LMFDB - Embedded newform 2880.2.b.b.2591.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2880,2,Mod(2591,2880)]

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([1, 1, 1, 0]))

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

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

chi := DirichletCharacter("2880.2591");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2880 = 2^{6} \cdot 3^{2} \cdot 5 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2880.b (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:	$8$
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:	$\Z[a_1, \ldots, a_{19}]$
Coefficient ring index:	$ 2^{14} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			2591.4
Root			$0.258819 + 0.965926i$ of defining polynomial
Character	$\chi$	$=$	2880.2591
Dual form			2880.2.b.b.2591.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000 q^{5} -0.449490i q^{7} +O(q^{10})$	$q-1.00000 q^{5} -0.449490i q^{7} +0.449490i q^{11} +1.41421i q^{13} -2.82843i q^{17} +2.82843 q^{19} -3.46410 q^{23} +1.00000 q^{25} +2.00000 q^{29} -4.89898i q^{31} +0.449490i q^{35} +5.51399i q^{37} -5.51399i q^{41} -4.09978 q^{47} +6.79796 q^{49} +4.00000 q^{53} -0.449490i q^{55} -5.34847i q^{59} +1.55708i q^{61} -1.41421i q^{65} +11.3137 q^{67} -5.65685 q^{71} +7.79796 q^{73} +0.202041 q^{77} -12.8990i q^{79} -8.89898i q^{83} +2.82843i q^{85} +0.142865i q^{89} +0.635674 q^{91} -2.82843 q^{95} +7.79796 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 8 q^{5}+O(q^{10}) $ 8 * q - 8 * q^5	$ 8 q - 8 q^{5} + 8 q^{25} + 16 q^{29} - 24 q^{49} + 32 q^{53} - 16 q^{73} + 80 q^{77} - 16 q^{97}+O(q^{100}) $ 8 * q - 8 * q^5 + 8 * q^25 + 16 * q^29 - 24 * q^49 + 32 * q^53 - 16 * q^73 + 80 * q^77 - 16 * 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
$5$	−1.00000	−0.447214
$5$	−1.00000	−0.447214
$6$	0	0
$7$	− 0.449490i	− 0.169891i	−0.996386	−	0.0849456i	$-0.972928\pi$
$7$	− 0.449490i	− 0.169891i	0.996386	−	0.0849456i	$-0.0270716\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	0.449490i	0.135526i	0.997701	+	0.0677631i	$0.0215862\pi$
$11$	0.449490i	0.135526i	−0.997701	+	0.0677631i	$0.978414\pi$
$12$	0	0
$13$	1.41421i	0.392232i	0.980581	+	0.196116i	$0.0628330\pi$
$13$	1.41421i	0.392232i	−0.980581	+	0.196116i	$0.937167\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	− 2.82843i	− 0.685994i	−0.939336	−	0.342997i	$-0.888558\pi$
$17$	− 2.82843i	− 0.685994i	0.939336	−	0.342997i	$-0.111442\pi$
$18$	0	0
$19$	2.82843	0.648886	0.324443	−	0.945905i	$-0.394823\pi$
$19$	2.82843	0.648886	0.324443	+	0.945905i	$0.394823\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−3.46410	−0.722315	−0.361158	−	0.932505i	$-0.617618\pi$
$23$	−3.46410	−0.722315	−0.361158	+	0.932505i	$0.617618\pi$
$24$	0	0
$25$	1.00000	0.200000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	2.00000	0.371391	0.185695	−	0.982607i	$-0.440546\pi$
$29$	2.00000	0.371391	0.185695	+	0.982607i	$0.440546\pi$
$30$	0	0
$31$	− 4.89898i	− 0.879883i	−0.898027	−	0.439941i	$-0.854999\pi$
$31$	− 4.89898i	− 0.879883i	0.898027	−	0.439941i	$-0.145001\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	0.449490i	0.0759776i
$36$	0	0
$37$	5.51399i	0.906494i	0.891385	+	0.453247i	$0.149734\pi$
$37$	5.51399i	0.906494i	−0.891385	+	0.453247i	$0.850266\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	− 5.51399i	− 0.861141i	−0.902557	−	0.430570i	$-0.858312\pi$
$41$	− 5.51399i	− 0.861141i	0.902557	−	0.430570i	$-0.141688\pi$
$42$	0	0
$43$	0	0		−	1.00000i	$-0.5\pi$
$43$	0	0			1.00000i	$0.5\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−4.09978	−0.598014	−0.299007	−	0.954251i	$-0.596655\pi$
$47$	−4.09978	−0.598014	−0.299007	+	0.954251i	$0.596655\pi$
$48$	0	0
$49$	6.79796	0.971137
$50$	0	0
$51$	0	0
$52$	0	0
$53$	4.00000	0.549442	0.274721	−	0.961524i	$-0.411414\pi$
$53$	4.00000	0.549442	0.274721	+	0.961524i	$0.411414\pi$
$54$	0	0
$55$	− 0.449490i	− 0.0606092i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 5.34847i	− 0.696311i	−0.937437	−	0.348156i	$-0.886808\pi$
$59$	− 5.34847i	− 0.696311i	0.937437	−	0.348156i	$-0.113192\pi$
$60$	0	0
$61$	1.55708i	0.199363i	0.995019	+	0.0996817i	$0.0317825\pi$
$61$	1.55708i	0.199363i	−0.995019	+	0.0996817i	$0.968218\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	− 1.41421i	− 0.175412i
$66$	0	0
$67$	11.3137	1.38219	0.691095	−	0.722764i	$-0.257129\pi$
$67$	11.3137	1.38219	0.691095	+	0.722764i	$0.257129\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−5.65685	−0.671345	−0.335673	−	0.941979i	$-0.608964\pi$
$71$	−5.65685	−0.671345	−0.335673	+	0.941979i	$0.608964\pi$
$72$	0	0
$73$	7.79796	0.912682	0.456341	−	0.889805i	$-0.349160\pi$
$73$	7.79796	0.912682	0.456341	+	0.889805i	$0.349160\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	0.202041	0.0230247
$78$	0	0
$79$	− 12.8990i	− 1.45125i	−0.688091	−	0.725624i	$-0.741551\pi$
$79$	− 12.8990i	− 1.45125i	0.688091	−	0.725624i	$-0.258449\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 8.89898i	− 0.976790i	−0.872623	−	0.488395i	$-0.837583\pi$
$83$	− 8.89898i	− 0.976790i	0.872623	−	0.488395i	$-0.162417\pi$
$84$	0	0
$85$	2.82843i	0.306786i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	0.142865i	0.0151436i	0.999971	+	0.00757181i	$0.00241020\pi$
$89$	0.142865i	0.0151436i	−0.999971	+	0.00757181i	$0.997590\pi$
$90$	0	0
$91$	0.635674	0.0666368
$92$	0	0
$93$	0	0
$94$	0	0
$95$	−2.82843	−0.290191
$96$	0	0
$97$	7.79796	0.791763	0.395881	−	0.918302i	$-0.370439\pi$
$97$	7.79796	0.791763	0.395881	+	0.918302i	$0.370439\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	3.79796	0.377911	0.188956	−	0.981986i	$-0.439490\pi$
$101$	3.79796	0.377911	0.188956	+	0.981986i	$0.439490\pi$
$102$	0	0
$103$	− 13.3485i	− 1.31526i	−0.753339	−	0.657632i	$-0.771558\pi$
$103$	− 13.3485i	− 1.31526i	0.753339	−	0.657632i	$-0.228442\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	8.89898i	0.860297i	0.902758	+	0.430148i	$0.141539\pi$
$107$	8.89898i	0.860297i	−0.902758	+	0.430148i	$0.858461\pi$
$108$	0	0
$109$	1.55708i	0.149141i	0.997216	+	0.0745705i	$0.0237586\pi$
$109$	1.55708i	0.149141i	−0.997216	+	0.0745705i	$0.976241\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 8.19955i	− 0.771349i	−0.922635	−	0.385674i	$-0.873969\pi$
$113$	− 8.19955i	− 0.771349i	0.922635	−	0.385674i	$-0.126031\pi$
$114$	0	0
$115$	3.46410	0.323029
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−1.27135	−0.116544
$120$	0	0
$121$	10.7980	0.981633
$122$	0	0
$123$	0	0
$124$	0	0
$125$	−1.00000	−0.0894427
$126$	0	0
$127$	− 5.34847i	− 0.474600i	−0.971436	−	0.237300i	$-0.923738\pi$
$127$	− 5.34847i	− 0.474600i	0.971436	−	0.237300i	$-0.0762624\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 15.5505i	− 1.35865i	−0.733836	−	0.679327i	$-0.762272\pi$
$131$	− 15.5505i	− 1.35865i	0.733836	−	0.679327i	$-0.237728\pi$
$132$	0	0
$133$	− 1.27135i	− 0.110240i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	− 11.3137i	− 0.966595i	−0.875456	−	0.483298i	$-0.839439\pi$
$137$	− 11.3137i	− 0.966595i	0.875456	−	0.483298i	$-0.160561\pi$
$138$	0	0
$139$	−10.3923	−0.881464	−0.440732	−	0.897639i	$-0.645281\pi$
$139$	−10.3923	−0.881464	−0.440732	+	0.897639i	$0.645281\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−0.635674	−0.0531578
$144$	0	0
$145$	−2.00000	−0.166091
$146$	0	0
$147$	0	0
$148$	0	0
$149$	19.7980	1.62191	0.810956	−	0.585107i	$-0.198948\pi$
$149$	19.7980	1.62191	0.810956	+	0.585107i	$0.198948\pi$
$150$	0	0
$151$	− 5.79796i	− 0.471831i	−0.971774	−	0.235916i	$-0.924191\pi$
$151$	− 5.79796i	− 0.471831i	0.971774	−	0.235916i	$-0.0758089\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	4.89898i	0.393496i
$156$	0	0
$157$	8.34242i	0.665797i	0.942963	+	0.332899i	$0.108027\pi$
$157$	8.34242i	0.665797i	−0.942963	+	0.332899i	$0.891973\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	1.55708i	0.122715i
$162$	0	0
$163$	10.0424	0.786578	0.393289	−	0.919415i	$-0.371337\pi$
$163$	10.0424	0.786578	0.393289	+	0.919415i	$0.371337\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−21.7060	−1.67966	−0.839831	−	0.542848i	$-0.817346\pi$
$167$	−21.7060	−1.67966	−0.839831	+	0.542848i	$0.817346\pi$
$168$	0	0
$169$	11.0000	0.846154
$170$	0	0
$171$	0	0
$172$	0	0
$173$	9.79796	0.744925	0.372463	−	0.928047i	$-0.378514\pi$
$173$	9.79796	0.744925	0.372463	+	0.928047i	$0.378514\pi$
$174$	0	0
$175$	− 0.449490i	− 0.0339782i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	− 18.2474i	− 1.36388i	−0.731409	−	0.681939i	$-0.761137\pi$
$179$	− 18.2474i	− 1.36388i	0.731409	−	0.681939i	$-0.238863\pi$
$180$	0	0
$181$	9.75663i	0.725205i	0.931944	+	0.362602i	$0.118112\pi$
$181$	9.75663i	0.725205i	−0.931944	+	0.362602i	$0.881888\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 5.51399i	− 0.405397i
$186$	0	0
$187$	1.27135	0.0929702
$188$	0	0
$189$	0	0
$190$	0	0
$191$	4.38551	0.317324	0.158662	−	0.987333i	$-0.449282\pi$
$191$	4.38551	0.317324	0.158662	+	0.987333i	$0.449282\pi$
$192$	0	0
$193$	10.0000	0.719816	0.359908	−	0.932988i	$-0.382808\pi$
$193$	10.0000	0.719816	0.359908	+	0.932988i	$0.382808\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	−21.5959	−1.53865	−0.769323	−	0.638860i	$-0.779406\pi$
$197$	−21.5959	−1.53865	−0.769323	+	0.638860i	$0.779406\pi$
$198$	0	0
$199$	− 5.79796i	− 0.411006i	−0.978656	−	0.205503i	$-0.934117\pi$
$199$	− 5.79796i	− 0.411006i	0.978656	−	0.205503i	$-0.0658831\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	− 0.898979i	− 0.0630960i
$204$	0	0
$205$	5.51399i	0.385114i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	1.27135i	0.0879410i
$210$	0	0
$211$	14.7778	1.01735	0.508673	−	0.860960i	$-0.330136\pi$
$211$	14.7778	1.01735	0.508673	+	0.860960i	$0.330136\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	0	0
$216$	0	0
$217$	−2.20204	−0.149484
$218$	0	0
$219$	0	0
$220$	0	0
$221$	4.00000	0.269069
$222$	0	0
$223$	− 2.24745i	− 0.150500i	−0.997165	−	0.0752501i	$-0.976024\pi$
$223$	− 2.24745i	− 0.150500i	0.997165	−	0.0752501i	$-0.0239755\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	11.5959i	0.769648i	0.922990	+	0.384824i	$0.125738\pi$
$227$	11.5959i	0.769648i	−0.922990	+	0.384824i	$0.874262\pi$
$228$	0	0
$229$	− 12.5851i	− 0.831644i	−0.909446	−	0.415822i	$-0.863494\pi$
$229$	− 12.5851i	− 0.831644i	0.909446	−	0.415822i	$-0.136506\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	5.37113i	0.351874i	0.984401	+	0.175937i	$0.0562955\pi$
$233$	5.37113i	0.351874i	−0.984401	+	0.175937i	$0.943704\pi$
$234$	0	0
$235$	4.09978	0.267440
$236$	0	0
$237$	0	0
$238$	0	0
$239$	−2.54270	−0.164473	−0.0822367	−	0.996613i	$-0.526206\pi$
$239$	−2.54270	−0.164473	−0.0822367	+	0.996613i	$0.526206\pi$
$240$	0	0
$241$	4.00000	0.257663	0.128831	−	0.991667i	$-0.458877\pi$
$241$	4.00000	0.257663	0.128831	+	0.991667i	$0.458877\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	−6.79796	−0.434306
$246$	0	0
$247$	4.00000i	0.254514i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	10.6515i	0.672319i	0.941805	+	0.336159i	$0.109128\pi$
$251$	10.6515i	0.672319i	−0.941805	+	0.336159i	$0.890872\pi$
$252$	0	0
$253$	− 1.55708i	− 0.0978927i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 19.5133i	− 1.21720i	−0.793476	−	0.608602i	$-0.791731\pi$
$257$	− 19.5133i	− 1.21720i	0.793476	−	0.608602i	$-0.208269\pi$
$258$	0	0
$259$	2.47848	0.154005
$260$	0	0
$261$	0	0
$262$	0	0
$263$	14.7778	0.911239	0.455619	−	0.890175i	$-0.349418\pi$
$263$	14.7778	0.911239	0.455619	+	0.890175i	$0.349418\pi$
$264$	0	0
$265$	−4.00000	−0.245718
$266$	0	0
$267$	0	0
$268$	0	0
$269$	13.5959	0.828958	0.414479	−	0.910059i	$-0.363964\pi$
$269$	13.5959	0.828958	0.414479	+	0.910059i	$0.363964\pi$
$270$	0	0
$271$	− 4.00000i	− 0.242983i	−0.992592	−	0.121491i	$-0.961232\pi$
$271$	− 4.00000i	− 0.242983i	0.992592	−	0.121491i	$-0.0387677\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	0.449490i	0.0271053i
$276$	0	0
$277$	19.6561i	1.18102i	0.807030	+	0.590511i	$0.201074\pi$
$277$	19.6561i	1.18102i	−0.807030	+	0.590511i	$0.798926\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	3.95691i	0.236050i	0.993011	+	0.118025i	$0.0376562\pi$
$281$	3.95691i	0.236050i	−0.993011	+	0.118025i	$0.962344\pi$
$282$	0	0
$283$	−23.8988	−1.42063	−0.710317	−	0.703882i	$-0.751448\pi$
$283$	−23.8988	−1.42063	−0.710317	+	0.703882i	$0.751448\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.47848	−0.146300
$288$	0	0
$289$	9.00000	0.529412
$290$	0	0
$291$	0	0
$292$	0	0
$293$	6.00000	0.350524	0.175262	−	0.984522i	$-0.443923\pi$
$293$	6.00000	0.350524	0.175262	+	0.984522i	$0.443923\pi$
$294$	0	0
$295$	5.34847i	0.311400i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	− 4.89898i	− 0.283315i
$300$	0	0
$301$	0	0
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 1.55708i	− 0.0891580i
$306$	0	0
$307$	33.3697	1.90451	0.952254	−	0.305308i	$-0.0987593\pi$
$307$	33.3697	1.90451	0.952254	+	0.305308i	$0.0987593\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−12.5851	−0.713633	−0.356817	−	0.934174i	$-0.616138\pi$
$311$	−12.5851	−0.713633	−0.356817	+	0.934174i	$0.616138\pi$
$312$	0	0
$313$	−2.00000	−0.113047	−0.0565233	−	0.998401i	$-0.518002\pi$
$313$	−2.00000	−0.113047	−0.0565233	+	0.998401i	$0.518002\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−25.5959	−1.43761	−0.718805	−	0.695212i	$-0.755311\pi$
$317$	−25.5959	−1.43761	−0.718805	+	0.695212i	$0.755311\pi$
$318$	0	0
$319$	0.898979i	0.0503332i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 8.00000i	− 0.445132i
$324$	0	0
$325$	1.41421i	0.0784465i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	1.84281i	0.101597i
$330$	0	0
$331$	−22.3417	−1.22801	−0.614005	−	0.789302i	$-0.710443\pi$
$331$	−22.3417	−1.22801	−0.614005	+	0.789302i	$0.710443\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−11.3137	−0.618134
$336$	0	0
$337$	−23.7980	−1.29636	−0.648179	−	0.761488i	$-0.724469\pi$
$337$	−23.7980	−1.29636	−0.648179	+	0.761488i	$0.724469\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	2.20204	0.119247
$342$	0	0
$343$	− 6.20204i	− 0.334879i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	29.3939i	1.57795i	0.614428	+	0.788973i	$0.289387\pi$
$347$	29.3939i	1.57795i	−0.614428	+	0.788973i	$0.710613\pi$
$348$	0	0
$349$	− 21.0703i	− 1.12787i	−0.825819	−	0.563935i	$-0.809287\pi$
$349$	− 21.0703i	− 1.12787i	0.825819	−	0.563935i	$-0.190713\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	19.7990i	1.05379i	0.849929	+	0.526897i	$0.176645\pi$
$353$	19.7990i	1.05379i	−0.849929	+	0.526897i	$0.823355\pi$
$354$	0	0
$355$	5.65685	0.300235
$356$	0	0
$357$	0	0
$358$	0	0
$359$	25.1701	1.32843	0.664214	−	0.747543i	$-0.268766\pi$
$359$	25.1701	1.32843	0.664214	+	0.747543i	$0.268766\pi$
$360$	0	0
$361$	−11.0000	−0.578947
$362$	0	0
$363$	0	0
$364$	0	0
$365$	−7.79796	−0.408164
$366$	0	0
$367$	14.2474i	0.743711i	0.928291	+	0.371855i	$0.121278\pi$
$367$	14.2474i	0.743711i	−0.928291	+	0.371855i	$0.878722\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 1.79796i	− 0.0933454i
$372$	0	0
$373$	− 5.22826i	− 0.270709i	−0.990797	−	0.135355i	$-0.956783\pi$
$373$	− 5.22826i	− 0.270709i	0.990797	−	0.135355i	$-0.0432173\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	2.82843i	0.145671i
$378$	0	0
$379$	−30.4770	−1.56550	−0.782750	−	0.622337i	$-0.786184\pi$
$379$	−30.4770	−1.56550	−0.782750	+	0.622337i	$0.786184\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−21.0703	−1.07664	−0.538322	−	0.842739i	$-0.680942\pi$
$383$	−21.0703	−1.07664	−0.538322	+	0.842739i	$0.680942\pi$
$384$	0	0
$385$	−0.202041	−0.0102970
$386$	0	0
$387$	0	0
$388$	0	0
$389$	1.59592	0.0809163	0.0404581	−	0.999181i	$-0.487118\pi$
$389$	1.59592	0.0809163	0.0404581	+	0.999181i	$0.487118\pi$
$390$	0	0
$391$	9.79796i	0.495504i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	12.8990i	0.649018i
$396$	0	0
$397$	18.0990i	0.908365i	0.890909	+	0.454183i	$0.150069\pi$
$397$	18.0990i	0.908365i	−0.890909	+	0.454183i	$0.849931\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 13.9993i	− 0.699090i	−0.936920	−	0.349545i	$-0.886336\pi$
$401$	− 13.9993i	− 0.699090i	0.936920	−	0.349545i	$-0.113664\pi$
$402$	0	0
$403$	6.92820	0.345118
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−2.47848	−0.122854
$408$	0	0
$409$	−5.59592	−0.276700	−0.138350	−	0.990383i	$-0.544180\pi$
$409$	−5.59592	−0.276700	−0.138350	+	0.990383i	$0.544180\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	−2.40408	−0.118297
$414$	0	0
$415$	8.89898i	0.436834i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	− 40.0454i	− 1.95635i	−0.207792	−	0.978173i	$-0.566628\pi$
$419$	− 40.0454i	− 1.95635i	0.207792	−	0.978173i	$-0.433372\pi$
$420$	0	0
$421$	− 29.5556i	− 1.44045i	−0.693739	−	0.720226i	$-0.744038\pi$
$421$	− 29.5556i	− 1.44045i	0.693739	−	0.720226i	$-0.255962\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	− 2.82843i	− 0.137199i
$426$	0	0
$427$	0.699891	0.0338701
$428$	0	0
$429$	0	0
$430$	0	0
$431$	26.4415	1.27364	0.636820	−	0.771012i	$-0.280249\pi$
$431$	26.4415	1.27364	0.636820	+	0.771012i	$0.280249\pi$
$432$	0	0
$433$	−3.79796	−0.182518	−0.0912591	−	0.995827i	$-0.529089\pi$
$433$	−3.79796	−0.182518	−0.0912591	+	0.995827i	$0.529089\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	−9.79796	−0.468700
$438$	0	0
$439$	− 2.20204i	− 0.105098i	−0.998618	−	0.0525488i	$-0.983265\pi$
$439$	− 2.20204i	− 0.105098i	0.998618	−	0.0525488i	$-0.0167345\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 3.59592i	− 0.170847i	−0.996345	−	0.0854236i	$-0.972776\pi$
$443$	− 3.59592i	− 0.170847i	0.996345	−	0.0854236i	$-0.0272244\pi$
$444$	0	0
$445$	− 0.142865i	− 0.00677243i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	34.7839i	1.64155i	0.571250	+	0.820776i	$0.306459\pi$
$449$	34.7839i	1.64155i	−0.571250	+	0.820776i	$0.693541\pi$
$450$	0	0
$451$	2.47848	0.116707
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−0.635674	−0.0298009
$456$	0	0
$457$	−15.7980	−0.738997	−0.369499	−	0.929231i	$-0.620471\pi$
$457$	−15.7980	−0.738997	−0.369499	+	0.929231i	$0.620471\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	0.202041	0.00940999	0.00470499	−	0.999989i	$-0.498502\pi$
$461$	0.202041	0.00940999	0.00470499	+	0.999989i	$0.498502\pi$
$462$	0	0
$463$	23.1464i	1.07571i	0.843039	+	0.537853i	$0.180764\pi$
$463$	23.1464i	1.07571i	−0.843039	+	0.537853i	$0.819236\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	20.4949i	0.948391i	0.880420	+	0.474195i	$0.157261\pi$
$467$	20.4949i	0.948391i	−0.880420	+	0.474195i	$0.842739\pi$
$468$	0	0
$469$	− 5.08540i	− 0.234822i
$470$	0	0
$471$	0	0
$472$	0	0
$473$	0	0
$474$	0	0
$475$	2.82843	0.129777
$476$	0	0
$477$	0	0
$478$	0	0
$479$	33.3697	1.52470	0.762350	−	0.647166i	$-0.224046\pi$
$479$	33.3697	1.52470	0.762350	+	0.647166i	$0.224046\pi$
$480$	0	0
$481$	−7.79796	−0.355556
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−7.79796	−0.354087
$486$	0	0
$487$	28.4495i	1.28917i	0.764533	+	0.644585i	$0.222970\pi$
$487$	28.4495i	1.28917i	−0.764533	+	0.644585i	$0.777030\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	10.6515i	0.480697i	0.970687	+	0.240348i	$0.0772617\pi$
$491$	10.6515i	0.480697i	−0.970687	+	0.240348i	$0.922738\pi$
$492$	0	0
$493$	− 5.65685i	− 0.254772i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	2.54270i	0.114056i
$498$	0	0
$499$	−5.37113	−0.240445	−0.120222	−	0.992747i	$-0.538361\pi$
$499$	−5.37113	−0.240445	−0.120222	+	0.992747i	$0.538361\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	43.6978	1.94839	0.974193	−	0.225715i	$-0.0724719\pi$
$503$	43.6978	1.94839	0.974193	+	0.225715i	$0.0724719\pi$
$504$	0	0
$505$	−3.79796	−0.169007
$506$	0	0
$507$	0	0
$508$	0	0
$509$	31.3939	1.39151	0.695755	−	0.718280i	$-0.255070\pi$
$509$	31.3939	1.39151	0.695755	+	0.718280i	$0.255070\pi$
$510$	0	0
$511$	− 3.50510i	− 0.155057i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	13.3485i	0.588204i
$516$	0	0
$517$	− 1.84281i	− 0.0810466i
$518$	0	0
$519$	0	0
$520$	0	0
$521$	31.9555i	1.39999i	0.714146	+	0.699997i	$0.246815\pi$
$521$	31.9555i	1.39999i	−0.714146	+	0.699997i	$0.753185\pi$
$522$	0	0
$523$	−32.0983	−1.40356	−0.701781	−	0.712393i	$-0.747611\pi$
$523$	−32.0983	−1.40356	−0.701781	+	0.712393i	$0.747611\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−13.8564	−0.603595
$528$	0	0
$529$	−11.0000	−0.478261
$530$	0	0
$531$	0	0
$532$	0	0
$533$	7.79796	0.337767
$534$	0	0
$535$	− 8.89898i	− 0.384736i
$536$	0	0
$537$	0	0
$538$	0	0
$539$	3.05561i	0.131615i
$540$	0	0
$541$	− 18.5276i	− 0.796565i	−0.917263	−	0.398283i	$-0.869606\pi$
$541$	− 18.5276i	− 0.796565i	0.917263	−	0.398283i	$-0.130394\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	− 1.55708i	− 0.0666979i
$546$	0	0
$547$	12.5851	0.538098	0.269049	−	0.963126i	$-0.413291\pi$
$547$	12.5851	0.538098	0.269049	+	0.963126i	$0.413291\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	5.65685	0.240990
$552$	0	0
$553$	−5.79796	−0.246554
$554$	0	0
$555$	0	0
$556$	0	0
$557$	−37.7980	−1.60155	−0.800775	−	0.598965i	$-0.795579\pi$
$557$	−37.7980	−1.60155	−0.800775	+	0.598965i	$0.795579\pi$
$558$	0	0
$559$	0	0
$560$	0	0
$561$	0	0
$562$	0	0
$563$	− 33.7980i	− 1.42441i	−0.701969	−	0.712207i	$-0.747696\pi$
$563$	− 33.7980i	− 1.42441i	0.701969	−	0.712207i	$-0.252304\pi$
$564$	0	0
$565$	8.19955i	0.344958i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	28.1414i	1.17975i	0.807495	+	0.589875i	$0.200823\pi$
$569$	28.1414i	1.17975i	−0.807495	+	0.589875i	$0.799177\pi$
$570$	0	0
$571$	−19.7990	−0.828562	−0.414281	−	0.910149i	$-0.635967\pi$
$571$	−19.7990	−0.828562	−0.414281	+	0.910149i	$0.635967\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−3.46410	−0.144463
$576$	0	0
$577$	−27.3939	−1.14042	−0.570211	−	0.821498i	$-0.693139\pi$
$577$	−27.3939	−1.14042	−0.570211	+	0.821498i	$0.693139\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	−4.00000	−0.165948
$582$	0	0
$583$	1.79796i	0.0744639i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	36.4949i	1.50631i	0.657846	+	0.753153i	$0.271468\pi$
$587$	36.4949i	1.50631i	−0.657846	+	0.753153i	$0.728532\pi$
$588$	0	0
$589$	− 13.8564i	− 0.570943i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	13.8564i	0.569014i	0.958674	+	0.284507i	$0.0918300\pi$
$593$	13.8564i	0.569014i	−0.958674	+	0.284507i	$0.908170\pi$
$594$	0	0
$595$	1.27135	0.0521202
$596$	0	0
$597$	0	0
$598$	0	0
$599$	−1.27135	−0.0519459	−0.0259730	−	0.999663i	$-0.508268\pi$
$599$	−1.27135	−0.0519459	−0.0259730	+	0.999663i	$0.508268\pi$
$600$	0	0
$601$	−3.59592	−0.146681	−0.0733403	−	0.997307i	$-0.523366\pi$
$601$	−3.59592	−0.146681	−0.0733403	+	0.997307i	$0.523366\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	−10.7980	−0.438999
$606$	0	0
$607$	23.5505i	0.955886i	0.878391	+	0.477943i	$0.158617\pi$
$607$	23.5505i	0.955886i	−0.878391	+	0.477943i	$0.841383\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	− 5.79796i	− 0.234560i
$612$	0	0
$613$	− 38.1838i	− 1.54223i	−0.636697	−	0.771114i	$-0.719700\pi$
$613$	− 38.1838i	− 1.54223i	0.636697	−	0.771114i	$-0.280300\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	− 36.7696i	− 1.48029i	−0.672449	−	0.740143i	$-0.734758\pi$
$617$	− 36.7696i	− 1.48029i	0.672449	−	0.740143i	$-0.265242\pi$
$618$	0	0
$619$	−42.4906	−1.70784	−0.853921	−	0.520402i	$-0.825782\pi$
$619$	−42.4906	−1.70784	−0.853921	+	0.520402i	$0.825782\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	0.0642162	0.00257277
$624$	0	0
$625$	1.00000	0.0400000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	15.5959	0.621850
$630$	0	0
$631$	40.4949i	1.61208i	0.591864	+	0.806038i	$0.298392\pi$
$631$	40.4949i	1.61208i	−0.591864	+	0.806038i	$0.701608\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	5.34847i	0.212248i
$636$	0	0
$637$	9.61377i	0.380911i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	20.9275i	0.826586i	0.910598	+	0.413293i	$0.135621\pi$
$641$	20.9275i	0.826586i	−0.910598	+	0.413293i	$0.864379\pi$
$642$	0	0
$643$	−2.54270	−0.100274	−0.0501371	−	0.998742i	$-0.515966\pi$
$643$	−2.54270	−0.100274	−0.0501371	+	0.998742i	$0.515966\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	0.985620	0.0387487	0.0193744	−	0.999812i	$-0.493833\pi$
$647$	0.985620	0.0387487	0.0193744	+	0.999812i	$0.493833\pi$
$648$	0	0
$649$	2.40408	0.0943685
$650$	0	0
$651$	0	0
$652$	0	0
$653$	26.0000	1.01746	0.508729	−	0.860927i	$-0.330115\pi$
$653$	26.0000	1.01746	0.508729	+	0.860927i	$0.330115\pi$
$654$	0	0
$655$	15.5505i	0.607609i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	− 6.24745i	− 0.243366i	−0.992569	−	0.121683i	$-0.961171\pi$
$659$	− 6.24745i	− 0.243366i	0.992569	−	0.121683i	$-0.0388291\pi$
$660$	0	0
$661$	21.0703i	0.819541i	0.912189	+	0.409771i	$0.134391\pi$
$661$	21.0703i	0.819541i	−0.912189	+	0.409771i	$0.865609\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	1.27135i	0.0493008i
$666$	0	0
$667$	−6.92820	−0.268261
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−0.699891	−0.0270190
$672$	0	0
$673$	−23.3939	−0.901768	−0.450884	−	0.892583i	$-0.648891\pi$
$673$	−23.3939	−0.901768	−0.450884	+	0.892583i	$0.648891\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	−37.5959	−1.44493	−0.722464	−	0.691408i	$-0.756991\pi$
$677$	−37.5959	−1.44493	−0.722464	+	0.691408i	$0.756991\pi$
$678$	0	0
$679$	− 3.50510i	− 0.134513i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	24.8990i	0.952733i	0.879247	+	0.476366i	$0.158046\pi$
$683$	24.8990i	0.952733i	−0.879247	+	0.476366i	$0.841954\pi$
$684$	0	0
$685$	11.3137i	0.432275i
$686$	0	0
$687$	0	0
$688$	0	0
$689$	5.65685i	0.215509i
$690$	0	0
$691$	2.82843	0.107598	0.0537992	−	0.998552i	$-0.482867\pi$
$691$	2.82843	0.107598	0.0537992	+	0.998552i	$0.482867\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	10.3923	0.394203
$696$	0	0
$697$	−15.5959	−0.590738
$698$	0	0
$699$	0	0
$700$	0	0
$701$	−8.20204	−0.309787	−0.154893	−	0.987931i	$-0.549503\pi$
$701$	−8.20204	−0.309787	−0.154893	+	0.987931i	$0.549503\pi$
$702$	0	0
$703$	15.5959i	0.588211i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 1.70714i	− 0.0642037i
$708$	0	0
$709$	− 35.2125i	− 1.32243i	−0.750195	−	0.661216i	$-0.770041\pi$
$709$	− 35.2125i	− 1.32243i	0.750195	−	0.661216i	$-0.229959\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	16.9706i	0.635553i
$714$	0	0
$715$	0.635674	0.0237729
$716$	0	0
$717$	0	0
$718$	0	0
$719$	30.8270	1.14965	0.574826	−	0.818276i	$-0.305070\pi$
$719$	30.8270	1.14965	0.574826	+	0.818276i	$0.305070\pi$
$720$	0	0
$721$	−6.00000	−0.223452
$722$	0	0
$723$	0	0
$724$	0	0
$725$	2.00000	0.0742781
$726$	0	0
$727$	21.3485i	0.791771i	0.918300	+	0.395885i	$0.129562\pi$
$727$	21.3485i	0.791771i	−0.918300	+	0.395885i	$0.870438\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	0	0
$732$	0	0
$733$	− 38.8837i	− 1.43620i	−0.695940	−	0.718100i	$-0.745012\pi$
$733$	− 38.8837i	− 1.43620i	0.695940	−	0.718100i	$-0.254988\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	5.08540i	0.187323i
$738$	0	0
$739$	−5.94258	−0.218601	−0.109301	−	0.994009i	$-0.534861\pi$
$739$	−5.94258	−0.218601	−0.109301	+	0.994009i	$0.534861\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	24.1845	0.887243	0.443622	−	0.896214i	$-0.353693\pi$
$743$	24.1845	0.887243	0.443622	+	0.896214i	$0.353693\pi$
$744$	0	0
$745$	−19.7980	−0.725341
$746$	0	0
$747$	0	0
$748$	0	0
$749$	4.00000	0.146157
$750$	0	0
$751$	− 6.69694i	− 0.244375i	−0.992507	−	0.122187i	$-0.961009\pi$
$751$	− 6.69694i	− 0.244375i	0.992507	−	0.122187i	$-0.0389909\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	5.79796i	0.211009i
$756$	0	0
$757$	22.1988i	0.806830i	0.915017	+	0.403415i	$0.132177\pi$
$757$	22.1988i	0.806830i	−0.915017	+	0.403415i	$0.867823\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	− 18.0990i	− 0.656090i	−0.944662	−	0.328045i	$-0.893610\pi$
$761$	− 18.0990i	− 0.656090i	0.944662	−	0.328045i	$-0.106390\pi$
$762$	0	0
$763$	0.699891	0.0253377
$764$	0	0
$765$	0	0
$766$	0	0
$767$	7.56388	0.273116
$768$	0	0
$769$	9.79796	0.353323	0.176662	−	0.984272i	$-0.443470\pi$
$769$	9.79796	0.353323	0.176662	+	0.984272i	$0.443470\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	−14.4041	−0.518079	−0.259039	−	0.965867i	$-0.583406\pi$
$773$	−14.4041	−0.518079	−0.259039	+	0.965867i	$0.583406\pi$
$774$	0	0
$775$	− 4.89898i	− 0.175977i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	− 15.5959i	− 0.558782i
$780$	0	0
$781$	− 2.54270i	− 0.0909849i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 8.34242i	− 0.297754i
$786$	0	0
$787$	19.5133	0.695573	0.347786	−	0.937574i	$-0.386933\pi$
$787$	19.5133	0.695573	0.347786	+	0.937574i	$0.386933\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−3.68561	−0.131045
$792$	0	0
$793$	−2.20204	−0.0781968
$794$	0	0
$795$	0	0
$796$	0	0
$797$	−31.1918	−1.10487	−0.552436	−	0.833555i	$-0.686302\pi$
$797$	−31.1918	−1.10487	−0.552436	+	0.833555i	$0.686302\pi$
$798$	0	0
$799$	11.5959i	0.410234i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	3.50510i	0.123692i
$804$	0	0
$805$	− 1.55708i	− 0.0548798i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	29.4128i	1.03410i	0.855956	+	0.517049i	$0.172969\pi$
$809$	29.4128i	1.03410i	−0.855956	+	0.517049i	$0.827031\pi$
$810$	0	0
$811$	−5.30691	−0.186351	−0.0931754	−	0.995650i	$-0.529702\pi$
$811$	−5.30691	−0.186351	−0.0931754	+	0.995650i	$0.529702\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−10.0424	−0.351769
$816$	0	0
$817$	0	0
$818$	0	0
$819$	0	0
$820$	0	0
$821$	9.59592	0.334900	0.167450	−	0.985881i	$-0.446447\pi$
$821$	9.59592	0.334900	0.167450	+	0.985881i	$0.446447\pi$
$822$	0	0
$823$	− 24.4495i	− 0.852256i	−0.904663	−	0.426128i	$-0.859877\pi$
$823$	− 24.4495i	− 0.852256i	0.904663	−	0.426128i	$-0.140123\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	53.3939i	1.85669i	0.371723	+	0.928344i	$0.378767\pi$
$827$	53.3939i	1.85669i	−0.371723	+	0.928344i	$0.621233\pi$
$828$	0	0
$829$	− 31.8126i	− 1.10490i	−0.833547	−	0.552448i	$-0.813694\pi$
$829$	− 31.8126i	− 1.10490i	0.833547	−	0.552448i	$-0.186306\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 19.2275i	− 0.666194i
$834$	0	0
$835$	21.7060	0.751168
$836$	0	0
$837$	0	0
$838$	0	0
$839$	2.54270	0.0877837	0.0438918	−	0.999036i	$-0.486024\pi$
$839$	2.54270	0.0877837	0.0438918	+	0.999036i	$0.486024\pi$
$840$	0	0
$841$	−25.0000	−0.862069
$842$	0	0
$843$	0	0
$844$	0	0
$845$	−11.0000	−0.378412
$846$	0	0
$847$	− 4.85357i	− 0.166771i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	− 19.1010i	− 0.654775i
$852$	0	0
$853$	51.7544i	1.77204i	0.463649	+	0.886019i	$0.346540\pi$
$853$	51.7544i	1.77204i	−0.463649	+	0.886019i	$0.653460\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	16.6848i	0.569943i	0.958536	+	0.284972i	$0.0919841\pi$
$857$	16.6848i	0.569943i	−0.958536	+	0.284972i	$0.908016\pi$
$858$	0	0
$859$	−10.3923	−0.354581	−0.177290	−	0.984159i	$-0.556733\pi$
$859$	−10.3923	−0.354581	−0.177290	+	0.984159i	$0.556733\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−12.2351	−0.416488	−0.208244	−	0.978077i	$-0.566775\pi$
$863$	−12.2351	−0.416488	−0.208244	+	0.978077i	$0.566775\pi$
$864$	0	0
$865$	−9.79796	−0.333141
$866$	0	0
$867$	0	0
$868$	0	0
$869$	5.79796	0.196682
$870$	0	0
$871$	16.0000i	0.542139i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	0.449490i	0.0151955i
$876$	0	0
$877$	25.0273i	0.845110i	0.906337	+	0.422555i	$0.138867\pi$
$877$	25.0273i	0.845110i	−0.906337	+	0.422555i	$0.861133\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	16.8277i	0.566940i	0.958981	+	0.283470i	$0.0914856\pi$
$881$	16.8277i	0.566940i	−0.958981	+	0.283470i	$0.908514\pi$
$882$	0	0
$883$	−17.6705	−0.594658	−0.297329	−	0.954775i	$-0.596096\pi$
$883$	−17.6705	−0.594658	−0.297329	+	0.954775i	$0.596096\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	−49.4188	−1.65932	−0.829661	−	0.558268i	$-0.811466\pi$
$887$	−49.4188	−1.65932	−0.829661	+	0.558268i	$0.811466\pi$
$888$	0	0
$889$	−2.40408	−0.0806303
$890$	0	0
$891$	0	0
$892$	0	0
$893$	−11.5959	−0.388043
$894$	0	0
$895$	18.2474i	0.609945i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 9.79796i	− 0.326780i
$900$	0	0
$901$	− 11.3137i	− 0.376914i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	− 9.75663i	− 0.324321i
$906$	0	0
$907$	9.47090	0.314476	0.157238	−	0.987561i	$-0.449741\pi$
$907$	9.47090	0.314476	0.157238	+	0.987561i	$0.449741\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	35.9124	1.18983	0.594915	−	0.803789i	$-0.297186\pi$
$911$	35.9124	1.18983	0.594915	+	0.803789i	$0.297186\pi$
$912$	0	0
$913$	4.00000	0.132381
$914$	0	0
$915$	0	0
$916$	0	0
$917$	−6.98979	−0.230823
$918$	0	0
$919$	− 12.8990i	− 0.425498i	−0.977107	−	0.212749i	$-0.931758\pi$
$919$	− 12.8990i	− 0.425498i	0.977107	−	0.212749i	$-0.0682417\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 8.00000i	− 0.263323i
$924$	0	0
$925$	5.51399i	0.181299i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	4.52837i	0.148571i	0.997237	+	0.0742855i	$0.0236676\pi$
$929$	4.52837i	0.148571i	−0.997237	+	0.0742855i	$0.976332\pi$
$930$	0	0
$931$	19.2275	0.630157
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−1.27135	−0.0415776
$936$	0	0
$937$	−6.40408	−0.209212	−0.104606	−	0.994514i	$-0.533358\pi$
$937$	−6.40408	−0.209212	−0.104606	+	0.994514i	$0.533358\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	25.5959	0.834403	0.417202	−	0.908814i	$-0.363011\pi$
$941$	25.5959	0.834403	0.417202	+	0.908814i	$0.363011\pi$
$942$	0	0
$943$	19.1010i	0.622015i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	32.0000i	1.03986i	0.854209	+	0.519930i	$0.174042\pi$
$947$	32.0000i	1.03986i	−0.854209	+	0.519930i	$0.825958\pi$
$948$	0	0
$949$	11.0280i	0.357983i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	30.2555i	0.980072i	0.871702	+	0.490036i	$0.163016\pi$
$953$	30.2555i	0.980072i	−0.871702	+	0.490036i	$0.836984\pi$
$954$	0	0
$955$	−4.38551	−0.141912
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.08540	−0.164216
$960$	0	0
$961$	7.00000	0.225806
$962$	0	0
$963$	0	0
$964$	0	0
$965$	−10.0000	−0.321911
$966$	0	0
$967$	18.2474i	0.586798i	0.955990	+	0.293399i	$0.0947865\pi$
$967$	18.2474i	0.586798i	−0.955990	+	0.293399i	$0.905213\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	− 2.65153i	− 0.0850917i	−0.999095	−	0.0425458i	$-0.986453\pi$
$971$	− 2.65153i	− 0.0850917i	0.999095	−	0.0425458i	$-0.0135468\pi$
$972$	0	0
$973$	4.67123i	0.149753i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	− 30.5412i	− 0.977101i	−0.872536	−	0.488550i	$-0.837526\pi$
$977$	− 30.5412i	− 0.977101i	0.872536	−	0.488550i	$-0.162474\pi$
$978$	0	0
$979$	−0.0642162	−0.00205236
$980$	0	0
$981$	0	0
$982$	0	0
$983$	32.3840	1.03289	0.516445	−	0.856320i	$-0.327255\pi$
$983$	32.3840	1.03289	0.516445	+	0.856320i	$0.327255\pi$
$984$	0	0
$985$	21.5959	0.688103
$986$	0	0
$987$	0	0
$988$	0	0
$989$	0	0
$990$	0	0
$991$	− 26.2020i	− 0.832335i	−0.909288	−	0.416168i	$-0.863373\pi$
$991$	− 26.2020i	− 0.832335i	0.909288	−	0.416168i	$-0.136627\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	5.79796i	0.183808i
$996$	0	0
$997$	− 29.6985i	− 0.940560i	−0.882517	−	0.470280i	$-0.844153\pi$
$997$	− 29.6985i	− 0.940560i	0.882517	−	0.470280i	$-0.155847\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2880.2.b.b.2591.4	yes	8
3.2	odd	2		2880.2.b.c.2591.4	yes	8
4.3	odd	2	inner	2880.2.b.b.2591.6	yes	8
8.3	odd	2		2880.2.b.c.2591.5	yes	8
8.5	even	2		2880.2.b.c.2591.3	yes	8
12.11	even	2		2880.2.b.c.2591.6	yes	8
24.5	odd	2	inner	2880.2.b.b.2591.3	✓	8
24.11	even	2	inner	2880.2.b.b.2591.5	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2880.2.b.b.2591.3	✓	8	24.5	odd	2	inner
2880.2.b.b.2591.4	yes	8	1.1	even	1	trivial
2880.2.b.b.2591.5	yes	8	24.11	even	2	inner
2880.2.b.b.2591.6	yes	8	4.3	odd	2	inner
2880.2.b.c.2591.3	yes	8	8.5	even	2
2880.2.b.c.2591.4	yes	8	3.2	odd	2
2880.2.b.c.2591.5	yes	8	8.3	odd	2
2880.2.b.c.2591.6	yes	8	12.11	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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-1.00000 q^{5} -0.449490i q^{7} +O(q^{10})\)	\(q-1.00000 q^{5} -0.449490i q^{7} +0.449490i q^{11} +1.41421i q^{13} -2.82843i q^{17} +2.82843 q^{19} -3.46410 q^{23} +1.00000 q^{25} +2.00000 q^{29} -4.89898i q^{31} +0.449490i q^{35} +5.51399i q^{37} -5.51399i q^{41} -4.09978 q^{47} +6.79796 q^{49} +4.00000 q^{53} -0.449490i q^{55} -5.34847i q^{59} +1.55708i q^{61} -1.41421i q^{65} +11.3137 q^{67} -5.65685 q^{71} +7.79796 q^{73} +0.202041 q^{77} -12.8990i q^{79} -8.89898i q^{83} +2.82843i q^{85} +0.142865i q^{89} +0.635674 q^{91} -2.82843 q^{95} +7.79796 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{5}+O(q^{10}) \) 8 * q - 8 * q^5	\( 8 q - 8 q^{5} + 8 q^{25} + 16 q^{29} - 24 q^{49} + 32 q^{53} - 16 q^{73} + 80 q^{77} - 16 q^{97}+O(q^{100}) \) 8 * q - 8 * q^5 + 8 * q^25 + 16 * q^29 - 24 * q^49 + 32 * q^53 - 16 * q^73 + 80 * q^77 - 16 * q^97

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