LMFDB - Embedded newform 540.2.j.b.377.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [540,2,Mod(53,540)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(540, base_ring=CyclotomicField(4)) chi = DirichletCharacter(H, H._module([0, 2, 3])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("540.53"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 540 = 2^{2} \cdot 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	540.j (of order $4$, degree $2$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,0,0,0,4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(7)] == traces)

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

Self dual:	no
Analytic conductor:	$4.31192170915$
Analytic rank:	$0$
Dimension:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	8.0.12745506816.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} + 23x^{4} + 1 $ x^8 + 23*x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			377.3
Root			$1.54779 - 1.54779i$ of defining polynomial
Character	$\chi$	$=$	540.377
Dual form			540.2.j.b.53.4

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.87083 - 1.22474i) q^{5} +(-1.79129 - 1.79129i) q^{7} -0.646084i q^{11} +(3.79129 - 3.79129i) q^{13} +(-4.96640 + 4.96640i) q^{17} -6.58258i q^{19} +(-1.87083 - 1.87083i) q^{23} +(2.00000 - 4.58258i) q^{25} +7.99455 q^{29} -0.582576 q^{31} +(-5.54506 - 1.15732i) q^{35} +(3.00000 + 3.00000i) q^{37} +4.25290i q^{41} +(-0.791288 + 0.791288i) q^{43} +(1.93825 - 1.93825i) q^{47} -0.582576i q^{49} +(-5.47764 - 5.47764i) q^{53} +(-0.791288 - 1.20871i) q^{55} +12.8935 q^{59} +6.16515 q^{61} +(2.44949 - 11.7362i) q^{65} +(-7.00000 - 7.00000i) q^{67} +14.3205i q^{71} +(-1.20871 + 1.20871i) q^{73} +(-1.15732 + 1.15732i) q^{77} +6.16515i q^{79} +(1.22474 + 1.22474i) q^{83} +(-3.20871 + 15.3739i) q^{85} -9.42157 q^{89} -13.5826 q^{91} +(-8.06198 - 12.3149i) q^{95} +(7.58258 + 7.58258i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 4 q^{7} + 12 q^{13} + 16 q^{25} + 32 q^{31} + 24 q^{37} + 12 q^{43} + 12 q^{55} - 24 q^{61} - 56 q^{67} - 28 q^{73} - 44 q^{85} - 72 q^{91} + 24 q^{97}+O(q^{100}) $ 8 * q + 4 * q^7 + 12 * q^13 + 16 * q^25 + 32 * q^31 + 24 * q^37 + 12 * q^43 + 12 * q^55 - 24 * q^61 - 56 * q^67 - 28 * q^73 - 44 * q^85 - 72 * q^91 + 24 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/540\mathbb{Z}\right)^\times$.

$n$	$217$	$271$	$461$
$\chi(n)$	$e\left(\frac{1}{4}\right)$	$1$	$-1$

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
$5$	1.87083	−	1.22474i	0.836660	−	0.547723i
$5$	1.87083	−	1.22474i	0.836660	−	0.547723i
$6$		0			0
$7$	−1.79129	−	1.79129i	−0.677043	−	0.677043i	0.282287	−	0.959330i	$-0.408907\pi$
$7$	−1.79129	−	1.79129i	−0.677043	−	0.677043i	−0.959330	+	0.282287i	$0.908907\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	0.646084i		−	0.194802i	−0.995245	−	0.0974008i	$-0.968947\pi$
$11$		−	0.646084i		−	0.194802i	0.995245	−	0.0974008i	$-0.0310529\pi$
$12$		0			0
$13$	3.79129	−	3.79129i	1.05151	−	1.05151i	0.0529150	−	0.998599i	$-0.483149\pi$
$13$	3.79129	−	3.79129i	1.05151	−	1.05151i	0.998599	−	0.0529150i	$-0.0168513\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.96640	+	4.96640i	−1.20453	+	1.20453i	−0.231755	+	0.972774i	$0.574447\pi$
$17$	−4.96640	+	4.96640i	−1.20453	+	1.20453i	−0.972774	+	0.231755i	$0.925553\pi$
$18$		0			0
$19$		−	6.58258i		−	1.51015i	−0.655640	−	0.755073i	$-0.727601\pi$
$19$		−	6.58258i		−	1.51015i	0.655640	−	0.755073i	$-0.272399\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−1.87083	−	1.87083i	−0.390095	−	0.390095i	0.484626	−	0.874721i	$-0.338956\pi$
$23$	−1.87083	−	1.87083i	−0.390095	−	0.390095i	−0.874721	+	0.484626i	$0.838956\pi$
$24$		0			0
$25$	2.00000	−	4.58258i	0.400000	−	0.916515i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	7.99455			1.48455			0.742276	−	0.670095i	$-0.233747\pi$
$29$	7.99455			1.48455			0.742276	+	0.670095i	$0.233747\pi$
$30$		0			0
$31$	−0.582576			−0.104634			−0.0523168	−	0.998631i	$-0.516661\pi$
$31$	−0.582576			−0.104634			−0.0523168	+	0.998631i	$0.516661\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−5.54506	−	1.15732i	−0.937287	−	0.195623i
$36$		0			0
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	0.909312	−	0.416115i	$-0.136609\pi$
$37$	3.00000	+	3.00000i	0.493197	+	0.493197i	−0.416115	+	0.909312i	$0.636609\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			4.25290i			0.664191i	0.943246	+	0.332095i	$0.107756\pi$
$41$			4.25290i			0.664191i	−0.943246	+	0.332095i	$0.892244\pi$
$42$		0			0
$43$	−0.791288	+	0.791288i	−0.120670	+	0.120670i	−0.764863	−	0.644193i	$-0.777193\pi$
$43$	−0.791288	+	0.791288i	−0.120670	+	0.120670i	0.644193	+	0.764863i	$0.277193\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.93825	−	1.93825i	0.282723	−	0.282723i	−0.551471	−	0.834194i	$-0.685933\pi$
$47$	1.93825	−	1.93825i	0.282723	−	0.282723i	0.834194	+	0.551471i	$0.185933\pi$
$48$		0			0
$49$		−	0.582576i		−	0.0832251i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−5.47764	−	5.47764i	−0.752412	−	0.752412i	0.222517	−	0.974929i	$-0.428573\pi$
$53$	−5.47764	−	5.47764i	−0.752412	−	0.752412i	−0.974929	+	0.222517i	$0.928573\pi$
$54$		0			0
$55$	−0.791288	−	1.20871i	−0.106697	−	0.162983i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	12.8935			1.67859			0.839297	−	0.543672i	$-0.182967\pi$
$59$	12.8935			1.67859			0.839297	+	0.543672i	$0.182967\pi$
$60$		0			0
$61$	6.16515			0.789367			0.394683	−	0.918817i	$-0.370854\pi$
$61$	6.16515			0.789367			0.394683	+	0.918817i	$0.370854\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	2.44949	−	11.7362i	0.303822	−	1.45570i
$66$		0			0
$67$	−7.00000	−	7.00000i	−0.855186	−	0.855186i	0.135580	−	0.990766i	$-0.456710\pi$
$67$	−7.00000	−	7.00000i	−0.855186	−	0.855186i	−0.990766	+	0.135580i	$0.956710\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			14.3205i			1.69954i	0.527157	+	0.849768i	$0.323258\pi$
$71$			14.3205i			1.69954i	−0.527157	+	0.849768i	$0.676742\pi$
$72$		0			0
$73$	−1.20871	+	1.20871i	−0.141469	+	0.141469i	−0.774294	−	0.632825i	$-0.781895\pi$
$73$	−1.20871	+	1.20871i	−0.141469	+	0.141469i	0.632825	+	0.774294i	$0.281895\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−1.15732	+	1.15732i	−0.131889	+	0.131889i
$78$		0			0
$79$			6.16515i			0.693634i	0.937933	+	0.346817i	$0.112737\pi$
$79$			6.16515i			0.693634i	−0.937933	+	0.346817i	$0.887263\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.22474	+	1.22474i	0.134433	+	0.134433i	0.771121	−	0.636688i	$-0.219696\pi$
$83$	1.22474	+	1.22474i	0.134433	+	0.134433i	−0.636688	+	0.771121i	$0.719696\pi$
$84$		0			0
$85$	−3.20871	+	15.3739i	−0.348034	+	1.66753i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−9.42157			−0.998684			−0.499342	−	0.866405i	$-0.666425\pi$
$89$	−9.42157			−0.998684			−0.499342	+	0.866405i	$0.666425\pi$
$90$		0			0
$91$	−13.5826			−1.42384
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−8.06198	−	12.3149i	−0.827141	−	1.26348i
$96$		0			0
$97$	7.58258	+	7.58258i	0.769894	+	0.769894i	0.978088	−	0.208194i	$-0.0667584\pi$
$97$	7.58258	+	7.58258i	0.769894	+	0.769894i	−0.208194	+	0.978088i	$0.566758\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$			8.50579i			0.846358i	0.906046	+	0.423179i	$0.139086\pi$
$101$			8.50579i			0.846358i	−0.906046	+	0.423179i	$0.860914\pi$
$102$		0			0
$103$	−3.58258	+	3.58258i	−0.353002	+	0.353002i	−0.861225	−	0.508224i	$-0.830302\pi$
$103$	−3.58258	+	3.58258i	−0.353002	+	0.353002i	0.508224	+	0.861225i	$0.330302\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−13.5396	+	13.5396i	−1.30892	+	1.30892i	−0.386732	+	0.922192i	$0.626396\pi$
$107$	−13.5396	+	13.5396i	−1.30892	+	1.30892i	−0.922192	+	0.386732i	$0.873604\pi$
$108$		0			0
$109$			15.7477i			1.50836i	0.656668	+	0.754179i	$0.271965\pi$
$109$			15.7477i			1.50836i	−0.656668	+	0.754179i	$0.728035\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	5.03383	+	5.03383i	0.473542	+	0.473542i	0.903059	−	0.429517i	$-0.141316\pi$
$113$	5.03383	+	5.03383i	0.473542	+	0.473542i	−0.429517	+	0.903059i	$0.641316\pi$
$114$		0			0
$115$	−5.79129	−	1.20871i	−0.540040	−	0.112713i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	17.7925			1.63104
$120$		0			0
$121$	10.5826			0.962052
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.87083	−	11.0227i	−0.167332	−	0.985901i
$126$		0			0
$127$	6.58258	+	6.58258i	0.584109	+	0.584109i	0.936030	−	0.351921i	$-0.114471\pi$
$127$	6.58258	+	6.58258i	0.584109	+	0.584109i	−0.351921	+	0.936030i	$0.614471\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	9.93280i		−	0.867833i	−0.900953	−	0.433917i	$-0.857131\pi$
$131$		−	9.93280i		−	0.867833i	0.900953	−	0.433917i	$-0.142869\pi$
$132$		0			0
$133$	−11.7913	+	11.7913i	−1.02243	+	1.02243i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−1.22474	+	1.22474i	−0.104637	+	0.104637i	−0.757487	−	0.652850i	$-0.773573\pi$
$137$	−1.22474	+	1.22474i	−0.104637	+	0.104637i	0.652850	+	0.757487i	$0.273573\pi$
$138$		0			0
$139$			11.1652i			0.947016i	0.880790	+	0.473508i	$0.157012\pi$
$139$			11.1652i			0.947016i	−0.880790	+	0.473508i	$0.842988\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−2.44949	−	2.44949i	−0.204837	−	0.204837i
$144$		0			0
$145$	14.9564	−	9.79129i	1.24206	−	0.813122i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	11.7362			0.961468			0.480734	−	0.876867i	$-0.340370\pi$
$149$	11.7362			0.961468			0.480734	+	0.876867i	$0.340370\pi$
$150$		0			0
$151$	−17.1652			−1.39688			−0.698440	−	0.715669i	$-0.746122\pi$
$151$	−17.1652			−1.39688			−0.698440	+	0.715669i	$0.746122\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.08990	+	0.713507i	−0.0875428	+	0.0573102i
$156$		0			0
$157$	−3.62614	−	3.62614i	−0.289397	−	0.289397i	0.547445	−	0.836842i	$-0.315601\pi$
$157$	−3.62614	−	3.62614i	−0.289397	−	0.289397i	−0.836842	+	0.547445i	$0.815601\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			6.70239i			0.528222i
$162$		0			0
$163$	−8.58258	+	8.58258i	−0.672239	+	0.672239i	−0.958232	−	0.285993i	$-0.907677\pi$
$163$	−8.58258	+	8.58258i	−0.672239	+	0.672239i	0.285993	+	0.958232i	$0.407677\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	−6.76981	+	6.76981i	−0.523863	+	0.523863i	−0.918736	−	0.394872i	$-0.870789\pi$
$167$	−6.76981	+	6.76981i	−0.523863	+	0.523863i	0.394872	+	0.918736i	$0.370789\pi$
$168$		0			0
$169$		−	15.7477i		−	1.21136i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−13.6070	−	13.6070i	−1.03452	−	1.03452i	−0.999382	−	0.0351417i	$-0.988812\pi$
$173$	−13.6070	−	13.6070i	−1.03452	−	1.03452i	−0.0351417	−	0.999382i	$-0.511188\pi$
$174$		0			0
$175$	−11.7913	+	4.62614i	−0.891338	+	0.349703i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	24.7646			1.85099			0.925496	−	0.378757i	$-0.123648\pi$
$179$	24.7646			1.85099			0.925496	+	0.378757i	$0.123648\pi$
$180$		0			0
$181$	−5.41742			−0.402674			−0.201337	−	0.979522i	$-0.564529\pi$
$181$	−5.41742			−0.402674			−0.201337	+	0.979522i	$0.564529\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	9.28672	+	1.93825i	0.682773	+	0.142503i
$186$		0			0
$187$	3.20871	+	3.20871i	0.234644	+	0.234644i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	3.74166i		−	0.270737i	−0.990795	−	0.135368i	$-0.956778\pi$
$191$		−	3.74166i		−	0.270737i	0.990795	−	0.135368i	$-0.0432218\pi$
$192$		0			0
$193$	15.7913	−	15.7913i	1.13668	−	1.13668i	0.147641	−	0.989041i	$-0.452832\pi$
$193$	15.7913	−	15.7913i	1.13668	−	1.13668i	0.989041	−	0.147641i	$-0.0471679\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.12372	−	6.12372i	0.436297	−	0.436297i	−0.454467	−	0.890764i	$-0.650170\pi$
$197$	6.12372	−	6.12372i	0.436297	−	0.436297i	0.890764	+	0.454467i	$0.150170\pi$
$198$		0			0
$199$		−	13.5826i		−	0.962843i	−0.876489	−	0.481422i	$-0.840121\pi$
$199$		−	13.5826i		−	0.962843i	0.876489	−	0.481422i	$-0.159879\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−14.3205	−	14.3205i	−1.00511	−	1.00511i
$204$		0			0
$205$	5.20871	+	7.95644i	0.363792	+	0.555702i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.25290			−0.294179
$210$		0			0
$211$	26.1652			1.80128			0.900642	−	0.434563i	$-0.143097\pi$
$211$	26.1652			1.80128			0.900642	+	0.434563i	$0.143097\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−0.511238	+	2.44949i	−0.0348662	+	0.167054i
$216$		0			0
$217$	1.04356	+	1.04356i	0.0708415	+	0.0708415i
$218$		0			0
$219$		0			0
$220$		0			0
$221$			37.6581i			2.53316i
$222$		0			0
$223$	12.1652	−	12.1652i	0.814639	−	0.814639i	−0.170687	−	0.985325i	$-0.554599\pi$
$223$	12.1652	−	12.1652i	0.814639	−	0.814639i	0.985325	+	0.170687i	$0.0545986\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−15.5453	+	15.5453i	−1.03178	+	1.03178i	−0.0322989	+	0.999478i	$0.510283\pi$
$227$	−15.5453	+	15.5453i	−1.03178	+	1.03178i	−0.999478	+	0.0322989i	$0.989717\pi$
$228$		0			0
$229$			15.0000i			0.991228i	0.868543	+	0.495614i	$0.165057\pi$
$229$			15.0000i			0.991228i	−0.868543	+	0.495614i	$0.834943\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−4.38774	−	4.38774i	−0.287450	−	0.287450i	0.548621	−	0.836071i	$-0.315153\pi$
$233$	−4.38774	−	4.38774i	−0.287450	−	0.287450i	−0.836071	+	0.548621i	$0.815153\pi$
$234$		0			0
$235$	1.25227	−	6.00000i	0.0816893	−	0.391397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−15.1015			−0.976833			−0.488417	−	0.872611i	$-0.662425\pi$
$239$	−15.1015			−0.976833			−0.488417	+	0.872611i	$0.662425\pi$
$240$		0			0
$241$	−3.74773			−0.241412			−0.120706	−	0.992688i	$-0.538516\pi$
$241$	−3.74773			−0.241412			−0.120706	+	0.992688i	$0.538516\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−0.713507	−	1.08990i	−0.0455843	−	0.0696311i
$246$		0			0
$247$	−24.9564	−	24.9564i	−1.58794	−	1.58794i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	27.3489i		−	1.72625i	−0.504991	−	0.863124i	$-0.668504\pi$
$251$		−	27.3489i		−	1.72625i	0.504991	−	0.863124i	$-0.331496\pi$
$252$		0			0
$253$	−1.20871	+	1.20871i	−0.0759911	+	0.0759911i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−1.87083	+	1.87083i	−0.116699	+	0.116699i	−0.763045	−	0.646346i	$-0.776296\pi$
$257$	−1.87083	+	1.87083i	−0.116699	+	0.116699i	0.646346	+	0.763045i	$0.276296\pi$
$258$		0			0
$259$		−	10.7477i		−	0.667831i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	16.7700	+	16.7700i	1.03408	+	1.03408i	0.999398	+	0.0346864i	$0.0110433\pi$
$263$	16.7700	+	16.7700i	1.03408	+	1.03408i	0.0346864	+	0.999398i	$0.488957\pi$
$264$		0			0
$265$	−16.9564	−	3.53901i	−1.04163	−	0.217400i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	14.8318			0.904310			0.452155	−	0.891939i	$-0.350655\pi$
$269$	14.8318			0.904310			0.452155	+	0.891939i	$0.350655\pi$
$270$		0			0
$271$	13.7477			0.835115			0.417557	−	0.908651i	$-0.362886\pi$
$271$	13.7477			0.835115			0.417557	+	0.908651i	$0.362886\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.96073	−	1.29217i	−0.178539	−	0.0779206i
$276$		0			0
$277$	0.791288	+	0.791288i	0.0475439	+	0.0475439i	0.730479	−	0.682935i	$-0.239297\pi$
$277$	0.791288	+	0.791288i	0.0475439	+	0.0475439i	−0.682935	+	0.730479i	$0.739297\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	0.511238i		−	0.0304979i	−0.999884	−	0.0152490i	$-0.995146\pi$
$281$		−	0.511238i		−	0.0304979i	0.999884	−	0.0152490i	$-0.00485408\pi$
$282$		0			0
$283$	14.7913	−	14.7913i	0.879251	−	0.879251i	−0.114206	−	0.993457i	$-0.536432\pi$
$283$	14.7913	−	14.7913i	0.879251	−	0.879251i	0.993457	+	0.114206i	$0.0364325\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	7.61816	−	7.61816i	0.449686	−	0.449686i
$288$		0			0
$289$		−	32.3303i		−	1.90178i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−1.73598	−	1.73598i	−0.101417	−	0.101417i	0.654578	−	0.755995i	$-0.272847\pi$
$293$	−1.73598	−	1.73598i	−0.101417	−	0.101417i	−0.755995	+	0.654578i	$0.772847\pi$
$294$		0			0
$295$	24.1216	−	15.7913i	1.40441	−	0.919404i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−14.1857			−0.820380
$300$		0			0
$301$	2.83485			0.163398
$302$		0			0
$303$		0			0
$304$		0			0
$305$	11.5339	−	7.55074i	0.660432	−	0.432354i
$306$		0			0
$307$	15.0000	+	15.0000i	0.856095	+	0.856095i	0.990876	−	0.134780i	$-0.0430329\pi$
$307$	15.0000	+	15.0000i	0.856095	+	0.856095i	−0.134780	+	0.990876i	$0.543033\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	26.2983i		−	1.49124i	−0.666371	−	0.745620i	$-0.732153\pi$
$311$		−	26.2983i		−	1.49124i	0.666371	−	0.745620i	$-0.267847\pi$
$312$		0			0
$313$	9.16515	−	9.16515i	0.518045	−	0.518045i	−0.398934	−	0.916979i	$-0.630620\pi$
$313$	9.16515	−	9.16515i	0.518045	−	0.518045i	0.916979	+	0.398934i	$0.130620\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−1.22474	+	1.22474i	−0.0687885	+	0.0687885i	−0.740664	−	0.671876i	$-0.765489\pi$
$317$	−1.22474	+	1.22474i	−0.0687885	+	0.0687885i	0.671876	+	0.740664i	$0.265489\pi$
$318$		0			0
$319$		−	5.16515i		−	0.289193i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	32.6917	+	32.6917i	1.81902	+	1.81902i
$324$		0			0
$325$	−9.79129	−	24.9564i	−0.543123	−	1.38433i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−6.94393			−0.382831
$330$		0			0
$331$	−9.58258			−0.526706			−0.263353	−	0.964700i	$-0.584828\pi$
$331$	−9.58258			−0.526706			−0.263353	+	0.964700i	$0.584828\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−21.6690	−	4.52259i	−1.18390	−	0.247095i
$336$		0			0
$337$	−18.9564	−	18.9564i	−1.03262	−	1.03262i	−0.999450	−	0.0331734i	$-0.989439\pi$
$337$	−18.9564	−	18.9564i	−1.03262	−	1.03262i	−0.0331734	−	0.999450i	$-0.510561\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$			0.376393i			0.0203828i
$342$		0			0
$343$	−13.5826	+	13.5826i	−0.733390	+	0.733390i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	8.50579	−	8.50579i	0.456615	−	0.456615i	−0.440928	−	0.897543i	$-0.645351\pi$
$347$	8.50579	−	8.50579i	0.456615	−	0.456615i	0.897543	+	0.440928i	$0.145351\pi$
$348$		0			0
$349$		−	0.252273i		−	0.0135039i	−0.999977	−	0.00675193i	$-0.997851\pi$
$349$		−	0.252273i		−	0.0135039i	0.999977	−	0.00675193i	$-0.00214922\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.79796	−	9.79796i	−0.521493	−	0.521493i	0.396529	−	0.918022i	$-0.370215\pi$
$353$	−9.79796	−	9.79796i	−0.521493	−	0.521493i	−0.918022	+	0.396529i	$0.870215\pi$
$354$		0			0
$355$	17.5390	+	26.7913i	0.930874	+	1.42193i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−0.134846			−0.00711688			−0.00355844	−	0.999994i	$-0.501133\pi$
$359$	−0.134846			−0.00711688			−0.00355844	+	0.999994i	$0.501133\pi$
$360$		0			0
$361$	−24.3303			−1.28054
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−0.780929	+	3.74166i	−0.0408757	+	0.195847i
$366$		0			0
$367$	17.7913	+	17.7913i	0.928698	+	0.928698i	0.997622	−	0.0689242i	$-0.0219567\pi$
$367$	17.7913	+	17.7913i	0.928698	+	0.928698i	−0.0689242	+	0.997622i	$0.521957\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			19.6241i			1.01883i
$372$		0			0
$373$	−6.74773	+	6.74773i	−0.349384	+	0.349384i	−0.859880	−	0.510496i	$-0.829462\pi$
$373$	−6.74773	+	6.74773i	−0.349384	+	0.349384i	0.510496	+	0.859880i	$0.329462\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	30.3097	−	30.3097i	1.56103	−	1.56103i
$378$		0			0
$379$			1.00000i			0.0513665i	0.999670	+	0.0256833i	$0.00817614\pi$
$379$			1.00000i			0.0513665i	−0.999670	+	0.0256833i	$0.991824\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	17.3487	+	17.3487i	0.886477	+	0.886477i	0.994183	−	0.107706i	$-0.0343505\pi$
$383$	17.3487	+	17.3487i	0.886477	+	0.886477i	−0.107706	+	0.994183i	$0.534350\pi$
$384$		0			0
$385$	−0.747727	+	3.58258i	−0.0381077	+	0.182585i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−39.0851			−1.98169			−0.990847	−	0.134986i	$-0.956901\pi$
$389$	−39.0851			−1.98169			−0.990847	+	0.134986i	$0.956901\pi$
$390$		0			0
$391$	18.5826			0.939761
$392$		0			0
$393$		0			0
$394$		0			0
$395$	7.55074	+	11.5339i	0.379919	+	0.580336i
$396$		0			0
$397$	16.9564	+	16.9564i	0.851019	+	0.851019i	0.990259	−	0.139239i	$-0.0444658\pi$
$397$	16.9564	+	16.9564i	0.851019	+	0.851019i	−0.139239	+	0.990259i	$0.544466\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			2.44949i			0.122322i	0.998128	+	0.0611608i	$0.0194803\pi$
$401$			2.44949i			0.122322i	−0.998128	+	0.0611608i	$0.980520\pi$
$402$		0			0
$403$	−2.20871	+	2.20871i	−0.110024	+	0.110024i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	1.93825	−	1.93825i	0.0960756	−	0.0960756i
$408$		0			0
$409$			12.1652i			0.601528i	0.953699	+	0.300764i	$0.0972417\pi$
$409$			12.1652i			0.601528i	−0.953699	+	0.300764i	$0.902758\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−23.0960	−	23.0960i	−1.13648	−	1.13648i
$414$		0			0
$415$	3.79129	+	0.791288i	0.186107	+	0.0388428i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−15.6127			−0.762731			−0.381365	−	0.924424i	$-0.624546\pi$
$419$	−15.6127			−0.762731			−0.381365	+	0.924424i	$0.624546\pi$
$420$		0			0
$421$	−11.4174			−0.556451			−0.278226	−	0.960516i	$-0.589746\pi$
$421$	−11.4174			−0.556451			−0.278226	+	0.960516i	$0.589746\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	12.8261	+	32.6917i	0.622158	+	1.58578i
$426$		0			0
$427$	−11.0436	−	11.0436i	−0.534435	−	0.534435i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			5.67991i			0.273592i	0.990599	+	0.136796i	$0.0436804\pi$
$431$			5.67991i			0.273592i	−0.990599	+	0.136796i	$0.956320\pi$
$432$		0			0
$433$	5.79129	−	5.79129i	0.278312	−	0.278312i	−0.554123	−	0.832435i	$-0.686946\pi$
$433$	5.79129	−	5.79129i	0.278312	−	0.278312i	0.832435	+	0.554123i	$0.186946\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−12.3149	+	12.3149i	−0.589100	+	0.589100i
$438$		0			0
$439$			26.1652i			1.24879i	0.781107	+	0.624397i	$0.214655\pi$
$439$			26.1652i			1.24879i	−0.781107	+	0.624397i	$0.785345\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−10.5115	−	10.5115i	−0.499415	−	0.499415i	0.411841	−	0.911256i	$-0.364886\pi$
$443$	−10.5115	−	10.5115i	−0.499415	−	0.499415i	−0.911256	+	0.411841i	$0.864886\pi$
$444$		0			0
$445$	−17.6261	+	11.5390i	−0.835559	+	0.547002i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	−12.6238			−0.595756			−0.297878	−	0.954604i	$-0.596279\pi$
$449$	−12.6238			−0.595756			−0.297878	+	0.954604i	$0.596279\pi$
$450$		0			0
$451$	2.74773			0.129385
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−25.4107	+	16.6352i	−1.19127	+	0.779870i
$456$		0			0
$457$	−22.7477	−	22.7477i	−1.06409	−	1.06409i	−0.997800	−	0.0662936i	$-0.978883\pi$
$457$	−22.7477	−	22.7477i	−1.06409	−	1.06409i	−0.0662936	−	0.997800i	$-0.521117\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			3.74166i			0.174266i	0.996197	+	0.0871332i	$0.0277706\pi$
$461$			3.74166i			0.174266i	−0.996197	+	0.0871332i	$0.972229\pi$
$462$		0			0
$463$	3.74773	−	3.74773i	0.174172	−	0.174172i	−0.614638	−	0.788809i	$-0.710698\pi$
$463$	3.74773	−	3.74773i	0.174172	−	0.174172i	0.788809	+	0.614638i	$0.210698\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	16.5678	−	16.5678i	0.766665	−	0.766665i	−0.210853	−	0.977518i	$-0.567624\pi$
$467$	16.5678	−	16.5678i	0.766665	−	0.766665i	0.977518	+	0.210853i	$0.0676241\pi$
$468$		0			0
$469$			25.0780i			1.15800i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	0.511238	+	0.511238i	0.0235068	+	0.0235068i
$474$		0			0
$475$	−30.1652	−	13.1652i	−1.38407	−	0.604059i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.2981			0.607604			0.303802	−	0.952735i	$-0.401744\pi$
$479$	13.2981			0.607604			0.303802	+	0.952735i	$0.401744\pi$
$480$		0			0
$481$	22.7477			1.03721
$482$		0			0
$483$		0			0
$484$		0			0
$485$	23.4724	+	4.89898i	1.06583	+	0.222451i
$486$		0			0
$487$	8.37386	+	8.37386i	0.379456	+	0.379456i	0.870906	−	0.491450i	$-0.163533\pi$
$487$	8.37386	+	8.37386i	0.379456	+	0.379456i	−0.491450	+	0.870906i	$0.663533\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	27.2141i		−	1.22815i	−0.789246	−	0.614077i	$-0.789528\pi$
$491$		−	27.2141i		−	1.22815i	0.789246	−	0.614077i	$-0.210472\pi$
$492$		0			0
$493$	−39.7042	+	39.7042i	−1.78819	+	1.78819i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	25.6522	−	25.6522i	1.15066	−	1.15066i
$498$		0			0
$499$			18.5826i			0.831870i	0.909394	+	0.415935i	$0.136546\pi$
$499$			18.5826i			0.831870i	−0.909394	+	0.415935i	$0.863454\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	12.9610	+	12.9610i	0.577900	+	0.577900i	0.934324	−	0.356424i	$-0.116004\pi$
$503$	12.9610	+	12.9610i	0.577900	+	0.577900i	−0.356424	+	0.934324i	$0.616004\pi$
$504$		0			0
$505$	10.4174	+	15.9129i	0.463569	+	0.708114i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−7.21362			−0.319738			−0.159869	−	0.987138i	$-0.551107\pi$
$509$	−7.21362			−0.319738			−0.159869	+	0.987138i	$0.551107\pi$
$510$		0			0
$511$	4.33030			0.191561
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−2.31464	+	11.0901i	−0.101995	+	0.488689i
$516$		0			0
$517$	−1.25227	−	1.25227i	−0.0550749	−	0.0550749i
$518$		0			0
$519$		0			0
$520$		0			0
$521$		−	26.4331i		−	1.15806i	−0.815307	−	0.579029i	$-0.803432\pi$
$521$		−	26.4331i		−	1.15806i	0.815307	−	0.579029i	$-0.196568\pi$
$522$		0			0
$523$	−5.04356	+	5.04356i	−0.220540	+	0.220540i	−0.808726	−	0.588186i	$-0.799842\pi$
$523$	−5.04356	+	5.04356i	−0.220540	+	0.220540i	0.588186	+	0.808726i	$0.299842\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	2.89331	−	2.89331i	0.126034	−	0.126034i
$528$		0			0
$529$		−	16.0000i		−	0.695652i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	16.1240	+	16.1240i	0.698406	+	0.698406i
$534$		0			0
$535$	−8.74773	+	41.9129i	−0.378197	+	1.81205i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−0.376393			−0.0162124
$540$		0			0
$541$	−41.1652			−1.76983			−0.884914	−	0.465754i	$-0.845783\pi$
$541$	−41.1652			−1.76983			−0.884914	+	0.465754i	$0.845783\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	19.2869	+	29.4613i	0.826162	+	1.26198i
$546$		0			0
$547$	29.5390	+	29.5390i	1.26300	+	1.26300i	0.949633	+	0.313364i	$0.101456\pi$
$547$	29.5390	+	29.5390i	1.26300	+	1.26300i	0.313364	+	0.949633i	$0.398544\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	52.6248i		−	2.24189i
$552$		0			0
$553$	11.0436	−	11.0436i	0.469620	−	0.469620i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−13.6745	+	13.6745i	−0.579406	+	0.579406i	−0.934739	−	0.355334i	$-0.884367\pi$
$557$	−13.6745	+	13.6745i	−0.579406	+	0.579406i	0.355334	+	0.934739i	$0.384367\pi$
$558$		0			0
$559$			6.00000i			0.253773i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	3.47197	+	3.47197i	0.146326	+	0.146326i	0.776475	−	0.630149i	$-0.217006\pi$
$563$	3.47197	+	3.47197i	0.146326	+	0.146326i	−0.630149	+	0.776475i	$0.717006\pi$
$564$		0			0
$565$	15.5826	+	3.25227i	0.655564	+	0.136824i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−32.4895			−1.36203			−0.681014	−	0.732270i	$-0.738461\pi$
$569$	−32.4895			−1.36203			−0.681014	+	0.732270i	$0.738461\pi$
$570$		0			0
$571$	−14.9129			−0.624085			−0.312042	−	0.950068i	$-0.601013\pi$
$571$	−14.9129			−0.624085			−0.312042	+	0.950068i	$0.601013\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−12.3149	+	4.83156i	−0.513566	+	0.201490i
$576$		0			0
$577$	−4.37386	−	4.37386i	−0.182086	−	0.182086i	0.610178	−	0.792264i	$-0.291098\pi$
$577$	−4.37386	−	4.37386i	−0.182086	−	0.182086i	−0.792264	+	0.610178i	$0.791098\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	4.38774i		−	0.182034i
$582$		0			0
$583$	−3.53901	+	3.53901i	−0.146571	+	0.146571i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−22.2477	+	22.2477i	−0.918260	+	0.918260i	−0.996903	−	0.0786430i	$-0.974941\pi$
$587$	−22.2477	+	22.2477i	−0.918260	+	0.918260i	0.0786430	+	0.996903i	$0.474941\pi$
$588$		0			0
$589$			3.83485i			0.158012i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	4.45516	+	4.45516i	0.182952	+	0.182952i	0.792641	−	0.609689i	$-0.208706\pi$
$593$	4.45516	+	4.45516i	0.182952	+	0.182952i	−0.609689	+	0.792641i	$0.708706\pi$
$594$		0			0
$595$	33.2867	−	21.7913i	1.36462	−	0.893356i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−2.82588			−0.115462			−0.0577312	−	0.998332i	$-0.518387\pi$
$599$	−2.82588			−0.115462			−0.0577312	+	0.998332i	$0.518387\pi$
$600$		0			0
$601$	15.7477			0.642363			0.321182	−	0.947018i	$-0.395920\pi$
$601$	15.7477			0.642363			0.321182	+	0.947018i	$0.395920\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	19.7982	−	12.9610i	0.804911	−	0.526938i
$606$		0			0
$607$	25.1216	+	25.1216i	1.01965	+	1.01965i	0.999803	+	0.0198510i	$0.00631917\pi$
$607$	25.1216	+	25.1216i	1.01965	+	1.01965i	0.0198510	+	0.999803i	$0.493681\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	14.6969i		−	0.594574i
$612$		0			0
$613$	−24.3303	+	24.3303i	−0.982692	+	0.982692i	−0.999853	−	0.0171611i	$-0.994537\pi$
$613$	−24.3303	+	24.3303i	−0.982692	+	0.982692i	0.0171611	+	0.999853i	$0.494537\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	23.2702	−	23.2702i	0.936821	−	0.936821i	−0.0612984	−	0.998119i	$-0.519524\pi$
$617$	23.2702	−	23.2702i	0.936821	−	0.936821i	0.998119	+	0.0612984i	$0.0195241\pi$
$618$		0			0
$619$			10.7477i			0.431988i	0.976395	+	0.215994i	$0.0692991\pi$
$619$			10.7477i			0.431988i	−0.976395	+	0.215994i	$0.930701\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	16.8767	+	16.8767i	0.676152	+	0.676152i
$624$		0			0
$625$	−17.0000	−	18.3303i	−0.680000	−	0.733212i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−29.7984			−1.18814
$630$		0			0
$631$	−2.16515			−0.0861933			−0.0430967	−	0.999071i	$-0.513722\pi$
$631$	−2.16515			−0.0861933			−0.0430967	+	0.999071i	$0.513722\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	20.3768	+	4.25290i	0.808631	+	0.168771i
$636$		0			0
$637$	−2.20871	−	2.20871i	−0.0875124	−	0.0875124i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	0.269691i		−	0.0106522i	−0.999986	−	0.00532608i	$-0.998305\pi$
$641$		−	0.269691i		−	0.0106522i	0.999986	−	0.00532608i	$-0.00169535\pi$
$642$		0			0
$643$	8.53901	−	8.53901i	0.336746	−	0.336746i	−0.518395	−	0.855141i	$-0.673470\pi$
$643$	8.53901	−	8.53901i	0.336746	−	0.336746i	0.855141	+	0.518395i	$0.173470\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−17.4835	+	17.4835i	−0.687349	+	0.687349i	−0.961645	−	0.274296i	$-0.911555\pi$
$647$	−17.4835	+	17.4835i	−0.687349	+	0.687349i	0.274296	+	0.961645i	$0.411555\pi$
$648$		0			0
$649$		−	8.33030i		−	0.326993i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	4.59001	+	4.59001i	0.179621	+	0.179621i	0.791191	−	0.611570i	$-0.209462\pi$
$653$	4.59001	+	4.59001i	0.179621	+	0.179621i	−0.611570	+	0.791191i	$0.709462\pi$
$654$		0			0
$655$	−12.1652	−	18.5826i	−0.475332	−	0.726081i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−16.7700			−0.653268			−0.326634	−	0.945151i	$-0.605914\pi$
$659$	−16.7700			−0.653268			−0.326634	+	0.945151i	$0.605914\pi$
$660$		0			0
$661$	−31.4955			−1.22503			−0.612516	−	0.790459i	$-0.709842\pi$
$661$	−31.4955			−1.22503			−0.612516	+	0.790459i	$0.709842\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−7.61816	+	36.5008i	−0.295420	+	1.41544i
$666$		0			0
$667$	−14.9564	−	14.9564i	−0.579116	−	0.579116i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	3.98320i		−	0.153770i
$672$		0			0
$673$	−7.04356	+	7.04356i	−0.271509	+	0.271509i	−0.829708	−	0.558198i	$-0.811493\pi$
$673$	−7.04356	+	7.04356i	−0.271509	+	0.271509i	0.558198	+	0.829708i	$0.311493\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	6.05630	−	6.05630i	0.232763	−	0.232763i	−0.581082	−	0.813845i	$-0.697371\pi$
$677$	6.05630	−	6.05630i	0.232763	−	0.232763i	0.813845	+	0.581082i	$0.197371\pi$
$678$		0			0
$679$		−	27.1652i		−	1.04250i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−12.4497	−	12.4497i	−0.476375	−	0.476375i	0.427595	−	0.903970i	$-0.359361\pi$
$683$	−12.4497	−	12.4497i	−0.476375	−	0.476375i	−0.903970	+	0.427595i	$0.859361\pi$
$684$		0			0
$685$	−0.791288	+	3.79129i	−0.0302336	+	0.144858i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−41.5346			−1.58234
$690$		0			0
$691$	39.6606			1.50876			0.754380	−	0.656438i	$-0.227938\pi$
$691$	39.6606			1.50876			0.754380	+	0.656438i	$0.227938\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	13.6745	+	20.8881i	0.518702	+	0.792330i
$696$		0			0
$697$	−21.1216	−	21.1216i	−0.800037	−	0.800037i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			27.2141i			1.02786i	0.857832	+	0.513931i	$0.171811\pi$
$701$			27.2141i			1.02786i	−0.857832	+	0.513931i	$0.828189\pi$
$702$		0			0
$703$	19.7477	−	19.7477i	0.744800	−	0.744800i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	15.2363	−	15.2363i	0.573021	−	0.573021i
$708$		0			0
$709$		−	26.0000i		−	0.976450i	−0.872718	−	0.488225i	$-0.837644\pi$
$709$		−	26.0000i		−	0.976450i	0.872718	−	0.488225i	$-0.162356\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1.08990	+	1.08990i	0.0408171	+	0.0408171i
$714$		0			0
$715$	−7.58258	−	1.58258i	−0.283572	−	0.0591850i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−37.0120			−1.38032			−0.690158	−	0.723659i	$-0.742459\pi$
$719$	−37.0120			−1.38032			−0.690158	+	0.723659i	$0.742459\pi$
$720$		0			0
$721$	12.8348			0.477995
$722$		0			0
$723$		0			0
$724$		0			0
$725$	15.9891	−	36.6356i	0.593820	−	1.36061i
$726$		0			0
$727$	−12.3303	−	12.3303i	−0.457306	−	0.457306i	0.440464	−	0.897770i	$-0.354814\pi$
$727$	−12.3303	−	12.3303i	−0.457306	−	0.457306i	−0.897770	+	0.440464i	$0.854814\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$		−	7.85971i		−	0.290702i
$732$		0			0
$733$	23.9129	−	23.9129i	0.883242	−	0.883242i	−0.110620	−	0.993863i	$-0.535284\pi$
$733$	23.9129	−	23.9129i	0.883242	−	0.883242i	0.993863	+	0.110620i	$0.0352838\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−4.52259	+	4.52259i	−0.166592	+	0.166592i
$738$		0			0
$739$		−	11.4174i		−	0.419997i	−0.977702	−	0.209998i	$-0.932654\pi$
$739$		−	11.4174i		−	0.419997i	0.977702	−	0.209998i	$-0.0673459\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−19.8656	−	19.8656i	−0.728799	−	0.728799i	0.241582	−	0.970380i	$-0.422334\pi$
$743$	−19.8656	−	19.8656i	−0.728799	−	0.728799i	−0.970380	+	0.241582i	$0.922334\pi$
$744$		0			0
$745$	21.9564	−	14.3739i	0.804422	−	0.526618i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	48.5067			1.77240
$750$		0			0
$751$	15.0000			0.547358			0.273679	−	0.961821i	$-0.411759\pi$
$751$	15.0000			0.547358			0.273679	+	0.961821i	$0.411759\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−32.1131	+	21.0229i	−1.16871	+	0.765103i
$756$		0			0
$757$	15.0000	+	15.0000i	0.545184	+	0.545184i	0.925044	−	0.379860i	$-0.124028\pi$
$757$	15.0000	+	15.0000i	0.545184	+	0.545184i	−0.379860	+	0.925044i	$0.624028\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			29.1523i			1.05677i	0.849005	+	0.528386i	$0.177202\pi$
$761$			29.1523i			1.05677i	−0.849005	+	0.528386i	$0.822798\pi$
$762$		0			0
$763$	28.2087	−	28.2087i	1.02122	−	1.02122i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	48.8831	−	48.8831i	1.76507	−	1.76507i
$768$		0			0
$769$			23.0000i			0.829401i	0.909958	+	0.414701i	$0.136114\pi$
$769$			23.0000i			0.829401i	−0.909958	+	0.414701i	$0.863886\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	25.6129	+	25.6129i	0.921233	+	0.921233i	0.997117	−	0.0758833i	$-0.0241776\pi$
$773$	25.6129	+	25.6129i	0.921233	+	0.921233i	−0.0758833	+	0.997117i	$0.524178\pi$
$774$		0			0
$775$	−1.16515	+	2.66970i	−0.0418535	+	0.0958984i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	27.9950			1.00303
$780$		0			0
$781$	9.25227			0.331072
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.2250	−	2.34279i	−0.400636	−	0.0836177i
$786$		0			0
$787$	−28.7042	−	28.7042i	−1.02319	−	1.02319i	−0.999725	−	0.0234684i	$-0.992529\pi$
$787$	−28.7042	−	28.7042i	−1.02319	−	1.02319i	−0.0234684	−	0.999725i	$-0.507471\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	18.0341i		−	0.641217i
$792$		0			0
$793$	23.3739	−	23.3739i	0.830030	−	0.830030i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	29.0849	−	29.0849i	1.03024	−	1.03024i	0.0307120	−	0.999528i	$-0.490223\pi$
$797$	29.0849	−	29.0849i	1.03024	−	1.03024i	0.999528	−	0.0307120i	$-0.00977747\pi$
$798$		0			0
$799$			19.2523i			0.681096i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	0.780929	+	0.780929i	0.0275584	+	0.0275584i
$804$		0			0
$805$	8.20871	+	12.5390i	0.289319	+	0.441942i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−39.8379			−1.40063			−0.700313	−	0.713836i	$-0.746956\pi$
$809$	−39.8379			−1.40063			−0.700313	+	0.713836i	$0.746956\pi$
$810$		0			0
$811$	−16.3303			−0.573434			−0.286717	−	0.958015i	$-0.592564\pi$
$811$	−16.3303			−0.573434			−0.286717	+	0.958015i	$0.592564\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−5.54506	+	26.5680i	−0.194235	+	0.930636i
$816$		0			0
$817$	5.20871	+	5.20871i	0.182230	+	0.182230i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	8.91033i		−	0.310973i	−0.987838	−	0.155486i	$-0.950306\pi$
$821$		−	8.91033i		−	0.310973i	0.987838	−	0.155486i	$-0.0496944\pi$
$822$		0			0
$823$	−11.1216	+	11.1216i	−0.387674	+	0.387674i	−0.873857	−	0.486183i	$-0.838389\pi$
$823$	−11.1216	+	11.1216i	−0.387674	+	0.387674i	0.486183	+	0.873857i	$0.338389\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	3.16300	−	3.16300i	0.109988	−	0.109988i	−0.649971	−	0.759959i	$-0.725219\pi$
$827$	3.16300	−	3.16300i	0.109988	−	0.109988i	0.759959	+	0.649971i	$0.225219\pi$
$828$		0			0
$829$		−	33.0780i		−	1.14885i	−0.818558	−	0.574424i	$-0.805226\pi$
$829$		−	33.0780i		−	1.14885i	0.818558	−	0.574424i	$-0.194774\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	2.89331	+	2.89331i	0.100247	+	0.100247i
$834$		0			0
$835$	−4.37386	+	20.9564i	−0.151364	+	0.725227i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	44.3605			1.53149			0.765747	−	0.643141i	$-0.222369\pi$
$839$	44.3605			1.53149			0.765747	+	0.643141i	$0.222369\pi$
$840$		0			0
$841$	34.9129			1.20389
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−19.2869	−	29.4613i	−0.663491	−	1.01350i
$846$		0			0
$847$	−18.9564	−	18.9564i	−0.651351	−	0.651351i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	11.2250i		−	0.384787i
$852$		0			0
$853$	16.3303	−	16.3303i	0.559139	−	0.559139i	−0.369923	−	0.929062i	$-0.620616\pi$
$853$	16.3303	−	16.3303i	0.559139	−	0.559139i	0.929062	+	0.369923i	$0.120616\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	4.59001	−	4.59001i	0.156792	−	0.156792i	−0.624352	−	0.781143i	$-0.714637\pi$
$857$	4.59001	−	4.59001i	0.156792	−	0.156792i	0.781143	+	0.624352i	$0.214637\pi$
$858$		0			0
$859$		−	44.9129i		−	1.53241i	−0.642598	−	0.766204i	$-0.722143\pi$
$859$		−	44.9129i		−	1.53241i	0.642598	−	0.766204i	$-0.277857\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−12.1800	−	12.1800i	−0.414613	−	0.414613i	0.468729	−	0.883342i	$-0.344712\pi$
$863$	−12.1800	−	12.1800i	−0.414613	−	0.414613i	−0.883342	+	0.468729i	$0.844712\pi$
$864$		0			0
$865$	−42.1216	−	8.79129i	−1.43218	−	0.298913i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	3.98320			0.135121
$870$		0			0
$871$	−53.0780			−1.79848
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−16.3936	+	23.0960i	−0.554206	+	0.780788i
$876$		0			0
$877$	17.2087	+	17.2087i	0.581097	+	0.581097i	0.935205	−	0.354108i	$-0.115215\pi$
$877$	17.2087	+	17.2087i	0.581097	+	0.581097i	−0.354108	+	0.935205i	$0.615215\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			18.3319i			0.617617i	0.951124	+	0.308809i	$0.0999303\pi$
$881$			18.3319i			0.617617i	−0.951124	+	0.308809i	$0.900070\pi$
$882$		0			0
$883$	−17.7913	+	17.7913i	−0.598725	+	0.598725i	−0.939973	−	0.341249i	$-0.889150\pi$
$883$	−17.7913	+	17.7913i	−0.598725	+	0.598725i	0.341249	+	0.939973i	$0.389150\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−32.0456	+	32.0456i	−1.07599	+	1.07599i	−0.0791222	+	0.996865i	$0.525212\pi$
$887$	−32.0456	+	32.0456i	−1.07599	+	1.07599i	−0.996865	+	0.0791222i	$0.974788\pi$
$888$		0			0
$889$		−	23.5826i		−	0.790934i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−12.7587	−	12.7587i	−0.426953	−	0.426953i
$894$		0			0
$895$	46.3303	−	30.3303i	1.54865	−	1.01383i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−4.65743			−0.155334
$900$		0			0
$901$	54.4083			1.81260
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−10.1351	+	6.63496i	−0.336901	+	0.220554i
$906$		0			0
$907$	−18.7477	−	18.7477i	−0.622508	−	0.622508i	0.323664	−	0.946172i	$-0.395085\pi$
$907$	−18.7477	−	18.7477i	−0.622508	−	0.622508i	−0.946172	+	0.323664i	$0.895085\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			9.01703i			0.298748i	0.988781	+	0.149374i	$0.0477258\pi$
$911$			9.01703i			0.298748i	−0.988781	+	0.149374i	$0.952274\pi$
$912$		0			0
$913$	0.791288	−	0.791288i	0.0261878	−	0.0261878i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−17.7925	+	17.7925i	−0.587561	+	0.587561i
$918$		0			0
$919$		−	16.0000i		−	0.527791i	−0.964551	−	0.263896i	$-0.914993\pi$
$919$		−	16.0000i		−	0.527791i	0.964551	−	0.263896i	$-0.0850075\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	54.2933	+	54.2933i	1.78709	+	1.78709i
$924$		0			0
$925$	19.7477	−	7.74773i	0.649301	−	0.254744i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−21.8039			−0.715361			−0.357681	−	0.933844i	$-0.616432\pi$
$929$	−21.8039			−0.715361			−0.357681	+	0.933844i	$0.616432\pi$
$930$		0			0
$931$	−3.83485			−0.125682
$932$		0			0
$933$		0			0
$934$		0			0
$935$	9.93280	+	2.07310i	0.324837	+	0.0677975i
$936$		0			0
$937$	27.0780	+	27.0780i	0.884601	+	0.884601i	0.993998	−	0.109397i	$-0.0348921\pi$
$937$	27.0780	+	27.0780i	0.884601	+	0.884601i	−0.109397	+	0.993998i	$0.534892\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	28.7759i		−	0.938069i	−0.883180	−	0.469034i	$-0.844602\pi$
$941$		−	28.7759i		−	0.938069i	0.883180	−	0.469034i	$-0.155398\pi$
$942$		0			0
$943$	7.95644	−	7.95644i	0.259097	−	0.259097i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.7644	−	14.7644i	0.479777	−	0.479777i	−0.425283	−	0.905060i	$-0.639825\pi$
$947$	14.7644	−	14.7644i	0.479777	−	0.479777i	0.905060	+	0.425283i	$0.139825\pi$
$948$		0			0
$949$			9.16515i			0.297513i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−21.5342	−	21.5342i	−0.697560	−	0.697560i	0.266324	−	0.963884i	$-0.414191\pi$
$953$	−21.5342	−	21.5342i	−0.697560	−	0.697560i	−0.963884	+	0.266324i	$0.914191\pi$
$954$		0			0
$955$	−4.58258	−	7.00000i	−0.148289	−	0.226515i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	4.38774			0.141688
$960$		0			0
$961$	−30.6606			−0.989052
$962$		0			0
$963$		0			0
$964$		0			0
$965$	10.2025	−	48.8831i	0.328430	−	1.57360i
$966$		0			0
$967$	−0.252273	−	0.252273i	−0.00811255	−	0.00811255i	0.703039	−	0.711151i	$-0.251826\pi$
$967$	−0.252273	−	0.252273i	−0.00811255	−	0.00811255i	−0.711151	+	0.703039i	$0.751826\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			23.6073i			0.757593i	0.925480	+	0.378797i	$0.123662\pi$
$971$			23.6073i			0.757593i	−0.925480	+	0.378797i	$0.876338\pi$
$972$		0			0
$973$	20.0000	−	20.0000i	0.641171	−	0.641171i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	1.02248	−	1.02248i	0.0327119	−	0.0327119i	−0.690562	−	0.723274i	$-0.742637\pi$
$977$	1.02248	−	1.02248i	0.0327119	−	0.0327119i	0.723274	+	0.690562i	$0.242637\pi$
$978$		0			0
$979$			6.08712i			0.194545i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	8.30352	+	8.30352i	0.264841	+	0.264841i	0.827017	−	0.562176i	$-0.190036\pi$
$983$	8.30352	+	8.30352i	0.264841	+	0.264841i	−0.562176	+	0.827017i	$0.690036\pi$
$984$		0			0
$985$	3.95644	−	18.9564i	0.126063	−	0.604002i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	2.96073			0.0941457
$990$		0			0
$991$	13.0000			0.412959			0.206479	−	0.978451i	$-0.433799\pi$
$991$	13.0000			0.412959			0.206479	+	0.978451i	$0.433799\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−16.6352	−	25.4107i	−0.527371	−	0.805572i
$996$		0			0
$997$	−10.0436	−	10.0436i	−0.318083	−	0.318083i	0.529948	−	0.848030i	$-0.322212\pi$
$997$	−10.0436	−	10.0436i	−0.318083	−	0.318083i	−0.848030	+	0.529948i	$0.822212\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	540.2.j.b.377.3	yes	8
3.2	odd	2	inner	540.2.j.b.377.2	yes	8
4.3	odd	2		2160.2.w.c.1457.3		8
5.2	odd	4		2700.2.j.i.593.3		8
5.3	odd	4	inner	540.2.j.b.53.1	✓	8
5.4	even	2		2700.2.j.i.1457.3		8
9.2	odd	6		1620.2.x.c.377.2		16
9.4	even	3		1620.2.x.c.917.1		16
9.5	odd	6		1620.2.x.c.917.4		16
9.7	even	3		1620.2.x.c.377.3		16
12.11	even	2		2160.2.w.c.1457.2		8
15.2	even	4		2700.2.j.i.593.4		8
15.8	even	4	inner	540.2.j.b.53.4	yes	8
15.14	odd	2		2700.2.j.i.1457.4		8
20.3	even	4		2160.2.w.c.593.1		8
45.13	odd	12		1620.2.x.c.593.2		16
45.23	even	12		1620.2.x.c.593.3		16
45.38	even	12		1620.2.x.c.53.1		16
45.43	odd	12		1620.2.x.c.53.4		16
60.23	odd	4		2160.2.w.c.593.4		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.2.j.b.53.1	✓	8	5.3	odd	4	inner
540.2.j.b.53.4	yes	8	15.8	even	4	inner
540.2.j.b.377.2	yes	8	3.2	odd	2	inner
540.2.j.b.377.3	yes	8	1.1	even	1	trivial
1620.2.x.c.53.1		16	45.38	even	12
1620.2.x.c.53.4		16	45.43	odd	12
1620.2.x.c.377.2		16	9.2	odd	6
1620.2.x.c.377.3		16	9.7	even	3
1620.2.x.c.593.2		16	45.13	odd	12
1620.2.x.c.593.3		16	45.23	even	12
1620.2.x.c.917.1		16	9.4	even	3
1620.2.x.c.917.4		16	9.5	odd	6
2160.2.w.c.593.1		8	20.3	even	4
2160.2.w.c.593.4		8	60.23	odd	4
2160.2.w.c.1457.2		8	12.11	even	2
2160.2.w.c.1457.3		8	4.3	odd	2
2700.2.j.i.593.3		8	5.2	odd	4
2700.2.j.i.593.4		8	15.2	even	4
2700.2.j.i.1457.3		8	5.4	even	2
2700.2.j.i.1457.4		8	15.14	odd	2

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:	\( N \)	\(=\)	\( 540 = 2^{2} \cdot 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	540.j (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(1.87083 - 1.22474i) q^{5} +(-1.79129 - 1.79129i) q^{7} -0.646084i q^{11} +(3.79129 - 3.79129i) q^{13} +(-4.96640 + 4.96640i) q^{17} -6.58258i q^{19} +(-1.87083 - 1.87083i) q^{23} +(2.00000 - 4.58258i) q^{25} +7.99455 q^{29} -0.582576 q^{31} +(-5.54506 - 1.15732i) q^{35} +(3.00000 + 3.00000i) q^{37} +4.25290i q^{41} +(-0.791288 + 0.791288i) q^{43} +(1.93825 - 1.93825i) q^{47} -0.582576i q^{49} +(-5.47764 - 5.47764i) q^{53} +(-0.791288 - 1.20871i) q^{55} +12.8935 q^{59} +6.16515 q^{61} +(2.44949 - 11.7362i) q^{65} +(-7.00000 - 7.00000i) q^{67} +14.3205i q^{71} +(-1.20871 + 1.20871i) q^{73} +(-1.15732 + 1.15732i) q^{77} +6.16515i q^{79} +(1.22474 + 1.22474i) q^{83} +(-3.20871 + 15.3739i) q^{85} -9.42157 q^{89} -13.5826 q^{91} +(-8.06198 - 12.3149i) q^{95} +(7.58258 + 7.58258i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7} + 12 q^{13} + 16 q^{25} + 32 q^{31} + 24 q^{37} + 12 q^{43} + 12 q^{55} - 24 q^{61} - 56 q^{67} - 28 q^{73} - 44 q^{85} - 72 q^{91} + 24 q^{97}+O(q^{100}) \) 8 * q + 4 * q^7 + 12 * q^13 + 16 * q^25 + 32 * q^31 + 24 * q^37 + 12 * q^43 + 12 * q^55 - 24 * q^61 - 56 * q^67 - 28 * q^73 - 44 * q^85 - 72 * q^91 + 24 * q^97

Introduction

L-functions

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