LMFDB - Embedded newform 1680.2.k.h.209.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(209,1680)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1680, 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("1680.209"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1680.k (of order $2$, degree $1$, not 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:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$24$
Twist minimal:	no (minimal twist has level 840)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.4
Character	$\chi$	$=$	1680.209
Dual form			1680.2.k.h.209.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.64602 + 0.539084i) q^{3} +(-1.30213 + 1.81782i) q^{5} +(-2.19974 - 1.47008i) q^{7} +(2.41878 - 1.77469i) q^{9} -0.958461i q^{11} -0.157315 q^{13} +(1.16337 - 3.69413i) q^{15} +2.51337i q^{17} -1.98260i q^{19} +(4.41332 + 1.23394i) q^{21} -2.67280 q^{23} +(-1.60893 - 4.73406i) q^{25} +(-3.02466 + 4.22510i) q^{27} +1.25028i q^{29} -8.66804i q^{31} +(0.516691 + 1.57765i) q^{33} +(5.53668 - 2.08450i) q^{35} -2.29909i q^{37} +(0.258945 - 0.0848062i) q^{39} +4.74507 q^{41} +6.58424i q^{43} +(0.0765078 + 6.70777i) q^{45} -5.60727i q^{47} +(2.67772 + 6.46760i) q^{49} +(-1.35492 - 4.13706i) q^{51} +8.59262 q^{53} +(1.74231 + 1.24804i) q^{55} +(1.06879 + 3.26340i) q^{57} +10.4488 q^{59} +4.27757i q^{61} +(-7.92962 + 0.348054i) q^{63} +(0.204845 - 0.285971i) q^{65} +13.8282i q^{67} +(4.39948 - 1.44086i) q^{69} +9.75994i q^{71} +4.43474 q^{73} +(5.20040 + 6.92502i) q^{75} +(-1.40902 + 2.10837i) q^{77} -0.517890 q^{79} +(2.70097 - 8.58515i) q^{81} +18.1293i q^{83} +(-4.56885 - 3.27272i) q^{85} +(-0.674006 - 2.05799i) q^{87} -0.954423 q^{89} +(0.346053 + 0.231267i) q^{91} +(4.67280 + 14.2678i) q^{93} +(3.60401 + 2.58160i) q^{95} -14.0907 q^{97} +(-1.70097 - 2.31830i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

$n$	$241$	$337$	$421$	$1121$	$1471$
$\chi(n)$	$-1$	$-1$	$1$	$-1$	$1$

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	$-0.953845\pi$
$11$		−	0.958461i		−	0.288987i	0.989506	−	0.144493i	$-0.0461553\pi$
$12$		0			0
$13$	−0.157315			−0.0436315			−0.0218157	−	0.999762i	$-0.506945\pi$
$13$	−0.157315			−0.0436315			−0.0218157	+	0.999762i	$0.506945\pi$
$14$		0			0
$15$	1.16337	−	3.69413i	0.300381	−	0.953819i
$16$		0			0
$17$			2.51337i			0.609581i	0.952419	+	0.304791i	$0.0985865\pi$
$17$			2.51337i			0.609581i	−0.952419	+	0.304791i	$0.901414\pi$
$18$		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
$21$	4.41332	+	1.23394i	0.963065	+	0.269268i
$22$		0			0
$23$	−2.67280			−0.557317			−0.278658	−	0.960390i	$-0.589890\pi$
$23$	−2.67280			−0.557317			−0.278658	+	0.960390i	$0.589890\pi$
$24$		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
$35$	5.53668	−	2.08450i	0.935870	−	0.352345i
$36$		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	$-0.379177\pi$
$41$	4.74507			0.741056			0.370528	+	0.928821i	$0.379177\pi$
$42$		0			0
$43$			6.58424i			1.00409i	0.864842	+	0.502043i	$0.167418\pi$
$43$			6.58424i			1.00409i	−0.864842	+	0.502043i	$0.832582\pi$
$44$		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	$0.0882949\pi$
$61$			4.27757i			0.547686i	−0.961774	+	0.273843i	$0.911705\pi$
$62$		0			0
$63$	−7.92962	+	0.348054i	−0.999038	+	0.0438506i
$64$		0			0
$65$	0.204845	−	0.285971i	0.0254078	−	0.0354704i
$66$		0			0
$67$			13.8282i			1.68938i	0.535255	+	0.844690i	$0.320215\pi$
$67$			13.8282i			1.68938i	−0.535255	+	0.844690i	$0.679785\pi$
$68$		0			0
$69$	4.39948	−	1.44086i	0.529635	−	0.173459i
$70$		0			0
$71$			9.75994i			1.15829i	0.815224	+	0.579146i	$0.196614\pi$
$71$			9.75994i			1.15829i	−0.815224	+	0.579146i	$0.803386\pi$
$72$		0			0
$73$	4.43474			0.519047			0.259524	−	0.965737i	$-0.416434\pi$
$73$	4.43474			0.519047			0.259524	+	0.965737i	$0.416434\pi$
$74$		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	$-0.509275\pi$
$79$	−0.517890			−0.0582671			−0.0291336	+	0.999576i	$0.509275\pi$
$80$		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	$-0.516108\pi$
$89$	−0.954423			−0.101169			−0.0505843	+	0.998720i	$0.516108\pi$
$90$		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	$-0.753732\pi$
$97$	−14.0907			−1.43070			−0.715348	+	0.698768i	$0.753732\pi$
$98$		0			0
$99$	−1.70097	−	2.31830i	−0.170954	−	0.232998i
$100$		0			0
$101$	8.87371			0.882967			0.441484	−	0.897269i	$-0.354452\pi$
$101$	8.87371			0.882967			0.441484	+	0.897269i	$0.354452\pi$
$102$		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
$105$	−7.98978	+	6.41587i	−0.779723	+	0.626125i
$106$		0			0
$107$	16.2677			1.57266			0.786331	−	0.617806i	$-0.211978\pi$
$107$	16.2677			1.57266			0.786331	+	0.617806i	$0.211978\pi$
$108$		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	$0.0480957\pi$
$127$			3.39261i			0.301046i	−0.988607	+	0.150523i	$0.951904\pi$
$128$		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
$147$	−7.89417	−	9.20229i	−0.651100	−	0.758992i
$148$		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	$-0.184825\pi$
$151$	20.5486			1.67222			0.836109	+	0.548563i	$0.184825\pi$
$152$		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	$-0.511130\pi$
$157$	−0.876066			−0.0699177			−0.0349588	+	0.999389i	$0.511130\pi$
$158$		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	$-0.670663\pi$
$163$		−	21.9513i		−	1.71936i	0.510833	−	0.859680i	$-0.329337\pi$
$164$		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	$-0.745226\pi$
$173$		−	18.8779i		−	1.43526i	0.696423	−	0.717632i	$-0.254774\pi$
$174$		0			0
$175$	−3.42021	+	12.7790i	−0.258544	+	0.966000i
$176$		0			0
$177$	−17.1989	+	5.63278i	−1.29275	+	0.423385i
$178$		0			0
$179$		−	12.3237i		−	0.921118i	−0.887629	−	0.460559i	$-0.847649\pi$
$179$		−	12.3237i		−	0.921118i	0.887629	−	0.460559i	$-0.152351\pi$
$180$		0			0
$181$		−	18.2694i		−	1.35795i	−0.734160	−	0.678976i	$-0.762424\pi$
$181$		−	18.2694i		−	1.35795i	0.734160	−	0.678976i	$-0.237576\pi$
$182$		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
$189$	12.8647	−	4.84763i	0.935769	−	0.352613i
$190$		0			0
$191$			12.4184i			0.898565i	0.893390	+	0.449283i	$0.148320\pi$
$191$			12.4184i			0.898565i	−0.893390	+	0.449283i	$0.851680\pi$
$192$		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	$-0.295578\pi$
$211$	17.4010			1.19794			0.598968	+	0.800773i	$0.295578\pi$
$212$		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	$0.152560\pi$
$229$			13.9567i			0.922286i	−0.887326	+	0.461143i	$0.847440\pi$
$230$		0			0
$231$	1.18269	−	4.23000i	0.0778150	−	0.278313i
$232$		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	$0.200869\pi$
$239$			18.2421i			1.17998i	−0.807410	+	0.589991i	$0.799131\pi$
$240$		0			0
$241$		−	5.36648i		−	0.345685i	−0.984949	−	0.172843i	$-0.944705\pi$
$241$		−	5.36648i		−	0.345685i	0.984949	−	0.172843i	$-0.0552952\pi$
$242$		0			0
$243$	0.182259	+	15.5874i	0.0116919	+	0.999932i
$244$		0			0
$245$	−15.2437	−	3.55401i	−0.973881	−	0.227057i
$246$		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	$-0.814673\pi$
$257$		−	17.6305i		−	1.09976i	0.835243	−	0.549881i	$-0.185327\pi$
$258$		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	$-0.814584\pi$
$263$	−27.0858			−1.67018			−0.835090	+	0.550113i	$0.814584\pi$
$264$		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	$-0.624068\pi$
$269$	−12.4642			−0.759954			−0.379977	+	0.924996i	$0.624068\pi$
$270$		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
$273$	−0.694283	−	0.194118i	−0.0420199	−	0.0117486i
$274$		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	$0.0823960\pi$
$293$			8.76313i			0.511947i	−0.966684	+	0.255974i	$0.917604\pi$
$294$		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	$-0.231812\pi$
$313$	26.4080			1.49267			0.746334	+	0.665572i	$0.231812\pi$
$314$		0			0
$315$	9.69267	−	14.8678i	0.546120	−	0.837707i
$316$		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	$-0.487288\pi$
$331$	1.45274			0.0798496			0.0399248	+	0.999203i	$0.487288\pi$
$332$		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	$-0.531050\pi$
$347$	−3.62840			−0.194783			−0.0973914	+	0.995246i	$0.531050\pi$
$348$		0			0
$349$			2.91682i			0.156134i	0.996948	+	0.0780670i	$0.0248748\pi$
$349$			2.91682i			0.156134i	−0.996948	+	0.0780670i	$0.975125\pi$
$350$		0			0
$351$	0.475825	−	0.664673i	0.0253977	−	0.0354776i
$352$		0			0
$353$		−	14.2988i		−	0.761049i	−0.924771	−	0.380525i	$-0.875743\pi$
$353$		−	14.2988i		−	0.761049i	0.924771	−	0.380525i	$-0.124257\pi$
$354$		0			0
$355$	−17.7418	−	12.7087i	−0.941638	−	0.674506i
$356$		0			0
$357$	−3.10135	+	11.0923i	−0.164141	+	0.587066i
$358$		0			0
$359$			5.66998i			0.299250i	0.988743	+	0.149625i	$0.0478067\pi$
$359$			5.66998i			0.299250i	−0.988743	+	0.149625i	$0.952193\pi$
$360$		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	$-0.763117\pi$
$379$	−28.6426			−1.47127			−0.735636	+	0.677377i	$0.763117\pi$
$380$		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
$385$	−1.99791	−	5.30670i	−0.101823	−	0.270454i
$386$		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	$-0.172851\pi$
$397$	34.1172			1.71230			0.856148	+	0.516731i	$0.172851\pi$
$398$		0			0
$399$	2.44641	−	8.74985i	0.122474	−	0.438040i
$400$		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	$0.113401\pi$
$409$			14.1069i			0.697540i	−0.937208	+	0.348770i	$0.886599\pi$
$410$		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	$0.0826757\pi$
$431$			10.6636i			0.513646i	−0.966458	+	0.256823i	$0.917324\pi$
$432$		0			0
$433$	28.7128			1.37985			0.689924	−	0.723882i	$-0.257644\pi$
$433$	28.7128			1.37985			0.689924	+	0.723882i	$0.257644\pi$
$434$		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
$441$	17.9548	+	10.8916i	0.854989	+	0.518646i
$442$		0			0
$443$	4.91751			0.233638			0.116819	−	0.993153i	$-0.462730\pi$
$443$	4.91751			0.233638			0.116819	+	0.993153i	$0.462730\pi$
$444$		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
$455$	−0.871006	+	0.327924i	−0.0408334	+	0.0153733i
$456$		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	$-0.759440\pi$
$461$	−31.2515			−1.45553			−0.727763	+	0.685828i	$0.759440\pi$
$462$		0			0
$463$			8.75214i			0.406746i	0.979101	+	0.203373i	$0.0651905\pi$
$463$			8.75214i			0.406746i	−0.979101	+	0.203373i	$0.934810\pi$
$464$		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
$483$	−11.7959	−	3.29807i	−0.536732	−	0.150068i
$484$		0			0
$485$	18.3479	−	25.6144i	0.833136	−	1.16309i
$486$		0			0
$487$		−	10.5659i		−	0.478787i	−0.970923	−	0.239393i	$-0.923051\pi$
$487$		−	10.5659i		−	0.478787i	0.970923	−	0.239393i	$-0.0769485\pi$
$488$		0			0
$489$	11.8336	+	36.1324i	0.535134	+	1.63396i
$490$		0			0
$491$			38.7844i			1.75032i	0.483836	+	0.875159i	$0.339243\pi$
$491$			38.7844i			1.75032i	−0.483836	+	0.875159i	$0.660757\pi$
$492$		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	$-0.542091\pi$
$499$	−5.89045			−0.263693			−0.131846	+	0.991270i	$0.542091\pi$
$500$		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	$-0.724045\pi$
$509$	−29.2013			−1.29433			−0.647163	+	0.762351i	$0.724045\pi$
$510$		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	$-0.167708\pi$
$521$	39.4599			1.72877			0.864385	+	0.502830i	$0.167708\pi$
$522$		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
$525$	−1.25919	−	22.8783i	−0.0549555	−	0.998489i
$526$		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	$-0.937735\pi$
$547$		−	9.09174i		−	0.388735i	0.980929	−	0.194367i	$-0.0622654\pi$
$548$		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
$567$	−18.5623	+	14.9145i	−0.779543	+	0.626348i
$568$		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	$-0.401662\pi$
$571$	14.5308			0.608097			0.304048	+	0.952657i	$0.401662\pi$
$572$		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	$-0.395500\pi$
$577$	15.4901			0.644863			0.322432	+	0.946593i	$0.395500\pi$
$578$		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	$0.0749397\pi$
$593$			11.3606i			0.466523i	−0.972414	+	0.233261i	$0.925060\pi$
$594$		0			0
$595$	5.23912	+	13.9157i	0.214783	+	0.570489i
$596$		0			0
$597$	−2.89054	−	8.82590i	−0.118302	−	0.361220i
$598$		0			0
$599$		−	35.8691i		−	1.46557i	−0.680460	−	0.732785i	$-0.738220\pi$
$599$		−	35.8691i		−	1.46557i	0.680460	−	0.732785i	$-0.261780\pi$
$600$		0			0
$601$		−	22.9788i		−	0.937326i	−0.883377	−	0.468663i	$-0.844736\pi$
$601$		−	22.9788i		−	0.937326i	0.883377	−	0.468663i	$-0.155264\pi$
$602$		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
$609$	−1.54277	+	5.51789i	−0.0625164	+	0.223596i
$610$		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	$-0.599386\pi$
$631$	−15.4326			−0.614363			−0.307181	+	0.951651i	$0.599386\pi$
$632$		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
$651$	10.6958	−	38.2548i	0.419203	−	1.49932i
$652$		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	$-0.799384\pi$
$659$		−	30.2584i		−	1.17870i	0.807878	−	0.589349i	$-0.200616\pi$
$660$		0			0
$661$			18.7051i			0.727542i	0.931488	+	0.363771i	$0.118511\pi$
$661$			18.7051i			0.727542i	−0.931488	+	0.363771i	$0.881489\pi$
$662$		0			0
$663$	0.213149	+	0.650823i	0.00827803	+	0.0252759i
$664$		0			0
$665$	−4.13273	−	10.9770i	−0.160260	−	0.425671i
$666$		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	$0.335754\pi$
$677$			45.2632i			1.73961i	−0.493400	+	0.869803i	$0.664246\pi$
$678$		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	$-0.528755\pi$
$683$	−4.71527			−0.180425			−0.0902124	+	0.995923i	$0.528755\pi$
$684$		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
$693$	0.333596	+	7.60023i	0.0126723	+	0.288709i
$694$		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	$-0.587897\pi$
$733$	−14.7630			−0.545284			−0.272642	+	0.962116i	$0.587897\pi$
$734$		0			0
$735$	27.0073	−	2.36762i	0.996179	−	0.0873312i
$736$		0			0
$737$	13.2538			0.488209
$738$		0			0
$739$	−42.3148			−1.55658			−0.778288	−	0.627908i	$-0.783912\pi$
$739$	−42.3148			−1.55658			−0.778288	+	0.627908i	$0.783912\pi$
$740$		0			0
$741$	−0.168137	−	0.513384i	−0.00617666	−	0.0188596i
$742$		0			0
$743$	38.3503			1.40693			0.703467	−	0.710728i	$-0.251634\pi$
$743$	38.3503			1.40693			0.703467	+	0.710728i	$0.251634\pi$
$744$		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	$-0.406019\pi$
$751$	15.9483			0.581961			0.290981	+	0.956729i	$0.406019\pi$
$752$		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	$-0.376176\pi$
$761$	20.9251			0.758536			0.379268	+	0.925287i	$0.376176\pi$
$762$		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	$-0.865205\pi$
$769$		−	22.7908i		−	0.821856i	0.911668	−	0.410928i	$-0.134795\pi$
$770$		0			0
$771$	9.50433	+	29.0202i	0.342290	+	1.04514i
$772$		0			0
$773$			35.5488i			1.27860i	0.768958	+	0.639300i	$0.220776\pi$
$773$			35.5488i			1.27860i	−0.768958	+	0.639300i	$0.779224\pi$
$774$		0			0
$775$	−41.0350	+	13.9463i	−1.47402	+	0.500966i
$776$		0			0
$777$	2.83695	−	10.1466i	0.101775	−	0.364008i
$778$		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	$-0.920030\pi$
$797$		−	14.0365i		−	0.497198i	0.968607	−	0.248599i	$-0.0799701\pi$
$798$		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
$805$	−14.7984	+	5.57145i	−0.521576	+	0.196368i
$806$		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
$819$	1.24745	−	0.0547542i	0.0435895	−	0.00191327i
$820$		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	$-0.847848\pi$
$823$		−	26.3931i		−	0.920006i	0.887917	−	0.460003i	$-0.152152\pi$
$824$		0			0
$825$	6.63736	−	4.98438i	0.231083	−	0.173534i
$826$		0			0
$827$	20.3363			0.707161			0.353580	−	0.935404i	$-0.384964\pi$
$827$	20.3363			0.707161			0.353580	+	0.935404i	$0.384964\pi$
$828$		0			0
$829$			5.14232i			0.178600i	0.996005	+	0.0893000i	$0.0284630\pi$
$829$			5.14232i			0.178600i	−0.996005	+	0.0893000i	$0.971537\pi$
$830$		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	$-0.635020\pi$
$853$	−24.0408			−0.823141			−0.411570	+	0.911378i	$0.635020\pi$
$854$		0			0
$855$	13.2988	−	0.151684i	0.454810	−	0.00518749i
$856$		0			0
$857$			52.0141i			1.77677i	0.459102	+	0.888384i	$0.348171\pi$
$857$			52.0141i			1.77677i	−0.459102	+	0.888384i	$0.651829\pi$
$858$		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
$861$	20.9415	+	5.85514i	0.713685	+	0.199543i
$862$		0			0
$863$	27.2461			0.927468			0.463734	−	0.885974i	$-0.346509\pi$
$863$	27.2461			0.927468			0.463734	+	0.885974i	$0.346509\pi$
$864$		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
$875$	−18.7763	−	22.8572i	−0.634755	−	0.772713i
$876$		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	$-0.724064\pi$
$881$	−38.4204			−1.29442			−0.647209	+	0.762313i	$0.724064\pi$
$882$		0			0
$883$		−	33.7562i		−	1.13599i	−0.823033	−	0.567993i	$-0.807720\pi$
$883$		−	33.7562i		−	1.13599i	0.823033	−	0.567993i	$-0.192280\pi$
$884$		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
$903$	−8.12456	+	29.0583i	−0.270369	+	0.967001i
$904$		0			0
$905$	33.2104	+	23.7890i	1.10395	+	0.790774i
$906$		0			0
$907$		−	5.51100i		−	0.182990i	−0.995806	−	0.0914949i	$-0.970835\pi$
$907$		−	5.51100i		−	0.182990i	0.995806	−	0.0914949i	$-0.0291645\pi$
$908$		0			0
$909$	21.4635	−	15.7481i	0.711900	−	0.522330i
$910$		0			0
$911$		−	32.8419i		−	1.08810i	−0.839052	−	0.544051i	$-0.816890\pi$
$911$		−	32.8419i		−	1.08810i	0.839052	−	0.544051i	$-0.183110\pi$
$912$		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	$-0.240016\pi$
$919$	44.1954			1.45787			0.728935	+	0.684583i	$0.240016\pi$
$920$		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	$-0.529615\pi$
$929$	−5.66334			−0.185808			−0.0929040	+	0.995675i	$0.529615\pi$
$930$		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	$-0.769809\pi$
$937$	−45.8981			−1.49943			−0.749713	+	0.661763i	$0.769809\pi$
$938$		0			0
$939$	−43.4681	+	14.2361i	−1.41853	+	0.464578i
$940$		0			0
$941$	50.0654			1.63209			0.816043	−	0.577991i	$-0.196163\pi$
$941$	50.0654			1.63209			0.816043	+	0.577991i	$0.196163\pi$
$942$		0			0
$943$	−12.6826			−0.413003
$944$		0			0
$945$	−7.93934	+	29.6979i	−0.258267	+	0.966074i
$946$		0			0
$947$	−27.8322			−0.904426			−0.452213	−	0.891910i	$-0.649365\pi$
$947$	−27.8322			−0.904426			−0.452213	+	0.891910i	$0.649365\pi$
$948$		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	$0.0989917\pi$
$967$			19.0313i			0.612006i	−0.952031	+	0.306003i	$0.901008\pi$
$968$		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
$987$	6.91904	−	24.7467i	0.220235	−	0.787695i
$988$		0			0
$989$		−	17.5983i		−	0.559594i
$990$		0			0
$991$	−25.8271			−0.820424			−0.410212	−	0.911990i	$-0.634545\pi$
$991$	−25.8271			−0.820424			−0.410212	+	0.911990i	$0.634545\pi$
$992$		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	$-0.523661\pi$
$997$	−4.68987			−0.148530			−0.0742649	+	0.997239i	$0.523661\pi$
$998$		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	1680.2.k.h.209.4		24
3.2	odd	2		1680.2.k.i.209.3		24
4.3	odd	2		840.2.k.b.209.21	yes	24
5.4	even	2		1680.2.k.i.209.21		24
7.6	odd	2	inner	1680.2.k.h.209.21		24
12.11	even	2		840.2.k.a.209.22	yes	24
15.14	odd	2	inner	1680.2.k.h.209.22		24
20.19	odd	2		840.2.k.a.209.4	yes	24
21.20	even	2		1680.2.k.i.209.22		24
28.27	even	2		840.2.k.b.209.4	yes	24
35.34	odd	2		1680.2.k.i.209.4		24
60.59	even	2		840.2.k.b.209.3	yes	24
84.83	odd	2		840.2.k.a.209.3	✓	24
105.104	even	2	inner	1680.2.k.h.209.3		24
140.139	even	2		840.2.k.a.209.21	yes	24
420.419	odd	2		840.2.k.b.209.22	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.k.a.209.3	✓	24	84.83	odd	2
840.2.k.a.209.4	yes	24	20.19	odd	2
840.2.k.a.209.21	yes	24	140.139	even	2
840.2.k.a.209.22	yes	24	12.11	even	2
840.2.k.b.209.3	yes	24	60.59	even	2
840.2.k.b.209.4	yes	24	28.27	even	2
840.2.k.b.209.21	yes	24	4.3	odd	2
840.2.k.b.209.22	yes	24	420.419	odd	2
1680.2.k.h.209.3		24	105.104	even	2	inner
1680.2.k.h.209.4		24	1.1	even	1	trivial
1680.2.k.h.209.21		24	7.6	odd	2	inner
1680.2.k.h.209.22		24	15.14	odd	2	inner
1680.2.k.i.209.3		24	3.2	odd	2
1680.2.k.i.209.4		24	35.34	odd	2
1680.2.k.i.209.21		24	5.4	even	2
1680.2.k.i.209.22		24	21.20	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:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.64602 + 0.539084i) q^{3} +(-1.30213 + 1.81782i) q^{5} +(-2.19974 - 1.47008i) q^{7} +(2.41878 - 1.77469i) q^{9} -0.958461i q^{11} -0.157315 q^{13} +(1.16337 - 3.69413i) q^{15} +2.51337i q^{17} -1.98260i q^{19} +(4.41332 + 1.23394i) q^{21} -2.67280 q^{23} +(-1.60893 - 4.73406i) q^{25} +(-3.02466 + 4.22510i) q^{27} +1.25028i q^{29} -8.66804i q^{31} +(0.516691 + 1.57765i) q^{33} +(5.53668 - 2.08450i) q^{35} -2.29909i q^{37} +(0.258945 - 0.0848062i) q^{39} +4.74507 q^{41} +6.58424i q^{43} +(0.0765078 + 6.70777i) q^{45} -5.60727i q^{47} +(2.67772 + 6.46760i) q^{49} +(-1.35492 - 4.13706i) q^{51} +8.59262 q^{53} +(1.74231 + 1.24804i) q^{55} +(1.06879 + 3.26340i) q^{57} +10.4488 q^{59} +4.27757i q^{61} +(-7.92962 + 0.348054i) q^{63} +(0.204845 - 0.285971i) q^{65} +13.8282i q^{67} +(4.39948 - 1.44086i) q^{69} +9.75994i q^{71} +4.43474 q^{73} +(5.20040 + 6.92502i) q^{75} +(-1.40902 + 2.10837i) q^{77} -0.517890 q^{79} +(2.70097 - 8.58515i) q^{81} +18.1293i q^{83} +(-4.56885 - 3.27272i) q^{85} +(-0.674006 - 2.05799i) q^{87} -0.954423 q^{89} +(0.346053 + 0.231267i) q^{91} +(4.67280 + 14.2678i) q^{93} +(3.60401 + 2.58160i) q^{95} -14.0907 q^{97} +(-1.70097 - 2.31830i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{9} - 2 q^{15} - 2 q^{21} + 16 q^{23} + 8 q^{25} - 8 q^{35} + 2 q^{39} - 6 q^{51} + 24 q^{53} + 8 q^{57} - 16 q^{63} + 16 q^{65} + 8 q^{77} - 4 q^{79} + 18 q^{81} - 12 q^{85} - 12 q^{91} + 32 q^{93}+ \cdots + 6 q^{99}+O(q^{100}) \) 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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