LMFDB - Embedded newform 2880.2.bj.c.1313.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(737,2880)]

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, 2, 2, 1]))

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

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

chi := DirichletCharacter("2880.737");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.bj (of order $4$, degree $2$, 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:	$8$
Relative dimension:	$4$ over $\Q(i)$
Coefficient field:	$\Q(\zeta_{24})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - x^{4} + 1 $ x^8 - x^4 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{6} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{4}]$

Embedding invariants

Embedding label			1313.4
Root			$-0.965926 - 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2880.1313
Dual form			2880.2.bj.c.737.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(2.22474 - 0.224745i) q^{5} +(1.41421 - 1.41421i) q^{7} +O(q^{10})$	$q+(2.22474 - 0.224745i) q^{5} +(1.41421 - 1.41421i) q^{7} +3.46410 q^{11} +(0.317837 - 0.317837i) q^{13} +(-2.04989 - 2.04989i) q^{17} +2.89898 q^{19} +(-0.449490 + 0.449490i) q^{23} +(4.89898 - 1.00000i) q^{25} -3.55051i q^{29} +4.09978 q^{31} +(2.82843 - 3.46410i) q^{35} +(-5.97469 - 5.97469i) q^{37} -1.41421i q^{41} +(-2.89898 + 2.89898i) q^{43} +(6.44949 + 6.44949i) q^{47} +3.00000i q^{49} +(2.89898 + 2.89898i) q^{53} +(7.70674 - 0.778539i) q^{55} +6.29253i q^{59} -13.2207i q^{61} +(0.635674 - 0.778539i) q^{65} +12.8990i q^{71} +(-7.89898 - 7.89898i) q^{73} +(4.89898 - 4.89898i) q^{77} -14.1421i q^{79} +(-9.12096 - 9.12096i) q^{83} +(-5.02118 - 4.09978i) q^{85} +4.24264 q^{89} -0.898979i q^{91} +(6.44949 - 0.651531i) q^{95} +(-7.89898 + 7.89898i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{5}+O(q^{10}) $ 8 * q + 8 * q^5	$ 8 q + 8 q^{5} - 16 q^{19} + 16 q^{23} + 16 q^{43} + 32 q^{47} - 16 q^{53} - 24 q^{73} + 32 q^{95} - 24 q^{97}+O(q^{100}) $ 8 * q + 8 * q^5 - 16 * q^19 + 16 * q^23 + 16 * q^43 + 32 * q^47 - 16 * q^53 - 24 * q^73 + 32 * q^95 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$e\left(\frac{3}{4}\right)$	$-1$	$-1$	$1$

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
$5$	2.22474	−	0.224745i	0.994936	−	0.100509i
$5$	2.22474	−	0.224745i	0.994936	−	0.100509i
$6$		0			0
$7$	1.41421	−	1.41421i	0.534522	−	0.534522i	−0.387392	−	0.921915i	$-0.626624\pi$
$7$	1.41421	−	1.41421i	0.534522	−	0.534522i	0.921915	+	0.387392i	$0.126624\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.46410			1.04447			0.522233	−	0.852803i	$-0.325099\pi$
$11$	3.46410			1.04447			0.522233	+	0.852803i	$0.325099\pi$
$12$		0			0
$13$	0.317837	−	0.317837i	0.0881522	−	0.0881522i	−0.661656	−	0.749808i	$-0.730146\pi$
$13$	0.317837	−	0.317837i	0.0881522	−	0.0881522i	0.749808	+	0.661656i	$0.230146\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−2.04989	−	2.04989i	−0.497171	−	0.497171i	0.413385	−	0.910556i	$-0.364346\pi$
$17$	−2.04989	−	2.04989i	−0.497171	−	0.497171i	−0.910556	+	0.413385i	$0.864346\pi$
$18$		0			0
$19$	2.89898			0.665072			0.332536	−	0.943091i	$-0.392096\pi$
$19$	2.89898			0.665072			0.332536	+	0.943091i	$0.392096\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−0.449490	+	0.449490i	−0.0937251	+	0.0937251i	−0.752415	−	0.658690i	$-0.771111\pi$
$23$	−0.449490	+	0.449490i	−0.0937251	+	0.0937251i	0.658690	+	0.752415i	$0.271111\pi$
$24$		0			0
$25$	4.89898	−	1.00000i	0.979796	−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	3.55051i		−	0.659313i	−0.944101	−	0.329657i	$-0.893067\pi$
$29$		−	3.55051i		−	0.659313i	0.944101	−	0.329657i	$-0.106933\pi$
$30$		0			0
$31$	4.09978			0.736342			0.368171	−	0.929758i	$-0.379984\pi$
$31$	4.09978			0.736342			0.368171	+	0.929758i	$0.379984\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	2.82843	−	3.46410i	0.478091	−	0.585540i
$36$		0			0
$37$	−5.97469	−	5.97469i	−0.982233	−	0.982233i	0.0176117	−	0.999845i	$-0.494394\pi$
$37$	−5.97469	−	5.97469i	−0.982233	−	0.982233i	−0.999845	+	0.0176117i	$0.994394\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	1.41421i		−	0.220863i	−0.993884	−	0.110432i	$-0.964777\pi$
$41$		−	1.41421i		−	0.220863i	0.993884	−	0.110432i	$-0.0352233\pi$
$42$		0			0
$43$	−2.89898	+	2.89898i	−0.442090	+	0.442090i	−0.892714	−	0.450624i	$-0.851202\pi$
$43$	−2.89898	+	2.89898i	−0.442090	+	0.442090i	0.450624	+	0.892714i	$0.351202\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	6.44949	+	6.44949i	0.940755	+	0.940755i	0.998341	−	0.0575858i	$-0.0183403\pi$
$47$	6.44949	+	6.44949i	0.940755	+	0.940755i	−0.0575858	+	0.998341i	$0.518340\pi$
$48$		0			0
$49$			3.00000i			0.428571i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	2.89898	+	2.89898i	0.398205	+	0.398205i	0.877600	−	0.479394i	$-0.159144\pi$
$53$	2.89898	+	2.89898i	0.398205	+	0.398205i	−0.479394	+	0.877600i	$0.659144\pi$
$54$		0			0
$55$	7.70674	−	0.778539i	1.03918	−	0.104978i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			6.29253i			0.819217i	0.912261	+	0.409609i	$0.134335\pi$
$59$			6.29253i			0.819217i	−0.912261	+	0.409609i	$0.865665\pi$
$60$		0			0
$61$		−	13.2207i		−	1.69274i	−0.532594	−	0.846371i	$-0.678783\pi$
$61$		−	13.2207i		−	1.69274i	0.532594	−	0.846371i	$-0.321217\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	0.635674	−	0.778539i	0.0788457	−	0.0965659i
$66$		0			0
$67$		0			0		0.707107	−	0.707107i	$-0.250000\pi$
$67$		0			0		−0.707107	+	0.707107i	$0.750000\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			12.8990i			1.53083i	0.643539	+	0.765414i	$0.277466\pi$
$71$			12.8990i			1.53083i	−0.643539	+	0.765414i	$0.722534\pi$
$72$		0			0
$73$	−7.89898	−	7.89898i	−0.924506	−	0.924506i	0.0728382	−	0.997344i	$-0.476794\pi$
$73$	−7.89898	−	7.89898i	−0.924506	−	0.924506i	−0.997344	+	0.0728382i	$0.976794\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	4.89898	−	4.89898i	0.558291	−	0.558291i
$78$		0			0
$79$		−	14.1421i		−	1.59111i	−0.605878	−	0.795557i	$-0.707178\pi$
$79$		−	14.1421i		−	1.59111i	0.605878	−	0.795557i	$-0.292822\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−9.12096	−	9.12096i	−1.00115	−	1.00115i	−0.999999	−	0.00115564i	$-0.999632\pi$
$83$	−9.12096	−	9.12096i	−1.00115	−	1.00115i	−0.00115564	−	0.999999i	$-0.500368\pi$
$84$		0			0
$85$	−5.02118	−	4.09978i	−0.544623	−	0.444683i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	4.24264			0.449719			0.224860	−	0.974391i	$-0.427808\pi$
$89$	4.24264			0.449719			0.224860	+	0.974391i	$0.427808\pi$
$90$		0			0
$91$		−	0.898979i		−	0.0942387i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	6.44949	−	0.651531i	0.661704	−	0.0668456i
$96$		0			0
$97$	−7.89898	+	7.89898i	−0.802020	+	0.802020i	−0.983411	−	0.181391i	$-0.941940\pi$
$97$	−7.89898	+	7.89898i	−0.802020	+	0.802020i	0.181391	+	0.983411i	$0.441940\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	9.34847			0.930207			0.465104	−	0.885256i	$-0.346017\pi$
$101$	9.34847			0.930207			0.465104	+	0.885256i	$0.346017\pi$
$102$		0			0
$103$	8.97809	+	8.97809i	0.884638	+	0.884638i	0.994002	−	0.109364i	$-0.0348815\pi$
$103$	8.97809	+	8.97809i	0.884638	+	0.884638i	−0.109364	+	0.994002i	$0.534881\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−9.12096	+	9.12096i	−0.881756	+	0.881756i	−0.993713	−	0.111957i	$-0.964288\pi$
$107$	−9.12096	+	9.12096i	−0.881756	+	0.881756i	0.111957	+	0.993713i	$0.464288\pi$
$108$		0			0
$109$	−5.02118			−0.480942			−0.240471	−	0.970656i	$-0.577302\pi$
$109$	−5.02118			−0.480942			−0.240471	+	0.970656i	$0.577302\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−5.02118	+	5.02118i	−0.472353	+	0.472353i	−0.902675	−	0.430322i	$-0.858400\pi$
$113$	−5.02118	+	5.02118i	−0.472353	+	0.472353i	0.430322	+	0.902675i	$0.358400\pi$
$114$		0			0
$115$	−0.898979	+	1.10102i	−0.0838303	+	0.102671i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	−5.79796			−0.531498
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	10.6742	−	3.32577i	0.954733	−	0.297465i
$126$		0			0
$127$	11.8065	−	11.8065i	1.04766	−	1.04766i	0.0488531	−	0.998806i	$-0.484443\pi$
$127$	11.8065	−	11.8065i	1.04766	−	1.04766i	0.998806	−	0.0488531i	$-0.0155566\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−4.73545			−0.413738			−0.206869	−	0.978369i	$-0.566327\pi$
$131$	−4.73545			−0.413738			−0.206869	+	0.978369i	$0.566327\pi$
$132$		0			0
$133$	4.09978	−	4.09978i	0.355496	−	0.355496i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	13.2207	+	13.2207i	1.12952	+	1.12952i	0.990255	+	0.139269i	$0.0444752\pi$
$137$	13.2207	+	13.2207i	1.12952	+	1.12952i	0.139269	+	0.990255i	$0.455525\pi$
$138$		0			0
$139$	−11.7980			−1.00069			−0.500345	−	0.865826i	$-0.666793\pi$
$139$	−11.7980			−1.00069			−0.500345	+	0.865826i	$0.666793\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	1.10102	−	1.10102i	0.0920720	−	0.0920720i
$144$		0			0
$145$	−0.797959	−	7.89898i	−0.0662669	−	0.655975i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	16.4495i		−	1.34759i	−0.738916	−	0.673797i	$-0.764662\pi$
$149$		−	16.4495i		−	1.34759i	0.738916	−	0.673797i	$-0.235338\pi$
$150$		0			0
$151$	18.2419			1.48451			0.742253	−	0.670120i	$-0.233757\pi$
$151$	18.2419			1.48451			0.742253	+	0.670120i	$0.233757\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	9.12096	−	0.921404i	0.732613	−	0.0740089i
$156$		0			0
$157$	−12.2672	−	12.2672i	−0.979031	−	0.979031i	0.0207539	−	0.999785i	$-0.493393\pi$
$157$	−12.2672	−	12.2672i	−0.979031	−	0.979031i	−0.999785	+	0.0207539i	$0.993393\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			1.27135i			0.100196i
$162$		0			0
$163$	15.7980	−	15.7980i	1.23739	−	1.23739i	0.276328	−	0.961063i	$-0.410882\pi$
$163$	15.7980	−	15.7980i	1.23739	−	1.23739i	0.961063	−	0.276328i	$-0.0891177\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	8.44949	+	8.44949i	0.653841	+	0.653841i	0.953916	−	0.300075i	$-0.0970116\pi$
$167$	8.44949	+	8.44949i	0.653841	+	0.653841i	−0.300075	+	0.953916i	$0.597012\pi$
$168$		0			0
$169$			12.7980i			0.984458i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	6.44949	+	6.44949i	0.490346	+	0.490346i	0.908415	−	0.418069i	$-0.137293\pi$
$173$	6.44949	+	6.44949i	0.490346	+	0.490346i	−0.418069	+	0.908415i	$0.637293\pi$
$174$		0			0
$175$	5.51399	−	8.34242i	0.416818	−	0.630627i
$176$		0			0
$177$		0			0
$178$		0			0
$179$			7.84961i			0.586707i	0.956004	+	0.293354i	$0.0947713\pi$
$179$			7.84961i			0.586707i	−0.956004	+	0.293354i	$0.905229\pi$
$180$		0			0
$181$		−	13.2207i		−	0.982689i	−0.870965	−	0.491345i	$-0.836506\pi$
$181$		−	13.2207i		−	0.982689i	0.870965	−	0.491345i	$-0.163494\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−14.6349	−	11.9494i	−1.07598	−	0.878536i
$186$		0			0
$187$	−7.10102	−	7.10102i	−0.519278	−	0.519278i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			5.79796i			0.419526i	0.977752	+	0.209763i	$0.0672692\pi$
$191$			5.79796i			0.419526i	−0.977752	+	0.209763i	$0.932731\pi$
$192$		0			0
$193$	0.797959	+	0.797959i	0.0574383	+	0.0574383i	0.735242	−	0.677804i	$-0.237068\pi$
$193$	0.797959	+	0.797959i	0.0574383	+	0.0574383i	−0.677804	+	0.735242i	$0.737068\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	1.55051	−	1.55051i	0.110469	−	0.110469i	−0.649712	−	0.760181i	$-0.725110\pi$
$197$	1.55051	−	1.55051i	0.110469	−	0.110469i	0.760181	+	0.649712i	$0.225110\pi$
$198$		0			0
$199$			8.19955i			0.581251i	0.956837	+	0.290625i	$0.0938633\pi$
$199$			8.19955i			0.581251i	−0.956837	+	0.290625i	$0.906137\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−5.02118	−	5.02118i	−0.352418	−	0.352418i
$204$		0			0
$205$	−0.317837	−	3.14626i	−0.0221987	−	0.219745i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	10.0424			0.694645
$210$		0			0
$211$			7.79796i			0.536834i	0.963303	+	0.268417i	$0.0865004\pi$
$211$			7.79796i			0.536834i	−0.963303	+	0.268417i	$0.913500\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	−5.79796	+	7.10102i	−0.395418	+	0.484286i
$216$		0			0
$217$	5.79796	−	5.79796i	0.393591	−	0.393591i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−1.30306			−0.0876534
$222$		0			0
$223$	−9.89949	−	9.89949i	−0.662919	−	0.662919i	0.293148	−	0.956067i	$-0.405297\pi$
$223$	−9.89949	−	9.89949i	−0.662919	−	0.662919i	−0.956067	+	0.293148i	$0.905297\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	4.09978	−	4.09978i	0.272112	−	0.272112i	−0.557838	−	0.829950i	$-0.688369\pi$
$227$	4.09978	−	4.09978i	0.272112	−	0.272112i	0.829950	+	0.557838i	$0.188369\pi$
$228$		0			0
$229$		0			0			−	1.00000i	$-0.5\pi$
$229$		0			0				1.00000i	$0.5\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−16.1920	+	16.1920i	−1.06077	+	1.06077i	−0.0627452	+	0.998030i	$0.519986\pi$
$233$	−16.1920	+	16.1920i	−1.06077	+	1.06077i	−0.998030	+	0.0627452i	$0.980014\pi$
$234$		0			0
$235$	15.7980	+	12.8990i	1.03055	+	0.841437i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	24.4949			1.58444			0.792222	−	0.610234i	$-0.208924\pi$
$239$	24.4949			1.58444			0.792222	+	0.610234i	$0.208924\pi$
$240$		0			0
$241$	15.7980			1.01764			0.508818	−	0.860874i	$-0.330083\pi$
$241$	15.7980			1.01764			0.508818	+	0.860874i	$0.330083\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	0.674235	+	6.67423i	0.0430753	+	0.426401i
$246$		0			0
$247$	0.921404	−	0.921404i	0.0586275	−	0.0586275i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	17.6062			1.11130			0.555648	−	0.831418i	$-0.312470\pi$
$251$	17.6062			1.11130			0.555648	+	0.831418i	$0.312470\pi$
$252$		0			0
$253$	−1.55708	+	1.55708i	−0.0978927	+	0.0978927i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	5.02118	+	5.02118i	0.313213	+	0.313213i	0.846153	−	0.532940i	$-0.178913\pi$
$257$	5.02118	+	5.02118i	0.313213	+	0.313213i	−0.532940	+	0.846153i	$0.678913\pi$
$258$		0			0
$259$	−16.8990			−1.05005
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−12.4495	+	12.4495i	−0.767668	+	0.767668i	−0.977696	−	0.210027i	$-0.932645\pi$
$263$	−12.4495	+	12.4495i	−0.767668	+	0.767668i	0.210027	+	0.977696i	$0.432645\pi$
$264$		0			0
$265$	7.10102	+	5.79796i	0.436212	+	0.356166i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			16.4495i			1.00294i	0.865174	+	0.501472i	$0.167208\pi$
$269$			16.4495i			1.00294i	−0.865174	+	0.501472i	$0.832792\pi$
$270$		0			0
$271$	−10.0424			−0.610030			−0.305015	−	0.952348i	$-0.598661\pi$
$271$	−10.0424			−0.610030			−0.305015	+	0.952348i	$0.598661\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	16.9706	−	3.46410i	1.02336	−	0.208893i
$276$		0			0
$277$	−3.14626	−	3.14626i	−0.189041	−	0.189041i	0.606241	−	0.795281i	$-0.292677\pi$
$277$	−3.14626	−	3.14626i	−0.189041	−	0.189041i	−0.795281	+	0.606241i	$0.792677\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	27.8557i		−	1.66173i	−0.556474	−	0.830865i	$-0.687846\pi$
$281$		−	27.8557i		−	1.66173i	0.556474	−	0.830865i	$-0.312154\pi$
$282$		0			0
$283$	12.8990	−	12.8990i	0.766765	−	0.766765i	−0.210771	−	0.977536i	$-0.567597\pi$
$283$	12.8990	−	12.8990i	0.766765	−	0.766765i	0.977536	+	0.210771i	$0.0675974\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.00000	−	2.00000i	−0.118056	−	0.118056i
$288$		0			0
$289$		−	8.59592i		−	0.505642i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−15.3485	−	15.3485i	−0.896667	−	0.896667i	0.0984726	−	0.995140i	$-0.468604\pi$
$293$	−15.3485	−	15.3485i	−0.896667	−	0.896667i	−0.995140	+	0.0984726i	$0.968604\pi$
$294$		0			0
$295$	1.41421	+	13.9993i	0.0823387	+	0.815069i
$296$		0			0
$297$		0			0
$298$		0			0
$299$			0.285729i			0.0165241i
$300$		0			0
$301$			8.19955i			0.472614i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	−2.97129	−	29.4128i	−0.170136	−	1.68417i
$306$		0			0
$307$	−2.89898	−	2.89898i	−0.165453	−	0.165453i	0.619524	−	0.784978i	$-0.287326\pi$
$307$	−2.89898	−	2.89898i	−0.165453	−	0.165453i	−0.784978	+	0.619524i	$0.787326\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	7.10102i		−	0.402662i	−0.979523	−	0.201331i	$-0.935473\pi$
$311$		−	7.10102i		−	0.402662i	0.979523	−	0.201331i	$-0.0645267\pi$
$312$		0			0
$313$	−5.00000	−	5.00000i	−0.282617	−	0.282617i	0.551535	−	0.834152i	$-0.314042\pi$
$313$	−5.00000	−	5.00000i	−0.282617	−	0.282617i	−0.834152	+	0.551535i	$0.814042\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	5.10102	−	5.10102i	0.286502	−	0.286502i	−0.549193	−	0.835695i	$-0.685065\pi$
$317$	5.10102	−	5.10102i	0.286502	−	0.286502i	0.835695	+	0.549193i	$0.185065\pi$
$318$		0			0
$319$		−	12.2993i		−	0.688630i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−5.94258	−	5.94258i	−0.330654	−	0.330654i
$324$		0			0
$325$	1.23924	−	1.87492i	0.0687407	−	0.104002i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	18.2419			1.00571
$330$		0			0
$331$			22.8990i			1.25864i	0.777146	+	0.629321i	$0.216667\pi$
$331$			22.8990i			1.25864i	−0.777146	+	0.629321i	$0.783333\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−2.10102	+	2.10102i	−0.114450	+	0.114450i	−0.762012	−	0.647562i	$-0.775788\pi$
$337$	−2.10102	+	2.10102i	−0.114450	+	0.114450i	0.647562	+	0.762012i	$0.275788\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	14.2020			0.769084
$342$		0			0
$343$	14.1421	+	14.1421i	0.763604	+	0.763604i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−10.0424	+	10.0424i	−0.539102	+	0.539102i	−0.923265	−	0.384163i	$-0.874490\pi$
$347$	−10.0424	+	10.0424i	−0.539102	+	0.539102i	0.384163	+	0.923265i	$0.374490\pi$
$348$		0			0
$349$	−5.02118			−0.268778			−0.134389	−	0.990929i	$-0.542907\pi$
$349$	−5.02118			−0.268778			−0.134389	+	0.990929i	$0.542907\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−20.2918	+	20.2918i	−1.08002	+	1.08002i	−0.0835172	+	0.996506i	$0.526615\pi$
$353$	−20.2918	+	20.2918i	−1.08002	+	1.08002i	−0.996506	+	0.0835172i	$0.973385\pi$
$354$		0			0
$355$	2.89898	+	28.6969i	0.153862	+	1.52308i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−24.4949			−1.29279			−0.646396	−	0.763002i	$-0.723724\pi$
$359$	−24.4949			−1.29279			−0.646396	+	0.763002i	$0.723724\pi$
$360$		0			0
$361$	−10.5959			−0.557680
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−19.3485	−	15.7980i	−1.01275	−	0.826903i
$366$		0			0
$367$	−2.33562	+	2.33562i	−0.121918	+	0.121918i	−0.765433	−	0.643515i	$-0.777475\pi$
$367$	−2.33562	+	2.33562i	−0.121918	+	0.121918i	0.643515	+	0.765433i	$0.277475\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	8.19955			0.425700
$372$		0			0
$373$	4.70334	−	4.70334i	0.243530	−	0.243530i	−0.574779	−	0.818309i	$-0.694912\pi$
$373$	4.70334	−	4.70334i	0.243530	−	0.243530i	0.818309	+	0.574779i	$0.194912\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−1.12848	−	1.12848i	−0.0581199	−	0.0581199i
$378$		0			0
$379$	−0.202041			−0.0103782			−0.00518908	−	0.999987i	$-0.501652\pi$
$379$	−0.202041			−0.0103782			−0.00518908	+	0.999987i	$0.501652\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	25.1464	−	25.1464i	1.28492	−	1.28492i	0.347091	−	0.937831i	$-0.387169\pi$
$383$	25.1464	−	25.1464i	1.28492	−	1.28492i	0.937831	−	0.347091i	$-0.112831\pi$
$384$		0			0
$385$	9.79796	−	12.0000i	0.499350	−	0.611577i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			16.4495i			0.834022i	0.908901	+	0.417011i	$0.136922\pi$
$389$			16.4495i			0.834022i	−0.908901	+	0.417011i	$0.863078\pi$
$390$		0			0
$391$	1.84281			0.0931948
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−3.17837	−	31.4626i	−0.159921	−	1.58306i
$396$		0			0
$397$	26.4094	+	26.4094i	1.32545	+	1.32545i	0.909295	+	0.416153i	$0.136622\pi$
$397$	26.4094	+	26.4094i	1.32545	+	1.32545i	0.416153	+	0.909295i	$0.363378\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$		−	8.62815i		−	0.430869i	−0.976518	−	0.215435i	$-0.930883\pi$
$401$		−	8.62815i		−	0.430869i	0.976518	−	0.215435i	$-0.0691168\pi$
$402$		0			0
$403$	1.30306	−	1.30306i	0.0649101	−	0.0649101i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−20.6969	−	20.6969i	−1.02591	−	1.02591i
$408$		0			0
$409$			10.2020i			0.504458i	0.967668	+	0.252229i	$0.0811637\pi$
$409$			10.2020i			0.504458i	−0.967668	+	0.252229i	$0.918836\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	8.89898	+	8.89898i	0.437890	+	0.437890i
$414$		0			0
$415$	−22.3417	−	18.2419i	−1.09671	−	0.895460i
$416$		0			0
$417$		0			0
$418$		0			0
$419$			30.1913i			1.47494i	0.675379	+	0.737471i	$0.263980\pi$
$419$			30.1913i			1.47494i	−0.675379	+	0.737471i	$0.736020\pi$
$420$		0			0
$421$			36.4838i			1.77811i	0.457798	+	0.889056i	$0.348638\pi$
$421$			36.4838i			1.77811i	−0.457798	+	0.889056i	$0.651362\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−12.0922	−	7.99247i	−0.586560	−	0.387692i
$426$		0			0
$427$	−18.6969	−	18.6969i	−0.904808	−	0.904808i
$428$		0			0
$429$		0			0
$430$		0			0
$431$		−	18.6969i		−	0.900600i	−0.892877	−	0.450300i	$-0.851317\pi$
$431$		−	18.6969i		−	0.900600i	0.892877	−	0.450300i	$-0.148683\pi$
$432$		0			0
$433$	−15.0000	−	15.0000i	−0.720854	−	0.720854i	0.247925	−	0.968779i	$-0.420251\pi$
$433$	−15.0000	−	15.0000i	−0.720854	−	0.720854i	−0.968779	+	0.247925i	$0.920251\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	−1.30306	+	1.30306i	−0.0623339	+	0.0623339i
$438$		0			0
$439$			36.4838i			1.74128i	0.491923	+	0.870639i	$0.336294\pi$
$439$			36.4838i			1.74128i	−0.491923	+	0.870639i	$0.663706\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−22.3417	−	22.3417i	−1.06149	−	1.06149i	−0.997982	−	0.0635040i	$-0.979772\pi$
$443$	−22.3417	−	22.3417i	−1.06149	−	1.06149i	−0.0635040	−	0.997982i	$-0.520228\pi$
$444$		0			0
$445$	9.43879	−	0.953512i	0.447442	−	0.0452008i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	34.7839			1.64155			0.820776	−	0.571250i	$-0.193541\pi$
$449$	34.7839			1.64155			0.820776	+	0.571250i	$0.193541\pi$
$450$		0			0
$451$		−	4.89898i		−	0.230684i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−0.202041	−	2.00000i	−0.00947183	−	0.0937614i
$456$		0			0
$457$	−17.8990	+	17.8990i	−0.837279	+	0.837279i	−0.988500	−	0.151221i	$-0.951679\pi$
$457$	−17.8990	+	17.8990i	−0.837279	+	0.837279i	0.151221	+	0.988500i	$0.451679\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−20.9444			−0.975477			−0.487739	−	0.872990i	$-0.662178\pi$
$461$	−20.9444			−0.975477			−0.487739	+	0.872990i	$0.662178\pi$
$462$		0			0
$463$	−27.2200	−	27.2200i	−1.26502	−	1.26502i	−0.948628	−	0.316392i	$-0.897528\pi$
$463$	−27.2200	−	27.2200i	−1.26502	−	1.26502i	−0.316392	−	0.948628i	$-0.602472\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−17.3205	+	17.3205i	−0.801498	+	0.801498i	−0.983330	−	0.181832i	$-0.941797\pi$
$467$	−17.3205	+	17.3205i	−0.801498	+	0.801498i	0.181832	+	0.983330i	$0.441797\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−10.0424	+	10.0424i	−0.461748	+	0.461748i
$474$		0			0
$475$	14.2020	−	2.89898i	0.651634	−	0.133014i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	12.8990			0.589369			0.294685	−	0.955595i	$-0.404785\pi$
$479$	12.8990			0.589369			0.294685	+	0.955595i	$0.404785\pi$
$480$		0			0
$481$	−3.79796			−0.173172
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−15.7980	+	19.3485i	−0.717348	+	0.878569i
$486$		0			0
$487$	−18.7347	+	18.7347i	−0.848951	+	0.848951i	−0.990002	−	0.141051i	$-0.954952\pi$
$487$	−18.7347	+	18.7347i	−0.848951	+	0.848951i	0.141051	+	0.990002i	$0.454952\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	9.40669			0.424518			0.212259	−	0.977213i	$-0.431918\pi$
$491$	9.40669			0.424518			0.212259	+	0.977213i	$0.431918\pi$
$492$		0			0
$493$	−7.27815	+	7.27815i	−0.327791	+	0.327791i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	18.2419	+	18.2419i	0.818262	+	0.818262i
$498$		0			0
$499$	−17.1010			−0.765547			−0.382773	−	0.923842i	$-0.625031\pi$
$499$	−17.1010			−0.765547			−0.382773	+	0.923842i	$0.625031\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−26.4495	+	26.4495i	−1.17932	+	1.17932i	−0.199408	+	0.979917i	$0.563902\pi$
$503$	−26.4495	+	26.4495i	−1.17932	+	1.17932i	−0.979917	+	0.199408i	$0.936098\pi$
$504$		0			0
$505$	20.7980	−	2.10102i	0.925497	−	0.0934942i
$506$		0			0
$507$		0			0
$508$		0			0
$509$		−	8.04541i		−	0.356606i	−0.983976	−	0.178303i	$-0.942939\pi$
$509$		−	8.04541i		−	0.356606i	0.983976	−	0.178303i	$-0.0570608\pi$
$510$		0			0
$511$	−22.3417			−0.988338
$512$		0			0
$513$		0			0
$514$		0			0
$515$	21.9917	+	17.9562i	0.969072	+	0.791244i
$516$		0			0
$517$	22.3417	+	22.3417i	0.982586	+	0.982586i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			17.8133i			0.780416i	0.920727	+	0.390208i	$0.127597\pi$
$521$			17.8133i			0.780416i	−0.920727	+	0.390208i	$0.872403\pi$
$522$		0			0
$523$	12.8990	−	12.8990i	0.564033	−	0.564033i	−0.366418	−	0.930451i	$-0.619416\pi$
$523$	12.8990	−	12.8990i	0.564033	−	0.564033i	0.930451	+	0.366418i	$0.119416\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−8.40408	−	8.40408i	−0.366088	−	0.366088i
$528$		0			0
$529$			22.5959i			0.982431i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−0.449490	−	0.449490i	−0.0194696	−	0.0194696i
$534$		0			0
$535$	−18.2419	+	22.3417i	−0.788667	+	0.965915i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			10.3923i			0.447628i
$540$		0			0
$541$		−	5.02118i		−	0.215877i	−0.994158	−	0.107939i	$-0.965575\pi$
$541$		−	5.02118i		−	0.215877i	0.994158	−	0.107939i	$-0.0344250\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−11.1708	+	1.12848i	−0.478506	+	0.0483390i
$546$		0			0
$547$	−10.0000	−	10.0000i	−0.427569	−	0.427569i	0.460230	−	0.887800i	$-0.347767\pi$
$547$	−10.0000	−	10.0000i	−0.427569	−	0.427569i	−0.887800	+	0.460230i	$0.847767\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		−	10.2929i		−	0.438490i
$552$		0			0
$553$	−20.0000	−	20.0000i	−0.850487	−	0.850487i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−21.5959	+	21.5959i	−0.915048	+	0.915048i	−0.996664	−	0.0816155i	$-0.973992\pi$
$557$	−21.5959	+	21.5959i	−0.915048	+	0.915048i	0.0816155	+	0.996664i	$0.473992\pi$
$558$		0			0
$559$			1.84281i			0.0779424i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−32.3840	−	32.3840i	−1.36482	−	1.36482i	−0.867651	−	0.497174i	$-0.834371\pi$
$563$	−32.3840	−	32.3840i	−1.36482	−	1.36482i	−0.497174	−	0.867651i	$-0.665629\pi$
$564$		0			0
$565$	−10.0424	+	12.2993i	−0.422485	+	0.517437i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−2.39983			−0.100606			−0.0503031	−	0.998734i	$-0.516019\pi$
$569$	−2.39983			−0.100606			−0.0503031	+	0.998734i	$0.516019\pi$
$570$		0			0
$571$		−	8.69694i		−	0.363956i	−0.983303	−	0.181978i	$-0.941750\pi$
$571$		−	8.69694i		−	0.363956i	0.983303	−	0.181978i	$-0.0582499\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−1.75255	+	2.65153i	−0.0730864	+	0.110576i
$576$		0			0
$577$	−16.5959	+	16.5959i	−0.690897	+	0.690897i	−0.962429	−	0.271532i	$-0.912470\pi$
$577$	−16.5959	+	16.5959i	−0.690897	+	0.690897i	0.271532	+	0.962429i	$0.412470\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−25.7980			−1.07028
$582$		0			0
$583$	10.0424	+	10.0424i	0.415912	+	0.415912i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−23.2631	+	23.2631i	−0.960171	+	0.960171i	−0.999237	−	0.0390661i	$-0.987562\pi$
$587$	−23.2631	+	23.2631i	−0.960171	+	0.960171i	0.0390661	+	0.999237i	$0.487562\pi$
$588$		0			0
$589$	11.8852			0.489720
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−23.2631	+	23.2631i	−0.955301	+	0.955301i	−0.999043	−	0.0437422i	$-0.986072\pi$
$593$	−23.2631	+	23.2631i	−0.955301	+	0.955301i	0.0437422	+	0.999043i	$0.486072\pi$
$594$		0			0
$595$	−12.8990	+	1.30306i	−0.528807	+	0.0534203i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	18.6969			0.763936			0.381968	−	0.924176i	$-0.375246\pi$
$599$	18.6969			0.763936			0.381968	+	0.924176i	$0.375246\pi$
$600$		0			0
$601$	25.7980			1.05232			0.526160	−	0.850385i	$-0.323631\pi$
$601$	25.7980			1.05232			0.526160	+	0.850385i	$0.323631\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	2.22474	−	0.224745i	0.0904487	−	0.00913718i
$606$		0			0
$607$	−12.7279	+	12.7279i	−0.516610	+	0.516610i	−0.916544	−	0.399934i	$-0.869033\pi$
$607$	−12.7279	+	12.7279i	−0.516610	+	0.516610i	0.399934	+	0.916544i	$0.369033\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	4.09978			0.165859
$612$		0			0
$613$	−11.9815	+	11.9815i	−0.483928	+	0.483928i	−0.906384	−	0.422456i	$-0.861168\pi$
$613$	−11.9815	+	11.9815i	−0.483928	+	0.483928i	0.422456	+	0.906384i	$0.361168\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	20.2918	+	20.2918i	0.816917	+	0.816917i	0.985660	−	0.168743i	$-0.0539708\pi$
$617$	20.2918	+	20.2918i	0.816917	+	0.816917i	−0.168743	+	0.985660i	$0.553971\pi$
$618$		0			0
$619$	17.5959			0.707240			0.353620	−	0.935389i	$-0.384951\pi$
$619$	17.5959			0.707240			0.353620	+	0.935389i	$0.384951\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	6.00000	−	6.00000i	0.240385	−	0.240385i
$624$		0			0
$625$	23.0000	−	9.79796i	0.920000	−	0.391918i
$626$		0			0
$627$		0			0
$628$		0			0
$629$			24.4949i			0.976676i
$630$		0			0
$631$	−22.3417			−0.889409			−0.444704	−	0.895677i	$-0.646691\pi$
$631$	−22.3417			−0.889409			−0.444704	+	0.895677i	$0.646691\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	23.6130	−	28.9199i	0.937055	−	1.14765i
$636$		0			0
$637$	0.953512	+	0.953512i	0.0377795	+	0.0377795i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	13.7135i		−	0.541652i	−0.962628	−	0.270826i	$-0.912703\pi$
$641$		−	13.7135i		−	0.541652i	0.962628	−	0.270826i	$-0.0872969\pi$
$642$		0			0
$643$	17.1010	−	17.1010i	0.674398	−	0.674398i	−0.284329	−	0.958727i	$-0.591771\pi$
$643$	17.1010	−	17.1010i	0.674398	−	0.674398i	0.958727	+	0.284329i	$0.0917706\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	−18.0454	−	18.0454i	−0.709438	−	0.709438i	0.256979	−	0.966417i	$-0.417273\pi$
$647$	−18.0454	−	18.0454i	−0.709438	−	0.709438i	−0.966417	+	0.256979i	$0.917273\pi$
$648$		0			0
$649$			21.7980i			0.855645i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−30.4949	−	30.4949i	−1.19336	−	1.19336i	−0.976118	−	0.217239i	$-0.930295\pi$
$653$	−30.4949	−	30.4949i	−1.19336	−	1.19336i	−0.217239	−	0.976118i	$-0.569705\pi$
$654$		0			0
$655$	−10.5352	+	1.06427i	−0.411643	+	0.0415844i
$656$		0			0
$657$		0			0
$658$		0			0
$659$			48.4332i			1.88669i	0.331814	+	0.943345i	$0.392339\pi$
$659$			48.4332i			1.88669i	−0.331814	+	0.943345i	$0.607661\pi$
$660$		0			0
$661$			23.2631i			0.904829i	0.891808	+	0.452415i	$0.149437\pi$
$661$			23.2631i			0.904829i	−0.891808	+	0.452415i	$0.850563\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	8.19955	−	10.0424i	0.317965	−	0.389426i
$666$		0			0
$667$	1.59592	+	1.59592i	0.0617942	+	0.0617942i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	45.7980i		−	1.76801i
$672$		0			0
$673$	−16.5959	−	16.5959i	−0.639726	−	0.639726i	0.310762	−	0.950488i	$-0.399416\pi$
$673$	−16.5959	−	16.5959i	−0.639726	−	0.639726i	−0.950488	+	0.310762i	$0.899416\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	−16.6969	+	16.6969i	−0.641715	+	0.641715i	−0.950977	−	0.309262i	$-0.899918\pi$
$677$	−16.6969	+	16.6969i	−0.641715	+	0.641715i	0.309262	+	0.950977i	$0.399918\pi$
$678$		0			0
$679$			22.3417i			0.857395i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−5.02118	−	5.02118i	−0.192130	−	0.192130i	0.604486	−	0.796616i	$-0.293379\pi$
$683$	−5.02118	−	5.02118i	−0.192130	−	0.192130i	−0.796616	+	0.604486i	$0.793379\pi$
$684$		0			0
$685$	32.3840	+	26.4415i	1.23733	+	1.01028i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	1.84281			0.0702054
$690$		0			0
$691$		−	14.4949i		−	0.551412i	−0.961242	−	0.275706i	$-0.911088\pi$
$691$		−	14.4949i		−	0.551412i	0.961242	−	0.275706i	$-0.0889116\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−26.2474	+	2.65153i	−0.995622	+	0.100578i
$696$		0			0
$697$	−2.89898	+	2.89898i	−0.109807	+	0.109807i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	3.55051			0.134101			0.0670505	−	0.997750i	$-0.478641\pi$
$701$	3.55051			0.134101			0.0670505	+	0.997750i	$0.478641\pi$
$702$		0			0
$703$	−17.3205	−	17.3205i	−0.653255	−	0.653255i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	13.2207	−	13.2207i	0.497217	−	0.497217i
$708$		0			0
$709$	28.2843			1.06224			0.531119	−	0.847297i	$-0.321772\pi$
$709$	28.2843			1.06224			0.531119	+	0.847297i	$0.321772\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−1.84281	+	1.84281i	−0.0690137	+	0.0690137i
$714$		0			0
$715$	2.20204	−	2.69694i	0.0823517	−	0.100860i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−25.7980			−0.962102			−0.481051	−	0.876693i	$-0.659745\pi$
$719$	−25.7980			−0.962102			−0.481051	+	0.876693i	$0.659745\pi$
$720$		0			0
$721$	25.3939			0.945717
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−3.55051	−	17.3939i	−0.131863	−	0.645992i
$726$		0			0
$727$	−15.9063	+	15.9063i	−0.589932	+	0.589932i	−0.937613	−	0.347681i	$-0.886969\pi$
$727$	−15.9063	+	15.9063i	−0.589932	+	0.589932i	0.347681	+	0.937613i	$0.386969\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	11.8852			0.439589
$732$		0			0
$733$	−0.603566	+	0.603566i	−0.0222932	+	0.0222932i	−0.718166	−	0.695872i	$-0.755018\pi$
$733$	−0.603566	+	0.603566i	−0.0222932	+	0.0222932i	0.695872	+	0.718166i	$0.255018\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	−40.2929			−1.48220			−0.741098	−	0.671396i	$-0.765695\pi$
$739$	−40.2929			−1.48220			−0.741098	+	0.671396i	$0.765695\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	26.4495	−	26.4495i	0.970338	−	0.970338i	−0.0292349	−	0.999573i	$-0.509307\pi$
$743$	26.4495	−	26.4495i	0.970338	−	0.970338i	0.999573	+	0.0292349i	$0.00930708\pi$
$744$		0			0
$745$	−3.69694	−	36.5959i	−0.135445	−	1.34077i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			25.7980i			0.942637i
$750$		0			0
$751$	−22.3417			−0.815260			−0.407630	−	0.913147i	$-0.633645\pi$
$751$	−22.3417			−0.815260			−0.407630	+	0.913147i	$0.633645\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	40.5836	−	4.09978i	1.47699	−	0.149206i
$756$		0			0
$757$	1.87492	+	1.87492i	0.0681450	+	0.0681450i	0.740358	−	0.672213i	$-0.234656\pi$
$757$	1.87492	+	1.87492i	0.0681450	+	0.0681450i	−0.672213	+	0.740358i	$0.734656\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	15.5563i		−	0.563917i	−0.959427	−	0.281959i	$-0.909016\pi$
$761$		−	15.5563i		−	0.563917i	0.959427	−	0.281959i	$-0.0909841\pi$
$762$		0			0
$763$	−7.10102	+	7.10102i	−0.257074	+	0.257074i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.00000	+	2.00000i	0.0722158	+	0.0722158i
$768$		0			0
$769$			31.5959i			1.13938i	0.821860	+	0.569689i	$0.192936\pi$
$769$			31.5959i			1.13938i	−0.821860	+	0.569689i	$0.807064\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	13.1010	+	13.1010i	0.471211	+	0.471211i	0.902306	−	0.431095i	$-0.141873\pi$
$773$	13.1010	+	13.1010i	0.471211	+	0.471211i	−0.431095	+	0.902306i	$0.641873\pi$
$774$		0			0
$775$	20.0847	−	4.09978i	0.721464	−	0.147268i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	4.09978i		−	0.146890i
$780$		0			0
$781$			44.6834i			1.59890i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−30.0484	−	24.5344i	−1.07247	−	0.875672i
$786$		0			0
$787$	37.3939	+	37.3939i	1.33295	+	1.33295i	0.902721	+	0.430227i	$0.141566\pi$
$787$	37.3939	+	37.3939i	1.33295	+	1.33295i	0.430227	+	0.902721i	$0.358434\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			14.2020i			0.504966i
$792$		0			0
$793$	−4.20204	−	4.20204i	−0.149219	−	0.149219i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	18.0454	−	18.0454i	0.639201	−	0.639201i	−0.311157	−	0.950358i	$-0.600717\pi$
$797$	18.0454	−	18.0454i	0.639201	−	0.639201i	0.950358	+	0.311157i	$0.100717\pi$
$798$		0			0
$799$		−	26.4415i		−	0.935432i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−27.3629	−	27.3629i	−0.965615	−	0.965615i
$804$		0			0
$805$	0.285729	+	2.82843i	0.0100706	+	0.0996890i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	11.7423			0.412837			0.206419	−	0.978464i	$-0.433819\pi$
$809$	11.7423			0.412837			0.206419	+	0.978464i	$0.433819\pi$
$810$		0			0
$811$			45.1918i			1.58690i	0.608635	+	0.793450i	$0.291717\pi$
$811$			45.1918i			1.58690i	−0.608635	+	0.793450i	$0.708283\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	31.5959	−	38.6969i	1.10676	−	1.35549i
$816$		0			0
$817$	−8.40408	+	8.40408i	−0.294022	+	0.294022i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	3.55051			0.123914			0.0619568	−	0.998079i	$-0.480266\pi$
$821$	3.55051			0.123914			0.0619568	+	0.998079i	$0.480266\pi$
$822$		0			0
$823$	4.24264	+	4.24264i	0.147889	+	0.147889i	0.777174	−	0.629285i	$-0.216652\pi$
$823$	4.24264	+	4.24264i	0.147889	+	0.147889i	−0.629285	+	0.777174i	$0.716652\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	−24.1845	+	24.1845i	−0.840977	+	0.840977i	−0.988986	−	0.148009i	$-0.952714\pi$
$827$	−24.1845	+	24.1845i	−0.840977	+	0.840977i	0.148009	+	0.988986i	$0.452714\pi$
$828$		0			0
$829$	−15.0635			−0.523178			−0.261589	−	0.965179i	$-0.584247\pi$
$829$	−15.0635			−0.523178			−0.261589	+	0.965179i	$0.584247\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	6.14966	−	6.14966i	0.213073	−	0.213073i
$834$		0			0
$835$	20.6969	+	16.8990i	0.716247	+	0.584813i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−52.8990			−1.82628			−0.913138	−	0.407651i	$-0.866348\pi$
$839$	−52.8990			−1.82628			−0.913138	+	0.407651i	$0.866348\pi$
$840$		0			0
$841$	16.3939			0.565306
$842$		0			0
$843$		0			0
$844$		0			0
$845$	2.87628	+	28.4722i	0.0989469	+	0.979473i
$846$		0			0
$847$	1.41421	−	1.41421i	0.0485930	−	0.0485930i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	5.37113			0.184120
$852$		0			0
$853$	9.43879	−	9.43879i	0.323178	−	0.323178i	−0.526807	−	0.849985i	$-0.676611\pi$
$853$	9.43879	−	9.43879i	0.323178	−	0.323178i	0.849985	+	0.526807i	$0.176611\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−6.14966	−	6.14966i	−0.210069	−	0.210069i	0.594228	−	0.804297i	$-0.297458\pi$
$857$	−6.14966	−	6.14966i	−0.210069	−	0.210069i	−0.804297	+	0.594228i	$0.797458\pi$
$858$		0			0
$859$	−6.00000			−0.204717			−0.102359	−	0.994748i	$-0.532639\pi$
$859$	−6.00000			−0.204717			−0.102359	+	0.994748i	$0.532639\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−14.6515	+	14.6515i	−0.498744	+	0.498744i	−0.911047	−	0.412303i	$-0.864725\pi$
$863$	−14.6515	+	14.6515i	−0.498744	+	0.498744i	0.412303	+	0.911047i	$0.364725\pi$
$864$		0			0
$865$	15.7980	+	12.8990i	0.537147	+	0.438578i
$866$		0			0
$867$		0			0
$868$		0			0
$869$		−	48.9898i		−	1.66186i
$870$		0			0
$871$		0			0
$872$		0			0
$873$		0			0
$874$		0			0
$875$	10.3923	−	19.7990i	0.351324	−	0.669328i
$876$		0			0
$877$	20.4668	+	20.4668i	0.691114	+	0.691114i	0.962477	−	0.271363i	$-0.0874745\pi$
$877$	20.4668	+	20.4668i	0.691114	+	0.691114i	−0.271363	+	0.962477i	$0.587474\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$		−	35.0696i		−	1.18153i	−0.806845	−	0.590763i	$-0.798827\pi$
$881$		−	35.0696i		−	1.18153i	0.806845	−	0.590763i	$-0.201173\pi$
$882$		0			0
$883$	−1.59592	+	1.59592i	−0.0537069	+	0.0537069i	−0.733450	−	0.679743i	$-0.762091\pi$
$883$	−1.59592	+	1.59592i	−0.0537069	+	0.0537069i	0.679743	+	0.733450i	$0.262091\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	14.2474	+	14.2474i	0.478383	+	0.478383i	0.904614	−	0.426232i	$-0.140159\pi$
$887$	14.2474	+	14.2474i	0.478383	+	0.478383i	−0.426232	+	0.904614i	$0.640159\pi$
$888$		0			0
$889$		−	33.3939i		−	1.11999i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	18.6969	+	18.6969i	0.625669	+	0.625669i
$894$		0			0
$895$	1.76416	+	17.4634i	0.0589693	+	0.583736i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	14.5563i		−	0.485480i
$900$		0			0
$901$		−	11.8852i		−	0.395952i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−2.97129	−	29.4128i	−0.0987691	−	0.977713i
$906$		0			0
$907$	−32.8990	−	32.8990i	−1.09239	−	1.09239i	−0.995273	−	0.0971200i	$-0.969037\pi$
$907$	−32.8990	−	32.8990i	−1.09239	−	1.09239i	−0.0971200	−	0.995273i	$-0.530963\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			24.4949i			0.811552i	0.913973	+	0.405776i	$0.132999\pi$
$911$			24.4949i			0.811552i	−0.913973	+	0.405776i	$0.867001\pi$
$912$		0			0
$913$	−31.5959	−	31.5959i	−1.04567	−	1.04567i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−6.69694	+	6.69694i	−0.221152	+	0.221152i
$918$		0			0
$919$			22.3417i			0.736984i	0.929631	+	0.368492i	$0.120126\pi$
$919$			22.3417i			0.736984i	−0.929631	+	0.368492i	$0.879874\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	4.09978	+	4.09978i	0.134946	+	0.134946i
$924$		0			0
$925$	−35.2446	−	23.2952i	−1.15883	−	0.765941i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−9.89949			−0.324792			−0.162396	−	0.986726i	$-0.551922\pi$
$929$	−9.89949			−0.324792			−0.162396	+	0.986726i	$0.551922\pi$
$930$		0			0
$931$			8.69694i			0.285031i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−17.3939	−	14.2020i	−0.568841	−	0.464456i
$936$		0			0
$937$	0.797959	−	0.797959i	0.0260682	−	0.0260682i	−0.693953	−	0.720021i	$-0.744132\pi$
$937$	0.797959	−	0.797959i	0.0260682	−	0.0260682i	0.720021	+	0.693953i	$0.244132\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−42.2474			−1.37723			−0.688614	−	0.725128i	$-0.741780\pi$
$941$	−42.2474			−1.37723			−0.688614	+	0.725128i	$0.741780\pi$
$942$		0			0
$943$	0.635674	+	0.635674i	0.0207004	+	0.0207004i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	14.1421	−	14.1421i	0.459558	−	0.459558i	−0.438953	−	0.898510i	$-0.644650\pi$
$947$	14.1421	−	14.1421i	0.459558	−	0.459558i	0.898510	+	0.438953i	$0.144650\pi$
$948$		0			0
$949$	−5.02118			−0.162994
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−13.2207	+	13.2207i	−0.428262	+	0.428262i	−0.888036	−	0.459774i	$-0.847930\pi$
$953$	−13.2207	+	13.2207i	−0.428262	+	0.428262i	0.459774	+	0.888036i	$0.347930\pi$
$954$		0			0
$955$	1.30306	+	12.8990i	0.0421661	+	0.417401i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	37.3939			1.20751
$960$		0			0
$961$	−14.1918			−0.457801
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.95459	+	1.59592i	0.0629206	+	0.0513744i
$966$		0			0
$967$	−7.35680	+	7.35680i	−0.236579	+	0.236579i	−0.815432	−	0.578853i	$-0.803500\pi$
$967$	−7.35680	+	7.35680i	−0.236579	+	0.236579i	0.578853	+	0.815432i	$0.303500\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	45.3191			1.45436			0.727179	−	0.686448i	$-0.240831\pi$
$971$	45.3191			1.45436			0.727179	+	0.686448i	$0.240831\pi$
$972$		0			0
$973$	−16.6848	+	16.6848i	−0.534891	+	0.534891i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	16.1920	+	16.1920i	0.518029	+	0.518029i	0.916975	−	0.398946i	$-0.130624\pi$
$977$	16.1920	+	16.1920i	0.518029	+	0.518029i	−0.398946	+	0.916975i	$0.630624\pi$
$978$		0			0
$979$	14.6969			0.469716
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−0.651531	+	0.651531i	−0.0207806	+	0.0207806i	−0.717421	−	0.696640i	$-0.754677\pi$
$983$	−0.651531	+	0.651531i	−0.0207806	+	0.0207806i	0.696640	+	0.717421i	$0.254677\pi$
$984$		0			0
$985$	3.10102	−	3.79796i	0.0988067	−	0.121013i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	2.60612i		−	0.0828699i
$990$		0			0
$991$	−10.0424			−0.319006			−0.159503	−	0.987197i	$-0.550989\pi$
$991$	−10.0424			−0.319006			−0.159503	+	0.987197i	$0.550989\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	1.84281	+	18.2419i	0.0584209	+	0.578307i
$996$		0			0
$997$	−5.05329	−	5.05329i	−0.160039	−	0.160039i	0.622545	−	0.782584i	$-0.286099\pi$
$997$	−5.05329	−	5.05329i	−0.160039	−	0.160039i	−0.782584	+	0.622545i	$0.786099\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.bj.c.1313.4	yes	8
3.2	odd	2		2880.2.bj.a.1313.2	yes	8
4.3	odd	2		2880.2.bj.d.1313.3	yes	8
5.2	odd	4		2880.2.bj.d.737.4	yes	8
8.3	odd	2		2880.2.bj.a.1313.1	yes	8
8.5	even	2		2880.2.bj.b.1313.2	yes	8
12.11	even	2		2880.2.bj.b.1313.1	yes	8
15.2	even	4		2880.2.bj.b.737.2	yes	8
20.7	even	4	inner	2880.2.bj.c.737.3	yes	8
24.5	odd	2		2880.2.bj.d.1313.4	yes	8
24.11	even	2	inner	2880.2.bj.c.1313.3	yes	8
40.27	even	4		2880.2.bj.b.737.1	yes	8
40.37	odd	4		2880.2.bj.a.737.2	yes	8
60.47	odd	4		2880.2.bj.a.737.1	✓	8
120.77	even	4	inner	2880.2.bj.c.737.4	yes	8
120.107	odd	4		2880.2.bj.d.737.3	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2880.2.bj.a.737.1	✓	8	60.47	odd	4
2880.2.bj.a.737.2	yes	8	40.37	odd	4
2880.2.bj.a.1313.1	yes	8	8.3	odd	2
2880.2.bj.a.1313.2	yes	8	3.2	odd	2
2880.2.bj.b.737.1	yes	8	40.27	even	4
2880.2.bj.b.737.2	yes	8	15.2	even	4
2880.2.bj.b.1313.1	yes	8	12.11	even	2
2880.2.bj.b.1313.2	yes	8	8.5	even	2
2880.2.bj.c.737.3	yes	8	20.7	even	4	inner
2880.2.bj.c.737.4	yes	8	120.77	even	4	inner
2880.2.bj.c.1313.3	yes	8	24.11	even	2	inner
2880.2.bj.c.1313.4	yes	8	1.1	even	1	trivial
2880.2.bj.d.737.3	yes	8	120.107	odd	4
2880.2.bj.d.737.4	yes	8	5.2	odd	4
2880.2.bj.d.1313.3	yes	8	4.3	odd	2
2880.2.bj.d.1313.4	yes	8	24.5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 2880 = 2^{6} \cdot 3^{2} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2880.bj (of order \(4\), degree \(2\), minimal)

\(f(q)\)	\(=\)	\(q+(2.22474 - 0.224745i) q^{5} +(1.41421 - 1.41421i) q^{7} +O(q^{10})\)	\(q+(2.22474 - 0.224745i) q^{5} +(1.41421 - 1.41421i) q^{7} +3.46410 q^{11} +(0.317837 - 0.317837i) q^{13} +(-2.04989 - 2.04989i) q^{17} +2.89898 q^{19} +(-0.449490 + 0.449490i) q^{23} +(4.89898 - 1.00000i) q^{25} -3.55051i q^{29} +4.09978 q^{31} +(2.82843 - 3.46410i) q^{35} +(-5.97469 - 5.97469i) q^{37} -1.41421i q^{41} +(-2.89898 + 2.89898i) q^{43} +(6.44949 + 6.44949i) q^{47} +3.00000i q^{49} +(2.89898 + 2.89898i) q^{53} +(7.70674 - 0.778539i) q^{55} +6.29253i q^{59} -13.2207i q^{61} +(0.635674 - 0.778539i) q^{65} +12.8990i q^{71} +(-7.89898 - 7.89898i) q^{73} +(4.89898 - 4.89898i) q^{77} -14.1421i q^{79} +(-9.12096 - 9.12096i) q^{83} +(-5.02118 - 4.09978i) q^{85} +4.24264 q^{89} -0.898979i q^{91} +(6.44949 - 0.651531i) q^{95} +(-7.89898 + 7.89898i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 8 * q + 8 * q^5	\( 8 q + 8 q^{5} - 16 q^{19} + 16 q^{23} + 16 q^{43} + 32 q^{47} - 16 q^{53} - 24 q^{73} + 32 q^{95} - 24 q^{97}+O(q^{100}) \) 8 * q + 8 * q^5 - 16 * q^19 + 16 * q^23 + 16 * q^43 + 32 * q^47 - 16 * q^53 - 24 * q^73 + 32 * q^95 - 24 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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