LMFDB - Embedded newform 725.2.a.l.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [725,2,Mod(1,725)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(725, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("725.1"); S:= CuspForms(chi, 2); N := Newforms(S);

Level:	$ N $	$=$	$ 725 = 5^{2} \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	725.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,14] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(4)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.78915414654$
Analytic rank:	$0$
Dimension:	$6$
Coefficient field:	6.6.337383424.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 13x^{4} + 41x^{2} - 1 $ x^6 - 13x^4 + 41x^2 - 1
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 145)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.156785$ of defining polynomial
Character	$\chi$	$=$	725.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.156785 q^{2} -2.56387 q^{3} -1.97542 q^{4} -0.401976 q^{6} -1.09364 q^{7} -0.623285 q^{8} +3.57344 q^{9} -4.40198 q^{11} +5.06472 q^{12} -3.97108 q^{13} -0.171466 q^{14} +3.85312 q^{16} -6.22138 q^{17} +0.560261 q^{18} +6.97542 q^{19} +2.80395 q^{21} -0.690163 q^{22} +0.780070 q^{23} +1.59802 q^{24} -0.622605 q^{26} -1.47023 q^{27} +2.16040 q^{28} -1.00000 q^{29} +6.40198 q^{31} +1.85068 q^{32} +11.2861 q^{33} -0.975419 q^{34} -7.05904 q^{36} +1.09364 q^{37} +1.09364 q^{38} +10.1813 q^{39} +0.439617 q^{42} +0.376593 q^{43} +8.69575 q^{44} +0.122303 q^{46} +4.75115 q^{47} -9.87890 q^{48} -5.80395 q^{49} +15.9508 q^{51} +7.84455 q^{52} +11.2861 q^{53} -0.230510 q^{54} +0.681649 q^{56} -17.8841 q^{57} -0.156785 q^{58} +10.0000 q^{59} -1.14688 q^{61} +1.00373 q^{62} -3.90806 q^{63} -7.41607 q^{64} +1.76949 q^{66} +5.90782 q^{67} +12.2898 q^{68} -2.00000 q^{69} +2.00000 q^{71} -2.22727 q^{72} -8.72223 q^{73} +0.171466 q^{74} -13.7794 q^{76} +4.81418 q^{77} +1.59628 q^{78} -5.54886 q^{79} -6.95084 q^{81} -6.22138 q^{83} -5.53898 q^{84} +0.0590441 q^{86} +2.56387 q^{87} +2.74369 q^{88} -10.8040 q^{89} +4.34293 q^{91} -1.54096 q^{92} -16.4139 q^{93} +0.744908 q^{94} -4.74491 q^{96} +17.4176 q^{97} -0.909972 q^{98} -15.7302 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 14 q^{4} + 14 q^{6} + 12 q^{9} - 10 q^{11} - 8 q^{14} + 42 q^{16} + 16 q^{19} - 16 q^{21} + 26 q^{24} - 46 q^{26} - 6 q^{29} + 22 q^{31} + 20 q^{34} - 12 q^{36} - 14 q^{39} - 2 q^{44} - 44 q^{46}+ \cdots + 16 q^{99}+O(q^{100}) $ 6 * q + 14 * q^4 + 14 * q^6 + 12 * q^9 - 10 * q^11 - 8 * q^14 + 42 * q^16 + 16 * q^19 - 16 * q^21 + 26 * q^24 - 46 * q^26 - 6 * q^29 + 22 * q^31 + 20 * q^34 - 12 * q^36 - 14 * q^39 - 2 * q^44 - 44 * q^46 - 2 * q^49 + 44 * q^51 + 22 * q^54 + 16 * q^56 + 60 * q^59 + 12 * q^61 + 38 * q^64 + 34 * q^66 - 12 * q^69 + 12 * q^71 + 8 * q^74 - 24 * q^76 + 2 * q^79 + 10 * q^81 - 80 * q^84 - 30 * q^86 - 32 * q^89 + 40 * q^91 + 2 * q^94 - 26 * q^96 + 16 * 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 $

$2$	0.156785	0.110864	0.0554318	−	0.998462i	$-0.482346\pi$
$2$	0.156785	0.110864	0.0554318	+	0.998462i	$0.482346\pi$
$3$	−2.56387	−1.48025	−0.740126	−	0.672468i	$-0.765234\pi$
$3$	−2.56387	−1.48025	−0.740126	+	0.672468i	$0.765234\pi$
$4$	−1.97542	−0.987709
$5$	0	0
$5$	0	0
$6$	−0.401976	−0.164106
$7$	−1.09364	−0.413357	−0.206678	−	0.978409i	$-0.566265\pi$
$7$	−1.09364	−0.413357	−0.206678	+	0.978409i	$0.566265\pi$
$8$	−0.623285	−0.220365
$9$	3.57344	1.19115
$10$	0	0
$11$	−4.40198	−1.32725	−0.663623	−	0.748067i	$-0.730982\pi$
$11$	−4.40198	−1.32725	−0.663623	+	0.748067i	$0.730982\pi$
$12$	5.06472	1.46206
$13$	−3.97108	−1.10138	−0.550690	−	0.834710i	$-0.685635\pi$
$13$	−3.97108	−1.10138	−0.550690	+	0.834710i	$0.685635\pi$
$14$	−0.171466	−0.0458262
$15$	0	0
$16$	3.85312	0.963279
$17$	−6.22138	−1.50891	−0.754454	−	0.656353i	$-0.772098\pi$
$17$	−6.22138	−1.50891	−0.754454	+	0.656353i	$0.772098\pi$
$18$	0.560261	0.132055
$19$	6.97542	1.60027	0.800135	−	0.599819i	$-0.204761\pi$
$19$	6.97542	1.60027	0.800135	+	0.599819i	$0.204761\pi$
$20$	0	0
$21$	2.80395	0.611873
$22$	−0.690163	−0.147143
$23$	0.780070	0.162656	0.0813279	−	0.996687i	$-0.474084\pi$
$23$	0.780070	0.162656	0.0813279	+	0.996687i	$0.474084\pi$
$24$	1.59802	0.326195
$25$	0	0
$26$	−0.622605	−0.122103
$27$	−1.47023	−0.282946
$28$	2.16040	0.408276
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	6.40198	1.14983	0.574914	−	0.818214i	$-0.305035\pi$
$31$	6.40198	1.14983	0.574914	+	0.818214i	$0.305035\pi$
$32$	1.85068	0.327157
$33$	11.2861	1.96466
$34$	−0.975419	−0.167283
$35$	0	0
$36$	−7.05904	−1.17651
$37$	1.09364	0.179793	0.0898966	−	0.995951i	$-0.471346\pi$
$37$	1.09364	0.179793	0.0898966	+	0.995951i	$0.471346\pi$
$38$	1.09364	0.177412
$39$	10.1813	1.63032
$40$	0	0
$41$	0	0		−	1.00000i	$-0.5\pi$
$41$	0	0			1.00000i	$0.5\pi$
$42$	0.439617	0.0678344
$43$	0.376593	0.0574299	0.0287150	−	0.999588i	$-0.490858\pi$
$43$	0.376593	0.0574299	0.0287150	+	0.999588i	$0.490858\pi$
$44$	8.69575	1.31093
$45$	0	0
$46$	0.122303	0.0180326
$47$	4.75115	0.693027	0.346513	−	0.938045i	$-0.387366\pi$
$47$	4.75115	0.693027	0.346513	+	0.938045i	$0.387366\pi$
$48$	−9.87890	−1.42590
$49$	−5.80395	−0.829136
$50$	0	0
$51$	15.9508	2.23356
$52$	7.84455	1.08784
$53$	11.2861	1.55027	0.775133	−	0.631798i	$-0.217683\pi$
$53$	11.2861	1.55027	0.775133	+	0.631798i	$0.217683\pi$
$54$	−0.230510	−0.0313685
$55$	0	0
$56$	0.681649	0.0910892
$57$	−17.8841	−2.36880
$58$	−0.156785	−0.0205869
$59$	10.0000	1.30189	0.650945	−	0.759125i	$-0.274373\pi$
$59$	10.0000	1.30189	0.650945	+	0.759125i	$0.274373\pi$
$60$	0	0
$61$	−1.14688	−0.146844	−0.0734218	−	0.997301i	$-0.523392\pi$
$61$	−1.14688	−0.146844	−0.0734218	+	0.997301i	$0.523392\pi$
$62$	1.00373	0.127474
$63$	−3.90806	−0.492369
$64$	−7.41607	−0.927009
$65$	0	0
$66$	1.76949	0.217809
$67$	5.90782	0.721754	0.360877	−	0.932613i	$-0.382477\pi$
$67$	5.90782	0.721754	0.360877	+	0.932613i	$0.382477\pi$
$68$	12.2898	1.49036
$69$	−2.00000	−0.240772
$70$	0	0
$71$	2.00000	0.237356	0.118678	−	0.992933i	$-0.462134\pi$
$71$	2.00000	0.237356	0.118678	+	0.992933i	$0.462134\pi$
$72$	−2.22727	−0.262487
$73$	−8.72223	−1.02086	−0.510430	−	0.859919i	$-0.670514\pi$
$73$	−8.72223	−1.02086	−0.510430	+	0.859919i	$0.670514\pi$
$74$	0.171466	0.0199325
$75$	0	0
$76$	−13.7794	−1.58060
$77$	4.81418	0.548626
$78$	1.59628	0.180743
$79$	−5.54886	−0.624296	−0.312148	−	0.950034i	$-0.601048\pi$
$79$	−5.54886	−0.624296	−0.312148	+	0.950034i	$0.601048\pi$
$80$	0	0
$81$	−6.95084	−0.772315
$82$	0	0
$83$	−6.22138	−0.682886	−0.341443	−	0.939903i	$-0.610916\pi$
$83$	−6.22138	−0.682886	−0.341443	+	0.939903i	$0.610916\pi$
$84$	−5.53898	−0.604352
$85$	0	0
$86$	0.0590441	0.00636689
$87$	2.56387	0.274876
$88$	2.74369	0.292478
$89$	−10.8040	−1.14522	−0.572608	−	0.819829i	$-0.694068\pi$
$89$	−10.8040	−1.14522	−0.572608	+	0.819829i	$0.694068\pi$
$90$	0	0
$91$	4.34293	0.455263
$92$	−1.54096	−0.160657
$93$	−16.4139	−1.70204
$94$	0.744908	0.0768314
$95$	0	0
$96$	−4.74491	−0.484275
$97$	17.4176	1.76849	0.884244	−	0.467026i	$-0.154674\pi$
$97$	17.4176	1.76849	0.884244	+	0.467026i	$0.154674\pi$
$98$	−0.909972	−0.0919210
$99$	−15.7302	−1.58095
$100$	0	0
$101$	−12.7548	−1.26915	−0.634574	−	0.772862i	$-0.718825\pi$
$101$	−12.7548	−1.26915	−0.634574	+	0.772862i	$0.718825\pi$
$102$	2.50085	0.247621
$103$	5.15463	0.507901	0.253950	−	0.967217i	$-0.418270\pi$
$103$	5.15463	0.507901	0.253950	+	0.967217i	$0.418270\pi$
$104$	2.47512	0.242705
$105$	0	0
$106$	1.76949	0.171868
$107$	1.09364	0.105726	0.0528631	−	0.998602i	$-0.483165\pi$
$107$	1.09364	0.105726	0.0528631	+	0.998602i	$0.483165\pi$
$108$	2.90433	0.279469
$109$	15.3282	1.46818	0.734089	−	0.679053i	$-0.237609\pi$
$109$	15.3282	1.46818	0.734089	+	0.679053i	$0.237609\pi$
$110$	0	0
$111$	−2.80395	−0.266139
$112$	−4.21392	−0.398178
$113$	−7.28814	−0.685611	−0.342805	−	0.939406i	$-0.611377\pi$
$113$	−7.28814	−0.685611	−0.342805	+	0.939406i	$0.611377\pi$
$114$	−2.80395	−0.262614
$115$	0	0
$116$	1.97542	0.183413
$117$	−14.1904	−1.31191
$118$	1.56785	0.144332
$119$	6.80395	0.623717
$120$	0	0
$121$	8.37739	0.761581
$122$	−0.179814	−0.0162796
$123$	0	0
$124$	−12.6466	−1.13570
$125$	0	0
$126$	−0.612724	−0.0545858
$127$	−17.7312	−1.57339	−0.786693	−	0.617345i	$-0.788208\pi$
$127$	−17.7312	−1.57339	−0.786693	+	0.617345i	$0.788208\pi$
$128$	−4.86409	−0.429929
$129$	−0.965537	−0.0850108
$130$	0	0
$131$	7.31835	0.639407	0.319704	−	0.947518i	$-0.396417\pi$
$131$	7.31835	0.639407	0.319704	+	0.947518i	$0.396417\pi$
$132$	−22.2948	−1.94051
$133$	−7.62859	−0.661483
$134$	0.926256	0.0800163
$135$	0	0
$136$	3.87770	0.332510
$137$	−9.78899	−0.836330	−0.418165	−	0.908371i	$-0.637327\pi$
$137$	−9.78899	−0.836330	−0.418165	+	0.908371i	$0.637327\pi$
$138$	−0.313570	−0.0266928
$139$	−2.75479	−0.233658	−0.116829	−	0.993152i	$-0.537273\pi$
$139$	−2.75479	−0.233658	−0.116829	+	0.993152i	$0.537273\pi$
$140$	0	0
$141$	−12.1813	−1.02585
$142$	0.313570	0.0263142
$143$	17.4806	1.46180
$144$	13.7689	1.14741
$145$	0	0
$146$	−1.36751	−0.113176
$147$	14.8806	1.22733
$148$	−2.16040	−0.177583
$149$	−4.52428	−0.370643	−0.185322	−	0.982678i	$-0.559333\pi$
$149$	−4.52428	−0.370643	−0.185322	+	0.982678i	$0.559333\pi$
$150$	0	0
$151$	2.80395	0.228182	0.114091	−	0.993470i	$-0.463604\pi$
$151$	2.80395	0.228182	0.114091	+	0.993470i	$0.463604\pi$
$152$	−4.34768	−0.352643
$153$	−22.2318	−1.79733
$154$	0.754790	0.0608227
$155$	0	0
$156$	−20.1124	−1.61028
$157$	4.97481	0.397033	0.198517	−	0.980098i	$-0.436388\pi$
$157$	4.97481	0.397033	0.198517	+	0.980098i	$0.436388\pi$
$158$	−0.869977	−0.0692117
$159$	−28.9361	−2.29478
$160$	0	0
$161$	−0.853115	−0.0672349
$162$	−1.08979	−0.0856216
$163$	20.7615	1.62617	0.813084	−	0.582146i	$-0.197787\pi$
$163$	20.7615	1.62617	0.813084	+	0.582146i	$0.197787\pi$
$164$	0	0
$165$	0	0
$166$	−0.975419	−0.0757072
$167$	21.9182	1.69608	0.848040	−	0.529932i	$-0.177783\pi$
$167$	21.9182	1.69608	0.848040	+	0.529932i	$0.177783\pi$
$168$	−1.74766	−0.134835
$169$	2.76949	0.213038
$170$	0	0
$171$	24.9263	1.90616
$172$	−0.743929	−0.0567241
$173$	−3.25404	−0.247400	−0.123700	−	0.992320i	$-0.539476\pi$
$173$	−3.25404	−0.247400	−0.123700	+	0.992320i	$0.539476\pi$
$174$	0.401976	0.0304737
$175$	0	0
$176$	−16.9613	−1.27851
$177$	−25.6387	−1.92712
$178$	−1.69390	−0.126963
$179$	−7.19605	−0.537858	−0.268929	−	0.963160i	$-0.586670\pi$
$179$	−7.19605	−0.537858	−0.268929	+	0.963160i	$0.586670\pi$
$180$	0	0
$181$	13.0345	0.968844	0.484422	−	0.874834i	$-0.339030\pi$
$181$	13.0345	0.968844	0.484422	+	0.874834i	$0.339030\pi$
$182$	0.680906	0.0504721
$183$	2.94047	0.217365
$184$	−0.486206	−0.0358436
$185$	0	0
$186$	−2.57344	−0.188694
$187$	27.3864	2.00269
$188$	−9.38551	−0.684509
$189$	1.60790	0.116958
$190$	0	0
$191$	−10.1223	−0.732424	−0.366212	−	0.930531i	$-0.619346\pi$
$191$	−10.1223	−0.732424	−0.366212	+	0.930531i	$0.619346\pi$
$192$	19.0139	1.37221
$193$	22.8589	1.64542	0.822710	−	0.568462i	$-0.192461\pi$
$193$	22.8589	1.64542	0.822710	+	0.568462i	$0.192461\pi$
$194$	2.73081	0.196061
$195$	0	0
$196$	11.4652	0.818945
$197$	22.0711	1.57250	0.786251	−	0.617907i	$-0.212019\pi$
$197$	22.0711	1.57250	0.786251	+	0.617907i	$0.212019\pi$
$198$	−2.46626	−0.175269
$199$	5.19605	0.368338	0.184169	−	0.982895i	$-0.441041\pi$
$199$	5.19605	0.368338	0.184169	+	0.982895i	$0.441041\pi$
$200$	0	0
$201$	−15.1469	−1.06838
$202$	−1.99976	−0.140702
$203$	1.09364	0.0767585
$204$	−31.5096	−2.20611
$205$	0	0
$206$	0.808167	0.0563077
$207$	2.78754	0.193747
$208$	−15.3010	−1.06094
$209$	−30.7056	−2.12395
$210$	0	0
$211$	1.64719	0.113397	0.0566985	−	0.998391i	$-0.481943\pi$
$211$	1.64719	0.113397	0.0566985	+	0.998391i	$0.481943\pi$
$212$	−22.2948	−1.53121
$213$	−5.12775	−0.351347
$214$	0.171466	0.0117212
$215$	0	0
$216$	0.916374	0.0623514
$217$	−7.00145	−0.475290
$218$	2.40323	0.162768
$219$	22.3627	1.51113
$220$	0	0
$221$	24.7056	1.66188
$222$	−0.439617	−0.0295052
$223$	9.47542	0.634521	0.317261	−	0.948338i	$-0.397237\pi$
$223$	9.47542	0.634521	0.317261	+	0.948338i	$0.397237\pi$
$224$	−2.02398	−0.135233
$225$	0	0
$226$	−1.14267	−0.0760093
$227$	5.72800	0.380181	0.190090	−	0.981767i	$-0.439122\pi$
$227$	5.72800	0.380181	0.190090	+	0.981767i	$0.439122\pi$
$228$	35.3286	2.33969
$229$	15.1469	1.00093	0.500467	−	0.865756i	$-0.333162\pi$
$229$	15.1469	1.00093	0.500467	+	0.865756i	$0.333162\pi$
$230$	0	0
$231$	−12.3429	−0.812105
$232$	0.623285	0.0409207
$233$	−0.717046	−0.0469753	−0.0234876	−	0.999724i	$-0.507477\pi$
$233$	−0.717046	−0.0469753	−0.0234876	+	0.999724i	$0.507477\pi$
$234$	−2.22484	−0.145443
$235$	0	0
$236$	−19.7542	−1.28589
$237$	14.2266	0.924115
$238$	1.06676	0.0691475
$239$	−2.00000	−0.129369	−0.0646846	−	0.997906i	$-0.520604\pi$
$239$	−2.00000	−0.129369	−0.0646846	+	0.997906i	$0.520604\pi$
$240$	0	0
$241$	17.3282	1.11621	0.558105	−	0.829770i	$-0.311529\pi$
$241$	17.3282	1.11621	0.558105	+	0.829770i	$0.311529\pi$
$242$	1.31345	0.0844316
$243$	22.2318	1.42617
$244$	2.26558	0.145039
$245$	0	0
$246$	0	0
$247$	−27.7000	−1.76251
$248$	−3.99026	−0.253382
$249$	15.9508	1.01084
$250$	0	0
$251$	26.0099	1.64173	0.820865	−	0.571123i	$-0.193492\pi$
$251$	26.0099	1.64173	0.820865	+	0.571123i	$0.193492\pi$
$252$	7.72005	0.486317
$253$	−3.43385	−0.215884
$254$	−2.77998	−0.174431
$255$	0	0
$256$	14.0695	0.879346
$257$	15.0335	0.937766	0.468883	−	0.883260i	$-0.344657\pi$
$257$	15.0335	0.937766	0.468883	+	0.883260i	$0.344657\pi$
$258$	−0.151382	−0.00942461
$259$	−1.19605	−0.0743188
$260$	0	0
$261$	−3.57344	−0.221191
$262$	1.14741	0.0708870
$263$	−12.0662	−0.744032	−0.372016	−	0.928226i	$-0.621333\pi$
$263$	−12.0662	−0.744032	−0.372016	+	0.928226i	$0.621333\pi$
$264$	−7.03446	−0.432941
$265$	0	0
$266$	−1.19605	−0.0733344
$267$	27.7000	1.69521
$268$	−11.6704	−0.712884
$269$	−10.8531	−0.661726	−0.330863	−	0.943679i	$-0.607340\pi$
$269$	−10.8531	−0.661726	−0.330863	+	0.943679i	$0.607340\pi$
$270$	0	0
$271$	−12.7449	−0.774198	−0.387099	−	0.922038i	$-0.626523\pi$
$271$	−12.7449	−0.774198	−0.387099	+	0.922038i	$0.626523\pi$
$272$	−23.9717	−1.45350
$273$	−11.1347	−0.673904
$274$	−1.53476	−0.0927185
$275$	0	0
$276$	3.95084	0.237812
$277$	−20.0714	−1.20597	−0.602986	−	0.797752i	$-0.706022\pi$
$277$	−20.0714	−1.20597	−0.602986	+	0.797752i	$0.706022\pi$
$278$	−0.431909	−0.0259042
$279$	22.8771	1.36962
$280$	0	0
$281$	−20.9361	−1.24895	−0.624473	−	0.781047i	$-0.714686\pi$
$281$	−20.9361	−1.24895	−0.624473	+	0.781047i	$0.714686\pi$
$282$	−1.90985	−0.113730
$283$	−16.9703	−1.00878	−0.504389	−	0.863477i	$-0.668282\pi$
$283$	−16.9703	−1.00878	−0.504389	+	0.863477i	$0.668282\pi$
$284$	−3.95084	−0.234439
$285$	0	0
$286$	2.74069	0.162061
$287$	0	0
$288$	6.61330	0.389692
$289$	21.7056	1.27680
$290$	0	0
$291$	−44.6565	−2.61781
$292$	17.2301	1.00831
$293$	9.47542	0.553560	0.276780	−	0.960933i	$-0.410733\pi$
$293$	9.47542	0.553560	0.276780	+	0.960933i	$0.410733\pi$
$294$	2.33305	0.136066
$295$	0	0
$296$	−0.681649	−0.0396201
$297$	6.47193	0.375539
$298$	−0.709338	−0.0410909
$299$	−3.09772	−0.179146
$300$	0	0
$301$	−0.411857	−0.0237391
$302$	0.439617	0.0252971
$303$	32.7017	1.87866
$304$	26.8771	1.54151
$305$	0	0
$306$	−3.48560	−0.199259
$307$	−19.3812	−1.10614	−0.553072	−	0.833134i	$-0.686544\pi$
$307$	−19.3812	−1.10614	−0.553072	+	0.833134i	$0.686544\pi$
$308$	−9.51001	−0.541883
$309$	−13.2158	−0.751821
$310$	0	0
$311$	−9.26919	−0.525607	−0.262804	−	0.964849i	$-0.584647\pi$
$311$	−9.26919	−0.525607	−0.262804	+	0.964849i	$0.584647\pi$
$312$	−6.34588	−0.359265
$313$	14.5324	0.821422	0.410711	−	0.911766i	$-0.365281\pi$
$313$	14.5324	0.821422	0.410711	+	0.911766i	$0.365281\pi$
$314$	0.779975	0.0440165
$315$	0	0
$316$	10.9613	0.616623
$317$	22.4116	1.25876	0.629380	−	0.777098i	$-0.283309\pi$
$317$	22.4116	1.25876	0.629380	+	0.777098i	$0.283309\pi$
$318$	−4.53675	−0.254408
$319$	4.40198	0.246463
$320$	0	0
$321$	−2.80395	−0.156501
$322$	−0.133756	−0.00745390
$323$	−43.3968	−2.41466
$324$	13.7308	0.762823
$325$	0	0
$326$	3.25509	0.180283
$327$	−39.2996	−2.17327
$328$	0	0
$329$	−5.19605	−0.286467
$330$	0	0
$331$	7.89179	0.433772	0.216886	−	0.976197i	$-0.430410\pi$
$331$	7.89179	0.433772	0.216886	+	0.976197i	$0.430410\pi$
$332$	12.2898	0.674493
$333$	3.90806	0.214160
$334$	3.43644	0.188034
$335$	0	0
$336$	10.8040	0.589404
$337$	−33.4280	−1.82094	−0.910468	−	0.413579i	$-0.864279\pi$
$337$	−33.4280	−1.82094	−0.910468	+	0.413579i	$0.864279\pi$
$338$	0.434214	0.0236181
$339$	18.6859	1.01488
$340$	0	0
$341$	−28.1813	−1.52611
$342$	3.90806	0.211324
$343$	14.0029	0.756086
$344$	−0.234725	−0.0126555
$345$	0	0
$346$	−0.510183	−0.0274276
$347$	−31.1069	−1.66991	−0.834954	−	0.550320i	$-0.814506\pi$
$347$	−31.1069	−1.66991	−0.834954	+	0.550320i	$0.814506\pi$
$348$	−5.06472	−0.271498
$349$	−5.76949	−0.308834	−0.154417	−	0.988006i	$-0.549350\pi$
$349$	−5.76949	−0.308834	−0.154417	+	0.988006i	$0.549350\pi$
$350$	0	0
$351$	5.83842	0.311632
$352$	−8.14665	−0.434218
$353$	13.5095	0.719039	0.359520	−	0.933137i	$-0.382941\pi$
$353$	13.5095	0.719039	0.359520	+	0.933137i	$0.382941\pi$
$354$	−4.01976	−0.213648
$355$	0	0
$356$	21.3423	1.13114
$357$	−17.4445	−0.923259
$358$	−1.12823	−0.0596289
$359$	−1.15677	−0.0610518	−0.0305259	−	0.999534i	$-0.509718\pi$
$359$	−1.15677	−0.0610518	−0.0305259	+	0.999534i	$0.509718\pi$
$360$	0	0
$361$	29.6565	1.56087
$362$	2.04361	0.107410
$363$	−21.4786	−1.12733
$364$	−8.57911	−0.449667
$365$	0	0
$366$	0.461020	0.0240979
$367$	10.5959	0.553104	0.276552	−	0.960999i	$-0.410808\pi$
$367$	10.5959	0.553104	0.276552	+	0.960999i	$0.410808\pi$
$368$	3.00570	0.156683
$369$	0	0
$370$	0	0
$371$	−12.3429	−0.640813
$372$	32.4242	1.68112
$373$	7.09907	0.367576	0.183788	−	0.982966i	$-0.441164\pi$
$373$	7.09907	0.367576	0.183788	+	0.982966i	$0.441164\pi$
$374$	4.29377	0.222026
$375$	0	0
$376$	−2.96132	−0.152719
$377$	3.97108	0.204521
$378$	0.252095	0.0129664
$379$	18.1223	0.930880	0.465440	−	0.885079i	$-0.345896\pi$
$379$	18.1223	0.930880	0.465440	+	0.885079i	$0.345896\pi$
$380$	0	0
$381$	45.4604	2.32901
$382$	−1.58702	−0.0811992
$383$	3.28092	0.167647	0.0838236	−	0.996481i	$-0.473287\pi$
$383$	3.28092	0.167647	0.0838236	+	0.996481i	$0.473287\pi$
$384$	12.4709	0.636403
$385$	0	0
$386$	3.58393	0.182417
$387$	1.34573	0.0684075
$388$	−34.4070	−1.74675
$389$	−12.0000	−0.608424	−0.304212	−	0.952604i	$-0.598393\pi$
$389$	−12.0000	−0.608424	−0.304212	+	0.952604i	$0.598393\pi$
$390$	0	0
$391$	−4.85312	−0.245433
$392$	3.61752	0.182712
$393$	−18.7633	−0.946484
$394$	3.46042	0.174333
$395$	0	0
$396$	31.0737	1.56151
$397$	36.3054	1.82212	0.911058	−	0.412278i	$-0.135267\pi$
$397$	36.3054	1.82212	0.911058	+	0.412278i	$0.135267\pi$
$398$	0.814661	0.0408353
$399$	19.5587	0.979162
$400$	0	0
$401$	26.0830	1.30252	0.651262	−	0.758853i	$-0.274240\pi$
$401$	26.0830	1.30252	0.651262	+	0.758853i	$0.274240\pi$
$402$	−2.37480	−0.118444
$403$	−25.4228	−1.26640
$404$	25.1960	1.25355
$405$	0	0
$406$	0.171466	0.00850972
$407$	−4.81418	−0.238630
$408$	−9.94192	−0.492198
$409$	13.0486	0.645210	0.322605	−	0.946534i	$-0.395442\pi$
$409$	13.0486	0.645210	0.322605	+	0.946534i	$0.395442\pi$
$410$	0	0
$411$	25.0977	1.23798
$412$	−10.1825	−0.501658
$413$	−10.9364	−0.538145
$414$	0.437043	0.0214795
$415$	0	0
$416$	−7.34920	−0.360324
$417$	7.06293	0.345873
$418$	−4.81418	−0.235469
$419$	35.8525	1.75151	0.875755	−	0.482756i	$-0.160364\pi$
$419$	35.8525	1.75151	0.875755	+	0.482756i	$0.160364\pi$
$420$	0	0
$421$	16.7056	0.814182	0.407091	−	0.913388i	$-0.366543\pi$
$421$	16.7056	0.814182	0.407091	+	0.913388i	$0.366543\pi$
$422$	0.258254	0.0125716
$423$	16.9780	0.825497
$424$	−7.03446	−0.341624
$425$	0	0
$426$	−0.803952	−0.0389516
$427$	1.25428	0.0606988
$428$	−2.16040	−0.104427
$429$	−44.8180	−2.16384
$430$	0	0
$431$	−26.3135	−1.26748	−0.633739	−	0.773547i	$-0.718481\pi$
$431$	−26.3135	−1.26748	−0.633739	+	0.773547i	$0.718481\pi$
$432$	−5.66498	−0.272556
$433$	−10.7220	−0.515266	−0.257633	−	0.966243i	$-0.582943\pi$
$433$	−10.7220	−0.515266	−0.257633	+	0.966243i	$0.582943\pi$
$434$	−1.09772	−0.0526923
$435$	0	0
$436$	−30.2797	−1.45013
$437$	5.44131	0.260293
$438$	3.50613	0.167529
$439$	29.5587	1.41076	0.705381	−	0.708828i	$-0.250776\pi$
$439$	29.5587	1.41076	0.705381	+	0.708828i	$0.250776\pi$
$440$	0	0
$441$	−20.7401	−0.987623
$442$	3.87347	0.184242
$443$	3.40697	0.161870	0.0809349	−	0.996719i	$-0.474209\pi$
$443$	3.40697	0.161870	0.0809349	+	0.996719i	$0.474209\pi$
$444$	5.53898	0.262868
$445$	0	0
$446$	1.48560	0.0703453
$447$	11.5997	0.548646
$448$	8.11051	0.383186
$449$	0.461020	0.0217569	0.0108784	−	0.999941i	$-0.496537\pi$
$449$	0.461020	0.0217569	0.0108784	+	0.999941i	$0.496537\pi$
$450$	0	0
$451$	0	0
$452$	14.3971	0.677184
$453$	−7.18898	−0.337768
$454$	0.898063	0.0421482
$455$	0	0
$456$	11.1469	0.522001
$457$	−1.37262	−0.0642084	−0.0321042	−	0.999485i	$-0.510221\pi$
$457$	−1.37262	−0.0642084	−0.0321042	+	0.999485i	$0.510221\pi$
$458$	2.37480	0.110967
$459$	9.14688	0.426940
$460$	0	0
$461$	14.5875	0.679409	0.339705	−	0.940532i	$-0.389673\pi$
$461$	14.5875	0.679409	0.339705	+	0.940532i	$0.389673\pi$
$462$	−1.93518	−0.0900329
$463$	18.8517	0.876112	0.438056	−	0.898948i	$-0.355667\pi$
$463$	18.8517	0.876112	0.438056	+	0.898948i	$0.355667\pi$
$464$	−3.85312	−0.178876
$465$	0	0
$466$	−0.112422	−0.00520785
$467$	−29.9580	−1.38629	−0.693145	−	0.720798i	$-0.743775\pi$
$467$	−29.9580	−1.38629	−0.693145	+	0.720798i	$0.743775\pi$
$468$	28.0320	1.29578
$469$	−6.46102	−0.298342
$470$	0	0
$471$	−12.7548	−0.587710
$472$	−6.23285	−0.286890
$473$	−1.65776	−0.0762237
$474$	2.23051	0.102451
$475$	0	0
$476$	−13.4407	−0.616051
$477$	40.3302	1.84660
$478$	−0.313570	−0.0143423
$479$	−32.7647	−1.49706	−0.748528	−	0.663103i	$-0.769239\pi$
$479$	−32.7647	−1.49706	−0.748528	+	0.663103i	$0.769239\pi$
$480$	0	0
$481$	−4.34293	−0.198021
$482$	2.71680	0.123747
$483$	2.18728	0.0995246
$484$	−16.5489	−0.752221
$485$	0	0
$486$	3.48560	0.158110
$487$	14.1098	0.639375	0.319688	−	0.947523i	$-0.396422\pi$
$487$	14.1098	0.639375	0.319688	+	0.947523i	$0.396422\pi$
$488$	0.714836	0.0323591
$489$	−53.2299	−2.40714
$490$	0	0
$491$	4.10821	0.185401	0.0927004	−	0.995694i	$-0.470450\pi$
$491$	4.10821	0.185401	0.0927004	+	0.995694i	$0.470450\pi$
$492$	0	0
$493$	6.22138	0.280197
$494$	−4.34293	−0.195398
$495$	0	0
$496$	24.6676	1.10761
$497$	−2.18728	−0.0981129
$498$	2.50085	0.112066
$499$	30.0689	1.34607	0.673035	−	0.739611i	$-0.264990\pi$
$499$	30.0689	1.34607	0.673035	+	0.739611i	$0.264990\pi$
$500$	0	0
$501$	−56.1954	−2.51063
$502$	4.07795	0.182008
$503$	14.8806	0.663493	0.331746	−	0.943369i	$-0.392362\pi$
$503$	14.8806	0.663493	0.331746	+	0.943369i	$0.392362\pi$
$504$	2.43583	0.108501
$505$	0	0
$506$	−0.538375	−0.0239337
$507$	−7.10062	−0.315350
$508$	35.0264	1.55405
$509$	1.93674	0.0858445	0.0429223	−	0.999078i	$-0.486333\pi$
$509$	1.93674	0.0858445	0.0429223	+	0.999078i	$0.486333\pi$
$510$	0	0
$511$	9.53898	0.421980
$512$	11.9341	0.527416
$513$	−10.2555	−0.452791
$514$	2.35703	0.103964
$515$	0	0
$516$	1.90734	0.0839660
$517$	−20.9145	−0.919817
$518$	−0.187522	−0.00823925
$519$	8.34293	0.366214
$520$	0	0
$521$	−4.23051	−0.185342	−0.0926710	−	0.995697i	$-0.529540\pi$
$521$	−4.23051	−0.185342	−0.0926710	+	0.995697i	$0.529540\pi$
$522$	−0.560261	−0.0245220
$523$	4.40144	0.192462	0.0962308	−	0.995359i	$-0.469321\pi$
$523$	4.40144	0.192462	0.0962308	+	0.995359i	$0.469321\pi$
$524$	−14.4568	−0.631548
$525$	0	0
$526$	−1.89179	−0.0824861
$527$	−39.8292	−1.73499
$528$	43.4867	1.89251
$529$	−22.3915	−0.973543
$530$	0	0
$531$	35.7344	1.55074
$532$	15.0697	0.653353
$533$	0	0
$534$	4.34293	0.187937
$535$	0	0
$536$	−3.68225	−0.159049
$537$	18.4497	0.796165
$538$	−1.70160	−0.0733613
$539$	25.5489	1.10047
$540$	0	0
$541$	−32.4119	−1.39349	−0.696747	−	0.717317i	$-0.745370\pi$
$541$	−32.4119	−1.39349	−0.696747	+	0.717317i	$0.745370\pi$
$542$	−1.99821	−0.0858304
$543$	−33.4187	−1.43413
$544$	−11.5138	−0.493650
$545$	0	0
$546$	−1.74576	−0.0747114
$547$	12.9093	0.551961	0.275980	−	0.961163i	$-0.410998\pi$
$547$	12.9093	0.551961	0.275980	+	0.961163i	$0.410998\pi$
$548$	19.3374	0.826051
$549$	−4.09833	−0.174912
$550$	0	0
$551$	−6.97542	−0.297163
$552$	1.24657	0.0530576
$553$	6.06845	0.258057
$554$	−3.14688	−0.133698
$555$	0	0
$556$	5.44186	0.230786
$557$	−23.0195	−0.975369	−0.487685	−	0.873020i	$-0.662158\pi$
$557$	−23.0195	−0.975369	−0.487685	+	0.873020i	$0.662158\pi$
$558$	3.58678	0.151841
$559$	−1.49548	−0.0632522
$560$	0	0
$561$	−70.2152	−2.96449
$562$	−3.28247	−0.138463
$563$	27.3234	1.15154	0.575771	−	0.817611i	$-0.304702\pi$
$563$	27.3234	1.15154	0.575771	+	0.817611i	$0.304702\pi$
$564$	24.0633	1.01325
$565$	0	0
$566$	−2.66068	−0.111837
$567$	7.60171	0.319242
$568$	−1.24657	−0.0523049
$569$	−0.803952	−0.0337034	−0.0168517	−	0.999858i	$-0.505364\pi$
$569$	−0.803952	−0.0337034	−0.0168517	+	0.999858i	$0.505364\pi$
$570$	0	0
$571$	43.4604	1.81876	0.909381	−	0.415964i	$-0.136556\pi$
$571$	43.4604	1.81876	0.909381	+	0.415964i	$0.136556\pi$
$572$	−34.5315	−1.44384
$573$	25.9523	1.08417
$574$	0	0
$575$	0	0
$576$	−26.5009	−1.10420
$577$	1.27345	0.0530146	0.0265073	−	0.999649i	$-0.491561\pi$
$577$	1.27345	0.0530146	0.0265073	+	0.999649i	$0.491561\pi$
$578$	3.40311	0.141551
$579$	−58.6073	−2.43564
$580$	0	0
$581$	6.80395	0.282276
$582$	−7.00145	−0.290220
$583$	−49.6812	−2.05758
$584$	5.43644	0.224961
$585$	0	0
$586$	1.48560	0.0613696
$587$	−31.8678	−1.31533	−0.657663	−	0.753312i	$-0.728455\pi$
$587$	−31.8678	−1.31533	−0.657663	+	0.753312i	$0.728455\pi$
$588$	−29.3954	−1.21225
$589$	44.6565	1.84004
$590$	0	0
$591$	−56.5875	−2.32770
$592$	4.21392	0.173191
$593$	−1.21043	−0.0497064	−0.0248532	−	0.999691i	$-0.507912\pi$
$593$	−1.21043	−0.0497064	−0.0248532	+	0.999691i	$0.507912\pi$
$594$	1.01470	0.0416337
$595$	0	0
$596$	8.93735	0.366088
$597$	−13.3220	−0.545233
$598$	−0.485676	−0.0198608
$599$	24.2053	0.989003	0.494501	−	0.869177i	$-0.335351\pi$
$599$	24.2053	0.989003	0.494501	+	0.869177i	$0.335351\pi$
$600$	0	0
$601$	4.68586	0.191140	0.0955702	−	0.995423i	$-0.469533\pi$
$601$	4.68586	0.191140	0.0955702	+	0.995423i	$0.469533\pi$
$602$	−0.0645730	−0.00263180
$603$	21.1112	0.859716
$604$	−5.53898	−0.225378
$605$	0	0
$606$	5.12712	0.208275
$607$	−0.502641	−0.0204016	−0.0102008	−	0.999948i	$-0.503247\pi$
$607$	−0.502641	−0.0204016	−0.0102008	+	0.999948i	$0.503247\pi$
$608$	12.9093	0.523540
$609$	−2.80395	−0.113622
$610$	0	0
$611$	−18.8672	−0.763286
$612$	43.9170	1.77524
$613$	22.9142	0.925496	0.462748	−	0.886490i	$-0.346863\pi$
$613$	22.9142	0.925496	0.462748	+	0.886490i	$0.346863\pi$
$614$	−3.03868	−0.122631
$615$	0	0
$616$	−3.00060	−0.120898
$617$	−15.8574	−0.638397	−0.319198	−	0.947688i	$-0.603414\pi$
$617$	−15.8574	−0.638397	−0.319198	+	0.947688i	$0.603414\pi$
$618$	−2.07204	−0.0833496
$619$	−35.4014	−1.42290	−0.711451	−	0.702736i	$-0.751961\pi$
$619$	−35.4014	−1.42290	−0.711451	+	0.702736i	$0.751961\pi$
$620$	0	0
$621$	−1.14688	−0.0460229
$622$	−1.45327	−0.0582707
$623$	11.8156	0.473383
$624$	39.2299	1.57045
$625$	0	0
$626$	2.27846	0.0910658
$627$	78.7253	3.14399
$628$	−9.82734	−0.392154
$629$	−6.80395	−0.271291
$630$	0	0
$631$	44.7056	1.77970	0.889851	−	0.456250i	$-0.150808\pi$
$631$	44.7056	1.77970	0.889851	+	0.456250i	$0.150808\pi$
$632$	3.45852	0.137573
$633$	−4.22318	−0.167856
$634$	3.51379	0.139551
$635$	0	0
$636$	57.1610	2.26658
$637$	23.0480	0.913194
$638$	0.690163	0.0273238
$639$	7.14688	0.282726
$640$	0	0
$641$	10.0689	0.397699	0.198849	−	0.980030i	$-0.436280\pi$
$641$	10.0689	0.397699	0.198849	+	0.980030i	$0.436280\pi$
$642$	−0.439617	−0.0173503
$643$	13.9760	0.551161	0.275580	−	0.961278i	$-0.411130\pi$
$643$	13.9760	0.551161	0.275580	+	0.961278i	$0.411130\pi$
$644$	1.68526	0.0664085
$645$	0	0
$646$	−6.80395	−0.267698
$647$	31.1607	1.22505	0.612527	−	0.790450i	$-0.290153\pi$
$647$	31.1607	1.22505	0.612527	+	0.790450i	$0.290153\pi$
$648$	4.33235	0.170191
$649$	−44.0198	−1.72793
$650$	0	0
$651$	17.9508	0.703549
$652$	−41.0127	−1.60618
$653$	−26.1129	−1.02188	−0.510939	−	0.859617i	$-0.670702\pi$
$653$	−26.1129	−1.02188	−0.510939	+	0.859617i	$0.670702\pi$
$654$	−6.16158	−0.240937
$655$	0	0
$656$	0	0
$657$	−31.1684	−1.21600
$658$	−0.814661	−0.0317588
$659$	−28.5974	−1.11400	−0.556999	−	0.830513i	$-0.688047\pi$
$659$	−28.5974	−1.11400	−0.556999	+	0.830513i	$0.688047\pi$
$660$	0	0
$661$	−38.9994	−1.51690	−0.758450	−	0.651731i	$-0.774043\pi$
$661$	−38.9994	−1.51690	−0.758450	+	0.651731i	$0.774043\pi$
$662$	1.23731	0.0480895
$663$	−63.3421	−2.46000
$664$	3.87770	0.150484
$665$	0	0
$666$	0.612724	0.0237426
$667$	−0.780070	−0.0302044
$668$	−43.2976	−1.67523
$669$	−24.2938	−0.939251
$670$	0	0
$671$	5.04856	0.194897
$672$	5.18922	0.200178
$673$	−38.6725	−1.49072	−0.745358	−	0.666665i	$-0.767721\pi$
$673$	−38.6725	−1.49072	−0.745358	+	0.666665i	$0.767721\pi$
$674$	−5.24100	−0.201876
$675$	0	0
$676$	−5.47090	−0.210419
$677$	28.4800	1.09458	0.547288	−	0.836944i	$-0.315660\pi$
$677$	28.4800	1.09458	0.547288	+	0.836944i	$0.315660\pi$
$678$	2.92966	0.112513
$679$	−19.0486	−0.731017
$680$	0	0
$681$	−14.6859	−0.562764
$682$	−4.41841	−0.169190
$683$	12.4159	0.475081	0.237540	−	0.971378i	$-0.423659\pi$
$683$	12.4159	0.475081	0.237540	+	0.971378i	$0.423659\pi$
$684$	−49.2398	−1.88273
$685$	0	0
$686$	2.19544	0.0838224
$687$	−38.8347	−1.48164
$688$	1.45106	0.0553211
$689$	−44.8180	−1.70743
$690$	0	0
$691$	10.8040	0.411002	0.205501	−	0.978657i	$-0.434118\pi$
$691$	10.8040	0.411002	0.205501	+	0.978657i	$0.434118\pi$
$692$	6.42808	0.244359
$693$	17.2032	0.653495
$694$	−4.87709	−0.185132
$695$	0	0
$696$	−1.59802	−0.0605729
$697$	0	0
$698$	−0.904568	−0.0342384
$699$	1.83842	0.0695352
$700$	0	0
$701$	8.47512	0.320101	0.160050	−	0.987109i	$-0.448834\pi$
$701$	8.47512	0.320101	0.160050	+	0.987109i	$0.448834\pi$
$702$	0.915375	0.0345486
$703$	7.62859	0.287718
$704$	32.6454	1.23037
$705$	0	0
$706$	2.11809	0.0797153
$707$	13.9491	0.524612
$708$	50.6472	1.90344
$709$	−32.8180	−1.23251	−0.616254	−	0.787548i	$-0.711350\pi$
$709$	−32.8180	−1.23251	−0.616254	+	0.787548i	$0.711350\pi$
$710$	0	0
$711$	−19.8285	−0.743628
$712$	6.73394	0.252365
$713$	4.99399	0.187026
$714$	−2.73503	−0.102356
$715$	0	0
$716$	14.2152	0.531247
$717$	5.12775	0.191499
$718$	−0.181363	−0.00676842
$719$	39.7542	1.48258	0.741290	−	0.671184i	$-0.234214\pi$
$719$	39.7542	1.48258	0.741290	+	0.671184i	$0.234214\pi$
$720$	0	0
$721$	−5.63731	−0.209944
$722$	4.64968	0.173043
$723$	−44.4274	−1.65227
$724$	−25.7485	−0.956936
$725$	0	0
$726$	−3.36751	−0.124980
$727$	−49.4844	−1.83527	−0.917637	−	0.397419i	$-0.869906\pi$
$727$	−49.4844	−1.83527	−0.917637	+	0.397419i	$0.869906\pi$
$728$	−2.70689	−0.100324
$729$	−36.1469	−1.33877
$730$	0	0
$731$	−2.34293	−0.0866565
$732$	−5.80865	−0.214694
$733$	46.2381	1.70784	0.853921	−	0.520403i	$-0.174218\pi$
$733$	46.2381	1.70784	0.853921	+	0.520403i	$0.174218\pi$
$734$	1.66128	0.0613191
$735$	0	0
$736$	1.44366	0.0532140
$737$	−26.0061	−0.957946
$738$	0	0
$739$	28.7155	1.05632	0.528159	−	0.849146i	$-0.322883\pi$
$739$	28.7155	1.05632	0.528159	+	0.849146i	$0.322883\pi$
$740$	0	0
$741$	71.0192	2.60895
$742$	−1.93518	−0.0710428
$743$	−27.0536	−0.992502	−0.496251	−	0.868179i	$-0.665290\pi$
$743$	−27.0536	−0.992502	−0.496251	+	0.868179i	$0.665290\pi$
$744$	10.2305	0.375069
$745$	0	0
$746$	1.11303	0.0407508
$747$	−22.2318	−0.813418
$748$	−54.0996	−1.97808
$749$	−1.19605	−0.0437026
$750$	0	0
$751$	−26.8771	−0.980759	−0.490380	−	0.871509i	$-0.663142\pi$
$751$	−26.8771	−0.980759	−0.490380	+	0.871509i	$0.663142\pi$
$752$	18.3067	0.667578
$753$	−66.6860	−2.43017
$754$	0.622605	0.0226739
$755$	0	0
$756$	−3.17629	−0.115520
$757$	7.28814	0.264892	0.132446	−	0.991190i	$-0.457717\pi$
$757$	7.28814	0.264892	0.132446	+	0.991190i	$0.457717\pi$
$758$	2.84130	0.103201
$759$	8.80395	0.319563
$760$	0	0
$761$	−31.6079	−1.14579	−0.572893	−	0.819630i	$-0.694179\pi$
$761$	−31.6079	−1.14579	−0.572893	+	0.819630i	$0.694179\pi$
$762$	7.12750	0.258202
$763$	−16.7636	−0.606882
$764$	19.9958	0.723422
$765$	0	0
$766$	0.514398	0.0185860
$767$	−39.7108	−1.43387
$768$	−36.0725	−1.30165
$769$	−0.293769	−0.0105936	−0.00529679	−	0.999986i	$-0.501686\pi$
$769$	−0.293769	−0.0105936	−0.00529679	+	0.999986i	$0.501686\pi$
$770$	0	0
$771$	−38.5440	−1.38813
$772$	−45.1559	−1.62520
$773$	−28.4877	−1.02463	−0.512316	−	0.858797i	$-0.671212\pi$
$773$	−28.4877	−1.02463	−0.512316	+	0.858797i	$0.671212\pi$
$774$	0.210991	0.00758391
$775$	0	0
$776$	−10.8561	−0.389712
$777$	3.06651	0.110011
$778$	−1.88142	−0.0674521
$779$	0	0
$780$	0	0
$781$	−8.80395	−0.315030
$782$	−0.760895	−0.0272095
$783$	1.47023	0.0525418
$784$	−22.3633	−0.798689
$785$	0	0
$786$	−2.94180	−0.104931
$787$	32.9883	1.17591	0.587954	−	0.808895i	$-0.299934\pi$
$787$	32.9883	1.17591	0.587954	+	0.808895i	$0.299934\pi$
$788$	−43.5997	−1.55317
$789$	30.9361	1.10136
$790$	0	0
$791$	7.97060	0.283402
$792$	9.80441	0.348384
$793$	4.55437	0.161731
$794$	5.69213	0.202006
$795$	0	0
$796$	−10.2644	−0.363811
$797$	30.9194	1.09522	0.547611	−	0.836733i	$-0.315537\pi$
$797$	30.9194	1.09522	0.547611	+	0.836733i	$0.315537\pi$
$798$	3.06651	0.108553
$799$	−29.5587	−1.04571
$800$	0	0
$801$	−38.6073	−1.36412
$802$	4.08942	0.144402
$803$	38.3951	1.35493
$804$	29.9214	1.05525
$805$	0	0
$806$	−3.98590	−0.140397
$807$	27.8260	0.979522
$808$	7.94987	0.279675
$809$	21.0977	0.741756	0.370878	−	0.928682i	$-0.379057\pi$
$809$	21.0977	0.741756	0.370878	+	0.928682i	$0.379057\pi$
$810$	0	0
$811$	28.0000	0.983213	0.491606	−	0.870817i	$-0.336410\pi$
$811$	28.0000	0.983213	0.491606	+	0.870817i	$0.336410\pi$
$812$	−2.16040	−0.0758150
$813$	32.6763	1.14601
$814$	−0.754790	−0.0264554
$815$	0	0
$816$	61.4604	2.15154
$817$	2.62690	0.0919035
$818$	2.04582	0.0715303
$819$	15.5192	0.542285
$820$	0	0
$821$	−15.2791	−0.533243	−0.266622	−	0.963801i	$-0.585907\pi$
$821$	−15.2791	−0.533243	−0.266622	+	0.963801i	$0.585907\pi$
$822$	3.93494	0.137247
$823$	−7.78152	−0.271247	−0.135623	−	0.990760i	$-0.543304\pi$
$823$	−7.78152	−0.271247	−0.135623	+	0.990760i	$0.543304\pi$
$824$	−3.21280	−0.111923
$825$	0	0
$826$	−1.71466	−0.0596607
$827$	−0.376593	−0.0130954	−0.00654772	−	0.999979i	$-0.502084\pi$
$827$	−0.376593	−0.0130954	−0.00654772	+	0.999979i	$0.502084\pi$
$828$	−5.50655	−0.191366
$829$	−49.8723	−1.73214	−0.866068	−	0.499927i	$-0.833360\pi$
$829$	−49.8723	−1.73214	−0.866068	+	0.499927i	$0.833360\pi$
$830$	0	0
$831$	51.4604	1.78514
$832$	29.4498	1.02099
$833$	36.1086	1.25109
$834$	1.10736	0.0383447
$835$	0	0
$836$	60.6565	2.09785
$837$	−9.41240	−0.325340
$838$	5.62113	0.194179
$839$	−0.0590441	−0.00203843	−0.00101921	−	0.999999i	$-0.500324\pi$
$839$	−0.0590441	−0.00203843	−0.00101921	+	0.999999i	$0.500324\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	2.61919	0.0902632
$843$	53.6776	1.84875
$844$	−3.25388	−0.112003
$845$	0	0
$846$	2.66189	0.0915176
$847$	−9.16185	−0.314805
$848$	43.4867	1.49334
$849$	43.5096	1.49324
$850$	0	0
$851$	0.853115	0.0292444
$852$	10.1294	0.347029
$853$	2.77983	0.0951795	0.0475897	−	0.998867i	$-0.484846\pi$
$853$	2.77983	0.0951795	0.0475897	+	0.998867i	$0.484846\pi$
$854$	0.196652	0.00672929
$855$	0	0
$856$	−0.681649	−0.0232983
$857$	0.850802	0.0290628	0.0145314	−	0.999894i	$-0.495374\pi$
$857$	0.850802	0.0290628	0.0145314	+	0.999894i	$0.495374\pi$
$858$	−7.02679	−0.239891
$859$	7.35342	0.250895	0.125448	−	0.992100i	$-0.459963\pi$
$859$	7.35342	0.250895	0.125448	+	0.992100i	$0.459963\pi$
$860$	0	0
$861$	0	0
$862$	−4.12556	−0.140517
$863$	42.6167	1.45069	0.725345	−	0.688386i	$-0.241680\pi$
$863$	42.6167	1.45069	0.725345	+	0.688386i	$0.241680\pi$
$864$	−2.72093	−0.0925680
$865$	0	0
$866$	−1.68105	−0.0571242
$867$	−55.6505	−1.88999
$868$	13.8308	0.469448
$869$	24.4260	0.828594
$870$	0	0
$871$	−23.4604	−0.794926
$872$	−9.55386	−0.323535
$873$	62.2407	2.10653
$874$	0.853115	0.0288571
$875$	0	0
$876$	−44.1757	−1.49256
$877$	13.4196	0.453148	0.226574	−	0.973994i	$-0.427247\pi$
$877$	13.4196	0.453148	0.226574	+	0.973994i	$0.427247\pi$
$878$	4.63436	0.156402
$879$	−24.2938	−0.819408
$880$	0	0
$881$	−38.3135	−1.29082	−0.645408	−	0.763838i	$-0.723313\pi$
$881$	−38.3135	−1.29082	−0.645408	+	0.763838i	$0.723313\pi$
$882$	−3.25173	−0.109491
$883$	−31.5542	−1.06188	−0.530942	−	0.847408i	$-0.678162\pi$
$883$	−31.5542	−1.06188	−0.530942	+	0.847408i	$0.678162\pi$
$884$	−48.8040	−1.64145
$885$	0	0
$886$	0.534161	0.0179455
$887$	23.3961	0.785565	0.392783	−	0.919631i	$-0.371512\pi$
$887$	23.3961	0.785565	0.392783	+	0.919631i	$0.371512\pi$
$888$	1.74766	0.0586477
$889$	19.3915	0.650370
$890$	0	0
$891$	30.5974	1.02505
$892$	−18.7179	−0.626722
$893$	33.1413	1.10903
$894$	1.81865	0.0608248
$895$	0	0
$896$	5.31956	0.177714
$897$	7.94216	0.265181
$898$	0.0722810	0.00241205
$899$	−6.40198	−0.213518
$900$	0	0
$901$	−70.2152	−2.33921
$902$	0	0
$903$	1.05595	0.0351398
$904$	4.54259	0.151084
$905$	0	0
$906$	−1.12712	−0.0374461
$907$	32.4150	1.07632	0.538161	−	0.842842i	$-0.319119\pi$
$907$	32.4150	1.07632	0.538161	+	0.842842i	$0.319119\pi$
$908$	−11.3152	−0.375508
$909$	−45.5785	−1.51174
$910$	0	0
$911$	−45.2749	−1.50002	−0.750011	−	0.661425i	$-0.769952\pi$
$911$	−45.2749	−1.50002	−0.750011	+	0.661425i	$0.769952\pi$
$912$	−68.9094	−2.28182
$913$	27.3864	0.906357
$914$	−0.215206	−0.00711837
$915$	0	0
$916$	−29.9214	−0.988632
$917$	−8.00364	−0.264303
$918$	1.43409	0.0473321
$919$	−5.90167	−0.194678	−0.0973391	−	0.995251i	$-0.531033\pi$
$919$	−5.90167	−0.194678	−0.0973391	+	0.995251i	$0.531033\pi$
$920$	0	0
$921$	49.6909	1.63737
$922$	2.28710	0.0753218
$923$	−7.94216	−0.261419
$924$	24.3825	0.802124
$925$	0	0
$926$	2.95566	0.0971289
$927$	18.4198	0.604985
$928$	−1.85068	−0.0607516
$929$	−6.00000	−0.196854	−0.0984268	−	0.995144i	$-0.531381\pi$
$929$	−6.00000	−0.196854	−0.0984268	+	0.995144i	$0.531381\pi$
$930$	0	0
$931$	−40.4850	−1.32684
$932$	1.41647	0.0463979
$933$	23.7650	0.778032
$934$	−4.69695	−0.153689
$935$	0	0
$936$	8.84469	0.289098
$937$	43.0755	1.40721	0.703607	−	0.710589i	$-0.251571\pi$
$937$	43.0755	1.40721	0.703607	+	0.710589i	$0.251571\pi$
$938$	−1.01299	−0.0330753
$939$	−37.2593	−1.21591
$940$	0	0
$941$	2.91637	0.0950711	0.0475355	−	0.998870i	$-0.484863\pi$
$941$	2.91637	0.0950711	0.0475355	+	0.998870i	$0.484863\pi$
$942$	−1.99976	−0.0651556
$943$	0	0
$944$	38.5312	1.25408
$945$	0	0
$946$	−0.259911	−0.00845043
$947$	0.556407	0.0180808	0.00904041	−	0.999959i	$-0.497122\pi$
$947$	0.556407	0.0180808	0.00904041	+	0.999959i	$0.497122\pi$
$948$	−28.1034	−0.912757
$949$	34.6367	1.12435
$950$	0	0
$951$	−57.4604	−1.86328
$952$	−4.24080	−0.137445
$953$	−37.1661	−1.20393	−0.601964	−	0.798523i	$-0.705615\pi$
$953$	−37.1661	−1.20393	−0.601964	+	0.798523i	$0.705615\pi$
$954$	6.32317	0.204720
$955$	0	0
$956$	3.95084	0.127779
$957$	−11.2861	−0.364828
$958$	−5.13700	−0.165969
$959$	10.7056	0.345703
$960$	0	0
$961$	9.98530	0.322106
$962$	−0.680906	−0.0219533
$963$	3.90806	0.125935
$964$	−34.2305	−1.10249
$965$	0	0
$966$	0.342932	0.0110337
$967$	−24.2754	−0.780643	−0.390322	−	0.920679i	$-0.627636\pi$
$967$	−24.2754	−0.780643	−0.390322	+	0.920679i	$0.627636\pi$
$968$	−5.22151	−0.167826
$969$	111.264	3.57431
$970$	0	0
$971$	−17.1709	−0.551039	−0.275520	−	0.961295i	$-0.588850\pi$
$971$	−17.1709	−0.551039	−0.275520	+	0.961295i	$0.588850\pi$
$972$	−43.9170	−1.40864
$973$	3.01275	0.0965842
$974$	2.21220	0.0708834
$975$	0	0
$976$	−4.41908	−0.141451
$977$	−4.95785	−0.158616	−0.0793078	−	0.996850i	$-0.525271\pi$
$977$	−4.95785	−0.158616	−0.0793078	+	0.996850i	$0.525271\pi$
$978$	−8.34564	−0.266864
$979$	47.5587	1.51998
$980$	0	0
$981$	54.7746	1.74882
$982$	0.644104	0.0205542
$983$	−11.5651	−0.368869	−0.184434	−	0.982845i	$-0.559045\pi$
$983$	−11.5651	−0.368869	−0.184434	+	0.982845i	$0.559045\pi$
$984$	0	0
$985$	0	0
$986$	0.975419	0.0310637
$987$	13.3220	0.424044
$988$	54.7190	1.74084
$989$	0.293769	0.00934132
$990$	0	0
$991$	−18.3429	−0.582682	−0.291341	−	0.956619i	$-0.594102\pi$
$991$	−18.3429	−0.582682	−0.291341	+	0.956619i	$0.594102\pi$
$992$	11.8480	0.376175
$993$	−20.2336	−0.642092
$994$	−0.342932	−0.0108771
$995$	0	0
$996$	−31.5096	−0.998419
$997$	32.2997	1.02294	0.511471	−	0.859300i	$-0.329101\pi$
$997$	32.2997	1.02294	0.511471	+	0.859300i	$0.329101\pi$
$998$	4.71435	0.149230
$999$	−1.60790	−0.0508719

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	725.2.a.l.1.4		6
3.2	odd	2		6525.2.a.bt.1.3		6
5.2	odd	4		145.2.b.c.59.4	yes	6
5.3	odd	4		145.2.b.c.59.3	✓	6
5.4	even	2	inner	725.2.a.l.1.3		6
15.2	even	4		1305.2.c.h.784.3		6
15.8	even	4		1305.2.c.h.784.4		6
15.14	odd	2		6525.2.a.bt.1.4		6
20.3	even	4		2320.2.d.g.929.5		6
20.7	even	4		2320.2.d.g.929.2		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
145.2.b.c.59.3	✓	6	5.3	odd	4
145.2.b.c.59.4	yes	6	5.2	odd	4
725.2.a.l.1.3		6	5.4	even	2	inner
725.2.a.l.1.4		6	1.1	even	1	trivial
1305.2.c.h.784.3		6	15.2	even	4
1305.2.c.h.784.4		6	15.8	even	4
2320.2.d.g.929.2		6	20.7	even	4
2320.2.d.g.929.5		6	20.3	even	4
6525.2.a.bt.1.3		6	3.2	odd	2
6525.2.a.bt.1.4		6	15.14	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 725 = 5^{2} \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	725.a (trivial)

\(f(q)\)	\(=\)	\(q+0.156785 q^{2} -2.56387 q^{3} -1.97542 q^{4} -0.401976 q^{6} -1.09364 q^{7} -0.623285 q^{8} +3.57344 q^{9} -4.40198 q^{11} +5.06472 q^{12} -3.97108 q^{13} -0.171466 q^{14} +3.85312 q^{16} -6.22138 q^{17} +0.560261 q^{18} +6.97542 q^{19} +2.80395 q^{21} -0.690163 q^{22} +0.780070 q^{23} +1.59802 q^{24} -0.622605 q^{26} -1.47023 q^{27} +2.16040 q^{28} -1.00000 q^{29} +6.40198 q^{31} +1.85068 q^{32} +11.2861 q^{33} -0.975419 q^{34} -7.05904 q^{36} +1.09364 q^{37} +1.09364 q^{38} +10.1813 q^{39} +0.439617 q^{42} +0.376593 q^{43} +8.69575 q^{44} +0.122303 q^{46} +4.75115 q^{47} -9.87890 q^{48} -5.80395 q^{49} +15.9508 q^{51} +7.84455 q^{52} +11.2861 q^{53} -0.230510 q^{54} +0.681649 q^{56} -17.8841 q^{57} -0.156785 q^{58} +10.0000 q^{59} -1.14688 q^{61} +1.00373 q^{62} -3.90806 q^{63} -7.41607 q^{64} +1.76949 q^{66} +5.90782 q^{67} +12.2898 q^{68} -2.00000 q^{69} +2.00000 q^{71} -2.22727 q^{72} -8.72223 q^{73} +0.171466 q^{74} -13.7794 q^{76} +4.81418 q^{77} +1.59628 q^{78} -5.54886 q^{79} -6.95084 q^{81} -6.22138 q^{83} -5.53898 q^{84} +0.0590441 q^{86} +2.56387 q^{87} +2.74369 q^{88} -10.8040 q^{89} +4.34293 q^{91} -1.54096 q^{92} -16.4139 q^{93} +0.744908 q^{94} -4.74491 q^{96} +17.4176 q^{97} -0.909972 q^{98} -15.7302 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 14 q^{4} + 14 q^{6} + 12 q^{9} - 10 q^{11} - 8 q^{14} + 42 q^{16} + 16 q^{19} - 16 q^{21} + 26 q^{24} - 46 q^{26} - 6 q^{29} + 22 q^{31} + 20 q^{34} - 12 q^{36} - 14 q^{39} - 2 q^{44} - 44 q^{46}+ \cdots + 16 q^{99}+O(q^{100}) \) 6 * q + 14 * q^4 + 14 * q^6 + 12 * q^9 - 10 * q^11 - 8 * q^14 + 42 * q^16 + 16 * q^19 - 16 * q^21 + 26 * q^24 - 46 * q^26 - 6 * q^29 + 22 * q^31 + 20 * q^34 - 12 * q^36 - 14 * q^39 - 2 * q^44 - 44 * q^46 - 2 * q^49 + 44 * q^51 + 22 * q^54 + 16 * q^56 + 60 * q^59 + 12 * q^61 + 38 * q^64 + 34 * q^66 - 12 * q^69 + 12 * q^71 + 8 * q^74 - 24 * q^76 + 2 * q^79 + 10 * q^81 - 80 * q^84 - 30 * q^86 - 32 * q^89 + 40 * q^91 + 2 * q^94 - 26 * q^96 + 16 * q^99

Introduction

L-functions

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Database

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