LMFDB - Embedded newform 1584.3.j.f.1297.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1584,3,Mod(1297,1584)]

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

N = Newforms(chi, 3, names="a")

//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);

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

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

$q$-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	$-0.848154\pi$
$7$		−	6.42810i		−	0.918300i	0.888359	−	0.459150i	$-0.151846\pi$
$8$		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	$0.0955630\pi$
$13$			7.68899i			0.591461i	−0.955271	+	0.295730i	$0.904437\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$			8.15051i			0.479442i	0.970842	+	0.239721i	$0.0770559\pi$
$17$			8.15051i			0.479442i	−0.970842	+	0.239721i	$0.922944\pi$
$18$		0			0
$19$			30.4181i			1.60095i	0.599364	+	0.800477i	$0.295420\pi$
$19$			30.4181i			1.60095i	−0.599364	+	0.800477i	$0.704580\pi$
$20$		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	$-0.962100\pi$
$29$		−	6.88962i		−	0.237573i	0.992920	−	0.118787i	$-0.0379004\pi$
$30$		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	$-0.801047\pi$
$41$		−	47.9800i		−	1.17024i	0.810945	−	0.585122i	$-0.198953\pi$
$42$		0			0
$43$			81.8429i			1.90332i	0.307147	+	0.951662i	$0.400626\pi$
$43$			81.8429i			1.90332i	−0.307147	+	0.951662i	$0.599374\pi$
$44$		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	$0.212119\pi$
$61$			75.4148i			1.23631i	−0.786057	+	0.618154i	$0.787881\pi$
$62$		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	$0.122639\pi$
$73$			54.8696i			0.751639i	−0.926693	+	0.375819i	$0.877361\pi$
$74$		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	$-0.950793\pi$
$79$		−	24.3279i		−	0.307948i	0.988075	−	0.153974i	$-0.0492071\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$			0.923034i			0.0111209i	0.999985	+	0.00556045i	$0.00176995\pi$
$83$			0.923034i			0.0111209i	−0.999985	+	0.00556045i	$0.998230\pi$
$84$		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	$0.0834417\pi$
$101$			52.3479i			0.518296i	−0.965838	+	0.259148i	$0.916558\pi$
$102$		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	$-0.757482\pi$
$107$		−	147.723i		−	1.38059i	0.723530	−	0.690293i	$-0.242518\pi$
$108$		0			0
$109$			3.56847i			0.0327383i	0.999866	+	0.0163691i	$0.00521069\pi$
$109$			3.56847i			0.0327383i	−0.999866	+	0.0163691i	$0.994789\pi$
$110$		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	$0.0351860\pi$
$127$			28.0200i			0.220630i	−0.993897	+	0.110315i	$0.964814\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$			113.184i			0.864001i	0.901873	+	0.432000i	$0.142192\pi$
$131$			113.184i			0.864001i	−0.901873	+	0.432000i	$0.857808\pi$
$132$		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	$0.155537\pi$
$139$			130.499i			0.938839i	−0.882975	+	0.469420i	$0.844463\pi$
$140$		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	$0.138715\pi$
$149$			125.793i			0.844248i	−0.906538	+	0.422124i	$0.861285\pi$
$150$		0			0
$151$		−	275.238i		−	1.82277i	−0.411556	−	0.911384i	$-0.635015\pi$
$151$		−	275.238i		−	1.82277i	0.411556	−	0.911384i	$-0.364985\pi$
$152$		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	$0.0458848\pi$
$167$			47.9800i			0.287305i	−0.989628	+	0.143653i	$0.954115\pi$
$168$		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	$0.135639\pi$
$173$			143.017i			0.826688i	−0.910575	+	0.413344i	$0.864361\pi$
$174$		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	$0.114723\pi$
$193$			136.127i			0.705323i	−0.935751	+	0.352662i	$0.885277\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			5.96659i			0.0302872i	0.999885	+	0.0151436i	$0.00482055\pi$
$197$			5.96659i			0.0302872i	−0.999885	+	0.0151436i	$0.995179\pi$
$198$		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	$0.129895\pi$
$211$			167.469i			0.793690i	−0.917886	+	0.396845i	$0.870105\pi$
$212$		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	$-0.971215\pi$
$227$		−	40.9999i		−	0.180616i	0.995914	−	0.0903081i	$-0.0287852\pi$
$228$		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	$-0.660573\pi$
$233$		−	407.954i		−	1.75087i	0.483332	−	0.875437i	$-0.339427\pi$
$234$		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	$0.139067\pi$
$239$			202.254i			0.846253i	−0.906071	+	0.423127i	$0.860933\pi$
$240$		0			0
$241$			26.8828i			0.111547i	0.998443	+	0.0557734i	$0.0177624\pi$
$241$			26.8828i			0.111547i	−0.998443	+	0.0557734i	$0.982238\pi$
$242$		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	$0.0823250\pi$
$263$			134.529i			0.511516i	−0.966741	+	0.255758i	$0.917675\pi$
$264$		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	$0.128749\pi$
$271$			213.298i			0.787077i	−0.919308	+	0.393538i	$0.871251\pi$
$272$		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	$0.0487870\pi$
$277$			84.5789i			0.305339i	−0.988277	+	0.152669i	$0.951213\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$			351.148i			1.24964i	0.780771	+	0.624818i	$0.214827\pi$
$281$			351.148i			1.24964i	−0.780771	+	0.624818i	$0.785173\pi$
$282$		0			0
$283$			285.943i			1.01040i	0.863002	+	0.505201i	$0.168581\pi$
$283$			285.943i			1.01040i	−0.863002	+	0.505201i	$0.831419\pi$
$284$		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	$-0.765336\pi$
$293$		−	393.927i		−	1.34446i	0.740342	−	0.672231i	$-0.234664\pi$
$294$		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	$-0.969770\pi$
$307$		−	58.2239i		−	0.189654i	0.995494	−	0.0948272i	$-0.0302299\pi$
$308$		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	$0.176022\pi$
$337$			354.007i			1.05047i	−0.850958	+	0.525233i	$0.823978\pi$
$338$		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	$-0.723523\pi$
$347$		−	529.807i		−	1.52682i	0.645913	−	0.763411i	$-0.276477\pi$
$348$		0			0
$349$		−	342.873i		−	0.982445i	−0.871034	−	0.491223i	$-0.836550\pi$
$349$		−	342.873i		−	0.982445i	0.871034	−	0.491223i	$-0.163450\pi$
$350$		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	$-0.683750\pi$
$359$		−	601.654i		−	1.67592i	0.545735	−	0.837958i	$-0.316250\pi$
$360$		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	$-0.866572\pi$
$373$		−	303.629i		−	0.814019i	0.913424	−	0.407009i	$-0.133428\pi$
$374$		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	$0.107140\pi$
$409$			270.161i			0.660541i	−0.943886	+	0.330271i	$0.892860\pi$
$410$		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	$-0.925637\pi$
$431$		−	199.552i		−	0.462997i	0.972835	−	0.231498i	$-0.0743628\pi$
$432$		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	$0.186124\pi$
$439$			484.630i			1.10394i	−0.833864	+	0.551970i	$0.813876\pi$
$440$		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	$0.135858\pi$
$457$			378.368i			0.827939i	−0.910291	+	0.413970i	$0.864142\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$			224.522i			0.487033i	0.969897	+	0.243516i	$0.0783010\pi$
$461$			224.522i			0.487033i	−0.969897	+	0.243516i	$0.921699\pi$
$462$		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	$0.0801663\pi$
$479$			238.730i			0.498392i	−0.968453	+	0.249196i	$0.919834\pi$
$480$		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	$0.338928\pi$
$491$			858.935i			1.74936i	−0.484703	+	0.874679i	$0.661072\pi$
$492$		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	$-0.731103\pi$
$503$		−	752.302i		−	1.49563i	0.663907	−	0.747815i	$-0.268897\pi$
$504$		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	$-0.887242\pi$
$523$		−	362.833i		−	0.693754i	0.937911	−	0.346877i	$-0.112758\pi$
$524$		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	$-0.721394\pi$
$541$		−	830.667i		−	1.53543i	0.640792	−	0.767715i	$-0.278606\pi$
$542$		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	$-0.863116\pi$
$547$		−	456.091i		−	0.833804i	0.908952	−	0.416902i	$-0.136884\pi$
$548$		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	$0.0983686\pi$
$557$			338.810i			0.608277i	−0.952628	+	0.304138i	$0.901631\pi$
$558$		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	$0.129955\pi$
$563$			447.041i			0.794034i	−0.917811	+	0.397017i	$0.870045\pi$
$564$		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	$0.151966\pi$
$569$			522.893i			0.918969i	−0.888186	+	0.459485i	$0.848034\pi$
$570$		0			0
$571$		−	904.574i		−	1.58419i	−0.610396	−	0.792096i	$-0.708990\pi$
$571$		−	904.574i		−	1.58419i	0.610396	−	0.792096i	$-0.291010\pi$
$572$		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	$0.0908314\pi$
$593$			333.857i			0.562997i	−0.959562	+	0.281499i	$0.909169\pi$
$594$		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	$-0.876425\pi$
$601$		−	455.011i		−	0.757089i	0.925583	−	0.378545i	$-0.123575\pi$
$602$		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	$0.204229\pi$
$607$			726.557i			1.19696i	−0.801137	+	0.598482i	$0.795771\pi$
$608$		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	$-0.844750\pi$
$613$		−	574.532i		−	0.937247i	0.883398	−	0.468624i	$-0.155250\pi$
$614$		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	$0.0782674\pi$
$659$			320.820i			0.486828i	−0.969922	+	0.243414i	$0.921733\pi$
$660$		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	$-0.798284\pi$
$673$		−	797.019i		−	1.18428i	0.805836	−	0.592139i	$-0.201716\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$		−	534.175i		−	0.789033i	−0.918889	−	0.394516i	$-0.870912\pi$
$677$		−	534.175i		−	0.789033i	0.918889	−	0.394516i	$-0.129088\pi$
$678$		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	$-0.896264\pi$
$701$		−	448.863i		−	0.640318i	0.947364	−	0.320159i	$-0.103736\pi$
$702$		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	$-0.961789\pi$
$733$		−	175.562i		−	0.239511i	0.992803	−	0.119756i	$-0.0382111\pi$
$734$		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	$-0.891402\pi$
$739$		−	494.527i		−	0.669184i	0.942363	−	0.334592i	$-0.108598\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$		−	1150.84i		−	1.54892i	−0.632625	−	0.774458i	$-0.718023\pi$
$743$		−	1150.84i		−	1.54892i	0.632625	−	0.774458i	$-0.281977\pi$
$744$		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	$-0.824194\pi$
$761$		−	798.527i		−	1.04931i	0.851314	−	0.524656i	$-0.175806\pi$
$762$		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	$0.0543455\pi$
$769$			261.311i			0.339806i	−0.985461	+	0.169903i	$0.945655\pi$
$770$		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	$-0.941142\pi$
$787$		−	289.388i		−	0.367711i	0.982953	−	0.183855i	$-0.0588578\pi$
$788$		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	$-0.822975\pi$
$809$		−	854.163i		−	1.05583i	0.849299	−	0.527913i	$-0.177025\pi$
$810$		0			0
$811$			486.533i			0.599917i	0.953952	+	0.299959i	$0.0969729\pi$
$811$			486.533i			0.599917i	−0.953952	+	0.299959i	$0.903027\pi$
$812$		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	$0.104024\pi$
$821$			527.104i			0.642027i	−0.947075	+	0.321014i	$0.895976\pi$
$822$		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	$0.0215035\pi$
$827$			111.652i			0.135008i	−0.997719	+	0.0675040i	$0.978496\pi$
$828$		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	$0.108542\pi$
$853$			570.527i			0.668847i	−0.942423	+	0.334424i	$0.891458\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$		−	315.553i		−	0.368207i	−0.982907	−	0.184103i	$-0.941062\pi$
$857$		−	315.553i		−	0.368207i	0.982907	−	0.184103i	$-0.0589382\pi$
$858$		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	$-0.895075\pi$
$877$		−	567.758i		−	0.647386i	0.946162	−	0.323693i	$-0.104925\pi$
$878$		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	$-0.813393\pi$
$887$		−	981.439i		−	1.10647i	0.833025	−	0.553235i	$-0.186607\pi$
$888$		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	$0.202012\pi$
$919$			1089.73i			1.18578i	−0.805285	+	0.592888i	$0.797988\pi$
$920$		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	$0.327925\pi$
$937$			1606.78i			1.71481i	−0.514641	+	0.857406i	$0.672075\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$			1279.34i			1.35955i	0.733419	+	0.679777i	$0.237923\pi$
$941$			1279.34i			1.35955i	−0.733419	+	0.679777i	$0.762077\pi$
$942$		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	$-0.869258\pi$
$953$		−	761.038i		−	0.798571i	0.916827	−	0.399285i	$-0.130742\pi$
$954$		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	$-0.881594\pi$
$967$		−	702.938i		−	0.726926i	0.931609	−	0.363463i	$-0.118406\pi$
$968$		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	$0.250797\pi$
$997$			1413.50i			1.41775i	−0.705333	+	0.708876i	$0.749203\pi$
$998$		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.1297.3		4
3.2	odd	2		528.3.j.c.241.1		4
4.3	odd	2		99.3.c.b.10.3		4
11.10	odd	2	inner	1584.3.j.f.1297.4		4
12.11	even	2		33.3.c.a.10.2	✓	4
24.5	odd	2		2112.3.j.d.769.3		4
24.11	even	2		2112.3.j.a.769.2		4
33.32	even	2		528.3.j.c.241.2		4
44.43	even	2		99.3.c.b.10.2		4
60.23	odd	4		825.3.h.a.274.4		8
60.47	odd	4		825.3.h.a.274.5		8
60.59	even	2		825.3.b.a.76.3		4
132.35	odd	10		363.3.g.e.40.3		16
132.47	even	10		363.3.g.e.112.2		16
132.59	even	10		363.3.g.e.94.3		16
132.71	even	10		363.3.g.e.118.3		16
132.83	odd	10		363.3.g.e.118.2		16
132.95	odd	10		363.3.g.e.94.2		16
132.107	odd	10		363.3.g.e.112.3		16
132.119	even	10		363.3.g.e.40.2		16
132.131	odd	2		33.3.c.a.10.3	yes	4
264.131	odd	2		2112.3.j.a.769.1		4
264.197	even	2		2112.3.j.d.769.4		4
660.263	even	4		825.3.h.a.274.6		8
660.527	even	4		825.3.h.a.274.3		8
660.659	odd	2		825.3.b.a.76.2		4

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

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 1584 = 2^{4} \cdot 3^{2} \cdot 11 \)
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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