LMFDB - Embedded newform 2880.2.k.k.1441.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(1441,2880)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2880, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 1, 0, 0]))

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

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

chi := DirichletCharacter("2880.1441");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.k (of order $2$, degree $1$, 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:	$4$
Coefficient field:	$\Q(\zeta_{12})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{2} + 1 $ x^4 - x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 960)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1441.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	2880.1441
Dual form			2880.2.k.k.1441.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} +2.00000 q^{7} +O(q^{10})$	$q-1.00000i q^{5} +2.00000 q^{7} +5.46410i q^{11} -5.46410i q^{13} -3.46410 q^{17} -6.92820i q^{19} -4.00000 q^{23} -1.00000 q^{25} -8.92820i q^{29} +0.535898 q^{31} -2.00000i q^{35} +9.46410i q^{37} -4.92820 q^{41} -6.92820i q^{43} +4.00000 q^{47} -3.00000 q^{49} +0.928203i q^{53} +5.46410 q^{55} -9.46410i q^{59} -4.00000i q^{61} -5.46410 q^{65} -6.92820i q^{67} +6.92820 q^{71} +10.0000 q^{73} +10.9282i q^{77} +14.3923 q^{79} +8.00000i q^{83} +3.46410i q^{85} -0.928203 q^{89} -10.9282i q^{91} -6.92820 q^{95} -12.9282 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{7}+O(q^{10}) $ 4 * q + 8 * q^7	$ 4 q + 8 q^{7} - 16 q^{23} - 4 q^{25} + 16 q^{31} + 8 q^{41} + 16 q^{47} - 12 q^{49} + 8 q^{55} - 8 q^{65} + 40 q^{73} + 16 q^{79} + 24 q^{89} - 24 q^{97}+O(q^{100}) $ 4 * q + 8 * q^7 - 16 * q^23 - 4 * q^25 + 16 * q^31 + 8 * q^41 + 16 * q^47 - 12 * q^49 + 8 * q^55 - 8 * q^65 + 40 * q^73 + 16 * q^79 + 24 * q^89 - 24 * q^97

Display 100 coefficients 10 coefficients

Character values

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

$n$	$577$	$641$	$901$	$2431$
$\chi(n)$	$1$	$1$	$-1$	$1$

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
$5$	− 1.00000i	− 0.447214i
$5$	− 1.00000i	− 0.447214i
$6$	0	0
$7$	2.00000	0.755929	0.377964	−	0.925820i	$-0.376624\pi$
$7$	2.00000	0.755929	0.377964	+	0.925820i	$0.376624\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	5.46410i	1.64749i	0.566961	+	0.823744i	$0.308119\pi$
$11$	5.46410i	1.64749i	−0.566961	+	0.823744i	$0.691881\pi$
$12$	0	0
$13$	− 5.46410i	− 1.51547i	−0.652563	−	0.757735i	$-0.726306\pi$
$13$	− 5.46410i	− 1.51547i	0.652563	−	0.757735i	$-0.273694\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−3.46410	−0.840168	−0.420084	−	0.907485i	$-0.637999\pi$
$17$	−3.46410	−0.840168	−0.420084	+	0.907485i	$0.637999\pi$
$18$	0	0
$19$	− 6.92820i	− 1.58944i	−0.606977	−	0.794719i	$-0.707618\pi$
$19$	− 6.92820i	− 1.58944i	0.606977	−	0.794719i	$-0.292382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−4.00000	−0.834058	−0.417029	−	0.908893i	$-0.636929\pi$
$23$	−4.00000	−0.834058	−0.417029	+	0.908893i	$0.636929\pi$
$24$	0	0
$25$	−1.00000	−0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.92820i	− 1.65793i	−0.559304	−	0.828963i	$-0.688931\pi$
$29$	− 8.92820i	− 1.65793i	0.559304	−	0.828963i	$-0.311069\pi$
$30$	0	0
$31$	0.535898	0.0962502	0.0481251	−	0.998841i	$-0.484675\pi$
$31$	0.535898	0.0962502	0.0481251	+	0.998841i	$0.484675\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	− 2.00000i	− 0.338062i
$36$	0	0
$37$	9.46410i	1.55589i	0.628333	+	0.777944i	$0.283737\pi$
$37$	9.46410i	1.55589i	−0.628333	+	0.777944i	$0.716263\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	−4.92820	−0.769656	−0.384828	−	0.922988i	$-0.625739\pi$
$41$	−4.92820	−0.769656	−0.384828	+	0.922988i	$0.625739\pi$
$42$	0	0
$43$	− 6.92820i	− 1.05654i	−0.849076	−	0.528271i	$-0.822841\pi$
$43$	− 6.92820i	− 1.05654i	0.849076	−	0.528271i	$-0.177159\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	4.00000	0.583460	0.291730	−	0.956501i	$-0.405769\pi$
$47$	4.00000	0.583460	0.291730	+	0.956501i	$0.405769\pi$
$48$	0	0
$49$	−3.00000	−0.428571
$50$	0	0
$51$	0	0
$52$	0	0
$53$	0.928203i	0.127499i	0.997966	+	0.0637493i	$0.0203058\pi$
$53$	0.928203i	0.127499i	−0.997966	+	0.0637493i	$0.979694\pi$
$54$	0	0
$55$	5.46410	0.736779
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 9.46410i	− 1.23212i	−0.787699	−	0.616061i	$-0.788728\pi$
$59$	− 9.46410i	− 1.23212i	0.787699	−	0.616061i	$-0.211272\pi$
$60$	0	0
$61$	− 4.00000i	− 0.512148i	−0.966657	−	0.256074i	$-0.917571\pi$
$61$	− 4.00000i	− 0.512148i	0.966657	−	0.256074i	$-0.0824290\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	−5.46410	−0.677738
$66$	0	0
$67$	− 6.92820i	− 0.846415i	−0.906033	−	0.423207i	$-0.860904\pi$
$67$	− 6.92820i	− 0.846415i	0.906033	−	0.423207i	$-0.139096\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	6.92820	0.822226	0.411113	−	0.911584i	$-0.365140\pi$
$71$	6.92820	0.822226	0.411113	+	0.911584i	$0.365140\pi$
$72$	0	0
$73$	10.0000	1.17041	0.585206	−	0.810885i	$-0.301014\pi$
$73$	10.0000	1.17041	0.585206	+	0.810885i	$0.301014\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	10.9282i	1.24538i
$78$	0	0
$79$	14.3923	1.61926	0.809630	−	0.586940i	$-0.199668\pi$
$79$	14.3923	1.61926	0.809630	+	0.586940i	$0.199668\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	8.00000i	0.878114i	0.898459	+	0.439057i	$0.144687\pi$
$83$	8.00000i	0.878114i	−0.898459	+	0.439057i	$0.855313\pi$
$84$	0	0
$85$	3.46410i	0.375735i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−0.928203	−0.0983893	−0.0491947	−	0.998789i	$-0.515665\pi$
$89$	−0.928203	−0.0983893	−0.0491947	+	0.998789i	$0.515665\pi$
$90$	0	0
$91$	− 10.9282i	− 1.14559i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−6.92820	−0.710819
$96$	0	0
$97$	−12.9282	−1.31266	−0.656330	−	0.754474i	$-0.727892\pi$
$97$	−12.9282	−1.31266	−0.656330	+	0.754474i	$0.727892\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 10.0000i	− 0.995037i	−0.867453	−	0.497519i	$-0.834245\pi$
$101$	− 10.0000i	− 0.995037i	0.867453	−	0.497519i	$-0.165755\pi$
$102$	0	0
$103$	−4.92820	−0.485590	−0.242795	−	0.970078i	$-0.578064\pi$
$103$	−4.92820	−0.485590	−0.242795	+	0.970078i	$0.578064\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	− 6.92820i	− 0.669775i	−0.942258	−	0.334887i	$-0.891302\pi$
$107$	− 6.92820i	− 0.669775i	0.942258	−	0.334887i	$-0.108698\pi$
$108$	0	0
$109$	− 5.07180i	− 0.485790i	−0.970053	−	0.242895i	$-0.921903\pi$
$109$	− 5.07180i	− 0.485790i	0.970053	−	0.242895i	$-0.0780971\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	−7.46410	−0.702164	−0.351082	−	0.936345i	$-0.614186\pi$
$113$	−7.46410	−0.702164	−0.351082	+	0.936345i	$0.614186\pi$
$114$	0	0
$115$	4.00000i	0.373002i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−6.92820	−0.635107
$120$	0	0
$121$	−18.8564	−1.71422
$122$	0	0
$123$	0	0
$124$	0	0
$125$	1.00000i	0.0894427i
$126$	0	0
$127$	7.07180	0.627520	0.313760	−	0.949502i	$-0.398411\pi$
$127$	7.07180	0.627520	0.313760	+	0.949502i	$0.398411\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 1.46410i	− 0.127919i	−0.997952	−	0.0639596i	$-0.979627\pi$
$131$	− 1.46410i	− 0.127919i	0.997952	−	0.0639596i	$-0.0203729\pi$
$132$	0	0
$133$	− 13.8564i	− 1.20150i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	−1.60770	−0.137355	−0.0686773	−	0.997639i	$-0.521878\pi$
$137$	−1.60770	−0.137355	−0.0686773	+	0.997639i	$0.521878\pi$
$138$	0	0
$139$	12.0000i	1.01783i	0.860818	+	0.508913i	$0.169953\pi$
$139$	12.0000i	1.01783i	−0.860818	+	0.508913i	$0.830047\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	29.8564	2.49672
$144$	0	0
$145$	−8.92820	−0.741447
$146$	0	0
$147$	0	0
$148$	0	0
$149$	− 22.7846i	− 1.86659i	−0.359113	−	0.933294i	$-0.616921\pi$
$149$	− 22.7846i	− 1.86659i	0.359113	−	0.933294i	$-0.383079\pi$
$150$	0	0
$151$	−11.4641	−0.932935	−0.466468	−	0.884538i	$-0.654474\pi$
$151$	−11.4641	−0.932935	−0.466468	+	0.884538i	$0.654474\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 0.535898i	− 0.0430444i
$156$	0	0
$157$	− 14.5359i	− 1.16009i	−0.814584	−	0.580045i	$-0.803035\pi$
$157$	− 14.5359i	− 1.16009i	0.814584	−	0.580045i	$-0.196965\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−8.00000	−0.630488
$162$	0	0
$163$	− 4.00000i	− 0.313304i	−0.987654	−	0.156652i	$-0.949930\pi$
$163$	− 4.00000i	− 0.313304i	0.987654	−	0.156652i	$-0.0500701\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−13.8564	−1.07224	−0.536120	−	0.844141i	$-0.680111\pi$
$167$	−13.8564	−1.07224	−0.536120	+	0.844141i	$0.680111\pi$
$168$	0	0
$169$	−16.8564	−1.29665
$170$	0	0
$171$	0	0
$172$	0	0
$173$	24.9282i	1.89526i	0.319376	+	0.947628i	$0.396527\pi$
$173$	24.9282i	1.89526i	−0.319376	+	0.947628i	$0.603473\pi$
$174$	0	0
$175$	−2.00000	−0.151186
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 4.39230i	− 0.328296i	−0.986436	−	0.164148i	$-0.947512\pi$
$179$	− 4.39230i	− 0.328296i	0.986436	−	0.164148i	$-0.0524875\pi$
$180$	0	0
$181$	− 10.9282i	− 0.812287i	−0.913809	−	0.406143i	$-0.866873\pi$
$181$	− 10.9282i	− 0.812287i	0.913809	−	0.406143i	$-0.133127\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	9.46410	0.695815
$186$	0	0
$187$	− 18.9282i	− 1.38417i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	−8.00000	−0.578860	−0.289430	−	0.957199i	$-0.593466\pi$
$191$	−8.00000	−0.578860	−0.289430	+	0.957199i	$0.593466\pi$
$192$	0	0
$193$	−22.7846	−1.64007	−0.820036	−	0.572312i	$-0.806047\pi$
$193$	−22.7846	−1.64007	−0.820036	+	0.572312i	$0.806047\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	3.85641i	0.274758i	0.990519	+	0.137379i	$0.0438678\pi$
$197$	3.85641i	0.274758i	−0.990519	+	0.137379i	$0.956132\pi$
$198$	0	0
$199$	9.32051	0.660713	0.330357	−	0.943856i	$-0.392831\pi$
$199$	9.32051	0.660713	0.330357	+	0.943856i	$0.392831\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 17.8564i	− 1.25327i
$204$	0	0
$205$	4.92820i	0.344201i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	37.8564	2.61858
$210$	0	0
$211$	− 4.00000i	− 0.275371i	−0.990476	−	0.137686i	$-0.956034\pi$
$211$	− 4.00000i	− 0.275371i	0.990476	−	0.137686i	$-0.0439664\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−6.92820	−0.472500
$216$	0	0
$217$	1.07180	0.0727583
$218$	0	0
$219$	0	0
$220$	0	0
$221$	18.9282i	1.27325i
$222$	0	0
$223$	7.07180	0.473563	0.236781	−	0.971563i	$-0.423908\pi$
$223$	7.07180	0.473563	0.236781	+	0.971563i	$0.423908\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	− 26.9282i	− 1.78729i	−0.448777	−	0.893644i	$-0.648140\pi$
$227$	− 26.9282i	− 1.78729i	0.448777	−	0.893644i	$-0.351860\pi$
$228$	0	0
$229$	29.8564i	1.97297i	0.163861	+	0.986483i	$0.447605\pi$
$229$	29.8564i	1.97297i	−0.163861	+	0.986483i	$0.552395\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−11.4641	−0.751038	−0.375519	−	0.926815i	$-0.622536\pi$
$233$	−11.4641	−0.751038	−0.375519	+	0.926815i	$0.622536\pi$
$234$	0	0
$235$	− 4.00000i	− 0.260931i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	16.0000	1.03495	0.517477	−	0.855697i	$-0.326871\pi$
$239$	16.0000	1.03495	0.517477	+	0.855697i	$0.326871\pi$
$240$	0	0
$241$	26.0000	1.67481	0.837404	−	0.546585i	$-0.184072\pi$
$241$	26.0000	1.67481	0.837404	+	0.546585i	$0.184072\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.00000i	0.191663i
$246$	0	0
$247$	−37.8564	−2.40875
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 11.3205i	− 0.714544i	−0.934000	−	0.357272i	$-0.883707\pi$
$251$	− 11.3205i	− 0.714544i	0.934000	−	0.357272i	$-0.116293\pi$
$252$	0	0
$253$	− 21.8564i	− 1.37410i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	0.535898	0.0334284	0.0167142	−	0.999860i	$-0.494679\pi$
$257$	0.535898	0.0334284	0.0167142	+	0.999860i	$0.494679\pi$
$258$	0	0
$259$	18.9282i	1.17614i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	27.7128	1.70885	0.854423	−	0.519579i	$-0.173911\pi$
$263$	27.7128	1.70885	0.854423	+	0.519579i	$0.173911\pi$
$264$	0	0
$265$	0.928203	0.0570191
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 14.7846i	− 0.901434i	−0.892667	−	0.450717i	$-0.851168\pi$
$269$	− 14.7846i	− 0.901434i	0.892667	−	0.450717i	$-0.148832\pi$
$270$	0	0
$271$	4.53590	0.275536	0.137768	−	0.990465i	$-0.456007\pi$
$271$	4.53590	0.275536	0.137768	+	0.990465i	$0.456007\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	− 5.46410i	− 0.329498i
$276$	0	0
$277$	− 13.4641i	− 0.808979i	−0.914543	−	0.404490i	$-0.867449\pi$
$277$	− 13.4641i	− 0.808979i	0.914543	−	0.404490i	$-0.132551\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	12.9282	0.771232	0.385616	−	0.922659i	$-0.373989\pi$
$281$	12.9282	0.771232	0.385616	+	0.922659i	$0.373989\pi$
$282$	0	0
$283$	− 9.07180i	− 0.539262i	−0.962964	−	0.269631i	$-0.913098\pi$
$283$	− 9.07180i	− 0.539262i	0.962964	−	0.269631i	$-0.0869018\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−9.85641	−0.581805
$288$	0	0
$289$	−5.00000	−0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	− 31.8564i	− 1.86107i	−0.366202	−	0.930536i	$-0.619342\pi$
$293$	− 31.8564i	− 1.86107i	0.366202	−	0.930536i	$-0.380658\pi$
$294$	0	0
$295$	−9.46410	−0.551021
$296$	0	0
$297$	0	0
$298$	0	0
$299$	21.8564i	1.26399i
$300$	0	0
$301$	− 13.8564i	− 0.798670i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	−4.00000	−0.229039
$306$	0	0
$307$	17.8564i	1.01912i	0.860435	+	0.509559i	$0.170192\pi$
$307$	17.8564i	1.01912i	−0.860435	+	0.509559i	$0.829808\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	29.8564	1.69300	0.846501	−	0.532388i	$-0.178705\pi$
$311$	29.8564	1.69300	0.846501	+	0.532388i	$0.178705\pi$
$312$	0	0
$313$	8.92820	0.504652	0.252326	−	0.967642i	$-0.418804\pi$
$313$	8.92820	0.504652	0.252326	+	0.967642i	$0.418804\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	15.8564i	0.890585i	0.895385	+	0.445292i	$0.146900\pi$
$317$	15.8564i	0.890585i	−0.895385	+	0.445292i	$0.853100\pi$
$318$	0	0
$319$	48.7846	2.73141
$320$	0	0
$321$	0	0
$322$	0	0
$323$	24.0000i	1.33540i
$324$	0	0
$325$	5.46410i	0.303094i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	8.00000	0.441054
$330$	0	0
$331$	22.9282i	1.26025i	0.776495	+	0.630124i	$0.216996\pi$
$331$	22.9282i	1.26025i	−0.776495	+	0.630124i	$0.783004\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−6.92820	−0.378528
$336$	0	0
$337$	3.85641	0.210072	0.105036	−	0.994468i	$-0.466504\pi$
$337$	3.85641	0.210072	0.105036	+	0.994468i	$0.466504\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.92820i	0.158571i
$342$	0	0
$343$	−20.0000	−1.07990
$344$	0	0
$345$	0	0
$346$	0	0
$347$	10.9282i	0.586657i	0.956012	+	0.293328i	$0.0947629\pi$
$347$	10.9282i	0.586657i	−0.956012	+	0.293328i	$0.905237\pi$
$348$	0	0
$349$	− 4.00000i	− 0.214115i	−0.994253	−	0.107058i	$-0.965857\pi$
$349$	− 4.00000i	− 0.214115i	0.994253	−	0.107058i	$-0.0341429\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	3.46410	0.184376	0.0921878	−	0.995742i	$-0.470614\pi$
$353$	3.46410	0.184376	0.0921878	+	0.995742i	$0.470614\pi$
$354$	0	0
$355$	− 6.92820i	− 0.367711i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−9.07180	−0.478791	−0.239396	−	0.970922i	$-0.576949\pi$
$359$	−9.07180	−0.478791	−0.239396	+	0.970922i	$0.576949\pi$
$360$	0	0
$361$	−29.0000	−1.52632
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 10.0000i	− 0.523424i
$366$	0	0
$367$	−2.00000	−0.104399	−0.0521996	−	0.998637i	$-0.516623\pi$
$367$	−2.00000	−0.104399	−0.0521996	+	0.998637i	$0.516623\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	1.85641i	0.0963798i
$372$	0	0
$373$	12.3923i	0.641649i	0.947139	+	0.320825i	$0.103960\pi$
$373$	12.3923i	0.641649i	−0.947139	+	0.320825i	$0.896040\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	−48.7846	−2.51254
$378$	0	0
$379$	20.0000i	1.02733i	0.857991	+	0.513665i	$0.171713\pi$
$379$	20.0000i	1.02733i	−0.857991	+	0.513665i	$0.828287\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−10.1436	−0.518313	−0.259157	−	0.965835i	$-0.583445\pi$
$383$	−10.1436	−0.518313	−0.259157	+	0.965835i	$0.583445\pi$
$384$	0	0
$385$	10.9282	0.556953
$386$	0	0
$387$	0	0
$388$	0	0
$389$	4.14359i	0.210089i	0.994468	+	0.105044i	$0.0334984\pi$
$389$	4.14359i	0.210089i	−0.994468	+	0.105044i	$0.966502\pi$
$390$	0	0
$391$	13.8564	0.700749
$392$	0	0
$393$	0	0
$394$	0	0
$395$	− 14.3923i	− 0.724155i
$396$	0	0
$397$	36.3923i	1.82648i	0.407425	+	0.913239i	$0.366427\pi$
$397$	36.3923i	1.82648i	−0.407425	+	0.913239i	$0.633573\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−22.0000	−1.09863	−0.549314	−	0.835616i	$-0.685111\pi$
$401$	−22.0000	−1.09863	−0.549314	+	0.835616i	$0.685111\pi$
$402$	0	0
$403$	− 2.92820i	− 0.145864i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−51.7128	−2.56331
$408$	0	0
$409$	23.8564	1.17962	0.589812	−	0.807541i	$-0.299202\pi$
$409$	23.8564	1.17962	0.589812	+	0.807541i	$0.299202\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 18.9282i	− 0.931396i
$414$	0	0
$415$	8.00000	0.392705
$416$	0	0
$417$	0	0
$418$	0	0
$419$	19.3205i	0.943869i	0.881634	+	0.471934i	$0.156444\pi$
$419$	19.3205i	0.943869i	−0.881634	+	0.471934i	$0.843556\pi$
$420$	0	0
$421$	16.0000i	0.779792i	0.920859	+	0.389896i	$0.127489\pi$
$421$	16.0000i	0.779792i	−0.920859	+	0.389896i	$0.872511\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	3.46410	0.168034
$426$	0	0
$427$	− 8.00000i	− 0.387147i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	17.0718	0.822320	0.411160	−	0.911563i	$-0.365124\pi$
$431$	17.0718	0.822320	0.411160	+	0.911563i	$0.365124\pi$
$432$	0	0
$433$	19.8564	0.954238	0.477119	−	0.878839i	$-0.341681\pi$
$433$	19.8564	0.954238	0.477119	+	0.878839i	$0.341681\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	27.7128i	1.32568i
$438$	0	0
$439$	1.32051	0.0630244	0.0315122	−	0.999503i	$-0.489968\pi$
$439$	1.32051	0.0630244	0.0315122	+	0.999503i	$0.489968\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	38.6410i	1.83589i	0.396708	+	0.917945i	$0.370153\pi$
$443$	38.6410i	1.83589i	−0.396708	+	0.917945i	$0.629847\pi$
$444$	0	0
$445$	0.928203i	0.0440011i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	−17.7128	−0.835919	−0.417960	−	0.908466i	$-0.637255\pi$
$449$	−17.7128	−0.835919	−0.417960	+	0.908466i	$0.637255\pi$
$450$	0	0
$451$	− 26.9282i	− 1.26800i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−10.9282	−0.512322
$456$	0	0
$457$	4.92820	0.230532	0.115266	−	0.993335i	$-0.463228\pi$
$457$	4.92820	0.230532	0.115266	+	0.993335i	$0.463228\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	3.07180i	0.143068i	0.997438	+	0.0715339i	$0.0227894\pi$
$461$	3.07180i	0.143068i	−0.997438	+	0.0715339i	$0.977211\pi$
$462$	0	0
$463$	−11.8564	−0.551014	−0.275507	−	0.961299i	$-0.588846\pi$
$463$	−11.8564	−0.551014	−0.275507	+	0.961299i	$0.588846\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	− 10.1436i	− 0.469390i	−0.972069	−	0.234695i	$-0.924591\pi$
$467$	− 10.1436i	− 0.469390i	0.972069	−	0.234695i	$-0.0754091\pi$
$468$	0	0
$469$	− 13.8564i	− 0.639829i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	37.8564	1.74064
$474$	0	0
$475$	6.92820i	0.317888i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	4.78461	0.218614	0.109307	−	0.994008i	$-0.465137\pi$
$479$	4.78461	0.218614	0.109307	+	0.994008i	$0.465137\pi$
$480$	0	0
$481$	51.7128	2.35790
$482$	0	0
$483$	0	0
$484$	0	0
$485$	12.9282i	0.587039i
$486$	0	0
$487$	−22.7846	−1.03247	−0.516235	−	0.856447i	$-0.672667\pi$
$487$	−22.7846	−1.03247	−0.516235	+	0.856447i	$0.672667\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	31.3205i	1.41347i	0.707476	+	0.706737i	$0.249834\pi$
$491$	31.3205i	1.41347i	−0.707476	+	0.706737i	$0.750166\pi$
$492$	0	0
$493$	30.9282i	1.39294i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	13.8564	0.621545
$498$	0	0
$499$	− 22.9282i	− 1.02641i	−0.858267	−	0.513204i	$-0.828459\pi$
$499$	− 22.9282i	− 1.02641i	0.858267	−	0.513204i	$-0.171541\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−4.00000	−0.178351	−0.0891756	−	0.996016i	$-0.528423\pi$
$503$	−4.00000	−0.178351	−0.0891756	+	0.996016i	$0.528423\pi$
$504$	0	0
$505$	−10.0000	−0.444994
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 43.8564i	− 1.94390i	−0.235185	−	0.971951i	$-0.575570\pi$
$509$	− 43.8564i	− 1.94390i	0.235185	−	0.971951i	$-0.424430\pi$
$510$	0	0
$511$	20.0000	0.884748
$512$	0	0
$513$	0	0
$514$	0	0
$515$	4.92820i	0.217163i
$516$	0	0
$517$	21.8564i	0.961244i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	30.0000	1.31432	0.657162	−	0.753749i	$-0.271757\pi$
$521$	30.0000	1.31432	0.657162	+	0.753749i	$0.271757\pi$
$522$	0	0
$523$	20.0000i	0.874539i	0.899331	+	0.437269i	$0.144054\pi$
$523$	20.0000i	0.874539i	−0.899331	+	0.437269i	$0.855946\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−1.85641	−0.0808663
$528$	0	0
$529$	−7.00000	−0.304348
$530$	0	0
$531$	0	0
$532$	0	0
$533$	26.9282i	1.16639i
$534$	0	0
$535$	−6.92820	−0.299532
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 16.3923i	− 0.706067i
$540$	0	0
$541$	10.9282i	0.469840i	0.972015	+	0.234920i	$0.0754828\pi$
$541$	10.9282i	0.469840i	−0.972015	+	0.234920i	$0.924517\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.07180	−0.217252
$546$	0	0
$547$	− 22.9282i	− 0.980339i	−0.871627	−	0.490170i	$-0.836935\pi$
$547$	− 22.9282i	− 0.980339i	0.871627	−	0.490170i	$-0.163065\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	−61.8564	−2.63517
$552$	0	0
$553$	28.7846	1.22405
$554$	0	0
$555$	0	0
$556$	0	0
$557$	14.7846i	0.626444i	0.949680	+	0.313222i	$0.101408\pi$
$557$	14.7846i	0.626444i	−0.949680	+	0.313222i	$0.898592\pi$
$558$	0	0
$559$	−37.8564	−1.60116
$560$	0	0
$561$	0	0
$562$	0	0
$563$	5.07180i	0.213751i	0.994272	+	0.106875i	$0.0340846\pi$
$563$	5.07180i	0.213751i	−0.994272	+	0.106875i	$0.965915\pi$
$564$	0	0
$565$	7.46410i	0.314017i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−4.14359	−0.173708	−0.0868542	−	0.996221i	$-0.527681\pi$
$569$	−4.14359	−0.173708	−0.0868542	+	0.996221i	$0.527681\pi$
$570$	0	0
$571$	14.9282i	0.624726i	0.949963	+	0.312363i	$0.101120\pi$
$571$	14.9282i	0.624726i	−0.949963	+	0.312363i	$0.898880\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	4.00000	0.166812
$576$	0	0
$577$	−15.0718	−0.627447	−0.313724	−	0.949514i	$-0.601577\pi$
$577$	−15.0718	−0.627447	−0.313724	+	0.949514i	$0.601577\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	16.0000i	0.663792i
$582$	0	0
$583$	−5.07180	−0.210052
$584$	0	0
$585$	0	0
$586$	0	0
$587$	34.6410i	1.42979i	0.699233	+	0.714894i	$0.253525\pi$
$587$	34.6410i	1.42979i	−0.699233	+	0.714894i	$0.746475\pi$
$588$	0	0
$589$	− 3.71281i	− 0.152984i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	2.67949	0.110034	0.0550168	−	0.998485i	$-0.482479\pi$
$593$	2.67949	0.110034	0.0550168	+	0.998485i	$0.482479\pi$
$594$	0	0
$595$	6.92820i	0.284029i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	34.6410	1.41539	0.707697	−	0.706516i	$-0.249734\pi$
$599$	34.6410	1.41539	0.707697	+	0.706516i	$0.249734\pi$
$600$	0	0
$601$	35.5692	1.45090	0.725449	−	0.688276i	$-0.241632\pi$
$601$	35.5692	1.45090	0.725449	+	0.688276i	$0.241632\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	18.8564i	0.766622i
$606$	0	0
$607$	18.0000	0.730597	0.365299	−	0.930890i	$-0.380967\pi$
$607$	18.0000	0.730597	0.365299	+	0.930890i	$0.380967\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 21.8564i	− 0.884216i
$612$	0	0
$613$	− 26.2487i	− 1.06018i	−0.847943	−	0.530088i	$-0.822159\pi$
$613$	− 26.2487i	− 1.06018i	0.847943	−	0.530088i	$-0.177841\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−49.3205	−1.98557	−0.992784	−	0.119913i	$-0.961739\pi$
$617$	−49.3205	−1.98557	−0.992784	+	0.119913i	$0.961739\pi$
$618$	0	0
$619$	− 6.14359i	− 0.246932i	−0.992349	−	0.123466i	$-0.960599\pi$
$619$	− 6.14359i	− 0.246932i	0.992349	−	0.123466i	$-0.0394010\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	−1.85641	−0.0743754
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 32.7846i	− 1.30721i
$630$	0	0
$631$	−30.3923	−1.20990	−0.604949	−	0.796264i	$-0.706807\pi$
$631$	−30.3923	−1.20990	−0.604949	+	0.796264i	$0.706807\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	− 7.07180i	− 0.280636i
$636$	0	0
$637$	16.3923i	0.649487i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	24.6410	0.973262	0.486631	−	0.873608i	$-0.338226\pi$
$641$	24.6410	0.973262	0.486631	+	0.873608i	$0.338226\pi$
$642$	0	0
$643$	39.7128i	1.56612i	0.621946	+	0.783060i	$0.286342\pi$
$643$	39.7128i	1.56612i	−0.621946	+	0.783060i	$0.713658\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	−17.8564	−0.702008	−0.351004	−	0.936374i	$-0.614160\pi$
$647$	−17.8564	−0.702008	−0.351004	+	0.936374i	$0.614160\pi$
$648$	0	0
$649$	51.7128	2.02991
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 7.85641i	− 0.307445i	−0.988114	−	0.153722i	$-0.950874\pi$
$653$	− 7.85641i	− 0.307445i	0.988114	−	0.153722i	$-0.0491262\pi$
$654$	0	0
$655$	−1.46410	−0.0572072
$656$	0	0
$657$	0	0
$658$	0	0
$659$	20.3923i	0.794371i	0.917738	+	0.397186i	$0.130013\pi$
$659$	20.3923i	0.794371i	−0.917738	+	0.397186i	$0.869987\pi$
$660$	0	0
$661$	− 4.00000i	− 0.155582i	−0.996970	−	0.0777910i	$-0.975213\pi$
$661$	− 4.00000i	− 0.155582i	0.996970	−	0.0777910i	$-0.0247867\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−13.8564	−0.537328
$666$	0	0
$667$	35.7128i	1.38281i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	21.8564	0.843757
$672$	0	0
$673$	25.7128	0.991156	0.495578	−	0.868563i	$-0.334956\pi$
$673$	25.7128	0.991156	0.495578	+	0.868563i	$0.334956\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	31.8564i	1.22434i	0.790726	+	0.612171i	$0.209703\pi$
$677$	31.8564i	1.22434i	−0.790726	+	0.612171i	$0.790297\pi$
$678$	0	0
$679$	−25.8564	−0.992278
$680$	0	0
$681$	0	0
$682$	0	0
$683$	− 45.8564i	− 1.75465i	−0.479900	−	0.877323i	$-0.659327\pi$
$683$	− 45.8564i	− 1.75465i	0.479900	−	0.877323i	$-0.340673\pi$
$684$	0	0
$685$	1.60770i	0.0614269i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.07180	0.193220
$690$	0	0
$691$	22.9282i	0.872230i	0.899891	+	0.436115i	$0.143646\pi$
$691$	22.9282i	0.872230i	−0.899891	+	0.436115i	$0.856354\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	12.0000	0.455186
$696$	0	0
$697$	17.0718	0.646640
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 51.8564i	− 1.95859i	−0.202440	−	0.979295i	$-0.564887\pi$
$701$	− 51.8564i	− 1.95859i	0.202440	−	0.979295i	$-0.435113\pi$
$702$	0	0
$703$	65.5692	2.47299
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 20.0000i	− 0.752177i
$708$	0	0
$709$	− 51.7128i	− 1.94212i	−0.238845	−	0.971058i	$-0.576769\pi$
$709$	− 51.7128i	− 1.94212i	0.238845	−	0.971058i	$-0.423231\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	−2.14359	−0.0802782
$714$	0	0
$715$	− 29.8564i	− 1.11657i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	44.7846	1.67018	0.835092	−	0.550110i	$-0.185414\pi$
$719$	44.7846	1.67018	0.835092	+	0.550110i	$0.185414\pi$
$720$	0	0
$721$	−9.85641	−0.367072
$722$	0	0
$723$	0	0
$724$	0	0
$725$	8.92820i	0.331585i
$726$	0	0
$727$	4.92820	0.182777	0.0913885	−	0.995815i	$-0.470870\pi$
$727$	4.92820	0.182777	0.0913885	+	0.995815i	$0.470870\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	24.0000i	0.887672i
$732$	0	0
$733$	5.46410i	0.201821i	0.994895	+	0.100911i	$0.0321756\pi$
$733$	5.46410i	0.201821i	−0.994895	+	0.100911i	$0.967824\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	37.8564	1.39446
$738$	0	0
$739$	42.6410i	1.56858i	0.620397	+	0.784288i	$0.286971\pi$
$739$	42.6410i	1.56858i	−0.620397	+	0.784288i	$0.713029\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−35.7128	−1.31018	−0.655088	−	0.755553i	$-0.727368\pi$
$743$	−35.7128	−1.31018	−0.655088	+	0.755553i	$0.727368\pi$
$744$	0	0
$745$	−22.7846	−0.834764
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 13.8564i	− 0.506302i
$750$	0	0
$751$	−25.6077	−0.934438	−0.467219	−	0.884142i	$-0.654744\pi$
$751$	−25.6077	−0.934438	−0.467219	+	0.884142i	$0.654744\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	11.4641i	0.417221i
$756$	0	0
$757$	28.1051i	1.02150i	0.859730	+	0.510749i	$0.170632\pi$
$757$	28.1051i	1.02150i	−0.859730	+	0.510749i	$0.829368\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	30.0000	1.08750	0.543750	−	0.839248i	$-0.317004\pi$
$761$	30.0000	1.08750	0.543750	+	0.839248i	$0.317004\pi$
$762$	0	0
$763$	− 10.1436i	− 0.367223i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	−51.7128	−1.86724
$768$	0	0
$769$	15.8564	0.571797	0.285898	−	0.958260i	$-0.407708\pi$
$769$	15.8564	0.571797	0.285898	+	0.958260i	$0.407708\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	− 13.7128i	− 0.493216i	−0.969115	−	0.246608i	$-0.920684\pi$
$773$	− 13.7128i	− 0.493216i	0.969115	−	0.246608i	$-0.0793159\pi$
$774$	0	0
$775$	−0.535898	−0.0192500
$776$	0	0
$777$	0	0
$778$	0	0
$779$	34.1436i	1.22332i
$780$	0	0
$781$	37.8564i	1.35461i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−14.5359	−0.518808
$786$	0	0
$787$	31.7128i	1.13044i	0.824940	+	0.565220i	$0.191209\pi$
$787$	31.7128i	1.13044i	−0.824940	+	0.565220i	$0.808791\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−14.9282	−0.530786
$792$	0	0
$793$	−21.8564	−0.776144
$794$	0	0
$795$	0	0
$796$	0	0
$797$	22.7846i	0.807072i	0.914964	+	0.403536i	$0.132219\pi$
$797$	22.7846i	0.807072i	−0.914964	+	0.403536i	$0.867781\pi$
$798$	0	0
$799$	−13.8564	−0.490204
$800$	0	0
$801$	0	0
$802$	0	0
$803$	54.6410i	1.92824i
$804$	0	0
$805$	8.00000i	0.281963i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	22.7846	0.801064	0.400532	−	0.916283i	$-0.368825\pi$
$809$	22.7846	0.801064	0.400532	+	0.916283i	$0.368825\pi$
$810$	0	0
$811$	9.85641i	0.346105i	0.984913	+	0.173053i	$0.0553631\pi$
$811$	9.85641i	0.346105i	−0.984913	+	0.173053i	$0.944637\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−4.00000	−0.140114
$816$	0	0
$817$	−48.0000	−1.67931
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 2.78461i	− 0.0971835i	−0.998819	−	0.0485918i	$-0.984527\pi$
$821$	− 2.78461i	− 0.0971835i	0.998819	−	0.0485918i	$-0.0154733\pi$
$822$	0	0
$823$	−13.2154	−0.460660	−0.230330	−	0.973113i	$-0.573980\pi$
$823$	−13.2154	−0.460660	−0.230330	+	0.973113i	$0.573980\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 45.5692i	− 1.58460i	−0.610134	−	0.792299i	$-0.708884\pi$
$827$	− 45.5692i	− 1.58460i	0.610134	−	0.792299i	$-0.291116\pi$
$828$	0	0
$829$	− 10.9282i	− 0.379552i	−0.981827	−	0.189776i	$-0.939224\pi$
$829$	− 10.9282i	− 0.379552i	0.981827	−	0.189776i	$-0.0607762\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	10.3923	0.360072
$834$	0	0
$835$	13.8564i	0.479521i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	26.6410	0.919750	0.459875	−	0.887984i	$-0.347894\pi$
$839$	26.6410	0.919750	0.459875	+	0.887984i	$0.347894\pi$
$840$	0	0
$841$	−50.7128	−1.74872
$842$	0	0
$843$	0	0
$844$	0	0
$845$	16.8564i	0.579878i
$846$	0	0
$847$	−37.7128	−1.29583
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 37.8564i	− 1.29770i
$852$	0	0
$853$	− 0.679492i	− 0.0232654i	−0.999932	−	0.0116327i	$-0.996297\pi$
$853$	− 0.679492i	− 0.0232654i	0.999932	−	0.0116327i	$-0.00370288\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	37.0333	1.26503	0.632517	−	0.774546i	$-0.282022\pi$
$857$	37.0333	1.26503	0.632517	+	0.774546i	$0.282022\pi$
$858$	0	0
$859$	− 15.7128i	− 0.536114i	−0.963403	−	0.268057i	$-0.913618\pi$
$859$	− 15.7128i	− 0.536114i	0.963403	−	0.268057i	$-0.0863816\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	24.0000	0.816970	0.408485	−	0.912765i	$-0.366057\pi$
$863$	24.0000	0.816970	0.408485	+	0.912765i	$0.366057\pi$
$864$	0	0
$865$	24.9282	0.847584
$866$	0	0
$867$	0	0
$868$	0	0
$869$	78.6410i	2.66771i
$870$	0	0
$871$	−37.8564	−1.28272
$872$	0	0
$873$	0	0
$874$	0	0
$875$	2.00000i	0.0676123i
$876$	0	0
$877$	25.4641i	0.859862i	0.902862	+	0.429931i	$0.141462\pi$
$877$	25.4641i	0.859862i	−0.902862	+	0.429931i	$0.858538\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−56.6410	−1.90828	−0.954142	−	0.299354i	$-0.903229\pi$
$881$	−56.6410	−1.90828	−0.954142	+	0.299354i	$0.903229\pi$
$882$	0	0
$883$	22.1436i	0.745191i	0.927994	+	0.372596i	$0.121532\pi$
$883$	22.1436i	0.745191i	−0.927994	+	0.372596i	$0.878468\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	37.8564	1.27109	0.635547	−	0.772062i	$-0.280775\pi$
$887$	37.8564	1.27109	0.635547	+	0.772062i	$0.280775\pi$
$888$	0	0
$889$	14.1436	0.474361
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 27.7128i	− 0.927374i
$894$	0	0
$895$	−4.39230	−0.146819
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 4.78461i	− 0.159576i
$900$	0	0
$901$	− 3.21539i	− 0.107120i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−10.9282	−0.363266
$906$	0	0
$907$	14.9282i	0.495683i	0.968801	+	0.247841i	$0.0797212\pi$
$907$	14.9282i	0.495683i	−0.968801	+	0.247841i	$0.920279\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	27.7128	0.918166	0.459083	−	0.888393i	$-0.348178\pi$
$911$	27.7128	0.918166	0.459083	+	0.888393i	$0.348178\pi$
$912$	0	0
$913$	−43.7128	−1.44668
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 2.92820i	− 0.0966978i
$918$	0	0
$919$	−6.39230	−0.210863	−0.105431	−	0.994427i	$-0.533622\pi$
$919$	−6.39230	−0.210863	−0.105431	+	0.994427i	$0.533622\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 37.8564i	− 1.24606i
$924$	0	0
$925$	− 9.46410i	− 0.311178i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	43.5692	1.42946	0.714730	−	0.699401i	$-0.246550\pi$
$929$	43.5692	1.42946	0.714730	+	0.699401i	$0.246550\pi$
$930$	0	0
$931$	20.7846i	0.681188i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−18.9282	−0.619018
$936$	0	0
$937$	27.8564	0.910029	0.455015	−	0.890484i	$-0.349634\pi$
$937$	27.8564	0.910029	0.455015	+	0.890484i	$0.349634\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	6.00000i	0.195594i	0.995206	+	0.0977972i	$0.0311797\pi$
$941$	6.00000i	0.195594i	−0.995206	+	0.0977972i	$0.968820\pi$
$942$	0	0
$943$	19.7128	0.641938
$944$	0	0
$945$	0	0
$946$	0	0
$947$	25.8564i	0.840220i	0.907473	+	0.420110i	$0.138009\pi$
$947$	25.8564i	0.840220i	−0.907473	+	0.420110i	$0.861991\pi$
$948$	0	0
$949$	− 54.6410i	− 1.77372i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−5.32051	−0.172348	−0.0861741	−	0.996280i	$-0.527464\pi$
$953$	−5.32051	−0.172348	−0.0861741	+	0.996280i	$0.527464\pi$
$954$	0	0
$955$	8.00000i	0.258874i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−3.21539	−0.103830
$960$	0	0
$961$	−30.7128	−0.990736
$962$	0	0
$963$	0	0
$964$	0	0
$965$	22.7846i	0.733463i
$966$	0	0
$967$	22.7846	0.732704	0.366352	−	0.930476i	$-0.380607\pi$
$967$	22.7846	0.732704	0.366352	+	0.930476i	$0.380607\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	16.3923i	0.526054i	0.964788	+	0.263027i	$0.0847208\pi$
$971$	16.3923i	0.526054i	−0.964788	+	0.263027i	$0.915279\pi$
$972$	0	0
$973$	24.0000i	0.769405i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	−13.6077	−0.435349	−0.217674	−	0.976021i	$-0.569847\pi$
$977$	−13.6077	−0.435349	−0.217674	+	0.976021i	$0.569847\pi$
$978$	0	0
$979$	− 5.07180i	− 0.162095i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	13.8564	0.441951	0.220975	−	0.975279i	$-0.429076\pi$
$983$	13.8564	0.441951	0.220975	+	0.975279i	$0.429076\pi$
$984$	0	0
$985$	3.85641	0.122875
$986$	0	0
$987$	0	0
$988$	0	0
$989$	27.7128i	0.881216i
$990$	0	0
$991$	49.3205	1.56672	0.783359	−	0.621570i	$-0.213505\pi$
$991$	49.3205	1.56672	0.783359	+	0.621570i	$0.213505\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 9.32051i	− 0.295480i
$996$	0	0
$997$	− 9.46410i	− 0.299731i	−0.988706	−	0.149866i	$-0.952116\pi$
$997$	− 9.46410i	− 0.299731i	0.988706	−	0.149866i	$-0.0478841\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.k.k.1441.2		4
3.2	odd	2		960.2.k.f.481.3	yes	4
4.3	odd	2		2880.2.k.f.1441.1		4
8.3	odd	2		2880.2.k.f.1441.4		4
8.5	even	2	inner	2880.2.k.k.1441.3		4
12.11	even	2		960.2.k.e.481.2	✓	4
15.2	even	4		4800.2.d.n.1249.3		4
15.8	even	4		4800.2.d.m.1249.1		4
15.14	odd	2		4800.2.k.i.2401.1		4
24.5	odd	2		960.2.k.f.481.2	yes	4
24.11	even	2		960.2.k.e.481.3	yes	4
48.5	odd	4		3840.2.a.bi.1.2		2
48.11	even	4		3840.2.a.be.1.1		2
48.29	odd	4		3840.2.a.bf.1.1		2
48.35	even	4		3840.2.a.bn.1.2		2
60.23	odd	4		4800.2.d.r.1249.4		4
60.47	odd	4		4800.2.d.i.1249.2		4
60.59	even	2		4800.2.k.o.2401.4		4
120.29	odd	2		4800.2.k.i.2401.4		4
120.53	even	4		4800.2.d.n.1249.2		4
120.59	even	2		4800.2.k.o.2401.1		4
120.77	even	4		4800.2.d.m.1249.4		4
120.83	odd	4		4800.2.d.i.1249.3		4
120.107	odd	4		4800.2.d.r.1249.1		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
960.2.k.e.481.2	✓	4	12.11	even	2
960.2.k.e.481.3	yes	4	24.11	even	2
960.2.k.f.481.2	yes	4	24.5	odd	2
960.2.k.f.481.3	yes	4	3.2	odd	2
2880.2.k.f.1441.1		4	4.3	odd	2
2880.2.k.f.1441.4		4	8.3	odd	2
2880.2.k.k.1441.2		4	1.1	even	1	trivial
2880.2.k.k.1441.3		4	8.5	even	2	inner
3840.2.a.be.1.1		2	48.11	even	4
3840.2.a.bf.1.1		2	48.29	odd	4
3840.2.a.bi.1.2		2	48.5	odd	4
3840.2.a.bn.1.2		2	48.35	even	4
4800.2.d.i.1249.2		4	60.47	odd	4
4800.2.d.i.1249.3		4	120.83	odd	4
4800.2.d.m.1249.1		4	15.8	even	4
4800.2.d.m.1249.4		4	120.77	even	4
4800.2.d.n.1249.2		4	120.53	even	4
4800.2.d.n.1249.3		4	15.2	even	4
4800.2.d.r.1249.1		4	120.107	odd	4
4800.2.d.r.1249.4		4	60.23	odd	4
4800.2.k.i.2401.1		4	15.14	odd	2
4800.2.k.i.2401.4		4	120.29	odd	2
4800.2.k.o.2401.1		4	120.59	even	2
4800.2.k.o.2401.4		4	60.59	even	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.k (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} +2.00000 q^{7} +O(q^{10})\)	\(q-1.00000i q^{5} +2.00000 q^{7} +5.46410i q^{11} -5.46410i q^{13} -3.46410 q^{17} -6.92820i q^{19} -4.00000 q^{23} -1.00000 q^{25} -8.92820i q^{29} +0.535898 q^{31} -2.00000i q^{35} +9.46410i q^{37} -4.92820 q^{41} -6.92820i q^{43} +4.00000 q^{47} -3.00000 q^{49} +0.928203i q^{53} +5.46410 q^{55} -9.46410i q^{59} -4.00000i q^{61} -5.46410 q^{65} -6.92820i q^{67} +6.92820 q^{71} +10.0000 q^{73} +10.9282i q^{77} +14.3923 q^{79} +8.00000i q^{83} +3.46410i q^{85} -0.928203 q^{89} -10.9282i q^{91} -6.92820 q^{95} -12.9282 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{7}+O(q^{10}) \) 4 * q + 8 * q^7	\( 4 q + 8 q^{7} - 16 q^{23} - 4 q^{25} + 16 q^{31} + 8 q^{41} + 16 q^{47} - 12 q^{49} + 8 q^{55} - 8 q^{65} + 40 q^{73} + 16 q^{79} + 24 q^{89} - 24 q^{97}+O(q^{100}) \) 4 * q + 8 * q^7 - 16 * q^23 - 4 * q^25 + 16 * q^31 + 8 * q^41 + 16 * q^47 - 12 * q^49 + 8 * q^55 - 8 * q^65 + 40 * q^73 + 16 * q^79 + 24 * q^89 - 24 * q^97

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