LMFDB - Embedded newform 2888.2.a.r.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2888,2,Mod(1,2888)] 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("2888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(2888, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 2888 = 2^{3} \cdot 19^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2888.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,0,1,0,1,0,-2,0,6,0,4,0,-1,0,5] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

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

Self dual:	yes
Analytic conductor:	$23.0607961037$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$2.34292$ of defining polynomial
Character	$\chi$	$=$	2888.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	$q-0.146365 q^{3} +2.34292 q^{5} +3.83221 q^{7} -2.97858 q^{9} +3.34292 q^{11} +6.17513 q^{13} -0.342923 q^{15} +5.19656 q^{17} -0.560904 q^{21} +2.34292 q^{23} +0.489289 q^{25} +0.875057 q^{27} +0.0501921 q^{29} -3.43910 q^{31} -0.489289 q^{33} +8.97858 q^{35} -5.43910 q^{37} -0.903827 q^{39} -7.29273 q^{41} -8.86098 q^{43} -6.97858 q^{45} +10.7360 q^{47} +7.68585 q^{49} -0.760597 q^{51} -3.19656 q^{53} +7.83221 q^{55} +3.85363 q^{59} +2.92839 q^{61} -11.4145 q^{63} +14.4679 q^{65} -11.5181 q^{67} -0.342923 q^{69} +1.48929 q^{71} -7.68585 q^{73} -0.0716150 q^{75} +12.8108 q^{77} +0.175135 q^{79} +8.80765 q^{81} -7.00735 q^{83} +12.1751 q^{85} -0.00734639 q^{87} -13.5468 q^{89} +23.6644 q^{91} +0.503365 q^{93} -3.68585 q^{97} -9.95715 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{5} - 2 q^{7} + 6 q^{9} + 4 q^{11} - q^{13} + 5 q^{15} + 11 q^{17} - 6 q^{21} + q^{23} - 6 q^{25} + 19 q^{27} - 3 q^{29} - 6 q^{31} + 6 q^{33} + 12 q^{35} - 12 q^{37} - q^{39} - 19 q^{41}+ \cdots + q^{97}+O(q^{100}) $ 3 * q + q^3 + q^5 - 2 * q^7 + 6 * q^9 + 4 * q^11 - q^13 + 5 * q^15 + 11 * q^17 - 6 * q^21 + q^23 - 6 * q^25 + 19 * q^27 - 3 * q^29 - 6 * q^31 + 6 * q^33 + 12 * q^35 - 12 * q^37 - q^39 - 19 * q^41 + 5 * q^43 - 6 * q^45 + 17 * q^47 + 11 * q^49 + 23 * q^51 - 5 * q^53 + 10 * q^55 + 13 * q^59 - 3 * q^61 - 40 * q^63 + 21 * q^65 - 9 * q^67 + 5 * q^69 - 3 * q^71 - 11 * q^73 - 12 * q^75 + 10 * q^77 - 19 * q^79 + 23 * q^81 + 12 * q^83 + 17 * q^85 + 33 * q^87 + 3 * q^89 + 44 * q^91 + 42 * q^93 + q^97

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
$3$	−0.146365	−0.0845042	−0.0422521	−	0.999107i	$-0.513453\pi$
$3$	−0.146365	−0.0845042	−0.0422521	+	0.999107i	$0.513453\pi$
$4$	0	0
$5$	2.34292	1.04779	0.523894	−	0.851784i	$-0.324479\pi$
$5$	2.34292	1.04779	0.523894	+	0.851784i	$0.324479\pi$
$6$	0	0
$7$	3.83221	1.44844	0.724220	−	0.689569i	$-0.242200\pi$
$7$	3.83221	1.44844	0.724220	+	0.689569i	$0.242200\pi$
$8$	0	0
$9$	−2.97858	−0.992859
$10$	0	0
$11$	3.34292	1.00793	0.503965	−	0.863724i	$-0.331874\pi$
$11$	3.34292	1.00793	0.503965	+	0.863724i	$0.331874\pi$
$12$	0	0
$13$	6.17513	1.71267	0.856337	−	0.516417i	$-0.172735\pi$
$13$	6.17513	1.71267	0.856337	+	0.516417i	$0.172735\pi$
$14$	0	0
$15$	−0.342923	−0.0885424
$16$	0	0
$17$	5.19656	1.26035	0.630175	−	0.776453i	$-0.282983\pi$
$17$	5.19656	1.26035	0.630175	+	0.776453i	$0.282983\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−0.560904	−0.122399
$22$	0	0
$23$	2.34292	0.488533	0.244267	−	0.969708i	$-0.421453\pi$
$23$	2.34292	0.488533	0.244267	+	0.969708i	$0.421453\pi$
$24$	0	0
$25$	0.489289	0.0978577
$26$	0	0
$27$	0.875057	0.168405
$28$	0	0
$29$	0.0501921	0.00932044	0.00466022	−	0.999989i	$-0.498517\pi$
$29$	0.0501921	0.00932044	0.00466022	+	0.999989i	$0.498517\pi$
$30$	0	0
$31$	−3.43910	−0.617680	−0.308840	−	0.951114i	$-0.599941\pi$
$31$	−3.43910	−0.617680	−0.308840	+	0.951114i	$0.599941\pi$
$32$	0	0
$33$	−0.489289	−0.0851742
$34$	0	0
$35$	8.97858	1.51766
$36$	0	0
$37$	−5.43910	−0.894182	−0.447091	−	0.894488i	$-0.647540\pi$
$37$	−5.43910	−0.894182	−0.447091	+	0.894488i	$0.647540\pi$
$38$	0	0
$39$	−0.903827	−0.144728
$40$	0	0
$41$	−7.29273	−1.13893	−0.569467	−	0.822014i	$-0.692850\pi$
$41$	−7.29273	−1.13893	−0.569467	+	0.822014i	$0.692850\pi$
$42$	0	0
$43$	−8.86098	−1.35129	−0.675643	−	0.737229i	$-0.736134\pi$
$43$	−8.86098	−1.35129	−0.675643	+	0.737229i	$0.736134\pi$
$44$	0	0
$45$	−6.97858	−1.04030
$46$	0	0
$47$	10.7360	1.56601	0.783006	−	0.622014i	$-0.213685\pi$
$47$	10.7360	1.56601	0.783006	+	0.622014i	$0.213685\pi$
$48$	0	0
$49$	7.68585	1.09798
$50$	0	0
$51$	−0.760597	−0.106505
$52$	0	0
$53$	−3.19656	−0.439081	−0.219540	−	0.975603i	$-0.570456\pi$
$53$	−3.19656	−0.439081	−0.219540	+	0.975603i	$0.570456\pi$
$54$	0	0
$55$	7.83221	1.05610
$56$	0	0
$57$	0	0
$58$	0	0
$59$	3.85363	0.501700	0.250850	−	0.968026i	$-0.419290\pi$
$59$	3.85363	0.501700	0.250850	+	0.968026i	$0.419290\pi$
$60$	0	0
$61$	2.92839	0.374941	0.187471	−	0.982270i	$-0.439971\pi$
$61$	2.92839	0.374941	0.187471	+	0.982270i	$0.439971\pi$
$62$	0	0
$63$	−11.4145	−1.43810
$64$	0	0
$65$	14.4679	1.79452
$66$	0	0
$67$	−11.5181	−1.40715	−0.703577	−	0.710619i	$-0.748415\pi$
$67$	−11.5181	−1.40715	−0.703577	+	0.710619i	$0.748415\pi$
$68$	0	0
$69$	−0.342923	−0.0412831
$70$	0	0
$71$	1.48929	0.176746	0.0883730	−	0.996087i	$-0.471833\pi$
$71$	1.48929	0.176746	0.0883730	+	0.996087i	$0.471833\pi$
$72$	0	0
$73$	−7.68585	−0.899560	−0.449780	−	0.893139i	$-0.648498\pi$
$73$	−7.68585	−0.899560	−0.449780	+	0.893139i	$0.648498\pi$
$74$	0	0
$75$	−0.0716150	−0.00826938
$76$	0	0
$77$	12.8108	1.45992
$78$	0	0
$79$	0.175135	0.0197042	0.00985210	−	0.999951i	$-0.496864\pi$
$79$	0.175135	0.0197042	0.00985210	+	0.999951i	$0.496864\pi$
$80$	0	0
$81$	8.80765	0.978628
$82$	0	0
$83$	−7.00735	−0.769156	−0.384578	−	0.923092i	$-0.625653\pi$
$83$	−7.00735	−0.769156	−0.384578	+	0.923092i	$0.625653\pi$
$84$	0	0
$85$	12.1751	1.32058
$86$	0	0
$87$	−0.00734639	−0.000787616 0
$88$	0	0
$89$	−13.5468	−1.43596	−0.717980	−	0.696063i	$-0.754933\pi$
$89$	−13.5468	−1.43596	−0.717980	+	0.696063i	$0.754933\pi$
$90$	0	0
$91$	23.6644	2.48071
$92$	0	0
$93$	0.503365	0.0521965
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−3.68585	−0.374241	−0.187120	−	0.982337i	$-0.559915\pi$
$97$	−3.68585	−0.374241	−0.187120	+	0.982337i	$0.559915\pi$
$98$	0	0
$99$	−9.95715	−1.00073
$100$	0	0
$101$	−7.61423	−0.757644	−0.378822	−	0.925469i	$-0.623671\pi$
$101$	−7.61423	−0.757644	−0.378822	+	0.925469i	$0.623671\pi$
$102$	0	0
$103$	16.6430	1.63988	0.819942	−	0.572447i	$-0.194006\pi$
$103$	16.6430	1.63988	0.819942	+	0.572447i	$0.194006\pi$
$104$	0	0
$105$	−1.31415	−0.128248
$106$	0	0
$107$	−6.29273	−0.608341	−0.304171	−	0.952618i	$-0.598379\pi$
$107$	−6.29273	−0.608341	−0.304171	+	0.952618i	$0.598379\pi$
$108$	0	0
$109$	11.1537	1.06833	0.534166	−	0.845380i	$-0.320626\pi$
$109$	11.1537	1.06833	0.534166	+	0.845380i	$0.320626\pi$
$110$	0	0
$111$	0.796096	0.0755621
$112$	0	0
$113$	−17.7606	−1.67078	−0.835388	−	0.549660i	$-0.814757\pi$
$113$	−17.7606	−1.67078	−0.835388	+	0.549660i	$0.814757\pi$
$114$	0	0
$115$	5.48929	0.511879
$116$	0	0
$117$	−18.3931	−1.70044
$118$	0	0
$119$	19.9143	1.82554
$120$	0	0
$121$	0.175135	0.0159213
$122$	0	0
$123$	1.06740	0.0962446
$124$	0	0
$125$	−10.5682	−0.945253
$126$	0	0
$127$	18.4250	1.63496	0.817478	−	0.575960i	$-0.195371\pi$
$127$	18.4250	1.63496	0.817478	+	0.575960i	$0.195371\pi$
$128$	0	0
$129$	1.29694	0.114189
$130$	0	0
$131$	−3.22533	−0.281798	−0.140899	−	0.990024i	$-0.544999\pi$
$131$	−3.22533	−0.281798	−0.140899	+	0.990024i	$0.544999\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.05019	0.176452
$136$	0	0
$137$	−5.97858	−0.510784	−0.255392	−	0.966838i	$-0.582205\pi$
$137$	−5.97858	−0.510784	−0.255392	+	0.966838i	$0.582205\pi$
$138$	0	0
$139$	−9.16779	−0.777602	−0.388801	−	0.921322i	$-0.627111\pi$
$139$	−9.16779	−0.777602	−0.388801	+	0.921322i	$0.627111\pi$
$140$	0	0
$141$	−1.57139	−0.132335
$142$	0	0
$143$	20.6430	1.72625
$144$	0	0
$145$	0.117596	0.00976584
$146$	0	0
$147$	−1.12494	−0.0927837
$148$	0	0
$149$	6.15058	0.503875	0.251937	−	0.967744i	$-0.418932\pi$
$149$	6.15058	0.503875	0.251937	+	0.967744i	$0.418932\pi$
$150$	0	0
$151$	5.14637	0.418805	0.209403	−	0.977830i	$-0.432848\pi$
$151$	5.14637	0.418805	0.209403	+	0.977830i	$0.432848\pi$
$152$	0	0
$153$	−15.4783	−1.25135
$154$	0	0
$155$	−8.05754	−0.647197
$156$	0	0
$157$	4.41033	0.351982	0.175991	−	0.984392i	$-0.443687\pi$
$157$	4.41033	0.351982	0.175991	+	0.984392i	$0.443687\pi$
$158$	0	0
$159$	0.467866	0.0371042
$160$	0	0
$161$	8.97858	0.707611
$162$	0	0
$163$	12.3215	0.965094	0.482547	−	0.875870i	$-0.339712\pi$
$163$	12.3215	0.965094	0.482547	+	0.875870i	$0.339712\pi$
$164$	0	0
$165$	−1.14637	−0.0892444
$166$	0	0
$167$	5.78202	0.447426	0.223713	−	0.974655i	$-0.428182\pi$
$167$	5.78202	0.447426	0.223713	+	0.974655i	$0.428182\pi$
$168$	0	0
$169$	25.1323	1.93325
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−1.82487	−0.138742	−0.0693710	−	0.997591i	$-0.522099\pi$
$173$	−1.82487	−0.138742	−0.0693710	+	0.997591i	$0.522099\pi$
$174$	0	0
$175$	1.87506	0.141741
$176$	0	0
$177$	−0.564039	−0.0423958
$178$	0	0
$179$	8.26396	0.617678	0.308839	−	0.951114i	$-0.400060\pi$
$179$	8.26396	0.617678	0.308839	+	0.951114i	$0.400060\pi$
$180$	0	0
$181$	−21.7146	−1.61403	−0.807017	−	0.590528i	$-0.798920\pi$
$181$	−21.7146	−1.61403	−0.807017	+	0.590528i	$0.798920\pi$
$182$	0	0
$183$	−0.428615	−0.0316841
$184$	0	0
$185$	−12.7434	−0.936912
$186$	0	0
$187$	17.3717	1.27034
$188$	0	0
$189$	3.35341	0.243924
$190$	0	0
$191$	−4.58546	−0.331792	−0.165896	−	0.986143i	$-0.553052\pi$
$191$	−4.58546	−0.331792	−0.165896	+	0.986143i	$0.553052\pi$
$192$	0	0
$193$	−12.1751	−0.876385	−0.438193	−	0.898881i	$-0.644381\pi$
$193$	−12.1751	−0.876385	−0.438193	+	0.898881i	$0.644381\pi$
$194$	0	0
$195$	−2.11760	−0.151644
$196$	0	0
$197$	−4.26817	−0.304095	−0.152047	−	0.988373i	$-0.548587\pi$
$197$	−4.26817	−0.304095	−0.152047	+	0.988373i	$0.548587\pi$
$198$	0	0
$199$	−2.86098	−0.202810	−0.101405	−	0.994845i	$-0.532334\pi$
$199$	−2.86098	−0.202810	−0.101405	+	0.994845i	$0.532334\pi$
$200$	0	0
$201$	1.68585	0.118910
$202$	0	0
$203$	0.192347	0.0135001
$204$	0	0
$205$	−17.0863	−1.19336
$206$	0	0
$207$	−6.97858	−0.485045
$208$	0	0
$209$	0	0
$210$	0	0
$211$	18.2755	1.25814	0.629069	−	0.777349i	$-0.283436\pi$
$211$	18.2755	1.25814	0.629069	+	0.777349i	$0.283436\pi$
$212$	0	0
$213$	−0.217980	−0.0149358
$214$	0	0
$215$	−20.7606	−1.41586
$216$	0	0
$217$	−13.1793	−0.894672
$218$	0	0
$219$	1.12494	0.0760166
$220$	0	0
$221$	32.0894	2.15857
$222$	0	0
$223$	−12.0502	−0.806941	−0.403470	−	0.914993i	$-0.632196\pi$
$223$	−12.0502	−0.806941	−0.403470	+	0.914993i	$0.632196\pi$
$224$	0	0
$225$	−1.45738	−0.0971589
$226$	0	0
$227$	−8.71462	−0.578409	−0.289205	−	0.957267i	$-0.593391\pi$
$227$	−8.71462	−0.578409	−0.289205	+	0.957267i	$0.593391\pi$
$228$	0	0
$229$	15.5640	1.02850	0.514250	−	0.857640i	$-0.328070\pi$
$229$	15.5640	1.02850	0.514250	+	0.857640i	$0.328070\pi$
$230$	0	0
$231$	−1.87506	−0.123370
$232$	0	0
$233$	−22.4292	−1.46939	−0.734694	−	0.678399i	$-0.762674\pi$
$233$	−22.4292	−1.46939	−0.734694	+	0.678399i	$0.762674\pi$
$234$	0	0
$235$	25.1537	1.64085
$236$	0	0
$237$	−0.0256337	−0.00166509
$238$	0	0
$239$	30.7434	1.98862	0.994312	−	0.106505i	$-0.0339662\pi$
$239$	30.7434	1.98862	0.994312	+	0.106505i	$0.0339662\pi$
$240$	0	0
$241$	4.95715	0.319318	0.159659	−	0.987172i	$-0.448960\pi$
$241$	4.95715	0.319318	0.159659	+	0.987172i	$0.448960\pi$
$242$	0	0
$243$	−3.91431	−0.251103
$244$	0	0
$245$	18.0073	1.15045
$246$	0	0
$247$	0	0
$248$	0	0
$249$	1.02563	0.0649969
$250$	0	0
$251$	27.5328	1.73785	0.868926	−	0.494942i	$-0.164811\pi$
$251$	27.5328	1.73785	0.868926	+	0.494942i	$0.164811\pi$
$252$	0	0
$253$	7.83221	0.492407
$254$	0	0
$255$	−1.78202	−0.111594
$256$	0	0
$257$	−11.8866	−0.741467	−0.370733	−	0.928739i	$-0.620894\pi$
$257$	−11.8866	−0.741467	−0.370733	+	0.928739i	$0.620894\pi$
$258$	0	0
$259$	−20.8438	−1.29517
$260$	0	0
$261$	−0.149501	−0.00925388
$262$	0	0
$263$	19.4219	1.19760	0.598802	−	0.800897i	$-0.295643\pi$
$263$	19.4219	1.19760	0.598802	+	0.800897i	$0.295643\pi$
$264$	0	0
$265$	−7.48929	−0.460063
$266$	0	0
$267$	1.98279	0.121345
$268$	0	0
$269$	18.4826	1.12690	0.563451	−	0.826150i	$-0.309473\pi$
$269$	18.4826	1.12690	0.563451	+	0.826150i	$0.309473\pi$
$270$	0	0
$271$	16.3429	0.992762	0.496381	−	0.868105i	$-0.334662\pi$
$271$	16.3429	0.992762	0.496381	+	0.868105i	$0.334662\pi$
$272$	0	0
$273$	−3.46365	−0.209630
$274$	0	0
$275$	1.63565	0.0986337
$276$	0	0
$277$	15.7894	0.948691	0.474346	−	0.880339i	$-0.342685\pi$
$277$	15.7894	0.948691	0.474346	+	0.880339i	$0.342685\pi$
$278$	0	0
$279$	10.2436	0.613269
$280$	0	0
$281$	8.42923	0.502846	0.251423	−	0.967877i	$-0.419102\pi$
$281$	8.42923	0.502846	0.251423	+	0.967877i	$0.419102\pi$
$282$	0	0
$283$	1.81079	0.107640	0.0538201	−	0.998551i	$-0.482860\pi$
$283$	1.81079	0.107640	0.0538201	+	0.998551i	$0.482860\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−27.9473	−1.64968
$288$	0	0
$289$	10.0042	0.588483
$290$	0	0
$291$	0.539481	0.0316249
$292$	0	0
$293$	24.2400	1.41612	0.708059	−	0.706154i	$-0.249571\pi$
$293$	24.2400	1.41612	0.708059	+	0.706154i	$0.249571\pi$
$294$	0	0
$295$	9.02877	0.525675
$296$	0	0
$297$	2.92525	0.169740
$298$	0	0
$299$	14.4679	0.836698
$300$	0	0
$301$	−33.9572	−1.95726
$302$	0	0
$303$	1.11446	0.0640241
$304$	0	0
$305$	6.86098	0.392859
$306$	0	0
$307$	22.0031	1.25579	0.627893	−	0.778300i	$-0.283918\pi$
$307$	22.0031	1.25579	0.627893	+	0.778300i	$0.283918\pi$
$308$	0	0
$309$	−2.43596	−0.138577
$310$	0	0
$311$	−32.4752	−1.84150	−0.920750	−	0.390153i	$-0.872422\pi$
$311$	−32.4752	−1.84150	−0.920750	+	0.390153i	$0.872422\pi$
$312$	0	0
$313$	−9.24989	−0.522834	−0.261417	−	0.965226i	$-0.584190\pi$
$313$	−9.24989	−0.522834	−0.261417	+	0.965226i	$0.584190\pi$
$314$	0	0
$315$	−26.7434	−1.50682
$316$	0	0
$317$	6.17513	0.346830	0.173415	−	0.984849i	$-0.444520\pi$
$317$	6.17513	0.346830	0.173415	+	0.984849i	$0.444520\pi$
$318$	0	0
$319$	0.167788	0.00939434
$320$	0	0
$321$	0.921039	0.0514074
$322$	0	0
$323$	0	0
$324$	0	0
$325$	3.02142	0.167598
$326$	0	0
$327$	−1.63252	−0.0902785
$328$	0	0
$329$	41.1428	2.26827
$330$	0	0
$331$	−10.3643	−0.569676	−0.284838	−	0.958576i	$-0.591940\pi$
$331$	−10.3643	−0.569676	−0.284838	+	0.958576i	$0.591940\pi$
$332$	0	0
$333$	16.2008	0.887797
$334$	0	0
$335$	−26.9859	−1.47440
$336$	0	0
$337$	20.6216	1.12333	0.561664	−	0.827365i	$-0.310161\pi$
$337$	20.6216	1.12333	0.561664	+	0.827365i	$0.310161\pi$
$338$	0	0
$339$	2.59954	0.141188
$340$	0	0
$341$	−11.4966	−0.622578
$342$	0	0
$343$	2.62831	0.141915
$344$	0	0
$345$	−0.803442	−0.0432559
$346$	0	0
$347$	−27.8108	−1.49296	−0.746481	−	0.665407i	$-0.768258\pi$
$347$	−27.8108	−1.49296	−0.746481	+	0.665407i	$0.768258\pi$
$348$	0	0
$349$	5.60688	0.300130	0.150065	−	0.988676i	$-0.452052\pi$
$349$	5.60688	0.300130	0.150065	+	0.988676i	$0.452052\pi$
$350$	0	0
$351$	5.40360	0.288423
$352$	0	0
$353$	−30.0105	−1.59730	−0.798648	−	0.601798i	$-0.794451\pi$
$353$	−30.0105	−1.59730	−0.798648	+	0.601798i	$0.794451\pi$
$354$	0	0
$355$	3.48929	0.185192
$356$	0	0
$357$	−2.91477	−0.154266
$358$	0	0
$359$	−8.97123	−0.473483	−0.236742	−	0.971573i	$-0.576080\pi$
$359$	−8.97123	−0.473483	−0.236742	+	0.971573i	$0.576080\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−0.0256337	−0.00134542
$364$	0	0
$365$	−18.0073	−0.942548
$366$	0	0
$367$	−10.4924	−0.547700	−0.273850	−	0.961772i	$-0.588297\pi$
$367$	−10.4924	−0.547700	−0.273850	+	0.961772i	$0.588297\pi$
$368$	0	0
$369$	21.7220	1.13080
$370$	0	0
$371$	−12.2499	−0.635982
$372$	0	0
$373$	−18.3074	−0.947922	−0.473961	−	0.880546i	$-0.657176\pi$
$373$	−18.3074	−0.947922	−0.473961	+	0.880546i	$0.657176\pi$
$374$	0	0
$375$	1.54683	0.0798778
$376$	0	0
$377$	0.309943	0.0159629
$378$	0	0
$379$	24.8438	1.27614	0.638069	−	0.769979i	$-0.279733\pi$
$379$	24.8438	1.27614	0.638069	+	0.769979i	$0.279733\pi$
$380$	0	0
$381$	−2.69679	−0.138161
$382$	0	0
$383$	3.17200	0.162082	0.0810408	−	0.996711i	$-0.474176\pi$
$383$	3.17200	0.162082	0.0810408	+	0.996711i	$0.474176\pi$
$384$	0	0
$385$	30.0147	1.52969
$386$	0	0
$387$	26.3931	1.34164
$388$	0	0
$389$	−11.3889	−0.577440	−0.288720	−	0.957414i	$-0.593230\pi$
$389$	−11.3889	−0.577440	−0.288720	+	0.957414i	$0.593230\pi$
$390$	0	0
$391$	12.1751	0.615723
$392$	0	0
$393$	0.472077	0.0238131
$394$	0	0
$395$	0.410327	0.0206458
$396$	0	0
$397$	−18.8855	−0.947838	−0.473919	−	0.880568i	$-0.657161\pi$
$397$	−18.8855	−0.947838	−0.473919	+	0.880568i	$0.657161\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−20.0214	−0.999822	−0.499911	−	0.866077i	$-0.666634\pi$
$401$	−20.0214	−0.999822	−0.499911	+	0.866077i	$0.666634\pi$
$402$	0	0
$403$	−21.2369	−1.05788
$404$	0	0
$405$	20.6357	1.02539
$406$	0	0
$407$	−18.1825	−0.901272
$408$	0	0
$409$	10.9143	0.539678	0.269839	−	0.962905i	$-0.413030\pi$
$409$	10.9143	0.539678	0.269839	+	0.962905i	$0.413030\pi$
$410$	0	0
$411$	0.875057	0.0431634
$412$	0	0
$413$	14.7679	0.726683
$414$	0	0
$415$	−16.4177	−0.805912
$416$	0	0
$417$	1.34185	0.0657106
$418$	0	0
$419$	−15.4145	−0.753049	−0.376525	−	0.926407i	$-0.622881\pi$
$419$	−15.4145	−0.753049	−0.376525	+	0.926407i	$0.622881\pi$
$420$	0	0
$421$	−5.96450	−0.290692	−0.145346	−	0.989381i	$-0.546430\pi$
$421$	−5.96450	−0.290692	−0.145346	+	0.989381i	$0.546430\pi$
$422$	0	0
$423$	−31.9781	−1.55483
$424$	0	0
$425$	2.54262	0.123335
$426$	0	0
$427$	11.2222	0.543080
$428$	0	0
$429$	−3.02142	−0.145876
$430$	0	0
$431$	−29.0189	−1.39779	−0.698896	−	0.715224i	$-0.746325\pi$
$431$	−29.0189	−1.39779	−0.698896	+	0.715224i	$0.746325\pi$
$432$	0	0
$433$	−18.3822	−0.883391	−0.441695	−	0.897165i	$-0.645623\pi$
$433$	−18.3822	−0.883391	−0.441695	+	0.897165i	$0.645623\pi$
$434$	0	0
$435$	−0.0172120	−0.000825254 0
$436$	0	0
$437$	0	0
$438$	0	0
$439$	7.61423	0.363407	0.181704	−	0.983353i	$-0.441839\pi$
$439$	7.61423	0.363407	0.181704	+	0.983353i	$0.441839\pi$
$440$	0	0
$441$	−22.8929	−1.09014
$442$	0	0
$443$	7.50337	0.356496	0.178248	−	0.983986i	$-0.442957\pi$
$443$	7.50337	0.356496	0.178248	+	0.983986i	$0.442957\pi$
$444$	0	0
$445$	−31.7392	−1.50458
$446$	0	0
$447$	−0.900232	−0.0425795
$448$	0	0
$449$	9.31836	0.439761	0.219880	−	0.975527i	$-0.429433\pi$
$449$	9.31836	0.439761	0.219880	+	0.975527i	$0.429433\pi$
$450$	0	0
$451$	−24.3790	−1.14796
$452$	0	0
$453$	−0.753250	−0.0353908
$454$	0	0
$455$	55.4439	2.59925
$456$	0	0
$457$	22.4893	1.05200	0.526002	−	0.850483i	$-0.323690\pi$
$457$	22.4893	1.05200	0.526002	+	0.850483i	$0.323690\pi$
$458$	0	0
$459$	4.54729	0.212249
$460$	0	0
$461$	3.34606	0.155841	0.0779207	−	0.996960i	$-0.475172\pi$
$461$	3.34606	0.155841	0.0779207	+	0.996960i	$0.475172\pi$
$462$	0	0
$463$	−26.0147	−1.20901	−0.604503	−	0.796603i	$-0.706628\pi$
$463$	−26.0147	−1.20901	−0.604503	+	0.796603i	$0.706628\pi$
$464$	0	0
$465$	1.17935	0.0546908
$466$	0	0
$467$	−11.3429	−0.524888	−0.262444	−	0.964947i	$-0.584528\pi$
$467$	−11.3429	−0.524888	−0.262444	+	0.964947i	$0.584528\pi$
$468$	0	0
$469$	−44.1396	−2.03818
$470$	0	0
$471$	−0.645520	−0.0297440
$472$	0	0
$473$	−29.6216	−1.36200
$474$	0	0
$475$	0	0
$476$	0	0
$477$	9.52119	0.435945
$478$	0	0
$479$	−11.5040	−0.525630	−0.262815	−	0.964846i	$-0.584651\pi$
$479$	−11.5040	−0.525630	−0.262815	+	0.964846i	$0.584651\pi$
$480$	0	0
$481$	−33.5872	−1.53144
$482$	0	0
$483$	−1.31415	−0.0597961
$484$	0	0
$485$	−8.63565	−0.392125
$486$	0	0
$487$	22.3832	1.01428	0.507141	−	0.861863i	$-0.330702\pi$
$487$	22.3832	1.01428	0.507141	+	0.861863i	$0.330702\pi$
$488$	0	0
$489$	−1.80344	−0.0815545
$490$	0	0
$491$	−22.0319	−0.994286	−0.497143	−	0.867669i	$-0.665618\pi$
$491$	−22.0319	−0.994286	−0.497143	+	0.867669i	$0.665618\pi$
$492$	0	0
$493$	0.260826	0.0117470
$494$	0	0
$495$	−23.3288	−1.04855
$496$	0	0
$497$	5.70727	0.256006
$498$	0	0
$499$	−1.21063	−0.0541954	−0.0270977	−	0.999633i	$-0.508627\pi$
$499$	−1.21063	−0.0541954	−0.0270977	+	0.999633i	$0.508627\pi$
$500$	0	0
$501$	−0.846288	−0.0378094
$502$	0	0
$503$	−30.4924	−1.35959	−0.679795	−	0.733402i	$-0.737931\pi$
$503$	−30.4924	−1.35959	−0.679795	+	0.733402i	$0.737931\pi$
$504$	0	0
$505$	−17.8396	−0.793850
$506$	0	0
$507$	−3.67850	−0.163368
$508$	0	0
$509$	−0.846288	−0.0375111	−0.0187555	−	0.999824i	$-0.505970\pi$
$509$	−0.846288	−0.0375111	−0.0187555	+	0.999824i	$0.505970\pi$
$510$	0	0
$511$	−29.4538	−1.30296
$512$	0	0
$513$	0	0
$514$	0	0
$515$	38.9933	1.71825
$516$	0	0
$517$	35.8898	1.57843
$518$	0	0
$519$	0.267097	0.0117243
$520$	0	0
$521$	−42.6044	−1.86653	−0.933266	−	0.359187i	$-0.883054\pi$
$521$	−42.6044	−1.86653	−0.933266	+	0.359187i	$0.883054\pi$
$522$	0	0
$523$	−11.7245	−0.512676	−0.256338	−	0.966587i	$-0.582516\pi$
$523$	−11.7245	−0.512676	−0.256338	+	0.966587i	$0.582516\pi$
$524$	0	0
$525$	−0.274444	−0.0119777
$526$	0	0
$527$	−17.8715	−0.778493
$528$	0	0
$529$	−17.5107	−0.761335
$530$	0	0
$531$	−11.4783	−0.498118
$532$	0	0
$533$	−45.0336	−1.95062
$534$	0	0
$535$	−14.7434	−0.637412
$536$	0	0
$537$	−1.20956	−0.0521963
$538$	0	0
$539$	25.6932	1.10668
$540$	0	0
$541$	−18.1323	−0.779568	−0.389784	−	0.920906i	$-0.627450\pi$
$541$	−18.1323	−0.779568	−0.389784	+	0.920906i	$0.627450\pi$
$542$	0	0
$543$	3.17827	0.136393
$544$	0	0
$545$	26.1323	1.11938
$546$	0	0
$547$	43.4036	1.85580	0.927902	−	0.372824i	$-0.121611\pi$
$547$	43.4036	1.85580	0.927902	+	0.372824i	$0.121611\pi$
$548$	0	0
$549$	−8.72242	−0.372264
$550$	0	0
$551$	0	0
$552$	0	0
$553$	0.671153	0.0285403
$554$	0	0
$555$	1.86519	0.0791730
$556$	0	0
$557$	9.19656	0.389671	0.194835	−	0.980836i	$-0.437583\pi$
$557$	9.19656	0.389671	0.194835	+	0.980836i	$0.437583\pi$
$558$	0	0
$559$	−54.7178	−2.31431
$560$	0	0
$561$	−2.54262	−0.107349
$562$	0	0
$563$	41.3576	1.74302	0.871508	−	0.490382i	$-0.163143\pi$
$563$	41.3576	1.74302	0.871508	+	0.490382i	$0.163143\pi$
$564$	0	0
$565$	−41.6117	−1.75062
$566$	0	0
$567$	33.7528	1.41748
$568$	0	0
$569$	1.22219	0.0512369	0.0256185	−	0.999672i	$-0.491844\pi$
$569$	1.22219	0.0512369	0.0256185	+	0.999672i	$0.491844\pi$
$570$	0	0
$571$	−4.32150	−0.180849	−0.0904246	−	0.995903i	$-0.528822\pi$
$571$	−4.32150	−0.180849	−0.0904246	+	0.995903i	$0.528822\pi$
$572$	0	0
$573$	0.671153	0.0280378
$574$	0	0
$575$	1.14637	0.0478067
$576$	0	0
$577$	2.03863	0.0848695	0.0424347	−	0.999099i	$-0.486489\pi$
$577$	2.03863	0.0848695	0.0424347	+	0.999099i	$0.486489\pi$
$578$	0	0
$579$	1.78202	0.0740582
$580$	0	0
$581$	−26.8536	−1.11408
$582$	0	0
$583$	−10.6858	−0.442563
$584$	0	0
$585$	−43.0937	−1.78170
$586$	0	0
$587$	−8.66021	−0.357445	−0.178723	−	0.983900i	$-0.557196\pi$
$587$	−8.66021	−0.357445	−0.178723	+	0.983900i	$0.557196\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	0.624713	0.0256973
$592$	0	0
$593$	10.9718	0.450560	0.225280	−	0.974294i	$-0.427670\pi$
$593$	10.9718	0.450560	0.225280	+	0.974294i	$0.427670\pi$
$594$	0	0
$595$	46.6577	1.91278
$596$	0	0
$597$	0.418749	0.0171383
$598$	0	0
$599$	45.4219	1.85589	0.927944	−	0.372720i	$-0.121575\pi$
$599$	45.4219	1.85589	0.927944	+	0.372720i	$0.121575\pi$
$600$	0	0
$601$	29.8757	1.21865	0.609327	−	0.792919i	$-0.291440\pi$
$601$	29.8757	1.21865	0.609327	+	0.792919i	$0.291440\pi$
$602$	0	0
$603$	34.3074	1.39711
$604$	0	0
$605$	0.410327	0.0166822
$606$	0	0
$607$	−34.7764	−1.41153	−0.705765	−	0.708446i	$-0.749396\pi$
$607$	−34.7764	−1.41153	−0.705765	+	0.708446i	$0.749396\pi$
$608$	0	0
$609$	−0.0281529	−0.00114081
$610$	0	0
$611$	66.2965	2.68207
$612$	0	0
$613$	14.7606	0.596175	0.298087	−	0.954539i	$-0.403651\pi$
$613$	14.7606	0.596175	0.298087	+	0.954539i	$0.403651\pi$
$614$	0	0
$615$	2.50085	0.100844
$616$	0	0
$617$	−26.0361	−1.04817	−0.524087	−	0.851665i	$-0.675593\pi$
$617$	−26.0361	−1.04817	−0.524087	+	0.851665i	$0.675593\pi$
$618$	0	0
$619$	−40.7497	−1.63787	−0.818933	−	0.573888i	$-0.805434\pi$
$619$	−40.7497	−1.63787	−0.818933	+	0.573888i	$0.805434\pi$
$620$	0	0
$621$	2.05019	0.0822714
$622$	0	0
$623$	−51.9143	−2.07990
$624$	0	0
$625$	−27.2070	−1.08828
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−28.2646	−1.12698
$630$	0	0
$631$	31.8213	1.26678	0.633392	−	0.773831i	$-0.281662\pi$
$631$	31.8213	1.26678	0.633392	+	0.773831i	$0.281662\pi$
$632$	0	0
$633$	−2.67491	−0.106318
$634$	0	0
$635$	43.1684	1.71309
$636$	0	0
$637$	47.4611	1.88048
$638$	0	0
$639$	−4.43596	−0.175484
$640$	0	0
$641$	−8.16465	−0.322484	−0.161242	−	0.986915i	$-0.551550\pi$
$641$	−8.16465	−0.322484	−0.161242	+	0.986915i	$0.551550\pi$
$642$	0	0
$643$	6.59702	0.260161	0.130081	−	0.991503i	$-0.458476\pi$
$643$	6.59702	0.260161	0.130081	+	0.991503i	$0.458476\pi$
$644$	0	0
$645$	3.03863	0.119646
$646$	0	0
$647$	2.60015	0.102223	0.0511113	−	0.998693i	$-0.483724\pi$
$647$	2.60015	0.102223	0.0511113	+	0.998693i	$0.483724\pi$
$648$	0	0
$649$	12.8824	0.505679
$650$	0	0
$651$	1.92900	0.0756035
$652$	0	0
$653$	−2.39312	−0.0936498	−0.0468249	−	0.998903i	$-0.514910\pi$
$653$	−2.39312	−0.0936498	−0.0468249	+	0.998903i	$0.514910\pi$
$654$	0	0
$655$	−7.55669	−0.295264
$656$	0	0
$657$	22.8929	0.893137
$658$	0	0
$659$	22.6833	0.883617	0.441808	−	0.897109i	$-0.354337\pi$
$659$	22.6833	0.883617	0.441808	+	0.897109i	$0.354337\pi$
$660$	0	0
$661$	40.5500	1.57721	0.788605	−	0.614900i	$-0.210803\pi$
$661$	40.5500	1.57721	0.788605	+	0.614900i	$0.210803\pi$
$662$	0	0
$663$	−4.69679	−0.182408
$664$	0	0
$665$	0	0
$666$	0	0
$667$	0.117596	0.00455334
$668$	0	0
$669$	1.76373	0.0681898
$670$	0	0
$671$	9.78937	0.377914
$672$	0	0
$673$	−20.1004	−0.774813	−0.387406	−	0.921909i	$-0.626629\pi$
$673$	−20.1004	−0.774813	−0.387406	+	0.921909i	$0.626629\pi$
$674$	0	0
$675$	0.428156	0.0164797
$676$	0	0
$677$	−19.3717	−0.744515	−0.372257	−	0.928130i	$-0.621416\pi$
$677$	−19.3717	−0.744515	−0.372257	+	0.928130i	$0.621416\pi$
$678$	0	0
$679$	−14.1249	−0.542066
$680$	0	0
$681$	1.27552	0.0488780
$682$	0	0
$683$	−16.1151	−0.616626	−0.308313	−	0.951285i	$-0.599764\pi$
$683$	−16.1151	−0.616626	−0.308313	+	0.951285i	$0.599764\pi$
$684$	0	0
$685$	−14.0073	−0.535193
$686$	0	0
$687$	−2.27804	−0.0869126
$688$	0	0
$689$	−19.7392	−0.752003
$690$	0	0
$691$	15.4145	0.586397	0.293198	−	0.956052i	$-0.405280\pi$
$691$	15.4145	0.586397	0.293198	+	0.956052i	$0.405280\pi$
$692$	0	0
$693$	−38.1579	−1.44950
$694$	0	0
$695$	−21.4794	−0.814761
$696$	0	0
$697$	−37.8971	−1.43545
$698$	0	0
$699$	3.28287	0.124169
$700$	0	0
$701$	35.8297	1.35327	0.676634	−	0.736319i	$-0.263438\pi$
$701$	35.8297	1.35327	0.676634	+	0.736319i	$0.263438\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−3.68164	−0.138658
$706$	0	0
$707$	−29.1793	−1.09740
$708$	0	0
$709$	8.82800	0.331543	0.165771	−	0.986164i	$-0.446989\pi$
$709$	8.82800	0.331543	0.165771	+	0.986164i	$0.446989\pi$
$710$	0	0
$711$	−0.521652	−0.0195635
$712$	0	0
$713$	−8.05754	−0.301757
$714$	0	0
$715$	48.3650	1.80875
$716$	0	0
$717$	−4.49977	−0.168047
$718$	0	0
$719$	−16.2671	−0.606660	−0.303330	−	0.952886i	$-0.598099\pi$
$719$	−16.2671	−0.606660	−0.303330	+	0.952886i	$0.598099\pi$
$720$	0	0
$721$	63.7795	2.37527
$722$	0	0
$723$	−0.725556	−0.0269837
$724$	0	0
$725$	0.0245584	0.000912077 0
$726$	0	0
$727$	40.7606	1.51173	0.755863	−	0.654729i	$-0.227217\pi$
$727$	40.7606	1.51173	0.755863	+	0.654729i	$0.227217\pi$
$728$	0	0
$729$	−25.8500	−0.957409
$730$	0	0
$731$	−46.0466	−1.70309
$732$	0	0
$733$	19.6827	0.726998	0.363499	−	0.931595i	$-0.381582\pi$
$733$	19.6827	0.726998	0.363499	+	0.931595i	$0.381582\pi$
$734$	0	0
$735$	−2.63565	−0.0972176
$736$	0	0
$737$	−38.5040	−1.41831
$738$	0	0
$739$	−42.7549	−1.57277	−0.786383	−	0.617739i	$-0.788049\pi$
$739$	−42.7549	−1.57277	−0.786383	+	0.617739i	$0.788049\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−16.6848	−0.612105	−0.306052	−	0.952015i	$-0.599008\pi$
$743$	−16.6848	−0.612105	−0.306052	+	0.952015i	$0.599008\pi$
$744$	0	0
$745$	14.4103	0.527954
$746$	0	0
$747$	20.8719	0.763664
$748$	0	0
$749$	−24.1151	−0.881146
$750$	0	0
$751$	−30.8610	−1.12613	−0.563067	−	0.826412i	$-0.690379\pi$
$751$	−30.8610	−1.12613	−0.563067	+	0.826412i	$0.690379\pi$
$752$	0	0
$753$	−4.02984	−0.146856
$754$	0	0
$755$	12.0575	0.438819
$756$	0	0
$757$	−15.9828	−0.580904	−0.290452	−	0.956890i	$-0.593806\pi$
$757$	−15.9828	−0.580904	−0.290452	+	0.956890i	$0.593806\pi$
$758$	0	0
$759$	−1.14637	−0.0416104
$760$	0	0
$761$	22.9975	0.833658	0.416829	−	0.908985i	$-0.363141\pi$
$761$	22.9975	0.833658	0.416829	+	0.908985i	$0.363141\pi$
$762$	0	0
$763$	42.7434	1.54741
$764$	0	0
$765$	−36.2646	−1.31115
$766$	0	0
$767$	23.7967	0.859249
$768$	0	0
$769$	29.9462	1.07989	0.539944	−	0.841701i	$-0.318445\pi$
$769$	29.9462	1.07989	0.539944	+	0.841701i	$0.318445\pi$
$770$	0	0
$771$	1.73979	0.0626570
$772$	0	0
$773$	5.50758	0.198094	0.0990469	−	0.995083i	$-0.468421\pi$
$773$	5.50758	0.198094	0.0990469	+	0.995083i	$0.468421\pi$
$774$	0	0
$775$	−1.68271	−0.0604447
$776$	0	0
$777$	3.05081	0.109447
$778$	0	0
$779$	0	0
$780$	0	0
$781$	4.97858	0.178147
$782$	0	0
$783$	0.0439210	0.00156961
$784$	0	0
$785$	10.3331	0.368803
$786$	0	0
$787$	−8.40719	−0.299684	−0.149842	−	0.988710i	$-0.547877\pi$
$787$	−8.40719	−0.299684	−0.149842	+	0.988710i	$0.547877\pi$
$788$	0	0
$789$	−2.84269	−0.101203
$790$	0	0
$791$	−68.0624	−2.42002
$792$	0	0
$793$	18.0832	0.642152
$794$	0	0
$795$	1.09617	0.0388773
$796$	0	0
$797$	33.2186	1.17666	0.588332	−	0.808620i	$-0.299785\pi$
$797$	33.2186	1.17666	0.588332	+	0.808620i	$0.299785\pi$
$798$	0	0
$799$	55.7904	1.97372
$800$	0	0
$801$	40.3503	1.42571
$802$	0	0
$803$	−25.6932	−0.906693
$804$	0	0
$805$	21.0361	0.741426
$806$	0	0
$807$	−2.70521	−0.0952279
$808$	0	0
$809$	12.8971	0.453438	0.226719	−	0.973960i	$-0.427200\pi$
$809$	12.8971	0.453438	0.226719	+	0.973960i	$0.427200\pi$
$810$	0	0
$811$	−2.66863	−0.0937084	−0.0468542	−	0.998902i	$-0.514920\pi$
$811$	−2.66863	−0.0937084	−0.0468542	+	0.998902i	$0.514920\pi$
$812$	0	0
$813$	−2.39204	−0.0838925
$814$	0	0
$815$	28.8683	1.01121
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−70.4863	−2.46299
$820$	0	0
$821$	−33.6535	−1.17451	−0.587257	−	0.809400i	$-0.699792\pi$
$821$	−33.6535	−1.17451	−0.587257	+	0.809400i	$0.699792\pi$
$822$	0	0
$823$	17.8396	0.621848	0.310924	−	0.950435i	$-0.399362\pi$
$823$	17.8396	0.621848	0.310924	+	0.950435i	$0.399362\pi$
$824$	0	0
$825$	−0.239403	−0.00833495
$826$	0	0
$827$	3.75325	0.130513	0.0652567	−	0.997869i	$-0.479213\pi$
$827$	3.75325	0.130513	0.0652567	+	0.997869i	$0.479213\pi$
$828$	0	0
$829$	2.77781	0.0964773	0.0482386	−	0.998836i	$-0.484639\pi$
$829$	2.77781	0.0964773	0.0482386	+	0.998836i	$0.484639\pi$
$830$	0	0
$831$	−2.31102	−0.0801683
$832$	0	0
$833$	39.9399	1.38384
$834$	0	0
$835$	13.5468	0.468807
$836$	0	0
$837$	−3.00941	−0.104020
$838$	0	0
$839$	−56.2228	−1.94103	−0.970513	−	0.241047i	$-0.922509\pi$
$839$	−56.2228	−1.94103	−0.970513	+	0.241047i	$0.922509\pi$
$840$	0	0
$841$	−28.9975	−0.999913
$842$	0	0
$843$	−1.23375	−0.0424926
$844$	0	0
$845$	58.8830	2.02564
$846$	0	0
$847$	0.671153	0.0230611
$848$	0	0
$849$	−0.265037	−0.00909605
$850$	0	0
$851$	−12.7434	−0.436838
$852$	0	0
$853$	−10.3246	−0.353509	−0.176754	−	0.984255i	$-0.556560\pi$
$853$	−10.3246	−0.353509	−0.176754	+	0.984255i	$0.556560\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	53.1789	1.81656	0.908278	−	0.418367i	$-0.137397\pi$
$857$	53.1789	1.81656	0.908278	+	0.418367i	$0.137397\pi$
$858$	0	0
$859$	−10.2530	−0.349829	−0.174914	−	0.984584i	$-0.555965\pi$
$859$	−10.2530	−0.349829	−0.174914	+	0.984584i	$0.555965\pi$
$860$	0	0
$861$	4.09052	0.139404
$862$	0	0
$863$	−10.7679	−0.366545	−0.183273	−	0.983062i	$-0.558669\pi$
$863$	−10.7679	−0.366545	−0.183273	+	0.983062i	$0.558669\pi$
$864$	0	0
$865$	−4.27552	−0.145372
$866$	0	0
$867$	−1.46427	−0.0497293
$868$	0	0
$869$	0.585462	0.0198604
$870$	0	0
$871$	−71.1256	−2.41000
$872$	0	0
$873$	10.9786	0.371569
$874$	0	0
$875$	−40.4998	−1.36914
$876$	0	0
$877$	22.7276	0.767457	0.383729	−	0.923446i	$-0.374640\pi$
$877$	22.7276	0.767457	0.383729	+	0.923446i	$0.374640\pi$
$878$	0	0
$879$	−3.54790	−0.119668
$880$	0	0
$881$	−8.44644	−0.284568	−0.142284	−	0.989826i	$-0.545445\pi$
$881$	−8.44644	−0.284568	−0.142284	+	0.989826i	$0.545445\pi$
$882$	0	0
$883$	45.0393	1.51569	0.757846	−	0.652434i	$-0.226252\pi$
$883$	45.0393	1.51569	0.757846	+	0.652434i	$0.226252\pi$
$884$	0	0
$885$	−1.32150	−0.0444217
$886$	0	0
$887$	−20.8463	−0.699950	−0.349975	−	0.936759i	$-0.613810\pi$
$887$	−20.8463	−0.699950	−0.349975	+	0.936759i	$0.613810\pi$
$888$	0	0
$889$	70.6086	2.36814
$890$	0	0
$891$	29.4433	0.986388
$892$	0	0
$893$	0	0
$894$	0	0
$895$	19.3618	0.647194
$896$	0	0
$897$	−2.11760	−0.0707045
$898$	0	0
$899$	−0.172615	−0.00575705
$900$	0	0
$901$	−16.6111	−0.553396
$902$	0	0
$903$	4.97016	0.165396
$904$	0	0
$905$	−50.8757	−1.69116
$906$	0	0
$907$	24.4109	0.810552	0.405276	−	0.914194i	$-0.367175\pi$
$907$	24.4109	0.810552	0.405276	+	0.914194i	$0.367175\pi$
$908$	0	0
$909$	22.6796	0.752234
$910$	0	0
$911$	24.2744	0.804248	0.402124	−	0.915585i	$-0.368272\pi$
$911$	24.2744	0.804248	0.402124	+	0.915585i	$0.368272\pi$
$912$	0	0
$913$	−23.4250	−0.775255
$914$	0	0
$915$	−1.00421	−0.0331982
$916$	0	0
$917$	−12.3601	−0.408168
$918$	0	0
$919$	−31.8568	−1.05086	−0.525429	−	0.850837i	$-0.676095\pi$
$919$	−31.8568	−1.05086	−0.525429	+	0.850837i	$0.676095\pi$
$920$	0	0
$921$	−3.22050	−0.106119
$922$	0	0
$923$	9.19656	0.302708
$924$	0	0
$925$	−2.66129	−0.0875026
$926$	0	0
$927$	−49.5725	−1.62817
$928$	0	0
$929$	−50.2087	−1.64730	−0.823648	−	0.567102i	$-0.808064\pi$
$929$	−50.2087	−1.64730	−0.823648	+	0.567102i	$0.808064\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	4.75325	0.155614
$934$	0	0
$935$	40.7005	1.33105
$936$	0	0
$937$	−4.65600	−0.152105	−0.0760525	−	0.997104i	$-0.524232\pi$
$937$	−4.65600	−0.152105	−0.0760525	+	0.997104i	$0.524232\pi$
$938$	0	0
$939$	1.35386	0.0441817
$940$	0	0
$941$	−23.3973	−0.762731	−0.381366	−	0.924424i	$-0.624546\pi$
$941$	−23.3973	−0.762731	−0.381366	+	0.924424i	$0.624546\pi$
$942$	0	0
$943$	−17.0863	−0.556407
$944$	0	0
$945$	7.85677	0.255581
$946$	0	0
$947$	−57.4267	−1.86612	−0.933059	−	0.359724i	$-0.882871\pi$
$947$	−57.4267	−1.86612	−0.933059	+	0.359724i	$0.882871\pi$
$948$	0	0
$949$	−47.4611	−1.54065
$950$	0	0
$951$	−0.903827	−0.0293086
$952$	0	0
$953$	−39.9933	−1.29551	−0.647755	−	0.761849i	$-0.724292\pi$
$953$	−39.9933	−1.29551	−0.647755	+	0.761849i	$0.724292\pi$
$954$	0	0
$955$	−10.7434	−0.347648
$956$	0	0
$957$	−0.0245584	−0.000793861 0
$958$	0	0
$959$	−22.9112	−0.739840
$960$	0	0
$961$	−19.1726	−0.618471
$962$	0	0
$963$	18.7434	0.603997
$964$	0	0
$965$	−28.5254	−0.918265
$966$	0	0
$967$	−3.88240	−0.124850	−0.0624248	−	0.998050i	$-0.519883\pi$
$967$	−3.88240	−0.124850	−0.0624248	+	0.998050i	$0.519883\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−1.37483	−0.0441203	−0.0220602	−	0.999757i	$-0.507023\pi$
$971$	−1.37483	−0.0441203	−0.0220602	+	0.999757i	$0.507023\pi$
$972$	0	0
$973$	−35.1329	−1.12631
$974$	0	0
$975$	−0.442232	−0.0141628
$976$	0	0
$977$	20.2541	0.647986	0.323993	−	0.946059i	$-0.394975\pi$
$977$	20.2541	0.647986	0.323993	+	0.946059i	$0.394975\pi$
$978$	0	0
$979$	−45.2860	−1.44735
$980$	0	0
$981$	−33.2222	−1.06070
$982$	0	0
$983$	12.7606	0.407000	0.203500	−	0.979075i	$-0.434768\pi$
$983$	12.7606	0.407000	0.203500	+	0.979075i	$0.434768\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	−6.02188	−0.191679
$988$	0	0
$989$	−20.7606	−0.660149
$990$	0	0
$991$	−18.9529	−0.602060	−0.301030	−	0.953615i	$-0.597330\pi$
$991$	−18.9529	−0.602060	−0.301030	+	0.953615i	$0.597330\pi$
$992$	0	0
$993$	1.51698	0.0481400
$994$	0	0
$995$	−6.70306	−0.212501
$996$	0	0
$997$	−12.0649	−0.382099	−0.191049	−	0.981580i	$-0.561189\pi$
$997$	−12.0649	−0.382099	−0.191049	+	0.981580i	$0.561189\pi$
$998$	0	0
$999$	−4.75952	−0.150585

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.r.1.2		3
4.3	odd	2		5776.2.a.bk.1.2		3
19.7	even	3		152.2.i.c.49.2	✓	6
19.11	even	3		152.2.i.c.121.2	yes	6
19.18	odd	2		2888.2.a.n.1.2		3
57.11	odd	6		1368.2.s.k.577.3		6
57.26	odd	6		1368.2.s.k.505.3		6
76.7	odd	6		304.2.i.f.49.2		6
76.11	odd	6		304.2.i.f.273.2		6
76.75	even	2		5776.2.a.bq.1.2		3
152.11	odd	6		1216.2.i.m.577.2		6
152.45	even	6		1216.2.i.n.961.2		6
152.83	odd	6		1216.2.i.m.961.2		6
152.125	even	6		1216.2.i.n.577.2		6
228.11	even	6		2736.2.s.y.577.3		6
228.83	even	6		2736.2.s.y.1873.3		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.i.c.49.2	✓	6	19.7	even	3
152.2.i.c.121.2	yes	6	19.11	even	3
304.2.i.f.49.2		6	76.7	odd	6
304.2.i.f.273.2		6	76.11	odd	6
1216.2.i.m.577.2		6	152.11	odd	6
1216.2.i.m.961.2		6	152.83	odd	6
1216.2.i.n.577.2		6	152.125	even	6
1216.2.i.n.961.2		6	152.45	even	6
1368.2.s.k.505.3		6	57.26	odd	6
1368.2.s.k.577.3		6	57.11	odd	6
2736.2.s.y.577.3		6	228.11	even	6
2736.2.s.y.1873.3		6	228.83	even	6
2888.2.a.n.1.2		3	19.18	odd	2
2888.2.a.r.1.2		3	1.1	even	1	trivial
5776.2.a.bk.1.2		3	4.3	odd	2
5776.2.a.bq.1.2		3	76.75	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:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2888.a (trivial)

\(f(q)\)	\(=\)	\(q-0.146365 q^{3} +2.34292 q^{5} +3.83221 q^{7} -2.97858 q^{9} +3.34292 q^{11} +6.17513 q^{13} -0.342923 q^{15} +5.19656 q^{17} -0.560904 q^{21} +2.34292 q^{23} +0.489289 q^{25} +0.875057 q^{27} +0.0501921 q^{29} -3.43910 q^{31} -0.489289 q^{33} +8.97858 q^{35} -5.43910 q^{37} -0.903827 q^{39} -7.29273 q^{41} -8.86098 q^{43} -6.97858 q^{45} +10.7360 q^{47} +7.68585 q^{49} -0.760597 q^{51} -3.19656 q^{53} +7.83221 q^{55} +3.85363 q^{59} +2.92839 q^{61} -11.4145 q^{63} +14.4679 q^{65} -11.5181 q^{67} -0.342923 q^{69} +1.48929 q^{71} -7.68585 q^{73} -0.0716150 q^{75} +12.8108 q^{77} +0.175135 q^{79} +8.80765 q^{81} -7.00735 q^{83} +12.1751 q^{85} -0.00734639 q^{87} -13.5468 q^{89} +23.6644 q^{91} +0.503365 q^{93} -3.68585 q^{97} -9.95715 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{5} - 2 q^{7} + 6 q^{9} + 4 q^{11} - q^{13} + 5 q^{15} + 11 q^{17} - 6 q^{21} + q^{23} - 6 q^{25} + 19 q^{27} - 3 q^{29} - 6 q^{31} + 6 q^{33} + 12 q^{35} - 12 q^{37} - q^{39} - 19 q^{41}+ \cdots + q^{97}+O(q^{100}) \) 3 * q + q^3 + q^5 - 2 * q^7 + 6 * q^9 + 4 * q^11 - q^13 + 5 * q^15 + 11 * q^17 - 6 * q^21 + q^23 - 6 * q^25 + 19 * q^27 - 3 * q^29 - 6 * q^31 + 6 * q^33 + 12 * q^35 - 12 * q^37 - q^39 - 19 * q^41 + 5 * q^43 - 6 * q^45 + 17 * q^47 + 11 * q^49 + 23 * q^51 - 5 * q^53 + 10 * q^55 + 13 * q^59 - 3 * q^61 - 40 * q^63 + 21 * q^65 - 9 * q^67 + 5 * q^69 - 3 * q^71 - 11 * q^73 - 12 * q^75 + 10 * q^77 - 19 * q^79 + 23 * q^81 + 12 * q^83 + 17 * q^85 + 33 * q^87 + 3 * q^89 + 44 * q^91 + 42 * q^93 + q^97

Introduction

L-functions

Modular forms

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Database

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