LMFDB - Embedded newform 3240.2.q.t.2161.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [3240,2,Mod(1081,3240)] 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("3240.1081"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(3240, base_ring=CyclotomicField(6)) chi = DirichletCharacter(H, H._module([0, 0, 2, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 3240 = 2^{3} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	3240.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment:select newform

sage:traces = [2,0,0,0,1,0,1,0,0,0,-2,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	no
Analytic conductor:	$25.8715302549$
Analytic rank:	$0$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{6})$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x + 1 $ x^2 - x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 1080)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			2161.1
Root			$0.500000 + 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.2161
Dual form			3240.2.q.t.1081.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} +(-1.00000 + 1.73205i) q^{11} +(2.50000 + 4.33013i) q^{13} +4.00000 q^{17} -5.00000 q^{19} +(-1.00000 - 1.73205i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(5.00000 - 8.66025i) q^{29} +(4.00000 + 6.92820i) q^{31} +1.00000 q^{35} -3.00000 q^{37} +(3.00000 + 5.19615i) q^{41} +(-2.00000 + 3.46410i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(3.00000 + 5.19615i) q^{49} -6.00000 q^{53} -2.00000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(2.50000 - 4.33013i) q^{61} +(-2.50000 + 4.33013i) q^{65} +(3.50000 + 6.06218i) q^{67} -6.00000 q^{71} -9.00000 q^{73} +(1.00000 + 1.73205i) q^{77} +(-1.50000 + 2.59808i) q^{79} +(1.00000 - 1.73205i) q^{83} +(2.00000 + 3.46410i) q^{85} +5.00000 q^{91} +(-2.50000 - 4.33013i) q^{95} +(-3.50000 + 6.06218i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + q^{5} + q^{7} - 2 q^{11} + 5 q^{13} + 8 q^{17} - 10 q^{19} - 2 q^{23} - q^{25} + 10 q^{29} + 8 q^{31} + 2 q^{35} - 6 q^{37} + 6 q^{41} - 4 q^{43} - 8 q^{47} + 6 q^{49} - 12 q^{53} - 4 q^{55} - 4 q^{59}+ \cdots - 7 q^{97}+O(q^{100}) $ 2 * q + q^5 + q^7 - 2 * q^11 + 5 * q^13 + 8 * q^17 - 10 * q^19 - 2 * q^23 - q^25 + 10 * q^29 + 8 * q^31 + 2 * q^35 - 6 * q^37 + 6 * q^41 - 4 * q^43 - 8 * q^47 + 6 * q^49 - 12 * q^53 - 4 * q^55 - 4 * q^59 + 5 * q^61 - 5 * q^65 + 7 * q^67 - 12 * q^71 - 18 * q^73 + 2 * q^77 - 3 * q^79 + 2 * q^83 + 4 * q^85 + 10 * q^91 - 5 * q^95 - 7 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/3240\mathbb{Z}\right)^\times$.

$n$	$1297$	$1621$	$2431$	$3161$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

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
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	−0.755929	−	0.654654i	$-0.772814\pi$
$7$	0.500000	−	0.866025i	0.188982	−	0.327327i	0.944911	+	0.327327i	$0.106148\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	−0.976478	−	0.215615i	$-0.930824\pi$
$11$	−1.00000	+	1.73205i	−0.301511	+	0.522233i	0.674967	+	0.737848i	$0.264158\pi$
$12$		0			0
$13$	2.50000	+	4.33013i	0.693375	+	1.20096i	0.970725	+	0.240192i	$0.0772105\pi$
$13$	2.50000	+	4.33013i	0.693375	+	1.20096i	−0.277350	+	0.960769i	$0.589456\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	4.00000			0.970143			0.485071	−	0.874475i	$-0.338794\pi$
$17$	4.00000			0.970143			0.485071	+	0.874475i	$0.338794\pi$
$18$		0			0
$19$	−5.00000			−1.14708			−0.573539	−	0.819178i	$-0.694430\pi$
$19$	−5.00000			−1.14708			−0.573539	+	0.819178i	$0.694430\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	0.742732	−	0.669588i	$-0.233529\pi$
$23$	−1.00000	−	1.73205i	−0.208514	−	0.361158i	−0.951247	+	0.308431i	$0.900196\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.00000	−	8.66025i	0.928477	−	1.60817i	0.142605	−	0.989780i	$-0.454452\pi$
$29$	5.00000	−	8.66025i	0.928477	−	1.60817i	0.785872	−	0.618389i	$-0.212214\pi$
$30$		0			0
$31$	4.00000	+	6.92820i	0.718421	+	1.24434i	0.961625	+	0.274367i	$0.0884683\pi$
$31$	4.00000	+	6.92820i	0.718421	+	1.24434i	−0.243204	+	0.969975i	$0.578198\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.00000			0.169031
$36$		0			0
$37$	−3.00000			−0.493197			−0.246598	−	0.969118i	$-0.579313\pi$
$37$	−3.00000			−0.493197			−0.246598	+	0.969118i	$0.579313\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	3.00000	+	5.19615i	0.468521	+	0.811503i	0.999353	−	0.0359748i	$-0.0114536\pi$
$41$	3.00000	+	5.19615i	0.468521	+	0.811503i	−0.530831	+	0.847477i	$0.678120\pi$
$42$		0			0
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	−0.977261	−	0.212041i	$-0.931989\pi$
$43$	−2.00000	+	3.46410i	−0.304997	+	0.528271i	0.672264	+	0.740312i	$0.265322\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	0.411606	+	0.911362i	$0.364968\pi$
$47$	−4.00000	+	6.92820i	−0.583460	+	1.01058i	−0.995066	+	0.0992202i	$0.968365\pi$
$48$		0			0
$49$	3.00000	+	5.19615i	0.428571	+	0.742307i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.00000			−0.824163			−0.412082	−	0.911147i	$-0.635198\pi$
$53$	−6.00000			−0.824163			−0.412082	+	0.911147i	$0.635198\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	0.705965	−	0.708247i	$-0.250514\pi$
$59$	−2.00000	−	3.46410i	−0.260378	−	0.450988i	−0.966342	+	0.257260i	$0.917180\pi$
$60$		0			0
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	−0.660415	−	0.750901i	$-0.729619\pi$
$61$	2.50000	−	4.33013i	0.320092	−	0.554416i	0.980507	+	0.196485i	$0.0629528\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.50000	+	4.33013i	−0.310087	+	0.537086i
$66$		0			0
$67$	3.50000	+	6.06218i	0.427593	+	0.740613i	0.996659	−	0.0816792i	$-0.0260283\pi$
$67$	3.50000	+	6.06218i	0.427593	+	0.740613i	−0.569066	+	0.822292i	$0.692695\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−6.00000			−0.712069			−0.356034	−	0.934473i	$-0.615871\pi$
$71$	−6.00000			−0.712069			−0.356034	+	0.934473i	$0.615871\pi$
$72$		0			0
$73$	−9.00000			−1.05337			−0.526685	−	0.850060i	$-0.676565\pi$
$73$	−9.00000			−1.05337			−0.526685	+	0.850060i	$0.676565\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	1.00000	+	1.73205i	0.113961	+	0.197386i
$78$		0			0
$79$	−1.50000	+	2.59808i	−0.168763	+	0.292306i	−0.937985	−	0.346675i	$-0.887311\pi$
$79$	−1.50000	+	2.59808i	−0.168763	+	0.292306i	0.769222	+	0.638982i	$0.220644\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.00000	−	1.73205i	0.109764	−	0.190117i	−0.805910	−	0.592037i	$-0.798324\pi$
$83$	1.00000	−	1.73205i	0.109764	−	0.190117i	0.915675	+	0.401920i	$0.131657\pi$
$84$		0			0
$85$	2.00000	+	3.46410i	0.216930	+	0.375735i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		0			0			−	1.00000i	$-0.5\pi$
$89$		0			0				1.00000i	$0.5\pi$
$90$		0			0
$91$	5.00000			0.524142
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.50000	−	4.33013i	−0.256495	−	0.444262i
$96$		0			0
$97$	−3.50000	+	6.06218i	−0.355371	+	0.615521i	−0.987181	−	0.159602i	$-0.948979\pi$
$97$	−3.50000	+	6.06218i	−0.355371	+	0.615521i	0.631810	+	0.775123i	$0.282312\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$101$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$102$		0			0
$103$	2.50000	+	4.33013i	0.246332	+	0.426660i	0.962505	−	0.271263i	$-0.0874412\pi$
$103$	2.50000	+	4.33013i	0.246332	+	0.426660i	−0.716173	+	0.697923i	$0.754108\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	18.0000			1.74013			0.870063	−	0.492941i	$-0.164078\pi$
$107$	18.0000			1.74013			0.870063	+	0.492941i	$0.164078\pi$
$108$		0			0
$109$	14.0000			1.34096			0.670478	−	0.741929i	$-0.266089\pi$
$109$	14.0000			1.34096			0.670478	+	0.741929i	$0.266089\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	0.971930	−	0.235269i	$-0.0755971\pi$
$113$	3.00000	+	5.19615i	0.282216	+	0.488813i	−0.689714	+	0.724082i	$0.742264\pi$
$114$		0			0
$115$	1.00000	−	1.73205i	0.0932505	−	0.161515i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	2.00000	−	3.46410i	0.183340	−	0.317554i
$120$		0			0
$121$	3.50000	+	6.06218i	0.318182	+	0.551107i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$		0			0			−	1.00000i	$-0.5\pi$
$127$		0			0				1.00000i	$0.5\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	−0.999602	−	0.0281993i	$-0.991023\pi$
$131$	−6.00000	−	10.3923i	−0.524222	−	0.907980i	0.475380	−	0.879781i	$-0.342311\pi$
$132$		0			0
$133$	−2.50000	+	4.33013i	−0.216777	+	0.375470i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	−0.905577	−	0.424182i	$-0.860562\pi$
$137$	−1.00000	+	1.73205i	−0.0854358	+	0.147979i	0.820141	+	0.572161i	$0.193895\pi$
$138$		0			0
$139$	7.50000	+	12.9904i	0.636142	+	1.10183i	0.986272	+	0.165129i	$0.0528040\pi$
$139$	7.50000	+	12.9904i	0.636142	+	1.10183i	−0.350130	+	0.936701i	$0.613863\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−10.0000			−0.836242
$144$		0			0
$145$	10.0000			0.830455
$146$		0			0
$147$		0			0
$148$		0			0
$149$	10.0000	+	17.3205i	0.819232	+	1.41895i	0.906249	+	0.422744i	$0.138933\pi$
$149$	10.0000	+	17.3205i	0.819232	+	1.41895i	−0.0870170	+	0.996207i	$0.527733\pi$
$150$		0			0
$151$	−11.5000	+	19.9186i	−0.935857	+	1.62095i	−0.162758	+	0.986666i	$0.552039\pi$
$151$	−11.5000	+	19.9186i	−0.935857	+	1.62095i	−0.773099	+	0.634285i	$0.781294\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−4.00000	+	6.92820i	−0.321288	+	0.556487i
$156$		0			0
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	0.997609	+	0.0691164i	$0.0220180\pi$
$157$	7.00000	+	12.1244i	0.558661	+	0.967629i	−0.438948	+	0.898513i	$0.644649\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.00000			−0.157622
$162$		0			0
$163$	19.0000			1.48819			0.744097	−	0.668071i	$-0.232880\pi$
$163$	19.0000			1.48819			0.744097	+	0.668071i	$0.232880\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−12.0000	−	20.7846i	−0.928588	−	1.60836i	−0.785687	−	0.618624i	$-0.787690\pi$
$167$	−12.0000	−	20.7846i	−0.928588	−	1.60836i	−0.142901	−	0.989737i	$-0.545643\pi$
$168$		0			0
$169$	−6.00000	+	10.3923i	−0.461538	+	0.799408i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.0000	−	17.3205i	0.760286	−	1.31685i	−0.182417	−	0.983221i	$-0.558392\pi$
$173$	10.0000	−	17.3205i	0.760286	−	1.31685i	0.942703	−	0.333633i	$-0.108275\pi$
$174$		0			0
$175$	0.500000	+	0.866025i	0.0377964	+	0.0654654i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−26.0000			−1.94333			−0.971666	−	0.236360i	$-0.924046\pi$
$179$	−26.0000			−1.94333			−0.971666	+	0.236360i	$0.924046\pi$
$180$		0			0
$181$	−7.00000			−0.520306			−0.260153	−	0.965567i	$-0.583773\pi$
$181$	−7.00000			−0.520306			−0.260153	+	0.965567i	$0.583773\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−1.50000	−	2.59808i	−0.110282	−	0.191014i
$186$		0			0
$187$	−4.00000	+	6.92820i	−0.292509	+	0.506640i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−8.00000	+	13.8564i	−0.578860	+	1.00261i	0.416751	+	0.909021i	$0.363169\pi$
$191$	−8.00000	+	13.8564i	−0.578860	+	1.00261i	−0.995610	+	0.0935936i	$0.970165\pi$
$192$		0			0
$193$	−0.500000	−	0.866025i	−0.0359908	−	0.0623379i	0.847469	−	0.530845i	$-0.178125\pi$
$193$	−0.500000	−	0.866025i	−0.0359908	−	0.0623379i	−0.883460	+	0.468507i	$0.844792\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	20.0000			1.42494			0.712470	−	0.701702i	$-0.247576\pi$
$197$	20.0000			1.42494			0.712470	+	0.701702i	$0.247576\pi$
$198$		0			0
$199$	−7.00000			−0.496217			−0.248108	−	0.968732i	$-0.579809\pi$
$199$	−7.00000			−0.496217			−0.248108	+	0.968732i	$0.579809\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.00000	−	8.66025i	−0.350931	−	0.607831i
$204$		0			0
$205$	−3.00000	+	5.19615i	−0.209529	+	0.362915i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	5.00000	−	8.66025i	0.345857	−	0.599042i
$210$		0			0
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	0.767049	−	0.641588i	$-0.221724\pi$
$211$	−2.50000	−	4.33013i	−0.172107	−	0.298098i	−0.939156	+	0.343490i	$0.888391\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−4.00000			−0.272798
$216$		0			0
$217$	8.00000			0.543075
$218$		0			0
$219$		0			0
$220$		0			0
$221$	10.0000	+	17.3205i	0.672673	+	1.16510i
$222$		0			0
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	−0.968309	−	0.249756i	$-0.919650\pi$
$223$	−4.00000	+	6.92820i	−0.267860	+	0.463947i	0.700449	+	0.713702i	$0.252983\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.00000	−	15.5885i	0.597351	−	1.03464i	−0.395860	−	0.918311i	$-0.629553\pi$
$227$	9.00000	−	15.5885i	0.597351	−	1.03464i	0.993210	−	0.116331i	$-0.0371134\pi$
$228$		0			0
$229$	−11.0000	−	19.0526i	−0.726900	−	1.25903i	−0.958187	−	0.286143i	$-0.907627\pi$
$229$	−11.0000	−	19.0526i	−0.726900	−	1.25903i	0.231287	−	0.972886i	$-0.425707\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	24.0000			1.57229			0.786146	−	0.618041i	$-0.212073\pi$
$233$	24.0000			1.57229			0.786146	+	0.618041i	$0.212073\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$	9.00000	+	15.5885i	0.582162	+	1.00833i	0.995223	+	0.0976302i	$0.0311262\pi$
$239$	9.00000	+	15.5885i	0.582162	+	1.00833i	−0.413061	+	0.910703i	$0.635540\pi$
$240$		0			0
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	0.450910	+	0.892570i	$0.351100\pi$
$241$	−8.50000	+	14.7224i	−0.547533	+	0.948355i	−0.998443	+	0.0557856i	$0.982234\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.00000	+	5.19615i	−0.191663	+	0.331970i
$246$		0			0
$247$	−12.5000	−	21.6506i	−0.795356	−	1.37760i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−18.0000			−1.13615			−0.568075	−	0.822977i	$-0.692312\pi$
$251$	−18.0000			−1.13615			−0.568075	+	0.822977i	$0.692312\pi$
$252$		0			0
$253$	4.00000			0.251478
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−12.0000	−	20.7846i	−0.748539	−	1.29651i	−0.948523	−	0.316709i	$-0.897422\pi$
$257$	−12.0000	−	20.7846i	−0.748539	−	1.29651i	0.199983	−	0.979799i	$-0.435911\pi$
$258$		0			0
$259$	−1.50000	+	2.59808i	−0.0932055	+	0.161437i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	−0.989561	−	0.144112i	$-0.953967\pi$
$263$	−6.00000	+	10.3923i	−0.369976	+	0.640817i	0.619586	+	0.784929i	$0.287301\pi$
$264$		0			0
$265$	−3.00000	−	5.19615i	−0.184289	−	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	4.00000			0.243884			0.121942	−	0.992537i	$-0.461088\pi$
$269$	4.00000			0.243884			0.121942	+	0.992537i	$0.461088\pi$
$270$		0			0
$271$	3.00000			0.182237			0.0911185	−	0.995840i	$-0.470956\pi$
$271$	3.00000			0.182237			0.0911185	+	0.995840i	$0.470956\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.00000	−	1.73205i	−0.0603023	−	0.104447i
$276$		0			0
$277$	−13.0000	+	22.5167i	−0.781094	+	1.35290i	0.150210	+	0.988654i	$0.452005\pi$
$277$	−13.0000	+	22.5167i	−0.781094	+	1.35290i	−0.931305	+	0.364241i	$0.881328\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−16.0000	+	27.7128i	−0.954480	+	1.65321i	−0.218926	+	0.975741i	$0.570255\pi$
$281$	−16.0000	+	27.7128i	−0.954480	+	1.65321i	−0.735554	+	0.677466i	$0.763078\pi$
$282$		0			0
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	0.919327	−	0.393494i	$-0.128734\pi$
$283$	2.00000	+	3.46410i	0.118888	+	0.205919i	−0.800439	+	0.599414i	$0.795400\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	6.00000			0.354169
$288$		0			0
$289$	−1.00000			−0.0588235
$290$		0			0
$291$		0			0
$292$		0			0
$293$	11.0000	+	19.0526i	0.642627	+	1.11306i	0.984844	+	0.173442i	$0.0554888\pi$
$293$	11.0000	+	19.0526i	0.642627	+	1.11306i	−0.342217	+	0.939621i	$0.611178\pi$
$294$		0			0
$295$	2.00000	−	3.46410i	0.116445	−	0.201688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	5.00000	−	8.66025i	0.289157	−	0.500835i
$300$		0			0
$301$	2.00000	+	3.46410i	0.115278	+	0.199667i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	5.00000			0.286299
$306$		0			0
$307$	16.0000			0.913168			0.456584	−	0.889680i	$-0.349073\pi$
$307$	16.0000			0.913168			0.456584	+	0.889680i	$0.349073\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	10.0000	+	17.3205i	0.567048	+	0.982156i	0.996856	+	0.0792356i	$0.0252479\pi$
$311$	10.0000	+	17.3205i	0.567048	+	0.982156i	−0.429808	+	0.902920i	$0.641419\pi$
$312$		0			0
$313$	9.50000	−	16.4545i	0.536972	−	0.930062i	−0.462093	−	0.886831i	$-0.652902\pi$
$313$	9.50000	−	16.4545i	0.536972	−	0.930062i	0.999065	−	0.0432311i	$-0.0137652\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	6.00000	−	10.3923i	0.336994	−	0.583690i	−0.646872	−	0.762598i	$-0.723923\pi$
$317$	6.00000	−	10.3923i	0.336994	−	0.583690i	0.983866	+	0.178908i	$0.0572566\pi$
$318$		0			0
$319$	10.0000	+	17.3205i	0.559893	+	0.969762i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−20.0000			−1.11283
$324$		0			0
$325$	−5.00000			−0.277350
$326$		0			0
$327$		0			0
$328$		0			0
$329$	4.00000	+	6.92820i	0.220527	+	0.381964i
$330$		0			0
$331$	7.50000	−	12.9904i	0.412237	−	0.714016i	−0.582897	−	0.812546i	$-0.698081\pi$
$331$	7.50000	−	12.9904i	0.412237	−	0.714016i	0.995134	+	0.0985303i	$0.0314141\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.50000	+	6.06218i	−0.191225	+	0.331212i
$336$		0			0
$337$	14.5000	+	25.1147i	0.789865	+	1.36809i	0.926049	+	0.377403i	$0.123183\pi$
$337$	14.5000	+	25.1147i	0.789865	+	1.36809i	−0.136184	+	0.990684i	$0.543484\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−16.0000			−0.866449
$342$		0			0
$343$	13.0000			0.701934
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−17.0000	−	29.4449i	−0.912608	−	1.58068i	−0.810366	−	0.585923i	$-0.800732\pi$
$347$	−17.0000	−	29.4449i	−0.912608	−	1.58068i	−0.102241	−	0.994760i	$-0.532601\pi$
$348$		0			0
$349$	−7.50000	+	12.9904i	−0.401466	+	0.695359i	−0.993903	−	0.110257i	$-0.964832\pi$
$349$	−7.50000	+	12.9904i	−0.401466	+	0.695359i	0.592437	+	0.805617i	$0.298166\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−2.00000	+	3.46410i	−0.106449	+	0.184376i	−0.914329	−	0.404971i	$-0.867282\pi$
$353$	−2.00000	+	3.46410i	−0.106449	+	0.184376i	0.807880	+	0.589347i	$0.200615\pi$
$354$		0			0
$355$	−3.00000	−	5.19615i	−0.159223	−	0.275783i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	18.0000			0.950004			0.475002	−	0.879985i	$-0.342447\pi$
$359$	18.0000			0.950004			0.475002	+	0.879985i	$0.342447\pi$
$360$		0			0
$361$	6.00000			0.315789
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.50000	−	7.79423i	−0.235541	−	0.407969i
$366$		0			0
$367$	12.5000	−	21.6506i	0.652495	−	1.13015i	−0.330021	−	0.943974i	$-0.607056\pi$
$367$	12.5000	−	21.6506i	0.652495	−	1.13015i	0.982516	−	0.186180i	$-0.0596109\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−3.00000	+	5.19615i	−0.155752	+	0.269771i
$372$		0			0
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	0.972556	−	0.232671i	$-0.0747464\pi$
$373$	5.50000	+	9.52628i	0.284779	+	0.493252i	−0.687776	+	0.725923i	$0.741413\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	50.0000			2.57513
$378$		0			0
$379$	7.00000			0.359566			0.179783	−	0.983706i	$-0.442460\pi$
$379$	7.00000			0.359566			0.179783	+	0.983706i	$0.442460\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000	+	20.7846i	0.613171	+	1.06204i	0.990702	+	0.136047i	$0.0434398\pi$
$383$	12.0000	+	20.7846i	0.613171	+	1.06204i	−0.377531	+	0.925997i	$0.623227\pi$
$384$		0			0
$385$	−1.00000	+	1.73205i	−0.0509647	+	0.0882735i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	6.00000	−	10.3923i	0.304212	−	0.526911i	−0.672874	−	0.739758i	$-0.734940\pi$
$389$	6.00000	−	10.3923i	0.304212	−	0.526911i	0.977086	+	0.212847i	$0.0682735\pi$
$390$		0			0
$391$	−4.00000	−	6.92820i	−0.202289	−	0.350374i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.00000			−0.150946
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	0.548911	−	0.835881i	$-0.315043\pi$
$401$	−9.00000	−	15.5885i	−0.449439	−	0.778450i	−0.998350	+	0.0574304i	$0.981709\pi$
$402$		0			0
$403$	−20.0000	+	34.6410i	−0.996271	+	1.72559i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	3.00000	−	5.19615i	0.148704	−	0.257564i
$408$		0			0
$409$	−12.5000	−	21.6506i	−0.618085	−	1.07056i	−0.989835	−	0.142222i	$-0.954575\pi$
$409$	−12.5000	−	21.6506i	−0.618085	−	1.07056i	0.371750	−	0.928333i	$-0.378758\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−4.00000			−0.196827
$414$		0			0
$415$	2.00000			0.0981761
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−6.00000	−	10.3923i	−0.293119	−	0.507697i	0.681426	−	0.731887i	$-0.261360\pi$
$419$	−6.00000	−	10.3923i	−0.293119	−	0.507697i	−0.974546	+	0.224189i	$0.928027\pi$
$420$		0			0
$421$	13.5000	−	23.3827i	0.657950	−	1.13960i	−0.323196	−	0.946332i	$-0.604757\pi$
$421$	13.5000	−	23.3827i	0.657950	−	1.13960i	0.981146	−	0.193270i	$-0.0619094\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−2.00000	+	3.46410i	−0.0970143	+	0.168034i
$426$		0			0
$427$	−2.50000	−	4.33013i	−0.120983	−	0.209550i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−38.0000			−1.83040			−0.915198	−	0.403005i	$-0.867966\pi$
$431$	−38.0000			−1.83040			−0.915198	+	0.403005i	$0.867966\pi$
$432$		0			0
$433$	−22.0000			−1.05725			−0.528626	−	0.848855i	$-0.677293\pi$
$433$	−22.0000			−1.05725			−0.528626	+	0.848855i	$0.677293\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	5.00000	+	8.66025i	0.239182	+	0.414276i
$438$		0			0
$439$	16.0000	−	27.7128i	0.763638	−	1.32266i	−0.177325	−	0.984152i	$-0.556744\pi$
$439$	16.0000	−	27.7128i	0.763638	−	1.32266i	0.940963	−	0.338508i	$-0.109922\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	18.0000	−	31.1769i	0.855206	−	1.48126i	−0.0212481	−	0.999774i	$-0.506764\pi$
$443$	18.0000	−	31.1769i	0.855206	−	1.48126i	0.876454	−	0.481486i	$-0.159903\pi$
$444$		0			0
$445$		0			0
$446$		0			0
$447$		0			0
$448$		0			0
$449$	24.0000			1.13263			0.566315	−	0.824189i	$-0.308369\pi$
$449$	24.0000			1.13263			0.566315	+	0.824189i	$0.308369\pi$
$450$		0			0
$451$	−12.0000			−0.565058
$452$		0			0
$453$		0			0
$454$		0			0
$455$	2.50000	+	4.33013i	0.117202	+	0.202999i
$456$		0			0
$457$	13.0000	−	22.5167i	0.608114	−	1.05328i	−0.383437	−	0.923567i	$-0.625260\pi$
$457$	13.0000	−	22.5167i	0.608114	−	1.05328i	0.991551	−	0.129718i	$-0.0414071\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$461$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$462$		0			0
$463$	−13.5000	−	23.3827i	−0.627398	−	1.08669i	−0.988072	−	0.153993i	$-0.950787\pi$
$463$	−13.5000	−	23.3827i	−0.627398	−	1.08669i	0.360674	−	0.932692i	$-0.382547\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−8.00000			−0.370196			−0.185098	−	0.982720i	$-0.559260\pi$
$467$	−8.00000			−0.370196			−0.185098	+	0.982720i	$0.559260\pi$
$468$		0			0
$469$	7.00000			0.323230
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−4.00000	−	6.92820i	−0.183920	−	0.318559i
$474$		0			0
$475$	2.50000	−	4.33013i	0.114708	−	0.198680i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	17.0000	−	29.4449i	0.776750	−	1.34537i	−0.157056	−	0.987590i	$-0.550200\pi$
$479$	17.0000	−	29.4449i	0.776750	−	1.34537i	0.933806	−	0.357780i	$-0.116466\pi$
$480$		0			0
$481$	−7.50000	−	12.9904i	−0.341971	−	0.592310i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−7.00000			−0.317854
$486$		0			0
$487$	−17.0000			−0.770344			−0.385172	−	0.922845i	$-0.625858\pi$
$487$	−17.0000			−0.770344			−0.385172	+	0.922845i	$0.625858\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−3.00000	−	5.19615i	−0.135388	−	0.234499i	0.790358	−	0.612646i	$-0.209895\pi$
$491$	−3.00000	−	5.19615i	−0.135388	−	0.234499i	−0.925746	+	0.378147i	$0.876561\pi$
$492$		0			0
$493$	20.0000	−	34.6410i	0.900755	−	1.56015i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−3.00000	+	5.19615i	−0.134568	+	0.233079i
$498$		0			0
$499$	20.0000	+	34.6410i	0.895323	+	1.55074i	0.833404	+	0.552664i	$0.186389\pi$
$499$	20.0000	+	34.6410i	0.895323	+	1.55074i	0.0619186	+	0.998081i	$0.480278\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	6.00000			0.267527			0.133763	−	0.991013i	$-0.457294\pi$
$503$	6.00000			0.267527			0.133763	+	0.991013i	$0.457294\pi$
$504$		0			0
$505$		0			0
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−1.00000	−	1.73205i	−0.0443242	−	0.0767718i	0.843012	−	0.537895i	$-0.180780\pi$
$509$	−1.00000	−	1.73205i	−0.0443242	−	0.0767718i	−0.887336	+	0.461123i	$0.847447\pi$
$510$		0			0
$511$	−4.50000	+	7.79423i	−0.199068	+	0.344796i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.50000	+	4.33013i	−0.110163	+	0.190808i
$516$		0			0
$517$	−8.00000	−	13.8564i	−0.351840	−	0.609404i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−10.0000			−0.438108			−0.219054	−	0.975713i	$-0.570297\pi$
$521$	−10.0000			−0.438108			−0.219054	+	0.975713i	$0.570297\pi$
$522$		0			0
$523$	−35.0000			−1.53044			−0.765222	−	0.643767i	$-0.777371\pi$
$523$	−35.0000			−1.53044			−0.765222	+	0.643767i	$0.777371\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.0000	+	27.7128i	0.696971	+	1.20719i
$528$		0			0
$529$	9.50000	−	16.4545i	0.413043	−	0.715412i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−15.0000	+	25.9808i	−0.649722	+	1.12535i
$534$		0			0
$535$	9.00000	+	15.5885i	0.389104	+	0.673948i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−12.0000			−0.516877
$540$		0			0
$541$	−11.0000			−0.472927			−0.236463	−	0.971640i	$-0.575988\pi$
$541$	−11.0000			−0.472927			−0.236463	+	0.971640i	$0.575988\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	7.00000	+	12.1244i	0.299847	+	0.519350i
$546$		0			0
$547$	18.5000	−	32.0429i	0.791003	−	1.37006i	−0.134344	−	0.990935i	$-0.542893\pi$
$547$	18.5000	−	32.0429i	0.791003	−	1.37006i	0.925347	−	0.379122i	$-0.123774\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−25.0000	+	43.3013i	−1.06504	+	1.84470i
$552$		0			0
$553$	1.50000	+	2.59808i	0.0637865	+	0.110481i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	16.0000			0.677942			0.338971	−	0.940797i	$-0.389921\pi$
$557$	16.0000			0.677942			0.338971	+	0.940797i	$0.389921\pi$
$558$		0			0
$559$	−20.0000			−0.845910
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−10.0000	−	17.3205i	−0.421450	−	0.729972i	0.574632	−	0.818412i	$-0.305145\pi$
$563$	−10.0000	−	17.3205i	−0.421450	−	0.729972i	−0.996082	+	0.0884397i	$0.971812\pi$
$564$		0			0
$565$	−3.00000	+	5.19615i	−0.126211	+	0.218604i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.00000	+	3.46410i	−0.0838444	+	0.145223i	−0.904898	−	0.425628i	$-0.860053\pi$
$569$	−2.00000	+	3.46410i	−0.0838444	+	0.145223i	0.821054	+	0.570851i	$0.193387\pi$
$570$		0			0
$571$	−15.5000	−	26.8468i	−0.648655	−	1.12350i	−0.983444	−	0.181210i	$-0.941999\pi$
$571$	−15.5000	−	26.8468i	−0.648655	−	1.12350i	0.334790	−	0.942293i	$-0.391335\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	2.00000			0.0834058
$576$		0			0
$577$	−19.0000			−0.790980			−0.395490	−	0.918470i	$-0.629425\pi$
$577$	−19.0000			−0.790980			−0.395490	+	0.918470i	$0.629425\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−1.00000	−	1.73205i	−0.0414870	−	0.0718576i
$582$		0			0
$583$	6.00000	−	10.3923i	0.248495	−	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	3.00000	−	5.19615i	0.123823	−	0.214468i	−0.797449	−	0.603386i	$-0.793818\pi$
$587$	3.00000	−	5.19615i	0.123823	−	0.214468i	0.921272	+	0.388918i	$0.127151\pi$
$588$		0			0
$589$	−20.0000	−	34.6410i	−0.824086	−	1.42736i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	6.00000			0.246390			0.123195	−	0.992382i	$-0.460686\pi$
$593$	6.00000			0.246390			0.123195	+	0.992382i	$0.460686\pi$
$594$		0			0
$595$	4.00000			0.163984
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−3.00000	−	5.19615i	−0.122577	−	0.212309i	0.798206	−	0.602384i	$-0.205782\pi$
$599$	−3.00000	−	5.19615i	−0.122577	−	0.212309i	−0.920783	+	0.390075i	$0.872449\pi$
$600$		0			0
$601$	13.0000	−	22.5167i	0.530281	−	0.918474i	−0.469095	−	0.883148i	$-0.655420\pi$
$601$	13.0000	−	22.5167i	0.530281	−	0.918474i	0.999376	−	0.0353259i	$-0.0112469\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.50000	+	6.06218i	−0.142295	+	0.246463i
$606$		0			0
$607$	−13.5000	−	23.3827i	−0.547948	−	0.949074i	−0.998415	−	0.0562808i	$-0.982076\pi$
$607$	−13.5000	−	23.3827i	−0.547948	−	0.949074i	0.450467	−	0.892793i	$-0.351258\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−40.0000			−1.61823
$612$		0			0
$613$	−15.0000			−0.605844			−0.302922	−	0.953015i	$-0.597962\pi$
$613$	−15.0000			−0.605844			−0.302922	+	0.953015i	$0.597962\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	0.920074	−	0.391745i	$-0.128129\pi$
$617$	3.00000	+	5.19615i	0.120775	+	0.209189i	−0.799298	+	0.600935i	$0.794795\pi$
$618$		0			0
$619$	0.500000	−	0.866025i	0.0200967	−	0.0348085i	−0.855802	−	0.517303i	$-0.826936\pi$
$619$	0.500000	−	0.866025i	0.0200967	−	0.0348085i	0.875899	+	0.482495i	$0.160269\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−12.0000			−0.478471
$630$		0			0
$631$	−5.00000			−0.199047			−0.0995234	−	0.995035i	$-0.531732\pi$
$631$	−5.00000			−0.199047			−0.0995234	+	0.995035i	$0.531732\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		0			0
$636$		0			0
$637$	−15.0000	+	25.9808i	−0.594322	+	1.02940i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0000	+	20.7846i	−0.473972	+	0.820943i	−0.999556	−	0.0297987i	$-0.990513\pi$
$641$	−12.0000	+	20.7846i	−0.473972	+	0.820943i	0.525584	+	0.850741i	$0.323847\pi$
$642$		0			0
$643$	−6.00000	−	10.3923i	−0.236617	−	0.409832i	0.723124	−	0.690718i	$-0.242705\pi$
$643$	−6.00000	−	10.3923i	−0.236617	−	0.409832i	−0.959741	+	0.280885i	$0.909372\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	18.0000			0.707653			0.353827	−	0.935311i	$-0.384880\pi$
$647$	18.0000			0.707653			0.353827	+	0.935311i	$0.384880\pi$
$648$		0			0
$649$	8.00000			0.314027
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−14.0000	−	24.2487i	−0.547862	−	0.948925i	−0.998421	−	0.0561784i	$-0.982108\pi$
$653$	−14.0000	−	24.2487i	−0.547862	−	0.948925i	0.450558	−	0.892747i	$-0.351225\pi$
$654$		0			0
$655$	6.00000	−	10.3923i	0.234439	−	0.406061i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.0000	−	20.7846i	0.467454	−	0.809653i	−0.531855	−	0.846836i	$-0.678505\pi$
$659$	12.0000	−	20.7846i	0.467454	−	0.809653i	0.999309	+	0.0371821i	$0.0118382\pi$
$660$		0			0
$661$	22.5000	+	38.9711i	0.875149	+	1.51580i	0.856604	+	0.515974i	$0.172570\pi$
$661$	22.5000	+	38.9711i	0.875149	+	1.51580i	0.0185442	+	0.999828i	$0.494097\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−5.00000			−0.193892
$666$		0			0
$667$	−20.0000			−0.774403
$668$		0			0
$669$		0			0
$670$		0			0
$671$	5.00000	+	8.66025i	0.193023	+	0.334325i
$672$		0			0
$673$	1.50000	−	2.59808i	0.0578208	−	0.100148i	−0.835666	−	0.549238i	$-0.814918\pi$
$673$	1.50000	−	2.59808i	0.0578208	−	0.100148i	0.893487	+	0.449089i	$0.148251\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	20.0000	−	34.6410i	0.768662	−	1.33136i	−0.169626	−	0.985509i	$-0.554256\pi$
$677$	20.0000	−	34.6410i	0.768662	−	1.33136i	0.938288	−	0.345854i	$-0.112411\pi$
$678$		0			0
$679$	3.50000	+	6.06218i	0.134318	+	0.232645i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	30.0000			1.14792			0.573959	−	0.818884i	$-0.305407\pi$
$683$	30.0000			1.14792			0.573959	+	0.818884i	$0.305407\pi$
$684$		0			0
$685$	−2.00000			−0.0764161
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−15.0000	−	25.9808i	−0.571454	−	0.989788i
$690$		0			0
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	−0.466691	−	0.884420i	$-0.654554\pi$
$691$	14.0000	−	24.2487i	0.532585	−	0.922464i	0.999276	−	0.0380440i	$-0.0121127\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−7.50000	+	12.9904i	−0.284491	+	0.492753i
$696$		0			0
$697$	12.0000	+	20.7846i	0.454532	+	0.787273i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−2.00000			−0.0755390			−0.0377695	−	0.999286i	$-0.512025\pi$
$701$	−2.00000			−0.0755390			−0.0377695	+	0.999286i	$0.512025\pi$
$702$		0			0
$703$	15.0000			0.565736
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	21.5000	−	37.2391i	0.807449	−	1.39854i	−0.107176	−	0.994240i	$-0.534181\pi$
$709$	21.5000	−	37.2391i	0.807449	−	1.39854i	0.914625	−	0.404303i	$-0.132486\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	8.00000	−	13.8564i	0.299602	−	0.518927i
$714$		0			0
$715$	−5.00000	−	8.66025i	−0.186989	−	0.323875i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−10.0000			−0.372937			−0.186469	−	0.982461i	$-0.559704\pi$
$719$	−10.0000			−0.372937			−0.186469	+	0.982461i	$0.559704\pi$
$720$		0			0
$721$	5.00000			0.186210
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.00000	+	8.66025i	0.185695	+	0.321634i
$726$		0			0
$727$	−4.00000	+	6.92820i	−0.148352	+	0.256953i	−0.930618	−	0.365991i	$-0.880730\pi$
$727$	−4.00000	+	6.92820i	−0.148352	+	0.256953i	0.782267	+	0.622944i	$0.214063\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.00000	+	13.8564i	−0.295891	+	0.512498i
$732$		0			0
$733$	−9.00000	−	15.5885i	−0.332423	−	0.575773i	0.650564	−	0.759452i	$-0.274533\pi$
$733$	−9.00000	−	15.5885i	−0.332423	−	0.575773i	−0.982986	+	0.183679i	$0.941199\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−14.0000			−0.515697
$738$		0			0
$739$	−12.0000			−0.441427			−0.220714	−	0.975339i	$-0.570839\pi$
$739$	−12.0000			−0.441427			−0.220714	+	0.975339i	$0.570839\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	6.00000	+	10.3923i	0.220119	+	0.381257i	0.954844	−	0.297108i	$-0.0960222\pi$
$743$	6.00000	+	10.3923i	0.220119	+	0.381257i	−0.734725	+	0.678365i	$0.762689\pi$
$744$		0			0
$745$	−10.0000	+	17.3205i	−0.366372	+	0.634574i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	9.00000	−	15.5885i	0.328853	−	0.569590i
$750$		0			0
$751$	16.5000	+	28.5788i	0.602094	+	1.04286i	0.992504	+	0.122216i	$0.0389999\pi$
$751$	16.5000	+	28.5788i	0.602094	+	1.04286i	−0.390410	+	0.920641i	$0.627667\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−23.0000			−0.837056
$756$		0			0
$757$	35.0000			1.27210			0.636048	−	0.771649i	$-0.280568\pi$
$757$	35.0000			1.27210			0.636048	+	0.771649i	$0.280568\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−3.00000	−	5.19615i	−0.108750	−	0.188360i	0.806514	−	0.591215i	$-0.201351\pi$
$761$	−3.00000	−	5.19615i	−0.108750	−	0.188360i	−0.915264	+	0.402854i	$0.868018\pi$
$762$		0			0
$763$	7.00000	−	12.1244i	0.253417	−	0.438931i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	10.0000	−	17.3205i	0.361079	−	0.625407i
$768$		0			0
$769$	−21.5000	−	37.2391i	−0.775310	−	1.34288i	−0.934620	−	0.355647i	$-0.884260\pi$
$769$	−21.5000	−	37.2391i	−0.775310	−	1.34288i	0.159310	−	0.987229i	$-0.449073\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	6.00000			0.215805			0.107903	−	0.994161i	$-0.465587\pi$
$773$	6.00000			0.215805			0.107903	+	0.994161i	$0.465587\pi$
$774$		0			0
$775$	−8.00000			−0.287368
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−15.0000	−	25.9808i	−0.537431	−	0.930857i
$780$		0			0
$781$	6.00000	−	10.3923i	0.214697	−	0.371866i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−7.00000	+	12.1244i	−0.249841	+	0.432737i
$786$		0			0
$787$	−20.5000	−	35.5070i	−0.730746	−	1.26569i	−0.956565	−	0.291520i	$-0.905839\pi$
$787$	−20.5000	−	35.5070i	−0.730746	−	1.26569i	0.225819	−	0.974169i	$-0.427494\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	6.00000			0.213335
$792$		0			0
$793$	25.0000			0.887776
$794$		0			0
$795$		0			0
$796$		0			0
$797$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$797$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$798$		0			0
$799$	−16.0000	+	27.7128i	−0.566039	+	0.980409i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	9.00000	−	15.5885i	0.317603	−	0.550105i
$804$		0			0
$805$	−1.00000	−	1.73205i	−0.0352454	−	0.0610468i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	24.0000			0.843795			0.421898	−	0.906644i	$-0.361364\pi$
$809$	24.0000			0.843795			0.421898	+	0.906644i	$0.361364\pi$
$810$		0			0
$811$	44.0000			1.54505			0.772524	−	0.634985i	$-0.218994\pi$
$811$	44.0000			1.54505			0.772524	+	0.634985i	$0.218994\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	9.50000	+	16.4545i	0.332770	+	0.576375i
$816$		0			0
$817$	10.0000	−	17.3205i	0.349856	−	0.605968i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−22.0000	+	38.1051i	−0.767805	+	1.32988i	0.170945	+	0.985281i	$0.445318\pi$
$821$	−22.0000	+	38.1051i	−0.767805	+	1.32988i	−0.938751	+	0.344597i	$0.888015\pi$
$822$		0			0
$823$	−0.500000	−	0.866025i	−0.0174289	−	0.0301877i	0.857179	−	0.515018i	$-0.172215\pi$
$823$	−0.500000	−	0.866025i	−0.0174289	−	0.0301877i	−0.874608	+	0.484830i	$0.838881\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		0			0			−	1.00000i	$-0.5\pi$
$827$		0			0				1.00000i	$0.5\pi$
$828$		0			0
$829$	−47.0000			−1.63238			−0.816189	−	0.577785i	$-0.803917\pi$
$829$	−47.0000			−1.63238			−0.816189	+	0.577785i	$0.803917\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	12.0000	+	20.7846i	0.415775	+	0.720144i
$834$		0			0
$835$	12.0000	−	20.7846i	0.415277	−	0.719281i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−26.0000	+	45.0333i	−0.897620	+	1.55472i	−0.0670911	+	0.997747i	$0.521372\pi$
$839$	−26.0000	+	45.0333i	−0.897620	+	1.55472i	−0.830529	+	0.556976i	$0.811962\pi$
$840$		0			0
$841$	−35.5000	−	61.4878i	−1.22414	−	2.12027i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−12.0000			−0.412813
$846$		0			0
$847$	7.00000			0.240523
$848$		0			0
$849$		0			0
$850$		0			0
$851$	3.00000	+	5.19615i	0.102839	+	0.178122i
$852$		0			0
$853$	9.50000	−	16.4545i	0.325274	−	0.563391i	−0.656294	−	0.754505i	$-0.727877\pi$
$853$	9.50000	−	16.4545i	0.325274	−	0.563391i	0.981568	+	0.191115i	$0.0612102\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	27.0000	−	46.7654i	0.922302	−	1.59747i	0.126459	−	0.991972i	$-0.459639\pi$
$857$	27.0000	−	46.7654i	0.922302	−	1.59747i	0.795843	−	0.605503i	$-0.207028\pi$
$858$		0			0
$859$	−2.50000	−	4.33013i	−0.0852989	−	0.147742i	0.820220	−	0.572049i	$-0.193851\pi$
$859$	−2.50000	−	4.33013i	−0.0852989	−	0.147742i	−0.905519	+	0.424307i	$0.860518\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$		0			0			−	1.00000i	$-0.5\pi$
$863$		0			0				1.00000i	$0.5\pi$
$864$		0			0
$865$	20.0000			0.680020
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−3.00000	−	5.19615i	−0.101768	−	0.176267i
$870$		0			0
$871$	−17.5000	+	30.3109i	−0.592965	+	1.02705i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−0.500000	+	0.866025i	−0.0169031	+	0.0292770i
$876$		0			0
$877$	−0.500000	−	0.866025i	−0.0168838	−	0.0292436i	0.857460	−	0.514551i	$-0.172041\pi$
$877$	−0.500000	−	0.866025i	−0.0168838	−	0.0292436i	−0.874344	+	0.485307i	$0.838708\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−30.0000			−1.01073			−0.505363	−	0.862907i	$-0.668641\pi$
$881$	−30.0000			−1.01073			−0.505363	+	0.862907i	$0.668641\pi$
$882$		0			0
$883$	11.0000			0.370179			0.185090	−	0.982722i	$-0.440742\pi$
$883$	11.0000			0.370179			0.185090	+	0.982722i	$0.440742\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	20.0000	+	34.6410i	0.671534	+	1.16313i	0.977469	+	0.211079i	$0.0676977\pi$
$887$	20.0000	+	34.6410i	0.671534	+	1.16313i	−0.305935	+	0.952052i	$0.598969\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$	20.0000	−	34.6410i	0.669274	−	1.15922i
$894$		0			0
$895$	−13.0000	−	22.5167i	−0.434542	−	0.752649i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	80.0000			2.66815
$900$		0			0
$901$	−24.0000			−0.799556
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−3.50000	−	6.06218i	−0.116344	−	0.201514i
$906$		0			0
$907$	1.50000	−	2.59808i	0.0498067	−	0.0862677i	−0.840047	−	0.542513i	$-0.817473\pi$
$907$	1.50000	−	2.59808i	0.0498067	−	0.0862677i	0.889854	+	0.456246i	$0.150806\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−4.00000	+	6.92820i	−0.132526	+	0.229542i	−0.924650	−	0.380819i	$-0.875642\pi$
$911$	−4.00000	+	6.92820i	−0.132526	+	0.229542i	0.792124	+	0.610361i	$0.208975\pi$
$912$		0			0
$913$	2.00000	+	3.46410i	0.0661903	+	0.114645i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−12.0000			−0.396275
$918$		0			0
$919$	8.00000			0.263896			0.131948	−	0.991257i	$-0.457877\pi$
$919$	8.00000			0.263896			0.131948	+	0.991257i	$0.457877\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−15.0000	−	25.9808i	−0.493731	−	0.855167i
$924$		0			0
$925$	1.50000	−	2.59808i	0.0493197	−	0.0854242i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13.0000	+	22.5167i	−0.426516	+	0.738748i	−0.996561	−	0.0828661i	$-0.973593\pi$
$929$	−13.0000	+	22.5167i	−0.426516	+	0.738748i	0.570045	+	0.821614i	$0.306926\pi$
$930$		0			0
$931$	−15.0000	−	25.9808i	−0.491605	−	0.851485i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−8.00000			−0.261628
$936$		0			0
$937$	−35.0000			−1.14340			−0.571700	−	0.820463i	$-0.693716\pi$
$937$	−35.0000			−1.14340			−0.571700	+	0.820463i	$0.693716\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.0000	+	39.8372i	0.749779	+	1.29865i	0.947929	+	0.318483i	$0.103173\pi$
$941$	23.0000	+	39.8372i	0.749779	+	1.29865i	−0.198150	+	0.980172i	$0.563493\pi$
$942$		0			0
$943$	6.00000	−	10.3923i	0.195387	−	0.338420i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	26.0000	−	45.0333i	0.844886	−	1.46339i	−0.0408333	−	0.999166i	$-0.513001\pi$
$947$	26.0000	−	45.0333i	0.844886	−	1.46339i	0.885720	−	0.464220i	$-0.153665\pi$
$948$		0			0
$949$	−22.5000	−	38.9711i	−0.730381	−	1.26506i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−12.0000			−0.388718			−0.194359	−	0.980930i	$-0.562263\pi$
$953$	−12.0000			−0.388718			−0.194359	+	0.980930i	$0.562263\pi$
$954$		0			0
$955$	−16.0000			−0.517748
$956$		0			0
$957$		0			0
$958$		0			0
$959$	1.00000	+	1.73205i	0.0322917	+	0.0559308i
$960$		0			0
$961$	−16.5000	+	28.5788i	−0.532258	+	0.921898i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	0.500000	−	0.866025i	0.0160956	−	0.0278783i
$966$		0			0
$967$	−16.5000	−	28.5788i	−0.530604	−	0.919033i	−0.999362	−	0.0357069i	$-0.988632\pi$
$967$	−16.5000	−	28.5788i	−0.530604	−	0.919033i	0.468758	−	0.883327i	$-0.344702\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	40.0000			1.28366			0.641831	−	0.766846i	$-0.278175\pi$
$971$	40.0000			1.28366			0.641831	+	0.766846i	$0.278175\pi$
$972$		0			0
$973$	15.0000			0.480878
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.00000	+	1.73205i	0.0319928	+	0.0554132i	0.881579	−	0.472037i	$-0.156481\pi$
$977$	1.00000	+	1.73205i	0.0319928	+	0.0554132i	−0.849586	+	0.527451i	$0.823148\pi$
$978$		0			0
$979$		0			0
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−21.0000	+	36.3731i	−0.669796	+	1.16012i	0.308165	+	0.951333i	$0.400285\pi$
$983$	−21.0000	+	36.3731i	−0.669796	+	1.16012i	−0.977961	+	0.208788i	$0.933048\pi$
$984$		0			0
$985$	10.0000	+	17.3205i	0.318626	+	0.551877i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	8.00000			0.254385
$990$		0			0
$991$	−7.00000			−0.222362			−0.111181	−	0.993800i	$-0.535463\pi$
$991$	−7.00000			−0.222362			−0.111181	+	0.993800i	$0.535463\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.50000	−	6.06218i	−0.110957	−	0.192184i
$996$		0			0
$997$	−15.0000	+	25.9808i	−0.475055	+	0.822819i	−0.999592	−	0.0285686i	$-0.990905\pi$
$997$	−15.0000	+	25.9808i	−0.475055	+	0.822819i	0.524537	+	0.851388i	$0.324238\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.q.t.2161.1		2
3.2	odd	2		3240.2.q.h.2161.1		2
9.2	odd	6		1080.2.a.i.1.1	yes	1
9.4	even	3	inner	3240.2.q.t.1081.1		2
9.5	odd	6		3240.2.q.h.1081.1		2
9.7	even	3		1080.2.a.c.1.1	✓	1
36.7	odd	6		2160.2.a.g.1.1		1
36.11	even	6		2160.2.a.t.1.1		1
45.2	even	12		5400.2.f.k.649.1		2
45.7	odd	12		5400.2.f.t.649.1		2
45.29	odd	6		5400.2.a.ba.1.1		1
45.34	even	6		5400.2.a.bc.1.1		1
45.38	even	12		5400.2.f.k.649.2		2
45.43	odd	12		5400.2.f.t.649.2		2
72.11	even	6		8640.2.a.q.1.1		1
72.29	odd	6		8640.2.a.n.1.1		1
72.43	odd	6		8640.2.a.bw.1.1		1
72.61	even	6		8640.2.a.bp.1.1		1

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1080.2.a.c.1.1	✓	1	9.7	even	3
1080.2.a.i.1.1	yes	1	9.2	odd	6
2160.2.a.g.1.1		1	36.7	odd	6
2160.2.a.t.1.1		1	36.11	even	6
3240.2.q.h.1081.1		2	9.5	odd	6
3240.2.q.h.2161.1		2	3.2	odd	2
3240.2.q.t.1081.1		2	9.4	even	3	inner
3240.2.q.t.2161.1		2	1.1	even	1	trivial
5400.2.a.ba.1.1		1	45.29	odd	6
5400.2.a.bc.1.1		1	45.34	even	6
5400.2.f.k.649.1		2	45.2	even	12
5400.2.f.k.649.2		2	45.38	even	12
5400.2.f.t.649.1		2	45.7	odd	12
5400.2.f.t.649.2		2	45.43	odd	12
8640.2.a.n.1.1		1	72.29	odd	6
8640.2.a.q.1.1		1	72.11	even	6
8640.2.a.bp.1.1		1	72.61	even	6
8640.2.a.bw.1.1		1	72.43	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:	\( N \)	\(=\)	\( 3240 = 2^{3} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	3240.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(0.500000 - 0.866025i) q^{7} +(-1.00000 + 1.73205i) q^{11} +(2.50000 + 4.33013i) q^{13} +4.00000 q^{17} -5.00000 q^{19} +(-1.00000 - 1.73205i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(5.00000 - 8.66025i) q^{29} +(4.00000 + 6.92820i) q^{31} +1.00000 q^{35} -3.00000 q^{37} +(3.00000 + 5.19615i) q^{41} +(-2.00000 + 3.46410i) q^{43} +(-4.00000 + 6.92820i) q^{47} +(3.00000 + 5.19615i) q^{49} -6.00000 q^{53} -2.00000 q^{55} +(-2.00000 - 3.46410i) q^{59} +(2.50000 - 4.33013i) q^{61} +(-2.50000 + 4.33013i) q^{65} +(3.50000 + 6.06218i) q^{67} -6.00000 q^{71} -9.00000 q^{73} +(1.00000 + 1.73205i) q^{77} +(-1.50000 + 2.59808i) q^{79} +(1.00000 - 1.73205i) q^{83} +(2.00000 + 3.46410i) q^{85} +5.00000 q^{91} +(-2.50000 - 4.33013i) q^{95} +(-3.50000 + 6.06218i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + q^{5} + q^{7} - 2 q^{11} + 5 q^{13} + 8 q^{17} - 10 q^{19} - 2 q^{23} - q^{25} + 10 q^{29} + 8 q^{31} + 2 q^{35} - 6 q^{37} + 6 q^{41} - 4 q^{43} - 8 q^{47} + 6 q^{49} - 12 q^{53} - 4 q^{55} - 4 q^{59}+ \cdots - 7 q^{97}+O(q^{100}) \) 2 * q + q^5 + q^7 - 2 * q^11 + 5 * q^13 + 8 * q^17 - 10 * q^19 - 2 * q^23 - q^25 + 10 * q^29 + 8 * q^31 + 2 * q^35 - 6 * q^37 + 6 * q^41 - 4 * q^43 - 8 * q^47 + 6 * q^49 - 12 * q^53 - 4 * q^55 - 4 * q^59 + 5 * q^61 - 5 * q^65 + 7 * q^67 - 12 * q^71 - 18 * q^73 + 2 * q^77 - 3 * q^79 + 2 * q^83 + 4 * q^85 + 10 * q^91 - 5 * q^95 - 7 * q^97

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