LMFDB - Embedded newform 1584.2.o.b.703.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,2,Mod(703,1584)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1584, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([1, 0, 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("1584.703");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1584.o (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:	$12.6483036802$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(\sqrt{-2}, \sqrt{-3})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{2} + 4 $ x^4 - 2*x^2 + 4
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2^{3} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			703.1
Root			$1.22474 - 0.707107i$ of defining polynomial
Character	$\chi$	$=$	1584.703
Dual form			1584.2.o.b.703.2

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.44949 q^{5} +2.44949 q^{7} +O(q^{10})$	$q-2.44949 q^{5} +2.44949 q^{7} +(-3.00000 - 1.41421i) q^{11} +4.24264i q^{13} -3.46410i q^{17} +1.41421i q^{23} +1.00000 q^{25} -6.92820i q^{29} -3.46410i q^{31} -6.00000 q^{35} -2.00000 q^{37} -6.92820i q^{41} -9.79796 q^{43} +7.07107i q^{47} -1.00000 q^{49} -7.34847 q^{53} +(7.34847 + 3.46410i) q^{55} +14.1421i q^{59} -12.7279i q^{61} -10.3923i q^{65} +3.46410i q^{67} -9.89949i q^{71} -8.48528i q^{73} +(-7.34847 - 3.46410i) q^{77} -2.44949 q^{79} -12.0000 q^{83} +8.48528i q^{85} -4.89898 q^{89} +10.3923i q^{91} -8.00000 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q+O(q^{10}) $ 4 * q	$ 4 q - 12 q^{11} + 4 q^{25} - 24 q^{35} - 8 q^{37} - 4 q^{49} - 48 q^{83} - 32 q^{97}+O(q^{100}) $ 4 * q - 12 * q^11 + 4 * q^25 - 24 * q^35 - 8 * q^37 - 4 * q^49 - 48 * q^83 - 32 * q^97

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1584\mathbb{Z}\right)^\times$.

$n$	$145$	$353$	$991$	$1189$
$\chi(n)$	$-1$	$1$	$-1$	$1$

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
$5$	−2.44949			−1.09545			−0.547723	−	0.836660i	$-0.684505\pi$
$5$	−2.44949			−1.09545			−0.547723	+	0.836660i	$0.684505\pi$
$6$		0			0
$7$	2.44949			0.925820			0.462910	−	0.886405i	$-0.346805\pi$
$7$	2.44949			0.925820			0.462910	+	0.886405i	$0.346805\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$11$	−3.00000	−	1.41421i	−0.904534	−	0.426401i
$12$		0			0
$13$			4.24264i			1.17670i	0.808608	+	0.588348i	$0.200222\pi$
$13$			4.24264i			1.17670i	−0.808608	+	0.588348i	$0.799778\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	3.46410i		−	0.840168i	−0.907485	−	0.420084i	$-0.862001\pi$
$17$		−	3.46410i		−	0.840168i	0.907485	−	0.420084i	$-0.137999\pi$
$18$		0			0
$19$		0			0			−	1.00000i	$-0.5\pi$
$19$		0			0				1.00000i	$0.5\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$			1.41421i			0.294884i	0.989071	+	0.147442i	$0.0471040\pi$
$23$			1.41421i			0.294884i	−0.989071	+	0.147442i	$0.952896\pi$
$24$		0			0
$25$	1.00000			0.200000
$26$		0			0
$27$		0			0
$28$		0			0
$29$		−	6.92820i		−	1.28654i	−0.765641	−	0.643268i	$-0.777578\pi$
$29$		−	6.92820i		−	1.28654i	0.765641	−	0.643268i	$-0.222422\pi$
$30$		0			0
$31$		−	3.46410i		−	0.622171i	−0.950382	−	0.311086i	$-0.899307\pi$
$31$		−	3.46410i		−	0.622171i	0.950382	−	0.311086i	$-0.100693\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$	−6.00000			−1.01419
$36$		0			0
$37$	−2.00000			−0.328798			−0.164399	−	0.986394i	$-0.552568\pi$
$37$	−2.00000			−0.328798			−0.164399	+	0.986394i	$0.552568\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$		−	6.92820i		−	1.08200i	−0.841021	−	0.541002i	$-0.818045\pi$
$41$		−	6.92820i		−	1.08200i	0.841021	−	0.541002i	$-0.181955\pi$
$42$		0			0
$43$	−9.79796			−1.49417			−0.747087	−	0.664726i	$-0.768548\pi$
$43$	−9.79796			−1.49417			−0.747087	+	0.664726i	$0.768548\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$			7.07107i			1.03142i	0.856763	+	0.515711i	$0.172472\pi$
$47$			7.07107i			1.03142i	−0.856763	+	0.515711i	$0.827528\pi$
$48$		0			0
$49$	−1.00000			−0.142857
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−7.34847			−1.00939			−0.504695	−	0.863298i	$-0.668395\pi$
$53$	−7.34847			−1.00939			−0.504695	+	0.863298i	$0.668395\pi$
$54$		0			0
$55$	7.34847	+	3.46410i	0.990867	+	0.467099i
$56$		0			0
$57$		0			0
$58$		0			0
$59$			14.1421i			1.84115i	0.390567	+	0.920575i	$0.372279\pi$
$59$			14.1421i			1.84115i	−0.390567	+	0.920575i	$0.627721\pi$
$60$		0			0
$61$		−	12.7279i		−	1.62964i	−0.579712	−	0.814822i	$-0.696835\pi$
$61$		−	12.7279i		−	1.62964i	0.579712	−	0.814822i	$-0.303165\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	10.3923i		−	1.28901i
$66$		0			0
$67$			3.46410i			0.423207i	0.977356	+	0.211604i	$0.0678686\pi$
$67$			3.46410i			0.423207i	−0.977356	+	0.211604i	$0.932131\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$		−	9.89949i		−	1.17485i	−0.809277	−	0.587427i	$-0.800141\pi$
$71$		−	9.89949i		−	1.17485i	0.809277	−	0.587427i	$-0.199859\pi$
$72$		0			0
$73$		−	8.48528i		−	0.993127i	−0.868000	−	0.496564i	$-0.834595\pi$
$73$		−	8.48528i		−	0.993127i	0.868000	−	0.496564i	$-0.165405\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	−7.34847	−	3.46410i	−0.837436	−	0.394771i
$78$		0			0
$79$	−2.44949			−0.275589			−0.137795	−	0.990461i	$-0.544001\pi$
$79$	−2.44949			−0.275589			−0.137795	+	0.990461i	$0.544001\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	−12.0000			−1.31717			−0.658586	−	0.752506i	$-0.728845\pi$
$83$	−12.0000			−1.31717			−0.658586	+	0.752506i	$0.728845\pi$
$84$		0			0
$85$			8.48528i			0.920358i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−4.89898			−0.519291			−0.259645	−	0.965704i	$-0.583606\pi$
$89$	−4.89898			−0.519291			−0.259645	+	0.965704i	$0.583606\pi$
$90$		0			0
$91$			10.3923i			1.08941i
$92$		0			0
$93$		0			0
$94$		0			0
$95$		0			0
$96$		0			0
$97$	−8.00000			−0.812277			−0.406138	−	0.913812i	$-0.633125\pi$
$97$	−8.00000			−0.812277			−0.406138	+	0.913812i	$0.633125\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	13.8564i		−	1.37876i	−0.724398	−	0.689382i	$-0.757882\pi$
$101$		−	13.8564i		−	1.37876i	0.724398	−	0.689382i	$-0.242118\pi$
$102$		0			0
$103$		−	10.3923i		−	1.02398i	−0.858990	−	0.511992i	$-0.828908\pi$
$103$		−	10.3923i		−	1.02398i	0.858990	−	0.511992i	$-0.171092\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	−6.00000			−0.580042			−0.290021	−	0.957020i	$-0.593662\pi$
$107$	−6.00000			−0.580042			−0.290021	+	0.957020i	$0.593662\pi$
$108$		0			0
$109$		−	4.24264i		−	0.406371i	−0.979140	−	0.203186i	$-0.934871\pi$
$109$		−	4.24264i		−	0.406371i	0.979140	−	0.203186i	$-0.0651295\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	4.89898			0.460857			0.230429	−	0.973089i	$-0.425987\pi$
$113$	4.89898			0.460857			0.230429	+	0.973089i	$0.425987\pi$
$114$		0			0
$115$		−	3.46410i		−	0.323029i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		−	8.48528i		−	0.777844i
$120$		0			0
$121$	7.00000	+	8.48528i	0.636364	+	0.771389i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	9.79796			0.876356
$126$		0			0
$127$	−7.34847			−0.652071			−0.326036	−	0.945357i	$-0.605713\pi$
$127$	−7.34847			−0.652071			−0.326036	+	0.945357i	$0.605713\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	−12.0000			−1.04844			−0.524222	−	0.851581i	$-0.675644\pi$
$131$	−12.0000			−1.04844			−0.524222	+	0.851581i	$0.675644\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$	−19.5959			−1.67419			−0.837096	−	0.547056i	$-0.815749\pi$
$137$	−19.5959			−1.67419			−0.837096	+	0.547056i	$0.815749\pi$
$138$		0			0
$139$	4.89898			0.415526			0.207763	−	0.978179i	$-0.433382\pi$
$139$	4.89898			0.415526			0.207763	+	0.978179i	$0.433382\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	6.00000	−	12.7279i	0.501745	−	1.06436i
$144$		0			0
$145$			16.9706i			1.40933i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		0			0		1.00000			$0$
$149$		0			0		−1.00000			$\pi$
$150$		0			0
$151$	7.34847			0.598010			0.299005	−	0.954252i	$-0.403345\pi$
$151$	7.34847			0.598010			0.299005	+	0.954252i	$0.403345\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$			8.48528i			0.681554i
$156$		0			0
$157$	14.0000			1.11732			0.558661	−	0.829396i	$-0.311315\pi$
$157$	14.0000			1.11732			0.558661	+	0.829396i	$0.311315\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$			3.46410i			0.273009i
$162$		0			0
$163$		−	20.7846i		−	1.62798i	−0.580881	−	0.813988i	$-0.697292\pi$
$163$		−	20.7846i		−	1.62798i	0.580881	−	0.813988i	$-0.302708\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	12.0000			0.928588			0.464294	−	0.885681i	$-0.346308\pi$
$167$	12.0000			0.928588			0.464294	+	0.885681i	$0.346308\pi$
$168$		0			0
$169$	−5.00000			−0.384615
$170$		0			0
$171$		0			0
$172$		0			0
$173$		0			0		1.00000			$0$
$173$		0			0		−1.00000			$\pi$
$174$		0			0
$175$	2.44949			0.185164
$176$		0			0
$177$		0			0
$178$		0			0
$179$			19.7990i			1.47985i	0.672692	+	0.739923i	$0.265138\pi$
$179$			19.7990i			1.47985i	−0.672692	+	0.739923i	$0.734862\pi$
$180$		0			0
$181$	2.00000			0.148659			0.0743294	−	0.997234i	$-0.476318\pi$
$181$	2.00000			0.148659			0.0743294	+	0.997234i	$0.476318\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	4.89898			0.360180
$186$		0			0
$187$	−4.89898	+	10.3923i	−0.358249	+	0.759961i
$188$		0			0
$189$		0			0
$190$		0			0
$191$			15.5563i			1.12562i	0.826587	+	0.562809i	$0.190279\pi$
$191$			15.5563i			1.12562i	−0.826587	+	0.562809i	$0.809721\pi$
$192$		0			0
$193$		0			0		1.00000			$0$
$193$		0			0		−1.00000			$\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			13.8564i			0.987228i	0.869681	+	0.493614i	$0.164324\pi$
$197$			13.8564i			0.987228i	−0.869681	+	0.493614i	$0.835676\pi$
$198$		0			0
$199$			10.3923i			0.736691i	0.929689	+	0.368345i	$0.120076\pi$
$199$			10.3923i			0.736691i	−0.929689	+	0.368345i	$0.879924\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		−	16.9706i		−	1.19110i
$204$		0			0
$205$			16.9706i			1.18528i
$206$		0			0
$207$		0			0
$208$		0			0
$209$		0			0
$210$		0			0
$211$	24.4949			1.68630			0.843149	−	0.537680i	$-0.180699\pi$
$211$	24.4949			1.68630			0.843149	+	0.537680i	$0.180699\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	24.0000			1.63679
$216$		0			0
$217$		−	8.48528i		−	0.576018i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	14.6969			0.988623
$222$		0			0
$223$			24.2487i			1.62381i	0.583787	+	0.811907i	$0.301570\pi$
$223$			24.2487i			1.62381i	−0.583787	+	0.811907i	$0.698430\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	−18.0000			−1.19470			−0.597351	−	0.801980i	$-0.703780\pi$
$227$	−18.0000			−1.19470			−0.597351	+	0.801980i	$0.703780\pi$
$228$		0			0
$229$	−14.0000			−0.925146			−0.462573	−	0.886581i	$-0.653074\pi$
$229$	−14.0000			−0.925146			−0.462573	+	0.886581i	$0.653074\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			20.7846i			1.36165i	0.732448	+	0.680823i	$0.238378\pi$
$233$			20.7846i			1.36165i	−0.732448	+	0.680823i	$0.761622\pi$
$234$		0			0
$235$		−	17.3205i		−	1.12987i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	12.0000			0.776215			0.388108	−	0.921614i	$-0.373129\pi$
$239$	12.0000			0.776215			0.388108	+	0.921614i	$0.373129\pi$
$240$		0			0
$241$		−	16.9706i		−	1.09317i	−0.837404	−	0.546585i	$-0.815928\pi$
$241$		−	16.9706i		−	1.09317i	0.837404	−	0.546585i	$-0.184072\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	2.44949			0.156492
$246$		0			0
$247$		0			0
$248$		0			0
$249$		0			0
$250$		0			0
$251$		−	5.65685i		−	0.357057i	−0.983935	−	0.178529i	$-0.942866\pi$
$251$		−	5.65685i		−	0.357057i	0.983935	−	0.178529i	$-0.0571337\pi$
$252$		0			0
$253$	2.00000	−	4.24264i	0.125739	−	0.266733i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	14.6969			0.916770			0.458385	−	0.888754i	$-0.348428\pi$
$257$	14.6969			0.916770			0.458385	+	0.888754i	$0.348428\pi$
$258$		0			0
$259$	−4.89898			−0.304408
$260$		0			0
$261$		0			0
$262$		0			0
$263$	24.0000			1.47990			0.739952	−	0.672660i	$-0.234848\pi$
$263$	24.0000			1.47990			0.739952	+	0.672660i	$0.234848\pi$
$264$		0			0
$265$	18.0000			1.10573
$266$		0			0
$267$		0			0
$268$		0			0
$269$	22.0454			1.34413			0.672066	−	0.740491i	$-0.265407\pi$
$269$	22.0454			1.34413			0.672066	+	0.740491i	$0.265407\pi$
$270$		0			0
$271$	−31.8434			−1.93435			−0.967173	−	0.254117i	$-0.918215\pi$
$271$	−31.8434			−1.93435			−0.967173	+	0.254117i	$0.918215\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−3.00000	−	1.41421i	−0.180907	−	0.0852803i
$276$		0			0
$277$		−	21.2132i		−	1.27458i	−0.770625	−	0.637289i	$-0.780056\pi$
$277$		−	21.2132i		−	1.27458i	0.770625	−	0.637289i	$-0.219944\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	3.46410i		−	0.206651i	−0.994648	−	0.103325i	$-0.967052\pi$
$281$		−	3.46410i		−	0.206651i	0.994648	−	0.103325i	$-0.0329483\pi$
$282$		0			0
$283$	−14.6969			−0.873642			−0.436821	−	0.899548i	$-0.643896\pi$
$283$	−14.6969			−0.873642			−0.436821	+	0.899548i	$0.643896\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		−	16.9706i		−	1.00174i
$288$		0			0
$289$	5.00000			0.294118
$290$		0			0
$291$		0			0
$292$		0			0
$293$			20.7846i			1.21425i	0.794606	+	0.607125i	$0.207677\pi$
$293$			20.7846i			1.21425i	−0.794606	+	0.607125i	$0.792323\pi$
$294$		0			0
$295$		−	34.6410i		−	2.01688i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−6.00000			−0.346989
$300$		0			0
$301$	−24.0000			−1.38334
$302$		0			0
$303$		0			0
$304$		0			0
$305$			31.1769i			1.78518i
$306$		0			0
$307$	24.4949			1.39800			0.698999	−	0.715123i	$-0.253629\pi$
$307$	24.4949			1.39800			0.698999	+	0.715123i	$0.253629\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$		−	15.5563i		−	0.882120i	−0.897478	−	0.441060i	$-0.854603\pi$
$311$		−	15.5563i		−	0.882120i	0.897478	−	0.441060i	$-0.145397\pi$
$312$		0			0
$313$	−14.0000			−0.791327			−0.395663	−	0.918396i	$-0.629485\pi$
$313$	−14.0000			−0.791327			−0.395663	+	0.918396i	$0.629485\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−12.2474			−0.687885			−0.343943	−	0.938991i	$-0.611763\pi$
$317$	−12.2474			−0.687885			−0.343943	+	0.938991i	$0.611763\pi$
$318$		0			0
$319$	−9.79796	+	20.7846i	−0.548580	+	1.16371i
$320$		0			0
$321$		0			0
$322$		0			0
$323$		0			0
$324$		0			0
$325$			4.24264i			0.235339i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			17.3205i			0.954911i
$330$		0			0
$331$			13.8564i			0.761617i	0.924654	+	0.380808i	$0.124354\pi$
$331$			13.8564i			0.761617i	−0.924654	+	0.380808i	$0.875646\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$		−	8.48528i		−	0.463600i
$336$		0			0
$337$		−	8.48528i		−	0.462223i	−0.972927	−	0.231111i	$-0.925764\pi$
$337$		−	8.48528i		−	0.462223i	0.972927	−	0.231111i	$-0.0742362\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−4.89898	+	10.3923i	−0.265295	+	0.562775i
$342$		0			0
$343$	−19.5959			−1.05808
$344$		0			0
$345$		0			0
$346$		0			0
$347$	12.0000			0.644194			0.322097	−	0.946707i	$-0.395612\pi$
$347$	12.0000			0.644194			0.322097	+	0.946707i	$0.395612\pi$
$348$		0			0
$349$			4.24264i			0.227103i	0.993532	+	0.113552i	$0.0362227\pi$
$349$			4.24264i			0.227103i	−0.993532	+	0.113552i	$0.963777\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−9.79796			−0.521493			−0.260746	−	0.965407i	$-0.583969\pi$
$353$	−9.79796			−0.521493			−0.260746	+	0.965407i	$0.583969\pi$
$354$		0			0
$355$			24.2487i			1.28699i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	36.0000			1.90001			0.950004	−	0.312239i	$-0.101079\pi$
$359$	36.0000			1.90001			0.950004	+	0.312239i	$0.101079\pi$
$360$		0			0
$361$	−19.0000			−1.00000
$362$		0			0
$363$		0			0
$364$		0			0
$365$			20.7846i			1.08792i
$366$		0			0
$367$		−	17.3205i		−	0.904123i	−0.891987	−	0.452062i	$-0.850689\pi$
$367$		−	17.3205i		−	0.904123i	0.891987	−	0.452062i	$-0.149311\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$	−18.0000			−0.934513
$372$		0			0
$373$			12.7279i			0.659027i	0.944151	+	0.329513i	$0.106885\pi$
$373$			12.7279i			0.659027i	−0.944151	+	0.329513i	$0.893115\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	29.3939			1.51386
$378$		0			0
$379$		−	6.92820i		−	0.355878i	−0.984042	−	0.177939i	$-0.943057\pi$
$379$		−	6.92820i		−	0.355878i	0.984042	−	0.177939i	$-0.0569430\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$		−	7.07107i		−	0.361315i	−0.983546	−	0.180657i	$-0.942177\pi$
$383$		−	7.07107i		−	0.361315i	0.983546	−	0.180657i	$-0.0578225\pi$
$384$		0			0
$385$	18.0000	+	8.48528i	0.917365	+	0.432450i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	−2.44949			−0.124194			−0.0620970	−	0.998070i	$-0.519779\pi$
$389$	−2.44949			−0.124194			−0.0620970	+	0.998070i	$0.519779\pi$
$390$		0			0
$391$	4.89898			0.247752
$392$		0			0
$393$		0			0
$394$		0			0
$395$	6.00000			0.301893
$396$		0			0
$397$	−2.00000			−0.100377			−0.0501886	−	0.998740i	$-0.515982\pi$
$397$	−2.00000			−0.100377			−0.0501886	+	0.998740i	$0.515982\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	−14.6969			−0.733930			−0.366965	−	0.930235i	$-0.619603\pi$
$401$	−14.6969			−0.733930			−0.366965	+	0.930235i	$0.619603\pi$
$402$		0			0
$403$	14.6969			0.732107
$404$		0			0
$405$		0			0
$406$		0			0
$407$	6.00000	+	2.82843i	0.297409	+	0.140200i
$408$		0			0
$409$		−	25.4558i		−	1.25871i	−0.777118	−	0.629355i	$-0.783319\pi$
$409$		−	25.4558i		−	1.25871i	0.777118	−	0.629355i	$-0.216681\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			34.6410i			1.70457i
$414$		0			0
$415$	29.3939			1.44289
$416$		0			0
$417$		0			0
$418$		0			0
$419$			11.3137i			0.552711i	0.961056	+	0.276355i	$0.0891267\pi$
$419$			11.3137i			0.552711i	−0.961056	+	0.276355i	$0.910873\pi$
$420$		0			0
$421$	38.0000			1.85201			0.926003	−	0.377515i	$-0.123221\pi$
$421$	38.0000			1.85201			0.926003	+	0.377515i	$0.123221\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$		−	3.46410i		−	0.168034i
$426$		0			0
$427$		−	31.1769i		−	1.50876i
$428$		0			0
$429$		0			0
$430$		0			0
$431$	12.0000			0.578020			0.289010	−	0.957326i	$-0.406674\pi$
$431$	12.0000			0.578020			0.289010	+	0.957326i	$0.406674\pi$
$432$		0			0
$433$	−2.00000			−0.0961139			−0.0480569	−	0.998845i	$-0.515303\pi$
$433$	−2.00000			−0.0961139			−0.0480569	+	0.998845i	$0.515303\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$		0			0
$438$		0			0
$439$	12.2474			0.584539			0.292269	−	0.956336i	$-0.405590\pi$
$439$	12.2474			0.584539			0.292269	+	0.956336i	$0.405590\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$		−	11.3137i		−	0.537531i	−0.963206	−	0.268765i	$-0.913384\pi$
$443$		−	11.3137i		−	0.537531i	0.963206	−	0.268765i	$-0.0866156\pi$
$444$		0			0
$445$	12.0000			0.568855
$446$		0			0
$447$		0			0
$448$		0			0
$449$	39.1918			1.84958			0.924789	−	0.380481i	$-0.124242\pi$
$449$	39.1918			1.84958			0.924789	+	0.380481i	$0.124242\pi$
$450$		0			0
$451$	−9.79796	+	20.7846i	−0.461368	+	0.978709i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		−	25.4558i		−	1.19339i
$456$		0			0
$457$			42.4264i			1.98462i	0.123763	+	0.992312i	$0.460504\pi$
$457$			42.4264i			1.98462i	−0.123763	+	0.992312i	$0.539496\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	20.7846i		−	0.968036i	−0.875058	−	0.484018i	$-0.839177\pi$
$461$		−	20.7846i		−	0.968036i	0.875058	−	0.484018i	$-0.160823\pi$
$462$		0			0
$463$			17.3205i			0.804952i	0.915430	+	0.402476i	$0.131850\pi$
$463$			17.3205i			0.804952i	−0.915430	+	0.402476i	$0.868150\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$			31.1127i			1.43972i	0.694117	+	0.719862i	$0.255795\pi$
$467$			31.1127i			1.43972i	−0.694117	+	0.719862i	$0.744205\pi$
$468$		0			0
$469$			8.48528i			0.391814i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	29.3939	+	13.8564i	1.35153	+	0.637118i
$474$		0			0
$475$		0			0
$476$		0			0
$477$		0			0
$478$		0			0
$479$	−24.0000			−1.09659			−0.548294	−	0.836286i	$-0.684723\pi$
$479$	−24.0000			−1.09659			−0.548294	+	0.836286i	$0.684723\pi$
$480$		0			0
$481$		−	8.48528i		−	0.386896i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	19.5959			0.889805
$486$		0			0
$487$			38.1051i			1.72671i	0.504599	+	0.863354i	$0.331640\pi$
$487$			38.1051i			1.72671i	−0.504599	+	0.863354i	$0.668360\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	6.00000			0.270776			0.135388	−	0.990793i	$-0.456772\pi$
$491$	6.00000			0.270776			0.135388	+	0.990793i	$0.456772\pi$
$492$		0			0
$493$	−24.0000			−1.08091
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	24.2487i		−	1.08770i
$498$		0			0
$499$		−	41.5692i		−	1.86089i	−0.366427	−	0.930447i	$-0.619419\pi$
$499$		−	41.5692i		−	1.86089i	0.366427	−	0.930447i	$-0.380581\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	36.0000			1.60516			0.802580	−	0.596544i	$-0.203460\pi$
$503$	36.0000			1.60516			0.802580	+	0.596544i	$0.203460\pi$
$504$		0			0
$505$			33.9411i			1.51036i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−31.8434			−1.41143			−0.705716	−	0.708495i	$-0.749375\pi$
$509$	−31.8434			−1.41143			−0.705716	+	0.708495i	$0.749375\pi$
$510$		0			0
$511$		−	20.7846i		−	0.919457i
$512$		0			0
$513$		0			0
$514$		0			0
$515$			25.4558i			1.12172i
$516$		0			0
$517$	10.0000	−	21.2132i	0.439799	−	0.932956i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−24.4949			−1.07314			−0.536570	−	0.843856i	$-0.680280\pi$
$521$	−24.4949			−1.07314			−0.536570	+	0.843856i	$0.680280\pi$
$522$		0			0
$523$	19.5959			0.856870			0.428435	−	0.903573i	$-0.359065\pi$
$523$	19.5959			0.856870			0.428435	+	0.903573i	$0.359065\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$	−12.0000			−0.522728
$528$		0			0
$529$	21.0000			0.913043
$530$		0			0
$531$		0			0
$532$		0			0
$533$	29.3939			1.27319
$534$		0			0
$535$	14.6969			0.635404
$536$		0			0
$537$		0			0
$538$		0			0
$539$	3.00000	+	1.41421i	0.129219	+	0.0609145i
$540$		0			0
$541$			12.7279i			0.547216i	0.961841	+	0.273608i	$0.0882171\pi$
$541$			12.7279i			0.547216i	−0.961841	+	0.273608i	$0.911783\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$			10.3923i			0.445157i
$546$		0			0
$547$	−39.1918			−1.67572			−0.837861	−	0.545884i	$-0.816194\pi$
$547$	−39.1918			−1.67572			−0.837861	+	0.545884i	$0.816194\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$		0			0
$552$		0			0
$553$	−6.00000			−0.255146
$554$		0			0
$555$		0			0
$556$		0			0
$557$			6.92820i			0.293557i	0.989169	+	0.146779i	$0.0468905\pi$
$557$			6.92820i			0.293557i	−0.989169	+	0.146779i	$0.953109\pi$
$558$		0			0
$559$		−	41.5692i		−	1.75819i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−6.00000			−0.252870			−0.126435	−	0.991975i	$-0.540353\pi$
$563$	−6.00000			−0.252870			−0.126435	+	0.991975i	$0.540353\pi$
$564$		0			0
$565$	−12.0000			−0.504844
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	31.1769i		−	1.30700i	−0.756925	−	0.653502i	$-0.773299\pi$
$569$		−	31.1769i		−	1.30700i	0.756925	−	0.653502i	$-0.226701\pi$
$570$		0			0
$571$	−34.2929			−1.43511			−0.717556	−	0.696501i	$-0.754739\pi$
$571$	−34.2929			−1.43511			−0.717556	+	0.696501i	$0.754739\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			1.41421i			0.0589768i
$576$		0			0
$577$	−16.0000			−0.666089			−0.333044	−	0.942911i	$-0.608076\pi$
$577$	−16.0000			−0.666089			−0.333044	+	0.942911i	$0.608076\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	−29.3939			−1.21946
$582$		0			0
$583$	22.0454	+	10.3923i	0.913027	+	0.430405i
$584$		0			0
$585$		0			0
$586$		0			0
$587$		−	19.7990i		−	0.817192i	−0.912715	−	0.408596i	$-0.866019\pi$
$587$		−	19.7990i		−	0.817192i	0.912715	−	0.408596i	$-0.133981\pi$
$588$		0			0
$589$		0			0
$590$		0			0
$591$		0			0
$592$		0			0
$593$			17.3205i			0.711268i	0.934625	+	0.355634i	$0.115735\pi$
$593$			17.3205i			0.711268i	−0.934625	+	0.355634i	$0.884265\pi$
$594$		0			0
$595$			20.7846i			0.852086i
$596$		0			0
$597$		0			0
$598$		0			0
$599$			9.89949i			0.404482i	0.979336	+	0.202241i	$0.0648225\pi$
$599$			9.89949i			0.404482i	−0.979336	+	0.202241i	$0.935178\pi$
$600$		0			0
$601$			8.48528i			0.346122i	0.984911	+	0.173061i	$0.0553658\pi$
$601$			8.48528i			0.346122i	−0.984911	+	0.173061i	$0.944634\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−17.1464	−	20.7846i	−0.697101	−	0.845015i
$606$		0			0
$607$	−22.0454			−0.894795			−0.447398	−	0.894335i	$-0.647649\pi$
$607$	−22.0454			−0.894795			−0.447398	+	0.894335i	$0.647649\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−30.0000			−1.21367
$612$		0			0
$613$		−	21.2132i		−	0.856793i	−0.903591	−	0.428397i	$-0.859079\pi$
$613$		−	21.2132i		−	0.856793i	0.903591	−	0.428397i	$-0.140921\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	−14.6969			−0.591676			−0.295838	−	0.955238i	$-0.595599\pi$
$617$	−14.6969			−0.591676			−0.295838	+	0.955238i	$0.595599\pi$
$618$		0			0
$619$			45.0333i			1.81004i	0.425367	+	0.905021i	$0.360145\pi$
$619$			45.0333i			1.81004i	−0.425367	+	0.905021i	$0.639855\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$	−12.0000			−0.480770
$624$		0			0
$625$	−29.0000			−1.16000
$626$		0			0
$627$		0			0
$628$		0			0
$629$			6.92820i			0.276246i
$630$		0			0
$631$		−	24.2487i		−	0.965326i	−0.875806	−	0.482663i	$-0.839670\pi$
$631$		−	24.2487i		−	0.965326i	0.875806	−	0.482663i	$-0.160330\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	18.0000			0.714308
$636$		0			0
$637$		−	4.24264i		−	0.168100i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−34.2929			−1.35449			−0.677243	−	0.735759i	$-0.736826\pi$
$641$	−34.2929			−1.35449			−0.677243	+	0.735759i	$0.736826\pi$
$642$		0			0
$643$			17.3205i			0.683054i	0.939872	+	0.341527i	$0.110944\pi$
$643$			17.3205i			0.683054i	−0.939872	+	0.341527i	$0.889056\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$			9.89949i			0.389189i	0.980884	+	0.194595i	$0.0623391\pi$
$647$			9.89949i			0.389189i	−0.980884	+	0.194595i	$0.937661\pi$
$648$		0			0
$649$	20.0000	−	42.4264i	0.785069	−	1.66538i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−2.44949			−0.0958559			−0.0479280	−	0.998851i	$-0.515262\pi$
$653$	−2.44949			−0.0958559			−0.0479280	+	0.998851i	$0.515262\pi$
$654$		0			0
$655$	29.3939			1.14851
$656$		0			0
$657$		0			0
$658$		0			0
$659$	−30.0000			−1.16863			−0.584317	−	0.811525i	$-0.698638\pi$
$659$	−30.0000			−1.16863			−0.584317	+	0.811525i	$0.698638\pi$
$660$		0			0
$661$	−22.0000			−0.855701			−0.427850	−	0.903850i	$-0.640729\pi$
$661$	−22.0000			−0.855701			−0.427850	+	0.903850i	$0.640729\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	9.79796			0.379378
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−18.0000	+	38.1838i	−0.694882	+	1.47407i
$672$		0			0
$673$		−	8.48528i		−	0.327084i	−0.986536	−	0.163542i	$-0.947708\pi$
$673$		−	8.48528i		−	0.327084i	0.986536	−	0.163542i	$-0.0522919\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	48.4974i		−	1.86391i	−0.362577	−	0.931954i	$-0.618103\pi$
$677$		−	48.4974i		−	1.86391i	0.362577	−	0.931954i	$-0.381897\pi$
$678$		0			0
$679$	−19.5959			−0.752022
$680$		0			0
$681$		0			0
$682$		0			0
$683$		−	39.5980i		−	1.51517i	−0.652734	−	0.757587i	$-0.726378\pi$
$683$		−	39.5980i		−	1.51517i	0.652734	−	0.757587i	$-0.273622\pi$
$684$		0			0
$685$	48.0000			1.83399
$686$		0			0
$687$		0			0
$688$		0			0
$689$		−	31.1769i		−	1.18775i
$690$		0			0
$691$		−	45.0333i		−	1.71315i	−0.516024	−	0.856574i	$-0.672588\pi$
$691$		−	45.0333i		−	1.71315i	0.516024	−	0.856574i	$-0.327412\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	−12.0000			−0.455186
$696$		0			0
$697$	−24.0000			−0.909065
$698$		0			0
$699$		0			0
$700$		0			0
$701$		−	13.8564i		−	0.523349i	−0.965156	−	0.261675i	$-0.915725\pi$
$701$		−	13.8564i		−	0.523349i	0.965156	−	0.261675i	$-0.0842747\pi$
$702$		0			0
$703$		0			0
$704$		0			0
$705$		0			0
$706$		0			0
$707$		−	33.9411i		−	1.27649i
$708$		0			0
$709$	10.0000			0.375558			0.187779	−	0.982211i	$-0.439871\pi$
$709$	10.0000			0.375558			0.187779	+	0.982211i	$0.439871\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	4.89898			0.183468
$714$		0			0
$715$	−14.6969	+	31.1769i	−0.549634	+	1.16595i
$716$		0			0
$717$		0			0
$718$		0			0
$719$		−	35.3553i		−	1.31853i	−0.751910	−	0.659266i	$-0.770867\pi$
$719$		−	35.3553i		−	1.31853i	0.751910	−	0.659266i	$-0.229133\pi$
$720$		0			0
$721$		−	25.4558i		−	0.948025i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	6.92820i		−	0.257307i
$726$		0			0
$727$			17.3205i			0.642382i	0.947014	+	0.321191i	$0.104083\pi$
$727$			17.3205i			0.642382i	−0.947014	+	0.321191i	$0.895917\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$			33.9411i			1.25536i
$732$		0			0
$733$		−	38.1838i		−	1.41035i	−0.709034	−	0.705175i	$-0.750869\pi$
$733$		−	38.1838i		−	1.41035i	0.709034	−	0.705175i	$-0.249131\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	4.89898	−	10.3923i	0.180456	−	0.382805i
$738$		0			0
$739$	−24.4949			−0.901059			−0.450530	−	0.892761i	$-0.648765\pi$
$739$	−24.4949			−0.901059			−0.450530	+	0.892761i	$0.648765\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	12.0000			0.440237			0.220119	−	0.975473i	$-0.429356\pi$
$743$	12.0000			0.440237			0.220119	+	0.975473i	$0.429356\pi$
$744$		0			0
$745$		0			0
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−14.6969			−0.537014
$750$		0			0
$751$			24.2487i			0.884848i	0.896806	+	0.442424i	$0.145881\pi$
$751$			24.2487i			0.884848i	−0.896806	+	0.442424i	$0.854119\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	−18.0000			−0.655087
$756$		0			0
$757$	−22.0000			−0.799604			−0.399802	−	0.916602i	$-0.630921\pi$
$757$	−22.0000			−0.799604			−0.399802	+	0.916602i	$0.630921\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$		−	31.1769i		−	1.13016i	−0.825035	−	0.565081i	$-0.808845\pi$
$761$		−	31.1769i		−	1.13016i	0.825035	−	0.565081i	$-0.191155\pi$
$762$		0			0
$763$		−	10.3923i		−	0.376227i
$764$		0			0
$765$		0			0
$766$		0			0
$767$	−60.0000			−2.16647
$768$		0			0
$769$			16.9706i			0.611974i	0.952036	+	0.305987i	$0.0989864\pi$
$769$			16.9706i			0.611974i	−0.952036	+	0.305987i	$0.901014\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	7.34847			0.264306			0.132153	−	0.991229i	$-0.457811\pi$
$773$	7.34847			0.264306			0.132153	+	0.991229i	$0.457811\pi$
$774$		0			0
$775$		−	3.46410i		−	0.124434i
$776$		0			0
$777$		0			0
$778$		0			0
$779$		0			0
$780$		0			0
$781$	−14.0000	+	29.6985i	−0.500959	+	1.06270i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−34.2929			−1.22396
$786$		0			0
$787$		0			0			−	1.00000i	$-0.5\pi$
$787$		0			0				1.00000i	$0.5\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$	12.0000			0.426671
$792$		0			0
$793$	54.0000			1.91760
$794$		0			0
$795$		0			0
$796$		0			0
$797$	17.1464			0.607358			0.303679	−	0.952774i	$-0.401785\pi$
$797$	17.1464			0.607358			0.303679	+	0.952774i	$0.401785\pi$
$798$		0			0
$799$	24.4949			0.866567
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−12.0000	+	25.4558i	−0.423471	+	0.898317i
$804$		0			0
$805$		−	8.48528i		−	0.299067i
$806$		0			0
$807$		0			0
$808$		0			0
$809$		−	6.92820i		−	0.243583i	−0.992556	−	0.121791i	$-0.961136\pi$
$809$		−	6.92820i		−	0.243583i	0.992556	−	0.121791i	$-0.0388639\pi$
$810$		0			0
$811$	39.1918			1.37621			0.688106	−	0.725610i	$-0.258443\pi$
$811$	39.1918			1.37621			0.688106	+	0.725610i	$0.258443\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$			50.9117i			1.78336i
$816$		0			0
$817$		0			0
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	55.4256i		−	1.93437i	−0.254078	−	0.967184i	$-0.581772\pi$
$821$		−	55.4256i		−	1.93437i	0.254078	−	0.967184i	$-0.418228\pi$
$822$		0			0
$823$		−	38.1051i		−	1.32826i	−0.747617	−	0.664130i	$-0.768802\pi$
$823$		−	38.1051i		−	1.32826i	0.747617	−	0.664130i	$-0.231198\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$	12.0000			0.417281			0.208640	−	0.977992i	$-0.433096\pi$
$827$	12.0000			0.417281			0.208640	+	0.977992i	$0.433096\pi$
$828$		0			0
$829$	−38.0000			−1.31979			−0.659897	−	0.751356i	$-0.729400\pi$
$829$	−38.0000			−1.31979			−0.659897	+	0.751356i	$0.729400\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$			3.46410i			0.120024i
$834$		0			0
$835$	−29.3939			−1.01722
$836$		0			0
$837$		0			0
$838$		0			0
$839$		−	15.5563i		−	0.537065i	−0.963271	−	0.268532i	$-0.913461\pi$
$839$		−	15.5563i		−	0.537065i	0.963271	−	0.268532i	$-0.0865386\pi$
$840$		0			0
$841$	−19.0000			−0.655172
$842$		0			0
$843$		0			0
$844$		0			0
$845$	12.2474			0.421325
$846$		0			0
$847$	17.1464	+	20.7846i	0.589158	+	0.714168i
$848$		0			0
$849$		0			0
$850$		0			0
$851$		−	2.82843i		−	0.0969572i
$852$		0			0
$853$			38.1838i			1.30739i	0.756759	+	0.653694i	$0.226781\pi$
$853$			38.1838i			1.30739i	−0.756759	+	0.653694i	$0.773219\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	48.4974i		−	1.65664i	−0.560255	−	0.828320i	$-0.689297\pi$
$857$		−	48.4974i		−	1.65664i	0.560255	−	0.828320i	$-0.310703\pi$
$858$		0			0
$859$			6.92820i			0.236387i	0.992991	+	0.118194i	$0.0377103\pi$
$859$			6.92820i			0.236387i	−0.992991	+	0.118194i	$0.962290\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$			52.3259i			1.78119i	0.454793	+	0.890597i	$0.349713\pi$
$863$			52.3259i			1.78119i	−0.454793	+	0.890597i	$0.650287\pi$
$864$		0			0
$865$		0			0
$866$		0			0
$867$		0			0
$868$		0			0
$869$	7.34847	+	3.46410i	0.249280	+	0.117512i
$870$		0			0
$871$	−14.6969			−0.497987
$872$		0			0
$873$		0			0
$874$		0			0
$875$	24.0000			0.811348
$876$		0			0
$877$			21.2132i			0.716319i	0.933660	+	0.358159i	$0.116596\pi$
$877$			21.2132i			0.716319i	−0.933660	+	0.358159i	$0.883404\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−48.9898			−1.65051			−0.825254	−	0.564762i	$-0.808968\pi$
$881$	−48.9898			−1.65051			−0.825254	+	0.564762i	$0.808968\pi$
$882$		0			0
$883$			31.1769i			1.04919i	0.851353	+	0.524593i	$0.175783\pi$
$883$			31.1769i			1.04919i	−0.851353	+	0.524593i	$0.824217\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−36.0000			−1.20876			−0.604381	−	0.796696i	$-0.706579\pi$
$887$	−36.0000			−1.20876			−0.604381	+	0.796696i	$0.706579\pi$
$888$		0			0
$889$	−18.0000			−0.603701
$890$		0			0
$891$		0			0
$892$		0			0
$893$		0			0
$894$		0			0
$895$		−	48.4974i		−	1.62109i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	−24.0000			−0.800445
$900$		0			0
$901$			25.4558i			0.848057i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−4.89898			−0.162848
$906$		0			0
$907$		−	31.1769i		−	1.03521i	−0.855619	−	0.517606i	$-0.826823\pi$
$907$		−	31.1769i		−	1.03521i	0.855619	−	0.517606i	$-0.173177\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$			32.5269i			1.07766i	0.842413	+	0.538832i	$0.181134\pi$
$911$			32.5269i			1.07766i	−0.842413	+	0.538832i	$0.818866\pi$
$912$		0			0
$913$	36.0000	+	16.9706i	1.19143	+	0.561644i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	−29.3939			−0.970671
$918$		0			0
$919$	−41.6413			−1.37362			−0.686810	−	0.726837i	$-0.740990\pi$
$919$	−41.6413			−1.37362			−0.686810	+	0.726837i	$0.740990\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	42.0000			1.38245
$924$		0			0
$925$	−2.00000			−0.0657596
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−29.3939			−0.964382			−0.482191	−	0.876066i	$-0.660159\pi$
$929$	−29.3939			−0.964382			−0.482191	+	0.876066i	$0.660159\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	12.0000	−	25.4558i	0.392442	−	0.832495i
$936$		0			0
$937$			16.9706i			0.554404i	0.960812	+	0.277202i	$0.0894071\pi$
$937$			16.9706i			0.554404i	−0.960812	+	0.277202i	$0.910593\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	48.4974i		−	1.58097i	−0.612481	−	0.790485i	$-0.709828\pi$
$941$		−	48.4974i		−	1.58097i	0.612481	−	0.790485i	$-0.290172\pi$
$942$		0			0
$943$	9.79796			0.319065
$944$		0			0
$945$		0			0
$946$		0			0
$947$			11.3137i			0.367646i	0.982959	+	0.183823i	$0.0588473\pi$
$947$			11.3137i			0.367646i	−0.982959	+	0.183823i	$0.941153\pi$
$948$		0			0
$949$	36.0000			1.16861
$950$		0			0
$951$		0			0
$952$		0			0
$953$			48.4974i			1.57099i	0.618871	+	0.785493i	$0.287590\pi$
$953$			48.4974i			1.57099i	−0.618871	+	0.785493i	$0.712410\pi$
$954$		0			0
$955$		−	38.1051i		−	1.23305i
$956$		0			0
$957$		0			0
$958$		0			0
$959$	−48.0000			−1.55000
$960$		0			0
$961$	19.0000			0.612903
$962$		0			0
$963$		0			0
$964$		0			0
$965$		0			0
$966$		0			0
$967$	26.9444			0.866473			0.433237	−	0.901280i	$-0.357371\pi$
$967$	26.9444			0.866473			0.433237	+	0.901280i	$0.357371\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$			22.6274i			0.726148i	0.931760	+	0.363074i	$0.118273\pi$
$971$			22.6274i			0.726148i	−0.931760	+	0.363074i	$0.881727\pi$
$972$		0			0
$973$	12.0000			0.384702
$974$		0			0
$975$		0			0
$976$		0			0
$977$	24.4949			0.783661			0.391831	−	0.920037i	$-0.371842\pi$
$977$	24.4949			0.783661			0.391831	+	0.920037i	$0.371842\pi$
$978$		0			0
$979$	14.6969	+	6.92820i	0.469716	+	0.221426i
$980$		0			0
$981$		0			0
$982$		0			0
$983$			1.41421i			0.0451064i	0.999746	+	0.0225532i	$0.00717952\pi$
$983$			1.41421i			0.0451064i	−0.999746	+	0.0225532i	$0.992820\pi$
$984$		0			0
$985$		−	33.9411i		−	1.08145i
$986$		0			0
$987$		0			0
$988$		0			0
$989$		−	13.8564i		−	0.440608i
$990$		0			0
$991$			10.3923i			0.330122i	0.986283	+	0.165061i	$0.0527822\pi$
$991$			10.3923i			0.330122i	−0.986283	+	0.165061i	$0.947218\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$		−	25.4558i		−	0.807005i
$996$		0			0
$997$			4.24264i			0.134366i	0.997741	+	0.0671829i	$0.0214011\pi$
$997$			4.24264i			0.134366i	−0.997741	+	0.0671829i	$0.978599\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.2.o.b.703.1	✓	4
3.2	odd	2		1584.2.o.d.703.4	yes	4
4.3	odd	2		1584.2.o.d.703.2	yes	4
11.10	odd	2		1584.2.o.d.703.1	yes	4
12.11	even	2	inner	1584.2.o.b.703.3	yes	4
33.32	even	2	inner	1584.2.o.b.703.4	yes	4
44.43	even	2	inner	1584.2.o.b.703.2	yes	4
132.131	odd	2		1584.2.o.d.703.3	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1584.2.o.b.703.1	✓	4	1.1	even	1	trivial
1584.2.o.b.703.2	yes	4	44.43	even	2	inner
1584.2.o.b.703.3	yes	4	12.11	even	2	inner
1584.2.o.b.703.4	yes	4	33.32	even	2	inner
1584.2.o.d.703.1	yes	4	11.10	odd	2
1584.2.o.d.703.2	yes	4	4.3	odd	2
1584.2.o.d.703.3	yes	4	132.131	odd	2
1584.2.o.d.703.4	yes	4	3.2	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1584.o (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q-2.44949 q^{5} +2.44949 q^{7} +O(q^{10})\)	\(q-2.44949 q^{5} +2.44949 q^{7} +(-3.00000 - 1.41421i) q^{11} +4.24264i q^{13} -3.46410i q^{17} +1.41421i q^{23} +1.00000 q^{25} -6.92820i q^{29} -3.46410i q^{31} -6.00000 q^{35} -2.00000 q^{37} -6.92820i q^{41} -9.79796 q^{43} +7.07107i q^{47} -1.00000 q^{49} -7.34847 q^{53} +(7.34847 + 3.46410i) q^{55} +14.1421i q^{59} -12.7279i q^{61} -10.3923i q^{65} +3.46410i q^{67} -9.89949i q^{71} -8.48528i q^{73} +(-7.34847 - 3.46410i) q^{77} -2.44949 q^{79} -12.0000 q^{83} +8.48528i q^{85} -4.89898 q^{89} +10.3923i q^{91} -8.00000 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q+O(q^{10}) \) 4 * q	\( 4 q - 12 q^{11} + 4 q^{25} - 24 q^{35} - 8 q^{37} - 4 q^{49} - 48 q^{83} - 32 q^{97}+O(q^{100}) \) 4 * q - 12 * q^11 + 4 * q^25 - 24 * q^35 - 8 * q^37 - 4 * q^49 - 48 * q^83 - 32 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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