LMFDB - Embedded newform 1080.2.q.e.721.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1080,2,Mod(361,1080)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1080, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 0, 2, 0]))

N = Newforms(chi, 2, names="a")

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

chi := DirichletCharacter("1080.361");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1080 = 2^{3} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1080.q (of order $3$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$8.62384341830$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(\zeta_{3})$
Coefficient field:	8.0.856615824.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 11x^{6} + 36x^{4} + 32x^{2} + 4 $ x^8 + 11x^6 + 36x^4 + 32*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 3^{4} $
Twist minimal:	no (minimal twist has level 360)
Sato-Tate group:	$\mathrm{SU}(2)[C_{3}]$

Embedding invariants

Embedding label			721.2
Root			$1.07834i$ of defining polynomial
Character	$\chi$	$=$	1080.721
Dual form			1080.2.q.e.361.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(0.500000 + 0.866025i) q^{5} +(-0.433868 + 0.751481i) q^{7} +(-2.04334 + 3.53916i) q^{11} +(-0.606223 - 1.05001i) q^{13} -7.82214 q^{17} -4.51156 q^{19} +(1.43387 + 2.48353i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(3.14956 - 5.45520i) q^{29} +(1.26151 + 2.18500i) q^{31} -0.867736 q^{35} -0.523026 q^{37} +(-4.06063 - 7.03322i) q^{41} +(-1.91107 + 3.31007i) q^{43} +(-0.695381 + 1.20444i) q^{47} +(3.12352 + 5.41009i) q^{49} -13.9088 q^{53} -4.08667 q^{55} +(3.43711 + 5.95325i) q^{59} +(-4.54334 + 7.86929i) q^{61} +(0.606223 - 1.05001i) q^{65} +(-1.68965 - 2.92656i) q^{67} -3.21245 q^{71} +8.60970 q^{73} +(-1.77308 - 3.07106i) q^{77} +(-8.12126 + 14.0664i) q^{79} +(3.22142 - 5.57967i) q^{83} +(-3.91107 - 6.77417i) q^{85} +10.2079 q^{89} +1.05208 q^{91} +(-2.25578 - 3.90713i) q^{95} +(4.38503 - 7.59510i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{5} + q^{7} + q^{11} - 4 q^{13} - 10 q^{17} + 2 q^{19} + 7 q^{23} - 4 q^{25} + 7 q^{29} + 2 q^{31} + 2 q^{35} + 12 q^{37} + 12 q^{41} + 11 q^{43} + 7 q^{47} - 3 q^{49} - 24 q^{53} + 2 q^{55}+ \cdots - q^{97}+O(q^{100}) $ 8 * q + 4 * q^5 + q^7 + q^11 - 4 * q^13 - 10 * q^17 + 2 * q^19 + 7 * q^23 - 4 * q^25 + 7 * q^29 + 2 * q^31 + 2 * q^35 + 12 * q^37 + 12 * q^41 + 11 * q^43 + 7 * q^47 - 3 * q^49 - 24 * q^53 + 2 * q^55 + 11 * q^59 - 19 * q^61 + 4 * q^65 + 10 * q^67 - 24 * q^71 + 18 * q^73 + 32 * q^77 + 24 * q^79 + 23 * q^83 - 5 * q^85 - 42 * q^89 + 28 * q^91 + q^95 - q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1080\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$541$	$1001$
$\chi(n)$	$1$	$1$	$1$	$e\left(\frac{2}{3}\right)$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$		0			0
$3$		0			0
$4$		0			0
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$5$	0.500000	+	0.866025i	0.223607	+	0.387298i
$6$		0			0
$7$	−0.433868	+	0.751481i	−0.163987	+	0.284033i	−0.936295	−	0.351215i	$-0.885769\pi$
$7$	−0.433868	+	0.751481i	−0.163987	+	0.284033i	0.772308	+	0.635248i	$0.219102\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−2.04334	+	3.53916i	−0.616089	+	1.06710i	0.374104	+	0.927387i	$0.377950\pi$
$11$	−2.04334	+	3.53916i	−0.616089	+	1.06710i	−0.990192	+	0.139710i	$0.955383\pi$
$12$		0			0
$13$	−0.606223	−	1.05001i	−0.168136	−	0.291220i	0.769629	−	0.638492i	$-0.220441\pi$
$13$	−0.606223	−	1.05001i	−0.168136	−	0.291220i	−0.937764	+	0.347272i	$0.887108\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−7.82214			−1.89715			−0.948574	−	0.316555i	$-0.897474\pi$
$17$	−7.82214			−1.89715			−0.948574	+	0.316555i	$0.897474\pi$
$18$		0			0
$19$	−4.51156			−1.03502			−0.517512	−	0.855676i	$-0.673142\pi$
$19$	−4.51156			−1.03502			−0.517512	+	0.855676i	$0.673142\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	1.43387	+	2.48353i	0.298982	+	0.517852i	0.975903	−	0.218203i	$-0.0700196\pi$
$23$	1.43387	+	2.48353i	0.298982	+	0.517852i	−0.676921	+	0.736055i	$0.736686\pi$
$24$		0			0
$25$	−0.500000	+	0.866025i	−0.100000	+	0.173205i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	3.14956	−	5.45520i	0.584858	−	1.01300i	−0.410035	−	0.912070i	$-0.634483\pi$
$29$	3.14956	−	5.45520i	0.584858	−	1.01300i	0.994893	−	0.100934i	$-0.0321832\pi$
$30$		0			0
$31$	1.26151	+	2.18500i	0.226574	+	0.392438i	0.956791	−	0.290778i	$-0.0939140\pi$
$31$	1.26151	+	2.18500i	0.226574	+	0.392438i	−0.730216	+	0.683216i	$0.760581\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−0.867736			−0.146674
$36$		0			0
$37$	−0.523026			−0.0859850			−0.0429925	−	0.999075i	$-0.513689\pi$
$37$	−0.523026			−0.0859850			−0.0429925	+	0.999075i	$0.513689\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	−4.06063	−	7.03322i	−0.634164	−	1.09840i	−0.986692	−	0.162603i	$-0.948011\pi$
$41$	−4.06063	−	7.03322i	−0.634164	−	1.09840i	0.352528	−	0.935801i	$-0.385322\pi$
$42$		0			0
$43$	−1.91107	+	3.31007i	−0.291436	+	0.504781i	−0.974149	−	0.225905i	$-0.927466\pi$
$43$	−1.91107	+	3.31007i	−0.291436	+	0.504781i	0.682714	+	0.730686i	$0.260800\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	−0.695381	+	1.20444i	−0.101432	+	0.175685i	−0.912275	−	0.409579i	$-0.865676\pi$
$47$	−0.695381	+	1.20444i	−0.101432	+	0.175685i	0.810843	+	0.585264i	$0.199009\pi$
$48$		0			0
$49$	3.12352	+	5.41009i	0.446217	+	0.772870i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−13.9088			−1.91052			−0.955261	−	0.295763i	$-0.904426\pi$
$53$	−13.9088			−1.91052			−0.955261	+	0.295763i	$0.904426\pi$
$54$		0			0
$55$	−4.08667			−0.551047
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.43711	+	5.95325i	0.447474	+	0.775048i	0.998221	−	0.0596246i	$-0.0189904\pi$
$59$	3.43711	+	5.95325i	0.447474	+	0.775048i	−0.550747	+	0.834672i	$0.685657\pi$
$60$		0			0
$61$	−4.54334	+	7.86929i	−0.581715	+	1.00756i	0.413562	+	0.910476i	$0.364285\pi$
$61$	−4.54334	+	7.86929i	−0.581715	+	1.00756i	−0.995276	+	0.0970830i	$0.969049\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.606223	−	1.05001i	0.0751927	−	0.130238i
$66$		0			0
$67$	−1.68965	−	2.92656i	−0.206424	−	0.357536i	0.744162	−	0.667999i	$-0.232849\pi$
$67$	−1.68965	−	2.92656i	−0.206424	−	0.357536i	−0.950585	+	0.310463i	$0.899516\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−3.21245			−0.381247			−0.190624	−	0.981663i	$-0.561051\pi$
$71$	−3.21245			−0.381247			−0.190624	+	0.981663i	$0.561051\pi$
$72$		0			0
$73$	8.60970			1.00769			0.503844	−	0.863794i	$-0.331918\pi$
$73$	8.60970			1.00769			0.503844	+	0.863794i	$0.331918\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.77308	−	3.07106i	−0.202061	−	0.349979i
$78$		0			0
$79$	−8.12126	+	14.0664i	−0.913713	+	1.58260i	−0.104939	+	0.994479i	$0.533465\pi$
$79$	−8.12126	+	14.0664i	−0.913713	+	1.58260i	−0.808774	+	0.588119i	$0.799869\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	3.22142	−	5.57967i	0.353597	−	0.612448i	−0.633280	−	0.773923i	$-0.718292\pi$
$83$	3.22142	−	5.57967i	0.353597	−	0.612448i	0.986877	+	0.161475i	$0.0516251\pi$
$84$		0			0
$85$	−3.91107	−	6.77417i	−0.424215	−	0.734762i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	10.2079			1.08204			0.541019	−	0.841010i	$-0.318039\pi$
$89$	10.2079			1.08204			0.541019	+	0.841010i	$0.318039\pi$
$90$		0			0
$91$	1.05208			0.110288
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−2.25578	−	3.90713i	−0.231438	−	0.400863i
$96$		0			0
$97$	4.38503	−	7.59510i	0.445232	−	0.771165i	−0.552836	−	0.833290i	$-0.686454\pi$
$97$	4.38503	−	7.59510i	0.445232	−	0.771165i	0.998068	+	0.0621250i	$0.0197877\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	1.13226	−	1.96114i	0.112664	−	0.195141i	−0.804179	−	0.594387i	$-0.797395\pi$
$101$	1.13226	−	1.96114i	0.112664	−	0.195141i	0.916844	+	0.399246i	$0.130728\pi$
$102$		0			0
$103$	7.69289	+	13.3245i	0.758003	+	1.31290i	0.943867	+	0.330325i	$0.107158\pi$
$103$	7.69289	+	13.3245i	0.758003	+	1.31290i	−0.185864	+	0.982575i	$0.559508\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−14.3402			−1.38632			−0.693160	−	0.720784i	$-0.743782\pi$
$107$	−14.3402			−1.38632			−0.693160	+	0.720784i	$0.743782\pi$
$108$		0			0
$109$	−5.35120			−0.512552			−0.256276	−	0.966604i	$-0.582496\pi$
$109$	−5.35120			−0.512552			−0.256276	+	0.966604i	$0.582496\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	1.34471	+	2.32911i	0.126500	+	0.219104i	0.922318	−	0.386431i	$-0.126292\pi$
$113$	1.34471	+	2.32911i	0.126500	+	0.219104i	−0.795818	+	0.605535i	$0.792959\pi$
$114$		0			0
$115$	−1.43387	+	2.48353i	−0.133709	+	0.231591i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	3.39378	−	5.87819i	0.311107	−	0.538853i
$120$		0			0
$121$	−2.85044	−	4.93711i	−0.259131	−	0.448828i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.00000			−0.0894427
$126$		0			0
$127$	−21.5641			−1.91350			−0.956752	−	0.290903i	$-0.906044\pi$
$127$	−21.5641			−1.91350			−0.956752	+	0.290903i	$0.906044\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	9.24703	+	16.0163i	0.807917	+	1.39935i	0.914304	+	0.405029i	$0.132738\pi$
$131$	9.24703	+	16.0163i	0.807917	+	1.39935i	−0.106387	+	0.994325i	$0.533928\pi$
$132$		0			0
$133$	1.95742	−	3.39036i	0.169730	−	0.293981i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	4.86200	−	8.42124i	0.415389	−	0.719475i	−0.580080	−	0.814559i	$-0.696979\pi$
$137$	4.86200	−	8.42124i	0.415389	−	0.719475i	0.995469	+	0.0950845i	$0.0303121\pi$
$138$		0			0
$139$	8.04032	+	13.9262i	0.681971	+	1.18121i	0.974378	+	0.224915i	$0.0722104\pi$
$139$	8.04032	+	13.9262i	0.681971	+	1.18121i	−0.292407	+	0.956294i	$0.594456\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	4.95487			0.414347
$144$		0			0
$145$	6.29912			0.523113
$146$		0			0
$147$		0			0
$148$		0			0
$149$	7.92263	+	13.7224i	0.649047	+	1.12418i	0.983351	+	0.181718i	$0.0581657\pi$
$149$	7.92263	+	13.7224i	0.649047	+	1.12418i	−0.334303	+	0.942466i	$0.608501\pi$
$150$		0			0
$151$	7.56063	−	13.0954i	0.615275	−	1.06569i	−0.375061	−	0.927000i	$-0.622378\pi$
$151$	7.56063	−	13.0954i	0.615275	−	1.06569i	0.990336	−	0.138688i	$-0.0442885\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.26151	+	2.18500i	−0.101327	+	0.175504i
$156$		0			0
$157$	−8.77655	−	15.2014i	−0.700445	−	1.21321i	−0.968310	−	0.249750i	$-0.919652\pi$
$157$	−8.77655	−	15.2014i	−0.700445	−	1.21321i	0.267865	−	0.963456i	$-0.413682\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	−2.48844			−0.196116
$162$		0			0
$163$	6.85673			0.537061			0.268530	−	0.963271i	$-0.413462\pi$
$163$	6.85673			0.537061			0.268530	+	0.963271i	$0.413462\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	1.95991	+	3.39466i	0.151662	+	0.262687i	0.931839	−	0.362873i	$-0.118204\pi$
$167$	1.95991	+	3.39466i	0.151662	+	0.262687i	−0.780176	+	0.625560i	$0.784871\pi$
$168$		0			0
$169$	5.76499	−	9.98525i	0.443461	−	0.768096i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−4.08667	+	7.07832i	−0.310704	+	0.538155i	−0.978515	−	0.206176i	$-0.933898\pi$
$173$	−4.08667	+	7.07832i	−0.310704	+	0.538155i	0.667811	+	0.744331i	$0.267231\pi$
$174$		0			0
$175$	−0.433868	−	0.751481i	−0.0327973	−	0.0568066i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	6.85673			0.512496			0.256248	−	0.966611i	$-0.417514\pi$
$179$	6.85673			0.512496			0.256248	+	0.966611i	$0.417514\pi$
$180$		0			0
$181$	16.2991			1.21150			0.605752	−	0.795654i	$-0.292872\pi$
$181$	16.2991			1.21150			0.605752	+	0.795654i	$0.292872\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−0.261513	−	0.452954i	−0.0192268	−	0.0333018i
$186$		0			0
$187$	15.9833	−	27.6838i	1.16881	−	2.02444i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	5.91680	−	10.2482i	0.428125	−	0.741534i	−0.568582	−	0.822627i	$-0.692508\pi$
$191$	5.91680	−	10.2482i	0.428125	−	0.741534i	0.996707	+	0.0810928i	$0.0258410\pi$
$192$		0			0
$193$	0.911072	+	1.57802i	0.0655804	+	0.113589i	0.896951	−	0.442129i	$-0.145777\pi$
$193$	0.911072	+	1.57802i	0.0655804	+	0.113589i	−0.831371	+	0.555718i	$0.812443\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−19.3798			−1.38075			−0.690375	−	0.723451i	$-0.742555\pi$
$197$	−19.3798			−1.38075			−0.690375	+	0.723451i	$0.742555\pi$
$198$		0			0
$199$	6.69637			0.474693			0.237347	−	0.971425i	$-0.423722\pi$
$199$	6.69637			0.474693			0.237347	+	0.971425i	$0.423722\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	2.73298	+	4.73367i	0.191818	+	0.332238i
$204$		0			0
$205$	4.06063	−	7.03322i	0.283607	−	0.491221i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	9.21864	−	15.9671i	0.637666	−	1.10447i
$210$		0			0
$211$	−4.26151	−	7.38116i	−0.293375	−	0.508140i	0.681231	−	0.732069i	$-0.261445\pi$
$211$	−4.26151	−	7.38116i	−0.293375	−	0.508140i	−0.974605	+	0.223929i	$0.928112\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−3.82214			−0.260668
$216$		0			0
$217$	−2.18932			−0.148621
$218$		0			0
$219$		0			0
$220$		0			0
$221$	4.74196	+	8.21332i	0.318979	+	0.552488i
$222$		0			0
$223$	5.43085	−	9.40651i	0.363677	−	0.629907i	−0.624886	−	0.780716i	$-0.714855\pi$
$223$	5.43085	−	9.40651i	0.363677	−	0.629907i	0.988563	+	0.150809i	$0.0481879\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−9.34245	+	16.1816i	−0.620080	+	1.07401i	0.369390	+	0.929274i	$0.379567\pi$
$227$	−9.34245	+	16.1816i	−0.620080	+	1.07401i	−0.989470	+	0.144736i	$0.953767\pi$
$228$		0			0
$229$	−4.79836	−	8.31100i	−0.317084	−	0.549206i	0.662794	−	0.748802i	$-0.269371\pi$
$229$	−4.79836	−	8.31100i	−0.317084	−	0.549206i	−0.979878	+	0.199595i	$0.936037\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	20.5807			1.34829			0.674145	−	0.738599i	$-0.264512\pi$
$233$	20.5807			1.34829			0.674145	+	0.738599i	$0.264512\pi$
$234$		0			0
$235$	−1.39076			−0.0907233
$236$		0			0
$237$		0			0
$238$		0			0
$239$	5.08018	+	8.79913i	0.328610	+	0.569169i	0.982236	−	0.187649i	$-0.0600866\pi$
$239$	5.08018	+	8.79913i	0.328610	+	0.569169i	−0.653627	+	0.756817i	$0.726753\pi$
$240$		0			0
$241$	−7.10622	+	12.3083i	−0.457752	+	0.792850i	−0.998842	−	0.0481150i	$-0.984679\pi$
$241$	−7.10622	+	12.3083i	−0.457752	+	0.792850i	0.541090	+	0.840965i	$0.318012\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−3.12352	+	5.41009i	−0.199554	+	0.345638i
$246$		0			0
$247$	2.73501	+	4.73718i	0.174025	+	0.301420i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	0.247034			0.0155927			0.00779634	−	0.999970i	$-0.497518\pi$
$251$	0.247034			0.0155927			0.00779634	+	0.999970i	$0.497518\pi$
$252$		0			0
$253$	−11.7195			−0.736798
$254$		0			0
$255$		0			0
$256$		0			0
$257$	2.43062	+	4.20996i	0.151618	+	0.262610i	0.931822	−	0.362914i	$-0.118218\pi$
$257$	2.43062	+	4.20996i	0.151618	+	0.262610i	−0.780204	+	0.625525i	$0.784885\pi$
$258$		0			0
$259$	0.226924	−	0.393044i	0.0141004	−	0.0244226i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	10.2159	−	17.6945i	0.629941	−	1.09109i	−0.357622	−	0.933866i	$-0.616413\pi$
$263$	10.2159	−	17.6945i	0.629941	−	1.09109i	0.987563	−	0.157223i	$-0.0502541\pi$
$264$		0			0
$265$	−6.95441	−	12.0454i	−0.427206	−	0.739942i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	25.5125			1.55552			0.777762	−	0.628559i	$-0.216355\pi$
$269$	25.5125			1.55552			0.777762	+	0.628559i	$0.216355\pi$
$270$		0			0
$271$	5.12126			0.311094			0.155547	−	0.987828i	$-0.450286\pi$
$271$	5.12126			0.311094			0.155547	+	0.987828i	$0.450286\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.04334	−	3.53916i	−0.123218	−	0.213419i
$276$		0			0
$277$	−5.70436	+	9.88024i	−0.342742	+	0.593646i	−0.984941	−	0.172892i	$-0.944689\pi$
$277$	−5.70436	+	9.88024i	−0.342742	+	0.593646i	0.642199	+	0.766538i	$0.278022\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	10.7969	−	18.7007i	0.644087	−	1.11559i	−0.340425	−	0.940272i	$-0.610571\pi$
$281$	10.7969	−	18.7007i	0.644087	−	1.11559i	0.984512	−	0.175319i	$-0.0560957\pi$
$282$		0			0
$283$	−9.96338	−	17.2571i	−0.592262	−	1.02583i	−0.993927	−	0.110041i	$-0.964902\pi$
$283$	−9.96338	−	17.2571i	−0.592262	−	1.02583i	0.401665	−	0.915787i	$-0.368432\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.04711			0.415978
$288$		0			0
$289$	44.1859			2.59917
$290$		0			0
$291$		0			0
$292$		0			0
$293$	0.916803	+	1.58795i	0.0535602	+	0.0927690i	0.891562	−	0.452898i	$-0.149610\pi$
$293$	0.916803	+	1.58795i	0.0535602	+	0.0927690i	−0.838002	+	0.545667i	$0.816276\pi$
$294$		0			0
$295$	−3.43711	+	5.95325i	−0.200116	+	0.346612i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.73849	−	3.01115i	0.100539	−	0.174139i
$300$		0			0
$301$	−1.65831	−	2.87227i	−0.0955831	−	0.165555i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−9.08667			−0.520301
$306$		0			0
$307$	9.02962			0.515347			0.257674	−	0.966232i	$-0.417044\pi$
$307$	9.02962			0.515347			0.257674	+	0.966232i	$0.417044\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	1.48045	+	2.56421i	0.0839485	+	0.145403i	0.904943	−	0.425533i	$-0.139914\pi$
$311$	1.48045	+	2.56421i	0.0839485	+	0.145403i	−0.820994	+	0.570937i	$0.806580\pi$
$312$		0			0
$313$	−9.64157	+	16.6997i	−0.544974	+	0.943922i	0.453635	+	0.891188i	$0.350127\pi$
$313$	−9.64157	+	16.6997i	−0.544974	+	0.943922i	−0.998609	+	0.0527345i	$0.983206\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−3.08366	+	5.34105i	−0.173195	+	0.299983i	−0.939535	−	0.342452i	$-0.888743\pi$
$317$	−3.08366	+	5.34105i	−0.173195	+	0.299983i	0.766340	+	0.642435i	$0.222076\pi$
$318$		0			0
$319$	12.8712	+	22.2936i	0.720649	+	1.24820i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	35.2901			1.96359
$324$		0			0
$325$	1.21245			0.0672544
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−0.603407	−	1.04513i	−0.0332669	−	0.0576200i
$330$		0			0
$331$	−10.3447	+	17.9176i	−0.568597	+	0.984838i	0.428108	+	0.903727i	$0.359180\pi$
$331$	−10.3447	+	17.9176i	−0.568597	+	0.984838i	−0.996705	+	0.0811109i	$0.974153\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	1.68965	−	2.92656i	0.0923154	−	0.159895i
$336$		0			0
$337$	0.862004	+	1.49303i	0.0469564	+	0.0813308i	0.888548	−	0.458783i	$-0.151714\pi$
$337$	0.862004	+	1.49303i	0.0469564	+	0.0813308i	−0.841592	+	0.540114i	$0.818381\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−10.3108			−0.558360
$342$		0			0
$343$	−11.4949			−0.620668
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−16.2448	−	28.1368i	−0.872065	−	1.51046i	−0.859856	−	0.510536i	$-0.829447\pi$
$347$	−16.2448	−	28.1368i	−0.872065	−	1.51046i	−0.0122090	−	0.999925i	$-0.503886\pi$
$348$		0			0
$349$	−2.59240	+	4.49017i	−0.138768	+	0.240354i	−0.927031	−	0.374986i	$-0.877648\pi$
$349$	−2.59240	+	4.49017i	−0.138768	+	0.240354i	0.788262	+	0.615339i	$0.210981\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−1.43062	+	2.47791i	−0.0761444	+	0.131886i	−0.901583	−	0.432605i	$-0.857594\pi$
$353$	−1.43062	+	2.47791i	−0.0761444	+	0.131886i	0.825439	+	0.564491i	$0.190928\pi$
$354$		0			0
$355$	−1.60622	−	2.78206i	−0.0852495	−	0.147656i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−37.0301			−1.95437			−0.977186	−	0.212384i	$-0.931877\pi$
$359$	−37.0301			−1.95437			−0.977186	+	0.212384i	$0.931877\pi$
$360$		0			0
$361$	1.35420			0.0712736
$362$		0			0
$363$		0			0
$364$		0			0
$365$	4.30485	+	7.45622i	0.225326	+	0.390276i
$366$		0			0
$367$	18.2159	−	31.5509i	0.950863	−	1.64694i	0.207301	−	0.978277i	$-0.433532\pi$
$367$	18.2159	−	31.5509i	0.950863	−	1.64694i	0.743563	−	0.668666i	$-0.233135\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	6.03459	−	10.4522i	0.313300	−	0.542652i
$372$		0			0
$373$	8.26499	+	14.3154i	0.427945	+	0.741222i	0.996690	−	0.0812913i	$-0.0259044\pi$
$373$	8.26499	+	14.3154i	0.427945	+	0.741222i	−0.568746	+	0.822514i	$0.692571\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−7.63734			−0.393343
$378$		0			0
$379$	−31.8062			−1.63377			−0.816886	−	0.576798i	$-0.804302\pi$
$379$	−31.8062			−1.63377			−0.816886	+	0.576798i	$0.804302\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	10.1178	+	17.5245i	0.516995	+	0.895461i	0.999805	+	0.0197362i	$0.00628264\pi$
$383$	10.1178	+	17.5245i	0.516995	+	0.895461i	−0.482811	+	0.875725i	$0.660384\pi$
$384$		0			0
$385$	1.77308	−	3.07106i	0.0903643	−	0.156516i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−12.0028	+	20.7895i	−0.608567	+	1.05407i	0.382910	+	0.923786i	$0.374922\pi$
$389$	−12.0028	+	20.7895i	−0.608567	+	1.05407i	−0.991477	+	0.130283i	$0.958411\pi$
$390$		0			0
$391$	−11.2159	−	19.4265i	−0.567213	−	0.982443i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−16.2425			−0.817250
$396$		0			0
$397$	−35.8868			−1.80111			−0.900554	−	0.434745i	$-0.856839\pi$
$397$	−35.8868			−1.80111			−0.900554	+	0.434745i	$0.856839\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−1.60894	−	2.78676i	−0.0803466	−	0.139164i	0.823052	−	0.567966i	$-0.192269\pi$
$401$	−1.60894	−	2.78676i	−0.0803466	−	0.139164i	−0.903399	+	0.428801i	$0.858936\pi$
$402$		0			0
$403$	1.52952	−	2.64920i	0.0761906	−	0.131966i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.06872	−	1.85107i	0.0529744	−	0.0917543i
$408$		0			0
$409$	13.1155	+	22.7168i	0.648521	+	1.12327i	0.983476	+	0.181037i	$0.0579455\pi$
$409$	13.1155	+	22.7168i	0.648521	+	1.12327i	−0.334955	+	0.942234i	$0.608721\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−5.96501			−0.293519
$414$		0			0
$415$	6.44284			0.316267
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−11.3828	−	19.7155i	−0.556085	−	0.963167i	−0.997818	−	0.0660209i	$-0.978970\pi$
$419$	−11.3828	−	19.7155i	−0.556085	−	0.963167i	0.441733	−	0.897146i	$-0.354364\pi$
$420$		0			0
$421$	−14.1213	+	24.4587i	−0.688228	+	1.19205i	0.284182	+	0.958770i	$0.408278\pi$
$421$	−14.1213	+	24.4587i	−0.688228	+	1.19205i	−0.972411	+	0.233276i	$0.925055\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	3.91107	−	6.77417i	0.189715	−	0.328596i
$426$		0			0
$427$	−3.94242	−	6.82846i	−0.190787	−	0.330453i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	21.4480			1.03311			0.516557	−	0.856253i	$-0.327213\pi$
$431$	21.4480			1.03311			0.516557	+	0.856253i	$0.327213\pi$
$432$		0			0
$433$	−0.498583			−0.0239604			−0.0119802	−	0.999928i	$-0.503814\pi$
$433$	−0.498583			−0.0239604			−0.0119802	+	0.999928i	$0.503814\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−6.46899	−	11.2046i	−0.309454	−	0.535989i
$438$		0			0
$439$	12.2075	−	21.1440i	0.582631	−	1.00915i	−0.412535	−	0.910942i	$-0.635357\pi$
$439$	12.2075	−	21.1440i	0.582631	−	1.00915i	0.995166	−	0.0982047i	$-0.0313100\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	8.07544	−	13.9871i	0.383676	−	0.664546i	−0.607909	−	0.794007i	$-0.707991\pi$
$443$	8.07544	−	13.9871i	0.383676	−	0.664546i	0.991584	+	0.129461i	$0.0413247\pi$
$444$		0			0
$445$	5.10397	+	8.84033i	0.241951	+	0.419072i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	30.8392			1.45539			0.727697	−	0.685899i	$-0.240591\pi$
$449$	30.8392			1.45539			0.727697	+	0.685899i	$0.240591\pi$
$450$		0			0
$451$	33.1889			1.56281
$452$		0			0
$453$		0			0
$454$		0			0
$455$	0.526041	+	0.911130i	0.0246612	+	0.0427144i
$456$		0			0
$457$	7.61844	−	13.1955i	0.356376	−	0.617261i	−0.630977	−	0.775802i	$-0.717346\pi$
$457$	7.61844	−	13.1955i	0.356376	−	0.617261i	0.987352	+	0.158541i	$0.0506790\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−20.8805	+	36.1661i	−0.972503	+	1.68442i	−0.284561	+	0.958658i	$0.591848\pi$
$461$	−20.8805	+	36.1661i	−0.972503	+	1.68442i	−0.687942	+	0.725766i	$0.741486\pi$
$462$		0			0
$463$	9.47396	+	16.4094i	0.440292	+	0.762608i	0.997711	−	0.0676230i	$-0.0215415\pi$
$463$	9.47396	+	16.4094i	0.440292	+	0.762608i	−0.557419	+	0.830232i	$0.688208\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	1.90035			0.0879377			0.0439688	−	0.999033i	$-0.486000\pi$
$467$	1.90035			0.0879377			0.0439688	+	0.999033i	$0.486000\pi$
$468$		0			0
$469$	2.93234			0.135403
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−7.80992	−	13.5272i	−0.359101	−	0.621980i
$474$		0			0
$475$	2.25578	−	3.90713i	0.103502	−	0.179271i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−14.9088	+	25.8228i	−0.681201	+	1.17987i	0.293413	+	0.955986i	$0.405209\pi$
$479$	−14.9088	+	25.8228i	−0.681201	+	1.17987i	−0.974615	+	0.223889i	$0.928125\pi$
$480$		0			0
$481$	0.317070	+	0.549182i	0.0144572	+	0.0250405i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	8.77006			0.398228
$486$		0			0
$487$	−43.1213			−1.95401			−0.977005	−	0.213215i	$-0.931607\pi$
$487$	−43.1213			−1.95401			−0.977005	+	0.213215i	$0.931607\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−5.89659	−	10.2132i	−0.266110	−	0.460915i	0.701744	−	0.712429i	$-0.252405\pi$
$491$	−5.89659	−	10.2132i	−0.266110	−	0.460915i	−0.967854	+	0.251514i	$0.919072\pi$
$492$		0			0
$493$	−24.6363	+	42.6713i	−1.10956	+	1.92182i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	1.39378	−	2.41409i	0.0625195	−	0.108287i
$498$		0			0
$499$	9.55837	+	16.5556i	0.427892	+	0.741130i	0.996686	−	0.0813501i	$-0.0259232\pi$
$499$	9.55837	+	16.5556i	0.427892	+	0.741130i	−0.568794	+	0.822480i	$0.692590\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−21.1522			−0.943130			−0.471565	−	0.881831i	$-0.656311\pi$
$503$	−21.1522			−0.943130			−0.471565	+	0.881831i	$0.656311\pi$
$504$		0			0
$505$	2.26453			0.100770
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−0.982706	−	1.70210i	−0.0435577	−	0.0754441i	0.843425	−	0.537247i	$-0.180536\pi$
$509$	−0.982706	−	1.70210i	−0.0435577	−	0.0754441i	−0.886982	+	0.461803i	$0.847203\pi$
$510$		0			0
$511$	−3.73547	+	6.47003i	−0.165248	+	0.286217i
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−7.69289	+	13.3245i	−0.338989	+	0.587147i
$516$		0			0
$517$	−2.84179	−	4.92213i	−0.124982	−	0.216475i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	14.1143			0.618359			0.309180	−	0.951004i	$-0.399946\pi$
$521$	14.1143			0.618359			0.309180	+	0.951004i	$0.399946\pi$
$522$		0			0
$523$	5.03413			0.220127			0.110064	−	0.993925i	$-0.464895\pi$
$523$	5.03413			0.220127			0.110064	+	0.993925i	$0.464895\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−9.86774	−	17.0914i	−0.429845	−	0.744514i
$528$		0			0
$529$	7.38805	−	12.7965i	0.321219	−	0.556368i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−4.92329	+	8.52739i	−0.213252	+	0.369362i
$534$		0			0
$535$	−7.17010	−	12.4190i	−0.309990	−	0.536919i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−25.5296			−1.09964
$540$		0			0
$541$	−9.30926			−0.400236			−0.200118	−	0.979772i	$-0.564133\pi$
$541$	−9.30926			−0.400236			−0.200118	+	0.979772i	$0.564133\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.67560	−	4.63427i	−0.114610	−	0.198511i
$546$		0			0
$547$	−15.1245	+	26.1964i	−0.646677	+	1.12008i	0.337234	+	0.941421i	$0.390509\pi$
$547$	−15.1245	+	26.1964i	−0.646677	+	1.12008i	−0.983911	+	0.178657i	$0.942825\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−14.2094	+	24.6115i	−0.605342	+	1.04848i
$552$		0			0
$553$	−7.04711	−	12.2060i	−0.299674	−	0.519050i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−14.6903			−0.622450			−0.311225	−	0.950336i	$-0.600739\pi$
$557$	−14.6903			−0.622450			−0.311225	+	0.950336i	$0.600739\pi$
$558$		0			0
$559$	4.63414			0.196003
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−1.64857	−	2.85541i	−0.0694790	−	0.120341i	0.829193	−	0.558962i	$-0.188800\pi$
$563$	−1.64857	−	2.85541i	−0.0694790	−	0.120341i	−0.898672	+	0.438621i	$0.855467\pi$
$564$		0			0
$565$	−1.34471	+	2.32911i	−0.0565724	+	0.0979862i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−11.9943	+	20.7747i	−0.502826	+	0.870920i	0.497169	+	0.867654i	$0.334373\pi$
$569$	−11.9943	+	20.7747i	−0.502826	+	0.870920i	−0.999995	+	0.00326613i	$0.998960\pi$
$570$		0			0
$571$	−10.0403	−	17.3903i	−0.420174	−	0.727763i	0.575782	−	0.817603i	$-0.304698\pi$
$571$	−10.0403	−	17.3903i	−0.420174	−	0.727763i	−0.995956	+	0.0898400i	$0.971364\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−2.86774			−0.119593
$576$		0			0
$577$	22.8622			0.951764			0.475882	−	0.879509i	$-0.342129\pi$
$577$	22.8622			0.951764			0.475882	+	0.879509i	$0.342129\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	2.79534	+	4.84168i	0.115970	+	0.200867i
$582$		0			0
$583$	28.4204	−	49.2255i	1.17705	−	2.03871i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−12.9447	+	22.4208i	−0.534284	+	0.925407i	0.464914	+	0.885356i	$0.346085\pi$
$587$	−12.9447	+	22.4208i	−0.534284	+	0.925407i	−0.999198	+	0.0400508i	$0.987248\pi$
$588$		0			0
$589$	−5.69140	−	9.85779i	−0.234510	−	0.406183i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−39.7887			−1.63392			−0.816962	−	0.576691i	$-0.804344\pi$
$593$	−39.7887			−1.63392			−0.816962	+	0.576691i	$0.804344\pi$
$594$		0			0
$595$	6.78755			0.278263
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−13.9434	−	24.1507i	−0.569712	−	0.986770i	−0.996594	−	0.0824629i	$-0.973721\pi$
$599$	−13.9434	−	24.1507i	−0.569712	−	0.986770i	0.426882	−	0.904307i	$-0.359612\pi$
$600$		0			0
$601$	−13.5633	+	23.4924i	−0.553260	+	0.958275i	0.444776	+	0.895642i	$0.353283\pi$
$601$	−13.5633	+	23.4924i	−0.553260	+	0.958275i	−0.998037	+	0.0626334i	$0.980050\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.85044	−	4.93711i	0.115887	−	0.200722i
$606$		0			0
$607$	6.43085	+	11.1386i	0.261020	+	0.452100i	0.966513	−	0.256616i	$-0.0826077\pi$
$607$	6.43085	+	11.1386i	0.261020	+	0.452100i	−0.705493	+	0.708717i	$0.749274\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	1.68622			0.0682173
$612$		0			0
$613$	−29.2726			−1.18231			−0.591154	−	0.806558i	$-0.701328\pi$
$613$	−29.2726			−1.18231			−0.591154	+	0.806558i	$0.701328\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	17.3865	+	30.1144i	0.699956	+	1.21236i	0.968481	+	0.249087i	$0.0801304\pi$
$617$	17.3865	+	30.1144i	0.699956	+	1.21236i	−0.268525	+	0.963273i	$0.586536\pi$
$618$		0			0
$619$	−17.7282	+	30.7062i	−0.712558	+	1.23419i	0.251336	+	0.967900i	$0.419130\pi$
$619$	−17.7282	+	30.7062i	−0.712558	+	1.23419i	−0.963894	+	0.266287i	$0.914203\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−4.42889	+	7.67107i	−0.177440	+	0.307335i
$624$		0			0
$625$	−0.500000	−	0.866025i	−0.0200000	−	0.0346410i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	4.09119			0.163126
$630$		0			0
$631$	6.68942			0.266302			0.133151	−	0.991096i	$-0.457491\pi$
$631$	6.68942			0.266302			0.133151	+	0.991096i	$0.457491\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−10.7821	−	18.6751i	−0.427873	−	0.741097i
$636$		0			0
$637$	3.78709	−	6.55944i	0.150050	−	0.259894i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−2.46889	+	4.27624i	−0.0975151	+	0.168901i	−0.910656	−	0.413166i	$-0.864423\pi$
$641$	−2.46889	+	4.27624i	−0.0975151	+	0.168901i	0.813140	+	0.582068i	$0.197756\pi$
$642$		0			0
$643$	7.64406	+	13.2399i	0.301452	+	0.522130i	0.976465	−	0.215675i	$-0.0691953\pi$
$643$	7.64406	+	13.2399i	0.301452	+	0.522130i	−0.675013	+	0.737806i	$0.735862\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	31.2304			1.22779			0.613897	−	0.789386i	$-0.289601\pi$
$647$	31.2304			1.22779			0.613897	+	0.789386i	$0.289601\pi$
$648$		0			0
$649$	−28.0927			−1.10273
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−3.82516	−	6.62537i	−0.149690	−	0.259271i	0.781423	−	0.624002i	$-0.214494\pi$
$653$	−3.82516	−	6.62537i	−0.149690	−	0.259271i	−0.931113	+	0.364731i	$0.881161\pi$
$654$		0			0
$655$	−9.24703	+	16.0163i	−0.361312	+	0.625810i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	20.9775	−	36.3342i	0.817169	−	1.41538i	−0.0905915	−	0.995888i	$-0.528876\pi$
$659$	20.9775	−	36.3342i	0.817169	−	1.41538i	0.907760	−	0.419489i	$-0.137791\pi$
$660$		0			0
$661$	−20.2991	−	35.1591i	−0.789544	−	1.36753i	−0.926246	−	0.376918i	$-0.876984\pi$
$661$	−20.2991	−	35.1591i	−0.789544	−	1.36753i	0.136702	−	0.990612i	$-0.456350\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	3.91484			0.151811
$666$		0			0
$667$	18.0642			0.699449
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−18.5671	−	32.1592i	−0.716776	−	1.24149i
$672$		0			0
$673$	−7.43486	+	12.8775i	−0.286593	+	0.496393i	−0.972994	−	0.230830i	$-0.925856\pi$
$673$	−7.43486	+	12.8775i	−0.286593	+	0.496393i	0.686402	+	0.727223i	$0.259189\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−5.61271	+	9.72150i	−0.215714	+	0.373628i	−0.953493	−	0.301414i	$-0.902541\pi$
$677$	−5.61271	+	9.72150i	−0.215714	+	0.373628i	0.737779	+	0.675042i	$0.235875\pi$
$678$		0			0
$679$	3.80505	+	6.59054i	0.146024	+	0.252922i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	6.60275			0.252647			0.126324	−	0.991989i	$-0.459682\pi$
$683$	6.60275			0.252647			0.126324	+	0.991989i	$0.459682\pi$
$684$		0			0
$685$	9.72401			0.371535
$686$		0			0
$687$		0			0
$688$		0			0
$689$	8.43184	+	14.6044i	0.321228	+	0.556382i
$690$		0			0
$691$	−5.87423	+	10.1745i	−0.223466	+	0.387055i	−0.955858	−	0.293829i	$-0.905071\pi$
$691$	−5.87423	+	10.1745i	−0.223466	+	0.387055i	0.732392	+	0.680883i	$0.238404\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−8.04032	+	13.9262i	−0.304987	+	0.528253i
$696$		0			0
$697$	31.7628	+	55.0148i	1.20310	+	2.08384i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	6.21396			0.234698			0.117349	−	0.993091i	$-0.462560\pi$
$701$	6.21396			0.234698			0.117349	+	0.993091i	$0.462560\pi$
$702$		0			0
$703$	2.35967			0.0889965
$704$		0			0
$705$		0			0
$706$		0			0
$707$	0.982506	+	1.70175i	0.0369509	+	0.0640009i
$708$		0			0
$709$	6.49473	−	11.2492i	0.243915	−	0.422473i	−0.717911	−	0.696135i	$-0.754902\pi$
$709$	6.49473	−	11.2492i	0.243915	−	0.422473i	0.961826	+	0.273662i	$0.0882350\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−3.61769	+	6.26602i	−0.135483	+	0.234664i
$714$		0			0
$715$	2.47743	+	4.29104i	0.0926508	+	0.160476i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−15.4610			−0.576598			−0.288299	−	0.957540i	$-0.593090\pi$
$719$	−15.4610			−0.576598			−0.288299	+	0.957540i	$0.593090\pi$
$720$		0			0
$721$	−13.3508			−0.497210
$722$		0			0
$723$		0			0
$724$		0			0
$725$	3.14956	+	5.45520i	0.116972	+	0.202601i
$726$		0			0
$727$	−6.42588	+	11.1299i	−0.238323	+	0.412787i	−0.960233	−	0.279200i	$-0.909931\pi$
$727$	−6.42588	+	11.1299i	−0.238323	+	0.412787i	0.721910	+	0.691986i	$0.243264\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	14.9487	−	25.8919i	0.552897	−	0.957645i
$732$		0			0
$733$	10.5636	+	18.2968i	0.390177	+	0.675807i	0.992473	−	0.122467i	$-0.0390804\pi$
$733$	10.5636	+	18.2968i	0.390177	+	0.675807i	−0.602295	+	0.798273i	$0.705747\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	13.8101			0.508701
$738$		0			0
$739$	−18.8652			−0.693968			−0.346984	−	0.937871i	$-0.612794\pi$
$739$	−18.8652			−0.693968			−0.346984	+	0.937871i	$0.612794\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−3.99751	−	6.92390i	−0.146655	−	0.254013i	0.783334	−	0.621600i	$-0.213517\pi$
$743$	−3.99751	−	6.92390i	−0.146655	−	0.254013i	−0.929989	+	0.367587i	$0.880184\pi$
$744$		0			0
$745$	−7.92263	+	13.7224i	−0.290263	+	0.502750i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	6.22175	−	10.7764i	0.227338	−	0.393761i
$750$		0			0
$751$	15.2390	+	26.3948i	0.556081	+	0.963160i	0.997819	+	0.0660158i	$0.0210288\pi$
$751$	15.2390	+	26.3948i	0.556081	+	0.963160i	−0.441738	+	0.897144i	$0.645638\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	15.1213			0.550319
$756$		0			0
$757$	−30.4318			−1.10606			−0.553032	−	0.833160i	$-0.686529\pi$
$757$	−30.4318			−1.10606			−0.553032	+	0.833160i	$0.686529\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−23.3575	−	40.4564i	−0.846708	−	1.46654i	−0.884129	−	0.467242i	$-0.845248\pi$
$761$	−23.3575	−	40.4564i	−0.846708	−	1.46654i	0.0374210	−	0.999300i	$-0.488086\pi$
$762$		0			0
$763$	2.32171	−	4.02133i	0.0840517	−	0.145582i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	4.16731	−	7.21800i	0.150473	−	0.260627i
$768$		0			0
$769$	−18.4947	−	32.0338i	−0.666937	−	1.15517i	−0.978756	−	0.205027i	$-0.934272\pi$
$769$	−18.4947	−	32.0338i	−0.666937	−	1.15517i	0.311819	−	0.950141i	$-0.399062\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	33.7485			1.21385			0.606924	−	0.794760i	$-0.292403\pi$
$773$	33.7485			1.21385			0.606924	+	0.794760i	$0.292403\pi$
$774$		0			0
$775$	−2.52303			−0.0906298
$776$		0			0
$777$		0			0
$778$		0			0
$779$	18.3198	+	31.7308i	0.656374	+	1.13687i
$780$		0			0
$781$	6.56410	−	11.3694i	0.234882	−	0.406828i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	8.77655	−	15.2014i	0.313249	−	0.542562i
$786$		0			0
$787$	20.9745	+	36.3289i	0.747661	+	1.29499i	0.948941	+	0.315453i	$0.102156\pi$
$787$	20.9745	+	36.3289i	0.747661	+	1.29499i	−0.201280	+	0.979534i	$0.564510\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	−2.33371			−0.0829770
$792$		0			0
$793$	11.0171			0.391229
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.56666	+	11.3738i	0.232603	+	0.402880i	0.958573	−	0.284846i	$-0.0919424\pi$
$797$	6.56666	+	11.3738i	0.232603	+	0.402880i	−0.725970	+	0.687726i	$0.758609\pi$
$798$		0			0
$799$	5.43937	−	9.42127i	0.192431	−	0.333300i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−17.5925	+	30.4711i	−0.620826	+	1.07530i
$804$		0			0
$805$	−1.24422	−	2.15505i	−0.0438529	−	0.0759555i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	26.0015			0.914165			0.457082	−	0.889424i	$-0.348894\pi$
$809$	26.0015			0.914165			0.457082	+	0.889424i	$0.348894\pi$
$810$		0			0
$811$	14.9425			0.524702			0.262351	−	0.964973i	$-0.415502\pi$
$811$	14.9425			0.524702			0.262351	+	0.964973i	$0.415502\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.42837	+	5.93810i	0.120090	+	0.208003i
$816$		0			0
$817$	8.62192	−	14.9336i	0.301643	−	0.522461i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	−10.8310	+	18.7598i	−0.378004	+	0.654722i	−0.990772	−	0.135541i	$-0.956723\pi$
$821$	−10.8310	+	18.7598i	−0.378004	+	0.654722i	0.612768	+	0.790263i	$0.290056\pi$
$822$		0			0
$823$	−19.9770	−	34.6012i	−0.696355	−	1.20612i	−0.969722	−	0.244212i	$-0.921471\pi$
$823$	−19.9770	−	34.6012i	−0.696355	−	1.20612i	0.273367	−	0.961910i	$-0.411863\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	7.04711			0.245052			0.122526	−	0.992465i	$-0.460901\pi$
$827$	7.04711			0.245052			0.122526	+	0.992465i	$0.460901\pi$
$828$		0			0
$829$	−7.21376			−0.250544			−0.125272	−	0.992122i	$-0.539980\pi$
$829$	−7.21376			−0.250544			−0.125272	+	0.992122i	$0.539980\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−24.4326	−	42.3185i	−0.846539	−	1.46625i
$834$		0			0
$835$	−1.95991	+	3.39466i	−0.0678255	+	0.117477i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	1.16338	−	2.01503i	0.0401643	−	0.0695666i	−0.845244	−	0.534380i	$-0.820545\pi$
$839$	1.16338	−	2.01503i	0.0401643	−	0.0695666i	0.885409	+	0.464813i	$0.153879\pi$
$840$		0			0
$841$	−5.33944	−	9.24818i	−0.184119	−	0.318903i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	11.5300			0.396643
$846$		0			0
$847$	4.94686			0.169976
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−0.749950	−	1.29895i	−0.0257080	−	0.0445275i
$852$		0			0
$853$	18.3713	−	31.8200i	0.629022	−	1.08950i	−0.358727	−	0.933443i	$-0.616789\pi$
$853$	18.3713	−	31.8200i	0.629022	−	1.08950i	0.987748	−	0.156055i	$-0.0498777\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.45732	−	7.72031i	0.152259	−	0.263721i	−0.779798	−	0.626031i	$-0.784678\pi$
$857$	4.45732	−	7.72031i	0.152259	−	0.263721i	0.932058	+	0.362310i	$0.118012\pi$
$858$		0			0
$859$	−19.4817	−	33.7432i	−0.664706	−	1.15130i	−0.979365	−	0.202100i	$-0.935224\pi$
$859$	−19.4817	−	33.7432i	−0.664706	−	1.15130i	0.314659	−	0.949205i	$-0.398110\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	15.2084			0.517700			0.258850	−	0.965918i	$-0.416657\pi$
$863$	15.2084			0.517700			0.258850	+	0.965918i	$0.416657\pi$
$864$		0			0
$865$	−8.17334			−0.277902
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−33.1889	−	57.4849i	−1.12586	−	1.95004i
$870$		0			0
$871$	−2.04861	+	3.54829i	−0.0694144	+	0.120229i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	0.433868	−	0.751481i	0.0146674	−	0.0254047i
$876$		0			0
$877$	5.21592	+	9.03424i	0.176129	+	0.305065i	0.940551	−	0.339651i	$-0.110309\pi$
$877$	5.21592	+	9.03424i	0.176129	+	0.305065i	−0.764422	+	0.644716i	$0.776976\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−22.9272			−0.772438			−0.386219	−	0.922407i	$-0.626219\pi$
$881$	−22.9272			−0.772438			−0.386219	+	0.922407i	$0.626219\pi$
$882$		0			0
$883$	10.9773			0.369417			0.184708	−	0.982793i	$-0.440866\pi$
$883$	10.9773			0.369417			0.184708	+	0.982793i	$0.440866\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−6.48496	−	11.2323i	−0.217744	−	0.377143i	0.736374	−	0.676575i	$-0.236536\pi$
$887$	−6.48496	−	11.2323i	−0.217744	−	0.377143i	−0.954118	+	0.299431i	$0.903203\pi$
$888$		0			0
$889$	9.35597	−	16.2050i	0.313789	−	0.543499i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	3.13726	−	5.43389i	0.104984	−	0.181838i
$894$		0			0
$895$	3.42837	+	5.93810i	0.114598	+	0.198489i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	15.8928			0.530056
$900$		0			0
$901$	108.797			3.62454
$902$		0			0
$903$		0			0
$904$		0			0
$905$	8.14956	+	14.1154i	0.270900	+	0.469213i
$906$		0			0
$907$	5.74173	−	9.94497i	0.190651	−	0.330217i	−0.754815	−	0.655938i	$-0.772273\pi$
$907$	5.74173	−	9.94497i	0.190651	−	0.330217i	0.945466	+	0.325720i	$0.105607\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−7.64383	+	13.2395i	−0.253251	+	0.438644i	−0.964419	−	0.264378i	$-0.914833\pi$
$911$	−7.64383	+	13.2395i	−0.253251	+	0.438644i	0.711168	+	0.703022i	$0.248167\pi$
$912$		0			0
$913$	13.1649	+	22.8023i	0.435694	+	0.754645i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−16.0480			−0.529951
$918$		0			0
$919$	−43.5890			−1.43787			−0.718934	−	0.695078i	$-0.755370\pi$
$919$	−43.5890			−1.43787			−0.718934	+	0.695078i	$0.755370\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	1.94746	+	3.37310i	0.0641014	+	0.111027i
$924$		0			0
$925$	0.261513	−	0.452954i	0.00859850	−	0.0148930i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	0.741502	−	1.28432i	0.0243279	−	0.0421371i	−0.853605	−	0.520921i	$-0.825589\pi$
$929$	0.741502	−	1.28432i	0.0243279	−	0.0421371i	0.877933	+	0.478783i	$0.158922\pi$
$930$		0			0
$931$	−14.0919	−	24.4080i	−0.461845	−	0.799939i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	31.9665			1.04542
$936$		0			0
$937$	5.12821			0.167531			0.0837657	−	0.996485i	$-0.473305\pi$
$937$	5.12821			0.167531			0.0837657	+	0.996485i	$0.473305\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	23.4381	+	40.5960i	0.764061	+	1.32339i	0.940741	+	0.339125i	$0.110131\pi$
$941$	23.4381	+	40.5960i	0.764061	+	1.32339i	−0.176680	+	0.984268i	$0.556536\pi$
$942$		0			0
$943$	11.6448	−	20.1694i	0.379207	−	0.656806i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	17.1850	−	29.7654i	0.558439	−	0.967244i	−0.439188	−	0.898395i	$-0.644734\pi$
$947$	17.1850	−	29.7654i	0.558439	−	0.967244i	0.997627	−	0.0688491i	$-0.0219327\pi$
$948$		0			0
$949$	−5.21940	−	9.04026i	−0.169429	−	0.293459i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	23.8092			0.771254			0.385627	−	0.922655i	$-0.373985\pi$
$953$	23.8092			0.771254			0.385627	+	0.922655i	$0.373985\pi$
$954$		0			0
$955$	11.8336			0.382927
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.21894	+	7.30741i	0.136237	+	0.235969i
$960$		0			0
$961$	12.3172	−	21.3340i	0.397328	−	0.688192i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−0.911072	+	1.57802i	−0.0293284	+	0.0507983i
$966$		0			0
$967$	−1.42889	−	2.47492i	−0.0459501	−	0.0795880i	0.842136	−	0.539266i	$-0.181298\pi$
$967$	−1.42889	−	2.47492i	−0.0459501	−	0.0795880i	−0.888086	+	0.459678i	$0.847965\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−3.55821			−0.114188			−0.0570942	−	0.998369i	$-0.518184\pi$
$971$	−3.55821			−0.114188			−0.0570942	+	0.998369i	$0.518184\pi$
$972$		0			0
$973$	−13.9537			−0.447337
$974$		0			0
$975$		0			0
$976$		0			0
$977$	14.8480	+	25.7175i	0.475029	+	0.822775i	0.999591	−	0.0285977i	$-0.00910419\pi$
$977$	14.8480	+	25.7175i	0.475029	+	0.822775i	−0.524562	+	0.851372i	$0.675771\pi$
$978$		0			0
$979$	−20.8582	+	36.1275i	−0.666632	+	1.15464i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−29.0557	+	50.3259i	−0.926732	+	1.60515i	−0.137982	+	0.990435i	$0.544061\pi$
$983$	−29.0557	+	50.3259i	−0.926732	+	1.60515i	−0.788751	+	0.614713i	$0.789272\pi$
$984$		0			0
$985$	−9.68988	−	16.7834i	−0.308745	−	0.534762i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−10.9609			−0.348536
$990$		0			0
$991$	−19.4489			−0.617816			−0.308908	−	0.951092i	$-0.599963\pi$
$991$	−19.4489			−0.617816			−0.308908	+	0.951092i	$0.599963\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	3.34818	+	5.79923i	0.106145	+	0.183848i
$996$		0			0
$997$	4.25201	−	7.36469i	0.134662	−	0.233242i	−0.790806	−	0.612067i	$-0.790338\pi$
$997$	4.25201	−	7.36469i	0.134662	−	0.233242i	0.925468	+	0.378825i	$0.123672\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1080.2.q.e.721.2		8
3.2	odd	2		360.2.q.e.241.4	yes	8
4.3	odd	2		2160.2.q.l.721.3		8
9.2	odd	6		3240.2.a.u.1.3		4
9.4	even	3	inner	1080.2.q.e.361.2		8
9.5	odd	6		360.2.q.e.121.4	✓	8
9.7	even	3		3240.2.a.s.1.3		4
12.11	even	2		720.2.q.l.241.1		8
36.7	odd	6		6480.2.a.bz.1.2		4
36.11	even	6		6480.2.a.cb.1.2		4
36.23	even	6		720.2.q.l.481.1		8
36.31	odd	6		2160.2.q.l.1441.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
360.2.q.e.121.4	✓	8	9.5	odd	6
360.2.q.e.241.4	yes	8	3.2	odd	2
720.2.q.l.241.1		8	12.11	even	2
720.2.q.l.481.1		8	36.23	even	6
1080.2.q.e.361.2		8	9.4	even	3	inner
1080.2.q.e.721.2		8	1.1	even	1	trivial
2160.2.q.l.721.3		8	4.3	odd	2
2160.2.q.l.1441.3		8	36.31	odd	6
3240.2.a.s.1.3		4	9.7	even	3
3240.2.a.u.1.3		4	9.2	odd	6
6480.2.a.bz.1.2		4	36.7	odd	6
6480.2.a.cb.1.2		4	36.11	even	6

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:	\( N \)	\(=\)	\( 1080 = 2^{3} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1080.q (of order \(3\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(0.500000 + 0.866025i) q^{5} +(-0.433868 + 0.751481i) q^{7} +(-2.04334 + 3.53916i) q^{11} +(-0.606223 - 1.05001i) q^{13} -7.82214 q^{17} -4.51156 q^{19} +(1.43387 + 2.48353i) q^{23} +(-0.500000 + 0.866025i) q^{25} +(3.14956 - 5.45520i) q^{29} +(1.26151 + 2.18500i) q^{31} -0.867736 q^{35} -0.523026 q^{37} +(-4.06063 - 7.03322i) q^{41} +(-1.91107 + 3.31007i) q^{43} +(-0.695381 + 1.20444i) q^{47} +(3.12352 + 5.41009i) q^{49} -13.9088 q^{53} -4.08667 q^{55} +(3.43711 + 5.95325i) q^{59} +(-4.54334 + 7.86929i) q^{61} +(0.606223 - 1.05001i) q^{65} +(-1.68965 - 2.92656i) q^{67} -3.21245 q^{71} +8.60970 q^{73} +(-1.77308 - 3.07106i) q^{77} +(-8.12126 + 14.0664i) q^{79} +(3.22142 - 5.57967i) q^{83} +(-3.91107 - 6.77417i) q^{85} +10.2079 q^{89} +1.05208 q^{91} +(-2.25578 - 3.90713i) q^{95} +(4.38503 - 7.59510i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{5} + q^{7} + q^{11} - 4 q^{13} - 10 q^{17} + 2 q^{19} + 7 q^{23} - 4 q^{25} + 7 q^{29} + 2 q^{31} + 2 q^{35} + 12 q^{37} + 12 q^{41} + 11 q^{43} + 7 q^{47} - 3 q^{49} - 24 q^{53} + 2 q^{55}+ \cdots - q^{97}+O(q^{100}) \) 8 * q + 4 * q^5 + q^7 + q^11 - 4 * q^13 - 10 * q^17 + 2 * q^19 + 7 * q^23 - 4 * q^25 + 7 * q^29 + 2 * q^31 + 2 * q^35 + 12 * q^37 + 12 * q^41 + 11 * q^43 + 7 * q^47 - 3 * q^49 - 24 * q^53 + 2 * q^55 + 11 * q^59 - 19 * q^61 + 4 * q^65 + 10 * q^67 - 24 * q^71 + 18 * q^73 + 32 * q^77 + 24 * q^79 + 23 * q^83 - 5 * q^85 - 42 * q^89 + 28 * q^91 + q^95 - q^97

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