LMFDB - Embedded newform 1680.2.k.h.209.15

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1680,2,Mod(209,1680)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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			209.15
Character	$\chi$	$=$	1680.209
Dual form			1680.2.k.h.209.16

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.726113 - 1.57250i) q^{3} +(-2.20987 - 0.341309i) q^{5} +(2.06855 + 1.64958i) q^{7} +(-1.94552 - 2.28363i) q^{9} +O(q^{10})$	\(q+(0.726113 - 1.57250i) q^{3} +(-2.20987 - 0.341309i) q^{5} +(2.06855 + 1.64958i) q^{7} +(-1.94552 - 2.28363i) q^{9} +1.06405i q^{11} +4.82475 q^{13} +(-2.14132 + 3.22719i) q^{15} +7.89197i q^{17} +4.02213i q^{19} +(4.09596 - 2.05502i) q^{21} -5.69760 q^{23} +(4.76702 + 1.50849i) q^{25} +(-5.00367 + 1.40116i) q^{27} +2.00415i q^{29} +4.89513i q^{31} +(1.67322 + 0.772622i) q^{33} +(-4.00820 - 4.35136i) q^{35} -2.56310i q^{37} +(3.50331 - 7.58693i) q^{39} +5.08131 q^{41} -6.15762i q^{43} +(3.51992 + 5.71053i) q^{45} -2.27201i q^{47} +(1.55779 + 6.82446i) q^{49} +(12.4101 + 5.73046i) q^{51} +9.84443 q^{53} +(0.363170 - 2.35141i) q^{55} +(6.32481 + 2.92052i) q^{57} +5.87025 q^{59} -7.02935i q^{61} +(-0.257389 - 7.93308i) q^{63} +(-10.6621 - 1.64673i) q^{65} +10.8619i q^{67} +(-4.13710 + 8.95948i) q^{69} +0.0512158i q^{71} +2.86622 q^{73} +(5.83350 - 6.40080i) q^{75} +(-1.75524 + 2.20104i) q^{77} -7.00663 q^{79} +(-1.42990 + 8.88568i) q^{81} +7.59344i q^{83} +(2.69360 - 17.4402i) q^{85} +(3.15153 + 1.45524i) q^{87} +9.72471 q^{89} +(9.98024 + 7.95880i) q^{91} +(7.69760 + 3.55442i) q^{93} +(1.37279 - 8.88838i) q^{95} -1.77183 q^{97} +(2.42990 - 2.07014i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 24 q - 2 q^{9}+O(q^{10}) $ 24 * q - 2 * q^9	$ 24 q - 2 q^{9} - 2 q^{15} - 2 q^{21} + 16 q^{23} + 8 q^{25} - 8 q^{35} + 2 q^{39} - 6 q^{51} + 24 q^{53} + 8 q^{57} - 16 q^{63} + 16 q^{65} + 8 q^{77} - 4 q^{79} + 18 q^{81} - 12 q^{85} - 12 q^{91} + 32 q^{93} + 24 q^{95} + 6 q^{99}+O(q^{100}) $ 24 * q - 2 * q^9 - 2 * q^15 - 2 * q^21 + 16 * q^23 + 8 * q^25 - 8 * q^35 + 2 * q^39 - 6 * q^51 + 24 * q^53 + 8 * q^57 - 16 * q^63 + 16 * q^65 + 8 * q^77 - 4 * q^79 + 18 * q^81 - 12 * q^85 - 12 * q^91 + 32 * q^93 + 24 * q^95 + 6 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1680\mathbb{Z}\right)^\times$.

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

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
$3$	0.726113	−	1.57250i	0.419221	−	0.907884i
$3$	0.726113	−	1.57250i	0.419221	−	0.907884i
$4$		0			0
$5$	−2.20987	−	0.341309i	−0.988282	−	0.152638i
$5$	−2.20987	−	0.341309i	−0.988282	−	0.152638i
$6$		0			0
$7$	2.06855	+	1.64958i	0.781838	+	0.623481i
$7$	2.06855	+	1.64958i	0.781838	+	0.623481i
$8$		0			0
$9$	−1.94552	−	2.28363i	−0.648507	−	0.761209i
$10$		0			0
$11$			1.06405i			0.320824i	0.987050	+	0.160412i	$0.0512822\pi$
$11$			1.06405i			0.320824i	−0.987050	+	0.160412i	$0.948718\pi$
$12$		0			0
$13$	4.82475			1.33815			0.669073	−	0.743197i	$-0.266691\pi$
$13$	4.82475			1.33815			0.669073	+	0.743197i	$0.266691\pi$
$14$		0			0
$15$	−2.14132	+	3.22719i	−0.552887	+	0.833257i
$16$		0			0
$17$			7.89197i			1.91408i	0.289949	+	0.957042i	$0.406362\pi$
$17$			7.89197i			1.91408i	−0.289949	+	0.957042i	$0.593638\pi$
$18$		0			0
$19$			4.02213i			0.922741i	0.887208	+	0.461370i	$0.152642\pi$
$19$			4.02213i			0.922741i	−0.887208	+	0.461370i	$0.847358\pi$
$20$		0			0
$21$	4.09596	−	2.05502i	0.893812	−	0.448442i
$22$		0			0
$23$	−5.69760			−1.18803			−0.594016	−	0.804453i	$-0.702458\pi$
$23$	−5.69760			−1.18803			−0.594016	+	0.804453i	$0.702458\pi$
$24$		0			0
$25$	4.76702	+	1.50849i	0.953403	+	0.301699i
$26$		0			0
$27$	−5.00367	+	1.40116i	−0.962957	+	0.269654i
$28$		0			0
$29$			2.00415i			0.372161i	0.982534	+	0.186081i	$0.0595786\pi$
$29$			2.00415i			0.372161i	−0.982534	+	0.186081i	$0.940421\pi$
$30$		0			0
$31$			4.89513i			0.879191i	0.898196	+	0.439596i	$0.144878\pi$
$31$			4.89513i			0.879191i	−0.898196	+	0.439596i	$0.855122\pi$
$32$		0			0
$33$	1.67322	+	0.772622i	0.291271	+	0.134496i
$34$		0			0
$35$	−4.00820	−	4.35136i	−0.677510	−	0.735514i
$36$		0			0
$37$		−	2.56310i		−	0.421371i	−0.977554	−	0.210686i	$-0.932430\pi$
$37$		−	2.56310i		−	0.421371i	0.977554	−	0.210686i	$-0.0675696\pi$
$38$		0			0
$39$	3.50331	−	7.58693i	0.560979	−	1.21488i
$40$		0			0
$41$	5.08131			0.793567			0.396783	−	0.917912i	$-0.370126\pi$
$41$	5.08131			0.793567			0.396783	+	0.917912i	$0.370126\pi$
$42$		0			0
$43$		−	6.15762i		−	0.939028i	−0.882925	−	0.469514i	$-0.844429\pi$
$43$		−	6.15762i		−	0.939028i	0.882925	−	0.469514i	$-0.155571\pi$
$44$		0			0
$45$	3.51992	+	5.71053i	0.524719	+	0.851276i
$46$		0			0
$47$		−	2.27201i		−	0.331406i	−0.986176	−	0.165703i	$-0.947011\pi$
$47$		−	2.27201i		−	0.331406i	0.986176	−	0.165703i	$-0.0529893\pi$
$48$		0			0
$49$	1.55779	+	6.82446i	0.222542	+	0.974923i
$50$		0			0
$51$	12.4101	+	5.73046i	1.73777	+	0.802425i
$52$		0			0
$53$	9.84443			1.35224			0.676118	−	0.736793i	$-0.263661\pi$
$53$	9.84443			1.35224			0.676118	+	0.736793i	$0.263661\pi$
$54$		0			0
$55$	0.363170	−	2.35141i	0.0489699	−	0.317064i
$56$		0			0
$57$	6.32481	+	2.92052i	0.837742	+	0.386833i
$58$		0			0
$59$	5.87025			0.764241			0.382120	−	0.924113i	$-0.375194\pi$
$59$	5.87025			0.764241			0.382120	+	0.924113i	$0.375194\pi$
$60$		0			0
$61$		−	7.02935i		−	0.900016i	−0.893025	−	0.450008i	$-0.851421\pi$
$61$		−	7.02935i		−	0.900016i	0.893025	−	0.450008i	$-0.148579\pi$
$62$		0			0
$63$	−0.257389	−	7.93308i	−0.0324280	−	0.999474i
$64$		0			0
$65$	−10.6621	−	1.64673i	−1.32247	−	0.204252i
$66$		0			0
$67$			10.8619i			1.32699i	0.748181	+	0.663494i	$0.230927\pi$
$67$			10.8619i			1.32699i	−0.748181	+	0.663494i	$0.769073\pi$
$68$		0			0
$69$	−4.13710	+	8.95948i	−0.498048	+	1.07859i
$70$		0			0
$71$			0.0512158i			0.00607819i	0.999995	+	0.00303910i	$0.000967376\pi$
$71$			0.0512158i			0.00607819i	−0.999995	+	0.00303910i	$0.999033\pi$
$72$		0			0
$73$	2.86622			0.335466			0.167733	−	0.985832i	$-0.446355\pi$
$73$	2.86622			0.335466			0.167733	+	0.985832i	$0.446355\pi$
$74$		0			0
$75$	5.83350	−	6.40080i	0.673594	−	0.739101i
$76$		0			0
$77$	−1.75524	+	2.20104i	−0.200028	+	0.250832i
$78$		0			0
$79$	−7.00663			−0.788307			−0.394154	−	0.919045i	$-0.628962\pi$
$79$	−7.00663			−0.788307			−0.394154	+	0.919045i	$0.628962\pi$
$80$		0			0
$81$	−1.42990	+	8.88568i	−0.158877	+	0.987298i
$82$		0			0
$83$			7.59344i			0.833489i	0.909024	+	0.416744i	$0.136829\pi$
$83$			7.59344i			0.833489i	−0.909024	+	0.416744i	$0.863171\pi$
$84$		0			0
$85$	2.69360	−	17.4402i	0.292162	−	1.89166i
$86$		0			0
$87$	3.15153	+	1.45524i	0.337879	+	0.156018i
$88$		0			0
$89$	9.72471			1.03082			0.515409	−	0.856945i	$-0.327640\pi$
$89$	9.72471			1.03082			0.515409	+	0.856945i	$0.327640\pi$
$90$		0			0
$91$	9.98024	+	7.95880i	1.04621	+	0.834309i
$92$		0			0
$93$	7.69760	+	3.55442i	0.798204	+	0.368576i
$94$		0			0
$95$	1.37279	−	8.88838i	0.140845	−	0.911928i
$96$		0			0
$97$	−1.77183			−0.179902			−0.0899512	−	0.995946i	$-0.528671\pi$
$97$	−1.77183			−0.179902			−0.0899512	+	0.995946i	$0.528671\pi$
$98$		0			0
$99$	2.42990	−	2.07014i	0.244214	−	0.208056i
$100$		0			0
$101$	11.1648			1.11094			0.555469	−	0.831537i	$-0.312539\pi$
$101$	11.1648			1.11094			0.555469	+	0.831537i	$0.312539\pi$
$102$		0			0
$103$	−11.1665			−1.10027			−0.550134	−	0.835076i	$-0.685423\pi$
$103$	−11.1665			−1.10027			−0.550134	+	0.835076i	$0.685423\pi$
$104$		0			0
$105$	−9.75292	+	3.14333i	−0.951788	+	0.306757i
$106$		0			0
$107$	0.0623499			0.00602759			0.00301379	−	0.999995i	$-0.499041\pi$
$107$	0.0623499			0.00602759			0.00301379	+	0.999995i	$0.499041\pi$
$108$		0			0
$109$	10.3095			0.987470			0.493735	−	0.869612i	$-0.335631\pi$
$109$	10.3095			0.987470			0.493735	+	0.869612i	$0.335631\pi$
$110$		0			0
$111$	−4.03048	−	1.86110i	−0.382556	−	0.176648i
$112$		0			0
$113$	−9.22363			−0.867686			−0.433843	−	0.900988i	$-0.642843\pi$
$113$	−9.22363			−0.867686			−0.433843	+	0.900988i	$0.642843\pi$
$114$		0			0
$115$	12.5909	+	1.94464i	1.17411	+	0.181339i
$116$		0			0
$117$	−9.38666	−	11.0179i	−0.867797	−	1.01861i
$118$		0			0
$119$	−13.0184	+	16.3249i	−1.19340	+	1.49650i
$120$		0			0
$121$	9.86779			0.897072
$122$		0			0
$123$	3.68960	−	7.99036i	0.332680	−	0.720466i
$124$		0			0
$125$	−10.0196	−	4.96059i	−0.896181	−	0.443689i
$126$		0			0
$127$		−	13.3225i		−	1.18218i	−0.806604	−	0.591092i	$-0.798697\pi$
$127$		−	13.3225i		−	1.18218i	0.806604	−	0.591092i	$-0.201303\pi$
$128$		0			0
$129$	−9.68286	−	4.47112i	−0.852528	−	0.393660i
$130$		0			0
$131$	15.6808			1.37004			0.685020	−	0.728524i	$-0.259793\pi$
$131$	15.6808			1.37004			0.685020	+	0.728524i	$0.259793\pi$
$132$		0			0
$133$	−6.63482	+	8.31998i	−0.575312	+	0.721434i
$134$		0			0
$135$	11.5357	−	1.38859i	0.992833	−	0.119511i
$136$		0			0
$137$	0.867682			0.0741311			0.0370655	−	0.999313i	$-0.488199\pi$
$137$	0.867682			0.0741311			0.0370655	+	0.999313i	$0.488199\pi$
$138$		0			0
$139$			22.8429i			1.93751i	0.248016	+	0.968756i	$0.420222\pi$
$139$			22.8429i			1.93751i	−0.248016	+	0.968756i	$0.579778\pi$
$140$		0			0
$141$	−3.57273	−	1.64973i	−0.300878	−	0.138932i
$142$		0			0
$143$			5.13379i			0.429309i
$144$		0			0
$145$	0.684034	−	4.42890i	0.0568060	−	0.367801i
$146$		0			0
$147$	11.8626	+	2.50570i	0.978411	+	0.206666i
$148$		0			0
$149$		−	17.9353i		−	1.46931i	−0.678438	−	0.734657i	$-0.737343\pi$
$149$		−	17.9353i		−	1.46931i	0.678438	−	0.734657i	$-0.262657\pi$
$150$		0			0
$151$	−17.6967			−1.44014			−0.720068	−	0.693903i	$-0.755889\pi$
$151$	−17.6967			−1.44014			−0.720068	+	0.693903i	$0.755889\pi$
$152$		0			0
$153$	18.0223	−	15.3540i	1.45702	−	1.24130i
$154$		0			0
$155$	1.67075	−	10.8176i	0.134198	−	0.868889i
$156$		0			0
$157$	−8.17120			−0.652133			−0.326066	−	0.945347i	$-0.605723\pi$
$157$	−8.17120			−0.652133			−0.326066	+	0.945347i	$0.605723\pi$
$158$		0			0
$159$	7.14817	−	15.4804i	0.566887	−	1.22767i
$160$		0			0
$161$	−11.7858	−	9.39863i	−0.928848	−	0.740716i
$162$		0			0
$163$		−	13.2664i		−	1.03911i	−0.854437	−	0.519554i	$-0.826098\pi$
$163$		−	13.2664i		−	1.03911i	0.854437	−	0.519554i	$-0.173902\pi$
$164$		0			0
$165$	−3.43390	−	2.27848i	−0.267328	−	0.177379i
$166$		0			0
$167$			12.0623i			0.933406i	0.884414	+	0.466703i	$0.154558\pi$
$167$			12.0623i			0.933406i	−0.884414	+	0.466703i	$0.845442\pi$
$168$		0			0
$169$	10.2782			0.790634
$170$		0			0
$171$	9.18505	−	7.82514i	0.702398	−	0.598404i
$172$		0			0
$173$		−	11.8101i		−	0.897904i	−0.893556	−	0.448952i	$-0.851797\pi$
$173$		−	11.8101i		−	0.897904i	0.893556	−	0.448952i	$-0.148203\pi$
$174$		0			0
$175$	7.37243	+	10.9840i	0.557304	+	0.830309i
$176$		0			0
$177$	4.26246	−	9.23097i	0.320386	−	0.693842i
$178$		0			0
$179$		−	23.8161i		−	1.78010i	−0.455860	−	0.890051i	$-0.650668\pi$
$179$		−	23.8161i		−	1.78010i	0.455860	−	0.890051i	$-0.349332\pi$
$180$		0			0
$181$			14.8182i			1.10143i	0.834693	+	0.550716i	$0.185645\pi$
$181$			14.8182i			1.10143i	−0.834693	+	0.550716i	$0.814355\pi$
$182$		0			0
$183$	−11.0537	−	5.10410i	−0.817110	−	0.377306i
$184$		0			0
$185$	−0.874809	+	5.66411i	−0.0643172	+	0.416434i
$186$		0			0
$187$	−8.39747			−0.614084
$188$		0			0
$189$	−12.6617	−	5.35556i	−0.921001	−	0.389560i
$190$		0			0
$191$			16.3904i			1.18597i	0.805213	+	0.592985i	$0.202051\pi$
$191$			16.3904i			1.18597i	−0.805213	+	0.592985i	$0.797949\pi$
$192$		0			0
$193$			22.8005i			1.64122i	0.571490	+	0.820609i	$0.306365\pi$
$193$			22.8005i			1.64122i	−0.571490	+	0.820609i	$0.693635\pi$
$194$		0			0
$195$	−10.3313	+	15.5704i	−0.739843	+	1.11502i
$196$		0			0
$197$	−21.8981			−1.56018			−0.780089	−	0.625669i	$-0.784826\pi$
$197$	−21.8981			−1.56018			−0.780089	+	0.625669i	$0.784826\pi$
$198$		0			0
$199$		−	11.5742i		−	0.820471i	−0.911980	−	0.410235i	$-0.865447\pi$
$199$		−	11.5742i		−	0.820471i	0.911980	−	0.410235i	$-0.134553\pi$
$200$		0			0
$201$	17.0803	+	7.88694i	1.20475	+	0.556302i
$202$		0			0
$203$	−3.30600	+	4.14568i	−0.232036	+	0.290970i
$204$		0			0
$205$	−11.2290	−	1.73429i	−0.784268	−	0.121128i
$206$		0			0
$207$	11.0848	+	13.0112i	0.770447	+	0.904340i
$208$		0			0
$209$	−4.27976			−0.296037
$210$		0			0
$211$	1.97012			0.135629			0.0678144	−	0.997698i	$-0.478397\pi$
$211$	1.97012			0.135629			0.0678144	+	0.997698i	$0.478397\pi$
$212$		0			0
$213$	0.0805368	+	0.0371884i	0.00551829	+	0.00254811i
$214$		0			0
$215$	−2.10165	+	13.6075i	−0.143331	+	0.928024i
$216$		0			0
$217$	−8.07489	+	10.1258i	−0.548159	+	0.687385i
$218$		0			0
$219$	2.08120	−	4.50713i	0.140634	−	0.304564i
$220$		0			0
$221$			38.0768i			2.56132i
$222$		0			0
$223$	2.88164			0.192969			0.0964844	−	0.995334i	$-0.469240\pi$
$223$	2.88164			0.192969			0.0964844	+	0.995334i	$0.469240\pi$
$224$		0			0
$225$	−5.82950	−	13.8209i	−0.388633	−	0.921393i
$226$		0			0
$227$			6.48259i			0.430265i	0.976585	+	0.215132i	$0.0690183\pi$
$227$			6.48259i			0.430265i	−0.976585	+	0.215132i	$0.930982\pi$
$228$		0			0
$229$		−	2.23823i		−	0.147907i	−0.997262	−	0.0739533i	$-0.976438\pi$
$229$		−	2.23823i		−	0.147907i	0.997262	−	0.0739533i	$-0.0235616\pi$
$230$		0			0
$231$	2.18665	+	4.35832i	0.143871	+	0.286756i
$232$		0			0
$233$	−23.1526			−1.51677			−0.758387	−	0.651805i	$-0.774012\pi$
$233$	−23.1526			−1.51677			−0.758387	+	0.651805i	$0.774012\pi$
$234$		0			0
$235$	−0.775455	+	5.02083i	−0.0505851	+	0.327523i
$236$		0			0
$237$	−5.08760	+	11.0179i	−0.330475	+	0.715692i
$238$		0			0
$239$			2.77822i			0.179708i	0.995955	+	0.0898541i	$0.0286401\pi$
$239$			2.77822i			0.179708i	−0.995955	+	0.0898541i	$0.971360\pi$
$240$		0			0
$241$		−	17.3711i		−	1.11897i	−0.828840	−	0.559486i	$-0.810998\pi$
$241$		−	17.3711i		−	1.11897i	0.828840	−	0.559486i	$-0.189002\pi$
$242$		0			0
$243$	12.9345	+	8.70052i	0.829748	+	0.558139i
$244$		0			0
$245$	−1.11327	−	15.6128i	−0.0711239	−	0.997467i
$246$		0			0
$247$			19.4058i			1.23476i
$248$		0			0
$249$	11.9407	+	5.51369i	0.756711	+	0.349416i
$250$		0			0
$251$	1.51403			0.0955645			0.0477822	−	0.998858i	$-0.484785\pi$
$251$	1.51403			0.0955645			0.0477822	+	0.998858i	$0.484785\pi$
$252$		0			0
$253$		−	6.06254i		−	0.381149i
$254$		0			0
$255$	−25.4689	−	16.8992i	−1.59492	−	1.05827i
$256$		0			0
$257$			11.6012i			0.723666i	0.932243	+	0.361833i	$0.117849\pi$
$257$			11.6012i			0.723666i	−0.932243	+	0.361833i	$0.882151\pi$
$258$		0			0
$259$	4.22803	−	5.30190i	0.262717	−	0.329444i
$260$		0			0
$261$	4.57673	−	3.89912i	0.283293	−	0.241349i
$262$		0			0
$263$	−11.5728			−0.713611			−0.356805	−	0.934179i	$-0.616134\pi$
$263$	−11.5728			−0.713611			−0.356805	+	0.934179i	$0.616134\pi$
$264$		0			0
$265$	−21.7549	−	3.35999i	−1.33639	−	0.206403i
$266$		0			0
$267$	7.06124	−	15.2921i	0.432141	−	0.935863i
$268$		0			0
$269$	5.94887			0.362709			0.181355	−	0.983418i	$-0.441952\pi$
$269$	5.94887			0.362709			0.181355	+	0.983418i	$0.441952\pi$
$270$		0			0
$271$		−	30.3950i		−	1.84636i	−0.384365	−	0.923181i	$-0.625580\pi$
$271$		−	30.3950i		−	1.84636i	0.384365	−	0.923181i	$-0.374420\pi$
$272$		0			0
$273$	19.7620	−	9.91495i	1.19605	−	0.600080i
$274$		0			0
$275$	−1.60512	+	5.07235i	−0.0967921	+	0.305874i
$276$		0			0
$277$		−	4.54573i		−	0.273126i	−0.990631	−	0.136563i	$-0.956394\pi$
$277$		−	4.54573i		−	0.273126i	0.990631	−	0.136563i	$-0.0436057\pi$
$278$		0			0
$279$	11.1786	−	9.52358i	0.669248	−	0.570162i
$280$		0			0
$281$			31.3045i			1.86747i	0.357968	+	0.933734i	$0.383470\pi$
$281$			31.3045i			1.86747i	−0.357968	+	0.933734i	$0.616530\pi$
$282$		0			0
$283$	−25.3109			−1.50458			−0.752290	−	0.658832i	$-0.771051\pi$
$283$	−25.3109			−1.50458			−0.752290	+	0.658832i	$0.771051\pi$
$284$		0			0
$285$	−12.9802	−	8.61268i	−0.768880	−	0.510171i
$286$		0			0
$287$	10.5109	+	8.38200i	0.620441	+	0.494774i
$288$		0			0
$289$	−45.2832			−2.66372
$290$		0			0
$291$	−1.28655	+	2.78621i	−0.0754190	+	0.163331i
$292$		0			0
$293$			22.4139i			1.30943i	0.755874	+	0.654717i	$0.227212\pi$
$293$			22.4139i			1.30943i	−0.755874	+	0.654717i	$0.772788\pi$
$294$		0			0
$295$	−12.9725	−	2.00357i	−0.755286	−	0.116652i
$296$		0			0
$297$	−1.49091	−	5.32417i	−0.0865115	−	0.308940i
$298$		0			0
$299$	−27.4895			−1.58976
$300$		0			0
$301$	10.1575	−	12.7373i	0.585466	−	0.734168i
$302$		0			0
$303$	8.10689	−	17.5566i	0.465729	−	1.00860i
$304$		0			0
$305$	−2.39918	+	15.5339i	−0.137377	+	0.889469i
$306$		0			0
$307$	25.9633			1.48181			0.740903	−	0.671612i	$-0.234398\pi$
$307$	25.9633			1.48181			0.740903	+	0.671612i	$0.234398\pi$
$308$		0			0
$309$	−8.10814	+	17.5593i	−0.461256	+	0.998916i
$310$		0			0
$311$	13.9994			0.793833			0.396917	−	0.917855i	$-0.370080\pi$
$311$	13.9994			0.793833			0.396917	+	0.917855i	$0.370080\pi$
$312$		0			0
$313$	23.7305			1.34133			0.670664	−	0.741761i	$-0.266009\pi$
$313$	23.7305			1.34133			0.670664	+	0.741761i	$0.266009\pi$
$314$		0			0
$315$	−2.13883	+	17.6189i	−0.120510	+	0.992712i
$316$		0			0
$317$	21.1149			1.18593			0.592966	−	0.805227i	$-0.297957\pi$
$317$	21.1149			1.18593			0.592966	+	0.805227i	$0.297957\pi$
$318$		0			0
$319$	−2.13252			−0.119398
$320$		0			0
$321$	0.0452730	−	0.0980452i	0.00252689	−	0.00547235i
$322$		0			0
$323$	−31.7426			−1.76620
$324$		0			0
$325$	22.9997	+	7.27811i	1.27579	+	0.403717i
$326$		0			0
$327$	7.48585	−	16.2117i	0.413968	−	0.896508i
$328$		0			0
$329$	3.74785	−	4.69976i	0.206626	−	0.259106i
$330$		0			0
$331$	12.6745			0.696653			0.348327	−	0.937373i	$-0.386750\pi$
$331$	12.6745			0.696653			0.348327	+	0.937373i	$0.386750\pi$
$332$		0			0
$333$	−5.85317	+	4.98657i	−0.320752	+	0.273262i
$334$		0			0
$335$	3.70725	−	24.0033i	0.202549	−	1.31144i
$336$		0			0
$337$		−	15.3722i		−	0.837376i	−0.908130	−	0.418688i	$-0.862490\pi$
$337$		−	15.3722i		−	0.837376i	0.908130	−	0.418688i	$-0.137510\pi$
$338$		0			0
$339$	−6.69740	+	14.5042i	−0.363753	+	0.787759i
$340$		0			0
$341$	−5.20867			−0.282065
$342$		0			0
$343$	−8.03510	+	16.6864i	−0.433855	+	0.900983i
$344$		0			0
$345$	12.2004	−	18.3872i	0.656847	−	0.989935i
$346$		0			0
$347$	−17.8577			−0.958651			−0.479326	−	0.877637i	$-0.659119\pi$
$347$	−17.8577			−0.958651			−0.479326	+	0.877637i	$0.659119\pi$
$348$		0			0
$349$		−	23.4696i		−	1.25630i	−0.778092	−	0.628150i	$-0.783812\pi$
$349$		−	23.4696i		−	1.25630i	0.778092	−	0.628150i	$-0.216188\pi$
$350$		0			0
$351$	−24.1415	+	6.76027i	−1.28858	+	0.360837i
$352$		0			0
$353$		−	8.57186i		−	0.456234i	−0.973634	−	0.228117i	$-0.926743\pi$
$353$		−	8.57186i		−	0.456234i	0.973634	−	0.228117i	$-0.0732569\pi$
$354$		0			0
$355$	0.0174804	−	0.113180i	0.000927763	−	0.00600697i
$356$		0			0
$357$	16.2181	+	32.3252i	0.858355	+	1.71083i
$358$		0			0
$359$			25.6758i			1.35512i	0.735469	+	0.677558i	$0.236962\pi$
$359$			25.6758i			1.35512i	−0.735469	+	0.677558i	$0.763038\pi$
$360$		0			0
$361$	2.82244			0.148550
$362$		0			0
$363$	7.16513	−	15.5171i	0.376072	−	0.814437i
$364$		0			0
$365$	−6.33396	−	0.978266i	−0.331535	−	0.0512048i
$366$		0			0
$367$	−11.7633			−0.614039			−0.307020	−	0.951703i	$-0.599332\pi$
$367$	−11.7633			−0.614039			−0.307020	+	0.951703i	$0.599332\pi$
$368$		0			0
$369$	−9.88579	−	11.6038i	−0.514633	−	0.604070i
$370$		0			0
$371$	20.3637	+	16.2391i	1.05723	+	0.843095i
$372$		0			0
$373$			32.3173i			1.67333i	0.547718	+	0.836663i	$0.315497\pi$
$373$			32.3173i			1.67333i	−0.547718	+	0.836663i	$0.684503\pi$
$374$		0			0
$375$	−15.0759	+	12.1539i	−0.778516	+	0.627624i
$376$		0			0
$377$			9.66953i			0.498006i
$378$		0			0
$379$	−1.40400			−0.0721185			−0.0360593	−	0.999350i	$-0.511481\pi$
$379$	−1.40400			−0.0721185			−0.0360593	+	0.999350i	$0.511481\pi$
$380$		0			0
$381$	−20.9497	−	9.67367i	−1.07329	−	0.495597i
$382$		0			0
$383$		−	5.26692i		−	0.269127i	−0.990905	−	0.134564i	$-0.957037\pi$
$383$		−	5.26692i		−	0.269127i	0.990905	−	0.134564i	$-0.0429632\pi$
$384$		0			0
$385$	4.63007	−	4.26494i	0.235970	−	0.217361i
$386$		0			0
$387$	−14.0617	+	11.9798i	−0.714796	+	0.608966i
$388$		0			0
$389$		−	18.6808i		−	0.947155i	−0.880752	−	0.473578i	$-0.842962\pi$
$389$		−	18.6808i		−	0.947155i	0.880752	−	0.473578i	$-0.157038\pi$
$390$		0			0
$391$		−	44.9653i		−	2.27399i
$392$		0			0
$393$	11.3860	−	24.6581i	0.574350	−	1.24384i
$394$		0			0
$395$	15.4837	+	2.39142i	0.779070	+	0.120326i
$396$		0			0
$397$	18.9468			0.950915			0.475457	−	0.879739i	$-0.342283\pi$
$397$	18.9468			0.950915			0.475457	+	0.879739i	$0.342283\pi$
$398$		0			0
$399$	8.26556	+	16.4745i	0.413795	+	0.824757i
$400$		0			0
$401$			0.701335i			0.0350230i	0.999847	+	0.0175115i	$0.00557436\pi$
$401$			0.701335i			0.0350230i	−0.999847	+	0.0175115i	$0.994426\pi$
$402$		0			0
$403$			23.6178i			1.17649i
$404$		0			0
$405$	6.19264	−	19.1481i	0.307715	−	0.951479i
$406$		0			0
$407$	2.72727			0.135186
$408$		0			0
$409$			12.6644i			0.626217i	0.949717	+	0.313108i	$0.101370\pi$
$409$			12.6644i			0.626217i	−0.949717	+	0.313108i	$0.898630\pi$
$410$		0			0
$411$	0.630035	−	1.36443i	0.0310773	−	0.0673024i
$412$		0			0
$413$	12.1429	+	9.68342i	0.597513	+	0.476490i
$414$		0			0
$415$	2.59171	−	16.7805i	0.127222	−	0.823722i
$416$		0			0
$417$	35.9205	+	16.5865i	1.75904	+	0.812246i
$418$		0			0
$419$	−24.1683			−1.18070			−0.590348	−	0.807149i	$-0.701010\pi$
$419$	−24.1683			−1.18070			−0.590348	+	0.807149i	$0.701010\pi$
$420$		0			0
$421$	8.24174			0.401678			0.200839	−	0.979624i	$-0.435633\pi$
$421$	8.24174			0.401678			0.200839	+	0.979624i	$0.435633\pi$
$422$		0			0
$423$	−5.18841	+	4.42023i	−0.252269	+	0.214919i
$424$		0			0
$425$	−11.9050	+	37.6212i	−0.577477	+	1.82489i
$426$		0			0
$427$	11.5954	−	14.5406i	0.561143	−	0.703667i
$428$		0			0
$429$	8.07289	+	3.72771i	0.389763	+	0.179975i
$430$		0			0
$431$		−	7.27482i		−	0.350416i	−0.984531	−	0.175208i	$-0.943940\pi$
$431$		−	7.27482i		−	0.350416i	0.984531	−	0.175208i	$-0.0560597\pi$
$432$		0			0
$433$	−28.1801			−1.35425			−0.677125	−	0.735868i	$-0.736774\pi$
$433$	−28.1801			−1.35425			−0.677125	+	0.735868i	$0.736774\pi$
$434$		0			0
$435$	−6.46777	−	4.29153i	−0.310106	−	0.205763i
$436$		0			0
$437$		−	22.9165i		−	1.09624i
$438$		0			0
$439$			8.99084i			0.429109i	0.976712	+	0.214555i	$0.0688300\pi$
$439$			8.99084i			0.429109i	−0.976712	+	0.214555i	$0.931170\pi$
$440$		0			0
$441$	12.5538	−	16.8346i	0.597800	−	0.801645i
$442$		0			0
$443$	23.6265			1.12253			0.561265	−	0.827636i	$-0.310315\pi$
$443$	23.6265			1.12253			0.561265	+	0.827636i	$0.310315\pi$
$444$		0			0
$445$	−21.4903	−	3.31913i	−1.01874	−	0.157342i
$446$		0			0
$447$	−28.2032	−	13.0230i	−1.33397	−	0.615968i
$448$		0			0
$449$		−	22.2130i		−	1.04830i	−0.851627	−	0.524148i	$-0.824384\pi$
$449$		−	22.2130i		−	1.04830i	0.851627	−	0.524148i	$-0.175616\pi$
$450$		0			0
$451$			5.40677i			0.254595i
$452$		0			0
$453$	−12.8498	+	27.8281i	−0.603736	+	1.30748i
$454$		0			0
$455$	−19.3386	−	20.9942i	−0.906607	−	0.984225i
$456$		0			0
$457$		−	38.7674i		−	1.81346i	−0.421710	−	0.906731i	$-0.638570\pi$
$457$		−	38.7674i		−	1.81346i	0.421710	−	0.906731i	$-0.361430\pi$
$458$		0			0
$459$	−11.0580	−	39.4888i	−0.516141	−	1.84318i
$460$		0			0
$461$	24.1613			1.12530			0.562652	−	0.826694i	$-0.309781\pi$
$461$	24.1613			1.12530			0.562652	+	0.826694i	$0.309781\pi$
$462$		0			0
$463$		−	19.7543i		−	0.918061i	−0.888421	−	0.459030i	$-0.848197\pi$
$463$		−	19.7543i		−	0.918061i	0.888421	−	0.459030i	$-0.151803\pi$
$464$		0			0
$465$	−15.7975	−	10.4820i	−0.732592	−	0.486093i
$466$		0			0
$467$		−	8.20343i		−	0.379609i	−0.981822	−	0.189805i	$-0.939215\pi$
$467$		−	8.20343i		−	0.379609i	0.981822	−	0.189805i	$-0.0607855\pi$
$468$		0			0
$469$	−17.9175	+	22.4683i	−0.827353	+	1.03749i
$470$		0			0
$471$	−5.93321	+	12.8492i	−0.273388	+	0.592061i
$472$		0			0
$473$	6.55202			0.301262
$474$		0			0
$475$	−6.06736	+	19.1736i	−0.278390	+	0.879744i
$476$		0			0
$477$	−19.1526	−	22.4810i	−0.876935	−	1.02933i
$478$		0			0
$479$	−16.4189			−0.750197			−0.375098	−	0.926985i	$-0.622391\pi$
$479$	−16.4189			−0.750197			−0.375098	+	0.926985i	$0.622391\pi$
$480$		0			0
$481$		−	12.3663i		−	0.563856i
$482$		0			0
$483$	−23.3371	+	11.7087i	−1.06188	+	0.532763i
$484$		0			0
$485$	3.91552	+	0.604742i	0.177794	+	0.0274599i
$486$		0			0
$487$		−	4.95785i		−	0.224662i	−0.993671	−	0.112331i	$-0.964168\pi$
$487$		−	4.95785i		−	0.224662i	0.993671	−	0.112331i	$-0.0358317\pi$
$488$		0			0
$489$	−20.8615	−	9.63294i	−0.943390	−	0.435616i
$490$		0			0
$491$			20.7222i			0.935178i	0.883946	+	0.467589i	$0.154877\pi$
$491$			20.7222i			0.935178i	−0.883946	+	0.467589i	$0.845123\pi$
$492$		0			0
$493$	−15.8167			−0.712348
$494$		0			0
$495$	−6.07630	+	3.74538i	−0.273109	+	0.168342i
$496$		0			0
$497$	−0.0844843	+	0.105942i	−0.00378964	+	0.00475216i
$498$		0			0
$499$	−21.8280			−0.977154			−0.488577	−	0.872521i	$-0.662484\pi$
$499$	−21.8280			−0.977154			−0.488577	+	0.872521i	$0.662484\pi$
$500$		0			0
$501$	18.9679	+	8.75856i	0.847425	+	0.391304i
$502$		0			0
$503$			4.31874i			0.192563i	0.995354	+	0.0962815i	$0.0306949\pi$
$503$			4.31874i			0.192563i	−0.995354	+	0.0962815i	$0.969305\pi$
$504$		0			0
$505$	−24.6727	−	3.81064i	−1.09792	−	0.169571i
$506$		0			0
$507$	7.46316	−	16.1625i	0.331451	−	0.717804i
$508$		0			0
$509$	−11.5168			−0.510474			−0.255237	−	0.966879i	$-0.582153\pi$
$509$	−11.5168			−0.510474			−0.255237	+	0.966879i	$0.582153\pi$
$510$		0			0
$511$	5.92892	+	4.72805i	0.262280	+	0.209157i
$512$		0			0
$513$	−5.63567	−	20.1254i	−0.248821	−	0.888560i
$514$		0			0
$515$	24.6765	+	3.81122i	1.08738	+	0.167943i
$516$		0			0
$517$	2.41753			0.106323
$518$		0			0
$519$	−18.5714	−	8.57545i	−0.815193	−	0.376421i
$520$		0			0
$521$	12.6788			0.555469			0.277734	−	0.960658i	$-0.410416\pi$
$521$	12.6788			0.555469			0.277734	+	0.960658i	$0.410416\pi$
$522$		0			0
$523$	−20.4499			−0.894212			−0.447106	−	0.894481i	$-0.647545\pi$
$523$	−20.4499			−0.894212			−0.447106	+	0.894481i	$0.647545\pi$
$524$		0			0
$525$	22.6255	−	3.61757i	0.987458	−	0.157884i
$526$		0			0
$527$	−38.6322			−1.68285
$528$		0			0
$529$	9.46264			0.411419
$530$		0			0
$531$	−11.4207	−	13.4054i	−0.495616	−	0.581747i
$532$		0			0
$533$	24.5160			1.06191
$534$		0			0
$535$	−0.137785	−	0.0212806i	−0.00595696	−	0.000920039i
$536$		0			0
$537$	−37.4509	−	17.2932i	−1.61613	−	0.746257i
$538$		0			0
$539$	−7.26158	+	1.65757i	−0.312778	+	0.0713967i
$540$		0			0
$541$	−27.3151			−1.17437			−0.587183	−	0.809454i	$-0.699763\pi$
$541$	−27.3151			−1.17437			−0.587183	+	0.809454i	$0.699763\pi$
$542$		0			0
$543$	23.3017	+	10.7597i	0.999972	+	0.461744i
$544$		0			0
$545$	−22.7826	−	3.51872i	−0.975899	−	0.150725i
$546$		0			0
$547$		−	24.0471i		−	1.02818i	−0.857736	−	0.514090i	$-0.828130\pi$
$547$		−	24.0471i		−	1.02818i	0.857736	−	0.514090i	$-0.171870\pi$
$548$		0			0
$549$	−16.0524	+	13.6757i	−0.685100	+	0.583666i
$550$		0			0
$551$	−8.06096			−0.343408
$552$		0			0
$553$	−14.4936	−	11.5580i	−0.616329	−	0.491495i
$554$		0			0
$555$	8.27161	+	5.48842i	0.351110	+	0.232971i
$556$		0			0
$557$	−6.33396			−0.268379			−0.134189	−	0.990956i	$-0.542843\pi$
$557$	−6.33396			−0.268379			−0.134189	+	0.990956i	$0.542843\pi$
$558$		0			0
$559$		−	29.7090i		−	1.25656i
$560$		0			0
$561$	−6.09751	+	13.2050i	−0.257437	+	0.557517i
$562$		0			0
$563$		−	18.3682i		−	0.774126i	−0.922053	−	0.387063i	$-0.873490\pi$
$563$		−	18.3682i		−	0.774126i	0.922053	−	0.387063i	$-0.126510\pi$
$564$		0			0
$565$	20.3830	+	3.14811i	0.857519	+	0.132442i
$566$		0			0
$567$	−17.6154	+	16.0218i	−0.739779	+	0.672850i
$568$		0			0
$569$			9.66392i			0.405133i	0.979269	+	0.202566i	$0.0649281\pi$
$569$			9.66392i			0.405133i	−0.979269	+	0.202566i	$0.935072\pi$
$570$		0			0
$571$	23.0841			0.966038			0.483019	−	0.875610i	$-0.339540\pi$
$571$	23.0841			0.966038			0.483019	+	0.875610i	$0.339540\pi$
$572$		0			0
$573$	25.7740	+	11.9013i	1.07672	+	0.497184i
$574$		0			0
$575$	−27.1606	−	8.59479i	−1.13267	−	0.358428i
$576$		0			0
$577$	25.7810			1.07328			0.536638	−	0.843813i	$-0.319694\pi$
$577$	25.7810			1.07328			0.536638	+	0.843813i	$0.319694\pi$
$578$		0			0
$579$	35.8539	+	16.5557i	1.49004	+	0.688033i
$580$		0			0
$581$	−12.5260	+	15.7074i	−0.519665	+	0.651653i
$582$		0			0
$583$			10.4750i			0.433830i
$584$		0			0
$585$	16.9827	+	27.5519i	0.702150	+	1.13913i
$586$		0			0
$587$		−	20.6620i		−	0.852813i	−0.904532	−	0.426406i	$-0.859779\pi$
$587$		−	20.6620i		−	0.852813i	0.904532	−	0.426406i	$-0.140221\pi$
$588$		0			0
$589$	−19.6889			−0.811266
$590$		0			0
$591$	−15.9005	+	34.4348i	−0.654060	+	1.41646i
$592$		0			0
$593$			15.2547i			0.626437i	0.949681	+	0.313219i	$0.101407\pi$
$593$			15.2547i			0.626437i	−0.949681	+	0.313219i	$0.898593\pi$
$594$		0			0
$595$	34.3408	−	31.6326i	1.40784	−	1.29681i
$596$		0			0
$597$	−18.2004	−	8.40415i	−0.744892	−	0.343959i
$598$		0			0
$599$			8.74189i			0.357184i	0.983923	+	0.178592i	$0.0571542\pi$
$599$			8.74189i			0.357184i	−0.983923	+	0.178592i	$0.942846\pi$
$600$		0			0
$601$			45.3152i			1.84845i	0.381851	+	0.924224i	$0.375287\pi$
$601$			45.3152i			1.84845i	−0.381851	+	0.924224i	$0.624713\pi$
$602$		0			0
$603$	24.8045	−	21.1320i	1.01012	−	0.860561i
$604$		0			0
$605$	−21.8065	−	3.36796i	−0.886560	−	0.136927i
$606$		0			0
$607$	−15.9521			−0.647477			−0.323738	−	0.946147i	$-0.604940\pi$
$607$	−15.9521			−0.647477			−0.323738	+	0.946147i	$0.604940\pi$
$608$		0			0
$609$	4.11857	+	8.20892i	0.166893	+	0.332642i
$610$		0			0
$611$		−	10.9619i		−	0.443470i
$612$		0			0
$613$		−	5.45597i		−	0.220365i	−0.993911	−	0.110182i	$-0.964857\pi$
$613$		−	5.45597i		−	0.220365i	0.993911	−	0.110182i	$-0.0351435\pi$
$614$		0			0
$615$	−10.8807	+	16.3983i	−0.438752	+	0.661245i
$616$		0			0
$617$	−24.7878			−0.997919			−0.498960	−	0.866625i	$-0.666284\pi$
$617$	−24.7878			−0.997919			−0.498960	+	0.866625i	$0.666284\pi$
$618$		0			0
$619$		−	18.7809i		−	0.754869i	−0.926036	−	0.377435i	$-0.876806\pi$
$619$		−	18.7809i		−	0.754869i	0.926036	−	0.377435i	$-0.123194\pi$
$620$		0			0
$621$	28.5089	−	7.98327i	1.14402	−	0.320358i
$622$		0			0
$623$	20.1160	+	16.0417i	0.805932	+	0.642695i
$624$		0			0
$625$	20.4489	+	14.3820i	0.817956	+	0.575281i
$626$		0			0
$627$	−3.10759	+	6.72993i	−0.124105	+	0.268767i
$628$		0			0
$629$	20.2279			0.806540
$630$		0			0
$631$	37.5631			1.49536			0.747682	−	0.664057i	$-0.231167\pi$
$631$	37.5631			1.49536			0.747682	+	0.664057i	$0.231167\pi$
$632$		0			0
$633$	1.43053	−	3.09802i	0.0568585	−	0.123135i
$634$		0			0
$635$	−4.54710	+	29.4410i	−0.180446	+	1.16833i
$636$		0			0
$637$	7.51597	+	32.9263i	0.297793	+	1.30459i
$638$		0			0
$639$	0.116958	−	0.0996413i	0.00462677	−	0.00394175i
$640$		0			0
$641$		−	6.57415i		−	0.259663i	−0.991536	−	0.129832i	$-0.958556\pi$
$641$		−	6.57415i		−	0.259663i	0.991536	−	0.129832i	$-0.0414437\pi$
$642$		0			0
$643$	10.2152			0.402849			0.201425	−	0.979504i	$-0.435443\pi$
$643$	10.2152			0.402849			0.201425	+	0.979504i	$0.435443\pi$
$644$		0			0
$645$	19.8718	+	13.1854i	0.782451	+	0.519176i
$646$		0			0
$647$		−	27.5779i		−	1.08420i	−0.840314	−	0.542100i	$-0.817629\pi$
$647$		−	27.5779i		−	1.08420i	0.840314	−	0.542100i	$-0.182371\pi$
$648$		0			0
$649$			6.24625i			0.245187i
$650$		0			0
$651$	10.0596	+	20.0503i	0.394266	+	0.785832i
$652$		0			0
$653$	4.89650			0.191615			0.0958074	−	0.995400i	$-0.469457\pi$
$653$	4.89650			0.191615			0.0958074	+	0.995400i	$0.469457\pi$
$654$		0			0
$655$	−34.6525	−	5.35200i	−1.35399	−	0.209120i
$656$		0			0
$657$	−5.57629	−	6.54538i	−0.217552	−	0.255359i
$658$		0			0
$659$		−	15.4272i		−	0.600958i	−0.953788	−	0.300479i	$-0.902853\pi$
$659$		−	15.4272i		−	0.600958i	0.953788	−	0.300479i	$-0.0971465\pi$
$660$		0			0
$661$		−	30.2132i		−	1.17516i	−0.809168	−	0.587578i	$-0.800082\pi$
$661$		−	30.2132i		−	1.17516i	0.809168	−	0.587578i	$-0.199918\pi$
$662$		0			0
$663$	59.8758	+	27.6481i	2.32539	+	1.07376i
$664$		0			0
$665$	17.5017	−	16.1215i	0.678688	−	0.625166i
$666$		0			0
$667$		−	11.4188i		−	0.442140i
$668$		0			0
$669$	2.09239	−	4.53138i	0.0808966	−	0.175193i
$670$		0			0
$671$	7.47959			0.288746
$672$		0			0
$673$		−	17.4276i		−	0.671784i	−0.941901	−	0.335892i	$-0.890962\pi$
$673$		−	17.4276i		−	0.671784i	0.941901	−	0.335892i	$-0.109038\pi$
$674$		0			0
$675$	−25.9662	−	0.868634i	−0.999441	−	0.0334337i
$676$		0			0
$677$		−	45.0367i		−	1.73090i	−0.500993	−	0.865451i	$-0.667032\pi$
$677$		−	45.0367i		−	1.73090i	0.500993	−	0.865451i	$-0.332968\pi$
$678$		0			0
$679$	−3.66513	−	2.92278i	−0.140655	−	0.112166i
$680$		0			0
$681$	10.1939	+	4.70709i	0.390631	+	0.180376i
$682$		0			0
$683$	15.8049			0.604758			0.302379	−	0.953188i	$-0.402219\pi$
$683$	15.8049			0.604758			0.302379	+	0.953188i	$0.402219\pi$
$684$		0			0
$685$	−1.91746	−	0.296147i	−0.0732624	−	0.0113152i
$686$		0			0
$687$	−3.51962	−	1.62521i	−0.134282	−	0.0620056i
$688$		0			0
$689$	47.4970			1.80949
$690$		0			0
$691$			37.7095i			1.43454i	0.696796	+	0.717269i	$0.254608\pi$
$691$			37.7095i			1.43454i	−0.696796	+	0.717269i	$0.745392\pi$
$692$		0			0
$693$	8.44121	−	0.273876i	0.320655	−	0.0104037i
$694$		0			0
$695$	7.79649	−	50.4798i	0.295738	−	1.91481i
$696$		0			0
$697$			40.1015i			1.51895i
$698$		0			0
$699$	−16.8114	+	36.4074i	−0.635864	+	1.37705i
$700$		0			0
$701$		−	17.3581i		−	0.655605i	−0.944746	−	0.327802i	$-0.893692\pi$
$701$		−	17.3581i		−	0.655605i	0.944746	−	0.327802i	$-0.106308\pi$
$702$		0			0
$703$	10.3091			0.388816
$704$		0			0
$705$	7.33219	+	4.86509i	0.276146	+	0.183230i
$706$		0			0
$707$	23.0949	+	18.4172i	0.868574	+	0.692649i
$708$		0			0
$709$	41.2298			1.54842			0.774210	−	0.632929i	$-0.218148\pi$
$709$	41.2298			1.54842			0.774210	+	0.632929i	$0.218148\pi$
$710$		0			0
$711$	13.6315	+	16.0005i	0.511223	+	0.600066i
$712$		0			0
$713$		−	27.8905i		−	1.04451i
$714$		0			0
$715$	1.75221	−	11.3450i	0.0655288	−	0.424278i
$716$		0			0
$717$	4.36876	+	2.01730i	0.163154	+	0.0753375i
$718$		0			0
$719$	12.2643			0.457380			0.228690	−	0.973499i	$-0.426556\pi$
$719$	12.2643			0.457380			0.228690	+	0.973499i	$0.426556\pi$
$720$		0			0
$721$	−23.0985	−	18.4200i	−0.860231	−	0.685997i
$722$		0			0
$723$	−27.3161	−	12.6134i	−1.01590	−	0.469097i
$724$		0			0
$725$	−3.02325	+	9.55382i	−0.112281	+	0.354820i
$726$		0			0
$727$	44.3194			1.64371			0.821857	−	0.569694i	$-0.192938\pi$
$727$	44.3194			1.64371			0.821857	+	0.569694i	$0.192938\pi$
$728$		0			0
$729$	23.0735	−	14.0219i	0.854573	−	0.519331i
$730$		0			0
$731$	48.5957			1.79738
$732$		0			0
$733$	−15.3583			−0.567270			−0.283635	−	0.958932i	$-0.591540\pi$
$733$	−15.3583			−0.567270			−0.283635	+	0.958932i	$0.591540\pi$
$734$		0			0
$735$	−25.3596	−	9.58607i	−0.935402	−	0.353587i
$736$		0			0
$737$	−11.5576			−0.425730
$738$		0			0
$739$	39.7697			1.46295			0.731476	−	0.681867i	$-0.238832\pi$
$739$	39.7697			1.46295			0.731476	+	0.681867i	$0.238832\pi$
$740$		0			0
$741$	30.5156	+	14.0908i	1.12102	+	0.517638i
$742$		0			0
$743$	19.9448			0.731703			0.365851	−	0.930673i	$-0.380778\pi$
$743$	19.9448			0.731703			0.365851	+	0.930673i	$0.380778\pi$
$744$		0			0
$745$	−6.12147	+	39.6346i	−0.224273	+	1.45210i
$746$		0			0
$747$	17.3406	−	14.7732i	0.634459	−	0.540523i
$748$		0			0
$749$	0.128974	+	0.102851i	0.00471260	+	0.00375809i
$750$		0			0
$751$	−10.7044			−0.390608			−0.195304	−	0.980743i	$-0.562569\pi$
$751$	−10.7044			−0.390608			−0.195304	+	0.980743i	$0.562569\pi$
$752$		0			0
$753$	1.09935	−	2.38081i	0.0400627	−	0.0867615i
$754$		0			0
$755$	39.1073	+	6.04004i	1.42326	+	0.219819i
$756$		0			0
$757$		−	13.5264i		−	0.491625i	−0.969317	−	0.245812i	$-0.920945\pi$
$757$		−	13.5264i		−	0.491625i	0.969317	−	0.245812i	$-0.0790547\pi$
$758$		0			0
$759$	−9.53336	−	4.40209i	−0.346039	−	0.159786i
$760$		0			0
$761$	−5.50054			−0.199394			−0.0996972	−	0.995018i	$-0.531787\pi$
$761$	−5.50054			−0.199394			−0.0996972	+	0.995018i	$0.531787\pi$
$762$		0			0
$763$	21.3257	+	17.0063i	0.772042	+	0.615669i
$764$		0			0
$765$	−45.0674	+	27.7791i	−1.62941	+	1.00436i
$766$		0			0
$767$	28.3225			1.02267
$768$		0			0
$769$			3.27831i			0.118219i	0.998252	+	0.0591094i	$0.0188261\pi$
$769$			3.27831i			0.118219i	−0.998252	+	0.0591094i	$0.981174\pi$
$770$		0			0
$771$	18.2430	+	8.42381i	0.657005	+	0.303376i
$772$		0			0
$773$		−	8.98512i		−	0.323172i	−0.986859	−	0.161586i	$-0.948339\pi$
$773$		−	8.98512i		−	0.323172i	0.986859	−	0.161586i	$-0.0516610\pi$
$774$		0			0
$775$	−7.38427	+	23.3352i	−0.265251	+	0.838224i
$776$		0			0
$777$	−5.26722	−	10.4984i	−0.188960	−	0.376627i
$778$		0			0
$779$			20.4377i			0.732256i
$780$		0			0
$781$	−0.0544962			−0.00195003
$782$		0			0
$783$	−2.80814	−	10.0281i	−0.100355	−	0.358376i
$784$		0			0
$785$	18.0573	+	2.78890i	0.644491	+	0.0995402i
$786$		0			0
$787$	−13.9746			−0.498142			−0.249071	−	0.968485i	$-0.580125\pi$
$787$	−13.9746			−0.498142			−0.249071	+	0.968485i	$0.580125\pi$
$788$		0			0
$789$	−8.40317	+	18.1983i	−0.299161	+	0.647876i
$790$		0			0
$791$	−19.0795	−	15.2151i	−0.678390	−	0.540986i
$792$		0			0
$793$		−	33.9149i		−	1.20435i
$794$		0			0
$795$	−21.0801	+	31.7698i	−0.747634	+	1.12676i
$796$		0			0
$797$		−	5.37816i		−	0.190504i	−0.995453	−	0.0952521i	$-0.969634\pi$
$797$		−	5.37816i		−	0.190504i	0.995453	−	0.0952521i	$-0.0303657\pi$
$798$		0			0
$799$	17.9306			0.634339
$800$		0			0
$801$	−18.9196	−	22.2076i	−0.668492	−	0.784667i
$802$		0			0
$803$			3.04981i			0.107625i
$804$		0			0
$805$	22.8371	+	24.7923i	0.804903	+	0.873814i
$806$		0			0
$807$	4.31955	−	9.35461i	0.152055	−	0.329298i
$808$		0			0
$809$		−	20.1832i		−	0.709602i	−0.934942	−	0.354801i	$-0.884549\pi$
$809$		−	20.1832i		−	0.709602i	0.934942	−	0.354801i	$-0.115451\pi$
$810$		0			0
$811$		−	12.4920i		−	0.438652i	−0.975652	−	0.219326i	$-0.929614\pi$
$811$		−	12.4920i		−	0.438652i	0.975652	−	0.219326i	$-0.0703859\pi$
$812$		0			0
$813$	−47.7961	−	22.0702i	−1.67628	−	0.774034i
$814$		0			0
$815$	−4.52796	+	29.3171i	−0.158607	+	1.02693i
$816$		0			0
$817$	24.7668			0.866479
$818$		0			0
$819$	−1.24184	−	38.2751i	−0.0433934	−	1.33744i
$820$		0			0
$821$		−	6.00612i		−	0.209615i	−0.994493	−	0.104807i	$-0.966577\pi$
$821$		−	6.00612i		−	0.209615i	0.994493	−	0.104807i	$-0.0334226\pi$
$822$		0			0
$823$			39.5492i			1.37860i	0.724476	+	0.689300i	$0.242082\pi$
$823$			39.5492i			1.37860i	−0.724476	+	0.689300i	$0.757918\pi$
$824$		0			0
$825$	6.81079	+	6.20715i	0.237121	+	0.216105i
$826$		0			0
$827$	−23.1841			−0.806191			−0.403095	−	0.915158i	$-0.632066\pi$
$827$	−23.1841			−0.806191			−0.403095	+	0.915158i	$0.632066\pi$
$828$		0			0
$829$			5.49703i			0.190920i	0.995433	+	0.0954599i	$0.0304322\pi$
$829$			5.49703i			0.190920i	−0.995433	+	0.0954599i	$0.969568\pi$
$830$		0			0
$831$	−7.14817	−	3.30071i	−0.247967	−	0.114500i
$832$		0			0
$833$	−53.8585	+	12.2941i	−1.86609	+	0.425964i
$834$		0			0
$835$	4.11696	−	26.6560i	0.142473	−	0.922469i
$836$		0			0
$837$	−6.85888	−	24.4936i	−0.237078	−	0.846624i
$838$		0			0
$839$	42.1549			1.45535			0.727674	−	0.685923i	$-0.240601\pi$
$839$	42.1549			1.45535			0.727674	+	0.685923i	$0.240601\pi$
$840$		0			0
$841$	24.9834			0.861496
$842$		0			0
$843$	49.2263	+	22.7306i	1.69544	+	0.782882i
$844$		0			0
$845$	−22.7135	−	3.50805i	−0.781369	−	0.120681i
$846$		0			0
$847$	20.4120	+	16.2777i	0.701365	+	0.559308i
$848$		0			0
$849$	−18.3786	+	39.8015i	−0.630752	+	1.36598i
$850$		0			0
$851$			14.6035i			0.500602i
$852$		0			0
$853$	−47.5307			−1.62742			−0.813711	−	0.581270i	$-0.802556\pi$
$853$	−47.5307			−1.62742			−0.813711	+	0.581270i	$0.802556\pi$
$854$		0			0
$855$	−22.9685	+	14.1576i	−0.785507	+	0.484179i
$856$		0			0
$857$			2.24556i			0.0767070i	0.999264	+	0.0383535i	$0.0122113\pi$
$857$			2.24556i			0.0767070i	−0.999264	+	0.0383535i	$0.987789\pi$
$858$		0			0
$859$			11.0803i			0.378055i	0.981972	+	0.189028i	$0.0605336\pi$
$859$			11.0803i			0.378055i	−0.981972	+	0.189028i	$0.939466\pi$
$860$		0			0
$861$	20.8128	−	10.4422i	0.709299	−	0.355868i
$862$		0			0
$863$	15.2792			0.520108			0.260054	−	0.965594i	$-0.416260\pi$
$863$	15.2792			0.520108			0.260054	+	0.965594i	$0.416260\pi$
$864$		0			0
$865$	−4.03089	+	26.0987i	−0.137054	+	0.887383i
$866$		0			0
$867$	−32.8807	+	71.2079i	−1.11669	+	2.41835i
$868$		0			0
$869$		−	7.45542i		−	0.252908i
$870$		0			0
$871$			52.4058i			1.77570i
$872$		0			0
$873$	3.44714	+	4.04621i	0.116668	+	0.136943i
$874$		0			0
$875$	−12.5432	−	26.7893i	−0.424037	−	0.905645i
$876$		0			0
$877$			8.67945i			0.293084i	0.989204	+	0.146542i	$0.0468144\pi$
$877$			8.67945i			0.293084i	−0.989204	+	0.146542i	$0.953186\pi$
$878$		0			0
$879$	35.2459	+	16.2750i	1.18881	+	0.548943i
$880$		0			0
$881$	25.6150			0.862992			0.431496	−	0.902115i	$-0.357986\pi$
$881$	25.6150			0.862992			0.431496	+	0.902115i	$0.357986\pi$
$882$		0			0
$883$			22.6666i			0.762791i	0.924412	+	0.381396i	$0.124556\pi$
$883$			22.6666i			0.762791i	−0.924412	+	0.381396i	$0.875444\pi$
$884$		0			0
$885$	−12.5701	+	18.9444i	−0.422539	+	0.636809i
$886$		0			0
$887$		−	13.5948i		−	0.456470i	−0.973606	−	0.228235i	$-0.926705\pi$
$887$		−	13.5948i		−	0.456470i	0.973606	−	0.228235i	$-0.0732954\pi$
$888$		0			0
$889$	21.9766	−	27.5583i	0.737070	−	0.924277i
$890$		0			0
$891$	−9.45483	−	1.52148i	−0.316749	−	0.0509717i
$892$		0			0
$893$	9.13831			0.305802
$894$		0			0
$895$	−8.12866	+	52.6305i	−0.271711	+	1.75924i
$896$		0			0
$897$	−19.9605	+	43.2273i	−0.666461	+	1.44332i
$898$		0			0
$899$	−9.81058			−0.327201
$900$		0			0
$901$			77.6920i			2.58830i
$902$		0			0
$903$	−12.6540	−	25.2214i	−0.421099	−	0.839314i
$904$		0			0
$905$	5.05760	−	32.7463i	0.168120	−	1.08852i
$906$		0			0
$907$			14.0054i			0.465043i	0.972591	+	0.232522i	$0.0746976\pi$
$907$			14.0054i			0.465043i	−0.972591	+	0.232522i	$0.925302\pi$
$908$		0			0
$909$	−21.7213	−	25.4962i	−0.720451	−	0.845656i
$910$		0			0
$911$		−	9.16683i		−	0.303711i	−0.988403	−	0.151855i	$-0.951475\pi$
$911$		−	9.16683i		−	0.303711i	0.988403	−	0.151855i	$-0.0485248\pi$
$912$		0			0
$913$	−8.07982			−0.267403
$914$		0			0
$915$	22.6850	+	15.0521i	0.749944	+	0.497606i
$916$		0			0
$917$	32.4366	+	25.8667i	1.07115	+	0.854195i
$918$		0			0
$919$	29.2538			0.964994			0.482497	−	0.875898i	$-0.339730\pi$
$919$	29.2538			0.964994			0.482497	+	0.875898i	$0.339730\pi$
$920$		0			0
$921$	18.8523	−	40.8274i	0.621205	−	1.34531i
$922$		0			0
$923$			0.247103i			0.00813351i
$924$		0			0
$925$	3.86642	−	12.2183i	0.127127	−	0.401737i
$926$		0			0
$927$	21.7247	+	25.5001i	0.713531	+	0.837533i
$928$		0			0
$929$	−36.3268			−1.19184			−0.595922	−	0.803042i	$-0.703213\pi$
$929$	−36.3268			−1.19184			−0.595922	+	0.803042i	$0.703213\pi$
$930$		0			0
$931$	−27.4489	+	6.26565i	−0.899601	+	0.205348i
$932$		0			0
$933$	10.1651	−	22.0141i	0.332792	−	0.720708i
$934$		0			0
$935$	18.5573	+	2.86613i	0.606888	+	0.0937325i
$936$		0			0
$937$	54.6383			1.78496			0.892478	−	0.451091i	$-0.148965\pi$
$937$	54.6383			1.78496			0.892478	+	0.451091i	$0.148965\pi$
$938$		0			0
$939$	17.2310	−	37.3163i	0.562313	−	1.21777i
$940$		0			0
$941$	−17.2428			−0.562098			−0.281049	−	0.959693i	$-0.590682\pi$
$941$	−17.2428			−0.562098			−0.281049	+	0.959693i	$0.590682\pi$
$942$		0			0
$943$	−28.9512			−0.942782
$944$		0			0
$945$	26.1527	+	16.1566i	0.850747	+	0.525575i
$946$		0			0
$947$	12.9432			0.420598			0.210299	−	0.977637i	$-0.432556\pi$
$947$	12.9432			0.420598			0.210299	+	0.977637i	$0.432556\pi$
$948$		0			0
$949$	13.8288			0.448902
$950$		0			0
$951$	15.3318	−	33.2033i	0.497168	−	1.07669i
$952$		0			0
$953$	−26.8909			−0.871081			−0.435541	−	0.900169i	$-0.643443\pi$
$953$	−26.8909			−0.871081			−0.435541	+	0.900169i	$0.643443\pi$
$954$		0			0
$955$	5.59420	−	36.2207i	0.181024	−	1.17207i
$956$		0			0
$957$	−1.54845	+	3.35339i	−0.0500543	+	0.108400i
$958$		0			0
$959$	1.79484	+	1.43131i	0.0579585	+	0.0462193i
$960$		0			0
$961$	7.03770			0.227022
$962$		0			0
$963$	−0.121303	−	0.142384i	−0.00390893	−	0.00458825i
$964$		0			0
$965$	7.78202	−	50.3861i	0.250512	−	1.62199i
$966$		0			0
$967$		−	53.5118i		−	1.72082i	−0.509600	−	0.860411i	$-0.670207\pi$
$967$		−	53.5118i		−	1.72082i	0.509600	−	0.860411i	$-0.329793\pi$
$968$		0			0
$969$	−23.0487	+	49.9152i	−0.740430	+	1.60351i
$970$		0			0
$971$	−55.3949			−1.77771			−0.888854	−	0.458191i	$-0.848497\pi$
$971$	−55.3949			−1.77771			−0.888854	+	0.458191i	$0.848497\pi$
$972$		0			0
$973$	−37.6812	+	47.2517i	−1.20800	+	1.51482i
$974$		0			0
$975$	28.1452	−	30.8823i	0.901368	−	0.989025i
$976$		0			0
$977$	−9.77243			−0.312648			−0.156324	−	0.987706i	$-0.549964\pi$
$977$	−9.77243			−0.312648			−0.156324	+	0.987706i	$0.549964\pi$
$978$		0			0
$979$			10.3476i			0.330711i
$980$		0			0
$981$	−20.0573	−	23.5430i	−0.640381	−	0.751671i
$982$		0			0
$983$		−	22.2706i		−	0.710321i	−0.934805	−	0.355160i	$-0.884426\pi$
$983$		−	22.2706i		−	0.710321i	0.934805	−	0.355160i	$-0.115574\pi$
$984$		0			0
$985$	48.3919	+	7.47402i	1.54190	+	0.238142i
$986$		0			0
$987$	−4.66901	−	9.30605i	−0.148616	−	0.296215i
$988$		0			0
$989$			35.0836i			1.11559i
$990$		0			0
$991$	13.9760			0.443964			0.221982	−	0.975051i	$-0.428747\pi$
$991$	13.9760			0.443964			0.221982	+	0.975051i	$0.428747\pi$
$992$		0			0
$993$	9.20311	−	19.9307i	0.292052	−	0.632480i
$994$		0			0
$995$	−3.95036	+	25.5773i	−0.125235	+	0.810856i
$996$		0			0
$997$	−45.0601			−1.42707			−0.713534	−	0.700620i	$-0.752907\pi$
$997$	−45.0601			−1.42707			−0.713534	+	0.700620i	$0.752907\pi$
$998$		0			0
$999$	3.59133	+	12.8249i	0.113625	+	0.405763i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.k.h.209.15		24
3.2	odd	2		1680.2.k.i.209.16		24
4.3	odd	2		840.2.k.b.209.10	yes	24
5.4	even	2		1680.2.k.i.209.10		24
7.6	odd	2	inner	1680.2.k.h.209.10		24
12.11	even	2		840.2.k.a.209.9	✓	24
15.14	odd	2	inner	1680.2.k.h.209.9		24
20.19	odd	2		840.2.k.a.209.15	yes	24
21.20	even	2		1680.2.k.i.209.9		24
28.27	even	2		840.2.k.b.209.15	yes	24
35.34	odd	2		1680.2.k.i.209.15		24
60.59	even	2		840.2.k.b.209.16	yes	24
84.83	odd	2		840.2.k.a.209.16	yes	24
105.104	even	2	inner	1680.2.k.h.209.16		24
140.139	even	2		840.2.k.a.209.10	yes	24
420.419	odd	2		840.2.k.b.209.9	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
840.2.k.a.209.9	✓	24	12.11	even	2
840.2.k.a.209.10	yes	24	140.139	even	2
840.2.k.a.209.15	yes	24	20.19	odd	2
840.2.k.a.209.16	yes	24	84.83	odd	2
840.2.k.b.209.9	yes	24	420.419	odd	2
840.2.k.b.209.10	yes	24	4.3	odd	2
840.2.k.b.209.15	yes	24	28.27	even	2
840.2.k.b.209.16	yes	24	60.59	even	2
1680.2.k.h.209.9		24	15.14	odd	2	inner
1680.2.k.h.209.10		24	7.6	odd	2	inner
1680.2.k.h.209.15		24	1.1	even	1	trivial
1680.2.k.h.209.16		24	105.104	even	2	inner
1680.2.k.i.209.9		24	21.20	even	2
1680.2.k.i.209.10		24	5.4	even	2
1680.2.k.i.209.15		24	35.34	odd	2
1680.2.k.i.209.16		24	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.k (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.726113 - 1.57250i) q^{3} +(-2.20987 - 0.341309i) q^{5} +(2.06855 + 1.64958i) q^{7} +(-1.94552 - 2.28363i) q^{9} +O(q^{10})\)	\(q+(0.726113 - 1.57250i) q^{3} +(-2.20987 - 0.341309i) q^{5} +(2.06855 + 1.64958i) q^{7} +(-1.94552 - 2.28363i) q^{9} +1.06405i q^{11} +4.82475 q^{13} +(-2.14132 + 3.22719i) q^{15} +7.89197i q^{17} +4.02213i q^{19} +(4.09596 - 2.05502i) q^{21} -5.69760 q^{23} +(4.76702 + 1.50849i) q^{25} +(-5.00367 + 1.40116i) q^{27} +2.00415i q^{29} +4.89513i q^{31} +(1.67322 + 0.772622i) q^{33} +(-4.00820 - 4.35136i) q^{35} -2.56310i q^{37} +(3.50331 - 7.58693i) q^{39} +5.08131 q^{41} -6.15762i q^{43} +(3.51992 + 5.71053i) q^{45} -2.27201i q^{47} +(1.55779 + 6.82446i) q^{49} +(12.4101 + 5.73046i) q^{51} +9.84443 q^{53} +(0.363170 - 2.35141i) q^{55} +(6.32481 + 2.92052i) q^{57} +5.87025 q^{59} -7.02935i q^{61} +(-0.257389 - 7.93308i) q^{63} +(-10.6621 - 1.64673i) q^{65} +10.8619i q^{67} +(-4.13710 + 8.95948i) q^{69} +0.0512158i q^{71} +2.86622 q^{73} +(5.83350 - 6.40080i) q^{75} +(-1.75524 + 2.20104i) q^{77} -7.00663 q^{79} +(-1.42990 + 8.88568i) q^{81} +7.59344i q^{83} +(2.69360 - 17.4402i) q^{85} +(3.15153 + 1.45524i) q^{87} +9.72471 q^{89} +(9.98024 + 7.95880i) q^{91} +(7.69760 + 3.55442i) q^{93} +(1.37279 - 8.88838i) q^{95} -1.77183 q^{97} +(2.42990 - 2.07014i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 24 q - 2 q^{9}+O(q^{10}) \) 24 * q - 2 * q^9	\( 24 q - 2 q^{9} - 2 q^{15} - 2 q^{21} + 16 q^{23} + 8 q^{25} - 8 q^{35} + 2 q^{39} - 6 q^{51} + 24 q^{53} + 8 q^{57} - 16 q^{63} + 16 q^{65} + 8 q^{77} - 4 q^{79} + 18 q^{81} - 12 q^{85} - 12 q^{91} + 32 q^{93} + 24 q^{95} + 6 q^{99}+O(q^{100}) \) 24 * q - 2 * q^9 - 2 * q^15 - 2 * q^21 + 16 * q^23 + 8 * q^25 - 8 * q^35 + 2 * q^39 - 6 * q^51 + 24 * q^53 + 8 * q^57 - 16 * q^63 + 16 * q^65 + 8 * q^77 - 4 * q^79 + 18 * q^81 - 12 * q^85 - 12 * q^91 + 32 * q^93 + 24 * q^95 + 6 * q^99

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