LMFDB - Embedded newform 1620.3.t.f.269.6

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1620,3,Mod(269,1620)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1620, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 1, 3]))

N = Newforms(chi, 3, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("1620.269");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 1620 = 2^{2} \cdot 3^{4} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1620.t (of order $6$, 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:	$44.1418028264$
Analytic rank:	$0$
Dimension:	$48$
Relative dimension:	$24$ over $\Q(\zeta_{6})$
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			269.6
Character	$\chi$	$=$	1620.269
Dual form			1620.3.t.f.1349.6

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.02456 + 2.96697i) q^{5} +(-9.28893 - 5.36297i) q^{7} +O(q^{10})$	$q+(-4.02456 + 2.96697i) q^{5} +(-9.28893 - 5.36297i) q^{7} +(-10.5523 - 6.09235i) q^{11} +(-20.2632 + 11.6990i) q^{13} +1.67925 q^{17} +0.234357 q^{19} +(-9.93181 - 17.2024i) q^{23} +(7.39413 - 23.8815i) q^{25} +(-29.5094 - 17.0373i) q^{29} +(9.53439 + 16.5140i) q^{31} +(53.2956 - 5.97645i) q^{35} +32.7402i q^{37} +(4.57102 - 2.63908i) q^{41} +(45.8845 + 26.4914i) q^{43} +(-26.8573 + 46.5182i) q^{47} +(33.0229 + 57.1973i) q^{49} -84.6464 q^{53} +(60.5441 - 6.78927i) q^{55} +(76.4925 - 44.1630i) q^{59} +(32.7982 - 56.8082i) q^{61} +(46.8399 - 107.204i) q^{65} +(86.1488 - 49.7380i) q^{67} +36.8639i q^{71} -79.0196i q^{73} +(65.3462 + 113.183i) q^{77} +(-17.6981 + 30.6540i) q^{79} +(13.9232 - 24.1156i) q^{83} +(-6.75825 + 4.98230i) q^{85} +152.513i q^{89} +250.965 q^{91} +(-0.943185 + 0.695332i) q^{95} +(-84.0834 - 48.5456i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 48 q+O(q^{10}) $ 48 * q	$ 48 q + 48 q^{25} + 288 q^{49} - 72 q^{55} - 120 q^{61} - 480 q^{79} + 24 q^{85} + 96 q^{91}+O(q^{100}) $ 48 * q + 48 * q^25 + 288 * q^49 - 72 * q^55 - 120 * q^61 - 480 * q^79 + 24 * q^85 + 96 * q^91

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1620\mathbb{Z}\right)^\times$.

$n$	$811$	$1297$	$1541$
$\chi(n)$	$1$	$-1$	$e\left(\frac{1}{6}\right)$

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
$5$	−4.02456	+	2.96697i	−0.804912	+	0.593395i
$5$	−4.02456	+	2.96697i	−0.804912	+	0.593395i
$6$		0			0
$7$	−9.28893	−	5.36297i	−1.32699	−	0.766138i	−0.342158	−	0.939643i	$-0.611158\pi$
$7$	−9.28893	−	5.36297i	−1.32699	−	0.766138i	−0.984833	+	0.173504i	$0.944491\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−10.5523	−	6.09235i	−0.959297	−	0.553850i	−0.0633403	−	0.997992i	$-0.520175\pi$
$11$	−10.5523	−	6.09235i	−0.959297	−	0.553850i	−0.895957	+	0.444142i	$0.853509\pi$
$12$		0			0
$13$	−20.2632	+	11.6990i	−1.55871	+	0.899920i	−0.561326	+	0.827595i	$0.689709\pi$
$13$	−20.2632	+	11.6990i	−1.55871	+	0.899920i	−0.997381	+	0.0723256i	$0.976958\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	1.67925			0.0987796			0.0493898	−	0.998780i	$-0.484272\pi$
$17$	1.67925			0.0987796			0.0493898	+	0.998780i	$0.484272\pi$
$18$		0			0
$19$	0.234357			0.0123346			0.00616730	−	0.999981i	$-0.498037\pi$
$19$	0.234357			0.0123346			0.00616730	+	0.999981i	$0.498037\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−9.93181	−	17.2024i	−0.431818	−	0.747930i	0.565212	−	0.824946i	$-0.308794\pi$
$23$	−9.93181	−	17.2024i	−0.431818	−	0.747930i	−0.997030	+	0.0770154i	$0.975461\pi$
$24$		0			0
$25$	7.39413	−	23.8815i	0.295765	−	0.955261i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−29.5094	−	17.0373i	−1.01757	−	0.587492i	−0.104168	−	0.994560i	$-0.533218\pi$
$29$	−29.5094	−	17.0373i	−1.01757	−	0.587492i	−0.913398	+	0.407068i	$0.866551\pi$
$30$		0			0
$31$	9.53439	+	16.5140i	0.307561	+	0.532711i	0.977828	−	0.209409i	$-0.0671539\pi$
$31$	9.53439	+	16.5140i	0.307561	+	0.532711i	−0.670267	+	0.742120i	$0.733821\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	53.2956	−	5.97645i	1.52273	−	0.170756i
$36$		0			0
$37$			32.7402i			0.884871i	0.896800	+	0.442436i	$0.145886\pi$
$37$			32.7402i			0.884871i	−0.896800	+	0.442436i	$0.854114\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.57102	−	2.63908i	0.111488	−	0.0643679i	−0.443219	−	0.896413i	$-0.646164\pi$
$41$	4.57102	−	2.63908i	0.111488	−	0.0643679i	0.554707	+	0.832046i	$0.312830\pi$
$42$		0			0
$43$	45.8845	+	26.4914i	1.06708	+	0.616080i	0.927383	−	0.374114i	$-0.122053\pi$
$43$	45.8845	+	26.4914i	1.06708	+	0.616080i	0.139699	+	0.990194i	$0.455386\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−26.8573	+	46.5182i	−0.571432	+	0.989749i	0.424987	+	0.905199i	$0.360279\pi$
$47$	−26.8573	+	46.5182i	−0.571432	+	0.989749i	−0.996419	+	0.0845497i	$0.973055\pi$
$48$		0			0
$49$	33.0229	+	57.1973i	0.673936	+	1.16729i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−84.6464			−1.59710			−0.798551	−	0.601927i	$-0.794400\pi$
$53$	−84.6464			−1.59710			−0.798551	+	0.601927i	$0.794400\pi$
$54$		0			0
$55$	60.5441	−	6.78927i	1.10080	−	0.123441i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	76.4925	−	44.1630i	1.29648	−	0.748525i	0.316688	−	0.948530i	$-0.397429\pi$
$59$	76.4925	−	44.1630i	1.29648	−	0.748525i	0.979795	+	0.200005i	$0.0640959\pi$
$60$		0			0
$61$	32.7982	−	56.8082i	0.537676	−	0.931282i	−0.461353	−	0.887217i	$-0.652636\pi$
$61$	32.7982	−	56.8082i	0.537676	−	0.931282i	0.999029	−	0.0440652i	$-0.0140309\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	46.8399	−	107.204i	0.720614	−	1.64929i
$66$		0			0
$67$	86.1488	−	49.7380i	1.28580	−	0.742359i	0.307900	−	0.951419i	$-0.400374\pi$
$67$	86.1488	−	49.7380i	1.28580	−	0.742359i	0.977903	+	0.209060i	$0.0670404\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			36.8639i			0.519210i	0.965715	+	0.259605i	$0.0835924\pi$
$71$			36.8639i			0.519210i	−0.965715	+	0.259605i	$0.916408\pi$
$72$		0			0
$73$		−	79.0196i		−	1.08246i	−0.840875	−	0.541230i	$-0.817959\pi$
$73$		−	79.0196i		−	1.08246i	0.840875	−	0.541230i	$-0.182041\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	65.3462	+	113.183i	0.848652	+	1.46991i
$78$		0			0
$79$	−17.6981	+	30.6540i	−0.224027	+	0.388025i	−0.956027	−	0.293279i	$-0.905254\pi$
$79$	−17.6981	+	30.6540i	−0.224027	+	0.388025i	0.732000	+	0.681304i	$0.238587\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	13.9232	−	24.1156i	0.167749	−	0.290550i	−0.769879	−	0.638190i	$-0.779684\pi$
$83$	13.9232	−	24.1156i	0.167749	−	0.290550i	0.937628	+	0.347640i	$0.113017\pi$
$84$		0			0
$85$	−6.75825	+	4.98230i	−0.0795088	+	0.0586153i
$86$		0			0
$87$		0			0
$88$		0			0
$89$			152.513i			1.71362i	0.515628	+	0.856812i	$0.327559\pi$
$89$			152.513i			1.71362i	−0.515628	+	0.856812i	$0.672441\pi$
$90$		0			0
$91$	250.965			2.75785
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−0.943185	+	0.695332i	−0.00992826	+	0.00731929i
$96$		0			0
$97$	−84.0834	−	48.5456i	−0.866839	−	0.500470i	−0.000542334	−	1.00000i	$-0.500173\pi$
$97$	−84.0834	−	48.5456i	−0.866839	−	0.500470i	−0.866296	+	0.499530i	$0.833506\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−24.7954	−	14.3156i	−0.245499	−	0.141739i	0.372203	−	0.928152i	$-0.378603\pi$
$101$	−24.7954	−	14.3156i	−0.245499	−	0.141739i	−0.617702	+	0.786413i	$0.711936\pi$
$102$		0			0
$103$	−37.5687	+	21.6903i	−0.364745	+	0.210585i	−0.671160	−	0.741312i	$-0.734204\pi$
$103$	−37.5687	+	21.6903i	−0.364745	+	0.210585i	0.306415	+	0.951898i	$0.400870\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−161.010			−1.50476			−0.752381	−	0.658728i	$-0.771095\pi$
$107$	−161.010			−1.50476			−0.752381	+	0.658728i	$0.771095\pi$
$108$		0			0
$109$	142.306			1.30556			0.652779	−	0.757548i	$-0.273603\pi$
$109$	142.306			1.30556			0.652779	+	0.757548i	$0.273603\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−42.0279	−	72.7944i	−0.371928	−	0.644198i	0.617934	−	0.786230i	$-0.287970\pi$
$113$	−42.0279	−	72.7944i	−0.371928	−	0.644198i	−0.989862	+	0.142032i	$0.954637\pi$
$114$		0			0
$115$	91.0102	+	39.7646i	0.791393	+	0.345779i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−15.5985	−	9.00578i	−0.131080	−	0.0756788i
$120$		0			0
$121$	13.7335	+	23.7872i	0.113500	+	0.196588i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	41.0977	+	118.051i	0.328782	+	0.944406i
$126$		0			0
$127$			188.651i			1.48544i	0.669602	+	0.742720i	$0.266465\pi$
$127$			188.651i			1.48544i	−0.669602	+	0.742720i	$0.733535\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	91.8070	−	53.0048i	0.700817	−	0.404617i	−0.106835	−	0.994277i	$-0.534072\pi$
$131$	91.8070	−	53.0048i	0.700817	−	0.404617i	0.807651	+	0.589660i	$0.200738\pi$
$132$		0			0
$133$	−2.17693	−	1.25685i	−0.0163679	−	0.00945001i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−128.067	+	221.818i	−0.934793	+	1.61911i	−0.159790	+	0.987151i	$0.551082\pi$
$137$	−128.067	+	221.818i	−0.934793	+	1.61911i	−0.775003	+	0.631958i	$0.782252\pi$
$138$		0			0
$139$	90.8442	+	157.347i	0.653555	+	1.13199i	0.982254	+	0.187556i	$0.0600567\pi$
$139$	90.8442	+	157.347i	0.653555	+	1.13199i	−0.328698	+	0.944435i	$0.606610\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	285.097			1.99368
$144$		0			0
$145$	169.311	−	18.9862i	1.16766	−	0.130939i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	249.248	−	143.904i	1.67281	−	0.965797i	0.706754	−	0.707460i	$-0.250159\pi$
$149$	249.248	−	143.904i	1.67281	−	0.965797i	0.966055	−	0.258337i	$-0.0831745\pi$
$150$		0			0
$151$	7.57042	−	13.1124i	0.0501353	−	0.0868368i	−0.839869	−	0.542790i	$-0.817368\pi$
$151$	7.57042	−	13.1124i	0.0501353	−	0.0868368i	0.890004	+	0.455953i	$0.150701\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−87.3684	−	38.1734i	−0.563667	−	0.246280i
$156$		0			0
$157$	−78.0606	+	45.0683i	−0.497202	+	0.287059i	−0.727557	−	0.686047i	$-0.759344\pi$
$157$	−78.0606	+	45.0683i	−0.497202	+	0.287059i	0.230356	+	0.973107i	$0.426011\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			213.056i			1.32333i
$162$		0			0
$163$			131.354i			0.805851i	0.915233	+	0.402925i	$0.132007\pi$
$163$			131.354i			0.805851i	−0.915233	+	0.402925i	$0.867993\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−77.9700	−	135.048i	−0.466886	−	0.808670i	0.532398	−	0.846494i	$-0.321291\pi$
$167$	−77.9700	−	135.048i	−0.466886	−	0.808670i	−0.999284	+	0.0378235i	$0.987958\pi$
$168$		0			0
$169$	189.231	−	327.758i	1.11971	−	1.93940i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	67.1594	−	116.323i	0.388205	−	0.672390i	−0.604004	−	0.796982i	$-0.706429\pi$
$173$	67.1594	−	116.323i	0.388205	−	0.672390i	0.992208	+	0.124592i	$0.0397621\pi$
$174$		0			0
$175$	−196.759	+	182.179i	−1.12434	+	1.04102i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	201.999i		−	1.12849i	−0.825608	−	0.564244i	$-0.809168\pi$
$179$		−	201.999i		−	1.12849i	0.825608	−	0.564244i	$-0.190832\pi$
$180$		0			0
$181$	6.27313			0.0346582			0.0173291	−	0.999850i	$-0.494484\pi$
$181$	6.27313			0.0346582			0.0173291	+	0.999850i	$0.494484\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−97.1394	−	131.765i	−0.525078	−	0.712243i
$186$		0			0
$187$	−17.7199	−	10.2306i	−0.0947589	−	0.0547091i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−257.787	−	148.834i	−1.34967	−	0.779233i	−0.361469	−	0.932384i	$-0.617725\pi$
$191$	−257.787	−	148.834i	−1.34967	−	0.779233i	−0.988203	+	0.153151i	$0.951058\pi$
$192$		0			0
$193$	48.2175	−	27.8384i	0.249831	−	0.144240i	−0.369856	−	0.929089i	$-0.620593\pi$
$193$	48.2175	−	27.8384i	0.249831	−	0.144240i	0.619687	+	0.784849i	$0.287260\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	22.9424			0.116459			0.0582296	−	0.998303i	$-0.481454\pi$
$197$	22.9424			0.116459			0.0582296	+	0.998303i	$0.481454\pi$
$198$		0			0
$199$	204.154			1.02590			0.512951	−	0.858418i	$-0.328552\pi$
$199$	204.154			1.02590			0.512951	+	0.858418i	$0.328552\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	182.741	+	316.516i	0.900200	+	1.55919i
$204$		0			0
$205$	−10.5663	+	24.1832i	−0.0515427	+	0.117967i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−2.47300	−	1.42779i	−0.0118325	−	0.00683152i
$210$		0			0
$211$	178.897	+	309.859i	0.847854	+	1.46853i	0.883120	+	0.469148i	$0.155439\pi$
$211$	178.897	+	309.859i	0.847854	+	1.46853i	−0.0352657	+	0.999378i	$0.511228\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−263.264	+	29.5219i	−1.22449	+	0.137311i
$216$		0			0
$217$		−	204.531i		−	0.942537i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−34.0270	+	19.6455i	−0.153968	+	0.0888937i
$222$		0			0
$223$	−211.390	−	122.046i	−0.947937	−	0.547292i	−0.0554978	−	0.998459i	$-0.517675\pi$
$223$	−211.390	−	122.046i	−0.947937	−	0.547292i	−0.892440	+	0.451167i	$0.851008\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	112.604	−	195.036i	0.496053	−	0.859189i	−0.503937	−	0.863741i	$-0.668115\pi$
$227$	112.604	−	195.036i	0.496053	−	0.859189i	0.999990	+	0.00455151i	$0.00144880\pi$
$228$		0			0
$229$	22.5813	+	39.1119i	0.0986082	+	0.170794i	0.911109	−	0.412166i	$-0.135228\pi$
$229$	22.5813	+	39.1119i	0.0986082	+	0.170794i	−0.812501	+	0.582960i	$0.801894\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−55.4732			−0.238083			−0.119041	−	0.992889i	$-0.537982\pi$
$233$	−55.4732			−0.238083			−0.119041	+	0.992889i	$0.537982\pi$
$234$		0			0
$235$	−29.9296	−	266.900i	−0.127360	−	1.13575i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−10.5094	+	6.06761i	−0.0439724	+	0.0253875i	−0.521825	−	0.853053i	$-0.674749\pi$
$239$	−10.5094	+	6.06761i	−0.0439724	+	0.0253875i	0.477853	+	0.878440i	$0.341415\pi$
$240$		0			0
$241$	−26.4497	+	45.8123i	−0.109750	+	0.190092i	−0.915669	−	0.401933i	$-0.868338\pi$
$241$	−26.4497	+	45.8123i	−0.109750	+	0.190092i	0.805919	+	0.592026i	$0.201672\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−302.605	−	132.216i	−1.23512	−	0.539656i
$246$		0			0
$247$	−4.74883	+	2.74174i	−0.0192260	+	0.0111002i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			44.3567i			0.176720i	0.996089	+	0.0883599i	$0.0281626\pi$
$251$			44.3567i			0.176720i	−0.996089	+	0.0883599i	$0.971837\pi$
$252$		0			0
$253$			242.032i			0.956649i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	85.6501	+	148.350i	0.333269	+	0.577239i	0.983151	−	0.182796i	$-0.0585149\pi$
$257$	85.6501	+	148.350i	0.333269	+	0.577239i	−0.649882	+	0.760035i	$0.725182\pi$
$258$		0			0
$259$	175.585	−	304.122i	0.677934	−	1.17422i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	61.5235	−	106.562i	0.233929	−	0.405178i	−0.725032	−	0.688716i	$-0.758175\pi$
$263$	61.5235	−	106.562i	0.233929	−	0.405178i	0.958961	+	0.283538i	$0.0915082\pi$
$264$		0			0
$265$	340.664	−	251.144i	1.28553	−	0.947712i
$266$		0			0
$267$		0			0
$268$		0			0
$269$		−	128.164i		−	0.476446i	−0.971210	−	0.238223i	$-0.923435\pi$
$269$		−	128.164i		−	0.476446i	0.971210	−	0.238223i	$-0.0765649\pi$
$270$		0			0
$271$	295.707			1.09117			0.545586	−	0.838055i	$-0.316307\pi$
$271$	295.707			1.09117			0.545586	+	0.838055i	$0.316307\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−223.519	+	206.956i	−0.812798	+	0.752569i
$276$		0			0
$277$	420.027	+	242.503i	1.51634	+	0.875462i	0.999816	+	0.0191923i	$0.00610947\pi$
$277$	420.027	+	242.503i	1.51634	+	0.875462i	0.516529	+	0.856270i	$0.327224\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−217.686	−	125.681i	−0.774684	−	0.447264i	0.0598587	−	0.998207i	$-0.480935\pi$
$281$	−217.686	−	125.681i	−0.774684	−	0.447264i	−0.834543	+	0.550943i	$0.814268\pi$
$282$		0			0
$283$	239.790	−	138.443i	0.847314	−	0.489197i	−0.0124298	−	0.999923i	$-0.503957\pi$
$283$	239.790	−	138.443i	0.847314	−	0.489197i	0.859744	+	0.510726i	$0.170623\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−56.6133			−0.197259
$288$		0			0
$289$	−286.180			−0.990243
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−87.0957	−	150.854i	−0.297255	−	0.514861i	0.678252	−	0.734829i	$-0.262738\pi$
$293$	−87.0957	−	150.854i	−0.297255	−	0.514861i	−0.975507	+	0.219969i	$0.929404\pi$
$294$		0			0
$295$	−176.818	+	404.688i	−0.599383	+	1.37182i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	402.500	+	232.384i	1.34615	+	0.777203i
$300$		0			0
$301$	−284.146	−	492.155i	−0.944005	−	1.63507i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	36.5501	+	325.939i	0.119836	+	1.06865i
$306$		0			0
$307$		−	343.344i		−	1.11838i	−0.829038	−	0.559192i	$-0.811112\pi$
$307$		−	343.344i		−	1.11838i	0.829038	−	0.559192i	$-0.188888\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−199.491	+	115.176i	−0.641449	+	0.370341i	−0.785172	−	0.619277i	$-0.787426\pi$
$311$	−199.491	+	115.176i	−0.641449	+	0.370341i	0.143724	+	0.989618i	$0.454092\pi$
$312$		0			0
$313$	347.732	+	200.763i	1.11097	+	0.641416i	0.939079	−	0.343701i	$-0.111681\pi$
$313$	347.732	+	200.763i	1.11097	+	0.641416i	0.171886	+	0.985117i	$0.445014\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	208.287	−	360.764i	0.657057	−	1.13806i	−0.324316	−	0.945949i	$-0.605134\pi$
$317$	208.287	−	360.764i	0.657057	−	1.13806i	0.981374	−	0.192108i	$-0.0615324\pi$
$318$		0			0
$319$	207.594	+	359.563i	0.650765	+	1.12716i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	0.393545			0.00121841
$324$		0			0
$325$	129.560	+	570.420i	0.398647	+	1.75514i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	498.951	−	288.070i	1.51657	−	0.875592i
$330$		0			0
$331$	−100.674	+	174.373i	−0.304151	+	0.526806i	−0.977072	−	0.212909i	$-0.931706\pi$
$331$	−100.674	+	174.373i	−0.304151	+	0.526806i	0.672921	+	0.739715i	$0.265040\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−199.139	+	455.775i	−0.594446	+	1.36052i
$336$		0			0
$337$	−38.8060	+	22.4046i	−0.115151	+	0.0664826i	−0.556469	−	0.830868i	$-0.687844\pi$
$337$	−38.8060	+	22.4046i	−0.115151	+	0.0664826i	0.441318	+	0.897351i	$0.354511\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	232.347i		−	0.681371i
$342$		0			0
$343$		−	182.832i		−	0.533037i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−227.065	−	393.288i	−0.654365	−	1.13339i	−0.982053	−	0.188607i	$-0.939603\pi$
$347$	−227.065	−	393.288i	−0.654365	−	1.13339i	0.327688	−	0.944786i	$-0.393731\pi$
$348$		0			0
$349$	29.2232	−	50.6161i	0.0837341	−	0.145032i	−0.821117	−	0.570760i	$-0.806649\pi$
$349$	29.2232	−	50.6161i	0.0837341	−	0.145032i	0.904851	+	0.425728i	$0.139982\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−280.574	+	485.968i	−0.794827	+	1.37668i	0.128122	+	0.991758i	$0.459105\pi$
$353$	−280.574	+	485.968i	−0.794827	+	1.37668i	−0.922949	+	0.384922i	$0.874228\pi$
$354$		0			0
$355$	−109.374	−	148.361i	−0.308097	−	0.417918i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	139.320i		−	0.388079i	−0.980994	−	0.194040i	$-0.937841\pi$
$359$		−	139.320i		−	0.388079i	0.980994	−	0.194040i	$-0.0621590\pi$
$360$		0			0
$361$	−360.945			−0.999848
$362$		0			0
$363$		0			0
$364$		0			0
$365$	234.449	+	318.019i	0.642326	+	0.871284i
$366$		0			0
$367$	−294.621	−	170.100i	−0.802782	−	0.463486i	0.0416610	−	0.999132i	$-0.486735\pi$
$367$	−294.621	−	170.100i	−0.802782	−	0.463486i	−0.844443	+	0.535645i	$0.820068\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	786.275	+	453.956i	2.11934	+	1.22360i
$372$		0			0
$373$	−335.661	+	193.794i	−0.899897	+	0.519556i	−0.877167	−	0.480186i	$-0.840569\pi$
$373$	−335.661	+	193.794i	−0.899897	+	0.519556i	−0.0227299	+	0.999742i	$0.507236\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	797.273			2.11478
$378$		0			0
$379$	−546.876			−1.44295			−0.721473	−	0.692443i	$-0.756535\pi$
$379$	−546.876			−1.44295			−0.721473	+	0.692443i	$0.756535\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−13.8567	−	24.0005i	−0.0361794	−	0.0626646i	0.847369	−	0.531005i	$-0.178185\pi$
$383$	−13.8567	−	24.0005i	−0.0361794	−	0.0626646i	−0.883548	+	0.468340i	$0.844852\pi$
$384$		0			0
$385$	−598.800	−	261.631i	−1.55533	−	0.679561i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	418.857	+	241.827i	1.07675	+	0.621664i	0.930019	−	0.367512i	$-0.119790\pi$
$389$	418.857	+	241.827i	1.07675	+	0.621664i	0.146735	+	0.989176i	$0.453123\pi$
$390$		0			0
$391$	−16.6780	−	28.8872i	−0.0426548	−	0.0738802i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−19.7226	−	175.879i	−0.0499307	−	0.445262i
$396$		0			0
$397$			347.456i			0.875204i	0.899169	+	0.437602i	$0.144172\pi$
$397$			347.456i			0.875204i	−0.899169	+	0.437602i	$0.855828\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−240.121	+	138.634i	−0.598806	+	0.345721i	−0.768572	−	0.639763i	$-0.779032\pi$
$401$	−240.121	+	138.634i	−0.598806	+	0.345721i	0.169765	+	0.985484i	$0.445699\pi$
$402$		0			0
$403$	−386.394	−	223.085i	−0.958795	−	0.553561i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	199.465	−	345.484i	0.490086	−	0.848854i
$408$		0			0
$409$	−263.838	−	456.981i	−0.645081	−	1.11731i	−0.984283	−	0.176599i	$-0.943490\pi$
$409$	−263.838	−	456.981i	−0.645081	−	1.11731i	0.339202	−	0.940713i	$-0.389843\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−947.378			−2.29389
$414$		0			0
$415$	15.5159	+	138.364i	0.0373876	+	0.333408i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	551.429	−	318.368i	1.31606	−	0.759827i	0.332967	−	0.942938i	$-0.391950\pi$
$419$	551.429	−	318.368i	1.31606	−	0.759827i	0.983092	+	0.183111i	$0.0586167\pi$
$420$		0			0
$421$	−15.3331	+	26.5578i	−0.0364207	+	0.0630825i	−0.883661	−	0.468127i	$-0.844929\pi$
$421$	−15.3331	+	26.5578i	−0.0364207	+	0.0630825i	0.847240	+	0.531210i	$0.178262\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.4166	−	40.1031i	0.0292156	−	0.0943602i
$426$		0			0
$427$	−609.321	+	351.792i	−1.42698	+	0.823868i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			381.687i			0.885584i	0.896624	+	0.442792i	$0.146012\pi$
$431$			381.687i			0.885584i	−0.896624	+	0.442792i	$0.853988\pi$
$432$		0			0
$433$		−	691.143i		−	1.59617i	−0.602543	−	0.798086i	$-0.705846\pi$
$433$		−	691.143i		−	1.59617i	0.602543	−	0.798086i	$-0.294154\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−2.32759	−	4.03151i	−0.00532630	−	0.00922542i
$438$		0			0
$439$	−245.061	+	424.458i	−0.558225	+	0.966874i	0.439420	+	0.898282i	$0.355184\pi$
$439$	−245.061	+	424.458i	−0.558225	+	0.966874i	−0.997645	+	0.0685920i	$0.978149\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−287.938	+	498.724i	−0.649974	+	1.12579i	0.333155	+	0.942872i	$0.391887\pi$
$443$	−287.938	+	498.724i	−0.649974	+	1.12579i	−0.983129	+	0.182916i	$0.941447\pi$
$444$		0			0
$445$	−452.501	−	613.796i	−1.01686	−	1.37932i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	515.232i		−	1.14751i	−0.819027	−	0.573755i	$-0.805486\pi$
$449$		−	515.232i		−	1.14751i	0.819027	−	0.573755i	$-0.194514\pi$
$450$		0			0
$451$	−64.3129			−0.142601
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−1010.02	+	744.606i	−2.21983	+	1.63650i
$456$		0			0
$457$	83.8428	+	48.4067i	0.183463	+	0.105923i	0.588919	−	0.808192i	$-0.299554\pi$
$457$	83.8428	+	48.4067i	0.183463	+	0.105923i	−0.405455	+	0.914115i	$0.632887\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−539.356	−	311.398i	−1.16997	−	0.675483i	−0.216297	−	0.976328i	$-0.569398\pi$
$461$	−539.356	−	311.398i	−1.16997	−	0.675483i	−0.953673	+	0.300845i	$0.902731\pi$
$462$		0			0
$463$	−87.0283	+	50.2458i	−0.187966	+	0.108522i	−0.591030	−	0.806650i	$-0.701279\pi$
$463$	−87.0283	+	50.2458i	−0.187966	+	0.108522i	0.403064	+	0.915172i	$0.367945\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	224.688			0.481131			0.240566	−	0.970633i	$-0.422667\pi$
$467$	224.688			0.481131			0.240566	+	0.970633i	$0.422667\pi$
$468$		0			0
$469$	−1066.97			−2.27500
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−322.790	−	559.090i	−0.682432	−	1.18201i
$474$		0			0
$475$	1.73287	−	5.59681i	0.00364815	−	0.0117828i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	605.459	+	349.562i	1.26401	+	0.729774i	0.973847	−	0.227205i	$-0.0729586\pi$
$479$	605.459	+	349.562i	1.26401	+	0.729774i	0.290159	+	0.956979i	$0.406292\pi$
$480$		0			0
$481$	−383.027	−	663.422i	−0.796314	−	1.37926i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	482.432	−	54.0988i	0.994705	−	0.111544i
$486$		0			0
$487$		−	222.666i		−	0.457220i	−0.973518	−	0.228610i	$-0.926582\pi$
$487$		−	222.666i		−	0.457220i	0.973518	−	0.228610i	$-0.0734181\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−789.824	+	456.005i	−1.60860	+	0.928728i	−0.618920	+	0.785454i	$0.712430\pi$
$491$	−789.824	+	456.005i	−1.60860	+	0.928728i	−0.989683	+	0.143274i	$0.954237\pi$
$492$		0			0
$493$	−49.5537	−	28.6099i	−0.100515	−	0.0580322i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	197.700	−	342.427i	0.397787	−	0.688987i
$498$		0			0
$499$	392.875	+	680.479i	0.787324	+	1.36369i	0.927601	+	0.373573i	$0.121868\pi$
$499$	392.875	+	680.479i	0.787324	+	1.36369i	−0.140276	+	0.990112i	$0.544799\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	288.196			0.572955			0.286477	−	0.958087i	$-0.407516\pi$
$503$	288.196			0.572955			0.286477	+	0.958087i	$0.407516\pi$
$504$		0			0
$505$	142.265	−	15.9532i	0.281712	−	0.0315905i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	412.825	−	238.345i	0.811052	−	0.468261i	−0.0362693	−	0.999342i	$-0.511547\pi$
$509$	412.825	−	238.345i	0.811052	−	0.468261i	0.847321	+	0.531081i	$0.178214\pi$
$510$		0			0
$511$	−423.779	+	734.007i	−0.829314	+	1.43641i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	86.8429	−	198.759i	0.168627	−	0.385940i
$516$		0			0
$517$	566.811	−	327.248i	1.09635	−	0.632975i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	337.377i		−	0.647557i	−0.946133	−	0.323778i	$-0.895047\pi$
$521$		−	337.377i		−	0.647557i	0.946133	−	0.323778i	$-0.104953\pi$
$522$		0			0
$523$			820.192i			1.56824i	0.620607	+	0.784122i	$0.286886\pi$
$523$			820.192i			1.56824i	−0.620607	+	0.784122i	$0.713114\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	16.0106	+	27.7312i	0.0303807	+	0.0526210i
$528$		0			0
$529$	67.2184	−	116.426i	0.127067	−	0.220086i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−61.7490	+	106.952i	−0.115852	+	0.200661i
$534$		0			0
$535$	647.992	−	477.711i	1.21120	−	0.892918i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	804.748i		−	1.49304i
$540$		0			0
$541$	−770.684			−1.42456			−0.712278	−	0.701898i	$-0.752336\pi$
$541$	−770.684			−1.42456			−0.712278	+	0.701898i	$0.752336\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−572.718	+	422.218i	−1.05086	+	0.774712i
$546$		0			0
$547$	750.002	+	433.014i	1.37112	+	0.791616i	0.991069	−	0.133350i	$-0.0425735\pi$
$547$	750.002	+	433.014i	1.37112	+	0.791616i	0.380050	+	0.924966i	$0.375907\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.91575	−	3.99281i	−0.0125513	−	0.00724647i
$552$		0			0
$553$	328.793	−	189.829i	0.594562	−	0.343271i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−245.790			−0.441275			−0.220637	−	0.975356i	$-0.570814\pi$
$557$	−245.790			−0.441275			−0.220637	+	0.975356i	$0.570814\pi$
$558$		0			0
$559$	−1239.69			−2.21769
$560$		0			0
$561$		0			0
$562$		0			0
$563$	169.294	+	293.226i	0.300700	+	0.520829i	0.976295	−	0.216445i	$-0.0694462\pi$
$563$	169.294	+	293.226i	0.300700	+	0.520829i	−0.675594	+	0.737274i	$0.736113\pi$
$564$		0			0
$565$	385.123	+	168.270i	0.681633	+	0.297823i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−265.584	−	153.335i	−0.466755	−	0.269481i	0.248125	−	0.968728i	$-0.420186\pi$
$569$	−265.584	−	153.335i	−0.466755	−	0.269481i	−0.714880	+	0.699247i	$0.753519\pi$
$570$		0			0
$571$	−9.09849	−	15.7590i	−0.0159343	−	0.0275990i	0.857948	−	0.513736i	$-0.171739\pi$
$571$	−9.09849	−	15.7590i	−0.0159343	−	0.0275990i	−0.873883	+	0.486137i	$0.838406\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−484.256	+	109.990i	−0.842185	+	0.191287i
$576$		0			0
$577$		−	766.613i		−	1.32862i	−0.747458	−	0.664309i	$-0.768726\pi$
$577$		−	766.613i		−	1.32862i	0.747458	−	0.664309i	$-0.231274\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−258.663	+	149.339i	−0.445203	+	0.257038i
$582$		0			0
$583$	893.211	+	515.696i	1.53209	+	0.884555i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−424.596	+	735.422i	−0.723332	+	1.25285i	0.236324	+	0.971674i	$0.424057\pi$
$587$	−424.596	+	735.422i	−0.723332	+	1.25285i	−0.959657	+	0.281174i	$0.909276\pi$
$588$		0			0
$589$	2.23445	+	3.87019i	0.00379364	+	0.00657078i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−91.0522			−0.153545			−0.0767725	−	0.997049i	$-0.524462\pi$
$593$	−91.0522			−0.153545			−0.0767725	+	0.997049i	$0.524462\pi$
$594$		0			0
$595$	89.4968	−	10.0360i	0.150415	−	0.0168672i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	49.1183	−	28.3585i	0.0820005	−	0.0473430i	−0.458439	−	0.888726i	$-0.651591\pi$
$599$	49.1183	−	28.3585i	0.0820005	−	0.0473430i	0.540440	+	0.841383i	$0.318258\pi$
$600$		0			0
$601$	107.324	−	185.890i	0.178575	−	0.309301i	−0.762818	−	0.646614i	$-0.776185\pi$
$601$	107.324	−	185.890i	0.178575	−	0.309301i	0.941393	+	0.337313i	$0.109518\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−125.847	−	54.9859i	−0.208012	−	0.0908857i
$606$		0			0
$607$	235.162	−	135.771i	0.387417	−	0.223675i	−0.293623	−	0.955921i	$-0.594861\pi$
$607$	235.162	−	135.771i	0.387417	−	0.223675i	0.681040	+	0.732246i	$0.261528\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	1256.81i		−	2.05697i
$612$		0			0
$613$		−	576.529i		−	0.940504i	−0.882532	−	0.470252i	$-0.844163\pi$
$613$		−	576.529i		−	0.940504i	0.882532	−	0.470252i	$-0.155837\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	171.771	+	297.517i	0.278398	+	0.482199i	0.970987	−	0.239134i	$-0.0768634\pi$
$617$	171.771	+	297.517i	0.278398	+	0.482199i	−0.692589	+	0.721332i	$0.743530\pi$
$618$		0			0
$619$	196.673	−	340.648i	0.317727	−	0.550319i	−0.662286	−	0.749251i	$-0.730414\pi$
$619$	196.673	−	340.648i	0.317727	−	0.550319i	0.980013	+	0.198931i	$0.0637471\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	817.920	−	1416.68i	1.31287	−	2.27396i
$624$		0			0
$625$	−515.654	−	353.166i	−0.825046	−	0.565066i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			54.9791i			0.0874072i
$630$		0			0
$631$	−224.300			−0.355467			−0.177734	−	0.984079i	$-0.556877\pi$
$631$	−224.300			−0.355467			−0.177734	+	0.984079i	$0.556877\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−559.722	−	759.237i	−0.881453	−	1.19565i
$636$		0			0
$637$	−1338.30	−	772.667i	−2.10094	−	1.21298i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	227.262	+	131.210i	0.354544	+	0.204696i	0.666685	−	0.745340i	$-0.267713\pi$
$641$	227.262	+	131.210i	0.354544	+	0.204696i	−0.312141	+	0.950036i	$0.601046\pi$
$642$		0			0
$643$	282.643	−	163.184i	0.439569	−	0.253785i	−0.263846	−	0.964565i	$-0.584991\pi$
$643$	282.643	−	163.184i	0.439569	−	0.253785i	0.703415	+	0.710779i	$0.251658\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−438.462			−0.677685			−0.338842	−	0.940843i	$-0.610035\pi$
$647$	−438.462			−0.677685			−0.338842	+	0.940843i	$0.610035\pi$
$648$		0			0
$649$	−1076.23			−1.65828
$650$		0			0
$651$		0			0
$652$		0			0
$653$	439.932	+	761.984i	0.673709	+	1.16690i	0.976845	+	0.213950i	$0.0686330\pi$
$653$	439.932	+	761.984i	0.673709	+	1.16690i	−0.303136	+	0.952947i	$0.598034\pi$
$654$		0			0
$655$	−212.219	+	485.710i	−0.323998	+	0.741542i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	888.871	+	513.190i	1.34882	+	0.778741i	0.988082	−	0.153927i	$-0.0491921\pi$
$659$	888.871	+	513.190i	1.34882	+	0.778741i	0.360736	+	0.932668i	$0.382525\pi$
$660$		0			0
$661$	−253.494	−	439.065i	−0.383501	−	0.664243i	0.608059	−	0.793892i	$-0.291948\pi$
$661$	−253.494	−	439.065i	−0.383501	−	0.664243i	−0.991560	+	0.129649i	$0.958615\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	12.4902	−	1.40063i	0.0187823	−	0.00210620i
$666$		0			0
$667$			676.843i			1.01476i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−692.191	+	399.637i	−1.03158	+	0.595584i
$672$		0			0
$673$	−454.331	−	262.308i	−0.675084	−	0.389760i	0.122916	−	0.992417i	$-0.460775\pi$
$673$	−454.331	−	262.308i	−0.675084	−	0.389760i	−0.798000	+	0.602657i	$0.794109\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−22.1441	+	38.3547i	−0.0327091	+	0.0566539i	−0.881917	−	0.471406i	$-0.843747\pi$
$677$	−22.1441	+	38.3547i	−0.0327091	+	0.0566539i	0.849207	+	0.528059i	$0.177080\pi$
$678$		0			0
$679$	520.697	+	901.873i	0.766858	+	1.32824i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	1020.38			1.49396			0.746982	−	0.664845i	$-0.231502\pi$
$683$	1020.38			1.49396			0.746982	+	0.664845i	$0.231502\pi$
$684$		0			0
$685$	−142.716	−	1272.69i	−0.208345	−	1.85794i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1715.21	−	990.275i	2.48941	−	1.43726i
$690$		0			0
$691$	−72.5955	+	125.739i	−0.105059	+	0.181967i	−0.913762	−	0.406250i	$-0.866836\pi$
$691$	−72.5955	+	125.739i	−0.105059	+	0.181967i	0.808704	+	0.588216i	$0.200170\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−832.451	−	363.719i	−1.19777	−	0.523336i
$696$		0			0
$697$	7.67590	−	4.43169i	0.0110128	−	0.00635823i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			1136.43i			1.62116i	0.585629	+	0.810579i	$0.300848\pi$
$701$			1136.43i			1.62116i	−0.585629	+	0.810579i	$0.699152\pi$
$702$		0			0
$703$			7.67292i			0.0109145i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	153.549	+	265.954i	0.217183	+	0.376172i
$708$		0			0
$709$	402.678	−	697.459i	0.567953	−	0.983723i	−0.428816	−	0.903392i	$-0.641069\pi$
$709$	402.678	−	697.459i	0.567953	−	0.983723i	0.996768	−	0.0803307i	$-0.0255976\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	189.387	−	328.029i	0.265620	−	0.460068i
$714$		0			0
$715$	−1147.39	+	845.875i	−1.60474	+	1.18304i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	1266.26i		−	1.76115i	−0.473910	−	0.880573i	$-0.657158\pi$
$719$		−	1266.26i		−	1.76115i	0.473910	−	0.880573i	$-0.342842\pi$
$720$		0			0
$721$	465.298			0.645351
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−625.072	+	578.753i	−0.862168	+	0.798281i
$726$		0			0
$727$	211.476	+	122.095i	0.290888	+	0.167944i	0.638342	−	0.769753i	$-0.279620\pi$
$727$	211.476	+	122.095i	0.290888	+	0.167944i	−0.347454	+	0.937697i	$0.612954\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	77.0517	+	44.4858i	0.105406	+	0.0608561i
$732$		0			0
$733$	−569.850	+	329.003i	−0.777421	+	0.448844i	−0.835516	−	0.549467i	$-0.814831\pi$
$733$	−569.850	+	329.003i	−0.777421	+	0.448844i	0.0580943	+	0.998311i	$0.481498\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−1212.09			−1.64462
$738$		0			0
$739$	178.113			0.241019			0.120509	−	0.992712i	$-0.461547\pi$
$739$	178.113			0.241019			0.120509	+	0.992712i	$0.461547\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−206.395	−	357.486i	−0.277786	−	0.481139i	0.693048	−	0.720891i	$-0.256267\pi$
$743$	−206.395	−	357.486i	−0.277786	−	0.481139i	−0.970834	+	0.239752i	$0.922934\pi$
$744$		0			0
$745$	−576.156	+	1318.66i	−0.773364	+	1.77002i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	1495.61	+	863.489i	1.99681	+	1.15286i
$750$		0			0
$751$	−199.806	−	346.075i	−0.266054	−	0.460819i	0.701785	−	0.712388i	$-0.252387\pi$
$751$	−199.806	−	346.075i	−0.266054	−	0.460819i	−0.967839	+	0.251570i	$0.919053\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	8.43642	+	75.2327i	0.0111741	+	0.0996460i
$756$		0			0
$757$			1032.23i			1.36358i	0.731546	+	0.681792i	$0.238799\pi$
$757$			1032.23i			1.36358i	−0.731546	+	0.681792i	$0.761201\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−248.216	+	143.308i	−0.326171	+	0.188315i	−0.654140	−	0.756373i	$-0.726969\pi$
$761$	−248.216	+	143.308i	−0.326171	+	0.188315i	0.327969	+	0.944689i	$0.393636\pi$
$762$		0			0
$763$	−1321.87	−	763.182i	−1.73246	−	1.00024i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−1033.32	+	1789.77i	−1.34722	+	2.33346i
$768$		0			0
$769$	−342.621	−	593.437i	−0.445541	−	0.771700i	0.552549	−	0.833481i	$-0.313655\pi$
$769$	−342.621	−	593.437i	−0.445541	−	0.771700i	−0.998090	+	0.0617809i	$0.980322\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	1155.89			1.49533			0.747663	−	0.664079i	$-0.231176\pi$
$773$	1155.89			1.49533			0.747663	+	0.664079i	$0.231176\pi$
$774$		0			0
$775$	464.879	−	105.589i	0.599844	−	0.136243i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1.07125	−	0.618488i	0.00137516	−	0.000793952i
$780$		0			0
$781$	224.588	−	388.998i	0.287565	−	0.498077i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	180.443	−	412.984i	0.229864	−	0.526094i
$786$		0			0
$787$	−8.38210	+	4.83941i	−0.0106507	+	0.00614919i	−0.505316	−	0.862934i	$-0.668624\pi$
$787$	−8.38210	+	4.83941i	−0.0106507	+	0.00614919i	0.494665	+	0.869084i	$0.335291\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			901.577i			1.13979i
$792$		0			0
$793$			1534.82i			1.93546i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−757.597	−	1312.20i	−0.950561	−	1.64642i	−0.744215	−	0.667940i	$-0.767176\pi$
$797$	−757.597	−	1312.20i	−0.950561	−	1.64642i	−0.206346	−	0.978479i	$-0.566157\pi$
$798$		0			0
$799$	−45.1002	+	78.1158i	−0.0564458	+	0.0977670i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−481.415	+	833.835i	−0.599521	+	1.03840i
$804$		0			0
$805$	−632.131	−	857.456i	−0.785256	−	1.06516i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			628.159i			0.776464i	0.921562	+	0.388232i	$0.126914\pi$
$809$			628.159i			0.776464i	−0.921562	+	0.388232i	$0.873086\pi$
$810$		0			0
$811$	1027.82			1.26735			0.633675	−	0.773600i	$-0.281546\pi$
$811$	1027.82			1.26735			0.633675	+	0.773600i	$0.281546\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−389.723	−	528.640i	−0.478188	−	0.648639i
$816$		0			0
$817$	10.7534	+	6.20847i	0.0131620	+	0.00759910i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−969.780	−	559.903i	−1.18122	−	0.681977i	−0.224922	−	0.974377i	$-0.572213\pi$
$821$	−969.780	−	559.903i	−1.18122	−	0.681977i	−0.956296	+	0.292400i	$0.905546\pi$
$822$		0			0
$823$	352.273	−	203.385i	0.428036	−	0.247127i	−0.270474	−	0.962727i	$-0.587180\pi$
$823$	352.273	−	203.385i	0.428036	−	0.247127i	0.698510	+	0.715601i	$0.253847\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	445.615			0.538833			0.269416	−	0.963024i	$-0.413169\pi$
$827$	445.615			0.538833			0.269416	+	0.963024i	$0.413169\pi$
$828$		0			0
$829$	763.809			0.921362			0.460681	−	0.887566i	$-0.347605\pi$
$829$	763.809			0.921362			0.460681	+	0.887566i	$0.347605\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	55.4537	+	96.0487i	0.0665711	+	0.115305i
$834$		0			0
$835$	714.479	+	312.173i	0.855663	+	0.373860i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−386.267	−	223.011i	−0.460389	−	0.265806i	0.251819	−	0.967774i	$-0.418971\pi$
$839$	−386.267	−	223.011i	−0.460389	−	0.265806i	−0.712208	+	0.701969i	$0.752305\pi$
$840$		0			0
$841$	160.036	+	277.191i	0.190293	+	0.329597i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	210.878	+	1880.53i	0.249560	+	2.22548i
$846$		0			0
$847$		−	294.610i		−	0.347828i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	563.211	−	325.170i	0.661822	−	0.382103i
$852$		0			0
$853$	922.445	+	532.574i	1.08141	+	0.624354i	0.931277	−	0.364312i	$-0.118696\pi$
$853$	922.445	+	532.574i	1.08141	+	0.624354i	0.150135	+	0.988665i	$0.452029\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−386.183	+	668.889i	−0.450622	+	0.780501i	−0.998425	−	0.0561069i	$-0.982131\pi$
$857$	−386.183	+	668.889i	−0.450622	+	0.780501i	0.547802	+	0.836608i	$0.315465\pi$
$858$		0			0
$859$	−34.0802	−	59.0286i	−0.0396742	−	0.0687178i	0.845506	−	0.533965i	$-0.179299\pi$
$859$	−34.0802	−	59.0286i	−0.0396742	−	0.0687178i	−0.885181	+	0.465247i	$0.845965\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	572.939			0.663892			0.331946	−	0.943298i	$-0.392295\pi$
$863$	572.939			0.663892			0.331946	+	0.943298i	$0.392295\pi$
$864$		0			0
$865$	74.8419	+	667.411i	0.0865224	+	0.771573i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	373.510	−	215.646i	0.429816	−	0.248154i
$870$		0			0
$871$	−1163.77	+	2015.70i	−1.33613	+	2.31424i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	251.348	−	1316.97i	0.287255	−	1.50511i
$876$		0			0
$877$	268.988	−	155.300i	0.306713	−	0.177081i	−0.338742	−	0.940879i	$-0.610001\pi$
$877$	268.988	−	155.300i	0.306713	−	0.177081i	0.645455	+	0.763799i	$0.276668\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	81.2933i		−	0.0922739i	−0.998935	−	0.0461369i	$-0.985309\pi$
$881$		−	81.2933i		−	0.0922739i	0.998935	−	0.0461369i	$-0.0146911\pi$
$882$		0			0
$883$		−	673.174i		−	0.762372i	−0.924498	−	0.381186i	$-0.875516\pi$
$883$		−	673.174i		−	0.762372i	0.924498	−	0.381186i	$-0.124484\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	871.614	+	1509.68i	0.982654	+	1.70201i	0.651931	+	0.758279i	$0.273959\pi$
$887$	871.614	+	1509.68i	0.982654	+	1.70201i	0.330723	+	0.943728i	$0.392707\pi$
$888$		0			0
$889$	1011.73	−	1752.37i	1.13805	−	1.97117i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−6.29421	+	10.9019i	−0.00704838	+	0.0122082i
$894$		0			0
$895$	599.327	+	812.958i	0.669639	+	0.908333i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	649.759i		−	0.722758i
$900$		0			0
$901$	−142.143			−0.157761
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−25.2466	+	18.6122i	−0.0278968	+	0.0205660i
$906$		0			0
$907$	−629.070	−	363.194i	−0.693573	−	0.400434i	0.111376	−	0.993778i	$-0.464474\pi$
$907$	−629.070	−	363.194i	−0.693573	−	0.400434i	−0.804949	+	0.593344i	$0.797807\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	991.195	+	572.267i	1.08803	+	0.628174i	0.933051	−	0.359743i	$-0.117136\pi$
$911$	991.195	+	572.267i	1.08803	+	0.628174i	0.154979	+	0.987918i	$0.450469\pi$
$912$		0			0
$913$	−293.842	+	169.650i	−0.321842	+	0.185816i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−1137.05			−1.23997
$918$		0			0
$919$	725.840			0.789815			0.394907	−	0.918721i	$-0.370777\pi$
$919$	725.840			0.789815			0.394907	+	0.918721i	$0.370777\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−431.270	−	746.981i	−0.467248	−	0.809297i
$924$		0			0
$925$	781.887	+	242.086i	0.845283	+	0.261714i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	623.881	+	360.198i	0.671562	+	0.387727i	0.796668	−	0.604417i	$-0.206594\pi$
$929$	623.881	+	360.198i	0.671562	+	0.387727i	−0.125106	+	0.992143i	$0.539927\pi$
$930$		0			0
$931$	7.73915	+	13.4046i	0.00831273	+	0.0143981i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	101.669	−	11.4009i	0.108737	−	0.0121935i
$936$		0			0
$937$			821.308i			0.876529i	0.898846	+	0.438265i	$0.144407\pi$
$937$			821.308i			0.876529i	−0.898846	+	0.438265i	$0.855593\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	150.992	−	87.1752i	0.160459	−	0.0926410i	−0.417620	−	0.908622i	$-0.637136\pi$
$941$	150.992	−	87.1752i	0.160459	−	0.0926410i	0.578079	+	0.815981i	$0.303802\pi$
$942$		0			0
$943$	−90.7971	−	52.4217i	−0.0962853	−	0.0555904i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	579.276	−	1003.34i	0.611696	−	1.05949i	−0.379258	−	0.925291i	$-0.623821\pi$
$947$	579.276	−	1003.34i	0.611696	−	1.05949i	0.990955	−	0.134198i	$-0.0428459\pi$
$948$		0			0
$949$	924.447	+	1601.19i	0.974127	+	1.68724i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−734.811			−0.771051			−0.385525	−	0.922697i	$-0.625980\pi$
$953$	−734.811			−0.771051			−0.385525	+	0.922697i	$0.625980\pi$
$954$		0			0
$955$	1479.06	−	165.859i	1.54876	−	0.173674i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	2379.21	−	1373.63i	2.48092	−	1.43236i
$960$		0			0
$961$	298.691	−	517.348i	0.310813	−	0.538343i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−111.458	+	255.097i	−0.115501	+	0.264349i
$966$		0			0
$967$	636.681	−	367.588i	0.658408	−	0.380132i	−0.133262	−	0.991081i	$-0.542545\pi$
$967$	636.681	−	367.588i	0.658408	−	0.380132i	0.791670	+	0.610949i	$0.209212\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			630.994i			0.649839i	0.945742	+	0.324920i	$0.105337\pi$
$971$			630.994i			0.649839i	−0.945742	+	0.324920i	$0.894663\pi$
$972$		0			0
$973$		−	1948.78i		−	2.00286i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	563.994	+	976.866i	0.577271	+	0.999863i	0.995791	+	0.0916552i	$0.0292157\pi$
$977$	563.994	+	976.866i	0.577271	+	0.999863i	−0.418520	+	0.908208i	$0.637451\pi$
$978$		0			0
$979$	929.161	−	1609.35i	0.949092	−	1.64388i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	467.565	−	809.846i	0.475651	−	0.823851i	−0.523960	−	0.851743i	$-0.675546\pi$
$983$	467.565	−	809.846i	0.475651	−	0.823851i	0.999611	+	0.0278915i	$0.00887928\pi$
$984$		0			0
$985$	−92.3332	+	68.0696i	−0.0937393	+	0.0691062i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	1052.43i		−	1.06414i
$990$		0			0
$991$	839.459			0.847083			0.423541	−	0.905877i	$-0.360787\pi$
$991$	839.459			0.847083			0.423541	+	0.905877i	$0.360787\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−821.631	+	605.721i	−0.825760	+	0.608764i
$996$		0			0
$997$	216.451	+	124.968i	0.217102	+	0.125344i	0.604608	−	0.796523i	$-0.293330\pi$
$997$	216.451	+	124.968i	0.217102	+	0.125344i	−0.387506	+	0.921867i	$0.626663\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1620.3.t.f.269.6		48
3.2	odd	2	inner	1620.3.t.f.269.19		48
5.4	even	2	inner	1620.3.t.f.269.4		48
9.2	odd	6		1620.3.b.a.809.14	yes	24
9.4	even	3	inner	1620.3.t.f.1349.21		48
9.5	odd	6	inner	1620.3.t.f.1349.4		48
9.7	even	3		1620.3.b.a.809.11	✓	24
15.14	odd	2	inner	1620.3.t.f.269.21		48
45.4	even	6	inner	1620.3.t.f.1349.19		48
45.14	odd	6	inner	1620.3.t.f.1349.6		48
45.29	odd	6		1620.3.b.a.809.12	yes	24
45.34	even	6		1620.3.b.a.809.13	yes	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.b.a.809.11	✓	24	9.7	even	3
1620.3.b.a.809.12	yes	24	45.29	odd	6
1620.3.b.a.809.13	yes	24	45.34	even	6
1620.3.b.a.809.14	yes	24	9.2	odd	6
1620.3.t.f.269.4		48	5.4	even	2	inner
1620.3.t.f.269.6		48	1.1	even	1	trivial
1620.3.t.f.269.19		48	3.2	odd	2	inner
1620.3.t.f.269.21		48	15.14	odd	2	inner
1620.3.t.f.1349.4		48	9.5	odd	6	inner
1620.3.t.f.1349.6		48	45.14	odd	6	inner
1620.3.t.f.1349.19		48	45.4	even	6	inner
1620.3.t.f.1349.21		48	9.4	even	3	inner

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:	\( N \)	\(=\)	\( 1620 = 2^{2} \cdot 3^{4} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1620.t (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-4.02456 + 2.96697i) q^{5} +(-9.28893 - 5.36297i) q^{7} +O(q^{10})\)	\(q+(-4.02456 + 2.96697i) q^{5} +(-9.28893 - 5.36297i) q^{7} +(-10.5523 - 6.09235i) q^{11} +(-20.2632 + 11.6990i) q^{13} +1.67925 q^{17} +0.234357 q^{19} +(-9.93181 - 17.2024i) q^{23} +(7.39413 - 23.8815i) q^{25} +(-29.5094 - 17.0373i) q^{29} +(9.53439 + 16.5140i) q^{31} +(53.2956 - 5.97645i) q^{35} +32.7402i q^{37} +(4.57102 - 2.63908i) q^{41} +(45.8845 + 26.4914i) q^{43} +(-26.8573 + 46.5182i) q^{47} +(33.0229 + 57.1973i) q^{49} -84.6464 q^{53} +(60.5441 - 6.78927i) q^{55} +(76.4925 - 44.1630i) q^{59} +(32.7982 - 56.8082i) q^{61} +(46.8399 - 107.204i) q^{65} +(86.1488 - 49.7380i) q^{67} +36.8639i q^{71} -79.0196i q^{73} +(65.3462 + 113.183i) q^{77} +(-17.6981 + 30.6540i) q^{79} +(13.9232 - 24.1156i) q^{83} +(-6.75825 + 4.98230i) q^{85} +152.513i q^{89} +250.965 q^{91} +(-0.943185 + 0.695332i) q^{95} +(-84.0834 - 48.5456i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 48 * q	\( 48 q + 48 q^{25} + 288 q^{49} - 72 q^{55} - 120 q^{61} - 480 q^{79} + 24 q^{85} + 96 q^{91}+O(q^{100}) \) 48 * q + 48 * q^25 + 288 * q^49 - 72 * q^55 - 120 * q^61 - 480 * q^79 + 24 * q^85 + 96 * q^91

Introduction

L-functions

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