LMFDB - Embedded newform 5184.2.c.f.5183.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5184,2,Mod(5183,5184)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5184, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("5184.5183");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5184 = 2^{6} \cdot 3^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5184.c (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:	$41.3944484078$
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:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 144)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			5183.2
Root			$-0.866025 + 0.500000i$ of defining polynomial
Character	$\chi$	$=$	5184.5183
Dual form			5184.2.c.f.5183.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-1.73205i q^{5} +3.00000i q^{7} +O(q^{10})$	$q-1.73205i q^{5} +3.00000i q^{7} -5.19615 q^{11} -1.00000 q^{13} +3.46410i q^{17} +6.00000i q^{19} -5.19615 q^{23} +2.00000 q^{25} -8.66025i q^{29} -3.00000i q^{31} +5.19615 q^{35} +4.00000 q^{37} +5.19615i q^{41} -3.00000i q^{43} -5.19615 q^{47} -2.00000 q^{49} -10.3923i q^{53} +9.00000i q^{55} -5.19615 q^{59} +7.00000 q^{61} +1.73205i q^{65} -9.00000i q^{67} +10.3923 q^{71} +4.00000 q^{73} -15.5885i q^{77} -15.0000i q^{79} +5.19615 q^{83} +6.00000 q^{85} -3.46410i q^{89} -3.00000i q^{91} +10.3923 q^{95} +1.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 4 q^{13} + 8 q^{25} + 16 q^{37} - 8 q^{49} + 28 q^{61} + 16 q^{73} + 24 q^{85} + 4 q^{97}+O(q^{100}) $ 4 * q - 4 * q^13 + 8 * q^25 + 16 * q^37 - 8 * q^49 + 28 * q^61 + 16 * q^73 + 24 * q^85 + 4 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5184\mathbb{Z}\right)^\times$.

$n$	$325$	$1217$	$2431$
$\chi(n)$	$1$	$-1$	$-1$

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
$5$	− 1.73205i	− 0.774597i	−0.921954	−	0.387298i	$-0.873408\pi$
$5$	− 1.73205i	− 0.774597i	0.921954	−	0.387298i	$-0.126592\pi$
$6$	0	0
$7$	3.00000i	1.13389i	0.823754	+	0.566947i	$0.191875\pi$
$7$	3.00000i	1.13389i	−0.823754	+	0.566947i	$0.808125\pi$
$8$	0	0
$9$	0	0
$10$	0	0
$11$	−5.19615	−1.56670	−0.783349	−	0.621582i	$-0.786490\pi$
$11$	−5.19615	−1.56670	−0.783349	+	0.621582i	$0.786490\pi$
$12$	0	0
$13$	−1.00000	−0.277350	−0.138675	−	0.990338i	$-0.544284\pi$
$13$	−1.00000	−0.277350	−0.138675	+	0.990338i	$0.544284\pi$
$14$	0	0
$15$	0	0
$16$	0	0
$17$	3.46410i	0.840168i	0.907485	+	0.420084i	$0.137999\pi$
$17$	3.46410i	0.840168i	−0.907485	+	0.420084i	$0.862001\pi$
$18$	0	0
$19$	6.00000i	1.37649i	0.725476	+	0.688247i	$0.241620\pi$
$19$	6.00000i	1.37649i	−0.725476	+	0.688247i	$0.758380\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−5.19615	−1.08347	−0.541736	−	0.840548i	$-0.682233\pi$
$23$	−5.19615	−1.08347	−0.541736	+	0.840548i	$0.682233\pi$
$24$	0	0
$25$	2.00000	0.400000
$26$	0	0
$27$	0	0
$28$	0	0
$29$	− 8.66025i	− 1.60817i	−0.594515	−	0.804084i	$-0.702656\pi$
$29$	− 8.66025i	− 1.60817i	0.594515	−	0.804084i	$-0.297344\pi$
$30$	0	0
$31$	− 3.00000i	− 0.538816i	−0.963026	−	0.269408i	$-0.913172\pi$
$31$	− 3.00000i	− 0.538816i	0.963026	−	0.269408i	$-0.0868280\pi$
$32$	0	0
$33$	0	0
$34$	0	0
$35$	5.19615	0.878310
$36$	0	0
$37$	4.00000	0.657596	0.328798	−	0.944400i	$-0.393356\pi$
$37$	4.00000	0.657596	0.328798	+	0.944400i	$0.393356\pi$
$38$	0	0
$39$	0	0
$40$	0	0
$41$	5.19615i	0.811503i	0.913984	+	0.405751i	$0.132990\pi$
$41$	5.19615i	0.811503i	−0.913984	+	0.405751i	$0.867010\pi$
$42$	0	0
$43$	− 3.00000i	− 0.457496i	−0.973486	−	0.228748i	$-0.926537\pi$
$43$	− 3.00000i	− 0.457496i	0.973486	−	0.228748i	$-0.0734631\pi$
$44$	0	0
$45$	0	0
$46$	0	0
$47$	−5.19615	−0.757937	−0.378968	−	0.925410i	$-0.623721\pi$
$47$	−5.19615	−0.757937	−0.378968	+	0.925410i	$0.623721\pi$
$48$	0	0
$49$	−2.00000	−0.285714
$50$	0	0
$51$	0	0
$52$	0	0
$53$	− 10.3923i	− 1.42749i	−0.700404	−	0.713746i	$-0.746997\pi$
$53$	− 10.3923i	− 1.42749i	0.700404	−	0.713746i	$-0.253003\pi$
$54$	0	0
$55$	9.00000i	1.21356i
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−5.19615	−0.676481	−0.338241	−	0.941060i	$-0.609832\pi$
$59$	−5.19615	−0.676481	−0.338241	+	0.941060i	$0.609832\pi$
$60$	0	0
$61$	7.00000	0.896258	0.448129	−	0.893969i	$-0.352090\pi$
$61$	7.00000	0.896258	0.448129	+	0.893969i	$0.352090\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	1.73205i	0.214834i
$66$	0	0
$67$	− 9.00000i	− 1.09952i	−0.835321	−	0.549762i	$-0.814718\pi$
$67$	− 9.00000i	− 1.09952i	0.835321	−	0.549762i	$-0.185282\pi$
$68$	0	0
$69$	0	0
$70$	0	0
$71$	10.3923	1.23334	0.616670	−	0.787222i	$-0.288481\pi$
$71$	10.3923	1.23334	0.616670	+	0.787222i	$0.288481\pi$
$72$	0	0
$73$	4.00000	0.468165	0.234082	−	0.972217i	$-0.424791\pi$
$73$	4.00000	0.468165	0.234082	+	0.972217i	$0.424791\pi$
$74$	0	0
$75$	0	0
$76$	0	0
$77$	− 15.5885i	− 1.77647i
$78$	0	0
$79$	− 15.0000i	− 1.68763i	−0.536633	−	0.843816i	$-0.680304\pi$
$79$	− 15.0000i	− 1.68763i	0.536633	−	0.843816i	$-0.319696\pi$
$80$	0	0
$81$	0	0
$82$	0	0
$83$	5.19615	0.570352	0.285176	−	0.958475i	$-0.407948\pi$
$83$	5.19615	0.570352	0.285176	+	0.958475i	$0.407948\pi$
$84$	0	0
$85$	6.00000	0.650791
$86$	0	0
$87$	0	0
$88$	0	0
$89$	− 3.46410i	− 0.367194i	−0.983002	−	0.183597i	$-0.941226\pi$
$89$	− 3.46410i	− 0.367194i	0.983002	−	0.183597i	$-0.0587741\pi$
$90$	0	0
$91$	− 3.00000i	− 0.314485i
$92$	0	0
$93$	0	0
$94$	0	0
$95$	10.3923	1.06623
$96$	0	0
$97$	1.00000	0.101535	0.0507673	−	0.998711i	$-0.483833\pi$
$97$	1.00000	0.101535	0.0507673	+	0.998711i	$0.483833\pi$
$98$	0	0
$99$	0	0
$100$	0	0
$101$	− 5.19615i	− 0.517036i	−0.966006	−	0.258518i	$-0.916766\pi$
$101$	− 5.19615i	− 0.517036i	0.966006	−	0.258518i	$-0.0832342\pi$
$102$	0	0
$103$	− 9.00000i	− 0.886796i	−0.896325	−	0.443398i	$-0.853773\pi$
$103$	− 9.00000i	− 0.886796i	0.896325	−	0.443398i	$-0.146227\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	0	0		−	1.00000i	$-0.5\pi$
$107$	0	0			1.00000i	$0.5\pi$
$108$	0	0
$109$	−4.00000	−0.383131	−0.191565	−	0.981480i	$-0.561356\pi$
$109$	−4.00000	−0.383131	−0.191565	+	0.981480i	$0.561356\pi$
$110$	0	0
$111$	0	0
$112$	0	0
$113$	− 12.1244i	− 1.14056i	−0.821449	−	0.570282i	$-0.806834\pi$
$113$	− 12.1244i	− 1.14056i	0.821449	−	0.570282i	$-0.193166\pi$
$114$	0	0
$115$	9.00000i	0.839254i
$116$	0	0
$117$	0	0
$118$	0	0
$119$	−10.3923	−0.952661
$120$	0	0
$121$	16.0000	1.45455
$122$	0	0
$123$	0	0
$124$	0	0
$125$	− 12.1244i	− 1.08444i
$126$	0	0
$127$	6.00000i	0.532414i	0.963916	+	0.266207i	$0.0857705\pi$
$127$	6.00000i	0.532414i	−0.963916	+	0.266207i	$0.914230\pi$
$128$	0	0
$129$	0	0
$130$	0	0
$131$	−5.19615	−0.453990	−0.226995	−	0.973896i	$-0.572890\pi$
$131$	−5.19615	−0.453990	−0.226995	+	0.973896i	$0.572890\pi$
$132$	0	0
$133$	−18.0000	−1.56080
$134$	0	0
$135$	0	0
$136$	0	0
$137$	1.73205i	0.147979i	0.997259	+	0.0739895i	$0.0235731\pi$
$137$	1.73205i	0.147979i	−0.997259	+	0.0739895i	$0.976427\pi$
$138$	0	0
$139$	− 3.00000i	− 0.254457i	−0.991873	−	0.127228i	$-0.959392\pi$
$139$	− 3.00000i	− 0.254457i	0.991873	−	0.127228i	$-0.0406081\pi$
$140$	0	0
$141$	0	0
$142$	0	0
$143$	5.19615	0.434524
$144$	0	0
$145$	−15.0000	−1.24568
$146$	0	0
$147$	0	0
$148$	0	0
$149$	1.73205i	0.141895i	0.997480	+	0.0709476i	$0.0226023\pi$
$149$	1.73205i	0.141895i	−0.997480	+	0.0709476i	$0.977398\pi$
$150$	0	0
$151$	9.00000i	0.732410i	0.930534	+	0.366205i	$0.119343\pi$
$151$	9.00000i	0.732410i	−0.930534	+	0.366205i	$0.880657\pi$
$152$	0	0
$153$	0	0
$154$	0	0
$155$	−5.19615	−0.417365
$156$	0	0
$157$	1.00000	0.0798087	0.0399043	−	0.999204i	$-0.487295\pi$
$157$	1.00000	0.0798087	0.0399043	+	0.999204i	$0.487295\pi$
$158$	0	0
$159$	0	0
$160$	0	0
$161$	− 15.5885i	− 1.22854i
$162$	0	0
$163$	6.00000i	0.469956i	0.972001	+	0.234978i	$0.0755019\pi$
$163$	6.00000i	0.469956i	−0.972001	+	0.234978i	$0.924498\pi$
$164$	0	0
$165$	0	0
$166$	0	0
$167$	−15.5885	−1.20627	−0.603136	−	0.797639i	$-0.706082\pi$
$167$	−15.5885	−1.20627	−0.603136	+	0.797639i	$0.706082\pi$
$168$	0	0
$169$	−12.0000	−0.923077
$170$	0	0
$171$	0	0
$172$	0	0
$173$	1.73205i	0.131685i	0.997830	+	0.0658427i	$0.0209736\pi$
$173$	1.73205i	0.131685i	−0.997830	+	0.0658427i	$0.979026\pi$
$174$	0	0
$175$	6.00000i	0.453557i
$176$	0	0
$177$	0	0
$178$	0	0
$179$	20.7846	1.55351	0.776757	−	0.629800i	$-0.216863\pi$
$179$	20.7846	1.55351	0.776757	+	0.629800i	$0.216863\pi$
$180$	0	0
$181$	−16.0000	−1.18927	−0.594635	−	0.803996i	$-0.702704\pi$
$181$	−16.0000	−1.18927	−0.594635	+	0.803996i	$0.702704\pi$
$182$	0	0
$183$	0	0
$184$	0	0
$185$	− 6.92820i	− 0.509372i
$186$	0	0
$187$	− 18.0000i	− 1.31629i
$188$	0	0
$189$	0	0
$190$	0	0
$191$	15.5885	1.12794	0.563971	−	0.825795i	$-0.309273\pi$
$191$	15.5885	1.12794	0.563971	+	0.825795i	$0.309273\pi$
$192$	0	0
$193$	−19.0000	−1.36765	−0.683825	−	0.729646i	$-0.739685\pi$
$193$	−19.0000	−1.36765	−0.683825	+	0.729646i	$0.739685\pi$
$194$	0	0
$195$	0	0
$196$	0	0
$197$	6.92820i	0.493614i	0.969065	+	0.246807i	$0.0793814\pi$
$197$	6.92820i	0.493614i	−0.969065	+	0.246807i	$0.920619\pi$
$198$	0	0
$199$	− 12.0000i	− 0.850657i	−0.905039	−	0.425329i	$-0.860158\pi$
$199$	− 12.0000i	− 0.850657i	0.905039	−	0.425329i	$-0.139842\pi$
$200$	0	0
$201$	0	0
$202$	0	0
$203$	25.9808	1.82349
$204$	0	0
$205$	9.00000	0.628587
$206$	0	0
$207$	0	0
$208$	0	0
$209$	− 31.1769i	− 2.15655i
$210$	0	0
$211$	3.00000i	0.206529i	0.994654	+	0.103264i	$0.0329287\pi$
$211$	3.00000i	0.206529i	−0.994654	+	0.103264i	$0.967071\pi$
$212$	0	0
$213$	0	0
$214$	0	0
$215$	−5.19615	−0.354375
$216$	0	0
$217$	9.00000	0.610960
$218$	0	0
$219$	0	0
$220$	0	0
$221$	− 3.46410i	− 0.233021i
$222$	0	0
$223$	15.0000i	1.00447i	0.864730	+	0.502237i	$0.167490\pi$
$223$	15.0000i	1.00447i	−0.864730	+	0.502237i	$0.832510\pi$
$224$	0	0
$225$	0	0
$226$	0	0
$227$	−5.19615	−0.344881	−0.172440	−	0.985020i	$-0.555165\pi$
$227$	−5.19615	−0.344881	−0.172440	+	0.985020i	$0.555165\pi$
$228$	0	0
$229$	−19.0000	−1.25556	−0.627778	−	0.778393i	$-0.716035\pi$
$229$	−19.0000	−1.25556	−0.627778	+	0.778393i	$0.716035\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	− 13.8564i	− 0.907763i	−0.891062	−	0.453882i	$-0.850039\pi$
$233$	− 13.8564i	− 0.907763i	0.891062	−	0.453882i	$-0.149961\pi$
$234$	0	0
$235$	9.00000i	0.587095i
$236$	0	0
$237$	0	0
$238$	0	0
$239$	15.5885	1.00833	0.504167	−	0.863606i	$-0.331800\pi$
$239$	15.5885	1.00833	0.504167	+	0.863606i	$0.331800\pi$
$240$	0	0
$241$	7.00000	0.450910	0.225455	−	0.974254i	$-0.427613\pi$
$241$	7.00000	0.450910	0.225455	+	0.974254i	$0.427613\pi$
$242$	0	0
$243$	0	0
$244$	0	0
$245$	3.46410i	0.221313i
$246$	0	0
$247$	− 6.00000i	− 0.381771i
$248$	0	0
$249$	0	0
$250$	0	0
$251$	−20.7846	−1.31191	−0.655956	−	0.754799i	$-0.727735\pi$
$251$	−20.7846	−1.31191	−0.655956	+	0.754799i	$0.727735\pi$
$252$	0	0
$253$	27.0000	1.69748
$254$	0	0
$255$	0	0
$256$	0	0
$257$	− 19.0526i	− 1.18847i	−0.804293	−	0.594233i	$-0.797456\pi$
$257$	− 19.0526i	− 1.18847i	0.804293	−	0.594233i	$-0.202544\pi$
$258$	0	0
$259$	12.0000i	0.745644i
$260$	0	0
$261$	0	0
$262$	0	0
$263$	25.9808	1.60204	0.801021	−	0.598636i	$-0.204290\pi$
$263$	25.9808	1.60204	0.801021	+	0.598636i	$0.204290\pi$
$264$	0	0
$265$	−18.0000	−1.10573
$266$	0	0
$267$	0	0
$268$	0	0
$269$	3.46410i	0.211210i	0.994408	+	0.105605i	$0.0336779\pi$
$269$	3.46410i	0.211210i	−0.994408	+	0.105605i	$0.966322\pi$
$270$	0	0
$271$	− 18.0000i	− 1.09342i	−0.837321	−	0.546711i	$-0.815880\pi$
$271$	− 18.0000i	− 1.09342i	0.837321	−	0.546711i	$-0.184120\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−10.3923	−0.626680
$276$	0	0
$277$	17.0000	1.02143	0.510716	−	0.859750i	$-0.329381\pi$
$277$	17.0000	1.02143	0.510716	+	0.859750i	$0.329381\pi$
$278$	0	0
$279$	0	0
$280$	0	0
$281$	− 12.1244i	− 0.723278i	−0.932318	−	0.361639i	$-0.882217\pi$
$281$	− 12.1244i	− 0.723278i	0.932318	−	0.361639i	$-0.117783\pi$
$282$	0	0
$283$	33.0000i	1.96165i	0.194900	+	0.980823i	$0.437562\pi$
$283$	33.0000i	1.96165i	−0.194900	+	0.980823i	$0.562438\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−15.5885	−0.920158
$288$	0	0
$289$	5.00000	0.294118
$290$	0	0
$291$	0	0
$292$	0	0
$293$	19.0526i	1.11306i	0.830827	+	0.556531i	$0.187868\pi$
$293$	19.0526i	1.11306i	−0.830827	+	0.556531i	$0.812132\pi$
$294$	0	0
$295$	9.00000i	0.524000i
$296$	0	0
$297$	0	0
$298$	0	0
$299$	5.19615	0.300501
$300$	0	0
$301$	9.00000	0.518751
$302$	0	0
$303$	0	0
$304$	0	0
$305$	− 12.1244i	− 0.694239i
$306$	0	0
$307$	− 18.0000i	− 1.02731i	−0.857996	−	0.513657i	$-0.828290\pi$
$307$	− 18.0000i	− 1.02731i	0.857996	−	0.513657i	$-0.171710\pi$
$308$	0	0
$309$	0	0
$310$	0	0
$311$	−5.19615	−0.294647	−0.147323	−	0.989088i	$-0.547066\pi$
$311$	−5.19615	−0.294647	−0.147323	+	0.989088i	$0.547066\pi$
$312$	0	0
$313$	11.0000	0.621757	0.310878	−	0.950450i	$-0.399377\pi$
$313$	11.0000	0.621757	0.310878	+	0.950450i	$0.399377\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	− 22.5167i	− 1.26466i	−0.774698	−	0.632331i	$-0.782098\pi$
$317$	− 22.5167i	− 1.26466i	0.774698	−	0.632331i	$-0.217902\pi$
$318$	0	0
$319$	45.0000i	2.51952i
$320$	0	0
$321$	0	0
$322$	0	0
$323$	−20.7846	−1.15649
$324$	0	0
$325$	−2.00000	−0.110940
$326$	0	0
$327$	0	0
$328$	0	0
$329$	− 15.5885i	− 0.859419i
$330$	0	0
$331$	3.00000i	0.164895i	0.996595	+	0.0824475i	$0.0262737\pi$
$331$	3.00000i	0.164895i	−0.996595	+	0.0824475i	$0.973726\pi$
$332$	0	0
$333$	0	0
$334$	0	0
$335$	−15.5885	−0.851688
$336$	0	0
$337$	31.0000	1.68868	0.844339	−	0.535810i	$-0.179994\pi$
$337$	31.0000	1.68868	0.844339	+	0.535810i	$0.179994\pi$
$338$	0	0
$339$	0	0
$340$	0	0
$341$	15.5885i	0.844162i
$342$	0	0
$343$	15.0000i	0.809924i
$344$	0	0
$345$	0	0
$346$	0	0
$347$	5.19615	0.278944	0.139472	−	0.990226i	$-0.455459\pi$
$347$	5.19615	0.278944	0.139472	+	0.990226i	$0.455459\pi$
$348$	0	0
$349$	−35.0000	−1.87351	−0.936754	−	0.349990i	$-0.886185\pi$
$349$	−35.0000	−1.87351	−0.936754	+	0.349990i	$0.886185\pi$
$350$	0	0
$351$	0	0
$352$	0	0
$353$	− 25.9808i	− 1.38282i	−0.722464	−	0.691408i	$-0.756991\pi$
$353$	− 25.9808i	− 1.38282i	0.722464	−	0.691408i	$-0.243009\pi$
$354$	0	0
$355$	− 18.0000i	− 0.955341i
$356$	0	0
$357$	0	0
$358$	0	0
$359$	−20.7846	−1.09697	−0.548485	−	0.836160i	$-0.684795\pi$
$359$	−20.7846	−1.09697	−0.548485	+	0.836160i	$0.684795\pi$
$360$	0	0
$361$	−17.0000	−0.894737
$362$	0	0
$363$	0	0
$364$	0	0
$365$	− 6.92820i	− 0.362639i
$366$	0	0
$367$	− 27.0000i	− 1.40939i	−0.709511	−	0.704694i	$-0.751084\pi$
$367$	− 27.0000i	− 1.40939i	0.709511	−	0.704694i	$-0.248916\pi$
$368$	0	0
$369$	0	0
$370$	0	0
$371$	31.1769	1.61862
$372$	0	0
$373$	−5.00000	−0.258890	−0.129445	−	0.991587i	$-0.541320\pi$
$373$	−5.00000	−0.258890	−0.129445	+	0.991587i	$0.541320\pi$
$374$	0	0
$375$	0	0
$376$	0	0
$377$	8.66025i	0.446026i
$378$	0	0
$379$	− 24.0000i	− 1.23280i	−0.787434	−	0.616399i	$-0.788591\pi$
$379$	− 24.0000i	− 1.23280i	0.787434	−	0.616399i	$-0.211409\pi$
$380$	0	0
$381$	0	0
$382$	0	0
$383$	5.19615	0.265511	0.132755	−	0.991149i	$-0.457617\pi$
$383$	5.19615	0.265511	0.132755	+	0.991149i	$0.457617\pi$
$384$	0	0
$385$	−27.0000	−1.37605
$386$	0	0
$387$	0	0
$388$	0	0
$389$	15.5885i	0.790366i	0.918602	+	0.395183i	$0.129319\pi$
$389$	15.5885i	0.790366i	−0.918602	+	0.395183i	$0.870681\pi$
$390$	0	0
$391$	− 18.0000i	− 0.910299i
$392$	0	0
$393$	0	0
$394$	0	0
$395$	−25.9808	−1.30723
$396$	0	0
$397$	−32.0000	−1.60603	−0.803017	−	0.595956i	$-0.796773\pi$
$397$	−32.0000	−1.60603	−0.803017	+	0.595956i	$0.796773\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	− 8.66025i	− 0.432472i	−0.976341	−	0.216236i	$-0.930622\pi$
$401$	− 8.66025i	− 0.432472i	0.976341	−	0.216236i	$-0.0693781\pi$
$402$	0	0
$403$	3.00000i	0.149441i
$404$	0	0
$405$	0	0
$406$	0	0
$407$	−20.7846	−1.03025
$408$	0	0
$409$	−17.0000	−0.840596	−0.420298	−	0.907386i	$-0.638074\pi$
$409$	−17.0000	−0.840596	−0.420298	+	0.907386i	$0.638074\pi$
$410$	0	0
$411$	0	0
$412$	0	0
$413$	− 15.5885i	− 0.767058i
$414$	0	0
$415$	− 9.00000i	− 0.441793i
$416$	0	0
$417$	0	0
$418$	0	0
$419$	−36.3731	−1.77694	−0.888470	−	0.458934i	$-0.848231\pi$
$419$	−36.3731	−1.77694	−0.888470	+	0.458934i	$0.848231\pi$
$420$	0	0
$421$	11.0000	0.536107	0.268054	−	0.963404i	$-0.413620\pi$
$421$	11.0000	0.536107	0.268054	+	0.963404i	$0.413620\pi$
$422$	0	0
$423$	0	0
$424$	0	0
$425$	6.92820i	0.336067i
$426$	0	0
$427$	21.0000i	1.01626i
$428$	0	0
$429$	0	0
$430$	0	0
$431$	−20.7846	−1.00116	−0.500580	−	0.865690i	$-0.666880\pi$
$431$	−20.7846	−1.00116	−0.500580	+	0.865690i	$0.666880\pi$
$432$	0	0
$433$	32.0000	1.53782	0.768911	−	0.639356i	$-0.220799\pi$
$433$	32.0000	1.53782	0.768911	+	0.639356i	$0.220799\pi$
$434$	0	0
$435$	0	0
$436$	0	0
$437$	− 31.1769i	− 1.49139i
$438$	0	0
$439$	− 27.0000i	− 1.28864i	−0.764756	−	0.644320i	$-0.777141\pi$
$439$	− 27.0000i	− 1.28864i	0.764756	−	0.644320i	$-0.222859\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	5.19615	0.246877	0.123438	−	0.992352i	$-0.460608\pi$
$443$	5.19615	0.246877	0.123438	+	0.992352i	$0.460608\pi$
$444$	0	0
$445$	−6.00000	−0.284427
$446$	0	0
$447$	0	0
$448$	0	0
$449$	− 31.1769i	− 1.47133i	−0.677346	−	0.735665i	$-0.736870\pi$
$449$	− 31.1769i	− 1.47133i	0.677346	−	0.735665i	$-0.263130\pi$
$450$	0	0
$451$	− 27.0000i	− 1.27138i
$452$	0	0
$453$	0	0
$454$	0	0
$455$	−5.19615	−0.243599
$456$	0	0
$457$	25.0000	1.16945	0.584725	−	0.811231i	$-0.301202\pi$
$457$	25.0000	1.16945	0.584725	+	0.811231i	$0.301202\pi$
$458$	0	0
$459$	0	0
$460$	0	0
$461$	5.19615i	0.242009i	0.992652	+	0.121004i	$0.0386115\pi$
$461$	5.19615i	0.242009i	−0.992652	+	0.121004i	$0.961388\pi$
$462$	0	0
$463$	− 3.00000i	− 0.139422i	−0.997567	−	0.0697109i	$-0.977792\pi$
$463$	− 3.00000i	− 0.139422i	0.997567	−	0.0697109i	$-0.0222077\pi$
$464$	0	0
$465$	0	0
$466$	0	0
$467$	0	0		−	1.00000i	$-0.5\pi$
$467$	0	0			1.00000i	$0.5\pi$
$468$	0	0
$469$	27.0000	1.24674
$470$	0	0
$471$	0	0
$472$	0	0
$473$	15.5885i	0.716758i
$474$	0	0
$475$	12.0000i	0.550598i
$476$	0	0
$477$	0	0
$478$	0	0
$479$	−15.5885	−0.712255	−0.356127	−	0.934437i	$-0.615903\pi$
$479$	−15.5885	−0.712255	−0.356127	+	0.934437i	$0.615903\pi$
$480$	0	0
$481$	−4.00000	−0.182384
$482$	0	0
$483$	0	0
$484$	0	0
$485$	− 1.73205i	− 0.0786484i
$486$	0	0
$487$	− 30.0000i	− 1.35943i	−0.733476	−	0.679715i	$-0.762104\pi$
$487$	− 30.0000i	− 1.35943i	0.733476	−	0.679715i	$-0.237896\pi$
$488$	0	0
$489$	0	0
$490$	0	0
$491$	15.5885	0.703497	0.351749	−	0.936094i	$-0.385587\pi$
$491$	15.5885	0.703497	0.351749	+	0.936094i	$0.385587\pi$
$492$	0	0
$493$	30.0000	1.35113
$494$	0	0
$495$	0	0
$496$	0	0
$497$	31.1769i	1.39848i
$498$	0	0
$499$	− 27.0000i	− 1.20869i	−0.796724	−	0.604343i	$-0.793436\pi$
$499$	− 27.0000i	− 1.20869i	0.796724	−	0.604343i	$-0.206564\pi$
$500$	0	0
$501$	0	0
$502$	0	0
$503$	31.1769	1.39011	0.695055	−	0.718957i	$-0.255380\pi$
$503$	31.1769	1.39011	0.695055	+	0.718957i	$0.255380\pi$
$504$	0	0
$505$	−9.00000	−0.400495
$506$	0	0
$507$	0	0
$508$	0	0
$509$	15.5885i	0.690946i	0.938429	+	0.345473i	$0.112282\pi$
$509$	15.5885i	0.690946i	−0.938429	+	0.345473i	$0.887718\pi$
$510$	0	0
$511$	12.0000i	0.530849i
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−15.5885	−0.686909
$516$	0	0
$517$	27.0000	1.18746
$518$	0	0
$519$	0	0
$520$	0	0
$521$	41.5692i	1.82118i	0.413310	+	0.910590i	$0.364373\pi$
$521$	41.5692i	1.82118i	−0.413310	+	0.910590i	$0.635627\pi$
$522$	0	0
$523$	0	0	1.00000			$0$
$523$	0	0	−1.00000			$\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	10.3923	0.452696
$528$	0	0
$529$	4.00000	0.173913
$530$	0	0
$531$	0	0
$532$	0	0
$533$	− 5.19615i	− 0.225070i
$534$	0	0
$535$	0	0
$536$	0	0
$537$	0	0
$538$	0	0
$539$	10.3923	0.447628
$540$	0	0
$541$	14.0000	0.601907	0.300954	−	0.953639i	$-0.402695\pi$
$541$	14.0000	0.601907	0.300954	+	0.953639i	$0.402695\pi$
$542$	0	0
$543$	0	0
$544$	0	0
$545$	6.92820i	0.296772i
$546$	0	0
$547$	− 9.00000i	− 0.384812i	−0.981315	−	0.192406i	$-0.938371\pi$
$547$	− 9.00000i	− 0.384812i	0.981315	−	0.192406i	$-0.0616291\pi$
$548$	0	0
$549$	0	0
$550$	0	0
$551$	51.9615	2.21364
$552$	0	0
$553$	45.0000	1.91359
$554$	0	0
$555$	0	0
$556$	0	0
$557$	− 45.0333i	− 1.90812i	−0.299611	−	0.954062i	$-0.596857\pi$
$557$	− 45.0333i	− 1.90812i	0.299611	−	0.954062i	$-0.403143\pi$
$558$	0	0
$559$	3.00000i	0.126886i
$560$	0	0
$561$	0	0
$562$	0	0
$563$	25.9808	1.09496	0.547479	−	0.836819i	$-0.315587\pi$
$563$	25.9808	1.09496	0.547479	+	0.836819i	$0.315587\pi$
$564$	0	0
$565$	−21.0000	−0.883477
$566$	0	0
$567$	0	0
$568$	0	0
$569$	19.0526i	0.798725i	0.916793	+	0.399362i	$0.130768\pi$
$569$	19.0526i	0.798725i	−0.916793	+	0.399362i	$0.869232\pi$
$570$	0	0
$571$	3.00000i	0.125546i	0.998028	+	0.0627730i	$0.0199944\pi$
$571$	3.00000i	0.125546i	−0.998028	+	0.0627730i	$0.980006\pi$
$572$	0	0
$573$	0	0
$574$	0	0
$575$	−10.3923	−0.433389
$576$	0	0
$577$	−40.0000	−1.66522	−0.832611	−	0.553858i	$-0.813155\pi$
$577$	−40.0000	−1.66522	−0.832611	+	0.553858i	$0.813155\pi$
$578$	0	0
$579$	0	0
$580$	0	0
$581$	15.5885i	0.646718i
$582$	0	0
$583$	54.0000i	2.23645i
$584$	0	0
$585$	0	0
$586$	0	0
$587$	−15.5885	−0.643404	−0.321702	−	0.946841i	$-0.604255\pi$
$587$	−15.5885	−0.643404	−0.321702	+	0.946841i	$0.604255\pi$
$588$	0	0
$589$	18.0000	0.741677
$590$	0	0
$591$	0	0
$592$	0	0
$593$	− 24.2487i	− 0.995775i	−0.867242	−	0.497888i	$-0.834109\pi$
$593$	− 24.2487i	− 0.995775i	0.867242	−	0.497888i	$-0.165891\pi$
$594$	0	0
$595$	18.0000i	0.737928i
$596$	0	0
$597$	0	0
$598$	0	0
$599$	25.9808	1.06155	0.530773	−	0.847514i	$-0.321902\pi$
$599$	25.9808	1.06155	0.530773	+	0.847514i	$0.321902\pi$
$600$	0	0
$601$	−7.00000	−0.285536	−0.142768	−	0.989756i	$-0.545600\pi$
$601$	−7.00000	−0.285536	−0.142768	+	0.989756i	$0.545600\pi$
$602$	0	0
$603$	0	0
$604$	0	0
$605$	− 27.7128i	− 1.12669i
$606$	0	0
$607$	3.00000i	0.121766i	0.998145	+	0.0608831i	$0.0193917\pi$
$607$	3.00000i	0.121766i	−0.998145	+	0.0608831i	$0.980608\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	5.19615	0.210214
$612$	0	0
$613$	−20.0000	−0.807792	−0.403896	−	0.914805i	$-0.632344\pi$
$613$	−20.0000	−0.807792	−0.403896	+	0.914805i	$0.632344\pi$
$614$	0	0
$615$	0	0
$616$	0	0
$617$	32.9090i	1.32487i	0.749122	+	0.662433i	$0.230476\pi$
$617$	32.9090i	1.32487i	−0.749122	+	0.662433i	$0.769524\pi$
$618$	0	0
$619$	− 9.00000i	− 0.361741i	−0.983507	−	0.180870i	$-0.942109\pi$
$619$	− 9.00000i	− 0.361741i	0.983507	−	0.180870i	$-0.0578914\pi$
$620$	0	0
$621$	0	0
$622$	0	0
$623$	10.3923	0.416359
$624$	0	0
$625$	−11.0000	−0.440000
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.8564i	0.552491i
$630$	0	0
$631$	6.00000i	0.238856i	0.992843	+	0.119428i	$0.0381061\pi$
$631$	6.00000i	0.238856i	−0.992843	+	0.119428i	$0.961894\pi$
$632$	0	0
$633$	0	0
$634$	0	0
$635$	10.3923	0.412406
$636$	0	0
$637$	2.00000	0.0792429
$638$	0	0
$639$	0	0
$640$	0	0
$641$	12.1244i	0.478883i	0.970911	+	0.239442i	$0.0769644\pi$
$641$	12.1244i	0.478883i	−0.970911	+	0.239442i	$0.923036\pi$
$642$	0	0
$643$	27.0000i	1.06478i	0.846500	+	0.532388i	$0.178705\pi$
$643$	27.0000i	1.06478i	−0.846500	+	0.532388i	$0.821295\pi$
$644$	0	0
$645$	0	0
$646$	0	0
$647$	20.7846	0.817127	0.408564	−	0.912730i	$-0.366030\pi$
$647$	20.7846	0.817127	0.408564	+	0.912730i	$0.366030\pi$
$648$	0	0
$649$	27.0000	1.05984
$650$	0	0
$651$	0	0
$652$	0	0
$653$	− 36.3731i	− 1.42339i	−0.702490	−	0.711694i	$-0.747928\pi$
$653$	− 36.3731i	− 1.42339i	0.702490	−	0.711694i	$-0.252072\pi$
$654$	0	0
$655$	9.00000i	0.351659i
$656$	0	0
$657$	0	0
$658$	0	0
$659$	−5.19615	−0.202413	−0.101207	−	0.994865i	$-0.532270\pi$
$659$	−5.19615	−0.202413	−0.101207	+	0.994865i	$0.532270\pi$
$660$	0	0
$661$	1.00000	0.0388955	0.0194477	−	0.999811i	$-0.493809\pi$
$661$	1.00000	0.0388955	0.0194477	+	0.999811i	$0.493809\pi$
$662$	0	0
$663$	0	0
$664$	0	0
$665$	31.1769i	1.20899i
$666$	0	0
$667$	45.0000i	1.74241i
$668$	0	0
$669$	0	0
$670$	0	0
$671$	−36.3731	−1.40417
$672$	0	0
$673$	−7.00000	−0.269830	−0.134915	−	0.990857i	$-0.543076\pi$
$673$	−7.00000	−0.269830	−0.134915	+	0.990857i	$0.543076\pi$
$674$	0	0
$675$	0	0
$676$	0	0
$677$	15.5885i	0.599113i	0.954079	+	0.299557i	$0.0968387\pi$
$677$	15.5885i	0.599113i	−0.954079	+	0.299557i	$0.903161\pi$
$678$	0	0
$679$	3.00000i	0.115129i
$680$	0	0
$681$	0	0
$682$	0	0
$683$	41.5692	1.59060	0.795301	−	0.606215i	$-0.207313\pi$
$683$	41.5692	1.59060	0.795301	+	0.606215i	$0.207313\pi$
$684$	0	0
$685$	3.00000	0.114624
$686$	0	0
$687$	0	0
$688$	0	0
$689$	10.3923i	0.395915i
$690$	0	0
$691$	− 15.0000i	− 0.570627i	−0.958434	−	0.285313i	$-0.907902\pi$
$691$	− 15.0000i	− 0.570627i	0.958434	−	0.285313i	$-0.0920977\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−5.19615	−0.197101
$696$	0	0
$697$	−18.0000	−0.681799
$698$	0	0
$699$	0	0
$700$	0	0
$701$	− 17.3205i	− 0.654187i	−0.944992	−	0.327093i	$-0.893931\pi$
$701$	− 17.3205i	− 0.654187i	0.944992	−	0.327093i	$-0.106069\pi$
$702$	0	0
$703$	24.0000i	0.905177i
$704$	0	0
$705$	0	0
$706$	0	0
$707$	15.5885	0.586264
$708$	0	0
$709$	29.0000	1.08912	0.544559	−	0.838723i	$-0.316697\pi$
$709$	29.0000	1.08912	0.544559	+	0.838723i	$0.316697\pi$
$710$	0	0
$711$	0	0
$712$	0	0
$713$	15.5885i	0.583792i
$714$	0	0
$715$	− 9.00000i	− 0.336581i
$716$	0	0
$717$	0	0
$718$	0	0
$719$	−20.7846	−0.775135	−0.387568	−	0.921841i	$-0.626685\pi$
$719$	−20.7846	−0.775135	−0.387568	+	0.921841i	$0.626685\pi$
$720$	0	0
$721$	27.0000	1.00553
$722$	0	0
$723$	0	0
$724$	0	0
$725$	− 17.3205i	− 0.643268i
$726$	0	0
$727$	− 27.0000i	− 1.00137i	−0.865628	−	0.500687i	$-0.833081\pi$
$727$	− 27.0000i	− 1.00137i	0.865628	−	0.500687i	$-0.166919\pi$
$728$	0	0
$729$	0	0
$730$	0	0
$731$	10.3923	0.384373
$732$	0	0
$733$	−13.0000	−0.480166	−0.240083	−	0.970752i	$-0.577175\pi$
$733$	−13.0000	−0.480166	−0.240083	+	0.970752i	$0.577175\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	46.7654i	1.72262i
$738$	0	0
$739$	48.0000i	1.76571i	0.469647	+	0.882854i	$0.344381\pi$
$739$	48.0000i	1.76571i	−0.469647	+	0.882854i	$0.655619\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−36.3731	−1.33440	−0.667199	−	0.744879i	$-0.732507\pi$
$743$	−36.3731	−1.33440	−0.667199	+	0.744879i	$0.732507\pi$
$744$	0	0
$745$	3.00000	0.109911
$746$	0	0
$747$	0	0
$748$	0	0
$749$	0	0
$750$	0	0
$751$	− 21.0000i	− 0.766301i	−0.923686	−	0.383150i	$-0.874839\pi$
$751$	− 21.0000i	− 0.766301i	0.923686	−	0.383150i	$-0.125161\pi$
$752$	0	0
$753$	0	0
$754$	0	0
$755$	15.5885	0.567322
$756$	0	0
$757$	2.00000	0.0726912	0.0363456	−	0.999339i	$-0.488428\pi$
$757$	2.00000	0.0726912	0.0363456	+	0.999339i	$0.488428\pi$
$758$	0	0
$759$	0	0
$760$	0	0
$761$	12.1244i	0.439508i	0.975555	+	0.219754i	$0.0705254\pi$
$761$	12.1244i	0.439508i	−0.975555	+	0.219754i	$0.929475\pi$
$762$	0	0
$763$	− 12.0000i	− 0.434429i
$764$	0	0
$765$	0	0
$766$	0	0
$767$	5.19615	0.187622
$768$	0	0
$769$	35.0000	1.26213	0.631066	−	0.775729i	$-0.282618\pi$
$769$	35.0000	1.26213	0.631066	+	0.775729i	$0.282618\pi$
$770$	0	0
$771$	0	0
$772$	0	0
$773$	38.1051i	1.37055i	0.728286	+	0.685273i	$0.240317\pi$
$773$	38.1051i	1.37055i	−0.728286	+	0.685273i	$0.759683\pi$
$774$	0	0
$775$	− 6.00000i	− 0.215526i
$776$	0	0
$777$	0	0
$778$	0	0
$779$	−31.1769	−1.11703
$780$	0	0
$781$	−54.0000	−1.93227
$782$	0	0
$783$	0	0
$784$	0	0
$785$	− 1.73205i	− 0.0618195i
$786$	0	0
$787$	9.00000i	0.320815i	0.987051	+	0.160408i	$0.0512809\pi$
$787$	9.00000i	0.320815i	−0.987051	+	0.160408i	$0.948719\pi$
$788$	0	0
$789$	0	0
$790$	0	0
$791$	36.3731	1.29328
$792$	0	0
$793$	−7.00000	−0.248577
$794$	0	0
$795$	0	0
$796$	0	0
$797$	− 19.0526i	− 0.674876i	−0.941348	−	0.337438i	$-0.890440\pi$
$797$	− 19.0526i	− 0.674876i	0.941348	−	0.337438i	$-0.109560\pi$
$798$	0	0
$799$	− 18.0000i	− 0.636794i
$800$	0	0
$801$	0	0
$802$	0	0
$803$	−20.7846	−0.733473
$804$	0	0
$805$	−27.0000	−0.951625
$806$	0	0
$807$	0	0
$808$	0	0
$809$	17.3205i	0.608957i	0.952519	+	0.304478i	$0.0984821\pi$
$809$	17.3205i	0.608957i	−0.952519	+	0.304478i	$0.901518\pi$
$810$	0	0
$811$	− 30.0000i	− 1.05344i	−0.850038	−	0.526721i	$-0.823421\pi$
$811$	− 30.0000i	− 1.05344i	0.850038	−	0.526721i	$-0.176579\pi$
$812$	0	0
$813$	0	0
$814$	0	0
$815$	10.3923	0.364027
$816$	0	0
$817$	18.0000	0.629740
$818$	0	0
$819$	0	0
$820$	0	0
$821$	− 1.73205i	− 0.0604490i	−0.999543	−	0.0302245i	$-0.990378\pi$
$821$	− 1.73205i	− 0.0604490i	0.999543	−	0.0302245i	$-0.00962222\pi$
$822$	0	0
$823$	21.0000i	0.732014i	0.930612	+	0.366007i	$0.119275\pi$
$823$	21.0000i	0.732014i	−0.930612	+	0.366007i	$0.880725\pi$
$824$	0	0
$825$	0	0
$826$	0	0
$827$	0	0		−	1.00000i	$-0.5\pi$
$827$	0	0			1.00000i	$0.5\pi$
$828$	0	0
$829$	−16.0000	−0.555703	−0.277851	−	0.960624i	$-0.589622\pi$
$829$	−16.0000	−0.555703	−0.277851	+	0.960624i	$0.589622\pi$
$830$	0	0
$831$	0	0
$832$	0	0
$833$	− 6.92820i	− 0.240048i
$834$	0	0
$835$	27.0000i	0.934374i
$836$	0	0
$837$	0	0
$838$	0	0
$839$	−25.9808	−0.896956	−0.448478	−	0.893794i	$-0.648034\pi$
$839$	−25.9808	−0.896956	−0.448478	+	0.893794i	$0.648034\pi$
$840$	0	0
$841$	−46.0000	−1.58621
$842$	0	0
$843$	0	0
$844$	0	0
$845$	20.7846i	0.715012i
$846$	0	0
$847$	48.0000i	1.64930i
$848$	0	0
$849$	0	0
$850$	0	0
$851$	−20.7846	−0.712487
$852$	0	0
$853$	19.0000	0.650548	0.325274	−	0.945620i	$-0.394544\pi$
$853$	19.0000	0.650548	0.325274	+	0.945620i	$0.394544\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	43.3013i	1.47914i	0.673078	+	0.739572i	$0.264972\pi$
$857$	43.3013i	1.47914i	−0.673078	+	0.739572i	$0.735028\pi$
$858$	0	0
$859$	− 57.0000i	− 1.94481i	−0.233289	−	0.972407i	$-0.574949\pi$
$859$	− 57.0000i	− 1.94481i	0.233289	−	0.972407i	$-0.425051\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	41.5692	1.41503	0.707516	−	0.706697i	$-0.249816\pi$
$863$	41.5692	1.41503	0.707516	+	0.706697i	$0.249816\pi$
$864$	0	0
$865$	3.00000	0.102003
$866$	0	0
$867$	0	0
$868$	0	0
$869$	77.9423i	2.64401i
$870$	0	0
$871$	9.00000i	0.304953i
$872$	0	0
$873$	0	0
$874$	0	0
$875$	36.3731	1.22963
$876$	0	0
$877$	−17.0000	−0.574049	−0.287025	−	0.957923i	$-0.592666\pi$
$877$	−17.0000	−0.574049	−0.287025	+	0.957923i	$0.592666\pi$
$878$	0	0
$879$	0	0
$880$	0	0
$881$	10.3923i	0.350126i	0.984557	+	0.175063i	$0.0560129\pi$
$881$	10.3923i	0.350126i	−0.984557	+	0.175063i	$0.943987\pi$
$882$	0	0
$883$	− 6.00000i	− 0.201916i	−0.994891	−	0.100958i	$-0.967809\pi$
$883$	− 6.00000i	− 0.201916i	0.994891	−	0.100958i	$-0.0321908\pi$
$884$	0	0
$885$	0	0
$886$	0	0
$887$	57.1577	1.91917	0.959583	−	0.281424i	$-0.0908068\pi$
$887$	57.1577	1.91917	0.959583	+	0.281424i	$0.0908068\pi$
$888$	0	0
$889$	−18.0000	−0.603701
$890$	0	0
$891$	0	0
$892$	0	0
$893$	− 31.1769i	− 1.04330i
$894$	0	0
$895$	− 36.0000i	− 1.20335i
$896$	0	0
$897$	0	0
$898$	0	0
$899$	−25.9808	−0.866507
$900$	0	0
$901$	36.0000	1.19933
$902$	0	0
$903$	0	0
$904$	0	0
$905$	27.7128i	0.921205i
$906$	0	0
$907$	− 3.00000i	− 0.0996134i	−0.998759	−	0.0498067i	$-0.984139\pi$
$907$	− 3.00000i	− 0.0996134i	0.998759	−	0.0498067i	$-0.0158605\pi$
$908$	0	0
$909$	0	0
$910$	0	0
$911$	−36.3731	−1.20509	−0.602547	−	0.798084i	$-0.705847\pi$
$911$	−36.3731	−1.20509	−0.602547	+	0.798084i	$0.705847\pi$
$912$	0	0
$913$	−27.0000	−0.893570
$914$	0	0
$915$	0	0
$916$	0	0
$917$	− 15.5885i	− 0.514776i
$918$	0	0
$919$	54.0000i	1.78130i	0.454694	+	0.890648i	$0.349749\pi$
$919$	54.0000i	1.78130i	−0.454694	+	0.890648i	$0.650251\pi$
$920$	0	0
$921$	0	0
$922$	0	0
$923$	−10.3923	−0.342067
$924$	0	0
$925$	8.00000	0.263038
$926$	0	0
$927$	0	0
$928$	0	0
$929$	− 53.6936i	− 1.76163i	−0.473461	−	0.880815i	$-0.656996\pi$
$929$	− 53.6936i	− 1.76163i	0.473461	−	0.880815i	$-0.343004\pi$
$930$	0	0
$931$	− 12.0000i	− 0.393284i
$932$	0	0
$933$	0	0
$934$	0	0
$935$	−31.1769	−1.01959
$936$	0	0
$937$	−40.0000	−1.30674	−0.653372	−	0.757037i	$-0.726646\pi$
$937$	−40.0000	−1.30674	−0.653372	+	0.757037i	$0.726646\pi$
$938$	0	0
$939$	0	0
$940$	0	0
$941$	12.1244i	0.395243i	0.980278	+	0.197621i	$0.0633216\pi$
$941$	12.1244i	0.395243i	−0.980278	+	0.197621i	$0.936678\pi$
$942$	0	0
$943$	− 27.0000i	− 0.879241i
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−25.9808	−0.844261	−0.422131	−	0.906535i	$-0.638718\pi$
$947$	−25.9808	−0.844261	−0.422131	+	0.906535i	$0.638718\pi$
$948$	0	0
$949$	−4.00000	−0.129845
$950$	0	0
$951$	0	0
$952$	0	0
$953$	− 24.2487i	− 0.785493i	−0.919647	−	0.392746i	$-0.871525\pi$
$953$	− 24.2487i	− 0.785493i	0.919647	−	0.392746i	$-0.128475\pi$
$954$	0	0
$955$	− 27.0000i	− 0.873699i
$956$	0	0
$957$	0	0
$958$	0	0
$959$	−5.19615	−0.167793
$960$	0	0
$961$	22.0000	0.709677
$962$	0	0
$963$	0	0
$964$	0	0
$965$	32.9090i	1.05938i
$966$	0	0
$967$	− 27.0000i	− 0.868261i	−0.900850	−	0.434131i	$-0.857056\pi$
$967$	− 27.0000i	− 0.868261i	0.900850	−	0.434131i	$-0.142944\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	31.1769	1.00051	0.500257	−	0.865877i	$-0.333239\pi$
$971$	31.1769	1.00051	0.500257	+	0.865877i	$0.333239\pi$
$972$	0	0
$973$	9.00000	0.288527
$974$	0	0
$975$	0	0
$976$	0	0
$977$	15.5885i	0.498719i	0.968411	+	0.249359i	$0.0802201\pi$
$977$	15.5885i	0.498719i	−0.968411	+	0.249359i	$0.919780\pi$
$978$	0	0
$979$	18.0000i	0.575282i
$980$	0	0
$981$	0	0
$982$	0	0
$983$	−57.1577	−1.82305	−0.911523	−	0.411248i	$-0.865093\pi$
$983$	−57.1577	−1.82305	−0.911523	+	0.411248i	$0.865093\pi$
$984$	0	0
$985$	12.0000	0.382352
$986$	0	0
$987$	0	0
$988$	0	0
$989$	15.5885i	0.495684i
$990$	0	0
$991$	− 54.0000i	− 1.71537i	−0.514178	−	0.857683i	$-0.671903\pi$
$991$	− 54.0000i	− 1.71537i	0.514178	−	0.857683i	$-0.328097\pi$
$992$	0	0
$993$	0	0
$994$	0	0
$995$	−20.7846	−0.658916
$996$	0	0
$997$	−11.0000	−0.348373	−0.174187	−	0.984713i	$-0.555730\pi$
$997$	−11.0000	−0.348373	−0.174187	+	0.984713i	$0.555730\pi$
$998$	0	0
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5184.2.c.f.5183.2		4
3.2	odd	2	inner	5184.2.c.f.5183.4		4
4.3	odd	2	inner	5184.2.c.f.5183.1		4
8.3	odd	2		1296.2.c.f.1295.3		4
8.5	even	2		1296.2.c.f.1295.4		4
9.2	odd	6		576.2.s.e.383.2		4
9.4	even	3		576.2.s.e.191.1		4
9.5	odd	6		1728.2.s.e.575.1		4
9.7	even	3		1728.2.s.e.1151.2		4
12.11	even	2	inner	5184.2.c.f.5183.3		4
24.5	odd	2		1296.2.c.f.1295.2		4
24.11	even	2		1296.2.c.f.1295.1		4
36.7	odd	6		1728.2.s.e.1151.1		4
36.11	even	6		576.2.s.e.383.1		4
36.23	even	6		1728.2.s.e.575.2		4
36.31	odd	6		576.2.s.e.191.2		4
72.5	odd	6		432.2.s.e.143.1		4
72.11	even	6		144.2.s.e.95.2	yes	4
72.13	even	6		144.2.s.e.47.2	yes	4
72.29	odd	6		144.2.s.e.95.1	yes	4
72.43	odd	6		432.2.s.e.287.1		4
72.59	even	6		432.2.s.e.143.2		4
72.61	even	6		432.2.s.e.287.2		4
72.67	odd	6		144.2.s.e.47.1	✓	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
144.2.s.e.47.1	✓	4	72.67	odd	6
144.2.s.e.47.2	yes	4	72.13	even	6
144.2.s.e.95.1	yes	4	72.29	odd	6
144.2.s.e.95.2	yes	4	72.11	even	6
432.2.s.e.143.1		4	72.5	odd	6
432.2.s.e.143.2		4	72.59	even	6
432.2.s.e.287.1		4	72.43	odd	6
432.2.s.e.287.2		4	72.61	even	6
576.2.s.e.191.1		4	9.4	even	3
576.2.s.e.191.2		4	36.31	odd	6
576.2.s.e.383.1		4	36.11	even	6
576.2.s.e.383.2		4	9.2	odd	6
1296.2.c.f.1295.1		4	24.11	even	2
1296.2.c.f.1295.2		4	24.5	odd	2
1296.2.c.f.1295.3		4	8.3	odd	2
1296.2.c.f.1295.4		4	8.5	even	2
1728.2.s.e.575.1		4	9.5	odd	6
1728.2.s.e.575.2		4	36.23	even	6
1728.2.s.e.1151.1		4	36.7	odd	6
1728.2.s.e.1151.2		4	9.7	even	3
5184.2.c.f.5183.1		4	4.3	odd	2	inner
5184.2.c.f.5183.2		4	1.1	even	1	trivial
5184.2.c.f.5183.3		4	12.11	even	2	inner
5184.2.c.f.5183.4		4	3.2	odd	2	inner

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:	\( N \)	\(=\)	\( 5184 = 2^{6} \cdot 3^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5184.c (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q-1.73205i q^{5} +3.00000i q^{7} +O(q^{10})\)	\(q-1.73205i q^{5} +3.00000i q^{7} -5.19615 q^{11} -1.00000 q^{13} +3.46410i q^{17} +6.00000i q^{19} -5.19615 q^{23} +2.00000 q^{25} -8.66025i q^{29} -3.00000i q^{31} +5.19615 q^{35} +4.00000 q^{37} +5.19615i q^{41} -3.00000i q^{43} -5.19615 q^{47} -2.00000 q^{49} -10.3923i q^{53} +9.00000i q^{55} -5.19615 q^{59} +7.00000 q^{61} +1.73205i q^{65} -9.00000i q^{67} +10.3923 q^{71} +4.00000 q^{73} -15.5885i q^{77} -15.0000i q^{79} +5.19615 q^{83} +6.00000 q^{85} -3.46410i q^{89} -3.00000i q^{91} +10.3923 q^{95} +1.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 4 q^{13} + 8 q^{25} + 16 q^{37} - 8 q^{49} + 28 q^{61} + 16 q^{73} + 24 q^{85} + 4 q^{97}+O(q^{100}) \) 4 * q - 4 * q^13 + 8 * q^25 + 16 * q^37 - 8 * q^49 + 28 * q^61 + 16 * q^73 + 24 * q^85 + 4 * q^97

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