LMFDB - Embedded newform 4020.2.f.a.401.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4020,2,Mod(401,4020)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4020, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("4020.401");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4020.f (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:	$32.0998616126$
Analytic rank:	$0$
Dimension:	$46$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			401.4
Character	$\chi$	$=$	4020.401
Dual form			4020.2.f.a.401.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.72437 + 0.162914i) q^{3} -1.00000 q^{5} -4.26819i q^{7} +(2.94692 - 0.561849i) q^{9} +O(q^{10})$	\(q+(-1.72437 + 0.162914i) q^{3} -1.00000 q^{5} -4.26819i q^{7} +(2.94692 - 0.561849i) q^{9} -2.35667 q^{11} +0.194362i q^{13} +(1.72437 - 0.162914i) q^{15} +0.326741i q^{17} +7.96669 q^{19} +(0.695347 + 7.35994i) q^{21} +6.28068i q^{23} +1.00000 q^{25} +(-4.99005 + 1.44893i) q^{27} -3.08809i q^{29} +1.32731i q^{31} +(4.06377 - 0.383934i) q^{33} +4.26819i q^{35} -3.97910 q^{37} +(-0.0316642 - 0.335152i) q^{39} +7.68993 q^{41} +12.0629i q^{43} +(-2.94692 + 0.561849i) q^{45} +10.6895i q^{47} -11.2174 q^{49} +(-0.0532307 - 0.563423i) q^{51} +7.78220 q^{53} +2.35667 q^{55} +(-13.7375 + 1.29789i) q^{57} -9.12746i q^{59} -8.50032i q^{61} +(-2.39808 - 12.5780i) q^{63} -0.194362i q^{65} +(-6.00330 - 5.56420i) q^{67} +(-1.02321 - 10.8302i) q^{69} -5.38056i q^{71} +9.17468 q^{73} +(-1.72437 + 0.162914i) q^{75} +10.0587i q^{77} +9.07939i q^{79} +(8.36865 - 3.31145i) q^{81} +12.1811i q^{83} -0.326741i q^{85} +(0.503093 + 5.32502i) q^{87} -6.98512i q^{89} +0.829571 q^{91} +(-0.216238 - 2.28878i) q^{93} -7.96669 q^{95} -4.00684i q^{97} +(-6.94491 + 1.32409i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) $ 46 * q - 46 * q^5 - 4 * q^9	$ 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) $ 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/4020\mathbb{Z}\right)^\times$.

$n$	$1141$	$2011$	$2681$	$3217$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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 $

$2$		0			0
$2$		0			0
$3$	−1.72437	+	0.162914i	−0.995567	+	0.0940585i
$3$	−1.72437	+	0.162914i	−0.995567	+	0.0940585i
$4$		0			0
$5$	−1.00000			−0.447214
$5$	−1.00000			−0.447214
$6$		0			0
$7$		−	4.26819i		−	1.61322i	−0.591082	−	0.806611i	$-0.701299\pi$
$7$		−	4.26819i		−	1.61322i	0.591082	−	0.806611i	$-0.298701\pi$
$8$		0			0
$9$	2.94692	−	0.561849i	0.982306	−	0.187283i
$10$		0			0
$11$	−2.35667			−0.710562			−0.355281	−	0.934760i	$-0.615615\pi$
$11$	−2.35667			−0.710562			−0.355281	+	0.934760i	$0.615615\pi$
$12$		0			0
$13$			0.194362i			0.0539062i	0.999637	+	0.0269531i	$0.00858047\pi$
$13$			0.194362i			0.0539062i	−0.999637	+	0.0269531i	$0.991420\pi$
$14$		0			0
$15$	1.72437	−	0.162914i	0.445231	−	0.0420642i
$16$		0			0
$17$			0.326741i			0.0792463i	0.999215	+	0.0396231i	$0.0126157\pi$
$17$			0.326741i			0.0792463i	−0.999215	+	0.0396231i	$0.987384\pi$
$18$		0			0
$19$	7.96669			1.82768			0.913842	−	0.406069i	$-0.133101\pi$
$19$	7.96669			1.82768			0.913842	+	0.406069i	$0.133101\pi$
$20$		0			0
$21$	0.695347	+	7.35994i	0.151737	+	1.60607i
$22$		0			0
$23$			6.28068i			1.30961i	0.755797	+	0.654806i	$0.227250\pi$
$23$			6.28068i			1.30961i	−0.755797	+	0.654806i	$0.772750\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$	−4.99005	+	1.44893i	−0.960336	+	0.278847i
$28$		0			0
$29$		−	3.08809i		−	0.573444i	−0.958014	−	0.286722i	$-0.907434\pi$
$29$		−	3.08809i		−	0.573444i	0.958014	−	0.286722i	$-0.0925656\pi$
$30$		0			0
$31$			1.32731i			0.238393i	0.992871	+	0.119196i	$0.0380318\pi$
$31$			1.32731i			0.238393i	−0.992871	+	0.119196i	$0.961968\pi$
$32$		0			0
$33$	4.06377	−	0.383934i	0.707412	−	0.0668344i
$34$		0			0
$35$			4.26819i			0.721455i
$36$		0			0
$37$	−3.97910			−0.654161			−0.327080	−	0.944997i	$-0.606065\pi$
$37$	−3.97910			−0.654161			−0.327080	+	0.944997i	$0.606065\pi$
$38$		0			0
$39$	−0.0316642	−	0.335152i	−0.00507033	−	0.0536672i
$40$		0			0
$41$	7.68993			1.20097			0.600483	−	0.799638i	$-0.294975\pi$
$41$	7.68993			1.20097			0.600483	+	0.799638i	$0.294975\pi$
$42$		0			0
$43$			12.0629i			1.83958i	0.392411	+	0.919790i	$0.371641\pi$
$43$			12.0629i			1.83958i	−0.392411	+	0.919790i	$0.628359\pi$
$44$		0			0
$45$	−2.94692	+	0.561849i	−0.439301	+	0.0837555i
$46$		0			0
$47$			10.6895i			1.55922i	0.626267	+	0.779609i	$0.284582\pi$
$47$			10.6895i			1.55922i	−0.626267	+	0.779609i	$0.715418\pi$
$48$		0			0
$49$	−11.2174			−1.60249
$50$		0			0
$51$	−0.0532307	−	0.563423i	−0.00745378	−	0.0788949i
$52$		0			0
$53$	7.78220			1.06897			0.534484	−	0.845179i	$-0.320506\pi$
$53$	7.78220			1.06897			0.534484	+	0.845179i	$0.320506\pi$
$54$		0			0
$55$	2.35667			0.317773
$56$		0			0
$57$	−13.7375	+	1.29789i	−1.81958	+	0.171909i
$58$		0			0
$59$		−	9.12746i		−	1.18829i	−0.804356	−	0.594147i	$-0.797490\pi$
$59$		−	9.12746i		−	1.18829i	0.804356	−	0.594147i	$-0.202510\pi$
$60$		0			0
$61$		−	8.50032i		−	1.08836i	−0.838970	−	0.544178i	$-0.816842\pi$
$61$		−	8.50032i		−	1.08836i	0.838970	−	0.544178i	$-0.183158\pi$
$62$		0			0
$63$	−2.39808	−	12.5780i	−0.302129	−	1.58468i
$64$		0			0
$65$		−	0.194362i		−	0.0241076i
$66$		0			0
$67$	−6.00330	−	5.56420i	−0.733420	−	0.679776i
$67$	−6.00330	−	5.56420i	−0.733420	−	0.679776i
$68$		0			0
$69$	−1.02321	−	10.8302i	−0.123180	−	1.30381i
$70$		0			0
$71$		−	5.38056i		−	0.638555i	−0.947661	−	0.319277i	$-0.896560\pi$
$71$		−	5.38056i		−	0.638555i	0.947661	−	0.319277i	$-0.103440\pi$
$72$		0			0
$73$	9.17468			1.07381			0.536907	−	0.843641i	$-0.319592\pi$
$73$	9.17468			1.07381			0.536907	+	0.843641i	$0.319592\pi$
$74$		0			0
$75$	−1.72437	+	0.162914i	−0.199113	+	0.0188117i
$76$		0			0
$77$			10.0587i			1.14630i
$78$		0			0
$79$			9.07939i			1.02151i	0.859726	+	0.510756i	$0.170634\pi$
$79$			9.07939i			1.02151i	−0.859726	+	0.510756i	$0.829366\pi$
$80$		0			0
$81$	8.36865	−	3.31145i	0.929850	−	0.367938i
$82$		0			0
$83$			12.1811i			1.33705i	0.743689	+	0.668526i	$0.233074\pi$
$83$			12.1811i			1.33705i	−0.743689	+	0.668526i	$0.766926\pi$
$84$		0			0
$85$		−	0.326741i		−	0.0354400i
$86$		0			0
$87$	0.503093	+	5.32502i	0.0539373	+	0.570902i
$88$		0			0
$89$		−	6.98512i		−	0.740421i	−0.928948	−	0.370211i	$-0.879285\pi$
$89$		−	6.98512i		−	0.740421i	0.928948	−	0.370211i	$-0.120715\pi$
$90$		0			0
$91$	0.829571			0.0869627
$92$		0			0
$93$	−0.216238	−	2.28878i	−0.0224228	−	0.237336i
$94$		0			0
$95$	−7.96669			−0.817365
$96$		0			0
$97$		−	4.00684i		−	0.406833i	−0.979092	−	0.203417i	$-0.934795\pi$
$97$		−	4.00684i		−	0.406833i	0.979092	−	0.203417i	$-0.0652046\pi$
$98$		0			0
$99$	−6.94491	+	1.32409i	−0.697990	+	0.133076i
$100$		0			0
$101$	−7.17057			−0.713498			−0.356749	−	0.934200i	$-0.616115\pi$
$101$	−7.17057			−0.713498			−0.356749	+	0.934200i	$0.616115\pi$
$102$		0			0
$103$	19.5052			1.92190			0.960950	−	0.276722i	$-0.0892482\pi$
$103$	19.5052			1.92190			0.960950	+	0.276722i	$0.0892482\pi$
$104$		0			0
$105$	−0.695347	−	7.35994i	−0.0678590	−	0.718257i
$106$		0			0
$107$		−	3.03255i		−	0.293168i	−0.989198	−	0.146584i	$-0.953172\pi$
$107$		−	3.03255i		−	0.293168i	0.989198	−	0.146584i	$-0.0468279\pi$
$108$		0			0
$109$		−	11.1335i		−	1.06639i	−0.845992	−	0.533196i	$-0.820991\pi$
$109$		−	11.1335i		−	1.06639i	0.845992	−	0.533196i	$-0.179009\pi$
$110$		0			0
$111$	6.86146	−	0.648252i	0.651261	−	0.0615294i
$112$		0			0
$113$	9.67007			0.909683			0.454842	−	0.890572i	$-0.349696\pi$
$113$	9.67007			0.909683			0.454842	+	0.890572i	$0.349696\pi$
$114$		0			0
$115$		−	6.28068i		−	0.585677i
$116$		0			0
$117$	0.109202	+	0.572767i	0.0100957	+	0.0529524i
$118$		0			0
$119$	1.39459			0.127842
$120$		0			0
$121$	−5.44611			−0.495101
$122$		0			0
$123$	−13.2603	+	1.25280i	−1.19564	+	0.112961i
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−5.83487			−0.517761			−0.258881	−	0.965909i	$-0.583354\pi$
$127$	−5.83487			−0.517761			−0.258881	+	0.965909i	$0.583354\pi$
$128$		0			0
$129$	−1.96522	−	20.8010i	−0.173028	−	1.83142i
$130$		0			0
$131$			2.17644i			0.190157i	0.995470	+	0.0950784i	$0.0303102\pi$
$131$			2.17644i			0.190157i	−0.995470	+	0.0950784i	$0.969690\pi$
$132$		0			0
$133$		−	34.0033i		−	2.94846i
$134$		0			0
$135$	4.99005	−	1.44893i	0.429475	−	0.124704i
$136$		0			0
$137$	−9.62109			−0.821985			−0.410993	−	0.911639i	$-0.634818\pi$
$137$	−9.62109			−0.821985			−0.410993	+	0.911639i	$0.634818\pi$
$138$		0			0
$139$			4.92907i			0.418078i	0.977907	+	0.209039i	$0.0670336\pi$
$139$			4.92907i			0.418078i	−0.977907	+	0.209039i	$0.932966\pi$
$140$		0			0
$141$	−1.74146	−	18.4326i	−0.146658	−	1.55230i
$142$		0			0
$143$		−	0.458046i		−	0.0383037i
$144$		0			0
$145$			3.08809i			0.256452i
$146$		0			0
$147$	19.3430	−	1.82747i	1.59538	−	0.150728i
$148$		0			0
$149$			0.0674479i			0.00552555i	0.999996	+	0.00276277i	$0.000879419\pi$
$149$			0.0674479i			0.00552555i	−0.999996	+	0.00276277i	$0.999121\pi$
$150$		0			0
$151$	4.30923			0.350680			0.175340	−	0.984508i	$-0.443897\pi$
$151$	4.30923			0.350680			0.175340	+	0.984508i	$0.443897\pi$
$152$		0			0
$153$	0.183579	+	0.962878i	0.0148415	+	0.0778441i
$154$		0			0
$155$		−	1.32731i		−	0.106612i
$156$		0			0
$157$	10.1636			0.811140			0.405570	−	0.914064i	$-0.367073\pi$
$157$	10.1636			0.811140			0.405570	+	0.914064i	$0.367073\pi$
$158$		0			0
$159$	−13.4194	+	1.26783i	−1.06423	+	0.100545i
$160$		0			0
$161$	26.8071			2.11270
$162$		0			0
$163$	−20.9216			−1.63870			−0.819352	−	0.573291i	$-0.805666\pi$
$163$	−20.9216			−1.63870			−0.819352	+	0.573291i	$0.805666\pi$
$164$		0			0
$165$	−4.06377	+	0.383934i	−0.316364	+	0.0298893i
$166$		0			0
$167$		−	11.2992i		−	0.874360i	−0.899374	−	0.437180i	$-0.855977\pi$
$167$		−	11.2992i		−	0.874360i	0.899374	−	0.437180i	$-0.144023\pi$
$168$		0			0
$169$	12.9622			0.997094
$170$		0			0
$171$	23.4772	−	4.47608i	1.79535	−	0.342294i
$172$		0			0
$173$			16.1555i			1.22828i	0.789198	+	0.614139i	$0.210496\pi$
$173$			16.1555i			1.22828i	−0.789198	+	0.614139i	$0.789504\pi$
$174$		0			0
$175$		−	4.26819i		−	0.322645i
$176$		0			0
$177$	1.48699	+	15.7391i	0.111769	+	1.18303i
$178$		0			0
$179$	−2.31394			−0.172952			−0.0864758	−	0.996254i	$-0.527561\pi$
$179$	−2.31394			−0.172952			−0.0864758	+	0.996254i	$0.527561\pi$
$180$		0			0
$181$	3.90459			0.290226			0.145113	−	0.989415i	$-0.453645\pi$
$181$	3.90459			0.290226			0.145113	+	0.989415i	$0.453645\pi$
$182$		0			0
$183$	1.38482	+	14.6577i	0.102369	+	1.08353i
$184$		0			0
$185$	3.97910			0.292550
$186$		0			0
$187$		−	0.770019i		−	0.0563094i
$188$		0			0
$189$	6.18431	+	21.2985i	0.449842	+	1.54924i
$190$		0			0
$191$	8.26032			0.597696			0.298848	−	0.954301i	$-0.403398\pi$
$191$	8.26032			0.597696			0.298848	+	0.954301i	$0.403398\pi$
$192$		0			0
$193$	9.35059			0.673071			0.336535	−	0.941671i	$-0.390745\pi$
$193$	9.35059			0.673071			0.336535	+	0.941671i	$0.390745\pi$
$194$		0			0
$195$	0.0316642	+	0.335152i	0.00226752	+	0.0240007i
$196$		0			0
$197$	23.7454			1.69179			0.845894	−	0.533351i	$-0.179067\pi$
$197$	23.7454			1.69179			0.845894	+	0.533351i	$0.179067\pi$
$198$		0			0
$199$	20.2934			1.43856			0.719281	−	0.694720i	$-0.244472\pi$
$199$	20.2934			1.43856			0.719281	+	0.694720i	$0.244472\pi$
$200$		0			0
$201$	11.2584	+	8.61674i	0.794107	+	0.607778i
$202$		0			0
$203$	−13.1805			−0.925093
$204$		0			0
$205$	−7.68993			−0.537088
$206$		0			0
$207$	3.52879	+	18.5087i	0.245268	+	1.28644i
$208$		0			0
$209$	−18.7749			−1.29868
$210$		0			0
$211$	−4.79395			−0.330029			−0.165015	−	0.986291i	$-0.552767\pi$
$211$	−4.79395			−0.330029			−0.165015	+	0.986291i	$0.552767\pi$
$212$		0			0
$213$	0.876569	+	9.27809i	0.0600615	+	0.635724i
$214$		0			0
$215$		−	12.0629i		−	0.822685i
$216$		0			0
$217$	5.66522			0.384580
$218$		0			0
$219$	−15.8206	+	1.49468i	−1.06905	+	0.101001i
$220$		0			0
$221$	−0.0635058			−0.00427186
$222$		0			0
$223$	−6.31435			−0.422840			−0.211420	−	0.977395i	$-0.567809\pi$
$223$	−6.31435			−0.422840			−0.211420	+	0.977395i	$0.567809\pi$
$224$		0			0
$225$	2.94692	−	0.561849i	0.196461	−	0.0374566i
$226$		0			0
$227$		−	8.35014i		−	0.554218i	−0.960838	−	0.277109i	$-0.910624\pi$
$227$		−	8.35014i		−	0.554218i	0.960838	−	0.277109i	$-0.0893763\pi$
$228$		0			0
$229$			5.86043i			0.387268i	0.981074	+	0.193634i	$0.0620275\pi$
$229$			5.86043i			0.387268i	−0.981074	+	0.193634i	$0.937973\pi$
$230$		0			0
$231$	−1.63870	−	17.3449i	−0.107819	−	1.14121i
$232$		0			0
$233$	9.83313			0.644190			0.322095	−	0.946707i	$-0.395613\pi$
$233$	9.83313			0.644190			0.322095	+	0.946707i	$0.395613\pi$
$234$		0			0
$235$		−	10.6895i		−	0.697303i
$236$		0			0
$237$	−1.47916	−	15.6562i	−0.0960818	−	1.01698i
$238$		0			0
$239$	15.9330			1.03062			0.515309	−	0.857004i	$-0.327677\pi$
$239$	15.9330			1.03062			0.515309	+	0.857004i	$0.327677\pi$
$240$		0			0
$241$	7.27476			0.468608			0.234304	−	0.972163i	$-0.424719\pi$
$241$	7.27476			0.468608			0.234304	+	0.972163i	$0.424719\pi$
$242$		0			0
$243$	−13.8912	+	7.07353i	−0.891120	+	0.453767i
$244$		0			0
$245$	11.2174			0.716654
$246$		0			0
$247$			1.54842i			0.0985235i
$248$		0			0
$249$	−1.98448	−	21.0048i	−0.125761	−	1.33112i
$250$		0			0
$251$	−12.2157			−0.771045			−0.385523	−	0.922698i	$-0.625979\pi$
$251$	−12.2157			−0.771045			−0.385523	+	0.922698i	$0.625979\pi$
$252$		0			0
$253$		−	14.8015i		−	0.930561i
$254$		0			0
$255$	0.0532307	+	0.563423i	0.00333343	+	0.0352829i
$256$		0			0
$257$		−	24.7994i		−	1.54694i	−0.633830	−	0.773472i	$-0.718518\pi$
$257$		−	24.7994i		−	1.54694i	0.633830	−	0.773472i	$-0.281482\pi$
$258$		0			0
$259$			16.9836i			1.05531i
$260$		0			0
$261$	−1.73504	−	9.10035i	−0.107396	−	0.563298i
$262$		0			0
$263$		−	17.7038i		−	1.09166i	−0.837895	−	0.545831i	$-0.816214\pi$
$263$		−	17.7038i		−	1.09166i	0.837895	−	0.545831i	$-0.183786\pi$
$264$		0			0
$265$	−7.78220			−0.478057
$266$		0			0
$267$	1.13797	+	12.0449i	0.0696429	+	0.737139i
$268$		0			0
$269$		−	24.8681i		−	1.51624i	−0.652118	−	0.758118i	$-0.726119\pi$
$269$		−	24.8681i		−	1.51624i	0.652118	−	0.758118i	$-0.273881\pi$
$270$		0			0
$271$		−	15.4244i		−	0.936965i	−0.883473	−	0.468483i	$-0.844801\pi$
$271$		−	15.4244i		−	0.936965i	0.883473	−	0.468483i	$-0.155199\pi$
$272$		0			0
$273$	−1.43049	+	0.135149i	−0.0865771	+	0.00817958i
$274$		0			0
$275$	−2.35667			−0.142112
$276$		0			0
$277$	23.1542			1.39120			0.695602	−	0.718427i	$-0.255138\pi$
$277$	23.1542			1.39120			0.695602	+	0.718427i	$0.255138\pi$
$278$		0			0
$279$	0.745750	+	3.91148i	0.0446469	+	0.234174i
$280$		0			0
$281$	19.7067			1.17560			0.587802	−	0.809005i	$-0.299993\pi$
$281$	19.7067			1.17560			0.587802	+	0.809005i	$0.299993\pi$
$282$		0			0
$283$	5.92893			0.352438			0.176219	−	0.984351i	$-0.443613\pi$
$283$	5.92893			0.352438			0.176219	+	0.984351i	$0.443613\pi$
$284$		0			0
$285$	13.7375	−	1.29789i	0.813742	−	0.0768801i
$286$		0			0
$287$		−	32.8221i		−	1.93742i
$288$		0			0
$289$	16.8932			0.993720
$290$		0			0
$291$	0.652771	+	6.90929i	0.0382661	+	0.405030i
$292$		0			0
$293$			24.3356i			1.42170i	0.703342	+	0.710851i	$0.251690\pi$
$293$			24.3356i			1.42170i	−0.703342	+	0.710851i	$0.748310\pi$
$294$		0			0
$295$			9.12746i			0.531421i
$296$		0			0
$297$	11.7599	−	3.41465i	0.682378	−	0.198138i
$298$		0			0
$299$	−1.22072			−0.0705962
$300$		0			0
$301$	51.4868			2.96765
$302$		0			0
$303$	12.3647	−	1.16819i	0.710335	−	0.0671106i
$304$		0			0
$305$			8.50032i			0.486727i
$306$		0			0
$307$	−8.07408			−0.460812			−0.230406	−	0.973095i	$-0.574005\pi$
$307$	−8.07408			−0.460812			−0.230406	+	0.973095i	$0.574005\pi$
$308$		0			0
$309$	−33.6341	+	3.17766i	−1.91338	+	0.180771i
$310$		0			0
$311$	10.5108			0.596014			0.298007	−	0.954564i	$-0.403678\pi$
$311$	10.5108			0.596014			0.298007	+	0.954564i	$0.403678\pi$
$312$		0			0
$313$			6.51535i			0.368270i	0.982901	+	0.184135i	$0.0589483\pi$
$313$			6.51535i			0.368270i	−0.982901	+	0.184135i	$0.941052\pi$
$314$		0			0
$315$	2.39808	+	12.5780i	0.135116	+	0.708690i
$316$		0			0
$317$		−	19.1735i		−	1.07689i	−0.842661	−	0.538444i	$-0.819012\pi$
$317$		−	19.1735i		−	1.07689i	0.842661	−	0.538444i	$-0.180988\pi$
$318$		0			0
$319$			7.27761i			0.407468i
$320$		0			0
$321$	0.494046	+	5.22925i	0.0275749	+	0.291868i
$322$		0			0
$323$			2.60304i			0.144837i
$324$		0			0
$325$			0.194362i			0.0107812i
$326$		0			0
$327$	1.81380	+	19.1982i	0.100303	+	1.06166i
$328$		0			0
$329$	45.6246			2.51536
$330$		0			0
$331$		−	4.06666i		−	0.223524i	−0.993735	−	0.111762i	$-0.964351\pi$
$331$		−	4.06666i		−	0.223524i	0.993735	−	0.111762i	$-0.0356494\pi$
$332$		0			0
$333$	−11.7261	+	2.23566i	−0.642586	+	0.122513i
$334$		0			0
$335$	6.00330	+	5.56420i	0.327995	+	0.304005i
$336$		0			0
$337$		−	0.274431i		−	0.0149492i	−0.999972	−	0.00747461i	$-0.997621\pi$
$337$		−	0.274431i		−	0.0149492i	0.999972	−	0.00747461i	$-0.00237926\pi$
$338$		0			0
$339$	−16.6748	+	1.57539i	−0.905650	+	0.0855634i
$340$		0			0
$341$		−	3.12804i		−	0.169393i
$342$		0			0
$343$			18.0007i			0.971946i
$344$		0			0
$345$	1.02321	+	10.8302i	0.0550879	+	0.583080i
$346$		0			0
$347$	24.3451			1.30691			0.653457	−	0.756964i	$-0.273318\pi$
$347$	24.3451			1.30691			0.653457	+	0.756964i	$0.273318\pi$
$348$		0			0
$349$	−19.0268			−1.01848			−0.509241	−	0.860624i	$-0.670074\pi$
$349$	−19.0268			−1.01848			−0.509241	+	0.860624i	$0.670074\pi$
$350$		0			0
$351$	−0.281616	−	0.969874i	−0.0150316	−	0.0517680i
$352$		0			0
$353$	−29.1855			−1.55339			−0.776695	−	0.629877i	$-0.783105\pi$
$353$	−29.1855			−1.55339			−0.776695	+	0.629877i	$0.783105\pi$
$354$		0			0
$355$			5.38056i			0.285570i
$356$		0			0
$357$	−2.40479	+	0.227198i	−0.127275	+	0.0120246i
$358$		0			0
$359$		−	19.7547i		−	1.04261i	−0.853369	−	0.521307i	$-0.825445\pi$
$359$		−	19.7547i		−	1.04261i	0.853369	−	0.521307i	$-0.174555\pi$
$360$		0			0
$361$	44.4682			2.34043
$362$		0			0
$363$	9.39113	−	0.887249i	0.492906	−	0.0465685i
$364$		0			0
$365$	−9.17468			−0.480225
$366$		0			0
$367$		−	16.9573i		−	0.885162i	−0.896729	−	0.442581i	$-0.854063\pi$
$367$		−	16.9573i		−	0.885162i	0.896729	−	0.442581i	$-0.145937\pi$
$368$		0			0
$369$	22.6616	−	4.32058i	1.17972	−	0.224920i
$370$		0			0
$371$		−	33.2159i		−	1.72448i
$372$		0			0
$373$		−	2.74393i		−	0.142076i	−0.997474	−	0.0710378i	$-0.977369\pi$
$373$		−	2.74393i		−	0.142076i	0.997474	−	0.0710378i	$-0.0226311\pi$
$374$		0			0
$375$	1.72437	−	0.162914i	0.0890462	−	0.00841285i
$376$		0			0
$377$	0.600206			0.0309122
$378$		0			0
$379$		−	34.4274i		−	1.76842i	−0.467092	−	0.884209i	$-0.654698\pi$
$379$		−	34.4274i		−	1.76842i	0.467092	−	0.884209i	$-0.345302\pi$
$380$		0			0
$381$	10.0615	−	0.950583i	0.515466	−	0.0486999i
$382$		0			0
$383$	−4.75114			−0.242772			−0.121386	−	0.992605i	$-0.538734\pi$
$383$	−4.75114			−0.242772			−0.121386	+	0.992605i	$0.538734\pi$
$384$		0			0
$385$		−	10.0587i		−	0.512639i
$386$		0			0
$387$	6.77754	+	35.5485i	0.344522	+	1.80703i
$388$		0			0
$389$		−	13.1287i		−	0.665652i	−0.942988	−	0.332826i	$-0.891998\pi$
$389$		−	13.1287i		−	0.665652i	0.942988	−	0.332826i	$-0.108002\pi$
$390$		0			0
$391$	−2.05215			−0.103782
$392$		0			0
$393$	−0.354573	−	3.75300i	−0.0178859	−	0.189314i
$394$		0			0
$395$		−	9.07939i		−	0.456834i
$396$		0			0
$397$	−35.1363			−1.76344			−0.881721	−	0.471772i	$-0.843615\pi$
$397$	−35.1363			−1.76344			−0.881721	+	0.471772i	$0.843615\pi$
$398$		0			0
$399$	5.53962	+	58.6344i	0.277328	+	2.93539i
$400$		0			0
$401$	1.64113			0.0819540			0.0409770	−	0.999160i	$-0.486953\pi$
$401$	1.64113			0.0819540			0.0409770	+	0.999160i	$0.486953\pi$
$402$		0			0
$403$	−0.257979			−0.0128508
$404$		0			0
$405$	−8.36865	+	3.31145i	−0.415842	+	0.164547i
$406$		0			0
$407$	9.37743			0.464822
$408$		0			0
$409$			27.4775i			1.35867i	0.733827	+	0.679336i	$0.237732\pi$
$409$			27.4775i			1.35867i	−0.733827	+	0.679336i	$0.762268\pi$
$410$		0			0
$411$	16.5903	−	1.56741i	0.818341	−	0.0773147i
$412$		0			0
$413$	−38.9577			−1.91698
$414$		0			0
$415$		−	12.1811i		−	0.597948i
$416$		0			0
$417$	−0.803014	−	8.49955i	−0.0393238	−	0.416225i
$418$		0			0
$419$			27.0245i			1.32023i	0.751163	+	0.660117i	$0.229493\pi$
$419$			27.0245i			1.32023i	−0.751163	+	0.660117i	$0.770507\pi$
$420$		0			0
$421$	−14.0556			−0.685029			−0.342515	−	0.939513i	$-0.611279\pi$
$421$	−14.0556			−0.685029			−0.342515	+	0.939513i	$0.611279\pi$
$422$		0			0
$423$	6.00586	+	31.5009i	0.292015	+	1.53163i
$424$		0			0
$425$			0.326741i			0.0158493i
$426$		0			0
$427$	−36.2810			−1.75576
$428$		0			0
$429$	0.0746221	+	0.789841i	0.00360279	+	0.0381339i
$430$		0			0
$431$		−	12.6236i		−	0.608060i	−0.952663	−	0.304030i	$-0.901668\pi$
$431$		−	12.6236i		−	0.608060i	0.952663	−	0.304030i	$-0.0983322\pi$
$432$		0			0
$433$			17.8977i			0.860107i	0.902803	+	0.430054i	$0.141505\pi$
$433$			17.8977i			0.860107i	−0.902803	+	0.430054i	$0.858495\pi$
$434$		0			0
$435$	−0.503093	−	5.32502i	−0.0241215	−	0.255315i
$436$		0			0
$437$			50.0363i			2.39356i
$438$		0			0
$439$	2.51766			0.120161			0.0600807	−	0.998194i	$-0.480864\pi$
$439$	2.51766			0.120161			0.0600807	+	0.998194i	$0.480864\pi$
$440$		0			0
$441$	−33.0568	+	6.30249i	−1.57413	+	0.300119i
$442$		0			0
$443$	13.0296			0.619053			0.309527	−	0.950891i	$-0.399829\pi$
$443$	13.0296			0.619053			0.309527	+	0.950891i	$0.399829\pi$
$444$		0			0
$445$			6.98512i			0.331126i
$446$		0			0
$447$	−0.0109882	−	0.116305i	−0.000519724	−	0.00550105i
$448$		0			0
$449$		−	20.9151i		−	0.987046i	−0.869733	−	0.493523i	$-0.835709\pi$
$449$		−	20.9151i		−	0.987046i	0.869733	−	0.493523i	$-0.164291\pi$
$450$		0			0
$451$	−18.1226			−0.853361
$452$		0			0
$453$	−7.43072	+	0.702035i	−0.349126	+	0.0329845i
$454$		0			0
$455$	−0.829571			−0.0388909
$456$		0			0
$457$	−18.7908			−0.878995			−0.439497	−	0.898244i	$-0.644843\pi$
$457$	−18.7908			−0.878995			−0.439497	+	0.898244i	$0.644843\pi$
$458$		0			0
$459$	−0.473425	−	1.63045i	−0.0220976	−	0.0761030i
$460$		0			0
$461$		−	31.0787i		−	1.44748i	−0.690073	−	0.723740i	$-0.742422\pi$
$461$		−	31.0787i		−	1.44748i	0.690073	−	0.723740i	$-0.257578\pi$
$462$		0			0
$463$			15.8173i			0.735091i	0.930006	+	0.367546i	$0.119802\pi$
$463$			15.8173i			0.735091i	−0.930006	+	0.367546i	$0.880198\pi$
$464$		0			0
$465$	0.216238	+	2.28878i	0.0100278	+	0.106140i
$466$		0			0
$467$			1.23979i			0.0573706i	0.999588	+	0.0286853i	$0.00913206\pi$
$467$			1.23979i			0.0573706i	−0.999588	+	0.0286853i	$0.990868\pi$
$468$		0			0
$469$	−23.7491	+	25.6232i	−1.09663	+	1.18317i
$470$		0			0
$471$	−17.5258	+	1.65579i	−0.807544	+	0.0762946i
$472$		0			0
$473$		−	28.4283i		−	1.30714i
$474$		0			0
$475$	7.96669			0.365537
$476$		0			0
$477$	22.9335	−	4.37242i	1.05005	−	0.200199i
$478$		0			0
$479$		−	10.0781i		−	0.460481i	−0.973134	−	0.230241i	$-0.926049\pi$
$479$		−	10.0781i		−	0.460481i	0.973134	−	0.230241i	$-0.0739513\pi$
$480$		0			0
$481$		−	0.773385i		−	0.0352633i
$482$		0			0
$483$	−46.2254	+	4.36726i	−2.10333	+	0.198717i
$484$		0			0
$485$			4.00684i			0.181941i
$486$		0			0
$487$		−	20.3842i		−	0.923695i	−0.886959	−	0.461847i	$-0.847187\pi$
$487$		−	20.3842i		−	0.923695i	0.886959	−	0.461847i	$-0.152813\pi$
$488$		0			0
$489$	36.0766	−	3.40842i	1.63144	−	0.154134i
$490$		0			0
$491$			14.3395i			0.647133i	0.946205	+	0.323566i	$0.104882\pi$
$491$			14.3395i			0.647133i	−0.946205	+	0.323566i	$0.895118\pi$
$492$		0			0
$493$	1.00900			0.0454433
$494$		0			0
$495$	6.94491	−	1.32409i	0.312150	−	0.0595135i
$496$		0			0
$497$	−22.9652			−1.03013
$498$		0			0
$499$		−	10.2381i		−	0.458320i	−0.973389	−	0.229160i	$-0.926402\pi$
$499$		−	10.2381i		−	0.458320i	0.973389	−	0.229160i	$-0.0735979\pi$
$500$		0			0
$501$	1.84080	+	19.4841i	0.0822410	+	0.870484i
$502$		0			0
$503$	−7.01864			−0.312946			−0.156473	−	0.987682i	$-0.550012\pi$
$503$	−7.01864			−0.312946			−0.156473	+	0.987682i	$0.550012\pi$
$504$		0			0
$505$	7.17057			0.319086
$506$		0			0
$507$	−22.3517	+	2.11173i	−0.992674	+	0.0937852i
$508$		0			0
$509$			33.3700i			1.47910i	0.673102	+	0.739550i	$0.264961\pi$
$509$			33.3700i			1.47910i	−0.673102	+	0.739550i	$0.735039\pi$
$510$		0			0
$511$		−	39.1592i		−	1.73230i
$512$		0			0
$513$	−39.7542	+	11.5432i	−1.75519	+	0.509644i
$514$		0			0
$515$	−19.5052			−0.859500
$516$		0			0
$517$		−	25.1915i		−	1.10792i
$518$		0			0
$519$	−2.63195	−	27.8580i	−0.115530	−	1.22283i
$520$		0			0
$521$	−21.2620			−0.931506			−0.465753	−	0.884915i	$-0.654217\pi$
$521$	−21.2620			−0.931506			−0.465753	+	0.884915i	$0.654217\pi$
$522$		0			0
$523$	6.32436			0.276545			0.138273	−	0.990394i	$-0.455845\pi$
$523$	6.32436			0.276545			0.138273	+	0.990394i	$0.455845\pi$
$524$		0			0
$525$	0.695347	+	7.35994i	0.0303475	+	0.321214i
$526$		0			0
$527$	−0.433687			−0.0188917
$528$		0			0
$529$	−16.4470			−0.715086
$530$		0			0
$531$	−5.12825	−	26.8979i	−0.222547	−	1.16727i
$532$		0			0
$533$			1.49463i			0.0647395i
$534$		0			0
$535$			3.03255i			0.131109i
$536$		0			0
$537$	3.99009	−	0.376973i	0.172185	−	0.0162676i
$538$		0			0
$539$	26.4357			1.13867
$540$		0			0
$541$			7.28423i			0.313174i	0.987664	+	0.156587i	$0.0500491\pi$
$541$			7.28423i			0.313174i	−0.987664	+	0.156587i	$0.949951\pi$
$542$		0			0
$543$	−6.73297	+	0.636113i	−0.288939	+	0.0272982i
$544$		0			0
$545$			11.1335i			0.476905i
$546$		0			0
$547$			38.1108i			1.62950i	0.579812	+	0.814751i	$0.303126\pi$
$547$			38.1108i			1.62950i	−0.579812	+	0.814751i	$0.696874\pi$
$548$		0			0
$549$	−4.77590	−	25.0498i	−0.203830	−	1.06910i
$550$		0			0
$551$		−	24.6019i		−	1.04807i
$552$		0			0
$553$	38.7525			1.64792
$554$		0			0
$555$	−6.86146	+	0.648252i	−0.291253	+	0.0275168i
$556$		0			0
$557$		−	45.2158i		−	1.91585i	−0.287013	−	0.957927i	$-0.592662\pi$
$557$		−	45.2158i		−	1.91585i	0.287013	−	0.957927i	$-0.407338\pi$
$558$		0			0
$559$	−2.34457			−0.0991647
$560$		0			0
$561$	0.125447	+	1.32780i	0.00529638	+	0.0560598i
$562$		0			0
$563$	−14.9039			−0.628124			−0.314062	−	0.949402i	$-0.601690\pi$
$563$	−14.9039			−0.628124			−0.314062	+	0.949402i	$0.601690\pi$
$564$		0			0
$565$	−9.67007			−0.406823
$566$		0			0
$567$	−14.1339	−	35.7190i	−0.593567	−	1.50006i
$568$		0			0
$569$			32.2516i			1.35206i	0.736875	+	0.676029i	$0.236301\pi$
$569$			32.2516i			1.35206i	−0.736875	+	0.676029i	$0.763699\pi$
$570$		0			0
$571$	−3.11137			−0.130207			−0.0651035	−	0.997879i	$-0.520738\pi$
$571$	−3.11137			−0.130207			−0.0651035	+	0.997879i	$0.520738\pi$
$572$		0			0
$573$	−14.2439	+	1.34572i	−0.595046	+	0.0562183i
$574$		0			0
$575$			6.28068i			0.261923i
$576$		0			0
$577$			18.4328i			0.767368i	0.923464	+	0.383684i	$0.125345\pi$
$577$			18.4328i			0.767368i	−0.923464	+	0.383684i	$0.874655\pi$
$578$		0			0
$579$	−16.1239	+	1.52334i	−0.670087	+	0.0633080i
$580$		0			0
$581$	51.9913			2.15696
$582$		0			0
$583$	−18.3401			−0.759568
$584$		0			0
$585$	−0.109202	−	0.572767i	−0.00451494	−	0.0236810i
$586$		0			0
$587$	14.9168			0.615681			0.307840	−	0.951438i	$-0.400394\pi$
$587$	14.9168			0.615681			0.307840	+	0.951438i	$0.400394\pi$
$588$		0			0
$589$			10.5743i			0.435706i
$590$		0			0
$591$	−40.9459	+	3.86846i	−1.68429	+	0.159127i
$592$		0			0
$593$	−19.4352			−0.798109			−0.399055	−	0.916927i	$-0.630662\pi$
$593$	−19.4352			−0.798109			−0.399055	+	0.916927i	$0.630662\pi$
$594$		0			0
$595$	−1.39459			−0.0571726
$596$		0			0
$597$	−34.9934	+	3.30608i	−1.43218	+	0.135309i
$598$		0			0
$599$	−35.5233			−1.45144			−0.725722	−	0.687988i	$-0.758494\pi$
$599$	−35.5233			−1.45144			−0.725722	+	0.687988i	$0.758494\pi$
$600$		0			0
$601$	23.9602			0.977358			0.488679	−	0.872464i	$-0.337479\pi$
$601$	23.9602			0.977358			0.488679	+	0.872464i	$0.337479\pi$
$602$		0			0
$603$	−20.8175	−	13.0243i	−0.847753	−	0.530391i
$604$		0			0
$605$	5.44611			0.221416
$606$		0			0
$607$	−38.9311			−1.58016			−0.790082	−	0.613001i	$-0.789962\pi$
$607$	−38.9311			−1.58016			−0.790082	+	0.613001i	$0.789962\pi$
$608$		0			0
$609$	22.7282	−	2.14730i	0.920992	−	0.0870128i
$610$		0			0
$611$	−2.07762			−0.0840514
$612$		0			0
$613$	19.4818			0.786862			0.393431	−	0.919354i	$-0.371288\pi$
$613$	19.4818			0.786862			0.393431	+	0.919354i	$0.371288\pi$
$614$		0			0
$615$	13.2603	−	1.25280i	0.534707	−	0.0505177i
$616$		0			0
$617$		−	13.0823i		−	0.526674i	−0.964704	−	0.263337i	$-0.915177\pi$
$617$		−	13.0823i		−	0.526674i	0.964704	−	0.263337i	$-0.0848230\pi$
$618$		0			0
$619$	−13.9849			−0.562100			−0.281050	−	0.959693i	$-0.590683\pi$
$619$	−13.9849			−0.562100			−0.281050	+	0.959693i	$0.590683\pi$
$620$		0			0
$621$	−9.10027	−	31.3409i	−0.365181	−	1.25767i
$622$		0			0
$623$	−29.8138			−1.19446
$624$		0			0
$625$	1.00000			0.0400000
$626$		0			0
$627$	32.3748	−	3.05869i	1.29293	−	0.122152i
$628$		0			0
$629$		−	1.30014i		−	0.0518398i
$630$		0			0
$631$			45.0562i			1.79366i	0.442376	+	0.896830i	$0.354136\pi$
$631$			45.0562i			1.79366i	−0.442376	+	0.896830i	$0.645864\pi$
$632$		0			0
$633$	8.26656	−	0.781003i	0.328566	−	0.0310421i
$634$		0			0
$635$	5.83487			0.231550
$636$		0			0
$637$		−	2.18023i		−	0.0863840i
$638$		0			0
$639$	−3.02306	−	15.8561i	−0.119590	−	0.627256i
$640$		0			0
$641$	45.0230			1.77830			0.889152	−	0.457613i	$-0.151295\pi$
$641$	45.0230			1.77830			0.889152	+	0.457613i	$0.151295\pi$
$642$		0			0
$643$	7.90426			0.311714			0.155857	−	0.987780i	$-0.450186\pi$
$643$	7.90426			0.311714			0.155857	+	0.987780i	$0.450186\pi$
$644$		0			0
$645$	1.96522	+	20.8010i	0.0773805	+	0.819038i
$646$		0			0
$647$	43.8866			1.72536			0.862679	−	0.505752i	$-0.168785\pi$
$647$	43.8866			1.72536			0.862679	+	0.505752i	$0.168785\pi$
$648$		0			0
$649$			21.5104i			0.844357i
$650$		0			0
$651$	−9.76895	+	0.922944i	−0.382875	+	0.0361730i
$652$		0			0
$653$	−31.7101			−1.24091			−0.620455	−	0.784242i	$-0.713052\pi$
$653$	−31.7101			−1.24091			−0.620455	+	0.784242i	$0.713052\pi$
$654$		0			0
$655$		−	2.17644i		−	0.0850407i
$656$		0			0
$657$	27.0370	−	5.15478i	1.05481	−	0.201107i
$658$		0			0
$659$			40.5889i			1.58112i	0.612386	+	0.790559i	$0.290210\pi$
$659$			40.5889i			1.58112i	−0.612386	+	0.790559i	$0.709790\pi$
$660$		0			0
$661$		−	35.6091i		−	1.38503i	−0.721402	−	0.692517i	$-0.756502\pi$
$661$		−	35.6091i		−	1.38503i	0.721402	−	0.692517i	$-0.243498\pi$
$662$		0			0
$663$	0.109508	−	0.0103460i	0.00425292	−	0.000401805i
$664$		0			0
$665$			34.0033i			1.31859i
$666$		0			0
$667$	19.3953			0.750990
$668$		0			0
$669$	10.8883	−	1.02870i	0.420965	−	0.0397717i
$670$		0			0
$671$			20.0324i			0.773344i
$672$		0			0
$673$		−	27.8588i		−	1.07388i	−0.843621	−	0.536940i	$-0.819580\pi$
$673$		−	27.8588i		−	1.07388i	0.843621	−	0.536940i	$-0.180420\pi$
$674$		0			0
$675$	−4.99005	+	1.44893i	−0.192067	+	0.0557694i
$676$		0			0
$677$	−12.1388			−0.466531			−0.233265	−	0.972413i	$-0.574941\pi$
$677$	−12.1388			−0.466531			−0.233265	+	0.972413i	$0.574941\pi$
$678$		0			0
$679$	−17.1019			−0.656312
$680$		0			0
$681$	1.36035	+	14.3987i	0.0521289	+	0.551761i
$682$		0			0
$683$	5.00694			0.191585			0.0957927	−	0.995401i	$-0.469461\pi$
$683$	5.00694			0.191585			0.0957927	+	0.995401i	$0.469461\pi$
$684$		0			0
$685$	9.62109			0.367603
$686$		0			0
$687$	−0.954747	−	10.1056i	−0.0364259	−	0.385551i
$688$		0			0
$689$			1.51256i			0.0576240i
$690$		0			0
$691$	1.51420			0.0576029			0.0288015	−	0.999585i	$-0.490831\pi$
$691$	1.51420			0.0576029			0.0288015	+	0.999585i	$0.490831\pi$
$692$		0			0
$693$	5.65147	+	29.6422i	0.214682	+	1.12601i
$694$		0			0
$695$		−	4.92907i		−	0.186970i
$696$		0			0
$697$			2.51261i			0.0951720i
$698$		0			0
$699$	−16.9560	+	1.60195i	−0.641334	+	0.0605915i
$700$		0			0
$701$	33.0867			1.24967			0.624834	−	0.780758i	$-0.285167\pi$
$701$	33.0867			1.24967			0.624834	+	0.780758i	$0.285167\pi$
$702$		0			0
$703$	−31.7003			−1.19560
$704$		0			0
$705$	1.74146	+	18.4326i	0.0655873	+	0.694212i
$706$		0			0
$707$			30.6053i			1.15103i
$708$		0			0
$709$	−25.3119			−0.950608			−0.475304	−	0.879822i	$-0.657662\pi$
$709$	−25.3119			−0.950608			−0.475304	+	0.879822i	$0.657662\pi$
$710$		0			0
$711$	5.10124	+	26.7562i	0.191312	+	1.00344i
$712$		0			0
$713$	−8.33644			−0.312202
$714$		0			0
$715$			0.458046i			0.0171299i
$716$		0			0
$717$	−27.4744	+	2.59571i	−1.02605	+	0.0969384i
$718$		0			0
$719$		−	10.7831i		−	0.402143i	−0.979577	−	0.201072i	$-0.935558\pi$
$719$		−	10.7831i		−	0.402143i	0.979577	−	0.201072i	$-0.0644424\pi$
$720$		0			0
$721$		−	83.2516i		−	3.10045i
$722$		0			0
$723$	−12.5444	+	1.18516i	−0.466531	+	0.0440766i
$724$		0			0
$725$		−	3.08809i		−	0.114689i
$726$		0			0
$727$			32.3247i			1.19886i	0.800428	+	0.599428i	$0.204605\pi$
$727$			32.3247i			1.19886i	−0.800428	+	0.599428i	$0.795395\pi$
$728$		0			0
$729$	22.8012	−	14.4605i	0.844489	−	0.535573i
$730$		0			0
$731$	−3.94145			−0.145780
$732$		0			0
$733$		−	35.2780i		−	1.30302i	−0.758639	−	0.651511i	$-0.774135\pi$
$733$		−	35.2780i		−	1.30302i	0.758639	−	0.651511i	$-0.225865\pi$
$734$		0			0
$735$	−19.3430	+	1.82747i	−0.713477	+	0.0674074i
$736$		0			0
$737$	14.1478	+	13.1130i	0.521141	+	0.483023i
$738$		0			0
$739$			49.3843i			1.81663i	0.418284	+	0.908316i	$0.362632\pi$
$739$			49.3843i			1.81663i	−0.418284	+	0.908316i	$0.637368\pi$
$740$		0			0
$741$	−0.252259	−	2.67005i	−0.00926697	−	0.0980867i
$742$		0			0
$743$			23.3351i			0.856083i	0.903759	+	0.428041i	$0.140796\pi$
$743$			23.3351i			0.856083i	−0.903759	+	0.428041i	$0.859204\pi$
$744$		0			0
$745$		−	0.0674479i		−	0.00247110i
$746$		0			0
$747$	6.84395	+	35.8968i	0.250407	+	1.31339i
$748$		0			0
$749$	−12.9435			−0.472945
$750$		0			0
$751$	−2.28274			−0.0832983			−0.0416492	−	0.999132i	$-0.513261\pi$
$751$	−2.28274			−0.0832983			−0.0416492	+	0.999132i	$0.513261\pi$
$752$		0			0
$753$	21.0643	−	1.99010i	0.767627	−	0.0725234i
$754$		0			0
$755$	−4.30923			−0.156829
$756$		0			0
$757$		−	46.5769i		−	1.69287i	−0.532496	−	0.846433i	$-0.678746\pi$
$757$		−	46.5769i		−	1.69287i	0.532496	−	0.846433i	$-0.321254\pi$
$758$		0			0
$759$	2.41137	+	25.5233i	0.0875272	+	0.926436i
$760$		0			0
$761$			2.06825i			0.0749741i	0.999297	+	0.0374870i	$0.0119353\pi$
$761$			2.06825i			0.0749741i	−0.999297	+	0.0374870i	$0.988065\pi$
$762$		0			0
$763$	−47.5197			−1.72033
$764$		0			0
$765$	−0.183579	−	0.962878i	−0.00663731	−	0.0348129i
$766$		0			0
$767$	1.77403			0.0640564
$768$		0			0
$769$			36.8336i			1.32825i	0.747620	+	0.664127i	$0.231196\pi$
$769$			36.8336i			1.32825i	−0.747620	+	0.664127i	$0.768804\pi$
$770$		0			0
$771$	4.04017	+	42.7634i	0.145503	+	1.54009i
$772$		0			0
$773$		−	10.1753i		−	0.365980i	−0.983115	−	0.182990i	$-0.941422\pi$
$773$		−	10.1753i		−	0.365980i	0.983115	−	0.182990i	$-0.0585776\pi$
$774$		0			0
$775$			1.32731i			0.0476785i
$776$		0			0
$777$	−2.76686	−	29.2860i	−0.0992606	−	1.05063i
$778$		0			0
$779$	61.2633			2.19499
$780$		0			0
$781$			12.6802i			0.453733i
$782$		0			0
$783$	4.47443	+	15.4097i	0.159903	+	0.550699i
$784$		0			0
$785$	−10.1636			−0.362753
$786$		0			0
$787$		−	5.57688i		−	0.198794i	−0.995048	−	0.0993972i	$-0.968309\pi$
$787$		−	5.57688i		−	0.198794i	0.995048	−	0.0993972i	$-0.0316914\pi$
$788$		0			0
$789$	2.88419	+	30.5279i	0.102680	+	1.08682i
$790$		0			0
$791$		−	41.2736i		−	1.46752i
$792$		0			0
$793$	1.65214			0.0586691
$794$		0			0
$795$	13.4194	−	1.26783i	0.475938	−	0.0449653i
$796$		0			0
$797$		−	20.0059i		−	0.708646i	−0.935123	−	0.354323i	$-0.884711\pi$
$797$		−	20.0059i		−	0.708646i	0.935123	−	0.354323i	$-0.115289\pi$
$798$		0			0
$799$	−3.49268			−0.123562
$800$		0			0
$801$	−3.92458	−	20.5846i	−0.138668	−	0.727320i
$802$		0			0
$803$	−21.6217			−0.763012
$804$		0			0
$805$	−26.8071			−0.944827
$806$		0			0
$807$	4.05136	+	42.8819i	0.142615	+	1.50951i
$808$		0			0
$809$	14.1204			0.496446			0.248223	−	0.968703i	$-0.420153\pi$
$809$	14.1204			0.496446			0.248223	+	0.968703i	$0.420153\pi$
$810$		0			0
$811$			37.3447i			1.31135i	0.755044	+	0.655674i	$0.227615\pi$
$811$			37.3447i			1.31135i	−0.755044	+	0.655674i	$0.772385\pi$
$812$		0			0
$813$	2.51285	+	26.5974i	0.0881295	+	0.932812i
$814$		0			0
$815$	20.9216			0.732851
$816$		0			0
$817$			96.1017i			3.36217i
$818$		0			0
$819$	2.44468	−	0.466094i	0.0854240	−	0.0162866i
$820$		0			0
$821$			47.6891i			1.66436i	0.554505	+	0.832180i	$0.312907\pi$
$821$			47.6891i			1.66436i	−0.554505	+	0.832180i	$0.687093\pi$
$822$		0			0
$823$	50.0925			1.74612			0.873058	−	0.487617i	$-0.162134\pi$
$823$	50.0925			1.74612			0.873058	+	0.487617i	$0.162134\pi$
$824$		0			0
$825$	4.06377	−	0.383934i	0.141482	−	0.0133669i
$826$		0			0
$827$		−	32.4916i		−	1.12984i	−0.825145	−	0.564922i	$-0.808906\pi$
$827$		−	32.4916i		−	1.12984i	0.825145	−	0.564922i	$-0.191094\pi$
$828$		0			0
$829$	29.6681			1.03042			0.515208	−	0.857065i	$-0.327715\pi$
$829$	29.6681			1.03042			0.515208	+	0.857065i	$0.327715\pi$
$830$		0			0
$831$	−39.9265	+	3.77215i	−1.38504	+	0.130855i
$832$		0			0
$833$		−	3.66518i		−	0.126991i
$834$		0			0
$835$			11.2992i			0.391026i
$836$		0			0
$837$	−1.92319	−	6.62336i	−0.0664750	−	0.228937i
$838$		0			0
$839$		−	33.0811i		−	1.14209i	−0.820920	−	0.571044i	$-0.806539\pi$
$839$		−	33.0811i		−	1.14209i	0.820920	−	0.571044i	$-0.193461\pi$
$840$		0			0
$841$	19.4637			0.671162
$842$		0			0
$843$	−33.9817	+	3.21050i	−1.17039	+	0.110576i
$844$		0			0
$845$	−12.9622			−0.445914
$846$		0			0
$847$			23.2450i			0.798709i
$848$		0			0
$849$	−10.2237	+	0.965906i	−0.350876	+	0.0331498i
$850$		0			0
$851$		−	24.9915i		−	0.856697i
$852$		0			0
$853$	−17.6524			−0.604407			−0.302203	−	0.953243i	$-0.597722\pi$
$853$	−17.6524			−0.604407			−0.302203	+	0.953243i	$0.597722\pi$
$854$		0			0
$855$	−23.4772	+	4.47608i	−0.802903	+	0.153079i
$856$		0			0
$857$	−18.5022			−0.632025			−0.316012	−	0.948755i	$-0.602344\pi$
$857$	−18.5022			−0.632025			−0.316012	+	0.948755i	$0.602344\pi$
$858$		0			0
$859$	−51.4130			−1.75419			−0.877094	−	0.480319i	$-0.840521\pi$
$859$	−51.4130			−1.75419			−0.877094	+	0.480319i	$0.840521\pi$
$860$		0			0
$861$	5.34717	+	56.5974i	0.182231	+	1.92884i
$862$		0			0
$863$		−	48.6194i		−	1.65502i	−0.561449	−	0.827511i	$-0.689756\pi$
$863$		−	48.6194i		−	1.65502i	0.561449	−	0.827511i	$-0.310244\pi$
$864$		0			0
$865$		−	16.1555i		−	0.549302i
$866$		0			0
$867$	−29.1302	+	2.75215i	−0.989315	+	0.0934678i
$868$		0			0
$869$		−	21.3971i		−	0.725847i
$870$		0			0
$871$	1.08147	−	1.16681i	0.0366441	−	0.0395359i
$872$		0			0
$873$	−2.25124	−	11.8078i	−0.0761929	−	0.399635i
$874$		0			0
$875$			4.26819i			0.144291i
$876$		0			0
$877$	−21.0830			−0.711921			−0.355961	−	0.934501i	$-0.615846\pi$
$877$	−21.0830			−0.711921			−0.355961	+	0.934501i	$0.615846\pi$
$878$		0			0
$879$	−3.96462	−	41.9637i	−0.133723	−	1.41540i
$880$		0			0
$881$		−	31.6626i		−	1.06674i	−0.845882	−	0.533371i	$-0.820925\pi$
$881$		−	31.6626i		−	1.06674i	0.845882	−	0.533371i	$-0.179075\pi$
$882$		0			0
$883$			50.5209i			1.70016i	0.526652	+	0.850081i	$0.323447\pi$
$883$			50.5209i			1.70016i	−0.526652	+	0.850081i	$0.676553\pi$
$884$		0			0
$885$	−1.48699	−	15.7391i	−0.0499847	−	0.529065i
$886$		0			0
$887$		−	13.2275i		−	0.444136i	−0.975031	−	0.222068i	$-0.928719\pi$
$887$		−	13.2275i		−	0.444136i	0.975031	−	0.222068i	$-0.0712807\pi$
$888$		0			0
$889$			24.9043i			0.835264i
$890$		0			0
$891$	−19.7221	+	7.80398i	−0.660716	+	0.261443i
$892$		0			0
$893$			85.1596i			2.84976i
$894$		0			0
$895$	2.31394			0.0773463
$896$		0			0
$897$	2.10498	−	0.198873i	0.0702832	−	0.00664017i
$898$		0			0
$899$	4.09887			0.136705
$900$		0			0
$901$			2.54276i			0.0847117i
$902$		0			0
$903$	−88.7825	+	8.38793i	−2.95450	+	0.279133i
$904$		0			0
$905$	−3.90459			−0.129793
$906$		0			0
$907$	36.8602			1.22392			0.611962	−	0.790887i	$-0.290381\pi$
$907$	36.8602			1.22392			0.611962	+	0.790887i	$0.290381\pi$
$908$		0			0
$909$	−21.1311	+	4.02878i	−0.700874	+	0.133626i
$910$		0			0
$911$		−	34.8522i		−	1.15471i	−0.816495	−	0.577353i	$-0.804086\pi$
$911$		−	34.8522i		−	1.15471i	0.816495	−	0.577353i	$-0.195914\pi$
$912$		0			0
$913$		−	28.7069i		−	0.950059i
$914$		0			0
$915$	−1.38482	−	14.6577i	−0.0457808	−	0.484569i
$916$		0			0
$917$	9.28947			0.306765
$918$		0			0
$919$		−	26.4669i		−	0.873062i	−0.899689	−	0.436531i	$-0.856207\pi$
$919$		−	26.4669i		−	0.873062i	0.899689	−	0.436531i	$-0.143793\pi$
$920$		0			0
$921$	13.9227	−	1.31538i	0.458769	−	0.0433433i
$922$		0			0
$923$	1.04577			0.0344221
$924$		0			0
$925$	−3.97910			−0.130832
$926$		0			0
$927$	57.4801	−	10.9589i	1.88789	−	0.359939i
$928$		0			0
$929$	52.9301			1.73658			0.868290	−	0.496057i	$-0.165219\pi$
$929$	52.9301			1.73658			0.868290	+	0.496057i	$0.165219\pi$
$930$		0			0
$931$	−89.3657			−2.92884
$932$		0			0
$933$	−18.1246	+	1.71236i	−0.593372	+	0.0560602i
$934$		0			0
$935$			0.770019i			0.0251823i
$936$		0			0
$937$		−	33.4278i		−	1.09204i	−0.837772	−	0.546020i	$-0.816142\pi$
$937$		−	33.4278i		−	1.09204i	0.837772	−	0.546020i	$-0.183858\pi$
$938$		0			0
$939$	−1.06144	−	11.2349i	−0.0346389	−	0.366637i
$940$		0			0
$941$	3.08211			0.100474			0.0502369	−	0.998737i	$-0.484002\pi$
$941$	3.08211			0.100474			0.0502369	+	0.998737i	$0.484002\pi$
$942$		0			0
$943$			48.2980i			1.57280i
$944$		0			0
$945$	−6.18431	−	21.2985i	−0.201176	−	0.692839i
$946$		0			0
$947$		−	22.6655i		−	0.736529i	−0.929721	−	0.368265i	$-0.879952\pi$
$947$		−	22.6655i		−	0.736529i	0.929721	−	0.368265i	$-0.120048\pi$
$948$		0			0
$949$			1.78320i			0.0578852i
$950$		0			0
$951$	3.12363	+	33.0622i	0.101291	+	1.07211i
$952$		0			0
$953$		−	9.86942i		−	0.319702i	−0.987141	−	0.159851i	$-0.948899\pi$
$953$		−	9.86942i		−	0.319702i	0.987141	−	0.159851i	$-0.0511014\pi$
$954$		0			0
$955$	−8.26032			−0.267298
$956$		0			0
$957$	−1.18562	−	12.5493i	−0.0383258	−	0.405661i
$958$		0			0
$959$			41.0646i			1.32604i
$960$		0			0
$961$	29.2382			0.943169
$962$		0			0
$963$	−1.70384	−	8.93669i	−0.0549054	−	0.287981i
$964$		0			0
$965$	−9.35059			−0.301006
$966$		0			0
$967$	−3.21025			−0.103235			−0.0516173	−	0.998667i	$-0.516438\pi$
$967$	−3.21025			−0.103235			−0.0516173	+	0.998667i	$0.516438\pi$
$968$		0			0
$969$	−0.424072	−	4.48861i	−0.0136232	−	0.144195i
$970$		0			0
$971$			26.3937i			0.847013i	0.905893	+	0.423507i	$0.139201\pi$
$971$			26.3937i			0.847013i	−0.905893	+	0.423507i	$0.860799\pi$
$972$		0			0
$973$	21.0382			0.674453
$974$		0			0
$975$	−0.0316642	−	0.335152i	−0.00101407	−	0.0107334i
$976$		0			0
$977$			31.6825i			1.01361i	0.862060	+	0.506807i	$0.169174\pi$
$977$			31.6825i			1.01361i	−0.862060	+	0.506807i	$0.830826\pi$
$978$		0			0
$979$			16.4616i			0.526115i
$980$		0			0
$981$	−6.25532	−	32.8094i	−0.199717	−	1.04752i
$982$		0			0
$983$	48.5360			1.54806			0.774029	−	0.633150i	$-0.218238\pi$
$983$	48.5360			1.54806			0.774029	+	0.633150i	$0.218238\pi$
$984$		0			0
$985$	−23.7454			−0.756591
$986$		0			0
$987$	−78.6737	+	7.43288i	−2.50421	+	0.236591i
$988$		0			0
$989$	−75.7634			−2.40914
$990$		0			0
$991$			35.4273i			1.12538i	0.826667	+	0.562692i	$0.190234\pi$
$991$			35.4273i			1.12538i	−0.826667	+	0.562692i	$0.809766\pi$
$992$		0			0
$993$	0.662517	+	7.01244i	0.0210243	+	0.222533i
$994$		0			0
$995$	−20.2934			−0.643344
$996$		0			0
$997$	41.2694			1.30702			0.653508	−	0.756920i	$-0.273297\pi$
$997$	41.2694			1.30702			0.653508	+	0.756920i	$0.273297\pi$
$998$		0			0
$999$	19.8559	−	5.76545i	0.628214	−	0.182411i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4020.2.f.a.401.4	yes	46
3.2	odd	2		4020.2.f.b.401.44	yes	46
67.66	odd	2		4020.2.f.b.401.43	yes	46
201.200	even	2	inner	4020.2.f.a.401.3	✓	46

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4020.2.f.a.401.3	✓	46	201.200	even	2	inner
4020.2.f.a.401.4	yes	46	1.1	even	1	trivial
4020.2.f.b.401.43	yes	46	67.66	odd	2
4020.2.f.b.401.44	yes	46	3.2	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:	\( N \)	\(=\)	\( 4020 = 2^{2} \cdot 3 \cdot 5 \cdot 67 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4020.f (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.72437 + 0.162914i) q^{3} -1.00000 q^{5} -4.26819i q^{7} +(2.94692 - 0.561849i) q^{9} +O(q^{10})\)	\(q+(-1.72437 + 0.162914i) q^{3} -1.00000 q^{5} -4.26819i q^{7} +(2.94692 - 0.561849i) q^{9} -2.35667 q^{11} +0.194362i q^{13} +(1.72437 - 0.162914i) q^{15} +0.326741i q^{17} +7.96669 q^{19} +(0.695347 + 7.35994i) q^{21} +6.28068i q^{23} +1.00000 q^{25} +(-4.99005 + 1.44893i) q^{27} -3.08809i q^{29} +1.32731i q^{31} +(4.06377 - 0.383934i) q^{33} +4.26819i q^{35} -3.97910 q^{37} +(-0.0316642 - 0.335152i) q^{39} +7.68993 q^{41} +12.0629i q^{43} +(-2.94692 + 0.561849i) q^{45} +10.6895i q^{47} -11.2174 q^{49} +(-0.0532307 - 0.563423i) q^{51} +7.78220 q^{53} +2.35667 q^{55} +(-13.7375 + 1.29789i) q^{57} -9.12746i q^{59} -8.50032i q^{61} +(-2.39808 - 12.5780i) q^{63} -0.194362i q^{65} +(-6.00330 - 5.56420i) q^{67} +(-1.02321 - 10.8302i) q^{69} -5.38056i q^{71} +9.17468 q^{73} +(-1.72437 + 0.162914i) q^{75} +10.0587i q^{77} +9.07939i q^{79} +(8.36865 - 3.31145i) q^{81} +12.1811i q^{83} -0.326741i q^{85} +(0.503093 + 5.32502i) q^{87} -6.98512i q^{89} +0.829571 q^{91} +(-0.216238 - 2.28878i) q^{93} -7.96669 q^{95} -4.00684i q^{97} +(-6.94491 + 1.32409i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 46 q - 46 q^{5} - 4 q^{9}+O(q^{10}) \) 46 * q - 46 * q^5 - 4 * q^9	\( 46 q - 46 q^{5} - 4 q^{9} + 8 q^{19} + 46 q^{25} - 18 q^{27} + 4 q^{33} - 8 q^{37} - 12 q^{39} - 4 q^{41} + 4 q^{45} - 62 q^{49} - 8 q^{51} - 8 q^{53} - 12 q^{57} - 10 q^{63} - 14 q^{67} - 40 q^{73} - 12 q^{81} - 4 q^{91} + 2 q^{93} - 8 q^{95}+O(q^{100}) \) 46 * q - 46 * q^5 - 4 * q^9 + 8 * q^19 + 46 * q^25 - 18 * q^27 + 4 * q^33 - 8 * q^37 - 12 * q^39 - 4 * q^41 + 4 * q^45 - 62 * q^49 - 8 * q^51 - 8 * q^53 - 12 * q^57 - 10 * q^63 - 14 * q^67 - 40 * q^73 - 12 * q^81 - 4 * q^91 + 2 * q^93 - 8 * q^95

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