LMFDB - Embedded newform 3240.2.q.n.1081.1

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)

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")

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);

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,-2,0,0,0,-2,0,-4] 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 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			1081.1
Root			$0.500000 - 0.866025i$ of defining polynomial
Character	$\chi$	$=$	3240.1081
Dual form			3240.2.q.n.2161.1

$q$ -expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 - 0.866025i) q^{5} +(-1.00000 - 1.73205i) q^{7} +(-1.00000 - 1.73205i) q^{11} +(-2.00000 + 3.46410i) q^{13} -2.00000 q^{17} +4.00000 q^{19} +(-4.00000 + 6.92820i) q^{23} +(-0.500000 - 0.866025i) q^{25} +(5.00000 + 8.66025i) q^{29} +(-2.00000 + 3.46410i) q^{31} -2.00000 q^{35} +(4.00000 + 6.92820i) q^{43} +(-4.00000 - 6.92820i) q^{47} +(1.50000 - 2.59808i) q^{49} +6.00000 q^{53} -2.00000 q^{55} +(7.00000 - 12.1244i) q^{59} +(7.00000 + 12.1244i) q^{61} +(2.00000 + 3.46410i) q^{65} +(2.00000 - 3.46410i) q^{67} +12.0000 q^{71} +6.00000 q^{73} +(-2.00000 + 3.46410i) q^{77} +(6.00000 + 10.3923i) q^{79} +(-2.00000 - 3.46410i) q^{83} +(-1.00000 + 1.73205i) q^{85} -12.0000 q^{89} +8.00000 q^{91} +(2.00000 - 3.46410i) q^{95} +(7.00000 + 12.1244i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$2 q + q^{5} - 2 q^{7} - 2 q^{11} - 4 q^{13} - 4 q^{17} + 8 q^{19} - 8 q^{23} - q^{25} + 10 q^{29} - 4 q^{31} - 4 q^{35} + 8 q^{43} - 8 q^{47} + 3 q^{49} + 12 q^{53} - 4 q^{55} + 14 q^{59} + 14 q^{61}+ \cdots + 14 q^{97}+O(q^{100})$ 2 * q + q^5 - 2 * q^7 - 2 * q^11 - 4 * q^13 - 4 * q^17 + 8 * q^19 - 8 * q^23 - q^25 + 10 * q^29 - 4 * q^31 - 4 * q^35 + 8 * q^43 - 8 * q^47 + 3 * q^49 + 12 * q^53 - 4 * q^55 + 14 * q^59 + 14 * q^61 + 4 * q^65 + 4 * q^67 + 24 * q^71 + 12 * q^73 - 4 * q^77 + 12 * q^79 - 4 * q^83 - 2 * q^85 - 24 * q^89 + 16 * q^91 + 4 * q^95 + 14 * q^97

Character values

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

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

Coefficient data

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

Display

a_p

with

p

up to: 50 250 1000 (See

a_n

instead) (See

a_n

instead) (See

a_n

instead) Display

a_n

with

n

up to: 50 250 1000 (See only

a_p

) (See only

a_p

) (See only

a_p

)

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

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$5$	0.500000	−	0.866025i	0.223607	−	0.387298i
$6$		0			0
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	0.612801	−	0.790237i	$-0.290043\pi$
$7$	−1.00000	−	1.73205i	−0.377964	−	0.654654i	−0.990766	+	0.135583i	$0.956709\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	0.674967	−	0.737848i	$-0.264158\pi$
$11$	−1.00000	−	1.73205i	−0.301511	−	0.522233i	−0.976478	+	0.215615i	$0.930824\pi$
$12$		0			0
$13$	−2.00000	+	3.46410i	−0.554700	+	0.960769i	0.443227	+	0.896410i	$0.353834\pi$
$13$	−2.00000	+	3.46410i	−0.554700	+	0.960769i	−0.997927	+	0.0643593i	$0.979500\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.00000			−0.485071			−0.242536	−	0.970143i	$-0.577979\pi$
$17$	−2.00000			−0.485071			−0.242536	+	0.970143i	$0.577979\pi$
$18$		0			0
$19$	4.00000			0.917663			0.458831	−	0.888523i	$-0.348268\pi$
$19$	4.00000			0.917663			0.458831	+	0.888523i	$0.348268\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−4.00000	+	6.92820i	−0.834058	+	1.44463i	0.0607377	+	0.998154i	$0.480655\pi$
$23$	−4.00000	+	6.92820i	−0.834058	+	1.44463i	−0.894795	+	0.446476i	$0.852679\pi$
$24$		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	0.785872	+	0.618389i	$0.212214\pi$
$29$	5.00000	+	8.66025i	0.928477	+	1.60817i	0.142605	+	0.989780i	$0.454452\pi$
$30$		0			0
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	−0.987829	−	0.155543i	$-0.950287\pi$
$31$	−2.00000	+	3.46410i	−0.359211	+	0.622171i	0.628619	+	0.777714i	$0.283621\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.00000			−0.338062
$36$		0			0
$37$		0			0			−	1.00000i	$-0.5\pi$
$37$		0			0				1.00000i	$0.5\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$41$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$42$		0			0
$43$	4.00000	+	6.92820i	0.609994	+	1.05654i	0.991241	+	0.132068i	$0.0421616\pi$
$43$	4.00000	+	6.92820i	0.609994	+	1.05654i	−0.381246	+	0.924473i	$0.624505\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−4.00000	−	6.92820i	−0.583460	−	1.01058i	−0.995066	−	0.0992202i	$-0.968365\pi$
$47$	−4.00000	−	6.92820i	−0.583460	−	1.01058i	0.411606	−	0.911362i	$-0.364968\pi$
$48$		0			0
$49$	1.50000	−	2.59808i	0.214286	−	0.371154i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	6.00000			0.824163			0.412082	−	0.911147i	$-0.364802\pi$
$53$	6.00000			0.824163			0.412082	+	0.911147i	$0.364802\pi$
$54$		0			0
$55$	−2.00000			−0.269680
$56$		0			0
$57$		0			0
$58$		0			0
$59$	7.00000	−	12.1244i	0.911322	−	1.57846i	0.0991242	−	0.995075i	$-0.468396\pi$
$59$	7.00000	−	12.1244i	0.911322	−	1.57846i	0.812198	−	0.583382i	$-0.198271\pi$
$60$		0			0
$61$	7.00000	+	12.1244i	0.896258	+	1.55236i	0.832240	+	0.554416i	$0.187058\pi$
$61$	7.00000	+	12.1244i	0.896258	+	1.55236i	0.0640184	+	0.997949i	$0.479608\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.00000	+	3.46410i	0.248069	+	0.429669i
$66$		0			0
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	−0.717607	−	0.696449i	$-0.754762\pi$
$67$	2.00000	−	3.46410i	0.244339	−	0.423207i	0.961946	+	0.273241i	$0.0880957\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	12.0000			1.42414			0.712069	−	0.702109i	$-0.247758\pi$
$71$	12.0000			1.42414			0.712069	+	0.702109i	$0.247758\pi$
$72$		0			0
$73$	6.00000			0.702247			0.351123	−	0.936329i	$-0.385800\pi$
$73$	6.00000			0.702247			0.351123	+	0.936329i	$0.385800\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−2.00000	+	3.46410i	−0.227921	+	0.394771i
$78$		0			0
$79$	6.00000	+	10.3923i	0.675053	+	1.16923i	0.976453	+	0.215728i	$0.0692125\pi$
$79$	6.00000	+	10.3923i	0.675053	+	1.16923i	−0.301401	+	0.953498i	$0.597454\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−2.00000	−	3.46410i	−0.219529	−	0.380235i	0.735135	−	0.677920i	$-0.237119\pi$
$83$	−2.00000	−	3.46410i	−0.219529	−	0.380235i	−0.954664	+	0.297686i	$0.903785\pi$
$84$		0			0
$85$	−1.00000	+	1.73205i	−0.108465	+	0.187867i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−12.0000			−1.27200			−0.635999	−	0.771690i	$-0.719412\pi$
$89$	−12.0000			−1.27200			−0.635999	+	0.771690i	$0.719412\pi$
$90$		0			0
$91$	8.00000			0.838628
$92$		0			0
$93$		0			0
$94$		0			0
$95$	2.00000	−	3.46410i	0.205196	−	0.355409i
$96$		0			0
$97$	7.00000	+	12.1244i	0.710742	+	1.23104i	0.964579	+	0.263795i	$0.0849741\pi$
$97$	7.00000	+	12.1244i	0.710742	+	1.23104i	−0.253837	+	0.967247i	$0.581693\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	0.677284	−	0.735721i	$-0.263157\pi$
$101$	−3.00000	−	5.19615i	−0.298511	−	0.517036i	−0.975796	+	0.218685i	$0.929823\pi$
$102$		0			0
$103$	7.00000	−	12.1244i	0.689730	−	1.19465i	−0.282194	−	0.959357i	$-0.591062\pi$
$103$	7.00000	−	12.1244i	0.689730	−	1.19465i	0.971925	−	0.235291i	$-0.0756043\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−12.0000			−1.16008			−0.580042	−	0.814587i	$-0.696964\pi$
$107$	−12.0000			−1.16008			−0.580042	+	0.814587i	$0.696964\pi$
$108$		0			0
$109$	2.00000			0.191565			0.0957826	−	0.995402i	$-0.469465\pi$
$109$	2.00000			0.191565			0.0957826	+	0.995402i	$0.469465\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−9.00000	+	15.5885i	−0.846649	+	1.46644i	0.0375328	+	0.999295i	$0.488050\pi$
$113$	−9.00000	+	15.5885i	−0.846649	+	1.46644i	−0.884182	+	0.467143i	$0.845283\pi$
$114$		0			0
$115$	4.00000	+	6.92820i	0.373002	+	0.646058i
$116$		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
$127$	18.0000			1.59724			0.798621	−	0.601834i	$-0.205563\pi$
$127$	18.0000			1.59724			0.798621	+	0.601834i	$0.205563\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.00000	−	15.5885i	0.786334	−	1.36197i	−0.141865	−	0.989886i	$-0.545310\pi$
$131$	9.00000	−	15.5885i	0.786334	−	1.36197i	0.928199	−	0.372084i	$-0.121357\pi$
$132$		0			0
$133$	−4.00000	−	6.92820i	−0.346844	−	0.600751i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.00000	−	1.73205i	−0.0854358	−	0.147979i	0.820141	−	0.572161i	$-0.193895\pi$
$137$	−1.00000	−	1.73205i	−0.0854358	−	0.147979i	−0.905577	+	0.424182i	$0.860562\pi$
$138$		0			0
$139$	−6.00000	+	10.3923i	−0.508913	+	0.881464i	0.491033	+	0.871141i	$0.336619\pi$
$139$	−6.00000	+	10.3923i	−0.508913	+	0.881464i	−0.999947	+	0.0103230i	$0.996714\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	8.00000			0.668994
$144$		0			0
$145$	10.0000			0.830455
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−5.00000	+	8.66025i	−0.409616	+	0.709476i	−0.994847	−	0.101391i	$-0.967671\pi$
$149$	−5.00000	+	8.66025i	−0.409616	+	0.709476i	0.585231	+	0.810867i	$0.301004\pi$
$150$		0			0
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	0.656101	−	0.754673i	$-0.272204\pi$
$151$	−4.00000	−	6.92820i	−0.325515	−	0.563809i	−0.981617	+	0.190864i	$0.938871\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	2.00000	+	3.46410i	0.160644	+	0.278243i
$156$		0			0
$157$	−2.00000	+	3.46410i	−0.159617	+	0.276465i	−0.934731	−	0.355357i	$-0.884359\pi$
$157$	−2.00000	+	3.46410i	−0.159617	+	0.276465i	0.775113	+	0.631822i	$0.217693\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	16.0000			1.26098
$162$		0			0
$163$	4.00000			0.313304			0.156652	−	0.987654i	$-0.449930\pi$
$163$	4.00000			0.313304			0.156652	+	0.987654i	$0.449930\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		0			0		−0.866025	−	0.500000i	$-0.833333\pi$
$167$		0			0		0.866025	+	0.500000i	$0.166667\pi$
$168$		0			0
$169$	−1.50000	−	2.59808i	−0.115385	−	0.199852i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.00000	+	1.73205i	0.0760286	+	0.131685i	0.901533	−	0.432710i	$-0.142443\pi$
$173$	1.00000	+	1.73205i	0.0760286	+	0.131685i	−0.825505	+	0.564396i	$0.809109\pi$
$174$		0			0
$175$	−1.00000	+	1.73205i	−0.0755929	+	0.130931i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−14.0000			−1.04641			−0.523205	−	0.852207i	$-0.675264\pi$
$179$	−14.0000			−1.04641			−0.523205	+	0.852207i	$0.675264\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$		0			0
$186$		0			0
$187$	2.00000	+	3.46410i	0.146254	+	0.253320i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−2.00000	−	3.46410i	−0.144715	−	0.250654i	0.784552	−	0.620063i	$-0.212893\pi$
$191$	−2.00000	−	3.46410i	−0.144715	−	0.250654i	−0.929267	+	0.369410i	$0.879560\pi$
$192$		0			0
$193$	−11.0000	+	19.0526i	−0.791797	+	1.37143i	0.133056	+	0.991109i	$0.457521\pi$
$193$	−11.0000	+	19.0526i	−0.791797	+	1.37143i	−0.924853	+	0.380325i	$0.875812\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	2.00000			0.142494			0.0712470	−	0.997459i	$-0.477302\pi$
$197$	2.00000			0.142494			0.0712470	+	0.997459i	$0.477302\pi$
$198$		0			0
$199$	8.00000			0.567105			0.283552	−	0.958957i	$-0.408487\pi$
$199$	8.00000			0.567105			0.283552	+	0.958957i	$0.408487\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	10.0000	−	17.3205i	0.701862	−	1.21566i
$204$		0			0
$205$		0			0
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.00000	−	6.92820i	−0.276686	−	0.479234i
$210$		0			0
$211$	−10.0000	+	17.3205i	−0.688428	+	1.19239i	0.283918	+	0.958849i	$0.408366\pi$
$211$	−10.0000	+	17.3205i	−0.688428	+	1.19239i	−0.972346	+	0.233544i	$0.924968\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	8.00000			0.545595
$216$		0			0
$217$	8.00000			0.543075
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.00000	−	6.92820i	0.269069	−	0.466041i
$222$		0			0
$223$	11.0000	+	19.0526i	0.736614	+	1.27585i	0.954011	+	0.299770i	$0.0969101\pi$
$223$	11.0000	+	19.0526i	0.736614	+	1.27585i	−0.217397	+	0.976083i	$0.569757\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$227$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$228$		0			0
$229$	1.00000	−	1.73205i	0.0660819	−	0.114457i	−0.831092	−	0.556136i	$-0.812283\pi$
$229$	1.00000	−	1.73205i	0.0660819	−	0.114457i	0.897173	+	0.441679i	$0.145617\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−18.0000			−1.17922			−0.589610	−	0.807688i	$-0.700718\pi$
$233$	−18.0000			−1.17922			−0.589610	+	0.807688i	$0.700718\pi$
$234$		0			0
$235$	−8.00000			−0.521862
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−6.00000	+	10.3923i	−0.388108	+	0.672222i	−0.992195	−	0.124696i	$-0.960204\pi$
$239$	−6.00000	+	10.3923i	−0.388108	+	0.672222i	0.604087	+	0.796918i	$0.293538\pi$
$240$		0			0
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	0.980917	−	0.194429i	$-0.0622852\pi$
$241$	5.00000	+	8.66025i	0.322078	+	0.557856i	−0.658838	+	0.752285i	$0.728952\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−1.50000	−	2.59808i	−0.0958315	−	0.165985i
$246$		0			0
$247$	−8.00000	+	13.8564i	−0.509028	+	0.881662i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	6.00000			0.378717			0.189358	−	0.981908i	$-0.439359\pi$
$251$	6.00000			0.378717			0.189358	+	0.981908i	$0.439359\pi$
$252$		0			0
$253$	16.0000			1.00591
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−9.00000	+	15.5885i	−0.561405	+	0.972381i	0.435970	+	0.899961i	$0.356405\pi$
$257$	−9.00000	+	15.5885i	−0.561405	+	0.972381i	−0.997374	+	0.0724199i	$0.976928\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		0			0		0.866025	−	0.500000i	$-0.166667\pi$
$263$		0			0		−0.866025	+	0.500000i	$0.833333\pi$
$264$		0			0
$265$	3.00000	−	5.19615i	0.184289	−	0.319197i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−2.00000			−0.121942			−0.0609711	−	0.998140i	$-0.519420\pi$
$269$	−2.00000			−0.121942			−0.0609711	+	0.998140i	$0.519420\pi$
$270$		0			0
$271$		0			0			−	1.00000i	$-0.5\pi$
$271$		0			0				1.00000i	$0.5\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−1.00000	+	1.73205i	−0.0603023	+	0.104447i
$276$		0			0
$277$	8.00000	+	13.8564i	0.480673	+	0.832551i	0.999754	−	0.0221745i	$-0.00705893\pi$
$277$	8.00000	+	13.8564i	0.480673	+	0.832551i	−0.519081	+	0.854725i	$0.673726\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	14.0000	+	24.2487i	0.835170	+	1.44656i	0.893892	+	0.448282i	$0.147964\pi$
$281$	14.0000	+	24.2487i	0.835170	+	1.44656i	−0.0587220	+	0.998274i	$0.518703\pi$
$282$		0			0
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	−0.800439	−	0.599414i	$-0.795400\pi$
$283$	2.00000	−	3.46410i	0.118888	−	0.205919i	0.919327	+	0.393494i	$0.128734\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−13.0000			−0.764706
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.00000	+	1.73205i	−0.0584206	+	0.101187i	−0.893757	−	0.448552i	$-0.851940\pi$
$293$	−1.00000	+	1.73205i	−0.0584206	+	0.101187i	0.835336	+	0.549740i	$0.185273\pi$
$294$		0			0
$295$	−7.00000	−	12.1244i	−0.407556	−	0.705907i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−16.0000	−	27.7128i	−0.925304	−	1.60267i
$300$		0			0
$301$	8.00000	−	13.8564i	0.461112	−	0.798670i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	14.0000			0.801638
$306$		0			0
$307$	−32.0000			−1.82634			−0.913168	−	0.407583i	$-0.866372\pi$
$307$	−32.0000			−1.82634			−0.913168	+	0.407583i	$0.866372\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	10.0000	−	17.3205i	0.567048	−	0.982156i	−0.429808	−	0.902920i	$-0.641419\pi$
$311$	10.0000	−	17.3205i	0.567048	−	0.982156i	0.996856	−	0.0792356i	$-0.0252479\pi$
$312$		0			0
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	0.972028	−	0.234863i	$-0.0754642\pi$
$313$	5.00000	+	8.66025i	0.282617	+	0.489506i	−0.689412	+	0.724370i	$0.742131\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	15.0000	+	25.9808i	0.842484	+	1.45922i	0.887788	+	0.460252i	$0.152241\pi$
$317$	15.0000	+	25.9808i	0.842484	+	1.45922i	−0.0453045	+	0.998973i	$0.514426\pi$
$318$		0			0
$319$	10.0000	−	17.3205i	0.559893	−	0.969762i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−8.00000			−0.445132
$324$		0			0
$325$	4.00000			0.221880
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−8.00000	+	13.8564i	−0.441054	+	0.763928i
$330$		0			0
$331$	6.00000	+	10.3923i	0.329790	+	0.571213i	0.982470	−	0.186421i	$-0.0596888\pi$
$331$	6.00000	+	10.3923i	0.329790	+	0.571213i	−0.652680	+	0.757634i	$0.726355\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−2.00000	−	3.46410i	−0.109272	−	0.189264i
$336$		0			0
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	−0.837503	−	0.546433i	$-0.815985\pi$
$337$	1.00000	−	1.73205i	0.0544735	−	0.0943508i	0.891976	+	0.452082i	$0.149319\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	8.00000			0.433224
$342$		0			0
$343$	−20.0000			−1.07990
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	−0.996826	−	0.0796169i	$-0.974630\pi$
$347$	−8.00000	+	13.8564i	−0.429463	+	0.743851i	0.567363	+	0.823468i	$0.307964\pi$
$348$		0			0
$349$	3.00000	+	5.19615i	0.160586	+	0.278144i	0.935079	−	0.354439i	$-0.115328\pi$
$349$	3.00000	+	5.19615i	0.160586	+	0.278144i	−0.774493	+	0.632583i	$0.781995\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	13.0000	+	22.5167i	0.691920	+	1.19844i	0.971208	+	0.238233i	$0.0765683\pi$
$353$	13.0000	+	22.5167i	0.691920	+	1.19844i	−0.279288	+	0.960207i	$0.590098\pi$
$354$		0			0
$355$	6.00000	−	10.3923i	0.318447	−	0.551566i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	24.0000			1.26667			0.633336	−	0.773877i	$-0.281685\pi$
$359$	24.0000			1.26667			0.633336	+	0.773877i	$0.281685\pi$
$360$		0			0
$361$	−3.00000			−0.157895
$362$		0			0
$363$		0			0
$364$		0			0
$365$	3.00000	−	5.19615i	0.157027	−	0.271979i
$366$		0			0
$367$	−13.0000	−	22.5167i	−0.678594	−	1.17536i	−0.975404	−	0.220423i	$-0.929256\pi$
$367$	−13.0000	−	22.5167i	−0.678594	−	1.17536i	0.296810	−	0.954937i	$-0.404077\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−6.00000	−	10.3923i	−0.311504	−	0.539542i
$372$		0			0
$373$	10.0000	−	17.3205i	0.517780	−	0.896822i	−0.482006	−	0.876168i	$-0.660092\pi$
$373$	10.0000	−	17.3205i	0.517780	−	0.896822i	0.999787	−	0.0206542i	$-0.00657489\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−40.0000			−2.06010
$378$		0			0
$379$	28.0000			1.43826			0.719132	−	0.694874i	$-0.244540\pi$
$379$	28.0000			1.43826			0.719132	+	0.694874i	$0.244540\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	−0.377531	−	0.925997i	$-0.623227\pi$
$383$	12.0000	−	20.7846i	0.613171	−	1.06204i	0.990702	−	0.136047i	$-0.0434398\pi$
$384$		0			0
$385$	2.00000	+	3.46410i	0.101929	+	0.176547i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−9.00000	−	15.5885i	−0.456318	−	0.790366i	0.542445	−	0.840091i	$-0.317499\pi$
$389$	−9.00000	−	15.5885i	−0.456318	−	0.790366i	−0.998763	+	0.0497253i	$0.984165\pi$
$390$		0			0
$391$	8.00000	−	13.8564i	0.404577	−	0.700749i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	12.0000			0.603786
$396$		0			0
$397$	−8.00000			−0.401508			−0.200754	−	0.979642i	$-0.564339\pi$
$397$	−8.00000			−0.401508			−0.200754	+	0.979642i	$0.564339\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	6.00000	−	10.3923i	0.299626	−	0.518967i	−0.676425	−	0.736512i	$-0.736472\pi$
$401$	6.00000	−	10.3923i	0.299626	−	0.518967i	0.976050	+	0.217545i	$0.0698049\pi$
$402$		0			0
$403$	−8.00000	−	13.8564i	−0.398508	−	0.690237i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		0			0
$408$		0			0
$409$	−5.00000	+	8.66025i	−0.247234	+	0.428222i	−0.962757	−	0.270367i	$-0.912855\pi$
$409$	−5.00000	+	8.66025i	−0.247234	+	0.428222i	0.715523	+	0.698589i	$0.246188\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−28.0000			−1.37779
$414$		0			0
$415$	−4.00000			−0.196352
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.0000	+	25.9808i	−0.732798	+	1.26924i	0.222885	+	0.974845i	$0.428453\pi$
$419$	−15.0000	+	25.9808i	−0.732798	+	1.26924i	−0.955683	+	0.294398i	$0.904881\pi$
$420$		0			0
$421$	−15.0000	−	25.9808i	−0.731055	−	1.26622i	−0.956433	−	0.291953i	$-0.905695\pi$
$421$	−15.0000	−	25.9808i	−0.731055	−	1.26622i	0.225377	−	0.974272i	$-0.427639\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	1.00000	+	1.73205i	0.0485071	+	0.0840168i
$426$		0			0
$427$	14.0000	−	24.2487i	0.677507	−	1.17348i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	16.0000			0.770693			0.385346	−	0.922772i	$-0.374082\pi$
$431$	16.0000			0.770693			0.385346	+	0.922772i	$0.374082\pi$
$432$		0			0
$433$	−10.0000			−0.480569			−0.240285	−	0.970702i	$-0.577241\pi$
$433$	−10.0000			−0.480569			−0.240285	+	0.970702i	$0.577241\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−16.0000	+	27.7128i	−0.765384	+	1.32568i
$438$		0			0
$439$	16.0000	+	27.7128i	0.763638	+	1.32266i	0.940963	+	0.338508i	$0.109922\pi$
$439$	16.0000	+	27.7128i	0.763638	+	1.32266i	−0.177325	+	0.984152i	$0.556744\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−12.0000	−	20.7846i	−0.570137	−	0.987507i	−0.996551	−	0.0829786i	$-0.973557\pi$
$443$	−12.0000	−	20.7846i	−0.570137	−	0.987507i	0.426414	−	0.904528i	$-0.359777\pi$
$444$		0			0
$445$	−6.00000	+	10.3923i	−0.284427	+	0.492642i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	12.0000			0.566315			0.283158	−	0.959073i	$-0.408618\pi$
$449$	12.0000			0.566315			0.283158	+	0.959073i	$0.408618\pi$
$450$		0			0
$451$		0			0
$452$		0			0
$453$		0			0
$454$		0			0
$455$	4.00000	−	6.92820i	0.187523	−	0.324799i
$456$		0			0
$457$	−5.00000	−	8.66025i	−0.233890	−	0.405110i	0.725059	−	0.688686i	$-0.241812\pi$
$457$	−5.00000	−	8.66025i	−0.233890	−	0.405110i	−0.958950	+	0.283577i	$0.908479\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	15.0000	+	25.9808i	0.698620	+	1.21004i	0.968945	+	0.247276i	$0.0795353\pi$
$461$	15.0000	+	25.9808i	0.698620	+	1.21004i	−0.270326	+	0.962769i	$0.587131\pi$
$462$		0			0
$463$	−15.0000	+	25.9808i	−0.697109	+	1.20743i	0.272355	+	0.962197i	$0.412197\pi$
$463$	−15.0000	+	25.9808i	−0.697109	+	1.20743i	−0.969465	+	0.245232i	$0.921136\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−20.0000			−0.925490			−0.462745	−	0.886492i	$-0.653135\pi$
$467$	−20.0000			−0.925490			−0.462745	+	0.886492i	$0.653135\pi$
$468$		0			0
$469$	−8.00000			−0.369406
$470$		0			0
$471$		0			0
$472$		0			0
$473$	8.00000	−	13.8564i	0.367840	−	0.637118i
$474$		0			0
$475$	−2.00000	−	3.46410i	−0.0917663	−	0.158944i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	2.00000	+	3.46410i	0.0913823	+	0.158279i	0.908093	−	0.418769i	$-0.137538\pi$
$479$	2.00000	+	3.46410i	0.0913823	+	0.158279i	−0.816711	+	0.577047i	$0.804205\pi$
$480$		0			0
$481$		0			0
$482$		0			0
$483$		0			0
$484$		0			0
$485$	14.0000			0.635707
$486$		0			0
$487$	22.0000			0.996915			0.498458	−	0.866914i	$-0.333900\pi$
$487$	22.0000			0.996915			0.498458	+	0.866914i	$0.333900\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.00000	−	15.5885i	0.406164	−	0.703497i	−0.588292	−	0.808649i	$-0.700199\pi$
$491$	9.00000	−	15.5885i	0.406164	−	0.703497i	0.994456	+	0.105151i	$0.0335327\pi$
$492$		0			0
$493$	−10.0000	−	17.3205i	−0.450377	−	0.780076i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−12.0000	−	20.7846i	−0.538274	−	0.932317i
$498$		0			0
$499$	−10.0000	+	17.3205i	−0.447661	+	0.775372i	−0.998233	−	0.0594153i	$-0.981076\pi$
$499$	−10.0000	+	17.3205i	−0.447661	+	0.775372i	0.550572	+	0.834788i	$0.314410\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−24.0000			−1.07011			−0.535054	−	0.844818i	$-0.679709\pi$
$503$	−24.0000			−1.07011			−0.535054	+	0.844818i	$0.679709\pi$
$504$		0			0
$505$	−6.00000			−0.266996
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.00000	+	12.1244i	−0.310270	+	0.537403i	−0.978421	−	0.206623i	$-0.933753\pi$
$509$	−7.00000	+	12.1244i	−0.310270	+	0.537403i	0.668151	+	0.744026i	$0.267086\pi$
$510$		0			0
$511$	−6.00000	−	10.3923i	−0.265424	−	0.459728i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.00000	−	12.1244i	−0.308457	−	0.534263i
$516$		0			0
$517$	−8.00000	+	13.8564i	−0.351840	+	0.609404i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	20.0000			0.876216			0.438108	−	0.898922i	$-0.355649\pi$
$521$	20.0000			0.876216			0.438108	+	0.898922i	$0.355649\pi$
$522$		0			0
$523$	28.0000			1.22435			0.612177	−	0.790721i	$-0.290294\pi$
$523$	28.0000			1.22435			0.612177	+	0.790721i	$0.290294\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	4.00000	−	6.92820i	0.174243	−	0.301797i
$528$		0			0
$529$	−20.5000	−	35.5070i	−0.891304	−	1.54378i
$530$		0			0
$531$		0			0
$532$		0			0
$533$		0			0
$534$		0			0
$535$	−6.00000	+	10.3923i	−0.259403	+	0.449299i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−6.00000			−0.258438
$540$		0			0
$541$	10.0000			0.429934			0.214967	−	0.976621i	$-0.431036\pi$
$541$	10.0000			0.429934			0.214967	+	0.976621i	$0.431036\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	1.00000	−	1.73205i	0.0428353	−	0.0741929i
$546$		0			0
$547$	−4.00000	−	6.92820i	−0.171028	−	0.296229i	0.767752	−	0.640747i	$-0.221375\pi$
$547$	−4.00000	−	6.92820i	−0.171028	−	0.296229i	−0.938779	+	0.344519i	$0.888042\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	20.0000	+	34.6410i	0.852029	+	1.47576i
$552$		0			0
$553$	12.0000	−	20.7846i	0.510292	−	0.883852i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−26.0000			−1.10166			−0.550828	−	0.834619i	$-0.685688\pi$
$557$	−26.0000			−1.10166			−0.550828	+	0.834619i	$0.685688\pi$
$558$		0			0
$559$	−32.0000			−1.35346
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.00000	+	6.92820i	−0.168580	+	0.291989i	−0.937921	−	0.346850i	$-0.887251\pi$
$563$	−4.00000	+	6.92820i	−0.168580	+	0.291989i	0.769341	+	0.638838i	$0.220585\pi$
$564$		0			0
$565$	9.00000	+	15.5885i	0.378633	+	0.655811i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	10.0000	+	17.3205i	0.419222	+	0.726113i	0.995861	−	0.0908852i	$-0.0289696\pi$
$569$	10.0000	+	17.3205i	0.419222	+	0.726113i	−0.576640	+	0.816999i	$0.695636\pi$
$570$		0			0
$571$	−2.00000	+	3.46410i	−0.0836974	+	0.144968i	−0.904835	−	0.425762i	$-0.860006\pi$
$571$	−2.00000	+	3.46410i	−0.0836974	+	0.144968i	0.821138	+	0.570730i	$0.193340\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	8.00000			0.333623
$576$		0			0
$577$	26.0000			1.08239			0.541197	−	0.840896i	$-0.317971\pi$
$577$	26.0000			1.08239			0.541197	+	0.840896i	$0.317971\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−4.00000	+	6.92820i	−0.165948	+	0.287430i
$582$		0			0
$583$	−6.00000	−	10.3923i	−0.248495	−	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−18.0000	−	31.1769i	−0.742940	−	1.28681i	−0.951151	−	0.308725i	$-0.900098\pi$
$587$	−18.0000	−	31.1769i	−0.742940	−	1.28681i	0.208212	−	0.978084i	$-0.433236\pi$
$588$		0			0
$589$	−8.00000	+	13.8564i	−0.329634	+	0.570943i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	42.0000			1.72473			0.862367	−	0.506284i	$-0.168981\pi$
$593$	42.0000			1.72473			0.862367	+	0.506284i	$0.168981\pi$
$594$		0			0
$595$	4.00000			0.163984
$596$		0			0
$597$		0			0
$598$		0			0
$599$	12.0000	−	20.7846i	0.490307	−	0.849236i	−0.509631	−	0.860393i	$-0.670218\pi$
$599$	12.0000	−	20.7846i	0.490307	−	0.849236i	0.999938	+	0.0111569i	$0.00355143\pi$
$600$		0			0
$601$	−11.0000	−	19.0526i	−0.448699	−	0.777170i	0.549602	−	0.835426i	$-0.314779\pi$
$601$	−11.0000	−	19.0526i	−0.448699	−	0.777170i	−0.998302	+	0.0582563i	$0.981446\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−3.50000	−	6.06218i	−0.142295	−	0.246463i
$606$		0			0
$607$	−3.00000	+	5.19615i	−0.121766	+	0.210905i	−0.920464	−	0.390827i	$-0.872189\pi$
$607$	−3.00000	+	5.19615i	−0.121766	+	0.210905i	0.798698	+	0.601732i	$0.205522\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	32.0000			1.29458
$612$		0			0
$613$	−48.0000			−1.93870			−0.969351	−	0.245680i	$-0.920989\pi$
$613$	−48.0000			−1.93870			−0.969351	+	0.245680i	$0.920989\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−3.00000	+	5.19615i	−0.120775	+	0.209189i	−0.920074	−	0.391745i	$-0.871871\pi$
$617$	−3.00000	+	5.19615i	−0.120775	+	0.209189i	0.799298	+	0.600935i	$0.205205\pi$
$618$		0			0
$619$	−22.0000	−	38.1051i	−0.884255	−	1.53157i	−0.846566	−	0.532284i	$-0.821334\pi$
$619$	−22.0000	−	38.1051i	−0.884255	−	1.53157i	−0.0376891	−	0.999290i	$-0.512000\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	12.0000	+	20.7846i	0.480770	+	0.832718i
$624$		0			0
$625$	−0.500000	+	0.866025i	−0.0200000	+	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		0			0
$630$		0			0
$631$	−20.0000			−0.796187			−0.398094	−	0.917345i	$-0.630328\pi$
$631$	−20.0000			−0.796187			−0.398094	+	0.917345i	$0.630328\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	9.00000	−	15.5885i	0.357154	−	0.618609i
$636$		0			0
$637$	6.00000	+	10.3923i	0.237729	+	0.411758i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−12.0000	−	20.7846i	−0.473972	−	0.820943i	0.525584	−	0.850741i	$-0.323847\pi$
$641$	−12.0000	−	20.7846i	−0.473972	−	0.820943i	−0.999556	+	0.0297987i	$0.990513\pi$
$642$		0			0
$643$	−6.00000	+	10.3923i	−0.236617	+	0.409832i	−0.959741	−	0.280885i	$-0.909372\pi$
$643$	−6.00000	+	10.3923i	−0.236617	+	0.409832i	0.723124	+	0.690718i	$0.242705\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−24.0000			−0.943537			−0.471769	−	0.881722i	$-0.656384\pi$
$647$	−24.0000			−0.943537			−0.471769	+	0.881722i	$0.656384\pi$
$648$		0			0
$649$	−28.0000			−1.09910
$650$		0			0
$651$		0			0
$652$		0			0
$653$	7.00000	−	12.1244i	0.273931	−	0.474463i	−0.695934	−	0.718106i	$-0.745009\pi$
$653$	7.00000	−	12.1244i	0.273931	−	0.474463i	0.969865	+	0.243643i	$0.0783426\pi$
$654$		0			0
$655$	−9.00000	−	15.5885i	−0.351659	−	0.609091i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−3.00000	−	5.19615i	−0.116863	−	0.202413i	0.801660	−	0.597781i	$-0.203951\pi$
$659$	−3.00000	−	5.19615i	−0.116863	−	0.202413i	−0.918523	+	0.395367i	$0.870617\pi$
$660$		0			0
$661$	−9.00000	+	15.5885i	−0.350059	+	0.606321i	−0.986260	−	0.165203i	$-0.947172\pi$
$661$	−9.00000	+	15.5885i	−0.350059	+	0.606321i	0.636200	+	0.771524i	$0.280505\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−8.00000			−0.310227
$666$		0			0
$667$	−80.0000			−3.09761
$668$		0			0
$669$		0			0
$670$		0			0
$671$	14.0000	−	24.2487i	0.540464	−	0.936111i
$672$		0			0
$673$	−15.0000	−	25.9808i	−0.578208	−	1.00148i	−0.995685	−	0.0927975i	$-0.970419\pi$
$673$	−15.0000	−	25.9808i	−0.578208	−	1.00148i	0.417477	−	0.908687i	$-0.362914\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.0000	−	22.5167i	−0.499631	−	0.865386i	0.500369	−	0.865812i	$-0.333198\pi$
$677$	−13.0000	−	22.5167i	−0.499631	−	0.865386i	−1.00000		0.000426509i	$0.999864\pi$
$678$		0			0
$679$	14.0000	−	24.2487i	0.537271	−	0.930580i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.0000			−0.459167			−0.229584	−	0.973289i	$-0.573736\pi$
$683$	−12.0000			−0.459167			−0.229584	+	0.973289i	$0.573736\pi$
$684$		0			0
$685$	−2.00000			−0.0764161
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−12.0000	+	20.7846i	−0.457164	+	0.791831i
$690$		0			0
$691$	14.0000	+	24.2487i	0.532585	+	0.922464i	0.999276	+	0.0380440i	$0.0121127\pi$
$691$	14.0000	+	24.2487i	0.532585	+	0.922464i	−0.466691	+	0.884420i	$0.654554\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	6.00000	+	10.3923i	0.227593	+	0.394203i
$696$		0			0
$697$		0			0
$698$		0			0
$699$		0			0
$700$		0			0
$701$	22.0000			0.830929			0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000			0.830929			0.415464	+	0.909610i	$0.363619\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−6.00000	+	10.3923i	−0.225653	+	0.390843i
$708$		0			0
$709$	−7.00000	−	12.1244i	−0.262891	−	0.455340i	0.704118	−	0.710083i	$-0.251342\pi$
$709$	−7.00000	−	12.1244i	−0.262891	−	0.455340i	−0.967009	+	0.254743i	$0.918009\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−16.0000	−	27.7128i	−0.599205	−	1.03785i
$714$		0			0
$715$	4.00000	−	6.92820i	0.149592	−	0.259100i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−40.0000			−1.49175			−0.745874	−	0.666087i	$-0.767968\pi$
$719$	−40.0000			−1.49175			−0.745874	+	0.666087i	$0.767968\pi$
$720$		0			0
$721$	−28.0000			−1.04277
$722$		0			0
$723$		0			0
$724$		0			0
$725$	5.00000	−	8.66025i	0.185695	−	0.321634i
$726$		0			0
$727$	−19.0000	−	32.9090i	−0.704671	−	1.22053i	−0.966810	−	0.255496i	$-0.917761\pi$
$727$	−19.0000	−	32.9090i	−0.704671	−	1.22053i	0.262139	−	0.965030i	$-0.415572\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−8.00000	−	13.8564i	−0.295891	−	0.512498i
$732$		0			0
$733$	12.0000	−	20.7846i	0.443230	−	0.767697i	−0.554697	−	0.832052i	$-0.687166\pi$
$733$	12.0000	−	20.7846i	0.443230	−	0.767697i	0.997927	+	0.0643554i	$0.0204991\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−8.00000			−0.294684
$738$		0			0
$739$	12.0000			0.441427			0.220714	−	0.975339i	$-0.429161\pi$
$739$	12.0000			0.441427			0.220714	+	0.975339i	$0.429161\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−12.0000	+	20.7846i	−0.440237	+	0.762513i	−0.997707	−	0.0676840i	$-0.978439\pi$
$743$	−12.0000	+	20.7846i	−0.440237	+	0.762513i	0.557470	+	0.830197i	$0.311772\pi$
$744$		0			0
$745$	5.00000	+	8.66025i	0.183186	+	0.317287i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	12.0000	+	20.7846i	0.438470	+	0.759453i
$750$		0			0
$751$	18.0000	−	31.1769i	0.656829	−	1.13766i	−0.324603	−	0.945851i	$-0.605231\pi$
$751$	18.0000	−	31.1769i	0.656829	−	1.13766i	0.981432	−	0.191811i	$-0.0614361\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−8.00000			−0.291150
$756$		0			0
$757$	32.0000			1.16306			0.581530	−	0.813525i	$-0.302454\pi$
$757$	32.0000			1.16306			0.581530	+	0.813525i	$0.302454\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	18.0000	−	31.1769i	0.652499	−	1.13016i	−0.330015	−	0.943976i	$-0.607054\pi$
$761$	18.0000	−	31.1769i	0.652499	−	1.13016i	0.982514	−	0.186187i	$-0.0596129\pi$
$762$		0			0
$763$	−2.00000	−	3.46410i	−0.0724049	−	0.125409i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	28.0000	+	48.4974i	1.01102	+	1.75114i
$768$		0			0
$769$	25.0000	−	43.3013i	0.901523	−	1.56148i	0.0760054	−	0.997107i	$-0.475783\pi$
$769$	25.0000	−	43.3013i	0.901523	−	1.56148i	0.825518	−	0.564376i	$-0.190883\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	42.0000			1.51064			0.755318	−	0.655359i	$-0.227483\pi$
$773$	42.0000			1.51064			0.755318	+	0.655359i	$0.227483\pi$
$774$		0			0
$775$	4.00000			0.143684
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−12.0000	−	20.7846i	−0.429394	−	0.743732i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	2.00000	+	3.46410i	0.0713831	+	0.123639i
$786$		0			0
$787$	14.0000	−	24.2487i	0.499046	−	0.864373i	−0.500953	−	0.865474i	$-0.667017\pi$
$787$	14.0000	−	24.2487i	0.499046	−	0.864373i	0.999999	+	0.00110111i	$0.000350496\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	36.0000			1.28001
$792$		0			0
$793$	−56.0000			−1.98862
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−15.0000	+	25.9808i	−0.531327	+	0.920286i	0.468004	+	0.883726i	$0.344973\pi$
$797$	−15.0000	+	25.9808i	−0.531327	+	0.920286i	−0.999331	+	0.0365596i	$0.988360\pi$
$798$		0			0
$799$	8.00000	+	13.8564i	0.283020	+	0.490204i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−6.00000	−	10.3923i	−0.211735	−	0.366736i
$804$		0			0
$805$	8.00000	−	13.8564i	0.281963	−	0.488374i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−24.0000			−0.843795			−0.421898	−	0.906644i	$-0.638636\pi$
$809$	−24.0000			−0.843795			−0.421898	+	0.906644i	$0.638636\pi$
$810$		0			0
$811$	−28.0000			−0.983213			−0.491606	−	0.870817i	$-0.663590\pi$
$811$	−28.0000			−0.983213			−0.491606	+	0.870817i	$0.663590\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	2.00000	−	3.46410i	0.0700569	−	0.121342i
$816$		0			0
$817$	16.0000	+	27.7128i	0.559769	+	0.969549i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−7.00000	−	12.1244i	−0.244302	−	0.423143i	0.717633	−	0.696421i	$-0.245225\pi$
$821$	−7.00000	−	12.1244i	−0.244302	−	0.423143i	−0.961935	+	0.273278i	$0.911892\pi$
$822$		0			0
$823$	1.00000	−	1.73205i	0.0348578	−	0.0603755i	−0.848070	−	0.529884i	$-0.822235\pi$
$823$	1.00000	−	1.73205i	0.0348578	−	0.0603755i	0.882928	+	0.469508i	$0.155569\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.0000			−0.834562			−0.417281	−	0.908778i	$-0.637017\pi$
$827$	−24.0000			−0.834562			−0.417281	+	0.908778i	$0.637017\pi$
$828$		0			0
$829$	−38.0000			−1.31979			−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000			−1.31979			−0.659897	+	0.751356i	$0.729400\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−3.00000	+	5.19615i	−0.103944	+	0.180036i
$834$		0			0
$835$		0			0
$836$		0			0
$837$		0			0
$838$		0			0
$839$	16.0000	+	27.7128i	0.552381	+	0.956753i	0.998102	+	0.0615805i	$0.0196141\pi$
$839$	16.0000	+	27.7128i	0.552381	+	0.956753i	−0.445721	+	0.895172i	$0.647053\pi$
$840$		0			0
$841$	−35.5000	+	61.4878i	−1.22414	+	2.12027i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.00000			−0.103203
$846$		0			0
$847$	−14.0000			−0.481046
$848$		0			0
$849$		0			0
$850$		0			0
$851$		0			0
$852$		0			0
$853$	8.00000	+	13.8564i	0.273915	+	0.474434i	0.969861	−	0.243660i	$-0.0783480\pi$
$853$	8.00000	+	13.8564i	0.273915	+	0.474434i	−0.695946	+	0.718094i	$0.745015\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	9.00000	+	15.5885i	0.307434	+	0.532492i	0.977800	−	0.209539i	$-0.0671963\pi$
$857$	9.00000	+	15.5885i	0.307434	+	0.532492i	−0.670366	+	0.742030i	$0.733863\pi$
$858$		0			0
$859$	2.00000	−	3.46410i	0.0682391	−	0.118194i	−0.829887	−	0.557931i	$-0.811595\pi$
$859$	2.00000	−	3.46410i	0.0682391	−	0.118194i	0.898126	+	0.439738i	$0.144929\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	48.0000			1.63394			0.816970	−	0.576681i	$-0.195652\pi$
$863$	48.0000			1.63394			0.816970	+	0.576681i	$0.195652\pi$
$864$		0			0
$865$	2.00000			0.0680020
$866$		0			0
$867$		0			0
$868$		0			0
$869$	12.0000	−	20.7846i	0.407072	−	0.705070i
$870$		0			0
$871$	8.00000	+	13.8564i	0.271070	+	0.469506i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1.00000	+	1.73205i	0.0338062	+	0.0585540i
$876$		0			0
$877$	10.0000	−	17.3205i	0.337676	−	0.584872i	−0.646319	−	0.763067i	$-0.723693\pi$
$877$	10.0000	−	17.3205i	0.337676	−	0.584872i	0.983995	+	0.178195i	$0.0570259\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	36.0000			1.21287			0.606435	−	0.795133i	$-0.292599\pi$
$881$	36.0000			1.21287			0.606435	+	0.795133i	$0.292599\pi$
$882$		0			0
$883$	44.0000			1.48072			0.740359	−	0.672212i	$-0.234656\pi$
$883$	44.0000			1.48072			0.740359	+	0.672212i	$0.234656\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−28.0000	+	48.4974i	−0.940148	+	1.62838i	−0.174962	+	0.984575i	$0.555980\pi$
$887$	−28.0000	+	48.4974i	−0.940148	+	1.62838i	−0.765186	+	0.643809i	$0.777353\pi$
$888$		0			0
$889$	−18.0000	−	31.1769i	−0.603701	−	1.04564i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−16.0000	−	27.7128i	−0.535420	−	0.927374i
$894$		0			0
$895$	−7.00000	+	12.1244i	−0.233984	+	0.405273i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−40.0000			−1.33407
$900$		0			0
$901$	−12.0000			−0.399778
$902$		0			0
$903$		0			0
$904$		0			0
$905$	1.00000	−	1.73205i	0.0332411	−	0.0575753i
$906$		0			0
$907$	−12.0000	−	20.7846i	−0.398453	−	0.690142i	0.595082	−	0.803665i	$-0.297120\pi$
$907$	−12.0000	−	20.7846i	−0.398453	−	0.690142i	−0.993535	+	0.113523i	$0.963786\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	8.00000	+	13.8564i	0.265052	+	0.459083i	0.967577	−	0.252575i	$-0.0812776\pi$
$911$	8.00000	+	13.8564i	0.265052	+	0.459083i	−0.702525	+	0.711659i	$0.747944\pi$
$912$		0			0
$913$	−4.00000	+	6.92820i	−0.132381	+	0.229290i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−36.0000			−1.18882
$918$		0			0
$919$	20.0000			0.659739			0.329870	−	0.944027i	$-0.392995\pi$
$919$	20.0000			0.659739			0.329870	+	0.944027i	$0.392995\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−24.0000	+	41.5692i	−0.789970	+	1.36827i
$924$		0			0
$925$		0			0
$926$		0			0
$927$		0			0
$928$		0			0
$929$	2.00000	+	3.46410i	0.0656179	+	0.113653i	0.896968	−	0.442096i	$-0.145765\pi$
$929$	2.00000	+	3.46410i	0.0656179	+	0.113653i	−0.831350	+	0.555749i	$0.812431\pi$
$930$		0			0
$931$	6.00000	−	10.3923i	0.196642	−	0.340594i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.00000			0.130814
$936$		0			0
$937$	10.0000			0.326686			0.163343	−	0.986569i	$-0.447772\pi$
$937$	10.0000			0.326686			0.163343	+	0.986569i	$0.447772\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	11.0000	−	19.0526i	0.358590	−	0.621096i	−0.629136	−	0.777295i	$-0.716591\pi$
$941$	11.0000	−	19.0526i	0.358590	−	0.621096i	0.987725	+	0.156200i	$0.0499244\pi$
$942$		0			0
$943$		0			0
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−28.0000	−	48.4974i	−0.909878	−	1.57595i	−0.814232	−	0.580539i	$-0.802842\pi$
$947$	−28.0000	−	48.4974i	−0.909878	−	1.57595i	−0.0956453	−	0.995415i	$-0.530491\pi$
$948$		0			0
$949$	−12.0000	+	20.7846i	−0.389536	+	0.674697i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−54.0000			−1.74923			−0.874616	−	0.484817i	$-0.838886\pi$
$953$	−54.0000			−1.74923			−0.874616	+	0.484817i	$0.838886\pi$
$954$		0			0
$955$	−4.00000			−0.129437
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−2.00000	+	3.46410i	−0.0645834	+	0.111862i
$960$		0			0
$961$	7.50000	+	12.9904i	0.241935	+	0.419045i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	11.0000	+	19.0526i	0.354103	+	0.613324i
$966$		0			0
$967$	−9.00000	+	15.5885i	−0.289420	+	0.501291i	−0.973672	−	0.227956i	$-0.926796\pi$
$967$	−9.00000	+	15.5885i	−0.289420	+	0.501291i	0.684251	+	0.729247i	$0.260129\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−2.00000			−0.0641831			−0.0320915	−	0.999485i	$-0.510217\pi$
$971$	−2.00000			−0.0641831			−0.0320915	+	0.999485i	$0.510217\pi$
$972$		0			0
$973$	24.0000			0.769405
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.00000	−	1.73205i	0.0319928	−	0.0554132i	−0.849586	−	0.527451i	$-0.823148\pi$
$977$	1.00000	−	1.73205i	0.0319928	−	0.0554132i	0.881579	+	0.472037i	$0.156481\pi$
$978$		0			0
$979$	12.0000	+	20.7846i	0.383522	+	0.664279i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	12.0000	+	20.7846i	0.382741	+	0.662926i	0.991453	−	0.130465i	$-0.0416470\pi$
$983$	12.0000	+	20.7846i	0.382741	+	0.662926i	−0.608712	+	0.793391i	$0.708314\pi$
$984$		0			0
$985$	1.00000	−	1.73205i	0.0318626	−	0.0551877i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−64.0000			−2.03508
$990$		0			0
$991$	8.00000			0.254128			0.127064	−	0.991894i	$-0.459445\pi$
$991$	8.00000			0.254128			0.127064	+	0.991894i	$0.459445\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	4.00000	−	6.92820i	0.126809	−	0.219639i
$996$		0			0
$997$	−18.0000	−	31.1769i	−0.570066	−	0.987383i	−0.996559	−	0.0828918i	$-0.973584\pi$
$997$	−18.0000	−	31.1769i	−0.570066	−	0.987383i	0.426493	−	0.904491i	$-0.359749\pi$
$998$		0			0
$999$		0			0

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

)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	3240.2.q.n.1081.1		2
3.2	odd	2		3240.2.q.d.1081.1		2
9.2	odd	6		3240.2.q.d.2161.1		2
9.4	even	3		360.2.a.c.1.1	✓	1
9.5	odd	6		360.2.a.d.1.1	yes	1
9.7	even	3	inner	3240.2.q.n.2161.1		2
36.23	even	6		720.2.a.i.1.1		1
36.31	odd	6		720.2.a.a.1.1		1
45.4	even	6		1800.2.a.i.1.1		1
45.13	odd	12		1800.2.f.h.649.1		2
45.14	odd	6		1800.2.a.f.1.1		1
45.22	odd	12		1800.2.f.h.649.2		2
45.23	even	12		1800.2.f.d.649.1		2
45.32	even	12		1800.2.f.d.649.2		2
72.5	odd	6		2880.2.a.n.1.1		1
72.13	even	6		2880.2.a.bd.1.1		1
72.59	even	6		2880.2.a.e.1.1		1
72.67	odd	6		2880.2.a.w.1.1		1
180.23	odd	12		3600.2.f.q.2449.2		2
180.59	even	6		3600.2.a.bh.1.1		1
180.67	even	12		3600.2.f.g.2449.1		2
180.103	even	12		3600.2.f.g.2449.2		2
180.139	odd	6		3600.2.a.bd.1.1		1
180.167	odd	12		3600.2.f.q.2449.1		2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.a.c.1.1	✓	1	9.4	even	3
360.2.a.d.1.1	yes	1	9.5	odd	6
720.2.a.a.1.1		1	36.31	odd	6
720.2.a.i.1.1		1	36.23	even	6
1800.2.a.f.1.1		1	45.14	odd	6
1800.2.a.i.1.1		1	45.4	even	6
1800.2.f.d.649.1		2	45.23	even	12
1800.2.f.d.649.2		2	45.32	even	12
1800.2.f.h.649.1		2	45.13	odd	12
1800.2.f.h.649.2		2	45.22	odd	12
2880.2.a.e.1.1		1	72.59	even	6
2880.2.a.n.1.1		1	72.5	odd	6
2880.2.a.w.1.1		1	72.67	odd	6
2880.2.a.bd.1.1		1	72.13	even	6
3240.2.q.d.1081.1		2	3.2	odd	2
3240.2.q.d.2161.1		2	9.2	odd	6
3240.2.q.n.1081.1		2	1.1	even	1	trivial
3240.2.q.n.2161.1		2	9.7	even	3	inner
3600.2.a.bd.1.1		1	180.139	odd	6
3600.2.a.bh.1.1		1	180.59	even	6
3600.2.f.g.2449.1		2	180.67	even	12
3600.2.f.g.2449.2		2	180.103	even	12
3600.2.f.q.2449.1		2	180.167	odd	12
3600.2.f.q.2449.2		2	180.23	odd	12

Overview	Random
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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

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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Newspace parameters

Newform invariants

Embedding invariants

$q$ -expansion

Character values

Coefficient data

Twists

This project is supported by grants from the US National Science Foundation, the UK Engineering and Physical Sciences Research Council, and the Simons Foundation.

Label	3240.2.q.n.1081.1

Level	$3240$
Weight	$2$
Character	3240.1081
Analytic conductor	$25.872$
Analytic rank	$0$
Dimension	$2$
Inner twists	$2$