LMFDB - Embedded newform 45.10.b.d.19.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [45,10,Mod(19,45)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(45, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1]))

N = Newforms(chi, 10, names="a")

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

chi := DirichletCharacter("45.19");

S:= CuspForms(chi, 10);

N := Newforms(S);

Level:	$ N $	$=$	$ 45 = 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 10 $
Character orbit:	$[\chi]$	$=$	45.b (of order $2$, degree $1$, 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:	$23.1766126274$
Analytic rank:	$0$
Dimension:	$8$
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} - 774x^{6} - 2822x^{5} + 257730x^{4} + 2421892x^{3} - 2488179x^{2} - 34986488x + 335502416 $ x^8 - 2x^7 - 774x^6 - 2822x^5 + 257730x^4 + 2421892x^3 - 2488179x^2 - 34986488*x + 335502416
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{18}\cdot 3^{8}\cdot 5^{7} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			19.3
Root			$-8.93777 - 3.05515i$ of defining polynomial
Character	$\chi$	$=$	45.19
Dual form			45.10.b.d.19.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-12.2206i q^{2} +362.657 q^{4} +(-1143.76 + 803.080i) q^{5} +4342.19i q^{7} -10688.8i q^{8} +(9814.11 + 13977.4i) q^{10} -36750.7 q^{11} -187538. i q^{13} +53064.1 q^{14} +55056.9 q^{16} +3942.12i q^{17} +237154. q^{19} +(-414793. + 291243. i) q^{20} +449115. i q^{22} -2.19106e6i q^{23} +(663249. - 1.83706e6i) q^{25} -2.29183e6 q^{26} +1.57473e6i q^{28} -6.47801e6 q^{29} -4.92268e6 q^{31} -6.14550e6i q^{32} +48175.0 q^{34} +(-3.48712e6 - 4.96642e6i) q^{35} +6.61220e6i q^{37} -2.89817e6i q^{38} +(8.58398e6 + 1.22254e7i) q^{40} -2.22036e7 q^{41} -1.25053e7i q^{43} -1.33279e7 q^{44} -2.67760e7 q^{46} -3.42169e7i q^{47} +2.14990e7 q^{49} +(-2.24500e7 - 8.10529e6i) q^{50} -6.80121e7i q^{52} +3.09549e7i q^{53} +(4.20340e7 - 2.95138e7i) q^{55} +4.64129e7 q^{56} +7.91651e7i q^{58} +7.50725e7 q^{59} +1.30229e7 q^{61} +6.01580e7i q^{62} -4.69125e7 q^{64} +(1.50608e8 + 2.14499e8i) q^{65} +1.37041e8i q^{67} +1.42964e6i q^{68} +(-6.06925e7 + 4.26147e7i) q^{70} +2.12149e8 q^{71} -2.65436e8i q^{73} +8.08050e7 q^{74} +8.60058e7 q^{76} -1.59578e8i q^{77} -2.69754e8 q^{79} +(-6.29719e7 + 4.42151e7i) q^{80} +2.71341e8i q^{82} +5.22793e8i q^{83} +(-3.16584e6 - 4.50884e6i) q^{85} -1.52822e8 q^{86} +3.92822e8i q^{88} +3.29338e8 q^{89} +8.14326e8 q^{91} -7.94603e8i q^{92} -4.18150e8 q^{94} +(-2.71248e8 + 1.90454e8i) q^{95} -1.12881e9i q^{97} -2.62731e8i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 144 q^{4} - 102680 q^{10} - 1009088 q^{16} - 2079872 q^{19} - 595720 q^{25} + 12077344 q^{31} + 25286480 q^{34} + 18248480 q^{40} + 3119840 q^{46} - 174619144 q^{49} - 105991200 q^{55} - 165845744 q^{61}+ \cdots - 4117219360 q^{94}+O(q^{100}) $ 8 * q - 144 * q^4 - 102680 * q^10 - 1009088 * q^16 - 2079872 * q^19 - 595720 * q^25 + 12077344 * q^31 + 25286480 * q^34 + 18248480 * q^40 + 3119840 * q^46 - 174619144 * q^49 - 105991200 * q^55 - 165845744 * q^61 + 91038464 * q^64 + 693705600 * q^70 + 1551352896 * q^76 - 1098932768 * q^79 + 965125360 * q^85 - 1465128000 * q^91 - 4117219360 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/45\mathbb{Z}\right)^\times$.

$n$	$11$	$37$
$\chi(n)$	$1$	$-1$

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 $

$2$		−	12.2206i		−	0.540079i	−0.962849	−	0.270039i	$-0.912963\pi$
$2$		−	12.2206i		−	0.540079i	0.962849	−	0.270039i	$-0.0870367\pi$
$3$		0			0
$3$		0			0
$4$	362.657			0.708315
$5$	−1143.76	+	803.080i	−0.818408	+	0.574638i
$5$	−1143.76	+	803.080i	−0.818408	+	0.574638i
$6$		0			0
$7$			4342.19i			0.683545i	0.939783	+	0.341773i	$0.111027\pi$
$7$			4342.19i			0.683545i	−0.939783	+	0.341773i	$0.888973\pi$
$8$		−	10688.8i		−	0.922624i
$9$		0			0
$10$	9814.11	+	13977.4i	0.310349	+	0.442005i
$11$	−36750.7			−0.756830			−0.378415	−	0.925636i	$-0.623531\pi$
$11$	−36750.7			−0.756830			−0.378415	+	0.925636i	$0.623531\pi$
$12$		0			0
$13$		−	187538.i		−	1.82115i	−0.413349	−	0.910573i	$-0.635641\pi$
$13$		−	187538.i		−	1.82115i	0.413349	−	0.910573i	$-0.364359\pi$
$14$	53064.1			0.369168
$15$		0			0
$16$	55056.9			0.210025
$17$			3942.12i			0.0114475i	0.999984	+	0.00572374i	$0.00182193\pi$
$17$			3942.12i			0.0114475i	−0.999984	+	0.00572374i	$0.998178\pi$
$18$		0			0
$19$	237154.			0.417484			0.208742	−	0.977971i	$-0.433063\pi$
$19$	237154.			0.417484			0.208742	+	0.977971i	$0.433063\pi$
$20$	−414793.	+	291243.i	−0.579691	+	0.407024i
$21$		0			0
$22$			449115.i			0.408748i
$23$		−	2.19106e6i		−	1.63260i	−0.577631	−	0.816298i	$-0.696023\pi$
$23$		−	2.19106e6i		−	1.63260i	0.577631	−	0.816298i	$-0.303977\pi$
$24$		0			0
$25$	663249.	−	1.83706e6i	0.339583	−	0.940576i
$26$	−2.29183e6			−0.983562
$27$		0			0
$28$			1.57473e6i			0.484165i
$29$	−6.47801e6			−1.70079			−0.850395	−	0.526145i	$-0.823637\pi$
$29$	−6.47801e6			−1.70079			−0.850395	+	0.526145i	$0.823637\pi$
$30$		0			0
$31$	−4.92268e6			−0.957357			−0.478679	−	0.877990i	$-0.658884\pi$
$31$	−4.92268e6			−0.957357			−0.478679	+	0.877990i	$0.658884\pi$
$32$		−	6.14550e6i		−	1.03605i
$33$		0			0
$34$	48175.0			0.00618254
$35$	−3.48712e6	−	4.96642e6i	−0.392791	−	0.559419i
$36$		0			0
$37$			6.61220e6i			0.580014i	0.957025	+	0.290007i	$0.0936576\pi$
$37$			6.61220e6i			0.580014i	−0.957025	+	0.290007i	$0.906342\pi$
$38$		−	2.89817e6i		−	0.225474i
$39$		0			0
$40$	8.58398e6	+	1.22254e7i	0.530175	+	0.755083i
$41$	−2.22036e7			−1.22715			−0.613573	−	0.789638i	$-0.710268\pi$
$41$	−2.22036e7			−1.22715			−0.613573	+	0.789638i	$0.710268\pi$
$42$		0			0
$43$		−	1.25053e7i		−	0.557808i	−0.960319	−	0.278904i	$-0.910029\pi$
$43$		−	1.25053e7i		−	0.557808i	0.960319	−	0.278904i	$-0.0899711\pi$
$44$	−1.33279e7			−0.536074
$45$		0			0
$46$	−2.67760e7			−0.881730
$47$		−	3.42169e7i		−	1.02282i	−0.859336	−	0.511411i	$-0.829123\pi$
$47$		−	3.42169e7i		−	1.02282i	0.859336	−	0.511411i	$-0.170877\pi$
$48$		0			0
$49$	2.14990e7			0.532766
$50$	−2.24500e7	−	8.10529e6i	−0.507985	−	0.183402i
$51$		0			0
$52$		−	6.80121e7i		−	1.28994i
$53$			3.09549e7i			0.538875i	0.963018	+	0.269438i	$0.0868378\pi$
$53$			3.09549e7i			0.538875i	−0.963018	+	0.269438i	$0.913162\pi$
$54$		0			0
$55$	4.20340e7	−	2.95138e7i	0.619396	−	0.434903i
$56$	4.64129e7			0.630656
$57$		0			0
$58$			7.91651e7i			0.918560i
$59$	7.50725e7			0.806579			0.403290	−	0.915072i	$-0.367867\pi$
$59$	7.50725e7			0.806579			0.403290	+	0.915072i	$0.367867\pi$
$60$		0			0
$61$	1.30229e7			0.120427			0.0602136	−	0.998186i	$-0.480822\pi$
$61$	1.30229e7			0.120427			0.0602136	+	0.998186i	$0.480822\pi$
$62$			6.01580e7i			0.517048i
$63$		0			0
$64$	−4.69125e7			−0.349526
$65$	1.50608e8	+	2.14499e8i	1.04650	+	1.49044i
$66$		0			0
$67$			1.37041e8i			0.830836i	0.909631	+	0.415418i	$0.136365\pi$
$67$			1.37041e8i			0.830836i	−0.909631	+	0.415418i	$0.863635\pi$
$68$			1.42964e6i			0.00810842i
$69$		0			0
$70$	−6.06925e7	+	4.26147e7i	−0.302130	+	0.212138i
$71$	2.12149e8			0.990784			0.495392	−	0.868670i	$-0.335024\pi$
$71$	2.12149e8			0.990784			0.495392	+	0.868670i	$0.335024\pi$
$72$		0			0
$73$		−	2.65436e8i		−	1.09397i	−0.837141	−	0.546987i	$-0.815775\pi$
$73$		−	2.65436e8i		−	1.09397i	0.837141	−	0.546987i	$-0.184225\pi$
$74$	8.08050e7			0.313253
$75$		0			0
$76$	8.60058e7			0.295710
$77$		−	1.59578e8i		−	0.517328i
$78$		0			0
$79$	−2.69754e8			−0.779194			−0.389597	−	0.920985i	$-0.627386\pi$
$79$	−2.69754e8			−0.779194			−0.389597	+	0.920985i	$0.627386\pi$
$80$	−6.29719e7	+	4.42151e7i	−0.171886	+	0.120688i
$81$		0			0
$82$			2.71341e8i			0.662755i
$83$			5.22793e8i			1.20915i	0.796550	+	0.604573i	$0.206656\pi$
$83$			5.22793e8i			1.20915i	−0.796550	+	0.604573i	$0.793344\pi$
$84$		0			0
$85$	−3.16584e6	−	4.50884e6i	−0.00657815	−	0.00936871i
$86$	−1.52822e8			−0.301260
$87$		0			0
$88$			3.92822e8i			0.698270i
$89$	3.29338e8			0.556400			0.278200	−	0.960523i	$-0.410262\pi$
$89$	3.29338e8			0.556400			0.278200	+	0.960523i	$0.410262\pi$
$90$		0			0
$91$	8.14326e8			1.24484
$92$		−	7.94603e8i		−	1.15639i
$93$		0			0
$94$	−4.18150e8			−0.552405
$95$	−2.71248e8	+	1.90454e8i	−0.341672	+	0.239902i
$96$		0			0
$97$		−	1.12881e9i		−	1.29464i	−0.762219	−	0.647319i	$-0.775890\pi$
$97$		−	1.12881e9i		−	1.29464i	0.762219	−	0.647319i	$-0.224110\pi$
$98$		−	2.62731e8i		−	0.287735i
$99$		0			0
$100$	2.40532e8	−	6.66224e8i	0.240532	−	0.666224i
$101$	−5.08135e8			−0.485884			−0.242942	−	0.970041i	$-0.578112\pi$
$101$	−5.08135e8			−0.485884			−0.242942	+	0.970041i	$0.578112\pi$
$102$		0			0
$103$			1.62711e7i			0.0142446i	0.999975	+	0.00712230i	$0.00226712\pi$
$103$			1.62711e7i			0.0142446i	−0.999975	+	0.00712230i	$0.997733\pi$
$104$	−2.00456e9			−1.68023
$105$		0			0
$106$	3.78287e8			0.291035
$107$		−	1.55057e9i		−	1.14357i	−0.820402	−	0.571787i	$-0.806250\pi$
$107$		−	1.55057e9i		−	1.14357i	0.820402	−	0.571787i	$-0.193750\pi$
$108$		0			0
$109$	−1.46179e9			−0.991893			−0.495947	−	0.868353i	$-0.665179\pi$
$109$	−1.46179e9			−0.991893			−0.495947	+	0.868353i	$0.665179\pi$
$110$	−3.60675e8	−	5.13680e8i	−0.234882	−	0.334522i
$111$		0			0
$112$			2.39067e8i			0.143562i
$113$		−	5.32165e8i		−	0.307039i	−0.988146	−	0.153519i	$-0.950939\pi$
$113$		−	5.32165e8i		−	0.307039i	0.988146	−	0.153519i	$-0.0490607\pi$
$114$		0			0
$115$	1.75960e9	+	2.50604e9i	0.938151	+	1.33613i
$116$	−2.34930e9			−1.20470
$117$		0			0
$118$		−	9.17430e8i		−	0.435616i
$119$	−1.71174e7			−0.00782487
$120$		0			0
$121$	−1.00733e9			−0.427208
$122$		−	1.59148e8i		−	0.0650401i
$123$		0			0
$124$	−1.78525e9			−0.678110
$125$	7.16711e8	+	2.63380e9i	0.262572	+	0.964912i
$126$		0			0
$127$			2.55988e9i			0.873177i	0.899661	+	0.436589i	$0.143813\pi$
$127$			2.55988e9i			0.873177i	−0.899661	+	0.436589i	$0.856187\pi$
$128$		−	2.57320e9i		−	0.847283i
$129$		0			0
$130$	2.62130e9	−	1.84052e9i	0.804955	−	0.565192i
$131$	5.60469e9			1.66276			0.831382	−	0.555702i	$-0.187550\pi$
$131$	5.60469e9			1.66276			0.831382	+	0.555702i	$0.187550\pi$
$132$		0			0
$133$			1.02977e9i			0.285369i
$134$	1.67473e9			0.448717
$135$		0			0
$136$	4.21367e7			0.0105617
$137$			5.27039e9i			1.27820i	0.769122	+	0.639102i	$0.220694\pi$
$137$			5.27039e9i			1.27820i	−0.769122	+	0.639102i	$0.779306\pi$
$138$		0			0
$139$	−7.90257e9			−1.79557			−0.897784	−	0.440436i	$-0.854824\pi$
$139$	−7.90257e9			−1.79557			−0.897784	+	0.440436i	$0.854824\pi$
$140$	−1.26463e9	−	1.80111e9i	−0.278220	−	0.396245i
$141$		0			0
$142$		−	2.59259e9i		−	0.535101i
$143$			6.89216e9i			1.37830i
$144$		0			0
$145$	7.40929e9	−	5.20236e9i	1.39194	−	0.977338i
$146$	−3.24379e9			−0.590832
$147$		0			0
$148$			2.39796e9i			0.410833i
$149$	8.88302e9			1.47646			0.738232	−	0.674547i	$-0.235661\pi$
$149$	8.88302e9			1.47646			0.738232	+	0.674547i	$0.235661\pi$
$150$		0			0
$151$	4.35498e9			0.681695			0.340847	−	0.940119i	$-0.389286\pi$
$151$	4.35498e9			0.681695			0.340847	+	0.940119i	$0.389286\pi$
$152$		−	2.53490e9i		−	0.385181i
$153$		0			0
$154$	−1.95014e9			−0.279398
$155$	5.63036e9	−	3.95331e9i	0.783509	−	0.550133i
$156$		0			0
$157$			2.10491e9i			0.276494i	0.990398	+	0.138247i	$0.0441467\pi$
$157$			2.10491e9i			0.276494i	−0.990398	+	0.138247i	$0.955853\pi$
$158$			3.29655e9i			0.420826i
$159$		0			0
$160$	4.93533e9	+	7.02898e9i	0.595356	+	0.847915i
$161$	9.51398e9			1.11595
$162$		0			0
$163$		−	7.77003e9i		−	0.862142i	−0.902318	−	0.431071i	$-0.858136\pi$
$163$		−	7.77003e9i		−	0.862142i	0.902318	−	0.431071i	$-0.141864\pi$
$164$	−8.05230e9			−0.869206
$165$		0			0
$166$	6.38884e9			0.653034
$167$			1.81537e10i			1.80610i	0.429540	+	0.903048i	$0.358676\pi$
$167$			1.81537e10i			1.80610i	−0.429540	+	0.903048i	$0.641324\pi$
$168$		0			0
$169$	−2.45661e10			−2.31657
$170$	−5.51007e7	+	3.86884e7i	−0.00505984	+	0.00355272i
$171$		0			0
$172$		−	4.53512e9i		−	0.395104i
$173$		−	8.80586e9i		−	0.747419i	−0.927546	−	0.373710i	$-0.878086\pi$
$173$		−	8.80586e9i		−	0.747419i	0.927546	−	0.373710i	$-0.121914\pi$
$174$		0			0
$175$	7.97687e9	+	2.87995e9i	0.642926	+	0.232121i
$176$	−2.02338e9			−0.158953
$177$		0			0
$178$		−	4.02470e9i		−	0.300500i
$179$	−7.00851e8			−0.0510255			−0.0255127	−	0.999674i	$-0.508122\pi$
$179$	−7.00851e8			−0.0510255			−0.0255127	+	0.999674i	$0.508122\pi$
$180$		0			0
$181$	1.87883e10			1.30117			0.650586	−	0.759433i	$-0.274523\pi$
$181$	1.87883e10			1.30117			0.650586	+	0.759433i	$0.274523\pi$
$182$		−	9.95154e9i		−	0.672309i
$183$		0			0
$184$	−2.34198e10			−1.50627
$185$	−5.31013e9	−	7.56277e9i	−0.333298	−	0.474688i
$186$		0			0
$187$		−	1.44876e8i		−	0.00866380i
$188$		−	1.24090e10i		−	0.724481i
$189$		0			0
$190$	2.32746e9	+	3.31481e9i	0.129566	+	0.184530i
$191$	−2.71653e10			−1.47695			−0.738473	−	0.674283i	$-0.764453\pi$
$191$	−2.71653e10			−1.47695			−0.738473	+	0.674283i	$0.764453\pi$
$192$		0			0
$193$			3.36849e10i			1.74754i	0.486338	+	0.873771i	$0.338332\pi$
$193$			3.36849e10i			1.74754i	−0.486338	+	0.873771i	$0.661668\pi$
$194$	−1.37947e10			−0.699207
$195$		0			0
$196$	7.79678e9			0.377366
$197$			3.52896e8i			0.0166936i	0.999965	+	0.00834678i	$0.00265689\pi$
$197$			3.52896e8i			0.0166936i	−0.999965	+	0.00834678i	$0.997343\pi$
$198$		0			0
$199$	−2.79906e10			−1.26524			−0.632620	−	0.774463i	$-0.718020\pi$
$199$	−2.79906e10			−1.26524			−0.632620	+	0.774463i	$0.718020\pi$
$200$	−1.96360e10	−	7.08935e9i	−0.867798	−	0.313308i
$201$		0			0
$202$			6.20970e9i			0.262416i
$203$		−	2.81287e10i		−	1.16257i
$204$		0			0
$205$	2.53956e10	−	1.78313e10i	1.00431	−	0.705164i
$206$	1.98843e8			0.00769320
$207$		0			0
$208$		−	1.03253e10i		−	0.382487i
$209$	−8.71559e9			−0.315965
$210$		0			0
$211$	1.91333e10			0.664536			0.332268	−	0.943185i	$-0.392186\pi$
$211$	1.91333e10			0.664536			0.332268	+	0.943185i	$0.392186\pi$
$212$			1.12260e10i			0.381694i
$213$		0			0
$214$	−1.89489e10			−0.617620
$215$	1.00427e10	+	1.43030e10i	0.320537	+	0.456514i
$216$		0			0
$217$		−	2.13752e10i		−	0.654397i
$218$			1.78639e10i			0.535700i
$219$		0			0
$220$	1.52439e10	−	1.07034e10i	0.438727	−	0.308048i
$221$	7.39299e8			0.0208475
$222$		0			0
$223$		−	5.66446e10i		−	1.53386i	−0.641728	−	0.766932i	$-0.721782\pi$
$223$		−	5.66446e10i		−	1.53386i	0.641728	−	0.766932i	$-0.278218\pi$
$224$	2.66849e10			0.708190
$225$		0			0
$226$	−6.50336e9			−0.165825
$227$		−	3.31522e10i		−	0.828698i	−0.910118	−	0.414349i	$-0.864009\pi$
$227$		−	3.31522e10i		−	0.828698i	0.910118	−	0.414349i	$-0.135991\pi$
$228$		0			0
$229$	−1.68693e10			−0.405356			−0.202678	−	0.979245i	$-0.564964\pi$
$229$	−1.68693e10			−0.405356			−0.202678	+	0.979245i	$0.564964\pi$
$230$	3.06253e10	−	2.15033e10i	0.721615	−	0.506675i
$231$		0			0
$232$			6.92423e10i			1.56919i
$233$			5.99796e10i			1.33322i	0.745407	+	0.666610i	$0.232255\pi$
$233$			5.99796e10i			1.33322i	−0.745407	+	0.666610i	$0.767745\pi$
$234$		0			0
$235$	2.74789e10	+	3.91359e10i	0.587752	+	0.837086i
$236$	2.72256e10			0.571312
$237$		0			0
$238$			2.09185e8i			0.00422605i
$239$	−1.06603e10			−0.211339			−0.105669	−	0.994401i	$-0.533699\pi$
$239$	−1.06603e10			−0.211339			−0.105669	+	0.994401i	$0.533699\pi$
$240$		0			0
$241$	−8.69990e9			−0.166126			−0.0830630	−	0.996544i	$-0.526470\pi$
$241$	−8.69990e9			−0.166126			−0.0830630	+	0.996544i	$0.526470\pi$
$242$			1.23102e10i			0.230726i
$243$		0			0
$244$	4.72286e9			0.0853004
$245$	−2.45897e10	+	1.72654e10i	−0.436020	+	0.306147i
$246$		0			0
$247$		−	4.44755e10i		−	0.760300i
$248$			5.26177e10i			0.883281i
$249$		0			0
$250$	3.21866e10	−	8.75863e9i	0.521128	−	0.141810i
$251$	6.50269e10			1.03410			0.517049	−	0.855956i	$-0.327031\pi$
$251$	6.50269e10			1.03410			0.517049	+	0.855956i	$0.327031\pi$
$252$		0			0
$253$			8.05229e10i			1.23560i
$254$	3.12832e10			0.471584
$255$		0			0
$256$	−5.54652e10			−0.807125
$257$		−	6.68013e9i		−	0.0955182i	−0.998859	−	0.0477591i	$-0.984792\pi$
$257$		−	6.68013e9i		−	0.0955182i	0.998859	−	0.0477591i	$-0.0152080\pi$
$258$		0			0
$259$	−2.87114e10			−0.396466
$260$	5.46192e10	+	7.77895e10i	0.741251	+	1.05570i
$261$		0			0
$262$		−	6.84925e10i		−	0.898023i
$263$		−	9.64903e10i		−	1.24361i	−0.783174	−	0.621803i	$-0.786401\pi$
$263$		−	9.64903e10i		−	1.24361i	0.783174	−	0.621803i	$-0.213599\pi$
$264$		0			0
$265$	−2.48593e10	−	3.54050e10i	−0.309658	−	0.441020i
$266$	1.25844e10			0.154122
$267$		0			0
$268$			4.96991e10i			0.588494i
$269$	7.10118e10			0.826886			0.413443	−	0.910530i	$-0.364326\pi$
$269$	7.10118e10			0.826886			0.413443	+	0.910530i	$0.364326\pi$
$270$		0			0
$271$	−4.09389e10			−0.461078			−0.230539	−	0.973063i	$-0.574049\pi$
$271$	−4.09389e10			−0.461078			−0.230539	+	0.973063i	$0.574049\pi$
$272$			2.17041e8i			0.00240426i
$273$		0			0
$274$	6.44072e10			0.690331
$275$	−2.43749e10	+	6.75133e10i	−0.257007	+	0.711856i
$276$		0			0
$277$			7.44596e10i			0.759909i	0.925005	+	0.379954i	$0.124060\pi$
$277$			7.44596e10i			0.759909i	−0.925005	+	0.379954i	$0.875940\pi$
$278$			9.65740e10i			0.969748i
$279$		0			0
$280$	−5.30852e10	+	3.72733e10i	−0.516134	+	0.362398i
$281$	−1.38658e11			−1.32668			−0.663340	−	0.748318i	$-0.730862\pi$
$281$	−1.38658e11			−1.32668			−0.663340	+	0.748318i	$0.730862\pi$
$282$		0			0
$283$		−	5.71956e10i		−	0.530059i	−0.964240	−	0.265029i	$-0.914618\pi$
$283$		−	5.71956e10i		−	0.530059i	0.964240	−	0.265029i	$-0.0853816\pi$
$284$	7.69375e10			0.701787
$285$		0			0
$286$	8.42262e10			0.744389
$287$		−	9.64122e10i		−	0.838810i
$288$		0			0
$289$	1.18572e11			0.999869
$290$	−6.35759e10	−	9.05458e10i	−0.527839	−	0.751757i
$291$		0			0
$292$		−	9.62624e10i		−	0.774879i
$293$		−	1.09734e11i		−	0.869832i	−0.900471	−	0.434916i	$-0.856778\pi$
$293$		−	1.09734e11i		−	0.869832i	0.900471	−	0.434916i	$-0.143222\pi$
$294$		0			0
$295$	−8.58649e10	+	6.02893e10i	−0.660111	+	0.463491i
$296$	7.06767e10			0.535135
$297$		0			0
$298$		−	1.08556e11i		−	0.797406i
$299$	−4.10907e11			−2.97319
$300$		0			0
$301$	5.43001e10			0.381287
$302$		−	5.32204e10i		−	0.368169i
$303$		0			0
$304$	1.30570e10			0.0876823
$305$	−1.48951e10	+	1.04585e10i	−0.0985585	+	0.0692020i
$306$		0			0
$307$			1.89369e11i			1.21670i	0.793667	+	0.608352i	$0.208169\pi$
$307$			1.89369e11i			1.21670i	−0.793667	+	0.608352i	$0.791831\pi$
$308$		−	5.78722e10i		−	0.366431i
$309$		0			0
$310$	−4.83117e10	−	6.88063e10i	−0.297115	−	0.423156i
$311$	−1.40583e11			−0.852139			−0.426070	−	0.904690i	$-0.640102\pi$
$311$	−1.40583e11			−0.852139			−0.426070	+	0.904690i	$0.640102\pi$
$312$		0			0
$313$		−	1.43754e11i		−	0.846585i	−0.905993	−	0.423292i	$-0.860874\pi$
$313$		−	1.43754e11i		−	0.846585i	0.905993	−	0.423292i	$-0.139126\pi$
$314$	2.57232e10			0.149328
$315$		0			0
$316$	−9.78282e10			−0.551915
$317$		−	1.67042e11i		−	0.929091i	−0.885549	−	0.464545i	$-0.846218\pi$
$317$		−	1.67042e11i		−	0.929091i	0.885549	−	0.464545i	$-0.153782\pi$
$318$		0			0
$319$	2.38071e11			1.28721
$320$	5.36567e10	−	3.76745e10i	0.286055	−	0.200851i
$321$		0			0
$322$		−	1.16266e11i		−	0.602702i
$323$			9.34892e8i			0.00477914i
$324$		0			0
$325$	−3.44519e11	−	1.24385e11i	−1.71293	−	0.618431i
$326$	−9.49544e10			−0.465624
$327$		0			0
$328$			2.37331e11i			1.13220i
$329$	1.48576e11			0.699145
$330$		0			0
$331$	−1.98423e10			−0.0908587			−0.0454294	−	0.998968i	$-0.514466\pi$
$331$	−1.98423e10			−0.0908587			−0.0454294	+	0.998968i	$0.514466\pi$
$332$			1.89595e11i			0.856456i
$333$		0			0
$334$	2.21849e11			0.975433
$335$	−1.10055e11	−	1.56742e11i	−0.477429	−	0.679963i
$336$		0			0
$337$			6.71805e10i			0.283732i	0.989886	+	0.141866i	$0.0453102\pi$
$337$			6.71805e10i			0.283732i	−0.989886	+	0.141866i	$0.954690\pi$
$338$			3.00212e11i			1.25113i
$339$		0			0
$340$	−1.14812e9	−	1.63516e9i	−0.00465941	−	0.00663600i
$341$	1.80912e11			0.724557
$342$		0			0
$343$			2.68576e11i			1.04771i
$344$	−1.33666e11			−0.514647
$345$		0			0
$346$	−1.07613e11			−0.403665
$347$			2.47932e11i			0.918014i	0.888433	+	0.459007i	$0.151795\pi$
$347$			2.47932e11i			0.918014i	−0.888433	+	0.459007i	$0.848205\pi$
$348$		0			0
$349$	1.62415e11			0.586018			0.293009	−	0.956110i	$-0.405343\pi$
$349$	1.62415e11			0.586018			0.293009	+	0.956110i	$0.405343\pi$
$350$	3.51947e10	−	9.74820e10i	0.125363	−	0.347231i
$351$		0			0
$352$			2.25852e11i			0.784117i
$353$		−	2.36285e11i		−	0.809935i	−0.914331	−	0.404968i	$-0.867283\pi$
$353$		−	2.36285e11i		−	0.809935i	0.914331	−	0.404968i	$-0.132717\pi$
$354$		0			0
$355$	−2.42648e11	+	1.70373e11i	−0.810865	+	0.569342i
$356$	1.19437e11			0.394106
$357$		0			0
$358$			8.56480e9i			0.0275578i
$359$	4.86202e11			1.54487			0.772434	−	0.635095i	$-0.219039\pi$
$359$	4.86202e11			1.54487			0.772434	+	0.635095i	$0.219039\pi$
$360$		0			0
$361$	−2.66445e11			−0.825707
$362$		−	2.29604e11i		−	0.702735i
$363$		0			0
$364$	2.95321e11			0.881736
$365$	2.13167e11	+	3.03595e11i	0.628639	+	0.895318i
$366$		0			0
$367$		−	6.91679e11i		−	1.99025i	−0.0986269	−	0.995124i	$-0.531445\pi$
$367$		−	6.91679e11i		−	1.99025i	0.0986269	−	0.995124i	$-0.468555\pi$
$368$		−	1.20633e11i		−	0.342886i
$369$		0			0
$370$	−9.24215e10	+	6.48929e10i	−0.256369	+	0.180007i
$371$	−1.34412e11			−0.368346
$372$		0			0
$373$		−	5.41625e11i		−	1.44880i	−0.689379	−	0.724401i	$-0.742116\pi$
$373$		−	5.41625e11i		−	1.44880i	0.689379	−	0.724401i	$-0.257884\pi$
$374$	−1.77047e9			−0.00467913
$375$		0			0
$376$	−3.65738e11			−0.943681
$377$			1.21487e12i			3.09739i
$378$		0			0
$379$	7.17554e11			1.78640			0.893199	−	0.449662i	$-0.148456\pi$
$379$	7.17554e11			1.78640			0.893199	+	0.449662i	$0.148456\pi$
$380$	−9.83700e10	+	6.90696e10i	−0.242012	+	0.169926i
$381$		0			0
$382$			3.31976e11i			0.797667i
$383$		−	1.81450e11i		−	0.430885i	−0.976516	−	0.215443i	$-0.930881\pi$
$383$		−	1.81450e11i		−	0.430885i	0.976516	−	0.215443i	$-0.0691195\pi$
$384$		0			0
$385$	1.28154e11	+	1.82519e11i	0.297276	+	0.423385i
$386$	4.11649e11			0.943810
$387$		0			0
$388$		−	4.09372e11i		−	0.917012i
$389$	1.56142e11			0.345737			0.172868	−	0.984945i	$-0.444696\pi$
$389$	1.56142e11			0.345737			0.172868	+	0.984945i	$0.444696\pi$
$390$		0			0
$391$	8.63742e9			0.0186891
$392$		−	2.29799e11i		−	0.491543i
$393$		0			0
$394$	4.31260e9			0.00901583
$395$	3.08534e11	−	2.16634e11i	0.637699	−	0.447754i
$396$		0			0
$397$			6.12411e9i			0.0123733i	0.999981	+	0.00618665i	$0.00196929\pi$
$397$			6.12411e9i			0.0123733i	−0.999981	+	0.00618665i	$0.998031\pi$
$398$			3.42061e11i			0.683329i
$399$		0			0
$400$	3.65164e10	−	1.01143e11i	0.0713211	−	0.197545i
$401$	7.63256e11			1.47408			0.737039	−	0.675851i	$-0.236224\pi$
$401$	7.63256e11			1.47408			0.737039	+	0.675851i	$0.236224\pi$
$402$		0			0
$403$			9.23190e11i			1.74349i
$404$	−1.84279e11			−0.344159
$405$		0			0
$406$	−3.43749e11			−0.627877
$407$		−	2.43003e11i		−	0.438972i
$408$		0			0
$409$	2.15366e11			0.380559			0.190279	−	0.981730i	$-0.439061\pi$
$409$	2.15366e11			0.380559			0.190279	+	0.981730i	$0.439061\pi$
$410$	−2.17909e11	−	3.10349e11i	−0.380844	−	0.542404i
$411$		0			0
$412$			5.90084e9i			0.0100897i
$413$			3.25979e11i			0.551333i
$414$		0			0
$415$	−4.19845e11	−	5.97950e11i	−0.694820	−	0.989574i
$416$	−1.15252e12			−1.88681
$417$		0			0
$418$			1.06510e11i			0.170646i
$419$	−5.13491e11			−0.813898			−0.406949	−	0.913451i	$-0.633407\pi$
$419$	−5.13491e11			−0.813898			−0.406949	+	0.913451i	$0.633407\pi$
$420$		0			0
$421$	4.25812e11			0.660615			0.330308	−	0.943873i	$-0.392848\pi$
$421$	4.25812e11			0.660615			0.330308	+	0.943873i	$0.392848\pi$
$422$		−	2.33820e11i		−	0.358902i
$423$		0			0
$424$	3.30872e11			0.497180
$425$	7.24193e9	+	2.61461e9i	0.0107672	+	0.00388738i
$426$		0			0
$427$			5.65480e10i			0.0823174i
$428$		−	5.62325e11i		−	0.810010i
$429$		0			0
$430$	1.74791e11	−	1.22728e11i	0.246554	−	0.173115i
$431$	−6.11767e11			−0.853962			−0.426981	−	0.904261i	$-0.640423\pi$
$431$	−6.11767e11			−0.853962			−0.426981	+	0.904261i	$0.640423\pi$
$432$		0			0
$433$		−	9.75502e10i		−	0.133362i	−0.997774	−	0.0666811i	$-0.978759\pi$
$433$		−	9.75502e10i		−	0.133362i	0.997774	−	0.0666811i	$-0.0212410\pi$
$434$	−2.61217e11			−0.353426
$435$		0			0
$436$	−5.30128e11			−0.702573
$437$		−	5.19619e11i		−	0.681583i
$438$		0			0
$439$	−1.10090e12			−1.41467			−0.707337	−	0.706877i	$-0.750104\pi$
$439$	−1.10090e12			−1.41467			−0.707337	+	0.706877i	$0.750104\pi$
$440$	−3.15467e11	−	4.49294e11i	−0.401252	−	0.571470i
$441$		0			0
$442$		−	9.03466e9i		−	0.0112593i
$443$		−	3.39709e11i		−	0.419073i	−0.977801	−	0.209537i	$-0.932804\pi$
$443$		−	3.39709e11i		−	0.419073i	0.977801	−	0.209537i	$-0.0671955\pi$
$444$		0			0
$445$	−3.76684e11	+	2.64485e11i	−0.455362	+	0.319728i
$446$	−6.92230e11			−0.828407
$447$		0			0
$448$		−	2.03703e11i		−	0.238917i
$449$	3.57373e11			0.414966			0.207483	−	0.978239i	$-0.433473\pi$
$449$	3.57373e11			0.414966			0.207483	+	0.978239i	$0.433473\pi$
$450$		0			0
$451$	8.15998e11			0.928741
$452$		−	1.92993e11i		−	0.217480i
$453$		0			0
$454$	−4.05140e11			−0.447562
$455$	−9.31393e11	+	6.53969e11i	−1.01878	+	0.715329i
$456$		0			0
$457$			1.48293e12i			1.59037i	0.606369	+	0.795184i	$0.292626\pi$
$457$			1.48293e12i			1.59037i	−0.606369	+	0.795184i	$0.707374\pi$
$458$			2.06152e11i			0.218924i
$459$		0			0
$460$	6.38130e11	+	9.08835e11i	0.664506	+	0.946401i
$461$	1.01555e12			1.04724			0.523622	−	0.851951i	$-0.324580\pi$
$461$	1.01555e12			1.04724			0.523622	+	0.851951i	$0.324580\pi$
$462$		0			0
$463$		−	6.56647e11i		−	0.664076i	−0.943266	−	0.332038i	$-0.892264\pi$
$463$		−	6.56647e11i		−	0.664076i	0.943266	−	0.332038i	$-0.107736\pi$
$464$	−3.56659e11			−0.357209
$465$		0			0
$466$	7.32986e11			0.720044
$467$		−	9.82330e11i		−	0.955721i	−0.878436	−	0.477861i	$-0.841412\pi$
$467$		−	9.82330e11i		−	0.955721i	0.878436	−	0.477861i	$-0.158588\pi$
$468$		0			0
$469$	−5.95059e11			−0.567914
$470$	4.78264e11	−	3.35808e11i	0.452092	−	0.317432i
$471$		0			0
$472$		−	8.02437e11i		−	0.744170i
$473$			4.59577e11i			0.422166i
$474$		0			0
$475$	1.57292e11	−	4.35668e11i	0.141771	−	0.392676i
$476$	−6.20776e9			−0.00554248
$477$		0			0
$478$			1.30275e11i			0.114140i
$479$	−1.43057e11			−0.124165			−0.0620826	−	0.998071i	$-0.519774\pi$
$479$	−1.43057e11			−0.124165			−0.0620826	+	0.998071i	$0.519774\pi$
$480$		0			0
$481$	1.24004e12			1.05629
$482$			1.06318e11i			0.0897211i
$483$		0			0
$484$	−3.65317e11			−0.302598
$485$	9.06526e11	+	1.29109e12i	0.743948	+	1.05954i
$486$		0			0
$487$		−	1.86986e12i		−	1.50636i	−0.657813	−	0.753181i	$-0.728518\pi$
$487$		−	1.86986e12i		−	1.50636i	0.657813	−	0.753181i	$-0.271482\pi$
$488$		−	1.39200e11i		−	0.111109i
$489$		0			0
$490$	2.10994e11	+	3.00501e11i	0.165344	+	0.235485i
$491$	−2.39101e11			−0.185658			−0.0928292	−	0.995682i	$-0.529591\pi$
$491$	−2.39101e11			−0.185658			−0.0928292	+	0.995682i	$0.529591\pi$
$492$		0			0
$493$		−	2.55371e10i		−	0.0194698i
$494$	−5.43517e11			−0.410622
$495$		0			0
$496$	−2.71027e11			−0.201069
$497$			9.21191e11i			0.677246i
$498$		0			0
$499$	−2.13098e12			−1.53861			−0.769304	−	0.638883i	$-0.779397\pi$
$499$	−2.13098e12			−1.53861			−0.769304	+	0.638883i	$0.779397\pi$
$500$	2.59921e11	+	9.55167e11i	0.185984	+	0.683462i
$501$		0			0
$502$		−	7.94667e11i		−	0.558494i
$503$		−	9.25247e11i		−	0.644469i	−0.946660	−	0.322234i	$-0.895566\pi$
$503$		−	9.25247e11i		−	0.644469i	0.946660	−	0.322234i	$-0.104434\pi$
$504$		0			0
$505$	5.81184e11	−	4.08073e11i	0.397651	−	0.279207i
$506$	9.84037e11			0.667320
$507$		0			0
$508$			9.28358e11i			0.618485i
$509$	1.48004e12			0.977336			0.488668	−	0.872470i	$-0.337483\pi$
$509$	1.48004e12			0.977336			0.488668	+	0.872470i	$0.337483\pi$
$510$		0			0
$511$	1.15257e12			0.747781
$512$		−	6.39661e11i		−	0.411372i
$513$		0			0
$514$	−8.16351e10			−0.0515873
$515$	−1.30670e10	−	1.86103e10i	−0.00818548	−	0.0116579i
$516$		0			0
$517$			1.25749e12i			0.774103i
$518$			3.50870e11i			0.214123i
$519$		0			0
$520$	2.29274e12	−	1.60982e12i	1.37512	−	0.965525i
$521$	8.54955e11			0.508363			0.254181	−	0.967157i	$-0.418194\pi$
$521$	8.54955e11			0.508363			0.254181	+	0.967157i	$0.418194\pi$
$522$		0			0
$523$			1.34309e12i			0.784960i	0.919761	+	0.392480i	$0.128383\pi$
$523$			1.34309e12i			0.784960i	−0.919761	+	0.392480i	$0.871617\pi$
$524$	2.03258e12			1.17776
$525$		0			0
$526$	−1.17917e12			−0.671645
$527$		−	1.94058e10i		−	0.0109593i
$528$		0			0
$529$	−2.99958e12			−1.66537
$530$	−4.32670e11	+	3.03795e11i	−0.238185	+	0.167240i
$531$		0			0
$532$			3.73453e11i			0.202131i
$533$			4.16403e12i			2.23481i
$534$		0			0
$535$	1.24523e12	+	1.77348e12i	0.657140	+	0.935910i
$536$	1.46481e12			0.766549
$537$		0			0
$538$		−	8.67806e11i		−	0.446583i
$539$	−7.90104e11			−0.403213
$540$		0			0
$541$	8.64727e11			0.434002			0.217001	−	0.976171i	$-0.430373\pi$
$541$	8.64727e11			0.434002			0.217001	+	0.976171i	$0.430373\pi$
$542$			5.00298e11i			0.249018i
$543$		0			0
$544$	2.42263e10			0.0118602
$545$	1.67193e12	−	1.17393e12i	0.811773	−	0.569979i
$546$		0			0
$547$			1.90338e12i			0.909041i	0.890736	+	0.454521i	$0.150189\pi$
$547$			1.90338e12i			0.909041i	−0.890736	+	0.454521i	$0.849811\pi$
$548$			1.91135e12i			0.905371i
$549$		0			0
$550$	8.25052e11	+	2.97875e11i	0.384458	+	0.138804i
$551$	−1.53629e12			−0.710053
$552$		0			0
$553$		−	1.17132e12i		−	0.532615i
$554$	9.09939e11			0.410411
$555$		0			0
$556$	−2.86593e12			−1.27183
$557$		−	1.55326e12i		−	0.683747i	−0.939746	−	0.341874i	$-0.888939\pi$
$557$		−	1.55326e12i		−	0.683747i	0.939746	−	0.341874i	$-0.111061\pi$
$558$		0			0
$559$	−2.34521e12			−1.01585
$560$	−1.91990e11	−	2.73436e11i	−0.0824960	−	0.117492i
$561$		0			0
$562$			1.69448e12i			0.716512i
$563$		−	3.94106e12i		−	1.65320i	−0.562789	−	0.826601i	$-0.690272\pi$
$563$		−	3.94106e12i		−	1.65320i	0.562789	−	0.826601i	$-0.309728\pi$
$564$		0			0
$565$	4.27371e11	+	6.08668e11i	0.176436	+	0.251283i
$566$	−6.98964e11			−0.286273
$567$		0			0
$568$		−	2.26763e12i		−	0.914121i
$569$	−1.03063e11			−0.0412189			−0.0206094	−	0.999788i	$-0.506561\pi$
$569$	−1.03063e11			−0.0412189			−0.0206094	+	0.999788i	$0.506561\pi$
$570$		0			0
$571$	2.92191e12			1.15028			0.575141	−	0.818054i	$-0.304947\pi$
$571$	2.92191e12			1.15028			0.575141	+	0.818054i	$0.304947\pi$
$572$			2.49949e12i			0.976269i
$573$		0			0
$574$	−1.17821e12			−0.453023
$575$	−4.02511e12	−	1.45322e12i	−1.53558	−	0.554402i
$576$		0			0
$577$		−	2.67123e12i		−	1.00327i	−0.865078	−	0.501637i	$-0.832731\pi$
$577$		−	2.67123e12i		−	1.00327i	0.865078	−	0.501637i	$-0.167269\pi$
$578$		−	1.44902e12i		−	0.540008i
$579$		0			0
$580$	2.68703e12	−	1.88667e12i	0.985932	−	0.692263i
$581$	−2.27007e12			−0.826506
$582$		0			0
$583$		−	1.13761e12i		−	0.407837i
$584$	−2.83720e12			−1.00933
$585$		0			0
$586$	−1.34101e12			−0.469777
$587$		−	2.14358e12i		−	0.745191i	−0.927994	−	0.372596i	$-0.878468\pi$
$587$		−	2.14358e12i		−	0.745191i	0.927994	−	0.372596i	$-0.121532\pi$
$588$		0			0
$589$	−1.16744e12			−0.399681
$590$	7.36770e11	+	1.04932e12i	0.250321	+	0.356512i
$591$		0			0
$592$			3.64047e11i			0.121818i
$593$		−	2.79629e12i		−	0.928616i	−0.885674	−	0.464308i	$-0.846303\pi$
$593$		−	2.79629e12i		−	0.928616i	0.885674	−	0.464308i	$-0.153697\pi$
$594$		0			0
$595$	1.95782e10	−	1.37467e10i	0.00640394	−	0.00449647i
$596$	3.22149e12			1.04580
$597$		0			0
$598$			5.02152e12i			1.60576i
$599$	−5.67415e12			−1.80086			−0.900430	−	0.435002i	$-0.856748\pi$
$599$	−5.67415e12			−1.80086			−0.900430	+	0.435002i	$0.856748\pi$
$600$		0			0
$601$	3.70560e12			1.15857			0.579287	−	0.815124i	$-0.303331\pi$
$601$	3.70560e12			1.15857			0.579287	+	0.815124i	$0.303331\pi$
$602$		−	6.63579e11i		−	0.205925i
$603$		0			0
$604$	1.57937e12			0.482855
$605$	1.15215e12	−	8.08971e11i	0.349631	−	0.245490i
$606$		0			0
$607$		−	3.05665e12i		−	0.913894i	−0.889494	−	0.456947i	$-0.848943\pi$
$607$		−	3.05665e12i		−	0.913894i	0.889494	−	0.456947i	$-0.151057\pi$
$608$		−	1.45743e12i		−	0.432537i
$609$		0			0
$610$	1.27808e11	+	1.82027e11i	0.0373745	+	0.0532294i
$611$	−6.41698e12			−1.86271
$612$		0			0
$613$			1.97493e12i			0.564911i	0.959280	+	0.282456i	$0.0911490\pi$
$613$			1.97493e12i			0.564911i	−0.959280	+	0.282456i	$0.908851\pi$
$614$	2.31419e12			0.657116
$615$		0			0
$616$	−1.70570e12			−0.477299
$617$			3.84845e12i			1.06906i	0.845149	+	0.534531i	$0.179512\pi$
$617$			3.84845e12i			1.06906i	−0.845149	+	0.534531i	$0.820488\pi$
$618$		0			0
$619$	4.09595e11			0.112136			0.0560682	−	0.998427i	$-0.482144\pi$
$619$	4.09595e11			0.112136			0.0560682	+	0.998427i	$0.482144\pi$
$620$	2.04189e12	−	1.43370e12i	0.554971	−	0.389668i
$621$		0			0
$622$			1.71800e12i			0.460222i
$623$			1.43005e12i			0.380324i
$624$		0			0
$625$	−2.93490e12	−	2.43686e12i	−0.769366	−	0.638808i
$626$	−1.75676e12			−0.457222
$627$		0			0
$628$			7.63361e11i			0.195845i
$629$	−2.60661e10			−0.00663970
$630$		0			0
$631$	2.20647e12			0.554072			0.277036	−	0.960860i	$-0.410648\pi$
$631$	2.20647e12			0.554072			0.277036	+	0.960860i	$0.410648\pi$
$632$			2.88335e12i			0.718904i
$633$		0			0
$634$	−2.04135e12			−0.501782
$635$	−2.05579e12	−	2.92788e12i	−0.501760	−	0.714615i
$636$		0			0
$637$		−	4.03189e12i		−	0.970244i
$638$		−	2.90937e12i		−	0.695194i
$639$		0			0
$640$	2.06649e12	+	2.94312e12i	0.486881	+	0.693424i
$641$	7.59163e12			1.77613			0.888064	−	0.459720i	$-0.152050\pi$
$641$	7.59163e12			1.77613			0.888064	+	0.459720i	$0.152050\pi$
$642$		0			0
$643$		−	6.12938e12i		−	1.41406i	−0.707184	−	0.707029i	$-0.750035\pi$
$643$		−	6.12938e12i		−	1.41406i	0.707184	−	0.707029i	$-0.249965\pi$
$644$	3.45032e12			0.790446
$645$		0			0
$646$	1.14249e10			0.00258111
$647$		−	1.67991e12i		−	0.376893i	−0.982084	−	0.188446i	$-0.939655\pi$
$647$		−	1.67991e12i		−	0.376893i	0.982084	−	0.188446i	$-0.0603451\pi$
$648$		0			0
$649$	−2.75897e12			−0.610443
$650$	−1.52005e12	+	4.21023e12i	−0.334001	+	0.925115i
$651$		0			0
$652$		−	2.81786e12i		−	0.610668i
$653$		−	2.09466e12i		−	0.450821i	−0.974264	−	0.225410i	$-0.927628\pi$
$653$		−	2.09466e12i		−	0.450821i	0.974264	−	0.225410i	$-0.0723723\pi$
$654$		0			0
$655$	−6.41042e12	+	4.50101e12i	−1.36082	+	0.955486i
$656$	−1.22246e12			−0.257732
$657$		0			0
$658$		−	1.81569e12i		−	0.377594i
$659$	7.17038e11			0.148101			0.0740505	−	0.997254i	$-0.476407\pi$
$659$	7.17038e11			0.148101			0.0740505	+	0.997254i	$0.476407\pi$
$660$		0			0
$661$	4.83658e12			0.985444			0.492722	−	0.870187i	$-0.336002\pi$
$661$	4.83658e12			0.985444			0.492722	+	0.870187i	$0.336002\pi$
$662$			2.42485e11i			0.0490708i
$663$		0			0
$664$	5.58804e12			1.11559
$665$	−8.26987e11	−	1.17781e12i	−0.163984	−	0.233549i
$666$		0			0
$667$			1.41937e13i			2.77670i
$668$			6.58357e12i			1.27928i
$669$		0			0
$670$	−1.91548e12	+	1.34494e12i	−0.367233	+	0.257849i
$671$	−4.78602e11			−0.0911429
$672$		0			0
$673$			4.63436e12i			0.870808i	0.900235	+	0.435404i	$0.143394\pi$
$673$			4.63436e12i			0.870808i	−0.900235	+	0.435404i	$0.856606\pi$
$674$	8.20985e11			0.153238
$675$		0			0
$676$	−8.90907e12			−1.64086
$677$			6.94095e12i			1.26990i	0.772553	+	0.634950i	$0.218979\pi$
$677$			6.94095e12i			1.26990i	−0.772553	+	0.634950i	$0.781021\pi$
$678$		0			0
$679$	4.90151e12			0.884944
$680$	−4.81942e10	+	3.38391e10i	−0.00864380	+	0.00606917i
$681$		0			0
$682$		−	2.21085e12i		−	0.391318i
$683$			6.91635e12i			1.21614i	0.793883	+	0.608070i	$0.208056\pi$
$683$			6.91635e12i			1.21614i	−0.793883	+	0.608070i	$0.791944\pi$
$684$		0			0
$685$	−4.23255e12	−	6.02806e12i	−0.734504	−	1.04609i
$686$	3.28215e12			0.565848
$687$		0			0
$688$		−	6.88500e11i		−	0.117154i
$689$	5.80523e12			0.981371
$690$		0			0
$691$	1.45798e12			0.243277			0.121638	−	0.992574i	$-0.461185\pi$
$691$	1.45798e12			0.243277			0.121638	+	0.992574i	$0.461185\pi$
$692$		−	3.19351e12i		−	0.529408i
$693$		0			0
$694$	3.02987e12			0.495800
$695$	9.03864e12	−	6.34640e12i	1.46951	−	1.03180i
$696$		0			0
$697$		−	8.75294e10i		−	0.0140477i
$698$		−	1.98480e12i		−	0.316496i
$699$		0			0
$700$	2.89287e12	+	1.04444e12i	0.455394	+	0.164415i
$701$	5.87637e12			0.919132			0.459566	−	0.888144i	$-0.348005\pi$
$701$	5.87637e12			0.919132			0.459566	+	0.888144i	$0.348005\pi$
$702$		0			0
$703$			1.56811e12i			0.242147i
$704$	1.72407e12			0.264531
$705$		0			0
$706$	−2.88754e12			−0.437429
$707$		−	2.20642e12i		−	0.332124i
$708$		0			0
$709$	7.62910e12			1.13388			0.566938	−	0.823761i	$-0.308128\pi$
$709$	7.62910e12			1.13388			0.566938	+	0.823761i	$0.308128\pi$
$710$	2.08206e12	+	2.96530e12i	0.307489	+	0.437931i
$711$		0			0
$712$		−	3.52024e12i		−	0.513348i
$713$			1.07859e13i			1.56298i
$714$		0			0
$715$	−5.53496e12	−	7.88297e12i	−0.792022	−	1.12801i
$716$	−2.54169e11			−0.0361421
$717$		0			0
$718$		−	5.94167e12i		−	0.834350i
$719$	−2.38732e12			−0.333142			−0.166571	−	0.986029i	$-0.553270\pi$
$719$	−2.38732e12			−0.333142			−0.166571	+	0.986029i	$0.553270\pi$
$720$		0			0
$721$	−7.06523e10			−0.00973682
$722$			3.25612e12i			0.445947i
$723$		0			0
$724$	6.81373e12			0.921640
$725$	−4.29653e12	+	1.19005e13i	−0.577560	+	1.59972i
$726$		0			0
$727$		−	5.38271e12i		−	0.714654i	−0.933979	−	0.357327i	$-0.883688\pi$
$727$		−	5.38271e12i		−	0.714654i	0.933979	−	0.357327i	$-0.116312\pi$
$728$		−	8.70419e12i		−	1.14852i
$729$		0			0
$730$	3.71011e12	−	2.60502e12i	0.483542	−	0.339514i
$731$	4.92972e10			0.00638549
$732$		0			0
$733$			7.93743e12i			1.01557i	0.861483	+	0.507787i	$0.169536\pi$
$733$			7.93743e12i			1.01557i	−0.861483	+	0.507787i	$0.830464\pi$
$734$	−8.45272e12			−1.07489
$735$		0			0
$736$	−1.34652e13			−1.69146
$737$		−	5.03637e12i		−	0.628801i
$738$		0			0
$739$	2.88800e12			0.356203			0.178101	−	0.984012i	$-0.443004\pi$
$739$	2.88800e12			0.356203			0.178101	+	0.984012i	$0.443004\pi$
$740$	−1.92576e12	−	2.74270e12i	−0.236080	−	0.336229i
$741$		0			0
$742$			1.64259e12i			0.198936i
$743$		−	9.68056e12i		−	1.16533i	−0.812711	−	0.582667i	$-0.802009\pi$
$743$		−	9.68056e12i		−	1.16533i	0.812711	−	0.582667i	$-0.197991\pi$
$744$		0			0
$745$	−1.01600e13	+	7.13378e12i	−1.20835	+	0.848431i
$746$	−6.61898e12			−0.782467
$747$		0			0
$748$		−	5.25402e10i		−	0.00613670i
$749$	6.73286e12			0.781684
$750$		0			0
$751$	1.55121e12			0.177947			0.0889735	−	0.996034i	$-0.471641\pi$
$751$	1.55121e12			0.177947			0.0889735	+	0.996034i	$0.471641\pi$
$752$		−	1.88388e12i		−	0.214819i
$753$		0			0
$754$	1.48465e13			1.67283
$755$	−4.98105e12	+	3.49740e12i	−0.557904	+	0.391727i
$756$		0			0
$757$		−	2.05679e12i		−	0.227646i	−0.993501	−	0.113823i	$-0.963690\pi$
$757$		−	2.05679e12i		−	0.227646i	0.993501	−	0.113823i	$-0.0363096\pi$
$758$		−	8.76893e12i		−	0.964795i
$759$		0			0
$760$	2.03573e12	+	2.89932e12i	0.221340	+	0.315235i
$761$	−4.01826e12			−0.434317			−0.217159	−	0.976136i	$-0.569679\pi$
$761$	−4.01826e12			−0.434317			−0.217159	+	0.976136i	$0.569679\pi$
$762$		0			0
$763$		−	6.34735e12i		−	0.678004i
$764$	−9.85171e12			−1.04614
$765$		0			0
$766$	−2.21742e12			−0.232712
$767$		−	1.40790e13i		−	1.46890i
$768$		0			0
$769$	7.34878e12			0.757786			0.378893	−	0.925441i	$-0.376305\pi$
$769$	7.34878e12			0.757786			0.378893	+	0.925441i	$0.376305\pi$
$770$	2.23049e12	−	1.56612e12i	0.228661	−	0.160552i
$771$		0			0
$772$			1.22161e13i			1.23781i
$773$			7.03720e12i			0.708912i	0.935073	+	0.354456i	$0.115334\pi$
$773$			7.03720e12i			0.708912i	−0.935073	+	0.354456i	$0.884666\pi$
$774$		0			0
$775$	−3.26496e12	+	9.04327e12i	−0.325103	+	0.900467i
$776$	−1.20657e13			−1.19447
$777$		0			0
$778$		−	1.90814e12i		−	0.186725i
$779$	−5.26569e12			−0.512314
$780$		0			0
$781$	−7.79663e12			−0.749855
$782$		−	1.05554e11i		−	0.0100936i
$783$		0			0
$784$	1.18367e12			0.111894
$785$	−1.69041e12	−	2.40751e12i	−0.158884	−	0.226285i
$786$		0			0
$787$		−	3.89415e12i		−	0.361848i	−0.983497	−	0.180924i	$-0.942091\pi$
$787$		−	3.89415e12i		−	0.361848i	0.983497	−	0.180924i	$-0.0579088\pi$
$788$			1.27980e11i			0.0118243i
$789$		0			0
$790$	−2.64739e12	−	3.77046e12i	−0.241823	−	0.344408i
$791$	2.31076e12			0.209875
$792$		0			0
$793$		−	2.44230e12i		−	0.219315i
$794$	7.48402e10			0.00668256
$795$		0			0
$796$	−1.01510e13			−0.896188
$797$		−	6.27475e12i		−	0.550851i	−0.961322	−	0.275425i	$-0.911181\pi$
$797$		−	6.27475e12i		−	0.550851i	0.961322	−	0.275425i	$-0.0888187\pi$
$798$		0			0
$799$	1.34887e11			0.0117087
$800$	−1.12897e13	−	4.07600e12i	−0.974488	−	0.351827i
$801$		0			0
$802$		−	9.32743e12i		−	0.796118i
$803$			9.75496e12i			0.827953i
$804$		0			0
$805$	−1.08817e13	+	7.64049e12i	−0.913305	+	0.641269i
$806$	1.12819e13			0.941620
$807$		0			0
$808$			5.43136e12i			0.448289i
$809$	8.39256e12			0.688852			0.344426	−	0.938813i	$-0.388074\pi$
$809$	8.39256e12			0.688852			0.344426	+	0.938813i	$0.388074\pi$
$810$		0			0
$811$	4.06885e12			0.330277			0.165138	−	0.986270i	$-0.447193\pi$
$811$	4.06885e12			0.330277			0.165138	+	0.986270i	$0.447193\pi$
$812$		−	1.02011e13i		−	0.823464i
$813$		0			0
$814$	−2.96964e12			−0.237079
$815$	6.23996e12	+	8.88705e12i	0.495419	+	0.705584i
$816$		0			0
$817$		−	2.96568e12i		−	0.232876i
$818$		−	2.63189e12i		−	0.205532i
$819$		0			0
$820$	9.20990e12	−	6.46665e12i	0.711365	−	0.499479i
$821$	−1.00927e13			−0.775287			−0.387643	−	0.921809i	$-0.626711\pi$
$821$	−1.00927e13			−0.775287			−0.387643	+	0.921809i	$0.626711\pi$
$822$		0			0
$823$			1.21960e13i			0.926654i	0.886187	+	0.463327i	$0.153344\pi$
$823$			1.21960e13i			0.926654i	−0.886187	+	0.463327i	$0.846656\pi$
$824$	1.73919e11			0.0131424
$825$		0			0
$826$	3.98365e12			0.297763
$827$			1.65725e13i			1.23201i	0.787743	+	0.616003i	$0.211249\pi$
$827$			1.65725e13i			1.23201i	−0.787743	+	0.616003i	$0.788751\pi$
$828$		0			0
$829$	7.37572e12			0.542387			0.271193	−	0.962525i	$-0.412582\pi$
$829$	7.37572e12			0.542387			0.271193	+	0.962525i	$0.412582\pi$
$830$	−7.30730e12	+	5.13075e12i	−0.534448	+	0.375258i
$831$		0			0
$832$			8.79789e12i			0.636537i
$833$			8.47518e10i			0.00609883i
$834$		0			0
$835$	−1.45789e13	−	2.07635e13i	−1.03785	−	1.47812i
$836$	−3.16077e12			−0.223803
$837$		0			0
$838$			6.27516e12i			0.439569i
$839$	−7.96014e12			−0.554615			−0.277308	−	0.960781i	$-0.589442\pi$
$839$	−7.96014e12			−0.554615			−0.277308	+	0.960781i	$0.589442\pi$
$840$		0			0
$841$	2.74575e13			1.89269
$842$		−	5.20367e12i		−	0.356784i
$843$		0			0
$844$	6.93883e12			0.470701
$845$	2.80977e13	−	1.97285e13i	1.89590	−	1.33119i
$846$		0			0
$847$		−	4.37404e12i		−	0.292016i
$848$			1.70428e12i			0.113178i
$849$		0			0
$850$	3.19520e10	−	8.85006e10i	0.00209949	−	0.00581515i
$851$	1.44877e13			0.946928
$852$		0			0
$853$		−	7.49408e12i		−	0.484672i	−0.970192	−	0.242336i	$-0.922086\pi$
$853$		−	7.49408e12i		−	0.484672i	0.970192	−	0.242336i	$-0.0779136\pi$
$854$	6.91049e11			0.0444579
$855$		0			0
$856$	−1.65738e13			−1.05509
$857$		−	2.06124e12i		−	0.130531i	−0.997868	−	0.0652656i	$-0.979211\pi$
$857$		−	2.06124e12i		−	0.130531i	0.997868	−	0.0652656i	$-0.0207895\pi$
$858$		0			0
$859$	−1.61145e13			−1.00983			−0.504914	−	0.863169i	$-0.668476\pi$
$859$	−1.61145e13			−1.00983			−0.504914	+	0.863169i	$0.668476\pi$
$860$	3.64207e12	+	5.18709e12i	0.227041	+	0.323356i
$861$		0			0
$862$			7.47615e12i			0.461207i
$863$			1.14800e13i			0.704521i	0.935902	+	0.352261i	$0.114587\pi$
$863$			1.14800e13i			0.704521i	−0.935902	+	0.352261i	$0.885413\pi$
$864$		0			0
$865$	7.07181e12	+	1.00718e13i	0.429495	+	0.611694i
$866$	−1.19212e12			−0.0720260
$867$		0			0
$868$		−	7.75187e12i		−	0.463519i
$869$	9.91364e12			0.589718
$870$		0			0
$871$	2.57005e13			1.51307
$872$			1.56248e13i			0.915145i
$873$		0			0
$874$	−6.35005e12			−0.368108
$875$	−1.14365e13	+	3.11209e12i	−0.659561	+	0.179480i
$876$		0			0
$877$			1.14652e13i			0.654460i	0.944945	+	0.327230i	$0.106115\pi$
$877$			1.14652e13i			0.654460i	−0.944945	+	0.327230i	$0.893885\pi$
$878$			1.34536e13i			0.764035i
$879$		0			0
$880$	2.31426e12	−	1.62494e12i	0.130089	−	0.0913406i
$881$	2.36703e13			1.32377			0.661885	−	0.749605i	$-0.269757\pi$
$881$	2.36703e13			1.32377			0.661885	+	0.749605i	$0.269757\pi$
$882$		0			0
$883$		−	2.01686e13i		−	1.11648i	−0.829678	−	0.558242i	$-0.811476\pi$
$883$		−	2.01686e13i		−	1.11648i	0.829678	−	0.558242i	$-0.188524\pi$
$884$	2.68112e11			0.0147666
$885$		0			0
$886$	−4.15144e12			−0.226332
$887$			2.58905e13i			1.40438i	0.711992	+	0.702188i	$0.247793\pi$
$887$			2.58905e13i			1.40438i	−0.711992	+	0.702188i	$0.752207\pi$
$888$		0			0
$889$	−1.11155e13			−0.596856
$890$	3.23216e12	+	4.60330e12i	0.172678	+	0.245931i
$891$		0			0
$892$		−	2.05426e13i		−	1.08646i
$893$		−	8.11469e12i		−	0.427012i
$894$		0			0
$895$	8.01605e11	−	5.62839e11i	0.0417596	−	0.0293211i
$896$	1.11733e13			0.579157
$897$		0			0
$898$		−	4.36730e12i		−	0.224115i
$899$	3.18892e13			1.62826
$900$		0			0
$901$	−1.22028e11			−0.00616877
$902$		−	9.97197e12i		−	0.501593i
$903$		0			0
$904$	−5.68821e12			−0.283281
$905$	−2.14893e13	+	1.50885e13i	−1.06489	+	0.747702i
$906$		0			0
$907$			2.09156e13i			1.02621i	0.858325	+	0.513107i	$0.171506\pi$
$907$			2.09156e13i			1.02621i	−0.858325	+	0.513107i	$0.828494\pi$
$908$		−	1.20229e13i		−	0.586979i
$909$		0			0
$910$	7.99188e12	+	1.13822e13i	0.386334	+	0.550223i
$911$	−4.47158e12			−0.215094			−0.107547	−	0.994200i	$-0.534300\pi$
$911$	−4.47158e12			−0.215094			−0.107547	+	0.994200i	$0.534300\pi$
$912$		0			0
$913$		−	1.92130e13i		−	0.915118i
$914$	1.81223e13			0.858923
$915$		0			0
$916$	−6.11776e12			−0.287120
$917$			2.43366e13i			1.13657i
$918$		0			0
$919$	−1.06549e13			−0.492753			−0.246377	−	0.969174i	$-0.579240\pi$
$919$	−1.06549e13			−0.492753			−0.246377	+	0.969174i	$0.579240\pi$
$920$	2.67867e13	−	1.88080e13i	1.23275	−	0.865561i
$921$		0			0
$922$		−	1.24106e13i		−	0.565594i
$923$		−	3.97861e13i		−	1.80436i
$924$		0			0
$925$	1.21470e13	+	4.38554e12i	0.545547	+	0.196963i
$926$	−8.02461e12			−0.358653
$927$		0			0
$928$			3.98106e13i			1.76211i
$929$	−3.87854e13			−1.70843			−0.854217	−	0.519917i	$-0.825963\pi$
$929$	−3.87854e13			−1.70843			−0.854217	+	0.519917i	$0.825963\pi$
$930$		0			0
$931$	5.09859e12			0.222421
$932$			2.17520e13i			0.944340i
$933$		0			0
$934$	−1.20046e13			−0.516165
$935$	1.16347e11	+	1.65703e11i	0.00497854	+	0.00709052i
$936$		0			0
$937$			6.15535e11i			0.0260870i	0.999915	+	0.0130435i	$0.00415199\pi$
$937$			6.15535e11i			0.0260870i	−0.999915	+	0.0130435i	$0.995848\pi$
$938$			7.27197e12i			0.306718i
$939$		0			0
$940$	9.96543e12	+	1.41929e13i	0.416314	+	0.592921i
$941$	−3.81077e13			−1.58438			−0.792190	−	0.610275i	$-0.791059\pi$
$941$	−3.81077e13			−1.58438			−0.792190	+	0.610275i	$0.791059\pi$
$942$		0			0
$943$			4.86494e13i			2.00343i
$944$	4.13326e12			0.169402
$945$		0			0
$946$	5.61629e12			0.228003
$947$		−	2.37836e13i		−	0.960955i	−0.877007	−	0.480478i	$-0.840463\pi$
$947$		−	2.37836e13i		−	0.960955i	0.877007	−	0.480478i	$-0.159537\pi$
$948$		0			0
$949$	−4.97794e13			−1.99229
$950$	−5.32411e12	−	1.92221e12i	−0.212076	−	0.0765673i
$951$		0			0
$952$			1.82965e11i			0.00721942i
$953$			4.19004e12i			0.164551i	0.996610	+	0.0822755i	$0.0262187\pi$
$953$			4.19004e12i			0.164551i	−0.996610	+	0.0822755i	$0.973781\pi$
$954$		0			0
$955$	3.10706e13	−	2.18159e13i	1.20874	−	0.848709i
$956$	−3.86604e12			−0.149694
$957$		0			0
$958$			1.74824e12i			0.0670589i
$959$	−2.28850e13			−0.873710
$960$		0			0
$961$	−2.20685e12			−0.0834675
$962$		−	1.51540e13i		−	0.570480i
$963$		0			0
$964$	−3.15508e12			−0.117670
$965$	−2.70517e13	−	3.85274e13i	−1.00420	−	1.43020i
$966$		0			0
$967$		−	2.08426e13i		−	0.766537i	−0.923637	−	0.383268i	$-0.874798\pi$
$967$		−	2.08426e13i		−	0.766537i	0.923637	−	0.383268i	$-0.125202\pi$
$968$			1.07672e13i			0.394153i
$969$		0			0
$970$	1.57779e13	−	1.10783e13i	0.572236	−	0.401790i
$971$	1.13186e13			0.408606			0.204303	−	0.978908i	$-0.434507\pi$
$971$	1.13186e13			0.408606			0.204303	+	0.978908i	$0.434507\pi$
$972$		0			0
$973$		−	3.43144e13i		−	1.22735i
$974$	−2.28508e13			−0.813554
$975$		0			0
$976$	7.17002e11			0.0252928
$977$		−	3.51597e13i		−	1.23458i	−0.786735	−	0.617291i	$-0.788230\pi$
$977$		−	3.51597e13i		−	1.23458i	0.786735	−	0.617291i	$-0.211770\pi$
$978$		0			0
$979$	−1.21034e13			−0.421100
$980$	−8.91764e12	+	6.26144e12i	−0.308839	+	0.216849i
$981$		0			0
$982$			2.92195e12i			0.100270i
$983$		−	2.63427e12i		−	0.0899850i	−0.998987	−	0.0449925i	$-0.985674\pi$
$983$		−	2.63427e12i		−	0.0899850i	0.998987	−	0.0449925i	$-0.0143264\pi$
$984$		0			0
$985$	−2.83404e11	−	4.03628e11i	−0.00959274	−	0.0136621i
$986$	−3.12078e11			−0.0105152
$987$		0			0
$988$		−	1.61294e13i		−	0.538532i
$989$	−2.73997e13			−0.910674
$990$		0			0
$991$	−9.08136e12			−0.299102			−0.149551	−	0.988754i	$-0.547783\pi$
$991$	−9.08136e12			−0.299102			−0.149551	+	0.988754i	$0.547783\pi$
$992$			3.02524e13i			0.991874i
$993$		0			0
$994$	1.12575e13			0.365766
$995$	3.20145e13	−	2.24787e13i	1.03548	−	0.727054i
$996$		0			0
$997$		−	3.23962e13i		−	1.03840i	−0.854652	−	0.519201i	$-0.826229\pi$
$997$		−	3.23962e13i		−	1.03840i	0.854652	−	0.519201i	$-0.173771\pi$
$998$			2.60419e13i			0.830969i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	45.10.b.d.19.3	✓	8
3.2	odd	2	inner	45.10.b.d.19.6	yes	8
5.2	odd	4		225.10.a.x.1.5		8
5.3	odd	4		225.10.a.x.1.4		8
5.4	even	2	inner	45.10.b.d.19.5	yes	8
15.2	even	4		225.10.a.x.1.3		8
15.8	even	4		225.10.a.x.1.6		8
15.14	odd	2	inner	45.10.b.d.19.4	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
45.10.b.d.19.3	✓	8	1.1	even	1	trivial
45.10.b.d.19.4	yes	8	15.14	odd	2	inner
45.10.b.d.19.5	yes	8	5.4	even	2	inner
45.10.b.d.19.6	yes	8	3.2	odd	2	inner
225.10.a.x.1.3		8	15.2	even	4
225.10.a.x.1.4		8	5.3	odd	4
225.10.a.x.1.5		8	5.2	odd	4
225.10.a.x.1.6		8	15.8	even	4

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:	\( N \)	\(=\)	\( 45 = 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 10 \)
Character orbit:	\([\chi]\)	\(=\)	45.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-12.2206i q^{2} +362.657 q^{4} +(-1143.76 + 803.080i) q^{5} +4342.19i q^{7} -10688.8i q^{8} +(9814.11 + 13977.4i) q^{10} -36750.7 q^{11} -187538. i q^{13} +53064.1 q^{14} +55056.9 q^{16} +3942.12i q^{17} +237154. q^{19} +(-414793. + 291243. i) q^{20} +449115. i q^{22} -2.19106e6i q^{23} +(663249. - 1.83706e6i) q^{25} -2.29183e6 q^{26} +1.57473e6i q^{28} -6.47801e6 q^{29} -4.92268e6 q^{31} -6.14550e6i q^{32} +48175.0 q^{34} +(-3.48712e6 - 4.96642e6i) q^{35} +6.61220e6i q^{37} -2.89817e6i q^{38} +(8.58398e6 + 1.22254e7i) q^{40} -2.22036e7 q^{41} -1.25053e7i q^{43} -1.33279e7 q^{44} -2.67760e7 q^{46} -3.42169e7i q^{47} +2.14990e7 q^{49} +(-2.24500e7 - 8.10529e6i) q^{50} -6.80121e7i q^{52} +3.09549e7i q^{53} +(4.20340e7 - 2.95138e7i) q^{55} +4.64129e7 q^{56} +7.91651e7i q^{58} +7.50725e7 q^{59} +1.30229e7 q^{61} +6.01580e7i q^{62} -4.69125e7 q^{64} +(1.50608e8 + 2.14499e8i) q^{65} +1.37041e8i q^{67} +1.42964e6i q^{68} +(-6.06925e7 + 4.26147e7i) q^{70} +2.12149e8 q^{71} -2.65436e8i q^{73} +8.08050e7 q^{74} +8.60058e7 q^{76} -1.59578e8i q^{77} -2.69754e8 q^{79} +(-6.29719e7 + 4.42151e7i) q^{80} +2.71341e8i q^{82} +5.22793e8i q^{83} +(-3.16584e6 - 4.50884e6i) q^{85} -1.52822e8 q^{86} +3.92822e8i q^{88} +3.29338e8 q^{89} +8.14326e8 q^{91} -7.94603e8i q^{92} -4.18150e8 q^{94} +(-2.71248e8 + 1.90454e8i) q^{95} -1.12881e9i q^{97} -2.62731e8i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 144 q^{4} - 102680 q^{10} - 1009088 q^{16} - 2079872 q^{19} - 595720 q^{25} + 12077344 q^{31} + 25286480 q^{34} + 18248480 q^{40} + 3119840 q^{46} - 174619144 q^{49} - 105991200 q^{55} - 165845744 q^{61}+ \cdots - 4117219360 q^{94}+O(q^{100}) \) 8 * q - 144 * q^4 - 102680 * q^10 - 1009088 * q^16 - 2079872 * q^19 - 595720 * q^25 + 12077344 * q^31 + 25286480 * q^34 + 18248480 * q^40 + 3119840 * q^46 - 174619144 * q^49 - 105991200 * q^55 - 165845744 * q^61 + 91038464 * q^64 + 693705600 * q^70 + 1551352896 * q^76 - 1098932768 * q^79 + 965125360 * q^85 - 1465128000 * q^91 - 4117219360 * q^94

Introduction

L-functions

Modular forms

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Database

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