LMFDB - Embedded newform 8619.2.a.br.1.14

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8619,2,Mod(1,8619)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8619, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("8619.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8619 = 3 \cdot 13^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8619.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:	$68.8230615021$
Analytic rank:	$0$
Dimension:	$22$
Twist minimal:	no (minimal twist has level 663)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.14
Character	$\chi$	$=$	8619.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.03403 q^{2} -1.00000 q^{3} -0.930781 q^{4} -3.65775 q^{5} -1.03403 q^{6} +3.14151 q^{7} -3.03052 q^{8} +1.00000 q^{9} +O(q^{10})$	\(q+1.03403 q^{2} -1.00000 q^{3} -0.930781 q^{4} -3.65775 q^{5} -1.03403 q^{6} +3.14151 q^{7} -3.03052 q^{8} +1.00000 q^{9} -3.78223 q^{10} +0.0720702 q^{11} +0.930781 q^{12} +3.24842 q^{14} +3.65775 q^{15} -1.27209 q^{16} -1.00000 q^{17} +1.03403 q^{18} -3.38749 q^{19} +3.40456 q^{20} -3.14151 q^{21} +0.0745228 q^{22} -1.02739 q^{23} +3.03052 q^{24} +8.37914 q^{25} -1.00000 q^{27} -2.92406 q^{28} -1.68466 q^{29} +3.78223 q^{30} +2.04412 q^{31} +4.74566 q^{32} -0.0720702 q^{33} -1.03403 q^{34} -11.4909 q^{35} -0.930781 q^{36} +0.667298 q^{37} -3.50276 q^{38} +11.0849 q^{40} -9.73361 q^{41} -3.24842 q^{42} -10.1145 q^{43} -0.0670816 q^{44} -3.65775 q^{45} -1.06235 q^{46} -9.26290 q^{47} +1.27209 q^{48} +2.86911 q^{49} +8.66429 q^{50} +1.00000 q^{51} +7.68378 q^{53} -1.03403 q^{54} -0.263615 q^{55} -9.52041 q^{56} +3.38749 q^{57} -1.74199 q^{58} +8.74749 q^{59} -3.40456 q^{60} +3.65869 q^{61} +2.11368 q^{62} +3.14151 q^{63} +7.45133 q^{64} -0.0745228 q^{66} +1.07568 q^{67} +0.930781 q^{68} +1.02739 q^{69} -11.8819 q^{70} -14.0472 q^{71} -3.03052 q^{72} +15.8379 q^{73} +0.690007 q^{74} -8.37914 q^{75} +3.15301 q^{76} +0.226410 q^{77} -7.62724 q^{79} +4.65298 q^{80} +1.00000 q^{81} -10.0648 q^{82} -7.47635 q^{83} +2.92406 q^{84} +3.65775 q^{85} -10.4587 q^{86} +1.68466 q^{87} -0.218410 q^{88} -2.74146 q^{89} -3.78223 q^{90} +0.956276 q^{92} -2.04412 q^{93} -9.57812 q^{94} +12.3906 q^{95} -4.74566 q^{96} -10.8696 q^{97} +2.96675 q^{98} +0.0720702 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 22 q - 22 q^{3} + 32 q^{4} + 22 q^{9}+O(q^{10}) $ 22 * q - 22 * q^3 + 32 * q^4 + 22 * q^9	$ 22 q - 22 q^{3} + 32 q^{4} + 22 q^{9} + 26 q^{10} - 32 q^{12} + 18 q^{14} + 36 q^{16} - 22 q^{17} + 22 q^{22} - 20 q^{23} + 46 q^{25} - 22 q^{27} + 6 q^{29} - 26 q^{30} - 14 q^{35} + 32 q^{36} + 38 q^{38} + 108 q^{40} - 18 q^{42} - 30 q^{43} - 36 q^{48} + 68 q^{49} + 22 q^{51} - 10 q^{53} - 18 q^{55} - 30 q^{56} + 92 q^{61} + 52 q^{62} + 96 q^{64} - 22 q^{66} - 32 q^{68} + 20 q^{69} + 42 q^{74} - 46 q^{75} + 28 q^{77} + 68 q^{79} + 22 q^{81} + 6 q^{82} - 6 q^{87} + 134 q^{88} + 26 q^{90} + 40 q^{92} - 94 q^{94} - 72 q^{95}+O(q^{100}) $ 22 * q - 22 * q^3 + 32 * q^4 + 22 * q^9 + 26 * q^10 - 32 * q^12 + 18 * q^14 + 36 * q^16 - 22 * q^17 + 22 * q^22 - 20 * q^23 + 46 * q^25 - 22 * q^27 + 6 * q^29 - 26 * q^30 - 14 * q^35 + 32 * q^36 + 38 * q^38 + 108 * q^40 - 18 * q^42 - 30 * q^43 - 36 * q^48 + 68 * q^49 + 22 * q^51 - 10 * q^53 - 18 * q^55 - 30 * q^56 + 92 * q^61 + 52 * q^62 + 96 * q^64 - 22 * q^66 - 32 * q^68 + 20 * q^69 + 42 * q^74 - 46 * q^75 + 28 * q^77 + 68 * q^79 + 22 * q^81 + 6 * q^82 - 6 * q^87 + 134 * q^88 + 26 * q^90 + 40 * q^92 - 94 * q^94 - 72 * q^95

Display 100 coefficients 10 coefficients

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 $

$2$	1.03403	0.731170	0.365585	−	0.930778i	$-0.380869\pi$
$2$	1.03403	0.731170	0.365585	+	0.930778i	$0.380869\pi$
$3$	−1.00000	−0.577350
$3$	−1.00000	−0.577350
$4$	−0.930781	−0.465390
$5$	−3.65775	−1.63580	−0.817898	−	0.575363i	$-0.804861\pi$
$5$	−3.65775	−1.63580	−0.817898	+	0.575363i	$0.804861\pi$
$6$	−1.03403	−0.422141
$7$	3.14151	1.18738	0.593690	−	0.804694i	$-0.297670\pi$
$7$	3.14151	1.18738	0.593690	+	0.804694i	$0.297670\pi$
$8$	−3.03052	−1.07145
$9$	1.00000	0.333333
$10$	−3.78223	−1.19605
$11$	0.0720702	0.0217300	0.0108650	−	0.999941i	$-0.496541\pi$
$11$	0.0720702	0.0217300	0.0108650	+	0.999941i	$0.496541\pi$
$12$	0.930781	0.268693
$13$	0	0
$13$	0	0
$14$	3.24842	0.868177
$15$	3.65775	0.944427
$16$	−1.27209	−0.318022
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	1.03403	0.243723
$19$	−3.38749	−0.777143	−0.388571	−	0.921419i	$-0.627031\pi$
$19$	−3.38749	−0.777143	−0.388571	+	0.921419i	$0.627031\pi$
$20$	3.40456	0.761283
$21$	−3.14151	−0.685535
$22$	0.0745228	0.0158883
$23$	−1.02739	−0.214226	−0.107113	−	0.994247i	$-0.534161\pi$
$23$	−1.02739	−0.214226	−0.107113	+	0.994247i	$0.534161\pi$
$24$	3.03052	0.618602
$25$	8.37914	1.67583
$26$	0	0
$27$	−1.00000	−0.192450
$28$	−2.92406	−0.552595
$29$	−1.68466	−0.312833	−0.156416	−	0.987691i	$-0.549994\pi$
$29$	−1.68466	−0.312833	−0.156416	+	0.987691i	$0.549994\pi$
$30$	3.78223	0.690537
$31$	2.04412	0.367134	0.183567	−	0.983007i	$-0.441235\pi$
$31$	2.04412	0.367134	0.183567	+	0.983007i	$0.441235\pi$
$32$	4.74566	0.838922
$33$	−0.0720702	−0.0125458
$34$	−1.03403	−0.177335
$35$	−11.4909	−1.94231
$36$	−0.930781	−0.155130
$37$	0.667298	0.109703	0.0548516	−	0.998495i	$-0.482531\pi$
$37$	0.667298	0.109703	0.0548516	+	0.998495i	$0.482531\pi$
$38$	−3.50276	−0.568223
$39$	0	0
$40$	11.0849	1.75267
$41$	−9.73361	−1.52013	−0.760067	−	0.649845i	$-0.774834\pi$
$41$	−9.73361	−1.52013	−0.760067	+	0.649845i	$0.774834\pi$
$42$	−3.24842	−0.501242
$43$	−10.1145	−1.54244	−0.771221	−	0.636568i	$-0.780354\pi$
$43$	−10.1145	−1.54244	−0.771221	+	0.636568i	$0.780354\pi$
$44$	−0.0670816	−0.0101129
$45$	−3.65775	−0.545265
$46$	−1.06235	−0.156636
$47$	−9.26290	−1.35113	−0.675566	−	0.737299i	$-0.736101\pi$
$47$	−9.26290	−1.35113	−0.675566	+	0.737299i	$0.736101\pi$
$48$	1.27209	0.183610
$49$	2.86911	0.409873
$50$	8.66429	1.22532
$51$	1.00000	0.140028
$52$	0	0
$53$	7.68378	1.05545	0.527724	−	0.849416i	$-0.323045\pi$
$53$	7.68378	1.05545	0.527724	+	0.849416i	$0.323045\pi$
$54$	−1.03403	−0.140714
$55$	−0.263615	−0.0355458
$56$	−9.52041	−1.27222
$57$	3.38749	0.448683
$58$	−1.74199	−0.228734
$59$	8.74749	1.13883	0.569413	−	0.822051i	$-0.307170\pi$
$59$	8.74749	1.13883	0.569413	+	0.822051i	$0.307170\pi$
$60$	−3.40456	−0.439527
$61$	3.65869	0.468448	0.234224	−	0.972183i	$-0.424745\pi$
$61$	3.65869	0.468448	0.234224	+	0.972183i	$0.424745\pi$
$62$	2.11368	0.268438
$63$	3.14151	0.395794
$64$	7.45133	0.931416
$65$	0	0
$66$	−0.0745228	−0.00917313
$67$	1.07568	0.131416	0.0657079	−	0.997839i	$-0.479069\pi$
$67$	1.07568	0.131416	0.0657079	+	0.997839i	$0.479069\pi$
$68$	0.930781	0.112874
$69$	1.02739	0.123683
$70$	−11.8819	−1.42016
$71$	−14.0472	−1.66710	−0.833548	−	0.552447i	$-0.813694\pi$
$71$	−14.0472	−1.66710	−0.833548	+	0.552447i	$0.813694\pi$
$72$	−3.03052	−0.357150
$73$	15.8379	1.85368	0.926841	−	0.375453i	$-0.122513\pi$
$73$	15.8379	1.85368	0.926841	+	0.375453i	$0.122513\pi$
$74$	0.690007	0.0802117
$75$	−8.37914	−0.967540
$76$	3.15301	0.361675
$77$	0.226410	0.0258018
$78$	0	0
$79$	−7.62724	−0.858132	−0.429066	−	0.903273i	$-0.641157\pi$
$79$	−7.62724	−0.858132	−0.429066	+	0.903273i	$0.641157\pi$
$80$	4.65298	0.520219
$81$	1.00000	0.111111
$82$	−10.0648	−1.11148
$83$	−7.47635	−0.820636	−0.410318	−	0.911942i	$-0.634582\pi$
$83$	−7.47635	−0.820636	−0.410318	+	0.911942i	$0.634582\pi$
$84$	2.92406	0.319041
$85$	3.65775	0.396739
$86$	−10.4587	−1.12779
$87$	1.68466	0.180614
$88$	−0.218410	−0.0232826
$89$	−2.74146	−0.290595	−0.145297	−	0.989388i	$-0.546414\pi$
$89$	−2.74146	−0.290595	−0.145297	+	0.989388i	$0.546414\pi$
$90$	−3.78223	−0.398682
$91$	0	0
$92$	0.956276	0.0996986
$93$	−2.04412	−0.211965
$94$	−9.57812	−0.987908
$95$	12.3906	1.27125
$96$	−4.74566	−0.484352
$97$	−10.8696	−1.10364	−0.551820	−	0.833963i	$-0.686066\pi$
$97$	−10.8696	−1.10364	−0.551820	+	0.833963i	$0.686066\pi$
$98$	2.96675	0.299687
$99$	0.0720702	0.00724333
$100$	−7.79914	−0.779914
$101$	6.70564	0.667236	0.333618	−	0.942708i	$-0.391730\pi$
$101$	6.70564	0.667236	0.333618	+	0.942708i	$0.391730\pi$
$102$	1.03403	0.102384
$103$	−1.53670	−0.151415	−0.0757077	−	0.997130i	$-0.524122\pi$
$103$	−1.53670	−0.151415	−0.0757077	+	0.997130i	$0.524122\pi$
$104$	0	0
$105$	11.4909	1.12139
$106$	7.94527	0.771713
$107$	−7.70227	−0.744607	−0.372303	−	0.928111i	$-0.621432\pi$
$107$	−7.70227	−0.744607	−0.372303	+	0.928111i	$0.621432\pi$
$108$	0.930781	0.0895644
$109$	−1.81527	−0.173872	−0.0869358	−	0.996214i	$-0.527708\pi$
$109$	−1.81527	−0.173872	−0.0869358	+	0.996214i	$0.527708\pi$
$110$	−0.272586	−0.0259901
$111$	−0.667298	−0.0633371
$112$	−3.99628	−0.377613
$113$	9.14896	0.860662	0.430331	−	0.902671i	$-0.358397\pi$
$113$	9.14896	0.860662	0.430331	+	0.902671i	$0.358397\pi$
$114$	3.50276	0.328064
$115$	3.75794	0.350430
$116$	1.56805	0.145589
$117$	0	0
$118$	9.04517	0.832676
$119$	−3.14151	−0.287982
$120$	−11.0849	−1.01191
$121$	−10.9948	−0.999528
$122$	3.78320	0.342515
$123$	9.73361	0.877650
$124$	−1.90263	−0.170861
$125$	−12.3601	−1.10552
$126$	3.24842	0.289392
$127$	−8.30738	−0.737161	−0.368580	−	0.929596i	$-0.620156\pi$
$127$	−8.30738	−0.737161	−0.368580	+	0.929596i	$0.620156\pi$
$128$	−1.78641	−0.157898
$129$	10.1145	0.890529
$130$	0	0
$131$	−11.4989	−1.00467	−0.502333	−	0.864674i	$-0.667525\pi$
$131$	−11.4989	−1.00467	−0.502333	+	0.864674i	$0.667525\pi$
$132$	0.0670816	0.00583870
$133$	−10.6418	−0.922764
$134$	1.11229	0.0960873
$135$	3.65775	0.314809
$136$	3.03052	0.259865
$137$	−15.2696	−1.30457	−0.652284	−	0.757974i	$-0.726189\pi$
$137$	−15.2696	−1.30457	−0.652284	+	0.757974i	$0.726189\pi$
$138$	1.06235	0.0904336
$139$	9.88131	0.838121	0.419061	−	0.907958i	$-0.362359\pi$
$139$	9.88131	0.838121	0.419061	+	0.907958i	$0.362359\pi$
$140$	10.6955	0.903933
$141$	9.26290	0.780077
$142$	−14.5252	−1.21893
$143$	0	0
$144$	−1.27209	−0.106007
$145$	6.16205	0.511731
$146$	16.3768	1.35536
$147$	−2.86911	−0.236640
$148$	−0.621108	−0.0510548
$149$	8.62227	0.706364	0.353182	−	0.935555i	$-0.385100\pi$
$149$	8.62227	0.706364	0.353182	+	0.935555i	$0.385100\pi$
$150$	−8.66429	−0.707436
$151$	1.83753	0.149536	0.0747680	−	0.997201i	$-0.476178\pi$
$151$	1.83753	0.149536	0.0747680	+	0.997201i	$0.476178\pi$
$152$	10.2658	0.832669
$153$	−1.00000	−0.0808452
$154$	0.234115	0.0188655
$155$	−7.47688	−0.600557
$156$	0	0
$157$	−0.805725	−0.0643039	−0.0321519	−	0.999483i	$-0.510236\pi$
$157$	−0.805725	−0.0643039	−0.0321519	+	0.999483i	$0.510236\pi$
$158$	−7.88680	−0.627440
$159$	−7.68378	−0.609364
$160$	−17.3584	−1.37230
$161$	−3.22756	−0.254368
$162$	1.03403	0.0812411
$163$	1.75326	0.137326	0.0686629	−	0.997640i	$-0.478127\pi$
$163$	1.75326	0.137326	0.0686629	+	0.997640i	$0.478127\pi$
$164$	9.05985	0.707456
$165$	0.263615	0.0205224
$166$	−7.73078	−0.600025
$167$	−14.5208	−1.12365	−0.561827	−	0.827255i	$-0.689901\pi$
$167$	−14.5208	−1.12365	−0.561827	+	0.827255i	$0.689901\pi$
$168$	9.52041	0.734516
$169$	0	0
$170$	3.78223	0.290084
$171$	−3.38749	−0.259048
$172$	9.41435	0.717837
$173$	5.90127	0.448666	0.224333	−	0.974513i	$-0.427980\pi$
$173$	5.90127	0.448666	0.224333	+	0.974513i	$0.427980\pi$
$174$	1.74199	0.132060
$175$	26.3232	1.98985
$176$	−0.0916796	−0.00691061
$177$	−8.74749	−0.657502
$178$	−2.83476	−0.212474
$179$	23.8803	1.78490	0.892449	−	0.451149i	$-0.148986\pi$
$179$	23.8803	1.78490	0.892449	+	0.451149i	$0.148986\pi$
$180$	3.40456	0.253761
$181$	14.1400	1.05102	0.525510	−	0.850787i	$-0.323874\pi$
$181$	14.1400	1.05102	0.525510	+	0.850787i	$0.323874\pi$
$182$	0	0
$183$	−3.65869	−0.270458
$184$	3.11353	0.229532
$185$	−2.44081	−0.179452
$186$	−2.11368	−0.154983
$187$	−0.0720702	−0.00527030
$188$	8.62173	0.628804
$189$	−3.14151	−0.228512
$190$	12.8122	0.929498
$191$	6.88248	0.497999	0.248999	−	0.968504i	$-0.419898\pi$
$191$	6.88248	0.497999	0.248999	+	0.968504i	$0.419898\pi$
$192$	−7.45133	−0.537753
$193$	18.9838	1.36648	0.683240	−	0.730193i	$-0.260570\pi$
$193$	18.9838	1.36648	0.683240	+	0.730193i	$0.260570\pi$
$194$	−11.2395	−0.806948
$195$	0	0
$196$	−2.67051	−0.190751
$197$	6.09875	0.434518	0.217259	−	0.976114i	$-0.430288\pi$
$197$	6.09875	0.434518	0.217259	+	0.976114i	$0.430288\pi$
$198$	0.0745228	0.00529611
$199$	14.2986	1.01360	0.506800	−	0.862064i	$-0.330828\pi$
$199$	14.2986	1.01360	0.506800	+	0.862064i	$0.330828\pi$
$200$	−25.3931	−1.79557
$201$	−1.07568	−0.0758729
$202$	6.93384	0.487863
$203$	−5.29237	−0.371452
$204$	−0.930781	−0.0651677
$205$	35.6031	2.48663
$206$	−1.58899	−0.110710
$207$	−1.02739	−0.0714086
$208$	0	0
$209$	−0.244137	−0.0168873
$210$	11.8819	0.819930
$211$	−3.70042	−0.254747	−0.127374	−	0.991855i	$-0.540655\pi$
$211$	−3.70042	−0.254747	−0.127374	+	0.991855i	$0.540655\pi$
$212$	−7.15192	−0.491196
$213$	14.0472	0.962498
$214$	−7.96438	−0.544434
$215$	36.9962	2.52312
$216$	3.03052	0.206201
$217$	6.42163	0.435928
$218$	−1.87705	−0.127130
$219$	−15.8379	−1.07022
$220$	0.245368	0.0165427
$221$	0	0
$222$	−0.690007	−0.0463102
$223$	23.0896	1.54620	0.773098	−	0.634287i	$-0.218706\pi$
$223$	23.0896	1.54620	0.773098	+	0.634287i	$0.218706\pi$
$224$	14.9086	0.996119
$225$	8.37914	0.558609
$226$	9.46031	0.629290
$227$	14.4296	0.957729	0.478864	−	0.877889i	$-0.341049\pi$
$227$	14.4296	0.957729	0.478864	+	0.877889i	$0.341049\pi$
$228$	−3.15301	−0.208813
$229$	−16.8020	−1.11031	−0.555153	−	0.831748i	$-0.687340\pi$
$229$	−16.8020	−1.11031	−0.555153	+	0.831748i	$0.687340\pi$
$230$	3.88583	0.256224
$231$	−0.226410	−0.0148967
$232$	5.10538	0.335185
$233$	−1.37750	−0.0902432	−0.0451216	−	0.998982i	$-0.514368\pi$
$233$	−1.37750	−0.0902432	−0.0451216	+	0.998982i	$0.514368\pi$
$234$	0	0
$235$	33.8814	2.21018
$236$	−8.14199	−0.529999
$237$	7.62724	0.495443
$238$	−3.24842	−0.210564
$239$	−1.24291	−0.0803971	−0.0401985	−	0.999192i	$-0.512799\pi$
$239$	−1.24291	−0.0803971	−0.0401985	+	0.999192i	$0.512799\pi$
$240$	−4.65298	−0.300348
$241$	28.6307	1.84427	0.922134	−	0.386871i	$-0.126444\pi$
$241$	28.6307	1.84427	0.922134	+	0.386871i	$0.126444\pi$
$242$	−11.3690	−0.730825
$243$	−1.00000	−0.0641500
$244$	−3.40544	−0.218011
$245$	−10.4945	−0.670469
$246$	10.0648	0.641711
$247$	0	0
$248$	−6.19474	−0.393366
$249$	7.47635	0.473795
$250$	−12.7807	−0.808321
$251$	21.9356	1.38457	0.692283	−	0.721626i	$-0.256605\pi$
$251$	21.9356	1.38457	0.692283	+	0.721626i	$0.256605\pi$
$252$	−2.92406	−0.184198
$253$	−0.0740443	−0.00465513
$254$	−8.59009	−0.538990
$255$	−3.65775	−0.229057
$256$	−16.7499	−1.04687
$257$	29.8519	1.86211	0.931055	−	0.364879i	$-0.118890\pi$
$257$	29.8519	1.86211	0.931055	+	0.364879i	$0.118890\pi$
$258$	10.4587	0.651128
$259$	2.09633	0.130259
$260$	0	0
$261$	−1.68466	−0.104278
$262$	−11.8903	−0.734582
$263$	−27.6156	−1.70285	−0.851427	−	0.524474i	$-0.824262\pi$
$263$	−27.6156	−1.70285	−0.851427	+	0.524474i	$0.824262\pi$
$264$	0.218410	0.0134422
$265$	−28.1054	−1.72650
$266$	−11.0040	−0.674698
$267$	2.74146	0.167775
$268$	−1.00123	−0.0611596
$269$	18.5650	1.13193	0.565964	−	0.824430i	$-0.308504\pi$
$269$	18.5650	1.13193	0.565964	+	0.824430i	$0.308504\pi$
$270$	3.78223	0.230179
$271$	25.1798	1.52956	0.764781	−	0.644290i	$-0.222847\pi$
$271$	25.1798	1.52956	0.764781	+	0.644290i	$0.222847\pi$
$272$	1.27209	0.0771316
$273$	0	0
$274$	−15.7892	−0.953862
$275$	0.603887	0.0364157
$276$	−0.956276	−0.0575610
$277$	24.2875	1.45930	0.729648	−	0.683823i	$-0.239684\pi$
$277$	24.2875	1.45930	0.729648	+	0.683823i	$0.239684\pi$
$278$	10.2176	0.612809
$279$	2.04412	0.122378
$280$	34.8233	2.08109
$281$	−21.4793	−1.28135	−0.640673	−	0.767814i	$-0.721345\pi$
$281$	−21.4793	−1.28135	−0.640673	+	0.767814i	$0.721345\pi$
$282$	9.57812	0.570369
$283$	21.0133	1.24911	0.624555	−	0.780981i	$-0.285280\pi$
$283$	21.0133	1.24911	0.624555	+	0.780981i	$0.285280\pi$
$284$	13.0749	0.775850
$285$	−12.3906	−0.733955
$286$	0	0
$287$	−30.5783	−1.80498
$288$	4.74566	0.279641
$289$	1.00000	0.0588235
$290$	6.37175	0.374162
$291$	10.8696	0.637187
$292$	−14.7416	−0.862686
$293$	22.9866	1.34289	0.671447	−	0.741053i	$-0.265673\pi$
$293$	22.9866	1.34289	0.671447	+	0.741053i	$0.265673\pi$
$294$	−2.96675	−0.173024
$295$	−31.9961	−1.86289
$296$	−2.02226	−0.117541
$297$	−0.0720702	−0.00418194
$298$	8.91569	0.516472
$299$	0	0
$300$	7.79914	0.450284
$301$	−31.7747	−1.83147
$302$	1.90006	0.109336
$303$	−6.70564	−0.385229
$304$	4.30918	0.247148
$305$	−13.3826	−0.766285
$306$	−1.03403	−0.0591116
$307$	28.9131	1.65016	0.825078	−	0.565018i	$-0.191131\pi$
$307$	28.9131	1.65016	0.825078	+	0.565018i	$0.191131\pi$
$308$	−0.210738	−0.0120079
$309$	1.53670	0.0874197
$310$	−7.73132	−0.439109
$311$	−18.5821	−1.05369	−0.526847	−	0.849960i	$-0.676626\pi$
$311$	−18.5821	−1.05369	−0.526847	+	0.849960i	$0.676626\pi$
$312$	0	0
$313$	−16.9400	−0.957504	−0.478752	−	0.877950i	$-0.658911\pi$
$313$	−16.9400	−0.957504	−0.478752	+	0.877950i	$0.658911\pi$
$314$	−0.833145	−0.0470171
$315$	−11.4909	−0.647438
$316$	7.09929	0.399366
$317$	10.7443	0.603459	0.301730	−	0.953394i	$-0.402436\pi$
$317$	10.7443	0.603459	0.301730	+	0.953394i	$0.402436\pi$
$318$	−7.94527	−0.445549
$319$	−0.121414	−0.00679785
$320$	−27.2551	−1.52361
$321$	7.70227	0.429899
$322$	−3.33740	−0.185986
$323$	3.38749	0.188485
$324$	−0.930781	−0.0517100
$325$	0	0
$326$	1.81292	0.100409
$327$	1.81527	0.100385
$328$	29.4979	1.62875
$329$	−29.0995	−1.60431
$330$	0.272586	0.0150054
$331$	26.3759	1.44975	0.724877	−	0.688879i	$-0.241897\pi$
$331$	26.3759	1.44975	0.724877	+	0.688879i	$0.241897\pi$
$332$	6.95884	0.381916
$333$	0.667298	0.0365677
$334$	−15.0150	−0.821582
$335$	−3.93458	−0.214969
$336$	3.99628	0.218015
$337$	22.4093	1.22071	0.610357	−	0.792127i	$-0.291026\pi$
$337$	22.4093	1.22071	0.610357	+	0.792127i	$0.291026\pi$
$338$	0	0
$339$	−9.14896	−0.496904
$340$	−3.40456	−0.184638
$341$	0.147320	0.00797783
$342$	−3.50276	−0.189408
$343$	−12.9772	−0.700705
$344$	30.6521	1.65265
$345$	−3.75794	−0.202321
$346$	6.10210	0.328051
$347$	1.20084	0.0644647	0.0322323	−	0.999480i	$-0.489738\pi$
$347$	1.20084	0.0644647	0.0322323	+	0.999480i	$0.489738\pi$
$348$	−1.56805	−0.0840560
$349$	−13.1503	−0.703920	−0.351960	−	0.936015i	$-0.614485\pi$
$349$	−13.1503	−0.703920	−0.351960	+	0.936015i	$0.614485\pi$
$350$	27.2190	1.45492
$351$	0	0
$352$	0.342021	0.0182298
$353$	17.4531	0.928937	0.464468	−	0.885590i	$-0.346245\pi$
$353$	17.4531	0.928937	0.464468	+	0.885590i	$0.346245\pi$
$354$	−9.04517	−0.480746
$355$	51.3812	2.72703
$356$	2.55170	0.135240
$357$	3.14151	0.166267
$358$	24.6930	1.30506
$359$	−32.4488	−1.71258	−0.856292	−	0.516492i	$-0.827238\pi$
$359$	−32.4488	−1.71258	−0.856292	+	0.516492i	$0.827238\pi$
$360$	11.0849	0.584224
$361$	−7.52494	−0.396049
$362$	14.6212	0.768475
$363$	10.9948	0.577078
$364$	0	0
$365$	−57.9310	−3.03225
$366$	−3.78320	−0.197751
$367$	−15.7594	−0.822633	−0.411317	−	0.911493i	$-0.634931\pi$
$367$	−15.7594	−0.822633	−0.411317	+	0.911493i	$0.634931\pi$
$368$	1.30693	0.0681285
$369$	−9.73361	−0.506711
$370$	−2.52387	−0.131210
$371$	24.1387	1.25322
$372$	1.90263	0.0986465
$373$	32.9919	1.70826	0.854128	−	0.520062i	$-0.174091\pi$
$373$	32.9919	1.70826	0.854128	+	0.520062i	$0.174091\pi$
$374$	−0.0745228	−0.00385348
$375$	12.3601	0.638271
$376$	28.0714	1.44767
$377$	0	0
$378$	−3.24842	−0.167081
$379$	−20.6649	−1.06149	−0.530743	−	0.847533i	$-0.678087\pi$
$379$	−20.6649	−1.06149	−0.530743	+	0.847533i	$0.678087\pi$
$380$	−11.5329	−0.591626
$381$	8.30738	0.425600
$382$	7.11670	0.364122
$383$	−25.8258	−1.31964	−0.659818	−	0.751425i	$-0.729367\pi$
$383$	−25.8258	−1.31964	−0.659818	+	0.751425i	$0.729367\pi$
$384$	1.78641	0.0911625
$385$	−0.828150	−0.0422064
$386$	19.6298	0.999130
$387$	−10.1145	−0.514147
$388$	10.1172	0.513623
$389$	−19.6366	−0.995618	−0.497809	−	0.867287i	$-0.665862\pi$
$389$	−19.6366	−0.995618	−0.497809	+	0.867287i	$0.665862\pi$
$390$	0	0
$391$	1.02739	0.0519574
$392$	−8.69489	−0.439159
$393$	11.4989	0.580045
$394$	6.30630	0.317707
$395$	27.8986	1.40373
$396$	−0.0670816	−0.00337098
$397$	8.67610	0.435441	0.217721	−	0.976011i	$-0.430138\pi$
$397$	8.67610	0.435441	0.217721	+	0.976011i	$0.430138\pi$
$398$	14.7852	0.741114
$399$	10.6418	0.532758
$400$	−10.6590	−0.532950
$401$	−34.9770	−1.74667	−0.873335	−	0.487121i	$-0.838047\pi$
$401$	−34.9770	−1.74667	−0.873335	+	0.487121i	$0.838047\pi$
$402$	−1.11229	−0.0554760
$403$	0	0
$404$	−6.24148	−0.310525
$405$	−3.65775	−0.181755
$406$	−5.47247	−0.271594
$407$	0.0480923	0.00238385
$408$	−3.03052	−0.150033
$409$	26.3816	1.30449	0.652244	−	0.758009i	$-0.273828\pi$
$409$	26.3816	1.30449	0.652244	+	0.758009i	$0.273828\pi$
$410$	36.8147	1.81815
$411$	15.2696	0.753193
$412$	1.43033	0.0704672
$413$	27.4804	1.35222
$414$	−1.06235	−0.0522119
$415$	27.3466	1.34239
$416$	0	0
$417$	−9.88131	−0.483890
$418$	−0.252445	−0.0123475
$419$	18.8529	0.921022	0.460511	−	0.887654i	$-0.347666\pi$
$419$	18.8529	0.921022	0.460511	+	0.887654i	$0.347666\pi$
$420$	−10.6955	−0.521886
$421$	25.1234	1.22444	0.612220	−	0.790688i	$-0.290277\pi$
$421$	25.1234	1.22444	0.612220	+	0.790688i	$0.290277\pi$
$422$	−3.82634	−0.186263
$423$	−9.26290	−0.450378
$424$	−23.2858	−1.13086
$425$	−8.37914	−0.406448
$426$	14.5252	0.703750
$427$	11.4938	0.556226
$428$	7.16912	0.346533
$429$	0	0
$430$	38.2552	1.84483
$431$	33.3809	1.60790	0.803950	−	0.594697i	$-0.202728\pi$
$431$	33.3809	1.60790	0.803950	+	0.594697i	$0.202728\pi$
$432$	1.27209	0.0612033
$433$	−2.58785	−0.124364	−0.0621820	−	0.998065i	$-0.519806\pi$
$433$	−2.58785	−0.124364	−0.0621820	+	0.998065i	$0.519806\pi$
$434$	6.64016	0.318738
$435$	−6.16205	−0.295448
$436$	1.68962	0.0809182
$437$	3.48027	0.166484
$438$	−16.3768	−0.782516
$439$	8.38319	0.400108	0.200054	−	0.979785i	$-0.435888\pi$
$439$	8.38319	0.400108	0.200054	+	0.979785i	$0.435888\pi$
$440$	0.798890	0.0380856
$441$	2.86911	0.136624
$442$	0	0
$443$	24.3139	1.15519	0.577595	−	0.816323i	$-0.303991\pi$
$443$	24.3139	1.15519	0.577595	+	0.816323i	$0.303991\pi$
$444$	0.621108	0.0294765
$445$	10.0276	0.475354
$446$	23.8754	1.13053
$447$	−8.62227	−0.407819
$448$	23.4085	1.10595
$449$	−31.0476	−1.46523	−0.732614	−	0.680644i	$-0.761700\pi$
$449$	−31.0476	−1.46523	−0.732614	+	0.680644i	$0.761700\pi$
$450$	8.66429	0.408439
$451$	−0.701503	−0.0330325
$452$	−8.51568	−0.400544
$453$	−1.83753	−0.0863347
$454$	14.9207	0.700262
$455$	0	0
$456$	−10.2658	−0.480742
$457$	−34.4206	−1.61013	−0.805065	−	0.593187i	$-0.797870\pi$
$457$	−34.4206	−1.61013	−0.805065	+	0.593187i	$0.797870\pi$
$458$	−17.3738	−0.811823
$459$	1.00000	0.0466760
$460$	−3.49782	−0.163087
$461$	2.12574	0.0990058	0.0495029	−	0.998774i	$-0.484236\pi$
$461$	2.12574	0.0990058	0.0495029	+	0.998774i	$0.484236\pi$
$462$	−0.234115	−0.0108920
$463$	23.7349	1.10305	0.551526	−	0.834158i	$-0.314046\pi$
$463$	23.7349	1.10305	0.551526	+	0.834158i	$0.314046\pi$
$464$	2.14303	0.0994876
$465$	7.47688	0.346732
$466$	−1.42438	−0.0659831
$467$	−18.2434	−0.844203	−0.422102	−	0.906549i	$-0.638707\pi$
$467$	−18.2434	−0.844203	−0.422102	+	0.906549i	$0.638707\pi$
$468$	0	0
$469$	3.37928	0.156041
$470$	35.0344	1.61602
$471$	0.805725	0.0371259
$472$	−26.5094	−1.22019
$473$	−0.728952	−0.0335172
$474$	7.88680	0.362253
$475$	−28.3842	−1.30236
$476$	2.92406	0.134024
$477$	7.68378	0.351816
$478$	−1.28521	−0.0587839
$479$	3.30367	0.150949	0.0754743	−	0.997148i	$-0.475953\pi$
$479$	3.30367	0.150949	0.0754743	+	0.997148i	$0.475953\pi$
$480$	17.3584	0.792300
$481$	0	0
$482$	29.6051	1.34847
$483$	3.22756	0.146859
$484$	10.2338	0.465171
$485$	39.7583	1.80533
$486$	−1.03403	−0.0469046
$487$	−1.74236	−0.0789537	−0.0394768	−	0.999220i	$-0.512569\pi$
$487$	−1.74236	−0.0789537	−0.0394768	+	0.999220i	$0.512569\pi$
$488$	−11.0877	−0.501918
$489$	−1.75326	−0.0792851
$490$	−10.8516	−0.490227
$491$	6.44662	0.290932	0.145466	−	0.989363i	$-0.453532\pi$
$491$	6.44662	0.290932	0.145466	+	0.989363i	$0.453532\pi$
$492$	−9.05985	−0.408450
$493$	1.68466	0.0758731
$494$	0	0
$495$	−0.263615	−0.0118486
$496$	−2.60030	−0.116757
$497$	−44.1295	−1.97948
$498$	7.73078	0.346424
$499$	−34.0925	−1.52619	−0.763095	−	0.646286i	$-0.776321\pi$
$499$	−34.0925	−1.52619	−0.763095	+	0.646286i	$0.776321\pi$
$500$	11.5045	0.514497
$501$	14.5208	0.648742
$502$	22.6821	1.01235
$503$	−21.3594	−0.952369	−0.476184	−	0.879345i	$-0.657981\pi$
$503$	−21.3594	−0.952369	−0.476184	+	0.879345i	$0.657981\pi$
$504$	−9.52041	−0.424073
$505$	−24.5276	−1.09146
$506$	−0.0765641	−0.00340369
$507$	0	0
$508$	7.73235	0.343067
$509$	−36.1270	−1.60130	−0.800652	−	0.599130i	$-0.795513\pi$
$509$	−36.1270	−1.60130	−0.800652	+	0.599130i	$0.795513\pi$
$510$	−3.78223	−0.167480
$511$	49.7549	2.20103
$512$	−13.7470	−0.607539
$513$	3.38749	0.149561
$514$	30.8678	1.36152
$515$	5.62086	0.247685
$516$	−9.41435	−0.414444
$517$	−0.667579	−0.0293601
$518$	2.16767	0.0952418
$519$	−5.90127	−0.259037
$520$	0	0
$521$	11.5341	0.505318	0.252659	−	0.967555i	$-0.418695\pi$
$521$	11.5341	0.505318	0.252659	+	0.967555i	$0.418695\pi$
$522$	−1.74199	−0.0762447
$523$	27.7501	1.21343	0.606714	−	0.794920i	$-0.292487\pi$
$523$	27.7501	1.21343	0.606714	+	0.794920i	$0.292487\pi$
$524$	10.7030	0.467562
$525$	−26.3232	−1.14884
$526$	−28.5554	−1.24508
$527$	−2.04412	−0.0890432
$528$	0.0916796	0.00398984
$529$	−21.9445	−0.954107
$530$	−29.0618	−1.26236
$531$	8.74749	0.379609
$532$	9.90521	0.429445
$533$	0	0
$534$	2.83476	0.122672
$535$	28.1730	1.21802
$536$	−3.25988	−0.140805
$537$	−23.8803	−1.03051
$538$	19.1968	0.827633
$539$	0.206778	0.00890654
$540$	−3.40456	−0.146509
$541$	−11.2590	−0.484063	−0.242031	−	0.970268i	$-0.577814\pi$
$541$	−11.2590	−0.484063	−0.242031	+	0.970268i	$0.577814\pi$
$542$	26.0367	1.11837
$543$	−14.1400	−0.606807
$544$	−4.74566	−0.203468
$545$	6.63982	0.284419
$546$	0	0
$547$	26.5727	1.13617	0.568084	−	0.822971i	$-0.307685\pi$
$547$	26.5727	1.13617	0.568084	+	0.822971i	$0.307685\pi$
$548$	14.2126	0.607134
$549$	3.65869	0.156149
$550$	0.624437	0.0266261
$551$	5.70675	0.243116
$552$	−3.11353	−0.132520
$553$	−23.9611	−1.01893
$554$	25.1141	1.06699
$555$	2.44081	0.103607
$556$	−9.19733	−0.390054
$557$	32.1579	1.36257	0.681287	−	0.732017i	$-0.261421\pi$
$557$	32.1579	1.36257	0.681287	+	0.732017i	$0.261421\pi$
$558$	2.11368	0.0894793
$559$	0	0
$560$	14.6174	0.617698
$561$	0.0720702	0.00304281
$562$	−22.2102	−0.936882
$563$	−12.5002	−0.526822	−0.263411	−	0.964684i	$-0.584847\pi$
$563$	−12.5002	−0.526822	−0.263411	+	0.964684i	$0.584847\pi$
$564$	−8.62173	−0.363040
$565$	−33.4646	−1.40787
$566$	21.7284	0.913312
$567$	3.14151	0.131931
$568$	42.5703	1.78621
$569$	35.8009	1.50085	0.750426	−	0.660954i	$-0.229848\pi$
$569$	35.8009	1.50085	0.750426	+	0.660954i	$0.229848\pi$
$570$	−12.8122	−0.536646
$571$	12.8584	0.538105	0.269053	−	0.963125i	$-0.413289\pi$
$571$	12.8584	0.538105	0.269053	+	0.963125i	$0.413289\pi$
$572$	0	0
$573$	−6.88248	−0.287520
$574$	−31.6189	−1.31975
$575$	−8.60866	−0.359006
$576$	7.45133	0.310472
$577$	−39.1643	−1.63043	−0.815215	−	0.579159i	$-0.803381\pi$
$577$	−39.1643	−1.63043	−0.815215	+	0.579159i	$0.803381\pi$
$578$	1.03403	0.0430100
$579$	−18.9838	−0.788938
$580$	−5.73552	−0.238154
$581$	−23.4871	−0.974408
$582$	11.2395	0.465892
$583$	0.553772	0.0229349
$584$	−47.9969	−1.98613
$585$	0	0
$586$	23.7689	0.981884
$587$	−22.5476	−0.930638	−0.465319	−	0.885143i	$-0.654060\pi$
$587$	−22.5476	−0.930638	−0.465319	+	0.885143i	$0.654060\pi$
$588$	2.67051	0.110130
$589$	−6.92442	−0.285316
$590$	−33.0850	−1.36209
$591$	−6.09875	−0.250869
$592$	−0.848861	−0.0348880
$593$	11.7905	0.484179	0.242089	−	0.970254i	$-0.422167\pi$
$593$	11.7905	0.484179	0.242089	+	0.970254i	$0.422167\pi$
$594$	−0.0745228	−0.00305771
$595$	11.4909	0.471080
$596$	−8.02544	−0.328735
$597$	−14.2986	−0.585202
$598$	0	0
$599$	−0.249561	−0.0101968	−0.00509840	−	0.999987i	$-0.501623\pi$
$599$	−0.249561	−0.0101968	−0.00509840	+	0.999987i	$0.501623\pi$
$600$	25.3931	1.03667
$601$	36.6027	1.49305	0.746527	−	0.665355i	$-0.231720\pi$
$601$	36.6027	1.49305	0.746527	+	0.665355i	$0.231720\pi$
$602$	−32.8561	−1.33911
$603$	1.07568	0.0438052
$604$	−1.71034	−0.0695926
$605$	40.2163	1.63502
$606$	−6.93384	−0.281668
$607$	35.0344	1.42200	0.711001	−	0.703191i	$-0.248242\pi$
$607$	35.0344	1.42200	0.711001	+	0.703191i	$0.248242\pi$
$608$	−16.0758	−0.651962
$609$	5.29237	0.214458
$610$	−13.8380	−0.560285
$611$	0	0
$612$	0.930781	0.0376246
$613$	−27.7816	−1.12209	−0.561045	−	0.827785i	$-0.689601\pi$
$613$	−27.7816	−1.12209	−0.561045	+	0.827785i	$0.689601\pi$
$614$	29.8970	1.20655
$615$	−35.6031	−1.43566
$616$	−0.686138	−0.0276453
$617$	−29.1803	−1.17476	−0.587378	−	0.809313i	$-0.699840\pi$
$617$	−29.1803	−1.17476	−0.587378	+	0.809313i	$0.699840\pi$
$618$	1.58899	0.0639187
$619$	−23.4477	−0.942442	−0.471221	−	0.882015i	$-0.656187\pi$
$619$	−23.4477	−0.942442	−0.471221	+	0.882015i	$0.656187\pi$
$620$	6.95933	0.279493
$621$	1.02739	0.0412278
$622$	−19.2145	−0.770430
$623$	−8.61235	−0.345047
$624$	0	0
$625$	3.31431	0.132572
$626$	−17.5165	−0.700098
$627$	0.244137	0.00974989
$628$	0.749954	0.0299264
$629$	−0.667298	−0.0266069
$630$	−11.8819	−0.473387
$631$	−31.4859	−1.25343	−0.626716	−	0.779248i	$-0.715601\pi$
$631$	−31.4859	−1.25343	−0.626716	+	0.779248i	$0.715601\pi$
$632$	23.1145	0.919445
$633$	3.70042	0.147078
$634$	11.1099	0.441231
$635$	30.3863	1.20584
$636$	7.15192	0.283592
$637$	0	0
$638$	−0.125545	−0.00497039
$639$	−14.0472	−0.555699
$640$	6.53425	0.258289
$641$	18.0307	0.712168	0.356084	−	0.934454i	$-0.384112\pi$
$641$	18.0307	0.712168	0.356084	+	0.934454i	$0.384112\pi$
$642$	7.96438	0.314329
$643$	−27.6036	−1.08858	−0.544290	−	0.838897i	$-0.683201\pi$
$643$	−27.6036	−1.08858	−0.544290	+	0.838897i	$0.683201\pi$
$644$	3.00415	0.118380
$645$	−36.9962	−1.45672
$646$	3.50276	0.137814
$647$	−25.5072	−1.00279	−0.501396	−	0.865218i	$-0.667180\pi$
$647$	−25.5072	−1.00279	−0.501396	+	0.865218i	$0.667180\pi$
$648$	−3.03052	−0.119050
$649$	0.630434	0.0247467
$650$	0	0
$651$	−6.42163	−0.251683
$652$	−1.63190	−0.0639101
$653$	−10.9995	−0.430443	−0.215221	−	0.976565i	$-0.569047\pi$
$653$	−10.9995	−0.430443	−0.215221	+	0.976565i	$0.569047\pi$
$654$	1.87705	0.0733984
$655$	42.0602	1.64343
$656$	12.3820	0.483436
$657$	15.8379	0.617894
$658$	−30.0898	−1.17302
$659$	31.1703	1.21422	0.607111	−	0.794617i	$-0.292329\pi$
$659$	31.1703	1.21422	0.607111	+	0.794617i	$0.292329\pi$
$660$	−0.245368	−0.00955092
$661$	−23.8569	−0.927925	−0.463963	−	0.885855i	$-0.653573\pi$
$661$	−23.8569	−0.927925	−0.463963	+	0.885855i	$0.653573\pi$
$662$	27.2735	1.06002
$663$	0	0
$664$	22.6572	0.879270
$665$	38.9252	1.50945
$666$	0.690007	0.0267372
$667$	1.73080	0.0670169
$668$	13.5157	0.522937
$669$	−23.0896	−0.892696
$670$	−4.06848	−0.157179
$671$	0.263683	0.0101794
$672$	−14.9086	−0.575110
$673$	12.2514	0.472256	0.236128	−	0.971722i	$-0.424122\pi$
$673$	12.2514	0.472256	0.236128	+	0.971722i	$0.424122\pi$
$674$	23.1719	0.892549
$675$	−8.37914	−0.322513
$676$	0	0
$677$	25.3461	0.974130	0.487065	−	0.873366i	$-0.338067\pi$
$677$	25.3461	0.974130	0.487065	+	0.873366i	$0.338067\pi$
$678$	−9.46031	−0.363321
$679$	−34.1470	−1.31044
$680$	−11.0849	−0.425086
$681$	−14.4296	−0.552945
$682$	0.152333	0.00583315
$683$	−26.1213	−0.999504	−0.499752	−	0.866169i	$-0.666576\pi$
$683$	−26.1213	−0.999504	−0.499752	+	0.866169i	$0.666576\pi$
$684$	3.15301	0.120558
$685$	55.8523	2.13401
$686$	−13.4189	−0.512335
$687$	16.8020	0.641036
$688$	12.8665	0.490530
$689$	0	0
$690$	−3.88583	−0.147931
$691$	47.0062	1.78820	0.894101	−	0.447866i	$-0.147816\pi$
$691$	47.0062	1.78820	0.894101	+	0.447866i	$0.147816\pi$
$692$	−5.49279	−0.208805
$693$	0.226410	0.00860059
$694$	1.24171	0.0471347
$695$	−36.1434	−1.37100
$696$	−5.10538	−0.193519
$697$	9.73361	0.368687
$698$	−13.5978	−0.514685
$699$	1.37750	0.0521019
$700$	−24.5011	−0.926055
$701$	−4.23609	−0.159995	−0.0799974	−	0.996795i	$-0.525491\pi$
$701$	−4.23609	−0.159995	−0.0799974	+	0.996795i	$0.525491\pi$
$702$	0	0
$703$	−2.26046	−0.0852550
$704$	0.537019	0.0202397
$705$	−33.8814	−1.27605
$706$	18.0471	0.679211
$707$	21.0659	0.792263
$708$	8.14199	0.305995
$709$	−1.61891	−0.0607994	−0.0303997	−	0.999538i	$-0.509678\pi$
$709$	−1.61891	−0.0607994	−0.0303997	+	0.999538i	$0.509678\pi$
$710$	53.1297	1.99392
$711$	−7.62724	−0.286044
$712$	8.30806	0.311358
$713$	−2.10011	−0.0786497
$714$	3.24842	0.121569
$715$	0	0
$716$	−22.2273	−0.830674
$717$	1.24291	0.0464173
$718$	−33.5531	−1.25219
$719$	25.3928	0.946993	0.473496	−	0.880796i	$-0.342992\pi$
$719$	25.3928	0.946993	0.473496	+	0.880796i	$0.342992\pi$
$720$	4.65298	0.173406
$721$	−4.82756	−0.179788
$722$	−7.78102	−0.289580
$723$	−28.6307	−1.06479
$724$	−13.1613	−0.489135
$725$	−14.1160	−0.524254
$726$	11.3690	0.421942
$727$	−30.6850	−1.13804	−0.569022	−	0.822322i	$-0.692678\pi$
$727$	−30.6850	−1.13804	−0.569022	+	0.822322i	$0.692678\pi$
$728$	0	0
$729$	1.00000	0.0370370
$730$	−59.9024	−2.21709
$731$	10.1145	0.374097
$732$	3.40544	0.125869
$733$	−40.8963	−1.51054	−0.755270	−	0.655413i	$-0.772494\pi$
$733$	−40.8963	−1.51054	−0.755270	+	0.655413i	$0.772494\pi$
$734$	−16.2957	−0.601485
$735$	10.4945	0.387095
$736$	−4.87565	−0.179719
$737$	0.0775248	0.00285566
$738$	−10.0648	−0.370492
$739$	22.5304	0.828793	0.414396	−	0.910097i	$-0.363993\pi$
$739$	22.5304	0.828793	0.414396	+	0.910097i	$0.363993\pi$
$740$	2.27186	0.0835152
$741$	0	0
$742$	24.9602	0.916317
$743$	30.6305	1.12372	0.561862	−	0.827231i	$-0.310085\pi$
$743$	30.6305	1.12372	0.561862	+	0.827231i	$0.310085\pi$
$744$	6.19474	0.227110
$745$	−31.5381	−1.15547
$746$	34.1147	1.24903
$747$	−7.47635	−0.273545
$748$	0.0670816	0.00245274
$749$	−24.1968	−0.884132
$750$	12.7807	0.466685
$751$	−5.27739	−0.192575	−0.0962873	−	0.995354i	$-0.530697\pi$
$751$	−5.27739	−0.192575	−0.0962873	+	0.995354i	$0.530697\pi$
$752$	11.7832	0.429690
$753$	−21.9356	−0.799379
$754$	0	0
$755$	−6.72122	−0.244610
$756$	2.92406	0.106347
$757$	−22.6067	−0.821654	−0.410827	−	0.911713i	$-0.634760\pi$
$757$	−22.6067	−0.821654	−0.410827	+	0.911713i	$0.634760\pi$
$758$	−21.3682	−0.776127
$759$	0.0740443	0.00268764
$760$	−37.5499	−1.36208
$761$	−7.68182	−0.278466	−0.139233	−	0.990260i	$-0.544464\pi$
$761$	−7.68182	−0.278466	−0.139233	+	0.990260i	$0.544464\pi$
$762$	8.59009	0.311186
$763$	−5.70271	−0.206452
$764$	−6.40608	−0.231764
$765$	3.65775	0.132246
$766$	−26.7047	−0.964879
$767$	0	0
$768$	16.7499	0.604409
$769$	−2.88923	−0.104188	−0.0520941	−	0.998642i	$-0.516590\pi$
$769$	−2.88923	−0.104188	−0.0520941	+	0.998642i	$0.516590\pi$
$770$	−0.856333	−0.0308601
$771$	−29.8519	−1.07509
$772$	−17.6697	−0.635947
$773$	−9.46193	−0.340322	−0.170161	−	0.985416i	$-0.554429\pi$
$773$	−9.46193	−0.340322	−0.170161	+	0.985416i	$0.554429\pi$
$774$	−10.4587	−0.375929
$775$	17.1280	0.615254
$776$	32.9405	1.18249
$777$	−2.09633	−0.0752053
$778$	−20.3049	−0.727966
$779$	32.9725	1.18136
$780$	0	0
$781$	−1.01238	−0.0362260
$782$	1.06235	0.0379897
$783$	1.68466	0.0602047
$784$	−3.64976	−0.130349
$785$	2.94714	0.105188
$786$	11.8903	0.424111
$787$	−36.7901	−1.31142	−0.655712	−	0.755011i	$-0.727631\pi$
$787$	−36.7901	−1.31142	−0.655712	+	0.755011i	$0.727631\pi$
$788$	−5.67660	−0.202221
$789$	27.6156	0.983143
$790$	28.8480	1.02636
$791$	28.7416	1.02193
$792$	−0.218410	−0.00776086
$793$	0	0
$794$	8.97136	0.318382
$795$	28.1054	0.996795
$796$	−13.3088	−0.471719
$797$	3.20568	0.113551	0.0567756	−	0.998387i	$-0.481918\pi$
$797$	3.20568	0.113551	0.0567756	+	0.998387i	$0.481918\pi$
$798$	11.0040	0.389537
$799$	9.26290	0.327698
$800$	39.7645	1.40589
$801$	−2.74146	−0.0968649
$802$	−36.1673	−1.27711
$803$	1.14144	0.0402805
$804$	1.00123	0.0353105
$805$	11.8056	0.416094
$806$	0	0
$807$	−18.5650	−0.653519
$808$	−20.3216	−0.714910
$809$	31.5768	1.11018	0.555091	−	0.831790i	$-0.312684\pi$
$809$	31.5768	1.11018	0.555091	+	0.831790i	$0.312684\pi$
$810$	−3.78223	−0.132894
$811$	20.4779	0.719075	0.359538	−	0.933131i	$-0.382934\pi$
$811$	20.4779	0.719075	0.359538	+	0.933131i	$0.382934\pi$
$812$	4.92604	0.172870
$813$	−25.1798	−0.883093
$814$	0.0497289	0.00174300
$815$	−6.41298	−0.224637
$816$	−1.27209	−0.0445319
$817$	34.2626	1.19870
$818$	27.2794	0.953802
$819$	0	0
$820$	−33.1387	−1.15725
$821$	32.4333	1.13193	0.565965	−	0.824430i	$-0.308504\pi$
$821$	32.4333	1.13193	0.565965	+	0.824430i	$0.308504\pi$
$822$	15.7892	0.550712
$823$	34.7299	1.21061	0.605304	−	0.795995i	$-0.293052\pi$
$823$	34.7299	1.21061	0.605304	+	0.795995i	$0.293052\pi$
$824$	4.65699	0.162234
$825$	−0.603887	−0.0210246
$826$	28.4155	0.988703
$827$	15.8057	0.549619	0.274809	−	0.961499i	$-0.411385\pi$
$827$	15.8057	0.549619	0.274809	+	0.961499i	$0.411385\pi$
$828$	0.956276	0.0332329
$829$	−12.8129	−0.445012	−0.222506	−	0.974931i	$-0.571424\pi$
$829$	−12.8129	−0.445012	−0.222506	+	0.974931i	$0.571424\pi$
$830$	28.2773	0.981518
$831$	−24.2875	−0.842525
$832$	0	0
$833$	−2.86911	−0.0994089
$834$	−10.2176	−0.353806
$835$	53.1135	1.83807
$836$	0.227238	0.00785919
$837$	−2.04412	−0.0706551
$838$	19.4944	0.673424
$839$	−11.6570	−0.402443	−0.201221	−	0.979546i	$-0.564491\pi$
$839$	−11.6570	−0.402443	−0.201221	+	0.979546i	$0.564491\pi$
$840$	−34.8233	−1.20152
$841$	−26.1619	−0.902136
$842$	25.9784	0.895274
$843$	21.4793	0.739786
$844$	3.44427	0.118557
$845$	0	0
$846$	−9.57812	−0.329303
$847$	−34.5403	−1.18682
$848$	−9.77444	−0.335656
$849$	−21.0133	−0.721174
$850$	−8.66429	−0.297183
$851$	−0.685576	−0.0235013
$852$	−13.0749	−0.447937
$853$	45.2472	1.54923	0.774617	−	0.632431i	$-0.217943\pi$
$853$	45.2472	1.54923	0.774617	+	0.632431i	$0.217943\pi$
$854$	11.8850	0.406696
$855$	12.3906	0.423749
$856$	23.3419	0.797808
$857$	2.58799	0.0884040	0.0442020	−	0.999023i	$-0.485925\pi$
$857$	2.58799	0.0884040	0.0442020	+	0.999023i	$0.485925\pi$
$858$	0	0
$859$	18.9943	0.648078	0.324039	−	0.946044i	$-0.394959\pi$
$859$	18.9943	0.648078	0.324039	+	0.946044i	$0.394959\pi$
$860$	−34.4353	−1.17424
$861$	30.5783	1.04210
$862$	34.5168	1.17565
$863$	6.14208	0.209079	0.104539	−	0.994521i	$-0.466663\pi$
$863$	6.14208	0.209079	0.104539	+	0.994521i	$0.466663\pi$
$864$	−4.74566	−0.161451
$865$	−21.5854	−0.733925
$866$	−2.67591	−0.0909312
$867$	−1.00000	−0.0339618
$868$	−5.97713	−0.202877
$869$	−0.549697	−0.0186472
$870$	−6.37175	−0.216023
$871$	0	0
$872$	5.50122	0.186295
$873$	−10.8696	−0.367880
$874$	3.59871	0.121728
$875$	−38.8293	−1.31267
$876$	14.7416	0.498072
$877$	−3.41633	−0.115361	−0.0576806	−	0.998335i	$-0.518370\pi$
$877$	−3.41633	−0.115361	−0.0576806	+	0.998335i	$0.518370\pi$
$878$	8.66847	0.292547
$879$	−22.9866	−0.775320
$880$	0.335341	0.0113043
$881$	10.6112	0.357499	0.178749	−	0.983895i	$-0.442795\pi$
$881$	10.6112	0.357499	0.178749	+	0.983895i	$0.442795\pi$
$882$	2.96675	0.0998957
$883$	−13.4501	−0.452630	−0.226315	−	0.974054i	$-0.572668\pi$
$883$	−13.4501	−0.452630	−0.226315	+	0.974054i	$0.572668\pi$
$884$	0	0
$885$	31.9961	1.07554
$886$	25.1414	0.844641
$887$	43.1205	1.44784	0.723922	−	0.689882i	$-0.242338\pi$
$887$	43.1205	1.44784	0.723922	+	0.689882i	$0.242338\pi$
$888$	2.02226	0.0678625
$889$	−26.0978	−0.875291
$890$	10.3688	0.347564
$891$	0.0720702	0.00241444
$892$	−21.4914	−0.719584
$893$	31.3779	1.05002
$894$	−8.91569	−0.298185
$895$	−87.3482	−2.91973
$896$	−5.61204	−0.187485
$897$	0	0
$898$	−32.1042	−1.07133
$899$	−3.44364	−0.114852
$900$	−7.79914	−0.259971
$901$	−7.68378	−0.255984
$902$	−0.725376	−0.0241524
$903$	31.7747	1.05740
$904$	−27.7261	−0.922156
$905$	−51.7207	−1.71926
$906$	−1.90006	−0.0631253
$907$	−29.9620	−0.994871	−0.497436	−	0.867501i	$-0.665725\pi$
$907$	−29.9620	−0.994871	−0.497436	+	0.867501i	$0.665725\pi$
$908$	−13.4308	−0.445718
$909$	6.70564	0.222412
$910$	0	0
$911$	1.32075	0.0437584	0.0218792	−	0.999761i	$-0.493035\pi$
$911$	1.32075	0.0437584	0.0218792	+	0.999761i	$0.493035\pi$
$912$	−4.30918	−0.142691
$913$	−0.538822	−0.0178324
$914$	−35.5920	−1.17728
$915$	13.3826	0.442415
$916$	15.6390	0.516726
$917$	−36.1241	−1.19292
$918$	1.03403	0.0341281
$919$	57.9024	1.91002	0.955012	−	0.296566i	$-0.0958414\pi$
$919$	57.9024	1.91002	0.955012	+	0.296566i	$0.0958414\pi$
$920$	−11.3885	−0.375468
$921$	−28.9131	−0.952718
$922$	2.19808	0.0723901
$923$	0	0
$924$	0.210738	0.00693276
$925$	5.59139	0.183844
$926$	24.5426	0.806519
$927$	−1.53670	−0.0504718
$928$	−7.99480	−0.262442
$929$	13.5832	0.445649	0.222825	−	0.974859i	$-0.428472\pi$
$929$	13.5832	0.445649	0.222825	+	0.974859i	$0.428472\pi$
$930$	7.73132	0.253520
$931$	−9.71908	−0.318530
$932$	1.28215	0.0419983
$933$	18.5821	0.608351
$934$	−18.8642	−0.617256
$935$	0.263615	0.00862113
$936$	0	0
$937$	45.2285	1.47755	0.738775	−	0.673953i	$-0.235405\pi$
$937$	45.2285	1.47755	0.738775	+	0.673953i	$0.235405\pi$
$938$	3.49428	0.114092
$939$	16.9400	0.552815
$940$	−31.5361	−1.02860
$941$	−47.2937	−1.54173	−0.770866	−	0.636998i	$-0.780176\pi$
$941$	−47.2937	−1.54173	−0.770866	+	0.636998i	$0.780176\pi$
$942$	0.833145	0.0271453
$943$	10.0002	0.325652
$944$	−11.1276	−0.362171
$945$	11.4909	0.373798
$946$	−0.753759	−0.0245068
$947$	−37.3693	−1.21434	−0.607169	−	0.794573i	$-0.707695\pi$
$947$	−37.3693	−1.21434	−0.607169	+	0.794573i	$0.707695\pi$
$948$	−7.09929	−0.230574
$949$	0	0
$950$	−29.3502	−0.952245
$951$	−10.7443	−0.348407
$952$	9.52041	0.308558
$953$	47.9564	1.55346	0.776731	−	0.629833i	$-0.216877\pi$
$953$	47.9564	1.55346	0.776731	+	0.629833i	$0.216877\pi$
$954$	7.94527	0.257238
$955$	−25.1744	−0.814625
$956$	1.15688	0.0374160
$957$	0.121414	0.00392474
$958$	3.41610	0.110369
$959$	−47.9696	−1.54902
$960$	27.2551	0.879655
$961$	−26.8216	−0.865212
$962$	0	0
$963$	−7.70227	−0.248202
$964$	−26.6489	−0.858304
$965$	−69.4379	−2.23528
$966$	3.33740	0.107379
$967$	−53.1149	−1.70806	−0.854030	−	0.520224i	$-0.825848\pi$
$967$	−53.1149	−1.70806	−0.854030	+	0.520224i	$0.825848\pi$
$968$	33.3199	1.07094
$969$	−3.38749	−0.108822
$970$	41.1113	1.32000
$971$	22.3963	0.718730	0.359365	−	0.933197i	$-0.382993\pi$
$971$	22.3963	0.718730	0.359365	+	0.933197i	$0.382993\pi$
$972$	0.930781	0.0298548
$973$	31.0423	0.995169
$974$	−1.80165	−0.0577286
$975$	0	0
$976$	−4.65418	−0.148977
$977$	−12.9932	−0.415690	−0.207845	−	0.978162i	$-0.566645\pi$
$977$	−12.9932	−0.415690	−0.207845	+	0.978162i	$0.566645\pi$
$978$	−1.81292	−0.0579709
$979$	−0.197578	−0.00631462
$980$	9.76808	0.312030
$981$	−1.81527	−0.0579572
$982$	6.66600	0.212721
$983$	−7.27708	−0.232103	−0.116051	−	0.993243i	$-0.537024\pi$
$983$	−7.27708	−0.232103	−0.116051	+	0.993243i	$0.537024\pi$
$984$	−29.4979	−0.940358
$985$	−22.3077	−0.710783
$986$	1.74199	0.0554761
$987$	29.0995	0.926248
$988$	0	0
$989$	10.3915	0.330431
$990$	−0.272586	−0.00866335
$991$	51.1442	1.62465	0.812324	−	0.583206i	$-0.198202\pi$
$991$	51.1442	1.62465	0.812324	+	0.583206i	$0.198202\pi$
$992$	9.70069	0.307997
$993$	−26.3759	−0.837015
$994$	−45.6312	−1.44733
$995$	−52.3006	−1.65804
$996$	−6.95884	−0.220499
$997$	30.4812	0.965350	0.482675	−	0.875799i	$-0.339665\pi$
$997$	30.4812	0.965350	0.482675	+	0.875799i	$0.339665\pi$
$998$	−35.2527	−1.11590
$999$	−0.667298	−0.0211124

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8619.2.a.br.1.14		22
13.6	odd	12		663.2.z.f.205.5	✓	22
13.11	odd	12		663.2.z.f.511.5	yes	22
13.12	even	2	inner	8619.2.a.br.1.9		22

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
663.2.z.f.205.5	✓	22	13.6	odd	12
663.2.z.f.511.5	yes	22	13.11	odd	12
8619.2.a.br.1.9		22	13.12	even	2	inner
8619.2.a.br.1.14		22	1.1	even	1	trivial

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:	\( N \)	\(=\)	\( 8619 = 3 \cdot 13^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8619.a (trivial)

\(f(q)\)	\(=\)	\(q+1.03403 q^{2} -1.00000 q^{3} -0.930781 q^{4} -3.65775 q^{5} -1.03403 q^{6} +3.14151 q^{7} -3.03052 q^{8} +1.00000 q^{9} +O(q^{10})\)	\(q+1.03403 q^{2} -1.00000 q^{3} -0.930781 q^{4} -3.65775 q^{5} -1.03403 q^{6} +3.14151 q^{7} -3.03052 q^{8} +1.00000 q^{9} -3.78223 q^{10} +0.0720702 q^{11} +0.930781 q^{12} +3.24842 q^{14} +3.65775 q^{15} -1.27209 q^{16} -1.00000 q^{17} +1.03403 q^{18} -3.38749 q^{19} +3.40456 q^{20} -3.14151 q^{21} +0.0745228 q^{22} -1.02739 q^{23} +3.03052 q^{24} +8.37914 q^{25} -1.00000 q^{27} -2.92406 q^{28} -1.68466 q^{29} +3.78223 q^{30} +2.04412 q^{31} +4.74566 q^{32} -0.0720702 q^{33} -1.03403 q^{34} -11.4909 q^{35} -0.930781 q^{36} +0.667298 q^{37} -3.50276 q^{38} +11.0849 q^{40} -9.73361 q^{41} -3.24842 q^{42} -10.1145 q^{43} -0.0670816 q^{44} -3.65775 q^{45} -1.06235 q^{46} -9.26290 q^{47} +1.27209 q^{48} +2.86911 q^{49} +8.66429 q^{50} +1.00000 q^{51} +7.68378 q^{53} -1.03403 q^{54} -0.263615 q^{55} -9.52041 q^{56} +3.38749 q^{57} -1.74199 q^{58} +8.74749 q^{59} -3.40456 q^{60} +3.65869 q^{61} +2.11368 q^{62} +3.14151 q^{63} +7.45133 q^{64} -0.0745228 q^{66} +1.07568 q^{67} +0.930781 q^{68} +1.02739 q^{69} -11.8819 q^{70} -14.0472 q^{71} -3.03052 q^{72} +15.8379 q^{73} +0.690007 q^{74} -8.37914 q^{75} +3.15301 q^{76} +0.226410 q^{77} -7.62724 q^{79} +4.65298 q^{80} +1.00000 q^{81} -10.0648 q^{82} -7.47635 q^{83} +2.92406 q^{84} +3.65775 q^{85} -10.4587 q^{86} +1.68466 q^{87} -0.218410 q^{88} -2.74146 q^{89} -3.78223 q^{90} +0.956276 q^{92} -2.04412 q^{93} -9.57812 q^{94} +12.3906 q^{95} -4.74566 q^{96} -10.8696 q^{97} +2.96675 q^{98} +0.0720702 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 22 q - 22 q^{3} + 32 q^{4} + 22 q^{9}+O(q^{10}) \) 22 * q - 22 * q^3 + 32 * q^4 + 22 * q^9	\( 22 q - 22 q^{3} + 32 q^{4} + 22 q^{9} + 26 q^{10} - 32 q^{12} + 18 q^{14} + 36 q^{16} - 22 q^{17} + 22 q^{22} - 20 q^{23} + 46 q^{25} - 22 q^{27} + 6 q^{29} - 26 q^{30} - 14 q^{35} + 32 q^{36} + 38 q^{38} + 108 q^{40} - 18 q^{42} - 30 q^{43} - 36 q^{48} + 68 q^{49} + 22 q^{51} - 10 q^{53} - 18 q^{55} - 30 q^{56} + 92 q^{61} + 52 q^{62} + 96 q^{64} - 22 q^{66} - 32 q^{68} + 20 q^{69} + 42 q^{74} - 46 q^{75} + 28 q^{77} + 68 q^{79} + 22 q^{81} + 6 q^{82} - 6 q^{87} + 134 q^{88} + 26 q^{90} + 40 q^{92} - 94 q^{94} - 72 q^{95}+O(q^{100}) \) 22 * q - 22 * q^3 + 32 * q^4 + 22 * q^9 + 26 * q^10 - 32 * q^12 + 18 * q^14 + 36 * q^16 - 22 * q^17 + 22 * q^22 - 20 * q^23 + 46 * q^25 - 22 * q^27 + 6 * q^29 - 26 * q^30 - 14 * q^35 + 32 * q^36 + 38 * q^38 + 108 * q^40 - 18 * q^42 - 30 * q^43 - 36 * q^48 + 68 * q^49 + 22 * q^51 - 10 * q^53 - 18 * q^55 - 30 * q^56 + 92 * q^61 + 52 * q^62 + 96 * q^64 - 22 * q^66 - 32 * q^68 + 20 * q^69 + 42 * q^74 - 46 * q^75 + 28 * q^77 + 68 * q^79 + 22 * q^81 + 6 * q^82 - 6 * q^87 + 134 * q^88 + 26 * q^90 + 40 * q^92 - 94 * q^94 - 72 * q^95

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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