LMFDB - Embedded newform 4928.2.a.ch.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [4928,2,Mod(1,4928)] 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("4928.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(4928, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 4928 = 2^{6} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4928.a (trivial)

Newform invariants

comment:select newform

sage:traces = [4,0,1,0,-5,0,4,0,9,0,4,0,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(13)] == traces)

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

Self dual:	yes
Analytic conductor:	$39.3502781161$
Analytic rank:	$0$
Dimension:	$4$
Coefficient field:	4.4.11348.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - x^{3} - 5x^{2} + x + 2 $ x^4 - x^3 - 5*x^2 + x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	no (minimal twist has level 616)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.77571$ of defining polynomial
Character	$\chi$	$=$	4928.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q+1.12631 q^{3} -4.42512 q^{5} +1.00000 q^{7} -1.73143 q^{9} +1.00000 q^{11} -5.85774 q^{13} -4.98405 q^{15} -1.55143 q^{17} -7.40916 q^{19} +1.12631 q^{21} -4.98405 q^{23} +14.5817 q^{25} -5.32905 q^{27} +9.10285 q^{29} +1.43262 q^{31} +1.12631 q^{33} -4.42512 q^{35} +0.731430 q^{37} -6.59762 q^{39} -1.55143 q^{41} +2.59762 q^{43} +7.66178 q^{45} +0.953807 q^{47} +1.00000 q^{49} -1.74738 q^{51} -2.00000 q^{53} -4.42512 q^{55} -8.34500 q^{57} -8.22916 q^{59} -3.60512 q^{61} -1.73143 q^{63} +25.9212 q^{65} +9.32905 q^{67} -5.61357 q^{69} +0.478813 q^{71} +6.16405 q^{73} +16.4234 q^{75} +1.00000 q^{77} +9.74738 q^{79} -0.807862 q^{81} +10.5589 q^{83} +6.86524 q^{85} +10.2526 q^{87} +4.11881 q^{89} -5.85774 q^{91} +1.61357 q^{93} +32.7864 q^{95} -11.3290 q^{97} -1.73143 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q + q^{3} - 5 q^{5} + 4 q^{7} + 9 q^{9} + 4 q^{11} - 4 q^{13} + 3 q^{15} + 10 q^{17} + 6 q^{19} + q^{21} + 3 q^{23} + 17 q^{25} + 13 q^{27} + 4 q^{29} - q^{31} + q^{33} - 5 q^{35} - 13 q^{37} - 8 q^{39}+ \cdots + 9 q^{99}+O(q^{100}) $ 4 * q + q^3 - 5 * q^5 + 4 * q^7 + 9 * q^9 + 4 * q^11 - 4 * q^13 + 3 * q^15 + 10 * q^17 + 6 * q^19 + q^21 + 3 * q^23 + 17 * q^25 + 13 * q^27 + 4 * q^29 - q^31 + q^33 - 5 * q^35 - 13 * q^37 - 8 * q^39 + 10 * q^41 - 8 * q^43 - 12 * q^45 + 6 * q^47 + 4 * q^49 - 14 * q^51 - 8 * q^53 - 5 * q^55 - 22 * q^57 + 3 * q^59 - 2 * q^61 + 9 * q^63 + 3 * q^67 - 27 * q^69 - 7 * q^71 + 2 * q^73 - 18 * q^75 + 4 * q^77 + 46 * q^79 + 40 * q^81 + 32 * q^83 + 14 * q^85 + 34 * q^87 + 7 * q^89 - 4 * q^91 + 11 * q^93 + 14 * q^95 - 11 * q^97 + 9 * 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
$3$	1.12631	0.650274	0.325137	−	0.945667i	$-0.394590\pi$
$3$	1.12631	0.650274	0.325137	+	0.945667i	$0.394590\pi$
$4$	0	0
$5$	−4.42512	−1.97897	−0.989486	−	0.144626i	$-0.953802\pi$
$5$	−4.42512	−1.97897	−0.989486	+	0.144626i	$0.953802\pi$
$6$	0	0
$7$	1.00000	0.377964
$7$	1.00000	0.377964
$8$	0	0
$9$	−1.73143	−0.577143
$10$	0	0
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−5.85774	−1.62464	−0.812322	−	0.583209i	$-0.801797\pi$
$13$	−5.85774	−1.62464	−0.812322	+	0.583209i	$0.801797\pi$
$14$	0	0
$15$	−4.98405	−1.28688
$16$	0	0
$17$	−1.55143	−0.376276	−0.188138	−	0.982143i	$-0.560245\pi$
$17$	−1.55143	−0.376276	−0.188138	+	0.982143i	$0.560245\pi$
$18$	0	0
$19$	−7.40916	−1.69978	−0.849889	−	0.526961i	$-0.823331\pi$
$19$	−7.40916	−1.69978	−0.849889	+	0.526961i	$0.823331\pi$
$20$	0	0
$21$	1.12631	0.245781
$22$	0	0
$23$	−4.98405	−1.03925	−0.519623	−	0.854396i	$-0.673927\pi$
$23$	−4.98405	−1.03925	−0.519623	+	0.854396i	$0.673927\pi$
$24$	0	0
$25$	14.5817	2.91633
$26$	0	0
$27$	−5.32905	−1.02558
$28$	0	0
$29$	9.10285	1.69036	0.845179	−	0.534484i	$-0.179494\pi$
$29$	9.10285	1.69036	0.845179	+	0.534484i	$0.179494\pi$
$30$	0	0
$31$	1.43262	0.257306	0.128653	−	0.991690i	$-0.458935\pi$
$31$	1.43262	0.257306	0.128653	+	0.991690i	$0.458935\pi$
$32$	0	0
$33$	1.12631	0.196065
$34$	0	0
$35$	−4.42512	−0.747981
$36$	0	0
$37$	0.731430	0.120246	0.0601232	−	0.998191i	$-0.480851\pi$
$37$	0.731430	0.120246	0.0601232	+	0.998191i	$0.480851\pi$
$38$	0	0
$39$	−6.59762	−1.05646
$40$	0	0
$41$	−1.55143	−0.242292	−0.121146	−	0.992635i	$-0.538657\pi$
$41$	−1.55143	−0.242292	−0.121146	+	0.992635i	$0.538657\pi$
$42$	0	0
$43$	2.59762	0.396133	0.198067	−	0.980189i	$-0.436534\pi$
$43$	2.59762	0.396133	0.198067	+	0.980189i	$0.436534\pi$
$44$	0	0
$45$	7.66178	1.14215
$46$	0	0
$47$	0.953807	0.139127	0.0695635	−	0.997578i	$-0.477839\pi$
$47$	0.953807	0.139127	0.0695635	+	0.997578i	$0.477839\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	0	0
$51$	−1.74738	−0.244683
$52$	0	0
$53$	−2.00000	−0.274721	−0.137361	−	0.990521i	$-0.543862\pi$
$53$	−2.00000	−0.274721	−0.137361	+	0.990521i	$0.543862\pi$
$54$	0	0
$55$	−4.42512	−0.596683
$56$	0	0
$57$	−8.34500	−1.10532
$58$	0	0
$59$	−8.22916	−1.07135	−0.535673	−	0.844426i	$-0.679942\pi$
$59$	−8.22916	−1.07135	−0.535673	+	0.844426i	$0.679942\pi$
$60$	0	0
$61$	−3.60512	−0.461589	−0.230794	−	0.973003i	$-0.574132\pi$
$61$	−3.60512	−0.461589	−0.230794	+	0.973003i	$0.574132\pi$
$62$	0	0
$63$	−1.73143	−0.218140
$64$	0	0
$65$	25.9212	3.21513
$66$	0	0
$67$	9.32905	1.13972	0.569862	−	0.821740i	$-0.306997\pi$
$67$	9.32905	1.13972	0.569862	+	0.821740i	$0.306997\pi$
$68$	0	0
$69$	−5.61357	−0.675795
$70$	0	0
$71$	0.478813	0.0568247	0.0284123	−	0.999596i	$-0.490955\pi$
$71$	0.478813	0.0568247	0.0284123	+	0.999596i	$0.490955\pi$
$72$	0	0
$73$	6.16405	0.721448	0.360724	−	0.932673i	$-0.382530\pi$
$73$	6.16405	0.721448	0.360724	+	0.932673i	$0.382530\pi$
$74$	0	0
$75$	16.4234	1.89642
$76$	0	0
$77$	1.00000	0.113961
$78$	0	0
$79$	9.74738	1.09667	0.548333	−	0.836260i	$-0.315263\pi$
$79$	9.74738	1.09667	0.548333	+	0.836260i	$0.315263\pi$
$80$	0	0
$81$	−0.807862	−0.0897624
$82$	0	0
$83$	10.5589	1.15899	0.579497	−	0.814975i	$-0.303249\pi$
$83$	10.5589	1.15899	0.579497	+	0.814975i	$0.303249\pi$
$84$	0	0
$85$	6.86524	0.744640
$86$	0	0
$87$	10.2526	1.09920
$88$	0	0
$89$	4.11881	0.436592	0.218296	−	0.975883i	$-0.429950\pi$
$89$	4.11881	0.436592	0.218296	+	0.975883i	$0.429950\pi$
$90$	0	0
$91$	−5.85774	−0.614058
$92$	0	0
$93$	1.61357	0.167320
$94$	0	0
$95$	32.7864	3.36382
$96$	0	0
$97$	−11.3290	−1.15029	−0.575145	−	0.818051i	$-0.695054\pi$
$97$	−11.3290	−1.15029	−0.575145	+	0.818051i	$0.695054\pi$
$98$	0	0
$99$	−1.73143	−0.174015
$100$	0	0
$101$	−5.24511	−0.521908	−0.260954	−	0.965351i	$-0.584037\pi$
$101$	−5.24511	−0.521908	−0.260954	+	0.965351i	$0.584037\pi$
$102$	0	0
$103$	18.6543	1.83806	0.919030	−	0.394187i	$-0.128974\pi$
$103$	18.6543	1.83806	0.919030	+	0.394187i	$0.128974\pi$
$104$	0	0
$105$	−4.98405	−0.486393
$106$	0	0
$107$	−13.3555	−1.29112	−0.645561	−	0.763709i	$-0.723376\pi$
$107$	−13.3555	−1.29112	−0.645561	+	0.763709i	$0.723376\pi$
$108$	0	0
$109$	9.44785	0.904940	0.452470	−	0.891780i	$-0.350543\pi$
$109$	9.44785	0.904940	0.452470	+	0.891780i	$0.350543\pi$
$110$	0	0
$111$	0.823815	0.0781931
$112$	0	0
$113$	−13.5817	−1.27766	−0.638828	−	0.769350i	$-0.720580\pi$
$113$	−13.5817	−1.27766	−0.638828	+	0.769350i	$0.720580\pi$
$114$	0	0
$115$	22.0550	2.05664
$116$	0	0
$117$	10.1423	0.937652
$118$	0	0
$119$	−1.55143	−0.142219
$120$	0	0
$121$	1.00000	0.0909091
$122$	0	0
$123$	−1.74738	−0.157556
$124$	0	0
$125$	−42.4000	−3.79237
$126$	0	0
$127$	−1.35547	−0.120278	−0.0601392	−	0.998190i	$-0.519154\pi$
$127$	−1.35547	−0.120278	−0.0601392	+	0.998190i	$0.519154\pi$
$128$	0	0
$129$	2.92572	0.257595
$130$	0	0
$131$	−10.0068	−0.874297	−0.437148	−	0.899389i	$-0.644012\pi$
$131$	−10.0068	−0.874297	−0.437148	+	0.899389i	$0.644012\pi$
$132$	0	0
$133$	−7.40916	−0.642456
$134$	0	0
$135$	23.5817	2.02959
$136$	0	0
$137$	18.4319	1.57474	0.787372	−	0.616479i	$-0.211441\pi$
$137$	18.4319	1.57474	0.787372	+	0.616479i	$0.211441\pi$
$138$	0	0
$139$	2.90393	0.246308	0.123154	−	0.992388i	$-0.460699\pi$
$139$	2.90393	0.246308	0.123154	+	0.992388i	$0.460699\pi$
$140$	0	0
$141$	1.07428	0.0904708
$142$	0	0
$143$	−5.85774	−0.489849
$144$	0	0
$145$	−40.2812	−3.34517
$146$	0	0
$147$	1.12631	0.0928963
$148$	0	0
$149$	16.2057	1.32762	0.663811	−	0.747900i	$-0.268938\pi$
$149$	16.2057	1.32762	0.663811	+	0.747900i	$0.268938\pi$
$150$	0	0
$151$	5.63999	0.458976	0.229488	−	0.973311i	$-0.426295\pi$
$151$	5.63999	0.458976	0.229488	+	0.973311i	$0.426295\pi$
$152$	0	0
$153$	2.68618	0.217165
$154$	0	0
$155$	−6.33951	−0.509202
$156$	0	0
$157$	0.692740	0.0552867	0.0276433	−	0.999618i	$-0.491200\pi$
$157$	0.692740	0.0552867	0.0276433	+	0.999618i	$0.491200\pi$
$158$	0	0
$159$	−2.25262	−0.178644
$160$	0	0
$161$	−4.98405	−0.392798
$162$	0	0
$163$	−16.2207	−1.27050	−0.635252	−	0.772305i	$-0.719104\pi$
$163$	−16.2207	−1.27050	−0.635252	+	0.772305i	$0.719104\pi$
$164$	0	0
$165$	−4.98405	−0.388007
$166$	0	0
$167$	−12.8502	−0.994381	−0.497191	−	0.867641i	$-0.665635\pi$
$167$	−12.8502	−0.994381	−0.497191	+	0.867641i	$0.665635\pi$
$168$	0	0
$169$	21.3131	1.63947
$170$	0	0
$171$	12.8284	0.981016
$172$	0	0
$173$	13.0680	0.993540	0.496770	−	0.867882i	$-0.334519\pi$
$173$	13.0680	0.993540	0.496770	+	0.867882i	$0.334519\pi$
$174$	0	0
$175$	14.5817	1.10227
$176$	0	0
$177$	−9.26857	−0.696668
$178$	0	0
$179$	11.8493	0.885657	0.442829	−	0.896606i	$-0.353975\pi$
$179$	11.8493	0.885657	0.442829	+	0.896606i	$0.353975\pi$
$180$	0	0
$181$	7.52797	0.559550	0.279775	−	0.960066i	$-0.409740\pi$
$181$	7.52797	0.559550	0.279775	+	0.960066i	$0.409740\pi$
$182$	0	0
$183$	−4.06048	−0.300159
$184$	0	0
$185$	−3.23666	−0.237964
$186$	0	0
$187$	−1.55143	−0.113451
$188$	0	0
$189$	−5.32905	−0.387631
$190$	0	0
$191$	−24.3245	−1.76006	−0.880030	−	0.474918i	$-0.842478\pi$
$191$	−24.3245	−1.76006	−0.880030	+	0.474918i	$0.842478\pi$
$192$	0	0
$193$	−8.31309	−0.598390	−0.299195	−	0.954192i	$-0.596718\pi$
$193$	−8.31309	−0.598390	−0.299195	+	0.954192i	$0.596718\pi$
$194$	0	0
$195$	29.1952	2.09071
$196$	0	0
$197$	10.6126	0.756118	0.378059	−	0.925781i	$-0.376592\pi$
$197$	10.6126	0.756118	0.378059	+	0.925781i	$0.376592\pi$
$198$	0	0
$199$	−7.04619	−0.499491	−0.249746	−	0.968311i	$-0.580347\pi$
$199$	−7.04619	−0.499491	−0.249746	+	0.968311i	$0.580347\pi$
$200$	0	0
$201$	10.5074	0.741134
$202$	0	0
$203$	9.10285	0.638895
$204$	0	0
$205$	6.86524	0.479489
$206$	0	0
$207$	8.62953	0.599794
$208$	0	0
$209$	−7.40916	−0.512503
$210$	0	0
$211$	9.86070	0.678839	0.339419	−	0.940635i	$-0.389769\pi$
$211$	9.86070	0.678839	0.339419	+	0.940635i	$0.389769\pi$
$212$	0	0
$213$	0.539291	0.0369516
$214$	0	0
$215$	−11.4948	−0.783937
$216$	0	0
$217$	1.43262	0.0972526
$218$	0	0
$219$	6.94262	0.469139
$220$	0	0
$221$	9.08785	0.611315
$222$	0	0
$223$	11.4176	0.764580	0.382290	−	0.924042i	$-0.375136\pi$
$223$	11.4176	0.764580	0.382290	+	0.924042i	$0.375136\pi$
$224$	0	0
$225$	−25.2471	−1.68314
$226$	0	0
$227$	14.1670	0.940298	0.470149	−	0.882587i	$-0.344200\pi$
$227$	14.1670	0.940298	0.470149	+	0.882587i	$0.344200\pi$
$228$	0	0
$229$	19.0227	1.25706	0.628529	−	0.777786i	$-0.283657\pi$
$229$	19.0227	1.25706	0.628529	+	0.777786i	$0.283657\pi$
$230$	0	0
$231$	1.12631	0.0741056
$232$	0	0
$233$	−13.1029	−0.858396	−0.429198	−	0.903210i	$-0.641204\pi$
$233$	−13.1029	−0.858396	−0.429198	+	0.903210i	$0.641204\pi$
$234$	0	0
$235$	−4.22071	−0.275329
$236$	0	0
$237$	10.9786	0.713134
$238$	0	0
$239$	−21.1179	−1.36600	−0.683000	−	0.730418i	$-0.739325\pi$
$239$	−21.1179	−1.36600	−0.683000	+	0.730418i	$0.739325\pi$
$240$	0	0
$241$	−24.7148	−1.59202	−0.796009	−	0.605285i	$-0.793059\pi$
$241$	−24.7148	−1.59202	−0.796009	+	0.605285i	$0.793059\pi$
$242$	0	0
$243$	15.0772	0.967206
$244$	0	0
$245$	−4.42512	−0.282710
$246$	0	0
$247$	43.4009	2.76154
$248$	0	0
$249$	11.8926	0.753663
$250$	0	0
$251$	17.1868	1.08482	0.542410	−	0.840114i	$-0.317512\pi$
$251$	17.1868	1.08482	0.542410	+	0.840114i	$0.317512\pi$
$252$	0	0
$253$	−4.98405	−0.313344
$254$	0	0
$255$	7.73238	0.484220
$256$	0	0
$257$	25.4310	1.58634	0.793170	−	0.609001i	$-0.208429\pi$
$257$	25.4310	1.58634	0.793170	+	0.609001i	$0.208429\pi$
$258$	0	0
$259$	0.731430	0.0454488
$260$	0	0
$261$	−15.7609	−0.975578
$262$	0	0
$263$	−0.957627	−0.0590498	−0.0295249	−	0.999564i	$-0.509399\pi$
$263$	−0.957627	−0.0590498	−0.0295249	+	0.999564i	$0.509399\pi$
$264$	0	0
$265$	8.85023	0.543666
$266$	0	0
$267$	4.63904	0.283905
$268$	0	0
$269$	−6.82845	−0.416338	−0.208169	−	0.978093i	$-0.566750\pi$
$269$	−6.82845	−0.416338	−0.208169	+	0.978093i	$0.566750\pi$
$270$	0	0
$271$	−18.0286	−1.09516	−0.547579	−	0.836754i	$-0.684450\pi$
$271$	−18.0286	−1.09516	−0.547579	+	0.836754i	$0.684450\pi$
$272$	0	0
$273$	−6.59762	−0.399306
$274$	0	0
$275$	14.5817	0.879307
$276$	0	0
$277$	4.53714	0.272610	0.136305	−	0.990667i	$-0.456477\pi$
$277$	4.53714	0.272610	0.136305	+	0.990667i	$0.456477\pi$
$278$	0	0
$279$	−2.48048	−0.148503
$280$	0	0
$281$	9.37047	0.558996	0.279498	−	0.960146i	$-0.409832\pi$
$281$	9.37047	0.558996	0.279498	+	0.960146i	$0.409832\pi$
$282$	0	0
$283$	2.23083	0.132609	0.0663045	−	0.997799i	$-0.478879\pi$
$283$	2.23083	0.132609	0.0663045	+	0.997799i	$0.478879\pi$
$284$	0	0
$285$	36.9276	2.18740
$286$	0	0
$287$	−1.55143	−0.0915778
$288$	0	0
$289$	−14.5931	−0.858416
$290$	0	0
$291$	−12.7600	−0.748004
$292$	0	0
$293$	−20.6156	−1.20438	−0.602188	−	0.798355i	$-0.705704\pi$
$293$	−20.6156	−1.20438	−0.602188	+	0.798355i	$0.705704\pi$
$294$	0	0
$295$	36.4150	2.12016
$296$	0	0
$297$	−5.32905	−0.309223
$298$	0	0
$299$	29.1952	1.68840
$300$	0	0
$301$	2.59762	0.149724
$302$	0	0
$303$	−5.90761	−0.339384
$304$	0	0
$305$	15.9531	0.913471
$306$	0	0
$307$	25.6618	1.46460	0.732298	−	0.680985i	$-0.238448\pi$
$307$	25.6618	1.46460	0.732298	+	0.680985i	$0.238448\pi$
$308$	0	0
$309$	21.0105	1.19524
$310$	0	0
$311$	18.2564	1.03523	0.517614	−	0.855614i	$-0.326820\pi$
$311$	18.2564	1.03523	0.517614	+	0.855614i	$0.326820\pi$
$312$	0	0
$313$	23.4424	1.32504	0.662520	−	0.749044i	$-0.269487\pi$
$313$	23.4424	1.32504	0.662520	+	0.749044i	$0.269487\pi$
$314$	0	0
$315$	7.66178	0.431692
$316$	0	0
$317$	5.07643	0.285121	0.142560	−	0.989786i	$-0.454466\pi$
$317$	5.07643	0.285121	0.142560	+	0.989786i	$0.454466\pi$
$318$	0	0
$319$	9.10285	0.509662
$320$	0	0
$321$	−15.0424	−0.839583
$322$	0	0
$323$	11.4948	0.639586
$324$	0	0
$325$	−85.4156	−4.73800
$326$	0	0
$327$	10.6412	0.588459
$328$	0	0
$329$	0.953807	0.0525851
$330$	0	0
$331$	−7.81928	−0.429786	−0.214893	−	0.976638i	$-0.568940\pi$
$331$	−7.81928	−0.429786	−0.214893	+	0.976638i	$0.568940\pi$
$332$	0	0
$333$	−1.26642	−0.0693993
$334$	0	0
$335$	−41.2821	−2.25548
$336$	0	0
$337$	−4.01501	−0.218711	−0.109356	−	0.994003i	$-0.534879\pi$
$337$	−4.01501	−0.218711	−0.109356	+	0.994003i	$0.534879\pi$
$338$	0	0
$339$	−15.2971	−0.830827
$340$	0	0
$341$	1.43262	0.0775808
$342$	0	0
$343$	1.00000	0.0539949
$344$	0	0
$345$	24.8407	1.33738
$346$	0	0
$347$	25.0709	1.34588	0.672939	−	0.739698i	$-0.265031\pi$
$347$	25.0709	1.34588	0.672939	+	0.739698i	$0.265031\pi$
$348$	0	0
$349$	5.69751	0.304981	0.152490	−	0.988305i	$-0.451271\pi$
$349$	5.69751	0.304981	0.152490	+	0.988305i	$0.451271\pi$
$350$	0	0
$351$	31.2162	1.66620
$352$	0	0
$353$	32.7450	1.74284	0.871420	−	0.490537i	$-0.163200\pi$
$353$	32.7450	1.74284	0.871420	+	0.490537i	$0.163200\pi$
$354$	0	0
$355$	−2.11881	−0.112455
$356$	0	0
$357$	−1.74738	−0.0924813
$358$	0	0
$359$	−2.09239	−0.110432	−0.0552159	−	0.998474i	$-0.517585\pi$
$359$	−2.09239	−0.110432	−0.0552159	+	0.998474i	$0.517585\pi$
$360$	0	0
$361$	35.8957	1.88925
$362$	0	0
$363$	1.12631	0.0591159
$364$	0	0
$365$	−27.2766	−1.42772
$366$	0	0
$367$	1.71714	0.0896342	0.0448171	−	0.998995i	$-0.485729\pi$
$367$	1.71714	0.0896342	0.0448171	+	0.998995i	$0.485729\pi$
$368$	0	0
$369$	2.68618	0.139837
$370$	0	0
$371$	−2.00000	−0.103835
$372$	0	0
$373$	−7.35547	−0.380852	−0.190426	−	0.981702i	$-0.560987\pi$
$373$	−7.35547	−0.380852	−0.190426	+	0.981702i	$0.560987\pi$
$374$	0	0
$375$	−47.7555	−2.46608
$376$	0	0
$377$	−53.3221	−2.74623
$378$	0	0
$379$	13.0914	0.672462	0.336231	−	0.941780i	$-0.390848\pi$
$379$	13.0914	0.672462	0.336231	+	0.941780i	$0.390848\pi$
$380$	0	0
$381$	−1.52667	−0.0782139
$382$	0	0
$383$	6.62786	0.338668	0.169334	−	0.985559i	$-0.445838\pi$
$383$	6.62786	0.338668	0.169334	+	0.985559i	$0.445838\pi$
$384$	0	0
$385$	−4.42512	−0.225525
$386$	0	0
$387$	−4.49759	−0.228626
$388$	0	0
$389$	−12.2262	−0.619893	−0.309946	−	0.950754i	$-0.600311\pi$
$389$	−12.2262	−0.619893	−0.309946	+	0.950754i	$0.600311\pi$
$390$	0	0
$391$	7.73238	0.391043
$392$	0	0
$393$	−11.2707	−0.568533
$394$	0	0
$395$	−43.1333	−2.17027
$396$	0	0
$397$	24.9644	1.25293	0.626464	−	0.779450i	$-0.284502\pi$
$397$	24.9644	1.25293	0.626464	+	0.779450i	$0.284502\pi$
$398$	0	0
$399$	−8.34500	−0.417773
$400$	0	0
$401$	−35.6367	−1.77961	−0.889805	−	0.456341i	$-0.849160\pi$
$401$	−35.6367	−1.77961	−0.889805	+	0.456341i	$0.849160\pi$
$402$	0	0
$403$	−8.39192	−0.418031
$404$	0	0
$405$	3.57488	0.177637
$406$	0	0
$407$	0.731430	0.0362556
$408$	0	0
$409$	−2.44857	−0.121074	−0.0605371	−	0.998166i	$-0.519281\pi$
$409$	−2.44857	−0.121074	−0.0605371	+	0.998166i	$0.519281\pi$
$410$	0	0
$411$	20.7600	1.02402
$412$	0	0
$413$	−8.22916	−0.404930
$414$	0	0
$415$	−46.7245	−2.29362
$416$	0	0
$417$	3.27072	0.160168
$418$	0	0
$419$	10.5023	0.513069	0.256535	−	0.966535i	$-0.417419\pi$
$419$	10.5023	0.513069	0.256535	+	0.966535i	$0.417419\pi$
$420$	0	0
$421$	−28.7109	−1.39929	−0.699643	−	0.714493i	$-0.746657\pi$
$421$	−28.7109	−1.39929	−0.699643	+	0.714493i	$0.746657\pi$
$422$	0	0
$423$	−1.65145	−0.0802963
$424$	0	0
$425$	−22.6224	−1.09735
$426$	0	0
$427$	−3.60512	−0.174464
$428$	0	0
$429$	−6.59762	−0.318536
$430$	0	0
$431$	−9.87760	−0.475787	−0.237894	−	0.971291i	$-0.576457\pi$
$431$	−9.87760	−0.475787	−0.237894	+	0.971291i	$0.576457\pi$
$432$	0	0
$433$	15.5967	0.749528	0.374764	−	0.927120i	$-0.377724\pi$
$433$	15.5967	0.749528	0.374764	+	0.927120i	$0.377724\pi$
$434$	0	0
$435$	−45.3690	−2.17528
$436$	0	0
$437$	36.9276	1.76649
$438$	0	0
$439$	−22.2662	−1.06271	−0.531353	−	0.847150i	$-0.678316\pi$
$439$	−22.2662	−1.06271	−0.531353	+	0.847150i	$0.678316\pi$
$440$	0	0
$441$	−1.73143	−0.0824490
$442$	0	0
$443$	−29.8174	−1.41667	−0.708333	−	0.705878i	$-0.750553\pi$
$443$	−29.8174	−1.41667	−0.708333	+	0.705878i	$0.750553\pi$
$444$	0	0
$445$	−18.2262	−0.864005
$446$	0	0
$447$	18.2526	0.863319
$448$	0	0
$449$	−9.23666	−0.435905	−0.217953	−	0.975959i	$-0.569938\pi$
$449$	−9.23666	−0.435905	−0.217953	+	0.975959i	$0.569938\pi$
$450$	0	0
$451$	−1.55143	−0.0730538
$452$	0	0
$453$	6.35237	0.298460
$454$	0	0
$455$	25.9212	1.21520
$456$	0	0
$457$	4.13930	0.193628	0.0968141	−	0.995302i	$-0.469135\pi$
$457$	4.13930	0.193628	0.0968141	+	0.995302i	$0.469135\pi$
$458$	0	0
$459$	8.26762	0.385900
$460$	0	0
$461$	−17.1982	−0.801000	−0.400500	−	0.916297i	$-0.631164\pi$
$461$	−17.1982	−0.801000	−0.400500	+	0.916297i	$0.631164\pi$
$462$	0	0
$463$	19.1897	0.891823	0.445912	−	0.895077i	$-0.352880\pi$
$463$	19.1897	0.891823	0.445912	+	0.895077i	$0.352880\pi$
$464$	0	0
$465$	−7.14025	−0.331121
$466$	0	0
$467$	21.0794	0.975438	0.487719	−	0.873001i	$-0.337829\pi$
$467$	21.0794	0.975438	0.487719	+	0.873001i	$0.337829\pi$
$468$	0	0
$469$	9.32905	0.430775
$470$	0	0
$471$	0.780239	0.0359515
$472$	0	0
$473$	2.59762	0.119439
$474$	0	0
$475$	−108.038	−4.95712
$476$	0	0
$477$	3.46286	0.158553
$478$	0	0
$479$	1.41595	0.0646962	0.0323481	−	0.999477i	$-0.489701\pi$
$479$	1.41595	0.0646962	0.0323481	+	0.999477i	$0.489701\pi$
$480$	0	0
$481$	−4.28452	−0.195357
$482$	0	0
$483$	−5.61357	−0.255426
$484$	0	0
$485$	50.1324	2.27639
$486$	0	0
$487$	4.24120	0.192187	0.0960936	−	0.995372i	$-0.469365\pi$
$487$	4.24120	0.192187	0.0960936	+	0.995372i	$0.469365\pi$
$488$	0	0
$489$	−18.2695	−0.826176
$490$	0	0
$491$	−23.4479	−1.05819	−0.529093	−	0.848564i	$-0.677468\pi$
$491$	−23.4479	−1.05819	−0.529093	+	0.848564i	$0.677468\pi$
$492$	0	0
$493$	−14.1224	−0.636041
$494$	0	0
$495$	7.66178	0.344371
$496$	0	0
$497$	0.478813	0.0214777
$498$	0	0
$499$	13.4159	0.600580	0.300290	−	0.953848i	$-0.402917\pi$
$499$	13.4159	0.600580	0.300290	+	0.953848i	$0.402917\pi$
$500$	0	0
$501$	−14.4733	−0.646620
$502$	0	0
$503$	−4.17713	−0.186249	−0.0931246	−	0.995654i	$-0.529685\pi$
$503$	−4.17713	−0.186249	−0.0931246	+	0.995654i	$0.529685\pi$
$504$	0	0
$505$	23.2102	1.03284
$506$	0	0
$507$	24.0051	1.06610
$508$	0	0
$509$	−10.1725	−0.450888	−0.225444	−	0.974256i	$-0.572383\pi$
$509$	−10.1725	−0.450888	−0.225444	+	0.974256i	$0.572383\pi$
$510$	0	0
$511$	6.16405	0.272682
$512$	0	0
$513$	39.4838	1.74325
$514$	0	0
$515$	−82.5474	−3.63747
$516$	0	0
$517$	0.953807	0.0419484
$518$	0	0
$519$	14.7186	0.646074
$520$	0	0
$521$	1.17618	0.0515296	0.0257648	−	0.999668i	$-0.491798\pi$
$521$	1.17618	0.0515296	0.0257648	+	0.999668i	$0.491798\pi$
$522$	0	0
$523$	35.3623	1.54628	0.773142	−	0.634233i	$-0.218684\pi$
$523$	35.3623	1.54628	0.773142	+	0.634233i	$0.218684\pi$
$524$	0	0
$525$	16.4234	0.716778
$526$	0	0
$527$	−2.22260	−0.0968182
$528$	0	0
$529$	1.84072	0.0800312
$530$	0	0
$531$	14.2482	0.618320
$532$	0	0
$533$	9.08785	0.393638
$534$	0	0
$535$	59.0995	2.55509
$536$	0	0
$537$	13.3459	0.575920
$538$	0	0
$539$	1.00000	0.0430730
$540$	0	0
$541$	22.2812	0.957943	0.478972	−	0.877830i	$-0.341010\pi$
$541$	22.2812	0.957943	0.478972	+	0.877830i	$0.341010\pi$
$542$	0	0
$543$	8.47881	0.363861
$544$	0	0
$545$	−41.8079	−1.79085
$546$	0	0
$547$	43.3371	1.85296	0.926481	−	0.376342i	$-0.122818\pi$
$547$	43.3371	1.85296	0.926481	+	0.376342i	$0.122818\pi$
$548$	0	0
$549$	6.24201	0.266403
$550$	0	0
$551$	−67.4445	−2.87323
$552$	0	0
$553$	9.74738	0.414501
$554$	0	0
$555$	−3.64548	−0.154742
$556$	0	0
$557$	−27.2388	−1.15415	−0.577073	−	0.816693i	$-0.695805\pi$
$557$	−27.2388	−1.15415	−0.577073	+	0.816693i	$0.695805\pi$
$558$	0	0
$559$	−15.2162	−0.643575
$560$	0	0
$561$	−1.74738	−0.0737746
$562$	0	0
$563$	9.37726	0.395204	0.197602	−	0.980282i	$-0.436685\pi$
$563$	9.37726	0.395204	0.197602	+	0.980282i	$0.436685\pi$
$564$	0	0
$565$	60.1005	2.52845
$566$	0	0
$567$	−0.807862	−0.0339270
$568$	0	0
$569$	0.0150061	0.000629087 0	0.000314544	−	1.00000i	$-0.499900\pi$
$569$	0.0150061	0.000629087 0	0.000314544		1.00000i	$0.499900\pi$
$570$	0	0
$571$	32.4114	1.35638	0.678188	−	0.734889i	$-0.262766\pi$
$571$	32.4114	1.35638	0.678188	+	0.734889i	$0.262766\pi$
$572$	0	0
$573$	−27.3969	−1.14452
$574$	0	0
$575$	−72.6757	−3.03079
$576$	0	0
$577$	−44.3831	−1.84769	−0.923846	−	0.382764i	$-0.874972\pi$
$577$	−44.3831	−1.84769	−0.923846	+	0.382764i	$0.874972\pi$
$578$	0	0
$579$	−9.36311	−0.389117
$580$	0	0
$581$	10.5589	0.438058
$582$	0	0
$583$	−2.00000	−0.0828315
$584$	0	0
$585$	−44.8807	−1.85559
$586$	0	0
$587$	−3.11489	−0.128565	−0.0642827	−	0.997932i	$-0.520476\pi$
$587$	−3.11489	−0.128565	−0.0642827	+	0.997932i	$0.520476\pi$
$588$	0	0
$589$	−10.6145	−0.437364
$590$	0	0
$591$	11.9531	0.491684
$592$	0	0
$593$	−9.26690	−0.380546	−0.190273	−	0.981731i	$-0.560937\pi$
$593$	−9.26690	−0.380546	−0.190273	+	0.981731i	$0.560937\pi$
$594$	0	0
$595$	6.86524	0.281447
$596$	0	0
$597$	−7.93618	−0.324806
$598$	0	0
$599$	−21.5233	−0.879420	−0.439710	−	0.898140i	$-0.644919\pi$
$599$	−21.5233	−0.879420	−0.439710	+	0.898140i	$0.644919\pi$
$600$	0	0
$601$	−4.70119	−0.191766	−0.0958828	−	0.995393i	$-0.530567\pi$
$601$	−4.70119	−0.191766	−0.0958828	+	0.995393i	$0.530567\pi$
$602$	0	0
$603$	−16.1526	−0.657784
$604$	0	0
$605$	−4.42512	−0.179907
$606$	0	0
$607$	28.2376	1.14613	0.573065	−	0.819510i	$-0.305754\pi$
$607$	28.2376	1.14613	0.573065	+	0.819510i	$0.305754\pi$
$608$	0	0
$609$	10.2526	0.415457
$610$	0	0
$611$	−5.58715	−0.226032
$612$	0	0
$613$	1.37047	0.0553529	0.0276765	−	0.999617i	$-0.491189\pi$
$613$	1.37047	0.0553529	0.0276765	+	0.999617i	$0.491189\pi$
$614$	0	0
$615$	7.73238	0.311800
$616$	0	0
$617$	−27.6231	−1.11206	−0.556032	−	0.831161i	$-0.687677\pi$
$617$	−27.6231	−1.11206	−0.556032	+	0.831161i	$0.687677\pi$
$618$	0	0
$619$	21.8222	0.877110	0.438555	−	0.898704i	$-0.355490\pi$
$619$	21.8222	0.877110	0.438555	+	0.898704i	$0.355490\pi$
$620$	0	0
$621$	26.5602	1.06583
$622$	0	0
$623$	4.11881	0.165016
$624$	0	0
$625$	114.717	4.58866
$626$	0	0
$627$	−8.34500	−0.333267
$628$	0	0
$629$	−1.13476	−0.0452458
$630$	0	0
$631$	−13.0974	−0.521398	−0.260699	−	0.965420i	$-0.583953\pi$
$631$	−13.0974	−0.521398	−0.260699	+	0.965420i	$0.583953\pi$
$632$	0	0
$633$	11.1062	0.441431
$634$	0	0
$635$	5.99810	0.238028
$636$	0	0
$637$	−5.85774	−0.232092
$638$	0	0
$639$	−0.829032	−0.0327960
$640$	0	0
$641$	−4.04143	−0.159627	−0.0798134	−	0.996810i	$-0.525432\pi$
$641$	−4.04143	−0.159627	−0.0798134	+	0.996810i	$0.525432\pi$
$642$	0	0
$643$	−32.6346	−1.28698	−0.643492	−	0.765453i	$-0.722515\pi$
$643$	−32.6346	−1.28698	−0.643492	+	0.765453i	$0.722515\pi$
$644$	0	0
$645$	−12.9467	−0.509774
$646$	0	0
$647$	−46.8621	−1.84234	−0.921170	−	0.389160i	$-0.872765\pi$
$647$	−46.8621	−1.84234	−0.921170	+	0.389160i	$0.872765\pi$
$648$	0	0
$649$	−8.22916	−0.323023
$650$	0	0
$651$	1.61357	0.0632409
$652$	0	0
$653$	−23.8948	−0.935074	−0.467537	−	0.883973i	$-0.654859\pi$
$653$	−23.8948	−0.935074	−0.467537	+	0.883973i	$0.654859\pi$
$654$	0	0
$655$	44.2812	1.73021
$656$	0	0
$657$	−10.6726	−0.416379
$658$	0	0
$659$	17.6400	0.687157	0.343578	−	0.939124i	$-0.388361\pi$
$659$	17.6400	0.687157	0.343578	+	0.939124i	$0.388361\pi$
$660$	0	0
$661$	−22.1256	−0.860586	−0.430293	−	0.902689i	$-0.641590\pi$
$661$	−22.1256	−0.860586	−0.430293	+	0.902689i	$0.641590\pi$
$662$	0	0
$663$	10.2357	0.397522
$664$	0	0
$665$	32.7864	1.27140
$666$	0	0
$667$	−45.3690	−1.75670
$668$	0	0
$669$	12.8598	0.497187
$670$	0	0
$671$	−3.60512	−0.139174
$672$	0	0
$673$	−7.63999	−0.294500	−0.147250	−	0.989099i	$-0.547042\pi$
$673$	−7.63999	−0.294500	−0.147250	+	0.989099i	$0.547042\pi$
$674$	0	0
$675$	−77.7064	−2.99092
$676$	0	0
$677$	−22.6006	−0.868611	−0.434305	−	0.900766i	$-0.643006\pi$
$677$	−22.6006	−0.868611	−0.434305	+	0.900766i	$0.643006\pi$
$678$	0	0
$679$	−11.3290	−0.434769
$680$	0	0
$681$	15.9564	0.611451
$682$	0	0
$683$	−44.4414	−1.70050	−0.850252	−	0.526376i	$-0.823550\pi$
$683$	−44.4414	−1.70050	−0.850252	+	0.526376i	$0.823550\pi$
$684$	0	0
$685$	−81.5633	−3.11637
$686$	0	0
$687$	21.4255	0.817432
$688$	0	0
$689$	11.7155	0.446324
$690$	0	0
$691$	−47.1249	−1.79271	−0.896357	−	0.443333i	$-0.853796\pi$
$691$	−47.1249	−1.79271	−0.896357	+	0.443333i	$0.853796\pi$
$692$	0	0
$693$	−1.73143	−0.0657716
$694$	0	0
$695$	−12.8502	−0.487437
$696$	0	0
$697$	2.40692	0.0911687
$698$	0	0
$699$	−14.7578	−0.558193
$700$	0	0
$701$	23.6536	0.893382	0.446691	−	0.894688i	$-0.352602\pi$
$701$	23.6536	0.893382	0.446691	+	0.894688i	$0.352602\pi$
$702$	0	0
$703$	−5.41928	−0.204392
$704$	0	0
$705$	−4.75382	−0.179039
$706$	0	0
$707$	−5.24511	−0.197263
$708$	0	0
$709$	4.73143	0.177693	0.0888463	−	0.996045i	$-0.471682\pi$
$709$	4.73143	0.177693	0.0888463	+	0.996045i	$0.471682\pi$
$710$	0	0
$711$	−16.8769	−0.632934
$712$	0	0
$713$	−7.14025	−0.267404
$714$	0	0
$715$	25.9212	0.969397
$716$	0	0
$717$	−23.7852	−0.888275
$718$	0	0
$719$	4.14356	0.154529	0.0772643	−	0.997011i	$-0.475381\pi$
$719$	4.14356	0.154529	0.0772643	+	0.997011i	$0.475381\pi$
$720$	0	0
$721$	18.6543	0.694722
$722$	0	0
$723$	−27.8364	−1.03525
$724$	0	0
$725$	132.735	4.92964
$726$	0	0
$727$	51.1331	1.89642	0.948211	−	0.317642i	$-0.102891\pi$
$727$	51.1331	1.89642	0.948211	+	0.317642i	$0.102891\pi$
$728$	0	0
$729$	19.4052	0.718711
$730$	0	0
$731$	−4.03001	−0.149055
$732$	0	0
$733$	−34.7685	−1.28420	−0.642101	−	0.766620i	$-0.721937\pi$
$733$	−34.7685	−1.28420	−0.642101	+	0.766620i	$0.721937\pi$
$734$	0	0
$735$	−4.98405	−0.183839
$736$	0	0
$737$	9.32905	0.343640
$738$	0	0
$739$	47.0995	1.73258	0.866292	−	0.499538i	$-0.166497\pi$
$739$	47.0995	1.73258	0.866292	+	0.499538i	$0.166497\pi$
$740$	0	0
$741$	48.8828	1.79576
$742$	0	0
$743$	−6.12832	−0.224826	−0.112413	−	0.993662i	$-0.535858\pi$
$743$	−6.12832	−0.224826	−0.112413	+	0.993662i	$0.535858\pi$
$744$	0	0
$745$	−71.7121	−2.62733
$746$	0	0
$747$	−18.2820	−0.668905
$748$	0	0
$749$	−13.3555	−0.487998
$750$	0	0
$751$	28.8902	1.05422	0.527110	−	0.849797i	$-0.323276\pi$
$751$	28.8902	1.05422	0.527110	+	0.849797i	$0.323276\pi$
$752$	0	0
$753$	19.3576	0.705431
$754$	0	0
$755$	−24.9576	−0.908301
$756$	0	0
$757$	31.7140	1.15267	0.576333	−	0.817215i	$-0.304483\pi$
$757$	31.7140	1.15267	0.576333	+	0.817215i	$0.304483\pi$
$758$	0	0
$759$	−5.61357	−0.203760
$760$	0	0
$761$	11.7981	0.427681	0.213841	−	0.976869i	$-0.431403\pi$
$761$	11.7981	0.427681	0.213841	+	0.976869i	$0.431403\pi$
$762$	0	0
$763$	9.44785	0.342035
$764$	0	0
$765$	−11.8867	−0.429764
$766$	0	0
$767$	48.2043	1.74055
$768$	0	0
$769$	42.2305	1.52287	0.761435	−	0.648242i	$-0.224495\pi$
$769$	42.2305	1.52287	0.761435	+	0.648242i	$0.224495\pi$
$770$	0	0
$771$	28.6431	1.03156
$772$	0	0
$773$	−5.86156	−0.210826	−0.105413	−	0.994429i	$-0.533616\pi$
$773$	−5.86156	−0.210826	−0.105413	+	0.994429i	$0.533616\pi$
$774$	0	0
$775$	20.8900	0.750391
$776$	0	0
$777$	0.823815	0.0295542
$778$	0	0
$779$	11.4948	0.411843
$780$	0	0
$781$	0.478813	0.0171333
$782$	0	0
$783$	−48.5095	−1.73359
$784$	0	0
$785$	−3.06546	−0.109411
$786$	0	0
$787$	6.32132	0.225331	0.112665	−	0.993633i	$-0.464061\pi$
$787$	6.32132	0.225331	0.112665	+	0.993633i	$0.464061\pi$
$788$	0	0
$789$	−1.07858	−0.0383986
$790$	0	0
$791$	−13.5817	−0.482908
$792$	0	0
$793$	21.1179	0.749917
$794$	0	0
$795$	9.96809	0.353532
$796$	0	0
$797$	−18.5534	−0.657197	−0.328598	−	0.944470i	$-0.606576\pi$
$797$	−18.5534	−0.657197	−0.328598	+	0.944470i	$0.606576\pi$
$798$	0	0
$799$	−1.47976	−0.0523502
$800$	0	0
$801$	−7.13142	−0.251976
$802$	0	0
$803$	6.16405	0.217525
$804$	0	0
$805$	22.0550	0.777336
$806$	0	0
$807$	−7.69094	−0.270734
$808$	0	0
$809$	−23.1817	−0.815024	−0.407512	−	0.913200i	$-0.633604\pi$
$809$	−23.1817	−0.815024	−0.407512	+	0.913200i	$0.633604\pi$
$810$	0	0
$811$	37.5451	1.31839	0.659194	−	0.751973i	$-0.270898\pi$
$811$	37.5451	1.31839	0.659194	+	0.751973i	$0.270898\pi$
$812$	0	0
$813$	−20.3057	−0.712153
$814$	0	0
$815$	71.7785	2.51429
$816$	0	0
$817$	−19.2462	−0.673339
$818$	0	0
$819$	10.1423	0.354399
$820$	0	0
$821$	20.7109	0.722817	0.361408	−	0.932408i	$-0.382296\pi$
$821$	20.7109	0.722817	0.361408	+	0.932408i	$0.382296\pi$
$822$	0	0
$823$	13.3988	0.467052	0.233526	−	0.972351i	$-0.424974\pi$
$823$	13.3988	0.467052	0.233526	+	0.972351i	$0.424974\pi$
$824$	0	0
$825$	16.4234	0.571791
$826$	0	0
$827$	−31.8364	−1.10706	−0.553531	−	0.832829i	$-0.686720\pi$
$827$	−31.8364	−1.10706	−0.553531	+	0.832829i	$0.686720\pi$
$828$	0	0
$829$	13.8444	0.480836	0.240418	−	0.970669i	$-0.422716\pi$
$829$	13.8444	0.480836	0.240418	+	0.970669i	$0.422716\pi$
$830$	0	0
$831$	5.11022	0.177272
$832$	0	0
$833$	−1.55143	−0.0537537
$834$	0	0
$835$	56.8638	1.96785
$836$	0	0
$837$	−7.63450	−0.263887
$838$	0	0
$839$	17.9078	0.618247	0.309124	−	0.951022i	$-0.399964\pi$
$839$	17.9078	0.618247	0.309124	+	0.951022i	$0.399964\pi$
$840$	0	0
$841$	53.8619	1.85731
$842$	0	0
$843$	10.5540	0.363501
$844$	0	0
$845$	−94.3129	−3.24446
$846$	0	0
$847$	1.00000	0.0343604
$848$	0	0
$849$	2.51260	0.0862322
$850$	0	0
$851$	−3.64548	−0.124965
$852$	0	0
$853$	−11.1058	−0.380256	−0.190128	−	0.981759i	$-0.560890\pi$
$853$	−11.1058	−0.380256	−0.190128	+	0.981759i	$0.560890\pi$
$854$	0	0
$855$	−56.7674	−1.94140
$856$	0	0
$857$	45.4048	1.55100	0.775499	−	0.631349i	$-0.217499\pi$
$857$	45.4048	1.55100	0.775499	+	0.631349i	$0.217499\pi$
$858$	0	0
$859$	19.6711	0.671169	0.335584	−	0.942010i	$-0.391066\pi$
$859$	19.6711	0.671169	0.335584	+	0.942010i	$0.391066\pi$
$860$	0	0
$861$	−1.74738	−0.0595507
$862$	0	0
$863$	−2.92572	−0.0995926	−0.0497963	−	0.998759i	$-0.515857\pi$
$863$	−2.92572	−0.0995926	−0.0497963	+	0.998759i	$0.515857\pi$
$864$	0	0
$865$	−57.8274	−1.96619
$866$	0	0
$867$	−16.4363	−0.558206
$868$	0	0
$869$	9.74738	0.330657
$870$	0	0
$871$	−54.6471	−1.85165
$872$	0	0
$873$	19.6155	0.663882
$874$	0	0
$875$	−42.4000	−1.43338
$876$	0	0
$877$	−27.3521	−0.923616	−0.461808	−	0.886980i	$-0.652799\pi$
$877$	−27.3521	−0.923616	−0.461808	+	0.886980i	$0.652799\pi$
$878$	0	0
$879$	−23.2195	−0.783174
$880$	0	0
$881$	−25.6722	−0.864917	−0.432458	−	0.901654i	$-0.642354\pi$
$881$	−25.6722	−0.864917	−0.432458	+	0.901654i	$0.642354\pi$
$882$	0	0
$883$	−9.63665	−0.324299	−0.162150	−	0.986766i	$-0.551843\pi$
$883$	−9.63665	−0.324299	−0.162150	+	0.986766i	$0.551843\pi$
$884$	0	0
$885$	41.0145	1.37869
$886$	0	0
$887$	43.2902	1.45354	0.726772	−	0.686879i	$-0.241020\pi$
$887$	43.2902	1.45354	0.726772	+	0.686879i	$0.241020\pi$
$888$	0	0
$889$	−1.35547	−0.0454609
$890$	0	0
$891$	−0.807862	−0.0270644
$892$	0	0
$893$	−7.06691	−0.236485
$894$	0	0
$895$	−52.4345	−1.75269
$896$	0	0
$897$	32.8828	1.09793
$898$	0	0
$899$	13.0409	0.434939
$900$	0	0
$901$	3.10285	0.103371
$902$	0	0
$903$	2.92572	0.0973618
$904$	0	0
$905$	−33.3121	−1.10733
$906$	0	0
$907$	−32.9107	−1.09278	−0.546391	−	0.837530i	$-0.683999\pi$
$907$	−32.9107	−1.09278	−0.546391	+	0.837530i	$0.683999\pi$
$908$	0	0
$909$	9.08155	0.301216
$910$	0	0
$911$	−29.3086	−0.971036	−0.485518	−	0.874227i	$-0.661369\pi$
$911$	−29.3086	−0.971036	−0.485518	+	0.874227i	$0.661369\pi$
$912$	0	0
$913$	10.5589	0.349450
$914$	0	0
$915$	17.9681	0.594007
$916$	0	0
$917$	−10.0068	−0.330453
$918$	0	0
$919$	25.1314	0.829009	0.414505	−	0.910047i	$-0.363955\pi$
$919$	25.1314	0.829009	0.414505	+	0.910047i	$0.363955\pi$
$920$	0	0
$921$	28.9031	0.952389
$922$	0	0
$923$	−2.80476	−0.0923199
$924$	0	0
$925$	10.6655	0.350678
$926$	0	0
$927$	−32.2986	−1.06082
$928$	0	0
$929$	37.4369	1.22826	0.614132	−	0.789203i	$-0.289506\pi$
$929$	37.4369	1.22826	0.614132	+	0.789203i	$0.289506\pi$
$930$	0	0
$931$	−7.40916	−0.242826
$932$	0	0
$933$	20.5624	0.673182
$934$	0	0
$935$	6.86524	0.224517
$936$	0	0
$937$	1.17641	0.0384317	0.0192158	−	0.999815i	$-0.493883\pi$
$937$	1.17641	0.0384317	0.0192158	+	0.999815i	$0.493883\pi$
$938$	0	0
$939$	26.4033	0.861640
$940$	0	0
$941$	−35.0361	−1.14214	−0.571072	−	0.820900i	$-0.693472\pi$
$941$	−35.0361	−1.14214	−0.571072	+	0.820900i	$0.693472\pi$
$942$	0	0
$943$	7.73238	0.251801
$944$	0	0
$945$	23.5817	0.767112
$946$	0	0
$947$	46.1459	1.49954	0.749771	−	0.661698i	$-0.230164\pi$
$947$	46.1459	1.49954	0.749771	+	0.661698i	$0.230164\pi$
$948$	0	0
$949$	−36.1074	−1.17210
$950$	0	0
$951$	5.71763	0.185407
$952$	0	0
$953$	3.52667	0.114240	0.0571201	−	0.998367i	$-0.481808\pi$
$953$	3.52667	0.114240	0.0571201	+	0.998367i	$0.481808\pi$
$954$	0	0
$955$	107.639	3.48311
$956$	0	0
$957$	10.2526	0.331420
$958$	0	0
$959$	18.4319	0.595197
$960$	0	0
$961$	−28.9476	−0.933793
$962$	0	0
$963$	23.1241	0.745162
$964$	0	0
$965$	36.7864	1.18420
$966$	0	0
$967$	7.34046	0.236053	0.118027	−	0.993010i	$-0.462343\pi$
$967$	7.34046	0.236053	0.118027	+	0.993010i	$0.462343\pi$
$968$	0	0
$969$	12.9467	0.415906
$970$	0	0
$971$	−42.5254	−1.36470	−0.682352	−	0.731024i	$-0.739043\pi$
$971$	−42.5254	−1.36470	−0.682352	+	0.731024i	$0.739043\pi$
$972$	0	0
$973$	2.90393	0.0930957
$974$	0	0
$975$	−96.2043	−3.08100
$976$	0	0
$977$	−43.8812	−1.40388	−0.701942	−	0.712234i	$-0.747683\pi$
$977$	−43.8812	−1.40388	−0.701942	+	0.712234i	$0.747683\pi$
$978$	0	0
$979$	4.11881	0.131638
$980$	0	0
$981$	−16.3583	−0.522280
$982$	0	0
$983$	12.5993	0.401855	0.200927	−	0.979606i	$-0.435604\pi$
$983$	12.5993	0.401855	0.200927	+	0.979606i	$0.435604\pi$
$984$	0	0
$985$	−46.9621	−1.49634
$986$	0	0
$987$	1.07428	0.0341947
$988$	0	0
$989$	−12.9467	−0.411680
$990$	0	0
$991$	23.5443	0.747908	0.373954	−	0.927447i	$-0.378002\pi$
$991$	23.5443	0.747908	0.373954	+	0.927447i	$0.378002\pi$
$992$	0	0
$993$	−8.80691	−0.279479
$994$	0	0
$995$	31.1802	0.988480
$996$	0	0
$997$	11.7654	0.372612	0.186306	−	0.982492i	$-0.440348\pi$
$997$	11.7654	0.372612	0.186306	+	0.982492i	$0.440348\pi$
$998$	0	0
$999$	−3.89782	−0.123322

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4928.2.a.ch.1.3		4
4.3	odd	2		4928.2.a.cc.1.2		4
8.3	odd	2		616.2.a.h.1.3	✓	4
8.5	even	2		1232.2.a.s.1.2		4
24.11	even	2		5544.2.a.bm.1.1		4
56.13	odd	2		8624.2.a.cy.1.3		4
56.27	even	2		4312.2.a.z.1.2		4
88.43	even	2		6776.2.a.bb.1.3		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
616.2.a.h.1.3	✓	4	8.3	odd	2
1232.2.a.s.1.2		4	8.5	even	2
4312.2.a.z.1.2		4	56.27	even	2
4928.2.a.cc.1.2		4	4.3	odd	2
4928.2.a.ch.1.3		4	1.1	even	1	trivial
5544.2.a.bm.1.1		4	24.11	even	2
6776.2.a.bb.1.3		4	88.43	even	2
8624.2.a.cy.1.3		4	56.13	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:	\( N \)	\(=\)	\( 4928 = 2^{6} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4928.a (trivial)

\(f(q)\)	\(=\)	\(q+1.12631 q^{3} -4.42512 q^{5} +1.00000 q^{7} -1.73143 q^{9} +1.00000 q^{11} -5.85774 q^{13} -4.98405 q^{15} -1.55143 q^{17} -7.40916 q^{19} +1.12631 q^{21} -4.98405 q^{23} +14.5817 q^{25} -5.32905 q^{27} +9.10285 q^{29} +1.43262 q^{31} +1.12631 q^{33} -4.42512 q^{35} +0.731430 q^{37} -6.59762 q^{39} -1.55143 q^{41} +2.59762 q^{43} +7.66178 q^{45} +0.953807 q^{47} +1.00000 q^{49} -1.74738 q^{51} -2.00000 q^{53} -4.42512 q^{55} -8.34500 q^{57} -8.22916 q^{59} -3.60512 q^{61} -1.73143 q^{63} +25.9212 q^{65} +9.32905 q^{67} -5.61357 q^{69} +0.478813 q^{71} +6.16405 q^{73} +16.4234 q^{75} +1.00000 q^{77} +9.74738 q^{79} -0.807862 q^{81} +10.5589 q^{83} +6.86524 q^{85} +10.2526 q^{87} +4.11881 q^{89} -5.85774 q^{91} +1.61357 q^{93} +32.7864 q^{95} -11.3290 q^{97} -1.73143 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q + q^{3} - 5 q^{5} + 4 q^{7} + 9 q^{9} + 4 q^{11} - 4 q^{13} + 3 q^{15} + 10 q^{17} + 6 q^{19} + q^{21} + 3 q^{23} + 17 q^{25} + 13 q^{27} + 4 q^{29} - q^{31} + q^{33} - 5 q^{35} - 13 q^{37} - 8 q^{39}+ \cdots + 9 q^{99}+O(q^{100}) \) 4 * q + q^3 - 5 * q^5 + 4 * q^7 + 9 * q^9 + 4 * q^11 - 4 * q^13 + 3 * q^15 + 10 * q^17 + 6 * q^19 + q^21 + 3 * q^23 + 17 * q^25 + 13 * q^27 + 4 * q^29 - q^31 + q^33 - 5 * q^35 - 13 * q^37 - 8 * q^39 + 10 * q^41 - 8 * q^43 - 12 * q^45 + 6 * q^47 + 4 * q^49 - 14 * q^51 - 8 * q^53 - 5 * q^55 - 22 * q^57 + 3 * q^59 - 2 * q^61 + 9 * q^63 + 3 * q^67 - 27 * q^69 - 7 * q^71 + 2 * q^73 - 18 * q^75 + 4 * q^77 + 46 * q^79 + 40 * q^81 + 32 * q^83 + 14 * q^85 + 34 * q^87 + 7 * q^89 - 4 * q^91 + 11 * q^93 + 14 * q^95 - 11 * q^97 + 9 * q^99

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