LMFDB - Embedded newform 425.2.a.j.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [425,2,Mod(1,425)] 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("425.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(425, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 425 = 5^{2} \cdot 17 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	425.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,1,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(3)] == traces)

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

Self dual:	yes
Analytic conductor:	$3.39364208590$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.1893456.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - x^{4} - 10x^{3} + 10x^{2} + 23x - 25 $ x^5 - x^4 - 10x^3 + 10x^2 + 23*x - 25
Coefficient ring:	$\Z[a_1, a_2, a_3]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.60789$ of defining polynomial
Character	$\chi$	$=$	425.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.80107 q^{2} -2.60789 q^{3} +5.84602 q^{4} -7.30489 q^{6} +1.33298 q^{7} +10.7730 q^{8} +3.80107 q^{9} +1.09485 q^{11} -15.2458 q^{12} -3.17083 q^{13} +3.73378 q^{14} +18.4839 q^{16} -1.00000 q^{17} +10.6471 q^{18} +2.75613 q^{19} -3.47626 q^{21} +3.06675 q^{22} -3.57111 q^{23} -28.0947 q^{24} -8.88173 q^{26} -2.08911 q^{27} +7.79262 q^{28} -0.180058 q^{29} +0.816039 q^{31} +30.2288 q^{32} -2.85524 q^{33} -2.80107 q^{34} +22.2211 q^{36} -8.44817 q^{37} +7.72013 q^{38} +8.26917 q^{39} -7.97191 q^{41} -9.73726 q^{42} -6.54798 q^{43} +6.40051 q^{44} -10.0029 q^{46} -0.576074 q^{47} -48.2039 q^{48} -5.22317 q^{49} +2.60789 q^{51} -18.5367 q^{52} -7.84602 q^{53} -5.85176 q^{54} +14.3602 q^{56} -7.18768 q^{57} -0.504355 q^{58} -9.76375 q^{59} -5.21577 q^{61} +2.28579 q^{62} +5.06675 q^{63} +47.7053 q^{64} -7.99775 q^{66} +2.64963 q^{67} -5.84602 q^{68} +9.31305 q^{69} +13.4114 q^{71} +40.9489 q^{72} +12.3299 q^{73} -23.6639 q^{74} +16.1124 q^{76} +1.45941 q^{77} +23.1626 q^{78} +3.74745 q^{79} -5.95506 q^{81} -22.3299 q^{82} +4.66596 q^{83} -20.3223 q^{84} -18.3414 q^{86} +0.469570 q^{87} +11.7948 q^{88} +3.00946 q^{89} -4.22665 q^{91} -20.8768 q^{92} -2.12814 q^{93} -1.61363 q^{94} -78.8332 q^{96} +12.2681 q^{97} -14.6305 q^{98} +4.16160 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + q^{2} - q^{3} + 11 q^{4} + 3 q^{6} - q^{7} + 9 q^{8} + 6 q^{9} + 4 q^{11} - 17 q^{12} + 3 q^{13} - 7 q^{14} + 27 q^{16} - 5 q^{17} + 22 q^{18} + 6 q^{19} - 5 q^{21} - 18 q^{22} - 4 q^{23} - 19 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) $ 5 * q + q^2 - q^3 + 11 * q^4 + 3 * q^6 - q^7 + 9 * q^8 + 6 * q^9 + 4 * q^11 - 17 * q^12 + 3 * q^13 - 7 * q^14 + 27 * q^16 - 5 * q^17 + 22 * q^18 + 6 * q^19 - 5 * q^21 - 18 * q^22 - 4 * q^23 - 19 * q^24 - 5 * q^26 + 5 * q^27 + 15 * q^28 + 2 * q^29 + 21 * q^31 + 9 * q^32 - 12 * q^33 - q^34 + 22 * q^36 + 2 * q^37 + 34 * q^38 + 23 * q^39 - 8 * q^41 - 25 * q^42 + 4 * q^43 - 4 * q^44 - 6 * q^46 + 2 * q^47 - 51 * q^48 + 10 * q^49 + q^51 + 13 * q^52 - 21 * q^53 - 25 * q^54 - 29 * q^56 + 20 * q^57 - 22 * q^58 + 12 * q^59 - 2 * q^61 - 3 * q^62 - 8 * q^63 + 43 * q^64 - 28 * q^66 + 12 * q^67 - 11 * q^68 + 21 * q^71 + 58 * q^72 - 22 * q^73 - 50 * q^74 - 14 * q^76 + 32 * q^77 + 53 * q^78 + 41 * q^79 - 35 * q^81 - 28 * q^82 + 8 * q^83 - 71 * q^84 - 36 * q^86 + 4 * q^87 - 14 * q^88 - 10 * q^89 + 13 * q^91 - 56 * q^92 - 9 * q^93 - 10 * q^94 - 85 * q^96 + 20 * q^97 - 38 * q^98 - 14 * 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 $

$2$	2.80107	1.98066	0.990329	−	0.138737i	$-0.0443042\pi$
$2$	2.80107	1.98066	0.990329	+	0.138737i	$0.0443042\pi$
$3$	−2.60789	−1.50566	−0.752832	−	0.658213i	$-0.771313\pi$
$3$	−2.60789	−1.50566	−0.752832	+	0.658213i	$0.771313\pi$
$4$	5.84602	2.92301
$5$	0	0
$5$	0	0
$6$	−7.30489	−2.98221
$7$	1.33298	0.503819	0.251909	−	0.967751i	$-0.418941\pi$
$7$	1.33298	0.503819	0.251909	+	0.967751i	$0.418941\pi$
$8$	10.7730	3.80882
$9$	3.80107	1.26702
$10$	0	0
$11$	1.09485	0.330110	0.165055	−	0.986284i	$-0.447220\pi$
$11$	1.09485	0.330110	0.165055	+	0.986284i	$0.447220\pi$
$12$	−15.2458	−4.40107
$13$	−3.17083	−0.879430	−0.439715	−	0.898137i	$-0.644921\pi$
$13$	−3.17083	−0.879430	−0.439715	+	0.898137i	$0.644921\pi$
$14$	3.73378	0.997893
$15$	0	0
$16$	18.4839	4.62097
$17$	−1.00000	−0.242536
$17$	−1.00000	−0.242536
$18$	10.6471	2.50954
$19$	2.75613	0.632300	0.316150	−	0.948709i	$-0.397610\pi$
$19$	2.75613	0.632300	0.316150	+	0.948709i	$0.397610\pi$
$20$	0	0
$21$	−3.47626	−0.758582
$22$	3.06675	0.653834
$23$	−3.57111	−0.744628	−0.372314	−	0.928107i	$-0.621436\pi$
$23$	−3.57111	−0.744628	−0.372314	+	0.928107i	$0.621436\pi$
$24$	−28.0947	−5.73481
$25$	0	0
$26$	−8.88173	−1.74185
$27$	−2.08911	−0.402050
$28$	7.79262	1.47267
$29$	−0.180058	−0.0334359	−0.0167179	−	0.999860i	$-0.505322\pi$
$29$	−0.180058	−0.0334359	−0.0167179	+	0.999860i	$0.505322\pi$
$30$	0	0
$31$	0.816039	0.146565	0.0732825	−	0.997311i	$-0.476653\pi$
$31$	0.816039	0.146565	0.0732825	+	0.997311i	$0.476653\pi$
$32$	30.2288	5.34374
$33$	−2.85524	−0.497034
$34$	−2.80107	−0.480380
$35$	0	0
$36$	22.2211	3.70352
$37$	−8.44817	−1.38887	−0.694435	−	0.719555i	$-0.744346\pi$
$37$	−8.44817	−1.38887	−0.694435	+	0.719555i	$0.744346\pi$
$38$	7.72013	1.25237
$39$	8.26917	1.32413
$40$	0	0
$41$	−7.97191	−1.24500	−0.622501	−	0.782619i	$-0.713883\pi$
$41$	−7.97191	−1.24500	−0.622501	+	0.782619i	$0.713883\pi$
$42$	−9.73726	−1.50249
$43$	−6.54798	−0.998557	−0.499279	−	0.866441i	$-0.666402\pi$
$43$	−6.54798	−0.998557	−0.499279	+	0.866441i	$0.666402\pi$
$44$	6.40051	0.964913
$45$	0	0
$46$	−10.0029	−1.47485
$47$	−0.576074	−0.0840290	−0.0420145	−	0.999117i	$-0.513378\pi$
$47$	−0.576074	−0.0840290	−0.0420145	+	0.999117i	$0.513378\pi$
$48$	−48.2039	−6.95763
$49$	−5.22317	−0.746166
$50$	0	0
$51$	2.60789	0.365177
$52$	−18.5367	−2.57058
$53$	−7.84602	−1.07773	−0.538867	−	0.842391i	$-0.681147\pi$
$53$	−7.84602	−1.07773	−0.538867	+	0.842391i	$0.681147\pi$
$54$	−5.85176	−0.796323
$55$	0	0
$56$	14.3602	1.91896
$57$	−7.18768	−0.952031
$58$	−0.504355	−0.0662251
$59$	−9.76375	−1.27113	−0.635566	−	0.772046i	$-0.719233\pi$
$59$	−9.76375	−1.27113	−0.635566	+	0.772046i	$0.719233\pi$
$60$	0	0
$61$	−5.21577	−0.667811	−0.333906	−	0.942606i	$-0.608367\pi$
$61$	−5.21577	−0.667811	−0.333906	+	0.942606i	$0.608367\pi$
$62$	2.28579	0.290295
$63$	5.06675	0.638351
$64$	47.7053	5.96316
$65$	0	0
$66$	−7.99775	−0.984455
$67$	2.64963	0.323704	0.161852	−	0.986815i	$-0.448253\pi$
$67$	2.64963	0.323704	0.161852	+	0.986815i	$0.448253\pi$
$68$	−5.84602	−0.708934
$69$	9.31305	1.12116
$70$	0	0
$71$	13.4114	1.59164	0.795820	−	0.605534i	$-0.207040\pi$
$71$	13.4114	1.59164	0.795820	+	0.605534i	$0.207040\pi$
$72$	40.9489	4.82587
$73$	12.3299	1.44311	0.721553	−	0.692359i	$-0.243429\pi$
$73$	12.3299	1.44311	0.721553	+	0.692359i	$0.243429\pi$
$74$	−23.6639	−2.75088
$75$	0	0
$76$	16.1124	1.84822
$77$	1.45941	0.166315
$78$	23.1626	2.62264
$79$	3.74745	0.421621	0.210810	−	0.977527i	$-0.432390\pi$
$79$	3.74745	0.421621	0.210810	+	0.977527i	$0.432390\pi$
$80$	0	0
$81$	−5.95506	−0.661673
$82$	−22.3299	−2.46592
$83$	4.66596	0.512156	0.256078	−	0.966656i	$-0.417570\pi$
$83$	4.66596	0.512156	0.256078	+	0.966656i	$0.417570\pi$
$84$	−20.3223	−2.21734
$85$	0	0
$86$	−18.3414	−1.97780
$87$	0.469570	0.0503432
$88$	11.7948	1.25733
$89$	3.00946	0.319002	0.159501	−	0.987198i	$-0.449012\pi$
$89$	3.00946	0.319002	0.159501	+	0.987198i	$0.449012\pi$
$90$	0	0
$91$	−4.22665	−0.443074
$92$	−20.8768	−2.17655
$93$	−2.12814	−0.220678
$94$	−1.61363	−0.166433
$95$	0	0
$96$	−78.8332	−8.04588
$97$	12.2681	1.24564	0.622819	−	0.782366i	$-0.285987\pi$
$97$	12.2681	1.24564	0.622819	+	0.782366i	$0.285987\pi$
$98$	−14.6305	−1.47790
$99$	4.16160	0.418257
$100$	0	0
$101$	2.98113	0.296634	0.148317	−	0.988940i	$-0.452614\pi$
$101$	2.98113	0.296634	0.148317	+	0.988940i	$0.452614\pi$
$102$	7.30489	0.723291
$103$	13.8440	1.36409	0.682045	−	0.731310i	$-0.261091\pi$
$103$	13.8440	1.36409	0.682045	+	0.731310i	$0.261091\pi$
$104$	−34.1593	−3.34959
$105$	0	0
$106$	−21.9773	−2.13462
$107$	7.87103	0.760921	0.380461	−	0.924797i	$-0.375765\pi$
$107$	7.87103	0.760921	0.380461	+	0.924797i	$0.375765\pi$
$108$	−12.2130	−1.17519
$109$	13.6739	1.30972	0.654859	−	0.755751i	$-0.272728\pi$
$109$	13.6739	1.30972	0.654859	+	0.755751i	$0.272728\pi$
$110$	0	0
$111$	22.0319	2.09117
$112$	24.6386	2.32813
$113$	−19.5722	−1.84120	−0.920600	−	0.390508i	$-0.872300\pi$
$113$	−19.5722	−1.84120	−0.920600	+	0.390508i	$0.872300\pi$
$114$	−20.1332	−1.88565
$115$	0	0
$116$	−1.05262	−0.0977334
$117$	−12.0526	−1.11426
$118$	−27.3490	−2.51768
$119$	−1.33298	−0.122194
$120$	0	0
$121$	−9.80130	−0.891028
$122$	−14.6098	−1.32271
$123$	20.7898	1.87456
$124$	4.77058	0.428411
$125$	0	0
$126$	14.1924	1.26436
$127$	5.00946	0.444517	0.222259	−	0.974988i	$-0.428657\pi$
$127$	5.00946	0.444517	0.222259	+	0.974988i	$0.428657\pi$
$128$	73.1684	6.46724
$129$	17.0764	1.50349
$130$	0	0
$131$	2.63893	0.230564	0.115282	−	0.993333i	$-0.463223\pi$
$131$	2.63893	0.230564	0.115282	+	0.993333i	$0.463223\pi$
$132$	−16.6918	−1.45284
$133$	3.67387	0.318565
$134$	7.42180	0.641146
$135$	0	0
$136$	−10.7730	−0.923775
$137$	3.27064	0.279430	0.139715	−	0.990192i	$-0.455381\pi$
$137$	3.27064	0.279430	0.139715	+	0.990192i	$0.455381\pi$
$138$	26.0865	2.22063
$139$	−10.3901	−0.881276	−0.440638	−	0.897685i	$-0.645248\pi$
$139$	−10.3901	−0.881276	−0.440638	+	0.897685i	$0.645248\pi$
$140$	0	0
$141$	1.50234	0.126519
$142$	37.5663	3.15249
$143$	−3.47158	−0.290308
$144$	70.2586	5.85488
$145$	0	0
$146$	34.5370	2.85830
$147$	13.6214	1.12348
$148$	−49.3881	−4.05968
$149$	−11.2566	−0.922179	−0.461090	−	0.887354i	$-0.652541\pi$
$149$	−11.2566	−0.922179	−0.461090	+	0.887354i	$0.652541\pi$
$150$	0	0
$151$	7.13736	0.580831	0.290415	−	0.956901i	$-0.406207\pi$
$151$	7.13736	0.580831	0.290415	+	0.956901i	$0.406207\pi$
$152$	29.6917	2.40832
$153$	−3.80107	−0.307299
$154$	4.08792	0.329414
$155$	0	0
$156$	48.3417	3.87043
$157$	8.30595	0.662887	0.331443	−	0.943475i	$-0.392464\pi$
$157$	8.30595	0.662887	0.331443	+	0.943475i	$0.392464\pi$
$158$	10.4969	0.835087
$159$	20.4615	1.62270
$160$	0	0
$161$	−4.76022	−0.375158
$162$	−16.6806	−1.31055
$163$	−11.6059	−0.909042	−0.454521	−	0.890736i	$-0.650189\pi$
$163$	−11.6059	−0.909042	−0.454521	+	0.890736i	$0.650189\pi$
$164$	−46.6039	−3.63915
$165$	0	0
$166$	13.0697	1.01441
$167$	19.6832	1.52313	0.761566	−	0.648087i	$-0.224431\pi$
$167$	19.6832	1.52313	0.761566	+	0.648087i	$0.224431\pi$
$168$	−37.4497	−2.88931
$169$	−2.94583	−0.226602
$170$	0	0
$171$	10.4763	0.801140
$172$	−38.2796	−2.91879
$173$	−19.8536	−1.50944	−0.754722	−	0.656045i	$-0.772228\pi$
$173$	−19.8536	−1.50944	−0.754722	+	0.656045i	$0.772228\pi$
$174$	1.31530	0.0997127
$175$	0	0
$176$	20.2371	1.52543
$177$	25.4628	1.91390
$178$	8.42971	0.631834
$179$	−6.71142	−0.501635	−0.250817	−	0.968034i	$-0.580699\pi$
$179$	−6.71142	−0.501635	−0.250817	+	0.968034i	$0.580699\pi$
$180$	0	0
$181$	1.49564	0.111170	0.0555852	−	0.998454i	$-0.482298\pi$
$181$	1.49564	0.111170	0.0555852	+	0.998454i	$0.482298\pi$
$182$	−11.8392	−0.877578
$183$	13.6021	1.00550
$184$	−38.4715	−2.83616
$185$	0	0
$186$	−5.96107	−0.437087
$187$	−1.09485	−0.0800633
$188$	−3.36774	−0.245617
$189$	−2.78474	−0.202560
$190$	0	0
$191$	−15.9320	−1.15280	−0.576401	−	0.817167i	$-0.695543\pi$
$191$	−15.9320	−1.15280	−0.576401	+	0.817167i	$0.695543\pi$
$192$	−124.410	−8.97851
$193$	17.4673	1.25732	0.628661	−	0.777680i	$-0.283603\pi$
$193$	17.4673	1.25732	0.628661	+	0.777680i	$0.283603\pi$
$194$	34.3639	2.46718
$195$	0	0
$196$	−30.5347	−2.18105
$197$	14.0436	1.00057	0.500283	−	0.865862i	$-0.333229\pi$
$197$	14.0436	1.00057	0.500283	+	0.865862i	$0.333229\pi$
$198$	11.6570	0.828424
$199$	−4.58571	−0.325072	−0.162536	−	0.986703i	$-0.551967\pi$
$199$	−4.58571	−0.325072	−0.162536	+	0.986703i	$0.551967\pi$
$200$	0	0
$201$	−6.90993	−0.487389
$202$	8.35037	0.587530
$203$	−0.240013	−0.0168456
$204$	15.2458	1.06742
$205$	0	0
$206$	38.7781	2.70180
$207$	−13.5741	−0.943462
$208$	−58.6093	−4.06382
$209$	3.01755	0.208728
$210$	0	0
$211$	7.23345	0.497971	0.248986	−	0.968507i	$-0.419903\pi$
$211$	7.23345	0.497971	0.248986	+	0.968507i	$0.419903\pi$
$212$	−45.8680	−3.15022
$213$	−34.9754	−2.39647
$214$	22.0473	1.50713
$215$	0	0
$216$	−22.5060	−1.53134
$217$	1.08776	0.0738422
$218$	38.3015	2.59411
$219$	−32.1550	−2.17283
$220$	0	0
$221$	3.17083	0.213293
$222$	61.7129	4.14190
$223$	22.2566	1.49041	0.745207	−	0.666833i	$-0.232351\pi$
$223$	22.2566	1.49041	0.745207	+	0.666833i	$0.232351\pi$
$224$	40.2943	2.69228
$225$	0	0
$226$	−54.8232	−3.64679
$227$	−13.3541	−0.886342	−0.443171	−	0.896437i	$-0.646147\pi$
$227$	−13.3541	−0.886342	−0.443171	+	0.896437i	$0.646147\pi$
$228$	−42.0193	−2.78280
$229$	−20.4647	−1.35235	−0.676174	−	0.736742i	$-0.736363\pi$
$229$	−20.4647	−1.35235	−0.676174	+	0.736742i	$0.736363\pi$
$230$	0	0
$231$	−3.80598	−0.250415
$232$	−1.93976	−0.127351
$233$	16.1239	1.05631	0.528155	−	0.849148i	$-0.322884\pi$
$233$	16.1239	1.05631	0.528155	+	0.849148i	$0.322884\pi$
$234$	−33.7601	−2.20697
$235$	0	0
$236$	−57.0791	−3.71553
$237$	−9.77292	−0.634820
$238$	−3.73378	−0.242025
$239$	−4.38637	−0.283731	−0.141865	−	0.989886i	$-0.545310\pi$
$239$	−4.38637	−0.283731	−0.141865	+	0.989886i	$0.545310\pi$
$240$	0	0
$241$	−14.8343	−0.955558	−0.477779	−	0.878480i	$-0.658558\pi$
$241$	−14.8343	−0.955558	−0.477779	+	0.878480i	$0.658558\pi$
$242$	−27.4542	−1.76482
$243$	21.7974	1.39831
$244$	−30.4915	−1.95202
$245$	0	0
$246$	58.2339	3.71285
$247$	−8.73923	−0.556064
$248$	8.79118	0.558240
$249$	−12.1683	−0.771134
$250$	0	0
$251$	14.5998	0.921534	0.460767	−	0.887521i	$-0.347574\pi$
$251$	14.5998	0.921534	0.460767	+	0.887521i	$0.347574\pi$
$252$	29.6203	1.86591
$253$	−3.90983	−0.245809
$254$	14.0319	0.880437
$255$	0	0
$256$	109.540	6.84623
$257$	−5.34722	−0.333550	−0.166775	−	0.985995i	$-0.553335\pi$
$257$	−5.34722	−0.333550	−0.166775	+	0.985995i	$0.553335\pi$
$258$	47.8322	2.97790
$259$	−11.2612	−0.699739
$260$	0	0
$261$	−0.684413	−0.0423641
$262$	7.39183	0.456669
$263$	−18.3754	−1.13307	−0.566537	−	0.824037i	$-0.691717\pi$
$263$	−18.3754	−1.13307	−0.566537	+	0.824037i	$0.691717\pi$
$264$	−30.7595	−1.89312
$265$	0	0
$266$	10.2908	0.630968
$267$	−7.84832	−0.480310
$268$	15.4898	0.946188
$269$	−15.0812	−0.919516	−0.459758	−	0.888044i	$-0.652064\pi$
$269$	−15.0812	−0.919516	−0.459758	+	0.888044i	$0.652064\pi$
$270$	0	0
$271$	20.6031	1.25155	0.625774	−	0.780004i	$-0.284783\pi$
$271$	20.6031	1.25155	0.625774	+	0.780004i	$0.284783\pi$
$272$	−18.4839	−1.12075
$273$	11.0226	0.667120
$274$	9.16132	0.553455
$275$	0	0
$276$	54.4443	3.27716
$277$	3.93417	0.236381	0.118191	−	0.992991i	$-0.462291\pi$
$277$	3.93417	0.236381	0.118191	+	0.992991i	$0.462291\pi$
$278$	−29.1034	−1.74551
$279$	3.10183	0.185701
$280$	0	0
$281$	24.7878	1.47872	0.739358	−	0.673312i	$-0.235129\pi$
$281$	24.7878	1.47872	0.739358	+	0.673312i	$0.235129\pi$
$282$	4.20815	0.250592
$283$	−10.2892	−0.611631	−0.305815	−	0.952091i	$-0.598929\pi$
$283$	−10.2892	−0.611631	−0.305815	+	0.952091i	$0.598929\pi$
$284$	78.4032	4.65237
$285$	0	0
$286$	−9.72416	−0.575002
$287$	−10.6264	−0.627256
$288$	114.902	6.77065
$289$	1.00000	0.0588235
$290$	0	0
$291$	−31.9938	−1.87551
$292$	72.0808	4.21821
$293$	4.20448	0.245628	0.122814	−	0.992430i	$-0.460808\pi$
$293$	4.20448	0.245628	0.122814	+	0.992430i	$0.460808\pi$
$294$	38.1546	2.22522
$295$	0	0
$296$	−91.0119	−5.28996
$297$	−2.28726	−0.132720
$298$	−31.5307	−1.82652
$299$	11.3234	0.654848
$300$	0	0
$301$	−8.72832	−0.503092
$302$	19.9923	1.15043
$303$	−7.77446	−0.446631
$304$	50.9440	2.92184
$305$	0	0
$306$	−10.6471	−0.608654
$307$	−20.2969	−1.15841	−0.579204	−	0.815183i	$-0.696637\pi$
$307$	−20.2969	−1.15841	−0.579204	+	0.815183i	$0.696637\pi$
$308$	8.53175	0.486141
$309$	−36.1036	−2.05386
$310$	0	0
$311$	8.03850	0.455822	0.227911	−	0.973682i	$-0.426811\pi$
$311$	8.03850	0.455822	0.227911	+	0.973682i	$0.426811\pi$
$312$	89.0836	5.04337
$313$	29.6525	1.67606	0.838028	−	0.545627i	$-0.183708\pi$
$313$	29.6525	1.67606	0.838028	+	0.545627i	$0.183708\pi$
$314$	23.2656	1.31295
$315$	0	0
$316$	21.9077	1.23240
$317$	−3.06410	−0.172097	−0.0860484	−	0.996291i	$-0.527424\pi$
$317$	−3.06410	−0.172097	−0.0860484	+	0.996291i	$0.527424\pi$
$318$	57.3143	3.21402
$319$	−0.197136	−0.0110375
$320$	0	0
$321$	−20.5268	−1.14569
$322$	−13.3337	−0.743059
$323$	−2.75613	−0.153355
$324$	−34.8134	−1.93408
$325$	0	0
$326$	−32.5089	−1.80050
$327$	−35.6599	−1.97200
$328$	−85.8812	−4.74199
$329$	−0.767895	−0.0423354
$330$	0	0
$331$	11.6185	0.638609	0.319305	−	0.947652i	$-0.396551\pi$
$331$	11.6185	0.638609	0.319305	+	0.947652i	$0.396551\pi$
$332$	27.2773	1.49704
$333$	−32.1121	−1.75973
$334$	55.1341	3.01681
$335$	0	0
$336$	−64.2548	−3.50539
$337$	7.19732	0.392063	0.196032	−	0.980598i	$-0.437195\pi$
$337$	7.19732	0.392063	0.196032	+	0.980598i	$0.437195\pi$
$338$	−8.25149	−0.448822
$339$	51.0421	2.77223
$340$	0	0
$341$	0.893440	0.0483825
$342$	29.3448	1.58678
$343$	−16.2932	−0.879752
$344$	−70.5412	−3.80333
$345$	0	0
$346$	−55.6115	−2.98969
$347$	−8.49365	−0.455963	−0.227982	−	0.973665i	$-0.573213\pi$
$347$	−8.49365	−0.455963	−0.227982	+	0.973665i	$0.573213\pi$
$348$	2.74512	0.147154
$349$	−0.281605	−0.0150739	−0.00753697	−	0.999972i	$-0.502399\pi$
$349$	−0.281605	−0.0150739	−0.00753697	+	0.999972i	$0.502399\pi$
$350$	0	0
$351$	6.62422	0.353575
$352$	33.0960	1.76402
$353$	−7.56460	−0.402623	−0.201311	−	0.979527i	$-0.564520\pi$
$353$	−7.56460	−0.402623	−0.201311	+	0.979527i	$0.564520\pi$
$354$	71.3231	3.79078
$355$	0	0
$356$	17.5933	0.932445
$357$	3.47626	0.183983
$358$	−18.7992	−0.993568
$359$	4.24214	0.223891	0.111946	−	0.993714i	$-0.464292\pi$
$359$	4.24214	0.223891	0.111946	+	0.993714i	$0.464292\pi$
$360$	0	0
$361$	−11.4037	−0.600197
$362$	4.18941	0.220191
$363$	25.5607	1.34159
$364$	−24.7091	−1.29511
$365$	0	0
$366$	38.1006	1.99155
$367$	27.9623	1.45962	0.729810	−	0.683650i	$-0.239608\pi$
$367$	27.9623	1.45962	0.729810	+	0.683650i	$0.239608\pi$
$368$	−66.0080	−3.44090
$369$	−30.3018	−1.57745
$370$	0	0
$371$	−10.4586	−0.542982
$372$	−12.4411	−0.645043
$373$	9.39715	0.486566	0.243283	−	0.969955i	$-0.421776\pi$
$373$	9.39715	0.486566	0.243283	+	0.969955i	$0.421776\pi$
$374$	−3.06675	−0.158578
$375$	0	0
$376$	−6.20603	−0.320052
$377$	0.570933	0.0294045
$378$	−7.80027	−0.401203
$379$	23.9952	1.23255	0.616275	−	0.787531i	$-0.288641\pi$
$379$	23.9952	1.23255	0.616275	+	0.787531i	$0.288641\pi$
$380$	0	0
$381$	−13.0641	−0.669294
$382$	−44.6268	−2.28331
$383$	−1.97888	−0.101116	−0.0505581	−	0.998721i	$-0.516100\pi$
$383$	−1.97888	−0.101116	−0.0505581	+	0.998721i	$0.516100\pi$
$384$	−190.815	−9.73749
$385$	0	0
$386$	48.9271	2.49032
$387$	−24.8894	−1.26520
$388$	71.7196	3.64101
$389$	−26.5739	−1.34735	−0.673674	−	0.739028i	$-0.735285\pi$
$389$	−26.5739	−1.34735	−0.673674	+	0.739028i	$0.735285\pi$
$390$	0	0
$391$	3.57111	0.180599
$392$	−56.2691	−2.84202
$393$	−6.88202	−0.347152
$394$	39.3372	1.98178
$395$	0	0
$396$	24.3288	1.22257
$397$	−6.32026	−0.317205	−0.158602	−	0.987343i	$-0.550699\pi$
$397$	−6.32026	−0.317205	−0.158602	+	0.987343i	$0.550699\pi$
$398$	−12.8449	−0.643857
$399$	−9.58103	−0.479651
$400$	0	0
$401$	−14.9995	−0.749041	−0.374520	−	0.927219i	$-0.622193\pi$
$401$	−14.9995	−0.749041	−0.374520	+	0.927219i	$0.622193\pi$
$402$	−19.3552	−0.965351
$403$	−2.58752	−0.128894
$404$	17.4277	0.867063
$405$	0	0
$406$	−0.672295	−0.0333654
$407$	−9.24947	−0.458479
$408$	28.0947	1.39090
$409$	18.0132	0.890697	0.445348	−	0.895357i	$-0.353080\pi$
$409$	18.0132	0.890697	0.445348	+	0.895357i	$0.353080\pi$
$410$	0	0
$411$	−8.52947	−0.420728
$412$	80.9322	3.98725
$413$	−13.0149	−0.640421
$414$	−38.0219	−1.86868
$415$	0	0
$416$	−95.8503	−4.69945
$417$	27.0962	1.32691
$418$	8.45238	0.413419
$419$	−11.2294	−0.548594	−0.274297	−	0.961645i	$-0.588445\pi$
$419$	−11.2294	−0.548594	−0.274297	+	0.961645i	$0.588445\pi$
$420$	0	0
$421$	37.8927	1.84678	0.923389	−	0.383866i	$-0.125408\pi$
$421$	37.8927	1.84678	0.923389	+	0.383866i	$0.125408\pi$
$422$	20.2614	0.986311
$423$	−2.18970	−0.106467
$424$	−84.5250	−4.10490
$425$	0	0
$426$	−97.9687	−4.74660
$427$	−6.95252	−0.336456
$428$	46.0142	2.22418
$429$	9.05350	0.437107
$430$	0	0
$431$	14.7151	0.708803	0.354402	−	0.935093i	$-0.384685\pi$
$431$	14.7151	0.708803	0.354402	+	0.935093i	$0.384685\pi$
$432$	−38.6149	−1.85786
$433$	37.5179	1.80299	0.901497	−	0.432786i	$-0.142469\pi$
$433$	37.5179	1.80299	0.901497	+	0.432786i	$0.142469\pi$
$434$	3.04691	0.146256
$435$	0	0
$436$	79.9377	3.82832
$437$	−9.84245	−0.470828
$438$	−90.0685	−4.30364
$439$	34.8342	1.66255	0.831273	−	0.555864i	$-0.187612\pi$
$439$	34.8342	1.66255	0.831273	+	0.555864i	$0.187612\pi$
$440$	0	0
$441$	−19.8536	−0.945411
$442$	8.88173	0.422461
$443$	−9.84325	−0.467667	−0.233833	−	0.972277i	$-0.575127\pi$
$443$	−9.84325	−0.467667	−0.233833	+	0.972277i	$0.575127\pi$
$444$	128.799	6.11251
$445$	0	0
$446$	62.3425	2.95200
$447$	29.3560	1.38849
$448$	63.5901	3.00435
$449$	27.7439	1.30932	0.654658	−	0.755925i	$-0.272813\pi$
$449$	27.7439	1.30932	0.654658	+	0.755925i	$0.272813\pi$
$450$	0	0
$451$	−8.72804	−0.410987
$452$	−114.420	−5.38184
$453$	−18.6134	−0.874536
$454$	−37.4058	−1.75554
$455$	0	0
$456$	−77.4327	−3.62612
$457$	−28.0953	−1.31424	−0.657120	−	0.753786i	$-0.728226\pi$
$457$	−28.0953	−1.31424	−0.657120	+	0.753786i	$0.728226\pi$
$458$	−57.3232	−2.67854
$459$	2.08911	0.0975114
$460$	0	0
$461$	5.20430	0.242388	0.121194	−	0.992629i	$-0.461328\pi$
$461$	5.20430	0.242388	0.121194	+	0.992629i	$0.461328\pi$
$462$	−10.6608	−0.495987
$463$	1.28264	0.0596092	0.0298046	−	0.999556i	$-0.490511\pi$
$463$	1.28264	0.0596092	0.0298046	+	0.999556i	$0.490511\pi$
$464$	−3.32817	−0.154506
$465$	0	0
$466$	45.1642	2.09219
$467$	−24.5006	−1.13375	−0.566876	−	0.823803i	$-0.691848\pi$
$467$	−24.5006	−1.13375	−0.566876	+	0.823803i	$0.691848\pi$
$468$	−70.4595	−3.25699
$469$	3.53190	0.163088
$470$	0	0
$471$	−21.6610	−0.998085
$472$	−105.185	−4.84152
$473$	−7.16905	−0.329633
$474$	−27.3747	−1.25736
$475$	0	0
$476$	−7.79262	−0.357174
$477$	−29.8233	−1.36551
$478$	−12.2866	−0.561974
$479$	−31.1171	−1.42178	−0.710888	−	0.703306i	$-0.751707\pi$
$479$	−31.1171	−1.42178	−0.710888	+	0.703306i	$0.751707\pi$
$480$	0	0
$481$	26.7877	1.22141
$482$	−41.5518	−1.89263
$483$	12.4141	0.564861
$484$	−57.2986	−2.60448
$485$	0	0
$486$	61.0563	2.76957
$487$	−19.8828	−0.900978	−0.450489	−	0.892782i	$-0.648750\pi$
$487$	−19.8828	−0.900978	−0.450489	+	0.892782i	$0.648750\pi$
$488$	−56.1894	−2.54358
$489$	30.2668	1.36871
$490$	0	0
$491$	19.8895	0.897602	0.448801	−	0.893632i	$-0.351851\pi$
$491$	19.8895	0.897602	0.448801	+	0.893632i	$0.351851\pi$
$492$	121.538	5.47934
$493$	0.180058	0.00810939
$494$	−24.4792	−1.10137
$495$	0	0
$496$	15.0836	0.677272
$497$	17.8771	0.801898
$498$	−34.0843	−1.52735
$499$	4.68299	0.209639	0.104820	−	0.994491i	$-0.466573\pi$
$499$	4.68299	0.209639	0.104820	+	0.994491i	$0.466573\pi$
$500$	0	0
$501$	−51.3316	−2.29333
$502$	40.8952	1.82524
$503$	−39.2233	−1.74888	−0.874440	−	0.485134i	$-0.838771\pi$
$503$	−39.2233	−1.74888	−0.874440	+	0.485134i	$0.838771\pi$
$504$	54.5840	2.43137
$505$	0	0
$506$	−10.9517	−0.486863
$507$	7.68239	0.341187
$508$	29.2854	1.29933
$509$	16.9770	0.752494	0.376247	−	0.926519i	$-0.377214\pi$
$509$	16.9770	0.752494	0.376247	+	0.926519i	$0.377214\pi$
$510$	0	0
$511$	16.4355	0.727064
$512$	160.492	7.09281
$513$	−5.75786	−0.254216
$514$	−14.9780	−0.660649
$515$	0	0
$516$	99.8289	4.39472
$517$	−0.630714	−0.0277388
$518$	−31.5436	−1.38594
$519$	51.7760	2.27272
$520$	0	0
$521$	−24.6927	−1.08181	−0.540903	−	0.841085i	$-0.681917\pi$
$521$	−24.6927	−1.08181	−0.540903	+	0.841085i	$0.681917\pi$
$522$	−1.91709	−0.0839088
$523$	41.2117	1.80206	0.901032	−	0.433753i	$-0.142811\pi$
$523$	41.2117	1.80206	0.901032	+	0.433753i	$0.142811\pi$
$524$	15.4272	0.673941
$525$	0	0
$526$	−51.4707	−2.24423
$527$	−0.816039	−0.0355472
$528$	−52.7760	−2.29678
$529$	−10.2472	−0.445529
$530$	0	0
$531$	−37.1128	−1.61056
$532$	21.4775	0.931167
$533$	25.2776	1.09489
$534$	−21.9837	−0.951329
$535$	0	0
$536$	28.5444	1.23293
$537$	17.5026	0.755294
$538$	−42.2435	−1.82125
$539$	−5.71858	−0.246317
$540$	0	0
$541$	−3.70168	−0.159147	−0.0795737	−	0.996829i	$-0.525356\pi$
$541$	−3.70168	−0.159147	−0.0795737	+	0.996829i	$0.525356\pi$
$542$	57.7108	2.47889
$543$	−3.90047	−0.167385
$544$	−30.2288	−1.29605
$545$	0	0
$546$	30.8752	1.32134
$547$	−31.0162	−1.32616	−0.663079	−	0.748550i	$-0.730750\pi$
$547$	−31.0162	−1.32616	−0.663079	+	0.748550i	$0.730750\pi$
$548$	19.1202	0.816776
$549$	−19.8255	−0.846134
$550$	0	0
$551$	−0.496263	−0.0211415
$552$	100.329	4.27030
$553$	4.99527	0.212421
$554$	11.0199	0.468191
$555$	0	0
$556$	−60.7407	−2.57598
$557$	40.6170	1.72100	0.860498	−	0.509453i	$-0.170152\pi$
$557$	40.6170	1.72100	0.860498	+	0.509453i	$0.170152\pi$
$558$	8.68845	0.367811
$559$	20.7625	0.878162
$560$	0	0
$561$	2.85524	0.120548
$562$	69.4325	2.92883
$563$	6.63255	0.279529	0.139764	−	0.990185i	$-0.455366\pi$
$563$	6.63255	0.279529	0.139764	+	0.990185i	$0.455366\pi$
$564$	8.78268	0.369817
$565$	0	0
$566$	−28.8209	−1.21143
$567$	−7.93797	−0.333363
$568$	144.481	6.06227
$569$	−20.9921	−0.880035	−0.440017	−	0.897989i	$-0.645028\pi$
$569$	−20.9921	−0.880035	−0.440017	+	0.897989i	$0.645028\pi$
$570$	0	0
$571$	9.22867	0.386208	0.193104	−	0.981178i	$-0.438145\pi$
$571$	9.22867	0.386208	0.193104	+	0.981178i	$0.438145\pi$
$572$	−20.2949	−0.848574
$573$	41.5490	1.73573
$574$	−29.7653	−1.24238
$575$	0	0
$576$	181.331	7.55547
$577$	1.17031	0.0487208	0.0243604	−	0.999703i	$-0.492245\pi$
$577$	1.17031	0.0487208	0.0243604	+	0.999703i	$0.492245\pi$
$578$	2.80107	0.116509
$579$	−45.5526	−1.89310
$580$	0	0
$581$	6.21963	0.258034
$582$	−89.6171	−3.71475
$583$	−8.59021	−0.355770
$584$	132.830	5.49653
$585$	0	0
$586$	11.7771	0.486506
$587$	−32.1314	−1.32620	−0.663102	−	0.748529i	$-0.730761\pi$
$587$	−32.1314	−1.32620	−0.663102	+	0.748529i	$0.730761\pi$
$588$	79.6311	3.28393
$589$	2.24911	0.0926730
$590$	0	0
$591$	−36.6242	−1.50652
$592$	−156.155	−6.41793
$593$	24.9096	1.02291	0.511456	−	0.859309i	$-0.329106\pi$
$593$	24.9096	1.02291	0.511456	+	0.859309i	$0.329106\pi$
$594$	−6.40679	−0.262874
$595$	0	0
$596$	−65.8065	−2.69554
$597$	11.9590	0.489450
$598$	31.7176	1.29703
$599$	−30.4971	−1.24608	−0.623039	−	0.782191i	$-0.714102\pi$
$599$	−30.4971	−1.24608	−0.623039	+	0.782191i	$0.714102\pi$
$600$	0	0
$601$	−4.43000	−0.180703	−0.0903517	−	0.995910i	$-0.528799\pi$
$601$	−4.43000	−0.180703	−0.0903517	+	0.995910i	$0.528799\pi$
$602$	−24.4487	−0.996454
$603$	10.0714	0.410140
$604$	41.7252	1.69777
$605$	0	0
$606$	−21.7768	−0.884623
$607$	−28.4929	−1.15649	−0.578246	−	0.815862i	$-0.696263\pi$
$607$	−28.4929	−1.15649	−0.578246	+	0.815862i	$0.696263\pi$
$608$	83.3145	3.37885
$609$	0.625927	0.0253639
$610$	0	0
$611$	1.82663	0.0738976
$612$	−22.2211	−0.898237
$613$	5.63970	0.227785	0.113893	−	0.993493i	$-0.463668\pi$
$613$	5.63970	0.227785	0.113893	+	0.993493i	$0.463668\pi$
$614$	−56.8533	−2.29441
$615$	0	0
$616$	15.7222	0.633466
$617$	−5.49860	−0.221365	−0.110683	−	0.993856i	$-0.535304\pi$
$617$	−5.49860	−0.221365	−0.110683	+	0.993856i	$0.535304\pi$
$618$	−101.129	−4.06800
$619$	−20.7770	−0.835096	−0.417548	−	0.908655i	$-0.637111\pi$
$619$	−20.7770	−0.835096	−0.417548	+	0.908655i	$0.637111\pi$
$620$	0	0
$621$	7.46045	0.299377
$622$	22.5164	0.902827
$623$	4.01154	0.160719
$624$	152.846	6.11875
$625$	0	0
$626$	83.0588	3.31970
$627$	−7.86943	−0.314275
$628$	48.5567	1.93762
$629$	8.44817	0.336850
$630$	0	0
$631$	−9.55265	−0.380285	−0.190142	−	0.981757i	$-0.560895\pi$
$631$	−9.55265	−0.380285	−0.190142	+	0.981757i	$0.560895\pi$
$632$	40.3712	1.60588
$633$	−18.8640	−0.749778
$634$	−8.58276	−0.340865
$635$	0	0
$636$	119.618	4.74318
$637$	16.5618	0.656201
$638$	−0.552193	−0.0218615
$639$	50.9777	2.01665
$640$	0	0
$641$	−38.2638	−1.51133	−0.755664	−	0.654959i	$-0.772686\pi$
$641$	−38.2638	−1.51133	−0.755664	+	0.654959i	$0.772686\pi$
$642$	−57.4970	−2.26922
$643$	−2.28036	−0.0899286	−0.0449643	−	0.998989i	$-0.514317\pi$
$643$	−2.28036	−0.0899286	−0.0449643	+	0.998989i	$0.514317\pi$
$644$	−27.8283	−1.09659
$645$	0	0
$646$	−7.72013	−0.303744
$647$	−15.3848	−0.604840	−0.302420	−	0.953175i	$-0.597794\pi$
$647$	−15.3848	−0.604840	−0.302420	+	0.953175i	$0.597794\pi$
$648$	−64.1537	−2.52020
$649$	−10.6898	−0.419613
$650$	0	0
$651$	−2.83677	−0.111182
$652$	−67.8481	−2.65714
$653$	−24.7373	−0.968046	−0.484023	−	0.875055i	$-0.660825\pi$
$653$	−24.7373	−0.968046	−0.484023	+	0.875055i	$0.660825\pi$
$654$	−99.8860	−3.90585
$655$	0	0
$656$	−147.352	−5.75312
$657$	46.8669	1.82845
$658$	−2.15093	−0.0838520
$659$	−17.3744	−0.676812	−0.338406	−	0.941000i	$-0.609888\pi$
$659$	−17.3744	−0.676812	−0.338406	+	0.941000i	$0.609888\pi$
$660$	0	0
$661$	4.18538	0.162793	0.0813963	−	0.996682i	$-0.474062\pi$
$661$	4.18538	0.162793	0.0813963	+	0.996682i	$0.474062\pi$
$662$	32.5442	1.26487
$663$	−8.26917	−0.321148
$664$	50.2663	1.95071
$665$	0	0
$666$	−89.9484	−3.48543
$667$	0.643006	0.0248973
$668$	115.068	4.45213
$669$	−58.0428	−2.24406
$670$	0	0
$671$	−5.71049	−0.220451
$672$	−105.083	−4.05367
$673$	−16.8948	−0.651246	−0.325623	−	0.945500i	$-0.605574\pi$
$673$	−16.8948	−0.651246	−0.325623	+	0.945500i	$0.605574\pi$
$674$	20.1602	0.776543
$675$	0	0
$676$	−17.2214	−0.662361
$677$	−6.13351	−0.235730	−0.117865	−	0.993030i	$-0.537605\pi$
$677$	−6.13351	−0.235730	−0.117865	+	0.993030i	$0.537605\pi$
$678$	142.973	5.49084
$679$	16.3531	0.627576
$680$	0	0
$681$	34.8260	1.33453
$682$	2.50259	0.0958292
$683$	26.7588	1.02390	0.511949	−	0.859016i	$-0.328924\pi$
$683$	26.7588	1.02390	0.511949	+	0.859016i	$0.328924\pi$
$684$	61.2444	2.34174
$685$	0	0
$686$	−45.6386	−1.74249
$687$	53.3697	2.03618
$688$	−121.032	−4.61430
$689$	24.8784	0.947791
$690$	0	0
$691$	23.1367	0.880160	0.440080	−	0.897959i	$-0.354950\pi$
$691$	23.1367	0.880160	0.440080	+	0.897959i	$0.354950\pi$
$692$	−116.065	−4.41212
$693$	5.54733	0.210726
$694$	−23.7914	−0.903107
$695$	0	0
$696$	5.05867	0.191748
$697$	7.97191	0.301957
$698$	−0.788795	−0.0298563
$699$	−42.0492	−1.59045
$700$	0	0
$701$	41.0017	1.54861	0.774307	−	0.632810i	$-0.218098\pi$
$701$	41.0017	1.54861	0.774307	+	0.632810i	$0.218098\pi$
$702$	18.5549	0.700311
$703$	−23.2843	−0.878182
$704$	52.2301	1.96850
$705$	0	0
$706$	−21.1890	−0.797458
$707$	3.97379	0.149450
$708$	148.856	5.59434
$709$	−10.7096	−0.402207	−0.201103	−	0.979570i	$-0.564453\pi$
$709$	−10.7096	−0.402207	−0.201103	+	0.979570i	$0.564453\pi$
$710$	0	0
$711$	14.2443	0.534204
$712$	32.4208	1.21502
$713$	−2.91417	−0.109136
$714$	9.73726	0.364408
$715$	0	0
$716$	−39.2351	−1.46628
$717$	11.4392	0.427204
$718$	11.8825	0.443452
$719$	−6.17046	−0.230119	−0.115060	−	0.993359i	$-0.536706\pi$
$719$	−6.17046	−0.230119	−0.115060	+	0.993359i	$0.536706\pi$
$720$	0	0
$721$	18.4538	0.687254
$722$	−31.9427	−1.18878
$723$	38.6861	1.43875
$724$	8.74357	0.324952
$725$	0	0
$726$	71.5974	2.65723
$727$	38.9277	1.44375	0.721875	−	0.692024i	$-0.243281\pi$
$727$	38.9277	1.44375	0.721875	+	0.692024i	$0.243281\pi$
$728$	−45.5337	−1.68759
$729$	−38.9801	−1.44371
$730$	0	0
$731$	6.54798	0.242186
$732$	79.5184	2.93908
$733$	−31.8680	−1.17707	−0.588535	−	0.808472i	$-0.700295\pi$
$733$	−31.8680	−1.17707	−0.588535	+	0.808472i	$0.700295\pi$
$734$	78.3245	2.89101
$735$	0	0
$736$	−107.950	−3.97910
$737$	2.90094	0.106858
$738$	−84.8776	−3.12439
$739$	29.7808	1.09551	0.547753	−	0.836640i	$-0.315483\pi$
$739$	29.7808	1.09551	0.547753	+	0.836640i	$0.315483\pi$
$740$	0	0
$741$	22.7909	0.837245
$742$	−29.2953	−1.07546
$743$	−33.4684	−1.22784	−0.613918	−	0.789370i	$-0.710408\pi$
$743$	−33.4684	−1.22784	−0.613918	+	0.789370i	$0.710408\pi$
$744$	−22.9264	−0.840522
$745$	0	0
$746$	26.3221	0.963721
$747$	17.7357	0.648914
$748$	−6.40051	−0.234026
$749$	10.4919	0.383367
$750$	0	0
$751$	15.9989	0.583808	0.291904	−	0.956448i	$-0.405711\pi$
$751$	15.9989	0.583808	0.291904	+	0.956448i	$0.405711\pi$
$752$	−10.6481	−0.388295
$753$	−38.0747	−1.38752
$754$	1.59922	0.0582403
$755$	0	0
$756$	−16.2797	−0.592085
$757$	20.6978	0.752275	0.376138	−	0.926564i	$-0.377252\pi$
$757$	20.6978	0.752275	0.376138	+	0.926564i	$0.377252\pi$
$758$	67.2123	2.44126
$759$	10.1964	0.370105
$760$	0	0
$761$	−22.4458	−0.813660	−0.406830	−	0.913504i	$-0.633366\pi$
$761$	−22.4458	−0.813660	−0.406830	+	0.913504i	$0.633366\pi$
$762$	−36.5935	−1.32564
$763$	18.2270	0.659861
$764$	−93.1390	−3.36965
$765$	0	0
$766$	−5.54300	−0.200277
$767$	30.9592	1.11787
$768$	−285.667	−10.3081
$769$	30.6260	1.10440	0.552202	−	0.833711i	$-0.313788\pi$
$769$	30.6260	1.10440	0.552202	+	0.833711i	$0.313788\pi$
$770$	0	0
$771$	13.9449	0.502215
$772$	102.114	3.67516
$773$	10.9578	0.394126	0.197063	−	0.980391i	$-0.436860\pi$
$773$	10.9578	0.394126	0.197063	+	0.980391i	$0.436860\pi$
$774$	−69.7169	−2.50592
$775$	0	0
$776$	132.164	4.74441
$777$	29.3680	1.05357
$778$	−74.4354	−2.66864
$779$	−21.9716	−0.787215
$780$	0	0
$781$	14.6835	0.525415
$782$	10.0029	0.357705
$783$	0.376161	0.0134429
$784$	−96.5444	−3.44801
$785$	0	0
$786$	−19.2770	−0.687590
$787$	−31.1539	−1.11052	−0.555259	−	0.831677i	$-0.687381\pi$
$787$	−31.1539	−1.11052	−0.555259	+	0.831677i	$0.687381\pi$
$788$	82.0993	2.92467
$789$	47.9209	1.70603
$790$	0	0
$791$	−26.0894	−0.927631
$792$	44.8329	1.59307
$793$	16.5383	0.587294
$794$	−17.7035	−0.628274
$795$	0	0
$796$	−26.8081	−0.950189
$797$	−15.2518	−0.540246	−0.270123	−	0.962826i	$-0.587064\pi$
$797$	−15.2518	−0.540246	−0.270123	+	0.962826i	$0.587064\pi$
$798$	−26.8372	−0.950026
$799$	0.576074	0.0203800
$800$	0	0
$801$	11.4392	0.404183
$802$	−42.0148	−1.48359
$803$	13.4994	0.476383
$804$	−40.3956	−1.42464
$805$	0	0
$806$	−7.24784	−0.255294
$807$	39.3300	1.38448
$808$	32.1157	1.12983
$809$	−2.66539	−0.0937100	−0.0468550	−	0.998902i	$-0.514920\pi$
$809$	−2.66539	−0.0937100	−0.0468550	+	0.998902i	$0.514920\pi$
$810$	0	0
$811$	5.54482	0.194705	0.0973525	−	0.995250i	$-0.468963\pi$
$811$	5.54482	0.194705	0.0973525	+	0.995250i	$0.468963\pi$
$812$	−1.40312	−0.0492399
$813$	−53.7305	−1.88441
$814$	−25.9085	−0.908091
$815$	0	0
$816$	48.2039	1.68747
$817$	−18.0471	−0.631388
$818$	50.4564	1.76417
$819$	−16.0658	−0.561385
$820$	0	0
$821$	31.2022	1.08896	0.544482	−	0.838773i	$-0.316726\pi$
$821$	31.2022	1.08896	0.544482	+	0.838773i	$0.316726\pi$
$822$	−23.8917	−0.833318
$823$	−50.4682	−1.75921	−0.879606	−	0.475703i	$-0.842194\pi$
$823$	−50.4682	−1.75921	−0.879606	+	0.475703i	$0.842194\pi$
$824$	149.141	5.19558
$825$	0	0
$826$	−36.4557	−1.26845
$827$	25.7704	0.896123	0.448062	−	0.894003i	$-0.352114\pi$
$827$	25.7704	0.896123	0.448062	+	0.894003i	$0.352114\pi$
$828$	−79.3542	−2.75775
$829$	4.34228	0.150814	0.0754068	−	0.997153i	$-0.475974\pi$
$829$	4.34228	0.150814	0.0754068	+	0.997153i	$0.475974\pi$
$830$	0	0
$831$	−10.2599	−0.355911
$832$	−151.265	−5.24418
$833$	5.22317	0.180972
$834$	75.8984	2.62815
$835$	0	0
$836$	17.6406	0.610114
$837$	−1.70480	−0.0589264
$838$	−31.4545	−1.08658
$839$	44.9482	1.55178	0.775892	−	0.630866i	$-0.217300\pi$
$839$	44.9482	1.55178	0.775892	+	0.630866i	$0.217300\pi$
$840$	0	0
$841$	−28.9676	−0.998882
$842$	106.140	3.65784
$843$	−64.6438	−2.22645
$844$	42.2869	1.45557
$845$	0	0
$846$	−6.13351	−0.210874
$847$	−13.0649	−0.448917
$848$	−145.025	−4.98017
$849$	26.8331	0.920910
$850$	0	0
$851$	30.1693	1.03419
$852$	−204.467	−7.00491
$853$	15.7009	0.537590	0.268795	−	0.963197i	$-0.413375\pi$
$853$	15.7009	0.537590	0.268795	+	0.963197i	$0.413375\pi$
$854$	−19.4745	−0.666405
$855$	0	0
$856$	84.7945	2.89821
$857$	−6.77401	−0.231396	−0.115698	−	0.993284i	$-0.536910\pi$
$857$	−6.77401	−0.231396	−0.115698	+	0.993284i	$0.536910\pi$
$858$	25.3595	0.865760
$859$	36.0071	1.22855	0.614273	−	0.789094i	$-0.289449\pi$
$859$	36.0071	1.22855	0.614273	+	0.789094i	$0.289449\pi$
$860$	0	0
$861$	27.7124	0.944437
$862$	41.2182	1.40390
$863$	35.5716	1.21087	0.605436	−	0.795894i	$-0.292999\pi$
$863$	35.5716	1.21087	0.605436	+	0.795894i	$0.292999\pi$
$864$	−63.1513	−2.14845
$865$	0	0
$866$	105.090	3.57111
$867$	−2.60789	−0.0885685
$868$	6.35909	0.215841
$869$	4.10289	0.139181
$870$	0	0
$871$	−8.40152	−0.284675
$872$	147.308	4.98849
$873$	46.6320	1.57825
$874$	−27.5694	−0.932550
$875$	0	0
$876$	−187.979	−6.35121
$877$	24.1451	0.815321	0.407660	−	0.913134i	$-0.366345\pi$
$877$	24.1451	0.815321	0.407660	+	0.913134i	$0.366345\pi$
$878$	97.5733	3.29294
$879$	−10.9648	−0.369834
$880$	0	0
$881$	32.9124	1.10885	0.554423	−	0.832235i	$-0.312939\pi$
$881$	32.9124	1.10885	0.554423	+	0.832235i	$0.312939\pi$
$882$	−55.6115	−1.87254
$883$	−15.5885	−0.524594	−0.262297	−	0.964987i	$-0.584480\pi$
$883$	−15.5885	−0.524594	−0.262297	+	0.964987i	$0.584480\pi$
$884$	18.5367	0.623458
$885$	0	0
$886$	−27.5717	−0.926289
$887$	−8.59951	−0.288743	−0.144372	−	0.989524i	$-0.546116\pi$
$887$	−8.59951	−0.288743	−0.144372	+	0.989524i	$0.546116\pi$
$888$	237.349	7.96490
$889$	6.67750	0.223956
$890$	0	0
$891$	−6.51989	−0.218425
$892$	130.113	4.35649
$893$	−1.58773	−0.0531315
$894$	82.2284	2.75013
$895$	0	0
$896$	97.5320	3.25832
$897$	−29.5301	−0.985982
$898$	77.7127	2.59331
$899$	−0.146934	−0.00490053
$900$	0	0
$901$	7.84602	0.261389
$902$	−24.4479	−0.814025
$903$	22.7625	0.757488
$904$	−210.851	−7.01280
$905$	0	0
$906$	−52.1376	−1.73216
$907$	44.9819	1.49360	0.746800	−	0.665049i	$-0.231589\pi$
$907$	44.9819	1.49360	0.746800	+	0.665049i	$0.231589\pi$
$908$	−78.0682	−2.59079
$909$	11.3315	0.375842
$910$	0	0
$911$	50.9800	1.68904	0.844522	−	0.535521i	$-0.179885\pi$
$911$	50.9800	1.68904	0.844522	+	0.535521i	$0.179885\pi$
$912$	−132.856	−4.39931
$913$	5.10852	0.169067
$914$	−78.6969	−2.60306
$915$	0	0
$916$	−119.637	−3.95292
$917$	3.51763	0.116163
$918$	5.85176	0.193137
$919$	−13.0346	−0.429973	−0.214986	−	0.976617i	$-0.568971\pi$
$919$	−13.0346	−0.429973	−0.214986	+	0.976617i	$0.568971\pi$
$920$	0	0
$921$	52.9321	1.74417
$922$	14.5776	0.480088
$923$	−42.5252	−1.39974
$924$	−22.2498	−0.731966
$925$	0	0
$926$	3.59276	0.118066
$927$	52.6221	1.72834
$928$	−5.44292	−0.178673
$929$	37.8748	1.24263	0.621316	−	0.783560i	$-0.286598\pi$
$929$	37.8748	1.24263	0.621316	+	0.783560i	$0.286598\pi$
$930$	0	0
$931$	−14.3957	−0.471801
$932$	94.2604	3.08760
$933$	−20.9635	−0.686314
$934$	−68.6280	−2.24558
$935$	0	0
$936$	−129.842	−4.24402
$937$	−22.9509	−0.749772	−0.374886	−	0.927071i	$-0.622318\pi$
$937$	−22.9509	−0.749772	−0.374886	+	0.927071i	$0.622318\pi$
$938$	9.89311	0.323022
$939$	−77.3303	−2.52358
$940$	0	0
$941$	−33.8175	−1.10242	−0.551209	−	0.834367i	$-0.685833\pi$
$941$	−33.8175	−1.10242	−0.551209	+	0.834367i	$0.685833\pi$
$942$	−60.6740	−1.97687
$943$	28.4685	0.927064
$944$	−180.472	−5.87387
$945$	0	0
$946$	−20.0810	−0.652891
$947$	−10.0614	−0.326953	−0.163476	−	0.986547i	$-0.552271\pi$
$947$	−10.0614	−0.326953	−0.163476	+	0.986547i	$0.552271\pi$
$948$	−57.1327	−1.85558
$949$	−39.0960	−1.26911
$950$	0	0
$951$	7.99082	0.259120
$952$	−14.3602	−0.465416
$953$	−38.8829	−1.25954	−0.629770	−	0.776781i	$-0.716851\pi$
$953$	−38.8829	−1.25954	−0.629770	+	0.776781i	$0.716851\pi$
$954$	−83.5373	−2.70462
$955$	0	0
$956$	−25.6428	−0.829348
$957$	0.514109	0.0166188
$958$	−87.1612	−2.81605
$959$	4.35970	0.140782
$960$	0	0
$961$	−30.3341	−0.978519
$962$	75.0344	2.41920
$963$	29.9184	0.964106
$964$	−86.7213	−2.79311
$965$	0	0
$966$	34.7728	1.11880
$967$	5.40033	0.173663	0.0868315	−	0.996223i	$-0.472326\pi$
$967$	5.40033	0.173663	0.0868315	+	0.996223i	$0.472326\pi$
$968$	−105.589	−3.39377
$969$	7.18768	0.230902
$970$	0	0
$971$	10.1469	0.325629	0.162815	−	0.986657i	$-0.447943\pi$
$971$	10.1469	0.325629	0.162815	+	0.986657i	$0.447943\pi$
$972$	127.428	4.08726
$973$	−13.8498	−0.444004
$974$	−55.6933	−1.78453
$975$	0	0
$976$	−96.4077	−3.08594
$977$	9.09855	0.291088	0.145544	−	0.989352i	$-0.453507\pi$
$977$	9.09855	0.291088	0.145544	+	0.989352i	$0.453507\pi$
$978$	84.7795	2.71095
$979$	3.29490	0.105306
$980$	0	0
$981$	51.9754	1.65945
$982$	55.7121	1.77784
$983$	−49.1437	−1.56744	−0.783721	−	0.621113i	$-0.786681\pi$
$983$	−49.1437	−1.56744	−0.783721	+	0.621113i	$0.786681\pi$
$984$	223.968	7.13985
$985$	0	0
$986$	0.504355	0.0160619
$987$	2.00258	0.0637429
$988$	−51.0897	−1.62538
$989$	23.3836	0.743554
$990$	0	0
$991$	43.2065	1.37250	0.686249	−	0.727366i	$-0.259256\pi$
$991$	43.2065	1.37250	0.686249	+	0.727366i	$0.259256\pi$
$992$	24.6679	0.783205
$993$	−30.2997	−0.961531
$994$	50.0751	1.58829
$995$	0	0
$996$	−71.1361	−2.25403
$997$	−48.5452	−1.53744	−0.768721	−	0.639584i	$-0.779107\pi$
$997$	−48.5452	−1.53744	−0.768721	+	0.639584i	$0.779107\pi$
$998$	13.1174	0.415224
$999$	17.6492	0.558395

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	425.2.a.j.1.5	yes	5
3.2	odd	2		3825.2.a.bl.1.1		5
4.3	odd	2		6800.2.a.cd.1.5		5
5.2	odd	4		425.2.b.f.324.10		10
5.3	odd	4		425.2.b.f.324.1		10
5.4	even	2		425.2.a.i.1.1	✓	5
15.14	odd	2		3825.2.a.bq.1.5		5
17.16	even	2		7225.2.a.y.1.5		5
20.19	odd	2		6800.2.a.bz.1.1		5
85.84	even	2		7225.2.a.x.1.1		5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
425.2.a.i.1.1	✓	5	5.4	even	2
425.2.a.j.1.5	yes	5	1.1	even	1	trivial
425.2.b.f.324.1		10	5.3	odd	4
425.2.b.f.324.10		10	5.2	odd	4
3825.2.a.bl.1.1		5	3.2	odd	2
3825.2.a.bq.1.5		5	15.14	odd	2
6800.2.a.bz.1.1		5	20.19	odd	2
6800.2.a.cd.1.5		5	4.3	odd	2
7225.2.a.x.1.1		5	85.84	even	2
7225.2.a.y.1.5		5	17.16	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:	\( N \)	\(=\)	\( 425 = 5^{2} \cdot 17 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	425.a (trivial)

\(f(q)\)	\(=\)	\(q+2.80107 q^{2} -2.60789 q^{3} +5.84602 q^{4} -7.30489 q^{6} +1.33298 q^{7} +10.7730 q^{8} +3.80107 q^{9} +1.09485 q^{11} -15.2458 q^{12} -3.17083 q^{13} +3.73378 q^{14} +18.4839 q^{16} -1.00000 q^{17} +10.6471 q^{18} +2.75613 q^{19} -3.47626 q^{21} +3.06675 q^{22} -3.57111 q^{23} -28.0947 q^{24} -8.88173 q^{26} -2.08911 q^{27} +7.79262 q^{28} -0.180058 q^{29} +0.816039 q^{31} +30.2288 q^{32} -2.85524 q^{33} -2.80107 q^{34} +22.2211 q^{36} -8.44817 q^{37} +7.72013 q^{38} +8.26917 q^{39} -7.97191 q^{41} -9.73726 q^{42} -6.54798 q^{43} +6.40051 q^{44} -10.0029 q^{46} -0.576074 q^{47} -48.2039 q^{48} -5.22317 q^{49} +2.60789 q^{51} -18.5367 q^{52} -7.84602 q^{53} -5.85176 q^{54} +14.3602 q^{56} -7.18768 q^{57} -0.504355 q^{58} -9.76375 q^{59} -5.21577 q^{61} +2.28579 q^{62} +5.06675 q^{63} +47.7053 q^{64} -7.99775 q^{66} +2.64963 q^{67} -5.84602 q^{68} +9.31305 q^{69} +13.4114 q^{71} +40.9489 q^{72} +12.3299 q^{73} -23.6639 q^{74} +16.1124 q^{76} +1.45941 q^{77} +23.1626 q^{78} +3.74745 q^{79} -5.95506 q^{81} -22.3299 q^{82} +4.66596 q^{83} -20.3223 q^{84} -18.3414 q^{86} +0.469570 q^{87} +11.7948 q^{88} +3.00946 q^{89} -4.22665 q^{91} -20.8768 q^{92} -2.12814 q^{93} -1.61363 q^{94} -78.8332 q^{96} +12.2681 q^{97} -14.6305 q^{98} +4.16160 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + q^{2} - q^{3} + 11 q^{4} + 3 q^{6} - q^{7} + 9 q^{8} + 6 q^{9} + 4 q^{11} - 17 q^{12} + 3 q^{13} - 7 q^{14} + 27 q^{16} - 5 q^{17} + 22 q^{18} + 6 q^{19} - 5 q^{21} - 18 q^{22} - 4 q^{23} - 19 q^{24}+ \cdots - 14 q^{99}+O(q^{100}) \) 5 * q + q^2 - q^3 + 11 * q^4 + 3 * q^6 - q^7 + 9 * q^8 + 6 * q^9 + 4 * q^11 - 17 * q^12 + 3 * q^13 - 7 * q^14 + 27 * q^16 - 5 * q^17 + 22 * q^18 + 6 * q^19 - 5 * q^21 - 18 * q^22 - 4 * q^23 - 19 * q^24 - 5 * q^26 + 5 * q^27 + 15 * q^28 + 2 * q^29 + 21 * q^31 + 9 * q^32 - 12 * q^33 - q^34 + 22 * q^36 + 2 * q^37 + 34 * q^38 + 23 * q^39 - 8 * q^41 - 25 * q^42 + 4 * q^43 - 4 * q^44 - 6 * q^46 + 2 * q^47 - 51 * q^48 + 10 * q^49 + q^51 + 13 * q^52 - 21 * q^53 - 25 * q^54 - 29 * q^56 + 20 * q^57 - 22 * q^58 + 12 * q^59 - 2 * q^61 - 3 * q^62 - 8 * q^63 + 43 * q^64 - 28 * q^66 + 12 * q^67 - 11 * q^68 + 21 * q^71 + 58 * q^72 - 22 * q^73 - 50 * q^74 - 14 * q^76 + 32 * q^77 + 53 * q^78 + 41 * q^79 - 35 * q^81 - 28 * q^82 + 8 * q^83 - 71 * q^84 - 36 * q^86 + 4 * q^87 - 14 * q^88 - 10 * q^89 + 13 * q^91 - 56 * q^92 - 9 * q^93 - 10 * q^94 - 85 * q^96 + 20 * q^97 - 38 * q^98 - 14 * q^99

Introduction

L-functions

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