LMFDB - Embedded newform 2880.2.t.d.2161.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(721,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(4))

chi = DirichletCharacter(H, H._module([0, 3, 0, 0]))

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

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

chi := DirichletCharacter("2880.721");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.t (of order $4$, degree $2$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$22.9969157821$
Analytic rank:	$0$
Dimension:	$20$
Relative dimension:	$10$ over $\Q(i)$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{18} - 4 x^{17} + 7 x^{16} + 16 x^{15} + 6 x^{14} - 36 x^{13} - 42 x^{12} + 40 x^{11} + \cdots + 1024 $ x^20 - 2x^18 - 4x^17 + 7x^16 + 16x^15 + 6x^14 - 36x^13 - 42x^12 + 40x^11 + 136x^10 + 80x^9 - 168x^8 - 288x^7 + 96x^6 + 512x^5 + 448x^4 - 512x^3 - 512*x^2 + 1024
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{20} $
Twist minimal:	no (minimal twist has level 240)
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			2161.5
Root			$1.18701 + 0.768775i$ of defining polynomial
Character	$\chi$	$=$	2880.2161
Dual form			2880.2.t.d.721.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.707107 - 0.707107i) q^{5} +4.92824i q^{7} +O(q^{10})$	$q+(-0.707107 - 0.707107i) q^{5} +4.92824i q^{7} +(2.45868 + 2.45868i) q^{11} +(-2.93127 + 2.93127i) q^{13} +5.77716 q^{17} +(0.984104 - 0.984104i) q^{19} -0.539543i q^{23} +1.00000i q^{25} +(-6.81092 + 6.81092i) q^{29} +2.63175 q^{31} +(3.48479 - 3.48479i) q^{35} +(-6.00637 - 6.00637i) q^{37} -5.17728i q^{41} +(0.180211 + 0.180211i) q^{43} +5.57785 q^{47} -17.2875 q^{49} +(0.146862 + 0.146862i) q^{53} -3.47711i q^{55} +(3.13789 + 3.13789i) q^{59} +(-1.87356 + 1.87356i) q^{61} +4.14544 q^{65} +(-8.02582 + 8.02582i) q^{67} -7.40711i q^{71} +11.1715i q^{73} +(-12.1170 + 12.1170i) q^{77} -7.71759 q^{79} +(-1.62773 + 1.62773i) q^{83} +(-4.08507 - 4.08507i) q^{85} -9.54484i q^{89} +(-14.4460 - 14.4460i) q^{91} -1.39173 q^{95} +4.39090 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q+O(q^{10}) $ 20 * q	$ 20 q + 8 q^{11} + 24 q^{17} + 4 q^{19} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 52 q^{49} + 16 q^{53} - 16 q^{59} - 4 q^{61} + 8 q^{67} + 40 q^{77} - 56 q^{79} - 48 q^{83} + 4 q^{85} + 8 q^{91} + 56 q^{97}+O(q^{100}) $ 20 * q + 8 * q^11 + 24 * q^17 + 4 * q^19 - 16 * q^29 + 16 * q^37 + 8 * q^43 - 52 * q^49 + 16 * q^53 - 16 * q^59 - 4 * q^61 + 8 * q^67 + 40 * q^77 - 56 * q^79 - 48 * q^83 + 4 * q^85 + 8 * q^91 + 56 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/2880\mathbb{Z}\right)^\times$.

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$e\left(\frac{1}{4}\right)$	$1$

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
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$5$	−0.707107	−	0.707107i	−0.316228	−	0.316228i
$6$		0			0
$7$			4.92824i			1.86270i	0.364127	+	0.931349i	$0.381368\pi$
$7$			4.92824i			1.86270i	−0.364127	+	0.931349i	$0.618632\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	2.45868	+	2.45868i	0.741321	+	0.741321i	0.972832	−	0.231511i	$-0.0743669\pi$
$11$	2.45868	+	2.45868i	0.741321	+	0.741321i	−0.231511	+	0.972832i	$0.574367\pi$
$12$		0			0
$13$	−2.93127	+	2.93127i	−0.812987	+	0.812987i	−0.985081	−	0.172093i	$-0.944947\pi$
$13$	−2.93127	+	2.93127i	−0.812987	+	0.812987i	0.172093	+	0.985081i	$0.444947\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	5.77716			1.40117			0.700584	−	0.713570i	$-0.252923\pi$
$17$	5.77716			1.40117			0.700584	+	0.713570i	$0.252923\pi$
$18$		0			0
$19$	0.984104	−	0.984104i	0.225769	−	0.225769i	−0.585154	−	0.810922i	$-0.698966\pi$
$19$	0.984104	−	0.984104i	0.225769	−	0.225769i	0.810922	+	0.585154i	$0.198966\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	0.539543i		−	0.112502i	−0.998417	−	0.0562512i	$-0.982085\pi$
$23$		−	0.539543i		−	0.112502i	0.998417	−	0.0562512i	$-0.0179148\pi$
$24$		0			0
$25$			1.00000i			0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$	−6.81092	+	6.81092i	−1.26476	+	1.26476i	−0.315995	+	0.948761i	$0.602338\pi$
$29$	−6.81092	+	6.81092i	−1.26476	+	1.26476i	−0.948761	+	0.315995i	$0.897662\pi$
$30$		0			0
$31$	2.63175			0.472677			0.236338	−	0.971671i	$-0.424053\pi$
$31$	2.63175			0.472677			0.236338	+	0.971671i	$0.424053\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	3.48479	−	3.48479i	0.589037	−	0.589037i
$36$		0			0
$37$	−6.00637	−	6.00637i	−0.987441	−	0.987441i	0.0124815	−	0.999922i	$-0.496027\pi$
$37$	−6.00637	−	6.00637i	−0.987441	−	0.987441i	−0.999922	+	0.0124815i	$0.996027\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	5.17728i		−	0.808555i	−0.914636	−	0.404277i	$-0.867523\pi$
$41$		−	5.17728i		−	0.808555i	0.914636	−	0.404277i	$-0.132477\pi$
$42$		0			0
$43$	0.180211	+	0.180211i	0.0274819	+	0.0274819i	0.720714	−	0.693232i	$-0.243814\pi$
$43$	0.180211	+	0.180211i	0.0274819	+	0.0274819i	−0.693232	+	0.720714i	$0.743814\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	5.57785			0.813613			0.406806	−	0.913514i	$-0.366642\pi$
$47$	5.57785			0.813613			0.406806	+	0.913514i	$0.366642\pi$
$48$		0			0
$49$	−17.2875			−2.46965
$50$		0			0
$51$		0			0
$52$		0			0
$53$	0.146862	+	0.146862i	0.0201730	+	0.0201730i	0.717121	−	0.696948i	$-0.245459\pi$
$53$	0.146862	+	0.146862i	0.0201730	+	0.0201730i	−0.696948	+	0.717121i	$0.745459\pi$
$54$		0			0
$55$		−	3.47711i		−	0.468853i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	3.13789	+	3.13789i	0.408518	+	0.408518i	0.881221	−	0.472704i	$-0.156722\pi$
$59$	3.13789	+	3.13789i	0.408518	+	0.408518i	−0.472704	+	0.881221i	$0.656722\pi$
$60$		0			0
$61$	−1.87356	+	1.87356i	−0.239884	+	0.239884i	−0.816802	−	0.576918i	$-0.804255\pi$
$61$	−1.87356	+	1.87356i	−0.239884	+	0.239884i	0.576918	+	0.816802i	$0.304255\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	4.14544			0.514178
$66$		0			0
$67$	−8.02582	+	8.02582i	−0.980510	+	0.980510i	−0.999814	−	0.0193039i	$-0.993855\pi$
$67$	−8.02582	+	8.02582i	−0.980510	+	0.980510i	0.0193039	+	0.999814i	$0.493855\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	7.40711i		−	0.879062i	−0.898227	−	0.439531i	$-0.855145\pi$
$71$		−	7.40711i		−	0.879062i	0.898227	−	0.439531i	$-0.144855\pi$
$72$		0			0
$73$			11.1715i			1.30753i	0.756699	+	0.653763i	$0.226811\pi$
$73$			11.1715i			1.30753i	−0.756699	+	0.653763i	$0.773189\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−12.1170	+	12.1170i	−1.38086	+	1.38086i
$78$		0			0
$79$	−7.71759			−0.868297			−0.434148	−	0.900841i	$-0.642951\pi$
$79$	−7.71759			−0.868297			−0.434148	+	0.900841i	$0.642951\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−1.62773	+	1.62773i	−0.178666	+	0.178666i	−0.790774	−	0.612108i	$-0.790322\pi$
$83$	−1.62773	+	1.62773i	−0.178666	+	0.178666i	0.612108	+	0.790774i	$0.290322\pi$
$84$		0			0
$85$	−4.08507	−	4.08507i	−0.443088	−	0.443088i
$86$		0			0
$87$		0			0
$88$		0			0
$89$		−	9.54484i		−	1.01175i	−0.862607	−	0.505875i	$-0.831170\pi$
$89$		−	9.54484i		−	1.01175i	0.862607	−	0.505875i	$-0.168830\pi$
$90$		0			0
$91$	−14.4460	−	14.4460i	−1.51435	−	1.51435i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.39173			−0.142789
$96$		0			0
$97$	4.39090			0.445828			0.222914	−	0.974838i	$-0.428443\pi$
$97$	4.39090			0.445828			0.222914	+	0.974838i	$0.428443\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−10.1289	−	10.1289i	−1.00787	−	1.00787i	−0.999969	−	0.00789816i	$-0.997486\pi$
$101$	−10.1289	−	10.1289i	−1.00787	−	1.00787i	−0.00789816	−	0.999969i	$-0.502514\pi$
$102$		0			0
$103$			9.89734i			0.975214i	0.873063	+	0.487607i	$0.162130\pi$
$103$			9.89734i			0.975214i	−0.873063	+	0.487607i	$0.837870\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	5.45556	+	5.45556i	0.527409	+	0.527409i	0.919799	−	0.392390i	$-0.128352\pi$
$107$	5.45556	+	5.45556i	0.527409	+	0.527409i	−0.392390	+	0.919799i	$0.628352\pi$
$108$		0			0
$109$	−6.08344	+	6.08344i	−0.582687	+	0.582687i	−0.935641	−	0.352953i	$-0.885177\pi$
$109$	−6.08344	+	6.08344i	−0.582687	+	0.582687i	0.352953	+	0.935641i	$0.385177\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.60003			−0.714951			−0.357475	−	0.933923i	$-0.616362\pi$
$113$	−7.60003			−0.714951			−0.357475	+	0.933923i	$0.616362\pi$
$114$		0			0
$115$	−0.381514	+	0.381514i	−0.0355764	+	0.0355764i
$116$		0			0
$117$		0			0
$118$		0			0
$119$			28.4712i			2.60995i
$120$		0			0
$121$			1.09026i			0.0991146i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	0.707107	−	0.707107i	0.0632456	−	0.0632456i
$126$		0			0
$127$	17.0209			1.51036			0.755179	−	0.655519i	$-0.227550\pi$
$127$	17.0209			1.51036			0.755179	+	0.655519i	$0.227550\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−0.449639	+	0.449639i	−0.0392851	+	0.0392851i	−0.726476	−	0.687191i	$-0.758843\pi$
$131$	−0.449639	+	0.449639i	−0.0392851	+	0.0392851i	0.687191	+	0.726476i	$0.258843\pi$
$132$		0			0
$133$	4.84990	+	4.84990i	0.420540	+	0.420540i
$134$		0			0
$135$		0			0
$136$		0			0
$137$			9.66718i			0.825923i	0.910749	+	0.412961i	$0.135506\pi$
$137$			9.66718i			0.825923i	−0.910749	+	0.412961i	$0.864494\pi$
$138$		0			0
$139$	−8.08375	−	8.08375i	−0.685655	−	0.685655i	0.275614	−	0.961268i	$-0.411119\pi$
$139$	−8.08375	−	8.08375i	−0.685655	−	0.685655i	−0.961268	+	0.275614i	$0.911119\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−14.4141			−1.20537
$144$		0			0
$145$	9.63209			0.799902
$146$		0			0
$147$		0			0
$148$		0			0
$149$	5.28504	+	5.28504i	0.432968	+	0.432968i	0.889637	−	0.456669i	$-0.150958\pi$
$149$	5.28504	+	5.28504i	0.432968	+	0.432968i	−0.456669	+	0.889637i	$0.650958\pi$
$150$		0			0
$151$			10.8929i			0.886451i	0.896410	+	0.443226i	$0.146166\pi$
$151$			10.8929i			0.886451i	−0.896410	+	0.443226i	$0.853834\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−1.86093	−	1.86093i	−0.149473	−	0.149473i
$156$		0			0
$157$	7.56692	−	7.56692i	0.603906	−	0.603906i	−0.337441	−	0.941347i	$-0.609561\pi$
$157$	7.56692	−	7.56692i	0.603906	−	0.603906i	0.941347	+	0.337441i	$0.109561\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$	2.65899			0.209558
$162$		0			0
$163$	−8.60466	+	8.60466i	−0.673969	+	0.673969i	−0.958629	−	0.284660i	$-0.908119\pi$
$163$	−8.60466	+	8.60466i	−0.673969	+	0.673969i	0.284660	+	0.958629i	$0.408119\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$			9.53289i			0.737677i	0.929494	+	0.368838i	$0.120244\pi$
$167$			9.53289i			0.737677i	−0.929494	+	0.368838i	$0.879756\pi$
$168$		0			0
$169$		−	4.18465i		−	0.321896i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	7.20532	−	7.20532i	0.547810	−	0.547810i	−0.377997	−	0.925807i	$-0.623387\pi$
$173$	7.20532	−	7.20532i	0.547810	−	0.547810i	0.925807	+	0.377997i	$0.123387\pi$
$174$		0			0
$175$	−4.92824			−0.372540
$176$		0			0
$177$		0			0
$178$		0			0
$179$	4.48245	−	4.48245i	0.335034	−	0.335034i	−0.519461	−	0.854494i	$-0.673867\pi$
$179$	4.48245	−	4.48245i	0.335034	−	0.335034i	0.854494	+	0.519461i	$0.173867\pi$
$180$		0			0
$181$	16.3260	+	16.3260i	1.21350	+	1.21350i	0.969867	+	0.243636i	$0.0783401\pi$
$181$	16.3260	+	16.3260i	1.21350	+	1.21350i	0.243636	+	0.969867i	$0.421660\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$			8.49429i			0.624512i
$186$		0			0
$187$	14.2042	+	14.2042i	1.03872	+	1.03872i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	−6.00369			−0.434412			−0.217206	−	0.976126i	$-0.569694\pi$
$191$	−6.00369			−0.434412			−0.217206	+	0.976126i	$0.569694\pi$
$192$		0			0
$193$	15.6641			1.12753			0.563763	−	0.825937i	$-0.309353\pi$
$193$	15.6641			1.12753			0.563763	+	0.825937i	$0.309353\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	−10.5309	−	10.5309i	−0.750299	−	0.750299i	0.224236	−	0.974535i	$-0.428011\pi$
$197$	−10.5309	−	10.5309i	−0.750299	−	0.750299i	−0.974535	+	0.224236i	$0.928011\pi$
$198$		0			0
$199$			25.4667i			1.80528i	0.430393	+	0.902642i	$0.358375\pi$
$199$			25.4667i			1.80528i	−0.430393	+	0.902642i	$0.641625\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−33.5658	−	33.5658i	−2.35586	−	2.35586i
$204$		0			0
$205$	−3.66089	+	3.66089i	−0.255687	+	0.255687i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	4.83920			0.334735
$210$		0			0
$211$	1.23633	−	1.23633i	0.0851128	−	0.0851128i	−0.663269	−	0.748381i	$-0.730831\pi$
$211$	1.23633	−	1.23633i	0.0851128	−	0.0851128i	0.748381	+	0.663269i	$0.230831\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	0.254857i		−	0.0173811i
$216$		0			0
$217$			12.9699i			0.880454i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−16.9344	+	16.9344i	−1.13913	+	1.13913i
$222$		0			0
$223$	−12.3805			−0.829061			−0.414531	−	0.910035i	$-0.636054\pi$
$223$	−12.3805			−0.829061			−0.414531	+	0.910035i	$0.636054\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	2.27494	−	2.27494i	0.150993	−	0.150993i	−0.627568	−	0.778561i	$-0.715950\pi$
$227$	2.27494	−	2.27494i	0.150993	−	0.150993i	0.778561	+	0.627568i	$0.215950\pi$
$228$		0			0
$229$	−9.28770	−	9.28770i	−0.613749	−	0.613749i	0.330172	−	0.943921i	$-0.392893\pi$
$229$	−9.28770	−	9.28770i	−0.613749	−	0.613749i	−0.943921	+	0.330172i	$0.892893\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$		−	7.00671i		−	0.459025i	−0.973306	−	0.229512i	$-0.926287\pi$
$233$		−	7.00671i		−	0.459025i	0.973306	−	0.229512i	$-0.0737131\pi$
$234$		0			0
$235$	−3.94413	−	3.94413i	−0.257287	−	0.257287i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−20.2607			−1.31055			−0.655276	−	0.755389i	$-0.727448\pi$
$239$	−20.2607			−1.31055			−0.655276	+	0.755389i	$0.727448\pi$
$240$		0			0
$241$	−22.3806			−1.44166			−0.720831	−	0.693110i	$-0.756240\pi$
$241$	−22.3806			−1.44166			−0.720831	+	0.693110i	$0.756240\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	12.2241	+	12.2241i	0.780971	+	0.780971i
$246$		0			0
$247$			5.76934i			0.367094i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−15.8288	−	15.8288i	−0.999108	−	0.999108i	0.000891670	−	1.00000i	$-0.499716\pi$
$251$	−15.8288	−	15.8288i	−0.999108	−	0.999108i	−1.00000		0.000891670i	$0.999716\pi$
$252$		0			0
$253$	1.32657	−	1.32657i	0.0834004	−	0.0834004i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	21.4018			1.33501			0.667503	−	0.744607i	$-0.267363\pi$
$257$	21.4018			1.33501			0.667503	+	0.744607i	$0.267363\pi$
$258$		0			0
$259$	29.6008	−	29.6008i	1.83930	−	1.83930i
$260$		0			0
$261$		0			0
$262$		0			0
$263$			29.4331i			1.81493i	0.420133	+	0.907463i	$0.361984\pi$
$263$			29.4331i			1.81493i	−0.420133	+	0.907463i	$0.638016\pi$
$264$		0			0
$265$		−	0.207694i		−	0.0127585i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	−4.28110	+	4.28110i	−0.261023	+	0.261023i	−0.825470	−	0.564446i	$-0.809090\pi$
$269$	−4.28110	+	4.28110i	−0.261023	+	0.261023i	0.564446	+	0.825470i	$0.309090\pi$
$270$		0			0
$271$	14.0914			0.855989			0.427995	−	0.903781i	$-0.359220\pi$
$271$	14.0914			0.855989			0.427995	+	0.903781i	$0.359220\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−2.45868	+	2.45868i	−0.148264	+	0.148264i
$276$		0			0
$277$	−7.97589	−	7.97589i	−0.479225	−	0.479225i	0.425659	−	0.904884i	$-0.360042\pi$
$277$	−7.97589	−	7.97589i	−0.479225	−	0.479225i	−0.904884	+	0.425659i	$0.860042\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			3.95952i			0.236205i	0.993001	+	0.118103i	$0.0376812\pi$
$281$			3.95952i			0.236205i	−0.993001	+	0.118103i	$0.962319\pi$
$282$		0			0
$283$	14.8192	+	14.8192i	0.880912	+	0.880912i	0.993627	−	0.112716i	$-0.0359549\pi$
$283$	14.8192	+	14.8192i	0.880912	+	0.880912i	−0.112716	+	0.993627i	$0.535955\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	25.5148			1.50609
$288$		0			0
$289$	16.3756			0.963270
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−3.34880	−	3.34880i	−0.195639	−	0.195639i	0.602488	−	0.798128i	$-0.294176\pi$
$293$	−3.34880	−	3.34880i	−0.195639	−	0.195639i	−0.798128	+	0.602488i	$0.794176\pi$
$294$		0			0
$295$		−	4.43764i		−	0.258369i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	1.58154	+	1.58154i	0.0914630	+	0.0914630i
$300$		0			0
$301$	−0.888123	+	0.888123i	−0.0511906	+	0.0511906i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	2.64961			0.151716
$306$		0			0
$307$	−1.58868	+	1.58868i	−0.0906710	+	0.0906710i	−0.750987	−	0.660316i	$-0.770422\pi$
$307$	−1.58868	+	1.58868i	−0.0906710	+	0.0906710i	0.660316	+	0.750987i	$0.270422\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	9.40357i		−	0.533228i	−0.963803	−	0.266614i	$-0.914095\pi$
$311$		−	9.40357i		−	0.533228i	0.963803	−	0.266614i	$-0.0859049\pi$
$312$		0			0
$313$		−	23.8228i		−	1.34654i	−0.739396	−	0.673271i	$-0.764889\pi$
$313$		−	23.8228i		−	1.34654i	0.739396	−	0.673271i	$-0.235111\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	14.2338	−	14.2338i	0.799450	−	0.799450i	−0.183559	−	0.983009i	$-0.558762\pi$
$317$	14.2338	−	14.2338i	0.799450	−	0.799450i	0.983009	+	0.183559i	$0.0587618\pi$
$318$		0			0
$319$	−33.4918			−1.87518
$320$		0			0
$321$		0			0
$322$		0			0
$323$	5.68533	−	5.68533i	0.316340	−	0.316340i
$324$		0			0
$325$	−2.93127	−	2.93127i	−0.162597	−	0.162597i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			27.4890i			1.51552i
$330$		0			0
$331$	7.54040	+	7.54040i	0.414458	+	0.414458i	0.883288	−	0.468831i	$-0.155324\pi$
$331$	7.54040	+	7.54040i	0.414458	+	0.414458i	−0.468831	+	0.883288i	$0.655324\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	11.3502			0.620129
$336$		0			0
$337$	2.57374			0.140201			0.0701004	−	0.997540i	$-0.477668\pi$
$337$	2.57374			0.140201			0.0701004	+	0.997540i	$0.477668\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	6.47065	+	6.47065i	0.350405	+	0.350405i
$342$		0			0
$343$		−	50.6994i		−	2.73751i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−1.35992	−	1.35992i	−0.0730044	−	0.0730044i	0.669662	−	0.742666i	$-0.266439\pi$
$347$	−1.35992	−	1.35992i	−0.0730044	−	0.0730044i	−0.742666	+	0.669662i	$0.766439\pi$
$348$		0			0
$349$	1.76754	−	1.76754i	0.0946143	−	0.0946143i	−0.658215	−	0.752830i	$-0.728688\pi$
$349$	1.76754	−	1.76754i	0.0946143	−	0.0946143i	0.752830	+	0.658215i	$0.228688\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−16.5420			−0.880442			−0.440221	−	0.897889i	$-0.645100\pi$
$353$	−16.5420			−0.880442			−0.440221	+	0.897889i	$0.645100\pi$
$354$		0			0
$355$	−5.23762	+	5.23762i	−0.277984	+	0.277984i
$356$		0			0
$357$		0			0
$358$		0			0
$359$		−	8.25343i		−	0.435599i	−0.975993	−	0.217800i	$-0.930112\pi$
$359$		−	8.25343i		−	0.435599i	0.975993	−	0.217800i	$-0.0698880\pi$
$360$		0			0
$361$			17.0631i			0.898057i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	7.89945	−	7.89945i	0.413476	−	0.413476i
$366$		0			0
$367$	−6.13754			−0.320377			−0.160188	−	0.987086i	$-0.551210\pi$
$367$	−6.13754			−0.320377			−0.160188	+	0.987086i	$0.551210\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−0.723770	+	0.723770i	−0.0375763	+	0.0375763i
$372$		0			0
$373$	7.29129	+	7.29129i	0.377529	+	0.377529i	0.870210	−	0.492681i	$-0.163983\pi$
$373$	7.29129	+	7.29129i	0.377529	+	0.377529i	−0.492681	+	0.870210i	$0.663983\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$		−	39.9292i		−	2.05646i
$378$		0			0
$379$	−18.6355	−	18.6355i	−0.957239	−	0.957239i	0.0418834	−	0.999123i	$-0.486664\pi$
$379$	−18.6355	−	18.6355i	−0.957239	−	0.957239i	−0.999123	+	0.0418834i	$0.986664\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	5.03829			0.257445			0.128722	−	0.991681i	$-0.458912\pi$
$383$	5.03829			0.257445			0.128722	+	0.991681i	$0.458912\pi$
$384$		0			0
$385$	17.1360			0.873332
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−20.6292	−	20.6292i	−1.04594	−	1.04594i	−0.998893	−	0.0470501i	$-0.985018\pi$
$389$	−20.6292	−	20.6292i	−1.04594	−	1.04594i	−0.0470501	−	0.998893i	$-0.514982\pi$
$390$		0			0
$391$		−	3.11703i		−	0.157635i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	5.45716	+	5.45716i	0.274580	+	0.274580i
$396$		0			0
$397$	−19.8413	+	19.8413i	−0.995804	+	0.995804i	−0.999991	−	0.00418691i	$-0.998667\pi$
$397$	−19.8413	+	19.8413i	−0.995804	+	0.995804i	0.00418691	+	0.999991i	$0.498667\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	23.0994			1.15353			0.576764	−	0.816911i	$-0.304315\pi$
$401$	23.0994			1.15353			0.576764	+	0.816911i	$0.304315\pi$
$402$		0			0
$403$	−7.71437	+	7.71437i	−0.384280	+	0.384280i
$404$		0			0
$405$		0			0
$406$		0			0
$407$		−	29.5355i		−	1.46402i
$408$		0			0
$409$		−	18.1354i		−	0.896740i	−0.893848	−	0.448370i	$-0.852005\pi$
$409$		−	18.1354i		−	0.896740i	0.893848	−	0.448370i	$-0.147995\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−15.4642	+	15.4642i	−0.760946	+	0.760946i
$414$		0			0
$415$	2.30195			0.112998
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−2.63245	+	2.63245i	−0.128604	+	0.128604i	−0.768479	−	0.639875i	$-0.778986\pi$
$419$	−2.63245	+	2.63245i	−0.128604	+	0.128604i	0.639875	+	0.768479i	$0.278986\pi$
$420$		0			0
$421$	8.09456	+	8.09456i	0.394505	+	0.394505i	0.876290	−	0.481785i	$-0.160011\pi$
$421$	8.09456	+	8.09456i	0.394505	+	0.394505i	−0.481785	+	0.876290i	$0.660011\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			5.77716i			0.280234i
$426$		0			0
$427$	−9.23334	−	9.23334i	−0.446832	−	0.446832i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	−0.645299			−0.0310829			−0.0155415	−	0.999879i	$-0.504947\pi$
$431$	−0.645299			−0.0310829			−0.0155415	+	0.999879i	$0.504947\pi$
$432$		0			0
$433$	−16.0436			−0.771005			−0.385503	−	0.922707i	$-0.625972\pi$
$433$	−16.0436			−0.771005			−0.385503	+	0.922707i	$0.625972\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−0.530966	−	0.530966i	−0.0253996	−	0.0253996i
$438$		0			0
$439$			3.81375i			0.182020i	0.995850	+	0.0910101i	$0.0290096\pi$
$439$			3.81375i			0.182020i	−0.995850	+	0.0910101i	$0.970990\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−2.79697	−	2.79697i	−0.132888	−	0.132888i	0.637534	−	0.770422i	$-0.279955\pi$
$443$	−2.79697	−	2.79697i	−0.132888	−	0.132888i	−0.770422	+	0.637534i	$0.779955\pi$
$444$		0			0
$445$	−6.74922	+	6.74922i	−0.319944	+	0.319944i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	2.93839			0.138671			0.0693357	−	0.997593i	$-0.477912\pi$
$449$	2.93839			0.138671			0.0693357	+	0.997593i	$0.477912\pi$
$450$		0			0
$451$	12.7293	−	12.7293i	0.599399	−	0.599399i
$452$		0			0
$453$		0			0
$454$		0			0
$455$			20.4297i			0.957759i
$456$		0			0
$457$		−	29.3145i		−	1.37127i	−0.727944	−	0.685637i	$-0.759524\pi$
$457$		−	29.3145i		−	1.37127i	0.727944	−	0.685637i	$-0.240476\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−2.24108	+	2.24108i	−0.104377	+	0.104377i	−0.757367	−	0.652989i	$-0.773515\pi$
$461$	−2.24108	+	2.24108i	−0.104377	+	0.104377i	0.652989	+	0.757367i	$0.273515\pi$
$462$		0			0
$463$	42.7521			1.98686			0.993429	−	0.114452i	$-0.0365113\pi$
$463$	42.7521			1.98686			0.993429	+	0.114452i	$0.0365113\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−10.8766	+	10.8766i	−0.503309	+	0.503309i	−0.912465	−	0.409155i	$-0.865823\pi$
$467$	−10.8766	+	10.8766i	−0.503309	+	0.503309i	0.409155	+	0.912465i	$0.365823\pi$
$468$		0			0
$469$	−39.5531	−	39.5531i	−1.82639	−	1.82639i
$470$		0			0
$471$		0			0
$472$		0			0
$473$			0.886165i			0.0407459i
$474$		0			0
$475$	0.984104	+	0.984104i	0.0451538	+	0.0451538i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	16.5881			0.757932			0.378966	−	0.925411i	$-0.376280\pi$
$479$	16.5881			0.757932			0.378966	+	0.925411i	$0.376280\pi$
$480$		0			0
$481$	35.2125			1.60555
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−3.10483	−	3.10483i	−0.140983	−	0.140983i
$486$		0			0
$487$		−	12.8494i		−	0.582260i	−0.956683	−	0.291130i	$-0.905969\pi$
$487$		−	12.8494i		−	0.582260i	0.956683	−	0.291130i	$-0.0940313\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	24.2617	+	24.2617i	1.09492	+	1.09492i	0.994995	+	0.0999218i	$0.0318593\pi$
$491$	24.2617	+	24.2617i	1.09492	+	1.09492i	0.0999218	+	0.994995i	$0.468141\pi$
$492$		0			0
$493$	−39.3478	+	39.3478i	−1.77213	+	1.77213i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	36.5040			1.63743
$498$		0			0
$499$	−27.1785	+	27.1785i	−1.21667	+	1.21667i	−0.247886	+	0.968789i	$0.579736\pi$
$499$	−27.1785	+	27.1785i	−1.21667	+	1.21667i	−0.968789	+	0.247886i	$0.920264\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$		−	2.96618i		−	0.132255i	−0.997811	−	0.0661276i	$-0.978936\pi$
$503$		−	2.96618i		−	0.132255i	0.997811	−	0.0661276i	$-0.0210644\pi$
$504$		0			0
$505$			14.3245i			0.637431i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	3.38314	−	3.38314i	0.149955	−	0.149955i	−0.628143	−	0.778098i	$-0.716185\pi$
$509$	3.38314	−	3.38314i	0.149955	−	0.149955i	0.778098	+	0.628143i	$0.216185\pi$
$510$		0			0
$511$	−55.0559			−2.43553
$512$		0			0
$513$		0			0
$514$		0			0
$515$	6.99848	−	6.99848i	0.308390	−	0.308390i
$516$		0			0
$517$	13.7142	+	13.7142i	0.603148	+	0.603148i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			19.6801i			0.862200i	0.902304	+	0.431100i	$0.141874\pi$
$521$			19.6801i			0.862200i	−0.902304	+	0.431100i	$0.858126\pi$
$522$		0			0
$523$	20.0799	+	20.0799i	0.878031	+	0.878031i	0.993331	−	0.115300i	$-0.0367829\pi$
$523$	20.0799	+	20.0799i	0.878031	+	0.878031i	−0.115300	+	0.993331i	$0.536783\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	15.2041			0.662299
$528$		0			0
$529$	22.7089			0.987343
$530$		0			0
$531$		0			0
$532$		0			0
$533$	15.1760	+	15.1760i	0.657344	+	0.657344i
$534$		0			0
$535$		−	7.71532i		−	0.333562i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	−42.5046	−	42.5046i	−1.83080	−	1.83080i
$540$		0			0
$541$	13.3406	−	13.3406i	0.573555	−	0.573555i	−0.359565	−	0.933120i	$-0.617075\pi$
$541$	13.3406	−	13.3406i	0.573555	−	0.573555i	0.933120	+	0.359565i	$0.117075\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	8.60328			0.368524
$546$		0			0
$547$	28.1198	−	28.1198i	1.20231	−	1.20231i	0.228854	−	0.973461i	$-0.426502\pi$
$547$	28.1198	−	28.1198i	1.20231	−	1.20231i	0.973461	−	0.228854i	$-0.0734979\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$			13.4053i			0.571085i
$552$		0			0
$553$		−	38.0341i		−	1.61738i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	24.3140	−	24.3140i	1.03022	−	1.03022i	0.0306885	−	0.999529i	$-0.490230\pi$
$557$	24.3140	−	24.3140i	1.03022	−	1.03022i	0.999529	−	0.0306885i	$-0.00977000\pi$
$558$		0			0
$559$	−1.05649			−0.0446849
$560$		0			0
$561$		0			0
$562$		0			0
$563$	15.7337	−	15.7337i	0.663098	−	0.663098i	−0.293011	−	0.956109i	$-0.594657\pi$
$563$	15.7337	−	15.7337i	0.663098	−	0.663098i	0.956109	+	0.293011i	$0.0946574\pi$
$564$		0			0
$565$	5.37403	+	5.37403i	0.226087	+	0.226087i
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	5.11372i		−	0.214378i	−0.994239	−	0.107189i	$-0.965815\pi$
$569$		−	5.11372i		−	0.214378i	0.994239	−	0.107189i	$-0.0341851\pi$
$570$		0			0
$571$	−17.8788	−	17.8788i	−0.748203	−	0.748203i	0.225938	−	0.974142i	$-0.427455\pi$
$571$	−17.8788	−	17.8788i	−0.748203	−	0.748203i	−0.974142	+	0.225938i	$0.927455\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	0.539543			0.0225005
$576$		0			0
$577$	1.02300			0.0425882			0.0212941	−	0.999773i	$-0.493221\pi$
$577$	1.02300			0.0425882			0.0212941	+	0.999773i	$0.493221\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−8.02183	−	8.02183i	−0.332802	−	0.332802i
$582$		0			0
$583$			0.722174i			0.0299094i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	27.3088	+	27.3088i	1.12716	+	1.12716i	0.990637	+	0.136519i	$0.0435915\pi$
$587$	27.3088	+	27.3088i	1.12716	+	1.12716i	0.136519	+	0.990637i	$0.456409\pi$
$588$		0			0
$589$	2.58992	−	2.58992i	0.106716	−	0.106716i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−21.9492			−0.901346			−0.450673	−	0.892689i	$-0.648816\pi$
$593$	−21.9492			−0.901346			−0.450673	+	0.892689i	$0.648816\pi$
$594$		0			0
$595$	20.1322	−	20.1322i	0.825340	−	0.825340i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			0.802273i			0.0327800i	0.999866	+	0.0163900i	$0.00521733\pi$
$599$			0.802273i			0.0327800i	−0.999866	+	0.0163900i	$0.994783\pi$
$600$		0			0
$601$			23.1634i			0.944853i	0.881370	+	0.472427i	$0.156622\pi$
$601$			23.1634i			0.944853i	−0.881370	+	0.472427i	$0.843378\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	0.770930	−	0.770930i	0.0313428	−	0.0313428i
$606$		0			0
$607$	45.0996			1.83054			0.915268	−	0.402846i	$-0.131979\pi$
$607$	45.0996			1.83054			0.915268	+	0.402846i	$0.131979\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−16.3502	+	16.3502i	−0.661457	+	0.661457i
$612$		0			0
$613$	−7.01756	−	7.01756i	−0.283437	−	0.283437i	0.551041	−	0.834478i	$-0.314231\pi$
$613$	−7.01756	−	7.01756i	−0.283437	−	0.283437i	−0.834478	+	0.551041i	$0.814231\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$		−	1.17606i		−	0.0473465i	−0.999720	−	0.0236733i	$-0.992464\pi$
$617$		−	1.17606i		−	0.0473465i	0.999720	−	0.0236733i	$-0.00753614\pi$
$618$		0			0
$619$	23.8296	+	23.8296i	0.957794	+	0.957794i	0.999145	−	0.0413509i	$-0.0131661\pi$
$619$	23.8296	+	23.8296i	0.957794	+	0.957794i	−0.0413509	+	0.999145i	$0.513166\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	47.0392			1.88459
$624$		0			0
$625$	−1.00000			−0.0400000
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−34.6998	−	34.6998i	−1.38357	−	1.38357i
$630$		0			0
$631$			7.89946i			0.314473i	0.987561	+	0.157236i	$0.0502584\pi$
$631$			7.89946i			0.314473i	−0.987561	+	0.157236i	$0.949742\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−12.0356	−	12.0356i	−0.477617	−	0.477617i
$636$		0			0
$637$	50.6744	−	50.6744i	2.00779	−	2.00779i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	31.2986			1.23622			0.618111	−	0.786091i	$-0.287898\pi$
$641$	31.2986			1.23622			0.618111	+	0.786091i	$0.287898\pi$
$642$		0			0
$643$	−23.7961	+	23.7961i	−0.938426	+	0.938426i	−0.998211	−	0.0597853i	$-0.980958\pi$
$643$	−23.7961	+	23.7961i	−0.938426	+	0.938426i	0.0597853	+	0.998211i	$0.480958\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			36.5905i			1.43852i	0.694741	+	0.719260i	$0.255519\pi$
$647$			36.5905i			1.43852i	−0.694741	+	0.719260i	$0.744481\pi$
$648$		0			0
$649$			15.4301i			0.605686i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	26.9694	−	26.9694i	1.05539	−	1.05539i	0.0570212	−	0.998373i	$-0.481840\pi$
$653$	26.9694	−	26.9694i	1.05539	−	1.05539i	0.998373	−	0.0570212i	$-0.0181603\pi$
$654$		0			0
$655$	0.635885			0.0248461
$656$		0			0
$657$		0			0
$658$		0			0
$659$	12.4626	−	12.4626i	0.485476	−	0.485476i	−0.421399	−	0.906875i	$-0.638461\pi$
$659$	12.4626	−	12.4626i	0.485476	−	0.485476i	0.906875	+	0.421399i	$0.138461\pi$
$660$		0			0
$661$	15.0063	+	15.0063i	0.583676	+	0.583676i	0.935911	−	0.352236i	$-0.114578\pi$
$661$	15.0063	+	15.0063i	0.583676	+	0.583676i	−0.352236	+	0.935911i	$0.614578\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		−	6.85879i		−	0.265973i
$666$		0			0
$667$	3.67478	+	3.67478i	0.142288	+	0.142288i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−9.21297			−0.355663
$672$		0			0
$673$	11.9004			0.458727			0.229364	−	0.973341i	$-0.426335\pi$
$673$	11.9004			0.458727			0.229364	+	0.973341i	$0.426335\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	33.6457	+	33.6457i	1.29311	+	1.29311i	0.932855	+	0.360253i	$0.117310\pi$
$677$	33.6457	+	33.6457i	1.29311	+	1.29311i	0.360253	+	0.932855i	$0.382690\pi$
$678$		0			0
$679$			21.6394i			0.830443i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	18.3777	+	18.3777i	0.703203	+	0.703203i	0.965097	−	0.261894i	$-0.0843470\pi$
$683$	18.3777	+	18.3777i	0.703203	+	0.703203i	−0.261894	+	0.965097i	$0.584347\pi$
$684$		0			0
$685$	6.83573	−	6.83573i	0.261180	−	0.261180i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.860982			−0.0328008
$690$		0			0
$691$	−0.774737	+	0.774737i	−0.0294724	+	0.0294724i	−0.721689	−	0.692217i	$-0.756634\pi$
$691$	−0.774737	+	0.774737i	−0.0294724	+	0.0294724i	0.692217	+	0.721689i	$0.256634\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$			11.4322i			0.433646i
$696$		0			0
$697$		−	29.9100i		−	1.13292i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	−7.34576	+	7.34576i	−0.277445	+	0.277445i	−0.832088	−	0.554643i	$-0.812855\pi$
$701$	−7.34576	+	7.34576i	−0.277445	+	0.277445i	0.554643	+	0.832088i	$0.312855\pi$
$702$		0			0
$703$	−11.8218			−0.445867
$704$		0			0
$705$		0			0
$706$		0			0
$707$	49.9178	−	49.9178i	1.87735	−	1.87735i
$708$		0			0
$709$	2.80617	+	2.80617i	0.105388	+	0.105388i	0.757835	−	0.652447i	$-0.226257\pi$
$709$	2.80617	+	2.80617i	0.105388	+	0.105388i	−0.652447	+	0.757835i	$0.726257\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$		−	1.41994i		−	0.0531773i
$714$		0			0
$715$	10.1923	+	10.1923i	0.381171	+	0.381171i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−22.1785			−0.827118			−0.413559	−	0.910477i	$-0.635714\pi$
$719$	−22.1785			−0.827118			−0.413559	+	0.910477i	$0.635714\pi$
$720$		0			0
$721$	−48.7764			−1.81653
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−6.81092	−	6.81092i	−0.252951	−	0.252951i
$726$		0			0
$727$			22.0297i			0.817038i	0.912750	+	0.408519i	$0.133955\pi$
$727$			22.0297i			0.817038i	−0.912750	+	0.408519i	$0.866045\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	1.04111	+	1.04111i	0.0385068	+	0.0385068i
$732$		0			0
$733$	11.6342	−	11.6342i	0.429720	−	0.429720i	−0.458813	−	0.888533i	$-0.651725\pi$
$733$	11.6342	−	11.6342i	0.429720	−	0.429720i	0.888533	+	0.458813i	$0.151725\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	−39.4659			−1.45375
$738$		0			0
$739$	9.97647	−	9.97647i	0.366990	−	0.366990i	−0.499388	−	0.866378i	$-0.666442\pi$
$739$	9.97647	−	9.97647i	0.366990	−	0.366990i	0.866378	+	0.499388i	$0.166442\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	19.9003i		−	0.730073i	−0.930993	−	0.365036i	$-0.881057\pi$
$743$		−	19.9003i		−	0.730073i	0.930993	−	0.365036i	$-0.118943\pi$
$744$		0			0
$745$		−	7.47418i		−	0.273833i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−26.8863	+	26.8863i	−0.982403	+	0.982403i
$750$		0			0
$751$	41.5710			1.51695			0.758473	−	0.651704i	$-0.225946\pi$
$751$	41.5710			1.51695			0.758473	+	0.651704i	$0.225946\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	7.70244	−	7.70244i	0.280320	−	0.280320i
$756$		0			0
$757$	27.0159	+	27.0159i	0.981909	+	0.981909i	0.999839	−	0.0179307i	$-0.00570783\pi$
$757$	27.0159	+	27.0159i	0.981909	+	0.981909i	−0.0179307	+	0.999839i	$0.505708\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	15.2812i		−	0.553944i	−0.960878	−	0.276972i	$-0.910669\pi$
$761$		−	15.2812i		−	0.553944i	0.960878	−	0.276972i	$-0.0893309\pi$
$762$		0			0
$763$	−29.9806	−	29.9806i	−1.08537	−	1.08537i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−18.3960			−0.664240
$768$		0			0
$769$	−11.9990			−0.432694			−0.216347	−	0.976317i	$-0.569414\pi$
$769$	−11.9990			−0.432694			−0.216347	+	0.976317i	$0.569414\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	19.2957	+	19.2957i	0.694020	+	0.694020i	0.963114	−	0.269094i	$-0.0867243\pi$
$773$	19.2957	+	19.2957i	0.694020	+	0.694020i	−0.269094	+	0.963114i	$0.586724\pi$
$774$		0			0
$775$			2.63175i			0.0945353i
$776$		0			0
$777$		0			0
$778$		0			0
$779$	−5.09498	−	5.09498i	−0.182546	−	0.182546i
$780$		0			0
$781$	18.2118	−	18.2118i	0.651668	−	0.651668i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−10.7012			−0.381944
$786$		0			0
$787$	26.5668	−	26.5668i	0.947006	−	0.947006i	−0.0516592	−	0.998665i	$-0.516451\pi$
$787$	26.5668	−	26.5668i	0.947006	−	0.947006i	0.998665	+	0.0516592i	$0.0164510\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		−	37.4548i		−	1.33174i
$792$		0			0
$793$		−	10.9838i		−	0.390046i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	4.94914	−	4.94914i	0.175307	−	0.175307i	−0.613999	−	0.789307i	$-0.710440\pi$
$797$	4.94914	−	4.94914i	0.175307	−	0.175307i	0.789307	+	0.613999i	$0.210440\pi$
$798$		0			0
$799$	32.2241			1.14001
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−27.4672	+	27.4672i	−0.969297	+	0.969297i
$804$		0			0
$805$	−1.88019	−	1.88019i	−0.0662681	−	0.0662681i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			26.1203i			0.918339i	0.888349	+	0.459170i	$0.151853\pi$
$809$			26.1203i			0.918339i	−0.888349	+	0.459170i	$0.848147\pi$
$810$		0			0
$811$	−7.57185	−	7.57185i	−0.265884	−	0.265884i	0.561555	−	0.827439i	$-0.310203\pi$
$811$	−7.57185	−	7.57185i	−0.265884	−	0.265884i	−0.827439	+	0.561555i	$0.810203\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	12.1688			0.426255
$816$		0			0
$817$	0.354693			0.0124091
$818$		0			0
$819$		0			0
$820$		0			0
$821$	35.4689	+	35.4689i	1.23787	+	1.23787i	0.960869	+	0.277004i	$0.0893416\pi$
$821$	35.4689	+	35.4689i	1.23787	+	1.23787i	0.277004	+	0.960869i	$0.410658\pi$
$822$		0			0
$823$		−	18.2252i		−	0.635290i	−0.948210	−	0.317645i	$-0.897108\pi$
$823$		−	18.2252i		−	0.635290i	0.948210	−	0.317645i	$-0.102892\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	6.88954	+	6.88954i	0.239573	+	0.239573i	0.816673	−	0.577100i	$-0.195816\pi$
$827$	6.88954	+	6.88954i	0.239573	+	0.239573i	−0.577100	+	0.816673i	$0.695816\pi$
$828$		0			0
$829$	−7.07707	+	7.07707i	−0.245797	+	0.245797i	−0.819243	−	0.573446i	$-0.805606\pi$
$829$	−7.07707	+	7.07707i	−0.245797	+	0.245797i	0.573446	+	0.819243i	$0.305606\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	−99.8729			−3.46039
$834$		0			0
$835$	6.74077	−	6.74077i	0.233274	−	0.233274i
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	19.0722i		−	0.658445i	−0.944252	−	0.329223i	$-0.893213\pi$
$839$		−	19.0722i		−	0.658445i	0.944252	−	0.329223i	$-0.106787\pi$
$840$		0			0
$841$		−	63.7772i		−	2.19921i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−2.95899	+	2.95899i	−0.101793	+	0.101793i
$846$		0			0
$847$	−5.37306			−0.184621
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−3.24069	+	3.24069i	−0.111089	+	0.111089i
$852$		0			0
$853$	−19.9330	−	19.9330i	−0.682494	−	0.682494i	0.278068	−	0.960562i	$-0.410306\pi$
$853$	−19.9330	−	19.9330i	−0.682494	−	0.682494i	−0.960562	+	0.278068i	$0.910306\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			32.3955i			1.10661i	0.832979	+	0.553304i	$0.186633\pi$
$857$			32.3955i			1.10661i	−0.832979	+	0.553304i	$0.813367\pi$
$858$		0			0
$859$	−13.5311	−	13.5311i	−0.461676	−	0.461676i	0.437529	−	0.899205i	$-0.355854\pi$
$859$	−13.5311	−	13.5311i	−0.461676	−	0.461676i	−0.899205	+	0.437529i	$0.855854\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−47.8745			−1.62967			−0.814833	−	0.579695i	$-0.803172\pi$
$863$	−47.8745			−1.62967			−0.814833	+	0.579695i	$0.803172\pi$
$864$		0			0
$865$	−10.1899			−0.346466
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−18.9751	−	18.9751i	−0.643687	−	0.643687i
$870$		0			0
$871$		−	47.0516i		−	1.59428i
$872$		0			0
$873$		0			0
$874$		0			0
$875$	3.48479	+	3.48479i	0.117807	+	0.117807i
$876$		0			0
$877$	−2.37481	+	2.37481i	−0.0801917	+	0.0801917i	−0.746065	−	0.665873i	$-0.768059\pi$
$877$	−2.37481	+	2.37481i	−0.0801917	+	0.0801917i	0.665873	+	0.746065i	$0.268059\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−57.1626			−1.92586			−0.962928	−	0.269759i	$-0.913056\pi$
$881$	−57.1626			−1.92586			−0.962928	+	0.269759i	$0.913056\pi$
$882$		0			0
$883$	−12.2888	+	12.2888i	−0.413550	+	0.413550i	−0.882973	−	0.469423i	$-0.844462\pi$
$883$	−12.2888	+	12.2888i	−0.413550	+	0.413550i	0.469423	+	0.882973i	$0.344462\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$		−	20.1999i		−	0.678248i	−0.940742	−	0.339124i	$-0.889869\pi$
$887$		−	20.1999i		−	0.678248i	0.940742	−	0.339124i	$-0.110131\pi$
$888$		0			0
$889$			83.8829i			2.81334i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	5.48918	−	5.48918i	0.183688	−	0.183688i
$894$		0			0
$895$	−6.33914			−0.211894
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−17.9246	+	17.9246i	−0.597820	+	0.597820i
$900$		0			0
$901$	0.848444	+	0.848444i	0.0282658	+	0.0282658i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	23.0885i		−	0.767486i
$906$		0			0
$907$	−5.76047	−	5.76047i	−0.191273	−	0.191273i	0.604973	−	0.796246i	$-0.293184\pi$
$907$	−5.76047	−	5.76047i	−0.191273	−	0.191273i	−0.796246	+	0.604973i	$0.793184\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	16.5721			0.549057			0.274529	−	0.961579i	$-0.411478\pi$
$911$	16.5721			0.549057			0.274529	+	0.961579i	$0.411478\pi$
$912$		0			0
$913$	−8.00414			−0.264898
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−2.21593	−	2.21593i	−0.0731764	−	0.0731764i
$918$		0			0
$919$		−	13.0931i		−	0.431903i	−0.976404	−	0.215952i	$-0.930715\pi$
$919$		−	13.0931i		−	0.431903i	0.976404	−	0.215952i	$-0.0692853\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	21.7122	+	21.7122i	0.714666	+	0.714666i
$924$		0			0
$925$	6.00637	−	6.00637i	0.197488	−	0.197488i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−17.2834			−0.567050			−0.283525	−	0.958965i	$-0.591504\pi$
$929$	−17.2834			−0.567050			−0.283525	+	0.958965i	$0.591504\pi$
$930$		0			0
$931$	−17.0127	+	17.0127i	−0.557570	+	0.557570i
$932$		0			0
$933$		0			0
$934$		0			0
$935$		−	20.0878i		−	0.656941i
$936$		0			0
$937$			30.8187i			1.00680i	0.864053	+	0.503401i	$0.167918\pi$
$937$			30.8187i			1.00680i	−0.864053	+	0.503401i	$0.832082\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	12.2305	−	12.2305i	0.398704	−	0.398704i	−0.479072	−	0.877776i	$-0.659027\pi$
$941$	12.2305	−	12.2305i	0.398704	−	0.398704i	0.877776	+	0.479072i	$0.159027\pi$
$942$		0			0
$943$	−2.79336			−0.0909643
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.880251	+	0.880251i	−0.0286043	+	0.0286043i	−0.721264	−	0.692660i	$-0.756439\pi$
$947$	−0.880251	+	0.880251i	−0.0286043	+	0.0286043i	0.692660	+	0.721264i	$0.256439\pi$
$948$		0			0
$949$	−32.7467	−	32.7467i	−1.06300	−	1.06300i
$950$		0			0
$951$		0			0
$952$		0			0
$953$		−	53.7271i		−	1.74039i	−0.492705	−	0.870196i	$-0.663992\pi$
$953$		−	53.7271i		−	1.74039i	0.492705	−	0.870196i	$-0.336008\pi$
$954$		0			0
$955$	4.24525	+	4.24525i	0.137373	+	0.137373i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−47.6422			−1.53845
$960$		0			0
$961$	−24.0739			−0.776577
$962$		0			0
$963$		0			0
$964$		0			0
$965$	−11.0762	−	11.0762i	−0.356555	−	0.356555i
$966$		0			0
$967$			46.3769i			1.49138i	0.666294	+	0.745690i	$0.267880\pi$
$967$			46.3769i			1.49138i	−0.666294	+	0.745690i	$0.732120\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−28.7246	−	28.7246i	−0.921815	−	0.921815i	0.0753424	−	0.997158i	$-0.475995\pi$
$971$	−28.7246	−	28.7246i	−0.921815	−	0.921815i	−0.997158	+	0.0753424i	$0.975995\pi$
$972$		0			0
$973$	39.8387	−	39.8387i	1.27717	−	1.27717i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	60.6613			1.94073			0.970364	−	0.241648i	$-0.0776878\pi$
$977$	60.6613			1.94073			0.970364	+	0.241648i	$0.0776878\pi$
$978$		0			0
$979$	23.4677	−	23.4677i	0.750032	−	0.750032i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			9.61371i			0.306630i	0.988177	+	0.153315i	$0.0489949\pi$
$983$			9.61371i			0.306630i	−0.988177	+	0.153315i	$0.951005\pi$
$984$		0			0
$985$			14.8930i			0.474531i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	0.0972316	−	0.0972316i	0.00309178	−	0.00309178i
$990$		0			0
$991$	28.3622			0.900956			0.450478	−	0.892788i	$-0.351254\pi$
$991$	28.3622			0.900956			0.450478	+	0.892788i	$0.351254\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	18.0076	−	18.0076i	0.570881	−	0.570881i
$996$		0			0
$997$	17.9440	+	17.9440i	0.568293	+	0.568293i	0.931650	−	0.363357i	$-0.118370\pi$
$997$	17.9440	+	17.9440i	0.568293	+	0.568293i	−0.363357	+	0.931650i	$0.618370\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.t.d.2161.5		20
3.2	odd	2		960.2.s.c.241.5		20
4.3	odd	2		720.2.t.d.181.3		20
12.11	even	2		240.2.s.c.181.8	yes	20
16.3	odd	4		720.2.t.d.541.3		20
16.13	even	4	inner	2880.2.t.d.721.1		20
24.5	odd	2		1920.2.s.f.481.10		20
24.11	even	2		1920.2.s.e.481.1		20
48.5	odd	4		1920.2.s.f.1441.6		20
48.11	even	4		1920.2.s.e.1441.5		20
48.29	odd	4		960.2.s.c.721.1		20
48.35	even	4		240.2.s.c.61.8	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
240.2.s.c.61.8	✓	20	48.35	even	4
240.2.s.c.181.8	yes	20	12.11	even	2
720.2.t.d.181.3		20	4.3	odd	2
720.2.t.d.541.3		20	16.3	odd	4
960.2.s.c.241.5		20	3.2	odd	2
960.2.s.c.721.1		20	48.29	odd	4
1920.2.s.e.481.1		20	24.11	even	2
1920.2.s.e.1441.5		20	48.11	even	4
1920.2.s.f.481.10		20	24.5	odd	2
1920.2.s.f.1441.6		20	48.5	odd	4
2880.2.t.d.721.1		20	16.13	even	4	inner
2880.2.t.d.2161.5		20	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.t (of order \(4\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+(-0.707107 - 0.707107i) q^{5} +4.92824i q^{7} +O(q^{10})\)	\(q+(-0.707107 - 0.707107i) q^{5} +4.92824i q^{7} +(2.45868 + 2.45868i) q^{11} +(-2.93127 + 2.93127i) q^{13} +5.77716 q^{17} +(0.984104 - 0.984104i) q^{19} -0.539543i q^{23} +1.00000i q^{25} +(-6.81092 + 6.81092i) q^{29} +2.63175 q^{31} +(3.48479 - 3.48479i) q^{35} +(-6.00637 - 6.00637i) q^{37} -5.17728i q^{41} +(0.180211 + 0.180211i) q^{43} +5.57785 q^{47} -17.2875 q^{49} +(0.146862 + 0.146862i) q^{53} -3.47711i q^{55} +(3.13789 + 3.13789i) q^{59} +(-1.87356 + 1.87356i) q^{61} +4.14544 q^{65} +(-8.02582 + 8.02582i) q^{67} -7.40711i q^{71} +11.1715i q^{73} +(-12.1170 + 12.1170i) q^{77} -7.71759 q^{79} +(-1.62773 + 1.62773i) q^{83} +(-4.08507 - 4.08507i) q^{85} -9.54484i q^{89} +(-14.4460 - 14.4460i) q^{91} -1.39173 q^{95} +4.39090 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q+O(q^{10}) \) 20 * q	\( 20 q + 8 q^{11} + 24 q^{17} + 4 q^{19} - 16 q^{29} + 16 q^{37} + 8 q^{43} - 52 q^{49} + 16 q^{53} - 16 q^{59} - 4 q^{61} + 8 q^{67} + 40 q^{77} - 56 q^{79} - 48 q^{83} + 4 q^{85} + 8 q^{91} + 56 q^{97}+O(q^{100}) \) 20 * q + 8 * q^11 + 24 * q^17 + 4 * q^19 - 16 * q^29 + 16 * q^37 + 8 * q^43 - 52 * q^49 + 16 * q^53 - 16 * q^59 - 4 * q^61 + 8 * q^67 + 40 * q^77 - 56 * q^79 - 48 * q^83 + 4 * q^85 + 8 * q^91 + 56 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more