LMFDB - Embedded newform 864.2.d.a.433.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [864,2,Mod(433,864)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(864, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("864.433");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 864 = 2^{5} \cdot 3^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	864.d (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$6.89907473464$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{7})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 3x^{2} + 4 $ x^4 - 3*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	no (minimal twist has level 216)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			433.2
Root			$-1.32288 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	864.433
Dual form			864.2.d.a.433.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.00000i q^{5} -1.00000 q^{7} +O(q^{10})$	$q-1.00000i q^{5} -1.00000 q^{7} +3.00000i q^{11} +5.29150i q^{13} -5.29150 q^{17} +5.29150i q^{19} +5.29150 q^{23} +4.00000 q^{25} -6.00000i q^{29} +7.00000 q^{31} +1.00000i q^{35} +5.29150i q^{37} +5.29150 q^{41} +10.5830i q^{43} -6.00000 q^{49} +9.00000i q^{53} +3.00000 q^{55} -4.00000i q^{59} +5.29150 q^{65} -15.8745 q^{71} +3.00000 q^{73} -3.00000i q^{77} -4.00000 q^{79} -7.00000i q^{83} +5.29150i q^{85} -10.5830 q^{89} -5.29150i q^{91} +5.29150 q^{95} +7.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{7}+O(q^{10}) $ 4 * q - 4 * q^7	$ 4 q - 4 q^{7} + 16 q^{25} + 28 q^{31} - 24 q^{49} + 12 q^{55} + 12 q^{73} - 16 q^{79} + 28 q^{97}+O(q^{100}) $ 4 * q - 4 * q^7 + 16 * q^25 + 28 * q^31 - 24 * q^49 + 12 * q^55 + 12 * q^73 - 16 * q^79 + 28 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/864\mathbb{Z}\right)^\times$.

$n$	$325$	$353$	$703$
$\chi(n)$	$-1$	$1$	$1$

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	−0.974679	−	0.223607i	$-0.928217\pi$
$5$	− 1.00000i	− 0.447214i	0.974679	−	0.223607i	$-0.0717831\pi$
$6$	0	0
$7$	−1.00000	−0.377964	−0.188982	−	0.981981i	$-0.560519\pi$
$7$	−1.00000	−0.377964	−0.188982	+	0.981981i	$0.560519\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	3.00000i	0.904534i	0.891883	+	0.452267i	$0.149385\pi$
$11$	3.00000i	0.904534i	−0.891883	+	0.452267i	$0.850615\pi$
$12$	0	0
$13$	5.29150i	1.46760i	0.679366	+	0.733799i	$0.262255\pi$
$13$	5.29150i	1.46760i	−0.679366	+	0.733799i	$0.737745\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	−5.29150	−1.28338	−0.641689	−	0.766965i	$-0.721766\pi$
$17$	−5.29150	−1.28338	−0.641689	+	0.766965i	$0.721766\pi$
$18$	0	0
$19$	5.29150i	1.21395i	0.794719	+	0.606977i	$0.207618\pi$
$19$	5.29150i	1.21395i	−0.794719	+	0.606977i	$0.792382\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	5.29150	1.10335	0.551677	−	0.834058i	$-0.313988\pi$
$23$	5.29150	1.10335	0.551677	+	0.834058i	$0.313988\pi$
$24$	0	0
$25$	4.00000	0.800000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 6.00000i	− 1.11417i	−0.830455	−	0.557086i	$-0.811919\pi$
$29$	− 6.00000i	− 1.11417i	0.830455	−	0.557086i	$-0.188081\pi$
$30$	0	0
$31$	7.00000	1.25724	0.628619	−	0.777714i	$-0.283621\pi$
$31$	7.00000	1.25724	0.628619	+	0.777714i	$0.283621\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	1.00000i	0.169031i
$36$	0	0
$37$	5.29150i	0.869918i	0.900450	+	0.434959i	$0.143237\pi$
$37$	5.29150i	0.869918i	−0.900450	+	0.434959i	$0.856763\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.29150	0.826394	0.413197	−	0.910642i	$-0.364412\pi$
$41$	5.29150	0.826394	0.413197	+	0.910642i	$0.364412\pi$
$42$	0	0
$43$	10.5830i	1.61389i	0.590624	+	0.806947i	$0.298881\pi$
$43$	10.5830i	1.61389i	−0.590624	+	0.806947i	$0.701119\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	0	0		−	1.00000i	$-0.5\pi$
$47$	0	0			1.00000i	$0.5\pi$
$48$	0	0
$49$	−6.00000	−0.857143
$50$	0	0
$51$	0	0
$52$	0	0
$53$	9.00000i	1.23625i	0.786082	+	0.618123i	$0.212106\pi$
$53$	9.00000i	1.23625i	−0.786082	+	0.618123i	$0.787894\pi$
$54$	0	0
$55$	3.00000	0.404520
$56$	0	0
$57$	0	0
$58$	0	0
$59$	− 4.00000i	− 0.520756i	−0.965507	−	0.260378i	$-0.916153\pi$
$59$	− 4.00000i	− 0.520756i	0.965507	−	0.260378i	$-0.0838471\pi$
$60$	0	0
$61$	0	0	1.00000			$0$
$61$	0	0	−1.00000			$\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	5.29150	0.656330
$66$	0	0
$67$	0	0	1.00000			$0$
$67$	0	0	−1.00000			$\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	−15.8745	−1.88396	−0.941979	−	0.335673i	$-0.891036\pi$
$71$	−15.8745	−1.88396	−0.941979	+	0.335673i	$0.891036\pi$
$72$	0	0
$73$	3.00000	0.351123	0.175562	−	0.984468i	$-0.443826\pi$
$73$	3.00000	0.351123	0.175562	+	0.984468i	$0.443826\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 3.00000i	− 0.341882i
$78$	0	0
$79$	−4.00000	−0.450035	−0.225018	−	0.974355i	$-0.572244\pi$
$79$	−4.00000	−0.450035	−0.225018	+	0.974355i	$0.572244\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	− 7.00000i	− 0.768350i	−0.923260	−	0.384175i	$-0.874486\pi$
$83$	− 7.00000i	− 0.768350i	0.923260	−	0.384175i	$-0.125514\pi$
$84$	0	0
$85$	5.29150i	0.573944i
$86$	0	0
$87$	0	0
$88$	0	0
$89$	−10.5830	−1.12180	−0.560898	−	0.827885i	$-0.689544\pi$
$89$	−10.5830	−1.12180	−0.560898	+	0.827885i	$0.689544\pi$
$90$	0	0
$91$	− 5.29150i	− 0.554700i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	5.29150	0.542897
$96$	0	0
$97$	7.00000	0.710742	0.355371	−	0.934725i	$-0.384354\pi$
$97$	7.00000	0.710742	0.355371	+	0.934725i	$0.384354\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	17.0000i	1.69156i	0.533529	+	0.845782i	$0.320865\pi$
$101$	17.0000i	1.69156i	−0.533529	+	0.845782i	$0.679135\pi$
$102$	0	0
$103$	0	0		−	1.00000i	$-0.5\pi$
$103$	0	0			1.00000i	$0.5\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	3.00000i	0.290021i	0.989430	+	0.145010i	$0.0463216\pi$
$107$	3.00000i	0.290021i	−0.989430	+	0.145010i	$0.953678\pi$
$108$	0	0
$109$	− 5.29150i	− 0.506834i	−0.967357	−	0.253417i	$-0.918446\pi$
$109$	− 5.29150i	− 0.506834i	0.967357	−	0.253417i	$-0.0815545\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	15.8745	1.49335	0.746674	−	0.665190i	$-0.231650\pi$
$113$	15.8745	1.49335	0.746674	+	0.665190i	$0.231650\pi$
$114$	0	0
$115$	− 5.29150i	− 0.493435i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	5.29150	0.485071
$120$	0	0
$121$	2.00000	0.181818
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 9.00000i	− 0.804984i
$126$	0	0
$127$	−13.0000	−1.15356	−0.576782	−	0.816898i	$-0.695692\pi$
$127$	−13.0000	−1.15356	−0.576782	+	0.816898i	$0.695692\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	− 7.00000i	− 0.611593i	−0.952097	−	0.305796i	$-0.901077\pi$
$131$	− 7.00000i	− 0.611593i	0.952097	−	0.305796i	$-0.0989227\pi$
$132$	0	0
$133$	− 5.29150i	− 0.458831i
$134$	0	0
$135$	0	0
$136$	0	0
$137$	0	0		−	1.00000i	$-0.5\pi$
$137$	0	0			1.00000i	$0.5\pi$
$138$	0	0
$139$	− 10.5830i	− 0.897639i	−0.893622	−	0.448819i	$-0.851845\pi$
$139$	− 10.5830i	− 0.897639i	0.893622	−	0.448819i	$-0.148155\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	−15.8745	−1.32749
$144$	0	0
$145$	−6.00000	−0.498273
$146$	0	0
$147$	0	0
$148$	0	0
$149$	3.00000i	0.245770i	0.992421	+	0.122885i	$0.0392146\pi$
$149$	3.00000i	0.245770i	−0.992421	+	0.122885i	$0.960785\pi$
$150$	0	0
$151$	−3.00000	−0.244137	−0.122068	−	0.992522i	$-0.538953\pi$
$151$	−3.00000	−0.244137	−0.122068	+	0.992522i	$0.538953\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	− 7.00000i	− 0.562254i
$156$	0	0
$157$	− 10.5830i	− 0.844616i	−0.906452	−	0.422308i	$-0.861220\pi$
$157$	− 10.5830i	− 0.844616i	0.906452	−	0.422308i	$-0.138780\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	−5.29150	−0.417029
$162$	0	0
$163$	− 5.29150i	− 0.414462i	−0.978292	−	0.207231i	$-0.933555\pi$
$163$	− 5.29150i	− 0.414462i	0.978292	−	0.207231i	$-0.0664452\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	10.5830	0.818938	0.409469	−	0.912324i	$-0.365714\pi$
$167$	10.5830	0.818938	0.409469	+	0.912324i	$0.365714\pi$
$168$	0	0
$169$	−15.0000	−1.15385
$170$	0	0
$171$	0	0
$172$	0	0
$173$	− 15.0000i	− 1.14043i	−0.821496	−	0.570214i	$-0.806860\pi$
$173$	− 15.0000i	− 1.14043i	0.821496	−	0.570214i	$-0.193140\pi$
$174$	0	0
$175$	−4.00000	−0.302372
$176$	0	0
$177$	0	0
$178$	0	0
$179$	23.0000i	1.71910i	0.511051	+	0.859550i	$0.329256\pi$
$179$	23.0000i	1.71910i	−0.511051	+	0.859550i	$0.670744\pi$
$180$	0	0
$181$	− 21.1660i	− 1.57326i	−0.617426	−	0.786629i	$-0.711825\pi$
$181$	− 21.1660i	− 1.57326i	0.617426	−	0.786629i	$-0.288175\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	5.29150	0.389039
$186$	0	0
$187$	− 15.8745i	− 1.16086i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	5.29150	0.382880	0.191440	−	0.981504i	$-0.438684\pi$
$191$	5.29150	0.382880	0.191440	+	0.981504i	$0.438684\pi$
$192$	0	0
$193$	−3.00000	−0.215945	−0.107972	−	0.994154i	$-0.534436\pi$
$193$	−3.00000	−0.215945	−0.107972	+	0.994154i	$0.534436\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	13.0000i	0.926212i	0.886303	+	0.463106i	$0.153265\pi$
$197$	13.0000i	0.926212i	−0.886303	+	0.463106i	$0.846735\pi$
$198$	0	0
$199$	17.0000	1.20510	0.602549	−	0.798082i	$-0.294152\pi$
$199$	17.0000	1.20510	0.602549	+	0.798082i	$0.294152\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	6.00000i	0.421117i
$204$	0	0
$205$	− 5.29150i	− 0.369575i
$206$	0	0
$207$	0	0
$208$	0	0
$209$	−15.8745	−1.09806
$210$	0	0
$211$	− 5.29150i	− 0.364282i	−0.983272	−	0.182141i	$-0.941697\pi$
$211$	− 5.29150i	− 0.364282i	0.983272	−	0.182141i	$-0.0583027\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	10.5830	0.721755
$216$	0	0
$217$	−7.00000	−0.475191
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 28.0000i	− 1.88348i
$222$	0	0
$223$	16.0000	1.07144	0.535720	−	0.844396i	$-0.320040\pi$
$223$	16.0000	1.07144	0.535720	+	0.844396i	$0.320040\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	4.00000i	0.265489i	0.991150	+	0.132745i	$0.0423790\pi$
$227$	4.00000i	0.265489i	−0.991150	+	0.132745i	$0.957621\pi$
$228$	0	0
$229$	− 21.1660i	− 1.39869i	−0.714785	−	0.699345i	$-0.753475\pi$
$229$	− 21.1660i	− 1.39869i	0.714785	−	0.699345i	$-0.246525\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−10.5830	−0.693316	−0.346658	−	0.937992i	$-0.612684\pi$
$233$	−10.5830	−0.693316	−0.346658	+	0.937992i	$0.612684\pi$
$234$	0	0
$235$	0	0
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.8745	1.02684	0.513418	−	0.858138i	$-0.328379\pi$
$239$	15.8745	1.02684	0.513418	+	0.858138i	$0.328379\pi$
$240$	0	0
$241$	−10.0000	−0.644157	−0.322078	−	0.946713i	$-0.604381\pi$
$241$	−10.0000	−0.644157	−0.322078	+	0.946713i	$0.604381\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	6.00000i	0.383326i
$246$	0	0
$247$	−28.0000	−1.78160
$248$	0	0
$249$	0	0
$250$	0	0
$251$	− 12.0000i	− 0.757433i	−0.925513	−	0.378717i	$-0.876365\pi$
$251$	− 12.0000i	− 0.757433i	0.925513	−	0.378717i	$-0.123635\pi$
$252$	0	0
$253$	15.8745i	0.998022i
$254$	0	0
$255$	0	0
$256$	0	0
$257$	−15.8745	−0.990225	−0.495112	−	0.868829i	$-0.664873\pi$
$257$	−15.8745	−0.990225	−0.495112	+	0.868829i	$0.664873\pi$
$258$	0	0
$259$	− 5.29150i	− 0.328798i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	−5.29150	−0.326288	−0.163144	−	0.986602i	$-0.552164\pi$
$263$	−5.29150	−0.326288	−0.163144	+	0.986602i	$0.552164\pi$
$264$	0	0
$265$	9.00000	0.552866
$266$	0	0
$267$	0	0
$268$	0	0
$269$	− 18.0000i	− 1.09748i	−0.835993	−	0.548740i	$-0.815108\pi$
$269$	− 18.0000i	− 1.09748i	0.835993	−	0.548740i	$-0.184892\pi$
$270$	0	0
$271$	−15.0000	−0.911185	−0.455593	−	0.890188i	$-0.650573\pi$
$271$	−15.0000	−0.911185	−0.455593	+	0.890188i	$0.650573\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	12.0000i	0.723627i
$276$	0	0
$277$	10.5830i	0.635871i	0.948112	+	0.317936i	$0.102990\pi$
$277$	10.5830i	0.635871i	−0.948112	+	0.317936i	$0.897010\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	15.8745	0.946994	0.473497	−	0.880795i	$-0.342992\pi$
$281$	15.8745	0.946994	0.473497	+	0.880795i	$0.342992\pi$
$282$	0	0
$283$	26.4575i	1.57274i	0.617758	+	0.786368i	$0.288041\pi$
$283$	26.4575i	1.57274i	−0.617758	+	0.786368i	$0.711959\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−5.29150	−0.312348
$288$	0	0
$289$	11.0000	0.647059
$290$	0	0
$291$	0	0
$292$	0	0
$293$	30.0000i	1.75262i	0.481749	+	0.876309i	$0.340002\pi$
$293$	30.0000i	1.75262i	−0.481749	+	0.876309i	$0.659998\pi$
$294$	0	0
$295$	−4.00000	−0.232889
$296$	0	0
$297$	0	0
$298$	0	0
$299$	28.0000i	1.61928i
$300$	0	0
$301$	− 10.5830i	− 0.609994i
$302$	0	0
$303$	0	0
$304$	0	0
$305$	0	0
$306$	0	0
$307$	− 15.8745i	− 0.906006i	−0.891509	−	0.453003i	$-0.850353\pi$
$307$	− 15.8745i	− 0.906006i	0.891509	−	0.453003i	$-0.149647\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−26.4575	−1.50027	−0.750134	−	0.661286i	$-0.770011\pi$
$311$	−26.4575	−1.50027	−0.750134	+	0.661286i	$0.770011\pi$
$312$	0	0
$313$	−1.00000	−0.0565233	−0.0282617	−	0.999601i	$-0.508997\pi$
$313$	−1.00000	−0.0565233	−0.0282617	+	0.999601i	$0.508997\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 13.0000i	− 0.730153i	−0.930978	−	0.365076i	$-0.881043\pi$
$317$	− 13.0000i	− 0.730153i	0.930978	−	0.365076i	$-0.118957\pi$
$318$	0	0
$319$	18.0000	1.00781
$320$	0	0
$321$	0	0
$322$	0	0
$323$	− 28.0000i	− 1.55796i
$324$	0	0
$325$	21.1660i	1.17408i
$326$	0	0
$327$	0	0
$328$	0	0
$329$	0	0
$330$	0	0
$331$	− 21.1660i	− 1.16339i	−0.813407	−	0.581695i	$-0.802390\pi$
$331$	− 21.1660i	− 1.16339i	0.813407	−	0.581695i	$-0.197610\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	0	0
$336$	0	0
$337$	34.0000	1.85210	0.926049	−	0.377403i	$-0.123183\pi$
$337$	34.0000	1.85210	0.926049	+	0.377403i	$0.123183\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	21.0000i	1.13721i
$342$	0	0
$343$	13.0000	0.701934
$344$	0	0
$345$	0	0
$346$	0	0
$347$	− 23.0000i	− 1.23470i	−0.786687	−	0.617352i	$-0.788205\pi$
$347$	− 23.0000i	− 1.23470i	0.786687	−	0.617352i	$-0.211795\pi$
$348$	0	0
$349$	5.29150i	0.283248i	0.989921	+	0.141624i	$0.0452323\pi$
$349$	5.29150i	0.283248i	−0.989921	+	0.141624i	$0.954768\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	15.8745	0.844915	0.422457	−	0.906383i	$-0.361168\pi$
$353$	15.8745	0.844915	0.422457	+	0.906383i	$0.361168\pi$
$354$	0	0
$355$	15.8745i	0.842531i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	21.1660	1.11710	0.558550	−	0.829471i	$-0.311358\pi$
$359$	21.1660	1.11710	0.558550	+	0.829471i	$0.311358\pi$
$360$	0	0
$361$	−9.00000	−0.473684
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 3.00000i	− 0.157027i
$366$	0	0
$367$	3.00000	0.156599	0.0782994	−	0.996930i	$-0.475051\pi$
$367$	3.00000	0.156599	0.0782994	+	0.996930i	$0.475051\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	− 9.00000i	− 0.467257i
$372$	0	0
$373$	26.4575i	1.36992i	0.728582	+	0.684959i	$0.240180\pi$
$373$	26.4575i	1.36992i	−0.728582	+	0.684959i	$0.759820\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	31.7490	1.63516
$378$	0	0
$379$	21.1660i	1.08722i	0.839336	+	0.543612i	$0.182944\pi$
$379$	21.1660i	1.08722i	−0.839336	+	0.543612i	$0.817056\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	−15.8745	−0.811149	−0.405575	−	0.914062i	$-0.632929\pi$
$383$	−15.8745	−0.811149	−0.405575	+	0.914062i	$0.632929\pi$
$384$	0	0
$385$	−3.00000	−0.152894
$386$	0	0
$387$	0	0
$388$	0	0
$389$	− 5.00000i	− 0.253510i	−0.991934	−	0.126755i	$-0.959544\pi$
$389$	− 5.00000i	− 0.253510i	0.991934	−	0.126755i	$-0.0404562\pi$
$390$	0	0
$391$	−28.0000	−1.41602
$392$	0	0
$393$	0	0
$394$	0	0
$395$	4.00000i	0.201262i
$396$	0	0
$397$	− 5.29150i	− 0.265573i	−0.991145	−	0.132786i	$-0.957608\pi$
$397$	− 5.29150i	− 0.265573i	0.991145	−	0.132786i	$-0.0423924\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	0	0		−	1.00000i	$-0.5\pi$
$401$	0	0			1.00000i	$0.5\pi$
$402$	0	0
$403$	37.0405i	1.84512i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−15.8745	−0.786870
$408$	0	0
$409$	−11.0000	−0.543915	−0.271957	−	0.962309i	$-0.587671\pi$
$409$	−11.0000	−0.543915	−0.271957	+	0.962309i	$0.587671\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	4.00000i	0.196827i
$414$	0	0
$415$	−7.00000	−0.343616
$416$	0	0
$417$	0	0
$418$	0	0
$419$	0	0	1.00000			$0$
$419$	0	0	−1.00000			$\pi$
$420$	0	0
$421$	10.5830i	0.515784i	0.966174	+	0.257892i	$0.0830279\pi$
$421$	10.5830i	0.515784i	−0.966174	+	0.257892i	$0.916972\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	−21.1660	−1.02670
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0	0
$430$	0	0
$431$	31.7490	1.52930	0.764648	−	0.644448i	$-0.222913\pi$
$431$	31.7490	1.52930	0.764648	+	0.644448i	$0.222913\pi$
$432$	0	0
$433$	5.00000	0.240285	0.120142	−	0.992757i	$-0.461665\pi$
$433$	5.00000	0.240285	0.120142	+	0.992757i	$0.461665\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	28.0000i	1.33942i
$438$	0	0
$439$	21.0000	1.00228	0.501138	−	0.865368i	$-0.332915\pi$
$439$	21.0000	1.00228	0.501138	+	0.865368i	$0.332915\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	− 24.0000i	− 1.14027i	−0.821549	−	0.570137i	$-0.806890\pi$
$443$	− 24.0000i	− 1.14027i	0.821549	−	0.570137i	$-0.193110\pi$
$444$	0	0
$445$	10.5830i	0.501683i
$446$	0	0
$447$	0	0
$448$	0	0
$449$	15.8745	0.749164	0.374582	−	0.927194i	$-0.377786\pi$
$449$	15.8745	0.749164	0.374582	+	0.927194i	$0.377786\pi$
$450$	0	0
$451$	15.8745i	0.747501i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.29150	−0.248069
$456$	0	0
$457$	17.0000	0.795226	0.397613	−	0.917553i	$-0.369839\pi$
$457$	17.0000	0.795226	0.397613	+	0.917553i	$0.369839\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	9.00000i	0.419172i	0.977790	+	0.209586i	$0.0672116\pi$
$461$	9.00000i	0.419172i	−0.977790	+	0.209586i	$0.932788\pi$
$462$	0	0
$463$	41.0000	1.90543	0.952716	−	0.303863i	$-0.0982765\pi$
$463$	41.0000	1.90543	0.952716	+	0.303863i	$0.0982765\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	1.00000i	0.0462745i	0.999732	+	0.0231372i	$0.00736547\pi$
$467$	1.00000i	0.0462745i	−0.999732	+	0.0231372i	$0.992635\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	0	0
$472$	0	0
$473$	−31.7490	−1.45982
$474$	0	0
$475$	21.1660i	0.971163i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−21.1660	−0.967100	−0.483550	−	0.875317i	$-0.660653\pi$
$479$	−21.1660	−0.967100	−0.483550	+	0.875317i	$0.660653\pi$
$480$	0	0
$481$	−28.0000	−1.27669
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 7.00000i	− 0.317854i
$486$	0	0
$487$	40.0000	1.81257	0.906287	−	0.422664i	$-0.138905\pi$
$487$	40.0000	1.81257	0.906287	+	0.422664i	$0.138905\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	29.0000i	1.30875i	0.756169	+	0.654376i	$0.227069\pi$
$491$	29.0000i	1.30875i	−0.756169	+	0.654376i	$0.772931\pi$
$492$	0	0
$493$	31.7490i	1.42990i
$494$	0	0
$495$	0	0
$496$	0	0
$497$	15.8745	0.712069
$498$	0	0
$499$	0	0	1.00000			$0$
$499$	0	0	−1.00000			$\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	−15.8745	−0.707809	−0.353905	−	0.935282i	$-0.615146\pi$
$503$	−15.8745	−0.707809	−0.353905	+	0.935282i	$0.615146\pi$
$504$	0	0
$505$	17.0000	0.756490
$506$	0	0
$507$	0	0
$508$	0	0
$509$	− 21.0000i	− 0.930809i	−0.885098	−	0.465404i	$-0.845909\pi$
$509$	− 21.0000i	− 0.930809i	0.885098	−	0.465404i	$-0.154091\pi$
$510$	0	0
$511$	−3.00000	−0.132712
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0	0
$516$	0	0
$517$	0	0
$518$	0	0
$519$	0	0
$520$	0	0
$521$	15.8745	0.695475	0.347737	−	0.937592i	$-0.386950\pi$
$521$	15.8745	0.695475	0.347737	+	0.937592i	$0.386950\pi$
$522$	0	0
$523$	15.8745i	0.694144i	0.937839	+	0.347072i	$0.112824\pi$
$523$	15.8745i	0.694144i	−0.937839	+	0.347072i	$0.887176\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−37.0405	−1.61351
$528$	0	0
$529$	5.00000	0.217391
$530$	0	0
$531$	0	0
$532$	0	0
$533$	28.0000i	1.21281i
$534$	0	0
$535$	3.00000	0.129701
$536$	0	0
$537$	0	0
$538$	0	0
$539$	− 18.0000i	− 0.775315i
$540$	0	0
$541$	− 10.5830i	− 0.454999i	−0.973778	−	0.227499i	$-0.926945\pi$
$541$	− 10.5830i	− 0.454999i	0.973778	−	0.227499i	$-0.0730550\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	−5.29150	−0.226663
$546$	0	0
$547$	5.29150i	0.226248i	0.993581	+	0.113124i	$0.0360858\pi$
$547$	5.29150i	0.226248i	−0.993581	+	0.113124i	$0.963914\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	31.7490	1.35255
$552$	0	0
$553$	4.00000	0.170097
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 39.0000i	− 1.65248i	−0.563316	−	0.826242i	$-0.690475\pi$
$557$	− 39.0000i	− 1.65248i	0.563316	−	0.826242i	$-0.309525\pi$
$558$	0	0
$559$	−56.0000	−2.36855
$560$	0	0
$561$	0	0
$562$	0	0
$563$	3.00000i	0.126435i	0.998000	+	0.0632175i	$0.0201362\pi$
$563$	3.00000i	0.126435i	−0.998000	+	0.0632175i	$0.979864\pi$
$564$	0	0
$565$	− 15.8745i	− 0.667846i
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−31.7490	−1.33099	−0.665494	−	0.746403i	$-0.731779\pi$
$569$	−31.7490	−1.33099	−0.665494	+	0.746403i	$0.731779\pi$
$570$	0	0
$571$	− 26.4575i	− 1.10721i	−0.832779	−	0.553606i	$-0.813251\pi$
$571$	− 26.4575i	− 1.10721i	0.832779	−	0.553606i	$-0.186749\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	21.1660	0.882684
$576$	0	0
$577$	−18.0000	−0.749350	−0.374675	−	0.927156i	$-0.622246\pi$
$577$	−18.0000	−0.749350	−0.374675	+	0.927156i	$0.622246\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	7.00000i	0.290409i
$582$	0	0
$583$	−27.0000	−1.11823
$584$	0	0
$585$	0	0
$586$	0	0
$587$	23.0000i	0.949312i	0.880172	+	0.474656i	$0.157427\pi$
$587$	23.0000i	0.949312i	−0.880172	+	0.474656i	$0.842573\pi$
$588$	0	0
$589$	37.0405i	1.52623i
$590$	0	0
$591$	0	0
$592$	0	0
$593$	31.7490	1.30378	0.651888	−	0.758315i	$-0.273977\pi$
$593$	31.7490	1.30378	0.651888	+	0.758315i	$0.273977\pi$
$594$	0	0
$595$	− 5.29150i	− 0.216930i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	21.1660	0.864820	0.432410	−	0.901677i	$-0.357663\pi$
$599$	21.1660	0.864820	0.432410	+	0.901677i	$0.357663\pi$
$600$	0	0
$601$	5.00000	0.203954	0.101977	−	0.994787i	$-0.467483\pi$
$601$	5.00000	0.203954	0.101977	+	0.994787i	$0.467483\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 2.00000i	− 0.0813116i
$606$	0	0
$607$	8.00000	0.324710	0.162355	−	0.986732i	$-0.448091\pi$
$607$	8.00000	0.324710	0.162355	+	0.986732i	$0.448091\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	0	0
$612$	0	0
$613$	10.5830i	0.427444i	0.976895	+	0.213722i	$0.0685586\pi$
$613$	10.5830i	0.427444i	−0.976895	+	0.213722i	$0.931441\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	−5.29150	−0.213028	−0.106514	−	0.994311i	$-0.533969\pi$
$617$	−5.29150	−0.213028	−0.106514	+	0.994311i	$0.533969\pi$
$618$	0	0
$619$	15.8745i	0.638050i	0.947746	+	0.319025i	$0.103355\pi$
$619$	15.8745i	0.638050i	−0.947746	+	0.319025i	$0.896645\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.5830	0.423999
$624$	0	0
$625$	11.0000	0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	− 28.0000i	− 1.11643i
$630$	0	0
$631$	23.0000	0.915616	0.457808	−	0.889051i	$-0.348635\pi$
$631$	23.0000	0.915616	0.457808	+	0.889051i	$0.348635\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	13.0000i	0.515889i
$636$	0	0
$637$	− 31.7490i	− 1.25794i
$638$	0	0
$639$	0	0
$640$	0	0
$641$	−37.0405	−1.46301	−0.731506	−	0.681835i	$-0.761182\pi$
$641$	−37.0405	−1.46301	−0.731506	+	0.681835i	$0.761182\pi$
$642$	0	0
$643$	− 21.1660i	− 0.834706i	−0.908744	−	0.417353i	$-0.862958\pi$
$643$	− 21.1660i	− 0.834706i	0.908744	−	0.417353i	$-0.137042\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	31.7490	1.24818	0.624091	−	0.781351i	$-0.285469\pi$
$647$	31.7490	1.24818	0.624091	+	0.781351i	$0.285469\pi$
$648$	0	0
$649$	12.0000	0.471041
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 3.00000i	− 0.117399i	−0.998276	−	0.0586995i	$-0.981305\pi$
$653$	− 3.00000i	− 0.117399i	0.998276	−	0.0586995i	$-0.0186954\pi$
$654$	0	0
$655$	−7.00000	−0.273513
$656$	0	0
$657$	0	0
$658$	0	0
$659$	15.0000i	0.584317i	0.956370	+	0.292159i	$0.0943735\pi$
$659$	15.0000i	0.584317i	−0.956370	+	0.292159i	$0.905627\pi$
$660$	0	0
$661$	− 26.4575i	− 1.02908i	−0.857467	−	0.514539i	$-0.827963\pi$
$661$	− 26.4575i	− 1.02908i	0.857467	−	0.514539i	$-0.172037\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	−5.29150	−0.205196
$666$	0	0
$667$	− 31.7490i	− 1.22933i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	0	0
$672$	0	0
$673$	29.0000	1.11787	0.558934	−	0.829212i	$-0.311211\pi$
$673$	29.0000	1.11787	0.558934	+	0.829212i	$0.311211\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	− 6.00000i	− 0.230599i	−0.993331	−	0.115299i	$-0.963217\pi$
$677$	− 6.00000i	− 0.230599i	0.993331	−	0.115299i	$-0.0367827\pi$
$678$	0	0
$679$	−7.00000	−0.268635
$680$	0	0
$681$	0	0
$682$	0	0
$683$	44.0000i	1.68361i	0.539779	+	0.841807i	$0.318508\pi$
$683$	44.0000i	1.68361i	−0.539779	+	0.841807i	$0.681492\pi$
$684$	0	0
$685$	0	0
$686$	0	0
$687$	0	0
$688$	0	0
$689$	−47.6235	−1.81431
$690$	0	0
$691$	− 42.3320i	− 1.61039i	−0.593013	−	0.805193i	$-0.702062\pi$
$691$	− 42.3320i	− 1.61039i	0.593013	−	0.805193i	$-0.297938\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−10.5830	−0.401436
$696$	0	0
$697$	−28.0000	−1.06058
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 1.00000i	− 0.0377695i	−0.999822	−	0.0188847i	$-0.993988\pi$
$701$	− 1.00000i	− 0.0377695i	0.999822	−	0.0188847i	$-0.00601156\pi$
$702$	0	0
$703$	−28.0000	−1.05604
$704$	0	0
$705$	0	0
$706$	0	0
$707$	− 17.0000i	− 0.639351i
$708$	0	0
$709$	− 21.1660i	− 0.794906i	−0.917622	−	0.397453i	$-0.869894\pi$
$709$	− 21.1660i	− 0.794906i	0.917622	−	0.397453i	$-0.130106\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	37.0405	1.38718
$714$	0	0
$715$	15.8745i	0.593673i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	21.1660	0.789359	0.394679	−	0.918819i	$-0.370856\pi$
$719$	21.1660	0.789359	0.394679	+	0.918819i	$0.370856\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 24.0000i	− 0.891338i
$726$	0	0
$727$	−33.0000	−1.22390	−0.611951	−	0.790896i	$-0.709615\pi$
$727$	−33.0000	−1.22390	−0.611951	+	0.790896i	$0.709615\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	− 56.0000i	− 2.07123i
$732$	0	0
$733$	5.29150i	0.195446i	0.995214	+	0.0977231i	$0.0311559\pi$
$733$	5.29150i	0.195446i	−0.995214	+	0.0977231i	$0.968844\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	0	0
$738$	0	0
$739$	− 5.29150i	− 0.194651i	−0.995253	−	0.0973255i	$-0.968971\pi$
$739$	− 5.29150i	− 0.194651i	0.995253	−	0.0973255i	$-0.0310288\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−37.0405	−1.35888	−0.679442	−	0.733729i	$-0.737778\pi$
$743$	−37.0405	−1.35888	−0.679442	+	0.733729i	$0.737778\pi$
$744$	0	0
$745$	3.00000	0.109911
$746$	0	0
$747$	0	0
$748$	0	0
$749$	− 3.00000i	− 0.109618i
$750$	0	0
$751$	31.0000	1.13121	0.565603	−	0.824678i	$-0.308643\pi$
$751$	31.0000	1.13121	0.565603	+	0.824678i	$0.308643\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	3.00000i	0.109181i
$756$	0	0
$757$	0	0	1.00000			$0$
$757$	0	0	−1.00000			$\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	10.5830	0.383634	0.191817	−	0.981431i	$-0.438562\pi$
$761$	10.5830	0.383634	0.191817	+	0.981431i	$0.438562\pi$
$762$	0	0
$763$	5.29150i	0.191565i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	21.1660	0.764260
$768$	0	0
$769$	−35.0000	−1.26213	−0.631066	−	0.775729i	$-0.717382\pi$
$769$	−35.0000	−1.26213	−0.631066	+	0.775729i	$0.717382\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	14.0000i	0.503545i	0.967786	+	0.251773i	$0.0810135\pi$
$773$	14.0000i	0.503545i	−0.967786	+	0.251773i	$0.918987\pi$
$774$	0	0
$775$	28.0000	1.00579
$776$	0	0
$777$	0	0
$778$	0	0
$779$	28.0000i	1.00320i
$780$	0	0
$781$	− 47.6235i	− 1.70410i
$782$	0	0
$783$	0	0
$784$	0	0
$785$	−10.5830	−0.377724
$786$	0	0
$787$	0	0	1.00000			$0$
$787$	0	0	−1.00000			$\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	−15.8745	−0.564433
$792$	0	0
$793$	0	0
$794$	0	0
$795$	0	0
$796$	0	0
$797$	49.0000i	1.73567i	0.496853	+	0.867835i	$0.334489\pi$
$797$	49.0000i	1.73567i	−0.496853	+	0.867835i	$0.665511\pi$
$798$	0	0
$799$	0	0
$800$	0	0
$801$	0	0
$802$	0	0
$803$	9.00000i	0.317603i
$804$	0	0
$805$	5.29150i	0.186501i
$806$	0	0
$807$	0	0
$808$	0	0
$809$	31.7490	1.11624	0.558118	−	0.829762i	$-0.311524\pi$
$809$	31.7490	1.11624	0.558118	+	0.829762i	$0.311524\pi$
$810$	0	0
$811$	− 37.0405i	− 1.30067i	−0.759648	−	0.650334i	$-0.774629\pi$
$811$	− 37.0405i	− 1.30067i	0.759648	−	0.650334i	$-0.225371\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	−5.29150	−0.185353
$816$	0	0
$817$	−56.0000	−1.95919
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 10.0000i	− 0.349002i	−0.984657	−	0.174501i	$-0.944169\pi$
$821$	− 10.0000i	− 0.349002i	0.984657	−	0.174501i	$-0.0558313\pi$
$822$	0	0
$823$	−23.0000	−0.801730	−0.400865	−	0.916137i	$-0.631290\pi$
$823$	−23.0000	−0.801730	−0.400865	+	0.916137i	$0.631290\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	− 36.0000i	− 1.25184i	−0.779886	−	0.625921i	$-0.784723\pi$
$827$	− 36.0000i	− 1.25184i	0.779886	−	0.625921i	$-0.215277\pi$
$828$	0	0
$829$	− 21.1660i	− 0.735126i	−0.929999	−	0.367563i	$-0.880192\pi$
$829$	− 21.1660i	− 0.735126i	0.929999	−	0.367563i	$-0.119808\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	31.7490	1.10004
$834$	0	0
$835$	− 10.5830i	− 0.366240i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−37.0405	−1.27878	−0.639390	−	0.768882i	$-0.720813\pi$
$839$	−37.0405	−1.27878	−0.639390	+	0.768882i	$0.720813\pi$
$840$	0	0
$841$	−7.00000	−0.241379
$842$	0	0
$843$	0	0
$844$	0	0
$845$	15.0000i	0.516016i
$846$	0	0
$847$	−2.00000	−0.0687208
$848$	0	0
$849$	0	0
$850$	0	0
$851$	28.0000i	0.959828i
$852$	0	0
$853$	0	0	1.00000			$0$
$853$	0	0	−1.00000			$\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	15.8745	0.542263	0.271131	−	0.962542i	$-0.412602\pi$
$857$	15.8745	0.542263	0.271131	+	0.962542i	$0.412602\pi$
$858$	0	0
$859$	− 52.9150i	− 1.80544i	−0.430231	−	0.902719i	$-0.641568\pi$
$859$	− 52.9150i	− 1.80544i	0.430231	−	0.902719i	$-0.358432\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	0	0		−	1.00000i	$-0.5\pi$
$863$	0	0			1.00000i	$0.5\pi$
$864$	0	0
$865$	−15.0000	−0.510015
$866$	0	0
$867$	0	0
$868$	0	0
$869$	− 12.0000i	− 0.407072i
$870$	0	0
$871$	0	0
$872$	0	0
$873$	0	0
$874$	0	0
$875$	9.00000i	0.304256i
$876$	0	0
$877$	− 52.9150i	− 1.78681i	−0.449249	−	0.893407i	$-0.648308\pi$
$877$	− 52.9150i	− 1.78681i	0.449249	−	0.893407i	$-0.351692\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	−26.4575	−0.891376	−0.445688	−	0.895188i	$-0.647041\pi$
$881$	−26.4575	−0.891376	−0.445688	+	0.895188i	$0.647041\pi$
$882$	0	0
$883$	58.2065i	1.95881i	0.201916	+	0.979403i	$0.435283\pi$
$883$	58.2065i	1.95881i	−0.201916	+	0.979403i	$0.564717\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	15.8745	0.533014	0.266507	−	0.963833i	$-0.414130\pi$
$887$	15.8745	0.533014	0.266507	+	0.963833i	$0.414130\pi$
$888$	0	0
$889$	13.0000	0.436006
$890$	0	0
$891$	0	0
$892$	0	0
$893$	0	0
$894$	0	0
$895$	23.0000	0.768805
$896$	0	0
$897$	0	0
$898$	0	0
$899$	− 42.0000i	− 1.40078i
$900$	0	0
$901$	− 47.6235i	− 1.58657i
$902$	0	0
$903$	0	0
$904$	0	0
$905$	−21.1660	−0.703582
$906$	0	0
$907$	5.29150i	0.175701i	0.996134	+	0.0878507i	$0.0279999\pi$
$907$	5.29150i	0.175701i	−0.996134	+	0.0878507i	$0.972000\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−31.7490	−1.05189	−0.525946	−	0.850518i	$-0.676289\pi$
$911$	−31.7490	−1.05189	−0.525946	+	0.850518i	$0.676289\pi$
$912$	0	0
$913$	21.0000	0.694999
$914$	0	0
$915$	0	0
$916$	0	0
$917$	7.00000i	0.231160i
$918$	0	0
$919$	−3.00000	−0.0989609	−0.0494804	−	0.998775i	$-0.515757\pi$
$919$	−3.00000	−0.0989609	−0.0494804	+	0.998775i	$0.515757\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	− 84.0000i	− 2.76489i
$924$	0	0
$925$	21.1660i	0.695934i
$926$	0	0
$927$	0	0
$928$	0	0
$929$	10.5830	0.347217	0.173609	−	0.984815i	$-0.444457\pi$
$929$	10.5830	0.347217	0.173609	+	0.984815i	$0.444457\pi$
$930$	0	0
$931$	− 31.7490i	− 1.04053i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−15.8745	−0.519152
$936$	0	0
$937$	−19.0000	−0.620703	−0.310351	−	0.950622i	$-0.600447\pi$
$937$	−19.0000	−0.620703	−0.310351	+	0.950622i	$0.600447\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	21.0000i	0.684580i	0.939594	+	0.342290i	$0.111203\pi$
$941$	21.0000i	0.684580i	−0.939594	+	0.342290i	$0.888797\pi$
$942$	0	0
$943$	28.0000	0.911805
$944$	0	0
$945$	0	0
$946$	0	0
$947$	39.0000i	1.26733i	0.773608	+	0.633665i	$0.218450\pi$
$947$	39.0000i	1.26733i	−0.773608	+	0.633665i	$0.781550\pi$
$948$	0	0
$949$	15.8745i	0.515308i
$950$	0	0
$951$	0	0
$952$	0	0
$953$	−31.7490	−1.02845	−0.514226	−	0.857655i	$-0.671921\pi$
$953$	−31.7490	−1.02845	−0.514226	+	0.857655i	$0.671921\pi$
$954$	0	0
$955$	− 5.29150i	− 0.171229i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	0	0
$960$	0	0
$961$	18.0000	0.580645
$962$	0	0
$963$	0	0
$964$	0	0
$965$	3.00000i	0.0965734i
$966$	0	0
$967$	−41.0000	−1.31847	−0.659236	−	0.751936i	$-0.729120\pi$
$967$	−41.0000	−1.31847	−0.659236	+	0.751936i	$0.729120\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	15.0000i	0.481373i	0.970603	+	0.240686i	$0.0773725\pi$
$971$	15.0000i	0.481373i	−0.970603	+	0.240686i	$0.922627\pi$
$972$	0	0
$973$	10.5830i	0.339276i
$974$	0	0
$975$	0	0
$976$	0	0
$977$	10.5830	0.338580	0.169290	−	0.985566i	$-0.445852\pi$
$977$	10.5830	0.338580	0.169290	+	0.985566i	$0.445852\pi$
$978$	0	0
$979$	− 31.7490i	− 1.01470i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	42.3320	1.35018	0.675091	−	0.737735i	$-0.264104\pi$
$983$	42.3320	1.35018	0.675091	+	0.737735i	$0.264104\pi$
$984$	0	0
$985$	13.0000	0.414214
$986$	0	0
$987$	0	0
$988$	0	0
$989$	56.0000i	1.78070i
$990$	0	0
$991$	−47.0000	−1.49300	−0.746502	−	0.665383i	$-0.768268\pi$
$991$	−47.0000	−1.49300	−0.746502	+	0.665383i	$0.768268\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	− 17.0000i	− 0.538936i
$996$	0	0
$997$	15.8745i	0.502751i	0.967890	+	0.251375i	$0.0808829\pi$
$997$	15.8745i	0.502751i	−0.967890	+	0.251375i	$0.919117\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	864.2.d.a.433.2		4
3.2	odd	2	inner	864.2.d.a.433.4		4
4.3	odd	2		216.2.d.b.109.4	yes	4
8.3	odd	2		216.2.d.b.109.3	yes	4
8.5	even	2	inner	864.2.d.a.433.3		4
9.2	odd	6		2592.2.r.p.433.1		8
9.4	even	3		2592.2.r.p.2161.2		8
9.5	odd	6		2592.2.r.p.2161.4		8
9.7	even	3		2592.2.r.p.433.3		8
12.11	even	2		216.2.d.b.109.1	✓	4
16.3	odd	4		6912.2.a.bc.1.1		2
16.5	even	4		6912.2.a.bv.1.2		2
16.11	odd	4		6912.2.a.bu.1.2		2
16.13	even	4		6912.2.a.bd.1.1		2
24.5	odd	2	inner	864.2.d.a.433.1		4
24.11	even	2		216.2.d.b.109.2	yes	4
36.7	odd	6		648.2.n.n.109.2		8
36.11	even	6		648.2.n.n.109.3		8
36.23	even	6		648.2.n.n.541.4		8
36.31	odd	6		648.2.n.n.541.1		8
48.5	odd	4		6912.2.a.bd.1.2		2
48.11	even	4		6912.2.a.bc.1.2		2
48.29	odd	4		6912.2.a.bv.1.1		2
48.35	even	4		6912.2.a.bu.1.1		2
72.5	odd	6		2592.2.r.p.2161.1		8
72.11	even	6		648.2.n.n.109.4		8
72.13	even	6		2592.2.r.p.2161.3		8
72.29	odd	6		2592.2.r.p.433.4		8
72.43	odd	6		648.2.n.n.109.1		8
72.59	even	6		648.2.n.n.541.3		8
72.61	even	6		2592.2.r.p.433.2		8
72.67	odd	6		648.2.n.n.541.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
216.2.d.b.109.1	✓	4	12.11	even	2
216.2.d.b.109.2	yes	4	24.11	even	2
216.2.d.b.109.3	yes	4	8.3	odd	2
216.2.d.b.109.4	yes	4	4.3	odd	2
648.2.n.n.109.1		8	72.43	odd	6
648.2.n.n.109.2		8	36.7	odd	6
648.2.n.n.109.3		8	36.11	even	6
648.2.n.n.109.4		8	72.11	even	6
648.2.n.n.541.1		8	36.31	odd	6
648.2.n.n.541.2		8	72.67	odd	6
648.2.n.n.541.3		8	72.59	even	6
648.2.n.n.541.4		8	36.23	even	6
864.2.d.a.433.1		4	24.5	odd	2	inner
864.2.d.a.433.2		4	1.1	even	1	trivial
864.2.d.a.433.3		4	8.5	even	2	inner
864.2.d.a.433.4		4	3.2	odd	2	inner
2592.2.r.p.433.1		8	9.2	odd	6
2592.2.r.p.433.2		8	72.61	even	6
2592.2.r.p.433.3		8	9.7	even	3
2592.2.r.p.433.4		8	72.29	odd	6
2592.2.r.p.2161.1		8	72.5	odd	6
2592.2.r.p.2161.2		8	9.4	even	3
2592.2.r.p.2161.3		8	72.13	even	6
2592.2.r.p.2161.4		8	9.5	odd	6
6912.2.a.bc.1.1		2	16.3	odd	4
6912.2.a.bc.1.2		2	48.11	even	4
6912.2.a.bd.1.1		2	16.13	even	4
6912.2.a.bd.1.2		2	48.5	odd	4
6912.2.a.bu.1.1		2	48.35	even	4
6912.2.a.bu.1.2		2	16.11	odd	4
6912.2.a.bv.1.1		2	48.29	odd	4
6912.2.a.bv.1.2		2	16.5	even	4

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:	\( N \)	\(=\)	\( 864 = 2^{5} \cdot 3^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	864.d (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.00000i q^{5} -1.00000 q^{7} +O(q^{10})\)	\(q-1.00000i q^{5} -1.00000 q^{7} +3.00000i q^{11} +5.29150i q^{13} -5.29150 q^{17} +5.29150i q^{19} +5.29150 q^{23} +4.00000 q^{25} -6.00000i q^{29} +7.00000 q^{31} +1.00000i q^{35} +5.29150i q^{37} +5.29150 q^{41} +10.5830i q^{43} -6.00000 q^{49} +9.00000i q^{53} +3.00000 q^{55} -4.00000i q^{59} +5.29150 q^{65} -15.8745 q^{71} +3.00000 q^{73} -3.00000i q^{77} -4.00000 q^{79} -7.00000i q^{83} +5.29150i q^{85} -10.5830 q^{89} -5.29150i q^{91} +5.29150 q^{95} +7.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{7}+O(q^{10}) \) 4 * q - 4 * q^7	\( 4 q - 4 q^{7} + 16 q^{25} + 28 q^{31} - 24 q^{49} + 12 q^{55} + 12 q^{73} - 16 q^{79} + 28 q^{97}+O(q^{100}) \) 4 * q - 4 * q^7 + 16 * q^25 + 28 * q^31 - 24 * q^49 + 12 * q^55 + 12 * q^73 - 16 * q^79 + 28 * q^97

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