LMFDB - Embedded newform 648.4.i.e.217.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [648,4,Mod(217,648)] 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("648.217"); S:= CuspForms(chi, 4); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(648, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 4])) N = Newforms(chi, 4, names="a")

Level:	$ N $	$=$	$ 648 = 2^{3} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	648.i (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,-2,0,-24,0,0,0,-44] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	no
Analytic conductor:	$38.2332376837$
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:	$\Z[a_1, \ldots, a_{25}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 8)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			217.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	648.217
Dual form			648.4.i.e.433.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(-1.00000 - 1.73205i) q^{5} +(-12.0000 + 20.7846i) q^{7} +(-22.0000 + 38.1051i) q^{11} +(-11.0000 - 19.0526i) q^{13} -50.0000 q^{17} +44.0000 q^{19} +(-28.0000 - 48.4974i) q^{23} +(60.5000 - 104.789i) q^{25} +(99.0000 - 171.473i) q^{29} +(80.0000 + 138.564i) q^{31} +48.0000 q^{35} -162.000 q^{37} +(-99.0000 - 171.473i) q^{41} +(-26.0000 + 45.0333i) q^{43} +(264.000 - 457.261i) q^{47} +(-116.500 - 201.784i) q^{49} +242.000 q^{53} +88.0000 q^{55} +(-334.000 - 578.505i) q^{59} +(-275.000 + 476.314i) q^{61} +(-22.0000 + 38.1051i) q^{65} +(-94.0000 - 162.813i) q^{67} -728.000 q^{71} +154.000 q^{73} +(-528.000 - 914.523i) q^{77} +(328.000 - 568.113i) q^{79} +(118.000 - 204.382i) q^{83} +(50.0000 + 86.6025i) q^{85} -714.000 q^{89} +528.000 q^{91} +(-44.0000 - 76.2102i) q^{95} +(239.000 - 413.960i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q - 2 q^{5} - 24 q^{7} - 44 q^{11} - 22 q^{13} - 100 q^{17} + 88 q^{19} - 56 q^{23} + 121 q^{25} + 198 q^{29} + 160 q^{31} + 96 q^{35} - 324 q^{37} - 198 q^{41} - 52 q^{43} + 528 q^{47} - 233 q^{49} + 484 q^{53}+ \cdots + 478 q^{97}+O(q^{100}) $ 2 * q - 2 * q^5 - 24 * q^7 - 44 * q^11 - 22 * q^13 - 100 * q^17 + 88 * q^19 - 56 * q^23 + 121 * q^25 + 198 * q^29 + 160 * q^31 + 96 * q^35 - 324 * q^37 - 198 * q^41 - 52 * q^43 + 528 * q^47 - 233 * q^49 + 484 * q^53 + 176 * q^55 - 668 * q^59 - 550 * q^61 - 44 * q^65 - 188 * q^67 - 1456 * q^71 + 308 * q^73 - 1056 * q^77 + 656 * q^79 + 236 * q^83 + 100 * q^85 - 1428 * q^89 + 1056 * q^91 - 88 * q^95 + 478 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/648\mathbb{Z}\right)^\times$.

$n$	$325$	$487$	$569$
$\chi(n)$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	−1.00000	−	1.73205i	−0.0894427	−	0.154919i	0.817833	−	0.575456i	$-0.195175\pi$
$5$	−1.00000	−	1.73205i	−0.0894427	−	0.154919i	−0.907276	+	0.420536i	$0.861842\pi$
$6$		0			0
$7$	−12.0000	+	20.7846i	−0.647939	+	1.12226i	0.335675	+	0.941978i	$0.391036\pi$
$7$	−12.0000	+	20.7846i	−0.647939	+	1.12226i	−0.983614	+	0.180286i	$0.942298\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−22.0000	+	38.1051i	−0.603023	+	1.04447i	0.389338	+	0.921095i	$0.372704\pi$
$11$	−22.0000	+	38.1051i	−0.603023	+	1.04447i	−0.992361	+	0.123371i	$0.960630\pi$
$12$		0			0
$13$	−11.0000	−	19.0526i	−0.234681	−	0.406479i	0.724499	−	0.689276i	$-0.242071\pi$
$13$	−11.0000	−	19.0526i	−0.234681	−	0.406479i	−0.959180	+	0.282797i	$0.908738\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−50.0000			−0.713340			−0.356670	−	0.934230i	$-0.616088\pi$
$17$	−50.0000			−0.713340			−0.356670	+	0.934230i	$0.616088\pi$
$18$		0			0
$19$	44.0000			0.531279			0.265639	−	0.964072i	$-0.414417\pi$
$19$	44.0000			0.531279			0.265639	+	0.964072i	$0.414417\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−28.0000	−	48.4974i	−0.253844	−	0.439670i	0.710737	−	0.703458i	$-0.248362\pi$
$23$	−28.0000	−	48.4974i	−0.253844	−	0.439670i	−0.964581	+	0.263788i	$0.915028\pi$
$24$		0			0
$25$	60.5000	−	104.789i	0.484000	−	0.838313i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	99.0000	−	171.473i	0.633925	−	1.09799i	−0.352816	−	0.935693i	$-0.614776\pi$
$29$	99.0000	−	171.473i	0.633925	−	1.09799i	0.986742	−	0.162298i	$-0.0518907\pi$
$30$		0			0
$31$	80.0000	+	138.564i	0.463498	+	0.802801i	0.999132	−	0.0416484i	$-0.0132609\pi$
$31$	80.0000	+	138.564i	0.463498	+	0.802801i	−0.535635	+	0.844450i	$0.679928\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	48.0000			0.231814
$36$		0			0
$37$	−162.000			−0.719801			−0.359900	−	0.932991i	$-0.617189\pi$
$37$	−162.000			−0.719801			−0.359900	+	0.932991i	$0.617189\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−99.0000	−	171.473i	−0.377102	−	0.653161i	0.613537	−	0.789666i	$-0.289746\pi$
$41$	−99.0000	−	171.473i	−0.377102	−	0.653161i	−0.990639	+	0.136505i	$0.956413\pi$
$42$		0			0
$43$	−26.0000	+	45.0333i	−0.0922084	+	0.159710i	−0.908440	−	0.418015i	$-0.862726\pi$
$43$	−26.0000	+	45.0333i	−0.0922084	+	0.159710i	0.816232	+	0.577725i	$0.196059\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	264.000	−	457.261i	0.819327	−	1.41912i	−0.0868522	−	0.996221i	$-0.527681\pi$
$47$	264.000	−	457.261i	0.819327	−	1.41912i	0.906179	−	0.422894i	$-0.138986\pi$
$48$		0			0
$49$	−116.500	−	201.784i	−0.339650	−	0.588291i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	242.000			0.627194			0.313597	−	0.949556i	$-0.398466\pi$
$53$	242.000			0.627194			0.313597	+	0.949556i	$0.398466\pi$
$54$		0			0
$55$	88.0000			0.215744
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−334.000	−	578.505i	−0.737002	−	1.27652i	−0.953840	−	0.300317i	$-0.902908\pi$
$59$	−334.000	−	578.505i	−0.737002	−	1.27652i	0.216838	−	0.976208i	$-0.430426\pi$
$60$		0			0
$61$	−275.000	+	476.314i	−0.577215	+	0.999766i	0.418582	+	0.908179i	$0.362527\pi$
$61$	−275.000	+	476.314i	−0.577215	+	0.999766i	−0.995797	+	0.0915873i	$0.970806\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−22.0000	+	38.1051i	−0.0419810	+	0.0727132i
$66$		0			0
$67$	−94.0000	−	162.813i	−0.171402	−	0.296877i	0.767508	−	0.641039i	$-0.221496\pi$
$67$	−94.0000	−	162.813i	−0.171402	−	0.296877i	−0.938910	+	0.344162i	$0.888163\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−728.000			−1.21687			−0.608435	−	0.793604i	$-0.708202\pi$
$71$	−728.000			−1.21687			−0.608435	+	0.793604i	$0.708202\pi$
$72$		0			0
$73$	154.000			0.246909			0.123454	−	0.992350i	$-0.460603\pi$
$73$	154.000			0.246909			0.123454	+	0.992350i	$0.460603\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−528.000	−	914.523i	−0.781444	−	1.35350i
$78$		0			0
$79$	328.000	−	568.113i	0.467125	−	0.809084i	−0.532170	−	0.846638i	$-0.678623\pi$
$79$	328.000	−	568.113i	0.467125	−	0.809084i	0.999295	+	0.0375534i	$0.0119564\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	118.000	−	204.382i	0.156050	−	0.270287i	−0.777391	−	0.629018i	$-0.783457\pi$
$83$	118.000	−	204.382i	0.156050	−	0.270287i	0.933441	+	0.358731i	$0.116790\pi$
$84$		0			0
$85$	50.0000	+	86.6025i	0.0638031	+	0.110510i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−714.000			−0.850380			−0.425190	−	0.905104i	$-0.639793\pi$
$89$	−714.000			−0.850380			−0.425190	+	0.905104i	$0.639793\pi$
$90$		0			0
$91$	528.000			0.608236
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−44.0000	−	76.2102i	−0.0475190	−	0.0823053i
$96$		0			0
$97$	239.000	−	413.960i	0.250173	−	0.433312i	−0.713400	−	0.700757i	$-0.752846\pi$
$97$	239.000	−	413.960i	0.250173	−	0.433312i	0.963573	+	0.267444i	$0.0861792\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	783.000	−	1356.20i	0.771400	−	1.33610i	−0.165396	−	0.986227i	$-0.552890\pi$
$101$	783.000	−	1356.20i	0.771400	−	1.33610i	0.936796	−	0.349877i	$-0.113777\pi$
$102$		0			0
$103$	484.000	+	838.313i	0.463009	+	0.801955i	0.999109	−	0.0421991i	$-0.0134364\pi$
$103$	484.000	+	838.313i	0.463009	+	0.801955i	−0.536100	+	0.844154i	$0.680103\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	780.000			0.704724			0.352362	−	0.935864i	$-0.385379\pi$
$107$	780.000			0.704724			0.352362	+	0.935864i	$0.385379\pi$
$108$		0			0
$109$	−1994.00			−1.75221			−0.876103	−	0.482123i	$-0.839866\pi$
$109$	−1994.00			−1.75221			−0.876103	+	0.482123i	$0.839866\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−471.000	−	815.796i	−0.392106	−	0.679147i	0.600621	−	0.799534i	$-0.294920\pi$
$113$	−471.000	−	815.796i	−0.392106	−	0.679147i	−0.992727	+	0.120387i	$0.961587\pi$
$114$		0			0
$115$	−56.0000	+	96.9948i	−0.0454089	+	0.0786506i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	600.000	−	1039.23i	0.462201	−	0.800555i
$120$		0			0
$121$	−302.500	−	523.945i	−0.227273	−	0.393648i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−492.000			−0.352047
$126$		0			0
$127$	1408.00			0.983778			0.491889	−	0.870658i	$-0.336307\pi$
$127$	1408.00			0.983778			0.491889	+	0.870658i	$0.336307\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−1346.00	−	2331.34i	−0.897714	−	1.55489i	−0.830409	−	0.557154i	$-0.811893\pi$
$131$	−1346.00	−	2331.34i	−0.897714	−	1.55489i	−0.0673052	−	0.997732i	$-0.521440\pi$
$132$		0			0
$133$	−528.000	+	914.523i	−0.344236	+	0.596234i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	813.000	−	1408.16i	0.507002	−	0.878153i	−0.492965	−	0.870049i	$-0.664087\pi$
$137$	813.000	−	1408.16i	0.507002	−	0.878153i	0.999967	−	0.00810420i	$-0.00257968\pi$
$138$		0			0
$139$	342.000	+	592.361i	0.208691	+	0.361464i	0.951302	−	0.308259i	$-0.0997463\pi$
$139$	342.000	+	592.361i	0.208691	+	0.361464i	−0.742611	+	0.669723i	$0.766413\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	968.000			0.566072
$144$		0			0
$145$	−396.000			−0.226800
$146$		0			0
$147$		0			0
$148$		0			0
$149$	151.000	+	261.540i	0.0830228	+	0.143800i	0.904547	−	0.426374i	$-0.140209\pi$
$149$	151.000	+	261.540i	0.0830228	+	0.143800i	−0.821524	+	0.570174i	$0.806876\pi$
$150$		0			0
$151$	−676.000	+	1170.87i	−0.364319	+	0.631018i	−0.988667	−	0.150128i	$-0.952031\pi$
$151$	−676.000	+	1170.87i	−0.364319	+	0.631018i	0.624348	+	0.781146i	$0.285365\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	160.000	−	277.128i	0.0829130	−	0.143609i
$156$		0			0
$157$	−1571.00	−	2721.05i	−0.798595	−	1.38321i	−0.920531	−	0.390670i	$-0.872244\pi$
$157$	−1571.00	−	2721.05i	−0.798595	−	1.38321i	0.121936	−	0.992538i	$-0.461090\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	1344.00			0.657901
$162$		0			0
$163$	3036.00			1.45888			0.729441	−	0.684043i	$-0.239780\pi$
$163$	3036.00			1.45888			0.729441	+	0.684043i	$0.239780\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−132.000	−	228.631i	−0.0611645	−	0.105940i	0.833822	−	0.552034i	$-0.186148\pi$
$167$	−132.000	−	228.631i	−0.0611645	−	0.105940i	−0.894986	+	0.446094i	$0.852815\pi$
$168$		0			0
$169$	856.500	−	1483.50i	0.389850	−	0.675240i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1413.00	+	2447.39i	−0.620973	+	1.07556i	0.368331	+	0.929695i	$0.379929\pi$
$173$	−1413.00	+	2447.39i	−0.620973	+	1.07556i	−0.989305	+	0.145863i	$0.953404\pi$
$174$		0			0
$175$	1452.00	+	2514.94i	0.627205	+	1.08635i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−3084.00			−1.28776			−0.643880	−	0.765127i	$-0.722676\pi$
$179$	−3084.00			−1.28776			−0.643880	+	0.765127i	$0.722676\pi$
$180$		0			0
$181$	−2418.00			−0.992975			−0.496488	−	0.868044i	$-0.665377\pi$
$181$	−2418.00			−0.992975			−0.496488	+	0.868044i	$0.665377\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	162.000	+	280.592i	0.0643810	+	0.111511i
$186$		0			0
$187$	1100.00	−	1905.26i	0.430160	−	0.745059i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−480.000	+	831.384i	−0.181841	+	0.314957i	−0.942507	−	0.334185i	$-0.891539\pi$
$191$	−480.000	+	831.384i	−0.181841	+	0.314957i	0.760667	+	0.649143i	$0.224872\pi$
$192$		0			0
$193$	−1441.00	−	2495.89i	−0.537438	−	0.930869i	−0.999041	−	0.0437828i	$-0.986059\pi$
$193$	−1441.00	−	2495.89i	−0.537438	−	0.930869i	0.461604	−	0.887086i	$-0.347274\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−1086.00			−0.392763			−0.196381	−	0.980528i	$-0.562919\pi$
$197$	−1086.00			−0.392763			−0.196381	+	0.980528i	$0.562919\pi$
$198$		0			0
$199$	88.0000			0.0313475			0.0156738	−	0.999877i	$-0.495011\pi$
$199$	88.0000			0.0313475			0.0156738	+	0.999877i	$0.495011\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2376.00	+	4115.35i	0.821490	+	1.42286i
$204$		0			0
$205$	−198.000	+	342.946i	−0.0674581	+	0.116841i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−968.000	+	1676.63i	−0.320373	+	0.554902i
$210$		0			0
$211$	1738.00	+	3010.30i	0.567056	+	0.982170i	0.996855	+	0.0792455i	$0.0252511\pi$
$211$	1738.00	+	3010.30i	0.567056	+	0.982170i	−0.429799	+	0.902925i	$0.641416\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	104.000			0.0329895
$216$		0			0
$217$	−3840.00			−1.20127
$218$		0			0
$219$		0			0
$220$		0			0
$221$	550.000	+	952.628i	0.167407	+	0.289958i
$222$		0			0
$223$	−464.000	+	803.672i	−0.139335	+	0.241336i	−0.927245	−	0.374455i	$-0.877830\pi$
$223$	−464.000	+	803.672i	−0.139335	+	0.241336i	0.787910	+	0.615790i	$0.211163\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	78.0000	−	135.100i	0.0228064	−	0.0395018i	−0.854397	−	0.519621i	$-0.826073\pi$
$227$	78.0000	−	135.100i	0.0228064	−	0.0395018i	0.877203	+	0.480119i	$0.159407\pi$
$228$		0			0
$229$	817.000	+	1415.09i	0.235759	+	0.408347i	0.959493	−	0.281732i	$-0.0909090\pi$
$229$	817.000	+	1415.09i	0.235759	+	0.408347i	−0.723734	+	0.690079i	$0.757576\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	902.000			0.253614			0.126807	−	0.991927i	$-0.459527\pi$
$233$	902.000			0.253614			0.126807	+	0.991927i	$0.459527\pi$
$234$		0			0
$235$	−1056.00			−0.293131
$236$		0			0
$237$		0			0
$238$		0			0
$239$	808.000	+	1399.50i	0.218683	+	0.378770i	0.954406	−	0.298513i	$-0.0964907\pi$
$239$	808.000	+	1399.50i	0.218683	+	0.378770i	−0.735723	+	0.677283i	$0.763157\pi$
$240$		0			0
$241$	−2409.00	+	4172.51i	−0.643889	+	1.11525i	0.340667	+	0.940184i	$0.389347\pi$
$241$	−2409.00	+	4172.51i	−0.643889	+	1.11525i	−0.984557	+	0.175065i	$0.943986\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−233.000	+	403.568i	−0.0607585	+	0.105237i
$246$		0			0
$247$	−484.000	−	838.313i	−0.124681	−	0.215954i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	2140.00			0.538150			0.269075	−	0.963119i	$-0.413282\pi$
$251$	2140.00			0.538150			0.269075	+	0.963119i	$0.413282\pi$
$252$		0			0
$253$	2464.00			0.612294
$254$		0			0
$255$		0			0
$256$		0			0
$257$	385.000	+	666.840i	0.0934461	+	0.161853i	0.908959	−	0.416886i	$-0.136878\pi$
$257$	385.000	+	666.840i	0.0934461	+	0.161853i	−0.815513	+	0.578739i	$0.803545\pi$
$258$		0			0
$259$	1944.00	−	3367.11i	0.466387	−	0.807806i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−3700.00	+	6408.59i	−0.867497	+	1.50255i	−0.00295121	+	0.999996i	$0.500939\pi$
$263$	−3700.00	+	6408.59i	−0.867497	+	1.50255i	−0.864546	+	0.502554i	$0.832394\pi$
$264$		0			0
$265$	−242.000	−	419.156i	−0.0560979	−	0.0971644i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	2794.00			0.633283			0.316642	−	0.948545i	$-0.397445\pi$
$269$	2794.00			0.633283			0.316642	+	0.948545i	$0.397445\pi$
$270$		0			0
$271$	8624.00			1.93310			0.966551	−	0.256474i	$-0.0825608\pi$
$271$	8624.00			1.93310			0.966551	+	0.256474i	$0.0825608\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	2662.00	+	4610.72i	0.583726	+	1.01104i
$276$		0			0
$277$	937.000	−	1622.93i	0.203245	−	0.352031i	−0.746327	−	0.665579i	$-0.768185\pi$
$277$	937.000	−	1622.93i	0.203245	−	0.352031i	0.949572	+	0.313549i	$0.101518\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	1669.00	−	2890.79i	0.354321	−	0.613702i	−0.632681	−	0.774413i	$-0.718045\pi$
$281$	1669.00	−	2890.79i	0.354321	−	0.613702i	0.987002	+	0.160711i	$0.0513787\pi$
$282$		0			0
$283$	−3586.00	−	6211.13i	−0.753235	−	1.30464i	−0.946247	−	0.323445i	$-0.895159\pi$
$283$	−3586.00	−	6211.13i	−0.753235	−	1.30464i	0.193012	−	0.981196i	$-0.438174\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	4752.00			0.977358
$288$		0			0
$289$	−2413.00			−0.491146
$290$		0			0
$291$		0			0
$292$		0			0
$293$	2607.00	+	4515.46i	0.519804	+	0.900327i	0.999735	+	0.0230207i	$0.00732835\pi$
$293$	2607.00	+	4515.46i	0.519804	+	0.900327i	−0.479931	+	0.877306i	$0.659338\pi$
$294$		0			0
$295$	−668.000	+	1157.01i	−0.131839	+	0.228352i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−616.000	+	1066.94i	−0.119144	+	0.206364i
$300$		0			0
$301$	−624.000	−	1080.80i	−0.119491	−	0.206964i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	1100.00			0.206511
$306$		0			0
$307$	396.000			0.0736186			0.0368093	−	0.999322i	$-0.488281\pi$
$307$	396.000			0.0736186			0.0368093	+	0.999322i	$0.488281\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−2028.00	−	3512.60i	−0.369766	−	0.640454i	0.619762	−	0.784789i	$-0.287229\pi$
$311$	−2028.00	−	3512.60i	−0.369766	−	0.640454i	−0.989529	+	0.144335i	$0.953896\pi$
$312$		0			0
$313$	−1077.00	+	1865.42i	−0.194491	+	0.336868i	−0.946733	−	0.322018i	$-0.895639\pi$
$313$	−1077.00	+	1865.42i	−0.194491	+	0.336868i	0.752243	+	0.658886i	$0.228972\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3693.00	+	6396.46i	−0.654320	+	1.13332i	0.327743	+	0.944767i	$0.393712\pi$
$317$	−3693.00	+	6396.46i	−0.654320	+	1.13332i	−0.982064	+	0.188549i	$0.939621\pi$
$318$		0			0
$319$	4356.00	+	7544.81i	0.764543	+	1.32423i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−2200.00			−0.378982
$324$		0			0
$325$	−2662.00			−0.454342
$326$		0			0
$327$		0			0
$328$		0			0
$329$	6336.00	+	10974.3i	1.06175	+	1.83900i
$330$		0			0
$331$	566.000	−	980.341i	0.0939884	−	0.162793i	−0.815198	−	0.579183i	$-0.803372\pi$
$331$	566.000	−	980.341i	0.0939884	−	0.162793i	0.909186	+	0.416390i	$0.136705\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−188.000	+	325.626i	−0.0306613	+	0.0531069i
$336$		0			0
$337$	1671.00	+	2894.26i	0.270104	+	0.467834i	0.968888	−	0.247498i	$-0.0796085\pi$
$337$	1671.00	+	2894.26i	0.270104	+	0.467834i	−0.698784	+	0.715333i	$0.746275\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−7040.00			−1.11800
$342$		0			0
$343$	−2640.00			−0.415588
$344$		0			0
$345$		0			0
$346$		0			0
$347$	1122.00	+	1943.36i	0.173580	+	0.300649i	0.939669	−	0.342086i	$-0.111133\pi$
$347$	1122.00	+	1943.36i	0.173580	+	0.300649i	−0.766089	+	0.642734i	$0.777800\pi$
$348$		0			0
$349$	3261.00	−	5648.22i	0.500164	−	0.866310i	−0.499836	−	0.866120i	$-0.666606\pi$
$349$	3261.00	−	5648.22i	0.500164	−	0.866310i	1.00000		0.000189686i	$-6.03788e-5\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−5615.00	+	9725.47i	−0.846618	+	1.46639i	0.0375899	+	0.999293i	$0.488032\pi$
$353$	−5615.00	+	9725.47i	−0.846618	+	1.46639i	−0.884208	+	0.467093i	$0.845301\pi$
$354$		0			0
$355$	728.000	+	1260.93i	0.108840	+	0.188517i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−1848.00			−0.271682			−0.135841	−	0.990731i	$-0.543374\pi$
$359$	−1848.00			−0.271682			−0.135841	+	0.990731i	$0.543374\pi$
$360$		0			0
$361$	−4923.00			−0.717743
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−154.000	−	266.736i	−0.0220842	−	0.0382509i
$366$		0			0
$367$	−3560.00	+	6166.10i	−0.506350	+	0.877024i	0.493623	+	0.869676i	$0.335672\pi$
$367$	−3560.00	+	6166.10i	−0.506350	+	0.877024i	−0.999973	+	0.00734805i	$0.997661\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−2904.00	+	5029.88i	−0.406383	+	0.703876i
$372$		0			0
$373$	−3175.00	−	5499.26i	−0.440738	−	0.763381i	0.557006	−	0.830508i	$-0.311950\pi$
$373$	−3175.00	−	5499.26i	−0.440738	−	0.763381i	−0.997744	+	0.0671276i	$0.978617\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−4356.00			−0.595081
$378$		0			0
$379$	−7900.00			−1.07070			−0.535351	−	0.844630i	$-0.679821\pi$
$379$	−7900.00			−1.07070			−0.535351	+	0.844630i	$0.679821\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5184.00	+	8978.95i	0.691619	+	1.19792i	0.971307	+	0.237828i	$0.0764355\pi$
$383$	5184.00	+	8978.95i	0.691619	+	1.19792i	−0.279688	+	0.960091i	$0.590231\pi$
$384$		0			0
$385$	−1056.00	+	1829.05i	−0.139789	+	0.242122i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	4415.00	−	7647.00i	0.575448	−	0.996706i	−0.420544	−	0.907272i	$-0.638161\pi$
$389$	4415.00	−	7647.00i	0.575448	−	0.996706i	0.995993	−	0.0894338i	$-0.0285057\pi$
$390$		0			0
$391$	1400.00	+	2424.87i	0.181077	+	0.313634i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−1312.00			−0.167124
$396$		0			0
$397$	9878.00			1.24877			0.624386	−	0.781116i	$-0.285349\pi$
$397$	9878.00			1.24877			0.624386	+	0.781116i	$0.285349\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−6567.00	−	11374.4i	−0.817806	−	1.41648i	−0.907295	−	0.420494i	$-0.861857\pi$
$401$	−6567.00	−	11374.4i	−0.817806	−	1.41648i	0.0894889	−	0.995988i	$-0.471477\pi$
$402$		0			0
$403$	1760.00	−	3048.41i	0.217548	−	0.376804i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3564.00	−	6173.03i	0.434056	−	0.751808i
$408$		0			0
$409$	−453.000	−	784.619i	−0.0547663	−	0.0948580i	0.837343	−	0.546679i	$-0.184108\pi$
$409$	−453.000	−	784.619i	−0.0547663	−	0.0948580i	−0.892109	+	0.451821i	$0.850775\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	16032.0			1.91013
$414$		0			0
$415$	−472.000			−0.0558303
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2706.00	−	4686.93i	−0.315505	−	0.546471i	0.664039	−	0.747698i	$-0.268841\pi$
$419$	−2706.00	−	4686.93i	−0.315505	−	0.546471i	−0.979545	+	0.201226i	$0.935507\pi$
$420$		0			0
$421$	2321.00	−	4020.09i	0.268690	−	0.465385i	−0.699834	−	0.714306i	$-0.746743\pi$
$421$	2321.00	−	4020.09i	0.268690	−	0.465385i	0.968524	+	0.248921i	$0.0800758\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−3025.00	+	5239.45i	−0.345257	+	0.598002i
$426$		0			0
$427$	−6600.00	−	11431.5i	−0.748001	−	1.29558i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−656.000			−0.0733142			−0.0366571	−	0.999328i	$-0.511671\pi$
$431$	−656.000			−0.0733142			−0.0366571	+	0.999328i	$0.511671\pi$
$432$		0			0
$433$	9490.00			1.05326			0.526629	−	0.850096i	$-0.323456\pi$
$433$	9490.00			1.05326			0.526629	+	0.850096i	$0.323456\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1232.00	−	2133.89i	−0.134862	−	0.233587i
$438$		0			0
$439$	−2772.00	+	4801.24i	−0.301368	+	0.521984i	−0.976446	−	0.215762i	$-0.930776\pi$
$439$	−2772.00	+	4801.24i	−0.301368	+	0.521984i	0.675078	+	0.737746i	$0.264110\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	3826.00	−	6626.83i	0.410336	−	0.710722i	−0.584591	−	0.811328i	$-0.698745\pi$
$443$	3826.00	−	6626.83i	0.410336	−	0.710722i	0.994926	+	0.100606i	$0.0320782\pi$
$444$		0			0
$445$	714.000	+	1236.68i	0.0760603	+	0.131740i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	446.000			0.0468776			0.0234388	−	0.999725i	$-0.492539\pi$
$449$	446.000			0.0468776			0.0234388	+	0.999725i	$0.492539\pi$
$450$		0			0
$451$	8712.00			0.909605
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−528.000	−	914.523i	−0.0544022	−	0.0942275i
$456$		0			0
$457$	−781.000	+	1352.73i	−0.0799423	+	0.138464i	−0.903225	−	0.429168i	$-0.858807\pi$
$457$	−781.000	+	1352.73i	−0.0799423	+	0.138464i	0.823283	+	0.567632i	$0.192140\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	5291.00	−	9164.28i	0.534548	−	0.925864i	−0.464638	−	0.885501i	$-0.653815\pi$
$461$	5291.00	−	9164.28i	0.534548	−	0.925864i	0.999185	−	0.0403626i	$-0.0128513\pi$
$462$		0			0
$463$	5384.00	+	9325.36i	0.540423	+	0.936040i	0.998880	+	0.0473229i	$0.0150690\pi$
$463$	5384.00	+	9325.36i	0.540423	+	0.936040i	−0.458457	+	0.888717i	$0.651598\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	9876.00			0.978601			0.489301	−	0.872115i	$-0.337252\pi$
$467$	9876.00			0.978601			0.489301	+	0.872115i	$0.337252\pi$
$468$		0			0
$469$	4512.00			0.444232
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−1144.00	−	1981.47i	−0.111208	−	0.192617i
$474$		0			0
$475$	2662.00	−	4610.72i	0.257139	−	0.445377i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−176.000	+	304.841i	−0.0167884	+	0.0290784i	−0.874298	−	0.485390i	$-0.838677\pi$
$479$	−176.000	+	304.841i	−0.0167884	+	0.0290784i	0.857509	+	0.514469i	$0.172011\pi$
$480$		0			0
$481$	1782.00	+	3086.51i	0.168924	+	0.292584i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−956.000			−0.0895046
$486$		0			0
$487$	−15176.0			−1.41209			−0.706047	−	0.708165i	$-0.749523\pi$
$487$	−15176.0			−1.41209			−0.706047	+	0.708165i	$0.749523\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−4422.00	−	7659.13i	−0.406440	−	0.703975i	0.588048	−	0.808826i	$-0.299897\pi$
$491$	−4422.00	−	7659.13i	−0.406440	−	0.703975i	−0.994488	+	0.104851i	$0.966563\pi$
$492$		0			0
$493$	−4950.00	+	8573.65i	−0.452204	+	0.783241i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	8736.00	−	15131.2i	0.788457	−	1.36565i
$498$		0			0
$499$	−9702.00	−	16804.4i	−0.870383	−	1.50755i	−0.861601	−	0.507586i	$-0.830538\pi$
$499$	−9702.00	−	16804.4i	−0.870383	−	1.50755i	−0.00878220	−	0.999961i	$-0.502795\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−16488.0			−1.46156			−0.730779	−	0.682614i	$-0.760843\pi$
$503$	−16488.0			−1.46156			−0.730779	+	0.682614i	$0.760843\pi$
$504$		0			0
$505$	−3132.00			−0.275984
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−6477.00	−	11218.5i	−0.564024	−	0.976917i	−0.997140	−	0.0755793i	$-0.975919\pi$
$509$	−6477.00	−	11218.5i	−0.564024	−	0.976917i	0.433116	−	0.901338i	$-0.357414\pi$
$510$		0			0
$511$	−1848.00	+	3200.83i	−0.159982	+	0.277097i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	968.000	−	1676.63i	0.0828256	−	0.143458i
$516$		0			0
$517$	11616.0	+	20119.5i	0.988145	+	1.71152i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10970.0			−0.922465			−0.461233	−	0.887279i	$-0.652593\pi$
$521$	−10970.0			−0.922465			−0.461233	+	0.887279i	$0.652593\pi$
$522$		0			0
$523$	−16940.0			−1.41632			−0.708159	−	0.706053i	$-0.750474\pi$
$523$	−16940.0			−1.41632			−0.708159	+	0.706053i	$0.750474\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−4000.00	−	6928.20i	−0.330631	−	0.572670i
$528$		0			0
$529$	4515.50	−	7821.08i	0.371127	−	0.642811i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2178.00	+	3772.41i	−0.176997	+	0.306569i
$534$		0			0
$535$	−780.000	−	1351.00i	−0.0630324	−	0.109175i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	10252.0			0.819267
$540$		0			0
$541$	198.000			0.0157351			0.00786755	−	0.999969i	$-0.497496\pi$
$541$	198.000			0.0157351			0.00786755	+	0.999969i	$0.497496\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1994.00	+	3453.71i	0.156722	+	0.271451i
$546$		0			0
$547$	7634.00	−	13222.5i	0.596721	−	1.03355i	−0.396581	−	0.918000i	$-0.629803\pi$
$547$	7634.00	−	13222.5i	0.596721	−	1.03355i	0.993302	−	0.115551i	$-0.0368633\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	4356.00	−	7544.81i	0.336791	−	0.583339i
$552$		0			0
$553$	7872.00	+	13634.7i	0.605337	+	1.04847i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−20854.0			−1.58638			−0.793189	−	0.608976i	$-0.791581\pi$
$557$	−20854.0			−1.58638			−0.793189	+	0.608976i	$0.791581\pi$
$558$		0			0
$559$	1144.00			0.0865582
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−9658.00	−	16728.1i	−0.722977	−	1.25223i	−0.959801	−	0.280681i	$-0.909440\pi$
$563$	−9658.00	−	16728.1i	−0.722977	−	1.25223i	0.236824	−	0.971553i	$-0.423894\pi$
$564$		0			0
$565$	−942.000	+	1631.59i	−0.0701420	+	0.121490i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	3509.00	−	6077.77i	0.258532	−	0.447791i	−0.707317	−	0.706897i	$-0.750094\pi$
$569$	3509.00	−	6077.77i	0.258532	−	0.447791i	0.965849	+	0.259106i	$0.0834278\pi$
$570$		0			0
$571$	−12210.0	−	21148.3i	−0.894873	−	1.54997i	−0.833961	−	0.551823i	$-0.813933\pi$
$571$	−12210.0	−	21148.3i	−0.894873	−	1.54997i	−0.0609117	−	0.998143i	$-0.519401\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−6776.00			−0.491441
$576$		0			0
$577$	23234.0			1.67633			0.838166	−	0.545415i	$-0.183628\pi$
$577$	23234.0			1.67633			0.838166	+	0.545415i	$0.183628\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2832.00	+	4905.17i	0.202222	+	0.350259i
$582$		0			0
$583$	−5324.00	+	9221.44i	−0.378212	+	0.655082i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−5302.00	+	9183.33i	−0.372806	+	0.645718i	−0.989996	−	0.141095i	$-0.954938\pi$
$587$	−5302.00	+	9183.33i	−0.372806	+	0.645718i	0.617190	+	0.786814i	$0.288271\pi$
$588$		0			0
$589$	3520.00	+	6096.82i	0.246246	+	0.426511i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	13838.0			0.958277			0.479139	−	0.877739i	$-0.340949\pi$
$593$	13838.0			0.958277			0.479139	+	0.877739i	$0.340949\pi$
$594$		0			0
$595$	−2400.00			−0.165362
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−1980.00	−	3429.46i	−0.135059	−	0.233930i	0.790561	−	0.612384i	$-0.209789\pi$
$599$	−1980.00	−	3429.46i	−0.135059	−	0.233930i	−0.925620	+	0.378454i	$0.876456\pi$
$600$		0			0
$601$	2971.00	−	5145.92i	0.201647	−	0.349262i	−0.747412	−	0.664360i	$-0.768704\pi$
$601$	2971.00	−	5145.92i	0.201647	−	0.349262i	0.949059	+	0.315098i	$0.102037\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−605.000	+	1047.89i	−0.0406558	+	0.0704179i
$606$		0			0
$607$	1520.00	+	2632.72i	0.101639	+	0.176044i	0.912360	−	0.409389i	$-0.134258\pi$
$607$	1520.00	+	2632.72i	0.101639	+	0.176044i	−0.810721	+	0.585433i	$0.800925\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−11616.0			−0.769121
$612$		0			0
$613$	−2530.00			−0.166698			−0.0833489	−	0.996520i	$-0.526562\pi$
$613$	−2530.00			−0.166698			−0.0833489	+	0.996520i	$0.526562\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−9603.00	−	16632.9i	−0.626584	−	1.08528i	−0.988232	−	0.152960i	$-0.951119\pi$
$617$	−9603.00	−	16632.9i	−0.626584	−	1.08528i	0.361648	−	0.932315i	$-0.382214\pi$
$618$		0			0
$619$	−5498.00	+	9522.82i	−0.357000	+	0.618343i	−0.987458	−	0.157881i	$-0.949534\pi$
$619$	−5498.00	+	9522.82i	−0.357000	+	0.618343i	0.630458	+	0.776224i	$0.282867\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	8568.00	−	14840.2i	0.550995	−	0.954351i
$624$		0			0
$625$	−7070.50	−	12246.5i	−0.452512	−	0.783774i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	8100.00			0.513463
$630$		0			0
$631$	−6680.00			−0.421437			−0.210718	−	0.977547i	$-0.567580\pi$
$631$	−6680.00			−0.421437			−0.210718	+	0.977547i	$0.567580\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1408.00	−	2438.73i	−0.0879918	−	0.152406i
$636$		0			0
$637$	−2563.00	+	4439.25i	−0.159419	+	0.276121i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	3137.00	−	5433.44i	0.193298	−	0.334802i	−0.753043	−	0.657971i	$-0.771415\pi$
$641$	3137.00	−	5433.44i	0.193298	−	0.334802i	0.946341	+	0.323169i	$0.104748\pi$
$642$		0			0
$643$	−4542.00	−	7866.97i	−0.278568	−	0.482493i	0.692461	−	0.721455i	$-0.256526\pi$
$643$	−4542.00	−	7866.97i	−0.278568	−	0.482493i	−0.971029	+	0.238962i	$0.923193\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	23656.0			1.43742			0.718712	−	0.695308i	$-0.244732\pi$
$647$	23656.0			1.43742			0.718712	+	0.695308i	$0.244732\pi$
$648$		0			0
$649$	29392.0			1.77771
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3381.00	−	5856.06i	−0.202617	−	0.350943i	0.746754	−	0.665100i	$-0.231611\pi$
$653$	−3381.00	−	5856.06i	−0.202617	−	0.350943i	−0.949371	+	0.314158i	$0.898278\pi$
$654$		0			0
$655$	−2692.00	+	4662.68i	−0.160588	+	0.278147i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	7638.00	−	13229.4i	0.451494	−	0.782010i	−0.546986	−	0.837142i	$-0.684225\pi$
$659$	7638.00	−	13229.4i	0.451494	−	0.782010i	0.998479	+	0.0551323i	$0.0175581\pi$
$660$		0			0
$661$	−5527.00	−	9573.04i	−0.325228	−	0.563311i	0.656331	−	0.754473i	$-0.272108\pi$
$661$	−5527.00	−	9573.04i	−0.325228	−	0.563311i	−0.981558	+	0.191163i	$0.938774\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	2112.00			0.123158
$666$		0			0
$667$	−11088.0			−0.643672
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−12100.0	−	20957.8i	−0.696148	−	1.20576i
$672$		0			0
$673$	10639.0	−	18427.3i	0.609366	−	1.05545i	−0.381979	−	0.924171i	$-0.624757\pi$
$673$	10639.0	−	18427.3i	0.609366	−	1.05545i	0.991345	−	0.131282i	$-0.0419093\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	4463.00	−	7730.14i	0.253363	−	0.438838i	−0.711086	−	0.703105i	$-0.751796\pi$
$677$	4463.00	−	7730.14i	0.253363	−	0.438838i	0.964450	+	0.264266i	$0.0851298\pi$
$678$		0			0
$679$	5736.00	+	9935.04i	0.324194	+	0.561520i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−8116.00			−0.454685			−0.227343	−	0.973815i	$-0.573004\pi$
$683$	−8116.00			−0.454685			−0.227343	+	0.973815i	$0.573004\pi$
$684$		0			0
$685$	−3252.00			−0.181391
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−2662.00	−	4610.72i	−0.147190	−	0.254941i
$690$		0			0
$691$	5882.00	−	10187.9i	0.323823	−	0.560878i	−0.657450	−	0.753498i	$-0.728365\pi$
$691$	5882.00	−	10187.9i	0.323823	−	0.560878i	0.981273	+	0.192620i	$0.0616984\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	684.000	−	1184.72i	0.0373318	−	0.0646606i
$696$		0			0
$697$	4950.00	+	8573.65i	0.269002	+	0.465926i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	4698.00			0.253126			0.126563	−	0.991959i	$-0.459605\pi$
$701$	4698.00			0.253126			0.126563	+	0.991959i	$0.459605\pi$
$702$		0			0
$703$	−7128.00			−0.382415
$704$		0			0
$705$		0			0
$706$		0			0
$707$	18792.0	+	32548.7i	0.999641	+	1.73143i
$708$		0			0
$709$	−12319.0	+	21337.1i	−0.652538	+	1.13023i	0.329966	+	0.943993i	$0.392963\pi$
$709$	−12319.0	+	21337.1i	−0.652538	+	1.13023i	−0.982505	+	0.186237i	$0.940371\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4480.00	−	7759.59i	0.235312	−	0.407572i
$714$		0			0
$715$	−968.000	−	1676.63i	−0.0506310	−	0.0876954i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−16624.0			−0.862268			−0.431134	−	0.902288i	$-0.641886\pi$
$719$	−16624.0			−0.862268			−0.431134	+	0.902288i	$0.641886\pi$
$720$		0			0
$721$	−23232.0			−1.20001
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−11979.0	−	20748.2i	−0.613640	−	1.06286i
$726$		0			0
$727$	−15108.0	+	26167.8i	−0.770735	+	1.33495i	0.166425	+	0.986054i	$0.446778\pi$
$727$	−15108.0	+	26167.8i	−0.770735	+	1.33495i	−0.937160	+	0.348899i	$0.886556\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1300.00	−	2251.67i	0.0657760	−	0.113927i
$732$		0			0
$733$	1661.00	+	2876.94i	0.0836977	+	0.144969i	0.904836	−	0.425761i	$-0.139994\pi$
$733$	1661.00	+	2876.94i	0.0836977	+	0.144969i	−0.821138	+	0.570730i	$0.806660\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	8272.00			0.413437
$738$		0			0
$739$	−14692.0			−0.731331			−0.365666	−	0.930746i	$-0.619159\pi$
$739$	−14692.0			−0.731331			−0.365666	+	0.930746i	$0.619159\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	14300.0	+	24768.3i	0.706078	+	1.22296i	0.966301	+	0.257415i	$0.0828707\pi$
$743$	14300.0	+	24768.3i	0.706078	+	1.22296i	−0.260223	+	0.965549i	$0.583796\pi$
$744$		0			0
$745$	302.000	−	523.079i	0.0148516	−	0.0257237i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−9360.00	+	16212.0i	−0.456618	+	0.790886i
$750$		0			0
$751$	14808.0	+	25648.2i	0.719509	+	1.24623i	0.961194	+	0.275872i	$0.0889666\pi$
$751$	14808.0	+	25648.2i	0.719509	+	1.24623i	−0.241685	+	0.970355i	$0.577700\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	2704.00			0.130343
$756$		0			0
$757$	2894.00			0.138949			0.0694744	−	0.997584i	$-0.477868\pi$
$757$	2894.00			0.138949			0.0694744	+	0.997584i	$0.477868\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	7381.00	+	12784.3i	0.351591	+	0.608974i	0.986528	−	0.163590i	$-0.0523073\pi$
$761$	7381.00	+	12784.3i	0.351591	+	0.608974i	−0.634937	+	0.772564i	$0.718974\pi$
$762$		0			0
$763$	23928.0	−	41444.5i	1.13532	−	1.96644i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−7348.00	+	12727.1i	−0.345920	+	0.599152i
$768$		0			0
$769$	3839.00	+	6649.34i	0.180023	+	0.311809i	0.941888	−	0.335926i	$-0.109049\pi$
$769$	3839.00	+	6649.34i	0.180023	+	0.311809i	−0.761865	+	0.647736i	$0.775716\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−27390.0			−1.27445			−0.637225	−	0.770678i	$-0.719918\pi$
$773$	−27390.0			−1.27445			−0.637225	+	0.770678i	$0.719918\pi$
$774$		0			0
$775$	19360.0			0.897331
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−4356.00	−	7544.81i	−0.200346	−	0.347010i
$780$		0			0
$781$	16016.0	−	27740.5i	0.733800	−	1.27098i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−3142.00	+	5442.10i	−0.142857	+	0.247436i
$786$		0			0
$787$	−9878.00	−	17109.2i	−0.447411	−	0.774939i	0.550805	−	0.834634i	$-0.314321\pi$
$787$	−9878.00	−	17109.2i	−0.447411	−	0.774939i	−0.998217	+	0.0596946i	$0.980987\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	22608.0			1.01624
$792$		0			0
$793$	12100.0			0.541846
$794$		0			0
$795$		0			0
$796$		0			0
$797$	19427.0	+	33648.6i	0.863412	+	1.49547i	0.868615	+	0.495488i	$0.165011\pi$
$797$	19427.0	+	33648.6i	0.863412	+	1.49547i	−0.00520266	+	0.999986i	$0.501656\pi$
$798$		0			0
$799$	−13200.0	+	22863.1i	−0.584459	+	1.01231i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−3388.00	+	5868.19i	−0.148892	+	0.257888i
$804$		0			0
$805$	−1344.00	−	2327.88i	−0.0588444	−	0.101922i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	14278.0			0.620504			0.310252	−	0.950654i	$-0.399587\pi$
$809$	14278.0			0.620504			0.310252	+	0.950654i	$0.399587\pi$
$810$		0			0
$811$	−716.000			−0.0310014			−0.0155007	−	0.999880i	$-0.504934\pi$
$811$	−716.000			−0.0310014			−0.0155007	+	0.999880i	$0.504934\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−3036.00	−	5258.51i	−0.130486	−	0.226009i
$816$		0			0
$817$	−1144.00	+	1981.47i	−0.0489884	+	0.0848503i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−11769.0	+	20384.5i	−0.500293	+	0.866534i	0.499706	+	0.866195i	$0.333441\pi$
$821$	−11769.0	+	20384.5i	−0.500293	+	0.866534i	−1.00000		0.000338894i	$0.999892\pi$
$822$		0			0
$823$	3308.00	+	5729.62i	0.140109	+	0.242676i	0.927537	−	0.373730i	$-0.121921\pi$
$823$	3308.00	+	5729.62i	0.140109	+	0.242676i	−0.787429	+	0.616406i	$0.788588\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−27236.0			−1.14521			−0.572605	−	0.819831i	$-0.694067\pi$
$827$	−27236.0			−1.14521			−0.572605	+	0.819831i	$0.694067\pi$
$828$		0			0
$829$	12070.0			0.505680			0.252840	−	0.967508i	$-0.418635\pi$
$829$	12070.0			0.505680			0.252840	+	0.967508i	$0.418635\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	5825.00	+	10089.2i	0.242286	+	0.419652i
$834$		0			0
$835$	−264.000	+	457.261i	−0.0109414	+	0.0189511i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−21012.0	+	36393.9i	−0.864618	+	1.49756i	0.00280727	+	0.999996i	$0.499106\pi$
$839$	−21012.0	+	36393.9i	−0.864618	+	1.49756i	−0.867426	+	0.497567i	$0.834227\pi$
$840$		0			0
$841$	−7407.50	−	12830.2i	−0.303723	−	0.526064i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3426.00			−0.139477
$846$		0			0
$847$	14520.0			0.589036
$848$		0			0
$849$		0			0
$850$		0			0
$851$	4536.00	+	7856.58i	0.182717	+	0.316475i
$852$		0			0
$853$	−1207.00	+	2090.59i	−0.0484489	+	0.0839159i	−0.889233	−	0.457455i	$-0.848761\pi$
$853$	−1207.00	+	2090.59i	−0.0484489	+	0.0839159i	0.840784	+	0.541371i	$0.182094\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−18843.0	+	32637.0i	−0.751067	+	1.30089i	0.196239	+	0.980556i	$0.437127\pi$
$857$	−18843.0	+	32637.0i	−0.751067	+	1.30089i	−0.947306	+	0.320330i	$0.896206\pi$
$858$		0			0
$859$	−20322.0	−	35198.7i	−0.807192	−	1.39810i	−0.914801	−	0.403904i	$-0.867653\pi$
$859$	−20322.0	−	35198.7i	−0.807192	−	1.39810i	0.107610	−	0.994193i	$-0.465680\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	18656.0			0.735872			0.367936	−	0.929851i	$-0.380065\pi$
$863$	18656.0			0.735872			0.367936	+	0.929851i	$0.380065\pi$
$864$		0			0
$865$	5652.00			0.222166
$866$		0			0
$867$		0			0
$868$		0			0
$869$	14432.0	+	24997.0i	0.563374	+	0.975793i
$870$		0			0
$871$	−2068.00	+	3581.88i	−0.0804495	+	0.139343i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	5904.00	−	10226.0i	0.228105	−	0.395089i
$876$		0			0
$877$	6501.00	+	11260.1i	0.250311	+	0.433552i	0.963612	−	0.267307i	$-0.0861336\pi$
$877$	6501.00	+	11260.1i	0.250311	+	0.433552i	−0.713300	+	0.700859i	$0.752800\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−49490.0			−1.89258			−0.946289	−	0.323323i	$-0.895200\pi$
$881$	−49490.0			−1.89258			−0.946289	+	0.323323i	$0.895200\pi$
$882$		0			0
$883$	1100.00			0.0419229			0.0209615	−	0.999780i	$-0.493327\pi$
$883$	1100.00			0.0419229			0.0209615	+	0.999780i	$0.493327\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−7052.00	−	12214.4i	−0.266948	−	0.462368i	0.701124	−	0.713039i	$-0.252682\pi$
$887$	−7052.00	−	12214.4i	−0.266948	−	0.462368i	−0.968072	+	0.250672i	$0.919349\pi$
$888$		0			0
$889$	−16896.0	+	29264.7i	−0.637428	+	1.10406i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	11616.0	−	20119.5i	0.435291	−	0.753946i
$894$		0			0
$895$	3084.00	+	5341.64i	0.115181	+	0.199499i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	31680.0			1.17529
$900$		0			0
$901$	−12100.0			−0.447402
$902$		0			0
$903$		0			0
$904$		0			0
$905$	2418.00	+	4188.10i	0.0888144	+	0.153831i
$906$		0			0
$907$	6358.00	−	11012.4i	0.232761	−	0.403153i	−0.725859	−	0.687844i	$-0.758557\pi$
$907$	6358.00	−	11012.4i	0.232761	−	0.403153i	0.958620	+	0.284690i	$0.0918908\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−19816.0	+	34322.3i	−0.720673	+	1.24824i	0.240057	+	0.970759i	$0.422834\pi$
$911$	−19816.0	+	34322.3i	−0.720673	+	1.24824i	−0.960730	+	0.277484i	$0.910499\pi$
$912$		0			0
$913$	5192.00	+	8992.81i	0.188204	+	0.325979i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	64608.0			2.32666
$918$		0			0
$919$	5704.00			0.204742			0.102371	−	0.994746i	$-0.467357\pi$
$919$	5704.00			0.204742			0.102371	+	0.994746i	$0.467357\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	8008.00	+	13870.3i	0.285576	+	0.494632i
$924$		0			0
$925$	−9801.00	+	16975.8i	−0.348384	+	0.603418i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	4081.00	−	7068.50i	0.144126	−	0.249634i	−0.784920	−	0.619597i	$-0.787296\pi$
$929$	4081.00	−	7068.50i	0.144126	−	0.249634i	0.929047	+	0.369963i	$0.120630\pi$
$930$		0			0
$931$	−5126.00	−	8878.49i	−0.180449	−	0.312547i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−4400.00			−0.153899
$936$		0			0
$937$	−55110.0			−1.92141			−0.960707	−	0.277564i	$-0.910473\pi$
$937$	−55110.0			−1.92141			−0.960707	+	0.277564i	$0.910473\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	8187.00	+	14180.3i	0.283622	+	0.491248i	0.972274	−	0.233844i	$-0.0751305\pi$
$941$	8187.00	+	14180.3i	0.283622	+	0.491248i	−0.688652	+	0.725092i	$0.741797\pi$
$942$		0			0
$943$	−5544.00	+	9602.49i	−0.191450	+	0.331601i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	4230.00	−	7326.57i	0.145149	−	0.251406i	−0.784279	−	0.620408i	$-0.786967\pi$
$947$	4230.00	−	7326.57i	0.145149	−	0.251406i	0.929429	+	0.369002i	$0.120300\pi$
$948$		0			0
$949$	−1694.00	−	2934.09i	−0.0579447	−	0.100363i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	20502.0			0.696878			0.348439	−	0.937331i	$-0.386712\pi$
$953$	20502.0			0.696878			0.348439	+	0.937331i	$0.386712\pi$
$954$		0			0
$955$	1920.00			0.0650573
$956$		0			0
$957$		0			0
$958$		0			0
$959$	19512.0	+	33795.8i	0.657013	+	1.13798i
$960$		0			0
$961$	2095.50	−	3629.51i	0.0703400	−	0.121833i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−2882.00	+	4991.77i	−0.0961398	+	0.166519i
$966$		0			0
$967$	18260.0	+	31627.2i	0.607241	+	1.05177i	0.991693	+	0.128627i	$0.0410571\pi$
$967$	18260.0	+	31627.2i	0.607241	+	1.05177i	−0.384452	+	0.923145i	$0.625610\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−20244.0			−0.669064			−0.334532	−	0.942384i	$-0.608578\pi$
$971$	−20244.0			−0.669064			−0.334532	+	0.942384i	$0.608578\pi$
$972$		0			0
$973$	−16416.0			−0.540876
$974$		0			0
$975$		0			0
$976$		0			0
$977$	25017.0	+	43330.7i	0.819206	+	1.41891i	0.906268	+	0.422704i	$0.138919\pi$
$977$	25017.0	+	43330.7i	0.819206	+	1.41891i	−0.0870612	+	0.996203i	$0.527748\pi$
$978$		0			0
$979$	15708.0	−	27207.1i	0.512799	−	0.888193i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	18564.0	−	32153.8i	0.602339	−	1.04328i	−0.390126	−	0.920761i	$-0.627569\pi$
$983$	18564.0	−	32153.8i	0.602339	−	1.04328i	0.992466	−	0.122521i	$-0.0390979\pi$
$984$		0			0
$985$	1086.00	+	1881.01i	0.0351298	+	0.0608466i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2912.00			0.0936261
$990$		0			0
$991$	27808.0			0.891373			0.445686	−	0.895189i	$-0.352960\pi$
$991$	27808.0			0.891373			0.445686	+	0.895189i	$0.352960\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−88.0000	−	152.420i	−0.00280381	−	0.00485634i
$996$		0			0
$997$	14257.0	−	24693.8i	0.452882	−	0.784415i	−0.545681	−	0.837993i	$-0.683729\pi$
$997$	14257.0	−	24693.8i	0.452882	−	0.784415i	0.998564	+	0.0535775i	$0.0170624\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	648.4.i.e.217.1		2
3.2	odd	2		648.4.i.h.217.1		2
9.2	odd	6		8.4.a.a.1.1	✓	1
9.4	even	3	inner	648.4.i.e.433.1		2
9.5	odd	6		648.4.i.h.433.1		2
9.7	even	3		72.4.a.c.1.1		1
36.7	odd	6		144.4.a.e.1.1		1
36.11	even	6		16.4.a.a.1.1		1
45.2	even	12		200.4.c.e.49.2		2
45.7	odd	12		1800.4.f.u.649.2		2
45.29	odd	6		200.4.a.g.1.1		1
45.34	even	6		1800.4.a.d.1.1		1
45.38	even	12		200.4.c.e.49.1		2
45.43	odd	12		1800.4.f.u.649.1		2
63.2	odd	6		392.4.i.g.361.1		2
63.11	odd	6		392.4.i.g.177.1		2
63.20	even	6		392.4.a.e.1.1		1
63.38	even	6		392.4.i.b.177.1		2
63.47	even	6		392.4.i.b.361.1		2
72.11	even	6		64.4.a.b.1.1		1
72.29	odd	6		64.4.a.d.1.1		1
72.43	odd	6		576.4.a.j.1.1		1
72.61	even	6		576.4.a.k.1.1		1
99.65	even	6		968.4.a.a.1.1		1
117.38	odd	6		1352.4.a.a.1.1		1
144.11	even	12		256.4.b.g.129.1		2
144.29	odd	12		256.4.b.a.129.1		2
144.83	even	12		256.4.b.g.129.2		2
144.101	odd	12		256.4.b.a.129.2		2
153.101	odd	6		2312.4.a.a.1.1		1
180.47	odd	12		400.4.c.i.49.1		2
180.83	odd	12		400.4.c.i.49.2		2
180.119	even	6		400.4.a.g.1.1		1
252.83	odd	6		784.4.a.e.1.1		1
360.29	odd	6		1600.4.a.o.1.1		1
360.299	even	6		1600.4.a.bm.1.1		1
396.263	odd	6		1936.4.a.l.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8.4.a.a.1.1	✓	1	9.2	odd	6
16.4.a.a.1.1		1	36.11	even	6
64.4.a.b.1.1		1	72.11	even	6
64.4.a.d.1.1		1	72.29	odd	6
72.4.a.c.1.1		1	9.7	even	3
144.4.a.e.1.1		1	36.7	odd	6
200.4.a.g.1.1		1	45.29	odd	6
200.4.c.e.49.1		2	45.38	even	12
200.4.c.e.49.2		2	45.2	even	12
256.4.b.a.129.1		2	144.29	odd	12
256.4.b.a.129.2		2	144.101	odd	12
256.4.b.g.129.1		2	144.11	even	12
256.4.b.g.129.2		2	144.83	even	12
392.4.a.e.1.1		1	63.20	even	6
392.4.i.b.177.1		2	63.38	even	6
392.4.i.b.361.1		2	63.47	even	6
392.4.i.g.177.1		2	63.11	odd	6
392.4.i.g.361.1		2	63.2	odd	6
400.4.a.g.1.1		1	180.119	even	6
400.4.c.i.49.1		2	180.47	odd	12
400.4.c.i.49.2		2	180.83	odd	12
576.4.a.j.1.1		1	72.43	odd	6
576.4.a.k.1.1		1	72.61	even	6
648.4.i.e.217.1		2	1.1	even	1	trivial
648.4.i.e.433.1		2	9.4	even	3	inner
648.4.i.h.217.1		2	3.2	odd	2
648.4.i.h.433.1		2	9.5	odd	6
784.4.a.e.1.1		1	252.83	odd	6
968.4.a.a.1.1		1	99.65	even	6
1352.4.a.a.1.1		1	117.38	odd	6
1600.4.a.o.1.1		1	360.29	odd	6
1600.4.a.bm.1.1		1	360.299	even	6
1800.4.a.d.1.1		1	45.34	even	6
1800.4.f.u.649.1		2	45.43	odd	12
1800.4.f.u.649.2		2	45.7	odd	12
1936.4.a.l.1.1		1	396.263	odd	6
2312.4.a.a.1.1		1	153.101	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 648 = 2^{3} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	648.i (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-1.00000 - 1.73205i) q^{5} +(-12.0000 + 20.7846i) q^{7} +(-22.0000 + 38.1051i) q^{11} +(-11.0000 - 19.0526i) q^{13} -50.0000 q^{17} +44.0000 q^{19} +(-28.0000 - 48.4974i) q^{23} +(60.5000 - 104.789i) q^{25} +(99.0000 - 171.473i) q^{29} +(80.0000 + 138.564i) q^{31} +48.0000 q^{35} -162.000 q^{37} +(-99.0000 - 171.473i) q^{41} +(-26.0000 + 45.0333i) q^{43} +(264.000 - 457.261i) q^{47} +(-116.500 - 201.784i) q^{49} +242.000 q^{53} +88.0000 q^{55} +(-334.000 - 578.505i) q^{59} +(-275.000 + 476.314i) q^{61} +(-22.0000 + 38.1051i) q^{65} +(-94.0000 - 162.813i) q^{67} -728.000 q^{71} +154.000 q^{73} +(-528.000 - 914.523i) q^{77} +(328.000 - 568.113i) q^{79} +(118.000 - 204.382i) q^{83} +(50.0000 + 86.6025i) q^{85} -714.000 q^{89} +528.000 q^{91} +(-44.0000 - 76.2102i) q^{95} +(239.000 - 413.960i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q - 2 q^{5} - 24 q^{7} - 44 q^{11} - 22 q^{13} - 100 q^{17} + 88 q^{19} - 56 q^{23} + 121 q^{25} + 198 q^{29} + 160 q^{31} + 96 q^{35} - 324 q^{37} - 198 q^{41} - 52 q^{43} + 528 q^{47} - 233 q^{49} + 484 q^{53}+ \cdots + 478 q^{97}+O(q^{100}) \) 2 * q - 2 * q^5 - 24 * q^7 - 44 * q^11 - 22 * q^13 - 100 * q^17 + 88 * q^19 - 56 * q^23 + 121 * q^25 + 198 * q^29 + 160 * q^31 + 96 * q^35 - 324 * q^37 - 198 * q^41 - 52 * q^43 + 528 * q^47 - 233 * q^49 + 484 * q^53 + 176 * q^55 - 668 * q^59 - 550 * q^61 - 44 * q^65 - 188 * q^67 - 1456 * q^71 + 308 * q^73 - 1056 * q^77 + 656 * q^79 + 236 * q^83 + 100 * q^85 - 1428 * q^89 + 1056 * q^91 - 88 * q^95 + 478 * q^97

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