LMFDB - Embedded newform 272.4.o.g.81.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [272,4,Mod(81,272)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(272, base_ring=CyclotomicField(4))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("272.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 272 = 2^{4} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	272.o (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	no
Analytic conductor:	$16.0485195216$
Analytic rank:	$0$
Dimension:	$14$
Relative dimension:	$7$ over $\Q(i)$
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} + 142x^{12} + 7785x^{10} + 208489x^{8} + 2822686x^{6} + 17977041x^{4} + 45020416x^{2} + 31719424 $ x^14 + 142x^12 + 7785x^10 + 208489x^8 + 2822686x^6 + 17977041x^4 + 45020416x^2 + 31719424
Coefficient ring:	$\Z[a_1, \ldots, a_{9}]$
Coefficient ring index:	$ 2^{17} $
Twist minimal:	no (minimal twist has level 136)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			81.7
Root			$-6.91794i$ of defining polynomial
Character	$\chi$	$=$	272.81
Dual form			272.4.o.g.225.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(6.91794 + 6.91794i) q^{3} +(8.19992 + 8.19992i) q^{5} +(18.1123 - 18.1123i) q^{7} +68.7157i q^{9} +O(q^{10})$	\(q+(6.91794 + 6.91794i) q^{3} +(8.19992 + 8.19992i) q^{5} +(18.1123 - 18.1123i) q^{7} +68.7157i q^{9} +(-44.7458 + 44.7458i) q^{11} +9.95609 q^{13} +113.453i q^{15} +(51.7734 - 47.2495i) q^{17} -108.484i q^{19} +250.599 q^{21} +(4.64582 - 4.64582i) q^{23} +9.47723i q^{25} +(-288.587 + 288.587i) q^{27} +(140.673 + 140.673i) q^{29} +(-158.302 - 158.302i) q^{31} -619.098 q^{33} +297.038 q^{35} +(33.6838 + 33.6838i) q^{37} +(68.8756 + 68.8756i) q^{39} +(-55.7328 + 55.7328i) q^{41} -193.194i q^{43} +(-563.463 + 563.463i) q^{45} -246.840 q^{47} -313.109i q^{49} +(685.034 + 31.2962i) q^{51} +208.770i q^{53} -733.824 q^{55} +(750.486 - 750.486i) q^{57} +163.237i q^{59} +(76.2796 - 76.2796i) q^{61} +(1244.60 + 1244.60i) q^{63} +(81.6391 + 81.6391i) q^{65} +292.899 q^{67} +64.2790 q^{69} +(-484.285 - 484.285i) q^{71} +(90.9637 + 90.9637i) q^{73} +(-65.5629 + 65.5629i) q^{75} +1620.90i q^{77} +(45.3370 - 45.3370i) q^{79} -2137.53 q^{81} -661.257i q^{83} +(811.979 + 37.0957i) q^{85} +1946.33i q^{87} -657.756 q^{89} +(180.327 - 180.327i) q^{91} -2190.25i q^{93} +(889.560 - 889.560i) q^{95} +(58.0769 + 58.0769i) q^{97} +(-3074.74 - 3074.74i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 14 q + 4 q^{3} + 8 q^{5} + 10 q^{7}+O(q^{10}) $ 14 * q + 4 * q^3 + 8 * q^5 + 10 * q^7	$ 14 q + 4 q^{3} + 8 q^{5} + 10 q^{7} - 16 q^{11} + 72 q^{13} - 40 q^{17} - 60 q^{21} - 246 q^{23} - 572 q^{27} + 260 q^{29} - 566 q^{31} - 904 q^{33} + 804 q^{35} - 28 q^{37} - 476 q^{39} - 786 q^{41} - 604 q^{45} - 448 q^{47} + 380 q^{51} - 1612 q^{55} + 1004 q^{57} + 660 q^{61} + 2250 q^{63} - 796 q^{65} + 284 q^{67} - 2532 q^{69} - 326 q^{71} + 730 q^{73} + 864 q^{75} - 342 q^{79} - 950 q^{81} + 4148 q^{85} - 4516 q^{89} + 3112 q^{91} - 3356 q^{95} - 1194 q^{97} - 2876 q^{99}+O(q^{100}) $ 14 * q + 4 * q^3 + 8 * q^5 + 10 * q^7 - 16 * q^11 + 72 * q^13 - 40 * q^17 - 60 * q^21 - 246 * q^23 - 572 * q^27 + 260 * q^29 - 566 * q^31 - 904 * q^33 + 804 * q^35 - 28 * q^37 - 476 * q^39 - 786 * q^41 - 604 * q^45 - 448 * q^47 + 380 * q^51 - 1612 * q^55 + 1004 * q^57 + 660 * q^61 + 2250 * q^63 - 796 * q^65 + 284 * q^67 - 2532 * q^69 - 326 * q^71 + 730 * q^73 + 864 * q^75 - 342 * q^79 - 950 * q^81 + 4148 * q^85 - 4516 * q^89 + 3112 * q^91 - 3356 * q^95 - 1194 * q^97 - 2876 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/272\mathbb{Z}\right)^\times$.

$n$	$69$	$239$	$241$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$

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
$3$	6.91794	+	6.91794i	1.33136	+	1.33136i	0.904157	+	0.427201i	$0.140500\pi$
$3$	6.91794	+	6.91794i	1.33136	+	1.33136i	0.427201	+	0.904157i	$0.359500\pi$
$4$		0			0
$5$	8.19992	+	8.19992i	0.733423	+	0.733423i	0.971296	−	0.237873i	$-0.0764503\pi$
$5$	8.19992	+	8.19992i	0.733423	+	0.733423i	−0.237873	+	0.971296i	$0.576450\pi$
$6$		0			0
$7$	18.1123	−	18.1123i	0.977971	−	0.977971i	−0.0217920	−	0.999763i	$-0.506937\pi$
$7$	18.1123	−	18.1123i	0.977971	−	0.977971i	0.999763	+	0.0217920i	$0.00693715\pi$
$8$		0			0
$9$			68.7157i			2.54503i
$10$		0			0
$11$	−44.7458	+	44.7458i	−1.22649	+	1.22649i	−0.261206	+	0.965283i	$0.584120\pi$
$11$	−44.7458	+	44.7458i	−1.22649	+	1.22649i	−0.965283	+	0.261206i	$0.915880\pi$
$12$		0			0
$13$	9.95609			0.212409			0.106205	−	0.994344i	$-0.466130\pi$
$13$	9.95609			0.212409			0.106205	+	0.994344i	$0.466130\pi$
$14$		0			0
$15$			113.453i			1.95290i
$16$		0			0
$17$	51.7734	−	47.2495i	0.738641	−	0.674099i
$17$	51.7734	−	47.2495i	0.738641	−	0.674099i
$18$		0			0
$19$		−	108.484i		−	1.30989i	−0.755676	−	0.654946i	$-0.772691\pi$
$19$		−	108.484i		−	1.30989i	0.755676	−	0.654946i	$-0.227309\pi$
$20$		0			0
$21$	250.599			2.60406
$22$		0			0
$23$	4.64582	−	4.64582i	0.0421183	−	0.0421183i	−0.685734	−	0.727852i	$-0.740519\pi$
$23$	4.64582	−	4.64582i	0.0421183	−	0.0421183i	0.727852	+	0.685734i	$0.240519\pi$
$24$		0			0
$25$			9.47723i			0.0758178i
$26$		0			0
$27$	−288.587	+	288.587i	−2.05698	+	2.05698i
$28$		0			0
$29$	140.673	+	140.673i	0.900768	+	0.900768i	0.995503	−	0.0947342i	$-0.0302001\pi$
$29$	140.673	+	140.673i	0.900768	+	0.900768i	−0.0947342	+	0.995503i	$0.530200\pi$
$30$		0			0
$31$	−158.302	−	158.302i	−0.917159	−	0.917159i	0.0796628	−	0.996822i	$-0.474616\pi$
$31$	−158.302	−	158.302i	−0.917159	−	0.917159i	−0.996822	+	0.0796628i	$0.974616\pi$
$32$		0			0
$33$	−619.098			−3.26579
$34$		0			0
$35$	297.038			1.43453
$36$		0			0
$37$	33.6838	+	33.6838i	0.149664	+	0.149664i	0.777968	−	0.628304i	$-0.216251\pi$
$37$	33.6838	+	33.6838i	0.149664	+	0.149664i	−0.628304	+	0.777968i	$0.716251\pi$
$38$		0			0
$39$	68.8756	+	68.8756i	0.282793	+	0.282793i
$40$		0			0
$41$	−55.7328	+	55.7328i	−0.212293	+	0.212293i	−0.805241	−	0.592948i	$-0.797964\pi$
$41$	−55.7328	+	55.7328i	−0.212293	+	0.212293i	0.592948	+	0.805241i	$0.297964\pi$
$42$		0			0
$43$		−	193.194i		−	0.685160i	−0.939489	−	0.342580i	$-0.888699\pi$
$43$		−	193.194i		−	0.685160i	0.939489	−	0.342580i	$-0.111301\pi$
$44$		0			0
$45$	−563.463	+	563.463i	−1.86658	+	1.86658i
$46$		0			0
$47$	−246.840			−0.766071			−0.383036	−	0.923734i	$-0.625121\pi$
$47$	−246.840			−0.766071			−0.383036	+	0.923734i	$0.625121\pi$
$48$		0			0
$49$		−	313.109i		−	0.912853i
$50$		0			0
$51$	685.034	+	31.2962i	1.88086	+	0.0859282i
$52$		0			0
$53$			208.770i			0.541071i	0.962710	+	0.270535i	$0.0872007\pi$
$53$			208.770i			0.541071i	−0.962710	+	0.270535i	$0.912799\pi$
$54$		0			0
$55$	−733.824			−1.79907
$56$		0			0
$57$	750.486	−	750.486i	1.74393	−	1.74393i
$58$		0			0
$59$			163.237i			0.360197i	0.983649	+	0.180098i	$0.0576416\pi$
$59$			163.237i			0.360197i	−0.983649	+	0.180098i	$0.942358\pi$
$60$		0			0
$61$	76.2796	−	76.2796i	0.160108	−	0.160108i	−0.622506	−	0.782615i	$-0.713886\pi$
$61$	76.2796	−	76.2796i	0.160108	−	0.160108i	0.782615	+	0.622506i	$0.213886\pi$
$62$		0			0
$63$	1244.60	+	1244.60i	2.48896	+	2.48896i
$64$		0			0
$65$	81.6391	+	81.6391i	0.155786	+	0.155786i
$66$		0			0
$67$	292.899			0.534079			0.267040	−	0.963686i	$-0.413955\pi$
$67$	292.899			0.534079			0.267040	+	0.963686i	$0.413955\pi$
$68$		0			0
$69$	64.2790			0.112149
$70$		0			0
$71$	−484.285	−	484.285i	−0.809494	−	0.809494i	0.175063	−	0.984557i	$-0.443987\pi$
$71$	−484.285	−	484.285i	−0.809494	−	0.809494i	−0.984557	+	0.175063i	$0.943987\pi$
$72$		0			0
$73$	90.9637	+	90.9637i	0.145842	+	0.145842i	0.776258	−	0.630415i	$-0.217115\pi$
$73$	90.9637	+	90.9637i	0.145842	+	0.145842i	−0.630415	+	0.776258i	$0.717115\pi$
$74$		0			0
$75$	−65.5629	+	65.5629i	−0.100941	+	0.100941i
$76$		0			0
$77$			1620.90i			2.39894i
$78$		0			0
$79$	45.3370	−	45.3370i	0.0645672	−	0.0645672i	−0.674086	−	0.738653i	$-0.735462\pi$
$79$	45.3370	−	45.3370i	0.0645672	−	0.0645672i	0.738653	+	0.674086i	$0.235462\pi$
$80$		0			0
$81$	−2137.53			−2.93214
$82$		0			0
$83$		−	661.257i		−	0.874487i	−0.899343	−	0.437243i	$-0.855955\pi$
$83$		−	661.257i		−	0.874487i	0.899343	−	0.437243i	$-0.144045\pi$
$84$		0			0
$85$	811.979	+	37.0957i	1.03614	+	0.0473364i
$86$		0			0
$87$			1946.33i			2.39849i
$88$		0			0
$89$	−657.756			−0.783394			−0.391697	−	0.920094i	$-0.628112\pi$
$89$	−657.756			−0.783394			−0.391697	+	0.920094i	$0.628112\pi$
$90$		0			0
$91$	180.327	−	180.327i	0.207730	−	0.207730i
$92$		0			0
$93$		−	2190.25i		−	2.44213i
$94$		0			0
$95$	889.560	−	889.560i	0.960704	−	0.960704i
$96$		0			0
$97$	58.0769	+	58.0769i	0.0607919	+	0.0607919i	0.736849	−	0.676057i	$-0.236313\pi$
$97$	58.0769	+	58.0769i	0.0607919	+	0.0607919i	−0.676057	+	0.736849i	$0.736313\pi$
$98$		0			0
$99$	−3074.74	−	3074.74i	−3.12145	−	3.12145i
$100$		0			0
$101$	711.196			0.700660			0.350330	−	0.936626i	$-0.386069\pi$
$101$	711.196			0.700660			0.350330	+	0.936626i	$0.386069\pi$
$102$		0			0
$103$	1784.81			1.70741			0.853703	−	0.520760i	$-0.174351\pi$
$103$	1784.81			1.70741			0.853703	+	0.520760i	$0.174351\pi$
$104$		0			0
$105$	2054.89	+	2054.89i	1.90988	+	1.90988i
$106$		0			0
$107$	905.237	+	905.237i	0.817874	+	0.817874i	0.985800	−	0.167926i	$-0.0537069\pi$
$107$	905.237	+	905.237i	0.817874	+	0.817874i	−0.167926	+	0.985800i	$0.553707\pi$
$108$		0			0
$109$	134.827	−	134.827i	0.118478	−	0.118478i	−0.645382	−	0.763860i	$-0.723302\pi$
$109$	134.827	−	134.827i	0.118478	−	0.118478i	0.763860	+	0.645382i	$0.223302\pi$
$110$		0			0
$111$			466.045i			0.398513i
$112$		0			0
$113$	956.503	−	956.503i	0.796286	−	0.796286i	−0.186222	−	0.982508i	$-0.559624\pi$
$113$	956.503	−	956.503i	0.796286	−	0.796286i	0.982508	+	0.186222i	$0.0596244\pi$
$114$		0			0
$115$	76.1907			0.0617811
$116$		0			0
$117$			684.140i			0.540588i
$118$		0			0
$119$	81.9383	−	1793.53i	0.0631200	−	1.38162i
$120$		0			0
$121$		−	2673.38i		−	2.00855i
$122$		0			0
$123$	−771.112			−0.565275
$124$		0			0
$125$	947.277	−	947.277i	0.677816	−	0.677816i
$126$		0			0
$127$			1930.69i			1.34898i	0.738283	+	0.674491i	$0.235637\pi$
$127$			1930.69i			1.34898i	−0.738283	+	0.674491i	$0.764363\pi$
$128$		0			0
$129$	1336.51	−	1336.51i	0.912193	−	0.912193i
$130$		0			0
$131$	−1896.70	−	1896.70i	−1.26500	−	1.26500i	−0.948638	−	0.316365i	$-0.897538\pi$
$131$	−1896.70	−	1896.70i	−1.26500	−	1.26500i	−0.316365	−	0.948638i	$-0.602462\pi$
$132$		0			0
$133$	−1964.89	−	1964.89i	−1.28104	−	1.28104i
$134$		0			0
$135$	−4732.78			−3.01728
$136$		0			0
$137$	−1582.49			−0.986871			−0.493435	−	0.869782i	$-0.664259\pi$
$137$	−1582.49			−0.986871			−0.493435	+	0.869782i	$0.664259\pi$
$138$		0			0
$139$	−767.315	−	767.315i	−0.468221	−	0.468221i	0.433116	−	0.901338i	$-0.357414\pi$
$139$	−767.315	−	767.315i	−0.468221	−	0.468221i	−0.901338	+	0.433116i	$0.857414\pi$
$140$		0			0
$141$	−1707.63	−	1707.63i	−1.01992	−	1.01992i
$142$		0			0
$143$	−445.494	+	445.494i	−0.260518	+	0.260518i
$144$		0			0
$145$			2307.01i			1.32129i
$146$		0			0
$147$	2166.07	−	2166.07i	1.21533	−	1.21533i
$148$		0			0
$149$	−218.582			−0.120181			−0.0600905	−	0.998193i	$-0.519139\pi$
$149$	−218.582			−0.120181			−0.0600905	+	0.998193i	$0.519139\pi$
$150$		0			0
$151$			188.561i			0.101622i	0.998708	+	0.0508109i	$0.0161806\pi$
$151$			188.561i			0.101622i	−0.998708	+	0.0508109i	$0.983819\pi$
$152$		0			0
$153$	3246.78	+	3557.65i	1.71560	+	1.87986i
$154$		0			0
$155$		−	2596.13i		−	1.34533i
$156$		0			0
$157$	1596.67			0.811646			0.405823	−	0.913952i	$-0.366985\pi$
$157$	1596.67			0.811646			0.405823	+	0.913952i	$0.366985\pi$
$158$		0			0
$159$	−1444.26	+	1444.26i	−0.720359	+	0.720359i
$160$		0			0
$161$		−	168.293i		−	0.0823809i
$162$		0			0
$163$	367.807	−	367.807i	0.176742	−	0.176742i	−0.613192	−	0.789934i	$-0.710115\pi$
$163$	367.807	−	367.807i	0.176742	−	0.176742i	0.789934	+	0.613192i	$0.210115\pi$
$164$		0			0
$165$	−5076.55	−	5076.55i	−2.39521	−	2.39521i
$166$		0			0
$167$	1037.76	+	1037.76i	0.480864	+	0.480864i	0.905407	−	0.424544i	$-0.139565\pi$
$167$	1037.76	+	1037.76i	0.480864	+	0.480864i	−0.424544	+	0.905407i	$0.639565\pi$
$168$		0			0
$169$	−2097.88			−0.954882
$170$		0			0
$171$	7454.56			3.33371
$172$		0			0
$173$	573.077	+	573.077i	0.251851	+	0.251851i	0.821729	−	0.569878i	$-0.193010\pi$
$173$	573.077	+	573.077i	0.251851	+	0.251851i	−0.569878	+	0.821729i	$0.693010\pi$
$174$		0			0
$175$	171.654	+	171.654i	0.0741476	+	0.0741476i
$176$		0			0
$177$	−1129.26	+	1129.26i	−0.479551	+	0.479551i
$178$		0			0
$179$			1394.23i			0.582179i	0.956696	+	0.291089i	$0.0940177\pi$
$179$			1394.23i			0.582179i	−0.956696	+	0.291089i	$0.905982\pi$
$180$		0			0
$181$	879.717	−	879.717i	0.361265	−	0.361265i	−0.503014	−	0.864278i	$-0.667776\pi$
$181$	879.717	−	879.717i	0.361265	−	0.361265i	0.864278	+	0.503014i	$0.167776\pi$
$182$		0			0
$183$	1055.40			0.426323
$184$		0			0
$185$			552.408i			0.219534i
$186$		0			0
$187$	−202.426	+	4430.86i	−0.0791598	+	1.73271i
$188$		0			0
$189$			10453.9i			4.02334i
$190$		0			0
$191$	4484.76			1.69898			0.849492	−	0.527602i	$-0.176909\pi$
$191$	4484.76			1.69898			0.849492	+	0.527602i	$0.176909\pi$
$192$		0			0
$193$	1765.05	−	1765.05i	0.658297	−	0.658297i	−0.296680	−	0.954977i	$-0.595879\pi$
$193$	1765.05	−	1765.05i	0.658297	−	0.658297i	0.954977	+	0.296680i	$0.0958794\pi$
$194$		0			0
$195$			1129.55i			0.414814i
$196$		0			0
$197$	28.7116	−	28.7116i	0.0103838	−	0.0103838i	−0.701896	−	0.712280i	$-0.747663\pi$
$197$	28.7116	−	28.7116i	0.0103838	−	0.0103838i	0.712280	+	0.701896i	$0.247663\pi$
$198$		0			0
$199$	−2675.79	−	2675.79i	−0.953173	−	0.953173i	0.0457787	−	0.998952i	$-0.485423\pi$
$199$	−2675.79	−	2675.79i	−0.953173	−	0.953173i	−0.998952	+	0.0457787i	$0.985423\pi$
$200$		0			0
$201$	2026.26	+	2026.26i	0.711050	+	0.711050i
$202$		0			0
$203$	5095.81			1.76185
$204$		0			0
$205$	−914.008			−0.311400
$206$		0			0
$207$	319.241	+	319.241i	0.107192	+	0.107192i
$208$		0			0
$209$	4854.21	+	4854.21i	1.60657	+	1.60657i
$210$		0			0
$211$	−1104.40	+	1104.40i	−0.360333	+	0.360333i	−0.863935	−	0.503603i	$-0.832008\pi$
$211$	−1104.40	+	1104.40i	−0.360333	+	0.360333i	0.503603	+	0.863935i	$0.332008\pi$
$212$		0			0
$213$		−	6700.51i		−	2.15545i
$214$		0			0
$215$	1584.18	−	1584.18i	0.502512	−	0.502512i
$216$		0			0
$217$	−5734.43			−1.79391
$218$		0			0
$219$			1258.56i			0.388337i
$220$		0			0
$221$	515.461	−	470.420i	0.156894	−	0.143185i
$222$		0			0
$223$			4998.81i			1.50110i	0.660814	+	0.750550i	$0.270211\pi$
$223$			4998.81i			1.50110i	−0.660814	+	0.750550i	$0.729789\pi$
$224$		0			0
$225$	−651.235			−0.192959
$226$		0			0
$227$	−1345.91	+	1345.91i	−0.393530	+	0.393530i	−0.875943	−	0.482414i	$-0.839760\pi$
$227$	−1345.91	+	1345.91i	−0.393530	+	0.393530i	0.482414	+	0.875943i	$0.339760\pi$
$228$		0			0
$229$			3619.52i			1.04447i	0.852801	+	0.522237i	$0.174902\pi$
$229$			3619.52i			1.04447i	−0.852801	+	0.522237i	$0.825098\pi$
$230$		0			0
$231$	−11213.3	+	11213.3i	−3.19385	+	3.19385i
$232$		0			0
$233$	−2423.33	−	2423.33i	−0.681362	−	0.681362i	0.278945	−	0.960307i	$-0.410015\pi$
$233$	−2423.33	−	2423.33i	−0.681362	−	0.681362i	−0.960307	+	0.278945i	$0.910015\pi$
$234$		0			0
$235$	−2024.07	−	2024.07i	−0.561854	−	0.561854i
$236$		0			0
$237$	627.277			0.171924
$238$		0			0
$239$	5797.35			1.56904			0.784518	−	0.620106i	$-0.212910\pi$
$239$	5797.35			1.56904			0.784518	+	0.620106i	$0.212910\pi$
$240$		0			0
$241$	−3138.64	−	3138.64i	−0.838912	−	0.838912i	0.149804	−	0.988716i	$-0.452136\pi$
$241$	−3138.64	−	3138.64i	−0.838912	−	0.838912i	−0.988716	+	0.149804i	$0.952136\pi$
$242$		0			0
$243$	−6995.44	−	6995.44i	−1.84674	−	1.84674i
$244$		0			0
$245$	2567.46	−	2567.46i	0.669507	−	0.669507i
$246$		0			0
$247$		−	1080.08i		−	0.278233i
$248$		0			0
$249$	4574.54	−	4574.54i	1.16425	−	1.16425i
$250$		0			0
$251$	−4172.42			−1.04925			−0.524623	−	0.851335i	$-0.675794\pi$
$251$	−4172.42			−1.04925			−0.524623	+	0.851335i	$0.675794\pi$
$252$		0			0
$253$			415.763i			0.103315i
$254$		0			0
$255$	5360.60	+	5873.85i	1.31645	+	1.44249i
$256$		0			0
$257$			6114.45i			1.48408i	0.670354	+	0.742041i	$0.266142\pi$
$257$			6114.45i			1.48408i	−0.670354	+	0.742041i	$0.733858\pi$
$258$		0			0
$259$	1220.18			0.292734
$260$		0			0
$261$	−9666.44	+	9666.44i	−2.29248	+	2.29248i
$262$		0			0
$263$		−	4931.92i		−	1.15633i	−0.815919	−	0.578166i	$-0.803769\pi$
$263$		−	4931.92i		−	1.15633i	0.815919	−	0.578166i	$-0.196231\pi$
$264$		0			0
$265$	−1711.90	+	1711.90i	−0.396834	+	0.396834i
$266$		0			0
$267$	−4550.32	−	4550.32i	−1.04298	−	1.04298i
$268$		0			0
$269$	−629.493	−	629.493i	−0.142680	−	0.142680i	0.632159	−	0.774839i	$-0.282169\pi$
$269$	−629.493	−	629.493i	−0.142680	−	0.142680i	−0.774839	+	0.632159i	$0.782169\pi$
$270$		0			0
$271$	−7325.27			−1.64199			−0.820993	−	0.570938i	$-0.806580\pi$
$271$	−7325.27			−1.64199			−0.820993	+	0.570938i	$0.806580\pi$
$272$		0			0
$273$	2494.99			0.553126
$274$		0			0
$275$	−424.067	−	424.067i	−0.0929897	−	0.0929897i
$276$		0			0
$277$	2047.56	+	2047.56i	0.444137	+	0.444137i	0.893400	−	0.449262i	$-0.148313\pi$
$277$	2047.56	+	2047.56i	0.444137	+	0.444137i	−0.449262	+	0.893400i	$0.648313\pi$
$278$		0			0
$279$	10877.9	−	10877.9i	2.33420	−	2.33420i
$280$		0			0
$281$			4743.17i			1.00695i	0.864009	+	0.503476i	$0.167946\pi$
$281$			4743.17i			1.00695i	−0.864009	+	0.503476i	$0.832054\pi$
$282$		0			0
$283$	6172.12	−	6172.12i	1.29645	−	1.29645i	0.365723	−	0.930724i	$-0.380822\pi$
$283$	6172.12	−	6172.12i	1.29645	−	1.29645i	0.930724	−	0.365723i	$-0.119178\pi$
$284$		0			0
$285$	12307.8			2.55808
$286$		0			0
$287$			2018.89i			0.415232i
$288$		0			0
$289$	447.971	−	4892.53i	0.0911808	−	0.995834i
$290$		0			0
$291$			803.545i			0.161872i
$292$		0			0
$293$	−8323.37			−1.65958			−0.829789	−	0.558077i	$-0.811539\pi$
$293$	−8323.37			−1.65958			−0.829789	+	0.558077i	$0.811539\pi$
$294$		0			0
$295$	−1338.53	+	1338.53i	−0.264176	+	0.264176i
$296$		0			0
$297$		−	25826.1i		−	5.04574i
$298$		0			0
$299$	46.2542	−	46.2542i	0.00894633	−	0.00894633i
$300$		0			0
$301$	−3499.19	−	3499.19i	−0.670066	−	0.670066i
$302$		0			0
$303$	4920.01	+	4920.01i	0.932829	+	0.932829i
$304$		0			0
$305$	1250.97			0.234854
$306$		0			0
$307$	−8916.96			−1.65771			−0.828856	−	0.559462i	$-0.811008\pi$
$307$	−8916.96			−1.65771			−0.828856	+	0.559462i	$0.811008\pi$
$308$		0			0
$309$	12347.2	+	12347.2i	2.27317	+	2.27317i
$310$		0			0
$311$	−5271.80	−	5271.80i	−0.961210	−	0.961210i	0.0380650	−	0.999275i	$-0.487881\pi$
$311$	−5271.80	−	5271.80i	−0.961210	−	0.961210i	−0.999275	+	0.0380650i	$0.987881\pi$
$312$		0			0
$313$	−2668.09	+	2668.09i	−0.481820	+	0.481820i	−0.905712	−	0.423893i	$-0.860663\pi$
$313$	−2668.09	+	2668.09i	−0.481820	+	0.481820i	0.423893	+	0.905712i	$0.360663\pi$
$314$		0			0
$315$			20411.2i			3.65092i
$316$		0			0
$317$	2431.92	−	2431.92i	0.430883	−	0.430883i	−0.458045	−	0.888929i	$-0.651450\pi$
$317$	2431.92	−	2431.92i	0.430883	−	0.430883i	0.888929	+	0.458045i	$0.151450\pi$
$318$		0			0
$319$	−12589.0			−2.20957
$320$		0			0
$321$			12524.7i			2.17777i
$322$		0			0
$323$	−5125.81	−	5616.59i	−0.882997	−	0.967540i
$324$		0			0
$325$			94.3562i			0.0161044i
$326$		0			0
$327$	1865.45			0.315473
$328$		0			0
$329$	−4470.84	+	4470.84i	−0.749195	+	0.749195i
$330$		0			0
$331$			8940.93i			1.48471i	0.670009	+	0.742353i	$0.266290\pi$
$331$			8940.93i			1.48471i	−0.670009	+	0.742353i	$0.733710\pi$
$332$		0			0
$333$	−2314.61	+	2314.61i	−0.380900	+	0.380900i
$334$		0			0
$335$	2401.75	+	2401.75i	0.391706	+	0.391706i
$336$		0			0
$337$	−2357.46	−	2357.46i	−0.381066	−	0.381066i	0.490420	−	0.871486i	$-0.336843\pi$
$337$	−2357.46	−	2357.46i	−0.381066	−	0.381066i	−0.871486	+	0.490420i	$0.836843\pi$
$338$		0			0
$339$	13234.1			2.12028
$340$		0			0
$341$	14166.7			2.24977
$342$		0			0
$343$	541.403	+	541.403i	0.0852274	+	0.0852274i
$344$		0			0
$345$	527.083	+	527.083i	0.0822527	+	0.0822527i
$346$		0			0
$347$	−5649.82	+	5649.82i	−0.874059	+	0.874059i	−0.992912	−	0.118853i	$-0.962078\pi$
$347$	−5649.82	+	5649.82i	−0.874059	+	0.874059i	0.118853	+	0.992912i	$0.462078\pi$
$348$		0			0
$349$		−	3588.04i		−	0.550326i	−0.961398	−	0.275163i	$-0.911268\pi$
$349$		−	3588.04i		−	0.550326i	0.961398	−	0.275163i	$-0.0887317\pi$
$350$		0			0
$351$	−2873.20	+	2873.20i	−0.436923	+	0.436923i
$352$		0			0
$353$	−150.268			−0.0226571			−0.0113286	−	0.999936i	$-0.503606\pi$
$353$	−150.268			−0.0226571			−0.0113286	+	0.999936i	$0.503606\pi$
$354$		0			0
$355$		−	7942.20i		−	1.18740i
$356$		0			0
$357$	12974.4	−	11840.7i	1.92346	−	1.75539i
$358$		0			0
$359$		−	2233.92i		−	0.328418i	−0.986426	−	0.164209i	$-0.947493\pi$
$359$		−	2233.92i		−	0.328418i	0.986426	−	0.164209i	$-0.0525071\pi$
$360$		0			0
$361$	−4909.78			−0.715816
$362$		0			0
$363$	18494.3	−	18494.3i	2.67410	−	2.67410i
$364$		0			0
$365$			1491.79i			0.213928i
$366$		0			0
$367$	−1974.23	+	1974.23i	−0.280801	+	0.280801i	−0.833428	−	0.552628i	$-0.813625\pi$
$367$	−1974.23	+	1974.23i	−0.280801	+	0.280801i	0.552628	+	0.833428i	$0.313625\pi$
$368$		0			0
$369$	−3829.72	−	3829.72i	−0.540290	−	0.540290i
$370$		0			0
$371$	3781.30	+	3781.30i	0.529151	+	0.529151i
$372$		0			0
$373$	10409.0			1.44492			0.722461	−	0.691411i	$-0.243011\pi$
$373$	10409.0			1.44492			0.722461	+	0.691411i	$0.243011\pi$
$374$		0			0
$375$	13106.4			1.80483
$376$		0			0
$377$	1400.55	+	1400.55i	0.191332	+	0.191332i
$378$		0			0
$379$	3345.96	+	3345.96i	0.453484	+	0.453484i	0.896509	−	0.443025i	$-0.146095\pi$
$379$	3345.96	+	3345.96i	0.453484	+	0.453484i	−0.443025	+	0.896509i	$0.646095\pi$
$380$		0			0
$381$	−13356.4	+	13356.4i	−1.79598	+	1.79598i
$382$		0			0
$383$			4845.96i			0.646519i	0.946310	+	0.323260i	$0.104779\pi$
$383$			4845.96i			0.646519i	−0.946310	+	0.323260i	$0.895221\pi$
$384$		0			0
$385$	−13291.2	+	13291.2i	−1.75944	+	1.75944i
$386$		0			0
$387$	13275.5			1.74375
$388$		0			0
$389$		−	8250.13i		−	1.07532i	−0.843163	−	0.537658i	$-0.819309\pi$
$389$		−	8250.13i		−	1.07532i	0.843163	−	0.537658i	$-0.180691\pi$
$390$		0			0
$391$	21.0173	−	460.043i	0.00271839	−	0.0595022i
$392$		0			0
$393$		−	26242.5i		−	3.36834i
$394$		0			0
$395$	743.519			0.0947102
$396$		0			0
$397$	−7050.98	+	7050.98i	−0.891381	+	0.891381i	−0.994653	−	0.103272i	$-0.967069\pi$
$397$	−7050.98	+	7050.98i	−0.891381	+	0.891381i	0.103272	+	0.994653i	$0.467069\pi$
$398$		0			0
$399$		−	27186.0i		−	3.41103i
$400$		0			0
$401$	−2335.54	+	2335.54i	−0.290851	+	0.290851i	−0.837416	−	0.546566i	$-0.815935\pi$
$401$	−2335.54	+	2335.54i	−0.290851	+	0.290851i	0.546566	+	0.837416i	$0.315935\pi$
$402$		0			0
$403$	−1576.07	−	1576.07i	−0.194813	−	0.194813i
$404$		0			0
$405$	−17527.6	−	17527.6i	−2.15050	−	2.15050i
$406$		0			0
$407$	−3014.42			−0.367123
$408$		0			0
$409$	−2024.08			−0.244705			−0.122352	−	0.992487i	$-0.539044\pi$
$409$	−2024.08			−0.244705			−0.122352	+	0.992487i	$0.539044\pi$
$410$		0			0
$411$	−10947.6	−	10947.6i	−1.31388	−	1.31388i
$412$		0			0
$413$	2956.59	+	2956.59i	0.352262	+	0.352262i
$414$		0			0
$415$	5422.25	−	5422.25i	0.641368	−	0.641368i
$416$		0			0
$417$		−	10616.5i		−	1.24674i
$418$		0			0
$419$	−6572.03	+	6572.03i	−0.766264	+	0.766264i	−0.977447	−	0.211183i	$-0.932269\pi$
$419$	−6572.03	+	6572.03i	−0.766264	+	0.766264i	0.211183	+	0.977447i	$0.432269\pi$
$420$		0			0
$421$	−9238.43			−1.06949			−0.534743	−	0.845015i	$-0.679592\pi$
$421$	−9238.43			−1.06949			−0.534743	+	0.845015i	$0.679592\pi$
$422$		0			0
$423$		−	16961.8i		−	1.94967i
$424$		0			0
$425$	447.794	+	490.668i	0.0511087	+	0.0560022i
$426$		0			0
$427$		−	2763.19i		−	0.313162i
$428$		0			0
$429$	−6163.79			−0.693685
$430$		0			0
$431$	1073.65	−	1073.65i	0.119991	−	0.119991i	−0.644562	−	0.764552i	$-0.722960\pi$
$431$	1073.65	−	1073.65i	0.119991	−	0.119991i	0.764552	+	0.644562i	$0.222960\pi$
$432$		0			0
$433$			3810.92i			0.422959i	0.977382	+	0.211479i	$0.0678281\pi$
$433$			3810.92i			0.422959i	−0.977382	+	0.211479i	$0.932172\pi$
$434$		0			0
$435$	−15959.8	+	15959.8i	−1.75911	+	1.75911i
$436$		0			0
$437$	−503.998	−	503.998i	−0.0551704	−	0.0551704i
$438$		0			0
$439$	−5637.63	−	5637.63i	−0.612915	−	0.612915i	0.330790	−	0.943704i	$-0.392685\pi$
$439$	−5637.63	−	5637.63i	−0.612915	−	0.612915i	−0.943704	+	0.330790i	$0.892685\pi$
$440$		0			0
$441$	21515.5			2.32324
$442$		0			0
$443$	−2524.64			−0.270765			−0.135383	−	0.990793i	$-0.543226\pi$
$443$	−2524.64			−0.270765			−0.135383	+	0.990793i	$0.543226\pi$
$444$		0			0
$445$	−5393.55	−	5393.55i	−0.574559	−	0.574559i
$446$		0			0
$447$	−1512.14	−	1512.14i	−0.160004	−	0.160004i
$448$		0			0
$449$	−276.973	+	276.973i	−0.0291117	+	0.0291117i	−0.721513	−	0.692401i	$-0.756553\pi$
$449$	−276.973	+	276.973i	−0.0291117	+	0.0291117i	0.692401	+	0.721513i	$0.256553\pi$
$450$		0			0
$451$		−	4987.62i		−	0.520749i
$452$		0			0
$453$	−1304.45	+	1304.45i	−0.135295	+	0.135295i
$454$		0			0
$455$	2957.34			0.304708
$456$		0			0
$457$			7463.84i			0.763990i	0.924164	+	0.381995i	$0.124763\pi$
$457$			7463.84i			0.763990i	−0.924164	+	0.381995i	$0.875237\pi$
$458$		0			0
$459$	−1305.54	+	28576.7i	−0.132761	+	2.90598i
$460$		0			0
$461$		−	4409.10i		−	0.445449i	−0.974881	−	0.222725i	$-0.928505\pi$
$461$		−	4409.10i		−	0.445449i	0.974881	−	0.222725i	$-0.0714951\pi$
$462$		0			0
$463$	−15821.7			−1.58811			−0.794055	−	0.607845i	$-0.792034\pi$
$463$	−15821.7			−1.58811			−0.794055	+	0.607845i	$0.792034\pi$
$464$		0			0
$465$	17959.9	−	17959.9i	1.79112	−	1.79112i
$466$		0			0
$467$			9150.74i			0.906736i	0.891323	+	0.453368i	$0.149778\pi$
$467$			9150.74i			0.906736i	−0.891323	+	0.453368i	$0.850222\pi$
$468$		0			0
$469$	5305.06	−	5305.06i	0.522314	−	0.522314i
$470$		0			0
$471$	11045.7	+	11045.7i	1.08059	+	1.08059i
$472$		0			0
$473$	8644.65	+	8644.65i	0.840341	+	0.840341i
$474$		0			0
$475$	1028.13			0.0993132
$476$		0			0
$477$	−14345.8			−1.37704
$478$		0			0
$479$	13585.6	+	13585.6i	1.29591	+	1.29591i	0.931071	+	0.364838i	$0.118876\pi$
$479$	13585.6	+	13585.6i	1.29591	+	1.29591i	0.364838	+	0.931071i	$0.381124\pi$
$480$		0			0
$481$	335.359	+	335.359i	0.0317901	+	0.0317901i
$482$		0			0
$483$	1164.24	−	1164.24i	0.109679	−	0.109679i
$484$		0			0
$485$			952.452i			0.0891724i
$486$		0			0
$487$	4236.87	−	4236.87i	0.394232	−	0.394232i	−0.481961	−	0.876193i	$-0.660075\pi$
$487$	4236.87	−	4236.87i	0.394232	−	0.394232i	0.876193	+	0.481961i	$0.160075\pi$
$488$		0			0
$489$	5088.94			0.470613
$490$		0			0
$491$			1595.37i			0.146636i	0.997309	+	0.0733178i	$0.0233588\pi$
$491$			1595.37i			0.146636i	−0.997309	+	0.0733178i	$0.976641\pi$
$492$		0			0
$493$	13929.8	+	636.392i	1.27255	+	0.0581372i
$494$		0			0
$495$		−	50425.3i		−	4.57868i
$496$		0			0
$497$	−17543.0			−1.58332
$498$		0			0
$499$	−3641.46	+	3641.46i	−0.326682	+	0.326682i	−0.851323	−	0.524642i	$-0.824199\pi$
$499$	−3641.46	+	3641.46i	−0.326682	+	0.326682i	0.524642	+	0.851323i	$0.324199\pi$
$500$		0			0
$501$			14358.3i			1.28040i
$502$		0			0
$503$	−10555.6	+	10555.6i	−0.935684	+	0.935684i	−0.998053	−	0.0623692i	$-0.980134\pi$
$503$	−10555.6	+	10555.6i	−0.935684	+	0.935684i	0.0623692	+	0.998053i	$0.480134\pi$
$504$		0			0
$505$	5831.75	+	5831.75i	0.513880	+	0.513880i
$506$		0			0
$507$	−14513.0	−	14513.0i	−1.27129	−	1.27129i
$508$		0			0
$509$	−18083.8			−1.57476			−0.787378	−	0.616471i	$-0.788562\pi$
$509$	−18083.8			−1.57476			−0.787378	+	0.616471i	$0.788562\pi$
$510$		0			0
$511$	3295.12			0.285259
$512$		0			0
$513$	31307.1	+	31307.1i	2.69443	+	2.69443i
$514$		0			0
$515$	14635.3	+	14635.3i	1.25225	+	1.25225i
$516$		0			0
$517$	11045.1	−	11045.1i	0.939578	−	0.939578i
$518$		0			0
$519$			7929.02i			0.670608i
$520$		0			0
$521$	11589.8	−	11589.8i	0.974587	−	0.974587i	−0.0250980	−	0.999685i	$-0.507990\pi$
$521$	11589.8	−	11589.8i	0.974587	−	0.974587i	0.999685	+	0.0250980i	$0.00798977\pi$
$522$		0			0
$523$	−17486.4			−1.46200			−0.731000	−	0.682377i	$-0.760946\pi$
$523$	−17486.4			−1.46200			−0.731000	+	0.682377i	$0.760946\pi$
$524$		0			0
$525$			2374.99i			0.197434i
$526$		0			0
$527$	−15675.6	−	716.146i	−1.29571	−	0.0591951i
$528$		0			0
$529$			12123.8i			0.996452i
$530$		0			0
$531$	−11216.9			−0.916711
$532$		0			0
$533$	−554.880	+	554.880i	−0.0450929	+	0.0450929i
$534$		0			0
$535$			14845.7i			1.19969i
$536$		0			0
$537$	−9645.23	+	9645.23i	−0.775088	+	0.775088i
$538$		0			0
$539$	14010.3	+	14010.3i	1.11960	+	1.11960i
$540$		0			0
$541$	−8874.41	−	8874.41i	−0.705251	−	0.705251i	0.260282	−	0.965533i	$-0.416184\pi$
$541$	−8874.41	−	8874.41i	−0.705251	−	0.705251i	−0.965533	+	0.260282i	$0.916184\pi$
$542$		0			0
$543$	12171.7			0.961945
$544$		0			0
$545$	2211.14			0.173789
$546$		0			0
$547$	17460.3	+	17460.3i	1.36480	+	1.36480i	0.867681	+	0.497121i	$0.165610\pi$
$547$	17460.3	+	17460.3i	1.36480	+	1.36480i	0.497121	+	0.867681i	$0.334390\pi$
$548$		0			0
$549$	5241.61	+	5241.61i	0.407480	+	0.407480i
$550$		0			0
$551$	15260.8	−	15260.8i	1.17991	−	1.17991i
$552$		0			0
$553$		−	1642.31i		−	0.126290i
$554$		0			0
$555$	−3821.53	+	3821.53i	−0.292279	+	0.292279i
$556$		0			0
$557$	1130.02			0.0859616			0.0429808	−	0.999076i	$-0.486315\pi$
$557$	1130.02			0.0859616			0.0429808	+	0.999076i	$0.486315\pi$
$558$		0			0
$559$		−	1923.46i		−	0.145534i
$560$		0			0
$561$	−32052.8	+	29252.1i	−2.41225	+	2.20147i
$562$		0			0
$563$		−	14758.6i		−	1.10480i	−0.833579	−	0.552400i	$-0.813712\pi$
$563$		−	14758.6i		−	1.10480i	0.833579	−	0.552400i	$-0.186288\pi$
$564$		0			0
$565$	15686.5			1.16803
$566$		0			0
$567$	−38715.5	+	38715.5i	−2.86754	+	2.86754i
$568$		0			0
$569$		−	22712.9i		−	1.67341i	−0.547651	−	0.836707i	$-0.684478\pi$
$569$		−	22712.9i		−	1.67341i	0.547651	−	0.836707i	$-0.315522\pi$
$570$		0			0
$571$	1704.72	−	1704.72i	0.124939	−	0.124939i	−0.641872	−	0.766811i	$-0.721842\pi$
$571$	1704.72	−	1704.72i	0.124939	−	0.124939i	0.766811	+	0.641872i	$0.221842\pi$
$572$		0			0
$573$	31025.3	+	31025.3i	2.26196	+	2.26196i
$574$		0			0
$575$	44.0295	+	44.0295i	0.00319332	+	0.00319332i
$576$		0			0
$577$	−7432.70			−0.536269			−0.268135	−	0.963381i	$-0.586407\pi$
$577$	−7432.70			−0.536269			−0.268135	+	0.963381i	$0.586407\pi$
$578$		0			0
$579$	24421.1			1.75286
$580$		0			0
$581$	−11976.9	−	11976.9i	−0.855222	−	0.855222i
$582$		0			0
$583$	−9341.58	−	9341.58i	−0.663617	−	0.663617i
$584$		0			0
$585$	−5609.89	+	5609.89i	−0.396479	+	0.396479i
$586$		0			0
$587$		−	18608.6i		−	1.30845i	−0.756299	−	0.654226i	$-0.772995\pi$
$587$		−	18608.6i		−	1.30845i	0.756299	−	0.654226i	$-0.227005\pi$
$588$		0			0
$589$	−17173.3	+	17173.3i	−1.20138	+	1.20138i
$590$		0			0
$591$	397.250			0.0276492
$592$		0			0
$593$			13926.8i			0.964423i	0.876055	+	0.482212i	$0.160166\pi$
$593$			13926.8i			0.964423i	−0.876055	+	0.482212i	$0.839834\pi$
$594$		0			0
$595$	15378.7	−	14034.9i	1.05960	−	0.967017i
$596$		0			0
$597$		−	37021.8i		−	2.53803i
$598$		0			0
$599$	4893.34			0.333784			0.166892	−	0.985975i	$-0.446627\pi$
$599$	4893.34			0.333784			0.166892	+	0.985975i	$0.446627\pi$
$600$		0			0
$601$	−5592.07	+	5592.07i	−0.379543	+	0.379543i	−0.870937	−	0.491394i	$-0.836487\pi$
$601$	−5592.07	+	5592.07i	−0.379543	+	0.379543i	0.491394	+	0.870937i	$0.336487\pi$
$602$		0			0
$603$			20126.8i			1.35925i
$604$		0			0
$605$	21921.5	−	21921.5i	1.47312	−	1.47312i
$606$		0			0
$607$	−522.294	−	522.294i	−0.0349246	−	0.0349246i	0.689429	−	0.724353i	$-0.257862\pi$
$607$	−522.294	−	522.294i	−0.0349246	−	0.0349246i	−0.724353	+	0.689429i	$0.757862\pi$
$608$		0			0
$609$	35252.5	+	35252.5i	2.34565	+	2.34565i
$610$		0			0
$611$	−2457.56			−0.162721
$612$		0			0
$613$	−27223.3			−1.79370			−0.896851	−	0.442333i	$-0.854151\pi$
$613$	−27223.3			−1.79370			−0.896851	+	0.442333i	$0.854151\pi$
$614$		0			0
$615$	−6323.05	−	6323.05i	−0.414585	−	0.414585i
$616$		0			0
$617$	−8374.06	−	8374.06i	−0.546397	−	0.546397i	0.379000	−	0.925397i	$-0.376268\pi$
$617$	−8374.06	−	8374.06i	−0.546397	−	0.546397i	−0.925397	+	0.379000i	$0.876268\pi$
$618$		0			0
$619$	4543.28	−	4543.28i	0.295008	−	0.295008i	−0.544047	−	0.839055i	$-0.683109\pi$
$619$	4543.28	−	4543.28i	0.295008	−	0.295008i	0.839055	+	0.544047i	$0.183109\pi$
$620$		0			0
$621$			2681.45i			0.173273i
$622$		0			0
$623$	−11913.5	+	11913.5i	−0.766136	+	0.766136i
$624$		0			0
$625$	16719.8			1.07007
$626$		0			0
$627$			67162.2i			4.27783i
$628$		0			0
$629$	3335.46	+	152.382i	0.211437	+	0.00965960i
$630$		0			0
$631$			24477.7i			1.54428i	0.635453	+	0.772140i	$0.280813\pi$
$631$			24477.7i			1.54428i	−0.635453	+	0.772140i	$0.719187\pi$
$632$		0			0
$633$	−15280.4			−0.959464
$634$		0			0
$635$	−15831.5	+	15831.5i	−0.989374	+	0.989374i
$636$		0			0
$637$		−	3117.34i		−	0.193899i
$638$		0			0
$639$	33278.0	−	33278.0i	2.06018	−	2.06018i
$640$		0			0
$641$	−19778.0	−	19778.0i	−1.21870	−	1.21870i	−0.968089	−	0.250607i	$-0.919370\pi$
$641$	−19778.0	−	19778.0i	−1.21870	−	1.21870i	−0.250607	−	0.968089i	$-0.580630\pi$
$642$		0			0
$643$	4303.81	+	4303.81i	0.263959	+	0.263959i	0.826660	−	0.562701i	$-0.190238\pi$
$643$	4303.81	+	4303.81i	0.263959	+	0.263959i	−0.562701	+	0.826660i	$0.690238\pi$
$644$		0			0
$645$	21918.5			1.33805
$646$		0			0
$647$	18583.9			1.12923			0.564613	−	0.825356i	$-0.309026\pi$
$647$	18583.9			1.12923			0.564613	+	0.825356i	$0.309026\pi$
$648$		0			0
$649$	−7304.16	−	7304.16i	−0.441777	−	0.441777i
$650$		0			0
$651$	−39670.4	−	39670.4i	−2.38834	−	2.38834i
$652$		0			0
$653$	17900.9	−	17900.9i	1.07277	−	1.07277i	0.0756324	−	0.997136i	$-0.475902\pi$
$653$	17900.9	−	17900.9i	1.07277	−	1.07277i	0.997136	−	0.0756324i	$-0.0240976\pi$
$654$		0			0
$655$		−	31105.5i		−	1.85556i
$656$		0			0
$657$	−6250.64	+	6250.64i	−0.371173	+	0.371173i
$658$		0			0
$659$	−4948.03			−0.292485			−0.146243	−	0.989249i	$-0.546718\pi$
$659$	−4948.03			−0.292485			−0.146243	+	0.989249i	$0.546718\pi$
$660$		0			0
$661$		−	1373.95i		−	0.0808476i	−0.999183	−	0.0404238i	$-0.987129\pi$
$661$		−	1373.95i		−	0.0808476i	0.999183	−	0.0404238i	$-0.0128708\pi$
$662$		0			0
$663$	6820.26	+	311.587i	0.399513	+	0.0182520i
$664$		0			0
$665$		−	32223.9i		−	1.87908i
$666$		0			0
$667$	1307.08			0.0758777
$668$		0			0
$669$	−34581.5	+	34581.5i	−1.99850	+	1.99850i
$670$		0			0
$671$			6826.39i			0.392742i
$672$		0			0
$673$	−4178.72	+	4178.72i	−0.239343	+	0.239343i	−0.816578	−	0.577235i	$-0.804132\pi$
$673$	−4178.72	+	4178.72i	−0.239343	+	0.239343i	0.577235	+	0.816578i	$0.304132\pi$
$674$		0			0
$675$	−2735.01	−	2735.01i	−0.155956	−	0.155956i
$676$		0			0
$677$	11478.1	+	11478.1i	0.651610	+	0.651610i	0.953381	−	0.301770i	$-0.0975776\pi$
$677$	11478.1	+	11478.1i	0.651610	+	0.651610i	−0.301770	+	0.953381i	$0.597578\pi$
$678$		0			0
$679$	2103.81			0.118905
$680$		0			0
$681$	−18621.8			−1.04786
$682$		0			0
$683$	−1275.88	−	1275.88i	−0.0714792	−	0.0714792i	0.670463	−	0.741943i	$-0.266095\pi$
$683$	−1275.88	−	1275.88i	−0.0714792	−	0.0714792i	−0.741943	+	0.670463i	$0.766095\pi$
$684$		0			0
$685$	−12976.3	−	12976.3i	−0.723794	−	0.723794i
$686$		0			0
$687$	−25039.6	+	25039.6i	−1.39057	+	1.39057i
$688$		0			0
$689$			2078.53i			0.114929i
$690$		0			0
$691$	6819.93	−	6819.93i	0.375459	−	0.375459i	−0.494002	−	0.869461i	$-0.664466\pi$
$691$	6819.93	−	6819.93i	0.375459	−	0.375459i	0.869461	+	0.494002i	$0.164466\pi$
$692$		0			0
$693$	−111381.			−6.10537
$694$		0			0
$695$		−	12583.8i		−	0.686809i
$696$		0			0
$697$	−252.130	+	5518.82i	−0.0137017	+	0.299914i
$698$		0			0
$699$		−	33528.8i		−	1.81427i
$700$		0			0
$701$	29610.2			1.59538			0.797689	−	0.603069i	$-0.206056\pi$
$701$	29610.2			1.59538			0.797689	+	0.603069i	$0.206056\pi$
$702$		0			0
$703$	3654.15	−	3654.15i	0.196044	−	0.196044i
$704$		0			0
$705$		−	28004.8i		−	1.49606i
$706$		0			0
$707$	12881.4	−	12881.4i	0.685225	−	0.685225i
$708$		0			0
$709$	−5734.76	−	5734.76i	−0.303771	−	0.303771i	0.538717	−	0.842487i	$-0.318909\pi$
$709$	−5734.76	−	5734.76i	−0.303771	−	0.303771i	−0.842487	+	0.538717i	$0.818909\pi$
$710$		0			0
$711$	3115.37	+	3115.37i	0.164325	+	0.164325i
$712$		0			0
$713$	−1470.89			−0.0772584
$714$		0			0
$715$	−7306.02			−0.382139
$716$		0			0
$717$	40105.7	+	40105.7i	2.08895	+	2.08895i
$718$		0			0
$719$	14823.8	+	14823.8i	0.768892	+	0.768892i	0.977911	−	0.209020i	$-0.0670273\pi$
$719$	14823.8	+	14823.8i	0.768892	+	0.768892i	−0.209020	+	0.977911i	$0.567027\pi$
$720$		0			0
$721$	32327.0	−	32327.0i	1.66979	−	1.66979i
$722$		0			0
$723$		−	43425.9i		−	2.23378i
$724$		0			0
$725$	−1333.19	+	1333.19i	−0.0682943	+	0.0682943i
$726$		0			0
$727$	35728.7			1.82270			0.911351	−	0.411630i	$-0.135040\pi$
$727$	35728.7			1.82270			0.911351	+	0.411630i	$0.135040\pi$
$728$		0			0
$729$		−	39074.8i		−	1.98521i
$730$		0			0
$731$	−9128.34	−	10002.3i	−0.461866	−	0.506087i
$732$		0			0
$733$			22697.4i			1.14372i	0.820350	+	0.571861i	$0.193778\pi$
$733$			22697.4i			1.14372i	−0.820350	+	0.571861i	$0.806222\pi$
$734$		0			0
$735$	35523.1			1.78271
$736$		0			0
$737$	−13106.0	+	13106.0i	−0.655042	+	0.655042i
$738$		0			0
$739$		−	5393.01i		−	0.268451i	−0.990951	−	0.134225i	$-0.957145\pi$
$739$		−	5393.01i		−	0.268451i	0.990951	−	0.134225i	$-0.0428546\pi$
$740$		0			0
$741$	7471.90	−	7471.90i	0.370428	−	0.370428i
$742$		0			0
$743$	11263.3	+	11263.3i	0.556139	+	0.556139i	0.928206	−	0.372067i	$-0.121351\pi$
$743$	11263.3	+	11263.3i	0.556139	+	0.556139i	−0.372067	+	0.928206i	$0.621351\pi$
$744$		0			0
$745$	−1792.36	−	1792.36i	−0.0881435	−	0.0881435i
$746$		0			0
$747$	45438.8			2.22559
$748$		0			0
$749$	32791.8			1.59971
$750$		0			0
$751$	−11947.6	−	11947.6i	−0.580523	−	0.580523i	0.354524	−	0.935047i	$-0.384643\pi$
$751$	−11947.6	−	11947.6i	−0.580523	−	0.580523i	−0.935047	+	0.354524i	$0.884643\pi$
$752$		0			0
$753$	−28864.5	−	28864.5i	−1.39692	−	1.39692i
$754$		0			0
$755$	−1546.19	+	1546.19i	−0.0745317	+	0.0745317i
$756$		0			0
$757$		−	19109.1i		−	0.917479i	−0.888571	−	0.458740i	$-0.848301\pi$
$757$		−	19109.1i		−	0.917479i	0.888571	−	0.458740i	$-0.151699\pi$
$758$		0			0
$759$	−2876.22	+	2876.22i	−0.137550	+	0.137550i
$760$		0			0
$761$	30188.7			1.43803			0.719014	−	0.694996i	$-0.244594\pi$
$761$	30188.7			1.43803			0.719014	+	0.694996i	$0.244594\pi$
$762$		0			0
$763$		−	4884.05i		−	0.231736i
$764$		0			0
$765$	−2549.06	+	55795.8i	−0.120472	+	2.63699i
$766$		0			0
$767$			1625.20i			0.0765092i
$768$		0			0
$769$	−6235.12			−0.292385			−0.146193	−	0.989256i	$-0.546702\pi$
$769$	−6235.12			−0.292385			−0.146193	+	0.989256i	$0.546702\pi$
$770$		0			0
$771$	−42299.4	+	42299.4i	−1.97584	+	1.97584i
$772$		0			0
$773$			23158.1i			1.07754i	0.842452	+	0.538771i	$0.181111\pi$
$773$			23158.1i			1.07754i	−0.842452	+	0.538771i	$0.818889\pi$
$774$		0			0
$775$	1500.27	−	1500.27i	0.0695370	−	0.0695370i
$776$		0			0
$777$	8441.12	+	8441.12i	0.389734	+	0.389734i
$778$		0			0
$779$	6046.11	+	6046.11i	0.278080	+	0.278080i
$780$		0			0
$781$	43339.5			1.98567
$782$		0			0
$783$	−81192.7			−3.70573
$784$		0			0
$785$	13092.6	+	13092.6i	0.595280	+	0.595280i
$786$		0			0
$787$	−6239.79	−	6239.79i	−0.282623	−	0.282623i	0.551531	−	0.834154i	$-0.314044\pi$
$787$	−6239.79	−	6239.79i	−0.282623	−	0.282623i	−0.834154	+	0.551531i	$0.814044\pi$
$788$		0			0
$789$	34118.7	−	34118.7i	1.53949	−	1.53949i
$790$		0			0
$791$		−	34648.9i		−	1.55749i
$792$		0			0
$793$	759.446	−	759.446i	0.0340085	−	0.0340085i
$794$		0			0
$795$	−23685.6			−1.05666
$796$		0			0
$797$			24954.4i			1.10907i	0.832160	+	0.554535i	$0.187104\pi$
$797$			24954.4i			1.10907i	−0.832160	+	0.554535i	$0.812896\pi$
$798$		0			0
$799$	−12779.8	+	11663.1i	−0.565852	+	0.516408i
$800$		0			0
$801$		−	45198.2i		−	1.99376i
$802$		0			0
$803$	−8140.50			−0.357748
$804$		0			0
$805$	1379.99	−	1379.99i	0.0604201	−	0.0604201i
$806$		0			0
$807$		−	8709.58i		−	0.379916i
$808$		0			0
$809$	−27734.1	+	27734.1i	−1.20529	+	1.20529i	−0.232754	+	0.972536i	$0.574774\pi$
$809$	−27734.1	+	27734.1i	−1.20529	+	1.20529i	−0.972536	+	0.232754i	$0.925226\pi$
$810$		0			0
$811$	13250.0	+	13250.0i	0.573701	+	0.573701i	0.933161	−	0.359459i	$-0.117039\pi$
$811$	13250.0	+	13250.0i	0.573701	+	0.573701i	−0.359459	+	0.933161i	$0.617039\pi$
$812$		0			0
$813$	−50675.7	−	50675.7i	−2.18607	−	2.18607i
$814$		0			0
$815$	6031.98			0.259253
$816$		0			0
$817$	−20958.5			−0.897485
$818$		0			0
$819$	12391.3	+	12391.3i	0.528679	+	0.528679i
$820$		0			0
$821$	9388.24	+	9388.24i	0.399089	+	0.399089i	0.877912	−	0.478823i	$-0.158936\pi$
$821$	9388.24	+	9388.24i	0.399089	+	0.399089i	−0.478823	+	0.877912i	$0.658936\pi$
$822$		0			0
$823$	−20495.3	+	20495.3i	−0.868069	+	0.868069i	−0.992258	−	0.124190i	$-0.960367\pi$
$823$	−20495.3	+	20495.3i	−0.868069	+	0.868069i	0.124190	+	0.992258i	$0.460367\pi$
$824$		0			0
$825$		−	5867.33i		−	0.247605i
$826$		0			0
$827$	616.048	−	616.048i	0.0259034	−	0.0259034i	−0.694036	−	0.719940i	$-0.744169\pi$
$827$	616.048	−	616.048i	0.0259034	−	0.0259034i	0.719940	+	0.694036i	$0.244169\pi$
$828$		0			0
$829$	25090.6			1.05118			0.525592	−	0.850737i	$-0.323844\pi$
$829$	25090.6			1.05118			0.525592	+	0.850737i	$0.323844\pi$
$830$		0			0
$831$			28329.8i			1.18261i
$832$		0			0
$833$	−14794.2	−	16210.7i	−0.615353	−	0.674270i
$834$		0			0
$835$			17019.1i			0.705353i
$836$		0			0
$837$	91368.0			3.77316
$838$		0			0
$839$	−3046.75	+	3046.75i	−0.125370	+	0.125370i	−0.767008	−	0.641638i	$-0.778255\pi$
$839$	−3046.75	+	3046.75i	−0.125370	+	0.125370i	0.641638	+	0.767008i	$0.278255\pi$
$840$		0			0
$841$			15188.7i			0.622768i
$842$		0			0
$843$	−32813.0	+	32813.0i	−1.34061	+	1.34061i
$844$		0			0
$845$	−17202.4	−	17202.4i	−0.700332	−	0.700332i
$846$		0			0
$847$	−48421.0	−	48421.0i	−1.96430	−	1.96430i
$848$		0			0
$849$	85396.7			3.45207
$850$		0			0
$851$	312.978			0.0126072
$852$		0			0
$853$	−24238.4	−	24238.4i	−0.972928	−	0.972928i	0.0267155	−	0.999643i	$-0.491495\pi$
$853$	−24238.4	−	24238.4i	−0.972928	−	0.972928i	−0.999643	+	0.0267155i	$0.991495\pi$
$854$		0			0
$855$	61126.8	+	61126.8i	2.44502	+	2.44502i
$856$		0			0
$857$	−2862.79	+	2862.79i	−0.114109	+	0.114109i	−0.761856	−	0.647747i	$-0.775711\pi$
$857$	−2862.79	+	2862.79i	−0.114109	+	0.114109i	0.647747	+	0.761856i	$0.275711\pi$
$858$		0			0
$859$		−	5033.78i		−	0.199942i	−0.994990	−	0.0999712i	$-0.968125\pi$
$859$		−	5033.78i		−	0.199942i	0.994990	−	0.0999712i	$-0.0318751\pi$
$860$		0			0
$861$	−13966.6	+	13966.6i	−0.552822	+	0.552822i
$862$		0			0
$863$	23465.3			0.925572			0.462786	−	0.886470i	$-0.346850\pi$
$863$	23465.3			0.925572			0.462786	+	0.886470i	$0.346850\pi$
$864$		0			0
$865$			9398.36i			0.369426i
$866$		0			0
$867$	36945.3	−	30747.2i	1.44721	−	1.20442i
$868$		0			0
$869$			4057.29i			0.158382i
$870$		0			0
$871$	2916.13			0.113443
$872$		0			0
$873$	−3990.80	+	3990.80i	−0.154717	+	0.154717i
$874$		0			0
$875$		−	34314.7i		−	1.32577i
$876$		0			0
$877$	16696.5	−	16696.5i	0.642873	−	0.642873i	−0.308388	−	0.951261i	$-0.599789\pi$
$877$	16696.5	−	16696.5i	0.642873	−	0.642873i	0.951261	+	0.308388i	$0.0997895\pi$
$878$		0			0
$879$	−57580.6	−	57580.6i	−2.20949	−	2.20949i
$880$		0			0
$881$	−20154.2	−	20154.2i	−0.770728	−	0.770728i	0.207506	−	0.978234i	$-0.433465\pi$
$881$	−20154.2	−	20154.2i	−0.770728	−	0.770728i	−0.978234	+	0.207506i	$0.933465\pi$
$882$		0			0
$883$	39621.1			1.51003			0.755014	−	0.655709i	$-0.227630\pi$
$883$	39621.1			1.51003			0.755014	+	0.655709i	$0.227630\pi$
$884$		0			0
$885$	−18519.7			−0.703427
$886$		0			0
$887$	−6952.20	−	6952.20i	−0.263170	−	0.263170i	0.563171	−	0.826341i	$-0.309581\pi$
$887$	−6952.20	−	6952.20i	−0.263170	−	0.263170i	−0.826341	+	0.563171i	$0.809581\pi$
$888$		0			0
$889$	34969.1	+	34969.1i	1.31926	+	1.31926i
$890$		0			0
$891$	95645.5	−	95645.5i	3.59623	−	3.59623i
$892$		0			0
$893$			26778.2i			1.00347i
$894$		0			0
$895$	−11432.6	+	11432.6i	−0.426983	+	0.426983i
$896$		0			0
$897$	639.968			0.0238215
$898$		0			0
$899$		−	44537.7i		−	1.65230i
$900$		0			0
$901$	9864.27	+	10808.7i	0.364735	+	0.399657i
$902$		0			0
$903$		−	48414.3i		−	1.78420i
$904$		0			0
$905$	14427.2			0.529919
$906$		0			0
$907$	4287.83	−	4287.83i	0.156974	−	0.156974i	−0.624251	−	0.781224i	$-0.714596\pi$
$907$	4287.83	−	4287.83i	0.156974	−	0.156974i	0.781224	+	0.624251i	$0.214596\pi$
$908$		0			0
$909$			48870.4i			1.78320i
$910$		0			0
$911$	−5371.57	+	5371.57i	−0.195355	+	0.195355i	−0.798005	−	0.602651i	$-0.794111\pi$
$911$	−5371.57	+	5371.57i	−0.195355	+	0.195355i	0.602651	+	0.798005i	$0.294111\pi$
$912$		0			0
$913$	29588.5	+	29588.5i	1.07255	+	1.07255i
$914$		0			0
$915$	8654.15	+	8654.15i	0.312675	+	0.312675i
$916$		0			0
$917$	−68707.0			−2.47427
$918$		0			0
$919$	36363.0			1.30523			0.652614	−	0.757691i	$-0.273673\pi$
$919$	36363.0			1.30523			0.652614	+	0.757691i	$0.273673\pi$
$920$		0			0
$921$	−61686.9	−	61686.9i	−2.20701	−	2.20701i
$922$		0			0
$923$	−4821.59	−	4821.59i	−0.171944	−	0.171944i
$924$		0			0
$925$	−319.229	+	319.229i	−0.0113472	+	0.0113472i
$926$		0			0
$927$			122645.i			4.34540i
$928$		0			0
$929$	−33206.6	+	33206.6i	−1.17274	+	1.17274i	−0.191185	+	0.981554i	$0.561233\pi$
$929$	−33206.6	+	33206.6i	−1.17274	+	1.17274i	−0.981554	+	0.191185i	$0.938767\pi$
$930$		0			0
$931$	−33967.3			−1.19574
$932$		0			0
$933$		−	72940.0i		−	2.55943i
$934$		0			0
$935$	−37992.6	+	34672.8i	−1.32887	+	1.21275i
$936$		0			0
$937$		−	36031.2i		−	1.25623i	−0.778120	−	0.628116i	$-0.783826\pi$
$937$		−	36031.2i		−	1.25623i	0.778120	−	0.628116i	$-0.216174\pi$
$938$		0			0
$939$	−36915.4			−1.28295
$940$		0			0
$941$	−1667.19	+	1667.19i	−0.0577565	+	0.0577565i	−0.735395	−	0.677639i	$-0.763003\pi$
$941$	−1667.19	+	1667.19i	−0.0577565	+	0.0577565i	0.677639	+	0.735395i	$0.263003\pi$
$942$		0			0
$943$			517.849i			0.0178828i
$944$		0			0
$945$	−85721.3	+	85721.3i	−2.95081	+	2.95081i
$946$		0			0
$947$	−7453.66	−	7453.66i	−0.255767	−	0.255767i	0.567563	−	0.823330i	$-0.307886\pi$
$947$	−7453.66	−	7453.66i	−0.255767	−	0.255767i	−0.823330	+	0.567563i	$0.807886\pi$
$948$		0			0
$949$	905.643	+	905.643i	0.0309783	+	0.0309783i
$950$		0			0
$951$	33647.7			1.14732
$952$		0			0
$953$	36771.6			1.24990			0.624948	−	0.780667i	$-0.285120\pi$
$953$	36771.6			1.24990			0.624948	+	0.780667i	$0.285120\pi$
$954$		0			0
$955$	36774.7	+	36774.7i	1.24607	+	1.24607i
$956$		0			0
$957$	−87090.2	−	87090.2i	−2.94172	−	2.94172i
$958$		0			0
$959$	−28662.5	+	28662.5i	−0.965131	+	0.965131i
$960$		0			0
$961$			20328.2i			0.682361i
$962$		0			0
$963$	−62204.0	+	62204.0i	−2.08151	+	2.08151i
$964$		0			0
$965$	28946.6			0.965620
$966$		0			0
$967$		−	38981.8i		−	1.29635i	−0.761492	−	0.648174i	$-0.775533\pi$
$967$		−	38981.8i		−	1.29635i	0.761492	−	0.648174i	$-0.224467\pi$
$968$		0			0
$969$	3395.13	−	74315.3i	0.112557	−	2.46373i
$970$		0			0
$971$			13355.0i			0.441383i	0.975344	+	0.220692i	$0.0708315\pi$
$971$			13355.0i			0.441383i	−0.975344	+	0.220692i	$0.929169\pi$
$972$		0			0
$973$	−27795.6			−0.915814
$974$		0			0
$975$	−652.750	+	652.750i	−0.0214408	+	0.0214408i
$976$		0			0
$977$		−	9868.81i		−	0.323164i	−0.986859	−	0.161582i	$-0.948340\pi$
$977$		−	9868.81i		−	0.323164i	0.986859	−	0.161582i	$-0.0516596\pi$
$978$		0			0
$979$	29431.9	−	29431.9i	0.960824	−	0.960824i
$980$		0			0
$981$	9264.75	+	9264.75i	0.301530	+	0.301530i
$982$		0			0
$983$	−28198.0	−	28198.0i	−0.914930	−	0.914930i	0.0817252	−	0.996655i	$-0.473957\pi$
$983$	−28198.0	−	28198.0i	−0.914930	−	0.914930i	−0.996655	+	0.0817252i	$0.973957\pi$
$984$		0			0
$985$	470.866			0.0152315
$986$		0			0
$987$	−61858.0			−1.99489
$988$		0			0
$989$	−897.547	−	897.547i	−0.0288578	−	0.0288578i
$990$		0			0
$991$	−37143.8	−	37143.8i	−1.19063	−	1.19063i	−0.976891	−	0.213736i	$-0.931437\pi$
$991$	−37143.8	−	37143.8i	−1.19063	−	1.19063i	−0.213736	−	0.976891i	$-0.568563\pi$
$992$		0			0
$993$	−61852.8	+	61852.8i	−1.97668	+	1.97668i
$994$		0			0
$995$		−	43882.4i		−	1.39816i
$996$		0			0
$997$	37045.7	−	37045.7i	1.17678	−	1.17678i	0.196217	−	0.980560i	$-0.437134\pi$
$997$	37045.7	−	37045.7i	1.17678	−	1.17678i	0.980560	−	0.196217i	$-0.0628659\pi$
$998$		0			0
$999$	−19441.4			−0.615714

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	272.4.o.g.81.7		14
4.3	odd	2		136.4.k.a.81.1	✓	14
17.4	even	4	inner	272.4.o.g.225.7		14
68.15	odd	8		2312.4.a.m.1.14		14
68.19	odd	8		2312.4.a.m.1.1		14
68.55	odd	4		136.4.k.a.89.1	yes	14

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.k.a.81.1	✓	14	4.3	odd	2
136.4.k.a.89.1	yes	14	68.55	odd	4
272.4.o.g.81.7		14	1.1	even	1	trivial
272.4.o.g.225.7		14	17.4	even	4	inner
2312.4.a.m.1.1		14	68.19	odd	8
2312.4.a.m.1.14		14	68.15	odd	8

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 272 = 2^{4} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	272.o (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(6.91794 + 6.91794i) q^{3} +(8.19992 + 8.19992i) q^{5} +(18.1123 - 18.1123i) q^{7} +68.7157i q^{9} +O(q^{10})\)	\(q+(6.91794 + 6.91794i) q^{3} +(8.19992 + 8.19992i) q^{5} +(18.1123 - 18.1123i) q^{7} +68.7157i q^{9} +(-44.7458 + 44.7458i) q^{11} +9.95609 q^{13} +113.453i q^{15} +(51.7734 - 47.2495i) q^{17} -108.484i q^{19} +250.599 q^{21} +(4.64582 - 4.64582i) q^{23} +9.47723i q^{25} +(-288.587 + 288.587i) q^{27} +(140.673 + 140.673i) q^{29} +(-158.302 - 158.302i) q^{31} -619.098 q^{33} +297.038 q^{35} +(33.6838 + 33.6838i) q^{37} +(68.8756 + 68.8756i) q^{39} +(-55.7328 + 55.7328i) q^{41} -193.194i q^{43} +(-563.463 + 563.463i) q^{45} -246.840 q^{47} -313.109i q^{49} +(685.034 + 31.2962i) q^{51} +208.770i q^{53} -733.824 q^{55} +(750.486 - 750.486i) q^{57} +163.237i q^{59} +(76.2796 - 76.2796i) q^{61} +(1244.60 + 1244.60i) q^{63} +(81.6391 + 81.6391i) q^{65} +292.899 q^{67} +64.2790 q^{69} +(-484.285 - 484.285i) q^{71} +(90.9637 + 90.9637i) q^{73} +(-65.5629 + 65.5629i) q^{75} +1620.90i q^{77} +(45.3370 - 45.3370i) q^{79} -2137.53 q^{81} -661.257i q^{83} +(811.979 + 37.0957i) q^{85} +1946.33i q^{87} -657.756 q^{89} +(180.327 - 180.327i) q^{91} -2190.25i q^{93} +(889.560 - 889.560i) q^{95} +(58.0769 + 58.0769i) q^{97} +(-3074.74 - 3074.74i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 14 q + 4 q^{3} + 8 q^{5} + 10 q^{7}+O(q^{10}) \) 14 * q + 4 * q^3 + 8 * q^5 + 10 * q^7	\( 14 q + 4 q^{3} + 8 q^{5} + 10 q^{7} - 16 q^{11} + 72 q^{13} - 40 q^{17} - 60 q^{21} - 246 q^{23} - 572 q^{27} + 260 q^{29} - 566 q^{31} - 904 q^{33} + 804 q^{35} - 28 q^{37} - 476 q^{39} - 786 q^{41} - 604 q^{45} - 448 q^{47} + 380 q^{51} - 1612 q^{55} + 1004 q^{57} + 660 q^{61} + 2250 q^{63} - 796 q^{65} + 284 q^{67} - 2532 q^{69} - 326 q^{71} + 730 q^{73} + 864 q^{75} - 342 q^{79} - 950 q^{81} + 4148 q^{85} - 4516 q^{89} + 3112 q^{91} - 3356 q^{95} - 1194 q^{97} - 2876 q^{99}+O(q^{100}) \) 14 * q + 4 * q^3 + 8 * q^5 + 10 * q^7 - 16 * q^11 + 72 * q^13 - 40 * q^17 - 60 * q^21 - 246 * q^23 - 572 * q^27 + 260 * q^29 - 566 * q^31 - 904 * q^33 + 804 * q^35 - 28 * q^37 - 476 * q^39 - 786 * q^41 - 604 * q^45 - 448 * q^47 + 380 * q^51 - 1612 * q^55 + 1004 * q^57 + 660 * q^61 + 2250 * q^63 - 796 * q^65 + 284 * q^67 - 2532 * q^69 - 326 * q^71 + 730 * q^73 + 864 * q^75 - 342 * q^79 - 950 * q^81 + 4148 * q^85 - 4516 * q^89 + 3112 * q^91 - 3356 * q^95 - 1194 * q^97 - 2876 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more