LMFDB - Embedded newform 1728.2.s.c.575.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1728,2,Mod(575,1728)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1728, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([3, 0, 1]))

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

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

chi := DirichletCharacter("1728.575");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1728 = 2^{6} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1728.s (of order $6$, 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:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			575.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	1728.575
Dual form			1728.2.s.c.1151.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.50000 - 0.866025i) q^{5} +(-1.50000 - 0.866025i) q^{7} +O(q^{10})$	$q+(1.50000 - 0.866025i) q^{5} +(-1.50000 - 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{11} +(-2.50000 - 4.33013i) q^{13} +6.92820i q^{17} -3.46410i q^{19} +(-4.50000 - 7.79423i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(1.50000 + 0.866025i) q^{29} +(-4.50000 + 2.59808i) q^{31} -3.00000 q^{35} -2.00000 q^{37} +(4.50000 - 2.59808i) q^{41} +(-4.50000 - 2.59808i) q^{43} +(-1.50000 + 2.59808i) q^{47} +(-2.00000 - 3.46410i) q^{49} -5.19615i q^{55} +(-1.50000 - 2.59808i) q^{59} +(-0.500000 + 0.866025i) q^{61} +(-7.50000 - 4.33013i) q^{65} +(-7.50000 + 4.33013i) q^{67} +12.0000 q^{71} -2.00000 q^{73} +(-4.50000 + 2.59808i) q^{77} +(-7.50000 - 4.33013i) q^{79} +(7.50000 - 12.9904i) q^{83} +(6.00000 + 10.3923i) q^{85} -6.92820i q^{89} +8.66025i q^{91} +(-3.00000 - 5.19615i) q^{95} +(2.50000 - 4.33013i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{5} - 3 q^{7}+O(q^{10}) $ 2 * q + 3 * q^5 - 3 * q^7	$ 2 q + 3 q^{5} - 3 q^{7} + 3 q^{11} - 5 q^{13} - 9 q^{23} - 2 q^{25} + 3 q^{29} - 9 q^{31} - 6 q^{35} - 4 q^{37} + 9 q^{41} - 9 q^{43} - 3 q^{47} - 4 q^{49} - 3 q^{59} - q^{61} - 15 q^{65} - 15 q^{67} + 24 q^{71} - 4 q^{73} - 9 q^{77} - 15 q^{79} + 15 q^{83} + 12 q^{85} - 6 q^{95} + 5 q^{97}+O(q^{100}) $ 2 * q + 3 * q^5 - 3 * q^7 + 3 * q^11 - 5 * q^13 - 9 * q^23 - 2 * q^25 + 3 * q^29 - 9 * q^31 - 6 * q^35 - 4 * q^37 + 9 * q^41 - 9 * q^43 - 3 * q^47 - 4 * q^49 - 3 * q^59 - q^61 - 15 * q^65 - 15 * q^67 + 24 * q^71 - 4 * q^73 - 9 * q^77 - 15 * q^79 + 15 * q^83 + 12 * q^85 - 6 * q^95 + 5 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1728\mathbb{Z}\right)^\times$.

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$5$	1.50000	−	0.866025i	0.670820	−	0.387298i	−0.125567	−	0.992085i	$-0.540075\pi$
$5$	1.50000	−	0.866025i	0.670820	−	0.387298i	0.796387	+	0.604787i	$0.206742\pi$
$6$		0			0
$7$	−1.50000	−	0.866025i	−0.566947	−	0.327327i	0.188982	−	0.981981i	$-0.439481\pi$
$7$	−1.50000	−	0.866025i	−0.566947	−	0.327327i	−0.755929	+	0.654654i	$0.772814\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	−0.546259	−	0.837616i	$-0.683949\pi$
$11$	1.50000	−	2.59808i	0.452267	−	0.783349i	0.998526	+	0.0542666i	$0.0172821\pi$
$12$		0			0
$13$	−2.50000	−	4.33013i	−0.693375	−	1.20096i	−0.970725	−	0.240192i	$-0.922790\pi$
$13$	−2.50000	−	4.33013i	−0.693375	−	1.20096i	0.277350	−	0.960769i	$-0.410544\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			6.92820i			1.68034i	0.542326	+	0.840168i	$0.317544\pi$
$17$			6.92820i			1.68034i	−0.542326	+	0.840168i	$0.682456\pi$
$18$		0			0
$19$		−	3.46410i		−	0.794719i	−0.917663	−	0.397360i	$-0.869927\pi$
$19$		−	3.46410i		−	0.794719i	0.917663	−	0.397360i	$-0.130073\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.50000	−	7.79423i	−0.938315	−	1.62521i	−0.768613	−	0.639713i	$-0.779053\pi$
$23$	−4.50000	−	7.79423i	−0.938315	−	1.62521i	−0.169701	−	0.985496i	$-0.554280\pi$
$24$		0			0
$25$	−1.00000	+	1.73205i	−0.200000	+	0.346410i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	1.50000	+	0.866025i	0.278543	+	0.160817i	0.632764	−	0.774345i	$-0.281920\pi$
$29$	1.50000	+	0.866025i	0.278543	+	0.160817i	−0.354221	+	0.935162i	$0.615254\pi$
$30$		0			0
$31$	−4.50000	+	2.59808i	−0.808224	+	0.466628i	−0.846339	−	0.532645i	$-0.821198\pi$
$31$	−4.50000	+	2.59808i	−0.808224	+	0.466628i	0.0381148	+	0.999273i	$0.487865\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−3.00000			−0.507093
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.50000	−	2.59808i	0.702782	−	0.405751i	−0.105601	−	0.994409i	$-0.533677\pi$
$41$	4.50000	−	2.59808i	0.702782	−	0.405751i	0.808383	+	0.588657i	$0.200343\pi$
$42$		0			0
$43$	−4.50000	−	2.59808i	−0.686244	−	0.396203i	0.115960	−	0.993254i	$-0.463006\pi$
$43$	−4.50000	−	2.59808i	−0.686244	−	0.396203i	−0.802203	+	0.597051i	$0.796339\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	−0.954441	−	0.298401i	$-0.903547\pi$
$47$	−1.50000	+	2.59808i	−0.218797	+	0.378968i	0.735643	+	0.677369i	$0.236880\pi$
$48$		0			0
$49$	−2.00000	−	3.46410i	−0.285714	−	0.494872i
$50$		0			0
$51$		0			0
$52$		0			0
$53$		0			0		1.00000			$0$
$53$		0			0		−1.00000			$\pi$
$54$		0			0
$55$		−	5.19615i		−	0.700649i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	0.751710	−	0.659494i	$-0.229229\pi$
$59$	−1.50000	−	2.59808i	−0.195283	−	0.338241i	−0.946993	+	0.321253i	$0.895896\pi$
$60$		0			0
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	−0.896258	−	0.443533i	$-0.853725\pi$
$61$	−0.500000	+	0.866025i	−0.0640184	+	0.110883i	0.832240	+	0.554416i	$0.187058\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−7.50000	−	4.33013i	−0.930261	−	0.537086i
$66$		0			0
$67$	−7.50000	+	4.33013i	−0.916271	+	0.529009i	−0.882443	−	0.470418i	$-0.844103\pi$
$67$	−7.50000	+	4.33013i	−0.916271	+	0.529009i	−0.0338274	+	0.999428i	$0.510770\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	−2.00000			−0.234082			−0.117041	−	0.993127i	$-0.537341\pi$
$73$	−2.00000			−0.234082			−0.117041	+	0.993127i	$0.537341\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−4.50000	+	2.59808i	−0.512823	+	0.296078i
$78$		0			0
$79$	−7.50000	−	4.33013i	−0.843816	−	0.487177i	0.0147436	−	0.999891i	$-0.495307\pi$
$79$	−7.50000	−	4.33013i	−0.843816	−	0.487177i	−0.858559	+	0.512714i	$0.828640\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	7.50000	−	12.9904i	0.823232	−	1.42588i	−0.0800311	−	0.996792i	$-0.525502\pi$
$83$	7.50000	−	12.9904i	0.823232	−	1.42588i	0.903263	−	0.429087i	$-0.141165\pi$
$84$		0			0
$85$	6.00000	+	10.3923i	0.650791	+	1.12720i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	6.92820i		−	0.734388i	−0.930144	−	0.367194i	$-0.880318\pi$
$89$		−	6.92820i		−	0.734388i	0.930144	−	0.367194i	$-0.119682\pi$
$90$		0			0
$91$			8.66025i			0.907841i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.00000	−	5.19615i	−0.307794	−	0.533114i
$96$		0			0
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	−0.710742	−	0.703452i	$-0.751641\pi$
$97$	2.50000	−	4.33013i	0.253837	−	0.439658i	0.964579	+	0.263795i	$0.0849741\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−4.50000	−	2.59808i	−0.447767	−	0.258518i	0.259120	−	0.965845i	$-0.416568\pi$
$101$	−4.50000	−	2.59808i	−0.447767	−	0.258518i	−0.706887	+	0.707327i	$0.749901\pi$
$102$		0			0
$103$	1.50000	−	0.866025i	0.147799	−	0.0853320i	−0.424277	−	0.905533i	$-0.639472\pi$
$103$	1.50000	−	0.866025i	0.147799	−	0.0853320i	0.572076	+	0.820201i	$0.306138\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	10.5000	−	6.06218i	0.987757	−	0.570282i	0.0831539	−	0.996537i	$-0.473501\pi$
$113$	10.5000	−	6.06218i	0.987757	−	0.570282i	0.904603	+	0.426255i	$0.140167\pi$
$114$		0			0
$115$	−13.5000	−	7.79423i	−1.25888	−	0.726816i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	6.00000	−	10.3923i	0.550019	−	0.952661i
$120$		0			0
$121$	1.00000	+	1.73205i	0.0909091	+	0.157459i
$122$		0			0
$123$		0			0
$124$		0			0
$125$			12.1244i			1.08444i
$126$		0			0
$127$			10.3923i			0.922168i	0.887357	+	0.461084i	$0.152539\pi$
$127$			10.3923i			0.922168i	−0.887357	+	0.461084i	$0.847461\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1.50000	−	2.59808i	−0.131056	−	0.226995i	0.793028	−	0.609185i	$-0.208503\pi$
$131$	−1.50000	−	2.59808i	−0.131056	−	0.226995i	−0.924084	+	0.382190i	$0.875170\pi$
$132$		0			0
$133$	−3.00000	+	5.19615i	−0.260133	+	0.450564i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−7.50000	−	4.33013i	−0.640768	−	0.369948i	0.144142	−	0.989557i	$-0.453958\pi$
$137$	−7.50000	−	4.33013i	−0.640768	−	0.369948i	−0.784910	+	0.619609i	$0.787291\pi$
$138$		0			0
$139$	−1.50000	+	0.866025i	−0.127228	+	0.0734553i	−0.562263	−	0.826958i	$-0.690069\pi$
$139$	−1.50000	+	0.866025i	−0.127228	+	0.0734553i	0.435035	+	0.900414i	$0.356736\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−15.0000			−1.25436
$144$		0			0
$145$	3.00000			0.249136
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.50000	−	4.33013i	0.614424	−	0.354738i	−0.160271	−	0.987073i	$-0.551237\pi$
$149$	7.50000	−	4.33013i	0.614424	−	0.354738i	0.774695	+	0.632335i	$0.217903\pi$
$150$		0			0
$151$	−7.50000	−	4.33013i	−0.610341	−	0.352381i	0.162758	−	0.986666i	$-0.447961\pi$
$151$	−7.50000	−	4.33013i	−0.610341	−	0.352381i	−0.773099	+	0.634285i	$0.781294\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.50000	+	7.79423i	−0.361449	+	0.626048i
$156$		0			0
$157$	−0.500000	−	0.866025i	−0.0399043	−	0.0691164i	0.845383	−	0.534160i	$-0.179372\pi$
$157$	−0.500000	−	0.866025i	−0.0399043	−	0.0691164i	−0.885288	+	0.465044i	$0.846039\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			15.5885i			1.22854i
$162$		0			0
$163$		−	17.3205i		−	1.35665i	−0.734763	−	0.678323i	$-0.762707\pi$
$163$		−	17.3205i		−	1.35665i	0.734763	−	0.678323i	$-0.237293\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.50000	+	2.59808i	0.116073	+	0.201045i	0.918208	−	0.396098i	$-0.129636\pi$
$167$	1.50000	+	2.59808i	0.116073	+	0.201045i	−0.802135	+	0.597143i	$0.796303\pi$
$168$		0			0
$169$	−6.00000	+	10.3923i	−0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−16.5000	−	9.52628i	−1.25447	−	0.724270i	−0.282477	−	0.959274i	$-0.591156\pi$
$173$	−16.5000	−	9.52628i	−1.25447	−	0.724270i	−0.971994	+	0.235004i	$0.924490\pi$
$174$		0			0
$175$	3.00000	−	1.73205i	0.226779	−	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	12.0000			0.896922			0.448461	−	0.893802i	$-0.351972\pi$
$179$	12.0000			0.896922			0.448461	+	0.893802i	$0.351972\pi$
$180$		0			0
$181$	−10.0000			−0.743294			−0.371647	−	0.928374i	$-0.621207\pi$
$181$	−10.0000			−0.743294			−0.371647	+	0.928374i	$0.621207\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−3.00000	+	1.73205i	−0.220564	+	0.127343i
$186$		0			0
$187$	18.0000	+	10.3923i	1.31629	+	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−1.50000	+	2.59808i	−0.108536	+	0.187990i	−0.915177	−	0.403051i	$-0.867950\pi$
$191$	−1.50000	+	2.59808i	−0.108536	+	0.187990i	0.806641	+	0.591041i	$0.201283\pi$
$192$		0			0
$193$	6.50000	+	11.2583i	0.467880	+	0.810392i	0.999326	−	0.0366998i	$-0.0116845\pi$
$193$	6.50000	+	11.2583i	0.467880	+	0.810392i	−0.531446	+	0.847092i	$0.678351\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	6.92820i		−	0.493614i	−0.969065	−	0.246807i	$-0.920619\pi$
$197$		−	6.92820i		−	0.493614i	0.969065	−	0.246807i	$-0.0793814\pi$
$198$		0			0
$199$		−	24.2487i		−	1.71895i	−0.511182	−	0.859473i	$-0.670792\pi$
$199$		−	24.2487i		−	1.71895i	0.511182	−	0.859473i	$-0.329208\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−1.50000	−	2.59808i	−0.105279	−	0.182349i
$204$		0			0
$205$	4.50000	−	7.79423i	0.314294	−	0.544373i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−9.00000	−	5.19615i	−0.622543	−	0.359425i
$210$		0			0
$211$	16.5000	−	9.52628i	1.13591	−	0.655816i	0.190493	−	0.981689i	$-0.438991\pi$
$211$	16.5000	−	9.52628i	1.13591	−	0.655816i	0.945414	+	0.325872i	$0.105658\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−9.00000			−0.613795
$216$		0			0
$217$	9.00000			0.610960
$218$		0			0
$219$		0			0
$220$		0			0
$221$	30.0000	−	17.3205i	2.01802	−	1.16510i
$222$		0			0
$223$	22.5000	+	12.9904i	1.50671	+	0.869900i	0.999970	+	0.00780243i	$0.00248362\pi$
$223$	22.5000	+	12.9904i	1.50671	+	0.869900i	0.506742	+	0.862098i	$0.330850\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	−0.811943	−	0.583736i	$-0.801590\pi$
$227$	1.50000	−	2.59808i	0.0995585	−	0.172440i	0.911502	+	0.411296i	$0.134924\pi$
$228$		0			0
$229$	−8.50000	−	14.7224i	−0.561696	−	0.972886i	−0.997349	−	0.0727709i	$-0.976816\pi$
$229$	−8.50000	−	14.7224i	−0.561696	−	0.972886i	0.435653	−	0.900115i	$-0.356518\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			13.8564i			0.907763i	0.891062	+	0.453882i	$0.149961\pi$
$233$			13.8564i			0.907763i	−0.891062	+	0.453882i	$0.850039\pi$
$234$		0			0
$235$			5.19615i			0.338960i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	7.50000	+	12.9904i	0.485135	+	0.840278i	0.999854	−	0.0170808i	$-0.00543724\pi$
$239$	7.50000	+	12.9904i	0.485135	+	0.840278i	−0.514719	+	0.857359i	$0.672104\pi$
$240$		0			0
$241$	8.50000	−	14.7224i	0.547533	−	0.948355i	−0.450910	−	0.892570i	$-0.648900\pi$
$241$	8.50000	−	14.7224i	0.547533	−	0.948355i	0.998443	−	0.0557856i	$-0.0177663\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.00000	−	3.46410i	−0.383326	−	0.221313i
$246$		0			0
$247$	−15.0000	+	8.66025i	−0.954427	+	0.551039i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	12.0000			0.757433			0.378717	−	0.925513i	$-0.376365\pi$
$251$	12.0000			0.757433			0.378717	+	0.925513i	$0.376365\pi$
$252$		0			0
$253$	−27.0000			−1.69748
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.50000	+	0.866025i	−0.0935674	+	0.0540212i	−0.546054	−	0.837750i	$-0.683871\pi$
$257$	−1.50000	+	0.866025i	−0.0935674	+	0.0540212i	0.452486	+	0.891771i	$0.350537\pi$
$258$		0			0
$259$	3.00000	+	1.73205i	0.186411	+	0.107624i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	4.50000	−	7.79423i	0.277482	−	0.480613i	−0.693276	−	0.720672i	$-0.743833\pi$
$263$	4.50000	−	7.79423i	0.277482	−	0.480613i	0.970758	+	0.240059i	$0.0771668\pi$
$264$		0			0
$265$		0			0
$266$		0			0
$267$		0			0
$268$		0			0
$269$			27.7128i			1.68968i	0.535019	+	0.844840i	$0.320304\pi$
$269$			27.7128i			1.68968i	−0.535019	+	0.844840i	$0.679696\pi$
$270$		0			0
$271$			24.2487i			1.47300i	0.676435	+	0.736502i	$0.263524\pi$
$271$			24.2487i			1.47300i	−0.676435	+	0.736502i	$0.736476\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	3.00000	+	5.19615i	0.180907	+	0.313340i
$276$		0			0
$277$	9.50000	−	16.4545i	0.570800	−	0.988654i	−0.425684	−	0.904872i	$-0.639967\pi$
$277$	9.50000	−	16.4545i	0.570800	−	0.988654i	0.996484	−	0.0837823i	$-0.0267000\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	16.5000	+	9.52628i	0.984307	+	0.568290i	0.903568	−	0.428445i	$-0.140938\pi$
$281$	16.5000	+	9.52628i	0.984307	+	0.568290i	0.0807396	+	0.996735i	$0.474272\pi$
$282$		0			0
$283$	−13.5000	+	7.79423i	−0.802492	+	0.463319i	−0.844342	−	0.535805i	$-0.820008\pi$
$283$	−13.5000	+	7.79423i	−0.802492	+	0.463319i	0.0418500	+	0.999124i	$0.486675\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−9.00000			−0.531253
$288$		0			0
$289$	−31.0000			−1.82353
$290$		0			0
$291$		0			0
$292$		0			0
$293$	1.50000	−	0.866025i	0.0876309	−	0.0505937i	−0.455544	−	0.890213i	$-0.650555\pi$
$293$	1.50000	−	0.866025i	0.0876309	−	0.0505937i	0.543175	+	0.839619i	$0.317222\pi$
$294$		0			0
$295$	−4.50000	−	2.59808i	−0.262000	−	0.151266i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−22.5000	+	38.9711i	−1.30121	+	2.25376i
$300$		0			0
$301$	4.50000	+	7.79423i	0.259376	+	0.449252i
$302$		0			0
$303$		0			0
$304$		0			0
$305$			1.73205i			0.0991769i
$306$		0			0
$307$			10.3923i			0.593120i	0.955014	+	0.296560i	$0.0958395\pi$
$307$			10.3923i			0.593120i	−0.955014	+	0.296560i	$0.904160\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	7.50000	+	12.9904i	0.425286	+	0.736617i	0.996447	−	0.0842210i	$-0.0268402\pi$
$311$	7.50000	+	12.9904i	0.425286	+	0.736617i	−0.571161	+	0.820838i	$0.693507\pi$
$312$		0			0
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	−0.851549	−	0.524276i	$-0.824336\pi$
$313$	0.500000	−	0.866025i	0.0282617	−	0.0489506i	0.879810	+	0.475325i	$0.157669\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−10.5000	−	6.06218i	−0.589739	−	0.340486i	0.175255	−	0.984523i	$-0.443925\pi$
$317$	−10.5000	−	6.06218i	−0.589739	−	0.340486i	−0.764994	+	0.644037i	$0.777258\pi$
$318$		0			0
$319$	4.50000	−	2.59808i	0.251952	−	0.145464i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	24.0000			1.33540
$324$		0			0
$325$	10.0000			0.554700
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.50000	−	2.59808i	0.248093	−	0.143237i
$330$		0			0
$331$	1.50000	+	0.866025i	0.0824475	+	0.0476011i	0.540657	−	0.841243i	$-0.318176\pi$
$331$	1.50000	+	0.866025i	0.0824475	+	0.0476011i	−0.458209	+	0.888844i	$0.651509\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−7.50000	+	12.9904i	−0.409769	+	0.709740i
$336$		0			0
$337$	−3.50000	−	6.06218i	−0.190657	−	0.330228i	0.754811	−	0.655942i	$-0.227729\pi$
$337$	−3.50000	−	6.06218i	−0.190657	−	0.330228i	−0.945468	+	0.325714i	$0.894395\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			15.5885i			0.844162i
$342$		0			0
$343$			19.0526i			1.02874i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	4.50000	+	7.79423i	0.241573	+	0.418416i	0.961162	−	0.275983i	$-0.0890035\pi$
$347$	4.50000	+	7.79423i	0.241573	+	0.418416i	−0.719590	+	0.694399i	$0.755670\pi$
$348$		0			0
$349$	−6.50000	+	11.2583i	−0.347937	+	0.602645i	−0.985883	−	0.167437i	$-0.946451\pi$
$349$	−6.50000	+	11.2583i	−0.347937	+	0.602645i	0.637946	+	0.770081i	$0.279784\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	4.50000	+	2.59808i	0.239511	+	0.138282i	0.614952	−	0.788565i	$-0.289175\pi$
$353$	4.50000	+	2.59808i	0.239511	+	0.138282i	−0.375441	+	0.926846i	$0.622509\pi$
$354$		0			0
$355$	18.0000	−	10.3923i	0.955341	−	0.551566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	7.00000			0.368421
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−3.00000	+	1.73205i	−0.157027	+	0.0906597i
$366$		0			0
$367$	16.5000	+	9.52628i	0.861293	+	0.497268i	0.864445	−	0.502727i	$-0.167670\pi$
$367$	16.5000	+	9.52628i	0.861293	+	0.497268i	−0.00315207	+	0.999995i	$0.501003\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	0.972556	−	0.232671i	$-0.0747464\pi$
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	−0.687776	+	0.725923i	$0.741413\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	8.66025i		−	0.446026i
$378$		0			0
$379$			17.3205i			0.889695i	0.895606	+	0.444847i	$0.146742\pi$
$379$			17.3205i			0.889695i	−0.895606	+	0.444847i	$0.853258\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	13.5000	+	23.3827i	0.689818	+	1.19480i	0.971897	+	0.235408i	$0.0756427\pi$
$383$	13.5000	+	23.3827i	0.689818	+	1.19480i	−0.282079	+	0.959391i	$0.591024\pi$
$384$		0			0
$385$	−4.50000	+	7.79423i	−0.229341	+	0.397231i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	31.5000	+	18.1865i	1.59711	+	0.922094i	0.992040	+	0.125924i	$0.0401896\pi$
$389$	31.5000	+	18.1865i	1.59711	+	0.922094i	0.605074	+	0.796170i	$0.293144\pi$
$390$		0			0
$391$	54.0000	−	31.1769i	2.73090	−	1.57668i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−15.0000			−0.754732
$396$		0			0
$397$	22.0000			1.10415			0.552074	−	0.833795i	$-0.313837\pi$
$397$	22.0000			1.10415			0.552074	+	0.833795i	$0.313837\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	16.5000	−	9.52628i	0.823971	−	0.475720i	−0.0278131	−	0.999613i	$-0.508854\pi$
$401$	16.5000	−	9.52628i	0.823971	−	0.475720i	0.851784	+	0.523893i	$0.175521\pi$
$402$		0			0
$403$	22.5000	+	12.9904i	1.12080	+	0.647097i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−3.00000	+	5.19615i	−0.148704	+	0.257564i
$408$		0			0
$409$	−15.5000	−	26.8468i	−0.766426	−	1.32749i	−0.939490	−	0.342578i	$-0.888700\pi$
$409$	−15.5000	−	26.8468i	−0.766426	−	1.32749i	0.173064	−	0.984911i	$-0.444633\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			5.19615i			0.255686i
$414$		0			0
$415$		−	25.9808i		−	1.27535i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−7.50000	−	12.9904i	−0.366399	−	0.634622i	0.622601	−	0.782540i	$-0.286076\pi$
$419$	−7.50000	−	12.9904i	−0.366399	−	0.634622i	−0.989000	+	0.147918i	$0.952743\pi$
$420$		0			0
$421$	−8.50000	+	14.7224i	−0.414265	+	0.717527i	−0.995351	−	0.0963145i	$-0.969295\pi$
$421$	−8.50000	+	14.7224i	−0.414265	+	0.717527i	0.581086	+	0.813842i	$0.302628\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.0000	−	6.92820i	−0.582086	−	0.336067i
$426$		0			0
$427$	1.50000	−	0.866025i	0.0725901	−	0.0419099i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		0			0			−	1.00000i	$-0.5\pi$
$431$		0			0				1.00000i	$0.5\pi$
$432$		0			0
$433$	14.0000			0.672797			0.336399	−	0.941720i	$-0.390791\pi$
$433$	14.0000			0.672797			0.336399	+	0.941720i	$0.390791\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−27.0000	+	15.5885i	−1.29159	+	0.745697i
$438$		0			0
$439$	4.50000	+	2.59808i	0.214773	+	0.123999i	0.603528	−	0.797342i	$-0.293761\pi$
$439$	4.50000	+	2.59808i	0.214773	+	0.123999i	−0.388755	+	0.921341i	$0.627095\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−4.50000	+	7.79423i	−0.213801	+	0.370315i	−0.952901	−	0.303281i	$-0.901918\pi$
$443$	−4.50000	+	7.79423i	−0.213801	+	0.370315i	0.739100	+	0.673596i	$0.235251\pi$
$444$		0			0
$445$	−6.00000	−	10.3923i	−0.284427	−	0.492642i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	20.7846i		−	0.980886i	−0.871473	−	0.490443i	$-0.836835\pi$
$449$		−	20.7846i		−	0.980886i	0.871473	−	0.490443i	$-0.163165\pi$
$450$		0			0
$451$		−	15.5885i		−	0.734032i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	7.50000	+	12.9904i	0.351605	+	0.608998i
$456$		0			0
$457$	14.5000	−	25.1147i	0.678281	−	1.17482i	−0.297217	−	0.954810i	$-0.596058\pi$
$457$	14.5000	−	25.1147i	0.678281	−	1.17482i	0.975498	−	0.220008i	$-0.0706083\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−22.5000	−	12.9904i	−1.04793	−	0.605022i	−0.125860	−	0.992048i	$-0.540169\pi$
$461$	−22.5000	−	12.9904i	−1.04793	−	0.605022i	−0.922069	+	0.387026i	$0.873503\pi$
$462$		0			0
$463$	−16.5000	+	9.52628i	−0.766820	+	0.442724i	−0.831739	−	0.555167i	$-0.812654\pi$
$463$	−16.5000	+	9.52628i	−0.766820	+	0.442724i	0.0649190	+	0.997891i	$0.479321\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.0000			−0.555294			−0.277647	−	0.960683i	$-0.589555\pi$
$467$	−12.0000			−0.555294			−0.277647	+	0.960683i	$0.589555\pi$
$468$		0			0
$469$	15.0000			0.692636
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−13.5000	+	7.79423i	−0.620731	+	0.358379i
$474$		0			0
$475$	6.00000	+	3.46410i	0.275299	+	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−7.50000	+	12.9904i	−0.342684	+	0.593546i	−0.984930	−	0.172953i	$-0.944669\pi$
$479$	−7.50000	+	12.9904i	−0.342684	+	0.593546i	0.642246	+	0.766498i	$0.278003\pi$
$480$		0			0
$481$	5.00000	+	8.66025i	0.227980	+	0.394874i
$482$		0			0
$483$		0			0
$484$		0			0
$485$		−	8.66025i		−	0.393242i
$486$		0			0
$487$		−	3.46410i		−	0.156973i	−0.996915	−	0.0784867i	$-0.974991\pi$
$487$		−	3.46410i		−	0.156973i	0.996915	−	0.0784867i	$-0.0250088\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	10.5000	+	18.1865i	0.473858	+	0.820747i	0.999552	−	0.0299272i	$-0.00952753\pi$
$491$	10.5000	+	18.1865i	0.473858	+	0.820747i	−0.525694	+	0.850674i	$0.676194\pi$
$492$		0			0
$493$	−6.00000	+	10.3923i	−0.270226	+	0.468046i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−18.0000	−	10.3923i	−0.807410	−	0.466159i
$498$		0			0
$499$	−13.5000	+	7.79423i	−0.604343	+	0.348918i	−0.770748	−	0.637140i	$-0.780117\pi$
$499$	−13.5000	+	7.79423i	−0.604343	+	0.348918i	0.166405	+	0.986057i	$0.446784\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−12.0000			−0.535054			−0.267527	−	0.963550i	$-0.586206\pi$
$503$	−12.0000			−0.535054			−0.267527	+	0.963550i	$0.586206\pi$
$504$		0			0
$505$	−9.00000			−0.400495
$506$		0			0
$507$		0			0
$508$		0			0
$509$	31.5000	−	18.1865i	1.39621	−	0.806104i	0.402219	−	0.915543i	$-0.368239\pi$
$509$	31.5000	−	18.1865i	1.39621	−	0.806104i	0.993993	+	0.109439i	$0.0349055\pi$
$510$		0			0
$511$	3.00000	+	1.73205i	0.132712	+	0.0766214i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	1.50000	−	2.59808i	0.0660979	−	0.114485i
$516$		0			0
$517$	4.50000	+	7.79423i	0.197910	+	0.342790i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		0			0		1.00000			$0$
$521$		0			0		−1.00000			$\pi$
$522$		0			0
$523$			17.3205i			0.757373i	0.925525	+	0.378686i	$0.123624\pi$
$523$			17.3205i			0.757373i	−0.925525	+	0.378686i	$0.876376\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−18.0000	−	31.1769i	−0.784092	−	1.35809i
$528$		0			0
$529$	−29.0000	+	50.2295i	−1.26087	+	2.18389i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−22.5000	−	12.9904i	−0.974583	−	0.562676i
$534$		0			0
$535$	−18.0000	+	10.3923i	−0.778208	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0000			−0.516877
$540$		0			0
$541$	2.00000			0.0859867			0.0429934	−	0.999075i	$-0.486311\pi$
$541$	2.00000			0.0859867			0.0429934	+	0.999075i	$0.486311\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	21.0000	−	12.1244i	0.899541	−	0.519350i
$546$		0			0
$547$	−34.5000	−	19.9186i	−1.47511	−	0.851657i	−0.475507	−	0.879712i	$-0.657735\pi$
$547$	−34.5000	−	19.9186i	−1.47511	−	0.851657i	−0.999606	+	0.0280547i	$0.991069\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	3.00000	−	5.19615i	0.127804	−	0.221364i
$552$		0			0
$553$	7.50000	+	12.9904i	0.318932	+	0.552407i
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	27.7128i		−	1.17423i	−0.809504	−	0.587115i	$-0.800264\pi$
$557$		−	27.7128i		−	1.17423i	0.809504	−	0.587115i	$-0.199736\pi$
$558$		0			0
$559$			25.9808i			1.09887i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.50000	+	7.79423i	0.189652	+	0.328488i	0.945134	−	0.326682i	$-0.105931\pi$
$563$	4.50000	+	7.79423i	0.189652	+	0.328488i	−0.755482	+	0.655169i	$0.772597\pi$
$564$		0			0
$565$	10.5000	−	18.1865i	0.441738	−	0.765113i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−1.50000	−	0.866025i	−0.0628833	−	0.0363057i	0.468229	−	0.883607i	$-0.344892\pi$
$569$	−1.50000	−	0.866025i	−0.0628833	−	0.0363057i	−0.531112	+	0.847302i	$0.678226\pi$
$570$		0			0
$571$	28.5000	−	16.4545i	1.19269	−	0.688599i	0.233773	−	0.972291i	$-0.424893\pi$
$571$	28.5000	−	16.4545i	1.19269	−	0.688599i	0.958915	+	0.283693i	$0.0915595\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	18.0000			0.750652
$576$		0			0
$577$	38.0000			1.58196			0.790980	−	0.611842i	$-0.209571\pi$
$577$	38.0000			1.58196			0.790980	+	0.611842i	$0.209571\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−22.5000	+	12.9904i	−0.933457	+	0.538932i
$582$		0			0
$583$		0			0
$584$		0			0
$585$		0			0
$586$		0			0
$587$	19.5000	−	33.7750i	0.804851	−	1.39404i	−0.111540	−	0.993760i	$-0.535578\pi$
$587$	19.5000	−	33.7750i	0.804851	−	1.39404i	0.916392	−	0.400283i	$-0.131088\pi$
$588$		0			0
$589$	9.00000	+	15.5885i	0.370839	+	0.642311i
$590$		0			0
$591$		0			0
$592$		0			0
$593$			34.6410i			1.42254i	0.702921	+	0.711268i	$0.251879\pi$
$593$			34.6410i			1.42254i	−0.702921	+	0.711268i	$0.748121\pi$
$594$		0			0
$595$		−	20.7846i		−	0.852086i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−22.5000	−	38.9711i	−0.919325	−	1.59232i	−0.800443	−	0.599409i	$-0.795402\pi$
$599$	−22.5000	−	38.9711i	−0.919325	−	1.59232i	−0.118882	−	0.992908i	$-0.537931\pi$
$600$		0			0
$601$	−5.50000	+	9.52628i	−0.224350	+	0.388585i	−0.956124	−	0.292962i	$-0.905359\pi$
$601$	−5.50000	+	9.52628i	−0.224350	+	0.388585i	0.731774	+	0.681547i	$0.238692\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	3.00000	+	1.73205i	0.121967	+	0.0704179i
$606$		0			0
$607$	1.50000	−	0.866025i	0.0608831	−	0.0351509i	−0.469249	−	0.883066i	$-0.655475\pi$
$607$	1.50000	−	0.866025i	0.0608831	−	0.0351509i	0.530133	+	0.847915i	$0.322142\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	15.0000			0.606835
$612$		0			0
$613$	−26.0000			−1.05013			−0.525065	−	0.851062i	$-0.675959\pi$
$613$	−26.0000			−1.05013			−0.525065	+	0.851062i	$0.675959\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−19.5000	+	11.2583i	−0.785040	+	0.453243i	−0.838214	−	0.545342i	$-0.816400\pi$
$617$	−19.5000	+	11.2583i	−0.785040	+	0.453243i	0.0531732	+	0.998585i	$0.483066\pi$
$618$		0			0
$619$	25.5000	+	14.7224i	1.02493	+	0.591744i	0.915529	−	0.402253i	$-0.131773\pi$
$619$	25.5000	+	14.7224i	1.02493	+	0.591744i	0.109403	+	0.993997i	$0.465106\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−6.00000	+	10.3923i	−0.240385	+	0.416359i
$624$		0			0
$625$	5.50000	+	9.52628i	0.220000	+	0.381051i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	13.8564i		−	0.552491i
$630$		0			0
$631$		−	17.3205i		−	0.689519i	−0.938691	−	0.344759i	$-0.887961\pi$
$631$		−	17.3205i		−	0.689519i	0.938691	−	0.344759i	$-0.112039\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.00000	+	15.5885i	0.357154	+	0.618609i
$636$		0			0
$637$	−10.0000	+	17.3205i	−0.396214	+	0.686264i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−25.5000	−	14.7224i	−1.00719	−	0.581501i	−0.0968219	−	0.995302i	$-0.530868\pi$
$641$	−25.5000	−	14.7224i	−1.00719	−	0.581501i	−0.910368	+	0.413801i	$0.864201\pi$
$642$		0			0
$643$	4.50000	−	2.59808i	0.177463	−	0.102458i	−0.408637	−	0.912697i	$-0.633996\pi$
$643$	4.50000	−	2.59808i	0.177463	−	0.102458i	0.586100	+	0.810239i	$0.300663\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	24.0000			0.943537			0.471769	−	0.881722i	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		0			0
$649$	−9.00000			−0.353281
$650$		0			0
$651$		0			0
$652$		0			0
$653$	13.5000	−	7.79423i	0.528296	−	0.305012i	−0.212026	−	0.977264i	$-0.568006\pi$
$653$	13.5000	−	7.79423i	0.528296	−	0.305012i	0.740322	+	0.672252i	$0.234673\pi$
$654$		0			0
$655$	−4.50000	−	2.59808i	−0.175830	−	0.101515i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.50000	−	2.59808i	0.0584317	−	0.101207i	−0.835330	−	0.549749i	$-0.814723\pi$
$659$	1.50000	−	2.59808i	0.0584317	−	0.101207i	0.893762	+	0.448542i	$0.148057\pi$
$660$		0			0
$661$	−24.5000	−	42.4352i	−0.952940	−	1.65054i	−0.739014	−	0.673690i	$-0.764708\pi$
$661$	−24.5000	−	42.4352i	−0.952940	−	1.65054i	−0.213925	−	0.976850i	$-0.568625\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$			10.3923i			0.402996i
$666$		0			0
$667$		−	15.5885i		−	0.603587i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	1.50000	+	2.59808i	0.0579069	+	0.100298i
$672$		0			0
$673$	18.5000	−	32.0429i	0.713123	−	1.23516i	−0.250557	−	0.968102i	$-0.580614\pi$
$673$	18.5000	−	32.0429i	0.713123	−	1.23516i	0.963679	−	0.267063i	$-0.0860531\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	31.5000	+	18.1865i	1.21064	+	0.698965i	0.962899	−	0.269860i	$-0.0869775\pi$
$677$	31.5000	+	18.1865i	1.21064	+	0.698965i	0.247744	+	0.968826i	$0.420311\pi$
$678$		0			0
$679$	−7.50000	+	4.33013i	−0.287824	+	0.166175i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	−15.0000			−0.573121
$686$		0			0
$687$		0			0
$688$		0			0
$689$		0			0
$690$		0			0
$691$	−4.50000	−	2.59808i	−0.171188	−	0.0988355i	0.411958	−	0.911203i	$-0.364845\pi$
$691$	−4.50000	−	2.59808i	−0.171188	−	0.0988355i	−0.583146	+	0.812367i	$0.698178\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.50000	+	2.59808i	−0.0568982	+	0.0985506i
$696$		0			0
$697$	18.0000	+	31.1769i	0.681799	+	1.18091i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	13.8564i		−	0.523349i	−0.965156	−	0.261675i	$-0.915725\pi$
$701$		−	13.8564i		−	0.523349i	0.965156	−	0.261675i	$-0.0842747\pi$
$702$		0			0
$703$			6.92820i			0.261302i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	4.50000	+	7.79423i	0.169240	+	0.293132i
$708$		0			0
$709$	9.50000	−	16.4545i	0.356780	−	0.617961i	−0.630641	−	0.776075i	$-0.717208\pi$
$709$	9.50000	−	16.4545i	0.356780	−	0.617961i	0.987421	+	0.158114i	$0.0505412\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	40.5000	+	23.3827i	1.51674	+	0.875688i
$714$		0			0
$715$	−22.5000	+	12.9904i	−0.841452	+	0.485813i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−24.0000			−0.895049			−0.447524	−	0.894272i	$-0.647694\pi$
$719$	−24.0000			−0.895049			−0.447524	+	0.894272i	$0.647694\pi$
$720$		0			0
$721$	−3.00000			−0.111726
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.00000	+	1.73205i	−0.111417	+	0.0643268i
$726$		0			0
$727$	−31.5000	−	18.1865i	−1.16827	−	0.674501i	−0.214998	−	0.976614i	$-0.568975\pi$
$727$	−31.5000	−	18.1865i	−1.16827	−	0.674501i	−0.953272	+	0.302113i	$0.902308\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	18.0000	−	31.1769i	0.665754	−	1.15312i
$732$		0			0
$733$	−2.50000	−	4.33013i	−0.0923396	−	0.159937i	0.816156	−	0.577832i	$-0.196101\pi$
$733$	−2.50000	−	4.33013i	−0.0923396	−	0.159937i	−0.908495	+	0.417895i	$0.862768\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$			25.9808i			0.957014i
$738$		0			0
$739$			31.1769i			1.14686i	0.819254	+	0.573431i	$0.194388\pi$
$739$			31.1769i			1.14686i	−0.819254	+	0.573431i	$0.805612\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	13.5000	+	23.3827i	0.495267	+	0.857828i	0.999985	−	0.00545664i	$-0.00173691\pi$
$743$	13.5000	+	23.3827i	0.495267	+	0.857828i	−0.504718	+	0.863284i	$0.668404\pi$
$744$		0			0
$745$	7.50000	−	12.9904i	0.274779	−	0.475931i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	18.0000	+	10.3923i	0.657706	+	0.379727i
$750$		0			0
$751$	−10.5000	+	6.06218i	−0.383150	+	0.221212i	−0.679188	−	0.733964i	$-0.737668\pi$
$751$	−10.5000	+	6.06218i	−0.383150	+	0.221212i	0.296038	+	0.955176i	$0.404335\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−15.0000			−0.545906
$756$		0			0
$757$	26.0000			0.944986			0.472493	−	0.881334i	$-0.343354\pi$
$757$	26.0000			0.944986			0.472493	+	0.881334i	$0.343354\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−19.5000	+	11.2583i	−0.706874	+	0.408114i	−0.809903	−	0.586564i	$-0.800480\pi$
$761$	−19.5000	+	11.2583i	−0.706874	+	0.408114i	0.103028	+	0.994678i	$0.467147\pi$
$762$		0			0
$763$	−21.0000	−	12.1244i	−0.760251	−	0.438931i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7.50000	+	12.9904i	−0.270809	+	0.469055i
$768$		0			0
$769$	−11.5000	−	19.9186i	−0.414701	−	0.718283i	0.580696	−	0.814120i	$-0.302780\pi$
$769$	−11.5000	−	19.9186i	−0.414701	−	0.718283i	−0.995397	+	0.0958377i	$0.969447\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$		−	27.7128i		−	0.996761i	−0.866959	−	0.498380i	$-0.833928\pi$
$773$		−	27.7128i		−	0.996761i	0.866959	−	0.498380i	$-0.166072\pi$
$774$		0			0
$775$		−	10.3923i		−	0.373303i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−9.00000	−	15.5885i	−0.322458	−	0.558514i
$780$		0			0
$781$	18.0000	−	31.1769i	0.644091	−	1.11560i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1.50000	−	0.866025i	−0.0535373	−	0.0309098i
$786$		0			0
$787$	−25.5000	+	14.7224i	−0.908977	+	0.524798i	−0.880102	−	0.474785i	$-0.842526\pi$
$787$	−25.5000	+	14.7224i	−0.908977	+	0.524798i	−0.0288750	+	0.999583i	$0.509192\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−21.0000			−0.746674
$792$		0			0
$793$	5.00000			0.177555
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−28.5000	+	16.4545i	−1.00952	+	0.582848i	−0.911052	−	0.412292i	$-0.864728\pi$
$797$	−28.5000	+	16.4545i	−1.00952	+	0.582848i	−0.0984702	+	0.995140i	$0.531395\pi$
$798$		0			0
$799$	−18.0000	−	10.3923i	−0.636794	−	0.367653i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3.00000	+	5.19615i	−0.105868	+	0.183368i
$804$		0			0
$805$	13.5000	+	23.3827i	0.475812	+	0.824131i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	48.4974i		−	1.70508i	−0.522663	−	0.852539i	$-0.675061\pi$
$809$		−	48.4974i		−	1.70508i	0.522663	−	0.852539i	$-0.324939\pi$
$810$		0			0
$811$		−	31.1769i		−	1.09477i	−0.836881	−	0.547385i	$-0.815623\pi$
$811$		−	31.1769i		−	1.09477i	0.836881	−	0.547385i	$-0.184377\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−15.0000	−	25.9808i	−0.525427	−	0.910066i
$816$		0			0
$817$	−9.00000	+	15.5885i	−0.314870	+	0.545371i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	25.5000	+	14.7224i	0.889956	+	0.513816i	0.873928	−	0.486055i	$-0.161565\pi$
$821$	25.5000	+	14.7224i	0.889956	+	0.513816i	0.0160280	+	0.999872i	$0.494898\pi$
$822$		0			0
$823$	−28.5000	+	16.4545i	−0.993448	+	0.573567i	−0.906303	−	0.422628i	$-0.861108\pi$
$823$	−28.5000	+	16.4545i	−0.993448	+	0.573567i	−0.0871445	+	0.996196i	$0.527774\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−36.0000			−1.25184			−0.625921	−	0.779886i	$-0.715277\pi$
$827$	−36.0000			−1.25184			−0.625921	+	0.779886i	$0.715277\pi$
$828$		0			0
$829$	−10.0000			−0.347314			−0.173657	−	0.984806i	$-0.555558\pi$
$829$	−10.0000			−0.347314			−0.173657	+	0.984806i	$0.555558\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	24.0000	−	13.8564i	0.831551	−	0.480096i
$834$		0			0
$835$	4.50000	+	2.59808i	0.155729	+	0.0899101i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	22.5000	−	38.9711i	0.776786	−	1.34543i	−0.156999	−	0.987599i	$-0.550182\pi$
$839$	22.5000	−	38.9711i	0.776786	−	1.34543i	0.933785	−	0.357834i	$-0.116485\pi$
$840$		0			0
$841$	−13.0000	−	22.5167i	−0.448276	−	0.776437i
$842$		0			0
$843$		0			0
$844$		0			0
$845$			20.7846i			0.715012i
$846$		0			0
$847$		−	3.46410i		−	0.119028i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	9.00000	+	15.5885i	0.308516	+	0.534365i
$852$		0			0
$853$	−24.5000	+	42.4352i	−0.838864	+	1.45296i	0.0519811	+	0.998648i	$0.483446\pi$
$853$	−24.5000	+	42.4352i	−0.838864	+	1.45296i	−0.890845	+	0.454307i	$0.849887\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	28.5000	+	16.4545i	0.973541	+	0.562074i	0.900314	−	0.435241i	$-0.143337\pi$
$857$	28.5000	+	16.4545i	0.973541	+	0.562074i	0.0732274	+	0.997315i	$0.476670\pi$
$858$		0			0
$859$	−19.5000	+	11.2583i	−0.665331	+	0.384129i	−0.794305	−	0.607519i	$-0.792165\pi$
$859$	−19.5000	+	11.2583i	−0.665331	+	0.384129i	0.128974	+	0.991648i	$0.458832\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	24.0000			0.816970			0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000			0.816970			0.408485	+	0.912765i	$0.366057\pi$
$864$		0			0
$865$	−33.0000			−1.12203
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−22.5000	+	12.9904i	−0.763260	+	0.440668i
$870$		0			0
$871$	37.5000	+	21.6506i	1.27064	+	0.733604i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.5000	−	18.1865i	0.354965	−	0.614817i
$876$		0			0
$877$	−6.50000	−	11.2583i	−0.219489	−	0.380167i	0.735163	−	0.677891i	$-0.237106\pi$
$877$	−6.50000	−	11.2583i	−0.219489	−	0.380167i	−0.954652	+	0.297724i	$0.903772\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	20.7846i		−	0.700251i	−0.936703	−	0.350126i	$-0.886139\pi$
$881$		−	20.7846i		−	0.700251i	0.936703	−	0.350126i	$-0.113861\pi$
$882$		0			0
$883$		−	17.3205i		−	0.582882i	−0.956589	−	0.291441i	$-0.905865\pi$
$883$		−	17.3205i		−	0.582882i	0.956589	−	0.291441i	$-0.0941346\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.5000	−	49.3634i	−0.956936	−	1.65746i	−0.729873	−	0.683582i	$-0.760421\pi$
$887$	−28.5000	−	49.3634i	−0.956936	−	1.65746i	−0.227063	−	0.973880i	$-0.572912\pi$
$888$		0			0
$889$	9.00000	−	15.5885i	0.301850	−	0.522820i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	9.00000	+	5.19615i	0.301174	+	0.173883i
$894$		0			0
$895$	18.0000	−	10.3923i	0.601674	−	0.347376i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−9.00000			−0.300167
$900$		0			0
$901$		0			0
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−15.0000	+	8.66025i	−0.498617	+	0.287877i
$906$		0			0
$907$	−28.5000	−	16.4545i	−0.946327	−	0.546362i	−0.0543890	−	0.998520i	$-0.517321\pi$
$907$	−28.5000	−	16.4545i	−0.946327	−	0.546362i	−0.891938	+	0.452158i	$0.850654\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16.5000	−	28.5788i	0.546669	−	0.946859i	−0.451830	−	0.892104i	$-0.649229\pi$
$911$	16.5000	−	28.5788i	0.546669	−	0.946859i	0.998500	−	0.0547553i	$-0.0174379\pi$
$912$		0			0
$913$	−22.5000	−	38.9711i	−0.744641	−	1.28976i
$914$		0			0
$915$		0			0
$916$		0			0
$917$			5.19615i			0.171592i
$918$		0			0
$919$			10.3923i			0.342811i	0.985201	+	0.171405i	$0.0548307\pi$
$919$			10.3923i			0.342811i	−0.985201	+	0.171405i	$0.945169\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−30.0000	−	51.9615i	−0.987462	−	1.71033i
$924$		0			0
$925$	2.00000	−	3.46410i	0.0657596	−	0.113899i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	16.5000	+	9.52628i	0.541347	+	0.312547i	0.745625	−	0.666366i	$-0.232151\pi$
$929$	16.5000	+	9.52628i	0.541347	+	0.312547i	−0.204277	+	0.978913i	$0.565484\pi$
$930$		0			0
$931$	−12.0000	+	6.92820i	−0.393284	+	0.227063i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	36.0000			1.17733
$936$		0			0
$937$	−34.0000			−1.11073			−0.555366	−	0.831606i	$-0.687422\pi$
$937$	−34.0000			−1.11073			−0.555366	+	0.831606i	$0.687422\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−46.5000	+	26.8468i	−1.51586	+	0.875180i	−0.516030	+	0.856571i	$0.672591\pi$
$941$	−46.5000	+	26.8468i	−1.51586	+	0.875180i	−0.999827	+	0.0186097i	$0.994076\pi$
$942$		0			0
$943$	−40.5000	−	23.3827i	−1.31886	−	0.761445i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	13.5000	−	23.3827i	0.438691	−	0.759835i	−0.558898	−	0.829237i	$-0.688776\pi$
$947$	13.5000	−	23.3827i	0.438691	−	0.759835i	0.997589	+	0.0694014i	$0.0221089\pi$
$948$		0			0
$949$	5.00000	+	8.66025i	0.162307	+	0.281124i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	48.4974i		−	1.57099i	−0.618871	−	0.785493i	$-0.712410\pi$
$953$		−	48.4974i		−	1.57099i	0.618871	−	0.785493i	$-0.287590\pi$
$954$		0			0
$955$			5.19615i			0.168144i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	7.50000	+	12.9904i	0.242188	+	0.419481i
$960$		0			0
$961$	−2.00000	+	3.46410i	−0.0645161	+	0.111745i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	19.5000	+	11.2583i	0.627727	+	0.362418i
$966$		0			0
$967$	7.50000	−	4.33013i	0.241184	−	0.139247i	−0.374537	−	0.927212i	$-0.622198\pi$
$967$	7.50000	−	4.33013i	0.241184	−	0.139247i	0.615721	+	0.787964i	$0.288865\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	48.0000			1.54039			0.770197	−	0.637806i	$-0.220158\pi$
$971$	48.0000			1.54039			0.770197	+	0.637806i	$0.220158\pi$
$972$		0			0
$973$	3.00000			0.0961756
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−13.5000	+	7.79423i	−0.431903	+	0.249359i	−0.700157	−	0.713989i	$-0.746887\pi$
$977$	−13.5000	+	7.79423i	−0.431903	+	0.249359i	0.268254	+	0.963348i	$0.413553\pi$
$978$		0			0
$979$	−18.0000	−	10.3923i	−0.575282	−	0.332140i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−7.50000	+	12.9904i	−0.239213	+	0.414329i	−0.960489	−	0.278319i	$-0.910223\pi$
$983$	−7.50000	+	12.9904i	−0.239213	+	0.414329i	0.721276	+	0.692648i	$0.243556\pi$
$984$		0			0
$985$	−6.00000	−	10.3923i	−0.191176	−	0.331126i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			46.7654i			1.48705i
$990$		0			0
$991$		−	45.0333i		−	1.43053i	−0.698853	−	0.715265i	$-0.746306\pi$
$991$		−	45.0333i		−	1.43053i	0.698853	−	0.715265i	$-0.253694\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−21.0000	−	36.3731i	−0.665745	−	1.15310i
$996$		0			0
$997$	5.50000	−	9.52628i	0.174187	−	0.301700i	−0.765693	−	0.643206i	$-0.777604\pi$
$997$	5.50000	−	9.52628i	0.174187	−	0.301700i	0.939880	+	0.341506i	$0.110937\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.s.c.575.1		2
3.2	odd	2		576.2.s.b.191.1		2
4.3	odd	2		1728.2.s.d.575.1		2
8.3	odd	2		432.2.s.b.143.1		2
8.5	even	2		432.2.s.a.143.1		2
9.2	odd	6		5184.2.c.d.5183.1		2
9.4	even	3		576.2.s.c.383.1		2
9.5	odd	6		1728.2.s.d.1151.1		2
9.7	even	3		5184.2.c.b.5183.2		2
12.11	even	2		576.2.s.c.191.1		2
24.5	odd	2		144.2.s.b.47.1	✓	2
24.11	even	2		144.2.s.c.47.1	yes	2
36.7	odd	6		5184.2.c.d.5183.2		2
36.11	even	6		5184.2.c.b.5183.1		2
36.23	even	6	inner	1728.2.s.c.1151.1		2
36.31	odd	6		576.2.s.b.383.1		2
72.5	odd	6		432.2.s.b.287.1		2
72.11	even	6		1296.2.c.c.1295.2		2
72.13	even	6		144.2.s.c.95.1	yes	2
72.29	odd	6		1296.2.c.a.1295.2		2
72.43	odd	6		1296.2.c.a.1295.1		2
72.59	even	6		432.2.s.a.287.1		2
72.61	even	6		1296.2.c.c.1295.1		2
72.67	odd	6		144.2.s.b.95.1	yes	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.s.b.47.1	✓	2	24.5	odd	2
144.2.s.b.95.1	yes	2	72.67	odd	6
144.2.s.c.47.1	yes	2	24.11	even	2
144.2.s.c.95.1	yes	2	72.13	even	6
432.2.s.a.143.1		2	8.5	even	2
432.2.s.a.287.1		2	72.59	even	6
432.2.s.b.143.1		2	8.3	odd	2
432.2.s.b.287.1		2	72.5	odd	6
576.2.s.b.191.1		2	3.2	odd	2
576.2.s.b.383.1		2	36.31	odd	6
576.2.s.c.191.1		2	12.11	even	2
576.2.s.c.383.1		2	9.4	even	3
1296.2.c.a.1295.1		2	72.43	odd	6
1296.2.c.a.1295.2		2	72.29	odd	6
1296.2.c.c.1295.1		2	72.61	even	6
1296.2.c.c.1295.2		2	72.11	even	6
1728.2.s.c.575.1		2	1.1	even	1	trivial
1728.2.s.c.1151.1		2	36.23	even	6	inner
1728.2.s.d.575.1		2	4.3	odd	2
1728.2.s.d.1151.1		2	9.5	odd	6
5184.2.c.b.5183.1		2	36.11	even	6
5184.2.c.b.5183.2		2	9.7	even	3
5184.2.c.d.5183.1		2	9.2	odd	6
5184.2.c.d.5183.2		2	36.7	odd	6

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:	\( N \)	\(=\)	\( 1728 = 2^{6} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1728.s (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(1.50000 - 0.866025i) q^{5} +(-1.50000 - 0.866025i) q^{7} +O(q^{10})\)	\(q+(1.50000 - 0.866025i) q^{5} +(-1.50000 - 0.866025i) q^{7} +(1.50000 - 2.59808i) q^{11} +(-2.50000 - 4.33013i) q^{13} +6.92820i q^{17} -3.46410i q^{19} +(-4.50000 - 7.79423i) q^{23} +(-1.00000 + 1.73205i) q^{25} +(1.50000 + 0.866025i) q^{29} +(-4.50000 + 2.59808i) q^{31} -3.00000 q^{35} -2.00000 q^{37} +(4.50000 - 2.59808i) q^{41} +(-4.50000 - 2.59808i) q^{43} +(-1.50000 + 2.59808i) q^{47} +(-2.00000 - 3.46410i) q^{49} -5.19615i q^{55} +(-1.50000 - 2.59808i) q^{59} +(-0.500000 + 0.866025i) q^{61} +(-7.50000 - 4.33013i) q^{65} +(-7.50000 + 4.33013i) q^{67} +12.0000 q^{71} -2.00000 q^{73} +(-4.50000 + 2.59808i) q^{77} +(-7.50000 - 4.33013i) q^{79} +(7.50000 - 12.9904i) q^{83} +(6.00000 + 10.3923i) q^{85} -6.92820i q^{89} +8.66025i q^{91} +(-3.00000 - 5.19615i) q^{95} +(2.50000 - 4.33013i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{5} - 3 q^{7}+O(q^{10}) \) 2 * q + 3 * q^5 - 3 * q^7	\( 2 q + 3 q^{5} - 3 q^{7} + 3 q^{11} - 5 q^{13} - 9 q^{23} - 2 q^{25} + 3 q^{29} - 9 q^{31} - 6 q^{35} - 4 q^{37} + 9 q^{41} - 9 q^{43} - 3 q^{47} - 4 q^{49} - 3 q^{59} - q^{61} - 15 q^{65} - 15 q^{67} + 24 q^{71} - 4 q^{73} - 9 q^{77} - 15 q^{79} + 15 q^{83} + 12 q^{85} - 6 q^{95} + 5 q^{97}+O(q^{100}) \) 2 * q + 3 * q^5 - 3 * q^7 + 3 * q^11 - 5 * q^13 - 9 * q^23 - 2 * q^25 + 3 * q^29 - 9 * q^31 - 6 * q^35 - 4 * q^37 + 9 * q^41 - 9 * q^43 - 3 * q^47 - 4 * q^49 - 3 * q^59 - q^61 - 15 * q^65 - 15 * q^67 + 24 * q^71 - 4 * q^73 - 9 * q^77 - 15 * q^79 + 15 * q^83 + 12 * q^85 - 6 * q^95 + 5 * q^97

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