LMFDB - Embedded newform 4025.2.a.be.1.19

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4025,2,Mod(1,4025)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4025, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("4025.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4025 = 5^{2} \cdot 7 \cdot 23 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4025.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:	$32.1397868136$
Analytic rank:	$0$
Dimension:	$21$
Twist minimal:	no (minimal twist has level 805)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.19
Character	$\chi$	$=$	4025.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.57149 q^{2} +0.285863 q^{3} +4.61257 q^{4} +0.735095 q^{6} +1.00000 q^{7} +6.71821 q^{8} -2.91828 q^{9} +O(q^{10})$	\(q+2.57149 q^{2} +0.285863 q^{3} +4.61257 q^{4} +0.735095 q^{6} +1.00000 q^{7} +6.71821 q^{8} -2.91828 q^{9} +6.14192 q^{11} +1.31856 q^{12} +1.02339 q^{13} +2.57149 q^{14} +8.05068 q^{16} +0.880458 q^{17} -7.50434 q^{18} -3.91439 q^{19} +0.285863 q^{21} +15.7939 q^{22} +1.00000 q^{23} +1.92049 q^{24} +2.63165 q^{26} -1.69182 q^{27} +4.61257 q^{28} +6.92254 q^{29} +0.390474 q^{31} +7.26584 q^{32} +1.75575 q^{33} +2.26409 q^{34} -13.4608 q^{36} -3.09439 q^{37} -10.0658 q^{38} +0.292551 q^{39} -3.45816 q^{41} +0.735095 q^{42} +0.0871065 q^{43} +28.3301 q^{44} +2.57149 q^{46} +6.66651 q^{47} +2.30139 q^{48} +1.00000 q^{49} +0.251691 q^{51} +4.72048 q^{52} +10.9506 q^{53} -4.35050 q^{54} +6.71821 q^{56} -1.11898 q^{57} +17.8013 q^{58} +14.1936 q^{59} -5.74679 q^{61} +1.00410 q^{62} -2.91828 q^{63} +2.58269 q^{64} +4.51490 q^{66} -3.65650 q^{67} +4.06118 q^{68} +0.285863 q^{69} -14.9617 q^{71} -19.6056 q^{72} -14.5236 q^{73} -7.95720 q^{74} -18.0554 q^{76} +6.14192 q^{77} +0.752291 q^{78} +3.40561 q^{79} +8.27122 q^{81} -8.89263 q^{82} +1.88029 q^{83} +1.31856 q^{84} +0.223994 q^{86} +1.97890 q^{87} +41.2627 q^{88} -13.2126 q^{89} +1.02339 q^{91} +4.61257 q^{92} +0.111622 q^{93} +17.1429 q^{94} +2.07704 q^{96} -16.5891 q^{97} +2.57149 q^{98} -17.9239 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9}+O(q^{10}) $ 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9	$ 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9} + 7 q^{11} - 22 q^{12} - 3 q^{13} + 2 q^{14} + 56 q^{16} + 7 q^{17} + 24 q^{19} - q^{21} + 4 q^{22} + 21 q^{23} + 24 q^{24} - 2 q^{26} - 19 q^{27} + 30 q^{28} + 11 q^{29} + 46 q^{31} - 6 q^{32} - 3 q^{33} + 28 q^{34} + 58 q^{36} + 24 q^{37} - 4 q^{38} + 31 q^{39} + 14 q^{41} + 6 q^{42} + 18 q^{43} + 12 q^{44} + 2 q^{46} - 25 q^{47} - 36 q^{48} + 21 q^{49} + 17 q^{51} - 8 q^{52} + 22 q^{53} - 6 q^{54} + 6 q^{56} + 40 q^{57} + 6 q^{58} + 10 q^{59} + 38 q^{61} - 54 q^{62} + 30 q^{63} + 100 q^{64} + 38 q^{66} + 12 q^{67} + 18 q^{68} - q^{69} + 56 q^{71} + 42 q^{72} - 40 q^{73} - 20 q^{74} + 60 q^{76} + 7 q^{77} + 38 q^{78} + 49 q^{79} + 57 q^{81} + 16 q^{82} - 2 q^{83} - 22 q^{84} + 16 q^{86} - 23 q^{87} + 12 q^{88} + 28 q^{89} - 3 q^{91} + 30 q^{92} - 30 q^{93} + 66 q^{94} + 46 q^{96} - q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) $ 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9 + 7 * q^11 - 22 * q^12 - 3 * q^13 + 2 * q^14 + 56 * q^16 + 7 * q^17 + 24 * q^19 - q^21 + 4 * q^22 + 21 * q^23 + 24 * q^24 - 2 * q^26 - 19 * q^27 + 30 * q^28 + 11 * q^29 + 46 * q^31 - 6 * q^32 - 3 * q^33 + 28 * q^34 + 58 * q^36 + 24 * q^37 - 4 * q^38 + 31 * q^39 + 14 * q^41 + 6 * q^42 + 18 * q^43 + 12 * q^44 + 2 * q^46 - 25 * q^47 - 36 * q^48 + 21 * q^49 + 17 * q^51 - 8 * q^52 + 22 * q^53 - 6 * q^54 + 6 * q^56 + 40 * q^57 + 6 * q^58 + 10 * q^59 + 38 * q^61 - 54 * q^62 + 30 * q^63 + 100 * q^64 + 38 * q^66 + 12 * q^67 + 18 * q^68 - q^69 + 56 * q^71 + 42 * q^72 - 40 * q^73 - 20 * q^74 + 60 * q^76 + 7 * q^77 + 38 * q^78 + 49 * q^79 + 57 * q^81 + 16 * q^82 - 2 * q^83 - 22 * q^84 + 16 * q^86 - 23 * q^87 + 12 * q^88 + 28 * q^89 - 3 * q^91 + 30 * q^92 - 30 * q^93 + 66 * q^94 + 46 * q^96 - q^97 + 2 * q^98 - 2 * 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 $

$2$	2.57149	1.81832	0.909160	−	0.416447i	$-0.136725\pi$
$2$	2.57149	1.81832	0.909160	+	0.416447i	$0.136725\pi$
$3$	0.285863	0.165043	0.0825216	−	0.996589i	$-0.473703\pi$
$3$	0.285863	0.165043	0.0825216	+	0.996589i	$0.473703\pi$
$4$	4.61257	2.30629
$5$	0	0
$5$	0	0
$6$	0.735095	0.300101
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	6.71821	2.37525
$9$	−2.91828	−0.972761
$10$	0	0
$11$	6.14192	1.85186	0.925930	−	0.377696i	$-0.123283\pi$
$11$	6.14192	1.85186	0.925930	+	0.377696i	$0.123283\pi$
$12$	1.31856	0.380637
$13$	1.02339	0.283838	0.141919	−	0.989878i	$-0.454673\pi$
$13$	1.02339	0.283838	0.141919	+	0.989878i	$0.454673\pi$
$14$	2.57149	0.687260
$15$	0	0
$16$	8.05068	2.01267
$17$	0.880458	0.213543	0.106771	−	0.994284i	$-0.465949\pi$
$17$	0.880458	0.213543	0.106771	+	0.994284i	$0.465949\pi$
$18$	−7.50434	−1.76879
$19$	−3.91439	−0.898024	−0.449012	−	0.893526i	$-0.648224\pi$
$19$	−3.91439	−0.898024	−0.449012	+	0.893526i	$0.648224\pi$
$20$	0	0
$21$	0.285863	0.0623804
$22$	15.7939	3.36727
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	1.92049	0.392018
$25$	0	0
$26$	2.63165	0.516109
$27$	−1.69182	−0.325591
$28$	4.61257	0.871694
$29$	6.92254	1.28548	0.642742	−	0.766083i	$-0.277797\pi$
$29$	6.92254	1.28548	0.642742	+	0.766083i	$0.277797\pi$
$30$	0	0
$31$	0.390474	0.0701312	0.0350656	−	0.999385i	$-0.488836\pi$
$31$	0.390474	0.0701312	0.0350656	+	0.999385i	$0.488836\pi$
$32$	7.26584	1.28443
$33$	1.75575	0.305637
$34$	2.26409	0.388289
$35$	0	0
$36$	−13.4608	−2.24346
$37$	−3.09439	−0.508714	−0.254357	−	0.967110i	$-0.581864\pi$
$37$	−3.09439	−0.508714	−0.254357	+	0.967110i	$0.581864\pi$
$38$	−10.0658	−1.63289
$39$	0.292551	0.0468456
$40$	0	0
$41$	−3.45816	−0.540074	−0.270037	−	0.962850i	$-0.587036\pi$
$41$	−3.45816	−0.540074	−0.270037	+	0.962850i	$0.587036\pi$
$42$	0.735095	0.113428
$43$	0.0871065	0.0132836	0.00664181	−	0.999978i	$-0.497886\pi$
$43$	0.0871065	0.0132836	0.00664181	+	0.999978i	$0.497886\pi$
$44$	28.3301	4.27092
$45$	0	0
$46$	2.57149	0.379146
$47$	6.66651	0.972410	0.486205	−	0.873845i	$-0.338381\pi$
$47$	6.66651	0.972410	0.486205	+	0.873845i	$0.338381\pi$
$48$	2.30139	0.332177
$49$	1.00000	0.142857
$50$	0	0
$51$	0.251691	0.0352437
$52$	4.72048	0.654613
$53$	10.9506	1.50418	0.752090	−	0.659060i	$-0.229046\pi$
$53$	10.9506	1.50418	0.752090	+	0.659060i	$0.229046\pi$
$54$	−4.35050	−0.592028
$55$	0	0
$56$	6.71821	0.897759
$57$	−1.11898	−0.148213
$58$	17.8013	2.33742
$59$	14.1936	1.84784	0.923922	−	0.382581i	$-0.124965\pi$
$59$	14.1936	1.84784	0.923922	+	0.382581i	$0.124965\pi$
$60$	0	0
$61$	−5.74679	−0.735801	−0.367900	−	0.929865i	$-0.619923\pi$
$61$	−5.74679	−0.735801	−0.367900	+	0.929865i	$0.619923\pi$
$62$	1.00410	0.127521
$63$	−2.91828	−0.367669
$64$	2.58269	0.322837
$65$	0	0
$66$	4.51490	0.555745
$67$	−3.65650	−0.446712	−0.223356	−	0.974737i	$-0.571701\pi$
$67$	−3.65650	−0.446712	−0.223356	+	0.974737i	$0.571701\pi$
$68$	4.06118	0.492490
$69$	0.285863	0.0344139
$70$	0	0
$71$	−14.9617	−1.77563	−0.887814	−	0.460202i	$-0.847777\pi$
$71$	−14.9617	−1.77563	−0.887814	+	0.460202i	$0.847777\pi$
$72$	−19.6056	−2.31055
$73$	−14.5236	−1.69986	−0.849930	−	0.526896i	$-0.823356\pi$
$73$	−14.5236	−1.69986	−0.849930	+	0.526896i	$0.823356\pi$
$74$	−7.95720	−0.925005
$75$	0	0
$76$	−18.0554	−2.07110
$77$	6.14192	0.699937
$78$	0.752291	0.0851802
$79$	3.40561	0.383161	0.191580	−	0.981477i	$-0.438639\pi$
$79$	3.40561	0.383161	0.191580	+	0.981477i	$0.438639\pi$
$80$	0	0
$81$	8.27122	0.919024
$82$	−8.89263	−0.982027
$83$	1.88029	0.206389	0.103195	−	0.994661i	$-0.467094\pi$
$83$	1.88029	0.206389	0.103195	+	0.994661i	$0.467094\pi$
$84$	1.31856	0.143867
$85$	0	0
$86$	0.223994	0.0241539
$87$	1.97890	0.212160
$88$	41.2627	4.39862
$89$	−13.2126	−1.40053	−0.700266	−	0.713882i	$-0.746935\pi$
$89$	−13.2126	−1.40053	−0.700266	+	0.713882i	$0.746935\pi$
$90$	0	0
$91$	1.02339	0.107281
$92$	4.61257	0.480894
$93$	0.111622	0.0115747
$94$	17.1429	1.76815
$95$	0	0
$96$	2.07704	0.211987
$97$	−16.5891	−1.68436	−0.842182	−	0.539194i	$-0.818729\pi$
$97$	−16.5891	−1.68436	−0.842182	+	0.539194i	$0.818729\pi$
$98$	2.57149	0.259760
$99$	−17.9239	−1.80142
$100$	0	0
$101$	15.0659	1.49911	0.749556	−	0.661941i	$-0.230267\pi$
$101$	15.0659	1.49911	0.749556	+	0.661941i	$0.230267\pi$
$102$	0.647220	0.0640844
$103$	−13.5684	−1.33694	−0.668469	−	0.743740i	$-0.733050\pi$
$103$	−13.5684	−1.33694	−0.668469	+	0.743740i	$0.733050\pi$
$104$	6.87537	0.674186
$105$	0	0
$106$	28.1594	2.73508
$107$	5.01686	0.484999	0.242499	−	0.970152i	$-0.422033\pi$
$107$	5.01686	0.484999	0.242499	+	0.970152i	$0.422033\pi$
$108$	−7.80364	−0.750905
$109$	0.946819	0.0906888	0.0453444	−	0.998971i	$-0.485561\pi$
$109$	0.946819	0.0906888	0.0453444	+	0.998971i	$0.485561\pi$
$110$	0	0
$111$	−0.884572	−0.0839598
$112$	8.05068	0.760718
$113$	13.9764	1.31479	0.657393	−	0.753548i	$-0.271659\pi$
$113$	13.9764	1.31479	0.657393	+	0.753548i	$0.271659\pi$
$114$	−2.87745	−0.269498
$115$	0	0
$116$	31.9307	2.96469
$117$	−2.98655	−0.276107
$118$	36.4986	3.35997
$119$	0.880458	0.0807115
$120$	0	0
$121$	26.7232	2.42938
$122$	−14.7778	−1.33792
$123$	−0.988560	−0.0891355
$124$	1.80109	0.161743
$125$	0	0
$126$	−7.50434	−0.668540
$127$	−7.83418	−0.695171	−0.347586	−	0.937648i	$-0.612998\pi$
$127$	−7.83418	−0.695171	−0.347586	+	0.937648i	$0.612998\pi$
$128$	−7.89030	−0.697411
$129$	0.0249005	0.00219237
$130$	0	0
$131$	−11.3276	−0.989694	−0.494847	−	0.868980i	$-0.664776\pi$
$131$	−11.3276	−0.989694	−0.494847	+	0.868980i	$0.664776\pi$
$132$	8.09852	0.704886
$133$	−3.91439	−0.339421
$134$	−9.40266	−0.812266
$135$	0	0
$136$	5.91510	0.507216
$137$	7.03539	0.601074	0.300537	−	0.953770i	$-0.402834\pi$
$137$	7.03539	0.601074	0.300537	+	0.953770i	$0.402834\pi$
$138$	0.735095	0.0625754
$139$	13.3584	1.13304	0.566522	−	0.824047i	$-0.308289\pi$
$139$	13.3584	1.13304	0.566522	+	0.824047i	$0.308289\pi$
$140$	0	0
$141$	1.90571	0.160490
$142$	−38.4739	−3.22866
$143$	6.28561	0.525629
$144$	−23.4942	−1.95785
$145$	0	0
$146$	−37.3473	−3.09089
$147$	0.285863	0.0235776
$148$	−14.2731	−1.17324
$149$	−0.0962328	−0.00788370	−0.00394185	−	0.999992i	$-0.501255\pi$
$149$	−0.0962328	−0.00788370	−0.00394185	+	0.999992i	$0.501255\pi$
$150$	0	0
$151$	−11.1272	−0.905518	−0.452759	−	0.891633i	$-0.649560\pi$
$151$	−11.1272	−0.905518	−0.452759	+	0.891633i	$0.649560\pi$
$152$	−26.2977	−2.13303
$153$	−2.56943	−0.207726
$154$	15.7939	1.27271
$155$	0	0
$156$	1.34941	0.108039
$157$	−6.74396	−0.538227	−0.269113	−	0.963109i	$-0.586731\pi$
$157$	−6.74396	−0.538227	−0.269113	+	0.963109i	$0.586731\pi$
$158$	8.75749	0.696708
$159$	3.13037	0.248255
$160$	0	0
$161$	1.00000	0.0788110
$162$	21.2694	1.67108
$163$	−0.364094	−0.0285181	−0.0142590	−	0.999898i	$-0.504539\pi$
$163$	−0.364094	−0.0285181	−0.0142590	+	0.999898i	$0.504539\pi$
$164$	−15.9510	−1.24556
$165$	0	0
$166$	4.83516	0.375281
$167$	−5.07724	−0.392888	−0.196444	−	0.980515i	$-0.562939\pi$
$167$	−5.07724	−0.392888	−0.196444	+	0.980515i	$0.562939\pi$
$168$	1.92049	0.148169
$169$	−11.9527	−0.919436
$170$	0	0
$171$	11.4233	0.873562
$172$	0.401785	0.0306358
$173$	−16.0673	−1.22157	−0.610787	−	0.791795i	$-0.709147\pi$
$173$	−16.0673	−1.22157	−0.610787	+	0.791795i	$0.709147\pi$
$174$	5.08872	0.385775
$175$	0	0
$176$	49.4467	3.72718
$177$	4.05741	0.304974
$178$	−33.9761	−2.54661
$179$	−5.22849	−0.390796	−0.195398	−	0.980724i	$-0.562600\pi$
$179$	−5.22849	−0.390796	−0.195398	+	0.980724i	$0.562600\pi$
$180$	0	0
$181$	8.20868	0.610146	0.305073	−	0.952329i	$-0.401319\pi$
$181$	8.20868	0.610146	0.305073	+	0.952329i	$0.401319\pi$
$182$	2.63165	0.195071
$183$	−1.64279	−0.121439
$184$	6.71821	0.495273
$185$	0	0
$186$	0.287035	0.0210465
$187$	5.40771	0.395451
$188$	30.7498	2.24266
$189$	−1.69182	−0.123062
$190$	0	0
$191$	−3.49029	−0.252548	−0.126274	−	0.991995i	$-0.540302\pi$
$191$	−3.49029	−0.252548	−0.126274	+	0.991995i	$0.540302\pi$
$192$	0.738297	0.0532820
$193$	18.8099	1.35396	0.676982	−	0.735999i	$-0.263287\pi$
$193$	18.8099	1.35396	0.676982	+	0.735999i	$0.263287\pi$
$194$	−42.6586	−3.06271
$195$	0	0
$196$	4.61257	0.329469
$197$	2.55570	0.182086	0.0910431	−	0.995847i	$-0.470980\pi$
$197$	2.55570	0.182086	0.0910431	+	0.995847i	$0.470980\pi$
$198$	−46.0911	−3.27555
$199$	5.01832	0.355739	0.177870	−	0.984054i	$-0.443079\pi$
$199$	5.01832	0.355739	0.177870	+	0.984054i	$0.443079\pi$
$200$	0	0
$201$	−1.04526	−0.0737268
$202$	38.7418	2.72587
$203$	6.92254	0.485867
$204$	1.16094	0.0812821
$205$	0	0
$206$	−34.8911	−2.43098
$207$	−2.91828	−0.202835
$208$	8.23902	0.571273
$209$	−24.0419	−1.66301
$210$	0	0
$211$	−14.2291	−0.979571	−0.489786	−	0.871843i	$-0.662925\pi$
$211$	−14.2291	−0.979571	−0.489786	+	0.871843i	$0.662925\pi$
$212$	50.5104	3.46907
$213$	−4.27700	−0.293055
$214$	12.9008	0.881882
$215$	0	0
$216$	−11.3660	−0.773358
$217$	0.390474	0.0265071
$218$	2.43474	0.164901
$219$	−4.15176	−0.280550
$220$	0	0
$221$	0.901056	0.0606116
$222$	−2.27467	−0.152666
$223$	−21.1228	−1.41449	−0.707245	−	0.706969i	$-0.750062\pi$
$223$	−21.1228	−1.41449	−0.707245	+	0.706969i	$0.750062\pi$
$224$	7.26584	0.485469
$225$	0	0
$226$	35.9401	2.39070
$227$	25.6431	1.70199	0.850995	−	0.525174i	$-0.176000\pi$
$227$	25.6431	1.70199	0.850995	+	0.525174i	$0.176000\pi$
$228$	−5.16138	−0.341821
$229$	10.3514	0.684041	0.342020	−	0.939693i	$-0.388889\pi$
$229$	10.3514	0.684041	0.342020	+	0.939693i	$0.388889\pi$
$230$	0	0
$231$	1.75575	0.115520
$232$	46.5071	3.05334
$233$	15.0916	0.988683	0.494341	−	0.869268i	$-0.335409\pi$
$233$	15.0916	0.988683	0.494341	+	0.869268i	$0.335409\pi$
$234$	−7.67990	−0.502051
$235$	0	0
$236$	65.4688	4.26166
$237$	0.973537	0.0632380
$238$	2.26409	0.146759
$239$	−6.86382	−0.443984	−0.221992	−	0.975049i	$-0.571256\pi$
$239$	−6.86382	−0.443984	−0.221992	+	0.975049i	$0.571256\pi$
$240$	0	0
$241$	−7.66803	−0.493941	−0.246971	−	0.969023i	$-0.579435\pi$
$241$	−7.66803	−0.493941	−0.246971	+	0.969023i	$0.579435\pi$
$242$	68.7186	4.41740
$243$	7.43989	0.477269
$244$	−26.5075	−1.69697
$245$	0	0
$246$	−2.54208	−0.162077
$247$	−4.00597	−0.254894
$248$	2.62329	0.166579
$249$	0.537507	0.0340631
$250$	0	0
$251$	11.2661	0.711110	0.355555	−	0.934655i	$-0.384292\pi$
$251$	11.2661	0.711110	0.355555	+	0.934655i	$0.384292\pi$
$252$	−13.4608	−0.847950
$253$	6.14192	0.386139
$254$	−20.1455	−1.26404
$255$	0	0
$256$	−25.4552	−1.59095
$257$	−20.6164	−1.28601	−0.643007	−	0.765860i	$-0.722313\pi$
$257$	−20.6164	−1.28601	−0.643007	+	0.765860i	$0.722313\pi$
$258$	0.0640316	0.00398643
$259$	−3.09439	−0.192276
$260$	0	0
$261$	−20.2019	−1.25047
$262$	−29.1287	−1.79958
$263$	−11.4258	−0.704543	−0.352272	−	0.935898i	$-0.614591\pi$
$263$	−11.4258	−0.704543	−0.352272	+	0.935898i	$0.614591\pi$
$264$	11.7955	0.725962
$265$	0	0
$266$	−10.0658	−0.617176
$267$	−3.77699	−0.231148
$268$	−16.8659	−1.03025
$269$	19.7562	1.20455	0.602277	−	0.798287i	$-0.294260\pi$
$269$	19.7562	1.20455	0.602277	+	0.798287i	$0.294260\pi$
$270$	0	0
$271$	−9.40553	−0.571345	−0.285673	−	0.958327i	$-0.592217\pi$
$271$	−9.40553	−0.571345	−0.285673	+	0.958327i	$0.592217\pi$
$272$	7.08829	0.429791
$273$	0.292551	0.0177060
$274$	18.0915	1.09295
$275$	0	0
$276$	1.31856	0.0793682
$277$	−26.4727	−1.59059	−0.795295	−	0.606223i	$-0.792684\pi$
$277$	−26.4727	−1.59059	−0.795295	+	0.606223i	$0.792684\pi$
$278$	34.3510	2.06023
$279$	−1.13951	−0.0682209
$280$	0	0
$281$	7.60606	0.453740	0.226870	−	0.973925i	$-0.427151\pi$
$281$	7.60606	0.453740	0.226870	+	0.973925i	$0.427151\pi$
$282$	4.90051	0.291821
$283$	1.27335	0.0756930	0.0378465	−	0.999284i	$-0.487950\pi$
$283$	1.27335	0.0756930	0.0378465	+	0.999284i	$0.487950\pi$
$284$	−69.0120	−4.09511
$285$	0	0
$286$	16.1634	0.955761
$287$	−3.45816	−0.204129
$288$	−21.2038	−1.24944
$289$	−16.2248	−0.954400
$290$	0	0
$291$	−4.74220	−0.277993
$292$	−66.9912	−3.92036
$293$	−2.64891	−0.154751	−0.0773755	−	0.997002i	$-0.524654\pi$
$293$	−2.64891	−0.154751	−0.0773755	+	0.997002i	$0.524654\pi$
$294$	0.735095	0.0428716
$295$	0	0
$296$	−20.7888	−1.20832
$297$	−10.3910	−0.602948
$298$	−0.247462	−0.0143351
$299$	1.02339	0.0591844
$300$	0	0
$301$	0.0871065	0.00502074
$302$	−28.6135	−1.64652
$303$	4.30678	0.247418
$304$	−31.5135	−1.80743
$305$	0	0
$306$	−6.60726	−0.377712
$307$	−32.1809	−1.83666	−0.918330	−	0.395816i	$-0.870462\pi$
$307$	−32.1809	−1.83666	−0.918330	+	0.395816i	$0.870462\pi$
$308$	28.3301	1.61426
$309$	−3.87872	−0.220652
$310$	0	0
$311$	−11.1772	−0.633799	−0.316899	−	0.948459i	$-0.602642\pi$
$311$	−11.1772	−0.633799	−0.316899	+	0.948459i	$0.602642\pi$
$312$	1.96542	0.111270
$313$	24.6707	1.39447	0.697234	−	0.716844i	$-0.254414\pi$
$313$	24.6707	1.39447	0.697234	+	0.716844i	$0.254414\pi$
$314$	−17.3420	−0.978668
$315$	0	0
$316$	15.7086	0.883678
$317$	−33.4901	−1.88099	−0.940496	−	0.339805i	$-0.889639\pi$
$317$	−33.4901	−1.88099	−0.940496	+	0.339805i	$0.889639\pi$
$318$	8.04973	0.451406
$319$	42.5177	2.38054
$320$	0	0
$321$	1.43414	0.0800457
$322$	2.57149	0.143304
$323$	−3.44646	−0.191766
$324$	38.1516	2.11953
$325$	0	0
$326$	−0.936265	−0.0518549
$327$	0.270661	0.0149676
$328$	−23.2326	−1.28281
$329$	6.66651	0.367536
$330$	0	0
$331$	23.2541	1.27816	0.639079	−	0.769141i	$-0.279316\pi$
$331$	23.2541	1.27816	0.639079	+	0.769141i	$0.279316\pi$
$332$	8.67300	0.475992
$333$	9.03030	0.494857
$334$	−13.0561	−0.714397
$335$	0	0
$336$	2.30139	0.125551
$337$	17.1987	0.936874	0.468437	−	0.883497i	$-0.344817\pi$
$337$	17.1987	0.936874	0.468437	+	0.883497i	$0.344817\pi$
$338$	−30.7362	−1.67183
$339$	3.99533	0.216997
$340$	0	0
$341$	2.39826	0.129873
$342$	29.3749	1.58842
$343$	1.00000	0.0539949
$344$	0.585200	0.0315519
$345$	0	0
$346$	−41.3170	−2.22121
$347$	−14.7588	−0.792293	−0.396146	−	0.918187i	$-0.629653\pi$
$347$	−14.7588	−0.792293	−0.396146	+	0.918187i	$0.629653\pi$
$348$	9.12782	0.489302
$349$	−0.697509	−0.0373368	−0.0186684	−	0.999826i	$-0.505943\pi$
$349$	−0.697509	−0.0373368	−0.0186684	+	0.999826i	$0.505943\pi$
$350$	0	0
$351$	−1.73140	−0.0924151
$352$	44.6262	2.37859
$353$	−32.3083	−1.71960	−0.859799	−	0.510633i	$-0.829411\pi$
$353$	−32.3083	−1.71960	−0.859799	+	0.510633i	$0.829411\pi$
$354$	10.4336	0.554540
$355$	0	0
$356$	−60.9440	−3.23003
$357$	0.251691	0.0133209
$358$	−13.4450	−0.710592
$359$	−25.1633	−1.32807	−0.664033	−	0.747703i	$-0.731157\pi$
$359$	−25.1633	−1.32807	−0.664033	+	0.747703i	$0.731157\pi$
$360$	0	0
$361$	−3.67752	−0.193553
$362$	21.1086	1.10944
$363$	7.63918	0.400953
$364$	4.72048	0.247420
$365$	0	0
$366$	−4.22443	−0.220815
$367$	−30.3693	−1.58526	−0.792632	−	0.609701i	$-0.791290\pi$
$367$	−30.3693	−1.58526	−0.792632	+	0.609701i	$0.791290\pi$
$368$	8.05068	0.419671
$369$	10.0919	0.525363
$370$	0	0
$371$	10.9506	0.568527
$372$	0.514865	0.0266945
$373$	−9.88041	−0.511588	−0.255794	−	0.966731i	$-0.582337\pi$
$373$	−9.88041	−0.511588	−0.255794	+	0.966731i	$0.582337\pi$
$374$	13.9059	0.719056
$375$	0	0
$376$	44.7870	2.30971
$377$	7.08449	0.364870
$378$	−4.35050	−0.223765
$379$	6.61691	0.339888	0.169944	−	0.985454i	$-0.445641\pi$
$379$	6.61691	0.339888	0.169944	+	0.985454i	$0.445641\pi$
$380$	0	0
$381$	−2.23950	−0.114733
$382$	−8.97525	−0.459214
$383$	−5.53793	−0.282975	−0.141487	−	0.989940i	$-0.545188\pi$
$383$	−5.53793	−0.282975	−0.141487	+	0.989940i	$0.545188\pi$
$384$	−2.25555	−0.115103
$385$	0	0
$386$	48.3695	2.46194
$387$	−0.254201	−0.0129218
$388$	−76.5182	−3.88462
$389$	32.5784	1.65179	0.825896	−	0.563823i	$-0.190670\pi$
$389$	32.5784	1.65179	0.825896	+	0.563823i	$0.190670\pi$
$390$	0	0
$391$	0.880458	0.0445267
$392$	6.71821	0.339321
$393$	−3.23813	−0.163342
$394$	6.57197	0.331091
$395$	0	0
$396$	−82.6751	−4.15458
$397$	−22.7575	−1.14217	−0.571083	−	0.820892i	$-0.693477\pi$
$397$	−22.7575	−1.14217	−0.571083	+	0.820892i	$0.693477\pi$
$398$	12.9046	0.646848
$399$	−1.11898	−0.0560191
$400$	0	0
$401$	28.3464	1.41555	0.707775	−	0.706438i	$-0.249699\pi$
$401$	28.3464	1.41555	0.707775	+	0.706438i	$0.249699\pi$
$402$	−2.68787	−0.134059
$403$	0.399609	0.0199059
$404$	69.4925	3.45738
$405$	0	0
$406$	17.8013	0.883462
$407$	−19.0055	−0.942068
$408$	1.69091	0.0837125
$409$	−7.89846	−0.390554	−0.195277	−	0.980748i	$-0.562561\pi$
$409$	−7.89846	−0.390554	−0.195277	+	0.980748i	$0.562561\pi$
$410$	0	0
$411$	2.01116	0.0992032
$412$	−62.5854	−3.08336
$413$	14.1936	0.698419
$414$	−7.50434	−0.368818
$415$	0	0
$416$	7.43582	0.364571
$417$	3.81867	0.187001
$418$	−61.8236	−3.02389
$419$	13.0465	0.637363	0.318682	−	0.947862i	$-0.396760\pi$
$419$	13.0465	0.637363	0.318682	+	0.947862i	$0.396760\pi$
$420$	0	0
$421$	17.8242	0.868697	0.434349	−	0.900745i	$-0.356979\pi$
$421$	17.8242	0.868697	0.434349	+	0.900745i	$0.356979\pi$
$422$	−36.5900	−1.78117
$423$	−19.4548	−0.945922
$424$	73.5684	3.57280
$425$	0	0
$426$	−10.9983	−0.532868
$427$	−5.74679	−0.278107
$428$	23.1406	1.11855
$429$	1.79682	0.0867514
$430$	0	0
$431$	6.69718	0.322592	0.161296	−	0.986906i	$-0.448433\pi$
$431$	6.69718	0.322592	0.161296	+	0.986906i	$0.448433\pi$
$432$	−13.6203	−0.655306
$433$	−15.5526	−0.747412	−0.373706	−	0.927547i	$-0.621913\pi$
$433$	−15.5526	−0.747412	−0.373706	+	0.927547i	$0.621913\pi$
$434$	1.00410	0.0481984
$435$	0	0
$436$	4.36727	0.209154
$437$	−3.91439	−0.187251
$438$	−10.6762	−0.510130
$439$	−16.6090	−0.792704	−0.396352	−	0.918099i	$-0.629724\pi$
$439$	−16.6090	−0.792704	−0.396352	+	0.918099i	$0.629724\pi$
$440$	0	0
$441$	−2.91828	−0.138966
$442$	2.31706	0.110211
$443$	−11.6526	−0.553630	−0.276815	−	0.960923i	$-0.589279\pi$
$443$	−11.6526	−0.553630	−0.276815	+	0.960923i	$0.589279\pi$
$444$	−4.08015	−0.193635
$445$	0	0
$446$	−54.3172	−2.57199
$447$	−0.0275094	−0.00130115
$448$	2.58269	0.122021
$449$	16.1924	0.764168	0.382084	−	0.924128i	$-0.375206\pi$
$449$	16.1924	0.764168	0.382084	+	0.924128i	$0.375206\pi$
$450$	0	0
$451$	−21.2398	−1.00014
$452$	64.4670	3.03227
$453$	−3.18085	−0.149450
$454$	65.9409	3.09476
$455$	0	0
$456$	−7.51755	−0.352041
$457$	−9.47239	−0.443100	−0.221550	−	0.975149i	$-0.571112\pi$
$457$	−9.47239	−0.443100	−0.221550	+	0.975149i	$0.571112\pi$
$458$	26.6186	1.24380
$459$	−1.48958	−0.0695274
$460$	0	0
$461$	6.80775	0.317068	0.158534	−	0.987353i	$-0.449323\pi$
$461$	6.80775	0.317068	0.158534	+	0.987353i	$0.449323\pi$
$462$	4.51490	0.210052
$463$	−12.2442	−0.569039	−0.284519	−	0.958670i	$-0.591834\pi$
$463$	−12.2442	−0.569039	−0.284519	+	0.958670i	$0.591834\pi$
$464$	55.7312	2.58725
$465$	0	0
$466$	38.8079	1.79774
$467$	−7.11405	−0.329199	−0.164600	−	0.986360i	$-0.552633\pi$
$467$	−7.11405	−0.329199	−0.164600	+	0.986360i	$0.552633\pi$
$468$	−13.7757	−0.636781
$469$	−3.65650	−0.168841
$470$	0	0
$471$	−1.92785	−0.0888306
$472$	95.3553	4.38908
$473$	0.535002	0.0245994
$474$	2.50344	0.114987
$475$	0	0
$476$	4.06118	0.186144
$477$	−31.9569	−1.46321
$478$	−17.6503	−0.807304
$479$	13.0420	0.595905	0.297952	−	0.954581i	$-0.403696\pi$
$479$	13.0420	0.595905	0.297952	+	0.954581i	$0.403696\pi$
$480$	0	0
$481$	−3.16678	−0.144393
$482$	−19.7183	−0.898143
$483$	0.285863	0.0130072
$484$	123.263	5.60285
$485$	0	0
$486$	19.1316	0.867828
$487$	17.6809	0.801200	0.400600	−	0.916253i	$-0.368802\pi$
$487$	17.6809	0.801200	0.400600	+	0.916253i	$0.368802\pi$
$488$	−38.6081	−1.74771
$489$	−0.104081	−0.00470671
$490$	0	0
$491$	29.4923	1.33097	0.665484	−	0.746412i	$-0.268225\pi$
$491$	29.4923	1.33097	0.665484	+	0.746412i	$0.268225\pi$
$492$	−4.55981	−0.205572
$493$	6.09501	0.274505
$494$	−10.3013	−0.463478
$495$	0	0
$496$	3.14358	0.141151
$497$	−14.9617	−0.671125
$498$	1.38219	0.0619376
$499$	−11.0638	−0.495284	−0.247642	−	0.968852i	$-0.579656\pi$
$499$	−11.0638	−0.495284	−0.247642	+	0.968852i	$0.579656\pi$
$500$	0	0
$501$	−1.45139	−0.0648435
$502$	28.9707	1.29302
$503$	27.0098	1.20431	0.602155	−	0.798379i	$-0.294309\pi$
$503$	27.0098	1.20431	0.602155	+	0.798379i	$0.294309\pi$
$504$	−19.6056	−0.873304
$505$	0	0
$506$	15.7939	0.702125
$507$	−3.41683	−0.151747
$508$	−36.1357	−1.60326
$509$	31.4405	1.39358	0.696789	−	0.717277i	$-0.254612\pi$
$509$	31.4405	1.39358	0.696789	+	0.717277i	$0.254612\pi$
$510$	0	0
$511$	−14.5236	−0.642486
$512$	−49.6774	−2.19545
$513$	6.62244	0.292388
$514$	−53.0149	−2.33839
$515$	0	0
$516$	0.114856	0.00505623
$517$	40.9452	1.80077
$518$	−7.95720	−0.349619
$519$	−4.59305	−0.201613
$520$	0	0
$521$	−33.6660	−1.47493	−0.737467	−	0.675383i	$-0.763978\pi$
$521$	−33.6660	−1.47493	−0.737467	+	0.675383i	$0.763978\pi$
$522$	−51.9491	−2.27375
$523$	−20.6357	−0.902335	−0.451168	−	0.892439i	$-0.648992\pi$
$523$	−20.6357	−0.902335	−0.451168	+	0.892439i	$0.648992\pi$
$524$	−52.2492	−2.28252
$525$	0	0
$526$	−29.3813	−1.28108
$527$	0.343796	0.0149760
$528$	14.1350	0.615146
$529$	1.00000	0.0434783
$530$	0	0
$531$	−41.4208	−1.79751
$532$	−18.0554	−0.782802
$533$	−3.53906	−0.153294
$534$	−9.71250	−0.420301
$535$	0	0
$536$	−24.5651	−1.06105
$537$	−1.49463	−0.0644981
$538$	50.8028	2.19026
$539$	6.14192	0.264551
$540$	0	0
$541$	8.09344	0.347964	0.173982	−	0.984749i	$-0.444337\pi$
$541$	8.09344	0.347964	0.173982	+	0.984749i	$0.444337\pi$
$542$	−24.1863	−1.03889
$543$	2.34656	0.100700
$544$	6.39727	0.274281
$545$	0	0
$546$	0.752291	0.0321951
$547$	−25.7441	−1.10074	−0.550369	−	0.834921i	$-0.685513\pi$
$547$	−25.7441	−1.10074	−0.550369	+	0.834921i	$0.685513\pi$
$548$	32.4513	1.38625
$549$	16.7708	0.715758
$550$	0	0
$551$	−27.0976	−1.15439
$552$	1.92049	0.0817414
$553$	3.40561	0.144821
$554$	−68.0743	−2.89220
$555$	0	0
$556$	61.6165	2.61312
$557$	15.9396	0.675383	0.337692	−	0.941257i	$-0.390354\pi$
$557$	15.9396	0.675383	0.337692	+	0.941257i	$0.390354\pi$
$558$	−2.93025	−0.124047
$559$	0.0891443	0.00377040
$560$	0	0
$561$	1.54586	0.0652664
$562$	19.5589	0.825044
$563$	−12.9073	−0.543977	−0.271989	−	0.962300i	$-0.587681\pi$
$563$	−12.9073	−0.543977	−0.271989	+	0.962300i	$0.587681\pi$
$564$	8.79022	0.370135
$565$	0	0
$566$	3.27442	0.137634
$567$	8.27122	0.347359
$568$	−100.516	−4.21756
$569$	25.8495	1.08367	0.541834	−	0.840485i	$-0.317730\pi$
$569$	25.8495	1.08367	0.541834	+	0.840485i	$0.317730\pi$
$570$	0	0
$571$	2.65336	0.111040	0.0555199	−	0.998458i	$-0.482318\pi$
$571$	2.65336	0.111040	0.0555199	+	0.998458i	$0.482318\pi$
$572$	28.9928	1.21225
$573$	−0.997744	−0.0416814
$574$	−8.89263	−0.371171
$575$	0	0
$576$	−7.53703	−0.314043
$577$	12.8442	0.534710	0.267355	−	0.963598i	$-0.413850\pi$
$577$	12.8442	0.534710	0.267355	+	0.963598i	$0.413850\pi$
$578$	−41.7219	−1.73540
$579$	5.37705	0.223463
$580$	0	0
$581$	1.88029	0.0780078
$582$	−12.1945	−0.505479
$583$	67.2578	2.78553
$584$	−97.5726	−4.03758
$585$	0	0
$586$	−6.81166	−0.281387
$587$	9.91770	0.409347	0.204674	−	0.978830i	$-0.434387\pi$
$587$	9.91770	0.409347	0.204674	+	0.978830i	$0.434387\pi$
$588$	1.31856	0.0543767
$589$	−1.52847	−0.0629795
$590$	0	0
$591$	0.730581	0.0300521
$592$	−24.9119	−1.02387
$593$	−31.1527	−1.27929	−0.639644	−	0.768671i	$-0.720918\pi$
$593$	−31.1527	−1.27929	−0.639644	+	0.768671i	$0.720918\pi$
$594$	−26.7204	−1.09635
$595$	0	0
$596$	−0.443881	−0.0181821
$597$	1.43455	0.0587123
$598$	2.63165	0.107616
$599$	−40.6804	−1.66216	−0.831079	−	0.556155i	$-0.812276\pi$
$599$	−40.6804	−1.66216	−0.831079	+	0.556155i	$0.812276\pi$
$600$	0	0
$601$	20.1729	0.822869	0.411435	−	0.911439i	$-0.365028\pi$
$601$	20.1729	0.822869	0.411435	+	0.911439i	$0.365028\pi$
$602$	0.223994	0.00912931
$603$	10.6707	0.434544
$604$	−51.3250	−2.08838
$605$	0	0
$606$	11.0749	0.449885
$607$	6.08065	0.246806	0.123403	−	0.992357i	$-0.460619\pi$
$607$	6.08065	0.246806	0.123403	+	0.992357i	$0.460619\pi$
$608$	−28.4414	−1.15345
$609$	1.97890	0.0801890
$610$	0	0
$611$	6.82246	0.276007
$612$	−11.8517	−0.479075
$613$	37.2909	1.50617	0.753084	−	0.657925i	$-0.228566\pi$
$613$	37.2909	1.50617	0.753084	+	0.657925i	$0.228566\pi$
$614$	−82.7529	−3.33963
$615$	0	0
$616$	41.2627	1.66252
$617$	41.6439	1.67652	0.838261	−	0.545270i	$-0.183573\pi$
$617$	41.6439	1.67652	0.838261	+	0.545270i	$0.183573\pi$
$618$	−9.97409	−0.401217
$619$	15.6115	0.627480	0.313740	−	0.949509i	$-0.398418\pi$
$619$	15.6115	0.627480	0.313740	+	0.949509i	$0.398418\pi$
$620$	0	0
$621$	−1.69182	−0.0678903
$622$	−28.7420	−1.15245
$623$	−13.2126	−0.529351
$624$	2.35523	0.0942847
$625$	0	0
$626$	63.4404	2.53559
$627$	−6.87269	−0.274469
$628$	−31.1070	−1.24130
$629$	−2.72448	−0.108632
$630$	0	0
$631$	26.3500	1.04898	0.524488	−	0.851418i	$-0.324257\pi$
$631$	26.3500	1.04898	0.524488	+	0.851418i	$0.324257\pi$
$632$	22.8796	0.910101
$633$	−4.06757	−0.161672
$634$	−86.1195	−3.42024
$635$	0	0
$636$	14.4391	0.572546
$637$	1.02339	0.0405483
$638$	109.334	4.32857
$639$	43.6625	1.72726
$640$	0	0
$641$	−9.38344	−0.370623	−0.185312	−	0.982680i	$-0.559329\pi$
$641$	−9.38344	−0.370623	−0.185312	+	0.982680i	$0.559329\pi$
$642$	3.68787	0.145549
$643$	−44.1627	−1.74161	−0.870803	−	0.491632i	$-0.836400\pi$
$643$	−44.1627	−1.74161	−0.870803	+	0.491632i	$0.836400\pi$
$644$	4.61257	0.181761
$645$	0	0
$646$	−8.86255	−0.348692
$647$	26.7049	1.04988	0.524939	−	0.851140i	$-0.324088\pi$
$647$	26.7049	1.04988	0.524939	+	0.851140i	$0.324088\pi$
$648$	55.5678	2.18291
$649$	87.1758	3.42195
$650$	0	0
$651$	0.111622	0.00437482
$652$	−1.67941	−0.0657708
$653$	−40.8546	−1.59876	−0.799382	−	0.600823i	$-0.794840\pi$
$653$	−40.8546	−1.59876	−0.799382	+	0.600823i	$0.794840\pi$
$654$	0.696001	0.0272158
$655$	0	0
$656$	−27.8405	−1.08699
$657$	42.3840	1.65356
$658$	17.1429	0.668299
$659$	−3.67292	−0.143077	−0.0715383	−	0.997438i	$-0.522791\pi$
$659$	−3.67292	−0.143077	−0.0715383	+	0.997438i	$0.522791\pi$
$660$	0	0
$661$	43.1813	1.67956	0.839779	−	0.542928i	$-0.182684\pi$
$661$	43.1813	1.67956	0.839779	+	0.542928i	$0.182684\pi$
$662$	59.7976	2.32410
$663$	0.257579	0.0100035
$664$	12.6322	0.490225
$665$	0	0
$666$	23.2213	0.899809
$667$	6.92254	0.268042
$668$	−23.4191	−0.906113
$669$	−6.03824	−0.233452
$670$	0	0
$671$	−35.2963	−1.36260
$672$	2.07704	0.0801234
$673$	−1.47909	−0.0570148	−0.0285074	−	0.999594i	$-0.509075\pi$
$673$	−1.47909	−0.0570148	−0.0285074	+	0.999594i	$0.509075\pi$
$674$	44.2264	1.70354
$675$	0	0
$676$	−55.1325	−2.12048
$677$	−3.13956	−0.120663	−0.0603316	−	0.998178i	$-0.519216\pi$
$677$	−3.13956	−0.120663	−0.0603316	+	0.998178i	$0.519216\pi$
$678$	10.2740	0.394569
$679$	−16.5891	−0.636630
$680$	0	0
$681$	7.33040	0.280902
$682$	6.16711	0.236151
$683$	0.168666	0.00645381	0.00322690	−	0.999995i	$-0.498973\pi$
$683$	0.168666	0.00645381	0.00322690	+	0.999995i	$0.498973\pi$
$684$	52.6908	2.01468
$685$	0	0
$686$	2.57149	0.0981800
$687$	2.95909	0.112896
$688$	0.701267	0.0267355
$689$	11.2068	0.426944
$690$	0	0
$691$	43.8940	1.66981	0.834903	−	0.550397i	$-0.185524\pi$
$691$	43.8940	1.66981	0.834903	+	0.550397i	$0.185524\pi$
$692$	−74.1116	−2.81730
$693$	−17.9239	−0.680871
$694$	−37.9521	−1.44064
$695$	0	0
$696$	13.2947	0.503933
$697$	−3.04477	−0.115329
$698$	−1.79364	−0.0678903
$699$	4.31413	0.163175
$700$	0	0
$701$	−1.91383	−0.0722843	−0.0361421	−	0.999347i	$-0.511507\pi$
$701$	−1.91383	−0.0722843	−0.0361421	+	0.999347i	$0.511507\pi$
$702$	−4.45227	−0.168040
$703$	12.1127	0.456838
$704$	15.8627	0.597848
$705$	0	0
$706$	−83.0805	−3.12678
$707$	15.0659	0.566611
$708$	18.7151	0.703357
$709$	−9.18131	−0.344811	−0.172406	−	0.985026i	$-0.555154\pi$
$709$	−9.18131	−0.344811	−0.172406	+	0.985026i	$0.555154\pi$
$710$	0	0
$711$	−9.93852	−0.372724
$712$	−88.7649	−3.32661
$713$	0.390474	0.0146234
$714$	0.647220	0.0242216
$715$	0	0
$716$	−24.1168	−0.901287
$717$	−1.96211	−0.0732765
$718$	−64.7071	−2.41485
$719$	38.6304	1.44067	0.720335	−	0.693626i	$-0.243988\pi$
$719$	38.6304	1.44067	0.720335	+	0.693626i	$0.243988\pi$
$720$	0	0
$721$	−13.5684	−0.505315
$722$	−9.45670	−0.351942
$723$	−2.19201	−0.0815216
$724$	37.8631	1.40717
$725$	0	0
$726$	19.6441	0.729061
$727$	−45.2391	−1.67782	−0.838912	−	0.544267i	$-0.816808\pi$
$727$	−45.2391	−1.67782	−0.838912	+	0.544267i	$0.816808\pi$
$728$	6.87537	0.254818
$729$	−22.6869	−0.840254
$730$	0	0
$731$	0.0766937	0.00283662
$732$	−7.57751	−0.280073
$733$	6.31250	0.233157	0.116579	−	0.993181i	$-0.462807\pi$
$733$	6.31250	0.233157	0.116579	+	0.993181i	$0.462807\pi$
$734$	−78.0944	−2.88252
$735$	0	0
$736$	7.26584	0.267822
$737$	−22.4579	−0.827248
$738$	25.9512	0.955277
$739$	33.8666	1.24580	0.622902	−	0.782300i	$-0.285953\pi$
$739$	33.8666	1.24580	0.622902	+	0.782300i	$0.285953\pi$
$740$	0	0
$741$	−1.14516	−0.0420684
$742$	28.1594	1.03376
$743$	−42.9413	−1.57536	−0.787682	−	0.616082i	$-0.788719\pi$
$743$	−42.9413	−1.57536	−0.787682	+	0.616082i	$0.788719\pi$
$744$	0.749901	0.0274927
$745$	0	0
$746$	−25.4074	−0.930231
$747$	−5.48723	−0.200767
$748$	24.9434	0.912023
$749$	5.01686	0.183312
$750$	0	0
$751$	−0.829615	−0.0302731	−0.0151365	−	0.999885i	$-0.504818\pi$
$751$	−0.829615	−0.0302731	−0.0151365	+	0.999885i	$0.504818\pi$
$752$	53.6699	1.95714
$753$	3.22056	0.117364
$754$	18.2177	0.663450
$755$	0	0
$756$	−7.80364	−0.283815
$757$	−24.2753	−0.882300	−0.441150	−	0.897434i	$-0.645429\pi$
$757$	−24.2753	−0.882300	−0.441150	+	0.897434i	$0.645429\pi$
$758$	17.0153	0.618024
$759$	1.75575	0.0637297
$760$	0	0
$761$	−33.5478	−1.21611	−0.608053	−	0.793896i	$-0.708049\pi$
$761$	−33.5478	−1.21611	−0.608053	+	0.793896i	$0.708049\pi$
$762$	−5.75886	−0.208622
$763$	0.946819	0.0342771
$764$	−16.0992	−0.582449
$765$	0	0
$766$	−14.2407	−0.514539
$767$	14.5256	0.524489
$768$	−7.27671	−0.262576
$769$	−35.7498	−1.28917	−0.644586	−	0.764532i	$-0.722970\pi$
$769$	−35.7498	−1.28917	−0.644586	+	0.764532i	$0.722970\pi$
$770$	0	0
$771$	−5.89346	−0.212248
$772$	86.7619	3.12263
$773$	14.7661	0.531101	0.265551	−	0.964097i	$-0.414446\pi$
$773$	14.7661	0.531101	0.265551	+	0.964097i	$0.414446\pi$
$774$	−0.653677	−0.0234959
$775$	0	0
$776$	−111.449	−4.00078
$777$	−0.884572	−0.0317338
$778$	83.7752	3.00348
$779$	13.5366	0.484999
$780$	0	0
$781$	−91.8937	−3.28822
$782$	2.26409	0.0809638
$783$	−11.7117	−0.418541
$784$	8.05068	0.287524
$785$	0	0
$786$	−8.32683	−0.297008
$787$	51.8562	1.84847	0.924236	−	0.381821i	$-0.124703\pi$
$787$	51.8562	1.84847	0.924236	+	0.381821i	$0.124703\pi$
$788$	11.7884	0.419943
$789$	−3.26621	−0.116280
$790$	0	0
$791$	13.9764	0.496943
$792$	−120.416	−4.27881
$793$	−5.88123	−0.208849
$794$	−58.5207	−2.07682
$795$	0	0
$796$	23.1474	0.820437
$797$	6.03809	0.213880	0.106940	−	0.994265i	$-0.465895\pi$
$797$	6.03809	0.213880	0.106940	+	0.994265i	$0.465895\pi$
$798$	−2.87745	−0.101861
$799$	5.86958	0.207651
$800$	0	0
$801$	38.5581	1.36238
$802$	72.8924	2.57392
$803$	−89.2029	−3.14790
$804$	−4.82133	−0.170035
$805$	0	0
$806$	1.02759	0.0361953
$807$	5.64756	0.198803
$808$	101.216	3.56076
$809$	−31.8270	−1.11898	−0.559489	−	0.828838i	$-0.689003\pi$
$809$	−31.8270	−1.11898	−0.559489	+	0.828838i	$0.689003\pi$
$810$	0	0
$811$	21.7616	0.764154	0.382077	−	0.924130i	$-0.375209\pi$
$811$	21.7616	0.764154	0.382077	+	0.924130i	$0.375209\pi$
$812$	31.9307	1.12055
$813$	−2.68869	−0.0942966
$814$	−48.8725	−1.71298
$815$	0	0
$816$	2.02628	0.0709340
$817$	−0.340969	−0.0119290
$818$	−20.3108	−0.710152
$819$	−2.98655	−0.104359
$820$	0	0
$821$	43.1303	1.50526	0.752630	−	0.658444i	$-0.228785\pi$
$821$	43.1303	1.50526	0.752630	+	0.658444i	$0.228785\pi$
$822$	5.17168	0.180383
$823$	−8.91338	−0.310701	−0.155350	−	0.987859i	$-0.549651\pi$
$823$	−8.91338	−0.310701	−0.155350	+	0.987859i	$0.549651\pi$
$824$	−91.1556	−3.17556
$825$	0	0
$826$	36.4986	1.26995
$827$	39.3167	1.36718	0.683588	−	0.729868i	$-0.260419\pi$
$827$	39.3167	1.36718	0.683588	+	0.729868i	$0.260419\pi$
$828$	−13.4608	−0.467795
$829$	9.43997	0.327864	0.163932	−	0.986472i	$-0.447582\pi$
$829$	9.43997	0.327864	0.163932	+	0.986472i	$0.447582\pi$
$830$	0	0
$831$	−7.56757	−0.262516
$832$	2.64311	0.0916334
$833$	0.880458	0.0305061
$834$	9.81967	0.340028
$835$	0	0
$836$	−110.895	−3.83539
$837$	−0.660611	−0.0228341
$838$	33.5490	1.15893
$839$	34.0428	1.17529	0.587644	−	0.809120i	$-0.300056\pi$
$839$	34.0428	1.17529	0.587644	+	0.809120i	$0.300056\pi$
$840$	0	0
$841$	18.9216	0.652468
$842$	45.8347	1.57957
$843$	2.17429	0.0748866
$844$	−65.6327	−2.25917
$845$	0	0
$846$	−50.0277	−1.71999
$847$	26.7232	0.918221
$848$	88.1598	3.02742
$849$	0.364005	0.0124926
$850$	0	0
$851$	−3.09439	−0.106074
$852$	−19.7280	−0.675870
$853$	10.6161	0.363490	0.181745	−	0.983346i	$-0.441825\pi$
$853$	10.6161	0.363490	0.181745	+	0.983346i	$0.441825\pi$
$854$	−14.7778	−0.505687
$855$	0	0
$856$	33.7043	1.15199
$857$	−13.1582	−0.449474	−0.224737	−	0.974419i	$-0.572152\pi$
$857$	−13.1582	−0.449474	−0.224737	+	0.974419i	$0.572152\pi$
$858$	4.62052	0.157742
$859$	27.5047	0.938447	0.469224	−	0.883079i	$-0.344534\pi$
$859$	27.5047	0.938447	0.469224	+	0.883079i	$0.344534\pi$
$860$	0	0
$861$	−0.988560	−0.0336900
$862$	17.2217	0.586575
$863$	55.1862	1.87856	0.939280	−	0.343151i	$-0.111494\pi$
$863$	55.1862	1.87856	0.939280	+	0.343151i	$0.111494\pi$
$864$	−12.2925	−0.418199
$865$	0	0
$866$	−39.9935	−1.35903
$867$	−4.63807	−0.157517
$868$	1.80109	0.0611330
$869$	20.9170	0.709560
$870$	0	0
$871$	−3.74204	−0.126794
$872$	6.36093	0.215408
$873$	48.4115	1.63848
$874$	−10.0658	−0.340482
$875$	0	0
$876$	−19.1503	−0.647029
$877$	26.8565	0.906880	0.453440	−	0.891287i	$-0.350197\pi$
$877$	26.8565	0.906880	0.453440	+	0.891287i	$0.350197\pi$
$878$	−42.7099	−1.44139
$879$	−0.757226	−0.0255406
$880$	0	0
$881$	21.0482	0.709131	0.354566	−	0.935031i	$-0.384629\pi$
$881$	21.0482	0.709131	0.354566	+	0.935031i	$0.384629\pi$
$882$	−7.50434	−0.252684
$883$	40.7345	1.37083	0.685413	−	0.728155i	$-0.259622\pi$
$883$	40.7345	1.37083	0.685413	+	0.728155i	$0.259622\pi$
$884$	4.15618	0.139788
$885$	0	0
$886$	−29.9645	−1.00668
$887$	32.8883	1.10428	0.552141	−	0.833751i	$-0.313811\pi$
$887$	32.8883	1.10428	0.552141	+	0.833751i	$0.313811\pi$
$888$	−5.94274	−0.199425
$889$	−7.83418	−0.262750
$890$	0	0
$891$	50.8012	1.70190
$892$	−97.4306	−3.26222
$893$	−26.0953	−0.873247
$894$	−0.0707402	−0.00236591
$895$	0	0
$896$	−7.89030	−0.263597
$897$	0.292551	0.00976798
$898$	41.6387	1.38950
$899$	2.70307	0.0901525
$900$	0	0
$901$	9.64155	0.321207
$902$	−54.6179	−1.81858
$903$	0.0249005	0.000828638 0
$904$	93.8962	3.12294
$905$	0	0
$906$	−8.17954	−0.271747
$907$	−57.5807	−1.91193	−0.955967	−	0.293473i	$-0.905189\pi$
$907$	−57.5807	−1.91193	−0.955967	+	0.293473i	$0.905189\pi$
$908$	118.280	3.92528
$909$	−43.9665	−1.45828
$910$	0	0
$911$	3.25171	0.107734	0.0538669	−	0.998548i	$-0.482845\pi$
$911$	3.25171	0.107734	0.0538669	+	0.998548i	$0.482845\pi$
$912$	−9.00856	−0.298303
$913$	11.5486	0.382204
$914$	−24.3582	−0.805697
$915$	0	0
$916$	47.7467	1.57759
$917$	−11.3276	−0.374069
$918$	−3.83043	−0.126423
$919$	43.5898	1.43790	0.718948	−	0.695064i	$-0.244624\pi$
$919$	43.5898	1.43790	0.718948	+	0.695064i	$0.244624\pi$
$920$	0	0
$921$	−9.19932	−0.303128
$922$	17.5061	0.576532
$923$	−15.3117	−0.503992
$924$	8.09852	0.266422
$925$	0	0
$926$	−31.4860	−1.03469
$927$	39.5965	1.30052
$928$	50.2981	1.65112
$929$	−30.8853	−1.01331	−0.506656	−	0.862148i	$-0.669119\pi$
$929$	−30.8853	−1.01331	−0.506656	+	0.862148i	$0.669119\pi$
$930$	0	0
$931$	−3.91439	−0.128289
$932$	69.6110	2.28019
$933$	−3.19514	−0.104604
$934$	−18.2937	−0.598589
$935$	0	0
$936$	−20.0643	−0.655822
$937$	−26.1035	−0.852763	−0.426381	−	0.904543i	$-0.640212\pi$
$937$	−26.1035	−0.852763	−0.426381	+	0.904543i	$0.640212\pi$
$938$	−9.40266	−0.307008
$939$	7.05243	0.230147
$940$	0	0
$941$	−24.9368	−0.812915	−0.406458	−	0.913670i	$-0.633236\pi$
$941$	−24.9368	−0.812915	−0.406458	+	0.913670i	$0.633236\pi$
$942$	−4.95745	−0.161522
$943$	−3.45816	−0.112613
$944$	114.268	3.71910
$945$	0	0
$946$	1.37575	0.0447296
$947$	−7.66115	−0.248954	−0.124477	−	0.992222i	$-0.539725\pi$
$947$	−7.66115	−0.248954	−0.124477	+	0.992222i	$0.539725\pi$
$948$	4.49051	0.145845
$949$	−14.8634	−0.482485
$950$	0	0
$951$	−9.57358	−0.310445
$952$	5.91510	0.191710
$953$	−11.1554	−0.361360	−0.180680	−	0.983542i	$-0.557830\pi$
$953$	−11.1554	−0.361360	−0.180680	+	0.983542i	$0.557830\pi$
$954$	−82.1770	−2.66058
$955$	0	0
$956$	−31.6599	−1.02395
$957$	12.1542	0.392891
$958$	33.5374	1.08354
$959$	7.03539	0.227185
$960$	0	0
$961$	−30.8475	−0.995082
$962$	−8.14335	−0.262552
$963$	−14.6406	−0.471788
$964$	−35.3693	−1.13917
$965$	0	0
$966$	0.735095	0.0236513
$967$	3.29023	0.105807	0.0529034	−	0.998600i	$-0.483152\pi$
$967$	3.29023	0.105807	0.0529034	+	0.998600i	$0.483152\pi$
$968$	179.532	5.77038
$969$	−0.985216	−0.0316497
$970$	0	0
$971$	50.2428	1.61237	0.806184	−	0.591665i	$-0.201529\pi$
$971$	50.2428	1.61237	0.806184	+	0.591665i	$0.201529\pi$
$972$	34.3170	1.10072
$973$	13.3584	0.428250
$974$	45.4664	1.45684
$975$	0	0
$976$	−46.2656	−1.48092
$977$	−13.4807	−0.431286	−0.215643	−	0.976472i	$-0.569185\pi$
$977$	−13.4807	−0.431286	−0.215643	+	0.976472i	$0.569185\pi$
$978$	−0.267644	−0.00855830
$979$	−81.1507	−2.59359
$980$	0	0
$981$	−2.76308	−0.0882185
$982$	75.8392	2.42013
$983$	−30.4473	−0.971118	−0.485559	−	0.874204i	$-0.661384\pi$
$983$	−30.4473	−0.971118	−0.485559	+	0.874204i	$0.661384\pi$
$984$	−6.64136	−0.211719
$985$	0	0
$986$	15.6733	0.499139
$987$	1.90571	0.0606594
$988$	−18.4778	−0.587858
$989$	0.0871065	0.00276983
$990$	0	0
$991$	−3.16595	−0.100570	−0.0502849	−	0.998735i	$-0.516013\pi$
$991$	−3.16595	−0.100570	−0.0502849	+	0.998735i	$0.516013\pi$
$992$	2.83712	0.0900787
$993$	6.64748	0.210951
$994$	−38.4739	−1.22032
$995$	0	0
$996$	2.47929	0.0785593
$997$	49.5209	1.56834	0.784171	−	0.620545i	$-0.213089\pi$
$997$	49.5209	1.56834	0.784171	+	0.620545i	$0.213089\pi$
$998$	−28.4505	−0.900585
$999$	5.23514	0.165633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4025.2.a.be.1.19		21
5.2	odd	4		805.2.c.c.484.38	yes	42
5.3	odd	4		805.2.c.c.484.5	✓	42
5.4	even	2		4025.2.a.bd.1.3		21

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
805.2.c.c.484.5	✓	42	5.3	odd	4
805.2.c.c.484.38	yes	42	5.2	odd	4
4025.2.a.bd.1.3		21	5.4	even	2
4025.2.a.be.1.19		21	1.1	even	1	trivial

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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:	\( N \)	\(=\)	\( 4025 = 5^{2} \cdot 7 \cdot 23 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4025.a (trivial)

\(f(q)\)	\(=\)	\(q+2.57149 q^{2} +0.285863 q^{3} +4.61257 q^{4} +0.735095 q^{6} +1.00000 q^{7} +6.71821 q^{8} -2.91828 q^{9} +O(q^{10})\)	\(q+2.57149 q^{2} +0.285863 q^{3} +4.61257 q^{4} +0.735095 q^{6} +1.00000 q^{7} +6.71821 q^{8} -2.91828 q^{9} +6.14192 q^{11} +1.31856 q^{12} +1.02339 q^{13} +2.57149 q^{14} +8.05068 q^{16} +0.880458 q^{17} -7.50434 q^{18} -3.91439 q^{19} +0.285863 q^{21} +15.7939 q^{22} +1.00000 q^{23} +1.92049 q^{24} +2.63165 q^{26} -1.69182 q^{27} +4.61257 q^{28} +6.92254 q^{29} +0.390474 q^{31} +7.26584 q^{32} +1.75575 q^{33} +2.26409 q^{34} -13.4608 q^{36} -3.09439 q^{37} -10.0658 q^{38} +0.292551 q^{39} -3.45816 q^{41} +0.735095 q^{42} +0.0871065 q^{43} +28.3301 q^{44} +2.57149 q^{46} +6.66651 q^{47} +2.30139 q^{48} +1.00000 q^{49} +0.251691 q^{51} +4.72048 q^{52} +10.9506 q^{53} -4.35050 q^{54} +6.71821 q^{56} -1.11898 q^{57} +17.8013 q^{58} +14.1936 q^{59} -5.74679 q^{61} +1.00410 q^{62} -2.91828 q^{63} +2.58269 q^{64} +4.51490 q^{66} -3.65650 q^{67} +4.06118 q^{68} +0.285863 q^{69} -14.9617 q^{71} -19.6056 q^{72} -14.5236 q^{73} -7.95720 q^{74} -18.0554 q^{76} +6.14192 q^{77} +0.752291 q^{78} +3.40561 q^{79} +8.27122 q^{81} -8.89263 q^{82} +1.88029 q^{83} +1.31856 q^{84} +0.223994 q^{86} +1.97890 q^{87} +41.2627 q^{88} -13.2126 q^{89} +1.02339 q^{91} +4.61257 q^{92} +0.111622 q^{93} +17.1429 q^{94} +2.07704 q^{96} -16.5891 q^{97} +2.57149 q^{98} -17.9239 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9}+O(q^{10}) \) 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9	\( 21 q + 2 q^{2} - q^{3} + 30 q^{4} + 6 q^{6} + 21 q^{7} + 6 q^{8} + 30 q^{9} + 7 q^{11} - 22 q^{12} - 3 q^{13} + 2 q^{14} + 56 q^{16} + 7 q^{17} + 24 q^{19} - q^{21} + 4 q^{22} + 21 q^{23} + 24 q^{24} - 2 q^{26} - 19 q^{27} + 30 q^{28} + 11 q^{29} + 46 q^{31} - 6 q^{32} - 3 q^{33} + 28 q^{34} + 58 q^{36} + 24 q^{37} - 4 q^{38} + 31 q^{39} + 14 q^{41} + 6 q^{42} + 18 q^{43} + 12 q^{44} + 2 q^{46} - 25 q^{47} - 36 q^{48} + 21 q^{49} + 17 q^{51} - 8 q^{52} + 22 q^{53} - 6 q^{54} + 6 q^{56} + 40 q^{57} + 6 q^{58} + 10 q^{59} + 38 q^{61} - 54 q^{62} + 30 q^{63} + 100 q^{64} + 38 q^{66} + 12 q^{67} + 18 q^{68} - q^{69} + 56 q^{71} + 42 q^{72} - 40 q^{73} - 20 q^{74} + 60 q^{76} + 7 q^{77} + 38 q^{78} + 49 q^{79} + 57 q^{81} + 16 q^{82} - 2 q^{83} - 22 q^{84} + 16 q^{86} - 23 q^{87} + 12 q^{88} + 28 q^{89} - 3 q^{91} + 30 q^{92} - 30 q^{93} + 66 q^{94} + 46 q^{96} - q^{97} + 2 q^{98} - 2 q^{99}+O(q^{100}) \) 21 * q + 2 * q^2 - q^3 + 30 * q^4 + 6 * q^6 + 21 * q^7 + 6 * q^8 + 30 * q^9 + 7 * q^11 - 22 * q^12 - 3 * q^13 + 2 * q^14 + 56 * q^16 + 7 * q^17 + 24 * q^19 - q^21 + 4 * q^22 + 21 * q^23 + 24 * q^24 - 2 * q^26 - 19 * q^27 + 30 * q^28 + 11 * q^29 + 46 * q^31 - 6 * q^32 - 3 * q^33 + 28 * q^34 + 58 * q^36 + 24 * q^37 - 4 * q^38 + 31 * q^39 + 14 * q^41 + 6 * q^42 + 18 * q^43 + 12 * q^44 + 2 * q^46 - 25 * q^47 - 36 * q^48 + 21 * q^49 + 17 * q^51 - 8 * q^52 + 22 * q^53 - 6 * q^54 + 6 * q^56 + 40 * q^57 + 6 * q^58 + 10 * q^59 + 38 * q^61 - 54 * q^62 + 30 * q^63 + 100 * q^64 + 38 * q^66 + 12 * q^67 + 18 * q^68 - q^69 + 56 * q^71 + 42 * q^72 - 40 * q^73 - 20 * q^74 + 60 * q^76 + 7 * q^77 + 38 * q^78 + 49 * q^79 + 57 * q^81 + 16 * q^82 - 2 * q^83 - 22 * q^84 + 16 * q^86 - 23 * q^87 + 12 * q^88 + 28 * q^89 - 3 * q^91 + 30 * q^92 - 30 * q^93 + 66 * q^94 + 46 * q^96 - q^97 + 2 * q^98 - 2 * q^99

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