LMFDB - Embedded newform 1620.3.t.f.269.5

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			269.5
Character	$\chi$	$=$	1620.269
Dual form			1620.3.t.f.1349.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-4.45514 - 2.26973i) q^{5} +(-2.97789 - 1.71928i) q^{7} +O(q^{10})$	$q+(-4.45514 - 2.26973i) q^{5} +(-2.97789 - 1.71928i) q^{7} +(-11.1039 - 6.41086i) q^{11} +(1.55354 - 0.896938i) q^{13} -30.9523 q^{17} -19.2201 q^{19} +(-1.19201 - 2.06461i) q^{23} +(14.6966 + 20.2240i) q^{25} +(31.0770 + 17.9423i) q^{29} +(20.4538 + 35.4270i) q^{31} +(9.36459 + 14.4187i) q^{35} -53.6302i q^{37} +(-2.14983 + 1.24120i) q^{41} +(47.7611 + 27.5749i) q^{43} +(28.5250 - 49.4067i) q^{47} +(-18.5881 - 32.1956i) q^{49} -19.4030 q^{53} +(34.9187 + 53.7643i) q^{55} +(-59.8830 + 34.5735i) q^{59} +(-8.80471 + 15.2502i) q^{61} +(-8.95707 + 0.469861i) q^{65} +(44.3133 - 25.5843i) q^{67} -53.9194i q^{71} +42.7832i q^{73} +(22.0442 + 38.1816i) q^{77} +(-44.4228 + 76.9425i) q^{79} +(14.0472 - 24.3305i) q^{83} +(137.897 + 70.2536i) q^{85} -68.3114i q^{89} -6.16836 q^{91} +(85.6282 + 43.6245i) q^{95} +(135.186 + 78.0498i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−4.45514	−	2.26973i	−0.891029	−	0.453947i
$5$	−4.45514	−	2.26973i	−0.891029	−	0.453947i
$6$		0			0
$7$	−2.97789	−	1.71928i	−0.425412	−	0.245612i	0.271978	−	0.962303i	$-0.412322\pi$
$7$	−2.97789	−	1.71928i	−0.425412	−	0.245612i	−0.697390	+	0.716692i	$0.745656\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−11.1039	−	6.41086i	−1.00945	−	0.582806i	−0.0984188	−	0.995145i	$-0.531378\pi$
$11$	−11.1039	−	6.41086i	−1.00945	−	0.582806i	−0.911030	+	0.412339i	$0.864712\pi$
$12$		0			0
$13$	1.55354	−	0.896938i	0.119503	−	0.0689952i	−0.439057	−	0.898459i	$-0.644687\pi$
$13$	1.55354	−	0.896938i	0.119503	−	0.0689952i	0.558560	+	0.829464i	$0.311354\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−30.9523			−1.82073			−0.910363	−	0.413811i	$-0.864197\pi$
$17$	−30.9523			−1.82073			−0.910363	+	0.413811i	$0.864197\pi$
$18$		0			0
$19$	−19.2201			−1.01158			−0.505792	−	0.862656i	$-0.668800\pi$
$19$	−19.2201			−1.01158			−0.505792	+	0.862656i	$0.668800\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.19201	−	2.06461i	−0.0518263	−	0.0897659i	0.838948	−	0.544211i	$-0.183171\pi$
$23$	−1.19201	−	2.06461i	−0.0518263	−	0.0897659i	−0.890775	+	0.454445i	$0.849838\pi$
$24$		0			0
$25$	14.6966	+	20.2240i	0.587865	+	0.808959i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	31.0770	+	17.9423i	1.07162	+	0.618701i	0.928624	−	0.371022i	$-0.120993\pi$
$29$	31.0770	+	17.9423i	1.07162	+	0.618701i	0.142998	+	0.989723i	$0.454326\pi$
$30$		0			0
$31$	20.4538	+	35.4270i	0.659800	+	1.14281i	0.980667	+	0.195683i	$0.0626924\pi$
$31$	20.4538	+	35.4270i	0.659800	+	1.14281i	−0.320867	+	0.947124i	$0.603974\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	9.36459	+	14.4187i	0.267560	+	0.411962i
$36$		0			0
$37$		−	53.6302i		−	1.44947i	−0.689030	−	0.724733i	$-0.741963\pi$
$37$		−	53.6302i		−	1.44947i	0.689030	−	0.724733i	$-0.258037\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−2.14983	+	1.24120i	−0.0524348	+	0.0302732i	−0.525988	−	0.850492i	$-0.676304\pi$
$41$	−2.14983	+	1.24120i	−0.0524348	+	0.0302732i	0.473553	+	0.880765i	$0.342971\pi$
$42$		0			0
$43$	47.7611	+	27.5749i	1.11072	+	0.641276i	0.939016	−	0.343873i	$-0.111739\pi$
$43$	47.7611	+	27.5749i	1.11072	+	0.641276i	0.171706	+	0.985148i	$0.445072\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	28.5250	−	49.4067i	0.606914	−	1.05121i	−0.384831	−	0.922987i	$-0.625740\pi$
$47$	28.5250	−	49.4067i	0.606914	−	1.05121i	0.991746	−	0.128220i	$-0.0409263\pi$
$48$		0			0
$49$	−18.5881	−	32.1956i	−0.379350	−	0.657053i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−19.4030			−0.366094			−0.183047	−	0.983104i	$-0.558596\pi$
$53$	−19.4030			−0.366094			−0.183047	+	0.983104i	$0.558596\pi$
$54$		0			0
$55$	34.9187	+	53.7643i	0.634886	+	0.977533i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−59.8830	+	34.5735i	−1.01497	+	0.585991i	−0.912642	−	0.408760i	$-0.865961\pi$
$59$	−59.8830	+	34.5735i	−1.01497	+	0.585991i	−0.102324	+	0.994751i	$0.532628\pi$
$60$		0			0
$61$	−8.80471	+	15.2502i	−0.144340	+	0.250003i	−0.929126	−	0.369762i	$-0.879439\pi$
$61$	−8.80471	+	15.2502i	−0.144340	+	0.250003i	0.784787	+	0.619766i	$0.212772\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−8.95707	+	0.469861i	−0.137801	+	0.00722864i
$66$		0			0
$67$	44.3133	−	25.5843i	0.661393	−	0.381855i	−0.131415	−	0.991327i	$-0.541952\pi$
$67$	44.3133	−	25.5843i	0.661393	−	0.381855i	0.792807	+	0.609472i	$0.208619\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	53.9194i		−	0.759428i	−0.925104	−	0.379714i	$-0.876022\pi$
$71$		−	53.9194i		−	0.759428i	0.925104	−	0.379714i	$-0.123978\pi$
$72$		0			0
$73$			42.7832i			0.586072i	0.956101	+	0.293036i	$0.0946655\pi$
$73$			42.7832i			0.586072i	−0.956101	+	0.293036i	$0.905334\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	22.0442	+	38.1816i	0.286288	+	0.495865i
$78$		0			0
$79$	−44.4228	+	76.9425i	−0.562314	+	0.973956i	0.434980	+	0.900440i	$0.356755\pi$
$79$	−44.4228	+	76.9425i	−0.562314	+	0.973956i	−0.997294	+	0.0735161i	$0.976578\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	14.0472	−	24.3305i	0.169243	−	0.293138i	−0.768911	−	0.639356i	$-0.779201\pi$
$83$	14.0472	−	24.3305i	0.169243	−	0.293138i	0.938154	+	0.346218i	$0.112534\pi$
$84$		0			0
$85$	137.897	+	70.2536i	1.62232	+	0.826512i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	68.3114i		−	0.767544i	−0.923428	−	0.383772i	$-0.874625\pi$
$89$		−	68.3114i		−	0.767544i	0.923428	−	0.383772i	$-0.125375\pi$
$90$		0			0
$91$	−6.16836			−0.0677842
$92$		0			0
$93$		0			0
$94$		0			0
$95$	85.6282	+	43.6245i	0.901350	+	0.459205i
$96$		0			0
$97$	135.186	+	78.0498i	1.39367	+	0.804637i	0.993720	−	0.111899i	$-0.0356933\pi$
$97$	135.186	+	78.0498i	1.39367	+	0.804637i	0.399952	+	0.916536i	$0.369027\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	60.5918	+	34.9827i	0.599919	+	0.346364i	0.769010	−	0.639237i	$-0.220750\pi$
$101$	60.5918	+	34.9827i	0.599919	+	0.346364i	−0.169091	+	0.985601i	$0.554083\pi$
$102$		0			0
$103$	166.405	−	96.0740i	1.61558	−	0.932757i	0.627539	−	0.778585i	$-0.284062\pi$
$103$	166.405	−	96.0740i	1.61558	−	0.932757i	0.988044	−	0.154172i	$-0.0492710\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−78.5826			−0.734417			−0.367209	−	0.930139i	$-0.619686\pi$
$107$	−78.5826			−0.734417			−0.367209	+	0.930139i	$0.619686\pi$
$108$		0			0
$109$	3.54619			0.0325339			0.0162669	−	0.999868i	$-0.494822\pi$
$109$	3.54619			0.0325339			0.0162669	+	0.999868i	$0.494822\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	106.243	+	184.018i	0.940201	+	1.62848i	0.765086	+	0.643928i	$0.222696\pi$
$113$	106.243	+	184.018i	0.940201	+	1.62848i	0.175115	+	0.984548i	$0.443970\pi$
$114$		0			0
$115$	0.624433	+	11.9037i	0.00542985	+	0.103510i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	92.1725	+	53.2158i	0.774559	+	0.447192i
$120$		0			0
$121$	21.6983	+	37.5826i	0.179325	+	0.310600i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−19.5725	−	123.458i	−0.156580	−	0.987665i
$126$		0			0
$127$			183.486i			1.44477i	0.691489	+	0.722387i	$0.256955\pi$
$127$			183.486i			1.44477i	−0.691489	+	0.722387i	$0.743045\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−52.5451	+	30.3369i	−0.401107	+	0.231579i	−0.686962	−	0.726694i	$-0.741056\pi$
$131$	−52.5451	+	30.3369i	−0.401107	+	0.231579i	0.285854	+	0.958273i	$0.407723\pi$
$132$		0			0
$133$	57.2352	+	33.0448i	0.430340	+	0.248457i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−6.23685	+	10.8025i	−0.0455244	+	0.0788506i	−0.887890	−	0.460056i	$-0.847829\pi$
$137$	−6.23685	+	10.8025i	−0.0455244	+	0.0788506i	0.842365	+	0.538907i	$0.181163\pi$
$138$		0			0
$139$	36.8400	+	63.8088i	0.265036	+	0.459056i	0.967573	−	0.252591i	$-0.0812828\pi$
$139$	36.8400	+	63.8088i	0.265036	+	0.459056i	−0.702537	+	0.711647i	$0.747949\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−23.0006			−0.160843
$144$		0			0
$145$	−97.7283	−	150.472i	−0.673989	−	1.03774i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	4.02350	−	2.32297i	0.0270033	−	0.0155904i	−0.486438	−	0.873715i	$-0.661704\pi$
$149$	4.02350	−	2.32297i	0.0270033	−	0.0155904i	0.513441	+	0.858125i	$0.328371\pi$
$150$		0			0
$151$	−118.510	+	205.265i	−0.784835	+	1.35937i	0.144263	+	0.989539i	$0.453919\pi$
$151$	−118.510	+	205.265i	−0.784835	+	1.35937i	−0.929098	+	0.369835i	$0.879414\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−10.7147	−	204.257i	−0.0691273	−	1.31779i
$156$		0			0
$157$	136.484	−	78.7991i	0.869325	−	0.501905i	0.00220122	−	0.999998i	$-0.499299\pi$
$157$	136.484	−	78.7991i	0.869325	−	0.501905i	0.867124	+	0.498092i	$0.165966\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			8.19758i			0.0509166i
$162$		0			0
$163$		−	82.9389i		−	0.508828i	−0.967095	−	0.254414i	$-0.918117\pi$
$163$		−	82.9389i		−	0.508828i	0.967095	−	0.254414i	$-0.0818826\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−45.2172	−	78.3185i	−0.270762	−	0.468973i	0.698295	−	0.715810i	$-0.253942\pi$
$167$	−45.2172	−	78.3185i	−0.270762	−	0.468973i	−0.969057	+	0.246837i	$0.920609\pi$
$168$		0			0
$169$	−82.8910	+	143.571i	−0.490479	+	0.849535i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	129.905	−	225.003i	0.750898	−	1.30059i	−0.196490	−	0.980506i	$-0.562954\pi$
$173$	129.905	−	225.003i	0.750898	−	1.30059i	0.947388	−	0.320088i	$-0.103712\pi$
$174$		0			0
$175$	−8.99411	−	85.4923i	−0.0513949	−	0.488528i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			228.604i			1.27712i	0.769573	+	0.638559i	$0.220469\pi$
$179$			228.604i			1.27712i	−0.769573	+	0.638559i	$0.779531\pi$
$180$		0			0
$181$	30.8901			0.170664			0.0853318	−	0.996353i	$-0.472805\pi$
$181$	30.8901			0.170664			0.0853318	+	0.996353i	$0.472805\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−121.726	+	238.930i	−0.657980	+	1.29152i
$186$		0			0
$187$	343.693	+	198.431i	1.83793	+	1.06113i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	2.27714	+	1.31471i	0.0119222	+	0.00688330i	0.505949	−	0.862563i	$-0.331142\pi$
$191$	2.27714	+	1.31471i	0.0119222	+	0.00688330i	−0.494027	+	0.869447i	$0.664476\pi$
$192$		0			0
$193$	18.9076	−	10.9163i	0.0979666	−	0.0565610i	−0.450216	−	0.892920i	$-0.648653\pi$
$193$	18.9076	−	10.9163i	0.0979666	−	0.0565610i	0.548183	+	0.836359i	$0.315320\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−68.8358			−0.349421			−0.174710	−	0.984620i	$-0.555899\pi$
$197$	−68.8358			−0.349421			−0.174710	+	0.984620i	$0.555899\pi$
$198$		0			0
$199$	116.185			0.583846			0.291923	−	0.956442i	$-0.405705\pi$
$199$	116.185			0.583846			0.291923	+	0.956442i	$0.405705\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−61.6959	−	106.860i	−0.303921	−	0.526406i
$204$		0			0
$205$	12.3950	−	0.650205i	0.0604633	−	0.00317173i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	213.419	+	123.217i	1.02114	+	0.589557i
$210$		0			0
$211$	112.307	+	194.521i	0.532258	+	0.921898i	0.999291	+	0.0376582i	$0.0119898\pi$
$211$	112.307	+	194.521i	0.532258	+	0.921898i	−0.467032	+	0.884240i	$0.654677\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−150.195	−	231.255i	−0.698580	−	1.07560i
$216$		0			0
$217$		−	140.664i		−	0.648219i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−48.0858	+	27.7623i	−0.217583	+	0.125621i
$222$		0			0
$223$	219.753	+	126.874i	0.985439	+	0.568944i	0.903908	−	0.427727i	$-0.140686\pi$
$223$	219.753	+	126.874i	0.985439	+	0.568944i	0.0815313	+	0.996671i	$0.474019\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	27.7996	−	48.1503i	0.122465	−	0.212116i	−0.798274	−	0.602294i	$-0.794253\pi$
$227$	27.7996	−	48.1503i	0.122465	−	0.212116i	0.920739	+	0.390179i	$0.127587\pi$
$228$		0			0
$229$	111.391	+	192.935i	0.486423	+	0.842509i	0.999878	−	0.0156074i	$-0.00496819\pi$
$229$	111.391	+	192.935i	0.486423	+	0.842509i	−0.513456	+	0.858116i	$0.671635\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−399.458			−1.71441			−0.857207	−	0.514972i	$-0.827802\pi$
$233$	−399.458			−1.71441			−0.857207	+	0.514972i	$0.827802\pi$
$234$		0			0
$235$	−239.223	+	155.370i	−1.01797	+	0.661149i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	17.4959	−	10.1013i	0.0732046	−	0.0422647i	−0.462951	−	0.886384i	$-0.653209\pi$
$239$	17.4959	−	10.1013i	0.0732046	−	0.0422647i	0.536156	+	0.844119i	$0.319876\pi$
$240$		0			0
$241$	168.299	−	291.502i	0.698336	−	1.20955i	−0.270707	−	0.962662i	$-0.587257\pi$
$241$	168.299	−	291.502i	0.698336	−	1.20955i	0.969043	−	0.246892i	$-0.0794092\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	9.73740	+	185.626i	0.0397445	+	0.757658i
$246$		0			0
$247$	−29.8592	+	17.2392i	−0.120887	+	0.0697944i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	280.569i		−	1.11781i	−0.829233	−	0.558903i	$-0.811222\pi$
$251$		−	280.569i		−	1.11781i	0.829233	−	0.558903i	$-0.188778\pi$
$252$		0			0
$253$			30.5671i			0.120819i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	82.1464	+	142.282i	0.319636	+	0.553625i	0.980412	−	0.196958i	$-0.0631062\pi$
$257$	82.1464	+	142.282i	0.319636	+	0.553625i	−0.660776	+	0.750583i	$0.729773\pi$
$258$		0			0
$259$	−92.2056	+	159.705i	−0.356006	+	0.616621i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−147.024	+	254.654i	−0.559028	+	0.968266i	0.438550	+	0.898707i	$0.355492\pi$
$263$	−147.024	+	254.654i	−0.559028	+	0.968266i	−0.997578	+	0.0695585i	$0.977841\pi$
$264$		0			0
$265$	86.4430	+	44.0396i	0.326200	+	0.166187i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			373.214i			1.38741i	0.720259	+	0.693706i	$0.244023\pi$
$269$			373.214i			1.38741i	−0.720259	+	0.693706i	$0.755977\pi$
$270$		0			0
$271$	−388.909			−1.43509			−0.717543	−	0.696514i	$-0.754733\pi$
$271$	−388.909			−1.43509			−0.717543	+	0.696514i	$0.754733\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−33.5372	−	318.784i	−0.121954	−	1.15921i
$276$		0			0
$277$	−321.698	−	185.732i	−1.16136	−	0.670514i	−0.209733	−	0.977759i	$-0.567260\pi$
$277$	−321.698	−	185.732i	−1.16136	−	0.670514i	−0.951630	+	0.307245i	$0.900593\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−234.725	−	135.519i	−0.835321	−	0.482273i	0.0203504	−	0.999793i	$-0.493522\pi$
$281$	−234.725	−	135.519i	−0.835321	−	0.482273i	−0.855671	+	0.517520i	$0.826855\pi$
$282$		0			0
$283$	28.8742	−	16.6705i	0.102029	−	0.0589065i	−0.448117	−	0.893975i	$-0.647905\pi$
$283$	28.8742	−	16.6705i	0.102029	−	0.0589065i	0.550146	+	0.835068i	$0.314572\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	8.53591			0.0297419
$288$		0			0
$289$	669.047			2.31504
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−59.5109	−	103.076i	−0.203109	−	0.351795i	0.746420	−	0.665476i	$-0.231771\pi$
$293$	−59.5109	−	103.076i	−0.203109	−	0.351795i	−0.949529	+	0.313681i	$0.898438\pi$
$294$		0			0
$295$	345.260	−	18.1113i	1.17037	−	0.0613943i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−3.70366	−	2.13831i	−0.0123868	−	0.00715154i
$300$		0			0
$301$	−94.8180	−	164.230i	−0.315010	−	0.545613i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	73.8402	−	47.9575i	0.242099	−	0.157238i
$306$		0			0
$307$		−	63.9942i		−	0.208450i	−0.994554	−	0.104225i	$-0.966764\pi$
$307$		−	63.9942i		−	0.208450i	0.994554	−	0.104225i	$-0.0332362\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−92.3321	+	53.3079i	−0.296888	+	0.171408i	−0.641044	−	0.767504i	$-0.721498\pi$
$311$	−92.3321	+	53.3079i	−0.296888	+	0.171408i	0.344156	+	0.938912i	$0.388165\pi$
$312$		0			0
$313$	−189.421	−	109.362i	−0.605179	−	0.349401i	0.165897	−	0.986143i	$-0.446948\pi$
$313$	−189.421	−	109.362i	−0.605179	−	0.349401i	−0.771076	+	0.636743i	$0.780281\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	176.345	−	305.438i	0.556292	−	0.963526i	−0.441510	−	0.897256i	$-0.645557\pi$
$317$	176.345	−	305.438i	0.556292	−	0.963526i	0.997802	−	0.0662695i	$-0.0211097\pi$
$318$		0			0
$319$	−230.052	−	398.461i	−0.721165	−	1.24909i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	594.906			1.84182
$324$		0			0
$325$	40.9715	+	18.2369i	0.126066	+	0.0561134i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−169.888	+	98.0850i	−0.516378	+	0.298131i
$330$		0			0
$331$	−30.3510	+	52.5695i	−0.0916949	+	0.158820i	−0.908224	−	0.418483i	$-0.862562\pi$
$331$	−30.3510	+	52.5695i	−0.0916949	+	0.158820i	0.816529	+	0.577304i	$0.195895\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−255.492	+	13.4024i	−0.762662	+	0.0400070i
$336$		0			0
$337$	19.1598	−	11.0619i	0.0568539	−	0.0328246i	−0.471304	−	0.881971i	$-0.656216\pi$
$337$	19.1598	−	11.0619i	0.0568539	−	0.0328246i	0.528158	+	0.849146i	$0.322883\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	524.506i		−	1.53814i
$342$		0			0
$343$			296.323i			0.863915i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	174.972	+	303.060i	0.504241	+	0.873371i	0.999988	+	0.00490371i	$0.00156091\pi$
$347$	174.972	+	303.060i	0.504241	+	0.873371i	−0.495747	+	0.868467i	$0.665106\pi$
$348$		0			0
$349$	−140.703	+	243.705i	−0.403161	+	0.698296i	−0.994106	−	0.108416i	$-0.965422\pi$
$349$	−140.703	+	243.705i	−0.403161	+	0.698296i	0.590944	+	0.806712i	$0.298755\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	79.9931	−	138.552i	0.226609	−	0.392499i	−0.730192	−	0.683242i	$-0.760569\pi$
$353$	79.9931	−	138.552i	0.226609	−	0.392499i	0.956801	+	0.290743i	$0.0939026\pi$
$354$		0			0
$355$	−122.383	+	240.219i	−0.344740	+	0.676673i
$356$		0			0
$357$		0			0
$358$		0			0
$359$			643.967i			1.79378i	0.442254	+	0.896890i	$0.354179\pi$
$359$			643.967i			1.79378i	−0.442254	+	0.896890i	$0.645821\pi$
$360$		0			0
$361$	8.41153			0.0233006
$362$		0			0
$363$		0			0
$364$		0			0
$365$	97.1065	−	190.605i	0.266045	−	0.522207i
$366$		0			0
$367$	−536.920	−	309.991i	−1.46300	−	0.844663i	−0.463850	−	0.885914i	$-0.653532\pi$
$367$	−536.920	−	309.991i	−1.46300	−	0.844663i	−0.999149	+	0.0412512i	$0.986866\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	57.7798	+	33.3592i	0.155741	+	0.0899169i
$372$		0			0
$373$	−123.864	+	71.5129i	−0.332075	+	0.191724i	−0.656762	−	0.754098i	$-0.728074\pi$
$373$	−123.864	+	71.5129i	−0.332075	+	0.191724i	0.324687	+	0.945822i	$0.394741\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	64.3726			0.170750
$378$		0			0
$379$	−57.1486			−0.150788			−0.0753940	−	0.997154i	$-0.524021\pi$
$379$	−57.1486			−0.150788			−0.0753940	+	0.997154i	$0.524021\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	145.454	+	251.933i	0.379775	+	0.657789i	0.991029	−	0.133645i	$-0.0426681\pi$
$383$	145.454	+	251.933i	0.379775	+	0.657789i	−0.611254	+	0.791434i	$0.709335\pi$
$384$		0			0
$385$	−11.5479	−	220.139i	−0.0299944	−	0.571790i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	566.472	+	327.053i	1.45623	+	0.840753i	0.998823	−	0.0485060i	$-0.0154460\pi$
$389$	566.472	+	327.053i	1.45623	+	0.840753i	0.457404	+	0.889259i	$0.348779\pi$
$390$		0			0
$391$	36.8954	+	63.9046i	0.0943615	+	0.163439i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	372.549	−	241.962i	0.943162	−	0.612563i
$396$		0			0
$397$		−	229.991i		−	0.579323i	−0.957129	−	0.289661i	$-0.906457\pi$
$397$		−	229.991i		−	0.579323i	0.957129	−	0.289661i	$-0.0935427\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	442.493	−	255.474i	1.10347	−	0.637091i	0.166343	−	0.986068i	$-0.446804\pi$
$401$	442.493	−	255.474i	1.10347	−	0.637091i	0.937131	+	0.348977i	$0.113471\pi$
$402$		0			0
$403$	63.5517	+	36.6916i	0.157697	+	0.0910461i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−343.816	+	595.507i	−0.844757	+	1.46316i
$408$		0			0
$409$	−299.632	−	518.979i	−0.732598	−	1.26890i	−0.955769	−	0.294117i	$-0.904974\pi$
$409$	−299.632	−	518.979i	−0.732598	−	1.26890i	0.223172	−	0.974779i	$-0.428359\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	237.766			0.575705
$414$		0			0
$415$	−117.806	+	76.5123i	−0.283870	+	0.184367i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	525.674	−	303.498i	1.25459	−	0.724339i	0.282574	−	0.959245i	$-0.408812\pi$
$419$	525.674	−	303.498i	1.25459	−	0.724339i	0.972018	+	0.234906i	$0.0754783\pi$
$420$		0			0
$421$	226.077	−	391.578i	0.537001	−	0.930113i	−0.462062	−	0.886847i	$-0.652890\pi$
$421$	226.077	−	391.578i	0.537001	−	0.930113i	0.999064	−	0.0432659i	$-0.0137763\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−454.895	−	625.979i	−1.07034	−	1.47289i
$426$		0			0
$427$	52.4389	−	30.2756i	0.122808	−	0.0709030i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			684.701i			1.58863i	0.607505	+	0.794316i	$0.292171\pi$
$431$			684.701i			1.58863i	−0.607505	+	0.794316i	$0.707829\pi$
$432$		0			0
$433$		−	458.058i		−	1.05787i	−0.848662	−	0.528936i	$-0.822591\pi$
$433$		−	458.058i		−	1.05787i	0.848662	−	0.528936i	$-0.177409\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	22.9104	+	39.6821i	0.0524267	+	0.0908056i
$438$		0			0
$439$	−6.07259	+	10.5180i	−0.0138328	+	0.0239591i	−0.872859	−	0.487973i	$-0.837737\pi$
$439$	−6.07259	+	10.5180i	−0.0138328	+	0.0239591i	0.859026	+	0.511932i	$0.171070\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−237.224	+	410.885i	−0.535495	+	0.927505i	0.463644	+	0.886022i	$0.346542\pi$
$443$	−237.224	+	410.885i	−0.535495	+	0.927505i	−0.999139	+	0.0414836i	$0.986792\pi$
$444$		0			0
$445$	−155.049	+	304.337i	−0.348424	+	0.683904i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			269.662i			0.600583i	0.953847	+	0.300291i	$0.0970840\pi$
$449$			269.662i			0.600583i	−0.953847	+	0.300291i	$0.902916\pi$
$450$		0			0
$451$	31.8287			0.0705737
$452$		0			0
$453$		0			0
$454$		0			0
$455$	27.4809	+	14.0005i	0.0603977	+	0.0307704i
$456$		0			0
$457$	−483.695	−	279.261i	−1.05841	−	0.611075i	−0.133420	−	0.991060i	$-0.542596\pi$
$457$	−483.695	−	279.261i	−1.05841	−	0.611075i	−0.924993	+	0.379985i	$0.875929\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−606.316	−	350.056i	−1.31522	−	0.759341i	−0.332263	−	0.943187i	$-0.607812\pi$
$461$	−606.316	−	350.056i	−1.31522	−	0.759341i	−0.982955	+	0.183845i	$0.941145\pi$
$462$		0			0
$463$	−694.612	+	401.035i	−1.50024	+	0.866165i	−0.500242	+	0.865885i	$0.666756\pi$
$463$	−694.612	+	401.035i	−1.50024	+	0.866165i	−1.00000		0.000279830i	$0.999911\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	868.960			1.86073			0.930364	−	0.366636i	$-0.119491\pi$
$467$	868.960			1.86073			0.930364	+	0.366636i	$0.119491\pi$
$468$		0			0
$469$	−175.947			−0.375153
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−353.557	−	612.379i	−0.747478	−	1.29467i
$474$		0			0
$475$	−282.470	−	388.707i	−0.594674	−	0.818330i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	610.797	+	352.644i	1.27515	+	0.736208i	0.975953	−	0.217983i	$-0.0699477\pi$
$479$	610.797	+	352.644i	1.27515	+	0.736208i	0.299197	+	0.954191i	$0.403281\pi$
$480$		0			0
$481$	−48.1030	−	83.3169i	−0.100006	−	0.173216i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−425.122	−	654.560i	−0.876540	−	1.34961i
$486$		0			0
$487$			715.753i			1.46972i	0.678220	+	0.734859i	$0.262752\pi$
$487$			715.753i			1.46972i	−0.678220	+	0.734859i	$0.737248\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	87.8066	−	50.6952i	0.178832	−	0.103249i	−0.407912	−	0.913021i	$-0.633743\pi$
$491$	87.8066	−	50.6952i	0.178832	−	0.103249i	0.586744	+	0.809773i	$0.300410\pi$
$492$		0			0
$493$	−961.907	−	555.357i	−1.95113	−	1.12648i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−92.7027	+	160.566i	−0.186525	+	0.323070i
$498$		0			0
$499$	−468.963	−	812.267i	−0.939805	−	1.62779i	−0.765833	−	0.643040i	$-0.777673\pi$
$499$	−468.963	−	812.267i	−0.939805	−	1.62779i	−0.173973	−	0.984751i	$-0.555660\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−766.791			−1.52444			−0.762218	−	0.647321i	$-0.775889\pi$
$503$	−766.791			−1.52444			−0.762218	+	0.647321i	$0.775889\pi$
$504$		0			0
$505$	−190.544	−	293.380i	−0.377315	−	0.580951i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−647.269	+	373.701i	−1.27165	+	0.734186i	−0.975298	−	0.220893i	$-0.929103\pi$
$509$	−647.269	+	373.701i	−1.27165	+	0.734186i	−0.296350	+	0.955079i	$0.595769\pi$
$510$		0			0
$511$	73.5565	−	127.404i	0.143946	−	0.249322i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−959.421	+	50.3284i	−1.86295	+	0.0977251i
$516$		0			0
$517$	−633.479	+	365.739i	−1.22530	+	0.707426i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	39.7011i		−	0.0762018i	−0.999274	−	0.0381009i	$-0.987869\pi$
$521$		−	39.7011i		−	0.0762018i	0.999274	−	0.0381009i	$-0.0121308\pi$
$522$		0			0
$523$			135.722i			0.259506i	0.991546	+	0.129753i	$0.0414184\pi$
$523$			135.722i			0.259506i	−0.991546	+	0.129753i	$0.958582\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−633.093	−	1096.55i	−1.20132	−	2.08074i
$528$		0			0
$529$	261.658	−	453.205i	0.494628	−	0.856721i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−2.22656	+	3.85652i	−0.00417742	+	0.00723550i
$534$		0			0
$535$	350.097	+	178.362i	0.654387	+	0.333386i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			476.664i			0.884349i
$540$		0			0
$541$	840.659			1.55390			0.776949	−	0.629563i	$-0.216766\pi$
$541$	840.659			1.55390			0.776949	+	0.629563i	$0.216766\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−15.7988	−	8.04892i	−0.0289886	−	0.0147687i
$546$		0			0
$547$	819.259	+	472.999i	1.49773	+	0.864715i	0.999997	−	0.00261458i	$-0.000832248\pi$
$547$	819.259	+	472.999i	1.49773	+	0.864715i	0.497734	+	0.867330i	$0.334166\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−597.303	−	344.853i	−1.08403	−	0.625868i
$552$		0			0
$553$	264.572	−	152.751i	0.478430	−	0.276222i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−78.6518			−0.141206			−0.0706031	−	0.997504i	$-0.522492\pi$
$557$	−78.6518			−0.141206			−0.0706031	+	0.997504i	$0.522492\pi$
$558$		0			0
$559$	98.9317			0.176980
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−139.427	−	241.495i	−0.247651	−	0.428944i	0.715223	−	0.698897i	$-0.246325\pi$
$563$	−139.427	−	241.495i	−0.247651	−	0.428944i	−0.962874	+	0.269953i	$0.912992\pi$
$564$		0			0
$565$	−55.6553	−	1060.97i	−0.0985049	−	1.87782i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	504.075	+	291.028i	0.885895	+	0.511472i	0.872598	−	0.488440i	$-0.162434\pi$
$569$	504.075	+	291.028i	0.885895	+	0.511472i	0.0132978	+	0.999912i	$0.495767\pi$
$570$		0			0
$571$	153.625	+	266.087i	0.269046	+	0.466001i	0.968616	−	0.248563i	$-0.0799583\pi$
$571$	153.625	+	266.087i	0.269046	+	0.466001i	−0.699570	+	0.714564i	$0.746625\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	24.2363	−	54.4500i	0.0421500	−	0.0946956i
$576$		0			0
$577$			24.6333i			0.0426921i	0.999772	+	0.0213460i	$0.00679517\pi$
$577$			24.6333i			0.0426921i	−0.999772	+	0.0213460i	$0.993205\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−83.6619	+	48.3022i	−0.143996	+	0.0831363i
$582$		0			0
$583$	215.449	+	124.390i	0.369553	+	0.213362i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−10.2262	+	17.7123i	−0.0174211	+	0.0301743i	−0.874605	−	0.484837i	$-0.838879\pi$
$587$	−10.2262	+	17.7123i	−0.0174211	+	0.0301743i	0.857183	+	0.515011i	$0.172212\pi$
$588$		0			0
$589$	−393.124	−	680.910i	−0.667443	−	1.15604i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	676.813			1.14134			0.570669	−	0.821180i	$-0.306684\pi$
$593$	676.813			1.14134			0.570669	+	0.821180i	$0.306684\pi$
$594$		0			0
$595$	−289.856	−	446.291i	−0.487153	−	0.750069i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	47.0075	−	27.1398i	0.0784766	−	0.0453085i	−0.460248	−	0.887790i	$-0.652240\pi$
$599$	47.0075	−	27.1398i	0.0784766	−	0.0453085i	0.538725	+	0.842482i	$0.318906\pi$
$600$		0			0
$601$	142.651	−	247.078i	0.237355	−	0.411111i	−0.722599	−	0.691267i	$-0.757053\pi$
$601$	142.651	−	247.078i	0.237355	−	0.411111i	0.959955	+	0.280156i	$0.0903861\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−11.3667	−	216.685i	−0.0187879	−	0.358158i
$606$		0			0
$607$	−850.741	+	491.175i	−1.40155	+	0.809185i	−0.994552	−	0.104244i	$-0.966758\pi$
$607$	−850.741	+	491.175i	−1.40155	+	0.809185i	−0.406998	+	0.913429i	$0.633424\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	102.341i		−	0.167497i
$612$		0			0
$613$		−	850.701i		−	1.38777i	−0.720087	−	0.693883i	$-0.755898\pi$
$613$		−	850.701i		−	1.38777i	0.720087	−	0.693883i	$-0.244102\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−64.4265	−	111.590i	−0.104419	−	0.180859i	0.809082	−	0.587696i	$-0.199965\pi$
$617$	−64.4265	−	111.590i	−0.104419	−	0.180859i	−0.913501	+	0.406837i	$0.866632\pi$
$618$		0			0
$619$	−452.830	+	784.325i	−0.731552	+	1.26708i	0.224668	+	0.974435i	$0.427870\pi$
$619$	−452.830	+	784.325i	−0.731552	+	1.26708i	−0.956220	+	0.292649i	$0.905463\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−117.447	+	203.424i	−0.188518	+	0.326523i
$624$		0			0
$625$	−193.019	+	594.448i	−0.308830	+	0.951117i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			1659.98i			2.63908i
$630$		0			0
$631$	−1083.23			−1.71669			−0.858345	−	0.513073i	$-0.828507\pi$
$631$	−1083.23			−1.71669			−0.858345	+	0.513073i	$0.828507\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	416.465	−	817.458i	0.655850	−	1.28734i
$636$		0			0
$637$	−57.7549	−	33.3448i	−0.0906670	−	0.0523466i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	875.815	+	505.652i	1.36633	+	0.788848i	0.990457	−	0.137825i	$-0.0440110\pi$
$641$	875.815	+	505.652i	1.36633	+	0.788848i	0.375869	+	0.926673i	$0.377344\pi$
$642$		0			0
$643$	−984.875	+	568.618i	−1.53169	+	0.884320i	−0.532403	+	0.846491i	$0.678711\pi$
$643$	−984.875	+	568.618i	−1.53169	+	0.884320i	−0.999284	+	0.0378292i	$0.987956\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	273.788			0.423166			0.211583	−	0.977360i	$-0.432138\pi$
$647$	273.788			0.423166			0.211583	+	0.977360i	$0.432138\pi$
$648$		0			0
$649$	886.583			1.36608
$650$		0			0
$651$		0			0
$652$		0			0
$653$	200.616	+	347.478i	0.307223	+	0.532125i	0.977754	−	0.209756i	$-0.0672670\pi$
$653$	200.616	+	347.478i	0.307223	+	0.532125i	−0.670531	+	0.741882i	$0.733934\pi$
$654$		0			0
$655$	302.953	−	15.8920i	0.462523	−	0.0242626i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	323.745	+	186.914i	0.491267	+	0.283633i	0.725100	−	0.688644i	$-0.241794\pi$
$659$	323.745	+	186.914i	0.491267	+	0.283633i	−0.233833	+	0.972277i	$0.575127\pi$
$660$		0			0
$661$	−65.4788	−	113.413i	−0.0990602	−	0.171577i	0.812236	−	0.583329i	$-0.198250\pi$
$661$	−65.4788	−	113.413i	−0.0990602	−	0.171577i	−0.911296	+	0.411752i	$0.864917\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−179.988	−	277.128i	−0.270659	−	0.416734i
$666$		0			0
$667$		−	85.5494i		−	0.128260i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	195.534	−	112.892i	0.291407	−	0.168244i
$672$		0			0
$673$	−603.505	−	348.434i	−0.896738	−	0.517732i	−0.0205979	−	0.999788i	$-0.506557\pi$
$673$	−603.505	−	348.434i	−0.896738	−	0.517732i	−0.876141	+	0.482056i	$0.839890\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−246.437	+	426.841i	−0.364013	+	0.630489i	−0.988617	−	0.150453i	$-0.951927\pi$
$677$	−246.437	+	426.841i	−0.364013	+	0.630489i	0.624604	+	0.780941i	$0.285260\pi$
$678$		0			0
$679$	−268.379	−	464.847i	−0.395257	−	0.684605i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	628.686			0.920477			0.460238	−	0.887795i	$-0.347764\pi$
$683$	628.686			0.920477			0.460238	+	0.887795i	$0.347764\pi$
$684$		0			0
$685$	52.3049	−	33.9709i	0.0763576	−	0.0495925i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−30.1433	+	17.4033i	−0.0437494	+	0.0252587i
$690$		0			0
$691$	617.313	−	1069.22i	0.893362	−	1.54735i	0.0575441	−	0.998343i	$-0.481673\pi$
$691$	617.313	−	1069.22i	0.893362	−	1.54735i	0.835818	−	0.549006i	$-0.184994\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−19.2987	−	367.894i	−0.0277679	−	0.529344i
$696$		0			0
$697$	66.5421	−	38.4181i	0.0954694	−	0.0551193i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	834.094i		−	1.18986i	−0.803777	−	0.594931i	$-0.797179\pi$
$701$		−	834.094i		−	1.18986i	0.803777	−	0.594931i	$-0.202821\pi$
$702$		0			0
$703$			1030.78i			1.46626i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−120.290	−	208.349i	−0.170142	−	0.294695i
$708$		0			0
$709$	−51.1022	+	88.5117i	−0.0720765	+	0.124840i	−0.899811	−	0.436279i	$-0.856296\pi$
$709$	−51.1022	+	88.5117i	−0.0720765	+	0.124840i	0.827735	+	0.561120i	$0.189629\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	48.7621	−	84.4584i	0.0683901	−	0.118455i
$714$		0			0
$715$	102.471	+	52.2052i	0.143316	+	0.0730143i
$716$		0			0
$717$		0			0
$718$		0			0
$719$			593.832i			0.825913i	0.910751	+	0.412957i	$0.135504\pi$
$719$			593.832i			0.825913i	−0.910751	+	0.412957i	$0.864496\pi$
$720$		0			0
$721$	−660.714			−0.916385
$722$		0			0
$723$		0			0
$724$		0			0
$725$	93.8619	+	892.193i	0.129465	+	1.23061i
$726$		0			0
$727$	−580.916	−	335.392i	−0.799059	−	0.461337i	0.0440827	−	0.999028i	$-0.485963\pi$
$727$	−580.916	−	335.392i	−0.799059	−	0.461337i	−0.843142	+	0.537691i	$0.819297\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−1478.32	−	853.506i	−2.02232	−	1.16759i
$732$		0			0
$733$	729.510	−	421.183i	0.995239	−	0.574602i	0.0884030	−	0.996085i	$-0.471824\pi$
$733$	729.510	−	421.183i	0.995239	−	0.574602i	0.906836	+	0.421483i	$0.138490\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−656.070			−0.890190
$738$		0			0
$739$	1423.99			1.92692			0.963460	−	0.267853i	$-0.0863141\pi$
$739$	1423.99			1.92692			0.963460	+	0.267853i	$0.0863141\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−275.646	−	477.432i	−0.370990	−	0.642574i	0.618728	−	0.785605i	$-0.287648\pi$
$743$	−275.646	−	477.432i	−0.370990	−	0.642574i	−0.989718	+	0.143032i	$0.954315\pi$
$744$		0			0
$745$	−23.1978	+	1.21689i	−0.0311380	+	0.00163341i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	234.010	+	135.106i	0.312430	+	0.180382i
$750$		0			0
$751$	−142.586	−	246.967i	−0.189862	−	0.328851i	0.755342	−	0.655331i	$-0.227471\pi$
$751$	−142.586	−	246.967i	−0.189862	−	0.328851i	−0.945204	+	0.326480i	$0.894137\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	993.877	−	645.501i	1.31639	−	0.854968i
$756$		0			0
$757$		−	787.273i		−	1.03999i	−0.854169	−	0.519995i	$-0.825934\pi$
$757$		−	787.273i		−	1.03999i	0.854169	−	0.519995i	$-0.174066\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	87.6204	−	50.5876i	0.115138	−	0.0664752i	−0.441325	−	0.897347i	$-0.645491\pi$
$761$	87.6204	−	50.5876i	0.115138	−	0.0664752i	0.556463	+	0.830872i	$0.312158\pi$
$762$		0			0
$763$	−10.5602	−	6.09691i	−0.0138403	−	0.00799071i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−62.0205	+	107.423i	−0.0808612	+	0.140056i
$768$		0			0
$769$	1.84498	+	3.19561i	0.00239920	+	0.00415554i	0.867223	−	0.497921i	$-0.165903\pi$
$769$	1.84498	+	3.19561i	0.00239920	+	0.00415554i	−0.864823	+	0.502076i	$0.832570\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−504.260			−0.652341			−0.326171	−	0.945311i	$-0.605758\pi$
$773$	−504.260			−0.652341			−0.326171	+	0.945311i	$0.605758\pi$
$774$		0			0
$775$	−415.874	+	934.315i	−0.536611	+	1.20557i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	41.3198	−	23.8560i	0.0530422	−	0.0306239i
$780$		0			0
$781$	−345.670	+	598.718i	−0.442599	+	0.766604i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−786.909	+	41.2789i	−1.00243	+	0.0525846i
$786$		0			0
$787$	1223.57	−	706.429i	1.55473	−	0.897622i	0.556981	−	0.830525i	$-0.311960\pi$
$787$	1223.57	−	706.429i	1.55473	−	0.897622i	0.997746	−	0.0670969i	$-0.0213737\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	730.645i		−	0.923698i
$792$		0			0
$793$			31.5891i			0.0398350i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	241.383	+	418.088i	0.302865	+	0.524577i	0.976784	−	0.214228i	$-0.0687236\pi$
$797$	241.383	+	418.088i	0.302865	+	0.524577i	−0.673919	+	0.738805i	$0.735390\pi$
$798$		0			0
$799$	−882.915	+	1529.25i	−1.10502	+	1.91396i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	274.277	−	475.063i	0.341566	−	0.591610i
$804$		0			0
$805$	18.6063	−	36.5214i	0.0231134	−	0.0453682i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			874.774i			1.08130i	0.841247	+	0.540652i	$0.181822\pi$
$809$			874.774i			1.08130i	−0.841247	+	0.540652i	$0.818178\pi$
$810$		0			0
$811$	−568.734			−0.701275			−0.350638	−	0.936511i	$-0.614035\pi$
$811$	−568.734			−0.701275			−0.350638	+	0.936511i	$0.614035\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−188.249	+	369.505i	−0.230981	+	0.453380i
$816$		0			0
$817$	−917.971	−	529.991i	−1.12359	−	0.648704i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−128.787	−	74.3550i	−0.156866	−	0.0905664i	0.419512	−	0.907750i	$-0.362201\pi$
$821$	−128.787	−	74.3550i	−0.156866	−	0.0905664i	−0.576378	+	0.817183i	$0.695534\pi$
$822$		0			0
$823$	437.829	−	252.780i	0.531991	−	0.307145i	−0.209836	−	0.977737i	$-0.567293\pi$
$823$	437.829	−	252.780i	0.531991	−	0.307145i	0.741827	+	0.670591i	$0.233960\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−980.267			−1.18533			−0.592664	−	0.805450i	$-0.701924\pi$
$827$	−980.267			−1.18533			−0.592664	+	0.805450i	$0.701924\pi$
$828$		0			0
$829$	190.269			0.229516			0.114758	−	0.993393i	$-0.463391\pi$
$829$	190.269			0.229516			0.114758	+	0.993393i	$0.463391\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	575.346	+	996.529i	0.690692	+	1.19631i
$834$		0			0
$835$	23.6871	+	451.551i	0.0283677	+	0.540780i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1021.86	+	589.969i	1.21795	+	0.703181i	0.964478	−	0.264163i	$-0.0850956\pi$
$839$	1021.86	+	589.969i	1.21795	+	0.703181i	0.253467	+	0.967344i	$0.418429\pi$
$840$		0			0
$841$	223.354	+	386.861i	0.265582	+	0.460002i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	695.160	−	451.491i	0.822675	−	0.534309i
$846$		0			0
$847$		−	149.222i		−	0.176177i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−110.726	+	63.9276i	−0.130113	+	0.0751205i
$852$		0			0
$853$	970.251	+	560.175i	1.13746	+	0.656711i	0.945800	−	0.324751i	$-0.105280\pi$
$853$	970.251	+	560.175i	1.13746	+	0.656711i	0.191657	+	0.981462i	$0.438614\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	650.362	−	1126.46i	0.758882	−	1.31442i	−0.184540	−	0.982825i	$-0.559079\pi$
$857$	650.362	−	1126.46i	0.758882	−	1.31442i	0.943421	−	0.331597i	$-0.107587\pi$
$858$		0			0
$859$	433.951	+	751.625i	0.505182	+	0.875000i	0.999982	+	0.00599363i	$0.00190784\pi$
$859$	433.951	+	751.625i	0.505182	+	0.875000i	−0.494800	+	0.869007i	$0.664759\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	258.845			0.299937			0.149968	−	0.988691i	$-0.452083\pi$
$863$	258.845			0.299937			0.149968	+	0.988691i	$0.452083\pi$
$864$		0			0
$865$	−1089.44	+	707.569i	−1.25947	+	0.817999i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	986.536	−	569.577i	1.13525	−	0.655439i
$870$		0			0
$871$	45.8951	−	79.4926i	0.0526924	−	0.0912659i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−153.975	+	401.295i	−0.175971	+	0.458623i
$876$		0			0
$877$	−16.4844	+	9.51730i	−0.0187964	+	0.0108521i	−0.509369	−	0.860548i	$-0.670121\pi$
$877$	−16.4844	+	9.51730i	−0.0187964	+	0.0108521i	0.490572	+	0.871400i	$0.336788\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	1688.86i		−	1.91699i	−0.285117	−	0.958493i	$-0.592032\pi$
$881$		−	1688.86i		−	1.91699i	0.285117	−	0.958493i	$-0.407968\pi$
$882$		0			0
$883$			373.276i			0.422736i	0.977407	+	0.211368i	$0.0677918\pi$
$883$			373.276i			0.422736i	−0.977407	+	0.211368i	$0.932208\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	598.296	+	1036.28i	0.674516	+	1.16830i	0.976610	+	0.215018i	$0.0689811\pi$
$887$	598.296	+	1036.28i	0.674516	+	1.16830i	−0.302094	+	0.953278i	$0.597686\pi$
$888$		0			0
$889$	315.465	−	546.401i	0.354854	−	0.614624i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−548.252	+	949.601i	−0.613944	+	1.06338i
$894$		0			0
$895$	518.870	−	1018.46i	0.579743	−	1.13795i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1467.96i			1.63288i
$900$		0			0
$901$	600.567			0.666556
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−137.620	−	70.1123i	−0.152066	−	0.0774722i
$906$		0			0
$907$	760.118	+	438.855i	0.838058	+	0.483853i	0.856604	−	0.515975i	$-0.172570\pi$
$907$	760.118	+	438.855i	0.838058	+	0.483853i	−0.0185458	+	0.999828i	$0.505904\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	212.780	+	122.849i	0.233567	+	0.134850i	0.612217	−	0.790690i	$-0.290278\pi$
$911$	212.780	+	122.849i	0.233567	+	0.134850i	−0.378649	+	0.925540i	$0.623611\pi$
$912$		0			0
$913$	−311.959	+	180.109i	−0.341685	+	0.197272i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	208.631			0.227515
$918$		0			0
$919$	66.4769			0.0723361			0.0361681	−	0.999346i	$-0.488485\pi$
$919$	66.4769			0.0723361			0.0361681	+	0.999346i	$0.488485\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−48.3624	−	83.7661i	−0.0523969	−	0.0907542i
$924$		0			0
$925$	1084.62	−	788.183i	1.17256	−	0.852090i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	1179.99	+	681.269i	1.27017	+	0.733336i	0.975021	−	0.222114i	$-0.0712957\pi$
$929$	1179.99	+	681.269i	1.27017	+	0.733336i	0.295154	+	0.955450i	$0.404629\pi$
$930$		0			0
$931$	357.265	+	618.802i	0.383744	+	0.664664i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−1080.82	−	1664.13i	−1.15595	−	1.77982i
$936$		0			0
$937$		−	750.141i		−	0.800578i	−0.916389	−	0.400289i	$-0.868910\pi$
$937$		−	750.141i		−	0.800578i	0.916389	−	0.400289i	$-0.131090\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−227.057	+	131.091i	−0.241293	+	0.139311i	−0.615771	−	0.787925i	$-0.711155\pi$
$941$	−227.057	+	131.091i	−0.241293	+	0.139311i	0.374478	+	0.927236i	$0.377822\pi$
$942$		0			0
$943$	5.12521	+	2.95904i	0.00543501	+	0.00313790i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	329.271	−	570.313i	0.347699	−	0.602232i	−0.638142	−	0.769919i	$-0.720297\pi$
$947$	329.271	−	570.313i	0.347699	−	0.602232i	0.985840	+	0.167687i	$0.0536299\pi$
$948$		0			0
$949$	38.3739	+	66.4656i	0.0404362	+	0.0700375i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−122.390			−0.128426			−0.0642131	−	0.997936i	$-0.520454\pi$
$953$	−122.390			−0.128426			−0.0642131	+	0.997936i	$0.520454\pi$
$954$		0			0
$955$	−7.16096	−	11.0257i	−0.00749839	−	0.0115453i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	37.1452	−	21.4458i	0.0387333	−	0.0223627i
$960$		0			0
$961$	−356.216	+	616.985i	−0.370672	+	0.642024i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−109.013	+	5.71850i	−0.112967	+	0.00592591i
$966$		0			0
$967$	−923.389	+	533.119i	−0.954901	+	0.551312i	−0.894600	−	0.446868i	$-0.852539\pi$
$967$	−923.389	+	533.119i	−0.954901	+	0.551312i	−0.0603010	+	0.998180i	$0.519206\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1798.30i			1.85201i	0.377508	+	0.926006i	$0.376781\pi$
$971$			1798.30i			1.85201i	−0.377508	+	0.926006i	$0.623219\pi$
$972$		0			0
$973$		−	253.354i		−	0.260384i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	287.036	+	497.161i	0.293794	+	0.508865i	0.974704	−	0.223502i	$-0.0717489\pi$
$977$	287.036	+	497.161i	0.293794	+	0.508865i	−0.680910	+	0.732367i	$0.738416\pi$
$978$		0			0
$979$	−437.935	+	758.526i	−0.447329	+	0.774797i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	382.134	−	661.875i	0.388742	−	0.673321i	−0.603538	−	0.797334i	$-0.706243\pi$
$983$	382.134	−	661.875i	0.388742	−	0.673321i	0.992281	+	0.124013i	$0.0395763\pi$
$984$		0			0
$985$	306.674	+	156.239i	0.311344	+	0.158618i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	131.478i		−	0.132940i
$990$		0			0
$991$	398.586			0.402206			0.201103	−	0.979570i	$-0.435547\pi$
$991$	398.586			0.402206			0.201103	+	0.979570i	$0.435547\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−517.622	−	263.710i	−0.520224	−	0.265035i
$996$		0			0
$997$	1278.30	+	738.025i	1.28214	+	0.740246i	0.977240	−	0.212138i	$-0.0680426\pi$
$997$	1278.30	+	738.025i	1.28214	+	0.740246i	0.304903	+	0.952383i	$0.401376\pi$
$998$		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.269.5		48
3.2	odd	2	inner	1620.3.t.f.269.20		48
5.4	even	2	inner	1620.3.t.f.269.12		48
9.2	odd	6		1620.3.b.a.809.4	yes	24
9.4	even	3	inner	1620.3.t.f.1349.13		48
9.5	odd	6	inner	1620.3.t.f.1349.12		48
9.7	even	3		1620.3.b.a.809.21	yes	24
15.14	odd	2	inner	1620.3.t.f.269.13		48
45.4	even	6	inner	1620.3.t.f.1349.20		48
45.14	odd	6	inner	1620.3.t.f.1349.5		48
45.29	odd	6		1620.3.b.a.809.22	yes	24
45.34	even	6		1620.3.b.a.809.3	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.b.a.809.3	✓	24	45.34	even	6
1620.3.b.a.809.4	yes	24	9.2	odd	6
1620.3.b.a.809.21	yes	24	9.7	even	3
1620.3.b.a.809.22	yes	24	45.29	odd	6
1620.3.t.f.269.5		48	1.1	even	1	trivial
1620.3.t.f.269.12		48	5.4	even	2	inner
1620.3.t.f.269.13		48	15.14	odd	2	inner
1620.3.t.f.269.20		48	3.2	odd	2	inner
1620.3.t.f.1349.5		48	45.14	odd	6	inner
1620.3.t.f.1349.12		48	9.5	odd	6	inner
1620.3.t.f.1349.13		48	9.4	even	3	inner
1620.3.t.f.1349.20		48	45.4	even	6	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)

\(f(q)\)	\(=\)	\(q+(-4.45514 - 2.26973i) q^{5} +(-2.97789 - 1.71928i) q^{7} +O(q^{10})\)	\(q+(-4.45514 - 2.26973i) q^{5} +(-2.97789 - 1.71928i) q^{7} +(-11.1039 - 6.41086i) q^{11} +(1.55354 - 0.896938i) q^{13} -30.9523 q^{17} -19.2201 q^{19} +(-1.19201 - 2.06461i) q^{23} +(14.6966 + 20.2240i) q^{25} +(31.0770 + 17.9423i) q^{29} +(20.4538 + 35.4270i) q^{31} +(9.36459 + 14.4187i) q^{35} -53.6302i q^{37} +(-2.14983 + 1.24120i) q^{41} +(47.7611 + 27.5749i) q^{43} +(28.5250 - 49.4067i) q^{47} +(-18.5881 - 32.1956i) q^{49} -19.4030 q^{53} +(34.9187 + 53.7643i) q^{55} +(-59.8830 + 34.5735i) q^{59} +(-8.80471 + 15.2502i) q^{61} +(-8.95707 + 0.469861i) q^{65} +(44.3133 - 25.5843i) q^{67} -53.9194i q^{71} +42.7832i q^{73} +(22.0442 + 38.1816i) q^{77} +(-44.4228 + 76.9425i) q^{79} +(14.0472 - 24.3305i) q^{83} +(137.897 + 70.2536i) q^{85} -68.3114i q^{89} -6.16836 q^{91} +(85.6282 + 43.6245i) q^{95} +(135.186 + 78.0498i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more