LMFDB - Embedded newform 1083.2.a.l.1.2

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1083,2,Mod(1,1083)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1083, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("1083.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1083 = 3 \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1083.a (trivial)

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:	yes
Analytic conductor:	$8.64779853890$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.564.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 5x + 3 $ x^3 - x^2 - 5*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 57)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$0.571993$ of defining polynomial
Character	$\chi$	$=$	1083.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	\(q-0.571993 q^{2} +1.00000 q^{3} -1.67282 q^{4} -2.67282 q^{5} -0.571993 q^{6} -3.67282 q^{7} +2.10083 q^{8} +1.00000 q^{9} +1.52884 q^{10} -3.81681 q^{11} -1.67282 q^{12} +0.143987 q^{13} +2.10083 q^{14} -2.67282 q^{15} +2.14399 q^{16} -0.571993 q^{18} +4.47116 q^{20} -3.67282 q^{21} +2.18319 q^{22} +7.52884 q^{23} +2.10083 q^{24} +2.14399 q^{25} -0.0823593 q^{26} +1.00000 q^{27} +6.14399 q^{28} -5.34565 q^{29} +1.52884 q^{30} +8.81681 q^{31} -5.42801 q^{32} -3.81681 q^{33} +9.81681 q^{35} -1.67282 q^{36} -1.00000 q^{37} +0.143987 q^{39} -5.61515 q^{40} +5.34565 q^{41} +2.10083 q^{42} +2.81681 q^{43} +6.38485 q^{44} -2.67282 q^{45} -4.30644 q^{46} -6.00000 q^{47} +2.14399 q^{48} +6.48963 q^{49} -1.22635 q^{50} -0.240864 q^{52} +8.01847 q^{53} -0.571993 q^{54} +10.2017 q^{55} -7.71598 q^{56} +3.05767 q^{58} +3.81681 q^{59} +4.47116 q^{60} +11.4896 q^{61} -5.04316 q^{62} -3.67282 q^{63} -1.18319 q^{64} -0.384851 q^{65} +2.18319 q^{66} +5.38485 q^{67} +7.52884 q^{69} -5.61515 q^{70} -13.6336 q^{71} +2.10083 q^{72} -0.345647 q^{73} +0.571993 q^{74} +2.14399 q^{75} +14.0185 q^{77} -0.0823593 q^{78} +6.52884 q^{79} -5.73050 q^{80} +1.00000 q^{81} -3.05767 q^{82} -2.28797 q^{83} +6.14399 q^{84} -1.61120 q^{86} -5.34565 q^{87} -8.01847 q^{88} -8.67282 q^{89} +1.52884 q^{90} -0.528837 q^{91} -12.5944 q^{92} +8.81681 q^{93} +3.43196 q^{94} -5.42801 q^{96} -5.91369 q^{97} -3.71203 q^{98} -3.81681 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 2 q^{5} - q^{6} - q^{7} - 3 q^{8} + 3 q^{9} - 4 q^{10} + 5 q^{12} - q^{13} - 3 q^{14} + 2 q^{15} + 5 q^{16} - q^{18} + 22 q^{20} - q^{21} + 18 q^{22} + 14 q^{23}+ \cdots - 14 q^{98}+O(q^{100}) $ 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 2 * q^5 - q^6 - q^7 - 3 * q^8 + 3 * q^9 - 4 * q^10 + 5 * q^12 - q^13 - 3 * q^14 + 2 * q^15 + 5 * q^16 - q^18 + 22 * q^20 - q^21 + 18 * q^22 + 14 * q^23 - 3 * q^24 + 5 * q^25 - 21 * q^26 + 3 * q^27 + 17 * q^28 + 4 * q^29 - 4 * q^30 + 15 * q^31 - 17 * q^32 + 18 * q^35 + 5 * q^36 - 3 * q^37 - q^39 - 24 * q^40 - 4 * q^41 - 3 * q^42 - 3 * q^43 + 12 * q^44 + 2 * q^45 + 20 * q^46 - 18 * q^47 + 5 * q^48 - 2 * q^49 - 23 * q^50 + 5 * q^52 - 6 * q^53 - q^54 + 12 * q^55 - 21 * q^56 - 8 * q^58 + 22 * q^60 + 13 * q^61 - 23 * q^62 - q^63 - 15 * q^64 + 6 * q^65 + 18 * q^66 + 9 * q^67 + 14 * q^69 - 24 * q^70 - 18 * q^71 - 3 * q^72 + 19 * q^73 + q^74 + 5 * q^75 + 12 * q^77 - 21 * q^78 + 11 * q^79 + 10 * q^80 + 3 * q^81 + 8 * q^82 - 4 * q^83 + 17 * q^84 - 17 * q^86 + 4 * q^87 + 6 * q^88 - 16 * q^89 - 4 * q^90 + 7 * q^91 - 2 * q^92 + 15 * q^93 + 6 * q^94 - 17 * q^96 - 2 * q^97 - 14 * q^98

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 $

$2$	−0.571993	−0.404460	−0.202230	−	0.979338i	$-0.564819\pi$
$2$	−0.571993	−0.404460	−0.202230	+	0.979338i	$0.564819\pi$
$3$	1.00000	0.577350
$3$	1.00000	0.577350
$4$	−1.67282	−0.836412
$5$	−2.67282	−1.19532	−0.597662	−	0.801749i	$-0.703903\pi$
$5$	−2.67282	−1.19532	−0.597662	+	0.801749i	$0.703903\pi$
$6$	−0.571993	−0.233515
$7$	−3.67282	−1.38820	−0.694098	−	0.719880i	$-0.744197\pi$
$7$	−3.67282	−1.38820	−0.694098	+	0.719880i	$0.744197\pi$
$8$	2.10083	0.742756
$9$	1.00000	0.333333
$10$	1.52884	0.483461
$11$	−3.81681	−1.15081	−0.575406	−	0.817868i	$-0.695156\pi$
$11$	−3.81681	−1.15081	−0.575406	+	0.817868i	$0.695156\pi$
$12$	−1.67282	−0.482903
$13$	0.143987	0.0399347	0.0199673	−	0.999801i	$-0.493644\pi$
$13$	0.143987	0.0399347	0.0199673	+	0.999801i	$0.493644\pi$
$14$	2.10083	0.561471
$15$	−2.67282	−0.690120
$16$	2.14399	0.535997
$17$	0	0		−	1.00000i	$-0.5\pi$
$17$	0	0			1.00000i	$0.5\pi$
$18$	−0.571993	−0.134820
$19$	0	0
$19$	0	0
$20$	4.47116	0.999782
$21$	−3.67282	−0.801476
$22$	2.18319	0.465458
$23$	7.52884	1.56987	0.784936	−	0.619577i	$-0.212696\pi$
$23$	7.52884	1.56987	0.784936	+	0.619577i	$0.212696\pi$
$24$	2.10083	0.428830
$25$	2.14399	0.428797
$26$	−0.0823593	−0.0161520
$27$	1.00000	0.192450
$28$	6.14399	1.16110
$29$	−5.34565	−0.992662	−0.496331	−	0.868133i	$-0.665320\pi$
$29$	−5.34565	−0.992662	−0.496331	+	0.868133i	$0.665320\pi$
$30$	1.52884	0.279126
$31$	8.81681	1.58355	0.791773	−	0.610816i	$-0.209158\pi$
$31$	8.81681	1.58355	0.791773	+	0.610816i	$0.209158\pi$
$32$	−5.42801	−0.959545
$33$	−3.81681	−0.664421
$34$	0	0
$35$	9.81681	1.65934
$36$	−1.67282	−0.278804
$37$	−1.00000	−0.164399	−0.0821995	−	0.996616i	$-0.526194\pi$
$37$	−1.00000	−0.164399	−0.0821995	+	0.996616i	$0.526194\pi$
$38$	0	0
$39$	0.143987	0.0230563
$40$	−5.61515	−0.887833
$41$	5.34565	0.834850	0.417425	−	0.908711i	$-0.362933\pi$
$41$	5.34565	0.834850	0.417425	+	0.908711i	$0.362933\pi$
$42$	2.10083	0.324165
$43$	2.81681	0.429560	0.214780	−	0.976663i	$-0.431097\pi$
$43$	2.81681	0.429560	0.214780	+	0.976663i	$0.431097\pi$
$44$	6.38485	0.962552
$45$	−2.67282	−0.398441
$46$	−4.30644	−0.634951
$47$	−6.00000	−0.875190	−0.437595	−	0.899172i	$-0.644170\pi$
$47$	−6.00000	−0.875190	−0.437595	+	0.899172i	$0.644170\pi$
$48$	2.14399	0.309458
$49$	6.48963	0.927091
$50$	−1.22635	−0.173431
$51$	0	0
$52$	−0.240864	−0.0334018
$53$	8.01847	1.10142	0.550711	−	0.834696i	$-0.314357\pi$
$53$	8.01847	1.10142	0.550711	+	0.834696i	$0.314357\pi$
$54$	−0.571993	−0.0778384
$55$	10.2017	1.37559
$56$	−7.71598	−1.03109
$57$	0	0
$58$	3.05767	0.401492
$59$	3.81681	0.496906	0.248453	−	0.968644i	$-0.420078\pi$
$59$	3.81681	0.496906	0.248453	+	0.968644i	$0.420078\pi$
$60$	4.47116	0.577225
$61$	11.4896	1.47110	0.735548	−	0.677472i	$-0.236925\pi$
$61$	11.4896	1.47110	0.735548	+	0.677472i	$0.236925\pi$
$62$	−5.04316	−0.640481
$63$	−3.67282	−0.462732
$64$	−1.18319	−0.147899
$65$	−0.384851	−0.0477348
$66$	2.18319	0.268732
$67$	5.38485	0.657864	0.328932	−	0.944354i	$-0.393311\pi$
$67$	5.38485	0.657864	0.328932	+	0.944354i	$0.393311\pi$
$68$	0	0
$69$	7.52884	0.906365
$70$	−5.61515	−0.671139
$71$	−13.6336	−1.61801	−0.809007	−	0.587800i	$-0.799994\pi$
$71$	−13.6336	−1.61801	−0.809007	+	0.587800i	$0.799994\pi$
$72$	2.10083	0.247585
$73$	−0.345647	−0.0404550	−0.0202275	−	0.999795i	$-0.506439\pi$
$73$	−0.345647	−0.0404550	−0.0202275	+	0.999795i	$0.506439\pi$
$74$	0.571993	0.0664929
$75$	2.14399	0.247566
$76$	0	0
$77$	14.0185	1.59755
$78$	−0.0823593	−0.00932536
$79$	6.52884	0.734552	0.367276	−	0.930112i	$-0.380291\pi$
$79$	6.52884	0.734552	0.367276	+	0.930112i	$0.380291\pi$
$80$	−5.73050	−0.640689
$81$	1.00000	0.111111
$82$	−3.05767	−0.337664
$83$	−2.28797	−0.251138	−0.125569	−	0.992085i	$-0.540076\pi$
$83$	−2.28797	−0.251138	−0.125569	+	0.992085i	$0.540076\pi$
$84$	6.14399	0.670364
$85$	0	0
$86$	−1.61120	−0.173740
$87$	−5.34565	−0.573114
$88$	−8.01847	−0.854772
$89$	−8.67282	−0.919317	−0.459659	−	0.888096i	$-0.652028\pi$
$89$	−8.67282	−0.919317	−0.459659	+	0.888096i	$0.652028\pi$
$90$	1.52884	0.161154
$91$	−0.528837	−0.0554372
$92$	−12.5944	−1.31306
$93$	8.81681	0.914261
$94$	3.43196	0.353980
$95$	0	0
$96$	−5.42801	−0.553994
$97$	−5.91369	−0.600444	−0.300222	−	0.953869i	$-0.597061\pi$
$97$	−5.91369	−0.600444	−0.300222	+	0.953869i	$0.597061\pi$
$98$	−3.71203	−0.374971
$99$	−3.81681	−0.383604
$100$	−3.58651	−0.358651
$101$	8.28797	0.824684	0.412342	−	0.911029i	$-0.364711\pi$
$101$	8.28797	0.824684	0.412342	+	0.911029i	$0.364711\pi$
$102$	0	0
$103$	14.6521	1.44371	0.721857	−	0.692043i	$-0.243289\pi$
$103$	14.6521	1.44371	0.721857	+	0.692043i	$0.243289\pi$
$104$	0.302491	0.0296617
$105$	9.81681	0.958023
$106$	−4.58651	−0.445481
$107$	−16.6913	−1.61361	−0.806804	−	0.590819i	$-0.798805\pi$
$107$	−16.6913	−1.61361	−0.806804	+	0.590819i	$0.798805\pi$
$108$	−1.67282	−0.160968
$109$	7.83528	0.750484	0.375242	−	0.926927i	$-0.377560\pi$
$109$	7.83528	0.750484	0.375242	+	0.926927i	$0.377560\pi$
$110$	−5.83528	−0.556372
$111$	−1.00000	−0.0949158
$112$	−7.87448	−0.744069
$113$	−6.38485	−0.600636	−0.300318	−	0.953839i	$-0.597093\pi$
$113$	−6.38485	−0.600636	−0.300318	+	0.953839i	$0.597093\pi$
$114$	0	0
$115$	−20.1233	−1.87650
$116$	8.94233	0.830274
$117$	0.143987	0.0133116
$118$	−2.18319	−0.200979
$119$	0	0
$120$	−5.61515	−0.512591
$121$	3.56804	0.324367
$122$	−6.57199	−0.595000
$123$	5.34565	0.482001
$124$	−14.7490	−1.32450
$125$	7.63362	0.682772
$126$	2.10083	0.187157
$127$	2.65435	0.235536	0.117768	−	0.993041i	$-0.462426\pi$
$127$	2.65435	0.235536	0.117768	+	0.993041i	$0.462426\pi$
$128$	11.5328	1.01936
$129$	2.81681	0.248006
$130$	0.220132	0.0193069
$131$	−11.3456	−0.991274	−0.495637	−	0.868530i	$-0.665065\pi$
$131$	−11.3456	−0.991274	−0.495637	+	0.868530i	$0.665065\pi$
$132$	6.38485	0.555730
$133$	0	0
$134$	−3.08010	−0.266080
$135$	−2.67282	−0.230040
$136$	0	0
$137$	−1.63362	−0.139570	−0.0697848	−	0.997562i	$-0.522231\pi$
$137$	−1.63362	−0.139570	−0.0697848	+	0.997562i	$0.522231\pi$
$138$	−4.30644	−0.366589
$139$	7.50811	0.636829	0.318415	−	0.947952i	$-0.396850\pi$
$139$	7.50811	0.636829	0.318415	+	0.947952i	$0.396850\pi$
$140$	−16.4218	−1.38789
$141$	−6.00000	−0.505291
$142$	7.79834	0.654422
$143$	−0.549569	−0.0459573
$144$	2.14399	0.178666
$145$	14.2880	1.18655
$146$	0.197708	0.0163624
$147$	6.48963	0.535256
$148$	1.67282	0.137505
$149$	14.0185	1.14844	0.574219	−	0.818702i	$-0.305306\pi$
$149$	14.0185	1.14844	0.574219	+	0.818702i	$0.305306\pi$
$150$	−1.22635	−0.100131
$151$	11.0577	0.899861	0.449930	−	0.893064i	$-0.351449\pi$
$151$	11.0577	0.899861	0.449930	+	0.893064i	$0.351449\pi$
$152$	0	0
$153$	0	0
$154$	−8.01847	−0.646147
$155$	−23.5658	−1.89285
$156$	−0.240864	−0.0192846
$157$	14.0577	1.12192	0.560962	−	0.827841i	$-0.310431\pi$
$157$	14.0577	1.12192	0.560962	+	0.827841i	$0.310431\pi$
$158$	−3.73445	−0.297097
$159$	8.01847	0.635906
$160$	14.5081	1.14697
$161$	−27.6521	−2.17929
$162$	−0.571993	−0.0449400
$163$	4.61515	0.361486	0.180743	−	0.983530i	$-0.442150\pi$
$163$	4.61515	0.361486	0.180743	+	0.983530i	$0.442150\pi$
$164$	−8.94233	−0.698278
$165$	10.2017	0.794198
$166$	1.30871	0.101575
$167$	−12.2201	−0.945622	−0.472811	−	0.881164i	$-0.656761\pi$
$167$	−12.2201	−0.945622	−0.472811	+	0.881164i	$0.656761\pi$
$168$	−7.71598	−0.595301
$169$	−12.9793	−0.998405
$170$	0	0
$171$	0	0
$172$	−4.71203	−0.359289
$173$	15.0577	1.14481	0.572407	−	0.819970i	$-0.306010\pi$
$173$	15.0577	1.14481	0.572407	+	0.819970i	$0.306010\pi$
$174$	3.05767	0.231802
$175$	−7.87448	−0.595255
$176$	−8.18319	−0.616831
$177$	3.81681	0.286889
$178$	4.96080	0.371827
$179$	−15.1625	−1.13330	−0.566648	−	0.823960i	$-0.691760\pi$
$179$	−15.1625	−1.13330	−0.566648	+	0.823960i	$0.691760\pi$
$180$	4.47116	0.333261
$181$	12.2017	0.906942	0.453471	−	0.891271i	$-0.350186\pi$
$181$	12.2017	0.906942	0.453471	+	0.891271i	$0.350186\pi$
$182$	0.302491	0.0224221
$183$	11.4896	0.849338
$184$	15.8168	1.16603
$185$	2.67282	0.196510
$186$	−5.04316	−0.369782
$187$	0	0
$188$	10.0369	0.732019
$189$	−3.67282	−0.267159
$190$	0	0
$191$	5.45043	0.394379	0.197190	−	0.980365i	$-0.436819\pi$
$191$	5.45043	0.394379	0.197190	+	0.980365i	$0.436819\pi$
$192$	−1.18319	−0.0853894
$193$	−0.510366	−0.0367370	−0.0183685	−	0.999831i	$-0.505847\pi$
$193$	−0.510366	−0.0367370	−0.0183685	+	0.999831i	$0.505847\pi$
$194$	3.38259	0.242856
$195$	−0.384851	−0.0275597
$196$	−10.8560	−0.775430
$197$	22.9608	1.63589	0.817945	−	0.575297i	$-0.195114\pi$
$197$	22.9608	1.63589	0.817945	+	0.575297i	$0.195114\pi$
$198$	2.18319	0.155153
$199$	−0.125515	−0.00889755	−0.00444878	−	0.999990i	$-0.501416\pi$
$199$	−0.125515	−0.00889755	−0.00444878	+	0.999990i	$0.501416\pi$
$200$	4.50415	0.318492
$201$	5.38485	0.379818
$202$	−4.74066	−0.333552
$203$	19.6336	1.37801
$204$	0	0
$205$	−14.2880	−0.997915
$206$	−8.38090	−0.583925
$207$	7.52884	0.523290
$208$	0.308705	0.0214049
$209$	0	0
$210$	−5.61515	−0.387482
$211$	24.8538	1.71100	0.855501	−	0.517800i	$-0.173249\pi$
$211$	24.8538	1.71100	0.855501	+	0.517800i	$0.173249\pi$
$212$	−13.4135	−0.921242
$213$	−13.6336	−0.934160
$214$	9.54731	0.652641
$215$	−7.52884	−0.513462
$216$	2.10083	0.142943
$217$	−32.3826	−2.19827
$218$	−4.48173	−0.303541
$219$	−0.345647	−0.0233567
$220$	−17.0656	−1.15056
$221$	0	0
$222$	0.571993	0.0383897
$223$	−23.7961	−1.59350	−0.796752	−	0.604307i	$-0.793450\pi$
$223$	−23.7961	−1.59350	−0.796752	+	0.604307i	$0.793450\pi$
$224$	19.9361	1.33204
$225$	2.14399	0.142932
$226$	3.65209	0.242934
$227$	9.81681	0.651565	0.325782	−	0.945445i	$-0.394372\pi$
$227$	9.81681	0.651565	0.325782	+	0.945445i	$0.394372\pi$
$228$	0	0
$229$	0.143987	0.00951490	0.00475745	−	0.999989i	$-0.498486\pi$
$229$	0.143987	0.00951490	0.00475745	+	0.999989i	$0.498486\pi$
$230$	11.5104	0.758971
$231$	14.0185	0.922348
$232$	−11.2303	−0.737305
$233$	10.5759	0.692853	0.346427	−	0.938077i	$-0.387395\pi$
$233$	10.5759	0.692853	0.346427	+	0.938077i	$0.387395\pi$
$234$	−0.0823593	−0.00538400
$235$	16.0369	1.04613
$236$	−6.38485	−0.415618
$237$	6.52884	0.424094
$238$	0	0
$239$	−9.81681	−0.634997	−0.317498	−	0.948259i	$-0.602843\pi$
$239$	−9.81681	−0.634997	−0.317498	+	0.948259i	$0.602843\pi$
$240$	−5.73050	−0.369902
$241$	1.94233	0.125116	0.0625581	−	0.998041i	$-0.480074\pi$
$241$	1.94233	0.125116	0.0625581	+	0.998041i	$0.480074\pi$
$242$	−2.04090	−0.131194
$243$	1.00000	0.0641500
$244$	−19.2201	−1.23044
$245$	−17.3456	−1.10817
$246$	−3.05767	−0.194950
$247$	0	0
$248$	18.5226	1.17619
$249$	−2.28797	−0.144994
$250$	−4.36638	−0.276154
$251$	13.6336	0.860546	0.430273	−	0.902699i	$-0.358417\pi$
$251$	13.6336	0.860546	0.430273	+	0.902699i	$0.358417\pi$
$252$	6.14399	0.387035
$253$	−28.7361	−1.80663
$254$	−1.51827	−0.0952648
$255$	0	0
$256$	−4.23030	−0.264394
$257$	−12.5944	−0.785618	−0.392809	−	0.919620i	$-0.628497\pi$
$257$	−12.5944	−0.785618	−0.392809	+	0.919620i	$0.628497\pi$
$258$	−1.61120	−0.100309
$259$	3.67282	0.228218
$260$	0.643787	0.0399260
$261$	−5.34565	−0.330887
$262$	6.48963	0.400931
$263$	−1.63362	−0.100733	−0.0503667	−	0.998731i	$-0.516039\pi$
$263$	−1.63362	−0.100733	−0.0503667	+	0.998731i	$0.516039\pi$
$264$	−8.01847	−0.493503
$265$	−21.4320	−1.31655
$266$	0	0
$267$	−8.67282	−0.530768
$268$	−9.00791	−0.550245
$269$	−21.6521	−1.32015	−0.660076	−	0.751199i	$-0.729476\pi$
$269$	−21.6521	−1.32015	−0.660076	+	0.751199i	$0.729476\pi$
$270$	1.52884	0.0930421
$271$	−21.5552	−1.30939	−0.654693	−	0.755895i	$-0.727202\pi$
$271$	−21.5552	−1.30939	−0.654693	+	0.755895i	$0.727202\pi$
$272$	0	0
$273$	−0.528837	−0.0320067
$274$	0.934420	0.0564504
$275$	−8.18319	−0.493465
$276$	−12.5944	−0.758095
$277$	7.62571	0.458185	0.229092	−	0.973405i	$-0.426424\pi$
$277$	7.62571	0.458185	0.229092	+	0.973405i	$0.426424\pi$
$278$	−4.29459	−0.257572
$279$	8.81681	0.527849
$280$	20.6235	1.23249
$281$	−1.69356	−0.101029	−0.0505145	−	0.998723i	$-0.516086\pi$
$281$	−1.69356	−0.101029	−0.0505145	+	0.998723i	$0.516086\pi$
$282$	3.43196	0.204370
$283$	10.2880	0.611557	0.305778	−	0.952103i	$-0.401083\pi$
$283$	10.2880	0.611557	0.305778	+	0.952103i	$0.401083\pi$
$284$	22.8066	1.35333
$285$	0	0
$286$	0.314350	0.0185879
$287$	−19.6336	−1.15894
$288$	−5.42801	−0.319848
$289$	−17.0000	−1.00000
$290$	−8.17262	−0.479913
$291$	−5.91369	−0.346667
$292$	0.578207	0.0338370
$293$	12.1153	0.707786	0.353893	−	0.935286i	$-0.384858\pi$
$293$	12.1153	0.707786	0.353893	+	0.935286i	$0.384858\pi$
$294$	−3.71203	−0.216490
$295$	−10.2017	−0.593964
$296$	−2.10083	−0.122108
$297$	−3.81681	−0.221474
$298$	−8.01847	−0.464498
$299$	1.08405	0.0626923
$300$	−3.58651	−0.207067
$301$	−10.3456	−0.596313
$302$	−6.32492	−0.363958
$303$	8.28797	0.476132
$304$	0	0
$305$	−30.7098	−1.75844
$306$	0	0
$307$	−4.48963	−0.256237	−0.128118	−	0.991759i	$-0.540894\pi$
$307$	−4.48963	−0.256237	−0.128118	+	0.991759i	$0.540894\pi$
$308$	−23.4504	−1.33621
$309$	14.6521	0.833528
$310$	13.4795	0.765582
$311$	0.759136	0.0430466	0.0215233	−	0.999768i	$-0.493148\pi$
$311$	0.759136	0.0430466	0.0215233	+	0.999768i	$0.493148\pi$
$312$	0.302491	0.0171252
$313$	11.4320	0.646173	0.323086	−	0.946370i	$-0.395280\pi$
$313$	11.4320	0.646173	0.323086	+	0.946370i	$0.395280\pi$
$314$	−8.04090	−0.453774
$315$	9.81681	0.553115
$316$	−10.9216	−0.614388
$317$	−11.6151	−0.652372	−0.326186	−	0.945306i	$-0.605764\pi$
$317$	−11.6151	−0.652372	−0.326186	+	0.945306i	$0.605764\pi$
$318$	−4.58651	−0.257199
$319$	20.4033	1.14237
$320$	3.16246	0.176787
$321$	−16.6913	−0.931617
$322$	15.8168	0.881436
$323$	0	0
$324$	−1.67282	−0.0929347
$325$	0.308705	0.0171239
$326$	−2.63983	−0.146207
$327$	7.83528	0.433292
$328$	11.2303	0.620090
$329$	22.0369	1.21494
$330$	−5.83528	−0.321222
$331$	−11.4712	−0.630512	−0.315256	−	0.949007i	$-0.602090\pi$
$331$	−11.4712	−0.630512	−0.315256	+	0.949007i	$0.602090\pi$
$332$	3.82738	0.210055
$333$	−1.00000	−0.0547997
$334$	6.98983	0.382467
$335$	−14.3928	−0.786360
$336$	−7.87448	−0.429588
$337$	7.56804	0.412257	0.206129	−	0.978525i	$-0.433913\pi$
$337$	7.56804	0.412257	0.206129	+	0.978525i	$0.433913\pi$
$338$	7.42405	0.403815
$339$	−6.38485	−0.346777
$340$	0	0
$341$	−33.6521	−1.82236
$342$	0	0
$343$	1.87448	0.101213
$344$	5.91764	0.319058
$345$	−20.1233	−1.08340
$346$	−8.61289	−0.463032
$347$	21.8168	1.17119	0.585594	−	0.810605i	$-0.300861\pi$
$347$	21.8168	1.17119	0.585594	+	0.810605i	$0.300861\pi$
$348$	8.94233	0.479359
$349$	−4.54731	−0.243412	−0.121706	−	0.992566i	$-0.538836\pi$
$349$	−4.54731	−0.243412	−0.121706	+	0.992566i	$0.538836\pi$
$350$	4.50415	0.240757
$351$	0.143987	0.00768543
$352$	20.7177	1.10426
$353$	−8.99774	−0.478901	−0.239451	−	0.970909i	$-0.576967\pi$
$353$	−8.99774	−0.478901	−0.239451	+	0.970909i	$0.576967\pi$
$354$	−2.18319	−0.116035
$355$	36.4403	1.93405
$356$	14.5081	0.768928
$357$	0	0
$358$	8.67282	0.458373
$359$	12.9793	0.685020	0.342510	−	0.939514i	$-0.388723\pi$
$359$	12.9793	0.685020	0.342510	+	0.939514i	$0.388723\pi$
$360$	−5.61515	−0.295944
$361$	0	0
$362$	−6.97927	−0.366822
$363$	3.56804	0.187274
$364$	0.884651	0.0463683
$365$	0.923855	0.0483568
$366$	−6.57199	−0.343524
$367$	−4.04711	−0.211257	−0.105629	−	0.994406i	$-0.533685\pi$
$367$	−4.04711	−0.211257	−0.105629	+	0.994406i	$0.533685\pi$
$368$	16.1417	0.841446
$369$	5.34565	0.278283
$370$	−1.52884	−0.0794805
$371$	−29.4504	−1.52899
$372$	−14.7490	−0.764698
$373$	3.79834	0.196671	0.0983353	−	0.995153i	$-0.468648\pi$
$373$	3.79834	0.196671	0.0983353	+	0.995153i	$0.468648\pi$
$374$	0	0
$375$	7.63362	0.394198
$376$	−12.6050	−0.650052
$377$	−0.769701	−0.0396416
$378$	2.10083	0.108055
$379$	4.89522	0.251450	0.125725	−	0.992065i	$-0.459874\pi$
$379$	4.89522	0.251450	0.125725	+	0.992065i	$0.459874\pi$
$380$	0	0
$381$	2.65435	0.135987
$382$	−3.11761	−0.159511
$383$	34.1417	1.74456	0.872280	−	0.489006i	$-0.162640\pi$
$383$	34.1417	1.74456	0.872280	+	0.489006i	$0.162640\pi$
$384$	11.5328	0.588530
$385$	−37.4689	−1.90959
$386$	0.291926	0.0148586
$387$	2.81681	0.143187
$388$	9.89256	0.502218
$389$	26.8824	1.36299	0.681496	−	0.731822i	$-0.261330\pi$
$389$	26.8824	1.36299	0.681496	+	0.731822i	$0.261330\pi$
$390$	0.220132	0.0111468
$391$	0	0
$392$	13.6336	0.688602
$393$	−11.3456	−0.572312
$394$	−13.1334	−0.661652
$395$	−17.4504	−0.878026
$396$	6.38485	0.320851
$397$	25.2386	1.26669	0.633345	−	0.773870i	$-0.281682\pi$
$397$	25.2386	1.26669	0.633345	+	0.773870i	$0.281682\pi$
$398$	0.0717940	0.00359871
$399$	0	0
$400$	4.59668	0.229834
$401$	−3.32718	−0.166151	−0.0830756	−	0.996543i	$-0.526474\pi$
$401$	−3.32718	−0.166151	−0.0830756	+	0.996543i	$0.526474\pi$
$402$	−3.08010	−0.153621
$403$	1.26950	0.0632384
$404$	−13.8643	−0.689776
$405$	−2.67282	−0.132814
$406$	−11.2303	−0.557350
$407$	3.81681	0.189192
$408$	0	0
$409$	34.1233	1.68729	0.843643	−	0.536904i	$-0.180406\pi$
$409$	34.1233	1.68729	0.843643	+	0.536904i	$0.180406\pi$
$410$	8.17262	0.403617
$411$	−1.63362	−0.0805806
$412$	−24.5104	−1.20754
$413$	−14.0185	−0.689804
$414$	−4.30644	−0.211650
$415$	6.11535	0.300191
$416$	−0.781560	−0.0383191
$417$	7.50811	0.367673
$418$	0	0
$419$	−9.16246	−0.447615	−0.223808	−	0.974633i	$-0.571849\pi$
$419$	−9.16246	−0.447615	−0.223808	+	0.974633i	$0.571849\pi$
$420$	−16.4218	−0.801301
$421$	−15.8353	−0.771764	−0.385882	−	0.922548i	$-0.626103\pi$
$421$	−15.8353	−0.771764	−0.385882	+	0.922548i	$0.626103\pi$
$422$	−14.2162	−0.692033
$423$	−6.00000	−0.291730
$424$	16.8454	0.818087
$425$	0	0
$426$	7.79834	0.377831
$427$	−42.1994	−2.04217
$428$	27.9216	1.34964
$429$	−0.549569	−0.0265335
$430$	4.30644	0.207675
$431$	−9.26724	−0.446387	−0.223194	−	0.974774i	$-0.571648\pi$
$431$	−9.26724	−0.446387	−0.223194	+	0.974774i	$0.571648\pi$
$432$	2.14399	0.103153
$433$	−19.8145	−0.952226	−0.476113	−	0.879384i	$-0.657955\pi$
$433$	−19.8145	−0.952226	−0.476113	+	0.879384i	$0.657955\pi$
$434$	18.5226	0.889114
$435$	14.2880	0.685056
$436$	−13.1070	−0.627714
$437$	0	0
$438$	0.197708	0.00944685
$439$	6.90312	0.329468	0.164734	−	0.986338i	$-0.447323\pi$
$439$	6.90312	0.329468	0.164734	+	0.986338i	$0.447323\pi$
$440$	21.4320	1.02173
$441$	6.48963	0.309030
$442$	0	0
$443$	−19.6336	−0.932821	−0.466411	−	0.884568i	$-0.654453\pi$
$443$	−19.6336	−0.932821	−0.466411	+	0.884568i	$0.654453\pi$
$444$	1.67282	0.0793887
$445$	23.1809	1.09888
$446$	13.6112	0.644509
$447$	14.0185	0.663051
$448$	4.34565	0.205313
$449$	29.0162	1.36936	0.684680	−	0.728844i	$-0.259942\pi$
$449$	29.0162	1.36936	0.684680	+	0.728844i	$0.259942\pi$
$450$	−1.22635	−0.0578105
$451$	−20.4033	−0.960755
$452$	10.6807	0.502379
$453$	11.0577	0.519535
$454$	−5.61515	−0.263532
$455$	1.41349	0.0662654
$456$	0	0
$457$	32.5473	1.52250	0.761249	−	0.648459i	$-0.224586\pi$
$457$	32.5473	1.52250	0.761249	+	0.648459i	$0.224586\pi$
$458$	−0.0823593	−0.00384840
$459$	0	0
$460$	33.6627	1.56953
$461$	−9.65209	−0.449543	−0.224771	−	0.974412i	$-0.572164\pi$
$461$	−9.65209	−0.449543	−0.224771	+	0.974412i	$0.572164\pi$
$462$	−8.01847	−0.373053
$463$	−31.0554	−1.44327	−0.721634	−	0.692275i	$-0.756608\pi$
$463$	−31.0554	−1.44327	−0.721634	+	0.692275i	$0.756608\pi$
$464$	−11.4610	−0.532063
$465$	−23.5658	−1.09284
$466$	−6.04937	−0.280232
$467$	8.61289	0.398557	0.199278	−	0.979943i	$-0.436140\pi$
$467$	8.61289	0.398557	0.199278	+	0.979943i	$0.436140\pi$
$468$	−0.240864	−0.0111339
$469$	−19.7776	−0.913245
$470$	−9.17302	−0.423120
$471$	14.0577	0.647743
$472$	8.01847	0.369080
$473$	−10.7512	−0.494342
$474$	−3.73445	−0.171529
$475$	0	0
$476$	0	0
$477$	8.01847	0.367141
$478$	5.61515	0.256831
$479$	17.4610	0.797813	0.398907	−	0.916992i	$-0.369390\pi$
$479$	17.4610	0.797813	0.398907	+	0.916992i	$0.369390\pi$
$480$	14.5081	0.662201
$481$	−0.143987	−0.00656522
$482$	−1.11100	−0.0506045
$483$	−27.6521	−1.25821
$484$	−5.96870	−0.271305
$485$	15.8062	0.717725
$486$	−0.571993	−0.0259461
$487$	−11.6336	−0.527170	−0.263585	−	0.964636i	$-0.584905\pi$
$487$	−11.6336	−0.527170	−0.263585	+	0.964636i	$0.584905\pi$
$488$	24.1378	1.09267
$489$	4.61515	0.208704
$490$	9.92159	0.448212
$491$	−24.3249	−1.09777	−0.548884	−	0.835899i	$-0.684947\pi$
$491$	−24.3249	−1.09777	−0.548884	+	0.835899i	$0.684947\pi$
$492$	−8.94233	−0.403151
$493$	0	0
$494$	0	0
$495$	10.2017	0.458531
$496$	18.9031	0.848775
$497$	50.0739	2.24612
$498$	1.30871	0.0586445
$499$	38.1131	1.70618	0.853088	−	0.521767i	$-0.174727\pi$
$499$	38.1131	1.70618	0.853088	+	0.521767i	$0.174727\pi$
$500$	−12.7697	−0.571078
$501$	−12.2201	−0.545955
$502$	−7.79834	−0.348057
$503$	4.36638	0.194687	0.0973436	−	0.995251i	$-0.468965\pi$
$503$	4.36638	0.194687	0.0973436	+	0.995251i	$0.468965\pi$
$504$	−7.71598	−0.343697
$505$	−22.1523	−0.985764
$506$	16.4369	0.730708
$507$	−12.9793	−0.576430
$508$	−4.44026	−0.197005
$509$	−15.7120	−0.696423	−0.348212	−	0.937416i	$-0.613211\pi$
$509$	−15.7120	−0.696423	−0.348212	+	0.937416i	$0.613211\pi$
$510$	0	0
$511$	1.26950	0.0561595
$512$	−20.6459	−0.912428
$513$	0	0
$514$	7.20392	0.317751
$515$	−39.1625	−1.72570
$516$	−4.71203	−0.207435
$517$	22.9009	1.00718
$518$	−2.10083	−0.0923052
$519$	15.0577	0.660959
$520$	−0.808506	−0.0354553
$521$	33.4425	1.46514	0.732572	−	0.680690i	$-0.238320\pi$
$521$	33.4425	1.46514	0.732572	+	0.680690i	$0.238320\pi$
$522$	3.05767	0.133831
$523$	−29.4218	−1.28653	−0.643263	−	0.765646i	$-0.722420\pi$
$523$	−29.4218	−1.28653	−0.643263	+	0.765646i	$0.722420\pi$
$524$	18.9793	0.829113
$525$	−7.87448	−0.343671
$526$	0.934420	0.0407426
$527$	0	0
$528$	−8.18319	−0.356128
$529$	33.6834	1.46450
$530$	12.2589	0.532494
$531$	3.81681	0.165635
$532$	0	0
$533$	0.769701	0.0333395
$534$	4.96080	0.214675
$535$	44.6129	1.92878
$536$	11.3127	0.488632
$537$	−15.1625	−0.654308
$538$	12.3849	0.533949
$539$	−24.7697	−1.06691
$540$	4.47116	0.192408
$541$	26.1730	1.12527	0.562633	−	0.826707i	$-0.309788\pi$
$541$	26.1730	1.12527	0.562633	+	0.826707i	$0.309788\pi$
$542$	12.3294	0.529595
$543$	12.2017	0.523623
$544$	0	0
$545$	−20.9423	−0.897071
$546$	0.302491	0.0129454
$547$	21.9114	0.936865	0.468432	−	0.883499i	$-0.344819\pi$
$547$	21.9114	0.936865	0.468432	+	0.883499i	$0.344819\pi$
$548$	2.73276	0.116738
$549$	11.4896	0.490366
$550$	4.68073	0.199587
$551$	0	0
$552$	15.8168	0.673208
$553$	−23.9793	−1.01970
$554$	−4.36186	−0.185318
$555$	2.67282	0.113455
$556$	−12.5597	−0.532651
$557$	34.0369	1.44219	0.721096	−	0.692835i	$-0.243639\pi$
$557$	34.0369	1.44219	0.721096	+	0.692835i	$0.243639\pi$
$558$	−5.04316	−0.213494
$559$	0.405583	0.0171543
$560$	21.0471	0.889403
$561$	0	0
$562$	0.968703	0.0408622
$563$	11.5552	0.486994	0.243497	−	0.969902i	$-0.421705\pi$
$563$	11.5552	0.486994	0.243497	+	0.969902i	$0.421705\pi$
$564$	10.0369	0.422632
$565$	17.0656	0.717954
$566$	−5.88465	−0.247350
$567$	−3.67282	−0.154244
$568$	−28.6419	−1.20179
$569$	39.7075	1.66463	0.832313	−	0.554307i	$-0.187016\pi$
$569$	39.7075	1.66463	0.832313	+	0.554307i	$0.187016\pi$
$570$	0	0
$571$	14.6521	0.613171	0.306585	−	0.951843i	$-0.400813\pi$
$571$	14.6521	0.613171	0.306585	+	0.951843i	$0.400813\pi$
$572$	0.919333	0.0384392
$573$	5.45043	0.227695
$574$	11.2303	0.468744
$575$	16.1417	0.673156
$576$	−1.18319	−0.0492996
$577$	20.8145	0.866521	0.433261	−	0.901269i	$-0.357363\pi$
$577$	20.8145	0.866521	0.433261	+	0.901269i	$0.357363\pi$
$578$	9.72389	0.404460
$579$	−0.510366	−0.0212101
$580$	−23.9013	−0.992446
$581$	8.40332	0.348629
$582$	3.38259	0.140213
$583$	−30.6050	−1.26753
$584$	−0.726147	−0.0300482
$585$	−0.384851	−0.0159116
$586$	−6.92990	−0.286271
$587$	19.6442	0.810802	0.405401	−	0.914139i	$-0.367132\pi$
$587$	19.6442	0.810802	0.405401	+	0.914139i	$0.367132\pi$
$588$	−10.8560	−0.447694
$589$	0	0
$590$	5.83528	0.240235
$591$	22.9608	0.944481
$592$	−2.14399	−0.0881173
$593$	21.6521	0.889145	0.444572	−	0.895743i	$-0.353356\pi$
$593$	21.6521	0.889145	0.444572	+	0.895743i	$0.353356\pi$
$594$	2.18319	0.0895774
$595$	0	0
$596$	−23.4504	−0.960567
$597$	−0.125515	−0.00513700
$598$	−0.620070	−0.0253565
$599$	−11.7799	−0.481312	−0.240656	−	0.970610i	$-0.577363\pi$
$599$	−11.7799	−0.481312	−0.240656	+	0.970610i	$0.577363\pi$
$600$	4.50415	0.183881
$601$	−11.4112	−0.465474	−0.232737	−	0.972540i	$-0.574768\pi$
$601$	−11.4112	−0.465474	−0.232737	+	0.972540i	$0.574768\pi$
$602$	5.91764	0.241185
$603$	5.38485	0.219288
$604$	−18.4975	−0.752654
$605$	−9.53674	−0.387724
$606$	−4.74066	−0.192576
$607$	−1.10478	−0.0448418	−0.0224209	−	0.999749i	$-0.507137\pi$
$607$	−1.10478	−0.0448418	−0.0224209	+	0.999749i	$0.507137\pi$
$608$	0	0
$609$	19.6336	0.795594
$610$	17.5658	0.711218
$611$	−0.863919	−0.0349504
$612$	0	0
$613$	4.77761	0.192966	0.0964829	−	0.995335i	$-0.469241\pi$
$613$	4.77761	0.192966	0.0964829	+	0.995335i	$0.469241\pi$
$614$	2.56804	0.103638
$615$	−14.2880	−0.576147
$616$	29.4504	1.18659
$617$	37.4795	1.50887	0.754433	−	0.656377i	$-0.227912\pi$
$617$	37.4795	1.50887	0.754433	+	0.656377i	$0.227912\pi$
$618$	−8.38090	−0.337129
$619$	−0.615149	−0.0247249	−0.0123625	−	0.999924i	$-0.503935\pi$
$619$	−0.615149	−0.0247249	−0.0123625	+	0.999924i	$0.503935\pi$
$620$	39.4214	1.58320
$621$	7.52884	0.302122
$622$	−0.434221	−0.0174107
$623$	31.8538	1.27619
$624$	0.308705	0.0123581
$625$	−31.1233	−1.24493
$626$	−6.53900	−0.261351
$627$	0	0
$628$	−23.5160	−0.938391
$629$	0	0
$630$	−5.61515	−0.223713
$631$	21.3064	0.848196	0.424098	−	0.905616i	$-0.360591\pi$
$631$	21.3064	0.848196	0.424098	+	0.905616i	$0.360591\pi$
$632$	13.7160	0.545592
$633$	24.8538	0.987848
$634$	6.64379	0.263858
$635$	−7.09462	−0.281541
$636$	−13.4135	−0.531879
$637$	0.934420	0.0370231
$638$	−11.6706	−0.462042
$639$	−13.6336	−0.539338
$640$	−30.8251	−1.21847
$641$	39.9216	1.57681	0.788404	−	0.615158i	$-0.210908\pi$
$641$	39.9216	1.57681	0.788404	+	0.615158i	$0.210908\pi$
$642$	9.54731	0.376802
$643$	−37.3849	−1.47431	−0.737157	−	0.675721i	$-0.763832\pi$
$643$	−37.3849	−1.47431	−0.737157	+	0.675721i	$0.763832\pi$
$644$	46.2571	1.82278
$645$	−7.52884	−0.296448
$646$	0	0
$647$	16.0369	0.630477	0.315239	−	0.949012i	$-0.397915\pi$
$647$	16.0369	0.630477	0.315239	+	0.949012i	$0.397915\pi$
$648$	2.10083	0.0825284
$649$	−14.5680	−0.571846
$650$	−0.176577	−0.00692593
$651$	−32.3826	−1.26917
$652$	−7.72033	−0.302352
$653$	13.4241	0.525324	0.262662	−	0.964888i	$-0.415400\pi$
$653$	13.4241	0.525324	0.262662	+	0.964888i	$0.415400\pi$
$654$	−4.48173	−0.175249
$655$	30.3249	1.18489
$656$	11.4610	0.447477
$657$	−0.345647	−0.0134850
$658$	−12.6050	−0.491393
$659$	−18.5450	−0.722412	−0.361206	−	0.932486i	$-0.617635\pi$
$659$	−18.5450	−0.722412	−0.361206	+	0.932486i	$0.617635\pi$
$660$	−17.0656	−0.664277
$661$	6.57595	0.255775	0.127887	−	0.991789i	$-0.459180\pi$
$661$	6.57595	0.255775	0.127887	+	0.991789i	$0.459180\pi$
$662$	6.56143	0.255017
$663$	0	0
$664$	−4.80664	−0.186534
$665$	0	0
$666$	0.571993	0.0221643
$667$	−40.2465	−1.55835
$668$	20.4421	0.790930
$669$	−23.7961	−0.920010
$670$	8.23256	0.318052
$671$	−43.8538	−1.69296
$672$	19.9361	0.769052
$673$	−39.2386	−1.51254	−0.756268	−	0.654261i	$-0.772980\pi$
$673$	−39.2386	−1.51254	−0.756268	+	0.654261i	$0.772980\pi$
$674$	−4.32887	−0.166742
$675$	2.14399	0.0825221
$676$	21.7120	0.835078
$677$	−25.8432	−0.993234	−0.496617	−	0.867970i	$-0.665425\pi$
$677$	−25.8432	−0.993234	−0.496617	+	0.867970i	$0.665425\pi$
$678$	3.65209	0.140258
$679$	21.7199	0.833535
$680$	0	0
$681$	9.81681	0.376181
$682$	19.2488	0.737073
$683$	42.9898	1.64496	0.822480	−	0.568794i	$-0.192590\pi$
$683$	42.9898	1.64496	0.822480	+	0.568794i	$0.192590\pi$
$684$	0	0
$685$	4.36638	0.166831
$686$	−1.07219	−0.0409365
$687$	0.143987	0.00549343
$688$	6.03920	0.230242
$689$	1.15455	0.0439849
$690$	11.5104	0.438192
$691$	−48.4033	−1.84135	−0.920675	−	0.390331i	$-0.872361\pi$
$691$	−48.4033	−1.84135	−0.920675	+	0.390331i	$0.872361\pi$
$692$	−25.1888	−0.957536
$693$	14.0185	0.532518
$694$	−12.4791	−0.473699
$695$	−20.0678	−0.761217
$696$	−11.2303	−0.425683
$697$	0	0
$698$	2.60103	0.0984504
$699$	10.5759	0.400019
$700$	13.1726	0.497878
$701$	−3.21183	−0.121309	−0.0606545	−	0.998159i	$-0.519319\pi$
$701$	−3.21183	−0.121309	−0.0606545	+	0.998159i	$0.519319\pi$
$702$	−0.0823593	−0.00310845
$703$	0	0
$704$	4.51601	0.170204
$705$	16.0369	0.603986
$706$	5.14665	0.193697
$707$	−30.4403	−1.14482
$708$	−6.38485	−0.239957
$709$	22.0656	0.828690	0.414345	−	0.910120i	$-0.364011\pi$
$709$	22.0656	0.828690	0.414345	+	0.910120i	$0.364011\pi$
$710$	−20.8436	−0.782246
$711$	6.52884	0.244851
$712$	−18.2201	−0.682828
$713$	66.3803	2.48596
$714$	0	0
$715$	1.46890	0.0549338
$716$	25.3641	0.947902
$717$	−9.81681	−0.366615
$718$	−7.42405	−0.277063
$719$	17.6600	0.658607	0.329303	−	0.944224i	$-0.393186\pi$
$719$	17.6600	0.658607	0.329303	+	0.944224i	$0.393186\pi$
$720$	−5.73050	−0.213563
$721$	−53.8145	−2.00416
$722$	0	0
$723$	1.94233	0.0722359
$724$	−20.4112	−0.758577
$725$	−11.4610	−0.425651
$726$	−2.04090	−0.0757447
$727$	−40.3932	−1.49810	−0.749050	−	0.662514i	$-0.769490\pi$
$727$	−40.3932	−1.49810	−0.749050	+	0.662514i	$0.769490\pi$
$728$	−1.11100	−0.0411763
$729$	1.00000	0.0370370
$730$	−0.528439	−0.0195584
$731$	0	0
$732$	−19.2201	−0.710397
$733$	−17.9137	−0.661657	−0.330829	−	0.943691i	$-0.607328\pi$
$733$	−17.9137	−0.661657	−0.330829	+	0.943691i	$0.607328\pi$
$734$	2.31492	0.0854452
$735$	−17.3456	−0.639804
$736$	−40.8666	−1.50636
$737$	−20.5530	−0.757078
$738$	−3.05767	−0.112555
$739$	−16.9770	−0.624509	−0.312255	−	0.949998i	$-0.601084\pi$
$739$	−16.9770	−0.624509	−0.312255	+	0.949998i	$0.601084\pi$
$740$	−4.47116	−0.164363
$741$	0	0
$742$	16.8454	0.618416
$743$	−50.7177	−1.86065	−0.930325	−	0.366735i	$-0.880476\pi$
$743$	−50.7177	−1.86065	−0.930325	+	0.366735i	$0.880476\pi$
$744$	18.5226	0.679072
$745$	−37.4689	−1.37275
$746$	−2.17262	−0.0795454
$747$	−2.28797	−0.0837126
$748$	0	0
$749$	61.3042	2.24001
$750$	−4.36638	−0.159438
$751$	−25.5944	−0.933954	−0.466977	−	0.884270i	$-0.654657\pi$
$751$	−25.5944	−0.933954	−0.466977	+	0.884270i	$0.654657\pi$
$752$	−12.8639	−0.469099
$753$	13.6336	0.496837
$754$	0.440264	0.0160335
$755$	−29.5552	−1.07562
$756$	6.14399	0.223455
$757$	45.7361	1.66231	0.831154	−	0.556042i	$-0.187681\pi$
$757$	45.7361	1.66231	0.831154	+	0.556042i	$0.187681\pi$
$758$	−2.80003	−0.101702
$759$	−28.7361	−1.04306
$760$	0	0
$761$	−44.1338	−1.59985	−0.799925	−	0.600100i	$-0.795127\pi$
$761$	−44.1338	−1.59985	−0.799925	+	0.600100i	$0.795127\pi$
$762$	−1.51827	−0.0550012
$763$	−28.7776	−1.04182
$764$	−9.11761	−0.329864
$765$	0	0
$766$	−19.5288	−0.705606
$767$	0.549569	0.0198438
$768$	−4.23030	−0.152648
$769$	−44.0946	−1.59009	−0.795046	−	0.606549i	$-0.792553\pi$
$769$	−44.0946	−1.59009	−0.795046	+	0.606549i	$0.792553\pi$
$770$	21.4320	0.772354
$771$	−12.5944	−0.453577
$772$	0.853752	0.0307272
$773$	4.90086	0.176272	0.0881359	−	0.996108i	$-0.471909\pi$
$773$	4.90086	0.176272	0.0881359	+	0.996108i	$0.471909\pi$
$774$	−1.61120	−0.0579133
$775$	18.9031	0.679020
$776$	−12.4237	−0.445983
$777$	3.67282	0.131762
$778$	−15.3765	−0.551276
$779$	0	0
$780$	0.643787	0.0230513
$781$	52.0369	1.86203
$782$	0	0
$783$	−5.34565	−0.191038
$784$	13.9137	0.496917
$785$	−37.5737	−1.34106
$786$	6.48963	0.231478
$787$	35.0554	1.24959	0.624795	−	0.780789i	$-0.285182\pi$
$787$	35.0554	1.24959	0.624795	+	0.780789i	$0.285182\pi$
$788$	−38.4094	−1.36828
$789$	−1.63362	−0.0581584
$790$	9.98153	0.355127
$791$	23.4504	0.833801
$792$	−8.01847	−0.284924
$793$	1.65435	0.0587478
$794$	−14.4363	−0.512326
$795$	−21.4320	−0.760113
$796$	0.209965	0.00744202
$797$	−5.67056	−0.200862	−0.100431	−	0.994944i	$-0.532022\pi$
$797$	−5.67056	−0.200862	−0.100431	+	0.994944i	$0.532022\pi$
$798$	0	0
$799$	0	0
$800$	−11.6376	−0.411450
$801$	−8.67282	−0.306439
$802$	1.90312	0.0672016
$803$	1.31927	0.0465560
$804$	−9.00791	−0.317684
$805$	73.9092	2.60496
$806$	−0.726147	−0.0255774
$807$	−21.6521	−0.762190
$808$	17.4116	0.612539
$809$	10.6359	0.373938	0.186969	−	0.982366i	$-0.440134\pi$
$809$	10.6359	0.373938	0.186969	+	0.982366i	$0.440134\pi$
$810$	1.52884	0.0537179
$811$	27.3042	0.958780	0.479390	−	0.877602i	$-0.340858\pi$
$811$	27.3042	0.958780	0.479390	+	0.877602i	$0.340858\pi$
$812$	−32.8436	−1.15258
$813$	−21.5552	−0.755974
$814$	−2.18319	−0.0765208
$815$	−12.3355	−0.432093
$816$	0	0
$817$	0	0
$818$	−19.5183	−0.682440
$819$	−0.528837	−0.0184791
$820$	23.9013	0.834668
$821$	21.1730	0.738944	0.369472	−	0.929242i	$-0.379539\pi$
$821$	21.1730	0.738944	0.369472	+	0.929242i	$0.379539\pi$
$822$	0.934420	0.0325916
$823$	46.7282	1.62884	0.814422	−	0.580273i	$-0.197054\pi$
$823$	46.7282	1.62884	0.814422	+	0.580273i	$0.197054\pi$
$824$	30.7816	1.07233
$825$	−8.18319	−0.284902
$826$	8.01847	0.278998
$827$	−5.67508	−0.197342	−0.0986710	−	0.995120i	$-0.531459\pi$
$827$	−5.67508	−0.197342	−0.0986710	+	0.995120i	$0.531459\pi$
$828$	−12.5944	−0.437686
$829$	−22.3377	−0.775822	−0.387911	−	0.921697i	$-0.626803\pi$
$829$	−22.3377	−0.775822	−0.387911	+	0.921697i	$0.626803\pi$
$830$	−3.49794	−0.121415
$831$	7.62571	0.264533
$832$	−0.170363	−0.00590629
$833$	0	0
$834$	−4.29459	−0.148709
$835$	32.6623	1.13032
$836$	0	0
$837$	8.81681	0.304754
$838$	5.24086	0.181043
$839$	−13.0841	−0.451712	−0.225856	−	0.974161i	$-0.572518\pi$
$839$	−13.0841	−0.451712	−0.225856	+	0.974161i	$0.572518\pi$
$840$	20.6235	0.711577
$841$	−0.424054	−0.0146225
$842$	9.05767	0.312148
$843$	−1.69356	−0.0583292
$844$	−41.5759	−1.43110
$845$	34.6913	1.19342
$846$	3.43196	0.117993
$847$	−13.1048	−0.450286
$848$	17.1915	0.590358
$849$	10.2880	0.353082
$850$	0	0
$851$	−7.52884	−0.258085
$852$	22.8066	0.781343
$853$	40.2258	1.37730	0.688652	−	0.725092i	$-0.258203\pi$
$853$	40.2258	1.37730	0.688652	+	0.725092i	$0.258203\pi$
$854$	24.1378	0.825978
$855$	0	0
$856$	−35.0656	−1.19852
$857$	−6.00000	−0.204956	−0.102478	−	0.994735i	$-0.532677\pi$
$857$	−6.00000	−0.204956	−0.102478	+	0.994735i	$0.532677\pi$
$858$	0.314350	0.0107317
$859$	−21.8824	−0.746618	−0.373309	−	0.927707i	$-0.621777\pi$
$859$	−21.8824	−0.746618	−0.373309	+	0.927707i	$0.621777\pi$
$860$	12.5944	0.429466
$861$	−19.6336	−0.669112
$862$	5.30080	0.180546
$863$	−10.5759	−0.360009	−0.180005	−	0.983666i	$-0.557611\pi$
$863$	−10.5759	−0.360009	−0.180005	+	0.983666i	$0.557611\pi$
$864$	−5.42801	−0.184665
$865$	−40.2465	−1.36842
$866$	11.3338	0.385138
$867$	−17.0000	−0.577350
$868$	54.1704	1.83866
$869$	−24.9193	−0.845330
$870$	−8.17262	−0.277078
$871$	0.775346	0.0262716
$872$	16.4606	0.557426
$873$	−5.91369	−0.200148
$874$	0	0
$875$	−28.0369	−0.947822
$876$	0.578207	0.0195358
$877$	−43.5345	−1.47006	−0.735028	−	0.678037i	$-0.762831\pi$
$877$	−43.5345	−1.47006	−0.735028	+	0.678037i	$0.762831\pi$
$878$	−3.94854	−0.133257
$879$	12.1153	0.408641
$880$	21.8722	0.737313
$881$	4.96080	0.167133	0.0835667	−	0.996502i	$-0.473369\pi$
$881$	4.96080	0.167133	0.0835667	+	0.996502i	$0.473369\pi$
$882$	−3.71203	−0.124990
$883$	−13.0106	−0.437840	−0.218920	−	0.975743i	$-0.570253\pi$
$883$	−13.0106	−0.437840	−0.218920	+	0.975743i	$0.570253\pi$
$884$	0	0
$885$	−10.2017	−0.342925
$886$	11.2303	0.377289
$887$	34.1417	1.14637	0.573183	−	0.819427i	$-0.305708\pi$
$887$	34.1417	1.14637	0.573183	+	0.819427i	$0.305708\pi$
$888$	−2.10083	−0.0704993
$889$	−9.74897	−0.326970
$890$	−13.2593	−0.444454
$891$	−3.81681	−0.127868
$892$	39.8066	1.33283
$893$	0	0
$894$	−8.01847	−0.268178
$895$	40.5266	1.35465
$896$	−42.3579	−1.41508
$897$	1.08405	0.0361954
$898$	−16.5971	−0.553852
$899$	−47.1316	−1.57193
$900$	−3.58651	−0.119550
$901$	0	0
$902$	11.6706	0.388587
$903$	−10.3456	−0.344282
$904$	−13.4135	−0.446126
$905$	−32.6129	−1.08409
$906$	−6.32492	−0.210131
$907$	47.4979	1.57714	0.788572	−	0.614943i	$-0.210821\pi$
$907$	47.4979	1.57714	0.788572	+	0.614943i	$0.210821\pi$
$908$	−16.4218	−0.544976
$909$	8.28797	0.274895
$910$	−0.808506	−0.0268017
$911$	−46.1523	−1.52909	−0.764547	−	0.644568i	$-0.777037\pi$
$911$	−46.1523	−1.52909	−0.764547	+	0.644568i	$0.777037\pi$
$912$	0	0
$913$	8.73276	0.289012
$914$	−18.6168	−0.615790
$915$	−30.7098	−1.01523
$916$	−0.240864	−0.00795837
$917$	41.6706	1.37608
$918$	0	0
$919$	−29.5160	−0.973643	−0.486822	−	0.873501i	$-0.661844\pi$
$919$	−29.5160	−0.973643	−0.486822	+	0.873501i	$0.661844\pi$
$920$	−42.2755	−1.39378
$921$	−4.48963	−0.147938
$922$	5.52093	0.181822
$923$	−1.96306	−0.0646148
$924$	−23.4504	−0.771463
$925$	−2.14399	−0.0704938
$926$	17.7635	0.583744
$927$	14.6521	0.481238
$928$	29.0162	0.952504
$929$	−39.4425	−1.29407	−0.647034	−	0.762461i	$-0.723991\pi$
$929$	−39.4425	−1.29407	−0.647034	+	0.762461i	$0.723991\pi$
$930$	13.4795	0.442009
$931$	0	0
$932$	−17.6917	−0.579511
$933$	0.759136	0.0248530
$934$	−4.92651	−0.161200
$935$	0	0
$936$	0.302491	0.00988724
$937$	27.7361	0.906100	0.453050	−	0.891485i	$-0.350336\pi$
$937$	27.7361	0.906100	0.453050	+	0.891485i	$0.350336\pi$
$938$	11.3127	0.369371
$939$	11.4320	0.373068
$940$	−26.8270	−0.875000
$941$	−16.2465	−0.529621	−0.264811	−	0.964300i	$-0.585309\pi$
$941$	−16.2465	−0.529621	−0.264811	+	0.964300i	$0.585309\pi$
$942$	−8.04090	−0.261987
$943$	40.2465	1.31061
$944$	8.18319	0.266340
$945$	9.81681	0.319341
$946$	6.14963	0.199942
$947$	−15.9216	−0.517382	−0.258691	−	0.965960i	$-0.583291\pi$
$947$	−15.9216	−0.517382	−0.258691	+	0.965960i	$0.583291\pi$
$948$	−10.9216	−0.354717
$949$	−0.0497686	−0.00161556
$950$	0	0
$951$	−11.6151	−0.376647
$952$	0	0
$953$	35.9401	1.16421	0.582106	−	0.813113i	$-0.302229\pi$
$953$	35.9401	1.16421	0.582106	+	0.813113i	$0.302229\pi$
$954$	−4.58651	−0.148494
$955$	−14.5680	−0.471411
$956$	16.4218	0.531119
$957$	20.4033	0.659546
$958$	−9.98757	−0.322684
$959$	6.00000	0.193750
$960$	3.16246	0.102068
$961$	46.7361	1.50762
$962$	0.0823593	0.00265537
$963$	−16.6913	−0.537869
$964$	−3.24917	−0.104649
$965$	1.36412	0.0439125
$966$	15.8168	0.508898
$967$	2.60724	0.0838433	0.0419217	−	0.999121i	$-0.486652\pi$
$967$	2.60724	0.0838433	0.0419217	+	0.999121i	$0.486652\pi$
$968$	7.49585	0.240926
$969$	0	0
$970$	−9.04107	−0.290291
$971$	16.1312	0.517674	0.258837	−	0.965921i	$-0.416661\pi$
$971$	16.1312	0.517674	0.258837	+	0.965921i	$0.416661\pi$
$972$	−1.67282	−0.0536558
$973$	−27.5759	−0.884044
$974$	6.65435	0.213219
$975$	0.308705	0.00988648
$976$	24.6336	0.788503
$977$	−3.17302	−0.101514	−0.0507570	−	0.998711i	$-0.516163\pi$
$977$	−3.17302	−0.101514	−0.0507570	+	0.998711i	$0.516163\pi$
$978$	−2.63983	−0.0844126
$979$	33.1025	1.05796
$980$	29.0162	0.926889
$981$	7.83528	0.250161
$982$	13.9137	0.444004
$983$	20.8330	0.664470	0.332235	−	0.943197i	$-0.392197\pi$
$983$	20.8330	0.664470	0.332235	+	0.943197i	$0.392197\pi$
$984$	11.2303	0.358009
$985$	−61.3702	−1.95542
$986$	0	0
$987$	22.0369	0.701444
$988$	0	0
$989$	21.2073	0.674353
$990$	−5.83528	−0.185457
$991$	−45.8330	−1.45593	−0.727967	−	0.685612i	$-0.759535\pi$
$991$	−45.8330	−1.45593	−0.727967	+	0.685612i	$0.759535\pi$
$992$	−47.8577	−1.51948
$993$	−11.4712	−0.364026
$994$	−28.6419	−0.908467
$995$	0.335481	0.0106355
$996$	3.82738	0.121275
$997$	54.6336	1.73026	0.865132	−	0.501544i	$-0.167235\pi$
$997$	54.6336	1.73026	0.865132	+	0.501544i	$0.167235\pi$
$998$	−21.8004	−0.690081
$999$	−1.00000	−0.0316386

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1083.2.a.l.1.2		3
3.2	odd	2		3249.2.a.y.1.2		3
19.7	even	3		57.2.e.b.49.2	yes	6
19.11	even	3		57.2.e.b.7.2	✓	6
19.18	odd	2		1083.2.a.o.1.2		3
57.11	odd	6		171.2.f.b.64.2		6
57.26	odd	6		171.2.f.b.163.2		6
57.56	even	2		3249.2.a.t.1.2		3
76.7	odd	6		912.2.q.l.49.3		6
76.11	odd	6		912.2.q.l.577.3		6
228.11	even	6		2736.2.s.z.577.1		6
228.83	even	6		2736.2.s.z.1873.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
57.2.e.b.7.2	✓	6	19.11	even	3
57.2.e.b.49.2	yes	6	19.7	even	3
171.2.f.b.64.2		6	57.11	odd	6
171.2.f.b.163.2		6	57.26	odd	6
912.2.q.l.49.3		6	76.7	odd	6
912.2.q.l.577.3		6	76.11	odd	6
1083.2.a.l.1.2		3	1.1	even	1	trivial
1083.2.a.o.1.2		3	19.18	odd	2
2736.2.s.z.577.1		6	228.11	even	6
2736.2.s.z.1873.1		6	228.83	even	6
3249.2.a.t.1.2		3	57.56	even	2
3249.2.a.y.1.2		3	3.2	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:	\( N \)	\(=\)	\( 1083 = 3 \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1083.a (trivial)

\(f(q)\)	\(=\)	\(q-0.571993 q^{2} +1.00000 q^{3} -1.67282 q^{4} -2.67282 q^{5} -0.571993 q^{6} -3.67282 q^{7} +2.10083 q^{8} +1.00000 q^{9} +1.52884 q^{10} -3.81681 q^{11} -1.67282 q^{12} +0.143987 q^{13} +2.10083 q^{14} -2.67282 q^{15} +2.14399 q^{16} -0.571993 q^{18} +4.47116 q^{20} -3.67282 q^{21} +2.18319 q^{22} +7.52884 q^{23} +2.10083 q^{24} +2.14399 q^{25} -0.0823593 q^{26} +1.00000 q^{27} +6.14399 q^{28} -5.34565 q^{29} +1.52884 q^{30} +8.81681 q^{31} -5.42801 q^{32} -3.81681 q^{33} +9.81681 q^{35} -1.67282 q^{36} -1.00000 q^{37} +0.143987 q^{39} -5.61515 q^{40} +5.34565 q^{41} +2.10083 q^{42} +2.81681 q^{43} +6.38485 q^{44} -2.67282 q^{45} -4.30644 q^{46} -6.00000 q^{47} +2.14399 q^{48} +6.48963 q^{49} -1.22635 q^{50} -0.240864 q^{52} +8.01847 q^{53} -0.571993 q^{54} +10.2017 q^{55} -7.71598 q^{56} +3.05767 q^{58} +3.81681 q^{59} +4.47116 q^{60} +11.4896 q^{61} -5.04316 q^{62} -3.67282 q^{63} -1.18319 q^{64} -0.384851 q^{65} +2.18319 q^{66} +5.38485 q^{67} +7.52884 q^{69} -5.61515 q^{70} -13.6336 q^{71} +2.10083 q^{72} -0.345647 q^{73} +0.571993 q^{74} +2.14399 q^{75} +14.0185 q^{77} -0.0823593 q^{78} +6.52884 q^{79} -5.73050 q^{80} +1.00000 q^{81} -3.05767 q^{82} -2.28797 q^{83} +6.14399 q^{84} -1.61120 q^{86} -5.34565 q^{87} -8.01847 q^{88} -8.67282 q^{89} +1.52884 q^{90} -0.528837 q^{91} -12.5944 q^{92} +8.81681 q^{93} +3.43196 q^{94} -5.42801 q^{96} -5.91369 q^{97} -3.71203 q^{98} -3.81681 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - q^{2} + 3 q^{3} + 5 q^{4} + 2 q^{5} - q^{6} - q^{7} - 3 q^{8} + 3 q^{9} - 4 q^{10} + 5 q^{12} - q^{13} - 3 q^{14} + 2 q^{15} + 5 q^{16} - q^{18} + 22 q^{20} - q^{21} + 18 q^{22} + 14 q^{23}+ \cdots - 14 q^{98}+O(q^{100}) \) 3 * q - q^2 + 3 * q^3 + 5 * q^4 + 2 * q^5 - q^6 - q^7 - 3 * q^8 + 3 * q^9 - 4 * q^10 + 5 * q^12 - q^13 - 3 * q^14 + 2 * q^15 + 5 * q^16 - q^18 + 22 * q^20 - q^21 + 18 * q^22 + 14 * q^23 - 3 * q^24 + 5 * q^25 - 21 * q^26 + 3 * q^27 + 17 * q^28 + 4 * q^29 - 4 * q^30 + 15 * q^31 - 17 * q^32 + 18 * q^35 + 5 * q^36 - 3 * q^37 - q^39 - 24 * q^40 - 4 * q^41 - 3 * q^42 - 3 * q^43 + 12 * q^44 + 2 * q^45 + 20 * q^46 - 18 * q^47 + 5 * q^48 - 2 * q^49 - 23 * q^50 + 5 * q^52 - 6 * q^53 - q^54 + 12 * q^55 - 21 * q^56 - 8 * q^58 + 22 * q^60 + 13 * q^61 - 23 * q^62 - q^63 - 15 * q^64 + 6 * q^65 + 18 * q^66 + 9 * q^67 + 14 * q^69 - 24 * q^70 - 18 * q^71 - 3 * q^72 + 19 * q^73 + q^74 + 5 * q^75 + 12 * q^77 - 21 * q^78 + 11 * q^79 + 10 * q^80 + 3 * q^81 + 8 * q^82 - 4 * q^83 + 17 * q^84 - 17 * q^86 + 4 * q^87 + 6 * q^88 - 16 * q^89 - 4 * q^90 + 7 * q^91 - 2 * q^92 + 15 * q^93 + 6 * q^94 - 17 * q^96 - 2 * q^97 - 14 * q^98

Introduction

L-functions

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