LMFDB - Embedded newform 171.3.c.b.37.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [171,3,Mod(37,171)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(171, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 3, names="a")

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

chi := DirichletCharacter("171.37");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 171 = 3^{2} \cdot 19 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	171.c (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$4.65941252056$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\sqrt{-13}) $
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} + 13 $ x^2 + 13
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 19)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			37.1
Root			$-3.60555i$ of defining polynomial
Character	$\chi$	$=$	171.37
Dual form			171.3.c.b.37.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-3.60555i q^{2} -9.00000 q^{4} -4.00000 q^{5} -5.00000 q^{7} +18.0278i q^{8} +O(q^{10})$	\(q-3.60555i q^{2} -9.00000 q^{4} -4.00000 q^{5} -5.00000 q^{7} +18.0278i q^{8} +14.4222i q^{10} +10.0000 q^{11} +3.60555i q^{13} +18.0278i q^{14} +29.0000 q^{16} -15.0000 q^{17} +(-6.00000 + 18.0278i) q^{19} +36.0000 q^{20} -36.0555i q^{22} -35.0000 q^{23} -9.00000 q^{25} +13.0000 q^{26} +45.0000 q^{28} -18.0278i q^{29} -36.0555i q^{31} -32.4500i q^{32} +54.0833i q^{34} +20.0000 q^{35} -21.6333i q^{37} +(65.0000 + 21.6333i) q^{38} -72.1110i q^{40} -36.0555i q^{41} -20.0000 q^{43} -90.0000 q^{44} +126.194i q^{46} -10.0000 q^{47} -24.0000 q^{49} +32.4500i q^{50} -32.4500i q^{52} +75.7166i q^{53} -40.0000 q^{55} -90.1388i q^{56} -65.0000 q^{58} -18.0278i q^{59} -40.0000 q^{61} -130.000 q^{62} -1.00000 q^{64} -14.4222i q^{65} +39.6611i q^{67} +135.000 q^{68} -72.1110i q^{70} -108.167i q^{71} +105.000 q^{73} -78.0000 q^{74} +(54.0000 - 162.250i) q^{76} -50.0000 q^{77} -36.0555i q^{79} -116.000 q^{80} -130.000 q^{82} +40.0000 q^{83} +60.0000 q^{85} +72.1110i q^{86} +180.278i q^{88} -18.0278i q^{91} +315.000 q^{92} +36.0555i q^{94} +(24.0000 - 72.1110i) q^{95} +122.589i q^{97} +86.5332i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 18 q^{4} - 8 q^{5} - 10 q^{7}+O(q^{10}) $ 2 * q - 18 * q^4 - 8 * q^5 - 10 * q^7	$ 2 q - 18 q^{4} - 8 q^{5} - 10 q^{7} + 20 q^{11} + 58 q^{16} - 30 q^{17} - 12 q^{19} + 72 q^{20} - 70 q^{23} - 18 q^{25} + 26 q^{26} + 90 q^{28} + 40 q^{35} + 130 q^{38} - 40 q^{43} - 180 q^{44} - 20 q^{47} - 48 q^{49} - 80 q^{55} - 130 q^{58} - 80 q^{61} - 260 q^{62} - 2 q^{64} + 270 q^{68} + 210 q^{73} - 156 q^{74} + 108 q^{76} - 100 q^{77} - 232 q^{80} - 260 q^{82} + 80 q^{83} + 120 q^{85} + 630 q^{92} + 48 q^{95}+O(q^{100}) $ 2 * q - 18 * q^4 - 8 * q^5 - 10 * q^7 + 20 * q^11 + 58 * q^16 - 30 * q^17 - 12 * q^19 + 72 * q^20 - 70 * q^23 - 18 * q^25 + 26 * q^26 + 90 * q^28 + 40 * q^35 + 130 * q^38 - 40 * q^43 - 180 * q^44 - 20 * q^47 - 48 * q^49 - 80 * q^55 - 130 * q^58 - 80 * q^61 - 260 * q^62 - 2 * q^64 + 270 * q^68 + 210 * q^73 - 156 * q^74 + 108 * q^76 - 100 * q^77 - 232 * q^80 - 260 * q^82 + 80 * q^83 + 120 * q^85 + 630 * q^92 + 48 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/171\mathbb{Z}\right)^\times$.

$n$	$20$	$154$
$\chi(n)$	$1$	$-1$

Coefficient data

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

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

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

$2$		−	3.60555i		−	1.80278i	−0.433013	−	0.901388i	$-0.642549\pi$
$2$		−	3.60555i		−	1.80278i	0.433013	−	0.901388i	$-0.357451\pi$
$3$		0			0
$3$		0			0
$4$	−9.00000			−2.25000
$5$	−4.00000			−0.800000			−0.400000	−	0.916515i	$-0.630990\pi$
$5$	−4.00000			−0.800000			−0.400000	+	0.916515i	$0.630990\pi$
$6$		0			0
$7$	−5.00000			−0.714286			−0.357143	−	0.934050i	$-0.616249\pi$
$7$	−5.00000			−0.714286			−0.357143	+	0.934050i	$0.616249\pi$
$8$			18.0278i			2.25347i
$9$		0			0
$10$			14.4222i			1.44222i
$11$	10.0000			0.909091			0.454545	−	0.890724i	$-0.349802\pi$
$11$	10.0000			0.909091			0.454545	+	0.890724i	$0.349802\pi$
$12$		0			0
$13$			3.60555i			0.277350i	0.990338	+	0.138675i	$0.0442844\pi$
$13$			3.60555i			0.277350i	−0.990338	+	0.138675i	$0.955716\pi$
$14$			18.0278i			1.28770i
$15$		0			0
$16$	29.0000			1.81250
$17$	−15.0000			−0.882353			−0.441176	−	0.897420i	$-0.645439\pi$
$17$	−15.0000			−0.882353			−0.441176	+	0.897420i	$0.645439\pi$
$18$		0			0
$19$	−6.00000	+	18.0278i	−0.315789	+	0.948829i
$19$	−6.00000	+	18.0278i	−0.315789	+	0.948829i
$20$	36.0000			1.80000
$21$		0			0
$22$		−	36.0555i		−	1.63889i
$23$	−35.0000			−1.52174			−0.760870	−	0.648905i	$-0.775227\pi$
$23$	−35.0000			−1.52174			−0.760870	+	0.648905i	$0.775227\pi$
$24$		0			0
$25$	−9.00000			−0.360000
$26$	13.0000			0.500000
$27$		0			0
$28$	45.0000			1.60714
$29$		−	18.0278i		−	0.621647i	−0.950468	−	0.310823i	$-0.899395\pi$
$29$		−	18.0278i		−	0.621647i	0.950468	−	0.310823i	$-0.100605\pi$
$30$		0			0
$31$		−	36.0555i		−	1.16308i	−0.813517	−	0.581541i	$-0.802450\pi$
$31$		−	36.0555i		−	1.16308i	0.813517	−	0.581541i	$-0.197550\pi$
$32$		−	32.4500i		−	1.01406i
$33$		0			0
$34$			54.0833i			1.59068i
$35$	20.0000			0.571429
$36$		0			0
$37$		−	21.6333i		−	0.584684i	−0.956314	−	0.292342i	$-0.905565\pi$
$37$		−	21.6333i		−	0.584684i	0.956314	−	0.292342i	$-0.0944346\pi$
$38$	65.0000	+	21.6333i	1.71053	+	0.569298i
$39$		0			0
$40$		−	72.1110i		−	1.80278i
$41$		−	36.0555i		−	0.879403i	−0.898144	−	0.439701i	$-0.855084\pi$
$41$		−	36.0555i		−	0.879403i	0.898144	−	0.439701i	$-0.144916\pi$
$42$		0			0
$43$	−20.0000			−0.465116			−0.232558	−	0.972582i	$-0.574710\pi$
$43$	−20.0000			−0.465116			−0.232558	+	0.972582i	$0.574710\pi$
$44$	−90.0000			−2.04545
$45$		0			0
$46$			126.194i			2.74335i
$47$	−10.0000			−0.212766			−0.106383	−	0.994325i	$-0.533927\pi$
$47$	−10.0000			−0.212766			−0.106383	+	0.994325i	$0.533927\pi$
$48$		0			0
$49$	−24.0000			−0.489796
$50$			32.4500i			0.648999i
$51$		0			0
$52$		−	32.4500i		−	0.624038i
$53$			75.7166i			1.42861i	0.699832	+	0.714307i	$0.253258\pi$
$53$			75.7166i			1.42861i	−0.699832	+	0.714307i	$0.746742\pi$
$54$		0			0
$55$	−40.0000			−0.727273
$56$		−	90.1388i		−	1.60962i
$57$		0			0
$58$	−65.0000			−1.12069
$59$		−	18.0278i		−	0.305555i	−0.988261	−	0.152778i	$-0.951178\pi$
$59$		−	18.0278i		−	0.305555i	0.988261	−	0.152778i	$-0.0488218\pi$
$60$		0			0
$61$	−40.0000			−0.655738			−0.327869	−	0.944723i	$-0.606330\pi$
$61$	−40.0000			−0.655738			−0.327869	+	0.944723i	$0.606330\pi$
$62$	−130.000			−2.09677
$63$		0			0
$64$	−1.00000			−0.0156250
$65$		−	14.4222i		−	0.221880i
$66$		0			0
$67$			39.6611i			0.591956i	0.955195	+	0.295978i	$0.0956455\pi$
$67$			39.6611i			0.591956i	−0.955195	+	0.295978i	$0.904354\pi$
$68$	135.000			1.98529
$69$		0			0
$70$		−	72.1110i		−	1.03016i
$71$		−	108.167i		−	1.52347i	−0.647887	−	0.761736i	$-0.724347\pi$
$71$		−	108.167i		−	1.52347i	0.647887	−	0.761736i	$-0.275653\pi$
$72$		0			0
$73$	105.000			1.43836			0.719178	−	0.694826i	$-0.244519\pi$
$73$	105.000			1.43836			0.719178	+	0.694826i	$0.244519\pi$
$74$	−78.0000			−1.05405
$75$		0			0
$76$	54.0000	−	162.250i	0.710526	−	2.13487i
$77$	−50.0000			−0.649351
$78$		0			0
$79$		−	36.0555i		−	0.456399i	−0.973614	−	0.228199i	$-0.926716\pi$
$79$		−	36.0555i		−	0.456399i	0.973614	−	0.228199i	$-0.0732838\pi$
$80$	−116.000			−1.45000
$81$		0			0
$82$	−130.000			−1.58537
$83$	40.0000			0.481928			0.240964	−	0.970534i	$-0.422536\pi$
$83$	40.0000			0.481928			0.240964	+	0.970534i	$0.422536\pi$
$84$		0			0
$85$	60.0000			0.705882
$86$			72.1110i			0.838500i
$87$		0			0
$88$			180.278i			2.04861i
$89$		0			0		1.00000			$0$
$89$		0			0		−1.00000			$\pi$
$90$		0			0
$91$		−	18.0278i		−	0.198107i
$92$	315.000			3.42391
$93$		0			0
$94$			36.0555i			0.383569i
$95$	24.0000	−	72.1110i	0.252632	−	0.759063i
$96$		0			0
$97$			122.589i			1.26380i	0.775049	+	0.631901i	$0.217725\pi$
$97$			122.589i			1.26380i	−0.775049	+	0.631901i	$0.782275\pi$
$98$			86.5332i			0.882992i
$99$		0			0
$100$	81.0000			0.810000
$101$	50.0000			0.495050			0.247525	−	0.968882i	$-0.420383\pi$
$101$	50.0000			0.495050			0.247525	+	0.968882i	$0.420383\pi$
$102$		0			0
$103$			57.6888i			0.560086i	0.959988	+	0.280043i	$0.0903487\pi$
$103$			57.6888i			0.560086i	−0.959988	+	0.280043i	$0.909651\pi$
$104$	−65.0000			−0.625000
$105$		0			0
$106$	273.000			2.57547
$107$		−	75.7166i		−	0.707632i	−0.935315	−	0.353816i	$-0.884884\pi$
$107$		−	75.7166i		−	0.707632i	0.935315	−	0.353816i	$-0.115116\pi$
$108$		0			0
$109$		−	198.305i		−	1.81931i	−0.415359	−	0.909657i	$-0.636344\pi$
$109$		−	198.305i		−	1.81931i	0.415359	−	0.909657i	$-0.363656\pi$
$110$			144.222i			1.31111i
$111$		0			0
$112$	−145.000			−1.29464
$113$			122.589i			1.08486i	0.840102	+	0.542428i	$0.182495\pi$
$113$			122.589i			1.08486i	−0.840102	+	0.542428i	$0.817505\pi$
$114$		0			0
$115$	140.000			1.21739
$116$			162.250i			1.39871i
$117$		0			0
$118$	−65.0000			−0.550847
$119$	75.0000			0.630252
$120$		0			0
$121$	−21.0000			−0.173554
$122$			144.222i			1.18215i
$123$		0			0
$124$			324.500i			2.61693i
$125$	136.000			1.08800
$126$		0			0
$127$		−	129.800i		−	1.02205i	−0.859567	−	0.511023i	$-0.829267\pi$
$127$		−	129.800i		−	1.02205i	0.859567	−	0.511023i	$-0.170733\pi$
$128$		−	126.194i		−	0.985893i
$129$		0			0
$130$	−52.0000			−0.400000
$131$	−112.000			−0.854962			−0.427481	−	0.904024i	$-0.640599\pi$
$131$	−112.000			−0.854962			−0.427481	+	0.904024i	$0.640599\pi$
$132$		0			0
$133$	30.0000	−	90.1388i	0.225564	−	0.677735i
$134$	143.000			1.06716
$135$		0			0
$136$		−	270.416i		−	1.98836i
$137$	−125.000			−0.912409			−0.456204	−	0.889875i	$-0.650791\pi$
$137$	−125.000			−0.912409			−0.456204	+	0.889875i	$0.650791\pi$
$138$		0			0
$139$	50.0000			0.359712			0.179856	−	0.983693i	$-0.442437\pi$
$139$	50.0000			0.359712			0.179856	+	0.983693i	$0.442437\pi$
$140$	−180.000			−1.28571
$141$		0			0
$142$	−390.000			−2.74648
$143$			36.0555i			0.252136i
$144$		0			0
$145$			72.1110i			0.497317i
$146$		−	378.583i		−	2.59303i
$147$		0			0
$148$			194.700i			1.31554i
$149$	−70.0000			−0.469799			−0.234899	−	0.972020i	$-0.575476\pi$
$149$	−70.0000			−0.469799			−0.234899	+	0.972020i	$0.575476\pi$
$150$		0			0
$151$			36.0555i			0.238778i	0.992848	+	0.119389i	$0.0380936\pi$
$151$			36.0555i			0.238778i	−0.992848	+	0.119389i	$0.961906\pi$
$152$	−325.000	−	108.167i	−2.13816	−	0.711622i
$153$		0			0
$154$			180.278i			1.17063i
$155$			144.222i			0.930465i
$156$		0			0
$157$	10.0000			0.0636943			0.0318471	−	0.999493i	$-0.489861\pi$
$157$	10.0000			0.0636943			0.0318471	+	0.999493i	$0.489861\pi$
$158$	−130.000			−0.822785
$159$		0			0
$160$			129.800i			0.811249i
$161$	175.000			1.08696
$162$		0			0
$163$	−270.000			−1.65644			−0.828221	−	0.560402i	$-0.810647\pi$
$163$	−270.000			−1.65644			−0.828221	+	0.560402i	$0.810647\pi$
$164$			324.500i			1.97866i
$165$		0			0
$166$		−	144.222i		−	0.868808i
$167$		−	122.589i		−	0.734064i	−0.930208	−	0.367032i	$-0.880374\pi$
$167$		−	122.589i		−	0.734064i	0.930208	−	0.367032i	$-0.119626\pi$
$168$		0			0
$169$	156.000			0.923077
$170$		−	216.333i		−	1.27255i
$171$		0			0
$172$	180.000			1.04651
$173$			122.589i			0.708605i	0.935131	+	0.354303i	$0.115282\pi$
$173$			122.589i			0.708605i	−0.935131	+	0.354303i	$0.884718\pi$
$174$		0			0
$175$	45.0000			0.257143
$176$	290.000			1.64773
$177$		0			0
$178$		0			0
$179$		−	36.0555i		−	0.201427i	−0.994915	−	0.100714i	$-0.967887\pi$
$179$		−	36.0555i		−	0.201427i	0.994915	−	0.100714i	$-0.0321126\pi$
$180$		0			0
$181$			108.167i			0.597605i	0.954315	+	0.298803i	$0.0965872\pi$
$181$			108.167i			0.597605i	−0.954315	+	0.298803i	$0.903413\pi$
$182$	−65.0000			−0.357143
$183$		0			0
$184$		−	630.971i		−	3.42919i
$185$			86.5332i			0.467747i
$186$		0			0
$187$	−150.000			−0.802139
$188$	90.0000			0.478723
$189$		0			0
$190$	−260.000	−	86.5332i	−1.36842	−	0.455438i
$191$	−193.000			−1.01047			−0.505236	−	0.862981i	$-0.668594\pi$
$191$	−193.000			−1.01047			−0.505236	+	0.862981i	$0.668594\pi$
$192$		0			0
$193$			266.811i			1.38244i	0.722645	+	0.691220i	$0.242926\pi$
$193$			266.811i			1.38244i	−0.722645	+	0.691220i	$0.757074\pi$
$194$	442.000			2.27835
$195$		0			0
$196$	216.000			1.10204
$197$	−90.0000			−0.456853			−0.228426	−	0.973561i	$-0.573358\pi$
$197$	−90.0000			−0.456853			−0.228426	+	0.973561i	$0.573358\pi$
$198$		0			0
$199$	123.000			0.618090			0.309045	−	0.951047i	$-0.399991\pi$
$199$	123.000			0.618090			0.309045	+	0.951047i	$0.399991\pi$
$200$		−	162.250i		−	0.811249i
$201$		0			0
$202$		−	180.278i		−	0.892463i
$203$			90.1388i			0.444033i
$204$		0			0
$205$			144.222i			0.703522i
$206$	208.000			1.00971
$207$		0			0
$208$			104.561i			0.502697i
$209$	−60.0000	+	180.278i	−0.287081	+	0.862572i
$210$		0			0
$211$		−	234.361i		−	1.11071i	−0.831612	−	0.555357i	$-0.812581\pi$
$211$		−	234.361i		−	1.11071i	0.831612	−	0.555357i	$-0.187419\pi$
$212$		−	681.449i		−	3.21438i
$213$		0			0
$214$	−273.000			−1.27570
$215$	80.0000			0.372093
$216$		0			0
$217$			180.278i			0.830772i
$218$	−715.000			−3.27982
$219$		0			0
$220$	360.000			1.63636
$221$		−	54.0833i		−	0.244721i
$222$		0			0
$223$		−	201.911i		−	0.905430i	−0.891655	−	0.452715i	$-0.850456\pi$
$223$		−	201.911i		−	0.905430i	0.891655	−	0.452715i	$-0.149544\pi$
$224$			162.250i			0.724329i
$225$		0			0
$226$	442.000			1.95575
$227$		−	255.994i		−	1.12773i	−0.825868	−	0.563864i	$-0.809314\pi$
$227$		−	255.994i		−	1.12773i	0.825868	−	0.563864i	$-0.190686\pi$
$228$		0			0
$229$	−160.000			−0.698690			−0.349345	−	0.936994i	$-0.613596\pi$
$229$	−160.000			−0.698690			−0.349345	+	0.936994i	$0.613596\pi$
$230$		−	504.777i		−	2.19468i
$231$		0			0
$232$	325.000			1.40086
$233$	270.000			1.15880			0.579399	−	0.815044i	$-0.303287\pi$
$233$	270.000			1.15880			0.579399	+	0.815044i	$0.303287\pi$
$234$		0			0
$235$	40.0000			0.170213
$236$			162.250i			0.687499i
$237$		0			0
$238$		−	270.416i		−	1.13620i
$239$	−197.000			−0.824268			−0.412134	−	0.911123i	$-0.635216\pi$
$239$	−197.000			−0.824268			−0.412134	+	0.911123i	$0.635216\pi$
$240$		0			0
$241$			396.611i			1.64569i	0.568268	+	0.822844i	$0.307614\pi$
$241$			396.611i			1.64569i	−0.568268	+	0.822844i	$0.692386\pi$
$242$			75.7166i			0.312878i
$243$		0			0
$244$	360.000			1.47541
$245$	96.0000			0.391837
$246$		0			0
$247$	−65.0000	−	21.6333i	−0.263158	−	0.0875842i
$248$	650.000			2.62097
$249$		0			0
$250$		−	490.355i		−	1.96142i
$251$	402.000			1.60159			0.800797	−	0.598936i	$-0.204410\pi$
$251$	402.000			1.60159			0.800797	+	0.598936i	$0.204410\pi$
$252$		0			0
$253$	−350.000			−1.38340
$254$	−468.000			−1.84252
$255$		0			0
$256$	−459.000			−1.79297
$257$			418.244i			1.62741i	0.581279	+	0.813704i	$0.302552\pi$
$257$			418.244i			1.62741i	−0.581279	+	0.813704i	$0.697448\pi$
$258$		0			0
$259$			108.167i			0.417631i
$260$			129.800i			0.499230i
$261$		0			0
$262$			403.822i			1.54130i
$263$	−310.000			−1.17871			−0.589354	−	0.807875i	$-0.700618\pi$
$263$	−310.000			−1.17871			−0.589354	+	0.807875i	$0.700618\pi$
$264$		0			0
$265$		−	302.866i		−	1.14289i
$266$	−325.000	−	108.167i	−1.22180	−	0.406641i
$267$		0			0
$268$		−	356.950i		−	1.33190i
$269$		−	108.167i		−	0.402106i	−0.979580	−	0.201053i	$-0.935564\pi$
$269$		−	108.167i		−	0.402106i	0.979580	−	0.201053i	$-0.0644364\pi$
$270$		0			0
$271$	105.000			0.387454			0.193727	−	0.981055i	$-0.437942\pi$
$271$	105.000			0.387454			0.193727	+	0.981055i	$0.437942\pi$
$272$	−435.000			−1.59926
$273$		0			0
$274$			450.694i			1.64487i
$275$	−90.0000			−0.327273
$276$		0			0
$277$	−50.0000			−0.180505			−0.0902527	−	0.995919i	$-0.528767\pi$
$277$	−50.0000			−0.180505			−0.0902527	+	0.995919i	$0.528767\pi$
$278$		−	180.278i		−	0.648480i
$279$		0			0
$280$			360.555i			1.28770i
$281$			288.444i			1.02649i	0.858242	+	0.513246i	$0.171557\pi$
$281$			288.444i			1.02649i	−0.858242	+	0.513246i	$0.828443\pi$
$282$		0			0
$283$	−320.000			−1.13074			−0.565371	−	0.824837i	$-0.691267\pi$
$283$	−320.000			−1.13074			−0.565371	+	0.824837i	$0.691267\pi$
$284$			973.499i			3.42781i
$285$		0			0
$286$	130.000			0.454545
$287$			180.278i			0.628145i
$288$		0			0
$289$	−64.0000			−0.221453
$290$	260.000			0.896552
$291$		0			0
$292$	−945.000			−3.23630
$293$			219.939i			0.750644i	0.926895	+	0.375322i	$0.122468\pi$
$293$			219.939i			0.750644i	−0.926895	+	0.375322i	$0.877532\pi$
$294$		0			0
$295$			72.1110i			0.244444i
$296$	390.000			1.31757
$297$		0			0
$298$			252.389i			0.846942i
$299$		−	126.194i		−	0.422054i
$300$		0			0
$301$	100.000			0.332226
$302$	130.000			0.430464
$303$		0			0
$304$	−174.000	+	522.805i	−0.572368	+	1.71975i
$305$	160.000			0.524590
$306$		0			0
$307$			237.966i			0.775135i	0.921841	+	0.387567i	$0.126685\pi$
$307$			237.966i			0.775135i	−0.921841	+	0.387567i	$0.873315\pi$
$308$	450.000			1.46104
$309$		0			0
$310$	520.000			1.67742
$311$	−395.000			−1.27010			−0.635048	−	0.772472i	$-0.719020\pi$
$311$	−395.000			−1.27010			−0.635048	+	0.772472i	$0.719020\pi$
$312$		0			0
$313$	125.000			0.399361			0.199681	−	0.979861i	$-0.436010\pi$
$313$	125.000			0.399361			0.199681	+	0.979861i	$0.436010\pi$
$314$		−	36.0555i		−	0.114826i
$315$		0			0
$316$			324.500i			1.02690i
$317$		−	3.60555i		−	0.0113740i	−0.999984	−	0.00568699i	$-0.998190\pi$
$317$		−	3.60555i		−	0.0113740i	0.999984	−	0.00568699i	$-0.00181023\pi$
$318$		0			0
$319$		−	180.278i		−	0.565133i
$320$	4.00000			0.0125000
$321$		0			0
$322$		−	630.971i		−	1.95954i
$323$	90.0000	−	270.416i	0.278638	−	0.837202i
$324$		0			0
$325$		−	32.4500i		−	0.0998460i
$326$			973.499i			2.98619i
$327$		0			0
$328$	650.000			1.98171
$329$	50.0000			0.151976
$330$		0			0
$331$		−	198.305i		−	0.599110i	−0.954079	−	0.299555i	$-0.903162\pi$
$331$		−	198.305i		−	0.599110i	0.954079	−	0.299555i	$-0.0968382\pi$
$332$	−360.000			−1.08434
$333$		0			0
$334$	−442.000			−1.32335
$335$		−	158.644i		−	0.473565i
$336$		0			0
$337$			57.6888i			0.171183i	0.996330	+	0.0855917i	$0.0272781\pi$
$337$			57.6888i			0.171183i	−0.996330	+	0.0855917i	$0.972722\pi$
$338$		−	562.466i		−	1.66410i
$339$		0			0
$340$	−540.000			−1.58824
$341$		−	360.555i		−	1.05735i
$342$		0			0
$343$	365.000			1.06414
$344$		−	360.555i		−	1.04813i
$345$		0			0
$346$	442.000			1.27746
$347$	40.0000			0.115274			0.0576369	−	0.998338i	$-0.481643\pi$
$347$	40.0000			0.115274			0.0576369	+	0.998338i	$0.481643\pi$
$348$		0			0
$349$	98.0000			0.280802			0.140401	−	0.990095i	$-0.455161\pi$
$349$	98.0000			0.280802			0.140401	+	0.990095i	$0.455161\pi$
$350$		−	162.250i		−	0.463571i
$351$		0			0
$352$		−	324.500i		−	0.921874i
$353$	185.000			0.524079			0.262040	−	0.965057i	$-0.415605\pi$
$353$	185.000			0.524079			0.262040	+	0.965057i	$0.415605\pi$
$354$		0			0
$355$			432.666i			1.21878i
$356$		0			0
$357$		0			0
$358$	−130.000			−0.363128
$359$	225.000			0.626741			0.313370	−	0.949631i	$-0.398542\pi$
$359$	225.000			0.626741			0.313370	+	0.949631i	$0.398542\pi$
$360$		0			0
$361$	−289.000	−	216.333i	−0.800554	−	0.599261i
$362$	390.000			1.07735
$363$		0			0
$364$			162.250i			0.445741i
$365$	−420.000			−1.15068
$366$		0			0
$367$	50.0000			0.136240			0.0681199	−	0.997677i	$-0.478300\pi$
$367$	50.0000			0.136240			0.0681199	+	0.997677i	$0.478300\pi$
$368$	−1015.00			−2.75815
$369$		0			0
$370$	312.000			0.843243
$371$		−	378.583i		−	1.02044i
$372$		0			0
$373$		−	436.272i		−	1.16963i	−0.811167	−	0.584815i	$-0.801167\pi$
$373$		−	436.272i		−	1.16963i	0.811167	−	0.584815i	$-0.198833\pi$
$374$			540.833i			1.44608i
$375$		0			0
$376$		−	180.278i		−	0.479462i
$377$	65.0000			0.172414
$378$		0			0
$379$		−	486.749i		−	1.28430i	−0.766579	−	0.642150i	$-0.778043\pi$
$379$		−	486.749i		−	1.28430i	0.766579	−	0.642150i	$-0.221957\pi$
$380$	−216.000	+	648.999i	−0.568421	+	1.70789i
$381$		0			0
$382$			695.871i			1.82165i
$383$		−	201.911i		−	0.527182i	−0.964634	−	0.263591i	$-0.915093\pi$
$383$		−	201.911i		−	0.527182i	0.964634	−	0.263591i	$-0.0849070\pi$
$384$		0			0
$385$	200.000			0.519481
$386$	962.000			2.49223
$387$		0			0
$388$		−	1103.30i		−	2.84355i
$389$	478.000			1.22879			0.614396	−	0.788998i	$-0.289400\pi$
$389$	478.000			1.22879			0.614396	+	0.788998i	$0.289400\pi$
$390$		0			0
$391$	525.000			1.34271
$392$		−	432.666i		−	1.10374i
$393$		0			0
$394$			324.500i			0.823603i
$395$			144.222i			0.365119i
$396$		0			0
$397$	750.000			1.88917			0.944584	−	0.328269i	$-0.106465\pi$
$397$	750.000			1.88917			0.944584	+	0.328269i	$0.106465\pi$
$398$		−	443.483i		−	1.11428i
$399$		0			0
$400$	−261.000			−0.652500
$401$			288.444i			0.719312i	0.933085	+	0.359656i	$0.117106\pi$
$401$			288.444i			0.719312i	−0.933085	+	0.359656i	$0.882894\pi$
$402$		0			0
$403$	130.000			0.322581
$404$	−450.000			−1.11386
$405$		0			0
$406$	325.000			0.800493
$407$		−	216.333i		−	0.531531i
$408$		0			0
$409$		−	36.0555i		−	0.0881553i	−0.999028	−	0.0440776i	$-0.985965\pi$
$409$		−	36.0555i		−	0.0881553i	0.999028	−	0.0440776i	$-0.0140349\pi$
$410$	520.000			1.26829
$411$		0			0
$412$		−	519.199i		−	1.26019i
$413$			90.1388i			0.218254i
$414$		0			0
$415$	−160.000			−0.385542
$416$	117.000			0.281250
$417$		0			0
$418$	650.000	+	216.333i	1.55502	+	0.517543i
$419$	−112.000			−0.267303			−0.133652	−	0.991028i	$-0.542670\pi$
$419$	−112.000			−0.267303			−0.133652	+	0.991028i	$0.542670\pi$
$420$		0			0
$421$			630.971i			1.49874i	0.662149	+	0.749372i	$0.269645\pi$
$421$			630.971i			1.49874i	−0.662149	+	0.749372i	$0.730355\pi$
$422$	−845.000			−2.00237
$423$		0			0
$424$	−1365.00			−3.21934
$425$	135.000			0.317647
$426$		0			0
$427$	200.000			0.468384
$428$			681.449i			1.59217i
$429$		0			0
$430$		−	288.444i		−	0.670800i
$431$			432.666i			1.00387i	0.864907	+	0.501933i	$0.167378\pi$
$431$			432.666i			1.00387i	−0.864907	+	0.501933i	$0.832622\pi$
$432$		0			0
$433$		−	735.532i		−	1.69869i	−0.527839	−	0.849345i	$-0.676997\pi$
$433$		−	735.532i		−	1.69869i	0.527839	−	0.849345i	$-0.323003\pi$
$434$	650.000			1.49770
$435$		0			0
$436$			1784.75i			4.09346i
$437$	210.000	−	630.971i	0.480549	−	1.44387i
$438$		0			0
$439$			793.221i			1.80688i	0.428712	+	0.903441i	$0.358967\pi$
$439$			793.221i			1.80688i	−0.428712	+	0.903441i	$0.641033\pi$
$440$		−	721.110i		−	1.63889i
$441$		0			0
$442$	−195.000			−0.441176
$443$	−670.000			−1.51242			−0.756208	−	0.654332i	$-0.772950\pi$
$443$	−670.000			−1.51242			−0.756208	+	0.654332i	$0.772950\pi$
$444$		0			0
$445$		0			0
$446$	−728.000			−1.63229
$447$		0			0
$448$	5.00000			0.0111607
$449$			36.0555i			0.0803018i	0.999194	+	0.0401509i	$0.0127839\pi$
$449$			36.0555i			0.0803018i	−0.999194	+	0.0401509i	$0.987216\pi$
$450$		0			0
$451$		−	360.555i		−	0.799457i
$452$		−	1103.30i		−	2.44093i
$453$		0			0
$454$	−923.000			−2.03304
$455$			72.1110i			0.158486i
$456$		0			0
$457$	−755.000			−1.65208			−0.826039	−	0.563612i	$-0.809411\pi$
$457$	−755.000			−1.65208			−0.826039	+	0.563612i	$0.809411\pi$
$458$			576.888i			1.25958i
$459$		0			0
$460$	−1260.00			−2.73913
$461$	−772.000			−1.67462			−0.837310	−	0.546728i	$-0.815873\pi$
$461$	−772.000			−1.67462			−0.837310	+	0.546728i	$0.815873\pi$
$462$		0			0
$463$	−350.000			−0.755940			−0.377970	−	0.925818i	$-0.623378\pi$
$463$	−350.000			−0.755940			−0.377970	+	0.925818i	$0.623378\pi$
$464$		−	522.805i		−	1.12673i
$465$		0			0
$466$		−	973.499i		−	2.08905i
$467$	−70.0000			−0.149893			−0.0749465	−	0.997188i	$-0.523879\pi$
$467$	−70.0000			−0.149893			−0.0749465	+	0.997188i	$0.523879\pi$
$468$		0			0
$469$		−	198.305i		−	0.422826i
$470$		−	144.222i		−	0.306855i
$471$		0			0
$472$	325.000			0.688559
$473$	−200.000			−0.422833
$474$		0			0
$475$	54.0000	−	162.250i	0.113684	−	0.341579i
$476$	−675.000			−1.41807
$477$		0			0
$478$			710.294i			1.48597i
$479$	370.000			0.772443			0.386221	−	0.922406i	$-0.373780\pi$
$479$	370.000			0.772443			0.386221	+	0.922406i	$0.373780\pi$
$480$		0			0
$481$	78.0000			0.162162
$482$	1430.00			2.96680
$483$		0			0
$484$	189.000			0.390496
$485$		−	490.355i		−	1.01104i
$486$		0			0
$487$		−	519.199i		−	1.06612i	−0.846078	−	0.533059i	$-0.821042\pi$
$487$		−	519.199i		−	1.06612i	0.846078	−	0.533059i	$-0.178958\pi$
$488$		−	721.110i		−	1.47768i
$489$		0			0
$490$		−	346.133i		−	0.706394i
$491$	632.000			1.28717			0.643585	−	0.765375i	$-0.277446\pi$
$491$	632.000			1.28717			0.643585	+	0.765375i	$0.277446\pi$
$492$		0			0
$493$			270.416i			0.548512i
$494$	−78.0000	+	234.361i	−0.157895	+	0.474415i
$495$		0			0
$496$		−	1045.61i		−	2.10808i
$497$			540.833i			1.08819i
$498$		0			0
$499$	380.000			0.761523			0.380762	−	0.924673i	$-0.375662\pi$
$499$	380.000			0.761523			0.380762	+	0.924673i	$0.375662\pi$
$500$	−1224.00			−2.44800
$501$		0			0
$502$		−	1449.43i		−	2.88731i
$503$	45.0000			0.0894632			0.0447316	−	0.998999i	$-0.485757\pi$
$503$	45.0000			0.0894632			0.0447316	+	0.998999i	$0.485757\pi$
$504$		0			0
$505$	−200.000			−0.396040
$506$			1261.94i			2.49396i
$507$		0			0
$508$			1168.20i			2.29960i
$509$		−	829.277i		−	1.62923i	−0.580004	−	0.814614i	$-0.696949\pi$
$509$		−	829.277i		−	1.62923i	0.580004	−	0.814614i	$-0.303051\pi$
$510$		0			0
$511$	−525.000			−1.02740
$512$			1150.17i			2.24643i
$513$		0			0
$514$	1508.00			2.93385
$515$		−	230.755i		−	0.448069i
$516$		0			0
$517$	−100.000			−0.193424
$518$	390.000			0.752896
$519$		0			0
$520$	260.000			0.500000
$521$		−	612.944i		−	1.17648i	−0.808688	−	0.588238i	$-0.799822\pi$
$521$		−	612.944i		−	1.17648i	0.808688	−	0.588238i	$-0.200178\pi$
$522$		0			0
$523$			465.116i			0.889323i	0.895699	+	0.444662i	$0.146676\pi$
$523$			465.116i			0.889323i	−0.895699	+	0.444662i	$0.853324\pi$
$524$	1008.00			1.92366
$525$		0			0
$526$			1117.72i			2.12494i
$527$			540.833i			1.02625i
$528$		0			0
$529$	696.000			1.31569
$530$	−1092.00			−2.06038
$531$		0			0
$532$	−270.000	+	811.249i	−0.507519	+	1.52490i
$533$	130.000			0.243902
$534$		0			0
$535$			302.866i			0.566105i
$536$	−715.000			−1.33396
$537$		0			0
$538$	−390.000			−0.724907
$539$	−240.000			−0.445269
$540$		0			0
$541$	−600.000			−1.10906			−0.554529	−	0.832165i	$-0.687101\pi$
$541$	−600.000			−1.10906			−0.554529	+	0.832165i	$0.687101\pi$
$542$		−	378.583i		−	0.698492i
$543$		0			0
$544$			486.749i			0.894760i
$545$			793.221i			1.45545i
$546$		0			0
$547$		−	598.522i		−	1.09419i	−0.837071	−	0.547095i	$-0.815734\pi$
$547$		−	598.522i		−	1.09419i	0.837071	−	0.547095i	$-0.184266\pi$
$548$	1125.00			2.05292
$549$		0			0
$550$			324.500i			0.589999i
$551$	325.000	+	108.167i	0.589837	+	0.196310i
$552$		0			0
$553$			180.278i			0.325999i
$554$			180.278i			0.325411i
$555$		0			0
$556$	−450.000			−0.809353
$557$	−380.000			−0.682226			−0.341113	−	0.940022i	$-0.610804\pi$
$557$	−380.000			−0.682226			−0.341113	+	0.940022i	$0.610804\pi$
$558$		0			0
$559$		−	72.1110i		−	0.129000i
$560$	580.000			1.03571
$561$		0			0
$562$	1040.00			1.85053
$563$		−	122.589i		−	0.217742i	−0.994056	−	0.108871i	$-0.965276\pi$
$563$		−	122.589i		−	0.217742i	0.994056	−	0.108871i	$-0.0347235\pi$
$564$		0			0
$565$		−	490.355i		−	0.867885i
$566$			1153.78i			2.03847i
$567$		0			0
$568$	1950.00			3.43310
$569$		−	36.0555i		−	0.0633665i	−0.999498	−	0.0316832i	$-0.989913\pi$
$569$		−	36.0555i		−	0.0633665i	0.999498	−	0.0316832i	$-0.0100868\pi$
$570$		0			0
$571$	−790.000			−1.38354			−0.691769	−	0.722119i	$-0.743168\pi$
$571$	−790.000			−1.38354			−0.691769	+	0.722119i	$0.743168\pi$
$572$		−	324.500i		−	0.567307i
$573$		0			0
$574$	650.000			1.13240
$575$	315.000			0.547826
$576$		0			0
$577$	675.000			1.16984			0.584922	−	0.811090i	$-0.301125\pi$
$577$	675.000			1.16984			0.584922	+	0.811090i	$0.301125\pi$
$578$			230.755i			0.399231i
$579$		0			0
$580$		−	648.999i		−	1.11896i
$581$	−200.000			−0.344234
$582$		0			0
$583$			757.166i			1.29874i
$584$			1892.91i			3.24129i
$585$		0			0
$586$	793.000			1.35324
$587$	280.000			0.477002			0.238501	−	0.971142i	$-0.423344\pi$
$587$	280.000			0.477002			0.238501	+	0.971142i	$0.423344\pi$
$588$		0			0
$589$	650.000	+	216.333i	1.10357	+	0.367289i
$590$	260.000			0.440678
$591$		0			0
$592$		−	627.366i		−	1.05974i
$593$	−750.000			−1.26476			−0.632378	−	0.774660i	$-0.717921\pi$
$593$	−750.000			−1.26476			−0.632378	+	0.774660i	$0.717921\pi$
$594$		0			0
$595$	−300.000			−0.504202
$596$	630.000			1.05705
$597$		0			0
$598$	−455.000			−0.760870
$599$		−	504.777i		−	0.842700i	−0.906898	−	0.421350i	$-0.861556\pi$
$599$		−	504.777i		−	0.842700i	0.906898	−	0.421350i	$-0.138444\pi$
$600$		0			0
$601$			612.944i			1.01987i	0.860212	+	0.509937i	$0.170331\pi$
$601$			612.944i			1.01987i	−0.860212	+	0.509937i	$0.829669\pi$
$602$		−	360.555i		−	0.598929i
$603$		0			0
$604$		−	324.500i		−	0.537251i
$605$	84.0000			0.138843
$606$		0			0
$607$		−	987.921i		−	1.62755i	−0.581182	−	0.813774i	$-0.697410\pi$
$607$		−	987.921i		−	1.62755i	0.581182	−	0.813774i	$-0.302590\pi$
$608$	585.000	+	194.700i	0.962171	+	0.320230i
$609$		0			0
$610$		−	576.888i		−	0.945718i
$611$		−	36.0555i		−	0.0590107i
$612$		0			0
$613$	−1200.00			−1.95759			−0.978793	−	0.204853i	$-0.934328\pi$
$613$	−1200.00			−1.95759			−0.978793	+	0.204853i	$0.934328\pi$
$614$	858.000			1.39739
$615$		0			0
$616$		−	901.388i		−	1.46329i
$617$	350.000			0.567261			0.283630	−	0.958934i	$-0.408461\pi$
$617$	350.000			0.567261			0.283630	+	0.958934i	$0.408461\pi$
$618$		0			0
$619$	560.000			0.904685			0.452342	−	0.891844i	$-0.350588\pi$
$619$	560.000			0.904685			0.452342	+	0.891844i	$0.350588\pi$
$620$		−	1298.00i		−	2.09355i
$621$		0			0
$622$			1424.19i			2.28970i
$623$		0			0
$624$		0			0
$625$	−319.000			−0.510400
$626$		−	450.694i		−	0.719958i
$627$		0			0
$628$	−90.0000			−0.143312
$629$			324.500i			0.515898i
$630$		0			0
$631$	−1050.00			−1.66403			−0.832013	−	0.554757i	$-0.812811\pi$
$631$	−1050.00			−1.66403			−0.832013	+	0.554757i	$0.812811\pi$
$632$	650.000			1.02848
$633$		0			0
$634$	−13.0000			−0.0205047
$635$			519.199i			0.817637i
$636$		0			0
$637$		−	86.5332i		−	0.135845i
$638$	−650.000			−1.01881
$639$		0			0
$640$			504.777i			0.788714i
$641$		−	1225.89i		−	1.91246i	−0.292615	−	0.956230i	$-0.594525\pi$
$641$		−	1225.89i		−	1.91246i	0.292615	−	0.956230i	$-0.405475\pi$
$642$		0			0
$643$	−1030.00			−1.60187			−0.800933	−	0.598754i	$-0.795663\pi$
$643$	−1030.00			−1.60187			−0.800933	+	0.598754i	$0.795663\pi$
$644$	−1575.00			−2.44565
$645$		0			0
$646$	−975.000	−	324.500i	−1.50929	−	0.502321i
$647$	−555.000			−0.857805			−0.428903	−	0.903351i	$-0.641100\pi$
$647$	−555.000			−0.857805			−0.428903	+	0.903351i	$0.641100\pi$
$648$		0			0
$649$		−	180.278i		−	0.277777i
$650$	−117.000			−0.180000
$651$		0			0
$652$	2430.00			3.72699
$653$	−50.0000			−0.0765697			−0.0382848	−	0.999267i	$-0.512189\pi$
$653$	−50.0000			−0.0765697			−0.0382848	+	0.999267i	$0.512189\pi$
$654$		0			0
$655$	448.000			0.683969
$656$		−	1045.61i		−	1.59392i
$657$		0			0
$658$		−	180.278i		−	0.273978i
$659$		−	198.305i		−	0.300919i	−0.988616	−	0.150459i	$-0.951925\pi$
$659$		−	198.305i		−	0.300919i	0.988616	−	0.150459i	$-0.0480752\pi$
$660$		0			0
$661$			198.305i			0.300008i	0.988685	+	0.150004i	$0.0479287\pi$
$661$			198.305i			0.300008i	−0.988685	+	0.150004i	$0.952071\pi$
$662$	−715.000			−1.08006
$663$		0			0
$664$			721.110i			1.08601i
$665$	−120.000	+	360.555i	−0.180451	+	0.542188i
$666$		0			0
$667$			630.971i			0.945984i
$668$			1103.30i			1.65164i
$669$		0			0
$670$	−572.000			−0.853731
$671$	−400.000			−0.596125
$672$		0			0
$673$			598.522i			0.889334i	0.895696	+	0.444667i	$0.146678\pi$
$673$			598.522i			0.889334i	−0.895696	+	0.444667i	$0.853322\pi$
$674$	208.000			0.308605
$675$		0			0
$676$	−1404.00			−2.07692
$677$		−	68.5055i		−	0.101190i	−0.998719	−	0.0505949i	$-0.983888\pi$
$677$		−	68.5055i		−	0.101190i	0.998719	−	0.0505949i	$-0.0161117\pi$
$678$		0			0
$679$		−	612.944i		−	0.902715i
$680$			1081.67i			1.59068i
$681$		0			0
$682$	−1300.00			−1.90616
$683$			237.966i			0.348413i	0.984709	+	0.174207i	$0.0557361\pi$
$683$			237.966i			0.348413i	−0.984709	+	0.174207i	$0.944264\pi$
$684$		0			0
$685$	500.000			0.729927
$686$		−	1316.03i		−	1.91841i
$687$		0			0
$688$	−580.000			−0.843023
$689$	−273.000			−0.396226
$690$		0			0
$691$	−820.000			−1.18669			−0.593343	−	0.804950i	$-0.702192\pi$
$691$	−820.000			−1.18669			−0.593343	+	0.804950i	$0.702192\pi$
$692$		−	1103.30i		−	1.59436i
$693$		0			0
$694$		−	144.222i		−	0.207813i
$695$	−200.000			−0.287770
$696$		0			0
$697$			540.833i			0.775944i
$698$		−	353.344i		−	0.506224i
$699$		0			0
$700$	−405.000			−0.578571
$701$	540.000			0.770328			0.385164	−	0.922848i	$-0.374145\pi$
$701$	540.000			0.770328			0.385164	+	0.922848i	$0.374145\pi$
$702$		0			0
$703$	390.000	+	129.800i	0.554765	+	0.184637i
$704$	−10.0000			−0.0142045
$705$		0			0
$706$		−	667.027i		−	0.944797i
$707$	−250.000			−0.353607
$708$		0			0
$709$	268.000			0.377997			0.188999	−	0.981977i	$-0.439476\pi$
$709$	268.000			0.377997			0.188999	+	0.981977i	$0.439476\pi$
$710$	1560.00			2.19718
$711$		0			0
$712$		0			0
$713$			1261.94i			1.76991i
$714$		0			0
$715$		−	144.222i		−	0.201709i
$716$			324.500i			0.453212i
$717$		0			0
$718$		−	811.249i		−	1.12987i
$719$	−105.000			−0.146036			−0.0730181	−	0.997331i	$-0.523263\pi$
$719$	−105.000			−0.146036			−0.0730181	+	0.997331i	$0.523263\pi$
$720$		0			0
$721$		−	288.444i		−	0.400061i
$722$	−780.000	+	1042.00i	−1.08033	+	1.44322i
$723$		0			0
$724$		−	973.499i		−	1.34461i
$725$			162.250i			0.223793i
$726$		0			0
$727$	695.000			0.955983			0.477992	−	0.878364i	$-0.341365\pi$
$727$	695.000			0.955983			0.477992	+	0.878364i	$0.341365\pi$
$728$	325.000			0.446429
$729$		0			0
$730$			1514.33i			2.07443i
$731$	300.000			0.410397
$732$		0			0
$733$	160.000			0.218281			0.109141	−	0.994026i	$-0.465190\pi$
$733$	160.000			0.218281			0.109141	+	0.994026i	$0.465190\pi$
$734$		−	180.278i		−	0.245610i
$735$		0			0
$736$			1135.75i			1.54314i
$737$			396.611i			0.538142i
$738$		0			0
$739$	1028.00			1.39107			0.695535	−	0.718493i	$-0.255168\pi$
$739$	1028.00			1.39107			0.695535	+	0.718493i	$0.255168\pi$
$740$		−	778.799i		−	1.05243i
$741$		0			0
$742$	−1365.00			−1.83962
$743$			526.410i			0.708493i	0.935152	+	0.354247i	$0.115263\pi$
$743$			526.410i			0.708493i	−0.935152	+	0.354247i	$0.884737\pi$
$744$		0			0
$745$	280.000			0.375839
$746$	−1573.00			−2.10858
$747$		0			0
$748$	1350.00			1.80481
$749$			378.583i			0.505451i
$750$		0			0
$751$			36.0555i			0.0480100i	0.999712	+	0.0240050i	$0.00764176\pi$
$751$			36.0555i			0.0480100i	−0.999712	+	0.0240050i	$0.992358\pi$
$752$	−290.000			−0.385638
$753$		0			0
$754$		−	234.361i		−	0.310823i
$755$		−	144.222i		−	0.191023i
$756$		0			0
$757$	60.0000			0.0792602			0.0396301	−	0.999214i	$-0.487382\pi$
$757$	60.0000			0.0792602			0.0396301	+	0.999214i	$0.487382\pi$
$758$	−1755.00			−2.31530
$759$		0			0
$760$	1300.00	+	432.666i	1.71053	+	0.569298i
$761$	655.000			0.860710			0.430355	−	0.902660i	$-0.358388\pi$
$761$	655.000			0.860710			0.430355	+	0.902660i	$0.358388\pi$
$762$		0			0
$763$			991.527i			1.29951i
$764$	1737.00			2.27356
$765$		0			0
$766$	−728.000			−0.950392
$767$	65.0000			0.0847458
$768$		0			0
$769$	−185.000			−0.240572			−0.120286	−	0.992739i	$-0.538381\pi$
$769$	−185.000			−0.240572			−0.120286	+	0.992739i	$0.538381\pi$
$770$		−	721.110i		−	0.936507i
$771$		0			0
$772$		−	2401.30i		−	3.11049i
$773$			320.894i			0.415128i	0.978221	+	0.207564i	$0.0665536\pi$
$773$			320.894i			0.415128i	−0.978221	+	0.207564i	$0.933446\pi$
$774$		0			0
$775$			324.500i			0.418709i
$776$	−2210.00			−2.84794
$777$		0			0
$778$		−	1723.45i		−	2.21524i
$779$	650.000	+	216.333i	0.834403	+	0.277706i
$780$		0			0
$781$		−	1081.67i		−	1.38497i
$782$		−	1892.91i		−	2.42061i
$783$		0			0
$784$	−696.000			−0.887755
$785$	−40.0000			−0.0509554
$786$		0			0
$787$			68.5055i			0.0870463i	0.999052	+	0.0435232i	$0.0138582\pi$
$787$			68.5055i			0.0870463i	−0.999052	+	0.0435232i	$0.986142\pi$
$788$	810.000			1.02792
$789$		0			0
$790$	520.000			0.658228
$791$		−	612.944i		−	0.774897i
$792$		0			0
$793$		−	144.222i		−	0.181869i
$794$		−	2704.16i		−	3.40575i
$795$		0			0
$796$	−1107.00			−1.39070
$797$		−	1258.34i		−	1.57884i	−0.613852	−	0.789421i	$-0.710381\pi$
$797$		−	1258.34i		−	1.57884i	0.613852	−	0.789421i	$-0.289619\pi$
$798$		0			0
$799$	150.000			0.187735
$800$			292.050i			0.365062i
$801$		0			0
$802$	1040.00			1.29676
$803$	1050.00			1.30760
$804$		0			0
$805$	−700.000			−0.869565
$806$		−	468.722i		−	0.581541i
$807$		0			0
$808$			901.388i			1.11558i
$809$	7.00000			0.00865266			0.00432633	−	0.999991i	$-0.498623\pi$
$809$	7.00000			0.00865266			0.00432633	+	0.999991i	$0.498623\pi$
$810$		0			0
$811$			1027.58i			1.26706i	0.773720	+	0.633528i	$0.218394\pi$
$811$			1027.58i			1.26706i	−0.773720	+	0.633528i	$0.781606\pi$
$812$		−	811.249i		−	0.999075i
$813$		0			0
$814$	−780.000			−0.958231
$815$	1080.00			1.32515
$816$		0			0
$817$	120.000	−	360.555i	0.146879	−	0.441316i
$818$	−130.000			−0.158924
$819$		0			0
$820$		−	1298.00i		−	1.58292i
$821$	−842.000			−1.02558			−0.512789	−	0.858515i	$-0.671388\pi$
$821$	−842.000			−1.02558			−0.512789	+	0.858515i	$0.671388\pi$
$822$		0			0
$823$	775.000			0.941677			0.470838	−	0.882219i	$-0.343951\pi$
$823$	775.000			0.941677			0.470838	+	0.882219i	$0.343951\pi$
$824$	−1040.00			−1.26214
$825$		0			0
$826$	325.000			0.393462
$827$			1265.55i			1.53029i	0.643859	+	0.765144i	$0.277332\pi$
$827$			1265.55i			1.53029i	−0.643859	+	0.765144i	$0.722668\pi$
$828$		0			0
$829$		−	1568.41i		−	1.89194i	−0.324260	−	0.945968i	$-0.605115\pi$
$829$		−	1568.41i		−	1.89194i	0.324260	−	0.945968i	$-0.394885\pi$
$830$			576.888i			0.695046i
$831$		0			0
$832$		−	3.60555i		−	0.00433360i
$833$	360.000			0.432173
$834$		0			0
$835$			490.355i			0.587251i
$836$	540.000	−	1622.50i	0.645933	−	1.94079i
$837$		0			0
$838$			403.822i			0.481888i
$839$			1153.78i			1.37518i	0.726099	+	0.687590i	$0.241331\pi$
$839$			1153.78i			1.37518i	−0.726099	+	0.687590i	$0.758669\pi$
$840$		0			0
$841$	516.000			0.613555
$842$	2275.00			2.70190
$843$		0			0
$844$			2109.25i			2.49911i
$845$	−624.000			−0.738462
$846$		0			0
$847$	105.000			0.123967
$848$			2195.78i			2.58936i
$849$		0			0
$850$		−	486.749i		−	0.572646i
$851$			757.166i			0.889737i
$852$		0			0
$853$	400.000			0.468933			0.234467	−	0.972124i	$-0.424666\pi$
$853$	400.000			0.468933			0.234467	+	0.972124i	$0.424666\pi$
$854$		−	721.110i		−	0.844391i
$855$		0			0
$856$	1365.00			1.59463
$857$		−	158.644i		−	0.185116i	−0.995707	−	0.0925579i	$-0.970496\pi$
$857$		−	158.644i		−	0.185116i	0.995707	−	0.0925579i	$-0.0295043\pi$
$858$		0			0
$859$	1432.00			1.66705			0.833527	−	0.552478i	$-0.186318\pi$
$859$	1432.00			1.66705			0.833527	+	0.552478i	$0.186318\pi$
$860$	−720.000			−0.837209
$861$		0			0
$862$	1560.00			1.80974
$863$			129.800i			0.150405i	0.997168	+	0.0752027i	$0.0239604\pi$
$863$			129.800i			0.150405i	−0.997168	+	0.0752027i	$0.976040\pi$
$864$		0			0
$865$		−	490.355i		−	0.566884i
$866$	−2652.00			−3.06236
$867$		0			0
$868$		−	1622.50i		−	1.86924i
$869$		−	360.555i		−	0.414908i
$870$		0			0
$871$	−143.000			−0.164179
$872$	3575.00			4.09977
$873$		0			0
$874$	−2275.00	−	757.166i	−2.60297	−	0.866322i
$875$	−680.000			−0.777143
$876$		0			0
$877$			393.005i			0.448124i	0.974575	+	0.224062i	$0.0719319\pi$
$877$			393.005i			0.448124i	−0.974575	+	0.224062i	$0.928068\pi$
$878$	2860.00			3.25740
$879$		0			0
$880$	−1160.00			−1.31818
$881$	−750.000			−0.851305			−0.425653	−	0.904887i	$-0.639955\pi$
$881$	−750.000			−0.851305			−0.425653	+	0.904887i	$0.639955\pi$
$882$		0			0
$883$	120.000			0.135900			0.0679502	−	0.997689i	$-0.478354\pi$
$883$	120.000			0.135900			0.0679502	+	0.997689i	$0.478354\pi$
$884$			486.749i			0.550622i
$885$		0			0
$886$			2415.72i			2.72655i
$887$		−	850.910i		−	0.959312i	−0.877457	−	0.479656i	$-0.840761\pi$
$887$		−	850.910i		−	0.959312i	0.877457	−	0.479656i	$-0.159239\pi$
$888$		0			0
$889$			648.999i			0.730033i
$890$		0			0
$891$		0			0
$892$			1817.20i			2.03722i
$893$	60.0000	−	180.278i	0.0671892	−	0.201879i
$894$		0			0
$895$			144.222i			0.161142i
$896$			630.971i			0.704209i
$897$		0			0
$898$	130.000			0.144766
$899$	−650.000			−0.723026
$900$		0			0
$901$		−	1135.75i		−	1.26054i
$902$	−1300.00			−1.44124
$903$		0			0
$904$	−2210.00			−2.44469
$905$		−	432.666i		−	0.478084i
$906$		0			0
$907$		−	645.394i		−	0.711570i	−0.934568	−	0.355785i	$-0.884214\pi$
$907$		−	645.394i		−	0.711570i	0.934568	−	0.355785i	$-0.115786\pi$
$908$			2303.95i			2.53739i
$909$		0			0
$910$	260.000			0.285714
$911$			1225.89i			1.34565i	0.739802	+	0.672825i	$0.234919\pi$
$911$			1225.89i			1.34565i	−0.739802	+	0.672825i	$0.765081\pi$
$912$		0			0
$913$	400.000			0.438116
$914$			2722.19i			2.97833i
$915$		0			0
$916$	1440.00			1.57205
$917$	560.000			0.610687
$918$		0			0
$919$	−513.000			−0.558215			−0.279108	−	0.960260i	$-0.590039\pi$
$919$	−513.000			−0.558215			−0.279108	+	0.960260i	$0.590039\pi$
$920$			2523.89i			2.74335i
$921$		0			0
$922$			2783.49i			3.01896i
$923$	390.000			0.422535
$924$		0			0
$925$			194.700i			0.210486i
$926$			1261.94i			1.36279i
$927$		0			0
$928$	−585.000			−0.630388
$929$	1217.00			1.31001			0.655005	−	0.755624i	$-0.272666\pi$
$929$	1217.00			1.31001			0.655005	+	0.755624i	$0.272666\pi$
$930$		0			0
$931$	144.000	−	432.666i	0.154672	−	0.464733i
$932$	−2430.00			−2.60730
$933$		0			0
$934$			252.389i			0.270223i
$935$	600.000			0.641711
$936$		0			0
$937$	−1095.00			−1.16862			−0.584312	−	0.811529i	$-0.698635\pi$
$937$	−1095.00			−1.16862			−0.584312	+	0.811529i	$0.698635\pi$
$938$	−715.000			−0.762260
$939$		0			0
$940$	−360.000			−0.382979
$941$		−	1099.69i		−	1.16864i	−0.811522	−	0.584322i	$-0.801361\pi$
$941$		−	1099.69i		−	1.16864i	0.811522	−	0.584322i	$-0.198639\pi$
$942$		0			0
$943$			1261.94i			1.33822i
$944$		−	522.805i		−	0.553819i
$945$		0			0
$946$			721.110i			0.762273i
$947$	−640.000			−0.675818			−0.337909	−	0.941179i	$-0.609720\pi$
$947$	−640.000			−0.675818			−0.337909	+	0.941179i	$0.609720\pi$
$948$		0			0
$949$			378.583i			0.398928i
$950$	−585.000	−	194.700i	−0.615789	−	0.204947i
$951$		0			0
$952$			1352.08i			1.42025i
$953$		−	1168.20i		−	1.22581i	−0.790156	−	0.612906i	$-0.790000\pi$
$953$		−	1168.20i		−	1.22581i	0.790156	−	0.612906i	$-0.210000\pi$
$954$		0			0
$955$	772.000			0.808377
$956$	1773.00			1.85460
$957$		0			0
$958$		−	1334.05i		−	1.39254i
$959$	625.000			0.651721
$960$		0			0
$961$	−339.000			−0.352758
$962$		−	281.233i		−	0.292342i
$963$		0			0
$964$		−	3569.50i		−	3.70280i
$965$		−	1067.24i		−	1.10595i
$966$		0			0
$967$	−1010.00			−1.04447			−0.522234	−	0.852802i	$-0.674901\pi$
$967$	−1010.00			−1.04447			−0.522234	+	0.852802i	$0.674901\pi$
$968$		−	378.583i		−	0.391098i
$969$		0			0
$970$	−1768.00			−1.82268
$971$			757.166i			0.779779i	0.920862	+	0.389890i	$0.127487\pi$
$971$			757.166i			0.779779i	−0.920862	+	0.389890i	$0.872513\pi$
$972$		0			0
$973$	−250.000			−0.256937
$974$	−1872.00			−1.92197
$975$		0			0
$976$	−1160.00			−1.18852
$977$		−	598.522i		−	0.612612i	−0.951933	−	0.306306i	$-0.900907\pi$
$977$		−	598.522i		−	0.612612i	0.951933	−	0.306306i	$-0.0990930\pi$
$978$		0			0
$979$		0			0
$980$	−864.000			−0.881633
$981$		0			0
$982$		−	2278.71i		−	2.32048i
$983$		−	57.6888i		−	0.0586865i	−0.999569	−	0.0293432i	$-0.990658\pi$
$983$		−	57.6888i		−	0.0586865i	0.999569	−	0.0293432i	$-0.00934159\pi$
$984$		0			0
$985$	360.000			0.365482
$986$	975.000			0.988844
$987$		0			0
$988$	585.000	+	194.700i	0.592105	+	0.197065i
$989$	700.000			0.707786
$990$		0			0
$991$		0			0		1.00000			$0$
$991$		0			0		−1.00000			$\pi$
$992$	−1170.00			−1.17944
$993$		0			0
$994$	1950.00			1.96177
$995$	−492.000			−0.494472
$996$		0			0
$997$	−170.000			−0.170512			−0.0852558	−	0.996359i	$-0.527171\pi$
$997$	−170.000			−0.170512			−0.0852558	+	0.996359i	$0.527171\pi$
$998$		−	1370.11i		−	1.37286i
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	171.3.c.b.37.1		2
3.2	odd	2		19.3.b.b.18.2	yes	2
4.3	odd	2		2736.3.o.d.721.2		2
12.11	even	2		304.3.e.d.113.2		2
15.2	even	4		475.3.d.b.474.1		4
15.8	even	4		475.3.d.b.474.4		4
15.14	odd	2		475.3.c.b.151.1		2
19.18	odd	2	inner	171.3.c.b.37.2		2
24.5	odd	2		1216.3.e.g.1025.2		2
24.11	even	2		1216.3.e.h.1025.1		2
57.2	even	18		361.3.f.d.262.1		12
57.5	odd	18		361.3.f.d.127.2		12
57.8	even	6		361.3.d.b.69.1		4
57.11	odd	6		361.3.d.b.69.2		4
57.14	even	18		361.3.f.d.127.1		12
57.17	odd	18		361.3.f.d.262.2		12
57.23	odd	18		361.3.f.d.307.1		12
57.26	odd	6		361.3.d.b.293.1		4
57.29	even	18		361.3.f.d.299.2		12
57.32	even	18		361.3.f.d.116.1		12
57.35	odd	18		361.3.f.d.333.1		12
57.41	even	18		361.3.f.d.333.2		12
57.44	odd	18		361.3.f.d.116.2		12
57.47	odd	18		361.3.f.d.299.1		12
57.50	even	6		361.3.d.b.293.2		4
57.53	even	18		361.3.f.d.307.2		12
57.56	even	2		19.3.b.b.18.1	✓	2
76.75	even	2		2736.3.o.d.721.1		2
228.227	odd	2		304.3.e.d.113.1		2
285.113	odd	4		475.3.d.b.474.2		4
285.227	odd	4		475.3.d.b.474.3		4
285.284	even	2		475.3.c.b.151.2		2
456.227	odd	2		1216.3.e.h.1025.2		2
456.341	even	2		1216.3.e.g.1025.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
19.3.b.b.18.1	✓	2	57.56	even	2
19.3.b.b.18.2	yes	2	3.2	odd	2
171.3.c.b.37.1		2	1.1	even	1	trivial
171.3.c.b.37.2		2	19.18	odd	2	inner
304.3.e.d.113.1		2	228.227	odd	2
304.3.e.d.113.2		2	12.11	even	2
361.3.d.b.69.1		4	57.8	even	6
361.3.d.b.69.2		4	57.11	odd	6
361.3.d.b.293.1		4	57.26	odd	6
361.3.d.b.293.2		4	57.50	even	6
361.3.f.d.116.1		12	57.32	even	18
361.3.f.d.116.2		12	57.44	odd	18
361.3.f.d.127.1		12	57.14	even	18
361.3.f.d.127.2		12	57.5	odd	18
361.3.f.d.262.1		12	57.2	even	18
361.3.f.d.262.2		12	57.17	odd	18
361.3.f.d.299.1		12	57.47	odd	18
361.3.f.d.299.2		12	57.29	even	18
361.3.f.d.307.1		12	57.23	odd	18
361.3.f.d.307.2		12	57.53	even	18
361.3.f.d.333.1		12	57.35	odd	18
361.3.f.d.333.2		12	57.41	even	18
475.3.c.b.151.1		2	15.14	odd	2
475.3.c.b.151.2		2	285.284	even	2
475.3.d.b.474.1		4	15.2	even	4
475.3.d.b.474.2		4	285.113	odd	4
475.3.d.b.474.3		4	285.227	odd	4
475.3.d.b.474.4		4	15.8	even	4
1216.3.e.g.1025.1		2	456.341	even	2
1216.3.e.g.1025.2		2	24.5	odd	2
1216.3.e.h.1025.1		2	24.11	even	2
1216.3.e.h.1025.2		2	456.227	odd	2
2736.3.o.d.721.1		2	76.75	even	2
2736.3.o.d.721.2		2	4.3	odd	2

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:	\( N \)	\(=\)	\( 171 = 3^{2} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	171.c (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-3.60555i q^{2} -9.00000 q^{4} -4.00000 q^{5} -5.00000 q^{7} +18.0278i q^{8} +O(q^{10})\)	\(q-3.60555i q^{2} -9.00000 q^{4} -4.00000 q^{5} -5.00000 q^{7} +18.0278i q^{8} +14.4222i q^{10} +10.0000 q^{11} +3.60555i q^{13} +18.0278i q^{14} +29.0000 q^{16} -15.0000 q^{17} +(-6.00000 + 18.0278i) q^{19} +36.0000 q^{20} -36.0555i q^{22} -35.0000 q^{23} -9.00000 q^{25} +13.0000 q^{26} +45.0000 q^{28} -18.0278i q^{29} -36.0555i q^{31} -32.4500i q^{32} +54.0833i q^{34} +20.0000 q^{35} -21.6333i q^{37} +(65.0000 + 21.6333i) q^{38} -72.1110i q^{40} -36.0555i q^{41} -20.0000 q^{43} -90.0000 q^{44} +126.194i q^{46} -10.0000 q^{47} -24.0000 q^{49} +32.4500i q^{50} -32.4500i q^{52} +75.7166i q^{53} -40.0000 q^{55} -90.1388i q^{56} -65.0000 q^{58} -18.0278i q^{59} -40.0000 q^{61} -130.000 q^{62} -1.00000 q^{64} -14.4222i q^{65} +39.6611i q^{67} +135.000 q^{68} -72.1110i q^{70} -108.167i q^{71} +105.000 q^{73} -78.0000 q^{74} +(54.0000 - 162.250i) q^{76} -50.0000 q^{77} -36.0555i q^{79} -116.000 q^{80} -130.000 q^{82} +40.0000 q^{83} +60.0000 q^{85} +72.1110i q^{86} +180.278i q^{88} -18.0278i q^{91} +315.000 q^{92} +36.0555i q^{94} +(24.0000 - 72.1110i) q^{95} +122.589i q^{97} +86.5332i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 18 q^{4} - 8 q^{5} - 10 q^{7}+O(q^{10}) \) 2 * q - 18 * q^4 - 8 * q^5 - 10 * q^7	\( 2 q - 18 q^{4} - 8 q^{5} - 10 q^{7} + 20 q^{11} + 58 q^{16} - 30 q^{17} - 12 q^{19} + 72 q^{20} - 70 q^{23} - 18 q^{25} + 26 q^{26} + 90 q^{28} + 40 q^{35} + 130 q^{38} - 40 q^{43} - 180 q^{44} - 20 q^{47} - 48 q^{49} - 80 q^{55} - 130 q^{58} - 80 q^{61} - 260 q^{62} - 2 q^{64} + 270 q^{68} + 210 q^{73} - 156 q^{74} + 108 q^{76} - 100 q^{77} - 232 q^{80} - 260 q^{82} + 80 q^{83} + 120 q^{85} + 630 q^{92} + 48 q^{95}+O(q^{100}) \) 2 * q - 18 * q^4 - 8 * q^5 - 10 * q^7 + 20 * q^11 + 58 * q^16 - 30 * q^17 - 12 * q^19 + 72 * q^20 - 70 * q^23 - 18 * q^25 + 26 * q^26 + 90 * q^28 + 40 * q^35 + 130 * q^38 - 40 * q^43 - 180 * q^44 - 20 * q^47 - 48 * q^49 - 80 * q^55 - 130 * q^58 - 80 * q^61 - 260 * q^62 - 2 * q^64 + 270 * q^68 + 210 * q^73 - 156 * q^74 + 108 * q^76 - 100 * q^77 - 232 * q^80 - 260 * q^82 + 80 * q^83 + 120 * q^85 + 630 * q^92 + 48 * q^95

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