LMFDB - Embedded newform 1680.2.k.g.209.6

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)

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);

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")

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 = [8,0,0,0,0,0,0,0,8,0,0,0,0,0,0] 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:	8.0.40960000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 7x^{4} + 1 $ x^8 + 7*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{8} $
Twist minimal:	no (minimal twist has level 420)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			209.6
Root			$1.14412 - 1.14412i$ of defining polynomial
Character	$\chi$	$=$	1680.209
Dual form			1680.2.k.g.209.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(1.41421 - 1.00000i) q^{3} +2.23607 q^{5} +(1.41421 + 2.23607i) q^{7} +(1.00000 - 2.82843i) q^{9} +2.82843i q^{11} +2.82843 q^{13} +(3.16228 - 2.23607i) q^{15} -4.00000i q^{17} -6.32456i q^{19} +(4.23607 + 1.74806i) q^{21} -6.32456 q^{23} +5.00000 q^{25} +(-1.41421 - 5.00000i) q^{27} -5.65685i q^{29} +6.32456i q^{31} +(2.82843 + 4.00000i) q^{33} +(3.16228 + 5.00000i) q^{35} +8.94427i q^{37} +(4.00000 - 2.82843i) q^{39} +4.47214 q^{41} +4.47214i q^{43} +(2.23607 - 6.32456i) q^{45} +6.00000i q^{47} +(-3.00000 + 6.32456i) q^{49} +(-4.00000 - 5.65685i) q^{51} +6.32456 q^{53} +6.32456i q^{55} +(-6.32456 - 8.94427i) q^{57} +8.94427 q^{59} +(7.73877 - 1.76393i) q^{63} +6.32456 q^{65} -4.47214i q^{67} +(-8.94427 + 6.32456i) q^{69} -14.1421i q^{71} -8.48528 q^{73} +(7.07107 - 5.00000i) q^{75} +(-6.32456 + 4.00000i) q^{77} +12.0000 q^{79} +(-7.00000 - 5.65685i) q^{81} +10.0000i q^{83} -8.94427i q^{85} +(-5.65685 - 8.00000i) q^{87} -4.47214 q^{89} +(4.00000 + 6.32456i) q^{91} +(6.32456 + 8.94427i) q^{93} -14.1421i q^{95} -8.48528 q^{97} +(8.00000 + 2.82843i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{9} + 16 q^{21} + 40 q^{25} + 32 q^{39} - 24 q^{49} - 32 q^{51} + 96 q^{79} - 56 q^{81} + 32 q^{91} + 64 q^{99}+O(q^{100}) $ 8 * q + 8 * q^9 + 16 * q^21 + 40 * q^25 + 32 * q^39 - 24 * q^49 - 32 * q^51 + 96 * q^79 - 56 * q^81 + 32 * q^91 + 64 * 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
$3$	1.41421	−	1.00000i	0.816497	−	0.577350i
$3$	1.41421	−	1.00000i	0.816497	−	0.577350i
$4$		0			0
$5$	2.23607			1.00000
$5$	2.23607			1.00000
$6$		0			0
$7$	1.41421	+	2.23607i	0.534522	+	0.845154i
$7$	1.41421	+	2.23607i	0.534522	+	0.845154i
$8$		0			0
$9$	1.00000	−	2.82843i	0.333333	−	0.942809i
$10$		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
$13$	2.82843			0.784465			0.392232	−	0.919866i	$-0.371703\pi$
$13$	2.82843			0.784465			0.392232	+	0.919866i	$0.371703\pi$
$14$		0			0
$15$	3.16228	−	2.23607i	0.816497	−	0.577350i
$16$		0			0
$17$		−	4.00000i		−	0.970143i	−0.874475	−	0.485071i	$-0.838794\pi$
$17$		−	4.00000i		−	0.970143i	0.874475	−	0.485071i	$-0.161206\pi$
$18$		0			0
$19$		−	6.32456i		−	1.45095i	−0.688247	−	0.725476i	$-0.741620\pi$
$19$		−	6.32456i		−	1.45095i	0.688247	−	0.725476i	$-0.258380\pi$
$20$		0			0
$21$	4.23607	+	1.74806i	0.924386	+	0.381459i
$22$		0			0
$23$	−6.32456			−1.31876			−0.659380	−	0.751809i	$-0.729181\pi$
$23$	−6.32456			−1.31876			−0.659380	+	0.751809i	$0.729181\pi$
$24$		0			0
$25$	5.00000			1.00000
$26$		0			0
$27$	−1.41421	−	5.00000i	−0.272166	−	0.962250i
$28$		0			0
$29$		−	5.65685i		−	1.05045i	−0.850963	−	0.525226i	$-0.823981\pi$
$29$		−	5.65685i		−	1.05045i	0.850963	−	0.525226i	$-0.176019\pi$
$30$		0			0
$31$			6.32456i			1.13592i	0.823055	+	0.567962i	$0.192268\pi$
$31$			6.32456i			1.13592i	−0.823055	+	0.567962i	$0.807732\pi$
$32$		0			0
$33$	2.82843	+	4.00000i	0.492366	+	0.696311i
$34$		0			0
$35$	3.16228	+	5.00000i	0.534522	+	0.845154i
$36$		0			0
$37$			8.94427i			1.47043i	0.677834	+	0.735215i	$0.262919\pi$
$37$			8.94427i			1.47043i	−0.677834	+	0.735215i	$0.737081\pi$
$38$		0			0
$39$	4.00000	−	2.82843i	0.640513	−	0.452911i
$40$		0			0
$41$	4.47214			0.698430			0.349215	−	0.937043i	$-0.386448\pi$
$41$	4.47214			0.698430			0.349215	+	0.937043i	$0.386448\pi$
$42$		0			0
$43$			4.47214i			0.681994i	0.940064	+	0.340997i	$0.110765\pi$
$43$			4.47214i			0.681994i	−0.940064	+	0.340997i	$0.889235\pi$
$44$		0			0
$45$	2.23607	−	6.32456i	0.333333	−	0.942809i
$46$		0			0
$47$			6.00000i			0.875190i	0.899172	+	0.437595i	$0.144170\pi$
$47$			6.00000i			0.875190i	−0.899172	+	0.437595i	$0.855830\pi$
$48$		0			0
$49$	−3.00000	+	6.32456i	−0.428571	+	0.903508i
$50$		0			0
$51$	−4.00000	−	5.65685i	−0.560112	−	0.792118i
$52$		0			0
$53$	6.32456			0.868744			0.434372	−	0.900733i	$-0.356970\pi$
$53$	6.32456			0.868744			0.434372	+	0.900733i	$0.356970\pi$
$54$		0			0
$55$			6.32456i			0.852803i
$56$		0			0
$57$	−6.32456	−	8.94427i	−0.837708	−	1.18470i
$58$		0			0
$59$	8.94427			1.16445			0.582223	−	0.813029i	$-0.302183\pi$
$59$	8.94427			1.16445			0.582223	+	0.813029i	$0.302183\pi$
$60$		0			0
$61$		0			0		1.00000			$0$
$61$		0			0		−1.00000			$\pi$
$62$		0			0
$63$	7.73877	−	1.76393i	0.974993	−	0.222235i
$64$		0			0
$65$	6.32456			0.784465
$66$		0			0
$67$		−	4.47214i		−	0.546358i	−0.961963	−	0.273179i	$-0.911925\pi$
$67$		−	4.47214i		−	0.546358i	0.961963	−	0.273179i	$-0.0880752\pi$
$68$		0			0
$69$	−8.94427	+	6.32456i	−1.07676	+	0.761387i
$70$		0			0
$71$		−	14.1421i		−	1.67836i	−0.543852	−	0.839181i	$-0.683035\pi$
$71$		−	14.1421i		−	1.67836i	0.543852	−	0.839181i	$-0.316965\pi$
$72$		0			0
$73$	−8.48528			−0.993127			−0.496564	−	0.868000i	$-0.665405\pi$
$73$	−8.48528			−0.993127			−0.496564	+	0.868000i	$0.665405\pi$
$74$		0			0
$75$	7.07107	−	5.00000i	0.816497	−	0.577350i
$76$		0			0
$77$	−6.32456	+	4.00000i	−0.720750	+	0.455842i
$78$		0			0
$79$	12.0000			1.35011			0.675053	−	0.737769i	$-0.264121\pi$
$79$	12.0000			1.35011			0.675053	+	0.737769i	$0.264121\pi$
$80$		0			0
$81$	−7.00000	−	5.65685i	−0.777778	−	0.628539i
$82$		0			0
$83$			10.0000i			1.09764i	0.835940	+	0.548821i	$0.184923\pi$
$83$			10.0000i			1.09764i	−0.835940	+	0.548821i	$0.815077\pi$
$84$		0			0
$85$		−	8.94427i		−	0.970143i
$86$		0			0
$87$	−5.65685	−	8.00000i	−0.606478	−	0.857690i
$88$		0			0
$89$	−4.47214			−0.474045			−0.237023	−	0.971504i	$-0.576172\pi$
$89$	−4.47214			−0.474045			−0.237023	+	0.971504i	$0.576172\pi$
$90$		0			0
$91$	4.00000	+	6.32456i	0.419314	+	0.662994i
$92$		0			0
$93$	6.32456	+	8.94427i	0.655826	+	0.927478i
$94$		0			0
$95$		−	14.1421i		−	1.45095i
$96$		0			0
$97$	−8.48528			−0.861550			−0.430775	−	0.902459i	$-0.641760\pi$
$97$	−8.48528			−0.861550			−0.430775	+	0.902459i	$0.641760\pi$
$98$		0			0
$99$	8.00000	+	2.82843i	0.804030	+	0.284268i
$100$		0			0
$101$	−4.47214			−0.444994			−0.222497	−	0.974933i	$-0.571421\pi$
$101$	−4.47214			−0.444994			−0.222497	+	0.974933i	$0.571421\pi$
$102$		0			0
$103$	−2.82843			−0.278693			−0.139347	−	0.990244i	$-0.544500\pi$
$103$	−2.82843			−0.278693			−0.139347	+	0.990244i	$0.544500\pi$
$104$		0			0
$105$	9.47214	+	3.90879i	0.924386	+	0.381459i
$106$		0			0
$107$	−18.9737			−1.83425			−0.917127	−	0.398596i	$-0.869498\pi$
$107$	−18.9737			−1.83425			−0.917127	+	0.398596i	$0.869498\pi$
$108$		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
$111$	8.94427	+	12.6491i	0.848953	+	1.20060i
$112$		0			0
$113$	−18.9737			−1.78489			−0.892446	−	0.451154i	$-0.851013\pi$
$113$	−18.9737			−1.78489			−0.892446	+	0.451154i	$0.851013\pi$
$114$		0			0
$115$	−14.1421			−1.31876
$116$		0			0
$117$	2.82843	−	8.00000i	0.261488	−	0.739600i
$118$		0			0
$119$	8.94427	−	5.65685i	0.819920	−	0.518563i
$120$		0			0
$121$	3.00000			0.272727
$122$		0			0
$123$	6.32456	−	4.47214i	0.570266	−	0.403239i
$124$		0			0
$125$	11.1803			1.00000
$126$		0			0
$127$		−	4.47214i		−	0.396838i	−0.980117	−	0.198419i	$-0.936419\pi$
$127$		−	4.47214i		−	0.396838i	0.980117	−	0.198419i	$-0.0635807\pi$
$128$		0			0
$129$	4.47214	+	6.32456i	0.393750	+	0.556846i
$130$		0			0
$131$	−8.94427			−0.781465			−0.390732	−	0.920504i	$-0.627778\pi$
$131$	−8.94427			−0.781465			−0.390732	+	0.920504i	$0.627778\pi$
$132$		0			0
$133$	14.1421	−	8.94427i	1.22628	−	0.775567i
$134$		0			0
$135$	−3.16228	−	11.1803i	−0.272166	−	0.962250i
$136$		0			0
$137$	6.32456			0.540343			0.270172	−	0.962812i	$-0.412920\pi$
$137$	6.32456			0.540343			0.270172	+	0.962812i	$0.412920\pi$
$138$		0			0
$139$		−	6.32456i		−	0.536442i	−0.963357	−	0.268221i	$-0.913564\pi$
$139$		−	6.32456i		−	0.536442i	0.963357	−	0.268221i	$-0.0864357\pi$
$140$		0			0
$141$	6.00000	+	8.48528i	0.505291	+	0.714590i
$142$		0			0
$143$			8.00000i			0.668994i
$144$		0			0
$145$		−	12.6491i		−	1.05045i
$146$		0			0
$147$	2.08191	+	11.9443i	0.171713	+	0.985147i
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	−4.00000			−0.325515			−0.162758	−	0.986666i	$-0.552039\pi$
$151$	−4.00000			−0.325515			−0.162758	+	0.986666i	$0.552039\pi$
$152$		0			0
$153$	−11.3137	−	4.00000i	−0.914659	−	0.323381i
$154$		0			0
$155$			14.1421i			1.13592i
$156$		0			0
$157$	2.82843			0.225733			0.112867	−	0.993610i	$-0.463997\pi$
$157$	2.82843			0.225733			0.112867	+	0.993610i	$0.463997\pi$
$158$		0			0
$159$	8.94427	−	6.32456i	0.709327	−	0.501570i
$160$		0			0
$161$	−8.94427	−	14.1421i	−0.704907	−	1.11456i
$162$		0			0
$163$		−	13.4164i		−	1.05085i	−0.850839	−	0.525427i	$-0.823906\pi$
$163$		−	13.4164i		−	1.05085i	0.850839	−	0.525427i	$-0.176094\pi$
$164$		0			0
$165$	6.32456	+	8.94427i	0.492366	+	0.696311i
$166$		0			0
$167$		−	6.00000i		−	0.464294i	−0.972681	−	0.232147i	$-0.925425\pi$
$167$		−	6.00000i		−	0.464294i	0.972681	−	0.232147i	$-0.0745750\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$	−17.8885	−	6.32456i	−1.36797	−	0.483651i
$172$		0			0
$173$			8.00000i			0.608229i	0.952636	+	0.304114i	$0.0983605\pi$
$173$			8.00000i			0.608229i	−0.952636	+	0.304114i	$0.901639\pi$
$174$		0			0
$175$	7.07107	+	11.1803i	0.534522	+	0.845154i
$176$		0			0
$177$	12.6491	−	8.94427i	0.950765	−	0.672293i
$178$		0			0
$179$			14.1421i			1.05703i	0.848923	+	0.528516i	$0.177252\pi$
$179$			14.1421i			1.05703i	−0.848923	+	0.528516i	$0.822748\pi$
$180$		0			0
$181$			12.6491i			0.940201i	0.882613	+	0.470100i	$0.155782\pi$
$181$			12.6491i			0.940201i	−0.882613	+	0.470100i	$0.844218\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			20.0000i			1.47043i
$186$		0			0
$187$	11.3137			0.827340
$188$		0			0
$189$	9.18034	−	10.2333i	0.667771	−	0.744366i
$190$		0			0
$191$		−	2.82843i		−	0.204658i	−0.994751	−	0.102329i	$-0.967371\pi$
$191$		−	2.82843i		−	0.204658i	0.994751	−	0.102329i	$-0.0326294\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$	8.94427	−	6.32456i	0.640513	−	0.452911i
$196$		0			0
$197$	−18.9737			−1.35182			−0.675909	−	0.736985i	$-0.736249\pi$
$197$	−18.9737			−1.35182			−0.675909	+	0.736985i	$0.736249\pi$
$198$		0			0
$199$		−	18.9737i		−	1.34501i	−0.740094	−	0.672504i	$-0.765219\pi$
$199$		−	18.9737i		−	1.34501i	0.740094	−	0.672504i	$-0.234781\pi$
$200$		0			0
$201$	−4.47214	−	6.32456i	−0.315440	−	0.446100i
$202$		0			0
$203$	12.6491	−	8.00000i	0.887794	−	0.561490i
$204$		0			0
$205$	10.0000			0.698430
$206$		0			0
$207$	−6.32456	+	17.8885i	−0.439587	+	1.24334i
$208$		0			0
$209$	17.8885			1.23738
$210$		0			0
$211$	−16.0000			−1.10149			−0.550743	−	0.834675i	$-0.685655\pi$
$211$	−16.0000			−1.10149			−0.550743	+	0.834675i	$0.685655\pi$
$212$		0			0
$213$	−14.1421	−	20.0000i	−0.969003	−	1.37038i
$214$		0			0
$215$			10.0000i			0.681994i
$216$		0			0
$217$	−14.1421	+	8.94427i	−0.960031	+	0.607177i
$218$		0			0
$219$	−12.0000	+	8.48528i	−0.810885	+	0.573382i
$220$		0			0
$221$		−	11.3137i		−	0.761042i
$222$		0			0
$223$	2.82843			0.189405			0.0947027	−	0.995506i	$-0.469810\pi$
$223$	2.82843			0.189405			0.0947027	+	0.995506i	$0.469810\pi$
$224$		0			0
$225$	5.00000	−	14.1421i	0.333333	−	0.942809i
$226$		0			0
$227$		−	6.00000i		−	0.398234i	−0.979976	−	0.199117i	$-0.936193\pi$
$227$		−	6.00000i		−	0.398234i	0.979976	−	0.199117i	$-0.0638074\pi$
$228$		0			0
$229$		−	12.6491i		−	0.835877i	−0.908475	−	0.417938i	$-0.862753\pi$
$229$		−	12.6491i		−	0.835877i	0.908475	−	0.417938i	$-0.137247\pi$
$230$		0			0
$231$	−4.94427	+	11.9814i	−0.325309	+	0.788319i
$232$		0			0
$233$	6.32456			0.414335			0.207168	−	0.978305i	$-0.433575\pi$
$233$	6.32456			0.414335			0.207168	+	0.978305i	$0.433575\pi$
$234$		0			0
$235$			13.4164i			0.875190i
$236$		0			0
$237$	16.9706	−	12.0000i	1.10236	−	0.779484i
$238$		0			0
$239$			8.48528i			0.548867i	0.961606	+	0.274434i	$0.0884904\pi$
$239$			8.48528i			0.548867i	−0.961606	+	0.274434i	$0.911510\pi$
$240$		0			0
$241$			12.6491i			0.814801i	0.913250	+	0.407400i	$0.133565\pi$
$241$			12.6491i			0.814801i	−0.913250	+	0.407400i	$0.866435\pi$
$242$		0			0
$243$	−15.5563	−	1.00000i	−0.997940	−	0.0641500i
$244$		0			0
$245$	−6.70820	+	14.1421i	−0.428571	+	0.903508i
$246$		0			0
$247$		−	17.8885i		−	1.13822i
$248$		0			0
$249$	10.0000	+	14.1421i	0.633724	+	0.896221i
$250$		0			0
$251$	−26.8328			−1.69367			−0.846836	−	0.531854i	$-0.821496\pi$
$251$	−26.8328			−1.69367			−0.846836	+	0.531854i	$0.821496\pi$
$252$		0			0
$253$		−	17.8885i		−	1.12464i
$254$		0			0
$255$	−8.94427	−	12.6491i	−0.560112	−	0.792118i
$256$		0			0
$257$			20.0000i			1.24757i	0.781598	+	0.623783i	$0.214405\pi$
$257$			20.0000i			1.24757i	−0.781598	+	0.623783i	$0.785595\pi$
$258$		0			0
$259$	−20.0000	+	12.6491i	−1.24274	+	0.785977i
$260$		0			0
$261$	−16.0000	−	5.65685i	−0.990375	−	0.350150i
$262$		0			0
$263$	6.32456			0.389989			0.194994	−	0.980804i	$-0.437531\pi$
$263$	6.32456			0.389989			0.194994	+	0.980804i	$0.437531\pi$
$264$		0			0
$265$	14.1421			0.868744
$266$		0			0
$267$	−6.32456	+	4.47214i	−0.387056	+	0.273690i
$268$		0			0
$269$	−4.47214			−0.272671			−0.136335	−	0.990663i	$-0.543533\pi$
$269$	−4.47214			−0.272671			−0.136335	+	0.990663i	$0.543533\pi$
$270$		0			0
$271$			6.32456i			0.384189i	0.981376	+	0.192095i	$0.0615281\pi$
$271$			6.32456i			0.384189i	−0.981376	+	0.192095i	$0.938472\pi$
$272$		0			0
$273$	11.9814	+	4.94427i	0.725148	+	0.299241i
$274$		0			0
$275$			14.1421i			0.852803i
$276$		0			0
$277$			26.8328i			1.61223i	0.591761	+	0.806114i	$0.298433\pi$
$277$			26.8328i			1.61223i	−0.591761	+	0.806114i	$0.701567\pi$
$278$		0			0
$279$	17.8885	+	6.32456i	1.07096	+	0.378641i
$280$		0			0
$281$			16.9706i			1.01238i	0.862422	+	0.506189i	$0.168946\pi$
$281$			16.9706i			1.01238i	−0.862422	+	0.506189i	$0.831054\pi$
$282$		0			0
$283$	25.4558			1.51319			0.756596	−	0.653882i	$-0.226861\pi$
$283$	25.4558			1.51319			0.756596	+	0.653882i	$0.226861\pi$
$284$		0			0
$285$	−14.1421	−	20.0000i	−0.837708	−	1.18470i
$286$		0			0
$287$	6.32456	+	10.0000i	0.373327	+	0.590281i
$288$		0			0
$289$	1.00000			0.0588235
$290$		0			0
$291$	−12.0000	+	8.48528i	−0.703452	+	0.497416i
$292$		0			0
$293$			32.0000i			1.86946i	0.355359	+	0.934730i	$0.384359\pi$
$293$			32.0000i			1.86946i	−0.355359	+	0.934730i	$0.615641\pi$
$294$		0			0
$295$	20.0000			1.16445
$296$		0			0
$297$	14.1421	−	4.00000i	0.820610	−	0.232104i
$298$		0			0
$299$	−17.8885			−1.03452
$300$		0			0
$301$	−10.0000	+	6.32456i	−0.576390	+	0.364541i
$302$		0			0
$303$	−6.32456	+	4.47214i	−0.363336	+	0.256917i
$304$		0			0
$305$		0			0
$306$		0			0
$307$	−8.48528			−0.484281			−0.242140	−	0.970241i	$-0.577849\pi$
$307$	−8.48528			−0.484281			−0.242140	+	0.970241i	$0.577849\pi$
$308$		0			0
$309$	−4.00000	+	2.82843i	−0.227552	+	0.160904i
$310$		0			0
$311$	17.8885			1.01437			0.507183	−	0.861838i	$-0.330687\pi$
$311$	17.8885			1.01437			0.507183	+	0.861838i	$0.330687\pi$
$312$		0			0
$313$	25.4558			1.43885			0.719425	−	0.694570i	$-0.244406\pi$
$313$	25.4558			1.43885			0.719425	+	0.694570i	$0.244406\pi$
$314$		0			0
$315$	17.3044	−	3.94427i	0.974993	−	0.222235i
$316$		0			0
$317$	−18.9737			−1.06567			−0.532834	−	0.846220i	$-0.678873\pi$
$317$	−18.9737			−1.06567			−0.532834	+	0.846220i	$0.678873\pi$
$318$		0			0
$319$	16.0000			0.895828
$320$		0			0
$321$	−26.8328	+	18.9737i	−1.49766	+	1.05901i
$322$		0			0
$323$	−25.2982			−1.40763
$324$		0			0
$325$	14.1421			0.784465
$326$		0			0
$327$	2.82843	−	2.00000i	0.156412	−	0.110600i
$328$		0			0
$329$	−13.4164	+	8.48528i	−0.739671	+	0.467809i
$330$		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
$333$	25.2982	+	8.94427i	1.38633	+	0.490143i
$334$		0			0
$335$		−	10.0000i		−	0.546358i
$336$		0			0
$337$		0			0		1.00000			$0$
$337$		0			0		−1.00000			$\pi$
$338$		0			0
$339$	−26.8328	+	18.9737i	−1.45736	+	1.03051i
$340$		0			0
$341$	−17.8885			−0.968719
$342$		0			0
$343$	−18.3848	+	2.23607i	−0.992685	+	0.120736i
$344$		0			0
$345$	−20.0000	+	14.1421i	−1.07676	+	0.761387i
$346$		0			0
$347$	18.9737			1.01856			0.509280	−	0.860601i	$-0.329912\pi$
$347$	18.9737			1.01856			0.509280	+	0.860601i	$0.329912\pi$
$348$		0			0
$349$		−	12.6491i		−	0.677091i	−0.940950	−	0.338546i	$-0.890065\pi$
$349$		−	12.6491i		−	0.677091i	0.940950	−	0.338546i	$-0.109935\pi$
$350$		0			0
$351$	−4.00000	−	14.1421i	−0.213504	−	0.754851i
$352$		0			0
$353$		−	28.0000i		−	1.49029i	−0.666903	−	0.745145i	$-0.732380\pi$
$353$		−	28.0000i		−	1.49029i	0.666903	−	0.745145i	$-0.267620\pi$
$354$		0			0
$355$		−	31.6228i		−	1.67836i
$356$		0			0
$357$	6.99226	−	16.9443i	0.370069	−	0.896786i
$358$		0			0
$359$		−	14.1421i		−	0.746393i	−0.927752	−	0.373197i	$-0.878262\pi$
$359$		−	14.1421i		−	0.746393i	0.927752	−	0.373197i	$-0.121738\pi$
$360$		0			0
$361$	−21.0000			−1.10526
$362$		0			0
$363$	4.24264	−	3.00000i	0.222681	−	0.157459i
$364$		0			0
$365$	−18.9737			−0.993127
$366$		0			0
$367$	8.48528			0.442928			0.221464	−	0.975169i	$-0.428916\pi$
$367$	8.48528			0.442928			0.221464	+	0.975169i	$0.428916\pi$
$368$		0			0
$369$	4.47214	−	12.6491i	0.232810	−	0.658486i
$370$		0			0
$371$	8.94427	+	14.1421i	0.464363	+	0.734223i
$372$		0			0
$373$		−	26.8328i		−	1.38935i	−0.719323	−	0.694675i	$-0.755548\pi$
$373$		−	26.8328i		−	1.38935i	0.719323	−	0.694675i	$-0.244452\pi$
$374$		0			0
$375$	15.8114	−	11.1803i	0.816497	−	0.577350i
$376$		0			0
$377$		−	16.0000i		−	0.824042i
$378$		0			0
$379$	−24.0000			−1.23280			−0.616399	−	0.787434i	$-0.711409\pi$
$379$	−24.0000			−1.23280			−0.616399	+	0.787434i	$0.711409\pi$
$380$		0			0
$381$	−4.47214	−	6.32456i	−0.229114	−	0.324017i
$382$		0			0
$383$		−	30.0000i		−	1.53293i	−0.642287	−	0.766464i	$-0.722014\pi$
$383$		−	30.0000i		−	1.53293i	0.642287	−	0.766464i	$-0.277986\pi$
$384$		0			0
$385$	−14.1421	+	8.94427i	−0.720750	+	0.455842i
$386$		0			0
$387$	12.6491	+	4.47214i	0.642990	+	0.227331i
$388$		0			0
$389$			33.9411i			1.72088i	0.509549	+	0.860442i	$0.329812\pi$
$389$			33.9411i			1.72088i	−0.509549	+	0.860442i	$0.670188\pi$
$390$		0			0
$391$			25.2982i			1.27939i
$392$		0			0
$393$	−12.6491	+	8.94427i	−0.638063	+	0.451179i
$394$		0			0
$395$	26.8328			1.35011
$396$		0			0
$397$	−19.7990			−0.993683			−0.496841	−	0.867841i	$-0.665507\pi$
$397$	−19.7990			−0.993683			−0.496841	+	0.867841i	$0.665507\pi$
$398$		0			0
$399$	11.0557	−	26.7912i	0.553479	−	1.34124i
$400$		0			0
$401$			11.3137i			0.564980i	0.959270	+	0.282490i	$0.0911603\pi$
$401$			11.3137i			0.564980i	−0.959270	+	0.282490i	$0.908840\pi$
$402$		0			0
$403$			17.8885i			0.891092i
$404$		0			0
$405$	−15.6525	−	12.6491i	−0.777778	−	0.628539i
$406$		0			0
$407$	−25.2982			−1.25399
$408$		0			0
$409$		−	37.9473i		−	1.87637i	−0.346128	−	0.938187i	$-0.612504\pi$
$409$		−	37.9473i		−	1.87637i	0.346128	−	0.938187i	$-0.387496\pi$
$410$		0			0
$411$	8.94427	−	6.32456i	0.441188	−	0.311967i
$412$		0			0
$413$	12.6491	+	20.0000i	0.622422	+	0.984136i
$414$		0			0
$415$			22.3607i			1.09764i
$416$		0			0
$417$	−6.32456	−	8.94427i	−0.309715	−	0.438003i
$418$		0			0
$419$	−26.8328			−1.31087			−0.655434	−	0.755252i	$-0.727514\pi$
$419$	−26.8328			−1.31087			−0.655434	+	0.755252i	$0.727514\pi$
$420$		0			0
$421$	−34.0000			−1.65706			−0.828529	−	0.559946i	$-0.810822\pi$
$421$	−34.0000			−1.65706			−0.828529	+	0.559946i	$0.810822\pi$
$422$		0			0
$423$	16.9706	+	6.00000i	0.825137	+	0.291730i
$424$		0			0
$425$		−	20.0000i		−	0.970143i
$426$		0			0
$427$		0			0
$428$		0			0
$429$	8.00000	+	11.3137i	0.386244	+	0.546231i
$430$		0			0
$431$		−	2.82843i		−	0.136241i	−0.997677	−	0.0681203i	$-0.978300\pi$
$431$		−	2.82843i		−	0.136241i	0.997677	−	0.0681203i	$-0.0217002\pi$
$432$		0			0
$433$	−8.48528			−0.407777			−0.203888	−	0.978994i	$-0.565358\pi$
$433$	−8.48528			−0.407777			−0.203888	+	0.978994i	$0.565358\pi$
$434$		0			0
$435$	−12.6491	−	17.8885i	−0.606478	−	0.857690i
$436$		0			0
$437$			40.0000i			1.91346i
$438$		0			0
$439$			6.32456i			0.301855i	0.988545	+	0.150927i	$0.0482259\pi$
$439$			6.32456i			0.301855i	−0.988545	+	0.150927i	$0.951774\pi$
$440$		0			0
$441$	14.8885	+	14.8098i	0.708978	+	0.705230i
$442$		0			0
$443$	6.32456			0.300489			0.150244	−	0.988649i	$-0.451994\pi$
$443$	6.32456			0.300489			0.150244	+	0.988649i	$0.451994\pi$
$444$		0			0
$445$	−10.0000			−0.474045
$446$		0			0
$447$		0			0
$448$		0			0
$449$			5.65685i			0.266963i	0.991051	+	0.133482i	$0.0426157\pi$
$449$			5.65685i			0.266963i	−0.991051	+	0.133482i	$0.957384\pi$
$450$		0			0
$451$			12.6491i			0.595623i
$452$		0			0
$453$	−5.65685	+	4.00000i	−0.265782	+	0.187936i
$454$		0			0
$455$	8.94427	+	14.1421i	0.419314	+	0.662994i
$456$		0			0
$457$		0			0		1.00000			$0$
$457$		0			0		−1.00000			$\pi$
$458$		0			0
$459$	−20.0000	+	5.65685i	−0.933520	+	0.264039i
$460$		0			0
$461$	4.47214			0.208288			0.104144	−	0.994562i	$-0.466790\pi$
$461$	4.47214			0.208288			0.104144	+	0.994562i	$0.466790\pi$
$462$		0			0
$463$		−	40.2492i		−	1.87054i	−0.353935	−	0.935270i	$-0.615157\pi$
$463$		−	40.2492i		−	1.87054i	0.353935	−	0.935270i	$-0.384843\pi$
$464$		0			0
$465$	14.1421	+	20.0000i	0.655826	+	0.927478i
$466$		0			0
$467$			26.0000i			1.20314i	0.798821	+	0.601568i	$0.205457\pi$
$467$			26.0000i			1.20314i	−0.798821	+	0.601568i	$0.794543\pi$
$468$		0			0
$469$	10.0000	−	6.32456i	0.461757	−	0.292041i
$470$		0			0
$471$	4.00000	−	2.82843i	0.184310	−	0.130327i
$472$		0			0
$473$	−12.6491			−0.581607
$474$		0			0
$475$		−	31.6228i		−	1.45095i
$476$		0			0
$477$	6.32456	−	17.8885i	0.289581	−	0.819060i
$478$		0			0
$479$	17.8885			0.817348			0.408674	−	0.912680i	$-0.365991\pi$
$479$	17.8885			0.817348			0.408674	+	0.912680i	$0.365991\pi$
$480$		0			0
$481$			25.2982i			1.15350i
$482$		0			0
$483$	−26.7912	−	11.0557i	−1.21904	−	0.503053i
$484$		0			0
$485$	−18.9737			−0.861550
$486$		0			0
$487$			13.4164i			0.607955i	0.952679	+	0.303978i	$0.0983148\pi$
$487$			13.4164i			0.607955i	−0.952679	+	0.303978i	$0.901685\pi$
$488$		0			0
$489$	−13.4164	−	18.9737i	−0.606711	−	0.858019i
$490$		0			0
$491$			2.82843i			0.127645i	0.997961	+	0.0638226i	$0.0203292\pi$
$491$			2.82843i			0.127645i	−0.997961	+	0.0638226i	$0.979671\pi$
$492$		0			0
$493$	−22.6274			−1.01909
$494$		0			0
$495$	17.8885	+	6.32456i	0.804030	+	0.284268i
$496$		0			0
$497$	31.6228	−	20.0000i	1.41848	−	0.897123i
$498$		0			0
$499$	8.00000			0.358129			0.179065	−	0.983837i	$-0.442693\pi$
$499$	8.00000			0.358129			0.179065	+	0.983837i	$0.442693\pi$
$500$		0			0
$501$	−6.00000	−	8.48528i	−0.268060	−	0.379094i
$502$		0			0
$503$			42.0000i			1.87269i	0.351085	+	0.936344i	$0.385813\pi$
$503$			42.0000i			1.87269i	−0.351085	+	0.936344i	$0.614187\pi$
$504$		0			0
$505$	−10.0000			−0.444994
$506$		0			0
$507$	−7.07107	+	5.00000i	−0.314037	+	0.222058i
$508$		0			0
$509$	13.4164			0.594672			0.297336	−	0.954773i	$-0.403902\pi$
$509$	13.4164			0.594672			0.297336	+	0.954773i	$0.403902\pi$
$510$		0			0
$511$	−12.0000	−	18.9737i	−0.530849	−	0.839346i
$512$		0			0
$513$	−31.6228	+	8.94427i	−1.39618	+	0.394899i
$514$		0			0
$515$	−6.32456			−0.278693
$516$		0			0
$517$	−16.9706			−0.746364
$518$		0			0
$519$	8.00000	+	11.3137i	0.351161	+	0.496617i
$520$		0			0
$521$	−13.4164			−0.587784			−0.293892	−	0.955839i	$-0.594951\pi$
$521$	−13.4164			−0.587784			−0.293892	+	0.955839i	$0.594951\pi$
$522$		0			0
$523$	−36.7696			−1.60782			−0.803910	−	0.594751i	$-0.797251\pi$
$523$	−36.7696			−1.60782			−0.803910	+	0.594751i	$0.797251\pi$
$524$		0			0
$525$	21.1803	+	8.74032i	0.924386	+	0.381459i
$526$		0			0
$527$	25.2982			1.10201
$528$		0			0
$529$	17.0000			0.739130
$530$		0			0
$531$	8.94427	−	25.2982i	0.388148	−	1.09785i
$532$		0			0
$533$	12.6491			0.547894
$534$		0			0
$535$	−42.4264			−1.83425
$536$		0			0
$537$	14.1421	+	20.0000i	0.610278	+	0.863064i
$538$		0			0
$539$	−17.8885	−	8.48528i	−0.770514	−	0.365487i
$540$		0			0
$541$	14.0000			0.601907			0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000			0.601907			0.300954	+	0.953639i	$0.402695\pi$
$542$		0			0
$543$	12.6491	+	17.8885i	0.542825	+	0.767671i
$544$		0			0
$545$	4.47214			0.191565
$546$		0			0
$547$		−	40.2492i		−	1.72093i	−0.509507	−	0.860466i	$-0.670172\pi$
$547$		−	40.2492i		−	1.72093i	0.509507	−	0.860466i	$-0.329828\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−35.7771			−1.52416
$552$		0			0
$553$	16.9706	+	26.8328i	0.721662	+	1.14105i
$554$		0			0
$555$	20.0000	+	28.2843i	0.848953	+	1.20060i
$556$		0			0
$557$	31.6228			1.33990			0.669950	−	0.742406i	$-0.266316\pi$
$557$	31.6228			1.33990			0.669950	+	0.742406i	$0.266316\pi$
$558$		0			0
$559$			12.6491i			0.535000i
$560$		0			0
$561$	16.0000	−	11.3137i	0.675521	−	0.477665i
$562$		0			0
$563$			30.0000i			1.26435i	0.774826	+	0.632175i	$0.217837\pi$
$563$			30.0000i			1.26435i	−0.774826	+	0.632175i	$0.782163\pi$
$564$		0			0
$565$	−42.4264			−1.78489
$566$		0			0
$567$	2.74962	−	23.6525i	0.115473	−	0.993311i
$568$		0			0
$569$		−	28.2843i		−	1.18574i	−0.805299	−	0.592869i	$-0.797995\pi$
$569$		−	28.2843i		−	1.18574i	0.805299	−	0.592869i	$-0.202005\pi$
$570$		0			0
$571$	8.00000			0.334790			0.167395	−	0.985890i	$-0.446465\pi$
$571$	8.00000			0.334790			0.167395	+	0.985890i	$0.446465\pi$
$572$		0			0
$573$	−2.82843	−	4.00000i	−0.118159	−	0.167102i
$574$		0			0
$575$	−31.6228			−1.31876
$576$		0			0
$577$	36.7696			1.53074			0.765368	−	0.643593i	$-0.222557\pi$
$577$	36.7696			1.53074			0.765368	+	0.643593i	$0.222557\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−22.3607	+	14.1421i	−0.927677	+	0.586715i
$582$		0			0
$583$			17.8885i			0.740868i
$584$		0			0
$585$	6.32456	−	17.8885i	0.261488	−	0.739600i
$586$		0			0
$587$		−	10.0000i		−	0.412744i	−0.978474	−	0.206372i	$-0.933834\pi$
$587$		−	10.0000i		−	0.412744i	0.978474	−	0.206372i	$-0.0661657\pi$
$588$		0			0
$589$	40.0000			1.64817
$590$		0			0
$591$	−26.8328	+	18.9737i	−1.10375	+	0.780472i
$592$		0			0
$593$		−	12.0000i		−	0.492781i	−0.969171	−	0.246390i	$-0.920755\pi$
$593$		−	12.0000i		−	0.492781i	0.969171	−	0.246390i	$-0.0792446\pi$
$594$		0			0
$595$	20.0000	−	12.6491i	0.819920	−	0.518563i
$596$		0			0
$597$	−18.9737	−	26.8328i	−0.776540	−	1.09819i
$598$		0			0
$599$			19.7990i			0.808965i	0.914546	+	0.404482i	$0.132548\pi$
$599$			19.7990i			0.808965i	−0.914546	+	0.404482i	$0.867452\pi$
$600$		0			0
$601$		−	12.6491i		−	0.515968i	−0.966149	−	0.257984i	$-0.916942\pi$
$601$		−	12.6491i		−	0.515968i	0.966149	−	0.257984i	$-0.0830582\pi$
$602$		0			0
$603$	−12.6491	−	4.47214i	−0.515112	−	0.182119i
$604$		0			0
$605$	6.70820			0.272727
$606$		0			0
$607$	19.7990			0.803616			0.401808	−	0.915724i	$-0.368382\pi$
$607$	19.7990			0.803616			0.401808	+	0.915724i	$0.368382\pi$
$608$		0			0
$609$	9.88854	−	23.9628i	0.400704	−	0.971022i
$610$		0			0
$611$			16.9706i			0.686555i
$612$		0			0
$613$			8.94427i			0.361256i	0.983552	+	0.180628i	$0.0578130\pi$
$613$			8.94427i			0.361256i	−0.983552	+	0.180628i	$0.942187\pi$
$614$		0			0
$615$	14.1421	−	10.0000i	0.570266	−	0.403239i
$616$		0			0
$617$	6.32456			0.254617			0.127309	−	0.991863i	$-0.459366\pi$
$617$	6.32456			0.254617			0.127309	+	0.991863i	$0.459366\pi$
$618$		0			0
$619$			18.9737i			0.762616i	0.924448	+	0.381308i	$0.124526\pi$
$619$			18.9737i			0.762616i	−0.924448	+	0.381308i	$0.875474\pi$
$620$		0			0
$621$	8.94427	+	31.6228i	0.358921	+	1.26898i
$622$		0			0
$623$	−6.32456	−	10.0000i	−0.253388	−	0.400642i
$624$		0			0
$625$	25.0000			1.00000
$626$		0			0
$627$	25.2982	−	17.8885i	1.01031	−	0.714400i
$628$		0			0
$629$	35.7771			1.42653
$630$		0			0
$631$	−4.00000			−0.159237			−0.0796187	−	0.996825i	$-0.525370\pi$
$631$	−4.00000			−0.159237			−0.0796187	+	0.996825i	$0.525370\pi$
$632$		0			0
$633$	−22.6274	+	16.0000i	−0.899359	+	0.635943i
$634$		0			0
$635$		−	10.0000i		−	0.396838i
$636$		0			0
$637$	−8.48528	+	17.8885i	−0.336199	+	0.708770i
$638$		0			0
$639$	−40.0000	−	14.1421i	−1.58238	−	0.559454i
$640$		0			0
$641$		−	28.2843i		−	1.11716i	−0.829450	−	0.558581i	$-0.811346\pi$
$641$		−	28.2843i		−	1.11716i	0.829450	−	0.558581i	$-0.188654\pi$
$642$		0			0
$643$	31.1127			1.22697			0.613483	−	0.789708i	$-0.289768\pi$
$643$	31.1127			1.22697			0.613483	+	0.789708i	$0.289768\pi$
$644$		0			0
$645$	10.0000	+	14.1421i	0.393750	+	0.556846i
$646$		0			0
$647$			10.0000i			0.393141i	0.980490	+	0.196570i	$0.0629804\pi$
$647$			10.0000i			0.393141i	−0.980490	+	0.196570i	$0.937020\pi$
$648$		0			0
$649$			25.2982i			0.993042i
$650$		0			0
$651$	−11.0557	+	26.7912i	−0.433308	+	1.05003i
$652$		0			0
$653$	−44.2719			−1.73249			−0.866246	−	0.499617i	$-0.833474\pi$
$653$	−44.2719			−1.73249			−0.866246	+	0.499617i	$0.833474\pi$
$654$		0			0
$655$	−20.0000			−0.781465
$656$		0			0
$657$	−8.48528	+	24.0000i	−0.331042	+	0.936329i
$658$		0			0
$659$		−	8.48528i		−	0.330540i	−0.986248	−	0.165270i	$-0.947151\pi$
$659$		−	8.48528i		−	0.330540i	0.986248	−	0.165270i	$-0.0528495\pi$
$660$		0			0
$661$			25.2982i			0.983987i	0.870599	+	0.491993i	$0.163732\pi$
$661$			25.2982i			0.983987i	−0.870599	+	0.491993i	$0.836268\pi$
$662$		0			0
$663$	−11.3137	−	16.0000i	−0.439388	−	0.621389i
$664$		0			0
$665$	31.6228	−	20.0000i	1.22628	−	0.775567i
$666$		0			0
$667$			35.7771i			1.38529i
$668$		0			0
$669$	4.00000	−	2.82843i	0.154649	−	0.109353i
$670$		0			0
$671$		0			0
$672$		0			0
$673$		−	35.7771i		−	1.37911i	−0.724236	−	0.689553i	$-0.757807\pi$
$673$		−	35.7771i		−	1.37911i	0.724236	−	0.689553i	$-0.242193\pi$
$674$		0			0
$675$	−7.07107	−	25.0000i	−0.272166	−	0.962250i
$676$		0			0
$677$		−	24.0000i		−	0.922395i	−0.887298	−	0.461197i	$-0.847420\pi$
$677$		−	24.0000i		−	0.922395i	0.887298	−	0.461197i	$-0.152580\pi$
$678$		0			0
$679$	−12.0000	−	18.9737i	−0.460518	−	0.728142i
$680$		0			0
$681$	−6.00000	−	8.48528i	−0.229920	−	0.325157i
$682$		0			0
$683$	6.32456			0.242002			0.121001	−	0.992652i	$-0.461390\pi$
$683$	6.32456			0.242002			0.121001	+	0.992652i	$0.461390\pi$
$684$		0			0
$685$	14.1421			0.540343
$686$		0			0
$687$	−12.6491	−	17.8885i	−0.482594	−	0.682491i
$688$		0			0
$689$	17.8885			0.681499
$690$		0			0
$691$			18.9737i			0.721792i	0.932606	+	0.360896i	$0.117529\pi$
$691$			18.9737i			0.721792i	−0.932606	+	0.360896i	$0.882471\pi$
$692$		0			0
$693$	4.98915	+	21.8885i	0.189522	+	0.831477i
$694$		0			0
$695$		−	14.1421i		−	0.536442i
$696$		0			0
$697$		−	17.8885i		−	0.677577i
$698$		0			0
$699$	8.94427	−	6.32456i	0.338303	−	0.239217i
$700$		0			0
$701$		−	45.2548i		−	1.70925i	−0.519244	−	0.854626i	$-0.673787\pi$
$701$		−	45.2548i		−	1.70925i	0.519244	−	0.854626i	$-0.326213\pi$
$702$		0			0
$703$	56.5685			2.13352
$704$		0			0
$705$	13.4164	+	18.9737i	0.505291	+	0.714590i
$706$		0			0
$707$	−6.32456	−	10.0000i	−0.237859	−	0.376089i
$708$		0			0
$709$	−42.0000			−1.57734			−0.788672	−	0.614815i	$-0.789231\pi$
$709$	−42.0000			−1.57734			−0.788672	+	0.614815i	$0.789231\pi$
$710$		0			0
$711$	12.0000	−	33.9411i	0.450035	−	1.27289i
$712$		0			0
$713$		−	40.0000i		−	1.49801i
$714$		0			0
$715$			17.8885i			0.668994i
$716$		0			0
$717$	8.48528	+	12.0000i	0.316889	+	0.448148i
$718$		0			0
$719$	17.8885			0.667130			0.333565	−	0.942727i	$-0.391748\pi$
$719$	17.8885			0.667130			0.333565	+	0.942727i	$0.391748\pi$
$720$		0			0
$721$	−4.00000	−	6.32456i	−0.148968	−	0.235539i
$722$		0			0
$723$	12.6491	+	17.8885i	0.470425	+	0.665282i
$724$		0			0
$725$		−	28.2843i		−	1.05045i
$726$		0			0
$727$	−25.4558			−0.944105			−0.472052	−	0.881570i	$-0.656487\pi$
$727$	−25.4558			−0.944105			−0.472052	+	0.881570i	$0.656487\pi$
$728$		0			0
$729$	−23.0000	+	14.1421i	−0.851852	+	0.523783i
$730$		0			0
$731$	17.8885			0.661632
$732$		0			0
$733$	25.4558			0.940233			0.470117	−	0.882604i	$-0.344212\pi$
$733$	25.4558			0.940233			0.470117	+	0.882604i	$0.344212\pi$
$734$		0			0
$735$	4.65530	+	26.7082i	0.171713	+	0.985147i
$736$		0			0
$737$	12.6491			0.465936
$738$		0			0
$739$	8.00000			0.294285			0.147142	−	0.989115i	$-0.452992\pi$
$739$	8.00000			0.294285			0.147142	+	0.989115i	$0.452992\pi$
$740$		0			0
$741$	−17.8885	−	25.2982i	−0.657152	−	0.929353i
$742$		0			0
$743$	−18.9737			−0.696076			−0.348038	−	0.937480i	$-0.613152\pi$
$743$	−18.9737			−0.696076			−0.348038	+	0.937480i	$0.613152\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$	28.2843	+	10.0000i	1.03487	+	0.365881i
$748$		0			0
$749$	−26.8328	−	42.4264i	−0.980450	−	1.55023i
$750$		0			0
$751$	−4.00000			−0.145962			−0.0729810	−	0.997333i	$-0.523251\pi$
$751$	−4.00000			−0.145962			−0.0729810	+	0.997333i	$0.523251\pi$
$752$		0			0
$753$	−37.9473	+	26.8328i	−1.38288	+	0.977842i
$754$		0			0
$755$	−8.94427			−0.325515
$756$		0			0
$757$			8.94427i			0.325085i	0.986702	+	0.162543i	$0.0519695\pi$
$757$			8.94427i			0.325085i	−0.986702	+	0.162543i	$0.948031\pi$
$758$		0			0
$759$	−17.8885	−	25.2982i	−0.649313	−	0.918267i
$760$		0			0
$761$	−4.47214			−0.162115			−0.0810574	−	0.996709i	$-0.525830\pi$
$761$	−4.47214			−0.162115			−0.0810574	+	0.996709i	$0.525830\pi$
$762$		0			0
$763$	2.82843	+	4.47214i	0.102396	+	0.161902i
$764$		0			0
$765$	−25.2982	−	8.94427i	−0.914659	−	0.323381i
$766$		0			0
$767$	25.2982			0.913466
$768$		0			0
$769$			25.2982i			0.912277i	0.889909	+	0.456139i	$0.150768\pi$
$769$			25.2982i			0.912277i	−0.889909	+	0.456139i	$0.849232\pi$
$770$		0			0
$771$	20.0000	+	28.2843i	0.720282	+	1.01863i
$772$		0			0
$773$		−	32.0000i		−	1.15096i	−0.817816	−	0.575480i	$-0.804815\pi$
$773$		−	32.0000i		−	1.15096i	0.817816	−	0.575480i	$-0.195185\pi$
$774$		0			0
$775$			31.6228i			1.13592i
$776$		0			0
$777$	−15.6352	+	37.8885i	−0.560908	+	1.35924i
$778$		0			0
$779$		−	28.2843i		−	1.01339i
$780$		0			0
$781$	40.0000			1.43131
$782$		0			0
$783$	−28.2843	+	8.00000i	−1.01080	+	0.285897i
$784$		0			0
$785$	6.32456			0.225733
$786$		0			0
$787$	−8.48528			−0.302468			−0.151234	−	0.988498i	$-0.548325\pi$
$787$	−8.48528			−0.302468			−0.151234	+	0.988498i	$0.548325\pi$
$788$		0			0
$789$	8.94427	−	6.32456i	0.318425	−	0.225160i
$790$		0			0
$791$	−26.8328	−	42.4264i	−0.954065	−	1.50851i
$792$		0			0
$793$		0			0
$794$		0			0
$795$	20.0000	−	14.1421i	0.709327	−	0.501570i
$796$		0			0
$797$			16.0000i			0.566749i	0.959009	+	0.283375i	$0.0914540\pi$
$797$			16.0000i			0.566749i	−0.959009	+	0.283375i	$0.908546\pi$
$798$		0			0
$799$	24.0000			0.849059
$800$		0			0
$801$	−4.47214	+	12.6491i	−0.158015	+	0.446934i
$802$		0			0
$803$		−	24.0000i		−	0.846942i
$804$		0			0
$805$	−20.0000	−	31.6228i	−0.704907	−	1.11456i
$806$		0			0
$807$	−6.32456	+	4.47214i	−0.222635	+	0.157427i
$808$		0			0
$809$			22.6274i			0.795538i	0.917486	+	0.397769i	$0.130215\pi$
$809$			22.6274i			0.795538i	−0.917486	+	0.397769i	$0.869785\pi$
$810$		0			0
$811$		−	31.6228i		−	1.11043i	−0.831708	−	0.555213i	$-0.812637\pi$
$811$		−	31.6228i		−	1.11043i	0.831708	−	0.555213i	$-0.187363\pi$
$812$		0			0
$813$	6.32456	+	8.94427i	0.221812	+	0.313689i
$814$		0			0
$815$		−	30.0000i		−	1.05085i
$816$		0			0
$817$	28.2843			0.989541
$818$		0			0
$819$	21.8885	−	4.98915i	0.764848	−	0.174335i
$820$		0			0
$821$			11.3137i			0.394851i	0.980318	+	0.197426i	$0.0632581\pi$
$821$			11.3137i			0.394851i	−0.980318	+	0.197426i	$0.936742\pi$
$822$		0			0
$823$		−	22.3607i		−	0.779444i	−0.920932	−	0.389722i	$-0.872571\pi$
$823$		−	22.3607i		−	0.779444i	0.920932	−	0.389722i	$-0.127429\pi$
$824$		0			0
$825$	14.1421	+	20.0000i	0.492366	+	0.696311i
$826$		0			0
$827$	−18.9737			−0.659779			−0.329890	−	0.944020i	$-0.607011\pi$
$827$	−18.9737			−0.659779			−0.329890	+	0.944020i	$0.607011\pi$
$828$		0			0
$829$			25.2982i			0.878644i	0.898330	+	0.439322i	$0.144781\pi$
$829$			25.2982i			0.878644i	−0.898330	+	0.439322i	$0.855219\pi$
$830$		0			0
$831$	26.8328	+	37.9473i	0.930820	+	1.31638i
$832$		0			0
$833$	25.2982	+	12.0000i	0.876531	+	0.415775i
$834$		0			0
$835$		−	13.4164i		−	0.464294i
$836$		0			0
$837$	31.6228	−	8.94427i	1.09304	−	0.309159i
$838$		0			0
$839$	35.7771			1.23516			0.617581	−	0.786507i	$-0.288113\pi$
$839$	35.7771			1.23516			0.617581	+	0.786507i	$0.288113\pi$
$840$		0			0
$841$	−3.00000			−0.103448
$842$		0			0
$843$	16.9706	+	24.0000i	0.584497	+	0.826604i
$844$		0			0
$845$	−11.1803			−0.384615
$846$		0			0
$847$	4.24264	+	6.70820i	0.145779	+	0.230497i
$848$		0			0
$849$	36.0000	−	25.4558i	1.23552	−	0.873642i
$850$		0			0
$851$		−	56.5685i		−	1.93914i
$852$		0			0
$853$	36.7696			1.25897			0.629483	−	0.777014i	$-0.283267\pi$
$853$	36.7696			1.25897			0.629483	+	0.777014i	$0.283267\pi$
$854$		0			0
$855$	−40.0000	−	14.1421i	−1.36797	−	0.483651i
$856$		0			0
$857$		−	20.0000i		−	0.683187i	−0.939848	−	0.341593i	$-0.889033\pi$
$857$		−	20.0000i		−	0.683187i	0.939848	−	0.341593i	$-0.110967\pi$
$858$		0			0
$859$			44.2719i			1.51054i	0.655415	+	0.755269i	$0.272494\pi$
$859$			44.2719i			1.51054i	−0.655415	+	0.755269i	$0.727506\pi$
$860$		0			0
$861$	18.9443	+	7.81758i	0.645619	+	0.266422i
$862$		0			0
$863$	18.9737			0.645871			0.322936	−	0.946421i	$-0.395330\pi$
$863$	18.9737			0.645871			0.322936	+	0.946421i	$0.395330\pi$
$864$		0			0
$865$			17.8885i			0.608229i
$866$		0			0
$867$	1.41421	−	1.00000i	0.0480292	−	0.0339618i
$868$		0			0
$869$			33.9411i			1.15137i
$870$		0			0
$871$		−	12.6491i		−	0.428599i
$872$		0			0
$873$	−8.48528	+	24.0000i	−0.287183	+	0.812277i
$874$		0			0
$875$	15.8114	+	25.0000i	0.534522	+	0.845154i
$876$		0			0
$877$		−	26.8328i		−	0.906080i	−0.891490	−	0.453040i	$-0.850340\pi$
$877$		−	26.8328i		−	0.906080i	0.891490	−	0.453040i	$-0.149660\pi$
$878$		0			0
$879$	32.0000	+	45.2548i	1.07933	+	1.52641i
$880$		0			0
$881$	31.3050			1.05469			0.527345	−	0.849651i	$-0.323187\pi$
$881$	31.3050			1.05469			0.527345	+	0.849651i	$0.323187\pi$
$882$		0			0
$883$		−	40.2492i		−	1.35449i	−0.735756	−	0.677247i	$-0.763173\pi$
$883$		−	40.2492i		−	1.35449i	0.735756	−	0.677247i	$-0.236827\pi$
$884$		0			0
$885$	28.2843	−	20.0000i	0.950765	−	0.672293i
$886$		0			0
$887$			30.0000i			1.00730i	0.863907	+	0.503651i	$0.168010\pi$
$887$			30.0000i			1.00730i	−0.863907	+	0.503651i	$0.831990\pi$
$888$		0			0
$889$	10.0000	−	6.32456i	0.335389	−	0.212119i
$890$		0			0
$891$	16.0000	−	19.7990i	0.536020	−	0.663291i
$892$		0			0
$893$	37.9473			1.26986
$894$		0			0
$895$			31.6228i			1.05703i
$896$		0			0
$897$	−25.2982	+	17.8885i	−0.844683	+	0.597281i
$898$		0			0
$899$	35.7771			1.19323
$900$		0			0
$901$		−	25.2982i		−	0.842806i
$902$		0			0
$903$	−7.81758	+	18.9443i	−0.260153	+	0.630426i
$904$		0			0
$905$			28.2843i			0.940201i
$906$		0			0
$907$			13.4164i			0.445485i	0.974877	+	0.222742i	$0.0715008\pi$
$907$			13.4164i			0.445485i	−0.974877	+	0.222742i	$0.928499\pi$
$908$		0			0
$909$	−4.47214	+	12.6491i	−0.148331	+	0.419545i
$910$		0			0
$911$		−	14.1421i		−	0.468550i	−0.972170	−	0.234275i	$-0.924728\pi$
$911$		−	14.1421i		−	0.468550i	0.972170	−	0.234275i	$-0.0752716\pi$
$912$		0			0
$913$	−28.2843			−0.936073
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.6491	−	20.0000i	−0.417710	−	0.660458i
$918$		0			0
$919$	52.0000			1.71532			0.857661	−	0.514216i	$-0.171917\pi$
$919$	52.0000			1.71532			0.857661	+	0.514216i	$0.171917\pi$
$920$		0			0
$921$	−12.0000	+	8.48528i	−0.395413	+	0.279600i
$922$		0			0
$923$		−	40.0000i		−	1.31662i
$924$		0			0
$925$			44.7214i			1.47043i
$926$		0			0
$927$	−2.82843	+	8.00000i	−0.0928977	+	0.262754i
$928$		0			0
$929$	−4.47214			−0.146726			−0.0733630	−	0.997305i	$-0.523373\pi$
$929$	−4.47214			−0.146726			−0.0733630	+	0.997305i	$0.523373\pi$
$930$		0			0
$931$	40.0000	+	18.9737i	1.31095	+	0.621837i
$932$		0			0
$933$	25.2982	−	17.8885i	0.828227	−	0.585645i
$934$		0			0
$935$	25.2982			0.827340
$936$		0			0
$937$	−8.48528			−0.277202			−0.138601	−	0.990348i	$-0.544261\pi$
$937$	−8.48528			−0.277202			−0.138601	+	0.990348i	$0.544261\pi$
$938$		0			0
$939$	36.0000	−	25.4558i	1.17482	−	0.830720i
$940$		0			0
$941$	40.2492			1.31209			0.656044	−	0.754723i	$-0.272229\pi$
$941$	40.2492			1.31209			0.656044	+	0.754723i	$0.272229\pi$
$942$		0			0
$943$	−28.2843			−0.921063
$944$		0			0
$945$	20.5279	−	22.8825i	0.667771	−	0.744366i
$946$		0			0
$947$	18.9737			0.616561			0.308281	−	0.951295i	$-0.400246\pi$
$947$	18.9737			0.616561			0.308281	+	0.951295i	$0.400246\pi$
$948$		0			0
$949$	−24.0000			−0.779073
$950$		0			0
$951$	−26.8328	+	18.9737i	−0.870114	+	0.615263i
$952$		0			0
$953$	−18.9737			−0.614617			−0.307309	−	0.951610i	$-0.599428\pi$
$953$	−18.9737			−0.614617			−0.307309	+	0.951610i	$0.599428\pi$
$954$		0			0
$955$		−	6.32456i		−	0.204658i
$956$		0			0
$957$	22.6274	−	16.0000i	0.731441	−	0.517207i
$958$		0			0
$959$	8.94427	+	14.1421i	0.288826	+	0.456673i
$960$		0			0
$961$	−9.00000			−0.290323
$962$		0			0
$963$	−18.9737	+	53.6656i	−0.611418	+	1.72935i
$964$		0			0
$965$		0			0
$966$		0			0
$967$			40.2492i			1.29433i	0.762351	+	0.647164i	$0.224045\pi$
$967$			40.2492i			1.29433i	−0.762351	+	0.647164i	$0.775955\pi$
$968$		0			0
$969$	−35.7771	+	25.2982i	−1.14933	+	0.812696i
$970$		0			0
$971$	44.7214			1.43518			0.717588	−	0.696467i	$-0.245246\pi$
$971$	44.7214			1.43518			0.717588	+	0.696467i	$0.245246\pi$
$972$		0			0
$973$	14.1421	−	8.94427i	0.453376	−	0.286740i
$974$		0			0
$975$	20.0000	−	14.1421i	0.640513	−	0.452911i
$976$		0			0
$977$	−18.9737			−0.607021			−0.303511	−	0.952828i	$-0.598159\pi$
$977$	−18.9737			−0.607021			−0.303511	+	0.952828i	$0.598159\pi$
$978$		0			0
$979$		−	12.6491i		−	0.404267i
$980$		0			0
$981$	2.00000	−	5.65685i	0.0638551	−	0.180609i
$982$		0			0
$983$		−	18.0000i		−	0.574111i	−0.957914	−	0.287055i	$-0.907324\pi$
$983$		−	18.0000i		−	0.574111i	0.957914	−	0.287055i	$-0.0926764\pi$
$984$		0			0
$985$	−42.4264			−1.35182
$986$		0			0
$987$	−10.4884	+	25.4164i	−0.333849	+	0.809013i
$988$		0			0
$989$		−	28.2843i		−	0.899388i
$990$		0			0
$991$	−28.0000			−0.889449			−0.444725	−	0.895667i	$-0.646698\pi$
$991$	−28.0000			−0.889449			−0.444725	+	0.895667i	$0.646698\pi$
$992$		0			0
$993$	−11.3137	+	8.00000i	−0.359030	+	0.253872i
$994$		0			0
$995$		−	42.4264i		−	1.34501i
$996$		0			0
$997$	−8.48528			−0.268732			−0.134366	−	0.990932i	$-0.542900\pi$
$997$	−8.48528			−0.268732			−0.134366	+	0.990932i	$0.542900\pi$
$998$		0			0
$999$	44.7214	−	12.6491i	1.41492	−	0.400200i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.k.g.209.6		8
3.2	odd	2	inner	1680.2.k.g.209.7		8
4.3	odd	2		420.2.f.b.209.4	yes	8
5.4	even	2	inner	1680.2.k.g.209.4		8
7.6	odd	2	inner	1680.2.k.g.209.3		8
12.11	even	2		420.2.f.b.209.1	✓	8
15.14	odd	2	inner	1680.2.k.g.209.1		8
20.3	even	4		2100.2.d.f.1301.1		4
20.7	even	4		2100.2.d.i.1301.4		4
20.19	odd	2		420.2.f.b.209.6	yes	8
21.20	even	2	inner	1680.2.k.g.209.2		8
28.27	even	2		420.2.f.b.209.5	yes	8
35.34	odd	2	inner	1680.2.k.g.209.5		8
60.23	odd	4		2100.2.d.i.1301.1		4
60.47	odd	4		2100.2.d.f.1301.4		4
60.59	even	2		420.2.f.b.209.7	yes	8
84.83	odd	2		420.2.f.b.209.8	yes	8
105.104	even	2	inner	1680.2.k.g.209.8		8
140.27	odd	4		2100.2.d.f.1301.2		4
140.83	odd	4		2100.2.d.i.1301.3		4
140.139	even	2		420.2.f.b.209.3	yes	8
420.83	even	4		2100.2.d.f.1301.3		4
420.167	even	4		2100.2.d.i.1301.2		4
420.419	odd	2		420.2.f.b.209.2	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
420.2.f.b.209.1	✓	8	12.11	even	2
420.2.f.b.209.2	yes	8	420.419	odd	2
420.2.f.b.209.3	yes	8	140.139	even	2
420.2.f.b.209.4	yes	8	4.3	odd	2
420.2.f.b.209.5	yes	8	28.27	even	2
420.2.f.b.209.6	yes	8	20.19	odd	2
420.2.f.b.209.7	yes	8	60.59	even	2
420.2.f.b.209.8	yes	8	84.83	odd	2
1680.2.k.g.209.1		8	15.14	odd	2	inner
1680.2.k.g.209.2		8	21.20	even	2	inner
1680.2.k.g.209.3		8	7.6	odd	2	inner
1680.2.k.g.209.4		8	5.4	even	2	inner
1680.2.k.g.209.5		8	35.34	odd	2	inner
1680.2.k.g.209.6		8	1.1	even	1	trivial
1680.2.k.g.209.7		8	3.2	odd	2	inner
1680.2.k.g.209.8		8	105.104	even	2	inner
2100.2.d.f.1301.1		4	20.3	even	4
2100.2.d.f.1301.2		4	140.27	odd	4
2100.2.d.f.1301.3		4	420.83	even	4
2100.2.d.f.1301.4		4	60.47	odd	4
2100.2.d.i.1301.1		4	60.23	odd	4
2100.2.d.i.1301.2		4	420.167	even	4
2100.2.d.i.1301.3		4	140.83	odd	4
2100.2.d.i.1301.4		4	20.7	even	4

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)

\(f(q)\)	\(=\)	\(q+(1.41421 - 1.00000i) q^{3} +2.23607 q^{5} +(1.41421 + 2.23607i) q^{7} +(1.00000 - 2.82843i) q^{9} +2.82843i q^{11} +2.82843 q^{13} +(3.16228 - 2.23607i) q^{15} -4.00000i q^{17} -6.32456i q^{19} +(4.23607 + 1.74806i) q^{21} -6.32456 q^{23} +5.00000 q^{25} +(-1.41421 - 5.00000i) q^{27} -5.65685i q^{29} +6.32456i q^{31} +(2.82843 + 4.00000i) q^{33} +(3.16228 + 5.00000i) q^{35} +8.94427i q^{37} +(4.00000 - 2.82843i) q^{39} +4.47214 q^{41} +4.47214i q^{43} +(2.23607 - 6.32456i) q^{45} +6.00000i q^{47} +(-3.00000 + 6.32456i) q^{49} +(-4.00000 - 5.65685i) q^{51} +6.32456 q^{53} +6.32456i q^{55} +(-6.32456 - 8.94427i) q^{57} +8.94427 q^{59} +(7.73877 - 1.76393i) q^{63} +6.32456 q^{65} -4.47214i q^{67} +(-8.94427 + 6.32456i) q^{69} -14.1421i q^{71} -8.48528 q^{73} +(7.07107 - 5.00000i) q^{75} +(-6.32456 + 4.00000i) q^{77} +12.0000 q^{79} +(-7.00000 - 5.65685i) q^{81} +10.0000i q^{83} -8.94427i q^{85} +(-5.65685 - 8.00000i) q^{87} -4.47214 q^{89} +(4.00000 + 6.32456i) q^{91} +(6.32456 + 8.94427i) q^{93} -14.1421i q^{95} -8.48528 q^{97} +(8.00000 + 2.82843i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{9} + 16 q^{21} + 40 q^{25} + 32 q^{39} - 24 q^{49} - 32 q^{51} + 96 q^{79} - 56 q^{81} + 32 q^{91} + 64 q^{99}+O(q^{100}) \) 8 * q + 8 * q^9 + 16 * q^21 + 40 * q^25 + 32 * q^39 - 24 * q^49 - 32 * q^51 + 96 * q^79 - 56 * q^81 + 32 * q^91 + 64 * q^99

Introduction

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