LMFDB - Embedded newform 2880.2.bj.c.1313.1

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			1313.1
Root			$0.965926 + 0.258819i$ of defining polynomial
Character	$\chi$	$=$	2880.1313
Dual form			2880.2.bj.c.737.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-0.224745 + 2.22474i) q^{5} +(-1.41421 + 1.41421i) q^{7} +O(q^{10})$	$q+(-0.224745 + 2.22474i) q^{5} +(-1.41421 + 1.41421i) q^{7} +3.46410 q^{11} +(3.14626 - 3.14626i) q^{13} +(-4.87832 - 4.87832i) q^{17} -6.89898 q^{19} +(4.44949 - 4.44949i) q^{23} +(-4.89898 - 1.00000i) q^{25} -8.44949i q^{29} +9.75663 q^{31} +(-2.82843 - 3.46410i) q^{35} +(2.51059 + 2.51059i) q^{37} +1.41421i q^{41} +(6.89898 - 6.89898i) q^{43} +(1.55051 + 1.55051i) q^{47} +3.00000i q^{49} +(-6.89898 - 6.89898i) q^{53} +(-0.778539 + 7.70674i) q^{55} +0.635674i q^{59} -7.56388i q^{61} +(6.29253 + 7.70674i) q^{65} +3.10102i q^{71} +(1.89898 + 1.89898i) q^{73} +(-4.89898 + 4.89898i) q^{77} +14.1421i q^{79} +(2.19275 + 2.19275i) q^{83} +(11.9494 - 9.75663i) q^{85} -4.24264 q^{89} +8.89898i q^{91} +(1.55051 - 15.3485i) q^{95} +(1.89898 - 1.89898i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−0.224745	+	2.22474i	−0.100509	+	0.994936i
$5$	−0.224745	+	2.22474i	−0.100509	+	0.994936i
$6$		0			0
$7$	−1.41421	+	1.41421i	−0.534522	+	0.534522i	−0.921915	−	0.387392i	$-0.873376\pi$
$7$	−1.41421	+	1.41421i	−0.534522	+	0.534522i	0.387392	+	0.921915i	$0.373376\pi$
$8$		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
$13$	3.14626	−	3.14626i	0.872617	−	0.872617i	−0.120140	−	0.992757i	$-0.538334\pi$
$13$	3.14626	−	3.14626i	0.872617	−	0.872617i	0.992757	+	0.120140i	$0.0383344\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	−4.87832	−	4.87832i	−1.18317	−	1.18317i	−0.978920	−	0.204246i	$-0.934526\pi$
$17$	−4.87832	−	4.87832i	−1.18317	−	1.18317i	−0.204246	−	0.978920i	$-0.565474\pi$
$18$		0			0
$19$	−6.89898			−1.58273			−0.791367	−	0.611341i	$-0.790630\pi$
$19$	−6.89898			−1.58273			−0.791367	+	0.611341i	$0.790630\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	4.44949	−	4.44949i	0.927783	−	0.927783i	−0.0697797	−	0.997562i	$-0.522230\pi$
$23$	4.44949	−	4.44949i	0.927783	−	0.927783i	0.997562	+	0.0697797i	$0.0222296\pi$
$24$		0			0
$25$	−4.89898	−	1.00000i	−0.979796	−	0.200000i
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	8.44949i		−	1.56903i	−0.620109	−	0.784515i	$-0.712912\pi$
$29$		−	8.44949i		−	1.56903i	0.620109	−	0.784515i	$-0.287088\pi$
$30$		0			0
$31$	9.75663			1.75234			0.876171	−	0.482000i	$-0.160089\pi$
$31$	9.75663			1.75234			0.876171	+	0.482000i	$0.160089\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−2.82843	−	3.46410i	−0.478091	−	0.585540i
$36$		0			0
$37$	2.51059	+	2.51059i	0.412738	+	0.412738i	0.882691	−	0.469953i	$-0.155729\pi$
$37$	2.51059	+	2.51059i	0.412738	+	0.412738i	−0.469953	+	0.882691i	$0.655729\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			1.41421i			0.220863i	0.993884	+	0.110432i	$0.0352233\pi$
$41$			1.41421i			0.220863i	−0.993884	+	0.110432i	$0.964777\pi$
$42$		0			0
$43$	6.89898	−	6.89898i	1.05208	−	1.05208i	0.0535176	−	0.998567i	$-0.482957\pi$
$43$	6.89898	−	6.89898i	1.05208	−	1.05208i	0.998567	−	0.0535176i	$-0.0170433\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	1.55051	+	1.55051i	0.226165	+	0.226165i	0.811089	−	0.584923i	$-0.198875\pi$
$47$	1.55051	+	1.55051i	0.226165	+	0.226165i	−0.584923	+	0.811089i	$0.698875\pi$
$48$		0			0
$49$			3.00000i			0.428571i
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−6.89898	−	6.89898i	−0.947648	−	0.947648i	0.0510485	−	0.998696i	$-0.483744\pi$
$53$	−6.89898	−	6.89898i	−0.947648	−	0.947648i	−0.998696	+	0.0510485i	$0.983744\pi$
$54$		0			0
$55$	−0.778539	+	7.70674i	−0.104978	+	1.03918i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			0.635674i			0.0827578i	0.999144	+	0.0413789i	$0.0131751\pi$
$59$			0.635674i			0.0827578i	−0.999144	+	0.0413789i	$0.986825\pi$
$60$		0			0
$61$		−	7.56388i		−	0.968455i	−0.874942	−	0.484228i	$-0.839101\pi$
$61$		−	7.56388i		−	0.968455i	0.874942	−	0.484228i	$-0.160899\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	6.29253	+	7.70674i	0.780492	+	0.955904i
$66$		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
$71$			3.10102i			0.368023i	0.982924	+	0.184012i	$0.0589084\pi$
$71$			3.10102i			0.368023i	−0.982924	+	0.184012i	$0.941092\pi$
$72$		0			0
$73$	1.89898	+	1.89898i	0.222259	+	0.222259i	0.809449	−	0.587190i	$-0.199766\pi$
$73$	1.89898	+	1.89898i	0.222259	+	0.222259i	−0.587190	+	0.809449i	$0.699766\pi$
$74$		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	$0.292822\pi$
$79$			14.1421i			1.59111i	−0.605878	+	0.795557i	$0.707178\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	2.19275	+	2.19275i	0.240686	+	0.240686i	0.817134	−	0.576448i	$-0.195562\pi$
$83$	2.19275	+	2.19275i	0.240686	+	0.240686i	−0.576448	+	0.817134i	$0.695562\pi$
$84$		0			0
$85$	11.9494	−	9.75663i	1.29609	−	1.05826i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.24264			−0.449719			−0.224860	−	0.974391i	$-0.572192\pi$
$89$	−4.24264			−0.449719			−0.224860	+	0.974391i	$0.572192\pi$
$90$		0			0
$91$			8.89898i			0.932867i
$92$		0			0
$93$		0			0
$94$		0			0
$95$	1.55051	−	15.3485i	0.159079	−	1.57472i
$96$		0			0
$97$	1.89898	−	1.89898i	0.192812	−	0.192812i	−0.604098	−	0.796910i	$-0.706466\pi$
$97$	1.89898	−	1.89898i	0.192812	−	0.192812i	0.796910	+	0.604098i	$0.206466\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−5.34847			−0.532193			−0.266096	−	0.963946i	$-0.585734\pi$
$101$	−5.34847			−0.532193			−0.266096	+	0.963946i	$0.585734\pi$
$102$		0			0
$103$	11.8065	+	11.8065i	1.16333	+	1.16333i	0.983742	+	0.179589i	$0.0574768\pi$
$103$	11.8065	+	11.8065i	1.16333	+	1.16333i	0.179589	+	0.983742i	$0.442523\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	2.19275	−	2.19275i	0.211981	−	0.211981i	−0.593127	−	0.805109i	$-0.702107\pi$
$107$	2.19275	−	2.19275i	0.211981	−	0.211981i	0.805109	+	0.593127i	$0.202107\pi$
$108$		0			0
$109$	11.9494			1.14454			0.572272	−	0.820064i	$-0.306062\pi$
$109$	11.9494			1.14454			0.572272	+	0.820064i	$0.306062\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	11.9494	−	11.9494i	1.12410	−	1.12410i	0.132985	−	0.991118i	$-0.457544\pi$
$113$	11.9494	−	11.9494i	1.12410	−	1.12410i	0.991118	−	0.132985i	$-0.0424563\pi$
$114$		0			0
$115$	8.89898	+	10.8990i	0.829834	+	1.01634i
$116$		0			0
$117$		0			0
$118$		0			0
$119$	13.7980			1.26486
$120$		0			0
$121$	1.00000			0.0909091
$122$		0			0
$123$		0			0
$124$		0			0
$125$	3.32577	−	10.6742i	0.297465	−	0.954733i
$126$		0			0
$127$	8.97809	−	8.97809i	0.796677	−	0.796677i	−0.185893	−	0.982570i	$-0.559518\pi$
$127$	8.97809	−	8.97809i	0.796677	−	0.796677i	0.982570	+	0.185893i	$0.0595178\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−16.0492			−1.40222			−0.701111	−	0.713052i	$-0.747312\pi$
$131$	−16.0492			−1.40222			−0.701111	+	0.713052i	$0.747312\pi$
$132$		0			0
$133$	9.75663	−	9.75663i	0.846007	−	0.846007i
$134$		0			0
$135$		0			0
$136$		0			0
$137$	7.56388	+	7.56388i	0.646226	+	0.646226i	0.952079	−	0.305853i	$-0.0989416\pi$
$137$	7.56388	+	7.56388i	0.646226	+	0.646226i	−0.305853	+	0.952079i	$0.598942\pi$
$138$		0			0
$139$	7.79796			0.661414			0.330707	−	0.943733i	$-0.392713\pi$
$139$	7.79796			0.661414			0.330707	+	0.943733i	$0.392713\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	10.8990	−	10.8990i	0.911418	−	0.911418i
$144$		0			0
$145$	18.7980	+	1.89898i	1.56109	+	0.157702i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	11.5505i		−	0.946255i	−0.880994	−	0.473127i	$-0.843125\pi$
$149$		−	11.5505i		−	0.946255i	0.880994	−	0.473127i	$-0.156875\pi$
$150$		0			0
$151$	−4.38551			−0.356887			−0.178444	−	0.983950i	$-0.557106\pi$
$151$	−4.38551			−0.356887			−0.178444	+	0.983950i	$0.557106\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−2.19275	+	21.7060i	−0.176126	+	1.74347i
$156$		0			0
$157$	1.87492	+	1.87492i	0.149635	+	0.149635i	0.777955	−	0.628320i	$-0.216257\pi$
$157$	1.87492	+	1.87492i	0.149635	+	0.149635i	−0.628320	+	0.777955i	$0.716257\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			12.5851i			0.991841i
$162$		0			0
$163$	−3.79796	+	3.79796i	−0.297479	+	0.297479i	−0.840026	−	0.542547i	$-0.817460\pi$
$163$	−3.79796	+	3.79796i	−0.297479	+	0.297479i	0.542547	+	0.840026i	$0.317460\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	3.55051	+	3.55051i	0.274747	+	0.274747i	0.831008	−	0.556261i	$-0.187764\pi$
$167$	3.55051	+	3.55051i	0.274747	+	0.274747i	−0.556261	+	0.831008i	$0.687764\pi$
$168$		0			0
$169$		−	6.79796i		−	0.522920i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	1.55051	+	1.55051i	0.117883	+	0.117883i	0.763587	−	0.645704i	$-0.223436\pi$
$173$	1.55051	+	1.55051i	0.117883	+	0.117883i	−0.645704	+	0.763587i	$0.723436\pi$
$174$		0			0
$175$	8.34242	−	5.51399i	0.630627	−	0.416818i
$176$		0			0
$177$		0			0
$178$		0			0
$179$		−	14.7778i		−	1.10455i	−0.833664	−	0.552273i	$-0.813761\pi$
$179$		−	14.7778i		−	1.10455i	0.833664	−	0.552273i	$-0.186239\pi$
$180$		0			0
$181$		−	7.56388i		−	0.562219i	−0.959676	−	0.281109i	$-0.909298\pi$
$181$		−	7.56388i		−	0.562219i	0.959676	−	0.281109i	$-0.0907023\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−6.14966	+	5.02118i	−0.452132	+	0.369164i
$186$		0			0
$187$	−16.8990	−	16.8990i	−1.23578	−	1.23578i
$188$		0			0
$189$		0			0
$190$		0			0
$191$		−	13.7980i		−	0.998385i	−0.866491	−	0.499193i	$-0.833630\pi$
$191$		−	13.7980i		−	0.998385i	0.866491	−	0.499193i	$-0.166370\pi$
$192$		0			0
$193$	−18.7980	−	18.7980i	−1.35311	−	1.35311i	−0.882164	−	0.470943i	$-0.843914\pi$
$193$	−18.7980	−	18.7980i	−1.35311	−	1.35311i	−0.470943	−	0.882164i	$-0.656086\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$	6.44949	−	6.44949i	0.459507	−	0.459507i	−0.438987	−	0.898494i	$-0.644662\pi$
$197$	6.44949	−	6.44949i	0.459507	−	0.459507i	0.898494	+	0.438987i	$0.144662\pi$
$198$		0			0
$199$			19.5133i			1.38326i	0.722253	+	0.691629i	$0.243107\pi$
$199$			19.5133i			1.38326i	−0.722253	+	0.691629i	$0.756893\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	11.9494	+	11.9494i	0.838682	+	0.838682i
$204$		0			0
$205$	−3.14626	−	0.317837i	−0.219745	−	0.0221987i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−23.8988			−1.65311
$210$		0			0
$211$		−	11.7980i		−	0.812205i	−0.913828	−	0.406102i	$-0.866888\pi$
$211$		−	11.7980i		−	0.812205i	0.913828	−	0.406102i	$-0.133112\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	13.7980	+	16.8990i	0.941013	+	1.15250i
$216$		0			0
$217$	−13.7980	+	13.7980i	−0.936666	+	0.936666i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−30.6969			−2.06490
$222$		0			0
$223$	9.89949	+	9.89949i	0.662919	+	0.662919i	0.956067	−	0.293148i	$-0.0947028\pi$
$223$	9.89949	+	9.89949i	0.662919	+	0.662919i	−0.293148	+	0.956067i	$0.594703\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	9.75663	−	9.75663i	0.647570	−	0.647570i	−0.304835	−	0.952405i	$-0.598601\pi$
$227$	9.75663	−	9.75663i	0.647570	−	0.647570i	0.952405	+	0.304835i	$0.0986013\pi$
$228$		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
$233$	9.26382	−	9.26382i	0.606893	−	0.606893i	−0.335240	−	0.942133i	$-0.608817\pi$
$233$	9.26382	−	9.26382i	0.606893	−	0.606893i	0.942133	+	0.335240i	$0.108817\pi$
$234$		0			0
$235$	−3.79796	+	3.10102i	−0.247752	+	0.202288i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−24.4949			−1.58444			−0.792222	−	0.610234i	$-0.791076\pi$
$239$	−24.4949			−1.58444			−0.792222	+	0.610234i	$0.791076\pi$
$240$		0			0
$241$	−3.79796			−0.244648			−0.122324	−	0.992490i	$-0.539035\pi$
$241$	−3.79796			−0.244648			−0.122324	+	0.992490i	$0.539035\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	−6.67423	−	0.674235i	−0.426401	−	0.0430753i
$246$		0			0
$247$	−21.7060	+	21.7060i	−1.38112	+	1.38112i
$248$		0			0
$249$		0			0
$250$		0			0
$251$	−10.6780			−0.673992			−0.336996	−	0.941506i	$-0.609411\pi$
$251$	−10.6780			−0.673992			−0.336996	+	0.941506i	$0.609411\pi$
$252$		0			0
$253$	15.4135	−	15.4135i	0.969037	−	0.969037i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	−11.9494	−	11.9494i	−0.745382	−	0.745382i	0.228226	−	0.973608i	$-0.426707\pi$
$257$	−11.9494	−	11.9494i	−0.745382	−	0.745382i	−0.973608	+	0.228226i	$0.926707\pi$
$258$		0			0
$259$	−7.10102			−0.441236
$260$		0			0
$261$		0			0
$262$		0			0
$263$	−7.55051	+	7.55051i	−0.465584	+	0.465584i	−0.900481	−	0.434896i	$-0.856785\pi$
$263$	−7.55051	+	7.55051i	−0.465584	+	0.465584i	0.434896	+	0.900481i	$0.356785\pi$
$264$		0			0
$265$	16.8990	−	13.7980i	1.03810	−	0.847602i
$266$		0			0
$267$		0			0
$268$		0			0
$269$			11.5505i			0.704247i	0.935954	+	0.352124i	$0.114540\pi$
$269$			11.5505i			0.704247i	−0.935954	+	0.352124i	$0.885460\pi$
$270$		0			0
$271$	23.8988			1.45175			0.725873	−	0.687828i	$-0.241436\pi$
$271$	23.8988			1.45175			0.725873	+	0.687828i	$0.241436\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−16.9706	−	3.46410i	−1.02336	−	0.208893i
$276$		0			0
$277$	−0.317837	−	0.317837i	−0.0190970	−	0.0190970i	0.697494	−	0.716591i	$-0.254298\pi$
$277$	−0.317837	−	0.317837i	−0.0190970	−	0.0190970i	−0.716591	+	0.697494i	$0.754298\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	13.7135i		−	0.818081i	−0.912516	−	0.409041i	$-0.865863\pi$
$281$		−	13.7135i		−	0.818081i	0.912516	−	0.409041i	$-0.134137\pi$
$282$		0			0
$283$	3.10102	−	3.10102i	0.184337	−	0.184337i	−0.608906	−	0.793242i	$-0.708391\pi$
$283$	3.10102	−	3.10102i	0.184337	−	0.184337i	0.793242	+	0.608906i	$0.208391\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−2.00000	−	2.00000i	−0.118056	−	0.118056i
$288$		0			0
$289$			30.5959i			1.79976i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	−0.651531	−	0.651531i	−0.0380628	−	0.0380628i	0.687819	−	0.725882i	$-0.258568\pi$
$293$	−0.651531	−	0.651531i	−0.0380628	−	0.0380628i	−0.725882	+	0.687819i	$0.758568\pi$
$294$		0			0
$295$	−1.41421	−	0.142865i	−0.0823387	−	0.00831790i
$296$		0			0
$297$		0			0
$298$		0			0
$299$		−	27.9985i		−	1.61920i
$300$		0			0
$301$			19.5133i			1.12473i
$302$		0			0
$303$		0			0
$304$		0			0
$305$	16.8277	+	1.69994i	0.963551	+	0.0973384i
$306$		0			0
$307$	6.89898	+	6.89898i	0.393746	+	0.393746i	0.876020	−	0.482275i	$-0.160189\pi$
$307$	6.89898	+	6.89898i	0.393746	+	0.393746i	−0.482275	+	0.876020i	$0.660189\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	16.8990i		−	0.958253i	−0.877746	−	0.479127i	$-0.840953\pi$
$311$		−	16.8990i		−	0.958253i	0.877746	−	0.479127i	$-0.159047\pi$
$312$		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
$317$	14.8990	−	14.8990i	0.836810	−	0.836810i	−0.151628	−	0.988438i	$-0.548451\pi$
$317$	14.8990	−	14.8990i	0.836810	−	0.836810i	0.988438	+	0.151628i	$0.0484515\pi$
$318$		0			0
$319$		−	29.2699i		−	1.63880i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	33.6554	+	33.6554i	1.87264	+	1.87264i
$324$		0			0
$325$	−18.5597	+	12.2672i	−1.02951	+	0.680463i
$326$		0			0
$327$		0			0
$328$		0			0
$329$	−4.38551			−0.241781
$330$		0			0
$331$			13.1010i			0.720097i	0.932934	+	0.360049i	$0.117240\pi$
$331$			13.1010i			0.720097i	−0.932934	+	0.360049i	$0.882760\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		0			0
$336$		0			0
$337$	−11.8990	+	11.8990i	−0.648179	+	0.648179i	−0.952553	−	0.304374i	$-0.901553\pi$
$337$	−11.8990	+	11.8990i	−0.648179	+	0.648179i	0.304374	+	0.952553i	$0.401553\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	33.7980			1.83026
$342$		0			0
$343$	−14.1421	−	14.1421i	−0.763604	−	0.763604i
$344$		0			0
$345$		0			0
$346$		0			0
$347$	23.8988	−	23.8988i	1.28295	−	1.28295i	0.343974	−	0.938979i	$-0.388227\pi$
$347$	23.8988	−	23.8988i	1.28295	−	1.28295i	0.938979	−	0.343974i	$-0.111773\pi$
$348$		0			0
$349$	11.9494			0.639636			0.319818	−	0.947479i	$-0.396378\pi$
$349$	11.9494			0.639636			0.319818	+	0.947479i	$0.396378\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−0.492810	+	0.492810i	−0.0262296	+	0.0262296i	−0.720100	−	0.693870i	$-0.755904\pi$
$353$	−0.492810	+	0.492810i	−0.0262296	+	0.0262296i	0.693870	+	0.720100i	$0.255904\pi$
$354$		0			0
$355$	−6.89898	−	0.696938i	−0.366160	−	0.0369897i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	24.4949			1.29279			0.646396	−	0.763002i	$-0.276276\pi$
$359$	24.4949			1.29279			0.646396	+	0.763002i	$0.276276\pi$
$360$		0			0
$361$	28.5959			1.50505
$362$		0			0
$363$		0			0
$364$		0			0
$365$	−4.65153	+	3.79796i	−0.243472	+	0.198794i
$366$		0			0
$367$	23.1202	−	23.1202i	1.20687	−	1.20687i	0.234829	−	0.972037i	$-0.424547\pi$
$367$	23.1202	−	23.1202i	1.20687	−	1.20687i	0.972037	−	0.234829i	$-0.0754532\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	19.5133			1.01308
$372$		0			0
$373$	−15.0956	+	15.0956i	−0.781623	+	0.781623i	−0.980105	−	0.198482i	$-0.936399\pi$
$373$	−15.0956	+	15.0956i	−0.781623	+	0.781623i	0.198482	+	0.980105i	$0.436399\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−26.5843	−	26.5843i	−1.36916	−	1.36916i
$378$		0			0
$379$	−19.7980			−1.01695			−0.508476	−	0.861076i	$-0.669791\pi$
$379$	−19.7980			−1.01695			−0.508476	+	0.861076i	$0.669791\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−9.14643	+	9.14643i	−0.467361	+	0.467361i	−0.901058	−	0.433698i	$-0.857209\pi$
$383$	−9.14643	+	9.14643i	−0.467361	+	0.467361i	0.433698	+	0.901058i	$0.357209\pi$
$384$		0			0
$385$	−9.79796	−	12.0000i	−0.499350	−	0.611577i
$386$		0			0
$387$		0			0
$388$		0			0
$389$			11.5505i			0.585634i	0.956168	+	0.292817i	$0.0945927\pi$
$389$			11.5505i			0.585634i	−0.956168	+	0.292817i	$0.905407\pi$
$390$		0			0
$391$	−43.4120			−2.19544
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−31.4626	−	3.17837i	−1.58306	−	0.159921i
$396$		0			0
$397$	−16.0171	−	16.0171i	−0.803873	−	0.803873i	0.179826	−	0.983698i	$-0.442447\pi$
$397$	−16.0171	−	16.0171i	−0.803873	−	0.803873i	−0.983698	+	0.179826i	$0.942447\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			22.4846i			1.12282i	0.827536	+	0.561412i	$0.189742\pi$
$401$			22.4846i			1.12282i	−0.827536	+	0.561412i	$0.810258\pi$
$402$		0			0
$403$	30.6969	−	30.6969i	1.52912	−	1.52912i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	8.69694	+	8.69694i	0.431091	+	0.431091i
$408$		0			0
$409$			29.7980i			1.47341i	0.676212	+	0.736707i	$0.263620\pi$
$409$			29.7980i			1.47341i	−0.676212	+	0.736707i	$0.736380\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$	−0.898979	−	0.898979i	−0.0442359	−	0.0442359i
$414$		0			0
$415$	−5.37113	+	4.38551i	−0.263658	+	0.215276i
$416$		0			0
$417$		0			0
$418$		0			0
$419$		−	9.40669i		−	0.459547i	−0.973244	−	0.229773i	$-0.926202\pi$
$419$		−	9.40669i		−	0.459547i	0.973244	−	0.229773i	$-0.0737985\pi$
$420$		0			0
$421$		−	8.77101i		−	0.427473i	−0.976891	−	0.213736i	$-0.931437\pi$
$421$		−	8.77101i		−	0.427473i	0.976891	−	0.213736i	$-0.0685634\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	19.0205	+	28.7771i	0.922627	+	1.39589i
$426$		0			0
$427$	10.6969	+	10.6969i	0.517661	+	0.517661i
$428$		0			0
$429$		0			0
$430$		0			0
$431$			10.6969i			0.515253i	0.966245	+	0.257627i	$0.0829405\pi$
$431$			10.6969i			0.515253i	−0.966245	+	0.257627i	$0.917060\pi$
$432$		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
$437$	−30.6969	+	30.6969i	−1.46843	+	1.46843i
$438$		0			0
$439$		−	8.77101i		−	0.418617i	−0.977850	−	0.209309i	$-0.932879\pi$
$439$		−	8.77101i		−	0.418617i	0.977850	−	0.209309i	$-0.0671214\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−5.37113	−	5.37113i	−0.255190	−	0.255190i	0.567905	−	0.823094i	$-0.307754\pi$
$443$	−5.37113	−	5.37113i	−0.255190	−	0.255190i	−0.823094	+	0.567905i	$0.807754\pi$
$444$		0			0
$445$	0.953512	−	9.43879i	0.0452008	−	0.447442i
$446$		0			0
$447$		0			0
$448$		0			0
$449$	20.6417			0.974144			0.487072	−	0.873362i	$-0.338065\pi$
$449$	20.6417			0.974144			0.487072	+	0.873362i	$0.338065\pi$
$450$		0			0
$451$			4.89898i			0.230684i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	−19.7980	−	2.00000i	−0.928143	−	0.0937614i
$456$		0			0
$457$	−8.10102	+	8.10102i	−0.378950	+	0.378950i	−0.870723	−	0.491773i	$-0.836349\pi$
$457$	−8.10102	+	8.10102i	−0.378950	+	0.378950i	0.491773	+	0.870723i	$0.336349\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	32.9444			1.53437			0.767187	−	0.641424i	$-0.221656\pi$
$461$	32.9444			1.53437			0.767187	+	0.641424i	$0.221656\pi$
$462$		0			0
$463$	−7.42101	−	7.42101i	−0.344884	−	0.344884i	0.513316	−	0.858200i	$-0.328417\pi$
$463$	−7.42101	−	7.42101i	−0.344884	−	0.344884i	−0.858200	+	0.513316i	$0.828417\pi$
$464$		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
$473$	23.8988	−	23.8988i	1.09887	−	1.09887i
$474$		0			0
$475$	33.7980	+	6.89898i	1.55076	+	0.316547i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	3.10102			0.141689			0.0708446	−	0.997487i	$-0.477431\pi$
$479$	3.10102			0.141689			0.0708446	+	0.997487i	$0.477431\pi$
$480$		0			0
$481$	15.7980			0.720325
$482$		0			0
$483$		0			0
$484$		0			0
$485$	3.79796	+	4.65153i	0.172456	+	0.211215i
$486$		0			0
$487$	−15.9063	+	15.9063i	−0.720783	+	0.720783i	−0.968765	−	0.247982i	$-0.920233\pi$
$487$	−15.9063	+	15.9063i	−0.720783	+	0.720783i	0.247982	+	0.968765i	$0.420233\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−30.1913			−1.36251			−0.681257	−	0.732044i	$-0.738566\pi$
$491$	−30.1913			−1.36251			−0.681257	+	0.732044i	$0.738566\pi$
$492$		0			0
$493$	−41.2193	+	41.2193i	−1.85642	+	1.85642i
$494$		0			0
$495$		0			0
$496$		0			0
$497$	−4.38551	−	4.38551i	−0.196717	−	0.196717i
$498$		0			0
$499$	−26.8990			−1.20416			−0.602082	−	0.798434i	$-0.705662\pi$
$499$	−26.8990			−1.20416			−0.602082	+	0.798434i	$0.705662\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	−21.5505	+	21.5505i	−0.960890	+	0.960890i	−0.999263	−	0.0383737i	$-0.987782\pi$
$503$	−21.5505	+	21.5505i	−0.960890	+	0.960890i	0.0383737	+	0.999263i	$0.487782\pi$
$504$		0			0
$505$	1.20204	−	11.8990i	0.0534901	−	0.529498i
$506$		0			0
$507$		0			0
$508$		0			0
$509$			36.0454i			1.59768i	0.601541	+	0.798842i	$0.294554\pi$
$509$			36.0454i			1.59768i	−0.601541	+	0.798842i	$0.705446\pi$
$510$		0			0
$511$	−5.37113			−0.237605
$512$		0			0
$513$		0			0
$514$		0			0
$515$	−28.9199	+	23.6130i	−1.27437	+	1.04051i
$516$		0			0
$517$	5.37113	+	5.37113i	0.236222	+	0.236222i
$518$		0			0
$519$		0			0
$520$		0			0
$521$			37.6123i			1.64783i	0.566717	+	0.823913i	$0.308213\pi$
$521$			37.6123i			1.64783i	−0.566717	+	0.823913i	$0.691787\pi$
$522$		0			0
$523$	3.10102	−	3.10102i	0.135598	−	0.135598i	−0.636050	−	0.771648i	$-0.719433\pi$
$523$	3.10102	−	3.10102i	0.135598	−	0.135598i	0.771648	+	0.636050i	$0.219433\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−47.5959	−	47.5959i	−2.07331	−	2.07331i
$528$		0			0
$529$		−	16.5959i		−	0.721562i
$530$		0			0
$531$		0			0
$532$		0			0
$533$	4.44949	+	4.44949i	0.192729	+	0.192729i
$534$		0			0
$535$	4.38551	+	5.37113i	0.189602	+	0.232214i
$536$		0			0
$537$		0			0
$538$		0			0
$539$			10.3923i			0.447628i
$540$		0			0
$541$			11.9494i			0.513744i	0.966445	+	0.256872i	$0.0826919\pi$
$541$			11.9494i			0.513744i	−0.966445	+	0.256872i	$0.917308\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	−2.68556	+	26.5843i	−0.115037	+	1.13875i
$546$		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
$551$			58.2929i			2.48336i
$552$		0			0
$553$	−20.0000	−	20.0000i	−0.850487	−	0.850487i
$554$		0			0
$555$		0			0
$556$		0			0
$557$	17.5959	−	17.5959i	0.745563	−	0.745563i	−0.228080	−	0.973643i	$-0.573245\pi$
$557$	17.5959	−	17.5959i	0.745563	−	0.745563i	0.973643	+	0.228080i	$0.0732446\pi$
$558$		0			0
$559$		−	43.4120i		−	1.83613i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	18.5276	+	18.5276i	0.780847	+	0.780847i	0.979974	−	0.199127i	$-0.0638106\pi$
$563$	18.5276	+	18.5276i	0.780847	+	0.780847i	−0.199127	+	0.979974i	$0.563811\pi$
$564$		0			0
$565$	23.8988	+	29.2699i	1.00543	+	1.23139i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	−39.1694			−1.64207			−0.821033	−	0.570881i	$-0.806602\pi$
$569$	−39.1694			−1.64207			−0.821033	+	0.570881i	$0.806602\pi$
$570$		0			0
$571$			20.6969i			0.866140i	0.901360	+	0.433070i	$0.142570\pi$
$571$			20.6969i			0.866140i	−0.901360	+	0.433070i	$0.857430\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	−26.2474	+	17.3485i	−1.09459	+	0.723481i
$576$		0			0
$577$	22.5959	−	22.5959i	0.940680	−	0.940680i	−0.0576561	−	0.998337i	$-0.518363\pi$
$577$	22.5959	−	22.5959i	0.940680	−	0.940680i	0.998337	+	0.0576561i	$0.0183627\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−6.20204			−0.257304
$582$		0			0
$583$	−23.8988	−	23.8988i	−0.989786	−	0.989786i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	16.3349	−	16.3349i	0.674213	−	0.674213i	−0.284472	−	0.958684i	$-0.591818\pi$
$587$	16.3349	−	16.3349i	0.674213	−	0.674213i	0.958684	+	0.284472i	$0.0918181\pi$
$588$		0			0
$589$	−67.3108			−2.77349
$590$		0			0
$591$		0			0
$592$		0			0
$593$	16.3349	−	16.3349i	0.670793	−	0.670793i	−0.287106	−	0.957899i	$-0.592693\pi$
$593$	16.3349	−	16.3349i	0.670793	−	0.670793i	0.957899	+	0.287106i	$0.0926930\pi$
$594$		0			0
$595$	−3.10102	+	30.6969i	−0.127129	+	1.25845i
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−10.6969			−0.437065			−0.218533	−	0.975830i	$-0.570127\pi$
$599$	−10.6969			−0.437065			−0.218533	+	0.975830i	$0.570127\pi$
$600$		0			0
$601$	6.20204			0.252987			0.126493	−	0.991967i	$-0.459628\pi$
$601$	6.20204			0.252987			0.126493	+	0.991967i	$0.459628\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−0.224745	+	2.22474i	−0.00913718	+	0.0904487i
$606$		0			0
$607$	12.7279	−	12.7279i	0.516610	−	0.516610i	−0.399934	−	0.916544i	$-0.630967\pi$
$607$	12.7279	−	12.7279i	0.516610	−	0.516610i	0.916544	+	0.399934i	$0.130967\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	9.75663			0.394711
$612$		0			0
$613$	−26.1236	+	26.1236i	−1.05512	+	1.05512i	−0.0567340	+	0.998389i	$0.518069\pi$
$613$	−26.1236	+	26.1236i	−1.05512	+	1.05512i	−0.998389	+	0.0567340i	$0.981931\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	0.492810	+	0.492810i	0.0198398	+	0.0198398i	0.716957	−	0.697117i	$-0.245534\pi$
$617$	0.492810	+	0.492810i	0.0198398	+	0.0198398i	−0.697117	+	0.716957i	$0.745534\pi$
$618$		0			0
$619$	−21.5959			−0.868013			−0.434007	−	0.900910i	$-0.642901\pi$
$619$	−21.5959			−0.868013			−0.434007	+	0.900910i	$0.642901\pi$
$620$		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
$631$	−5.37113			−0.213821			−0.106911	−	0.994269i	$-0.534096\pi$
$631$	−5.37113			−0.213821			−0.106911	+	0.994269i	$0.534096\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	17.9562	+	21.9917i	0.712569	+	0.872716i
$636$		0			0
$637$	9.43879	+	9.43879i	0.373979	+	0.373979i
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	27.8557i		−	1.10023i	−0.835088	−	0.550117i	$-0.814583\pi$
$641$		−	27.8557i		−	1.10023i	0.835088	−	0.550117i	$-0.185417\pi$
$642$		0			0
$643$	26.8990	−	26.8990i	1.06079	−	1.06079i	0.0627638	−	0.998028i	$-0.480009\pi$
$643$	26.8990	−	26.8990i	1.06079	−	1.06079i	0.998028	−	0.0627638i	$-0.0199915\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	26.0454	+	26.0454i	1.02395	+	1.02395i	0.999706	+	0.0242446i	$0.00771804\pi$
$647$	26.0454	+	26.0454i	1.02395	+	1.02395i	0.0242446	+	0.999706i	$0.492282\pi$
$648$		0			0
$649$			2.20204i			0.0864377i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	18.4949	+	18.4949i	0.723761	+	0.723761i	0.969369	−	0.245608i	$-0.0789876\pi$
$653$	18.4949	+	18.4949i	0.723761	+	0.723761i	−0.245608	+	0.969369i	$0.578988\pi$
$654$		0			0
$655$	3.60697	−	35.7053i	0.140936	−	1.39512i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	13.7922i		−	0.537267i	−0.963242	−	0.268634i	$-0.913428\pi$
$659$		−	13.7922i		−	0.537267i	0.963242	−	0.268634i	$-0.0865721\pi$
$660$		0			0
$661$		−	16.3349i		−	0.635354i	−0.948199	−	0.317677i	$-0.897097\pi$
$661$		−	16.3349i		−	0.635354i	0.948199	−	0.317677i	$-0.102903\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	19.5133	+	23.8988i	0.756692	+	0.926754i
$666$		0			0
$667$	−37.5959	−	37.5959i	−1.45572	−	1.45572i
$668$		0			0
$669$		0			0
$670$		0			0
$671$		−	26.2020i		−	1.01152i
$672$		0			0
$673$	22.5959	+	22.5959i	0.871009	+	0.871009i	0.992582	−	0.121574i	$-0.0387941\pi$
$673$	22.5959	+	22.5959i	0.871009	+	0.871009i	−0.121574	+	0.992582i	$0.538794\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	12.6969	−	12.6969i	0.487983	−	0.487983i	−0.419686	−	0.907669i	$-0.637860\pi$
$677$	12.6969	−	12.6969i	0.487983	−	0.487983i	0.907669	+	0.419686i	$0.137860\pi$
$678$		0			0
$679$			5.37113i			0.206125i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	11.9494	+	11.9494i	0.457230	+	0.457230i	0.897745	−	0.440515i	$-0.145204\pi$
$683$	11.9494	+	11.9494i	0.457230	+	0.457230i	−0.440515	+	0.897745i	$0.645204\pi$
$684$		0			0
$685$	−18.5276	+	15.1278i	−0.707905	+	0.578002i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−43.4120			−1.65387
$690$		0			0
$691$			34.4949i			1.31225i	0.754653	+	0.656124i	$0.227805\pi$
$691$			34.4949i			1.31225i	−0.754653	+	0.656124i	$0.772195\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−1.75255	+	17.3485i	−0.0664781	+	0.658065i
$696$		0			0
$697$	6.89898	−	6.89898i	0.261317	−	0.261317i
$698$		0			0
$699$		0			0
$700$		0			0
$701$	8.44949			0.319133			0.159566	−	0.987187i	$-0.448990\pi$
$701$	8.44949			0.319133			0.159566	+	0.987187i	$0.448990\pi$
$702$		0			0
$703$	−17.3205	−	17.3205i	−0.653255	−	0.653255i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	7.56388	−	7.56388i	0.284469	−	0.284469i
$708$		0			0
$709$	−28.2843			−1.06224			−0.531119	−	0.847297i	$-0.678228\pi$
$709$	−28.2843			−1.06224			−0.531119	+	0.847297i	$0.678228\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	43.4120	−	43.4120i	1.62579	−	1.62579i
$714$		0			0
$715$	21.7980	+	26.6969i	0.815197	+	0.998409i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−6.20204			−0.231297			−0.115649	−	0.993290i	$-0.536895\pi$
$719$	−6.20204			−0.231297			−0.115649	+	0.993290i	$0.536895\pi$
$720$		0			0
$721$	−33.3939			−1.24365
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−8.44949	+	41.3939i	−0.313806	+	1.53733i
$726$		0			0
$727$	−18.7347	+	18.7347i	−0.694832	+	0.694832i	−0.963291	−	0.268459i	$-0.913486\pi$
$727$	−18.7347	+	18.7347i	−0.694832	+	0.694832i	0.268459	+	0.963291i	$0.413486\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−67.3108			−2.48958
$732$		0			0
$733$	24.8523	−	24.8523i	0.917940	−	0.917940i	−0.0789396	−	0.996879i	$-0.525153\pi$
$733$	24.8523	−	24.8523i	0.917940	−	0.917940i	0.996879	+	0.0789396i	$0.0251534\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$		0			0
$738$		0			0
$739$	28.2929			1.04077			0.520385	−	0.853932i	$-0.325789\pi$
$739$	28.2929			1.04077			0.520385	+	0.853932i	$0.325789\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	21.5505	−	21.5505i	0.790612	−	0.790612i	−0.190982	−	0.981594i	$-0.561167\pi$
$743$	21.5505	−	21.5505i	0.790612	−	0.790612i	0.981594	+	0.190982i	$0.0611672\pi$
$744$		0			0
$745$	25.6969	+	2.59592i	0.941463	+	0.0951071i
$746$		0			0
$747$		0			0
$748$		0			0
$749$			6.20204i			0.226618i
$750$		0			0
$751$	−5.37113			−0.195995			−0.0979976	−	0.995187i	$-0.531244\pi$
$751$	−5.37113			−0.195995			−0.0979976	+	0.995187i	$0.531244\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	0.985620	−	9.75663i	0.0358704	−	0.355080i
$756$		0			0
$757$	−12.2672	−	12.2672i	−0.445860	−	0.445860i	0.448116	−	0.893976i	$-0.352095\pi$
$757$	−12.2672	−	12.2672i	−0.445860	−	0.445860i	−0.893976	+	0.448116i	$0.852095\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			15.5563i			0.563917i	0.959427	+	0.281959i	$0.0909841\pi$
$761$			15.5563i			0.563917i	−0.959427	+	0.281959i	$0.909016\pi$
$762$		0			0
$763$	−16.8990	+	16.8990i	−0.611784	+	0.611784i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	2.00000	+	2.00000i	0.0722158	+	0.0722158i
$768$		0			0
$769$		−	7.59592i		−	0.273916i	−0.990577	−	0.136958i	$-0.956267\pi$
$769$		−	7.59592i		−	0.273916i	0.990577	−	0.136958i	$-0.0437325\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	22.8990	+	22.8990i	0.823619	+	0.823619i	0.986625	−	0.163006i	$-0.0521190\pi$
$773$	22.8990	+	22.8990i	0.823619	+	0.823619i	−0.163006	+	0.986625i	$0.552119\pi$
$774$		0			0
$775$	−47.7975	−	9.75663i	−1.71694	−	0.350469i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		−	9.75663i		−	0.349568i
$780$		0			0
$781$			10.7423i			0.384388i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−4.59259	+	3.74983i	−0.163916	+	0.133837i
$786$		0			0
$787$	−21.3939	−	21.3939i	−0.762609	−	0.762609i	0.214184	−	0.976793i	$-0.431291\pi$
$787$	−21.3939	−	21.3939i	−0.762609	−	0.762609i	−0.976793	+	0.214184i	$0.931291\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			33.7980i			1.20172i
$792$		0			0
$793$	−23.7980	−	23.7980i	−0.845090	−	0.845090i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−26.0454	+	26.0454i	−0.922576	+	0.922576i	−0.997211	−	0.0746352i	$-0.976221\pi$
$797$	−26.0454	+	26.0454i	−0.922576	+	0.922576i	0.0746352	+	0.997211i	$0.476221\pi$
$798$		0			0
$799$		−	15.1278i		−	0.535182i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	6.57826	+	6.57826i	0.232142	+	0.232142i
$804$		0			0
$805$	−27.9985	−	2.82843i	−0.986819	−	0.0996890i
$806$		0			0
$807$		0			0
$808$		0			0
$809$	−53.3115			−1.87433			−0.937167	−	0.348882i	$-0.886561\pi$
$809$	−53.3115			−1.87433			−0.937167	+	0.348882i	$0.886561\pi$
$810$		0			0
$811$		−	33.1918i		−	1.16552i	−0.812643	−	0.582761i	$-0.801972\pi$
$811$		−	33.1918i		−	1.16552i	0.812643	−	0.582761i	$-0.198028\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−7.59592	−	9.30306i	−0.266073	−	0.325872i
$816$		0			0
$817$	−47.5959	+	47.5959i	−1.66517	+	1.66517i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	8.44949			0.294889			0.147445	−	0.989070i	$-0.452895\pi$
$821$	8.44949			0.294889			0.147445	+	0.989070i	$0.452895\pi$
$822$		0			0
$823$	−4.24264	−	4.24264i	−0.147889	−	0.147889i	0.629285	−	0.777174i	$-0.283348\pi$
$823$	−4.24264	−	4.24264i	−0.147889	−	0.147889i	−0.777174	+	0.629285i	$0.783348\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	38.0409	−	38.0409i	1.32281	−	1.32281i	0.411321	−	0.911490i	$-0.365067\pi$
$827$	38.0409	−	38.0409i	1.32281	−	1.32281i	0.911490	−	0.411321i	$-0.134933\pi$
$828$		0			0
$829$	35.8481			1.24506			0.622529	−	0.782597i	$-0.286105\pi$
$829$	35.8481			1.24506			0.622529	+	0.782597i	$0.286105\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$	14.6349	−	14.6349i	0.507071	−	0.507071i
$834$		0			0
$835$	−8.69694	+	7.10102i	−0.300970	+	0.245741i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−43.1010			−1.48801			−0.744006	−	0.668173i	$-0.767077\pi$
$839$	−43.1010			−1.48801			−0.744006	+	0.668173i	$0.767077\pi$
$840$		0			0
$841$	−42.3939			−1.46186
$842$		0			0
$843$		0			0
$844$		0			0
$845$	15.1237	+	1.52781i	0.520272	+	0.0525581i
$846$		0			0
$847$	−1.41421	+	1.41421i	−0.0485930	+	0.0485930i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	22.3417			0.765863
$852$		0			0
$853$	0.953512	−	0.953512i	0.0326476	−	0.0326476i	−0.690595	−	0.723242i	$-0.742651\pi$
$853$	0.953512	−	0.953512i	0.0326476	−	0.0326476i	0.723242	+	0.690595i	$0.242651\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−14.6349	−	14.6349i	−0.499920	−	0.499920i	0.411493	−	0.911413i	$-0.365008\pi$
$857$	−14.6349	−	14.6349i	−0.499920	−	0.499920i	−0.911413	+	0.411493i	$0.865008\pi$
$858$		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
$863$	−29.3485	+	29.3485i	−0.999034	+	0.999034i	−1.00000		0.000966022i	$-0.999693\pi$
$863$	−29.3485	+	29.3485i	−0.999034	+	0.999034i	0.000966022		1.00000i	$0.499693\pi$
$864$		0			0
$865$	−3.79796	+	3.10102i	−0.129134	+	0.105438i
$866$		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
$877$	17.6383	+	17.6383i	0.595605	+	0.595605i	0.939140	−	0.343535i	$-0.111624\pi$
$877$	17.6383	+	17.6383i	0.595605	+	0.595605i	−0.343535	+	0.939140i	$0.611624\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$			7.35680i			0.247857i	0.992291	+	0.123928i	$0.0395493\pi$
$881$			7.35680i			0.247857i	−0.992291	+	0.123928i	$0.960451\pi$
$882$		0			0
$883$	37.5959	−	37.5959i	1.26520	−	1.26520i	0.316666	−	0.948537i	$-0.397436\pi$
$883$	37.5959	−	37.5959i	1.26520	−	1.26520i	0.948537	−	0.316666i	$-0.102564\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−10.2474	−	10.2474i	−0.344076	−	0.344076i	0.513821	−	0.857897i	$-0.328229\pi$
$887$	−10.2474	−	10.2474i	−0.344076	−	0.344076i	−0.857897	+	0.513821i	$0.828229\pi$
$888$		0			0
$889$			25.3939i			0.851683i
$890$		0			0
$891$		0			0
$892$		0			0
$893$	−10.6969	−	10.6969i	−0.357959	−	0.357959i
$894$		0			0
$895$	32.8769	+	3.32124i	1.09895	+	0.111017i
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	82.4385i		−	2.74948i
$900$		0			0
$901$			67.3108i			2.24245i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	16.8277	+	1.69994i	0.559372	+	0.0565080i
$906$		0			0
$907$	−23.1010	−	23.1010i	−0.767057	−	0.767057i	0.210530	−	0.977587i	$-0.432481\pi$
$907$	−23.1010	−	23.1010i	−0.767057	−	0.767057i	−0.977587	+	0.210530i	$0.932481\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$		−	24.4949i		−	0.811552i	−0.913973	−	0.405776i	$-0.867001\pi$
$911$		−	24.4949i		−	0.811552i	0.913973	−	0.405776i	$-0.132999\pi$
$912$		0			0
$913$	7.59592	+	7.59592i	0.251388	+	0.251388i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	22.6969	−	22.6969i	0.749519	−	0.749519i
$918$		0			0
$919$			5.37113i			0.177177i	0.996068	+	0.0885885i	$0.0282356\pi$
$919$			5.37113i			0.177177i	−0.996068	+	0.0885885i	$0.971764\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	9.75663	+	9.75663i	0.321143	+	0.321143i
$924$		0			0
$925$	−9.78874	−	14.8099i	−0.321852	−	0.486947i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	9.89949			0.324792			0.162396	−	0.986726i	$-0.448078\pi$
$929$	9.89949			0.324792			0.162396	+	0.986726i	$0.448078\pi$
$930$		0			0
$931$		−	20.6969i		−	0.678315i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	41.3939	−	33.7980i	1.35372	−	1.10531i
$936$		0			0
$937$	−18.7980	+	18.7980i	−0.614103	+	0.614103i	−0.944012	−	0.329910i	$-0.892982\pi$
$937$	−18.7980	+	18.7980i	−0.614103	+	0.614103i	0.329910	+	0.944012i	$0.392982\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−17.7526			−0.578717			−0.289358	−	0.957221i	$-0.593442\pi$
$941$	−17.7526			−0.578717			−0.289358	+	0.957221i	$0.593442\pi$
$942$		0			0
$943$	6.29253	+	6.29253i	0.204913	+	0.204913i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−14.1421	+	14.1421i	−0.459558	+	0.459558i	−0.898510	−	0.438953i	$-0.855350\pi$
$947$	−14.1421	+	14.1421i	−0.459558	+	0.459558i	0.438953	+	0.898510i	$0.355350\pi$
$948$		0			0
$949$	11.9494			0.387893
$950$		0			0
$951$		0			0
$952$		0			0
$953$	−7.56388	+	7.56388i	−0.245018	+	0.245018i	−0.818922	−	0.573904i	$-0.805428\pi$
$953$	−7.56388	+	7.56388i	−0.245018	+	0.245018i	0.573904	+	0.818922i	$0.305428\pi$
$954$		0			0
$955$	30.6969	+	3.10102i	0.993330	+	0.100347i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−21.3939			−0.690844
$960$		0			0
$961$	64.1918			2.07070
$962$		0			0
$963$		0			0
$964$		0			0
$965$	46.0454	−	37.5959i	1.48225	−	1.21026i
$966$		0			0
$967$	35.0696	−	35.0696i	1.12776	−	1.12776i	0.137222	−	0.990540i	$-0.456183\pi$
$967$	35.0696	−	35.0696i	1.12776	−	1.12776i	0.990540	−	0.137222i	$-0.0438175\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	17.0348			0.546672			0.273336	−	0.961919i	$-0.411873\pi$
$971$	17.0348			0.546672			0.273336	+	0.961919i	$0.411873\pi$
$972$		0			0
$973$	−11.0280	+	11.0280i	−0.353541	+	0.353541i
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−9.26382	−	9.26382i	−0.296376	−	0.296376i	0.543217	−	0.839593i	$-0.317206\pi$
$977$	−9.26382	−	9.26382i	−0.296376	−	0.296376i	−0.839593	+	0.543217i	$0.817206\pi$
$978$		0			0
$979$	−14.6969			−0.469716
$980$		0			0
$981$		0			0
$982$		0			0
$983$	−15.3485	+	15.3485i	−0.489540	+	0.489540i	−0.908161	−	0.418621i	$-0.862514\pi$
$983$	−15.3485	+	15.3485i	−0.489540	+	0.489540i	0.418621	+	0.908161i	$0.362514\pi$
$984$		0			0
$985$	12.8990	+	15.7980i	0.410996	+	0.503365i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	61.3939i		−	1.95221i
$990$		0			0
$991$	23.8988			0.759169			0.379585	−	0.925157i	$-0.376067\pi$
$991$	23.8988			0.759169			0.379585	+	0.925157i	$0.376067\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−43.4120	−	4.38551i	−1.37625	−	0.139030i
$996$		0			0
$997$	−19.1954	−	19.1954i	−0.607925	−	0.607925i	0.334478	−	0.942403i	$-0.391440\pi$
$997$	−19.1954	−	19.1954i	−0.607925	−	0.607925i	−0.942403	+	0.334478i	$0.891440\pi$
$998$		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.1313.1	yes	8
3.2	odd	2		2880.2.bj.a.1313.3	yes	8
4.3	odd	2		2880.2.bj.d.1313.2	yes	8
5.2	odd	4		2880.2.bj.d.737.1	yes	8
8.3	odd	2		2880.2.bj.a.1313.4	yes	8
8.5	even	2		2880.2.bj.b.1313.3	yes	8
12.11	even	2		2880.2.bj.b.1313.4	yes	8
15.2	even	4		2880.2.bj.b.737.3	yes	8
20.7	even	4	inner	2880.2.bj.c.737.2	yes	8
24.5	odd	2		2880.2.bj.d.1313.1	yes	8
24.11	even	2	inner	2880.2.bj.c.1313.2	yes	8
40.27	even	4		2880.2.bj.b.737.4	yes	8
40.37	odd	4		2880.2.bj.a.737.3	✓	8
60.47	odd	4		2880.2.bj.a.737.4	yes	8
120.77	even	4	inner	2880.2.bj.c.737.1	yes	8
120.107	odd	4		2880.2.bj.d.737.2	yes	8

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

\(f(q)\)	\(=\)	\(q+(-0.224745 + 2.22474i) q^{5} +(-1.41421 + 1.41421i) q^{7} +O(q^{10})\)	\(q+(-0.224745 + 2.22474i) q^{5} +(-1.41421 + 1.41421i) q^{7} +3.46410 q^{11} +(3.14626 - 3.14626i) q^{13} +(-4.87832 - 4.87832i) q^{17} -6.89898 q^{19} +(4.44949 - 4.44949i) q^{23} +(-4.89898 - 1.00000i) q^{25} -8.44949i q^{29} +9.75663 q^{31} +(-2.82843 - 3.46410i) q^{35} +(2.51059 + 2.51059i) q^{37} +1.41421i q^{41} +(6.89898 - 6.89898i) q^{43} +(1.55051 + 1.55051i) q^{47} +3.00000i q^{49} +(-6.89898 - 6.89898i) q^{53} +(-0.778539 + 7.70674i) q^{55} +0.635674i q^{59} -7.56388i q^{61} +(6.29253 + 7.70674i) q^{65} +3.10102i q^{71} +(1.89898 + 1.89898i) q^{73} +(-4.89898 + 4.89898i) q^{77} +14.1421i q^{79} +(2.19275 + 2.19275i) q^{83} +(11.9494 - 9.75663i) q^{85} -4.24264 q^{89} +8.89898i q^{91} +(1.55051 - 15.3485i) q^{95} +(1.89898 - 1.89898i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{5}+O(q^{10}) \) 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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