LMFDB - Embedded newform 2268.2.i.l.865.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2268,2,Mod(865,2268)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2268, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 4, 4]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("2268.865");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2268 = 2^{2} \cdot 3^{4} \cdot 7 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2268.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$18.1100711784$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.310217769.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 4x^{6} - 2x^{5} + 15x^{4} - 4x^{3} + 5x^{2} + x + 1 $ x^8 + 4x^6 - 2x^5 + 15x^4 - 4x^3 + 5*x^2 + x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			865.3
Root			$-0.198169 + 0.343239i$ of defining polynomial
Character	$\chi$	$=$	2268.865
Dual form			2268.2.i.l.2053.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.705299 - 1.22161i) q^{5} +(-0.779537 + 2.52830i) q^{7} +O(q^{10})$	$q+(0.705299 - 1.22161i) q^{5} +(-0.779537 + 2.52830i) q^{7} +(2.48484 + 4.30386i) q^{11} +(1.48484 + 2.57181i) q^{13} +(-1.29981 + 2.25133i) q^{17} +(-3.76437 - 6.52009i) q^{19} +(-4.06418 + 7.03937i) q^{23} +(1.50511 + 2.60692i) q^{25} +(3.46967 - 6.00965i) q^{29} -10.5390 q^{31} +(2.53880 + 2.73550i) q^{35} +(-0.0945078 - 0.163692i) q^{37} +(-1.02027 - 1.76716i) q^{41} +(2.19014 - 3.79343i) q^{43} -9.38027 q^{47} +(-5.78464 - 3.94181i) q^{49} +(-6.95339 + 12.0436i) q^{53} +7.01021 q^{55} -6.78863 q^{59} +5.89657 q^{61} +4.18902 q^{65} +10.0125 q^{67} -7.86174 q^{71} +(0.894315 - 1.54900i) q^{73} +(-12.8185 + 2.92740i) q^{77} +1.70755 q^{79} +(0.0100574 - 0.0174200i) q^{83} +(1.83351 + 3.17572i) q^{85} +(7.33366 + 12.7023i) q^{89} +(-7.65981 + 1.74929i) q^{91} -10.6200 q^{95} +(-4.33636 + 7.51080i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 2 q^{5} + q^{7}+O(q^{10}) $ 8 * q - 2 * q^5 + q^7	$ 8 q - 2 q^{5} + q^{7} + 5 q^{11} - 3 q^{13} + 2 q^{17} - 8 q^{19} + 2 q^{23} - 8 q^{25} - 2 q^{29} + 11 q^{35} + 4 q^{37} - 3 q^{41} - 5 q^{43} - 30 q^{47} - 19 q^{49} - 24 q^{53} + 16 q^{55} - 20 q^{59} + 24 q^{61} + 24 q^{65} + 14 q^{67} - 22 q^{71} - 10 q^{73} - 11 q^{77} + 35 q^{83} + 13 q^{85} - 18 q^{89} - 9 q^{91} + 20 q^{95} - 19 q^{97}+O(q^{100}) $ 8 * q - 2 * q^5 + q^7 + 5 * q^11 - 3 * q^13 + 2 * q^17 - 8 * q^19 + 2 * q^23 - 8 * q^25 - 2 * q^29 + 11 * q^35 + 4 * q^37 - 3 * q^41 - 5 * q^43 - 30 * q^47 - 19 * q^49 - 24 * q^53 + 16 * q^55 - 20 * q^59 + 24 * q^61 + 24 * q^65 + 14 * q^67 - 22 * q^71 - 10 * q^73 - 11 * q^77 + 35 * q^83 + 13 * q^85 - 18 * q^89 - 9 * q^91 + 20 * q^95 - 19 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2268\mathbb{Z}\right)^\times$.

$n$	$325$	$1135$	$1541$
$\chi(n)$	$e\left(\frac{2}{3}\right)$	$1$	$e\left(\frac{2}{3}\right)$

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			0
$3$		0			0
$4$		0			0
$5$	0.705299	−	1.22161i	0.315419	−	0.546322i	−0.664107	−	0.747637i	$-0.731188\pi$
$5$	0.705299	−	1.22161i	0.315419	−	0.546322i	0.979527	+	0.201315i	$0.0645215\pi$
$6$		0			0
$7$	−0.779537	+	2.52830i	−0.294637	+	0.955609i
$7$	−0.779537	+	2.52830i	−0.294637	+	0.955609i
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.48484	+	4.30386i	0.749206	+	1.29766i	0.948204	+	0.317663i	$0.102898\pi$
$11$	2.48484	+	4.30386i	0.749206	+	1.29766i	−0.198997	+	0.980000i	$0.563768\pi$
$12$		0			0
$13$	1.48484	+	2.57181i	0.411820	+	0.713292i	0.995089	−	0.0989865i	$-0.0315601\pi$
$13$	1.48484	+	2.57181i	0.411820	+	0.713292i	−0.583269	+	0.812279i	$0.698227\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.29981	+	2.25133i	−0.315249	+	0.546028i	−0.979491	−	0.201490i	$-0.935422\pi$
$17$	−1.29981	+	2.25133i	−0.315249	+	0.546028i	0.664241	+	0.747518i	$0.268755\pi$
$18$		0			0
$19$	−3.76437	−	6.52009i	−0.863607	−	1.49581i	−0.868424	−	0.495822i	$-0.834867\pi$
$19$	−3.76437	−	6.52009i	−0.863607	−	1.49581i	0.00481762	−	0.999988i	$-0.498466\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.06418	+	7.03937i	−0.847440	+	1.46781i	0.0360448	+	0.999350i	$0.488524\pi$
$23$	−4.06418	+	7.03937i	−0.847440	+	1.46781i	−0.883485	+	0.468459i	$0.844809\pi$
$24$		0			0
$25$	1.50511	+	2.60692i	0.301021	+	0.521384i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.46967	−	6.00965i	0.644302	−	1.11596i	−0.340160	−	0.940368i	$-0.610481\pi$
$29$	3.46967	−	6.00965i	0.644302	−	1.11596i	0.984462	−	0.175596i	$-0.0561854\pi$
$30$		0			0
$31$	−10.5390			−1.89285			−0.946427	−	0.322919i	$-0.895336\pi$
$31$	−10.5390			−1.89285			−0.946427	+	0.322919i	$0.895336\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.53880	+	2.73550i	0.429136	+	0.462385i
$36$		0			0
$37$	−0.0945078	−	0.163692i	−0.0155370	−	0.0269108i	0.858152	−	0.513395i	$-0.171612\pi$
$37$	−0.0945078	−	0.163692i	−0.0155370	−	0.0269108i	−0.873689	+	0.486484i	$0.838279\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−1.02027	−	1.76716i	−0.159339	−	0.275984i	0.775291	−	0.631604i	$-0.217603\pi$
$41$	−1.02027	−	1.76716i	−0.159339	−	0.275984i	−0.934631	+	0.355620i	$0.884270\pi$
$42$		0			0
$43$	2.19014	−	3.79343i	0.333993	−	0.578492i	−0.649298	−	0.760534i	$-0.724937\pi$
$43$	2.19014	−	3.79343i	0.333993	−	0.578492i	0.983291	+	0.182042i	$0.0582706\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−9.38027			−1.36825			−0.684127	−	0.729363i	$-0.739816\pi$
$47$	−9.38027			−1.36825			−0.684127	+	0.729363i	$0.739816\pi$
$48$		0			0
$49$	−5.78464	−	3.94181i	−0.826378	−	0.563116i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.95339	+	12.0436i	−0.955121	+	1.65432i	−0.221032	+	0.975267i	$0.570942\pi$
$53$	−6.95339	+	12.0436i	−0.955121	+	1.65432i	−0.734090	+	0.679052i	$0.762391\pi$
$54$		0			0
$55$	7.01021			0.945257
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−6.78863			−0.883804			−0.441902	−	0.897063i	$-0.645696\pi$
$59$	−6.78863			−0.883804			−0.441902	+	0.897063i	$0.645696\pi$
$60$		0			0
$61$	5.89657			0.754978			0.377489	−	0.926014i	$-0.376788\pi$
$61$	5.89657			0.754978			0.377489	+	0.926014i	$0.376788\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.18902			0.519583
$66$		0			0
$67$	10.0125			1.22322			0.611608	−	0.791161i	$-0.290523\pi$
$67$	10.0125			1.22322			0.611608	+	0.791161i	$0.290523\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−7.86174			−0.933016			−0.466508	−	0.884517i	$-0.654488\pi$
$71$	−7.86174			−0.933016			−0.466508	+	0.884517i	$0.654488\pi$
$72$		0			0
$73$	0.894315	−	1.54900i	0.104672	−	0.181297i	−0.808932	−	0.587902i	$-0.799954\pi$
$73$	0.894315	−	1.54900i	0.104672	−	0.181297i	0.913604	+	0.406605i	$0.133288\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.8185	+	2.92740i	−1.46080	+	0.333608i
$78$		0			0
$79$	1.70755			0.192114			0.0960572	−	0.995376i	$-0.469377\pi$
$79$	1.70755			0.192114			0.0960572	+	0.995376i	$0.469377\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.0100574	−	0.0174200i	0.00110395	−	0.00191209i	−0.865473	−	0.500956i	$-0.832982\pi$
$83$	0.0100574	−	0.0174200i	0.00110395	−	0.00191209i	0.866577	+	0.499044i	$0.166315\pi$
$84$		0			0
$85$	1.83351	+	3.17572i	0.198872	+	0.344456i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	7.33366	+	12.7023i	0.777366	+	1.34644i	0.933455	+	0.358695i	$0.116778\pi$
$89$	7.33366	+	12.7023i	0.777366	+	1.34644i	−0.156088	+	0.987743i	$0.549888\pi$
$90$		0			0
$91$	−7.65981	+	1.74929i	−0.802966	+	0.183376i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−10.6200			−1.08959
$96$		0			0
$97$	−4.33636	+	7.51080i	−0.440291	+	0.762606i	−0.997711	−	0.0676247i	$-0.978458\pi$
$97$	−4.33636	+	7.51080i	−0.440291	+	0.762606i	0.557420	+	0.830231i	$0.311791\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	5.56418	+	9.63744i	0.553657	+	0.958961i	0.998007	+	0.0631082i	$0.0201013\pi$
$101$	5.56418	+	9.63744i	0.553657	+	0.958961i	−0.444350	+	0.895853i	$0.646565\pi$
$102$		0			0
$103$	−5.40054	+	9.35401i	−0.532131	+	0.921678i	0.467165	+	0.884170i	$0.345275\pi$
$103$	−5.40054	+	9.35401i	−0.532131	+	0.921678i	−0.999296	+	0.0375081i	$0.988058\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	8.21776	+	14.2336i	0.794441	+	1.37601i	0.923193	+	0.384336i	$0.125569\pi$
$107$	8.21776	+	14.2336i	0.794441	+	1.37601i	−0.128752	+	0.991677i	$0.541097\pi$
$108$		0			0
$109$	5.77459	−	10.0019i	0.553105	−	0.958006i	−0.444943	−	0.895559i	$-0.646776\pi$
$109$	5.77459	−	10.0019i	0.553105	−	0.958006i	0.998048	−	0.0624472i	$-0.0198905\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	0.590522	+	1.02281i	0.0555516	+	0.0962182i	0.892464	−	0.451119i	$-0.148975\pi$
$113$	0.590522	+	1.02281i	0.0555516	+	0.0962182i	−0.836912	+	0.547337i	$0.815642\pi$
$114$		0			0
$115$	5.73293	+	9.92972i	0.534598	+	0.925951i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−4.67880	−	5.04130i	−0.428905	−	0.462136i
$120$		0			0
$121$	−6.84882	+	11.8625i	−0.622620	+	1.07841i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	11.2992			1.01063
$126$		0			0
$127$	9.16890			0.813608			0.406804	−	0.913515i	$-0.366643\pi$
$127$	9.16890			0.813608			0.406804	+	0.913515i	$0.366643\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−3.84260	+	6.65557i	−0.335729	+	0.581500i	−0.983625	−	0.180229i	$-0.942316\pi$
$131$	−3.84260	+	6.65557i	−0.335729	+	0.581500i	0.647895	+	0.761729i	$0.275649\pi$
$132$		0			0
$133$	19.4192	−	4.43483i	1.68386	−	0.384549i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	2.38696	+	4.13433i	0.203932	+	0.353220i	0.949792	−	0.312883i	$-0.101295\pi$
$137$	2.38696	+	4.13433i	0.203932	+	0.353220i	−0.745860	+	0.666102i	$0.767961\pi$
$138$		0			0
$139$	−10.4382	−	18.0795i	−0.885359	−	1.53349i	−0.845302	−	0.534289i	$-0.820579\pi$
$139$	−10.4382	−	18.0795i	−0.885359	−	1.53349i	−0.0400568	−	0.999197i	$-0.512754\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−7.37915	+	12.7811i	−0.617076	+	1.06881i
$144$		0			0
$145$	−4.89431	−	8.47720i	−0.406451	−	0.703993i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.90054	−	10.2200i	0.483391	−	0.837258i	−0.516427	−	0.856331i	$-0.672738\pi$
$149$	5.90054	−	10.2200i	0.483391	−	0.837258i	0.999818	+	0.0190732i	$0.00607156\pi$
$150$		0			0
$151$	3.51133	+	6.08181i	0.285748	+	0.494930i	0.972790	−	0.231687i	$-0.0744246\pi$
$151$	3.51133	+	6.08181i	0.285748	+	0.494930i	−0.687042	+	0.726618i	$0.741091\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−7.43312	+	12.8745i	−0.597043	+	1.03411i
$156$		0			0
$157$	6.67722			0.532900			0.266450	−	0.963849i	$-0.414149\pi$
$157$	6.67722			0.532900			0.266450	+	0.963849i	$0.414149\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−14.6295	−	15.7629i	−1.15296	−	1.24229i
$162$		0			0
$163$	−0.169866	−	0.294216i	−0.0133049	−	0.0230448i	0.859296	−	0.511478i	$-0.170902\pi$
$163$	−0.169866	−	0.294216i	−0.0133049	−	0.0230448i	−0.872601	+	0.488433i	$0.837569\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	0.448283	+	0.776449i	0.0346892	+	0.0600834i	0.882849	−	0.469657i	$-0.155623\pi$
$167$	0.448283	+	0.776449i	0.0346892	+	0.0600834i	−0.848160	+	0.529741i	$0.822289\pi$
$168$		0			0
$169$	2.09052	−	3.62089i	0.160809	−	0.278530i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	4.83591			0.367668			0.183834	−	0.982957i	$-0.441149\pi$
$173$	4.83591			0.367668			0.183834	+	0.982957i	$0.441149\pi$
$174$		0			0
$175$	−7.76437	+	1.77317i	−0.586931	+	0.134039i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−1.42464	+	2.46755i	−0.106483	+	0.184434i	−0.914343	−	0.404941i	$-0.867292\pi$
$179$	−1.42464	+	2.46755i	−0.106483	+	0.184434i	0.807860	+	0.589374i	$0.200626\pi$
$180$		0			0
$181$	1.46104			0.108598			0.0542991	−	0.998525i	$-0.482708\pi$
$181$	1.46104			0.108598			0.0542991	+	0.998525i	$0.482708\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.266625			−0.0196027
$186$		0			0
$187$	−12.9192			−0.944748
$188$		0			0
$189$		0			0
$190$		0			0
$191$	4.58143			0.331501			0.165750	−	0.986168i	$-0.446995\pi$
$191$	4.58143			0.331501			0.165750	+	0.986168i	$0.446995\pi$
$192$		0			0
$193$	−13.5840			−0.977798			−0.488899	−	0.872340i	$-0.662601\pi$
$193$	−13.5840			−0.977798			−0.488899	+	0.872340i	$0.662601\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−13.3905			−0.954032			−0.477016	−	0.878894i	$-0.658282\pi$
$197$	−13.3905			−0.954032			−0.477016	+	0.878894i	$0.658282\pi$
$198$		0			0
$199$	0.385222	−	0.667225i	0.0273077	−	0.0472983i	−0.852049	−	0.523463i	$-0.824640\pi$
$199$	0.385222	−	0.667225i	0.0273077	−	0.0472983i	0.879356	+	0.476164i	$0.157973\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	12.4895	+	13.4571i	0.876590	+	0.944506i
$204$		0			0
$205$	−2.87838			−0.201035
$206$		0			0
$207$		0			0
$208$		0			0
$209$	18.7077	−	32.4027i	1.29404	−	2.24134i
$210$		0			0
$211$	−2.28576	−	3.95906i	−0.157358	−	0.272553i	0.776557	−	0.630047i	$-0.216964\pi$
$211$	−2.28576	−	3.95906i	−0.157358	−	0.272553i	−0.933915	+	0.357494i	$0.883631\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.08940	−	5.35100i	−0.210695	−	0.364935i
$216$		0			0
$217$	8.21551	−	26.6457i	0.557705	−	1.80883i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−7.72000			−0.519304
$222$		0			0
$223$	−5.78352	+	10.0174i	−0.387293	+	0.670812i	−0.992084	−	0.125572i	$-0.959923\pi$
$223$	−5.78352	+	10.0174i	−0.387293	+	0.670812i	0.604791	+	0.796384i	$0.293257\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−2.90054	−	5.02388i	−0.192516	−	0.333447i	0.753568	−	0.657370i	$-0.228331\pi$
$227$	−2.90054	−	5.02388i	−0.192516	−	0.333447i	−0.946083	+	0.323923i	$0.894998\pi$
$228$		0			0
$229$	−6.43087	+	11.1386i	−0.424964	+	0.736059i	−0.996417	−	0.0845758i	$-0.973046\pi$
$229$	−6.43087	+	11.1386i	−0.424964	+	0.736059i	0.571453	+	0.820635i	$0.306380\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	6.33636	+	10.9749i	0.415109	+	0.718989i	0.995440	−	0.0953912i	$-0.0304102\pi$
$233$	6.33636	+	10.9749i	0.415109	+	0.718989i	−0.580331	+	0.814381i	$0.697077\pi$
$234$		0			0
$235$	−6.61590	+	11.4591i	−0.431574	+	0.747507i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	4.14352	+	7.17680i	0.268022	+	0.464228i	0.968351	−	0.249592i	$-0.0802966\pi$
$239$	4.14352	+	7.17680i	0.268022	+	0.464228i	−0.700329	+	0.713821i	$0.746963\pi$
$240$		0			0
$241$	−8.50848	−	14.7371i	−0.548079	−	0.949301i	−0.998406	−	0.0564373i	$-0.982026\pi$
$241$	−8.50848	−	14.7371i	−0.548079	−	0.949301i	0.450327	−	0.892864i	$-0.351307\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−8.89528	+	4.28644i	−0.568299	+	0.273851i
$246$		0			0
$247$	11.1790	−	19.3625i	0.711300	−	1.23201i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	13.3228			0.840926			0.420463	−	0.907310i	$-0.361868\pi$
$251$	13.3228			0.840926			0.420463	+	0.907310i	$0.361868\pi$
$252$		0			0
$253$	−40.3953			−2.53963
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−0.196207	+	0.339840i	−0.0122390	+	0.0211987i	−0.872080	−	0.489363i	$-0.837229\pi$
$257$	−0.196207	+	0.339840i	−0.0122390	+	0.0211987i	0.859841	+	0.510562i	$0.170563\pi$
$258$		0			0
$259$	0.487536	−	0.111340i	0.0302940	−	0.00691834i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	12.4342	+	21.5367i	0.766728	+	1.32801i	0.939328	+	0.343020i	$0.111450\pi$
$263$	12.4342	+	21.5367i	0.766728	+	1.32801i	−0.172600	+	0.984992i	$0.555217\pi$
$264$		0			0
$265$	9.80844	+	16.9887i	0.602528	+	1.04361i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−1.60073	+	2.77255i	−0.0975985	+	0.169046i	−0.910690	−	0.413090i	$-0.864449\pi$
$269$	−1.60073	+	2.77255i	−0.0975985	+	0.169046i	0.813092	+	0.582136i	$0.197783\pi$
$270$		0			0
$271$	12.2443	+	21.2077i	0.743786	+	1.28827i	0.950760	+	0.309928i	$0.100305\pi$
$271$	12.2443	+	21.2077i	0.743786	+	1.28827i	−0.206974	+	0.978346i	$0.566362\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−7.47989	+	12.9555i	−0.451054	+	0.781249i
$276$		0			0
$277$	−4.42576	−	7.66564i	−0.265918	−	0.460584i	0.701886	−	0.712290i	$-0.252342\pi$
$277$	−4.42576	−	7.66564i	−0.265918	−	0.460584i	−0.967804	+	0.251706i	$0.919008\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−8.47079	+	14.6718i	−0.505325	+	0.875249i	0.494656	+	0.869089i	$0.335294\pi$
$281$	−8.47079	+	14.6718i	−0.505325	+	0.875249i	−0.999981	+	0.00615989i	$0.998039\pi$
$282$		0			0
$283$	14.5390			0.864251			0.432126	−	0.901813i	$-0.357764\pi$
$283$	14.5390			0.864251			0.432126	+	0.901813i	$0.357764\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	5.26325	−	1.20199i	0.310680	−	0.0709510i
$288$		0			0
$289$	5.12100	+	8.86984i	0.301236	+	0.521755i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−12.9619	−	22.4506i	−0.757240	−	1.31158i	−0.944253	−	0.329221i	$-0.893214\pi$
$293$	−12.9619	−	22.4506i	−0.757240	−	1.31158i	0.187012	−	0.982358i	$-0.440120\pi$
$294$		0			0
$295$	−4.78801	+	8.29308i	−0.278769	+	0.482842i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−24.1386			−1.39597
$300$		0			0
$301$	7.88364	+	8.49445i	0.454406	+	0.489612i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.15884	−	7.20333i	0.238135	−	0.412461i
$306$		0			0
$307$	13.1204			0.748820			0.374410	−	0.927263i	$-0.377845\pi$
$307$	13.1204			0.748820			0.374410	+	0.927263i	$0.377845\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−8.30490			−0.470928			−0.235464	−	0.971883i	$-0.575661\pi$
$311$	−8.30490			−0.470928			−0.235464	+	0.971883i	$0.575661\pi$
$312$		0			0
$313$	23.1587			1.30901			0.654503	−	0.756059i	$-0.272878\pi$
$313$	23.1587			1.30901			0.654503	+	0.756059i	$0.272878\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	11.7155			0.658009			0.329005	−	0.944328i	$-0.393287\pi$
$317$	11.7155			0.658009			0.329005	+	0.944328i	$0.393287\pi$
$318$		0			0
$319$	34.4863			1.93086
$320$		0			0
$321$		0			0
$322$		0			0
$323$	19.5718			1.08901
$324$		0			0
$325$	−4.46967	+	7.74170i	−0.247933	+	0.429432i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	7.31227	−	23.7162i	0.403139	−	1.30752i
$330$		0			0
$331$	−16.5019			−0.907026			−0.453513	−	0.891250i	$-0.649829\pi$
$331$	−16.5019			−0.907026			−0.453513	+	0.891250i	$0.649829\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	7.06177	−	12.2314i	0.385826	−	0.668270i
$336$		0			0
$337$	−8.97815	−	15.5506i	−0.489071	−	0.847096i	0.510850	−	0.859670i	$-0.329331\pi$
$337$	−8.97815	−	15.5506i	−0.489071	−	0.847096i	−0.999921	+	0.0125741i	$0.995997\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−26.1876	−	45.3582i	−1.41814	−	2.45629i
$342$		0			0
$343$	14.4755	−	11.5525i	0.781601	−	0.623779i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	22.7079			1.21902			0.609511	−	0.792778i	$-0.291366\pi$
$347$	22.7079			1.21902			0.609511	+	0.792778i	$0.291366\pi$
$348$		0			0
$349$	−9.39145	+	16.2665i	−0.502713	+	0.870724i	0.497282	+	0.867589i	$0.334331\pi$
$349$	−9.39145	+	16.2665i	−0.502713	+	0.870724i	−0.999995	+	0.00313522i	$0.999002\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−14.8421	−	25.7073i	−0.789967	−	1.36826i	−0.925986	−	0.377557i	$-0.876764\pi$
$353$	−14.8421	−	25.7073i	−0.789967	−	1.36826i	0.136019	−	0.990706i	$-0.456569\pi$
$354$		0			0
$355$	−5.54488	+	9.60401i	−0.294291	+	0.509728i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−12.5829	−	21.7942i	−0.664099	−	1.15025i	−0.979529	−	0.201304i	$-0.935482\pi$
$359$	−12.5829	−	21.7942i	−0.664099	−	1.15025i	0.315430	−	0.948949i	$-0.397851\pi$
$360$		0			0
$361$	−18.8410	+	32.6336i	−0.991633	+	1.71756i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−1.26152	−	2.18501i	−0.0660309	−	0.114369i
$366$		0			0
$367$	−13.2534	−	22.9555i	−0.691819	−	1.19827i	−0.971241	−	0.238098i	$-0.923476\pi$
$367$	−13.2534	−	22.9555i	−0.691819	−	1.19827i	0.279422	−	0.960168i	$-0.409857\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−25.0295	−	26.9687i	−1.29947	−	1.40015i
$372$		0			0
$373$	6.97815	−	12.0865i	0.361315	−	0.625816i	−0.626863	−	0.779130i	$-0.715661\pi$
$373$	6.97815	−	12.0865i	0.361315	−	0.625816i	0.988178	+	0.153314i	$0.0489946\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	20.6076			1.06134
$378$		0			0
$379$	10.5537			0.542106			0.271053	−	0.962564i	$-0.412628\pi$
$379$	10.5537			0.542106			0.271053	+	0.962564i	$0.412628\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−4.72557	+	8.18493i	−0.241465	+	0.418230i	−0.961132	−	0.276090i	$-0.910961\pi$
$383$	−4.72557	+	8.18493i	−0.241465	+	0.418230i	0.719667	+	0.694320i	$0.244295\pi$
$384$		0			0
$385$	−5.46472	+	17.7239i	−0.278508	+	0.903296i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	14.3740	+	24.8966i	0.728793	+	1.26231i	0.957394	+	0.288786i	$0.0932517\pi$
$389$	14.3740	+	24.8966i	0.728793	+	1.26231i	−0.228601	+	0.973520i	$0.573415\pi$
$390$		0			0
$391$	−10.5653	−	18.2996i	−0.534310	−	0.925452i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	1.20433	−	2.08597i	0.0605966	−	0.104956i
$396$		0			0
$397$	−4.34642	−	7.52822i	−0.218140	−	0.377830i	0.736099	−	0.676874i	$-0.236666\pi$
$397$	−4.34642	−	7.52822i	−0.218140	−	0.377830i	−0.954239	+	0.299044i	$0.903332\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	17.0514	−	29.5339i	0.851507	−	1.47485i	−0.0283402	−	0.999598i	$-0.509022\pi$
$401$	17.0514	−	29.5339i	0.851507	−	1.47485i	0.879848	−	0.475256i	$-0.157644\pi$
$402$		0			0
$403$	−15.6486	−	27.1042i	−0.779514	−	1.35016i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	0.469673	−	0.813497i	0.0232808	−	0.0403236i
$408$		0			0
$409$	−7.96427			−0.393808			−0.196904	−	0.980423i	$-0.563089\pi$
$409$	−7.96427			−0.393808			−0.196904	+	0.980423i	$0.563089\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	5.29199	−	17.1637i	0.260402	−	0.844571i
$414$		0			0
$415$	−0.0141870	−	0.0245726i	−0.000696413	−	0.00120622i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	11.2178	+	19.4297i	0.548024	+	0.949205i	0.998410	+	0.0563709i	$0.0179529\pi$
$419$	11.2178	+	19.4297i	0.548024	+	0.949205i	−0.450386	+	0.892834i	$0.648714\pi$
$420$		0			0
$421$	16.2910	−	28.2169i	0.793977	−	1.37521i	−0.129511	−	0.991578i	$-0.541341\pi$
$421$	16.2910	−	28.2169i	0.793977	−	1.37521i	0.923487	−	0.383630i	$-0.125326\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−7.82539			−0.379587
$426$		0			0
$427$	−4.59659	+	14.9083i	−0.222445	+	0.721464i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−11.2666	+	19.5143i	−0.542694	+	0.939973i	0.456054	+	0.889952i	$0.349262\pi$
$431$	−11.2666	+	19.5143i	−0.542694	+	0.939973i	−0.998748	+	0.0500212i	$0.984071\pi$
$432$		0			0
$433$	11.1868			0.537602			0.268801	−	0.963196i	$-0.413373\pi$
$433$	11.1868			0.537602			0.268801	+	0.963196i	$0.413373\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	61.1964			2.92742
$438$		0			0
$439$	7.56355			0.360989			0.180494	−	0.983576i	$-0.442230\pi$
$439$	7.56355			0.360989			0.180494	+	0.983576i	$0.442230\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3.62768			0.172356			0.0861781	−	0.996280i	$-0.472535\pi$
$443$	3.62768			0.172356			0.0861781	+	0.996280i	$0.472535\pi$
$444$		0			0
$445$	20.6897			0.980786
$446$		0			0
$447$		0			0
$448$		0			0
$449$	5.04851			0.238254			0.119127	−	0.992879i	$-0.461990\pi$
$449$	5.04851			0.238254			0.119127	+	0.992879i	$0.461990\pi$
$450$		0			0
$451$	5.07041	−	8.78220i	0.238756	−	0.413538i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−3.26549	+	10.5911i	−0.153089	+	0.496519i
$456$		0			0
$457$	24.1127			1.12795			0.563973	−	0.825793i	$-0.309272\pi$
$457$	24.1127			1.12795			0.563973	+	0.825793i	$0.309272\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−5.25162	+	9.09607i	−0.244592	+	0.423646i	−0.962017	−	0.272990i	$-0.911987\pi$
$461$	−5.25162	+	9.09607i	−0.244592	+	0.423646i	0.717425	+	0.696636i	$0.245321\pi$
$462$		0			0
$463$	−1.82121	−	3.15443i	−0.0846387	−	0.146599i	0.820598	−	0.571505i	$-0.193640\pi$
$463$	−1.82121	−	3.15443i	−0.0846387	−	0.146599i	−0.905237	+	0.424907i	$0.860307\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−15.9933	−	27.7012i	−0.740082	−	1.28186i	−0.952457	−	0.304672i	$-0.901453\pi$
$467$	−15.9933	−	27.7012i	−0.740082	−	1.28186i	0.212375	−	0.977188i	$-0.431880\pi$
$468$		0			0
$469$	−7.80508	+	25.3145i	−0.360405	+	1.16892i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	21.7685			1.00092
$474$		0			0
$475$	11.3316	−	19.6268i	0.519928	−	0.900541i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.3613	+	30.0706i	0.793257	+	1.37396i	0.923940	+	0.382537i	$0.124950\pi$
$479$	17.3613	+	30.0706i	0.793257	+	1.37396i	−0.130683	+	0.991424i	$0.541717\pi$
$480$		0			0
$481$	0.280657	−	0.486113i	0.0127969	−	0.0221648i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	6.11686	+	10.5947i	0.277752	+	0.481081i
$486$		0			0
$487$	−17.9411	+	31.0749i	−0.812988	+	1.40814i	0.0977757	+	0.995208i	$0.468827\pi$
$487$	−17.9411	+	31.0749i	−0.812988	+	1.40814i	−0.910764	+	0.412928i	$0.864506\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−0.0427795	−	0.0740962i	−0.00193061	−	0.00334391i	0.865058	−	0.501671i	$-0.167281\pi$
$491$	−0.0427795	−	0.0740962i	−0.00193061	−	0.00334391i	−0.866989	+	0.498327i	$0.833948\pi$
$492$		0			0
$493$	9.01981	+	15.6228i	0.406232	+	0.703614i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	6.12852	−	19.8769i	0.274902	−	0.891599i
$498$		0			0
$499$	6.27842	−	10.8745i	0.281061	−	0.486811i	−0.690586	−	0.723251i	$-0.742647\pi$
$499$	6.27842	−	10.8745i	0.281061	−	0.486811i	0.971646	+	0.236439i	$0.0759805\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	42.8822			1.91202			0.956011	−	0.293332i	$-0.0947643\pi$
$503$	42.8822			1.91202			0.956011	+	0.293332i	$0.0947643\pi$
$504$		0			0
$505$	15.6976			0.698536
$506$		0			0
$507$		0			0
$508$		0			0
$509$	7.48102	−	12.9575i	0.331590	−	0.574331i	−0.651234	−	0.758877i	$-0.725748\pi$
$509$	7.48102	−	12.9575i	0.331590	−	0.574331i	0.982824	+	0.184546i	$0.0590815\pi$
$510$		0			0
$511$	3.21919	+	3.46860i	0.142408	+	0.153442i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	7.61799	+	13.1948i	0.335689	+	0.581430i
$516$		0			0
$517$	−23.3084	−	40.3714i	−1.02510	−	1.77553i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	18.8479	−	32.6454i	0.825740	−	1.43022i	−0.0756130	−	0.997137i	$-0.524091\pi$
$521$	18.8479	−	32.6454i	0.825740	−	1.43022i	0.901353	−	0.433086i	$-0.142575\pi$
$522$		0			0
$523$	−0.707999	−	1.22629i	−0.0309586	−	0.0536219i	0.850131	−	0.526571i	$-0.176523\pi$
$523$	−0.707999	−	1.22629i	−0.0309586	−	0.0536219i	−0.881090	+	0.472949i	$0.843189\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	13.6986	−	23.7267i	0.596721	−	1.03355i
$528$		0			0
$529$	−21.5351	−	37.2999i	−0.936310	−	1.62174i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	3.02987	−	5.24788i	0.131238	−	0.227311i
$534$		0			0
$535$	23.1839			1.00233
$536$		0			0
$537$		0			0
$538$		0			0
$539$	2.59114	−	34.6911i	0.111608	−	1.49425i
$540$		0			0
$541$	−3.35266	−	5.80697i	−0.144142	−	0.249661i	0.784911	−	0.619609i	$-0.212709\pi$
$541$	−3.35266	−	5.80697i	−0.144142	−	0.249661i	−0.929052	+	0.369948i	$0.879376\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−8.14562	−	14.1086i	−0.348920	−	0.604347i
$546$		0			0
$547$	−15.8175	+	27.3968i	−0.676309	+	1.17140i	0.299776	+	0.954010i	$0.403088\pi$
$547$	−15.8175	+	27.3968i	−0.676309	+	1.17140i	−0.976084	+	0.217392i	$0.930245\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−52.2446			−2.22569
$552$		0			0
$553$	−1.33110	+	4.31721i	−0.0566041	+	0.183586i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.6007	+	23.5572i	−0.576282	+	0.998149i	0.419620	+	0.907700i	$0.362164\pi$
$557$	−13.6007	+	23.5572i	−0.576282	+	0.998149i	−0.995901	+	0.0904489i	$0.971170\pi$
$558$		0			0
$559$	13.0080			0.550179
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−2.30235			−0.0970326			−0.0485163	−	0.998822i	$-0.515449\pi$
$563$	−2.30235			−0.0970326			−0.0485163	+	0.998822i	$0.515449\pi$
$564$		0			0
$565$	1.66598			0.0700882
$566$		0			0
$567$		0			0
$568$		0			0
$569$	33.9473			1.42315			0.711573	−	0.702612i	$-0.247983\pi$
$569$	33.9473			1.42315			0.711573	+	0.702612i	$0.247983\pi$
$570$		0			0
$571$	0.229245			0.00959362			0.00479681	−	0.999988i	$-0.498473\pi$
$571$	0.229245			0.00959362			0.00479681	+	0.999988i	$0.498473\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−24.4681			−1.02039
$576$		0			0
$577$	−11.5733	+	20.0455i	−0.481802	+	0.834505i	−0.999782	−	0.0208877i	$-0.993351\pi$
$577$	−11.5733	+	20.0455i	−0.481802	+	0.834505i	0.517980	+	0.855393i	$0.326684\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	0.0362029	+	0.0390078i	0.00150195	+	0.00161832i
$582$		0			0
$583$	−69.1121			−2.86233
$584$		0			0
$585$		0			0
$586$		0			0
$587$	18.6739	−	32.3441i	0.770752	−	1.33498i	−0.166399	−	0.986059i	$-0.553214\pi$
$587$	18.6739	−	32.3441i	0.770752	−	1.33498i	0.937151	−	0.348924i	$-0.113453\pi$
$588$		0			0
$589$	39.6726	+	68.7149i	1.63468	+	2.83135i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−10.1975	−	17.6626i	−0.418761	−	0.725315i	0.577054	−	0.816706i	$-0.304202\pi$
$593$	−10.1975	−	17.6626i	−0.418761	−	0.725315i	−0.995815	+	0.0913907i	$0.970869\pi$
$594$		0			0
$595$	−9.45848	+	2.16006i	−0.387760	+	0.0885540i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	24.7912			1.01294			0.506470	−	0.862258i	$-0.330950\pi$
$599$	24.7912			1.01294			0.506470	+	0.862258i	$0.330950\pi$
$600$		0			0
$601$	−23.0441	+	39.9135i	−0.939987	+	1.62811i	−0.174497	+	0.984658i	$0.555830\pi$
$601$	−23.0441	+	39.9135i	−0.939987	+	1.62811i	−0.765490	+	0.643448i	$0.777503\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	9.66094	+	16.7332i	0.392773	+	0.680303i
$606$		0			0
$607$	11.8399	−	20.5073i	0.480566	−	0.832365i	−0.519185	−	0.854662i	$-0.673765\pi$
$607$	11.8399	−	20.5073i	0.480566	−	0.832365i	0.999751	+	0.0222967i	$0.00709786\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−13.9282	−	24.1243i	−0.563473	−	0.975965i
$612$		0			0
$613$	11.2993	−	19.5710i	0.456376	−	0.790467i	−0.542390	−	0.840127i	$-0.682480\pi$
$613$	11.2993	−	19.5710i	0.456376	−	0.790467i	0.998766	+	0.0496599i	$0.0158138\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9.62964	−	16.6790i	−0.387675	−	0.671472i	0.604462	−	0.796634i	$-0.293388\pi$
$617$	−9.62964	−	16.6790i	−0.387675	−	0.671472i	−0.992136	+	0.125162i	$0.960055\pi$
$618$		0			0
$619$	1.28066	+	2.21816i	0.0514740	+	0.0891555i	0.890614	−	0.454759i	$-0.150275\pi$
$619$	1.28066	+	2.21816i	0.0514740	+	0.0891555i	−0.839140	+	0.543915i	$0.816941\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−37.8321	+	8.63983i	−1.51571	+	0.346147i
$624$		0			0
$625$	0.443780	−	0.768650i	0.0177512	−	0.0307460i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	0.491368			0.0195921
$630$		0			0
$631$	−23.3806			−0.930767			−0.465384	−	0.885109i	$-0.654084\pi$
$631$	−23.3806			−0.930767			−0.465384	+	0.885109i	$0.654084\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	6.46682	−	11.2009i	0.256628	−	0.444492i
$636$		0			0
$637$	1.54836	−	20.7300i	0.0613481	−	0.821351i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−5.39784	−	9.34933i	−0.213202	−	0.369277i	0.739513	−	0.673142i	$-0.235056\pi$
$641$	−5.39784	−	9.34933i	−0.213202	−	0.369277i	−0.952715	+	0.303866i	$0.901723\pi$
$642$		0			0
$643$	0.783523	+	1.35710i	0.0308991	+	0.0535189i	0.881061	−	0.473002i	$-0.156830\pi$
$643$	0.783523	+	1.35710i	0.0308991	+	0.0535189i	−0.850162	+	0.526521i	$0.823496\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	14.7966	−	25.6285i	0.581716	−	1.00756i	−0.413560	−	0.910477i	$-0.635715\pi$
$647$	14.7966	−	25.6285i	0.581716	−	1.00756i	0.995276	−	0.0970851i	$-0.0309519\pi$
$648$		0			0
$649$	−16.8686	−	29.2173i	−0.662152	−	1.14688i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−19.3696	+	33.5491i	−0.757991	+	1.31288i	0.185883	+	0.982572i	$0.440486\pi$
$653$	−19.3696	+	33.5491i	−0.757991	+	1.31288i	−0.943874	+	0.330307i	$0.892848\pi$
$654$		0			0
$655$	5.42036	+	9.38834i	0.211791	+	0.366833i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−14.0733	+	24.3756i	−0.548216	+	0.949539i	0.450180	+	0.892938i	$0.351360\pi$
$659$	−14.0733	+	24.3756i	−0.548216	+	0.949539i	−0.998397	+	0.0566011i	$0.981974\pi$
$660$		0			0
$661$	27.4863			1.06909			0.534546	−	0.845139i	$-0.320483\pi$
$661$	27.4863			1.06909			0.534546	+	0.845139i	$0.320483\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.27872	−	26.8507i	0.321035	−	1.04122i
$666$		0			0
$667$	28.2028	+	48.8486i	1.09202	+	1.89143i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.6520	+	25.3780i	0.565634	+	0.979707i
$672$		0			0
$673$	5.45834	−	9.45412i	0.210404	−	0.364430i	−0.741437	−	0.671022i	$-0.765856\pi$
$673$	5.45834	−	9.45412i	0.210404	−	0.364430i	0.951841	+	0.306592i	$0.0991889\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	19.1456			0.735826			0.367913	−	0.929860i	$-0.380072\pi$
$677$	19.1456			0.735826			0.367913	+	0.929860i	$0.380072\pi$
$678$		0			0
$679$	−15.6092	−	16.8186i	−0.599027	−	0.645438i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−2.60854	+	4.51812i	−0.0998130	+	0.172881i	−0.911607	−	0.411063i	$-0.865158\pi$
$683$	−2.60854	+	4.51812i	−0.0998130	+	0.172881i	0.811794	+	0.583944i	$0.198491\pi$
$684$		0			0
$685$	6.73408			0.257296
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−41.2986			−1.57335
$690$		0			0
$691$	32.4367			1.23395			0.616976	−	0.786982i	$-0.288358\pi$
$691$	32.4367			1.23395			0.616976	+	0.786982i	$0.288358\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−29.4483			−1.11704
$696$		0			0
$697$	5.30461			0.200927
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−15.0981			−0.570246			−0.285123	−	0.958491i	$-0.592034\pi$
$701$	−15.0981			−0.570246			−0.285123	+	0.958491i	$0.592034\pi$
$702$		0			0
$703$	−0.711525	+	1.23240i	−0.0268357	+	0.0464808i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−28.7039	+	6.55519i	−1.07952	+	0.246533i
$708$		0			0
$709$	−9.01472			−0.338555			−0.169277	−	0.985568i	$-0.554143\pi$
$709$	−9.01472			−0.338555			−0.169277	+	0.985568i	$0.554143\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	42.8322	−	74.1876i	1.60408	−	2.77835i
$714$		0			0
$715$	10.4090	+	18.0289i	0.389275	+	0.674244i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	13.2753	+	22.9934i	0.495084	+	0.857510i	0.999984	−	0.00566774i	$-0.00180411\pi$
$719$	13.2753	+	22.9934i	0.495084	+	0.857510i	−0.504900	+	0.863178i	$0.668471\pi$
$720$		0			0
$721$	−19.4399	−	20.9460i	−0.723978	−	0.780070i
$722$		0			0
$723$		0			0
$724$		0			0
$725$	20.8889			0.775794
$726$		0			0
$727$	4.84259	−	8.38761i	0.179602	−	0.311079i	−0.762143	−	0.647409i	$-0.775852\pi$
$727$	4.84259	−	8.38761i	0.179602	−	0.311079i	0.941744	+	0.336330i	$0.109186\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	5.69351	+	9.86144i	0.210582	+	0.364739i
$732$		0			0
$733$	−4.23676	+	7.33828i	−0.156488	+	0.271046i	−0.933600	−	0.358317i	$-0.883351\pi$
$733$	−4.23676	+	7.33828i	−0.156488	+	0.271046i	0.777112	+	0.629363i	$0.216684\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	24.8793	+	43.0922i	0.916441	+	1.58732i
$738$		0			0
$739$	−4.58445	+	7.94050i	−0.168642	+	0.292096i	−0.937943	−	0.346791i	$-0.887271\pi$
$739$	−4.58445	+	7.94050i	−0.168642	+	0.292096i	0.769301	+	0.638887i	$0.220605\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	5.80108	+	10.0478i	0.212821	+	0.368617i	0.952596	−	0.304237i	$-0.0984015\pi$
$743$	5.80108	+	10.0478i	0.212821	+	0.368617i	−0.739775	+	0.672854i	$0.765068\pi$
$744$		0			0
$745$	−8.32329	−	14.4164i	−0.304942	−	0.528175i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−42.3929	+	9.68140i	−1.54900	+	0.353751i
$750$		0			0
$751$	1.92515	−	3.33445i	0.0702496	−	0.121676i	−0.828761	−	0.559603i	$-0.810954\pi$
$751$	1.92515	−	3.33445i	0.0702496	−	0.121676i	0.899011	+	0.437927i	$0.144287\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	9.90616			0.360522
$756$		0			0
$757$	2.92720			0.106391			0.0531955	−	0.998584i	$-0.483059\pi$
$757$	2.92720			0.106391			0.0531955	+	0.998584i	$0.483059\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	3.15470	−	5.46410i	0.114358	−	0.198074i	−0.803165	−	0.595757i	$-0.796852\pi$
$761$	3.15470	−	5.46410i	0.114358	−	0.198074i	0.917523	+	0.397683i	$0.130186\pi$
$762$		0			0
$763$	20.7863	+	22.3967i	0.752514	+	0.810817i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−10.0800	−	17.4591i	−0.363968	−	0.630411i
$768$		0			0
$769$	−12.4309	−	21.5309i	−0.448269	−	0.776424i	0.550005	−	0.835161i	$-0.314626\pi$
$769$	−12.4309	−	21.5309i	−0.448269	−	0.776424i	−0.998273	+	0.0587375i	$0.981293\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−7.01982	+	12.1587i	−0.252485	+	0.437318i	−0.964209	−	0.265142i	$-0.914581\pi$
$773$	−7.01982	+	12.1587i	−0.252485	+	0.437318i	0.711724	+	0.702459i	$0.247915\pi$
$774$		0			0
$775$	−15.8623	−	27.4742i	−0.569789	−	0.986903i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−7.68135	+	13.3045i	−0.275213	+	0.476683i
$780$		0			0
$781$	−19.5351	−	33.8358i	−0.699022	−	1.21074i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.70944	−	8.15699i	0.168087	−	0.291135i
$786$		0			0
$787$	−25.3794			−0.904677			−0.452338	−	0.891846i	$-0.649410\pi$
$787$	−25.3794			−0.904677			−0.452338	+	0.891846i	$0.649410\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−3.04632	+	0.695697i	−0.108315	+	0.0247361i
$792$		0			0
$793$	8.75544	+	15.1649i	0.310915	+	0.538520i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−9.59451	−	16.6182i	−0.339855	−	0.588646i	0.644550	−	0.764562i	$-0.277045\pi$
$797$	−9.59451	−	16.6182i	−0.339855	−	0.588646i	−0.984405	+	0.175916i	$0.943711\pi$
$798$		0			0
$799$	12.1925	−	21.1181i	0.431341	−	0.747105i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	8.88890			0.313683
$804$		0			0
$805$	−29.5744	+	6.75400i	−1.04236	+	0.238047i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	0.221134	−	0.383016i	0.00777467	−	0.0134661i	−0.862112	−	0.506718i	$-0.830859\pi$
$809$	0.221134	−	0.383016i	0.00777467	−	0.0134661i	0.869887	+	0.493252i	$0.164192\pi$
$810$		0			0
$811$	28.1510			0.988516			0.494258	−	0.869315i	$-0.335440\pi$
$811$	28.1510			0.988516			0.494258	+	0.869315i	$0.335440\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−0.479225			−0.0167865
$816$		0			0
$817$	−32.9780			−1.15375
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.53292			0.297801			0.148900	−	0.988852i	$-0.452427\pi$
$821$	8.53292			0.297801			0.148900	+	0.988852i	$0.452427\pi$
$822$		0			0
$823$	−29.5179			−1.02893			−0.514465	−	0.857511i	$-0.672009\pi$
$823$	−29.5179			−1.02893			−0.514465	+	0.857511i	$0.672009\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−35.7222			−1.24218			−0.621092	−	0.783738i	$-0.713311\pi$
$827$	−35.7222			−1.24218			−0.621092	+	0.783738i	$0.713311\pi$
$828$		0			0
$829$	−25.8399	+	44.7560i	−0.897456	+	1.55444i	−0.0667215	+	0.997772i	$0.521254\pi$
$829$	−25.8399	+	44.7560i	−0.897456	+	1.55444i	−0.830735	+	0.556668i	$0.812079\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	16.3933	−	7.89955i	0.567993	−	0.273703i
$834$		0			0
$835$	1.26469			0.0437666
$836$		0			0
$837$		0			0
$838$		0			0
$839$	18.6284	−	32.2653i	0.643122	−	1.11392i	−0.341609	−	0.939842i	$-0.610972\pi$
$839$	18.6284	−	32.2653i	0.643122	−	1.11392i	0.984732	−	0.174079i	$-0.0556947\pi$
$840$		0			0
$841$	−9.57726	−	16.5883i	−0.330250	−	0.572010i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.94889	−	5.10762i	−0.101445	−	0.175708i
$846$		0			0
$847$	−24.6531	−	26.5632i	−0.847091	−	0.912722i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1.53639			0.0526667
$852$		0			0
$853$	6.03115	−	10.4463i	0.206503	−	0.357673i	−0.744108	−	0.668060i	$-0.767125\pi$
$853$	6.03115	−	10.4463i	0.206503	−	0.357673i	0.950610	+	0.310386i	$0.100458\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	14.1047	+	24.4301i	0.481808	+	0.834515i	0.999782	−	0.0208808i	$-0.00664705\pi$
$857$	14.1047	+	24.4301i	0.481808	+	0.834515i	−0.517974	+	0.855396i	$0.673314\pi$
$858$		0			0
$859$	25.7641	−	44.6247i	0.879059	−	1.52257i	0.0266833	−	0.999644i	$-0.491505\pi$
$859$	25.7641	−	44.6247i	0.879059	−	1.52257i	0.852375	−	0.522930i	$-0.175161\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−1.57423	−	2.72664i	−0.0535873	−	0.0928159i	0.837987	−	0.545690i	$-0.183732\pi$
$863$	−1.57423	−	2.72664i	−0.0535873	−	0.0928159i	−0.891575	+	0.452874i	$0.850399\pi$
$864$		0			0
$865$	3.41076	−	5.90762i	0.115969	−	0.200865i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	4.24298	+	7.34906i	0.143933	+	0.249300i
$870$		0			0
$871$	14.8669	+	25.7501i	0.503744	+	0.872510i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−8.80814	+	28.5678i	−0.297770	+	0.965768i
$876$		0			0
$877$	−1.27555	+	2.20932i	−0.0430723	+	0.0746034i	−0.886758	−	0.462234i	$-0.847048\pi$
$877$	−1.27555	+	2.20932i	−0.0430723	+	0.0746034i	0.843686	+	0.536838i	$0.180381\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−14.4378			−0.486421			−0.243210	−	0.969974i	$-0.578201\pi$
$881$	−14.4378			−0.486421			−0.243210	+	0.969974i	$0.578201\pi$
$882$		0			0
$883$	0.478915			0.0161168			0.00805839	−	0.999968i	$-0.497435\pi$
$883$	0.478915			0.0161168			0.00805839	+	0.999968i	$0.497435\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.62821	+	9.74834i	−0.188977	+	0.327317i	−0.944909	−	0.327332i	$-0.893850\pi$
$887$	−5.62821	+	9.74834i	−0.188977	+	0.327317i	0.755933	+	0.654649i	$0.227184\pi$
$888$		0			0
$889$	−7.14750	+	23.1818i	−0.239719	+	0.777492i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	35.3108	+	61.1602i	1.18163	+	2.04665i
$894$		0			0
$895$	2.00960	+	3.48073i	0.0671734	+	0.116348i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−36.5667	+	63.3355i	−1.21957	+	2.11236i
$900$		0			0
$901$	−18.0761	−	31.3088i	−0.602203	−	1.04305i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.03047	−	1.78483i	0.0342540	−	0.0593297i
$906$		0			0
$907$	−0.359496	−	0.622666i	−0.0119369	−	0.0206753i	0.859995	−	0.510302i	$-0.170466\pi$
$907$	−0.359496	−	0.622666i	−0.0119369	−	0.0206753i	−0.871932	+	0.489627i	$0.837133\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−10.5890	+	18.3406i	−0.350828	+	0.607651i	−0.986395	−	0.164394i	$-0.947433\pi$
$911$	−10.5890	+	18.3406i	−0.350828	+	0.607651i	0.635567	+	0.772046i	$0.280766\pi$
$912$		0			0
$913$	0.0999644			0.00330834
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−13.8319	−	14.9035i	−0.456769	−	0.492158i
$918$		0			0
$919$	7.17784	+	12.4324i	0.236775	+	0.410106i	0.959787	−	0.280729i	$-0.0905762\pi$
$919$	7.17784	+	12.4324i	0.236775	+	0.410106i	−0.723012	+	0.690835i	$0.757243\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−11.6734	−	20.2189i	−0.384234	−	0.665513i
$924$		0			0
$925$	0.284489	−	0.492749i	0.00935393	−	0.0162015i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−37.3085			−1.22405			−0.612026	−	0.790838i	$-0.709645\pi$
$929$	−37.3085			−1.22405			−0.612026	+	0.790838i	$0.709645\pi$
$930$		0			0
$931$	−3.92541	+	52.5548i	−0.128650	+	1.72242i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−9.11192	+	15.7823i	−0.297992	+	0.516137i
$936$		0			0
$937$	50.9094			1.66314			0.831568	−	0.555423i	$-0.187444\pi$
$937$	50.9094			1.66314			0.831568	+	0.555423i	$0.187444\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.8503			0.777497			0.388749	−	0.921344i	$-0.372907\pi$
$941$	23.8503			0.777497			0.388749	+	0.921344i	$0.372907\pi$
$942$		0			0
$943$	16.5862			0.540122
$944$		0			0
$945$		0			0
$946$		0			0
$947$	58.7152			1.90799			0.953994	−	0.299825i	$-0.0969283\pi$
$947$	58.7152			1.90799			0.953994	+	0.299825i	$0.0969283\pi$
$948$		0			0
$949$	5.31164			0.172423
$950$		0			0
$951$		0			0
$952$		0			0
$953$	28.4324			0.921015			0.460507	−	0.887656i	$-0.347667\pi$
$953$	28.4324			0.921015			0.460507	+	0.887656i	$0.347667\pi$
$954$		0			0
$955$	3.23128	−	5.59674i	0.104562	−	0.181106i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−12.3136	+	2.81209i	−0.397626	+	0.0908071i
$960$		0			0
$961$	80.0697			2.58289
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−9.58079	+	16.5944i	−0.308416	+	0.534193i
$966$		0			0
$967$	13.8466	+	23.9830i	0.445276	+	0.771240i	0.998071	−	0.0620766i	$-0.0197723\pi$
$967$	13.8466	+	23.9830i	0.445276	+	0.771240i	−0.552796	+	0.833317i	$0.686439\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−0.510672	−	0.884510i	−0.0163882	−	0.0283853i	0.857715	−	0.514125i	$-0.171883\pi$
$971$	−0.510672	−	0.884510i	−0.0163882	−	0.0283853i	−0.874103	+	0.485740i	$0.838550\pi$
$972$		0			0
$973$	53.8476	−	12.2973i	1.72627	−	0.394235i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−37.8037			−1.20945			−0.604723	−	0.796436i	$-0.706716\pi$
$977$	−37.8037			−1.20945			−0.604723	+	0.796436i	$0.706716\pi$
$978$		0			0
$979$	−36.4459	+	63.1261i	−1.16482	+	2.01752i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−8.19366	−	14.1918i	−0.261337	−	0.452649i	0.705260	−	0.708948i	$-0.250830\pi$
$983$	−8.19366	−	14.1918i	−0.261337	−	0.452649i	−0.966598	+	0.256299i	$0.917497\pi$
$984$		0			0
$985$	−9.44430	+	16.3580i	−0.300920	+	0.521209i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	17.8022	+	30.8343i	0.566077	+	0.980475i
$990$		0			0
$991$	13.7067	−	23.7408i	0.435409	−	0.754150i	−0.561920	−	0.827192i	$-0.689937\pi$
$991$	13.7067	−	23.7408i	0.435409	−	0.754150i	0.997329	+	0.0730412i	$0.0232704\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−0.543394	−	0.941186i	−0.0172268	−	0.0298376i
$996$		0			0
$997$	7.67099	+	13.2865i	0.242943	+	0.420789i	0.961551	−	0.274626i	$-0.0885540\pi$
$997$	7.67099	+	13.2865i	0.242943	+	0.420789i	−0.718609	+	0.695415i	$0.755221\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2268.2.i.l.865.3		8
3.2	odd	2		2268.2.i.m.865.2		8
7.2	even	3		2268.2.l.m.541.2		8
9.2	odd	6		2268.2.k.d.1621.2	yes	8
9.4	even	3		2268.2.l.m.109.2		8
9.5	odd	6		2268.2.l.l.109.3		8
9.7	even	3		2268.2.k.c.1621.3	yes	8
21.2	odd	6		2268.2.l.l.541.3		8
63.2	odd	6		2268.2.k.d.1297.2	yes	8
63.16	even	3		2268.2.k.c.1297.3	✓	8
63.23	odd	6		2268.2.i.m.2053.2		8
63.58	even	3	inner	2268.2.i.l.2053.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2268.2.i.l.865.3		8	1.1	even	1	trivial
2268.2.i.l.2053.3		8	63.58	even	3	inner
2268.2.i.m.865.2		8	3.2	odd	2
2268.2.i.m.2053.2		8	63.23	odd	6
2268.2.k.c.1297.3	✓	8	63.16	even	3
2268.2.k.c.1621.3	yes	8	9.7	even	3
2268.2.k.d.1297.2	yes	8	63.2	odd	6
2268.2.k.d.1621.2	yes	8	9.2	odd	6
2268.2.l.l.109.3		8	9.5	odd	6
2268.2.l.l.541.3		8	21.2	odd	6
2268.2.l.m.109.2		8	9.4	even	3
2268.2.l.m.541.2		8	7.2	even	3

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2268 = 2^{2} \cdot 3^{4} \cdot 7 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2268.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.705299 - 1.22161i) q^{5} +(-0.779537 + 2.52830i) q^{7} +O(q^{10})\)	\(q+(0.705299 - 1.22161i) q^{5} +(-0.779537 + 2.52830i) q^{7} +(2.48484 + 4.30386i) q^{11} +(1.48484 + 2.57181i) q^{13} +(-1.29981 + 2.25133i) q^{17} +(-3.76437 - 6.52009i) q^{19} +(-4.06418 + 7.03937i) q^{23} +(1.50511 + 2.60692i) q^{25} +(3.46967 - 6.00965i) q^{29} -10.5390 q^{31} +(2.53880 + 2.73550i) q^{35} +(-0.0945078 - 0.163692i) q^{37} +(-1.02027 - 1.76716i) q^{41} +(2.19014 - 3.79343i) q^{43} -9.38027 q^{47} +(-5.78464 - 3.94181i) q^{49} +(-6.95339 + 12.0436i) q^{53} +7.01021 q^{55} -6.78863 q^{59} +5.89657 q^{61} +4.18902 q^{65} +10.0125 q^{67} -7.86174 q^{71} +(0.894315 - 1.54900i) q^{73} +(-12.8185 + 2.92740i) q^{77} +1.70755 q^{79} +(0.0100574 - 0.0174200i) q^{83} +(1.83351 + 3.17572i) q^{85} +(7.33366 + 12.7023i) q^{89} +(-7.65981 + 1.74929i) q^{91} -10.6200 q^{95} +(-4.33636 + 7.51080i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 2 q^{5} + q^{7}+O(q^{10}) \) 8 * q - 2 * q^5 + q^7	\( 8 q - 2 q^{5} + q^{7} + 5 q^{11} - 3 q^{13} + 2 q^{17} - 8 q^{19} + 2 q^{23} - 8 q^{25} - 2 q^{29} + 11 q^{35} + 4 q^{37} - 3 q^{41} - 5 q^{43} - 30 q^{47} - 19 q^{49} - 24 q^{53} + 16 q^{55} - 20 q^{59} + 24 q^{61} + 24 q^{65} + 14 q^{67} - 22 q^{71} - 10 q^{73} - 11 q^{77} + 35 q^{83} + 13 q^{85} - 18 q^{89} - 9 q^{91} + 20 q^{95} - 19 q^{97}+O(q^{100}) \) 8 * q - 2 * q^5 + q^7 + 5 * q^11 - 3 * q^13 + 2 * q^17 - 8 * q^19 + 2 * q^23 - 8 * q^25 - 2 * q^29 + 11 * q^35 + 4 * q^37 - 3 * q^41 - 5 * q^43 - 30 * q^47 - 19 * q^49 - 24 * q^53 + 16 * q^55 - 20 * q^59 + 24 * q^61 + 24 * q^65 + 14 * q^67 - 22 * q^71 - 10 * q^73 - 11 * q^77 + 35 * q^83 + 13 * q^85 - 18 * q^89 - 9 * q^91 + 20 * q^95 - 19 * q^97

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