LMFDB - Embedded newform 1680.2.v.a.239.6

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			239.6
Root			$-2.15605 + 0.592845i$ of defining polynomial
Character	$\chi$	$=$	1680.239
Dual form			1680.2.v.a.239.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+(-0.393401 - 1.68678i) q^{3} +(2.15605 + 0.592845i) q^{5} -1.00000 q^{7} +(-2.69047 + 1.32716i) q^{9} -4.95259 q^{11} +1.78105i q^{13} +(0.151809 - 3.87001i) q^{15} -1.69638 q^{17} -3.94181i q^{19} +(0.393401 + 1.68678i) q^{21} +5.56144i q^{23} +(4.29707 + 2.55640i) q^{25} +(3.29707 + 4.01613i) q^{27} +7.03008i q^{29} +5.49252i q^{31} +(1.94835 + 8.35394i) q^{33} +(-2.15605 - 0.592845i) q^{35} +9.43432i q^{37} +(3.00424 - 0.700665i) q^{39} -10.6564i q^{41} -6.16774 q^{43} +(-6.58758 + 1.26639i) q^{45} -1.72142i q^{47} +1.00000 q^{49} +(0.667358 + 2.86143i) q^{51} -6.64897 q^{53} +(-10.6780 - 2.93612i) q^{55} +(-6.64897 + 1.55071i) q^{57} -12.0169 q^{59} +3.57360 q^{61} +(2.69047 - 1.32716i) q^{63} +(-1.05588 + 3.84002i) q^{65} +10.7619 q^{67} +(9.38094 - 2.18788i) q^{69} +2.33688 q^{71} +7.04323i q^{73} +(2.62162 - 8.25391i) q^{75} +4.95259 q^{77} +9.28494i q^{79} +(5.47727 - 7.14139i) q^{81} -2.93728i q^{83} +(-3.65748 - 1.00569i) q^{85} +(11.8582 - 2.76564i) q^{87} +2.11859i q^{89} -1.78105i q^{91} +(9.26468 - 2.16076i) q^{93} +(2.33688 - 8.49872i) q^{95} +16.3282i q^{97} +(13.3248 - 6.57290i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−0.393401	−	1.68678i	−0.227130	−	0.973864i
$3$	−0.393401	−	1.68678i	−0.227130	−	0.973864i
$4$		0			0
$5$	2.15605	+	0.592845i	0.964213	+	0.265128i
$5$	2.15605	+	0.592845i	0.964213	+	0.265128i
$6$		0			0
$7$	−1.00000			−0.377964
$7$	−1.00000			−0.377964
$8$		0			0
$9$	−2.69047	+	1.32716i	−0.896824	+	0.442388i
$10$		0			0
$11$	−4.95259			−1.49326			−0.746631	−	0.665238i	$-0.768330\pi$
$11$	−4.95259			−1.49326			−0.746631	+	0.665238i	$0.768330\pi$
$12$		0			0
$13$			1.78105i			0.493973i	0.969019	+	0.246987i	$0.0794404\pi$
$13$			1.78105i			0.493973i	−0.969019	+	0.246987i	$0.920560\pi$
$14$		0			0
$15$	0.151809	−	3.87001i	0.0391970	−	0.999232i
$16$		0			0
$17$	−1.69638			−0.411433			−0.205716	−	0.978612i	$-0.565952\pi$
$17$	−1.69638			−0.411433			−0.205716	+	0.978612i	$0.565952\pi$
$18$		0			0
$19$		−	3.94181i		−	0.904312i	−0.891939	−	0.452156i	$-0.850655\pi$
$19$		−	3.94181i		−	0.904312i	0.891939	−	0.452156i	$-0.149345\pi$
$20$		0			0
$21$	0.393401	+	1.68678i	0.0858471	+	0.368086i
$22$		0			0
$23$			5.56144i			1.15964i	0.814744	+	0.579820i	$0.196877\pi$
$23$			5.56144i			1.15964i	−0.814744	+	0.579820i	$0.803123\pi$
$24$		0			0
$25$	4.29707	+	2.55640i	0.859414	+	0.511280i
$26$		0			0
$27$	3.29707	+	4.01613i	0.634522	+	0.772905i
$28$		0			0
$29$			7.03008i			1.30545i	0.757594	+	0.652726i	$0.226375\pi$
$29$			7.03008i			1.30545i	−0.757594	+	0.652726i	$0.773625\pi$
$30$		0			0
$31$			5.49252i			0.986485i	0.869892	+	0.493243i	$0.164189\pi$
$31$			5.49252i			0.986485i	−0.869892	+	0.493243i	$0.835811\pi$
$32$		0			0
$33$	1.94835	+	8.35394i	0.339165	+	1.45423i
$34$		0			0
$35$	−2.15605	−	0.592845i	−0.364438	−	0.100209i
$36$		0			0
$37$			9.43432i			1.55099i	0.631352	+	0.775497i	$0.282500\pi$
$37$			9.43432i			1.55099i	−0.631352	+	0.775497i	$0.717500\pi$
$38$		0			0
$39$	3.00424	−	0.700665i	0.481063	−	0.112196i
$40$		0			0
$41$		−	10.6564i		−	1.66426i	−0.554584	−	0.832128i	$-0.687123\pi$
$41$		−	10.6564i		−	1.66426i	0.554584	−	0.832128i	$-0.312877\pi$
$42$		0			0
$43$	−6.16774			−0.940572			−0.470286	−	0.882514i	$-0.655849\pi$
$43$	−6.16774			−0.940572			−0.470286	+	0.882514i	$0.655849\pi$
$44$		0			0
$45$	−6.58758	+	1.26639i	−0.982019	+	0.188783i
$46$		0			0
$47$		−	1.72142i		−	0.251096i	−0.992088	−	0.125548i	$-0.959931\pi$
$47$		−	1.72142i		−	0.251096i	0.992088	−	0.125548i	$-0.0400688\pi$
$48$		0			0
$49$	1.00000			0.142857
$50$		0			0
$51$	0.667358	+	2.86143i	0.0934488	+	0.400680i
$52$		0			0
$53$	−6.64897			−0.913306			−0.456653	−	0.889645i	$-0.650952\pi$
$53$	−6.64897			−0.913306			−0.456653	+	0.889645i	$0.650952\pi$
$54$		0			0
$55$	−10.6780	−	2.93612i	−1.43982	−	0.395906i
$56$		0			0
$57$	−6.64897	+	1.55071i	−0.880678	+	0.205397i
$58$		0			0
$59$	−12.0169			−1.56447			−0.782237	−	0.622981i	$-0.785921\pi$
$59$	−12.0169			−1.56447			−0.782237	+	0.622981i	$0.785921\pi$
$60$		0			0
$61$	3.57360			0.457553			0.228777	−	0.973479i	$-0.426527\pi$
$61$	3.57360			0.457553			0.228777	+	0.973479i	$0.426527\pi$
$62$		0			0
$63$	2.69047	−	1.32716i	0.338968	−	0.167207i
$64$		0			0
$65$	−1.05588	+	3.84002i	−0.130966	+	0.476295i
$66$		0			0
$67$	10.7619			1.31477			0.657387	−	0.753553i	$-0.271662\pi$
$67$	10.7619			1.31477			0.657387	+	0.753553i	$0.271662\pi$
$68$		0			0
$69$	9.38094	−	2.18788i	1.12933	−	0.263389i
$70$		0			0
$71$	2.33688			0.277337			0.138668	−	0.990339i	$-0.455718\pi$
$71$	2.33688			0.277337			0.138668	+	0.990339i	$0.455718\pi$
$72$		0			0
$73$			7.04323i			0.824347i	0.911105	+	0.412174i	$0.135230\pi$
$73$			7.04323i			0.824347i	−0.911105	+	0.412174i	$0.864770\pi$
$74$		0			0
$75$	2.62162	−	8.25391i	0.302719	−	0.953080i
$76$		0			0
$77$	4.95259			0.564400
$78$		0			0
$79$			9.28494i			1.04464i	0.852750	+	0.522319i	$0.174933\pi$
$79$			9.28494i			1.04464i	−0.852750	+	0.522319i	$0.825067\pi$
$80$		0			0
$81$	5.47727	−	7.14139i	0.608586	−	0.793488i
$82$		0			0
$83$		−	2.93728i		−	0.322408i	−0.986921	−	0.161204i	$-0.948462\pi$
$83$		−	2.93728i		−	0.322408i	0.986921	−	0.161204i	$-0.0515377\pi$
$84$		0			0
$85$	−3.65748	−	1.00569i	−0.396709	−	0.109082i
$86$		0			0
$87$	11.8582	−	2.76564i	1.27133	−	0.296508i
$88$		0			0
$89$			2.11859i			0.224570i	0.993676	+	0.112285i	$0.0358170\pi$
$89$			2.11859i			0.224570i	−0.993676	+	0.112285i	$0.964183\pi$
$90$		0			0
$91$		−	1.78105i		−	0.186704i
$92$		0			0
$93$	9.26468	−	2.16076i	0.960703	−	0.224061i
$94$		0			0
$95$	2.33688	−	8.49872i	0.239759	−	0.871950i
$96$		0			0
$97$			16.3282i			1.65787i	0.559342	+	0.828937i	$0.311054\pi$
$97$			16.3282i			1.65787i	−0.559342	+	0.828937i	$0.688946\pi$
$98$		0			0
$99$	13.3248	−	6.57290i	1.33919	−	0.660601i
$100$		0			0
$101$		−	8.65206i		−	0.860912i	−0.902612	−	0.430456i	$-0.858353\pi$
$101$		−	8.65206i		−	0.860912i	0.902612	−	0.430456i	$-0.141647\pi$
$102$		0			0
$103$	4.78680			0.471658			0.235829	−	0.971795i	$-0.424220\pi$
$103$	4.78680			0.471658			0.235829	+	0.971795i	$0.424220\pi$
$104$		0			0
$105$	−0.151809	+	3.87001i	−0.0148151	+	0.377674i
$106$		0			0
$107$			13.2415i			1.28010i	0.768333	+	0.640051i	$0.221087\pi$
$107$			13.2415i			1.28010i	−0.768333	+	0.640051i	$0.778913\pi$
$108$		0			0
$109$	−15.5487			−1.48929			−0.744647	−	0.667458i	$-0.767382\pi$
$109$	−15.5487			−1.48929			−0.744647	+	0.667458i	$0.767382\pi$
$110$		0			0
$111$	15.9137	−	3.71147i	1.51046	−	0.352277i
$112$		0			0
$113$	−19.9469			−1.87645			−0.938224	−	0.346029i	$-0.887530\pi$
$113$	−19.9469			−1.87645			−0.938224	+	0.346029i	$0.887530\pi$
$114$		0			0
$115$	−3.29707	+	11.9907i	−0.307453	+	1.11814i
$116$		0			0
$117$	−2.36374	−	4.79185i	−0.218528	−	0.443007i
$118$		0			0
$119$	1.69638			0.155507
$120$		0			0
$121$	13.5282			1.22983
$122$		0			0
$123$	−17.9751	+	4.19225i	−1.62076	+	0.378002i
$124$		0			0
$125$	7.74913	+	8.05921i	0.693104	+	0.720838i
$126$		0			0
$127$	3.74135			0.331991			0.165995	−	0.986127i	$-0.446916\pi$
$127$	3.74135			0.331991			0.165995	+	0.986127i	$0.446916\pi$
$128$		0			0
$129$	2.42640	+	10.4036i	0.213632	+	0.915990i
$130$		0			0
$131$	−7.34319			−0.641577			−0.320789	−	0.947151i	$-0.603948\pi$
$131$	−7.34319			−0.641577			−0.320789	+	0.947151i	$0.603948\pi$
$132$		0			0
$133$			3.94181i			0.341798i
$134$		0			0
$135$	4.72769	+	10.6136i	0.406895	+	0.913475i
$136$		0			0
$137$	3.25621			0.278197			0.139098	−	0.990279i	$-0.455580\pi$
$137$	3.25621			0.278197			0.139098	+	0.990279i	$0.455580\pi$
$138$		0			0
$139$		−	4.73309i		−	0.401455i	−0.979647	−	0.200728i	$-0.935669\pi$
$139$		−	4.73309i		−	0.401455i	0.979647	−	0.200728i	$-0.0643306\pi$
$140$		0			0
$141$	−2.90367	+	0.677210i	−0.244533	+	0.0570314i
$142$		0			0
$143$		−	8.82079i		−	0.737631i
$144$		0			0
$145$	−4.16774	+	15.1572i	−0.346112	+	1.25873i
$146$		0			0
$147$	−0.393401	−	1.68678i	−0.0324472	−	0.139123i
$148$		0			0
$149$		−	7.31303i		−	0.599107i	−0.954080	−	0.299553i	$-0.903162\pi$
$149$		−	7.31303i		−	0.599107i	0.954080	−	0.299553i	$-0.0968376\pi$
$150$		0			0
$151$		−	16.7079i		−	1.35967i	−0.733366	−	0.679834i	$-0.762052\pi$
$151$		−	16.7079i		−	1.35967i	0.733366	−	0.679834i	$-0.237948\pi$
$152$		0			0
$153$	4.56406	−	2.25138i	0.368983	−	0.182013i
$154$		0			0
$155$	−3.25621	+	11.8421i	−0.261545	+	0.951182i
$156$		0			0
$157$		−	14.9268i		−	1.19129i	−0.803247	−	0.595646i	$-0.796896\pi$
$157$		−	14.9268i		−	1.19129i	0.803247	−	0.595646i	$-0.203104\pi$
$158$		0			0
$159$	2.61571	+	11.2154i	0.207439	+	0.889437i
$160$		0			0
$161$		−	5.56144i		−	0.438303i
$162$		0			0
$163$	3.40586			0.266767			0.133384	−	0.991064i	$-0.457416\pi$
$163$	3.40586			0.266767			0.133384	+	0.991064i	$0.457416\pi$
$164$		0			0
$165$	−0.751850	+	19.1666i	−0.0585314	+	1.49211i
$166$		0			0
$167$			10.1207i			0.783163i	0.920143	+	0.391581i	$0.128072\pi$
$167$			10.1207i			0.783163i	−0.920143	+	0.391581i	$0.871928\pi$
$168$		0			0
$169$	9.82788			0.755991
$170$		0			0
$171$	5.23142	+	10.6053i	0.400057	+	0.811009i
$172$		0			0
$173$	13.7133			1.04261			0.521303	−	0.853372i	$-0.325446\pi$
$173$	13.7133			1.04261			0.521303	+	0.853372i	$0.325446\pi$
$174$		0			0
$175$	−4.29707	−	2.55640i	−0.324828	−	0.193246i
$176$		0			0
$177$	4.72748	+	20.2700i	0.355339	+	1.52358i
$178$		0			0
$179$	−6.28731			−0.469935			−0.234968	−	0.972003i	$-0.575498\pi$
$179$	−6.28731			−0.469935			−0.234968	+	0.972003i	$0.575498\pi$
$180$		0			0
$181$	−12.0411			−0.895006			−0.447503	−	0.894282i	$-0.647687\pi$
$181$	−12.0411			−0.895006			−0.447503	+	0.894282i	$0.647687\pi$
$182$		0			0
$183$	−1.40586	−	6.02789i	−0.103924	−	0.445595i
$184$		0			0
$185$	−5.59309	+	20.3408i	−0.411212	+	1.49549i
$186$		0			0
$187$	8.40148			0.614377
$188$		0			0
$189$	−3.29707	−	4.01613i	−0.239827	−	0.292131i
$190$		0			0
$191$	1.55983			0.112865			0.0564326	−	0.998406i	$-0.482027\pi$
$191$	1.55983			0.112865			0.0564326	+	0.998406i	$0.482027\pi$
$192$		0			0
$193$			12.9964i			0.935502i	0.883860	+	0.467751i	$0.154936\pi$
$193$			12.9964i			0.935502i	−0.883860	+	0.467751i	$0.845064\pi$
$194$		0			0
$195$	6.89266	+	0.270380i	0.493594	+	0.0193623i
$196$		0			0
$197$	−13.1614			−0.937710			−0.468855	−	0.883275i	$-0.655333\pi$
$197$	−13.1614			−0.937710			−0.468855	+	0.883275i	$0.655333\pi$
$198$		0			0
$199$			21.5904i			1.53050i	0.643733	+	0.765250i	$0.277385\pi$
$199$			21.5904i			1.53050i	−0.643733	+	0.765250i	$0.722615\pi$
$200$		0			0
$201$	−4.23374	−	18.1530i	−0.298625	−	1.28041i
$202$		0			0
$203$		−	7.03008i		−	0.493415i
$204$		0			0
$205$	6.31761	−	22.9758i	0.441241	−	1.60470i
$206$		0			0
$207$	−7.38094	−	14.9629i	−0.513011	−	1.03999i
$208$		0			0
$209$			19.5222i			1.35038i
$210$		0			0
$211$		−	23.3714i		−	1.60895i	−0.593984	−	0.804477i	$-0.702446\pi$
$211$		−	23.3714i		−	1.60895i	0.593984	−	0.804477i	$-0.297554\pi$
$212$		0			0
$213$	−0.919330	−	3.94181i	−0.0629915	−	0.270088i
$214$		0			0
$215$	−13.2979	−	3.65651i	−0.906912	−	0.249372i
$216$		0			0
$217$		−	5.49252i		−	0.372856i
$218$		0			0
$219$	11.8804	−	2.77081i	0.802803	−	0.187234i
$220$		0			0
$221$		−	3.02133i		−	0.203237i
$222$		0			0
$223$	11.3809			0.762124			0.381062	−	0.924549i	$-0.375558\pi$
$223$	11.3809			0.762124			0.381062	+	0.924549i	$0.375558\pi$
$224$		0			0
$225$	−14.9539	−	1.17501i	−0.996927	−	0.0783338i
$226$		0			0
$227$			4.52013i			0.300012i	0.988685	+	0.150006i	$0.0479293\pi$
$227$			4.52013i			0.300012i	−0.988685	+	0.150006i	$0.952071\pi$
$228$		0			0
$229$	19.5238			1.29017			0.645084	−	0.764112i	$-0.276823\pi$
$229$	19.5238			1.29017			0.645084	+	0.764112i	$0.276823\pi$
$230$		0			0
$231$	−1.94835	−	8.35394i	−0.128192	−	0.549649i
$232$		0			0
$233$	−0.694218			−0.0454797			−0.0227399	−	0.999741i	$-0.507239\pi$
$233$	−0.694218			−0.0454797			−0.0227399	+	0.999741i	$0.507239\pi$
$234$		0			0
$235$	1.02054	−	3.71147i	0.0665725	−	0.242110i
$236$		0			0
$237$	15.6617	−	3.65271i	1.01734	−	0.237269i
$238$		0			0
$239$	−10.1840			−0.658749			−0.329375	−	0.944199i	$-0.606838\pi$
$239$	−10.1840			−0.658749			−0.329375	+	0.944199i	$0.606838\pi$
$240$		0			0
$241$	5.14721			0.331561			0.165780	−	0.986163i	$-0.446986\pi$
$241$	5.14721			0.331561			0.165780	+	0.986163i	$0.446986\pi$
$242$		0			0
$243$	−14.2007	−	6.42954i	−0.910978	−	0.412455i
$244$		0			0
$245$	2.15605	+	0.592845i	0.137745	+	0.0378755i
$246$		0			0
$247$	7.02054			0.446706
$248$		0			0
$249$	−4.95455	+	1.15553i	−0.313982	+	0.0732286i
$250$		0			0
$251$	−25.0418			−1.58062			−0.790312	−	0.612705i	$-0.790081\pi$
$251$	−25.0418			−1.58062			−0.790312	+	0.612705i	$0.790081\pi$
$252$		0			0
$253$		−	27.5435i		−	1.73165i
$254$		0			0
$255$	−0.257527	+	6.56501i	−0.0161269	+	0.411117i
$256$		0			0
$257$	24.1225			1.50472			0.752359	−	0.658754i	$-0.228916\pi$
$257$	24.1225			1.50472			0.752359	+	0.658754i	$0.228916\pi$
$258$		0			0
$259$		−	9.43432i		−	0.586220i
$260$		0			0
$261$	−9.33007	−	18.9142i	−0.577517	−	1.17076i
$262$		0			0
$263$			16.3173i			1.00617i	0.864237	+	0.503085i	$0.167802\pi$
$263$			16.3173i			1.00617i	−0.864237	+	0.503085i	$0.832198\pi$
$264$		0			0
$265$	−14.3355	−	3.94181i	−0.880622	−	0.242143i
$266$		0			0
$267$	3.57360	−	0.833456i	0.218701	−	0.0510067i
$268$		0			0
$269$		−	11.0838i		−	0.675789i	−0.941184	−	0.337894i	$-0.890285\pi$
$269$		−	11.0838i		−	0.675789i	0.941184	−	0.337894i	$-0.109715\pi$
$270$		0			0
$271$			27.9233i			1.69622i	0.529823	+	0.848109i	$0.322259\pi$
$271$			27.9233i			1.69622i	−0.529823	+	0.848109i	$0.677741\pi$
$272$		0			0
$273$	−3.00424	+	0.700665i	−0.181825	+	0.0424062i
$274$		0			0
$275$	−21.2816	−	12.6608i	−1.28333	−	0.763475i
$276$		0			0
$277$		−	24.4421i		−	1.46858i	−0.678833	−	0.734292i	$-0.737514\pi$
$277$		−	24.4421i		−	1.46858i	0.678833	−	0.734292i	$-0.262486\pi$
$278$		0			0
$279$	−7.28947	−	14.7775i	−0.436409	−	0.884703i
$280$		0			0
$281$		−	25.2671i		−	1.50731i	−0.657271	−	0.753654i	$-0.728289\pi$
$281$		−	25.2671i		−	1.50731i	0.657271	−	0.753654i	$-0.271711\pi$
$282$		0			0
$283$	25.8073			1.53409			0.767044	−	0.641595i	$-0.221727\pi$
$283$	25.8073			1.53409			0.767044	+	0.641595i	$0.221727\pi$
$284$		0			0
$285$	−15.2548	−	0.598404i	−0.903618	−	0.0354464i
$286$		0			0
$287$			10.6564i			0.629029i
$288$		0			0
$289$	−14.1223			−0.830723
$290$		0			0
$291$	27.5421	−	6.42352i	1.61455	−	0.376553i
$292$		0			0
$293$	6.37014			0.372147			0.186074	−	0.982536i	$-0.440424\pi$
$293$	6.37014			0.372147			0.186074	+	0.982536i	$0.440424\pi$
$294$		0			0
$295$	−25.9091	−	7.12418i	−1.50849	−	0.414786i
$296$		0			0
$297$	−16.3290	−	19.8903i	−0.947507	−	1.15415i
$298$		0			0
$299$	−9.90518			−0.572831
$300$		0			0
$301$	6.16774			0.355503
$302$		0			0
$303$	−14.5941	+	3.40373i	−0.838411	+	0.195539i
$304$		0			0
$305$	7.70485	+	2.11859i	0.441179	+	0.121310i
$306$		0			0
$307$	−20.9545			−1.19594			−0.597970	−	0.801519i	$-0.704026\pi$
$307$	−20.9545			−1.19594			−0.597970	+	0.801519i	$0.704026\pi$
$308$		0			0
$309$	−1.88313	−	8.07429i	−0.107128	−	0.459331i
$310$		0			0
$311$	−28.7077			−1.62786			−0.813931	−	0.580962i	$-0.802677\pi$
$311$	−28.7077			−1.62786			−0.813931	+	0.580962i	$0.802677\pi$
$312$		0			0
$313$		−	9.20399i		−	0.520240i	−0.965576	−	0.260120i	$-0.916238\pi$
$313$		−	9.20399i		−	0.520240i	0.965576	−	0.260120i	$-0.0837622\pi$
$314$		0			0
$315$	6.58758	−	1.26639i	0.371168	−	0.0713533i
$316$		0			0
$317$	−18.3928			−1.03304			−0.516522	−	0.856274i	$-0.672773\pi$
$317$	−18.3928			−1.03304			−0.516522	+	0.856274i	$0.672773\pi$
$318$		0			0
$319$		−	34.8171i		−	1.94938i
$320$		0			0
$321$	22.3355	−	5.20921i	1.24665	−	0.290750i
$322$		0			0
$323$			6.68681i			0.372064i
$324$		0			0
$325$	−4.55307	+	7.65328i	−0.252559	+	0.424528i
$326$		0			0
$327$	6.11687	+	26.2273i	0.338264	+	1.45037i
$328$		0			0
$329$			1.72142i			0.0949052i
$330$		0			0
$331$			17.6485i			0.970052i	0.874500	+	0.485026i	$0.161190\pi$
$331$			17.6485i			0.970052i	−0.874500	+	0.485026i	$0.838810\pi$
$332$		0			0
$333$	−12.5209	−	25.3828i	−0.686141	−	1.39097i
$334$		0			0
$335$	23.2031	+	6.38013i	1.26772	+	0.348584i
$336$		0			0
$337$		−	3.56209i		−	0.194039i	−0.995282	−	0.0970197i	$-0.969069\pi$
$337$		−	3.56209i		−	0.194039i	0.995282	−	0.0970197i	$-0.0309310\pi$
$338$		0			0
$339$	7.84713	+	33.6461i	0.426198	+	1.82741i
$340$		0			0
$341$		−	27.2022i		−	1.47308i
$342$		0			0
$343$	−1.00000			−0.0539949
$344$		0			0
$345$	21.5228	+	0.844279i	1.15875	+	0.0454545i
$346$		0			0
$347$		−	0.313111i		−	0.0168087i	−0.999965	−	0.00840434i	$-0.997325\pi$
$347$		−	0.313111i		−	0.0168087i	0.999965	−	0.00840434i	$-0.00267522\pi$
$348$		0			0
$349$	−8.42640			−0.451055			−0.225527	−	0.974237i	$-0.572410\pi$
$349$	−8.42640			−0.451055			−0.225527	+	0.974237i	$0.572410\pi$
$350$		0			0
$351$	−7.15292	+	5.87223i	−0.381794	+	0.313437i
$352$		0			0
$353$	5.64681			0.300549			0.150275	−	0.988644i	$-0.451984\pi$
$353$	5.64681			0.300549			0.150275	+	0.988644i	$0.451984\pi$
$354$		0			0
$355$	5.03842	+	1.38541i	0.267412	+	0.0735297i
$356$		0			0
$357$	−0.667358	−	2.86143i	−0.0353203	−	0.151443i
$358$		0			0
$359$	−10.9611			−0.578503			−0.289251	−	0.957253i	$-0.593406\pi$
$359$	−10.9611			−0.578503			−0.289251	+	0.957253i	$0.593406\pi$
$360$		0			0
$361$	3.46216			0.182219
$362$		0			0
$363$	−5.32199	−	22.8190i	−0.279332	−	1.19769i
$364$		0			0
$365$	−4.17554	+	15.1855i	−0.218558	+	0.794847i
$366$		0			0
$367$	−6.99562			−0.365168			−0.182584	−	0.983190i	$-0.558446\pi$
$367$	−6.99562			−0.365168			−0.182584	+	0.983190i	$0.558446\pi$
$368$		0			0
$369$	14.1428	+	28.6708i	0.736246	+	1.49254i
$370$		0			0
$371$	6.64897			0.345197
$372$		0			0
$373$		−	10.2256i		−	0.529462i	−0.964322	−	0.264731i	$-0.914717\pi$
$373$		−	10.2256i		−	0.529462i	0.964322	−	0.264731i	$-0.0852831\pi$
$374$		0			0
$375$	10.5456	−	16.2416i	0.544574	−	0.838713i
$376$		0			0
$377$	−12.5209			−0.644859
$378$		0			0
$379$		−	25.5322i		−	1.31150i	−0.754979	−	0.655749i	$-0.772353\pi$
$379$		−	25.5322i		−	1.31150i	0.754979	−	0.655749i	$-0.227647\pi$
$380$		0			0
$381$	−1.47185	−	6.31084i	−0.0754052	−	0.323314i
$382$		0			0
$383$			21.0210i			1.07412i	0.843544	+	0.537060i	$0.180465\pi$
$383$			21.0210i			1.07412i	−0.843544	+	0.537060i	$0.819535\pi$
$384$		0			0
$385$	10.6780	+	2.93612i	0.544202	+	0.149638i
$386$		0			0
$387$	16.5941	−	8.18561i	0.843528	−	0.416098i
$388$		0			0
$389$		−	1.58286i		−	0.0802540i	−0.999195	−	0.0401270i	$-0.987224\pi$
$389$		−	1.58286i		−	0.0802540i	0.999195	−	0.0401270i	$-0.0127762\pi$
$390$		0			0
$391$		−	9.43432i		−	0.477114i
$392$		0			0
$393$	2.88882	+	12.3864i	0.145722	+	0.624809i
$394$		0			0
$395$	−5.50453	+	20.0188i	−0.276963	+	1.00725i
$396$		0			0
$397$		−	26.5538i		−	1.33270i	−0.745641	−	0.666348i	$-0.767857\pi$
$397$		−	26.5538i		−	1.33270i	0.745641	−	0.666348i	$-0.232143\pi$
$398$		0			0
$399$	6.64897	−	1.55071i	0.332865	−	0.0776326i
$400$		0			0
$401$		−	17.2201i		−	0.859929i	−0.902846	−	0.429964i	$-0.858526\pi$
$401$		−	17.2201i		−	0.859929i	0.902846	−	0.429964i	$-0.141474\pi$
$402$		0			0
$403$	−9.78242			−0.487297
$404$		0			0
$405$	16.0430	−	12.1500i	0.797183	−	0.603738i
$406$		0			0
$407$		−	46.7243i		−	2.31604i
$408$		0			0
$409$	−36.2446			−1.79218			−0.896090	−	0.443873i	$-0.853604\pi$
$409$	−36.2446			−1.79218			−0.896090	+	0.443873i	$0.853604\pi$
$410$		0			0
$411$	−1.28100	−	5.49252i	−0.0631869	−	0.270926i
$412$		0			0
$413$	12.0169			0.591315
$414$		0			0
$415$	1.74135	−	6.33290i	0.0854794	−	0.310870i
$416$		0			0
$417$	−7.98369	+	1.86200i	−0.390963	+	0.0911825i
$418$		0			0
$419$	−2.66943			−0.130410			−0.0652051	−	0.997872i	$-0.520770\pi$
$419$	−2.66943			−0.130410			−0.0652051	+	0.997872i	$0.520770\pi$
$420$		0			0
$421$	−2.06599			−0.100690			−0.0503451	−	0.998732i	$-0.516032\pi$
$421$	−2.06599			−0.100690			−0.0503451	+	0.998732i	$0.516032\pi$
$422$		0			0
$423$	2.28461	+	4.63144i	0.111082	+	0.225189i
$424$		0			0
$425$	−7.28947	−	4.33663i	−0.353591	−	0.210357i
$426$		0			0
$427$	−3.57360			−0.172939
$428$		0			0
$429$	−14.8788	+	3.47011i	−0.718353	+	0.167538i
$430$		0			0
$431$	5.22570			0.251713			0.125856	−	0.992048i	$-0.459832\pi$
$431$	5.22570			0.251713			0.125856	+	0.992048i	$0.459832\pi$
$432$		0			0
$433$		−	11.8254i		−	0.568293i	−0.958781	−	0.284147i	$-0.908290\pi$
$433$		−	11.8254i		−	0.568293i	0.958781	−	0.284147i	$-0.0917103\pi$
$434$		0			0
$435$	27.2065	+	1.06723i	1.30445	+	0.0511699i
$436$		0			0
$437$	21.9221			1.04868
$438$		0			0
$439$			17.7295i			0.846183i	0.906087	+	0.423091i	$0.139055\pi$
$439$			17.7295i			0.846183i	−0.906087	+	0.423091i	$0.860945\pi$
$440$		0			0
$441$	−2.69047	+	1.32716i	−0.128118	+	0.0631983i
$442$		0			0
$443$		−	21.5656i		−	1.02461i	−0.858802	−	0.512307i	$-0.828791\pi$
$443$		−	21.5656i		−	1.02461i	0.858802	−	0.512307i	$-0.171209\pi$
$444$		0			0
$445$	−1.25600	+	4.56778i	−0.0595399	+	0.216534i
$446$		0			0
$447$	−12.3355	+	2.87695i	−0.583449	+	0.136075i
$448$		0			0
$449$			36.9559i			1.74406i	0.489456	+	0.872028i	$0.337195\pi$
$449$			36.9559i			1.74406i	−0.489456	+	0.872028i	$0.662805\pi$
$450$		0			0
$451$			52.7769i			2.48517i
$452$		0			0
$453$	−28.1826	+	6.57290i	−1.32413	+	0.308822i
$454$		0			0
$455$	1.05588	−	3.84002i	0.0495006	−	0.180023i
$456$		0			0
$457$			4.32152i			0.202152i	0.994879	+	0.101076i	$0.0322286\pi$
$457$			4.32152i			0.202152i	−0.994879	+	0.101076i	$0.967771\pi$
$458$		0			0
$459$	−5.59309	−	6.81289i	−0.261063	−	0.317999i
$460$		0			0
$461$		−	21.4874i		−	1.00077i	−0.865804	−	0.500384i	$-0.833192\pi$
$461$		−	21.4874i		−	1.00077i	0.865804	−	0.500384i	$-0.166808\pi$
$462$		0			0
$463$	36.5443			1.69836			0.849179	−	0.528105i	$-0.177097\pi$
$463$	36.5443			1.69836			0.849179	+	0.528105i	$0.177097\pi$
$464$		0			0
$465$	21.2561	+	0.833816i	0.985727	+	0.0386673i
$466$		0			0
$467$			1.52253i			0.0704544i	0.999379	+	0.0352272i	$0.0112155\pi$
$467$			1.52253i			0.0704544i	−0.999379	+	0.0352272i	$0.988785\pi$
$468$		0			0
$469$	−10.7619			−0.496938
$470$		0			0
$471$	−25.1783	+	5.87223i	−1.16016	+	0.270578i
$472$		0			0
$473$	30.5463			1.40452
$474$		0			0
$475$	10.0768	−	16.9382i	0.462357	−	0.777179i
$476$		0			0
$477$	17.8889	−	8.82427i	0.819075	−	0.404036i
$478$		0			0
$479$	20.3680			0.930639			0.465319	−	0.885143i	$-0.345939\pi$
$479$	20.3680			0.930639			0.465319	+	0.885143i	$0.345939\pi$
$480$		0			0
$481$	−16.8030			−0.766149
$482$		0			0
$483$	−9.38094	+	2.18788i	−0.426848	+	0.0995518i
$484$		0			0
$485$	−9.68007	+	35.2043i	−0.439549	+	1.59854i
$486$		0			0
$487$	−29.7824			−1.34957			−0.674785	−	0.738014i	$-0.735764\pi$
$487$	−29.7824			−1.34957			−0.674785	+	0.738014i	$0.735764\pi$
$488$		0			0
$489$	−1.33987	−	5.74494i	−0.0605909	−	0.259795i
$490$		0			0
$491$	13.0191			0.587544			0.293772	−	0.955875i	$-0.405089\pi$
$491$	13.0191			0.587544			0.293772	+	0.955875i	$0.405089\pi$
$492$		0			0
$493$		−	11.9257i		−	0.537106i
$494$		0			0
$495$	32.6256	−	6.27193i	1.46641	−	0.281902i
$496$		0			0
$497$	−2.33688			−0.104823
$498$		0			0
$499$			42.9995i			1.92492i	0.271422	+	0.962460i	$0.412506\pi$
$499$			42.9995i			1.92492i	−0.271422	+	0.962460i	$0.587494\pi$
$500$		0			0
$501$	17.0714	−	3.98149i	0.762695	−	0.177880i
$502$		0			0
$503$		−	21.2436i		−	0.947204i	−0.880739	−	0.473602i	$-0.842953\pi$
$503$		−	21.2436i		−	0.947204i	0.880739	−	0.473602i	$-0.157047\pi$
$504$		0			0
$505$	5.12933	−	18.6542i	0.228252	−	0.830103i
$506$		0			0
$507$	−3.86630	−	16.5775i	−0.171708	−	0.736232i
$508$		0			0
$509$			23.8440i			1.05687i	0.848975	+	0.528434i	$0.177220\pi$
$509$			23.8440i			1.05687i	−0.848975	+	0.528434i	$0.822780\pi$
$510$		0			0
$511$		−	7.04323i		−	0.311574i
$512$		0			0
$513$	15.8308	−	12.9964i	0.698948	−	0.573806i
$514$		0			0
$515$	10.3206	+	2.83783i	0.454778	+	0.125050i
$516$		0			0
$517$			8.52551i			0.374952i
$518$		0			0
$519$	−5.39484	−	23.1314i	−0.236807	−	1.01536i
$520$		0			0
$521$		−	15.7662i		−	0.690729i	−0.938469	−	0.345365i	$-0.887755\pi$
$521$		−	15.7662i		−	0.690729i	0.938469	−	0.345365i	$-0.112245\pi$
$522$		0			0
$523$	0.217577			0.00951397			0.00475698	−	0.999989i	$-0.498486\pi$
$523$	0.217577			0.00951397			0.00475698	+	0.999989i	$0.498486\pi$
$524$		0			0
$525$	−2.62162	+	8.25391i	−0.114417	+	0.360230i
$526$		0			0
$527$		−	9.31740i		−	0.405872i
$528$		0			0
$529$	−7.92963			−0.344767
$530$		0			0
$531$	32.3313	−	15.9485i	1.40306	−	0.692104i
$532$		0			0
$533$	18.9796			0.822097
$534$		0			0
$535$	−7.85014	+	28.5492i	−0.339391	+	1.23429i
$536$		0			0
$537$	2.47343	+	10.6053i	0.106736	+	0.457653i
$538$		0			0
$539$	−4.95259			−0.213323
$540$		0			0
$541$	11.9751			0.514849			0.257425	−	0.966298i	$-0.417126\pi$
$541$	11.9751			0.514849			0.257425	+	0.966298i	$0.417126\pi$
$542$		0			0
$543$	4.73697	+	20.3107i	0.203283	+	0.871615i
$544$		0			0
$545$	−33.5237	−	9.21796i	−1.43600	−	0.394854i
$546$		0			0
$547$	−32.0821			−1.37173			−0.685867	−	0.727727i	$-0.740577\pi$
$547$	−32.0821			−1.37173			−0.685867	+	0.727727i	$0.740577\pi$
$548$		0			0
$549$	−9.61468	+	4.74276i	−0.410344	+	0.202416i
$550$		0			0
$551$	27.7112			1.18054
$552$		0			0
$553$		−	9.28494i		−	0.394836i
$554$		0			0
$555$	36.5109	+	1.43222i	1.54980	+	0.0607943i
$556$		0			0
$557$	−1.14444			−0.0484916			−0.0242458	−	0.999706i	$-0.507718\pi$
$557$	−1.14444			−0.0484916			−0.0242458	+	0.999706i	$0.507718\pi$
$558$		0			0
$559$		−	10.9850i		−	0.464617i
$560$		0			0
$561$	−3.30515	−	14.1715i	−0.139544	−	0.598320i
$562$		0			0
$563$			30.5065i			1.28569i	0.765994	+	0.642847i	$0.222247\pi$
$563$			30.5065i			1.28569i	−0.765994	+	0.642847i	$0.777753\pi$
$564$		0			0
$565$	−43.0065	−	11.8254i	−1.80930	−	0.497499i
$566$		0			0
$567$	−5.47727	+	7.14139i	−0.230024	+	0.299910i
$568$		0			0
$569$			23.4853i			0.984557i	0.870438	+	0.492279i	$0.163836\pi$
$569$			23.4853i			0.984557i	−0.870438	+	0.492279i	$0.836164\pi$
$570$		0			0
$571$			1.22010i			0.0510597i	0.999674	+	0.0255298i	$0.00812728\pi$
$571$			1.22010i			0.0510597i	−0.999674	+	0.0255298i	$0.991873\pi$
$572$		0			0
$573$	−0.613638	−	2.63109i	−0.0256351	−	0.109915i
$574$		0			0
$575$	−14.2173	+	23.8979i	−0.592901	+	0.996612i
$576$		0			0
$577$			15.5687i			0.648135i	0.946034	+	0.324068i	$0.105051\pi$
$577$			15.5687i			0.648135i	−0.946034	+	0.324068i	$0.894949\pi$
$578$		0			0
$579$	21.9221	−	5.11280i	0.911053	−	0.212481i
$580$		0			0
$581$			2.93728i			0.121859i
$582$		0			0
$583$	32.9296			1.36381
$584$		0			0
$585$	−2.25551	−	11.7328i	−0.0932537	−	0.485091i
$586$		0			0
$587$			8.53784i			0.352394i	0.984355	+	0.176197i	$0.0563796\pi$
$587$			8.53784i			0.352394i	−0.984355	+	0.176197i	$0.943620\pi$
$588$		0			0
$589$	21.6504			0.892091
$590$		0			0
$591$	5.17770	+	22.2004i	0.212982	+	0.913203i
$592$		0			0
$593$	8.20880			0.337095			0.168547	−	0.985694i	$-0.446092\pi$
$593$	8.20880			0.337095			0.168547	+	0.985694i	$0.446092\pi$
$594$		0			0
$595$	3.65748	+	1.00569i	0.149942	+	0.0412293i
$596$		0			0
$597$	36.4182	−	8.49366i	1.49050	−	0.347623i
$598$		0			0
$599$	19.0813			0.779641			0.389820	−	0.920891i	$-0.372537\pi$
$599$	19.0813			0.779641			0.389820	+	0.920891i	$0.372537\pi$
$600$		0			0
$601$	7.90909			0.322619			0.161309	−	0.986904i	$-0.448428\pi$
$601$	7.90909			0.322619			0.161309	+	0.986904i	$0.448428\pi$
$602$		0			0
$603$	−28.9545	+	14.2828i	−1.17912	+	0.581640i
$604$		0			0
$605$	29.1673	+	8.02009i	1.18582	+	0.326063i
$606$		0			0
$607$	−40.3874			−1.63927			−0.819637	−	0.572883i	$-0.805825\pi$
$607$	−40.3874			−1.63927			−0.819637	+	0.572883i	$0.805825\pi$
$608$		0			0
$609$	−11.8582	+	2.76564i	−0.480519	+	0.112069i
$610$		0			0
$611$	3.06594			0.124034
$612$		0			0
$613$			2.34199i			0.0945921i	0.998881	+	0.0472960i	$0.0150604\pi$
$613$			2.34199i			0.0945921i	−0.998881	+	0.0472960i	$0.984940\pi$
$614$		0			0
$615$	−41.2405	−	1.61775i	−1.66298	−	0.0652339i
$616$		0			0
$617$	17.8351			0.718016			0.359008	−	0.933334i	$-0.383115\pi$
$617$	17.8351			0.718016			0.359008	+	0.933334i	$0.383115\pi$
$618$		0			0
$619$			33.8273i			1.35964i	0.733381	+	0.679818i	$0.237941\pi$
$619$			33.8273i			1.35964i	−0.733381	+	0.679818i	$0.762059\pi$
$620$		0			0
$621$	−22.3355	+	18.3365i	−0.896292	+	0.735817i
$622$		0			0
$623$		−	2.11859i		−	0.0848796i
$624$		0			0
$625$	11.9296	+	21.9701i	0.477185	+	0.878803i
$626$		0			0
$627$	32.9296	−	7.68003i	1.31508	−	0.306711i
$628$		0			0
$629$		−	16.0042i		−	0.638130i
$630$		0			0
$631$		−	32.4751i		−	1.29281i	−0.762993	−	0.646407i	$-0.776271\pi$
$631$		−	32.4751i		−	1.29281i	0.762993	−	0.646407i	$-0.223729\pi$
$632$		0			0
$633$	−39.4225	+	9.19433i	−1.56690	+	0.365442i
$634$		0			0
$635$	8.06652	+	2.21804i	0.320110	+	0.0880202i
$636$		0			0
$637$			1.78105i			0.0705676i
$638$		0			0
$639$	−6.28731	+	3.10142i	−0.248722	+	0.122690i
$640$		0			0
$641$		−	31.5028i		−	1.24429i	−0.782903	−	0.622144i	$-0.786262\pi$
$641$		−	31.5028i		−	1.24429i	0.782903	−	0.622144i	$-0.213738\pi$
$642$		0			0
$643$	8.61906			0.339902			0.169951	−	0.985452i	$-0.445639\pi$
$643$	8.61906			0.339902			0.169951	+	0.985452i	$0.445639\pi$
$644$		0			0
$645$	−0.936322	+	23.8692i	−0.0368676	+	0.939849i
$646$		0			0
$647$		−	6.90047i		−	0.271286i	−0.990758	−	0.135643i	$-0.956690\pi$
$647$		−	6.90047i		−	0.271286i	0.990758	−	0.135643i	$-0.0433099\pi$
$648$		0			0
$649$	59.5150			2.33617
$650$		0			0
$651$	−9.26468	+	2.16076i	−0.363112	+	0.0846869i
$652$		0			0
$653$	−40.3149			−1.57765			−0.788823	−	0.614621i	$-0.789309\pi$
$653$	−40.3149			−1.57765			−0.788823	+	0.614621i	$0.789309\pi$
$654$		0			0
$655$	−15.8323	−	4.35337i	−0.618617	−	0.170100i
$656$		0			0
$657$	−9.34752	−	18.9496i	−0.364681	−	0.739294i
$658$		0			0
$659$	22.4741			0.875465			0.437733	−	0.899105i	$-0.355782\pi$
$659$	22.4741			0.875465			0.437733	+	0.899105i	$0.355782\pi$
$660$		0			0
$661$	39.9091			1.55228			0.776142	−	0.630558i	$-0.217174\pi$
$661$	39.9091			1.55228			0.776142	+	0.630558i	$0.217174\pi$
$662$		0			0
$663$	−5.09633	+	1.18859i	−0.197925	+	0.0461612i
$664$		0			0
$665$	−2.33688	+	8.49872i	−0.0906203	+	0.329566i
$666$		0			0
$667$	−39.0974			−1.51386
$668$		0			0
$669$	−4.47727	−	19.1972i	−0.173101	−	0.742206i
$670$		0			0
$671$	−17.6986			−0.683247
$672$		0			0
$673$			42.8501i			1.65175i	0.563853	+	0.825875i	$0.309318\pi$
$673$			42.8501i			1.65175i	−0.563853	+	0.825875i	$0.690682\pi$
$674$		0			0
$675$	3.90090	+	25.6862i	0.150146	+	0.988664i
$676$		0			0
$677$	−20.9491			−0.805138			−0.402569	−	0.915390i	$-0.631883\pi$
$677$	−20.9491			−0.805138			−0.402569	+	0.915390i	$0.631883\pi$
$678$		0			0
$679$		−	16.3282i		−	0.626618i
$680$		0			0
$681$	7.62448	−	1.77822i	0.292171	−	0.0681417i
$682$		0			0
$683$			9.00429i			0.344540i	0.985050	+	0.172270i	$0.0551101\pi$
$683$			9.00429i			0.344540i	−0.985050	+	0.172270i	$0.944890\pi$
$684$		0			0
$685$	7.02054	+	1.93043i	0.268241	+	0.0737578i
$686$		0			0
$687$	−7.68067	−	32.9324i	−0.293036	−	1.25645i
$688$		0			0
$689$		−	11.8421i		−	0.451149i
$690$		0			0
$691$		−	35.8069i		−	1.36216i	−0.732210	−	0.681079i	$-0.761511\pi$
$691$		−	35.8069i		−	1.36216i	0.732210	−	0.681079i	$-0.238489\pi$
$692$		0			0
$693$	−13.3248	+	6.57290i	−0.506167	+	0.249684i
$694$		0			0
$695$	2.80598	−	10.2048i	0.106437	−	0.387088i
$696$		0			0
$697$			18.0774i			0.684729i
$698$		0			0
$699$	0.273106	+	1.17099i	0.0103298	+	0.0442911i
$700$		0			0
$701$		−	13.0432i		−	0.492635i	−0.969189	−	0.246317i	$-0.920779\pi$
$701$		−	13.0432i		−	0.492635i	0.969189	−	0.246317i	$-0.0792206\pi$
$702$		0			0
$703$	37.1883			1.40258
$704$		0			0
$705$	−6.66193	−	0.261329i	−0.250903	−	0.00984220i
$706$		0			0
$707$			8.65206i			0.325394i
$708$		0			0
$709$	−4.14283			−0.155587			−0.0777936	−	0.996969i	$-0.524788\pi$
$709$	−4.14283			−0.155587			−0.0777936	+	0.996969i	$0.524788\pi$
$710$		0			0
$711$	−12.3226	−	24.9809i	−0.462135	−	0.936856i
$712$		0			0
$713$	−30.5463			−1.14397
$714$		0			0
$715$	5.22936	−	19.0180i	0.195567	−	0.711234i
$716$		0			0
$717$	4.00640	+	17.1782i	0.149622	+	0.641532i
$718$		0			0
$719$	1.00789			0.0375879			0.0187940	−	0.999823i	$-0.494017\pi$
$719$	1.00789			0.0375879			0.0187940	+	0.999823i	$0.494017\pi$
$720$		0			0
$721$	−4.78680			−0.178270
$722$		0			0
$723$	−2.02492	−	8.68222i	−0.0753074	−	0.322895i
$724$		0			0
$725$	−17.9717	+	30.2087i	−0.667452	+	1.12192i
$726$		0			0
$727$	17.4880			0.648594			0.324297	−	0.945955i	$-0.394872\pi$
$727$	17.4880			0.648594			0.324297	+	0.945955i	$0.394872\pi$
$728$		0			0
$729$	−5.25865	+	26.4829i	−0.194765	+	0.980850i
$730$		0			0
$731$	10.4628			0.386982
$732$		0			0
$733$			20.1890i			0.745699i	0.927892	+	0.372849i	$0.121619\pi$
$733$			20.1890i			0.745699i	−0.927892	+	0.372849i	$0.878381\pi$
$734$		0			0
$735$	0.151809	−	3.87001i	0.00559958	−	0.142747i
$736$		0			0
$737$	−53.2992			−1.96330
$738$		0			0
$739$			8.82427i			0.324606i	0.986741	+	0.162303i	$0.0518922\pi$
$739$			8.82427i			0.324606i	−0.986741	+	0.162303i	$0.948108\pi$
$740$		0			0
$741$	−2.76189	−	11.8421i	−0.101460	−	0.435031i
$742$		0			0
$743$			12.8745i			0.472318i	0.971714	+	0.236159i	$0.0758887\pi$
$743$			12.8745i			0.472318i	−0.971714	+	0.236159i	$0.924111\pi$
$744$		0			0
$745$	4.33549	−	15.7672i	0.158840	−	0.577667i
$746$		0			0
$747$	3.89825	+	7.90266i	0.142629	+	0.289143i
$748$		0			0
$749$		−	13.2415i		−	0.483833i
$750$		0			0
$751$			5.72285i			0.208830i	0.994534	+	0.104415i	$0.0332970\pi$
$751$			5.72285i			0.208830i	−0.994534	+	0.104415i	$0.966703\pi$
$752$		0			0
$753$	9.85146	+	42.2400i	0.359007	+	1.53931i
$754$		0			0
$755$	9.90518	−	36.0230i	0.360486	−	1.31101i
$756$		0			0
$757$			41.0006i			1.49019i	0.666957	+	0.745097i	$0.267597\pi$
$757$			41.0006i			1.49019i	−0.666957	+	0.745097i	$0.732403\pi$
$758$		0			0
$759$	−46.4600	+	10.8357i	−1.68639	+	0.393309i
$760$		0			0
$761$		−	27.1027i		−	0.982474i	−0.871026	−	0.491237i	$-0.836545\pi$
$761$		−	27.1027i		−	0.982474i	0.871026	−	0.491237i	$-0.163455\pi$
$762$		0			0
$763$	15.5487			0.562900
$764$		0			0
$765$	11.1751	−	2.14829i	0.404035	−	0.0776715i
$766$		0			0
$767$		−	21.4027i		−	0.772808i
$768$		0			0
$769$	2.13198			0.0768812			0.0384406	−	0.999261i	$-0.487761\pi$
$769$	2.13198			0.0768812			0.0384406	+	0.999261i	$0.487761\pi$
$770$		0			0
$771$	−9.48980	−	40.6893i	−0.341767	−	1.46539i
$772$		0			0
$773$	19.9526			0.717647			0.358823	−	0.933406i	$-0.383178\pi$
$773$	19.9526			0.717647			0.358823	+	0.933406i	$0.383178\pi$
$774$		0			0
$775$	−14.0411	+	23.6017i	−0.504370	+	0.847799i
$776$		0			0
$777$	−15.9137	+	3.71147i	−0.570899	+	0.133148i
$778$		0			0
$779$	−42.0056			−1.50501
$780$		0			0
$781$	−11.5736			−0.414136
$782$		0			0
$783$	−28.2337	+	23.1787i	−1.00899	+	0.828338i
$784$		0			0
$785$	8.84930	−	32.1830i	0.315845	−	1.14866i
$786$		0			0
$787$	45.7933			1.63235			0.816177	−	0.577802i	$-0.196089\pi$
$787$	45.7933			1.63235			0.816177	+	0.577802i	$0.196089\pi$
$788$		0			0
$789$	27.5238	−	6.41925i	0.979872	−	0.228531i
$790$		0			0
$791$	19.9469			0.709231
$792$		0			0
$793$			6.36475i			0.226019i
$794$		0			0
$795$	−1.00938	+	25.7316i	−0.0357989	+	0.912605i
$796$		0			0
$797$	−14.5441			−0.515178			−0.257589	−	0.966255i	$-0.582928\pi$
$797$	−14.5441			−0.515178			−0.257589	+	0.966255i	$0.582928\pi$
$798$		0			0
$799$			2.92019i			0.103309i
$800$		0			0
$801$	−2.81172	−	5.70001i	−0.0993472	−	0.201400i
$802$		0			0
$803$		−	34.8822i		−	1.23097i
$804$		0			0
$805$	3.29707	−	11.9907i	0.116206	−	0.422618i
$806$		0			0
$807$	−18.6959	+	4.36036i	−0.658127	+	0.153492i
$808$		0			0
$809$			4.96538i			0.174574i	0.996183	+	0.0872868i	$0.0278196\pi$
$809$			4.96538i			0.174574i	−0.996183	+	0.0872868i	$0.972180\pi$
$810$		0			0
$811$		−	38.6095i		−	1.35576i	−0.735171	−	0.677882i	$-0.762898\pi$
$811$		−	38.6095i		−	1.35576i	0.735171	−	0.677882i	$-0.237102\pi$
$812$		0			0
$813$	47.1005	−	10.9850i	1.65189	−	0.385262i
$814$		0			0
$815$	7.34319	+	2.01915i	0.257221	+	0.0707276i
$816$		0			0
$817$			24.3121i			0.850571i
$818$		0			0
$819$	2.36374	+	4.79185i	0.0825957	+	0.167441i
$820$		0			0
$821$			36.8173i			1.28493i	0.766314	+	0.642466i	$0.222089\pi$
$821$			36.8173i			1.28493i	−0.766314	+	0.642466i	$0.777911\pi$
$822$		0			0
$823$	−6.88856			−0.240120			−0.120060	−	0.992767i	$-0.538309\pi$
$823$	−6.88856			−0.240120			−0.120060	+	0.992767i	$0.538309\pi$
$824$		0			0
$825$	−12.9838	+	40.8783i	−0.452038	+	1.42320i
$826$		0			0
$827$		−	50.2814i		−	1.74846i	−0.485516	−	0.874228i	$-0.661368\pi$
$827$		−	50.2814i		−	1.74846i	0.485516	−	0.874228i	$-0.338632\pi$
$828$		0			0
$829$	−0.893868			−0.0310453			−0.0155227	−	0.999880i	$-0.504941\pi$
$829$	−0.893868			−0.0310453			−0.0155227	+	0.999880i	$0.504941\pi$
$830$		0			0
$831$	−41.2285	+	9.61555i	−1.43020	+	0.333560i
$832$		0			0
$833$	−1.69638			−0.0587761
$834$		0			0
$835$	−6.00000	+	21.8207i	−0.207639	+	0.755136i
$836$		0			0
$837$	−22.0587	+	18.1092i	−0.762460	+	0.625946i
$838$		0			0
$839$	41.8399			1.44447			0.722237	−	0.691645i	$-0.243114\pi$
$839$	41.8399			1.44447			0.722237	+	0.691645i	$0.243114\pi$
$840$		0			0
$841$	−20.4220			−0.704208
$842$		0			0
$843$	−42.6201	+	9.94010i	−1.46791	+	0.342355i
$844$		0			0
$845$	21.1894	+	5.82640i	0.728936	+	0.200434i
$846$		0			0
$847$	−13.5282			−0.464833
$848$		0			0
$849$	−10.1526	−	43.5314i	−0.348437	−	1.49399i
$850$		0			0
$851$	−52.4684			−1.79859
$852$		0			0
$853$			36.8969i			1.26333i	0.775243	+	0.631663i	$0.217627\pi$
$853$			36.8969i			1.26333i	−0.775243	+	0.631663i	$0.782373\pi$
$854$		0			0
$855$	4.99188	+	25.9670i	0.170719	+	0.888052i
$856$		0			0
$857$	32.2964			1.10322			0.551612	−	0.834101i	$-0.314013\pi$
$857$	32.2964			1.10322			0.551612	+	0.834101i	$0.314013\pi$
$858$		0			0
$859$		−	21.2916i		−	0.726460i	−0.931700	−	0.363230i	$-0.881674\pi$
$859$		−	21.2916i		−	0.726460i	0.931700	−	0.363230i	$-0.118326\pi$
$860$		0			0
$861$	17.9751	−	4.19225i	0.612589	−	0.142871i
$862$		0			0
$863$			11.4360i			0.389286i	0.980874	+	0.194643i	$0.0623548\pi$
$863$			11.4360i			0.389286i	−0.980874	+	0.194643i	$0.937645\pi$
$864$		0			0
$865$	29.5666	+	8.12987i	1.00529	+	0.276424i
$866$		0			0
$867$	5.55572	+	23.8212i	0.188682	+	0.809012i
$868$		0			0
$869$		−	45.9845i		−	1.55992i
$870$		0			0
$871$			19.1674i			0.649463i
$872$		0			0
$873$	−21.6702	−	43.9305i	−0.733424	−	1.48682i
$874$		0			0
$875$	−7.74913	−	8.05921i	−0.261969	−	0.272451i
$876$		0			0
$877$		−	33.8764i		−	1.14393i	−0.820279	−	0.571963i	$-0.806182\pi$
$877$		−	33.8764i		−	1.14393i	0.820279	−	0.571963i	$-0.193818\pi$
$878$		0			0
$879$	−2.50602	−	10.7450i	−0.0845259	−	0.362421i
$880$		0			0
$881$			51.0462i			1.71979i	0.510471	+	0.859895i	$0.329471\pi$
$881$			51.0462i			1.71979i	−0.510471	+	0.859895i	$0.670529\pi$
$882$		0			0
$883$	−2.62990			−0.0885033			−0.0442517	−	0.999020i	$-0.514090\pi$
$883$	−2.62990			−0.0885033			−0.0442517	+	0.999020i	$0.514090\pi$
$884$		0			0
$885$	−1.82429	+	46.5057i	−0.0613227	+	1.56327i
$886$		0			0
$887$			47.8561i			1.60685i	0.595405	+	0.803426i	$0.296992\pi$
$887$			47.8561i			1.60685i	−0.595405	+	0.803426i	$0.703008\pi$
$888$		0			0
$889$	−3.74135			−0.125481
$890$		0			0
$891$	−27.1267	+	35.3684i	−0.908778	+	1.18489i
$892$		0			0
$893$	−6.78552			−0.227069
$894$		0			0
$895$	−13.5557	−	3.72740i	−0.453118	−	0.124593i
$896$		0			0
$897$	3.89671	+	16.7079i	0.130107	+	0.557860i
$898$		0			0
$899$	−38.6128			−1.28781
$900$		0			0
$901$	11.2792			0.375764
$902$		0			0
$903$	−2.42640	−	10.4036i	−0.0807454	−	0.346212i
$904$		0			0
$905$	−25.9611	−	7.13849i	−0.862977	−	0.237291i
$906$		0			0
$907$	−31.9502			−1.06089			−0.530444	−	0.847720i	$-0.677975\pi$
$907$	−31.9502			−1.06089			−0.530444	+	0.847720i	$0.677975\pi$
$908$		0			0
$909$	11.4827	+	23.2781i	0.380857	+	0.772086i
$910$		0			0
$911$	51.9702			1.72185			0.860925	−	0.508732i	$-0.169885\pi$
$911$	51.9702			1.72185			0.860925	+	0.508732i	$0.169885\pi$
$912$		0			0
$913$			14.5471i			0.481440i
$914$		0			0
$915$	0.542507	−	13.8299i	0.0179347	−	0.457201i
$916$		0			0
$917$	7.34319			0.242493
$918$		0			0
$919$		−	23.3714i		−	0.770951i	−0.922718	−	0.385476i	$-0.874037\pi$
$919$		−	23.3714i		−	0.770951i	0.922718	−	0.385476i	$-0.125963\pi$
$920$		0			0
$921$	8.24354	+	35.3458i	0.271634	+	1.16468i
$922$		0			0
$923$			4.16209i			0.136997i
$924$		0			0
$925$	−24.1179	+	40.5400i	−0.792992	+	1.33295i
$926$		0			0
$927$	−12.8788	+	6.35287i	−0.422994	+	0.208656i
$928$		0			0
$929$			25.3576i			0.831955i	0.909375	+	0.415978i	$0.136561\pi$
$929$			25.3576i			0.831955i	−0.909375	+	0.415978i	$0.863439\pi$
$930$		0			0
$931$		−	3.94181i		−	0.129187i
$932$		0			0
$933$	11.2936	+	48.4236i	0.369736	+	1.58532i
$934$		0			0
$935$	18.1140	+	4.98077i	0.592390	+	0.162889i
$936$		0			0
$937$		−	23.4524i		−	0.766155i	−0.923716	−	0.383077i	$-0.874864\pi$
$937$		−	23.4524i		−	0.766155i	0.923716	−	0.383077i	$-0.125136\pi$
$938$		0			0
$939$	−15.5251	+	3.62086i	−0.506643	+	0.118162i
$940$		0			0
$941$			17.0513i			0.555857i	0.960602	+	0.277929i	$0.0896479\pi$
$941$			17.0513i			0.555857i	−0.960602	+	0.277929i	$0.910352\pi$
$942$		0			0
$943$	59.2651			1.92994
$944$		0			0
$945$	−4.72769	−	10.6136i	−0.153792	−	0.345261i
$946$		0			0
$947$			52.1177i			1.69360i	0.531914	+	0.846798i	$0.321473\pi$
$947$			52.1177i			1.69360i	−0.531914	+	0.846798i	$0.678527\pi$
$948$		0			0
$949$	−12.5443			−0.407206
$950$		0			0
$951$	7.23575	+	31.0247i	0.234635	+	1.00604i
$952$		0			0
$953$	17.3849			0.563153			0.281576	−	0.959539i	$-0.409143\pi$
$953$	17.3849			0.563153			0.281576	+	0.959539i	$0.409143\pi$
$954$		0			0
$955$	3.36306	+	0.924736i	0.108826	+	0.0299237i
$956$		0			0
$957$	−58.7289	+	13.6971i	−1.89844	+	0.442764i
$958$		0			0
$959$	−3.25621			−0.105148
$960$		0			0
$961$	0.832255			0.0268469
$962$		0			0
$963$	−17.5736	−	35.6258i	−0.566301	−	1.14803i
$964$		0			0
$965$	−7.70485	+	28.0209i	−0.248028	+	0.902024i
$966$		0			0
$967$	33.7771			1.08620			0.543099	−	0.839668i	$-0.317251\pi$
$967$	33.7771			1.08620			0.543099	+	0.839668i	$0.317251\pi$
$968$		0			0
$969$	11.2792	−	2.63060i	0.362340	−	0.0845069i
$970$		0			0
$971$	14.1287			0.453412			0.226706	−	0.973963i	$-0.427204\pi$
$971$	14.1287			0.453412			0.226706	+	0.973963i	$0.427204\pi$
$972$		0			0
$973$			4.73309i			0.151736i
$974$		0			0
$975$	14.7006	+	4.66923i	0.470796	+	0.149535i
$976$		0			0
$977$	57.5633			1.84161			0.920807	−	0.390019i	$-0.127532\pi$
$977$	57.5633			1.84161			0.920807	+	0.390019i	$0.127532\pi$
$978$		0			0
$979$		−	10.4925i		−	0.335342i
$980$		0			0
$981$	41.8333	−	20.6357i	1.33563	−	0.658846i
$982$		0			0
$983$		−	13.0124i		−	0.415032i	−0.978232	−	0.207516i	$-0.933462\pi$
$983$		−	13.0124i		−	0.415032i	0.978232	−	0.207516i	$-0.0665379\pi$
$984$		0			0
$985$	−28.3766	−	7.80266i	−0.904153	−	0.248613i
$986$		0			0
$987$	2.90367	−	0.677210i	0.0924248	−	0.0215558i
$988$		0			0
$989$		−	34.3016i		−	1.09073i
$990$		0			0
$991$		−	21.6713i		−	0.688412i	−0.938894	−	0.344206i	$-0.888148\pi$
$991$		−	21.6713i		−	0.688412i	0.938894	−	0.344206i	$-0.111852\pi$
$992$		0			0
$993$	29.7693	−	6.94295i	0.944699	−	0.220328i
$994$		0			0
$995$	−12.7997	+	46.5498i	−0.405779	+	1.47573i
$996$		0			0
$997$			62.7085i			1.98600i	0.118121	+	0.992999i	$0.462313\pi$
$997$			62.7085i			1.98600i	−0.118121	+	0.992999i	$0.537687\pi$
$998$		0			0
$999$	−37.8895	+	31.1056i	−1.19877	+	0.984139i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1680.2.v.a.239.6	yes	12
3.2	odd	2	inner	1680.2.v.a.239.7	yes	12
4.3	odd	2		1680.2.v.b.239.8	yes	12
5.4	even	2		1680.2.v.b.239.7	yes	12
12.11	even	2		1680.2.v.b.239.5	yes	12
15.14	odd	2		1680.2.v.b.239.6	yes	12
20.19	odd	2	inner	1680.2.v.a.239.5	✓	12
60.59	even	2	inner	1680.2.v.a.239.8	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1680.2.v.a.239.5	✓	12	20.19	odd	2	inner
1680.2.v.a.239.6	yes	12	1.1	even	1	trivial
1680.2.v.a.239.7	yes	12	3.2	odd	2	inner
1680.2.v.a.239.8	yes	12	60.59	even	2	inner
1680.2.v.b.239.5	yes	12	12.11	even	2
1680.2.v.b.239.6	yes	12	15.14	odd	2
1680.2.v.b.239.7	yes	12	5.4	even	2
1680.2.v.b.239.8	yes	12	4.3	odd	2

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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)

\(f(q)\)	\(=\)	\(q+(-0.393401 - 1.68678i) q^{3} +(2.15605 + 0.592845i) q^{5} -1.00000 q^{7} +(-2.69047 + 1.32716i) q^{9} -4.95259 q^{11} +1.78105i q^{13} +(0.151809 - 3.87001i) q^{15} -1.69638 q^{17} -3.94181i q^{19} +(0.393401 + 1.68678i) q^{21} +5.56144i q^{23} +(4.29707 + 2.55640i) q^{25} +(3.29707 + 4.01613i) q^{27} +7.03008i q^{29} +5.49252i q^{31} +(1.94835 + 8.35394i) q^{33} +(-2.15605 - 0.592845i) q^{35} +9.43432i q^{37} +(3.00424 - 0.700665i) q^{39} -10.6564i q^{41} -6.16774 q^{43} +(-6.58758 + 1.26639i) q^{45} -1.72142i q^{47} +1.00000 q^{49} +(0.667358 + 2.86143i) q^{51} -6.64897 q^{53} +(-10.6780 - 2.93612i) q^{55} +(-6.64897 + 1.55071i) q^{57} -12.0169 q^{59} +3.57360 q^{61} +(2.69047 - 1.32716i) q^{63} +(-1.05588 + 3.84002i) q^{65} +10.7619 q^{67} +(9.38094 - 2.18788i) q^{69} +2.33688 q^{71} +7.04323i q^{73} +(2.62162 - 8.25391i) q^{75} +4.95259 q^{77} +9.28494i q^{79} +(5.47727 - 7.14139i) q^{81} -2.93728i q^{83} +(-3.65748 - 1.00569i) q^{85} +(11.8582 - 2.76564i) q^{87} +2.11859i q^{89} -1.78105i q^{91} +(9.26468 - 2.16076i) q^{93} +(2.33688 - 8.49872i) q^{95} +16.3282i q^{97} +(13.3248 - 6.57290i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 2 q^{3} - 12 q^{7} + 6 q^{9} + 12 q^{15} + 2 q^{21} + 16 q^{25} + 4 q^{27} + 8 q^{43} - 4 q^{45} + 12 q^{49} - 16 q^{55} + 32 q^{61} - 6 q^{63} - 24 q^{67} + 36 q^{69} + 18 q^{75} + 22 q^{81} + 8 q^{85}+ \cdots + 22 q^{87}+O(q^{100}) \) 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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