LMFDB - Embedded newform 112.8.i.a.65.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [112,8,Mod(65,112)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://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);

sage: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")

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:traces = [4,0,-56] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	no
Analytic conductor:	$34.9871228542$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{2389})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} + 598x^{2} + 597x + 356409 $ x^4 - x^3 + 598x^2 + 597x + 356409
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 14)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			65.1
Root			$12.4693 + 21.5975i$ of defining polynomial
Character	$\chi$	$=$	112.65
Dual form			112.8.i.a.81.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-38.4387 - 66.5778i) q^{3} +(10.6226 - 18.3989i) q^{5} +(642.284 - 641.104i) q^{7} +(-1861.57 + 3224.33i) q^{9} +(1633.07 + 2828.56i) q^{11} +10238.0 q^{13} -1633.28 q^{15} +(18264.6 + 31635.1i) q^{17} +(-16200.6 + 28060.3i) q^{19} +(-67371.9 - 18118.6i) q^{21} +(8370.58 - 14498.3i) q^{23} +(38836.8 + 67267.3i) q^{25} +118094. q^{27} +43984.4 q^{29} +(66921.6 + 115912. i) q^{31} +(125546. - 217452. i) q^{33} +(-4972.88 - 18627.5i) q^{35} +(289939. - 502188. i) q^{37} +(-393534. - 681621. i) q^{39} -532967. q^{41} +365375. q^{43} +(39549.4 + 68501.5i) q^{45} +(-103432. + 179149. i) q^{47} +(1513.36 - 823542. i) q^{49} +(1.40413e6 - 2.43203e6i) q^{51} +(544104. + 942416. i) q^{53} +69389.9 q^{55} +2.49092e6 q^{57} +(80826.2 + 139995. i) q^{59} +(422154. - 731192. i) q^{61} +(871477. + 3.26439e6i) q^{63} +(108754. - 188367. i) q^{65} +(613923. + 1.06335e6i) q^{67} -1.28702e6 q^{69} -1.10759e6 q^{71} +(-819423. - 1.41928e6i) q^{73} +(2.98567e6 - 5.17134e6i) q^{75} +(2.86230e6 + 769770. i) q^{77} +(-2.49711e6 + 4.32512e6i) q^{79} +(-468133. - 810829. i) q^{81} +3.38355e6 q^{83} +776069. q^{85} +(-1.69070e6 - 2.92838e6i) q^{87} +(-3.26848e6 + 5.66118e6i) q^{89} +(6.57568e6 - 6.56361e6i) q^{91} +(5.14476e6 - 8.91098e6i) q^{93} +(344185. + 596146. i) q^{95} +1.35914e6 q^{97} -1.21603e7 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 56 q^{3} + 238 q^{5} - 168 q^{7} - 1972 q^{9} + 5848 q^{11} + 2632 q^{13} + 5784 q^{15} + 47642 q^{17} - 41048 q^{19} - 169582 q^{21} - 49316 q^{23} + 108816 q^{25} + 400624 q^{27} + 345640 q^{29}+ \cdots - 15278208 q^{99}+O(q^{100}) $ 4 * q - 56 * q^3 + 238 * q^5 - 168 * q^7 - 1972 * q^9 + 5848 * q^11 + 2632 * q^13 + 5784 * q^15 + 47642 * q^17 - 41048 * q^19 - 169582 * q^21 - 49316 * q^23 + 108816 * q^25 + 400624 * q^27 + 345640 * q^29 - 70252 * q^31 + 197190 * q^33 + 36904 * q^35 + 88166 * q^37 - 973336 * q^39 - 2312520 * q^41 - 115088 * q^43 - 300468 * q^45 - 1412292 * q^47 + 465892 * q^49 + 2576256 * q^51 + 2361174 * q^53 + 1258040 * q^55 + 4620796 * q^57 + 1842512 * q^59 - 1278242 * q^61 + 2121448 * q^63 - 1716372 * q^65 - 2121480 * q^67 - 5332236 * q^69 - 7200320 * q^71 + 1634682 * q^73 + 5321176 * q^75 + 1416226 * q^77 - 9192604 * q^79 - 3049498 * q^81 + 56560 * q^83 + 6369676 * q^85 - 691656 * q^87 - 8936550 * q^89 + 26111120 * q^91 + 14550490 * q^93 + 2562604 * q^95 + 11748408 * q^97 - 15278208 * q^99

Character values

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

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

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
$3$	−38.4387	−	66.5778i	−0.821948	−	1.42366i	−0.904230	−	0.427046i	$-0.859554\pi$
$3$	−38.4387	−	66.5778i	−0.821948	−	1.42366i	0.0822818	−	0.996609i	$-0.473779\pi$
$4$		0			0
$5$	10.6226	−	18.3989i	0.0380046	−	0.0658259i	−0.846397	−	0.532552i	$-0.821233\pi$
$5$	10.6226	−	18.3989i	0.0380046	−	0.0658259i	0.884402	+	0.466726i	$0.154567\pi$
$6$		0			0
$7$	642.284	−	641.104i	0.707756	−	0.706457i
$7$	642.284	−	641.104i	0.707756	−	0.706457i
$8$		0			0
$9$	−1861.57	+	3224.33i	−0.851197	+	1.47432i
$10$		0			0
$11$	1633.07	+	2828.56i	0.369939	+	0.640754i	0.989556	−	0.144151i	$-0.0460451\pi$
$11$	1633.07	+	2828.56i	0.369939	+	0.640754i	−0.619616	+	0.784905i	$0.712712\pi$
$12$		0			0
$13$	10238.0			1.29245			0.646223	−	0.763149i	$-0.276348\pi$
$13$	10238.0			1.29245			0.646223	+	0.763149i	$0.276348\pi$
$14$		0			0
$15$	−1633.28			−0.124951
$16$		0			0
$17$	18264.6	+	31635.1i	0.901650	+	1.56170i	0.825352	+	0.564619i	$0.190977\pi$
$17$	18264.6	+	31635.1i	0.901650	+	1.56170i	0.0762982	+	0.997085i	$0.475690\pi$
$18$		0			0
$19$	−16200.6	+	28060.3i	−0.541868	+	0.938543i	0.456929	+	0.889503i	$0.348949\pi$
$19$	−16200.6	+	28060.3i	−0.541868	+	0.938543i	−0.998797	+	0.0490398i	$0.984384\pi$
$20$		0			0
$21$	−67371.9	−	18118.6i	−1.58749	−	0.426930i
$22$		0			0
$23$	8370.58	−	14498.3i	0.143452	−	0.248467i	−0.785342	−	0.619062i	$-0.787513\pi$
$23$	8370.58	−	14498.3i	0.143452	−	0.248467i	0.928795	+	0.370595i	$0.120846\pi$
$24$		0			0
$25$	38836.8	+	67267.3i	0.497111	+	0.861022i
$26$		0			0
$27$	118094.			1.15466
$28$		0			0
$29$	43984.4			0.334893			0.167446	−	0.985881i	$-0.446448\pi$
$29$	43984.4			0.334893			0.167446	+	0.985881i	$0.446448\pi$
$30$		0			0
$31$	66921.6	+	115912.i	0.403460	+	0.698813i	0.994141	−	0.108092i	$-0.0344742\pi$
$31$	66921.6	+	115912.i	0.403460	+	0.698813i	−0.590681	+	0.806905i	$0.701141\pi$
$32$		0			0
$33$	125546.	−	217452.i	0.608142	−	1.05333i
$34$		0			0
$35$	−4972.88	−	18627.5i	−0.0196052	−	0.0734373i
$36$		0			0
$37$	289939.	−	502188.i	0.941022	−	1.62990i	0.177496	−	0.984122i	$-0.443200\pi$
$37$	289939.	−	502188.i	0.941022	−	1.62990i	0.763526	−	0.645777i	$-0.223466\pi$
$38$		0			0
$39$	−393534.	−	681621.i	−1.06232	−	1.84000i
$40$		0			0
$41$	−532967.			−1.20769			−0.603847	−	0.797100i	$-0.706366\pi$
$41$	−532967.			−1.20769			−0.603847	+	0.797100i	$0.706366\pi$
$42$		0			0
$43$	365375.			0.700809			0.350404	−	0.936598i	$-0.386044\pi$
$43$	365375.			0.700809			0.350404	+	0.936598i	$0.386044\pi$
$44$		0			0
$45$	39549.4	+	68501.5i	0.0646987	+	0.112062i
$46$		0			0
$47$	−103432.	+	179149.i	−0.145315	+	0.251693i	−0.929491	−	0.368846i	$-0.879753\pi$
$47$	−103432.	+	179149.i	−0.145315	+	0.251693i	0.784175	+	0.620539i	$0.213086\pi$
$48$		0			0
$49$	1513.36	−	823542.i	0.00183762	−	0.999998i
$50$		0			0
$51$	1.40413e6	−	2.43203e6i	1.48222	−	2.56728i
$52$		0			0
$53$	544104.	+	942416.i	0.502015	+	0.869515i	0.999997	+	0.00232824i	$0.000741104\pi$
$53$	544104.	+	942416.i	0.502015	+	0.869515i	−0.497982	+	0.867187i	$0.665926\pi$
$54$		0			0
$55$	69389.9			0.0562376
$56$		0			0
$57$	2.49092e6			1.78155
$58$		0			0
$59$	80826.2	+	139995.i	0.0512354	+	0.0887423i	0.890506	−	0.454972i	$-0.150351\pi$
$59$	80826.2	+	139995.i	0.0512354	+	0.0887423i	−0.839270	+	0.543714i	$0.817017\pi$
$60$		0			0
$61$	422154.	−	731192.i	0.238131	−	0.412456i	−0.722047	−	0.691844i	$-0.756798\pi$
$61$	422154.	−	731192.i	0.238131	−	0.412456i	0.960178	+	0.279389i	$0.0901318\pi$
$62$		0			0
$63$	871477.	+	3.26439e6i	0.439101	+	1.64479i
$64$		0			0
$65$	108754.	−	188367.i	0.0491188	−	0.0850763i
$66$		0			0
$67$	613923.	+	1.06335e6i	0.249375	+	0.431929i	0.963352	−	0.268239i	$-0.0864416\pi$
$67$	613923.	+	1.06335e6i	0.249375	+	0.431929i	−0.713978	+	0.700168i	$0.753108\pi$
$68$		0			0
$69$	−1.28702e6			−0.471642
$70$		0			0
$71$	−1.10759e6			−0.367259			−0.183630	−	0.982995i	$-0.558785\pi$
$71$	−1.10759e6			−0.367259			−0.183630	+	0.982995i	$0.558785\pi$
$72$		0			0
$73$	−819423.	−	1.41928e6i	−0.246535	−	0.427011i	0.716027	−	0.698072i	$-0.245959\pi$
$73$	−819423.	−	1.41928e6i	−0.246535	−	0.427011i	−0.962562	+	0.271062i	$0.912625\pi$
$74$		0			0
$75$	2.98567e6	−	5.17134e6i	0.817199	−	1.41543i
$76$		0			0
$77$	2.86230e6	+	769770.i	0.714492	+	0.192151i
$78$		0			0
$79$	−2.49711e6	+	4.32512e6i	−0.569826	+	0.986968i	0.426757	+	0.904366i	$0.359656\pi$
$79$	−2.49711e6	+	4.32512e6i	−0.569826	+	0.986968i	−0.996583	+	0.0826011i	$0.973677\pi$
$80$		0			0
$81$	−468133.	−	810829.i	−0.0978749	−	0.169524i
$82$		0			0
$83$	3.38355e6			0.649531			0.324765	−	0.945795i	$-0.394715\pi$
$83$	3.38355e6			0.649531			0.324765	+	0.945795i	$0.394715\pi$
$84$		0			0
$85$	776069.			0.137067
$86$		0			0
$87$	−1.69070e6	−	2.92838e6i	−0.275264	−	0.476772i
$88$		0			0
$89$	−3.26848e6	+	5.66118e6i	−0.491452	+	0.851220i	−0.999952	−	0.00984263i	$-0.996867\pi$
$89$	−3.26848e6	+	5.66118e6i	−0.491452	+	0.851220i	0.508500	+	0.861062i	$0.330200\pi$
$90$		0			0
$91$	6.57568e6	−	6.56361e6i	0.914736	−	0.913057i
$92$		0			0
$93$	5.14476e6	−	8.91098e6i	0.663246	−	1.14878i
$94$		0			0
$95$	344185.	+	596146.i	0.0411869	+	0.0713379i
$96$		0			0
$97$	1.35914e6			0.151204			0.0756022	−	0.997138i	$-0.475912\pi$
$97$	1.35914e6			0.151204			0.0756022	+	0.997138i	$0.475912\pi$
$98$		0			0
$99$	−1.21603e7			−1.25956
$100$		0			0
$101$	−5.84618e6	−	1.01259e7i	−0.564608	−	0.977930i	−0.997086	−	0.0762853i	$-0.975694\pi$
$101$	−5.84618e6	−	1.01259e7i	−0.564608	−	0.977930i	0.432478	−	0.901644i	$-0.357639\pi$
$102$		0			0
$103$	−1.08658e6	+	1.88202e6i	−0.0979790	+	0.169705i	−0.910848	−	0.412742i	$-0.864571\pi$
$103$	−1.08658e6	+	1.88202e6i	−0.0979790	+	0.169705i	0.812869	+	0.582447i	$0.197904\pi$
$104$		0			0
$105$	−1.04903e6	+	1.04710e6i	−0.0884349	+	0.0882726i
$106$		0			0
$107$	5.38290e6	−	9.32345e6i	0.424789	−	0.735756i	−0.571612	−	0.820524i	$-0.693682\pi$
$107$	5.38290e6	−	9.32345e6i	0.424789	−	0.735756i	0.996401	+	0.0847685i	$0.0270151\pi$
$108$		0			0
$109$	6.18768e6	+	1.07174e7i	0.457652	+	0.792676i	0.998836	−	0.0482280i	$-0.0153574\pi$
$109$	6.18768e6	+	1.07174e7i	0.457652	+	0.792676i	−0.541185	+	0.840904i	$0.682024\pi$
$110$		0			0
$111$	−4.45794e7			−3.09388
$112$		0			0
$113$	−1.93909e7			−1.26422			−0.632112	−	0.774877i	$-0.717812\pi$
$113$	−1.93909e7			−1.26422			−0.632112	+	0.774877i	$0.717812\pi$
$114$		0			0
$115$	−177835.	−	308019.i	−0.0109037	−	0.0188858i
$116$		0			0
$117$	−1.90587e7	+	3.30106e7i	−1.10013	+	1.90547i
$118$		0			0
$119$	3.20125e7	+	8.60925e6i	1.74142	+	0.468329i
$120$		0			0
$121$	4.40974e6	−	7.63790e6i	0.226290	−	0.391945i
$122$		0			0
$123$	2.04866e7	+	3.54838e7i	0.992662	+	1.71934i
$124$		0			0
$125$	3.30997e6			0.151579
$126$		0			0
$127$	1.41475e6			0.0612868			0.0306434	−	0.999530i	$-0.490244\pi$
$127$	1.41475e6			0.0612868			0.0306434	+	0.999530i	$0.490244\pi$
$128$		0			0
$129$	−1.40446e7	−	2.43259e7i	−0.576028	−	0.997711i
$130$		0			0
$131$	−1.77615e7	+	3.07638e7i	−0.690287	+	1.19561i	0.281457	+	0.959574i	$0.409182\pi$
$131$	−1.77615e7	+	3.07638e7i	−0.690287	+	1.19561i	−0.971744	+	0.236038i	$0.924151\pi$
$132$		0			0
$133$	7.58418e6	+	2.84089e7i	0.279530	+	1.04707i
$134$		0			0
$135$	1.25447e6	−	2.17280e6i	0.0438824	−	0.0760066i
$136$		0			0
$137$	2.21508e7	+	3.83663e7i	0.735982	+	1.27476i	0.954291	+	0.298878i	$0.0966125\pi$
$137$	2.21508e7	+	3.83663e7i	0.735982	+	1.27476i	−0.218309	+	0.975880i	$0.570054\pi$
$138$		0			0
$139$	−2.97789e7			−0.940496			−0.470248	−	0.882534i	$-0.655836\pi$
$139$	−2.97789e7			−0.940496			−0.470248	+	0.882534i	$0.655836\pi$
$140$		0			0
$141$	1.59031e7			0.477766
$142$		0			0
$143$	1.67193e7	+	2.89587e7i	0.478126	+	0.828139i
$144$		0			0
$145$	467229.	−	809264.i	0.0127275	−	0.0220446i
$146$		0			0
$147$	−5.48877e7	+	3.15551e7i	−1.42516	+	0.819330i
$148$		0			0
$149$	2.04028e7	−	3.53387e7i	0.505286	−	0.875181i	−0.494695	−	0.869067i	$-0.664720\pi$
$149$	2.04028e7	−	3.53387e7i	0.505286	−	0.875181i	0.999981	−	0.00611484i	$-0.00194642\pi$
$150$		0			0
$151$	−1.78291e7	−	3.08810e7i	−0.421416	−	0.729914i	0.574662	−	0.818391i	$-0.305133\pi$
$151$	−1.78291e7	−	3.08810e7i	−0.421416	−	0.729914i	−0.996078	+	0.0884767i	$0.971800\pi$
$152$		0			0
$153$	−1.36003e8			−3.06993
$154$		0			0
$155$	2.84353e6			0.0613333
$156$		0			0
$157$	2.60341e7	+	4.50923e7i	0.536899	+	0.929937i	0.999069	+	0.0431455i	$0.0137379\pi$
$157$	2.60341e7	+	4.50923e7i	0.536899	+	0.929937i	−0.462169	+	0.886792i	$0.652929\pi$
$158$		0			0
$159$	4.18293e7	−	7.24505e7i	0.825260	−	1.42939i
$160$		0			0
$161$	−3.91862e6	−	1.46784e7i	−0.0740018	−	0.277197i
$162$		0			0
$163$	2.72230e7	−	4.71516e7i	0.492356	−	0.852785i	−0.507605	−	0.861590i	$-0.669469\pi$
$163$	2.72230e7	−	4.71516e7i	0.492356	−	0.852785i	0.999961	+	0.00880425i	$0.00280252\pi$
$164$		0			0
$165$	−2.66726e6	−	4.61982e6i	−0.0462244	−	0.0800629i
$166$		0			0
$167$	3.69146e7			0.613325			0.306662	−	0.951818i	$-0.400788\pi$
$167$	3.69146e7			0.613325			0.306662	+	0.951818i	$0.400788\pi$
$168$		0			0
$169$	4.20675e7			0.670414
$170$		0			0
$171$	−6.03170e7	−	1.04472e8i	−0.922473	−	1.59777i
$172$		0			0
$173$	3.84918e7	−	6.66697e7i	0.565206	−	0.978965i	−0.431825	−	0.901958i	$-0.642130\pi$
$173$	3.84918e7	−	6.66697e7i	0.565206	−	0.978965i	0.997031	−	0.0770076i	$-0.0245366\pi$
$174$		0			0
$175$	6.80696e7	+	1.83063e7i	0.960108	+	0.258206i
$176$		0			0
$177$	6.21371e6	−	1.07625e7i	0.0842256	−	0.145883i
$178$		0			0
$179$	2.50183e7	+	4.33329e7i	0.326041	+	0.564719i	0.981722	−	0.190318i	$-0.0609520\pi$
$179$	2.50183e7	+	4.33329e7i	0.326041	+	0.564719i	−0.655682	+	0.755037i	$0.727619\pi$
$180$		0			0
$181$	1.63228e7			0.204606			0.102303	−	0.994753i	$-0.467379\pi$
$181$	1.63228e7			0.204606			0.102303	+	0.994753i	$0.467379\pi$
$182$		0			0
$183$	−6.49082e7			−0.782926
$184$		0			0
$185$	−6.15980e6	−	1.06691e7i	−0.0715263	−	0.123887i
$186$		0			0
$187$	−5.96546e7	+	1.03325e8i	−0.667112	+	1.15547i
$188$		0			0
$189$	7.58498e7	−	7.57106e7i	0.817219	−	0.815718i
$190$		0			0
$191$	7.45240e7	−	1.29079e8i	0.773890	−	1.34042i	−0.161526	−	0.986869i	$-0.551641\pi$
$191$	7.45240e7	−	1.29079e8i	0.773890	−	1.34042i	0.935416	−	0.353549i	$-0.115025\pi$
$192$		0			0
$193$	−3.16993e7	−	5.49049e7i	−0.317395	−	0.549744i	0.662549	−	0.749019i	$-0.269475\pi$
$193$	−3.16993e7	−	5.49049e7i	−0.317395	−	0.549744i	−0.979944	+	0.199275i	$0.936141\pi$
$194$		0			0
$195$	−1.67214e7			−0.161492
$196$		0			0
$197$	2.93791e7			0.273783			0.136892	−	0.990586i	$-0.456289\pi$
$197$	2.93791e7			0.273783			0.136892	+	0.990586i	$0.456289\pi$
$198$		0			0
$199$	2.45848e7	+	4.25821e7i	0.221147	+	0.383037i	0.955156	−	0.296102i	$-0.0956867\pi$
$199$	2.45848e7	+	4.25821e7i	0.221147	+	0.383037i	−0.734010	+	0.679139i	$0.762353\pi$
$200$		0			0
$201$	4.71968e7	−	8.17473e7i	0.409946	−	0.710047i
$202$		0			0
$203$	2.82505e7	−	2.81986e7i	0.237022	−	0.236587i
$204$		0			0
$205$	−5.66150e6	+	9.80600e6i	−0.0458979	+	0.0794975i
$206$		0			0
$207$	3.11648e7	+	5.39790e7i	0.244213	+	0.422988i
$208$		0			0
$209$	−1.05827e8			−0.801834
$210$		0			0
$211$	2.55314e8			1.87105			0.935526	−	0.353258i	$-0.114926\pi$
$211$	2.55314e8			1.87105			0.935526	+	0.353258i	$0.114926\pi$
$212$		0			0
$213$	4.25741e7	+	7.37406e7i	0.301868	+	0.522851i
$214$		0			0
$215$	3.88124e6	−	6.72250e6i	0.0266339	−	0.0461314i
$216$		0			0
$217$	1.17294e8	+	3.15444e7i	0.779232	+	0.209562i
$218$		0			0
$219$	−6.29951e7	+	1.09111e8i	−0.405277	+	0.701961i
$220$		0			0
$221$	1.86992e8	+	3.23880e8i	1.16533	+	2.01842i
$222$		0			0
$223$	3.35491e7			0.202588			0.101294	−	0.994857i	$-0.467702\pi$
$223$	3.35491e7			0.202588			0.101294	+	0.994857i	$0.467702\pi$
$224$		0			0
$225$	−2.89189e8			−1.69256
$226$		0			0
$227$	7.55391e6	+	1.30838e7i	0.0428629	+	0.0742407i	0.886661	−	0.462420i	$-0.153019\pi$
$227$	7.55391e6	+	1.30838e7i	0.0428629	+	0.0742407i	−0.843798	+	0.536661i	$0.819685\pi$
$228$		0			0
$229$	5.15808e7	−	8.93406e7i	0.283834	−	0.491615i	−0.688492	−	0.725244i	$-0.741727\pi$
$229$	5.15808e7	−	8.93406e7i	0.283834	−	0.491615i	0.972326	+	0.233629i	$0.0750603\pi$
$230$		0			0
$231$	−5.87734e7	−	2.20154e8i	−0.313718	−	1.17513i
$232$		0			0
$233$	−5.19814e7	+	9.00345e7i	−0.269217	+	0.466298i	−0.968660	−	0.248391i	$-0.920098\pi$
$233$	−5.19814e7	+	9.00345e7i	−0.269217	+	0.466298i	0.699443	+	0.714689i	$0.253432\pi$
$234$		0			0
$235$	2.19743e6	+	3.80606e6i	0.0110453	+	0.0191310i
$236$		0			0
$237$	3.83942e8			1.87347
$238$		0			0
$239$	−1.40545e8			−0.665919			−0.332960	−	0.942941i	$-0.608047\pi$
$239$	−1.40545e8			−0.665919			−0.332960	+	0.942941i	$0.608047\pi$
$240$		0			0
$241$	−1.52501e8	−	2.64139e8i	−0.701798	−	1.21555i	−0.967835	−	0.251587i	$-0.919047\pi$
$241$	−1.52501e8	−	2.64139e8i	−0.701798	−	1.21555i	0.266036	−	0.963963i	$-0.414286\pi$
$242$		0			0
$243$	9.31470e7	−	1.61335e8i	0.416435	−	0.721286i
$244$		0			0
$245$	−1.51362e7	−	8.77600e6i	−0.0657559	−	0.0381255i
$246$		0			0
$247$	−1.65861e8	+	2.87280e8i	−0.700335	+	1.21302i
$248$		0			0
$249$	−1.30059e8	−	2.25269e8i	−0.533880	−	0.924708i
$250$		0			0
$251$	−7.20544e7			−0.287609			−0.143804	−	0.989606i	$-0.545934\pi$
$251$	−7.20544e7			−0.287609			−0.143804	+	0.989606i	$0.545934\pi$
$252$		0			0
$253$	5.46790e7			0.212275
$254$		0			0
$255$	−2.98311e7	−	5.16689e7i	−0.112662	−	0.195137i
$256$		0			0
$257$	2.89343e6	−	5.01156e6i	0.0106328	−	0.0184165i	−0.860660	−	0.509180i	$-0.829949\pi$
$257$	2.89343e6	−	5.01156e6i	0.0106328	−	0.0184165i	0.871293	+	0.490763i	$0.163282\pi$
$258$		0			0
$259$	−1.35732e8	−	5.08428e8i	−0.485438	−	1.81836i
$260$		0			0
$261$	−8.18799e7	+	1.41820e8i	−0.285060	+	0.493738i
$262$		0			0
$263$	1.16636e8	+	2.02020e8i	0.395356	+	0.684778i	0.993147	−	0.116875i	$-0.0372877\pi$
$263$	1.16636e8	+	2.02020e8i	0.395356	+	0.684778i	−0.597790	+	0.801653i	$0.703954\pi$
$264$		0			0
$265$	2.31192e7			0.0763155
$266$		0			0
$267$	5.02545e8			1.61579
$268$		0			0
$269$	2.06500e8	+	3.57668e8i	0.646825	+	1.12033i	0.983877	+	0.178848i	$0.0572371\pi$
$269$	2.06500e8	+	3.57668e8i	0.646825	+	1.12033i	−0.337051	+	0.941486i	$0.609430\pi$
$270$		0			0
$271$	1.45930e7	−	2.52758e7i	0.0445401	−	0.0771457i	−0.842896	−	0.538077i	$-0.819151\pi$
$271$	1.45930e7	−	2.52758e7i	0.0445401	−	0.0771457i	0.887436	+	0.460931i	$0.152484\pi$
$272$		0			0
$273$	−6.89751e8	−	1.85498e8i	−2.05174	−	0.551784i
$274$		0			0
$275$	−1.26847e8	+	2.19705e8i	−0.367802	+	0.637052i
$276$		0			0
$277$	−3.26538e8	−	5.65581e8i	−0.923113	−	1.59888i	−0.794568	−	0.607175i	$-0.792303\pi$
$277$	−3.26538e8	−	5.65581e8i	−0.923113	−	1.59888i	−0.128545	−	0.991704i	$-0.541031\pi$
$278$		0			0
$279$	−4.98316e8			−1.37369
$280$		0			0
$281$	5.17554e8			1.39150			0.695750	−	0.718284i	$-0.255072\pi$
$281$	5.17554e8			1.39150			0.695750	+	0.718284i	$0.255072\pi$
$282$		0			0
$283$	7.16532e6	+	1.24107e7i	0.0187924	+	0.0325494i	0.875269	−	0.483637i	$-0.160685\pi$
$283$	7.16532e6	+	1.24107e7i	0.0187924	+	0.0325494i	−0.856476	+	0.516186i	$0.827351\pi$
$284$		0			0
$285$	2.64601e7	−	4.58302e7i	0.0677070	−	0.117272i
$286$		0			0
$287$	−3.42316e8	+	3.41688e8i	−0.854753	+	0.853184i
$288$		0			0
$289$	−4.62019e8	+	8.00241e8i	−1.12595	+	1.95020i
$290$		0			0
$291$	−5.22437e7	−	9.04888e7i	−0.124282	−	0.215263i
$292$		0			0
$293$	−4.59597e8			−1.06743			−0.533716	−	0.845664i	$-0.679205\pi$
$293$	−4.59597e8			−1.06743			−0.533716	+	0.845664i	$0.679205\pi$
$294$		0			0
$295$	3.43434e6			0.00778872
$296$		0			0
$297$	1.92856e8	+	3.34036e8i	0.427155	+	0.739854i
$298$		0			0
$299$	8.56977e7	−	1.48433e8i	0.185404	−	0.321130i
$300$		0			0
$301$	2.34675e8	−	2.34244e8i	0.496002	−	0.495091i
$302$		0			0
$303$	−4.49439e8	+	7.78451e8i	−0.928157	+	1.60761i
$304$		0			0
$305$	−8.96875e6	−	1.55343e7i	−0.0181002	−	0.0313504i
$306$		0			0
$307$	2.28161e8			0.450046			0.225023	−	0.974354i	$-0.427754\pi$
$307$	2.28161e8			0.450046			0.225023	+	0.974354i	$0.427754\pi$
$308$		0			0
$309$	1.67068e8			0.322135
$310$		0			0
$311$	3.56536e8	+	6.17538e8i	0.672112	+	1.16413i	0.977304	+	0.211842i	$0.0679462\pi$
$311$	3.56536e8	+	6.17538e8i	0.672112	+	1.16413i	−0.305192	+	0.952291i	$0.598720\pi$
$312$		0			0
$313$	6.97714e7	−	1.20848e8i	0.128609	−	0.222758i	−0.794529	−	0.607226i	$-0.792282\pi$
$313$	6.97714e7	−	1.20848e8i	0.128609	−	0.222758i	0.923138	+	0.384469i	$0.125615\pi$
$314$		0			0
$315$	6.93185e7	+	1.86421e7i	0.124958	+	0.0336054i
$316$		0			0
$317$	7.69441e7	−	1.33271e8i	0.135665	−	0.234979i	−0.790186	−	0.612867i	$-0.790016\pi$
$317$	7.69441e7	−	1.33271e8i	0.135665	−	0.234979i	0.925851	+	0.377888i	$0.123350\pi$
$318$		0			0
$319$	7.18297e7	+	1.24413e8i	0.123890	+	0.214584i
$320$		0			0
$321$	−8.27646e8			−1.39662
$322$		0			0
$323$	−1.18359e9			−1.95430
$324$		0			0
$325$	3.97610e8	+	6.88681e8i	0.642489	+	1.11282i
$326$		0			0
$327$	4.75693e8	−	8.23924e8i	0.752331	−	1.30308i
$328$		0			0
$329$	4.84207e7	+	1.81375e8i	0.0749627	+	0.280796i
$330$		0			0
$331$	6.99907e7	−	1.21227e8i	0.106082	−	0.183740i	−0.808098	−	0.589048i	$-0.799503\pi$
$331$	6.99907e7	−	1.21227e8i	0.106082	−	0.183740i	0.914180	+	0.405309i	$0.132836\pi$
$332$		0			0
$333$	1.07948e9	+	1.86971e9i	1.60199	+	2.77473i
$334$		0			0
$335$	2.60858e7			0.0379095
$336$		0			0
$337$	−9.36806e7			−0.133335			−0.0666676	−	0.997775i	$-0.521237\pi$
$337$	−9.36806e7			−0.133335			−0.0666676	+	0.997775i	$0.521237\pi$
$338$		0			0
$339$	7.45363e8	+	1.29101e9i	1.03913	+	1.79982i
$340$		0			0
$341$	−2.18575e8	+	3.78584e8i	−0.298511	+	0.517037i
$342$		0			0
$343$	−5.27004e8	−	5.29917e8i	−0.705155	−	0.709053i
$344$		0			0
$345$	−1.36715e7	+	2.36797e7i	−0.0179245	+	0.0310462i
$346$		0			0
$347$	−2.85722e8	−	4.94886e8i	−0.367106	−	0.635846i	0.622006	−	0.783012i	$-0.286318\pi$
$347$	−2.85722e8	−	4.94886e8i	−0.367106	−	0.635846i	−0.989112	+	0.147167i	$0.952985\pi$
$348$		0			0
$349$	−2.32123e8			−0.292300			−0.146150	−	0.989262i	$-0.546688\pi$
$349$	−2.32123e8			−0.292300			−0.146150	+	0.989262i	$0.546688\pi$
$350$		0			0
$351$	1.20904e9			1.49234
$352$		0			0
$353$	−5.79874e8	−	1.00437e9i	−0.701653	−	1.21530i	−0.967886	−	0.251390i	$-0.919112\pi$
$353$	−5.79874e8	−	1.00437e9i	−0.701653	−	1.21530i	0.266233	−	0.963909i	$-0.414221\pi$
$354$		0			0
$355$	−1.17654e7	+	2.03783e7i	−0.0139575	+	0.0241752i
$356$		0			0
$357$	−6.57333e8	−	2.46225e9i	−0.764622	−	2.86413i
$358$		0			0
$359$	4.85816e8	−	8.41458e8i	0.554168	−	0.959847i	−0.443800	−	0.896126i	$-0.646370\pi$
$359$	4.85816e8	−	8.41458e8i	0.554168	−	0.959847i	0.997968	−	0.0637210i	$-0.0202968\pi$
$360$		0			0
$361$	−7.79833e7	−	1.35071e8i	−0.0872421	−	0.151108i
$362$		0			0
$363$	−6.78019e8			−0.743993
$364$		0			0
$365$	−3.48176e7			−0.0374778
$366$		0			0
$367$	−8.51399e8	−	1.47467e9i	−0.899088	−	1.55727i	−0.828663	−	0.559749i	$-0.810898\pi$
$367$	−8.51399e8	−	1.47467e9i	−0.899088	−	1.55727i	−0.0704252	−	0.997517i	$-0.522436\pi$
$368$		0			0
$369$	9.92154e8	−	1.71846e9i	1.02799	−	1.78052i
$370$		0			0
$371$	9.53657e8	+	2.56471e8i	0.969579	+	0.260753i
$372$		0			0
$373$	−1.88494e8	+	3.26481e8i	−0.188068	+	0.325744i	−0.944606	−	0.328206i	$-0.893556\pi$
$373$	−1.88494e8	+	3.26481e8i	−0.188068	+	0.325744i	0.756538	+	0.653950i	$0.226889\pi$
$374$		0			0
$375$	−1.27231e8	−	2.20371e8i	−0.124590	−	0.215797i
$376$		0			0
$377$	4.50311e8			0.432831
$378$		0			0
$379$	−8.86999e8			−0.836924			−0.418462	−	0.908234i	$-0.637431\pi$
$379$	−8.86999e8			−0.836924			−0.418462	+	0.908234i	$0.637431\pi$
$380$		0			0
$381$	−5.43812e7	−	9.41910e7i	−0.0503746	−	0.0872513i
$382$		0			0
$383$	−2.17148e8	+	3.76111e8i	−0.197497	+	0.342074i	−0.947716	−	0.319115i	$-0.896614\pi$
$383$	−2.17148e8	+	3.76111e8i	−0.197497	+	0.342074i	0.750220	+	0.661189i	$0.229948\pi$
$384$		0			0
$385$	4.45680e7	−	4.44861e7i	0.0398025	−	0.0397294i
$386$		0			0
$387$	−6.80171e8	+	1.17809e9i	−0.596526	+	1.03321i
$388$		0			0
$389$	−7.57986e7	−	1.31287e8i	−0.0652886	−	0.113083i	0.831533	−	0.555475i	$-0.187463\pi$
$389$	−7.57986e7	−	1.31287e8i	−0.0652886	−	0.113083i	−0.896822	+	0.442392i	$0.854130\pi$
$390$		0			0
$391$	6.11540e8			0.517376
$392$		0			0
$393$	2.73091e9			2.26952
$394$		0			0
$395$	5.30515e7	+	9.18880e7i	0.0433120	+	0.0750186i
$396$		0			0
$397$	8.52585e8	−	1.47672e9i	0.683866	−	1.18449i	−0.289926	−	0.957049i	$-0.593631\pi$
$397$	8.52585e8	−	1.47672e9i	0.683866	−	1.18449i	0.973792	−	0.227441i	$-0.0730360\pi$
$398$		0			0
$399$	1.59988e9	−	1.59694e9i	1.26090	−	1.25859i
$400$		0			0
$401$	−5.75883e8	+	9.97459e8i	−0.445994	+	0.772484i	−0.998121	−	0.0612759i	$-0.980483\pi$
$401$	−5.75883e8	+	9.97459e8i	−0.445994	+	0.772484i	0.552127	+	0.833760i	$0.313816\pi$
$402$		0			0
$403$	6.85141e8	+	1.18670e9i	0.521450	+	0.903177i
$404$		0			0
$405$	−1.98911e7			−0.0148788
$406$		0			0
$407$	1.89396e9			1.39248
$408$		0			0
$409$	−5.71985e8	−	9.90706e8i	−0.413383	−	0.716001i	0.581874	−	0.813279i	$-0.302320\pi$
$409$	−5.71985e8	−	9.90706e8i	−0.413383	−	0.716001i	−0.995257	+	0.0972783i	$0.968986\pi$
$410$		0			0
$411$	1.70289e9	−	2.94950e9i	1.20988	−	2.09557i
$412$		0			0
$413$	1.41665e8	+	3.80985e7i	0.0989548	+	0.0266123i
$414$		0			0
$415$	3.59421e7	−	6.22536e7i	0.0246851	−	0.0427559i
$416$		0			0
$417$	1.14466e9	+	1.98261e9i	0.773039	+	1.33894i
$418$		0			0
$419$	−1.72282e9			−1.14417			−0.572084	−	0.820195i	$-0.693865\pi$
$419$	−1.72282e9			−1.14417			−0.572084	+	0.820195i	$0.693865\pi$
$420$		0			0
$421$	−2.50496e9			−1.63611			−0.818056	−	0.575139i	$-0.804948\pi$
$421$	−2.50496e9			−1.63611			−0.818056	+	0.575139i	$0.804948\pi$
$422$		0			0
$423$	−3.85090e8	−	6.66996e8i	−0.247384	−	0.428481i
$424$		0			0
$425$	−1.41868e9	+	2.45722e9i	−0.896441	+	1.55268i
$426$		0			0
$427$	−1.97628e8	−	7.40278e8i	−0.122843	−	0.460147i
$428$		0			0
$429$	1.28534e9	−	2.22627e9i	0.785990	−	1.36137i
$430$		0			0
$431$	1.14303e9	+	1.97978e9i	0.687679	+	1.19110i	0.972587	+	0.232541i	$0.0747039\pi$
$431$	1.14303e9	+	1.97978e9i	0.687679	+	1.19110i	−0.284907	+	0.958555i	$0.591963\pi$
$432$		0			0
$433$	−2.07332e9			−1.22732			−0.613660	−	0.789570i	$-0.710303\pi$
$433$	−2.07332e9			−1.22732			−0.613660	+	0.789570i	$0.710303\pi$
$434$		0			0
$435$	−7.18387e7			−0.0418452
$436$		0			0
$437$	2.71217e8	+	4.69761e8i	0.155465	+	0.269273i
$438$		0			0
$439$	−8.79323e8	+	1.52303e9i	−0.496047	+	0.859178i	−0.999990	−	0.00455873i	$-0.998549\pi$
$439$	−8.79323e8	+	1.52303e9i	−0.496047	+	0.859178i	0.503943	+	0.863737i	$0.331882\pi$
$440$		0			0
$441$	2.65255e9	+	1.53796e9i	1.47275	+	0.853904i
$442$		0			0
$443$	−9.64736e8	+	1.67097e9i	−0.527224	+	0.913179i	0.472272	+	0.881453i	$0.343434\pi$
$443$	−9.64736e8	+	1.67097e9i	−0.527224	+	0.913179i	−0.999497	+	0.0317264i	$0.989899\pi$
$444$		0			0
$445$	6.94395e7	+	1.20273e8i	0.0373548	+	0.0647005i
$446$		0			0
$447$	−3.13703e9			−1.66128
$448$		0			0
$449$	3.29325e9			1.71697			0.858484	−	0.512840i	$-0.171407\pi$
$449$	3.29325e9			1.71697			0.858484	+	0.512840i	$0.171407\pi$
$450$		0			0
$451$	−8.70373e8	−	1.50753e9i	−0.446774	−	0.773835i
$452$		0			0
$453$	−1.37066e9	+	2.37405e9i	−0.692764	+	1.19990i
$454$		0			0
$455$	−5.09122e7	−	1.90708e8i	−0.0253386	−	0.0949136i
$456$		0			0
$457$	9.57310e8	−	1.65811e9i	0.469187	−	0.812655i	−0.530193	−	0.847877i	$-0.677880\pi$
$457$	9.57310e8	−	1.65811e9i	0.469187	−	0.812655i	0.999380	+	0.0352218i	$0.0112138\pi$
$458$		0			0
$459$	2.15694e9	+	3.73592e9i	1.04110	+	1.80324i
$460$		0			0
$461$	−1.84068e9			−0.875032			−0.437516	−	0.899211i	$-0.644142\pi$
$461$	−1.84068e9			−0.875032			−0.437516	+	0.899211i	$0.644142\pi$
$462$		0			0
$463$	1.67942e8			0.0786367			0.0393184	−	0.999227i	$-0.487481\pi$
$463$	1.67942e8			0.0786367			0.0393184	+	0.999227i	$0.487481\pi$
$464$		0			0
$465$	−1.09301e8	−	1.89316e8i	−0.0504128	−	0.0873175i
$466$		0			0
$467$	6.89538e8	−	1.19432e9i	0.313292	−	0.542638i	−0.665781	−	0.746147i	$-0.731901\pi$
$467$	6.89538e8	−	1.19432e9i	0.313292	−	0.542638i	0.979073	+	0.203510i	$0.0652348\pi$
$468$		0			0
$469$	1.07603e9	+	2.89381e8i	0.481636	+	0.129528i
$470$		0			0
$471$	2.00143e9	−	3.46658e9i	0.882607	−	1.52872i
$472$		0			0
$473$	5.96684e8	+	1.03349e9i	0.259257	+	0.449046i
$474$		0			0
$475$	−2.51672e9			−1.07748
$476$		0			0
$477$	−4.05155e9			−1.70925
$478$		0			0
$479$	2.26505e9	+	3.92318e9i	0.941679	+	1.63104i	0.762268	+	0.647261i	$0.224086\pi$
$479$	2.26505e9	+	3.92318e9i	0.941679	+	1.63104i	0.179411	+	0.983774i	$0.442581\pi$
$480$		0			0
$481$	2.96838e9	−	5.14139e9i	1.21622	−	2.10655i
$482$		0			0
$483$	−8.26630e8	+	8.25112e8i	−0.333807	+	0.333195i
$484$		0			0
$485$	1.44376e7	−	2.50067e7i	0.00574646	−	0.00995316i
$486$		0			0
$487$	7.69774e8	+	1.33329e9i	0.302003	+	0.523085i	0.976590	−	0.215111i	$-0.0690112\pi$
$487$	7.69774e8	+	1.33329e9i	0.302003	+	0.523085i	−0.674586	+	0.738196i	$0.735678\pi$
$488$		0			0
$489$	−4.18566e9			−1.61876
$490$		0			0
$491$	1.85780e9			0.708295			0.354148	−	0.935190i	$-0.384771\pi$
$491$	1.85780e9			0.708295			0.354148	+	0.935190i	$0.384771\pi$
$492$		0			0
$493$	8.03356e8	+	1.39145e9i	0.301956	+	0.523003i
$494$		0			0
$495$	−1.29174e8	+	2.23736e8i	−0.0478692	+	0.0829119i
$496$		0			0
$497$	−7.11384e8	+	7.10078e8i	−0.259930	+	0.259453i
$498$		0			0
$499$	2.67677e9	−	4.63630e9i	0.964405	−	1.67040i	0.253198	−	0.967414i	$-0.418518\pi$
$499$	2.67677e9	−	4.63630e9i	0.964405	−	1.67040i	0.711206	−	0.702983i	$-0.248149\pi$
$500$		0			0
$501$	−1.41895e9	−	2.45769e9i	−0.504121	−	0.873163i
$502$		0			0
$503$	−5.43979e9			−1.90587			−0.952936	−	0.303170i	$-0.901955\pi$
$503$	−5.43979e9			−1.90587			−0.952936	+	0.303170i	$0.901955\pi$
$504$		0			0
$505$	−2.48406e8			−0.0858308
$506$		0			0
$507$	−1.61702e9	−	2.80076e9i	−0.551046	−	0.954439i
$508$		0			0
$509$	−9.12084e8	+	1.57978e9i	−0.306565	+	0.530986i	−0.977609	−	0.210432i	$-0.932513\pi$
$509$	−9.12084e8	+	1.57978e9i	−0.306565	+	0.530986i	0.671044	+	0.741418i	$0.265846\pi$
$510$		0			0
$511$	−1.43621e9	−	3.86246e8i	−0.476151	−	0.128053i
$512$		0			0
$513$	−1.91319e9	+	3.31375e9i	−0.625674	+	1.08370i
$514$		0			0
$515$	2.30847e7	+	3.99839e7i	0.00744730	+	0.0128991i
$516$		0			0
$517$	−6.75645e8			−0.215031
$518$		0			0
$519$	−5.91830e9			−1.85828
$520$		0			0
$521$	2.07430e8	+	3.59280e8i	0.0642599	+	0.111301i	0.896365	−	0.443316i	$-0.146198\pi$
$521$	2.07430e8	+	3.59280e8i	0.0642599	+	0.111301i	−0.832106	+	0.554617i	$0.812865\pi$
$522$		0			0
$523$	−1.60836e9	+	2.78576e9i	−0.491616	+	0.851505i	−0.999953	−	0.00965354i	$-0.996927\pi$
$523$	−1.60836e9	+	2.78576e9i	−0.491616	+	0.851505i	0.508337	+	0.861158i	$0.330260\pi$
$524$		0			0
$525$	−1.39772e9	−	5.23559e9i	−0.421563	−	1.57910i
$526$		0			0
$527$	−2.44459e9	+	4.23415e9i	−0.727559	+	1.26017i
$528$		0			0
$529$	1.56228e9	+	2.70595e9i	0.458843	+	0.794739i
$530$		0			0
$531$	−6.01853e8			−0.174446
$532$		0			0
$533$	−5.45650e9			−1.56088
$534$		0			0
$535$	−1.14361e8	−	1.98079e8i	−0.0322878	−	0.0559242i
$536$		0			0
$537$	1.92334e9	−	3.33132e9i	0.535977	−	0.928339i
$538$		0			0
$539$	2.33191e9	−	1.34062e9i	0.641433	−	0.368761i
$540$		0			0
$541$	1.14907e9	−	1.99025e9i	0.312002	−	0.540403i	−0.666794	−	0.745242i	$-0.732334\pi$
$541$	1.14907e9	−	1.99025e9i	0.312002	−	0.540403i	0.978796	+	0.204839i	$0.0656672\pi$
$542$		0			0
$543$	−6.27426e8	−	1.08673e9i	−0.168176	−	0.291289i
$544$		0			0
$545$	2.62917e8			0.0695714
$546$		0			0
$547$	6.33362e9			1.65461			0.827306	−	0.561752i	$-0.189872\pi$
$547$	6.33362e9			1.65461			0.827306	+	0.561752i	$0.189872\pi$
$548$		0			0
$549$	1.57174e9	+	2.72233e9i	0.405393	+	0.702162i
$550$		0			0
$551$	−7.12574e8	+	1.23421e9i	−0.181468	+	0.314311i
$552$		0			0
$553$	1.16900e9	+	4.37886e9i	0.293952	+	1.10109i
$554$		0			0
$555$	−4.73550e8	+	8.20212e8i	−0.117582	+	0.203658i
$556$		0			0
$557$	3.75684e9	+	6.50705e9i	0.921149	+	1.59548i	0.797640	+	0.603134i	$0.206082\pi$
$557$	3.75684e9	+	6.50705e9i	0.921149	+	1.59548i	0.123510	+	0.992343i	$0.460585\pi$
$558$		0			0
$559$	3.74070e9			0.905757
$560$		0			0
$561$	9.17219e9			2.19332
$562$		0			0
$563$	2.39314e9	+	4.14504e9i	0.565183	+	0.978926i	0.997033	+	0.0769801i	$0.0245278\pi$
$563$	2.39314e9	+	4.14504e9i	0.565183	+	0.978926i	−0.431850	+	0.901946i	$0.642139\pi$
$564$		0			0
$565$	−2.05982e8	+	3.56772e8i	−0.0480463	+	0.0832187i
$566$		0			0
$567$	−8.20500e8	−	2.20661e8i	−0.189033	−	0.0508375i
$568$		0			0
$569$	−2.60548e9	+	4.51283e9i	−0.592919	+	1.02697i	0.400918	+	0.916114i	$0.368691\pi$
$569$	−2.60548e9	+	4.51283e9i	−0.592919	+	1.02697i	−0.993837	+	0.110852i	$0.964642\pi$
$570$		0			0
$571$	−3.04516e9	−	5.27437e9i	−0.684516	−	1.18562i	−0.973589	−	0.228310i	$-0.926680\pi$
$571$	−3.04516e9	−	5.27437e9i	−0.684516	−	1.18562i	0.289072	−	0.957307i	$-0.406653\pi$
$572$		0			0
$573$	−1.14584e10			−2.54439
$574$		0			0
$575$	1.30035e9			0.285247
$576$		0			0
$577$	1.01575e9	+	1.75934e9i	0.220127	+	0.381271i	0.954846	−	0.297100i	$-0.0960195\pi$
$577$	1.01575e9	+	1.75934e9i	0.220127	+	0.381271i	−0.734719	+	0.678371i	$0.762686\pi$
$578$		0			0
$579$	−2.43696e9	+	4.22094e9i	−0.521764	+	0.903721i
$580$		0			0
$581$	2.17320e9	−	2.16921e9i	0.459709	−	0.458865i
$582$		0			0
$583$	−1.77712e9	+	3.07807e9i	−0.371430	+	0.643336i
$584$		0			0
$585$	4.04905e8	+	7.01317e8i	0.0836196	+	0.144833i
$586$		0			0
$587$	7.84037e9			1.59994			0.799970	−	0.600041i	$-0.204849\pi$
$587$	7.84037e9			1.59994			0.799970	+	0.600041i	$0.204849\pi$
$588$		0			0
$589$	−4.33668e9			−0.874488
$590$		0			0
$591$	−1.12930e9	−	1.95600e9i	−0.225036	−	0.389773i
$592$		0			0
$593$	7.96578e8	−	1.37971e9i	0.156869	−	0.271705i	−0.776869	−	0.629662i	$-0.783193\pi$
$593$	7.96578e8	−	1.37971e9i	0.156869	−	0.271705i	0.933738	+	0.357957i	$0.116527\pi$
$594$		0			0
$595$	4.98456e8	−	4.97541e8i	0.0970102	−	0.0968321i
$596$		0			0
$597$	1.89001e9	−	3.27360e9i	0.363542	−	0.629674i
$598$		0			0
$599$	3.97790e8	+	6.88992e8i	0.0756240	+	0.130985i	0.901357	−	0.433076i	$-0.142572\pi$
$599$	3.97790e8	+	6.88992e8i	0.0756240	+	0.130985i	−0.825733	+	0.564060i	$0.809238\pi$
$600$		0			0
$601$	6.44884e8			0.121177			0.0605886	−	0.998163i	$-0.480702\pi$
$601$	6.44884e8			0.121177			0.0605886	+	0.998163i	$0.480702\pi$
$602$		0			0
$603$	−4.57144e9			−0.849067
$604$		0			0
$605$	−9.36859e7	−	1.62269e8i	−0.0172001	−	0.0297914i
$606$		0			0
$607$	1.62737e9	−	2.81869e9i	0.295343	−	0.511549i	−0.679721	−	0.733470i	$-0.737899\pi$
$607$	1.62737e9	−	2.81869e9i	0.295343	−	0.511549i	0.975065	+	0.221921i	$0.0712327\pi$
$608$		0			0
$609$	−2.96331e9	−	7.96936e8i	−0.531639	−	0.142976i
$610$		0			0
$611$	−1.05893e9	+	1.83412e9i	−0.187812	+	0.325300i
$612$		0			0
$613$	−1.95906e9	−	3.39319e9i	−0.343508	−	0.594973i	0.641574	−	0.767061i	$-0.278282\pi$
$613$	−1.95906e9	−	3.39319e9i	−0.343508	−	0.594973i	−0.985081	+	0.172089i	$0.944948\pi$
$614$		0			0
$615$	8.70483e8			0.150903
$616$		0			0
$617$	6.38504e9			1.09437			0.547187	−	0.837010i	$-0.315699\pi$
$617$	6.38504e9			1.09437			0.547187	+	0.837010i	$0.315699\pi$
$618$		0			0
$619$	−5.48911e9	−	9.50741e9i	−0.930217	−	1.61118i	−0.782948	−	0.622087i	$-0.786285\pi$
$619$	−5.48911e9	−	9.50741e9i	−0.930217	−	1.61118i	−0.147269	−	0.989096i	$-0.547048\pi$
$620$		0			0
$621$	9.88515e8	−	1.71216e9i	0.165639	−	0.286895i
$622$		0			0
$623$	1.53011e9	+	5.73152e9i	0.253522	+	0.949645i
$624$		0			0
$625$	−2.99897e9	+	5.19436e9i	−0.491351	+	0.851044i
$626$		0			0
$627$	4.06785e9	+	7.04572e9i	0.659065	+	1.14153i
$628$		0			0
$629$	2.11824e10			3.39389
$630$		0			0
$631$	6.75136e9			1.06977			0.534883	−	0.844926i	$-0.320356\pi$
$631$	6.75136e9			1.06977			0.534883	+	0.844926i	$0.320356\pi$
$632$		0			0
$633$	−9.81393e9	−	1.69982e10i	−1.53791	−	2.66373i
$634$		0			0
$635$	1.50283e7	−	2.60299e7i	0.00232918	−	0.00403426i
$636$		0			0
$637$	1.54937e7	−	8.43139e9i	0.00237503	−	1.29244i
$638$		0			0
$639$	2.06184e9	−	3.57122e9i	0.312610	−	0.541456i
$640$		0			0
$641$	−2.66451e9	−	4.61507e9i	−0.399590	−	0.692110i	0.594085	−	0.804402i	$-0.297514\pi$
$641$	−2.66451e9	−	4.61507e9i	−0.399590	−	0.692110i	−0.993675	+	0.112292i	$0.964181\pi$
$642$		0			0
$643$	−3.64486e9			−0.540683			−0.270342	−	0.962764i	$-0.587137\pi$
$643$	−3.64486e9			−0.540683			−0.270342	+	0.962764i	$0.587137\pi$
$644$		0			0
$645$	−5.96759e8			−0.0875669
$646$		0			0
$647$	−4.98373e8	−	8.63207e8i	−0.0723418	−	0.125300i	0.827585	−	0.561340i	$-0.189714\pi$
$647$	−4.98373e8	−	8.63207e8i	−0.0723418	−	0.125300i	−0.899927	+	0.436040i	$0.856381\pi$
$648$		0			0
$649$	−2.63990e8	+	4.57244e8i	−0.0379080	+	0.0656585i
$650$		0			0
$651$	−2.40848e9	−	9.02170e9i	−0.342144	−	1.28161i
$652$		0			0
$653$	−1.25807e9	+	2.17905e9i	−0.176811	+	0.306246i	−0.940787	−	0.338999i	$-0.889912\pi$
$653$	−1.25807e9	+	2.17905e9i	−0.176811	+	0.306246i	0.763975	+	0.645245i	$0.223245\pi$
$654$		0			0
$655$	3.77346e8	+	6.53583e8i	0.0524681	+	0.0908774i
$656$		0			0
$657$	6.10164e9			0.839398
$658$		0			0
$659$	−2.14485e9			−0.291944			−0.145972	−	0.989289i	$-0.546631\pi$
$659$	−2.14485e9			−0.291944			−0.145972	+	0.989289i	$0.546631\pi$
$660$		0			0
$661$	2.15616e9	+	3.73457e9i	0.290386	+	0.502962i	0.973901	−	0.226974i	$-0.0728831\pi$
$661$	2.15616e9	+	3.73457e9i	0.290386	+	0.502962i	−0.683515	+	0.729936i	$0.739550\pi$
$662$		0			0
$663$	1.43755e10	−	2.48990e10i	1.91569	−	3.31807i
$664$		0			0
$665$	6.03256e8	+	1.62236e8i	0.0795474	+	0.0213930i
$666$		0			0
$667$	3.68175e8	−	6.37698e8i	0.0480412	−	0.0832098i
$668$		0			0
$669$	−1.28958e9	−	2.23363e9i	−0.166517	−	0.288416i
$670$		0			0
$671$	2.75763e9			0.352377
$672$		0			0
$673$	8.49626e9			1.07442			0.537211	−	0.843448i	$-0.319478\pi$
$673$	8.49626e9			1.07442			0.537211	+	0.843448i	$0.319478\pi$
$674$		0			0
$675$	4.58640e9	+	7.94387e9i	0.573995	+	0.994189i
$676$		0			0
$677$	−3.85081e9	+	6.66980e9i	−0.476971	+	0.826138i	−0.999652	−	0.0263906i	$-0.991599\pi$
$677$	−3.85081e9	+	6.66980e9i	−0.476971	+	0.826138i	0.522681	+	0.852528i	$0.324932\pi$
$678$		0			0
$679$	8.72956e8	−	8.71353e8i	0.107016	−	0.106819i
$680$		0			0
$681$	5.80725e8	−	1.00584e9i	0.0704621	−	0.122044i
$682$		0			0
$683$	−4.85022e9	−	8.40083e9i	−0.582491	−	1.00890i	−0.995183	−	0.0980333i	$-0.968745\pi$
$683$	−4.85022e9	−	8.40083e9i	−0.582491	−	1.00890i	0.412692	−	0.910871i	$-0.364588\pi$
$684$		0			0
$685$	9.41196e8			0.111883
$686$		0			0
$687$	−7.93080e9			−0.933186
$688$		0			0
$689$	5.57052e9	+	9.64843e9i	0.648827	+	1.12380i
$690$		0			0
$691$	−1.42444e9	+	2.46720e9i	−0.164237	+	0.284466i	−0.936384	−	0.350977i	$-0.885849\pi$
$691$	−1.42444e9	+	2.46720e9i	−0.164237	+	0.284466i	0.772147	+	0.635444i	$0.219183\pi$
$692$		0			0
$693$	−7.81035e9	+	7.79601e9i	−0.891465	+	0.889828i
$694$		0			0
$695$	−3.16329e8	+	5.47899e8i	−0.0357432	+	0.0619090i
$696$		0			0
$697$	−9.73441e9	−	1.68605e10i	−1.08892	−	1.88606i
$698$		0			0
$699$	7.99240e9			0.885130
$700$		0			0
$701$	−4.47816e9			−0.491006			−0.245503	−	0.969396i	$-0.578953\pi$
$701$	−4.47816e9			−0.491006			−0.245503	+	0.969396i	$0.578953\pi$
$702$		0			0
$703$	9.39436e9	+	1.62715e10i	1.01982	+	1.76638i
$704$		0			0
$705$	1.68933e8	−	2.92600e8i	0.0181573	−	0.0314494i
$706$		0			0
$707$	−1.02466e10	−	2.75567e9i	−1.09047	−	0.293265i
$708$		0			0
$709$	4.19450e9	−	7.26510e9i	0.441997	−	0.765560i	−0.555841	−	0.831289i	$-0.687604\pi$
$709$	4.19450e9	−	7.26510e9i	0.441997	−	0.765560i	0.997838	+	0.0657281i	$0.0209370\pi$
$710$		0			0
$711$	−9.29706e9	−	1.61030e10i	−0.970068	−	1.68021i
$712$		0			0
$713$	2.24069e9			0.231509
$714$		0			0
$715$	7.10411e8			0.0726840
$716$		0			0
$717$	5.40235e9	+	9.35715e9i	0.547351	+	0.948040i
$718$		0			0
$719$	−6.19091e8	+	1.07230e9i	−0.0621160	+	0.107588i	−0.895411	−	0.445240i	$-0.853118\pi$
$719$	−6.19091e8	+	1.07230e9i	−0.0621160	+	0.107588i	0.833295	+	0.552829i	$0.186452\pi$
$720$		0			0
$721$	5.08676e8	+	1.90540e9i	0.0505437	+	0.189327i
$722$		0			0
$723$	−1.17239e10	+	2.03063e10i	−1.15368	+	1.99824i
$724$		0			0
$725$	1.70821e9	+	2.95872e9i	0.166479	+	0.288350i
$726$		0			0
$727$	3.52132e9			0.339888			0.169944	−	0.985454i	$-0.445641\pi$
$727$	3.52132e9			0.339888			0.169944	+	0.985454i	$0.445641\pi$
$728$		0			0
$729$	−1.63694e10			−1.56490
$730$		0			0
$731$	6.67342e9	+	1.15587e10i	0.631884	+	1.09446i
$732$		0			0
$733$	−3.95436e9	+	6.84915e9i	−0.370862	+	0.642352i	−0.989698	−	0.143168i	$-0.954271\pi$
$733$	−3.95436e9	+	6.84915e9i	−0.370862	+	0.642352i	0.618836	+	0.785520i	$0.287604\pi$
$734$		0			0
$735$	−2.47174e6	+	1.34507e9i	−0.000229613	+	0.124951i
$736$		0			0
$737$	−2.00516e9	+	3.47304e9i	−0.184507	+	0.319576i
$738$		0			0
$739$	2.24356e9	+	3.88597e9i	0.204495	+	0.354196i	0.949972	−	0.312336i	$-0.101111\pi$
$739$	2.24356e9	+	3.88597e9i	0.204495	+	0.354196i	−0.745477	+	0.666532i	$0.767778\pi$
$740$		0			0
$741$	2.55020e10			2.30255
$742$		0			0
$743$	4.33508e9			0.387736			0.193868	−	0.981028i	$-0.437897\pi$
$743$	4.33508e9			0.387736			0.193868	+	0.981028i	$0.437897\pi$
$744$		0			0
$745$	−4.33461e8	−	7.50777e8i	−0.0384064	−	0.0665218i
$746$		0			0
$747$	−6.29871e9	+	1.09097e10i	−0.552878	+	0.957613i
$748$		0			0
$749$	−2.51996e9	−	9.43930e9i	−0.219133	−	0.820830i
$750$		0			0
$751$	−5.94199e9	+	1.02918e10i	−0.511908	+	0.886651i	0.487997	+	0.872845i	$0.337728\pi$
$751$	−5.94199e9	+	1.02918e10i	−0.511908	+	0.886651i	−0.999905	+	0.0138052i	$0.995606\pi$
$752$		0			0
$753$	2.76968e9	+	4.79722e9i	0.236400	+	0.409456i
$754$		0			0
$755$	−7.57567e8			−0.0640630
$756$		0			0
$757$	1.37796e10			1.15452			0.577260	−	0.816560i	$-0.304122\pi$
$757$	1.37796e10			1.15452			0.577260	+	0.816560i	$0.304122\pi$
$758$		0			0
$759$	−2.10179e9	−	3.64041e9i	−0.174479	−	0.302206i
$760$		0			0
$761$	−4.26500e9	+	7.38719e9i	−0.350810	+	0.607622i	−0.986392	−	0.164412i	$-0.947427\pi$
$761$	−4.26500e9	+	7.38719e9i	−0.350810	+	0.607622i	0.635581	+	0.772034i	$0.280761\pi$
$762$		0			0
$763$	1.08452e10	+	2.91664e9i	0.883897	+	0.237710i
$764$		0			0
$765$	−1.44470e9	+	2.50230e9i	−0.116671	+	0.202081i
$766$		0			0
$767$	8.27496e8	+	1.43327e9i	0.0662189	+	0.114695i
$768$		0			0
$769$	−1.57605e10			−1.24976			−0.624881	−	0.780720i	$-0.714853\pi$
$769$	−1.57605e10			−1.24976			−0.624881	+	0.780720i	$0.714853\pi$
$770$		0			0
$771$	−4.44878e8			−0.0349583
$772$		0			0
$773$	1.87506e9	+	3.24770e9i	0.146011	+	0.252899i	0.929750	−	0.368192i	$-0.120023\pi$
$773$	1.87506e9	+	3.24770e9i	0.146011	+	0.252899i	−0.783738	+	0.621091i	$0.786690\pi$
$774$		0			0
$775$	−5.19804e9	+	9.00327e9i	−0.401129	+	0.694776i
$776$		0			0
$777$	−2.86326e10	+	2.85801e10i	−2.18972	+	2.18570i
$778$		0			0
$779$	8.63439e9	−	1.49552e10i	0.654411	−	1.13347i
$780$		0			0
$781$	−1.80876e9	−	3.13287e9i	−0.135864	−	0.235323i
$782$		0			0
$783$	5.19430e9			0.386688
$784$		0			0
$785$	1.10620e9			0.0816186
$786$		0			0
$787$	−1.01205e10	−	1.75292e10i	−0.740099	−	1.28189i	−0.952450	−	0.304696i	$-0.901445\pi$
$787$	−1.01205e10	−	1.75292e10i	−0.740099	−	1.28189i	0.212350	−	0.977194i	$-0.431888\pi$
$788$		0			0
$789$	8.96670e9	−	1.55308e10i	0.649925	−	1.12570i
$790$		0			0
$791$	−1.24545e10	+	1.24316e10i	−0.894763	+	0.893120i
$792$		0			0
$793$	4.32200e9	−	7.48592e9i	0.307772	−	0.533076i
$794$		0			0
$795$	−8.88673e8	−	1.53923e9i	−0.0627273	−	0.108647i
$796$		0			0
$797$	−1.96658e10			−1.37597			−0.687983	−	0.725727i	$-0.741504\pi$
$797$	−1.96658e10			−1.37597			−0.687983	+	0.725727i	$0.741504\pi$
$798$		0			0
$799$	−7.55654e9			−0.524094
$800$		0			0
$801$	−1.21690e10	−	2.10773e10i	−0.836644	−	1.44911i
$802$		0			0
$803$	2.67635e9	−	4.63558e9i	0.182406	−	0.315936i
$804$		0			0
$805$	−3.11692e8	−	8.38247e7i	−0.0210591	−	0.00566352i
$806$		0			0
$807$	1.58752e10	−	2.74966e10i	1.06331	−	1.84171i
$808$		0			0
$809$	−2.91367e9	−	5.04662e9i	−0.193473	−	0.335105i	0.752926	−	0.658105i	$-0.228642\pi$
$809$	−2.91367e9	−	5.04662e9i	−0.193473	−	0.335105i	−0.946399	+	0.323000i	$0.895308\pi$
$810$		0			0
$811$	−1.13903e10			−0.749829			−0.374914	−	0.927059i	$-0.622328\pi$
$811$	−1.13903e10			−0.749829			−0.374914	+	0.927059i	$0.622328\pi$
$812$		0			0
$813$	−2.24374e9			−0.146439
$814$		0			0
$815$	−5.78358e8	−	1.00175e9i	−0.0374236	−	0.0648195i
$816$		0			0
$817$	−5.91930e9	+	1.02525e10i	−0.379746	+	0.657739i
$818$		0			0
$819$	8.92216e9	+	3.34207e10i	0.567514	+	2.12580i
$820$		0			0
$821$	−5.98364e9	+	1.03640e10i	−0.377367	+	0.653619i	−0.990678	−	0.136222i	$-0.956504\pi$
$821$	−5.98364e9	+	1.03640e10i	−0.377367	+	0.653619i	0.613311	+	0.789841i	$0.289837\pi$
$822$		0			0
$823$	−1.24348e10	−	2.15376e10i	−0.777568	−	1.34679i	−0.933340	−	0.358994i	$-0.883120\pi$
$823$	−1.24348e10	−	2.15376e10i	−0.777568	−	1.34679i	0.155772	−	0.987793i	$-0.450213\pi$
$824$		0			0
$825$	1.95033e10			1.20926
$826$		0			0
$827$	−1.05586e10			−0.649137			−0.324569	−	0.945862i	$-0.605219\pi$
$827$	−1.05586e10			−0.649137			−0.324569	+	0.945862i	$0.605219\pi$
$828$		0			0
$829$	5.93269e9	+	1.02757e10i	0.361668	+	0.626428i	0.988236	−	0.152939i	$-0.0488739\pi$
$829$	5.93269e9	+	1.02757e10i	0.361668	+	0.626428i	−0.626567	+	0.779367i	$0.715541\pi$
$830$		0			0
$831$	−2.51034e10	+	4.34804e10i	−1.51750	+	2.62839i
$832$		0			0
$833$	2.60805e10	−	1.49938e10i	1.56336	−	0.898779i
$834$		0			0
$835$	3.92129e8	−	6.79188e8i	0.0233092	−	0.0403726i
$836$		0			0
$837$	7.90304e9	+	1.36885e10i	0.465859	+	0.806892i
$838$		0			0
$839$	−4.68285e9			−0.273743			−0.136872	−	0.990589i	$-0.543705\pi$
$839$	−4.68285e9			−0.273743			−0.136872	+	0.990589i	$0.543705\pi$
$840$		0			0
$841$	−1.53152e10			−0.887847
$842$		0			0
$843$	−1.98941e10	−	3.44576e10i	−1.14374	−	1.98102i
$844$		0			0
$845$	4.46866e8	−	7.73995e8i	0.0254788	−	0.0441306i
$846$		0			0
$847$	−2.06439e9	−	7.73280e9i	−0.116734	−	0.437265i
$848$		0			0
$849$	5.50851e8	−	9.54102e8i	0.0308928	−	0.0535079i
$850$		0			0
$851$	−4.85391e9	−	8.40721e9i	−0.269984	−	0.467626i
$852$		0			0
$853$	−5.76473e9			−0.318022			−0.159011	−	0.987277i	$-0.550831\pi$
$853$	−5.76473e9			−0.318022			−0.159011	+	0.987277i	$0.550831\pi$
$854$		0			0
$855$	−2.56290e9			−0.140233
$856$		0			0
$857$	−1.23111e10	−	2.13235e10i	−0.668135	−	1.15724i	−0.978425	−	0.206602i	$-0.933759\pi$
$857$	−1.23111e10	−	2.13235e10i	−0.668135	−	1.15724i	0.310290	−	0.950642i	$-0.399574\pi$
$858$		0			0
$859$	1.49943e10	−	2.59709e10i	0.807144	−	1.39801i	−0.107691	−	0.994184i	$-0.534346\pi$
$859$	1.49943e10	−	2.59709e10i	0.807144	−	1.39801i	0.914834	−	0.403829i	$-0.132321\pi$
$860$		0			0
$861$	3.59070e10	+	9.65662e9i	1.91720	+	0.515601i
$862$		0			0
$863$	4.28994e8	−	7.43040e8i	0.0227203	−	0.0393527i	−0.854442	−	0.519547i	$-0.826101\pi$
$863$	4.28994e8	−	7.43040e8i	0.0227203	−	0.0393527i	0.877162	+	0.480195i	$0.159434\pi$
$864$		0			0
$865$	−8.17766e8	−	1.41641e9i	−0.0429608	−	0.0744103i
$866$		0			0
$867$	7.10377e10			3.70187
$868$		0			0
$869$	−1.63118e10			−0.843204
$870$		0			0
$871$	6.28533e9	+	1.08865e10i	0.322303	+	0.558245i
$872$		0			0
$873$	−2.53014e9	+	4.38233e9i	−0.128705	+	0.222923i
$874$		0			0
$875$	2.12594e9	−	2.12204e9i	0.107281	−	0.107084i
$876$		0			0
$877$	2.82487e9	−	4.89282e9i	0.141416	−	0.244940i	−0.786614	−	0.617445i	$-0.788168\pi$
$877$	2.82487e9	−	4.89282e9i	0.141416	−	0.244940i	0.928030	+	0.372505i	$0.121501\pi$
$878$		0			0
$879$	1.76663e10	+	3.05989e10i	0.877373	+	1.51966i
$880$		0			0
$881$	−2.87116e10			−1.41462			−0.707312	−	0.706902i	$-0.750092\pi$
$881$	−2.87116e10			−1.41462			−0.707312	+	0.706902i	$0.750092\pi$
$882$		0			0
$883$	−1.38023e10			−0.674668			−0.337334	−	0.941385i	$-0.609525\pi$
$883$	−1.38023e10			−0.674668			−0.337334	+	0.941385i	$0.609525\pi$
$884$		0			0
$885$	−1.32011e8	−	2.28651e8i	−0.00640192	−	0.0110884i
$886$		0			0
$887$	−1.86125e10	+	3.22378e10i	−0.895513	+	1.55107i	−0.0623449	+	0.998055i	$0.519858\pi$
$887$	−1.86125e10	+	3.22378e10i	−0.895513	+	1.55107i	−0.833168	+	0.553020i	$0.813475\pi$
$888$		0			0
$889$	9.08672e8	−	9.07003e8i	0.0433761	−	0.0432965i
$890$		0			0
$891$	1.52899e9	−	2.64828e9i	0.0724156	−	0.125427i
$892$		0			0
$893$	−3.35131e9	−	5.80464e9i	−0.157483	−	0.272769i
$894$		0			0
$895$	1.06304e9			0.0495642
$896$		0			0
$897$	−1.31764e10			−0.609571
$898$		0			0
$899$	2.94351e9	+	5.09830e9i	0.135116	+	0.234027i
$900$		0			0
$901$	−1.98757e10	+	3.44256e10i	−0.905284	+	1.56800i
$902$		0			0
$903$	−2.46160e10	−	6.62009e9i	−1.11253	−	0.299197i
$904$		0			0
$905$	1.73390e8	−	3.00321e8i	0.00777597	−	0.0134684i
$906$		0			0
$907$	−2.80129e9	−	4.85198e9i	−0.124662	−	0.215920i	0.796939	−	0.604060i	$-0.206451\pi$
$907$	−2.80129e9	−	4.85198e9i	−0.124662	−	0.215920i	−0.921601	+	0.388139i	$0.873118\pi$
$908$		0			0
$909$	4.35322e10			1.92237
$910$		0			0
$911$	3.57046e10			1.56462			0.782312	−	0.622886i	$-0.214040\pi$
$911$	3.57046e10			1.56462			0.782312	+	0.622886i	$0.214040\pi$
$912$		0			0
$913$	5.52558e9	+	9.57059e9i	0.240287	+	0.416189i
$914$		0			0
$915$	−6.89494e8	+	1.19424e9i	−0.0297548	+	0.0515368i
$916$		0			0
$917$	8.31489e9	+	3.11460e10i	0.356093	+	1.33386i
$918$		0			0
$919$	2.19020e9	−	3.79354e9i	0.0930850	−	0.161228i	−0.815723	−	0.578443i	$-0.803660\pi$
$919$	2.19020e9	−	3.79354e9i	0.0930850	−	0.161228i	0.908808	+	0.417215i	$0.136994\pi$
$920$		0			0
$921$	−8.77020e9	−	1.51904e10i	−0.369914	−	0.640710i
$922$		0			0
$923$	−1.13394e10			−0.474663
$924$		0			0
$925$	4.50412e10			1.87117
$926$		0			0
$927$	−4.04550e9	−	7.00701e9i	−0.166799	−	0.288904i
$928$		0			0
$929$	9.08246e9	−	1.57313e10i	0.371662	−	0.643738i	−0.618159	−	0.786053i	$-0.712121\pi$
$929$	9.08246e9	−	1.57313e10i	0.371662	−	0.643738i	0.989821	+	0.142315i	$0.0454546\pi$
$930$		0			0
$931$	2.30843e10	+	1.33843e10i	0.937546	+	0.543592i
$932$		0			0
$933$	2.74096e10	−	4.74747e10i	1.10488	−	1.91371i
$934$		0			0
$935$	1.26738e9	+	2.19516e9i	0.0507066	+	0.0878264i
$936$		0			0
$937$	2.55925e10			1.01631			0.508153	−	0.861267i	$-0.330328\pi$
$937$	2.55925e10			1.01631			0.508153	+	0.861267i	$0.330328\pi$
$938$		0			0
$939$	−1.07277e10			−0.422840
$940$		0			0
$941$	−1.53193e10	−	2.65338e10i	−0.599343	−	1.03809i	−0.992918	−	0.118800i	$-0.962095\pi$
$941$	−1.53193e10	−	2.65338e10i	−0.599343	−	1.03809i	0.393576	−	0.919292i	$-0.371238\pi$
$942$		0			0
$943$	−4.46124e9	+	7.72710e9i	−0.173247	+	0.300072i
$944$		0			0
$945$	−5.87268e8	−	2.19980e9i	−0.0226373	−	0.0847952i
$946$		0			0
$947$	1.13440e10	−	1.96484e10i	0.434052	−	0.751800i	−0.563166	−	0.826344i	$-0.690417\pi$
$947$	1.13440e10	−	1.96484e10i	0.434052	−	0.751800i	0.997218	+	0.0745437i	$0.0237500\pi$
$948$		0			0
$949$	−8.38923e9	−	1.45306e10i	−0.318633	−	0.551888i
$950$		0			0
$951$	−1.18305e10			−0.446038
$952$		0			0
$953$	−2.31130e10			−0.865030			−0.432515	−	0.901627i	$-0.642374\pi$
$953$	−2.31130e10			−0.865030			−0.432515	+	0.901627i	$0.642374\pi$
$954$		0			0
$955$	−1.58328e9	−	2.74232e9i	−0.0588228	−	0.101884i
$956$		0			0
$957$	5.52208e9	−	9.56452e9i	0.203662	−	0.352754i
$958$		0			0
$959$	3.88239e10	+	1.04411e10i	1.42146	+	0.382278i
$960$		0			0
$961$	4.79931e9	−	8.31265e9i	0.174440	−	0.302140i
$962$		0			0
$963$	2.00413e10	+	3.47125e10i	0.723157	+	1.25255i
$964$		0			0
$965$	−1.34692e9			−0.0482498
$966$		0			0
$967$	−1.63381e10			−0.581043			−0.290521	−	0.956868i	$-0.593829\pi$
$967$	−1.63381e10			−0.581043			−0.290521	+	0.956868i	$0.593829\pi$
$968$		0			0
$969$	4.54956e10	+	7.88006e10i	1.60633	+	2.78225i
$970$		0			0
$971$	−2.36893e9	+	4.10311e9i	−0.0830396	+	0.143829i	−0.904554	−	0.426359i	$-0.859796\pi$
$971$	−2.36893e9	+	4.10311e9i	−0.0830396	+	0.143829i	0.821515	+	0.570188i	$0.193129\pi$
$972$		0			0
$973$	−1.91265e10	+	1.90914e10i	−0.665642	+	0.664420i
$974$		0			0
$975$	3.05672e10	−	5.29440e10i	1.05619	−	1.82937i
$976$		0			0
$977$	−8.57370e9	−	1.48501e10i	−0.294129	−	0.509446i	0.680653	−	0.732606i	$-0.261696\pi$
$977$	−8.57370e9	−	1.48501e10i	−0.294129	−	0.509446i	−0.974782	+	0.223160i	$0.928363\pi$
$978$		0			0
$979$	−2.13506e10			−0.727230
$980$		0			0
$981$	−4.60751e10			−1.55821
$982$		0			0
$983$	−2.27209e10	−	3.93537e10i	−0.762935	−	1.32144i	−0.941332	−	0.337483i	$-0.890425\pi$
$983$	−2.27209e10	−	3.93537e10i	−0.762935	−	1.32144i	0.178397	−	0.983959i	$-0.442909\pi$
$984$		0			0
$985$	3.12083e8	−	5.40543e8i	0.0104050	−	0.0180220i
$986$		0			0
$987$	1.02143e10	−	1.01956e10i	0.338142	−	0.337521i
$988$		0			0
$989$	3.05840e9	−	5.29731e9i	0.100533	−	0.174128i
$990$		0			0
$991$	−1.63460e10	−	2.83122e10i	−0.533525	−	0.924092i	−0.999233	−	0.0391537i	$-0.987534\pi$
$991$	−1.63460e10	−	2.83122e10i	−0.533525	−	0.924092i	0.465708	−	0.884938i	$-0.345800\pi$
$992$		0			0
$993$	−1.07614e10			−0.348776
$994$		0			0
$995$	1.04462e9			0.0336184
$996$		0			0
$997$	−2.70222e10	−	4.68038e10i	−0.863550	−	1.49571i	−0.868480	−	0.495724i	$-0.834903\pi$
$997$	−2.70222e10	−	4.68038e10i	−0.863550	−	1.49571i	0.00493036	−	0.999988i	$-0.498431\pi$
$998$		0			0
$999$	3.42400e10	−	5.93054e10i	1.08656	−	1.88198i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	112.8.i.a.65.1		4
4.3	odd	2		14.8.c.a.9.2	✓	4
7.4	even	3	inner	112.8.i.a.81.1		4
12.11	even	2		126.8.g.e.37.2		4
28.3	even	6		98.8.c.h.67.1		4
28.11	odd	6		14.8.c.a.11.2	yes	4
28.19	even	6		98.8.a.k.1.2		2
28.23	odd	6		98.8.a.h.1.1		2
28.27	even	2		98.8.c.h.79.1		4
84.11	even	6		126.8.g.e.109.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
14.8.c.a.9.2	✓	4	4.3	odd	2
14.8.c.a.11.2	yes	4	28.11	odd	6
98.8.a.h.1.1		2	28.23	odd	6
98.8.a.k.1.2		2	28.19	even	6
98.8.c.h.67.1		4	28.3	even	6
98.8.c.h.79.1		4	28.27	even	2
112.8.i.a.65.1		4	1.1	even	1	trivial
112.8.i.a.81.1		4	7.4	even	3	inner
126.8.g.e.37.2		4	12.11	even	2
126.8.g.e.109.2		4	84.11	even	6

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q+(-38.4387 - 66.5778i) q^{3} +(10.6226 - 18.3989i) q^{5} +(642.284 - 641.104i) q^{7} +(-1861.57 + 3224.33i) q^{9} +(1633.07 + 2828.56i) q^{11} +10238.0 q^{13} -1633.28 q^{15} +(18264.6 + 31635.1i) q^{17} +(-16200.6 + 28060.3i) q^{19} +(-67371.9 - 18118.6i) q^{21} +(8370.58 - 14498.3i) q^{23} +(38836.8 + 67267.3i) q^{25} +118094. q^{27} +43984.4 q^{29} +(66921.6 + 115912. i) q^{31} +(125546. - 217452. i) q^{33} +(-4972.88 - 18627.5i) q^{35} +(289939. - 502188. i) q^{37} +(-393534. - 681621. i) q^{39} -532967. q^{41} +365375. q^{43} +(39549.4 + 68501.5i) q^{45} +(-103432. + 179149. i) q^{47} +(1513.36 - 823542. i) q^{49} +(1.40413e6 - 2.43203e6i) q^{51} +(544104. + 942416. i) q^{53} +69389.9 q^{55} +2.49092e6 q^{57} +(80826.2 + 139995. i) q^{59} +(422154. - 731192. i) q^{61} +(871477. + 3.26439e6i) q^{63} +(108754. - 188367. i) q^{65} +(613923. + 1.06335e6i) q^{67} -1.28702e6 q^{69} -1.10759e6 q^{71} +(-819423. - 1.41928e6i) q^{73} +(2.98567e6 - 5.17134e6i) q^{75} +(2.86230e6 + 769770. i) q^{77} +(-2.49711e6 + 4.32512e6i) q^{79} +(-468133. - 810829. i) q^{81} +3.38355e6 q^{83} +776069. q^{85} +(-1.69070e6 - 2.92838e6i) q^{87} +(-3.26848e6 + 5.66118e6i) q^{89} +(6.57568e6 - 6.56361e6i) q^{91} +(5.14476e6 - 8.91098e6i) q^{93} +(344185. + 596146. i) q^{95} +1.35914e6 q^{97} -1.21603e7 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 56 q^{3} + 238 q^{5} - 168 q^{7} - 1972 q^{9} + 5848 q^{11} + 2632 q^{13} + 5784 q^{15} + 47642 q^{17} - 41048 q^{19} - 169582 q^{21} - 49316 q^{23} + 108816 q^{25} + 400624 q^{27} + 345640 q^{29}+ \cdots - 15278208 q^{99}+O(q^{100}) \) 4 * q - 56 * q^3 + 238 * q^5 - 168 * q^7 - 1972 * q^9 + 5848 * q^11 + 2632 * q^13 + 5784 * q^15 + 47642 * q^17 - 41048 * q^19 - 169582 * q^21 - 49316 * q^23 + 108816 * q^25 + 400624 * q^27 + 345640 * q^29 - 70252 * q^31 + 197190 * q^33 + 36904 * q^35 + 88166 * q^37 - 973336 * q^39 - 2312520 * q^41 - 115088 * q^43 - 300468 * q^45 - 1412292 * q^47 + 465892 * q^49 + 2576256 * q^51 + 2361174 * q^53 + 1258040 * q^55 + 4620796 * q^57 + 1842512 * q^59 - 1278242 * q^61 + 2121448 * q^63 - 1716372 * q^65 - 2121480 * q^67 - 5332236 * q^69 - 7200320 * q^71 + 1634682 * q^73 + 5321176 * q^75 + 1416226 * q^77 - 9192604 * q^79 - 3049498 * q^81 + 56560 * q^83 + 6369676 * q^85 - 691656 * q^87 - 8936550 * q^89 + 26111120 * q^91 + 14550490 * q^93 + 2562604 * q^95 + 11748408 * q^97 - 15278208 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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