LMFDB - Embedded newform 360.3.v.f.73.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [360,3,Mod(73,360)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("360.73"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(360, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 0, 0, 3])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 360 = 2^{3} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	360.v (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	no
Analytic conductor:	$9.80928951697$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 4x^{7} + 8x^{6} + 269x^{4} - 1116x^{3} + 2312x^{2} + 680x + 100 $ x^8 - 4x^7 + 8x^6 + 269x^4 - 1116x^3 + 2312x^2 + 680x + 100
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{5}\cdot 5 $
Twist minimal:	no (minimal twist has level 120)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			73.3
Root			$2.36263 - 2.36263i$ of defining polynomial
Character	$\chi$	$=$	360.73
Dual form			360.3.v.f.217.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(3.36263 - 3.70037i) q^{5} +(-8.78825 + 8.78825i) q^{7} -13.7482 q^{11} +(-4.88927 - 4.88927i) q^{13} +(-5.99029 + 5.99029i) q^{17} +25.5765i q^{19} +(-18.4257 - 18.4257i) q^{23} +(-2.38547 - 24.8859i) q^{25} +37.2222i q^{29} +31.6955 q^{31} +(2.96816 + 62.0714i) q^{35} +(-32.5046 + 32.5046i) q^{37} -36.7295 q^{41} +(-24.5780 - 24.5780i) q^{43} +(-20.4063 + 20.4063i) q^{47} -105.467i q^{49} +(-33.5559 - 33.5559i) q^{53} +(-46.2300 + 50.8734i) q^{55} -7.54615i q^{59} +43.6183 q^{61} +(-34.5329 + 1.65131i) q^{65} +(-60.1739 + 60.1739i) q^{67} +18.1632 q^{71} +(68.8218 + 68.8218i) q^{73} +(120.823 - 120.823i) q^{77} -22.2628i q^{79} +(0.00221809 + 0.00221809i) q^{83} +(2.02317 + 42.3094i) q^{85} -77.1943i q^{89} +85.9363 q^{91} +(94.6425 + 86.0043i) q^{95} +(29.4309 - 29.4309i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 12 q^{5} + 4 q^{7} - 32 q^{11} - 4 q^{13} - 52 q^{17} + 40 q^{23} - 84 q^{25} + 96 q^{31} - 24 q^{35} - 60 q^{37} + 152 q^{41} - 88 q^{43} + 16 q^{47} - 108 q^{53} + 116 q^{55} + 264 q^{61} - 164 q^{65}+ \cdots - 208 q^{97}+O(q^{100}) $ 8 * q + 12 * q^5 + 4 * q^7 - 32 * q^11 - 4 * q^13 - 52 * q^17 + 40 * q^23 - 84 * q^25 + 96 * q^31 - 24 * q^35 - 60 * q^37 + 152 * q^41 - 88 * q^43 + 16 * q^47 - 108 * q^53 + 116 * q^55 + 264 * q^61 - 164 * q^65 - 216 * q^67 + 240 * q^71 - 208 * q^73 - 168 * q^77 - 336 * q^83 - 12 * q^85 + 592 * q^91 + 128 * q^95 - 208 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/360\mathbb{Z}\right)^\times$.

$n$	$181$	$217$	$271$	$281$
$\chi(n)$	$1$	$e\left(\frac{3}{4}\right)$	$1$	$1$

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
$5$	3.36263	−	3.70037i	0.672526	−	0.740074i
$5$	3.36263	−	3.70037i	0.672526	−	0.740074i
$6$		0			0
$7$	−8.78825	+	8.78825i	−1.25546	+	1.25546i	−0.302229	+	0.953235i	$0.597731\pi$
$7$	−8.78825	+	8.78825i	−1.25546	+	1.25546i	−0.953235	+	0.302229i	$0.902269\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−13.7482			−1.24984			−0.624918	−	0.780691i	$-0.714867\pi$
$11$	−13.7482			−1.24984			−0.624918	+	0.780691i	$0.714867\pi$
$12$		0			0
$13$	−4.88927	−	4.88927i	−0.376098	−	0.376098i	0.493594	−	0.869692i	$-0.335683\pi$
$13$	−4.88927	−	4.88927i	−0.376098	−	0.376098i	−0.869692	+	0.493594i	$0.835683\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−5.99029	+	5.99029i	−0.352370	+	0.352370i	−0.860991	−	0.508621i	$-0.830156\pi$
$17$	−5.99029	+	5.99029i	−0.352370	+	0.352370i	0.508621	+	0.860991i	$0.330156\pi$
$18$		0			0
$19$			25.5765i			1.34613i	0.739583	+	0.673066i	$0.235023\pi$
$19$			25.5765i			1.34613i	−0.739583	+	0.673066i	$0.764977\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−18.4257	−	18.4257i	−0.801118	−	0.801118i	0.182152	−	0.983270i	$-0.441694\pi$
$23$	−18.4257	−	18.4257i	−0.801118	−	0.801118i	−0.983270	+	0.182152i	$0.941694\pi$
$24$		0			0
$25$	−2.38547	−	24.8859i	−0.0954189	−	0.995437i
$26$		0			0
$27$		0			0
$28$		0			0
$29$			37.2222i			1.28352i	0.766904	+	0.641762i	$0.221796\pi$
$29$			37.2222i			1.28352i	−0.766904	+	0.641762i	$0.778204\pi$
$30$		0			0
$31$	31.6955			1.02243			0.511217	−	0.859452i	$-0.329195\pi$
$31$	31.6955			1.02243			0.511217	+	0.859452i	$0.329195\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.96816	+	62.0714i	0.0848047	+	1.77347i
$36$		0			0
$37$	−32.5046	+	32.5046i	−0.878503	+	0.878503i	−0.993380	−	0.114877i	$-0.963353\pi$
$37$	−32.5046	+	32.5046i	−0.878503	+	0.878503i	0.114877	+	0.993380i	$0.463353\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−36.7295			−0.895842			−0.447921	−	0.894073i	$-0.647835\pi$
$41$	−36.7295			−0.895842			−0.447921	+	0.894073i	$0.647835\pi$
$42$		0			0
$43$	−24.5780	−	24.5780i	−0.571581	−	0.571581i	0.360989	−	0.932570i	$-0.382439\pi$
$43$	−24.5780	−	24.5780i	−0.571581	−	0.571581i	−0.932570	+	0.360989i	$0.882439\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−20.4063	+	20.4063i	−0.434177	+	0.434177i	−0.890046	−	0.455870i	$-0.849328\pi$
$47$	−20.4063	+	20.4063i	−0.434177	+	0.434177i	0.455870	+	0.890046i	$0.349328\pi$
$48$		0			0
$49$		−	105.467i		−	2.15238i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−33.5559	−	33.5559i	−0.633129	−	0.633129i	0.315722	−	0.948852i	$-0.397753\pi$
$53$	−33.5559	−	33.5559i	−0.633129	−	0.633129i	−0.948852	+	0.315722i	$0.897753\pi$
$54$		0			0
$55$	−46.2300	+	50.8734i	−0.840546	+	0.924971i
$56$		0			0
$57$		0			0
$58$		0			0
$59$		−	7.54615i		−	0.127901i	−0.997953	−	0.0639504i	$-0.979630\pi$
$59$		−	7.54615i		−	0.127901i	0.997953	−	0.0639504i	$-0.0203699\pi$
$60$		0			0
$61$	43.6183			0.715054			0.357527	−	0.933903i	$-0.383620\pi$
$61$	43.6183			0.715054			0.357527	+	0.933903i	$0.383620\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−34.5329	+	1.65131i	−0.531276	+	0.0254048i
$66$		0			0
$67$	−60.1739	+	60.1739i	−0.898118	+	0.898118i	−0.995270	−	0.0971517i	$-0.969027\pi$
$67$	−60.1739	+	60.1739i	−0.898118	+	0.898118i	0.0971517	+	0.995270i	$0.469027\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	18.1632			0.255820			0.127910	−	0.991786i	$-0.459173\pi$
$71$	18.1632			0.255820			0.127910	+	0.991786i	$0.459173\pi$
$72$		0			0
$73$	68.8218	+	68.8218i	0.942764	+	0.942764i	0.998448	−	0.0556839i	$-0.0177339\pi$
$73$	68.8218	+	68.8218i	0.942764	+	0.942764i	−0.0556839	+	0.998448i	$0.517734\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	120.823	−	120.823i	1.56912	−	1.56912i
$78$		0			0
$79$		−	22.2628i		−	0.281807i	−0.990023	−	0.140904i	$-0.954999\pi$
$79$		−	22.2628i		−	0.281807i	0.990023	−	0.140904i	$-0.0450007\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	0.00221809	+	0.00221809i	2.67240e−5	+	2.67240e-5i	0.707120	−	0.707093i	$-0.249994\pi$
$83$	0.00221809	+	0.00221809i	2.67240e−5	+	2.67240e-5i	−0.707093	+	0.707120i	$0.749994\pi$
$84$		0			0
$85$	2.02317	+	42.3094i	0.0238021	+	0.497758i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	77.1943i		−	0.867352i	−0.901069	−	0.433676i	$-0.857216\pi$
$89$		−	77.1943i		−	0.867352i	0.901069	−	0.433676i	$-0.142784\pi$
$90$		0			0
$91$	85.9363			0.944355
$92$		0			0
$93$		0			0
$94$		0			0
$95$	94.6425	+	86.0043i	0.996237	+	0.905308i
$96$		0			0
$97$	29.4309	−	29.4309i	0.303411	−	0.303411i	−0.538936	−	0.842347i	$-0.681173\pi$
$97$	29.4309	−	29.4309i	0.303411	−	0.303411i	0.842347	+	0.538936i	$0.181173\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	178.867			1.77096			0.885479	−	0.464679i	$-0.153830\pi$
$101$	178.867			1.77096			0.885479	+	0.464679i	$0.153830\pi$
$102$		0			0
$103$	41.0972	+	41.0972i	0.399002	+	0.399002i	0.877881	−	0.478879i	$-0.158957\pi$
$103$	41.0972	+	41.0972i	0.399002	+	0.399002i	−0.478879	+	0.877881i	$0.658957\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	36.8104	−	36.8104i	0.344022	−	0.344022i	−0.513855	−	0.857877i	$-0.671783\pi$
$107$	36.8104	−	36.8104i	0.344022	−	0.344022i	0.857877	+	0.513855i	$0.171783\pi$
$108$		0			0
$109$		−	85.4905i		−	0.784316i	−0.919898	−	0.392158i	$-0.871729\pi$
$109$		−	85.4905i		−	0.784316i	0.919898	−	0.392158i	$-0.128271\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	15.6917	+	15.6917i	0.138864	+	0.138864i	0.773122	−	0.634258i	$-0.218694\pi$
$113$	15.6917	+	15.6917i	0.138864	+	0.138864i	−0.634258	+	0.773122i	$0.718694\pi$
$114$		0			0
$115$	−130.141	+	6.22314i	−1.13166	+	0.0541143i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	105.288i		−	0.884777i
$120$		0			0
$121$	68.0127			0.562088
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−100.109	−	74.8550i	−0.800869	−	0.598840i
$126$		0			0
$127$	127.554	−	127.554i	1.00436	−	1.00436i	0.00436887	−	0.999990i	$-0.498609\pi$
$127$	127.554	−	127.554i	1.00436	−	1.00436i	0.999990	−	0.00436887i	$-0.00139066\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−123.810			−0.945117			−0.472559	−	0.881299i	$-0.656670\pi$
$131$	−123.810			−0.945117			−0.472559	+	0.881299i	$0.656670\pi$
$132$		0			0
$133$	−224.773	−	224.773i	−1.69002	−	1.69002i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−173.145	+	173.145i	−1.26383	+	1.26383i	−0.314609	+	0.949221i	$0.601873\pi$
$137$	−173.145	+	173.145i	−1.26383	+	1.26383i	−0.949221	+	0.314609i	$0.898127\pi$
$138$		0			0
$139$			1.06696i			0.00767601i	0.999993	+	0.00383800i	$0.00122168\pi$
$139$			1.06696i			0.00767601i	−0.999993	+	0.00383800i	$0.998778\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	67.2186	+	67.2186i	0.470060	+	0.470060i
$144$		0			0
$145$	137.736	+	125.164i	0.949903	+	0.863203i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			65.2505i			0.437923i	0.975734	+	0.218961i	$0.0702669\pi$
$149$			65.2505i			0.437923i	−0.975734	+	0.218961i	$0.929733\pi$
$150$		0			0
$151$	157.183			1.04095			0.520473	−	0.853878i	$-0.325756\pi$
$151$	157.183			1.04095			0.520473	+	0.853878i	$0.325756\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	106.580	−	117.285i	0.687613	−	0.756677i
$156$		0			0
$157$	−121.763	+	121.763i	−0.775561	+	0.775561i	−0.979073	−	0.203512i	$-0.934765\pi$
$157$	−121.763	+	121.763i	−0.775561	+	0.775561i	0.203512	+	0.979073i	$0.434765\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	323.860			2.01155
$162$		0			0
$163$	−196.095	−	196.095i	−1.20304	−	1.20304i	−0.973238	−	0.229801i	$-0.926193\pi$
$163$	−196.095	−	196.095i	−1.20304	−	1.20304i	−0.229801	−	0.973238i	$-0.573807\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	26.4088	−	26.4088i	0.158136	−	0.158136i	−0.623604	−	0.781740i	$-0.714332\pi$
$167$	26.4088	−	26.4088i	0.158136	−	0.158136i	0.781740	+	0.623604i	$0.214332\pi$
$168$		0			0
$169$		−	121.190i		−	0.717101i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	8.69052	+	8.69052i	0.0502342	+	0.0502342i	0.731778	−	0.681543i	$-0.238691\pi$
$173$	8.69052	+	8.69052i	0.0502342	+	0.0502342i	−0.681543	+	0.731778i	$0.738691\pi$
$174$		0			0
$175$	239.668	+	197.740i	1.36953	+	1.12994i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	9.67002i		−	0.0540225i	−0.999635	−	0.0270112i	$-0.991401\pi$
$179$		−	9.67002i		−	0.0540225i	0.999635	−	0.0270112i	$-0.00859899\pi$
$180$		0			0
$181$	249.048			1.37595			0.687977	−	0.725733i	$-0.258499\pi$
$181$	249.048			1.37595			0.687977	+	0.725733i	$0.258499\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	10.9782	+	229.580i	0.0593415	+	1.24097i
$186$		0			0
$187$	82.3557	−	82.3557i	0.440405	−	0.440405i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	162.349			0.849996			0.424998	−	0.905194i	$-0.360275\pi$
$191$	162.349			0.849996			0.424998	+	0.905194i	$0.360275\pi$
$192$		0			0
$193$	−8.11418	−	8.11418i	−0.0420424	−	0.0420424i	0.685773	−	0.727815i	$-0.259464\pi$
$193$	−8.11418	−	8.11418i	−0.0420424	−	0.0420424i	−0.727815	+	0.685773i	$0.759464\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−229.057	+	229.057i	−1.16272	+	1.16272i	−0.178847	+	0.983877i	$0.557237\pi$
$197$	−229.057	+	229.057i	−1.16272	+	1.16272i	−0.983877	+	0.178847i	$0.942763\pi$
$198$		0			0
$199$			296.892i			1.49192i	0.665992	+	0.745959i	$0.268008\pi$
$199$			296.892i			1.49192i	−0.665992	+	0.745959i	$0.731992\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−327.118	−	327.118i	−1.61142	−	1.61142i
$204$		0			0
$205$	−123.508	+	135.913i	−0.602476	+	0.662989i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		−	351.631i		−	1.68244i
$210$		0			0
$211$	−273.424			−1.29585			−0.647923	−	0.761706i	$-0.724362\pi$
$211$	−273.424			−1.29585			−0.647923	+	0.761706i	$0.724362\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−173.594	+	8.30102i	−0.807415	+	0.0386094i
$216$		0			0
$217$	−278.548	+	278.548i	−1.28363	+	1.28363i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	58.5764			0.265051
$222$		0			0
$223$	−232.809	−	232.809i	−1.04399	−	1.04399i	−0.998987	−	0.0450002i	$-0.985671\pi$
$223$	−232.809	−	232.809i	−1.04399	−	1.04399i	−0.0450002	−	0.998987i	$-0.514329\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−105.206	+	105.206i	−0.463465	+	0.463465i	−0.899789	−	0.436325i	$-0.856280\pi$
$227$	−105.206	+	105.206i	−0.463465	+	0.463465i	0.436325	+	0.899789i	$0.356280\pi$
$228$		0			0
$229$			154.353i			0.674032i	0.941499	+	0.337016i	$0.109418\pi$
$229$			154.353i			0.674032i	−0.941499	+	0.337016i	$0.890582\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.6237	−	16.6237i	−0.0713464	−	0.0713464i	0.670533	−	0.741880i	$-0.266065\pi$
$233$	−16.6237	−	16.6237i	−0.0713464	−	0.0713464i	−0.741880	+	0.670533i	$0.766065\pi$
$234$		0			0
$235$	6.89207	+	144.130i	0.0293280	+	0.613318i
$236$		0			0
$237$		0			0
$238$		0			0
$239$			97.2929i			0.407083i	0.979066	+	0.203542i	$0.0652452\pi$
$239$			97.2929i			0.407083i	−0.979066	+	0.203542i	$0.934755\pi$
$240$		0			0
$241$	−308.012			−1.27806			−0.639029	−	0.769183i	$-0.720663\pi$
$241$	−308.012			−1.27806			−0.639029	+	0.769183i	$0.720663\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−390.266	−	354.645i	−1.59292	−	1.44753i
$246$		0			0
$247$	125.051	−	125.051i	0.506277	−	0.506277i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−155.213			−0.618379			−0.309190	−	0.951000i	$-0.600058\pi$
$251$	−155.213			−0.618379			−0.309190	+	0.951000i	$0.600058\pi$
$252$		0			0
$253$	253.320	+	253.320i	1.00127	+	1.00127i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−206.538	+	206.538i	−0.803648	+	0.803648i	−0.983664	−	0.180015i	$-0.942385\pi$
$257$	−206.538	+	206.538i	−0.803648	+	0.803648i	0.180015	+	0.983664i	$0.442385\pi$
$258$		0			0
$259$		−	571.317i		−	2.20586i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	236.194	+	236.194i	0.898075	+	0.898075i	0.995266	−	0.0971908i	$-0.0309857\pi$
$263$	236.194	+	236.194i	0.898075	+	0.898075i	−0.0971908	+	0.995266i	$0.530986\pi$
$264$		0			0
$265$	−237.005	+	11.3332i	−0.894358	+	0.0427669i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			376.885i			1.40106i	0.713623	+	0.700530i	$0.247053\pi$
$269$			376.885i			1.40106i	−0.713623	+	0.700530i	$0.752947\pi$
$270$		0			0
$271$	196.495			0.725074			0.362537	−	0.931969i	$-0.381911\pi$
$271$	196.495			0.725074			0.362537	+	0.931969i	$0.381911\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	32.7959	+	342.136i	0.119258	+	1.24413i
$276$		0			0
$277$	90.3781	−	90.3781i	0.326275	−	0.326275i	−0.524893	−	0.851168i	$-0.675895\pi$
$277$	90.3781	−	90.3781i	0.326275	−	0.326275i	0.851168	+	0.524893i	$0.175895\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	−124.506			−0.443083			−0.221542	−	0.975151i	$-0.571109\pi$
$281$	−124.506			−0.443083			−0.221542	+	0.975151i	$0.571109\pi$
$282$		0			0
$283$	−91.3284	−	91.3284i	−0.322715	−	0.322715i	0.527093	−	0.849808i	$-0.323282\pi$
$283$	−91.3284	−	91.3284i	−0.322715	−	0.322715i	−0.849808	+	0.527093i	$0.823282\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	322.788	−	322.788i	1.12470	−	1.12470i
$288$		0			0
$289$			217.233i			0.751670i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	311.623	+	311.623i	1.06356	+	1.06356i	0.997838	+	0.0657234i	$0.0209355\pi$
$293$	311.623	+	311.623i	1.06356	+	1.06356i	0.0657234	+	0.997838i	$0.479065\pi$
$294$		0			0
$295$	−27.9235	−	25.3749i	−0.0946560	−	0.0860166i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			180.177i			0.602598i
$300$		0			0
$301$	431.995			1.43520
$302$		0			0
$303$		0			0
$304$		0			0
$305$	146.672	−	161.404i	0.480892	−	0.529193i
$306$		0			0
$307$	−98.1938	+	98.1938i	−0.319850	+	0.319850i	−0.848709	−	0.528860i	$-0.822620\pi$
$307$	−98.1938	+	98.1938i	−0.319850	+	0.319850i	0.528860	+	0.848709i	$0.322620\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−429.889			−1.38228			−0.691141	−	0.722720i	$-0.742891\pi$
$311$	−429.889			−1.38228			−0.691141	+	0.722720i	$0.742891\pi$
$312$		0			0
$313$	65.4997	+	65.4997i	0.209264	+	0.209264i	0.803955	−	0.594691i	$-0.202725\pi$
$313$	65.4997	+	65.4997i	0.209264	+	0.209264i	−0.594691	+	0.803955i	$0.702725\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−174.272	+	174.272i	−0.549755	+	0.549755i	−0.926370	−	0.376615i	$-0.877088\pi$
$317$	−174.272	+	174.272i	−0.549755	+	0.549755i	0.376615	+	0.926370i	$0.377088\pi$
$318$		0			0
$319$		−	511.738i		−	1.60419i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−153.211	−	153.211i	−0.474337	−	0.474337i
$324$		0			0
$325$	−110.011	+	133.337i	−0.338495	+	0.410269i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		−	358.671i		−	1.09019i
$330$		0			0
$331$	−380.789			−1.15042			−0.575210	−	0.818006i	$-0.695080\pi$
$331$	−380.789			−1.15042			−0.575210	+	0.818006i	$0.695080\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	20.3233	+	425.008i	0.0606665	+	1.26868i
$336$		0			0
$337$	245.960	−	245.960i	0.729851	−	0.729851i	−0.240739	−	0.970590i	$-0.577390\pi$
$337$	245.960	−	245.960i	0.729851	−	0.729851i	0.970590	+	0.240739i	$0.0773897\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−435.755			−1.27787
$342$		0			0
$343$	496.244	+	496.244i	1.44678	+	1.44678i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	172.320	−	172.320i	0.496600	−	0.496600i	−0.413778	−	0.910378i	$-0.635791\pi$
$347$	172.320	−	172.320i	0.496600	−	0.496600i	0.910378	+	0.413778i	$0.135791\pi$
$348$		0			0
$349$		−	221.895i		−	0.635803i	−0.948124	−	0.317902i	$-0.897022\pi$
$349$		−	221.895i		−	0.635803i	0.948124	−	0.317902i	$-0.102978\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−53.4623	−	53.4623i	−0.151451	−	0.151451i	0.627315	−	0.778766i	$-0.284154\pi$
$353$	−53.4623	−	53.4623i	−0.151451	−	0.151451i	−0.778766	+	0.627315i	$0.784154\pi$
$354$		0			0
$355$	61.0761	−	67.2106i	0.172045	−	0.189326i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	49.7577i		−	0.138601i	−0.997596	−	0.0693004i	$-0.977923\pi$
$359$		−	49.7577i		−	0.138601i	0.997596	−	0.0693004i	$-0.0220767\pi$
$360$		0			0
$361$	−293.158			−0.812071
$362$		0			0
$363$		0			0
$364$		0			0
$365$	486.088	−	23.2440i	1.33175	−	0.0636823i
$366$		0			0
$367$	−56.5205	+	56.5205i	−0.154007	+	0.154007i	−0.779905	−	0.625898i	$-0.784733\pi$
$367$	−56.5205	+	56.5205i	−0.154007	+	0.154007i	0.625898	+	0.779905i	$0.284733\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	589.795			1.58974
$372$		0			0
$373$	−143.430	−	143.430i	−0.384531	−	0.384531i	0.488200	−	0.872732i	$-0.337654\pi$
$373$	−143.430	−	143.430i	−0.384531	−	0.384531i	−0.872732	+	0.488200i	$0.837654\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	181.989	−	181.989i	0.482731	−	0.482731i
$378$		0			0
$379$			659.728i			1.74071i	0.492427	+	0.870354i	$0.336110\pi$
$379$			659.728i			1.74071i	−0.492427	+	0.870354i	$0.663890\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−402.216	−	402.216i	−1.05017	−	1.05017i	−0.998673	−	0.0514981i	$-0.983600\pi$
$383$	−402.216	−	402.216i	−1.05017	−	1.05017i	−0.0514981	−	0.998673i	$-0.516400\pi$
$384$		0			0
$385$	−40.8069	−	853.369i	−0.105992	−	2.21654i
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	206.222i		−	0.530133i	−0.964230	−	0.265067i	$-0.914606\pi$
$389$		−	206.222i		−	0.530133i	0.964230	−	0.265067i	$-0.0853940\pi$
$390$		0			0
$391$	220.751			0.564580
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−82.3804	−	74.8614i	−0.208558	−	0.189522i
$396$		0			0
$397$	−72.3916	+	72.3916i	−0.182347	+	0.182347i	−0.792378	−	0.610031i	$-0.791157\pi$
$397$	−72.3916	+	72.3916i	−0.182347	+	0.182347i	0.610031	+	0.792378i	$0.291157\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	489.191			1.21993			0.609964	−	0.792429i	$-0.291184\pi$
$401$	489.191			1.21993			0.609964	+	0.792429i	$0.291184\pi$
$402$		0			0
$403$	−154.968	−	154.968i	−0.384535	−	0.384535i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	446.879	−	446.879i	1.09798	−	1.09798i
$408$		0			0
$409$			724.079i			1.77037i	0.465244	+	0.885183i	$0.345967\pi$
$409$			724.079i			1.77037i	−0.465244	+	0.885183i	$0.654033\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	66.3174	+	66.3174i	0.160575	+	0.160575i
$414$		0			0
$415$	0.0156664		0.000749143i	3.77503e−5		1.80516e-6i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	20.3800i		−	0.0486395i	−0.999704	−	0.0243198i	$-0.992258\pi$
$419$		−	20.3800i		−	0.0486395i	0.999704	−	0.0243198i	$-0.00774199\pi$
$420$		0			0
$421$	471.915			1.12094			0.560469	−	0.828176i	$-0.310621\pi$
$421$	471.915			1.12094			0.560469	+	0.828176i	$0.310621\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	163.364	+	134.784i	0.384385	+	0.317140i
$426$		0			0
$427$	−383.329	+	383.329i	−0.897725	+	0.897725i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	433.846			1.00660			0.503302	−	0.864110i	$-0.332118\pi$
$431$	433.846			1.00660			0.503302	+	0.864110i	$0.332118\pi$
$432$		0			0
$433$	143.893	+	143.893i	0.332317	+	0.332317i	0.853466	−	0.521149i	$-0.174496\pi$
$433$	143.893	+	143.893i	0.332317	+	0.332317i	−0.521149	+	0.853466i	$0.674496\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	471.265	−	471.265i	1.07841	−	1.07841i
$438$		0			0
$439$		−	21.7905i		−	0.0496368i	−0.999692	−	0.0248184i	$-0.992099\pi$
$439$		−	21.7905i		−	0.0496368i	0.999692	−	0.0248184i	$-0.00790075\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−186.194	−	186.194i	−0.420302	−	0.420302i	0.465005	−	0.885308i	$-0.346052\pi$
$443$	−186.194	−	186.194i	−0.420302	−	0.420302i	−0.885308	+	0.465005i	$0.846052\pi$
$444$		0			0
$445$	−285.647	−	259.576i	−0.641904	−	0.583316i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			173.546i			0.386517i	0.981148	+	0.193258i	$0.0619056\pi$
$449$			173.546i			0.386517i	−0.981148	+	0.193258i	$0.938094\pi$
$450$		0			0
$451$	504.964			1.11965
$452$		0			0
$453$		0			0
$454$		0			0
$455$	288.972	−	317.996i	0.635103	−	0.698893i
$456$		0			0
$457$	−195.883	+	195.883i	−0.428628	+	0.428628i	−0.888161	−	0.459533i	$-0.848017\pi$
$457$	−195.883	+	195.883i	−0.428628	+	0.428628i	0.459533	+	0.888161i	$0.348017\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−710.784			−1.54183			−0.770916	−	0.636937i	$-0.780201\pi$
$461$	−710.784			−1.54183			−0.770916	+	0.636937i	$0.780201\pi$
$462$		0			0
$463$	407.194	+	407.194i	0.879468	+	0.879468i	0.993479	−	0.114011i	$-0.0363700\pi$
$463$	407.194	+	407.194i	0.879468	+	0.879468i	−0.114011	+	0.993479i	$0.536370\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	46.4766	−	46.4766i	0.0995215	−	0.0995215i	−0.655593	−	0.755114i	$-0.727581\pi$
$467$	46.4766	−	46.4766i	0.0995215	−	0.0995215i	0.755114	+	0.655593i	$0.227581\pi$
$468$		0			0
$469$		−	1057.65i		−	2.25511i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	337.903	+	337.903i	0.714382	+	0.714382i
$474$		0			0
$475$	636.495	−	61.0120i	1.33999	−	0.128446i
$476$		0			0
$477$		0			0
$478$		0			0
$479$			632.568i			1.32060i	0.751001	+	0.660301i	$0.229571\pi$
$479$			632.568i			1.32060i	−0.751001	+	0.660301i	$0.770429\pi$
$480$		0			0
$481$	317.848			0.660806
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−9.94005	−	207.870i	−0.0204950	−	0.428599i
$486$		0			0
$487$	−24.5000	+	24.5000i	−0.0503080	+	0.0503080i	−0.731813	−	0.681505i	$-0.761326\pi$
$487$	−24.5000	+	24.5000i	−0.0503080	+	0.0503080i	0.681505	+	0.731813i	$0.261326\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−888.550			−1.80967			−0.904837	−	0.425759i	$-0.860007\pi$
$491$	−888.550			−1.80967			−0.904837	+	0.425759i	$0.860007\pi$
$492$		0			0
$493$	−222.972	−	222.972i	−0.452276	−	0.452276i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−159.623	+	159.623i	−0.321173	+	0.321173i
$498$		0			0
$499$			313.223i			0.627702i	0.949472	+	0.313851i	$0.101619\pi$
$499$			313.223i			0.627702i	−0.949472	+	0.313851i	$0.898381\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	22.1824	+	22.1824i	0.0441001	+	0.0441001i	0.728813	−	0.684713i	$-0.240072\pi$
$503$	22.1824	+	22.1824i	0.0441001	+	0.0441001i	−0.684713	+	0.728813i	$0.740072\pi$
$504$		0			0
$505$	601.462	−	661.873i	1.19101	−	1.31064i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	510.456i		−	1.00286i	−0.865198	−	0.501431i	$-0.832807\pi$
$509$		−	510.456i		−	1.00286i	0.865198	−	0.501431i	$-0.167193\pi$
$510$		0			0
$511$	−1209.65			−2.36721
$512$		0			0
$513$		0			0
$514$		0			0
$515$	290.270	−	13.8803i	0.563630	−	0.0269520i
$516$		0			0
$517$	280.550	−	280.550i	0.542649	−	0.542649i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	765.379			1.46906			0.734529	−	0.678578i	$-0.237403\pi$
$521$	765.379			1.46906			0.734529	+	0.678578i	$0.237403\pi$
$522$		0			0
$523$	−57.8350	−	57.8350i	−0.110583	−	0.110583i	0.649650	−	0.760233i	$-0.274915\pi$
$523$	−57.8350	−	57.8350i	−0.110583	−	0.110583i	−0.760233	+	0.649650i	$0.774915\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−189.865	+	189.865i	−0.360275	+	0.360275i
$528$		0			0
$529$			150.014i			0.283580i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	179.581	+	179.581i	0.336924	+	0.336924i
$534$		0			0
$535$	−12.4324	−	259.992i	−0.0232382	−	0.485966i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			1449.98i			2.69012i
$540$		0			0
$541$	−650.947			−1.20323			−0.601615	−	0.798786i	$-0.705476\pi$
$541$	−650.947			−1.20323			−0.601615	+	0.798786i	$0.705476\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−316.346	−	287.473i	−0.580452	−	0.527473i
$546$		0			0
$547$	200.024	−	200.024i	0.365674	−	0.365674i	−0.500223	−	0.865897i	$-0.666749\pi$
$547$	200.024	−	200.024i	0.365674	−	0.365674i	0.865897	+	0.500223i	$0.166749\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−952.014			−1.72779
$552$		0			0
$553$	195.651	+	195.651i	0.353799	+	0.353799i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−169.939	+	169.939i	−0.305097	+	0.305097i	−0.843004	−	0.537907i	$-0.819215\pi$
$557$	−169.939	+	169.939i	−0.305097	+	0.305097i	0.537907	+	0.843004i	$0.319215\pi$
$558$		0			0
$559$			240.337i			0.429941i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	288.641	+	288.641i	0.512684	+	0.512684i	0.915348	−	0.402664i	$-0.131916\pi$
$563$	288.641	+	288.641i	0.512684	+	0.512684i	−0.402664	+	0.915348i	$0.631916\pi$
$564$		0			0
$565$	110.830	−	5.29974i	0.196160	−	0.00938007i
$566$		0			0
$567$		0			0
$568$		0			0
$569$			119.228i			0.209540i	0.994496	+	0.104770i	$0.0334107\pi$
$569$			119.228i			0.209540i	−0.994496	+	0.104770i	$0.966589\pi$
$570$		0			0
$571$	−1070.64			−1.87503			−0.937513	−	0.347951i	$-0.886878\pi$
$571$	−1070.64			−1.87503			−0.937513	+	0.347951i	$0.886878\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−414.587	+	502.495i	−0.721021	+	0.873904i
$576$		0			0
$577$	572.086	−	572.086i	0.991484	−	0.991484i	−0.00848018	−	0.999964i	$-0.502699\pi$
$577$	572.086	−	572.086i	0.991484	−	0.991484i	0.999964	+	0.00848018i	$0.00269936\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−0.0389863			−6.71020e−5
$582$		0			0
$583$	461.332	+	461.332i	0.791307	+	0.791307i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	314.886	−	314.886i	0.536433	−	0.536433i	−0.386047	−	0.922479i	$-0.626160\pi$
$587$	314.886	−	314.886i	0.536433	−	0.536433i	0.922479	+	0.386047i	$0.126160\pi$
$588$		0			0
$589$			810.659i			1.37633i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−781.658	−	781.658i	−1.31814	−	1.31814i	−0.915246	−	0.402896i	$-0.868004\pi$
$593$	−781.658	−	781.658i	−1.31814	−	1.31814i	−0.402896	−	0.915246i	$-0.631996\pi$
$594$		0			0
$595$	−389.606	−	354.046i	−0.654800	−	0.595035i
$596$		0			0
$597$		0			0
$598$		0			0
$599$		−	423.280i		−	0.706645i	−0.935502	−	0.353322i	$-0.885052\pi$
$599$		−	423.280i		−	0.706645i	0.935502	−	0.353322i	$-0.114948\pi$
$600$		0			0
$601$	−68.1617			−0.113414			−0.0567069	−	0.998391i	$-0.518060\pi$
$601$	−68.1617			−0.113414			−0.0567069	+	0.998391i	$0.518060\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	228.701	−	251.672i	0.378019	−	0.415987i
$606$		0			0
$607$	186.390	−	186.390i	0.307068	−	0.307068i	−0.536703	−	0.843771i	$-0.680331\pi$
$607$	186.390	−	186.390i	0.307068	−	0.307068i	0.843771	+	0.536703i	$0.180331\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	199.544			0.326586
$612$		0			0
$613$	337.566	+	337.566i	0.550679	+	0.550679i	0.926637	−	0.375958i	$-0.122686\pi$
$613$	337.566	+	337.566i	0.550679	+	0.550679i	−0.375958	+	0.926637i	$0.622686\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−858.797	+	858.797i	−1.39189	+	1.39189i	−0.570808	+	0.821084i	$0.693370\pi$
$617$	−858.797	+	858.797i	−1.39189	+	1.39189i	−0.821084	+	0.570808i	$0.806630\pi$
$618$		0			0
$619$			202.653i			0.327388i	0.986511	+	0.163694i	$0.0523410\pi$
$619$			202.653i			0.327388i	−0.986511	+	0.163694i	$0.947659\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	678.403	+	678.403i	1.08893	+	1.08893i
$624$		0			0
$625$	−613.619	+	118.729i	−0.981790	+	0.189967i
$626$		0			0
$627$		0			0
$628$		0			0
$629$		−	389.424i		−	0.619116i
$630$		0			0
$631$	−536.853			−0.850797			−0.425398	−	0.905006i	$-0.639866\pi$
$631$	−536.853			−0.850797			−0.425398	+	0.905006i	$0.639866\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−43.0802	−	900.911i	−0.0678429	−	1.41876i
$636$		0			0
$637$	−515.656	+	515.656i	−0.809507	+	0.809507i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−665.993			−1.03899			−0.519495	−	0.854473i	$-0.673880\pi$
$641$	−665.993			−1.03899			−0.519495	+	0.854473i	$0.673880\pi$
$642$		0			0
$643$	−7.83317	−	7.83317i	−0.0121822	−	0.0121822i	0.700989	−	0.713172i	$-0.252742\pi$
$643$	−7.83317	−	7.83317i	−0.0121822	−	0.0121822i	−0.713172	+	0.700989i	$0.752742\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	287.426	−	287.426i	0.444244	−	0.444244i	−0.449191	−	0.893436i	$-0.648288\pi$
$647$	287.426	−	287.426i	0.444244	−	0.444244i	0.893436	+	0.449191i	$0.148288\pi$
$648$		0			0
$649$			103.746i			0.159855i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	587.685	+	587.685i	0.899976	+	0.899976i	0.995434	−	0.0954571i	$-0.0304313\pi$
$653$	587.685	+	587.685i	0.899976	+	0.899976i	−0.0954571	+	0.995434i	$0.530431\pi$
$654$		0			0
$655$	−416.328	+	458.144i	−0.635616	+	0.699457i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	754.028i		−	1.14420i	−0.820184	−	0.572101i	$-0.806129\pi$
$659$		−	754.028i		−	1.14420i	0.820184	−	0.572101i	$-0.193871\pi$
$660$		0			0
$661$	−818.290			−1.23796			−0.618978	−	0.785408i	$-0.712453\pi$
$661$	−818.290			−1.23796			−0.618978	+	0.785408i	$0.712453\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−1587.57	+	75.9152i	−2.38732	+	0.114158i
$666$		0			0
$667$	685.845	−	685.845i	1.02825	−	1.02825i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−599.672			−0.893700
$672$		0			0
$673$	202.080	+	202.080i	0.300268	+	0.300268i	0.841119	−	0.540851i	$-0.181898\pi$
$673$	202.080	+	202.080i	0.300268	+	0.300268i	−0.540851	+	0.841119i	$0.681898\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	47.2645	−	47.2645i	0.0698147	−	0.0698147i	−0.671337	−	0.741152i	$-0.734280\pi$
$677$	47.2645	−	47.2645i	0.0698147	−	0.0698147i	0.741152	+	0.671337i	$0.234280\pi$
$678$		0			0
$679$			517.292i			0.761844i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	126.090	+	126.090i	0.184612	+	0.184612i	0.793362	−	0.608750i	$-0.208329\pi$
$683$	126.090	+	126.090i	0.184612	+	0.184612i	−0.608750	+	0.793362i	$0.708329\pi$
$684$		0			0
$685$	58.4783	+	1222.92i	0.0853698	+	1.78529i
$686$		0			0
$687$		0			0
$688$		0			0
$689$			328.127i			0.476237i
$690$		0			0
$691$	485.927			0.703223			0.351612	−	0.936146i	$-0.385634\pi$
$691$	485.927			0.703223			0.351612	+	0.936146i	$0.385634\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	3.94816	+	3.58781i	0.00568081	+	0.00516231i
$696$		0			0
$697$	220.021	−	220.021i	0.315668	−	0.315668i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	94.5850			0.134929			0.0674643	−	0.997722i	$-0.478509\pi$
$701$	94.5850			0.134929			0.0674643	+	0.997722i	$0.478509\pi$
$702$		0			0
$703$	−831.354	−	831.354i	−1.18258	−	1.18258i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	−1571.93	+	1571.93i	−2.22338	+	2.22338i
$708$		0			0
$709$			24.3499i			0.0343440i	0.999853	+	0.0171720i	$0.00546629\pi$
$709$			24.3499i			0.0343440i	−0.999853	+	0.0171720i	$0.994534\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−584.011	−	584.011i	−0.819090	−	0.819090i
$714$		0			0
$715$	474.765	−	22.7026i	0.664007	−	0.0317518i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	534.841i		−	0.743868i	−0.928259	−	0.371934i	$-0.878695\pi$
$719$		−	534.841i		−	0.743868i	0.928259	−	0.371934i	$-0.121305\pi$
$720$		0			0
$721$	−722.346			−1.00187
$722$		0			0
$723$		0			0
$724$		0			0
$725$	926.309	−	88.7925i	1.27767	−	0.122472i
$726$		0			0
$727$	435.834	−	435.834i	0.599497	−	0.599497i	−0.340682	−	0.940179i	$-0.610658\pi$
$727$	435.834	−	435.834i	0.599497	−	0.599497i	0.940179	+	0.340682i	$0.110658\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	294.459			0.402816
$732$		0			0
$733$	224.087	+	224.087i	0.305712	+	0.305712i	0.843244	−	0.537531i	$-0.180643\pi$
$733$	224.087	+	224.087i	0.305712	+	0.305712i	−0.537531	+	0.843244i	$0.680643\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	827.282	−	827.282i	1.12250	−	1.12250i
$738$		0			0
$739$			730.510i			0.988512i	0.869317	+	0.494256i	$0.164559\pi$
$739$			730.510i			0.988512i	−0.869317	+	0.494256i	$0.835441\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	941.961	+	941.961i	1.26778	+	1.26778i	0.947232	+	0.320549i	$0.103867\pi$
$743$	941.961	+	941.961i	1.26778	+	1.26778i	0.320549	+	0.947232i	$0.396133\pi$
$744$		0			0
$745$	241.451	+	219.413i	0.324095	+	0.294514i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			646.998i			0.863816i
$750$		0			0
$751$	231.169			0.307814			0.153907	−	0.988085i	$-0.450814\pi$
$751$	231.169			0.307814			0.153907	+	0.988085i	$0.450814\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	528.547	−	581.634i	0.700062	−	0.770377i
$756$		0			0
$757$	−382.911	+	382.911i	−0.505826	+	0.505826i	−0.913243	−	0.407416i	$-0.866430\pi$
$757$	−382.911	+	382.911i	−0.505826	+	0.505826i	0.407416	+	0.913243i	$0.366430\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	941.999			1.23784			0.618922	−	0.785453i	$-0.287570\pi$
$761$	941.999			1.23784			0.618922	+	0.785453i	$0.287570\pi$
$762$		0			0
$763$	751.312	+	751.312i	0.984681	+	0.984681i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−36.8952	+	36.8952i	−0.0481032	+	0.0481032i
$768$		0			0
$769$		−	1388.76i		−	1.80593i	−0.429718	−	0.902963i	$-0.641387\pi$
$769$		−	1388.76i		−	1.80593i	0.429718	−	0.902963i	$-0.358613\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	185.158	+	185.158i	0.239532	+	0.239532i	0.816656	−	0.577124i	$-0.195825\pi$
$773$	185.158	+	185.158i	0.239532	+	0.239532i	−0.577124	+	0.816656i	$0.695825\pi$
$774$		0			0
$775$	−75.6086	−	788.771i	−0.0975595	−	1.01777i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	939.413i		−	1.20592i
$780$		0			0
$781$	−249.711			−0.319733
$782$		0			0
$783$		0			0
$784$		0			0
$785$	41.1245	+	860.012i	0.0523879	+	1.09556i
$786$		0			0
$787$	194.815	−	194.815i	0.247542	−	0.247542i	−0.572419	−	0.819961i	$-0.693995\pi$
$787$	194.815	−	194.815i	0.247542	−	0.247542i	0.819961	+	0.572419i	$0.193995\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−275.805			−0.348679
$792$		0			0
$793$	−213.262	−	213.262i	−0.268930	−	0.268930i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−694.112	+	694.112i	−0.870906	+	0.870906i	−0.992571	−	0.121665i	$-0.961177\pi$
$797$	−694.112	+	694.112i	−0.870906	+	0.870906i	0.121665	+	0.992571i	$0.461177\pi$
$798$		0			0
$799$		−	244.479i		−	0.305982i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−946.175	−	946.175i	−1.17830	−	1.17830i
$804$		0			0
$805$	1089.02	−	1198.40i	1.35282	−	1.48870i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			1492.22i			1.84453i	0.386563	+	0.922263i	$0.373662\pi$
$809$			1492.22i			1.84453i	−0.386563	+	0.922263i	$0.626338\pi$
$810$		0			0
$811$	−80.5721			−0.0993490			−0.0496745	−	0.998765i	$-0.515818\pi$
$811$	−80.5721			−0.0993490			−0.0496745	+	0.998765i	$0.515818\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−1385.02	+	66.2296i	−1.69941	+	0.0812634i
$816$		0			0
$817$	628.619	−	628.619i	0.769423	−	0.769423i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	221.422			0.269698			0.134849	−	0.990866i	$-0.456945\pi$
$821$	221.422			0.269698			0.134849	+	0.990866i	$0.456945\pi$
$822$		0			0
$823$	−815.236	−	815.236i	−0.990566	−	0.990566i	0.00938979	−	0.999956i	$-0.497011\pi$
$823$	−815.236	−	815.236i	−0.990566	−	0.990566i	−0.999956	+	0.00938979i	$0.997011\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	29.5640	−	29.5640i	0.0357485	−	0.0357485i	−0.689007	−	0.724755i	$-0.741953\pi$
$827$	29.5640	−	29.5640i	0.0357485	−	0.0357485i	0.724755	+	0.689007i	$0.241953\pi$
$828$		0			0
$829$			558.092i			0.673211i	0.941646	+	0.336606i	$0.109279\pi$
$829$			558.092i			0.673211i	−0.941646	+	0.336606i	$0.890721\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	631.777	+	631.777i	0.758436	+	0.758436i
$834$		0			0
$835$	−8.91936	−	186.525i	−0.0106819	−	0.223383i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	623.755i		−	0.743451i	−0.928343	−	0.371725i	$-0.878766\pi$
$839$		−	623.755i		−	0.743451i	0.928343	−	0.371725i	$-0.121234\pi$
$840$		0			0
$841$	−544.492			−0.647434
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−448.448	−	407.517i	−0.530708	−	0.482269i
$846$		0			0
$847$	−597.713	+	597.713i	−0.705682	+	0.705682i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	1197.84			1.40757
$852$		0			0
$853$	−454.178	−	454.178i	−0.532448	−	0.532448i	0.388852	−	0.921300i	$-0.372872\pi$
$853$	−454.178	−	454.178i	−0.532448	−	0.532448i	−0.921300	+	0.388852i	$0.872872\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	769.104	−	769.104i	0.897438	−	0.897438i	−0.0977709	−	0.995209i	$-0.531171\pi$
$857$	769.104	−	769.104i	0.897438	−	0.897438i	0.995209	+	0.0977709i	$0.0311712\pi$
$858$		0			0
$859$		−	536.071i		−	0.624064i	−0.950072	−	0.312032i	$-0.898990\pi$
$859$		−	536.071i		−	0.624064i	0.950072	−	0.312032i	$-0.101010\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−500.177	−	500.177i	−0.579579	−	0.579579i	0.355208	−	0.934787i	$-0.384410\pi$
$863$	−500.177	−	500.177i	−0.579579	−	0.579579i	−0.934787	+	0.355208i	$0.884410\pi$
$864$		0			0
$865$	61.3812	−	2.93516i	0.0709609	−	0.00339324i
$866$		0			0
$867$		0			0
$868$		0			0
$869$			306.073i			0.352212i
$870$		0			0
$871$	588.413			0.675561
$872$		0			0
$873$		0			0
$874$		0			0
$875$	1537.62	−	221.935i	1.75728	−	0.253640i
$876$		0			0
$877$	−102.540	+	102.540i	−0.116922	+	0.116922i	−0.763147	−	0.646225i	$-0.776347\pi$
$877$	−102.540	+	102.540i	−0.116922	+	0.116922i	0.646225	+	0.763147i	$0.276347\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−649.515			−0.737247			−0.368624	−	0.929579i	$-0.620171\pi$
$881$	−649.515			−0.737247			−0.368624	+	0.929579i	$0.620171\pi$
$882$		0			0
$883$	−341.476	−	341.476i	−0.386723	−	0.386723i	0.486794	−	0.873517i	$-0.338166\pi$
$883$	−341.476	−	341.476i	−0.386723	−	0.386723i	−0.873517	+	0.486794i	$0.838166\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−516.110	+	516.110i	−0.581860	+	0.581860i	−0.935414	−	0.353554i	$-0.884973\pi$
$887$	−516.110	+	516.110i	−0.581860	+	0.581860i	0.353554	+	0.935414i	$0.384973\pi$
$888$		0			0
$889$			2241.95i			2.52188i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−521.922	−	521.922i	−0.584459	−	0.584459i
$894$		0			0
$895$	−35.7827	−	32.5167i	−0.0399806	−	0.0363315i
$896$		0			0
$897$		0			0
$898$		0			0
$899$			1179.77i			1.31232i
$900$		0			0
$901$	402.019			0.446192
$902$		0			0
$903$		0			0
$904$		0			0
$905$	837.454	−	921.568i	0.925364	−	1.01831i
$906$		0			0
$907$	−62.4116	+	62.4116i	−0.0688111	+	0.0688111i	−0.740675	−	0.671864i	$-0.765494\pi$
$907$	−62.4116	+	62.4116i	−0.0688111	+	0.0688111i	0.671864	+	0.740675i	$0.265494\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	1140.91			1.25237			0.626183	−	0.779676i	$-0.284616\pi$
$911$	1140.91			1.25237			0.626183	+	0.779676i	$0.284616\pi$
$912$		0			0
$913$	−0.0304947	−	0.0304947i	−3.34006e−5	−	3.34006e-5i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	1088.08	−	1088.08i	1.18656	−	1.18656i
$918$		0			0
$919$		−	1200.77i		−	1.30661i	−0.757097	−	0.653303i	$-0.773383\pi$
$919$		−	1200.77i		−	1.30661i	0.757097	−	0.653303i	$-0.226617\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	−88.8049	−	88.8049i	−0.0962133	−	0.0962133i
$924$		0			0
$925$	886.446	+	731.368i	0.958320	+	0.790669i
$926$		0			0
$927$		0			0
$928$		0			0
$929$			87.8294i			0.0945419i	0.998882	+	0.0472709i	$0.0150524\pi$
$929$			87.8294i			0.0945419i	−0.998882	+	0.0472709i	$0.984948\pi$
$930$		0			0
$931$	2697.47			2.89739
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−27.8150	−	581.678i	−0.0297486	−	0.622115i
$936$		0			0
$937$	−215.447	+	215.447i	−0.229933	+	0.229933i	−0.812665	−	0.582732i	$-0.801984\pi$
$937$	−215.447	+	215.447i	−0.229933	+	0.229933i	0.582732	+	0.812665i	$0.301984\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−805.908			−0.856438			−0.428219	−	0.903675i	$-0.640859\pi$
$941$	−805.908			−0.856438			−0.428219	+	0.903675i	$0.640859\pi$
$942$		0			0
$943$	676.767	+	676.767i	0.717675	+	0.717675i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	739.218	−	739.218i	0.780589	−	0.780589i	−0.199341	−	0.979930i	$-0.563880\pi$
$947$	739.218	−	739.218i	0.780589	−	0.780589i	0.979930	+	0.199341i	$0.0638801\pi$
$948$		0			0
$949$		−	672.977i		−	0.709144i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	790.131	+	790.131i	0.829098	+	0.829098i	0.987392	−	0.158294i	$-0.0505993\pi$
$953$	790.131	+	790.131i	0.829098	+	0.829098i	−0.158294	+	0.987392i	$0.550599\pi$
$954$		0			0
$955$	545.920	−	600.752i	0.571644	−	0.629060i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		−	3043.28i		−	3.17339i
$960$		0			0
$961$	43.6021			0.0453716
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−57.3105	+	2.74050i	−0.0593891	+	0.00283990i
$966$		0			0
$967$	−213.927	+	213.927i	−0.221227	+	0.221227i	−0.809015	−	0.587788i	$-0.799999\pi$
$967$	−213.927	+	213.927i	−0.221227	+	0.221227i	0.587788	+	0.809015i	$0.299999\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	1356.34			1.39685			0.698423	−	0.715686i	$-0.253886\pi$
$971$	1356.34			1.39685			0.698423	+	0.715686i	$0.253886\pi$
$972$		0			0
$973$	−9.37676	−	9.37676i	−0.00963695	−	0.00963695i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1047.93	−	1047.93i	1.07260	−	1.07260i	0.0754457	−	0.997150i	$-0.475962\pi$
$977$	1047.93	−	1047.93i	1.07260	−	1.07260i	0.997150	−	0.0754457i	$-0.0240380\pi$
$978$		0			0
$979$			1061.28i			1.08405i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	741.601	+	741.601i	0.754426	+	0.754426i	0.975302	−	0.220876i	$-0.0708915\pi$
$983$	741.601	+	741.601i	0.754426	+	0.754426i	−0.220876	+	0.975302i	$0.570891\pi$
$984$		0			0
$985$	77.3621	+	1617.83i	0.0785402	+	1.64246i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			905.734i			0.915808i
$990$		0			0
$991$	304.849			0.307618			0.153809	−	0.988101i	$-0.450846\pi$
$991$	304.849			0.307618			0.153809	+	0.988101i	$0.450846\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1098.61	+	998.336i	1.10413	+	1.00335i
$996$		0			0
$997$	−542.650	+	542.650i	−0.544283	+	0.544283i	−0.924782	−	0.380499i	$-0.875752\pi$
$997$	−542.650	+	542.650i	−0.544283	+	0.544283i	0.380499	+	0.924782i	$0.375752\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	360.3.v.f.73.3		8
3.2	odd	2		120.3.u.b.73.3	✓	8
4.3	odd	2		720.3.bh.o.433.3		8
5.2	odd	4	inner	360.3.v.f.217.3		8
5.3	odd	4		1800.3.v.t.1657.4		8
5.4	even	2		1800.3.v.t.793.4		8
12.11	even	2		240.3.bg.e.193.1		8
15.2	even	4		120.3.u.b.97.3	yes	8
15.8	even	4		600.3.u.h.457.2		8
15.14	odd	2		600.3.u.h.193.2		8
20.7	even	4		720.3.bh.o.577.3		8
24.5	odd	2		960.3.bg.l.193.2		8
24.11	even	2		960.3.bg.k.193.4		8
60.23	odd	4		1200.3.bg.q.1057.3		8
60.47	odd	4		240.3.bg.e.97.1		8
60.59	even	2		1200.3.bg.q.193.3		8
120.77	even	4		960.3.bg.l.577.2		8
120.107	odd	4		960.3.bg.k.577.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
120.3.u.b.73.3	✓	8	3.2	odd	2
120.3.u.b.97.3	yes	8	15.2	even	4
240.3.bg.e.97.1		8	60.47	odd	4
240.3.bg.e.193.1		8	12.11	even	2
360.3.v.f.73.3		8	1.1	even	1	trivial
360.3.v.f.217.3		8	5.2	odd	4	inner
600.3.u.h.193.2		8	15.14	odd	2
600.3.u.h.457.2		8	15.8	even	4
720.3.bh.o.433.3		8	4.3	odd	2
720.3.bh.o.577.3		8	20.7	even	4
960.3.bg.k.193.4		8	24.11	even	2
960.3.bg.k.577.4		8	120.107	odd	4
960.3.bg.l.193.2		8	24.5	odd	2
960.3.bg.l.577.2		8	120.77	even	4
1200.3.bg.q.193.3		8	60.59	even	2
1200.3.bg.q.1057.3		8	60.23	odd	4
1800.3.v.t.793.4		8	5.4	even	2
1800.3.v.t.1657.4		8	5.3	odd	4

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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 360 = 2^{3} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	360.v (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(3.36263 - 3.70037i) q^{5} +(-8.78825 + 8.78825i) q^{7} -13.7482 q^{11} +(-4.88927 - 4.88927i) q^{13} +(-5.99029 + 5.99029i) q^{17} +25.5765i q^{19} +(-18.4257 - 18.4257i) q^{23} +(-2.38547 - 24.8859i) q^{25} +37.2222i q^{29} +31.6955 q^{31} +(2.96816 + 62.0714i) q^{35} +(-32.5046 + 32.5046i) q^{37} -36.7295 q^{41} +(-24.5780 - 24.5780i) q^{43} +(-20.4063 + 20.4063i) q^{47} -105.467i q^{49} +(-33.5559 - 33.5559i) q^{53} +(-46.2300 + 50.8734i) q^{55} -7.54615i q^{59} +43.6183 q^{61} +(-34.5329 + 1.65131i) q^{65} +(-60.1739 + 60.1739i) q^{67} +18.1632 q^{71} +(68.8218 + 68.8218i) q^{73} +(120.823 - 120.823i) q^{77} -22.2628i q^{79} +(0.00221809 + 0.00221809i) q^{83} +(2.02317 + 42.3094i) q^{85} -77.1943i q^{89} +85.9363 q^{91} +(94.6425 + 86.0043i) q^{95} +(29.4309 - 29.4309i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 12 q^{5} + 4 q^{7} - 32 q^{11} - 4 q^{13} - 52 q^{17} + 40 q^{23} - 84 q^{25} + 96 q^{31} - 24 q^{35} - 60 q^{37} + 152 q^{41} - 88 q^{43} + 16 q^{47} - 108 q^{53} + 116 q^{55} + 264 q^{61} - 164 q^{65}+ \cdots - 208 q^{97}+O(q^{100}) \) 8 * q + 12 * q^5 + 4 * q^7 - 32 * q^11 - 4 * q^13 - 52 * q^17 + 40 * q^23 - 84 * q^25 + 96 * q^31 - 24 * q^35 - 60 * q^37 + 152 * q^41 - 88 * q^43 + 16 * q^47 - 108 * q^53 + 116 * q^55 + 264 * q^61 - 164 * q^65 - 216 * q^67 + 240 * q^71 - 208 * q^73 - 168 * q^77 - 336 * q^83 - 12 * q^85 + 592 * q^91 + 128 * q^95 - 208 * q^97

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