LMFDB - Embedded newform 1680.2.k.e.209.1

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 = [8,0,0,0,0,0,0,0,0,0,0,0,0,0,0] 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:	$8$
Coefficient field:	8.0.10070523904.11
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 10x^{4} + 81 $ x^8 - 10*x^4 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 210)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.1
Root			$1.68014 - 0.420861i$ of defining polynomial
Character	$\chi$	$=$	1680.209
Dual form			1680.2.k.e.209.2

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.68014 - 0.420861i) q^{3} +(1.08495 - 1.95522i) q^{5} +(0.595188 + 2.57794i) q^{7} +(2.64575 + 1.41421i) q^{9} +2.82843i q^{11} -3.36028 q^{13} +(-2.64575 + 2.82843i) q^{15} +4.75216i q^{17} -5.59388i q^{19} +(0.0849536 - 4.58179i) q^{21} -7.29150 q^{23} +(-2.64575 - 4.24264i) q^{25} +(-3.85005 - 3.48957i) q^{27} +0.500983i q^{29} +3.06871i q^{31} +(1.19038 - 4.75216i) q^{33} +(5.68618 + 1.63322i) q^{35} +3.32941i q^{37} +(5.64575 + 1.41421i) q^{39} -4.33981 q^{41} -10.3117i q^{43} +(5.63561 - 3.63866i) q^{45} -7.82087i q^{47} +(-6.29150 + 3.06871i) q^{49} +(2.00000 - 7.98430i) q^{51} -8.58301 q^{53} +(5.53019 + 3.06871i) q^{55} +(-2.35425 + 9.39851i) q^{57} -2.16991 q^{59} -2.52517i q^{61} +(-2.07103 + 7.66230i) q^{63} +(-3.64575 + 6.57008i) q^{65} +10.3117i q^{67} +(12.2508 + 3.06871i) q^{69} +9.81076i q^{71} -5.53019 q^{73} +(2.65967 + 8.24173i) q^{75} +(-7.29150 + 1.68345i) q^{77} -3.29150 q^{79} +(5.00000 + 7.48331i) q^{81} +6.97915i q^{83} +(9.29150 + 5.15587i) q^{85} +(0.210845 - 0.841723i) q^{87} -15.8219 q^{89} +(-2.00000 - 8.66259i) q^{91} +(1.29150 - 5.15587i) q^{93} +(-10.9373 - 6.06910i) q^{95} +14.6315 q^{97} +(-4.00000 + 7.48331i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{21} - 16 q^{23} - 16 q^{35} + 24 q^{39} - 8 q^{49} + 16 q^{51} + 16 q^{53} - 40 q^{57} - 8 q^{63} - 8 q^{65} - 16 q^{77} + 16 q^{79} + 40 q^{81} + 32 q^{85} - 16 q^{91} - 32 q^{93} - 24 q^{95}+ \cdots - 32 q^{99}+O(q^{100}) $ 8 * q - 8 * q^21 - 16 * q^23 - 16 * q^35 + 24 * q^39 - 8 * q^49 + 16 * q^51 + 16 * q^53 - 40 * q^57 - 8 * q^63 - 8 * q^65 - 16 * q^77 + 16 * q^79 + 40 * q^81 + 32 * q^85 - 16 * q^91 - 32 * q^93 - 24 * q^95 - 32 * 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
$3$	−1.68014	−	0.420861i	−0.970030	−	0.242984i
$3$	−1.68014	−	0.420861i	−0.970030	−	0.242984i
$4$		0			0
$5$	1.08495	−	1.95522i	0.485206	−	0.874400i
$5$	1.08495	−	1.95522i	0.485206	−	0.874400i
$6$		0			0
$7$	0.595188	+	2.57794i	0.224960	+	0.974368i
$7$	0.595188	+	2.57794i	0.224960	+	0.974368i
$8$		0			0
$9$	2.64575	+	1.41421i	0.881917	+	0.471405i
$10$		0			0
$11$			2.82843i			0.852803i	0.904534	+	0.426401i	$0.140219\pi$
$11$			2.82843i			0.852803i	−0.904534	+	0.426401i	$0.859781\pi$
$12$		0			0
$13$	−3.36028			−0.931975			−0.465987	−	0.884791i	$-0.654301\pi$
$13$	−3.36028			−0.931975			−0.465987	+	0.884791i	$0.654301\pi$
$14$		0			0
$15$	−2.64575	+	2.82843i	−0.683130	+	0.730297i
$16$		0			0
$17$			4.75216i			1.15257i	0.817250	+	0.576284i	$0.195498\pi$
$17$			4.75216i			1.15257i	−0.817250	+	0.576284i	$0.804502\pi$
$18$		0			0
$19$		−	5.59388i		−	1.28332i	−0.766987	−	0.641662i	$-0.778245\pi$
$19$		−	5.59388i		−	1.28332i	0.766987	−	0.641662i	$-0.221755\pi$
$20$		0			0
$21$	0.0849536	−	4.58179i	0.0185384	−	0.999828i
$22$		0			0
$23$	−7.29150			−1.52038			−0.760192	−	0.649699i	$-0.774895\pi$
$23$	−7.29150			−1.52038			−0.760192	+	0.649699i	$0.774895\pi$
$24$		0			0
$25$	−2.64575	−	4.24264i	−0.529150	−	0.848528i
$26$		0			0
$27$	−3.85005	−	3.48957i	−0.740942	−	0.671569i
$28$		0			0
$29$			0.500983i			0.0930303i	0.998918	+	0.0465151i	$0.0148116\pi$
$29$			0.500983i			0.0930303i	−0.998918	+	0.0465151i	$0.985188\pi$
$30$		0			0
$31$			3.06871i			0.551157i	0.961279	+	0.275578i	$0.0888694\pi$
$31$			3.06871i			0.551157i	−0.961279	+	0.275578i	$0.911131\pi$
$32$		0			0
$33$	1.19038	−	4.75216i	0.207218	−	0.827245i
$34$		0			0
$35$	5.68618	+	1.63322i	0.961139	+	0.276064i
$36$		0			0
$37$			3.32941i			0.547352i	0.961822	+	0.273676i	$0.0882396\pi$
$37$			3.32941i			0.547352i	−0.961822	+	0.273676i	$0.911760\pi$
$38$		0			0
$39$	5.64575	+	1.41421i	0.904044	+	0.226455i
$40$		0			0
$41$	−4.33981			−0.677765			−0.338883	−	0.940829i	$-0.610049\pi$
$41$	−4.33981			−0.677765			−0.338883	+	0.940829i	$0.610049\pi$
$42$		0			0
$43$		−	10.3117i		−	1.57253i	−0.617892	−	0.786263i	$-0.712013\pi$
$43$		−	10.3117i		−	1.57253i	0.617892	−	0.786263i	$-0.287987\pi$
$44$		0			0
$45$	5.63561	−	3.63866i	0.840108	−	0.542420i
$46$		0			0
$47$		−	7.82087i		−	1.14079i	−0.821370	−	0.570396i	$-0.806790\pi$
$47$		−	7.82087i		−	1.14079i	0.821370	−	0.570396i	$-0.193210\pi$
$48$		0			0
$49$	−6.29150	+	3.06871i	−0.898786	+	0.438387i
$50$		0			0
$51$	2.00000	−	7.98430i	0.280056	−	1.11803i
$52$		0			0
$53$	−8.58301			−1.17897			−0.589483	−	0.807781i	$-0.700669\pi$
$53$	−8.58301			−1.17897			−0.589483	+	0.807781i	$0.700669\pi$
$54$		0			0
$55$	5.53019	+	3.06871i	0.745691	+	0.413785i
$56$		0			0
$57$	−2.35425	+	9.39851i	−0.311828	+	1.24486i
$58$		0			0
$59$	−2.16991			−0.282498			−0.141249	−	0.989974i	$-0.545112\pi$
$59$	−2.16991			−0.282498			−0.141249	+	0.989974i	$0.545112\pi$
$60$		0			0
$61$		−	2.52517i		−	0.323315i	−0.986847	−	0.161657i	$-0.948316\pi$
$61$		−	2.52517i		−	0.323315i	0.986847	−	0.161657i	$-0.0516839\pi$
$62$		0			0
$63$	−2.07103	+	7.66230i	−0.260926	+	0.965359i
$64$		0			0
$65$	−3.64575	+	6.57008i	−0.452200	+	0.814919i
$66$		0			0
$67$			10.3117i			1.25978i	0.776684	+	0.629890i	$0.216900\pi$
$67$			10.3117i			1.25978i	−0.776684	+	0.629890i	$0.783100\pi$
$68$		0			0
$69$	12.2508	+	3.06871i	1.47482	+	0.369430i
$70$		0			0
$71$			9.81076i			1.16432i	0.813073	+	0.582161i	$0.197793\pi$
$71$			9.81076i			1.16432i	−0.813073	+	0.582161i	$0.802207\pi$
$72$		0			0
$73$	−5.53019			−0.647260			−0.323630	−	0.946184i	$-0.604903\pi$
$73$	−5.53019			−0.647260			−0.323630	+	0.946184i	$0.604903\pi$
$74$		0			0
$75$	2.65967	+	8.24173i	0.307113	+	0.951673i
$76$		0			0
$77$	−7.29150	+	1.68345i	−0.830944	+	0.191846i
$78$		0			0
$79$	−3.29150			−0.370323			−0.185161	−	0.982708i	$-0.559281\pi$
$79$	−3.29150			−0.370323			−0.185161	+	0.982708i	$0.559281\pi$
$80$		0			0
$81$	5.00000	+	7.48331i	0.555556	+	0.831479i
$82$		0			0
$83$			6.97915i			0.766061i	0.923736	+	0.383030i	$0.125120\pi$
$83$			6.97915i			0.766061i	−0.923736	+	0.383030i	$0.874880\pi$
$84$		0			0
$85$	9.29150	+	5.15587i	1.00780	+	0.559233i
$86$		0			0
$87$	0.210845	−	0.841723i	0.0226049	−	0.0902422i
$88$		0			0
$89$	−15.8219			−1.67712			−0.838558	−	0.544812i	$-0.816601\pi$
$89$	−15.8219			−1.67712			−0.838558	+	0.544812i	$0.816601\pi$
$90$		0			0
$91$	−2.00000	−	8.66259i	−0.209657	−	0.908086i
$92$		0			0
$93$	1.29150	−	5.15587i	0.133923	−	0.534639i
$94$		0			0
$95$	−10.9373	−	6.06910i	−1.12214	−	0.622677i
$96$		0			0
$97$	14.6315			1.48560			0.742802	−	0.669511i	$-0.233496\pi$
$97$	14.6315			1.48560			0.742802	+	0.669511i	$0.233496\pi$
$98$		0			0
$99$	−4.00000	+	7.48331i	−0.402015	+	0.752101i
$100$		0			0
$101$	−2.16991			−0.215914			−0.107957	−	0.994156i	$-0.534431\pi$
$101$	−2.16991			−0.215914			−0.107957	+	0.994156i	$0.534431\pi$
$102$		0			0
$103$	−3.14944			−0.310323			−0.155162	−	0.987889i	$-0.549590\pi$
$103$	−3.14944			−0.310323			−0.155162	+	0.987889i	$0.549590\pi$
$104$		0			0
$105$	−8.86622	−	5.13713i	−0.865255	−	0.501333i
$106$		0			0
$107$		0			0			−	1.00000i	$-0.5\pi$
$107$		0			0				1.00000i	$0.5\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$	1.40122	−	5.59388i	0.132998	−	0.530948i
$112$		0			0
$113$	−6.00000			−0.564433			−0.282216	−	0.959351i	$-0.591070\pi$
$113$	−6.00000			−0.564433			−0.282216	+	0.959351i	$0.591070\pi$
$114$		0			0
$115$	−7.91094	+	14.2565i	−0.737699	+	1.32942i
$116$		0			0
$117$	−8.89047	−	4.75216i	−0.821925	−	0.439337i
$118$		0			0
$119$	−12.2508	+	2.82843i	−1.12303	+	0.259281i
$120$		0			0
$121$	3.00000			0.272727
$122$		0			0
$123$	7.29150	+	1.82646i	0.657453	+	0.164686i
$124$		0			0
$125$	−11.1658	+	0.569951i	−0.998700	+	0.0509780i
$126$		0			0
$127$			3.32941i			0.295437i	0.989029	+	0.147719i	$0.0471930\pi$
$127$			3.32941i			0.295437i	−0.989029	+	0.147719i	$0.952807\pi$
$128$		0			0
$129$	−4.33981	+	17.3252i	−0.382099	+	1.52540i
$130$		0			0
$131$	−17.9918			−1.57195			−0.785975	−	0.618258i	$-0.787839\pi$
$131$	−17.9918			−1.57195			−0.785975	+	0.618258i	$0.787839\pi$
$132$		0			0
$133$	14.4207	−	3.32941i	1.25043	−	0.288696i
$134$		0			0
$135$	−11.0000	+	3.74166i	−0.946729	+	0.322031i
$136$		0			0
$137$	1.29150			0.110341			0.0551703	−	0.998477i	$-0.482430\pi$
$137$	1.29150			0.110341			0.0551703	+	0.998477i	$0.482430\pi$
$138$		0			0
$139$		−	0.543544i		−	0.0461028i	−0.999734	−	0.0230514i	$-0.992662\pi$
$139$		−	0.543544i		−	0.0461028i	0.999734	−	0.0230514i	$-0.00733813\pi$
$140$		0			0
$141$	−3.29150	+	13.1402i	−0.277195	+	1.10660i
$142$		0			0
$143$		−	9.50432i		−	0.794791i
$144$		0			0
$145$	0.979531	+	0.543544i	0.0813457	+	0.0451388i
$146$		0			0
$147$	11.8621	−	2.50802i	0.978371	−	0.206858i
$148$		0			0
$149$			10.8127i			0.885813i	0.896568	+	0.442906i	$0.146053\pi$
$149$			10.8127i			0.885813i	−0.896568	+	0.442906i	$0.853947\pi$
$150$		0			0
$151$	1.41699			0.115313			0.0576567	−	0.998336i	$-0.481637\pi$
$151$	1.41699			0.115313			0.0576567	+	0.998336i	$0.481637\pi$
$152$		0			0
$153$	−6.72057	+	12.5730i	−0.543326	+	1.01647i
$154$		0			0
$155$	6.00000	+	3.32941i	0.481932	+	0.267425i
$156$		0			0
$157$	−16.3797			−1.30724			−0.653622	−	0.756821i	$-0.726751\pi$
$157$	−16.3797			−1.30724			−0.653622	+	0.756821i	$0.726751\pi$
$158$		0			0
$159$	14.4207	+	3.61226i	1.14363	+	0.286471i
$160$		0			0
$161$	−4.33981	−	18.7970i	−0.342025	−	1.48141i
$162$		0			0
$163$		0			0		1.00000			$0$
$163$		0			0		−1.00000			$\pi$
$164$		0			0
$165$	−8.00000	−	7.48331i	−0.622799	−	0.582575i
$166$		0			0
$167$			20.6921i			1.60120i	0.599198	+	0.800601i	$0.295486\pi$
$167$			20.6921i			1.60120i	−0.599198	+	0.800601i	$0.704514\pi$
$168$		0			0
$169$	−1.70850			−0.131423
$170$		0			0
$171$	7.91094	−	14.8000i	0.604965	−	1.13179i
$172$		0			0
$173$			2.22699i			0.169315i	0.996410	+	0.0846574i	$0.0269796\pi$
$173$			2.22699i			0.169315i	−0.996410	+	0.0846574i	$0.973020\pi$
$174$		0			0
$175$	9.36253	−	9.34574i	0.707741	−	0.706472i
$176$		0			0
$177$	3.64575	+	0.913230i	0.274031	+	0.0686426i
$178$		0			0
$179$		−	24.4539i		−	1.82777i	−0.405975	−	0.913884i	$-0.633068\pi$
$179$		−	24.4539i		−	1.82777i	0.405975	−	0.913884i	$-0.366932\pi$
$180$		0			0
$181$		−	14.8000i		−	1.10008i	−0.835139	−	0.550038i	$-0.814613\pi$
$181$		−	14.8000i		−	1.10008i	0.835139	−	0.550038i	$-0.185387\pi$
$182$		0			0
$183$	−1.06275	+	4.24264i	−0.0785604	+	0.313625i
$184$		0			0
$185$	6.50972	+	3.61226i	0.478604	+	0.265578i
$186$		0			0
$187$	−13.4411			−0.982913
$188$		0			0
$189$	6.70439	−	12.0021i	0.487673	−	0.873026i
$190$		0			0
$191$		−	0.500983i		−	0.0362499i	−0.999836	−	0.0181249i	$-0.994230\pi$
$191$		−	0.500983i		−	0.0362499i	0.999836	−	0.0181249i	$-0.00576966\pi$
$192$		0			0
$193$			8.48528i			0.610784i	0.952227	+	0.305392i	$0.0987875\pi$
$193$			8.48528i			0.610784i	−0.952227	+	0.305392i	$0.901213\pi$
$194$		0			0
$195$	8.89047	−	9.50432i	0.636660	−	0.680618i
$196$		0			0
$197$	18.0000			1.28245			0.641223	−	0.767354i	$-0.278427\pi$
$197$	18.0000			1.28245			0.641223	+	0.767354i	$0.278427\pi$
$198$		0			0
$199$			9.20614i			0.652606i	0.945265	+	0.326303i	$0.105803\pi$
$199$			9.20614i			0.652606i	−0.945265	+	0.326303i	$0.894197\pi$
$200$		0			0
$201$	4.33981	−	17.3252i	0.306107	−	1.22202i
$202$		0			0
$203$	−1.29150	+	0.298179i	−0.0906457	+	0.0209281i
$204$		0			0
$205$	−4.70850	+	8.48528i	−0.328856	+	0.592638i
$206$		0			0
$207$	−19.2915	−	10.3117i	−1.34085	−	0.716716i
$208$		0			0
$209$	15.8219			1.09442
$210$		0			0
$211$	21.1660			1.45713			0.728564	−	0.684978i	$-0.240188\pi$
$211$	21.1660			1.45713			0.728564	+	0.684978i	$0.240188\pi$
$212$		0			0
$213$	4.12897	−	16.4835i	0.282912	−	1.12943i
$214$		0			0
$215$	−20.1617	−	11.1878i	−1.37502	−	0.762999i
$216$		0			0
$217$	−7.91094	+	1.82646i	−0.537030	+	0.123988i
$218$		0			0
$219$	9.29150	+	2.32744i	0.627862	+	0.157274i
$220$		0			0
$221$		−	15.9686i		−	1.07416i
$222$		0			0
$223$	1.19038			0.0797135			0.0398567	−	0.999205i	$-0.487310\pi$
$223$	1.19038			0.0797135			0.0398567	+	0.999205i	$0.487310\pi$
$224$		0			0
$225$	−1.00000	−	14.9666i	−0.0666667	−	0.997775i
$226$		0			0
$227$			18.1669i			1.20578i	0.797824	+	0.602890i	$0.205984\pi$
$227$			18.1669i			1.20578i	−0.797824	+	0.602890i	$0.794016\pi$
$228$		0			0
$229$			24.9007i			1.64548i	0.568415	+	0.822742i	$0.307557\pi$
$229$			24.9007i			1.64548i	−0.568415	+	0.822742i	$0.692443\pi$
$230$		0			0
$231$	12.9593	+	0.240285i	0.852656	+	0.0158096i
$232$		0			0
$233$	13.2915			0.870755			0.435378	−	0.900248i	$-0.356615\pi$
$233$	13.2915			0.870755			0.435378	+	0.900248i	$0.356615\pi$
$234$		0			0
$235$	−15.2915	−	8.48528i	−0.997508	−	0.553519i
$236$		0			0
$237$	5.53019	+	1.38527i	0.359224	+	0.0899827i
$238$		0			0
$239$			14.6431i			0.947185i	0.880744	+	0.473592i	$0.157043\pi$
$239$			14.6431i			0.947185i	−0.880744	+	0.473592i	$0.842957\pi$
$240$		0			0
$241$		−	17.3252i		−	1.11601i	−0.829836	−	0.558007i	$-0.811566\pi$
$241$		−	17.3252i		−	1.11601i	0.829836	−	0.558007i	$-0.188434\pi$
$242$		0			0
$243$	−5.25127	−	14.6773i	−0.336869	−	0.941551i
$244$		0			0
$245$	−0.825989	+	15.6307i	−0.0527705	+	0.998607i
$246$		0			0
$247$			18.7970i			1.19603i
$248$		0			0
$249$	2.93725	−	11.7260i	0.186141	−	0.743102i
$250$		0			0
$251$	6.50972			0.410890			0.205445	−	0.978669i	$-0.434136\pi$
$251$	6.50972			0.410890			0.205445	+	0.978669i	$0.434136\pi$
$252$		0			0
$253$		−	20.6235i		−	1.29659i
$254$		0			0
$255$	−13.4411	−	12.5730i	−0.841716	−	0.787354i
$256$		0			0
$257$		−	18.7105i		−	1.16713i	−0.812068	−	0.583563i	$-0.801658\pi$
$257$		−	18.7105i		−	1.16713i	0.812068	−	0.583563i	$-0.198342\pi$
$258$		0			0
$259$	−8.58301	+	1.98162i	−0.533322	+	0.123132i
$260$		0			0
$261$	−0.708497	+	1.32548i	−0.0438549	+	0.0820450i
$262$		0			0
$263$	−14.5830			−0.899227			−0.449613	−	0.893223i	$-0.648438\pi$
$263$	−14.5830			−0.899227			−0.449613	+	0.893223i	$0.648438\pi$
$264$		0			0
$265$	−9.31216	+	16.7816i	−0.572042	+	1.03089i
$266$		0			0
$267$	26.5830	+	6.65882i	1.62685	+	0.407513i
$268$		0			0
$269$	−9.31216			−0.567773			−0.283886	−	0.958858i	$-0.591624\pi$
$269$	−9.31216			−0.567773			−0.283886	+	0.958858i	$0.591624\pi$
$270$		0			0
$271$		−	20.3939i		−	1.23884i	−0.785059	−	0.619421i	$-0.787368\pi$
$271$		−	20.3939i		−	1.23884i	0.785059	−	0.619421i	$-0.212632\pi$
$272$		0			0
$273$	−0.285468	+	15.3961i	−0.0172773	+	0.931815i
$274$		0			0
$275$	12.0000	−	7.48331i	0.723627	−	0.451261i
$276$		0			0
$277$			6.98233i			0.419528i	0.977752	+	0.209764i	$0.0672695\pi$
$277$			6.98233i			0.419528i	−0.977752	+	0.209764i	$0.932730\pi$
$278$		0			0
$279$	−4.33981	+	8.11905i	−0.259818	+	0.486075i
$280$		0			0
$281$			9.30978i			0.555375i	0.960672	+	0.277687i	$0.0895679\pi$
$281$			9.30978i			0.555375i	−0.960672	+	0.277687i	$0.910432\pi$
$282$		0			0
$283$	32.2016			1.91419			0.957094	−	0.289779i	$-0.0935819\pi$
$283$	32.2016			1.91419			0.957094	+	0.289779i	$0.0935819\pi$
$284$		0			0
$285$	15.8219	+	14.8000i	0.937208	+	0.876677i
$286$		0			0
$287$	−2.58301	−	11.1878i	−0.152470	−	0.660393i
$288$		0			0
$289$	−5.58301			−0.328412
$290$		0			0
$291$	−24.5830	−	6.15784i	−1.44108	−	0.360979i
$292$		0			0
$293$			7.27733i			0.425146i	0.977145	+	0.212573i	$0.0681843\pi$
$293$			7.27733i			0.425146i	−0.977145	+	0.212573i	$0.931816\pi$
$294$		0			0
$295$	−2.35425	+	4.24264i	−0.137070	+	0.247016i
$296$		0			0
$297$	9.87000	−	10.8896i	0.572716	−	0.631878i
$298$		0			0
$299$	24.5015			1.41696
$300$		0			0
$301$	26.5830	−	6.13742i	1.53222	−	0.353755i
$302$		0			0
$303$	3.64575	+	0.913230i	0.209443	+	0.0524637i
$304$		0			0
$305$	−4.93725	−	2.73969i	−0.282706	−	0.156874i
$306$		0			0
$307$	12.0399			0.687154			0.343577	−	0.939124i	$-0.388361\pi$
$307$	12.0399			0.687154			0.343577	+	0.939124i	$0.388361\pi$
$308$		0			0
$309$	5.29150	+	1.32548i	0.301023	+	0.0754038i
$310$		0			0
$311$	−24.5015			−1.38935			−0.694677	−	0.719322i	$-0.744453\pi$
$311$	−24.5015			−1.38935			−0.694677	+	0.719322i	$0.744453\pi$
$312$		0			0
$313$	14.6315			0.827022			0.413511	−	0.910499i	$-0.364302\pi$
$313$	14.6315			0.827022			0.413511	+	0.910499i	$0.364302\pi$
$314$		0			0
$315$	12.7345	+	12.3626i	0.717507	+	0.696551i
$316$		0			0
$317$	6.00000			0.336994			0.168497	−	0.985702i	$-0.446109\pi$
$317$	6.00000			0.336994			0.168497	+	0.985702i	$0.446109\pi$
$318$		0			0
$319$	−1.41699			−0.0793365
$320$		0			0
$321$		0			0
$322$		0			0
$323$	26.5830			1.47912
$324$		0			0
$325$	8.89047	+	14.2565i	0.493155	+	0.790807i
$326$		0			0
$327$	−3.36028	−	0.841723i	−0.185824	−	0.0465474i
$328$		0			0
$329$	20.1617	−	4.65489i	1.11155	−	0.256632i
$330$		0			0
$331$	−8.00000			−0.439720			−0.219860	−	0.975531i	$-0.570560\pi$
$331$	−8.00000			−0.439720			−0.219860	+	0.975531i	$0.570560\pi$
$332$		0			0
$333$	−4.70850	+	8.80879i	−0.258024	+	0.482719i
$334$		0			0
$335$	20.1617	+	11.1878i	1.10155	+	0.611253i
$336$		0			0
$337$		−	22.4499i		−	1.22293i	−0.791273	−	0.611463i	$-0.790581\pi$
$337$		−	22.4499i		−	1.22293i	0.791273	−	0.611463i	$-0.209419\pi$
$338$		0			0
$339$	10.0808	+	2.52517i	0.547517	+	0.137148i
$340$		0			0
$341$	−8.67963			−0.470028
$342$		0			0
$343$	−11.6556	−	14.3926i	−0.629342	−	0.777129i
$344$		0			0
$345$	19.2915	−	20.6235i	1.03862	−	1.11033i
$346$		0			0
$347$	24.0000			1.28839			0.644194	−	0.764862i	$-0.277193\pi$
$347$	24.0000			1.28839			0.644194	+	0.764862i	$0.277193\pi$
$348$		0			0
$349$		−	13.7129i		−	0.734036i	−0.930214	−	0.367018i	$-0.880379\pi$
$349$		−	13.7129i		−	0.734036i	0.930214	−	0.367018i	$-0.119621\pi$
$350$		0			0
$351$	12.9373	+	11.7260i	0.690540	+	0.625885i
$352$		0			0
$353$			9.80250i			0.521734i	0.965375	+	0.260867i	$0.0840084\pi$
$353$			9.80250i			0.521734i	−0.965375	+	0.260867i	$0.915992\pi$
$354$		0			0
$355$	19.1822	+	10.6442i	1.01808	+	0.564936i
$356$		0			0
$357$	21.7734	+	0.403713i	1.15237	+	0.0213668i
$358$		0			0
$359$			4.33138i			0.228601i	0.993446	+	0.114301i	$0.0364627\pi$
$359$			4.33138i			0.228601i	−0.993446	+	0.114301i	$0.963537\pi$
$360$		0			0
$361$	−12.2915			−0.646921
$362$		0			0
$363$	−5.04042	−	1.26258i	−0.264554	−	0.0662685i
$364$		0			0
$365$	−6.00000	+	10.8127i	−0.314054	+	0.565964i
$366$		0			0
$367$	−14.6315			−0.763759			−0.381879	−	0.924212i	$-0.624723\pi$
$367$	−14.6315			−0.763759			−0.381879	+	0.924212i	$0.624723\pi$
$368$		0			0
$369$	−11.4821	−	6.13742i	−0.597733	−	0.319502i
$370$		0			0
$371$	−5.10850	−	22.1264i	−0.265220	−	1.14875i
$372$		0			0
$373$		−	6.98233i		−	0.361531i	−0.983526	−	0.180766i	$-0.942142\pi$
$373$		−	6.98233i		−	0.361531i	0.983526	−	0.180766i	$-0.0578576\pi$
$374$		0			0
$375$	19.0000	+	3.74166i	0.981156	+	0.193218i
$376$		0			0
$377$		−	1.68345i		−	0.0867019i
$378$		0			0
$379$	6.58301			0.338146			0.169073	−	0.985604i	$-0.445923\pi$
$379$	6.58301			0.338146			0.169073	+	0.985604i	$0.445923\pi$
$380$		0			0
$381$	1.40122	−	5.59388i	0.0717867	−	0.286583i
$382$		0			0
$383$			20.6921i			1.05732i	0.848835	+	0.528658i	$0.177305\pi$
$383$			20.6921i			1.05732i	−0.848835	+	0.528658i	$0.822695\pi$
$384$		0			0
$385$	−4.61944	+	16.0829i	−0.235428	+	0.819662i
$386$		0			0
$387$	14.5830	−	27.2823i	0.741296	−	1.38684i
$388$		0			0
$389$		−	37.0931i		−	1.88069i	−0.340218	−	0.940346i	$-0.610501\pi$
$389$		−	37.0931i		−	1.88069i	0.340218	−	0.940346i	$-0.389499\pi$
$390$		0			0
$391$		−	34.6504i		−	1.75234i
$392$		0			0
$393$	30.2288	+	7.57205i	1.52484	+	0.381959i
$394$		0			0
$395$	−3.57113	+	6.43560i	−0.179683	+	0.323810i
$396$		0			0
$397$	−14.8424			−0.744916			−0.372458	−	0.928049i	$-0.621485\pi$
$397$	−14.8424			−0.744916			−0.372458	+	0.928049i	$0.621485\pi$
$398$		0			0
$399$	−25.6300	−	0.475220i	−1.28310	−	0.0237908i
$400$		0			0
$401$			7.48331i			0.373699i	0.982389	+	0.186849i	$0.0598277\pi$
$401$			7.48331i			0.373699i	−0.982389	+	0.186849i	$0.940172\pi$
$402$		0			0
$403$		−	10.3117i		−	0.513664i
$404$		0			0
$405$	20.0563	−	1.65704i	0.996604	−	0.0823389i
$406$		0			0
$407$	−9.41699			−0.466783
$408$		0			0
$409$		−	23.4626i		−	1.16015i	−0.814563	−	0.580076i	$-0.803023\pi$
$409$		−	23.4626i		−	1.16015i	0.814563	−	0.580076i	$-0.196977\pi$
$410$		0			0
$411$	−2.16991	−	0.543544i	−0.107034	−	0.0268110i
$412$		0			0
$413$	−1.29150	−	5.59388i	−0.0635507	−	0.275257i
$414$		0			0
$415$	13.6458	+	7.57205i	0.669844	+	0.371697i
$416$		0			0
$417$	−0.228757	+	0.913230i	−0.0112023	+	0.0447211i
$418$		0			0
$419$	−33.8137			−1.65191			−0.825953	−	0.563739i	$-0.809362\pi$
$419$	−33.8137			−1.65191			−0.825953	+	0.563739i	$0.809362\pi$
$420$		0			0
$421$	−10.0000			−0.487370			−0.243685	−	0.969854i	$-0.578356\pi$
$421$	−10.0000			−0.487370			−0.243685	+	0.969854i	$0.578356\pi$
$422$		0			0
$423$	11.0604	−	20.6921i	0.537774	−	1.00608i
$424$		0			0
$425$	20.1617	−	12.5730i	0.977986	−	0.609881i
$426$		0			0
$427$	6.50972	−	1.50295i	0.315028	−	0.0727328i
$428$		0			0
$429$	−4.00000	+	15.9686i	−0.193122	+	0.770971i
$430$		0			0
$431$		−	8.98626i		−	0.432853i	−0.976299	−	0.216427i	$-0.930560\pi$
$431$		−	8.98626i		−	0.432853i	0.976299	−	0.216427i	$-0.0694402\pi$
$432$		0			0
$433$	−18.5496			−0.891439			−0.445719	−	0.895173i	$-0.647052\pi$
$433$	−18.5496			−0.891439			−0.445719	+	0.895173i	$0.647052\pi$
$434$		0			0
$435$	−1.41699	−	1.32548i	−0.0679397	−	0.0635518i
$436$		0			0
$437$			40.7878i			1.95114i
$438$		0			0
$439$			26.5313i			1.26627i	0.774041	+	0.633135i	$0.218232\pi$
$439$			26.5313i			1.26627i	−0.774041	+	0.633135i	$0.781768\pi$
$440$		0			0
$441$	−20.9856	−	0.778479i	−0.999313	−	0.0370704i
$442$		0			0
$443$	29.1660			1.38572			0.692859	−	0.721073i	$-0.256351\pi$
$443$	29.1660			1.38572			0.692859	+	0.721073i	$0.256351\pi$
$444$		0			0
$445$	−17.1660	+	30.9352i	−0.813747	+	1.46647i
$446$		0			0
$447$	4.55066	−	18.1669i	0.215239	−	0.859265i
$448$		0			0
$449$			36.5921i			1.72689i	0.504446	+	0.863444i	$0.331697\pi$
$449$			36.5921i			1.72689i	−0.504446	+	0.863444i	$0.668303\pi$
$450$		0			0
$451$		−	12.2748i		−	0.578000i
$452$		0			0
$453$	−2.38075	−	0.596358i	−0.111857	−	0.0280194i
$454$		0			0
$455$	−19.1072	−	5.48808i	−0.895758	−	0.257285i
$456$		0			0
$457$		−	8.48528i		−	0.396925i	−0.980109	−	0.198462i	$-0.936405\pi$
$457$		−	8.48528i		−	0.396925i	0.980109	−	0.198462i	$-0.0635948\pi$
$458$		0			0
$459$	16.5830	−	18.2960i	0.774028	−	0.853986i
$460$		0			0
$461$	−17.9918			−0.837961			−0.418981	−	0.907995i	$-0.637612\pi$
$461$	−17.9918			−0.837961			−0.418981	+	0.907995i	$0.637612\pi$
$462$		0			0
$463$			1.50295i			0.0698480i	0.999390	+	0.0349240i	$0.0111189\pi$
$463$			1.50295i			0.0698480i	−0.999390	+	0.0349240i	$0.988881\pi$
$464$		0			0
$465$	−8.67963	−	8.11905i	−0.402508	−	0.376512i
$466$		0			0
$467$			0.841723i			0.0389503i	0.999810	+	0.0194751i	$0.00619952\pi$
$467$			0.841723i			0.0389503i	−0.999810	+	0.0194751i	$0.993800\pi$
$468$		0			0
$469$	−26.5830	+	6.13742i	−1.22749	+	0.283400i
$470$		0			0
$471$	27.5203	+	6.89360i	1.26807	+	0.317640i
$472$		0			0
$473$	29.1660			1.34105
$474$		0			0
$475$	−23.7328	+	14.8000i	−1.08894	+	0.679071i
$476$		0			0
$477$	−22.7085	−	12.1382i	−1.03975	−	0.555770i
$478$		0			0
$479$	21.6991			0.991456			0.495728	−	0.868478i	$-0.334901\pi$
$479$	21.6991			0.991456			0.495728	+	0.868478i	$0.334901\pi$
$480$		0			0
$481$		−	11.1878i		−	0.510118i
$482$		0			0
$483$	−0.619439	+	33.4081i	−0.0281855	+	1.52012i
$484$		0			0
$485$	15.8745	−	28.6078i	0.720824	−	1.29901i
$486$		0			0
$487$		−	9.98823i		−	0.452610i	−0.974056	−	0.226305i	$-0.927335\pi$
$487$		−	9.98823i		−	0.452610i	0.974056	−	0.226305i	$-0.0726646\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	14.1421i		−	0.638226i	−0.947717	−	0.319113i	$-0.896615\pi$
$491$		−	14.1421i		−	0.638226i	0.947717	−	0.319113i	$-0.103385\pi$
$492$		0			0
$493$	−2.38075			−0.107224
$494$		0			0
$495$	10.2917	+	15.9399i	0.462577	+	0.716446i
$496$		0			0
$497$	−25.2915	+	5.83925i	−1.13448	+	0.261926i
$498$		0			0
$499$	−22.5830			−1.01095			−0.505477	−	0.862840i	$-0.668683\pi$
$499$	−22.5830			−1.01095			−0.505477	+	0.862840i	$0.668683\pi$
$500$		0			0
$501$	8.70850	−	34.7656i	0.389067	−	1.55321i
$502$		0			0
$503$			15.6417i			0.697431i	0.937229	+	0.348715i	$0.113382\pi$
$503$			15.6417i			0.697431i	−0.937229	+	0.348715i	$0.886618\pi$
$504$		0			0
$505$	−2.35425	+	4.24264i	−0.104763	+	0.188795i
$506$		0			0
$507$	2.87052	+	0.719041i	0.127484	+	0.0319337i
$508$		0			0
$509$	−39.6909			−1.75927			−0.879633	−	0.475652i	$-0.842212\pi$
$509$	−39.6909			−1.75927			−0.879633	+	0.475652i	$0.842212\pi$
$510$		0			0
$511$	−3.29150	−	14.2565i	−0.145608	−	0.630669i
$512$		0			0
$513$	−19.5203	+	21.5367i	−0.861840	+	0.950869i
$514$		0			0
$515$	−3.41699	+	6.15784i	−0.150571	+	0.271347i
$516$		0			0
$517$	22.1208			0.972870
$518$		0			0
$519$	0.937254	−	3.74166i	0.0411409	−	0.164241i
$520$		0			0
$521$	13.0194			0.570392			0.285196	−	0.958469i	$-0.407941\pi$
$521$	13.0194			0.570392			0.285196	+	0.958469i	$0.407941\pi$
$522$		0			0
$523$	−16.8014			−0.734675			−0.367337	−	0.930088i	$-0.619731\pi$
$523$	−16.8014			−0.734675			−0.367337	+	0.930088i	$0.619731\pi$
$524$		0			0
$525$	−19.6636	+	11.7618i	−0.858192	+	0.513329i
$526$		0			0
$527$	−14.5830			−0.635246
$528$		0			0
$529$	30.1660			1.31157
$530$		0			0
$531$	−5.74103	−	3.06871i	−0.249140	−	0.133171i
$532$		0			0
$533$	14.5830			0.631660
$534$		0			0
$535$		0			0
$536$		0			0
$537$	−10.2917	+	41.0860i	−0.444119	+	1.77299i
$538$		0			0
$539$	−8.67963	−	17.7951i	−0.373858	−	0.766487i
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		0			0
$543$	−6.22876	+	24.8661i	−0.267302	+	1.06711i
$544$		0			0
$545$	2.16991	−	3.91044i	0.0929486	−	0.167505i
$546$		0			0
$547$		−	16.9706i		−	0.725609i	−0.931865	−	0.362804i	$-0.881819\pi$
$547$		−	16.9706i		−	0.725609i	0.931865	−	0.362804i	$-0.118181\pi$
$548$		0			0
$549$	3.57113	−	6.68097i	0.152412	−	0.285137i
$550$		0			0
$551$	2.80244			0.119388
$552$		0			0
$553$	−1.95906	−	8.48528i	−0.0833078	−	0.360831i
$554$		0			0
$555$	−9.41699	−	8.80879i	−0.399729	−	0.373912i
$556$		0			0
$557$	−35.1660			−1.49003			−0.745016	−	0.667047i	$-0.767558\pi$
$557$	−35.1660			−1.49003			−0.745016	+	0.667047i	$0.767558\pi$
$558$		0			0
$559$			34.6504i			1.46555i
$560$		0			0
$561$	22.5830	+	5.65685i	0.953455	+	0.238833i
$562$		0			0
$563$			13.1166i			0.552798i	0.961043	+	0.276399i	$0.0891411\pi$
$563$			13.1166i			0.552798i	−0.961043	+	0.276399i	$0.910859\pi$
$564$		0			0
$565$	−6.50972	+	11.7313i	−0.273866	+	0.493540i
$566$		0			0
$567$	−16.3156	+	17.3437i	−0.685189	+	0.728365i
$568$		0			0
$569$		−	33.7637i		−	1.41545i	−0.706490	−	0.707723i	$-0.749722\pi$
$569$		−	33.7637i		−	1.41545i	0.706490	−	0.707723i	$-0.250278\pi$
$570$		0			0
$571$	13.4170			0.561484			0.280742	−	0.959783i	$-0.409420\pi$
$571$	13.4170			0.561484			0.280742	+	0.959783i	$0.409420\pi$
$572$		0			0
$573$	−0.210845	+	0.841723i	−0.00880816	+	0.0351635i
$574$		0			0
$575$	19.2915	+	30.9352i	0.804511	+	1.29009i
$576$		0			0
$577$	36.3306			1.51246			0.756231	−	0.654305i	$-0.227039\pi$
$577$	36.3306			1.51246			0.756231	+	0.654305i	$0.227039\pi$
$578$		0			0
$579$	3.57113	−	14.2565i	0.148411	−	0.592479i
$580$		0			0
$581$	−17.9918	+	4.15390i	−0.746425	+	0.172333i
$582$		0			0
$583$		−	24.2764i		−	1.00543i
$584$		0			0
$585$	−18.9373	+	12.2269i	−0.782959	+	0.505522i
$586$		0			0
$587$		−	37.7719i		−	1.55901i	−0.626394	−	0.779507i	$-0.715470\pi$
$587$		−	37.7719i		−	1.55901i	0.626394	−	0.779507i	$-0.284530\pi$
$588$		0			0
$589$	17.1660			0.707313
$590$		0			0
$591$	−30.2425	−	7.57551i	−1.24401	−	0.311615i
$592$		0			0
$593$		−	6.43560i		−	0.264279i	−0.991231	−	0.132139i	$-0.957815\pi$
$593$		−	6.43560i		−	0.264279i	0.991231	−	0.132139i	$-0.0421846\pi$
$594$		0			0
$595$	−7.76131	+	27.0216i	−0.318183	+	1.10778i
$596$		0			0
$597$	3.87451	−	15.4676i	0.158573	−	0.633047i
$598$		0			0
$599$		−	4.15390i		−	0.169724i	−0.996393	−	0.0848620i	$-0.972955\pi$
$599$		−	4.15390i		−	0.169724i	0.996393	−	0.0848620i	$-0.0270449\pi$
$600$		0			0
$601$		−	6.13742i		−	0.250351i	−0.992135	−	0.125175i	$-0.960051\pi$
$601$		−	6.13742i		−	0.250351i	0.992135	−	0.125175i	$-0.0399494\pi$
$602$		0			0
$603$	−14.5830	+	27.2823i	−0.593866	+	1.11102i
$604$		0			0
$605$	3.25486	−	5.86565i	0.132329	−	0.238473i
$606$		0			0
$607$	−5.95188			−0.241579			−0.120790	−	0.992678i	$-0.538543\pi$
$607$	−5.95188			−0.241579			−0.120790	+	0.992678i	$0.538543\pi$
$608$		0			0
$609$	2.29540	+	0.0425603i	0.0930143	+	0.00172463i
$610$		0			0
$611$			26.2803i			1.06319i
$612$		0			0
$613$		−	47.5823i		−	1.92183i	−0.276843	−	0.960915i	$-0.589288\pi$
$613$		−	47.5823i		−	1.92183i	0.276843	−	0.960915i	$-0.410712\pi$
$614$		0			0
$615$	11.4821	−	12.2748i	0.463002	−	0.494970i
$616$		0			0
$617$	47.1660			1.89883			0.949416	−	0.314021i	$-0.101676\pi$
$617$	47.1660			1.89883			0.949416	+	0.314021i	$0.101676\pi$
$618$		0			0
$619$		−	40.2443i		−	1.61755i	−0.588116	−	0.808777i	$-0.700130\pi$
$619$		−	40.2443i		−	1.61755i	0.588116	−	0.808777i	$-0.299870\pi$
$620$		0			0
$621$	28.0726	+	25.4442i	1.12652	+	1.02104i
$622$		0			0
$623$	−9.41699	−	40.7878i	−0.377284	−	1.63413i
$624$		0			0
$625$	−11.0000	+	22.4499i	−0.440000	+	0.897998i
$626$		0			0
$627$	−26.5830	−	6.65882i	−1.06162	−	0.265928i
$628$		0			0
$629$	−15.8219			−0.630860
$630$		0			0
$631$	−32.4575			−1.29211			−0.646057	−	0.763290i	$-0.723583\pi$
$631$	−32.4575			−1.29211			−0.646057	+	0.763290i	$0.723583\pi$
$632$		0			0
$633$	−35.5619	−	8.90796i	−1.41346	−	0.354060i
$634$		0			0
$635$	6.50972	+	3.61226i	0.258330	+	0.143348i
$636$		0			0
$637$	21.1412	−	10.3117i	0.837646	−	0.408566i
$638$		0			0
$639$	−13.8745	+	25.9568i	−0.548867	+	1.02684i
$640$		0			0
$641$			28.1068i			1.11015i	0.831800	+	0.555076i	$0.187311\pi$
$641$			28.1068i			1.11015i	−0.831800	+	0.555076i	$0.812689\pi$
$642$		0			0
$643$	−22.6786			−0.894357			−0.447178	−	0.894445i	$-0.647571\pi$
$643$	−22.6786			−0.894357			−0.447178	+	0.894445i	$0.647571\pi$
$644$		0			0
$645$	29.1660	+	27.2823i	1.14841	+	1.07424i
$646$		0			0
$647$		−	13.9583i		−	0.548757i	−0.961622	−	0.274379i	$-0.911528\pi$
$647$		−	13.9583i		−	0.548757i	0.961622	−	0.274379i	$-0.0884721\pi$
$648$		0			0
$649$		−	6.13742i		−	0.240915i
$650$		0			0
$651$	14.0602	+	0.260698i	0.551062	+	0.0102176i
$652$		0			0
$653$	−23.1660			−0.906556			−0.453278	−	0.891369i	$-0.649746\pi$
$653$	−23.1660			−0.906556			−0.453278	+	0.891369i	$0.649746\pi$
$654$		0			0
$655$	−19.5203	+	35.1779i	−0.762720	+	1.37451i
$656$		0			0
$657$	−14.6315	−	7.82087i	−0.570830	−	0.305121i
$658$		0			0
$659$			27.1048i			1.05585i	0.849290	+	0.527927i	$0.177031\pi$
$659$			27.1048i			1.05585i	−0.849290	+	0.527927i	$0.822969\pi$
$660$		0			0
$661$		−	8.66259i		−	0.336936i	−0.985707	−	0.168468i	$-0.946118\pi$
$661$		−	8.66259i		−	0.336936i	0.985707	−	0.168468i	$-0.0538820\pi$
$662$		0			0
$663$	−6.72057	+	26.8295i	−0.261005	+	1.04197i
$664$		0			0
$665$	9.13603	−	31.8078i	0.354280	−	1.23345i
$666$		0			0
$667$		−	3.65292i		−	0.141442i
$668$		0			0
$669$	−2.00000	−	0.500983i	−0.0773245	−	0.0193691i
$670$		0			0
$671$	7.14226			0.275724
$672$		0			0
$673$			23.6294i			0.910846i	0.890275	+	0.455423i	$0.150512\pi$
$673$			23.6294i			0.910846i	−0.890275	+	0.455423i	$0.849488\pi$
$674$		0			0
$675$	−4.61874	+	25.5669i	−0.177775	+	0.984071i
$676$		0			0
$677$			6.19024i			0.237910i	0.992900	+	0.118955i	$0.0379545\pi$
$677$			6.19024i			0.237910i	−0.992900	+	0.118955i	$0.962046\pi$
$678$		0			0
$679$	8.70850	+	37.7191i	0.334201	+	1.44753i
$680$		0			0
$681$	7.64575	−	30.5230i	0.292986	−	1.16964i
$682$		0			0
$683$	−26.5830			−1.01717			−0.508585	−	0.861012i	$-0.669831\pi$
$683$	−26.5830			−1.01717			−0.508585	+	0.861012i	$0.669831\pi$
$684$		0			0
$685$	1.40122	−	2.52517i	0.0535379	−	0.0964817i
$686$		0			0
$687$	10.4797	−	41.8367i	0.399827	−	1.59617i
$688$		0			0
$689$	28.8413			1.09877
$690$		0			0
$691$			11.7313i			0.446280i	0.974786	+	0.223140i	$0.0716307\pi$
$691$			11.7313i			0.446280i	−0.974786	+	0.223140i	$0.928369\pi$
$692$		0			0
$693$	−21.6723	−	5.85776i	−0.823261	−	0.222518i
$694$		0			0
$695$	−1.06275	−	0.589720i	−0.0403123	−	0.0223693i
$696$		0			0
$697$		−	20.6235i		−	0.781170i
$698$		0			0
$699$	−22.3316	−	5.59388i	−0.844659	−	0.211580i
$700$		0			0
$701$			21.1245i			0.797860i	0.916981	+	0.398930i	$0.130618\pi$
$701$			21.1245i			0.797860i	−0.916981	+	0.398930i	$0.869382\pi$
$702$		0			0
$703$	18.6243			0.702430
$704$		0			0
$705$	22.1208	+	20.6921i	0.833116	+	0.779309i
$706$		0			0
$707$	−1.29150	−	5.59388i	−0.0485720	−	0.210380i
$708$		0			0
$709$	−0.583005			−0.0218952			−0.0109476	−	0.999940i	$-0.503485\pi$
$709$	−0.583005			−0.0218952			−0.0109476	+	0.999940i	$0.503485\pi$
$710$		0			0
$711$	−8.70850	−	4.65489i	−0.326594	−	0.174572i
$712$		0			0
$713$		−	22.3755i		−	0.837970i
$714$		0			0
$715$	−18.5830	−	10.3117i	−0.694965	−	0.385637i
$716$		0			0
$717$	6.16272	−	24.6025i	0.230151	−	0.918798i
$718$		0			0
$719$	2.80244			0.104513			0.0522567	−	0.998634i	$-0.483359\pi$
$719$	2.80244			0.104513			0.0522567	+	0.998634i	$0.483359\pi$
$720$		0			0
$721$	−1.87451	−	8.11905i	−0.0698103	−	0.302369i
$722$		0			0
$723$	−7.29150	+	29.1088i	−0.271174	+	1.08257i
$724$		0			0
$725$	2.12549	−	1.32548i	0.0789388	−	0.0492270i
$726$		0			0
$727$	45.8536			1.70062			0.850308	−	0.526286i	$-0.176416\pi$
$727$	45.8536			1.70062			0.850308	+	0.526286i	$0.176416\pi$
$728$		0			0
$729$	2.64575	+	26.8701i	0.0979908	+	0.995187i
$730$		0			0
$731$	49.0030			1.81244
$732$		0			0
$733$	28.2835			1.04467			0.522337	−	0.852739i	$-0.325060\pi$
$733$	28.2835			1.04467			0.522337	+	0.852739i	$0.325060\pi$
$734$		0			0
$735$	7.96612	−	25.9141i	0.293835	−	0.955856i
$736$		0			0
$737$	−29.1660			−1.07434
$738$		0			0
$739$	33.1660			1.22003			0.610016	−	0.792389i	$-0.291163\pi$
$739$	33.1660			1.22003			0.610016	+	0.792389i	$0.291163\pi$
$740$		0			0
$741$	7.91094	−	31.5817i	0.290616	−	1.16018i
$742$		0			0
$743$	2.12549			0.0779767			0.0389884	−	0.999240i	$-0.487586\pi$
$743$	2.12549			0.0779767			0.0389884	+	0.999240i	$0.487586\pi$
$744$		0			0
$745$	21.1412	+	11.7313i	0.774555	+	0.429802i
$746$		0			0
$747$	−9.87000	+	18.4651i	−0.361125	+	0.675602i
$748$		0			0
$749$		0			0
$750$		0			0
$751$	21.1660			0.772359			0.386179	−	0.922424i	$-0.373795\pi$
$751$	21.1660			0.772359			0.386179	+	0.922424i	$0.373795\pi$
$752$		0			0
$753$	−10.9373	−	2.73969i	−0.398576	−	0.0998399i
$754$		0			0
$755$	1.53737	−	2.77053i	0.0559508	−	0.100830i
$756$		0			0
$757$			34.2646i			1.24537i	0.782473	+	0.622685i	$0.213958\pi$
$757$			34.2646i			1.24537i	−0.782473	+	0.622685i	$0.786042\pi$
$758$		0			0
$759$	−8.67963	+	34.6504i	−0.315051	+	1.25773i
$760$		0			0
$761$	11.4821			0.416225			0.208112	−	0.978105i	$-0.433268\pi$
$761$	11.4821			0.416225			0.208112	+	0.978105i	$0.433268\pi$
$762$		0			0
$763$	1.19038	+	5.15587i	0.0430945	+	0.186655i
$764$		0			0
$765$	17.2915	+	26.7813i	0.625176	+	0.968281i
$766$		0			0
$767$	7.29150			0.263281
$768$		0			0
$769$			35.7375i			1.28873i	0.764720	+	0.644363i	$0.222877\pi$
$769$			35.7375i			1.28873i	−0.764720	+	0.644363i	$0.777123\pi$
$770$		0			0
$771$	−7.87451	+	31.4362i	−0.283593	+	1.13215i
$772$		0			0
$773$			41.9277i			1.50803i	0.656855	+	0.754017i	$0.271887\pi$
$773$			41.9277i			1.50803i	−0.656855	+	0.754017i	$0.728113\pi$
$774$		0			0
$775$	13.0194	−	8.11905i	0.467672	−	0.291645i
$776$		0			0
$777$	15.2547	+	0.282845i	0.547258	+	0.0101470i
$778$		0			0
$779$			24.2764i			0.869792i
$780$		0			0
$781$	−27.7490			−0.992938
$782$		0			0
$783$	1.74822	−	1.92881i	0.0624762	−	0.0689301i
$784$		0			0
$785$	−17.7712	+	32.0259i	−0.634283	+	1.14305i
$786$		0			0
$787$	−28.2835			−1.00820			−0.504099	−	0.863646i	$-0.668175\pi$
$787$	−28.2835			−1.00820			−0.504099	+	0.863646i	$0.668175\pi$
$788$		0			0
$789$	24.5015	+	6.13742i	0.872277	+	0.218498i
$790$		0			0
$791$	−3.57113	−	15.4676i	−0.126975	−	0.549965i
$792$		0			0
$793$			8.48528i			0.301321i
$794$		0			0
$795$	22.7085	−	24.2764i	0.805387	−	0.860995i
$796$		0			0
$797$		−	15.0982i		−	0.534806i	−0.963585	−	0.267403i	$-0.913835\pi$
$797$		−	15.0982i		−	0.534806i	0.963585	−	0.267403i	$-0.0861654\pi$
$798$		0			0
$799$	37.1660			1.31484
$800$		0			0
$801$	−41.8608	−	22.3755i	−1.47908	−	0.790600i
$802$		0			0
$803$		−	15.6417i		−	0.551985i
$804$		0			0
$805$	−41.4608	−	11.9086i	−1.46130	−	0.419724i
$806$		0			0
$807$	15.6458	+	3.91913i	0.550757	+	0.137960i
$808$		0			0
$809$			11.1362i			0.391529i	0.980651	+	0.195765i	$0.0627188\pi$
$809$			11.1362i			0.391529i	−0.980651	+	0.195765i	$0.937281\pi$
$810$		0			0
$811$			24.0062i			0.842970i	0.906835	+	0.421485i	$0.138491\pi$
$811$			24.0062i			0.842970i	−0.906835	+	0.421485i	$0.861509\pi$
$812$		0			0
$813$	−8.58301	+	34.2646i	−0.301019	+	1.20171i
$814$		0			0
$815$		0			0
$816$		0			0
$817$	−57.6827			−2.01806
$818$		0			0
$819$	6.95925	−	25.7475i	0.243176	−	0.899690i
$820$		0			0
$821$			31.4362i			1.09713i	0.836107	+	0.548566i	$0.184826\pi$
$821$			31.4362i			1.09713i	−0.836107	+	0.548566i	$0.815174\pi$
$822$		0			0
$823$		−	11.8147i		−	0.411834i	−0.978569	−	0.205917i	$-0.933982\pi$
$823$		−	11.8147i		−	0.411834i	0.978569	−	0.205917i	$-0.0660177\pi$
$824$		0			0
$825$	−23.3111	+	7.52269i	−0.811590	+	0.261906i
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$			33.2123i			1.15351i	0.816917	+	0.576755i	$0.195681\pi$
$829$			33.2123i			1.15351i	−0.816917	+	0.576755i	$0.804319\pi$
$830$		0			0
$831$	2.93859	−	11.7313i	0.101939	−	0.406954i
$832$		0			0
$833$	−14.5830	−	29.8982i	−0.505271	−	1.03591i
$834$		0			0
$835$	40.4575	+	22.4499i	1.40009	+	0.776912i
$836$		0			0
$837$	10.7085	−	11.8147i	0.370140	−	0.408375i
$838$		0			0
$839$	37.5210			1.29537			0.647684	−	0.761909i	$-0.275738\pi$
$839$	37.5210			1.29537			0.647684	+	0.761909i	$0.275738\pi$
$840$		0			0
$841$	28.7490			0.991345
$842$		0			0
$843$	3.91813	−	15.6417i	0.134947	−	0.538730i
$844$		0			0
$845$	−1.85364	+	3.34048i	−0.0637672	+	0.114916i
$846$		0			0
$847$	1.78556	+	7.73381i	0.0613527	+	0.265737i
$848$		0			0
$849$	−54.1033	−	13.5524i	−1.85682	−	0.465118i
$850$		0			0
$851$		−	24.2764i		−	0.832184i
$852$		0			0
$853$	−23.5220			−0.805377			−0.402689	−	0.915337i	$-0.631924\pi$
$853$	−23.5220			−0.805377			−0.402689	+	0.915337i	$0.631924\pi$
$854$		0			0
$855$	−20.3542	−	31.5249i	−0.696101	−	1.07813i
$856$		0			0
$857$		−	46.1363i		−	1.57599i	−0.615684	−	0.787993i	$-0.711120\pi$
$857$		−	46.1363i		−	1.57599i	0.615684	−	0.787993i	$-0.288880\pi$
$858$		0			0
$859$		−	22.9191i		−	0.781988i	−0.920393	−	0.390994i	$-0.872131\pi$
$859$		−	22.9191i		−	0.781988i	0.920393	−	0.390994i	$-0.127869\pi$
$860$		0			0
$861$	−0.368683	+	19.8841i	−0.0125647	+	0.677649i
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	4.35425	+	2.41618i	0.148049	+	0.0821526i
$866$		0			0
$867$	9.38024	+	2.34967i	0.318570	+	0.0797990i
$868$		0			0
$869$		−	9.30978i		−	0.315812i
$870$		0			0
$871$		−	34.6504i		−	1.17408i
$872$		0			0
$873$	38.7113	+	20.6921i	1.31018	+	0.700321i
$874$		0			0
$875$	−8.11505	−	28.4455i	−0.274339	−	0.961633i
$876$		0			0
$877$			23.9529i			0.808832i	0.914575	+	0.404416i	$0.132525\pi$
$877$			23.9529i			0.808832i	−0.914575	+	0.404416i	$0.867475\pi$
$878$		0			0
$879$	3.06275	−	12.2269i	0.103304	−	0.412404i
$880$		0			0
$881$	15.8219			0.533053			0.266526	−	0.963828i	$-0.414124\pi$
$881$	15.8219			0.533053			0.266526	+	0.963828i	$0.414124\pi$
$882$		0			0
$883$		−	30.9352i		−	1.04105i	−0.853845	−	0.520527i	$-0.825736\pi$
$883$		−	30.9352i		−	1.04105i	0.853845	−	0.520527i	$-0.174264\pi$
$884$		0			0
$885$	5.74103	−	6.13742i	0.192983	−	0.206307i
$886$		0			0
$887$			2.27980i			0.0765483i	0.999267	+	0.0382742i	$0.0121860\pi$
$887$			2.27980i			0.0765483i	−0.999267	+	0.0382742i	$0.987814\pi$
$888$		0			0
$889$	−8.58301	+	1.98162i	−0.287865	+	0.0664616i
$890$		0			0
$891$	−21.1660	+	14.1421i	−0.709088	+	0.473779i
$892$		0			0
$893$	−43.7490			−1.46400
$894$		0			0
$895$	−47.8127	−	26.5313i	−1.59820	−	0.886844i
$896$		0			0
$897$	−41.1660	−	10.3117i	−1.37449	−	0.344299i
$898$		0			0
$899$	−1.53737			−0.0512743
$900$		0			0
$901$		−	40.7878i		−	1.35884i
$902$		0			0
$903$	−47.2462	−	0.876020i	−1.57226	−	0.0291521i
$904$		0			0
$905$	−28.9373	−	16.0573i	−0.961907	−	0.533764i
$906$		0			0
$907$			33.9411i			1.12700i	0.826117	+	0.563498i	$0.190545\pi$
$907$			33.9411i			1.12700i	−0.826117	+	0.563498i	$0.809455\pi$
$908$		0			0
$909$	−5.74103	−	3.06871i	−0.190418	−	0.101783i
$910$		0			0
$911$			18.2960i			0.606175i	0.952963	+	0.303087i	$0.0980174\pi$
$911$			18.2960i			0.606175i	−0.952963	+	0.303087i	$0.901983\pi$
$912$		0			0
$913$	−19.7400			−0.653299
$914$		0			0
$915$	7.14226	+	6.68097i	0.236116	+	0.220866i
$916$		0			0
$917$	−10.7085	−	46.3817i	−0.353626	−	1.53166i
$918$		0			0
$919$	−41.8745			−1.38131			−0.690656	−	0.723183i	$-0.742678\pi$
$919$	−41.8745			−1.38131			−0.690656	+	0.723183i	$0.742678\pi$
$920$		0			0
$921$	−20.2288	−	5.06713i	−0.666560	−	0.166968i
$922$		0			0
$923$		−	32.9669i		−	1.08512i
$924$		0			0
$925$	14.1255	−	8.80879i	0.464443	−	0.289631i
$926$		0			0
$927$	−8.33263	−	4.45398i	−0.273680	−	0.146288i
$928$		0			0
$929$	51.8055			1.69968			0.849841	−	0.527039i	$-0.176698\pi$
$929$	51.8055			1.69968			0.849841	+	0.527039i	$0.176698\pi$
$930$		0			0
$931$	17.1660	+	35.1939i	0.562593	+	1.15343i
$932$		0			0
$933$	41.1660	+	10.3117i	1.34771	+	0.337591i
$934$		0			0
$935$	−14.5830	+	26.2803i	−0.476915	+	0.859459i
$936$		0			0
$937$	−5.53019			−0.180663			−0.0903317	−	0.995912i	$-0.528793\pi$
$937$	−5.53019			−0.180663			−0.0903317	+	0.995912i	$0.528793\pi$
$938$		0			0
$939$	−24.5830	−	6.15784i	−0.802236	−	0.200953i
$940$		0			0
$941$	−0.632534			−0.0206200			−0.0103100	−	0.999947i	$-0.503282\pi$
$941$	−0.632534			−0.0206200			−0.0103100	+	0.999947i	$0.503282\pi$
$942$		0			0
$943$	31.6438			1.03046
$944$		0			0
$945$	−16.1928	−	26.1303i	−0.526752	−	0.850019i
$946$		0			0
$947$	−24.0000			−0.779895			−0.389948	−	0.920837i	$-0.627507\pi$
$947$	−24.0000			−0.779895			−0.389948	+	0.920837i	$0.627507\pi$
$948$		0			0
$949$	18.5830			0.603230
$950$		0			0
$951$	−10.0808	−	2.52517i	−0.326894	−	0.0818842i
$952$		0			0
$953$	3.87451			0.125508			0.0627538	−	0.998029i	$-0.480012\pi$
$953$	3.87451			0.125508			0.0627538	+	0.998029i	$0.480012\pi$
$954$		0			0
$955$	−0.979531	−	0.543544i	−0.0316969	−	0.0175887i
$956$		0			0
$957$	2.38075	+	0.596358i	0.0769588	+	0.0192775i
$958$		0			0
$959$	0.768687	+	3.32941i	0.0248222	+	0.107512i
$960$		0			0
$961$	21.5830			0.696226
$962$		0			0
$963$		0			0
$964$		0			0
$965$	16.5906	+	9.20614i	0.534069	+	0.296356i
$966$		0			0
$967$			26.9588i			0.866936i	0.901169	+	0.433468i	$0.142710\pi$
$967$			26.9588i			0.866936i	−0.901169	+	0.433468i	$0.857290\pi$
$968$		0			0
$969$	−44.6632	−	11.1878i	−1.43479	−	0.359403i
$970$		0			0
$971$	9.31216			0.298842			0.149421	−	0.988774i	$-0.452259\pi$
$971$	9.31216			0.298842			0.149421	+	0.988774i	$0.452259\pi$
$972$		0			0
$973$	1.40122	−	0.323511i	0.0449211	−	0.0103713i
$974$		0			0
$975$	−8.93725	−	27.6946i	−0.286221	−	0.886935i
$976$		0			0
$977$	−20.1255			−0.643872			−0.321936	−	0.946762i	$-0.604334\pi$
$977$	−20.1255			−0.643872			−0.321936	+	0.946762i	$0.604334\pi$
$978$		0			0
$979$		−	44.7510i		−	1.43025i
$980$		0			0
$981$	5.29150	+	2.82843i	0.168945	+	0.0903047i
$982$		0			0
$983$			29.0037i			0.925074i	0.886600	+	0.462537i	$0.153061\pi$
$983$			29.0037i			0.925074i	−0.886600	+	0.462537i	$0.846939\pi$
$984$		0			0
$985$	19.5292	−	35.1939i	0.622251	−	1.12137i
$986$		0			0
$987$	−35.8336	−	0.664411i	−1.14060	−	0.0211484i
$988$		0			0
$989$			75.1881i			2.39084i
$990$		0			0
$991$	11.2915			0.358686			0.179343	−	0.983787i	$-0.442603\pi$
$991$	11.2915			0.358686			0.179343	+	0.983787i	$0.442603\pi$
$992$		0			0
$993$	13.4411	+	3.36689i	0.426541	+	0.106845i
$994$		0			0
$995$	18.0000	+	9.98823i	0.570638	+	0.316648i
$996$		0			0
$997$	−26.3244			−0.833703			−0.416851	−	0.908975i	$-0.636866\pi$
$997$	−26.3244			−0.833703			−0.416851	+	0.908975i	$0.636866\pi$
$998$		0			0
$999$	11.6182	−	12.8184i	0.367584	−	0.405556i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.k.e.209.1		8
3.2	odd	2		1680.2.k.f.209.2		8
4.3	odd	2		210.2.d.a.209.8	yes	8
5.4	even	2		1680.2.k.f.209.8		8
7.6	odd	2	inner	1680.2.k.e.209.8		8
12.11	even	2		210.2.d.b.209.7	yes	8
15.14	odd	2	inner	1680.2.k.e.209.7		8
20.3	even	4		1050.2.b.f.251.12		16
20.7	even	4		1050.2.b.f.251.5		16
20.19	odd	2		210.2.d.b.209.1	yes	8
21.20	even	2		1680.2.k.f.209.7		8
28.27	even	2		210.2.d.a.209.1	✓	8
35.34	odd	2		1680.2.k.f.209.1		8
60.23	odd	4		1050.2.b.f.251.6		16
60.47	odd	4		1050.2.b.f.251.11		16
60.59	even	2		210.2.d.a.209.2	yes	8
84.83	odd	2		210.2.d.b.209.2	yes	8
105.104	even	2	inner	1680.2.k.e.209.2		8
140.27	odd	4		1050.2.b.f.251.4		16
140.83	odd	4		1050.2.b.f.251.13		16
140.139	even	2		210.2.d.b.209.8	yes	8
420.83	even	4		1050.2.b.f.251.3		16
420.167	even	4		1050.2.b.f.251.14		16
420.419	odd	2		210.2.d.a.209.7	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
210.2.d.a.209.1	✓	8	28.27	even	2
210.2.d.a.209.2	yes	8	60.59	even	2
210.2.d.a.209.7	yes	8	420.419	odd	2
210.2.d.a.209.8	yes	8	4.3	odd	2
210.2.d.b.209.1	yes	8	20.19	odd	2
210.2.d.b.209.2	yes	8	84.83	odd	2
210.2.d.b.209.7	yes	8	12.11	even	2
210.2.d.b.209.8	yes	8	140.139	even	2
1050.2.b.f.251.3		16	420.83	even	4
1050.2.b.f.251.4		16	140.27	odd	4
1050.2.b.f.251.5		16	20.7	even	4
1050.2.b.f.251.6		16	60.23	odd	4
1050.2.b.f.251.11		16	60.47	odd	4
1050.2.b.f.251.12		16	20.3	even	4
1050.2.b.f.251.13		16	140.83	odd	4
1050.2.b.f.251.14		16	420.167	even	4
1680.2.k.e.209.1		8	1.1	even	1	trivial
1680.2.k.e.209.2		8	105.104	even	2	inner
1680.2.k.e.209.7		8	15.14	odd	2	inner
1680.2.k.e.209.8		8	7.6	odd	2	inner
1680.2.k.f.209.1		8	35.34	odd	2
1680.2.k.f.209.2		8	3.2	odd	2
1680.2.k.f.209.7		8	21.20	even	2
1680.2.k.f.209.8		8	5.4	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)

\(f(q)\)	\(=\)	\(q+(-1.68014 - 0.420861i) q^{3} +(1.08495 - 1.95522i) q^{5} +(0.595188 + 2.57794i) q^{7} +(2.64575 + 1.41421i) q^{9} +2.82843i q^{11} -3.36028 q^{13} +(-2.64575 + 2.82843i) q^{15} +4.75216i q^{17} -5.59388i q^{19} +(0.0849536 - 4.58179i) q^{21} -7.29150 q^{23} +(-2.64575 - 4.24264i) q^{25} +(-3.85005 - 3.48957i) q^{27} +0.500983i q^{29} +3.06871i q^{31} +(1.19038 - 4.75216i) q^{33} +(5.68618 + 1.63322i) q^{35} +3.32941i q^{37} +(5.64575 + 1.41421i) q^{39} -4.33981 q^{41} -10.3117i q^{43} +(5.63561 - 3.63866i) q^{45} -7.82087i q^{47} +(-6.29150 + 3.06871i) q^{49} +(2.00000 - 7.98430i) q^{51} -8.58301 q^{53} +(5.53019 + 3.06871i) q^{55} +(-2.35425 + 9.39851i) q^{57} -2.16991 q^{59} -2.52517i q^{61} +(-2.07103 + 7.66230i) q^{63} +(-3.64575 + 6.57008i) q^{65} +10.3117i q^{67} +(12.2508 + 3.06871i) q^{69} +9.81076i q^{71} -5.53019 q^{73} +(2.65967 + 8.24173i) q^{75} +(-7.29150 + 1.68345i) q^{77} -3.29150 q^{79} +(5.00000 + 7.48331i) q^{81} +6.97915i q^{83} +(9.29150 + 5.15587i) q^{85} +(0.210845 - 0.841723i) q^{87} -15.8219 q^{89} +(-2.00000 - 8.66259i) q^{91} +(1.29150 - 5.15587i) q^{93} +(-10.9373 - 6.06910i) q^{95} +14.6315 q^{97} +(-4.00000 + 7.48331i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{21} - 16 q^{23} - 16 q^{35} + 24 q^{39} - 8 q^{49} + 16 q^{51} + 16 q^{53} - 40 q^{57} - 8 q^{63} - 8 q^{65} - 16 q^{77} + 16 q^{79} + 40 q^{81} + 32 q^{85} - 16 q^{91} - 32 q^{93} - 24 q^{95}+ \cdots - 32 q^{99}+O(q^{100}) \) 8 * q - 8 * q^21 - 16 * q^23 - 16 * q^35 + 24 * q^39 - 8 * q^49 + 16 * q^51 + 16 * q^53 - 40 * q^57 - 8 * q^63 - 8 * q^65 - 16 * q^77 + 16 * q^79 + 40 * q^81 + 32 * q^85 - 16 * q^91 - 32 * q^93 - 24 * q^95 - 32 * q^99

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