LMFDB - Embedded newform 135.3.d.g.134.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [135,3,Mod(134,135)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(135, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("135.134");

S:= CuspForms(chi, 3);

N := Newforms(S);

Level:	$ N $	$=$	$ 135 = 3^{3} \cdot 5 $
Weight:	$ k $	$=$	$ 3 $
Character orbit:	$[\chi]$	$=$	135.d (of order $2$, degree $1$, minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$3.67848356886$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	$\Q(i, \sqrt{10})$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} + 25 $ x^4 + 25
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3^{2} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			134.3
Root			$-1.58114 + 1.58114i$ of defining polynomial
Character	$\chi$	$=$	135.134
Dual form			135.3.d.g.134.4

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q+3.16228 q^{2} +6.00000 q^{4} +(1.58114 - 4.74342i) q^{5} -3.00000i q^{7} +6.32456 q^{8} +(5.00000 - 15.0000i) q^{10} +9.48683i q^{11} +21.0000i q^{13} -9.48683i q^{14} -4.00000 q^{16} +12.6491 q^{17} -31.0000 q^{19} +(9.48683 - 28.4605i) q^{20} +30.0000i q^{22} +22.1359 q^{23} +(-20.0000 - 15.0000i) q^{25} +66.4078i q^{26} -18.0000i q^{28} -47.4342i q^{29} -16.0000 q^{31} -37.9473 q^{32} +40.0000 q^{34} +(-14.2302 - 4.74342i) q^{35} +27.0000i q^{37} -98.0306 q^{38} +(10.0000 - 30.0000i) q^{40} +47.4342i q^{41} -48.0000i q^{43} +56.9210i q^{44} +70.0000 q^{46} +12.6491 q^{47} +40.0000 q^{49} +(-63.2456 - 47.4342i) q^{50} +126.000i q^{52} +41.1096 q^{53} +(45.0000 + 15.0000i) q^{55} -18.9737i q^{56} -150.000i q^{58} -37.9473i q^{59} -1.00000 q^{61} -50.5964 q^{62} -104.000 q^{64} +(99.6117 + 33.2039i) q^{65} +21.0000i q^{67} +75.8947 q^{68} +(-45.0000 - 15.0000i) q^{70} -28.4605i q^{71} +27.0000i q^{73} +85.3815i q^{74} -186.000 q^{76} +28.4605 q^{77} -1.00000 q^{79} +(-6.32456 + 18.9737i) q^{80} +150.000i q^{82} -110.680 q^{83} +(20.0000 - 60.0000i) q^{85} -151.789i q^{86} +60.0000i q^{88} -113.842i q^{89} +63.0000 q^{91} +132.816 q^{92} +40.0000 q^{94} +(-49.0153 + 147.046i) q^{95} -93.0000i q^{97} +126.491 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 24 q^{4} + 20 q^{10} - 16 q^{16} - 124 q^{19} - 80 q^{25} - 64 q^{31} + 160 q^{34} + 40 q^{40} + 280 q^{46} + 160 q^{49} + 180 q^{55} - 4 q^{61} - 416 q^{64} - 180 q^{70} - 744 q^{76} - 4 q^{79} + 80 q^{85}+ \cdots + 160 q^{94}+O(q^{100}) $ 4 * q + 24 * q^4 + 20 * q^10 - 16 * q^16 - 124 * q^19 - 80 * q^25 - 64 * q^31 + 160 * q^34 + 40 * q^40 + 280 * q^46 + 160 * q^49 + 180 * q^55 - 4 * q^61 - 416 * q^64 - 180 * q^70 - 744 * q^76 - 4 * q^79 + 80 * q^85 + 252 * q^91 + 160 * q^94

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/135\mathbb{Z}\right)^\times$.

$n$	$56$	$82$
$\chi(n)$	$-1$	$-1$

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 $

$2$	3.16228			1.58114			0.790569	−	0.612372i	$-0.209785\pi$
$2$	3.16228			1.58114			0.790569	+	0.612372i	$0.209785\pi$
$3$		0			0
$3$		0			0
$4$	6.00000			1.50000
$5$	1.58114	−	4.74342i	0.316228	−	0.948683i
$5$	1.58114	−	4.74342i	0.316228	−	0.948683i
$6$		0			0
$7$		−	3.00000i		−	0.428571i	−0.976771	−	0.214286i	$-0.931258\pi$
$7$		−	3.00000i		−	0.428571i	0.976771	−	0.214286i	$-0.0687424\pi$
$8$	6.32456			0.790569
$9$		0			0
$10$	5.00000	−	15.0000i	0.500000	−	1.50000i
$11$			9.48683i			0.862439i	0.902247	+	0.431220i	$0.141917\pi$
$11$			9.48683i			0.862439i	−0.902247	+	0.431220i	$0.858083\pi$
$12$		0			0
$13$			21.0000i			1.61538i	0.589604	+	0.807692i	$0.299284\pi$
$13$			21.0000i			1.61538i	−0.589604	+	0.807692i	$0.700716\pi$
$14$		−	9.48683i		−	0.677631i
$15$		0			0
$16$	−4.00000			−0.250000
$17$	12.6491			0.744065			0.372033	−	0.928220i	$-0.378661\pi$
$17$	12.6491			0.744065			0.372033	+	0.928220i	$0.378661\pi$
$18$		0			0
$19$	−31.0000			−1.63158			−0.815789	−	0.578349i	$-0.803697\pi$
$19$	−31.0000			−1.63158			−0.815789	+	0.578349i	$0.803697\pi$
$20$	9.48683	−	28.4605i	0.474342	−	1.42302i
$21$		0			0
$22$			30.0000i			1.36364i
$23$	22.1359			0.962432			0.481216	−	0.876602i	$-0.340195\pi$
$23$	22.1359			0.962432			0.481216	+	0.876602i	$0.340195\pi$
$24$		0			0
$25$	−20.0000	−	15.0000i	−0.800000	−	0.600000i
$26$			66.4078i			2.55415i
$27$		0			0
$28$		−	18.0000i		−	0.642857i
$29$		−	47.4342i		−	1.63566i	−0.575459	−	0.817830i	$-0.695177\pi$
$29$		−	47.4342i		−	1.63566i	0.575459	−	0.817830i	$-0.304823\pi$
$30$		0			0
$31$	−16.0000			−0.516129			−0.258065	−	0.966128i	$-0.583085\pi$
$31$	−16.0000			−0.516129			−0.258065	+	0.966128i	$0.583085\pi$
$32$	−37.9473			−1.18585
$33$		0			0
$34$	40.0000			1.17647
$35$	−14.2302	−	4.74342i	−0.406579	−	0.135526i
$36$		0			0
$37$			27.0000i			0.729730i	0.931060	+	0.364865i	$0.118885\pi$
$37$			27.0000i			0.729730i	−0.931060	+	0.364865i	$0.881115\pi$
$38$	−98.0306			−2.57975
$39$		0			0
$40$	10.0000	−	30.0000i	0.250000	−	0.750000i
$41$			47.4342i			1.15693i	0.815707	+	0.578465i	$0.196348\pi$
$41$			47.4342i			1.15693i	−0.815707	+	0.578465i	$0.803652\pi$
$42$		0			0
$43$		−	48.0000i		−	1.11628i	−0.829747	−	0.558140i	$-0.811515\pi$
$43$		−	48.0000i		−	1.11628i	0.829747	−	0.558140i	$-0.188485\pi$
$44$			56.9210i			1.29366i
$45$		0			0
$46$	70.0000			1.52174
$47$	12.6491			0.269130			0.134565	−	0.990905i	$-0.457036\pi$
$47$	12.6491			0.269130			0.134565	+	0.990905i	$0.457036\pi$
$48$		0			0
$49$	40.0000			0.816327
$50$	−63.2456	−	47.4342i	−1.26491	−	0.948683i
$51$		0			0
$52$			126.000i			2.42308i
$53$	41.1096			0.775653			0.387827	−	0.921732i	$-0.373226\pi$
$53$	41.1096			0.775653			0.387827	+	0.921732i	$0.373226\pi$
$54$		0			0
$55$	45.0000	+	15.0000i	0.818182	+	0.272727i
$56$		−	18.9737i		−	0.338815i
$57$		0			0
$58$		−	150.000i		−	2.58621i
$59$		−	37.9473i		−	0.643175i	−0.946880	−	0.321588i	$-0.895784\pi$
$59$		−	37.9473i		−	0.643175i	0.946880	−	0.321588i	$-0.104216\pi$
$60$		0			0
$61$	−1.00000			−0.0163934			−0.00819672	−	0.999966i	$-0.502609\pi$
$61$	−1.00000			−0.0163934			−0.00819672	+	0.999966i	$0.502609\pi$
$62$	−50.5964			−0.816072
$63$		0			0
$64$	−104.000			−1.62500
$65$	99.6117	+	33.2039i	1.53249	+	0.510829i
$66$		0			0
$67$			21.0000i			0.313433i	0.987644	+	0.156716i	$0.0500909\pi$
$67$			21.0000i			0.313433i	−0.987644	+	0.156716i	$0.949909\pi$
$68$	75.8947			1.11610
$69$		0			0
$70$	−45.0000	−	15.0000i	−0.642857	−	0.214286i
$71$		−	28.4605i		−	0.400852i	−0.979709	−	0.200426i	$-0.935767\pi$
$71$		−	28.4605i		−	0.400852i	0.979709	−	0.200426i	$-0.0642326\pi$
$72$		0			0
$73$			27.0000i			0.369863i	0.982751	+	0.184932i	$0.0592063\pi$
$73$			27.0000i			0.369863i	−0.982751	+	0.184932i	$0.940794\pi$
$74$			85.3815i			1.15380i
$75$		0			0
$76$	−186.000			−2.44737
$77$	28.4605			0.369617
$78$		0			0
$79$	−1.00000			−0.0126582			−0.00632911	−	0.999980i	$-0.502015\pi$
$79$	−1.00000			−0.0126582			−0.00632911	+	0.999980i	$0.502015\pi$
$80$	−6.32456	+	18.9737i	−0.0790569	+	0.237171i
$81$		0			0
$82$			150.000i			1.82927i
$83$	−110.680			−1.33349			−0.666745	−	0.745286i	$-0.732313\pi$
$83$	−110.680			−1.33349			−0.666745	+	0.745286i	$0.732313\pi$
$84$		0			0
$85$	20.0000	−	60.0000i	0.235294	−	0.705882i
$86$		−	151.789i		−	1.76499i
$87$		0			0
$88$			60.0000i			0.681818i
$89$		−	113.842i		−	1.27912i	−0.768740	−	0.639562i	$-0.779116\pi$
$89$		−	113.842i		−	1.27912i	0.768740	−	0.639562i	$-0.220884\pi$
$90$		0			0
$91$	63.0000			0.692308
$92$	132.816			1.44365
$93$		0			0
$94$	40.0000			0.425532
$95$	−49.0153	+	147.046i	−0.515951	+	1.54785i
$96$		0			0
$97$		−	93.0000i		−	0.958763i	−0.877607	−	0.479381i	$-0.840861\pi$
$97$		−	93.0000i		−	0.958763i	0.877607	−	0.479381i	$-0.159139\pi$
$98$	126.491			1.29073
$99$		0			0
$100$	−120.000	−	90.0000i	−1.20000	−	0.900000i
$101$		−	75.8947i		−	0.751432i	−0.926735	−	0.375716i	$-0.877397\pi$
$101$		−	75.8947i		−	0.751432i	0.926735	−	0.375716i	$-0.122603\pi$
$102$		0			0
$103$			147.000i			1.42718i	0.700561	+	0.713592i	$0.252933\pi$
$103$			147.000i			1.42718i	−0.700561	+	0.713592i	$0.747067\pi$
$104$			132.816i			1.27707i
$105$		0			0
$106$	130.000			1.22642
$107$	22.1359			0.206878			0.103439	−	0.994636i	$-0.467015\pi$
$107$	22.1359			0.206878			0.103439	+	0.994636i	$0.467015\pi$
$108$		0			0
$109$	104.000			0.954128			0.477064	−	0.878868i	$-0.341701\pi$
$109$	104.000			0.954128			0.477064	+	0.878868i	$0.341701\pi$
$110$	142.302	+	47.4342i	1.29366	+	0.431220i
$111$		0			0
$112$			12.0000i			0.107143i
$113$	173.925			1.53916			0.769581	−	0.638549i	$-0.220465\pi$
$113$	173.925			1.53916			0.769581	+	0.638549i	$0.220465\pi$
$114$		0			0
$115$	35.0000	−	105.000i	0.304348	−	0.913043i
$116$		−	284.605i		−	2.45349i
$117$		0			0
$118$		−	120.000i		−	1.01695i
$119$		−	37.9473i		−	0.318885i
$120$		0			0
$121$	31.0000			0.256198
$122$	−3.16228			−0.0259203
$123$		0			0
$124$	−96.0000			−0.774194
$125$	−102.774	+	71.1512i	−0.822192	+	0.569210i
$126$		0			0
$127$		−	144.000i		−	1.13386i	−0.823767	−	0.566929i	$-0.808131\pi$
$127$		−	144.000i		−	1.13386i	0.823767	−	0.566929i	$-0.191869\pi$
$128$	−177.088			−1.38350
$129$		0			0
$130$	315.000	+	105.000i	2.42308	+	0.807692i
$131$			75.8947i			0.579349i	0.957125	+	0.289674i	$0.0935470\pi$
$131$			75.8947i			0.579349i	−0.957125	+	0.289674i	$0.906453\pi$
$132$		0			0
$133$			93.0000i			0.699248i
$134$			66.4078i			0.495581i
$135$		0			0
$136$	80.0000			0.588235
$137$	−53.7587			−0.392399			−0.196200	−	0.980564i	$-0.562860\pi$
$137$	−53.7587			−0.392399			−0.196200	+	0.980564i	$0.562860\pi$
$138$		0			0
$139$	−1.00000			−0.00719424			−0.00359712	−	0.999994i	$-0.501145\pi$
$139$	−1.00000			−0.00719424			−0.00359712	+	0.999994i	$0.501145\pi$
$140$	−85.3815	−	28.4605i	−0.609868	−	0.203289i
$141$		0			0
$142$		−	90.0000i		−	0.633803i
$143$	−199.223			−1.39317
$144$		0			0
$145$	−225.000	−	75.0000i	−1.55172	−	0.517241i
$146$			85.3815i			0.584805i
$147$		0			0
$148$			162.000i			1.09459i
$149$			75.8947i			0.509360i	0.967025	+	0.254680i	$0.0819702\pi$
$149$			75.8947i			0.509360i	−0.967025	+	0.254680i	$0.918030\pi$
$150$		0			0
$151$	−31.0000			−0.205298			−0.102649	−	0.994718i	$-0.532732\pi$
$151$	−31.0000			−0.205298			−0.102649	+	0.994718i	$0.532732\pi$
$152$	−196.061			−1.28988
$153$		0			0
$154$	90.0000			0.584416
$155$	−25.2982	+	75.8947i	−0.163214	+	0.489643i
$156$		0			0
$157$			102.000i			0.649682i	0.945769	+	0.324841i	$0.105311\pi$
$157$			102.000i			0.649682i	−0.945769	+	0.324841i	$0.894689\pi$
$158$	−3.16228			−0.0200144
$159$		0			0
$160$	−60.0000	+	180.000i	−0.375000	+	1.12500i
$161$		−	66.4078i		−	0.412471i
$162$		0			0
$163$			171.000i			1.04908i	0.851386	+	0.524540i	$0.175763\pi$
$163$			171.000i			1.04908i	−0.851386	+	0.524540i	$0.824237\pi$
$164$			284.605i			1.73540i
$165$		0			0
$166$	−350.000			−2.10843
$167$	−290.930			−1.74209			−0.871047	−	0.491200i	$-0.836558\pi$
$167$	−290.930			−1.74209			−0.871047	+	0.491200i	$0.836558\pi$
$168$		0			0
$169$	−272.000			−1.60947
$170$	63.2456	−	189.737i	0.372033	−	1.11610i
$171$		0			0
$172$		−	288.000i		−	1.67442i
$173$	−25.2982			−0.146232			−0.0731162	−	0.997323i	$-0.523294\pi$
$173$	−25.2982			−0.146232			−0.0731162	+	0.997323i	$0.523294\pi$
$174$		0			0
$175$	−45.0000	+	60.0000i	−0.257143	+	0.342857i
$176$		−	37.9473i		−	0.215610i
$177$		0			0
$178$		−	360.000i		−	2.02247i
$179$		−	199.223i		−	1.11298i	−0.830854	−	0.556490i	$-0.812148\pi$
$179$		−	199.223i		−	1.11298i	0.830854	−	0.556490i	$-0.187852\pi$
$180$		0			0
$181$	−103.000			−0.569061			−0.284530	−	0.958667i	$-0.591838\pi$
$181$	−103.000			−0.569061			−0.284530	+	0.958667i	$0.591838\pi$
$182$	199.223			1.09463
$183$		0			0
$184$	140.000			0.760870
$185$	128.072	+	42.6907i	0.692282	+	0.230761i
$186$		0			0
$187$			120.000i			0.641711i
$188$	75.8947			0.403695
$189$		0			0
$190$	−155.000	+	465.000i	−0.815789	+	2.44737i
$191$			265.631i			1.39074i	0.718652	+	0.695370i	$0.244759\pi$
$191$			265.631i			1.39074i	−0.718652	+	0.695370i	$0.755241\pi$
$192$		0			0
$193$		−	213.000i		−	1.10363i	−0.833968	−	0.551813i	$-0.813936\pi$
$193$		−	213.000i		−	1.10363i	0.833968	−	0.551813i	$-0.186064\pi$
$194$		−	294.092i		−	1.51594i
$195$		0			0
$196$	240.000			1.22449
$197$	240.333			1.21996			0.609982	−	0.792415i	$-0.291176\pi$
$197$	240.333			1.21996			0.609982	+	0.792415i	$0.291176\pi$
$198$		0			0
$199$	47.0000			0.236181			0.118090	−	0.993003i	$-0.462323\pi$
$199$	47.0000			0.236181			0.118090	+	0.993003i	$0.462323\pi$
$200$	−126.491	−	94.8683i	−0.632456	−	0.474342i
$201$		0			0
$202$		−	240.000i		−	1.18812i
$203$	−142.302			−0.700998
$204$		0			0
$205$	225.000	+	75.0000i	1.09756	+	0.365854i
$206$			464.855i			2.25658i
$207$		0			0
$208$		−	84.0000i		−	0.403846i
$209$		−	294.092i		−	1.40714i
$210$		0			0
$211$	−223.000			−1.05687			−0.528436	−	0.848973i	$-0.677221\pi$
$211$	−223.000			−1.05687			−0.528436	+	0.848973i	$0.677221\pi$
$212$	246.658			1.16348
$213$		0			0
$214$	70.0000			0.327103
$215$	−227.684	−	75.8947i	−1.05900	−	0.352998i
$216$		0			0
$217$			48.0000i			0.221198i
$218$	328.877			1.50861
$219$		0			0
$220$	270.000	+	90.0000i	1.22727	+	0.409091i
$221$			265.631i			1.20195i
$222$		0			0
$223$			312.000i			1.39910i	0.714582	+	0.699552i	$0.246617\pi$
$223$			312.000i			1.39910i	−0.714582	+	0.699552i	$0.753383\pi$
$224$			113.842i			0.508223i
$225$		0			0
$226$	550.000			2.43363
$227$	−338.364			−1.49059			−0.745295	−	0.666735i	$-0.767691\pi$
$227$	−338.364			−1.49059			−0.745295	+	0.666735i	$0.767691\pi$
$228$		0			0
$229$	−16.0000			−0.0698690			−0.0349345	−	0.999390i	$-0.511122\pi$
$229$	−16.0000			−0.0698690			−0.0349345	+	0.999390i	$0.511122\pi$
$230$	110.680	−	332.039i	0.481216	−	1.44365i
$231$		0			0
$232$		−	300.000i		−	1.29310i
$233$	50.5964			0.217152			0.108576	−	0.994088i	$-0.465371\pi$
$233$	50.5964			0.217152			0.108576	+	0.994088i	$0.465371\pi$
$234$		0			0
$235$	20.0000	−	60.0000i	0.0851064	−	0.255319i
$236$		−	227.684i		−	0.964763i
$237$		0			0
$238$		−	120.000i		−	0.504202i
$239$			104.355i			0.436632i	0.975878	+	0.218316i	$0.0700564\pi$
$239$			104.355i			0.436632i	−0.975878	+	0.218316i	$0.929944\pi$
$240$		0			0
$241$	287.000			1.19087			0.595436	−	0.803403i	$-0.296979\pi$
$241$	287.000			1.19087			0.595436	+	0.803403i	$0.296979\pi$
$242$	98.0306			0.405085
$243$		0			0
$244$	−6.00000			−0.0245902
$245$	63.2456	−	189.737i	0.258145	−	0.774435i
$246$		0			0
$247$		−	651.000i		−	2.63563i
$248$	−101.193			−0.408036
$249$		0			0
$250$	−325.000	+	225.000i	−1.30000	+	0.900000i
$251$			256.144i			1.02050i	0.860027	+	0.510248i	$0.170446\pi$
$251$			256.144i			1.02050i	−0.860027	+	0.510248i	$0.829554\pi$
$252$		0			0
$253$			210.000i			0.830040i
$254$		−	455.368i		−	1.79279i
$255$		0			0
$256$	−144.000			−0.562500
$257$	278.280			1.08280			0.541402	−	0.840764i	$-0.317894\pi$
$257$	278.280			1.08280			0.541402	+	0.840764i	$0.317894\pi$
$258$		0			0
$259$	81.0000			0.312741
$260$	597.670	+	199.223i	2.29873	+	0.766244i
$261$		0			0
$262$			240.000i			0.916031i
$263$	354.175			1.34667			0.673337	−	0.739336i	$-0.264860\pi$
$263$	354.175			1.34667			0.673337	+	0.739336i	$0.264860\pi$
$264$		0			0
$265$	65.0000	−	195.000i	0.245283	−	0.735849i
$266$			294.092i			1.10561i
$267$		0			0
$268$			126.000i			0.470149i
$269$			113.842i			0.423204i	0.977356	+	0.211602i	$0.0678681\pi$
$269$			113.842i			0.423204i	−0.977356	+	0.211602i	$0.932132\pi$
$270$		0			0
$271$	−121.000			−0.446494			−0.223247	−	0.974762i	$-0.571666\pi$
$271$	−121.000			−0.446494			−0.223247	+	0.974762i	$0.571666\pi$
$272$	−50.5964			−0.186016
$273$		0			0
$274$	−170.000			−0.620438
$275$	142.302	−	189.737i	0.517464	−	0.689951i
$276$		0			0
$277$		−	48.0000i		−	0.173285i	−0.996239	−	0.0866426i	$-0.972386\pi$
$277$		−	48.0000i		−	0.173285i	0.996239	−	0.0866426i	$-0.0276138\pi$
$278$	−3.16228			−0.0113751
$279$		0			0
$280$	−90.0000	−	30.0000i	−0.321429	−	0.107143i
$281$		−	303.579i		−	1.08035i	−0.841552	−	0.540176i	$-0.818358\pi$
$281$		−	303.579i		−	1.08035i	0.841552	−	0.540176i	$-0.181642\pi$
$282$		0			0
$283$			192.000i			0.678445i	0.940706	+	0.339223i	$0.110164\pi$
$283$			192.000i			0.678445i	−0.940706	+	0.339223i	$0.889836\pi$
$284$		−	170.763i		−	0.601278i
$285$		0			0
$286$	−630.000			−2.20280
$287$	142.302			0.495828
$288$		0			0
$289$	−129.000			−0.446367
$290$	−711.512	−	237.171i	−2.45349	−	0.817830i
$291$		0			0
$292$			162.000i			0.554795i
$293$	−205.548			−0.701529			−0.350765	−	0.936464i	$-0.614078\pi$
$293$	−205.548			−0.701529			−0.350765	+	0.936464i	$0.614078\pi$
$294$		0			0
$295$	−180.000	−	60.0000i	−0.610169	−	0.203390i
$296$			170.763i			0.576902i
$297$		0			0
$298$			240.000i			0.805369i
$299$			464.855i			1.55470i
$300$		0			0
$301$	−144.000			−0.478405
$302$	−98.0306			−0.324605
$303$		0			0
$304$	124.000			0.407895
$305$	−1.58114	+	4.74342i	−0.00518406	+	0.0155522i
$306$		0			0
$307$		−	198.000i		−	0.644951i	−0.946578	−	0.322476i	$-0.895485\pi$
$307$		−	198.000i		−	0.644951i	0.946578	−	0.322476i	$-0.104515\pi$
$308$	170.763			0.554425
$309$		0			0
$310$	−80.0000	+	240.000i	−0.258065	+	0.774194i
$311$		−	493.315i		−	1.58622i	−0.609077	−	0.793111i	$-0.708460\pi$
$311$		−	493.315i		−	1.58622i	0.609077	−	0.793111i	$-0.291540\pi$
$312$		0			0
$313$			411.000i			1.31310i	0.754283	+	0.656550i	$0.227985\pi$
$313$			411.000i			1.31310i	−0.754283	+	0.656550i	$0.772015\pi$
$314$			322.552i			1.02724i
$315$		0			0
$316$	−6.00000			−0.0189873
$317$	−252.982			−0.798051			−0.399026	−	0.916940i	$-0.630652\pi$
$317$	−252.982			−0.798051			−0.399026	+	0.916940i	$0.630652\pi$
$318$		0			0
$319$	450.000			1.41066
$320$	−164.438	+	493.315i	−0.513870	+	1.54161i
$321$		0			0
$322$		−	210.000i		−	0.652174i
$323$	−392.122			−1.21400
$324$		0			0
$325$	315.000	−	420.000i	0.969231	−	1.29231i
$326$			540.749i			1.65874i
$327$		0			0
$328$			300.000i			0.914634i
$329$		−	37.9473i		−	0.115341i
$330$		0			0
$331$	329.000			0.993958			0.496979	−	0.867763i	$-0.334443\pi$
$331$	329.000			0.993958			0.496979	+	0.867763i	$0.334443\pi$
$332$	−664.078			−2.00024
$333$		0			0
$334$	−920.000			−2.75449
$335$	99.6117	+	33.2039i	0.297348	+	0.0991162i
$336$		0			0
$337$			381.000i			1.13056i	0.824898	+	0.565282i	$0.191233\pi$
$337$			381.000i			1.13056i	−0.824898	+	0.565282i	$0.808767\pi$
$338$	−860.140			−2.54479
$339$		0			0
$340$	120.000	−	360.000i	0.352941	−	1.05882i
$341$		−	151.789i		−	0.445130i
$342$		0			0
$343$		−	267.000i		−	0.778426i
$344$		−	303.579i		−	0.882496i
$345$		0			0
$346$	−80.0000			−0.231214
$347$	117.004			0.337188			0.168594	−	0.985686i	$-0.446077\pi$
$347$	117.004			0.337188			0.168594	+	0.985686i	$0.446077\pi$
$348$		0			0
$349$	89.0000			0.255014			0.127507	−	0.991838i	$-0.459302\pi$
$349$	89.0000			0.255014			0.127507	+	0.991838i	$0.459302\pi$
$350$	−142.302	+	189.737i	−0.406579	+	0.542105i
$351$		0			0
$352$		−	360.000i		−	1.02273i
$353$	581.859			1.64833			0.824163	−	0.566353i	$-0.191646\pi$
$353$	581.859			1.64833			0.824163	+	0.566353i	$0.191646\pi$
$354$		0			0
$355$	−135.000	−	45.0000i	−0.380282	−	0.126761i
$356$		−	683.052i		−	1.91869i
$357$		0			0
$358$		−	630.000i		−	1.75978i
$359$			85.3815i			0.237831i	0.992904	+	0.118916i	$0.0379418\pi$
$359$			85.3815i			0.237831i	−0.992904	+	0.118916i	$0.962058\pi$
$360$		0			0
$361$	600.000			1.66205
$362$	−325.715			−0.899764
$363$		0			0
$364$	378.000			1.03846
$365$	128.072	+	42.6907i	0.350883	+	0.116961i
$366$		0			0
$367$			51.0000i			0.138965i	0.997583	+	0.0694823i	$0.0221347\pi$
$367$			51.0000i			0.138965i	−0.997583	+	0.0694823i	$0.977865\pi$
$368$	−88.5438			−0.240608
$369$		0			0
$370$	405.000	+	135.000i	1.09459	+	0.364865i
$371$		−	123.329i		−	0.332423i
$372$		0			0
$373$			237.000i			0.635389i	0.948193	+	0.317694i	$0.102909\pi$
$373$			237.000i			0.635389i	−0.948193	+	0.317694i	$0.897091\pi$
$374$			379.473i			1.01463i
$375$		0			0
$376$	80.0000			0.212766
$377$	996.117			2.64222
$378$		0			0
$379$	−103.000			−0.271768			−0.135884	−	0.990725i	$-0.543387\pi$
$379$	−103.000			−0.271768			−0.135884	+	0.990725i	$0.543387\pi$
$380$	−294.092	+	882.275i	−0.773926	+	2.32178i
$381$		0			0
$382$			840.000i			2.19895i
$383$	50.5964			0.132106			0.0660528	−	0.997816i	$-0.478959\pi$
$383$	50.5964			0.132106			0.0660528	+	0.997816i	$0.478959\pi$
$384$		0			0
$385$	45.0000	−	135.000i	0.116883	−	0.350649i
$386$		−	673.565i		−	1.74499i
$387$		0			0
$388$		−	558.000i		−	1.43814i
$389$			265.631i			0.682857i	0.939908	+	0.341428i	$0.110911\pi$
$389$			265.631i			0.682857i	−0.939908	+	0.341428i	$0.889089\pi$
$390$		0			0
$391$	280.000			0.716113
$392$	252.982			0.645363
$393$		0			0
$394$	760.000			1.92893
$395$	−1.58114	+	4.74342i	−0.00400288	+	0.0120086i
$396$		0			0
$397$			72.0000i			0.181360i	0.995880	+	0.0906801i	$0.0289041\pi$
$397$			72.0000i			0.181360i	−0.995880	+	0.0906801i	$0.971096\pi$
$398$	148.627			0.373435
$399$		0			0
$400$	80.0000	+	60.0000i	0.200000	+	0.150000i
$401$			104.355i			0.260237i	0.991498	+	0.130119i	$0.0415358\pi$
$401$			104.355i			0.260237i	−0.991498	+	0.130119i	$0.958464\pi$
$402$		0			0
$403$		−	336.000i		−	0.833747i
$404$		−	455.368i		−	1.12715i
$405$		0			0
$406$	−450.000			−1.10837
$407$	−256.144			−0.629348
$408$		0			0
$409$	137.000			0.334963			0.167482	−	0.985875i	$-0.446437\pi$
$409$	137.000			0.334963			0.167482	+	0.985875i	$0.446437\pi$
$410$	711.512	+	237.171i	1.73540	+	0.578465i
$411$		0			0
$412$			882.000i			2.14078i
$413$	−113.842			−0.275646
$414$		0			0
$415$	−175.000	+	525.000i	−0.421687	+	1.26506i
$416$		−	796.894i		−	1.91561i
$417$		0			0
$418$		−	930.000i		−	2.22488i
$419$		−	37.9473i		−	0.0905664i	−0.998974	−	0.0452832i	$-0.985581\pi$
$419$		−	37.9473i		−	0.0905664i	0.998974	−	0.0452832i	$-0.0144190\pi$
$420$		0			0
$421$	−631.000			−1.49881			−0.749406	−	0.662111i	$-0.769661\pi$
$421$	−631.000			−1.49881			−0.749406	+	0.662111i	$0.769661\pi$
$422$	−705.188			−1.67106
$423$		0			0
$424$	260.000			0.613208
$425$	−252.982	−	189.737i	−0.595252	−	0.446439i
$426$		0			0
$427$			3.00000i			0.00702576i
$428$	132.816			0.310317
$429$		0			0
$430$	−720.000	−	240.000i	−1.67442	−	0.558140i
$431$		−	540.749i		−	1.25464i	−0.778762	−	0.627320i	$-0.784152\pi$
$431$		−	540.749i		−	1.25464i	0.778762	−	0.627320i	$-0.215848\pi$
$432$		0			0
$433$		−	288.000i		−	0.665127i	−0.943081	−	0.332564i	$-0.892086\pi$
$433$		−	288.000i		−	0.665127i	0.943081	−	0.332564i	$-0.107914\pi$
$434$			151.789i			0.349745i
$435$		0			0
$436$	624.000			1.43119
$437$	−686.214			−1.57028
$438$		0			0
$439$	−856.000			−1.94989			−0.974943	−	0.222455i	$-0.928593\pi$
$439$	−856.000			−1.94989			−0.974943	+	0.222455i	$0.928593\pi$
$440$	284.605	+	94.8683i	0.646830	+	0.215610i
$441$		0			0
$442$			840.000i			1.90045i
$443$	430.070			0.970812			0.485406	−	0.874289i	$-0.338672\pi$
$443$	430.070			0.970812			0.485406	+	0.874289i	$0.338672\pi$
$444$		0			0
$445$	−540.000	−	180.000i	−1.21348	−	0.404494i
$446$			986.631i			2.21218i
$447$		0			0
$448$			312.000i			0.696429i
$449$			796.894i			1.77482i	0.460981	+	0.887410i	$0.347497\pi$
$449$			796.894i			1.77482i	−0.460981	+	0.887410i	$0.652503\pi$
$450$		0			0
$451$	−450.000			−0.997783
$452$	1043.55			2.30874
$453$		0			0
$454$	−1070.00			−2.35683
$455$	99.6117	−	298.835i	0.218927	−	0.656781i
$456$		0			0
$457$			96.0000i			0.210066i	0.994469	+	0.105033i	$0.0334948\pi$
$457$			96.0000i			0.210066i	−0.994469	+	0.105033i	$0.966505\pi$
$458$	−50.5964			−0.110473
$459$		0			0
$460$	210.000	−	630.000i	0.456522	−	1.36957i
$461$		−	417.421i		−	0.905468i	−0.891646	−	0.452734i	$-0.850449\pi$
$461$		−	417.421i		−	0.905468i	0.891646	−	0.452734i	$-0.149551\pi$
$462$		0			0
$463$			237.000i			0.511879i	0.966693	+	0.255940i	$0.0823848\pi$
$463$			237.000i			0.511879i	−0.966693	+	0.255940i	$0.917615\pi$
$464$			189.737i			0.408915i
$465$		0			0
$466$	160.000			0.343348
$467$	468.017			1.00218			0.501089	−	0.865396i	$-0.332933\pi$
$467$	468.017			1.00218			0.501089	+	0.865396i	$0.332933\pi$
$468$		0			0
$469$	63.0000			0.134328
$470$	63.2456	−	189.737i	0.134565	−	0.403695i
$471$		0			0
$472$		−	240.000i		−	0.508475i
$473$	455.368			0.962723
$474$		0			0
$475$	620.000	+	465.000i	1.30526	+	0.978947i
$476$		−	227.684i		−	0.478328i
$477$		0			0
$478$			330.000i			0.690377i
$479$			407.934i			0.851636i	0.904809	+	0.425818i	$0.140014\pi$
$479$			407.934i			0.851636i	−0.904809	+	0.425818i	$0.859986\pi$
$480$		0			0
$481$	−567.000			−1.17879
$482$	907.574			1.88293
$483$		0			0
$484$	186.000			0.384298
$485$	−441.138	−	147.046i	−0.909562	−	0.303187i
$486$		0			0
$487$			261.000i			0.535934i	0.963428	+	0.267967i	$0.0863519\pi$
$487$			261.000i			0.535934i	−0.963428	+	0.267967i	$0.913648\pi$
$488$	−6.32456			−0.0129602
$489$		0			0
$490$	200.000	−	600.000i	0.408163	−	1.22449i
$491$		−	749.460i		−	1.52639i	−0.646165	−	0.763197i	$-0.723628\pi$
$491$		−	749.460i		−	1.52639i	0.646165	−	0.763197i	$-0.276372\pi$
$492$		0			0
$493$		−	600.000i		−	1.21704i
$494$		−	2058.64i		−	4.16729i
$495$		0			0
$496$	64.0000			0.129032
$497$	−85.3815			−0.171794
$498$		0			0
$499$	−106.000			−0.212425			−0.106212	−	0.994343i	$-0.533872\pi$
$499$	−106.000			−0.212425			−0.106212	+	0.994343i	$0.533872\pi$
$500$	−616.644	+	426.907i	−1.23329	+	0.853815i
$501$		0			0
$502$			810.000i			1.61355i
$503$	382.636			0.760707			0.380353	−	0.924841i	$-0.375802\pi$
$503$	382.636			0.760707			0.380353	+	0.924841i	$0.375802\pi$
$504$		0			0
$505$	−360.000	−	120.000i	−0.712871	−	0.237624i
$506$			664.078i			1.31241i
$507$		0			0
$508$		−	864.000i		−	1.70079i
$509$			161.276i			0.316849i	0.987371	+	0.158425i	$0.0506415\pi$
$509$			161.276i			0.316849i	−0.987371	+	0.158425i	$0.949359\pi$
$510$		0			0
$511$	81.0000			0.158513
$512$	252.982			0.494106
$513$		0			0
$514$	880.000			1.71206
$515$	697.282	+	232.427i	1.35395	+	0.451315i
$516$		0			0
$517$			120.000i			0.232108i
$518$	256.144			0.494487
$519$		0			0
$520$	630.000	+	210.000i	1.21154	+	0.403846i
$521$			369.986i			0.710147i	0.934838	+	0.355073i	$0.115544\pi$
$521$			369.986i			0.710147i	−0.934838	+	0.355073i	$0.884456\pi$
$522$		0			0
$523$			837.000i			1.60038i	0.599745	+	0.800191i	$0.295269\pi$
$523$			837.000i			1.60038i	−0.599745	+	0.800191i	$0.704731\pi$
$524$			455.368i			0.869023i
$525$		0			0
$526$	1120.00			2.12928
$527$	−202.386			−0.384034
$528$		0			0
$529$	−39.0000			−0.0737240
$530$	205.548	−	616.644i	0.387827	−	1.16348i
$531$		0			0
$532$			558.000i			1.04887i
$533$	−996.117			−1.86889
$534$		0			0
$535$	35.0000	−	105.000i	0.0654206	−	0.196262i
$536$			132.816i			0.247790i
$537$		0			0
$538$			360.000i			0.669145i
$539$			379.473i			0.704032i
$540$		0			0
$541$	−511.000			−0.944547			−0.472274	−	0.881452i	$-0.656567\pi$
$541$	−511.000			−0.944547			−0.472274	+	0.881452i	$0.656567\pi$
$542$	−382.636			−0.705970
$543$		0			0
$544$	−480.000			−0.882353
$545$	164.438	−	493.315i	0.301722	−	0.905166i
$546$		0			0
$547$		−	1083.00i		−	1.97989i	−0.141452	−	0.989945i	$-0.545177\pi$
$547$		−	1083.00i		−	1.97989i	0.141452	−	0.989945i	$-0.454823\pi$
$548$	−322.552			−0.588599
$549$		0			0
$550$	450.000	−	600.000i	0.818182	−	1.09091i
$551$			1470.46i			2.66871i
$552$		0			0
$553$			3.00000i			0.00542495i
$554$		−	151.789i		−	0.273988i
$555$		0			0
$556$	−6.00000			−0.0107914
$557$	−973.982			−1.74862			−0.874310	−	0.485368i	$-0.838686\pi$
$557$	−973.982			−1.74862			−0.874310	+	0.485368i	$0.838686\pi$
$558$		0			0
$559$	1008.00			1.80322
$560$	56.9210	+	18.9737i	0.101645	+	0.0338815i
$561$		0			0
$562$		−	960.000i		−	1.70819i
$563$	−480.666			−0.853759			−0.426879	−	0.904309i	$-0.640387\pi$
$563$	−480.666			−0.853759			−0.426879	+	0.904309i	$0.640387\pi$
$564$		0			0
$565$	275.000	−	825.000i	0.486726	−	1.46018i
$566$			607.157i			1.07272i
$567$		0			0
$568$		−	180.000i		−	0.316901i
$569$		−	645.105i		−	1.13375i	−0.823803	−	0.566876i	$-0.808152\pi$
$569$		−	645.105i		−	1.13375i	0.823803	−	0.566876i	$-0.191848\pi$
$570$		0			0
$571$	569.000			0.996497			0.498249	−	0.867034i	$-0.333977\pi$
$571$	569.000			0.996497			0.498249	+	0.867034i	$0.333977\pi$
$572$	−1195.34			−2.08976
$573$		0			0
$574$	450.000			0.783972
$575$	−442.719	−	332.039i	−0.769946	−	0.577459i
$576$		0			0
$577$			837.000i			1.45061i	0.688429	+	0.725303i	$0.258300\pi$
$577$			837.000i			1.45061i	−0.688429	+	0.725303i	$0.741700\pi$
$578$	−407.934			−0.705768
$579$		0			0
$580$	−1350.00	−	450.000i	−2.32759	−	0.775862i
$581$			332.039i			0.571496i
$582$		0			0
$583$			390.000i			0.668954i
$584$			170.763i			0.292402i
$585$		0			0
$586$	−650.000			−1.10922
$587$	629.293			1.07205			0.536025	−	0.844202i	$-0.319925\pi$
$587$	629.293			1.07205			0.536025	+	0.844202i	$0.319925\pi$
$588$		0			0
$589$	496.000			0.842105
$590$	−569.210	−	189.737i	−0.964763	−	0.321588i
$591$		0			0
$592$		−	108.000i		−	0.182432i
$593$	−641.942			−1.08253			−0.541267	−	0.840851i	$-0.682055\pi$
$593$	−641.942			−1.08253			−0.541267	+	0.840851i	$0.682055\pi$
$594$		0			0
$595$	−180.000	−	60.0000i	−0.302521	−	0.100840i
$596$			455.368i			0.764040i
$597$		0			0
$598$			1470.00i			2.45819i
$599$		−	635.618i		−	1.06113i	−0.847644	−	0.530566i	$-0.821979\pi$
$599$		−	635.618i		−	1.06113i	0.847644	−	0.530566i	$-0.178021\pi$
$600$		0			0
$601$	224.000			0.372712			0.186356	−	0.982482i	$-0.440332\pi$
$601$	224.000			0.372712			0.186356	+	0.982482i	$0.440332\pi$
$602$	−455.368			−0.756425
$603$		0			0
$604$	−186.000			−0.307947
$605$	49.0153	−	147.046i	0.0810170	−	0.243051i
$606$		0			0
$607$			147.000i			0.242175i	0.992642	+	0.121087i	$0.0386381\pi$
$607$			147.000i			0.242175i	−0.992642	+	0.121087i	$0.961362\pi$
$608$	1176.37			1.93481
$609$		0			0
$610$	−5.00000	+	15.0000i	−0.00819672	+	0.0245902i
$611$			265.631i			0.434748i
$612$		0			0
$613$		−	243.000i		−	0.396411i	−0.980160	−	0.198206i	$-0.936489\pi$
$613$		−	243.000i		−	0.396411i	0.980160	−	0.198206i	$-0.0635114\pi$
$614$		−	626.131i		−	1.01976i
$615$		0			0
$616$	180.000			0.292208
$617$	79.0569			0.128131			0.0640656	−	0.997946i	$-0.479593\pi$
$617$	79.0569			0.128131			0.0640656	+	0.997946i	$0.479593\pi$
$618$		0			0
$619$	−1201.00			−1.94023			−0.970113	−	0.242653i	$-0.921982\pi$
$619$	−1201.00			−1.94023			−0.970113	+	0.242653i	$0.921982\pi$
$620$	−151.789	+	455.368i	−0.244821	+	0.734464i
$621$		0			0
$622$		−	1560.00i		−	2.50804i
$623$	−341.526			−0.548196
$624$		0			0
$625$	175.000	+	600.000i	0.280000	+	0.960000i
$626$			1299.70i			2.07619i
$627$		0			0
$628$			612.000i			0.974522i
$629$			341.526i			0.542967i
$630$		0			0
$631$	−193.000			−0.305864			−0.152932	−	0.988237i	$-0.548872\pi$
$631$	−193.000			−0.305864			−0.152932	+	0.988237i	$0.548872\pi$
$632$	−6.32456			−0.0100072
$633$		0			0
$634$	−800.000			−1.26183
$635$	−683.052	−	227.684i	−1.07567	−	0.358557i
$636$		0			0
$637$			840.000i			1.31868i
$638$	1423.02			2.23045
$639$		0			0
$640$	−280.000	+	840.000i	−0.437500	+	1.31250i
$641$		−	303.579i		−	0.473602i	−0.971558	−	0.236801i	$-0.923901\pi$
$641$		−	303.579i		−	0.473602i	0.971558	−	0.236801i	$-0.0760989\pi$
$642$		0			0
$643$		−	24.0000i		−	0.0373250i	−0.999826	−	0.0186625i	$-0.994059\pi$
$643$		−	24.0000i		−	0.0373250i	0.999826	−	0.0186625i	$-0.00594081\pi$
$644$		−	398.447i		−	0.618706i
$645$		0			0
$646$	−1240.00			−1.91950
$647$	667.241			1.03128			0.515642	−	0.856804i	$-0.327554\pi$
$647$	667.241			1.03128			0.515642	+	0.856804i	$0.327554\pi$
$648$		0			0
$649$	360.000			0.554700
$650$	996.117	−	1328.16i	1.53249	−	2.04332i
$651$		0			0
$652$			1026.00i			1.57362i
$653$	−404.772			−0.619865			−0.309932	−	0.950759i	$-0.600306\pi$
$653$	−404.772			−0.619865			−0.309932	+	0.950759i	$0.600306\pi$
$654$		0			0
$655$	360.000	+	120.000i	0.549618	+	0.183206i
$656$		−	189.737i		−	0.289233i
$657$		0			0
$658$		−	120.000i		−	0.182371i
$659$			151.789i			0.230333i	0.993346	+	0.115166i	$0.0367401\pi$
$659$			151.789i			0.230333i	−0.993346	+	0.115166i	$0.963260\pi$
$660$		0			0
$661$	719.000			1.08775			0.543873	−	0.839168i	$-0.316957\pi$
$661$	719.000			1.08775			0.543873	+	0.839168i	$0.316957\pi$
$662$	1040.39			1.57159
$663$		0			0
$664$	−700.000			−1.05422
$665$	441.138	+	147.046i	0.663365	+	0.221122i
$666$		0			0
$667$		−	1050.00i		−	1.57421i
$668$	−1745.58			−2.61314
$669$		0			0
$670$	315.000	+	105.000i	0.470149	+	0.156716i
$671$		−	9.48683i		−	0.0141384i
$672$		0			0
$673$		−	1173.00i		−	1.74294i	−0.490447	−	0.871471i	$-0.663166\pi$
$673$		−	1173.00i		−	1.74294i	0.490447	−	0.871471i	$-0.336834\pi$
$674$			1204.83i			1.78758i
$675$		0			0
$676$	−1632.00			−2.41420
$677$	−1125.77			−1.66288			−0.831441	−	0.555613i	$-0.812483\pi$
$677$	−1125.77			−1.66288			−0.831441	+	0.555613i	$0.812483\pi$
$678$		0			0
$679$	−279.000			−0.410898
$680$	126.491	−	379.473i	0.186016	−	0.558049i
$681$		0			0
$682$		−	480.000i		−	0.703812i
$683$	894.925			1.31028			0.655142	−	0.755505i	$-0.272609\pi$
$683$	894.925			1.31028			0.655142	+	0.755505i	$0.272609\pi$
$684$		0			0
$685$	−85.0000	+	255.000i	−0.124088	+	0.372263i
$686$		−	844.328i		−	1.23080i
$687$		0			0
$688$			192.000i			0.279070i
$689$			863.302i			1.25298i
$690$		0			0
$691$	872.000			1.26194			0.630970	−	0.775808i	$-0.282657\pi$
$691$	872.000			1.26194			0.630970	+	0.775808i	$0.282657\pi$
$692$	−151.789			−0.219349
$693$		0			0
$694$	370.000			0.533141
$695$	−1.58114	+	4.74342i	−0.00227502	+	0.00682506i
$696$		0			0
$697$			600.000i			0.860832i
$698$	281.443			0.403213
$699$		0			0
$700$	−270.000	+	360.000i	−0.385714	+	0.514286i
$701$			1280.72i			1.82699i	0.406846	+	0.913497i	$0.366629\pi$
$701$			1280.72i			1.82699i	−0.406846	+	0.913497i	$0.633371\pi$
$702$		0			0
$703$		−	837.000i		−	1.19061i
$704$		−	986.631i		−	1.40146i
$705$		0			0
$706$	1840.00			2.60623
$707$	−227.684			−0.322042
$708$		0			0
$709$	−481.000			−0.678420			−0.339210	−	0.940711i	$-0.610160\pi$
$709$	−481.000			−0.678420			−0.339210	+	0.940711i	$0.610160\pi$
$710$	−426.907	−	142.302i	−0.601278	−	0.200426i
$711$		0			0
$712$		−	720.000i		−	1.01124i
$713$	−354.175			−0.496739
$714$		0			0
$715$	−315.000	+	945.000i	−0.440559	+	1.32168i
$716$		−	1195.34i		−	1.66947i
$717$		0			0
$718$			270.000i			0.376045i
$719$		−	939.196i		−	1.30625i	−0.757248	−	0.653127i	$-0.773457\pi$
$719$		−	939.196i		−	1.30625i	0.757248	−	0.653127i	$-0.226543\pi$
$720$		0			0
$721$	441.000			0.611650
$722$	1897.37			2.62793
$723$		0			0
$724$	−618.000			−0.853591
$725$	−711.512	+	948.683i	−0.981397	+	1.30853i
$726$		0			0
$727$		−	768.000i		−	1.05640i	−0.849121	−	0.528198i	$-0.822868\pi$
$727$		−	768.000i		−	1.05640i	0.849121	−	0.528198i	$-0.177132\pi$
$728$	398.447			0.547317
$729$		0			0
$730$	405.000	+	135.000i	0.554795	+	0.184932i
$731$		−	607.157i		−	0.830585i
$732$		0			0
$733$		−	1104.00i		−	1.50614i	−0.657941	−	0.753070i	$-0.728572\pi$
$733$		−	1104.00i		−	1.50614i	0.657941	−	0.753070i	$-0.271428\pi$
$734$			161.276i			0.219722i
$735$		0			0
$736$	−840.000			−1.14130
$737$	−199.223			−0.270317
$738$		0			0
$739$	392.000			0.530447			0.265223	−	0.964187i	$-0.414554\pi$
$739$	392.000			0.530447			0.265223	+	0.964187i	$0.414554\pi$
$740$	768.433	+	256.144i	1.03842	+	0.346141i
$741$		0			0
$742$		−	390.000i		−	0.525606i
$743$	999.280			1.34493			0.672463	−	0.740131i	$-0.265236\pi$
$743$	999.280			1.34493			0.672463	+	0.740131i	$0.265236\pi$
$744$		0			0
$745$	360.000	+	120.000i	0.483221	+	0.161074i
$746$			749.460i			1.00464i
$747$		0			0
$748$			720.000i			0.962567i
$749$		−	66.4078i		−	0.0886620i
$750$		0			0
$751$	−241.000			−0.320905			−0.160453	−	0.987044i	$-0.551295\pi$
$751$	−241.000			−0.320905			−0.160453	+	0.987044i	$0.551295\pi$
$752$	−50.5964			−0.0672825
$753$		0			0
$754$	3150.00			4.17772
$755$	−49.0153	+	147.046i	−0.0649209	+	0.194763i
$756$		0			0
$757$			27.0000i			0.0356671i	0.999841	+	0.0178336i	$0.00567690\pi$
$757$			27.0000i			0.0356671i	−0.999841	+	0.0178336i	$0.994323\pi$
$758$	−325.715			−0.429703
$759$		0			0
$760$	−310.000	+	930.000i	−0.407895	+	1.22368i
$761$		−	692.539i		−	0.910038i	−0.890482	−	0.455019i	$-0.849632\pi$
$761$		−	692.539i		−	0.910038i	0.890482	−	0.455019i	$-0.150368\pi$
$762$		0			0
$763$		−	312.000i		−	0.408912i
$764$			1593.79i			2.08611i
$765$		0			0
$766$	160.000			0.208877
$767$	796.894			1.03898
$768$		0			0
$769$	1127.00			1.46554			0.732770	−	0.680477i	$-0.238227\pi$
$769$	1127.00			1.46554			0.732770	+	0.680477i	$0.238227\pi$
$770$	142.302	−	426.907i	0.184808	−	0.554425i
$771$		0			0
$772$		−	1278.00i		−	1.65544i
$773$	−243.495			−0.315000			−0.157500	−	0.987519i	$-0.550344\pi$
$773$	−243.495			−0.315000			−0.157500	+	0.987519i	$0.550344\pi$
$774$		0			0
$775$	320.000	+	240.000i	0.412903	+	0.309677i
$776$		−	588.184i		−	0.757969i
$777$		0			0
$778$			840.000i			1.07969i
$779$		−	1470.46i		−	1.88762i
$780$		0			0
$781$	270.000			0.345711
$782$	885.438			1.13227
$783$		0			0
$784$	−160.000			−0.204082
$785$	483.828	+	161.276i	0.616342	+	0.205447i
$786$		0			0
$787$		−	573.000i		−	0.728081i	−0.931383	−	0.364041i	$-0.881397\pi$
$787$		−	573.000i		−	0.728081i	0.931383	−	0.364041i	$-0.118603\pi$
$788$	1442.00			1.82995
$789$		0			0
$790$	−5.00000	+	15.0000i	−0.00632911	+	0.0189873i
$791$		−	521.776i		−	0.659641i
$792$		0			0
$793$		−	21.0000i		−	0.0264817i
$794$			227.684i			0.286756i
$795$		0			0
$796$	282.000			0.354271
$797$	−632.456			−0.793545			−0.396773	−	0.917917i	$-0.629870\pi$
$797$	−632.456			−0.793545			−0.396773	+	0.917917i	$0.629870\pi$
$798$		0			0
$799$	160.000			0.200250
$800$	758.947	+	569.210i	0.948683	+	0.711512i
$801$		0			0
$802$			330.000i			0.411471i
$803$	−256.144			−0.318984
$804$		0			0
$805$	−315.000	−	105.000i	−0.391304	−	0.130435i
$806$		−	1062.53i		−	1.31827i
$807$		0			0
$808$		−	480.000i		−	0.594059i
$809$		−	1138.42i		−	1.40719i	−0.710599	−	0.703597i	$-0.751576\pi$
$809$		−	1138.42i		−	1.40719i	0.710599	−	0.703597i	$-0.248424\pi$
$810$		0			0
$811$	−376.000			−0.463625			−0.231813	−	0.972760i	$-0.574466\pi$
$811$	−376.000			−0.463625			−0.231813	+	0.972760i	$0.574466\pi$
$812$	−853.815			−1.05150
$813$		0			0
$814$	−810.000			−0.995086
$815$	811.124	+	270.375i	0.995244	+	0.331748i
$816$		0			0
$817$			1488.00i			1.82130i
$818$	433.232			0.529624
$819$		0			0
$820$	1350.00	+	450.000i	1.64634	+	0.548780i
$821$			37.9473i			0.0462209i	0.999733	+	0.0231104i	$0.00735693\pi$
$821$			37.9473i			0.0462209i	−0.999733	+	0.0231104i	$0.992643\pi$
$822$		0			0
$823$		−	339.000i		−	0.411908i	−0.978562	−	0.205954i	$-0.933970\pi$
$823$		−	339.000i		−	0.411908i	0.978562	−	0.205954i	$-0.0660297\pi$
$824$			929.710i			1.12829i
$825$		0			0
$826$	−360.000			−0.435835
$827$	−442.719			−0.535331			−0.267666	−	0.963512i	$-0.586252\pi$
$827$	−442.719			−0.535331			−0.267666	+	0.963512i	$0.586252\pi$
$828$		0			0
$829$	−961.000			−1.15923			−0.579614	−	0.814891i	$-0.696797\pi$
$829$	−961.000			−1.15923			−0.579614	+	0.814891i	$0.696797\pi$
$830$	−553.399	+	1660.20i	−0.666745	+	2.00024i
$831$		0			0
$832$		−	2184.00i		−	2.62500i
$833$	505.964			0.607400
$834$		0			0
$835$	−460.000	+	1380.00i	−0.550898	+	1.65269i
$836$		−	1764.55i		−	2.11071i
$837$		0			0
$838$		−	120.000i		−	0.143198i
$839$			1062.53i			1.26642i	0.773981	+	0.633209i	$0.218263\pi$
$839$			1062.53i			1.26642i	−0.773981	+	0.633209i	$0.781737\pi$
$840$		0			0
$841$	−1409.00			−1.67539
$842$	−1995.40			−2.36983
$843$		0			0
$844$	−1338.00			−1.58531
$845$	−430.070	+	1290.21i	−0.508958	+	1.52687i
$846$		0			0
$847$		−	93.0000i		−	0.109799i
$848$	−164.438			−0.193913
$849$		0			0
$850$	−800.000	−	600.000i	−0.941176	−	0.705882i
$851$			597.670i			0.702315i
$852$		0			0
$853$			1221.00i			1.43142i	0.698398	+	0.715709i	$0.253896\pi$
$853$			1221.00i			1.43142i	−0.698398	+	0.715709i	$0.746104\pi$
$854$			9.48683i			0.0111087i
$855$		0			0
$856$	140.000			0.163551
$857$	572.372			0.667879			0.333939	−	0.942595i	$-0.391622\pi$
$857$	572.372			0.667879			0.333939	+	0.942595i	$0.391622\pi$
$858$		0			0
$859$	119.000			0.138533			0.0692666	−	0.997598i	$-0.477934\pi$
$859$	119.000			0.138533			0.0692666	+	0.997598i	$0.477934\pi$
$860$	−1366.10	−	455.368i	−1.58849	−	0.529498i
$861$		0			0
$862$		−	1710.00i		−	1.98376i
$863$	−442.719			−0.513000			−0.256500	−	0.966544i	$-0.582569\pi$
$863$	−442.719			−0.513000			−0.256500	+	0.966544i	$0.582569\pi$
$864$		0			0
$865$	−40.0000	+	120.000i	−0.0462428	+	0.138728i
$866$		−	910.736i		−	1.05166i
$867$		0			0
$868$			288.000i			0.331797i
$869$		−	9.48683i		−	0.0109170i
$870$		0			0
$871$	−441.000			−0.506315
$872$	657.754			0.754305
$873$		0			0
$874$	−2170.00			−2.48284
$875$	213.454	+	308.322i	0.243947	+	0.352368i
$876$		0			0
$877$		−	123.000i		−	0.140251i	−0.997538	−	0.0701254i	$-0.977660\pi$
$877$		−	123.000i		−	0.140251i	0.997538	−	0.0701254i	$-0.0223400\pi$
$878$	−2706.91			−3.08304
$879$		0			0
$880$	−180.000	−	60.0000i	−0.204545	−	0.0681818i
$881$		−	996.117i		−	1.13067i	−0.824862	−	0.565333i	$-0.808748\pi$
$881$		−	996.117i		−	1.13067i	0.824862	−	0.565333i	$-0.191252\pi$
$882$		0			0
$883$			387.000i			0.438279i	0.975694	+	0.219139i	$0.0703249\pi$
$883$			387.000i			0.438279i	−0.975694	+	0.219139i	$0.929675\pi$
$884$			1593.79i			1.80293i
$885$		0			0
$886$	1360.00			1.53499
$887$	961.332			1.08380			0.541901	−	0.840442i	$-0.317705\pi$
$887$	961.332			1.08380			0.541901	+	0.840442i	$0.317705\pi$
$888$		0			0
$889$	−432.000			−0.485939
$890$	−1707.63	−	569.210i	−1.91869	−	0.639562i
$891$		0			0
$892$			1872.00i			2.09865i
$893$	−392.122			−0.439107
$894$		0			0
$895$	−945.000	−	315.000i	−1.05587	−	0.351955i
$896$			531.263i			0.592927i
$897$		0			0
$898$			2520.00i			2.80624i
$899$			758.947i			0.844212i
$900$		0			0
$901$	520.000			0.577137
$902$	−1423.02			−1.57763
$903$		0			0
$904$	1100.00			1.21681
$905$	−162.857	+	488.572i	−0.179953	+	0.539858i
$906$		0			0
$907$		−	93.0000i		−	0.102536i	−0.998685	−	0.0512679i	$-0.983674\pi$
$907$		−	93.0000i		−	0.102536i	0.998685	−	0.0512679i	$-0.0163262\pi$
$908$	−2030.18			−2.23588
$909$		0			0
$910$	315.000	−	945.000i	0.346154	−	1.03846i
$911$		−	531.263i		−	0.583164i	−0.956546	−	0.291582i	$-0.905818\pi$
$911$		−	531.263i		−	0.583164i	0.956546	−	0.291582i	$-0.0941817\pi$
$912$		0			0
$913$		−	1050.00i		−	1.15005i
$914$			303.579i			0.332143i
$915$		0			0
$916$	−96.0000			−0.104803
$917$	227.684			0.248292
$918$		0			0
$919$	−376.000			−0.409140			−0.204570	−	0.978852i	$-0.565580\pi$
$919$	−376.000			−0.409140			−0.204570	+	0.978852i	$0.565580\pi$
$920$	221.359	−	664.078i	0.240608	−	0.721824i
$921$		0			0
$922$		−	1320.00i		−	1.43167i
$923$	597.670			0.647530
$924$		0			0
$925$	405.000	−	540.000i	0.437838	−	0.583784i
$926$			749.460i			0.809352i
$927$		0			0
$928$			1800.00i			1.93966i
$929$		−	1413.54i		−	1.52157i	−0.649004	−	0.760785i	$-0.724814\pi$
$929$		−	1413.54i		−	1.52157i	0.649004	−	0.760785i	$-0.275186\pi$
$930$		0			0
$931$	−1240.00			−1.33190
$932$	303.579			0.325728
$933$		0			0
$934$	1480.00			1.58458
$935$	569.210	+	189.737i	0.608781	+	0.202927i
$936$		0			0
$937$		−	243.000i		−	0.259338i	−0.991557	−	0.129669i	$-0.958608\pi$
$937$		−	243.000i		−	0.259338i	0.991557	−	0.129669i	$-0.0413915\pi$
$938$	199.223			0.212392
$939$		0			0
$940$	120.000	−	360.000i	0.127660	−	0.382979i
$941$			388.960i			0.413348i	0.978410	+	0.206674i	$0.0662639\pi$
$941$			388.960i			0.413348i	−0.978410	+	0.206674i	$0.933736\pi$
$942$		0			0
$943$			1050.00i			1.11347i
$944$			151.789i			0.160794i
$945$		0			0
$946$	1440.00			1.52220
$947$	316.228			0.333926			0.166963	−	0.985963i	$-0.446604\pi$
$947$	316.228			0.333926			0.166963	+	0.985963i	$0.446604\pi$
$948$		0			0
$949$	−567.000			−0.597471
$950$	1960.61	+	1470.46i	2.06380	+	1.54785i
$951$		0			0
$952$		−	240.000i		−	0.252101i
$953$	1530.54			1.60603			0.803013	−	0.595962i	$-0.203229\pi$
$953$	1530.54			1.60603			0.803013	+	0.595962i	$0.203229\pi$
$954$		0			0
$955$	1260.00	+	420.000i	1.31937	+	0.439791i
$956$			626.131i			0.654949i
$957$		0			0
$958$			1290.00i			1.34656i
$959$			161.276i			0.168171i
$960$		0			0
$961$	−705.000			−0.733611
$962$	−1793.01			−1.86384
$963$		0			0
$964$	1722.00			1.78631
$965$	−1010.35	−	336.783i	−1.04699	−	0.348997i
$966$		0			0
$967$			237.000i			0.245088i	0.992463	+	0.122544i	$0.0391052\pi$
$967$			237.000i			0.245088i	−0.992463	+	0.122544i	$0.960895\pi$
$968$	196.061			0.202543
$969$		0			0
$970$	−1395.00	−	465.000i	−1.43814	−	0.479381i
$971$		−	569.210i		−	0.586210i	−0.956080	−	0.293105i	$-0.905311\pi$
$971$		−	569.210i		−	0.586210i	0.956080	−	0.293105i	$-0.0946886\pi$
$972$		0			0
$973$			3.00000i			0.00308325i
$974$			825.354i			0.847387i
$975$		0			0
$976$	4.00000			0.00409836
$977$	515.451			0.527586			0.263793	−	0.964579i	$-0.415026\pi$
$977$	515.451			0.527586			0.263793	+	0.964579i	$0.415026\pi$
$978$		0			0
$979$	1080.00			1.10317
$980$	379.473	−	1138.42i	0.387218	−	1.16165i
$981$		0			0
$982$		−	2370.00i		−	2.41344i
$983$	629.293			0.640176			0.320088	−	0.947388i	$-0.396287\pi$
$983$	629.293			0.640176			0.320088	+	0.947388i	$0.396287\pi$
$984$		0			0
$985$	380.000	−	1140.00i	0.385787	−	1.15736i
$986$		−	1897.37i		−	1.92431i
$987$		0			0
$988$		−	3906.00i		−	3.95344i
$989$		−	1062.53i		−	1.07434i
$990$		0			0
$991$	−751.000			−0.757820			−0.378910	−	0.925433i	$-0.623701\pi$
$991$	−751.000			−0.757820			−0.378910	+	0.925433i	$0.623701\pi$
$992$	607.157			0.612054
$993$		0			0
$994$	−270.000			−0.271630
$995$	74.3135	−	222.941i	0.0746870	−	0.224061i
$996$		0			0
$997$			1536.00i			1.54062i	0.637668	+	0.770311i	$0.279899\pi$
$997$			1536.00i			1.54062i	−0.637668	+	0.770311i	$0.720101\pi$
$998$	−335.201			−0.335873
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	135.3.d.g.134.3	yes	4
3.2	odd	2	inner	135.3.d.g.134.2	yes	4
4.3	odd	2		2160.3.c.k.1889.3		4
5.2	odd	4		675.3.c.g.26.2		2
5.3	odd	4		675.3.c.f.26.1		2
5.4	even	2	inner	135.3.d.g.134.1	✓	4
9.2	odd	6		405.3.h.i.134.3		8
9.4	even	3		405.3.h.i.269.1		8
9.5	odd	6		405.3.h.i.269.4		8
9.7	even	3		405.3.h.i.134.2		8
12.11	even	2		2160.3.c.k.1889.2		4
15.2	even	4		675.3.c.g.26.1		2
15.8	even	4		675.3.c.f.26.2		2
15.14	odd	2	inner	135.3.d.g.134.4	yes	4
20.19	odd	2		2160.3.c.k.1889.1		4
45.4	even	6		405.3.h.i.269.3		8
45.14	odd	6		405.3.h.i.269.2		8
45.29	odd	6		405.3.h.i.134.1		8
45.34	even	6		405.3.h.i.134.4		8
60.59	even	2		2160.3.c.k.1889.4		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
135.3.d.g.134.1	✓	4	5.4	even	2	inner
135.3.d.g.134.2	yes	4	3.2	odd	2	inner
135.3.d.g.134.3	yes	4	1.1	even	1	trivial
135.3.d.g.134.4	yes	4	15.14	odd	2	inner
405.3.h.i.134.1		8	45.29	odd	6
405.3.h.i.134.2		8	9.7	even	3
405.3.h.i.134.3		8	9.2	odd	6
405.3.h.i.134.4		8	45.34	even	6
405.3.h.i.269.1		8	9.4	even	3
405.3.h.i.269.2		8	45.14	odd	6
405.3.h.i.269.3		8	45.4	even	6
405.3.h.i.269.4		8	9.5	odd	6
675.3.c.f.26.1		2	5.3	odd	4
675.3.c.f.26.2		2	15.8	even	4
675.3.c.g.26.1		2	15.2	even	4
675.3.c.g.26.2		2	5.2	odd	4
2160.3.c.k.1889.1		4	20.19	odd	2
2160.3.c.k.1889.2		4	12.11	even	2
2160.3.c.k.1889.3		4	4.3	odd	2
2160.3.c.k.1889.4		4	60.59	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:	\( N \)	\(=\)	\( 135 = 3^{3} \cdot 5 \)
Weight:	\( k \)	\(=\)	\( 3 \)
Character orbit:	\([\chi]\)	\(=\)	135.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+3.16228 q^{2} +6.00000 q^{4} +(1.58114 - 4.74342i) q^{5} -3.00000i q^{7} +6.32456 q^{8} +(5.00000 - 15.0000i) q^{10} +9.48683i q^{11} +21.0000i q^{13} -9.48683i q^{14} -4.00000 q^{16} +12.6491 q^{17} -31.0000 q^{19} +(9.48683 - 28.4605i) q^{20} +30.0000i q^{22} +22.1359 q^{23} +(-20.0000 - 15.0000i) q^{25} +66.4078i q^{26} -18.0000i q^{28} -47.4342i q^{29} -16.0000 q^{31} -37.9473 q^{32} +40.0000 q^{34} +(-14.2302 - 4.74342i) q^{35} +27.0000i q^{37} -98.0306 q^{38} +(10.0000 - 30.0000i) q^{40} +47.4342i q^{41} -48.0000i q^{43} +56.9210i q^{44} +70.0000 q^{46} +12.6491 q^{47} +40.0000 q^{49} +(-63.2456 - 47.4342i) q^{50} +126.000i q^{52} +41.1096 q^{53} +(45.0000 + 15.0000i) q^{55} -18.9737i q^{56} -150.000i q^{58} -37.9473i q^{59} -1.00000 q^{61} -50.5964 q^{62} -104.000 q^{64} +(99.6117 + 33.2039i) q^{65} +21.0000i q^{67} +75.8947 q^{68} +(-45.0000 - 15.0000i) q^{70} -28.4605i q^{71} +27.0000i q^{73} +85.3815i q^{74} -186.000 q^{76} +28.4605 q^{77} -1.00000 q^{79} +(-6.32456 + 18.9737i) q^{80} +150.000i q^{82} -110.680 q^{83} +(20.0000 - 60.0000i) q^{85} -151.789i q^{86} +60.0000i q^{88} -113.842i q^{89} +63.0000 q^{91} +132.816 q^{92} +40.0000 q^{94} +(-49.0153 + 147.046i) q^{95} -93.0000i q^{97} +126.491 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 24 q^{4} + 20 q^{10} - 16 q^{16} - 124 q^{19} - 80 q^{25} - 64 q^{31} + 160 q^{34} + 40 q^{40} + 280 q^{46} + 160 q^{49} + 180 q^{55} - 4 q^{61} - 416 q^{64} - 180 q^{70} - 744 q^{76} - 4 q^{79} + 80 q^{85}+ \cdots + 160 q^{94}+O(q^{100}) \) 4 * q + 24 * q^4 + 20 * q^10 - 16 * q^16 - 124 * q^19 - 80 * q^25 - 64 * q^31 + 160 * q^34 + 40 * q^40 + 280 * q^46 + 160 * q^49 + 180 * q^55 - 4 * q^61 - 416 * q^64 - 180 * q^70 - 744 * q^76 - 4 * q^79 + 80 * q^85 + 252 * q^91 + 160 * q^94

Introduction

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