LMFDB - Embedded newform 208.8.w.a.49.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [208,8,Mod(17,208)] 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("208.17"); S:= CuspForms(chi, 8); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(208, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 1])) N = Newforms(chi, 8, names="a")

Level:	$ N $	$=$	$ 208 = 2^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 8 $
Character orbit:	$[\chi]$	$=$	208.w (of order $6$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(1)] == traces)

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

Self dual:	no
Analytic conductor:	$64.9760853007$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{14} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{14} + 1279 x^{12} + 629380 x^{10} + 148562016 x^{8} + 16872573312 x^{6} + 790180980480 x^{4} + \cdots + 4669637050368 $ x^14 + 1279x^12 + 629380x^10 + 148562016x^8 + 16872573312x^6 + 790180980480x^4 + 10484997239808x^2 + 4669637050368
Coefficient ring:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{22}\cdot 3^{3}\cdot 13^{4} $
Twist minimal:	no (minimal twist has level 13)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			49.1
Root			$-16.7657i$ of defining polynomial
Character	$\chi$	$=$	208.49
Dual form			208.8.w.a.17.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-40.5114 - 70.1679i) q^{3} +223.318i q^{5} +(577.559 + 333.454i) q^{7} +(-2188.85 + 3791.20i) q^{9} +(-6573.82 + 3795.40i) q^{11} +(-6519.11 + 4499.97i) q^{13} +(15669.7 - 9046.93i) q^{15} +(-7051.44 + 12213.4i) q^{17} +(7226.10 + 4171.99i) q^{19} -54034.7i q^{21} +(-29072.6 - 50355.3i) q^{23} +28254.0 q^{25} +177497. q^{27} +(16522.5 + 28617.8i) q^{29} -91450.2i q^{31} +(532630. + 307514. i) q^{33} +(-74466.2 + 128979. i) q^{35} +(-307911. + 177773. i) q^{37} +(579852. + 275132. i) q^{39} +(504092. - 291038. i) q^{41} +(52248.9 - 90497.7i) q^{43} +(-846644. - 488810. i) q^{45} -398024. i q^{47} +(-189389. - 328031. i) q^{49} +1.14266e6 q^{51} +1.49881e6 q^{53} +(-847581. - 1.46805e6i) q^{55} -676053. i q^{57} +(-286332. - 165314. i) q^{59} +(721365. - 1.24944e6i) q^{61} +(-2.52838e6 + 1.45976e6i) q^{63} +(-1.00493e6 - 1.45583e6i) q^{65} +(-2.22292e6 + 1.28340e6i) q^{67} +(-2.35555e6 + 4.07993e6i) q^{69} +(-1.12386e6 - 648863. i) q^{71} -2.66195e6i q^{73} +(-1.14461e6 - 1.98253e6i) q^{75} -5.06236e6 q^{77} -2.13906e6 q^{79} +(-2.40364e6 - 4.16323e6i) q^{81} -1.63158e6i q^{83} +(-2.72748e6 - 1.57471e6i) q^{85} +(1.33870e6 - 2.31870e6i) q^{87} +(-347455. + 200603. i) q^{89} +(-5.26570e6 + 425178. i) q^{91} +(-6.41686e6 + 3.70478e6i) q^{93} +(-931681. + 1.61372e6i) q^{95} +(-9.56294e6 - 5.52117e6i) q^{97} -3.32303e7i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q - 26 q^{3} + 2772 q^{7} - 3491 q^{9} - 6516 q^{11} + 5109 q^{13} + 19266 q^{15} - 38403 q^{17} - 43254 q^{19} + 68550 q^{23} + 39380 q^{25} + 432400 q^{27} + 221583 q^{29} + 219756 q^{33} - 659616 q^{35}+ \cdots - 68556288 q^{97}+O(q^{100}) $ 14 * q - 26 * q^3 + 2772 * q^7 - 3491 * q^9 - 6516 * q^11 + 5109 * q^13 + 19266 * q^15 - 38403 * q^17 - 43254 * q^19 + 68550 * q^23 + 39380 * q^25 + 432400 * q^27 + 221583 * q^29 + 219756 * q^33 - 659616 * q^35 - 565347 * q^37 - 30082 * q^39 + 1716057 * q^41 + 882692 * q^43 - 4616853 * q^45 + 2653979 * q^49 + 38964 * q^51 - 1791210 * q^53 - 2453536 * q^55 + 1189032 * q^59 + 3858525 * q^61 - 7718520 * q^63 + 10424661 * q^65 - 4674330 * q^67 - 13743840 * q^69 + 25203126 * q^71 - 10316152 * q^75 + 19599336 * q^77 + 30688440 * q^79 + 13886221 * q^81 - 22436313 * q^85 - 12769686 * q^87 + 2123160 * q^89 - 37205532 * q^91 + 36638964 * q^93 + 43420386 * q^95 - 68556288 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/208\mathbb{Z}\right)^\times$.

$n$	$53$	$79$	$145$
$\chi(n)$	$1$	$1$	$e\left(\frac{5}{6}\right)$

Coefficient data

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

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

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

$2$		0			0
$2$		0			0
$3$	−40.5114	−	70.1679i	−0.866270	−	1.50042i	−0.865781	−	0.500423i	$-0.833178\pi$
$3$	−40.5114	−	70.1679i	−0.866270	−	1.50042i	−0.000489031	−	1.00000i	$-0.500156\pi$
$4$		0			0
$5$			223.318i			0.798967i	0.916740	+	0.399483i	$0.130811\pi$
$5$			223.318i			0.798967i	−0.916740	+	0.399483i	$0.869189\pi$
$6$		0			0
$7$	577.559	+	333.454i	0.636433	+	0.367445i	0.783239	−	0.621720i	$-0.213566\pi$
$7$	577.559	+	333.454i	0.636433	+	0.367445i	−0.146806	+	0.989165i	$0.546899\pi$
$8$		0			0
$9$	−2188.85	+	3791.20i	−1.00085	+	1.73352i
$10$		0			0
$11$	−6573.82	+	3795.40i	−1.48917	+	0.859771i	−0.999923	−	0.0123755i	$-0.996061\pi$
$11$	−6573.82	+	3795.40i	−1.48917	+	0.859771i	−0.489244	+	0.872147i	$0.662727\pi$
$12$		0			0
$13$	−6519.11	+	4499.97i	−0.822975	+	0.568078i
$13$	−6519.11	+	4499.97i	−0.822975	+	0.568078i
$14$		0			0
$15$	15669.7	−	9046.93i	1.19879	−	0.692121i
$16$		0			0
$17$	−7051.44	+	12213.4i	−0.348102	+	0.602930i	−0.985912	−	0.167263i	$-0.946507\pi$
$17$	−7051.44	+	12213.4i	−0.348102	+	0.602930i	0.637810	+	0.770193i	$0.279840\pi$
$18$		0			0
$19$	7226.10	+	4171.99i	0.241694	+	0.139542i	0.615955	−	0.787781i	$-0.288770\pi$
$19$	7226.10	+	4171.99i	0.241694	+	0.139542i	−0.374261	+	0.927323i	$0.622104\pi$
$20$		0			0
$21$		−	54034.7i		−	1.27323i
$22$		0			0
$23$	−29072.6	−	50355.3i	−0.498238	−	0.862974i	0.501760	−	0.865007i	$-0.332686\pi$
$23$	−29072.6	−	50355.3i	−0.498238	−	0.862974i	−0.999998	+	0.00203315i	$0.999353\pi$
$24$		0			0
$25$	28254.0			0.361652
$26$		0			0
$27$	177497.			1.73547
$28$		0			0
$29$	16522.5	+	28617.8i	0.125801	+	0.217893i	0.922046	−	0.387081i	$-0.126517\pi$
$29$	16522.5	+	28617.8i	0.125801	+	0.217893i	−0.796245	+	0.604974i	$0.793183\pi$
$30$		0			0
$31$		−	91450.2i		−	0.551339i	−0.961252	−	0.275669i	$-0.911100\pi$
$31$		−	91450.2i		−	0.551339i	0.961252	−	0.275669i	$-0.0888995\pi$
$32$		0			0
$33$	532630.	+	307514.i	2.58004	+	1.48959i
$34$		0			0
$35$	−74466.2	+	128979.i	−0.293576	+	0.508489i
$36$		0			0
$37$	−307911.	+	177773.i	−0.999354	+	0.576977i	−0.908057	−	0.418846i	$-0.862435\pi$
$37$	−307911.	+	177773.i	−0.999354	+	0.576977i	−0.0912968	+	0.995824i	$0.529101\pi$
$38$		0			0
$39$	579852.	+	275132.i	1.56528	+	0.742701i
$40$		0			0
$41$	504092.	−	291038.i	1.14226	−	0.659486i	0.195273	−	0.980749i	$-0.437441\pi$
$41$	504092.	−	291038.i	1.14226	−	0.659486i	0.946990	+	0.321263i	$0.104107\pi$
$42$		0			0
$43$	52248.9	−	90497.7i	0.100216	−	0.173579i	−0.811558	−	0.584273i	$-0.801380\pi$
$43$	52248.9	−	90497.7i	0.100216	−	0.173579i	0.911774	+	0.410693i	$0.134713\pi$
$44$		0			0
$45$	−846644.	−	488810.i	−1.38502	−	0.799644i
$46$		0			0
$47$		−	398024.i		−	0.559199i	−0.960117	−	0.279600i	$-0.909798\pi$
$47$		−	398024.i		−	0.559199i	0.960117	−	0.279600i	$-0.0902017\pi$
$48$		0			0
$49$	−189389.	−	328031.i	−0.229968	−	0.398317i
$50$		0			0
$51$	1.14266e6			1.20620
$52$		0			0
$53$	1.49881e6			1.38287			0.691433	−	0.722441i	$-0.256980\pi$
$53$	1.49881e6			1.38287			0.691433	+	0.722441i	$0.256980\pi$
$54$		0			0
$55$	−847581.	−	1.46805e6i	−0.686929	−	1.18980i
$56$		0			0
$57$		−	676053.i		−	0.483525i
$58$		0			0
$59$	−286332.	−	165314.i	−0.181505	−	0.104792i	0.406495	−	0.913653i	$-0.366751\pi$
$59$	−286332.	−	165314.i	−0.181505	−	0.104792i	−0.588000	+	0.808861i	$0.700084\pi$
$60$		0			0
$61$	721365.	−	1.24944e6i	0.406912	−	0.704793i	−0.587630	−	0.809130i	$-0.699939\pi$
$61$	721365.	−	1.24944e6i	0.406912	−	0.704793i	0.994542	+	0.104337i	$0.0332721\pi$
$62$		0			0
$63$	−2.52838e6	+	1.45976e6i	−1.27394	+	0.735512i
$64$		0			0
$65$	−1.00493e6	−	1.45583e6i	−0.453876	−	0.657530i
$66$		0			0
$67$	−2.22292e6	+	1.28340e6i	−0.902945	+	0.521316i	−0.878155	−	0.478377i	$-0.841225\pi$
$67$	−2.22292e6	+	1.28340e6i	−0.902945	+	0.521316i	−0.0247906	+	0.999693i	$0.507892\pi$
$68$		0			0
$69$	−2.35555e6	+	4.07993e6i	−0.863217	+	1.49514i
$70$		0			0
$71$	−1.12386e6	−	648863.i	−0.372657	−	0.215154i	0.301962	−	0.953320i	$-0.402359\pi$
$71$	−1.12386e6	−	648863.i	−0.372657	−	0.215154i	−0.674619	+	0.738166i	$0.735692\pi$
$72$		0			0
$73$		−	2.66195e6i		−	0.800885i	−0.916322	−	0.400442i	$-0.868856\pi$
$73$		−	2.66195e6i		−	0.800885i	0.916322	−	0.400442i	$-0.131144\pi$
$74$		0			0
$75$	−1.14461e6	−	1.98253e6i	−0.313288	−	0.542631i
$76$		0			0
$77$	−5.06236e6			−1.26367
$78$		0			0
$79$	−2.13906e6			−0.488121			−0.244060	−	0.969760i	$-0.578480\pi$
$79$	−2.13906e6			−0.488121			−0.244060	+	0.969760i	$0.578480\pi$
$80$		0			0
$81$	−2.40364e6	−	4.16323e6i	−0.502542	−	0.870428i
$82$		0			0
$83$		−	1.63158e6i		−	0.313211i	−0.987661	−	0.156605i	$-0.949945\pi$
$83$		−	1.63158e6i		−	0.313211i	0.987661	−	0.156605i	$-0.0500550\pi$
$84$		0			0
$85$	−2.72748e6	−	1.57471e6i	−0.481721	−	0.278122i
$86$		0			0
$87$	1.33870e6	−	2.31870e6i	0.217955	−	0.377509i
$88$		0			0
$89$	−347455.	+	200603.i	−0.0522436	+	0.0301629i	−0.525894	−	0.850550i	$-0.676269\pi$
$89$	−347455.	+	200603.i	−0.0522436	+	0.0301629i	0.473651	+	0.880713i	$0.342936\pi$
$90$		0			0
$91$	−5.26570e6	+	425178.i	−0.732506	+	0.0591460i
$92$		0			0
$93$	−6.41686e6	+	3.70478e6i	−0.827242	+	0.477608i
$94$		0			0
$95$	−931681.	+	1.61372e6i	−0.111490	+	0.193106i
$96$		0			0
$97$	−9.56294e6	−	5.52117e6i	−1.06387	−	0.614228i	−0.137373	−	0.990519i	$-0.543866\pi$
$97$	−9.56294e6	−	5.52117e6i	−1.06387	−	0.614228i	−0.926501	+	0.376291i	$0.877199\pi$
$98$		0			0
$99$		−	3.32303e7i		−	3.44200i
$100$		0			0
$101$	3.84159e6	+	6.65383e6i	0.371011	+	0.642610i	0.989721	−	0.143010i	$-0.0456779\pi$
$101$	3.84159e6	+	6.65383e6i	0.371011	+	0.642610i	−0.618711	+	0.785619i	$0.712345\pi$
$102$		0			0
$103$	−485826.			−0.0438077			−0.0219039	−	0.999760i	$-0.506973\pi$
$103$	−485826.			−0.0438077			−0.0219039	+	0.999760i	$0.506973\pi$
$104$		0			0
$105$	1.20669e7			1.01727
$106$		0			0
$107$	−34987.1	−	60599.5i	−0.00276099	−	0.00478218i	0.864642	−	0.502389i	$-0.167546\pi$
$107$	−34987.1	−	60599.5i	−0.00276099	−	0.00478218i	−0.867403	+	0.497607i	$0.834212\pi$
$108$		0			0
$109$			4.06366e6i			0.300556i	0.988644	+	0.150278i	$0.0480168\pi$
$109$			4.06366e6i			0.300556i	−0.988644	+	0.150278i	$0.951983\pi$
$110$		0			0
$111$	2.49478e7	+	1.44036e7i	1.73142	+	0.999636i
$112$		0			0
$113$	1.04977e7	−	1.81826e7i	0.684417	−	1.18544i	−0.289203	−	0.957268i	$-0.593390\pi$
$113$	1.04977e7	−	1.81826e7i	0.684417	−	1.18544i	0.973620	−	0.228177i	$-0.0732764\pi$
$114$		0			0
$115$	1.12452e7	−	6.49245e6i	0.689488	−	0.398076i
$116$		0			0
$117$	−2.79094e6	−	3.45650e7i	−0.161102	−	1.99520i
$118$		0			0
$119$	−8.14524e6	+	4.70266e6i	−0.443087	+	0.255817i
$120$		0			0
$121$	1.90665e7	−	3.30242e7i	0.978414	−	1.69466i
$122$		0			0
$123$	−4.08430e7	−	2.35807e7i	−1.97902	−	1.14259i
$124$		0			0
$125$			2.37564e7i			1.08791i
$126$		0			0
$127$	−4.66784e6	−	8.08493e6i	−0.202210	−	0.350238i	0.747030	−	0.664790i	$-0.231479\pi$
$127$	−4.66784e6	−	8.08493e6i	−0.202210	−	0.350238i	−0.949240	+	0.314552i	$0.898146\pi$
$128$		0			0
$129$	−8.46671e6			−0.347257
$130$		0			0
$131$	4.07546e7			1.58390			0.791948	−	0.610588i	$-0.209067\pi$
$131$	4.07546e7			1.58390			0.791948	+	0.610588i	$0.209067\pi$
$132$		0			0
$133$	2.78233e6	+	4.81914e6i	0.102548	+	0.177619i
$134$		0			0
$135$			3.96383e7i			1.38659i
$136$		0			0
$137$	4.21961e7	+	2.43619e7i	1.40201	+	0.809449i	0.994598	−	0.103797i	$-0.0330993\pi$
$137$	4.21961e7	+	2.43619e7i	1.40201	+	0.809449i	0.407408	+	0.913246i	$0.366433\pi$
$138$		0			0
$139$	−2.66447e7	+	4.61501e7i	−0.841511	+	1.45754i	0.0471059	+	0.998890i	$0.485000\pi$
$139$	−2.66447e7	+	4.61501e7i	−0.841511	+	1.45754i	−0.888617	+	0.458650i	$0.848333\pi$
$140$		0			0
$141$	−2.79285e7	+	1.61245e7i	−0.839036	+	0.484417i
$142$		0			0
$143$	2.57763e7	−	5.43246e7i	0.737130	−	1.55353i
$144$		0			0
$145$	−6.39088e6	+	3.68977e6i	−0.174089	+	0.100511i
$146$		0			0
$147$	−1.53448e7	+	2.65780e7i	−0.398429	+	0.690100i
$148$		0			0
$149$	−3.13839e7	−	1.81195e7i	−0.777241	−	0.448740i	0.0582108	−	0.998304i	$-0.481460\pi$
$149$	−3.13839e7	−	1.81195e7i	−0.777241	−	0.448740i	−0.835451	+	0.549564i	$0.814794\pi$
$150$		0			0
$151$		−	2.55676e7i		−	0.604326i	−0.953256	−	0.302163i	$-0.902291\pi$
$151$		−	2.55676e7i		−	0.604326i	0.953256	−	0.302163i	$-0.0977087\pi$
$152$		0			0
$153$	−3.08691e7	−	5.34669e7i	−0.696793	−	1.20688i
$154$		0			0
$155$	2.04225e7			0.440502
$156$		0			0
$157$	4.23295e7			0.872960			0.436480	−	0.899714i	$-0.356225\pi$
$157$	4.23295e7			0.872960			0.436480	+	0.899714i	$0.356225\pi$
$158$		0			0
$159$	−6.07188e7	−	1.05168e8i	−1.19793	−	2.07488i
$160$		0			0
$161$		−	3.87775e7i		−	0.732301i
$162$		0			0
$163$	2.98690e7	+	1.72449e7i	0.540213	+	0.311892i	0.745165	−	0.666880i	$-0.232371\pi$
$163$	2.98690e7	+	1.72449e7i	0.540213	+	0.311892i	−0.204952	+	0.978772i	$0.565704\pi$
$164$		0			0
$165$	−6.86734e7	+	1.18946e8i	−1.19013	+	2.06137i
$166$		0			0
$167$	−6.93649e6	+	4.00479e6i	−0.115248	+	0.0665383i	−0.556516	−	0.830837i	$-0.687862\pi$
$167$	−6.93649e6	+	4.00479e6i	−0.115248	+	0.0665383i	0.441268	+	0.897375i	$0.354529\pi$
$168$		0			0
$169$	2.22490e7	−	5.86716e7i	0.354574	−	0.935028i
$170$		0			0
$171$	−3.16337e7	+	1.82637e7i	−0.483798	+	0.279321i
$172$		0			0
$173$	1.86570e7	−	3.23149e7i	0.273956	−	0.474506i	−0.695915	−	0.718124i	$-0.745001\pi$
$173$	1.86570e7	−	3.23149e7i	0.273956	−	0.474506i	0.969871	+	0.243618i	$0.0783343\pi$
$174$		0			0
$175$	1.63184e7	+	9.42142e6i	0.230167	+	0.132887i
$176$		0			0
$177$			2.67884e7i			0.363112i
$178$		0			0
$179$	3.42603e7	+	5.93407e7i	0.446484	+	0.773333i	0.998154	−	0.0607291i	$-0.0193426\pi$
$179$	3.42603e7	+	5.93407e7i	0.446484	+	0.773333i	−0.551670	+	0.834062i	$0.686009\pi$
$180$		0			0
$181$	−1.20860e8			−1.51498			−0.757489	−	0.652848i	$-0.773574\pi$
$181$	−1.20860e8			−1.51498			−0.757489	+	0.652848i	$0.773574\pi$
$182$		0			0
$183$	−1.16894e8			−1.40998
$184$		0			0
$185$	−3.96998e7	−	6.87621e7i	−0.460986	−	0.798451i
$186$		0			0
$187$		−	1.07052e8i		−	1.19715i
$188$		0			0
$189$	1.02515e8	+	5.91871e7i	1.10451	+	0.637691i
$190$		0			0
$191$	−4.20222e7	+	7.27846e7i	−0.436377	+	0.755828i	−0.997407	−	0.0719679i	$-0.977072\pi$
$191$	−4.20222e7	+	7.27846e7i	−0.436377	+	0.755828i	0.561029	+	0.827796i	$0.310405\pi$
$192$		0			0
$193$	−3.85466e7	+	2.22549e7i	−0.385954	+	0.222830i	−0.680405	−	0.732836i	$-0.738196\pi$
$193$	−3.85466e7	+	2.22549e7i	−0.385954	+	0.222830i	0.294452	+	0.955666i	$0.404863\pi$
$194$		0			0
$195$	−6.14418e7	+	1.29491e8i	−0.593394	+	1.25060i
$196$		0			0
$197$	−1.09554e8	+	6.32512e7i	−1.02093	+	0.589437i	−0.914374	−	0.404870i	$-0.867317\pi$
$197$	−1.09554e8	+	6.32512e7i	−1.02093	+	0.589437i	−0.106560	+	0.994306i	$0.533984\pi$
$198$		0			0
$199$	4.81092e7	−	8.33276e7i	0.432755	−	0.749554i	−0.564354	−	0.825533i	$-0.690875\pi$
$199$	4.81092e7	−	8.33276e7i	0.432755	−	0.749554i	0.997109	+	0.0759785i	$0.0242080\pi$
$200$		0			0
$201$	1.80107e8	+	1.03985e8i	1.56439	+	0.903200i
$202$		0			0
$203$			2.20380e7i			0.184899i
$204$		0			0
$205$	6.49939e7	+	1.12573e8i	0.526908	+	0.912631i
$206$		0			0
$207$	2.54543e8			1.99464
$208$		0			0
$209$	−6.33375e7			−0.479898
$210$		0			0
$211$	−2.51766e7	−	4.36072e7i	−0.184505	−	0.319573i	0.758904	−	0.651202i	$-0.225735\pi$
$211$	−2.51766e7	−	4.36072e7i	−0.184505	−	0.319573i	−0.943410	+	0.331629i	$0.892402\pi$
$212$		0			0
$213$			1.05145e8i			0.745524i
$214$		0			0
$215$	2.02098e7	+	1.16681e7i	0.138684	+	0.0800693i
$216$		0			0
$217$	3.04944e7	−	5.28178e7i	0.202587	−	0.350890i
$218$		0			0
$219$	−1.86783e8	+	1.07839e8i	−1.20167	+	0.693782i
$220$		0			0
$221$	−8.99109e6	−	1.11352e8i	−0.0560324	−	0.693945i
$222$		0			0
$223$	2.12315e8	−	1.22580e8i	1.28208	−	0.740208i	0.304851	−	0.952400i	$-0.401393\pi$
$223$	2.12315e8	−	1.22580e8i	1.28208	−	0.740208i	0.977228	+	0.212192i	$0.0680602\pi$
$224$		0			0
$225$	−6.18439e7	+	1.07117e8i	−0.361958	+	0.626930i
$226$		0			0
$227$	1.65319e6	+	954470.i	0.00938064	+	0.00541591i	0.504683	−	0.863305i	$-0.331609\pi$
$227$	1.65319e6	+	954470.i	0.00938064	+	0.00541591i	−0.495302	+	0.868721i	$0.664943\pi$
$228$		0			0
$229$		−	2.07407e8i		−	1.14130i	−0.821194	−	0.570650i	$-0.806691\pi$
$229$		−	2.07407e8i		−	1.14130i	0.821194	−	0.570650i	$-0.193309\pi$
$230$		0			0
$231$	2.05083e8	+	3.55215e8i	1.09468	+	1.89605i
$232$		0			0
$233$	3.42955e8			1.77620			0.888099	−	0.459652i	$-0.152026\pi$
$233$	3.42955e8			1.77620			0.888099	+	0.459652i	$0.152026\pi$
$234$		0			0
$235$	8.88859e7			0.446782
$236$		0			0
$237$	8.66562e7	+	1.50093e8i	0.422844	+	0.732388i
$238$		0			0
$239$			3.94963e8i			1.87139i	0.352815	+	0.935693i	$0.385225\pi$
$239$			3.94963e8i			1.87139i	−0.352815	+	0.935693i	$0.614775\pi$
$240$		0			0
$241$	6.73226e7	+	3.88687e7i	0.309814	+	0.178871i	0.646843	−	0.762623i	$-0.276089\pi$
$241$	6.73226e7	+	3.88687e7i	0.309814	+	0.178871i	−0.337029	+	0.941494i	$0.609422\pi$
$242$		0			0
$243$	−656866.	+	1.13773e6i	−0.00293667	+	0.00508646i
$244$		0			0
$245$	7.32552e7	−	4.22939e7i	0.318242	−	0.183737i
$246$		0			0
$247$	−6.58816e7	+	5.31959e6i	−0.278179	+	0.0224615i
$248$		0			0
$249$	−1.14485e8	+	6.60978e7i	−0.469948	+	0.271325i
$250$		0			0
$251$	1.75438e8	−	3.03868e8i	0.700272	−	1.21291i	−0.268099	−	0.963391i	$-0.586396\pi$
$251$	1.75438e8	−	3.03868e8i	0.700272	−	1.21291i	0.968371	−	0.249515i	$-0.0802711\pi$
$252$		0			0
$253$	3.82237e8	+	2.20684e8i	1.48392	+	0.856742i
$254$		0			0
$255$			2.55176e8i			0.963715i
$256$		0			0
$257$	−1.79600e8	−	3.11076e8i	−0.659995	−	1.14314i	−0.980616	−	0.195937i	$-0.937225\pi$
$257$	−1.79600e8	−	3.11076e8i	−0.659995	−	1.14314i	0.320622	−	0.947207i	$-0.396108\pi$
$258$		0			0
$259$	−2.37116e8			−0.848030
$260$		0			0
$261$	−1.44661e8			−0.503629
$262$		0			0
$263$	−9.97562e7	−	1.72783e8i	−0.338139	−	0.585673i	0.645944	−	0.763385i	$-0.276464\pi$
$263$	−9.97562e7	−	1.72783e8i	−0.338139	−	0.585673i	−0.984083	+	0.177712i	$0.943131\pi$
$264$		0			0
$265$			3.34711e8i			1.10486i
$266$		0			0
$267$	2.81518e7	+	1.62534e7i	0.0905142	+	0.0522584i
$268$		0			0
$269$	3.22428e7	−	5.58463e7i	0.100995	−	0.174929i	−0.811100	−	0.584908i	$-0.801131\pi$
$269$	3.22428e7	−	5.58463e7i	0.100995	−	0.174929i	0.912095	+	0.409979i	$0.134464\pi$
$270$		0			0
$271$	4.15254e7	−	2.39747e7i	0.126742	−	0.0731747i	−0.435288	−	0.900291i	$-0.643354\pi$
$271$	4.15254e7	−	2.39747e7i	0.126742	−	0.0731747i	0.562031	+	0.827116i	$0.310020\pi$
$272$		0			0
$273$	2.43155e8	+	3.52258e8i	0.723292	+	1.04783i
$274$		0			0
$275$	−1.85737e8	+	1.07235e8i	−0.538560	+	0.310938i
$276$		0			0
$277$	2.96509e8	−	5.13568e8i	0.838221	−	1.45184i	−0.0531601	−	0.998586i	$-0.516929\pi$
$277$	2.96509e8	−	5.13568e8i	0.838221	−	1.45184i	0.891381	−	0.453255i	$-0.149737\pi$
$278$		0			0
$279$	3.46706e8	+	2.00171e8i	0.955756	+	0.551806i
$280$		0			0
$281$			1.74758e8i			0.469856i	0.972013	+	0.234928i	$0.0754854\pi$
$281$			1.74758e8i			0.469856i	−0.972013	+	0.234928i	$0.924515\pi$
$282$		0			0
$283$	−3.30559e8	−	5.72544e8i	−0.866954	−	1.50161i	−0.865094	−	0.501609i	$-0.832741\pi$
$283$	−3.30559e8	−	5.72544e8i	−0.866954	−	1.50161i	−0.00185936	−	0.999998i	$-0.500592\pi$
$284$		0			0
$285$	1.50975e8			0.386321
$286$		0			0
$287$	3.88190e8			0.969299
$288$		0			0
$289$	1.05724e8	+	1.83119e8i	0.257650	+	0.446263i
$290$		0			0
$291$			8.94681e8i			2.12835i
$292$		0			0
$293$	−4.64833e8	−	2.68372e8i	−1.07959	−	0.623304i	−0.148807	−	0.988866i	$-0.547543\pi$
$293$	−4.64833e8	−	2.68372e8i	−1.07959	−	0.623304i	−0.930787	+	0.365562i	$0.880877\pi$
$294$		0			0
$295$	3.69176e7	−	6.39432e7i	0.0837253	−	0.145016i
$296$		0			0
$297$	−1.16683e9	+	6.73672e8i	−2.58441	+	1.49211i
$298$		0			0
$299$	4.16125e8	+	1.97445e8i	0.900274	+	0.427167i
$300$		0			0
$301$	6.03536e7	−	3.48452e7i	0.127562	−	0.0736478i
$302$		0			0
$303$	3.11257e8	−	5.39113e8i	0.642791	−	1.11335i
$304$		0			0
$305$	2.79023e8	+	1.61094e8i	0.563106	+	0.325110i
$306$		0			0
$307$		−	3.58135e8i		−	0.706419i	−0.935544	−	0.353209i	$-0.885090\pi$
$307$		−	3.58135e8i		−	0.706419i	0.935544	−	0.353209i	$-0.114910\pi$
$308$		0			0
$309$	1.96815e7	+	3.40894e7i	0.0379493	+	0.0657301i
$310$		0			0
$311$	−4.95093e8			−0.933309			−0.466655	−	0.884440i	$-0.654541\pi$
$311$	−4.95093e8			−0.933309			−0.466655	+	0.884440i	$0.654541\pi$
$312$		0			0
$313$	−3.14797e8			−0.580263			−0.290131	−	0.956987i	$-0.593699\pi$
$313$	−3.14797e8			−0.580263			−0.290131	+	0.956987i	$0.593699\pi$
$314$		0			0
$315$	−3.25991e8	−	5.64633e8i	−0.587650	−	1.01784i
$316$		0			0
$317$		−	6.24177e8i		−	1.10053i	−0.834991	−	0.550263i	$-0.814527\pi$
$317$		−	6.24177e8i		−	1.10053i	0.834991	−	0.550263i	$-0.185473\pi$
$318$		0			0
$319$	−2.17232e8	−	1.25419e8i	−0.374677	−	0.216320i
$320$		0			0
$321$	−2.83476e6	+	4.90994e6i	−0.00478353	+	0.00828531i
$322$		0			0
$323$	−1.01909e8	+	5.88371e7i	−0.168269	+	0.0971499i
$324$		0			0
$325$	−1.84191e8	+	1.27142e8i	−0.297630	+	0.205446i
$326$		0			0
$327$	2.85139e8	−	1.64625e8i	0.450961	−	0.260362i
$328$		0			0
$329$	1.32722e8	−	2.29882e8i	0.205475	−	0.355893i
$330$		0			0
$331$	4.36304e8	+	2.51900e8i	0.661289	+	0.381795i	0.792768	−	0.609524i	$-0.208639\pi$
$331$	4.36304e8	+	2.51900e8i	0.661289	+	0.381795i	−0.131479	+	0.991319i	$0.541973\pi$
$332$		0			0
$333$		−	1.55647e9i		−	2.30986i
$334$		0			0
$335$	−2.86607e8	−	4.96417e8i	−0.416514	−	0.721423i
$336$		0			0
$337$	−9.64189e8			−1.37233			−0.686164	−	0.727447i	$-0.740707\pi$
$337$	−9.64189e8			−1.37233			−0.686164	+	0.727447i	$0.740707\pi$
$338$		0			0
$339$	−1.70111e9			−2.37156
$340$		0			0
$341$	3.47090e8	+	6.01177e8i	0.474025	+	0.821036i
$342$		0			0
$343$		−	8.01836e8i		−	1.07289i
$344$		0			0
$345$	−9.11122e8	−	5.26036e8i	−1.19456	−	0.689682i
$346$		0			0
$347$	2.30234e8	−	3.98777e8i	0.295812	−	0.512362i	−0.679361	−	0.733804i	$-0.737743\pi$
$347$	2.30234e8	−	3.98777e8i	0.295812	−	0.512362i	0.975173	+	0.221442i	$0.0710764\pi$
$348$		0			0
$349$	−2.11799e8	+	1.22282e8i	−0.266707	+	0.153984i	−0.627390	−	0.778705i	$-0.715877\pi$
$349$	−2.11799e8	+	1.22282e8i	−0.266707	+	0.153984i	0.360683	+	0.932688i	$0.382544\pi$
$350$		0			0
$351$	−1.15712e9	+	7.98732e8i	−1.42825	+	0.985885i
$352$		0			0
$353$	4.69439e8	−	2.71031e8i	0.568025	−	0.327949i	−0.188335	−	0.982105i	$-0.560309\pi$
$353$	4.69439e8	−	2.71031e8i	0.568025	−	0.327949i	0.756360	+	0.654155i	$0.226976\pi$
$354$		0			0
$355$	1.44903e8	−	2.50979e8i	0.171901	−	0.297741i
$356$		0			0
$357$	6.59950e8	+	3.81023e8i	0.767667	+	0.443212i
$358$		0			0
$359$			5.16223e8i			0.588853i	0.955674	+	0.294427i	$0.0951286\pi$
$359$			5.16223e8i			0.588853i	−0.955674	+	0.294427i	$0.904871\pi$
$360$		0			0
$361$	−4.12125e8	−	7.13821e8i	−0.461056	−	0.798572i
$362$		0			0
$363$	−3.08965e9			−3.39028
$364$		0			0
$365$	5.94462e8			0.639880
$366$		0			0
$367$	5.16064e8	+	8.93849e8i	0.544969	+	0.943915i	0.998609	+	0.0527295i	$0.0167921\pi$
$367$	5.16064e8	+	8.93849e8i	0.544969	+	0.943915i	−0.453639	+	0.891185i	$0.649875\pi$
$368$		0			0
$369$			2.54815e9i			2.64018i
$370$		0			0
$371$	8.65649e8	+	4.99783e8i	0.880102	+	0.508127i
$372$		0			0
$373$	9.00042e8	−	1.55892e9i	0.898012	−	1.55540i	0.0679796	−	0.997687i	$-0.478345\pi$
$373$	9.00042e8	−	1.55892e9i	0.898012	−	1.55540i	0.830032	−	0.557715i	$-0.188322\pi$
$374$		0			0
$375$	1.66693e9	−	9.62404e8i	1.63233	−	0.942428i
$376$		0			0
$377$	−2.36491e8	−	1.12212e8i	−0.227311	−	0.107856i
$378$		0			0
$379$	−6.87995e8	+	3.97214e8i	−0.649155	+	0.374790i	−0.788132	−	0.615506i	$-0.788952\pi$
$379$	−6.87995e8	+	3.97214e8i	−0.649155	+	0.374790i	0.138978	+	0.990296i	$0.455618\pi$
$380$		0			0
$381$	−3.78201e8	+	6.55064e8i	−0.350337	+	0.606801i
$382$		0			0
$383$	1.63844e9	+	9.45952e8i	1.49016	+	0.860346i	0.999937	−	0.0112491i	$-0.00358078\pi$
$383$	1.63844e9	+	9.45952e8i	1.49016	+	0.860346i	0.490226	+	0.871595i	$0.336914\pi$
$384$		0			0
$385$		−	1.13052e9i		−	1.00963i
$386$		0			0
$387$	2.28730e8	+	3.96172e8i	0.200602	+	0.347453i
$388$		0			0
$389$	−9.79194e8			−0.843422			−0.421711	−	0.906730i	$-0.638570\pi$
$389$	−9.79194e8			−0.843422			−0.421711	+	0.906730i	$0.638570\pi$
$390$		0			0
$391$	8.20016e8			0.693751
$392$		0			0
$393$	−1.65103e9	−	2.85966e9i	−1.37208	−	2.37651i
$394$		0			0
$395$		−	4.77690e8i		−	0.389993i
$396$		0			0
$397$	1.02057e9	+	5.89224e8i	0.818605	+	0.472622i	0.849935	−	0.526887i	$-0.176641\pi$
$397$	1.02057e9	+	5.89224e8i	0.818605	+	0.472622i	−0.0313301	+	0.999509i	$0.509974\pi$
$398$		0			0
$399$	2.25433e8	−	3.90461e8i	0.177669	−	0.307732i
$400$		0			0
$401$	8.16601e8	−	4.71465e8i	0.632418	−	0.365127i	−0.149270	−	0.988797i	$-0.547692\pi$
$401$	8.16601e8	−	4.71465e8i	0.632418	−	0.365127i	0.781688	+	0.623670i	$0.214359\pi$
$402$		0			0
$403$	4.11523e8	+	5.96174e8i	0.313204	+	0.453738i
$404$		0			0
$405$	9.29724e8	−	5.36777e8i	0.695443	−	0.401514i
$406$		0			0
$407$	1.34944e9	−	2.33729e9i	0.992137	−	1.71843i
$408$		0			0
$409$	1.70476e9	+	9.84245e8i	1.23206	+	0.711331i	0.967459	−	0.253027i	$-0.0814262\pi$
$409$	1.70476e9	+	9.84245e8i	1.23206	+	0.711331i	0.264602	+	0.964358i	$0.414760\pi$
$410$		0			0
$411$		−	3.94774e9i		−	2.80480i
$412$		0			0
$413$	−1.10249e8	−	1.90957e8i	−0.0770105	−	0.133386i
$414$		0			0
$415$	3.64362e8			0.250245
$416$		0			0
$417$	4.31767e9			2.91590
$418$		0			0
$419$	−9.78364e8	−	1.69458e9i	−0.649758	−	1.12541i	−0.983181	−	0.182636i	$-0.941537\pi$
$419$	−9.78364e8	−	1.69458e9i	−0.649758	−	1.12541i	0.333423	−	0.942777i	$-0.391796\pi$
$420$		0			0
$421$		−	2.92602e8i		−	0.191113i	−0.995424	−	0.0955563i	$-0.969537\pi$
$421$		−	2.92602e8i		−	0.191113i	0.995424	−	0.0955563i	$-0.0304630\pi$
$422$		0			0
$423$	1.50899e9	+	8.71215e8i	0.969382	+	0.559673i
$424$		0			0
$425$	−1.99232e8	+	3.45079e8i	−0.125892	+	0.218051i
$426$		0			0
$427$	8.33262e8	−	4.81084e8i	0.517945	−	0.299036i
$428$		0			0
$429$	−4.85608e9	+	3.92103e8i	−2.96951	+	0.239772i
$430$		0			0
$431$	8.46643e7	−	4.88810e7i	0.0509366	−	0.0294083i	−0.474315	−	0.880355i	$-0.657304\pi$
$431$	8.46643e7	−	4.88810e7i	0.0509366	−	0.0294083i	0.525252	+	0.850947i	$0.323971\pi$
$432$		0			0
$433$	−3.26639e8	+	5.65755e8i	−0.193357	+	0.334904i	−0.946361	−	0.323112i	$-0.895271\pi$
$433$	−3.26639e8	+	5.65755e8i	−0.193357	+	0.334904i	0.753004	+	0.658016i	$0.228604\pi$
$434$		0			0
$435$	5.17807e8	+	2.98956e8i	0.301617	+	0.174139i
$436$		0			0
$437$		−	4.85163e8i		−	0.278101i
$438$		0			0
$439$	−7.87825e8	−	1.36455e9i	−0.444431	−	0.769777i	0.553582	−	0.832795i	$-0.313261\pi$
$439$	−7.87825e8	−	1.36455e9i	−0.444431	−	0.769777i	−0.998012	+	0.0630183i	$0.979927\pi$
$440$		0			0
$441$	1.65818e9			0.920652
$442$		0			0
$443$	−2.36999e9			−1.29519			−0.647594	−	0.761985i	$-0.724225\pi$
$443$	−2.36999e9			−1.29519			−0.647594	+	0.761985i	$0.724225\pi$
$444$		0			0
$445$	−4.47983e7	−	7.75930e7i	−0.0240991	−	0.0417409i
$446$		0			0
$447$			2.93619e9i			1.55492i
$448$		0			0
$449$	1.77974e8	+	1.02754e8i	0.0927887	+	0.0535716i	0.545676	−	0.837996i	$-0.316273\pi$
$449$	1.77974e8	+	1.02754e8i	0.0927887	+	0.0535716i	−0.452888	+	0.891568i	$0.649606\pi$
$450$		0			0
$451$	−2.20921e9	+	3.82646e9i	−1.13401	+	1.96417i
$452$		0			0
$453$	−1.79403e9	+	1.03578e9i	−0.906745	+	0.523509i
$454$		0			0
$455$	−9.49498e7	−	1.17593e9i	−0.0472557	−	0.585248i
$456$		0			0
$457$	−2.33582e9	+	1.34858e9i	−1.14481	+	0.660954i	−0.947616	−	0.319411i	$-0.896515\pi$
$457$	−2.33582e9	+	1.34858e9i	−1.14481	+	0.660954i	−0.197190	+	0.980365i	$0.563182\pi$
$458$		0			0
$459$	−1.25161e9	+	2.16785e9i	−0.604122	+	1.04637i
$460$		0			0
$461$	1.66805e9	+	9.63051e8i	0.792970	+	0.457821i	0.841007	−	0.541024i	$-0.181963\pi$
$461$	1.66805e9	+	9.63051e8i	0.792970	+	0.457821i	−0.0480374	+	0.998846i	$0.515297\pi$
$462$		0			0
$463$			2.34776e9i			1.09931i	0.835391	+	0.549655i	$0.185241\pi$
$463$			2.34776e9i			1.09931i	−0.835391	+	0.549655i	$0.814759\pi$
$464$		0			0
$465$	−8.27344e8	−	1.43300e9i	−0.381593	−	0.660939i
$466$		0			0
$467$	1.71342e9			0.778493			0.389246	−	0.921134i	$-0.372736\pi$
$467$	1.71342e9			0.778493			0.389246	+	0.921134i	$0.372736\pi$
$468$		0			0
$469$	−1.71182e9			−0.766219
$470$		0			0
$471$	−1.71483e9	−	2.97017e9i	−0.756219	−	1.30981i
$472$		0			0
$473$			7.93221e8i			0.344652i
$474$		0			0
$475$	2.04167e8	+	1.17876e8i	0.0874092	+	0.0504657i
$476$		0			0
$477$	−3.28067e9	+	5.68228e9i	−1.38404	+	2.39722i
$478$		0			0
$479$	−9.86746e7	+	5.69698e7i	−0.0410234	+	0.0236848i	−0.520371	−	0.853940i	$-0.674206\pi$
$479$	−9.86746e7	+	5.69698e7i	−0.0410234	+	0.0236848i	0.479348	+	0.877625i	$0.340873\pi$
$480$		0			0
$481$	1.20733e9	−	2.54451e9i	0.494675	−	1.04255i
$482$		0			0
$483$	−2.72093e9	+	1.57093e9i	−1.09876	+	0.634370i
$484$		0			0
$485$	1.23298e9	−	2.13558e9i	0.490748	−	0.850001i
$486$		0			0
$487$	−6.89579e8	−	3.98129e8i	−0.270541	−	0.156197i	0.358593	−	0.933494i	$-0.383257\pi$
$487$	−6.89579e8	−	3.98129e8i	−0.270541	−	0.156197i	−0.629133	+	0.777297i	$0.716590\pi$
$488$		0			0
$489$		−	2.79446e9i		−	1.08073i
$490$		0			0
$491$	−1.82811e9	−	3.16637e9i	−0.696973	−	1.20719i	−0.969511	−	0.245048i	$-0.921196\pi$
$491$	−1.82811e9	−	3.16637e9i	−0.696973	−	1.20719i	0.272538	−	0.962145i	$-0.412137\pi$
$492$		0			0
$493$	−4.66030e8			−0.175166
$494$		0			0
$495$	7.42092e9			2.75004
$496$		0			0
$497$	−4.32731e8	−	7.49513e8i	−0.158114	−	0.273862i
$498$		0			0
$499$			7.46131e8i			0.268821i	0.990926	+	0.134411i	$0.0429141\pi$
$499$			7.46131e8i			0.268821i	−0.990926	+	0.134411i	$0.957086\pi$
$500$		0			0
$501$	5.62015e8	+	3.24479e8i	0.199671	+	0.115280i
$502$		0			0
$503$	−2.34080e8	+	4.05438e8i	−0.0820117	+	0.142048i	−0.904114	−	0.427291i	$-0.859468\pi$
$503$	−2.34080e8	+	4.05438e8i	−0.0820117	+	0.142048i	0.822102	+	0.569340i	$0.192801\pi$
$504$		0			0
$505$	−1.48592e9	+	8.57897e8i	−0.513424	+	0.296425i
$506$		0			0
$507$	−5.01820e9	+	8.15705e8i	−1.71009	+	0.277975i
$508$		0			0
$509$	−5.13887e8	+	2.96693e8i	−0.172725	+	0.0997229i	−0.583870	−	0.811847i	$-0.698462\pi$
$509$	−5.13887e8	+	2.96693e8i	−0.172725	+	0.0997229i	0.411145	+	0.911570i	$0.365129\pi$
$510$		0			0
$511$	8.87637e8	−	1.53743e9i	0.294281	−	0.509710i
$512$		0			0
$513$	1.28261e9	+	7.40516e8i	0.419454	+	0.242172i
$514$		0			0
$515$		−	1.08494e8i		−	0.0350009i
$516$		0			0
$517$	1.51066e9	+	2.61654e9i	0.480783	+	0.832741i
$518$		0			0
$519$	−3.02329e9			−0.949280
$520$		0			0
$521$	2.35950e9			0.730950			0.365475	−	0.930821i	$-0.380907\pi$
$521$	2.35950e9			0.730950			0.365475	+	0.930821i	$0.380907\pi$
$522$		0			0
$523$	2.74632e8	+	4.75677e8i	0.0839451	+	0.145397i	0.904941	−	0.425537i	$-0.139915\pi$
$523$	2.74632e8	+	4.75677e8i	0.0839451	+	0.145397i	−0.820996	+	0.570934i	$0.806581\pi$
$524$		0			0
$525$		−	1.52670e9i		−	0.460465i
$526$		0			0
$527$	1.11692e9	+	6.44855e8i	0.332419	+	0.191922i
$528$		0			0
$529$	1.19760e7	−	2.07431e7i	0.00351737	−	0.00609227i
$530$		0			0
$531$	1.25348e9	−	7.23696e8i	0.363317	−	0.209761i
$532$		0			0
$533$	−1.97657e9	+	4.16570e9i	−0.565414	+	1.19164i
$534$		0			0
$535$	1.35330e7	−	7.81326e6i	0.00382080	−	0.00220594i
$536$		0			0
$537$	2.77587e9	−	4.80795e9i	0.773552	−	1.33983i
$538$		0			0
$539$	2.49002e9	+	1.43761e9i	0.684923	+	0.395440i
$540$		0			0
$541$		−	6.19192e7i		−	0.0168126i	−0.999965	−	0.00840630i	$-0.997324\pi$
$541$		−	6.19192e7i		−	0.0168126i	0.999965	−	0.00840630i	$-0.00267584\pi$
$542$		0			0
$543$	4.89620e9	+	8.48047e9i	1.31238	+	2.27311i
$544$		0			0
$545$	−9.07489e8			−0.240134
$546$		0			0
$547$	4.11747e9			1.07566			0.537830	−	0.843053i	$-0.319244\pi$
$547$	4.11747e9			1.07566			0.537830	+	0.843053i	$0.319244\pi$
$548$		0			0
$549$	3.15792e9	+	5.46968e9i	0.814514	+	1.41078i
$550$		0			0
$551$			2.75727e8i			0.0702181i
$552$		0			0
$553$	−1.23543e9	−	7.13276e8i	−0.310656	−	0.179358i
$554$		0			0
$555$	−3.21659e9	+	5.57130e9i	−0.798676	+	1.38335i
$556$		0			0
$557$	3.49708e9	−	2.01904e9i	0.857456	−	0.495052i	−0.00570362	−	0.999984i	$-0.501816\pi$
$557$	3.49708e9	−	2.01904e9i	0.857456	−	0.495052i	0.863160	+	0.504931i	$0.168482\pi$
$558$		0			0
$559$	6.66211e7	+	8.25083e8i	0.0161313	+	0.199782i
$560$		0			0
$561$	−7.51161e9	+	4.33683e9i	−1.79624	+	1.03706i
$562$		0			0
$563$	3.26975e8	−	5.66337e8i	0.0772209	−	0.133751i	−0.824829	−	0.565382i	$-0.808729\pi$
$563$	3.26975e8	−	5.66337e8i	0.0772209	−	0.133751i	0.902050	+	0.431632i	$0.142062\pi$
$564$		0			0
$565$	4.06050e9	+	2.34433e9i	0.947131	+	0.546826i
$566$		0			0
$567$		−	3.20601e9i		−	0.738626i
$568$		0			0
$569$	1.89049e9	+	3.27443e9i	0.430211	+	0.745148i	0.996891	−	0.0787902i	$-0.0251057\pi$
$569$	1.89049e9	+	3.27443e9i	0.430211	+	0.745148i	−0.566680	+	0.823938i	$0.691772\pi$
$570$		0			0
$571$	1.41244e9			0.317499			0.158750	−	0.987319i	$-0.449254\pi$
$571$	1.41244e9			0.317499			0.158750	+	0.987319i	$0.449254\pi$
$572$		0			0
$573$	6.80952e9			1.51208
$574$		0			0
$575$	−8.21420e8	−	1.42274e9i	−0.180189	−	0.312096i
$576$		0			0
$577$		−	2.80901e9i		−	0.608749i	−0.952553	−	0.304374i	$-0.901553\pi$
$577$		−	2.80901e9i		−	0.608749i	0.952553	−	0.304374i	$-0.0984474\pi$
$578$		0			0
$579$	3.12315e9	+	1.80315e9i	0.668680	+	0.386063i
$580$		0			0
$581$	5.44058e8	−	9.42336e8i	0.115088	−	0.199338i
$582$		0			0
$583$	−9.85289e9	+	5.68857e9i	−2.05932	+	1.18895i
$584$		0			0
$585$	7.71900e9	−	6.23268e8i	1.59410	−	0.128715i
$586$		0			0
$587$	−4.64252e9	+	2.68036e9i	−0.947371	+	0.546965i	−0.892263	−	0.451516i	$-0.850884\pi$
$587$	−4.64252e9	+	2.68036e9i	−0.947371	+	0.546965i	−0.0551076	+	0.998480i	$0.517550\pi$
$588$		0			0
$589$	3.81529e8	−	6.60828e8i	0.0769351	−	0.133255i
$590$		0			0
$591$	8.87641e9	+	5.12480e9i	1.76881	+	1.02122i
$592$		0			0
$593$			2.19757e9i			0.432765i	0.976309	+	0.216382i	$0.0694258\pi$
$593$			2.19757e9i			0.432765i	−0.976309	+	0.216382i	$0.930574\pi$
$594$		0			0
$595$	−1.05019e9	−	1.81898e9i	−0.204389	−	0.354012i
$596$		0			0
$597$	−7.79589e9			−1.49953
$598$		0			0
$599$	9.75825e9			1.85515			0.927573	−	0.373642i	$-0.121891\pi$
$599$	9.75825e9			1.85515			0.927573	+	0.373642i	$0.121891\pi$
$600$		0			0
$601$	−1.84366e9	−	3.19331e9i	−0.346434	−	0.600041i	0.639179	−	0.769058i	$-0.279274\pi$
$601$	−1.84366e9	−	3.19331e9i	−0.346434	−	0.600041i	−0.985613	+	0.169017i	$0.945941\pi$
$602$		0			0
$603$		−	1.12367e10i		−	2.08703i
$604$		0			0
$605$	7.37489e9	+	4.25790e9i	1.35398	+	0.781720i
$606$		0			0
$607$	2.95249e9	−	5.11386e9i	0.535831	−	0.928086i	−0.463292	−	0.886206i	$-0.653332\pi$
$607$	2.95249e9	−	5.11386e9i	0.535831	−	0.928086i	0.999123	−	0.0418806i	$-0.0133349\pi$
$608$		0			0
$609$	1.54636e9	−	8.92789e8i	0.277427	−	0.160173i
$610$		0			0
$611$	1.79110e9	+	2.59476e9i	0.317669	+	0.460207i
$612$		0			0
$613$	−7.28326e9	+	4.20499e9i	−1.27707	+	0.737316i	−0.976308	−	0.216384i	$-0.930574\pi$
$613$	−7.28326e9	+	4.20499e9i	−1.27707	+	0.737316i	−0.300760	+	0.953700i	$0.597240\pi$
$614$		0			0
$615$	5.26600e9	−	9.12097e9i	0.912888	−	1.58117i
$616$		0			0
$617$	7.37210e9	+	4.25629e9i	1.26355	+	0.729512i	0.973760	−	0.227577i	$-0.0730803\pi$
$617$	7.37210e9	+	4.25629e9i	1.26355	+	0.729512i	0.289793	+	0.957089i	$0.406414\pi$
$618$		0			0
$619$		−	8.70392e9i		−	1.47502i	−0.675337	−	0.737509i	$-0.736002\pi$
$619$		−	8.70392e9i		−	1.47502i	0.675337	−	0.737509i	$-0.263998\pi$
$620$		0			0
$621$	−5.16031e9	−	8.93792e9i	−0.864679	−	1.49767i
$622$		0			0
$623$	−2.67568e8			−0.0443328
$624$		0			0
$625$	−3.09788e9			−0.507556
$626$		0			0
$627$	2.56589e9	+	4.44426e9i	0.415721	+	0.720050i
$628$		0			0
$629$		−	5.01421e9i		−	0.803388i
$630$		0			0
$631$	−8.56201e9	−	4.94328e9i	−1.35667	−	0.783272i	−0.367494	−	0.930026i	$-0.619784\pi$
$631$	−8.56201e9	−	4.94328e9i	−1.35667	−	0.783272i	−0.989173	+	0.146754i	$0.953117\pi$
$632$		0			0
$633$	−2.03988e9	+	3.53318e9i	−0.319663	+	0.553672i
$634$		0			0
$635$	1.80551e9	−	1.04241e9i	0.279829	−	0.161559i
$636$		0			0
$637$	2.71078e9	+	1.28623e9i	0.415533	+	0.197165i
$638$		0			0
$639$	4.91994e9	−	2.84053e9i	0.745945	−	0.430672i
$640$		0			0
$641$	−3.38085e9	+	5.85580e9i	−0.507017	+	0.878179i	0.492950	+	0.870058i	$0.335918\pi$
$641$	−3.38085e9	+	5.85580e9i	−0.507017	+	0.878179i	−0.999967	+	0.00812132i	$0.997415\pi$
$642$		0			0
$643$	−2.43813e9	−	1.40766e9i	−0.361676	−	0.208814i	0.308140	−	0.951341i	$-0.400294\pi$
$643$	−2.43813e9	−	1.40766e9i	−0.361676	−	0.208814i	−0.669816	+	0.742528i	$0.733627\pi$
$644$		0			0
$645$		−	1.89077e9i		−	0.277447i
$646$		0			0
$647$	5.96746e9	+	1.03360e10i	0.866213	+	1.50033i	0.865837	+	0.500326i	$0.166786\pi$
$647$	5.96746e9	+	1.03360e10i	0.866213	+	1.50033i	0.000375912		1.00000i	$0.499880\pi$
$648$		0			0
$649$	2.50973e9			0.360388
$650$		0			0
$651$	−4.94149e9			−0.701979
$652$		0			0
$653$	−6.41202e8	−	1.11059e9i	−0.0901153	−	0.156084i	0.817444	−	0.576008i	$-0.195390\pi$
$653$	−6.41202e8	−	1.11059e9i	−0.0901153	−	0.156084i	−0.907559	+	0.419924i	$0.862057\pi$
$654$		0			0
$655$			9.10123e9i			1.26548i
$656$		0			0
$657$	1.00920e10	+	5.82662e9i	1.38835	+	0.801563i
$658$		0			0
$659$	−2.42623e9	+	4.20235e9i	−0.330242	+	0.571996i	−0.982559	−	0.185950i	$-0.940464\pi$
$659$	−2.42623e9	+	4.20235e9i	−0.330242	+	0.571996i	0.652317	+	0.757946i	$0.273797\pi$
$660$		0			0
$661$	−4.12321e9	+	2.38054e9i	−0.555304	+	0.320605i	−0.751258	−	0.660008i	$-0.770553\pi$
$661$	−4.12321e9	+	2.38054e9i	−0.555304	+	0.320605i	0.195955	+	0.980613i	$0.437219\pi$
$662$		0			0
$663$	−7.44909e9	+	5.14192e9i	−0.992673	+	0.685216i
$664$		0			0
$665$	−1.07620e9	+	6.21345e8i	−0.141912	+	0.0819327i
$666$		0			0
$667$	9.60706e8	−	1.66399e9i	0.125357	−	0.217125i
$668$		0			0
$669$	−1.72024e10	−	9.93181e9i	−2.22125	−	1.28244i
$670$		0			0
$671$			1.09515e10i			1.39941i
$672$		0			0
$673$	−5.17406e9	−	8.96173e9i	−0.654303	−	1.13329i	−0.982068	−	0.188526i	$-0.939629\pi$
$673$	−5.17406e9	−	8.96173e9i	−0.654303	−	1.13329i	0.327766	−	0.944759i	$-0.393704\pi$
$674$		0			0
$675$	5.01501e9			0.627637
$676$		0			0
$677$	−1.09230e9			−0.135295			−0.0676477	−	0.997709i	$-0.521549\pi$
$677$	−1.09230e9			−0.135295			−0.0676477	+	0.997709i	$0.521549\pi$
$678$		0			0
$679$	−3.68211e9	−	6.37760e9i	−0.451390	−	0.781831i
$680$		0			0
$681$		−	1.54668e8i		−	0.0187666i
$682$		0			0
$683$	−6.69361e9	−	3.86456e9i	−0.803875	−	0.464117i	0.0409497	−	0.999161i	$-0.486962\pi$
$683$	−6.69361e9	−	3.86456e9i	−0.803875	−	0.464117i	−0.844824	+	0.535044i	$0.820295\pi$
$684$		0			0
$685$	−5.44046e9	+	9.42314e9i	−0.646723	+	1.12016i
$686$		0			0
$687$	−1.45533e10	+	8.40236e9i	−1.71243	+	0.988673i
$688$		0			0
$689$	−9.77088e9	+	6.74459e9i	−1.13806	+	0.785576i
$690$		0			0
$691$	−7.44583e9	+	4.29885e9i	−0.858500	+	0.495655i	−0.863510	−	0.504332i	$-0.831739\pi$
$691$	−7.44583e9	+	4.29885e9i	−0.858500	+	0.495655i	0.00500987	+	0.999987i	$0.498405\pi$
$692$		0			0
$693$	1.10808e10	−	1.91924e10i	1.26474	−	2.19060i
$694$		0			0
$695$	−1.03061e10	−	5.95025e9i	−1.16453	−	0.672340i
$696$		0			0
$697$			8.20893e9i			0.918273i
$698$		0			0
$699$	−1.38936e10	−	2.40644e10i	−1.53867	−	2.66505i
$700$		0			0
$701$	4.34708e9			0.476633			0.238317	−	0.971188i	$-0.423404\pi$
$701$	4.34708e9			0.476633			0.238317	+	0.971188i	$0.423404\pi$
$702$		0			0
$703$	−2.96666e9			−0.322051
$704$		0			0
$705$	−3.60089e9	−	6.23693e9i	−0.387033	−	0.670362i
$706$		0			0
$707$			5.12397e9i			0.545304i
$708$		0			0
$709$	−1.06035e10	−	6.12194e9i	−1.11735	−	0.645100i	−0.176624	−	0.984278i	$-0.556518\pi$
$709$	−1.06035e10	−	6.12194e9i	−1.11735	−	0.645100i	−0.940722	+	0.339178i	$0.889851\pi$
$710$		0			0
$711$	4.68208e9	−	8.10960e9i	0.488534	−	0.846166i
$712$		0			0
$713$	−4.60500e9	+	2.65870e9i	−0.475791	+	0.274698i
$714$		0			0
$715$	1.21317e10	+	5.75631e9i	1.24122	+	0.588942i
$716$		0			0
$717$	2.77137e10	−	1.60005e10i	2.80787	−	1.62113i
$718$		0			0
$719$	1.14736e9	−	1.98729e9i	0.115119	−	0.199393i	−0.802708	−	0.596372i	$-0.796608\pi$
$719$	1.14736e9	−	1.98729e9i	0.115119	−	0.199393i	0.917828	+	0.396979i	$0.129942\pi$
$720$		0			0
$721$	−2.80593e8	−	1.62001e8i	−0.0278807	−	0.0160969i
$722$		0			0
$723$		−	6.29851e9i		−	0.619803i
$724$		0			0
$725$	4.66828e8	+	8.08569e8i	0.0454960	+	0.0788015i
$726$		0			0
$727$	5.74619e9			0.554638			0.277319	−	0.960778i	$-0.410554\pi$
$727$	5.74619e9			0.554638			0.277319	+	0.960778i	$0.410554\pi$
$728$		0			0
$729$	−1.04071e10			−0.994908
$730$		0			0
$731$	7.36860e8	+	1.27628e9i	0.0697708	+	0.120847i
$732$		0			0
$733$		−	1.26955e9i		−	0.119065i	−0.998226	−	0.0595326i	$-0.981039\pi$
$733$		−	1.26955e9i		−	0.119065i	0.998226	−	0.0595326i	$-0.0189610\pi$
$734$		0			0
$735$	−5.93535e9	−	3.42678e9i	−0.551367	−	0.318332i
$736$		0			0
$737$	9.74204e9	−	1.68737e10i	0.896424	−	1.55265i
$738$		0			0
$739$	−2.22921e9	+	1.28703e9i	−0.203187	+	0.117310i	−0.598141	−	0.801391i	$-0.704094\pi$
$739$	−2.22921e9	+	1.28703e9i	−0.203187	+	0.117310i	0.394954	+	0.918701i	$0.370760\pi$
$740$		0			0
$741$	3.04222e9	+	4.40727e9i	0.274680	+	0.397929i
$742$		0			0
$743$	4.19446e9	−	2.42167e9i	0.375159	−	0.216598i	−0.300551	−	0.953766i	$-0.597171\pi$
$743$	4.19446e9	−	2.42167e9i	0.375159	−	0.216598i	0.675710	+	0.737168i	$0.263837\pi$
$744$		0			0
$745$	4.04642e9	−	7.00860e9i	0.358529	−	0.620990i
$746$		0			0
$747$	6.18567e9	+	3.57130e9i	0.542956	+	0.313476i
$748$		0			0
$749$		−	4.66664e7i		−	0.00405805i
$750$		0			0
$751$	−1.03699e10	−	1.79612e10i	−0.893378	−	1.54738i	−0.835800	−	0.549034i	$-0.814996\pi$
$751$	−1.03699e10	−	1.79612e10i	−0.893378	−	1.54738i	−0.0575779	−	0.998341i	$-0.518338\pi$
$752$		0			0
$753$	−2.84290e10			−2.42650
$754$		0			0
$755$	5.70971e9			0.482836
$756$		0			0
$757$	−1.49204e9	−	2.58429e9i	−0.125010	−	0.216524i	0.796727	−	0.604340i	$-0.206563\pi$
$757$	−1.49204e9	−	2.58429e9i	−0.125010	−	0.216524i	−0.921737	+	0.387816i	$0.873230\pi$
$758$		0			0
$759$		−	3.57610e10i		−	2.96868i
$760$		0			0
$761$	−1.83097e10	−	1.05711e10i	−1.50603	−	0.869507i	−0.999975	−	0.00700633i	$-0.997770\pi$
$761$	−1.83097e10	−	1.05711e10i	−1.50603	−	0.869507i	−0.506055	−	0.862501i	$-0.668897\pi$
$762$		0			0
$763$	−1.35504e9	+	2.34700e9i	−0.110438	+	0.191284i
$764$		0			0
$765$	1.19401e10	−	6.89363e9i	0.964258	−	0.556715i
$766$		0			0
$767$	2.61054e9	−	2.10787e8i	0.208904	−	0.0168679i
$768$		0			0
$769$	1.19184e8	−	6.88110e7i	0.00945097	−	0.00545652i	−0.495267	−	0.868741i	$-0.664930\pi$
$769$	1.19184e8	−	6.88110e7i	0.00945097	−	0.00545652i	0.504718	+	0.863284i	$0.331596\pi$
$770$		0			0
$771$	−1.45517e10	+	2.52043e10i	−1.14347	+	1.98054i
$772$		0			0
$773$	3.53984e9	+	2.04373e9i	0.275649	+	0.159146i	0.631452	−	0.775415i	$-0.282459\pi$
$773$	3.53984e9	+	2.04373e9i	0.275649	+	0.159146i	−0.355803	+	0.934561i	$0.615793\pi$
$774$		0			0
$775$		−	2.58384e9i		−	0.199393i
$776$		0			0
$777$	9.60589e9	+	1.66379e10i	0.734622	+	1.27240i
$778$		0			0
$779$	4.85683e9			0.368105
$780$		0			0
$781$	9.85077e9			0.739932
$782$		0			0
$783$	2.93270e9	+	5.07958e9i	0.218324	+	0.378148i
$784$		0			0
$785$			9.45295e9i			0.697466i
$786$		0			0
$787$	1.59259e10	+	9.19480e9i	1.16464	+	0.672404i	0.952411	−	0.304816i	$-0.0985950\pi$
$787$	1.59259e10	+	9.19480e9i	1.16464	+	0.672404i	0.212227	+	0.977220i	$0.431928\pi$
$788$		0			0
$789$	−8.08253e9	+	1.39994e10i	−0.585839	+	1.01470i
$790$		0			0
$791$	1.21261e10	−	7.00101e9i	0.871171	−	0.502971i
$792$		0			0
$793$	9.19793e8	+	1.13914e10i	0.0654989	+	0.811185i
$794$		0			0
$795$	2.34859e10	−	1.35596e10i	1.65776	−	0.957110i
$796$		0			0
$797$	−5.37705e9	+	9.31332e9i	−0.376218	+	0.651629i	−0.990509	−	0.137451i	$-0.956109\pi$
$797$	−5.37705e9	+	9.31332e9i	−0.376218	+	0.651629i	0.614290	+	0.789080i	$0.289442\pi$
$798$		0			0
$799$	4.86124e9	+	2.80664e9i	0.337158	+	0.194658i
$800$		0			0
$801$		−	1.75636e9i		−	0.120754i
$802$		0			0
$803$	1.01032e10	+	1.74992e10i	0.688578	+	1.19265i
$804$		0			0
$805$	8.65972e9			0.585084
$806$		0			0
$807$	−5.22482e9			−0.349956
$808$		0			0
$809$	−3.96695e9	−	6.87096e9i	−0.263413	−	0.456245i	0.703734	−	0.710464i	$-0.251515\pi$
$809$	−3.96695e9	−	6.87096e9i	−0.263413	−	0.456245i	−0.967147	+	0.254219i	$0.918182\pi$
$810$		0			0
$811$			2.50497e10i			1.64903i	0.565837	+	0.824517i	$0.308553\pi$
$811$			2.50497e10i			1.64903i	−0.565837	+	0.824517i	$0.691447\pi$
$812$		0			0
$813$	−3.36451e9	−	1.94250e9i	−0.219586	−	0.126778i
$814$		0			0
$815$	−3.85110e9	+	6.67030e9i	−0.249191	+	0.431612i
$816$		0			0
$817$	7.55112e8	−	4.35964e8i	0.0484433	−	0.0279688i
$818$		0			0
$819$	9.91390e9	−	2.08940e10i	0.630596	−	1.32901i
$820$		0			0
$821$	4.79566e9	−	2.76877e9i	0.302445	−	0.174617i	−0.341095	−	0.940029i	$-0.610798\pi$
$821$	4.79566e9	−	2.76877e9i	0.302445	−	0.174617i	0.643541	+	0.765412i	$0.277465\pi$
$822$		0			0
$823$	−1.47330e10	+	2.55183e10i	−0.921282	+	1.59571i	−0.123847	+	0.992301i	$0.539523\pi$
$823$	−1.47330e10	+	2.55183e10i	−0.921282	+	1.59571i	−0.797435	+	0.603405i	$0.793810\pi$
$824$		0			0
$825$	1.50490e10	+	8.68852e9i	0.933077	+	0.538712i
$826$		0			0
$827$		−	2.47966e10i		−	1.52448i	−0.647294	−	0.762241i	$-0.724099\pi$
$827$		−	2.47966e10i		−	1.52448i	0.647294	−	0.762241i	$-0.275901\pi$
$828$		0			0
$829$	−5.43179e9	−	9.40814e9i	−0.331133	−	0.573539i	0.651601	−	0.758562i	$-0.274098\pi$
$829$	−5.43179e9	−	9.40814e9i	−0.331133	−	0.573539i	−0.982734	+	0.185022i	$0.940764\pi$
$830$		0			0
$831$	−4.80480e10			−2.90450
$832$		0			0
$833$	5.34185e9			0.320210
$834$		0			0
$835$	−8.94341e8	−	1.54904e9i	−0.0531619	−	0.0920791i
$836$		0			0
$837$		−	1.62321e10i		−	0.956834i
$838$		0			0
$839$	−1.24298e10	−	7.17633e9i	−0.726602	−	0.419504i	0.0905759	−	0.995890i	$-0.471129\pi$
$839$	−1.24298e10	−	7.17633e9i	−0.726602	−	0.419504i	−0.817178	+	0.576386i	$0.804463\pi$
$840$		0			0
$841$	8.07895e9	−	1.39932e10i	0.468348	−	0.811203i
$842$		0			0
$843$	1.22624e10	−	7.07969e9i	0.704982	−	0.407022i
$844$		0			0
$845$	1.31024e10	+	4.96861e9i	0.747056	+	0.283293i
$846$		0			0
$847$	2.20241e10	−	1.27156e10i	1.24539	−	0.719026i
$848$		0			0
$849$	−2.67828e10	+	4.63892e10i	−1.50203	+	2.60159i
$850$		0			0
$851$	1.79036e10	+	1.03366e10i	0.995833	+	0.574944i
$852$		0			0
$853$			2.42734e10i			1.33909i	0.742774	+	0.669543i	$0.233510\pi$
$853$			2.42734e10i			1.33909i	−0.742774	+	0.669543i	$0.766490\pi$
$854$		0			0
$855$	−4.07862e9	−	7.06438e9i	−0.223168	−	0.386539i
$856$		0			0
$857$	9.63070e9			0.522667			0.261334	−	0.965249i	$-0.415838\pi$
$857$	9.63070e9			0.522667			0.261334	+	0.965249i	$0.415838\pi$
$858$		0			0
$859$	−2.19480e10			−1.18146			−0.590729	−	0.806870i	$-0.701160\pi$
$859$	−2.19480e10			−1.18146			−0.590729	+	0.806870i	$0.701160\pi$
$860$		0			0
$861$	−1.57261e10	−	2.72385e10i	−0.839675	−	1.45436i
$862$		0			0
$863$		−	1.90983e10i		−	1.01148i	−0.862686	−	0.505740i	$-0.831220\pi$
$863$		−	1.90983e10i		−	1.01148i	0.862686	−	0.505740i	$-0.168780\pi$
$864$		0			0
$865$	7.21651e9	+	4.16645e9i	0.379115	+	0.218882i
$866$		0			0
$867$	8.56604e9	−	1.48368e10i	0.446389	−	0.773168i
$868$		0			0
$869$	1.40618e10	−	8.11857e9i	0.726894	−	0.419672i
$870$		0			0
$871$	8.71616e9	−	1.83697e10i	0.446953	−	0.941973i
$872$		0			0
$873$	4.18637e10	−	2.41700e10i	2.12955	−	1.22950i
$874$		0			0
$875$	−7.92165e9	+	1.37207e10i	−0.399749	+	0.692385i
$876$		0			0
$877$	−2.41084e10	−	1.39190e10i	−1.20690	−	0.696802i	−0.244817	−	0.969569i	$-0.578728\pi$
$877$	−2.41084e10	−	1.39190e10i	−1.20690	−	0.696802i	−0.962080	+	0.272767i	$0.912061\pi$
$878$		0			0
$879$			4.34885e10i			2.15980i
$880$		0			0
$881$	1.12237e10	+	1.94400e10i	0.552992	+	0.957811i	0.998057	+	0.0623123i	$0.0198475\pi$
$881$	1.12237e10	+	1.94400e10i	0.552992	+	0.957811i	−0.445064	+	0.895499i	$0.646819\pi$
$882$		0			0
$883$	−1.41508e10			−0.691700			−0.345850	−	0.938290i	$-0.612409\pi$
$883$	−1.41508e10			−0.691700			−0.345850	+	0.938290i	$0.612409\pi$
$884$		0			0
$885$	−5.98234e9			−0.290115
$886$		0			0
$887$	1.63692e10	+	2.83523e10i	0.787582	+	1.36413i	0.927445	+	0.373961i	$0.122000\pi$
$887$	1.63692e10	+	2.83523e10i	0.787582	+	1.36413i	−0.139863	+	0.990171i	$0.544666\pi$
$888$		0			0
$889$		−	6.22603e9i		−	0.297204i
$890$		0			0
$891$	3.16022e10	+	1.82456e10i	1.49674	+	0.864142i
$892$		0			0
$893$	1.66055e9	−	2.87616e9i	0.0780319	−	0.135155i
$894$		0			0
$895$	−1.32518e10	+	7.65095e9i	−0.617868	+	0.356726i
$896$		0			0
$897$	−3.00349e9	−	3.71974e10i	−0.138948	−	1.72083i
$898$		0			0
$899$	2.61710e9	−	1.51099e9i	0.120133	−	0.0693588i
$900$		0			0
$901$	−1.05687e10	+	1.83056e10i	−0.481378	+	0.833772i
$902$		0			0
$903$	−4.89002e9	−	2.82326e9i	−0.221006	−	0.127598i
$904$		0			0
$905$		−	2.69902e10i		−	1.21042i
$906$		0			0
$907$	−2.32474e9	−	4.02657e9i	−0.103454	−	0.179188i	0.809651	−	0.586911i	$-0.199656\pi$
$907$	−2.32474e9	−	4.02657e9i	−0.103454	−	0.179188i	−0.913106	+	0.407723i	$0.866323\pi$
$908$		0			0
$909$	−3.36347e10			−1.48530
$910$		0			0
$911$	1.91286e9			0.0838242			0.0419121	−	0.999121i	$-0.486655\pi$
$911$	1.91286e9			0.0838242			0.0419121	+	0.999121i	$0.486655\pi$
$912$		0			0
$913$	6.19251e9	+	1.07258e10i	0.269289	+	0.466423i
$914$		0			0
$915$		−	2.61046e10i		−	1.12653i
$916$		0			0
$917$	2.35382e10	+	1.35898e10i	1.00804	+	0.581995i
$918$		0			0
$919$	−2.25653e10	+	3.90842e10i	−0.959039	+	1.66110i	−0.234196	+	0.972189i	$0.575246\pi$
$919$	−2.25653e10	+	3.90842e10i	−0.959039	+	1.66110i	−0.724843	+	0.688914i	$0.758088\pi$
$920$		0			0
$921$	−2.51296e10	+	1.45086e10i	−1.05993	+	0.611949i
$922$		0			0
$923$	1.02464e10	−	8.27347e8i	0.428911	−	0.0346323i
$924$		0			0
$925$	−8.69973e9	+	5.02279e9i	−0.361418	+	0.208665i
$926$		0			0
$927$	1.06340e9	−	1.84187e9i	0.0438448	−	0.0759415i
$928$		0			0
$929$	−5.68298e9	−	3.28107e9i	−0.232553	−	0.134264i	0.379196	−	0.925316i	$-0.376200\pi$
$929$	−5.68298e9	−	3.28107e9i	−0.232553	−	0.134264i	−0.611749	+	0.791052i	$0.709534\pi$
$930$		0			0
$931$		−	3.16051e9i		−	0.128361i
$932$		0			0
$933$	2.00569e10	+	3.47396e10i	0.808498	+	1.40036i
$934$		0			0
$935$	2.39067e10			0.956485
$936$		0			0
$937$	−1.55199e10			−0.616314			−0.308157	−	0.951336i	$-0.599712\pi$
$937$	−1.55199e10			−0.616314			−0.308157	+	0.951336i	$0.599712\pi$
$938$		0			0
$939$	1.27529e10	+	2.20886e10i	0.502664	+	0.870640i
$940$		0			0
$941$			1.49766e10i			0.585933i	0.956123	+	0.292967i	$0.0946425\pi$
$941$			1.49766e10i			0.585933i	−0.956123	+	0.292967i	$0.905358\pi$
$942$		0			0
$943$	−2.93106e10	−	1.69225e10i	−1.13824	−	0.657162i
$944$		0			0
$945$	−1.32175e10	+	2.28934e10i	−0.509494	+	0.882470i
$946$		0			0
$947$	−1.03810e10	+	5.99345e9i	−0.397203	+	0.229325i	−0.685276	−	0.728283i	$-0.740319\pi$
$947$	−1.03810e10	+	5.99345e9i	−0.397203	+	0.229325i	0.288073	+	0.957608i	$0.406985\pi$
$948$		0			0
$949$	1.19787e10	+	1.73535e10i	0.454965	+	0.659108i
$950$		0			0
$951$	−4.37972e10	+	2.52863e10i	−1.65126	+	0.953353i
$952$		0			0
$953$	−3.02214e9	+	5.23451e9i	−0.113107	+	0.195907i	−0.917022	−	0.398838i	$-0.869414\pi$
$953$	−3.02214e9	+	5.23451e9i	−0.113107	+	0.195907i	0.803914	+	0.594745i	$0.202747\pi$
$954$		0			0
$955$	−1.62541e10	−	9.38432e9i	−0.603882	−	0.348651i
$956$		0			0
$957$			2.03236e10i			0.749565i
$958$		0			0
$959$	1.62471e10	+	2.81409e10i	0.594856	+	1.03032i
$960$		0			0
$961$	1.91495e10			0.696025
$962$		0			0
$963$	3.06327e8			0.0110533
$964$		0			0
$965$	−4.96991e9	−	8.60815e9i	−0.178034	−	0.308364i
$966$		0			0
$967$			4.09241e9i			0.145541i	0.997349	+	0.0727706i	$0.0231841\pi$
$967$			4.09241e9i			0.145541i	−0.997349	+	0.0727706i	$0.976816\pi$
$968$		0			0
$969$	8.25694e9	+	4.76715e9i	0.291532	+	0.168316i
$970$		0			0
$971$	2.47218e10	−	4.28194e10i	0.866587	−	1.50097i	0.00112478	−	0.999999i	$-0.499642\pi$
$971$	2.47218e10	−	4.28194e10i	0.866587	−	1.50097i	0.865462	−	0.500974i	$-0.167025\pi$
$972$		0			0
$973$	−3.07778e10	+	1.77696e10i	−1.07113	+	0.618418i
$974$		0			0
$975$	1.63832e10	+	7.77358e9i	0.566085	+	0.268599i
$976$		0			0
$977$	4.23697e10	−	2.44622e10i	1.45353	−	0.839197i	0.454852	−	0.890567i	$-0.349692\pi$
$977$	4.23697e10	−	2.44622e10i	1.45353	−	0.839197i	0.998680	+	0.0513706i	$0.0163590\pi$
$978$		0			0
$979$	1.52274e9	−	2.63746e9i	0.0518664	−	0.0898352i
$980$		0			0
$981$	−1.54062e10	−	8.89476e9i	−0.521019	−	0.300810i
$982$		0			0
$983$			1.74370e10i			0.585511i	0.956187	+	0.292755i	$0.0945722\pi$
$983$			1.74370e10i			0.585511i	−0.956187	+	0.292755i	$0.905428\pi$
$984$		0			0
$985$	−1.41251e10	−	2.44655e10i	−0.470940	−	0.815693i
$986$		0			0
$987$	−2.15071e10			−0.711987
$988$		0			0
$989$	−6.07605e9			−0.199726
$990$		0			0
$991$	−2.42425e10	−	4.19893e10i	−0.791262	−	1.37051i	−0.925186	−	0.379514i	$-0.876091\pi$
$991$	−2.42425e10	−	4.19893e10i	−0.791262	−	1.37051i	0.133924	−	0.990992i	$-0.457242\pi$
$992$		0			0
$993$		−	4.08193e10i		−	1.32295i
$994$		0			0
$995$	1.86086e10	+	1.07437e10i	0.598869	+	0.345757i
$996$		0			0
$997$	−1.09796e10	+	1.90172e10i	−0.350876	+	0.607735i	−0.986403	−	0.164343i	$-0.947450\pi$
$997$	−1.09796e10	+	1.90172e10i	−0.350876	+	0.607735i	0.635527	+	0.772079i	$0.280783\pi$
$998$		0			0
$999$	−5.46533e10	+	3.15541e10i	−1.73435	+	1.00133i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	208.8.w.a.49.1		14
4.3	odd	2		13.8.e.a.10.1	yes	14
12.11	even	2		117.8.q.b.10.7		14
13.4	even	6	inner	208.8.w.a.17.1		14
52.3	odd	6		169.8.b.d.168.2		14
52.11	even	12		169.8.a.g.1.13		14
52.15	even	12		169.8.a.g.1.2		14
52.23	odd	6		169.8.b.d.168.13		14
52.43	odd	6		13.8.e.a.4.1	✓	14
156.95	even	6		117.8.q.b.82.7		14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
13.8.e.a.4.1	✓	14	52.43	odd	6
13.8.e.a.10.1	yes	14	4.3	odd	2
117.8.q.b.10.7		14	12.11	even	2
117.8.q.b.82.7		14	156.95	even	6
169.8.a.g.1.2		14	52.15	even	12
169.8.a.g.1.13		14	52.11	even	12
169.8.b.d.168.2		14	52.3	odd	6
169.8.b.d.168.13		14	52.23	odd	6
208.8.w.a.17.1		14	13.4	even	6	inner
208.8.w.a.49.1		14	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 208 = 2^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 8 \)
Character orbit:	\([\chi]\)	\(=\)	208.w (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-40.5114 - 70.1679i) q^{3} +223.318i q^{5} +(577.559 + 333.454i) q^{7} +(-2188.85 + 3791.20i) q^{9} +(-6573.82 + 3795.40i) q^{11} +(-6519.11 + 4499.97i) q^{13} +(15669.7 - 9046.93i) q^{15} +(-7051.44 + 12213.4i) q^{17} +(7226.10 + 4171.99i) q^{19} -54034.7i q^{21} +(-29072.6 - 50355.3i) q^{23} +28254.0 q^{25} +177497. q^{27} +(16522.5 + 28617.8i) q^{29} -91450.2i q^{31} +(532630. + 307514. i) q^{33} +(-74466.2 + 128979. i) q^{35} +(-307911. + 177773. i) q^{37} +(579852. + 275132. i) q^{39} +(504092. - 291038. i) q^{41} +(52248.9 - 90497.7i) q^{43} +(-846644. - 488810. i) q^{45} -398024. i q^{47} +(-189389. - 328031. i) q^{49} +1.14266e6 q^{51} +1.49881e6 q^{53} +(-847581. - 1.46805e6i) q^{55} -676053. i q^{57} +(-286332. - 165314. i) q^{59} +(721365. - 1.24944e6i) q^{61} +(-2.52838e6 + 1.45976e6i) q^{63} +(-1.00493e6 - 1.45583e6i) q^{65} +(-2.22292e6 + 1.28340e6i) q^{67} +(-2.35555e6 + 4.07993e6i) q^{69} +(-1.12386e6 - 648863. i) q^{71} -2.66195e6i q^{73} +(-1.14461e6 - 1.98253e6i) q^{75} -5.06236e6 q^{77} -2.13906e6 q^{79} +(-2.40364e6 - 4.16323e6i) q^{81} -1.63158e6i q^{83} +(-2.72748e6 - 1.57471e6i) q^{85} +(1.33870e6 - 2.31870e6i) q^{87} +(-347455. + 200603. i) q^{89} +(-5.26570e6 + 425178. i) q^{91} +(-6.41686e6 + 3.70478e6i) q^{93} +(-931681. + 1.61372e6i) q^{95} +(-9.56294e6 - 5.52117e6i) q^{97} -3.32303e7i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q - 26 q^{3} + 2772 q^{7} - 3491 q^{9} - 6516 q^{11} + 5109 q^{13} + 19266 q^{15} - 38403 q^{17} - 43254 q^{19} + 68550 q^{23} + 39380 q^{25} + 432400 q^{27} + 221583 q^{29} + 219756 q^{33} - 659616 q^{35}+ \cdots - 68556288 q^{97}+O(q^{100}) \) 14 * q - 26 * q^3 + 2772 * q^7 - 3491 * q^9 - 6516 * q^11 + 5109 * q^13 + 19266 * q^15 - 38403 * q^17 - 43254 * q^19 + 68550 * q^23 + 39380 * q^25 + 432400 * q^27 + 221583 * q^29 + 219756 * q^33 - 659616 * q^35 - 565347 * q^37 - 30082 * q^39 + 1716057 * q^41 + 882692 * q^43 - 4616853 * q^45 + 2653979 * q^49 + 38964 * q^51 - 1791210 * q^53 - 2453536 * q^55 + 1189032 * q^59 + 3858525 * q^61 - 7718520 * q^63 + 10424661 * q^65 - 4674330 * q^67 - 13743840 * q^69 + 25203126 * q^71 - 10316152 * q^75 + 19599336 * q^77 + 30688440 * q^79 + 13886221 * q^81 - 22436313 * q^85 - 12769686 * q^87 + 2123160 * q^89 - 37205532 * q^91 + 36638964 * q^93 + 43420386 * q^95 - 68556288 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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