LMFDB - Embedded newform 4925.2.a.q.1.29

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4925,2,Mod(1,4925)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("4925.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4925, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4925 = 5^{2} \cdot 197 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4925.a (trivial)

Newform invariants

comment:select newform

sage:traces = [37,2,1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$39.3263229955$
Analytic rank:	$0$
Dimension:	$37$
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.29
Character	$\chi$	$=$	4925.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.86497 q^{2} +0.0285304 q^{3} +1.47810 q^{4} +0.0532083 q^{6} +2.61269 q^{7} -0.973320 q^{8} -2.99919 q^{9} -2.95031 q^{11} +0.0421709 q^{12} +5.30110 q^{13} +4.87258 q^{14} -4.77142 q^{16} -6.23067 q^{17} -5.59338 q^{18} +7.70215 q^{19} +0.0745411 q^{21} -5.50222 q^{22} +4.37513 q^{23} -0.0277693 q^{24} +9.88639 q^{26} -0.171159 q^{27} +3.86182 q^{28} -6.90125 q^{29} +4.74310 q^{31} -6.95190 q^{32} -0.0841735 q^{33} -11.6200 q^{34} -4.43311 q^{36} +7.34312 q^{37} +14.3643 q^{38} +0.151243 q^{39} +12.3379 q^{41} +0.139017 q^{42} +11.2845 q^{43} -4.36086 q^{44} +8.15948 q^{46} +8.16281 q^{47} -0.136131 q^{48} -0.173865 q^{49} -0.177764 q^{51} +7.83558 q^{52} +5.90870 q^{53} -0.319207 q^{54} -2.54298 q^{56} +0.219746 q^{57} -12.8706 q^{58} -9.78424 q^{59} -0.306027 q^{61} +8.84573 q^{62} -7.83594 q^{63} -3.42223 q^{64} -0.156981 q^{66} +12.8152 q^{67} -9.20957 q^{68} +0.124824 q^{69} -3.43459 q^{71} +2.91917 q^{72} +14.0877 q^{73} +13.6947 q^{74} +11.3846 q^{76} -7.70823 q^{77} +0.282063 q^{78} -6.51002 q^{79} +8.99267 q^{81} +23.0098 q^{82} -6.17514 q^{83} +0.110179 q^{84} +21.0453 q^{86} -0.196896 q^{87} +2.87159 q^{88} -0.892411 q^{89} +13.8501 q^{91} +6.46690 q^{92} +0.135323 q^{93} +15.2234 q^{94} -0.198341 q^{96} +2.91287 q^{97} -0.324252 q^{98} +8.84851 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 37 q + 2 q^{2} + q^{3} + 50 q^{4} + 8 q^{6} - 4 q^{7} + 50 q^{9} + 17 q^{11} - 2 q^{12} - 3 q^{13} + 14 q^{14} + 72 q^{16} - 10 q^{17} - 4 q^{18} + 54 q^{19} + 15 q^{21} - 11 q^{22} + 4 q^{23} + 28 q^{24}+ \cdots + 40 q^{99}+O(q^{100}) $ 37 * q + 2 * q^2 + q^3 + 50 * q^4 + 8 * q^6 - 4 * q^7 + 50 * q^9 + 17 * q^11 - 2 * q^12 - 3 * q^13 + 14 * q^14 + 72 * q^16 - 10 * q^17 - 4 * q^18 + 54 * q^19 + 15 * q^21 - 11 * q^22 + 4 * q^23 + 28 * q^24 + 19 * q^26 - 2 * q^27 + 3 * q^28 - 4 * q^29 + 58 * q^31 + 22 * q^32 - 7 * q^33 + 63 * q^34 + 66 * q^36 + 7 * q^37 + 12 * q^38 + 23 * q^39 + 17 * q^41 - 17 * q^42 + 11 * q^43 + 34 * q^44 + 41 * q^46 - 19 * q^47 + 10 * q^48 + 81 * q^49 + 39 * q^51 + 31 * q^52 - 14 * q^53 + 5 * q^54 + 21 * q^56 + 24 * q^57 - 15 * q^58 + 56 * q^59 + 40 * q^61 + 4 * q^62 + 5 * q^63 + 78 * q^64 + 22 * q^66 + 2 * q^67 - 3 * q^68 + 24 * q^69 + 31 * q^71 + 85 * q^72 + 4 * q^73 - 32 * q^74 + 137 * q^76 + 16 * q^77 - 113 * q^78 + 17 * q^79 + 85 * q^81 + 52 * q^82 - 18 * q^83 - 32 * q^84 + 35 * q^86 + 39 * q^87 - 63 * q^88 + 23 * q^89 + 152 * q^91 + 75 * q^92 + 8 * q^93 + 32 * q^94 + 72 * q^96 + 34 * q^97 - 40 * q^98 + 40 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	1.86497	1.31873	0.659366	−	0.751823i	$-0.270825\pi$
$2$	1.86497	1.31873	0.659366	+	0.751823i	$0.270825\pi$
$3$	0.0285304	0.0164721	0.00823603	−	0.999966i	$-0.497378\pi$
$3$	0.0285304	0.0164721	0.00823603	+	0.999966i	$0.497378\pi$
$4$	1.47810	0.739052
$5$	0	0
$5$	0	0
$6$	0.0532083	0.0217222
$7$	2.61269	0.987503	0.493752	−	0.869603i	$-0.335625\pi$
$7$	2.61269	0.987503	0.493752	+	0.869603i	$0.335625\pi$
$8$	−0.973320	−0.344121
$9$	−2.99919	−0.999729
$10$	0	0
$11$	−2.95031	−0.889551	−0.444775	−	0.895642i	$-0.646716\pi$
$11$	−2.95031	−0.889551	−0.444775	+	0.895642i	$0.646716\pi$
$12$	0.0421709	0.0121737
$13$	5.30110	1.47026	0.735131	−	0.677925i	$-0.237121\pi$
$13$	5.30110	1.47026	0.735131	+	0.677925i	$0.237121\pi$
$14$	4.87258	1.30225
$15$	0	0
$16$	−4.77142	−1.19285
$17$	−6.23067	−1.51116	−0.755579	−	0.655057i	$-0.772644\pi$
$17$	−6.23067	−1.51116	−0.755579	+	0.655057i	$0.772644\pi$
$18$	−5.59338	−1.31837
$19$	7.70215	1.76699	0.883497	−	0.468437i	$-0.155183\pi$
$19$	7.70215	1.76699	0.883497	+	0.468437i	$0.155183\pi$
$20$	0	0
$21$	0.0745411	0.0162662
$22$	−5.50222	−1.17308
$23$	4.37513	0.912278	0.456139	−	0.889908i	$-0.349232\pi$
$23$	4.37513	0.912278	0.456139	+	0.889908i	$0.349232\pi$
$24$	−0.0277693	−0.00566837
$25$	0	0
$26$	9.88639	1.93888
$27$	−0.171159	−0.0329396
$28$	3.86182	0.729816
$29$	−6.90125	−1.28153	−0.640765	−	0.767737i	$-0.721383\pi$
$29$	−6.90125	−1.28153	−0.640765	+	0.767737i	$0.721383\pi$
$30$	0	0
$31$	4.74310	0.851887	0.425943	−	0.904750i	$-0.359942\pi$
$31$	4.74310	0.851887	0.425943	+	0.904750i	$0.359942\pi$
$32$	−6.95190	−1.22893
$33$	−0.0841735	−0.0146527
$34$	−11.6200	−1.99281
$35$	0	0
$36$	−4.43311	−0.738851
$37$	7.34312	1.20720	0.603600	−	0.797287i	$-0.293732\pi$
$37$	7.34312	1.20720	0.603600	+	0.797287i	$0.293732\pi$
$38$	14.3643	2.33019
$39$	0.151243	0.0242182
$40$	0	0
$41$	12.3379	1.92685	0.963427	−	0.267971i	$-0.0863530\pi$
$41$	12.3379	1.92685	0.963427	+	0.267971i	$0.0863530\pi$
$42$	0.139017	0.0214508
$43$	11.2845	1.72088	0.860438	−	0.509555i	$-0.170190\pi$
$43$	11.2845	1.72088	0.860438	+	0.509555i	$0.170190\pi$
$44$	−4.36086	−0.657424
$45$	0	0
$46$	8.15948	1.20305
$47$	8.16281	1.19067	0.595334	−	0.803478i	$-0.297020\pi$
$47$	8.16281	1.19067	0.595334	+	0.803478i	$0.297020\pi$
$48$	−0.136131	−0.0196488
$49$	−0.173865	−0.0248378
$50$	0	0
$51$	−0.177764	−0.0248919
$52$	7.83558	1.08660
$53$	5.90870	0.811623	0.405812	−	0.913957i	$-0.366989\pi$
$53$	5.90870	0.811623	0.405812	+	0.913957i	$0.366989\pi$
$54$	−0.319207	−0.0434385
$55$	0	0
$56$	−2.54298	−0.339820
$57$	0.219746	0.0291060
$58$	−12.8706	−1.68999
$59$	−9.78424	−1.27380	−0.636900	−	0.770947i	$-0.719783\pi$
$59$	−9.78424	−1.27380	−0.636900	+	0.770947i	$0.719783\pi$
$60$	0	0
$61$	−0.306027	−0.0391828	−0.0195914	−	0.999808i	$-0.506237\pi$
$61$	−0.306027	−0.0391828	−0.0195914	+	0.999808i	$0.506237\pi$
$62$	8.84573	1.12341
$63$	−7.83594	−0.987235
$64$	−3.42223	−0.427778
$65$	0	0
$66$	−0.156981	−0.0193230
$67$	12.8152	1.56563	0.782815	−	0.622255i	$-0.213783\pi$
$67$	12.8152	1.56563	0.782815	+	0.622255i	$0.213783\pi$
$68$	−9.20957	−1.11682
$69$	0.124824	0.0150271
$70$	0	0
$71$	−3.43459	−0.407611	−0.203805	−	0.979011i	$-0.565331\pi$
$71$	−3.43459	−0.407611	−0.203805	+	0.979011i	$0.565331\pi$
$72$	2.91917	0.344027
$73$	14.0877	1.64884	0.824422	−	0.565975i	$-0.191500\pi$
$73$	14.0877	1.64884	0.824422	+	0.565975i	$0.191500\pi$
$74$	13.6947	1.59197
$75$	0	0
$76$	11.3846	1.30590
$77$	−7.70823	−0.878434
$78$	0.282063	0.0319373
$79$	−6.51002	−0.732435	−0.366217	−	0.930529i	$-0.619347\pi$
$79$	−6.51002	−0.732435	−0.366217	+	0.930529i	$0.619347\pi$
$80$	0	0
$81$	8.99267	0.999186
$82$	23.0098	2.54100
$83$	−6.17514	−0.677809	−0.338905	−	0.940821i	$-0.610056\pi$
$83$	−6.17514	−0.677809	−0.338905	+	0.940821i	$0.610056\pi$
$84$	0.110179	0.0120216
$85$	0	0
$86$	21.0453	2.26937
$87$	−0.196896	−0.0211094
$88$	2.87159	0.306113
$89$	−0.892411	−0.0945954	−0.0472977	−	0.998881i	$-0.515061\pi$
$89$	−0.892411	−0.0945954	−0.0472977	+	0.998881i	$0.515061\pi$
$90$	0	0
$91$	13.8501	1.45189
$92$	6.46690	0.674221
$93$	0.135323	0.0140323
$94$	15.2234	1.57017
$95$	0	0
$96$	−0.198341	−0.0202431
$97$	2.91287	0.295757	0.147879	−	0.989006i	$-0.452756\pi$
$97$	2.91287	0.295757	0.147879	+	0.989006i	$0.452756\pi$
$98$	−0.324252	−0.0327544
$99$	8.84851	0.889309
$100$	0	0
$101$	−4.43051	−0.440852	−0.220426	−	0.975404i	$-0.570745\pi$
$101$	−4.43051	−0.440852	−0.220426	+	0.975404i	$0.570745\pi$
$102$	−0.331523	−0.0328257
$103$	−16.8911	−1.66433	−0.832166	−	0.554526i	$-0.812900\pi$
$103$	−16.8911	−1.66433	−0.832166	+	0.554526i	$0.812900\pi$
$104$	−5.15967	−0.505947
$105$	0	0
$106$	11.0195	1.07031
$107$	−13.2657	−1.28244	−0.641220	−	0.767357i	$-0.721571\pi$
$107$	−13.2657	−1.28244	−0.641220	+	0.767357i	$0.721571\pi$
$108$	−0.252991	−0.0243441
$109$	1.13961	0.109155	0.0545775	−	0.998510i	$-0.482619\pi$
$109$	1.13961	0.109155	0.0545775	+	0.998510i	$0.482619\pi$
$110$	0	0
$111$	0.209502	0.0198851
$112$	−12.4662	−1.17795
$113$	9.11889	0.857833	0.428917	−	0.903344i	$-0.358895\pi$
$113$	9.11889	0.857833	0.428917	+	0.903344i	$0.358895\pi$
$114$	0.409818	0.0383830
$115$	0	0
$116$	−10.2008	−0.947117
$117$	−15.8990	−1.46986
$118$	−18.2473	−1.67980
$119$	−16.2788	−1.49227
$120$	0	0
$121$	−2.29570	−0.208700
$122$	−0.570731	−0.0516715
$123$	0.352005	0.0317393
$124$	7.01080	0.629588
$125$	0	0
$126$	−14.6138	−1.30190
$127$	−18.2177	−1.61656	−0.808278	−	0.588802i	$-0.799600\pi$
$127$	−18.2177	−1.61656	−0.808278	+	0.588802i	$0.799600\pi$
$128$	7.52145	0.664809
$129$	0.321953	0.0283464
$130$	0	0
$131$	8.39706	0.733654	0.366827	−	0.930289i	$-0.380444\pi$
$131$	8.39706	0.733654	0.366827	+	0.930289i	$0.380444\pi$
$132$	−0.124417	−0.0108291
$133$	20.1233	1.74491
$134$	23.9000	2.06464
$135$	0	0
$136$	6.06443	0.520021
$137$	7.67056	0.655340	0.327670	−	0.944792i	$-0.393737\pi$
$137$	7.67056	0.655340	0.327670	+	0.944792i	$0.393737\pi$
$138$	0.232794	0.0198167
$139$	−17.6010	−1.49289	−0.746447	−	0.665445i	$-0.768242\pi$
$139$	−17.6010	−1.49289	−0.746447	+	0.665445i	$0.768242\pi$
$140$	0	0
$141$	0.232888	0.0196127
$142$	−6.40540	−0.537529
$143$	−15.6399	−1.30787
$144$	14.3104	1.19253
$145$	0	0
$146$	26.2732	2.17438
$147$	−0.00496044	−0.000409130 0
$148$	10.8539	0.892184
$149$	20.7583	1.70059	0.850295	−	0.526307i	$-0.176424\pi$
$149$	20.7583	1.70059	0.850295	+	0.526307i	$0.176424\pi$
$150$	0	0
$151$	8.24865	0.671265	0.335633	−	0.941993i	$-0.391050\pi$
$151$	8.24865	0.671265	0.335633	+	0.941993i	$0.391050\pi$
$152$	−7.49665	−0.608059
$153$	18.6869	1.51075
$154$	−14.3756	−1.15842
$155$	0	0
$156$	0.223553	0.0178985
$157$	−13.4770	−1.07558	−0.537791	−	0.843079i	$-0.680741\pi$
$157$	−13.4770	−1.07558	−0.537791	+	0.843079i	$0.680741\pi$
$158$	−12.1410	−0.965884
$159$	0.168578	0.0133691
$160$	0	0
$161$	11.4309	0.900878
$162$	16.7710	1.31766
$163$	3.90935	0.306204	0.153102	−	0.988210i	$-0.451074\pi$
$163$	3.90935	0.306204	0.153102	+	0.988210i	$0.451074\pi$
$164$	18.2367	1.42404
$165$	0	0
$166$	−11.5164	−0.893848
$167$	10.0354	0.776559	0.388280	−	0.921542i	$-0.373069\pi$
$167$	10.0354	0.776559	0.388280	+	0.921542i	$0.373069\pi$
$168$	−0.0725524	−0.00559754
$169$	15.1017	1.16167
$170$	0	0
$171$	−23.1002	−1.76651
$172$	16.6797	1.27182
$173$	1.77803	0.135181	0.0675906	−	0.997713i	$-0.478469\pi$
$173$	1.77803	0.135181	0.0675906	+	0.997713i	$0.478469\pi$
$174$	−0.367204	−0.0278377
$175$	0	0
$176$	14.0771	1.06110
$177$	−0.279149	−0.0209821
$178$	−1.66432	−0.124746
$179$	11.0579	0.826503	0.413252	−	0.910617i	$-0.364393\pi$
$179$	11.0579	0.826503	0.413252	+	0.910617i	$0.364393\pi$
$180$	0	0
$181$	8.38065	0.622929	0.311464	−	0.950258i	$-0.399181\pi$
$181$	8.38065	0.622929	0.311464	+	0.950258i	$0.399181\pi$
$182$	25.8300	1.91465
$183$	−0.00873109	−0.000645421 0
$184$	−4.25841	−0.313934
$185$	0	0
$186$	0.252373	0.0185049
$187$	18.3824	1.34425
$188$	12.0655	0.879965
$189$	−0.447186	−0.0325280
$190$	0	0
$191$	−11.1126	−0.804081	−0.402040	−	0.915622i	$-0.631699\pi$
$191$	−11.1126	−0.804081	−0.402040	+	0.915622i	$0.631699\pi$
$192$	−0.0976376	−0.00704639
$193$	11.0641	0.796412	0.398206	−	0.917296i	$-0.369633\pi$
$193$	11.0641	0.796412	0.398206	+	0.917296i	$0.369633\pi$
$194$	5.43241	0.390024
$195$	0	0
$196$	−0.256990	−0.0183564
$197$	1.00000	0.0712470
$197$	1.00000	0.0712470
$198$	16.5022	1.17276
$199$	−4.67613	−0.331482	−0.165741	−	0.986169i	$-0.553002\pi$
$199$	−4.67613	−0.331482	−0.165741	+	0.986169i	$0.553002\pi$
$200$	0	0
$201$	0.365624	0.0257891
$202$	−8.26275	−0.581365
$203$	−18.0308	−1.26551
$204$	−0.262753	−0.0183964
$205$	0	0
$206$	−31.5014	−2.19481
$207$	−13.1218	−0.912031
$208$	−25.2938	−1.75381
$209$	−22.7237	−1.57183
$210$	0	0
$211$	16.2179	1.11649	0.558243	−	0.829677i	$-0.311476\pi$
$211$	16.2179	1.11649	0.558243	+	0.829677i	$0.311476\pi$
$212$	8.73368	0.599831
$213$	−0.0979904	−0.00671419
$214$	−24.7400	−1.69119
$215$	0	0
$216$	0.166593	0.0113352
$217$	12.3922	0.841241
$218$	2.12534	0.143946
$219$	0.401929	0.0271599
$220$	0	0
$221$	−33.0294	−2.22180
$222$	0.390715	0.0262231
$223$	12.5711	0.841820	0.420910	−	0.907102i	$-0.361711\pi$
$223$	12.5711	0.841820	0.420910	+	0.907102i	$0.361711\pi$
$224$	−18.1631	−1.21358
$225$	0	0
$226$	17.0064	1.13125
$227$	−16.7032	−1.10863	−0.554314	−	0.832307i	$-0.687019\pi$
$227$	−16.7032	−1.10863	−0.554314	+	0.832307i	$0.687019\pi$
$228$	0.324807	0.0215108
$229$	24.0153	1.58698	0.793488	−	0.608586i	$-0.208263\pi$
$229$	24.0153	1.58698	0.793488	+	0.608586i	$0.208263\pi$
$230$	0	0
$231$	−0.219919	−0.0144696
$232$	6.71712	0.441001
$233$	−10.5658	−0.692187	−0.346093	−	0.938200i	$-0.612492\pi$
$233$	−10.5658	−0.692187	−0.346093	+	0.938200i	$0.612492\pi$
$234$	−29.6511	−1.93835
$235$	0	0
$236$	−14.4621	−0.941404
$237$	−0.185734	−0.0120647
$238$	−30.3594	−1.96791
$239$	−11.5049	−0.744190	−0.372095	−	0.928195i	$-0.621360\pi$
$239$	−11.5049	−0.744190	−0.372095	+	0.928195i	$0.621360\pi$
$240$	0	0
$241$	−1.34956	−0.0869325	−0.0434662	−	0.999055i	$-0.513840\pi$
$241$	−1.34956	−0.0869325	−0.0434662	+	0.999055i	$0.513840\pi$
$242$	−4.28140	−0.275219
$243$	0.770043	0.0493983
$244$	−0.452340	−0.0289581
$245$	0	0
$246$	0.656478	0.0418555
$247$	40.8299	2.59794
$248$	−4.61656	−0.293152
$249$	−0.176179	−0.0111649
$250$	0	0
$251$	3.75459	0.236988	0.118494	−	0.992955i	$-0.462193\pi$
$251$	3.75459	0.236988	0.118494	+	0.992955i	$0.462193\pi$
$252$	−11.5823	−0.729618
$253$	−12.9080	−0.811518
$254$	−33.9753	−2.13180
$255$	0	0
$256$	20.8717	1.30448
$257$	−2.39224	−0.149224	−0.0746118	−	0.997213i	$-0.523772\pi$
$257$	−2.39224	−0.149224	−0.0746118	+	0.997213i	$0.523772\pi$
$258$	0.600432	0.0373812
$259$	19.1853	1.19211
$260$	0	0
$261$	20.6981	1.28118
$262$	15.6602	0.967493
$263$	−4.79320	−0.295561	−0.147781	−	0.989020i	$-0.547213\pi$
$263$	−4.79320	−0.295561	−0.147781	+	0.989020i	$0.547213\pi$
$264$	0.0819278	0.00504231
$265$	0	0
$266$	37.5293	2.30107
$267$	−0.0254609	−0.00155818
$268$	18.9422	1.15708
$269$	−20.9599	−1.27795	−0.638975	−	0.769227i	$-0.720641\pi$
$269$	−20.9599	−1.27795	−0.638975	+	0.769227i	$0.720641\pi$
$270$	0	0
$271$	5.01429	0.304597	0.152298	−	0.988335i	$-0.451333\pi$
$271$	5.01429	0.304597	0.152298	+	0.988335i	$0.451333\pi$
$272$	29.7291	1.80259
$273$	0.395150	0.0239156
$274$	14.3053	0.864217
$275$	0	0
$276$	0.184504	0.0111058
$277$	−18.1690	−1.09167	−0.545835	−	0.837892i	$-0.683788\pi$
$277$	−18.1690	−1.09167	−0.545835	+	0.837892i	$0.683788\pi$
$278$	−32.8252	−1.96873
$279$	−14.2255	−0.851655
$280$	0	0
$281$	−1.63371	−0.0974590	−0.0487295	−	0.998812i	$-0.515517\pi$
$281$	−1.63371	−0.0974590	−0.0487295	+	0.998812i	$0.515517\pi$
$282$	0.434329	0.0258639
$283$	−22.2328	−1.32160	−0.660800	−	0.750562i	$-0.729783\pi$
$283$	−22.2328	−1.32160	−0.660800	+	0.750562i	$0.729783\pi$
$284$	−5.07668	−0.301245
$285$	0	0
$286$	−29.1679	−1.72473
$287$	32.2350	1.90277
$288$	20.8500	1.22860
$289$	21.8212	1.28360
$290$	0	0
$291$	0.0831054	0.00487173
$292$	20.8231	1.21858
$293$	−12.4690	−0.728449	−0.364224	−	0.931311i	$-0.618666\pi$
$293$	−12.4690	−0.728449	−0.364224	+	0.931311i	$0.618666\pi$
$294$	−0.00925105	−0.000539532 0
$295$	0	0
$296$	−7.14720	−0.415423
$297$	0.504973	0.0293015
$298$	38.7136	2.24262
$299$	23.1930	1.34129
$300$	0	0
$301$	29.4830	1.69937
$302$	15.3835	0.885219
$303$	−0.126404	−0.00726174
$304$	−36.7501	−2.10777
$305$	0	0
$306$	34.8505	1.99227
$307$	−26.6488	−1.52093	−0.760463	−	0.649381i	$-0.775028\pi$
$307$	−26.6488	−1.52093	−0.760463	+	0.649381i	$0.775028\pi$
$308$	−11.3936	−0.649208
$309$	−0.481911	−0.0274150
$310$	0	0
$311$	−11.5838	−0.656858	−0.328429	−	0.944529i	$-0.606519\pi$
$311$	−11.5838	−0.656858	−0.328429	+	0.944529i	$0.606519\pi$
$312$	−0.147208	−0.00833399
$313$	−2.66092	−0.150404	−0.0752019	−	0.997168i	$-0.523960\pi$
$313$	−2.66092	−0.150404	−0.0752019	+	0.997168i	$0.523960\pi$
$314$	−25.1341	−1.41840
$315$	0	0
$316$	−9.62249	−0.541307
$317$	13.5020	0.758347	0.379174	−	0.925326i	$-0.376208\pi$
$317$	13.5020	0.758347	0.379174	+	0.925326i	$0.376208\pi$
$318$	0.314392	0.0176302
$319$	20.3608	1.13999
$320$	0	0
$321$	−0.378475	−0.0211244
$322$	21.3182	1.18802
$323$	−47.9895	−2.67021
$324$	13.2921	0.738450
$325$	0	0
$326$	7.29080	0.403800
$327$	0.0325136	0.00179801
$328$	−12.0087	−0.663070
$329$	21.3269	1.17579
$330$	0	0
$331$	1.11883	0.0614967	0.0307484	−	0.999527i	$-0.490211\pi$
$331$	1.11883	0.0614967	0.0307484	+	0.999527i	$0.490211\pi$
$332$	−9.12749	−0.500936
$333$	−22.0234	−1.20687
$334$	18.7156	1.02407
$335$	0	0
$336$	−0.355667	−0.0194032
$337$	−17.0812	−0.930471	−0.465236	−	0.885187i	$-0.654030\pi$
$337$	−17.0812	−0.930471	−0.465236	+	0.885187i	$0.654030\pi$
$338$	28.1642	1.53193
$339$	0.260166	0.0141303
$340$	0	0
$341$	−13.9936	−0.757796
$342$	−43.0811	−2.32956
$343$	−18.7431	−1.01203
$344$	−10.9835	−0.592189
$345$	0	0
$346$	3.31597	0.178268
$347$	7.08597	0.380395	0.190197	−	0.981746i	$-0.439087\pi$
$347$	7.08597	0.380395	0.190197	+	0.981746i	$0.439087\pi$
$348$	−0.291032	−0.0156010
$349$	4.05841	0.217241	0.108621	−	0.994083i	$-0.465357\pi$
$349$	4.05841	0.217241	0.108621	+	0.994083i	$0.465357\pi$
$350$	0	0
$351$	−0.907334	−0.0484299
$352$	20.5102	1.09320
$353$	−6.49646	−0.345772	−0.172886	−	0.984942i	$-0.555309\pi$
$353$	−6.49646	−0.345772	−0.172886	+	0.984942i	$0.555309\pi$
$354$	−0.520603	−0.0276697
$355$	0	0
$356$	−1.31908	−0.0699109
$357$	−0.464441	−0.0245808
$358$	20.6226	1.08994
$359$	9.81942	0.518249	0.259125	−	0.965844i	$-0.416566\pi$
$359$	9.81942	0.518249	0.259125	+	0.965844i	$0.416566\pi$
$360$	0	0
$361$	40.3230	2.12227
$362$	15.6296	0.821476
$363$	−0.0654973	−0.00343772
$364$	20.4719	1.07302
$365$	0	0
$366$	−0.0162832	−0.000851137 0
$367$	37.1694	1.94023	0.970113	−	0.242655i	$-0.0780184\pi$
$367$	37.1694	1.94023	0.970113	+	0.242655i	$0.0780184\pi$
$368$	−20.8756	−1.08822
$369$	−37.0036	−1.92633
$370$	0	0
$371$	15.4376	0.801480
$372$	0.200021	0.0103706
$373$	24.4554	1.26625	0.633126	−	0.774049i	$-0.281772\pi$
$373$	24.4554	1.26625	0.633126	+	0.774049i	$0.281772\pi$
$374$	34.2825	1.77271
$375$	0	0
$376$	−7.94502	−0.409733
$377$	−36.5842	−1.88418
$378$	−0.833987	−0.0428957
$379$	−20.3558	−1.04561	−0.522804	−	0.852453i	$-0.675114\pi$
$379$	−20.3558	−1.04561	−0.522804	+	0.852453i	$0.675114\pi$
$380$	0	0
$381$	−0.519758	−0.0266280
$382$	−20.7247	−1.06037
$383$	34.3333	1.75435	0.877176	−	0.480169i	$-0.159425\pi$
$383$	34.3333	1.75435	0.877176	+	0.480169i	$0.159425\pi$
$384$	0.214590	0.0109508
$385$	0	0
$386$	20.6342	1.05025
$387$	−33.8444	−1.72041
$388$	4.30552	0.218580
$389$	−5.12671	−0.259935	−0.129967	−	0.991518i	$-0.541487\pi$
$389$	−5.12671	−0.259935	−0.129967	+	0.991518i	$0.541487\pi$
$390$	0	0
$391$	−27.2600	−1.37860
$392$	0.169226	0.00854720
$393$	0.239572	0.0120848
$394$	1.86497	0.0939557
$395$	0	0
$396$	13.0790	0.657245
$397$	−38.6708	−1.94083	−0.970417	−	0.241437i	$-0.922381\pi$
$397$	−38.6708	−1.94083	−0.970417	+	0.241437i	$0.922381\pi$
$398$	−8.72083	−0.437136
$399$	0.574127	0.0287423
$400$	0	0
$401$	−15.8108	−0.789555	−0.394777	−	0.918777i	$-0.629178\pi$
$401$	−15.8108	−0.789555	−0.394777	+	0.918777i	$0.629178\pi$
$402$	0.681877	0.0340089
$403$	25.1437	1.25250
$404$	−6.54875	−0.325812
$405$	0	0
$406$	−33.6269	−1.66887
$407$	−21.6644	−1.07387
$408$	0.173021	0.00856581
$409$	−14.0860	−0.696506	−0.348253	−	0.937400i	$-0.613225\pi$
$409$	−14.0860	−0.696506	−0.348253	+	0.937400i	$0.613225\pi$
$410$	0	0
$411$	0.218844	0.0107948
$412$	−24.9668	−1.23003
$413$	−25.5632	−1.25788
$414$	−24.4718	−1.20272
$415$	0	0
$416$	−36.8527	−1.80685
$417$	−0.502163	−0.0245910
$418$	−42.3789	−2.07282
$419$	11.2511	0.549653	0.274827	−	0.961494i	$-0.411380\pi$
$419$	11.2511	0.549653	0.274827	+	0.961494i	$0.411380\pi$
$420$	0	0
$421$	34.8165	1.69685	0.848427	−	0.529312i	$-0.177550\pi$
$421$	34.8165	1.69685	0.848427	+	0.529312i	$0.177550\pi$
$422$	30.2459	1.47235
$423$	−24.4818	−1.19034
$424$	−5.75106	−0.279296
$425$	0	0
$426$	−0.182749	−0.00885421
$427$	−0.799554	−0.0386931
$428$	−19.6080	−0.947789
$429$	−0.446213	−0.0215433
$430$	0	0
$431$	10.5202	0.506742	0.253371	−	0.967369i	$-0.418461\pi$
$431$	10.5202	0.506742	0.253371	+	0.967369i	$0.418461\pi$
$432$	0.816673	0.0392922
$433$	5.25119	0.252356	0.126178	−	0.992008i	$-0.459729\pi$
$433$	5.25119	0.252356	0.126178	+	0.992008i	$0.459729\pi$
$434$	23.1111	1.10937
$435$	0	0
$436$	1.68446	0.0806712
$437$	33.6979	1.61199
$438$	0.749585	0.0358166
$439$	10.3858	0.495688	0.247844	−	0.968800i	$-0.420278\pi$
$439$	10.3858	0.495688	0.247844	+	0.968800i	$0.420278\pi$
$440$	0	0
$441$	0.521452	0.0248311
$442$	−61.5988	−2.92995
$443$	−18.1023	−0.860067	−0.430033	−	0.902813i	$-0.641498\pi$
$443$	−18.1023	−0.860067	−0.430033	+	0.902813i	$0.641498\pi$
$444$	0.309666	0.0146961
$445$	0	0
$446$	23.4446	1.11013
$447$	0.592245	0.0280122
$448$	−8.94121	−0.422432
$449$	−15.5560	−0.734134	−0.367067	−	0.930195i	$-0.619638\pi$
$449$	−15.5560	−0.734134	−0.367067	+	0.930195i	$0.619638\pi$
$450$	0	0
$451$	−36.4005	−1.71403
$452$	13.4787	0.633983
$453$	0.235338	0.0110571
$454$	−31.1509	−1.46198
$455$	0	0
$456$	−0.213883	−0.0100160
$457$	−29.7196	−1.39022	−0.695111	−	0.718902i	$-0.744645\pi$
$457$	−29.7196	−1.39022	−0.695111	+	0.718902i	$0.744645\pi$
$458$	44.7878	2.09280
$459$	1.06644	0.0497770
$460$	0	0
$461$	−19.3372	−0.900623	−0.450312	−	0.892872i	$-0.648687\pi$
$461$	−19.3372	−0.900623	−0.450312	+	0.892872i	$0.648687\pi$
$462$	−0.410142	−0.0190815
$463$	−0.900459	−0.0418479	−0.0209239	−	0.999781i	$-0.506661\pi$
$463$	−0.900459	−0.0418479	−0.0209239	+	0.999781i	$0.506661\pi$
$464$	32.9287	1.52868
$465$	0	0
$466$	−19.7048	−0.912808
$467$	−4.60643	−0.213160	−0.106580	−	0.994304i	$-0.533990\pi$
$467$	−4.60643	−0.213160	−0.106580	+	0.994304i	$0.533990\pi$
$468$	−23.5004	−1.08630
$469$	33.4822	1.54606
$470$	0	0
$471$	−0.384504	−0.0177170
$472$	9.52320	0.438341
$473$	−33.2928	−1.53081
$474$	−0.346387	−0.0159101
$475$	0	0
$476$	−24.0617	−1.10287
$477$	−17.7213	−0.811403
$478$	−21.4563	−0.981386
$479$	−3.24393	−0.148219	−0.0741095	−	0.997250i	$-0.523611\pi$
$479$	−3.24393	−0.148219	−0.0741095	+	0.997250i	$0.523611\pi$
$480$	0	0
$481$	38.9266	1.77490
$482$	−2.51688	−0.114641
$483$	0.326127	0.0148393
$484$	−3.39328	−0.154240
$485$	0	0
$486$	1.43611	0.0651431
$487$	6.70261	0.303724	0.151862	−	0.988402i	$-0.451473\pi$
$487$	6.70261	0.303724	0.151862	+	0.988402i	$0.451473\pi$
$488$	0.297862	0.0134836
$489$	0.111535	0.00504380
$490$	0	0
$491$	13.2883	0.599694	0.299847	−	0.953987i	$-0.403064\pi$
$491$	13.2883	0.599694	0.299847	+	0.953987i	$0.403064\pi$
$492$	0.520300	0.0234569
$493$	42.9994	1.93659
$494$	76.1464	3.42599
$495$	0	0
$496$	−22.6313	−1.01618
$497$	−8.97351	−0.402517
$498$	−0.328569	−0.0147235
$499$	2.82098	0.126284	0.0631422	−	0.998005i	$-0.479888\pi$
$499$	2.82098	0.126284	0.0631422	+	0.998005i	$0.479888\pi$
$500$	0	0
$501$	0.286313	0.0127915
$502$	7.00220	0.312523
$503$	21.7164	0.968286	0.484143	−	0.874989i	$-0.339131\pi$
$503$	21.7164	0.968286	0.484143	+	0.874989i	$0.339131\pi$
$504$	7.62687	0.339728
$505$	0	0
$506$	−24.0730	−1.07017
$507$	0.430858	0.0191351
$508$	−26.9276	−1.19472
$509$	15.8153	0.700999	0.350500	−	0.936563i	$-0.386012\pi$
$509$	15.8153	0.700999	0.350500	+	0.936563i	$0.386012\pi$
$510$	0	0
$511$	36.8069	1.62824
$512$	23.8822	1.05545
$513$	−1.31829	−0.0582041
$514$	−4.46144	−0.196786
$515$	0	0
$516$	0.475880	0.0209494
$517$	−24.0828	−1.05916
$518$	35.7799	1.57208
$519$	0.0507280	0.00222671
$520$	0	0
$521$	3.85636	0.168950	0.0844751	−	0.996426i	$-0.473079\pi$
$521$	3.85636	0.168950	0.0844751	+	0.996426i	$0.473079\pi$
$522$	38.6013	1.68953
$523$	−15.3629	−0.671774	−0.335887	−	0.941902i	$-0.609036\pi$
$523$	−15.3629	−0.671774	−0.335887	+	0.941902i	$0.609036\pi$
$524$	12.4117	0.542208
$525$	0	0
$526$	−8.93915	−0.389766
$527$	−29.5527	−1.28734
$528$	0.401627	0.0174786
$529$	−3.85820	−0.167748
$530$	0	0
$531$	29.3447	1.27345
$532$	29.7443	1.28958
$533$	65.4044	2.83298
$534$	−0.0474837	−0.00205482
$535$	0	0
$536$	−12.4733	−0.538765
$537$	0.315486	0.0136142
$538$	−39.0896	−1.68527
$539$	0.512954	0.0220945
$540$	0	0
$541$	19.8614	0.853907	0.426953	−	0.904274i	$-0.359587\pi$
$541$	19.8614	0.853907	0.426953	+	0.904274i	$0.359587\pi$
$542$	9.35149	0.401681
$543$	0.239104	0.0102609
$544$	43.3149	1.85711
$545$	0	0
$546$	0.736942	0.0315382
$547$	−36.7007	−1.56921	−0.784605	−	0.619997i	$-0.787134\pi$
$547$	−36.7007	−1.56921	−0.784605	+	0.619997i	$0.787134\pi$
$548$	11.3379	0.484330
$549$	0.917833	0.0391721
$550$	0	0
$551$	−53.1544	−2.26445
$552$	−0.121494	−0.00517114
$553$	−17.0087	−0.723282
$554$	−33.8846	−1.43962
$555$	0	0
$556$	−26.0160	−1.10333
$557$	16.1457	0.684116	0.342058	−	0.939679i	$-0.388876\pi$
$557$	16.1457	0.684116	0.342058	+	0.939679i	$0.388876\pi$
$558$	−26.5300	−1.12310
$559$	59.8205	2.53014
$560$	0	0
$561$	0.524457	0.0221426
$562$	−3.04682	−0.128522
$563$	3.67718	0.154975	0.0774873	−	0.996993i	$-0.475310\pi$
$563$	3.67718	0.154975	0.0774873	+	0.996993i	$0.475310\pi$
$564$	0.344233	0.0144948
$565$	0	0
$566$	−41.4634	−1.74283
$567$	23.4950	0.986699
$568$	3.34296	0.140267
$569$	−46.5398	−1.95105	−0.975526	−	0.219886i	$-0.929431\pi$
$569$	−46.5398	−1.95105	−0.975526	+	0.219886i	$0.929431\pi$
$570$	0	0
$571$	35.9928	1.50625	0.753125	−	0.657877i	$-0.228546\pi$
$571$	35.9928	1.50625	0.753125	+	0.657877i	$0.228546\pi$
$572$	−23.1174	−0.966585
$573$	−0.317048	−0.0132449
$574$	60.1173	2.50925
$575$	0	0
$576$	10.2639	0.427662
$577$	38.4625	1.60122	0.800608	−	0.599189i	$-0.204510\pi$
$577$	38.4625	1.60122	0.800608	+	0.599189i	$0.204510\pi$
$578$	40.6958	1.69272
$579$	0.315664	0.0131186
$580$	0	0
$581$	−16.1337	−0.669339
$582$	0.154989	0.00642450
$583$	−17.4325	−0.721980
$584$	−13.7119	−0.567402
$585$	0	0
$586$	−23.2543	−0.960628
$587$	6.00361	0.247796	0.123898	−	0.992295i	$-0.460461\pi$
$587$	6.00361	0.247796	0.123898	+	0.992295i	$0.460461\pi$
$588$	−0.00733204	−0.000302368 0
$589$	36.5321	1.50528
$590$	0	0
$591$	0.0285304	0.00117359
$592$	−35.0371	−1.44001
$593$	−31.8022	−1.30596	−0.652981	−	0.757375i	$-0.726482\pi$
$593$	−31.8022	−1.30596	−0.652981	+	0.757375i	$0.726482\pi$
$594$	0.941757	0.0386408
$595$	0	0
$596$	30.6830	1.25682
$597$	−0.133412	−0.00546019
$598$	43.2543	1.76880
$599$	18.0162	0.736121	0.368061	−	0.929802i	$-0.380022\pi$
$599$	18.0162	0.736121	0.368061	+	0.929802i	$0.380022\pi$
$600$	0	0
$601$	21.4146	0.873520	0.436760	−	0.899578i	$-0.356126\pi$
$601$	21.4146	0.873520	0.436760	+	0.899578i	$0.356126\pi$
$602$	54.9848	2.24101
$603$	−38.4353	−1.56520
$604$	12.1924	0.496100
$605$	0	0
$606$	−0.235740	−0.00957628
$607$	38.4079	1.55893	0.779463	−	0.626448i	$-0.215492\pi$
$607$	38.4079	1.55893	0.779463	+	0.626448i	$0.215492\pi$
$608$	−53.5445	−2.17152
$609$	−0.514427	−0.0208456
$610$	0	0
$611$	43.2719	1.75059
$612$	27.6212	1.11652
$613$	28.9559	1.16952	0.584759	−	0.811207i	$-0.301189\pi$
$613$	28.9559	1.16952	0.584759	+	0.811207i	$0.301189\pi$
$614$	−49.6991	−2.00569
$615$	0	0
$616$	7.50257	0.302287
$617$	6.39381	0.257405	0.128702	−	0.991683i	$-0.458919\pi$
$617$	6.39381	0.257405	0.128702	+	0.991683i	$0.458919\pi$
$618$	−0.898749	−0.0361530
$619$	3.90869	0.157104	0.0785518	−	0.996910i	$-0.474970\pi$
$619$	3.90869	0.157104	0.0785518	+	0.996910i	$0.474970\pi$
$620$	0	0
$621$	−0.748845	−0.0300501
$622$	−21.6034	−0.866218
$623$	−2.33159	−0.0934133
$624$	−0.721643	−0.0288888
$625$	0	0
$626$	−4.96252	−0.198342
$627$	−0.648317	−0.0258913
$628$	−19.9204	−0.794910
$629$	−45.7525	−1.82427
$630$	0	0
$631$	−29.9929	−1.19400	−0.596998	−	0.802243i	$-0.703640\pi$
$631$	−29.9929	−1.19400	−0.596998	+	0.802243i	$0.703640\pi$
$632$	6.33634	0.252046
$633$	0.462704	0.0183908
$634$	25.1808	1.00006
$635$	0	0
$636$	0.249176	0.00988046
$637$	−0.921675	−0.0365181
$638$	37.9722	1.50333
$639$	10.3010	0.407500
$640$	0	0
$641$	19.5696	0.772952	0.386476	−	0.922299i	$-0.373692\pi$
$641$	19.5696	0.772952	0.386476	+	0.922299i	$0.373692\pi$
$642$	−0.705844	−0.0278574
$643$	−23.7483	−0.936541	−0.468271	−	0.883585i	$-0.655123\pi$
$643$	−23.7483	−0.936541	−0.468271	+	0.883585i	$0.655123\pi$
$644$	16.8960	0.665795
$645$	0	0
$646$	−89.4988	−3.52128
$647$	−14.3567	−0.564419	−0.282209	−	0.959353i	$-0.591067\pi$
$647$	−14.3567	−0.564419	−0.282209	+	0.959353i	$0.591067\pi$
$648$	−8.75275	−0.343841
$649$	28.8665	1.13311
$650$	0	0
$651$	0.353556	0.0138570
$652$	5.77842	0.226300
$653$	48.0337	1.87970	0.939852	−	0.341581i	$-0.110962\pi$
$653$	48.0337	1.87970	0.939852	+	0.341581i	$0.110962\pi$
$654$	0.0606368	0.00237109
$655$	0	0
$656$	−58.8692	−2.29846
$657$	−42.2517	−1.64840
$658$	39.7739	1.55055
$659$	35.6174	1.38746	0.693729	−	0.720237i	$-0.255967\pi$
$659$	35.6174	1.38746	0.693729	+	0.720237i	$0.255967\pi$
$660$	0	0
$661$	−17.0841	−0.664493	−0.332247	−	0.943193i	$-0.607807\pi$
$661$	−17.0841	−0.664493	−0.332247	+	0.943193i	$0.607807\pi$
$662$	2.08659	0.0810976
$663$	−0.942343	−0.0365976
$664$	6.01038	0.233248
$665$	0	0
$666$	−41.0729	−1.59154
$667$	−30.1939	−1.16911
$668$	14.8333	0.573918
$669$	0.358658	0.0138665
$670$	0	0
$671$	0.902874	0.0348551
$672$	−0.518202	−0.0199901
$673$	−37.3974	−1.44156	−0.720781	−	0.693163i	$-0.756217\pi$
$673$	−37.3974	−1.44156	−0.720781	+	0.693163i	$0.756217\pi$
$674$	−31.8559	−1.22704
$675$	0	0
$676$	22.3219	0.858534
$677$	−16.5709	−0.636870	−0.318435	−	0.947945i	$-0.603157\pi$
$677$	−16.5709	−0.636870	−0.318435	+	0.947945i	$0.603157\pi$
$678$	0.485201	0.0186340
$679$	7.61042	0.292061
$680$	0	0
$681$	−0.476549	−0.0182614
$682$	−26.0976	−0.999329
$683$	8.22248	0.314625	0.157312	−	0.987549i	$-0.449717\pi$
$683$	8.22248	0.314625	0.157312	+	0.987549i	$0.449717\pi$
$684$	−34.1444	−1.30555
$685$	0	0
$686$	−34.9552	−1.33460
$687$	0.685167	0.0261408
$688$	−53.8432	−2.05275
$689$	31.3227	1.19330
$690$	0	0
$691$	37.2988	1.41891	0.709457	−	0.704749i	$-0.248940\pi$
$691$	37.2988	1.41891	0.709457	+	0.704749i	$0.248940\pi$
$692$	2.62811	0.0999059
$693$	23.1184	0.878195
$694$	13.2151	0.501638
$695$	0	0
$696$	0.191642	0.00726419
$697$	−76.8732	−2.91178
$698$	7.56880	0.286483
$699$	−0.301446	−0.0114017
$700$	0	0
$701$	−40.5905	−1.53308	−0.766542	−	0.642194i	$-0.778024\pi$
$701$	−40.5905	−1.53308	−0.766542	+	0.642194i	$0.778024\pi$
$702$	−1.69215	−0.0638660
$703$	56.5578	2.13312
$704$	10.0966	0.380531
$705$	0	0
$706$	−12.1157	−0.455980
$707$	−11.5755	−0.435343
$708$	−0.412611	−0.0155069
$709$	−44.6773	−1.67789	−0.838945	−	0.544216i	$-0.816827\pi$
$709$	−44.6773	−1.67789	−0.838945	+	0.544216i	$0.816827\pi$
$710$	0	0
$711$	19.5248	0.732236
$712$	0.868602	0.0325522
$713$	20.7517	0.777158
$714$	−0.866167	−0.0324155
$715$	0	0
$716$	16.3447	0.610829
$717$	−0.328240	−0.0122583
$718$	18.3129	0.683431
$719$	17.1234	0.638597	0.319298	−	0.947654i	$-0.396553\pi$
$719$	17.1234	0.638597	0.319298	+	0.947654i	$0.396553\pi$
$720$	0	0
$721$	−44.1313	−1.64353
$722$	75.2012	2.79870
$723$	−0.0385034	−0.00143196
$724$	12.3875	0.460377
$725$	0	0
$726$	−0.122150	−0.00453342
$727$	−2.63180	−0.0976079	−0.0488040	−	0.998808i	$-0.515541\pi$
$727$	−2.63180	−0.0976079	−0.0488040	+	0.998808i	$0.515541\pi$
$728$	−13.4806	−0.499625
$729$	−26.9561	−0.998372
$730$	0	0
$731$	−70.3102	−2.60052
$732$	−0.0129055	−0.000476999 0
$733$	−41.3253	−1.52638	−0.763192	−	0.646171i	$-0.776369\pi$
$733$	−41.3253	−1.52638	−0.763192	+	0.646171i	$0.776369\pi$
$734$	69.3197	2.55864
$735$	0	0
$736$	−30.4155	−1.12113
$737$	−37.8088	−1.39271
$738$	−69.0105	−2.54031
$739$	7.96028	0.292824	0.146412	−	0.989224i	$-0.453228\pi$
$739$	7.96028	0.292824	0.146412	+	0.989224i	$0.453228\pi$
$740$	0	0
$741$	1.16489	0.0427935
$742$	28.7906	1.05694
$743$	1.80849	0.0663470	0.0331735	−	0.999450i	$-0.489439\pi$
$743$	1.80849	0.0663470	0.0331735	+	0.999450i	$0.489439\pi$
$744$	−0.131712	−0.00482881
$745$	0	0
$746$	45.6085	1.66984
$747$	18.5204	0.677625
$748$	27.1710	0.993471
$749$	−34.6590	−1.26641
$750$	0	0
$751$	31.6702	1.15566	0.577831	−	0.816157i	$-0.303899\pi$
$751$	31.6702	1.15566	0.577831	+	0.816157i	$0.303899\pi$
$752$	−38.9482	−1.42029
$753$	0.107120	0.00390368
$754$	−68.2284	−2.48473
$755$	0	0
$756$	−0.660987	−0.0240399
$757$	−39.2174	−1.42538	−0.712690	−	0.701479i	$-0.752523\pi$
$757$	−39.2174	−1.42538	−0.712690	+	0.701479i	$0.752523\pi$
$758$	−37.9629	−1.37888
$759$	−0.368270	−0.0133674
$760$	0	0
$761$	−44.0014	−1.59505	−0.797524	−	0.603287i	$-0.793857\pi$
$761$	−44.0014	−1.59505	−0.797524	+	0.603287i	$0.793857\pi$
$762$	−0.969331	−0.0351152
$763$	2.97745	0.107791
$764$	−16.4256	−0.594257
$765$	0	0
$766$	64.0306	2.31352
$767$	−51.8673	−1.87282
$768$	0.595479	0.0214875
$769$	−36.9417	−1.33215	−0.666076	−	0.745884i	$-0.732027\pi$
$769$	−36.9417	−1.33215	−0.666076	+	0.745884i	$0.732027\pi$
$770$	0	0
$771$	−0.0682516	−0.00245802
$772$	16.3539	0.588590
$773$	23.1250	0.831748	0.415874	−	0.909422i	$-0.363476\pi$
$773$	23.1250	0.831748	0.415874	+	0.909422i	$0.363476\pi$
$774$	−63.1188	−2.26876
$775$	0	0
$776$	−2.83515	−0.101776
$777$	0.547364	0.0196366
$778$	−9.56115	−0.342784
$779$	95.0282	3.40474
$780$	0	0
$781$	10.1331	0.362590
$782$	−50.8390	−1.81800
$783$	1.18121	0.0422131
$784$	0.829581	0.0296279
$785$	0	0
$786$	0.446793	0.0159366
$787$	−39.2036	−1.39746	−0.698728	−	0.715387i	$-0.746250\pi$
$787$	−39.2036	−1.39746	−0.698728	+	0.715387i	$0.746250\pi$
$788$	1.47810	0.0526553
$789$	−0.136752	−0.00486850
$790$	0	0
$791$	23.8248	0.847113
$792$	−8.61244	−0.306030
$793$	−1.62228	−0.0576089
$794$	−72.1198	−2.55944
$795$	0	0
$796$	−6.91180	−0.244982
$797$	−10.1373	−0.359081	−0.179540	−	0.983751i	$-0.557461\pi$
$797$	−10.1373	−0.359081	−0.179540	+	0.983751i	$0.557461\pi$
$798$	1.07073	0.0379033
$799$	−50.8597	−1.79929
$800$	0	0
$801$	2.67651	0.0945697
$802$	−29.4867	−1.04121
$803$	−41.5631	−1.46673
$804$	0.540430	0.0190595
$805$	0	0
$806$	46.8922	1.65171
$807$	−0.597996	−0.0210505
$808$	4.31230	0.151706
$809$	20.5503	0.722512	0.361256	−	0.932467i	$-0.382348\pi$
$809$	20.5503	0.722512	0.361256	+	0.932467i	$0.382348\pi$
$810$	0	0
$811$	−5.73011	−0.201211	−0.100606	−	0.994926i	$-0.532078\pi$
$811$	−5.73011	−0.201211	−0.100606	+	0.994926i	$0.532078\pi$
$812$	−26.6514	−0.935280
$813$	0.143060	0.00501733
$814$	−40.4035	−1.41614
$815$	0	0
$816$	0.848184	0.0296924
$817$	86.9152	3.04078
$818$	−26.2699	−0.918505
$819$	−41.5391	−1.45149
$820$	0	0
$821$	21.0194	0.733582	0.366791	−	0.930303i	$-0.380456\pi$
$821$	21.0194	0.733582	0.366791	+	0.930303i	$0.380456\pi$
$822$	0.408138	0.0142354
$823$	−11.2514	−0.392199	−0.196099	−	0.980584i	$-0.562828\pi$
$823$	−11.2514	−0.392199	−0.196099	+	0.980584i	$0.562828\pi$
$824$	16.4405	0.572731
$825$	0	0
$826$	−47.6745	−1.65881
$827$	−23.1918	−0.806456	−0.403228	−	0.915099i	$-0.632112\pi$
$827$	−23.1918	−0.806456	−0.403228	+	0.915099i	$0.632112\pi$
$828$	−19.3954	−0.674038
$829$	−35.2673	−1.22489	−0.612443	−	0.790515i	$-0.709813\pi$
$829$	−35.2673	−1.22489	−0.612443	+	0.790515i	$0.709813\pi$
$830$	0	0
$831$	−0.518370	−0.0179821
$832$	−18.1416	−0.628946
$833$	1.08329	0.0375339
$834$	−0.936518	−0.0324290
$835$	0	0
$836$	−33.5880	−1.16166
$837$	−0.811827	−0.0280608
$838$	20.9830	0.724845
$839$	23.8919	0.824840	0.412420	−	0.910994i	$-0.364684\pi$
$839$	23.8919	0.824840	0.412420	+	0.910994i	$0.364684\pi$
$840$	0	0
$841$	18.6272	0.642318
$842$	64.9317	2.23769
$843$	−0.0466105	−0.00160535
$844$	23.9717	0.825141
$845$	0	0
$846$	−45.6577	−1.56974
$847$	−5.99794	−0.206092
$848$	−28.1929	−0.968148
$849$	−0.634310	−0.0217695
$850$	0	0
$851$	32.1271	1.10130
$852$	−0.144840	−0.00496213
$853$	−53.3791	−1.82767	−0.913834	−	0.406089i	$-0.866892\pi$
$853$	−53.3791	−1.82767	−0.913834	+	0.406089i	$0.866892\pi$
$854$	−1.49114	−0.0510258
$855$	0	0
$856$	12.9117	0.441314
$857$	12.6788	0.433099	0.216550	−	0.976272i	$-0.430520\pi$
$857$	12.6788	0.433099	0.216550	+	0.976272i	$0.430520\pi$
$858$	−0.832172	−0.0284099
$859$	34.1078	1.16374	0.581871	−	0.813281i	$-0.302321\pi$
$859$	34.1078	1.16374	0.581871	+	0.813281i	$0.302321\pi$
$860$	0	0
$861$	0.919680	0.0313426
$862$	19.6199	0.668257
$863$	11.6070	0.395108	0.197554	−	0.980292i	$-0.436700\pi$
$863$	11.6070	0.395108	0.197554	+	0.980292i	$0.436700\pi$
$864$	1.18988	0.0404806
$865$	0	0
$866$	9.79331	0.332790
$867$	0.622568	0.0211435
$868$	18.3170	0.621720
$869$	19.2066	0.651538
$870$	0	0
$871$	67.9349	2.30189
$872$	−1.10921	−0.0375625
$873$	−8.73624	−0.295677
$874$	62.8455	2.12578
$875$	0	0
$876$	0.594093	0.0200725
$877$	−16.2373	−0.548294	−0.274147	−	0.961688i	$-0.588395\pi$
$877$	−16.2373	−0.548294	−0.274147	+	0.961688i	$0.588395\pi$
$878$	19.3692	0.653680
$879$	−0.355747	−0.0119991
$880$	0	0
$881$	−35.6574	−1.20133	−0.600665	−	0.799501i	$-0.705097\pi$
$881$	−35.6574	−1.20133	−0.600665	+	0.799501i	$0.705097\pi$
$882$	0.972492	0.0327455
$883$	9.20448	0.309756	0.154878	−	0.987934i	$-0.450502\pi$
$883$	9.20448	0.309756	0.154878	+	0.987934i	$0.450502\pi$
$884$	−48.8209	−1.64202
$885$	0	0
$886$	−33.7602	−1.13420
$887$	12.8124	0.430197	0.215098	−	0.976592i	$-0.430993\pi$
$887$	12.8124	0.430197	0.215098	+	0.976592i	$0.430993\pi$
$888$	−0.203913	−0.00684287
$889$	−47.5970	−1.59635
$890$	0	0
$891$	−26.5311	−0.888827
$892$	18.5813	0.622149
$893$	62.8711	2.10390
$894$	1.10452	0.0369406
$895$	0	0
$896$	19.6512	0.656501
$897$	0.661708	0.0220938
$898$	−29.0115	−0.968125
$899$	−32.7333	−1.09172
$900$	0	0
$901$	−36.8152	−1.22649
$902$	−67.8858	−2.26035
$903$	0.841162	0.0279921
$904$	−8.87560	−0.295198
$905$	0	0
$906$	0.438897	0.0145814
$907$	−48.6838	−1.61652	−0.808260	−	0.588825i	$-0.799591\pi$
$907$	−48.6838	−1.61652	−0.808260	+	0.588825i	$0.799591\pi$
$908$	−24.6890	−0.819334
$909$	13.2879	0.440732
$910$	0	0
$911$	25.7556	0.853320	0.426660	−	0.904412i	$-0.359690\pi$
$911$	25.7556	0.853320	0.426660	+	0.904412i	$0.359690\pi$
$912$	−1.04850	−0.0347192
$913$	18.2185	0.602946
$914$	−55.4260	−1.83333
$915$	0	0
$916$	35.4971	1.17286
$917$	21.9389	0.724486
$918$	1.98887	0.0656425
$919$	−17.7142	−0.584337	−0.292169	−	0.956367i	$-0.594377\pi$
$919$	−17.7142	−0.584337	−0.292169	+	0.956367i	$0.594377\pi$
$920$	0	0
$921$	−0.760301	−0.0250528
$922$	−36.0632	−1.18768
$923$	−18.2071	−0.599295
$924$	−0.325063	−0.0106938
$925$	0	0
$926$	−1.67933	−0.0551861
$927$	50.6597	1.66388
$928$	47.9768	1.57491
$929$	3.51259	0.115244	0.0576221	−	0.998338i	$-0.481648\pi$
$929$	3.51259	0.115244	0.0576221	+	0.998338i	$0.481648\pi$
$930$	0	0
$931$	−1.33913	−0.0438882
$932$	−15.6173	−0.511562
$933$	−0.330491	−0.0108198
$934$	−8.59085	−0.281101
$935$	0	0
$936$	15.4748	0.505810
$937$	−12.5179	−0.408942	−0.204471	−	0.978873i	$-0.565547\pi$
$937$	−12.5179	−0.408942	−0.204471	+	0.978873i	$0.565547\pi$
$938$	62.4432	2.03884
$939$	−0.0759171	−0.00247746
$940$	0	0
$941$	−19.0252	−0.620202	−0.310101	−	0.950704i	$-0.600363\pi$
$941$	−19.0252	−0.620202	−0.310101	+	0.950704i	$0.600363\pi$
$942$	−0.717088	−0.0233640
$943$	53.9799	1.75783
$944$	46.6847	1.51946
$945$	0	0
$946$	−62.0901	−2.01872
$947$	−30.0889	−0.977757	−0.488878	−	0.872352i	$-0.662594\pi$
$947$	−30.0889	−0.977757	−0.488878	+	0.872352i	$0.662594\pi$
$948$	−0.274534	−0.00891644
$949$	74.6806	2.42423
$950$	0	0
$951$	0.385218	0.0124915
$952$	15.8445	0.513522
$953$	−35.3452	−1.14494	−0.572471	−	0.819925i	$-0.694015\pi$
$953$	−35.3452	−1.14494	−0.572471	+	0.819925i	$0.694015\pi$
$954$	−33.0497	−1.07002
$955$	0	0
$956$	−17.0054	−0.549995
$957$	0.580902	0.0187779
$958$	−6.04982	−0.195461
$959$	20.0408	0.647150
$960$	0	0
$961$	−8.50297	−0.274289
$962$	72.5969	2.34062
$963$	39.7862	1.28209
$964$	−1.99478	−0.0642476
$965$	0	0
$966$	0.608217	0.0195691
$967$	20.9472	0.673617	0.336808	−	0.941573i	$-0.390653\pi$
$967$	20.9472	0.673617	0.336808	+	0.941573i	$0.390653\pi$
$968$	2.23445	0.0718179
$969$	−1.36916	−0.0439838
$970$	0	0
$971$	−53.8204	−1.72718	−0.863590	−	0.504195i	$-0.831789\pi$
$971$	−53.8204	−1.72718	−0.863590	+	0.504195i	$0.831789\pi$
$972$	1.13820	0.0365079
$973$	−45.9858	−1.47424
$974$	12.5002	0.400531
$975$	0	0
$976$	1.46018	0.0467393
$977$	33.0686	1.05796	0.528980	−	0.848634i	$-0.322575\pi$
$977$	33.0686	1.05796	0.528980	+	0.848634i	$0.322575\pi$
$978$	0.208010	0.00665142
$979$	2.63289	0.0841474
$980$	0	0
$981$	−3.41791	−0.109125
$982$	24.7823	0.790836
$983$	20.3930	0.650437	0.325219	−	0.945639i	$-0.394562\pi$
$983$	20.3930	0.650437	0.325219	+	0.945639i	$0.394562\pi$
$984$	−0.342614	−0.0109221
$985$	0	0
$986$	80.1924	2.55385
$987$	0.608465	0.0193676
$988$	60.3508	1.92001
$989$	49.3714	1.56992
$990$	0	0
$991$	11.9938	0.380997	0.190499	−	0.981687i	$-0.438990\pi$
$991$	11.9938	0.380997	0.190499	+	0.981687i	$0.438990\pi$
$992$	−32.9736	−1.04691
$993$	0.0319208	0.00101298
$994$	−16.7353	−0.530812
$995$	0	0
$996$	−0.260411	−0.00825145
$997$	40.4595	1.28136	0.640682	−	0.767806i	$-0.278652\pi$
$997$	40.4595	1.28136	0.640682	+	0.767806i	$0.278652\pi$
$998$	5.26104	0.166535
$999$	−1.25684	−0.0397648

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4925.2.a.q.1.29	yes	37
5.4	even	2		4925.2.a.p.1.9	✓	37

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
4925.2.a.p.1.9	✓	37	5.4	even	2
4925.2.a.q.1.29	yes	37	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 4925 = 5^{2} \cdot 197 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4925.a (trivial)

\(f(q)\)	\(=\)	\(q+1.86497 q^{2} +0.0285304 q^{3} +1.47810 q^{4} +0.0532083 q^{6} +2.61269 q^{7} -0.973320 q^{8} -2.99919 q^{9} -2.95031 q^{11} +0.0421709 q^{12} +5.30110 q^{13} +4.87258 q^{14} -4.77142 q^{16} -6.23067 q^{17} -5.59338 q^{18} +7.70215 q^{19} +0.0745411 q^{21} -5.50222 q^{22} +4.37513 q^{23} -0.0277693 q^{24} +9.88639 q^{26} -0.171159 q^{27} +3.86182 q^{28} -6.90125 q^{29} +4.74310 q^{31} -6.95190 q^{32} -0.0841735 q^{33} -11.6200 q^{34} -4.43311 q^{36} +7.34312 q^{37} +14.3643 q^{38} +0.151243 q^{39} +12.3379 q^{41} +0.139017 q^{42} +11.2845 q^{43} -4.36086 q^{44} +8.15948 q^{46} +8.16281 q^{47} -0.136131 q^{48} -0.173865 q^{49} -0.177764 q^{51} +7.83558 q^{52} +5.90870 q^{53} -0.319207 q^{54} -2.54298 q^{56} +0.219746 q^{57} -12.8706 q^{58} -9.78424 q^{59} -0.306027 q^{61} +8.84573 q^{62} -7.83594 q^{63} -3.42223 q^{64} -0.156981 q^{66} +12.8152 q^{67} -9.20957 q^{68} +0.124824 q^{69} -3.43459 q^{71} +2.91917 q^{72} +14.0877 q^{73} +13.6947 q^{74} +11.3846 q^{76} -7.70823 q^{77} +0.282063 q^{78} -6.51002 q^{79} +8.99267 q^{81} +23.0098 q^{82} -6.17514 q^{83} +0.110179 q^{84} +21.0453 q^{86} -0.196896 q^{87} +2.87159 q^{88} -0.892411 q^{89} +13.8501 q^{91} +6.46690 q^{92} +0.135323 q^{93} +15.2234 q^{94} -0.198341 q^{96} +2.91287 q^{97} -0.324252 q^{98} +8.84851 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 37 q + 2 q^{2} + q^{3} + 50 q^{4} + 8 q^{6} - 4 q^{7} + 50 q^{9} + 17 q^{11} - 2 q^{12} - 3 q^{13} + 14 q^{14} + 72 q^{16} - 10 q^{17} - 4 q^{18} + 54 q^{19} + 15 q^{21} - 11 q^{22} + 4 q^{23} + 28 q^{24}+ \cdots + 40 q^{99}+O(q^{100}) \) 37 * q + 2 * q^2 + q^3 + 50 * q^4 + 8 * q^6 - 4 * q^7 + 50 * q^9 + 17 * q^11 - 2 * q^12 - 3 * q^13 + 14 * q^14 + 72 * q^16 - 10 * q^17 - 4 * q^18 + 54 * q^19 + 15 * q^21 - 11 * q^22 + 4 * q^23 + 28 * q^24 + 19 * q^26 - 2 * q^27 + 3 * q^28 - 4 * q^29 + 58 * q^31 + 22 * q^32 - 7 * q^33 + 63 * q^34 + 66 * q^36 + 7 * q^37 + 12 * q^38 + 23 * q^39 + 17 * q^41 - 17 * q^42 + 11 * q^43 + 34 * q^44 + 41 * q^46 - 19 * q^47 + 10 * q^48 + 81 * q^49 + 39 * q^51 + 31 * q^52 - 14 * q^53 + 5 * q^54 + 21 * q^56 + 24 * q^57 - 15 * q^58 + 56 * q^59 + 40 * q^61 + 4 * q^62 + 5 * q^63 + 78 * q^64 + 22 * q^66 + 2 * q^67 - 3 * q^68 + 24 * q^69 + 31 * q^71 + 85 * q^72 + 4 * q^73 - 32 * q^74 + 137 * q^76 + 16 * q^77 - 113 * q^78 + 17 * q^79 + 85 * q^81 + 52 * q^82 - 18 * q^83 - 32 * q^84 + 35 * q^86 + 39 * q^87 - 63 * q^88 + 23 * q^89 + 152 * q^91 + 75 * q^92 + 8 * q^93 + 32 * q^94 + 72 * q^96 + 34 * q^97 - 40 * q^98 + 40 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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