LMFDB - Embedded newform 1620.3.t.f.269.7

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			269.7
Character	$\chi$	$=$	1620.269
Dual form			1620.3.t.f.1349.7

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-2.76464 - 4.16614i) q^{5} +(-4.03229 - 2.32805i) q^{7} +O(q^{10})$	$q+(-2.76464 - 4.16614i) q^{5} +(-4.03229 - 2.32805i) q^{7} +(10.4161 + 6.01375i) q^{11} +(7.33373 - 4.23413i) q^{13} -1.25172 q^{17} +16.4797 q^{19} +(-1.19409 - 2.06822i) q^{23} +(-9.71349 + 23.0358i) q^{25} +(20.7820 + 11.9985i) q^{29} +(2.62707 + 4.55023i) q^{31} +(1.44888 + 23.2353i) q^{35} -25.1935i q^{37} +(25.8444 - 14.9212i) q^{41} +(-22.0927 - 12.7552i) q^{43} +(-32.4891 + 56.2728i) q^{47} +(-13.6604 - 23.6605i) q^{49} +71.4673 q^{53} +(-3.74272 - 60.0210i) q^{55} +(-16.0229 + 9.25082i) q^{59} +(-9.55124 + 16.5432i) q^{61} +(-37.9151 - 18.8475i) q^{65} +(70.2262 - 40.5451i) q^{67} +21.9875i q^{71} -109.701i q^{73} +(-28.0006 - 48.4984i) q^{77} +(67.0965 - 116.214i) q^{79} +(-7.80612 + 13.5206i) q^{83} +(3.46055 + 5.21483i) q^{85} -79.4211i q^{89} -39.4290 q^{91} +(-45.5606 - 68.6569i) q^{95} +(-112.846 - 65.1518i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.76464	−	4.16614i	−0.552929	−	0.833229i
$5$	−2.76464	−	4.16614i	−0.552929	−	0.833229i
$6$		0			0
$7$	−4.03229	−	2.32805i	−0.576042	−	0.332578i	0.183517	−	0.983017i	$-0.441252\pi$
$7$	−4.03229	−	2.32805i	−0.576042	−	0.332578i	−0.759559	+	0.650439i	$0.774585\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	10.4161	+	6.01375i	0.946921	+	0.546705i	0.892123	−	0.451793i	$-0.149215\pi$
$11$	10.4161	+	6.01375i	0.946921	+	0.546705i	0.0547977	+	0.998497i	$0.482549\pi$
$12$		0			0
$13$	7.33373	−	4.23413i	0.564133	−	0.325702i	−0.190670	−	0.981654i	$-0.561066\pi$
$13$	7.33373	−	4.23413i	0.564133	−	0.325702i	0.754803	+	0.655952i	$0.227733\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−1.25172			−0.0736304			−0.0368152	−	0.999322i	$-0.511721\pi$
$17$	−1.25172			−0.0736304			−0.0368152	+	0.999322i	$0.511721\pi$
$18$		0			0
$19$	16.4797			0.867354			0.433677	−	0.901068i	$-0.357216\pi$
$19$	16.4797			0.867354			0.433677	+	0.901068i	$0.357216\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.19409	−	2.06822i	−0.0519169	−	0.0899227i	0.838899	−	0.544287i	$-0.183200\pi$
$23$	−1.19409	−	2.06822i	−0.0519169	−	0.0899227i	−0.890816	+	0.454364i	$0.849866\pi$
$24$		0			0
$25$	−9.71349	+	23.0358i	−0.388540	+	0.921432i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	20.7820	+	11.9985i	0.716619	+	0.413740i	0.813507	−	0.581555i	$-0.197555\pi$
$29$	20.7820	+	11.9985i	0.716619	+	0.413740i	−0.0968879	+	0.995295i	$0.530889\pi$
$30$		0			0
$31$	2.62707	+	4.55023i	0.0847443	+	0.146781i	0.905282	−	0.424811i	$-0.139659\pi$
$31$	2.62707	+	4.55023i	0.0847443	+	0.146781i	−0.820538	+	0.571592i	$0.806326\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.44888	+	23.2353i	0.0413967	+	0.663867i
$36$		0			0
$37$		−	25.1935i		−	0.680906i	−0.940262	−	0.340453i	$-0.889420\pi$
$37$		−	25.1935i		−	0.680906i	0.940262	−	0.340453i	$-0.110580\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	25.8444	−	14.9212i	0.630350	−	0.363933i	−0.150538	−	0.988604i	$-0.548100\pi$
$41$	25.8444	−	14.9212i	0.630350	−	0.363933i	0.780888	+	0.624671i	$0.214767\pi$
$42$		0			0
$43$	−22.0927	−	12.7552i	−0.513783	−	0.296633i	0.220604	−	0.975363i	$-0.429197\pi$
$43$	−22.0927	−	12.7552i	−0.513783	−	0.296633i	−0.734387	+	0.678731i	$0.762530\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−32.4891	+	56.2728i	−0.691258	+	1.19729i	0.280168	+	0.959951i	$0.409610\pi$
$47$	−32.4891	+	56.2728i	−0.691258	+	1.19729i	−0.971426	+	0.237343i	$0.923723\pi$
$48$		0			0
$49$	−13.6604	−	23.6605i	−0.278784	−	0.482868i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	71.4673			1.34844			0.674219	−	0.738531i	$-0.264480\pi$
$53$	71.4673			1.34844			0.674219	+	0.738531i	$0.264480\pi$
$54$		0			0
$55$	−3.74272	−	60.0210i	−0.0680495	−	1.09129i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−16.0229	+	9.25082i	−0.271574	+	0.156794i	−0.629603	−	0.776917i	$-0.716782\pi$
$59$	−16.0229	+	9.25082i	−0.271574	+	0.156794i	0.358028	+	0.933711i	$0.383449\pi$
$60$		0			0
$61$	−9.55124	+	16.5432i	−0.156578	+	0.271200i	−0.933632	−	0.358233i	$-0.883379\pi$
$61$	−9.55124	+	16.5432i	−0.156578	+	0.271200i	0.777055	+	0.629433i	$0.216713\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−37.9151	−	18.8475i	−0.583310	−	0.289962i
$66$		0			0
$67$	70.2262	−	40.5451i	1.04815	−	0.605151i	0.126021	−	0.992028i	$-0.459779\pi$
$67$	70.2262	−	40.5451i	1.04815	−	0.605151i	0.922132	+	0.386876i	$0.126446\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			21.9875i			0.309683i	0.987939	+	0.154842i	$0.0494867\pi$
$71$			21.9875i			0.309683i	−0.987939	+	0.154842i	$0.950513\pi$
$72$		0			0
$73$		−	109.701i		−	1.50276i	−0.659871	−	0.751379i	$-0.729389\pi$
$73$		−	109.701i		−	1.50276i	0.659871	−	0.751379i	$-0.270611\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−28.0006	−	48.4984i	−0.363644	−	0.629850i
$78$		0			0
$79$	67.0965	−	116.214i	0.849322	−	1.47107i	−0.0324916	−	0.999472i	$-0.510344\pi$
$79$	67.0965	−	116.214i	0.849322	−	1.47107i	0.881814	−	0.471597i	$-0.156322\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−7.80612	+	13.5206i	−0.0940496	+	0.162899i	−0.909212	−	0.416334i	$-0.863315\pi$
$83$	−7.80612	+	13.5206i	−0.0940496	+	0.162899i	0.815162	+	0.579233i	$0.196648\pi$
$84$		0			0
$85$	3.46055	+	5.21483i	0.0407123	+	0.0613509i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	79.4211i		−	0.892372i	−0.894940	−	0.446186i	$-0.852782\pi$
$89$		−	79.4211i		−	0.892372i	0.894940	−	0.446186i	$-0.147218\pi$
$90$		0			0
$91$	−39.4290			−0.433286
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−45.5606	−	68.6569i	−0.479585	−	0.722704i
$96$		0			0
$97$	−112.846	−	65.1518i	−1.16336	−	0.671668i	−0.211256	−	0.977431i	$-0.567755\pi$
$97$	−112.846	−	65.1518i	−1.16336	−	0.671668i	−0.952108	+	0.305763i	$0.901089\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−112.530	−	64.9693i	−1.11416	−	0.643260i	−0.174256	−	0.984700i	$-0.555752\pi$
$101$	−112.530	−	64.9693i	−1.11416	−	0.643260i	−0.939904	+	0.341440i	$0.889085\pi$
$102$		0			0
$103$	110.447	−	63.7664i	1.07230	−	0.619091i	0.143489	−	0.989652i	$-0.454168\pi$
$103$	110.447	−	63.7664i	1.07230	−	0.619091i	0.928808	+	0.370561i	$0.120835\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	28.0983			0.262601			0.131301	−	0.991343i	$-0.458085\pi$
$107$	28.0983			0.262601			0.131301	+	0.991343i	$0.458085\pi$
$108$		0			0
$109$	−14.6527			−0.134429			−0.0672144	−	0.997739i	$-0.521411\pi$
$109$	−14.6527			−0.134429			−0.0672144	+	0.997739i	$0.521411\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−66.5423	−	115.255i	−0.588870	−	1.01995i	−0.994381	−	0.105863i	$-0.966240\pi$
$113$	−66.5423	−	115.255i	−0.588870	−	1.01995i	0.405510	−	0.914090i	$-0.367094\pi$
$114$		0			0
$115$	−5.31528	+	10.6926i	−0.0462198	+	0.0929795i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	5.04729	+	2.91405i	0.0424142	+	0.0244878i
$120$		0			0
$121$	11.8305	+	20.4910i	0.0977725	+	0.169347i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	122.825	−	23.2180i	0.982598	−	0.185744i
$126$		0			0
$127$		−	219.646i		−	1.72950i	−0.502206	−	0.864748i	$-0.667478\pi$
$127$		−	219.646i		−	1.72950i	0.502206	−	0.864748i	$-0.332522\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−131.640	+	76.0022i	−1.00488	+	0.580170i	−0.909689	−	0.415289i	$-0.863680\pi$
$131$	−131.640	+	76.0022i	−1.00488	+	0.580170i	−0.0951939	+	0.995459i	$0.530347\pi$
$132$		0			0
$133$	−66.4511	−	38.3656i	−0.499632	−	0.288463i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−100.848	+	174.675i	−0.736120	+	1.27500i	0.218110	+	0.975924i	$0.430011\pi$
$137$	−100.848	+	174.675i	−0.736120	+	1.27500i	−0.954230	+	0.299073i	$0.903323\pi$
$138$		0			0
$139$	83.1778	+	144.068i	0.598401	+	1.03646i	0.993057	+	0.117633i	$0.0375305\pi$
$139$	83.1778	+	144.068i	0.598401	+	1.03646i	−0.394656	+	0.918829i	$0.629136\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	101.852			0.712252
$144$		0			0
$145$	−7.46737	−	119.752i	−0.0514991	−	0.825876i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	163.074	−	94.1507i	1.09446	−	0.631884i	0.159696	−	0.987166i	$-0.448949\pi$
$149$	163.074	−	94.1507i	1.09446	−	0.631884i	0.934759	+	0.355282i	$0.115615\pi$
$150$		0			0
$151$	−26.3466	+	45.6336i	−0.174481	+	0.302209i	−0.939981	−	0.341226i	$-0.889158\pi$
$151$	−26.3466	+	45.6336i	−0.174481	+	0.302209i	0.765501	+	0.643435i	$0.222491\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	11.6940	−	23.5245i	0.0754449	−	0.151771i
$156$		0			0
$157$	138.614	−	80.0290i	0.882893	−	0.509739i	0.0112819	−	0.999936i	$-0.496409\pi$
$157$	138.614	−	80.0290i	0.882893	−	0.509739i	0.871611	+	0.490198i	$0.163075\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			11.1196i			0.0690657i
$162$		0			0
$163$		−	300.512i		−	1.84363i	−0.387630	−	0.921815i	$-0.626706\pi$
$163$		−	300.512i		−	1.84363i	0.387630	−	0.921815i	$-0.373294\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.42644	+	5.93476i	0.0205176	+	0.0355375i	0.876102	−	0.482126i	$-0.160135\pi$
$167$	3.42644	+	5.93476i	0.0205176	+	0.0355375i	−0.855584	+	0.517664i	$0.826802\pi$
$168$		0			0
$169$	−48.6443	+	84.2543i	−0.287836	+	0.498546i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−104.155	+	180.401i	−0.602049	+	1.04278i	0.390461	+	0.920620i	$0.372316\pi$
$173$	−104.155	+	180.401i	−0.602049	+	1.04278i	−0.992510	+	0.122161i	$0.961018\pi$
$174$		0			0
$175$	92.7961	−	70.2737i	0.530263	−	0.401564i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			23.2298i			0.129776i	0.997893	+	0.0648878i	$0.0206690\pi$
$179$			23.2298i			0.129776i	−0.997893	+	0.0648878i	$0.979331\pi$
$180$		0			0
$181$	259.043			1.43118			0.715588	−	0.698522i	$-0.246159\pi$
$181$	259.043			1.43118			0.715588	+	0.698522i	$0.246159\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−104.960	+	69.6511i	−0.567350	+	0.376492i
$186$		0			0
$187$	−13.0380	−	7.52751i	−0.0697221	−	0.0402541i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	147.220	+	84.9977i	0.770788	+	0.445014i	0.833155	−	0.553039i	$-0.186532\pi$
$191$	147.220	+	84.9977i	0.770788	+	0.445014i	−0.0623679	+	0.998053i	$0.519865\pi$
$192$		0			0
$193$	−69.2103	+	39.9586i	−0.358603	+	0.207039i	−0.668468	−	0.743741i	$-0.733049\pi$
$193$	−69.2103	+	39.9586i	−0.358603	+	0.207039i	0.309865	+	0.950781i	$0.399716\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	211.199			1.07208			0.536039	−	0.844193i	$-0.319920\pi$
$197$	211.199			1.07208			0.536039	+	0.844193i	$0.319920\pi$
$198$		0			0
$199$	−175.943			−0.884133			−0.442067	−	0.896982i	$-0.645755\pi$
$199$	−175.943			−0.884133			−0.442067	+	0.896982i	$0.645755\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−55.8660	−	96.7627i	−0.275202	−	0.476664i
$204$		0			0
$205$	−133.614	−	66.4193i	−0.651778	−	0.323997i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	171.655	+	99.1050i	0.821315	+	0.474187i
$210$		0			0
$211$	−112.879	−	195.512i	−0.534971	−	0.926597i	−0.999165	−	0.0408634i	$-0.986989\pi$
$211$	−112.879	−	195.512i	−0.534971	−	0.926597i	0.464194	−	0.885734i	$-0.346344\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	7.93834	+	127.305i	0.0369225	+	0.592115i
$216$		0			0
$217$		−	24.4638i		−	0.112736i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−9.17975	+	5.29993i	−0.0415373	+	0.0239816i
$222$		0			0
$223$	4.07966	+	2.35539i	0.0182944	+	0.0105623i	0.509119	−	0.860696i	$-0.329971\pi$
$223$	4.07966	+	2.35539i	0.0182944	+	0.0105623i	−0.490825	+	0.871258i	$0.663305\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.24897	−	7.35943i	0.0187179	−	0.0324204i	−0.856515	−	0.516123i	$-0.827375\pi$
$227$	4.24897	−	7.35943i	0.0187179	−	0.0324204i	0.875233	+	0.483702i	$0.160708\pi$
$228$		0			0
$229$	17.7879	+	30.8096i	0.0776766	+	0.134540i	0.902247	−	0.431219i	$-0.141917\pi$
$229$	17.7879	+	30.8096i	0.0776766	+	0.134540i	−0.824571	+	0.565759i	$0.808583\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−110.611			−0.474726			−0.237363	−	0.971421i	$-0.576283\pi$
$233$	−110.611			−0.474726			−0.237363	+	0.971421i	$0.576283\pi$
$234$		0			0
$235$	324.261	−	20.2200i	1.37984	−	0.0860424i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	384.544	−	222.016i	1.60897	−	0.928939i	0.619367	−	0.785101i	$-0.287389\pi$
$239$	384.544	−	222.016i	1.60897	−	0.928939i	0.989601	−	0.143837i	$-0.0459442\pi$
$240$		0			0
$241$	83.9964	−	145.486i	0.348533	−	0.603676i	−0.637456	−	0.770486i	$-0.720013\pi$
$241$	83.9964	−	145.486i	0.348533	−	0.603676i	0.985989	+	0.166810i	$0.0533467\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−60.8069	+	122.324i	−0.248192	+	0.499282i
$246$		0			0
$247$	120.858	−	69.7773i	0.489303	−	0.282499i
$248$		0			0
$249$		0			0
$250$		0			0
$251$			54.9639i			0.218980i	0.993988	+	0.109490i	$0.0349217\pi$
$251$			54.9639i			0.218980i	−0.993988	+	0.109490i	$0.965078\pi$
$252$		0			0
$253$		−	28.7238i		−	0.113533i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−141.201	−	244.567i	−0.549419	−	0.951622i	−0.998314	−	0.0580379i	$-0.981516\pi$
$257$	−141.201	−	244.567i	−0.549419	−	0.951622i	0.448895	−	0.893585i	$-0.351818\pi$
$258$		0			0
$259$	−58.6517	+	101.588i	−0.226454	+	0.392230i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	57.9154	−	100.312i	0.220211	−	0.381416i	−0.734661	−	0.678434i	$-0.762659\pi$
$263$	57.9154	−	100.312i	0.220211	−	0.381416i	0.954872	+	0.297018i	$0.0959922\pi$
$264$		0			0
$265$	−197.581	−	297.743i	−0.745591	−	1.12356i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			97.7513i			0.363388i	0.983355	+	0.181694i	$0.0581580\pi$
$269$			97.7513i			0.363388i	−0.983355	+	0.181694i	$0.941842\pi$
$270$		0			0
$271$	−110.971			−0.409486			−0.204743	−	0.978816i	$-0.565636\pi$
$271$	−110.971			−0.409486			−0.204743	+	0.978816i	$0.565636\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−239.709	+	181.529i	−0.871668	+	0.660107i
$276$		0			0
$277$	−164.743	−	95.1143i	−0.594740	−	0.343373i	0.172230	−	0.985057i	$-0.444903\pi$
$277$	−164.743	−	95.1143i	−0.594740	−	0.343373i	−0.766969	+	0.641684i	$0.778236\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−57.3035	−	33.0842i	−0.203927	−	0.117737i	0.394559	−	0.918871i	$-0.370897\pi$
$281$	−57.3035	−	33.0842i	−0.203927	−	0.117737i	−0.598486	+	0.801133i	$0.704231\pi$
$282$		0			0
$283$	300.296	−	173.376i	1.06112	−	0.612635i	0.135375	−	0.990794i	$-0.456776\pi$
$283$	300.296	−	173.376i	1.06112	−	0.612635i	0.925740	+	0.378159i	$0.123443\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−138.949			−0.484144
$288$		0			0
$289$	−287.433			−0.994579
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−11.5247	−	19.9614i	−0.0393334	−	0.0681275i	0.845688	−	0.533677i	$-0.179190\pi$
$293$	−11.5247	−	19.9614i	−0.0393334	−	0.0681275i	−0.885022	+	0.465549i	$0.845857\pi$
$294$		0			0
$295$	82.8378	+	41.1784i	0.280806	+	0.139588i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−17.5143	−	10.1119i	−0.0585761	−	0.0338189i
$300$		0			0
$301$	59.3894	+	102.865i	0.197307	+	0.341746i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	95.3272	−	5.94432i	0.312548	−	0.0194896i
$306$		0			0
$307$		−	169.526i		−	0.552200i	−0.961129	−	0.276100i	$-0.910958\pi$
$307$		−	169.526i		−	0.552200i	0.961129	−	0.276100i	$-0.0890422\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	63.3291	−	36.5630i	0.203630	−	0.117566i	−0.394717	−	0.918803i	$-0.629157\pi$
$311$	63.3291	−	36.5630i	0.203630	−	0.117566i	0.598348	+	0.801237i	$0.295824\pi$
$312$		0			0
$313$	−4.69315	−	2.70959i	−0.0149941	−	0.00865685i	0.492484	−	0.870321i	$-0.336089\pi$
$313$	−4.69315	−	2.70959i	−0.0149941	−	0.00865685i	−0.507478	+	0.861665i	$0.669422\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	169.585	−	293.731i	0.534970	−	0.926595i	−0.464195	−	0.885733i	$-0.653656\pi$
$317$	169.585	−	293.731i	0.534970	−	0.926595i	0.999165	−	0.0408617i	$-0.0130103\pi$
$318$		0			0
$319$	144.312	+	249.955i	0.452388	+	0.783558i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−20.6279			−0.0638636
$324$		0			0
$325$	26.3005	+	210.067i	0.0809246	+	0.646359i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	262.011	−	151.272i	0.796387	−	0.459794i
$330$		0			0
$331$	−127.851	+	221.444i	−0.386256	+	0.669016i	−0.991943	−	0.126688i	$-0.959565\pi$
$331$	−127.851	+	221.444i	−0.386256	+	0.669016i	0.605686	+	0.795704i	$0.292899\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−363.067	−	180.480i	−1.08378	−	0.538745i
$336$		0			0
$337$	−538.292	+	310.783i	−1.59731	+	0.922205i	−0.605303	+	0.795996i	$0.706948\pi$
$337$	−538.292	+	310.783i	−1.59731	+	0.922205i	−0.992004	+	0.126210i	$0.959719\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			63.1943i			0.185321i
$342$		0			0
$343$			355.357i			1.03603i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	173.858	+	301.131i	0.501032	+	0.867813i	0.999999	+	0.00119211i	$0.000379461\pi$
$347$	173.858	+	301.131i	0.501032	+	0.867813i	−0.498967	+	0.866621i	$0.666287\pi$
$348$		0			0
$349$	280.376	−	485.625i	0.803368	−	1.39147i	−0.114019	−	0.993479i	$-0.536372\pi$
$349$	280.376	−	485.625i	0.803368	−	1.39147i	0.917387	−	0.397996i	$-0.130294\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−132.761	+	229.949i	−0.376094	+	0.651415i	−0.990490	−	0.137583i	$-0.956067\pi$
$353$	−132.761	+	229.949i	−0.376094	+	0.651415i	0.614396	+	0.788998i	$0.289400\pi$
$354$		0			0
$355$	91.6031	−	60.7876i	0.258037	−	0.171233i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	502.476i		−	1.39965i	−0.714313	−	0.699827i	$-0.753261\pi$
$359$		−	502.476i		−	1.39965i	0.714313	−	0.699827i	$-0.246739\pi$
$360$		0			0
$361$	−89.4187			−0.247697
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−457.031	+	303.285i	−1.25214	+	0.830918i
$366$		0			0
$367$	569.949	+	329.060i	1.55299	+	0.896622i	0.997896	+	0.0648367i	$0.0206527\pi$
$367$	569.949	+	329.060i	1.55299	+	0.896622i	0.555098	+	0.831785i	$0.312681\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−288.177	−	166.379i	−0.776757	−	0.448461i
$372$		0			0
$373$	−552.690	+	319.096i	−1.48174	+	0.855484i	−0.999785	−	0.0207114i	$-0.993407\pi$
$373$	−552.690	+	319.096i	−1.48174	+	0.855484i	−0.481956	+	0.876195i	$0.660074\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	203.212			0.539025
$378$		0			0
$379$	−126.043			−0.332567			−0.166284	−	0.986078i	$-0.553177\pi$
$379$	−126.043			−0.332567			−0.166284	+	0.986078i	$0.553177\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	319.637	+	553.628i	0.834562	+	1.44550i	0.894386	+	0.447295i	$0.147613\pi$
$383$	319.637	+	553.628i	0.834562	+	1.44550i	−0.0598244	+	0.998209i	$0.519054\pi$
$384$		0			0
$385$	−124.640	+	250.735i	−0.323740	+	0.651261i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	243.635	+	140.663i	0.626311	+	0.361601i	0.779322	−	0.626624i	$-0.215564\pi$
$389$	243.635	+	140.663i	0.626311	+	0.361601i	−0.153011	+	0.988224i	$0.548897\pi$
$390$		0			0
$391$	1.49466	+	2.58883i	0.00382266	+	0.00662104i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−669.664	+	41.7582i	−1.69535	+	0.105717i
$396$		0			0
$397$			408.350i			1.02859i	0.857613	+	0.514295i	$0.171946\pi$
$397$			408.350i			1.02859i	−0.857613	+	0.514295i	$0.828054\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−515.033	+	297.355i	−1.28437	+	0.741533i	−0.977645	−	0.210264i	$-0.932568\pi$
$401$	−515.033	+	297.355i	−1.28437	+	0.741533i	−0.306728	+	0.951797i	$0.599234\pi$
$402$		0			0
$403$	38.5325	+	22.2467i	0.0956141	+	0.0552029i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	151.508	−	262.419i	0.372255	−	0.644764i
$408$		0			0
$409$	262.478	+	454.624i	0.641754	+	1.11155i	0.985041	+	0.172320i	$0.0551264\pi$
$409$	262.478	+	454.624i	0.641754	+	1.11155i	−0.343287	+	0.939231i	$0.611540\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	86.1454			0.208584
$414$		0			0
$415$	77.9099	−	4.85822i	0.187735	−	0.0117066i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	307.311	−	177.426i	0.733440	−	0.423452i	−0.0862393	−	0.996274i	$-0.527485\pi$
$419$	307.311	−	177.426i	0.733440	−	0.423452i	0.819679	+	0.572823i	$0.194152\pi$
$420$		0			0
$421$	−9.44753	+	16.3636i	−0.0224407	+	0.0388684i	−0.877028	−	0.480440i	$-0.840477\pi$
$421$	−9.44753	+	16.3636i	−0.0224407	+	0.0388684i	0.854587	+	0.519308i	$0.173810\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.1585	−	28.8343i	0.0286083	−	0.0678454i
$426$		0			0
$427$	77.0268	−	44.4714i	0.180391	−	0.104149i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	294.610i		−	0.683549i	−0.939782	−	0.341774i	$-0.888972\pi$
$431$		−	294.610i		−	0.683549i	0.939782	−	0.341774i	$-0.111028\pi$
$432$		0			0
$433$			541.170i			1.24982i	0.780698	+	0.624908i	$0.214864\pi$
$433$			541.170i			1.24982i	−0.780698	+	0.624908i	$0.785136\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−19.6783	−	34.0837i	−0.0450303	−	0.0779948i
$438$		0			0
$439$	266.126	−	460.944i	0.606209	−	1.04999i	−0.385650	−	0.922645i	$-0.626023\pi$
$439$	266.126	−	460.944i	0.606209	−	1.04999i	0.991859	−	0.127340i	$-0.0406440\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	370.480	−	641.691i	0.836299	−	1.44851i	−0.0566700	−	0.998393i	$-0.518048\pi$
$443$	370.480	−	641.691i	0.836299	−	1.44851i	0.892969	−	0.450119i	$-0.148618\pi$
$444$		0			0
$445$	−330.880	+	219.571i	−0.743549	+	0.493418i
$446$		0			0
$447$		0			0
$448$		0			0
$449$		−	575.484i		−	1.28170i	−0.767665	−	0.640851i	$-0.778582\pi$
$449$		−	575.484i		−	1.28170i	0.767665	−	0.640851i	$-0.221418\pi$
$450$		0			0
$451$	358.931			0.795855
$452$		0			0
$453$		0			0
$454$		0			0
$455$	109.007	+	164.267i	0.239576	+	0.361026i
$456$		0			0
$457$	143.659	+	82.9416i	0.314353	+	0.181492i	0.648873	−	0.760897i	$-0.275241\pi$
$457$	143.659	+	82.9416i	0.314353	+	0.181492i	−0.334520	+	0.942389i	$0.608574\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−309.050	−	178.430i	−0.670390	−	0.387050i	0.125834	−	0.992051i	$-0.459839\pi$
$461$	−309.050	−	178.430i	−0.670390	−	0.387050i	−0.796224	+	0.605001i	$0.793173\pi$
$462$		0			0
$463$	340.737	−	196.725i	0.735933	−	0.424891i	−0.0846557	−	0.996410i	$-0.526979\pi$
$463$	340.737	−	196.725i	0.735933	−	0.424891i	0.820589	+	0.571519i	$0.193646\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−413.760			−0.885997			−0.442998	−	0.896522i	$-0.646085\pi$
$467$	−413.760			−0.885997			−0.442998	+	0.896522i	$0.646085\pi$
$468$		0			0
$469$	−377.564			−0.805040
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−153.413	−	265.720i	−0.324341	−	0.561775i
$474$		0			0
$475$	−160.076	+	379.624i	−0.337001	+	0.799208i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	761.561	+	439.687i	1.58990	+	0.917928i	0.993323	+	0.115370i	$0.0368054\pi$
$479$	761.561	+	439.687i	1.58990	+	0.917928i	0.596575	+	0.802558i	$0.296528\pi$
$480$		0			0
$481$	−106.673	−	184.762i	−0.221773	−	0.384122i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	40.5479	+	650.255i	0.0836040	+	1.34073i
$486$		0			0
$487$			492.034i			1.01034i	0.863021	+	0.505169i	$0.168570\pi$
$487$			492.034i			1.01034i	−0.863021	+	0.505169i	$0.831430\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−386.931	+	223.395i	−0.788046	+	0.454979i	−0.839274	−	0.543708i	$-0.817020\pi$
$491$	−386.931	+	223.395i	−0.788046	+	0.454979i	0.0512279	+	0.998687i	$0.483687\pi$
$492$		0			0
$493$	−26.0131	−	15.0187i	−0.0527649	−	0.0304638i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	51.1879	−	88.6601i	0.102994	−	0.178390i
$498$		0			0
$499$	−72.4939	−	125.563i	−0.145278	−	0.251629i	0.784198	−	0.620510i	$-0.213074\pi$
$499$	−72.4939	−	125.563i	−0.145278	−	0.251629i	−0.929477	+	0.368881i	$0.879741\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−542.902			−1.07933			−0.539664	−	0.841881i	$-0.681449\pi$
$503$	−542.902			−1.07933			−0.539664	+	0.841881i	$0.681449\pi$
$504$		0			0
$505$	40.4343	+	648.434i	0.0800680	+	1.28403i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	688.989	−	397.788i	1.35361	−	0.781509i	0.364860	−	0.931062i	$-0.381117\pi$
$509$	688.989	−	397.788i	1.35361	−	0.781509i	0.988754	+	0.149553i	$0.0477835\pi$
$510$		0			0
$511$	−255.390	+	442.348i	−0.499784	+	0.865652i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−571.005	−	283.845i	−1.10875	−	0.551155i
$516$		0			0
$517$	−676.822	+	390.763i	−1.30913	+	0.755828i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			857.089i			1.64508i	0.568705	+	0.822542i	$0.307445\pi$
$521$			857.089i			1.64508i	−0.568705	+	0.822542i	$0.692555\pi$
$522$		0			0
$523$			616.196i			1.17819i	0.808062	+	0.589097i	$0.200517\pi$
$523$			616.196i			1.17819i	−0.808062	+	0.589097i	$0.799483\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−3.28835	−	5.69559i	−0.00623975	−	0.0108076i
$528$		0			0
$529$	261.648	−	453.188i	0.494609	−	0.856688i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	126.357	−	218.857i	0.237068	−	0.410613i
$534$		0			0
$535$	−77.6818	−	117.062i	−0.145200	−	0.218807i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		−	328.601i		−	0.609650i
$540$		0			0
$541$	145.542			0.269025			0.134512	−	0.990912i	$-0.457053\pi$
$541$	145.542			0.269025			0.134512	+	0.990912i	$0.457053\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	40.5096	+	61.0454i	0.0743295	+	0.112010i
$546$		0			0
$547$	51.9794	+	30.0103i	0.0950264	+	0.0548635i	0.546760	−	0.837289i	$-0.315861\pi$
$547$	51.9794	+	30.0103i	0.0950264	+	0.0548635i	−0.451734	+	0.892153i	$0.649194\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	342.481	+	197.731i	0.621562	+	0.358859i
$552$		0			0
$553$	−541.105	+	312.407i	−0.978491	+	0.564932i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−557.225			−1.00040			−0.500202	−	0.865909i	$-0.666741\pi$
$557$	−557.225			−1.00040			−0.500202	+	0.865909i	$0.666741\pi$
$558$		0			0
$559$	−216.029			−0.386456
$560$		0			0
$561$		0			0
$562$		0			0
$563$	489.429	+	847.716i	0.869324	+	1.50571i	0.862689	+	0.505735i	$0.168779\pi$
$563$	489.429	+	847.716i	0.869324	+	1.50571i	0.00663478	+	0.999978i	$0.497888\pi$
$564$		0			0
$565$	−296.202	+	595.863i	−0.524251	+	1.05463i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	679.468	+	392.291i	1.19414	+	0.689440i	0.959244	−	0.282580i	$-0.0911902\pi$
$569$	679.468	+	392.291i	1.19414	+	0.689440i	0.234901	+	0.972019i	$0.424524\pi$
$570$		0			0
$571$	−369.407	−	639.831i	−0.646947	−	1.12054i	−0.983848	−	0.179005i	$-0.942712\pi$
$571$	−369.407	−	639.831i	−0.646947	−	1.12054i	0.336902	−	0.941540i	$-0.390621\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	59.2419	−	7.41713i	0.103029	−	0.0128994i
$576$		0			0
$577$			707.354i			1.22592i	0.790115	+	0.612959i	$0.210021\pi$
$577$			707.354i			1.22592i	−0.790115	+	0.612959i	$0.789979\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	62.9531	−	36.3460i	0.108353	−	0.0625577i
$582$		0			0
$583$	744.412	+	429.786i	1.27686	+	0.737198i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	456.999	−	791.545i	0.778533	−	1.34846i	−0.154254	−	0.988031i	$-0.549297\pi$
$587$	456.999	−	791.545i	0.778533	−	1.34846i	0.932787	−	0.360428i	$-0.117369\pi$
$588$		0			0
$589$	43.2934	+	74.9864i	0.0735033	+	0.127311i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−97.8573			−0.165021			−0.0825103	−	0.996590i	$-0.526294\pi$
$593$	−97.8573			−0.165021			−0.0825103	+	0.996590i	$0.526294\pi$
$594$		0			0
$595$	−1.81359	−	29.0840i	−0.00304805	−	0.0488807i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	991.487	−	572.435i	1.65524	−	0.955651i	0.680368	−	0.732871i	$-0.261820\pi$
$599$	991.487	−	572.435i	1.65524	−	0.955651i	0.974869	−	0.222780i	$-0.0715132\pi$
$600$		0			0
$601$	−155.414	+	269.185i	−0.258593	+	0.447896i	−0.965865	−	0.259045i	$-0.916592\pi$
$601$	−155.414	+	269.185i	−0.258593	+	0.447896i	0.707272	+	0.706941i	$0.249925\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	52.6613	−	105.938i	0.0870435	−	0.175104i
$606$		0			0
$607$	221.431	−	127.843i	0.364796	−	0.210615i	−0.306386	−	0.951907i	$-0.599120\pi$
$607$	221.431	−	127.843i	0.364796	−	0.210615i	0.671183	+	0.741292i	$0.265787\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			550.253i			0.900578i
$612$		0			0
$613$			423.661i			0.691127i	0.938395	+	0.345564i	$0.112312\pi$
$613$			423.661i			0.691127i	−0.938395	+	0.345564i	$0.887688\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	393.873	+	682.208i	0.638368	+	1.10569i	0.985791	+	0.167977i	$0.0537234\pi$
$617$	393.873	+	682.208i	0.638368	+	1.10569i	−0.347423	+	0.937708i	$0.612943\pi$
$618$		0			0
$619$	297.478	−	515.247i	0.480578	−	0.832386i	−0.519174	−	0.854669i	$-0.673760\pi$
$619$	297.478	−	515.247i	0.480578	−	0.832386i	0.999752	+	0.0222831i	$0.00709353\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−184.896	+	320.249i	−0.296783	+	0.514043i
$624$		0			0
$625$	−436.296	−	447.516i	−0.698074	−	0.716026i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			31.5351i			0.0501353i
$630$		0			0
$631$	265.641			0.420984			0.210492	−	0.977596i	$-0.432493\pi$
$631$	265.641			0.420984			0.210492	+	0.977596i	$0.432493\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−915.077	+	607.243i	−1.44107	+	0.956288i
$636$		0			0
$637$	−200.363	−	115.680i	−0.314542	−	0.181601i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−467.619	−	269.980i	−0.729515	−	0.421185i	0.0887300	−	0.996056i	$-0.471719\pi$
$641$	−467.619	−	269.980i	−0.729515	−	0.421185i	−0.818245	+	0.574870i	$0.805053\pi$
$642$		0			0
$643$	−311.044	+	179.581i	−0.483738	+	0.279286i	−0.721973	−	0.691921i	$-0.756765\pi$
$643$	−311.044	+	179.581i	−0.483738	+	0.279286i	0.238235	+	0.971208i	$0.423431\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−210.526			−0.325387			−0.162694	−	0.986677i	$-0.552018\pi$
$647$	−210.526			−0.325387			−0.162694	+	0.986677i	$0.552018\pi$
$648$		0			0
$649$	−222.529			−0.342879
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−76.7446	−	132.926i	−0.117526	−	0.203561i	0.801261	−	0.598316i	$-0.204163\pi$
$653$	−76.7446	−	132.926i	−0.117526	−	0.203561i	−0.918787	+	0.394754i	$0.870830\pi$
$654$		0			0
$655$	680.573	+	338.311i	1.03904	+	0.516505i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	491.886	+	283.991i	0.746413	+	0.430942i	0.824397	−	0.566013i	$-0.191515\pi$
$659$	491.886	+	283.991i	0.746413	+	0.430942i	−0.0779831	+	0.996955i	$0.524848\pi$
$660$		0			0
$661$	−1.42969	−	2.47629i	−0.00216292	−	0.00374628i	0.864942	−	0.501872i	$-0.167355\pi$
$661$	−1.42969	−	2.47629i	−0.00216292	−	0.00374628i	−0.867105	+	0.498126i	$0.834022\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	23.8772	+	382.912i	0.0359056	+	0.575807i
$666$		0			0
$667$		−	57.3090i		−	0.0859205i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−198.974	+	114.878i	−0.296533	+	0.171204i
$672$		0			0
$673$	−224.499	−	129.614i	−0.333579	−	0.192592i	0.323850	−	0.946108i	$-0.395023\pi$
$673$	−224.499	−	129.614i	−0.333579	−	0.192592i	−0.657429	+	0.753517i	$0.728356\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−587.168	+	1017.00i	−0.867308	+	1.50222i	−0.00257142	+	0.999997i	$0.500819\pi$
$677$	−587.168	+	1017.00i	−0.867308	+	1.50222i	−0.864737	+	0.502225i	$0.832515\pi$
$678$		0			0
$679$	303.353	+	525.423i	0.446764	+	0.773818i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−118.941			−0.174145			−0.0870725	−	0.996202i	$-0.527751\pi$
$683$	−118.941			−0.174145			−0.0870725	+	0.996202i	$0.527751\pi$
$684$		0			0
$685$	1006.53	−	62.7641i	1.46939	−	0.0916264i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	524.122	−	302.602i	0.760699	−	0.439190i
$690$		0			0
$691$	24.8875	−	43.1064i	0.0360166	−	0.0623826i	−0.847455	−	0.530867i	$-0.821866\pi$
$691$	24.8875	−	43.1064i	0.0360166	−	0.0623826i	0.883472	+	0.468484i	$0.155200\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	370.252	−	744.828i	0.532736	−	1.07169i
$696$		0			0
$697$	−32.3498	+	18.6772i	−0.0464129	+	0.0267965i
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	1120.97i		−	1.59910i	−0.600600	−	0.799550i	$-0.705071\pi$
$701$		−	1120.97i		−	1.59910i	0.600600	−	0.799550i	$-0.294929\pi$
$702$		0			0
$703$		−	415.182i		−	0.590586i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	302.503	+	523.951i	0.427869	+	0.741090i
$708$		0			0
$709$	138.892	−	240.568i	0.195899	−	0.339307i	−0.751296	−	0.659965i	$-0.770571\pi$
$709$	138.892	−	240.568i	0.195899	−	0.339307i	0.947195	+	0.320659i	$0.103904\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	6.27392	−	10.8667i	0.00879933	−	0.0152409i
$714$		0			0
$715$	−281.585	−	424.330i	−0.393825	−	0.593469i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	450.600i		−	0.626704i	−0.949637	−	0.313352i	$-0.898548\pi$
$719$		−	450.600i		−	0.626704i	0.949637	−	0.313352i	$-0.101452\pi$
$720$		0			0
$721$	−593.804			−0.823584
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−478.260	+	362.182i	−0.659668	+	0.499561i
$726$		0			0
$727$	404.860	+	233.746i	0.556891	+	0.321521i	0.751897	−	0.659281i	$-0.229139\pi$
$727$	404.860	+	233.746i	0.556891	+	0.321521i	−0.195006	+	0.980802i	$0.562473\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	27.6537	+	15.9659i	0.0378300	+	0.0218412i
$732$		0			0
$733$	−602.440	+	347.819i	−0.821883	+	0.474514i	−0.851065	−	0.525060i	$-0.824043\pi$
$733$	−602.440	+	347.819i	−0.821883	+	0.474514i	0.0291824	+	0.999574i	$0.490710\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	975.314			1.32336
$738$		0			0
$739$	−521.831			−0.706132			−0.353066	−	0.935598i	$-0.614861\pi$
$739$	−521.831			−0.706132			−0.353066	+	0.935598i	$0.614861\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−470.653	−	815.194i	−0.633449	−	1.09717i	−0.986841	−	0.161691i	$-0.948305\pi$
$743$	−470.653	−	815.194i	−0.633449	−	1.09717i	0.353393	−	0.935475i	$-0.385028\pi$
$744$		0			0
$745$	−843.086	−	419.096i	−1.13166	−	0.562545i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−113.301	−	65.4142i	−0.151269	−	0.0873353i
$750$		0			0
$751$	359.733	+	623.075i	0.479005	+	0.829661i	0.999710	−	0.0240757i	$-0.00766428\pi$
$751$	359.733	+	623.075i	0.479005	+	0.829661i	−0.520705	+	0.853737i	$0.674331\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	262.955	−	16.3971i	0.348285	−	0.0217180i
$756$		0			0
$757$			578.171i			0.763767i	0.924211	+	0.381883i	$0.124724\pi$
$757$			578.171i			0.763767i	−0.924211	+	0.381883i	$0.875276\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−319.176	+	184.276i	−0.419416	+	0.242150i	−0.694828	−	0.719176i	$-0.744519\pi$
$761$	−319.176	+	184.276i	−0.419416	+	0.242150i	0.275411	+	0.961327i	$0.411186\pi$
$762$		0			0
$763$	59.0841	+	34.1122i	0.0774366	+	0.0447080i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−78.3384	+	135.686i	−0.102136	+	0.176905i
$768$		0			0
$769$	517.820	+	896.891i	0.673369	+	1.16631i	0.976943	+	0.213501i	$0.0684866\pi$
$769$	517.820	+	896.891i	0.673369	+	1.16631i	−0.303574	+	0.952808i	$0.598180\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	−778.625			−1.00728			−0.503638	−	0.863915i	$-0.668006\pi$
$773$	−778.625			−1.00728			−0.503638	+	0.863915i	$0.668006\pi$
$774$		0			0
$775$	−130.336	+	16.3182i	−0.168176	+	0.0210557i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	425.908	−	245.898i	0.546737	−	0.315659i
$780$		0			0
$781$	−132.227	+	229.025i	−0.169305	+	0.293245i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−716.631	−	356.235i	−0.912906	−	0.453803i
$786$		0			0
$787$	511.363	−	295.236i	0.649763	−	0.375141i	−0.138603	−	0.990348i	$-0.544261\pi$
$787$	511.363	−	295.236i	0.649763	−	0.375141i	0.788365	+	0.615207i	$0.210928\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			619.655i			0.783381i
$792$		0			0
$793$			161.765i			0.203991i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−512.749	−	888.107i	−0.643348	−	1.11431i	−0.984680	−	0.174369i	$-0.944211\pi$
$797$	−512.749	−	888.107i	−0.643348	−	1.11431i	0.341332	−	0.939943i	$-0.389122\pi$
$798$		0			0
$799$	40.6672	−	70.4376i	0.0508976	−	0.0881572i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	659.717	−	1142.66i	0.821565	−	1.42299i
$804$		0			0
$805$	46.3257	−	30.7417i	0.0575475	−	0.0381884i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	769.002i		−	0.950559i	−0.879835	−	0.475280i	$-0.842347\pi$
$809$		−	769.002i		−	0.950559i	0.879835	−	0.475280i	$-0.157653\pi$
$810$		0			0
$811$	699.204			0.862150			0.431075	−	0.902316i	$-0.358134\pi$
$811$	699.204			0.862150			0.431075	+	0.902316i	$0.358134\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1251.97	+	830.808i	−1.53616	+	1.01940i
$816$		0			0
$817$	−364.081	−	210.202i	−0.445631	−	0.257285i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−278.412	−	160.741i	−0.339113	−	0.195787i	0.320767	−	0.947158i	$-0.396059\pi$
$821$	−278.412	−	160.741i	−0.339113	−	0.195787i	−0.659880	+	0.751371i	$0.729393\pi$
$822$		0			0
$823$	1136.63	−	656.236i	1.38109	−	0.797371i	0.388799	−	0.921323i	$-0.372890\pi$
$823$	1136.63	−	656.236i	1.38109	−	0.797371i	0.992288	+	0.123951i	$0.0395567\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−1407.89			−1.70241			−0.851205	−	0.524834i	$-0.824127\pi$
$827$	−1407.89			−1.70241			−0.851205	+	0.524834i	$0.824127\pi$
$828$		0			0
$829$	−1307.92			−1.57771			−0.788855	−	0.614579i	$-0.789326\pi$
$829$	−1307.92			−1.57771			−0.788855	+	0.614579i	$0.789326\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	17.0989	+	29.6162i	0.0205269	+	0.0355537i
$834$		0			0
$835$	15.2522	−	30.6825i	0.0182661	−	0.0367456i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−239.370	−	138.200i	−0.285304	−	0.164720i	0.350518	−	0.936556i	$-0.386005\pi$
$839$	−239.370	−	138.200i	−0.285304	−	0.164720i	−0.635822	+	0.771836i	$0.719339\pi$
$840$		0			0
$841$	−132.574	−	229.624i	−0.157638	−	0.273037i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	485.500	−	30.2743i	0.574556	−	0.0358275i
$846$		0			0
$847$		−	110.168i		−	0.130068i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−52.1058	+	30.0833i	−0.0612289	+	0.0353505i
$852$		0			0
$853$	172.115	+	99.3707i	0.201776	+	0.116496i	0.597484	−	0.801881i	$-0.296167\pi$
$853$	172.115	+	99.3707i	0.201776	+	0.116496i	−0.395708	+	0.918377i	$0.629501\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−62.5994	+	108.425i	−0.0730448	+	0.126517i	−0.900234	−	0.435406i	$-0.856605\pi$
$857$	−62.5994	+	108.425i	−0.0730448	+	0.126517i	0.827190	+	0.561923i	$0.189938\pi$
$858$		0			0
$859$	707.245	+	1224.98i	0.823335	+	1.42606i	0.903185	+	0.429251i	$0.141222\pi$
$859$	707.245	+	1224.98i	0.823335	+	1.42606i	−0.0798501	+	0.996807i	$0.525444\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	753.685			0.873331			0.436665	−	0.899624i	$-0.356159\pi$
$863$	753.685			0.873331			0.436665	+	0.899624i	$0.356159\pi$
$864$		0			0
$865$	1039.53	−	64.8217i	1.20176	−	0.0749384i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	1397.77	−	807.003i	1.60848	−	0.928657i
$870$		0			0
$871$	343.347	−	594.694i	0.394198	−	0.682772i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−549.318	−	192.320i	−0.627792	−	0.219794i
$876$		0			0
$877$	−173.628	+	100.244i	−0.197980	+	0.114304i	−0.595713	−	0.803198i	$-0.703130\pi$
$877$	−173.628	+	100.244i	−0.197980	+	0.114304i	0.397733	+	0.917501i	$0.369797\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	755.616i		−	0.857680i	−0.903381	−	0.428840i	$-0.858922\pi$
$881$		−	755.616i		−	0.857680i	0.903381	−	0.428840i	$-0.141078\pi$
$882$		0			0
$883$		−	1582.03i		−	1.79166i	−0.444398	−	0.895829i	$-0.646582\pi$
$883$		−	1582.03i		−	1.79166i	0.444398	−	0.895829i	$-0.353418\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	0.533099	+	0.923355i	0.000601014	+	0.00104099i	0.866326	−	0.499479i	$-0.166475\pi$
$887$	0.533099	+	0.923355i	0.000601014	+	0.00104099i	−0.865725	+	0.500520i	$0.833142\pi$
$888$		0			0
$889$	−511.346	+	885.677i	−0.575192	+	0.996263i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−535.412	+	927.360i	−0.599565	+	1.03848i
$894$		0			0
$895$	96.7789	−	64.2223i	0.108133	−	0.0717567i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			126.083i			0.140249i
$900$		0			0
$901$	−89.4567			−0.0992860
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−716.162	−	1079.21i	−0.791339	−	1.19250i
$906$		0			0
$907$	−1031.97	−	595.808i	−1.13778	−	0.656900i	−0.191903	−	0.981414i	$-0.561466\pi$
$907$	−1031.97	−	595.808i	−1.13778	−	0.656900i	−0.945881	+	0.324514i	$0.894799\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−154.947	−	89.4589i	−0.170085	−	0.0981986i	0.412541	−	0.910939i	$-0.364641\pi$
$911$	−154.947	−	89.4589i	−0.170085	−	0.0981986i	−0.582626	+	0.812740i	$0.697975\pi$
$912$		0			0
$913$	−162.619	+	93.8882i	−0.178115	+	0.102835i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	707.747			0.771807
$918$		0			0
$919$	−1400.32			−1.52374			−0.761870	−	0.647729i	$-0.775719\pi$
$919$	−1400.32			−1.52374			−0.761870	+	0.647729i	$0.775719\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	93.0979	+	161.250i	0.100865	+	0.174702i
$924$		0			0
$925$	580.353	+	244.717i	0.627408	+	0.264559i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	361.308	+	208.601i	0.388922	+	0.224544i	0.681693	−	0.731638i	$-0.261244\pi$
$929$	361.308	+	208.601i	0.388922	+	0.224544i	−0.292771	+	0.956183i	$0.594577\pi$
$930$		0			0
$931$	−225.120	−	389.919i	−0.241804	−	0.418817i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	4.68483	+	75.1292i	0.00501051	+	0.0803521i
$936$		0			0
$937$		−	1300.03i		−	1.38744i	−0.720246	−	0.693719i	$-0.755971\pi$
$937$		−	1300.03i		−	1.38744i	0.720246	−	0.693719i	$-0.244029\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−1373.74	+	793.127i	−1.45987	+	0.842856i	−0.999004	−	0.0446150i	$-0.985794\pi$
$941$	−1373.74	+	793.127i	−1.45987	+	0.842856i	−0.460864	+	0.887471i	$0.652461\pi$
$942$		0			0
$943$	−61.7209	−	35.6346i	−0.0654517	−	0.0377885i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−512.429	+	887.553i	−0.541108	+	0.937226i	0.457733	+	0.889090i	$0.348662\pi$
$947$	−512.429	+	887.553i	−0.541108	+	0.937226i	−0.998841	+	0.0481364i	$0.984672\pi$
$948$		0			0
$949$	−464.490	−	804.520i	−0.489452	−	0.847755i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	1740.92			1.82678			0.913388	−	0.407091i	$-0.133457\pi$
$953$	1740.92			1.82678			0.913388	+	0.407091i	$0.133457\pi$
$954$		0			0
$955$	−52.8993	−	848.330i	−0.0553919	−	0.888303i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	813.301	−	469.560i	0.848072	−	0.489635i
$960$		0			0
$961$	466.697	−	808.343i	0.485637	−	0.841148i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	357.815	+	177.869i	0.370793	+	0.184320i
$966$		0			0
$967$	1051.97	−	607.356i	1.08787	−	0.628082i	0.154862	−	0.987936i	$-0.450507\pi$
$967$	1051.97	−	607.356i	1.08787	−	0.628082i	0.933009	+	0.359854i	$0.117173\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			1150.91i			1.18529i	0.805465	+	0.592644i	$0.201916\pi$
$971$			1150.91i			1.18529i	−0.805465	+	0.592644i	$0.798084\pi$
$972$		0			0
$973$		−	774.567i		−	0.796061i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	272.265	+	471.577i	0.278674	+	0.482678i	0.971056	−	0.238854i	$-0.0767716\pi$
$977$	272.265	+	471.577i	0.278674	+	0.482678i	−0.692381	+	0.721532i	$0.743438\pi$
$978$		0			0
$979$	477.619	−	827.260i	0.487864	−	0.845005i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−719.925	+	1246.95i	−0.732375	+	1.26851i	0.223491	+	0.974706i	$0.428255\pi$
$983$	−719.925	+	1246.95i	−0.732375	+	1.26851i	−0.955866	+	0.293804i	$0.905079\pi$
$984$		0			0
$985$	−583.891	−	879.886i	−0.592783	−	0.893286i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			60.9234i			0.0616010i
$990$		0			0
$991$	1104.64			1.11467			0.557334	−	0.830288i	$-0.311824\pi$
$991$	1104.64			1.11467			0.557334	+	0.830288i	$0.311824\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	486.418	+	733.002i	0.488863	+	0.736685i
$996$		0			0
$997$	−529.515	−	305.716i	−0.531108	−	0.306636i	0.210359	−	0.977624i	$-0.432537\pi$
$997$	−529.515	−	305.716i	−0.531108	−	0.306636i	−0.741468	+	0.670989i	$0.765870\pi$
$998$		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.269.7		48
3.2	odd	2	inner	1620.3.t.f.269.18		48
5.4	even	2	inner	1620.3.t.f.269.17		48
9.2	odd	6		1620.3.b.a.809.2	yes	24
9.4	even	3	inner	1620.3.t.f.1349.8		48
9.5	odd	6	inner	1620.3.t.f.1349.17		48
9.7	even	3		1620.3.b.a.809.23	yes	24
15.14	odd	2	inner	1620.3.t.f.269.8		48
45.4	even	6	inner	1620.3.t.f.1349.18		48
45.14	odd	6	inner	1620.3.t.f.1349.7		48
45.29	odd	6		1620.3.b.a.809.24	yes	24
45.34	even	6		1620.3.b.a.809.1	✓	24

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1620.3.b.a.809.1	✓	24	45.34	even	6
1620.3.b.a.809.2	yes	24	9.2	odd	6
1620.3.b.a.809.23	yes	24	9.7	even	3
1620.3.b.a.809.24	yes	24	45.29	odd	6
1620.3.t.f.269.7		48	1.1	even	1	trivial
1620.3.t.f.269.8		48	15.14	odd	2	inner
1620.3.t.f.269.17		48	5.4	even	2	inner
1620.3.t.f.269.18		48	3.2	odd	2	inner
1620.3.t.f.1349.7		48	45.14	odd	6	inner
1620.3.t.f.1349.8		48	9.4	even	3	inner
1620.3.t.f.1349.17		48	9.5	odd	6	inner
1620.3.t.f.1349.18		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)

\(f(q)\)	\(=\)	\(q+(-2.76464 - 4.16614i) q^{5} +(-4.03229 - 2.32805i) q^{7} +O(q^{10})\)	\(q+(-2.76464 - 4.16614i) q^{5} +(-4.03229 - 2.32805i) q^{7} +(10.4161 + 6.01375i) q^{11} +(7.33373 - 4.23413i) q^{13} -1.25172 q^{17} +16.4797 q^{19} +(-1.19409 - 2.06822i) q^{23} +(-9.71349 + 23.0358i) q^{25} +(20.7820 + 11.9985i) q^{29} +(2.62707 + 4.55023i) q^{31} +(1.44888 + 23.2353i) q^{35} -25.1935i q^{37} +(25.8444 - 14.9212i) q^{41} +(-22.0927 - 12.7552i) q^{43} +(-32.4891 + 56.2728i) q^{47} +(-13.6604 - 23.6605i) q^{49} +71.4673 q^{53} +(-3.74272 - 60.0210i) q^{55} +(-16.0229 + 9.25082i) q^{59} +(-9.55124 + 16.5432i) q^{61} +(-37.9151 - 18.8475i) q^{65} +(70.2262 - 40.5451i) q^{67} +21.9875i q^{71} -109.701i q^{73} +(-28.0006 - 48.4984i) q^{77} +(67.0965 - 116.214i) q^{79} +(-7.80612 + 13.5206i) q^{83} +(3.46055 + 5.21483i) q^{85} -79.4211i q^{89} -39.4290 q^{91} +(-45.5606 - 68.6569i) q^{95} +(-112.846 - 65.1518i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 48 q+O(q^{10}) \) 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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