LMFDB - Embedded newform 136.4.k.a.89.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [136,4,Mod(81,136)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(136, 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("136.81");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 136 = 2^{3} \cdot 17 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	136.k (of order $4$, degree $2$, 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:	$8.02425976078$
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:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			89.1
Root			$-6.91794i$ of defining polynomial
Character	$\chi$	$=$	136.89
Dual form			136.4.k.a.81.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-6.91794 + 6.91794i) q^{3} +(8.19992 - 8.19992i) q^{5} +(-18.1123 - 18.1123i) q^{7} -68.7157i q^{9} +O(q^{10})$	\(q+(-6.91794 + 6.91794i) q^{3} +(8.19992 - 8.19992i) q^{5} +(-18.1123 - 18.1123i) q^{7} -68.7157i q^{9} +(44.7458 + 44.7458i) q^{11} +9.95609 q^{13} +113.453i q^{15} +(51.7734 + 47.2495i) q^{17} -108.484i q^{19} +250.599 q^{21} +(-4.64582 - 4.64582i) q^{23} -9.47723i q^{25} +(288.587 + 288.587i) q^{27} +(140.673 - 140.673i) q^{29} +(158.302 - 158.302i) q^{31} -619.098 q^{33} -297.038 q^{35} +(33.6838 - 33.6838i) q^{37} +(-68.8756 + 68.8756i) q^{39} +(-55.7328 - 55.7328i) q^{41} -193.194i q^{43} +(-563.463 - 563.463i) q^{45} +246.840 q^{47} +313.109i q^{49} +(-685.034 + 31.2962i) q^{51} -208.770i q^{53} +733.824 q^{55} +(750.486 + 750.486i) q^{57} +163.237i q^{59} +(76.2796 + 76.2796i) q^{61} +(-1244.60 + 1244.60i) q^{63} +(81.6391 - 81.6391i) q^{65} -292.899 q^{67} +64.2790 q^{69} +(484.285 - 484.285i) q^{71} +(90.9637 - 90.9637i) q^{73} +(65.5629 + 65.5629i) q^{75} -1620.90i q^{77} +(-45.3370 - 45.3370i) q^{79} -2137.53 q^{81} -661.257i q^{83} +(811.979 - 37.0957i) q^{85} +1946.33i q^{87} -657.756 q^{89} +(-180.327 - 180.327i) q^{91} +2190.25i q^{93} +(-889.560 - 889.560i) q^{95} +(58.0769 - 58.0769i) q^{97} +(3074.74 - 3074.74i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/136\mathbb{Z}\right)^\times$.

$n$	$69$	$103$	$105$
$\chi(n)$	$1$	$1$	$e\left(\frac{3}{4}\right)$

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	−0.427201	+	0.904157i	$0.640500\pi$
$3$	−6.91794	+	6.91794i	−1.33136	+	1.33136i	−0.904157	+	0.427201i	$0.859500\pi$
$4$		0			0
$5$	8.19992	−	8.19992i	0.733423	−	0.733423i	−0.237873	−	0.971296i	$-0.576450\pi$
$5$	8.19992	−	8.19992i	0.733423	−	0.733423i	0.971296	+	0.237873i	$0.0764503\pi$
$6$		0			0
$7$	−18.1123	−	18.1123i	−0.977971	−	0.977971i	0.0217920	−	0.999763i	$-0.493063\pi$
$7$	−18.1123	−	18.1123i	−0.977971	−	0.977971i	−0.999763	+	0.0217920i	$0.993063\pi$
$8$		0			0
$9$		−	68.7157i		−	2.54503i
$10$		0			0
$11$	44.7458	+	44.7458i	1.22649	+	1.22649i	0.965283	+	0.261206i	$0.0841201\pi$
$11$	44.7458	+	44.7458i	1.22649	+	1.22649i	0.261206	+	0.965283i	$0.415880\pi$
$12$		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	$-0.259481\pi$
$23$	−4.64582	−	4.64582i	−0.0421183	−	0.0421183i	−0.727852	+	0.685734i	$0.759481\pi$
$24$		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	−0.0947342	−	0.995503i	$-0.530200\pi$
$29$	140.673	−	140.673i	0.900768	−	0.900768i	0.995503	+	0.0947342i	$0.0302001\pi$
$30$		0			0
$31$	158.302	−	158.302i	0.917159	−	0.917159i	−0.0796628	−	0.996822i	$-0.525384\pi$
$31$	158.302	−	158.302i	0.917159	−	0.917159i	0.996822	+	0.0796628i	$0.0253844\pi$
$32$		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	−0.628304	−	0.777968i	$-0.716251\pi$
$37$	33.6838	−	33.6838i	0.149664	−	0.149664i	0.777968	+	0.628304i	$0.216251\pi$
$38$		0			0
$39$	−68.8756	+	68.8756i	−0.282793	+	0.282793i
$40$		0			0
$41$	−55.7328	−	55.7328i	−0.212293	−	0.212293i	0.592948	−	0.805241i	$-0.297964\pi$
$41$	−55.7328	−	55.7328i	−0.212293	−	0.212293i	−0.805241	+	0.592948i	$0.797964\pi$
$42$		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	$-0.374879\pi$
$47$	246.840			0.766071			0.383036	+	0.923734i	$0.374879\pi$
$48$		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	$-0.912799\pi$
$53$		−	208.770i		−	0.541071i	0.962710	−	0.270535i	$-0.0872007\pi$
$54$		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	0.782615	−	0.622506i	$-0.213886\pi$
$61$	76.2796	+	76.2796i	0.160108	+	0.160108i	−0.622506	+	0.782615i	$0.713886\pi$
$62$		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	$-0.586045\pi$
$67$	−292.899			−0.534079			−0.267040	+	0.963686i	$0.586045\pi$
$68$		0			0
$69$	64.2790			0.112149
$70$		0			0
$71$	484.285	−	484.285i	0.809494	−	0.809494i	−0.175063	−	0.984557i	$-0.556013\pi$
$71$	484.285	−	484.285i	0.809494	−	0.809494i	0.984557	+	0.175063i	$0.0560129\pi$
$72$		0			0
$73$	90.9637	−	90.9637i	0.145842	−	0.145842i	−0.630415	−	0.776258i	$-0.717115\pi$
$73$	90.9637	−	90.9637i	0.145842	−	0.145842i	0.776258	+	0.630415i	$0.217115\pi$
$74$		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	$-0.264538\pi$
$79$	−45.3370	−	45.3370i	−0.0645672	−	0.0645672i	−0.738653	+	0.674086i	$0.764538\pi$
$80$		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	−0.676057	−	0.736849i	$-0.736313\pi$
$97$	58.0769	−	58.0769i	0.0607919	−	0.0607919i	0.736849	+	0.676057i	$0.236313\pi$
$98$		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	$-0.825649\pi$
$103$	−1784.81			−1.70741			−0.853703	+	0.520760i	$0.825649\pi$
$104$		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	$-0.946293\pi$
$107$	−905.237	+	905.237i	−0.817874	+	0.817874i	0.167926	+	0.985800i	$0.446293\pi$
$108$		0			0
$109$	134.827	+	134.827i	0.118478	+	0.118478i	0.763860	−	0.645382i	$-0.223302\pi$
$109$	134.827	+	134.827i	0.118478	+	0.118478i	−0.645382	+	0.763860i	$0.723302\pi$
$110$		0			0
$111$			466.045i			0.398513i
$112$		0			0
$113$	956.503	+	956.503i	0.796286	+	0.796286i	0.982508	−	0.186222i	$-0.0596244\pi$
$113$	956.503	+	956.503i	0.796286	+	0.796286i	−0.186222	+	0.982508i	$0.559624\pi$
$114$		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	0.316365	−	0.948638i	$-0.397538\pi$
$131$	1896.70	−	1896.70i	1.26500	−	1.26500i	0.948638	−	0.316365i	$-0.102462\pi$
$132$		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	$-0.642586\pi$
$139$	767.315	−	767.315i	0.468221	−	0.468221i	0.901338	+	0.433116i	$0.142586\pi$
$140$		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	$-0.289885\pi$
$163$	−367.807	−	367.807i	−0.176742	−	0.176742i	−0.789934	+	0.613192i	$0.789885\pi$
$164$		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	$-0.860435\pi$
$167$	−1037.76	+	1037.76i	−0.480864	+	0.480864i	0.424544	+	0.905407i	$0.360435\pi$
$168$		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	−0.569878	−	0.821729i	$-0.693010\pi$
$173$	573.077	−	573.077i	0.251851	−	0.251851i	0.821729	+	0.569878i	$0.193010\pi$
$174$		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	0.864278	−	0.503014i	$-0.167776\pi$
$181$	879.717	+	879.717i	0.361265	+	0.361265i	−0.503014	+	0.864278i	$0.667776\pi$
$182$		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	$-0.823091\pi$
$191$	−4484.76			−1.69898			−0.849492	+	0.527602i	$0.823091\pi$
$192$		0			0
$193$	1765.05	+	1765.05i	0.658297	+	0.658297i	0.954977	−	0.296680i	$-0.0958794\pi$
$193$	1765.05	+	1765.05i	0.658297	+	0.658297i	−0.296680	+	0.954977i	$0.595879\pi$
$194$		0			0
$195$			1129.55i			0.414814i
$196$		0			0
$197$	28.7116	+	28.7116i	0.0103838	+	0.0103838i	0.712280	−	0.701896i	$-0.247663\pi$
$197$	28.7116	+	28.7116i	0.0103838	+	0.0103838i	−0.701896	+	0.712280i	$0.747663\pi$
$198$		0			0
$199$	2675.79	−	2675.79i	0.953173	−	0.953173i	−0.0457787	−	0.998952i	$-0.514577\pi$
$199$	2675.79	−	2675.79i	0.953173	−	0.953173i	0.998952	+	0.0457787i	$0.0145769\pi$
$200$		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	$-0.167992\pi$
$211$	1104.40	+	1104.40i	0.360333	+	0.360333i	−0.503603	+	0.863935i	$0.667992\pi$
$212$		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	$-0.160240\pi$
$227$	1345.91	+	1345.91i	0.393530	+	0.393530i	−0.482414	+	0.875943i	$0.660240\pi$
$228$		0			0
$229$		−	3619.52i		−	1.04447i	−0.852801	−	0.522237i	$-0.825098\pi$
$229$		−	3619.52i		−	1.04447i	0.852801	−	0.522237i	$-0.174902\pi$
$230$		0			0
$231$	11213.3	+	11213.3i	3.19385	+	3.19385i
$232$		0			0
$233$	−2423.33	+	2423.33i	−0.681362	+	0.681362i	−0.960307	−	0.278945i	$-0.910015\pi$
$233$	−2423.33	+	2423.33i	−0.681362	+	0.681362i	0.278945	+	0.960307i	$0.410015\pi$
$234$		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	$-0.787090\pi$
$239$	−5797.35			−1.56904			−0.784518	+	0.620106i	$0.787090\pi$
$240$		0			0
$241$	−3138.64	+	3138.64i	−0.838912	+	0.838912i	−0.988716	−	0.149804i	$-0.952136\pi$
$241$	−3138.64	+	3138.64i	−0.838912	+	0.838912i	0.149804	+	0.988716i	$0.452136\pi$
$242$		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	$-0.324206\pi$
$251$	4172.42			1.04925			0.524623	+	0.851335i	$0.324206\pi$
$252$		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	$-0.733858\pi$
$257$		−	6114.45i		−	1.48408i	0.670354	−	0.742041i	$-0.266142\pi$
$258$		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	−0.774839	−	0.632159i	$-0.782169\pi$
$269$	−629.493	+	629.493i	−0.142680	+	0.142680i	0.632159	+	0.774839i	$0.282169\pi$
$270$		0			0
$271$	7325.27			1.64199			0.820993	−	0.570938i	$-0.193420\pi$
$271$	7325.27			1.64199			0.820993	+	0.570938i	$0.193420\pi$
$272$		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	−0.449262	−	0.893400i	$-0.648313\pi$
$277$	2047.56	−	2047.56i	0.444137	−	0.444137i	0.893400	+	0.449262i	$0.148313\pi$
$278$		0			0
$279$	−10877.9	−	10877.9i	−2.33420	−	2.33420i
$280$		0			0
$281$		−	4743.17i		−	1.00695i	−0.864009	−	0.503476i	$-0.832054\pi$
$281$		−	4743.17i		−	1.00695i	0.864009	−	0.503476i	$-0.167946\pi$
$282$		0			0
$283$	−6172.12	−	6172.12i	−1.29645	−	1.29645i	−0.930724	−	0.365723i	$-0.880822\pi$
$283$	−6172.12	−	6172.12i	−1.29645	−	1.29645i	−0.365723	−	0.930724i	$-0.619178\pi$
$284$		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	$-0.188992\pi$
$307$	8916.96			1.65771			0.828856	+	0.559462i	$0.188992\pi$
$308$		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	$-0.512119\pi$
$311$	5271.80	−	5271.80i	0.961210	−	0.961210i	0.999275	+	0.0380650i	$0.0121194\pi$
$312$		0			0
$313$	−2668.09	−	2668.09i	−0.481820	−	0.481820i	0.423893	−	0.905712i	$-0.360663\pi$
$313$	−2668.09	−	2668.09i	−0.481820	−	0.481820i	−0.905712	+	0.423893i	$0.860663\pi$
$314$		0			0
$315$			20411.2i			3.65092i
$316$		0			0
$317$	2431.92	+	2431.92i	0.430883	+	0.430883i	0.888929	−	0.458045i	$-0.151450\pi$
$317$	2431.92	+	2431.92i	0.430883	+	0.430883i	−0.458045	+	0.888929i	$0.651450\pi$
$318$		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	−0.871486	−	0.490420i	$-0.836843\pi$
$337$	−2357.46	+	2357.46i	−0.381066	+	0.381066i	0.490420	+	0.871486i	$0.336843\pi$
$338$		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	$-0.0379217\pi$
$347$	5649.82	+	5649.82i	0.874059	+	0.874059i	−0.118853	+	0.992912i	$0.537922\pi$
$348$		0			0
$349$			3588.04i			0.550326i	0.961398	+	0.275163i	$0.0887317\pi$
$349$			3588.04i			0.550326i	−0.961398	+	0.275163i	$0.911268\pi$
$350$		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	$-0.186375\pi$
$367$	1974.23	+	1974.23i	0.280801	+	0.280801i	−0.552628	+	0.833428i	$0.686375\pi$
$368$		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	$-0.853905\pi$
$379$	−3345.96	+	3345.96i	−0.453484	+	0.453484i	0.443025	+	0.896509i	$0.353905\pi$
$380$		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	$0.180691\pi$
$389$			8250.13i			1.07532i	−0.843163	+	0.537658i	$0.819309\pi$
$390$		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	0.103272	−	0.994653i	$-0.467069\pi$
$397$	−7050.98	−	7050.98i	−0.891381	−	0.891381i	−0.994653	+	0.103272i	$0.967069\pi$
$398$		0			0
$399$		−	27186.0i		−	3.41103i
$400$		0			0
$401$	−2335.54	−	2335.54i	−0.290851	−	0.290851i	0.546566	−	0.837416i	$-0.315935\pi$
$401$	−2335.54	−	2335.54i	−0.290851	−	0.290851i	−0.837416	+	0.546566i	$0.815935\pi$
$402$		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	$-0.0677315\pi$
$419$	6572.03	+	6572.03i	0.766264	+	0.766264i	−0.211183	+	0.977447i	$0.567731\pi$
$420$		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	$-0.277040\pi$
$431$	−1073.65	−	1073.65i	−0.119991	−	0.119991i	−0.764552	+	0.644562i	$0.777040\pi$
$432$		0			0
$433$		−	3810.92i		−	0.422959i	−0.977382	−	0.211479i	$-0.932172\pi$
$433$		−	3810.92i		−	0.422959i	0.977382	−	0.211479i	$-0.0678281\pi$
$434$		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	$-0.607315\pi$
$439$	5637.63	−	5637.63i	0.612915	−	0.612915i	0.943704	+	0.330790i	$0.107315\pi$
$440$		0			0
$441$	21515.5			2.32324
$442$		0			0
$443$	2524.64			0.270765			0.135383	−	0.990793i	$-0.456774\pi$
$443$	2524.64			0.270765			0.135383	+	0.990793i	$0.456774\pi$
$444$		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	0.692401	−	0.721513i	$-0.256553\pi$
$449$	−276.973	−	276.973i	−0.0291117	−	0.0291117i	−0.721513	+	0.692401i	$0.756553\pi$
$450$		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	$-0.875237\pi$
$457$		−	7463.84i		−	0.763990i	0.924164	−	0.381995i	$-0.124763\pi$
$458$		0			0
$459$	1305.54	+	28576.7i	0.132761	+	2.90598i
$460$		0			0
$461$			4409.10i			0.445449i	0.974881	+	0.222725i	$0.0714951\pi$
$461$			4409.10i			0.445449i	−0.974881	+	0.222725i	$0.928505\pi$
$462$		0			0
$463$	15821.7			1.58811			0.794055	−	0.607845i	$-0.207966\pi$
$463$	15821.7			1.58811			0.794055	+	0.607845i	$0.207966\pi$
$464$		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	−0.364838	+	0.931071i	$0.618876\pi$
$479$	−13585.6	+	13585.6i	−1.29591	+	1.29591i	−0.931071	+	0.364838i	$0.881124\pi$
$480$		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	$-0.339925\pi$
$487$	−4236.87	−	4236.87i	−0.394232	−	0.394232i	−0.876193	+	0.481961i	$0.839925\pi$
$488$		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	$-0.175801\pi$
$499$	3641.46	+	3641.46i	0.326682	+	0.326682i	−0.524642	+	0.851323i	$0.675801\pi$
$500$		0			0
$501$		−	14358.3i		−	1.28040i
$502$		0			0
$503$	10555.6	+	10555.6i	0.935684	+	0.935684i	0.998053	−	0.0623692i	$-0.0198656\pi$
$503$	10555.6	+	10555.6i	0.935684	+	0.935684i	−0.0623692	+	0.998053i	$0.519866\pi$
$504$		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	0.999685	−	0.0250980i	$-0.00798977\pi$
$521$	11589.8	+	11589.8i	0.974587	+	0.974587i	−0.0250980	+	0.999685i	$0.507990\pi$
$522$		0			0
$523$	17486.4			1.46200			0.731000	−	0.682377i	$-0.239054\pi$
$523$	17486.4			1.46200			0.731000	+	0.682377i	$0.239054\pi$
$524$		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	−0.965533	−	0.260282i	$-0.916184\pi$
$541$	−8874.41	+	8874.41i	−0.705251	+	0.705251i	0.260282	+	0.965533i	$0.416184\pi$
$542$		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	−0.497121	+	0.867681i	$0.665610\pi$
$547$	−17460.3	+	17460.3i	−1.36480	+	1.36480i	−0.867681	+	0.497121i	$0.834390\pi$
$548$		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	$0.315522\pi$
$569$			22712.9i			1.67341i	−0.547651	+	0.836707i	$0.684478\pi$
$570$		0			0
$571$	−1704.72	−	1704.72i	−0.124939	−	0.124939i	0.641872	−	0.766811i	$-0.278158\pi$
$571$	−1704.72	−	1704.72i	−0.124939	−	0.124939i	−0.766811	+	0.641872i	$0.778158\pi$
$572$		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	$-0.839834\pi$
$593$		−	13926.8i		−	0.964423i	0.876055	−	0.482212i	$-0.160166\pi$
$594$		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	$-0.553373\pi$
$599$	−4893.34			−0.333784			−0.166892	+	0.985975i	$0.553373\pi$
$600$		0			0
$601$	−5592.07	−	5592.07i	−0.379543	−	0.379543i	0.491394	−	0.870937i	$-0.336487\pi$
$601$	−5592.07	−	5592.07i	−0.379543	−	0.379543i	−0.870937	+	0.491394i	$0.836487\pi$
$602$		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	$-0.742138\pi$
$607$	522.294	−	522.294i	0.0349246	−	0.0349246i	0.724353	+	0.689429i	$0.242138\pi$
$608$		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	−0.925397	−	0.379000i	$-0.876268\pi$
$617$	−8374.06	+	8374.06i	−0.546397	+	0.546397i	0.379000	+	0.925397i	$0.376268\pi$
$618$		0			0
$619$	−4543.28	−	4543.28i	−0.295008	−	0.295008i	0.544047	−	0.839055i	$-0.316891\pi$
$619$	−4543.28	−	4543.28i	−0.295008	−	0.295008i	−0.839055	+	0.544047i	$0.816891\pi$
$620$		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	−0.250607	+	0.968089i	$0.580630\pi$
$641$	−19778.0	+	19778.0i	−1.21870	+	1.21870i	−0.968089	+	0.250607i	$0.919370\pi$
$642$		0			0
$643$	−4303.81	+	4303.81i	−0.263959	+	0.263959i	−0.826660	−	0.562701i	$-0.809762\pi$
$643$	−4303.81	+	4303.81i	−0.263959	+	0.263959i	0.562701	+	0.826660i	$0.309762\pi$
$644$		0			0
$645$	21918.5			1.33805
$646$		0			0
$647$	−18583.9			−1.12923			−0.564613	−	0.825356i	$-0.690974\pi$
$647$	−18583.9			−1.12923			−0.564613	+	0.825356i	$0.690974\pi$
$648$		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	0.997136	+	0.0756324i	$0.0240976\pi$
$653$	17900.9	+	17900.9i	1.07277	+	1.07277i	0.0756324	+	0.997136i	$0.475902\pi$
$654$		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	$-0.453282\pi$
$659$	4948.03			0.292485			0.146243	+	0.989249i	$0.453282\pi$
$660$		0			0
$661$			1373.95i			0.0808476i	0.999183	+	0.0404238i	$0.0128708\pi$
$661$			1373.95i			0.0808476i	−0.999183	+	0.0404238i	$0.987129\pi$
$662$		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	0.577235	−	0.816578i	$-0.304132\pi$
$673$	−4178.72	−	4178.72i	−0.239343	−	0.239343i	−0.816578	+	0.577235i	$0.804132\pi$
$674$		0			0
$675$	2735.01	−	2735.01i	0.155956	−	0.155956i
$676$		0			0
$677$	11478.1	−	11478.1i	0.651610	−	0.651610i	−0.301770	−	0.953381i	$-0.597578\pi$
$677$	11478.1	−	11478.1i	0.651610	−	0.651610i	0.953381	+	0.301770i	$0.0975776\pi$
$678$		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	$-0.733905\pi$
$683$	1275.88	−	1275.88i	0.0714792	−	0.0714792i	0.741943	+	0.670463i	$0.233905\pi$
$684$		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	$-0.335534\pi$
$691$	−6819.93	−	6819.93i	−0.375459	−	0.375459i	−0.869461	+	0.494002i	$0.835534\pi$
$692$		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	−0.842487	−	0.538717i	$-0.818909\pi$
$709$	−5734.76	+	5734.76i	−0.303771	+	0.303771i	0.538717	+	0.842487i	$0.318909\pi$
$710$		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	$-0.932973\pi$
$719$	−14823.8	+	14823.8i	−0.768892	+	0.768892i	0.209020	+	0.977911i	$0.432973\pi$
$720$		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	$-0.864960\pi$
$727$	−35728.7			−1.82270			−0.911351	+	0.411630i	$0.864960\pi$
$728$		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	$-0.806222\pi$
$733$		−	22697.4i		−	1.14372i	0.820350	−	0.571861i	$-0.193778\pi$
$734$		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	$-0.878649\pi$
$743$	−11263.3	+	11263.3i	−0.556139	+	0.556139i	0.372067	+	0.928206i	$0.378649\pi$
$744$		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	$-0.615357\pi$
$751$	11947.6	−	11947.6i	0.580523	−	0.580523i	0.935047	+	0.354524i	$0.115357\pi$
$752$		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	$0.151699\pi$
$757$			19109.1i			0.917479i	−0.888571	+	0.458740i	$0.848301\pi$
$758$		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	$-0.818889\pi$
$773$		−	23158.1i		−	1.07754i	0.842452	−	0.538771i	$-0.181111\pi$
$774$		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	$-0.685956\pi$
$787$	6239.79	−	6239.79i	0.282623	−	0.282623i	0.834154	+	0.551531i	$0.185956\pi$
$788$		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	$-0.812896\pi$
$797$		−	24954.4i		−	1.10907i	0.832160	−	0.554535i	$-0.187104\pi$
$798$		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	−0.972536	−	0.232754i	$-0.925226\pi$
$809$	−27734.1	−	27734.1i	−1.20529	−	1.20529i	−0.232754	−	0.972536i	$-0.574774\pi$
$810$		0			0
$811$	−13250.0	+	13250.0i	−0.573701	+	0.573701i	−0.933161	−	0.359459i	$-0.882961\pi$
$811$	−13250.0	+	13250.0i	−0.573701	+	0.573701i	0.359459	+	0.933161i	$0.382961\pi$
$812$		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	−0.478823	−	0.877912i	$-0.658936\pi$
$821$	9388.24	−	9388.24i	0.399089	−	0.399089i	0.877912	+	0.478823i	$0.158936\pi$
$822$		0			0
$823$	20495.3	+	20495.3i	0.868069	+	0.868069i	0.992258	−	0.124190i	$-0.0396331\pi$
$823$	20495.3	+	20495.3i	0.868069	+	0.868069i	−0.124190	+	0.992258i	$0.539633\pi$
$824$		0			0
$825$			5867.33i			0.247605i
$826$		0			0
$827$	−616.048	−	616.048i	−0.0259034	−	0.0259034i	0.694036	−	0.719940i	$-0.255831\pi$
$827$	−616.048	−	616.048i	−0.0259034	−	0.0259034i	−0.719940	+	0.694036i	$0.755831\pi$
$828$		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	$-0.221745\pi$
$839$	3046.75	+	3046.75i	0.125370	+	0.125370i	−0.641638	+	0.767008i	$0.721745\pi$
$840$		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	−0.999643	−	0.0267155i	$-0.991495\pi$
$853$	−24238.4	+	24238.4i	−0.972928	+	0.972928i	0.0267155	+	0.999643i	$0.491495\pi$
$854$		0			0
$855$	−61126.8	+	61126.8i	−2.44502	+	2.44502i
$856$		0			0
$857$	−2862.79	−	2862.79i	−0.114109	−	0.114109i	0.647747	−	0.761856i	$-0.275711\pi$
$857$	−2862.79	−	2862.79i	−0.114109	−	0.114109i	−0.761856	+	0.647747i	$0.775711\pi$
$858$		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	$-0.653150\pi$
$863$	−23465.3			−0.925572			−0.462786	+	0.886470i	$0.653150\pi$
$864$		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	0.951261	−	0.308388i	$-0.0997895\pi$
$877$	16696.5	+	16696.5i	0.642873	+	0.642873i	−0.308388	+	0.951261i	$0.599789\pi$
$878$		0			0
$879$	57580.6	−	57580.6i	2.20949	−	2.20949i
$880$		0			0
$881$	−20154.2	+	20154.2i	−0.770728	+	0.770728i	−0.978234	−	0.207506i	$-0.933465\pi$
$881$	−20154.2	+	20154.2i	−0.770728	+	0.770728i	0.207506	+	0.978234i	$0.433465\pi$
$882$		0			0
$883$	−39621.1			−1.51003			−0.755014	−	0.655709i	$-0.772370\pi$
$883$	−39621.1			−1.51003			−0.755014	+	0.655709i	$0.772370\pi$
$884$		0			0
$885$	−18519.7			−0.703427
$886$		0			0
$887$	6952.20	−	6952.20i	0.263170	−	0.263170i	−0.563171	−	0.826341i	$-0.690419\pi$
$887$	6952.20	−	6952.20i	0.263170	−	0.263170i	0.826341	+	0.563171i	$0.190419\pi$
$888$		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	$-0.285404\pi$
$907$	−4287.83	−	4287.83i	−0.156974	−	0.156974i	−0.781224	+	0.624251i	$0.785404\pi$
$908$		0			0
$909$		−	48870.4i		−	1.78320i
$910$		0			0
$911$	5371.57	+	5371.57i	0.195355	+	0.195355i	0.798005	−	0.602651i	$-0.205889\pi$
$911$	5371.57	+	5371.57i	0.195355	+	0.195355i	−0.602651	+	0.798005i	$0.705889\pi$
$912$		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	$-0.726327\pi$
$919$	−36363.0			−1.30523			−0.652614	+	0.757691i	$0.726327\pi$
$920$		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	−0.981554	−	0.191185i	$-0.938767\pi$
$929$	−33206.6	−	33206.6i	−1.17274	−	1.17274i	−0.191185	−	0.981554i	$-0.561233\pi$
$930$		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	$0.216174\pi$
$937$			36031.2i			1.25623i	−0.778120	+	0.628116i	$0.783826\pi$
$938$		0			0
$939$	36915.4			1.28295
$940$		0			0
$941$	−1667.19	−	1667.19i	−0.0577565	−	0.0577565i	0.677639	−	0.735395i	$-0.263003\pi$
$941$	−1667.19	−	1667.19i	−0.0577565	−	0.0577565i	−0.735395	+	0.677639i	$0.763003\pi$
$942$		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	$-0.692114\pi$
$947$	7453.66	−	7453.66i	0.255767	−	0.255767i	0.823330	+	0.567563i	$0.192114\pi$
$948$		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	$0.0516596\pi$
$977$			9868.81i			0.323164i	−0.986859	+	0.161582i	$0.948340\pi$
$978$		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	$-0.526043\pi$
$983$	28198.0	−	28198.0i	0.914930	−	0.914930i	0.996655	+	0.0817252i	$0.0260430\pi$
$984$		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	0.213736	−	0.976891i	$-0.431437\pi$
$991$	37143.8	−	37143.8i	1.19063	−	1.19063i	0.976891	−	0.213736i	$-0.0685633\pi$
$992$		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	0.980560	+	0.196217i	$0.0628659\pi$
$997$	37045.7	+	37045.7i	1.17678	+	1.17678i	0.196217	+	0.980560i	$0.437134\pi$
$998$		0			0
$999$	19441.4			0.615714

Twists

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

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
136.4.k.a.81.1	✓	14	17.13	even	4	inner
136.4.k.a.89.1	yes	14	1.1	even	1	trivial
272.4.o.g.81.7		14	68.47	odd	4
272.4.o.g.225.7		14	4.3	odd	2
2312.4.a.m.1.1		14	17.9	even	8
2312.4.a.m.1.14		14	17.8	even	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:	\( N \)	\(=\)	\( 136 = 2^{3} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	136.k (of order \(4\), degree \(2\), 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