LMFDB - Embedded newform 5292.2.bm.b.2285.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [5292,2,Mod(2285,5292)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(5292, base_ring=CyclotomicField(6))

chi = DirichletCharacter(H, H._module([0, 5, 1]))

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

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

chi := DirichletCharacter("5292.2285");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	5292.bm (of order $6$, degree $2$, 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:	$42.2568327497$
Analytic rank:	$0$
Dimension:	$16$
Relative dimension:	$8$ over $\Q(\zeta_{6})$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 3x^{14} - 9x^{12} - 9x^{10} + 225x^{8} - 81x^{6} - 729x^{4} - 2187x^{2} + 6561 $ x^16 - 3x^14 - 9x^12 - 9x^10 + 225x^8 - 81x^6 - 729x^4 - 2187*x^2 + 6561
Coefficient ring:	$\Z[a_1, \ldots, a_{37}]$
Coefficient ring index:	$ 2^{2}\cdot 3^{6} $
Twist minimal:	no (minimal twist has level 252)
Sato-Tate group:	$\mathrm{SU}(2)[C_{6}]$

Embedding invariants

Embedding label			2285.5
Root			$-0.604587 - 1.62311i$ of defining polynomial
Character	$\chi$	$=$	5292.2285
Dual form			5292.2.bm.b.4625.5

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.533560 q^{5} +O(q^{10})$	$q+0.533560 q^{5} +3.92524i q^{11} +(-0.116911 - 0.0674987i) q^{13} +(2.16266 - 3.74584i) q^{17} +(-1.93067 + 1.11467i) q^{19} -1.96732i q^{23} -4.71531 q^{25} +(5.16548 - 2.98229i) q^{29} +(0.800341 - 0.462077i) q^{31} +(-3.89936 - 6.75388i) q^{37} +(4.59027 - 7.95059i) q^{41} +(3.24544 + 5.62127i) q^{43} +(3.04329 - 5.27114i) q^{47} +(9.54072 + 5.50834i) q^{53} +2.09435i q^{55} +(1.89588 + 3.28377i) q^{59} +(9.35116 + 5.39889i) q^{61} +(-0.0623791 - 0.0360146i) q^{65} +(-5.75701 - 9.97144i) q^{67} +3.22884i q^{71} +(0.329991 + 0.190521i) q^{73} +(-4.60310 + 7.97280i) q^{79} +(1.28020 + 2.21737i) q^{83} +(1.15391 - 1.99863i) q^{85} +(8.56670 + 14.8380i) q^{89} +(-1.03013 + 0.594746i) q^{95} +(13.6747 - 7.89507i) q^{97} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q+O(q^{10}) $ 16 * q	$ 16 q + 16 q^{25} + 12 q^{29} - 2 q^{37} + 4 q^{43} + 36 q^{53} - 24 q^{65} + 14 q^{67} + 20 q^{79} + 6 q^{85} - 60 q^{95}+O(q^{100}) $ 16 * q + 16 * q^25 + 12 * q^29 - 2 * q^37 + 4 * q^43 + 36 * q^53 - 24 * q^65 + 14 * q^67 + 20 * q^79 + 6 * q^85 - 60 * q^95

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/5292\mathbb{Z}\right)^\times$.

$n$	$785$	$1081$	$2647$
$\chi(n)$	$e\left(\frac{5}{6}\right)$	$e\left(\frac{1}{6}\right)$	$1$

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
$5$	0.533560			0.238615			0.119308	−	0.992857i	$-0.461932\pi$
$5$	0.533560			0.238615			0.119308	+	0.992857i	$0.461932\pi$
$6$		0			0
$7$		0			0
$7$		0			0
$8$		0			0
$9$		0			0
$10$		0			0
$11$			3.92524i			1.18350i	0.806120	+	0.591752i	$0.201563\pi$
$11$			3.92524i			1.18350i	−0.806120	+	0.591752i	$0.798437\pi$
$12$		0			0
$13$	−0.116911	−	0.0674987i	−0.0324253	−	0.0187208i	0.483700	−	0.875234i	$-0.339293\pi$
$13$	−0.116911	−	0.0674987i	−0.0324253	−	0.0187208i	−0.516125	+	0.856513i	$0.672626\pi$
$14$		0			0
$15$		0			0
$16$		0			0
$17$	2.16266	−	3.74584i	0.524523	−	0.908500i	−0.475069	−	0.879948i	$-0.657577\pi$
$17$	2.16266	−	3.74584i	0.524523	−	0.908500i	0.999592	−	0.0285519i	$-0.00908959\pi$
$18$		0			0
$19$	−1.93067	+	1.11467i	−0.442927	+	0.255724i	−0.704838	−	0.709368i	$-0.748980\pi$
$19$	−1.93067	+	1.11467i	−0.442927	+	0.255724i	0.261912	+	0.965092i	$0.415647\pi$
$20$		0			0
$21$		0			0
$22$		0			0
$23$		−	1.96732i		−	0.410214i	−0.978740	−	0.205107i	$-0.934246\pi$
$23$		−	1.96732i		−	0.410214i	0.978740	−	0.205107i	$-0.0657542\pi$
$24$		0			0
$25$	−4.71531			−0.943063
$26$		0			0
$27$		0			0
$28$		0			0
$29$	5.16548	−	2.98229i	0.959205	−	0.553798i	0.0632771	−	0.997996i	$-0.479845\pi$
$29$	5.16548	−	2.98229i	0.959205	−	0.553798i	0.895928	+	0.444198i	$0.146511\pi$
$30$		0			0
$31$	0.800341	−	0.462077i	0.143745	−	0.0829915i	−0.426402	−	0.904534i	$-0.640219\pi$
$31$	0.800341	−	0.462077i	0.143745	−	0.0829915i	0.570148	+	0.821542i	$0.306886\pi$
$32$		0			0
$33$		0			0
$34$		0			0
$35$		0			0
$36$		0			0
$37$	−3.89936	−	6.75388i	−0.641050	−	1.11033i	−0.985199	−	0.171416i	$-0.945166\pi$
$37$	−3.89936	−	6.75388i	−0.641050	−	1.11033i	0.344149	−	0.938915i	$-0.388168\pi$
$38$		0			0
$39$		0			0
$40$		0			0
$41$	4.59027	−	7.95059i	0.716880	−	1.24167i	−0.245349	−	0.969435i	$-0.578903\pi$
$41$	4.59027	−	7.95059i	0.716880	−	1.24167i	0.962230	−	0.272239i	$-0.0877640\pi$
$42$		0			0
$43$	3.24544	+	5.62127i	0.494926	+	0.857236i	0.999983	−	0.00584958i	$-0.00186199\pi$
$43$	3.24544	+	5.62127i	0.494926	+	0.857236i	−0.505057	+	0.863086i	$0.668529\pi$
$44$		0			0
$45$		0			0
$46$		0			0
$47$	3.04329	−	5.27114i	0.443910	−	0.768874i	−0.554066	−	0.832473i	$-0.686924\pi$
$47$	3.04329	−	5.27114i	0.443910	−	0.768874i	0.997976	+	0.0635985i	$0.0202577\pi$
$48$		0			0
$49$		0			0
$50$		0			0
$51$		0			0
$52$		0			0
$53$	9.54072	+	5.50834i	1.31052	+	0.756628i	0.982182	−	0.187932i	$-0.0601785\pi$
$53$	9.54072	+	5.50834i	1.31052	+	0.756628i	0.328337	+	0.944561i	$0.393512\pi$
$54$		0			0
$55$			2.09435i			0.282402i
$56$		0			0
$57$		0			0
$58$		0			0
$59$	1.89588	+	3.28377i	0.246823	+	0.427510i	0.962643	−	0.270775i	$-0.0872801\pi$
$59$	1.89588	+	3.28377i	0.246823	+	0.427510i	−0.715820	+	0.698285i	$0.753947\pi$
$60$		0			0
$61$	9.35116	+	5.39889i	1.19729	+	0.691258i	0.959951	−	0.280168i	$-0.0903903\pi$
$61$	9.35116	+	5.39889i	1.19729	+	0.691258i	0.237342	+	0.971426i	$0.423724\pi$
$62$		0			0
$63$		0			0
$64$		0			0
$65$	−0.0623791	−	0.0360146i	−0.00773718	−	0.00446706i
$66$		0			0
$67$	−5.75701	−	9.97144i	−0.703331	−	1.21820i	−0.967291	−	0.253671i	$-0.918362\pi$
$67$	−5.75701	−	9.97144i	−0.703331	−	1.21820i	0.263960	−	0.964534i	$-0.414971\pi$
$68$		0			0
$69$		0			0
$70$		0			0
$71$			3.22884i			0.383192i	0.981474	+	0.191596i	$0.0613664\pi$
$71$			3.22884i			0.383192i	−0.981474	+	0.191596i	$0.938634\pi$
$72$		0			0
$73$	0.329991	+	0.190521i	0.0386226	+	0.0222988i	0.519187	−	0.854661i	$-0.326235\pi$
$73$	0.329991	+	0.190521i	0.0386226	+	0.0222988i	−0.480564	+	0.876959i	$0.659568\pi$
$74$		0			0
$75$		0			0
$76$		0			0
$77$		0			0
$78$		0			0
$79$	−4.60310	+	7.97280i	−0.517889	+	0.897011i	0.481895	+	0.876229i	$0.339949\pi$
$79$	−4.60310	+	7.97280i	−0.517889	+	0.897011i	−0.999784	+	0.0207814i	$0.993385\pi$
$80$		0			0
$81$		0			0
$82$		0			0
$83$	1.28020	+	2.21737i	0.140520	+	0.243388i	0.927693	−	0.373345i	$-0.121789\pi$
$83$	1.28020	+	2.21737i	0.140520	+	0.243388i	−0.787172	+	0.616733i	$0.788456\pi$
$84$		0			0
$85$	1.15391	−	1.99863i	0.125159	−	0.216782i
$86$		0			0
$87$		0			0
$88$		0			0
$89$	8.56670	+	14.8380i	0.908068	+	1.57282i	0.816745	+	0.576999i	$0.195776\pi$
$89$	8.56670	+	14.8380i	0.908068	+	1.57282i	0.0913236	+	0.995821i	$0.470890\pi$
$90$		0			0
$91$		0			0
$92$		0			0
$93$		0			0
$94$		0			0
$95$	−1.03013	+	0.594746i	−0.105689	+	0.0610197i
$96$		0			0
$97$	13.6747	−	7.89507i	1.38845	−	0.801622i	0.395310	−	0.918548i	$-0.370637\pi$
$97$	13.6747	−	7.89507i	1.38845	−	0.801622i	0.993141	+	0.116925i	$0.0373038\pi$
$98$		0			0
$99$		0			0
$100$		0			0
$101$	−14.7372			−1.46641			−0.733205	−	0.680008i	$-0.761976\pi$
$101$	−14.7372			−1.46641			−0.733205	+	0.680008i	$0.761976\pi$
$102$		0			0
$103$			12.8682i			1.26794i	0.773357	+	0.633970i	$0.218576\pi$
$103$			12.8682i			1.26794i	−0.773357	+	0.633970i	$0.781424\pi$
$104$		0			0
$105$		0			0
$106$		0			0
$107$	13.6679	−	7.89119i	1.32133	−	0.762870i	0.337389	−	0.941365i	$-0.390456\pi$
$107$	13.6679	−	7.89119i	1.32133	−	0.762870i	0.983941	+	0.178496i	$0.0571231\pi$
$108$		0			0
$109$	1.54170	−	2.67030i	0.147668	−	0.255768i	−0.782697	−	0.622403i	$-0.786157\pi$
$109$	1.54170	−	2.67030i	0.147668	−	0.255768i	0.930365	+	0.366634i	$0.119490\pi$
$110$		0			0
$111$		0			0
$112$		0			0
$113$	−7.96173	−	4.59671i	−0.748977	−	0.432422i	0.0763472	−	0.997081i	$-0.475674\pi$
$113$	−7.96173	−	4.59671i	−0.748977	−	0.432422i	−0.825324	+	0.564659i	$0.809008\pi$
$114$		0			0
$115$		−	1.04968i		−	0.0978833i
$116$		0			0
$117$		0			0
$118$		0			0
$119$		0			0
$120$		0			0
$121$	−4.40749			−0.400681
$122$		0			0
$123$		0			0
$124$		0			0
$125$	−5.18371			−0.463645
$126$		0			0
$127$	−10.1065			−0.896810			−0.448405	−	0.893831i	$-0.648008\pi$
$127$	−10.1065			−0.896810			−0.448405	+	0.893831i	$0.648008\pi$
$128$		0			0
$129$		0			0
$130$		0			0
$131$	15.6365			1.36616			0.683082	−	0.730342i	$-0.260639\pi$
$131$	15.6365			1.36616			0.683082	+	0.730342i	$0.260639\pi$
$132$		0			0
$133$		0			0
$134$		0			0
$135$		0			0
$136$		0			0
$137$		−	2.46980i		−	0.211009i	−0.994419	−	0.105505i	$-0.966354\pi$
$137$		−	2.46980i		−	0.211009i	0.994419	−	0.105505i	$-0.0336458\pi$
$138$		0			0
$139$	16.8526	+	9.72984i	1.42942	+	0.825274i	0.997074	−	0.0764359i	$-0.0243540\pi$
$139$	16.8526	+	9.72984i	1.42942	+	0.825274i	0.432342	+	0.901710i	$0.357687\pi$
$140$		0			0
$141$		0			0
$142$		0			0
$143$	0.264948	−	0.458904i	0.0221561	−	0.0383755i
$144$		0			0
$145$	2.75610	−	1.59123i	0.228881	−	0.132145i
$146$		0			0
$147$		0			0
$148$		0			0
$149$			15.8544i			1.29884i	0.760429	+	0.649422i	$0.224989\pi$
$149$			15.8544i			1.29884i	−0.760429	+	0.649422i	$0.775011\pi$
$150$		0			0
$151$	8.33096			0.677964			0.338982	−	0.940793i	$-0.389917\pi$
$151$	8.33096			0.677964			0.338982	+	0.940793i	$0.389917\pi$
$152$		0			0
$153$		0			0
$154$		0			0
$155$	0.427030	−	0.246546i	0.0342999	−	0.0198031i
$156$		0			0
$157$	7.73794	−	4.46750i	0.617555	−	0.356545i	−0.158362	−	0.987381i	$-0.550621\pi$
$157$	7.73794	−	4.46750i	0.617555	−	0.356545i	0.775916	+	0.630836i	$0.217288\pi$
$158$		0			0
$159$		0			0
$160$		0			0
$161$		0			0
$162$		0			0
$163$	7.10310	+	12.3029i	0.556358	+	0.963640i	0.997797	+	0.0663485i	$0.0211349\pi$
$163$	7.10310	+	12.3029i	0.556358	+	0.963640i	−0.441439	+	0.897291i	$0.645532\pi$
$164$		0			0
$165$		0			0
$166$		0			0
$167$	6.27308	−	10.8653i	0.485425	−	0.840781i	−0.514434	−	0.857530i	$-0.671998\pi$
$167$	6.27308	−	10.8653i	0.485425	−	0.840781i	0.999860	+	0.0167485i	$0.00533145\pi$
$168$		0			0
$169$	−6.49089	−	11.2425i	−0.499299	−	0.864811i
$170$		0			0
$171$		0			0
$172$		0			0
$173$	10.6787	−	18.4960i	0.811886	−	1.40623i	−0.0996566	−	0.995022i	$-0.531774\pi$
$173$	10.6787	−	18.4960i	0.811886	−	1.40623i	0.911543	−	0.411206i	$-0.134892\pi$
$174$		0			0
$175$		0			0
$176$		0			0
$177$		0			0
$178$		0			0
$179$	−5.03259	−	2.90557i	−0.376153	−	0.217172i	0.299990	−	0.953942i	$-0.403017\pi$
$179$	−5.03259	−	2.90557i	−0.376153	−	0.217172i	−0.676143	+	0.736770i	$0.736350\pi$
$180$		0			0
$181$		−	7.38877i		−	0.549203i	−0.961558	−	0.274602i	$-0.911454\pi$
$181$		−	7.38877i		−	0.549203i	0.961558	−	0.274602i	$-0.0885460\pi$
$182$		0			0
$183$		0			0
$184$		0			0
$185$	−2.08054	−	3.60360i	−0.152964	−	0.264942i
$186$		0			0
$187$	14.7033	+	8.48897i	1.07521	+	0.620775i
$188$		0			0
$189$		0			0
$190$		0			0
$191$	6.86109	+	3.96125i	0.496451	+	0.286626i	0.727247	−	0.686376i	$-0.240800\pi$
$191$	6.86109	+	3.96125i	0.496451	+	0.286626i	−0.230796	+	0.973002i	$0.574133\pi$
$192$		0			0
$193$	3.16548	+	5.48277i	0.227856	+	0.394659i	0.957173	−	0.289518i	$-0.0934951\pi$
$193$	3.16548	+	5.48277i	0.227856	+	0.394659i	−0.729316	+	0.684177i	$0.760162\pi$
$194$		0			0
$195$		0			0
$196$		0			0
$197$			15.1580i			1.07996i	0.841677	+	0.539981i	$0.181569\pi$
$197$			15.1580i			1.07996i	−0.841677	+	0.539981i	$0.818431\pi$
$198$		0			0
$199$	−7.40524	−	4.27542i	−0.524944	−	0.303076i	0.214011	−	0.976831i	$-0.431347\pi$
$199$	−7.40524	−	4.27542i	−0.524944	−	0.303076i	−0.738955	+	0.673755i	$0.764680\pi$
$200$		0			0
$201$		0			0
$202$		0			0
$203$		0			0
$204$		0			0
$205$	2.44919	−	4.24212i	0.171059	−	0.296282i
$206$		0			0
$207$		0			0
$208$		0			0
$209$	−4.37536	−	7.57835i	−0.302650	−	0.524205i
$210$		0			0
$211$	−2.80782	+	4.86329i	−0.193299	+	0.334803i	−0.946341	−	0.323169i	$-0.895252\pi$
$211$	−2.80782	+	4.86329i	−0.193299	+	0.334803i	0.753043	+	0.657971i	$0.228585\pi$
$212$		0			0
$213$		0			0
$214$		0			0
$215$	1.73164	+	2.99929i	0.118097	+	0.204550i
$216$		0			0
$217$		0			0
$218$		0			0
$219$		0			0
$220$		0			0
$221$	−0.505679	+	0.291954i	−0.0340156	+	0.0196389i
$222$		0			0
$223$	−6.00510	+	3.46705i	−0.402131	+	0.232171i	−0.687403	−	0.726276i	$-0.741249\pi$
$223$	−6.00510	+	3.46705i	−0.402131	+	0.232171i	0.285272	+	0.958447i	$0.407916\pi$
$224$		0			0
$225$		0			0
$226$		0			0
$227$	14.5767			0.967487			0.483743	−	0.875210i	$-0.339277\pi$
$227$	14.5767			0.967487			0.483743	+	0.875210i	$0.339277\pi$
$228$		0			0
$229$			24.5631i			1.62317i	0.584233	+	0.811586i	$0.301396\pi$
$229$			24.5631i			1.62317i	−0.584233	+	0.811586i	$0.698604\pi$
$230$		0			0
$231$		0			0
$232$		0			0
$233$	−9.10556	+	5.25710i	−0.596525	+	0.344404i	−0.767673	−	0.640841i	$-0.778586\pi$
$233$	−9.10556	+	5.25710i	−0.596525	+	0.344404i	0.171148	+	0.985245i	$0.445252\pi$
$234$		0			0
$235$	1.62378	−	2.81247i	0.105924	−	0.183465i
$236$		0			0
$237$		0			0
$238$		0			0
$239$	−16.9075	−	9.76154i	−1.09365	−	0.631422i	−0.159108	−	0.987261i	$-0.550862\pi$
$239$	−16.9075	−	9.76154i	−1.09365	−	0.631422i	−0.934547	+	0.355839i	$0.884195\pi$
$240$		0			0
$241$		−	13.1382i		−	0.846305i	−0.906059	−	0.423152i	$-0.860923\pi$
$241$		−	13.1382i		−	0.846305i	0.906059	−	0.423152i	$-0.139077\pi$
$242$		0			0
$243$		0			0
$244$		0			0
$245$		0			0
$246$		0			0
$247$	0.300956			0.0191494
$248$		0			0
$249$		0			0
$250$		0			0
$251$	22.6864			1.43195			0.715977	−	0.698124i	$-0.245981\pi$
$251$	22.6864			1.43195			0.715977	+	0.698124i	$0.245981\pi$
$252$		0			0
$253$	7.72218			0.485489
$254$		0			0
$255$		0			0
$256$		0			0
$257$	13.6096			0.848944			0.424472	−	0.905441i	$-0.360460\pi$
$257$	13.6096			0.848944			0.424472	+	0.905441i	$0.360460\pi$
$258$		0			0
$259$		0			0
$260$		0			0
$261$		0			0
$262$		0			0
$263$		−	22.8550i		−	1.40930i	−0.709554	−	0.704651i	$-0.751104\pi$
$263$		−	22.8550i		−	1.40930i	0.709554	−	0.704651i	$-0.248896\pi$
$264$		0			0
$265$	5.09055	+	2.93903i	0.312710	+	0.180543i
$266$		0			0
$267$		0			0
$268$		0			0
$269$	5.33875	−	9.24698i	0.325509	−	0.563798i	−0.656106	−	0.754669i	$-0.727798\pi$
$269$	5.33875	−	9.24698i	0.325509	−	0.563798i	0.981615	+	0.190870i	$0.0611310\pi$
$270$		0			0
$271$	3.90987	−	2.25737i	0.237508	−	0.137125i	−0.376523	−	0.926407i	$-0.622880\pi$
$271$	3.90987	−	2.25737i	0.237508	−	0.137125i	0.614031	+	0.789282i	$0.289547\pi$
$272$		0			0
$273$		0			0
$274$		0			0
$275$		−	18.5087i		−	1.11612i
$276$		0			0
$277$	−6.69904			−0.402507			−0.201253	−	0.979539i	$-0.564501\pi$
$277$	−6.69904			−0.402507			−0.201253	+	0.979539i	$0.564501\pi$
$278$		0			0
$279$		0			0
$280$		0			0
$281$	15.1414	−	8.74187i	0.903258	−	0.521496i	0.0250023	−	0.999687i	$-0.492041\pi$
$281$	15.1414	−	8.74187i	0.903258	−	0.521496i	0.878256	+	0.478191i	$0.158707\pi$
$282$		0			0
$283$	7.42049	−	4.28422i	0.441102	−	0.254670i	−0.262963	−	0.964806i	$-0.584700\pi$
$283$	7.42049	−	4.28422i	0.441102	−	0.254670i	0.704065	+	0.710135i	$0.251366\pi$
$284$		0			0
$285$		0			0
$286$		0			0
$287$		0			0
$288$		0			0
$289$	−0.854223	−	1.47956i	−0.0502484	−	0.0870328i
$290$		0			0
$291$		0			0
$292$		0			0
$293$	12.1436	−	21.0333i	0.709434	−	1.22878i	−0.255633	−	0.966774i	$-0.582284\pi$
$293$	12.1436	−	21.0333i	0.709434	−	1.22878i	0.965067	−	0.262002i	$-0.0843826\pi$
$294$		0			0
$295$	1.01157	+	1.75209i	0.0588958	+	0.102010i
$296$		0			0
$297$		0			0
$298$		0			0
$299$	−0.132791	+	0.230001i	−0.00767951	+	0.0133013i
$300$		0			0
$301$		0			0
$302$		0			0
$303$		0			0
$304$		0			0
$305$	4.98941	+	2.88064i	0.285693	+	0.164945i
$306$		0			0
$307$			12.4777i			0.712139i	0.934460	+	0.356069i	$0.115883\pi$
$307$			12.4777i			0.712139i	−0.934460	+	0.356069i	$0.884117\pi$
$308$		0			0
$309$		0			0
$310$		0			0
$311$	−9.07984	−	15.7267i	−0.514871	−	0.891782i	−0.999851	−	0.0172571i	$-0.994507\pi$
$311$	−9.07984	−	15.7267i	−0.514871	−	0.891782i	0.484980	−	0.874525i	$-0.338827\pi$
$312$		0			0
$313$	2.76700	+	1.59753i	0.156400	+	0.0902977i	0.576157	−	0.817339i	$-0.304551\pi$
$313$	2.76700	+	1.59753i	0.156400	+	0.0902977i	−0.419757	+	0.907636i	$0.637885\pi$
$314$		0			0
$315$		0			0
$316$		0			0
$317$	−22.5893	−	13.0419i	−1.26874	−	0.732508i	−0.293991	−	0.955808i	$-0.594984\pi$
$317$	−22.5893	−	13.0419i	−1.26874	−	0.732508i	−0.974750	+	0.223300i	$0.928317\pi$
$318$		0			0
$319$	11.7062	+	20.2757i	0.655421	+	1.13522i
$320$		0			0
$321$		0			0
$322$		0			0
$323$			9.64266i			0.536532i
$324$		0			0
$325$	0.551272	+	0.318277i	0.0305791	+	0.0176548i
$326$		0			0
$327$		0			0
$328$		0			0
$329$		0			0
$330$		0			0
$331$	8.06484	−	13.9687i	0.443283	−	0.767789i	−0.554647	−	0.832085i	$-0.687147\pi$
$331$	8.06484	−	13.9687i	0.443283	−	0.767789i	0.997931	+	0.0642960i	$0.0204802\pi$
$332$		0			0
$333$		0			0
$334$		0			0
$335$	−3.07171	−	5.32036i	−0.167826	−	0.290683i
$336$		0			0
$337$	−4.16548	+	7.21482i	−0.226908	+	0.393016i	−0.956890	−	0.290450i	$-0.906195\pi$
$337$	−4.16548	+	7.21482i	−0.226908	+	0.393016i	0.729982	+	0.683466i	$0.239528\pi$
$338$		0			0
$339$		0			0
$340$		0			0
$341$	1.81376	+	3.14153i	0.0982207	+	0.170123i
$342$		0			0
$343$		0			0
$344$		0			0
$345$		0			0
$346$		0			0
$347$	−30.1403	+	17.4015i	−1.61801	+	0.934161i	−0.630581	+	0.776124i	$0.717183\pi$
$347$	−30.1403	+	17.4015i	−1.61801	+	0.934161i	−0.987433	+	0.158037i	$0.949483\pi$
$348$		0			0
$349$	19.6825	−	11.3637i	1.05358	−	0.608283i	0.129929	−	0.991523i	$-0.458525\pi$
$349$	19.6825	−	11.3637i	1.05358	−	0.608283i	0.923649	+	0.383240i	$0.125192\pi$
$350$		0			0
$351$		0			0
$352$		0			0
$353$	−8.05659			−0.428809			−0.214404	−	0.976745i	$-0.568781\pi$
$353$	−8.05659			−0.428809			−0.214404	+	0.976745i	$0.568781\pi$
$354$		0			0
$355$			1.72278i			0.0914357i
$356$		0			0
$357$		0			0
$358$		0			0
$359$	−15.4341	+	8.91086i	−0.814579	+	0.470297i	−0.848543	−	0.529126i	$-0.822520\pi$
$359$	−15.4341	+	8.91086i	−0.814579	+	0.470297i	0.0339648	+	0.999423i	$0.489187\pi$
$360$		0			0
$361$	−7.01500	+	12.1503i	−0.369211	+	0.639492i
$362$		0			0
$363$		0			0
$364$		0			0
$365$	0.176070	+	0.101654i	0.00921594	+	0.00532083i
$366$		0			0
$367$			23.7739i			1.24099i	0.784211	+	0.620494i	$0.213068\pi$
$367$			23.7739i			1.24099i	−0.784211	+	0.620494i	$0.786932\pi$
$368$		0			0
$369$		0			0
$370$		0			0
$371$		0			0
$372$		0			0
$373$	10.5372			0.545593			0.272797	−	0.962072i	$-0.412051\pi$
$373$	10.5372			0.545593			0.272797	+	0.962072i	$0.412051\pi$
$374$		0			0
$375$		0			0
$376$		0			0
$377$	−0.805203			−0.0414700
$378$		0			0
$379$	24.0049			1.23305			0.616525	−	0.787336i	$-0.288540\pi$
$379$	24.0049			1.23305			0.616525	+	0.787336i	$0.288540\pi$
$380$		0			0
$381$		0			0
$382$		0			0
$383$	36.1960			1.84953			0.924764	−	0.380542i	$-0.124263\pi$
$383$	36.1960			1.84953			0.924764	+	0.380542i	$0.124263\pi$
$384$		0			0
$385$		0			0
$386$		0			0
$387$		0			0
$388$		0			0
$389$		−	21.4810i		−	1.08913i	−0.838718	−	0.544565i	$-0.816695\pi$
$389$		−	21.4810i		−	1.08913i	0.838718	−	0.544565i	$-0.183305\pi$
$390$		0			0
$391$	−7.36925	−	4.25464i	−0.372679	−	0.215166i
$392$		0			0
$393$		0			0
$394$		0			0
$395$	−2.45603	+	4.25397i	−0.123576	+	0.214041i
$396$		0			0
$397$	−18.4505	+	10.6524i	−0.926006	+	0.534630i	−0.885546	−	0.464551i	$-0.846216\pi$
$397$	−18.4505	+	10.6524i	−0.926006	+	0.534630i	−0.0404601	+	0.999181i	$0.512882\pi$
$398$		0			0
$399$		0			0
$400$		0			0
$401$			18.1081i			0.904276i	0.891948	+	0.452138i	$0.149339\pi$
$401$			18.1081i			0.904276i	−0.891948	+	0.452138i	$0.850661\pi$
$402$		0			0
$403$	−0.124758			−0.00621465
$404$		0			0
$405$		0			0
$406$		0			0
$407$	26.5106	−	15.3059i	1.31408	−	0.758685i
$408$		0			0
$409$	−17.6807	+	10.2080i	−0.874254	+	0.504751i	−0.868760	−	0.495234i	$-0.835082\pi$
$409$	−17.6807	+	10.2080i	−0.874254	+	0.504751i	−0.00549461	+	0.999985i	$0.501749\pi$
$410$		0			0
$411$		0			0
$412$		0			0
$413$		0			0
$414$		0			0
$415$	0.683065	+	1.18310i	0.0335303	+	0.0580762i
$416$		0			0
$417$		0			0
$418$		0			0
$419$	−12.6789	+	21.9606i	−0.619407	+	1.07284i	0.370187	+	0.928957i	$0.379294\pi$
$419$	−12.6789	+	21.9606i	−0.619407	+	1.07284i	−0.989594	+	0.143887i	$0.954040\pi$
$420$		0			0
$421$	−3.21875	−	5.57503i	−0.156872	−	0.271710i	0.776867	−	0.629665i	$-0.216808\pi$
$421$	−3.21875	−	5.57503i	−0.156872	−	0.271710i	−0.933739	+	0.357954i	$0.883474\pi$
$422$		0			0
$423$		0			0
$424$		0			0
$425$	−10.1976	+	17.6628i	−0.494658	+	0.856773i
$426$		0			0
$427$		0			0
$428$		0			0
$429$		0			0
$430$		0			0
$431$	13.1109	+	7.56961i	0.631532	+	0.364615i	0.781345	−	0.624099i	$-0.214534\pi$
$431$	13.1109	+	7.56961i	0.631532	+	0.364615i	−0.149813	+	0.988714i	$0.547867\pi$
$432$		0			0
$433$		−	8.44792i		−	0.405981i	−0.979181	−	0.202991i	$-0.934934\pi$
$433$		−	8.44792i		−	0.405981i	0.979181	−	0.202991i	$-0.0650661\pi$
$434$		0			0
$435$		0			0
$436$		0			0
$437$	2.19292	+	3.79824i	0.104901	+	0.181695i
$438$		0			0
$439$	23.6831	+	13.6734i	1.13033	+	0.652598i	0.944018	−	0.329893i	$-0.107013\pi$
$439$	23.6831	+	13.6734i	1.13033	+	0.652598i	0.186314	+	0.982490i	$0.440346\pi$
$440$		0			0
$441$		0			0
$442$		0			0
$443$	−14.6520	−	8.45931i	−0.696135	−	0.401914i	0.109771	−	0.993957i	$-0.464988\pi$
$443$	−14.6520	−	8.45931i	−0.696135	−	0.401914i	−0.805906	+	0.592043i	$0.798322\pi$
$444$		0			0
$445$	4.57085	+	7.91695i	0.216679	+	0.375299i
$446$		0			0
$447$		0			0
$448$		0			0
$449$			22.5985i			1.06649i	0.845962	+	0.533244i	$0.179027\pi$
$449$			22.5985i			1.06649i	−0.845962	+	0.533244i	$0.820973\pi$
$450$		0			0
$451$	31.2079	+	18.0179i	1.46952	+	0.848431i
$452$		0			0
$453$		0			0
$454$		0			0
$455$		0			0
$456$		0			0
$457$	17.4018	−	30.1408i	0.814022	−	1.40993i	−0.0960053	−	0.995381i	$-0.530607\pi$
$457$	17.4018	−	30.1408i	0.814022	−	1.40993i	0.910028	−	0.414547i	$-0.136060\pi$
$458$		0			0
$459$		0			0
$460$		0			0
$461$	−13.8264	−	23.9479i	−0.643958	−	1.11537i	−0.984541	−	0.175153i	$-0.943958\pi$
$461$	−13.8264	−	23.9479i	−0.643958	−	1.11537i	0.340584	−	0.940214i	$-0.389375\pi$
$462$		0			0
$463$	10.6272	−	18.4069i	0.493889	−	0.855440i	−0.506087	−	0.862483i	$-0.668908\pi$
$463$	10.6272	−	18.4069i	0.493889	−	0.855440i	0.999975	+	0.00704260i	$0.00224175\pi$
$464$		0			0
$465$		0			0
$466$		0			0
$467$	−4.40378	−	7.62758i	−0.203783	−	0.352962i	0.745961	−	0.665989i	$-0.231990\pi$
$467$	−4.40378	−	7.62758i	−0.203783	−	0.352962i	−0.949744	+	0.313027i	$0.898657\pi$
$468$		0			0
$469$		0			0
$470$		0			0
$471$		0			0
$472$		0			0
$473$	−22.0648	+	12.7391i	−1.01454	+	0.585746i
$474$		0			0
$475$	9.10373	−	5.25604i	0.417708	−	0.241164i
$476$		0			0
$477$		0			0
$478$		0			0
$479$	13.6628			0.624269			0.312134	−	0.950038i	$-0.398956\pi$
$479$	13.6628			0.624269			0.312134	+	0.950038i	$0.398956\pi$
$480$		0			0
$481$			1.05281i			0.0480038i
$482$		0			0
$483$		0			0
$484$		0			0
$485$	7.29625	−	4.21249i	0.331306	−	0.191280i
$486$		0			0
$487$	−8.31028	+	14.3938i	−0.376575	+	0.652246i	−0.990561	−	0.137070i	$-0.956232\pi$
$487$	−8.31028	+	14.3938i	−0.376575	+	0.652246i	0.613987	+	0.789316i	$0.289565\pi$
$488$		0			0
$489$		0			0
$490$		0			0
$491$	−17.8129	−	10.2843i	−0.803883	−	0.464122i	0.0409440	−	0.999161i	$-0.486963\pi$
$491$	−17.8129	−	10.2843i	−0.803883	−	0.464122i	−0.844827	+	0.535039i	$0.820297\pi$
$492$		0			0
$493$		−	25.7988i		−	1.16192i
$494$		0			0
$495$		0			0
$496$		0			0
$497$		0			0
$498$		0			0
$499$	−2.69218			−0.120518			−0.0602592	−	0.998183i	$-0.519193\pi$
$499$	−2.69218			−0.120518			−0.0602592	+	0.998183i	$0.519193\pi$
$500$		0			0
$501$		0			0
$502$		0			0
$503$	27.3871			1.22113			0.610566	−	0.791965i	$-0.290942\pi$
$503$	27.3871			1.22113			0.610566	+	0.791965i	$0.290942\pi$
$504$		0			0
$505$	−7.86321			−0.349908
$506$		0			0
$507$		0			0
$508$		0			0
$509$	−5.92234			−0.262503			−0.131252	−	0.991349i	$-0.541900\pi$
$509$	−5.92234			−0.262503			−0.131252	+	0.991349i	$0.541900\pi$
$510$		0			0
$511$		0			0
$512$		0			0
$513$		0			0
$514$		0			0
$515$			6.86596i			0.302550i
$516$		0			0
$517$	20.6905	+	11.9456i	0.909966	+	0.525369i
$518$		0			0
$519$		0			0
$520$		0			0
$521$	−19.5943	+	33.9383i	−0.858442	+	1.48686i	0.0149735	+	0.999888i	$0.495234\pi$
$521$	−19.5943	+	33.9383i	−0.858442	+	1.48686i	−0.873415	+	0.486976i	$0.838100\pi$
$522$		0			0
$523$	19.9496	−	11.5179i	0.872333	−	0.503642i	0.00421050	−	0.999991i	$-0.498660\pi$
$523$	19.9496	−	11.5179i	0.872333	−	0.503642i	0.868123	+	0.496349i	$0.165326\pi$
$524$		0			0
$525$		0			0
$526$		0			0
$527$		−	3.99727i		−	0.174124i
$528$		0			0
$529$	19.1297			0.831725
$530$		0			0
$531$		0			0
$532$		0			0
$533$	−1.07331	+	0.619675i	−0.0464901	+	0.0268411i
$534$		0			0
$535$	7.29267	−	4.21043i	0.315290	−	0.182033i
$536$		0			0
$537$		0			0
$538$		0			0
$539$		0			0
$540$		0			0
$541$	1.66302	+	2.88044i	0.0714990	+	0.123840i	0.899558	−	0.436800i	$-0.143888\pi$
$541$	1.66302	+	2.88044i	0.0714990	+	0.123840i	−0.828059	+	0.560640i	$0.810555\pi$
$542$		0			0
$543$		0			0
$544$		0			0
$545$	0.822590	−	1.42477i	0.0352359	−	0.0610303i
$546$		0			0
$547$	13.8937	+	24.0646i	0.594051	+	1.02893i	0.993680	+	0.112249i	$0.0358055\pi$
$547$	13.8937	+	24.0646i	0.594051	+	1.02893i	−0.399629	+	0.916677i	$0.630861\pi$
$548$		0			0
$549$		0			0
$550$		0			0
$551$	−6.64857	+	11.5157i	−0.283238	+	0.490583i
$552$		0			0
$553$		0			0
$554$		0			0
$555$		0			0
$556$		0			0
$557$	−3.13213	−	1.80833i	−0.132712	−	0.0766216i	0.432174	−	0.901790i	$-0.357747\pi$
$557$	−3.13213	−	1.80833i	−0.132712	−	0.0766216i	−0.564886	+	0.825169i	$0.691080\pi$
$558$		0			0
$559$		−	0.876252i		−	0.0370615i
$560$		0			0
$561$		0			0
$562$		0			0
$563$	−4.75452	−	8.23506i	−0.200379	−	0.347067i	0.748272	−	0.663393i	$-0.230884\pi$
$563$	−4.75452	−	8.23506i	−0.200379	−	0.347067i	−0.948651	+	0.316326i	$0.897551\pi$
$564$		0			0
$565$	−4.24807	−	2.45262i	−0.178718	−	0.103183i
$566$		0			0
$567$		0			0
$568$		0			0
$569$	9.09742	+	5.25240i	0.381384	+	0.220192i	0.678420	−	0.734674i	$-0.262665\pi$
$569$	9.09742	+	5.25240i	0.381384	+	0.220192i	−0.297036	+	0.954866i	$0.595998\pi$
$570$		0			0
$571$	−2.24201	−	3.88328i	−0.0938252	−	0.162510i	0.815292	−	0.579049i	$-0.196576\pi$
$571$	−2.24201	−	3.88328i	−0.0938252	−	0.162510i	−0.909118	+	0.416539i	$0.863243\pi$
$572$		0			0
$573$		0			0
$574$		0			0
$575$			9.27651i			0.386857i
$576$		0			0
$577$	−40.8602	−	23.5906i	−1.70103	−	0.982090i	−0.944720	−	0.327877i	$-0.893667\pi$
$577$	−40.8602	−	23.5906i	−1.70103	−	0.982090i	−0.756310	−	0.654213i	$-0.773000\pi$
$578$		0			0
$579$		0			0
$580$		0			0
$581$		0			0
$582$		0			0
$583$	−21.6215	+	37.4496i	−0.895473	+	1.55100i
$584$		0			0
$585$		0			0
$586$		0			0
$587$	5.65373	+	9.79255i	0.233354	+	0.404182i	0.958793	−	0.284105i	$-0.0916964\pi$
$587$	5.65373	+	9.79255i	0.233354	+	0.404182i	−0.725439	+	0.688287i	$0.758363\pi$
$588$		0			0
$589$	−1.03013	+	1.78424i	−0.0424458	+	0.0735183i
$590$		0			0
$591$		0			0
$592$		0			0
$593$	−4.72490	−	8.18376i	−0.194028	−	0.336067i	0.752553	−	0.658531i	$-0.228822\pi$
$593$	−4.72490	−	8.18376i	−0.194028	−	0.336067i	−0.946582	+	0.322465i	$0.895489\pi$
$594$		0			0
$595$		0			0
$596$		0			0
$597$		0			0
$598$		0			0
$599$	31.6406	−	18.2677i	1.29280	−	0.746398i	0.313650	−	0.949539i	$-0.398448\pi$
$599$	31.6406	−	18.2677i	1.29280	−	0.746398i	0.979150	+	0.203140i	$0.0651147\pi$
$600$		0			0
$601$	−1.92247	+	1.10994i	−0.0784193	+	0.0452754i	−0.538697	−	0.842500i	$-0.681083\pi$
$601$	−1.92247	+	1.10994i	−0.0784193	+	0.0452754i	0.460278	+	0.887775i	$0.347750\pi$
$602$		0			0
$603$		0			0
$604$		0			0
$605$	−2.35166			−0.0956087
$606$		0			0
$607$			1.98331i			0.0804999i	0.999190	+	0.0402499i	$0.0128154\pi$
$607$			1.98331i			0.0804999i	−0.999190	+	0.0402499i	$0.987185\pi$
$608$		0			0
$609$		0			0
$610$		0			0
$611$	−0.711589	+	0.410836i	−0.0287878	+	0.0166207i
$612$		0			0
$613$	11.5683	−	20.0368i	0.467238	−	0.809280i	−0.532061	−	0.846706i	$-0.678582\pi$
$613$	11.5683	−	20.0368i	0.467238	−	0.809280i	0.999299	+	0.0374258i	$0.0119158\pi$
$614$		0			0
$615$		0			0
$616$		0			0
$617$	1.19807	+	0.691704i	0.0482323	+	0.0278470i	0.523922	−	0.851766i	$-0.324468\pi$
$617$	1.19807	+	0.691704i	0.0482323	+	0.0278470i	−0.475690	+	0.879613i	$0.657802\pi$
$618$		0			0
$619$		−	6.03389i		−	0.242522i	−0.992621	−	0.121261i	$-0.961306\pi$
$619$		−	6.03389i		−	0.242522i	0.992621	−	0.121261i	$-0.0386939\pi$
$620$		0			0
$621$		0			0
$622$		0			0
$623$		0			0
$624$		0			0
$625$	20.8107			0.832430
$626$		0			0
$627$		0			0
$628$		0			0
$629$	−33.7320			−1.34498
$630$		0			0
$631$	−20.4727			−0.815004			−0.407502	−	0.913204i	$-0.633600\pi$
$631$	−20.4727			−0.815004			−0.407502	+	0.913204i	$0.633600\pi$
$632$		0			0
$633$		0			0
$634$		0			0
$635$	−5.39245			−0.213993
$636$		0			0
$637$		0			0
$638$		0			0
$639$		0			0
$640$		0			0
$641$		−	46.4474i		−	1.83456i	−0.398240	−	0.917281i	$-0.630379\pi$
$641$		−	46.4474i		−	1.83456i	0.398240	−	0.917281i	$-0.369621\pi$
$642$		0			0
$643$	−9.74133	−	5.62416i	−0.384161	−	0.221795i	0.295466	−	0.955353i	$-0.404525\pi$
$643$	−9.74133	−	5.62416i	−0.384161	−	0.221795i	−0.679627	+	0.733558i	$0.737858\pi$
$644$		0			0
$645$		0			0
$646$		0			0
$647$	2.32507	−	4.02715i	0.0914081	−	0.158323i	−0.816696	−	0.577068i	$-0.804197\pi$
$647$	2.32507	−	4.02715i	0.0914081	−	0.158323i	0.908104	+	0.418745i	$0.137530\pi$
$648$		0			0
$649$	−12.8896	+	7.44179i	−0.505959	+	0.292116i
$650$		0			0
$651$		0			0
$652$		0			0
$653$			3.46371i			0.135545i	0.997701	+	0.0677727i	$0.0215893\pi$
$653$			3.46371i			0.135545i	−0.997701	+	0.0677727i	$0.978411\pi$
$654$		0			0
$655$	8.34299			0.325988
$656$		0			0
$657$		0			0
$658$		0			0
$659$	1.59819	−	0.922715i	0.0622566	−	0.0359439i	−0.468549	−	0.883438i	$-0.655223\pi$
$659$	1.59819	−	0.922715i	0.0622566	−	0.0359439i	0.530805	+	0.847494i	$0.321890\pi$
$660$		0			0
$661$	17.5196	−	10.1149i	0.681433	−	0.393426i	−0.118962	−	0.992899i	$-0.537957\pi$
$661$	17.5196	−	10.1149i	0.681433	−	0.393426i	0.800395	+	0.599473i	$0.204623\pi$
$662$		0			0
$663$		0			0
$664$		0			0
$665$		0			0
$666$		0			0
$667$	−5.86711	−	10.1621i	−0.227175	−	0.393479i
$668$		0			0
$669$		0			0
$670$		0			0
$671$	−21.1919	+	36.7055i	−0.818106	+	1.41700i
$672$		0			0
$673$	−7.31596	−	12.6716i	−0.282009	−	0.488455i	0.689870	−	0.723933i	$-0.257668\pi$
$673$	−7.31596	−	12.6716i	−0.282009	−	0.488455i	−0.971880	+	0.235478i	$0.924334\pi$
$674$		0			0
$675$		0			0
$676$		0			0
$677$	7.71449	−	13.3619i	0.296492	−	0.513539i	−0.678839	−	0.734287i	$-0.737517\pi$
$677$	7.71449	−	13.3619i	0.296492	−	0.513539i	0.975331	+	0.220748i	$0.0708499\pi$
$678$		0			0
$679$		0			0
$680$		0			0
$681$		0			0
$682$		0			0
$683$	13.6137	+	7.85987i	0.520913	+	0.300750i	0.737308	−	0.675556i	$-0.236097\pi$
$683$	13.6137	+	7.85987i	0.520913	+	0.300750i	−0.216395	+	0.976306i	$0.569430\pi$
$684$		0			0
$685$		−	1.31779i		−	0.0503501i
$686$		0			0
$687$		0			0
$688$		0			0
$689$	−0.743611	−	1.28797i	−0.0283293	−	0.0490678i
$690$		0			0
$691$	−41.5878	−	24.0107i	−1.58207	−	0.913411i	−0.994556	−	0.104199i	$-0.966772\pi$
$691$	−41.5878	−	24.0107i	−1.58207	−	0.913411i	−0.587517	−	0.809212i	$-0.699894\pi$
$692$		0			0
$693$		0			0
$694$		0			0
$695$	8.99187	+	5.19146i	0.341081	+	0.196923i
$696$		0			0
$697$	−19.8544	−	34.3889i	−0.752040	−	1.30257i
$698$		0			0
$699$		0			0
$700$		0			0
$701$			24.7005i			0.932923i	0.884541	+	0.466462i	$0.154471\pi$
$701$			24.7005i			0.932923i	−0.884541	+	0.466462i	$0.845529\pi$
$702$		0			0
$703$	15.0568	+	8.69302i	0.567876	+	0.327864i
$704$		0			0
$705$		0			0
$706$		0			0
$707$		0			0
$708$		0			0
$709$	−17.0432	+	29.5196i	−0.640070	+	1.10863i	0.345347	+	0.938475i	$0.387761\pi$
$709$	−17.0432	+	29.5196i	−0.640070	+	1.10863i	−0.985417	+	0.170159i	$0.945572\pi$
$710$		0			0
$711$		0			0
$712$		0			0
$713$	−0.909051	−	1.57452i	−0.0340442	−	0.0589663i
$714$		0			0
$715$	0.141366	−	0.244853i	0.00528679	−	0.00915698i
$716$		0			0
$717$		0			0
$718$		0			0
$719$	−9.23791	−	16.0005i	−0.344516	−	0.596719i	0.640750	−	0.767750i	$-0.278624\pi$
$719$	−9.23791	−	16.0005i	−0.344516	−	0.596719i	−0.985266	+	0.171031i	$0.945290\pi$
$720$		0			0
$721$		0			0
$722$		0			0
$723$		0			0
$724$		0			0
$725$	−24.3569	+	14.0624i	−0.904591	+	0.522266i
$726$		0			0
$727$	−39.2911	+	22.6847i	−1.45723	+	0.841330i	−0.998874	−	0.0474398i	$-0.984894\pi$
$727$	−39.2911	+	22.6847i	−1.45723	+	0.841330i	−0.458353	+	0.888770i	$0.651560\pi$
$728$		0			0
$729$		0			0
$730$		0			0
$731$	28.0752			1.03840
$732$		0			0
$733$		−	50.0773i		−	1.84965i	−0.380394	−	0.924825i	$-0.624212\pi$
$733$		−	50.0773i		−	1.84965i	0.380394	−	0.924825i	$-0.375788\pi$
$734$		0			0
$735$		0			0
$736$		0			0
$737$	39.1403	−	22.5976i	1.44175	−	0.832395i
$738$		0			0
$739$	8.47021	−	14.6708i	0.311582	−	0.539675i	−0.667123	−	0.744947i	$-0.732475\pi$
$739$	8.47021	−	14.6708i	0.311582	−	0.539675i	0.978705	+	0.205272i	$0.0658079\pi$
$740$		0			0
$741$		0			0
$742$		0			0
$743$	−34.4723	−	19.9026i	−1.26467	−	0.730156i	−0.290693	−	0.956816i	$-0.593886\pi$
$743$	−34.4723	−	19.9026i	−1.26467	−	0.730156i	−0.973974	+	0.226661i	$0.927219\pi$
$744$		0			0
$745$			8.45928i			0.309924i
$746$		0			0
$747$		0			0
$748$		0			0
$749$		0			0
$750$		0			0
$751$	−29.4055			−1.07302			−0.536512	−	0.843893i	$-0.680258\pi$
$751$	−29.4055			−1.07302			−0.536512	+	0.843893i	$0.680258\pi$
$752$		0			0
$753$		0			0
$754$		0			0
$755$	4.44507			0.161773
$756$		0			0
$757$	−27.4010			−0.995908			−0.497954	−	0.867203i	$-0.665915\pi$
$757$	−27.4010			−0.995908			−0.497954	+	0.867203i	$0.665915\pi$
$758$		0			0
$759$		0			0
$760$		0			0
$761$	−12.2213			−0.443023			−0.221511	−	0.975158i	$-0.571099\pi$
$761$	−12.2213			−0.443023			−0.221511	+	0.975158i	$0.571099\pi$
$762$		0			0
$763$		0			0
$764$		0			0
$765$		0			0
$766$		0			0
$767$		−	0.511878i		−	0.0184829i
$768$		0			0
$769$	−29.4039	−	16.9764i	−1.06033	−	0.612184i	−0.134809	−	0.990872i	$-0.543042\pi$
$769$	−29.4039	−	16.9764i	−1.06033	−	0.612184i	−0.925524	+	0.378688i	$0.876375\pi$
$770$		0			0
$771$		0			0
$772$		0			0
$773$	15.1047	−	26.1622i	0.543279	−	0.940987i	−0.455434	−	0.890270i	$-0.650516\pi$
$773$	15.1047	−	26.1622i	0.543279	−	0.940987i	0.998713	−	0.0507175i	$-0.0161508\pi$
$774$		0			0
$775$	−3.77386	+	2.17884i	−0.135561	+	0.0782662i
$776$		0			0
$777$		0			0
$778$		0			0
$779$			20.4666i			0.733294i
$780$		0			0
$781$	−12.6740			−0.453510
$782$		0			0
$783$		0			0
$784$		0			0
$785$	4.12866	−	2.38368i	0.147358	−	0.0850772i
$786$		0			0
$787$	−33.3310	+	19.2436i	−1.18812	+	0.685962i	−0.957879	−	0.287172i	$-0.907285\pi$
$787$	−33.3310	+	19.2436i	−1.18812	+	0.685962i	−0.230241	+	0.973134i	$0.573952\pi$
$788$		0			0
$789$		0			0
$790$		0			0
$791$		0			0
$792$		0			0
$793$	−0.728836	−	1.26238i	−0.0258817	−	0.0448285i
$794$		0			0
$795$		0			0
$796$		0			0
$797$	6.72949	−	11.6558i	0.238371	−	0.412870i	−0.721876	−	0.692022i	$-0.756720\pi$
$797$	6.72949	−	11.6558i	0.238371	−	0.412870i	0.960247	+	0.279152i	$0.0900533\pi$
$798$		0			0
$799$	−13.1632	−	22.7994i	−0.465682	−	0.806584i
$800$		0			0
$801$		0			0
$802$		0			0
$803$	−0.747839	+	1.29529i	−0.0263907	+	0.0457100i
$804$		0			0
$805$		0			0
$806$		0			0
$807$		0			0
$808$		0			0
$809$	44.8465	+	25.8921i	1.57672	+	0.910318i	0.995313	+	0.0967036i	$0.0308299\pi$
$809$	44.8465	+	25.8921i	1.57672	+	0.910318i	0.581404	+	0.813615i	$0.302503\pi$
$810$		0			0
$811$			27.2471i			0.956775i	0.878149	+	0.478387i	$0.158779\pi$
$811$			27.2471i			0.956775i	−0.878149	+	0.478387i	$0.841221\pi$
$812$		0			0
$813$		0			0
$814$		0			0
$815$	3.78993	+	6.56436i	0.132756	+	0.229939i
$816$		0			0
$817$	−12.5318	−	7.23523i	−0.438432	−	0.253129i
$818$		0			0
$819$		0			0
$820$		0			0
$821$	22.3465	+	12.9017i	0.779897	+	0.450274i	0.836394	−	0.548129i	$-0.184660\pi$
$821$	22.3465	+	12.9017i	0.779897	+	0.450274i	−0.0564968	+	0.998403i	$0.517993\pi$
$822$		0			0
$823$	0.570514	+	0.988159i	0.0198869	+	0.0344451i	0.875798	−	0.482679i	$-0.160336\pi$
$823$	0.570514	+	0.988159i	0.0198869	+	0.0344451i	−0.855911	+	0.517124i	$0.827003\pi$
$824$		0			0
$825$		0			0
$826$		0			0
$827$			23.1713i			0.805746i	0.915256	+	0.402873i	$0.131988\pi$
$827$			23.1713i			0.805746i	−0.915256	+	0.402873i	$0.868012\pi$
$828$		0			0
$829$	8.31700	+	4.80182i	0.288861	+	0.166774i	0.637428	−	0.770510i	$-0.279998\pi$
$829$	8.31700	+	4.80182i	0.288861	+	0.166774i	−0.348567	+	0.937284i	$0.613332\pi$
$830$		0			0
$831$		0			0
$832$		0			0
$833$		0			0
$834$		0			0
$835$	3.34707	−	5.79729i	0.115830	−	0.200623i
$836$		0			0
$837$		0			0
$838$		0			0
$839$	15.7821	+	27.3354i	0.544859	+	0.943723i	0.998616	+	0.0525978i	$0.0167501\pi$
$839$	15.7821	+	27.3354i	0.544859	+	0.943723i	−0.453757	+	0.891126i	$0.649917\pi$
$840$		0			0
$841$	3.28812	−	5.69519i	0.113383	−	0.196386i
$842$		0			0
$843$		0			0
$844$		0			0
$845$	−3.46328	−	5.99858i	−0.119140	−	0.206357i
$846$		0			0
$847$		0			0
$848$		0			0
$849$		0			0
$850$		0			0
$851$	−13.2870	+	7.67126i	−0.455473	+	0.262967i
$852$		0			0
$853$	7.50412	−	4.33250i	0.256936	−	0.148342i	−0.366000	−	0.930615i	$-0.619273\pi$
$853$	7.50412	−	4.33250i	0.256936	−	0.148342i	0.622936	+	0.782273i	$0.285940\pi$
$854$		0			0
$855$		0			0
$856$		0			0
$857$	−22.8878			−0.781834			−0.390917	−	0.920426i	$-0.627842\pi$
$857$	−22.8878			−0.781834			−0.390917	+	0.920426i	$0.627842\pi$
$858$		0			0
$859$			13.2701i			0.452769i	0.974038	+	0.226384i	$0.0726905\pi$
$859$			13.2701i			0.452769i	−0.974038	+	0.226384i	$0.927309\pi$
$860$		0			0
$861$		0			0
$862$		0			0
$863$	−0.646706	+	0.373376i	−0.0220141	+	0.0127099i	−0.510967	−	0.859601i	$-0.670712\pi$
$863$	−0.646706	+	0.373376i	−0.0220141	+	0.0127099i	0.488953	+	0.872310i	$0.337379\pi$
$864$		0			0
$865$	5.69773	−	9.86875i	0.193729	−	0.335548i
$866$		0			0
$867$		0			0
$868$		0			0
$869$	−31.2951	−	18.0683i	−1.06162	−	0.612924i
$870$		0			0
$871$			1.55436i			0.0526676i
$872$		0			0
$873$		0			0
$874$		0			0
$875$		0			0
$876$		0			0
$877$	4.37919			0.147875			0.0739373	−	0.997263i	$-0.476444\pi$
$877$	4.37919			0.147875			0.0739373	+	0.997263i	$0.476444\pi$
$878$		0			0
$879$		0			0
$880$		0			0
$881$	14.1505			0.476742			0.238371	−	0.971174i	$-0.423387\pi$
$881$	14.1505			0.476742			0.238371	+	0.971174i	$0.423387\pi$
$882$		0			0
$883$	23.0261			0.774890			0.387445	−	0.921893i	$-0.373358\pi$
$883$	23.0261			0.774890			0.387445	+	0.921893i	$0.373358\pi$
$884$		0			0
$885$		0			0
$886$		0			0
$887$	−52.3623			−1.75815			−0.879077	−	0.476679i	$-0.841840\pi$
$887$	−52.3623			−1.75815			−0.879077	+	0.476679i	$0.841840\pi$
$888$		0			0
$889$		0			0
$890$		0			0
$891$		0			0
$892$		0			0
$893$			13.5691i			0.454073i
$894$		0			0
$895$	−2.68519	−	1.55029i	−0.0897560	−	0.0518206i
$896$		0			0
$897$		0			0
$898$		0			0
$899$	2.75610	−	4.77370i	0.0919209	−	0.159212i
$900$		0			0
$901$	41.2667	−	23.8254i	1.37479	−	0.793738i
$902$		0			0
$903$		0			0
$904$		0			0
$905$		−	3.94236i		−	0.131048i
$906$		0			0
$907$	−24.3804			−0.809539			−0.404770	−	0.914419i	$-0.632648\pi$
$907$	−24.3804			−0.809539			−0.404770	+	0.914419i	$0.632648\pi$
$908$		0			0
$909$		0			0
$910$		0			0
$911$	46.8606	−	27.0550i	1.55256	−	0.896372i	0.554629	−	0.832098i	$-0.312860\pi$
$911$	46.8606	−	27.0550i	1.55256	−	0.896372i	0.997932	−	0.0642741i	$-0.0204732\pi$
$912$		0			0
$913$	−8.70372	+	5.02509i	−0.288051	+	0.166306i
$914$		0			0
$915$		0			0
$916$		0			0
$917$		0			0
$918$		0			0
$919$	−3.25947	−	5.64556i	−0.107520	−	0.186230i	0.807245	−	0.590216i	$-0.200958\pi$
$919$	−3.25947	−	5.64556i	−0.107520	−	0.186230i	−0.914765	+	0.403986i	$0.867624\pi$
$920$		0			0
$921$		0			0
$922$		0			0
$923$	0.217942	−	0.377487i	0.00717365	−	0.0124251i
$924$		0			0
$925$	18.3867	+	31.8467i	0.604550	+	1.04711i
$926$		0			0
$927$		0			0
$928$		0			0
$929$	−13.9048	+	24.0838i	−0.456202	+	0.790165i	−0.998756	−	0.0498555i	$-0.984124\pi$
$929$	−13.9048	+	24.0838i	−0.456202	+	0.790165i	0.542554	+	0.840021i	$0.317457\pi$
$930$		0			0
$931$		0			0
$932$		0			0
$933$		0			0
$934$		0			0
$935$	7.84511	+	4.52938i	0.256563	+	0.148126i
$936$		0			0
$937$		−	54.8174i		−	1.79081i	−0.445256	−	0.895403i	$-0.646887\pi$
$937$		−	54.8174i		−	1.79081i	0.445256	−	0.895403i	$-0.353113\pi$
$938$		0			0
$939$		0			0
$940$		0			0
$941$	−2.56526	−	4.44317i	−0.0836252	−	0.144843i	0.821179	−	0.570670i	$-0.193317\pi$
$941$	−2.56526	−	4.44317i	−0.0836252	−	0.144843i	−0.904805	+	0.425827i	$0.859983\pi$
$942$		0			0
$943$	−15.6413	−	9.03052i	−0.509351	−	0.294074i
$944$		0			0
$945$		0			0
$946$		0			0
$947$	−0.606033	−	0.349893i	−0.0196934	−	0.0113700i	0.490121	−	0.871654i	$-0.336953\pi$
$947$	−0.606033	−	0.349893i	−0.0196934	−	0.0113700i	−0.509814	+	0.860284i	$0.670286\pi$
$948$		0			0
$949$	−0.0257198	−	0.0445480i	−0.000834899	−	0.00144609i
$950$		0			0
$951$		0			0
$952$		0			0
$953$			0.162845i			0.00527506i	0.999997	+	0.00263753i	$0.000839552\pi$
$953$			0.162845i			0.00527506i	−0.999997	+	0.00263753i	$0.999160\pi$
$954$		0			0
$955$	3.66081	+	2.11357i	0.118461	+	0.0683934i
$956$		0			0
$957$		0			0
$958$		0			0
$959$		0			0
$960$		0			0
$961$	−15.0730	+	26.1071i	−0.486225	+	0.842166i
$962$		0			0
$963$		0			0
$964$		0			0
$965$	1.68897	+	2.92539i	0.0543700	+	0.0941716i
$966$		0			0
$967$	−15.6968	+	27.1876i	−0.504773	+	0.874293i	0.495211	+	0.868773i	$0.335091\pi$
$967$	−15.6968	+	27.1876i	−0.504773	+	0.874293i	−0.999985	+	0.00552073i	$0.998243\pi$
$968$		0			0
$969$		0			0
$970$		0			0
$971$	−1.46120	−	2.53087i	−0.0468920	−	0.0812194i	0.841627	−	0.540060i	$-0.181598\pi$
$971$	−1.46120	−	2.53087i	−0.0468920	−	0.0812194i	−0.888519	+	0.458840i	$0.848265\pi$
$972$		0			0
$973$		0			0
$974$		0			0
$975$		0			0
$976$		0			0
$977$	−24.7013	+	14.2613i	−0.790264	+	0.456259i	−0.840056	−	0.542500i	$-0.817478\pi$
$977$	−24.7013	+	14.2613i	−0.790264	+	0.456259i	0.0497913	+	0.998760i	$0.484144\pi$
$978$		0			0
$979$	−58.2425	+	33.6263i	−1.86144	+	1.07470i
$980$		0			0
$981$		0			0
$982$		0			0
$983$	11.2579			0.359073			0.179536	−	0.983751i	$-0.442540\pi$
$983$	11.2579			0.359073			0.179536	+	0.983751i	$0.442540\pi$
$984$		0			0
$985$			8.08771i			0.257696i
$986$		0			0
$987$		0			0
$988$		0			0
$989$	11.0588	−	6.38481i	0.351650	−	0.203025i
$990$		0			0
$991$	−10.5811	+	18.3270i	−0.336120	+	0.582177i	−0.983699	−	0.179821i	$-0.942448\pi$
$991$	−10.5811	+	18.3270i	−0.336120	+	0.582177i	0.647579	+	0.761998i	$0.275781\pi$
$992$		0			0
$993$		0			0
$994$		0			0
$995$	−3.95114	−	2.28119i	−0.125260	−	0.0723187i
$996$		0			0
$997$			16.9251i			0.536024i	0.963416	+	0.268012i	$0.0863667\pi$
$997$			16.9251i			0.536024i	−0.963416	+	0.268012i	$0.913633\pi$
$998$		0			0
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	5292.2.bm.b.2285.5		16
3.2	odd	2		1764.2.bm.b.1697.5		16
7.2	even	3		5292.2.w.a.1097.4		16
7.3	odd	6		756.2.x.a.125.5		16
7.4	even	3		756.2.x.a.125.4		16
7.5	odd	6		5292.2.w.a.1097.5		16
7.6	odd	2	inner	5292.2.bm.b.2285.4		16
9.2	odd	6		5292.2.w.a.521.5		16
9.7	even	3		1764.2.w.a.1109.8		16
21.2	odd	6		1764.2.w.a.509.1		16
21.5	even	6		1764.2.w.a.509.8		16
21.11	odd	6		252.2.x.a.41.6	yes	16
21.17	even	6		252.2.x.a.41.3	✓	16
21.20	even	2		1764.2.bm.b.1697.4		16
28.3	even	6		3024.2.cc.c.881.5		16
28.11	odd	6		3024.2.cc.c.881.4		16
63.2	odd	6	inner	5292.2.bm.b.4625.4		16
63.4	even	3		2268.2.f.b.1133.10		16
63.11	odd	6		756.2.x.a.629.5		16
63.16	even	3		1764.2.bm.b.1685.4		16
63.20	even	6		5292.2.w.a.521.4		16
63.25	even	3		252.2.x.a.209.3	yes	16
63.31	odd	6		2268.2.f.b.1133.7		16
63.32	odd	6		2268.2.f.b.1133.8		16
63.34	odd	6		1764.2.w.a.1109.1		16
63.38	even	6		756.2.x.a.629.4		16
63.47	even	6	inner	5292.2.bm.b.4625.5		16
63.52	odd	6		252.2.x.a.209.6	yes	16
63.59	even	6		2268.2.f.b.1133.9		16
63.61	odd	6		1764.2.bm.b.1685.5		16
84.11	even	6		1008.2.cc.c.545.3		16
84.59	odd	6		1008.2.cc.c.545.6		16
252.11	even	6		3024.2.cc.c.2897.5		16
252.115	even	6		1008.2.cc.c.209.3		16
252.151	odd	6		1008.2.cc.c.209.6		16
252.227	odd	6		3024.2.cc.c.2897.4		16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
252.2.x.a.41.3	✓	16	21.17	even	6
252.2.x.a.41.6	yes	16	21.11	odd	6
252.2.x.a.209.3	yes	16	63.25	even	3
252.2.x.a.209.6	yes	16	63.52	odd	6
756.2.x.a.125.4		16	7.4	even	3
756.2.x.a.125.5		16	7.3	odd	6
756.2.x.a.629.4		16	63.38	even	6
756.2.x.a.629.5		16	63.11	odd	6
1008.2.cc.c.209.3		16	252.115	even	6
1008.2.cc.c.209.6		16	252.151	odd	6
1008.2.cc.c.545.3		16	84.11	even	6
1008.2.cc.c.545.6		16	84.59	odd	6
1764.2.w.a.509.1		16	21.2	odd	6
1764.2.w.a.509.8		16	21.5	even	6
1764.2.w.a.1109.1		16	63.34	odd	6
1764.2.w.a.1109.8		16	9.7	even	3
1764.2.bm.b.1685.4		16	63.16	even	3
1764.2.bm.b.1685.5		16	63.61	odd	6
1764.2.bm.b.1697.4		16	21.20	even	2
1764.2.bm.b.1697.5		16	3.2	odd	2
2268.2.f.b.1133.7		16	63.31	odd	6
2268.2.f.b.1133.8		16	63.32	odd	6
2268.2.f.b.1133.9		16	63.59	even	6
2268.2.f.b.1133.10		16	63.4	even	3
3024.2.cc.c.881.4		16	28.11	odd	6
3024.2.cc.c.881.5		16	28.3	even	6
3024.2.cc.c.2897.4		16	252.227	odd	6
3024.2.cc.c.2897.5		16	252.11	even	6
5292.2.w.a.521.4		16	63.20	even	6
5292.2.w.a.521.5		16	9.2	odd	6
5292.2.w.a.1097.4		16	7.2	even	3
5292.2.w.a.1097.5		16	7.5	odd	6
5292.2.bm.b.2285.4		16	7.6	odd	2	inner
5292.2.bm.b.2285.5		16	1.1	even	1	trivial
5292.2.bm.b.4625.4		16	63.2	odd	6	inner
5292.2.bm.b.4625.5		16	63.47	even	6	inner

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 5292 = 2^{2} \cdot 3^{3} \cdot 7^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	5292.bm (of order \(6\), degree \(2\), not minimal)

\(f(q)\)	\(=\)	\(q+0.533560 q^{5} +O(q^{10})\)	\(q+0.533560 q^{5} +3.92524i q^{11} +(-0.116911 - 0.0674987i) q^{13} +(2.16266 - 3.74584i) q^{17} +(-1.93067 + 1.11467i) q^{19} -1.96732i q^{23} -4.71531 q^{25} +(5.16548 - 2.98229i) q^{29} +(0.800341 - 0.462077i) q^{31} +(-3.89936 - 6.75388i) q^{37} +(4.59027 - 7.95059i) q^{41} +(3.24544 + 5.62127i) q^{43} +(3.04329 - 5.27114i) q^{47} +(9.54072 + 5.50834i) q^{53} +2.09435i q^{55} +(1.89588 + 3.28377i) q^{59} +(9.35116 + 5.39889i) q^{61} +(-0.0623791 - 0.0360146i) q^{65} +(-5.75701 - 9.97144i) q^{67} +3.22884i q^{71} +(0.329991 + 0.190521i) q^{73} +(-4.60310 + 7.97280i) q^{79} +(1.28020 + 2.21737i) q^{83} +(1.15391 - 1.99863i) q^{85} +(8.56670 + 14.8380i) q^{89} +(-1.03013 + 0.594746i) q^{95} +(13.6747 - 7.89507i) q^{97} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q+O(q^{10}) \) 16 * q	\( 16 q + 16 q^{25} + 12 q^{29} - 2 q^{37} + 4 q^{43} + 36 q^{53} - 24 q^{65} + 14 q^{67} + 20 q^{79} + 6 q^{85} - 60 q^{95}+O(q^{100}) \) 16 * q + 16 * q^25 + 12 * q^29 - 2 * q^37 + 4 * q^43 + 36 * q^53 - 24 * q^65 + 14 * q^67 + 20 * q^79 + 6 * q^85 - 60 * q^95

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