LMFDB - Embedded newform 125.2.a.c.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [125,2,Mod(1,125)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(125, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("125.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 125 = 5^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	125.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:	$0.998130025266$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.4400.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 7x^{2} + 11 $ x^4 - 7*x^2 + 11
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$1.54336$ of defining polynomial
Character	$\chi$	$=$	125.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.49721 q^{2} -1.54336 q^{3} +4.23607 q^{4} -3.85410 q^{6} -0.953850 q^{7} +5.58394 q^{8} -0.618034 q^{9} +O(q^{10})$	\(q+2.49721 q^{2} -1.54336 q^{3} +4.23607 q^{4} -3.85410 q^{6} -0.953850 q^{7} +5.58394 q^{8} -0.618034 q^{9} +2.00000 q^{11} -6.53779 q^{12} -4.99442 q^{13} -2.38197 q^{14} +5.47214 q^{16} -3.08672 q^{17} -1.54336 q^{18} +2.76393 q^{19} +1.47214 q^{21} +4.99442 q^{22} +4.04057 q^{23} -8.61803 q^{24} -12.4721 q^{26} +5.58394 q^{27} -4.04057 q^{28} -5.85410 q^{29} +2.00000 q^{31} +2.49721 q^{32} -3.08672 q^{33} -7.70820 q^{34} -2.61803 q^{36} +8.08115 q^{37} +6.90212 q^{38} +7.70820 q^{39} +5.09017 q^{41} +3.67624 q^{42} +9.62451 q^{43} +8.47214 q^{44} +10.0902 q^{46} -6.53779 q^{47} -8.44549 q^{48} -6.09017 q^{49} +4.76393 q^{51} -21.1567 q^{52} +6.17345 q^{53} +13.9443 q^{54} -5.32624 q^{56} -4.26575 q^{57} -14.6189 q^{58} +4.47214 q^{59} -6.09017 q^{61} +4.99442 q^{62} +0.589512 q^{63} -4.70820 q^{64} -7.70820 q^{66} -3.08672 q^{67} -13.0756 q^{68} -6.23607 q^{69} -14.1803 q^{71} -3.45106 q^{72} +8.80982 q^{73} +20.1803 q^{74} +11.7082 q^{76} -1.90770 q^{77} +19.2490 q^{78} +7.23607 q^{79} -6.76393 q^{81} +12.7112 q^{82} -7.12730 q^{83} +6.23607 q^{84} +24.0344 q^{86} +9.03500 q^{87} +11.1679 q^{88} -6.38197 q^{89} +4.76393 q^{91} +17.1161 q^{92} -3.08672 q^{93} -16.3262 q^{94} -3.85410 q^{96} +1.17902 q^{97} -15.2084 q^{98} -1.23607 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + 8 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10}) $ 4 * q + 8 * q^4 - 2 * q^6 + 2 * q^9	$ 4 q + 8 q^{4} - 2 q^{6} + 2 q^{9} + 8 q^{11} - 14 q^{14} + 4 q^{16} + 20 q^{19} - 12 q^{21} - 30 q^{24} - 32 q^{26} - 10 q^{29} + 8 q^{31} - 4 q^{34} - 6 q^{36} + 4 q^{39} - 2 q^{41} + 16 q^{44} + 18 q^{46} - 2 q^{49} + 28 q^{51} + 20 q^{54} + 10 q^{56} - 2 q^{61} + 8 q^{64} - 4 q^{66} - 16 q^{69} - 12 q^{71} + 36 q^{74} + 20 q^{76} + 20 q^{79} - 36 q^{81} + 16 q^{84} + 38 q^{86} - 30 q^{89} + 28 q^{91} - 34 q^{94} - 2 q^{96} + 4 q^{99}+O(q^{100}) $ 4 * q + 8 * q^4 - 2 * q^6 + 2 * q^9 + 8 * q^11 - 14 * q^14 + 4 * q^16 + 20 * q^19 - 12 * q^21 - 30 * q^24 - 32 * q^26 - 10 * q^29 + 8 * q^31 - 4 * q^34 - 6 * q^36 + 4 * q^39 - 2 * q^41 + 16 * q^44 + 18 * q^46 - 2 * q^49 + 28 * q^51 + 20 * q^54 + 10 * q^56 - 2 * q^61 + 8 * q^64 - 4 * q^66 - 16 * q^69 - 12 * q^71 + 36 * q^74 + 20 * q^76 + 20 * q^79 - 36 * q^81 + 16 * q^84 + 38 * q^86 - 30 * q^89 + 28 * q^91 - 34 * q^94 - 2 * q^96 + 4 * q^99

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 $

$2$	2.49721	1.76580	0.882898	−	0.469565i	$-0.155589\pi$
$2$	2.49721	1.76580	0.882898	+	0.469565i	$0.155589\pi$
$3$	−1.54336	−0.891060	−0.445530	−	0.895267i	$-0.646985\pi$
$3$	−1.54336	−0.891060	−0.445530	+	0.895267i	$0.646985\pi$
$4$	4.23607	2.11803
$5$	0	0
$5$	0	0
$6$	−3.85410	−1.57343
$7$	−0.953850	−0.360521	−0.180261	−	0.983619i	$-0.557694\pi$
$7$	−0.953850	−0.360521	−0.180261	+	0.983619i	$0.557694\pi$
$8$	5.58394	1.97422
$9$	−0.618034	−0.206011
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	−6.53779	−1.88730
$13$	−4.99442	−1.38520	−0.692602	−	0.721320i	$-0.743536\pi$
$13$	−4.99442	−1.38520	−0.692602	+	0.721320i	$0.743536\pi$
$14$	−2.38197	−0.636607
$15$	0	0
$16$	5.47214	1.36803
$17$	−3.08672	−0.748640	−0.374320	−	0.927299i	$-0.622124\pi$
$17$	−3.08672	−0.748640	−0.374320	+	0.927299i	$0.622124\pi$
$18$	−1.54336	−0.363774
$19$	2.76393	0.634089	0.317045	−	0.948411i	$-0.397309\pi$
$19$	2.76393	0.634089	0.317045	+	0.948411i	$0.397309\pi$
$20$	0	0
$21$	1.47214	0.321246
$22$	4.99442	1.06481
$23$	4.04057	0.842518	0.421259	−	0.906940i	$-0.361588\pi$
$23$	4.04057	0.842518	0.421259	+	0.906940i	$0.361588\pi$
$24$	−8.61803	−1.75915
$25$	0	0
$26$	−12.4721	−2.44599
$27$	5.58394	1.07463
$28$	−4.04057	−0.763597
$29$	−5.85410	−1.08708	−0.543540	−	0.839383i	$-0.682916\pi$
$29$	−5.85410	−1.08708	−0.543540	+	0.839383i	$0.682916\pi$
$30$	0	0
$31$	2.00000	0.359211	0.179605	−	0.983739i	$-0.442518\pi$
$31$	2.00000	0.359211	0.179605	+	0.983739i	$0.442518\pi$
$32$	2.49721	0.441449
$33$	−3.08672	−0.537330
$34$	−7.70820	−1.32195
$35$	0	0
$36$	−2.61803	−0.436339
$37$	8.08115	1.32853	0.664266	−	0.747496i	$-0.268744\pi$
$37$	8.08115	1.32853	0.664266	+	0.747496i	$0.268744\pi$
$38$	6.90212	1.11967
$39$	7.70820	1.23430
$40$	0	0
$41$	5.09017	0.794951	0.397475	−	0.917613i	$-0.369886\pi$
$41$	5.09017	0.794951	0.397475	+	0.917613i	$0.369886\pi$
$42$	3.67624	0.567255
$43$	9.62451	1.46772	0.733862	−	0.679299i	$-0.237716\pi$
$43$	9.62451	1.46772	0.733862	+	0.679299i	$0.237716\pi$
$44$	8.47214	1.27722
$45$	0	0
$46$	10.0902	1.48771
$47$	−6.53779	−0.953634	−0.476817	−	0.879003i	$-0.658210\pi$
$47$	−6.53779	−0.953634	−0.476817	+	0.879003i	$0.658210\pi$
$48$	−8.44549	−1.21900
$49$	−6.09017	−0.870024
$50$	0	0
$51$	4.76393	0.667084
$52$	−21.1567	−2.93391
$53$	6.17345	0.847988	0.423994	−	0.905665i	$-0.360628\pi$
$53$	6.17345	0.847988	0.423994	+	0.905665i	$0.360628\pi$
$54$	13.9443	1.89758
$55$	0	0
$56$	−5.32624	−0.711748
$57$	−4.26575	−0.565012
$58$	−14.6189	−1.91956
$59$	4.47214	0.582223	0.291111	−	0.956689i	$-0.405975\pi$
$59$	4.47214	0.582223	0.291111	+	0.956689i	$0.405975\pi$
$60$	0	0
$61$	−6.09017	−0.779766	−0.389883	−	0.920864i	$-0.627485\pi$
$61$	−6.09017	−0.779766	−0.389883	+	0.920864i	$0.627485\pi$
$62$	4.99442	0.634292
$63$	0.589512	0.0742715
$64$	−4.70820	−0.588525
$65$	0	0
$66$	−7.70820	−0.948814
$67$	−3.08672	−0.377103	−0.188552	−	0.982063i	$-0.560379\pi$
$67$	−3.08672	−0.377103	−0.188552	+	0.982063i	$0.560379\pi$
$68$	−13.0756	−1.58565
$69$	−6.23607	−0.750734
$70$	0	0
$71$	−14.1803	−1.68290	−0.841448	−	0.540338i	$-0.818297\pi$
$71$	−14.1803	−1.68290	−0.841448	+	0.540338i	$0.818297\pi$
$72$	−3.45106	−0.406712
$73$	8.80982	1.03111	0.515556	−	0.856856i	$-0.327585\pi$
$73$	8.80982	1.03111	0.515556	+	0.856856i	$0.327585\pi$
$74$	20.1803	2.34592
$75$	0	0
$76$	11.7082	1.34302
$77$	−1.90770	−0.217403
$78$	19.2490	2.17952
$79$	7.23607	0.814121	0.407061	−	0.913401i	$-0.366554\pi$
$79$	7.23607	0.814121	0.407061	+	0.913401i	$0.366554\pi$
$80$	0	0
$81$	−6.76393	−0.751548
$82$	12.7112	1.40372
$83$	−7.12730	−0.782323	−0.391161	−	0.920322i	$-0.627927\pi$
$83$	−7.12730	−0.782323	−0.391161	+	0.920322i	$0.627927\pi$
$84$	6.23607	0.680411
$85$	0	0
$86$	24.0344	2.59170
$87$	9.03500	0.968653
$88$	11.1679	1.19050
$89$	−6.38197	−0.676487	−0.338244	−	0.941059i	$-0.609833\pi$
$89$	−6.38197	−0.676487	−0.338244	+	0.941059i	$0.609833\pi$
$90$	0	0
$91$	4.76393	0.499396
$92$	17.1161	1.78448
$93$	−3.08672	−0.320078
$94$	−16.3262	−1.68392
$95$	0	0
$96$	−3.85410	−0.393358
$97$	1.17902	0.119712	0.0598559	−	0.998207i	$-0.480936\pi$
$97$	1.17902	0.119712	0.0598559	+	0.998207i	$0.480936\pi$
$98$	−15.2084	−1.53629
$99$	−1.23607	−0.124230
$100$	0	0
$101$	0.0901699	0.00897224	0.00448612	−	0.999990i	$-0.498572\pi$
$101$	0.0901699	0.00897224	0.00448612	+	0.999990i	$0.498572\pi$
$102$	11.8965	1.17793
$103$	−9.26017	−0.912432	−0.456216	−	0.889869i	$-0.650796\pi$
$103$	−9.26017	−0.912432	−0.456216	+	0.889869i	$0.650796\pi$
$104$	−27.8885	−2.73470
$105$	0	0
$106$	15.4164	1.49737
$107$	−13.4399	−1.29929	−0.649643	−	0.760240i	$-0.725081\pi$
$107$	−13.4399	−1.29929	−0.649643	+	0.760240i	$0.725081\pi$
$108$	23.6539	2.27610
$109$	7.56231	0.724338	0.362169	−	0.932113i	$-0.382036\pi$
$109$	7.56231	0.724338	0.362169	+	0.932113i	$0.382036\pi$
$110$	0	0
$111$	−12.4721	−1.18380
$112$	−5.21960	−0.493206
$113$	−4.99442	−0.469836	−0.234918	−	0.972015i	$-0.575482\pi$
$113$	−4.99442	−0.469836	−0.234918	+	0.972015i	$0.575482\pi$
$114$	−10.6525	−0.997696
$115$	0	0
$116$	−24.7984	−2.30247
$117$	3.08672	0.285368
$118$	11.1679	1.02809
$119$	2.94427	0.269901
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−15.2084	−1.37691
$123$	−7.85597	−0.708349
$124$	8.47214	0.760820
$125$	0	0
$126$	1.47214	0.131148
$127$	7.26646	0.644794	0.322397	−	0.946605i	$-0.395511\pi$
$127$	7.26646	0.644794	0.322397	+	0.946605i	$0.395511\pi$
$128$	−16.7518	−1.48066
$129$	−14.8541	−1.30783
$130$	0	0
$131$	8.18034	0.714720	0.357360	−	0.933967i	$-0.383677\pi$
$131$	8.18034	0.714720	0.357360	+	0.933967i	$0.383677\pi$
$132$	−13.0756	−1.13808
$133$	−2.63638	−0.228603
$134$	−7.70820	−0.665887
$135$	0	0
$136$	−17.2361	−1.47798
$137$	−16.8910	−1.44309	−0.721547	−	0.692366i	$-0.756568\pi$
$137$	−16.8910	−1.44309	−0.721547	+	0.692366i	$0.756568\pi$
$138$	−15.5728	−1.32564
$139$	21.7082	1.84127	0.920633	−	0.390429i	$-0.127673\pi$
$139$	21.7082	1.84127	0.920633	+	0.390429i	$0.127673\pi$
$140$	0	0
$141$	10.0902	0.849746
$142$	−35.4113	−2.97165
$143$	−9.98885	−0.835309
$144$	−3.38197	−0.281831
$145$	0	0
$146$	22.0000	1.82073
$147$	9.39934	0.775244
$148$	34.2323	2.81388
$149$	−14.2705	−1.16909	−0.584543	−	0.811363i	$-0.698726\pi$
$149$	−14.2705	−1.16909	−0.584543	+	0.811363i	$0.698726\pi$
$150$	0	0
$151$	−4.18034	−0.340191	−0.170096	−	0.985428i	$-0.554408\pi$
$151$	−4.18034	−0.340191	−0.170096	+	0.985428i	$0.554408\pi$
$152$	15.4336	1.25183
$153$	1.90770	0.154228
$154$	−4.76393	−0.383889
$155$	0	0
$156$	32.6525	2.61429
$157$	19.2490	1.53624	0.768120	−	0.640307i	$-0.221193\pi$
$157$	19.2490	1.53624	0.768120	+	0.640307i	$0.221193\pi$
$158$	18.0700	1.43757
$159$	−9.52786	−0.755609
$160$	0	0
$161$	−3.85410	−0.303746
$162$	−16.8910	−1.32708
$163$	−0.225173	−0.0176369	−0.00881847	−	0.999961i	$-0.502807\pi$
$163$	−0.225173	−0.0176369	−0.00881847	+	0.999961i	$0.502807\pi$
$164$	21.5623	1.68373
$165$	0	0
$166$	−17.7984	−1.38142
$167$	4.63009	0.358287	0.179143	−	0.983823i	$-0.442667\pi$
$167$	4.63009	0.358287	0.179143	+	0.983823i	$0.442667\pi$
$168$	8.22031	0.634211
$169$	11.9443	0.918790
$170$	0	0
$171$	−1.70820	−0.130630
$172$	40.7701	3.10869
$173$	−0.728677	−0.0554003	−0.0277001	−	0.999616i	$-0.508818\pi$
$173$	−0.728677	−0.0554003	−0.0277001	+	0.999616i	$0.508818\pi$
$174$	22.5623	1.71044
$175$	0	0
$176$	10.9443	0.824956
$177$	−6.90212	−0.518795
$178$	−15.9371	−1.19454
$179$	1.05573	0.0789088	0.0394544	−	0.999221i	$-0.487438\pi$
$179$	1.05573	0.0789088	0.0394544	+	0.999221i	$0.487438\pi$
$180$	0	0
$181$	−11.0902	−0.824326	−0.412163	−	0.911110i	$-0.635227\pi$
$181$	−11.0902	−0.824326	−0.412163	+	0.911110i	$0.635227\pi$
$182$	11.8965	0.881831
$183$	9.39934	0.694819
$184$	22.5623	1.66332
$185$	0	0
$186$	−7.70820	−0.565193
$187$	−6.17345	−0.451447
$188$	−27.6945	−2.01983
$189$	−5.32624	−0.387427
$190$	0	0
$191$	−14.1803	−1.02605	−0.513027	−	0.858373i	$-0.671476\pi$
$191$	−14.1803	−1.02605	−0.513027	+	0.858373i	$0.671476\pi$
$192$	7.26646	0.524412
$193$	1.90770	0.137319	0.0686596	−	0.997640i	$-0.478128\pi$
$193$	1.90770	0.137319	0.0686596	+	0.997640i	$0.478128\pi$
$194$	2.94427	0.211386
$195$	0	0
$196$	−25.7984	−1.84274
$197$	10.7175	0.763592	0.381796	−	0.924247i	$-0.375306\pi$
$197$	10.7175	0.763592	0.381796	+	0.924247i	$0.375306\pi$
$198$	−3.08672	−0.219364
$199$	6.18034	0.438113	0.219056	−	0.975712i	$-0.429702\pi$
$199$	6.18034	0.438113	0.219056	+	0.975712i	$0.429702\pi$
$200$	0	0
$201$	4.76393	0.336022
$202$	0.225173	0.0158432
$203$	5.58394	0.391915
$204$	20.1803	1.41291
$205$	0	0
$206$	−23.1246	−1.61117
$207$	−2.49721	−0.173568
$208$	−27.3302	−1.89501
$209$	5.52786	0.382370
$210$	0	0
$211$	28.1803	1.94001	0.970007	−	0.243076i	$-0.0781564\pi$
$211$	28.1803	1.94001	0.970007	+	0.243076i	$0.0781564\pi$
$212$	26.1511	1.79607
$213$	21.8854	1.49956
$214$	−33.5623	−2.29427
$215$	0	0
$216$	31.1803	2.12155
$217$	−1.90770	−0.129503
$218$	18.8847	1.27903
$219$	−13.5967	−0.918783
$220$	0	0
$221$	15.4164	1.03702
$222$	−31.1456	−2.09035
$223$	−8.44549	−0.565552	−0.282776	−	0.959186i	$-0.591255\pi$
$223$	−8.44549	−0.565552	−0.282776	+	0.959186i	$0.591255\pi$
$224$	−2.38197	−0.159152
$225$	0	0
$226$	−12.4721	−0.829634
$227$	−5.21960	−0.346437	−0.173218	−	0.984883i	$-0.555417\pi$
$227$	−5.21960	−0.346437	−0.173218	+	0.984883i	$0.555417\pi$
$228$	−18.0700	−1.19671
$229$	−4.67376	−0.308851	−0.154425	−	0.988004i	$-0.549353\pi$
$229$	−4.67376	−0.308851	−0.154425	+	0.988004i	$0.549353\pi$
$230$	0	0
$231$	2.94427	0.193719
$232$	−32.6889	−2.14613
$233$	13.0756	0.856609	0.428305	−	0.903634i	$-0.359111\pi$
$233$	13.0756	0.856609	0.428305	+	0.903634i	$0.359111\pi$
$234$	7.70820	0.503901
$235$	0	0
$236$	18.9443	1.23317
$237$	−11.1679	−0.725431
$238$	7.35247	0.476590
$239$	1.70820	0.110495	0.0552473	−	0.998473i	$-0.482405\pi$
$239$	1.70820	0.110495	0.0552473	+	0.998473i	$0.482405\pi$
$240$	0	0
$241$	10.0902	0.649965	0.324982	−	0.945720i	$-0.394642\pi$
$241$	10.0902	0.649965	0.324982	+	0.945720i	$0.394642\pi$
$242$	−17.4805	−1.12369
$243$	−6.31261	−0.404954
$244$	−25.7984	−1.65157
$245$	0	0
$246$	−19.6180	−1.25080
$247$	−13.8042	−0.878343
$248$	11.1679	0.709161
$249$	11.0000	0.697097
$250$	0	0
$251$	24.3607	1.53763	0.768816	−	0.639470i	$-0.220846\pi$
$251$	24.3607	1.53763	0.768816	+	0.639470i	$0.220846\pi$
$252$	2.49721	0.157310
$253$	8.08115	0.508057
$254$	18.1459	1.13857
$255$	0	0
$256$	−32.4164	−2.02603
$257$	3.81540	0.237998	0.118999	−	0.992894i	$-0.462031\pi$
$257$	3.81540	0.237998	0.118999	+	0.992894i	$0.462031\pi$
$258$	−37.0938	−2.30936
$259$	−7.70820	−0.478964
$260$	0	0
$261$	3.61803	0.223951
$262$	20.4280	1.26205
$263$	27.6945	1.70772	0.853858	−	0.520506i	$-0.174257\pi$
$263$	27.6945	1.70772	0.853858	+	0.520506i	$0.174257\pi$
$264$	−17.2361	−1.06081
$265$	0	0
$266$	−6.58359	−0.403666
$267$	9.84968	0.602791
$268$	−13.0756	−0.798718
$269$	18.9443	1.15505	0.577526	−	0.816372i	$-0.304018\pi$
$269$	18.9443	1.15505	0.577526	+	0.816372i	$0.304018\pi$
$270$	0	0
$271$	−8.00000	−0.485965	−0.242983	−	0.970031i	$-0.578126\pi$
$271$	−8.00000	−0.485965	−0.242983	+	0.970031i	$0.578126\pi$
$272$	−16.8910	−1.02417
$273$	−7.35247	−0.444992
$274$	−42.1803	−2.54821
$275$	0	0
$276$	−26.4164	−1.59008
$277$	−25.4225	−1.52749	−0.763744	−	0.645519i	$-0.776641\pi$
$277$	−25.4225	−1.52749	−0.763744	+	0.645519i	$0.776641\pi$
$278$	54.2100	3.25130
$279$	−1.23607	−0.0740015
$280$	0	0
$281$	−2.27051	−0.135447	−0.0677236	−	0.997704i	$-0.521574\pi$
$281$	−2.27051	−0.135447	−0.0677236	+	0.997704i	$0.521574\pi$
$282$	25.1973	1.50048
$283$	13.0756	0.777262	0.388631	−	0.921393i	$-0.372948\pi$
$283$	13.0756	0.777262	0.388631	+	0.921393i	$0.372948\pi$
$284$	−60.0689	−3.56443
$285$	0	0
$286$	−24.9443	−1.47499
$287$	−4.85526	−0.286597
$288$	−1.54336	−0.0909435
$289$	−7.47214	−0.439537
$290$	0	0
$291$	−1.81966	−0.106670
$292$	37.3190	2.18393
$293$	−13.5259	−0.790193	−0.395096	−	0.918640i	$-0.629289\pi$
$293$	−13.5259	−0.790193	−0.395096	+	0.918640i	$0.629289\pi$
$294$	23.4721	1.36892
$295$	0	0
$296$	45.1246	2.62281
$297$	11.1679	0.648026
$298$	−35.6365	−2.06437
$299$	−20.1803	−1.16706
$300$	0	0
$301$	−9.18034	−0.529146
$302$	−10.4392	−0.600708
$303$	−0.139165	−0.00799481
$304$	15.1246	0.867456
$305$	0	0
$306$	4.76393	0.272336
$307$	24.0183	1.37080	0.685398	−	0.728169i	$-0.259628\pi$
$307$	24.0183	1.37080	0.685398	+	0.728169i	$0.259628\pi$
$308$	−8.08115	−0.460466
$309$	14.2918	0.813032
$310$	0	0
$311$	−18.0000	−1.02069	−0.510343	−	0.859971i	$-0.670482\pi$
$311$	−18.0000	−1.02069	−0.510343	+	0.859971i	$0.670482\pi$
$312$	43.0421	2.43678
$313$	−29.9665	−1.69381	−0.846905	−	0.531745i	$-0.821537\pi$
$313$	−29.9665	−1.69381	−0.846905	+	0.531745i	$0.821537\pi$
$314$	48.0689	2.71268
$315$	0	0
$316$	30.6525	1.72434
$317$	21.8854	1.22921	0.614603	−	0.788836i	$-0.289316\pi$
$317$	21.8854	1.22921	0.614603	+	0.788836i	$0.289316\pi$
$318$	−23.7931	−1.33425
$319$	−11.7082	−0.655534
$320$	0	0
$321$	20.7426	1.15774
$322$	−9.62451	−0.536353
$323$	−8.53149	−0.474705
$324$	−28.6525	−1.59180
$325$	0	0
$326$	−0.562306	−0.0311432
$327$	−11.6714	−0.645429
$328$	28.4232	1.56941
$329$	6.23607	0.343806
$330$	0	0
$331$	28.1803	1.54893	0.774466	−	0.632616i	$-0.218019\pi$
$331$	28.1803	1.54893	0.774466	+	0.632616i	$0.218019\pi$
$332$	−30.1917	−1.65699
$333$	−4.99442	−0.273693
$334$	11.5623	0.632661
$335$	0	0
$336$	8.05573	0.439476
$337$	8.08115	0.440208	0.220104	−	0.975476i	$-0.429360\pi$
$337$	8.08115	0.440208	0.220104	+	0.975476i	$0.429360\pi$
$338$	29.8274	1.62240
$339$	7.70820	0.418652
$340$	0	0
$341$	4.00000	0.216612
$342$	−4.26575	−0.230665
$343$	12.4861	0.674184
$344$	53.7426	2.89761
$345$	0	0
$346$	−1.81966	−0.0978255
$347$	−3.59023	−0.192733	−0.0963667	−	0.995346i	$-0.530722\pi$
$347$	−3.59023	−0.192733	−0.0963667	+	0.995346i	$0.530722\pi$
$348$	38.2729	2.05164
$349$	24.2705	1.29917	0.649585	−	0.760289i	$-0.274943\pi$
$349$	24.2705	1.29917	0.649585	+	0.760289i	$0.274943\pi$
$350$	0	0
$351$	−27.8885	−1.48858
$352$	4.99442	0.266204
$353$	−18.7987	−1.00055	−0.500276	−	0.865866i	$-0.666768\pi$
$353$	−18.7987	−1.00055	−0.500276	+	0.865866i	$0.666768\pi$
$354$	−17.2361	−0.916087
$355$	0	0
$356$	−27.0344	−1.43282
$357$	−4.54408	−0.240498
$358$	2.63638	0.139337
$359$	−27.8885	−1.47190	−0.735951	−	0.677035i	$-0.763264\pi$
$359$	−27.8885	−1.47190	−0.735951	+	0.677035i	$0.763264\pi$
$360$	0	0
$361$	−11.3607	−0.597931
$362$	−27.6945	−1.45559
$363$	10.8035	0.567038
$364$	20.1803	1.05774
$365$	0	0
$366$	23.4721	1.22691
$367$	−7.85597	−0.410079	−0.205039	−	0.978754i	$-0.565732\pi$
$367$	−7.85597	−0.410079	−0.205039	+	0.978754i	$0.565732\pi$
$368$	22.1106	1.15259
$369$	−3.14590	−0.163769
$370$	0	0
$371$	−5.88854	−0.305718
$372$	−13.0756	−0.677937
$373$	−25.7008	−1.33074	−0.665368	−	0.746515i	$-0.731726\pi$
$373$	−25.7008	−1.33074	−0.665368	+	0.746515i	$0.731726\pi$
$374$	−15.4164	−0.797163
$375$	0	0
$376$	−36.5066	−1.88268
$377$	29.2379	1.50583
$378$	−13.3007	−0.684117
$379$	11.0557	0.567895	0.283947	−	0.958840i	$-0.408356\pi$
$379$	11.0557	0.567895	0.283947	+	0.958840i	$0.408356\pi$
$380$	0	0
$381$	−11.2148	−0.574551
$382$	−35.4113	−1.81180
$383$	−19.6134	−1.00220	−0.501098	−	0.865391i	$-0.667070\pi$
$383$	−19.6134	−1.00220	−0.501098	+	0.865391i	$0.667070\pi$
$384$	25.8541	1.31936
$385$	0	0
$386$	4.76393	0.242478
$387$	−5.94827	−0.302368
$388$	4.99442	0.253553
$389$	−21.3820	−1.08411	−0.542054	−	0.840343i	$-0.682353\pi$
$389$	−21.3820	−1.08411	−0.542054	+	0.840343i	$0.682353\pi$
$390$	0	0
$391$	−12.4721	−0.630743
$392$	−34.0071	−1.71762
$393$	−12.6252	−0.636858
$394$	26.7639	1.34835
$395$	0	0
$396$	−5.23607	−0.263122
$397$	−29.6882	−1.49001	−0.745004	−	0.667060i	$-0.767553\pi$
$397$	−29.6882	−1.49001	−0.745004	+	0.667060i	$0.767553\pi$
$398$	15.4336	0.773617
$399$	4.06888	0.203699
$400$	0	0
$401$	−3.72949	−0.186242	−0.0931209	−	0.995655i	$-0.529684\pi$
$401$	−3.72949	−0.186242	−0.0931209	+	0.995655i	$0.529684\pi$
$402$	11.8965	0.593346
$403$	−9.98885	−0.497580
$404$	0.381966	0.0190035
$405$	0	0
$406$	13.9443	0.692043
$407$	16.1623	0.801135
$408$	26.6015	1.31697
$409$	28.7426	1.42123	0.710616	−	0.703580i	$-0.248416\pi$
$409$	28.7426	1.42123	0.710616	+	0.703580i	$0.248416\pi$
$410$	0	0
$411$	26.0689	1.28588
$412$	−39.2267	−1.93256
$413$	−4.26575	−0.209904
$414$	−6.23607	−0.306486
$415$	0	0
$416$	−12.4721	−0.611497
$417$	−33.5036	−1.64068
$418$	13.8042	0.675188
$419$	−23.4164	−1.14397	−0.571983	−	0.820265i	$-0.693826\pi$
$419$	−23.4164	−1.14397	−0.571983	+	0.820265i	$0.693826\pi$
$420$	0	0
$421$	−1.09017	−0.0531316	−0.0265658	−	0.999647i	$-0.508457\pi$
$421$	−1.09017	−0.0531316	−0.0265658	+	0.999647i	$0.508457\pi$
$422$	70.3723	3.42567
$423$	4.04057	0.196459
$424$	34.4721	1.67411
$425$	0	0
$426$	54.6525	2.64792
$427$	5.80911	0.281123
$428$	−56.9324	−2.75193
$429$	15.4164	0.744311
$430$	0	0
$431$	8.18034	0.394033	0.197017	−	0.980400i	$-0.436875\pi$
$431$	8.18034	0.394033	0.197017	+	0.980400i	$0.436875\pi$
$432$	30.5561	1.47013
$433$	38.0477	1.82846	0.914228	−	0.405201i	$-0.132798\pi$
$433$	38.0477	1.82846	0.914228	+	0.405201i	$0.132798\pi$
$434$	−4.76393	−0.228676
$435$	0	0
$436$	32.0344	1.53417
$437$	11.1679	0.534232
$438$	−33.9540	−1.62238
$439$	−20.6525	−0.985689	−0.492844	−	0.870117i	$-0.664043\pi$
$439$	−20.6525	−0.985689	−0.492844	+	0.870117i	$0.664043\pi$
$440$	0	0
$441$	3.76393	0.179235
$442$	38.4980	1.83116
$443$	6.67695	0.317232	0.158616	−	0.987340i	$-0.449297\pi$
$443$	6.67695	0.317232	0.158616	+	0.987340i	$0.449297\pi$
$444$	−52.8328	−2.50733
$445$	0	0
$446$	−21.0902	−0.998648
$447$	22.0246	1.04173
$448$	4.49092	0.212176
$449$	10.0000	0.471929	0.235965	−	0.971762i	$-0.424175\pi$
$449$	10.0000	0.471929	0.235965	+	0.971762i	$0.424175\pi$
$450$	0	0
$451$	10.1803	0.479373
$452$	−21.1567	−0.995128
$453$	6.45178	0.303131
$454$	−13.0344	−0.611737
$455$	0	0
$456$	−23.8197	−1.11546
$457$	3.81540	0.178477	0.0892385	−	0.996010i	$-0.471557\pi$
$457$	3.81540	0.178477	0.0892385	+	0.996010i	$0.471557\pi$
$458$	−11.6714	−0.545368
$459$	−17.2361	−0.804511
$460$	0	0
$461$	−12.2705	−0.571495	−0.285747	−	0.958305i	$-0.592242\pi$
$461$	−12.2705	−0.571495	−0.285747	+	0.958305i	$0.592242\pi$
$462$	7.35247	0.342068
$463$	−40.6309	−1.88828	−0.944139	−	0.329547i	$-0.893104\pi$
$463$	−40.6309	−1.88828	−0.944139	+	0.329547i	$0.893104\pi$
$464$	−32.0344	−1.48716
$465$	0	0
$466$	32.6525	1.51260
$467$	1.68253	0.0778581	0.0389290	−	0.999242i	$-0.487605\pi$
$467$	1.68253	0.0778581	0.0389290	+	0.999242i	$0.487605\pi$
$468$	13.0756	0.604419
$469$	2.94427	0.135954
$470$	0	0
$471$	−29.7082	−1.36888
$472$	24.9721	1.14944
$473$	19.2490	0.885071
$474$	−27.8885	−1.28096
$475$	0	0
$476$	12.4721	0.571659
$477$	−3.81540	−0.174695
$478$	4.26575	0.195111
$479$	19.5967	0.895398	0.447699	−	0.894184i	$-0.352244\pi$
$479$	19.5967	0.895398	0.447699	+	0.894184i	$0.352244\pi$
$480$	0	0
$481$	−40.3607	−1.84029
$482$	25.1973	1.14771
$483$	5.94827	0.270656
$484$	−29.6525	−1.34784
$485$	0	0
$486$	−15.7639	−0.715066
$487$	−9.17416	−0.415721	−0.207861	−	0.978158i	$-0.566650\pi$
$487$	−9.17416	−0.415721	−0.207861	+	0.978158i	$0.566650\pi$
$488$	−34.0071	−1.53943
$489$	0.347524	0.0157156
$490$	0	0
$491$	−34.1803	−1.54254	−0.771269	−	0.636510i	$-0.780377\pi$
$491$	−34.1803	−1.54254	−0.771269	+	0.636510i	$0.780377\pi$
$492$	−33.2784	−1.50031
$493$	18.0700	0.813832
$494$	−34.4721	−1.55097
$495$	0	0
$496$	10.9443	0.491412
$497$	13.5259	0.606720
$498$	27.4693	1.23093
$499$	−6.18034	−0.276670	−0.138335	−	0.990385i	$-0.544175\pi$
$499$	−6.18034	−0.276670	−0.138335	+	0.990385i	$0.544175\pi$
$500$	0	0
$501$	−7.14590	−0.319255
$502$	60.8338	2.71514
$503$	10.9427	0.487911	0.243955	−	0.969786i	$-0.421555\pi$
$503$	10.9427	0.487911	0.243955	+	0.969786i	$0.421555\pi$
$504$	3.29180	0.146628
$505$	0	0
$506$	20.1803	0.897126
$507$	−18.4343	−0.818698
$508$	30.7812	1.36570
$509$	24.4721	1.08471	0.542354	−	0.840150i	$-0.317533\pi$
$509$	24.4721	1.08471	0.542354	+	0.840150i	$0.317533\pi$
$510$	0	0
$511$	−8.40325	−0.371738
$512$	−47.4470	−2.09688
$513$	15.4336	0.681411
$514$	9.52786	0.420256
$515$	0	0
$516$	−62.9230	−2.77003
$517$	−13.0756	−0.575063
$518$	−19.2490	−0.845753
$519$	1.12461	0.0493650
$520$	0	0
$521$	35.0902	1.53733	0.768664	−	0.639653i	$-0.220922\pi$
$521$	35.0902	1.53733	0.768664	+	0.639653i	$0.220922\pi$
$522$	9.03500	0.395451
$523$	31.9603	1.39752	0.698762	−	0.715354i	$-0.253735\pi$
$523$	31.9603	1.39752	0.698762	+	0.715354i	$0.253735\pi$
$524$	34.6525	1.51380
$525$	0	0
$526$	69.1591	3.01548
$527$	−6.17345	−0.268920
$528$	−16.8910	−0.735085
$529$	−6.67376	−0.290164
$530$	0	0
$531$	−2.76393	−0.119944
$532$	−11.1679	−0.484189
$533$	−25.4225	−1.10117
$534$	24.5967	1.06441
$535$	0	0
$536$	−17.2361	−0.744485
$537$	−1.62937	−0.0703125
$538$	47.3079	2.03959
$539$	−12.1803	−0.524644
$540$	0	0
$541$	−11.0902	−0.476804	−0.238402	−	0.971167i	$-0.576624\pi$
$541$	−11.0902	−0.476804	−0.238402	+	0.971167i	$0.576624\pi$
$542$	−19.9777	−0.858116
$543$	17.1161	0.734524
$544$	−7.70820	−0.330487
$545$	0	0
$546$	−18.3607	−0.785765
$547$	−34.4575	−1.47329	−0.736647	−	0.676277i	$-0.763592\pi$
$547$	−34.4575	−1.47329	−0.736647	+	0.676277i	$0.763592\pi$
$548$	−71.5513	−3.05652
$549$	3.76393	0.160641
$550$	0	0
$551$	−16.1803	−0.689306
$552$	−34.8218	−1.48211
$553$	−6.90212	−0.293508
$554$	−63.4853	−2.69723
$555$	0	0
$556$	91.9574	3.89986
$557$	−30.6952	−1.30060	−0.650299	−	0.759678i	$-0.725356\pi$
$557$	−30.6952	−1.30060	−0.650299	+	0.759678i	$0.725356\pi$
$558$	−3.08672	−0.130671
$559$	−48.0689	−2.03310
$560$	0	0
$561$	9.52786	0.402267
$562$	−5.66994	−0.239172
$563$	−14.5329	−0.612490	−0.306245	−	0.951953i	$-0.599073\pi$
$563$	−14.5329	−0.612490	−0.306245	+	0.951953i	$0.599073\pi$
$564$	42.7426	1.79979
$565$	0	0
$566$	32.6525	1.37249
$567$	6.45178	0.270949
$568$	−79.1821	−3.32241
$569$	28.2148	1.18283	0.591413	−	0.806369i	$-0.298570\pi$
$569$	28.2148	1.18283	0.591413	+	0.806369i	$0.298570\pi$
$570$	0	0
$571$	32.0000	1.33916	0.669579	−	0.742741i	$-0.266474\pi$
$571$	32.0000	1.33916	0.669579	+	0.742741i	$0.266474\pi$
$572$	−42.3134	−1.76921
$573$	21.8854	0.914276
$574$	−12.1246	−0.506072
$575$	0	0
$576$	2.90983	0.121243
$577$	39.9554	1.66336	0.831682	−	0.555252i	$-0.187378\pi$
$577$	39.9554	1.66336	0.831682	+	0.555252i	$0.187378\pi$
$578$	−18.6595	−0.776133
$579$	−2.94427	−0.122360
$580$	0	0
$581$	6.79837	0.282044
$582$	−4.54408	−0.188358
$583$	12.3469	0.511356
$584$	49.1935	2.03564
$585$	0	0
$586$	−33.7771	−1.39532
$587$	33.0533	1.36425	0.682127	−	0.731234i	$-0.261055\pi$
$587$	33.0533	1.36425	0.682127	+	0.731234i	$0.261055\pi$
$588$	39.8162	1.64199
$589$	5.52786	0.227772
$590$	0	0
$591$	−16.5410	−0.680407
$592$	44.2211	1.81748
$593$	32.7749	1.34591	0.672953	−	0.739686i	$-0.265026\pi$
$593$	32.7749	1.34591	0.672953	+	0.739686i	$0.265026\pi$
$594$	27.8885	1.14428
$595$	0	0
$596$	−60.4508	−2.47616
$597$	−9.53850	−0.390385
$598$	−50.3946	−2.06079
$599$	28.5410	1.16615	0.583077	−	0.812417i	$-0.301848\pi$
$599$	28.5410	1.16615	0.583077	+	0.812417i	$0.301848\pi$
$600$	0	0
$601$	−27.2705	−1.11239	−0.556194	−	0.831053i	$-0.687739\pi$
$601$	−27.2705	−1.11239	−0.556194	+	0.831053i	$0.687739\pi$
$602$	−22.9253	−0.934364
$603$	1.90770	0.0776876
$604$	−17.7082	−0.720537
$605$	0	0
$606$	−0.347524	−0.0141172
$607$	−11.6182	−0.471569	−0.235784	−	0.971805i	$-0.575766\pi$
$607$	−11.6182	−0.471569	−0.235784	+	0.971805i	$0.575766\pi$
$608$	6.90212	0.279918
$609$	−8.61803	−0.349220
$610$	0	0
$611$	32.6525	1.32098
$612$	8.08115	0.326661
$613$	19.9777	0.806892	0.403446	−	0.915004i	$-0.367812\pi$
$613$	19.9777	0.806892	0.403446	+	0.915004i	$0.367812\pi$
$614$	59.9787	2.42054
$615$	0	0
$616$	−10.6525	−0.429200
$617$	27.7805	1.11840	0.559201	−	0.829032i	$-0.311108\pi$
$617$	27.7805	1.11840	0.559201	+	0.829032i	$0.311108\pi$
$618$	35.6896	1.43565
$619$	−9.59675	−0.385726	−0.192863	−	0.981226i	$-0.561777\pi$
$619$	−9.59675	−0.385726	−0.192863	+	0.981226i	$0.561777\pi$
$620$	0	0
$621$	22.5623	0.905394
$622$	−44.9498	−1.80232
$623$	6.08744	0.243888
$624$	42.1803	1.68856
$625$	0	0
$626$	−74.8328	−2.99092
$627$	−8.53149	−0.340715
$628$	81.5402	3.25381
$629$	−24.9443	−0.994593
$630$	0	0
$631$	34.3607	1.36788	0.683939	−	0.729540i	$-0.260266\pi$
$631$	34.3607	1.36788	0.683939	+	0.729540i	$0.260266\pi$
$632$	40.4057	1.60725
$633$	−43.4925	−1.72867
$634$	54.6525	2.17053
$635$	0	0
$636$	−40.3607	−1.60041
$637$	30.4169	1.20516
$638$	−29.2379	−1.15754
$639$	8.76393	0.346696
$640$	0	0
$641$	−7.27051	−0.287168	−0.143584	−	0.989638i	$-0.545863\pi$
$641$	−7.27051	−0.287168	−0.143584	+	0.989638i	$0.545863\pi$
$642$	51.7988	2.04433
$643$	−43.2673	−1.70630	−0.853148	−	0.521669i	$-0.825309\pi$
$643$	−43.2673	−1.70630	−0.853148	+	0.521669i	$0.825309\pi$
$644$	−16.3262	−0.643344
$645$	0	0
$646$	−21.3050	−0.838232
$647$	41.5848	1.63487	0.817433	−	0.576024i	$-0.195396\pi$
$647$	41.5848	1.63487	0.817433	+	0.576024i	$0.195396\pi$
$648$	−37.7694	−1.48372
$649$	8.94427	0.351093
$650$	0	0
$651$	2.94427	0.115395
$652$	−0.953850	−0.0373557
$653$	15.7119	0.614856	0.307428	−	0.951571i	$-0.400532\pi$
$653$	15.7119	0.614856	0.307428	+	0.951571i	$0.400532\pi$
$654$	−29.1459	−1.13969
$655$	0	0
$656$	27.8541	1.08752
$657$	−5.44477	−0.212421
$658$	15.5728	0.607090
$659$	20.6525	0.804506	0.402253	−	0.915528i	$-0.368227\pi$
$659$	20.6525	0.804506	0.402253	+	0.915528i	$0.368227\pi$
$660$	0	0
$661$	11.2705	0.438372	0.219186	−	0.975683i	$-0.429660\pi$
$661$	11.2705	0.438372	0.219186	+	0.975683i	$0.429660\pi$
$662$	70.3723	2.73510
$663$	−23.7931	−0.924047
$664$	−39.7984	−1.54448
$665$	0	0
$666$	−12.4721	−0.483285
$667$	−23.6539	−0.915884
$668$	19.6134	0.758864
$669$	13.0344	0.503941
$670$	0	0
$671$	−12.1803	−0.470217
$672$	3.67624	0.141814
$673$	8.80982	0.339594	0.169797	−	0.985479i	$-0.445689\pi$
$673$	8.80982	0.339594	0.169797	+	0.985479i	$0.445689\pi$
$674$	20.1803	0.777318
$675$	0	0
$676$	50.5967	1.94603
$677$	−9.98885	−0.383903	−0.191951	−	0.981404i	$-0.561482\pi$
$677$	−9.98885	−0.383903	−0.191951	+	0.981404i	$0.561482\pi$
$678$	19.2490	0.739254
$679$	−1.12461	−0.0431586
$680$	0	0
$681$	8.05573	0.308696
$682$	9.98885	0.382493
$683$	−29.1519	−1.11546	−0.557732	−	0.830021i	$-0.688328\pi$
$683$	−29.1519	−1.11546	−0.557732	+	0.830021i	$0.688328\pi$
$684$	−7.23607	−0.276678
$685$	0	0
$686$	31.1803	1.19047
$687$	7.21331	0.275205
$688$	52.6666	2.00790
$689$	−30.8328	−1.17464
$690$	0	0
$691$	8.18034	0.311195	0.155597	−	0.987821i	$-0.450270\pi$
$691$	8.18034	0.311195	0.155597	+	0.987821i	$0.450270\pi$
$692$	−3.08672	−0.117340
$693$	1.17902	0.0447874
$694$	−8.96556	−0.340328
$695$	0	0
$696$	50.4508	1.91233
$697$	−15.7119	−0.595133
$698$	60.6086	2.29407
$699$	−20.1803	−0.763291
$700$	0	0
$701$	22.0000	0.830929	0.415464	−	0.909610i	$-0.363619\pi$
$701$	22.0000	0.830929	0.415464	+	0.909610i	$0.363619\pi$
$702$	−69.6436	−2.62853
$703$	22.3357	0.842409
$704$	−9.41641	−0.354894
$705$	0	0
$706$	−46.9443	−1.76677
$707$	−0.0860086	−0.00323469
$708$	−29.2379	−1.09883
$709$	−25.9787	−0.975651	−0.487826	−	0.872941i	$-0.662210\pi$
$709$	−25.9787	−0.975651	−0.487826	+	0.872941i	$0.662210\pi$
$710$	0	0
$711$	−4.47214	−0.167718
$712$	−35.6365	−1.33553
$713$	8.08115	0.302641
$714$	−11.3475	−0.424670
$715$	0	0
$716$	4.47214	0.167132
$717$	−2.63638	−0.0984573
$718$	−69.6436	−2.59908
$719$	12.7639	0.476014	0.238007	−	0.971263i	$-0.423506\pi$
$719$	12.7639	0.476014	0.238007	+	0.971263i	$0.423506\pi$
$720$	0	0
$721$	8.83282	0.328951
$722$	−28.3700	−1.05582
$723$	−15.5728	−0.579158
$724$	−46.9787	−1.74595
$725$	0	0
$726$	26.9787	1.00127
$727$	13.1616	0.488136	0.244068	−	0.969758i	$-0.421518\pi$
$727$	13.1616	0.488136	0.244068	+	0.969758i	$0.421518\pi$
$728$	26.6015	0.985917
$729$	30.0344	1.11239
$730$	0	0
$731$	−29.7082	−1.09880
$732$	39.8162	1.47165
$733$	−27.3302	−1.00946	−0.504731	−	0.863276i	$-0.668408\pi$
$733$	−27.3302	−1.00946	−0.504731	+	0.863276i	$0.668408\pi$
$734$	−19.6180	−0.724115
$735$	0	0
$736$	10.0902	0.371929
$737$	−6.17345	−0.227402
$738$	−7.85597	−0.289182
$739$	41.7082	1.53426	0.767131	−	0.641491i	$-0.221684\pi$
$739$	41.7082	1.53426	0.767131	+	0.641491i	$0.221684\pi$
$740$	0	0
$741$	21.3050	0.782657
$742$	−14.7049	−0.539835
$743$	−23.0644	−0.846152	−0.423076	−	0.906094i	$-0.639050\pi$
$743$	−23.0644	−0.846152	−0.423076	+	0.906094i	$0.639050\pi$
$744$	−17.2361	−0.631905
$745$	0	0
$746$	−64.1803	−2.34981
$747$	4.40491	0.161167
$748$	−26.1511	−0.956181
$749$	12.8197	0.468420
$750$	0	0
$751$	8.18034	0.298505	0.149252	−	0.988799i	$-0.452313\pi$
$751$	8.18034	0.298505	0.149252	+	0.988799i	$0.452313\pi$
$752$	−35.7757	−1.30460
$753$	−37.5973	−1.37012
$754$	73.0132	2.65898
$755$	0	0
$756$	−22.5623	−0.820583
$757$	−21.1567	−0.768954	−0.384477	−	0.923135i	$-0.625618\pi$
$757$	−21.1567	−0.768954	−0.384477	+	0.923135i	$0.625618\pi$
$758$	27.6085	1.00279
$759$	−12.4721	−0.452710
$760$	0	0
$761$	−3.72949	−0.135194	−0.0675970	−	0.997713i	$-0.521533\pi$
$761$	−3.72949	−0.135194	−0.0675970	+	0.997713i	$0.521533\pi$
$762$	−28.0057	−1.01454
$763$	−7.21331	−0.261139
$764$	−60.0689	−2.17322
$765$	0	0
$766$	−48.9787	−1.76967
$767$	−22.3357	−0.806497
$768$	50.0302	1.80531
$769$	−46.5066	−1.67707	−0.838535	−	0.544848i	$-0.816587\pi$
$769$	−46.5066	−1.67707	−0.838535	+	0.544848i	$0.816587\pi$
$770$	0	0
$771$	−5.88854	−0.212071
$772$	8.08115	0.290847
$773$	−16.1623	−0.581317	−0.290659	−	0.956827i	$-0.593874\pi$
$773$	−16.1623	−0.581317	−0.290659	+	0.956827i	$0.593874\pi$
$774$	−14.8541	−0.533920
$775$	0	0
$776$	6.58359	0.236337
$777$	11.8965	0.426786
$778$	−53.3953	−1.91431
$779$	14.0689	0.504070
$780$	0	0
$781$	−28.3607	−1.01482
$782$	−31.1456	−1.11376
$783$	−32.6889	−1.16821
$784$	−33.3262	−1.19022
$785$	0	0
$786$	−31.5279	−1.12456
$787$	6.25946	0.223126	0.111563	−	0.993757i	$-0.464414\pi$
$787$	6.25946	0.223126	0.111563	+	0.993757i	$0.464414\pi$
$788$	45.4002	1.61731
$789$	−42.7426	−1.52168
$790$	0	0
$791$	4.76393	0.169386
$792$	−6.90212	−0.245256
$793$	30.4169	1.08014
$794$	−74.1378	−2.63105
$795$	0	0
$796$	26.1803	0.927938
$797$	−29.6882	−1.05161	−0.525805	−	0.850605i	$-0.676236\pi$
$797$	−29.6882	−1.05161	−0.525805	+	0.850605i	$0.676236\pi$
$798$	10.1609	0.359691
$799$	20.1803	0.713929
$800$	0	0
$801$	3.94427	0.139364
$802$	−9.31333	−0.328865
$803$	17.6196	0.621784
$804$	20.1803	0.711706
$805$	0	0
$806$	−24.9443	−0.878625
$807$	−29.2379	−1.02922
$808$	0.503503	0.0177132
$809$	30.2016	1.06183	0.530916	−	0.847424i	$-0.321848\pi$
$809$	30.2016	1.06183	0.530916	+	0.847424i	$0.321848\pi$
$810$	0	0
$811$	22.0000	0.772524	0.386262	−	0.922389i	$-0.373766\pi$
$811$	22.0000	0.772524	0.386262	+	0.922389i	$0.373766\pi$
$812$	23.6539	0.830090
$813$	12.3469	0.433025
$814$	40.3607	1.41464
$815$	0	0
$816$	26.0689	0.912593
$817$	26.6015	0.930668
$818$	71.7765	2.50961
$819$	−2.94427	−0.102881
$820$	0	0
$821$	6.27051	0.218842	0.109421	−	0.993995i	$-0.465100\pi$
$821$	6.27051	0.218842	0.109421	+	0.993995i	$0.465100\pi$
$822$	65.0995	2.27061
$823$	−0.728677	−0.0254001	−0.0127000	−	0.999919i	$-0.504043\pi$
$823$	−0.728677	−0.0254001	−0.0127000	+	0.999919i	$0.504043\pi$
$824$	−51.7082	−1.80134
$825$	0	0
$826$	−10.6525	−0.370647
$827$	5.44477	0.189333	0.0946666	−	0.995509i	$-0.469821\pi$
$827$	5.44477	0.189333	0.0946666	+	0.995509i	$0.469821\pi$
$828$	−10.5784	−0.367623
$829$	4.14590	0.143993	0.0719965	−	0.997405i	$-0.477063\pi$
$829$	4.14590	0.143993	0.0719965	+	0.997405i	$0.477063\pi$
$830$	0	0
$831$	39.2361	1.36108
$832$	23.5148	0.815228
$833$	18.7987	0.651335
$834$	−83.6656	−2.89710
$835$	0	0
$836$	23.4164	0.809873
$837$	11.1679	0.386018
$838$	−58.4757	−2.02001
$839$	−18.2918	−0.631503	−0.315751	−	0.948842i	$-0.602257\pi$
$839$	−18.2918	−0.631503	−0.315751	+	0.948842i	$0.602257\pi$
$840$	0	0
$841$	5.27051	0.181742
$842$	−2.72239	−0.0938196
$843$	3.50422	0.120692
$844$	119.374	4.10902
$845$	0	0
$846$	10.0902	0.346907
$847$	6.67695	0.229423
$848$	33.7819	1.16008
$849$	−20.1803	−0.692587
$850$	0	0
$851$	32.6525	1.11931
$852$	92.7080	3.17612
$853$	−24.6938	−0.845499	−0.422750	−	0.906247i	$-0.638935\pi$
$853$	−24.6938	−0.845499	−0.422750	+	0.906247i	$0.638935\pi$
$854$	14.5066	0.496405
$855$	0	0
$856$	−75.0476	−2.56507
$857$	−21.1567	−0.722700	−0.361350	−	0.932430i	$-0.617684\pi$
$857$	−21.1567	−0.722700	−0.361350	+	0.932430i	$0.617684\pi$
$858$	38.4980	1.31430
$859$	26.8328	0.915524	0.457762	−	0.889075i	$-0.348651\pi$
$859$	26.8328	0.915524	0.457762	+	0.889075i	$0.348651\pi$
$860$	0	0
$861$	7.49342	0.255375
$862$	20.4280	0.695782
$863$	9.31333	0.317029	0.158515	−	0.987357i	$-0.449329\pi$
$863$	9.31333	0.317029	0.158515	+	0.987357i	$0.449329\pi$
$864$	13.9443	0.474394
$865$	0	0
$866$	95.0132	3.22868
$867$	11.5322	0.391654
$868$	−8.08115	−0.274292
$869$	14.4721	0.490934
$870$	0	0
$871$	15.4164	0.522365
$872$	42.2274	1.43000
$873$	−0.728677	−0.0246620
$874$	27.8885	0.943344
$875$	0	0
$876$	−57.5967	−1.94601
$877$	−0.450347	−0.0152071	−0.00760357	−	0.999971i	$-0.502420\pi$
$877$	−0.450347	−0.0152071	−0.00760357	+	0.999971i	$0.502420\pi$
$878$	−51.5736	−1.74053
$879$	20.8754	0.704109
$880$	0	0
$881$	42.4508	1.43021	0.715103	−	0.699019i	$-0.246380\pi$
$881$	42.4508	1.43021	0.715103	+	0.699019i	$0.246380\pi$
$882$	9.39934	0.316492
$883$	23.7399	0.798913	0.399456	−	0.916752i	$-0.369199\pi$
$883$	23.7399	0.798913	0.399456	+	0.916752i	$0.369199\pi$
$884$	65.3050	2.19644
$885$	0	0
$886$	16.6738	0.560166
$887$	−52.5275	−1.76370	−0.881850	−	0.471530i	$-0.843702\pi$
$887$	−52.5275	−1.76370	−0.881850	+	0.471530i	$0.843702\pi$
$888$	−69.6436	−2.33709
$889$	−6.93112	−0.232462
$890$	0	0
$891$	−13.5279	−0.453200
$892$	−35.7757	−1.19786
$893$	−18.0700	−0.604689
$894$	55.0000	1.83948
$895$	0	0
$896$	15.9787	0.533811
$897$	31.1456	1.03992
$898$	24.9721	0.833330
$899$	−11.7082	−0.390490
$900$	0	0
$901$	−19.0557	−0.634838
$902$	25.4225	0.846476
$903$	14.1686	0.471501
$904$	−27.8885	−0.927559
$905$	0	0
$906$	16.1115	0.535267
$907$	39.4519	1.30998	0.654989	−	0.755638i	$-0.272673\pi$
$907$	39.4519	1.30998	0.654989	+	0.755638i	$0.272673\pi$
$908$	−22.1106	−0.733765
$909$	−0.0557281	−0.00184838
$910$	0	0
$911$	−42.7214	−1.41542	−0.707711	−	0.706502i	$-0.750272\pi$
$911$	−42.7214	−1.41542	−0.707711	+	0.706502i	$0.750272\pi$
$912$	−23.3427	−0.772956
$913$	−14.2546	−0.471758
$914$	9.52786	0.315154
$915$	0	0
$916$	−19.7984	−0.654157
$917$	−7.80282	−0.257672
$918$	−43.0421	−1.42060
$919$	−17.2361	−0.568565	−0.284283	−	0.958740i	$-0.591755\pi$
$919$	−17.2361	−0.568565	−0.284283	+	0.958740i	$0.591755\pi$
$920$	0	0
$921$	−37.0689	−1.22146
$922$	−30.6421	−1.00914
$923$	70.8226	2.33116
$924$	12.4721	0.410303
$925$	0	0
$926$	−101.464	−3.33431
$927$	5.72310	0.187971
$928$	−14.6189	−0.479890
$929$	−47.0344	−1.54315	−0.771575	−	0.636138i	$-0.780531\pi$
$929$	−47.0344	−1.54315	−0.771575	+	0.636138i	$0.780531\pi$
$930$	0	0
$931$	−16.8328	−0.551673
$932$	55.3890	1.81433
$933$	27.7805	0.909493
$934$	4.20163	0.137481
$935$	0	0
$936$	17.2361	0.563379
$937$	−32.3246	−1.05600	−0.527999	−	0.849245i	$-0.677058\pi$
$937$	−32.3246	−1.05600	−0.527999	+	0.849245i	$0.677058\pi$
$938$	7.35247	0.240067
$939$	46.2492	1.50929
$940$	0	0
$941$	9.63932	0.314233	0.157116	−	0.987580i	$-0.449780\pi$
$941$	9.63932	0.314233	0.157116	+	0.987580i	$0.449780\pi$
$942$	−74.1877	−2.41717
$943$	20.5672	0.669760
$944$	24.4721	0.796500
$945$	0	0
$946$	48.0689	1.56285
$947$	33.8680	1.10056	0.550280	−	0.834980i	$-0.314521\pi$
$947$	33.8680	1.10056	0.550280	+	0.834980i	$0.314521\pi$
$948$	−47.3079	−1.53649
$949$	−44.0000	−1.42830
$950$	0	0
$951$	−33.7771	−1.09530
$952$	16.4406	0.532844
$953$	6.17345	0.199978	0.0999888	−	0.994989i	$-0.468119\pi$
$953$	6.17345	0.199978	0.0999888	+	0.994989i	$0.468119\pi$
$954$	−9.52786	−0.308476
$955$	0	0
$956$	7.23607	0.234031
$957$	18.0700	0.584120
$958$	48.9372	1.58109
$959$	16.1115	0.520266
$960$	0	0
$961$	−27.0000	−0.870968
$962$	−100.789	−3.24957
$963$	8.30632	0.267667
$964$	42.7426	1.37665
$965$	0	0
$966$	14.8541	0.477923
$967$	4.63009	0.148894	0.0744468	−	0.997225i	$-0.476281\pi$
$967$	4.63009	0.148894	0.0744468	+	0.997225i	$0.476281\pi$
$968$	−39.0876	−1.25632
$969$	13.1672	0.422991
$970$	0	0
$971$	−24.1803	−0.775984	−0.387992	−	0.921663i	$-0.626831\pi$
$971$	−24.1803	−0.775984	−0.387992	+	0.921663i	$0.626831\pi$
$972$	−26.7407	−0.857707
$973$	−20.7064	−0.663816
$974$	−22.9098	−0.734078
$975$	0	0
$976$	−33.3262	−1.06675
$977$	−25.4225	−0.813337	−0.406668	−	0.913576i	$-0.633310\pi$
$977$	−25.4225	−0.813337	−0.406668	+	0.913576i	$0.633310\pi$
$978$	0.867842	0.0277505
$979$	−12.7639	−0.407937
$980$	0	0
$981$	−4.67376	−0.149222
$982$	−85.3556	−2.72381
$983$	43.9428	1.40156	0.700779	−	0.713378i	$-0.252836\pi$
$983$	43.9428	1.40156	0.700779	+	0.713378i	$0.252836\pi$
$984$	−43.8673	−1.39844
$985$	0	0
$986$	45.1246	1.43706
$987$	−9.62451	−0.306352
$988$	−58.4757	−1.86036
$989$	38.8885	1.23658
$990$	0	0
$991$	44.3607	1.40916	0.704582	−	0.709623i	$-0.251135\pi$
$991$	44.3607	1.40916	0.704582	+	0.709623i	$0.251135\pi$
$992$	4.99442	0.158573
$993$	−43.4925	−1.38019
$994$	33.7771	1.07134
$995$	0	0
$996$	46.5967	1.47647
$997$	10.7175	0.339427	0.169714	−	0.985493i	$-0.445716\pi$
$997$	10.7175	0.339427	0.169714	+	0.985493i	$0.445716\pi$
$998$	−15.4336	−0.488543
$999$	45.1246	1.42768

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	125.2.a.c.1.4	yes	4
3.2	odd	2		1125.2.a.k.1.1		4
4.3	odd	2		2000.2.a.o.1.3		4
5.2	odd	4		125.2.b.a.124.4		4
5.3	odd	4		125.2.b.a.124.1		4
5.4	even	2	inner	125.2.a.c.1.1	✓	4
7.6	odd	2		6125.2.a.o.1.4		4
8.3	odd	2		8000.2.a.bk.1.2		4
8.5	even	2		8000.2.a.bj.1.3		4
15.2	even	4		1125.2.b.a.874.1		4
15.8	even	4		1125.2.b.a.874.4		4
15.14	odd	2		1125.2.a.k.1.4		4
20.3	even	4		2000.2.c.c.1249.3		4
20.7	even	4		2000.2.c.c.1249.2		4
20.19	odd	2		2000.2.a.o.1.2		4
25.2	odd	20		625.2.e.h.124.2		8
25.3	odd	20		625.2.e.b.249.1		8
25.4	even	10		625.2.d.l.376.2		8
25.6	even	5		625.2.d.l.251.1		8
25.8	odd	20		625.2.e.b.374.2		8
25.9	even	10		625.2.d.k.126.1		8
25.11	even	5		625.2.d.k.501.2		8
25.12	odd	20		625.2.e.h.499.1		8
25.13	odd	20		625.2.e.h.499.2		8
25.14	even	10		625.2.d.k.501.1		8
25.16	even	5		625.2.d.k.126.2		8
25.17	odd	20		625.2.e.b.374.1		8
25.19	even	10		625.2.d.l.251.2		8
25.21	even	5		625.2.d.l.376.1		8
25.22	odd	20		625.2.e.b.249.2		8
25.23	odd	20		625.2.e.h.124.1		8
35.34	odd	2		6125.2.a.o.1.1		4
40.19	odd	2		8000.2.a.bk.1.3		4
40.29	even	2		8000.2.a.bj.1.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
125.2.a.c.1.1	✓	4	5.4	even	2	inner
125.2.a.c.1.4	yes	4	1.1	even	1	trivial
125.2.b.a.124.1		4	5.3	odd	4
125.2.b.a.124.4		4	5.2	odd	4
625.2.d.k.126.1		8	25.9	even	10
625.2.d.k.126.2		8	25.16	even	5
625.2.d.k.501.1		8	25.14	even	10
625.2.d.k.501.2		8	25.11	even	5
625.2.d.l.251.1		8	25.6	even	5
625.2.d.l.251.2		8	25.19	even	10
625.2.d.l.376.1		8	25.21	even	5
625.2.d.l.376.2		8	25.4	even	10
625.2.e.b.249.1		8	25.3	odd	20
625.2.e.b.249.2		8	25.22	odd	20
625.2.e.b.374.1		8	25.17	odd	20
625.2.e.b.374.2		8	25.8	odd	20
625.2.e.h.124.1		8	25.23	odd	20
625.2.e.h.124.2		8	25.2	odd	20
625.2.e.h.499.1		8	25.12	odd	20
625.2.e.h.499.2		8	25.13	odd	20
1125.2.a.k.1.1		4	3.2	odd	2
1125.2.a.k.1.4		4	15.14	odd	2
1125.2.b.a.874.1		4	15.2	even	4
1125.2.b.a.874.4		4	15.8	even	4
2000.2.a.o.1.2		4	20.19	odd	2
2000.2.a.o.1.3		4	4.3	odd	2
2000.2.c.c.1249.2		4	20.7	even	4
2000.2.c.c.1249.3		4	20.3	even	4
6125.2.a.o.1.1		4	35.34	odd	2
6125.2.a.o.1.4		4	7.6	odd	2
8000.2.a.bj.1.2		4	40.29	even	2
8000.2.a.bj.1.3		4	8.5	even	2
8000.2.a.bk.1.2		4	8.3	odd	2
8000.2.a.bk.1.3		4	40.19	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:	\( N \)	\(=\)	\( 125 = 5^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	125.a (trivial)

\(f(q)\)	\(=\)	\(q+2.49721 q^{2} -1.54336 q^{3} +4.23607 q^{4} -3.85410 q^{6} -0.953850 q^{7} +5.58394 q^{8} -0.618034 q^{9} +O(q^{10})\)	\(q+2.49721 q^{2} -1.54336 q^{3} +4.23607 q^{4} -3.85410 q^{6} -0.953850 q^{7} +5.58394 q^{8} -0.618034 q^{9} +2.00000 q^{11} -6.53779 q^{12} -4.99442 q^{13} -2.38197 q^{14} +5.47214 q^{16} -3.08672 q^{17} -1.54336 q^{18} +2.76393 q^{19} +1.47214 q^{21} +4.99442 q^{22} +4.04057 q^{23} -8.61803 q^{24} -12.4721 q^{26} +5.58394 q^{27} -4.04057 q^{28} -5.85410 q^{29} +2.00000 q^{31} +2.49721 q^{32} -3.08672 q^{33} -7.70820 q^{34} -2.61803 q^{36} +8.08115 q^{37} +6.90212 q^{38} +7.70820 q^{39} +5.09017 q^{41} +3.67624 q^{42} +9.62451 q^{43} +8.47214 q^{44} +10.0902 q^{46} -6.53779 q^{47} -8.44549 q^{48} -6.09017 q^{49} +4.76393 q^{51} -21.1567 q^{52} +6.17345 q^{53} +13.9443 q^{54} -5.32624 q^{56} -4.26575 q^{57} -14.6189 q^{58} +4.47214 q^{59} -6.09017 q^{61} +4.99442 q^{62} +0.589512 q^{63} -4.70820 q^{64} -7.70820 q^{66} -3.08672 q^{67} -13.0756 q^{68} -6.23607 q^{69} -14.1803 q^{71} -3.45106 q^{72} +8.80982 q^{73} +20.1803 q^{74} +11.7082 q^{76} -1.90770 q^{77} +19.2490 q^{78} +7.23607 q^{79} -6.76393 q^{81} +12.7112 q^{82} -7.12730 q^{83} +6.23607 q^{84} +24.0344 q^{86} +9.03500 q^{87} +11.1679 q^{88} -6.38197 q^{89} +4.76393 q^{91} +17.1161 q^{92} -3.08672 q^{93} -16.3262 q^{94} -3.85410 q^{96} +1.17902 q^{97} -15.2084 q^{98} -1.23607 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + 8 q^{4} - 2 q^{6} + 2 q^{9}+O(q^{10}) \) 4 * q + 8 * q^4 - 2 * q^6 + 2 * q^9	\( 4 q + 8 q^{4} - 2 q^{6} + 2 q^{9} + 8 q^{11} - 14 q^{14} + 4 q^{16} + 20 q^{19} - 12 q^{21} - 30 q^{24} - 32 q^{26} - 10 q^{29} + 8 q^{31} - 4 q^{34} - 6 q^{36} + 4 q^{39} - 2 q^{41} + 16 q^{44} + 18 q^{46} - 2 q^{49} + 28 q^{51} + 20 q^{54} + 10 q^{56} - 2 q^{61} + 8 q^{64} - 4 q^{66} - 16 q^{69} - 12 q^{71} + 36 q^{74} + 20 q^{76} + 20 q^{79} - 36 q^{81} + 16 q^{84} + 38 q^{86} - 30 q^{89} + 28 q^{91} - 34 q^{94} - 2 q^{96} + 4 q^{99}+O(q^{100}) \) 4 * q + 8 * q^4 - 2 * q^6 + 2 * q^9 + 8 * q^11 - 14 * q^14 + 4 * q^16 + 20 * q^19 - 12 * q^21 - 30 * q^24 - 32 * q^26 - 10 * q^29 + 8 * q^31 - 4 * q^34 - 6 * q^36 + 4 * q^39 - 2 * q^41 + 16 * q^44 + 18 * q^46 - 2 * q^49 + 28 * q^51 + 20 * q^54 + 10 * q^56 - 2 * q^61 + 8 * q^64 - 4 * q^66 - 16 * q^69 - 12 * q^71 + 36 * q^74 + 20 * q^76 + 20 * q^79 - 36 * q^81 + 16 * q^84 + 38 * q^86 - 30 * q^89 + 28 * q^91 - 34 * q^94 - 2 * q^96 + 4 * q^99

Introduction

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