LMFDB - Embedded newform 840.2.k.b.209.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [840,2,Mod(209,840)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(840, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("840.209");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	840.k (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:	$6.70743376979$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.6
Character	$\chi$	$=$	840.209
Dual form			840.2.k.b.209.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.37873 + 1.04838i) q^{3} +(1.84099 + 1.26915i) q^{5} +(2.63201 - 0.269297i) q^{7} +(0.801780 - 2.89087i) q^{9} +O(q^{10})$	\(q+(-1.37873 + 1.04838i) q^{3} +(1.84099 + 1.26915i) q^{5} +(2.63201 - 0.269297i) q^{7} +(0.801780 - 2.89087i) q^{9} -3.01442i q^{11} +4.39041 q^{13} +(-3.86879 + 0.180252i) q^{15} -2.71017i q^{17} -8.23502i q^{19} +(-3.34650 + 3.13065i) q^{21} -3.81803 q^{23} +(1.77850 + 4.67300i) q^{25} +(1.92531 + 4.82630i) q^{27} +1.17337i q^{29} +1.73412i q^{31} +(3.16027 + 4.15606i) q^{33} +(5.18729 + 2.84465i) q^{35} -4.60838i q^{37} +(-6.05318 + 4.60284i) q^{39} +10.6600 q^{41} -9.18515i q^{43} +(5.14503 - 4.30449i) q^{45} +12.0659i q^{47} +(6.85496 - 1.41759i) q^{49} +(2.84130 + 3.73659i) q^{51} -7.14027 q^{53} +(3.82575 - 5.54952i) q^{55} +(8.63346 + 11.3538i) q^{57} -9.11464 q^{59} +13.5606i q^{61} +(1.33179 - 7.82473i) q^{63} +(8.08271 + 5.57210i) q^{65} -0.494447i q^{67} +(5.26402 - 4.00276i) q^{69} +5.15821i q^{71} +4.76038 q^{73} +(-7.35117 - 4.57824i) q^{75} +(-0.811774 - 7.93398i) q^{77} +12.1064 q^{79} +(-7.71430 - 4.63569i) q^{81} +8.16745i q^{83} +(3.43962 - 4.98941i) q^{85} +(-1.23014 - 1.61776i) q^{87} +2.26858 q^{89} +(11.5556 - 1.18232i) q^{91} +(-1.81803 - 2.39088i) q^{93} +(10.4515 - 15.1606i) q^{95} +2.92728 q^{97} +(-8.71430 - 2.41690i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{9}+O(q^{10}) $ 24 * q - 2 * q^9	$ 24 q - 2 q^{9} + 2 q^{15} - 2 q^{21} - 16 q^{23} + 8 q^{25} + 8 q^{35} - 2 q^{39} + 6 q^{51} + 24 q^{53} + 8 q^{57} + 16 q^{63} + 16 q^{65} + 8 q^{77} + 4 q^{79} + 18 q^{81} - 12 q^{85} + 12 q^{91} + 32 q^{93} - 24 q^{95} - 6 q^{99}+O(q^{100}) $ 24 * q - 2 * q^9 + 2 * q^15 - 2 * q^21 - 16 * q^23 + 8 * q^25 + 8 * q^35 - 2 * q^39 + 6 * q^51 + 24 * q^53 + 8 * q^57 + 16 * q^63 + 16 * q^65 + 8 * q^77 + 4 * q^79 + 18 * q^81 - 12 * q^85 + 12 * q^91 + 32 * q^93 - 24 * q^95 - 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/840\mathbb{Z}\right)^\times$.

$n$	$241$	$281$	$337$	$421$	$631$
$\chi(n)$	$-1$	$-1$	$-1$	$1$	$1$

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
$3$	−1.37873	+	1.04838i	−0.796009	+	0.605285i
$3$	−1.37873	+	1.04838i	−0.796009	+	0.605285i
$4$		0			0
$5$	1.84099	+	1.26915i	0.823317	+	0.567582i
$5$	1.84099	+	1.26915i	0.823317	+	0.567582i
$6$		0			0
$7$	2.63201	−	0.269297i	0.994806	−	0.101785i
$7$	2.63201	−	0.269297i	0.994806	−	0.101785i
$8$		0			0
$9$	0.801780	−	2.89087i	0.267260	−	0.963624i
$10$		0			0
$11$		−	3.01442i		−	0.908881i	−0.890777	−	0.454440i	$-0.849839\pi$
$11$		−	3.01442i		−	0.908881i	0.890777	−	0.454440i	$-0.150161\pi$
$12$		0			0
$13$	4.39041			1.21768			0.608840	−	0.793293i	$-0.291635\pi$
$13$	4.39041			1.21768			0.608840	+	0.793293i	$0.291635\pi$
$14$		0			0
$15$	−3.86879	+	0.180252i	−0.998916	+	0.0465408i
$16$		0			0
$17$		−	2.71017i		−	0.657314i	−0.944449	−	0.328657i	$-0.893404\pi$
$17$		−	2.71017i		−	0.657314i	0.944449	−	0.328657i	$-0.106596\pi$
$18$		0			0
$19$		−	8.23502i		−	1.88924i	−0.328163	−	0.944621i	$-0.606430\pi$
$19$		−	8.23502i		−	1.88924i	0.328163	−	0.944621i	$-0.393570\pi$
$20$		0			0
$21$	−3.34650	+	3.13065i	−0.730266	+	0.683163i
$22$		0			0
$23$	−3.81803			−0.796114			−0.398057	−	0.917361i	$-0.630315\pi$
$23$	−3.81803			−0.796114			−0.398057	+	0.917361i	$0.630315\pi$
$24$		0			0
$25$	1.77850	+	4.67300i	0.355701	+	0.934600i
$26$		0			0
$27$	1.92531	+	4.82630i	0.370526	+	0.928822i
$28$		0			0
$29$			1.17337i			0.217889i	0.994048	+	0.108944i	$0.0347471\pi$
$29$			1.17337i			0.217889i	−0.994048	+	0.108944i	$0.965253\pi$
$30$		0			0
$31$			1.73412i			0.311458i	0.987800	+	0.155729i	$0.0497726\pi$
$31$			1.73412i			0.311458i	−0.987800	+	0.155729i	$0.950227\pi$
$32$		0			0
$33$	3.16027	+	4.15606i	0.550132	+	0.723477i
$34$		0			0
$35$	5.18729	+	2.84465i	0.876812	+	0.480833i
$36$		0			0
$37$		−	4.60838i		−	0.757613i	−0.925476	−	0.378806i	$-0.876335\pi$
$37$		−	4.60838i		−	0.757613i	0.925476	−	0.378806i	$-0.123665\pi$
$38$		0			0
$39$	−6.05318	+	4.60284i	−0.969284	+	0.737044i
$40$		0			0
$41$	10.6600			1.66482			0.832408	−	0.554163i	$-0.186962\pi$
$41$	10.6600			1.66482			0.832408	+	0.554163i	$0.186962\pi$
$42$		0			0
$43$		−	9.18515i		−	1.40072i	−0.713789	−	0.700361i	$-0.753022\pi$
$43$		−	9.18515i		−	1.40072i	0.713789	−	0.700361i	$-0.246978\pi$
$44$		0			0
$45$	5.14503	−	4.30449i	0.766976	−	0.641676i
$46$		0			0
$47$			12.0659i			1.75999i	0.474980	+	0.879997i	$0.342455\pi$
$47$			12.0659i			1.75999i	−0.474980	+	0.879997i	$0.657545\pi$
$48$		0			0
$49$	6.85496	−	1.41759i	0.979280	−	0.202512i
$50$		0			0
$51$	2.84130	+	3.73659i	0.397862	+	0.523227i
$52$		0			0
$53$	−7.14027			−0.980791			−0.490396	−	0.871500i	$-0.663148\pi$
$53$	−7.14027			−0.980791			−0.490396	+	0.871500i	$0.663148\pi$
$54$		0			0
$55$	3.82575	−	5.54952i	0.515865	−	0.748297i
$56$		0			0
$57$	8.63346	+	11.3538i	1.14353	+	1.50385i
$58$		0			0
$59$	−9.11464			−1.18663			−0.593313	−	0.804972i	$-0.702180\pi$
$59$	−9.11464			−1.18663			−0.593313	+	0.804972i	$0.702180\pi$
$60$		0			0
$61$			13.5606i			1.73626i	0.496340	+	0.868128i	$0.334677\pi$
$61$			13.5606i			1.73626i	−0.496340	+	0.868128i	$0.665323\pi$
$62$		0			0
$63$	1.33179	−	7.82473i	0.167790	−	0.985823i
$64$		0			0
$65$	8.08271	+	5.57210i	1.00254	+	0.691134i
$66$		0			0
$67$		−	0.494447i		−	0.0604063i	−0.999544	−	0.0302031i	$-0.990385\pi$
$67$		−	0.494447i		−	0.0604063i	0.999544	−	0.0302031i	$-0.00961542\pi$
$68$		0			0
$69$	5.26402	−	4.00276i	0.633714	−	0.481876i
$70$		0			0
$71$			5.15821i			0.612167i	0.952005	+	0.306084i	$0.0990187\pi$
$71$			5.15821i			0.612167i	−0.952005	+	0.306084i	$0.900981\pi$
$72$		0			0
$73$	4.76038			0.557160			0.278580	−	0.960413i	$-0.410136\pi$
$73$	4.76038			0.557160			0.278580	+	0.960413i	$0.410136\pi$
$74$		0			0
$75$	−7.35117	−	4.57824i	−0.848840	−	0.528649i
$76$		0			0
$77$	−0.811774	−	7.93398i	−0.0925102	−	0.904161i
$78$		0			0
$79$	12.1064			1.36207			0.681036	−	0.732250i	$-0.261530\pi$
$79$	12.1064			1.36207			0.681036	+	0.732250i	$0.261530\pi$
$80$		0			0
$81$	−7.71430	−	4.63569i	−0.857144	−	0.515077i
$82$		0			0
$83$			8.16745i			0.896494i	0.893910	+	0.448247i	$0.147952\pi$
$83$			8.16745i			0.896494i	−0.893910	+	0.448247i	$0.852048\pi$
$84$		0			0
$85$	3.43962	−	4.98941i	0.373080	−	0.541177i
$86$		0			0
$87$	−1.23014	−	1.61776i	−0.131885	−	0.173442i
$88$		0			0
$89$	2.26858			0.240469			0.120235	−	0.992745i	$-0.461635\pi$
$89$	2.26858			0.240469			0.120235	+	0.992745i	$0.461635\pi$
$90$		0			0
$91$	11.5556	−	1.18232i	1.21136	−	0.123941i
$92$		0			0
$93$	−1.81803	−	2.39088i	−0.188521	−	0.247923i
$94$		0			0
$95$	10.4515	−	15.1606i	1.07230	−	1.55544i
$96$		0			0
$97$	2.92728			0.297220			0.148610	−	0.988896i	$-0.452520\pi$
$97$	2.92728			0.297220			0.148610	+	0.988896i	$0.452520\pi$
$98$		0			0
$99$	−8.71430	−	2.41690i	−0.875820	−	0.242908i
$100$		0			0
$101$	−0.416078			−0.0414013			−0.0207007	−	0.999786i	$-0.506590\pi$
$101$	−0.416078			−0.0414013			−0.0207007	+	0.999786i	$0.506590\pi$
$102$		0			0
$103$	−10.4552			−1.03019			−0.515093	−	0.857134i	$-0.672243\pi$
$103$	−10.4552			−1.03019			−0.515093	+	0.857134i	$0.672243\pi$
$104$		0			0
$105$	−10.1341	+	1.51628i	−0.988991	+	0.147974i
$106$		0			0
$107$	5.93315			0.573579			0.286789	−	0.957994i	$-0.407412\pi$
$107$	5.93315			0.573579			0.286789	+	0.957994i	$0.407412\pi$
$108$		0			0
$109$	−11.7565			−1.12607			−0.563033	−	0.826435i	$-0.690365\pi$
$109$	−11.7565			−1.12607			−0.563033	+	0.826435i	$0.690365\pi$
$110$		0			0
$111$	4.83135	+	6.35370i	0.458572	+	0.603066i
$112$		0			0
$113$	−14.2543			−1.34093			−0.670465	−	0.741941i	$-0.733905\pi$
$113$	−14.2543			−1.34093			−0.670465	+	0.741941i	$0.733905\pi$
$114$		0			0
$115$	−7.02896	−	4.84566i	−0.655454	−	0.451860i
$116$		0			0
$117$	3.52014	−	12.6921i	0.325437	−	1.17339i
$118$		0			0
$119$	−0.729842	−	7.13321i	−0.0669045	−	0.653900i
$120$		0			0
$121$	1.91329			0.173936
$122$		0			0
$123$	−14.6973	+	11.1758i	−1.32521	+	1.00769i
$124$		0			0
$125$	−2.65654	+	10.8601i	−0.237608	+	0.971361i
$126$		0			0
$127$			15.4193i			1.36824i	0.729371	+	0.684119i	$0.239813\pi$
$127$			15.4193i			1.36824i	−0.729371	+	0.684119i	$0.760187\pi$
$128$		0			0
$129$	9.62956	+	12.6638i	0.847836	+	1.11499i
$130$		0			0
$131$	−7.07988			−0.618572			−0.309286	−	0.950969i	$-0.600090\pi$
$131$	−7.07988			−0.618572			−0.309286	+	0.950969i	$0.600090\pi$
$132$		0			0
$133$	−2.21767	−	21.6746i	−0.192296	−	1.87943i
$134$		0			0
$135$	−2.58083	+	11.3287i	−0.222123	+	0.975019i
$136$		0			0
$137$	−13.6571			−1.16681			−0.583403	−	0.812183i	$-0.698279\pi$
$137$	−13.6571			−1.16681			−0.583403	+	0.812183i	$0.698279\pi$
$138$		0			0
$139$			7.97668i			0.676573i	0.941043	+	0.338286i	$0.109847\pi$
$139$			7.97668i			0.676573i	−0.941043	+	0.338286i	$0.890153\pi$
$140$		0			0
$141$	−12.6497	−	16.6356i	−1.06530	−	1.40097i
$142$		0			0
$143$		−	13.2345i		−	1.10673i
$144$		0			0
$145$	−1.48918	+	2.16016i	−0.123670	+	0.179392i
$146$		0			0
$147$	−7.96495	+	9.14110i	−0.656938	+	0.753945i
$148$		0			0
$149$			11.2515i			0.921755i	0.887464	+	0.460878i	$0.152465\pi$
$149$			11.2515i			0.921755i	−0.887464	+	0.460878i	$0.847535\pi$
$150$		0			0
$151$	−2.87722			−0.234145			−0.117072	−	0.993123i	$-0.537351\pi$
$151$	−2.87722			−0.234145			−0.117072	+	0.993123i	$0.537351\pi$
$152$		0			0
$153$	−7.83477	−	2.17296i	−0.633404	−	0.175674i
$154$		0			0
$155$	−2.20087	+	3.19251i	−0.176778	+	0.256428i
$156$		0			0
$157$	−10.7109			−0.854826			−0.427413	−	0.904056i	$-0.640575\pi$
$157$	−10.7109			−0.854826			−0.427413	+	0.904056i	$0.640575\pi$
$158$		0			0
$159$	9.84448	−	7.48574i	0.780718	−	0.593658i
$160$		0			0
$161$	−10.0491	+	1.02818i	−0.791979	+	0.0810323i
$162$		0			0
$163$		−	13.9669i		−	1.09397i	−0.837141	−	0.546987i	$-0.815775\pi$
$163$		−	13.9669i		−	1.09397i	0.837141	−	0.546987i	$-0.184225\pi$
$164$		0			0
$165$	0.543354	+	11.6621i	0.0423001	+	0.907896i
$166$		0			0
$167$		−	8.59766i		−	0.665307i	−0.943049	−	0.332653i	$-0.892056\pi$
$167$		−	8.59766i		−	0.665307i	0.943049	−	0.332653i	$-0.107944\pi$
$168$		0			0
$169$	6.27569			0.482745
$170$		0			0
$171$	−23.8064	−	6.60267i	−1.82052	−	0.504919i
$172$		0			0
$173$		−	8.74412i		−	0.664803i	−0.943138	−	0.332401i	$-0.892141\pi$
$173$		−	8.74412i		−	0.664803i	0.943138	−	0.332401i	$-0.107859\pi$
$174$		0			0
$175$	5.93947	+	11.8204i	0.448982	+	0.893541i
$176$		0			0
$177$	12.5666	−	9.55565i	0.944564	−	0.718246i
$178$		0			0
$179$		−	17.0676i		−	1.27569i	−0.770165	−	0.637845i	$-0.779826\pi$
$179$		−	17.0676i		−	1.27569i	0.770165	−	0.637845i	$-0.220174\pi$
$180$		0			0
$181$		−	6.65867i		−	0.494935i	−0.968896	−	0.247468i	$-0.920402\pi$
$181$		−	6.65867i		−	0.494935i	0.968896	−	0.247468i	$-0.0795984\pi$
$182$		0			0
$183$	−14.2167	−	18.6964i	−1.05093	−	1.38208i
$184$		0			0
$185$	5.84873	−	8.48399i	0.430008	−	0.623755i
$186$		0			0
$187$	−8.16959			−0.597420
$188$		0			0
$189$	6.36715	+	12.1844i	0.463142	+	0.886284i
$190$		0			0
$191$			13.1256i			0.949733i	0.880058	+	0.474866i	$0.157504\pi$
$191$			13.1256i			0.949733i	−0.880058	+	0.474866i	$0.842496\pi$
$192$		0			0
$193$		−	10.7880i		−	0.776538i	−0.921546	−	0.388269i	$-0.873073\pi$
$193$		−	10.7880i		−	0.776538i	0.921546	−	0.388269i	$-0.126927\pi$
$194$		0			0
$195$	−16.9856	+	0.791379i	−1.21636	+	0.0566719i
$196$		0			0
$197$	15.1781			1.08139			0.540696	−	0.841218i	$-0.318161\pi$
$197$	15.1781			1.08139			0.540696	+	0.841218i	$0.318161\pi$
$198$		0			0
$199$			19.8125i			1.40447i	0.711944	+	0.702236i	$0.247815\pi$
$199$			19.8125i			1.40447i	−0.711944	+	0.702236i	$0.752185\pi$
$200$		0			0
$201$	0.518370	+	0.681707i	0.0365630	+	0.0480839i
$202$		0			0
$203$	0.315985	+	3.08832i	0.0221778	+	0.216757i
$204$		0			0
$205$	19.6250	+	13.5292i	1.37067	+	0.944920i
$206$		0			0
$207$	−3.06122	+	11.0374i	−0.212769	+	0.767155i
$208$		0			0
$209$	−24.8238			−1.71710
$210$		0			0
$211$	5.58950			0.384797			0.192399	−	0.981317i	$-0.438373\pi$
$211$	5.58950			0.384797			0.192399	+	0.981317i	$0.438373\pi$
$212$		0			0
$213$	−5.40779	−	7.11177i	−0.370536	−	0.487291i
$214$		0			0
$215$	11.6573	−	16.9098i	0.795025	−	1.15324i
$216$		0			0
$217$	0.466995	+	4.56423i	0.0317017	+	0.309840i
$218$		0			0
$219$	−6.56326	+	4.99070i	−0.443504	+	0.337241i
$220$		0			0
$221$		−	11.8988i		−	0.800398i
$222$		0			0
$223$	2.13609			0.143043			0.0715216	−	0.997439i	$-0.477215\pi$
$223$	2.13609			0.143043			0.0715216	+	0.997439i	$0.477215\pi$
$224$		0			0
$225$	14.9350	−	1.39471i	0.995668	−	0.0929808i
$226$		0			0
$227$			3.92566i			0.260555i	0.991478	+	0.130277i	$0.0415868\pi$
$227$			3.92566i			0.260555i	−0.991478	+	0.130277i	$0.958413\pi$
$228$		0			0
$229$			15.0457i			0.994251i	0.867679	+	0.497126i	$0.165611\pi$
$229$			15.0457i			0.994251i	−0.867679	+	0.497126i	$0.834389\pi$
$230$		0			0
$231$	9.43707	+	10.0877i	0.620914	+	0.663725i
$232$		0			0
$233$	−9.72492			−0.637101			−0.318550	−	0.947906i	$-0.603196\pi$
$233$	−9.72492			−0.637101			−0.318550	+	0.947906i	$0.603196\pi$
$234$		0			0
$235$	−15.3135	+	22.2132i	−0.998941	+	1.44903i
$236$		0			0
$237$	−16.6914	+	12.6921i	−1.08422	+	0.824441i
$238$		0			0
$239$			2.52992i			0.163647i	0.996647	+	0.0818233i	$0.0260743\pi$
$239$			2.52992i			0.163647i	−0.996647	+	0.0818233i	$0.973926\pi$
$240$		0			0
$241$		−	9.87223i		−	0.635926i	−0.948103	−	0.317963i	$-0.897001\pi$
$241$		−	9.87223i		−	0.635926i	0.948103	−	0.317963i	$-0.102999\pi$
$242$		0			0
$243$	15.4959	−	1.69620i	0.994062	−	0.108811i
$244$		0			0
$245$	14.4191	+	6.09022i	0.921200	+	0.389090i
$246$		0			0
$247$		−	36.1551i		−	2.30049i
$248$		0			0
$249$	−8.56263	−	11.2607i	−0.542635	−	0.713617i
$250$		0			0
$251$	17.2182			1.08680			0.543402	−	0.839473i	$-0.317136\pi$
$251$	17.2182			1.08680			0.543402	+	0.839473i	$0.317136\pi$
$252$		0			0
$253$			11.5091i			0.723573i
$254$		0			0
$255$	0.488514	+	10.4851i	0.0305919	+	0.656601i
$256$		0			0
$257$		−	6.40758i		−	0.399694i	−0.979827	−	0.199847i	$-0.935955\pi$
$257$		−	6.40758i		−	0.399694i	0.979827	−	0.199847i	$-0.0640445\pi$
$258$		0			0
$259$	−1.24102	−	12.1293i	−0.0771135	−	0.753678i
$260$		0			0
$261$	3.39206	+	0.940783i	0.209963	+	0.0582330i
$262$		0			0
$263$	3.69040			0.227560			0.113780	−	0.993506i	$-0.463704\pi$
$263$	3.69040			0.227560			0.113780	+	0.993506i	$0.463704\pi$
$264$		0			0
$265$	−13.1452	−	9.06209i	−0.807502	−	0.556680i
$266$		0			0
$267$	−3.12776	+	2.37835i	−0.191416	+	0.145553i
$268$		0			0
$269$	−17.4759			−1.06553			−0.532763	−	0.846264i	$-0.678846\pi$
$269$	−17.4759			−1.06553			−0.532763	+	0.846264i	$0.678846\pi$
$270$		0			0
$271$			3.60083i			0.218735i	0.994001	+	0.109367i	$0.0348825\pi$
$271$			3.60083i			0.218735i	−0.994001	+	0.109367i	$0.965118\pi$
$272$		0			0
$273$	−14.6925	+	13.7448i	−0.889230	+	0.831874i
$274$		0			0
$275$	14.0864	−	5.36115i	0.849440	−	0.323290i
$276$		0			0
$277$			9.39015i			0.564199i	0.959385	+	0.282100i	$0.0910309\pi$
$277$			9.39015i			0.564199i	−0.959385	+	0.282100i	$0.908969\pi$
$278$		0			0
$279$	5.01313	+	1.39039i	0.300128	+	0.0832402i
$280$		0			0
$281$		−	2.80219i		−	0.167165i	−0.996501	−	0.0835824i	$-0.973364\pi$
$281$		−	2.80219i		−	0.167165i	0.996501	−	0.0835824i	$-0.0266362\pi$
$282$		0			0
$283$	−11.8686			−0.705518			−0.352759	−	0.935714i	$-0.614756\pi$
$283$	−11.8686			−0.705518			−0.352759	+	0.935714i	$0.614756\pi$
$284$		0			0
$285$	1.48438	+	31.8595i	0.0879269	+	1.88719i
$286$		0			0
$287$	28.0573	−	2.87072i	1.65617	−	0.169453i
$288$		0			0
$289$	9.65496			0.567939
$290$		0			0
$291$	−4.03592	+	3.06891i	−0.236590	+	0.179903i
$292$		0			0
$293$		−	15.5145i		−	0.906366i	−0.891418	−	0.453183i	$-0.850288\pi$
$293$		−	15.5145i		−	0.906366i	0.891418	−	0.453183i	$-0.149712\pi$
$294$		0			0
$295$	−16.7800	−	11.5679i	−0.976968	−	0.673507i
$296$		0			0
$297$	14.5485	−	5.80369i	0.844189	−	0.336764i
$298$		0			0
$299$	−16.7627			−0.969412
$300$		0			0
$301$	−2.47353	−	24.1754i	−0.142572	−	1.39345i
$302$		0			0
$303$	0.573658	−	0.436210i	0.0329558	−	0.0250596i
$304$		0			0
$305$	−17.2105	+	24.9650i	−0.985468	+	1.42949i
$306$		0			0
$307$	4.98662			0.284601			0.142301	−	0.989823i	$-0.454550\pi$
$307$	4.98662			0.284601			0.142301	+	0.989823i	$0.454550\pi$
$308$		0			0
$309$	14.4149	−	10.9611i	0.820037	−	0.623556i
$310$		0			0
$311$	23.2945			1.32091			0.660455	−	0.750865i	$-0.270363\pi$
$311$	23.2945			1.32091			0.660455	+	0.750865i	$0.270363\pi$
$312$		0			0
$313$	7.16856			0.405191			0.202595	−	0.979263i	$-0.435062\pi$
$313$	7.16856			0.405191			0.202595	+	0.979263i	$0.435062\pi$
$314$		0			0
$315$	12.3826	−	12.7150i	0.697680	−	0.716410i
$316$		0			0
$317$	−2.91003			−0.163444			−0.0817218	−	0.996655i	$-0.526042\pi$
$317$	−2.91003			−0.163444			−0.0817218	+	0.996655i	$0.526042\pi$
$318$		0			0
$319$	3.53702			0.198035
$320$		0			0
$321$	−8.18019	+	6.22022i	−0.456574	+	0.347179i
$322$		0			0
$323$	−22.3183			−1.24182
$324$		0			0
$325$	7.80836	+	20.5164i	0.433130	+	1.13804i
$326$		0			0
$327$	16.2090	−	12.3253i	0.896358	−	0.681591i
$328$		0			0
$329$	3.24932	+	31.7576i	0.179141	+	1.75085i
$330$		0			0
$331$	29.4323			1.61775			0.808874	−	0.587982i	$-0.200078\pi$
$331$	29.4323			1.61775			0.808874	+	0.587982i	$0.200078\pi$
$332$		0			0
$333$	−13.3222	−	3.69491i	−0.730054	−	0.202480i
$334$		0			0
$335$	0.627528	−	0.910273i	0.0342855	−	0.0497335i
$336$		0			0
$337$			15.8598i			0.863940i	0.901888	+	0.431970i	$0.142181\pi$
$337$			15.8598i			0.863940i	−0.901888	+	0.431970i	$0.857819\pi$
$338$		0			0
$339$	19.6528	−	14.9440i	1.06739	−	0.811645i
$340$		0			0
$341$	5.22737			0.283078
$342$		0			0
$343$	17.6606	−	5.57712i	0.953581	−	0.301136i
$344$		0			0
$345$	14.7711	−	0.688207i	0.795251	−	0.0370518i
$346$		0			0
$347$	11.0724			0.594400			0.297200	−	0.954815i	$-0.403947\pi$
$347$	11.0724			0.594400			0.297200	+	0.954815i	$0.403947\pi$
$348$		0			0
$349$		−	13.9422i		−	0.746306i	−0.927770	−	0.373153i	$-0.878277\pi$
$349$		−	13.9422i		−	0.746306i	0.927770	−	0.373153i	$-0.121723\pi$
$350$		0			0
$351$	8.45290	+	21.1894i	0.451182	+	1.13101i
$352$		0			0
$353$			31.2855i			1.66516i	0.553904	+	0.832580i	$0.313137\pi$
$353$			31.2855i			1.66516i	−0.553904	+	0.832580i	$0.686863\pi$
$354$		0			0
$355$	−6.54656	+	9.49623i	−0.347455	+	0.504008i
$356$		0			0
$357$	8.48460	+	9.06959i	0.449052	+	0.480014i
$358$		0			0
$359$			2.58454i			0.136407i	0.997671	+	0.0682033i	$0.0217267\pi$
$359$			2.58454i			0.136407i	−0.997671	+	0.0682033i	$0.978273\pi$
$360$		0			0
$361$	−48.8155			−2.56924
$362$		0			0
$363$	−2.63791	+	2.00586i	−0.138454	+	0.105281i
$364$		0			0
$365$	8.76381	+	6.04164i	0.458719	+	0.316234i
$366$		0			0
$367$	25.8803			1.35094			0.675470	−	0.737388i	$-0.263941\pi$
$367$	25.8803			1.35094			0.675470	+	0.737388i	$0.263941\pi$
$368$		0			0
$369$	8.54700	−	30.8168i	0.444939	−	1.60426i
$370$		0			0
$371$	−18.7933	+	1.92285i	−0.975697	+	0.0998296i
$372$		0			0
$373$			5.37044i			0.278071i	0.990287	+	0.139035i	$0.0444002\pi$
$373$			5.37044i			0.278071i	−0.990287	+	0.139035i	$0.955600\pi$
$374$		0			0
$375$	−7.72297	−	17.7583i	−0.398812	−	0.917033i
$376$		0			0
$377$			5.15157i			0.265319i
$378$		0			0
$379$	−33.6626			−1.72913			−0.864565	−	0.502521i	$-0.832406\pi$
$379$	−33.6626			−1.72913			−0.864565	+	0.502521i	$0.832406\pi$
$380$		0			0
$381$	−16.1653	−	21.2590i	−0.828174	−	1.08913i
$382$		0			0
$383$		−	5.70589i		−	0.291557i	−0.989317	−	0.145779i	$-0.953431\pi$
$383$		−	5.70589i		−	0.291557i	0.989317	−	0.145779i	$-0.0465687\pi$
$384$		0			0
$385$	8.57496	−	15.6367i	0.437020	−	0.796918i
$386$		0			0
$387$	−26.5531	−	7.36447i	−1.34977	−	0.374357i
$388$		0			0
$389$		−	30.8578i		−	1.56455i	−0.622930	−	0.782277i	$-0.714058\pi$
$389$		−	30.8578i		−	1.56455i	0.622930	−	0.782277i	$-0.285942\pi$
$390$		0			0
$391$			10.3475i			0.523297i
$392$		0			0
$393$	9.76122	−	7.42243i	0.492388	−	0.374412i
$394$		0			0
$395$	22.2877	+	15.3648i	1.12142	+	0.773088i
$396$		0			0
$397$	−22.7345			−1.14101			−0.570505	−	0.821294i	$-0.693253\pi$
$397$	−22.7345			−1.14101			−0.570505	+	0.821294i	$0.693253\pi$
$398$		0			0
$399$	25.7809	+	27.5585i	1.29066	+	1.37965i
$400$		0			0
$401$		−	13.7170i		−	0.684992i	−0.939519	−	0.342496i	$-0.888728\pi$
$401$		−	13.7170i		−	0.684992i	0.939519	−	0.342496i	$-0.111272\pi$
$402$		0			0
$403$			7.61351i			0.379256i
$404$		0			0
$405$	−8.31856	−	18.3249i	−0.413353	−	0.910571i
$406$		0			0
$407$	−13.8916			−0.688580
$408$		0			0
$409$		−	12.6202i		−	0.624030i	−0.950077	−	0.312015i	$-0.898996\pi$
$409$		−	12.6202i		−	0.624030i	0.950077	−	0.312015i	$-0.101004\pi$
$410$		0			0
$411$	18.8294	−	14.3179i	0.928788	−	0.706250i
$412$		0			0
$413$	−23.9898	+	2.45455i	−1.18046	+	0.120780i
$414$		0			0
$415$	−10.3657	+	15.0362i	−0.508834	+	0.738099i
$416$		0			0
$417$	−8.36262	−	10.9977i	−0.409519	−	0.538558i
$418$		0			0
$419$	−23.6389			−1.15484			−0.577418	−	0.816449i	$-0.695940\pi$
$419$	−23.6389			−1.15484			−0.577418	+	0.816449i	$0.695940\pi$
$420$		0			0
$421$	−8.42017			−0.410374			−0.205187	−	0.978723i	$-0.565780\pi$
$421$	−8.42017			−0.410374			−0.205187	+	0.978723i	$0.565780\pi$
$422$		0			0
$423$	34.8810	+	9.67421i	1.69597	+	0.470376i
$424$		0			0
$425$	12.6646	−	4.82006i	0.614325	−	0.233807i
$426$		0			0
$427$	3.65183	+	35.6916i	0.176725	+	1.72724i
$428$		0			0
$429$	13.8749	+	18.2468i	0.669885	+	0.880964i
$430$		0			0
$431$		−	18.5310i		−	0.892606i	−0.894882	−	0.446303i	$-0.852740\pi$
$431$		−	18.5310i		−	0.892606i	0.894882	−	0.446303i	$-0.147260\pi$
$432$		0			0
$433$	9.12614			0.438574			0.219287	−	0.975660i	$-0.429627\pi$
$433$	9.12614			0.438574			0.219287	+	0.975660i	$0.429627\pi$
$434$		0			0
$435$	−0.211502	−	4.53951i	−0.0101407	−	0.217653i
$436$		0			0
$437$			31.4415i			1.50405i
$438$		0			0
$439$			16.9640i			0.809645i	0.914395	+	0.404823i	$0.132667\pi$
$439$			16.9640i			0.809645i	−0.914395	+	0.404823i	$0.867333\pi$
$440$		0			0
$441$	1.39811	−	20.9534i	0.0665765	−	0.997781i
$442$		0			0
$443$	4.34739			0.206551			0.103275	−	0.994653i	$-0.467068\pi$
$443$	4.34739			0.206551			0.103275	+	0.994653i	$0.467068\pi$
$444$		0			0
$445$	4.17644	+	2.87918i	0.197982	+	0.136486i
$446$		0			0
$447$	−11.7959	−	15.5127i	−0.557925	−	0.733725i
$448$		0			0
$449$		−	15.9781i		−	0.754053i	−0.926202	−	0.377027i	$-0.876947\pi$
$449$		−	15.9781i		−	0.754053i	0.926202	−	0.377027i	$-0.123053\pi$
$450$		0			0
$451$		−	32.1338i		−	1.51312i
$452$		0			0
$453$	3.96690	−	3.01643i	0.186381	−	0.141724i
$454$		0			0
$455$	22.7743	+	12.4892i	1.06768	+	0.585501i
$456$		0			0
$457$		−	16.4839i		−	0.771085i	−0.922690	−	0.385542i	$-0.874014\pi$
$457$		−	16.4839i		−	0.771085i	0.922690	−	0.385542i	$-0.125986\pi$
$458$		0			0
$459$	13.0801	−	5.21792i	0.610527	−	0.243552i
$460$		0			0
$461$	20.8741			0.972203			0.486102	−	0.873902i	$-0.338419\pi$
$461$	20.8741			0.972203			0.486102	+	0.873902i	$0.338419\pi$
$462$		0			0
$463$			9.46270i			0.439769i	0.975526	+	0.219885i	$0.0705681\pi$
$463$			9.46270i			0.439769i	−0.975526	+	0.219885i	$0.929432\pi$
$464$		0			0
$465$	−0.312579	−	6.70895i	−0.0144955	−	0.311120i
$466$		0			0
$467$			19.1519i			0.886245i	0.896461	+	0.443122i	$0.146129\pi$
$467$			19.1519i			0.886245i	−0.896461	+	0.443122i	$0.853871\pi$
$468$		0			0
$469$	−0.133153	−	1.30139i	−0.00614844	−	0.0600926i
$470$		0			0
$471$	14.7675	−	11.2292i	0.680449	−	0.517414i
$472$		0			0
$473$	−27.6879			−1.27309
$474$		0			0
$475$	38.4822	−	14.6460i	1.76569	−	0.672005i
$476$		0			0
$477$	−5.72492	+	20.6416i	−0.262126	+	0.945114i
$478$		0			0
$479$	22.9310			1.04774			0.523871	−	0.851798i	$-0.324487\pi$
$479$	22.9310			1.04774			0.523871	+	0.851798i	$0.324487\pi$
$480$		0			0
$481$		−	20.2327i		−	0.922530i
$482$		0			0
$483$	12.7770	−	11.9529i	0.581375	−	0.543876i
$484$		0			0
$485$	5.38909	+	3.71516i	0.244706	+	0.168697i
$486$		0			0
$487$			31.0733i			1.40807i	0.710167	+	0.704033i	$0.248619\pi$
$487$			31.0733i			1.40807i	−0.710167	+	0.704033i	$0.751381\pi$
$488$		0			0
$489$	14.6427	+	19.2566i	0.662166	+	0.870812i
$490$		0			0
$491$			5.50641i			0.248501i	0.992251	+	0.124250i	$0.0396526\pi$
$491$			5.50641i			0.248501i	−0.992251	+	0.124250i	$0.960347\pi$
$492$		0			0
$493$	3.18003			0.143221
$494$		0			0
$495$	−12.9755	−	15.5093i	−0.583207	−	0.697090i
$496$		0			0
$497$	1.38909	+	13.5765i	0.0623093	+	0.608988i
$498$		0			0
$499$	−22.6482			−1.01387			−0.506937	−	0.861983i	$-0.669222\pi$
$499$	−22.6482			−1.01387			−0.506937	+	0.861983i	$0.669222\pi$
$500$		0			0
$501$	9.01365	+	11.8538i	0.402700	+	0.529590i
$502$		0			0
$503$			3.10345i			0.138376i	0.997604	+	0.0691881i	$0.0220409\pi$
$503$			3.10345i			0.138376i	−0.997604	+	0.0691881i	$0.977959\pi$
$504$		0			0
$505$	−0.765997	−	0.528067i	−0.0340864	−	0.0234987i
$506$		0			0
$507$	−8.65246	+	6.57933i	−0.384269	+	0.292198i
$508$		0			0
$509$	−22.9387			−1.01674			−0.508371	−	0.861138i	$-0.669752\pi$
$509$	−22.9387			−1.01674			−0.508371	+	0.861138i	$0.669752\pi$
$510$		0			0
$511$	12.5294	−	1.28196i	0.554266	−	0.0567104i
$512$		0			0
$513$	39.7447	−	15.8550i	1.75477	−	0.700014i
$514$		0			0
$515$	−19.2480	−	13.2693i	−0.848170	−	0.584715i
$516$		0			0
$517$	36.3717			1.59962
$518$		0			0
$519$	9.16719	+	12.0558i	0.402395	+	0.529189i
$520$		0			0
$521$	−17.6343			−0.772572			−0.386286	−	0.922379i	$-0.626242\pi$
$521$	−17.6343			−0.772572			−0.386286	+	0.922379i	$0.626242\pi$
$522$		0			0
$523$	−31.8408			−1.39230			−0.696151	−	0.717895i	$-0.745106\pi$
$523$	−31.8408			−1.39230			−0.696151	+	0.717895i	$0.745106\pi$
$524$		0			0
$525$	−20.5813	−	10.0703i	−0.898240	−	0.439505i
$526$		0			0
$527$	4.69978			0.204725
$528$		0			0
$529$	−8.42266			−0.366203
$530$		0			0
$531$	−7.30794	+	26.3493i	−0.317138	+	1.14346i
$532$		0			0
$533$	46.8019			2.02721
$534$		0			0
$535$	10.9229	+	7.53007i	0.472237	+	0.325553i
$536$		0			0
$537$	17.8934	+	23.5315i	0.772156	+	1.01546i
$538$		0			0
$539$	−4.27320	−	20.6637i	−0.184060	−	0.890049i
$540$		0			0
$541$	−29.4219			−1.26494			−0.632472	−	0.774583i	$-0.717960\pi$
$541$	−29.4219			−1.26494			−0.632472	+	0.774583i	$0.717960\pi$
$542$		0			0
$543$	6.98085	+	9.18050i	0.299577	+	0.393973i
$544$		0			0
$545$	−21.6436	−	14.9207i	−0.927108	−	0.639135i
$546$		0			0
$547$		−	20.7168i		−	0.885784i	−0.896575	−	0.442892i	$-0.853952\pi$
$547$		−	20.7168i		−	0.885784i	0.896575	−	0.442892i	$-0.146048\pi$
$548$		0			0
$549$	39.2020	+	10.8726i	1.67310	+	0.464032i
$550$		0			0
$551$	9.66270			0.411645
$552$		0			0
$553$	31.8641	−	3.26021i	1.35500	−	0.138638i
$554$		0			0
$555$	0.830669	+	17.8288i	0.0352599	+	0.756792i
$556$		0			0
$557$	8.76381			0.371335			0.185667	−	0.982613i	$-0.440555\pi$
$557$	8.76381			0.371335			0.185667	+	0.982613i	$0.440555\pi$
$558$		0			0
$559$		−	40.3265i		−	1.70563i
$560$		0			0
$561$	11.2636	−	8.56487i	0.475551	−	0.361609i
$562$		0			0
$563$		−	1.85064i		−	0.0779952i	−0.999239	−	0.0389976i	$-0.987584\pi$
$563$		−	1.85064i		−	0.0779952i	0.999239	−	0.0389976i	$-0.0124165\pi$
$564$		0			0
$565$	−26.2420	−	18.0909i	−1.10401	−	0.761088i
$566$		0			0
$567$	−21.5525	−	10.1237i	−0.905119	−	0.425157i
$568$		0			0
$569$			11.8626i			0.497307i	0.968593	+	0.248653i	$0.0799880\pi$
$569$			11.8626i			0.497307i	−0.968593	+	0.248653i	$0.920012\pi$
$570$		0			0
$571$	29.9166			1.25197			0.625985	−	0.779835i	$-0.284697\pi$
$571$	29.9166			1.25197			0.625985	+	0.779835i	$0.284697\pi$
$572$		0			0
$573$	−13.7606	−	18.0966i	−0.574859	−	0.755996i
$574$		0			0
$575$	−6.79038	−	17.8416i	−0.283178	−	0.744048i
$576$		0			0
$577$	−30.5778			−1.27297			−0.636486	−	0.771288i	$-0.719613\pi$
$577$	−30.5778			−1.27297			−0.636486	+	0.771288i	$0.719613\pi$
$578$		0			0
$579$	11.3100	+	14.8737i	0.470027	+	0.618131i
$580$		0			0
$581$	2.19947	+	21.4968i	0.0912495	+	0.891838i
$582$		0			0
$583$			21.5237i			0.891422i
$584$		0			0
$585$	22.5888	−	18.8985i	0.933931	−	0.781356i
$586$		0			0
$587$		−	39.6667i		−	1.63722i	−0.574351	−	0.818609i	$-0.694745\pi$
$587$		−	39.6667i		−	1.63722i	0.574351	−	0.818609i	$-0.305255\pi$
$588$		0			0
$589$	14.2805			0.588419
$590$		0			0
$591$	−20.9264	+	15.9124i	−0.860797	+	0.654550i
$592$		0			0
$593$			23.8526i			0.979509i	0.871860	+	0.489755i	$0.162914\pi$
$593$			23.8526i			0.979509i	−0.871860	+	0.489755i	$0.837086\pi$
$594$		0			0
$595$	7.70949	−	14.0585i	0.316058	−	0.576341i
$596$		0			0
$597$	−20.7711	−	27.3161i	−0.850106	−	1.11797i
$598$		0			0
$599$			37.7489i			1.54238i	0.636605	+	0.771190i	$0.280338\pi$
$599$			37.7489i			1.54238i	−0.636605	+	0.771190i	$0.719662\pi$
$600$		0			0
$601$			43.9087i			1.79107i	0.444986	+	0.895537i	$0.353209\pi$
$601$			43.9087i			1.79107i	−0.444986	+	0.895537i	$0.646791\pi$
$602$		0			0
$603$	−1.42938	−	0.396438i	−0.0582090	−	0.0161442i
$604$		0			0
$605$	3.52235	+	2.42826i	0.143204	+	0.0987227i
$606$		0			0
$607$	4.62053			0.187542			0.0937709	−	0.995594i	$-0.470108\pi$
$607$	4.62053			0.187542			0.0937709	+	0.995594i	$0.470108\pi$
$608$		0			0
$609$	−3.67340	−	3.92667i	−0.148854	−	0.159117i
$610$		0			0
$611$			52.9743i			2.14311i
$612$		0			0
$613$			28.2008i			1.13902i	0.821985	+	0.569510i	$0.192867\pi$
$613$			28.2008i			1.13902i	−0.821985	+	0.569510i	$0.807133\pi$
$614$		0			0
$615$	−41.2414	+	1.92149i	−1.66301	+	0.0774820i
$616$		0			0
$617$	−7.84004			−0.315628			−0.157814	−	0.987469i	$-0.550445\pi$
$617$	−7.84004			−0.315628			−0.157814	+	0.987469i	$0.550445\pi$
$618$		0			0
$619$		−	9.24487i		−	0.371583i	−0.982589	−	0.185791i	$-0.940515\pi$
$619$		−	9.24487i		−	0.371583i	0.982589	−	0.185791i	$-0.0594849\pi$
$620$		0			0
$621$	−7.35089	−	18.4270i	−0.294981	−	0.739448i
$622$		0			0
$623$	5.97094	−	0.610923i	0.239220	−	0.0244761i
$624$		0			0
$625$	−18.6738	+	16.6219i	−0.746954	+	0.664876i
$626$		0			0
$627$	34.2252	−	26.0249i	1.36682	−	1.03933i
$628$		0			0
$629$	−12.4895			−0.497989
$630$		0			0
$631$	−34.6577			−1.37970			−0.689851	−	0.723951i	$-0.742324\pi$
$631$	−34.6577			−1.37970			−0.689851	+	0.723951i	$0.742324\pi$
$632$		0			0
$633$	−7.70640	+	5.85995i	−0.306302	+	0.232912i
$634$		0			0
$635$	−19.5694	+	28.3867i	−0.776587	+	1.12649i
$636$		0			0
$637$	30.0961	−	6.22378i	1.19245	−	0.246595i
$638$		0			0
$639$	14.9117	+	4.13575i	0.589899	+	0.163608i
$640$		0			0
$641$			23.1085i			0.912732i	0.889792	+	0.456366i	$0.150849\pi$
$641$			23.1085i			0.912732i	−0.889792	+	0.456366i	$0.849151\pi$
$642$		0			0
$643$	6.11558			0.241175			0.120588	−	0.992703i	$-0.461522\pi$
$643$	6.11558			0.241175			0.120588	+	0.992703i	$0.461522\pi$
$644$		0			0
$645$	1.65564	+	35.5354i	0.0651908	+	1.39920i
$646$		0			0
$647$		−	22.7956i		−	0.896188i	−0.893987	−	0.448094i	$-0.852103\pi$
$647$		−	22.7956i		−	0.896188i	0.893987	−	0.448094i	$-0.147897\pi$
$648$		0			0
$649$			27.4753i			1.07850i
$650$		0			0
$651$	−5.42893	−	5.80324i	−0.212776	−	0.227447i
$652$		0			0
$653$	17.0044			0.665432			0.332716	−	0.943027i	$-0.392035\pi$
$653$	17.0044			0.665432			0.332716	+	0.943027i	$0.392035\pi$
$654$		0			0
$655$	−13.0340	−	8.98544i	−0.509280	−	0.351090i
$656$		0			0
$657$	3.81678	−	13.7616i	0.148907	−	0.536893i
$658$		0			0
$659$			13.4782i			0.525038i	0.964927	+	0.262519i	$0.0845532\pi$
$659$			13.4782i			0.525038i	−0.964927	+	0.262519i	$0.915447\pi$
$660$		0			0
$661$			29.4321i			1.14478i	0.819983	+	0.572388i	$0.193983\pi$
$661$			29.4321i			1.14478i	−0.819983	+	0.572388i	$0.806017\pi$
$662$		0			0
$663$	12.4745	+	16.4052i	0.484469	+	0.637124i
$664$		0			0
$665$	23.4257	−	42.7174i	0.908411	−	1.65651i
$666$		0			0
$667$		−	4.47995i		−	0.173464i
$668$		0			0
$669$	−2.94509	+	2.23944i	−0.113864	+	0.0865819i
$670$		0			0
$671$	40.8773			1.57805
$672$		0			0
$673$			4.23323i			0.163179i	0.996666	+	0.0815895i	$0.0259996\pi$
$673$			4.23323i			0.163179i	−0.996666	+	0.0815895i	$0.974000\pi$
$674$		0			0
$675$	−19.1291	+	17.5806i	−0.736281	+	0.676676i
$676$		0			0
$677$		−	18.8318i		−	0.723764i	−0.932224	−	0.361882i	$-0.882134\pi$
$677$		−	18.8318i		−	0.723764i	0.932224	−	0.361882i	$-0.117866\pi$
$678$		0			0
$679$	7.70462	−	0.788307i	0.295676	−	0.0302525i
$680$		0			0
$681$	−4.11560	−	5.41241i	−0.157710	−	0.207404i
$682$		0			0
$683$	44.2515			1.69324			0.846618	−	0.532202i	$-0.178635\pi$
$683$	44.2515			1.69324			0.846618	+	0.532202i	$0.178635\pi$
$684$		0			0
$685$	−25.1426	−	17.3330i	−0.960651	−	0.662259i
$686$		0			0
$687$	−15.7737	−	20.7440i	−0.601805	−	0.791433i
$688$		0			0
$689$	−31.3487			−1.19429
$690$		0			0
$691$			50.4812i			1.92039i	0.279322	+	0.960197i	$0.409890\pi$
$691$			50.4812i			1.92039i	−0.279322	+	0.960197i	$0.590110\pi$
$692$		0			0
$693$	−23.5870	−	4.01457i	−0.895995	−	0.152501i
$694$		0			0
$695$	−10.1236	+	14.6850i	−0.384011	+	0.557034i
$696$		0			0
$697$		−	28.8905i		−	1.09431i
$698$		0			0
$699$	13.4080	−	10.1955i	0.507138	−	0.385628i
$700$		0			0
$701$		−	44.6207i		−	1.68530i	−0.538462	−	0.842650i	$-0.680994\pi$
$701$		−	44.6207i		−	1.68530i	0.538462	−	0.842650i	$-0.319006\pi$
$702$		0			0
$703$	−37.9501			−1.43131
$704$		0			0
$705$	−2.17490	−	46.6804i	−0.0819116	−	1.75809i
$706$		0			0
$707$	−1.09512	+	0.112049i	−0.0411863	+	0.00421403i
$708$		0			0
$709$	−0.166857			−0.00626645			−0.00313323	−	0.999995i	$-0.500997\pi$
$709$	−0.166857			−0.00626645			−0.00313323	+	0.999995i	$0.500997\pi$
$710$		0			0
$711$	9.70664	−	34.9979i	0.364027	−	1.31253i
$712$		0			0
$713$		−	6.62093i		−	0.247956i
$714$		0			0
$715$	16.7966	−	24.3646i	0.628158	−	0.911186i
$716$		0			0
$717$	−2.65232	−	3.48806i	−0.0990528	−	0.130264i
$718$		0			0
$719$	−32.9774			−1.22985			−0.614925	−	0.788586i	$-0.710814\pi$
$719$	−32.9774			−1.22985			−0.614925	+	0.788586i	$0.710814\pi$
$720$		0			0
$721$	−27.5183	+	2.81557i	−1.02484	+	0.104857i
$722$		0			0
$723$	10.3499	+	13.6111i	0.384916	+	0.506203i
$724$		0			0
$725$	−5.48315	+	2.08684i	−0.203639	+	0.0775033i
$726$		0			0
$727$	−35.8146			−1.32829			−0.664144	−	0.747604i	$-0.731204\pi$
$727$	−35.8146			−1.32829			−0.664144	+	0.747604i	$0.731204\pi$
$728$		0			0
$729$	−19.5864	+	18.5843i	−0.725421	+	0.688306i
$730$		0			0
$731$	−24.8933			−0.920713
$732$		0			0
$733$	−14.7825			−0.546003			−0.273002	−	0.962014i	$-0.588016\pi$
$733$	−14.7825			−0.546003			−0.273002	+	0.962014i	$0.588016\pi$
$734$		0			0
$735$	−26.2648	+	6.71996i	−0.968793	+	0.247869i
$736$		0			0
$737$	−1.49047			−0.0549021
$738$		0			0
$739$	−44.5145			−1.63749			−0.818746	−	0.574156i	$-0.805330\pi$
$739$	−44.5145			−1.63749			−0.818746	+	0.574156i	$0.805330\pi$
$740$		0			0
$741$	37.9044	+	49.8480i	1.39245	+	1.83121i
$742$		0			0
$743$	9.42880			0.345909			0.172955	−	0.984930i	$-0.444669\pi$
$743$	9.42880			0.345909			0.172955	+	0.984930i	$0.444669\pi$
$744$		0			0
$745$	−14.2798	+	20.7138i	−0.523172	+	0.758897i
$746$		0			0
$747$	23.6111	+	6.54850i	0.863884	+	0.239597i
$748$		0			0
$749$	15.6161	−	1.59778i	0.570600	−	0.0583816i
$750$		0			0
$751$	−23.8428			−0.870037			−0.435019	−	0.900421i	$-0.643258\pi$
$751$	−23.8428			−0.870037			−0.435019	+	0.900421i	$0.643258\pi$
$752$		0			0
$753$	−23.7392	+	18.0513i	−0.865105	+	0.657826i
$754$		0			0
$755$	−5.29694	−	3.65163i	−0.192775	−	0.132896i
$756$		0			0
$757$			35.2632i			1.28166i	0.767682	+	0.640831i	$0.221410\pi$
$757$			35.2632i			1.28166i	−0.767682	+	0.640831i	$0.778590\pi$
$758$		0			0
$759$	−12.0660	−	15.8680i	−0.437968	−	0.575970i
$760$		0			0
$761$	52.0087			1.88531			0.942657	−	0.333764i	$-0.108319\pi$
$761$	52.0087			1.88531			0.942657	+	0.333764i	$0.108319\pi$
$762$		0			0
$763$	−30.9431	+	3.16598i	−1.12022	+	0.114616i
$764$		0			0
$765$	−11.6659	−	13.9439i	−0.421782	−	0.504144i
$766$		0			0
$767$	−40.0170			−1.44493
$768$		0			0
$769$		−	28.0310i		−	1.01082i	−0.862879	−	0.505411i	$-0.831341\pi$
$769$		−	28.0310i		−	1.01082i	0.862879	−	0.505411i	$-0.168659\pi$
$770$		0			0
$771$	6.71761	+	8.83431i	0.241929	+	0.318160i
$772$		0			0
$773$		−	46.6884i		−	1.67927i	−0.543155	−	0.839633i	$-0.682770\pi$
$773$		−	46.6884i		−	1.67927i	0.543155	−	0.839633i	$-0.317230\pi$
$774$		0			0
$775$	−8.10356	+	3.08415i	−0.291088	+	0.110786i
$776$		0			0
$777$	14.4272	+	15.4219i	0.517573	+	0.553259i
$778$		0			0
$779$		−	87.7855i		−	3.14524i
$780$		0			0
$781$	15.5490			0.556387
$782$		0			0
$783$	−5.66303	+	2.25910i	−0.202380	+	0.0807336i
$784$		0			0
$785$	−19.7188	−	13.5938i	−0.703793	−	0.485184i
$786$		0			0
$787$	−16.1183			−0.574554			−0.287277	−	0.957847i	$-0.592750\pi$
$787$	−16.1183			−0.574554			−0.287277	+	0.957847i	$0.592750\pi$
$788$		0			0
$789$	−5.08806	+	3.86896i	−0.181140	+	0.137739i
$790$		0			0
$791$	−37.5174	+	3.83864i	−1.33397	+	0.136486i
$792$		0			0
$793$			59.5366i			2.11421i
$794$		0			0
$795$	27.6242	−	1.28705i	0.979728	−	0.0456468i
$796$		0			0
$797$			35.4492i			1.25567i	0.778345	+	0.627837i	$0.216060\pi$
$797$			35.4492i			1.25567i	−0.778345	+	0.627837i	$0.783940\pi$
$798$		0			0
$799$	32.7007			1.15687
$800$		0			0
$801$	1.81891	−	6.55819i	0.0642679	−	0.231722i
$802$		0			0
$803$		−	14.3498i		−	0.506392i
$804$		0			0
$805$	−19.8052	−	10.8609i	−0.698042	−	0.382798i
$806$		0			0
$807$	24.0946	−	18.3215i	0.848168	−	0.644947i
$808$		0			0
$809$			31.8061i			1.11824i	0.829086	+	0.559121i	$0.188861\pi$
$809$			31.8061i			1.11824i	−0.829086	+	0.559121i	$0.811139\pi$
$810$		0			0
$811$			40.0124i			1.40503i	0.711671	+	0.702513i	$0.247939\pi$
$811$			40.0124i			1.40503i	−0.711671	+	0.702513i	$0.752061\pi$
$812$		0			0
$813$	−3.77505	−	4.96456i	−0.132397	−	0.174115i
$814$		0			0
$815$	17.7261	−	25.7130i	0.620920	−	0.900686i
$816$		0			0
$817$	−75.6398			−2.64630
$818$		0			0
$819$	5.84710	−	34.3537i	0.204314	−	1.20042i
$820$		0			0
$821$		−	46.2101i		−	1.61274i	−0.591409	−	0.806371i	$-0.701428\pi$
$821$		−	46.2101i		−	1.61274i	0.591409	−	0.806371i	$-0.298572\pi$
$822$		0			0
$823$		−	12.6943i		−	0.442494i	−0.975218	−	0.221247i	$-0.928987\pi$
$823$		−	12.6943i		−	0.442494i	0.975218	−	0.221247i	$-0.0710128\pi$
$824$		0			0
$825$	−13.8007	+	22.1595i	−0.480479	+	0.771495i
$826$		0			0
$827$	−14.8569			−0.516626			−0.258313	−	0.966061i	$-0.583167\pi$
$827$	−14.8569			−0.516626			−0.258313	+	0.966061i	$0.583167\pi$
$828$		0			0
$829$		−	5.46791i		−	0.189908i	−0.995482	−	0.0949542i	$-0.969730\pi$
$829$		−	5.46791i		−	0.189908i	0.995482	−	0.0949542i	$-0.0302705\pi$
$830$		0			0
$831$	−9.84448	−	12.9465i	−0.341501	−	0.449108i
$832$		0			0
$833$	−3.84191	−	18.5781i	−0.133114	−	0.643694i
$834$		0			0
$835$	10.9117	−	15.8282i	0.377616	−	0.547758i
$836$		0			0
$837$	−8.36940	+	3.33873i	−0.289289	+	0.115403i
$838$		0			0
$839$	−47.5202			−1.64058			−0.820290	−	0.571948i	$-0.806188\pi$
$839$	−47.5202			−1.64058			−0.820290	+	0.571948i	$0.806188\pi$
$840$		0			0
$841$	27.6232			0.952524
$842$		0			0
$843$	2.93777	+	3.86346i	0.101182	+	0.133065i
$844$		0			0
$845$	11.5535	+	7.96480i	0.397452	+	0.273998i
$846$		0			0
$847$	5.03580	−	0.515244i	0.173032	−	0.0177040i
$848$		0			0
$849$	16.3636	−	12.4429i	0.561598	−	0.427039i
$850$		0			0
$851$			17.5949i			0.603146i
$852$		0			0
$853$	31.8095			1.08914			0.544569	−	0.838716i	$-0.316693\pi$
$853$	31.8095			1.08914			0.544569	+	0.838716i	$0.316693\pi$
$854$		0			0
$855$	−35.4476	−	42.3694i	−1.21228	−	1.44900i
$856$		0			0
$857$		−	47.8666i		−	1.63509i	−0.575864	−	0.817545i	$-0.695334\pi$
$857$		−	47.8666i		−	1.63509i	0.575864	−	0.817545i	$-0.304666\pi$
$858$		0			0
$859$			3.56261i			0.121555i	0.998151	+	0.0607774i	$0.0193580\pi$
$859$			3.56261i			0.121555i	−0.998151	+	0.0607774i	$0.980642\pi$
$860$		0			0
$861$	−35.6738	+	33.3728i	−1.21576	+	1.13734i
$862$		0			0
$863$	−22.3349			−0.760288			−0.380144	−	0.924927i	$-0.624126\pi$
$863$	−22.3349			−0.760288			−0.380144	+	0.924927i	$0.624126\pi$
$864$		0			0
$865$	11.0976	−	16.0978i	0.377330	−	0.547343i
$866$		0			0
$867$	−13.3116	+	10.1221i	−0.452084	+	0.343765i
$868$		0			0
$869$		−	36.4936i		−	1.23796i
$870$		0			0
$871$		−	2.17082i		−	0.0735555i
$872$		0			0
$873$	2.34703	−	8.46239i	0.0794350	−	0.286408i
$874$		0			0
$875$	−4.06742	+	29.2994i	−0.137504	+	0.990501i
$876$		0			0
$877$		−	18.6566i		−	0.629989i	−0.949094	−	0.314994i	$-0.897997\pi$
$877$		−	18.6566i		−	0.629989i	0.949094	−	0.314994i	$-0.102003\pi$
$878$		0			0
$879$	16.2651	+	21.3903i	0.548610	+	0.721475i
$880$		0			0
$881$	26.8648			0.905097			0.452549	−	0.891740i	$-0.350515\pi$
$881$	26.8648			0.905097			0.452549	+	0.891740i	$0.350515\pi$
$882$		0			0
$883$			12.0169i			0.404400i	0.979344	+	0.202200i	$0.0648092\pi$
$883$			12.0169i			0.404400i	−0.979344	+	0.202200i	$0.935191\pi$
$884$		0			0
$885$	35.2626	−	1.64293i	1.18534	−	0.0552265i
$886$		0			0
$887$		−	13.9283i		−	0.467666i	−0.972277	−	0.233833i	$-0.924873\pi$
$887$		−	13.9283i		−	0.467666i	0.972277	−	0.233833i	$-0.0751268\pi$
$888$		0			0
$889$	4.15236	+	40.5836i	0.139266	+	1.36113i
$890$		0			0
$891$	−13.9739	+	23.2541i	−0.468143	+	0.779042i
$892$		0			0
$893$	99.3629			3.32505
$894$		0			0
$895$	21.6613	−	31.4213i	0.724059	−	1.05030i
$896$		0			0
$897$	23.1112	−	17.5738i	0.771661	−	0.586771i
$898$		0			0
$899$	−2.03477			−0.0678632
$900$		0			0
$901$			19.3514i			0.644687i
$902$		0			0
$903$	28.7554	+	30.7381i	0.956921	+	1.02290i
$904$		0			0
$905$	8.45087	−	12.2586i	0.280916	−	0.407488i
$906$		0			0
$907$		−	23.7097i		−	0.787269i	−0.919267	−	0.393634i	$-0.871218\pi$
$907$		−	23.7097i		−	0.787269i	0.919267	−	0.393634i	$-0.128782\pi$
$908$		0			0
$909$	−0.333603	+	1.20283i	−0.0110649	+	0.0398953i
$910$		0			0
$911$			0.247204i			0.00819025i	0.999992	+	0.00409512i	$0.00130352\pi$
$911$			0.247204i			0.00819025i	−0.999992	+	0.00409512i	$0.998696\pi$
$912$		0			0
$913$	24.6201			0.814807
$914$		0			0
$915$	−2.44432	−	52.4631i	−0.0808068	−	1.73438i
$916$		0			0
$917$	−18.6343	+	1.90659i	−0.615359	+	0.0629612i
$918$		0			0
$919$	−14.4956			−0.478165			−0.239083	−	0.970999i	$-0.576847\pi$
$919$	−14.4956			−0.478165			−0.239083	+	0.970999i	$0.576847\pi$
$920$		0			0
$921$	−6.87519	+	5.22790i	−0.226545	+	0.172265i
$922$		0			0
$923$			22.6467i			0.745424i
$924$		0			0
$925$	21.5349	−	8.19602i	0.708065	−	0.269483i
$926$		0			0
$927$	−8.38281	+	30.2248i	−0.275328	+	0.992713i
$928$		0			0
$929$	−3.97297			−0.130349			−0.0651745	−	0.997874i	$-0.520760\pi$
$929$	−3.97297			−0.130349			−0.0651745	+	0.997874i	$0.520760\pi$
$930$		0			0
$931$	−11.6738	−	56.4507i	−0.382595	−	1.85010i
$932$		0			0
$933$	−32.1168	+	24.4216i	−1.05146	+	0.799527i
$934$		0			0
$935$	−15.0402	−	10.3685i	−0.491866	−	0.339085i
$936$		0			0
$937$	32.7664			1.07043			0.535216	−	0.844715i	$-0.320230\pi$
$937$	32.7664			1.07043			0.535216	+	0.844715i	$0.320230\pi$
$938$		0			0
$939$	−9.88349	+	7.51540i	−0.322535	+	0.245256i
$940$		0			0
$941$	−34.4787			−1.12397			−0.561987	−	0.827146i	$-0.689963\pi$
$941$	−34.4787			−1.12397			−0.561987	+	0.827146i	$0.689963\pi$
$942$		0			0
$943$	−40.7003			−1.32538
$944$		0			0
$945$	−3.74199	+	30.5123i	−0.121727	+	0.992564i
$946$		0			0
$947$	−43.1115			−1.40093			−0.700467	−	0.713684i	$-0.747025\pi$
$947$	−43.1115			−1.40093			−0.700467	+	0.713684i	$0.747025\pi$
$948$		0			0
$949$	20.9000			0.678443
$950$		0			0
$951$	4.01214	−	3.05083i	0.130103	−	0.0989300i
$952$		0			0
$953$	53.6918			1.73925			0.869624	−	0.493714i	$-0.164361\pi$
$953$	53.6918			1.73925			0.869624	+	0.493714i	$0.164361\pi$
$954$		0			0
$955$	−16.6584	+	24.1641i	−0.539052	+	0.781931i
$956$		0			0
$957$	−4.87659	+	3.70816i	−0.157638	+	0.119868i
$958$		0			0
$959$	−35.9457	+	3.67782i	−1.16075	+	0.118763i
$960$		0			0
$961$	27.9928			0.902994
$962$		0			0
$963$	4.75708	−	17.1520i	0.153295	−	0.552715i
$964$		0			0
$965$	13.6916	−	19.8606i	0.440749	−	0.639337i
$966$		0			0
$967$			1.49312i			0.0480155i	0.999712	+	0.0240077i	$0.00764263\pi$
$967$			1.49312i			0.0480155i	−0.999712	+	0.0240077i	$0.992357\pi$
$968$		0			0
$969$	30.7709	−	23.3982i	0.988503	−	0.751658i
$970$		0			0
$971$	3.95684			0.126981			0.0634905	−	0.997982i	$-0.479777\pi$
$971$	3.95684			0.126981			0.0634905	+	0.997982i	$0.479777\pi$
$972$		0			0
$973$	2.14810	+	20.9947i	0.0688648	+	0.673059i
$974$		0			0
$975$	−32.2747	−	20.1003i	−1.03362	−	0.643726i
$976$		0			0
$977$	−47.6921			−1.52580			−0.762902	−	0.646514i	$-0.776226\pi$
$977$	−47.6921			−1.52580			−0.762902	+	0.646514i	$0.776226\pi$
$978$		0			0
$979$		−	6.83846i		−	0.218558i
$980$		0			0
$981$	−9.42610	+	33.9865i	−0.300952	+	1.08510i
$982$		0			0
$983$			13.0582i			0.416491i	0.978077	+	0.208245i	$0.0667752\pi$
$983$			13.0582i			0.416491i	−0.978077	+	0.208245i	$0.933225\pi$
$984$		0			0
$985$	27.9427	+	19.2633i	0.890328	+	0.613779i
$986$		0			0
$987$	−37.7741	−	40.3785i	−1.20236	−	1.28526i
$988$		0			0
$989$			35.0691i			1.11513i
$990$		0			0
$991$	12.6624			0.402236			0.201118	−	0.979567i	$-0.435543\pi$
$991$	12.6624			0.402236			0.201118	+	0.979567i	$0.435543\pi$
$992$		0			0
$993$	−40.5792	+	30.8564i	−1.28774	+	0.979198i
$994$		0			0
$995$	−25.1451	+	36.4747i	−0.797153	+	1.15633i
$996$		0			0
$997$	22.6851			0.718444			0.359222	−	0.933252i	$-0.383042\pi$
$997$	22.6851			0.718444			0.359222	+	0.933252i	$0.383042\pi$
$998$		0			0
$999$	22.2414	−	8.87256i	0.703687	−	0.280715i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.k.b.209.6	yes	24
3.2	odd	2		840.2.k.a.209.5	✓	24
4.3	odd	2		1680.2.k.h.209.19		24
5.4	even	2		840.2.k.a.209.19	yes	24
7.6	odd	2	inner	840.2.k.b.209.19	yes	24
12.11	even	2		1680.2.k.i.209.20		24
15.14	odd	2	inner	840.2.k.b.209.20	yes	24
20.19	odd	2		1680.2.k.i.209.6		24
21.20	even	2		840.2.k.a.209.20	yes	24
28.27	even	2		1680.2.k.h.209.6		24
35.34	odd	2		840.2.k.a.209.6	yes	24
60.59	even	2		1680.2.k.h.209.5		24
84.83	odd	2		1680.2.k.i.209.5		24
105.104	even	2	inner	840.2.k.b.209.5	yes	24
140.139	even	2		1680.2.k.i.209.19		24
420.419	odd	2		1680.2.k.h.209.20		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.k.a.209.5	✓	24	3.2	odd	2
840.2.k.a.209.6	yes	24	35.34	odd	2
840.2.k.a.209.19	yes	24	5.4	even	2
840.2.k.a.209.20	yes	24	21.20	even	2
840.2.k.b.209.5	yes	24	105.104	even	2	inner
840.2.k.b.209.6	yes	24	1.1	even	1	trivial
840.2.k.b.209.19	yes	24	7.6	odd	2	inner
840.2.k.b.209.20	yes	24	15.14	odd	2	inner
1680.2.k.h.209.5		24	60.59	even	2
1680.2.k.h.209.6		24	28.27	even	2
1680.2.k.h.209.19		24	4.3	odd	2
1680.2.k.h.209.20		24	420.419	odd	2
1680.2.k.i.209.5		24	84.83	odd	2
1680.2.k.i.209.6		24	20.19	odd	2
1680.2.k.i.209.19		24	140.139	even	2
1680.2.k.i.209.20		24	12.11	even	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:	\( N \)	\(=\)	\( 840 = 2^{3} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	840.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.37873 + 1.04838i) q^{3} +(1.84099 + 1.26915i) q^{5} +(2.63201 - 0.269297i) q^{7} +(0.801780 - 2.89087i) q^{9} +O(q^{10})\)	\(q+(-1.37873 + 1.04838i) q^{3} +(1.84099 + 1.26915i) q^{5} +(2.63201 - 0.269297i) q^{7} +(0.801780 - 2.89087i) q^{9} -3.01442i q^{11} +4.39041 q^{13} +(-3.86879 + 0.180252i) q^{15} -2.71017i q^{17} -8.23502i q^{19} +(-3.34650 + 3.13065i) q^{21} -3.81803 q^{23} +(1.77850 + 4.67300i) q^{25} +(1.92531 + 4.82630i) q^{27} +1.17337i q^{29} +1.73412i q^{31} +(3.16027 + 4.15606i) q^{33} +(5.18729 + 2.84465i) q^{35} -4.60838i q^{37} +(-6.05318 + 4.60284i) q^{39} +10.6600 q^{41} -9.18515i q^{43} +(5.14503 - 4.30449i) q^{45} +12.0659i q^{47} +(6.85496 - 1.41759i) q^{49} +(2.84130 + 3.73659i) q^{51} -7.14027 q^{53} +(3.82575 - 5.54952i) q^{55} +(8.63346 + 11.3538i) q^{57} -9.11464 q^{59} +13.5606i q^{61} +(1.33179 - 7.82473i) q^{63} +(8.08271 + 5.57210i) q^{65} -0.494447i q^{67} +(5.26402 - 4.00276i) q^{69} +5.15821i q^{71} +4.76038 q^{73} +(-7.35117 - 4.57824i) q^{75} +(-0.811774 - 7.93398i) q^{77} +12.1064 q^{79} +(-7.71430 - 4.63569i) q^{81} +8.16745i q^{83} +(3.43962 - 4.98941i) q^{85} +(-1.23014 - 1.61776i) q^{87} +2.26858 q^{89} +(11.5556 - 1.18232i) q^{91} +(-1.81803 - 2.39088i) q^{93} +(10.4515 - 15.1606i) q^{95} +2.92728 q^{97} +(-8.71430 - 2.41690i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{9}+O(q^{10}) \) 24 * q - 2 * q^9	\( 24 q - 2 q^{9} + 2 q^{15} - 2 q^{21} - 16 q^{23} + 8 q^{25} + 8 q^{35} - 2 q^{39} + 6 q^{51} + 24 q^{53} + 8 q^{57} + 16 q^{63} + 16 q^{65} + 8 q^{77} + 4 q^{79} + 18 q^{81} - 12 q^{85} + 12 q^{91} + 32 q^{93} - 24 q^{95} - 6 q^{99}+O(q^{100}) \) 24 * q - 2 * q^9 + 2 * q^15 - 2 * q^21 - 16 * q^23 + 8 * q^25 + 8 * q^35 - 2 * q^39 + 6 * q^51 + 24 * q^53 + 8 * q^57 + 16 * q^63 + 16 * q^65 + 8 * q^77 + 4 * q^79 + 18 * q^81 - 12 * q^85 + 12 * q^91 + 32 * q^93 - 24 * q^95 - 6 * q^99

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