LMFDB - Embedded newform 768.4.f.g.383.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [768,4,Mod(383,768)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(768, base_ring=CyclotomicField(2))

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

N = Newforms(chi, 4, names="a")

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

chi := DirichletCharacter("768.383");

S:= CuspForms(chi, 4);

N := Newforms(S);

Level:	$ N $	$=$	$ 768 = 2^{8} \cdot 3 $
Weight:	$ k $	$=$	$ 4 $
Character orbit:	$[\chi]$	$=$	768.f (of order $2$, degree $1$, not minimal)

Newform invariants

comment: select newform

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

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

Self dual:	no
Analytic conductor:	$45.3134668844$
Analytic rank:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} + \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} + 39x^{10} + 549x^{8} + 3500x^{6} + 10236x^{4} + 11952x^{2} + 4624 $ x^12 + 39x^10 + 549x^8 + 3500x^6 + 10236x^4 + 11952*x^2 + 4624
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{26}\cdot 3^{2} $
Twist minimal:	no (minimal twist has level 384)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			383.4
Root			$3.14286i$ of defining polynomial
Character	$\chi$	$=$	768.383
Dual form			768.4.f.g.383.3

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+(-3.16369 + 4.12202i) q^{3} -21.4043 q^{5} +20.9034i q^{7} +(-6.98212 - 26.0816i) q^{9} +O(q^{10})$	\(q+(-3.16369 + 4.12202i) q^{3} -21.4043 q^{5} +20.9034i q^{7} +(-6.98212 - 26.0816i) q^{9} -9.94564i q^{11} -67.8290i q^{13} +(67.7167 - 88.2292i) q^{15} -7.97348i q^{17} +62.4014 q^{19} +(-86.1644 - 66.1320i) q^{21} +101.816 q^{23} +333.146 q^{25} +(129.598 + 53.7337i) q^{27} -122.445 q^{29} +87.5072i q^{31} +(40.9961 + 31.4649i) q^{33} -447.424i q^{35} -106.136i q^{37} +(279.593 + 214.590i) q^{39} -90.3818i q^{41} -451.366 q^{43} +(149.448 + 558.260i) q^{45} -428.890 q^{47} -93.9530 q^{49} +(32.8669 + 25.2256i) q^{51} -362.452 q^{53} +212.880i q^{55} +(-197.419 + 257.220i) q^{57} -801.483i q^{59} +647.903i q^{61} +(545.195 - 145.950i) q^{63} +1451.84i q^{65} +957.240 q^{67} +(-322.115 + 419.688i) q^{69} -224.090 q^{71} +108.474 q^{73} +(-1053.97 + 1373.24i) q^{75} +207.898 q^{77} -615.569i q^{79} +(-631.500 + 364.210i) q^{81} +204.385i q^{83} +170.667i q^{85} +(387.378 - 504.721i) q^{87} +454.067i q^{89} +1417.86 q^{91} +(-360.707 - 276.846i) q^{93} -1335.66 q^{95} +740.490 q^{97} +(-259.398 + 69.4417i) q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q + 2 q^{3} - 12 q^{5}+O(q^{10}) $ 12 * q + 2 * q^3 - 12 * q^5	$ 12 q + 2 q^{3} - 12 q^{5} + 84 q^{15} + 180 q^{19} - 156 q^{21} + 120 q^{23} + 300 q^{25} - 130 q^{27} - 588 q^{29} - 116 q^{33} - 620 q^{39} - 372 q^{43} + 740 q^{45} - 1248 q^{47} - 948 q^{49} - 360 q^{51} + 948 q^{53} + 172 q^{57} + 2744 q^{63} + 2292 q^{67} - 3280 q^{69} + 2040 q^{71} + 216 q^{73} - 2522 q^{75} - 4824 q^{77} - 1076 q^{81} - 4156 q^{87} + 3480 q^{91} + 4180 q^{93} - 5448 q^{95} - 48 q^{97} - 3048 q^{99}+O(q^{100}) $ 12 * q + 2 * q^3 - 12 * q^5 + 84 * q^15 + 180 * q^19 - 156 * q^21 + 120 * q^23 + 300 * q^25 - 130 * q^27 - 588 * q^29 - 116 * q^33 - 620 * q^39 - 372 * q^43 + 740 * q^45 - 1248 * q^47 - 948 * q^49 - 360 * q^51 + 948 * q^53 + 172 * q^57 + 2744 * q^63 + 2292 * q^67 - 3280 * q^69 + 2040 * q^71 + 216 * q^73 - 2522 * q^75 - 4824 * q^77 - 1076 * q^81 - 4156 * q^87 + 3480 * q^91 + 4180 * q^93 - 5448 * q^95 - 48 * q^97 - 3048 * q^99

Display 100 coefficients 10 coefficients

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/768\mathbb{Z}\right)^\times$.

$n$	$257$	$511$	$517$
$\chi(n)$	$-1$	$-1$	$-1$

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$			$a_n / n^{(k-1)/2}$			$ \alpha_n $			$ \theta_n $
$p$	$a_p$			$a_p / p^{(k-1)/2}$			$ \alpha_p$			$ \theta_p $

$2$		0			0
$2$		0			0
$3$	−3.16369	+	4.12202i	−0.608853	+	0.793283i
$3$	−3.16369	+	4.12202i	−0.608853	+	0.793283i
$4$		0			0
$5$	−21.4043			−1.91446			−0.957232	−	0.289323i	$-0.906570\pi$
$5$	−21.4043			−1.91446			−0.957232	+	0.289323i	$0.906570\pi$
$6$		0			0
$7$			20.9034i			1.12868i	0.825543	+	0.564339i	$0.190869\pi$
$7$			20.9034i			1.12868i	−0.825543	+	0.564339i	$0.809131\pi$
$8$		0			0
$9$	−6.98212	−	26.0816i	−0.258597	−	0.965985i
$10$		0			0
$11$		−	9.94564i		−	0.272611i	−0.990667	−	0.136306i	$-0.956477\pi$
$11$		−	9.94564i		−	0.272611i	0.990667	−	0.136306i	$-0.0435229\pi$
$12$		0			0
$13$		−	67.8290i		−	1.44711i	−0.690268	−	0.723554i	$-0.742507\pi$
$13$		−	67.8290i		−	1.44711i	0.690268	−	0.723554i	$-0.257493\pi$
$14$		0			0
$15$	67.7167	−	88.2292i	1.16563	−	1.51871i
$16$		0			0
$17$		−	7.97348i		−	0.113756i	−0.998381	−	0.0568780i	$-0.981885\pi$
$17$		−	7.97348i		−	0.113756i	0.998381	−	0.0568780i	$-0.0181146\pi$
$18$		0			0
$19$	62.4014			0.753466			0.376733	−	0.926322i	$-0.377047\pi$
$19$	62.4014			0.753466			0.376733	+	0.926322i	$0.377047\pi$
$20$		0			0
$21$	−86.1644	−	66.1320i	−0.895362	−	0.687199i
$22$		0			0
$23$	101.816			0.923049			0.461525	−	0.887127i	$-0.347303\pi$
$23$	101.816			0.923049			0.461525	+	0.887127i	$0.347303\pi$
$24$		0			0
$25$	333.146			2.66517
$26$		0			0
$27$	129.598	+	53.7337i	0.923748	+	0.383002i
$28$		0			0
$29$	−122.445			−0.784051			−0.392025	−	0.919954i	$-0.628225\pi$
$29$	−122.445			−0.784051			−0.392025	+	0.919954i	$0.628225\pi$
$30$		0			0
$31$			87.5072i			0.506992i	0.967336	+	0.253496i	$0.0815805\pi$
$31$			87.5072i			0.506992i	−0.967336	+	0.253496i	$0.918419\pi$
$32$		0			0
$33$	40.9961	+	31.4649i	0.216258	+	0.165980i
$34$		0			0
$35$		−	447.424i		−	2.16081i
$36$		0			0
$37$		−	106.136i		−	0.471587i	−0.971803	−	0.235794i	$-0.924231\pi$
$37$		−	106.136i		−	0.471587i	0.971803	−	0.235794i	$-0.0757689\pi$
$38$		0			0
$39$	279.593	+	214.590i	1.14797	+	0.881075i
$40$		0			0
$41$		−	90.3818i		−	0.344275i	−0.985073	−	0.172137i	$-0.944933\pi$
$41$		−	90.3818i		−	0.344275i	0.985073	−	0.172137i	$-0.0550673\pi$
$42$		0			0
$43$	−451.366			−1.60076			−0.800380	−	0.599494i	$-0.795369\pi$
$43$	−451.366			−1.60076			−0.800380	+	0.599494i	$0.795369\pi$
$44$		0			0
$45$	149.448	+	558.260i	0.495075	+	1.84934i
$46$		0			0
$47$	−428.890			−1.33106			−0.665532	−	0.746369i	$-0.731795\pi$
$47$	−428.890			−1.33106			−0.665532	+	0.746369i	$0.731795\pi$
$48$		0			0
$49$	−93.9530			−0.273916
$50$		0			0
$51$	32.8669	+	25.2256i	0.0902408	+	0.0692607i
$52$		0			0
$53$	−362.452			−0.939370			−0.469685	−	0.882834i	$-0.655633\pi$
$53$	−362.452			−0.939370			−0.469685	+	0.882834i	$0.655633\pi$
$54$		0			0
$55$			212.880i			0.521904i
$56$		0			0
$57$	−197.419	+	257.220i	−0.458750	+	0.597712i
$58$		0			0
$59$		−	801.483i		−	1.76855i	−0.466970	−	0.884273i	$-0.654654\pi$
$59$		−	801.483i		−	1.76855i	0.466970	−	0.884273i	$-0.345346\pi$
$60$		0			0
$61$			647.903i			1.35993i	0.733247	+	0.679963i	$0.238004\pi$
$61$			647.903i			1.35993i	−0.733247	+	0.679963i	$0.761996\pi$
$62$		0			0
$63$	545.195	−	145.950i	1.09029	−	0.291873i
$64$		0			0
$65$			1451.84i			2.77043i
$66$		0			0
$67$	957.240			1.74545			0.872727	−	0.488208i	$-0.162349\pi$
$67$	957.240			1.74545			0.872727	+	0.488208i	$0.162349\pi$
$68$		0			0
$69$	−322.115	+	419.688i	−0.562001	+	0.732240i
$70$		0			0
$71$	−224.090			−0.374572			−0.187286	−	0.982305i	$-0.559969\pi$
$71$	−224.090			−0.374572			−0.187286	+	0.982305i	$0.559969\pi$
$72$		0			0
$73$	108.474			0.173916			0.0869582	−	0.996212i	$-0.472285\pi$
$73$	108.474			0.173916			0.0869582	+	0.996212i	$0.472285\pi$
$74$		0			0
$75$	−1053.97	+	1373.24i	−1.62269	+	2.11423i
$76$		0			0
$77$	207.898			0.307690
$78$		0			0
$79$		−	615.569i		−	0.876670i	−0.898812	−	0.438335i	$-0.855568\pi$
$79$		−	615.569i		−	0.876670i	0.898812	−	0.438335i	$-0.144432\pi$
$80$		0			0
$81$	−631.500	+	364.210i	−0.866255	+	0.499602i
$82$		0			0
$83$			204.385i			0.270291i	0.990826	+	0.135145i	$0.0431501\pi$
$83$			204.385i			0.270291i	−0.990826	+	0.135145i	$0.956850\pi$
$84$		0			0
$85$			170.667i			0.217782i
$86$		0			0
$87$	387.378	−	504.721i	0.477371	−	0.621974i
$88$		0			0
$89$			454.067i			0.540798i	0.962748	+	0.270399i	$0.0871556\pi$
$89$			454.067i			0.540798i	−0.962748	+	0.270399i	$0.912844\pi$
$90$		0			0
$91$	1417.86			1.63332
$92$		0			0
$93$	−360.707	−	276.846i	−0.402189	−	0.308684i
$94$		0			0
$95$	−1335.66			−1.44248
$96$		0			0
$97$	740.490			0.775106			0.387553	−	0.921847i	$-0.373320\pi$
$97$	740.490			0.775106			0.387553	+	0.921847i	$0.373320\pi$
$98$		0			0
$99$	−259.398	+	69.4417i	−0.263338	+	0.0704965i
$100$		0			0
$101$	−52.9454			−0.0521610			−0.0260805	−	0.999660i	$-0.508303\pi$
$101$	−52.9454			−0.0521610			−0.0260805	+	0.999660i	$0.508303\pi$
$102$		0			0
$103$			808.292i			0.773236i	0.922240	+	0.386618i	$0.126357\pi$
$103$			808.292i			0.773236i	−0.922240	+	0.386618i	$0.873643\pi$
$104$		0			0
$105$	1844.29	+	1415.51i	1.71414	+	1.31562i
$106$		0			0
$107$			728.059i			0.657796i	0.944366	+	0.328898i	$0.106677\pi$
$107$			728.059i			0.657796i	−0.944366	+	0.328898i	$0.893323\pi$
$108$		0			0
$109$			1488.85i			1.30831i	0.756358	+	0.654157i	$0.226977\pi$
$109$			1488.85i			1.30831i	−0.756358	+	0.654157i	$0.773023\pi$
$110$		0			0
$111$	437.497	+	335.783i	0.374102	+	0.287127i
$112$		0			0
$113$		−	38.0071i		−	0.0316408i	−0.999875	−	0.0158204i	$-0.994964\pi$
$113$		−	38.0071i		−	0.0316408i	0.999875	−	0.0158204i	$-0.00503600\pi$
$114$		0			0
$115$	−2179.31			−1.76714
$116$		0			0
$117$	−1769.09	+	473.591i	−1.39788	+	0.374218i
$118$		0			0
$119$	166.673			0.128394
$120$		0			0
$121$	1232.08			0.925683
$122$		0			0
$123$	372.556	+	285.940i	0.273107	+	0.209613i
$124$		0			0
$125$	−4455.23			−3.18790
$126$		0			0
$127$		−	226.671i		−	0.158376i	−0.996860	−	0.0791882i	$-0.974767\pi$
$127$		−	226.671i		−	0.158376i	0.996860	−	0.0791882i	$-0.0252328\pi$
$128$		0			0
$129$	1427.98	−	1860.54i	0.974626	−	1.26986i
$130$		0			0
$131$			1580.10i			1.05385i	0.849912	+	0.526925i	$0.176655\pi$
$131$			1580.10i			1.05385i	−0.849912	+	0.526925i	$0.823345\pi$
$132$		0			0
$133$			1304.40i			0.850421i
$134$		0			0
$135$	−2773.96	−	1150.13i	−1.76848	−	0.733243i
$136$		0			0
$137$		−	543.435i		−	0.338896i	−0.985539	−	0.169448i	$-0.945801\pi$
$137$		−	543.435i		−	0.338896i	0.985539	−	0.169448i	$-0.0541985\pi$
$138$		0			0
$139$	−29.7527			−0.0181554			−0.00907768	−	0.999959i	$-0.502890\pi$
$139$	−29.7527			−0.0181554			−0.00907768	+	0.999959i	$0.502890\pi$
$140$		0			0
$141$	1356.87	−	1767.89i	0.810422	−	1.05591i
$142$		0			0
$143$	−674.603			−0.394498
$144$		0			0
$145$	2620.86			1.50104
$146$		0			0
$147$	297.238	−	387.276i	0.166774	−	0.217293i
$148$		0			0
$149$	−489.426			−0.269096			−0.134548	−	0.990907i	$-0.542958\pi$
$149$	−489.426			−0.269096			−0.134548	+	0.990907i	$0.542958\pi$
$150$		0			0
$151$		−	396.852i		−	0.213877i	−0.994266	−	0.106938i	$-0.965895\pi$
$151$		−	396.852i		−	0.213877i	0.994266	−	0.106938i	$-0.0341047\pi$
$152$		0			0
$153$	−207.961	+	55.6718i	−0.109887	+	0.0294170i
$154$		0			0
$155$		−	1873.04i		−	0.970618i
$156$		0			0
$157$		−	2542.51i		−	1.29245i	−0.763148	−	0.646224i	$-0.776347\pi$
$157$		−	2542.51i		−	1.29245i	0.763148	−	0.646224i	$-0.223653\pi$
$158$		0			0
$159$	1146.69	−	1494.03i	0.571938	−	0.745187i
$160$		0			0
$161$			2128.31i			1.04183i
$162$		0			0
$163$	1807.94			0.868765			0.434383	−	0.900728i	$-0.356967\pi$
$163$	1807.94			0.868765			0.434383	+	0.900728i	$0.356967\pi$
$164$		0			0
$165$	−877.496	−	673.486i	−0.414018	−	0.317763i
$166$		0			0
$167$	25.5944			0.0118596			0.00592980	−	0.999982i	$-0.498112\pi$
$167$	25.5944			0.0118596			0.00592980	+	0.999982i	$0.498112\pi$
$168$		0			0
$169$	−2403.78			−1.09412
$170$		0			0
$171$	−435.694	−	1627.53i	−0.194844	−	0.727837i
$172$		0			0
$173$	−677.278			−0.297644			−0.148822	−	0.988864i	$-0.547548\pi$
$173$	−677.278			−0.297644			−0.148822	+	0.988864i	$0.547548\pi$
$174$		0			0
$175$			6963.89i			3.00812i
$176$		0			0
$177$	3303.73	+	2535.64i	1.40296	+	1.07678i
$178$		0			0
$179$			3240.41i			1.35307i	0.736411	+	0.676535i	$0.236519\pi$
$179$			3240.41i			1.35307i	−0.736411	+	0.676535i	$0.763481\pi$
$180$		0			0
$181$			2123.69i			0.872114i	0.899919	+	0.436057i	$0.143625\pi$
$181$			2123.69i			0.872114i	−0.899919	+	0.436057i	$0.856375\pi$
$182$		0			0
$183$	−2670.67	−	2049.76i	−1.07881	−	0.827994i
$184$		0			0
$185$			2271.78i			0.902836i
$186$		0			0
$187$	−79.3014			−0.0310112
$188$		0			0
$189$	−1123.22	+	2709.05i	−0.432286	+	1.04261i
$190$		0			0
$191$	3080.73			1.16709			0.583543	−	0.812082i	$-0.301666\pi$
$191$	3080.73			1.16709			0.583543	+	0.812082i	$0.301666\pi$
$192$		0			0
$193$	−2614.09			−0.974955			−0.487478	−	0.873135i	$-0.662083\pi$
$193$	−2614.09			−0.974955			−0.487478	+	0.873135i	$0.662083\pi$
$194$		0			0
$195$	−5984.50	−	4593.16i	−2.19774	−	1.68679i
$196$		0			0
$197$	4223.51			1.52748			0.763738	−	0.645527i	$-0.223362\pi$
$197$	4223.51			1.52748			0.763738	+	0.645527i	$0.223362\pi$
$198$		0			0
$199$			783.071i			0.278947i	0.990226	+	0.139474i	$0.0445410\pi$
$199$			783.071i			0.278947i	−0.990226	+	0.139474i	$0.955459\pi$
$200$		0			0
$201$	−3028.41	+	3945.76i	−1.06272	+	1.38464i
$202$		0			0
$203$		−	2559.52i		−	0.884941i
$204$		0			0
$205$			1934.56i			0.659101i
$206$		0			0
$207$	−710.893	−	2655.53i	−0.238698	−	0.891652i
$208$		0			0
$209$		−	620.622i		−	0.205403i
$210$		0			0
$211$	3515.60			1.14703			0.573516	−	0.819195i	$-0.305579\pi$
$211$	3515.60			1.14703			0.573516	+	0.819195i	$0.305579\pi$
$212$		0			0
$213$	708.952	−	923.705i	0.228059	−	0.297142i
$214$		0			0
$215$	9661.19			3.06459
$216$		0			0
$217$	−1829.20			−0.572232
$218$		0			0
$219$	−343.177	+	447.131i	−0.105889	+	0.137965i
$220$		0			0
$221$	−540.834			−0.164617
$222$		0			0
$223$			1864.10i			0.559772i	0.960033	+	0.279886i	$0.0902967\pi$
$223$			1864.10i			0.559772i	−0.960033	+	0.279886i	$0.909703\pi$
$224$		0			0
$225$	−2326.07	−	8688.98i	−0.689205	−	2.57451i
$226$		0			0
$227$			2871.51i			0.839600i	0.907617	+	0.419800i	$0.137900\pi$
$227$			2871.51i			0.839600i	−0.907617	+	0.419800i	$0.862100\pi$
$228$		0			0
$229$			3048.58i			0.879720i	0.898066	+	0.439860i	$0.144972\pi$
$229$			3048.58i			0.879720i	−0.898066	+	0.439860i	$0.855028\pi$
$230$		0			0
$231$	−657.725	+	856.960i	−0.187338	+	0.244086i
$232$		0			0
$233$		−	2078.46i		−	0.584398i	−0.956358	−	0.292199i	$-0.905613\pi$
$233$		−	2078.46i		−	0.584398i	0.956358	−	0.292199i	$-0.0943870\pi$
$234$		0			0
$235$	9180.11			2.54827
$236$		0			0
$237$	2537.39	+	1947.47i	0.695448	+	0.533763i
$238$		0			0
$239$	2778.79			0.752072			0.376036	−	0.926605i	$-0.377287\pi$
$239$	2778.79			0.752072			0.376036	+	0.926605i	$0.377287\pi$
$240$		0			0
$241$	−2838.97			−0.758815			−0.379408	−	0.925230i	$-0.623872\pi$
$241$	−2838.97			−0.758815			−0.379408	+	0.925230i	$0.623872\pi$
$242$		0			0
$243$	496.589	−	3755.30i	0.131095	−	0.991370i
$244$		0			0
$245$	2011.00			0.524401
$246$		0			0
$247$		−	4232.63i		−	1.09035i
$248$		0			0
$249$	−842.478	−	646.610i	−0.214417	−	0.164567i
$250$		0			0
$251$			5339.24i			1.34267i	0.741155	+	0.671334i	$0.234278\pi$
$251$			5339.24i			1.34267i	−0.741155	+	0.671334i	$0.765722\pi$
$252$		0			0
$253$		−	1012.63i		−	0.251634i
$254$		0			0
$255$	−703.494	−	539.938i	−0.172763	−	0.132597i
$256$		0			0
$257$			6223.99i			1.51067i	0.655339	+	0.755335i	$0.272526\pi$
$257$			6223.99i			1.51067i	−0.655339	+	0.755335i	$0.727474\pi$
$258$		0			0
$259$	2218.61			0.532270
$260$		0			0
$261$	854.926	+	3193.56i	0.202753	+	0.757381i
$262$		0			0
$263$	−4779.81			−1.12067			−0.560334	−	0.828267i	$-0.689327\pi$
$263$	−4779.81			−1.12067			−0.560334	+	0.828267i	$0.689327\pi$
$264$		0			0
$265$	7758.05			1.79839
$266$		0			0
$267$	−1871.67	−	1436.53i	−0.429006	−	0.329266i
$268$		0			0
$269$	3916.03			0.887601			0.443800	−	0.896126i	$-0.353630\pi$
$269$	3916.03			0.887601			0.443800	+	0.896126i	$0.353630\pi$
$270$		0			0
$271$		−	6412.78i		−	1.43745i	−0.695295	−	0.718725i	$-0.744726\pi$
$271$		−	6412.78i		−	1.43745i	0.695295	−	0.718725i	$-0.255274\pi$
$272$		0			0
$273$	−4485.67	+	5844.45i	−0.994450	+	1.29568i
$274$		0			0
$275$		−	3313.35i		−	0.726555i
$276$		0			0
$277$		−	3745.22i		−	0.812376i	−0.913789	−	0.406188i	$-0.866858\pi$
$277$		−	3745.22i		−	0.812376i	0.913789	−	0.406188i	$-0.133142\pi$
$278$		0			0
$279$	2282.33	−	610.986i	0.489747	−	0.131107i
$280$		0			0
$281$		−	6656.43i		−	1.41313i	−0.707648	−	0.706565i	$-0.750244\pi$
$281$		−	6656.43i		−	1.41313i	0.707648	−	0.706565i	$-0.249756\pi$
$282$		0			0
$283$	−6460.77			−1.35708			−0.678539	−	0.734564i	$-0.737387\pi$
$283$	−6460.77			−1.35708			−0.678539	+	0.734564i	$0.737387\pi$
$284$		0			0
$285$	4225.62	−	5505.62i	0.878260	−	1.14430i
$286$		0			0
$287$	1889.29			0.388576
$288$		0			0
$289$	4849.42			0.987060
$290$		0			0
$291$	−2342.68	+	3052.31i	−0.471925	+	0.614879i
$292$		0			0
$293$	−2620.84			−0.522564			−0.261282	−	0.965263i	$-0.584145\pi$
$293$	−2620.84			−0.522564			−0.261282	+	0.965263i	$0.584145\pi$
$294$		0			0
$295$			17155.2i			3.38582i
$296$		0			0
$297$	534.416	−	1288.94i	0.104411	−	0.251824i
$298$		0			0
$299$		−	6906.09i		−	1.33575i
$300$		0			0
$301$		−	9435.09i		−	1.80674i
$302$		0			0
$303$	167.503	−	218.242i	0.0317584	−	0.0413785i
$304$		0			0
$305$		−	13867.9i		−	2.60353i
$306$		0			0
$307$	952.063			0.176994			0.0884969	−	0.996076i	$-0.471794\pi$
$307$	952.063			0.176994			0.0884969	+	0.996076i	$0.471794\pi$
$308$		0			0
$309$	−3331.80	−	2557.18i	−0.613395	−	0.470787i
$310$		0			0
$311$	464.965			0.0847773			0.0423887	−	0.999101i	$-0.486503\pi$
$311$	464.965			0.0847773			0.0423887	+	0.999101i	$0.486503\pi$
$312$		0			0
$313$	−5656.16			−1.02142			−0.510711	−	0.859752i	$-0.670618\pi$
$313$	−5656.16			−1.02142			−0.510711	+	0.859752i	$0.670618\pi$
$314$		0			0
$315$	−11669.5	+	3123.97i	−2.08731	+	0.558780i
$316$		0			0
$317$	4593.57			0.813883			0.406941	−	0.913454i	$-0.366595\pi$
$317$	4593.57			0.813883			0.406941	+	0.913454i	$0.366595\pi$
$318$		0			0
$319$			1217.79i			0.213741i
$320$		0			0
$321$	−3001.08	−	2303.35i	−0.521818	−	0.400501i
$322$		0			0
$323$		−	497.556i		−	0.0857113i
$324$		0			0
$325$		−	22597.0i		−	3.85678i
$326$		0			0
$327$	−6137.09	−	4710.27i	−1.03786	−	0.796571i
$328$		0			0
$329$		−	8965.26i		−	1.50234i
$330$		0			0
$331$	6709.02			1.11408			0.557041	−	0.830485i	$-0.311937\pi$
$331$	6709.02			1.11408			0.557041	+	0.830485i	$0.311937\pi$
$332$		0			0
$333$	−2768.21	+	741.058i	−0.455546	+	0.121951i
$334$		0			0
$335$	−20489.1			−3.34161
$336$		0			0
$337$	−3712.30			−0.600064			−0.300032	−	0.953929i	$-0.596997\pi$
$337$	−3712.30			−0.600064			−0.300032	+	0.953929i	$0.596997\pi$
$338$		0			0
$339$	156.666	+	120.243i	0.0251001	+	0.0192646i
$340$		0			0
$341$	870.316			0.138212
$342$		0			0
$343$			5205.93i			0.819516i
$344$		0			0
$345$	6894.66	−	8983.16i	1.07593	−	1.40185i
$346$		0			0
$347$			6331.12i			0.979459i	0.871874	+	0.489729i	$0.162904\pi$
$347$			6331.12i			0.979459i	−0.871874	+	0.489729i	$0.837096\pi$
$348$		0			0
$349$			8328.66i			1.27743i	0.769444	+	0.638714i	$0.220533\pi$
$349$			8328.66i			1.27743i	−0.769444	+	0.638714i	$0.779467\pi$
$350$		0			0
$351$	3644.70	−	8790.52i	0.554245	−	1.33676i
$352$		0			0
$353$			1494.95i			0.225406i	0.993629	+	0.112703i	$0.0359508\pi$
$353$			1494.95i			0.225406i	−0.993629	+	0.112703i	$0.964049\pi$
$354$		0			0
$355$	4796.51			0.717104
$356$		0			0
$357$	−527.302	+	687.030i	−0.0781730	+	0.101853i
$358$		0			0
$359$	−2701.95			−0.397223			−0.198612	−	0.980078i	$-0.563643\pi$
$359$	−2701.95			−0.397223			−0.198612	+	0.980078i	$0.563643\pi$
$360$		0			0
$361$	−2965.07			−0.432289
$362$		0			0
$363$	−3897.93	+	5078.68i	−0.563605	+	0.734329i
$364$		0			0
$365$	−2321.81			−0.332956
$366$		0			0
$367$			11089.3i			1.57727i	0.614861	+	0.788636i	$0.289212\pi$
$367$			11089.3i			1.57727i	−0.614861	+	0.788636i	$0.710788\pi$
$368$		0			0
$369$	−2357.30	+	631.057i	−0.332564	+	0.0890285i
$370$		0			0
$371$		−	7576.49i		−	1.06025i
$372$		0			0
$373$		−	1299.98i		−	0.180456i	−0.995921	−	0.0902282i	$-0.971240\pi$
$373$		−	1299.98i		−	0.180456i	0.995921	−	0.0902282i	$-0.0287597\pi$
$374$		0			0
$375$	14095.0	−	18364.6i	1.94096	−	2.52891i
$376$		0			0
$377$			8305.33i			1.13461i
$378$		0			0
$379$	2953.18			0.400250			0.200125	−	0.979770i	$-0.435865\pi$
$379$	2953.18			0.400250			0.200125	+	0.979770i	$0.435865\pi$
$380$		0			0
$381$	934.343	+	717.117i	0.125637	+	0.0964279i
$382$		0			0
$383$	−2647.22			−0.353176			−0.176588	−	0.984285i	$-0.556506\pi$
$383$	−2647.22			−0.353176			−0.176588	+	0.984285i	$0.556506\pi$
$384$		0			0
$385$	−4449.92			−0.589062
$386$		0			0
$387$	3151.49	+	11772.3i	0.413952	+	1.54631i
$388$		0			0
$389$	3585.34			0.467310			0.233655	−	0.972320i	$-0.424931\pi$
$389$	3585.34			0.467310			0.233655	+	0.972320i	$0.424931\pi$
$390$		0			0
$391$		−	811.829i		−	0.105002i
$392$		0			0
$393$	−6513.22	−	4998.96i	−0.836001	−	0.641639i
$394$		0			0
$395$			13175.9i			1.67835i
$396$		0			0
$397$			6502.91i			0.822094i	0.911614	+	0.411047i	$0.134837\pi$
$397$			6502.91i			0.822094i	−0.911614	+	0.411047i	$0.865163\pi$
$398$		0			0
$399$	−5376.77	−	4126.73i	−0.674625	−	0.517781i
$400$		0			0
$401$		−	6673.49i		−	0.831068i	−0.909578	−	0.415534i	$-0.863595\pi$
$401$		−	6673.49i		−	0.831068i	0.909578	−	0.415534i	$-0.136405\pi$
$402$		0			0
$403$	5935.53			0.733672
$404$		0			0
$405$	13516.8	−	7795.68i	1.65841	−	0.956470i
$406$		0			0
$407$	−1055.59			−0.128560
$408$		0			0
$409$	2660.34			0.321627			0.160814	−	0.986985i	$-0.448588\pi$
$409$	2660.34			0.321627			0.160814	+	0.986985i	$0.448588\pi$
$410$		0			0
$411$	2240.05	+	1719.26i	0.268841	+	0.206338i
$412$		0			0
$413$	16753.7			1.99612
$414$		0			0
$415$		−	4374.72i		−	0.517461i
$416$		0			0
$417$	94.1285	−	122.641i	0.0110539	−	0.0144023i
$418$		0			0
$419$		−	8957.94i		−	1.04445i	−0.852808	−	0.522224i	$-0.825102\pi$
$419$		−	8957.94i		−	1.04445i	0.852808	−	0.522224i	$-0.174898\pi$
$420$		0			0
$421$			12070.2i			1.39730i	0.715461	+	0.698652i	$0.246217\pi$
$421$			12070.2i			1.39730i	−0.715461	+	0.698652i	$0.753783\pi$
$422$		0			0
$423$	2994.56	+	11186.1i	0.344209	+	1.28579i
$424$		0			0
$425$		−	2656.33i		−	0.303179i
$426$		0			0
$427$	−13543.4			−1.53492
$428$		0			0
$429$	2134.24	−	2780.73i	0.240191	−	0.312948i
$430$		0			0
$431$	14197.2			1.58667			0.793334	−	0.608787i	$-0.208344\pi$
$431$	14197.2			1.58667			0.793334	+	0.608787i	$0.208344\pi$
$432$		0			0
$433$	−7129.67			−0.791294			−0.395647	−	0.918403i	$-0.629480\pi$
$433$	−7129.67			−0.791294			−0.395647	+	0.918403i	$0.629480\pi$
$434$		0			0
$435$	−8291.58	+	10803.2i	−0.913909	+	1.19075i
$436$		0			0
$437$	6353.47			0.695487
$438$		0			0
$439$		−	5206.56i		−	0.566049i	−0.959113	−	0.283025i	$-0.908662\pi$
$439$		−	5206.56i		−	0.566049i	0.959113	−	0.283025i	$-0.0913378\pi$
$440$		0			0
$441$	655.992	+	2450.45i	0.0708338	+	0.264598i
$442$		0			0
$443$		−	17034.3i		−	1.82692i	−0.406930	−	0.913459i	$-0.633401\pi$
$443$		−	17034.3i		−	1.82692i	0.406930	−	0.913459i	$-0.366599\pi$
$444$		0			0
$445$		−	9719.01i		−	1.03534i
$446$		0			0
$447$	1548.39	−	2017.42i	0.163840	−	0.213470i
$448$		0			0
$449$			9312.78i			0.978836i	0.872049	+	0.489418i	$0.162791\pi$
$449$			9312.78i			0.978836i	−0.872049	+	0.489418i	$0.837209\pi$
$450$		0			0
$451$	−898.905			−0.0938532
$452$		0			0
$453$	1635.83	+	1255.52i	0.169665	+	0.130219i
$454$		0			0
$455$	−30348.4			−3.12693
$456$		0			0
$457$	−13276.6			−1.35898			−0.679488	−	0.733687i	$-0.737798\pi$
$457$	−13276.6			−1.35898			−0.679488	+	0.733687i	$0.737798\pi$
$458$		0			0
$459$	428.444	−	1033.35i	0.0435688	−	0.105082i
$460$		0			0
$461$	507.329			0.0512552			0.0256276	−	0.999672i	$-0.491842\pi$
$461$	507.329			0.0512552			0.0256276	+	0.999672i	$0.491842\pi$
$462$		0			0
$463$			4468.90i			0.448569i	0.974524	+	0.224284i	$0.0720044\pi$
$463$			4468.90i			0.448569i	−0.974524	+	0.224284i	$0.927996\pi$
$464$		0			0
$465$	7720.69	+	5925.70i	0.769975	+	0.590963i
$466$		0			0
$467$			12258.9i			1.21472i	0.794428	+	0.607358i	$0.207771\pi$
$467$			12258.9i			1.21472i	−0.794428	+	0.607358i	$0.792229\pi$
$468$		0			0
$469$			20009.6i			1.97006i
$470$		0			0
$471$	10480.3	+	8043.71i	1.02528	+	0.786910i
$472$		0			0
$473$			4489.12i			0.436385i
$474$		0			0
$475$	20788.8			2.00811
$476$		0			0
$477$	2530.68	+	9453.33i	0.242918	+	0.907418i
$478$		0			0
$479$	5781.56			0.551495			0.275747	−	0.961230i	$-0.411075\pi$
$479$	5781.56			0.551495			0.275747	+	0.961230i	$0.411075\pi$
$480$		0			0
$481$	−7199.13			−0.682437
$482$		0			0
$483$	−8772.92	−	6733.30i	−0.826463	−	0.634318i
$484$		0			0
$485$	−15849.7			−1.48391
$486$		0			0
$487$			8699.77i			0.809495i	0.914429	+	0.404747i	$0.132641\pi$
$487$			8699.77i			0.809495i	−0.914429	+	0.404747i	$0.867359\pi$
$488$		0			0
$489$	−5719.76	+	7452.36i	−0.528950	+	0.689177i
$490$		0			0
$491$		−	6084.95i		−	0.559287i	−0.960104	−	0.279643i	$-0.909784\pi$
$491$		−	6084.95i		−	0.559287i	0.960104	−	0.279643i	$-0.0902163\pi$
$492$		0			0
$493$			976.313i			0.0891905i
$494$		0			0
$495$	5552.25	−	1486.35i	0.504152	−	0.134963i
$496$		0			0
$497$		−	4684.25i		−	0.422772i
$498$		0			0
$499$	6970.12			0.625301			0.312651	−	0.949868i	$-0.398783\pi$
$499$	6970.12			0.625301			0.312651	+	0.949868i	$0.398783\pi$
$500$		0			0
$501$	−80.9728	+	105.501i	−0.00722075	+	0.00940803i
$502$		0			0
$503$	15721.2			1.39359			0.696795	−	0.717271i	$-0.254609\pi$
$503$	15721.2			1.39359			0.696795	+	0.717271i	$0.254609\pi$
$504$		0			0
$505$	1133.26			0.0998604
$506$		0			0
$507$	7604.82	−	9908.43i	0.666157	−	0.867947i
$508$		0			0
$509$	11249.3			0.979602			0.489801	−	0.871834i	$-0.337069\pi$
$509$	11249.3			0.979602			0.489801	+	0.871834i	$0.337069\pi$
$510$		0			0
$511$			2267.47i			0.196296i
$512$		0			0
$513$	8087.11	+	3353.05i	0.696013	+	0.288579i
$514$		0			0
$515$		−	17301.0i		−	1.48033i
$516$		0			0
$517$			4265.58i			0.362863i
$518$		0			0
$519$	2142.70	−	2791.75i	0.181222	−	0.236116i
$520$		0			0
$521$			12276.2i			1.03231i	0.856496	+	0.516153i	$0.172636\pi$
$521$			12276.2i			1.03231i	−0.856496	+	0.516153i	$0.827364\pi$
$522$		0			0
$523$	770.278			0.0644013			0.0322007	−	0.999481i	$-0.489748\pi$
$523$	770.278			0.0644013			0.0322007	+	0.999481i	$0.489748\pi$
$524$		0			0
$525$	−28705.3	−	22031.6i	−2.38629	−	1.83150i
$526$		0			0
$527$	697.737			0.0576735
$528$		0			0
$529$	−1800.47			−0.147980
$530$		0			0
$531$	−20904.0	+	5596.05i	−1.70839	+	0.457341i
$532$		0			0
$533$	−6130.51			−0.498202
$534$		0			0
$535$		−	15583.6i		−	1.25933i
$536$		0			0
$537$	−13357.0	−	10251.6i	−1.07337	−	0.823820i
$538$		0			0
$539$			934.423i			0.0746725i
$540$		0			0
$541$		−	14469.7i		−	1.14991i	−0.818184	−	0.574956i	$-0.805019\pi$
$541$		−	14469.7i		−	1.14991i	0.818184	−	0.574956i	$-0.194981\pi$
$542$		0			0
$543$	−8753.90	−	6718.70i	−0.691834	−	0.530989i
$544$		0			0
$545$		−	31867.9i		−	2.50472i
$546$		0			0
$547$	1150.14			0.0899021			0.0449511	−	0.998989i	$-0.485687\pi$
$547$	1150.14			0.0899021			0.0449511	+	0.998989i	$0.485687\pi$
$548$		0			0
$549$	16898.3	−	4523.74i	1.31367	−	0.351673i
$550$		0			0
$551$	−7640.74			−0.590756
$552$		0			0
$553$	12867.5			0.989479
$554$		0			0
$555$	−9364.33	−	7187.21i	−0.716205	−	0.549694i
$556$		0			0
$557$	−24347.5			−1.85213			−0.926066	−	0.377361i	$-0.876832\pi$
$557$	−24347.5			−1.85213			−0.926066	+	0.377361i	$0.876832\pi$
$558$		0			0
$559$			30615.7i			2.31647i
$560$		0			0
$561$	250.885	−	326.882i	0.0188812	−	0.0246007i
$562$		0			0
$563$		−	15515.2i		−	1.16144i	−0.814105	−	0.580718i	$-0.802772\pi$
$563$		−	15515.2i		−	1.16144i	0.814105	−	0.580718i	$-0.197228\pi$
$564$		0			0
$565$			813.518i			0.0605752i
$566$		0			0
$567$	−7613.23	−	13200.5i	−0.563890	−	0.977724i
$568$		0			0
$569$			10960.4i			0.807531i	0.914863	+	0.403765i	$0.132299\pi$
$569$			10960.4i			0.807531i	−0.914863	+	0.403765i	$0.867701\pi$
$570$		0			0
$571$	24367.3			1.78588			0.892941	−	0.450174i	$-0.148638\pi$
$571$	24367.3			1.78588			0.892941	+	0.450174i	$0.148638\pi$
$572$		0			0
$573$	−9746.46	+	12698.8i	−0.710583	+	0.925830i
$574$		0			0
$575$	33919.7			2.46008
$576$		0			0
$577$	17109.4			1.23445			0.617223	−	0.786788i	$-0.288258\pi$
$577$	17109.4			1.23445			0.617223	+	0.786788i	$0.288258\pi$
$578$		0			0
$579$	8270.17	−	10775.3i	0.593604	−	0.773416i
$580$		0			0
$581$	−4272.34			−0.305071
$582$		0			0
$583$			3604.82i			0.256083i
$584$		0			0
$585$	37866.2	−	10136.9i	2.67620	−	0.716426i
$586$		0			0
$587$			13267.9i			0.932921i	0.884542	+	0.466460i	$0.154471\pi$
$587$			13267.9i			0.932921i	−0.884542	+	0.466460i	$0.845529\pi$
$588$		0			0
$589$			5460.57i			0.382002i
$590$		0			0
$591$	−13361.9	+	17409.4i	−0.930007	+	1.21172i
$592$		0			0
$593$			16312.5i			1.12963i	0.825216	+	0.564817i	$0.191053\pi$
$593$			16312.5i			1.12963i	−0.825216	+	0.564817i	$0.808947\pi$
$594$		0			0
$595$	−3567.53			−0.245806
$596$		0			0
$597$	−3227.84	−	2477.40i	−0.221284	−	0.169838i
$598$		0			0
$599$	15038.9			1.02583			0.512915	−	0.858440i	$-0.328566\pi$
$599$	15038.9			1.02583			0.512915	+	0.858440i	$0.328566\pi$
$600$		0			0
$601$	4408.70			0.299226			0.149613	−	0.988745i	$-0.452197\pi$
$601$	4408.70			0.299226			0.149613	+	0.988745i	$0.452197\pi$
$602$		0			0
$603$	−6683.57	−	24966.3i	−0.451370	−	1.68608i
$604$		0			0
$605$	−26372.0			−1.77219
$606$		0			0
$607$			1271.09i			0.0849950i	0.999097	+	0.0424975i	$0.0135315\pi$
$607$			1271.09i			0.0849950i	−0.999097	+	0.0424975i	$0.986469\pi$
$608$		0			0
$609$	10550.4	+	8097.53i	0.702009	+	0.538799i
$610$		0			0
$611$			29091.2i			1.92619i
$612$		0			0
$613$		−	21306.8i		−	1.40387i	−0.712239	−	0.701937i	$-0.752319\pi$
$613$		−	21306.8i		−	1.40387i	0.712239	−	0.701937i	$-0.247681\pi$
$614$		0			0
$615$	−7974.31	−	6120.36i	−0.522854	−	0.401295i
$616$		0			0
$617$			26964.4i			1.75940i	0.475533	+	0.879698i	$0.342255\pi$
$617$			26964.4i			1.75940i	−0.475533	+	0.879698i	$0.657745\pi$
$618$		0			0
$619$	−17633.6			−1.14500			−0.572500	−	0.819905i	$-0.694026\pi$
$619$	−17633.6			−1.14500			−0.572500	+	0.819905i	$0.694026\pi$
$620$		0			0
$621$	13195.2	+	5470.95i	0.852665	+	0.353529i
$622$		0			0
$623$	−9491.56			−0.610387
$624$		0			0
$625$	53718.1			3.43796
$626$		0			0
$627$	2558.22	+	1963.46i	0.162943	+	0.125060i
$628$		0			0
$629$	−846.277			−0.0536459
$630$		0			0
$631$		−	168.115i		−	0.0106062i	−0.999986	−	0.00530312i	$-0.998312\pi$
$631$		−	168.115i		−	0.0106062i	0.999986	−	0.00530312i	$-0.00168804\pi$
$632$		0			0
$633$	−11122.3	+	14491.4i	−0.698373	+	0.909921i
$634$		0			0
$635$			4851.75i			0.303206i
$636$		0			0
$637$			6372.74i			0.396385i
$638$		0			0
$639$	1564.63	+	5844.63i	0.0968633	+	0.361831i
$640$		0			0
$641$		−	823.334i		−	0.0507328i	−0.999678	−	0.0253664i	$-0.991925\pi$
$641$		−	823.334i		−	0.0507328i	0.999678	−	0.0253664i	$-0.00807525\pi$
$642$		0			0
$643$	−17860.7			−1.09543			−0.547713	−	0.836666i	$-0.684501\pi$
$643$	−17860.7			−1.09543			−0.547713	+	0.836666i	$0.684501\pi$
$644$		0			0
$645$	−30565.0	+	39823.6i	−1.86589	+	2.43109i
$646$		0			0
$647$	2518.14			0.153011			0.0765057	−	0.997069i	$-0.475624\pi$
$647$	2518.14			0.153011			0.0765057	+	0.997069i	$0.475624\pi$
$648$		0			0
$649$	−7971.26			−0.482126
$650$		0			0
$651$	5787.03	−	7540.01i	0.348405	−	0.453942i
$652$		0			0
$653$	18937.3			1.13488			0.567439	−	0.823416i	$-0.307934\pi$
$653$	18937.3			1.13488			0.567439	+	0.823416i	$0.307934\pi$
$654$		0			0
$655$		−	33821.1i		−	2.01756i
$656$		0			0
$657$	−757.377	−	2829.17i	−0.0449743	−	0.168001i
$658$		0			0
$659$			10400.8i			0.614807i	0.951579	+	0.307404i	$0.0994602\pi$
$659$			10400.8i			0.614807i	−0.951579	+	0.307404i	$0.900540\pi$
$660$		0			0
$661$			3225.88i			0.189822i	0.995486	+	0.0949108i	$0.0302566\pi$
$661$			3225.88i			0.189822i	−0.995486	+	0.0949108i	$0.969743\pi$
$662$		0			0
$663$	1711.03	−	2229.33i	0.100228	−	0.130588i
$664$		0			0
$665$		−	27919.9i		−	1.62810i
$666$		0			0
$667$	−12466.9			−0.723717
$668$		0			0
$669$	−7683.85	−	5897.43i	−0.444058	−	0.340819i
$670$		0			0
$671$	6443.81			0.370731
$672$		0			0
$673$	1215.38			0.0696128			0.0348064	−	0.999394i	$-0.488919\pi$
$673$	1215.38			0.0696128			0.0348064	+	0.999394i	$0.488919\pi$
$674$		0			0
$675$	43175.1	+	17901.2i	2.46194	+	1.02076i
$676$		0			0
$677$	3901.92			0.221511			0.110755	−	0.993848i	$-0.464673\pi$
$677$	3901.92			0.221511			0.110755	+	0.993848i	$0.464673\pi$
$678$		0			0
$679$			15478.8i			0.874846i
$680$		0			0
$681$	−11836.4	−	9084.58i	−0.666040	−	0.511192i
$682$		0			0
$683$			31547.8i			1.76742i	0.468039	+	0.883708i	$0.344960\pi$
$683$			31547.8i			1.76742i	−0.468039	+	0.883708i	$0.655040\pi$
$684$		0			0
$685$			11631.9i			0.648805i
$686$		0			0
$687$	−12566.3	−	9644.76i	−0.697867	−	0.535620i
$688$		0			0
$689$			24584.8i			1.35937i
$690$		0			0
$691$	−2791.51			−0.153682			−0.0768408	−	0.997043i	$-0.524483\pi$
$691$	−2791.51			−0.153682			−0.0768408	+	0.997043i	$0.524483\pi$
$692$		0			0
$693$	−1451.57	−	5422.31i	−0.0795679	−	0.297224i
$694$		0			0
$695$	636.838			0.0347578
$696$		0			0
$697$	−720.657			−0.0391633
$698$		0			0
$699$	8567.48	+	6575.62i	0.463593	+	0.355812i
$700$		0			0
$701$	9830.35			0.529653			0.264827	−	0.964296i	$-0.414685\pi$
$701$	9830.35			0.529653			0.264827	+	0.964296i	$0.414685\pi$
$702$		0			0
$703$		−	6623.06i		−	0.355325i
$704$		0			0
$705$	−29043.0	+	37840.6i	−1.55152	+	2.02150i
$706$		0			0
$707$		−	1106.74i		−	0.0588730i
$708$		0			0
$709$			3029.84i			0.160491i	0.996775	+	0.0802455i	$0.0255704\pi$
$709$			3029.84i			0.160491i	−0.996775	+	0.0802455i	$0.974430\pi$
$710$		0			0
$711$	−16055.0	+	4297.98i	−0.846850	+	0.226704i
$712$		0			0
$713$			8909.65i			0.467979i
$714$		0			0
$715$	14439.4			0.755251
$716$		0			0
$717$	−8791.24	+	11454.2i	−0.457901	+	0.596606i
$718$		0			0
$719$	28359.2			1.47096			0.735479	−	0.677548i	$-0.236957\pi$
$719$	28359.2			1.47096			0.735479	+	0.677548i	$0.236957\pi$
$720$		0			0
$721$	−16896.1			−0.872735
$722$		0			0
$723$	8981.64	−	11702.3i	0.462006	−	0.601955i
$724$		0			0
$725$	−40792.1			−2.08963
$726$		0			0
$727$		−	24702.5i		−	1.26020i	−0.776514	−	0.630100i	$-0.783014\pi$
$727$		−	24702.5i		−	1.26020i	0.776514	−	0.630100i	$-0.216986\pi$
$728$		0			0
$729$	13908.4	+	13927.6i	0.706619	+	0.707594i
$730$		0			0
$731$			3598.96i			0.182096i
$732$		0			0
$733$			30507.1i			1.53725i	0.639698	+	0.768627i	$0.279060\pi$
$733$			30507.1i			1.53725i	−0.639698	+	0.768627i	$0.720940\pi$
$734$		0			0
$735$	−6362.19	+	8289.40i	−0.319283	+	0.415999i
$736$		0			0
$737$		−	9520.36i		−	0.475831i
$738$		0			0
$739$	−4105.52			−0.204363			−0.102181	−	0.994766i	$-0.532582\pi$
$739$	−4105.52			−0.204363			−0.102181	+	0.994766i	$0.532582\pi$
$740$		0			0
$741$	17447.0	+	13390.7i	0.864954	+	0.663860i
$742$		0			0
$743$	−4140.87			−0.204460			−0.102230	−	0.994761i	$-0.532598\pi$
$743$	−4140.87			−0.204460			−0.102230	+	0.994761i	$0.532598\pi$
$744$		0			0
$745$	10475.8			0.515175
$746$		0			0
$747$	5330.68	−	1427.04i	0.261097	−	0.0698964i
$748$		0			0
$749$	−15218.9			−0.742440
$750$		0			0
$751$		−	22033.0i		−	1.07057i	−0.844672	−	0.535284i	$-0.820204\pi$
$751$		−	22033.0i		−	1.07057i	0.844672	−	0.535284i	$-0.179796\pi$
$752$		0			0
$753$	−22008.5	−	16891.7i	−1.06512	−	0.817487i
$754$		0			0
$755$			8494.36i			0.409459i
$756$		0			0
$757$			37525.3i			1.80169i	0.434137	+	0.900847i	$0.357053\pi$
$757$			37525.3i			1.80169i	−0.434137	+	0.900847i	$0.642947\pi$
$758$		0			0
$759$	4174.07	+	3203.64i	0.199617	+	0.153208i
$760$		0			0
$761$			32680.8i			1.55674i	0.627806	+	0.778369i	$0.283953\pi$
$761$			32680.8i			1.55674i	−0.627806	+	0.778369i	$0.716047\pi$
$762$		0			0
$763$	−31122.1			−1.47667
$764$		0			0
$765$	4451.27	−	1191.62i	0.210374	−	0.0563177i
$766$		0			0
$767$	−54363.8			−2.55928
$768$		0			0
$769$	−27457.2			−1.28756			−0.643779	−	0.765212i	$-0.722634\pi$
$769$	−27457.2			−1.28756			−0.643779	+	0.765212i	$0.722634\pi$
$770$		0			0
$771$	−25655.4	−	19690.8i	−1.19839	−	0.919775i
$772$		0			0
$773$	−8851.25			−0.411847			−0.205923	−	0.978568i	$-0.566020\pi$
$773$	−8851.25			−0.411847			−0.205923	+	0.978568i	$0.566020\pi$
$774$		0			0
$775$			29152.7i			1.35122i
$776$		0			0
$777$	−7019.01	+	9145.18i	−0.324074	+	0.422241i
$778$		0			0
$779$		−	5639.95i		−	0.259399i
$780$		0			0
$781$			2228.72i			0.102113i
$782$		0			0
$783$	−15868.7	−	6579.42i	−0.724265	−	0.300293i
$784$		0			0
$785$			54420.8i			2.47434i
$786$		0			0
$787$	−3892.16			−0.176290			−0.0881452	−	0.996108i	$-0.528094\pi$
$787$	−3892.16			−0.176290			−0.0881452	+	0.996108i	$0.528094\pi$
$788$		0			0
$789$	15121.8	−	19702.5i	0.682321	−	0.889007i
$790$		0			0
$791$	794.479			0.0357123
$792$		0			0
$793$	43946.6			1.96796
$794$		0			0
$795$	−24544.1	+	31978.8i	−1.09495	+	1.42663i
$796$		0			0
$797$	−16462.8			−0.731672			−0.365836	−	0.930679i	$-0.619217\pi$
$797$	−16462.8			−0.731672			−0.365836	+	0.930679i	$0.619217\pi$
$798$		0			0
$799$			3419.74i			0.151417i
$800$		0			0
$801$	11842.8	−	3170.35i	0.522403	−	0.139849i
$802$		0			0
$803$		−	1078.84i		−	0.0474116i
$804$		0			0
$805$		−	45555.0i		−	1.99454i
$806$		0			0
$807$	−12389.1	+	16142.0i	−0.540418	+	0.704119i
$808$		0			0
$809$			2287.89i			0.0994287i	0.998763	+	0.0497144i	$0.0158311\pi$
$809$			2287.89i			0.0994287i	−0.998763	+	0.0497144i	$0.984169\pi$
$810$		0			0
$811$	−2892.28			−0.125230			−0.0626150	−	0.998038i	$-0.519944\pi$
$811$	−2892.28			−0.125230			−0.0626150	+	0.998038i	$0.519944\pi$
$812$		0			0
$813$	26433.6	+	20288.1i	1.14031	+	0.875195i
$814$		0			0
$815$	−38697.8			−1.66322
$816$		0			0
$817$	−28165.8			−1.20612
$818$		0			0
$819$	−9899.67	−	36980.0i	−0.422372	−	1.57776i
$820$		0			0
$821$	37351.7			1.58780			0.793899	−	0.608049i	$-0.208048\pi$
$821$	37351.7			1.58780			0.793899	+	0.608049i	$0.208048\pi$
$822$		0			0
$823$			25859.6i			1.09527i	0.836717	+	0.547636i	$0.184472\pi$
$823$			25859.6i			1.09527i	−0.836717	+	0.547636i	$0.815528\pi$
$824$		0			0
$825$	13657.7	+	10482.4i	0.576364	+	0.442365i
$826$		0			0
$827$			32182.0i			1.35318i	0.736361	+	0.676589i	$0.236543\pi$
$827$			32182.0i			1.35318i	−0.736361	+	0.676589i	$0.763457\pi$
$828$		0			0
$829$		−	16710.7i		−	0.700105i	−0.936730	−	0.350053i	$-0.886164\pi$
$829$		−	16710.7i		−	0.700105i	0.936730	−	0.350053i	$-0.113836\pi$
$830$		0			0
$831$	15437.9	+	11848.7i	0.644445	+	0.494617i
$832$		0			0
$833$			749.133i			0.0311595i
$834$		0			0
$835$	−547.831			−0.0227048
$836$		0			0
$837$	−4702.08	+	11340.8i	−0.194179	+	0.468333i
$838$		0			0
$839$	19182.4			0.789334			0.394667	−	0.918824i	$-0.370860\pi$
$839$	19182.4			0.789334			0.394667	+	0.918824i	$0.370860\pi$
$840$		0			0
$841$	−9396.22			−0.385265
$842$		0			0
$843$	27438.0	+	21058.9i	1.12101	+	0.860388i
$844$		0			0
$845$	51451.3			2.09465
$846$		0			0
$847$			25754.8i			1.04480i
$848$		0			0
$849$	20439.9	−	26631.4i	0.826260	−	1.07655i
$850$		0			0
$851$		−	10806.4i		−	0.435298i
$852$		0			0
$853$			16306.8i			0.654552i	0.944929	+	0.327276i	$0.106131\pi$
$853$			16306.8i			0.654552i	−0.944929	+	0.327276i	$0.893869\pi$
$854$		0			0
$855$	9325.75	+	34836.2i	0.373022	+	1.39342i
$856$		0			0
$857$		−	42374.8i		−	1.68903i	−0.535534	−	0.844514i	$-0.679890\pi$
$857$		−	42374.8i		−	1.68903i	0.535534	−	0.844514i	$-0.320110\pi$
$858$		0			0
$859$	−21561.9			−0.856442			−0.428221	−	0.903674i	$-0.640860\pi$
$859$	−21561.9			−0.856442			−0.428221	+	0.903674i	$0.640860\pi$
$860$		0			0
$861$	−5977.12	+	7787.69i	−0.236585	+	0.308251i
$862$		0			0
$863$	16679.1			0.657894			0.328947	−	0.944348i	$-0.393306\pi$
$863$	16679.1			0.657894			0.328947	+	0.944348i	$0.393306\pi$
$864$		0			0
$865$	14496.7			0.569829
$866$		0			0
$867$	−15342.1	+	19989.4i	−0.600974	+	0.783018i
$868$		0			0
$869$	−6122.23			−0.238990
$870$		0			0
$871$		−	64928.7i		−	2.52586i
$872$		0			0
$873$	−5170.19	−	19313.2i	−0.200440	−	0.748741i
$874$		0			0
$875$		−	93129.6i		−	3.59812i
$876$		0			0
$877$			14094.6i			0.542690i	0.962482	+	0.271345i	$0.0874685\pi$
$877$			14094.6i			0.542690i	−0.962482	+	0.271345i	$0.912532\pi$
$878$		0			0
$879$	8291.53	−	10803.2i	0.318164	−	0.414541i
$880$		0			0
$881$		−	12553.3i		−	0.480057i	−0.970766	−	0.240028i	$-0.922843\pi$
$881$		−	12553.3i		−	0.480057i	0.970766	−	0.240028i	$-0.0771568\pi$
$882$		0			0
$883$	43192.3			1.64613			0.823067	−	0.567944i	$-0.192261\pi$
$883$	43192.3			1.64613			0.823067	+	0.567944i	$0.192261\pi$
$884$		0			0
$885$	−70714.2	−	54273.8i	−2.68591	−	2.06146i
$886$		0			0
$887$	−17930.0			−0.678725			−0.339362	−	0.940656i	$-0.610211\pi$
$887$	−17930.0			−0.678725			−0.339362	+	0.940656i	$0.610211\pi$
$888$		0			0
$889$	4738.20			0.178756
$890$		0			0
$891$	3622.30	+	6280.67i	0.136197	+	0.236151i
$892$		0			0
$893$	−26763.3			−1.00291
$894$		0			0
$895$		−	69358.8i		−	2.59040i
$896$		0			0
$897$	28467.1	+	21848.7i	1.05963	+	0.813276i
$898$		0			0
$899$		−	10714.8i		−	0.397508i
$900$		0			0
$901$			2890.00i			0.106859i
$902$		0			0
$903$	38891.6	+	29849.7i	1.43326	+	1.10004i
$904$		0			0
$905$		−	45456.2i		−	1.66963i
$906$		0			0
$907$	5289.85			0.193657			0.0968283	−	0.995301i	$-0.469130\pi$
$907$	5289.85			0.193657			0.0968283	+	0.995301i	$0.469130\pi$
$908$		0			0
$909$	369.671	+	1380.90i	0.0134887	+	0.0503868i
$910$		0			0
$911$	−4585.72			−0.166774			−0.0833872	−	0.996517i	$-0.526574\pi$
$911$	−4585.72			−0.166774			−0.0833872	+	0.996517i	$0.526574\pi$
$912$		0			0
$913$	2032.74			0.0736843
$914$		0			0
$915$	57163.9	+	43873.9i	2.06533	+	1.58516i
$916$		0			0
$917$	−33029.6			−1.18946
$918$		0			0
$919$		−	43234.5i		−	1.55188i	−0.630809	−	0.775939i	$-0.717277\pi$
$919$		−	43234.5i		−	1.55188i	0.630809	−	0.775939i	$-0.282723\pi$
$920$		0			0
$921$	−3012.03	+	3924.42i	−0.107763	+	0.140406i
$922$		0			0
$923$			15199.8i			0.542046i
$924$		0			0
$925$		−	35358.9i		−	1.25686i
$926$		0			0
$927$	21081.5	−	5643.59i	0.746935	−	0.199957i
$928$		0			0
$929$			695.704i			0.0245697i	0.999925	+	0.0122849i	$0.00391050\pi$
$929$			695.704i			0.0245697i	−0.999925	+	0.0122849i	$0.996090\pi$
$930$		0			0
$931$	−5862.80			−0.206386
$932$		0			0
$933$	−1471.01	+	1916.60i	−0.0516169	+	0.0672525i
$934$		0			0
$935$	1697.39			0.0593698
$936$		0			0
$937$	21092.2			0.735381			0.367691	−	0.929948i	$-0.380149\pi$
$937$	21092.2			0.735381			0.367691	+	0.929948i	$0.380149\pi$
$938$		0			0
$939$	17894.3	−	23314.8i	0.621896	−	0.810277i
$940$		0			0
$941$	−11392.6			−0.394676			−0.197338	−	0.980336i	$-0.563230\pi$
$941$	−11392.6			−0.394676			−0.197338	+	0.980336i	$0.563230\pi$
$942$		0			0
$943$		−	9202.33i		−	0.317783i
$944$		0			0
$945$	24041.7	−	57985.4i	0.827595	−	1.99605i
$946$		0			0
$947$		−	3540.41i		−	0.121487i	−0.998153	−	0.0607433i	$-0.980653\pi$
$947$		−	3540.41i		−	0.121487i	0.998153	−	0.0607433i	$-0.0193471\pi$
$948$		0			0
$949$		−	7357.67i		−	0.251676i
$950$		0			0
$951$	−14532.6	+	18934.8i	−0.495535	+	0.645640i
$952$		0			0
$953$		−	29360.7i		−	0.997992i	−0.866604	−	0.498996i	$-0.833702\pi$
$953$		−	29360.7i		−	0.997992i	0.866604	−	0.498996i	$-0.166298\pi$
$954$		0			0
$955$	−65940.9			−2.23434
$956$		0			0
$957$	−5019.77	−	3852.72i	−0.169557	−	0.130137i
$958$		0			0
$959$	11359.7			0.382505
$960$		0			0
$961$	22133.5			0.742959
$962$		0			0
$963$	18989.0	−	5083.40i	0.635421	−	0.170104i
$964$		0			0
$965$	55952.9			1.86652
$966$		0			0
$967$		−	27094.7i		−	0.901043i	−0.892766	−	0.450521i	$-0.851238\pi$
$967$		−	27094.7i		−	0.901043i	0.892766	−	0.450521i	$-0.148762\pi$
$968$		0			0
$969$	2050.94	+	1574.11i	0.0679934	+	0.0521856i
$970$		0			0
$971$		−	10806.5i		−	0.357154i	−0.983926	−	0.178577i	$-0.942851\pi$
$971$		−	10806.5i		−	0.357154i	0.983926	−	0.178577i	$-0.0571494\pi$
$972$		0			0
$973$		−	621.934i		−	0.0204916i
$974$		0			0
$975$	93145.2	+	71489.9i	3.05952	+	2.34821i
$976$		0			0
$977$		−	14327.6i		−	0.469172i	−0.972095	−	0.234586i	$-0.924627\pi$
$977$		−	14327.6i		−	0.469172i	0.972095	−	0.234586i	$-0.0753734\pi$
$978$		0			0
$979$	4515.99			0.147428
$980$		0			0
$981$	38831.7	−	10395.4i	1.26381	−	0.338327i
$982$		0			0
$983$	25691.9			0.833617			0.416808	−	0.908994i	$-0.363149\pi$
$983$	25691.9			0.833617			0.416808	+	0.908994i	$0.363149\pi$
$984$		0			0
$985$	−90401.5			−2.92430
$986$		0			0
$987$	36955.0	+	28363.3i	1.19178	+	0.914706i
$988$		0			0
$989$	−45956.3			−1.47758
$990$		0			0
$991$			49378.3i			1.58280i	0.611300	+	0.791399i	$0.290647\pi$
$991$			49378.3i			1.58280i	−0.611300	+	0.791399i	$0.709353\pi$
$992$		0			0
$993$	−21225.3	+	27654.7i	−0.678311	+	0.883782i
$994$		0			0
$995$		−	16761.1i		−	0.534034i
$996$		0			0
$997$		−	22705.1i		−	0.721241i	−0.932713	−	0.360621i	$-0.882565\pi$
$997$		−	22705.1i		−	0.721241i	0.932713	−	0.360621i	$-0.117435\pi$
$998$		0			0
$999$	5703.10	−	13755.1i	0.180619	−	0.435627i

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	768.4.f.g.383.4		12
3.2	odd	2		768.4.f.h.383.3		12
4.3	odd	2		768.4.f.e.383.9		12
8.3	odd	2		768.4.f.h.383.4		12
8.5	even	2		768.4.f.f.383.9		12
12.11	even	2		768.4.f.f.383.10		12
16.3	odd	4		384.4.c.c.383.10	yes	12
16.5	even	4		384.4.c.d.383.10	yes	12
16.11	odd	4		384.4.c.a.383.3	✓	12
16.13	even	4		384.4.c.b.383.3	yes	12
24.5	odd	2		768.4.f.e.383.10		12
24.11	even	2	inner	768.4.f.g.383.3		12
48.5	odd	4		384.4.c.a.383.4	yes	12
48.11	even	4		384.4.c.d.383.9	yes	12
48.29	odd	4		384.4.c.c.383.9	yes	12
48.35	even	4		384.4.c.b.383.4	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
384.4.c.a.383.3	✓	12	16.11	odd	4
384.4.c.a.383.4	yes	12	48.5	odd	4
384.4.c.b.383.3	yes	12	16.13	even	4
384.4.c.b.383.4	yes	12	48.35	even	4
384.4.c.c.383.9	yes	12	48.29	odd	4
384.4.c.c.383.10	yes	12	16.3	odd	4
384.4.c.d.383.9	yes	12	48.11	even	4
384.4.c.d.383.10	yes	12	16.5	even	4
768.4.f.e.383.9		12	4.3	odd	2
768.4.f.e.383.10		12	24.5	odd	2
768.4.f.f.383.9		12	8.5	even	2
768.4.f.f.383.10		12	12.11	even	2
768.4.f.g.383.3		12	24.11	even	2	inner
768.4.f.g.383.4		12	1.1	even	1	trivial
768.4.f.h.383.3		12	3.2	odd	2
768.4.f.h.383.4		12	8.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 768 = 2^{8} \cdot 3 \)
Weight:	\( k \)	\(=\)	\( 4 \)
Character orbit:	\([\chi]\)	\(=\)	768.f (of order \(2\), degree \(1\), not minimal)

\(f(q)\)	\(=\)	\(q+(-3.16369 + 4.12202i) q^{3} -21.4043 q^{5} +20.9034i q^{7} +(-6.98212 - 26.0816i) q^{9} +O(q^{10})\)	\(q+(-3.16369 + 4.12202i) q^{3} -21.4043 q^{5} +20.9034i q^{7} +(-6.98212 - 26.0816i) q^{9} -9.94564i q^{11} -67.8290i q^{13} +(67.7167 - 88.2292i) q^{15} -7.97348i q^{17} +62.4014 q^{19} +(-86.1644 - 66.1320i) q^{21} +101.816 q^{23} +333.146 q^{25} +(129.598 + 53.7337i) q^{27} -122.445 q^{29} +87.5072i q^{31} +(40.9961 + 31.4649i) q^{33} -447.424i q^{35} -106.136i q^{37} +(279.593 + 214.590i) q^{39} -90.3818i q^{41} -451.366 q^{43} +(149.448 + 558.260i) q^{45} -428.890 q^{47} -93.9530 q^{49} +(32.8669 + 25.2256i) q^{51} -362.452 q^{53} +212.880i q^{55} +(-197.419 + 257.220i) q^{57} -801.483i q^{59} +647.903i q^{61} +(545.195 - 145.950i) q^{63} +1451.84i q^{65} +957.240 q^{67} +(-322.115 + 419.688i) q^{69} -224.090 q^{71} +108.474 q^{73} +(-1053.97 + 1373.24i) q^{75} +207.898 q^{77} -615.569i q^{79} +(-631.500 + 364.210i) q^{81} +204.385i q^{83} +170.667i q^{85} +(387.378 - 504.721i) q^{87} +454.067i q^{89} +1417.86 q^{91} +(-360.707 - 276.846i) q^{93} -1335.66 q^{95} +740.490 q^{97} +(-259.398 + 69.4417i) q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q + 2 q^{3} - 12 q^{5}+O(q^{10}) \) 12 * q + 2 * q^3 - 12 * q^5	\( 12 q + 2 q^{3} - 12 q^{5} + 84 q^{15} + 180 q^{19} - 156 q^{21} + 120 q^{23} + 300 q^{25} - 130 q^{27} - 588 q^{29} - 116 q^{33} - 620 q^{39} - 372 q^{43} + 740 q^{45} - 1248 q^{47} - 948 q^{49} - 360 q^{51} + 948 q^{53} + 172 q^{57} + 2744 q^{63} + 2292 q^{67} - 3280 q^{69} + 2040 q^{71} + 216 q^{73} - 2522 q^{75} - 4824 q^{77} - 1076 q^{81} - 4156 q^{87} + 3480 q^{91} + 4180 q^{93} - 5448 q^{95} - 48 q^{97} - 3048 q^{99}+O(q^{100}) \) 12 * q + 2 * q^3 - 12 * q^5 + 84 * q^15 + 180 * q^19 - 156 * q^21 + 120 * q^23 + 300 * q^25 - 130 * q^27 - 588 * q^29 - 116 * q^33 - 620 * q^39 - 372 * q^43 + 740 * q^45 - 1248 * q^47 - 948 * q^49 - 360 * q^51 + 948 * q^53 + 172 * q^57 + 2744 * q^63 + 2292 * q^67 - 3280 * q^69 + 2040 * q^71 + 216 * q^73 - 2522 * q^75 - 4824 * q^77 - 1076 * q^81 - 4156 * q^87 + 3480 * q^91 + 4180 * q^93 - 5448 * q^95 - 48 * q^97 - 3048 * q^99

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