LMFDB - Embedded newform 1584.3.j.f.1297.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1584,3,Mod(1297,1584)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("1584.1297"); S:= CuspForms(chi, 3); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1584, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 1])) N = Newforms(chi, 3, names="a")

Level:	$ N $	$=$	$ 1584 = 2^{4} \cdot 3^{2} \cdot 11 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	1584.j (of order $2$, degree $1$, not minimal)

Newform invariants

comment:select newform

sage:traces = [4,0,0,0,-4,0,0,0,0,0,20,0,0,0,0,0,0,0,0,0,0,0,-92,0,12] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(25)] == traces)

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

Self dual:	no
Analytic conductor:	$43.1608738747$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.0.39744.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} + 12x^{2} + 4x + 22 $ x^4 - 2x^3 + 12x^2 + 4*x + 22
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 2 $
Twist minimal:	no (minimal twist has level 33)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			1297.4
Root			$1.36603 - 3.21405i$ of defining polynomial
Character	$\chi$	$=$	1584.1297
Dual form			1584.3.j.f.1297.3

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+4.19615 q^{5} +6.42810i q^{7} +(3.26795 - 10.5034i) q^{11} -7.68899i q^{13} -8.15051i q^{17} -30.4181i q^{19} -31.6603 q^{23} -7.39230 q^{25} +6.88962i q^{29} -51.1769 q^{31} +26.9733i q^{35} -19.4641 q^{37} +47.9800i q^{41} -81.8429i q^{43} +30.1962 q^{47} +7.67949 q^{49} -26.0526 q^{53} +(13.7128 - 44.0737i) q^{55} +82.7461 q^{59} -75.4148i q^{61} -32.2642i q^{65} +34.0000 q^{67} -72.7321 q^{71} -54.8696i q^{73} +(67.5167 + 21.0067i) q^{77} +24.3279i q^{79} -0.923034i q^{83} -34.2008i q^{85} -44.8231 q^{89} +49.4256 q^{91} -127.639i q^{95} -21.2539 q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 4 q^{5} + 20 q^{11} - 92 q^{23} + 12 q^{25} - 80 q^{31} - 64 q^{37} + 100 q^{47} + 100 q^{49} - 28 q^{53} - 56 q^{55} + 40 q^{59} + 136 q^{67} - 284 q^{71} + 180 q^{77} - 304 q^{89} - 24 q^{91} - 376 q^{97}+O(q^{100}) $ 4 * q - 4 * q^5 + 20 * q^11 - 92 * q^23 + 12 * q^25 - 80 * q^31 - 64 * q^37 + 100 * q^47 + 100 * q^49 - 28 * q^53 - 56 * q^55 + 40 * q^59 + 136 * q^67 - 284 * q^71 + 180 * q^77 - 304 * q^89 - 24 * q^91 - 376 * q^97

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
$5$	4.19615			0.839230			0.419615	−	0.907702i	$-0.362165\pi$
$5$	4.19615			0.839230			0.419615	+	0.907702i	$0.362165\pi$
$6$		0			0
$7$			6.42810i			0.918300i	0.888359	+	0.459150i	$0.151846\pi$
$7$			6.42810i			0.918300i	−0.888359	+	0.459150i	$0.848154\pi$
$8$		0			0
$9$		0			0
$10$		0			0
$11$	3.26795	−	10.5034i	0.297086	−	0.954851i
$11$	3.26795	−	10.5034i	0.297086	−	0.954851i
$12$		0			0
$13$		−	7.68899i		−	0.591461i	−0.955271	−	0.295730i	$-0.904437\pi$
$13$		−	7.68899i		−	0.591461i	0.955271	−	0.295730i	$-0.0955630\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$		−	8.15051i		−	0.479442i	−0.970842	−	0.239721i	$-0.922944\pi$
$17$		−	8.15051i		−	0.479442i	0.970842	−	0.239721i	$-0.0770559\pi$
$18$		0			0
$19$		−	30.4181i		−	1.60095i	−0.599364	−	0.800477i	$-0.704580\pi$
$19$		−	30.4181i		−	1.60095i	0.599364	−	0.800477i	$-0.295420\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$	−31.6603			−1.37653			−0.688266	−	0.725458i	$-0.741628\pi$
$23$	−31.6603			−1.37653			−0.688266	+	0.725458i	$0.741628\pi$
$24$		0			0
$25$	−7.39230			−0.295692
$26$		0			0
$27$		0			0
$28$		0			0
$29$			6.88962i			0.237573i	0.992920	+	0.118787i	$0.0379004\pi$
$29$			6.88962i			0.237573i	−0.992920	+	0.118787i	$0.962100\pi$
$30$		0			0
$31$	−51.1769			−1.65087			−0.825434	−	0.564498i	$-0.809070\pi$
$31$	−51.1769			−1.65087			−0.825434	+	0.564498i	$0.809070\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$			26.9733i			0.770666i
$36$		0			0
$37$	−19.4641			−0.526057			−0.263028	−	0.964788i	$-0.584721\pi$
$37$	−19.4641			−0.526057			−0.263028	+	0.964788i	$0.584721\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$			47.9800i			1.17024i	0.810945	+	0.585122i	$0.198953\pi$
$41$			47.9800i			1.17024i	−0.810945	+	0.585122i	$0.801047\pi$
$42$		0			0
$43$		−	81.8429i		−	1.90332i	−0.307147	−	0.951662i	$-0.599374\pi$
$43$		−	81.8429i		−	1.90332i	0.307147	−	0.951662i	$-0.400626\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	30.1962			0.642471			0.321236	−	0.946999i	$-0.395902\pi$
$47$	30.1962			0.642471			0.321236	+	0.946999i	$0.395902\pi$
$48$		0			0
$49$	7.67949			0.156724
$50$		0			0
$51$		0			0
$52$		0			0
$53$	−26.0526			−0.491558			−0.245779	−	0.969326i	$-0.579044\pi$
$53$	−26.0526			−0.491558			−0.245779	+	0.969326i	$0.579044\pi$
$54$		0			0
$55$	13.7128	−	44.0737i	0.249324	−	0.801340i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	82.7461			1.40248			0.701238	−	0.712927i	$-0.252631\pi$
$59$	82.7461			1.40248			0.701238	+	0.712927i	$0.252631\pi$
$60$		0			0
$61$		−	75.4148i		−	1.23631i	−0.786057	−	0.618154i	$-0.787881\pi$
$61$		−	75.4148i		−	1.23631i	0.786057	−	0.618154i	$-0.212119\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$		−	32.2642i		−	0.496372i
$66$		0			0
$67$	34.0000			0.507463			0.253731	−	0.967275i	$-0.418342\pi$
$67$	34.0000			0.507463			0.253731	+	0.967275i	$0.418342\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$	−72.7321			−1.02440			−0.512198	−	0.858868i	$-0.671168\pi$
$71$	−72.7321			−1.02440			−0.512198	+	0.858868i	$0.671168\pi$
$72$		0			0
$73$		−	54.8696i		−	0.751639i	−0.926693	−	0.375819i	$-0.877361\pi$
$73$		−	54.8696i		−	0.751639i	0.926693	−	0.375819i	$-0.122639\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$	67.5167	+	21.0067i	0.876840	+	0.272814i
$78$		0			0
$79$			24.3279i			0.307948i	0.988075	+	0.153974i	$0.0492071\pi$
$79$			24.3279i			0.307948i	−0.988075	+	0.153974i	$0.950793\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$		−	0.923034i		−	0.0111209i	−0.999985	−	0.00556045i	$-0.998230\pi$
$83$		−	0.923034i		−	0.0111209i	0.999985	−	0.00556045i	$-0.00176995\pi$
$84$		0			0
$85$		−	34.2008i		−	0.402362i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	−44.8231			−0.503630			−0.251815	−	0.967775i	$-0.581027\pi$
$89$	−44.8231			−0.503630			−0.251815	+	0.967775i	$0.581027\pi$
$90$		0			0
$91$	49.4256			0.543139
$92$		0			0
$93$		0			0
$94$		0			0
$95$		−	127.639i		−	1.34357i
$96$		0			0
$97$	−21.2539			−0.219112			−0.109556	−	0.993981i	$-0.534943\pi$
$97$	−21.2539			−0.219112			−0.109556	+	0.993981i	$0.534943\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$		−	52.3479i		−	0.518296i	−0.965838	−	0.259148i	$-0.916558\pi$
$101$		−	52.3479i		−	0.518296i	0.965838	−	0.259148i	$-0.0834417\pi$
$102$		0			0
$103$	4.74613			0.0460790			0.0230395	−	0.999735i	$-0.492666\pi$
$103$	4.74613			0.0460790			0.0230395	+	0.999735i	$0.492666\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$			147.723i			1.38059i	0.723530	+	0.690293i	$0.242518\pi$
$107$			147.723i			1.38059i	−0.723530	+	0.690293i	$0.757482\pi$
$108$		0			0
$109$		−	3.56847i		−	0.0327383i	−0.999866	−	0.0163691i	$-0.994789\pi$
$109$		−	3.56847i		−	0.0327383i	0.999866	−	0.0163691i	$-0.00521069\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	62.3154			0.551463			0.275732	−	0.961235i	$-0.411080\pi$
$113$	62.3154			0.551463			0.275732	+	0.961235i	$0.411080\pi$
$114$		0			0
$115$	−132.851			−1.15523
$116$		0			0
$117$		0			0
$118$		0			0
$119$	52.3923			0.440271
$120$		0			0
$121$	−99.6410	−	68.6489i	−0.823479	−	0.567346i
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−135.923			−1.08738
$126$		0			0
$127$		−	28.0200i		−	0.220630i	−0.993897	−	0.110315i	$-0.964814\pi$
$127$		−	28.0200i		−	0.220630i	0.993897	−	0.110315i	$-0.0351860\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$		−	113.184i		−	0.864001i	−0.901873	−	0.432000i	$-0.857808\pi$
$131$		−	113.184i		−	0.864001i	0.901873	−	0.432000i	$-0.142192\pi$
$132$		0			0
$133$	195.531			1.47016
$134$		0			0
$135$		0			0
$136$		0			0
$137$	70.7846			0.516676			0.258338	−	0.966055i	$-0.416825\pi$
$137$	70.7846			0.516676			0.258338	+	0.966055i	$0.416825\pi$
$138$		0			0
$139$		−	130.499i		−	0.938839i	−0.882975	−	0.469420i	$-0.844463\pi$
$139$		−	130.499i		−	0.938839i	0.882975	−	0.469420i	$-0.155537\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	−80.7602	−	25.1272i	−0.564757	−	0.175715i
$144$		0			0
$145$			28.9099i			0.199379i
$146$		0			0
$147$		0			0
$148$		0			0
$149$		−	125.793i		−	0.844248i	−0.906538	−	0.422124i	$-0.861285\pi$
$149$		−	125.793i		−	0.844248i	0.906538	−	0.422124i	$-0.138715\pi$
$150$		0			0
$151$			275.238i			1.82277i	0.411556	+	0.911384i	$0.364985\pi$
$151$			275.238i			1.82277i	−0.411556	+	0.911384i	$0.635015\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	−214.746			−1.38546
$156$		0			0
$157$	−273.138			−1.73974			−0.869868	−	0.493285i	$-0.835796\pi$
$157$	−273.138			−1.73974			−0.869868	+	0.493285i	$0.835796\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		−	203.515i		−	1.26407i
$162$		0			0
$163$	−62.0666			−0.380777			−0.190388	−	0.981709i	$-0.560975\pi$
$163$	−62.0666			−0.380777			−0.190388	+	0.981709i	$0.560975\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$		−	47.9800i		−	0.287305i	−0.989628	−	0.143653i	$-0.954115\pi$
$167$		−	47.9800i		−	0.287305i	0.989628	−	0.143653i	$-0.0458848\pi$
$168$		0			0
$169$	109.879			0.650174
$170$		0			0
$171$		0			0
$172$		0			0
$173$		−	143.017i		−	0.826688i	−0.910575	−	0.413344i	$-0.864361\pi$
$173$		−	143.017i		−	0.826688i	0.910575	−	0.413344i	$-0.135639\pi$
$174$		0			0
$175$		−	47.5185i		−	0.271534i
$176$		0			0
$177$		0			0
$178$		0			0
$179$	36.6025			0.204483			0.102242	−	0.994760i	$-0.467398\pi$
$179$	36.6025			0.204483			0.102242	+	0.994760i	$0.467398\pi$
$180$		0			0
$181$	−84.1718			−0.465037			−0.232519	−	0.972592i	$-0.574697\pi$
$181$	−84.1718			−0.465037			−0.232519	+	0.972592i	$0.574697\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−81.6743			−0.441483
$186$		0			0
$187$	−85.6077	−	26.6354i	−0.457795	−	0.142436i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	44.2346			0.231595			0.115797	−	0.993273i	$-0.463058\pi$
$191$	44.2346			0.231595			0.115797	+	0.993273i	$0.463058\pi$
$192$		0			0
$193$		−	136.127i		−	0.705323i	−0.935751	−	0.352662i	$-0.885277\pi$
$193$		−	136.127i		−	0.705323i	0.935751	−	0.352662i	$-0.114723\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$		−	5.96659i		−	0.0302872i	−0.999885	−	0.0151436i	$-0.995179\pi$
$197$		−	5.96659i		−	0.0302872i	0.999885	−	0.0151436i	$-0.00482055\pi$
$198$		0			0
$199$	234.056			1.17616			0.588081	−	0.808802i	$-0.299884\pi$
$199$	234.056			1.17616			0.588081	+	0.808802i	$0.299884\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$	−44.2872			−0.218163
$204$		0			0
$205$			201.331i			0.982105i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−319.492	−	99.4048i	−1.52867	−	0.475621i
$210$		0			0
$211$		−	167.469i		−	0.793690i	−0.917886	−	0.396845i	$-0.870105\pi$
$211$		−	167.469i		−	0.793690i	0.917886	−	0.396845i	$-0.129895\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$		−	343.425i		−	1.59733i
$216$		0			0
$217$		−	328.970i		−	1.51599i
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−62.6692			−0.283571
$222$		0			0
$223$	−82.2872			−0.369001			−0.184500	−	0.982832i	$-0.559067\pi$
$223$	−82.2872			−0.369001			−0.184500	+	0.982832i	$0.559067\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$			40.9999i			0.180616i	0.995914	+	0.0903081i	$0.0287852\pi$
$227$			40.9999i			0.180616i	−0.995914	+	0.0903081i	$0.971215\pi$
$228$		0			0
$229$	160.718			0.701825			0.350913	−	0.936408i	$-0.385871\pi$
$229$	160.718			0.701825			0.350913	+	0.936408i	$0.385871\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$			407.954i			1.75087i	0.483332	+	0.875437i	$0.339427\pi$
$233$			407.954i			1.75087i	−0.483332	+	0.875437i	$0.660573\pi$
$234$		0			0
$235$	126.708			0.539182
$236$		0			0
$237$		0			0
$238$		0			0
$239$		−	202.254i		−	0.846253i	−0.906071	−	0.423127i	$-0.860933\pi$
$239$		−	202.254i		−	0.846253i	0.906071	−	0.423127i	$-0.139067\pi$
$240$		0			0
$241$		−	26.8828i		−	0.111547i	−0.998443	−	0.0557734i	$-0.982238\pi$
$241$		−	26.8828i		−	0.111547i	0.998443	−	0.0557734i	$-0.0177624\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$	32.2243			0.131528
$246$		0			0
$247$	−233.885			−0.946901
$248$		0			0
$249$		0			0
$250$		0			0
$251$	398.277			1.58676			0.793380	−	0.608726i	$-0.208319\pi$
$251$	398.277			1.58676			0.793380	+	0.608726i	$0.208319\pi$
$252$		0			0
$253$	−103.464	+	332.539i	−0.408949	+	1.31438i
$254$		0			0
$255$		0			0
$256$		0			0
$257$	337.818			1.31447			0.657233	−	0.753687i	$-0.271727\pi$
$257$	337.818			1.31447			0.657233	+	0.753687i	$0.271727\pi$
$258$		0			0
$259$		−	125.117i		−	0.483078i
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	134.529i		−	0.511516i	−0.966741	−	0.255758i	$-0.917675\pi$
$263$		−	134.529i		−	0.511516i	0.966741	−	0.255758i	$-0.0823250\pi$
$264$		0			0
$265$	−109.321			−0.412530
$266$		0			0
$267$		0			0
$268$		0			0
$269$	341.870			1.27089			0.635447	−	0.772144i	$-0.280816\pi$
$269$	341.870			1.27089			0.635447	+	0.772144i	$0.280816\pi$
$270$		0			0
$271$		−	213.298i		−	0.787077i	−0.919308	−	0.393538i	$-0.871251\pi$
$271$		−	213.298i		−	0.787077i	0.919308	−	0.393538i	$-0.128749\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$	−24.1577	+	77.6440i	−0.0878461	+	0.282342i
$276$		0			0
$277$		−	84.5789i		−	0.305339i	−0.988277	−	0.152669i	$-0.951213\pi$
$277$		−	84.5789i		−	0.305339i	0.988277	−	0.152669i	$-0.0487870\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$		−	351.148i		−	1.24964i	−0.780771	−	0.624818i	$-0.785173\pi$
$281$		−	351.148i		−	1.24964i	0.780771	−	0.624818i	$-0.214827\pi$
$282$		0			0
$283$		−	285.943i		−	1.01040i	−0.863002	−	0.505201i	$-0.831419\pi$
$283$		−	285.943i		−	1.01040i	0.863002	−	0.505201i	$-0.168581\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$	−308.420			−1.07464
$288$		0			0
$289$	222.569			0.770136
$290$		0			0
$291$		0			0
$292$		0			0
$293$			393.927i			1.34446i	0.740342	+	0.672231i	$0.234664\pi$
$293$			393.927i			1.34446i	−0.740342	+	0.672231i	$0.765336\pi$
$294$		0			0
$295$	347.215			1.17700
$296$		0			0
$297$		0			0
$298$		0			0
$299$			243.435i			0.814165i
$300$		0			0
$301$	526.095			1.74782
$302$		0			0
$303$		0			0
$304$		0			0
$305$		−	316.452i		−	1.03755i
$306$		0			0
$307$			58.2239i			0.189654i	0.995494	+	0.0948272i	$0.0302299\pi$
$307$			58.2239i			0.189654i	−0.995494	+	0.0948272i	$0.969770\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	207.611			0.667561			0.333780	−	0.942651i	$-0.391676\pi$
$311$	207.611			0.667561			0.333780	+	0.942651i	$0.391676\pi$
$312$		0			0
$313$	−376.697			−1.20351			−0.601753	−	0.798682i	$-0.705531\pi$
$313$	−376.697			−1.20351			−0.601753	+	0.798682i	$0.705531\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−246.809			−0.778577			−0.389289	−	0.921116i	$-0.627279\pi$
$317$	−246.809			−0.778577			−0.389289	+	0.921116i	$0.627279\pi$
$318$		0			0
$319$	72.3641	+	22.5149i	0.226847	+	0.0705797i
$320$		0			0
$321$		0			0
$322$		0			0
$323$	−247.923			−0.767564
$324$		0			0
$325$			56.8394i			0.174890i
$326$		0			0
$327$		0			0
$328$		0			0
$329$			194.104i			0.589982i
$330$		0			0
$331$	−146.431			−0.442389			−0.221195	−	0.975230i	$-0.570996\pi$
$331$	−146.431			−0.442389			−0.221195	+	0.975230i	$0.570996\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	142.669			0.425878
$336$		0			0
$337$		−	354.007i		−	1.05047i	−0.850958	−	0.525233i	$-0.823978\pi$
$337$		−	354.007i		−	1.05047i	0.850958	−	0.525233i	$-0.176022\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	−167.244	+	537.529i	−0.490450	+	1.57633i
$342$		0			0
$343$			364.342i			1.06222i
$344$		0			0
$345$		0			0
$346$		0			0
$347$			529.807i			1.52682i	0.645913	+	0.763411i	$0.276477\pi$
$347$			529.807i			1.52682i	−0.645913	+	0.763411i	$0.723523\pi$
$348$		0			0
$349$			342.873i			0.982445i	0.871034	+	0.491223i	$0.163450\pi$
$349$			342.873i			0.982445i	−0.871034	+	0.491223i	$0.836550\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−191.990			−0.543880			−0.271940	−	0.962314i	$-0.587665\pi$
$353$	−191.990			−0.543880			−0.271940	+	0.962314i	$0.587665\pi$
$354$		0			0
$355$	−305.195			−0.859704
$356$		0			0
$357$		0			0
$358$		0			0
$359$			601.654i			1.67592i	0.545735	+	0.837958i	$0.316250\pi$
$359$			601.654i			1.67592i	−0.545735	+	0.837958i	$0.683750\pi$
$360$		0			0
$361$	−564.261			−1.56305
$362$		0			0
$363$		0			0
$364$		0			0
$365$		−	230.241i		−	0.630798i
$366$		0			0
$367$	139.415			0.379878			0.189939	−	0.981796i	$-0.439171\pi$
$367$	139.415			0.379878			0.189939	+	0.981796i	$0.439171\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		−	167.469i		−	0.451398i
$372$		0			0
$373$			303.629i			0.814019i	0.913424	+	0.407009i	$0.133428\pi$
$373$			303.629i			0.814019i	−0.913424	+	0.407009i	$0.866572\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	52.9742			0.140515
$378$		0			0
$379$	−690.046			−1.82070			−0.910351	−	0.413837i	$-0.864188\pi$
$379$	−690.046			−1.82070			−0.910351	+	0.413837i	$0.864188\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	−53.2576			−0.139054			−0.0695269	−	0.997580i	$-0.522149\pi$
$383$	−53.2576			−0.139054			−0.0695269	+	0.997580i	$0.522149\pi$
$384$		0			0
$385$	283.310	+	88.1474i	0.735871	+	0.228954i
$386$		0			0
$387$		0			0
$388$		0			0
$389$	587.027			1.50907			0.754533	−	0.656262i	$-0.227863\pi$
$389$	587.027			1.50907			0.754533	+	0.656262i	$0.227863\pi$
$390$		0			0
$391$			258.047i			0.659967i
$392$		0			0
$393$		0			0
$394$		0			0
$395$			102.083i			0.258439i
$396$		0			0
$397$	485.797			1.22367			0.611835	−	0.790985i	$-0.290431\pi$
$397$	485.797			1.22367			0.611835	+	0.790985i	$0.290431\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$	166.469			0.415135			0.207568	−	0.978221i	$-0.433445\pi$
$401$	166.469			0.415135			0.207568	+	0.978221i	$0.433445\pi$
$402$		0			0
$403$			393.499i			0.976424i
$404$		0			0
$405$		0			0
$406$		0			0
$407$	−63.6077	+	204.438i	−0.156284	+	0.502306i
$408$		0			0
$409$		−	270.161i		−	0.660541i	−0.943886	−	0.330271i	$-0.892860\pi$
$409$		−	270.161i		−	0.660541i	0.943886	−	0.330271i	$-0.107140\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$			531.901i			1.28790i
$414$		0			0
$415$		−	3.87319i		−	0.00933299i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−356.631			−0.851147			−0.425574	−	0.904924i	$-0.639928\pi$
$419$	−356.631			−0.851147			−0.425574	+	0.904924i	$0.639928\pi$
$420$		0			0
$421$	−462.238			−1.09795			−0.548977	−	0.835838i	$-0.684982\pi$
$421$	−462.238			−1.09795			−0.548977	+	0.835838i	$0.684982\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$			60.2510i			0.141767i
$426$		0			0
$427$	484.774			1.13530
$428$		0			0
$429$		0			0
$430$		0			0
$431$			199.552i			0.462997i	0.972835	+	0.231498i	$0.0743628\pi$
$431$			199.552i			0.462997i	−0.972835	+	0.231498i	$0.925637\pi$
$432$		0			0
$433$	−45.3693			−0.104779			−0.0523895	−	0.998627i	$-0.516684\pi$
$433$	−45.3693			−0.104779			−0.0523895	+	0.998627i	$0.516684\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$			963.045i			2.20376i
$438$		0			0
$439$		−	484.630i		−	1.10394i	−0.833864	−	0.551970i	$-0.813876\pi$
$439$		−	484.630i		−	1.10394i	0.833864	−	0.551970i	$-0.186124\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	375.538			0.847716			0.423858	−	0.905729i	$-0.360676\pi$
$443$	375.538			0.847716			0.423858	+	0.905729i	$0.360676\pi$
$444$		0			0
$445$	−188.084			−0.422662
$446$		0			0
$447$		0			0
$448$		0			0
$449$	385.636			0.858877			0.429439	−	0.903096i	$-0.358711\pi$
$449$	385.636			0.858877			0.429439	+	0.903096i	$0.358711\pi$
$450$		0			0
$451$	503.951	+	156.796i	1.11741	+	0.347664i
$452$		0			0
$453$		0			0
$454$		0			0
$455$	207.397			0.455819
$456$		0			0
$457$		−	378.368i		−	0.827939i	−0.910291	−	0.413970i	$-0.864142\pi$
$457$		−	378.368i		−	0.827939i	0.910291	−	0.413970i	$-0.135858\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$		−	224.522i		−	0.487033i	−0.969897	−	0.243516i	$-0.921699\pi$
$461$		−	224.522i		−	0.487033i	0.969897	−	0.243516i	$-0.0783010\pi$
$462$		0			0
$463$	52.3820			0.113136			0.0565680	−	0.998399i	$-0.481984\pi$
$463$	52.3820			0.113136			0.0565680	+	0.998399i	$0.481984\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	513.387			1.09933			0.549665	−	0.835385i	$-0.314755\pi$
$467$	513.387			1.09933			0.549665	+	0.835385i	$0.314755\pi$
$468$		0			0
$469$			218.556i			0.466003i
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−859.626	−	267.459i	−1.81739	−	0.565451i
$474$		0			0
$475$			224.860i			0.473389i
$476$		0			0
$477$		0			0
$478$		0			0
$479$		−	238.730i		−	0.498392i	−0.968453	−	0.249196i	$-0.919834\pi$
$479$		−	238.730i		−	0.498392i	0.968453	−	0.249196i	$-0.0801663\pi$
$480$		0			0
$481$			149.659i			0.311142i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	−89.1845			−0.183885
$486$		0			0
$487$	−593.251			−1.21817			−0.609087	−	0.793103i	$-0.708464\pi$
$487$	−593.251			−1.21817			−0.609087	+	0.793103i	$0.708464\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$		−	858.935i		−	1.74936i	−0.484703	−	0.874679i	$-0.661072\pi$
$491$		−	858.935i		−	1.74936i	0.484703	−	0.874679i	$-0.338928\pi$
$492$		0			0
$493$	56.1539			0.113902
$494$		0			0
$495$		0			0
$496$		0			0
$497$		−	467.529i		−	0.940702i
$498$		0			0
$499$	803.692			1.61061			0.805303	−	0.592864i	$-0.202003\pi$
$499$	803.692			1.61061			0.805303	+	0.592864i	$0.202003\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$			752.302i			1.49563i	0.663907	+	0.747815i	$0.268897\pi$
$503$			752.302i			1.49563i	−0.663907	+	0.747815i	$0.731103\pi$
$504$		0			0
$505$		−	219.660i		−	0.434969i
$506$		0			0
$507$		0			0
$508$		0			0
$509$	249.268			0.489721			0.244860	−	0.969558i	$-0.421258\pi$
$509$	249.268			0.489721			0.244860	+	0.969558i	$0.421258\pi$
$510$		0			0
$511$	352.708			0.690230
$512$		0			0
$513$		0			0
$514$		0			0
$515$	19.9155			0.0386709
$516$		0			0
$517$	98.6795	−	317.161i	0.190869	−	0.613464i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−475.864			−0.913367			−0.456683	−	0.889629i	$-0.650963\pi$
$521$	−475.864			−0.913367			−0.456683	+	0.889629i	$0.650963\pi$
$522$		0			0
$523$			362.833i			0.693754i	0.937911	+	0.346877i	$0.112758\pi$
$523$			362.833i			0.693754i	−0.937911	+	0.346877i	$0.887242\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$			417.118i			0.791495i
$528$		0			0
$529$	473.372			0.894843
$530$		0			0
$531$		0			0
$532$		0			0
$533$	368.918			0.692154
$534$		0			0
$535$			619.867i			1.15863i
$536$		0			0
$537$		0			0
$538$		0			0
$539$	25.0962	−	80.6604i	0.0465606	−	0.149648i
$540$		0			0
$541$			830.667i			1.53543i	0.640792	+	0.767715i	$0.278606\pi$
$541$			830.667i			1.53543i	−0.640792	+	0.767715i	$0.721394\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$		−	14.9739i		−	0.0274750i
$546$		0			0
$547$			456.091i			0.833804i	0.908952	+	0.416902i	$0.136884\pi$
$547$			456.091i			0.833804i	−0.908952	+	0.416902i	$0.863116\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	209.569			0.380343
$552$		0			0
$553$	−156.382			−0.282788
$554$		0			0
$555$		0			0
$556$		0			0
$557$		−	338.810i		−	0.608277i	−0.952628	−	0.304138i	$-0.901631\pi$
$557$		−	338.810i		−	0.608277i	0.952628	−	0.304138i	$-0.0983686\pi$
$558$		0			0
$559$	−629.290			−1.12574
$560$		0			0
$561$		0			0
$562$		0			0
$563$		−	447.041i		−	0.794034i	−0.917811	−	0.397017i	$-0.870045\pi$
$563$		−	447.041i		−	0.794034i	0.917811	−	0.397017i	$-0.129955\pi$
$564$		0			0
$565$	261.485			0.462805
$566$		0			0
$567$		0			0
$568$		0			0
$569$		−	522.893i		−	0.918969i	−0.888186	−	0.459485i	$-0.848034\pi$
$569$		−	522.893i		−	0.918969i	0.888186	−	0.459485i	$-0.151966\pi$
$570$		0			0
$571$			904.574i			1.58419i	0.610396	+	0.792096i	$0.291010\pi$
$571$			904.574i			1.58419i	−0.610396	+	0.792096i	$0.708990\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$	234.042			0.407030
$576$		0			0
$577$	237.674			0.411914			0.205957	−	0.978561i	$-0.433969\pi$
$577$	237.674			0.411914			0.205957	+	0.978561i	$0.433969\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$	5.93336			0.0102123
$582$		0			0
$583$	−85.1384	+	273.639i	−0.146035	+	0.469364i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	−578.515			−0.985546			−0.492773	−	0.870158i	$-0.664017\pi$
$587$	−578.515			−0.985546			−0.492773	+	0.870158i	$0.664017\pi$
$588$		0			0
$589$			1556.71i			2.64296i
$590$		0			0
$591$		0			0
$592$		0			0
$593$		−	333.857i		−	0.562997i	−0.959562	−	0.281499i	$-0.909169\pi$
$593$		−	333.857i		−	0.562997i	0.959562	−	0.281499i	$-0.0908314\pi$
$594$		0			0
$595$	219.846			0.369489
$596$		0			0
$597$		0			0
$598$		0			0
$599$	−846.483			−1.41316			−0.706580	−	0.707633i	$-0.749763\pi$
$599$	−846.483			−1.41316			−0.706580	+	0.707633i	$0.749763\pi$
$600$		0			0
$601$			455.011i			0.757089i	0.925583	+	0.378545i	$0.123575\pi$
$601$			455.011i			0.757089i	−0.925583	+	0.378545i	$0.876425\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−418.109	−	288.061i	−0.691089	−	0.476134i
$606$		0			0
$607$		−	726.557i		−	1.19696i	−0.801137	−	0.598482i	$-0.795771\pi$
$607$		−	726.557i		−	1.19696i	0.801137	−	0.598482i	$-0.204229\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$		−	232.178i		−	0.379997i
$612$		0			0
$613$			574.532i			0.937247i	0.883398	+	0.468624i	$0.155250\pi$
$613$			574.532i			0.937247i	−0.883398	+	0.468624i	$0.844750\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	278.946			0.452101			0.226050	−	0.974116i	$-0.427419\pi$
$617$	278.946			0.452101			0.226050	+	0.974116i	$0.427419\pi$
$618$		0			0
$619$	593.061			0.958096			0.479048	−	0.877789i	$-0.340982\pi$
$619$	593.061			0.958096			0.479048	+	0.877789i	$0.340982\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		−	288.127i		−	0.462484i
$624$		0			0
$625$	−385.546			−0.616874
$626$		0			0
$627$		0			0
$628$		0			0
$629$			158.642i			0.252214i
$630$		0			0
$631$	361.664			0.573160			0.286580	−	0.958056i	$-0.407482\pi$
$631$	361.664			0.573160			0.286580	+	0.958056i	$0.407482\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$		−	117.576i		−	0.185159i
$636$		0			0
$637$		−	59.0475i		−	0.0926963i
$638$		0			0
$639$		0			0
$640$		0			0
$641$	−784.382			−1.22368			−0.611842	−	0.790980i	$-0.709571\pi$
$641$	−784.382			−1.22368			−0.611842	+	0.790980i	$0.709571\pi$
$642$		0			0
$643$	−362.764			−0.564174			−0.282087	−	0.959389i	$-0.591027\pi$
$643$	−362.764			−0.564174			−0.282087	+	0.959389i	$0.591027\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	921.727			1.42462			0.712308	−	0.701867i	$-0.247650\pi$
$647$	921.727			1.42462			0.712308	+	0.701867i	$0.247650\pi$
$648$		0			0
$649$	270.410	−	869.112i	0.416657	−	1.33916i
$650$		0			0
$651$		0			0
$652$		0			0
$653$	−26.8653			−0.0411414			−0.0205707	−	0.999788i	$-0.506548\pi$
$653$	−26.8653			−0.0411414			−0.0205707	+	0.999788i	$0.506548\pi$
$654$		0			0
$655$		−	474.938i		−	0.725096i
$656$		0			0
$657$		0			0
$658$		0			0
$659$		−	320.820i		−	0.486828i	−0.969922	−	0.243414i	$-0.921733\pi$
$659$		−	320.820i		−	0.486828i	0.969922	−	0.243414i	$-0.0782674\pi$
$660$		0			0
$661$	331.969			0.502222			0.251111	−	0.967958i	$-0.419204\pi$
$661$	331.969			0.502222			0.251111	+	0.967958i	$0.419204\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$	820.477			1.23380
$666$		0			0
$667$		−	218.127i		−	0.327027i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−792.109	−	246.452i	−1.18049	−	0.367290i
$672$		0			0
$673$			797.019i			1.18428i	0.805836	+	0.592139i	$0.201716\pi$
$673$			797.019i			1.18428i	−0.805836	+	0.592139i	$0.798284\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$			534.175i			0.789033i	0.918889	+	0.394516i	$0.129088\pi$
$677$			534.175i			0.789033i	−0.918889	+	0.394516i	$0.870912\pi$
$678$		0			0
$679$		−	136.622i		−	0.201211i
$680$		0			0
$681$		0			0
$682$		0			0
$683$	−843.097			−1.23440			−0.617201	−	0.786805i	$-0.711734\pi$
$683$	−843.097			−1.23440			−0.617201	+	0.786805i	$0.711734\pi$
$684$		0			0
$685$	297.023			0.433610
$686$		0			0
$687$		0			0
$688$		0			0
$689$			200.318i			0.290737i
$690$		0			0
$691$	193.615			0.280196			0.140098	−	0.990138i	$-0.455258\pi$
$691$	193.615			0.280196			0.140098	+	0.990138i	$0.455258\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$		−	547.592i		−	0.787903i
$696$		0			0
$697$	391.061			0.561064
$698$		0			0
$699$		0			0
$700$		0			0
$701$			448.863i			0.640318i	0.947364	+	0.320159i	$0.103736\pi$
$701$			448.863i			0.640318i	−0.947364	+	0.320159i	$0.896264\pi$
$702$		0			0
$703$			592.061i			0.842192i
$704$		0			0
$705$		0			0
$706$		0			0
$707$	336.497			0.475951
$708$		0			0
$709$	−311.031			−0.438689			−0.219345	−	0.975647i	$-0.570392\pi$
$709$	−311.031			−0.438689			−0.219345	+	0.975647i	$0.570392\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	1620.27			2.27247
$714$		0			0
$715$	−338.882	−	105.438i	−0.473961	−	0.147465i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	486.014			0.675958			0.337979	−	0.941154i	$-0.390257\pi$
$719$	486.014			0.675958			0.337979	+	0.941154i	$0.390257\pi$
$720$		0			0
$721$			30.5086i			0.0423143i
$722$		0			0
$723$		0			0
$724$		0			0
$725$		−	50.9302i		−	0.0702485i
$726$		0			0
$727$	−38.8616			−0.0534547			−0.0267273	−	0.999643i	$-0.508509\pi$
$727$	−38.8616			−0.0534547			−0.0267273	+	0.999643i	$0.508509\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	−667.061			−0.912533
$732$		0			0
$733$			175.562i			0.239511i	0.992803	+	0.119756i	$0.0382111\pi$
$733$			175.562i			0.239511i	−0.992803	+	0.119756i	$0.961789\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	111.110	−	357.114i	0.150760	−	0.484551i
$738$		0			0
$739$			494.527i			0.669184i	0.942363	+	0.334592i	$0.108598\pi$
$739$			494.527i			0.669184i	−0.942363	+	0.334592i	$0.891402\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$			1150.84i			1.54892i	0.632625	+	0.774458i	$0.281977\pi$
$743$			1150.84i			1.54892i	−0.632625	+	0.774458i	$0.718023\pi$
$744$		0			0
$745$		−	527.846i		−	0.708519i
$746$		0			0
$747$		0			0
$748$		0			0
$749$	−949.577			−1.26779
$750$		0			0
$751$	−1343.73			−1.78926			−0.894628	−	0.446813i	$-0.852559\pi$
$751$	−1343.73			−1.78926			−0.894628	+	0.446813i	$0.852559\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$			1154.94i			1.52972i
$756$		0			0
$757$	1034.58			1.36669			0.683343	−	0.730097i	$-0.260525\pi$
$757$	1034.58			1.36669			0.683343	+	0.730097i	$0.260525\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$			798.527i			1.04931i	0.851314	+	0.524656i	$0.175806\pi$
$761$			798.527i			1.04931i	−0.851314	+	0.524656i	$0.824194\pi$
$762$		0			0
$763$	22.9385			0.0300636
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	636.234i		−	0.829510i
$768$		0			0
$769$		−	261.311i		−	0.339806i	−0.985461	−	0.169903i	$-0.945655\pi$
$769$		−	261.311i		−	0.339806i	0.985461	−	0.169903i	$-0.0543455\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	327.734			0.423977			0.211989	−	0.977272i	$-0.432006\pi$
$773$	327.734			0.423977			0.211989	+	0.977272i	$0.432006\pi$
$774$		0			0
$775$	378.315			0.488149
$776$		0			0
$777$		0			0
$778$		0			0
$779$	1459.46			1.87351
$780$		0			0
$781$	−237.685	+	763.931i	−0.304334	+	0.978144i
$782$		0			0
$783$		0			0
$784$		0			0
$785$	−1146.13			−1.46004
$786$		0			0
$787$			289.388i			0.367711i	0.982953	+	0.183855i	$0.0588578\pi$
$787$			289.388i			0.367711i	−0.982953	+	0.183855i	$0.941142\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$			400.570i			0.506409i
$792$		0			0
$793$	−579.864			−0.731228
$794$		0			0
$795$		0			0
$796$		0			0
$797$	−113.681			−0.142636			−0.0713180	−	0.997454i	$-0.522721\pi$
$797$	−113.681			−0.142636			−0.0713180	+	0.997454i	$0.522721\pi$
$798$		0			0
$799$		−	246.114i		−	0.308028i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−576.315	−	179.311i	−0.717703	−	0.223302i
$804$		0			0
$805$		−	853.982i		−	1.06085i
$806$		0			0
$807$		0			0
$808$		0			0
$809$			854.163i			1.05583i	0.849299	+	0.527913i	$0.177025\pi$
$809$			854.163i			1.05583i	−0.849299	+	0.527913i	$0.822975\pi$
$810$		0			0
$811$		−	486.533i		−	0.599917i	−0.953952	−	0.299959i	$-0.903027\pi$
$811$		−	486.533i		−	0.599917i	0.953952	−	0.299959i	$-0.0969729\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	−260.441			−0.319560
$816$		0			0
$817$	−2489.51			−3.04713
$818$		0			0
$819$		0			0
$820$		0			0
$821$		−	527.104i		−	0.642027i	−0.947075	−	0.321014i	$-0.895976\pi$
$821$		−	527.104i		−	0.642027i	0.947075	−	0.321014i	$-0.104024\pi$
$822$		0			0
$823$	314.805			0.382509			0.191255	−	0.981540i	$-0.438744\pi$
$823$	314.805			0.382509			0.191255	+	0.981540i	$0.438744\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$		−	111.652i		−	0.135008i	−0.997719	−	0.0675040i	$-0.978496\pi$
$827$		−	111.652i		−	0.135008i	0.997719	−	0.0675040i	$-0.0215035\pi$
$828$		0			0
$829$	−695.395			−0.838836			−0.419418	−	0.907793i	$-0.637766\pi$
$829$	−695.395			−0.838836			−0.419418	+	0.907793i	$0.637766\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		−	62.5918i		−	0.0751402i
$834$		0			0
$835$		−	201.331i		−	0.241116i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	−383.968			−0.457650			−0.228825	−	0.973468i	$-0.573488\pi$
$839$	−383.968			−0.457650			−0.228825	+	0.973468i	$0.573488\pi$
$840$		0			0
$841$	793.533			0.943559
$842$		0			0
$843$		0			0
$844$		0			0
$845$	461.071			0.545646
$846$		0			0
$847$	441.282	−	640.503i	0.520994	−	0.756202i
$848$		0			0
$849$		0			0
$850$		0			0
$851$	616.238			0.724134
$852$		0			0
$853$		−	570.527i		−	0.668847i	−0.942423	−	0.334424i	$-0.891458\pi$
$853$		−	570.527i		−	0.668847i	0.942423	−	0.334424i	$-0.108542\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$			315.553i			0.368207i	0.982907	+	0.184103i	$0.0589382\pi$
$857$			315.553i			0.368207i	−0.982907	+	0.184103i	$0.941062\pi$
$858$		0			0
$859$	107.923			0.125638			0.0628190	−	0.998025i	$-0.479991\pi$
$859$	107.923			0.125638			0.0628190	+	0.998025i	$0.479991\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	1431.55			1.65881			0.829403	−	0.558651i	$-0.188681\pi$
$863$	1431.55			1.65881			0.829403	+	0.558651i	$0.188681\pi$
$864$		0			0
$865$		−	600.121i		−	0.693782i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	255.524	+	79.5022i	0.294044	+	0.0914870i
$870$		0			0
$871$		−	261.426i		−	0.300144i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		−	873.727i		−	0.998546i
$876$		0			0
$877$			567.758i			0.647386i	0.946162	+	0.323693i	$0.104925\pi$
$877$			567.758i			0.647386i	−0.946162	+	0.323693i	$0.895075\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	−488.641			−0.554644			−0.277322	−	0.960777i	$-0.589447\pi$
$881$	−488.641			−0.554644			−0.277322	+	0.960777i	$0.589447\pi$
$882$		0			0
$883$	−840.144			−0.951465			−0.475732	−	0.879590i	$-0.657817\pi$
$883$	−840.144			−0.951465			−0.475732	+	0.879590i	$0.657817\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$			981.439i			1.10647i	0.833025	+	0.553235i	$0.186607\pi$
$887$			981.439i			1.10647i	−0.833025	+	0.553235i	$0.813393\pi$
$888$		0			0
$889$	180.115			0.202605
$890$		0			0
$891$		0			0
$892$		0			0
$893$		−	918.510i		−	1.02857i
$894$		0			0
$895$	153.590			0.171609
$896$		0			0
$897$		0			0
$898$		0			0
$899$		−	352.589i		−	0.392202i
$900$		0			0
$901$			212.342i			0.235673i
$902$		0			0
$903$		0			0
$904$		0			0
$905$	−353.198			−0.390274
$906$		0			0
$907$	−438.610			−0.483583			−0.241792	−	0.970328i	$-0.577735\pi$
$907$	−438.610			−0.483583			−0.241792	+	0.970328i	$0.577735\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	−170.042			−0.186654			−0.0933272	−	0.995635i	$-0.529750\pi$
$911$	−170.042			−0.186654			−0.0933272	+	0.995635i	$0.529750\pi$
$912$		0			0
$913$	−9.69496	−	3.01643i	−0.0106188	−	0.00330386i
$914$		0			0
$915$		0			0
$916$		0			0
$917$	727.559			0.793412
$918$		0			0
$919$		−	1089.73i		−	1.18578i	−0.805285	−	0.592888i	$-0.797988\pi$
$919$		−	1089.73i		−	1.18578i	0.805285	−	0.592888i	$-0.202012\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$			559.236i			0.605890i
$924$		0			0
$925$	143.885			0.155551
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−769.446			−0.828252			−0.414126	−	0.910220i	$-0.635913\pi$
$929$	−769.446			−0.828252			−0.414126	+	0.910220i	$0.635913\pi$
$930$		0			0
$931$		−	233.596i		−	0.250908i
$932$		0			0
$933$		0			0
$934$		0			0
$935$	−359.223	−	111.766i	−0.384196	−	0.119536i
$936$		0			0
$937$		−	1606.78i		−	1.71481i	−0.514641	−	0.857406i	$-0.672075\pi$
$937$		−	1606.78i		−	1.71481i	0.514641	−	0.857406i	$-0.327925\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$		−	1279.34i		−	1.35955i	−0.733419	−	0.679777i	$-0.762077\pi$
$941$		−	1279.34i		−	1.35955i	0.733419	−	0.679777i	$-0.237923\pi$
$942$		0			0
$943$		−	1519.06i		−	1.61088i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−985.800			−1.04097			−0.520486	−	0.853870i	$-0.674249\pi$
$947$	−985.800			−1.04097			−0.520486	+	0.853870i	$0.674249\pi$
$948$		0			0
$949$	−421.892			−0.444565
$950$		0			0
$951$		0			0
$952$		0			0
$953$			761.038i			0.798571i	0.916827	+	0.399285i	$0.130742\pi$
$953$			761.038i			0.798571i	−0.916827	+	0.399285i	$0.869258\pi$
$954$		0			0
$955$	185.615			0.194362
$956$		0			0
$957$		0			0
$958$		0			0
$959$			455.011i			0.474464i
$960$		0			0
$961$	1658.08			1.72537
$962$		0			0
$963$		0			0
$964$		0			0
$965$		−	571.211i		−	0.591929i
$966$		0			0
$967$			702.938i			0.726926i	0.931609	+	0.363463i	$0.118406\pi$
$967$			702.938i			0.726926i	−0.931609	+	0.363463i	$0.881594\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1289.42			−1.32793			−0.663963	−	0.747766i	$-0.731127\pi$
$971$	−1289.42			−1.32793			−0.663963	+	0.747766i	$0.731127\pi$
$972$		0			0
$973$	838.859			0.862136
$974$		0			0
$975$		0			0
$976$		0			0
$977$	690.526			0.706782			0.353391	−	0.935476i	$-0.385029\pi$
$977$	690.526			0.706782			0.353391	+	0.935476i	$0.385029\pi$
$978$		0			0
$979$	−146.480	+	470.793i	−0.149622	+	0.480892i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	1080.40			1.09908			0.549540	−	0.835467i	$-0.314803\pi$
$983$	1080.40			1.09908			0.549540	+	0.835467i	$0.314803\pi$
$984$		0			0
$985$		−	25.0367i		−	0.0254180i
$986$		0			0
$987$		0			0
$988$		0			0
$989$			2591.17i			2.61999i
$990$		0			0
$991$	686.561			0.692796			0.346398	−	0.938088i	$-0.387405\pi$
$991$	686.561			0.692796			0.346398	+	0.938088i	$0.387405\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	982.136			0.987071
$996$		0			0
$997$		−	1413.50i		−	1.41775i	−0.705333	−	0.708876i	$-0.749203\pi$
$997$		−	1413.50i		−	1.41775i	0.705333	−	0.708876i	$-0.250797\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1584.3.j.f.1297.4		4
3.2	odd	2		528.3.j.c.241.2		4
4.3	odd	2		99.3.c.b.10.2		4
11.10	odd	2	inner	1584.3.j.f.1297.3		4
12.11	even	2		33.3.c.a.10.3	yes	4
24.5	odd	2		2112.3.j.d.769.4		4
24.11	even	2		2112.3.j.a.769.1		4
33.32	even	2		528.3.j.c.241.1		4
44.43	even	2		99.3.c.b.10.3		4
60.23	odd	4		825.3.h.a.274.6		8
60.47	odd	4		825.3.h.a.274.3		8
60.59	even	2		825.3.b.a.76.2		4
132.35	odd	10		363.3.g.e.40.2		16
132.47	even	10		363.3.g.e.112.3		16
132.59	even	10		363.3.g.e.94.2		16
132.71	even	10		363.3.g.e.118.2		16
132.83	odd	10		363.3.g.e.118.3		16
132.95	odd	10		363.3.g.e.94.3		16
132.107	odd	10		363.3.g.e.112.2		16
132.119	even	10		363.3.g.e.40.3		16
132.131	odd	2		33.3.c.a.10.2	✓	4
264.131	odd	2		2112.3.j.a.769.2		4
264.197	even	2		2112.3.j.d.769.3		4
660.263	even	4		825.3.h.a.274.4		8
660.527	even	4		825.3.h.a.274.5		8
660.659	odd	2		825.3.b.a.76.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
33.3.c.a.10.2	✓	4	132.131	odd	2
33.3.c.a.10.3	yes	4	12.11	even	2
99.3.c.b.10.2		4	4.3	odd	2
99.3.c.b.10.3		4	44.43	even	2
363.3.g.e.40.2		16	132.35	odd	10
363.3.g.e.40.3		16	132.119	even	10
363.3.g.e.94.2		16	132.59	even	10
363.3.g.e.94.3		16	132.95	odd	10
363.3.g.e.112.2		16	132.107	odd	10
363.3.g.e.112.3		16	132.47	even	10
363.3.g.e.118.2		16	132.71	even	10
363.3.g.e.118.3		16	132.83	odd	10
528.3.j.c.241.1		4	33.32	even	2
528.3.j.c.241.2		4	3.2	odd	2
825.3.b.a.76.2		4	60.59	even	2
825.3.b.a.76.3		4	660.659	odd	2
825.3.h.a.274.3		8	60.47	odd	4
825.3.h.a.274.4		8	660.263	even	4
825.3.h.a.274.5		8	660.527	even	4
825.3.h.a.274.6		8	60.23	odd	4
1584.3.j.f.1297.3		4	11.10	odd	2	inner
1584.3.j.f.1297.4		4	1.1	even	1	trivial
2112.3.j.a.769.1		4	24.11	even	2
2112.3.j.a.769.2		4	264.131	odd	2
2112.3.j.d.769.3		4	264.197	even	2
2112.3.j.d.769.4		4	24.5	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	1584.j (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+4.19615 q^{5} +6.42810i q^{7} +(3.26795 - 10.5034i) q^{11} -7.68899i q^{13} -8.15051i q^{17} -30.4181i q^{19} -31.6603 q^{23} -7.39230 q^{25} +6.88962i q^{29} -51.1769 q^{31} +26.9733i q^{35} -19.4641 q^{37} +47.9800i q^{41} -81.8429i q^{43} +30.1962 q^{47} +7.67949 q^{49} -26.0526 q^{53} +(13.7128 - 44.0737i) q^{55} +82.7461 q^{59} -75.4148i q^{61} -32.2642i q^{65} +34.0000 q^{67} -72.7321 q^{71} -54.8696i q^{73} +(67.5167 + 21.0067i) q^{77} +24.3279i q^{79} -0.923034i q^{83} -34.2008i q^{85} -44.8231 q^{89} +49.4256 q^{91} -127.639i q^{95} -21.2539 q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 4 q^{5} + 20 q^{11} - 92 q^{23} + 12 q^{25} - 80 q^{31} - 64 q^{37} + 100 q^{47} + 100 q^{49} - 28 q^{53} - 56 q^{55} + 40 q^{59} + 136 q^{67} - 284 q^{71} + 180 q^{77} - 304 q^{89} - 24 q^{91} - 376 q^{97}+O(q^{100}) \) 4 * q - 4 * q^5 + 20 * q^11 - 92 * q^23 + 12 * q^25 - 80 * q^31 - 64 * q^37 + 100 * q^47 + 100 * q^49 - 28 * q^53 - 56 * q^55 + 40 * q^59 + 136 * q^67 - 284 * q^71 + 180 * q^77 - 304 * q^89 - 24 * q^91 - 376 * q^97

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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