LMFDB - Embedded newform 1680.2.v.a.239.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1680,2,Mod(239,1680)] 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([1, 0, 1, 1, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1680.239"); 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.v (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [12,0,-2] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$13.4148675396$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 8x^{10} + 55x^{8} - 272x^{6} + 1375x^{4} - 5000x^{2} + 15625 $ x^12 - 8x^10 + 55x^8 - 272x^6 + 1375x^4 - 5000*x^2 + 15625
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			239.3
Root			$1.20750 - 1.88200i$ of defining polynomial
Character	$\chi$	$=$	1680.239
Dual form			1680.2.v.a.239.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-1.64854 + 0.531349i) q^{3} +(-1.20750 - 1.88200i) q^{5} -1.00000 q^{7} +(2.43534 - 1.75189i) q^{9} -4.93129 q^{11} +6.01886i q^{13} +(2.99061 + 2.46094i) q^{15} +3.98122 q^{17} +1.78803i q^{19} +(1.64854 - 0.531349i) q^{21} +1.63861i q^{23} +(-2.08387 + 4.54505i) q^{25} +(-3.08387 + 4.18207i) q^{27} +1.89883i q^{29} -4.73565i q^{31} +(8.12940 - 2.62023i) q^{33} +(1.20750 + 1.88200i) q^{35} -6.52368i q^{37} +(-3.19811 - 9.92231i) q^{39} -9.48232i q^{41} +1.57360 q^{43} +(-6.23775 - 2.46789i) q^{45} -8.90641i q^{47} +1.00000 q^{49} +(-6.56319 + 2.11542i) q^{51} -0.950066 q^{53} +(5.95455 + 9.28070i) q^{55} +(-0.950066 - 2.94762i) q^{57} +12.7925 q^{59} +8.59414 q^{61} +(-2.43534 + 1.75189i) q^{63} +(11.3275 - 7.26779i) q^{65} -9.74135 q^{67} +(-0.870674 - 2.70131i) q^{69} +3.36507 q^{71} -7.68328i q^{73} +(1.02033 - 8.59994i) q^{75} +4.93129 q^{77} +16.2686i q^{79} +(2.86173 - 8.53291i) q^{81} -0.520436i q^{83} +(-4.80734 - 7.49267i) q^{85} +(-1.00894 - 3.13029i) q^{87} -16.1742i q^{89} -6.01886i q^{91} +(2.51628 + 7.80689i) q^{93} +(3.36507 - 2.15905i) q^{95} +8.58528i q^{97} +(-12.0094 + 8.63910i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 2 q^{3} - 12 q^{7} + 6 q^{9} + 12 q^{15} + 2 q^{21} + 16 q^{25} + 4 q^{27} + 8 q^{43} - 4 q^{45} + 12 q^{49} - 16 q^{55} + 32 q^{61} - 6 q^{63} - 24 q^{67} + 36 q^{69} + 18 q^{75} + 22 q^{81} + 8 q^{85}+ \cdots + 22 q^{87}+O(q^{100}) $ 12 * q - 2 * q^3 - 12 * q^7 + 6 * q^9 + 12 * q^15 + 2 * q^21 + 16 * q^25 + 4 * q^27 + 8 * q^43 - 4 * q^45 + 12 * q^49 - 16 * q^55 + 32 * q^61 - 6 * q^63 - 24 * q^67 + 36 * q^69 + 18 * q^75 + 22 * q^81 + 8 * q^85 + 22 * q^87

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
$3$	−1.64854	+	0.531349i	−0.951782	+	0.306774i
$3$	−1.64854	+	0.531349i	−0.951782	+	0.306774i
$4$		0			0
$5$	−1.20750	−	1.88200i	−0.540012	−	0.841657i
$5$	−1.20750	−	1.88200i	−0.540012	−	0.841657i
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$	2.43534	−	1.75189i	0.811779	−	0.583965i
$10$		0			0
$11$	−4.93129			−1.48684			−0.743420	−	0.668825i	$-0.766797\pi$
$11$	−4.93129			−1.48684			−0.743420	+	0.668825i	$0.766797\pi$
$12$		0			0
$13$			6.01886i			1.66933i	0.550756	+	0.834666i	$0.314339\pi$
$13$			6.01886i			1.66933i	−0.550756	+	0.834666i	$0.685661\pi$
$14$		0			0
$15$	2.99061	+	2.46094i	0.772173	+	0.635413i
$16$		0			0
$17$	3.98122			0.965588			0.482794	−	0.875734i	$-0.339622\pi$
$17$	3.98122			0.965588			0.482794	+	0.875734i	$0.339622\pi$
$18$		0			0
$19$			1.78803i			0.410201i	0.978741	+	0.205101i	$0.0657522\pi$
$19$			1.78803i			0.410201i	−0.978741	+	0.205101i	$0.934248\pi$
$20$		0			0
$21$	1.64854	−	0.531349i	0.359740	−	0.115950i
$22$		0			0
$23$			1.63861i			0.341674i	0.985299	+	0.170837i	$0.0546472\pi$
$23$			1.63861i			0.341674i	−0.985299	+	0.170837i	$0.945353\pi$
$24$		0			0
$25$	−2.08387	+	4.54505i	−0.416774	+	0.909010i
$26$		0			0
$27$	−3.08387	+	4.18207i	−0.593492	+	0.804840i
$28$		0			0
$29$			1.89883i			0.352604i	0.984336	+	0.176302i	$0.0564135\pi$
$29$			1.89883i			0.352604i	−0.984336	+	0.176302i	$0.943586\pi$
$30$		0			0
$31$		−	4.73565i		−	0.850548i	−0.905065	−	0.425274i	$-0.860178\pi$
$31$		−	4.73565i		−	0.850548i	0.905065	−	0.425274i	$-0.139822\pi$
$32$		0			0
$33$	8.12940	−	2.62023i	1.41515	−	0.456124i
$34$		0			0
$35$	1.20750	+	1.88200i	0.204105	+	0.318117i
$36$		0			0
$37$		−	6.52368i		−	1.07249i	−0.844064	−	0.536243i	$-0.819843\pi$
$37$		−	6.52368i		−	1.07249i	0.844064	−	0.536243i	$-0.180157\pi$
$38$		0			0
$39$	−3.19811	−	9.92231i	−0.512108	−	1.58884i
$40$		0			0
$41$		−	9.48232i		−	1.48089i	−0.672117	−	0.740445i	$-0.734615\pi$
$41$		−	9.48232i		−	1.48089i	0.672117	−	0.740445i	$-0.265385\pi$
$42$		0			0
$43$	1.57360			0.239972			0.119986	−	0.992776i	$-0.461715\pi$
$43$	1.57360			0.239972			0.119986	+	0.992776i	$0.461715\pi$
$44$		0			0
$45$	−6.23775	−	2.46789i	−0.929869	−	0.367892i
$46$		0			0
$47$		−	8.90641i		−	1.29913i	−0.760305	−	0.649566i	$-0.774950\pi$
$47$		−	8.90641i		−	1.29913i	0.760305	−	0.649566i	$-0.225050\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	−6.56319	+	2.11542i	−0.919030	+	0.296218i
$52$		0			0
$53$	−0.950066			−0.130502			−0.0652508	−	0.997869i	$-0.520785\pi$
$53$	−0.950066			−0.130502			−0.0652508	+	0.997869i	$0.520785\pi$
$54$		0			0
$55$	5.95455	+	9.28070i	0.802911	+	1.25141i
$56$		0			0
$57$	−0.950066	−	2.94762i	−0.125839	−	0.390422i
$58$		0			0
$59$	12.7925			1.66544			0.832718	−	0.553697i	$-0.186783\pi$
$59$	12.7925			1.66544			0.832718	+	0.553697i	$0.186783\pi$
$60$		0			0
$61$	8.59414			1.10037			0.550184	−	0.835044i	$-0.314558\pi$
$61$	8.59414			1.10037			0.550184	+	0.835044i	$0.314558\pi$
$62$		0			0
$63$	−2.43534	+	1.75189i	−0.306824	+	0.220718i
$64$		0			0
$65$	11.3275	−	7.26779i	1.40501	−	0.901459i
$66$		0			0
$67$	−9.74135			−1.19010			−0.595048	−	0.803690i	$-0.702867\pi$
$67$	−9.74135			−1.19010			−0.595048	+	0.803690i	$0.702867\pi$
$68$		0			0
$69$	−0.870674	−	2.70131i	−0.104817	−	0.325199i
$70$		0			0
$71$	3.36507			0.399361			0.199680	−	0.979861i	$-0.436010\pi$
$71$	3.36507			0.399361			0.199680	+	0.979861i	$0.436010\pi$
$72$		0			0
$73$		−	7.68328i		−	0.899260i	−0.893215	−	0.449630i	$-0.851556\pi$
$73$		−	7.68328i		−	0.899260i	0.893215	−	0.449630i	$-0.148444\pi$
$74$		0			0
$75$	1.02033	−	8.59994i	0.117818	−	0.993035i
$76$		0			0
$77$	4.93129			0.561973
$78$		0			0
$79$			16.2686i			1.83036i	0.403050	+	0.915178i	$0.367950\pi$
$79$			16.2686i			1.83036i	−0.403050	+	0.915178i	$0.632050\pi$
$80$		0			0
$81$	2.86173	−	8.53291i	0.317970	−	0.948101i
$82$		0			0
$83$		−	0.520436i		−	0.0571253i	−0.999592	−	0.0285626i	$-0.990907\pi$
$83$		−	0.520436i		−	0.0571253i	0.999592	−	0.0285626i	$-0.00909301\pi$
$84$		0			0
$85$	−4.80734	−	7.49267i	−0.521429	−	0.812695i
$86$		0			0
$87$	−1.00894	−	3.13029i	−0.108170	−	0.335602i
$88$		0			0
$89$		−	16.1742i		−	1.71446i	−0.514932	−	0.857231i	$-0.672183\pi$
$89$		−	16.1742i		−	1.71446i	0.514932	−	0.857231i	$-0.327817\pi$
$90$		0			0
$91$		−	6.01886i		−	0.630948i
$92$		0			0
$93$	2.51628	+	7.80689i	0.260926	+	0.809537i
$94$		0			0
$95$	3.36507	−	2.15905i	0.345249	−	0.221514i
$96$		0			0
$97$			8.58528i			0.871703i	0.900018	+	0.435852i	$0.143553\pi$
$97$			8.58528i			0.871703i	−0.900018	+	0.435852i	$0.856447\pi$
$98$		0			0
$99$	−12.0094	+	8.63910i	−1.20699	+	0.868262i
$100$		0			0
$101$			3.44831i			0.343120i	0.985174	+	0.171560i	$0.0548807\pi$
$101$			3.44831i			0.343120i	−0.985174	+	0.171560i	$0.945119\pi$
$102$		0			0
$103$	7.29707			0.719002			0.359501	−	0.933145i	$-0.382947\pi$
$103$	7.29707			0.719002			0.359501	+	0.933145i	$0.382947\pi$
$104$		0			0
$105$	−2.99061	−	2.46094i	−0.291854	−	0.240163i
$106$		0			0
$107$		−	12.8970i		−	1.24680i	−0.781904	−	0.623399i	$-0.785751\pi$
$107$		−	12.8970i		−	1.24680i	0.781904	−	0.623399i	$-0.214249\pi$
$108$		0			0
$109$	2.44428			0.234119			0.117060	−	0.993125i	$-0.462653\pi$
$109$	2.44428			0.234119			0.117060	+	0.993125i	$0.462653\pi$
$110$		0			0
$111$	3.46635	+	10.7545i	0.329011	+	1.02077i
$112$		0			0
$113$	−2.85020			−0.268124			−0.134062	−	0.990973i	$-0.542802\pi$
$113$	−2.85020			−0.268124			−0.134062	+	0.990973i	$0.542802\pi$
$114$		0			0
$115$	3.08387	−	1.97863i	0.287573	−	0.184508i
$116$		0			0
$117$	10.5444	+	14.6580i	0.974831	+	1.35513i
$118$		0			0
$119$	−3.98122			−0.364958
$120$		0			0
$121$	13.3176			1.21069
$122$		0			0
$123$	5.03842	+	15.6319i	0.454299	+	1.40948i
$124$		0			0
$125$	11.0701	−	1.56631i	0.990138	−	0.140095i
$126$		0			0
$127$	1.02054			0.0905581			0.0452790	−	0.998974i	$-0.485582\pi$
$127$	1.02054			0.0905581			0.0452790	+	0.998974i	$0.485582\pi$
$128$		0			0
$129$	−2.59414	+	0.836132i	−0.228401	+	0.0736173i
$130$		0			0
$131$	19.5226			1.70570			0.852849	−	0.522158i	$-0.174873\pi$
$131$	19.5226			1.70570			0.852849	+	0.522158i	$0.174873\pi$
$132$		0			0
$133$		−	1.78803i		−	0.155042i
$134$		0			0
$135$	11.5945	+	0.753992i	0.997892	+	0.0648933i
$136$		0			0
$137$	8.91251			0.761447			0.380724	−	0.924689i	$-0.375675\pi$
$137$	8.91251			0.761447			0.380724	+	0.924689i	$0.375675\pi$
$138$		0			0
$139$		−	22.9159i		−	1.94370i	−0.235607	−	0.971849i	$-0.575708\pi$
$139$		−	22.9159i		−	1.94370i	0.235607	−	0.971849i	$-0.424292\pi$
$140$		0			0
$141$	4.73241	+	14.6825i	0.398541	+	1.23649i
$142$		0			0
$143$		−	29.6807i		−	2.48203i
$144$		0			0
$145$	3.57360	−	2.29284i	0.296772	−	0.190410i
$146$		0			0
$147$	−1.64854	+	0.531349i	−0.135969	+	0.0438249i
$148$		0			0
$149$		−	5.92305i		−	0.485236i	−0.970122	−	0.242618i	$-0.921994\pi$
$149$		−	5.92305i		−	0.485236i	0.970122	−	0.242618i	$-0.0780061\pi$
$150$		0			0
$151$			5.24047i			0.426463i	0.977002	+	0.213232i	$0.0683989\pi$
$151$			5.24047i			0.426463i	−0.977002	+	0.213232i	$0.931601\pi$
$152$		0			0
$153$	9.69562	−	6.97468i	0.783844	−	0.563870i
$154$		0			0
$155$	−8.91251	+	5.71831i	−0.715870	+	0.459306i
$156$		0			0
$157$			11.2593i			0.898592i	0.893383	+	0.449296i	$0.148325\pi$
$157$			11.2593i			0.898592i	−0.893383	+	0.449296i	$0.851675\pi$
$158$		0			0
$159$	1.56622	−	0.504816i	0.124209	−	0.0400345i
$160$		0			0
$161$		−	1.63861i		−	0.129141i
$162$		0			0
$163$	16.1677			1.26636			0.633178	−	0.774006i	$-0.281750\pi$
$163$	16.1677			1.26636			0.633178	+	0.774006i	$0.281750\pi$
$164$		0			0
$165$	−14.7476	−	12.1356i	−1.14810	−	0.944757i
$166$		0			0
$167$		−	3.18809i		−	0.246702i	−0.992363	−	0.123351i	$-0.960636\pi$
$167$		−	3.18809i		−	0.246702i	0.992363	−	0.123351i	$-0.0393641\pi$
$168$		0			0
$169$	−23.2267			−1.78667
$170$		0			0
$171$	3.13243	+	4.35445i	0.239543	+	0.332993i
$172$		0			0
$173$	−16.7737			−1.27528			−0.637640	−	0.770335i	$-0.720089\pi$
$173$	−16.7737			−1.27528			−0.637640	+	0.770335i	$0.720089\pi$
$174$		0			0
$175$	2.08387	−	4.54505i	0.157526	−	0.343573i
$176$		0			0
$177$	−21.0888	+	6.79726i	−1.58513	+	0.510913i
$178$		0			0
$179$	8.19508			0.612529			0.306265	−	0.951946i	$-0.400921\pi$
$179$	8.19508			0.612529			0.306265	+	0.951946i	$0.400921\pi$
$180$		0			0
$181$	23.5238			1.74851			0.874254	−	0.485469i	$-0.161351\pi$
$181$	23.5238			1.74851			0.874254	+	0.485469i	$0.161351\pi$
$182$		0			0
$183$	−14.1677	+	4.56649i	−1.04731	+	0.337564i
$184$		0			0
$185$	−12.2776	+	7.87736i	−0.902666	+	0.579155i
$186$		0			0
$187$	−19.6326			−1.43567
$188$		0			0
$189$	3.08387	−	4.18207i	0.224319	−	0.304201i
$190$		0			0
$191$	12.8937			0.932958			0.466479	−	0.884532i	$-0.345522\pi$
$191$	12.8937			0.932958			0.466479	+	0.884532i	$0.345522\pi$
$192$		0			0
$193$			5.51405i			0.396910i	0.980110	+	0.198455i	$0.0635923\pi$
$193$			5.51405i			0.396910i	−0.980110	+	0.198455i	$0.936408\pi$
$194$		0			0
$195$	−14.8121	+	18.0001i	−1.06072	+	1.28901i
$196$		0			0
$197$	−18.7751			−1.33767			−0.668835	−	0.743411i	$-0.733207\pi$
$197$	−18.7751			−1.33767			−0.668835	+	0.743411i	$0.733207\pi$
$198$		0			0
$199$		−	5.11686i		−	0.362724i	−0.983416	−	0.181362i	$-0.941949\pi$
$199$		−	5.11686i		−	0.362724i	0.983416	−	0.181362i	$-0.0580506\pi$
$200$		0			0
$201$	16.0590	−	5.17605i	1.13271	−	0.365091i
$202$		0			0
$203$		−	1.89883i		−	0.133272i
$204$		0			0
$205$	−17.8458	+	11.4499i	−1.24640	+	0.799698i
$206$		0			0
$207$	2.87067	+	3.99057i	0.199526	+	0.277364i
$208$		0			0
$209$		−	8.81728i		−	0.609904i
$210$		0			0
$211$		−	0.902006i		−	0.0620967i	−0.999518	−	0.0310483i	$-0.990115\pi$
$211$		−	0.902006i		−	0.0620967i	0.999518	−	0.0310483i	$-0.00988458\pi$
$212$		0			0
$213$	−5.54744	+	1.78803i	−0.380104	+	0.122514i
$214$		0			0
$215$	−1.90013	−	2.96153i	−0.129588	−	0.201974i
$216$		0			0
$217$			4.73565i			0.321477i
$218$		0			0
$219$	4.08250	+	12.6662i	0.275870	+	0.855899i
$220$		0			0
$221$			23.9624i			1.61189i
$222$		0			0
$223$	1.12933			0.0756253			0.0378126	−	0.999285i	$-0.487961\pi$
$223$	1.12933			0.0756253			0.0378126	+	0.999285i	$0.487961\pi$
$224$		0			0
$225$	2.88751	+	14.7195i	0.192501	+	0.981297i
$226$		0			0
$227$		−	28.3242i		−	1.87994i	−0.341255	−	0.939971i	$-0.610852\pi$
$227$		−	28.3242i		−	1.87994i	0.341255	−	0.939971i	$-0.389148\pi$
$228$		0			0
$229$	−21.4827			−1.41962			−0.709808	−	0.704395i	$-0.751218\pi$
$229$	−21.4827			−1.41962			−0.709808	+	0.704395i	$0.751218\pi$
$230$		0			0
$231$	−8.12940	+	2.62023i	−0.534875	+	0.172399i
$232$		0			0
$233$	20.4727			1.34121			0.670605	−	0.741815i	$-0.266035\pi$
$233$	20.4727			1.34121			0.670605	+	0.741815i	$0.266035\pi$
$234$		0			0
$235$	−16.7619	+	10.7545i	−1.09342	+	0.701547i
$236$		0			0
$237$	−8.64428	−	26.8193i	−0.561506	−	1.74210i
$238$		0			0
$239$	−8.06372			−0.521599			−0.260799	−	0.965393i	$-0.583986\pi$
$239$	−8.06372			−0.521599			−0.260799	+	0.965393i	$0.583986\pi$
$240$		0			0
$241$	15.1883			0.978363			0.489182	−	0.872182i	$-0.337296\pi$
$241$	15.1883			0.978363			0.489182	+	0.872182i	$0.337296\pi$
$242$		0			0
$243$	−0.183720	+	15.5874i	−0.0117856	+	0.999931i
$244$		0			0
$245$	−1.20750	−	1.88200i	−0.0771445	−	0.120237i
$246$		0			0
$247$	−10.7619			−0.684762
$248$		0			0
$249$	0.276533	+	0.857957i	0.0175246	+	0.0543708i
$250$		0			0
$251$	−22.8576			−1.44276			−0.721379	−	0.692540i	$-0.756491\pi$
$251$	−22.8576			−1.44276			−0.721379	+	0.692540i	$0.756491\pi$
$252$		0			0
$253$		−	8.08047i		−	0.508015i
$254$		0			0
$255$	11.9063	+	9.79756i	0.745601	+	0.613547i
$256$		0			0
$257$	17.3101			1.07978			0.539889	−	0.841736i	$-0.318466\pi$
$257$	17.3101			1.07978			0.539889	+	0.841736i	$0.318466\pi$
$258$		0			0
$259$			6.52368i			0.405362i
$260$		0			0
$261$	3.32655	+	4.62429i	0.205908	+	0.286236i
$262$		0			0
$263$			25.3745i			1.56466i	0.622866	+	0.782329i	$0.285968\pi$
$263$			25.3745i			1.56466i	−0.622866	+	0.782329i	$0.714032\pi$
$264$		0			0
$265$	1.14721	+	1.78803i	0.0704724	+	0.109838i
$266$		0			0
$267$	8.59414	+	26.6637i	0.525953	+	1.63179i
$268$		0			0
$269$			20.2203i			1.23285i	0.787413	+	0.616425i	$0.211420\pi$
$269$			20.2203i			1.23285i	−0.787413	+	0.616425i	$0.788580\pi$
$270$		0			0
$271$		−	5.74528i		−	0.349001i	−0.984657	−	0.174501i	$-0.944169\pi$
$271$		−	5.74528i		−	0.349001i	0.984657	−	0.174501i	$-0.0558311\pi$
$272$		0			0
$273$	3.19811	+	9.92231i	0.193559	+	0.600525i
$274$		0			0
$275$	10.2762	−	22.4130i	0.619677	−	1.35155i
$276$		0			0
$277$		−	13.9757i		−	0.839719i	−0.907589	−	0.419860i	$-0.862079\pi$
$277$		−	13.9757i		−	0.839719i	0.907589	−	0.419860i	$-0.137921\pi$
$278$		0			0
$279$	−8.29636	−	11.5329i	−0.496690	−	0.690457i
$280$		0			0
$281$			17.4080i			1.03847i	0.854630	+	0.519237i	$0.173784\pi$
$281$			17.4080i			1.03847i	−0.854630	+	0.519237i	$0.826216\pi$
$282$		0			0
$283$	10.5352			0.626252			0.313126	−	0.949712i	$-0.398624\pi$
$283$	10.5352			0.626252			0.313126	+	0.949712i	$0.398624\pi$
$284$		0			0
$285$	−4.40023	+	5.34729i	−0.260647	+	0.316746i
$286$		0			0
$287$			9.48232i			0.559724i
$288$		0			0
$289$	−1.14986			−0.0676390
$290$		0			0
$291$	−4.56178	−	14.1531i	−0.267416	−	0.829672i
$292$		0			0
$293$	2.74892			0.160594			0.0802968	−	0.996771i	$-0.474413\pi$
$293$	2.74892			0.160594			0.0802968	+	0.996771i	$0.474413\pi$
$294$		0			0
$295$	−15.4469	−	24.0754i	−0.899355	−	1.40173i
$296$		0			0
$297$	15.2075	−	20.6230i	0.882427	−	1.19667i
$298$		0			0
$299$	−9.86258			−0.570368
$300$		0			0
$301$	−1.57360			−0.0907010
$302$		0			0
$303$	−1.83226	−	5.68466i	−0.105260	−	0.326575i
$304$		0			0
$305$	−10.3775	−	16.1742i	−0.594211	−	0.926132i
$306$		0			0
$307$	−15.7235			−0.897386			−0.448693	−	0.893686i	$-0.648110\pi$
$307$	−15.7235			−0.897386			−0.448693	+	0.893686i	$0.648110\pi$
$308$		0			0
$309$	−12.0295	+	3.87729i	−0.684333	+	0.220571i
$310$		0			0
$311$	18.8548			1.06916			0.534578	−	0.845119i	$-0.320471\pi$
$311$	18.8548			1.06916			0.534578	+	0.845119i	$0.320471\pi$
$312$		0			0
$313$			15.4902i			0.875556i	0.899083	+	0.437778i	$0.144234\pi$
$313$			15.4902i			0.875556i	−0.899083	+	0.437778i	$0.855766\pi$
$314$		0			0
$315$	6.23775	+	2.46789i	0.351457	+	0.139050i
$316$		0			0
$317$	−21.9075			−1.23045			−0.615225	−	0.788352i	$-0.710935\pi$
$317$	−21.9075			−1.23045			−0.615225	+	0.788352i	$0.710935\pi$
$318$		0			0
$319$		−	9.36368i		−	0.524265i
$320$		0			0
$321$	6.85279	+	21.2611i	0.382486	+	1.18668i
$322$		0			0
$323$			7.11853i			0.396086i
$324$		0			0
$325$	−27.3560	−	12.5425i	−1.51744	−	0.695735i
$326$		0			0
$327$	−4.02948	+	1.29876i	−0.222831	+	0.0718218i
$328$		0			0
$329$			8.90641i			0.491026i
$330$		0			0
$331$		−	3.32883i		−	0.182969i	−0.995807	−	0.0914845i	$-0.970839\pi$
$331$		−	3.32883i		−	0.182969i	0.995807	−	0.0914845i	$-0.0291612\pi$
$332$		0			0
$333$	−11.4288	−	15.8874i	−0.626294	−	0.870622i
$334$		0			0
$335$	11.7627	+	18.3332i	0.642665	+	1.00165i
$336$		0			0
$337$		−	12.0377i		−	0.655737i	−0.944723	−	0.327868i	$-0.893670\pi$
$337$		−	12.0377i		−	0.655737i	0.944723	−	0.327868i	$-0.106330\pi$
$338$		0			0
$339$	4.69865	−	1.51445i	0.255196	−	0.0822536i
$340$		0			0
$341$			23.3529i			1.26463i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$	−4.03253	+	4.90045i	−0.217104	+	0.263831i
$346$		0			0
$347$			0.597739i			0.0320883i	0.999871	+	0.0160442i	$0.00510723\pi$
$347$			0.597739i			0.0320883i	−0.999871	+	0.0160442i	$0.994893\pi$
$348$		0			0
$349$	−3.40586			−0.182311			−0.0911557	−	0.995837i	$-0.529056\pi$
$349$	−3.40586			−0.182311			−0.0911557	+	0.995837i	$0.529056\pi$
$350$		0			0
$351$	−25.1713	−	18.5614i	−1.34355	−	0.990734i
$352$		0			0
$353$	−15.5414			−0.827184			−0.413592	−	0.910462i	$-0.635726\pi$
$353$	−15.5414			−0.827184			−0.413592	+	0.910462i	$0.635726\pi$
$354$		0			0
$355$	−4.06333	−	6.33308i	−0.215659	−	0.336125i
$356$		0			0
$357$	6.56319	−	2.11542i	0.347361	−	0.111960i
$358$		0			0
$359$	1.46494			0.0773166			0.0386583	−	0.999252i	$-0.487692\pi$
$359$	1.46494			0.0773166			0.0386583	+	0.999252i	$0.487692\pi$
$360$		0			0
$361$	15.8030			0.831735
$362$		0			0
$363$	−21.9545	+	7.07629i	−1.15231	+	0.371409i
$364$		0			0
$365$	−14.4600	+	9.27758i	−0.756868	+	0.485611i
$366$		0			0
$367$	33.8003			1.76436			0.882181	−	0.470911i	$-0.156075\pi$
$367$	33.8003			1.76436			0.882181	+	0.470911i	$0.156075\pi$
$368$		0			0
$369$	−16.6120	−	23.0926i	−0.864787	−	1.20216i
$370$		0			0
$371$	0.950066			0.0493250
$372$		0			0
$373$		−	18.1802i		−	0.941335i	−0.882311	−	0.470667i	$-0.844013\pi$
$373$		−	18.1802i		−	0.941335i	0.882311	−	0.470667i	$-0.155987\pi$
$374$		0			0
$375$	−17.4172	+	8.46419i	−0.899418	+	0.437089i
$376$		0			0
$377$	−11.4288			−0.588613
$378$		0			0
$379$			6.90488i			0.354680i	0.984150	+	0.177340i	$0.0567492\pi$
$379$			6.90488i			0.354680i	−0.984150	+	0.177340i	$0.943251\pi$
$380$		0			0
$381$	−1.68239	+	0.542261i	−0.0861916	+	0.0277809i
$382$		0			0
$383$		−	13.1790i		−	0.673417i	−0.941609	−	0.336708i	$-0.890686\pi$
$383$		−	13.1790i		−	0.673417i	0.941609	−	0.336708i	$-0.109314\pi$
$384$		0			0
$385$	−5.95455	−	9.28070i	−0.303472	−	0.472988i
$386$		0			0
$387$	3.83226	−	2.75679i	0.194804	−	0.140135i
$388$		0			0
$389$			28.8446i			1.46248i	0.682120	+	0.731240i	$0.261058\pi$
$389$			28.8446i			1.46248i	−0.682120	+	0.731240i	$0.738942\pi$
$390$		0			0
$391$			6.52368i			0.329917i
$392$		0			0
$393$	−32.1837	+	10.3733i	−1.62345	+	0.523264i
$394$		0			0
$395$	30.6175	−	19.6443i	1.54053	−	0.988414i
$396$		0			0
$397$		−	26.7655i		−	1.34332i	−0.740859	−	0.671661i	$-0.765581\pi$
$397$		−	26.7655i		−	1.34332i	0.740859	−	0.671661i	$-0.234419\pi$
$398$		0			0
$399$	0.950066	+	2.94762i	0.0475628	+	0.147566i
$400$		0			0
$401$		−	17.5862i		−	0.878215i	−0.898434	−	0.439108i	$-0.855295\pi$
$401$		−	17.5862i		−	0.878215i	0.898434	−	0.439108i	$-0.144705\pi$
$402$		0			0
$403$	28.5032			1.41985
$404$		0			0
$405$	−19.5145	+	4.91772i	−0.969684	+	0.244363i
$406$		0			0
$407$			32.1701i			1.59461i
$408$		0			0
$409$	−10.2997			−0.509289			−0.254644	−	0.967035i	$-0.581958\pi$
$409$	−10.2997			−0.509289			−0.254644	+	0.967035i	$0.581958\pi$
$410$		0			0
$411$	−14.6926	+	4.73565i	−0.724732	+	0.233592i
$412$		0			0
$413$	−12.7925			−0.629476
$414$		0			0
$415$	−0.979463	+	0.628428i	−0.0480799	+	0.0308483i
$416$		0			0
$417$	12.1763	+	37.7776i	0.596276	+	1.84998i
$418$		0			0
$419$	26.2527			1.28253			0.641265	−	0.767319i	$-0.278410\pi$
$419$	26.2527			1.28253			0.641265	+	0.767319i	$0.278410\pi$
$420$		0			0
$421$	10.4854			0.511025			0.255512	−	0.966806i	$-0.417756\pi$
$421$	10.4854			0.511025			0.255512	+	0.966806i	$0.417756\pi$
$422$		0			0
$423$	−15.6031	−	21.6901i	−0.758648	−	1.05461i
$424$		0			0
$425$	−8.29636	+	18.0949i	−0.402433	+	0.877729i
$426$		0			0
$427$	−8.59414			−0.415900
$428$		0			0
$429$	15.7708	+	48.9298i	0.761423	+	2.36235i
$430$		0			0
$431$	−28.8186			−1.38814			−0.694072	−	0.719905i	$-0.744185\pi$
$431$	−28.8186			−1.38814			−0.694072	+	0.719905i	$0.744185\pi$
$432$		0			0
$433$			5.36408i			0.257781i	0.991659	+	0.128891i	$0.0411416\pi$
$433$			5.36408i			0.257781i	−0.991659	+	0.128891i	$0.958858\pi$
$434$		0			0
$435$	−4.67291	+	5.67866i	−0.224049	+	0.272271i
$436$		0			0
$437$	−2.92988			−0.140155
$438$		0			0
$439$			28.4299i			1.35688i	0.734654	+	0.678442i	$0.237345\pi$
$439$			28.4299i			1.35688i	−0.734654	+	0.678442i	$0.762655\pi$
$440$		0			0
$441$	2.43534	−	1.75189i	0.115968	−	0.0834235i
$442$		0			0
$443$		−	27.6108i		−	1.31183i	−0.754835	−	0.655915i	$-0.772283\pi$
$443$		−	27.6108i		−	1.31183i	0.754835	−	0.655915i	$-0.227717\pi$
$444$		0			0
$445$	−30.4399	+	19.5304i	−1.44299	+	0.925830i
$446$		0			0
$447$	3.14721	+	9.76436i	0.148858	+	0.461839i
$448$		0			0
$449$		−	6.08231i		−	0.287042i	−0.989647	−	0.143521i	$-0.954158\pi$
$449$		−	6.08231i		−	0.287042i	0.989647	−	0.143521i	$-0.0458425\pi$
$450$		0			0
$451$			46.7601i			2.20185i
$452$		0			0
$453$	−2.78452	−	8.63910i	−0.130828	−	0.405900i
$454$		0			0
$455$	−11.3275	+	7.26779i	−0.531042	+	0.340719i
$456$		0			0
$457$		−	15.6138i		−	0.730382i	−0.930933	−	0.365191i	$-0.881004\pi$
$457$		−	15.6138i		−	0.730382i	0.930933	−	0.365191i	$-0.118996\pi$
$458$		0			0
$459$	−12.2776	+	16.6498i	−0.573069	+	0.777144i
$460$		0			0
$461$			19.3841i			0.902808i	0.892320	+	0.451404i	$0.149077\pi$
$461$			19.3841i			0.902808i	−0.892320	+	0.451404i	$0.850923\pi$
$462$		0			0
$463$	−22.2446			−1.03379			−0.516897	−	0.856048i	$-0.672913\pi$
$463$	−22.2446			−1.03379			−0.516897	+	0.856048i	$0.672913\pi$
$464$		0			0
$465$	11.6542	−	14.1625i	0.540449	−	0.656770i
$466$		0			0
$467$		−	19.6007i		−	0.907012i	−0.891253	−	0.453506i	$-0.850173\pi$
$467$		−	19.6007i		−	0.907012i	0.891253	−	0.453506i	$-0.149827\pi$
$468$		0			0
$469$	9.74135			0.449814
$470$		0			0
$471$	−5.98263	−	18.5614i	−0.275665	−	0.855264i
$472$		0			0
$473$	−7.75989			−0.356800
$474$		0			0
$475$	−8.12667	−	3.72602i	−0.372877	−	0.170961i
$476$		0			0
$477$	−2.31373	+	1.66441i	−0.105938	+	0.0762083i
$478$		0			0
$479$	16.1274			0.736882			0.368441	−	0.929651i	$-0.379892\pi$
$479$	16.1274			0.736882			0.368441	+	0.929651i	$0.379892\pi$
$480$		0			0
$481$	39.2651			1.79034
$482$		0			0
$483$	0.870674	+	2.70131i	0.0396170	+	0.122914i
$484$		0			0
$485$	16.1575	−	10.3668i	0.733676	−	0.470730i
$486$		0			0
$487$	8.50323			0.385318			0.192659	−	0.981266i	$-0.438289\pi$
$487$	8.50323			0.385318			0.192659	+	0.981266i	$0.438289\pi$
$488$		0			0
$489$	−26.6531	+	8.59071i	−1.20529	+	0.388485i
$490$		0			0
$491$	3.69899			0.166933			0.0834665	−	0.996511i	$-0.473401\pi$
$491$	3.69899			0.166933			0.0834665	+	0.996511i	$0.473401\pi$
$492$		0			0
$493$			7.55966i			0.340470i
$494$		0			0
$495$	30.7601	+	12.1699i	1.38257	+	0.546996i
$496$		0			0
$497$	−3.36507			−0.150944
$498$		0			0
$499$		−	39.7969i		−	1.78155i	−0.454443	−	0.890776i	$-0.650162\pi$
$499$		−	39.7969i		−	1.78155i	0.454443	−	0.890776i	$-0.349838\pi$
$500$		0			0
$501$	1.69399	+	5.25568i	0.0756818	+	0.234807i
$502$		0			0
$503$		−	0.0891309i		−	0.00397415i	−0.999998	−	0.00198708i	$-0.999367\pi$
$503$		−	0.0891309i		−	0.00397415i	0.999998	−	0.00198708i	$-0.000632506\pi$
$504$		0			0
$505$	6.48973	−	4.16385i	0.288789	−	0.185289i
$506$		0			0
$507$	38.2900	−	12.3415i	1.70052	−	0.548104i
$508$		0			0
$509$			16.4462i			0.728967i	0.931210	+	0.364484i	$0.118754\pi$
$509$			16.4462i			0.728967i	−0.931210	+	0.364484i	$0.881246\pi$
$510$		0			0
$511$			7.68328i			0.339888i
$512$		0			0
$513$	−7.47766	−	5.51405i	−0.330147	−	0.243451i
$514$		0			0
$515$	−8.81124	−	13.7331i	−0.388269	−	0.605153i
$516$		0			0
$517$			43.9201i			1.93160i
$518$		0			0
$519$	27.6520	−	8.91267i	1.21379	−	0.391223i
$520$		0			0
$521$			26.0323i			1.14050i	0.821472	+	0.570249i	$0.193153\pi$
$521$			26.0323i			1.14050i	−0.821472	+	0.570249i	$0.806847\pi$
$522$		0			0
$523$	38.5032			1.68363			0.841814	−	0.539767i	$-0.181488\pi$
$523$	38.5032			1.68363			0.841814	+	0.539767i	$0.181488\pi$
$524$		0			0
$525$	−1.02033	+	8.59994i	−0.0445309	+	0.375332i
$526$		0			0
$527$		−	18.8537i		−	0.821279i
$528$		0			0
$529$	20.3150			0.883259
$530$		0			0
$531$	31.1539	−	22.4110i	1.35197	−	0.972556i
$532$		0			0
$533$	57.0728			2.47210
$534$		0			0
$535$	−24.2722	+	15.5731i	−1.04938	+	0.673286i
$536$		0			0
$537$	−13.5099	+	4.35445i	−0.582995	+	0.187908i
$538$		0			0
$539$	−4.93129			−0.212406
$540$		0			0
$541$	−11.0384			−0.474579			−0.237289	−	0.971439i	$-0.576259\pi$
$541$	−11.0384			−0.474579			−0.237289	+	0.971439i	$0.576259\pi$
$542$		0			0
$543$	−38.7798	+	12.4993i	−1.66420	+	0.536397i
$544$		0			0
$545$	−2.95147	−	4.60014i	−0.126427	−	0.197048i
$546$		0			0
$547$	39.0475			1.66955			0.834776	−	0.550589i	$-0.185597\pi$
$547$	39.0475			1.66955			0.834776	+	0.550589i	$0.185597\pi$
$548$		0			0
$549$	20.9296	−	15.0560i	0.893255	−	0.642576i
$550$		0			0
$551$	−3.39516			−0.144639
$552$		0			0
$553$		−	16.2686i		−	0.691810i
$554$		0			0
$555$	16.0544	−	19.5098i	0.681471	−	0.828144i
$556$		0			0
$557$	−31.5675			−1.33756			−0.668780	−	0.743460i	$-0.733183\pi$
$557$	−31.5675			−1.33756			−0.668780	+	0.743460i	$0.733183\pi$
$558$		0			0
$559$			9.47130i			0.400593i
$560$		0			0
$561$	32.3650	−	10.4317i	1.36645	−	0.440428i
$562$		0			0
$563$		−	43.2911i		−	1.82450i	−0.409633	−	0.912251i	$-0.634343\pi$
$563$		−	43.2911i		−	1.82450i	0.409633	−	0.912251i	$-0.365657\pi$
$564$		0			0
$565$	3.44162	+	5.36408i	0.144790	+	0.225669i
$566$		0			0
$567$	−2.86173	+	8.53291i	−0.120182	+	0.358348i
$568$		0			0
$569$		−	17.0705i		−	0.715631i	−0.933792	−	0.357816i	$-0.883522\pi$
$569$		−	17.0705i		−	0.715631i	0.933792	−	0.357816i	$-0.116478\pi$
$570$		0			0
$571$		−	9.71853i		−	0.406708i	−0.979105	−	0.203354i	$-0.934816\pi$
$571$		−	9.71853i		−	0.406708i	0.979105	−	0.203354i	$-0.0651842\pi$
$572$		0			0
$573$	−21.2558	+	6.85107i	−0.887973	+	0.286208i
$574$		0			0
$575$	−7.44757	−	3.41466i	−0.310585	−	0.142401i
$576$		0			0
$577$			36.2368i			1.50856i	0.656555	+	0.754279i	$0.272013\pi$
$577$			36.2368i			1.50856i	−0.656555	+	0.754279i	$0.727987\pi$
$578$		0			0
$579$	−2.92988	−	9.09010i	−0.121762	−	0.377772i
$580$		0			0
$581$			0.520436i			0.0215913i
$582$		0			0
$583$	4.68505			0.194035
$584$		0			0
$585$	14.8539	−	37.5441i	0.614134	−	1.55226i
$586$		0			0
$587$			25.6565i			1.05896i	0.848323	+	0.529479i	$0.177613\pi$
$587$			25.6565i			1.05896i	−0.848323	+	0.529479i	$0.822387\pi$
$588$		0			0
$589$	8.46747			0.348896
$590$		0			0
$591$	30.9514	−	9.97612i	1.27317	−	0.410363i
$592$		0			0
$593$	13.8438			0.568497			0.284248	−	0.958751i	$-0.408256\pi$
$593$	13.8438			0.568497			0.284248	+	0.958751i	$0.408256\pi$
$594$		0			0
$595$	4.80734	+	7.49267i	0.197082	+	0.307170i
$596$		0			0
$597$	2.71883	+	8.43532i	0.111274	+	0.345235i
$598$		0			0
$599$	−30.5162			−1.24686			−0.623429	−	0.781880i	$-0.714261\pi$
$599$	−30.5162			−1.24686			−0.623429	+	0.781880i	$0.714261\pi$
$600$		0			0
$601$	−2.55307			−0.104142			−0.0520709	−	0.998643i	$-0.516582\pi$
$601$	−2.55307			−0.104142			−0.0520709	+	0.998643i	$0.516582\pi$
$602$		0			0
$603$	−23.7235	+	17.0658i	−0.966094	+	0.694974i
$604$		0			0
$605$	−16.0811	−	25.0638i	−0.653788	−	1.01899i
$606$		0			0
$607$	16.3123			0.662096			0.331048	−	0.943614i	$-0.392598\pi$
$607$	16.3123			0.662096			0.331048	+	0.943614i	$0.392598\pi$
$608$		0			0
$609$	1.00894	+	3.13029i	0.0408843	+	0.126846i
$610$		0			0
$611$	53.6064			2.16868
$612$		0			0
$613$			21.7563i			0.878727i	0.898309	+	0.439363i	$0.144796\pi$
$613$			21.7563i			0.878727i	−0.898309	+	0.439363i	$0.855204\pi$
$614$		0			0
$615$	23.3355	−	28.3579i	0.940976	−	1.14350i
$616$		0			0
$617$	25.5052			1.02680			0.513401	−	0.858149i	$-0.328385\pi$
$617$	25.5052			1.02680			0.513401	+	0.858149i	$0.328385\pi$
$618$		0			0
$619$			28.0487i			1.12737i	0.825989	+	0.563686i	$0.190617\pi$
$619$			28.0487i			1.12737i	−0.825989	+	0.563686i	$0.809383\pi$
$620$		0			0
$621$	−6.85279	−	5.05327i	−0.274993	−	0.202781i
$622$		0			0
$623$			16.1742i			0.648006i
$624$		0			0
$625$	−16.3150	−	18.9426i	−0.652598	−	0.757704i
$626$		0			0
$627$	4.68505	+	14.5356i	0.187103	+	0.580496i
$628$		0			0
$629$		−	25.9722i		−	1.03558i
$630$		0			0
$631$			12.3926i			0.493341i	0.969099	+	0.246670i	$0.0793365\pi$
$631$			12.3926i			0.493341i	−0.969099	+	0.246670i	$0.920664\pi$
$632$		0			0
$633$	0.479280	+	1.48699i	0.0190497	+	0.0591025i
$634$		0			0
$635$	−1.23230	−	1.92065i	−0.0489024	−	0.0762189i
$636$		0			0
$637$			6.01886i			0.238476i
$638$		0			0
$639$	8.19508	−	5.89525i	0.324193	−	0.233212i
$640$		0			0
$641$		−	34.6521i		−	1.36867i	−0.729166	−	0.684337i	$-0.760092\pi$
$641$		−	34.6521i		−	1.36867i	0.729166	−	0.684337i	$-0.239908\pi$
$642$		0			0
$643$	18.8707			0.744187			0.372093	−	0.928195i	$-0.378640\pi$
$643$	18.8707			0.744187			0.372093	+	0.928195i	$0.378640\pi$
$644$		0			0
$645$	4.70604	+	3.87255i	0.185300	+	0.152481i
$646$		0			0
$647$			7.73275i			0.304006i	0.988380	+	0.152003i	$0.0485723\pi$
$647$			7.73275i			0.304006i	−0.988380	+	0.152003i	$0.951428\pi$
$648$		0			0
$649$	−63.0833			−2.47624
$650$		0			0
$651$	−2.51628	−	7.80689i	−0.0986209	−	0.305976i
$652$		0			0
$653$	−18.9776			−0.742652			−0.371326	−	0.928502i	$-0.621097\pi$
$653$	−18.9776			−0.742652			−0.371326	+	0.928502i	$0.621097\pi$
$654$		0			0
$655$	−23.5736	−	36.7416i	−0.921097	−	1.43561i
$656$		0			0
$657$	−13.4603	−	18.7114i	−0.525136	−	0.730000i
$658$		0			0
$659$	−38.4787			−1.49892			−0.749458	−	0.662052i	$-0.769686\pi$
$659$	−38.4787			−1.49892			−0.749458	+	0.662052i	$0.769686\pi$
$660$		0			0
$661$	29.4469			1.14535			0.572676	−	0.819781i	$-0.305905\pi$
$661$	29.4469			1.14535			0.572676	+	0.819781i	$0.305905\pi$
$662$		0			0
$663$	−12.7324	−	39.5029i	−0.494486	−	1.53417i
$664$		0			0
$665$	−3.36507	+	2.15905i	−0.130492	+	0.0837243i
$666$		0			0
$667$	−3.11144			−0.120476
$668$		0			0
$669$	−1.86173	+	0.600066i	−0.0719788	+	0.0231999i
$670$		0			0
$671$	−42.3802			−1.63607
$672$		0			0
$673$		−	17.0046i		−	0.655480i	−0.944768	−	0.327740i	$-0.893713\pi$
$673$		−	17.0046i		−	0.655480i	0.944768	−	0.327740i	$-0.106287\pi$
$674$		0			0
$675$	−12.5813	−	22.7313i	−0.484256	−	0.874927i
$676$		0			0
$677$	−19.3416			−0.743360			−0.371680	−	0.928361i	$-0.621218\pi$
$677$	−19.3416			−0.743360			−0.371680	+	0.928361i	$0.621218\pi$
$678$		0			0
$679$		−	8.58528i		−	0.329473i
$680$		0			0
$681$	15.0500	+	46.6934i	0.576718	+	1.78930i
$682$		0			0
$683$			19.4514i			0.744288i	0.928175	+	0.372144i	$0.121377\pi$
$683$			19.4514i			0.744288i	−0.928175	+	0.372144i	$0.878623\pi$
$684$		0			0
$685$	−10.7619	−	16.7734i	−0.411191	−	0.640878i
$686$		0			0
$687$	35.4150	−	11.4148i	1.35117	−	0.435502i
$688$		0			0
$689$		−	5.71831i		−	0.217850i
$690$		0			0
$691$			9.32134i			0.354600i	0.984157	+	0.177300i	$0.0567363\pi$
$691$			9.32134i			0.354600i	−0.984157	+	0.177300i	$0.943264\pi$
$692$		0			0
$693$	12.0094	−	8.63910i	0.456197	−	0.328172i
$694$		0			0
$695$	−43.1277	+	27.6710i	−1.63593	+	1.04962i
$696$		0			0
$697$		−	37.7512i		−	1.42993i
$698$		0			0
$699$	−33.7499	+	10.8781i	−1.27654	+	0.411449i
$700$		0			0
$701$		−	40.6907i		−	1.53687i	−0.639929	−	0.768434i	$-0.721036\pi$
$701$		−	40.6907i		−	1.53687i	0.639929	−	0.768434i	$-0.278964\pi$
$702$		0			0
$703$	11.6645			0.439935
$704$		0			0
$705$	21.9182	−	26.6356i	0.825486	−	1.00315i
$706$		0			0
$707$		−	3.44831i		−	0.129687i
$708$		0			0
$709$	26.6120			0.999435			0.499718	−	0.866188i	$-0.333437\pi$
$709$	26.6120			0.999435			0.499718	+	0.866188i	$0.333437\pi$
$710$		0			0
$711$	28.5008	+	39.6194i	1.06886	+	1.48584i
$712$		0			0
$713$	7.75989			0.290610
$714$		0			0
$715$	−55.8593	+	35.8396i	−2.08902	+	1.34032i
$716$		0			0
$717$	13.2933	−	4.28465i	0.496449	−	0.160013i
$718$		0			0
$719$	48.4425			1.80660			0.903300	−	0.429009i	$-0.141137\pi$
$719$	48.4425			1.80660			0.903300	+	0.429009i	$0.141137\pi$
$720$		0			0
$721$	−7.29707			−0.271757
$722$		0			0
$723$	−25.0384	+	8.07027i	−0.931189	+	0.300137i
$724$		0			0
$725$	−8.63028	−	3.95692i	−0.320520	−	0.146956i
$726$		0			0
$727$	−40.8798			−1.51615			−0.758074	−	0.652169i	$-0.773859\pi$
$727$	−40.8798			−1.51615			−0.758074	+	0.652169i	$0.773859\pi$
$728$		0			0
$729$	−7.97946	−	25.7940i	−0.295536	−	0.955332i
$730$		0			0
$731$	6.26487			0.231714
$732$		0			0
$733$		−	24.9615i		−	0.921973i	−0.887407	−	0.460986i	$-0.847496\pi$
$733$		−	24.9615i		−	0.921973i	0.887407	−	0.460986i	$-0.152504\pi$
$734$		0			0
$735$	2.99061	+	2.46094i	0.110310	+	0.0907733i
$736$		0			0
$737$	48.0374			1.76948
$738$		0			0
$739$		−	1.66441i		−	0.0612265i	−0.999531	−	0.0306132i	$-0.990254\pi$
$739$		−	1.66441i		−	0.0612265i	0.999531	−	0.0306132i	$-0.00974602\pi$
$740$		0			0
$741$	17.7413	−	5.71831i	0.651745	−	0.210068i
$742$		0			0
$743$			7.56167i			0.277411i	0.990334	+	0.138705i	$0.0442941\pi$
$743$			7.56167i			0.277411i	−0.990334	+	0.138705i	$0.955706\pi$
$744$		0			0
$745$	−11.1472	+	7.15211i	−0.408402	+	0.262033i
$746$		0			0
$747$	−0.911749	−	1.26744i	−0.0333592	−	0.0463731i
$748$		0			0
$749$			12.8970i			0.471245i
$750$		0			0
$751$			4.23084i			0.154385i	0.997016	+	0.0771927i	$0.0245957\pi$
$751$			4.23084i			0.154385i	−0.997016	+	0.0771927i	$0.975404\pi$
$752$		0			0
$753$	37.6815	−	12.1454i	1.37319	−	0.442601i
$754$		0			0
$755$	9.86258	−	6.32788i	0.358936	−	0.230295i
$756$		0			0
$757$			31.5275i			1.14589i	0.819595	+	0.572943i	$0.194198\pi$
$757$			31.5275i			1.14589i	−0.819595	+	0.572943i	$0.805802\pi$
$758$		0			0
$759$	4.29355	+	13.3209i	0.155846	+	0.483519i
$760$		0			0
$761$			37.6064i			1.36323i	0.731710	+	0.681616i	$0.238722\pi$
$761$			37.6064i			1.36323i	−0.731710	+	0.681616i	$0.761278\pi$
$762$		0			0
$763$	−2.44428			−0.0884888
$764$		0			0
$765$	−24.8339	−	9.82524i	−0.897870	−	0.355232i
$766$		0			0
$767$			76.9960i			2.78017i
$768$		0			0
$769$	−22.9707			−0.828345			−0.414172	−	0.910198i	$-0.635929\pi$
$769$	−22.9707			−0.828345			−0.414172	+	0.910198i	$0.635929\pi$
$770$		0			0
$771$	−28.5364	+	9.19773i	−1.02771	+	0.331248i
$772$		0			0
$773$	34.8013			1.25171			0.625857	−	0.779938i	$-0.284749\pi$
$773$	34.8013			1.25171			0.625857	+	0.779938i	$0.284749\pi$
$774$		0			0
$775$	21.5238	+	9.86849i	0.773157	+	0.354487i
$776$		0			0
$777$	−3.46635	−	10.7545i	−0.124355	−	0.385816i
$778$		0			0
$779$	16.9546			0.607463
$780$		0			0
$781$	−16.5941			−0.593785
$782$		0			0
$783$	−7.94104	−	5.85575i	−0.283790	−	0.209267i
$784$		0			0
$785$	21.1901	−	13.5957i	0.756307	−	0.485250i
$786$		0			0
$787$	1.85545			0.0661396			0.0330698	−	0.999453i	$-0.489472\pi$
$787$	1.85545			0.0661396			0.0330698	+	0.999453i	$0.489472\pi$
$788$		0			0
$789$	−13.4827	−	41.8307i	−0.479997	−	1.48921i
$790$		0			0
$791$	2.85020			0.101341
$792$		0			0
$793$			51.7270i			1.83688i
$794$		0			0
$795$	−2.84128	−	2.33806i	−0.100770	−	0.0829224i
$796$		0			0
$797$	54.1213			1.91708			0.958538	−	0.284966i	$-0.0919824\pi$
$797$	54.1213			1.91708			0.958538	+	0.284966i	$0.0919824\pi$
$798$		0			0
$799$		−	35.4584i		−	1.25443i
$800$		0			0
$801$	−28.3355	−	39.3896i	−1.00119	−	1.39176i
$802$		0			0
$803$			37.8885i			1.33705i
$804$		0			0
$805$	−3.08387	+	1.97863i	−0.108692	+	0.0697375i
$806$		0			0
$807$	−10.7440	−	33.3338i	−0.378207	−	1.17341i
$808$		0			0
$809$		−	1.78787i		−	0.0628583i	−0.999506	−	0.0314291i	$-0.989994\pi$
$809$		−	1.78787i		−	0.0628583i	0.999506	−	0.0314291i	$-0.0100059\pi$
$810$		0			0
$811$		−	30.3679i		−	1.06636i	−0.846002	−	0.533180i	$-0.820997\pi$
$811$		−	30.3679i		−	1.06636i	0.846002	−	0.533180i	$-0.179003\pi$
$812$		0			0
$813$	3.05275	+	9.47130i	0.107065	+	0.332173i
$814$		0			0
$815$	−19.5226	−	30.4277i	−0.683847	−	1.06584i
$816$		0			0
$817$			2.81365i			0.0984370i
$818$		0			0
$819$	−10.5444	−	14.6580i	−0.368451	−	0.512191i
$820$		0			0
$821$		−	43.8333i		−	1.52979i	−0.644153	−	0.764897i	$-0.722790\pi$
$821$		−	43.8333i		−	1.52979i	0.644153	−	0.764897i	$-0.277210\pi$
$822$		0			0
$823$	−14.2088			−0.495288			−0.247644	−	0.968851i	$-0.579656\pi$
$823$	−14.2088			−0.495288			−0.247644	+	0.968851i	$0.579656\pi$
$824$		0			0
$825$	−5.03155	+	42.4088i	−0.175176	+	1.47648i
$826$		0			0
$827$			43.4622i			1.51133i	0.654960	+	0.755664i	$0.272686\pi$
$827$			43.4622i			1.51133i	−0.654960	+	0.755664i	$0.727314\pi$
$828$		0			0
$829$	44.7121			1.55291			0.776457	−	0.630170i	$-0.217015\pi$
$829$	44.7121			1.55291			0.776457	+	0.630170i	$0.217015\pi$
$830$		0			0
$831$	7.42598	+	23.0395i	0.257604	+	0.799230i
$832$		0			0
$833$	3.98122			0.137941
$834$		0			0
$835$	−6.00000	+	3.84963i	−0.207639	+	0.133222i
$836$		0			0
$837$	19.8048	+	14.6041i	0.684555	+	0.504793i
$838$		0			0
$839$	−38.8427			−1.34100			−0.670499	−	0.741911i	$-0.733920\pi$
$839$	−38.8427			−1.34100			−0.670499	+	0.741911i	$0.733920\pi$
$840$		0			0
$841$	25.3944			0.875671
$842$		0			0
$843$	−9.24971	−	28.6977i	−0.318577	−	0.988401i
$844$		0			0
$845$	28.0463	+	43.7127i	0.964823	+	1.50376i
$846$		0			0
$847$	−13.3176			−0.457598
$848$		0			0
$849$	−17.3676	+	5.59786i	−0.596055	+	0.192118i
$850$		0			0
$851$	10.6898			0.366441
$852$		0			0
$853$		−	30.2019i		−	1.03409i	−0.855957	−	0.517047i	$-0.827031\pi$
$853$		−	30.2019i		−	1.03409i	0.855957	−	0.517047i	$-0.172969\pi$
$854$		0			0
$855$	4.41266	−	11.1533i	0.150910	−	0.381433i
$856$		0			0
$857$	−39.5601			−1.35135			−0.675673	−	0.737201i	$-0.736147\pi$
$857$	−39.5601			−1.35135			−0.675673	+	0.737201i	$0.736147\pi$
$858$		0			0
$859$		−	40.4676i		−	1.38074i	−0.723458	−	0.690369i	$-0.757448\pi$
$859$		−	40.4676i		−	1.38074i	0.723458	−	0.690369i	$-0.242552\pi$
$860$		0			0
$861$	−5.03842	−	15.6319i	−0.171709	−	0.532735i
$862$		0			0
$863$			2.67948i			0.0912107i	0.998960	+	0.0456053i	$0.0145217\pi$
$863$			2.67948i			0.0912107i	−0.998960	+	0.0456053i	$0.985478\pi$
$864$		0			0
$865$	20.2543	+	31.5681i	0.688666	+	1.07335i
$866$		0			0
$867$	1.89559	−	0.610978i	0.0643776	−	0.0207499i
$868$		0			0
$869$		−	80.2250i		−	2.72145i
$870$		0			0
$871$		−	58.6318i		−	1.98666i
$872$		0			0
$873$	15.0405	+	20.9081i	0.509044	+	0.707631i
$874$		0			0
$875$	−11.0701	+	1.56631i	−0.374237	+	0.0529508i
$876$		0			0
$877$		−	7.45204i		−	0.251637i	−0.992053	−	0.125819i	$-0.959844\pi$
$877$		−	7.45204i		−	0.251637i	0.992053	−	0.125819i	$-0.0401558\pi$
$878$		0			0
$879$	−4.53169	+	1.46064i	−0.152850	+	0.0492660i
$880$		0			0
$881$		−	6.90662i		−	0.232690i	−0.993209	−	0.116345i	$-0.962882\pi$
$881$		−	6.90662i		−	0.232690i	0.993209	−	0.116345i	$-0.0371178\pi$
$882$		0			0
$883$	−7.22936			−0.243287			−0.121644	−	0.992574i	$-0.538816\pi$
$883$	−7.22936			−0.243287			−0.121644	+	0.992574i	$0.538816\pi$
$884$		0			0
$885$	38.2573	+	31.4815i	1.28600	+	1.05824i
$886$		0			0
$887$		−	16.0732i		−	0.539687i	−0.962904	−	0.269843i	$-0.913028\pi$
$887$		−	16.0732i		−	0.539687i	0.962904	−	0.269843i	$-0.0869719\pi$
$888$		0			0
$889$	−1.02054			−0.0342277
$890$		0			0
$891$	−14.1120	+	42.0782i	−0.472771	+	1.40967i
$892$		0			0
$893$	15.9249			0.532906
$894$		0			0
$895$	−9.89559	−	15.4232i	−0.330773	−	0.515540i
$896$		0			0
$897$	16.2588	−	5.24047i	0.542866	−	0.174974i
$898$		0			0
$899$	8.99220			0.299907
$900$		0			0
$901$	−3.78242			−0.126011
$902$		0			0
$903$	2.59414	−	0.836132i	0.0863276	−	0.0278247i
$904$		0			0
$905$	−28.4050	−	44.2718i	−0.944215	−	1.47164i
$906$		0			0
$907$	14.0768			0.467414			0.233707	−	0.972307i	$-0.424914\pi$
$907$	14.0768			0.467414			0.233707	+	0.972307i	$0.424914\pi$
$908$		0			0
$909$	6.04107	+	8.39780i	0.200370	+	0.278537i
$910$		0			0
$911$	−2.95997			−0.0980681			−0.0490340	−	0.998797i	$-0.515614\pi$
$911$	−2.95997			−0.0980681			−0.0490340	+	0.998797i	$0.515614\pi$
$912$		0			0
$913$			2.56642i			0.0849361i
$914$		0			0
$915$	25.7017	+	21.1497i	0.849673	+	0.699187i
$916$		0			0
$917$	−19.5226			−0.644693
$918$		0			0
$919$		−	0.902006i		−	0.0297544i	−0.999889	−	0.0148772i	$-0.995264\pi$
$919$		−	0.902006i		−	0.0297544i	0.999889	−	0.0148772i	$-0.00473574\pi$
$920$		0			0
$921$	25.9207	−	8.35464i	0.854116	−	0.275295i
$922$		0			0
$923$			20.2539i			0.666665i
$924$		0			0
$925$	29.6504	+	13.5945i	0.974901	+	0.446985i
$926$		0			0
$927$	17.7708	−	12.7837i	0.583671	−	0.419872i
$928$		0			0
$929$		−	31.2739i		−	1.02606i	−0.858370	−	0.513031i	$-0.828522\pi$
$929$		−	31.2739i		−	1.02606i	0.858370	−	0.513031i	$-0.171478\pi$
$930$		0			0
$931$			1.78803i			0.0586002i
$932$		0			0
$933$	−31.0828	+	10.0185i	−1.01760	+	0.327990i
$934$		0			0
$935$	23.7064	+	36.9485i	0.775281	+	1.20835i
$936$		0			0
$937$		−	32.6607i		−	1.06698i	−0.845807	−	0.533490i	$-0.820880\pi$
$937$		−	32.6607i		−	1.06698i	0.845807	−	0.533490i	$-0.179120\pi$
$938$		0			0
$939$	−8.23068	−	25.5361i	−0.268598	−	0.833339i
$940$		0			0
$941$		−	15.5428i		−	0.506681i	−0.967377	−	0.253341i	$-0.918471\pi$
$941$		−	15.5428i		−	0.506681i	0.967377	−	0.253341i	$-0.0815293\pi$
$942$		0			0
$943$	15.5378			0.505982
$944$		0			0
$945$	−11.5945	−	0.753992i	−0.377168	−	0.0245274i
$946$		0			0
$947$			18.4342i			0.599031i	0.954091	+	0.299516i	$0.0968251\pi$
$947$			18.4342i			0.599031i	−0.954091	+	0.299516i	$0.903175\pi$
$948$		0			0
$949$	46.2446			1.50116
$950$		0			0
$951$	36.1153	−	11.6405i	1.17112	−	0.377470i
$952$		0			0
$953$	−26.5350			−0.859552			−0.429776	−	0.902935i	$-0.641408\pi$
$953$	−26.5350			−0.859552			−0.429776	+	0.902935i	$0.641408\pi$
$954$		0			0
$955$	−15.5692	−	24.2661i	−0.503808	−	0.785231i
$956$		0			0
$957$	4.97538	+	15.4364i	0.160831	+	0.498986i
$958$		0			0
$959$	−8.91251			−0.287800
$960$		0			0
$961$	8.57360			0.276568
$962$		0			0
$963$	−22.5941	−	31.4085i	−0.728086	−	1.01212i
$964$		0			0
$965$	10.3775	−	6.65823i	0.334062	−	0.214336i
$966$		0			0
$967$	48.4176			1.55701			0.778503	−	0.627641i	$-0.215979\pi$
$967$	48.4176			1.55701			0.778503	+	0.627641i	$0.215979\pi$
$968$		0			0
$969$	−3.78242	−	11.7352i	−0.121509	−	0.376987i
$970$		0			0
$971$	−35.4475			−1.13756			−0.568782	−	0.822488i	$-0.692585\pi$
$971$	−35.4475			−1.13756			−0.568782	+	0.822488i	$0.692585\pi$
$972$		0			0
$973$			22.9159i			0.734648i
$974$		0			0
$975$	51.7618	+	6.14123i	1.65771	+	0.196677i
$976$		0			0
$977$	9.31762			0.298097			0.149048	−	0.988830i	$-0.452379\pi$
$977$	9.31762			0.298097			0.149048	+	0.988830i	$0.452379\pi$
$978$		0			0
$979$			79.7597i			2.54913i
$980$		0			0
$981$	5.95264	−	4.28212i	0.190053	−	0.136717i
$982$		0			0
$983$			36.7821i			1.17317i	0.809889	+	0.586584i	$0.199527\pi$
$983$			36.7821i			1.17317i	−0.809889	+	0.586584i	$0.800473\pi$
$984$		0			0
$985$	22.6710	+	35.3348i	0.722357	+	1.12586i
$986$		0			0
$987$	−4.73241	−	14.6825i	−0.150634	−	0.467350i
$988$		0			0
$989$			2.57853i			0.0819923i
$990$		0			0
$991$		−	26.6419i		−	0.846307i	−0.906058	−	0.423153i	$-0.860923\pi$
$991$		−	26.6419i		−	0.846307i	0.906058	−	0.423153i	$-0.139077\pi$
$992$		0			0
$993$	1.76877	+	5.48769i	0.0561302	+	0.174147i
$994$		0			0
$995$	−9.62994	+	6.17862i	−0.305290	+	0.195875i
$996$		0			0
$997$		−	48.7370i		−	1.54352i	−0.635916	−	0.771758i	$-0.719378\pi$
$997$		−	48.7370i		−	1.54352i	0.635916	−	0.771758i	$-0.280622\pi$
$998$		0			0
$999$	27.2825	+	20.1182i	0.863180	+	0.636511i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.v.a.239.3	yes	12
3.2	odd	2	inner	1680.2.v.a.239.2	yes	12
4.3	odd	2		1680.2.v.b.239.9	yes	12
5.4	even	2		1680.2.v.b.239.10	yes	12
12.11	even	2		1680.2.v.b.239.12	yes	12
15.14	odd	2		1680.2.v.b.239.11	yes	12
20.19	odd	2	inner	1680.2.v.a.239.4	yes	12
60.59	even	2	inner	1680.2.v.a.239.1	✓	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.v.a.239.1	✓	12	60.59	even	2	inner
1680.2.v.a.239.2	yes	12	3.2	odd	2	inner
1680.2.v.a.239.3	yes	12	1.1	even	1	trivial
1680.2.v.a.239.4	yes	12	20.19	odd	2	inner
1680.2.v.b.239.9	yes	12	4.3	odd	2
1680.2.v.b.239.10	yes	12	5.4	even	2
1680.2.v.b.239.11	yes	12	15.14	odd	2
1680.2.v.b.239.12	yes	12	12.11	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1680 = 2^{4} \cdot 3 \cdot 5 \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1680.v (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+(-1.64854 + 0.531349i) q^{3} +(-1.20750 - 1.88200i) q^{5} -1.00000 q^{7} +(2.43534 - 1.75189i) q^{9} -4.93129 q^{11} +6.01886i q^{13} +(2.99061 + 2.46094i) q^{15} +3.98122 q^{17} +1.78803i q^{19} +(1.64854 - 0.531349i) q^{21} +1.63861i q^{23} +(-2.08387 + 4.54505i) q^{25} +(-3.08387 + 4.18207i) q^{27} +1.89883i q^{29} -4.73565i q^{31} +(8.12940 - 2.62023i) q^{33} +(1.20750 + 1.88200i) q^{35} -6.52368i q^{37} +(-3.19811 - 9.92231i) q^{39} -9.48232i q^{41} +1.57360 q^{43} +(-6.23775 - 2.46789i) q^{45} -8.90641i q^{47} +1.00000 q^{49} +(-6.56319 + 2.11542i) q^{51} -0.950066 q^{53} +(5.95455 + 9.28070i) q^{55} +(-0.950066 - 2.94762i) q^{57} +12.7925 q^{59} +8.59414 q^{61} +(-2.43534 + 1.75189i) q^{63} +(11.3275 - 7.26779i) q^{65} -9.74135 q^{67} +(-0.870674 - 2.70131i) q^{69} +3.36507 q^{71} -7.68328i q^{73} +(1.02033 - 8.59994i) q^{75} +4.93129 q^{77} +16.2686i q^{79} +(2.86173 - 8.53291i) q^{81} -0.520436i q^{83} +(-4.80734 - 7.49267i) q^{85} +(-1.00894 - 3.13029i) q^{87} -16.1742i q^{89} -6.01886i q^{91} +(2.51628 + 7.80689i) q^{93} +(3.36507 - 2.15905i) q^{95} +8.58528i q^{97} +(-12.0094 + 8.63910i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{3} - 12 q^{7} + 6 q^{9} + 12 q^{15} + 2 q^{21} + 16 q^{25} + 4 q^{27} + 8 q^{43} - 4 q^{45} + 12 q^{49} - 16 q^{55} + 32 q^{61} - 6 q^{63} - 24 q^{67} + 36 q^{69} + 18 q^{75} + 22 q^{81} + 8 q^{85}+ \cdots + 22 q^{87}+O(q^{100}) \) 12 * q - 2 * q^3 - 12 * q^7 + 6 * q^9 + 12 * q^15 + 2 * q^21 + 16 * q^25 + 4 * q^27 + 8 * q^43 - 4 * q^45 + 12 * q^49 - 16 * q^55 + 32 * q^61 - 6 * q^63 - 24 * q^67 + 36 * q^69 + 18 * q^75 + 22 * q^81 + 8 * q^85 + 22 * q^87

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