LMFDB - Embedded newform 1728.2.i.j.577.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1728,2,Mod(577,1728)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1728.577"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1728, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2])) N = Newforms(chi, 2, names="a")

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

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,1,0,3] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$13.7981494693$
Analytic rank:	$0$
Dimension:	$4$
Relative dimension:	$2$ over $\Q(\zeta_{3})$
Coefficient field:	$\Q(\sqrt{-3}, \sqrt{-11})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 2x^{2} - 3x + 9 $ x^4 - x^3 - 2x^2 - 3x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 72)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			577.1
Root			$-1.18614 + 1.26217i$ of defining polynomial
Character	$\chi$	$=$	1728.577
Dual form			1728.2.i.j.1153.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.18614 + 2.05446i) q^{5} +(2.18614 + 3.78651i) q^{7} +(0.500000 + 0.866025i) q^{11} +(-0.186141 + 0.322405i) q^{13} +5.37228 q^{17} +0.627719 q^{19} +(0.186141 - 0.322405i) q^{23} +(-0.313859 - 0.543620i) q^{25} +(2.18614 + 3.78651i) q^{29} +(3.18614 - 5.51856i) q^{31} -10.3723 q^{35} -8.74456 q^{37} +(-5.87228 + 10.1711i) q^{41} +(0.872281 + 1.51084i) q^{43} +(-2.18614 - 3.78651i) q^{47} +(-6.05842 + 10.4935i) q^{49} +0.744563 q^{53} -2.37228 q^{55} +(3.50000 - 6.06218i) q^{59} +(1.18614 + 2.05446i) q^{61} +(-0.441578 - 0.764836i) q^{65} +(1.87228 - 3.24289i) q^{67} -4.00000 q^{71} -12.1168 q^{73} +(-2.18614 + 3.78651i) q^{77} +(-3.18614 - 5.51856i) q^{79} +(4.81386 + 8.33785i) q^{83} +(-6.37228 + 11.0371i) q^{85} -6.00000 q^{89} -1.62772 q^{91} +(-0.744563 + 1.28962i) q^{95} +(-0.872281 - 1.51084i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{5} + 3 q^{7} + 2 q^{11} + 5 q^{13} + 10 q^{17} + 14 q^{19} - 5 q^{23} - 7 q^{25} + 3 q^{29} + 7 q^{31} - 30 q^{35} - 12 q^{37} - 12 q^{41} - 8 q^{43} - 3 q^{47} - 7 q^{49} - 20 q^{53} + 2 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) $ 4 * q + q^5 + 3 * q^7 + 2 * q^11 + 5 * q^13 + 10 * q^17 + 14 * q^19 - 5 * q^23 - 7 * q^25 + 3 * q^29 + 7 * q^31 - 30 * q^35 - 12 * q^37 - 12 * q^41 - 8 * q^43 - 3 * q^47 - 7 * q^49 - 20 * q^53 + 2 * q^55 + 14 * q^59 - q^61 - 19 * q^65 - 4 * q^67 - 16 * q^71 - 14 * q^73 - 3 * q^77 - 7 * q^79 + 25 * q^83 - 14 * q^85 - 24 * q^89 - 18 * q^91 + 20 * q^95 + 8 * q^97

Character values

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

$n$	$325$	$703$	$1217$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{3}\right)$

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
$5$	−1.18614	+	2.05446i	−0.530458	+	0.918781i	0.468910	+	0.883246i	$0.344647\pi$
$5$	−1.18614	+	2.05446i	−0.530458	+	0.918781i	−0.999368	+	0.0355348i	$0.988687\pi$
$6$		0			0
$7$	2.18614	+	3.78651i	0.826284	+	1.43117i	0.900934	+	0.433955i	$0.142882\pi$
$7$	2.18614	+	3.78651i	0.826284	+	1.43117i	−0.0746509	+	0.997210i	$0.523784\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	0.931505	−	0.363727i	$-0.118496\pi$
$11$	0.500000	+	0.866025i	0.150756	+	0.261116i	−0.780750	+	0.624844i	$0.785163\pi$
$12$		0			0
$13$	−0.186141	+	0.322405i	−0.0516261	+	0.0894191i	−0.890684	−	0.454624i	$-0.849774\pi$
$13$	−0.186141	+	0.322405i	−0.0516261	+	0.0894191i	0.839057	+	0.544043i	$0.183107\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.37228			1.30297			0.651485	−	0.758662i	$-0.274146\pi$
$17$	5.37228			1.30297			0.651485	+	0.758662i	$0.274146\pi$
$18$		0			0
$19$	0.627719			0.144009			0.0720043	−	0.997404i	$-0.477060\pi$
$19$	0.627719			0.144009			0.0720043	+	0.997404i	$0.477060\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	0.186141	−	0.322405i	0.0388130	−	0.0672261i	−0.845966	−	0.533236i	$-0.820976\pi$
$23$	0.186141	−	0.322405i	0.0388130	−	0.0672261i	0.884779	+	0.466010i	$0.154309\pi$
$24$		0			0
$25$	−0.313859	−	0.543620i	−0.0627719	−	0.108724i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	2.18614	+	3.78651i	0.405956	+	0.703137i	0.994432	−	0.105378i	$-0.0336052\pi$
$29$	2.18614	+	3.78651i	0.405956	+	0.703137i	−0.588476	+	0.808515i	$0.700272\pi$
$30$		0			0
$31$	3.18614	−	5.51856i	0.572248	−	0.991162i	−0.424087	−	0.905621i	$-0.639405\pi$
$31$	3.18614	−	5.51856i	0.572248	−	0.991162i	0.996335	−	0.0855407i	$-0.0272618\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−10.3723			−1.75324
$36$		0			0
$37$	−8.74456			−1.43760			−0.718799	−	0.695218i	$-0.755308\pi$
$37$	−8.74456			−1.43760			−0.718799	+	0.695218i	$0.755308\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−5.87228	+	10.1711i	−0.917096	+	1.58846i	−0.113293	+	0.993562i	$0.536140\pi$
$41$	−5.87228	+	10.1711i	−0.917096	+	1.58846i	−0.803803	+	0.594896i	$0.797193\pi$
$42$		0			0
$43$	0.872281	+	1.51084i	0.133022	+	0.230400i	0.924840	−	0.380356i	$-0.124199\pi$
$43$	0.872281	+	1.51084i	0.133022	+	0.230400i	−0.791818	+	0.610757i	$0.790865\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−2.18614	−	3.78651i	−0.318881	−	0.552319i	0.661374	−	0.750057i	$-0.269974\pi$
$47$	−2.18614	−	3.78651i	−0.318881	−	0.552319i	−0.980255	+	0.197738i	$0.936640\pi$
$48$		0			0
$49$	−6.05842	+	10.4935i	−0.865489	+	1.49907i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.744563			0.102274			0.0511368	−	0.998692i	$-0.483716\pi$
$53$	0.744563			0.102274			0.0511368	+	0.998692i	$0.483716\pi$
$54$		0			0
$55$	−2.37228			−0.319878
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.50000	−	6.06218i	0.455661	−	0.789228i	−0.543065	−	0.839691i	$-0.682736\pi$
$59$	3.50000	−	6.06218i	0.455661	−	0.789228i	0.998726	+	0.0504625i	$0.0160695\pi$
$60$		0			0
$61$	1.18614	+	2.05446i	0.151870	+	0.263046i	0.931915	−	0.362677i	$-0.118137\pi$
$61$	1.18614	+	2.05446i	0.151870	+	0.263046i	−0.780045	+	0.625723i	$0.784804\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.441578	−	0.764836i	−0.0547710	−	0.0948662i
$66$		0			0
$67$	1.87228	−	3.24289i	0.228736	−	0.396182i	−0.728698	−	0.684835i	$-0.759874\pi$
$67$	1.87228	−	3.24289i	0.228736	−	0.396182i	0.957434	+	0.288653i	$0.0932076\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−4.00000			−0.474713			−0.237356	−	0.971423i	$-0.576281\pi$
$71$	−4.00000			−0.474713			−0.237356	+	0.971423i	$0.576281\pi$
$72$		0			0
$73$	−12.1168			−1.41817			−0.709085	−	0.705123i	$-0.750892\pi$
$73$	−12.1168			−1.41817			−0.709085	+	0.705123i	$0.750892\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.18614	+	3.78651i	−0.249134	+	0.431512i
$78$		0			0
$79$	−3.18614	−	5.51856i	−0.358469	−	0.620886i	0.629236	−	0.777214i	$-0.283368\pi$
$79$	−3.18614	−	5.51856i	−0.358469	−	0.620886i	−0.987705	+	0.156328i	$0.950034\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	4.81386	+	8.33785i	0.528390	+	0.915198i	0.999452	+	0.0330979i	$0.0105373\pi$
$83$	4.81386	+	8.33785i	0.528390	+	0.915198i	−0.471062	+	0.882100i	$0.656129\pi$
$84$		0			0
$85$	−6.37228	+	11.0371i	−0.691171	+	1.19714i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−6.00000			−0.635999			−0.317999	−	0.948091i	$-0.603011\pi$
$89$	−6.00000			−0.635999			−0.317999	+	0.948091i	$0.603011\pi$
$90$		0			0
$91$	−1.62772			−0.170631
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.744563	+	1.28962i	−0.0763905	+	0.132312i
$96$		0			0
$97$	−0.872281	−	1.51084i	−0.0885667	−	0.153402i	0.818339	−	0.574736i	$-0.194895\pi$
$97$	−0.872281	−	1.51084i	−0.0885667	−	0.153402i	−0.906906	+	0.421334i	$0.861562\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	3.55842	+	6.16337i	0.354076	+	0.613278i	0.986959	−	0.160969i	$-0.0514620\pi$
$101$	3.55842	+	6.16337i	0.354076	+	0.613278i	−0.632883	+	0.774247i	$0.718129\pi$
$102$		0			0
$103$	−6.18614	+	10.7147i	−0.609539	+	1.05575i	0.381778	+	0.924254i	$0.375312\pi$
$103$	−6.18614	+	10.7147i	−0.609539	+	1.05575i	−0.991316	+	0.131498i	$0.958021\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	12.8614			1.24336			0.621680	−	0.783272i	$-0.286451\pi$
$107$	12.8614			1.24336			0.621680	+	0.783272i	$0.286451\pi$
$108$		0			0
$109$	4.74456			0.454447			0.227223	−	0.973843i	$-0.427035\pi$
$109$	4.74456			0.454447			0.227223	+	0.973843i	$0.427035\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−2.18614	+	3.78651i	−0.205655	+	0.356205i	−0.950341	−	0.311210i	$-0.899266\pi$
$113$	−2.18614	+	3.78651i	−0.205655	+	0.356205i	0.744686	+	0.667415i	$0.232599\pi$
$114$		0			0
$115$	0.441578	+	0.764836i	0.0411774	+	0.0713213i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	11.7446	+	20.3422i	1.07662	+	1.86476i
$120$		0			0
$121$	5.00000	−	8.66025i	0.454545	−	0.787296i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−10.3723			−0.927725
$126$		0			0
$127$	−6.74456			−0.598483			−0.299242	−	0.954177i	$-0.596734\pi$
$127$	−6.74456			−0.598483			−0.299242	+	0.954177i	$0.596734\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.18614	−	15.9109i	0.802597	−	1.39014i	−0.115305	−	0.993330i	$-0.536784\pi$
$131$	9.18614	−	15.9109i	0.802597	−	1.39014i	0.917901	−	0.396808i	$-0.129882\pi$
$132$		0			0
$133$	1.37228	+	2.37686i	0.118992	+	0.206100i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	8.87228	+	15.3672i	0.758010	+	1.31291i	0.943864	+	0.330335i	$0.107162\pi$
$137$	8.87228	+	15.3672i	0.758010	+	1.31291i	−0.185854	+	0.982577i	$0.559505\pi$
$138$		0			0
$139$	−2.87228	+	4.97494i	−0.243624	+	0.421969i	−0.961744	−	0.273951i	$-0.911670\pi$
$139$	−2.87228	+	4.97494i	−0.243624	+	0.421969i	0.718120	+	0.695919i	$0.245003\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−0.372281			−0.0311317
$144$		0			0
$145$	−10.3723			−0.861371
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.18614	−	7.25061i	0.342942	−	0.593993i	−0.642036	−	0.766675i	$-0.721910\pi$
$149$	4.18614	−	7.25061i	0.342942	−	0.593993i	0.984978	+	0.172682i	$0.0552432\pi$
$150$		0			0
$151$	0.186141	+	0.322405i	0.0151479	+	0.0262370i	0.873500	−	0.486824i	$-0.161845\pi$
$151$	0.186141	+	0.322405i	0.0151479	+	0.0262370i	−0.858352	+	0.513061i	$0.828511\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	7.55842	+	13.0916i	0.607107	+	1.05154i
$156$		0			0
$157$	−1.55842	+	2.69927i	−0.124376	+	0.215425i	−0.921489	−	0.388405i	$-0.873026\pi$
$157$	−1.55842	+	2.69927i	−0.124376	+	0.215425i	0.797113	+	0.603830i	$0.206359\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1.62772			0.128282
$162$		0			0
$163$	−12.0000			−0.939913			−0.469956	−	0.882690i	$-0.655730\pi$
$163$	−12.0000			−0.939913			−0.469956	+	0.882690i	$0.655730\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	9.55842	−	16.5557i	0.739653	−	1.28112i	−0.212999	−	0.977052i	$-0.568323\pi$
$167$	9.55842	−	16.5557i	0.739653	−	1.28112i	0.952652	−	0.304064i	$-0.0983436\pi$
$168$		0			0
$169$	6.43070	+	11.1383i	0.494669	+	0.856793i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.18614	+	14.1788i	0.622381	+	1.07800i	0.989041	+	0.147640i	$0.0471678\pi$
$173$	8.18614	+	14.1788i	0.622381	+	1.07800i	−0.366660	+	0.930355i	$0.619499\pi$
$174$		0			0
$175$	1.37228	−	2.37686i	0.103735	−	0.179674i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−22.9783			−1.71748			−0.858738	−	0.512416i	$-0.828751\pi$
$179$	−22.9783			−1.71748			−0.858738	+	0.512416i	$0.828751\pi$
$180$		0			0
$181$	0.510875			0.0379730			0.0189865	−	0.999820i	$-0.493956\pi$
$181$	0.510875			0.0379730			0.0189865	+	0.999820i	$0.493956\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.3723	−	17.9653i	0.762585	−	1.32084i
$186$		0			0
$187$	2.68614	+	4.65253i	0.196430	+	0.340227i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−4.93070	−	8.54023i	−0.356773	−	0.617949i	0.630647	−	0.776070i	$-0.282790\pi$
$191$	−4.93070	−	8.54023i	−0.356773	−	0.617949i	−0.987420	+	0.158121i	$0.949456\pi$
$192$		0			0
$193$	−0.872281	+	1.51084i	−0.0627882	+	0.108752i	−0.895711	−	0.444637i	$-0.853333\pi$
$193$	−0.872281	+	1.51084i	−0.0627882	+	0.108752i	0.832923	+	0.553390i	$0.186666\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	15.2554			1.08690			0.543452	−	0.839440i	$-0.317117\pi$
$197$	15.2554			1.08690			0.543452	+	0.839440i	$0.317117\pi$
$198$		0			0
$199$	16.2337			1.15078			0.575388	−	0.817881i	$-0.304851\pi$
$199$	16.2337			1.15078			0.575388	+	0.817881i	$0.304851\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−9.55842	+	16.5557i	−0.670870	+	1.16198i
$204$		0			0
$205$	−13.9307	−	24.1287i	−0.972963	−	1.68522i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	0.313859	+	0.543620i	0.0217101	+	0.0376030i
$210$		0			0
$211$	4.81386	−	8.33785i	0.331400	−	0.574001i	−0.651387	−	0.758746i	$-0.725812\pi$
$211$	4.81386	−	8.33785i	0.331400	−	0.574001i	0.982787	+	0.184745i	$0.0591458\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.13859			−0.282250
$216$		0			0
$217$	27.8614			1.89136
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.00000	+	1.73205i	−0.0672673	+	0.116510i
$222$		0			0
$223$	−7.30298	−	12.6491i	−0.489044	−	0.847049i	0.510877	−	0.859654i	$-0.329321\pi$
$223$	−7.30298	−	12.6491i	−0.489044	−	0.847049i	−0.999921	+	0.0126050i	$0.995988\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−7.50000	−	12.9904i	−0.497792	−	0.862202i	0.502204	−	0.864749i	$-0.332523\pi$
$227$	−7.50000	−	12.9904i	−0.497792	−	0.862202i	−0.999997	+	0.00254715i	$0.999189\pi$
$228$		0			0
$229$	11.3030	−	19.5773i	0.746922	−	1.29371i	−0.202369	−	0.979309i	$-0.564864\pi$
$229$	11.3030	−	19.5773i	0.746922	−	1.29371i	0.949291	−	0.314398i	$-0.101803\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−5.37228			−0.351950			−0.175975	−	0.984395i	$-0.556308\pi$
$233$	−5.37228			−0.351950			−0.175975	+	0.984395i	$0.556308\pi$
$234$		0			0
$235$	10.3723			0.676613
$236$		0			0
$237$		0			0
$238$		0			0
$239$	6.93070	−	12.0043i	0.448310	−	0.776496i	−0.549966	−	0.835187i	$-0.685359\pi$
$239$	6.93070	−	12.0043i	0.448310	−	0.776496i	0.998276	+	0.0586913i	$0.0186928\pi$
$240$		0			0
$241$	−2.87228	−	4.97494i	−0.185020	−	0.320464i	0.758563	−	0.651599i	$-0.225902\pi$
$241$	−2.87228	−	4.97494i	−0.185020	−	0.320464i	−0.943583	+	0.331135i	$0.892568\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−14.3723	−	24.8935i	−0.918211	−	1.59039i
$246$		0			0
$247$	−0.116844	+	0.202380i	−0.00743460	+	0.0128771i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−9.88316			−0.623819			−0.311910	−	0.950112i	$-0.600969\pi$
$251$	−9.88316			−0.623819			−0.311910	+	0.950112i	$0.600969\pi$
$252$		0			0
$253$	0.372281			0.0234051
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.24456	−	3.88770i	0.140012	−	0.242508i	−0.787489	−	0.616329i	$-0.788619\pi$
$257$	2.24456	−	3.88770i	0.140012	−	0.242508i	0.927501	+	0.373821i	$0.121953\pi$
$258$		0			0
$259$	−19.1168	−	33.1113i	−1.18786	−	2.05744i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.5584	+	18.2877i	0.651060	+	1.12767i	0.982866	+	0.184322i	$0.0590088\pi$
$263$	10.5584	+	18.2877i	0.651060	+	1.12767i	−0.331806	+	0.943348i	$0.607658\pi$
$264$		0			0
$265$	−0.883156	+	1.52967i	−0.0542518	+	0.0939669i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−12.7446			−0.777050			−0.388525	−	0.921438i	$-0.627015\pi$
$269$	−12.7446			−0.777050			−0.388525	+	0.921438i	$0.627015\pi$
$270$		0			0
$271$	21.4891			1.30537			0.652686	−	0.757629i	$-0.273642\pi$
$271$	21.4891			1.30537			0.652686	+	0.757629i	$0.273642\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	0.313859	−	0.543620i	0.0189264	−	0.0327815i
$276$		0			0
$277$	14.5584	+	25.2159i	0.874731	+	1.51508i	0.857049	+	0.515235i	$0.172295\pi$
$277$	14.5584	+	25.2159i	0.874731	+	1.51508i	0.0176816	+	0.999844i	$0.494371\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−6.93070	−	12.0043i	−0.413451	−	0.716118i	0.581813	−	0.813322i	$-0.302343\pi$
$281$	−6.93070	−	12.0043i	−0.413451	−	0.716118i	−0.995264	+	0.0972040i	$0.969010\pi$
$282$		0			0
$283$	−11.9307	+	20.6646i	−0.709207	+	1.22838i	0.255945	+	0.966691i	$0.417613\pi$
$283$	−11.9307	+	20.6646i	−0.709207	+	1.22838i	−0.965152	+	0.261691i	$0.915720\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−51.3505			−3.03113
$288$		0			0
$289$	11.8614			0.697730
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−9.81386	+	16.9981i	−0.573332	+	0.993040i	0.422889	+	0.906182i	$0.361016\pi$
$293$	−9.81386	+	16.9981i	−0.573332	+	0.993040i	−0.996221	+	0.0868582i	$0.972317\pi$
$294$		0			0
$295$	8.30298	+	14.3812i	0.483418	+	0.837305i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	0.0692967	+	0.120025i	0.00400753	+	0.00694125i
$300$		0			0
$301$	−3.81386	+	6.60580i	−0.219827	+	0.380752i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−5.62772			−0.322242
$306$		0			0
$307$	31.3723			1.79051			0.895255	−	0.445553i	$-0.146993\pi$
$307$	31.3723			1.79051			0.895255	+	0.445553i	$0.146993\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	5.44158	−	9.42509i	0.308564	−	0.534448i	−0.669485	−	0.742826i	$-0.733485\pi$
$311$	5.44158	−	9.42509i	0.308564	−	0.534448i	0.978048	+	0.208378i	$0.0668184\pi$
$312$		0			0
$313$	−5.61684	−	9.72866i	−0.317483	−	0.549896i	0.662479	−	0.749080i	$-0.269504\pi$
$313$	−5.61684	−	9.72866i	−0.317483	−	0.549896i	−0.979962	+	0.199184i	$0.936171\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−15.3030	−	26.5055i	−0.859501	−	1.48870i	−0.872405	−	0.488783i	$-0.837441\pi$
$317$	−15.3030	−	26.5055i	−0.859501	−	1.48870i	0.0129041	−	0.999917i	$-0.495892\pi$
$318$		0			0
$319$	−2.18614	+	3.78651i	−0.122400	+	0.212004i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	3.37228			0.187639
$324$		0			0
$325$	0.233688			0.0129627
$326$		0			0
$327$		0			0
$328$		0			0
$329$	9.55842	−	16.5557i	0.526973	−	0.912744i
$330$		0			0
$331$	15.9307	+	27.5928i	0.875631	+	1.51664i	0.856089	+	0.516828i	$0.172887\pi$
$331$	15.9307	+	27.5928i	0.875631	+	1.51664i	0.0195412	+	0.999809i	$0.493779\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	4.44158	+	7.69304i	0.242669	+	0.420316i
$336$		0			0
$337$	9.98913	−	17.3017i	0.544142	−	0.942482i	−0.454518	−	0.890738i	$-0.650188\pi$
$337$	9.98913	−	17.3017i	0.544142	−	0.942482i	0.998660	−	0.0517446i	$-0.0164782\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.37228			0.345078
$342$		0			0
$343$	−22.3723			−1.20799
$344$		0			0
$345$		0			0
$346$		0			0
$347$	14.3614	−	24.8747i	0.770961	−	1.33534i	−0.166076	−	0.986113i	$-0.553110\pi$
$347$	14.3614	−	24.8747i	0.770961	−	1.33534i	0.937037	−	0.349230i	$-0.113557\pi$
$348$		0			0
$349$	8.44158	+	14.6212i	0.451867	+	0.782657i	0.998502	−	0.0547140i	$-0.0174247\pi$
$349$	8.44158	+	14.6212i	0.451867	+	0.782657i	−0.546635	+	0.837371i	$0.684091\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−11.9891	−	20.7658i	−0.638117	−	1.10525i	−0.985846	−	0.167656i	$-0.946380\pi$
$353$	−11.9891	−	20.7658i	−0.638117	−	1.10525i	0.347729	−	0.937595i	$-0.386953\pi$
$354$		0			0
$355$	4.74456	−	8.21782i	0.251815	−	0.436157i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	20.2337			1.06789			0.533947	−	0.845518i	$-0.320708\pi$
$359$	20.2337			1.06789			0.533947	+	0.845518i	$0.320708\pi$
$360$		0			0
$361$	−18.6060			−0.979262
$362$		0			0
$363$		0			0
$364$		0			0
$365$	14.3723	−	24.8935i	0.752280	−	1.30299i
$366$		0			0
$367$	8.81386	+	15.2661i	0.460080	+	0.796881i	0.998964	−	0.0454981i	$-0.0144875\pi$
$367$	8.81386	+	15.2661i	0.460080	+	0.796881i	−0.538885	+	0.842380i	$0.681154\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	1.62772	+	2.81929i	0.0845069	+	0.146370i
$372$		0			0
$373$	10.5584	−	18.2877i	0.546694	−	0.946902i	−0.451804	−	0.892117i	$-0.649219\pi$
$373$	10.5584	−	18.2877i	0.546694	−	0.946902i	0.998498	−	0.0547851i	$-0.0174474\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.62772			−0.0838318
$378$		0			0
$379$	−5.88316			−0.302197			−0.151099	−	0.988519i	$-0.548281\pi$
$379$	−5.88316			−0.302197			−0.151099	+	0.988519i	$0.548281\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−10.6753	+	18.4901i	−0.545481	+	0.944800i	0.453096	+	0.891462i	$0.350320\pi$
$383$	−10.6753	+	18.4901i	−0.545481	+	0.944800i	−0.998576	+	0.0533383i	$0.983014\pi$
$384$		0			0
$385$	−5.18614	−	8.98266i	−0.264310	−	0.457799i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−5.30298	−	9.18504i	−0.268872	−	0.465700i	0.699699	−	0.714438i	$-0.253318\pi$
$389$	−5.30298	−	9.18504i	−0.268872	−	0.465700i	−0.968571	+	0.248738i	$0.919984\pi$
$390$		0			0
$391$	1.00000	−	1.73205i	0.0505722	−	0.0875936i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	15.1168			0.760611
$396$		0			0
$397$	18.2337			0.915123			0.457561	−	0.889178i	$-0.348723\pi$
$397$	18.2337			0.915123			0.457561	+	0.889178i	$0.348723\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−8.61684	+	14.9248i	−0.430305	+	0.745310i	−0.996899	−	0.0786871i	$-0.974927\pi$
$401$	−8.61684	+	14.9248i	−0.430305	+	0.745310i	0.566595	+	0.823997i	$0.308261\pi$
$402$		0			0
$403$	1.18614	+	2.05446i	0.0590859	+	0.102340i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−4.37228	−	7.57301i	−0.216726	−	0.375380i
$408$		0			0
$409$	−2.87228	+	4.97494i	−0.142025	+	0.245995i	−0.928259	−	0.371934i	$-0.878695\pi$
$409$	−2.87228	+	4.97494i	−0.142025	+	0.245995i	0.786234	+	0.617929i	$0.212028\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	30.6060			1.50602
$414$		0			0
$415$	−22.8397			−1.12115
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.30298	+	10.9171i	−0.307921	+	0.533335i	−0.977907	−	0.209039i	$-0.932967\pi$
$419$	−6.30298	+	10.9171i	−0.307921	+	0.533335i	0.669986	+	0.742373i	$0.266300\pi$
$420$		0			0
$421$	−17.5584	−	30.4121i	−0.855745	−	1.48219i	−0.875952	−	0.482398i	$-0.839766\pi$
$421$	−17.5584	−	30.4121i	−0.855745	−	1.48219i	0.0202069	−	0.999796i	$-0.493568\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−1.68614	−	2.92048i	−0.0817898	−	0.141664i
$426$		0			0
$427$	−5.18614	+	8.98266i	−0.250975	+	0.434701i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−1.25544			−0.0604723			−0.0302361	−	0.999543i	$-0.509626\pi$
$431$	−1.25544			−0.0604723			−0.0302361	+	0.999543i	$0.509626\pi$
$432$		0			0
$433$	32.1168			1.54344			0.771719	−	0.635964i	$-0.219397\pi$
$433$	32.1168			1.54344			0.771719	+	0.635964i	$0.219397\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	0.116844	−	0.202380i	0.00558941	−	0.00968113i
$438$		0			0
$439$	3.55842	+	6.16337i	0.169834	+	0.294161i	0.938361	−	0.345656i	$-0.112343\pi$
$439$	3.55842	+	6.16337i	0.169834	+	0.294161i	−0.768527	+	0.639817i	$0.779010\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	15.3614	+	26.6067i	0.729842	+	1.26412i	0.956950	+	0.290254i	$0.0937399\pi$
$443$	15.3614	+	26.6067i	0.729842	+	1.26412i	−0.227107	+	0.973870i	$0.572927\pi$
$444$		0			0
$445$	7.11684	−	12.3267i	0.337371	−	0.584343i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	15.8832			0.749572			0.374786	−	0.927111i	$-0.377716\pi$
$449$	15.8832			0.749572			0.374786	+	0.927111i	$0.377716\pi$
$450$		0			0
$451$	−11.7446			−0.553030
$452$		0			0
$453$		0			0
$454$		0			0
$455$	1.93070	−	3.34408i	0.0905128	−	0.156773i
$456$		0			0
$457$	1.87228	+	3.24289i	0.0875816	+	0.151696i	0.906488	−	0.422231i	$-0.138753\pi$
$457$	1.87228	+	3.24289i	0.0875816	+	0.151696i	−0.818907	+	0.573927i	$0.805419\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	3.44158	+	5.96099i	0.160290	+	0.277631i	0.934973	−	0.354720i	$-0.115424\pi$
$461$	3.44158	+	5.96099i	0.160290	+	0.277631i	−0.774682	+	0.632350i	$0.782090\pi$
$462$		0			0
$463$	20.6753	−	35.8106i	0.960861	−	1.66426i	0.240515	−	0.970645i	$-0.422684\pi$
$463$	20.6753	−	35.8106i	0.960861	−	1.66426i	0.720346	−	0.693615i	$-0.243983\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−7.37228			−0.341148			−0.170574	−	0.985345i	$-0.554562\pi$
$467$	−7.37228			−0.341148			−0.170574	+	0.985345i	$0.554562\pi$
$468$		0			0
$469$	16.3723			0.756002
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−0.872281	+	1.51084i	−0.0401075	+	0.0694683i
$474$		0			0
$475$	−0.197015	−	0.341241i	−0.00903969	−	0.0156572i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	5.30298	+	9.18504i	0.242300	+	0.419675i	0.961369	−	0.275263i	$-0.0887650\pi$
$479$	5.30298	+	9.18504i	0.242300	+	0.419675i	−0.719069	+	0.694938i	$0.755432\pi$
$480$		0			0
$481$	1.62772	−	2.81929i	0.0742176	−	0.128549i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	4.13859			0.187924
$486$		0			0
$487$	−6.74456			−0.305625			−0.152813	−	0.988255i	$-0.548833\pi$
$487$	−6.74456			−0.305625			−0.152813	+	0.988255i	$0.548833\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	0.127719	−	0.221215i	0.00576386	−	0.00998330i	−0.863129	−	0.504983i	$-0.831499\pi$
$491$	0.127719	−	0.221215i	0.00576386	−	0.00998330i	0.868893	+	0.495000i	$0.164832\pi$
$492$		0			0
$493$	11.7446	+	20.3422i	0.528948	+	0.916166i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−8.74456	−	15.1460i	−0.392247	−	0.679392i
$498$		0			0
$499$	9.98913	−	17.3017i	0.447175	−	0.774529i	−0.551026	−	0.834488i	$-0.685764\pi$
$499$	9.98913	−	17.3017i	0.447175	−	0.774529i	0.998201	+	0.0599587i	$0.0190969\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.51087			0.290306			0.145153	−	0.989409i	$-0.453633\pi$
$503$	6.51087			0.290306			0.145153	+	0.989409i	$0.453633\pi$
$504$		0			0
$505$	−16.8832			−0.751291
$506$		0			0
$507$		0			0
$508$		0			0
$509$	15.5584	−	26.9480i	0.689615	−	1.19445i	−0.282348	−	0.959312i	$-0.591113\pi$
$509$	15.5584	−	26.9480i	0.689615	−	1.19445i	0.971962	−	0.235136i	$-0.0755535\pi$
$510$		0			0
$511$	−26.4891	−	45.8805i	−1.17181	−	2.02963i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−14.6753	−	25.4183i	−0.646669	−	1.12006i
$516$		0			0
$517$	2.18614	−	3.78651i	0.0961464	−	0.166530i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−12.1168			−0.530849			−0.265424	−	0.964132i	$-0.585512\pi$
$521$	−12.1168			−0.530849			−0.265424	+	0.964132i	$0.585512\pi$
$522$		0			0
$523$	−13.4891			−0.589838			−0.294919	−	0.955522i	$-0.595293\pi$
$523$	−13.4891			−0.589838			−0.294919	+	0.955522i	$0.595293\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	17.1168	−	29.6472i	0.745621	−	1.29145i
$528$		0			0
$529$	11.4307	+	19.7986i	0.496987	+	0.860807i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.18614	−	3.78651i	−0.0946923	−	0.164012i
$534$		0			0
$535$	−15.2554	+	26.4232i	−0.659550	+	1.14237i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.1168			−0.521909
$540$		0			0
$541$	−2.23369			−0.0960337			−0.0480169	−	0.998847i	$-0.515290\pi$
$541$	−2.23369			−0.0960337			−0.0480169	+	0.998847i	$0.515290\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−5.62772	+	9.74749i	−0.241065	+	0.417537i
$546$		0			0
$547$	8.12772	+	14.0776i	0.347516	+	0.601916i	0.985808	−	0.167879i	$-0.0536918\pi$
$547$	8.12772	+	14.0776i	0.347516	+	0.601916i	−0.638291	+	0.769795i	$0.720359\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	1.37228	+	2.37686i	0.0584611	+	0.101258i
$552$		0			0
$553$	13.9307	−	24.1287i	0.592394	−	1.02606i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.0000			−0.593199			−0.296600	−	0.955002i	$-0.595853\pi$
$557$	−14.0000			−0.593199			−0.296600	+	0.955002i	$0.595853\pi$
$558$		0			0
$559$	−0.649468			−0.0274696
$560$		0			0
$561$		0			0
$562$		0			0
$563$	4.87228	−	8.43904i	0.205342	−	0.355663i	−0.744900	−	0.667177i	$-0.767503\pi$
$563$	4.87228	−	8.43904i	0.205342	−	0.355663i	0.950242	+	0.311514i	$0.100836\pi$
$564$		0			0
$565$	−5.18614	−	8.98266i	−0.218183	−	0.377903i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	13.6168	+	23.5851i	0.570848	+	0.988737i	0.996479	+	0.0838407i	$0.0267187\pi$
$569$	13.6168	+	23.5851i	0.570848	+	0.988737i	−0.425631	+	0.904897i	$0.639948\pi$
$570$		0			0
$571$	13.2446	−	22.9403i	0.554268	−	0.960020i	−0.443692	−	0.896179i	$-0.646332\pi$
$571$	13.2446	−	22.9403i	0.554268	−	0.960020i	0.997960	−	0.0638407i	$-0.0203349\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−0.233688			−0.00974546
$576$		0			0
$577$	14.8614			0.618688			0.309344	−	0.950950i	$-0.399890\pi$
$577$	14.8614			0.618688			0.309344	+	0.950950i	$0.399890\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−21.0475	+	36.4554i	−0.873199	+	1.51243i
$582$		0			0
$583$	0.372281	+	0.644810i	0.0154183	+	0.0267053i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	4.61684	+	7.99661i	0.190558	+	0.330055i	0.945435	−	0.325810i	$-0.105637\pi$
$587$	4.61684	+	7.99661i	0.190558	+	0.330055i	−0.754878	+	0.655866i	$0.772304\pi$
$588$		0			0
$589$	2.00000	−	3.46410i	0.0824086	−	0.142736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	26.0000			1.06769			0.533846	−	0.845582i	$-0.320746\pi$
$593$	26.0000			1.06769			0.533846	+	0.845582i	$0.320746\pi$
$594$		0			0
$595$	−55.7228			−2.28441
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−19.9307	+	34.5210i	−0.814346	+	1.41049i	0.0954498	+	0.995434i	$0.469571\pi$
$599$	−19.9307	+	34.5210i	−0.814346	+	1.41049i	−0.909796	+	0.415055i	$0.863762\pi$
$600$		0			0
$601$	−2.98913	−	5.17732i	−0.121929	−	0.211187i	0.798599	−	0.601863i	$-0.205575\pi$
$601$	−2.98913	−	5.17732i	−0.121929	−	0.211187i	−0.920528	+	0.390676i	$0.872241\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	11.8614	+	20.5446i	0.482235	+	0.835255i
$606$		0			0
$607$	−5.55842	+	9.62747i	−0.225609	+	0.390767i	−0.956502	−	0.291725i	$-0.905771\pi$
$607$	−5.55842	+	9.62747i	−0.225609	+	0.390767i	0.730893	+	0.682492i	$0.239104\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.62772			0.0658504
$612$		0			0
$613$	12.7446			0.514748			0.257374	−	0.966312i	$-0.417143\pi$
$613$	12.7446			0.514748			0.257374	+	0.966312i	$0.417143\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	18.9891	−	32.8901i	0.764473	−	1.32411i	−0.176051	−	0.984381i	$-0.556332\pi$
$617$	18.9891	−	32.8901i	0.764473	−	1.32411i	0.940525	−	0.339726i	$-0.110334\pi$
$618$		0			0
$619$	−10.6168	−	18.3889i	−0.426727	−	0.739113i	0.569853	−	0.821747i	$-0.307000\pi$
$619$	−10.6168	−	18.3889i	−0.426727	−	0.739113i	−0.996580	+	0.0826338i	$0.973667\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−13.1168	−	22.7190i	−0.525515	−	0.910219i
$624$		0			0
$625$	13.8723	−	24.0275i	0.554891	−	0.961100i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−46.9783			−1.87315
$630$		0			0
$631$	32.0000			1.27390			0.636950	−	0.770905i	$-0.280196\pi$
$631$	32.0000			1.27390			0.636950	+	0.770905i	$0.280196\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	8.00000	−	13.8564i	0.317470	−	0.549875i
$636$		0			0
$637$	−2.25544	−	3.90653i	−0.0893637	−	0.154782i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	1.61684	+	2.80046i	0.0638615	+	0.110611i	0.896188	−	0.443674i	$-0.146325\pi$
$641$	1.61684	+	2.80046i	0.0638615	+	0.110611i	−0.832327	+	0.554285i	$0.812992\pi$
$642$		0			0
$643$	−1.50000	+	2.59808i	−0.0591542	+	0.102458i	−0.894086	−	0.447895i	$-0.852174\pi$
$643$	−1.50000	+	2.59808i	−0.0591542	+	0.102458i	0.834932	+	0.550353i	$0.185507\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	33.7228			1.32578			0.662890	−	0.748717i	$-0.269330\pi$
$647$	33.7228			1.32578			0.662890	+	0.748717i	$0.269330\pi$
$648$		0			0
$649$	7.00000			0.274774
$650$		0			0
$651$		0			0
$652$		0			0
$653$	11.4416	−	19.8174i	0.447744	−	0.775515i	−0.550495	−	0.834838i	$-0.685561\pi$
$653$	11.4416	−	19.8174i	0.447744	−	0.775515i	0.998239	+	0.0593237i	$0.0188944\pi$
$654$		0			0
$655$	21.7921	+	37.7450i	0.851488	+	1.47482i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	9.55842	+	16.5557i	0.372343	+	0.644917i	0.989926	−	0.141589i	$-0.0452211\pi$
$659$	9.55842	+	16.5557i	0.372343	+	0.644917i	−0.617582	+	0.786506i	$0.711888\pi$
$660$		0			0
$661$	−23.0475	+	39.9195i	−0.896446	+	1.55269i	−0.0644406	+	0.997922i	$0.520526\pi$
$661$	−23.0475	+	39.9195i	−0.896446	+	1.55269i	−0.832005	+	0.554768i	$0.812807\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−6.51087			−0.252481
$666$		0			0
$667$	1.62772			0.0630255
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−1.18614	+	2.05446i	−0.0457905	+	0.0793114i
$672$		0			0
$673$	0.186141	+	0.322405i	0.00717520	+	0.0124278i	0.869591	−	0.493773i	$-0.164383\pi$
$673$	0.186141	+	0.322405i	0.00717520	+	0.0124278i	−0.862416	+	0.506201i	$0.831049\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	10.3030	+	17.8453i	0.395976	+	0.685850i	0.993225	−	0.116205i	$-0.0370731\pi$
$677$	10.3030	+	17.8453i	0.395976	+	0.685850i	−0.597249	+	0.802056i	$0.703740\pi$
$678$		0			0
$679$	3.81386	−	6.60580i	0.146362	−	0.253507i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	15.3723			0.588204			0.294102	−	0.955774i	$-0.404979\pi$
$683$	15.3723			0.588204			0.294102	+	0.955774i	$0.404979\pi$
$684$		0			0
$685$	−42.0951			−1.60837
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.138593	+	0.240051i	−0.00527999	+	0.00914521i
$690$		0			0
$691$	−7.55842	−	13.0916i	−0.287536	−	0.498027i	0.685685	−	0.727898i	$-0.259503\pi$
$691$	−7.55842	−	13.0916i	−0.287536	−	0.498027i	−0.973221	+	0.229872i	$0.926169\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−6.81386	−	11.8020i	−0.258464	−	0.447674i
$696$		0			0
$697$	−31.5475	+	54.6420i	−1.19495	+	2.06971i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−35.4891			−1.34041			−0.670203	−	0.742178i	$-0.733793\pi$
$701$	−35.4891			−1.34041			−0.670203	+	0.742178i	$0.733793\pi$
$702$		0			0
$703$	−5.48913			−0.207026
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−15.5584	+	26.9480i	−0.585135	+	1.01348i
$708$		0			0
$709$	−14.3030	−	24.7735i	−0.537160	−	0.930388i	−0.999055	−	0.0434539i	$-0.986164\pi$
$709$	−14.3030	−	24.7735i	−0.537160	−	0.930388i	0.461896	−	0.886934i	$-0.347169\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.18614	−	2.05446i	−0.0444213	−	0.0769400i
$714$		0			0
$715$	0.441578	−	0.764836i	0.0165141	−	0.0286032i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	45.4891			1.69646			0.848229	−	0.529630i	$-0.177669\pi$
$719$	45.4891			1.69646			0.848229	+	0.529630i	$0.177669\pi$
$720$		0			0
$721$	−54.0951			−2.01461
$722$		0			0
$723$		0			0
$724$		0			0
$725$	1.37228	−	2.37686i	0.0509652	−	0.0882744i
$726$		0			0
$727$	5.44158	+	9.42509i	0.201817	+	0.349557i	0.949114	−	0.314933i	$-0.101982\pi$
$727$	5.44158	+	9.42509i	0.201817	+	0.349557i	−0.747297	+	0.664490i	$0.768649\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	4.68614	+	8.11663i	0.173323	+	0.300205i
$732$		0			0
$733$	−18.9307	+	32.7889i	−0.699221	+	1.21109i	0.269515	+	0.962996i	$0.413137\pi$
$733$	−18.9307	+	32.7889i	−0.699221	+	1.21109i	−0.968737	+	0.248091i	$0.920197\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	3.74456			0.137933
$738$		0			0
$739$	42.1168			1.54929			0.774647	−	0.632394i	$-0.217928\pi$
$739$	42.1168			1.54929			0.774647	+	0.632394i	$0.217928\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−17.8139	+	30.8545i	−0.653527	+	1.13194i	0.328734	+	0.944423i	$0.393378\pi$
$743$	−17.8139	+	30.8545i	−0.653527	+	1.13194i	−0.982261	+	0.187520i	$0.939955\pi$
$744$		0			0
$745$	9.93070	+	17.2005i	0.363833	+	0.630177i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	28.1168	+	48.6998i	1.02737	+	1.77945i
$750$		0			0
$751$	−18.8139	+	32.5866i	−0.686527	+	1.18910i	0.286427	+	0.958102i	$0.407533\pi$
$751$	−18.8139	+	32.5866i	−0.686527	+	1.18910i	−0.972954	+	0.230998i	$0.925801\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−0.883156			−0.0321413
$756$		0			0
$757$	−8.51087			−0.309333			−0.154667	−	0.987967i	$-0.549430\pi$
$757$	−8.51087			−0.309333			−0.154667	+	0.987967i	$0.549430\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−9.67527	+	16.7581i	−0.350728	+	0.607479i	−0.986377	−	0.164499i	$-0.947399\pi$
$761$	−9.67527	+	16.7581i	−0.350728	+	0.607479i	0.635649	+	0.771978i	$0.280733\pi$
$762$		0			0
$763$	10.3723	+	17.9653i	0.375502	+	0.650388i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	1.30298	+	2.25684i	0.0470480	+	0.0814896i
$768$		0			0
$769$	−16.5584	+	28.6800i	−0.597112	+	1.03423i	0.396133	+	0.918193i	$0.370352\pi$
$769$	−16.5584	+	28.6800i	−0.597112	+	1.03423i	−0.993245	+	0.116035i	$0.962981\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	28.9783			1.04228			0.521138	−	0.853473i	$-0.325508\pi$
$773$	28.9783			1.04228			0.521138	+	0.853473i	$0.325508\pi$
$774$		0			0
$775$	−4.00000			−0.143684
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−3.68614	+	6.38458i	−0.132070	+	0.228751i
$780$		0			0
$781$	−2.00000	−	3.46410i	−0.0715656	−	0.123955i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3.69702	−	6.40342i	−0.131952	−	0.228548i
$786$		0			0
$787$	−21.4198	+	37.1002i	−0.763534	+	1.32248i	0.177484	+	0.984124i	$0.443204\pi$
$787$	−21.4198	+	37.1002i	−0.763534	+	1.32248i	−0.941018	+	0.338357i	$0.890129\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−19.1168			−0.679717
$792$		0			0
$793$	−0.883156			−0.0313618
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−1.69702	+	2.93932i	−0.0601114	+	0.104116i	−0.894515	−	0.447038i	$-0.852479\pi$
$797$	−1.69702	+	2.93932i	−0.0601114	+	0.104116i	0.834404	+	0.551154i	$0.185812\pi$
$798$		0			0
$799$	−11.7446	−	20.3422i	−0.415493	−	0.719655i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.05842	−	10.4935i	−0.213797	−	0.370307i
$804$		0			0
$805$	−1.93070	+	3.34408i	−0.0680484	+	0.117863i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−13.1386			−0.461928			−0.230964	−	0.972962i	$-0.574188\pi$
$809$	−13.1386			−0.461928			−0.230964	+	0.972962i	$0.574188\pi$
$810$		0			0
$811$	10.3505			0.363456			0.181728	−	0.983349i	$-0.441831\pi$
$811$	10.3505			0.363456			0.181728	+	0.983349i	$0.441831\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	14.2337	−	24.6535i	0.498584	−	0.863573i
$816$		0			0
$817$	0.547547	+	0.948380i	0.0191563	+	0.0331796i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−9.81386	−	16.9981i	−0.342506	−	0.593238i	0.642391	−	0.766377i	$-0.277942\pi$
$821$	−9.81386	−	16.9981i	−0.342506	−	0.593238i	−0.984897	+	0.173139i	$0.944609\pi$
$822$		0			0
$823$	12.4416	−	21.5494i	0.433686	−	0.751166i	−0.563501	−	0.826115i	$-0.690546\pi$
$823$	12.4416	−	21.5494i	0.433686	−	0.751166i	0.997187	+	0.0749488i	$0.0238793\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−30.9783			−1.07722			−0.538610	−	0.842555i	$-0.681050\pi$
$827$	−30.9783			−1.07722			−0.538610	+	0.842555i	$0.681050\pi$
$828$		0			0
$829$	−15.4891			−0.537960			−0.268980	−	0.963146i	$-0.586686\pi$
$829$	−15.4891			−0.537960			−0.268980	+	0.963146i	$0.586686\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−32.5475	+	56.3740i	−1.12771	+	1.95324i
$834$		0			0
$835$	22.6753	+	39.2747i	0.784710	+	1.35916i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−15.0475	−	26.0631i	−0.519499	−	0.899799i	−0.999743	−	0.0226638i	$-0.992785\pi$
$839$	−15.0475	−	26.0631i	−0.519499	−	0.899799i	0.480244	−	0.877135i	$-0.340548\pi$
$840$		0			0
$841$	4.94158	−	8.55906i	0.170399	−	0.295140i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−30.5109			−1.04961
$846$		0			0
$847$	43.7228			1.50233
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−1.62772	+	2.81929i	−0.0557975	+	0.0966441i
$852$		0			0
$853$	16.5584	+	28.6800i	0.566950	+	0.981985i	0.996865	+	0.0791165i	$0.0252099\pi$
$853$	16.5584	+	28.6800i	0.566950	+	0.981985i	−0.429916	+	0.902869i	$0.641457\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	18.5584	+	32.1441i	0.633944	+	1.09802i	0.986738	+	0.162322i	$0.0518982\pi$
$857$	18.5584	+	32.1441i	0.633944	+	1.09802i	−0.352794	+	0.935701i	$0.614768\pi$
$858$		0			0
$859$	1.24456	−	2.15565i	0.0424639	−	0.0735497i	−0.844012	−	0.536324i	$-0.819813\pi$
$859$	1.24456	−	2.15565i	0.0424639	−	0.0735497i	0.886476	+	0.462774i	$0.153146\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−18.9783			−0.646027			−0.323014	−	0.946394i	$-0.604696\pi$
$863$	−18.9783			−0.646027			−0.323014	+	0.946394i	$0.604696\pi$
$864$		0			0
$865$	−38.8397			−1.32059
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.18614	−	5.51856i	0.108082	−	0.187204i
$870$		0			0
$871$	0.697015	+	1.20727i	0.0236175	+	0.0409066i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−22.6753	−	39.2747i	−0.766564	−	1.32773i
$876$		0			0
$877$	13.9307	−	24.1287i	0.470406	−	0.814768i	−0.529021	−	0.848609i	$-0.677441\pi$
$877$	13.9307	−	24.1287i	0.470406	−	0.814768i	0.999427	+	0.0338410i	$0.0107740\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−24.9783			−0.841539			−0.420769	−	0.907168i	$-0.638240\pi$
$881$	−24.9783			−0.841539			−0.420769	+	0.907168i	$0.638240\pi$
$882$		0			0
$883$	31.8397			1.07149			0.535745	−	0.844380i	$-0.320031\pi$
$883$	31.8397			1.07149			0.535745	+	0.844380i	$0.320031\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−5.93070	+	10.2723i	−0.199134	+	0.344909i	−0.948248	−	0.317531i	$-0.897146\pi$
$887$	−5.93070	+	10.2723i	−0.199134	+	0.344909i	0.749114	+	0.662441i	$0.230479\pi$
$888$		0			0
$889$	−14.7446	−	25.5383i	−0.494517	−	0.856528i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−1.37228	−	2.37686i	−0.0459216	−	0.0795386i
$894$		0			0
$895$	27.2554	−	47.2078i	0.911049	−	1.57798i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	27.8614			0.929230
$900$		0			0
$901$	4.00000			0.133259
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−0.605969	+	1.04957i	−0.0201431	+	0.0348889i
$906$		0			0
$907$	−18.5000	−	32.0429i	−0.614282	−	1.06397i	−0.990510	−	0.137441i	$-0.956112\pi$
$907$	−18.5000	−	32.0429i	−0.614282	−	1.06397i	0.376228	−	0.926527i	$-0.377221\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	0.441578	+	0.764836i	0.0146301	+	0.0253401i	0.873248	−	0.487276i	$-0.162010\pi$
$911$	0.441578	+	0.764836i	0.0146301	+	0.0253401i	−0.858618	+	0.512617i	$0.828676\pi$
$912$		0			0
$913$	−4.81386	+	8.33785i	−0.159315	+	0.275943i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	80.3288			2.65269
$918$		0			0
$919$	−36.2337			−1.19524			−0.597620	−	0.801780i	$-0.703887\pi$
$919$	−36.2337			−1.19524			−0.597620	+	0.801780i	$0.703887\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.744563	−	1.28962i	0.0245076	−	0.0424484i
$924$		0			0
$925$	2.74456	+	4.75372i	0.0902407	+	0.156301i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	18.7921	+	32.5489i	0.616549	+	1.06789i	0.990111	+	0.140289i	$0.0448032\pi$
$929$	18.7921	+	32.5489i	0.616549	+	1.06789i	−0.373561	+	0.927605i	$0.621864\pi$
$930$		0			0
$931$	−3.80298	+	6.58696i	−0.124638	+	0.215879i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−12.7446			−0.416792
$936$		0			0
$937$	30.0000			0.980057			0.490029	−	0.871706i	$-0.336986\pi$
$937$	30.0000			0.980057			0.490029	+	0.871706i	$0.336986\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−13.4198	+	23.2438i	−0.437474	+	0.757727i	−0.997494	−	0.0707520i	$-0.977460\pi$
$941$	−13.4198	+	23.2438i	−0.437474	+	0.757727i	0.560020	+	0.828479i	$0.310793\pi$
$942$		0			0
$943$	2.18614	+	3.78651i	0.0711905	+	0.123306i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−26.8723	−	46.5442i	−0.873232	−	1.51248i	−0.858634	−	0.512588i	$-0.828687\pi$
$947$	−26.8723	−	46.5442i	−0.873232	−	1.51248i	−0.0145974	−	0.999893i	$-0.504647\pi$
$948$		0			0
$949$	2.25544	−	3.90653i	0.0732146	−	0.126811i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	7.88316			0.255360			0.127680	−	0.991815i	$-0.459247\pi$
$953$	7.88316			0.255360			0.127680	+	0.991815i	$0.459247\pi$
$954$		0			0
$955$	23.3940			0.757013
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−38.7921	+	67.1899i	−1.25266	+	2.16968i
$960$		0			0
$961$	−4.80298	−	8.31901i	−0.154935	−	0.268355i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2.06930	−	3.58413i	−0.0666130	−	0.115377i
$966$		0			0
$967$	−4.18614	+	7.25061i	−0.134617	+	0.233164i	−0.925451	−	0.378867i	$-0.876314\pi$
$967$	−4.18614	+	7.25061i	−0.134617	+	0.233164i	0.790834	+	0.612031i	$0.209647\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	5.48913			0.176154			0.0880772	−	0.996114i	$-0.471928\pi$
$971$	5.48913			0.176154			0.0880772	+	0.996114i	$0.471928\pi$
$972$		0			0
$973$	−25.1168			−0.805209
$974$		0			0
$975$		0			0
$976$		0			0
$977$	0.127719	−	0.221215i	0.00408608	−	0.00707730i	−0.863975	−	0.503534i	$-0.832033\pi$
$977$	0.127719	−	0.221215i	0.00408608	−	0.00707730i	0.868061	+	0.496457i	$0.165366\pi$
$978$		0			0
$979$	−3.00000	−	5.19615i	−0.0958804	−	0.166070i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−27.5584	−	47.7326i	−0.878977	−	1.52243i	−0.852465	−	0.522785i	$-0.824893\pi$
$983$	−27.5584	−	47.7326i	−0.878977	−	1.52243i	−0.0265123	−	0.999648i	$-0.508440\pi$
$984$		0			0
$985$	−18.0951	+	31.3416i	−0.576558	+	0.998627i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.649468			0.0206519
$990$		0			0
$991$	−50.9783			−1.61938			−0.809689	−	0.586860i	$-0.800364\pi$
$991$	−50.9783			−1.61938			−0.809689	+	0.586860i	$0.800364\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−19.2554	+	33.3514i	−0.610438	+	1.05731i
$996$		0			0
$997$	24.6753	+	42.7388i	0.781474	+	1.35355i	0.931083	+	0.364807i	$0.118865\pi$
$997$	24.6753	+	42.7388i	0.781474	+	1.35355i	−0.149610	+	0.988745i	$0.547802\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1728.2.i.j.577.1		4
3.2	odd	2		576.2.i.l.193.2		4
4.3	odd	2		1728.2.i.i.577.1		4
8.3	odd	2		216.2.i.b.145.2		4
8.5	even	2		432.2.i.d.145.2		4
9.2	odd	6		576.2.i.l.385.2		4
9.4	even	3		5184.2.a.bo.1.2		2
9.5	odd	6		5184.2.a.bs.1.1		2
9.7	even	3	inner	1728.2.i.j.1153.1		4
12.11	even	2		576.2.i.j.193.1		4
24.5	odd	2		144.2.i.d.49.1		4
24.11	even	2		72.2.i.b.49.2	yes	4
36.7	odd	6		1728.2.i.i.1153.1		4
36.11	even	6		576.2.i.j.385.1		4
36.23	even	6		5184.2.a.bt.1.1		2
36.31	odd	6		5184.2.a.bp.1.2		2
72.5	odd	6		1296.2.a.n.1.2		2
72.11	even	6		72.2.i.b.25.2	✓	4
72.13	even	6		1296.2.a.p.1.1		2
72.29	odd	6		144.2.i.d.97.1		4
72.43	odd	6		216.2.i.b.73.2		4
72.59	even	6		648.2.a.f.1.2		2
72.61	even	6		432.2.i.d.289.2		4
72.67	odd	6		648.2.a.g.1.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
72.2.i.b.25.2	✓	4	72.11	even	6
72.2.i.b.49.2	yes	4	24.11	even	2
144.2.i.d.49.1		4	24.5	odd	2
144.2.i.d.97.1		4	72.29	odd	6
216.2.i.b.73.2		4	72.43	odd	6
216.2.i.b.145.2		4	8.3	odd	2
432.2.i.d.145.2		4	8.5	even	2
432.2.i.d.289.2		4	72.61	even	6
576.2.i.j.193.1		4	12.11	even	2
576.2.i.j.385.1		4	36.11	even	6
576.2.i.l.193.2		4	3.2	odd	2
576.2.i.l.385.2		4	9.2	odd	6
648.2.a.f.1.2		2	72.59	even	6
648.2.a.g.1.1		2	72.67	odd	6
1296.2.a.n.1.2		2	72.5	odd	6
1296.2.a.p.1.1		2	72.13	even	6
1728.2.i.i.577.1		4	4.3	odd	2
1728.2.i.i.1153.1		4	36.7	odd	6
1728.2.i.j.577.1		4	1.1	even	1	trivial
1728.2.i.j.1153.1		4	9.7	even	3	inner
5184.2.a.bo.1.2		2	9.4	even	3
5184.2.a.bp.1.2		2	36.31	odd	6
5184.2.a.bs.1.1		2	9.5	odd	6
5184.2.a.bt.1.1		2	36.23	even	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 $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+(-1.18614 + 2.05446i) q^{5} +(2.18614 + 3.78651i) q^{7} +(0.500000 + 0.866025i) q^{11} +(-0.186141 + 0.322405i) q^{13} +5.37228 q^{17} +0.627719 q^{19} +(0.186141 - 0.322405i) q^{23} +(-0.313859 - 0.543620i) q^{25} +(2.18614 + 3.78651i) q^{29} +(3.18614 - 5.51856i) q^{31} -10.3723 q^{35} -8.74456 q^{37} +(-5.87228 + 10.1711i) q^{41} +(0.872281 + 1.51084i) q^{43} +(-2.18614 - 3.78651i) q^{47} +(-6.05842 + 10.4935i) q^{49} +0.744563 q^{53} -2.37228 q^{55} +(3.50000 - 6.06218i) q^{59} +(1.18614 + 2.05446i) q^{61} +(-0.441578 - 0.764836i) q^{65} +(1.87228 - 3.24289i) q^{67} -4.00000 q^{71} -12.1168 q^{73} +(-2.18614 + 3.78651i) q^{77} +(-3.18614 - 5.51856i) q^{79} +(4.81386 + 8.33785i) q^{83} +(-6.37228 + 11.0371i) q^{85} -6.00000 q^{89} -1.62772 q^{91} +(-0.744563 + 1.28962i) q^{95} +(-0.872281 - 1.51084i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{5} + 3 q^{7} + 2 q^{11} + 5 q^{13} + 10 q^{17} + 14 q^{19} - 5 q^{23} - 7 q^{25} + 3 q^{29} + 7 q^{31} - 30 q^{35} - 12 q^{37} - 12 q^{41} - 8 q^{43} - 3 q^{47} - 7 q^{49} - 20 q^{53} + 2 q^{55}+ \cdots + 8 q^{97}+O(q^{100}) \) 4 * q + q^5 + 3 * q^7 + 2 * q^11 + 5 * q^13 + 10 * q^17 + 14 * q^19 - 5 * q^23 - 7 * q^25 + 3 * q^29 + 7 * q^31 - 30 * q^35 - 12 * q^37 - 12 * q^41 - 8 * q^43 - 3 * q^47 - 7 * q^49 - 20 * q^53 + 2 * q^55 + 14 * q^59 - q^61 - 19 * q^65 - 4 * q^67 - 16 * q^71 - 14 * q^73 - 3 * q^77 - 7 * q^79 + 25 * q^83 - 14 * q^85 - 24 * q^89 - 18 * q^91 + 20 * q^95 + 8 * q^97

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