LMFDB - Embedded newform 1372.2.a.b.1.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1372,2,Mod(1,1372)] 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("1372.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1372, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1372 = 2^{2} \cdot 7^{3} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1372.a (trivial)

Newform invariants

comment:select newform

sage:traces = [6,0,0,0,0,0,0,0,2,0,-18] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(11)] == traces)

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

Self dual:	yes
Analytic conductor:	$10.9554751573$
Analytic rank:	$1$
Dimension:	$6$
Coefficient field:	6.6.1229312.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 10x^{4} + 24x^{2} - 8 $ x^6 - 10x^4 + 24x^2 - 8
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.6
Root			$2.54832$ of defining polynomial
Character	$\chi$	$=$	1372.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+2.54832 q^{3} -3.96254 q^{5} +3.49396 q^{9} -4.69202 q^{11} -0.0691796 q^{13} -10.0978 q^{15} +6.19997 q^{17} -3.02226 q^{19} -9.15883 q^{23} +10.7017 q^{25} +1.25877 q^{27} -6.93900 q^{29} -2.39288 q^{31} -11.9568 q^{33} -2.02177 q^{37} -0.176292 q^{39} -7.07107 q^{41} +0.137063 q^{43} -13.8449 q^{45} +5.91987 q^{47} +15.7995 q^{51} -4.75302 q^{53} +18.5923 q^{55} -7.70171 q^{57} -3.73791 q^{59} +0.224625 q^{61} +0.274127 q^{65} -5.82908 q^{67} -23.3397 q^{69} -6.85086 q^{71} -3.61326 q^{73} +27.2714 q^{75} -4.57673 q^{79} -7.27413 q^{81} +6.03692 q^{83} -24.5676 q^{85} -17.6828 q^{87} +7.45114 q^{89} -6.09783 q^{93} +11.9758 q^{95} +15.8331 q^{97} -16.3937 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 2 q^{9} - 18 q^{11} - 24 q^{15} - 38 q^{23} + 10 q^{25} - 22 q^{29} - 6 q^{37} - 16 q^{39} - 10 q^{43} + 4 q^{51} - 38 q^{53} + 8 q^{57} - 20 q^{65} - 14 q^{67} - 14 q^{71} - 22 q^{79} - 22 q^{81}+ \cdots - 34 q^{99}+O(q^{100}) $ 6 * q + 2 * q^9 - 18 * q^11 - 24 * q^15 - 38 * q^23 + 10 * q^25 - 22 * q^29 - 6 * q^37 - 16 * q^39 - 10 * q^43 + 4 * q^51 - 38 * q^53 + 8 * q^57 - 20 * q^65 - 14 * q^67 - 14 * q^71 - 22 * q^79 - 22 * q^81 - 16 * q^85 - 4 * q^95 - 34 * 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	0
$2$	0	0
$3$	2.54832	1.47128	0.735638	−	0.677375i	$-0.236882\pi$
$3$	2.54832	1.47128	0.735638	+	0.677375i	$0.236882\pi$
$4$	0	0
$5$	−3.96254	−1.77210	−0.886051	−	0.463589i	$-0.846562\pi$
$5$	−3.96254	−1.77210	−0.886051	+	0.463589i	$0.846562\pi$
$6$	0	0
$7$	0	0
$7$	0	0
$8$	0	0
$9$	3.49396	1.16465
$10$	0	0
$11$	−4.69202	−1.41470	−0.707349	−	0.706865i	$-0.750109\pi$
$11$	−4.69202	−1.41470	−0.707349	+	0.706865i	$0.750109\pi$
$12$	0	0
$13$	−0.0691796	−0.0191870	−0.00959348	−	0.999954i	$-0.503054\pi$
$13$	−0.0691796	−0.0191870	−0.00959348	+	0.999954i	$0.503054\pi$
$14$	0	0
$15$	−10.0978	−2.60725
$16$	0	0
$17$	6.19997	1.50371	0.751857	−	0.659326i	$-0.229158\pi$
$17$	6.19997	1.50371	0.751857	+	0.659326i	$0.229158\pi$
$18$	0	0
$19$	−3.02226	−0.693355	−0.346677	−	0.937984i	$-0.612690\pi$
$19$	−3.02226	−0.693355	−0.346677	+	0.937984i	$0.612690\pi$
$20$	0	0
$21$	0	0
$22$	0	0
$23$	−9.15883	−1.90975	−0.954874	−	0.297010i	$-0.904011\pi$
$23$	−9.15883	−1.90975	−0.954874	+	0.297010i	$0.904011\pi$
$24$	0	0
$25$	10.7017	2.14034
$26$	0	0
$27$	1.25877	0.242250
$28$	0	0
$29$	−6.93900	−1.28854	−0.644270	−	0.764798i	$-0.722839\pi$
$29$	−6.93900	−1.28854	−0.644270	+	0.764798i	$0.722839\pi$
$30$	0	0
$31$	−2.39288	−0.429774	−0.214887	−	0.976639i	$-0.568938\pi$
$31$	−2.39288	−0.429774	−0.214887	+	0.976639i	$0.568938\pi$
$32$	0	0
$33$	−11.9568	−2.08141
$34$	0	0
$35$	0	0
$36$	0	0
$37$	−2.02177	−0.332377	−0.166188	−	0.986094i	$-0.553146\pi$
$37$	−2.02177	−0.332377	−0.166188	+	0.986094i	$0.553146\pi$
$38$	0	0
$39$	−0.176292	−0.0282293
$40$	0	0
$41$	−7.07107	−1.10432	−0.552158	−	0.833740i	$-0.686195\pi$
$41$	−7.07107	−1.10432	−0.552158	+	0.833740i	$0.686195\pi$
$42$	0	0
$43$	0.137063	0.0209020	0.0104510	−	0.999945i	$-0.496673\pi$
$43$	0.137063	0.0209020	0.0104510	+	0.999945i	$0.496673\pi$
$44$	0	0
$45$	−13.8449	−2.06388
$46$	0	0
$47$	5.91987	0.863502	0.431751	−	0.901993i	$-0.357896\pi$
$47$	5.91987	0.863502	0.431751	+	0.901993i	$0.357896\pi$
$48$	0	0
$49$	0	0
$50$	0	0
$51$	15.7995	2.21238
$52$	0	0
$53$	−4.75302	−0.652878	−0.326439	−	0.945218i	$-0.605849\pi$
$53$	−4.75302	−0.652878	−0.326439	+	0.945218i	$0.605849\pi$
$54$	0	0
$55$	18.5923	2.50699
$56$	0	0
$57$	−7.70171	−1.02012
$58$	0	0
$59$	−3.73791	−0.486635	−0.243317	−	0.969947i	$-0.578236\pi$
$59$	−3.73791	−0.486635	−0.243317	+	0.969947i	$0.578236\pi$
$60$	0	0
$61$	0.224625	0.0287602	0.0143801	−	0.999897i	$-0.495423\pi$
$61$	0.224625	0.0287602	0.0143801	+	0.999897i	$0.495423\pi$
$62$	0	0
$63$	0	0
$64$	0	0
$65$	0.274127	0.0340012
$66$	0	0
$67$	−5.82908	−0.712136	−0.356068	−	0.934460i	$-0.615883\pi$
$67$	−5.82908	−0.712136	−0.356068	+	0.934460i	$0.615883\pi$
$68$	0	0
$69$	−23.3397	−2.80977
$70$	0	0
$71$	−6.85086	−0.813047	−0.406523	−	0.913640i	$-0.633259\pi$
$71$	−6.85086	−0.813047	−0.406523	+	0.913640i	$0.633259\pi$
$72$	0	0
$73$	−3.61326	−0.422900	−0.211450	−	0.977389i	$-0.567819\pi$
$73$	−3.61326	−0.422900	−0.211450	+	0.977389i	$0.567819\pi$
$74$	0	0
$75$	27.2714	3.14903
$76$	0	0
$77$	0	0
$78$	0	0
$79$	−4.57673	−0.514922	−0.257461	−	0.966289i	$-0.582886\pi$
$79$	−4.57673	−0.514922	−0.257461	+	0.966289i	$0.582886\pi$
$80$	0	0
$81$	−7.27413	−0.808236
$82$	0	0
$83$	6.03692	0.662638	0.331319	−	0.943519i	$-0.392506\pi$
$83$	6.03692	0.662638	0.331319	+	0.943519i	$0.392506\pi$
$84$	0	0
$85$	−24.5676	−2.66473
$86$	0	0
$87$	−17.6828	−1.89580
$88$	0	0
$89$	7.45114	0.789819	0.394909	−	0.918720i	$-0.370776\pi$
$89$	7.45114	0.789819	0.394909	+	0.918720i	$0.370776\pi$
$90$	0	0
$91$	0	0
$92$	0	0
$93$	−6.09783	−0.632316
$94$	0	0
$95$	11.9758	1.22869
$96$	0	0
$97$	15.8331	1.60760	0.803802	−	0.594897i	$-0.202807\pi$
$97$	15.8331	1.60760	0.803802	+	0.594897i	$0.202807\pi$
$98$	0	0
$99$	−16.3937	−1.64763
$100$	0	0
$101$	13.9867	1.39173	0.695864	−	0.718174i	$-0.255022\pi$
$101$	13.9867	1.39173	0.695864	+	0.718174i	$0.255022\pi$
$102$	0	0
$103$	5.23839	0.516154	0.258077	−	0.966124i	$-0.416911\pi$
$103$	5.23839	0.516154	0.258077	+	0.966124i	$0.416911\pi$
$104$	0	0
$105$	0	0
$106$	0	0
$107$	−5.73125	−0.554061	−0.277030	−	0.960861i	$-0.589350\pi$
$107$	−5.73125	−0.554061	−0.277030	+	0.960861i	$0.589350\pi$
$108$	0	0
$109$	5.78448	0.554053	0.277026	−	0.960862i	$-0.410651\pi$
$109$	5.78448	0.554053	0.277026	+	0.960862i	$0.410651\pi$
$110$	0	0
$111$	−5.15213	−0.489018
$112$	0	0
$113$	12.2174	1.14932	0.574660	−	0.818392i	$-0.305134\pi$
$113$	12.2174	1.14932	0.574660	+	0.818392i	$0.305134\pi$
$114$	0	0
$115$	36.2922	3.38427
$116$	0	0
$117$	−0.241711	−0.0223462
$118$	0	0
$119$	0	0
$120$	0	0
$121$	11.0151	1.00137
$122$	0	0
$123$	−18.0194	−1.62475
$124$	0	0
$125$	−22.5932	−2.02080
$126$	0	0
$127$	16.7657	1.48771	0.743857	−	0.668338i	$-0.232994\pi$
$127$	16.7657	1.48771	0.743857	+	0.668338i	$0.232994\pi$
$128$	0	0
$129$	0.349282	0.0307526
$130$	0	0
$131$	−12.4170	−1.08488	−0.542441	−	0.840094i	$-0.682500\pi$
$131$	−12.4170	−1.08488	−0.542441	+	0.840094i	$0.682500\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.98792	−0.429292
$136$	0	0
$137$	12.7627	1.09039	0.545196	−	0.838309i	$-0.316455\pi$
$137$	12.7627	1.09039	0.545196	+	0.838309i	$0.316455\pi$
$138$	0	0
$139$	12.0431	1.02148	0.510740	−	0.859736i	$-0.329372\pi$
$139$	12.0431	1.02148	0.510740	+	0.859736i	$0.329372\pi$
$140$	0	0
$141$	15.0858	1.27045
$142$	0	0
$143$	0.324592	0.0271437
$144$	0	0
$145$	27.4961	2.28342
$146$	0	0
$147$	0	0
$148$	0	0
$149$	−16.9855	−1.39151	−0.695754	−	0.718280i	$-0.744930\pi$
$149$	−16.9855	−1.39151	−0.695754	+	0.718280i	$0.744930\pi$
$150$	0	0
$151$	−11.7235	−0.954043	−0.477022	−	0.878892i	$-0.658284\pi$
$151$	−11.7235	−0.954043	−0.477022	+	0.878892i	$0.658284\pi$
$152$	0	0
$153$	21.6625	1.75131
$154$	0	0
$155$	9.48188	0.761603
$156$	0	0
$157$	11.6459	0.929444	0.464722	−	0.885457i	$-0.346154\pi$
$157$	11.6459	0.929444	0.464722	+	0.885457i	$0.346154\pi$
$158$	0	0
$159$	−12.1122	−0.960563
$160$	0	0
$161$	0	0
$162$	0	0
$163$	15.9801	1.25166	0.625831	−	0.779959i	$-0.284760\pi$
$163$	15.9801	1.25166	0.625831	+	0.779959i	$0.284760\pi$
$164$	0	0
$165$	47.3793	3.68847
$166$	0	0
$167$	−14.4496	−1.11815	−0.559073	−	0.829118i	$-0.688843\pi$
$167$	−14.4496	−1.11815	−0.559073	+	0.829118i	$0.688843\pi$
$168$	0	0
$169$	−12.9952	−0.999632
$170$	0	0
$171$	−10.5597	−0.807518
$172$	0	0
$173$	9.37768	0.712972	0.356486	−	0.934301i	$-0.383975\pi$
$173$	9.37768	0.712972	0.356486	+	0.934301i	$0.383975\pi$
$174$	0	0
$175$	0	0
$176$	0	0
$177$	−9.52542	−0.715974
$178$	0	0
$179$	−13.2470	−0.990126	−0.495063	−	0.868857i	$-0.664855\pi$
$179$	−13.2470	−0.990126	−0.495063	+	0.868857i	$0.664855\pi$
$180$	0	0
$181$	−12.5478	−0.932670	−0.466335	−	0.884608i	$-0.654426\pi$
$181$	−12.5478	−0.932670	−0.466335	+	0.884608i	$0.654426\pi$
$182$	0	0
$183$	0.572417	0.0423142
$184$	0	0
$185$	8.01134	0.589006
$186$	0	0
$187$	−29.0904	−2.12730
$188$	0	0
$189$	0	0
$190$	0	0
$191$	19.7235	1.42714	0.713570	−	0.700583i	$-0.247077\pi$
$191$	19.7235	1.42714	0.713570	+	0.700583i	$0.247077\pi$
$192$	0	0
$193$	−0.860544	−0.0619433	−0.0309716	−	0.999520i	$-0.509860\pi$
$193$	−0.860544	−0.0619433	−0.0309716	+	0.999520i	$0.509860\pi$
$194$	0	0
$195$	0.698564	0.0500252
$196$	0	0
$197$	−18.3773	−1.30933	−0.654666	−	0.755919i	$-0.727191\pi$
$197$	−18.3773	−1.30933	−0.654666	+	0.755919i	$0.727191\pi$
$198$	0	0
$199$	7.65258	0.542477	0.271238	−	0.962512i	$-0.412567\pi$
$199$	7.65258	0.542477	0.271238	+	0.962512i	$0.412567\pi$
$200$	0	0
$201$	−14.8544	−1.04775
$202$	0	0
$203$	0	0
$204$	0	0
$205$	28.0194	1.95696
$206$	0	0
$207$	−32.0006	−2.22419
$208$	0	0
$209$	14.1805	0.980888
$210$	0	0
$211$	0.323044	0.0222393	0.0111196	−	0.999938i	$-0.496460\pi$
$211$	0.323044	0.0222393	0.0111196	+	0.999938i	$0.496460\pi$
$212$	0	0
$213$	−17.4582	−1.19622
$214$	0	0
$215$	−0.543119	−0.0370404
$216$	0	0
$217$	0	0
$218$	0	0
$219$	−9.20775	−0.622202
$220$	0	0
$221$	−0.428911	−0.0288517
$222$	0	0
$223$	8.66203	0.580053	0.290026	−	0.957019i	$-0.406336\pi$
$223$	8.66203	0.580053	0.290026	+	0.957019i	$0.406336\pi$
$224$	0	0
$225$	37.3913	2.49276
$226$	0	0
$227$	8.10521	0.537962	0.268981	−	0.963146i	$-0.413313\pi$
$227$	8.10521	0.537962	0.268981	+	0.963146i	$0.413313\pi$
$228$	0	0
$229$	−17.1226	−1.13149	−0.565747	−	0.824579i	$-0.691412\pi$
$229$	−17.1226	−1.13149	−0.565747	+	0.824579i	$0.691412\pi$
$230$	0	0
$231$	0	0
$232$	0	0
$233$	−9.93661	−0.650969	−0.325484	−	0.945547i	$-0.605527\pi$
$233$	−9.93661	−0.650969	−0.325484	+	0.945547i	$0.605527\pi$
$234$	0	0
$235$	−23.4577	−1.53021
$236$	0	0
$237$	−11.6630	−0.757593
$238$	0	0
$239$	12.9584	0.838208	0.419104	−	0.907938i	$-0.362344\pi$
$239$	12.9584	0.838208	0.419104	+	0.907938i	$0.362344\pi$
$240$	0	0
$241$	−23.2534	−1.49788	−0.748942	−	0.662635i	$-0.769438\pi$
$241$	−23.2534	−1.49788	−0.748942	+	0.662635i	$0.769438\pi$
$242$	0	0
$243$	−22.3131	−1.43139
$244$	0	0
$245$	0	0
$246$	0	0
$247$	0.209079	0.0133034
$248$	0	0
$249$	15.3840	0.974924
$250$	0	0
$251$	−19.2833	−1.21715	−0.608575	−	0.793497i	$-0.708258\pi$
$251$	−19.2833	−1.21715	−0.608575	+	0.793497i	$0.708258\pi$
$252$	0	0
$253$	42.9734	2.70172
$254$	0	0
$255$	−62.6063	−3.92056
$256$	0	0
$257$	−25.6155	−1.59785	−0.798926	−	0.601430i	$-0.794598\pi$
$257$	−25.6155	−1.59785	−0.798926	+	0.601430i	$0.794598\pi$
$258$	0	0
$259$	0	0
$260$	0	0
$261$	−24.2446	−1.50070
$262$	0	0
$263$	−8.61356	−0.531135	−0.265568	−	0.964092i	$-0.585559\pi$
$263$	−8.61356	−0.531135	−0.265568	+	0.964092i	$0.585559\pi$
$264$	0	0
$265$	18.8340	1.15696
$266$	0	0
$267$	18.9879	1.16204
$268$	0	0
$269$	−1.45261	−0.0885669	−0.0442835	−	0.999019i	$-0.514100\pi$
$269$	−1.45261	−0.0885669	−0.0442835	+	0.999019i	$0.514100\pi$
$270$	0	0
$271$	−27.2330	−1.65429	−0.827145	−	0.561989i	$-0.810036\pi$
$271$	−27.2330	−1.65429	−0.827145	+	0.561989i	$0.810036\pi$
$272$	0	0
$273$	0	0
$274$	0	0
$275$	−50.2127	−3.02794
$276$	0	0
$277$	−6.11960	−0.367691	−0.183846	−	0.982955i	$-0.558855\pi$
$277$	−6.11960	−0.367691	−0.183846	+	0.982955i	$0.558855\pi$
$278$	0	0
$279$	−8.36062	−0.500537
$280$	0	0
$281$	12.7356	0.759740	0.379870	−	0.925040i	$-0.375969\pi$
$281$	12.7356	0.759740	0.379870	+	0.925040i	$0.375969\pi$
$282$	0	0
$283$	18.1397	1.07829	0.539146	−	0.842212i	$-0.318747\pi$
$283$	18.1397	1.07829	0.539146	+	0.842212i	$0.318747\pi$
$284$	0	0
$285$	30.5183	1.80775
$286$	0	0
$287$	0	0
$288$	0	0
$289$	21.4397	1.26116
$290$	0	0
$291$	40.3478	2.36523
$292$	0	0
$293$	28.9103	1.68896	0.844478	−	0.535590i	$-0.179911\pi$
$293$	28.9103	1.68896	0.844478	+	0.535590i	$0.179911\pi$
$294$	0	0
$295$	14.8116	0.862366
$296$	0	0
$297$	−5.90617	−0.342711
$298$	0	0
$299$	0.633604	0.0366423
$300$	0	0
$301$	0	0
$302$	0	0
$303$	35.6426	2.04762
$304$	0	0
$305$	−0.890084	−0.0509660
$306$	0	0
$307$	−25.2799	−1.44280	−0.721401	−	0.692518i	$-0.756501\pi$
$307$	−25.2799	−1.44280	−0.721401	+	0.692518i	$0.756501\pi$
$308$	0	0
$309$	13.3491	0.759405
$310$	0	0
$311$	−4.39048	−0.248961	−0.124481	−	0.992222i	$-0.539726\pi$
$311$	−4.39048	−0.248961	−0.124481	+	0.992222i	$0.539726\pi$
$312$	0	0
$313$	23.7889	1.34463	0.672315	−	0.740265i	$-0.265300\pi$
$313$	23.7889	1.34463	0.672315	+	0.740265i	$0.265300\pi$
$314$	0	0
$315$	0	0
$316$	0	0
$317$	−26.8092	−1.50576	−0.752878	−	0.658160i	$-0.771335\pi$
$317$	−26.8092	−1.50576	−0.752878	+	0.658160i	$0.771335\pi$
$318$	0	0
$319$	32.5579	1.82289
$320$	0	0
$321$	−14.6051	−0.815176
$322$	0	0
$323$	−18.7380	−1.04261
$324$	0	0
$325$	−0.740340	−0.0410667
$326$	0	0
$327$	14.7407	0.815164
$328$	0	0
$329$	0	0
$330$	0	0
$331$	−19.5066	−1.07218	−0.536091	−	0.844160i	$-0.680100\pi$
$331$	−19.5066	−1.07218	−0.536091	+	0.844160i	$0.680100\pi$
$332$	0	0
$333$	−7.06398	−0.387104
$334$	0	0
$335$	23.0980	1.26198
$336$	0	0
$337$	8.92261	0.486045	0.243023	−	0.970021i	$-0.421861\pi$
$337$	8.92261	0.486045	0.243023	+	0.970021i	$0.421861\pi$
$338$	0	0
$339$	31.1340	1.69097
$340$	0	0
$341$	11.2274	0.608000
$342$	0	0
$343$	0	0
$344$	0	0
$345$	92.4844	4.97919
$346$	0	0
$347$	2.50471	0.134460	0.0672300	−	0.997738i	$-0.478584\pi$
$347$	2.50471	0.134460	0.0672300	+	0.997738i	$0.478584\pi$
$348$	0	0
$349$	−12.4341	−0.665583	−0.332792	−	0.943000i	$-0.607991\pi$
$349$	−12.4341	−0.665583	−0.332792	+	0.943000i	$0.607991\pi$
$350$	0	0
$351$	−0.0870811	−0.00464804
$352$	0	0
$353$	13.0525	0.694715	0.347358	−	0.937733i	$-0.387079\pi$
$353$	13.0525	0.694715	0.347358	+	0.937733i	$0.387079\pi$
$354$	0	0
$355$	27.1468	1.44080
$356$	0	0
$357$	0	0
$358$	0	0
$359$	0.0978347	0.00516352	0.00258176	−	0.999997i	$-0.499178\pi$
$359$	0.0978347	0.00516352	0.00258176	+	0.999997i	$0.499178\pi$
$360$	0	0
$361$	−9.86592	−0.519259
$362$	0	0
$363$	28.0700	1.47329
$364$	0	0
$365$	14.3177	0.749421
$366$	0	0
$367$	−11.4734	−0.598905	−0.299453	−	0.954111i	$-0.596804\pi$
$367$	−11.4734	−0.598905	−0.299453	+	0.954111i	$0.596804\pi$
$368$	0	0
$369$	−24.7060	−1.28614
$370$	0	0
$371$	0	0
$372$	0	0
$373$	−11.8726	−0.614741	−0.307371	−	0.951590i	$-0.599449\pi$
$373$	−11.8726	−0.614741	−0.307371	+	0.951590i	$0.599449\pi$
$374$	0	0
$375$	−57.5749	−2.97316
$376$	0	0
$377$	0.480037	0.0247232
$378$	0	0
$379$	−6.07367	−0.311984	−0.155992	−	0.987758i	$-0.549857\pi$
$379$	−6.07367	−0.311984	−0.155992	+	0.987758i	$0.549857\pi$
$380$	0	0
$381$	42.7244	2.18884
$382$	0	0
$383$	−26.5311	−1.35568	−0.677838	−	0.735211i	$-0.737083\pi$
$383$	−26.5311	−1.35568	−0.677838	+	0.735211i	$0.737083\pi$
$384$	0	0
$385$	0	0
$386$	0	0
$387$	0.478894	0.0243435
$388$	0	0
$389$	27.5512	1.39690	0.698452	−	0.715657i	$-0.253873\pi$
$389$	27.5512	1.39690	0.698452	+	0.715657i	$0.253873\pi$
$390$	0	0
$391$	−56.7845	−2.87172
$392$	0	0
$393$	−31.6426	−1.59616
$394$	0	0
$395$	18.1355	0.912494
$396$	0	0
$397$	9.87481	0.495602	0.247801	−	0.968811i	$-0.420292\pi$
$397$	9.87481	0.495602	0.247801	+	0.968811i	$0.420292\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−18.9594	−0.946789	−0.473395	−	0.880850i	$-0.656972\pi$
$401$	−18.9594	−0.946789	−0.473395	+	0.880850i	$0.656972\pi$
$402$	0	0
$403$	0.165538	0.00824605
$404$	0	0
$405$	28.8240	1.43228
$406$	0	0
$407$	9.48619	0.470213
$408$	0	0
$409$	−24.5643	−1.21463	−0.607313	−	0.794463i	$-0.707752\pi$
$409$	−24.5643	−1.21463	−0.607313	+	0.794463i	$0.707752\pi$
$410$	0	0
$411$	32.5235	1.60427
$412$	0	0
$413$	0	0
$414$	0	0
$415$	−23.9215	−1.17426
$416$	0	0
$417$	30.6896	1.50288
$418$	0	0
$419$	4.05490	0.198095	0.0990475	−	0.995083i	$-0.468420\pi$
$419$	4.05490	0.198095	0.0990475	+	0.995083i	$0.468420\pi$
$420$	0	0
$421$	2.10215	0.102452	0.0512262	−	0.998687i	$-0.483687\pi$
$421$	2.10215	0.102452	0.0512262	+	0.998687i	$0.483687\pi$
$422$	0	0
$423$	20.6838	1.00568
$424$	0	0
$425$	66.3503	3.21846
$426$	0	0
$427$	0	0
$428$	0	0
$429$	0.827166	0.0399359
$430$	0	0
$431$	−5.50173	−0.265009	−0.132504	−	0.991182i	$-0.542302\pi$
$431$	−5.50173	−0.265009	−0.132504	+	0.991182i	$0.542302\pi$
$432$	0	0
$433$	26.8914	1.29232	0.646158	−	0.763203i	$-0.276375\pi$
$433$	26.8914	1.29232	0.646158	+	0.763203i	$0.276375\pi$
$434$	0	0
$435$	70.0689	3.35955
$436$	0	0
$437$	27.6804	1.32413
$438$	0	0
$439$	13.1601	0.628097	0.314048	−	0.949407i	$-0.398315\pi$
$439$	13.1601	0.628097	0.314048	+	0.949407i	$0.398315\pi$
$440$	0	0
$441$	0	0
$442$	0	0
$443$	−4.01075	−0.190557	−0.0952783	−	0.995451i	$-0.530374\pi$
$443$	−4.01075	−0.190557	−0.0952783	+	0.995451i	$0.530374\pi$
$444$	0	0
$445$	−29.5254	−1.39964
$446$	0	0
$447$	−43.2846	−2.04729
$448$	0	0
$449$	17.0164	0.803053	0.401527	−	0.915847i	$-0.368480\pi$
$449$	17.0164	0.803053	0.401527	+	0.915847i	$0.368480\pi$
$450$	0	0
$451$	33.1776	1.56227
$452$	0	0
$453$	−29.8752	−1.40366
$454$	0	0
$455$	0	0
$456$	0	0
$457$	13.0911	0.612377	0.306189	−	0.951971i	$-0.400946\pi$
$457$	13.0911	0.612377	0.306189	+	0.951971i	$0.400946\pi$
$458$	0	0
$459$	7.80433	0.364275
$460$	0	0
$461$	25.6695	1.19555	0.597773	−	0.801665i	$-0.296052\pi$
$461$	25.6695	1.19555	0.597773	+	0.801665i	$0.296052\pi$
$462$	0	0
$463$	−7.00969	−0.325768	−0.162884	−	0.986645i	$-0.552080\pi$
$463$	−7.00969	−0.325768	−0.162884	+	0.986645i	$0.552080\pi$
$464$	0	0
$465$	24.1629	1.12053
$466$	0	0
$467$	11.0815	0.512790	0.256395	−	0.966572i	$-0.417465\pi$
$467$	11.0815	0.512790	0.256395	+	0.966572i	$0.417465\pi$
$468$	0	0
$469$	0	0
$470$	0	0
$471$	29.6775	1.36747
$472$	0	0
$473$	−0.643104	−0.0295700
$474$	0	0
$475$	−32.3434	−1.48402
$476$	0	0
$477$	−16.6069	−0.760376
$478$	0	0
$479$	−39.3145	−1.79633	−0.898163	−	0.439664i	$-0.855098\pi$
$479$	−39.3145	−1.79633	−0.898163	+	0.439664i	$0.855098\pi$
$480$	0	0
$481$	0.139865	0.00637730
$482$	0	0
$483$	0	0
$484$	0	0
$485$	−62.7391	−2.84884
$486$	0	0
$487$	−8.85517	−0.401266	−0.200633	−	0.979666i	$-0.564300\pi$
$487$	−8.85517	−0.401266	−0.200633	+	0.979666i	$0.564300\pi$
$488$	0	0
$489$	40.7226	1.84154
$490$	0	0
$491$	12.2000	0.550577	0.275289	−	0.961362i	$-0.411227\pi$
$491$	12.2000	0.550577	0.275289	+	0.961362i	$0.411227\pi$
$492$	0	0
$493$	−43.0216	−1.93760
$494$	0	0
$495$	64.9608	2.91977
$496$	0	0
$497$	0	0
$498$	0	0
$499$	32.3260	1.44711	0.723556	−	0.690266i	$-0.242506\pi$
$499$	32.3260	1.44711	0.723556	+	0.690266i	$0.242506\pi$
$500$	0	0
$501$	−36.8224	−1.64510
$502$	0	0
$503$	20.1662	0.899166	0.449583	−	0.893239i	$-0.351573\pi$
$503$	20.1662	0.899166	0.449583	+	0.893239i	$0.351573\pi$
$504$	0	0
$505$	−55.4228	−2.46628
$506$	0	0
$507$	−33.1160	−1.47073
$508$	0	0
$509$	−9.24542	−0.409796	−0.204898	−	0.978783i	$-0.565686\pi$
$509$	−9.24542	−0.409796	−0.204898	+	0.978783i	$0.565686\pi$
$510$	0	0
$511$	0	0
$512$	0	0
$513$	−3.80433	−0.167965
$514$	0	0
$515$	−20.7573	−0.914677
$516$	0	0
$517$	−27.7762	−1.22159
$518$	0	0
$519$	23.8974	1.04898
$520$	0	0
$521$	−9.98576	−0.437484	−0.218742	−	0.975783i	$-0.570195\pi$
$521$	−9.98576	−0.437484	−0.218742	+	0.975783i	$0.570195\pi$
$522$	0	0
$523$	13.4231	0.586951	0.293476	−	0.955967i	$-0.405188\pi$
$523$	13.4231	0.586951	0.293476	+	0.955967i	$0.405188\pi$
$524$	0	0
$525$	0	0
$526$	0	0
$527$	−14.8358	−0.646257
$528$	0	0
$529$	60.8842	2.64714
$530$	0	0
$531$	−13.0601	−0.566761
$532$	0	0
$533$	0.489173	0.0211885
$534$	0	0
$535$	22.7103	0.981852
$536$	0	0
$537$	−33.7576	−1.45675
$538$	0	0
$539$	0	0
$540$	0	0
$541$	−17.9511	−0.771777	−0.385889	−	0.922545i	$-0.626105\pi$
$541$	−17.9511	−0.771777	−0.385889	+	0.922545i	$0.626105\pi$
$542$	0	0
$543$	−31.9758	−1.37221
$544$	0	0
$545$	−22.9212	−0.981837
$546$	0	0
$547$	−34.0605	−1.45632	−0.728161	−	0.685406i	$-0.759625\pi$
$547$	−34.0605	−1.45632	−0.728161	+	0.685406i	$0.759625\pi$
$548$	0	0
$549$	0.784829	0.0334957
$550$	0	0
$551$	20.9715	0.893416
$552$	0	0
$553$	0	0
$554$	0	0
$555$	20.4155	0.866590
$556$	0	0
$557$	36.3279	1.53926	0.769632	−	0.638487i	$-0.220440\pi$
$557$	36.3279	1.53926	0.769632	+	0.638487i	$0.220440\pi$
$558$	0	0
$559$	−0.00948198	−0.000401045 0
$560$	0	0
$561$	−74.1318	−3.12985
$562$	0	0
$563$	2.96679	0.125035	0.0625176	−	0.998044i	$-0.480087\pi$
$563$	2.96679	0.125035	0.0625176	+	0.998044i	$0.480087\pi$
$564$	0	0
$565$	−48.4121	−2.03671
$566$	0	0
$567$	0	0
$568$	0	0
$569$	−32.0877	−1.34519	−0.672593	−	0.740013i	$-0.734819\pi$
$569$	−32.0877	−1.34519	−0.672593	+	0.740013i	$0.734819\pi$
$570$	0	0
$571$	3.73019	0.156103	0.0780517	−	0.996949i	$-0.475130\pi$
$571$	3.73019	0.156103	0.0780517	+	0.996949i	$0.475130\pi$
$572$	0	0
$573$	50.2618	2.09972
$574$	0	0
$575$	−98.0152	−4.08752
$576$	0	0
$577$	3.63034	0.151133	0.0755666	−	0.997141i	$-0.475923\pi$
$577$	3.63034	0.151133	0.0755666	+	0.997141i	$0.475923\pi$
$578$	0	0
$579$	−2.19294	−0.0911357
$580$	0	0
$581$	0	0
$582$	0	0
$583$	22.3013	0.923624
$584$	0	0
$585$	0.957787	0.0395996
$586$	0	0
$587$	−21.3440	−0.880959	−0.440480	−	0.897763i	$-0.645192\pi$
$587$	−21.3440	−0.880959	−0.440480	+	0.897763i	$0.645192\pi$
$588$	0	0
$589$	7.23191	0.297986
$590$	0	0
$591$	−46.8314	−1.92639
$592$	0	0
$593$	−46.5007	−1.90956	−0.954778	−	0.297319i	$-0.903908\pi$
$593$	−46.5007	−1.90956	−0.954778	+	0.297319i	$0.903908\pi$
$594$	0	0
$595$	0	0
$596$	0	0
$597$	19.5013	0.798133
$598$	0	0
$599$	9.57971	0.391416	0.195708	−	0.980662i	$-0.437299\pi$
$599$	9.57971	0.391416	0.195708	+	0.980662i	$0.437299\pi$
$600$	0	0
$601$	−42.2539	−1.72357	−0.861786	−	0.507272i	$-0.830654\pi$
$601$	−42.2539	−1.72357	−0.861786	+	0.507272i	$0.830654\pi$
$602$	0	0
$603$	−20.3666	−0.829391
$604$	0	0
$605$	−43.6476	−1.77453
$606$	0	0
$607$	−25.3510	−1.02896	−0.514482	−	0.857501i	$-0.672016\pi$
$607$	−25.3510	−1.02896	−0.514482	+	0.857501i	$0.672016\pi$
$608$	0	0
$609$	0	0
$610$	0	0
$611$	−0.409534	−0.0165680
$612$	0	0
$613$	12.3696	0.499602	0.249801	−	0.968297i	$-0.419635\pi$
$613$	12.3696	0.499602	0.249801	+	0.968297i	$0.419635\pi$
$614$	0	0
$615$	71.4025	2.87923
$616$	0	0
$617$	48.9004	1.96865	0.984327	−	0.176352i	$-0.0564295\pi$
$617$	48.9004	1.96865	0.984327	+	0.176352i	$0.0564295\pi$
$618$	0	0
$619$	−15.9117	−0.639546	−0.319773	−	0.947494i	$-0.603607\pi$
$619$	−15.9117	−0.639546	−0.319773	+	0.947494i	$0.603607\pi$
$620$	0	0
$621$	−11.5289	−0.462637
$622$	0	0
$623$	0	0
$624$	0	0
$625$	36.0180	1.44072
$626$	0	0
$627$	36.1366	1.44316
$628$	0	0
$629$	−12.5349	−0.499800
$630$	0	0
$631$	8.18167	0.325707	0.162854	−	0.986650i	$-0.447930\pi$
$631$	8.18167	0.325707	0.162854	+	0.986650i	$0.447930\pi$
$632$	0	0
$633$	0.823221	0.0327201
$634$	0	0
$635$	−66.4347	−2.63638
$636$	0	0
$637$	0	0
$638$	0	0
$639$	−23.9366	−0.946918
$640$	0	0
$641$	−27.9463	−1.10381	−0.551906	−	0.833906i	$-0.686099\pi$
$641$	−27.9463	−1.10381	−0.551906	+	0.833906i	$0.686099\pi$
$642$	0	0
$643$	15.6886	0.618698	0.309349	−	0.950949i	$-0.399889\pi$
$643$	15.6886	0.618698	0.309349	+	0.950949i	$0.399889\pi$
$644$	0	0
$645$	−1.38404	−0.0544966
$646$	0	0
$647$	−16.0885	−0.632503	−0.316252	−	0.948675i	$-0.602424\pi$
$647$	−16.0885	−0.632503	−0.316252	+	0.948675i	$0.602424\pi$
$648$	0	0
$649$	17.5384	0.688441
$650$	0	0
$651$	0	0
$652$	0	0
$653$	17.8950	0.700285	0.350142	−	0.936696i	$-0.386133\pi$
$653$	17.8950	0.700285	0.350142	+	0.936696i	$0.386133\pi$
$654$	0	0
$655$	49.2030	1.92252
$656$	0	0
$657$	−12.6246	−0.492531
$658$	0	0
$659$	3.62266	0.141119	0.0705594	−	0.997508i	$-0.477522\pi$
$659$	3.62266	0.141119	0.0705594	+	0.997508i	$0.477522\pi$
$660$	0	0
$661$	−38.4187	−1.49431	−0.747157	−	0.664647i	$-0.768582\pi$
$661$	−38.4187	−1.49431	−0.747157	+	0.664647i	$0.768582\pi$
$662$	0	0
$663$	−1.09301	−0.0424488
$664$	0	0
$665$	0	0
$666$	0	0
$667$	63.5532	2.46079
$668$	0	0
$669$	22.0737	0.853417
$670$	0	0
$671$	−1.05394	−0.0406870
$672$	0	0
$673$	19.3448	0.745688	0.372844	−	0.927894i	$-0.378383\pi$
$673$	19.3448	0.745688	0.372844	+	0.927894i	$0.378383\pi$
$674$	0	0
$675$	13.4710	0.518498
$676$	0	0
$677$	27.3193	1.04997	0.524983	−	0.851113i	$-0.324072\pi$
$677$	27.3193	1.04997	0.524983	+	0.851113i	$0.324072\pi$
$678$	0	0
$679$	0	0
$680$	0	0
$681$	20.6547	0.791490
$682$	0	0
$683$	−32.1002	−1.22828	−0.614141	−	0.789197i	$-0.710497\pi$
$683$	−32.1002	−1.22828	−0.614141	+	0.789197i	$0.710497\pi$
$684$	0	0
$685$	−50.5727	−1.93228
$686$	0	0
$687$	−43.6340	−1.66474
$688$	0	0
$689$	0.328812	0.0125267
$690$	0	0
$691$	0.249315	0.00948437	0.00474219	−	0.999989i	$-0.498491\pi$
$691$	0.249315	0.00948437	0.00474219	+	0.999989i	$0.498491\pi$
$692$	0	0
$693$	0	0
$694$	0	0
$695$	−47.7211	−1.81016
$696$	0	0
$697$	−43.8404	−1.66057
$698$	0	0
$699$	−25.3217	−0.957755
$700$	0	0
$701$	17.4989	0.660923	0.330461	−	0.943819i	$-0.392796\pi$
$701$	17.4989	0.660923	0.330461	+	0.943819i	$0.392796\pi$
$702$	0	0
$703$	6.11032	0.230455
$704$	0	0
$705$	−59.7779	−2.25137
$706$	0	0
$707$	0	0
$708$	0	0
$709$	−7.65173	−0.287367	−0.143683	−	0.989624i	$-0.545895\pi$
$709$	−7.65173	−0.287367	−0.143683	+	0.989624i	$0.545895\pi$
$710$	0	0
$711$	−15.9909	−0.599706
$712$	0	0
$713$	21.9160	0.820760
$714$	0	0
$715$	−1.28621	−0.0481015
$716$	0	0
$717$	33.0222	1.23324
$718$	0	0
$719$	2.39288	0.0892394	0.0446197	−	0.999004i	$-0.485792\pi$
$719$	2.39288	0.0892394	0.0446197	+	0.999004i	$0.485792\pi$
$720$	0	0
$721$	0	0
$722$	0	0
$723$	−59.2573	−2.20380
$724$	0	0
$725$	−74.2592	−2.75792
$726$	0	0
$727$	24.5924	0.912080	0.456040	−	0.889959i	$-0.349267\pi$
$727$	24.5924	0.912080	0.456040	+	0.889959i	$0.349267\pi$
$728$	0	0
$729$	−35.0388	−1.29773
$730$	0	0
$731$	0.849789	0.0314306
$732$	0	0
$733$	−0.387674	−0.0143191	−0.00715953	−	0.999974i	$-0.502279\pi$
$733$	−0.387674	−0.0143191	−0.00715953	+	0.999974i	$0.502279\pi$
$734$	0	0
$735$	0	0
$736$	0	0
$737$	27.3502	1.00746
$738$	0	0
$739$	10.2446	0.376853	0.188427	−	0.982087i	$-0.439661\pi$
$739$	10.2446	0.376853	0.188427	+	0.982087i	$0.439661\pi$
$740$	0	0
$741$	0.532801	0.0195729
$742$	0	0
$743$	−36.2500	−1.32988	−0.664941	−	0.746896i	$-0.731543\pi$
$743$	−36.2500	−1.32988	−0.664941	+	0.746896i	$0.731543\pi$
$744$	0	0
$745$	67.3058	2.46589
$746$	0	0
$747$	21.0928	0.771744
$748$	0	0
$749$	0	0
$750$	0	0
$751$	−32.4916	−1.18563	−0.592817	−	0.805337i	$-0.701984\pi$
$751$	−32.4916	−1.18563	−0.592817	+	0.805337i	$0.701984\pi$
$752$	0	0
$753$	−49.1400	−1.79076
$754$	0	0
$755$	46.4547	1.69066
$756$	0	0
$757$	−11.5985	−0.421555	−0.210777	−	0.977534i	$-0.567599\pi$
$757$	−11.5985	−0.421555	−0.210777	+	0.977534i	$0.567599\pi$
$758$	0	0
$759$	109.510	3.97497
$760$	0	0
$761$	−20.3269	−0.736850	−0.368425	−	0.929658i	$-0.620103\pi$
$761$	−20.3269	−0.736850	−0.368425	+	0.929658i	$0.620103\pi$
$762$	0	0
$763$	0	0
$764$	0	0
$765$	−85.8383	−3.10349
$766$	0	0
$767$	0.258587	0.00933704
$768$	0	0
$769$	−6.28624	−0.226688	−0.113344	−	0.993556i	$-0.536156\pi$
$769$	−6.28624	−0.226688	−0.113344	+	0.993556i	$0.536156\pi$
$770$	0	0
$771$	−65.2766	−2.35088
$772$	0	0
$773$	−20.1080	−0.723235	−0.361617	−	0.932327i	$-0.617775\pi$
$773$	−20.1080	−0.723235	−0.361617	+	0.932327i	$0.617775\pi$
$774$	0	0
$775$	−25.6079	−0.919863
$776$	0	0
$777$	0	0
$778$	0	0
$779$	21.3706	0.765682
$780$	0	0
$781$	32.1444	1.15022
$782$	0	0
$783$	−8.73460	−0.312149
$784$	0	0
$785$	−46.1473	−1.64707
$786$	0	0
$787$	−4.22894	−0.150745	−0.0753727	−	0.997155i	$-0.524015\pi$
$787$	−4.22894	−0.150745	−0.0753727	+	0.997155i	$0.524015\pi$
$788$	0	0
$789$	−21.9502	−0.781446
$790$	0	0
$791$	0	0
$792$	0	0
$793$	−0.0155394	−0.000551822 0
$794$	0	0
$795$	47.9952	1.70221
$796$	0	0
$797$	33.1563	1.17446	0.587228	−	0.809421i	$-0.300219\pi$
$797$	33.1563	1.17446	0.587228	+	0.809421i	$0.300219\pi$
$798$	0	0
$799$	36.7030	1.29846
$800$	0	0
$801$	26.0340	0.919865
$802$	0	0
$803$	16.9535	0.598275
$804$	0	0
$805$	0	0
$806$	0	0
$807$	−3.70171	−0.130306
$808$	0	0
$809$	10.5265	0.370091	0.185046	−	0.982730i	$-0.440757\pi$
$809$	10.5265	0.370091	0.185046	+	0.982730i	$0.440757\pi$
$810$	0	0
$811$	8.94063	0.313948	0.156974	−	0.987603i	$-0.449826\pi$
$811$	8.94063	0.313948	0.156974	+	0.987603i	$0.449826\pi$
$812$	0	0
$813$	−69.3986	−2.43392
$814$	0	0
$815$	−63.3220	−2.21807
$816$	0	0
$817$	−0.414242	−0.0144925
$818$	0	0
$819$	0	0
$820$	0	0
$821$	−10.9293	−0.381436	−0.190718	−	0.981645i	$-0.561082\pi$
$821$	−10.9293	−0.381436	−0.190718	+	0.981645i	$0.561082\pi$
$822$	0	0
$823$	12.4547	0.434145	0.217072	−	0.976156i	$-0.430349\pi$
$823$	12.4547	0.434145	0.217072	+	0.976156i	$0.430349\pi$
$824$	0	0
$825$	−127.958	−4.45493
$826$	0	0
$827$	−39.2922	−1.36632	−0.683161	−	0.730267i	$-0.739395\pi$
$827$	−39.2922	−1.36632	−0.683161	+	0.730267i	$0.739395\pi$
$828$	0	0
$829$	−21.2695	−0.738721	−0.369360	−	0.929286i	$-0.620423\pi$
$829$	−21.2695	−0.738721	−0.369360	+	0.929286i	$0.620423\pi$
$830$	0	0
$831$	−15.5947	−0.540976
$832$	0	0
$833$	0	0
$834$	0	0
$835$	57.2573	1.98147
$836$	0	0
$837$	−3.01208	−0.104113
$838$	0	0
$839$	−24.2781	−0.838172	−0.419086	−	0.907946i	$-0.637649\pi$
$839$	−24.2781	−0.838172	−0.419086	+	0.907946i	$0.637649\pi$
$840$	0	0
$841$	19.1497	0.660336
$842$	0	0
$843$	32.4543	1.11779
$844$	0	0
$845$	51.4940	1.77145
$846$	0	0
$847$	0	0
$848$	0	0
$849$	46.2258	1.58647
$850$	0	0
$851$	18.5171	0.634757
$852$	0	0
$853$	30.2382	1.03534	0.517668	−	0.855581i	$-0.326800\pi$
$853$	30.2382	1.03534	0.517668	+	0.855581i	$0.326800\pi$
$854$	0	0
$855$	41.8431	1.43100
$856$	0	0
$857$	33.1947	1.13391	0.566954	−	0.823749i	$-0.308122\pi$
$857$	33.1947	1.13391	0.566954	+	0.823749i	$0.308122\pi$
$858$	0	0
$859$	2.38866	0.0815000	0.0407500	−	0.999169i	$-0.487025\pi$
$859$	2.38866	0.0815000	0.0407500	+	0.999169i	$0.487025\pi$
$860$	0	0
$861$	0	0
$862$	0	0
$863$	−41.9017	−1.42635	−0.713175	−	0.700986i	$-0.752744\pi$
$863$	−41.9017	−1.42635	−0.713175	+	0.700986i	$0.752744\pi$
$864$	0	0
$865$	−37.1594	−1.26346
$866$	0	0
$867$	54.6352	1.85551
$868$	0	0
$869$	21.4741	0.728459
$870$	0	0
$871$	0.403254	0.0136637
$872$	0	0
$873$	55.3201	1.87230
$874$	0	0
$875$	0	0
$876$	0	0
$877$	21.8377	0.737407	0.368704	−	0.929547i	$-0.379802\pi$
$877$	21.8377	0.737407	0.368704	+	0.929547i	$0.379802\pi$
$878$	0	0
$879$	73.6728	2.48492
$880$	0	0
$881$	−23.1842	−0.781097	−0.390548	−	0.920582i	$-0.627715\pi$
$881$	−23.1842	−0.781097	−0.390548	+	0.920582i	$0.627715\pi$
$882$	0	0
$883$	−35.0968	−1.18110	−0.590550	−	0.807001i	$-0.701089\pi$
$883$	−35.0968	−1.18110	−0.590550	+	0.807001i	$0.701089\pi$
$884$	0	0
$885$	37.7448	1.26878
$886$	0	0
$887$	−55.8537	−1.87538	−0.937692	−	0.347467i	$-0.887042\pi$
$887$	−55.8537	−1.87538	−0.937692	+	0.347467i	$0.887042\pi$
$888$	0	0
$889$	0	0
$890$	0	0
$891$	34.1304	1.14341
$892$	0	0
$893$	−17.8914	−0.598713
$894$	0	0
$895$	52.4917	1.75460
$896$	0	0
$897$	1.61463	0.0539109
$898$	0	0
$899$	16.6042	0.553781
$900$	0	0
$901$	−29.4686	−0.981741
$902$	0	0
$903$	0	0
$904$	0	0
$905$	49.7211	1.65278
$906$	0	0
$907$	−47.7187	−1.58447	−0.792237	−	0.610214i	$-0.791083\pi$
$907$	−47.7187	−1.58447	−0.792237	+	0.610214i	$0.791083\pi$
$908$	0	0
$909$	48.8689	1.62088
$910$	0	0
$911$	8.80061	0.291577	0.145789	−	0.989316i	$-0.453428\pi$
$911$	8.80061	0.291577	0.145789	+	0.989316i	$0.453428\pi$
$912$	0	0
$913$	−28.3254	−0.937433
$914$	0	0
$915$	−2.26822	−0.0749851
$916$	0	0
$917$	0	0
$918$	0	0
$919$	19.3297	0.637630	0.318815	−	0.947817i	$-0.396715\pi$
$919$	19.3297	0.637630	0.318815	+	0.947817i	$0.396715\pi$
$920$	0	0
$921$	−64.4215	−2.12276
$922$	0	0
$923$	0.473939	0.0155999
$924$	0	0
$925$	−21.6364	−0.711400
$926$	0	0
$927$	18.3027	0.601141
$928$	0	0
$929$	52.6129	1.72617	0.863087	−	0.505055i	$-0.168528\pi$
$929$	52.6129	1.72617	0.863087	+	0.505055i	$0.168528\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−11.1884	−0.366291
$934$	0	0
$935$	115.272	3.76979
$936$	0	0
$937$	−23.5445	−0.769166	−0.384583	−	0.923091i	$-0.625655\pi$
$937$	−23.5445	−0.769166	−0.384583	+	0.923091i	$0.625655\pi$
$938$	0	0
$939$	60.6219	1.97832
$940$	0	0
$941$	34.9198	1.13835	0.569176	−	0.822215i	$-0.307262\pi$
$941$	34.9198	1.13835	0.569176	+	0.822215i	$0.307262\pi$
$942$	0	0
$943$	64.7627	2.10896
$944$	0	0
$945$	0	0
$946$	0	0
$947$	−41.1685	−1.33780	−0.668899	−	0.743354i	$-0.733234\pi$
$947$	−41.1685	−1.33780	−0.668899	+	0.743354i	$0.733234\pi$
$948$	0	0
$949$	0.249964	0.00811416
$950$	0	0
$951$	−68.3186	−2.21538
$952$	0	0
$953$	48.3126	1.56500	0.782500	−	0.622651i	$-0.213944\pi$
$953$	48.3126	1.56500	0.782500	+	0.622651i	$0.213944\pi$
$954$	0	0
$955$	−78.1550	−2.52904
$956$	0	0
$957$	82.9682	2.68198
$958$	0	0
$959$	0	0
$960$	0	0
$961$	−25.2741	−0.815294
$962$	0	0
$963$	−20.0248	−0.645289
$964$	0	0
$965$	3.40994	0.109770
$966$	0	0
$967$	60.2747	1.93830	0.969152	−	0.246463i	$-0.0792685\pi$
$967$	60.2747	1.93830	0.969152	+	0.246463i	$0.0792685\pi$
$968$	0	0
$969$	−47.7504	−1.53396
$970$	0	0
$971$	9.80713	0.314726	0.157363	−	0.987541i	$-0.449701\pi$
$971$	9.80713	0.314726	0.157363	+	0.987541i	$0.449701\pi$
$972$	0	0
$973$	0	0
$974$	0	0
$975$	−1.88663	−0.0604204
$976$	0	0
$977$	12.8140	0.409957	0.204978	−	0.978766i	$-0.434288\pi$
$977$	12.8140	0.409957	0.204978	+	0.978766i	$0.434288\pi$
$978$	0	0
$979$	−34.9609	−1.11736
$980$	0	0
$981$	20.2107	0.645279
$982$	0	0
$983$	9.40087	0.299841	0.149921	−	0.988698i	$-0.452098\pi$
$983$	9.40087	0.299841	0.149921	+	0.988698i	$0.452098\pi$
$984$	0	0
$985$	72.8209	2.32027
$986$	0	0
$987$	0	0
$988$	0	0
$989$	−1.25534	−0.0399175
$990$	0	0
$991$	−36.7338	−1.16689	−0.583443	−	0.812154i	$-0.698295\pi$
$991$	−36.7338	−1.16689	−0.583443	+	0.812154i	$0.698295\pi$
$992$	0	0
$993$	−49.7092	−1.57747
$994$	0	0
$995$	−30.3236	−0.961324
$996$	0	0
$997$	−4.19664	−0.132909	−0.0664545	−	0.997789i	$-0.521169\pi$
$997$	−4.19664	−0.132909	−0.0664545	+	0.997789i	$0.521169\pi$
$998$	0	0
$999$	−2.54494	−0.0805184

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1372.2.a.b.1.6	yes	6
4.3	odd	2		5488.2.a.l.1.1		6
7.2	even	3		1372.2.e.c.361.1		12
7.3	odd	6		1372.2.e.c.1353.6		12
7.4	even	3		1372.2.e.c.1353.1		12
7.5	odd	6		1372.2.e.c.361.6		12
7.6	odd	2	inner	1372.2.a.b.1.1	✓	6
28.27	even	2		5488.2.a.l.1.6		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1372.2.a.b.1.1	✓	6	7.6	odd	2	inner
1372.2.a.b.1.6	yes	6	1.1	even	1	trivial
1372.2.e.c.361.1		12	7.2	even	3
1372.2.e.c.361.6		12	7.5	odd	6
1372.2.e.c.1353.1		12	7.4	even	3
1372.2.e.c.1353.6		12	7.3	odd	6
5488.2.a.l.1.1		6	4.3	odd	2
5488.2.a.l.1.6		6	28.27	even	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:	\( N \)	\(=\)	\( 1372 = 2^{2} \cdot 7^{3} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1372.a (trivial)

\(f(q)\)	\(=\)	\(q+2.54832 q^{3} -3.96254 q^{5} +3.49396 q^{9} -4.69202 q^{11} -0.0691796 q^{13} -10.0978 q^{15} +6.19997 q^{17} -3.02226 q^{19} -9.15883 q^{23} +10.7017 q^{25} +1.25877 q^{27} -6.93900 q^{29} -2.39288 q^{31} -11.9568 q^{33} -2.02177 q^{37} -0.176292 q^{39} -7.07107 q^{41} +0.137063 q^{43} -13.8449 q^{45} +5.91987 q^{47} +15.7995 q^{51} -4.75302 q^{53} +18.5923 q^{55} -7.70171 q^{57} -3.73791 q^{59} +0.224625 q^{61} +0.274127 q^{65} -5.82908 q^{67} -23.3397 q^{69} -6.85086 q^{71} -3.61326 q^{73} +27.2714 q^{75} -4.57673 q^{79} -7.27413 q^{81} +6.03692 q^{83} -24.5676 q^{85} -17.6828 q^{87} +7.45114 q^{89} -6.09783 q^{93} +11.9758 q^{95} +15.8331 q^{97} -16.3937 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 2 q^{9} - 18 q^{11} - 24 q^{15} - 38 q^{23} + 10 q^{25} - 22 q^{29} - 6 q^{37} - 16 q^{39} - 10 q^{43} + 4 q^{51} - 38 q^{53} + 8 q^{57} - 20 q^{65} - 14 q^{67} - 14 q^{71} - 22 q^{79} - 22 q^{81}+ \cdots - 34 q^{99}+O(q^{100}) \) 6 * q + 2 * q^9 - 18 * q^11 - 24 * q^15 - 38 * q^23 + 10 * q^25 - 22 * q^29 - 6 * q^37 - 16 * q^39 - 10 * q^43 + 4 * q^51 - 38 * q^53 + 8 * q^57 - 20 * q^65 - 14 * q^67 - 14 * q^71 - 22 * q^79 - 22 * q^81 - 16 * q^85 - 4 * q^95 - 34 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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