+	1.28999i	−0.950299	+	0.311339i	$0.899223\pi$
$410$		0			0
$411$	3.40230e8	+	5.89296e8i	0.241728	+	0.418685i
$412$		0			0
$413$	−9.34910e8	+	1.63621e9i	−0.653047	+	1.14292i
$414$		0			0
$415$	−5.33673e7	−	9.24349e7i	−0.0366528	−	0.0634845i
$416$		0			0
$417$	1.65364e9	−	2.86418e9i	1.11677	−	1.93430i
$418$		0			0
$419$	2.50238e8			0.166190			0.0830948	−	0.996542i	$-0.473520\pi$
$419$	2.50238e8			0.166190			0.0830948	+	0.996542i	$0.473520\pi$
$420$		0			0
$421$	−8.67705e8			−0.566741			−0.283370	−	0.959011i	$-0.591453\pi$
$421$	−8.67705e8			−0.566741			−0.283370	+	0.959011i	$0.591453\pi$
$422$		0			0
$423$	−2.71460e8	+	4.70183e8i	−0.174387	+	0.302048i
$424$		0			0
$425$	−6.70926e8	−	1.16208e9i	−0.423949	−	0.734301i
$426$		0			0
$427$	−4.35180e8	−	7.46006e8i	−0.270502	−	0.463708i
$428$		0			0
$429$	6.61107e8	+	1.14507e9i	0.404270	+	0.700216i
$430$		0			0
$431$	−6.03709e7	+	1.04565e8i	−0.0363209	+	0.0629097i	−0.883614	−	0.468215i	$-0.844897\pi$
$431$	−6.03709e7	+	1.04565e8i	−0.0363209	+	0.0629097i	0.847294	+	0.531125i	$0.178231\pi$
$432$		0			0
$433$	−8.11123e8			−0.480153			−0.240076	−	0.970754i	$-0.577172\pi$
$433$	−8.11123e8			−0.480153			−0.240076	+	0.970754i	$0.577172\pi$
$434$		0			0
$435$	−4.91944e9			−2.86552
$436$		0			0
$437$	−6.86663e7	+	1.18934e8i	−0.0393603	+	0.0681741i
$438$		0			0
$439$	−1.15817e9	−	2.00601e9i	−0.653352	−	1.13164i	−0.982304	−	0.187293i	$-0.940029\pi$
$439$	−1.15817e9	−	2.00601e9i	−0.653352	−	1.13164i	0.328952	−	0.944347i	$-0.393305\pi$
$440$		0			0
$441$	−1.48931e9	−	2.52695e9i	−0.826897	−	1.40301i
$442$		0			0
$443$	1.25883e9	+	2.18036e9i	0.687947	+	1.19156i	0.972501	+	0.232899i	$0.0748213\pi$
$443$	1.25883e9	+	2.18036e9i	0.687947	+	1.19156i	−0.284554	+	0.958660i	$0.591845\pi$
$444$		0			0
$445$	1.56694e9	−	2.71401e9i	0.842930	−	1.46000i
$446$		0			0
$447$	−3.97968e9			−2.10752
$448$		0			0
$449$	1.98108e7			0.0103286			0.00516429	−	0.999987i	$-0.498356\pi$
$449$	1.98108e7			0.0103286			0.00516429	+	0.999987i	$0.498356\pi$
$450$		0			0
$451$	1.02222e9	−	1.77053e9i	0.524716	−	0.908835i
$452$		0			0
$453$	1.59111e9	+	2.75588e9i	0.804184	+	1.39289i
$454$		0			0
$455$	9.84189e8	+	1.68715e9i	0.489823	+	0.839678i
$456$		0			0
$457$	3.90833e8	+	6.76943e8i	0.191551	+	0.331776i	0.945764	−	0.324853i	$-0.105315\pi$
$457$	3.90833e8	+	6.76943e8i	0.191551	+	0.331776i	−0.754213	+	0.656630i	$0.771982\pi$
$458$		0			0
$459$	−3.12840e8	+	5.41855e8i	−0.151000	+	0.261540i
$460$		0			0
$461$	−6.02108e8			−0.286234			−0.143117	−	0.989706i	$-0.545713\pi$
$461$	−6.02108e8			−0.286234			−0.143117	+	0.989706i	$0.545713\pi$
$462$		0			0
$463$	1.26552e9			0.592565			0.296282	−	0.955100i	$-0.404253\pi$
$463$	1.26552e9			0.592565			0.296282	+	0.955100i	$0.404253\pi$
$464$		0			0
$465$	−1.77915e9	+	3.08157e9i	−0.820591	+	1.42130i
$466$		0			0
$467$	−2.01050e8	−	3.48229e8i	−0.0913471	−	0.158218i	0.816731	−	0.577018i	$-0.195784\pi$
$467$	−2.01050e8	−	3.48229e8i	−0.0913471	−	0.158218i	−0.908078	+	0.418801i	$0.862451\pi$
$468$		0			0
$469$	5.03065e8	−	8.80429e8i	0.225174	−	0.394084i
$470$		0			0
$471$	1.33795e9	+	2.31740e9i	0.590020	+	1.02195i
$472$		0			0
$473$	−1.13871e9	+	1.97230e9i	−0.494763	+	0.856955i
$474$		0			0
$475$	−4.73095e8			−0.202545
$476$		0			0
$477$	−2.17421e8			−0.0917250
$478$		0			0
$479$	8.08815e8	−	1.40091e9i	0.336260	−	0.582419i	−0.647466	−	0.762094i	$-0.724171\pi$
$479$	8.08815e8	−	1.40091e9i	0.336260	−	0.582419i	0.983726	+	0.179675i	$0.0575046\pi$
$480$		0			0
$481$	−5.16347e7	−	8.94340e7i	−0.0211560	−	0.0366433i
$482$		0			0
$483$	4.46454e9	−	2.00232e7i	1.80286	−	0.00808570i
$484$		0			0
$485$	2.79725e9	+	4.84498e9i	1.11336	+	1.92839i
$486$		0			0
$487$	1.52284e8	−	2.63764e8i	0.0597453	−	0.103482i	−0.834606	−	0.550848i	$-0.814304\pi$
$487$	1.52284e8	−	2.63764e8i	0.0597453	−	0.103482i	0.894351	+	0.447366i	$0.147638\pi$
$488$		0			0
$489$	−3.85454e9			−1.49071
$490$		0			0
$491$	2.14359e9			0.817252			0.408626	−	0.912702i	$-0.366008\pi$
$491$	2.14359e9			0.817252			0.408626	+	0.912702i	$0.366008\pi$
$492$		0			0
$493$	−3.54588e8	+	6.14165e8i	−0.133279	+	0.230845i
$494$		0			0
$495$	4.35248e9	+	7.53872e9i	1.61294	+	2.79370i
$496$		0			0
$497$	−4.19505e9	+	1.88145e7i	−1.53281	+	0.00687457i
$498$		0			0
$499$	−1.24815e9	−	2.16187e9i	−0.449693	−	0.778891i	0.548673	−	0.836037i	$-0.315133\pi$
$499$	−1.24815e9	−	2.16187e9i	−0.449693	−	0.778891i	−0.998366	+	0.0571461i	$0.981800\pi$
$500$		0			0
$501$	2.57017e9	−	4.45166e9i	0.913124	−	1.58158i
$502$		0			0
$503$	−2.89416e9			−1.01399			−0.506997	−	0.861948i	$-0.669244\pi$
$503$	−2.89416e9			−1.01399			−0.506997	+	0.861948i	$0.669244\pi$
$504$		0			0
$505$	8.27966e9			2.86084
$506$		0			0
$507$	1.79660e9	−	3.11180e9i	0.612243	−	1.06044i
$508$		0			0
$509$	−1.24625e9	−	2.15857e9i	−0.418883	−	0.725526i	0.576945	−	0.816783i	$-0.304245\pi$
$509$	−1.24625e9	−	2.15857e9i	−0.418883	−	0.725526i	−0.995827	+	0.0912573i	$0.970911\pi$
$510$		0			0
$511$	1.12847e9	−	1.97498e9i	0.374126	−	0.654770i
$512$		0			0
$513$	1.10298e8	+	1.91041e8i	0.0360708	+	0.0624764i
$514$		0			0
$515$	1.00115e9	−	1.73404e9i	0.322978	−	0.559414i
$516$		0			0
$517$	6.78339e8			0.215889
$518$		0			0
$519$	3.85458e9			1.21029
$520$		0			0
$521$	1.06085e9	−	1.83745e9i	0.328641	−	0.569223i	−0.653601	−	0.756839i	$-0.726743\pi$
$521$	1.06085e9	−	1.83745e9i	0.328641	−	0.569223i	0.982242	+	0.187616i	$0.0600760\pi$
$522$		0			0
$523$	−4.49299e8	−	7.78208e8i	−0.137334	−	0.237870i	0.789152	−	0.614197i	$-0.210520\pi$
$523$	−4.49299e8	−	7.78208e8i	−0.137334	−	0.237870i	−0.926487	+	0.376327i	$0.877187\pi$
$524$		0			0
$525$	7.74965e9	+	1.32848e10i	2.33735	+	4.00681i
$526$		0			0
$527$	2.56478e8	+	4.44234e8i	0.0763333	+	0.132213i
$528$		0			0
$529$	−4.02719e8	+	6.97530e8i	−0.118279	+	0.204865i
$530$		0			0
$531$	−7.39605e9			−2.14372
$532$		0			0
$533$	1.80039e9			0.515017
$534$		0			0
$535$	−2.50816e9	+	4.34425e9i	−0.708135	+	1.22653i
$536$		0			0
$537$	−1.76830e9	−	3.06278e9i	−0.492771	−	0.853504i
$538$		0			0
$539$	−1.80385e9	+	3.19011e9i	−0.496181	+	0.877494i
$540$		0			0
$541$	−1.77642e9	−	3.07685e9i	−0.482342	−	0.835441i	0.517452	−	0.855712i	$-0.326880\pi$
$541$	−1.77642e9	−	3.07685e9i	−0.482342	−	0.835441i	−0.999795	+	0.0202709i	$0.993547\pi$
$542$		0			0
$543$	−2.04287e9	+	3.53836e9i	−0.547572	+	0.948423i
$544$		0			0
$545$	−1.32482e10			−3.50565
$546$		0			0
$547$	1.53131e9			0.400043			0.200021	−	0.979792i	$-0.435899\pi$
$547$	1.53131e9			0.400043			0.200021	+	0.979792i	$0.435899\pi$
$548$		0			0
$549$	1.69481e9	−	2.93550e9i	0.437137	−	0.757144i
$550$		0			0
$551$	1.25017e8	+	2.16536e8i	0.0318374	+	0.0551440i
$552$		0			0
$553$	−1.66287e9	−	2.85058e9i	−0.418140	−	0.716796i
$554$		0			0
$555$	−5.48683e8	−	9.50347e8i	−0.136237	−	0.235970i
$556$		0			0
$557$	−9.29428e8	+	1.60982e9i	−0.227889	+	0.394715i	−0.957182	−	0.289486i	$-0.906516\pi$
$557$	−9.29428e8	+	1.60982e9i	−0.227889	+	0.394715i	0.729294	+	0.684201i	$0.239849\pi$
$558$		0			0
$559$	−2.00556e9			−0.485617
$560$		0			0
$561$	2.02545e9			0.484340
$562$		0			0
$563$	−3.22008e9	+	5.57735e9i	−0.760480	+	1.31719i	0.182124	+	0.983276i	$0.441703\pi$
$563$	−3.22008e9	+	5.57735e9i	−0.760480	+	1.31719i	−0.942604	+	0.333914i	$0.891631\pi$
$564$		0			0
$565$	−4.16717e9	−	7.21775e9i	−0.972013	−	1.68358i
$566$		0			0
$567$	5.09133e7	−	8.91050e7i	0.0117298	−	0.0205287i
$568$		0			0
$569$	2.37325e9	+	4.11058e9i	0.540070	+	0.935429i	0.998899	+	0.0469042i	$0.0149355\pi$
$569$	2.37325e9	+	4.11058e9i	0.540070	+	0.935429i	−0.458829	+	0.888524i	$0.651731\pi$
$570$		0			0
$571$	2.87568e9	−	4.98083e9i	0.646420	−	1.11963i	−0.337552	−	0.941307i	$-0.609599\pi$
$571$	2.87568e9	−	4.98083e9i	0.646420	−	1.11963i	0.983972	−	0.178325i	$-0.0570679\pi$
$572$		0			0
$573$	6.43754e9			1.42948
$574$		0			0
$575$	−1.45039e10			−3.18161
$576$		0			0
$577$	−3.45836e9	+	5.99005e9i	−0.749471	+	1.29812i	0.198606	+	0.980079i	$0.436359\pi$
$577$	−3.45836e9	+	5.99005e9i	−0.749471	+	1.29812i	−0.948077	+	0.318042i	$0.896975\pi$
$578$		0			0
$579$	2.38402e9	+	4.12925e9i	0.510429	+	0.884089i
$580$		0			0
$581$	1.76356e8	−	790947.i	0.0373056	−	0.000167313i
$582$		0			0
$583$	1.35826e8	+	2.35257e8i	0.0283885	+	0.0491703i
$584$		0			0
$585$	−3.83293e9	+	6.63884e9i	−0.791564	+	1.37103i
$586$		0			0
$587$	2.21194e9			0.451378			0.225689	−	0.974199i	$-0.427537\pi$
$587$	2.21194e9			0.451378			0.225689	+	0.974199i	$0.427537\pi$
$588$		0			0
$589$	1.80853e8			0.0364688
$590$		0			0
$591$	6.25893e9	−	1.08408e10i	1.24722	−	2.16025i
$592$		0			0
$593$	1.71060e9	+	2.96285e9i	0.336866	+	0.583470i	0.983842	−	0.179041i	$-0.0572996\pi$
$593$	1.71060e9	+	2.96285e9i	0.336866	+	0.583470i	−0.646975	+	0.762511i	$0.723966\pi$
$594$		0			0
$595$	2.99204e9	−	1.34191e7i	0.582314	−	0.00261164i
$596$		0			0
$597$	2.16626e8	+	3.75208e8i	0.0416678	+	0.0721708i
$598$		0			0
$599$	−4.61146e9	+	7.98728e9i	−0.876687	+	1.51847i	−0.0217322	+	0.999764i	$0.506918\pi$
$599$	−4.61146e9	+	7.98728e9i	−0.876687	+	1.51847i	−0.854955	+	0.518703i	$0.826415\pi$
$600$		0			0
$601$	6.81256e9			1.28012			0.640059	−	0.768326i	$-0.278910\pi$
$601$	6.81256e9			1.28012			0.640059	+	0.768326i	$0.278910\pi$
$602$		0			0
$603$	3.97973e9			0.739168
$604$		0			0
$605$	8.66499e7	−	1.50082e8i	0.0159083	−	0.0275540i
$606$		0			0
$607$	−2.00349e9	−	3.47015e9i	−0.363603	−	0.629778i	0.624948	−	0.780666i	$-0.285120\pi$
$607$	−2.00349e9	−	3.47015e9i	−0.363603	−	0.629778i	−0.988551	+	0.150888i	$0.951787\pi$
$608$		0			0
$609$	4.03260e9	−	7.05757e9i	0.723476	−	1.26618i
$610$		0			0
$611$	2.98683e8	+	5.17335e8i	0.0529745	+	0.0917545i
$612$		0			0
$613$	−4.14048e9	+	7.17153e9i	−0.726005	+	1.25748i	0.232555	+	0.972583i	$0.425292\pi$
$613$	−4.14048e9	+	7.17153e9i	−0.726005	+	1.25748i	−0.958559	+	0.284893i	$0.908042\pi$
$614$		0			0
$615$	1.91314e10			3.31653
$616$		0			0
$617$	3.35578e7			0.00575169			0.00287584	−	0.999996i	$-0.499085\pi$
$617$	3.35578e7			0.00575169			0.00287584	+	0.999996i	$0.499085\pi$
$618$		0			0
$619$	1.45841e9	−	2.52604e9i	0.247151	−	0.428079i	−0.715583	−	0.698528i	$-0.753839\pi$
$619$	1.45841e9	−	2.52604e9i	0.247151	−	0.428079i	0.962734	+	0.270449i	$0.0871722\pi$
$620$		0			0
$621$	3.38144e9	+	5.85683e9i	0.566606	+	0.981391i
$622$		0			0
$623$	2.60914e9	+	4.47272e9i	0.432305	+	0.741078i
$624$		0			0
$625$	−1.31988e10	−	2.28610e10i	−2.16250	−	3.74555i
$626$		0			0
$627$	3.57054e8	−	6.18437e8i	0.0578493	−	0.100198i
$628$		0			0
$629$	−1.58194e8			−0.0253462
$630$		0			0
$631$	7.72632e9			1.22425			0.612125	−	0.790761i	$-0.290315\pi$
$631$	7.72632e9			1.22425			0.612125	+	0.790761i	$0.290315\pi$
$632$		0			0
$633$	4.48948e9	−	7.77601e9i	0.703531	−	1.21855i
$634$		0			0
$635$	−8.57726e9	−	1.48562e10i	−1.32935	−	2.30251i
$636$		0			0
$637$	−3.22720e9	+	2.89481e7i	−0.494695	+	0.00443744i
$638$		0			0
$639$	−8.23228e9	−	1.42587e10i	−1.24815	−	2.16186i
$640$		0			0
$641$	3.01543e9	−	5.22288e9i	0.452216	−	0.783261i	−0.546307	−	0.837585i	$-0.683967\pi$
$641$	3.01543e9	−	5.22288e9i	0.452216	−	0.783261i	0.998523	+	0.0543235i	$0.0173002\pi$
$642$		0			0
$643$	−3.27128e9			−0.485265			−0.242633	−	0.970118i	$-0.578011\pi$
$643$	−3.27128e9			−0.485265			−0.242633	+	0.970118i	$0.578011\pi$
$644$		0			0
$645$	−2.13116e10			−3.12720
$646$		0			0
$647$	3.36141e9	−	5.82214e9i	0.487929	−	0.845118i	−0.511974	−	0.859001i	$-0.671086\pi$
$647$	3.36141e9	−	5.82214e9i	0.487929	−	0.845118i	0.999904	+	0.0138825i	$0.00441906\pi$
$648$		0			0
$649$	4.62041e9	+	8.00278e9i	0.663474	+	1.14917i
$650$		0			0
$651$	−2.96250e9	−	5.07846e9i	−0.420848	−	0.721438i
$652$		0			0
$653$	2.47194e9	+	4.28152e9i	0.347409	+	0.601730i	0.985788	−	0.167992i	$-0.0537282\pi$
$653$	2.47194e9	+	4.28152e9i	0.347409	+	0.601730i	−0.638379	+	0.769722i	$0.720395\pi$
$654$		0			0
$655$	−2.55123e9	+	4.41887e9i	−0.354736	+	0.614421i
$656$		0			0
$657$	8.92733e9			1.22813
$658$		0			0
$659$	−3.36078e9			−0.457448			−0.228724	−	0.973491i	$-0.573455\pi$
$659$	−3.36078e9			−0.457448			−0.228724	+	0.973491i	$0.573455\pi$
$660$		0			0
$661$	1.05592e9	−	1.82890e9i	0.142208	−	0.246312i	−0.786120	−	0.618074i	$-0.787913\pi$
$661$	1.05592e9	−	1.82890e9i	0.142208	−	0.246312i	0.928328	+	0.371762i	$0.121246\pi$
$662$		0			0
$663$	8.91835e8	+	1.54470e9i	0.118847	+	0.205849i
$664$		0			0
$665$	5.23352e8	−	9.15935e8i	0.0690110	−	0.120778i
$666$		0			0
$667$	3.83269e9	+	6.63842e9i	0.500108	+	0.866212i
$668$		0			0
$669$	1.10131e10	−	1.90752e10i	1.42206	−	2.46308i
$670$		0			0
$671$	−4.23508e9			−0.541168
$672$		0			0
$673$	1.12547e10			1.42324			0.711622	−	0.702562i	$-0.247961\pi$
$673$	1.12547e10			1.42324			0.711622	+	0.702562i	$0.247961\pi$
$674$		0			0
$675$	−1.16487e10	+	2.01761e10i	−1.45785	+	2.52507i
$676$		0			0
$677$	−1.57599e9	−	2.72970e9i	−0.195206	−	0.338107i	0.751762	−	0.659435i	$-0.229204\pi$
$677$	−1.57599e9	−	2.72970e9i	−0.195206	−	0.338107i	−0.946968	+	0.321327i	$0.895871\pi$
$678$		0			0
$679$	−9.24373e9	+	4.14575e7i	−1.13319	+	0.00508228i
$680$		0			0
$681$	2.59912e9	+	4.50181e9i	0.315363	+	0.546225i
$682$		0			0
$683$	4.30544e9	−	7.45724e9i	0.517065	−	0.895583i	−0.482739	−	0.875764i	$-0.660358\pi$
$683$	4.30544e9	−	7.45724e9i	0.517065	−	0.895583i	0.999804	−	0.0198183i	$-0.00630877\pi$
$684$		0			0
$685$	4.92916e9			0.585943
$686$		0			0
$687$	1.93540e10			2.27731
$688$		0			0
$689$	−1.19612e8	+	2.07175e8i	−0.0139319	+	0.0241307i
$690$		0			0
$691$	−6.39246e8	−	1.10721e9i	−0.0737046	−	0.127660i	0.826818	−	0.562470i	$-0.190149\pi$
$691$	−6.39246e8	−	1.10721e9i	−0.0737046	−	0.127660i	−0.900522	+	0.434810i	$0.856816\pi$
$692$		0			0
$693$	−1.43831e10	+	6.45073e7i	−1.64167	+	0.00736280i
$694$		0			0
$695$	−1.19787e10	−	2.07477e10i	−1.35351	−	2.34436i
$696$		0			0
$697$	1.37897e9	−	2.38845e9i	0.154255	−	0.267178i
$698$		0			0
$699$	4.52459e9			0.501082
$700$		0			0
$701$	−7.24407e9			−0.794273			−0.397136	−	0.917760i	$-0.629996\pi$
$701$	−7.24407e9			−0.794273			−0.397136	+	0.917760i	$0.629996\pi$
$702$		0			0
$703$	−2.78872e7	+	4.83020e7i	−0.00302734	+	0.00524350i
$704$		0			0
$705$	3.17388e9	+	5.49732e9i	0.341137	+	0.590866i
$706$		0			0
$707$	−6.78706e9	+	1.18782e10i	−0.722294	+	1.26411i
$708$		0			0
$709$	−2.12572e9	−	3.68185e9i	−0.223998	−	0.387976i	0.732020	−	0.681283i	$-0.238578\pi$
$709$	−2.12572e9	−	3.68185e9i	−0.223998	−	0.387976i	−0.956018	+	0.293307i	$0.905244\pi$
$710$		0			0
$711$	6.47608e9	−	1.12169e10i	0.675723	−	1.17039i
$712$		0			0
$713$	5.54448e9			0.572858
$714$		0			0
$715$	9.57793e9			0.979942
$716$		0			0
$717$	1.17954e10	−	2.04302e10i	1.19508	−	2.06993i
$718$		0			0
$719$	3.48927e9	+	6.04360e9i	0.350093	+	0.606379i	0.986265	−	0.165168i	$-0.0528166\pi$
$719$	3.48927e9	+	6.04360e9i	0.350093	+	0.606379i	−0.636172	+	0.771547i	$0.719483\pi$
$720$		0			0
$721$	1.66703e9	+	2.85771e9i	0.165642	+	0.283952i
$722$		0			0
$723$	1.36636e10	+	2.36660e10i	1.34456	+	2.32885i
$724$		0			0
$725$	−1.32032e10	+	2.28686e10i	−1.28676	+	2.22873i
$726$		0			0
$727$	1.21909e9			0.117670			0.0588349	−	0.998268i	$-0.481261\pi$
$727$	1.21909e9			0.117670			0.0588349	+	0.998268i	$0.481261\pi$
$728$		0			0
$729$	−1.68798e10			−1.61370
$730$		0			0
$731$	−1.53612e9	+	2.66063e9i	−0.145450	+	0.251927i
$732$		0			0
$733$	8.20330e9	+	1.42085e10i	0.769351	+	1.33255i	0.937915	+	0.346864i	$0.112754\pi$
$733$	8.20330e9	+	1.42085e10i	0.769351	+	1.33255i	−0.168564	+	0.985691i	$0.553913\pi$
$734$		0			0
$735$	−3.42930e10	+	3.07609e8i	−3.18566	+	0.0285755i
$736$		0			0
$737$	−2.48619e9	−	4.30620e9i	−0.228769	−	0.396240i
$738$		0			0
$739$	−1.86533e9	+	3.23084e9i	−0.170020	+	0.294483i	−0.938427	−	0.345479i	$-0.887717\pi$
$739$	−1.86533e9	+	3.23084e9i	−0.170020	+	0.294483i	0.768407	+	0.639962i	$0.221050\pi$
$740$		0			0
$741$	6.28867e8			0.0567799
$742$		0			0
$743$	−1.82498e10			−1.63229			−0.816144	−	0.577849i	$-0.803892\pi$
$743$	−1.82498e10			−1.63229			−0.816144	+	0.577849i	$0.803892\pi$
$744$		0			0
$745$	−1.44141e10	+	2.49660e10i	−1.27715	+	2.21208i
$746$		0			0
$747$	3.46078e8	+	5.99425e8i	0.0303775	+	0.0526154i
$748$		0			0
$749$	−4.17639e9	−	7.15937e9i	−0.363174	−	0.622571i
$750$		0			0
$751$	5.97037e9	+	1.03410e10i	0.514353	+	0.890886i	0.999861	+	0.0166536i	$0.00530125\pi$
$751$	5.97037e9	+	1.03410e10i	0.514353	+	0.890886i	−0.485508	+	0.874232i	$0.661365\pi$
$752$		0			0
$753$	−5.93582e9	+	1.02811e10i	−0.506639	+	0.877524i
$754$		0			0
$755$	2.30515e10			1.94933
$756$		0			0
$757$	1.34863e10			1.12994			0.564971	−	0.825111i	$-0.308887\pi$
$757$	1.34863e10			1.12994			0.564971	+	0.825111i	$0.308887\pi$
$758$		0			0
$759$	1.09464e10	−	1.89597e10i	0.908707	−	1.57393i
$760$		0			0
$761$	5.63097e9	+	9.75313e9i	0.463166	+	0.802228i	0.999117	−	0.0420218i	$-0.0133799\pi$
$761$	5.63097e9	+	9.75313e9i	0.463166	+	0.802228i	−0.535950	+	0.844250i	$0.680047\pi$
$762$		0			0
$763$	1.08599e10	−	1.90062e10i	0.885095	−	1.54903i
$764$		0			0
$765$	5.87151e9	+	1.01698e10i	0.474171	+	0.821288i
$766$		0			0
$767$	−4.06887e9	+	7.04750e9i	−0.325605	+	0.563964i
$768$		0			0
$769$	−2.13869e10			−1.69592			−0.847961	−	0.530059i	$-0.822170\pi$
$769$	−2.13869e10			−1.69592			−0.847961	+	0.530059i	$0.822170\pi$
$770$		0			0
$771$	−2.58129e10			−2.02837
$772$		0			0
$773$	8.36012e9	−	1.44802e10i	0.651005	−	1.12757i	−0.331874	−	0.943324i	$-0.607681\pi$
$773$	8.36012e9	−	1.44802e10i	0.651005	−	1.12757i	0.982879	−	0.184250i	$-0.0589857\pi$
$774$		0			0
$775$	9.55004e9	+	1.65412e10i	0.736969	+	1.27647i
$776$		0			0
$777$	1.81317e9	−	8.13193e6i	0.138664	−	0.000621899i
$778$		0			0
$779$	−4.86182e8	−	8.42093e8i	−0.0368484	−	0.0638232i
$780$		0			0
$781$	−1.02856e10	+	1.78152e10i	−0.772595	+	1.33817i
$782$		0			0
$783$	1.23128e10			0.916623
$784$		0			0
$785$	1.93838e10			1.43020
$786$		0			0
$787$	−4.72932e9	+	8.19142e9i	−0.345849	+	0.599029i	−0.985508	−	0.169631i	$-0.945743\pi$
$787$	−4.72932e9	+	8.19142e9i	−0.345849	+	0.599029i	0.639658	+	0.768659i	$0.279076\pi$
$788$		0			0
$789$	−1.47874e10	−	2.56126e10i	−1.07182	−	1.85645i
$790$		0			0
$791$	1.37707e10	−	6.17609e7i	0.989327	−	0.00443706i
$792$		0			0
$793$	−1.86477e9	−	3.22988e9i	−0.132791	−	0.230001i
$794$		0			0
$795$	−1.27103e9	+	2.20149e9i	−0.0897162	+	0.155393i
$796$		0			0
$797$	−4.25156e9			−0.297471			−0.148736	−	0.988877i	$-0.547520\pi$
$797$	−4.25156e9			−0.297471			−0.148736	+	0.988877i	$0.547520\pi$
$798$		0			0
$799$	9.15081e8			0.0634667
$800$		0			0
$801$	−1.01613e10	+	1.75999e10i	−0.698613	+	1.21003i
$802$		0			0
$803$	−5.57701e9	−	9.65967e9i	−0.380100	−	0.658352i
$804$		0			0
$805$	1.60446e10	−	2.80802e10i	1.08404	−	1.89721i
$806$		0			0
$807$	1.36594e10	+	2.36587e10i	0.914899	+	1.58465i
$808$		0			0
$809$	−2.76390e9	+	4.78722e9i	−0.183528	+	0.317880i	−0.943080	−	0.332567i	$-0.892085\pi$
$809$	−2.76390e9	+	4.78722e9i	−0.183528	+	0.317880i	0.759551	+	0.650447i	$0.225419\pi$
$810$		0			0
$811$	−4.99197e9			−0.328624			−0.164312	−	0.986408i	$-0.552540\pi$
$811$	−4.99197e9			−0.328624			−0.164312	+	0.986408i	$0.552540\pi$
$812$		0			0
$813$	−1.88544e10			−1.23054
$814$		0			0
$815$	−1.39609e10	+	2.41810e10i	−0.903361	+	1.56467i
$816$		0			0
$817$	5.41587e8	+	9.38056e8i	0.0347449	+	0.0601799i
$818$		0			0
$819$	−6.38231e9	−	1.09409e10i	−0.405961	−	0.695918i
$820$		0			0
$821$	5.29502e9	+	9.17125e9i	0.333939	+	0.578399i	0.983280	−	0.182098i	$-0.0582888\pi$
$821$	5.29502e9	+	9.17125e9i	0.333939	+	0.578399i	−0.649342	+	0.760497i	$0.724955\pi$
$822$		0			0
$823$	−1.00055e10	+	1.73301e10i	−0.625663	+	1.08368i	0.362749	+	0.931887i	$0.381838\pi$
$823$	−1.00055e10	+	1.73301e10i	−0.625663	+	1.08368i	−0.988412	+	0.151793i	$0.951495\pi$
$824$		0			0
$825$	7.54180e10			4.67612
$826$		0			0
$827$	−2.82024e9			−0.173387			−0.0866935	−	0.996235i	$-0.527630\pi$
$827$	−2.82024e9			−0.173387			−0.0866935	+	0.996235i	$0.527630\pi$
$828$		0			0
$829$	−2.35107e9	+	4.07217e9i	−0.143326	+	0.248247i	−0.928747	−	0.370714i	$-0.879113\pi$
$829$	−2.35107e9	+	4.07217e9i	−0.143326	+	0.248247i	0.785421	+	0.618961i	$0.212446\pi$
$830$		0			0
$831$	−1.46401e10	−	2.53574e10i	−0.884994	−	1.53285i
$832$		0			0
$833$	−2.43340e9	+	4.30346e9i	−0.145867	+	0.257965i
$834$		0			0
$835$	−1.86179e10	−	3.22472e10i	−1.10670	−	1.91686i
$836$		0			0
$837$	4.45301e9	−	7.71283e9i	0.262491	−	0.454648i
$838$		0			0
$839$	2.73548e10			1.59907			0.799534	−	0.600621i	$-0.205080\pi$
$839$	2.73548e10			1.59907			0.799534	+	0.600621i	$0.205080\pi$
$840$		0			0
$841$	−3.29394e9			−0.190954
$842$		0			0
$843$	5.25692e9	−	9.10525e9i	0.302228	−	0.523474i
$844$		0			0
$845$	−1.30143e10	−	2.25415e10i	−0.742032	−	1.28524i
$846$		0			0
$847$	1.44283e8	+	2.47337e8i	0.00815873	+	0.0139861i
$848$		0			0
$849$	1.61882e10	+	2.80388e10i	0.907868	+	1.57247i
$850$		0			0
$851$	−8.54949e8	+	1.48081e9i	−0.0475540	+	0.0823659i
$852$		0			0
$853$	1.99998e10			1.10333			0.551665	−	0.834066i	$-0.313993\pi$
$853$	1.99998e10			1.10333			0.551665	+	0.834066i	$0.313993\pi$
$854$		0			0
$855$	4.14023e9			0.226539
$856$		0			0
$857$	1.79869e10	−	3.11543e10i	0.976167	−	1.69077i	0.300139	−	0.953896i	$-0.402967\pi$
$857$	1.79869e10	−	3.11543e10i	0.976167	−	1.69077i	0.676028	−	0.736876i	$-0.263700\pi$
$858$		0			0
$859$	1.05362e10	+	1.82492e10i	0.567161	+	0.982351i	0.996845	+	0.0793721i	$0.0252915\pi$
$859$	1.05362e10	+	1.82492e10i	0.567161	+	0.982351i	−0.429684	+	0.902979i	$0.641375\pi$
$860$		0			0
$861$	−1.56825e10	+	2.74464e10i	−0.837345	+	1.46546i
$862$		0			0
$863$	1.22382e10	+	2.11971e10i	0.648154	+	1.12264i	0.983563	+	0.180563i	$0.0577919\pi$
$863$	1.22382e10	+	2.11971e10i	0.648154	+	1.12264i	−0.335410	+	0.942072i	$0.608875\pi$
$864$		0			0
$865$	1.39610e10	−	2.41812e10i	0.733433	−	1.27034i
$866$		0			0
$867$	−2.83795e10			−1.47890
$868$		0			0
$869$	−1.61828e10			−0.836533
$870$		0			0
$871$	2.18942e9	−	3.79218e9i	0.112270	−	0.194458i
$872$		0			0
$873$	−1.81397e10	−	3.14189e10i	−0.922743	−	1.59824i
$874$		0			0
$875$	7.24705e10	−	3.25025e8i	3.65707	−	0.0164017i
$876$		0			0
$877$	1.65279e9	+	2.86271e9i	0.0827404	+	0.143311i	0.904426	−	0.426630i	$-0.140299\pi$
$877$	1.65279e9	+	2.86271e9i	0.0827404	+	0.143311i	−0.821686	+	0.569941i	$0.806966\pi$
$878$		0			0
$879$	−2.12208e10	+	3.67555e10i	−1.05390	+	1.82541i
$880$		0			0
$881$	−3.05557e10			−1.50549			−0.752743	−	0.658315i	$-0.771270\pi$
$881$	−3.05557e10			−1.50549			−0.752743	+	0.658315i	$0.771270\pi$
$882$		0			0
$883$	1.20403e10			0.588540			0.294270	−	0.955722i	$-0.404923\pi$
$883$	1.20403e10			0.588540			0.294270	+	0.955722i	$0.404923\pi$
$884$		0			0
$885$	−4.32368e10	+	7.48884e10i	−2.09678	+	3.63173i
$886$		0			0
$887$	−1.16411e9	−	2.01629e9i	−0.0560093	−	0.0970109i	0.836661	−	0.547721i	$-0.184504\pi$
$887$	−1.16411e9	−	2.01629e9i	−0.0560093	−	0.0970109i	−0.892671	+	0.450710i	$0.851171\pi$
$888$		0			0
$889$	2.83442e10	−	1.27122e8i	1.35303	−	0.00606826i
$890$		0			0
$891$	−2.51618e8	−	4.35815e8i	−0.0119171	−	0.0206410i
$892$		0			0
$893$	1.61315e8	−	2.79405e8i	0.00758042	−	0.0131297i
$894$		0			0
$895$	−2.56186e10			−1.19447
$896$		0			0
$897$	1.92794e10			0.891909
$898$		0			0
$899$	5.04726e9	−	8.74211e9i	0.231684	−	0.401289i
$900$		0			0
$901$	1.83229e8	+	3.17362e8i	0.00834561	+	0.0144550i
$902$		0			0
$903$	1.74697e10	−	3.05742e10i	0.789545	−	1.38181i
$904$		0			0
$905$	1.47983e10	+	2.56313e10i	0.663652	+	1.14948i
$906$		0			0
$907$	−9.46293e9	+	1.63903e10i	−0.421114	+	0.729391i	−0.996049	−	0.0888082i	$-0.971694\pi$
$907$	−9.46293e9	+	1.63903e10i	−0.421114	+	0.729391i	0.574934	+	0.818199i	$0.305028\pi$
$908$		0			0
$909$	−5.36923e10			−2.37104
$910$		0			0
$911$	1.12249e10			0.491891			0.245945	−	0.969284i	$-0.420902\pi$
$911$	1.12249e10			0.491891			0.245945	+	0.969284i	$0.420902\pi$
$912$		0			0
$913$	4.32399e8	−	7.48937e8i	0.0188034	−	0.0325685i
$914$		0			0
$915$	−1.98155e10	−	3.43215e10i	−0.855128	−	1.48113i
$916$		0			0
$917$	−4.24812e9	−	7.28234e9i	−0.181930	−	0.311873i
$918$		0			0
$919$	−1.64634e10	−	2.85155e10i	−0.699706	−	1.21193i	−0.968568	−	0.248747i	$-0.919981\pi$
$919$	−1.64634e10	−	2.85155e10i	−0.699706	−	1.21193i	0.268863	−	0.963179i	$-0.413352\pi$
$920$		0			0
$921$	−5.70465e9	+	9.88074e9i	−0.240614	+	0.416755i
$922$		0			0
$923$	−1.81157e10			−0.758313
$924$		0			0
$925$	−5.89040e9			−0.244708
$926$		0			0
$927$	−6.49227e9	+	1.12449e10i	−0.267681	+	0.463637i
$928$		0			0
$929$	4.16396e9	+	7.21219e9i	0.170393	+	0.295129i	0.938557	−	0.345124i	$-0.112163\pi$
$929$	4.16396e9	+	7.21219e9i	0.170393	+	0.295129i	−0.768164	+	0.640253i	$0.778830\pi$
$930$		0			0
$931$	8.85021e8	+	1.50163e9i	0.0359443	+	0.0609873i
$932$		0			0
$933$	−3.26847e10	−	5.66116e10i	−1.31753	−	2.28202i
$934$		0			0
$935$	7.33601e9	−	1.27064e10i	0.293508	−	0.508370i
$936$		0			0
$937$	−3.35209e10			−1.33115			−0.665576	−	0.746330i	$-0.731814\pi$
$937$	−3.35209e10			−1.33115			−0.665576	+	0.746330i	$0.731814\pi$
$938$		0			0
$939$	−7.27116e10			−2.86599
$940$		0			0
$941$	1.21961e10	−	2.11243e10i	0.477152	−	0.826452i	−0.522505	−	0.852636i	$-0.675002\pi$
$941$	1.21961e10	−	2.11243e10i	0.477152	−	0.826452i	0.999657	+	0.0261843i	$0.00833566\pi$
$942$		0			0
$943$	−1.49051e10	−	2.58164e10i	−0.578821	−	1.00255i
$944$		0			0
$945$	−2.61758e10	−	4.48718e10i	−1.00899	−	1.72967i
$946$		0			0
$947$	5.80141e8	+	1.00483e9i	0.0221977	+	0.0384476i	0.876911	−	0.480653i	$-0.159600\pi$
$947$	5.80141e8	+	1.00483e9i	0.0221977	+	0.0384476i	−0.854713	+	0.519101i	$0.826267\pi$
$948$		0			0
$949$	4.91129e9	−	8.50661e9i	0.186537	−	0.323091i
$950$		0			0
$951$	−1.15905e10			−0.436990
$952$		0			0
$953$	1.31548e10			0.492333			0.246167	−	0.969228i	$-0.420829\pi$
$953$	1.31548e10			0.492333			0.246167	+	0.969228i	$0.420829\pi$
$954$		0			0
$955$	2.33163e10	−	4.03850e10i	0.866259	−	1.50040i
$956$		0			0
$957$	−1.99294e10	−	3.45188e10i	−0.735027	−	1.27310i
$958$		0			0
$959$	−4.04056e9	+	7.07151e9i	−0.147937	+	0.258909i
$960$		0			0
$961$	1.01056e10	+	1.75033e10i	0.367306	+	0.636193i
$962$		0			0
$963$	1.62650e10	−	2.81718e10i	0.586897	−	1.01653i
$964$		0			0
$965$	3.45390e10			1.23727
$966$		0			0
$967$	1.67834e10			0.596880			0.298440	−	0.954428i	$-0.403534\pi$
$967$	1.67834e10			0.596880			0.298440	+	0.954428i	$0.403534\pi$
$968$		0			0
$969$	4.81667e8	−	8.34272e8i	0.0170065	−	0.0294561i
$970$		0			0
$971$	1.69538e10	+	2.93648e10i	0.594292	+	1.02934i	0.993646	+	0.112547i	$0.0359009\pi$
$971$	1.69538e10	+	2.93648e10i	0.594292	+	1.02934i	−0.399355	+	0.916796i	$0.630766\pi$
$972$		0			0
$973$	3.95845e10	−	1.77534e8i	1.37762	−	0.00617855i
$974$		0			0
$975$	3.32077e10	+	5.75175e10i	1.14742	+	1.98739i
$976$		0			0
$977$	−1.65238e10	+	2.86200e10i	−0.566863	+	0.981835i	0.430011	+	0.902824i	$0.358510\pi$
$977$	−1.65238e10	+	2.86200e10i	−0.566863	+	0.981835i	−0.996874	+	0.0790114i	$0.974824\pi$
$978$		0			0
$979$	2.53916e10			0.864871
$980$		0			0
$981$	8.59124e10			2.90546
$982$		0			0
$983$	1.60564e10	−	2.78105e10i	0.539151	−	0.933837i	−0.459799	−	0.888023i	$-0.652079\pi$
$983$	1.60564e10	−	2.78105e10i	0.539151	−	0.933837i	0.998950	−	0.0458138i	$-0.0145881\pi$
$984$		0			0
$985$	−4.53388e10	−	7.85291e10i	−1.51162	−	2.61820i
$986$		0			0
$987$	−1.04883e10	+	4.70395e7i	−0.347213	+	0.00155723i
$988$		0			0
$989$	1.66037e10	+	2.87584e10i	0.545779	+	0.945317i
$990$		0			0
$991$	4.68129e9	−	8.10824e9i	0.152795	−	0.264648i	−0.779459	−	0.626453i	$-0.784506\pi$
$991$	4.68129e9	−	8.10824e9i	0.152795	−	0.264648i	0.932254	+	0.361805i	$0.117839\pi$
$992$		0			0
$993$	5.52949e10			1.79210
$994$		0			0
$995$	3.13842e9			0.101002
$996$		0			0
$997$	−2.62304e10	+	4.54323e10i	−0.838246	+	1.45188i	0.0531148	+	0.998588i	$0.483085\pi$
$997$	−2.62304e10	+	4.54323e10i	−0.838246	+	1.45188i	−0.891360	+	0.453295i	$0.850248\pi$
$998$		0			0
$999$	1.37329e9	+	2.37861e9i	0.0435796	+	0.0754822i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.8.i.d.81.1		10
4.3	odd	2		28.8.e.a.25.5	yes	10
7.2	even	3	inner	112.8.i.d.65.1		10
12.11	even	2		252.8.k.c.109.1		10
28.3	even	6		196.8.a.e.1.5		5
28.11	odd	6		196.8.a.d.1.1		5
28.19	even	6		196.8.e.f.177.1		10
28.23	odd	6		28.8.e.a.9.5	✓	10
28.27	even	2		196.8.e.f.165.1		10
84.23	even	6		252.8.k.c.37.1		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
28.8.e.a.9.5	✓	10	28.23	odd	6
28.8.e.a.25.5	yes	10	4.3	odd	2
112.8.i.d.65.1		10	7.2	even	3	inner
112.8.i.d.81.1		10	1.1	even	1	trivial
196.8.a.d.1.1		5	28.11	odd	6
196.8.a.e.1.5		5	28.3	even	6
196.8.e.f.165.1		10	28.27	even	2
196.8.e.f.177.1		10	28.19	even	6
252.8.k.c.37.1		10	84.23	even	6
252.8.k.c.109.1		10	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 112 = 2^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	112.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-37.9099 + 65.6620i) q^{3} +(274.614 + 475.646i) q^{5} +(-907.484 + 4.07000i) q^{7} +(-1780.83 - 3084.49i) q^{9} +(-2225.01 + 3853.83i) q^{11} -3918.83 q^{13} -41642.4 q^{15} +(-3001.55 + 5198.83i) q^{17} +(1058.25 + 1832.95i) q^{19} +(34135.4 - 59741.4i) q^{21} +(32443.3 + 56193.4i) q^{23} +(-111763. + 193580. i) q^{25} +104226. q^{27} +118135. q^{29} +(42724.4 - 74000.8i) q^{31} +(-168700. - 292197. i) q^{33} +(-251144. - 430523. i) q^{35} +(13176.1 + 22821.6i) q^{37} +(148563. - 257318. i) q^{39} -459420. q^{41} +511775. q^{43} +(978081. - 1.69409e6i) q^{45} +(-76217.4 - 132012. i) q^{47} +(823510. - 7386.92i) q^{49} +(-227577. - 394175. i) q^{51} +(30522.5 - 52866.5i) q^{53} -2.44408e6 q^{55} -160473. q^{57} +(1.03829e6 - 1.79837e6i) q^{59} +(475849. + 824194. i) q^{61} +(1.62863e6 + 2.79187e6i) q^{63} +(-1.07617e6 - 1.86397e6i) q^{65} +(-558691. + 967681. i) q^{67} -4.91969e6 q^{69} +4.62273e6 q^{71} +(-1.25326e6 + 2.17070e6i) q^{73} +(-8.47389e6 - 1.46772e7i) q^{75} +(2.00348e6 - 3.50635e6i) q^{77} +(1.81828e6 + 3.14935e6i) q^{79} +(-56543.1 + 97935.5i) q^{81} -194336. q^{83} -3.29707e6 q^{85} +(-4.47850e6 + 7.75699e6i) q^{87} +(-2.85298e6 - 4.94150e6i) q^{89} +(3.55627e6 - 15949.7i) q^{91} +(3.23936e6 + 5.61073e6i) q^{93} +(-581222. + 1.00671e6i) q^{95} +1.01861e7 q^{97} +1.58495e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 27 q^{3} + 249 q^{5} - 332 q^{7} - 5702 q^{9} - 6399 q^{11} - 26988 q^{13} - 19294 q^{15} + 3609 q^{17} + 12403 q^{19} + 16099 q^{21} + 13959 q^{23} - 162364 q^{25} - 161550 q^{27} + 26148 q^{29}+ \cdots + 119277812 q^{99}+O(q^{100}) \) 10 * q - 27 * q^3 + 249 * q^5 - 332 * q^7 - 5702 * q^9 - 6399 * q^11 - 26988 * q^13 - 19294 * q^15 + 3609 * q^17 + 12403 * q^19 + 16099 * q^21 + 13959 * q^23 - 162364 * q^25 - 161550 * q^27 + 26148 * q^29 + 20181 * q^31 + 264239 * q^33 - 791715 * q^35 - 54763 * q^37 - 4750 * q^39 - 1824468 * q^41 + 1938424 * q^43 + 1936430 * q^45 - 1217253 * q^47 + 1722058 * q^49 + 1353871 * q^51 - 2349159 * q^53 - 3123494 * q^55 - 9823194 * q^57 - 2188611 * q^59 + 6107445 * q^61 - 10275990 * q^63 - 3022014 * q^65 + 3171581 * q^67 - 11255938 * q^69 + 18938976 * q^71 + 8034853 * q^73 - 29173588 * q^75 + 24100191 * q^77 + 7589559 * q^79 - 9846485 * q^81 + 13452408 * q^83 - 48717670 * q^85 - 30033142 * q^87 + 29864757 * q^89 - 34267512 * q^91 - 20217029 * q^93 + 25812009 * q^95 + 285580 * q^97 + 119277812 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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