LMFDB - Embedded newform 840.2.k.a.209.17

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [840,2,Mod(209,840)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage: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")

magma://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:traces = [24,0,0,0,0,0,0,0,-2,0,0,0,0,0,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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			209.17
Character	$\chi$	$=$	840.209
Dual form			840.2.k.a.209.18

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(0.773053 - 1.54996i) q^{3} +(-0.194052 - 2.22763i) q^{5} +(0.942148 - 2.47232i) q^{7} +(-1.80478 - 2.39641i) q^{9} -1.45449i q^{11} +4.42393 q^{13} +(-3.60276 - 1.42131i) q^{15} +0.440293i q^{17} +6.54815i q^{19} +(-3.10367 - 3.37153i) q^{21} -2.43747 q^{23} +(-4.92469 + 0.864552i) q^{25} +(-5.10954 + 0.944786i) q^{27} -4.30871i q^{29} -2.86295i q^{31} +(-2.25441 - 1.12440i) q^{33} +(-5.69024 - 1.61900i) q^{35} +9.10153i q^{37} +(3.41994 - 6.85694i) q^{39} +4.55874 q^{41} -8.57261i q^{43} +(-4.98810 + 4.48541i) q^{45} +6.31118i q^{47} +(-5.22471 - 4.65858i) q^{49} +(0.682438 + 0.340370i) q^{51} +9.37352 q^{53} +(-3.24008 + 0.282247i) q^{55} +(10.1494 + 5.06207i) q^{57} -8.33478 q^{59} +7.14436i q^{61} +(-7.62506 + 2.20421i) q^{63} +(-0.858472 - 9.85489i) q^{65} -11.3488i q^{67} +(-1.88430 + 3.77800i) q^{69} +1.12511i q^{71} +11.2423 q^{73} +(-2.46702 + 8.30143i) q^{75} +(-3.59597 - 1.37035i) q^{77} -6.83987 q^{79} +(-2.48556 + 8.64997i) q^{81} -5.75536i q^{83} +(0.980811 - 0.0854397i) q^{85} +(-6.67834 - 3.33086i) q^{87} +11.7153 q^{89} +(4.16800 - 10.9374i) q^{91} +(-4.43747 - 2.21322i) q^{93} +(14.5869 - 1.27068i) q^{95} +6.27856 q^{97} +(-3.48556 + 2.62504i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{9} - 6 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}+ \cdots - 6 q^{99}+O(q^{100}) $ 24 * q - 2 * q^9 - 6 * 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

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
$3$	0.773053	−	1.54996i	0.446323	−	0.894872i
$3$	0.773053	−	1.54996i	0.446323	−	0.894872i
$4$		0			0
$5$	−0.194052	−	2.22763i	−0.0867826	−	0.996227i
$5$	−0.194052	−	2.22763i	−0.0867826	−	0.996227i
$6$		0			0
$7$	0.942148	−	2.47232i	0.356099	−	0.934448i
$7$	0.942148	−	2.47232i	0.356099	−	0.934448i
$8$		0			0
$9$	−1.80478	−	2.39641i	−0.601592	−	0.798803i
$10$		0			0
$11$		−	1.45449i		−	0.438546i	−0.975664	−	0.219273i	$-0.929631\pi$
$11$		−	1.45449i		−	0.438546i	0.975664	−	0.219273i	$-0.0703686\pi$
$12$		0			0
$13$	4.42393			1.22698			0.613489	−	0.789703i	$-0.289766\pi$
$13$	4.42393			1.22698			0.613489	+	0.789703i	$0.289766\pi$
$14$		0			0
$15$	−3.60276	−	1.42131i	−0.930229	−	0.366979i
$16$		0			0
$17$			0.440293i			0.106787i	0.998574	+	0.0533934i	$0.0170037\pi$
$17$			0.440293i			0.106787i	−0.998574	+	0.0533934i	$0.982996\pi$
$18$		0			0
$19$			6.54815i			1.50225i	0.660160	+	0.751125i	$0.270488\pi$
$19$			6.54815i			1.50225i	−0.660160	+	0.751125i	$0.729512\pi$
$20$		0			0
$21$	−3.10367	−	3.37153i	−0.677277	−	0.735728i
$22$		0			0
$23$	−2.43747			−0.508248			−0.254124	−	0.967172i	$-0.581787\pi$
$23$	−2.43747			−0.508248			−0.254124	+	0.967172i	$0.581787\pi$
$24$		0			0
$25$	−4.92469	+	0.864552i	−0.984938	+	0.172910i
$26$		0			0
$27$	−5.10954	+	0.944786i	−0.983331	+	0.181824i
$28$		0			0
$29$		−	4.30871i		−	0.800107i	−0.916492	−	0.400054i	$-0.868991\pi$
$29$		−	4.30871i		−	0.800107i	0.916492	−	0.400054i	$-0.131009\pi$
$30$		0			0
$31$		−	2.86295i		−	0.514201i	−0.966385	−	0.257101i	$-0.917233\pi$
$31$		−	2.86295i		−	0.514201i	0.966385	−	0.257101i	$-0.0827672\pi$
$32$		0			0
$33$	−2.25441	−	1.12440i	−0.392443	−	0.195733i
$34$		0			0
$35$	−5.69024	−	1.61900i	−0.961826	−	0.273661i
$36$		0			0
$37$			9.10153i			1.49628i	0.663540	+	0.748141i	$0.269053\pi$
$37$			9.10153i			1.49628i	−0.663540	+	0.748141i	$0.730947\pi$
$38$		0			0
$39$	3.41994	−	6.85694i	0.547628	−	1.09799i
$40$		0			0
$41$	4.55874			0.711955			0.355978	−	0.934494i	$-0.384148\pi$
$41$	4.55874			0.711955			0.355978	+	0.934494i	$0.384148\pi$
$42$		0			0
$43$		−	8.57261i		−	1.30731i	−0.756793	−	0.653655i	$-0.773235\pi$
$43$		−	8.57261i		−	1.30731i	0.756793	−	0.653655i	$-0.226765\pi$
$44$		0			0
$45$	−4.98810	+	4.48541i	−0.743582	+	0.668645i
$46$		0			0
$47$			6.31118i			0.920580i	0.887769	+	0.460290i	$0.152255\pi$
$47$			6.31118i			0.920580i	−0.887769	+	0.460290i	$0.847745\pi$
$48$		0			0
$49$	−5.22471	−	4.65858i	−0.746388	−	0.665512i
$50$		0			0
$51$	0.682438	+	0.340370i	0.0955605	+	0.0476613i
$52$		0			0
$53$	9.37352			1.28755			0.643776	−	0.765214i	$-0.277367\pi$
$53$	9.37352			1.28755			0.643776	+	0.765214i	$0.277367\pi$
$54$		0			0
$55$	−3.24008	+	0.282247i	−0.436892	+	0.0380582i
$56$		0			0
$57$	10.1494	+	5.06207i	1.34432	+	0.670488i
$58$		0			0
$59$	−8.33478			−1.08510			−0.542548	−	0.840025i	$-0.682540\pi$
$59$	−8.33478			−1.08510			−0.542548	+	0.840025i	$0.682540\pi$
$60$		0			0
$61$			7.14436i			0.914741i	0.889276	+	0.457371i	$0.151209\pi$
$61$			7.14436i			0.914741i	−0.889276	+	0.457371i	$0.848791\pi$
$62$		0			0
$63$	−7.62506	+	2.20421i	−0.960667	+	0.277704i
$64$		0			0
$65$	−0.858472	−	9.85489i	−0.106480	−	1.22235i
$66$		0			0
$67$		−	11.3488i		−	1.38648i	−0.720706	−	0.693241i	$-0.756182\pi$
$67$		−	11.3488i		−	1.38648i	0.720706	−	0.693241i	$-0.243818\pi$
$68$		0			0
$69$	−1.88430	+	3.77800i	−0.226843	+	0.454817i
$70$		0			0
$71$			1.12511i			0.133526i	0.997769	+	0.0667631i	$0.0212672\pi$
$71$			1.12511i			0.133526i	−0.997769	+	0.0667631i	$0.978733\pi$
$72$		0			0
$73$	11.2423			1.31581			0.657904	−	0.753102i	$-0.271443\pi$
$73$	11.2423			1.31581			0.657904	+	0.753102i	$0.271443\pi$
$74$		0			0
$75$	−2.46702	+	8.30143i	−0.284867	+	0.958567i
$76$		0			0
$77$	−3.59597	−	1.37035i	−0.409799	−	0.156166i
$78$		0			0
$79$	−6.83987			−0.769546			−0.384773	−	0.923011i	$-0.625720\pi$
$79$	−6.83987			−0.769546			−0.384773	+	0.923011i	$0.625720\pi$
$80$		0			0
$81$	−2.48556	+	8.64997i	−0.276174	+	0.961108i
$82$		0			0
$83$		−	5.75536i		−	0.631733i	−0.948804	−	0.315867i	$-0.897705\pi$
$83$		−	5.75536i		−	0.631733i	0.948804	−	0.315867i	$-0.102295\pi$
$84$		0			0
$85$	0.980811	−	0.0854397i	0.106384	−	0.00926723i
$86$		0			0
$87$	−6.67834	−	3.33086i	−0.715994	−	0.357106i
$88$		0			0
$89$	11.7153			1.24182			0.620908	−	0.783883i	$-0.286764\pi$
$89$	11.7153			1.24182			0.620908	+	0.783883i	$0.286764\pi$
$90$		0			0
$91$	4.16800	−	10.9374i	0.436925	−	1.14655i
$92$		0			0
$93$	−4.43747	−	2.21322i	−0.460145	−	0.229500i
$94$		0			0
$95$	14.5869	−	1.27068i	1.49658	−	0.130369i
$96$		0			0
$97$	6.27856			0.637491			0.318746	−	0.947840i	$-0.396738\pi$
$97$	6.27856			0.637491			0.318746	+	0.947840i	$0.396738\pi$
$98$		0			0
$99$	−3.48556	+	2.62504i	−0.350312	+	0.263826i
$100$		0			0
$101$	−11.5520			−1.14946			−0.574732	−	0.818342i	$-0.694894\pi$
$101$	−11.5520			−1.14946			−0.574732	+	0.818342i	$0.694894\pi$
$102$		0			0
$103$	17.9525			1.76892			0.884458	−	0.466619i	$-0.154528\pi$
$103$	17.9525			1.76892			0.884458	+	0.466619i	$0.154528\pi$
$104$		0			0
$105$	−6.90825	+	7.56809i	−0.674177	+	0.738570i
$106$		0			0
$107$	−18.5926			−1.79742			−0.898709	−	0.438546i	$-0.855494\pi$
$107$	−18.5926			−1.79742			−0.898709	+	0.438546i	$0.855494\pi$
$108$		0			0
$109$	4.20960			0.403207			0.201603	−	0.979467i	$-0.435385\pi$
$109$	4.20960			0.403207			0.201603	+	0.979467i	$0.435385\pi$
$110$		0			0
$111$	14.1070	+	7.03597i	1.33898	+	0.667825i
$112$		0			0
$113$	−10.3252			−0.971316			−0.485658	−	0.874149i	$-0.661420\pi$
$113$	−10.3252			−0.971316			−0.485658	+	0.874149i	$0.661420\pi$
$114$		0			0
$115$	0.472996	+	5.42979i	0.0441071	+	0.506331i
$116$		0			0
$117$	−7.98421	−	10.6016i	−0.738140	−	0.980114i
$118$		0			0
$119$	1.08854	+	0.414821i	0.0997867	+	0.0380266i
$120$		0			0
$121$	8.88445			0.807677
$122$		0			0
$123$	3.52415	−	7.06588i	0.317762	−	0.637109i
$124$		0			0
$125$	2.88155	+	10.8026i	0.257734	+	0.966216i
$126$		0			0
$127$		−	0.00291325i		−	0.000258509i	−1.00000		0.000129255i	$-0.999959\pi$
$127$		−	0.00291325i		−	0.000258509i	1.00000		0.000129255i	$-4.11430e-5\pi$
$128$		0			0
$129$	−13.2872	−	6.62708i	−1.16988	−	0.583482i
$130$		0			0
$131$	4.00085			0.349556			0.174778	−	0.984608i	$-0.444079\pi$
$131$	4.00085			0.349556			0.174778	+	0.984608i	$0.444079\pi$
$132$		0			0
$133$	16.1891	+	6.16933i	1.40377	+	0.534949i
$134$		0			0
$135$	3.09615	+	11.1988i	0.266474	+	0.963842i
$136$		0			0
$137$	17.6484			1.50781			0.753903	−	0.656986i	$-0.228169\pi$
$137$	17.6484			1.50781			0.753903	+	0.656986i	$0.228169\pi$
$138$		0			0
$139$		−	4.22249i		−	0.358147i	−0.983836	−	0.179073i	$-0.942690\pi$
$139$		−	4.22249i		−	0.358147i	0.983836	−	0.179073i	$-0.0573099\pi$
$140$		0			0
$141$	9.78210	+	4.87888i	0.823801	+	0.410876i
$142$		0			0
$143$		−	6.43458i		−	0.538087i
$144$		0			0
$145$	−9.59822	+	0.836113i	−0.797089	+	0.0694354i
$146$		0			0
$147$	−11.2596	+	4.49678i	−0.928677	+	0.370889i
$148$		0			0
$149$			17.7877i			1.45723i	0.684926	+	0.728613i	$0.259835\pi$
$149$			17.7877i			1.45723i	−0.684926	+	0.728613i	$0.740165\pi$
$150$		0			0
$151$	8.17122			0.664965			0.332482	−	0.943109i	$-0.392114\pi$
$151$	8.17122			0.664965			0.332482	+	0.943109i	$0.392114\pi$
$152$		0			0
$153$	1.05512	−	0.794630i	0.0853016	−	0.0642421i
$154$		0			0
$155$	−6.37760	+	0.555561i	−0.512261	+	0.0446237i
$156$		0			0
$157$	0.0848924			0.00677515			0.00338758	−	0.999994i	$-0.498922\pi$
$157$	0.0848924			0.00677515			0.00338758	+	0.999994i	$0.498922\pi$
$158$		0			0
$159$	7.24624	−	14.5286i	0.574664	−	1.15219i
$160$		0			0
$161$	−2.29646	+	6.02621i	−0.180986	+	0.474932i
$162$		0			0
$163$		−	12.9990i		−	1.01816i	−0.860718	−	0.509082i	$-0.829985\pi$
$163$		−	12.9990i		−	1.01816i	0.860718	−	0.509082i	$-0.170015\pi$
$164$		0			0
$165$	−2.06728	+	5.24019i	−0.160937	+	0.407948i
$166$		0			0
$167$		−	0.585272i		−	0.0452897i	−0.999744	−	0.0226449i	$-0.992791\pi$
$167$		−	0.585272i		−	0.0452897i	0.999744	−	0.0226449i	$-0.00720870\pi$
$168$		0			0
$169$	6.57117			0.505475
$170$		0			0
$171$	15.6921	−	11.8180i	1.20000	−	0.903741i
$172$		0			0
$173$		−	17.4963i		−	1.33022i	−0.746745	−	0.665111i	$-0.768384\pi$
$173$		−	17.4963i		−	1.33022i	0.746745	−	0.665111i	$-0.231616\pi$
$174$		0			0
$175$	−2.50234	+	12.9899i	−0.189159	+	0.981946i
$176$		0			0
$177$	−6.44323	+	12.9186i	−0.484303	+	0.971022i
$178$		0			0
$179$			20.2719i			1.51519i	0.652724	+	0.757596i	$0.273626\pi$
$179$			20.2719i			1.51519i	−0.652724	+	0.757596i	$0.726374\pi$
$180$		0			0
$181$			12.9003i			0.958871i	0.877577	+	0.479435i	$0.159159\pi$
$181$			12.9003i			0.958871i	−0.877577	+	0.479435i	$0.840841\pi$
$182$		0			0
$183$	11.0735	+	5.52297i	0.818577	+	0.408270i
$184$		0			0
$185$	20.2749	−	1.76617i	1.49064	−	0.129851i
$186$		0			0
$187$	0.640403			0.0468309
$188$		0			0
$189$	−2.47813	+	13.5225i	−0.180258	+	0.983619i
$190$		0			0
$191$		−	1.96182i		−	0.141952i	−0.997478	−	0.0709760i	$-0.977389\pi$
$191$		−	1.96182i		−	0.141952i	0.997478	−	0.0709760i	$-0.0226114\pi$
$192$		0			0
$193$		−	17.9119i		−	1.28933i	−0.764466	−	0.644664i	$-0.776997\pi$
$193$		−	17.9119i		−	1.28933i	0.764466	−	0.644664i	$-0.223003\pi$
$194$		0			0
$195$	−15.9384	−	6.28776i	−1.14137	−	0.450276i
$196$		0			0
$197$	12.2566			0.873249			0.436624	−	0.899644i	$-0.356174\pi$
$197$	12.2566			0.873249			0.436624	+	0.899644i	$0.356174\pi$
$198$		0			0
$199$		−	1.32283i		−	0.0937726i	−0.998900	−	0.0468863i	$-0.985070\pi$
$199$		−	1.32283i		−	0.0937726i	0.998900	−	0.0468863i	$-0.0149298\pi$
$200$		0			0
$201$	−17.5903	−	8.77326i	−1.24072	−	0.618818i
$202$		0			0
$203$	−10.6525	−	4.05944i	−0.747659	−	0.284917i
$204$		0			0
$205$	−0.884632	−	10.1552i	−0.0617854	−	0.709269i
$206$		0			0
$207$	4.39909	+	5.84118i	0.305758	+	0.405990i
$208$		0			0
$209$	9.52424			0.658806
$210$		0			0
$211$	−15.1148			−1.04054			−0.520272	−	0.854001i	$-0.674169\pi$
$211$	−15.1148			−1.04054			−0.520272	+	0.854001i	$0.674169\pi$
$212$		0			0
$213$	1.74388	+	0.869771i	0.119489	+	0.0595958i
$214$		0			0
$215$	−19.0966	+	1.66353i	−1.30238	+	0.113452i
$216$		0			0
$217$	−7.07813	−	2.69733i	−0.480495	−	0.183106i
$218$		0			0
$219$	8.69088	−	17.4251i	0.587275	−	1.17748i
$220$		0			0
$221$			1.94783i			0.131025i
$222$		0			0
$223$	22.8929			1.53302			0.766511	−	0.642231i	$-0.221991\pi$
$223$	22.8929			1.53302			0.766511	+	0.642231i	$0.221991\pi$
$224$		0			0
$225$	10.9598	+	10.2412i	0.730652	+	0.682750i
$226$		0			0
$227$			12.5749i			0.834629i	0.908762	+	0.417314i	$0.137029\pi$
$227$			12.5749i			0.834629i	−0.908762	+	0.417314i	$0.862971\pi$
$228$		0			0
$229$		−	0.500577i		−	0.0330791i	−0.999863	−	0.0165395i	$-0.994735\pi$
$229$		−	0.500577i		−	0.0330791i	0.999863	−	0.0165395i	$-0.00526494\pi$
$230$		0			0
$231$	−4.90387	+	4.51427i	−0.322651	+	0.297017i
$232$		0			0
$233$	−12.9171			−0.846228			−0.423114	−	0.906076i	$-0.639063\pi$
$233$	−12.9171			−0.846228			−0.423114	+	0.906076i	$0.639063\pi$
$234$		0			0
$235$	14.0590	−	1.22470i	0.917107	−	0.0798903i
$236$		0			0
$237$	−5.28759	+	10.6016i	−0.343466	+	0.688645i
$238$		0			0
$239$			27.1112i			1.75368i	0.480783	+	0.876840i	$0.340353\pi$
$239$			27.1112i			1.75368i	−0.480783	+	0.876840i	$0.659647\pi$
$240$		0			0
$241$		−	4.75478i		−	0.306282i	−0.988204	−	0.153141i	$-0.951061\pi$
$241$		−	4.75478i		−	0.306282i	0.988204	−	0.153141i	$-0.0489389\pi$
$242$		0			0
$243$	11.4857	+	10.5394i	0.736806	+	0.676104i
$244$		0			0
$245$	−9.36374	+	12.5427i	−0.598227	+	0.801326i
$246$		0			0
$247$			28.9686i			1.84323i
$248$		0			0
$249$	−8.92061	−	4.44920i	−0.565321	−	0.281957i
$250$		0			0
$251$	10.8100			0.682318			0.341159	−	0.940006i	$-0.389181\pi$
$251$	10.8100			0.682318			0.341159	+	0.940006i	$0.389181\pi$
$252$		0			0
$253$			3.54529i			0.222890i
$254$		0			0
$255$	0.625791	−	1.58627i	0.0391885	−	0.0993361i
$256$		0			0
$257$			22.4592i			1.40097i	0.713669	+	0.700483i	$0.247032\pi$
$257$			22.4592i			1.40097i	−0.713669	+	0.700483i	$0.752968\pi$
$258$		0			0
$259$	22.5019	+	8.57499i	1.39820	+	0.532824i
$260$		0			0
$261$	−10.3254	+	7.77626i	−0.639128	+	0.481338i
$262$		0			0
$263$	17.7846			1.09664			0.548322	−	0.836267i	$-0.315267\pi$
$263$	17.7846			1.09664			0.548322	+	0.836267i	$0.315267\pi$
$264$		0			0
$265$	−1.81895	−	20.8808i	−0.111737	−	1.28269i
$266$		0			0
$267$	9.05653	−	18.1582i	0.554251	−	1.11127i
$268$		0			0
$269$	16.0968			0.981438			0.490719	−	0.871318i	$-0.336734\pi$
$269$	16.0968			0.981438			0.490719	+	0.871318i	$0.336734\pi$
$270$		0			0
$271$			9.44781i			0.573914i	0.957943	+	0.286957i	$0.0926437\pi$
$271$			9.44781i			0.573914i	−0.957943	+	0.286957i	$0.907356\pi$
$272$		0			0
$273$	−13.7304	−	14.9154i	−0.831004	−	0.902722i
$274$		0			0
$275$	1.25749	+	7.16293i	0.0758292	+	0.431941i
$276$		0			0
$277$		−	4.67510i		−	0.280899i	−0.990088	−	0.140450i	$-0.955145\pi$
$277$		−	4.67510i		−	0.280899i	0.990088	−	0.140450i	$-0.0448548\pi$
$278$		0			0
$279$	−6.86081	+	5.16699i	−0.410746	+	0.309340i
$280$		0			0
$281$		−	11.8520i		−	0.707032i	−0.935429	−	0.353516i	$-0.884986\pi$
$281$		−	11.8520i		−	0.707032i	0.935429	−	0.353516i	$-0.115014\pi$
$282$		0			0
$283$	5.84917			0.347697			0.173849	−	0.984772i	$-0.444380\pi$
$283$	5.84917			0.347697			0.173849	+	0.984772i	$0.444380\pi$
$284$		0			0
$285$	9.30692	−	23.5914i	0.551295	−	1.39744i
$286$		0			0
$287$	4.29501	−	11.2707i	0.253526	−	0.665285i
$288$		0			0
$289$	16.8061			0.988597
$290$		0			0
$291$	4.85366	−	9.73154i	0.284527	−	0.570473i
$292$		0			0
$293$			0.346350i			0.0202340i	0.999949	+	0.0101170i	$0.00322040\pi$
$293$			0.346350i			0.0202340i	−0.999949	+	0.0101170i	$0.996780\pi$
$294$		0			0
$295$	1.61738	+	18.5668i	0.0941675	+	1.08100i
$296$		0			0
$297$	1.37418	+	7.43179i	0.0797383	+	0.431236i
$298$		0			0
$299$	−10.7832			−0.623609
$300$		0			0
$301$	−21.1942	−	8.07667i	−1.22161	−	0.465531i
$302$		0			0
$303$	−8.93029	+	17.9051i	−0.513032	+	1.02862i
$304$		0			0
$305$	15.9150	−	1.38638i	0.911290	−	0.0793837i
$306$		0			0
$307$	−24.7866			−1.41465			−0.707323	−	0.706890i	$-0.750098\pi$
$307$	−24.7866			−1.41465			−0.707323	+	0.706890i	$0.750098\pi$
$308$		0			0
$309$	13.8783	−	27.8258i	0.789508	−	1.58295i
$310$		0			0
$311$	−13.0969			−0.742659			−0.371330	−	0.928501i	$-0.621098\pi$
$311$	−13.0969			−0.742659			−0.371330	+	0.928501i	$0.621098\pi$
$312$		0			0
$313$	1.69327			0.0957092			0.0478546	−	0.998854i	$-0.484762\pi$
$313$	1.69327			0.0957092			0.0478546	+	0.998854i	$0.484762\pi$
$314$		0			0
$315$	6.38982	+	16.5581i	0.360026	+	0.932942i
$316$		0			0
$317$	−32.6867			−1.83587			−0.917934	−	0.396734i	$-0.870143\pi$
$317$	−32.6867			−1.83587			−0.917934	+	0.396734i	$0.870143\pi$
$318$		0			0
$319$	−6.26699			−0.350884
$320$		0			0
$321$	−14.3731	+	28.8179i	−0.802228	+	1.60846i
$322$		0			0
$323$	−2.88311			−0.160420
$324$		0			0
$325$	−21.7865	+	3.82472i	−1.20850	+	0.212157i
$326$		0			0
$327$	3.25425	−	6.52473i	0.179960	−	0.360819i
$328$		0			0
$329$	15.6032	+	5.94607i	0.860234	+	0.327817i
$330$		0			0
$331$	−22.5819			−1.24121			−0.620606	−	0.784123i	$-0.713113\pi$
$331$	−22.5819			−1.24121			−0.620606	+	0.784123i	$0.713113\pi$
$332$		0			0
$333$	21.8110	−	16.4262i	1.19524	−	0.900152i
$334$		0			0
$335$	−25.2810	+	2.20226i	−1.38125	+	0.120322i
$336$		0			0
$337$		−	26.8892i		−	1.46475i	−0.680901	−	0.732375i	$-0.738412\pi$
$337$		−	26.8892i		−	1.46475i	0.680901	−	0.732375i	$-0.261588\pi$
$338$		0			0
$339$	−7.98195	+	16.0037i	−0.433520	+	0.869203i
$340$		0			0
$341$	−4.16414			−0.225501
$342$		0			0
$343$	−16.4399	+	8.52808i	−0.887674	+	0.460473i
$344$		0			0
$345$	8.78163	+	3.46439i	0.472787	+	0.186517i
$346$		0			0
$347$	29.0533			1.55966			0.779831	−	0.625991i	$-0.215305\pi$
$347$	29.0533			1.55966			0.779831	+	0.625991i	$0.215305\pi$
$348$		0			0
$349$		−	27.1001i		−	1.45063i	−0.688416	−	0.725317i	$-0.741693\pi$
$349$		−	27.1001i		−	1.45063i	0.688416	−	0.725317i	$-0.258307\pi$
$350$		0			0
$351$	−22.6042	+	4.17967i	−1.20653	+	0.223094i
$352$		0			0
$353$		−	11.7715i		−	0.626536i	−0.949665	−	0.313268i	$-0.898576\pi$
$353$		−	11.7715i		−	0.626536i	0.949665	−	0.313268i	$-0.101424\pi$
$354$		0			0
$355$	2.50633	−	0.218330i	0.133022	−	0.0115878i
$356$		0			0
$357$	1.48446	−	1.36653i	0.0785660	−	0.0723242i
$358$		0			0
$359$		−	8.39177i		−	0.442901i	−0.975172	−	0.221450i	$-0.928921\pi$
$359$		−	8.39177i		−	0.442901i	0.975172	−	0.221450i	$-0.0710791\pi$
$360$		0			0
$361$	−23.8783			−1.25675
$362$		0			0
$363$	6.86815	−	13.7706i	0.360485	−	0.722768i
$364$		0			0
$365$	−2.18158	−	25.0436i	−0.114189	−	1.31084i
$366$		0			0
$367$	3.13112			0.163443			0.0817216	−	0.996655i	$-0.473958\pi$
$367$	3.13112			0.163443			0.0817216	+	0.996655i	$0.473958\pi$
$368$		0			0
$369$	−8.22750	−	10.9246i	−0.428307	−	0.568712i
$370$		0			0
$371$	8.83125	−	23.1743i	0.458496	−	1.20315i
$372$		0			0
$373$			9.27241i			0.480107i	0.970760	+	0.240054i	$0.0771650\pi$
$373$			9.27241i			0.480107i	−0.970760	+	0.240054i	$0.922835\pi$
$374$		0			0
$375$	18.9713	+	3.88471i	0.979672	+	0.200605i
$376$		0			0
$377$		−	19.0614i		−	0.981714i
$378$		0			0
$379$	−19.4784			−1.00054			−0.500268	−	0.865871i	$-0.666765\pi$
$379$	−19.4784			−1.00054			−0.500268	+	0.865871i	$0.666765\pi$
$380$		0			0
$381$	−0.00451543	−	0.00225210i	−0.000231333	−	0.000115378i
$382$		0			0
$383$		−	1.95349i		−	0.0998186i	−0.998754	−	0.0499093i	$-0.984107\pi$
$383$		−	1.95349i		−	0.0998186i	0.998754	−	0.0499093i	$-0.0158932\pi$
$384$		0			0
$385$	−2.35483	+	8.27642i	−0.120013	+	0.421805i
$386$		0			0
$387$	−20.5435	+	15.4716i	−1.04428	+	0.786468i
$388$		0			0
$389$			1.48098i			0.0750884i	0.999295	+	0.0375442i	$0.0119535\pi$
$389$			1.48098i			0.0750884i	−0.999295	+	0.0375442i	$0.988046\pi$
$390$		0			0
$391$		−	1.07320i		−	0.0542742i
$392$		0			0
$393$	3.09287	−	6.20117i	0.156015	−	0.312808i
$394$		0			0
$395$	1.32729	+	15.2367i	0.0667832	+	0.766642i
$396$		0			0
$397$	−12.9737			−0.651132			−0.325566	−	0.945519i	$-0.605555\pi$
$397$	−12.9737			−0.651132			−0.325566	+	0.945519i	$0.605555\pi$
$398$		0			0
$399$	22.0773	−	20.3233i	1.10525	−	1.01744i
$400$		0			0
$401$		−	31.1453i		−	1.55532i	−0.628683	−	0.777662i	$-0.716406\pi$
$401$		−	31.1453i		−	1.55532i	0.628683	−	0.777662i	$-0.283594\pi$
$402$		0			0
$403$		−	12.6655i		−	0.630914i
$404$		0			0
$405$	19.7513	+	3.85838i	0.981449	+	0.191724i
$406$		0			0
$407$	13.2381			0.656189
$408$		0			0
$409$			27.3261i			1.35119i	0.737273	+	0.675594i	$0.236113\pi$
$409$			27.3261i			1.35119i	−0.737273	+	0.675594i	$0.763887\pi$
$410$		0			0
$411$	13.6432	−	27.3544i	0.672968	−	1.34929i
$412$		0			0
$413$	−7.85260	+	20.6062i	−0.386401	+	1.01397i
$414$		0			0
$415$	−12.8208	+	1.11684i	−0.629350	+	0.0548235i
$416$		0			0
$417$	−6.54470	−	3.26421i	−0.320495	−	0.159849i
$418$		0			0
$419$	−24.6777			−1.20559			−0.602793	−	0.797897i	$-0.705946\pi$
$419$	−24.6777			−1.20559			−0.602793	+	0.797897i	$0.705946\pi$
$420$		0			0
$421$	31.5258			1.53647			0.768237	−	0.640165i	$-0.221134\pi$
$421$	31.5258			1.53647			0.768237	+	0.640165i	$0.221134\pi$
$422$		0			0
$423$	15.1242	−	11.3903i	0.735362	−	0.553814i
$424$		0			0
$425$	−0.380656	−	2.16831i	−0.0184645	−	0.105178i
$426$		0			0
$427$	17.6631	+	6.73105i	0.854779	+	0.325738i
$428$		0			0
$429$	−9.97337	−	4.97427i	−0.481519	−	0.240160i
$430$		0			0
$431$		−	24.6470i		−	1.18720i	−0.804759	−	0.593602i	$-0.797705\pi$
$431$		−	24.6470i		−	1.18720i	0.804759	−	0.593602i	$-0.202295\pi$
$432$		0			0
$433$	−15.3862			−0.739412			−0.369706	−	0.929149i	$-0.620542\pi$
$433$	−15.3862			−0.739412			−0.369706	+	0.929149i	$0.620542\pi$
$434$		0			0
$435$	−6.12399	+	15.5233i	−0.293623	+	0.744283i
$436$		0			0
$437$		−	15.9609i		−	0.763515i
$438$		0			0
$439$			23.8686i			1.13919i	0.821926	+	0.569594i	$0.192899\pi$
$439$			23.8686i			1.13919i	−0.821926	+	0.569594i	$0.807101\pi$
$440$		0			0
$441$	−1.73443	+	20.9283i	−0.0825920	+	0.996583i
$442$		0			0
$443$	3.84558			0.182709			0.0913546	−	0.995818i	$-0.470880\pi$
$443$	3.84558			0.182709			0.0913546	+	0.995818i	$0.470880\pi$
$444$		0			0
$445$	−2.27337	−	26.0973i	−0.107768	−	1.23713i
$446$		0			0
$447$	27.5703	+	13.7508i	1.30403	+	0.650393i
$448$		0			0
$449$			2.50182i			0.118068i	0.998256	+	0.0590340i	$0.0188020\pi$
$449$			2.50182i			0.118068i	−0.998256	+	0.0590340i	$0.981198\pi$
$450$		0			0
$451$		−	6.63065i		−	0.312225i
$452$		0			0
$453$	6.31679	−	12.6651i	0.296789	−	0.595058i
$454$		0			0
$455$	−25.1732	−	7.16235i	−1.18014	−	0.335776i
$456$		0			0
$457$			10.4851i			0.490472i	0.969463	+	0.245236i	$0.0788655\pi$
$457$			10.4851i			0.490472i	−0.969463	+	0.245236i	$0.921134\pi$
$458$		0			0
$459$	−0.415982	−	2.24969i	−0.0194164	−	0.105007i
$460$		0			0
$461$	−24.6007			−1.14577			−0.572885	−	0.819636i	$-0.694176\pi$
$461$	−24.6007			−1.14577			−0.572885	+	0.819636i	$0.694176\pi$
$462$		0			0
$463$			1.28981i			0.0599424i	0.999551	+	0.0299712i	$0.00954156\pi$
$463$			1.28981i			0.0599424i	−0.999551	+	0.0299712i	$0.990458\pi$
$464$		0			0
$465$	−4.06913	+	10.3145i	−0.188701	+	0.478325i
$466$		0			0
$467$		−	40.9656i		−	1.89566i	−0.318771	−	0.947832i	$-0.603270\pi$
$467$		−	40.9656i		−	1.89566i	0.318771	−	0.947832i	$-0.396730\pi$
$468$		0			0
$469$	−28.0579	−	10.6923i	−1.29560	−	0.493724i
$470$		0			0
$471$	0.0656264	−	0.131580i	0.00302390	−	0.00606290i
$472$		0			0
$473$	−12.4688			−0.573316
$474$		0			0
$475$	−5.66122	−	32.2476i	−0.259755	−	1.47962i
$476$		0			0
$477$	−16.9171	−	22.4628i	−0.774582	−	1.02850i
$478$		0			0
$479$	28.8388			1.31768			0.658838	−	0.752285i	$-0.271048\pi$
$479$	28.8388			1.31768			0.658838	+	0.752285i	$0.271048\pi$
$480$		0			0
$481$			40.2645i			1.83591i
$482$		0			0
$483$	7.56512	+	8.21801i	0.344225	+	0.373933i
$484$		0			0
$485$	−1.21837	−	13.9863i	−0.0553232	−	0.635086i
$486$		0			0
$487$			30.1405i			1.36580i	0.730514	+	0.682898i	$0.239280\pi$
$487$			30.1405i			1.36580i	−0.730514	+	0.682898i	$0.760720\pi$
$488$		0			0
$489$	−20.1480	−	10.0489i	−0.911126	−	0.454429i
$490$		0			0
$491$		−	15.9029i		−	0.717690i	−0.933397	−	0.358845i	$-0.883171\pi$
$491$		−	15.9029i		−	0.717690i	0.933397	−	0.358845i	$-0.116829\pi$
$492$		0			0
$493$	1.89709			0.0854408
$494$		0			0
$495$	6.52399	+	7.25516i	0.293232	+	0.326095i
$496$		0			0
$497$	2.78163	+	1.06002i	0.124773	+	0.0475485i
$498$		0			0
$499$	1.93099			0.0864429			0.0432214	−	0.999066i	$-0.486238\pi$
$499$	1.93099			0.0864429			0.0432214	+	0.999066i	$0.486238\pi$
$500$		0			0
$501$	−0.907151	−	0.452447i	−0.0405285	−	0.0202138i
$502$		0			0
$503$			10.3348i			0.460807i	0.973095	+	0.230404i	$0.0740046\pi$
$503$			10.3348i			0.460807i	−0.973095	+	0.230404i	$0.925995\pi$
$504$		0			0
$505$	2.24168	+	25.7335i	0.0997535	+	1.14513i
$506$		0			0
$507$	5.07987	−	10.1851i	0.225605	−	0.452335i
$508$		0			0
$509$	−9.90113			−0.438860			−0.219430	−	0.975628i	$-0.570420\pi$
$509$	−9.90113			−0.438860			−0.219430	+	0.975628i	$0.570420\pi$
$510$		0			0
$511$	10.5919	−	27.7945i	0.468558	−	1.22956i
$512$		0			0
$513$	−6.18660	−	33.4580i	−0.273145	−	1.47721i
$514$		0			0
$515$	−3.48372	−	39.9917i	−0.153511	−	1.76224i
$516$		0			0
$517$	9.17956			0.403717
$518$		0			0
$519$	−27.1187	−	13.5256i	−1.19038	−	0.593708i
$520$		0			0
$521$	−22.3619			−0.979694			−0.489847	−	0.871808i	$-0.662947\pi$
$521$	−22.3619			−0.979694			−0.489847	+	0.871808i	$0.662947\pi$
$522$		0			0
$523$	−39.1612			−1.71240			−0.856201	−	0.516643i	$-0.827181\pi$
$523$	−39.1612			−1.71240			−0.856201	+	0.516643i	$0.827181\pi$
$524$		0			0
$525$	18.1995	+	13.9204i	0.794291	+	0.607538i
$526$		0			0
$527$	1.26054			0.0549099
$528$		0			0
$529$	−17.0587			−0.741684
$530$		0			0
$531$	15.0424	+	19.9735i	0.652785	+	0.866778i
$532$		0			0
$533$	20.1675			0.873553
$534$		0			0
$535$	3.60793	+	41.4175i	0.155985	+	1.79064i
$536$		0			0
$537$	31.4207	+	15.6712i	1.35590	+	0.676264i
$538$		0			0
$539$	−6.77587	+	7.59931i	−0.291858	+	0.327325i
$540$		0			0
$541$	12.7660			0.548853			0.274427	−	0.961608i	$-0.411512\pi$
$541$	12.7660			0.548853			0.274427	+	0.961608i	$0.411512\pi$
$542$		0			0
$543$	19.9950	+	9.97261i	0.858067	+	0.427966i
$544$		0			0
$545$	−0.816881	−	9.37745i	−0.0349913	−	0.401686i
$546$		0			0
$547$		−	18.1815i		−	0.777383i	−0.921368	−	0.388691i	$-0.872927\pi$
$547$		−	18.1815i		−	0.777383i	0.921368	−	0.388691i	$-0.127073\pi$
$548$		0			0
$549$	17.1208	−	12.8940i	0.730699	−	0.550301i
$550$		0			0
$551$	28.2141			1.20196
$552$		0			0
$553$	−6.44417	+	16.9103i	−0.274034	+	0.719101i
$554$		0			0
$555$	12.9361	−	32.7906i	0.549105	−	1.39189i
$556$		0			0
$557$	−2.18158			−0.0924367			−0.0462183	−	0.998931i	$-0.514717\pi$
$557$	−2.18158			−0.0924367			−0.0462183	+	0.998931i	$0.514717\pi$
$558$		0			0
$559$		−	37.9246i		−	1.60404i
$560$		0			0
$561$	0.495066	−	0.992602i	0.0209017	−	0.0419077i
$562$		0			0
$563$			36.6727i			1.54557i	0.634667	+	0.772786i	$0.281137\pi$
$563$			36.6727i			1.54557i	−0.634667	+	0.772786i	$0.718863\pi$
$564$		0			0
$565$	2.00363	+	23.0008i	0.0842933	+	0.967651i
$566$		0			0
$567$	19.0437	+	14.2947i	0.799761	+	0.600319i
$568$		0			0
$569$			6.87817i			0.288348i	0.989552	+	0.144174i	$0.0460525\pi$
$569$			6.87817i			0.288348i	−0.989552	+	0.144174i	$0.953948\pi$
$570$		0			0
$571$	21.8721			0.915319			0.457659	−	0.889128i	$-0.348688\pi$
$571$	21.8721			0.915319			0.457659	+	0.889128i	$0.348688\pi$
$572$		0			0
$573$	−3.04074	−	1.51659i	−0.127029	−	0.0633564i
$574$		0			0
$575$	12.0038	−	2.10732i	0.500593	−	0.0878814i
$576$		0			0
$577$	−40.0096			−1.66562			−0.832810	−	0.553559i	$-0.813269\pi$
$577$	−40.0096			−1.66562			−0.832810	+	0.553559i	$0.813269\pi$
$578$		0			0
$579$	−27.7628	−	13.8469i	−1.15378	−	0.575456i
$580$		0			0
$581$	−14.2291	−	5.42241i	−0.590322	−	0.224959i
$582$		0			0
$583$		−	13.6337i		−	0.564651i
$584$		0			0
$585$	−22.0670	+	19.8431i	−0.912359	+	0.820412i
$586$		0			0
$587$			15.0493i			0.621151i	0.950549	+	0.310576i	$0.100522\pi$
$587$			15.0493i			0.621151i	−0.950549	+	0.310576i	$0.899478\pi$
$588$		0			0
$589$	18.7470			0.772459
$590$		0			0
$591$	9.47503	−	18.9973i	0.389751	−	0.781446i
$592$		0			0
$593$			39.0931i			1.60536i	0.596408	+	0.802681i	$0.296594\pi$
$593$			39.0931i			1.60536i	−0.596408	+	0.802681i	$0.703406\pi$
$594$		0			0
$595$	0.712835	−	2.50537i	0.0292234	−	0.102710i
$596$		0			0
$597$	−2.05033	−	1.02261i	−0.0839145	−	0.0418528i
$598$		0			0
$599$			14.1576i			0.578463i	0.957259	+	0.289231i	$0.0933997\pi$
$599$			14.1576i			0.578463i	−0.957259	+	0.289231i	$0.906600\pi$
$600$		0			0
$601$			4.57952i			0.186803i	0.995629	+	0.0934013i	$0.0297740\pi$
$601$			4.57952i			0.186803i	−0.995629	+	0.0934013i	$0.970226\pi$
$602$		0			0
$603$	−27.1965	+	20.4821i	−1.10753	+	0.834096i
$604$		0			0
$605$	−1.72404	−	19.7913i	−0.0700923	−	0.804630i
$606$		0			0
$607$	−22.3014			−0.905188			−0.452594	−	0.891717i	$-0.649501\pi$
$607$	−22.3014			−0.905188			−0.452594	+	0.891717i	$0.649501\pi$
$608$		0			0
$609$	−14.5269	+	13.3728i	−0.588661	+	0.541894i
$610$		0			0
$611$			27.9202i			1.12953i
$612$		0			0
$613$			44.5311i			1.79859i	0.437340	+	0.899296i	$0.355921\pi$
$613$			44.5311i			1.79859i	−0.437340	+	0.899296i	$0.644079\pi$
$614$		0			0
$615$	−16.4240	−	6.47936i	−0.662282	−	0.261273i
$616$		0			0
$617$	4.11299			0.165583			0.0827913	−	0.996567i	$-0.473617\pi$
$617$	4.11299			0.165583			0.0827913	+	0.996567i	$0.473617\pi$
$618$		0			0
$619$			8.36511i			0.336222i	0.985768	+	0.168111i	$0.0537667\pi$
$619$			8.36511i			0.336222i	−0.985768	+	0.168111i	$0.946233\pi$
$620$		0			0
$621$	12.4544	−	2.30289i	0.499776	−	0.0924118i
$622$		0			0
$623$	11.0375	−	28.9639i	0.442209	−	1.16041i
$624$		0			0
$625$	23.5051	−	8.51530i	0.940204	−	0.340612i
$626$		0			0
$627$	7.36275	−	14.7622i	0.294040	−	0.589547i
$628$		0			0
$629$	−4.00734			−0.159783
$630$		0			0
$631$	−16.3025			−0.648991			−0.324496	−	0.945887i	$-0.605195\pi$
$631$	−16.3025			−0.648991			−0.324496	+	0.945887i	$0.605195\pi$
$632$		0			0
$633$	−11.6845	+	23.4273i	−0.464418	+	0.931153i
$634$		0			0
$635$	−0.00648965		0.000565321i	−0.000257534		2.24341e-5i
$636$		0			0
$637$	−23.1138	−	20.6092i	−0.915801	−	0.816568i
$638$		0			0
$639$	2.69623	−	2.03058i	0.106661	−	0.0803283i
$640$		0			0
$641$			45.8483i			1.81090i	0.424455	+	0.905449i	$0.360466\pi$
$641$			45.8483i			1.81090i	−0.424455	+	0.905449i	$0.639534\pi$
$642$		0			0
$643$	−26.3467			−1.03901			−0.519505	−	0.854467i	$-0.673884\pi$
$643$	−26.3467			−1.03901			−0.519505	+	0.854467i	$0.673884\pi$
$644$		0			0
$645$	−12.1843	+	30.8851i	−0.479756	+	1.21610i
$646$		0			0
$647$			4.41576i			0.173601i	0.996226	+	0.0868007i	$0.0276643\pi$
$647$			4.41576i			0.173601i	−0.996226	+	0.0868007i	$0.972336\pi$
$648$		0			0
$649$			12.1229i			0.475865i
$650$		0			0
$651$	−9.65253	+	8.88567i	−0.378312	+	0.348257i
$652$		0			0
$653$	46.8989			1.83530			0.917648	−	0.397393i	$-0.130085\pi$
$653$	46.8989			1.83530			0.917648	+	0.397393i	$0.130085\pi$
$654$		0			0
$655$	−0.776372	−	8.91242i	−0.0303354	−	0.348237i
$656$		0			0
$657$	−20.2898	−	26.9411i	−0.791580	−	1.05107i
$658$		0			0
$659$			26.9930i			1.05150i	0.850640	+	0.525748i	$0.176215\pi$
$659$			26.9930i			1.05150i	−0.850640	+	0.525748i	$0.823785\pi$
$660$		0			0
$661$		−	22.4053i		−	0.871467i	−0.900076	−	0.435733i	$-0.856489\pi$
$661$		−	22.4053i		−	0.871467i	0.900076	−	0.435733i	$-0.143511\pi$
$662$		0			0
$663$	3.01906	+	1.50577i	0.117251	+	0.0584794i
$664$		0			0
$665$	10.6015	−	37.2606i	0.411107	−	1.44490i
$666$		0			0
$667$			10.5024i			0.406653i
$668$		0			0
$669$	17.6974	−	35.4832i	0.684222	−	1.37186i
$670$		0			0
$671$	10.3914			0.401156
$672$		0			0
$673$			13.7897i			0.531554i	0.964035	+	0.265777i	$0.0856284\pi$
$673$			13.7897i			0.531554i	−0.964035	+	0.265777i	$0.914372\pi$
$674$		0			0
$675$	24.3461	−	9.07024i	0.937080	−	0.349114i
$676$		0			0
$677$			36.5086i			1.40314i	0.712600	+	0.701570i	$0.247517\pi$
$677$			36.5086i			1.40314i	−0.712600	+	0.701570i	$0.752483\pi$
$678$		0			0
$679$	5.91534	−	15.5226i	0.227010	−	0.595703i
$680$		0			0
$681$	19.4907	+	9.72111i	0.746886	+	0.372514i
$682$		0			0
$683$	−31.7095			−1.21333			−0.606666	−	0.794957i	$-0.707493\pi$
$683$	−31.7095			−1.21333			−0.606666	+	0.794957i	$0.707493\pi$
$684$		0			0
$685$	−3.42471	−	39.3142i	−0.130851	−	1.50212i
$686$		0			0
$687$	−0.775877	−	0.386973i	−0.0296015	−	0.0147639i
$688$		0			0
$689$	41.4678			1.57980
$690$		0			0
$691$			30.2906i			1.15231i	0.817340	+	0.576155i	$0.195448\pi$
$691$			30.2906i			1.15231i	−0.817340	+	0.576155i	$0.804552\pi$
$692$		0			0
$693$	3.20601	+	11.0906i	0.121786	+	0.421297i
$694$		0			0
$695$	−9.40615	+	0.819381i	−0.356795	+	0.0310809i
$696$		0			0
$697$			2.00718i			0.0760274i
$698$		0			0
$699$	−9.98562	+	20.0211i	−0.377691	+	0.757266i
$700$		0			0
$701$			23.8010i			0.898952i	0.893293	+	0.449476i	$0.148389\pi$
$701$			23.8010i			0.898952i	−0.893293	+	0.449476i	$0.851611\pi$
$702$		0			0
$703$	−59.5982			−2.24779
$704$		0			0
$705$	8.97011	−	22.7377i	0.337834	−	0.856350i
$706$		0			0
$707$	−10.8837	+	28.5602i	−0.409323	+	1.07412i
$708$		0			0
$709$	−39.8188			−1.49543			−0.747714	−	0.664021i	$-0.768849\pi$
$709$	−39.8188			−1.49543			−0.747714	+	0.664021i	$0.768849\pi$
$710$		0			0
$711$	12.3444	+	16.3911i	0.462953	+	0.614716i
$712$		0			0
$713$			6.97837i			0.261342i
$714$		0			0
$715$	−14.3339	+	1.24864i	−0.536057	+	0.0466966i
$716$		0			0
$717$	42.0214	+	20.9584i	1.56932	+	0.782707i
$718$		0			0
$719$	9.97913			0.372159			0.186079	−	0.982535i	$-0.440422\pi$
$719$	9.97913			0.372159			0.186079	+	0.982535i	$0.440422\pi$
$720$		0			0
$721$	16.9140	−	44.3844i	0.629909	−	1.65296i
$722$		0			0
$723$	−7.36973	−	3.67570i	−0.274083	−	0.136701i
$724$		0			0
$725$	3.72510	+	21.2190i	0.138347	+	0.788056i
$726$		0			0
$727$	31.2440			1.15878			0.579388	−	0.815052i	$-0.303291\pi$
$727$	31.2440			1.15878			0.579388	+	0.815052i	$0.303291\pi$
$728$		0			0
$729$	25.2148	−	9.65484i	0.933880	−	0.357587i
$730$		0			0
$731$	3.77446			0.139603
$732$		0			0
$733$	−10.4386			−0.385558			−0.192779	−	0.981242i	$-0.561750\pi$
$733$	−10.4386			−0.385558			−0.192779	+	0.981242i	$0.561750\pi$
$734$		0			0
$735$	12.2021	+	24.2097i	0.450082	+	0.892987i
$736$		0			0
$737$	−16.5068			−0.608036
$738$		0			0
$739$	23.7381			0.873218			0.436609	−	0.899651i	$-0.356179\pi$
$739$	23.7381			0.873218			0.436609	+	0.899651i	$0.356179\pi$
$740$		0			0
$741$	44.9003	+	22.3943i	1.64945	+	0.822674i
$742$		0			0
$743$	−16.6347			−0.610269			−0.305135	−	0.952309i	$-0.598701\pi$
$743$	−16.6347			−0.610269			−0.305135	+	0.952309i	$0.598701\pi$
$744$		0			0
$745$	39.6245	−	3.45174i	1.45173	−	0.126462i
$746$		0			0
$747$	−13.7922	+	10.3871i	−0.504631	+	0.380046i
$748$		0			0
$749$	−17.5170	+	45.9669i	−0.640058	+	1.67959i
$750$		0			0
$751$	7.46709			0.272478			0.136239	−	0.990676i	$-0.456499\pi$
$751$	7.46709			0.272478			0.136239	+	0.990676i	$0.456499\pi$
$752$		0			0
$753$	8.35667	−	16.7550i	0.304534	−	0.610587i
$754$		0			0
$755$	−1.58564	−	18.2025i	−0.0577074	−	0.662456i
$756$		0			0
$757$			6.54945i			0.238044i	0.992892	+	0.119022i	$0.0379759\pi$
$757$			6.54945i			0.238044i	−0.992892	+	0.119022i	$0.962024\pi$
$758$		0			0
$759$	5.49507	+	2.74070i	0.199458	+	0.0994810i
$760$		0			0
$761$	12.5194			0.453826			0.226913	−	0.973915i	$-0.427137\pi$
$761$	12.5194			0.453826			0.226913	+	0.973915i	$0.427137\pi$
$762$		0			0
$763$	3.96607	−	10.4075i	0.143581	−	0.376776i
$764$		0			0
$765$	−1.97489	−	2.19622i	−0.0714024	−	0.0794047i
$766$		0			0
$767$	−36.8725			−1.33139
$768$		0			0
$769$		−	15.0198i		−	0.541629i	−0.962632	−	0.270815i	$-0.912707\pi$
$769$		−	15.0198i		−	0.541629i	0.962632	−	0.270815i	$-0.0872931\pi$
$770$		0			0
$771$	34.8110	+	17.3622i	1.25369	+	0.625283i
$772$		0			0
$773$		−	31.2676i		−	1.12462i	−0.826927	−	0.562309i	$-0.809913\pi$
$773$		−	31.2676i		−	1.12462i	0.826927	−	0.562309i	$-0.190087\pi$
$774$		0			0
$775$	2.47517	+	14.0991i	0.0889108	+	0.506456i
$776$		0			0
$777$	30.6861	−	28.2482i	1.10086	−	1.01340i
$778$		0			0
$779$			29.8513i			1.06953i
$780$		0			0
$781$	1.63647			0.0585574
$782$		0			0
$783$	4.07081	+	22.0155i	0.145479	+	0.786770i
$784$		0			0
$785$	−0.0164735	−	0.189109i	−0.000587966	−	0.00674959i
$786$		0			0
$787$	22.0233			0.785048			0.392524	−	0.919742i	$-0.371602\pi$
$787$	22.0233			0.785048			0.392524	+	0.919742i	$0.371602\pi$
$788$		0			0
$789$	13.7484	−	27.5654i	0.489457	−	0.981356i
$790$		0			0
$791$	−9.72790	+	25.5272i	−0.345884	+	0.907644i
$792$		0			0
$793$			31.6062i			1.12237i
$794$		0			0
$795$	−33.7706	−	13.3226i	−1.19772	−	0.472505i
$796$		0			0
$797$		−	8.33314i		−	0.295175i	−0.989049	−	0.147587i	$-0.952849\pi$
$797$		−	8.33314i		−	0.295175i	0.989049	−	0.147587i	$-0.0471508\pi$
$798$		0			0
$799$	−2.77877			−0.0983057
$800$		0			0
$801$	−21.1434	−	28.0746i	−0.747067	−	0.991967i
$802$		0			0
$803$		−	16.3518i		−	0.577043i
$804$		0			0
$805$	13.8698	+	3.94627i	0.488846	+	0.139088i
$806$		0			0
$807$	12.4437	−	24.9494i	0.438038	−	0.878261i
$808$		0			0
$809$			5.19293i			0.182574i	0.995825	+	0.0912868i	$0.0290980\pi$
$809$			5.19293i			0.182574i	−0.995825	+	0.0912868i	$0.970902\pi$
$810$		0			0
$811$			30.8998i			1.08504i	0.840044	+	0.542519i	$0.182529\pi$
$811$			30.8998i			1.08504i	−0.840044	+	0.542519i	$0.817471\pi$
$812$		0			0
$813$	14.6438	+	7.30367i	0.513579	+	0.256151i
$814$		0			0
$815$	−28.9571	+	2.52249i	−1.01432	+	0.0883589i
$816$		0			0
$817$	56.1347			1.96391
$818$		0			0
$819$	−33.7327	+	9.75127i	−1.17872	+	0.340737i
$820$		0			0
$821$			16.6956i			0.582679i	0.956620	+	0.291340i	$0.0941010\pi$
$821$			16.6956i			0.582679i	−0.956620	+	0.291340i	$0.905899\pi$
$822$		0			0
$823$		−	30.9576i		−	1.07911i	−0.841949	−	0.539557i	$-0.818592\pi$
$823$		−	30.9576i		−	1.07911i	0.841949	−	0.539557i	$-0.181408\pi$
$824$		0			0
$825$	12.0744	+	3.58827i	0.420376	+	0.124927i
$826$		0			0
$827$	24.6612			0.857555			0.428777	−	0.903410i	$-0.358944\pi$
$827$	24.6612			0.857555			0.428777	+	0.903410i	$0.358944\pi$
$828$		0			0
$829$			26.6977i			0.927249i	0.886032	+	0.463625i	$0.153451\pi$
$829$			26.6977i			0.927249i	−0.886032	+	0.463625i	$0.846549\pi$
$830$		0			0
$831$	−7.24624	−	3.61410i	−0.251369	−	0.125372i
$832$		0			0
$833$	2.05114	−	2.30040i	0.0710678	−	0.0797043i
$834$		0			0
$835$	−1.30377	+	0.113573i	−0.0451189	+	0.00393036i
$836$		0			0
$837$	2.70488	+	14.6284i	0.0934942	+	0.505630i
$838$		0			0
$839$	51.2367			1.76889			0.884443	−	0.466647i	$-0.154538\pi$
$839$	51.2367			1.76889			0.884443	+	0.466647i	$0.154538\pi$
$840$		0			0
$841$	10.4350			0.359828
$842$		0			0
$843$	−18.3702	−	9.16224i	−0.632703	−	0.315564i
$844$		0			0
$845$	−1.27515	−	14.6382i	−0.0438664	−	0.503568i
$846$		0			0
$847$	8.37047	−	21.9652i	0.287613	−	0.754733i
$848$		0			0
$849$	4.52172	−	9.06600i	0.155185	−	0.311144i
$850$		0			0
$851$		−	22.1847i		−	0.760483i
$852$		0			0
$853$	−15.3303			−0.524901			−0.262450	−	0.964946i	$-0.584531\pi$
$853$	−15.3303			−0.524901			−0.262450	+	0.964946i	$0.584531\pi$
$854$		0			0
$855$	−29.3711	−	32.6628i	−1.00447	−	1.11705i
$856$		0			0
$857$		−	10.2905i		−	0.351518i	−0.984433	−	0.175759i	$-0.943762\pi$
$857$		−	10.2905i		−	0.351518i	0.984433	−	0.175759i	$-0.0562379\pi$
$858$		0			0
$859$			19.8890i			0.678603i	0.940678	+	0.339301i	$0.110191\pi$
$859$			19.8890i			0.678603i	−0.940678	+	0.339301i	$0.889809\pi$
$860$		0			0
$861$	−14.1488	−	15.3699i	−0.482191	−	0.523806i
$862$		0			0
$863$	17.8374			0.607193			0.303596	−	0.952801i	$-0.401813\pi$
$863$	17.8374			0.607193			0.303596	+	0.952801i	$0.401813\pi$
$864$		0			0
$865$	−38.9754	+	3.39520i	−1.32520	+	0.115440i
$866$		0			0
$867$	12.9920	−	26.0489i	0.441233	−	0.884668i
$868$		0			0
$869$			9.94855i			0.337481i
$870$		0			0
$871$		−	50.2065i		−	1.70118i
$872$		0			0
$873$	−11.3314	−	15.0460i	−0.383510	−	0.509230i
$874$		0			0
$875$	29.4224	+	3.05357i	0.994658	+	0.103230i
$876$		0			0
$877$			8.09995i			0.273516i	0.990604	+	0.136758i	$0.0436682\pi$
$877$			8.09995i			0.273516i	−0.990604	+	0.136758i	$0.956332\pi$
$878$		0			0
$879$	0.536831	+	0.267747i	0.0181068	+	0.00903089i
$880$		0			0
$881$	46.3728			1.56234			0.781169	−	0.624319i	$-0.214624\pi$
$881$	46.3728			1.56234			0.781169	+	0.624319i	$0.214624\pi$
$882$		0			0
$883$		−	8.51933i		−	0.286698i	−0.989672	−	0.143349i	$-0.954213\pi$
$883$		−	8.51933i		−	0.286698i	0.989672	−	0.143349i	$-0.0457872\pi$
$884$		0			0
$885$	30.0282	+	11.8463i	1.00939	+	0.398208i
$886$		0			0
$887$		−	54.7644i		−	1.83881i	−0.393312	−	0.919405i	$-0.628671\pi$
$887$		−	54.7644i		−	1.83881i	0.393312	−	0.919405i	$-0.371329\pi$
$888$		0			0
$889$	−0.00720248	−	0.00274471i	−0.000241563	−	9.20547e-5i
$890$		0			0
$891$	12.5813	+	3.61523i	0.421490	+	0.121115i
$892$		0			0
$893$	−41.3265			−1.38294
$894$		0			0
$895$	45.1583	−	3.93380i	1.50947	−	0.131492i
$896$		0			0
$897$	−8.33600	+	16.7136i	−0.278331	+	0.558051i
$898$		0			0
$899$	−12.3356			−0.411416
$900$		0			0
$901$			4.12710i			0.137494i
$902$		0			0
$903$	−28.9028	+	26.6066i	−0.961825	+	0.885411i
$904$		0			0
$905$	28.7371	−	2.50332i	0.955253	−	0.0832133i
$906$		0			0
$907$			20.6308i			0.685036i	0.939511	+	0.342518i	$0.111280\pi$
$907$			20.6308i			0.685036i	−0.939511	+	0.342518i	$0.888720\pi$
$908$		0			0
$909$	20.8487	+	27.6833i	0.691509	+	0.918196i
$910$		0			0
$911$			25.4837i			0.844313i	0.906523	+	0.422156i	$0.138727\pi$
$911$			25.4837i			0.844313i	−0.906523	+	0.422156i	$0.861273\pi$
$912$		0			0
$913$	−8.37114			−0.277044
$914$		0			0
$915$	10.1543	−	25.7394i	0.335691	−	0.850919i
$916$		0			0
$917$	3.76939	−	9.89137i	0.124476	−	0.326642i
$918$		0			0
$919$	17.9121			0.590865			0.295432	−	0.955364i	$-0.404536\pi$
$919$	17.9121			0.590865			0.295432	+	0.955364i	$0.404536\pi$
$920$		0			0
$921$	−19.1614	+	38.4184i	−0.631389	+	1.26593i
$922$		0			0
$923$			4.97742i			0.163834i
$924$		0			0
$925$	−7.86875	−	44.8222i	−0.258723	−	1.47374i
$926$		0			0
$927$	−32.4003	−	43.0217i	−1.06417	−	1.41302i
$928$		0			0
$929$	−10.9000			−0.357616			−0.178808	−	0.983884i	$-0.557224\pi$
$929$	−10.9000			−0.357616			−0.178808	+	0.983884i	$0.557224\pi$
$930$		0			0
$931$	30.5051	−	34.2122i	0.999764	−	1.12126i
$932$		0			0
$933$	−10.1246	+	20.2998i	−0.331466	+	0.664585i
$934$		0			0
$935$	−0.124271	−	1.42658i	−0.00406411	−	0.0466542i
$936$		0			0
$937$	−26.9791			−0.881368			−0.440684	−	0.897662i	$-0.645264\pi$
$937$	−26.9791			−0.881368			−0.440684	+	0.897662i	$0.645264\pi$
$938$		0			0
$939$	1.30899	−	2.62451i	0.0427172	−	0.0856475i
$940$		0			0
$941$	−37.6275			−1.22662			−0.613310	−	0.789842i	$-0.710162\pi$
$941$	−37.6275			−1.22662			−0.613310	+	0.789842i	$0.710162\pi$
$942$		0			0
$943$	−11.1118			−0.361850
$944$		0			0
$945$	30.6041	+	2.89630i	0.995552	+	0.0942165i
$946$		0			0
$947$	−2.38298			−0.0774364			−0.0387182	−	0.999250i	$-0.512327\pi$
$947$	−2.38298			−0.0774364			−0.0387182	+	0.999250i	$0.512327\pi$
$948$		0			0
$949$	49.7351			1.61447
$950$		0			0
$951$	−25.2686	+	50.6632i	−0.819389	+	1.64287i
$952$		0			0
$953$	10.3582			0.335534			0.167767	−	0.985827i	$-0.446344\pi$
$953$	10.3582			0.335534			0.167767	+	0.985827i	$0.446344\pi$
$954$		0			0
$955$	−4.37020	+	0.380694i	−0.141416	+	0.0123190i
$956$		0			0
$957$	−4.84472	+	9.71361i	−0.156607	+	0.313996i
$958$		0			0
$959$	16.6274	−	43.6325i	0.536928	−	1.40897i
$960$		0			0
$961$	22.8035			0.735597
$962$		0			0
$963$	33.5555	+	44.5556i	1.08131	+	1.43578i
$964$		0			0
$965$	−39.9012	+	3.47584i	−1.28446	+	0.111891i
$966$		0			0
$967$		−	10.3006i		−	0.331245i	−0.986189	−	0.165622i	$-0.947037\pi$
$967$		−	10.3006i		−	0.331245i	0.986189	−	0.165622i	$-0.0529633\pi$
$968$		0			0
$969$	−2.22879	+	4.46871i	−0.0715992	+	0.143556i
$970$		0			0
$971$	18.2487			0.585629			0.292814	−	0.956169i	$-0.405408\pi$
$971$	18.2487			0.585629			0.292814	+	0.956169i	$0.405408\pi$
$972$		0			0
$973$	−10.4393	−	3.97821i	−0.334670	−	0.127536i
$974$		0			0
$975$	−10.9139	+	36.7250i	−0.349526	+	1.17614i
$976$		0			0
$977$	−34.9393			−1.11781			−0.558904	−	0.829233i	$-0.688778\pi$
$977$	−34.9393			−1.11781			−0.558904	+	0.829233i	$0.688778\pi$
$978$		0			0
$979$		−	17.0398i		−	0.544594i
$980$		0			0
$981$	−7.59739	−	10.0879i	−0.242566	−	0.322083i
$982$		0			0
$983$			37.9493i			1.21039i	0.796076	+	0.605197i	$0.206906\pi$
$983$			37.9493i			1.21039i	−0.796076	+	0.605197i	$0.793094\pi$
$984$		0			0
$985$	−2.37842	−	27.3033i	−0.0757828	−	0.869954i
$986$		0			0
$987$	21.2783	−	19.5878i	0.677296	−	0.623487i
$988$		0			0
$989$			20.8955i			0.664438i
$990$		0			0
$991$	19.9068			0.632359			0.316180	−	0.948699i	$-0.397600\pi$
$991$	19.9068			0.632359			0.316180	+	0.948699i	$0.397600\pi$
$992$		0			0
$993$	−17.4570	+	35.0011i	−0.553981	+	1.11073i
$994$		0			0
$995$	−2.94677	+	0.256697i	−0.0934188	+	0.00813783i
$996$		0			0
$997$	43.8198			1.38779			0.693893	−	0.720078i	$-0.255894\pi$
$997$	43.8198			1.38779			0.693893	+	0.720078i	$0.255894\pi$
$998$		0			0
$999$	−8.59899	−	46.5046i	−0.272060	−	1.47134i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.k.a.209.17	yes	24
3.2	odd	2		840.2.k.b.209.18	yes	24
4.3	odd	2		1680.2.k.i.209.8		24
5.4	even	2		840.2.k.b.209.8	yes	24
7.6	odd	2	inner	840.2.k.a.209.8	yes	24
12.11	even	2		1680.2.k.h.209.7		24
15.14	odd	2	inner	840.2.k.a.209.7	✓	24
20.19	odd	2		1680.2.k.h.209.17		24
21.20	even	2		840.2.k.b.209.7	yes	24
28.27	even	2		1680.2.k.i.209.17		24
35.34	odd	2		840.2.k.b.209.17	yes	24
60.59	even	2		1680.2.k.i.209.18		24
84.83	odd	2		1680.2.k.h.209.18		24
105.104	even	2	inner	840.2.k.a.209.18	yes	24
140.139	even	2		1680.2.k.h.209.8		24
420.419	odd	2		1680.2.k.i.209.7		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.k.a.209.7	✓	24	15.14	odd	2	inner
840.2.k.a.209.8	yes	24	7.6	odd	2	inner
840.2.k.a.209.17	yes	24	1.1	even	1	trivial
840.2.k.a.209.18	yes	24	105.104	even	2	inner
840.2.k.b.209.7	yes	24	21.20	even	2
840.2.k.b.209.8	yes	24	5.4	even	2
840.2.k.b.209.17	yes	24	35.34	odd	2
840.2.k.b.209.18	yes	24	3.2	odd	2
1680.2.k.h.209.7		24	12.11	even	2
1680.2.k.h.209.8		24	140.139	even	2
1680.2.k.h.209.17		24	20.19	odd	2
1680.2.k.h.209.18		24	84.83	odd	2
1680.2.k.i.209.7		24	420.419	odd	2
1680.2.k.i.209.8		24	4.3	odd	2
1680.2.k.i.209.17		24	28.27	even	2
1680.2.k.i.209.18		24	60.59	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 $\F_{q}$
Belyi maps

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)

\(f(q)\)	\(=\)	\(q+(0.773053 - 1.54996i) q^{3} +(-0.194052 - 2.22763i) q^{5} +(0.942148 - 2.47232i) q^{7} +(-1.80478 - 2.39641i) q^{9} -1.45449i q^{11} +4.42393 q^{13} +(-3.60276 - 1.42131i) q^{15} +0.440293i q^{17} +6.54815i q^{19} +(-3.10367 - 3.37153i) q^{21} -2.43747 q^{23} +(-4.92469 + 0.864552i) q^{25} +(-5.10954 + 0.944786i) q^{27} -4.30871i q^{29} -2.86295i q^{31} +(-2.25441 - 1.12440i) q^{33} +(-5.69024 - 1.61900i) q^{35} +9.10153i q^{37} +(3.41994 - 6.85694i) q^{39} +4.55874 q^{41} -8.57261i q^{43} +(-4.98810 + 4.48541i) q^{45} +6.31118i q^{47} +(-5.22471 - 4.65858i) q^{49} +(0.682438 + 0.340370i) q^{51} +9.37352 q^{53} +(-3.24008 + 0.282247i) q^{55} +(10.1494 + 5.06207i) q^{57} -8.33478 q^{59} +7.14436i q^{61} +(-7.62506 + 2.20421i) q^{63} +(-0.858472 - 9.85489i) q^{65} -11.3488i q^{67} +(-1.88430 + 3.77800i) q^{69} +1.12511i q^{71} +11.2423 q^{73} +(-2.46702 + 8.30143i) q^{75} +(-3.59597 - 1.37035i) q^{77} -6.83987 q^{79} +(-2.48556 + 8.64997i) q^{81} -5.75536i q^{83} +(0.980811 - 0.0854397i) q^{85} +(-6.67834 - 3.33086i) q^{87} +11.7153 q^{89} +(4.16800 - 10.9374i) q^{91} +(-4.43747 - 2.21322i) q^{93} +(14.5869 - 1.27068i) q^{95} +6.27856 q^{97} +(-3.48556 + 2.62504i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{9} - 6 q^{15} - 2 q^{21} + 16 q^{23} + 8 q^{25} - 8 q^{35} - 2 q^{39} + 6 q^{51} - 24 q^{53} - 8 q^{57} - 16 q^{63} - 16 q^{65} - 8 q^{77} + 4 q^{79} + 18 q^{81} - 12 q^{85} + 12 q^{91} - 32 q^{93}+ \cdots - 6 q^{99}+O(q^{100}) \) 24 * q - 2 * q^9 - 6 * 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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