LMFDB - Embedded newform 62.3.b.a.61.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [62,3,Mod(61,62)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(62, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1])) N = Newforms(chi, 3, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("62.61"); S:= CuspForms(chi, 3); N := Newforms(S);

Level:	$ N $	$=$	$ 62 = 2 \cdot 31 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	62.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(0)] == traces)

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

Self dual:	no
Analytic conductor:	$1.68937763903$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.48128.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 14x^{2} + 47 $ x^4 + 14*x^2 + 47
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			61.1
Root			$2.90073i$ of defining polynomial
Character	$\chi$	$=$	62.61
Dual form			62.3.b.a.61.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q-1.41421 q^{2} -4.10225i q^{3} +2.00000 q^{4} -5.82843 q^{5} +5.80145i q^{6} -4.41421 q^{7} -2.82843 q^{8} -7.82843 q^{9} +8.24264 q^{10} -19.8074i q^{11} -8.20449i q^{12} +6.50529i q^{13} +6.24264 q^{14} +23.9096i q^{15} +4.00000 q^{16} -13.3021i q^{17} +11.0711 q^{18} +30.2132 q^{19} -11.6569 q^{20} +18.1082i q^{21} +28.0119i q^{22} +19.1036i q^{23} +11.6029i q^{24} +8.97056 q^{25} -9.19987i q^{26} -4.80608i q^{27} -8.82843 q^{28} -6.50529i q^{29} -33.8133i q^{30} +(24.4142 - 19.1036i) q^{31} -5.65685 q^{32} -81.2548 q^{33} +18.8120i q^{34} +25.7279 q^{35} -15.6569 q^{36} -59.4222i q^{37} -42.7279 q^{38} +26.6863 q^{39} +16.4853 q^{40} +16.6569 q^{41} -25.6089i q^{42} +65.5152i q^{43} -39.6148i q^{44} +45.6274 q^{45} -27.0165i q^{46} -67.7401 q^{47} -16.4090i q^{48} -29.5147 q^{49} -12.6863 q^{50} -54.5685 q^{51} +13.0106i q^{52} +48.4024i q^{53} +6.79682i q^{54} +115.446i q^{55} +12.4853 q^{56} -123.942i q^{57} +9.19987i q^{58} +56.2132 q^{59} +47.8193i q^{60} +4.51454i q^{61} +(-34.5269 + 27.0165i) q^{62} +34.5563 q^{63} +8.00000 q^{64} -37.9156i q^{65} +114.912 q^{66} +21.0538 q^{67} -26.6042i q^{68} +78.3675 q^{69} -36.3848 q^{70} +11.2426 q^{71} +22.1421 q^{72} -25.1966i q^{73} +84.0357i q^{74} -36.7995i q^{75} +60.4264 q^{76} +87.4341i q^{77} -37.7401 q^{78} -36.0956i q^{79} -23.3137 q^{80} -90.1716 q^{81} -23.5563 q^{82} -159.284i q^{83} +36.2164i q^{84} +77.5304i q^{85} -92.6525i q^{86} -26.6863 q^{87} +56.0238i q^{88} +143.166i q^{89} -64.5269 q^{90} -28.7157i q^{91} +38.2071i q^{92} +(-78.3675 - 100.153i) q^{93} +95.7990 q^{94} -176.095 q^{95} +23.2058i q^{96} +27.7107 q^{97} +41.7401 q^{98} +155.061i q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 12 q^{5} - 12 q^{7} - 20 q^{9} + 16 q^{10} + 8 q^{14} + 16 q^{16} + 16 q^{18} + 36 q^{19} - 24 q^{20} - 32 q^{25} - 24 q^{28} + 92 q^{31} - 144 q^{33} + 52 q^{35} - 40 q^{36} - 120 q^{38}+ \cdots - 48 q^{98}+O(q^{100}) $ 4 * q + 8 * q^4 - 12 * q^5 - 12 * q^7 - 20 * q^9 + 16 * q^10 + 8 * q^14 + 16 * q^16 + 16 * q^18 + 36 * q^19 - 24 * q^20 - 32 * q^25 - 24 * q^28 + 92 * q^31 - 144 * q^33 + 52 * q^35 - 40 * q^36 - 120 * q^38 + 152 * q^39 + 32 * q^40 + 44 * q^41 + 92 * q^45 - 56 * q^47 - 152 * q^49 - 96 * q^50 + 8 * q^51 + 16 * q^56 + 140 * q^59 - 8 * q^62 + 76 * q^63 + 32 * q^64 + 256 * q^66 - 176 * q^67 + 8 * q^69 - 72 * q^70 + 28 * q^71 + 32 * q^72 + 72 * q^76 + 64 * q^78 - 48 * q^80 - 372 * q^81 - 32 * q^82 - 152 * q^87 - 128 * q^90 - 8 * q^93 + 304 * q^94 - 348 * q^95 - 172 * q^97 - 48 * q^98

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/62\mathbb{Z}\right)^\times$.

$n$	$3$
$\chi(n)$	$-1$

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 $

$2$	−1.41421			−0.707107
$2$	−1.41421			−0.707107
$3$		−	4.10225i		−	1.36742i	−0.729756	−	0.683708i	$-0.760366\pi$
$3$		−	4.10225i		−	1.36742i	0.729756	−	0.683708i	$-0.239634\pi$
$4$	2.00000			0.500000
$5$	−5.82843			−1.16569			−0.582843	−	0.812585i	$-0.698060\pi$
$5$	−5.82843			−1.16569			−0.582843	+	0.812585i	$0.698060\pi$
$6$			5.80145i			0.966909i
$7$	−4.41421			−0.630602			−0.315301	−	0.948992i	$-0.602105\pi$
$7$	−4.41421			−0.630602			−0.315301	+	0.948992i	$0.602105\pi$
$8$	−2.82843			−0.353553
$9$	−7.82843			−0.869825
$10$	8.24264			0.824264
$11$		−	19.8074i		−	1.80067i	−0.435195	−	0.900336i	$-0.643321\pi$
$11$		−	19.8074i		−	1.80067i	0.435195	−	0.900336i	$-0.356679\pi$
$12$		−	8.20449i		−	0.683708i
$13$			6.50529i			0.500407i	0.968193	+	0.250203i	$0.0804975\pi$
$13$			6.50529i			0.500407i	−0.968193	+	0.250203i	$0.919503\pi$
$14$	6.24264			0.445903
$15$			23.9096i			1.59398i
$16$	4.00000			0.250000
$17$		−	13.3021i		−	0.782477i	−0.920289	−	0.391239i	$-0.872047\pi$
$17$		−	13.3021i		−	0.782477i	0.920289	−	0.391239i	$-0.127953\pi$
$18$	11.0711			0.615059
$19$	30.2132			1.59017			0.795084	−	0.606499i	$-0.207427\pi$
$19$	30.2132			1.59017			0.795084	+	0.606499i	$0.207427\pi$
$20$	−11.6569			−0.582843
$21$			18.1082i			0.862295i
$22$			28.0119i			1.27327i
$23$			19.1036i			0.830590i	0.909687	+	0.415295i	$0.136322\pi$
$23$			19.1036i			0.830590i	−0.909687	+	0.415295i	$0.863678\pi$
$24$			11.6029i			0.483454i
$25$	8.97056			0.358823
$26$		−	9.19987i		−	0.353841i
$27$		−	4.80608i		−	0.178003i
$28$	−8.82843			−0.315301
$29$		−	6.50529i		−	0.224320i	−0.993690	−	0.112160i	$-0.964223\pi$
$29$		−	6.50529i		−	0.224320i	0.993690	−	0.112160i	$-0.0357770\pi$
$30$		−	33.8133i		−	1.12711i
$31$	24.4142	−	19.1036i	0.787555	−	0.616244i
$31$	24.4142	−	19.1036i	0.787555	−	0.616244i
$32$	−5.65685			−0.176777
$33$	−81.2548			−2.46227
$34$			18.8120i			0.553295i
$35$	25.7279			0.735083
$36$	−15.6569			−0.434913
$37$		−	59.4222i		−	1.60601i	−0.595975	−	0.803003i	$-0.703234\pi$
$37$		−	59.4222i		−	1.60601i	0.595975	−	0.803003i	$-0.296766\pi$
$38$	−42.7279			−1.12442
$39$	26.6863			0.684264
$40$	16.4853			0.412132
$41$	16.6569			0.406265			0.203132	−	0.979151i	$-0.434888\pi$
$41$	16.6569			0.406265			0.203132	+	0.979151i	$0.434888\pi$
$42$		−	25.6089i		−	0.609735i
$43$			65.5152i			1.52361i	0.647807	+	0.761804i	$0.275686\pi$
$43$			65.5152i			1.52361i	−0.647807	+	0.761804i	$0.724314\pi$
$44$		−	39.6148i		−	0.900336i
$45$	45.6274			1.01394
$46$		−	27.0165i		−	0.587316i
$47$	−67.7401			−1.44128			−0.720640	−	0.693310i	$-0.756152\pi$
$47$	−67.7401			−1.44128			−0.720640	+	0.693310i	$0.756152\pi$
$48$		−	16.4090i		−	0.341854i
$49$	−29.5147			−0.602341
$50$	−12.6863			−0.253726
$51$	−54.5685			−1.06997
$52$			13.0106i			0.250203i
$53$			48.4024i			0.913252i	0.889659	+	0.456626i	$0.150942\pi$
$53$			48.4024i			0.913252i	−0.889659	+	0.456626i	$0.849058\pi$
$54$			6.79682i			0.125867i
$55$			115.446i			2.09902i
$56$	12.4853			0.222951
$57$		−	123.942i		−	2.17442i
$58$			9.19987i			0.158618i
$59$	56.2132			0.952766			0.476383	−	0.879238i	$-0.341948\pi$
$59$	56.2132			0.952766			0.476383	+	0.879238i	$0.341948\pi$
$60$			47.8193i			0.796988i
$61$			4.51454i			0.0740089i	0.999315	+	0.0370045i	$0.0117816\pi$
$61$			4.51454i			0.0740089i	−0.999315	+	0.0370045i	$0.988218\pi$
$62$	−34.5269	+	27.0165i	−0.556886	+	0.435750i
$63$	34.5563			0.548513
$64$	8.00000			0.125000
$65$		−	37.9156i		−	0.583317i
$66$	114.912			1.74109
$67$	21.0538			0.314236			0.157118	−	0.987580i	$-0.449780\pi$
$67$	21.0538			0.314236			0.157118	+	0.987580i	$0.449780\pi$
$68$		−	26.6042i		−	0.391239i
$69$	78.3675			1.13576
$70$	−36.3848			−0.519783
$71$	11.2426			0.158347			0.0791735	−	0.996861i	$-0.474772\pi$
$71$	11.2426			0.158347			0.0791735	+	0.996861i	$0.474772\pi$
$72$	22.1421			0.307530
$73$		−	25.1966i		−	0.345158i	−0.984996	−	0.172579i	$-0.944790\pi$
$73$		−	25.1966i		−	0.345158i	0.984996	−	0.172579i	$-0.0552101\pi$
$74$			84.0357i			1.13562i
$75$		−	36.7995i		−	0.490659i
$76$	60.4264			0.795084
$77$			87.4341i			1.13551i
$78$	−37.7401			−0.483848
$79$		−	36.0956i		−	0.456907i	−0.973555	−	0.228453i	$-0.926633\pi$
$79$		−	36.0956i		−	0.456907i	0.973555	−	0.228453i	$-0.0733668\pi$
$80$	−23.3137			−0.291421
$81$	−90.1716			−1.11323
$82$	−23.5563			−0.287273
$83$		−	159.284i		−	1.91908i	−0.281569	−	0.959541i	$-0.590855\pi$
$83$		−	159.284i		−	1.91908i	0.281569	−	0.959541i	$-0.409145\pi$
$84$			36.2164i			0.431147i
$85$			77.5304i			0.912122i
$86$		−	92.6525i		−	1.07735i
$87$	−26.6863			−0.306739
$88$			56.0238i			0.636634i
$89$			143.166i			1.60861i	0.594216	+	0.804305i	$0.297462\pi$
$89$			143.166i			1.60861i	−0.594216	+	0.804305i	$0.702538\pi$
$90$	−64.5269			−0.716966
$91$		−	28.7157i		−	0.315557i
$92$			38.2071i			0.415295i
$93$	−78.3675	−	100.153i	−0.842662	−	1.07692i
$94$	95.7990			1.01914
$95$	−176.095			−1.85364
$96$			23.2058i			0.241727i
$97$	27.7107			0.285677			0.142839	−	0.989746i	$-0.454377\pi$
$97$	27.7107			0.285677			0.142839	+	0.989746i	$0.454377\pi$
$98$	41.7401			0.425920
$99$			155.061i			1.56627i
$100$	17.9411			0.179411
$101$	81.7452			0.809358			0.404679	−	0.914459i	$-0.367383\pi$
$101$	81.7452			0.809358			0.404679	+	0.914459i	$0.367383\pi$
$102$	77.1716			0.756584
$103$	185.752			1.80342			0.901710	−	0.432341i	$-0.142312\pi$
$103$	185.752			1.80342			0.901710	+	0.432341i	$0.142312\pi$
$104$		−	18.3997i		−	0.176920i
$105$		−	105.542i		−	1.00516i
$106$		−	68.4513i		−	0.645767i
$107$	81.1838			0.758727			0.379363	−	0.925248i	$-0.376143\pi$
$107$	81.1838			0.758727			0.379363	+	0.925248i	$0.376143\pi$
$108$		−	9.61216i		−	0.0890015i
$109$	−94.6812			−0.868635			−0.434318	−	0.900760i	$-0.643010\pi$
$109$	−94.6812			−0.868635			−0.434318	+	0.900760i	$0.643010\pi$
$110$		−	163.265i		−	1.48423i
$111$	−243.765			−2.19608
$112$	−17.6569			−0.157650
$113$	6.14719			0.0543999			0.0271999	−	0.999630i	$-0.491341\pi$
$113$	6.14719			0.0543999			0.0271999	+	0.999630i	$0.491341\pi$
$114$			175.280i			1.53755i
$115$		−	111.344i		−	0.968206i
$116$		−	13.0106i		−	0.112160i
$117$		−	50.9262i		−	0.435266i
$118$	−79.4975			−0.673707
$119$			58.7184i			0.493432i
$120$		−	67.6267i		−	0.563556i
$121$	−271.333			−2.24242
$122$		−	6.38453i		−	0.0523322i
$123$		−	68.3305i		−	0.555533i
$124$	48.8284	−	38.2071i	0.393778	−	0.308122i
$125$	93.4264			0.747411
$126$	−48.8701			−0.387858
$127$		−	83.4526i		−	0.657107i	−0.944485	−	0.328554i	$-0.893439\pi$
$127$		−	83.4526i		−	0.657107i	0.944485	−	0.328554i	$-0.106561\pi$
$128$	−11.3137			−0.0883883
$129$	268.759			2.08341
$130$			53.6207i			0.412467i
$131$	197.848			1.51029			0.755144	−	0.655559i	$-0.227567\pi$
$131$	197.848			1.51029			0.755144	+	0.655559i	$0.227567\pi$
$132$	−162.510			−1.23113
$133$	−133.368			−1.00276
$134$	−29.7746			−0.222199
$135$			28.0119i			0.207495i
$136$			37.6241i			0.276647i
$137$			125.350i			0.914961i	0.889220	+	0.457481i	$0.151248\pi$
$137$			125.350i			0.914961i	−0.889220	+	0.457481i	$0.848752\pi$
$138$	−110.828			−0.803105
$139$		−	62.8206i		−	0.451947i	−0.974133	−	0.225973i	$-0.927444\pi$
$139$		−	62.8206i		−	0.451947i	0.974133	−	0.225973i	$-0.0725562\pi$
$140$	51.4558			0.367542
$141$			277.887i			1.97083i
$142$	−15.8995			−0.111968
$143$	128.853			0.901069
$144$	−31.3137			−0.217456
$145$			37.9156i			0.261487i
$146$			35.6333i			0.244064i
$147$			121.077i			0.823651i
$148$		−	118.844i		−	0.803003i
$149$	102.024			0.684727			0.342364	−	0.939568i	$-0.388773\pi$
$149$	102.024			0.684727			0.342364	+	0.939568i	$0.388773\pi$
$150$			52.0423i			0.346949i
$151$		−	174.164i		−	1.15341i	−0.816954	−	0.576703i	$-0.804339\pi$
$151$		−	174.164i		−	1.15341i	0.816954	−	0.576703i	$-0.195661\pi$
$152$	−85.4558			−0.562210
$153$			104.135i			0.680618i
$154$		−	123.650i		−	0.802925i
$155$	−142.296	+	111.344i	−0.918042	+	0.718347i
$156$	53.3726			0.342132
$157$	66.4315			0.423130			0.211565	−	0.977364i	$-0.432144\pi$
$157$	66.4315			0.423130			0.211565	+	0.977364i	$0.432144\pi$
$158$			51.0469i			0.323082i
$159$	198.558			1.24880
$160$	32.9706			0.206066
$161$		−	84.3272i		−	0.523772i
$162$	127.522			0.787172
$163$	−157.718			−0.967594			−0.483797	−	0.875180i	$-0.660743\pi$
$163$	−157.718			−0.967594			−0.483797	+	0.875180i	$0.660743\pi$
$164$	33.3137			0.203132
$165$	473.588			2.87023
$166$			225.261i			1.35700i
$167$		−	55.1992i		−	0.330534i	−0.986249	−	0.165267i	$-0.947151\pi$
$167$		−	55.1992i		−	0.330534i	0.986249	−	0.165267i	$-0.0528486\pi$
$168$		−	51.2177i		−	0.304867i
$169$	126.681			0.749593
$170$		−	109.645i		−	0.644968i
$171$	−236.522			−1.38317
$172$			131.030i			0.761804i
$173$	−6.71068			−0.0387900			−0.0193950	−	0.999812i	$-0.506174\pi$
$173$	−6.71068			−0.0387900			−0.0193950	+	0.999812i	$0.506174\pi$
$174$	37.7401			0.216897
$175$	−39.5980			−0.226274
$176$		−	79.2296i		−	0.450168i
$177$		−	230.600i		−	1.30283i
$178$		−	202.468i		−	1.13746i
$179$			63.1829i			0.352977i	0.984303	+	0.176489i	$0.0564738\pi$
$179$			63.1829i			0.352977i	−0.984303	+	0.176489i	$0.943526\pi$
$180$	91.2548			0.506971
$181$			2.04076i			0.0112749i	0.999984	+	0.00563747i	$0.00179447\pi$
$181$			2.04076i			0.0112749i	−0.999984	+	0.00563747i	$0.998206\pi$
$182$			40.6102i			0.223133i
$183$	18.5198			0.101201
$184$		−	54.0330i		−	0.293658i
$185$			346.338i			1.87210i
$186$	110.828	+	141.638i	0.595852	+	0.761494i
$187$	−263.480			−1.40899
$188$	−135.480			−0.720640
$189$			21.2151i			0.112249i
$190$	249.037			1.31072
$191$	−207.787			−1.08789			−0.543944	−	0.839121i	$-0.683070\pi$
$191$	−207.787			−1.08789			−0.543944	+	0.839121i	$0.683070\pi$
$192$		−	32.8180i		−	0.170927i
$193$	48.3970			0.250762			0.125381	−	0.992109i	$-0.459985\pi$
$193$	48.3970			0.250762			0.125381	+	0.992109i	$0.459985\pi$
$194$	−39.1888			−0.202004
$195$	−155.539			−0.797636
$196$	−59.0294			−0.301171
$197$			115.788i			0.587754i	0.955843	+	0.293877i	$0.0949456\pi$
$197$			115.788i			0.587754i	−0.955843	+	0.293877i	$0.905054\pi$
$198$		−	219.289i		−	1.10752i
$199$			262.885i			1.32103i	0.750812	+	0.660516i	$0.229662\pi$
$199$			262.885i			1.32103i	−0.750812	+	0.660516i	$0.770338\pi$
$200$	−25.3726			−0.126863
$201$		−	86.3680i		−	0.429691i
$202$	−115.605			−0.572303
$203$			28.7157i			0.141457i
$204$	−109.137			−0.534986
$205$	−97.0833			−0.473577
$206$	−262.693			−1.27521
$207$		−	149.551i		−	0.722468i
$208$			26.0211i			0.125102i
$209$		−	598.445i		−	2.86337i
$210$			149.259i			0.710759i
$211$	−170.983			−0.810345			−0.405172	−	0.914240i	$-0.632789\pi$
$211$	−170.983			−0.810345			−0.405172	+	0.914240i	$0.632789\pi$
$212$			96.8047i			0.456626i
$213$		−	46.1201i		−	0.216526i
$214$	−114.811			−0.536501
$215$		−	381.850i		−	1.77605i
$216$			13.5936i			0.0629336i
$217$	−107.770	+	84.3272i	−0.496634	+	0.388605i
$218$	133.899			0.614218
$219$	−103.362			−0.471975
$220$			230.892i			1.04951i
$221$	86.5341			0.391557
$222$	344.735			1.55286
$223$		−	349.978i		−	1.56941i	−0.619871	−	0.784704i	$-0.712815\pi$
$223$		−	349.978i		−	1.56941i	0.619871	−	0.784704i	$-0.287185\pi$
$224$	24.9706			0.111476
$225$	−70.2254			−0.312113
$226$	−8.69343			−0.0384665
$227$	−190.485			−0.839142			−0.419571	−	0.907722i	$-0.637820\pi$
$227$	−190.485			−0.839142			−0.419571	+	0.907722i	$0.637820\pi$
$228$		−	247.884i		−	1.08721i
$229$		−	116.854i		−	0.510278i	−0.966904	−	0.255139i	$-0.917879\pi$
$229$		−	116.854i		−	0.510278i	0.966904	−	0.255139i	$-0.0821212\pi$
$230$			157.464i			0.684625i
$231$	358.676			1.55271
$232$			18.3997i			0.0793092i
$233$	−226.019			−0.970040			−0.485020	−	0.874503i	$-0.661188\pi$
$233$	−226.019			−0.970040			−0.485020	+	0.874503i	$0.661188\pi$
$234$			72.0205i			0.307780i
$235$	394.818			1.68008
$236$	112.426			0.476383
$237$	−148.073			−0.624781
$238$		−	83.0403i		−	0.348909i
$239$			46.1908i			0.193267i	0.995320	+	0.0966335i	$0.0308075\pi$
$239$			46.1908i			0.193267i	−0.995320	+	0.0966335i	$0.969193\pi$
$240$			95.6386i			0.398494i
$241$		−	137.827i		−	0.571897i	−0.958245	−	0.285949i	$-0.907691\pi$
$241$		−	137.827i		−	0.571897i	0.958245	−	0.285949i	$-0.0923086\pi$
$242$	383.723			1.58563
$243$			326.651i			1.34424i
$244$			9.02909i			0.0370045i
$245$	172.024			0.702140
$246$			96.6340i			0.392821i
$247$			196.546i			0.795731i
$248$	−69.0538	+	54.0330i	−0.278443	+	0.217875i
$249$	−653.421			−2.62418
$250$	−132.125			−0.528500
$251$			26.4835i			0.105512i	0.998607	+	0.0527559i	$0.0168005\pi$
$251$			26.4835i			0.105512i	−0.998607	+	0.0527559i	$0.983199\pi$
$252$	69.1127			0.274257
$253$	378.392			1.49562
$254$			118.020i			0.464645i
$255$	318.049			1.24725
$256$	16.0000			0.0625000
$257$	17.8183			0.0693320			0.0346660	−	0.999399i	$-0.488963\pi$
$257$	17.8183			0.0693320			0.0346660	+	0.999399i	$0.488963\pi$
$258$	−380.083			−1.47319
$259$			262.302i			1.01275i
$260$		−	75.8312i		−	0.291658i
$261$			50.9262i			0.195119i
$262$	−279.799			−1.06794
$263$			132.317i			0.503108i	0.967843	+	0.251554i	$0.0809415\pi$
$263$			132.317i			0.503108i	−0.967843	+	0.251554i	$0.919058\pi$
$264$	229.823			0.870543
$265$		−	282.110i		−	1.06456i
$266$	188.610			0.709061
$267$	587.304			2.19964
$268$	42.1076			0.157118
$269$			500.233i			1.85960i	0.368064	+	0.929800i	$0.380021\pi$
$269$			500.233i			1.85960i	−0.368064	+	0.929800i	$0.619979\pi$
$270$		−	39.6148i		−	0.146721i
$271$			13.7144i			0.0506067i	0.999680	+	0.0253033i	$0.00805516\pi$
$271$			13.7144i			0.0506067i	−0.999680	+	0.0253033i	$0.991945\pi$
$272$		−	53.2084i		−	0.195619i
$273$	−117.799			−0.431498
$274$		−	177.271i		−	0.646975i
$275$		−	177.684i		−	0.646122i
$276$	156.735			0.567881
$277$			174.235i			0.629008i	0.949256	+	0.314504i	$0.101838\pi$
$277$			174.235i			0.629008i	−0.949256	+	0.314504i	$0.898162\pi$
$278$			88.8418i			0.319575i
$279$	−191.125	+	149.551i	−0.685035	+	0.536025i
$280$	−72.7696			−0.259891
$281$	−181.333			−0.645313			−0.322657	−	0.946516i	$-0.604576\pi$
$281$	−181.333			−0.645313			−0.322657	+	0.946516i	$0.604576\pi$
$282$		−	392.991i		−	1.39359i
$283$	87.7889			0.310208			0.155104	−	0.987898i	$-0.450429\pi$
$283$	87.7889			0.310208			0.155104	+	0.987898i	$0.450429\pi$
$284$	22.4853			0.0791735
$285$			722.387i			2.53469i
$286$	−182.225			−0.637152
$287$	−73.5269			−0.256191
$288$	44.2843			0.153765
$289$	112.054			0.387729
$290$		−	53.6207i		−	0.184899i
$291$		−	113.676i		−	0.390639i
$292$		−	50.3931i		−	0.172579i
$293$	350.593			1.19656			0.598281	−	0.801286i	$-0.295850\pi$
$293$	350.593			1.19656			0.598281	+	0.801286i	$0.295850\pi$
$294$		−	171.228i		−	0.582409i
$295$	−327.635			−1.11063
$296$			168.071i			0.567809i
$297$	−95.1960			−0.320525
$298$	−144.284			−0.484175
$299$	−124.274			−0.415633
$300$		−	73.5989i		−	0.245330i
$301$		−	289.198i		−	0.960791i
$302$			246.306i			0.815581i
$303$		−	335.339i		−	1.10673i
$304$	120.853			0.397542
$305$		−	26.3127i		−	0.0862711i
$306$		−	147.269i		−	0.481270i
$307$	473.468			1.54224			0.771121	−	0.636689i	$-0.219697\pi$
$307$	473.468			1.54224			0.771121	+	0.636689i	$0.219697\pi$
$308$			174.868i			0.567754i
$309$		−	762.002i		−	2.46603i
$310$	201.238	−	157.464i	0.649154	−	0.507948i
$311$	448.252			1.44132			0.720662	−	0.693286i	$-0.243838\pi$
$311$	448.252			1.44132			0.720662	+	0.693286i	$0.243838\pi$
$312$	−75.4802			−0.241924
$313$			362.114i			1.15691i	0.815713	+	0.578457i	$0.196345\pi$
$313$			362.114i			1.15691i	−0.815713	+	0.578457i	$0.803655\pi$
$314$	−93.9483			−0.299198
$315$	−201.409			−0.639394
$316$		−	72.1913i		−	0.228453i
$317$	−226.470			−0.714417			−0.357208	−	0.934025i	$-0.616271\pi$
$317$	−226.470			−0.714417			−0.357208	+	0.934025i	$0.616271\pi$
$318$	−280.804			−0.883032
$319$	−128.853			−0.403927
$320$	−46.6274			−0.145711
$321$		−	333.036i		−	1.03749i
$322$			119.257i			0.370362i
$323$		−	401.899i		−	1.24427i
$324$	−180.343			−0.556615
$325$			58.3561i			0.179557i
$326$	223.047			0.684192
$327$			388.406i			1.18779i
$328$	−47.1127			−0.143636
$329$	299.019			0.908873
$330$	−669.754			−2.02956
$331$			124.575i			0.376360i	0.982135	+	0.188180i	$0.0602588\pi$
$331$			124.575i			0.376360i	−0.982135	+	0.188180i	$0.939741\pi$
$332$		−	318.568i		−	0.959541i
$333$			465.182i			1.39694i
$334$			78.0634i			0.233723i
$335$	−122.711			−0.366301
$336$			72.4328i			0.215574i
$337$			197.199i			0.585161i	0.956241	+	0.292581i	$0.0945140\pi$
$337$			197.199i			0.585161i	−0.956241	+	0.292581i	$0.905486\pi$
$338$	−179.154			−0.530042
$339$		−	25.2173i		−	0.0743872i
$340$			155.061i			0.456061i
$341$	−378.392	−	483.582i	−1.10965	−	1.41813i
$342$	334.492			0.978048
$343$	346.581			1.01044
$344$		−	185.305i		−	0.538677i
$345$	−456.759			−1.32394
$346$	9.49033			0.0274287
$347$		−	381.026i		−	1.09806i	−0.835804	−	0.549029i	$-0.814998\pi$
$347$		−	381.026i		−	1.09806i	0.835804	−	0.549029i	$-0.185002\pi$
$348$	−53.3726			−0.153369
$349$	−212.936			−0.610132			−0.305066	−	0.952331i	$-0.598679\pi$
$349$	−212.936			−0.610132			−0.305066	+	0.952331i	$0.598679\pi$
$350$	56.0000			0.160000
$351$	31.2649			0.0890739
$352$			112.048i			0.318317i
$353$			193.368i			0.547785i	0.961760	+	0.273892i	$0.0883112\pi$
$353$			193.368i			0.547785i	−0.961760	+	0.273892i	$0.911689\pi$
$354$			326.118i			0.921238i
$355$	−65.5269			−0.184583
$356$			286.333i			0.804305i
$357$	240.877			0.674726
$358$		−	89.3541i		−	0.249592i
$359$	51.1939			0.142601			0.0713007	−	0.997455i	$-0.477285\pi$
$359$	51.1939			0.142601			0.0713007	+	0.997455i	$0.477285\pi$
$360$	−129.054			−0.358483
$361$	551.838			1.52864
$362$		−	2.88608i		−	0.00797259i
$363$			1113.08i			3.06632i
$364$		−	57.4315i		−	0.157779i
$365$			146.856i			0.402346i
$366$	−26.1909			−0.0715599
$367$		−	26.1212i		−	0.0711749i	−0.999367	−	0.0355874i	$-0.988670\pi$
$367$		−	26.1212i		−	0.0711749i	0.999367	−	0.0355874i	$-0.0113302\pi$
$368$			76.4143i			0.207647i
$369$	−130.397			−0.353379
$370$		−	489.796i		−	1.32377i
$371$		−	213.658i		−	0.575899i
$372$	−156.735	−	200.306i	−0.421331	−	0.538458i
$373$	−570.539			−1.52960			−0.764798	−	0.644270i	$-0.777161\pi$
$373$	−570.539			−1.52960			−0.764798	+	0.644270i	$0.777161\pi$
$374$	372.617			0.996303
$375$		−	383.258i		−	1.02202i
$376$	191.598			0.509569
$377$	42.3188			0.112251
$378$		−	30.0026i		−	0.0793721i
$379$	67.2893			0.177544			0.0887722	−	0.996052i	$-0.471706\pi$
$379$	67.2893			0.177544			0.0887722	+	0.996052i	$0.471706\pi$
$380$	−352.191			−0.926818
$381$	−342.343			−0.898538
$382$	293.855			0.769254
$383$			252.832i			0.660135i	0.943957	+	0.330067i	$0.107071\pi$
$383$			252.832i			0.660135i	−0.943957	+	0.330067i	$0.892929\pi$
$384$			46.4116i			0.120864i
$385$		−	509.603i		−	1.32364i
$386$	−68.4437			−0.177315
$387$		−	512.881i		−	1.32527i
$388$	55.4214			0.142839
$389$		−	726.077i		−	1.86652i	−0.359199	−	0.933261i	$-0.616950\pi$
$389$		−	726.077i		−	1.86652i	0.359199	−	0.933261i	$-0.383050\pi$
$390$	219.966			0.564014
$391$	254.118			0.649918
$392$	83.4802			0.212960
$393$		−	811.620i		−	2.06519i
$394$		−	163.748i		−	0.415605i
$395$			210.381i			0.532609i
$396$			310.122i			0.783135i
$397$	250.696			0.631475			0.315737	−	0.948847i	$-0.397748\pi$
$397$	250.696			0.631475			0.315737	+	0.948847i	$0.397748\pi$
$398$		−	371.776i		−	0.934111i
$399$			547.107i			1.37119i
$400$	35.8823			0.0897056
$401$			602.618i			1.50279i	0.659854	+	0.751394i	$0.270618\pi$
$401$			602.618i			1.50279i	−0.659854	+	0.751394i	$0.729382\pi$
$402$			122.143i			0.303838i
$403$	124.274	+	158.821i	0.308373	+	0.394098i
$404$	163.490			0.404679
$405$	525.558			1.29768
$406$		−	40.6102i		−	0.100025i
$407$	−1177.00			−2.89189
$408$	154.343			0.378292
$409$			152.678i			0.373297i	0.982427	+	0.186648i	$0.0597625\pi$
$409$			152.678i			0.373297i	−0.982427	+	0.186648i	$0.940237\pi$
$410$	137.296			0.334869
$411$	514.215			1.25113
$412$	371.505			0.901710
$413$	−248.137			−0.600816
$414$			211.497i			0.510862i
$415$			928.374i			2.23705i
$416$		−	36.7995i		−	0.0884602i
$417$	−257.706			−0.617999
$418$			846.329i			2.02471i
$419$	−275.963			−0.658624			−0.329312	−	0.944221i	$-0.606817\pi$
$419$	−275.963			−0.658624			−0.329312	+	0.944221i	$0.606817\pi$
$420$		−	211.085i		−	0.502582i
$421$	38.1270			0.0905629			0.0452815	−	0.998974i	$-0.485582\pi$
$421$	38.1270			0.0905629			0.0452815	+	0.998974i	$0.485582\pi$
$422$	241.806			0.573000
$423$	530.299			1.25366
$424$		−	136.903i		−	0.322883i
$425$		−	119.327i		−	0.280770i
$426$			65.2236i			0.153107i
$427$		−	19.9282i		−	0.0466702i
$428$	162.368			0.379363
$429$		−	528.586i		−	1.23214i
$430$			540.018i			1.25586i
$431$	−406.828			−0.943917			−0.471959	−	0.881621i	$-0.656453\pi$
$431$	−406.828			−0.943917			−0.471959	+	0.881621i	$0.656453\pi$
$432$		−	19.2243i		−	0.0445007i
$433$			788.606i			1.82126i	0.413222	+	0.910630i	$0.364403\pi$
$433$			788.606i			1.82126i	−0.413222	+	0.910630i	$0.635597\pi$
$434$	152.409	−	119.257i	0.351173	−	0.274785i
$435$	155.539			0.357561
$436$	−189.362			−0.434318
$437$			577.180i			1.32078i
$438$	146.177			0.333737
$439$	−175.434			−0.399621			−0.199810	−	0.979835i	$-0.564033\pi$
$439$	−175.434			−0.399621			−0.199810	+	0.979835i	$0.564033\pi$
$440$		−	326.531i		−	0.742115i
$441$	231.054			0.523932
$442$	−122.378			−0.276872
$443$	395.115			0.891907			0.445953	−	0.895056i	$-0.352865\pi$
$443$	395.115			0.891907			0.445953	+	0.895056i	$0.352865\pi$
$444$	−487.529			−1.09804
$445$		−	834.435i		−	1.87513i
$446$			494.943i			1.10974i
$447$		−	418.529i		−	0.936307i
$448$	−35.3137			−0.0788252
$449$		−	293.854i		−	0.654463i	−0.944944	−	0.327232i	$-0.893884\pi$
$449$		−	293.854i		−	0.654463i	0.944944	−	0.327232i	$-0.106116\pi$
$450$	99.3137			0.220697
$451$		−	329.929i		−	0.731550i
$452$	12.2944			0.0271999
$453$	−714.465			−1.57719
$454$	269.387			0.593363
$455$			167.368i			0.367841i
$456$			350.561i			0.768774i
$457$		−	220.014i		−	0.481430i	−0.970596	−	0.240715i	$-0.922618\pi$
$457$		−	220.014i		−	0.481430i	0.970596	−	0.240715i	$-0.0773819\pi$
$458$			165.256i			0.360821i
$459$	−63.9310			−0.139283
$460$		−	222.687i		−	0.484103i
$461$		−	319.492i		−	0.693042i	−0.938042	−	0.346521i	$-0.887363\pi$
$461$		−	319.492i		−	0.693042i	0.938042	−	0.346521i	$-0.112637\pi$
$462$	−507.245			−1.09793
$463$			261.136i			0.564009i	0.959413	+	0.282004i	$0.0909993\pi$
$463$			261.136i			0.564009i	−0.959413	+	0.282004i	$0.909001\pi$
$464$		−	26.0211i		−	0.0560801i
$465$	456.759	+	583.735i	0.982278	+	1.25534i
$466$	319.640			0.685922
$467$	−271.856			−0.582132			−0.291066	−	0.956703i	$-0.594010\pi$
$467$	−271.856			−0.582132			−0.291066	+	0.956703i	$0.594010\pi$
$468$		−	101.852i		−	0.217633i
$469$	−92.9361			−0.198158
$470$	−558.357			−1.18799
$471$		−	272.518i		−	0.578595i
$472$	−158.995			−0.336854
$473$	1297.69			2.74352
$474$	209.407			0.441787
$475$	271.029			0.570588
$476$			117.437i			0.246716i
$477$		−	378.914i		−	0.794370i
$478$		−	65.3237i		−	0.136660i
$479$	−192.335			−0.401535			−0.200767	−	0.979639i	$-0.564344\pi$
$479$	−192.335			−0.401535			−0.200767	+	0.979639i	$0.564344\pi$
$480$		−	135.253i		−	0.281778i
$481$	386.558			0.803656
$482$			194.917i			0.404392i
$483$	−345.931			−0.716213
$484$	−542.666			−1.12121
$485$	−161.510			−0.333010
$486$		−	461.955i		−	0.950524i
$487$			653.665i			1.34223i	0.741354	+	0.671114i	$0.234184\pi$
$487$			653.665i			1.34223i	−0.741354	+	0.671114i	$0.765816\pi$
$488$		−	12.7691i		−	0.0261661i
$489$			646.997i			1.32310i
$490$	−243.279			−0.496488
$491$		−	76.7558i		−	0.156325i	−0.996941	−	0.0781627i	$-0.975095\pi$
$491$		−	76.7558i		−	0.156325i	0.996941	−	0.0781627i	$-0.0249054\pi$
$492$		−	136.661i		−	0.277766i
$493$	−86.5341			−0.175525
$494$		−	277.957i		−	0.562667i
$495$		−	903.760i		−	1.82578i
$496$	97.6569	−	76.4143i	0.196889	−	0.154061i
$497$	−49.6274			−0.0998540
$498$	924.077			1.85558
$499$		−	668.304i		−	1.33929i	−0.742683	−	0.669643i	$-0.766447\pi$
$499$		−	668.304i		−	1.33929i	0.742683	−	0.669643i	$-0.233553\pi$
$500$	186.853			0.373706
$501$	−226.441			−0.451977
$502$		−	37.4533i		−	0.0746081i
$503$	410.899			0.816898			0.408449	−	0.912781i	$-0.366070\pi$
$503$	410.899			0.816898			0.408449	+	0.912781i	$0.366070\pi$
$504$	−97.7401			−0.193929
$505$	−476.446			−0.943457
$506$	−535.127			−1.05756
$507$		−	519.678i		−	1.02501i
$508$		−	166.905i		−	0.328554i
$509$		−	232.833i		−	0.457432i	−0.973493	−	0.228716i	$-0.926547\pi$
$509$		−	232.833i		−	0.457432i	0.973493	−	0.228716i	$-0.0734527\pi$
$510$	−449.789			−0.881939
$511$			111.223i			0.217657i
$512$	−22.6274			−0.0441942
$513$		−	145.207i		−	0.283055i
$514$	−25.1989			−0.0490251
$515$	−1082.64			−2.10222
$516$	537.519			1.04170
$517$			1341.76i			2.59527i
$518$		−	370.951i		−	0.716122i
$519$			27.5289i			0.0530421i
$520$			107.241i			0.206234i
$521$	−472.701			−0.907295			−0.453647	−	0.891181i	$-0.649877\pi$
$521$	−472.701			−0.907295			−0.453647	+	0.891181i	$0.649877\pi$
$522$		−	72.0205i		−	0.137970i
$523$		−	352.814i		−	0.674596i	−0.941398	−	0.337298i	$-0.890487\pi$
$523$		−	352.814i		−	0.674596i	0.941398	−	0.337298i	$-0.109513\pi$
$524$	395.696			0.755144
$525$			162.441i			0.309411i
$526$		−	187.125i		−	0.355751i
$527$	−254.118	−	324.761i	−0.482197	−	0.616244i
$528$	−325.019			−0.615567
$529$	164.054			0.310121
$530$			398.963i			0.752761i
$531$	−440.061			−0.828740
$532$	−266.735			−0.501382
$533$			108.358i			0.203298i
$534$	−830.573			−1.55538
$535$	−473.174			−0.884437
$536$	−59.5492			−0.111099
$537$	259.192			0.482666
$538$		−	707.436i		−	1.31494i
$539$			584.610i			1.08462i
$540$			56.0238i			0.103748i
$541$	−685.337			−1.26680			−0.633399	−	0.773826i	$-0.718341\pi$
$541$	−685.337			−1.26680			−0.633399	+	0.773826i	$0.718341\pi$
$542$		−	19.3951i		−	0.0357843i
$543$	8.37172			0.0154175
$544$			75.2481i			0.138324i
$545$	551.843			1.01256
$546$	166.593			0.305115
$547$	839.947			1.53555			0.767776	−	0.640718i	$-0.221363\pi$
$547$	839.947			1.53555			0.767776	+	0.640718i	$0.221363\pi$
$548$			250.699i			0.457481i
$549$		−	35.3418i		−	0.0643748i
$550$			251.282i			0.456877i
$551$		−	196.546i		−	0.356707i
$552$	−221.657			−0.401552
$553$			159.334i			0.288126i
$554$		−	246.406i		−	0.444775i
$555$	1420.76			2.55993
$556$		−	125.641i		−	0.225973i
$557$		−	336.143i		−	0.603488i	−0.953389	−	0.301744i	$-0.902431\pi$
$557$		−	336.143i		−	0.603488i	0.953389	−	0.301744i	$-0.0975688\pi$
$558$	270.291	−	211.497i	0.484393	−	0.379027i
$559$	−426.195			−0.762424
$560$	102.912			0.183771
$561$			1080.86i			1.92667i
$562$	256.444			0.456305
$563$	−455.444			−0.808959			−0.404479	−	0.914547i	$-0.632547\pi$
$563$	−455.444			−0.808959			−0.404479	+	0.914547i	$0.632547\pi$
$564$			555.773i			0.985414i
$565$	−35.8284			−0.0634131
$566$	−124.152			−0.219350
$567$	398.037			0.702005
$568$	−31.7990			−0.0559841
$569$			370.468i			0.651087i	0.945527	+	0.325543i	$0.105547\pi$
$569$			370.468i			0.651087i	−0.945527	+	0.325543i	$0.894453\pi$
$570$		−	1021.61i		−	1.79230i
$571$			558.126i			0.977454i	0.872437	+	0.488727i	$0.162539\pi$
$571$			558.126i			0.977454i	−0.872437	+	0.488727i	$0.837461\pi$
$572$	257.706			0.450534
$573$			852.393i			1.48760i
$574$	103.983			0.181155
$575$			171.370i			0.298034i
$576$	−62.6274			−0.108728
$577$	−49.6224			−0.0860006			−0.0430003	−	0.999075i	$-0.513692\pi$
$577$	−49.6224			−0.0860006			−0.0430003	+	0.999075i	$0.513692\pi$
$578$	−158.468			−0.274166
$579$		−	198.536i		−	0.342895i
$580$			75.8312i			0.130743i
$581$			703.113i			1.21018i
$582$			160.762i			0.276224i
$583$	958.725			1.64447
$584$			71.2666i			0.122032i
$585$			296.819i			0.507384i
$586$	−495.813			−0.846098
$587$		−	608.882i		−	1.03728i	−0.854994	−	0.518639i	$-0.826439\pi$
$587$		−	608.882i		−	1.03728i	0.854994	−	0.518639i	$-0.173561\pi$
$588$			242.153i			0.411825i
$589$	737.632	−	577.180i	1.25235	−	0.979932i
$590$	463.345			0.785331
$591$	474.989			0.803704
$592$		−	237.689i		−	0.401501i
$593$	719.093			1.21264			0.606318	−	0.795222i	$-0.292646\pi$
$593$	719.093			1.21264			0.606318	+	0.795222i	$0.292646\pi$
$594$	134.627			0.226645
$595$		−	342.236i		−	0.575186i
$596$	204.049			0.342364
$597$	1078.42			1.80640
$598$	175.750			0.293897
$599$	−199.090			−0.332371			−0.166186	−	0.986094i	$-0.553145\pi$
$599$	−199.090			−0.332371			−0.166186	+	0.986094i	$0.553145\pi$
$600$			104.085i			0.173474i
$601$		−	797.535i		−	1.32701i	−0.748170	−	0.663507i	$-0.769067\pi$
$601$		−	797.535i		−	1.32701i	0.748170	−	0.663507i	$-0.230933\pi$
$602$			408.988i			0.679382i
$603$	−164.818			−0.273331
$604$		−	348.329i		−	0.576703i
$605$	1581.44			2.61396
$606$			474.241i			0.782575i
$607$	−197.417			−0.325234			−0.162617	−	0.986689i	$-0.551994\pi$
$607$	−197.417			−0.325234			−0.162617	+	0.986689i	$0.551994\pi$
$608$	−170.912			−0.281105
$609$	117.799			0.193430
$610$			37.2118i			0.0610029i
$611$		−	440.669i		−	0.721226i
$612$			208.269i			0.340309i
$613$		−	766.958i		−	1.25115i	−0.780162	−	0.625577i	$-0.784863\pi$
$613$		−	766.958i		−	1.25115i	0.780162	−	0.625577i	$-0.215137\pi$
$614$	−669.585			−1.09053
$615$			398.259i			0.647576i
$616$		−	247.301i		−	0.401463i
$617$	314.975			0.510494			0.255247	−	0.966876i	$-0.417843\pi$
$617$	314.975			0.510494			0.255247	+	0.966876i	$0.417843\pi$
$618$			1077.63i			1.74374i
$619$			6.67607i			0.0107852i	0.999985	+	0.00539262i	$0.00171653\pi$
$619$			6.67607i			0.0107852i	−0.999985	+	0.00539262i	$0.998283\pi$
$620$	−284.593	+	222.687i	−0.459021	+	0.359173i
$621$	91.8133			0.147847
$622$	−633.924			−1.01917
$623$		−	631.967i		−	1.01439i
$624$	106.745			0.171066
$625$	−768.793			−1.23007
$626$		−	512.106i		−	0.818061i
$627$	−2454.97			−3.91542
$628$	132.863			0.211565
$629$	−790.441			−1.25666
$630$	284.836			0.452120
$631$			1058.10i			1.67686i	0.545012	+	0.838428i	$0.316525\pi$
$631$			1058.10i			1.67686i	−0.545012	+	0.838428i	$0.683475\pi$
$632$			102.094i			0.161541i
$633$			701.413i			1.10808i
$634$	320.277			0.505169
$635$			486.397i			0.765980i
$636$	397.117			0.624398
$637$		−	192.002i		−	0.301416i
$638$	182.225			0.285620
$639$	−88.0122			−0.137734
$640$	65.9411			0.103033
$641$			796.861i			1.24315i	0.783354	+	0.621576i	$0.213507\pi$
$641$			796.861i			1.24315i	−0.783354	+	0.621576i	$0.786493\pi$
$642$			470.984i			0.733620i
$643$		−	875.949i		−	1.36228i	−0.732151	−	0.681142i	$-0.761484\pi$
$643$		−	875.949i		−	1.36228i	0.732151	−	0.681142i	$-0.238516\pi$
$644$		−	168.654i		−	0.261886i
$645$	−1566.44			−2.42860
$646$			568.372i			0.879832i
$647$		−	702.288i		−	1.08545i	−0.839910	−	0.542726i	$-0.817392\pi$
$647$		−	702.288i		−	1.08545i	0.839910	−	0.542726i	$-0.182608\pi$
$648$	255.044			0.393586
$649$		−	1113.44i		−	1.71562i
$650$		−	82.5280i		−	0.126966i
$651$	345.931	+	442.097i	0.531384	+	0.679105i
$652$	−315.436			−0.483797
$653$	−139.653			−0.213863			−0.106932	−	0.994266i	$-0.534103\pi$
$653$	−139.653			−0.213863			−0.106932	+	0.994266i	$0.534103\pi$
$654$		−	549.289i		−	0.839891i
$655$	−1153.14			−1.76052
$656$	66.6274			0.101566
$657$			197.249i			0.300227i
$658$	−422.877			−0.642671
$659$	1241.56			1.88400			0.942000	−	0.335614i	$-0.108944\pi$
$659$	1241.56			1.88400			0.942000	+	0.335614i	$0.108944\pi$
$660$	947.176			1.43511
$661$	−520.200			−0.786990			−0.393495	−	0.919327i	$-0.628734\pi$
$661$	−520.200			−0.786990			−0.393495	+	0.919327i	$0.628734\pi$
$662$		−	176.176i		−	0.266127i
$663$		−	354.984i		−	0.535421i
$664$			450.523i			0.678498i
$665$	777.323			1.16891
$666$		−	657.867i		−	0.987789i
$667$	124.274			0.186318
$668$		−	110.398i		−	0.165267i
$669$	−1435.70			−2.14603
$670$	173.539			0.259014
$671$	89.4214			0.133266
$672$		−	102.435i		−	0.152434i
$673$		−	295.895i		−	0.439665i	−0.975538	−	0.219833i	$-0.929449\pi$
$673$		−	295.895i		−	0.439665i	0.975538	−	0.219833i	$-0.0705511\pi$
$674$		−	278.882i		−	0.413772i
$675$		−	43.1132i		−	0.0638715i
$676$	253.362			0.374797
$677$			130.889i			0.193337i	0.995317	+	0.0966683i	$0.0308186\pi$
$677$			130.889i			0.193337i	−0.995317	+	0.0966683i	$0.969181\pi$
$678$			35.6626i			0.0525997i
$679$	−122.321			−0.180149
$680$		−	219.289i		−	0.322484i
$681$			781.418i			1.14746i
$682$	535.127	+	683.888i	0.784644	+	1.00277i
$683$	731.066			1.07037			0.535187	−	0.844733i	$-0.320241\pi$
$683$	731.066			1.07037			0.535187	+	0.844733i	$0.320241\pi$
$684$	−473.044			−0.691584
$685$		−	730.591i		−	1.06656i
$686$	−490.139			−0.714489
$687$	−479.362			−0.697762
$688$			262.061i			0.380902i
$689$	−314.871			−0.456998
$690$	645.955			0.936167
$691$	−287.247			−0.415697			−0.207849	−	0.978161i	$-0.566646\pi$
$691$	−287.247			−0.415697			−0.207849	+	0.978161i	$0.566646\pi$
$692$	−13.4214			−0.0193950
$693$		−	684.471i		−	0.987693i
$694$			538.852i			0.776444i
$695$			366.145i			0.526828i
$696$	75.4802			0.108449
$697$		−	221.571i		−	0.317893i
$698$	301.137			0.431428
$699$			927.187i			1.32645i
$700$	−79.1960			−0.113137
$701$	379.382			0.541201			0.270600	−	0.962692i	$-0.412778\pi$
$701$	379.382			0.541201			0.270600	+	0.962692i	$0.412778\pi$
$702$	−44.2153			−0.0629848
$703$		−	1795.33i		−	2.55382i
$704$		−	158.459i		−	0.225084i
$705$		−	1619.64i		−	2.29736i
$706$		−	273.464i		−	0.387342i
$707$	−360.841			−0.510383
$708$		−	461.201i		−	0.651414i
$709$			427.358i			0.602762i	0.953504	+	0.301381i	$0.0974476\pi$
$709$			427.358i			0.602762i	−0.953504	+	0.301381i	$0.902552\pi$
$710$	92.6690			0.130520
$711$			282.572i			0.397429i
$712$		−	404.936i		−	0.568730i
$713$	364.946	+	466.399i	0.511846	+	0.654135i
$714$	−340.652			−0.477103
$715$	−751.009			−1.05036
$716$			126.366i			0.176489i
$717$	189.486			0.264276
$718$	−72.3991			−0.100834
$719$			165.377i			0.230009i	0.993365	+	0.115005i	$0.0366883\pi$
$719$			165.377i			0.230009i	−0.993365	+	0.115005i	$0.963312\pi$
$720$	182.510			0.253486
$721$	−819.950			−1.13724
$722$	−780.416			−1.08091
$723$	−565.401			−0.782021
$724$			4.08153i			0.00563747i
$725$		−	58.3561i		−	0.0804912i
$726$		−	1574.13i		−	2.16822i
$727$	−376.217			−0.517493			−0.258746	−	0.965945i	$-0.583309\pi$
$727$	−376.217			−0.517493			−0.258746	+	0.965945i	$0.583309\pi$
$728$			81.2203i			0.111566i
$729$	528.460			0.724911
$730$		−	207.686i		−	0.284502i
$731$	871.490			1.19219
$732$	37.0395			0.0506005
$733$	389.965			0.532012			0.266006	−	0.963971i	$-0.414296\pi$
$733$	389.965			0.532012			0.266006	+	0.963971i	$0.414296\pi$
$734$			36.9409i			0.0503283i
$735$		−	705.686i		−	0.960118i
$736$		−	108.066i		−	0.146829i
$737$		−	417.021i		−	0.565836i
$738$	184.409			0.249877
$739$			1079.09i			1.46020i	0.683338	+	0.730102i	$0.260528\pi$
$739$			1079.09i			1.46020i	−0.683338	+	0.730102i	$0.739472\pi$
$740$			692.676i			0.936048i
$741$	806.278			1.08809
$742$			302.159i			0.407222i
$743$			1176.58i			1.58355i	0.610812	+	0.791776i	$0.290843\pi$
$743$			1176.58i			1.58355i	−0.610812	+	0.791776i	$0.709157\pi$
$744$	221.657	+	283.276i	0.297926	+	0.380747i
$745$	−594.642			−0.798177
$746$	806.864			1.08159
$747$			1246.94i			1.66927i
$748$	−526.960			−0.704493
$749$	−358.362			−0.478455
$750$			542.009i			0.722679i
$751$	−88.5421			−0.117899			−0.0589494	−	0.998261i	$-0.518775\pi$
$751$	−88.5421			−0.117899			−0.0589494	+	0.998261i	$0.518775\pi$
$752$	−270.960			−0.360320
$753$	108.642			0.144278
$754$	−59.8478			−0.0793737
$755$			1015.10i			1.34451i
$756$			42.4301i			0.0561245i
$757$		−	457.411i		−	0.604242i	−0.953270	−	0.302121i	$-0.902305\pi$
$757$		−	457.411i		−	0.604242i	0.953270	−	0.302121i	$-0.0976946\pi$
$758$	−95.1615			−0.125543
$759$		−	1552.26i		−	2.04513i
$760$	498.073			0.655359
$761$			878.422i			1.15430i	0.816638	+	0.577150i	$0.195835\pi$
$761$			878.422i			1.15430i	−0.816638	+	0.577150i	$0.804165\pi$
$762$	484.146			0.635363
$763$	417.943			0.547763
$764$	−415.574			−0.543944
$765$		−	606.941i		−	0.793387i
$766$		−	357.558i		−	0.466786i
$767$			365.683i			0.476771i
$768$		−	65.6359i		−	0.0854635i
$769$	772.734			1.00486			0.502428	−	0.864619i	$-0.332440\pi$
$769$	772.734			1.00486			0.502428	+	0.864619i	$0.332440\pi$
$770$			720.688i			0.935958i
$771$		−	73.0952i		−	0.0948057i
$772$	96.7939			0.125381
$773$			243.752i			0.315333i	0.987492	+	0.157667i	$0.0503971\pi$
$773$			243.752i			0.315333i	−0.987492	+	0.157667i	$0.949603\pi$
$774$			725.323i			0.937110i
$775$	219.009	−	171.370i	0.282593	−	0.221122i
$776$	−78.3776			−0.101002
$777$	1076.03			1.38485
$778$			1026.83i			1.31983i
$779$	503.257			0.646029
$780$	−311.078			−0.398818
$781$		−	222.687i		−	0.285131i
$782$	−359.377			−0.459561
$783$	−31.2649			−0.0399297
$784$	−118.059			−0.150585
$785$	−387.191			−0.493237
$786$			1147.80i			1.46031i
$787$		−	210.039i		−	0.266886i	−0.991056	−	0.133443i	$-0.957397\pi$
$787$		−	210.039i		−	0.266886i	0.991056	−	0.133443i	$-0.0426033\pi$
$788$			231.575i			0.293877i
$789$	542.798			0.687957
$790$		−	297.523i		−	0.376612i
$791$	−27.1350			−0.0343047
$792$		−	438.578i		−	0.553760i
$793$	−29.3684			−0.0370346
$794$	−354.537			−0.446520
$795$	−1157.28			−1.45570
$796$			525.771i			0.660516i
$797$		−	451.447i		−	0.566433i	−0.959056	−	0.283217i	$-0.908598\pi$
$797$		−	451.447i		−	0.566433i	0.959056	−	0.283217i	$-0.0914015\pi$
$798$		−	773.725i		−	0.969581i
$799$			901.087i			1.12777i
$800$	−50.7452			−0.0634315
$801$		−	1120.77i		−	1.39921i
$802$		−	852.230i		−	1.06263i
$803$	−499.078			−0.621517
$804$		−	172.736i		−	0.214846i
$805$			491.495i			0.610553i
$806$	−175.750	−	224.607i	−0.218052	−	0.278669i
$807$	2052.08			2.54285
$808$	−231.210			−0.286151
$809$		−	439.303i		−	0.543019i	−0.962436	−	0.271510i	$-0.912477\pi$
$809$		−	439.303i		−	0.543019i	0.962436	−	0.271510i	$-0.0875229\pi$
$810$	−743.252			−0.917595
$811$	−1336.32			−1.64774			−0.823870	−	0.566779i	$-0.808189\pi$
$811$	−1336.32			−1.64774			−0.823870	+	0.566779i	$0.808189\pi$
$812$			57.4315i			0.0707284i
$813$	56.2599			0.0692004
$814$	1664.53			2.04487
$815$	919.247			1.12791
$816$	−218.274			−0.267493
$817$			1979.42i			2.42280i
$818$		−	215.920i		−	0.263961i
$819$			224.799i			0.274480i
$820$	−194.167			−0.236788
$821$		−	437.079i		−	0.532374i	−0.963921	−	0.266187i	$-0.914236\pi$
$821$		−	437.079i		−	0.532374i	0.963921	−	0.266187i	$-0.0857639\pi$
$822$	−727.210			−0.884684
$823$		−	405.739i		−	0.493001i	−0.969143	−	0.246500i	$-0.920719\pi$
$823$		−	405.739i		−	0.493001i	0.969143	−	0.246500i	$-0.0792806\pi$
$824$	−525.387			−0.637605
$825$	−728.902			−0.883517
$826$	350.919			0.424841
$827$		−	729.979i		−	0.882683i	−0.897339	−	0.441342i	$-0.854503\pi$
$827$		−	729.979i		−	0.882683i	0.897339	−	0.441342i	$-0.145497\pi$
$828$		−	299.102i		−	0.361234i
$829$			847.054i			1.02178i	0.859647	+	0.510889i	$0.170684\pi$
$829$			847.054i			1.02178i	−0.859647	+	0.510889i	$0.829316\pi$
$830$		−	1312.92i		−	1.58183i
$831$	714.755			0.860115
$832$			52.0423i			0.0625508i
$833$			392.608i			0.471318i
$834$	364.451			0.436991
$835$			321.724i			0.385299i
$836$		−	1196.89i		−	1.43169i
$837$	−91.8133	−	117.337i	−0.109693	−	0.140187i
$838$	390.271			0.465717
$839$	407.720			0.485959			0.242980	−	0.970031i	$-0.421875\pi$
$839$	407.720			0.485959			0.242980	+	0.970031i	$0.421875\pi$
$840$			298.519i			0.355379i
$841$	798.681			0.949680
$842$	−53.9197			−0.0640376
$843$			743.873i			0.882411i
$844$	−341.966			−0.405172
$845$	−738.352			−0.873790
$846$	−749.955			−0.886472
$847$	1197.72			1.41408
$848$			193.609i			0.228313i
$849$		−	360.132i		−	0.424183i
$850$			168.754i			0.198535i
$851$	1135.18			1.33393
$852$		−	92.2402i		−	0.108263i
$853$	−1231.52			−1.44376			−0.721878	−	0.692021i	$-0.756721\pi$
$853$	−1231.52			−1.44376			−0.721878	+	0.692021i	$0.756721\pi$
$854$			28.1827i			0.0330008i
$855$	1378.55			1.61234
$856$	−229.622			−0.268250
$857$	563.750			0.657818			0.328909	−	0.944362i	$-0.393319\pi$
$857$	563.750			0.657818			0.328909	+	0.944362i	$0.393319\pi$
$858$			747.534i			0.871251i
$859$		−	1092.81i		−	1.27218i	−0.771614	−	0.636092i	$-0.780550\pi$
$859$		−	1092.81i		−	1.27218i	0.771614	−	0.636092i	$-0.219450\pi$
$860$		−	763.701i		−	0.888024i
$861$			301.626i			0.350320i
$862$	575.342			0.667450
$863$			512.739i			0.594136i	0.954856	+	0.297068i	$0.0960088\pi$
$863$			512.739i			0.594136i	−0.954856	+	0.297068i	$0.903991\pi$
$864$			27.1873i			0.0314668i
$865$	39.1127			0.0452170
$866$		−	1115.26i		−	1.28783i
$867$		−	459.672i		−	0.530187i
$868$	−215.539	+	168.654i	−0.248317	+	0.194302i
$869$	−714.960			−0.822739
$870$	−219.966			−0.252834
$871$			136.961i			0.157246i
$872$	267.799			0.307109
$873$	−216.931			−0.248489
$874$		−	816.256i		−	0.933931i
$875$	−412.404			−0.471319
$876$	−206.725			−0.235987
$877$	1361.26			1.55218			0.776088	−	0.630625i	$-0.217201\pi$
$877$	1361.26			1.55218			0.776088	+	0.630625i	$0.217201\pi$
$878$	248.101			0.282575
$879$		−	1438.22i		−	1.63620i
$880$			461.784i			0.524754i
$881$			980.799i			1.11328i	0.830754	+	0.556640i	$0.187910\pi$
$881$			980.799i			1.11328i	−0.830754	+	0.556640i	$0.812090\pi$
$882$	−326.759			−0.370476
$883$			879.489i			0.996023i	0.867170	+	0.498012i	$0.165936\pi$
$883$			879.489i			0.996023i	−0.867170	+	0.498012i	$0.834064\pi$
$884$	173.068			0.195778
$885$			1344.04i			1.51869i
$886$	−558.777			−0.630673
$887$	−798.079			−0.899751			−0.449876	−	0.893091i	$-0.648532\pi$
$887$	−798.079			−0.899751			−0.449876	+	0.893091i	$0.648532\pi$
$888$	689.470			0.776430
$889$			368.378i			0.414373i
$890$			1180.07i			1.32592i
$891$			1786.06i			2.00456i
$892$		−	699.956i		−	0.784704i
$893$	−2046.65			−2.29188
$894$			591.890i			0.662069i
$895$		−	368.257i		−	0.411460i
$896$	49.9411			0.0557379
$897$			509.803i			0.568343i
$898$			415.572i			0.462775i
$899$	−124.274	−	158.821i	−0.138236	−	0.176665i
$900$	−140.451			−0.156056
$901$	643.854			0.714599
$902$			466.590i			0.517284i
$903$	−1186.36			−1.31380
$904$	−17.3869			−0.0192333
$905$		−	11.8944i		−	0.0131430i
$906$	1010.41			1.11524
$907$	586.301			0.646417			0.323209	−	0.946328i	$-0.395238\pi$
$907$	586.301			0.646417			0.323209	+	0.946328i	$0.395238\pi$
$908$	−380.971			−0.419571
$909$	−639.936			−0.704000
$910$		−	236.693i		−	0.260103i
$911$		−	1086.27i		−	1.19239i	−0.802838	−	0.596197i	$-0.796678\pi$
$911$		−	1086.27i		−	1.19239i	0.802838	−	0.596197i	$-0.203322\pi$
$912$		−	495.768i		−	0.543605i
$913$	−3155.00			−3.45564
$914$			311.146i			0.340423i
$915$	−107.941			−0.117968
$916$		−	233.707i		−	0.255139i
$917$	−873.342			−0.952391
$918$	90.4121			0.0984881
$919$	362.396			0.394337			0.197169	−	0.980370i	$-0.436825\pi$
$919$	362.396			0.394337			0.197169	+	0.980370i	$0.436825\pi$
$920$			314.928i			0.342313i
$921$		−	1942.28i		−	2.10888i
$922$			451.830i			0.490054i
$923$			73.1366i			0.0792379i
$924$	717.352			0.776355
$925$		−	533.051i		−	0.576271i
$926$		−	369.302i		−	0.398815i
$927$	−1454.15			−1.56866
$928$			36.7995i			0.0396546i
$929$			66.6020i			0.0716922i	0.999357	+	0.0358461i	$0.0114126\pi$
$929$			66.6020i			0.0716922i	−0.999357	+	0.0358461i	$0.988587\pi$
$930$	−645.955	−	825.526i	−0.694576	−	0.887663i
$931$	−891.734			−0.957824
$932$	−452.039			−0.485020
$933$		−	1838.84i		−	1.97089i
$934$	384.462			0.411630
$935$	1535.68			1.64243
$936$			144.041i			0.153890i
$937$	803.024			0.857015			0.428508	−	0.903538i	$-0.359039\pi$
$937$	803.024			0.857015			0.428508	+	0.903538i	$0.359039\pi$
$938$	131.431			0.140119
$939$	1485.48			1.58198
$940$	789.637			0.840039
$941$		−	1229.02i		−	1.30608i	−0.757322	−	0.653042i	$-0.773493\pi$
$941$		−	1229.02i		−	1.30608i	0.757322	−	0.653042i	$-0.226507\pi$
$942$			385.399i			0.409128i
$943$			318.205i			0.337439i
$944$	224.853			0.238192
$945$		−	123.650i		−	0.130847i
$946$	−1835.20			−1.93996
$947$		−	755.838i		−	0.798139i	−0.916921	−	0.399070i	$-0.869333\pi$
$947$		−	755.838i		−	0.798139i	0.916921	−	0.399070i	$-0.130667\pi$
$948$	−296.146			−0.312391
$949$	163.911			0.172720
$950$	−383.294			−0.403467
$951$			929.036i			0.976905i
$952$		−	166.081i		−	0.174454i
$953$		−	831.719i		−	0.872738i	−0.899768	−	0.436369i	$-0.856264\pi$
$953$		−	831.719i		−	0.872738i	0.899768	−	0.436369i	$-0.143736\pi$
$954$			535.866i			0.561704i
$955$	1211.07			1.26814
$956$			92.3816i			0.0966335i
$957$			528.586i			0.552336i
$958$	272.003			0.283928
$959$		−	553.320i		−	0.576976i
$960$			191.277i			0.199247i
$961$	231.108	−	932.797i	0.240487	−	0.970652i
$962$	−546.676			−0.568270
$963$	−635.541			−0.659960
$964$		−	275.654i		−	0.285949i
$965$	−282.078			−0.292309
$966$	489.220			0.506439
$967$			1276.50i			1.32006i	0.751237	+	0.660032i	$0.229457\pi$
$967$			1276.50i			1.32006i	−0.751237	+	0.660032i	$0.770543\pi$
$968$	767.446			0.792816
$969$	−1648.69			−1.70143
$970$	228.409			0.235473
$971$	−1899.15			−1.95587			−0.977936	−	0.208906i	$-0.933010\pi$
$971$	−1899.15			−1.95587			−0.977936	+	0.208906i	$0.933010\pi$
$972$			653.303i			0.672122i
$973$			277.304i			0.284999i
$974$		−	924.422i		−	0.949098i
$975$	239.391			0.245529
$976$			18.0582i			0.0185022i
$977$	902.996			0.924254			0.462127	−	0.886814i	$-0.347086\pi$
$977$	902.996			0.924254			0.462127	+	0.886814i	$0.347086\pi$
$978$		−	914.992i		−	0.935575i
$979$	2835.75			2.89658
$980$	344.049			0.351070
$981$	741.205			0.755561
$982$			108.549i			0.110539i
$983$			1559.25i			1.58622i	0.609079	+	0.793110i	$0.291539\pi$
$983$			1559.25i			1.58622i	−0.609079	+	0.793110i	$0.708461\pi$
$984$			193.268i			0.196410i
$985$		−	674.859i		−	0.685136i
$986$	122.378			0.124115
$987$		−	1226.65i		−	1.24281i
$988$			393.091i			0.397866i
$989$	−1251.57			−1.26549
$990$			1278.11i			1.29102i
$991$		−	1385.77i		−	1.39836i	−0.714947	−	0.699179i	$-0.753549\pi$
$991$		−	1385.77i		−	1.39836i	0.714947	−	0.699179i	$-0.246451\pi$
$992$	−138.108	+	108.066i	−0.139221	+	0.108938i
$993$	511.038			0.514640
$994$	70.1838			0.0706074
$995$		−	1532.21i		−	1.53991i
$996$	−1306.84			−1.31209
$997$	1199.12			1.20273			0.601363	−	0.798976i	$-0.294625\pi$
$997$	1199.12			1.20273			0.601363	+	0.798976i	$0.294625\pi$
$998$			945.124i			0.947019i
$999$	−285.588			−0.285874

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	62.3.b.a.61.1	✓	4
3.2	odd	2		558.3.d.a.433.4		4
4.3	odd	2		496.3.e.d.433.4		4
5.2	odd	4		1550.3.d.a.1549.1		8
5.3	odd	4		1550.3.d.a.1549.8		8
5.4	even	2		1550.3.c.a.1301.4		4
31.30	odd	2	inner	62.3.b.a.61.2	yes	4
93.92	even	2		558.3.d.a.433.3		4
124.123	even	2		496.3.e.d.433.1		4
155.92	even	4		1550.3.d.a.1549.4		8
155.123	even	4		1550.3.d.a.1549.5		8
155.154	odd	2		1550.3.c.a.1301.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
62.3.b.a.61.1	✓	4	1.1	even	1	trivial
62.3.b.a.61.2	yes	4	31.30	odd	2	inner
496.3.e.d.433.1		4	124.123	even	2
496.3.e.d.433.4		4	4.3	odd	2
558.3.d.a.433.3		4	93.92	even	2
558.3.d.a.433.4		4	3.2	odd	2
1550.3.c.a.1301.3		4	155.154	odd	2
1550.3.c.a.1301.4		4	5.4	even	2
1550.3.d.a.1549.1		8	5.2	odd	4
1550.3.d.a.1549.4		8	155.92	even	4
1550.3.d.a.1549.5		8	155.123	even	4
1550.3.d.a.1549.8		8	5.3	odd	4

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 62 = 2 \cdot 31 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	62.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.41421 q^{2} -4.10225i q^{3} +2.00000 q^{4} -5.82843 q^{5} +5.80145i q^{6} -4.41421 q^{7} -2.82843 q^{8} -7.82843 q^{9} +8.24264 q^{10} -19.8074i q^{11} -8.20449i q^{12} +6.50529i q^{13} +6.24264 q^{14} +23.9096i q^{15} +4.00000 q^{16} -13.3021i q^{17} +11.0711 q^{18} +30.2132 q^{19} -11.6569 q^{20} +18.1082i q^{21} +28.0119i q^{22} +19.1036i q^{23} +11.6029i q^{24} +8.97056 q^{25} -9.19987i q^{26} -4.80608i q^{27} -8.82843 q^{28} -6.50529i q^{29} -33.8133i q^{30} +(24.4142 - 19.1036i) q^{31} -5.65685 q^{32} -81.2548 q^{33} +18.8120i q^{34} +25.7279 q^{35} -15.6569 q^{36} -59.4222i q^{37} -42.7279 q^{38} +26.6863 q^{39} +16.4853 q^{40} +16.6569 q^{41} -25.6089i q^{42} +65.5152i q^{43} -39.6148i q^{44} +45.6274 q^{45} -27.0165i q^{46} -67.7401 q^{47} -16.4090i q^{48} -29.5147 q^{49} -12.6863 q^{50} -54.5685 q^{51} +13.0106i q^{52} +48.4024i q^{53} +6.79682i q^{54} +115.446i q^{55} +12.4853 q^{56} -123.942i q^{57} +9.19987i q^{58} +56.2132 q^{59} +47.8193i q^{60} +4.51454i q^{61} +(-34.5269 + 27.0165i) q^{62} +34.5563 q^{63} +8.00000 q^{64} -37.9156i q^{65} +114.912 q^{66} +21.0538 q^{67} -26.6042i q^{68} +78.3675 q^{69} -36.3848 q^{70} +11.2426 q^{71} +22.1421 q^{72} -25.1966i q^{73} +84.0357i q^{74} -36.7995i q^{75} +60.4264 q^{76} +87.4341i q^{77} -37.7401 q^{78} -36.0956i q^{79} -23.3137 q^{80} -90.1716 q^{81} -23.5563 q^{82} -159.284i q^{83} +36.2164i q^{84} +77.5304i q^{85} -92.6525i q^{86} -26.6863 q^{87} +56.0238i q^{88} +143.166i q^{89} -64.5269 q^{90} -28.7157i q^{91} +38.2071i q^{92} +(-78.3675 - 100.153i) q^{93} +95.7990 q^{94} -176.095 q^{95} +23.2058i q^{96} +27.7107 q^{97} +41.7401 q^{98} +155.061i q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 12 q^{5} - 12 q^{7} - 20 q^{9} + 16 q^{10} + 8 q^{14} + 16 q^{16} + 16 q^{18} + 36 q^{19} - 24 q^{20} - 32 q^{25} - 24 q^{28} + 92 q^{31} - 144 q^{33} + 52 q^{35} - 40 q^{36} - 120 q^{38}+ \cdots - 48 q^{98}+O(q^{100}) \) 4 * q + 8 * q^4 - 12 * q^5 - 12 * q^7 - 20 * q^9 + 16 * q^10 + 8 * q^14 + 16 * q^16 + 16 * q^18 + 36 * q^19 - 24 * q^20 - 32 * q^25 - 24 * q^28 + 92 * q^31 - 144 * q^33 + 52 * q^35 - 40 * q^36 - 120 * q^38 + 152 * q^39 + 32 * q^40 + 44 * q^41 + 92 * q^45 - 56 * q^47 - 152 * q^49 - 96 * q^50 + 8 * q^51 + 16 * q^56 + 140 * q^59 - 8 * q^62 + 76 * q^63 + 32 * q^64 + 256 * q^66 - 176 * q^67 + 8 * q^69 - 72 * q^70 + 28 * q^71 + 32 * q^72 + 72 * q^76 + 64 * q^78 - 48 * q^80 - 372 * q^81 - 32 * q^82 - 152 * q^87 - 128 * q^90 - 8 * q^93 + 304 * q^94 - 348 * q^95 - 172 * q^97 - 48 * q^98

Introduction

L-functions

Modular forms

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