LMFDB - Embedded newform 3240.2.q.d.1081.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [3240,2,Mod(1081,3240)]

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

lf = mfeigenbasis(mf)

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")

//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: f = N[0] # Warning: the index may be different

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.d.2161.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

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

Display 100 coefficients 10 coefficients

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	0.976478	−	0.215615i	$-0.0691756\pi$
$11$	1.00000	+	1.73205i	0.301511	+	0.522233i	−0.674967	+	0.737848i	$0.735842\pi$
$12$		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	$-0.422021\pi$
$17$	2.00000			0.485071			0.242536	+	0.970143i	$0.422021\pi$
$18$		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	$-0.519345\pi$
$23$	4.00000	−	6.92820i	0.834058	−	1.44463i	0.894795	−	0.446476i	$-0.147321\pi$
$24$		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	$-0.787786\pi$
$29$	−5.00000	−	8.66025i	−0.928477	−	1.60817i	−0.142605	−	0.989780i	$-0.545548\pi$
$30$		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	$0.0316348\pi$
$47$	4.00000	+	6.92820i	0.583460	+	1.01058i	−0.411606	+	0.911362i	$0.635032\pi$
$48$		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	$-0.635198\pi$
$53$	−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
$59$	−7.00000	+	12.1244i	−0.911322	+	1.57846i	−0.0991242	+	0.995075i	$0.531604\pi$
$59$	−7.00000	+	12.1244i	−0.911322	+	1.57846i	−0.812198	+	0.583382i	$0.801729\pi$
$60$		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	$-0.752242\pi$
$71$	−12.0000			−1.42414			−0.712069	+	0.702109i	$0.752242\pi$
$72$		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	0.954664	−	0.297686i	$-0.0962148\pi$
$83$	2.00000	+	3.46410i	0.219529	+	0.380235i	−0.735135	+	0.677920i	$0.762881\pi$
$84$		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	$-0.280588\pi$
$89$	12.0000			1.27200			0.635999	+	0.771690i	$0.280588\pi$
$90$		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	0.975796	−	0.218685i	$-0.0701767\pi$
$101$	3.00000	+	5.19615i	0.298511	+	0.517036i	−0.677284	+	0.735721i	$0.736843\pi$
$102$		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	$-0.303036\pi$
$107$	12.0000			1.16008			0.580042	+	0.814587i	$0.303036\pi$
$108$		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	$-0.511950\pi$
$113$	9.00000	−	15.5885i	0.846649	−	1.46644i	0.884182	−	0.467143i	$-0.154717\pi$
$114$		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	$0.454690\pi$
$131$	−9.00000	+	15.5885i	−0.786334	+	1.36197i	−0.928199	+	0.372084i	$0.878643\pi$
$132$		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	0.905577	−	0.424182i	$-0.139438\pi$
$137$	1.00000	+	1.73205i	0.0854358	+	0.147979i	−0.820141	+	0.572161i	$0.806105\pi$
$138$		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	−0.585231	−	0.810867i	$-0.698996\pi$
$149$	5.00000	−	8.66025i	0.409616	−	0.709476i	0.994847	+	0.101391i	$0.0323294\pi$
$150$		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	0.825505	−	0.564396i	$-0.190891\pi$
$173$	−1.00000	−	1.73205i	−0.0760286	−	0.131685i	−0.901533	+	0.432710i	$0.857557\pi$
$174$		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	$-0.324736\pi$
$179$	14.0000			1.04641			0.523205	+	0.852207i	$0.324736\pi$
$180$		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	0.929267	−	0.369410i	$-0.120440\pi$
$191$	2.00000	+	3.46410i	0.144715	+	0.250654i	−0.784552	+	0.620063i	$0.787107\pi$
$192$		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	$-0.522698\pi$
$197$	−2.00000			−0.142494			−0.0712470	+	0.997459i	$0.522698\pi$
$198$		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	$-0.299282\pi$
$233$	18.0000			1.17922			0.589610	+	0.807688i	$0.299282\pi$
$234$		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	−0.604087	−	0.796918i	$-0.706462\pi$
$239$	6.00000	−	10.3923i	0.388108	−	0.672222i	0.992195	+	0.124696i	$0.0397955\pi$
$240$		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	$-0.560641\pi$
$251$	−6.00000			−0.378717			−0.189358	+	0.981908i	$0.560641\pi$
$252$		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	$-0.643595\pi$
$257$	9.00000	−	15.5885i	0.561405	−	0.972381i	0.997374	−	0.0724199i	$-0.0230722\pi$
$258$		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	$-0.480580\pi$
$269$	2.00000			0.121942			0.0609711	+	0.998140i	$0.480580\pi$
$270$		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	$-0.852036\pi$
$281$	−14.0000	−	24.2487i	−0.835170	−	1.44656i	0.0587220	−	0.998274i	$-0.481297\pi$
$282$		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	−0.835336	−	0.549740i	$-0.814727\pi$
$293$	1.00000	−	1.73205i	0.0584206	−	0.101187i	0.893757	+	0.448552i	$0.148060\pi$
$294$		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	$0.358581\pi$
$311$	−10.0000	+	17.3205i	−0.567048	+	0.982156i	−0.996856	+	0.0792356i	$0.974752\pi$
$312$		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	$-0.847759\pi$
$317$	−15.0000	−	25.9808i	−0.842484	−	1.45922i	0.0453045	−	0.998973i	$-0.485574\pi$
$318$		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	−0.567363	−	0.823468i	$-0.692036\pi$
$347$	8.00000	−	13.8564i	0.429463	−	0.743851i	0.996826	+	0.0796169i	$0.0253697\pi$
$348$		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	$-0.923432\pi$
$353$	−13.0000	−	22.5167i	−0.691920	−	1.19844i	0.279288	−	0.960207i	$-0.409902\pi$
$354$		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	$-0.718315\pi$
$359$	−24.0000			−1.26667			−0.633336	+	0.773877i	$0.718315\pi$
$360$		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	$0.376773\pi$
$383$	−12.0000	+	20.7846i	−0.613171	+	1.06204i	−0.990702	+	0.136047i	$0.956560\pi$
$384$		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	0.998763	−	0.0497253i	$-0.0158346\pi$
$389$	9.00000	+	15.5885i	0.456318	+	0.790366i	−0.542445	+	0.840091i	$0.682501\pi$
$390$		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	−0.976050	−	0.217545i	$-0.930195\pi$
$401$	−6.00000	+	10.3923i	−0.299626	+	0.518967i	0.676425	+	0.736512i	$0.263528\pi$
$402$		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	$-0.571547\pi$
$419$	15.0000	−	25.9808i	0.732798	−	1.26924i	0.955683	−	0.294398i	$-0.0951193\pi$
$420$		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	$-0.625918\pi$
$431$	−16.0000			−0.770693			−0.385346	+	0.922772i	$0.625918\pi$
$432$		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	$0.0264433\pi$
$443$	12.0000	+	20.7846i	0.570137	+	0.987507i	−0.426414	+	0.904528i	$0.640223\pi$
$444$		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	$-0.591382\pi$
$449$	−12.0000			−0.566315			−0.283158	+	0.959073i	$0.591382\pi$
$450$		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	$-0.920465\pi$
$461$	−15.0000	−	25.9808i	−0.698620	−	1.21004i	0.270326	−	0.962769i	$-0.412869\pi$
$462$		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	$-0.346865\pi$
$467$	20.0000			0.925490			0.462745	+	0.886492i	$0.346865\pi$
$468$		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	0.816711	−	0.577047i	$-0.195795\pi$
$479$	−2.00000	−	3.46410i	−0.0913823	−	0.158279i	−0.908093	+	0.418769i	$0.862462\pi$
$480$		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	−0.994456	−	0.105151i	$-0.966467\pi$
$491$	−9.00000	+	15.5885i	−0.406164	+	0.703497i	0.588292	+	0.808649i	$0.299801\pi$
$492$		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	$-0.320291\pi$
$503$	24.0000			1.07011			0.535054	+	0.844818i	$0.320291\pi$
$504$		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	−0.668151	−	0.744026i	$-0.732914\pi$
$509$	7.00000	−	12.1244i	0.310270	−	0.537403i	0.978421	+	0.206623i	$0.0662474\pi$
$510$		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	$-0.644351\pi$
$521$	−20.0000			−0.876216			−0.438108	+	0.898922i	$0.644351\pi$
$522$		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	$-0.314312\pi$
$557$	26.0000			1.10166			0.550828	+	0.834619i	$0.314312\pi$
$558$		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	−0.769341	−	0.638838i	$-0.779415\pi$
$563$	4.00000	−	6.92820i	0.168580	−	0.291989i	0.937921	+	0.346850i	$0.112749\pi$
$564$		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	0.576640	−	0.816999i	$-0.304364\pi$
$569$	−10.0000	−	17.3205i	−0.419222	−	0.726113i	−0.995861	+	0.0908852i	$0.971030\pi$
$570$		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	$0.0999023\pi$
$587$	18.0000	+	31.1769i	0.742940	+	1.28681i	−0.208212	+	0.978084i	$0.566764\pi$
$588$		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	$-0.831019\pi$
$593$	−42.0000			−1.72473			−0.862367	+	0.506284i	$0.831019\pi$
$594$		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	−0.999938	−	0.0111569i	$-0.996449\pi$
$599$	−12.0000	+	20.7846i	−0.490307	+	0.849236i	0.509631	+	0.860393i	$0.329782\pi$
$600$		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	−0.799298	−	0.600935i	$-0.794795\pi$
$617$	3.00000	−	5.19615i	0.120775	−	0.209189i	0.920074	+	0.391745i	$0.128129\pi$
$618$		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	0.999556	−	0.0297987i	$-0.00948663\pi$
$641$	12.0000	+	20.7846i	0.473972	+	0.820943i	−0.525584	+	0.850741i	$0.676153\pi$
$642$		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	$-0.343616\pi$
$647$	24.0000			0.943537			0.471769	+	0.881722i	$0.343616\pi$
$648$		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	−0.969865	−	0.243643i	$-0.921657\pi$
$653$	−7.00000	+	12.1244i	−0.273931	+	0.474463i	0.695934	+	0.718106i	$0.254991\pi$
$654$		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	0.918523	−	0.395367i	$-0.129383\pi$
$659$	3.00000	+	5.19615i	0.116863	+	0.202413i	−0.801660	+	0.597781i	$0.796049\pi$
$660$		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	1.00000		0.000426509i	$-0.000135762\pi$
$677$	13.0000	+	22.5167i	0.499631	+	0.865386i	−0.500369	+	0.865812i	$0.666802\pi$
$678$		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	$-0.426264\pi$
$683$	12.0000			0.459167			0.229584	+	0.973289i	$0.426264\pi$
$684$		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	$-0.636381\pi$
$701$	−22.0000			−0.830929			−0.415464	+	0.909610i	$0.636381\pi$
$702$		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	$-0.232032\pi$
$719$	40.0000			1.49175			0.745874	+	0.666087i	$0.232032\pi$
$720$		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	−0.557470	−	0.830197i	$-0.688228\pi$
$743$	12.0000	−	20.7846i	0.440237	−	0.762513i	0.997707	+	0.0676840i	$0.0215610\pi$
$744$		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	$0.392946\pi$
$761$	−18.0000	+	31.1769i	−0.652499	+	1.13016i	−0.982514	+	0.186187i	$0.940387\pi$
$762$		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	$-0.772517\pi$
$773$	−42.0000			−1.51064			−0.755318	+	0.655359i	$0.772517\pi$
$774$		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	$-0.655027\pi$
$797$	15.0000	−	25.9808i	0.531327	−	0.920286i	0.999331	−	0.0365596i	$-0.0116399\pi$
$798$		0			0

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

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

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

Introduction

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