LMFDB - Embedded newform 540.2.j.b.377.4

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			377.4
Root			$0.323042 - 0.323042i$ of defining polynomial
Character	$\chi$	$=$	540.377
Dual form			540.2.j.b.53.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+(1.87083 + 1.22474i) q^{5} +(2.79129 + 2.79129i) q^{7} -3.09557i q^{11} +(-0.791288 + 0.791288i) q^{13} +(-2.51691 + 2.51691i) q^{17} +2.58258i q^{19} +(-1.87083 - 1.87083i) q^{23} +(2.00000 + 4.58258i) q^{25} -4.25290 q^{29} +8.58258 q^{31} +(1.80341 + 8.64064i) q^{35} +(3.00000 + 3.00000i) q^{37} -7.99455i q^{41} +(3.79129 - 3.79129i) q^{43} +(9.28672 - 9.28672i) q^{47} +8.58258i q^{49} +(9.21930 + 9.21930i) q^{53} +(3.79129 - 5.79129i) q^{55} -9.15188 q^{59} -12.1652 q^{61} +(-2.44949 + 0.511238i) q^{65} +(-7.00000 - 7.00000i) q^{67} +11.8711i q^{71} +(-5.79129 + 5.79129i) q^{73} +(8.64064 - 8.64064i) q^{77} -12.1652i q^{79} +(-1.22474 - 1.22474i) q^{83} +(-7.79129 + 1.62614i) q^{85} -16.7700 q^{89} -4.41742 q^{91} +(-3.16300 + 4.83156i) q^{95} +(-1.58258 - 1.58258i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$7$	2.79129	+	2.79129i	1.05501	+	1.05501i	0.998396	+	0.0566113i	$0.0180296\pi$
$7$	2.79129	+	2.79129i	1.05501	+	1.05501i	0.0566113	+	0.998396i	$0.481970\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$		−	3.09557i		−	0.933351i	−0.884429	−	0.466675i	$-0.845452\pi$
$11$		−	3.09557i		−	0.933351i	0.884429	−	0.466675i	$-0.154548\pi$
$12$		0			0
$13$	−0.791288	+	0.791288i	−0.219464	+	0.219464i	−0.808272	−	0.588809i	$-0.799597\pi$
$13$	−0.791288	+	0.791288i	−0.219464	+	0.219464i	0.588809	+	0.808272i	$0.299597\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.51691	+	2.51691i	−0.610441	+	0.610441i	−0.943061	−	0.332620i	$-0.892067\pi$
$17$	−2.51691	+	2.51691i	−0.610441	+	0.610441i	0.332620	+	0.943061i	$0.392067\pi$
$18$		0			0
$19$			2.58258i			0.592483i	0.955113	+	0.296242i	$0.0957334\pi$
$19$			2.58258i			0.592483i	−0.955113	+	0.296242i	$0.904267\pi$
$20$		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
$29$	−4.25290			−0.789743			−0.394871	−	0.918736i	$-0.629211\pi$
$29$	−4.25290			−0.789743			−0.394871	+	0.918736i	$0.629211\pi$
$30$		0			0
$31$	8.58258			1.54148			0.770738	−	0.637152i	$-0.219888\pi$
$31$	8.58258			1.54148			0.770738	+	0.637152i	$0.219888\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	1.80341	+	8.64064i	0.304831	+	1.46053i
$36$		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
$41$		−	7.99455i		−	1.24854i	−0.781209	−	0.624270i	$-0.785397\pi$
$41$		−	7.99455i		−	1.24854i	0.781209	−	0.624270i	$-0.214603\pi$
$42$		0			0
$43$	3.79129	−	3.79129i	0.578166	−	0.578166i	−0.356232	−	0.934398i	$-0.615939\pi$
$43$	3.79129	−	3.79129i	0.578166	−	0.578166i	0.934398	+	0.356232i	$0.115939\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	9.28672	−	9.28672i	1.35461	−	1.35461i	0.474179	−	0.880428i	$-0.342745\pi$
$47$	9.28672	−	9.28672i	1.35461	−	1.35461i	0.880428	−	0.474179i	$-0.157255\pi$
$48$		0			0
$49$			8.58258i			1.22608i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.21930	+	9.21930i	1.26637	+	1.26637i	0.947951	+	0.318417i	$0.103151\pi$
$53$	9.21930	+	9.21930i	1.26637	+	1.26637i	0.318417	+	0.947951i	$0.396849\pi$
$54$		0			0
$55$	3.79129	−	5.79129i	0.511217	−	0.780897i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	−9.15188			−1.19147			−0.595736	−	0.803180i	$-0.703140\pi$
$59$	−9.15188			−1.19147			−0.595736	+	0.803180i	$0.703140\pi$
$60$		0			0
$61$	−12.1652			−1.55759			−0.778794	−	0.627280i	$-0.784168\pi$
$61$	−12.1652			−1.55759			−0.778794	+	0.627280i	$0.784168\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−2.44949	+	0.511238i	−0.303822	+	0.0634113i
$66$		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
$71$			11.8711i			1.40884i	0.709786	+	0.704418i	$0.248792\pi$
$71$			11.8711i			1.40884i	−0.709786	+	0.704418i	$0.751208\pi$
$72$		0			0
$73$	−5.79129	+	5.79129i	−0.677819	+	0.677819i	−0.959506	−	0.281687i	$-0.909106\pi$
$73$	−5.79129	+	5.79129i	−0.677819	+	0.677819i	0.281687	+	0.959506i	$0.409106\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	8.64064	−	8.64064i	0.984692	−	0.984692i
$78$		0			0
$79$		−	12.1652i		−	1.36869i	−0.729160	−	0.684343i	$-0.760089\pi$
$79$		−	12.1652i		−	1.36869i	0.729160	−	0.684343i	$-0.239911\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.22474	−	1.22474i	−0.134433	−	0.134433i	0.636688	−	0.771121i	$-0.280304\pi$
$83$	−1.22474	−	1.22474i	−0.134433	−	0.134433i	−0.771121	+	0.636688i	$0.780304\pi$
$84$		0			0
$85$	−7.79129	+	1.62614i	−0.845084	+	0.176379i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−16.7700			−1.77762			−0.888810	−	0.458276i	$-0.848467\pi$
$89$	−16.7700			−1.77762			−0.888810	+	0.458276i	$0.848467\pi$
$90$		0			0
$91$	−4.41742			−0.463072
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−3.16300	+	4.83156i	−0.324517	+	0.495707i
$96$		0			0
$97$	−1.58258	−	1.58258i	−0.160686	−	0.160686i	0.622184	−	0.782871i	$-0.286245\pi$
$97$	−1.58258	−	1.58258i	−0.160686	−	0.160686i	−0.782871	+	0.622184i	$0.786245\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	15.9891i		−	1.59098i	−0.605970	−	0.795488i	$-0.707215\pi$
$101$		−	15.9891i		−	1.59098i	0.605970	−	0.795488i	$-0.292785\pi$
$102$		0			0
$103$	5.58258	−	5.58258i	0.550068	−	0.550068i	−0.376393	−	0.926460i	$-0.622836\pi$
$103$	5.58258	−	5.58258i	0.550068	−	0.550068i	0.926460	+	0.376393i	$0.122836\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	6.05630	−	6.05630i	0.585485	−	0.585485i	−0.350920	−	0.936405i	$-0.614131\pi$
$107$	6.05630	−	6.05630i	0.585485	−	0.585485i	0.936405	+	0.350920i	$0.114131\pi$
$108$		0			0
$109$		−	11.7477i		−	1.12523i	−0.826720	−	0.562614i	$-0.809796\pi$
$109$		−	11.7477i		−	1.12523i	0.826720	−	0.562614i	$-0.190204\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	9.93280	+	9.93280i	0.934400	+	0.934400i	0.997977	−	0.0635773i	$-0.0202509\pi$
$113$	9.93280	+	9.93280i	0.934400	+	0.934400i	−0.0635773	+	0.997977i	$0.520251\pi$
$114$		0			0
$115$	−1.20871	−	5.79129i	−0.112713	−	0.540040i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−14.0509			−1.28804
$120$		0			0
$121$	1.41742			0.128857
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−1.87083	+	11.0227i	−0.167332	+	0.985901i
$126$		0			0
$127$	−2.58258	−	2.58258i	−0.229167	−	0.229167i	0.583178	−	0.812344i	$-0.301809\pi$
$127$	−2.58258	−	2.58258i	−0.229167	−	0.229167i	−0.812344	+	0.583178i	$0.801809\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	5.03383i		−	0.439807i	−0.975522	−	0.219904i	$-0.929426\pi$
$131$		−	5.03383i		−	0.439807i	0.975522	−	0.219904i	$-0.0705743\pi$
$132$		0			0
$133$	−7.20871	+	7.20871i	−0.625075	+	0.625075i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	1.22474	−	1.22474i	0.104637	−	0.104637i	−0.652850	−	0.757487i	$-0.726427\pi$
$137$	1.22474	−	1.22474i	0.104637	−	0.104637i	0.757487	+	0.652850i	$0.226427\pi$
$138$		0			0
$139$		−	7.16515i		−	0.607740i	−0.952713	−	0.303870i	$-0.901721\pi$
$139$		−	7.16515i		−	0.607740i	0.952713	−	0.303870i	$-0.0982789\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	2.44949	+	2.44949i	0.204837	+	0.204837i
$144$		0			0
$145$	−7.95644	−	5.20871i	−0.660746	−	0.432560i
$146$		0			0
$147$		0			0
$148$		0			0
$149$	−0.511238			−0.0418823			−0.0209411	−	0.999781i	$-0.506666\pi$
$149$	−0.511238			−0.0418823			−0.0209411	+	0.999781i	$0.506666\pi$
$150$		0			0
$151$	1.16515			0.0948187			0.0474093	−	0.998876i	$-0.484903\pi$
$151$	1.16515			0.0948187			0.0474093	+	0.998876i	$0.484903\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	16.0565	+	10.5115i	1.28969	+	0.844301i
$156$		0			0
$157$	−17.3739	−	17.3739i	−1.38659	−	1.38659i	−0.832380	−	0.554205i	$-0.813022\pi$
$157$	−17.3739	−	17.3739i	−1.38659	−	1.38659i	−0.554205	−	0.832380i	$-0.686978\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	10.4440i		−	0.823106i
$162$		0			0
$163$	0.582576	−	0.582576i	0.0456309	−	0.0456309i	−0.683923	−	0.729554i	$-0.739728\pi$
$163$	0.582576	−	0.582576i	0.0456309	−	0.0456309i	0.729554	+	0.683923i	$0.239728\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.02815	−	3.02815i	0.234325	−	0.234325i	−0.580170	−	0.814495i	$-0.697014\pi$
$167$	3.02815	−	3.02815i	0.234325	−	0.234325i	0.814495	+	0.580170i	$0.197014\pi$
$168$		0			0
$169$			11.7477i			0.903671i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	−1.35959	−	1.35959i	−0.103368	−	0.103368i	0.653532	−	0.756899i	$-0.273287\pi$
$173$	−1.35959	−	1.35959i	−0.103368	−	0.103368i	−0.756899	+	0.653532i	$0.773287\pi$
$174$		0			0
$175$	−7.20871	+	18.3739i	−0.544927	+	1.38893i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	5.16867			0.386325			0.193162	−	0.981167i	$-0.438126\pi$
$179$	5.16867			0.386325			0.193162	+	0.981167i	$0.438126\pi$
$180$		0			0
$181$	−14.5826			−1.08391			−0.541957	−	0.840406i	$-0.682317\pi$
$181$	−14.5826			−1.08391			−0.541957	+	0.840406i	$0.682317\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	1.93825	+	9.28672i	0.142503	+	0.682773i
$186$		0			0
$187$	7.79129	+	7.79129i	0.569755	+	0.569755i
$188$		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
$193$	11.2087	−	11.2087i	0.806821	−	0.806821i	−0.177331	−	0.984151i	$-0.556746\pi$
$193$	11.2087	−	11.2087i	0.806821	−	0.806821i	0.984151	+	0.177331i	$0.0567462\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−6.12372	+	6.12372i	−0.436297	+	0.436297i	−0.890764	−	0.454467i	$-0.849830\pi$
$197$	−6.12372	+	6.12372i	−0.436297	+	0.436297i	0.454467	+	0.890764i	$0.349830\pi$
$198$		0			0
$199$		−	4.41742i		−	0.313143i	−0.987667	−	0.156571i	$-0.949956\pi$
$199$		−	4.41742i		−	0.313143i	0.987667	−	0.156571i	$-0.0500441\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−11.8711	−	11.8711i	−0.833185	−	0.833185i
$204$		0			0
$205$	9.79129	−	14.9564i	0.683853	−	1.04460i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	7.99455			0.552995
$210$		0			0
$211$	7.83485			0.539373			0.269687	−	0.962948i	$-0.413080\pi$
$211$	7.83485			0.539373			0.269687	+	0.962948i	$0.413080\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	11.7362	−	2.44949i	0.800403	−	0.167054i
$216$		0			0
$217$	23.9564	+	23.9564i	1.62627	+	1.62627i
$218$		0			0
$219$		0			0
$220$		0			0
$221$		−	3.98320i		−	0.267939i
$222$		0			0
$223$	−6.16515	+	6.16515i	−0.412849	+	0.412849i	−0.882730	−	0.469881i	$-0.844297\pi$
$223$	−6.16515	+	6.16515i	−0.412849	+	0.412849i	0.469881	+	0.882730i	$0.344297\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−10.6463	+	10.6463i	−0.706620	+	0.706620i	−0.965823	−	0.259203i	$-0.916540\pi$
$227$	−10.6463	+	10.6463i	−0.706620	+	0.706620i	0.259203	+	0.965823i	$0.416540\pi$
$228$		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
$233$	−6.83723	−	6.83723i	−0.447922	−	0.447922i	0.446741	−	0.894663i	$-0.352584\pi$
$233$	−6.83723	−	6.83723i	−0.447922	−	0.447922i	−0.894663	+	0.446741i	$0.852584\pi$
$234$		0			0
$235$	28.7477	−	6.00000i	1.87529	−	0.391397i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−29.7984			−1.92750			−0.963750	−	0.266808i	$-0.914031\pi$
$239$	−29.7984			−1.92750			−0.963750	+	0.266808i	$0.914031\pi$
$240$		0			0
$241$	23.7477			1.52973			0.764863	−	0.644193i	$-0.222807\pi$
$241$	23.7477			1.52973			0.764863	+	0.644193i	$0.222807\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−10.5115	+	16.0565i	−0.671553	+	1.02581i
$246$		0			0
$247$	−2.04356	−	2.04356i	−0.130029	−	0.130029i
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	17.5510i		−	1.10781i	−0.832581	−	0.553904i	$-0.813138\pi$
$251$		−	17.5510i		−	1.10781i	0.832581	−	0.553904i	$-0.186862\pi$
$252$		0			0
$253$	−5.79129	+	5.79129i	−0.364095	+	0.364095i
$254$		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
$259$			16.7477i			1.04065i
$260$		0			0
$261$		0			0
$262$		0			0
$263$	9.42157	+	9.42157i	0.580959	+	0.580959i	0.935167	−	0.354208i	$-0.115250\pi$
$263$	9.42157	+	9.42157i	0.580959	+	0.580959i	−0.354208	+	0.935167i	$0.615250\pi$
$264$		0			0
$265$	5.95644	+	28.5390i	0.365901	+	1.75314i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	0.134846			0.00822168			0.00411084	−	0.999992i	$-0.498691\pi$
$269$	0.134846			0.00822168			0.00411084	+	0.999992i	$0.498691\pi$
$270$		0			0
$271$	−13.7477			−0.835115			−0.417557	−	0.908651i	$-0.637114\pi$
$271$	−13.7477			−0.835115			−0.417557	+	0.908651i	$0.637114\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	14.1857	−	6.19115i	0.855430	−	0.373340i
$276$		0			0
$277$	−3.79129	−	3.79129i	−0.227796	−	0.227796i	0.583975	−	0.811772i	$-0.301497\pi$
$277$	−3.79129	−	3.79129i	−0.227796	−	0.227796i	−0.811772	+	0.583975i	$0.801497\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			11.7362i			0.700124i	0.936727	+	0.350062i	$0.113839\pi$
$281$			11.7362i			0.700124i	−0.936727	+	0.350062i	$0.886161\pi$
$282$		0			0
$283$	10.2087	−	10.2087i	0.606845	−	0.606845i	−0.335275	−	0.942120i	$-0.608829\pi$
$283$	10.2087	−	10.2087i	0.606845	−	0.606845i	0.942120	+	0.335275i	$0.108829\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	22.3151	−	22.3151i	1.31722	−	1.31722i
$288$		0			0
$289$			4.33030i			0.254724i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.9610	+	12.9610i	0.757187	+	0.757187i	0.975810	−	0.218622i	$-0.0701563\pi$
$293$	12.9610	+	12.9610i	0.757187	+	0.757187i	−0.218622	+	0.975810i	$0.570156\pi$
$294$		0			0
$295$	−17.1216	−	11.2087i	−0.996858	−	0.652596i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	2.96073			0.171223
$300$		0			0
$301$	21.1652			1.21994
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−22.7589	−	14.8992i	−1.30317	−	0.853126i
$306$		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
$311$			30.0400i			1.70341i	0.524022	+	0.851705i	$0.324431\pi$
$311$			30.0400i			1.70341i	−0.524022	+	0.851705i	$0.675569\pi$
$312$		0			0
$313$	−9.16515	+	9.16515i	−0.518045	+	0.518045i	−0.916979	−	0.398934i	$-0.869380\pi$
$313$	−9.16515	+	9.16515i	−0.518045	+	0.518045i	0.398934	+	0.916979i	$0.369380\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	1.22474	−	1.22474i	0.0687885	−	0.0687885i	−0.671876	−	0.740664i	$-0.734511\pi$
$317$	1.22474	−	1.22474i	0.0687885	−	0.0687885i	0.740664	+	0.671876i	$0.234511\pi$
$318$		0			0
$319$			13.1652i			0.737107i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−6.50012	−	6.50012i	−0.361676	−	0.361676i
$324$		0			0
$325$	−5.20871	−	2.04356i	−0.288927	−	0.113356i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	51.8438			2.85824
$330$		0			0
$331$	−0.417424			−0.0229437			−0.0114719	−	0.999934i	$-0.503652\pi$
$331$	−0.417424			−0.0229437			−0.0114719	+	0.999934i	$0.503652\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−4.52259	−	21.6690i	−0.247095	−	1.18390i
$336$		0			0
$337$	3.95644	+	3.95644i	0.215521	+	0.215521i	0.806608	−	0.591087i	$-0.201301\pi$
$337$	3.95644	+	3.95644i	0.215521	+	0.215521i	−0.591087	+	0.806608i	$0.701301\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$		−	26.5680i		−	1.43874i
$342$		0			0
$343$	−4.41742	+	4.41742i	−0.238518	+	0.238518i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−15.9891	+	15.9891i	−0.858340	+	0.858340i	−0.991143	−	0.132802i	$-0.957602\pi$
$347$	−15.9891	+	15.9891i	−0.858340	+	0.858340i	0.132802	+	0.991143i	$0.457602\pi$
$348$		0			0
$349$		−	27.7477i		−	1.48530i	−0.669679	−	0.742651i	$-0.733568\pi$
$349$		−	27.7477i		−	1.48530i	0.669679	−	0.742651i	$-0.266432\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	9.79796	+	9.79796i	0.521493	+	0.521493i	0.918022	−	0.396529i	$-0.129785\pi$
$353$	9.79796	+	9.79796i	0.521493	+	0.521493i	−0.396529	+	0.918022i	$0.629785\pi$
$354$		0			0
$355$	−14.5390	+	22.2087i	−0.771651	+	1.17872i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−14.8318			−0.782792			−0.391396	−	0.920222i	$-0.628008\pi$
$359$	−14.8318			−0.782792			−0.391396	+	0.920222i	$0.628008\pi$
$360$		0			0
$361$	12.3303			0.648963
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−17.9274	+	3.74166i	−0.938361	+	0.195847i
$366$		0			0
$367$	13.2087	+	13.2087i	0.689489	+	0.689489i	0.962119	−	0.272630i	$-0.0878934\pi$
$367$	13.2087	+	13.2087i	0.689489	+	0.689489i	−0.272630	+	0.962119i	$0.587893\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$			51.4674i			2.67206i
$372$		0			0
$373$	20.7477	−	20.7477i	1.07428	−	1.07428i	0.0772661	−	0.997011i	$-0.475381\pi$
$373$	20.7477	−	20.7477i	1.07428	−	1.07428i	0.997011	−	0.0772661i	$-0.0246191\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	3.36526	−	3.36526i	0.173320	−	0.173320i
$378$		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
$383$	5.10125	+	5.10125i	0.260662	+	0.260662i	0.825323	−	0.564661i	$-0.190993\pi$
$383$	5.10125	+	5.10125i	0.260662	+	0.260662i	−0.564661	+	0.825323i	$0.690993\pi$
$384$		0			0
$385$	26.7477	−	5.58258i	1.36319	−	0.284514i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−17.0397			−0.863948			−0.431974	−	0.901886i	$-0.642183\pi$
$389$	−17.0397			−0.863948			−0.431974	+	0.901886i	$0.642183\pi$
$390$		0			0
$391$	9.41742			0.476260
$392$		0			0
$393$		0			0
$394$		0			0
$395$	14.8992	−	22.7589i	0.749660	−	1.14513i
$396$		0			0
$397$	−5.95644	−	5.95644i	−0.298945	−	0.298945i	0.541655	−	0.840601i	$-0.317798\pi$
$397$	−5.95644	−	5.95644i	−0.298945	−	0.298945i	−0.840601	+	0.541655i	$0.817798\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	2.44949i		−	0.122322i	−0.998128	−	0.0611608i	$-0.980520\pi$
$401$		−	2.44949i		−	0.122322i	0.998128	−	0.0611608i	$-0.0194803\pi$
$402$		0			0
$403$	−6.79129	+	6.79129i	−0.338298	+	0.338298i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	9.28672	−	9.28672i	0.460326	−	0.460326i
$408$		0			0
$409$		−	6.16515i		−	0.304847i	−0.988315	−	0.152424i	$-0.951292\pi$
$409$		−	6.16515i		−	0.304847i	0.988315	−	0.152424i	$-0.0487078\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−25.5455	−	25.5455i	−1.25701	−	1.25701i
$414$		0			0
$415$	−0.791288	−	3.79129i	−0.0388428	−	0.186107i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−18.0622			−0.882396			−0.441198	−	0.897410i	$-0.645446\pi$
$419$	−18.0622			−0.882396			−0.441198	+	0.897410i	$0.645446\pi$
$420$		0			0
$421$	−20.5826			−1.00313			−0.501567	−	0.865119i	$-0.667243\pi$
$421$	−20.5826			−1.00313			−0.501567	+	0.865119i	$0.667243\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−16.5678	−	6.50012i	−0.803655	−	0.315302i
$426$		0			0
$427$	−33.9564	−	33.9564i	−1.64327	−	1.64327i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			13.0284i			0.627555i	0.949497	+	0.313777i	$0.101595\pi$
$431$			13.0284i			0.627555i	−0.949497	+	0.313777i	$0.898405\pi$
$432$		0			0
$433$	1.20871	−	1.20871i	0.0580870	−	0.0580870i	−0.677467	−	0.735554i	$-0.736922\pi$
$433$	1.20871	−	1.20871i	0.0580870	−	0.0580870i	0.735554	+	0.677467i	$0.236922\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	4.83156	−	4.83156i	0.231125	−	0.231125i
$438$		0			0
$439$			7.83485i			0.373937i	0.982366	+	0.186968i	$0.0598662\pi$
$439$			7.83485i			0.373937i	−0.982366	+	0.186968i	$0.940134\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−0.713507	−	0.713507i	−0.0338997	−	0.0338997i	0.689954	−	0.723853i	$-0.257631\pi$
$443$	−0.713507	−	0.713507i	−0.0338997	−	0.0338997i	−0.723853	+	0.689954i	$0.757631\pi$
$444$		0			0
$445$	−31.3739	−	20.5390i	−1.48726	−	0.973643i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	38.8154			1.83181			0.915907	−	0.401391i	$-0.131473\pi$
$449$	38.8154			1.83181			0.915907	+	0.401391i	$0.131473\pi$
$450$		0			0
$451$	−24.7477			−1.16532
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−8.26424	−	5.41022i	−0.387434	−	0.253635i
$456$		0			0
$457$	4.74773	+	4.74773i	0.222089	+	0.222089i	0.809378	−	0.587288i	$-0.199805\pi$
$457$	4.74773	+	4.74773i	0.222089	+	0.222089i	−0.587288	+	0.809378i	$0.699805\pi$
$458$		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
$463$	−23.7477	+	23.7477i	−1.10365	+	1.10365i	−0.109684	+	0.993967i	$0.534984\pi$
$463$	−23.7477	+	23.7477i	−1.10365	+	1.10365i	−0.993967	+	0.109684i	$0.965016\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−12.8261	+	12.8261i	−0.593522	+	0.593522i	−0.938581	−	0.345059i	$-0.887859\pi$
$467$	−12.8261	+	12.8261i	−0.593522	+	0.593522i	0.345059	+	0.938581i	$0.387859\pi$
$468$		0			0
$469$		−	39.0780i		−	1.80446i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−11.7362	−	11.7362i	−0.539632	−	0.539632i
$474$		0			0
$475$	−11.8348	+	5.16515i	−0.543020	+	0.236993i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	35.3435			1.61488			0.807442	−	0.589946i	$-0.200851\pi$
$479$	35.3435			1.61488			0.807442	+	0.589946i	$0.200851\pi$
$480$		0			0
$481$	−4.74773			−0.216478
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−1.02248	−	4.89898i	−0.0464283	−	0.222451i
$486$		0			0
$487$	−5.37386	−	5.37386i	−0.243513	−	0.243513i	0.574789	−	0.818302i	$-0.305084\pi$
$487$	−5.37386	−	5.37386i	−0.243513	−	0.243513i	−0.818302	+	0.574789i	$0.805084\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	2.71918i		−	0.122715i	−0.998116	−	0.0613575i	$-0.980457\pi$
$491$		−	2.71918i		−	0.122715i	0.998116	−	0.0613575i	$-0.0195430\pi$
$492$		0			0
$493$	10.7042	−	10.7042i	0.482091	−	0.482091i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−33.1355	+	33.1355i	−1.48633	+	1.48633i
$498$		0			0
$499$			9.41742i			0.421582i	0.977531	+	0.210791i	$0.0676039\pi$
$499$			9.41742i			0.421582i	−0.977531	+	0.210791i	$0.932396\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−1.73598	−	1.73598i	−0.0774037	−	0.0774037i	0.667345	−	0.744749i	$-0.267431\pi$
$503$	−1.73598	−	1.73598i	−0.0774037	−	0.0774037i	−0.744749	+	0.667345i	$0.767431\pi$
$504$		0			0
$505$	19.5826	−	29.9129i	0.871413	−	1.33111i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	22.1803			0.983122			0.491561	−	0.870843i	$-0.336426\pi$
$509$	22.1803			0.983122			0.491561	+	0.870843i	$0.336426\pi$
$510$		0			0
$511$	−32.3303			−1.43021
$512$		0			0
$513$		0			0
$514$		0			0
$515$	17.2813	−	3.60681i	0.761504	−	0.158935i
$516$		0			0
$517$	−28.7477	−	28.7477i	−1.26432	−	1.26432i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			15.2082i			0.666282i	0.942877	+	0.333141i	$0.108109\pi$
$521$			15.2082i			0.666282i	−0.942877	+	0.333141i	$0.891891\pi$
$522$		0			0
$523$	−27.9564	+	27.9564i	−1.22245	+	1.22245i	−0.255691	+	0.966758i	$0.582303\pi$
$523$	−27.9564	+	27.9564i	−1.22245	+	1.22245i	−0.966758	+	0.255691i	$0.917697\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−21.6016	+	21.6016i	−0.940980	+	0.940980i
$528$		0			0
$529$		−	16.0000i		−	0.695652i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	6.32599	+	6.32599i	0.274009	+	0.274009i
$534$		0			0
$535$	18.7477	−	3.91288i	0.810535	−	0.169168i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	26.5680			1.14436
$540$		0			0
$541$	−22.8348			−0.981747			−0.490873	−	0.871231i	$-0.663322\pi$
$541$	−22.8348			−0.981747			−0.490873	+	0.871231i	$0.663322\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	14.3880	−	21.9780i	0.616313	−	0.941433i
$546$		0			0
$547$	−2.53901	−	2.53901i	−0.108560	−	0.108560i	0.650740	−	0.759301i	$-0.274459\pi$
$547$	−2.53901	−	2.53901i	−0.108560	−	0.108560i	−0.759301	+	0.650740i	$0.774459\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	10.9834i		−	0.467910i
$552$		0			0
$553$	33.9564	−	33.9564i	1.44397	−	1.44397i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−8.77548	+	8.77548i	−0.371829	+	0.371829i	−0.868143	−	0.496314i	$-0.834686\pi$
$557$	−8.77548	+	8.77548i	−0.371829	+	0.371829i	0.496314	+	0.868143i	$0.334686\pi$
$558$		0			0
$559$			6.00000i			0.253773i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−25.9219	−	25.9219i	−1.09248	−	1.09248i	−0.995263	−	0.0972149i	$-0.969007\pi$
$563$	−25.9219	−	25.9219i	−1.09248	−	1.09248i	−0.0972149	−	0.995263i	$-0.530993\pi$
$564$		0			0
$565$	6.41742	+	30.7477i	0.269983	+	1.29357i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	28.7478			1.20517			0.602585	−	0.798055i	$-0.294137\pi$
$569$	28.7478			1.20517			0.602585	+	0.798055i	$0.294137\pi$
$570$		0			0
$571$	30.9129			1.29366			0.646832	−	0.762633i	$-0.276094\pi$
$571$	30.9129			1.29366			0.646832	+	0.762633i	$0.276094\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	4.83156	−	12.3149i	0.201490	−	0.513566i
$576$		0			0
$577$	9.37386	+	9.37386i	0.390239	+	0.390239i	0.874773	−	0.484534i	$-0.161011\pi$
$577$	9.37386	+	9.37386i	0.390239	+	0.390239i	−0.484534	+	0.874773i	$0.661011\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		−	6.83723i		−	0.283656i
$582$		0			0
$583$	28.5390	−	28.5390i	1.18197	−	1.18197i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−0.202268	+	0.202268i	−0.00834851	+	0.00834851i	−0.711269	−	0.702920i	$-0.751879\pi$
$587$	−0.202268	+	0.202268i	−0.00834851	+	0.00834851i	0.702920	+	0.711269i	$0.251879\pi$
$588$		0			0
$589$			22.1652i			0.913299i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	14.2531	+	14.2531i	0.585306	+	0.585306i	0.936356	−	0.351051i	$-0.114176\pi$
$593$	14.2531	+	14.2531i	0.585306	+	0.585306i	−0.351051	+	0.936356i	$0.614176\pi$
$594$		0			0
$595$	−26.2867	−	17.2087i	−1.07765	−	0.705488i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	29.0175			1.18562			0.592811	−	0.805342i	$-0.298018\pi$
$599$	29.0175			1.18562			0.592811	+	0.805342i	$0.298018\pi$
$600$		0			0
$601$	−11.7477			−0.479200			−0.239600	−	0.970872i	$-0.577016\pi$
$601$	−11.7477			−0.479200			−0.239600	+	0.970872i	$0.577016\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.65176	+	1.73598i	0.107809	+	0.0705778i
$606$		0			0
$607$	−16.1216	−	16.1216i	−0.654355	−	0.654355i	0.299684	−	0.954039i	$-0.403119\pi$
$607$	−16.1216	−	16.1216i	−0.654355	−	0.654355i	−0.954039	+	0.299684i	$0.903119\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$			14.6969i			0.594574i
$612$		0			0
$613$	12.3303	−	12.3303i	0.498016	−	0.498016i	−0.412804	−	0.910820i	$-0.635450\pi$
$613$	12.3303	−	12.3303i	0.498016	−	0.498016i	0.910820	+	0.412804i	$0.135450\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−23.2702	+	23.2702i	−0.936821	+	0.936821i	−0.998119	−	0.0612984i	$-0.980476\pi$
$617$	−23.2702	+	23.2702i	−0.936821	+	0.936821i	0.0612984	+	0.998119i	$0.480476\pi$
$618$		0			0
$619$		−	16.7477i		−	0.673148i	−0.941657	−	0.336574i	$-0.890732\pi$
$619$		−	16.7477i		−	0.673148i	0.941657	−	0.336574i	$-0.109268\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−46.8100	−	46.8100i	−1.87540	−	1.87540i
$624$		0			0
$625$	−17.0000	+	18.3303i	−0.680000	+	0.733212i
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−15.1015			−0.602135
$630$		0			0
$631$	16.1652			0.643525			0.321762	−	0.946820i	$-0.395725\pi$
$631$	16.1652			0.643525			0.321762	+	0.946820i	$0.395725\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−1.66856	−	7.99455i	−0.0662148	−	0.317254i
$636$		0			0
$637$	−6.79129	−	6.79129i	−0.269081	−	0.269081i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	29.6636i		−	1.17164i	−0.810441	−	0.585820i	$-0.800772\pi$
$641$		−	29.6636i		−	1.17164i	0.810441	−	0.585820i	$-0.199228\pi$
$642$		0			0
$643$	−23.5390	+	23.5390i	−0.928288	+	0.928288i	−0.997595	−	0.0693072i	$-0.977921\pi$
$643$	−23.5390	+	23.5390i	−0.928288	+	0.928288i	0.0693072	+	0.997595i	$0.477921\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−19.9330	+	19.9330i	−0.783648	+	0.783648i	−0.980444	−	0.196796i	$-0.936946\pi$
$647$	−19.9330	+	19.9330i	−0.783648	+	0.783648i	0.196796	+	0.980444i	$0.436946\pi$
$648$		0			0
$649$			28.3303i			1.11206i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	29.0849	+	29.0849i	1.13818	+	1.13818i	0.988776	+	0.149404i	$0.0477355\pi$
$653$	29.0849	+	29.0849i	1.13818	+	1.13818i	0.149404	+	0.988776i	$0.452264\pi$
$654$		0			0
$655$	6.16515	−	9.41742i	0.240892	−	0.367969i
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−9.42157			−0.367012			−0.183506	−	0.983019i	$-0.558745\pi$
$659$	−9.42157			−0.367012			−0.183506	+	0.983019i	$0.558745\pi$
$660$		0			0
$661$	23.4955			0.913867			0.456934	−	0.889501i	$-0.348948\pi$
$661$	23.4955			0.913867			0.456934	+	0.889501i	$0.348948\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	−22.3151	+	4.65743i	−0.865342	+	0.180607i
$666$		0			0
$667$	7.95644	+	7.95644i	0.308075	+	0.308075i
$668$		0			0
$669$		0			0
$670$		0			0
$671$			37.6581i			1.45378i
$672$		0			0
$673$	−29.9564	+	29.9564i	−1.15474	+	1.15474i	−0.169145	+	0.985591i	$0.554100\pi$
$673$	−29.9564	+	29.9564i	−1.15474	+	1.15474i	−0.985591	+	0.169145i	$0.945900\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−13.5396	+	13.5396i	−0.520370	+	0.520370i	−0.917683	−	0.397313i	$-0.869943\pi$
$677$	−13.5396	+	13.5396i	−0.520370	+	0.520370i	0.397313	+	0.917683i	$0.369943\pi$
$678$		0			0
$679$		−	8.83485i		−	0.339050i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−10.0002	−	10.0002i	−0.382648	−	0.382648i	0.489407	−	0.872055i	$-0.337213\pi$
$683$	−10.0002	−	10.0002i	−0.382648	−	0.382648i	−0.872055	+	0.489407i	$0.837213\pi$
$684$		0			0
$685$	3.79129	−	0.791288i	0.144858	−	0.0302336i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−14.5902			−0.555844
$690$		0			0
$691$	−33.6606			−1.28051			−0.640255	−	0.768163i	$-0.721171\pi$
$691$	−33.6606			−1.28051			−0.640255	+	0.768163i	$0.721171\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.77548	−	13.4048i	0.332873	−	0.508472i
$696$		0			0
$697$	20.1216	+	20.1216i	0.762160	+	0.762160i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			2.71918i			0.102702i	0.998681	+	0.0513510i	$0.0163527\pi$
$701$			2.71918i			0.102702i	−0.998681	+	0.0513510i	$0.983647\pi$
$702$		0			0
$703$	−7.74773	+	7.74773i	−0.292211	+	0.292211i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	44.6302	−	44.6302i	1.67849	−	1.67849i
$708$		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
$713$	−16.0565	−	16.0565i	−0.601322	−	0.601322i
$714$		0			0
$715$	1.58258	+	7.58258i	0.0591850	+	0.283572i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	7.07878			0.263994			0.131997	−	0.991250i	$-0.457861\pi$
$719$	7.07878			0.263994			0.131997	+	0.991250i	$0.457861\pi$
$720$		0			0
$721$	31.1652			1.16065
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.50579	−	19.4892i	−0.315897	−	0.723811i
$726$		0			0
$727$	24.3303	+	24.3303i	0.902361	+	0.902361i	0.995640	−	0.0932790i	$-0.0297349\pi$
$727$	24.3303	+	24.3303i	0.902361	+	0.902361i	−0.0932790	+	0.995640i	$0.529735\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			19.0847i			0.705872i
$732$		0			0
$733$	−21.9129	+	21.9129i	−0.809371	+	0.809371i	−0.984539	−	0.175168i	$-0.943953\pi$
$733$	−21.9129	+	21.9129i	−0.809371	+	0.809371i	0.175168	+	0.984539i	$0.443953\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−21.6690	+	21.6690i	−0.798188	+	0.798188i
$738$		0			0
$739$		−	20.5826i		−	0.757142i	−0.925572	−	0.378571i	$-0.876416\pi$
$739$		−	20.5826i		−	0.757142i	0.925572	−	0.378571i	$-0.123584\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−10.0677	−	10.0677i	−0.369346	−	0.369346i	0.497892	−	0.867239i	$-0.334107\pi$
$743$	−10.0677	−	10.0677i	−0.369346	−	0.369346i	−0.867239	+	0.497892i	$0.834107\pi$
$744$		0			0
$745$	−0.956439	−	0.626136i	−0.0350412	−	0.0229399i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	33.8098			1.23538
$750$		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
$755$	2.17980	+	1.42701i	0.0793310	+	0.0519343i
$756$		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
$761$			12.0059i			0.435214i	0.976037	+	0.217607i	$0.0698250\pi$
$761$			12.0059i			0.435214i	−0.976037	+	0.217607i	$0.930175\pi$
$762$		0			0
$763$	32.7913	−	32.7913i	1.18712	−	1.18712i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	7.24177	−	7.24177i	0.261485	−	0.261485i
$768$		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
$773$	30.5119	+	30.5119i	1.09744	+	1.09744i	0.994709	+	0.102728i	$0.0327571\pi$
$773$	30.5119	+	30.5119i	1.09744	+	1.09744i	0.102728	+	0.994709i	$0.467243\pi$
$774$		0			0
$775$	17.1652	+	39.3303i	0.616590	+	1.41279i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	20.6465			0.739739
$780$		0			0
$781$	36.7477			1.31494
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−11.2250	−	53.7821i	−0.400636	−	1.91956i
$786$		0			0
$787$	21.7042	+	21.7042i	0.773670	+	0.773670i	0.978746	−	0.205076i	$-0.0657442\pi$
$787$	21.7042	+	21.7042i	0.773670	+	0.773670i	−0.205076	+	0.978746i	$0.565744\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			55.4506i			1.97160i
$792$		0			0
$793$	9.62614	−	9.62614i	0.341834	−	0.341834i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.59001	−	4.59001i	0.162586	−	0.162586i	−0.621125	−	0.783711i	$-0.713324\pi$
$797$	4.59001	−	4.59001i	0.162586	−	0.162586i	0.783711	+	0.621125i	$0.213324\pi$
$798$		0			0
$799$			46.7477i			1.65382i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	17.9274	+	17.9274i	0.632643	+	0.632643i
$804$		0			0
$805$	12.7913	−	19.5390i	0.450834	−	0.688660i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	36.0963			1.26908			0.634539	−	0.772891i	$-0.281190\pi$
$809$	36.0963			1.26908			0.634539	+	0.772891i	$0.281190\pi$
$810$		0			0
$811$	20.3303			0.713893			0.356947	−	0.934125i	$-0.383818\pi$
$811$	20.3303			0.713893			0.356947	+	0.934125i	$0.383818\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	1.80341	−	0.376393i	0.0631706	−	0.0131845i
$816$		0			0
$817$	9.79129	+	9.79129i	0.342554	+	0.342554i
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	28.5062i		−	0.994875i	−0.867500	−	0.497437i	$-0.834274\pi$
$821$		−	28.5062i		−	0.994875i	0.867500	−	0.497437i	$-0.165726\pi$
$822$		0			0
$823$	30.1216	−	30.1216i	1.04997	−	1.04997i	0.0512888	−	0.998684i	$-0.483667\pi$
$823$	30.1216	−	30.1216i	1.04997	−	1.04997i	0.998684	−	0.0512888i	$-0.0163329\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	8.06198	−	8.06198i	0.280342	−	0.280342i	−0.552903	−	0.833246i	$-0.686480\pi$
$827$	8.06198	−	8.06198i	0.280342	−	0.280342i	0.833246	+	0.552903i	$0.186480\pi$
$828$		0			0
$829$			31.0780i			1.07938i	0.841862	+	0.539692i	$0.181459\pi$
$829$			31.0780i			1.07938i	−0.841862	+	0.539692i	$0.818541\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−21.6016	−	21.6016i	−0.748451	−	0.748451i
$834$		0			0
$835$	9.37386	−	1.95644i	0.324396	−	0.0677054i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−14.4272			−0.498084			−0.249042	−	0.968493i	$-0.580116\pi$
$839$	−14.4272			−0.498084			−0.249042	+	0.968493i	$0.580116\pi$
$840$		0			0
$841$	−10.9129			−0.376306
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−14.3880	+	21.9780i	−0.494961	+	0.756066i
$846$		0			0
$847$	3.95644	+	3.95644i	0.135945	+	0.135945i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	11.2250i		−	0.384787i
$852$		0			0
$853$	−20.3303	+	20.3303i	−0.696096	+	0.696096i	−0.963566	−	0.267470i	$-0.913812\pi$
$853$	−20.3303	+	20.3303i	−0.696096	+	0.696096i	0.267470	+	0.963566i	$0.413812\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	29.0849	−	29.0849i	0.993521	−	0.993521i	−0.00645765	−	0.999979i	$-0.502056\pi$
$857$	29.0849	−	29.0849i	0.993521	−	0.993521i	0.999979	+	0.00645765i	$0.00205555\pi$
$858$		0			0
$859$			0.912878i			0.0311470i	0.999879	+	0.0155735i	$0.00495740\pi$
$859$			0.912878i			0.0311470i	−0.999879	+	0.0155735i	$0.995043\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	19.6633	+	19.6633i	0.669348	+	0.669348i	0.957565	−	0.288217i	$-0.0930625\pi$
$863$	19.6633	+	19.6633i	0.669348	+	0.669348i	−0.288217	+	0.957565i	$0.593063\pi$
$864$		0			0
$865$	−0.878409	−	4.20871i	−0.0298668	−	0.143101i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−37.6581			−1.27746
$870$		0			0
$871$	11.0780			0.375365
$872$		0			0
$873$		0			0
$874$		0			0
$875$	−35.9896	+	25.5455i	−1.21667	+	0.863596i
$876$		0			0
$877$	21.7913	+	21.7913i	0.735840	+	0.735840i	0.971770	−	0.235930i	$-0.0758138\pi$
$877$	21.7913	+	21.7913i	0.735840	+	0.735840i	−0.235930	+	0.971770i	$0.575814\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			45.2763i			1.52540i	0.646754	+	0.762698i	$0.276126\pi$
$881$			45.2763i			1.52540i	−0.646754	+	0.762698i	$0.723874\pi$
$882$		0			0
$883$	−13.2087	+	13.2087i	−0.444509	+	0.444509i	−0.893524	−	0.449015i	$-0.851775\pi$
$883$	−13.2087	+	13.2087i	−0.444509	+	0.444509i	0.449015	+	0.893524i	$0.351775\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	9.59569	−	9.59569i	0.322192	−	0.322192i	−0.527416	−	0.849607i	$-0.676839\pi$
$887$	9.59569	−	9.59569i	0.322192	−	0.322192i	0.849607	+	0.527416i	$0.176839\pi$
$888$		0			0
$889$		−	14.4174i		−	0.483545i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	23.9837	+	23.9837i	0.802583	+	0.802583i
$894$		0			0
$895$	9.66970	+	6.33030i	0.323222	+	0.211599i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−36.5008			−1.21737
$900$		0			0
$901$	−46.4083			−1.54609
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−27.2815	−	17.8599i	−0.906868	−	0.593684i
$906$		0			0
$907$	8.74773	+	8.74773i	0.290464	+	0.290464i	0.837263	−	0.546800i	$-0.184154\pi$
$907$	8.74773	+	8.74773i	0.290464	+	0.290464i	−0.546800	+	0.837263i	$0.684154\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	27.7253i		−	0.918581i	−0.888286	−	0.459290i	$-0.848104\pi$
$911$		−	27.7253i		−	0.918581i	0.888286	−	0.459290i	$-0.151896\pi$
$912$		0			0
$913$	−3.79129	+	3.79129i	−0.125473	+	0.125473i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	14.0509	−	14.0509i	0.464000	−	0.464000i
$918$		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
$923$	−9.39342	−	9.39342i	−0.309188	−	0.309188i
$924$		0			0
$925$	−7.74773	+	19.7477i	−0.254744	+	0.649301i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−19.3544			−0.634996			−0.317498	−	0.948259i	$-0.602843\pi$
$929$	−19.3544			−0.634996			−0.317498	+	0.948259i	$0.602843\pi$
$930$		0			0
$931$	−22.1652			−0.726433
$932$		0			0
$933$		0			0
$934$		0			0
$935$	5.03383	+	24.1185i	0.164624	+	0.788759i
$936$		0			0
$937$	−37.0780	−	37.0780i	−1.21129	−	1.21129i	−0.970604	−	0.240683i	$-0.922629\pi$
$937$	−37.0780	−	37.0780i	−1.21129	−	1.21129i	−0.240683	−	0.970604i	$-0.577371\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	38.5739i		−	1.25747i	−0.777618	−	0.628737i	$-0.783572\pi$
$941$		−	38.5739i		−	1.25747i	0.777618	−	0.628737i	$-0.216428\pi$
$942$		0			0
$943$	−14.9564	+	14.9564i	−0.487049	+	0.487049i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−7.28105	+	7.28105i	−0.236602	+	0.236602i	−0.815442	−	0.578839i	$-0.803506\pi$
$947$	−7.28105	+	7.28105i	−0.236602	+	0.236602i	0.578839	+	0.815442i	$0.303506\pi$
$948$		0			0
$949$		−	9.16515i		−	0.297513i
$950$		0			0
$951$		0			0
$952$		0			0
$953$	10.3092	+	10.3092i	0.333948	+	0.333948i	0.854084	−	0.520136i	$-0.174119\pi$
$953$	10.3092	+	10.3092i	0.333948	+	0.333948i	−0.520136	+	0.854084i	$0.674119\pi$
$954$		0			0
$955$	4.58258	−	7.00000i	0.148289	−	0.226515i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	6.83723			0.220786
$960$		0			0
$961$	42.6606			1.37615
$962$		0			0
$963$		0			0
$964$		0			0
$965$	34.6974	−	7.24177i	1.11695	−	0.233121i
$966$		0			0
$967$	−27.7477	−	27.7477i	−0.892307	−	0.892307i	0.102433	−	0.994740i	$-0.467337\pi$
$967$	−27.7477	−	27.7477i	−0.892307	−	0.892307i	−0.994740	+	0.102433i	$0.967337\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			13.8093i			0.443162i	0.975142	+	0.221581i	$0.0711217\pi$
$971$			13.8093i			0.443162i	−0.975142	+	0.221581i	$0.928878\pi$
$972$		0			0
$973$	20.0000	−	20.0000i	0.641171	−	0.641171i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−23.4724	+	23.4724i	−0.750949	+	0.750949i	−0.974656	−	0.223707i	$-0.928184\pi$
$977$	−23.4724	+	23.4724i	−0.750949	+	0.750949i	0.223707	+	0.974656i	$0.428184\pi$
$978$		0			0
$979$			51.9129i			1.65914i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−38.2368	−	38.2368i	−1.21956	−	1.21956i	−0.967785	−	0.251779i	$-0.918984\pi$
$983$	−38.2368	−	38.2368i	−1.21956	−	1.21956i	−0.251779	−	0.967785i	$-0.581016\pi$
$984$		0			0
$985$	−18.9564	+	3.95644i	−0.604002	+	0.126063i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	−14.1857			−0.451079
$990$		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
$995$	5.41022	−	8.26424i	0.171515	−	0.261994i
$996$		0			0
$997$	−32.9564	−	32.9564i	−1.04374	−	1.04374i	−0.998999	−	0.0447423i	$-0.985753\pi$
$997$	−32.9564	−	32.9564i	−1.04374	−	1.04374i	−0.0447423	−	0.998999i	$-0.514247\pi$
$998$		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.377.4	yes	8
3.2	odd	2	inner	540.2.j.b.377.1	yes	8
4.3	odd	2		2160.2.w.c.1457.4		8
5.2	odd	4		2700.2.j.i.593.1		8
5.3	odd	4	inner	540.2.j.b.53.2	✓	8
5.4	even	2		2700.2.j.i.1457.1		8
9.2	odd	6		1620.2.x.c.377.4		16
9.4	even	3		1620.2.x.c.917.3		16
9.5	odd	6		1620.2.x.c.917.2		16
9.7	even	3		1620.2.x.c.377.1		16
12.11	even	2		2160.2.w.c.1457.1		8
15.2	even	4		2700.2.j.i.593.2		8
15.8	even	4	inner	540.2.j.b.53.3	yes	8
15.14	odd	2		2700.2.j.i.1457.2		8
20.3	even	4		2160.2.w.c.593.2		8
45.13	odd	12		1620.2.x.c.593.4		16
45.23	even	12		1620.2.x.c.593.1		16
45.38	even	12		1620.2.x.c.53.3		16
45.43	odd	12		1620.2.x.c.53.2		16
60.23	odd	4		2160.2.w.c.593.3		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
540.2.j.b.53.2	✓	8	5.3	odd	4	inner
540.2.j.b.53.3	yes	8	15.8	even	4	inner
540.2.j.b.377.1	yes	8	3.2	odd	2	inner
540.2.j.b.377.4	yes	8	1.1	even	1	trivial
1620.2.x.c.53.2		16	45.43	odd	12
1620.2.x.c.53.3		16	45.38	even	12
1620.2.x.c.377.1		16	9.7	even	3
1620.2.x.c.377.4		16	9.2	odd	6
1620.2.x.c.593.1		16	45.23	even	12
1620.2.x.c.593.4		16	45.13	odd	12
1620.2.x.c.917.2		16	9.5	odd	6
1620.2.x.c.917.3		16	9.4	even	3
2160.2.w.c.593.2		8	20.3	even	4
2160.2.w.c.593.3		8	60.23	odd	4
2160.2.w.c.1457.1		8	12.11	even	2
2160.2.w.c.1457.4		8	4.3	odd	2
2700.2.j.i.593.1		8	5.2	odd	4
2700.2.j.i.593.2		8	15.2	even	4
2700.2.j.i.1457.1		8	5.4	even	2
2700.2.j.i.1457.2		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)

\(f(q)\)	\(=\)	\(q+(1.87083 + 1.22474i) q^{5} +(2.79129 + 2.79129i) q^{7} -3.09557i q^{11} +(-0.791288 + 0.791288i) q^{13} +(-2.51691 + 2.51691i) q^{17} +2.58258i q^{19} +(-1.87083 - 1.87083i) q^{23} +(2.00000 + 4.58258i) q^{25} -4.25290 q^{29} +8.58258 q^{31} +(1.80341 + 8.64064i) q^{35} +(3.00000 + 3.00000i) q^{37} -7.99455i q^{41} +(3.79129 - 3.79129i) q^{43} +(9.28672 - 9.28672i) q^{47} +8.58258i q^{49} +(9.21930 + 9.21930i) q^{53} +(3.79129 - 5.79129i) q^{55} -9.15188 q^{59} -12.1652 q^{61} +(-2.44949 + 0.511238i) q^{65} +(-7.00000 - 7.00000i) q^{67} +11.8711i q^{71} +(-5.79129 + 5.79129i) q^{73} +(8.64064 - 8.64064i) q^{77} -12.1652i q^{79} +(-1.22474 - 1.22474i) q^{83} +(-7.79129 + 1.62614i) q^{85} -16.7700 q^{89} -4.41742 q^{91} +(-3.16300 + 4.83156i) q^{95} +(-1.58258 - 1.58258i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 4 q^{7} + 12 q^{13} + 16 q^{25} + 32 q^{31} + 24 q^{37} + 12 q^{43} + 12 q^{55} - 24 q^{61} - 56 q^{67} - 28 q^{73} - 44 q^{85} - 72 q^{91} + 24 q^{97}+O(q^{100}) \) 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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