LMFDB - Embedded newform 840.2.k.b.209.4

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,-2,0,0,0,0,0,2] 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			209.4
Character	$\chi$	$=$	840.209
Dual form			840.2.k.b.209.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.64602 + 0.539084i) q^{3} +(1.30213 - 1.81782i) q^{5} +(-2.19974 + 1.47008i) q^{7} +(2.41878 - 1.77469i) q^{9} +0.958461i q^{11} +0.157315 q^{13} +(-1.16337 + 3.69413i) q^{15} -2.51337i q^{17} -1.98260i q^{19} +(2.82833 - 3.60563i) q^{21} +2.67280 q^{23} +(-1.60893 - 4.73406i) q^{25} +(-3.02466 + 4.22510i) q^{27} +1.25028i q^{29} -8.66804i q^{31} +(-0.516691 - 1.57765i) q^{33} +(-0.191998 + 5.91296i) q^{35} -2.29909i q^{37} +(-0.258945 + 0.0848062i) q^{39} -4.74507 q^{41} -6.58424i q^{43} +(-0.0765078 - 6.70777i) q^{45} -5.60727i q^{47} +(2.67772 - 6.46760i) q^{49} +(1.35492 + 4.13706i) q^{51} +8.59262 q^{53} +(1.74231 + 1.24804i) q^{55} +(1.06879 + 3.26340i) q^{57} +10.4488 q^{59} -4.27757i q^{61} +(-2.71175 + 7.45965i) q^{63} +(0.204845 - 0.285971i) q^{65} -13.8282i q^{67} +(-4.39948 + 1.44086i) q^{69} -9.75994i q^{71} -4.43474 q^{73} +(5.20040 + 6.92502i) q^{75} +(-1.40902 - 2.10837i) q^{77} +0.517890 q^{79} +(2.70097 - 8.58515i) q^{81} +18.1293i q^{83} +(-4.56885 - 3.27272i) q^{85} +(-0.674006 - 2.05799i) q^{87} +0.954423 q^{89} +(-0.346053 + 0.231267i) q^{91} +(4.67280 + 14.2678i) q^{93} +(-3.60401 - 2.58160i) q^{95} +14.0907 q^{97} +(1.70097 + 2.31830i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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}+ \cdots - 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

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
$3$	−1.64602	+	0.539084i	−0.950331	+	0.311240i
$3$	−1.64602	+	0.539084i	−0.950331	+	0.311240i
$4$		0			0
$5$	1.30213	−	1.81782i	0.582329	−	0.812954i
$5$	1.30213	−	1.81782i	0.582329	−	0.812954i
$6$		0			0
$7$	−2.19974	+	1.47008i	−0.831424	+	0.555638i
$7$	−2.19974	+	1.47008i	−0.831424	+	0.555638i
$8$		0			0
$9$	2.41878	−	1.77469i	0.806259	−	0.591562i
$10$		0			0
$11$			0.958461i			0.288987i	0.989506	+	0.144493i	$0.0461553\pi$
$11$			0.958461i			0.288987i	−0.989506	+	0.144493i	$0.953845\pi$
$12$		0			0
$13$	0.157315			0.0436315			0.0218157	−	0.999762i	$-0.493055\pi$
$13$	0.157315			0.0436315			0.0218157	+	0.999762i	$0.493055\pi$
$14$		0			0
$15$	−1.16337	+	3.69413i	−0.300381	+	0.953819i
$16$		0			0
$17$		−	2.51337i		−	0.609581i	−0.952419	−	0.304791i	$-0.901414\pi$
$17$		−	2.51337i		−	0.609581i	0.952419	−	0.304791i	$-0.0985865\pi$
$18$		0			0
$19$		−	1.98260i		−	0.454840i	−0.973797	−	0.227420i	$-0.926971\pi$
$19$		−	1.98260i		−	0.454840i	0.973797	−	0.227420i	$-0.0730290\pi$
$20$		0			0
$21$	2.82833	−	3.60563i	0.617191	−	0.786813i
$22$		0			0
$23$	2.67280			0.557317			0.278658	−	0.960390i	$-0.410110\pi$
$23$	2.67280			0.557317			0.278658	+	0.960390i	$0.410110\pi$
$24$		0			0
$25$	−1.60893	−	4.73406i	−0.321787	−	0.946812i
$26$		0			0
$27$	−3.02466	+	4.22510i	−0.582095	+	0.813121i
$28$		0			0
$29$			1.25028i			0.232171i	0.993239	+	0.116086i	$0.0370347\pi$
$29$			1.25028i			0.232171i	−0.993239	+	0.116086i	$0.962965\pi$
$30$		0			0
$31$		−	8.66804i		−	1.55683i	−0.627753	−	0.778413i	$-0.716025\pi$
$31$		−	8.66804i		−	1.55683i	0.627753	−	0.778413i	$-0.283975\pi$
$32$		0			0
$33$	−0.516691	−	1.57765i	−0.0899443	−	0.274633i
$34$		0			0
$35$	−0.191998	+	5.91296i	−0.0324536	+	0.999473i
$36$		0			0
$37$		−	2.29909i		−	0.377969i	−0.981980	−	0.188984i	$-0.939480\pi$
$37$		−	2.29909i		−	0.377969i	0.981980	−	0.188984i	$-0.0605195\pi$
$38$		0			0
$39$	−0.258945	+	0.0848062i	−0.0414643	+	0.0135799i
$40$		0			0
$41$	−4.74507			−0.741056			−0.370528	−	0.928821i	$-0.620823\pi$
$41$	−4.74507			−0.741056			−0.370528	+	0.928821i	$0.620823\pi$
$42$		0			0
$43$		−	6.58424i		−	1.00409i	−0.864842	−	0.502043i	$-0.832582\pi$
$43$		−	6.58424i		−	1.00409i	0.864842	−	0.502043i	$-0.167418\pi$
$44$		0			0
$45$	−0.0765078	−	6.70777i	−0.0114051	−	0.999935i
$46$		0			0
$47$		−	5.60727i		−	0.817904i	−0.912556	−	0.408952i	$-0.865894\pi$
$47$		−	5.60727i		−	0.817904i	0.912556	−	0.408952i	$-0.134106\pi$
$48$		0			0
$49$	2.67772	−	6.46760i	0.382532	−	0.923942i
$50$		0			0
$51$	1.35492	+	4.13706i	0.189726	+	0.579304i
$52$		0			0
$53$	8.59262			1.18029			0.590144	−	0.807298i	$-0.299071\pi$
$53$	8.59262			1.18029			0.590144	+	0.807298i	$0.299071\pi$
$54$		0			0
$55$	1.74231	+	1.24804i	0.234933	+	0.168285i
$56$		0			0
$57$	1.06879	+	3.26340i	0.141564	+	0.432248i
$58$		0			0
$59$	10.4488			1.36032			0.680159	−	0.733065i	$-0.261911\pi$
$59$	10.4488			1.36032			0.680159	+	0.733065i	$0.261911\pi$
$60$		0			0
$61$		−	4.27757i		−	0.547686i	−0.961774	−	0.273843i	$-0.911705\pi$
$61$		−	4.27757i		−	0.547686i	0.961774	−	0.273843i	$-0.0882949\pi$
$62$		0			0
$63$	−2.71175	+	7.45965i	−0.341648	+	0.939828i
$64$		0			0
$65$	0.204845	−	0.285971i	0.0254078	−	0.0354704i
$66$		0			0
$67$		−	13.8282i		−	1.68938i	−0.535255	−	0.844690i	$-0.679785\pi$
$67$		−	13.8282i		−	1.68938i	0.535255	−	0.844690i	$-0.320215\pi$
$68$		0			0
$69$	−4.39948	+	1.44086i	−0.529635	+	0.173459i
$70$		0			0
$71$		−	9.75994i		−	1.15829i	−0.815224	−	0.579146i	$-0.803386\pi$
$71$		−	9.75994i		−	1.15829i	0.815224	−	0.579146i	$-0.196614\pi$
$72$		0			0
$73$	−4.43474			−0.519047			−0.259524	−	0.965737i	$-0.583566\pi$
$73$	−4.43474			−0.519047			−0.259524	+	0.965737i	$0.583566\pi$
$74$		0			0
$75$	5.20040	+	6.92502i	0.600490	+	0.799632i
$76$		0			0
$77$	−1.40902	−	2.10837i	−0.160572	−	0.240271i
$78$		0			0
$79$	0.517890			0.0582671			0.0291336	−	0.999576i	$-0.490725\pi$
$79$	0.517890			0.0582671			0.0291336	+	0.999576i	$0.490725\pi$
$80$		0			0
$81$	2.70097	−	8.58515i	0.300108	−	0.953905i
$82$		0			0
$83$			18.1293i			1.98995i	0.100105	+	0.994977i	$0.468082\pi$
$83$			18.1293i			1.98995i	−0.100105	+	0.994977i	$0.531918\pi$
$84$		0			0
$85$	−4.56885	−	3.27272i	−0.495561	−	0.354977i
$86$		0			0
$87$	−0.674006	−	2.05799i	−0.0722611	−	0.220640i
$88$		0			0
$89$	0.954423			0.101169			0.0505843	−	0.998720i	$-0.483892\pi$
$89$	0.954423			0.101169			0.0505843	+	0.998720i	$0.483892\pi$
$90$		0			0
$91$	−0.346053	+	0.231267i	−0.0362762	+	0.0242433i
$92$		0			0
$93$	4.67280	+	14.2678i	0.484546	+	1.47950i
$94$		0			0
$95$	−3.60401	−	2.58160i	−0.369764	−	0.264866i
$96$		0			0
$97$	14.0907			1.43070			0.715348	−	0.698768i	$-0.246268\pi$
$97$	14.0907			1.43070			0.715348	+	0.698768i	$0.246268\pi$
$98$		0			0
$99$	1.70097	+	2.31830i	0.170954	+	0.232998i
$100$		0			0
$101$	−8.87371			−0.882967			−0.441484	−	0.897269i	$-0.645548\pi$
$101$	−8.87371			−0.882967			−0.441484	+	0.897269i	$0.645548\pi$
$102$		0			0
$103$	2.26288			0.222968			0.111484	−	0.993766i	$-0.464440\pi$
$103$	2.26288			0.222968			0.111484	+	0.993766i	$0.464440\pi$
$104$		0			0
$105$	−2.87155	−	9.83637i	−0.280234	−	0.959932i
$106$		0			0
$107$	−16.2677			−1.57266			−0.786331	−	0.617806i	$-0.788022\pi$
$107$	−16.2677			−1.57266			−0.786331	+	0.617806i	$0.788022\pi$
$108$		0			0
$109$	−13.4109			−1.28453			−0.642264	−	0.766483i	$-0.722005\pi$
$109$	−13.4109			−1.28453			−0.642264	+	0.766483i	$0.722005\pi$
$110$		0			0
$111$	1.23940	+	3.78436i	0.117639	+	0.359195i
$112$		0			0
$113$	15.0284			1.41375			0.706875	−	0.707339i	$-0.250104\pi$
$113$	15.0284			1.41375			0.706875	+	0.707339i	$0.250104\pi$
$114$		0			0
$115$	3.48032	−	4.85866i	0.324541	−	0.453073i
$116$		0			0
$117$	0.380511	−	0.279186i	0.0351783	−	0.0258107i
$118$		0			0
$119$	3.69485	+	5.52876i	0.338707	+	0.506820i
$120$		0			0
$121$	10.0814			0.916487
$122$		0			0
$123$	7.81049	−	2.55799i	0.704248	−	0.230646i
$124$		0			0
$125$	−10.7007	−	3.23959i	−0.957100	−	0.289758i
$126$		0			0
$127$		−	3.39261i		−	0.301046i	−0.988607	−	0.150523i	$-0.951904\pi$
$127$		−	3.39261i		−	0.301046i	0.988607	−	0.150523i	$-0.0480957\pi$
$128$		0			0
$129$	3.54945	+	10.8378i	0.312512	+	0.954215i
$130$		0			0
$131$	−0.388687			−0.0339598			−0.0169799	−	0.999856i	$-0.505405\pi$
$131$	−0.388687			−0.0339598			−0.0169799	+	0.999856i	$0.505405\pi$
$132$		0			0
$133$	2.91458	+	4.36121i	0.252726	+	0.378165i
$134$		0			0
$135$	3.74198	+	10.9999i	0.322059	+	0.946720i
$136$		0			0
$137$	−9.32628			−0.796798			−0.398399	−	0.917212i	$-0.630434\pi$
$137$	−9.32628			−0.796798			−0.398399	+	0.917212i	$0.630434\pi$
$138$		0			0
$139$			14.5321i			1.23260i	0.787511	+	0.616300i	$0.211369\pi$
$139$			14.5321i			1.23260i	−0.787511	+	0.616300i	$0.788631\pi$
$140$		0			0
$141$	3.02279	+	9.22969i	0.254565	+	0.777280i
$142$		0			0
$143$			0.150781i			0.0126089i
$144$		0			0
$145$	2.27279	+	1.62802i	0.188745	+	0.135200i
$146$		0			0
$147$	−0.921014	+	12.0893i	−0.0759640	+	0.997111i
$148$		0			0
$149$			10.9872i			0.900110i	0.893001	+	0.450055i	$0.148596\pi$
$149$			10.9872i			0.900110i	−0.893001	+	0.450055i	$0.851404\pi$
$150$		0			0
$151$	−20.5486			−1.67222			−0.836109	−	0.548563i	$-0.815175\pi$
$151$	−20.5486			−1.67222			−0.836109	+	0.548563i	$0.815175\pi$
$152$		0			0
$153$	−4.46044	−	6.07928i	−0.360605	−	0.491480i
$154$		0			0
$155$	−15.7569	−	11.2869i	−1.26563	−	0.906584i
$156$		0			0
$157$	0.876066			0.0699177			0.0349588	−	0.999389i	$-0.488870\pi$
$157$	0.876066			0.0699177			0.0349588	+	0.999389i	$0.488870\pi$
$158$		0			0
$159$	−14.1436	+	4.63214i	−1.12166	+	0.367353i
$160$		0			0
$161$	−5.87946	+	3.92923i	−0.463366	+	0.309667i
$162$		0			0
$163$			21.9513i			1.71936i	0.510833	+	0.859680i	$0.329337\pi$
$163$			21.9513i			1.71936i	−0.510833	+	0.859680i	$0.670663\pi$
$164$		0			0
$165$	−3.54068	−	1.11505i	−0.275641	−	0.0868063i
$166$		0			0
$167$		−	11.7288i		−	0.907602i	−0.891103	−	0.453801i	$-0.850068\pi$
$167$		−	11.7288i		−	0.907602i	0.891103	−	0.453801i	$-0.149932\pi$
$168$		0			0
$169$	−12.9753			−0.998096
$170$		0			0
$171$	−3.51850	−	4.79547i	−0.269066	−	0.366719i
$172$		0			0
$173$			18.8779i			1.43526i	0.696423	+	0.717632i	$0.254774\pi$
$173$			18.8779i			1.43526i	−0.696423	+	0.717632i	$0.745226\pi$
$174$		0			0
$175$	10.4987	+	8.04844i	0.793627	+	0.608405i
$176$		0			0
$177$	−17.1989	+	5.63278i	−1.29275	+	0.423385i
$178$		0			0
$179$			12.3237i			0.921118i	0.887629	+	0.460559i	$0.152351\pi$
$179$			12.3237i			0.921118i	−0.887629	+	0.460559i	$0.847649\pi$
$180$		0			0
$181$			18.2694i			1.35795i	0.734160	+	0.678976i	$0.237576\pi$
$181$			18.2694i			1.35795i	−0.734160	+	0.678976i	$0.762424\pi$
$182$		0			0
$183$	2.30597	+	7.04097i	0.170462	+	0.520484i
$184$		0			0
$185$	−4.17934	−	2.99371i	−0.307271	−	0.220102i
$186$		0			0
$187$	2.40897			0.176161
$188$		0			0
$189$	0.442223	−	13.7406i	0.0321670	−	0.999483i
$190$		0			0
$191$		−	12.4184i		−	0.898565i	−0.893390	−	0.449283i	$-0.851680\pi$
$191$		−	12.4184i		−	0.898565i	0.893390	−	0.449283i	$-0.148320\pi$
$192$		0			0
$193$		−	20.0849i		−	1.44574i	−0.690983	−	0.722871i	$-0.742822\pi$
$193$		−	20.0849i		−	1.44574i	0.690983	−	0.722871i	$-0.257178\pi$
$194$		0			0
$195$	−0.183016	+	0.581143i	−0.0131061	+	0.0416165i
$196$		0			0
$197$	13.5756			0.967223			0.483612	−	0.875283i	$-0.339325\pi$
$197$	13.5756			0.967223			0.483612	+	0.875283i	$0.339325\pi$
$198$		0			0
$199$			5.36196i			0.380099i	0.981774	+	0.190050i	$0.0608649\pi$
$199$			5.36196i			0.380099i	−0.981774	+	0.190050i	$0.939135\pi$
$200$		0			0
$201$	7.45455	+	22.7615i	0.525803	+	1.60547i
$202$		0			0
$203$	−1.83802	−	2.75030i	−0.129003	−	0.193033i
$204$		0			0
$205$	−6.17868	+	8.62568i	−0.431538	+	0.602444i
$206$		0			0
$207$	6.46490	−	4.74338i	0.449342	−	0.329688i
$208$		0			0
$209$	1.90025			0.131443
$210$		0			0
$211$	−17.4010			−1.19794			−0.598968	−	0.800773i	$-0.704422\pi$
$211$	−17.4010			−1.19794			−0.598968	+	0.800773i	$0.704422\pi$
$212$		0			0
$213$	5.26143	+	16.0651i	0.360507	+	1.10076i
$214$		0			0
$215$	−11.9690	−	8.57351i	−0.816276	−	0.584708i
$216$		0			0
$217$	12.7427	+	19.0674i	0.865032	+	1.29438i
$218$		0			0
$219$	7.29968	−	2.39070i	0.493267	−	0.161548i
$220$		0			0
$221$		−	0.395392i		−	0.0265969i
$222$		0			0
$223$	15.7794			1.05666			0.528332	−	0.849038i	$-0.322818\pi$
$223$	15.7794			1.05666			0.528332	+	0.849038i	$0.322818\pi$
$224$		0			0
$225$	−12.2931	−	8.59528i	−0.819542	−	0.573019i
$226$		0			0
$227$			3.69704i			0.245381i	0.992445	+	0.122690i	$0.0391522\pi$
$227$			3.69704i			0.245381i	−0.992445	+	0.122690i	$0.960848\pi$
$228$		0			0
$229$		−	13.9567i		−	0.922286i	−0.887326	−	0.461143i	$-0.847440\pi$
$229$		−	13.9567i		−	0.922286i	0.887326	−	0.461143i	$-0.152560\pi$
$230$		0			0
$231$	3.45586	+	2.71084i	0.227379	+	0.178360i
$232$		0			0
$233$	16.7836			1.09953			0.549767	−	0.835318i	$-0.314717\pi$
$233$	16.7836			1.09953			0.549767	+	0.835318i	$0.314717\pi$
$234$		0			0
$235$	−10.1930	−	7.30137i	−0.664918	−	0.476289i
$236$		0			0
$237$	−0.852458	+	0.279186i	−0.0553731	+	0.0181351i
$238$		0			0
$239$		−	18.2421i		−	1.17998i	−0.807410	−	0.589991i	$-0.799131\pi$
$239$		−	18.2421i		−	1.17998i	0.807410	−	0.589991i	$-0.200869\pi$
$240$		0			0
$241$			5.36648i			0.345685i	0.984949	+	0.172843i	$0.0552952\pi$
$241$			5.36648i			0.345685i	−0.984949	+	0.172843i	$0.944705\pi$
$242$		0			0
$243$	0.182259	+	15.5874i	0.0116919	+	0.999932i
$244$		0			0
$245$	−8.27019	−	13.2892i	−0.528363	−	0.849019i
$246$		0			0
$247$		−	0.311894i		−	0.0198453i
$248$		0			0
$249$	−9.77323	−	29.8413i	−0.619353	−	1.89112i
$250$		0			0
$251$	30.5862			1.93058			0.965292	−	0.261172i	$-0.0841091\pi$
$251$	30.5862			1.93058			0.965292	+	0.261172i	$0.0841091\pi$
$252$		0			0
$253$			2.56177i			0.161057i
$254$		0			0
$255$	9.28470	+	2.92398i	0.581430	+	0.183107i
$256$		0			0
$257$			17.6305i			1.09976i	0.835243	+	0.549881i	$0.185327\pi$
$257$			17.6305i			1.09976i	−0.835243	+	0.549881i	$0.814673\pi$
$258$		0			0
$259$	3.37985	+	5.05741i	0.210014	+	0.314252i
$260$		0			0
$261$	2.21886	+	3.02415i	0.137344	+	0.187190i
$262$		0			0
$263$	27.0858			1.67018			0.835090	−	0.550113i	$-0.185416\pi$
$263$	27.0858			1.67018			0.835090	+	0.550113i	$0.185416\pi$
$264$		0			0
$265$	11.1887	−	15.6198i	0.687315	−	0.959519i
$266$		0			0
$267$	−1.57100	+	0.514514i	−0.0961437	+	0.0314877i
$268$		0			0
$269$	12.4642			0.759954			0.379977	−	0.924996i	$-0.375932\pi$
$269$	12.4642			0.759954			0.379977	+	0.924996i	$0.375932\pi$
$270$		0			0
$271$		−	11.1528i		−	0.677484i	−0.940879	−	0.338742i	$-0.889999\pi$
$271$		−	11.1528i		−	0.677484i	0.940879	−	0.338742i	$-0.110001\pi$
$272$		0			0
$273$	0.444939	−	0.567222i	0.0269290	−	0.0343298i
$274$		0			0
$275$	4.53741	−	1.54210i	0.273616	−	0.0929922i
$276$		0			0
$277$		−	26.2365i		−	1.57640i	−0.615422	−	0.788198i	$-0.711014\pi$
$277$		−	26.2365i		−	1.57640i	0.615422	−	0.788198i	$-0.288986\pi$
$278$		0			0
$279$	−15.3831	−	20.9660i	−0.920959	−	1.25520i
$280$		0			0
$281$			14.9648i			0.892722i	0.894853	+	0.446361i	$0.147280\pi$
$281$			14.9648i			0.892722i	−0.894853	+	0.446361i	$0.852720\pi$
$282$		0			0
$283$	0.613047			0.0364418			0.0182209	−	0.999834i	$-0.494200\pi$
$283$	0.613047			0.0364418			0.0182209	+	0.999834i	$0.494200\pi$
$284$		0			0
$285$	7.32397	+	2.30650i	0.433835	+	0.136625i
$286$		0			0
$287$	10.4379	−	6.97564i	0.616132	−	0.411759i
$288$		0			0
$289$	10.6830			0.628411
$290$		0			0
$291$	−23.1937	+	7.59608i	−1.35964	+	0.445290i
$292$		0			0
$293$		−	8.76313i		−	0.511947i	−0.966684	−	0.255974i	$-0.917604\pi$
$293$		−	8.76313i		−	0.511947i	0.966684	−	0.255974i	$-0.0823960\pi$
$294$		0			0
$295$	13.6056	−	18.9940i	0.792152	−	1.10587i
$296$		0			0
$297$	−4.04959	−	2.89902i	−0.234981	−	0.168218i
$298$		0			0
$299$	0.420472			0.0243165
$300$		0			0
$301$	9.67936	+	14.4836i	0.557909	+	0.834822i
$302$		0			0
$303$	14.6063	−	4.78367i	0.839111	−	0.274815i
$304$		0			0
$305$	−7.77585	−	5.56993i	−0.445244	−	0.318933i
$306$		0			0
$307$	−17.8865			−1.02084			−0.510418	−	0.859926i	$-0.670509\pi$
$307$	−17.8865			−1.02084			−0.510418	+	0.859926i	$0.670509\pi$
$308$		0			0
$309$	−3.72474	+	1.21988i	−0.211893	+	0.0693965i
$310$		0			0
$311$	−12.9594			−0.734859			−0.367430	−	0.930051i	$-0.619762\pi$
$311$	−12.9594			−0.734859			−0.367430	+	0.930051i	$0.619762\pi$
$312$		0			0
$313$	−26.4080			−1.49267			−0.746334	−	0.665572i	$-0.768188\pi$
$313$	−26.4080			−1.49267			−0.746334	+	0.665572i	$0.768188\pi$
$314$		0			0
$315$	10.0293	+	14.6429i	0.565085	+	0.825033i
$316$		0			0
$317$	−18.5973			−1.04453			−0.522263	−	0.852785i	$-0.674912\pi$
$317$	−18.5973			−1.04453			−0.522263	+	0.852785i	$0.674912\pi$
$318$		0			0
$319$	−1.19835			−0.0670945
$320$		0			0
$321$	26.7770	−	8.76967i	1.49455	−	0.489475i
$322$		0			0
$323$	−4.98300			−0.277262
$324$		0			0
$325$	−0.253110	−	0.744741i	−0.0140400	−	0.0413108i
$326$		0			0
$327$	22.0746	−	7.22958i	1.22073	−	0.399797i
$328$		0			0
$329$	8.24314	+	12.3345i	0.454459	+	0.680025i
$330$		0			0
$331$	−1.45274			−0.0798496			−0.0399248	−	0.999203i	$-0.512712\pi$
$331$	−1.45274			−0.0798496			−0.0399248	+	0.999203i	$0.512712\pi$
$332$		0			0
$333$	−4.08017	−	5.56100i	−0.223592	−	0.304741i
$334$		0			0
$335$	−25.1371	−	18.0060i	−1.37339	−	0.983775i
$336$		0			0
$337$			13.2863i			0.723753i	0.932226	+	0.361876i	$0.117864\pi$
$337$			13.2863i			0.723753i	−0.932226	+	0.361876i	$0.882136\pi$
$338$		0			0
$339$	−24.7370	+	8.10155i	−1.34353	+	0.440016i
$340$		0			0
$341$	8.30797			0.449902
$342$		0			0
$343$	3.61760	+	18.1635i	0.195332	+	0.980737i
$344$		0			0
$345$	−3.10946	+	9.87365i	−0.167407	+	0.531579i
$346$		0			0
$347$	3.62840			0.194783			0.0973914	−	0.995246i	$-0.468950\pi$
$347$	3.62840			0.194783			0.0973914	+	0.995246i	$0.468950\pi$
$348$		0			0
$349$		−	2.91682i		−	0.156134i	−0.996948	−	0.0780670i	$-0.975125\pi$
$349$		−	2.91682i		−	0.156134i	0.996948	−	0.0780670i	$-0.0248748\pi$
$350$		0			0
$351$	−0.475825	+	0.664673i	−0.0253977	+	0.0354776i
$352$		0			0
$353$			14.2988i			0.761049i	0.924771	+	0.380525i	$0.124257\pi$
$353$			14.2988i			0.761049i	−0.924771	+	0.380525i	$0.875743\pi$
$354$		0			0
$355$	−17.7418	−	12.7087i	−0.941638	−	0.674506i
$356$		0			0
$357$	−9.06228	−	7.10862i	−0.479627	−	0.376228i
$358$		0			0
$359$		−	5.66998i		−	0.299250i	−0.988743	−	0.149625i	$-0.952193\pi$
$359$		−	5.66998i		−	0.299250i	0.988743	−	0.149625i	$-0.0478067\pi$
$360$		0			0
$361$	15.0693			0.793121
$362$		0			0
$363$	−16.5941	+	5.43469i	−0.870966	+	0.285247i
$364$		0			0
$365$	−5.77459	+	8.06156i	−0.302256	+	0.421961i
$366$		0			0
$367$	1.46586			0.0765170			0.0382585	−	0.999268i	$-0.487819\pi$
$367$	1.46586			0.0765170			0.0382585	+	0.999268i	$0.487819\pi$
$368$		0			0
$369$	−11.4773	+	8.42102i	−0.597483	+	0.438381i
$370$		0			0
$371$	−18.9015	+	12.6319i	−0.981320	+	0.655813i
$372$		0			0
$373$			4.24635i			0.219868i	0.993939	+	0.109934i	$0.0350639\pi$
$373$			4.24635i			0.219868i	−0.993939	+	0.109934i	$0.964936\pi$
$374$		0			0
$375$	19.3600	−	0.436136i	0.999746	−	0.0225220i
$376$		0			0
$377$			0.196689i			0.0101300i
$378$		0			0
$379$	28.6426			1.47127			0.735636	−	0.677377i	$-0.236883\pi$
$379$	28.6426			1.47127			0.735636	+	0.677377i	$0.236883\pi$
$380$		0			0
$381$	1.82890	+	5.58431i	0.0936975	+	0.286093i
$382$		0			0
$383$		−	19.9108i		−	1.01740i	−0.860945	−	0.508698i	$-0.830127\pi$
$383$		−	19.9108i		−	1.01740i	0.860945	−	0.508698i	$-0.169873\pi$
$384$		0			0
$385$	−5.66735	−	0.184023i	−0.288835	−	0.00937868i
$386$		0			0
$387$	−11.6850	−	15.9258i	−0.593980	−	0.809554i
$388$		0			0
$389$		−	14.0461i		−	0.712168i	−0.934454	−	0.356084i	$-0.884112\pi$
$389$		−	14.0461i		−	0.712168i	0.934454	−	0.356084i	$-0.115888\pi$
$390$		0			0
$391$		−	6.71772i		−	0.339730i
$392$		0			0
$393$	0.639788	−	0.209535i	0.0322730	−	0.0105696i
$394$		0			0
$395$	0.674358	−	0.941430i	0.0339306	−	0.0473685i
$396$		0			0
$397$	−34.1172			−1.71230			−0.856148	−	0.516731i	$-0.827149\pi$
$397$	−34.1172			−1.71230			−0.856148	+	0.516731i	$0.827149\pi$
$398$		0			0
$399$	−7.14852	−	5.60744i	−0.357874	−	0.280723i
$400$		0			0
$401$			29.9190i			1.49408i	0.664778	+	0.747041i	$0.268526\pi$
$401$			29.9190i			1.49408i	−0.664778	+	0.747041i	$0.731474\pi$
$402$		0			0
$403$		−	1.36362i		−	0.0679266i
$404$		0			0
$405$	−12.0892	−	16.0888i	−0.600719	−	0.799460i
$406$		0			0
$407$	2.20359			0.109228
$408$		0			0
$409$		−	14.1069i		−	0.697540i	−0.937208	−	0.348770i	$-0.886599\pi$
$409$		−	14.1069i		−	0.697540i	0.937208	−	0.348770i	$-0.113401\pi$
$410$		0			0
$411$	15.3513	−	5.02765i	0.757222	−	0.247996i
$412$		0			0
$413$	−22.9846	+	15.3606i	−1.13100	+	0.755845i
$414$		0			0
$415$	32.9559	+	23.6067i	1.61774	+	1.15881i
$416$		0			0
$417$	−7.83404	−	23.9202i	−0.383635	−	1.17138i
$418$		0			0
$419$	15.5078			0.757606			0.378803	−	0.925477i	$-0.376336\pi$
$419$	15.5078			0.757606			0.378803	+	0.925477i	$0.376336\pi$
$420$		0			0
$421$	6.90788			0.336669			0.168335	−	0.985730i	$-0.446161\pi$
$421$	6.90788			0.336669			0.168335	+	0.985730i	$0.446161\pi$
$422$		0			0
$423$	−9.95115	−	13.5627i	−0.483841	−	0.659443i
$424$		0			0
$425$	−11.8984	+	4.04384i	−0.577159	+	0.196155i
$426$		0			0
$427$	6.28837	+	9.40954i	0.304316	+	0.455360i
$428$		0			0
$429$	−0.0812835	−	0.248188i	−0.00392440	−	0.0119827i
$430$		0			0
$431$		−	10.6636i		−	0.513646i	−0.966458	−	0.256823i	$-0.917324\pi$
$431$		−	10.6636i		−	0.513646i	0.966458	−	0.256823i	$-0.0826757\pi$
$432$		0			0
$433$	−28.7128			−1.37985			−0.689924	−	0.723882i	$-0.742356\pi$
$433$	−28.7128			−1.37985			−0.689924	+	0.723882i	$0.742356\pi$
$434$		0			0
$435$	−4.61870	−	1.45454i	−0.221450	−	0.0697399i
$436$		0			0
$437$		−	5.29909i		−	0.253490i
$438$		0			0
$439$		−	30.4819i		−	1.45482i	−0.686203	−	0.727410i	$-0.740724\pi$
$439$		−	30.4819i		−	1.45482i	0.686203	−	0.727410i	$-0.259276\pi$
$440$		0			0
$441$	−5.00115	−	20.3958i	−0.238150	−	0.971228i
$442$		0			0
$443$	−4.91751			−0.233638			−0.116819	−	0.993153i	$-0.537270\pi$
$443$	−4.91751			−0.233638			−0.116819	+	0.993153i	$0.537270\pi$
$444$		0			0
$445$	1.24278	−	1.73497i	0.0589134	−	0.0822454i
$446$		0			0
$447$	−5.92304	−	18.0852i	−0.280150	−	0.855402i
$448$		0			0
$449$			37.5082i			1.77012i	0.465477	+	0.885060i	$0.345883\pi$
$449$			37.5082i			1.77012i	−0.465477	+	0.885060i	$0.654117\pi$
$450$		0			0
$451$		−	4.54797i		−	0.214155i
$452$		0			0
$453$	33.8234	−	11.0774i	1.58916	−	0.520462i
$454$		0			0
$455$	−0.0302043	+	0.930201i	−0.00141600	+	0.0436085i
$456$		0			0
$457$		−	5.50935i		−	0.257717i	−0.991663	−	0.128858i	$-0.958869\pi$
$457$		−	5.50935i		−	0.257717i	0.991663	−	0.128858i	$-0.0411313\pi$
$458$		0			0
$459$	10.6192	+	7.60207i	0.495663	+	0.354834i
$460$		0			0
$461$	31.2515			1.45553			0.727763	−	0.685828i	$-0.240560\pi$
$461$	31.2515			1.45553			0.727763	+	0.685828i	$0.240560\pi$
$462$		0			0
$463$		−	8.75214i		−	0.406746i	−0.979101	−	0.203373i	$-0.934810\pi$
$463$		−	8.75214i		−	0.406746i	0.979101	−	0.203373i	$-0.0651905\pi$
$464$		0			0
$465$	32.0208	+	10.0841i	1.48493	+	0.467641i
$466$		0			0
$467$			0.527427i			0.0244064i	0.999926	+	0.0122032i	$0.00388450\pi$
$467$			0.527427i			0.0244064i	−0.999926	+	0.0122032i	$0.996116\pi$
$468$		0			0
$469$	20.3285	+	30.4184i	0.938685	+	1.40459i
$470$		0			0
$471$	−1.44202	+	0.472273i	−0.0664450	+	0.0217612i
$472$		0			0
$473$	6.31073			0.290168
$474$		0			0
$475$	−9.38575	+	3.18987i	−0.430648	+	0.146361i
$476$		0			0
$477$	20.7836	−	15.2492i	0.951618	−	0.698214i
$478$		0			0
$479$	20.2855			0.926869			0.463435	−	0.886131i	$-0.346617\pi$
$479$	20.2855			0.926869			0.463435	+	0.886131i	$0.346617\pi$
$480$		0			0
$481$		−	0.361683i		−	0.0164913i
$482$		0			0
$483$	7.55954	−	9.63712i	0.343971	−	0.438504i
$484$		0			0
$485$	18.3479	−	25.6144i	0.833136	−	1.16309i
$486$		0			0
$487$			10.5659i			0.478787i	0.970923	+	0.239393i	$0.0769485\pi$
$487$			10.5659i			0.478787i	−0.970923	+	0.239393i	$0.923051\pi$
$488$		0			0
$489$	−11.8336	−	36.1324i	−0.535134	−	1.63396i
$490$		0			0
$491$		−	38.7844i		−	1.75032i	−0.483836	−	0.875159i	$-0.660757\pi$
$491$		−	38.7844i		−	1.75032i	0.483836	−	0.875159i	$-0.339243\pi$
$492$		0			0
$493$	3.14242			0.141527
$494$		0			0
$495$	6.42913	−	0.0733297i	0.288968	−	0.00329593i
$496$		0			0
$497$	14.3479	+	21.4693i	0.643592	+	0.963032i
$498$		0			0
$499$	5.89045			0.263693			0.131846	−	0.991270i	$-0.457909\pi$
$499$	5.89045			0.263693			0.131846	+	0.991270i	$0.457909\pi$
$500$		0			0
$501$	6.32281	+	19.3059i	0.282482	+	0.862523i
$502$		0			0
$503$		−	14.4764i		−	0.645471i	−0.946489	−	0.322736i	$-0.895398\pi$
$503$		−	14.4764i		−	0.645471i	0.946489	−	0.322736i	$-0.104602\pi$
$504$		0			0
$505$	−11.5547	+	16.1308i	−0.514177	+	0.717811i
$506$		0			0
$507$	21.3576	−	6.99475i	0.948522	−	0.310648i
$508$		0			0
$509$	29.2013			1.29433			0.647163	−	0.762351i	$-0.275955\pi$
$509$	29.2013			1.29433			0.647163	+	0.762351i	$0.275955\pi$
$510$		0			0
$511$	9.75528	−	6.51943i	0.431548	−	0.288403i
$512$		0			0
$513$	8.37668	+	5.99668i	0.369839	+	0.264760i
$514$		0			0
$515$	2.94655	−	4.11350i	0.129841	−	0.181262i
$516$		0			0
$517$	5.37435			0.236364
$518$		0			0
$519$	−10.1768	−	31.0735i	−0.446712	−	1.36398i
$520$		0			0
$521$	−39.4599			−1.72877			−0.864385	−	0.502830i	$-0.832292\pi$
$521$	−39.4599			−1.72877			−0.864385	+	0.502830i	$0.832292\pi$
$522$		0			0
$523$	−16.9475			−0.741061			−0.370531	−	0.928820i	$-0.620824\pi$
$523$	−16.9475			−0.741061			−0.370531	+	0.928820i	$0.620824\pi$
$524$		0			0
$525$	−21.6199	−	7.58824i	−0.943568	−	0.331178i
$526$		0			0
$527$	−21.7860			−0.949011
$528$		0			0
$529$	−15.8562			−0.689398
$530$		0			0
$531$	25.2733	−	18.5433i	1.09677	−	0.804713i
$532$		0			0
$533$	−0.746473			−0.0323333
$534$		0			0
$535$	−21.1826	+	29.5718i	−0.915805	+	1.27850i
$536$		0			0
$537$	−6.64352	−	20.2851i	−0.286689	−	0.875367i
$538$		0			0
$539$	6.19894	+	2.56649i	0.267007	+	0.110547i
$540$		0			0
$541$	20.9754			0.901802			0.450901	−	0.892574i	$-0.351103\pi$
$541$	20.9754			0.901802			0.450901	+	0.892574i	$0.351103\pi$
$542$		0			0
$543$	−9.84873	−	30.0718i	−0.422649	−	1.29050i
$544$		0			0
$545$	−17.4626	+	24.3785i	−0.748017	+	1.04426i
$546$		0			0
$547$			9.09174i			0.388735i	0.980929	+	0.194367i	$0.0622654\pi$
$547$			9.09174i			0.388735i	−0.980929	+	0.194367i	$0.937735\pi$
$548$		0			0
$549$	−7.59135	−	10.3465i	−0.323991	−	0.441577i
$550$		0			0
$551$	2.47881			0.105601
$552$		0			0
$553$	−1.13922	+	0.761340i	−0.0484447	+	0.0323755i
$554$		0			0
$555$	8.49314	+	2.67470i	0.360514	+	0.113535i
$556$		0			0
$557$	−5.77459			−0.244677			−0.122339	−	0.992488i	$-0.539039\pi$
$557$	−5.77459			−0.244677			−0.122339	+	0.992488i	$0.539039\pi$
$558$		0			0
$559$		−	1.03580i		−	0.0438098i
$560$		0			0
$561$	−3.96521	+	1.29863i	−0.167411	+	0.0548284i
$562$		0			0
$563$			0.378414i			0.0159483i	0.999968	+	0.00797413i	$0.00253827\pi$
$563$			0.378414i			0.0159483i	−0.999968	+	0.00797413i	$0.997462\pi$
$564$		0			0
$565$	19.5688	−	27.3188i	0.823267	−	1.14931i
$566$		0			0
$567$	6.67943	+	22.8557i	0.280510	+	0.959851i
$568$		0			0
$569$			25.5314i			1.07033i	0.844747	+	0.535166i	$0.179751\pi$
$569$			25.5314i			1.07033i	−0.844747	+	0.535166i	$0.820249\pi$
$570$		0			0
$571$	−14.5308			−0.608097			−0.304048	−	0.952657i	$-0.598338\pi$
$571$	−14.5308			−0.608097			−0.304048	+	0.952657i	$0.598338\pi$
$572$		0			0
$573$	6.69457	+	20.4410i	0.279670	+	0.853935i
$574$		0			0
$575$	−4.30036	−	12.6532i	−0.179337	−	0.527674i
$576$		0			0
$577$	−15.4901			−0.644863			−0.322432	−	0.946593i	$-0.604500\pi$
$577$	−15.4901			−0.644863			−0.322432	+	0.946593i	$0.604500\pi$
$578$		0			0
$579$	10.8274	+	33.0602i	0.449973	+	1.37393i
$580$		0			0
$581$	−26.6516	−	39.8799i	−1.10569	−	1.65450i
$582$		0			0
$583$			8.23570i			0.341088i
$584$		0			0
$585$	−0.0120359	−	1.05524i	−0.000497621	−	0.0436286i
$586$		0			0
$587$			21.9975i			0.907933i	0.891019	+	0.453967i	$0.149991\pi$
$587$			21.9975i			0.907933i	−0.891019	+	0.453967i	$0.850009\pi$
$588$		0			0
$589$	−17.1852			−0.708106
$590$		0			0
$591$	−22.3458	+	7.31840i	−0.919183	+	0.301039i
$592$		0			0
$593$		−	11.3606i		−	0.466523i	−0.972414	−	0.233261i	$-0.925060\pi$
$593$		−	11.3606i		−	0.466523i	0.972414	−	0.233261i	$-0.0749397\pi$
$594$		0			0
$595$	14.8615	+	0.482562i	0.609260	+	0.0197831i
$596$		0			0
$597$	−2.89054	−	8.82590i	−0.118302	−	0.361220i
$598$		0			0
$599$			35.8691i			1.46557i	0.680460	+	0.732785i	$0.261780\pi$
$599$			35.8691i			1.46557i	−0.680460	+	0.732785i	$0.738220\pi$
$600$		0			0
$601$			22.9788i			0.937326i	0.883377	+	0.468663i	$0.155264\pi$
$601$			22.9788i			0.937326i	−0.883377	+	0.468663i	$0.844736\pi$
$602$		0			0
$603$	−24.5407	−	33.4473i	−0.999374	−	1.36208i
$604$		0			0
$605$	13.1272	−	18.3261i	0.533696	−	0.745061i
$606$		0			0
$607$	3.58776			0.145623			0.0728114	−	0.997346i	$-0.476803\pi$
$607$	3.58776			0.145623			0.0728114	+	0.997346i	$0.476803\pi$
$608$		0			0
$609$	4.50805	+	3.53620i	0.182676	+	0.143294i
$610$		0			0
$611$		−	0.882110i		−	0.0356864i
$612$		0			0
$613$			20.9840i			0.847535i	0.905771	+	0.423768i	$0.139293\pi$
$613$			20.9840i			0.847535i	−0.905771	+	0.423768i	$0.860707\pi$
$614$		0			0
$615$	5.52028	−	17.5289i	0.222599	−	0.706833i
$616$		0			0
$617$	34.3786			1.38403			0.692015	−	0.721883i	$-0.256723\pi$
$617$	34.3786			1.38403			0.692015	+	0.721883i	$0.256723\pi$
$618$		0			0
$619$			6.34530i			0.255039i	0.991836	+	0.127520i	$0.0407016\pi$
$619$			6.34530i			0.255039i	−0.991836	+	0.127520i	$0.959298\pi$
$620$		0			0
$621$	−8.08429	+	11.2928i	−0.324411	+	0.453166i
$622$		0			0
$623$	−2.09948	+	1.40308i	−0.0841140	+	0.0562132i
$624$		0			0
$625$	−19.8227	+	15.2336i	−0.792906	+	0.609344i
$626$		0			0
$627$	−3.12785	+	1.02439i	−0.124914	+	0.0409102i
$628$		0			0
$629$	−5.77847			−0.230403
$630$		0			0
$631$	15.4326			0.614363			0.307181	−	0.951651i	$-0.400614\pi$
$631$	15.4326			0.614363			0.307181	+	0.951651i	$0.400614\pi$
$632$		0			0
$633$	28.6425	−	9.38061i	1.13844	−	0.372846i
$634$		0			0
$635$	−6.16715	−	4.41761i	−0.244736	−	0.175307i
$636$		0			0
$637$	0.421247	−	1.01745i	0.0166904	−	0.0403130i
$638$		0			0
$639$	−17.3208	−	23.6071i	−0.685202	−	0.933883i
$640$		0			0
$641$		−	23.5106i		−	0.928614i	−0.885674	−	0.464307i	$-0.846304\pi$
$641$		−	23.5106i		−	0.928614i	0.885674	−	0.464307i	$-0.153696\pi$
$642$		0			0
$643$	23.9541			0.944658			0.472329	−	0.881422i	$-0.343413\pi$
$643$	23.9541			0.944658			0.472329	+	0.881422i	$0.343413\pi$
$644$		0			0
$645$	24.3230	+	7.65991i	0.957717	+	0.301609i
$646$		0			0
$647$			29.3039i			1.15205i	0.817431	+	0.576027i	$0.195398\pi$
$647$			29.3039i			1.15205i	−0.817431	+	0.576027i	$0.804602\pi$
$648$		0			0
$649$			10.0148i			0.393114i
$650$		0			0
$651$	−31.2537	−	24.5160i	−1.22493	−	0.960859i
$652$		0			0
$653$	−38.8097			−1.51874			−0.759370	−	0.650659i	$-0.774493\pi$
$653$	−38.8097			−1.51874			−0.759370	+	0.650659i	$0.774493\pi$
$654$		0			0
$655$	−0.506120	+	0.706564i	−0.0197758	+	0.0276077i
$656$		0			0
$657$	−10.7267	+	7.87028i	−0.418487	+	0.307049i
$658$		0			0
$659$			30.2584i			1.17870i	0.807878	+	0.589349i	$0.200616\pi$
$659$			30.2584i			1.17870i	−0.807878	+	0.589349i	$0.799384\pi$
$660$		0			0
$661$		−	18.7051i		−	0.727542i	−0.931488	−	0.363771i	$-0.881489\pi$
$661$		−	18.7051i		−	0.727542i	0.931488	−	0.363771i	$-0.118511\pi$
$662$		0			0
$663$	0.213149	+	0.650823i	0.00827803	+	0.0252759i
$664$		0			0
$665$	11.7230	+	0.380656i	0.454600	+	0.0147612i
$666$		0			0
$667$			3.34175i			0.129393i
$668$		0			0
$669$	−25.9732	+	8.50639i	−1.00418	+	0.328876i
$670$		0			0
$671$	4.09988			0.158274
$672$		0			0
$673$		−	22.1399i		−	0.853432i	−0.904386	−	0.426716i	$-0.859670\pi$
$673$		−	22.1399i		−	0.853432i	0.904386	−	0.426716i	$-0.140330\pi$
$674$		0			0
$675$	24.8683	+	7.52100i	0.957183	+	0.289483i
$676$		0			0
$677$		−	45.2632i		−	1.73961i	−0.493400	−	0.869803i	$-0.664246\pi$
$677$		−	45.2632i		−	1.73961i	0.493400	−	0.869803i	$-0.335754\pi$
$678$		0			0
$679$	−30.9960	+	20.7145i	−1.18952	+	0.794950i
$680$		0			0
$681$	−1.99301	−	6.08540i	−0.0763724	−	0.233193i
$682$		0			0
$683$	4.71527			0.180425			0.0902124	−	0.995923i	$-0.471245\pi$
$683$	4.71527			0.180425			0.0902124	+	0.995923i	$0.471245\pi$
$684$		0			0
$685$	−12.1440	+	16.9535i	−0.463998	+	0.647760i
$686$		0			0
$687$	7.52384	+	22.9731i	0.287052	+	0.876477i
$688$		0			0
$689$	1.35175			0.0514977
$690$		0			0
$691$		−	19.5451i		−	0.743531i	−0.928327	−	0.371766i	$-0.878753\pi$
$691$		−	19.5451i		−	0.743531i	0.928327	−	0.371766i	$-0.121247\pi$
$692$		0			0
$693$	−7.14979	−	2.59911i	−0.271598	−	0.0987319i
$694$		0			0
$695$	26.4168	+	18.9227i	1.00205	+	0.717778i
$696$		0			0
$697$			11.9261i			0.451734i
$698$		0			0
$699$	−27.6263	+	9.04779i	−1.04492	+	0.342219i
$700$		0			0
$701$			27.0912i			1.02322i	0.859217	+	0.511611i	$0.170951\pi$
$701$			27.0912i			1.02322i	−0.859217	+	0.511611i	$0.829049\pi$
$702$		0			0
$703$	−4.55818			−0.171915
$704$		0			0
$705$	20.7140	+	6.52334i	0.780133	+	0.245683i
$706$		0			0
$707$	19.5199	−	13.0451i	0.734120	−	0.490611i
$708$		0			0
$709$	16.3683			0.614723			0.307362	−	0.951593i	$-0.400554\pi$
$709$	16.3683			0.614723			0.307362	+	0.951593i	$0.400554\pi$
$710$		0			0
$711$	1.25266	−	0.919092i	0.0469784	−	0.0344686i
$712$		0			0
$713$		−	23.1679i		−	0.867644i
$714$		0			0
$715$	0.274092	+	0.196336i	0.0102505	+	0.00734254i
$716$		0			0
$717$	9.83401	+	30.0269i	0.367258	+	1.12137i
$718$		0			0
$719$	21.4591			0.800289			0.400145	−	0.916452i	$-0.368960\pi$
$719$	21.4591			0.800289			0.400145	+	0.916452i	$0.368960\pi$
$720$		0			0
$721$	−4.97774	+	3.32661i	−0.185381	+	0.123890i
$722$		0			0
$723$	−2.89298	−	8.83334i	−0.107591	−	0.328515i
$724$		0			0
$725$	5.91891	−	2.01162i	0.219823	−	0.0747097i
$726$		0			0
$727$	14.9291			0.553690			0.276845	−	0.960915i	$-0.410711\pi$
$727$	14.9291			0.553690			0.276845	+	0.960915i	$0.410711\pi$
$728$		0			0
$729$	−8.70291	−	25.5589i	−0.322330	−	0.946627i
$730$		0			0
$731$	−16.5486			−0.612072
$732$		0			0
$733$	14.7630			0.545284			0.272642	−	0.962116i	$-0.412103\pi$
$733$	14.7630			0.545284			0.272642	+	0.962116i	$0.412103\pi$
$734$		0			0
$735$	20.7769	+	17.4161i	0.766369	+	0.642401i
$736$		0			0
$737$	13.2538			0.488209
$738$		0			0
$739$	42.3148			1.55658			0.778288	−	0.627908i	$-0.216088\pi$
$739$	42.3148			1.55658			0.778288	+	0.627908i	$0.216088\pi$
$740$		0			0
$741$	0.168137	+	0.513384i	0.00617666	+	0.0188596i
$742$		0			0
$743$	−38.3503			−1.40693			−0.703467	−	0.710728i	$-0.748366\pi$
$743$	−38.3503			−1.40693			−0.703467	+	0.710728i	$0.748366\pi$
$744$		0			0
$745$	19.9728	+	14.3068i	0.731747	+	0.524160i
$746$		0			0
$747$	32.1739	+	43.8508i	1.17718	+	1.60442i
$748$		0			0
$749$	35.7848	−	23.9149i	1.30755	−	0.873831i
$750$		0			0
$751$	−15.9483			−0.581961			−0.290981	−	0.956729i	$-0.593981\pi$
$751$	−15.9483			−0.581961			−0.290981	+	0.956729i	$0.593981\pi$
$752$		0			0
$753$	−50.3456	+	16.4885i	−1.83469	+	0.600875i
$754$		0			0
$755$	−26.7568	+	37.3536i	−0.973781	+	1.35944i
$756$		0			0
$757$			37.3428i			1.35725i	0.734486	+	0.678624i	$0.237423\pi$
$757$			37.3428i			1.35725i	−0.734486	+	0.678624i	$0.762577\pi$
$758$		0			0
$759$	−1.38101	−	4.21673i	−0.0501275	−	0.153058i
$760$		0			0
$761$	−20.9251			−0.758536			−0.379268	−	0.925287i	$-0.623824\pi$
$761$	−20.9251			−0.758536			−0.379268	+	0.925287i	$0.623824\pi$
$762$		0			0
$763$	29.5004	−	19.7151i	1.06799	−	0.713733i
$764$		0			0
$765$	−16.8591	+	0.192292i	−0.609542	+	0.00695234i
$766$		0			0
$767$	1.64376			0.0593526
$768$		0			0
$769$			22.7908i			0.821856i	0.911668	+	0.410928i	$0.134795\pi$
$769$			22.7908i			0.821856i	−0.911668	+	0.410928i	$0.865205\pi$
$770$		0			0
$771$	−9.50433	−	29.0202i	−0.342290	−	1.04514i
$772$		0			0
$773$		−	35.5488i		−	1.27860i	−0.768958	−	0.639300i	$-0.779224\pi$
$773$		−	35.5488i		−	1.27860i	0.768958	−	0.639300i	$-0.220776\pi$
$774$		0			0
$775$	−41.0350	+	13.9463i	−1.47402	+	0.500966i
$776$		0			0
$777$	−8.28968	−	6.50259i	−0.297391	−	0.233279i
$778$		0			0
$779$			9.40758i			0.337062i
$780$		0			0
$781$	9.35452			0.334731
$782$		0			0
$783$	−5.28256	−	3.78167i	−0.188783	−	0.135146i
$784$		0			0
$785$	1.14075	−	1.59253i	0.0407151	−	0.0568398i
$786$		0			0
$787$	−3.01968			−0.107640			−0.0538201	−	0.998551i	$-0.517140\pi$
$787$	−3.01968			−0.107640			−0.0538201	+	0.998551i	$0.517140\pi$
$788$		0			0
$789$	−44.5838	+	14.6015i	−1.58722	+	0.519827i
$790$		0			0
$791$	−33.0585	+	22.0929i	−1.17543	+	0.785534i
$792$		0			0
$793$		−	0.672928i		−	0.0238964i
$794$		0			0
$795$	−9.99642	+	31.7422i	−0.354536	+	1.12578i
$796$		0			0
$797$			14.0365i			0.497198i	0.968607	+	0.248599i	$0.0799701\pi$
$797$			14.0365i			0.497198i	−0.968607	+	0.248599i	$0.920030\pi$
$798$		0			0
$799$	−14.0931			−0.498579
$800$		0			0
$801$	2.30854	−	1.69380i	0.0815681	−	0.0598475i
$802$		0			0
$803$		−	4.25053i		−	0.149998i
$804$		0			0
$805$	−0.513173	+	15.8041i	−0.0180870	+	0.557023i
$806$		0			0
$807$	−20.5163	+	6.71923i	−0.722208	+	0.236528i
$808$		0			0
$809$		−	46.9966i		−	1.65231i	−0.563442	−	0.826156i	$-0.690523\pi$
$809$		−	46.9966i		−	1.65231i	0.563442	−	0.826156i	$-0.309477\pi$
$810$		0			0
$811$			52.8375i			1.85537i	0.373358	+	0.927687i	$0.378206\pi$
$811$			52.8375i			1.85537i	−0.373358	+	0.927687i	$0.621794\pi$
$812$		0			0
$813$	6.01229	+	18.3577i	0.210860	+	0.643834i
$814$		0			0
$815$	39.9035	+	28.5834i	1.39776	+	1.00123i
$816$		0			0
$817$	−13.0539			−0.456698
$818$		0			0
$819$	−0.426600	+	1.17352i	−0.0149066	+	0.0410061i
$820$		0			0
$821$		−	53.1985i		−	1.85664i	−0.371783	−	0.928320i	$-0.621253\pi$
$821$		−	53.1985i		−	1.85664i	0.371783	−	0.928320i	$-0.378747\pi$
$822$		0			0
$823$			26.3931i			0.920006i	0.887917	+	0.460003i	$0.152152\pi$
$823$			26.3931i			0.920006i	−0.887917	+	0.460003i	$0.847848\pi$
$824$		0			0
$825$	−6.63736	+	4.98438i	−0.231083	+	0.173534i
$826$		0			0
$827$	−20.3363			−0.707161			−0.353580	−	0.935404i	$-0.615036\pi$
$827$	−20.3363			−0.707161			−0.353580	+	0.935404i	$0.615036\pi$
$828$		0			0
$829$		−	5.14232i		−	0.178600i	−0.996005	−	0.0893000i	$-0.971537\pi$
$829$		−	5.14232i		−	0.178600i	0.996005	−	0.0893000i	$-0.0284630\pi$
$830$		0			0
$831$	14.1436	+	43.1858i	0.490638	+	1.49810i
$832$		0			0
$833$	−16.2554	−	6.73010i	−0.563218	−	0.233184i
$834$		0			0
$835$	−21.3208	−	15.2724i	−0.737838	−	0.528523i
$836$		0			0
$837$	36.6233	+	26.2178i	1.26589	+	0.906221i
$838$		0			0
$839$	−25.1161			−0.867103			−0.433552	−	0.901129i	$-0.642740\pi$
$839$	−25.1161			−0.867103			−0.433552	+	0.901129i	$0.642740\pi$
$840$		0			0
$841$	27.4368			0.946096
$842$		0			0
$843$	−8.06725	−	24.6323i	−0.277851	−	0.848382i
$844$		0			0
$845$	−16.8954	+	23.5867i	−0.581220	+	0.811406i
$846$		0			0
$847$	−22.1764	+	14.8204i	−0.761989	+	0.509235i
$848$		0			0
$849$	−1.00909	+	0.330483i	−0.0346318	+	0.0113422i
$850$		0			0
$851$		−	6.14501i		−	0.210648i
$852$		0			0
$853$	24.0408			0.823141			0.411570	−	0.911378i	$-0.364980\pi$
$853$	24.0408			0.823141			0.411570	+	0.911378i	$0.364980\pi$
$854$		0			0
$855$	−13.2988	+	0.151684i	−0.454810	+	0.00518749i
$856$		0			0
$857$		−	52.0141i		−	1.77677i	−0.459102	−	0.888384i	$-0.651829\pi$
$857$		−	52.0141i		−	1.77677i	0.459102	−	0.888384i	$-0.348171\pi$
$858$		0			0
$859$			19.8217i			0.676309i	0.941091	+	0.338154i	$0.109803\pi$
$859$			19.8217i			0.676309i	−0.941091	+	0.338154i	$0.890197\pi$
$860$		0			0
$861$	−13.4206	+	17.1090i	−0.457373	+	0.583072i
$862$		0			0
$863$	−27.2461			−0.927468			−0.463734	−	0.885974i	$-0.653491\pi$
$863$	−27.2461			−0.927468			−0.463734	+	0.885974i	$0.653491\pi$
$864$		0			0
$865$	34.3167	+	24.5815i	1.16680	+	0.835795i
$866$		0			0
$867$	−17.5844	+	5.75902i	−0.597198	+	0.195587i
$868$		0			0
$869$			0.496377i			0.0168384i
$870$		0			0
$871$		−	2.17539i		−	0.0737102i
$872$		0			0
$873$	34.0823	−	25.0066i	1.15351	−	0.846346i
$874$		0			0
$875$	28.3012	−	8.60464i	0.956757	−	0.290890i
$876$		0			0
$877$		−	21.6226i		−	0.730145i	−0.930979	−	0.365072i	$-0.881044\pi$
$877$		−	21.6226i		−	0.730145i	0.930979	−	0.365072i	$-0.118956\pi$
$878$		0			0
$879$	4.72406	+	14.4243i	0.159339	+	0.486520i
$880$		0			0
$881$	38.4204			1.29442			0.647209	−	0.762313i	$-0.275936\pi$
$881$	38.4204			1.29442			0.647209	+	0.762313i	$0.275936\pi$
$882$		0			0
$883$			33.7562i			1.13599i	0.823033	+	0.567993i	$0.192280\pi$
$883$			33.7562i			1.13599i	−0.823033	+	0.567993i	$0.807720\pi$
$884$		0			0
$885$	−12.1558	+	38.5992i	−0.408614	+	1.29750i
$886$		0			0
$887$			37.9418i			1.27396i	0.770879	+	0.636981i	$0.219817\pi$
$887$			37.9418i			1.27396i	−0.770879	+	0.636981i	$0.780183\pi$
$888$		0			0
$889$	4.98741	+	7.46287i	0.167273	+	0.250297i
$890$		0			0
$891$	8.22853	+	2.58877i	0.275666	+	0.0867272i
$892$		0			0
$893$	−11.1170			−0.372015
$894$		0			0
$895$	22.4023	+	16.0470i	0.748826	+	0.536393i
$896$		0			0
$897$	−0.692107	+	0.226670i	−0.0231088	+	0.00756828i
$898$		0			0
$899$	10.8375			0.361450
$900$		0			0
$901$		−	21.5964i		−	0.719481i
$902$		0			0
$903$	−23.7403	−	18.6224i	−0.790029	−	0.619713i
$904$		0			0
$905$	33.2104	+	23.7890i	1.10395	+	0.790774i
$906$		0			0
$907$			5.51100i			0.182990i	0.995806	+	0.0914949i	$0.0291645\pi$
$907$			5.51100i			0.182990i	−0.995806	+	0.0914949i	$0.970835\pi$
$908$		0			0
$909$	−21.4635	+	15.7481i	−0.711900	+	0.522330i
$910$		0			0
$911$			32.8419i			1.08810i	0.839052	+	0.544051i	$0.183110\pi$
$911$			32.8419i			1.08810i	−0.839052	+	0.544051i	$0.816890\pi$
$912$		0			0
$913$	−17.3763			−0.575071
$914$		0			0
$915$	15.8019	+	4.97640i	0.522394	+	0.164515i
$916$		0			0
$917$	0.855012	−	0.571402i	0.0282350	−	0.0188694i
$918$		0			0
$919$	−44.1954			−1.45787			−0.728935	−	0.684583i	$-0.759984\pi$
$919$	−44.1954			−1.45787			−0.728935	+	0.684583i	$0.759984\pi$
$920$		0			0
$921$	29.4416	−	9.64232i	0.970133	−	0.317725i
$922$		0			0
$923$		−	1.53539i		−	0.0505380i
$924$		0			0
$925$	−10.8840	+	3.69909i	−0.357865	+	0.121625i
$926$		0			0
$927$	5.47339	−	4.01590i	0.179770	−	0.131899i
$928$		0			0
$929$	5.66334			0.185808			0.0929040	−	0.995675i	$-0.470385\pi$
$929$	5.66334			0.185808			0.0929040	+	0.995675i	$0.470385\pi$
$930$		0			0
$931$	−12.8227	−	5.30885i	−0.420246	−	0.173991i
$932$		0			0
$933$	21.3314	−	6.98619i	0.698360	−	0.228718i
$934$		0			0
$935$	3.13678	−	4.37906i	0.102584	−	0.143211i
$936$		0			0
$937$	45.8981			1.49943			0.749713	−	0.661763i	$-0.230191\pi$
$937$	45.8981			1.49943			0.749713	+	0.661763i	$0.230191\pi$
$938$		0			0
$939$	43.4681	−	14.2361i	1.41853	−	0.464578i
$940$		0			0
$941$	−50.0654			−1.63209			−0.816043	−	0.577991i	$-0.803837\pi$
$941$	−50.0654			−1.63209			−0.816043	+	0.577991i	$0.803837\pi$
$942$		0			0
$943$	−12.6826			−0.413003
$944$		0			0
$945$	−24.4021	−	18.6959i	−0.793801	−	0.608177i
$946$		0			0
$947$	27.8322			0.904426			0.452213	−	0.891910i	$-0.350635\pi$
$947$	27.8322			0.904426			0.452213	+	0.891910i	$0.350635\pi$
$948$		0			0
$949$	−0.697653			−0.0226468
$950$		0			0
$951$	30.6115	−	10.0255i	0.992645	−	0.325098i
$952$		0			0
$953$	−17.3099			−0.560721			−0.280361	−	0.959895i	$-0.590454\pi$
$953$	−17.3099			−0.560721			−0.280361	+	0.959895i	$0.590454\pi$
$954$		0			0
$955$	−22.5744	−	16.1703i	−0.730492	−	0.523260i
$956$		0			0
$957$	1.97250	−	0.646009i	0.0637620	−	0.0208825i
$958$		0			0
$959$	20.5154	−	13.7104i	0.662477	−	0.442732i
$960$		0			0
$961$	−44.1348			−1.42370
$962$		0			0
$963$	−39.3480	+	28.8701i	−1.26797	+	0.930327i
$964$		0			0
$965$	−36.5107	−	26.1531i	−1.17532	−	0.841897i
$966$		0			0
$967$		−	19.0313i		−	0.612006i	−0.952031	−	0.306003i	$-0.901008\pi$
$967$		−	19.0313i		−	0.612006i	0.952031	−	0.306003i	$-0.0989917\pi$
$968$		0			0
$969$	8.20213	−	2.68626i	0.263490	−	0.0862950i
$970$		0			0
$971$	−0.307918			−0.00988156			−0.00494078	−	0.999988i	$-0.501573\pi$
$971$	−0.307918			−0.00988156			−0.00494078	+	0.999988i	$0.501573\pi$
$972$		0			0
$973$	−21.3634	−	31.9670i	−0.684880	−	1.02481i
$974$		0			0
$975$	0.818103	+	1.08941i	0.0262003	+	0.0348891i
$976$		0			0
$977$	−33.3493			−1.06694			−0.533469	−	0.845820i	$-0.679112\pi$
$977$	−33.3493			−1.06694			−0.533469	+	0.845820i	$0.679112\pi$
$978$		0			0
$979$			0.914777i			0.0292364i
$980$		0			0
$981$	−32.4379	+	23.8001i	−1.03566	+	0.759879i
$982$		0			0
$983$			38.9932i			1.24369i	0.783140	+	0.621845i	$0.213617\pi$
$983$			38.9932i			1.24369i	−0.783140	+	0.621845i	$0.786383\pi$
$984$		0			0
$985$	17.6772	−	24.6780i	0.563242	−	0.786308i
$986$		0			0
$987$	−20.2177	−	15.8592i	−0.643538	−	0.504803i
$988$		0			0
$989$		−	17.5983i		−	0.559594i
$990$		0			0
$991$	25.8271			0.820424			0.410212	−	0.911990i	$-0.365455\pi$
$991$	25.8271			0.820424			0.410212	+	0.911990i	$0.365455\pi$
$992$		0			0
$993$	2.39124	−	0.783146i	0.0758836	−	0.0248524i
$994$		0			0
$995$	9.74707	+	6.98195i	0.309003	+	0.221343i
$996$		0			0
$997$	4.68987			0.148530			0.0742649	−	0.997239i	$-0.476339\pi$
$997$	4.68987			0.148530			0.0742649	+	0.997239i	$0.476339\pi$
$998$		0			0
$999$	9.71390	+	6.95397i	0.307334	+	0.220014i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	840.2.k.b.209.4	yes	24
3.2	odd	2		840.2.k.a.209.3	✓	24
4.3	odd	2		1680.2.k.h.209.21		24
5.4	even	2		840.2.k.a.209.21	yes	24
7.6	odd	2	inner	840.2.k.b.209.21	yes	24
12.11	even	2		1680.2.k.i.209.22		24
15.14	odd	2	inner	840.2.k.b.209.22	yes	24
20.19	odd	2		1680.2.k.i.209.4		24
21.20	even	2		840.2.k.a.209.22	yes	24
28.27	even	2		1680.2.k.h.209.4		24
35.34	odd	2		840.2.k.a.209.4	yes	24
60.59	even	2		1680.2.k.h.209.3		24
84.83	odd	2		1680.2.k.i.209.3		24
105.104	even	2	inner	840.2.k.b.209.3	yes	24
140.139	even	2		1680.2.k.i.209.21		24
420.419	odd	2		1680.2.k.h.209.22		24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.k.a.209.3	✓	24	3.2	odd	2
840.2.k.a.209.4	yes	24	35.34	odd	2
840.2.k.a.209.21	yes	24	5.4	even	2
840.2.k.a.209.22	yes	24	21.20	even	2
840.2.k.b.209.3	yes	24	105.104	even	2	inner
840.2.k.b.209.4	yes	24	1.1	even	1	trivial
840.2.k.b.209.21	yes	24	7.6	odd	2	inner
840.2.k.b.209.22	yes	24	15.14	odd	2	inner
1680.2.k.h.209.3		24	60.59	even	2
1680.2.k.h.209.4		24	28.27	even	2
1680.2.k.h.209.21		24	4.3	odd	2
1680.2.k.h.209.22		24	420.419	odd	2
1680.2.k.i.209.3		24	84.83	odd	2
1680.2.k.i.209.4		24	20.19	odd	2
1680.2.k.i.209.21		24	140.139	even	2
1680.2.k.i.209.22		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 $\F_{q}$
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)

\(f(q)\)	\(=\)	\(q+(-1.64602 + 0.539084i) q^{3} +(1.30213 - 1.81782i) q^{5} +(-2.19974 + 1.47008i) q^{7} +(2.41878 - 1.77469i) q^{9} +0.958461i q^{11} +0.157315 q^{13} +(-1.16337 + 3.69413i) q^{15} -2.51337i q^{17} -1.98260i q^{19} +(2.82833 - 3.60563i) q^{21} +2.67280 q^{23} +(-1.60893 - 4.73406i) q^{25} +(-3.02466 + 4.22510i) q^{27} +1.25028i q^{29} -8.66804i q^{31} +(-0.516691 - 1.57765i) q^{33} +(-0.191998 + 5.91296i) q^{35} -2.29909i q^{37} +(-0.258945 + 0.0848062i) q^{39} -4.74507 q^{41} -6.58424i q^{43} +(-0.0765078 - 6.70777i) q^{45} -5.60727i q^{47} +(2.67772 - 6.46760i) q^{49} +(1.35492 + 4.13706i) q^{51} +8.59262 q^{53} +(1.74231 + 1.24804i) q^{55} +(1.06879 + 3.26340i) q^{57} +10.4488 q^{59} -4.27757i q^{61} +(-2.71175 + 7.45965i) q^{63} +(0.204845 - 0.285971i) q^{65} -13.8282i q^{67} +(-4.39948 + 1.44086i) q^{69} -9.75994i q^{71} -4.43474 q^{73} +(5.20040 + 6.92502i) q^{75} +(-1.40902 - 2.10837i) q^{77} +0.517890 q^{79} +(2.70097 - 8.58515i) q^{81} +18.1293i q^{83} +(-4.56885 - 3.27272i) q^{85} +(-0.674006 - 2.05799i) q^{87} +0.954423 q^{89} +(-0.346053 + 0.231267i) q^{91} +(4.67280 + 14.2678i) q^{93} +(-3.60401 - 2.58160i) q^{95} +14.0907 q^{97} +(1.70097 + 2.31830i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{9} + 2 q^{15} - 2 q^{21} - 16 q^{23} + 8 q^{25} + 8 q^{35} - 2 q^{39} + 6 q^{51} + 24 q^{53} + 8 q^{57} + 16 q^{63} + 16 q^{65} + 8 q^{77} + 4 q^{79} + 18 q^{81} - 12 q^{85} + 12 q^{91} + 32 q^{93}+ \cdots - 6 q^{99}+O(q^{100}) \) 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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