LMFDB - Embedded newform 625.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [625,2,Mod(1,625)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(625, 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("625.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 625 = 5^{4} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	625.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:	$4.99065012633$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.8.14884000000.2
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 11x^{6} + 36x^{4} - 31x^{2} + 1 $ x^8 - 11x^6 + 36x^4 - 31*x^2 + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{13}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 25)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-0.183172$ of defining polynomial
Character	$\chi$	$=$	625.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.183172 q^{2} -1.47195 q^{3} -1.96645 q^{4} +0.269620 q^{6} -3.26086 q^{7} +0.726543 q^{8} -0.833366 q^{9} +O(q^{10})$	\(q-0.183172 q^{2} -1.47195 q^{3} -1.96645 q^{4} +0.269620 q^{6} -3.26086 q^{7} +0.726543 q^{8} -0.833366 q^{9} +2.00000 q^{11} +2.89451 q^{12} -0.296379 q^{13} +0.597298 q^{14} +3.79981 q^{16} -5.16297 q^{17} +0.152649 q^{18} +1.73038 q^{19} +4.79981 q^{21} -0.366344 q^{22} +0.879192 q^{23} -1.06943 q^{24} +0.0542883 q^{26} +5.64252 q^{27} +6.41230 q^{28} +5.91216 q^{29} +6.09953 q^{31} -2.14910 q^{32} -2.94390 q^{33} +0.945712 q^{34} +1.63877 q^{36} +8.08800 q^{37} -0.316957 q^{38} +0.436254 q^{39} -1.01515 q^{41} -0.879192 q^{42} -3.24199 q^{43} -3.93290 q^{44} -0.161043 q^{46} +4.21996 q^{47} -5.59313 q^{48} +3.63318 q^{49} +7.59963 q^{51} +0.582813 q^{52} -8.10072 q^{53} -1.03355 q^{54} -2.36915 q^{56} -2.54703 q^{57} -1.08294 q^{58} +5.93635 q^{59} +0.915615 q^{61} -1.11726 q^{62} +2.71749 q^{63} -7.20597 q^{64} +0.539240 q^{66} +6.88806 q^{67} +10.1527 q^{68} -1.29413 q^{69} -5.96878 q^{71} -0.605476 q^{72} +8.83341 q^{73} -1.48150 q^{74} -3.40270 q^{76} -6.52171 q^{77} -0.0799096 q^{78} -7.76067 q^{79} -5.80540 q^{81} +0.185946 q^{82} +14.5154 q^{83} -9.43858 q^{84} +0.593842 q^{86} -8.70240 q^{87} +1.45309 q^{88} +7.52642 q^{89} +0.966448 q^{91} -1.72889 q^{92} -8.97820 q^{93} -0.772978 q^{94} +3.16337 q^{96} -6.72649 q^{97} -0.665497 q^{98} -1.66673 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) $ 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9	$ 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9} + 16 q^{11} + 12 q^{14} - 2 q^{16} + 10 q^{19} + 6 q^{21} + 20 q^{24} + 6 q^{26} + 20 q^{29} + 16 q^{31} + 2 q^{34} - 12 q^{36} + 18 q^{39} + 26 q^{41} + 12 q^{44} + 6 q^{46} - 14 q^{49} - 4 q^{51} - 30 q^{54} + 10 q^{56} + 30 q^{59} + 6 q^{61} - 44 q^{64} + 12 q^{66} + 8 q^{69} + 46 q^{71} + 12 q^{74} - 20 q^{76} + 10 q^{79} - 32 q^{81} - 18 q^{84} - 14 q^{86} + 30 q^{89} - 14 q^{91} - 68 q^{94} - 54 q^{96} + 8 q^{99}+O(q^{100}) $ 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9 + 16 * q^11 + 12 * q^14 - 2 * q^16 + 10 * q^19 + 6 * q^21 + 20 * q^24 + 6 * q^26 + 20 * q^29 + 16 * q^31 + 2 * q^34 - 12 * q^36 + 18 * q^39 + 26 * q^41 + 12 * q^44 + 6 * q^46 - 14 * q^49 - 4 * q^51 - 30 * q^54 + 10 * q^56 + 30 * q^59 + 6 * q^61 - 44 * q^64 + 12 * q^66 + 8 * q^69 + 46 * q^71 + 12 * q^74 - 20 * q^76 + 10 * q^79 - 32 * q^81 - 18 * q^84 - 14 * q^86 + 30 * q^89 - 14 * q^91 - 68 * q^94 - 54 * q^96 + 8 * 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 $

$2$	−0.183172	−0.129522	−0.0647611	−	0.997901i	$-0.520629\pi$
$2$	−0.183172	−0.129522	−0.0647611	+	0.997901i	$0.520629\pi$
$3$	−1.47195	−0.849830	−0.424915	−	0.905233i	$-0.639696\pi$
$3$	−1.47195	−0.849830	−0.424915	+	0.905233i	$0.639696\pi$
$4$	−1.96645	−0.983224
$5$	0	0
$5$	0	0
$6$	0.269620	0.110072
$7$	−3.26086	−1.23249	−0.616244	−	0.787555i	$-0.711346\pi$
$7$	−3.26086	−1.23249	−0.616244	+	0.787555i	$0.711346\pi$
$8$	0.726543	0.256872
$9$	−0.833366	−0.277789
$10$	0	0
$11$	2.00000	0.603023	0.301511	−	0.953463i	$-0.402509\pi$
$11$	2.00000	0.603023	0.301511	+	0.953463i	$0.402509\pi$
$12$	2.89451	0.835573
$13$	−0.296379	−0.0822006	−0.0411003	−	0.999155i	$-0.513086\pi$
$13$	−0.296379	−0.0822006	−0.0411003	+	0.999155i	$0.513086\pi$
$14$	0.597298	0.159635
$15$	0	0
$16$	3.79981	0.949953
$17$	−5.16297	−1.25220	−0.626102	−	0.779741i	$-0.715351\pi$
$17$	−5.16297	−1.25220	−0.626102	+	0.779741i	$0.715351\pi$
$18$	0.152649	0.0359798
$19$	1.73038	0.396976	0.198488	−	0.980103i	$-0.436397\pi$
$19$	1.73038	0.396976	0.198488	+	0.980103i	$0.436397\pi$
$20$	0	0
$21$	4.79981	1.04741
$22$	−0.366344	−0.0781048
$23$	0.879192	0.183324	0.0916621	−	0.995790i	$-0.470782\pi$
$23$	0.879192	0.183324	0.0916621	+	0.995790i	$0.470782\pi$
$24$	−1.06943	−0.218297
$25$	0	0
$26$	0.0542883	0.0106468
$27$	5.64252	1.08590
$28$	6.41230	1.21181
$29$	5.91216	1.09786	0.548930	−	0.835868i	$-0.315035\pi$
$29$	5.91216	1.09786	0.548930	+	0.835868i	$0.315035\pi$
$30$	0	0
$31$	6.09953	1.09551	0.547754	−	0.836639i	$-0.315483\pi$
$31$	6.09953	1.09551	0.547754	+	0.836639i	$0.315483\pi$
$32$	−2.14910	−0.379912
$33$	−2.94390	−0.512467
$34$	0.945712	0.162188
$35$	0	0
$36$	1.63877	0.273128
$37$	8.08800	1.32966	0.664830	−	0.746995i	$-0.268504\pi$
$37$	8.08800	1.32966	0.664830	+	0.746995i	$0.268504\pi$
$38$	−0.316957	−0.0514173
$39$	0.436254	0.0698566
$40$	0	0
$41$	−1.01515	−0.158539	−0.0792695	−	0.996853i	$-0.525259\pi$
$41$	−1.01515	−0.158539	−0.0792695	+	0.996853i	$0.525259\pi$
$42$	−0.879192	−0.135662
$43$	−3.24199	−0.494399	−0.247200	−	0.968965i	$-0.579510\pi$
$43$	−3.24199	−0.494399	−0.247200	+	0.968965i	$0.579510\pi$
$44$	−3.93290	−0.592906
$45$	0	0
$46$	−0.161043	−0.0237446
$47$	4.21996	0.615544	0.307772	−	0.951460i	$-0.400417\pi$
$47$	4.21996	0.615544	0.307772	+	0.951460i	$0.400417\pi$
$48$	−5.59313	−0.807299
$49$	3.63318	0.519026
$50$	0	0
$51$	7.59963	1.06416
$52$	0.582813	0.0808216
$53$	−8.10072	−1.11272	−0.556360	−	0.830941i	$-0.687802\pi$
$53$	−8.10072	−1.11272	−0.556360	+	0.830941i	$0.687802\pi$
$54$	−1.03355	−0.140649
$55$	0	0
$56$	−2.36915	−0.316591
$57$	−2.54703	−0.337363
$58$	−1.08294	−0.142197
$59$	5.93635	0.772847	0.386424	−	0.922321i	$-0.373710\pi$
$59$	5.93635	0.772847	0.386424	+	0.922321i	$0.373710\pi$
$60$	0	0
$61$	0.915615	0.117232	0.0586162	−	0.998281i	$-0.481331\pi$
$61$	0.915615	0.117232	0.0586162	+	0.998281i	$0.481331\pi$
$62$	−1.11726	−0.141893
$63$	2.71749	0.342371
$64$	−7.20597	−0.900746
$65$	0	0
$66$	0.539240	0.0663759
$67$	6.88806	0.841510	0.420755	−	0.907174i	$-0.361765\pi$
$67$	6.88806	0.841510	0.420755	+	0.907174i	$0.361765\pi$
$68$	10.1527	1.23120
$69$	−1.29413	−0.155794
$70$	0	0
$71$	−5.96878	−0.708364	−0.354182	−	0.935177i	$-0.615241\pi$
$71$	−5.96878	−0.708364	−0.354182	+	0.935177i	$0.615241\pi$
$72$	−0.605476	−0.0713560
$73$	8.83341	1.03387	0.516936	−	0.856024i	$-0.327072\pi$
$73$	8.83341	1.03387	0.516936	+	0.856024i	$0.327072\pi$
$74$	−1.48150	−0.172220
$75$	0	0
$76$	−3.40270	−0.390317
$77$	−6.52171	−0.743218
$78$	−0.0799096	−0.00904798
$79$	−7.76067	−0.873144	−0.436572	−	0.899669i	$-0.643808\pi$
$79$	−7.76067	−0.873144	−0.436572	+	0.899669i	$0.643808\pi$
$80$	0	0
$81$	−5.80540	−0.645045
$82$	0.185946	0.0205343
$83$	14.5154	1.59327	0.796635	−	0.604461i	$-0.206612\pi$
$83$	14.5154	1.59327	0.796635	+	0.604461i	$0.206612\pi$
$84$	−9.43858	−1.02983
$85$	0	0
$86$	0.593842	0.0640357
$87$	−8.70240	−0.932995
$88$	1.45309	0.154899
$89$	7.52642	0.797799	0.398900	−	0.916995i	$-0.369392\pi$
$89$	7.52642	0.797799	0.398900	+	0.916995i	$0.369392\pi$
$90$	0	0
$91$	0.966448	0.101311
$92$	−1.72889	−0.180249
$93$	−8.97820	−0.930996
$94$	−0.772978	−0.0797266
$95$	0	0
$96$	3.16337	0.322860
$97$	−6.72649	−0.682971	−0.341486	−	0.939887i	$-0.610930\pi$
$97$	−6.72649	−0.682971	−0.341486	+	0.939887i	$0.610930\pi$
$98$	−0.665497	−0.0672254
$99$	−1.66673	−0.167513
$100$	0	0
$101$	−12.1955	−1.21350	−0.606748	−	0.794894i	$-0.707526\pi$
$101$	−12.1955	−1.21350	−0.606748	+	0.794894i	$0.707526\pi$
$102$	−1.39204	−0.137832
$103$	1.38140	0.136114	0.0680569	−	0.997681i	$-0.478320\pi$
$103$	1.38140	0.136114	0.0680569	+	0.997681i	$0.478320\pi$
$104$	−0.215332	−0.0211150
$105$	0	0
$106$	1.48383	0.144122
$107$	15.8285	1.53020	0.765101	−	0.643911i	$-0.222689\pi$
$107$	15.8285	1.53020	0.765101	+	0.643911i	$0.222689\pi$
$108$	−11.0957	−1.06769
$109$	−2.00377	−0.191926	−0.0959632	−	0.995385i	$-0.530593\pi$
$109$	−2.00377	−0.191926	−0.0959632	+	0.995385i	$0.530593\pi$
$110$	0	0
$111$	−11.9051	−1.12998
$112$	−12.3906	−1.17081
$113$	−10.4275	−0.980935	−0.490468	−	0.871459i	$-0.663174\pi$
$113$	−10.4275	−0.980935	−0.490468	+	0.871459i	$0.663174\pi$
$114$	0.466545	0.0436959
$115$	0	0
$116$	−11.6260	−1.07944
$117$	0.246992	0.0228344
$118$	−1.08737	−0.100101
$119$	16.8357	1.54333
$120$	0	0
$121$	−7.00000	−0.636364
$122$	−0.167715	−0.0151842
$123$	1.49424	0.134731
$124$	−11.9944	−1.07713
$125$	0	0
$126$	−0.497767	−0.0443446
$127$	−5.85007	−0.519110	−0.259555	−	0.965728i	$-0.583576\pi$
$127$	−5.85007	−0.519110	−0.259555	+	0.965728i	$0.583576\pi$
$128$	5.61814	0.496578
$129$	4.77205	0.420155
$130$	0	0
$131$	−1.49664	−0.130762	−0.0653811	−	0.997860i	$-0.520826\pi$
$131$	−1.49664	−0.130762	−0.0653811	+	0.997860i	$0.520826\pi$
$132$	5.78902	0.503870
$133$	−5.64252	−0.489268
$134$	−1.26170	−0.108994
$135$	0	0
$136$	−3.75112	−0.321656
$137$	−7.84887	−0.670574	−0.335287	−	0.942116i	$-0.608833\pi$
$137$	−7.84887	−0.670574	−0.335287	+	0.942116i	$0.608833\pi$
$138$	0.237048	0.0201788
$139$	5.39152	0.457303	0.228651	−	0.973508i	$-0.426568\pi$
$139$	5.39152	0.457303	0.228651	+	0.973508i	$0.426568\pi$
$140$	0	0
$141$	−6.21156	−0.523108
$142$	1.09331	0.0917488
$143$	−0.592757	−0.0495689
$144$	−3.16663	−0.263886
$145$	0	0
$146$	−1.61803	−0.133909
$147$	−5.34786	−0.441084
$148$	−15.9046	−1.30735
$149$	18.8229	1.54203	0.771015	−	0.636817i	$-0.219749\pi$
$149$	18.8229	1.54203	0.771015	+	0.636817i	$0.219749\pi$
$150$	0	0
$151$	−3.88797	−0.316398	−0.158199	−	0.987407i	$-0.550569\pi$
$151$	−3.88797	−0.316398	−0.158199	+	0.987407i	$0.550569\pi$
$152$	1.25719	0.101972
$153$	4.30264	0.347848
$154$	1.19460	0.0962632
$155$	0	0
$156$	−0.857871	−0.0686847
$157$	4.28378	0.341883	0.170941	−	0.985281i	$-0.445319\pi$
$157$	4.28378	0.341883	0.170941	+	0.985281i	$0.445319\pi$
$158$	1.42154	0.113092
$159$	11.9238	0.945623
$160$	0	0
$161$	−2.86692	−0.225945
$162$	1.06339	0.0835477
$163$	15.7263	1.23178	0.615890	−	0.787832i	$-0.288796\pi$
$163$	15.7263	1.23178	0.615890	+	0.787832i	$0.288796\pi$
$164$	1.99623	0.155879
$165$	0	0
$166$	−2.65881	−0.206364
$167$	21.0330	1.62758	0.813792	−	0.581156i	$-0.197399\pi$
$167$	21.0330	1.62758	0.813792	+	0.581156i	$0.197399\pi$
$168$	3.48727	0.269049
$169$	−12.9122	−0.993243
$170$	0	0
$171$	−1.44204	−0.110276
$172$	6.37521	0.486105
$173$	7.16663	0.544869	0.272434	−	0.962174i	$-0.412171\pi$
$173$	7.16663	0.544869	0.272434	+	0.962174i	$0.412171\pi$
$174$	1.59404	0.120844
$175$	0	0
$176$	7.59963	0.572843
$177$	−8.73801	−0.656789
$178$	−1.37863	−0.103333
$179$	8.03934	0.600888	0.300444	−	0.953799i	$-0.402865\pi$
$179$	8.03934	0.600888	0.300444	+	0.953799i	$0.402865\pi$
$180$	0	0
$181$	−20.6171	−1.53246	−0.766229	−	0.642568i	$-0.777869\pi$
$181$	−20.6171	−1.53246	−0.766229	+	0.642568i	$0.777869\pi$
$182$	−0.177026	−0.0131221
$183$	−1.34774	−0.0996277
$184$	0.638770	0.0470908
$185$	0	0
$186$	1.64456	0.120585
$187$	−10.3259	−0.755107
$188$	−8.29833	−0.605218
$189$	−18.3994	−1.33836
$190$	0	0
$191$	18.0303	1.30463	0.652313	−	0.757950i	$-0.273799\pi$
$191$	18.0303	1.30463	0.652313	+	0.757950i	$0.273799\pi$
$192$	10.6068	0.765482
$193$	6.78859	0.488653	0.244327	−	0.969693i	$-0.421433\pi$
$193$	6.78859	0.488653	0.244327	+	0.969693i	$0.421433\pi$
$194$	1.23210	0.0884599
$195$	0	0
$196$	−7.14446	−0.510318
$197$	7.99537	0.569647	0.284823	−	0.958580i	$-0.408065\pi$
$197$	7.99537	0.569647	0.284823	+	0.958580i	$0.408065\pi$
$198$	0.305299	0.0216966
$199$	−5.20485	−0.368962	−0.184481	−	0.982836i	$-0.559060\pi$
$199$	−5.20485	−0.368962	−0.184481	+	0.982836i	$0.559060\pi$
$200$	0	0
$201$	−10.1389	−0.715141
$202$	2.23387	0.157175
$203$	−19.2787	−1.35310
$204$	−14.9443	−1.04631
$205$	0	0
$206$	−0.253035	−0.0176298
$207$	−0.732688	−0.0509254
$208$	−1.12618	−0.0780868
$209$	3.46076	0.239386
$210$	0	0
$211$	16.6020	1.14293	0.571463	−	0.820628i	$-0.306376\pi$
$211$	16.6020	1.14293	0.571463	+	0.820628i	$0.306376\pi$
$212$	15.9296	1.09405
$213$	8.78574	0.601989
$214$	−2.89934	−0.198195
$215$	0	0
$216$	4.09953	0.278938
$217$	−19.8897	−1.35020
$218$	0.367035	0.0248587
$219$	−13.0023	−0.878616
$220$	0	0
$221$	1.53019	0.102932
$222$	2.18069	0.146358
$223$	6.62808	0.443849	0.221925	−	0.975064i	$-0.428766\pi$
$223$	6.62808	0.443849	0.221925	+	0.975064i	$0.428766\pi$
$224$	7.00792	0.468236
$225$	0	0
$226$	1.91002	0.127053
$227$	13.3919	0.888853	0.444427	−	0.895815i	$-0.353407\pi$
$227$	13.3919	0.888853	0.444427	+	0.895815i	$0.353407\pi$
$228$	5.00860	0.331703
$229$	10.0867	0.666549	0.333274	−	0.942830i	$-0.391846\pi$
$229$	10.0867	0.666549	0.333274	+	0.942830i	$0.391846\pi$
$230$	0	0
$231$	9.59963	0.631609
$232$	4.29544	0.282009
$233$	22.0055	1.44163	0.720814	−	0.693129i	$-0.243768\pi$
$233$	22.0055	1.44163	0.720814	+	0.693129i	$0.243768\pi$
$234$	−0.0452420	−0.00295756
$235$	0	0
$236$	−11.6735	−0.759882
$237$	11.4233	0.742024
$238$	−3.08383	−0.199895
$239$	7.56029	0.489035	0.244517	−	0.969645i	$-0.421371\pi$
$239$	7.56029	0.489035	0.244517	+	0.969645i	$0.421371\pi$
$240$	0	0
$241$	−20.3945	−1.31373	−0.656864	−	0.754009i	$-0.728117\pi$
$241$	−20.3945	−1.31373	−0.656864	+	0.754009i	$0.728117\pi$
$242$	1.28220	0.0824232
$243$	−8.38230	−0.537725
$244$	−1.80051	−0.115266
$245$	0	0
$246$	−0.273703	−0.0174507
$247$	−0.512848	−0.0326317
$248$	4.43157	0.281405
$249$	−21.3659	−1.35401
$250$	0	0
$251$	10.5717	0.667278	0.333639	−	0.942701i	$-0.391723\pi$
$251$	10.5717	0.667278	0.333639	+	0.942701i	$0.391723\pi$
$252$	−5.34379	−0.336627
$253$	1.75838	0.110549
$254$	1.07157	0.0672362
$255$	0	0
$256$	13.3829	0.836428
$257$	−20.2700	−1.26441	−0.632205	−	0.774801i	$-0.717850\pi$
$257$	−20.2700	−1.26441	−0.632205	+	0.774801i	$0.717850\pi$
$258$	−0.874106	−0.0544194
$259$	−26.3738	−1.63879
$260$	0	0
$261$	−4.92699	−0.304973
$262$	0.274143	0.0169366
$263$	−28.0909	−1.73216	−0.866079	−	0.499906i	$-0.833368\pi$
$263$	−28.0909	−1.73216	−0.866079	+	0.499906i	$0.833368\pi$
$264$	−2.13887	−0.131638
$265$	0	0
$266$	1.03355	0.0633711
$267$	−11.0785	−0.677994
$268$	−13.5450	−0.827393
$269$	20.3230	1.23911	0.619557	−	0.784952i	$-0.287312\pi$
$269$	20.3230	1.23911	0.619557	+	0.784952i	$0.287312\pi$
$270$	0	0
$271$	31.4467	1.91025	0.955125	−	0.296202i	$-0.0957200\pi$
$271$	31.4467	1.91025	0.955125	+	0.296202i	$0.0957200\pi$
$272$	−19.6183	−1.18954
$273$	−1.42256	−0.0860974
$274$	1.43769	0.0868543
$275$	0	0
$276$	2.54483	0.153181
$277$	13.9305	0.837001	0.418501	−	0.908217i	$-0.362556\pi$
$277$	13.9305	0.837001	0.418501	+	0.908217i	$0.362556\pi$
$278$	−0.987576	−0.0592309
$279$	−5.08314	−0.304320
$280$	0	0
$281$	25.8777	1.54373	0.771866	−	0.635785i	$-0.219323\pi$
$281$	25.8777	1.54373	0.771866	+	0.635785i	$0.219323\pi$
$282$	1.13778	0.0677541
$283$	−23.7316	−1.41070	−0.705350	−	0.708859i	$-0.749210\pi$
$283$	−23.7316	−1.41070	−0.705350	+	0.708859i	$0.749210\pi$
$284$	11.7373	0.696480
$285$	0	0
$286$	0.108577	0.00642027
$287$	3.31024	0.195397
$288$	1.79099	0.105535
$289$	9.65625	0.568014
$290$	0	0
$291$	9.90105	0.580410
$292$	−17.3704	−1.01653
$293$	12.3029	0.718742	0.359371	−	0.933195i	$-0.382991\pi$
$293$	12.3029	0.718742	0.359371	+	0.933195i	$0.382991\pi$
$294$	0.979578	0.0571301
$295$	0	0
$296$	5.87628	0.341552
$297$	11.2850	0.654824
$298$	−3.44783	−0.199727
$299$	−0.260574	−0.0150694
$300$	0	0
$301$	10.5717	0.609341
$302$	0.712167	0.0409806
$303$	17.9511	1.03127
$304$	6.57512	0.377109
$305$	0	0
$306$	−0.788124	−0.0450540
$307$	−4.28249	−0.244415	−0.122207	−	0.992505i	$-0.538997\pi$
$307$	−4.28249	−0.244415	−0.122207	+	0.992505i	$0.538997\pi$
$308$	12.8246	0.730750
$309$	−2.03336	−0.115674
$310$	0	0
$311$	25.6467	1.45429	0.727145	−	0.686484i	$-0.240847\pi$
$311$	25.6467	1.45429	0.727145	+	0.686484i	$0.240847\pi$
$312$	0.316957	0.0179442
$313$	22.3513	1.26337	0.631684	−	0.775226i	$-0.282364\pi$
$313$	22.3513	1.26337	0.631684	+	0.775226i	$0.282364\pi$
$314$	−0.784668	−0.0442814
$315$	0	0
$316$	15.2610	0.858496
$317$	−21.9228	−1.23131	−0.615655	−	0.788016i	$-0.711108\pi$
$317$	−21.9228	−1.23131	−0.615655	+	0.788016i	$0.711108\pi$
$318$	−2.18412	−0.122479
$319$	11.8243	0.662035
$320$	0	0
$321$	−23.2988	−1.30041
$322$	0.525139	0.0292649
$323$	−8.93390	−0.497095
$324$	11.4160	0.634224
$325$	0	0
$326$	−2.88062	−0.159543
$327$	2.94945	0.163105
$328$	−0.737546	−0.0407242
$329$	−13.7607	−0.758650
$330$	0	0
$331$	−8.96299	−0.492651	−0.246325	−	0.969187i	$-0.579223\pi$
$331$	−8.96299	−0.492651	−0.246325	+	0.969187i	$0.579223\pi$
$332$	−28.5437	−1.56654
$333$	−6.74026	−0.369364
$334$	−3.85266	−0.210808
$335$	0	0
$336$	18.2384	0.994986
$337$	29.0836	1.58428	0.792141	−	0.610338i	$-0.208966\pi$
$337$	29.0836	1.58428	0.792141	+	0.610338i	$0.208966\pi$
$338$	2.36515	0.128647
$339$	15.3487	0.833628
$340$	0	0
$341$	12.1991	0.660616
$342$	0.264141	0.0142831
$343$	10.9787	0.592795
$344$	−2.35544	−0.126997
$345$	0	0
$346$	−1.31273	−0.0705726
$347$	−14.2972	−0.767513	−0.383757	−	0.923434i	$-0.625370\pi$
$347$	−14.2972	−0.767513	−0.383757	+	0.923434i	$0.625370\pi$
$348$	17.1128	0.917343
$349$	5.62382	0.301036	0.150518	−	0.988607i	$-0.451906\pi$
$349$	5.62382	0.301036	0.150518	+	0.988607i	$0.451906\pi$
$350$	0	0
$351$	−1.67232	−0.0892620
$352$	−4.29821	−0.229095
$353$	−1.90997	−0.101658	−0.0508288	−	0.998707i	$-0.516186\pi$
$353$	−1.90997	−0.101658	−0.0508288	+	0.998707i	$0.516186\pi$
$354$	1.60056	0.0850688
$355$	0	0
$356$	−14.8003	−0.784415
$357$	−24.7813	−1.31156
$358$	−1.47258	−0.0778284
$359$	22.2047	1.17192	0.585958	−	0.810341i	$-0.300718\pi$
$359$	22.2047	1.17192	0.585958	+	0.810341i	$0.300718\pi$
$360$	0	0
$361$	−16.0058	−0.842410
$362$	3.77648	0.198487
$363$	10.3036	0.540801
$364$	−1.90047	−0.0996117
$365$	0	0
$366$	0.246868	0.0129040
$367$	21.4450	1.11942	0.559709	−	0.828689i	$-0.310913\pi$
$367$	21.4450	1.11942	0.559709	+	0.828689i	$0.310913\pi$
$368$	3.34077	0.174149
$369$	0.845987	0.0440403
$370$	0	0
$371$	26.4153	1.37141
$372$	17.6552	0.915377
$373$	−23.1399	−1.19814	−0.599070	−	0.800697i	$-0.704463\pi$
$373$	−23.1399	−1.19814	−0.599070	+	0.800697i	$0.704463\pi$
$374$	1.89142	0.0978032
$375$	0	0
$376$	3.06598	0.158116
$377$	−1.75224	−0.0902448
$378$	3.37026	0.173348
$379$	−21.6501	−1.11209	−0.556047	−	0.831151i	$-0.687682\pi$
$379$	−21.6501	−1.11209	−0.556047	+	0.831151i	$0.687682\pi$
$380$	0	0
$381$	8.61100	0.441155
$382$	−3.30265	−0.168978
$383$	−5.65481	−0.288947	−0.144474	−	0.989509i	$-0.546149\pi$
$383$	−5.65481	−0.288947	−0.144474	+	0.989509i	$0.546149\pi$
$384$	−8.26962	−0.422007
$385$	0	0
$386$	−1.24348	−0.0632915
$387$	2.70176	0.137338
$388$	13.2273	0.671514
$389$	−8.12605	−0.412007	−0.206004	−	0.978551i	$-0.566046\pi$
$389$	−8.12605	−0.412007	−0.206004	+	0.978551i	$0.566046\pi$
$390$	0	0
$391$	−4.53924	−0.229559
$392$	2.63966	0.133323
$393$	2.20298	0.111126
$394$	−1.46453	−0.0737819
$395$	0	0
$396$	3.27754	0.164703
$397$	20.7689	1.04236	0.521180	−	0.853447i	$-0.325492\pi$
$397$	20.7689	1.04236	0.521180	+	0.853447i	$0.325492\pi$
$398$	0.953382	0.0477887
$399$	8.30550	0.415795
$400$	0	0
$401$	30.1195	1.50410	0.752049	−	0.659107i	$-0.229066\pi$
$401$	30.1195	1.50410	0.752049	+	0.659107i	$0.229066\pi$
$402$	1.85716	0.0926266
$403$	−1.80777	−0.0900515
$404$	23.9818	1.19314
$405$	0	0
$406$	3.53132	0.175256
$407$	16.1760	0.801815
$408$	5.52145	0.273353
$409$	11.0452	0.546152	0.273076	−	0.961992i	$-0.411959\pi$
$409$	11.0452	0.546152	0.273076	+	0.961992i	$0.411959\pi$
$410$	0	0
$411$	11.5531	0.569874
$412$	−2.71646	−0.133830
$413$	−19.3576	−0.952524
$414$	0.134208	0.00659597
$415$	0	0
$416$	0.636949	0.0312290
$417$	−7.93604	−0.388630
$418$	−0.633915	−0.0310058
$419$	−32.9212	−1.60830	−0.804152	−	0.594424i	$-0.797380\pi$
$419$	−32.9212	−1.60830	−0.804152	+	0.594424i	$0.797380\pi$
$420$	0	0
$421$	−20.2647	−0.987642	−0.493821	−	0.869564i	$-0.664400\pi$
$421$	−20.2647	−0.987642	−0.493821	+	0.869564i	$0.664400\pi$
$422$	−3.04101	−0.148034
$423$	−3.51677	−0.170991
$424$	−5.88552	−0.285826
$425$	0	0
$426$	−1.60930	−0.0779709
$427$	−2.98569	−0.144488
$428$	−31.1260	−1.50453
$429$	0.872509	0.0421251
$430$	0	0
$431$	7.15526	0.344657	0.172328	−	0.985040i	$-0.444871\pi$
$431$	7.15526	0.344657	0.172328	+	0.985040i	$0.444871\pi$
$432$	21.4405	1.03156
$433$	5.48264	0.263479	0.131739	−	0.991284i	$-0.457944\pi$
$433$	5.48264	0.263479	0.131739	+	0.991284i	$0.457944\pi$
$434$	3.64324	0.174881
$435$	0	0
$436$	3.94031	0.188707
$437$	1.52134	0.0727754
$438$	2.38166	0.113800
$439$	−30.7561	−1.46791	−0.733954	−	0.679199i	$-0.762327\pi$
$439$	−30.7561	−1.46791	−0.733954	+	0.679199i	$0.762327\pi$
$440$	0	0
$441$	−3.02777	−0.144179
$442$	−0.280289	−0.0133320
$443$	−11.3527	−0.539381	−0.269691	−	0.962947i	$-0.586921\pi$
$443$	−11.3527	−0.539381	−0.269691	+	0.962947i	$0.586921\pi$
$444$	23.4108	1.11103
$445$	0	0
$446$	−1.21408	−0.0574883
$447$	−27.7063	−1.31046
$448$	23.4976	1.11016
$449$	−15.7661	−0.744050	−0.372025	−	0.928223i	$-0.621336\pi$
$449$	−15.7661	−0.744050	−0.372025	+	0.928223i	$0.621336\pi$
$450$	0	0
$451$	−2.03029	−0.0956027
$452$	20.5051	0.964479
$453$	5.72289	0.268885
$454$	−2.45303	−0.115126
$455$	0	0
$456$	−1.85053	−0.0866588
$457$	4.16714	0.194931	0.0974653	−	0.995239i	$-0.468926\pi$
$457$	4.16714	0.194931	0.0974653	+	0.995239i	$0.468926\pi$
$458$	−1.84760	−0.0863329
$459$	−29.1322	−1.35977
$460$	0	0
$461$	−23.9714	−1.11646	−0.558230	−	0.829686i	$-0.688519\pi$
$461$	−23.9714	−1.11646	−0.558230	+	0.829686i	$0.688519\pi$
$462$	−1.75838	−0.0818074
$463$	−41.3247	−1.92052	−0.960260	−	0.279107i	$-0.909962\pi$
$463$	−41.3247	−1.92052	−0.960260	+	0.279107i	$0.909962\pi$
$464$	22.4651	1.04292
$465$	0	0
$466$	−4.03079	−0.186723
$467$	10.3966	0.481097	0.240548	−	0.970637i	$-0.422673\pi$
$467$	10.3966	0.481097	0.240548	+	0.970637i	$0.422673\pi$
$468$	−0.485697	−0.0224513
$469$	−22.4610	−1.03715
$470$	0	0
$471$	−6.30550	−0.290542
$472$	4.31301	0.198522
$473$	−6.48398	−0.298134
$474$	−2.09243	−0.0961086
$475$	0	0
$476$	−33.1065	−1.51743
$477$	6.75086	0.309101
$478$	−1.38483	−0.0633408
$479$	36.0081	1.64525	0.822626	−	0.568582i	$-0.192508\pi$
$479$	36.0081	1.64525	0.822626	+	0.568582i	$0.192508\pi$
$480$	0	0
$481$	−2.39711	−0.109299
$482$	3.73571	0.170157
$483$	4.21996	0.192015
$484$	13.7651	0.625688
$485$	0	0
$486$	1.53540	0.0696473
$487$	−10.6378	−0.482045	−0.241023	−	0.970520i	$-0.577483\pi$
$487$	−10.6378	−0.482045	−0.241023	+	0.970520i	$0.577483\pi$
$488$	0.665233	0.0301137
$489$	−23.1483	−1.04680
$490$	0	0
$491$	−17.6693	−0.797402	−0.398701	−	0.917081i	$-0.630539\pi$
$491$	−17.6693	−0.797402	−0.398701	+	0.917081i	$0.630539\pi$
$492$	−2.93835	−0.132471
$493$	−30.5243	−1.37475
$494$	0.0939394	0.00422653
$495$	0	0
$496$	23.1771	1.04068
$497$	19.4633	0.873049
$498$	3.91363	0.175374
$499$	−9.41734	−0.421578	−0.210789	−	0.977532i	$-0.567603\pi$
$499$	−9.41734	−0.421578	−0.210789	+	0.977532i	$0.567603\pi$
$500$	0	0
$501$	−30.9595	−1.38317
$502$	−1.93643	−0.0864273
$503$	18.0133	0.803172	0.401586	−	0.915821i	$-0.368459\pi$
$503$	18.0133	0.803172	0.401586	+	0.915821i	$0.368459\pi$
$504$	1.97437	0.0879454
$505$	0	0
$506$	−0.322087	−0.0143185
$507$	19.0060	0.844088
$508$	11.5039	0.510401
$509$	−16.0485	−0.711337	−0.355669	−	0.934612i	$-0.615747\pi$
$509$	−16.0485	−0.711337	−0.355669	+	0.934612i	$0.615747\pi$
$510$	0	0
$511$	−28.8045	−1.27423
$512$	−13.6877	−0.604914
$513$	9.76370	0.431078
$514$	3.71291	0.163769
$515$	0	0
$516$	−9.38398	−0.413107
$517$	8.43991	0.371187
$518$	4.83095	0.212260
$519$	−10.5489	−0.463046
$520$	0	0
$521$	−2.20069	−0.0964142	−0.0482071	−	0.998837i	$-0.515351\pi$
$521$	−2.20069	−0.0964142	−0.0482071	+	0.998837i	$0.515351\pi$
$522$	0.902487	0.0395008
$523$	7.43588	0.325148	0.162574	−	0.986696i	$-0.448020\pi$
$523$	7.43588	0.325148	0.162574	+	0.986696i	$0.448020\pi$
$524$	2.94307	0.128569
$525$	0	0
$526$	5.14547	0.224353
$527$	−31.4917	−1.37180
$528$	−11.1863	−0.486820
$529$	−22.2270	−0.966392
$530$	0	0
$531$	−4.94715	−0.214688
$532$	11.0957	0.481061
$533$	0.300867	0.0130320
$534$	2.02927	0.0878153
$535$	0	0
$536$	5.00447	0.216160
$537$	−11.8335	−0.510653
$538$	−3.72260	−0.160493
$539$	7.26636	0.312984
$540$	0	0
$541$	20.6474	0.887701	0.443850	−	0.896101i	$-0.353612\pi$
$541$	20.6474	0.887701	0.443850	+	0.896101i	$0.353612\pi$
$542$	−5.76016	−0.247420
$543$	30.3473	1.30233
$544$	11.0958	0.475727
$545$	0	0
$546$	0.260574	0.0111515
$547$	−31.3762	−1.34155	−0.670774	−	0.741662i	$-0.734038\pi$
$547$	−31.3762	−1.34155	−0.670774	+	0.741662i	$0.734038\pi$
$548$	15.4344	0.659325
$549$	−0.763042	−0.0325658
$550$	0	0
$551$	10.2303	0.435825
$552$	−0.940237	−0.0400192
$553$	25.3064	1.07614
$554$	−2.55167	−0.108410
$555$	0	0
$556$	−10.6021	−0.449631
$557$	22.3515	0.947064	0.473532	−	0.880776i	$-0.342979\pi$
$557$	22.3515	0.947064	0.473532	+	0.880776i	$0.342979\pi$
$558$	0.931089	0.0394161
$559$	0.960857	0.0406399
$560$	0	0
$561$	15.1993	0.641713
$562$	−4.74007	−0.199948
$563$	34.3663	1.44837	0.724184	−	0.689607i	$-0.242217\pi$
$563$	34.3663	1.44837	0.724184	+	0.689607i	$0.242217\pi$
$564$	12.2147	0.514332
$565$	0	0
$566$	4.34697	0.182717
$567$	18.9306	0.795010
$568$	−4.33657	−0.181958
$569$	32.1662	1.34848	0.674239	−	0.738513i	$-0.264472\pi$
$569$	32.1662	1.34848	0.674239	+	0.738513i	$0.264472\pi$
$570$	0	0
$571$	−27.1668	−1.13690	−0.568448	−	0.822719i	$-0.692456\pi$
$571$	−27.1668	−1.13690	−0.568448	+	0.822719i	$0.692456\pi$
$572$	1.16563	0.0487373
$573$	−26.5397	−1.10871
$574$	−0.606344	−0.0253083
$575$	0	0
$576$	6.00521	0.250217
$577$	−13.8503	−0.576596	−0.288298	−	0.957541i	$-0.593089\pi$
$577$	−13.8503	−0.576596	−0.288298	+	0.957541i	$0.593089\pi$
$578$	−1.76875	−0.0735705
$579$	−9.99246	−0.415273
$580$	0	0
$581$	−47.3325	−1.96368
$582$	−1.81360	−0.0751759
$583$	−16.2014	−0.670995
$584$	6.41785	0.265572
$585$	0	0
$586$	−2.25354	−0.0930930
$587$	−44.1499	−1.82226	−0.911131	−	0.412118i	$-0.864789\pi$
$587$	−44.1499	−1.82226	−0.911131	+	0.412118i	$0.864789\pi$
$588$	10.5163	0.433684
$589$	10.5545	0.434891
$590$	0	0
$591$	−11.7688	−0.484103
$592$	30.7329	1.26311
$593$	16.2531	0.667437	0.333718	−	0.942673i	$-0.391697\pi$
$593$	16.2531	0.667437	0.333718	+	0.942673i	$0.391697\pi$
$594$	−2.06710	−0.0848143
$595$	0	0
$596$	−37.0142	−1.51616
$597$	7.66127	0.313555
$598$	0.0477298	0.00195182
$599$	30.4822	1.24547	0.622734	−	0.782433i	$-0.286022\pi$
$599$	30.4822	1.24547	0.622734	+	0.782433i	$0.286022\pi$
$600$	0	0
$601$	−28.9162	−1.17952	−0.589758	−	0.807580i	$-0.700777\pi$
$601$	−28.9162	−1.17952	−0.589758	+	0.807580i	$0.700777\pi$
$602$	−1.93643	−0.0789232
$603$	−5.74027	−0.233762
$604$	7.64549	0.311090
$605$	0	0
$606$	−3.28815	−0.133572
$607$	−8.23276	−0.334157	−0.167079	−	0.985944i	$-0.553433\pi$
$607$	−8.23276	−0.334157	−0.167079	+	0.985944i	$0.553433\pi$
$608$	−3.71877	−0.150816
$609$	28.3773	1.14990
$610$	0	0
$611$	−1.25071	−0.0505981
$612$	−8.46092	−0.342012
$613$	−4.79811	−0.193794	−0.0968969	−	0.995294i	$-0.530892\pi$
$613$	−4.79811	−0.193794	−0.0968969	+	0.995294i	$0.530892\pi$
$614$	0.784432	0.0316571
$615$	0	0
$616$	−4.73830	−0.190912
$617$	2.03184	0.0817986	0.0408993	−	0.999163i	$-0.486978\pi$
$617$	2.03184	0.0817986	0.0408993	+	0.999163i	$0.486978\pi$
$618$	0.372454	0.0149823
$619$	8.14100	0.327215	0.163607	−	0.986526i	$-0.447687\pi$
$619$	8.14100	0.327215	0.163607	+	0.986526i	$0.447687\pi$
$620$	0	0
$621$	4.96086	0.199072
$622$	−4.69776	−0.188363
$623$	−24.5426	−0.983278
$624$	1.65769	0.0663605
$625$	0	0
$626$	−4.09413	−0.163634
$627$	−5.09406	−0.203437
$628$	−8.42382	−0.336147
$629$	−41.7581	−1.66500
$630$	0	0
$631$	33.2847	1.32504	0.662522	−	0.749042i	$-0.269486\pi$
$631$	33.2847	1.32504	0.662522	+	0.749042i	$0.269486\pi$
$632$	−5.63846	−0.224286
$633$	−24.4372	−0.971293
$634$	4.01565	0.159482
$635$	0	0
$636$	−23.4476	−0.929759
$637$	−1.07680	−0.0426642
$638$	−2.16589	−0.0857482
$639$	4.97417	0.196775
$640$	0	0
$641$	−40.0481	−1.58180	−0.790902	−	0.611943i	$-0.790388\pi$
$641$	−40.0481	−1.58180	−0.790902	+	0.611943i	$0.790388\pi$
$642$	4.26769	0.168432
$643$	11.6870	0.460890	0.230445	−	0.973085i	$-0.425982\pi$
$643$	11.6870	0.460890	0.230445	+	0.973085i	$0.425982\pi$
$644$	5.63764	0.222154
$645$	0	0
$646$	1.63644	0.0643849
$647$	7.94936	0.312522	0.156261	−	0.987716i	$-0.450056\pi$
$647$	7.94936	0.312522	0.156261	+	0.987716i	$0.450056\pi$
$648$	−4.21787	−0.165694
$649$	11.8727	0.466044
$650$	0	0
$651$	29.2766	1.14744
$652$	−30.9250	−1.21112
$653$	−3.33736	−0.130601	−0.0653005	−	0.997866i	$-0.520801\pi$
$653$	−3.33736	−0.130601	−0.0653005	+	0.997866i	$0.520801\pi$
$654$	−0.540256	−0.0211257
$655$	0	0
$656$	−3.85736	−0.150605
$657$	−7.36146	−0.287198
$658$	2.52057	0.0982621
$659$	−30.0508	−1.17061	−0.585306	−	0.810812i	$-0.699026\pi$
$659$	−30.0508	−1.17061	−0.585306	+	0.810812i	$0.699026\pi$
$660$	0	0
$661$	−6.58417	−0.256094	−0.128047	−	0.991768i	$-0.540871\pi$
$661$	−6.58417	−0.256094	−0.128047	+	0.991768i	$0.540871\pi$
$662$	1.64177	0.0638092
$663$	−2.25237	−0.0874747
$664$	10.5460	0.409266
$665$	0	0
$666$	1.23463	0.0478409
$667$	5.19792	0.201264
$668$	−41.3603	−1.60028
$669$	−9.75620	−0.377196
$670$	0	0
$671$	1.83123	0.0706939
$672$	−10.3153	−0.397921
$673$	−6.73140	−0.259476	−0.129738	−	0.991548i	$-0.541414\pi$
$673$	−6.73140	−0.259476	−0.129738	+	0.991548i	$0.541414\pi$
$674$	−5.32730	−0.205200
$675$	0	0
$676$	25.3911	0.976580
$677$	−13.6478	−0.524529	−0.262265	−	0.964996i	$-0.584469\pi$
$677$	−13.6478	−0.524529	−0.262265	+	0.964996i	$0.584469\pi$
$678$	−2.81146	−0.107973
$679$	21.9341	0.841753
$680$	0	0
$681$	−19.7122	−0.755374
$682$	−2.23453	−0.0855645
$683$	−1.17220	−0.0448529	−0.0224265	−	0.999748i	$-0.507139\pi$
$683$	−1.17220	−0.0448529	−0.0224265	+	0.999748i	$0.507139\pi$
$684$	2.83570	0.108426
$685$	0	0
$686$	−2.01099	−0.0767801
$687$	−14.8471	−0.566453
$688$	−12.3190	−0.469656
$689$	2.40088	0.0914663
$690$	0	0
$691$	12.2674	0.466673	0.233336	−	0.972396i	$-0.425036\pi$
$691$	12.2674	0.466673	0.233336	+	0.972396i	$0.425036\pi$
$692$	−14.0928	−0.535728
$693$	5.43497	0.206457
$694$	2.61885	0.0994100
$695$	0	0
$696$	−6.32266	−0.239660
$697$	5.24116	0.198523
$698$	−1.03013	−0.0389909
$699$	−32.3910	−1.22514
$700$	0	0
$701$	−20.0271	−0.756415	−0.378207	−	0.925721i	$-0.623459\pi$
$701$	−20.0271	−0.756415	−0.378207	+	0.925721i	$0.623459\pi$
$702$	0.306323	0.0115614
$703$	13.9953	0.527843
$704$	−14.4119	−0.543171
$705$	0	0
$706$	0.349854	0.0131669
$707$	39.7677	1.49562
$708$	17.1828	0.645771
$709$	−4.39510	−0.165061	−0.0825306	−	0.996589i	$-0.526300\pi$
$709$	−4.39510	−0.165061	−0.0825306	+	0.996589i	$0.526300\pi$
$710$	0	0
$711$	6.46748	0.242549
$712$	5.46827	0.204932
$713$	5.36266	0.200833
$714$	4.53924	0.169877
$715$	0	0
$716$	−15.8089	−0.590808
$717$	−11.1284	−0.415596
$718$	−4.06727	−0.151789
$719$	−10.9283	−0.407557	−0.203779	−	0.979017i	$-0.565322\pi$
$719$	−10.9283	−0.407557	−0.203779	+	0.979017i	$0.565322\pi$
$720$	0	0
$721$	−4.50456	−0.167759
$722$	2.93181	0.109111
$723$	30.0197	1.11645
$724$	40.5425	1.50675
$725$	0	0
$726$	−1.88734	−0.0700458
$727$	32.9032	1.22031	0.610156	−	0.792281i	$-0.291107\pi$
$727$	32.9032	1.22031	0.610156	+	0.792281i	$0.291107\pi$
$728$	0.702166	0.0260240
$729$	29.7545	1.10202
$730$	0	0
$731$	16.7383	0.619088
$732$	2.65026	0.0979564
$733$	13.7498	0.507859	0.253929	−	0.967223i	$-0.418277\pi$
$733$	13.7498	0.507859	0.253929	+	0.967223i	$0.418277\pi$
$734$	−3.92812	−0.144990
$735$	0	0
$736$	−1.88948	−0.0696470
$737$	13.7761	0.507450
$738$	−0.154961	−0.00570420
$739$	−43.1893	−1.58874	−0.794372	−	0.607432i	$-0.792200\pi$
$739$	−43.1893	−1.58874	−0.794372	+	0.607432i	$0.792200\pi$
$740$	0	0
$741$	0.754886	0.0277314
$742$	−4.83854	−0.177628
$743$	31.8479	1.16838	0.584192	−	0.811615i	$-0.301411\pi$
$743$	31.8479	1.16838	0.584192	+	0.811615i	$0.301411\pi$
$744$	−6.52304	−0.239146
$745$	0	0
$746$	4.23859	0.155186
$747$	−12.0966	−0.442592
$748$	20.3054	0.742440
$749$	−51.6145	−1.88595
$750$	0	0
$751$	−29.5952	−1.07995	−0.539973	−	0.841682i	$-0.681565\pi$
$751$	−29.5952	−1.07995	−0.539973	+	0.841682i	$0.681565\pi$
$752$	16.0351	0.584738
$753$	−15.5610	−0.567073
$754$	0.320961	0.0116887
$755$	0	0
$756$	36.1815	1.31591
$757$	−0.0984401	−0.00357786	−0.00178893	−	0.999998i	$-0.500569\pi$
$757$	−0.0984401	−0.00357786	−0.00178893	+	0.999998i	$0.500569\pi$
$758$	3.96570	0.144041
$759$	−2.58825	−0.0939476
$760$	0	0
$761$	−3.54402	−0.128471	−0.0642353	−	0.997935i	$-0.520461\pi$
$761$	−3.54402	−0.128471	−0.0642353	+	0.997935i	$0.520461\pi$
$762$	−1.57730	−0.0571394
$763$	6.53400	0.236547
$764$	−35.4556	−1.28274
$765$	0	0
$766$	1.03580	0.0374251
$767$	−1.75941	−0.0635285
$768$	−19.6989	−0.710822
$769$	1.42759	0.0514802	0.0257401	−	0.999669i	$-0.491806\pi$
$769$	1.42759	0.0514802	0.0257401	+	0.999669i	$0.491806\pi$
$770$	0	0
$771$	29.8365	1.07453
$772$	−13.3494	−0.480456
$773$	33.7807	1.21501	0.607504	−	0.794317i	$-0.292171\pi$
$773$	33.7807	1.21501	0.607504	+	0.794317i	$0.292171\pi$
$774$	−0.494888	−0.0177884
$775$	0	0
$776$	−4.88708	−0.175436
$777$	38.8209	1.39269
$778$	1.48847	0.0533641
$779$	−1.75659	−0.0629363
$780$	0	0
$781$	−11.9376	−0.427159
$782$	0.831462	0.0297330
$783$	33.3595	1.19217
$784$	13.8054	0.493050
$785$	0	0
$786$	−0.403525	−0.0143932
$787$	2.18030	0.0777192	0.0388596	−	0.999245i	$-0.487627\pi$
$787$	2.18030	0.0777192	0.0388596	+	0.999245i	$0.487627\pi$
$788$	−15.7225	−0.560090
$789$	41.3484	1.47204
$790$	0	0
$791$	34.0025	1.20899
$792$	−1.21095	−0.0430293
$793$	−0.271369	−0.00963659
$794$	−3.80428	−0.135009
$795$	0	0
$796$	10.2351	0.362772
$797$	23.4919	0.832124	0.416062	−	0.909336i	$-0.363410\pi$
$797$	23.4919	0.832124	0.416062	+	0.909336i	$0.363410\pi$
$798$	−1.52134	−0.0538547
$799$	−21.7875	−0.770787
$800$	0	0
$801$	−6.27226	−0.221620
$802$	−5.51706	−0.194814
$803$	17.6668	0.623449
$804$	19.9376	0.703143
$805$	0	0
$806$	0.331133	0.0116637
$807$	−29.9144	−1.05304
$808$	−8.86054	−0.311713
$809$	−38.1509	−1.34132	−0.670658	−	0.741767i	$-0.733988\pi$
$809$	−38.1509	−1.34132	−0.670658	+	0.741767i	$0.733988\pi$
$810$	0	0
$811$	47.9069	1.68224	0.841120	−	0.540849i	$-0.181897\pi$
$811$	47.9069	1.68224	0.841120	+	0.540849i	$0.181897\pi$
$812$	37.9106	1.33040
$813$	−46.2879	−1.62339
$814$	−2.96299	−0.103853
$815$	0	0
$816$	28.8772	1.01090
$817$	−5.60988	−0.196265
$818$	−2.02318	−0.0707388
$819$	−0.805405	−0.0281431
$820$	0	0
$821$	18.9808	0.662434	0.331217	−	0.943555i	$-0.392541\pi$
$821$	18.9808	0.662434	0.331217	+	0.943555i	$0.392541\pi$
$822$	−2.11621	−0.0738114
$823$	−22.1681	−0.772732	−0.386366	−	0.922345i	$-0.626270\pi$
$823$	−22.1681	−0.772732	−0.386366	+	0.922345i	$0.626270\pi$
$824$	1.00365	0.0349638
$825$	0	0
$826$	3.54577	0.123373
$827$	−4.72938	−0.164457	−0.0822283	−	0.996614i	$-0.526204\pi$
$827$	−4.72938	−0.164457	−0.0822283	+	0.996614i	$0.526204\pi$
$828$	1.44079	0.0500710
$829$	16.4400	0.570986	0.285493	−	0.958381i	$-0.407843\pi$
$829$	16.4400	0.570986	0.285493	+	0.958381i	$0.407843\pi$
$830$	0	0
$831$	−20.5050	−0.711309
$832$	2.13570	0.0740419
$833$	−18.7580	−0.649926
$834$	1.45366	0.0503362
$835$	0	0
$836$	−6.80540	−0.235370
$837$	34.4167	1.18962
$838$	6.03024	0.208311
$839$	−5.60083	−0.193362	−0.0966811	−	0.995315i	$-0.530823\pi$
$839$	−5.60083	−0.193362	−0.0966811	+	0.995315i	$0.530823\pi$
$840$	0	0
$841$	5.95363	0.205298
$842$	3.71193	0.127922
$843$	−38.0906	−1.31191
$844$	−32.6469	−1.12375
$845$	0	0
$846$	0.644174	0.0221471
$847$	22.8260	0.784310
$848$	−30.7812	−1.05703
$849$	34.9318	1.19886
$850$	0	0
$851$	7.11091	0.243759
$852$	−17.2767	−0.591890
$853$	−17.8905	−0.612559	−0.306279	−	0.951942i	$-0.599084\pi$
$853$	−17.8905	−0.612559	−0.306279	+	0.951942i	$0.599084\pi$
$854$	0.546895	0.0187144
$855$	0	0
$856$	11.5001	0.393065
$857$	3.19536	0.109151	0.0545757	−	0.998510i	$-0.482619\pi$
$857$	3.19536	0.109151	0.0545757	+	0.998510i	$0.482619\pi$
$858$	−0.159819	−0.00545614
$859$	43.4196	1.48146	0.740728	−	0.671805i	$-0.234481\pi$
$859$	43.4196	1.48146	0.740728	+	0.671805i	$0.234481\pi$
$860$	0	0
$861$	−4.87251	−0.166055
$862$	−1.31064	−0.0446407
$863$	43.2724	1.47301	0.736504	−	0.676433i	$-0.236475\pi$
$863$	43.2724	1.47301	0.736504	+	0.676433i	$0.236475\pi$
$864$	−12.1264	−0.412547
$865$	0	0
$866$	−1.00427	−0.0341264
$867$	−14.2135	−0.482716
$868$	39.1120	1.32755
$869$	−15.5213	−0.526525
$870$	0	0
$871$	−2.04147	−0.0691727
$872$	−1.45582	−0.0493004
$873$	5.60562	0.189722
$874$	−0.278666	−0.00942603
$875$	0	0
$876$	25.5684	0.863876
$877$	34.2339	1.15600	0.577999	−	0.816038i	$-0.303834\pi$
$877$	34.2339	1.15600	0.577999	+	0.816038i	$0.303834\pi$
$878$	5.63366	0.190127
$879$	−18.1092	−0.610808
$880$	0	0
$881$	27.6270	0.930779	0.465389	−	0.885106i	$-0.345914\pi$
$881$	27.6270	0.930779	0.465389	+	0.885106i	$0.345914\pi$
$882$	0.554602	0.0186744
$883$	25.8990	0.871571	0.435786	−	0.900051i	$-0.356471\pi$
$883$	25.8990	0.871571	0.435786	+	0.900051i	$0.356471\pi$
$884$	−3.00905	−0.101205
$885$	0	0
$886$	2.07949	0.0698618
$887$	17.2449	0.579027	0.289514	−	0.957174i	$-0.406506\pi$
$887$	17.2449	0.579027	0.289514	+	0.957174i	$0.406506\pi$
$888$	−8.64958	−0.290261
$889$	19.0762	0.639796
$890$	0	0
$891$	−11.6108	−0.388977
$892$	−13.0338	−0.436403
$893$	7.30213	0.244356
$894$	5.07502	0.169734
$895$	0	0
$896$	−18.3200	−0.612027
$897$	0.383551	0.0128064
$898$	2.88792	0.0963710
$899$	36.0614	1.20271
$900$	0	0
$901$	41.8238	1.39335
$902$	0.371893	0.0123827
$903$	−15.5610	−0.517836
$904$	−7.57601	−0.251974
$905$	0	0
$906$	−1.04827	−0.0348266
$907$	−57.0465	−1.89420	−0.947099	−	0.320940i	$-0.896001\pi$
$907$	−57.0465	−1.89420	−0.947099	+	0.320940i	$0.896001\pi$
$908$	−26.3345	−0.873942
$909$	10.1633	0.337095
$910$	0	0
$911$	19.8683	0.658267	0.329133	−	0.944283i	$-0.393243\pi$
$911$	19.8683	0.658267	0.329133	+	0.944283i	$0.393243\pi$
$912$	−9.67824	−0.320479
$913$	29.0307	0.960777
$914$	−0.763304	−0.0252478
$915$	0	0
$916$	−19.8350	−0.655367
$917$	4.88033	0.161163
$918$	5.33620	0.176121
$919$	27.9785	0.922924	0.461462	−	0.887160i	$-0.347325\pi$
$919$	27.9785	0.922924	0.461462	+	0.887160i	$0.347325\pi$
$920$	0	0
$921$	6.30361	0.207711
$922$	4.39090	0.144606
$923$	1.76902	0.0582279
$924$	−18.8772	−0.621013
$925$	0	0
$926$	7.56952	0.248750
$927$	−1.15122	−0.0378109
$928$	−12.7059	−0.417090
$929$	−14.1249	−0.463424	−0.231712	−	0.972784i	$-0.574433\pi$
$929$	−14.1249	−0.463424	−0.231712	+	0.972784i	$0.574433\pi$
$930$	0	0
$931$	6.28678	0.206041
$932$	−43.2727	−1.41744
$933$	−37.7506	−1.23590
$934$	−1.90437	−0.0623127
$935$	0	0
$936$	0.179450	0.00586551
$937$	35.9546	1.17459	0.587293	−	0.809374i	$-0.300194\pi$
$937$	35.9546	1.17459	0.587293	+	0.809374i	$0.300194\pi$
$938$	4.11422	0.134334
$939$	−32.8999	−1.07365
$940$	0	0
$941$	26.6979	0.870327	0.435164	−	0.900351i	$-0.356691\pi$
$941$	26.6979	0.870327	0.435164	+	0.900351i	$0.356691\pi$
$942$	1.15499	0.0376317
$943$	−0.892508	−0.0290640
$944$	22.5570	0.734169
$945$	0	0
$946$	1.18768	0.0386150
$947$	17.3102	0.562506	0.281253	−	0.959634i	$-0.409250\pi$
$947$	17.3102	0.562506	0.281253	+	0.959634i	$0.409250\pi$
$948$	−22.4634	−0.729576
$949$	−2.61803	−0.0849850
$950$	0	0
$951$	32.2693	1.04640
$952$	12.2318	0.396436
$953$	24.2622	0.785931	0.392965	−	0.919553i	$-0.371449\pi$
$953$	24.2622	0.785931	0.392965	+	0.919553i	$0.371449\pi$
$954$	−1.23657	−0.0400354
$955$	0	0
$956$	−14.8669	−0.480830
$957$	−17.4048	−0.562617
$958$	−6.59568	−0.213097
$959$	25.5940	0.826475
$960$	0	0
$961$	6.20427	0.200138
$962$	0.439084	0.0141566
$963$	−13.1910	−0.425072
$964$	40.1048	1.29169
$965$	0	0
$966$	−0.772978	−0.0248702
$967$	−31.1143	−1.00057	−0.500284	−	0.865861i	$-0.666771\pi$
$967$	−31.1143	−1.00057	−0.500284	+	0.865861i	$0.666771\pi$
$968$	−5.08580	−0.163464
$969$	13.1502	0.422447
$970$	0	0
$971$	−17.3721	−0.557497	−0.278749	−	0.960364i	$-0.589920\pi$
$971$	−17.3721	−0.557497	−0.278749	+	0.960364i	$0.589920\pi$
$972$	16.4834	0.528704
$973$	−17.5810	−0.563620
$974$	1.94855	0.0624356
$975$	0	0
$976$	3.47917	0.111365
$977$	−42.8929	−1.37227	−0.686133	−	0.727476i	$-0.740693\pi$
$977$	−42.8929	−1.37227	−0.686133	+	0.727476i	$0.740693\pi$
$978$	4.24013	0.135584
$979$	15.0528	0.481091
$980$	0	0
$981$	1.66987	0.0533149
$982$	3.23651	0.103281
$983$	−37.4533	−1.19457	−0.597287	−	0.802028i	$-0.703755\pi$
$983$	−37.4533	−1.19457	−0.597287	+	0.802028i	$0.703755\pi$
$984$	1.08563	0.0346086
$985$	0	0
$986$	5.59120	0.178060
$987$	20.2550	0.644724
$988$	1.00849	0.0320843
$989$	−2.85033	−0.0906353
$990$	0	0
$991$	16.7624	0.532476	0.266238	−	0.963907i	$-0.414219\pi$
$991$	16.7624	0.532476	0.266238	+	0.963907i	$0.414219\pi$
$992$	−13.1085	−0.416196
$993$	13.1931	0.418669
$994$	−3.56514	−0.113079
$995$	0	0
$996$	42.0149	1.33129
$997$	−59.8595	−1.89577	−0.947884	−	0.318615i	$-0.896782\pi$
$997$	−59.8595	−1.89577	−0.947884	+	0.318615i	$0.896782\pi$
$998$	1.72499	0.0546037
$999$	45.6367	1.44388

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	625.2.a.f.1.4		8
3.2	odd	2		5625.2.a.x.1.5		8
4.3	odd	2		10000.2.a.bj.1.6		8
5.2	odd	4		625.2.b.c.624.4		8
5.3	odd	4		625.2.b.c.624.5		8
5.4	even	2	inner	625.2.a.f.1.5		8
15.14	odd	2		5625.2.a.x.1.4		8
20.19	odd	2		10000.2.a.bj.1.3		8
25.2	odd	20		125.2.e.b.24.1		8
25.3	odd	20		625.2.e.a.249.2		8
25.4	even	10		625.2.d.o.376.2		16
25.6	even	5		625.2.d.o.251.3		16
25.8	odd	20		625.2.e.i.374.1		8
25.9	even	10		125.2.d.b.26.3		16
25.11	even	5		125.2.d.b.101.2		16
25.12	odd	20		25.2.e.a.19.2	yes	8
25.13	odd	20		125.2.e.b.99.1		8
25.14	even	10		125.2.d.b.101.3		16
25.16	even	5		125.2.d.b.26.2		16
25.17	odd	20		625.2.e.a.374.2		8
25.19	even	10		625.2.d.o.251.2		16
25.21	even	5		625.2.d.o.376.3		16
25.22	odd	20		625.2.e.i.249.1		8
25.23	odd	20		25.2.e.a.4.2	✓	8
75.23	even	20		225.2.m.a.154.1		8
75.62	even	20		225.2.m.a.19.1		8
100.23	even	20		400.2.y.c.129.2		8
100.87	even	20		400.2.y.c.369.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
25.2.e.a.4.2	✓	8	25.23	odd	20
25.2.e.a.19.2	yes	8	25.12	odd	20
125.2.d.b.26.2		16	25.16	even	5
125.2.d.b.26.3		16	25.9	even	10
125.2.d.b.101.2		16	25.11	even	5
125.2.d.b.101.3		16	25.14	even	10
125.2.e.b.24.1		8	25.2	odd	20
125.2.e.b.99.1		8	25.13	odd	20
225.2.m.a.19.1		8	75.62	even	20
225.2.m.a.154.1		8	75.23	even	20
400.2.y.c.129.2		8	100.23	even	20
400.2.y.c.369.2		8	100.87	even	20
625.2.a.f.1.4		8	1.1	even	1	trivial
625.2.a.f.1.5		8	5.4	even	2	inner
625.2.b.c.624.4		8	5.2	odd	4
625.2.b.c.624.5		8	5.3	odd	4
625.2.d.o.251.2		16	25.19	even	10
625.2.d.o.251.3		16	25.6	even	5
625.2.d.o.376.2		16	25.4	even	10
625.2.d.o.376.3		16	25.21	even	5
625.2.e.a.249.2		8	25.3	odd	20
625.2.e.a.374.2		8	25.17	odd	20
625.2.e.i.249.1		8	25.22	odd	20
625.2.e.i.374.1		8	25.8	odd	20
5625.2.a.x.1.4		8	15.14	odd	2
5625.2.a.x.1.5		8	3.2	odd	2
10000.2.a.bj.1.3		8	20.19	odd	2
10000.2.a.bj.1.6		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 625 = 5^{4} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	625.a (trivial)

\(f(q)\)	\(=\)	\(q-0.183172 q^{2} -1.47195 q^{3} -1.96645 q^{4} +0.269620 q^{6} -3.26086 q^{7} +0.726543 q^{8} -0.833366 q^{9} +O(q^{10})\)	\(q-0.183172 q^{2} -1.47195 q^{3} -1.96645 q^{4} +0.269620 q^{6} -3.26086 q^{7} +0.726543 q^{8} -0.833366 q^{9} +2.00000 q^{11} +2.89451 q^{12} -0.296379 q^{13} +0.597298 q^{14} +3.79981 q^{16} -5.16297 q^{17} +0.152649 q^{18} +1.73038 q^{19} +4.79981 q^{21} -0.366344 q^{22} +0.879192 q^{23} -1.06943 q^{24} +0.0542883 q^{26} +5.64252 q^{27} +6.41230 q^{28} +5.91216 q^{29} +6.09953 q^{31} -2.14910 q^{32} -2.94390 q^{33} +0.945712 q^{34} +1.63877 q^{36} +8.08800 q^{37} -0.316957 q^{38} +0.436254 q^{39} -1.01515 q^{41} -0.879192 q^{42} -3.24199 q^{43} -3.93290 q^{44} -0.161043 q^{46} +4.21996 q^{47} -5.59313 q^{48} +3.63318 q^{49} +7.59963 q^{51} +0.582813 q^{52} -8.10072 q^{53} -1.03355 q^{54} -2.36915 q^{56} -2.54703 q^{57} -1.08294 q^{58} +5.93635 q^{59} +0.915615 q^{61} -1.11726 q^{62} +2.71749 q^{63} -7.20597 q^{64} +0.539240 q^{66} +6.88806 q^{67} +10.1527 q^{68} -1.29413 q^{69} -5.96878 q^{71} -0.605476 q^{72} +8.83341 q^{73} -1.48150 q^{74} -3.40270 q^{76} -6.52171 q^{77} -0.0799096 q^{78} -7.76067 q^{79} -5.80540 q^{81} +0.185946 q^{82} +14.5154 q^{83} -9.43858 q^{84} +0.593842 q^{86} -8.70240 q^{87} +1.45309 q^{88} +7.52642 q^{89} +0.966448 q^{91} -1.72889 q^{92} -8.97820 q^{93} -0.772978 q^{94} +3.16337 q^{96} -6.72649 q^{97} -0.665497 q^{98} -1.66673 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9}+O(q^{10}) \) 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9	\( 8 q + 6 q^{4} + 6 q^{6} + 4 q^{9} + 16 q^{11} + 12 q^{14} - 2 q^{16} + 10 q^{19} + 6 q^{21} + 20 q^{24} + 6 q^{26} + 20 q^{29} + 16 q^{31} + 2 q^{34} - 12 q^{36} + 18 q^{39} + 26 q^{41} + 12 q^{44} + 6 q^{46} - 14 q^{49} - 4 q^{51} - 30 q^{54} + 10 q^{56} + 30 q^{59} + 6 q^{61} - 44 q^{64} + 12 q^{66} + 8 q^{69} + 46 q^{71} + 12 q^{74} - 20 q^{76} + 10 q^{79} - 32 q^{81} - 18 q^{84} - 14 q^{86} + 30 q^{89} - 14 q^{91} - 68 q^{94} - 54 q^{96} + 8 q^{99}+O(q^{100}) \) 8 * q + 6 * q^4 + 6 * q^6 + 4 * q^9 + 16 * q^11 + 12 * q^14 - 2 * q^16 + 10 * q^19 + 6 * q^21 + 20 * q^24 + 6 * q^26 + 20 * q^29 + 16 * q^31 + 2 * q^34 - 12 * q^36 + 18 * q^39 + 26 * q^41 + 12 * q^44 + 6 * q^46 - 14 * q^49 - 4 * q^51 - 30 * q^54 + 10 * q^56 + 30 * q^59 + 6 * q^61 - 44 * q^64 + 12 * q^66 + 8 * q^69 + 46 * q^71 + 12 * q^74 - 20 * q^76 + 10 * q^79 - 32 * q^81 - 18 * q^84 - 14 * q^86 + 30 * q^89 - 14 * q^91 - 68 * q^94 - 54 * q^96 + 8 * q^99

Introduction

L-functions

Modular forms

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Database

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