LMFDB - Embedded newform 2888.2.a.v.1.5

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 = [8,0,-6,0,-2,0,-2,0,2,0,12,0,-6,0,0] 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:	$1$
Dimension:	$8$
Coefficient field:	8.8.10564000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 14x^{5} + 24x^{4} - 28x^{3} - 21x^{2} + 16x + 1 $ x^8 - 2x^7 - 9x^6 + 14x^5 + 24x^4 - 28x^3 - 21x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$0.650108$ 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)

$f(q)$	$=$	$q-0.349892 q^{3} -2.14090 q^{5} -3.07223 q^{7} -2.87758 q^{9} +3.54580 q^{11} +4.22215 q^{13} +0.749083 q^{15} +5.61700 q^{17} +1.07495 q^{21} -2.09445 q^{23} -0.416553 q^{25} +2.05652 q^{27} -6.63658 q^{29} +10.2277 q^{31} -1.24065 q^{33} +6.57734 q^{35} +1.96752 q^{37} -1.47730 q^{39} -4.26670 q^{41} -8.29763 q^{43} +6.16060 q^{45} +7.96820 q^{47} +2.43860 q^{49} -1.96534 q^{51} -1.96367 q^{53} -7.59119 q^{55} -2.80294 q^{59} -8.86636 q^{61} +8.84058 q^{63} -9.03920 q^{65} -3.36917 q^{67} +0.732832 q^{69} -10.5187 q^{71} +12.1063 q^{73} +0.145749 q^{75} -10.8935 q^{77} -15.0842 q^{79} +7.91317 q^{81} +10.3147 q^{83} -12.0254 q^{85} +2.32209 q^{87} +6.56315 q^{89} -12.9714 q^{91} -3.57859 q^{93} -15.4547 q^{97} -10.2033 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 12 q^{11} - 6 q^{13} - 10 q^{17} + 4 q^{21} - 12 q^{23} + 2 q^{25} - 12 q^{27} - 18 q^{29} - 14 q^{31} - 40 q^{33} + 18 q^{35} - 16 q^{37} + 28 q^{39} - 12 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) $ 8 * q - 6 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 12 * q^11 - 6 * q^13 - 10 * q^17 + 4 * q^21 - 12 * q^23 + 2 * q^25 - 12 * q^27 - 18 * q^29 - 14 * q^31 - 40 * q^33 + 18 * q^35 - 16 * q^37 + 28 * q^39 - 12 * q^41 + 8 * q^43 - 2 * q^45 + 24 * q^47 + 12 * q^49 + 22 * q^51 - 12 * q^53 - 2 * q^55 + 16 * q^59 + 12 * q^63 - 12 * q^65 - 42 * q^67 + 12 * q^69 - 36 * q^71 - 10 * q^73 - 32 * q^75 - 38 * q^77 - 6 * q^79 + 4 * q^81 + 12 * q^83 - 42 * q^85 - 12 * q^87 - 8 * q^89 - 16 * q^91 - 14 * q^93 - 28 * q^97 + 32 * 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
$3$	−0.349892	−0.202010	−0.101005	−	0.994886i	$-0.532206\pi$
$3$	−0.349892	−0.202010	−0.101005	+	0.994886i	$0.532206\pi$
$4$	0	0
$5$	−2.14090	−0.957439	−0.478719	−	0.877968i	$-0.658899\pi$
$5$	−2.14090	−0.957439	−0.478719	+	0.877968i	$0.658899\pi$
$6$	0	0
$7$	−3.07223	−1.16119	−0.580597	−	0.814191i	$-0.697181\pi$
$7$	−3.07223	−1.16119	−0.580597	+	0.814191i	$0.697181\pi$
$8$	0	0
$9$	−2.87758	−0.959192
$10$	0	0
$11$	3.54580	1.06910	0.534549	−	0.845138i	$-0.320482\pi$
$11$	3.54580	1.06910	0.534549	+	0.845138i	$0.320482\pi$
$12$	0	0
$13$	4.22215	1.17101	0.585507	−	0.810667i	$-0.300895\pi$
$13$	4.22215	1.17101	0.585507	+	0.810667i	$0.300895\pi$
$14$	0	0
$15$	0.749083	0.193412
$16$	0	0
$17$	5.61700	1.36232	0.681161	−	0.732133i	$-0.261475\pi$
$17$	5.61700	1.36232	0.681161	+	0.732133i	$0.261475\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	1.07495	0.234573
$22$	0	0
$23$	−2.09445	−0.436724	−0.218362	−	0.975868i	$-0.570071\pi$
$23$	−2.09445	−0.436724	−0.218362	+	0.975868i	$0.570071\pi$
$24$	0	0
$25$	−0.416553	−0.0833106
$26$	0	0
$27$	2.05652	0.395777
$28$	0	0
$29$	−6.63658	−1.23238	−0.616191	−	0.787597i	$-0.711325\pi$
$29$	−6.63658	−1.23238	−0.616191	+	0.787597i	$0.711325\pi$
$30$	0	0
$31$	10.2277	1.83695	0.918474	−	0.395482i	$-0.129422\pi$
$31$	10.2277	1.83695	0.918474	+	0.395482i	$0.129422\pi$
$32$	0	0
$33$	−1.24065	−0.215969
$34$	0	0
$35$	6.57734	1.11177
$36$	0	0
$37$	1.96752	0.323458	0.161729	−	0.986835i	$-0.448293\pi$
$37$	1.96752	0.323458	0.161729	+	0.986835i	$0.448293\pi$
$38$	0	0
$39$	−1.47730	−0.236557
$40$	0	0
$41$	−4.26670	−0.666346	−0.333173	−	0.942866i	$-0.608119\pi$
$41$	−4.26670	−0.666346	−0.333173	+	0.942866i	$0.608119\pi$
$42$	0	0
$43$	−8.29763	−1.26538	−0.632688	−	0.774407i	$-0.718049\pi$
$43$	−8.29763	−1.26538	−0.632688	+	0.774407i	$0.718049\pi$
$44$	0	0
$45$	6.16060	0.918368
$46$	0	0
$47$	7.96820	1.16228	0.581141	−	0.813803i	$-0.302607\pi$
$47$	7.96820	1.16228	0.581141	+	0.813803i	$0.302607\pi$
$48$	0	0
$49$	2.43860	0.348372
$50$	0	0
$51$	−1.96534	−0.275203
$52$	0	0
$53$	−1.96367	−0.269730	−0.134865	−	0.990864i	$-0.543060\pi$
$53$	−1.96367	−0.269730	−0.134865	+	0.990864i	$0.543060\pi$
$54$	0	0
$55$	−7.59119	−1.02360
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−2.80294	−0.364912	−0.182456	−	0.983214i	$-0.558405\pi$
$59$	−2.80294	−0.364912	−0.182456	+	0.983214i	$0.558405\pi$
$60$	0	0
$61$	−8.86636	−1.13522	−0.567611	−	0.823297i	$-0.692132\pi$
$61$	−8.86636	−1.13522	−0.567611	+	0.823297i	$0.692132\pi$
$62$	0	0
$63$	8.84058	1.11381
$64$	0	0
$65$	−9.03920	−1.12117
$66$	0	0
$67$	−3.36917	−0.411609	−0.205805	−	0.978593i	$-0.565981\pi$
$67$	−3.36917	−0.411609	−0.205805	+	0.978593i	$0.565981\pi$
$68$	0	0
$69$	0.732832	0.0882227
$70$	0	0
$71$	−10.5187	−1.24834	−0.624171	−	0.781288i	$-0.714563\pi$
$71$	−10.5187	−1.24834	−0.624171	+	0.781288i	$0.714563\pi$
$72$	0	0
$73$	12.1063	1.41694	0.708470	−	0.705741i	$-0.249386\pi$
$73$	12.1063	1.41694	0.708470	+	0.705741i	$0.249386\pi$
$74$	0	0
$75$	0.145749	0.0168296
$76$	0	0
$77$	−10.8935	−1.24143
$78$	0	0
$79$	−15.0842	−1.69711	−0.848555	−	0.529107i	$-0.822527\pi$
$79$	−15.0842	−1.69711	−0.848555	+	0.529107i	$0.822527\pi$
$80$	0	0
$81$	7.91317	0.879241
$82$	0	0
$83$	10.3147	1.13218	0.566092	−	0.824342i	$-0.308455\pi$
$83$	10.3147	1.13218	0.566092	+	0.824342i	$0.308455\pi$
$84$	0	0
$85$	−12.0254	−1.30434
$86$	0	0
$87$	2.32209	0.248954
$88$	0	0
$89$	6.56315	0.695692	0.347846	−	0.937552i	$-0.386913\pi$
$89$	6.56315	0.695692	0.347846	+	0.937552i	$0.386913\pi$
$90$	0	0
$91$	−12.9714	−1.35978
$92$	0	0
$93$	−3.57859	−0.371082
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−15.4547	−1.56919	−0.784593	−	0.620012i	$-0.787128\pi$
$97$	−15.4547	−1.56919	−0.784593	+	0.620012i	$0.787128\pi$
$98$	0	0
$99$	−10.2033	−1.02547
$100$	0	0
$101$	−8.11824	−0.807795	−0.403898	−	0.914804i	$-0.632345\pi$
$101$	−8.11824	−0.807795	−0.403898	+	0.914804i	$0.632345\pi$
$102$	0	0
$103$	−8.19991	−0.807961	−0.403980	−	0.914768i	$-0.632374\pi$
$103$	−8.19991	−0.807961	−0.403980	+	0.914768i	$0.632374\pi$
$104$	0	0
$105$	−2.30136	−0.224589
$106$	0	0
$107$	−15.0156	−1.45161	−0.725807	−	0.687898i	$-0.758534\pi$
$107$	−15.0156	−1.45161	−0.725807	+	0.687898i	$0.758534\pi$
$108$	0	0
$109$	−9.57657	−0.917269	−0.458634	−	0.888625i	$-0.651661\pi$
$109$	−9.57657	−0.917269	−0.458634	+	0.888625i	$0.651661\pi$
$110$	0	0
$111$	−0.688419	−0.0653418
$112$	0	0
$113$	−8.24448	−0.775576	−0.387788	−	0.921749i	$-0.626761\pi$
$113$	−8.24448	−0.775576	−0.387788	+	0.921749i	$0.626761\pi$
$114$	0	0
$115$	4.48401	0.418136
$116$	0	0
$117$	−12.1496	−1.12323
$118$	0	0
$119$	−17.2567	−1.58192
$120$	0	0
$121$	1.57267	0.142970
$122$	0	0
$123$	1.49288	0.134609
$124$	0	0
$125$	11.5963	1.03720
$126$	0	0
$127$	−8.03977	−0.713414	−0.356707	−	0.934216i	$-0.616101\pi$
$127$	−8.03977	−0.713414	−0.356707	+	0.934216i	$0.616101\pi$
$128$	0	0
$129$	2.90327	0.255619
$130$	0	0
$131$	18.5660	1.62212	0.811061	−	0.584962i	$-0.198891\pi$
$131$	18.5660	1.62212	0.811061	+	0.584962i	$0.198891\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.40279	−0.378932
$136$	0	0
$137$	−21.2462	−1.81518	−0.907591	−	0.419856i	$-0.862081\pi$
$137$	−21.2462	−1.81518	−0.907591	+	0.419856i	$0.862081\pi$
$138$	0	0
$139$	13.9946	1.18701	0.593504	−	0.804831i	$-0.297744\pi$
$139$	13.9946	1.18701	0.593504	+	0.804831i	$0.297744\pi$
$140$	0	0
$141$	−2.78801	−0.234793
$142$	0	0
$143$	14.9709	1.25193
$144$	0	0
$145$	14.2083	1.17993
$146$	0	0
$147$	−0.853248	−0.0703747
$148$	0	0
$149$	−23.4095	−1.91778	−0.958891	−	0.283775i	$-0.908413\pi$
$149$	−23.4095	−1.91778	−0.958891	+	0.283775i	$0.908413\pi$
$150$	0	0
$151$	10.8277	0.881143	0.440571	−	0.897718i	$-0.354776\pi$
$151$	10.8277	0.881143	0.440571	+	0.897718i	$0.354776\pi$
$152$	0	0
$153$	−16.1633	−1.30673
$154$	0	0
$155$	−21.8965	−1.75877
$156$	0	0
$157$	−6.54371	−0.522245	−0.261122	−	0.965306i	$-0.584093\pi$
$157$	−6.54371	−0.522245	−0.261122	+	0.965306i	$0.584093\pi$
$158$	0	0
$159$	0.687071	0.0544883
$160$	0	0
$161$	6.43465	0.507121
$162$	0	0
$163$	16.4252	1.28652	0.643259	−	0.765649i	$-0.277582\pi$
$163$	16.4252	1.28652	0.643259	+	0.765649i	$0.277582\pi$
$164$	0	0
$165$	2.65610	0.206777
$166$	0	0
$167$	9.09617	0.703883	0.351941	−	0.936022i	$-0.385522\pi$
$167$	9.09617	0.703883	0.351941	+	0.936022i	$0.385522\pi$
$168$	0	0
$169$	4.82657	0.371275
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−8.52045	−0.647798	−0.323899	−	0.946092i	$-0.604994\pi$
$173$	−8.52045	−0.647798	−0.323899	+	0.946092i	$0.604994\pi$
$174$	0	0
$175$	1.27975	0.0967398
$176$	0	0
$177$	0.980728	0.0737160
$178$	0	0
$179$	15.4193	1.15249	0.576245	−	0.817277i	$-0.304517\pi$
$179$	15.4193	1.15249	0.576245	+	0.817277i	$0.304517\pi$
$180$	0	0
$181$	−20.0713	−1.49189	−0.745945	−	0.666007i	$-0.768002\pi$
$181$	−20.0713	−1.49189	−0.745945	+	0.666007i	$0.768002\pi$
$182$	0	0
$183$	3.10227	0.229326
$184$	0	0
$185$	−4.21226	−0.309691
$186$	0	0
$187$	19.9167	1.45646
$188$	0	0
$189$	−6.31809	−0.459574
$190$	0	0
$191$	−9.19053	−0.665003	−0.332502	−	0.943103i	$-0.607893\pi$
$191$	−9.19053	−0.665003	−0.332502	+	0.943103i	$0.607893\pi$
$192$	0	0
$193$	16.3507	1.17695	0.588476	−	0.808515i	$-0.299728\pi$
$193$	16.3507	1.17695	0.588476	+	0.808515i	$0.299728\pi$
$194$	0	0
$195$	3.16274	0.226489
$196$	0	0
$197$	3.55555	0.253322	0.126661	−	0.991946i	$-0.459574\pi$
$197$	3.55555	0.253322	0.126661	+	0.991946i	$0.459574\pi$
$198$	0	0
$199$	−4.82595	−0.342103	−0.171051	−	0.985262i	$-0.554716\pi$
$199$	−4.82595	−0.342103	−0.171051	+	0.985262i	$0.554716\pi$
$200$	0	0
$201$	1.17884	0.0831493
$202$	0	0
$203$	20.3891	1.43104
$204$	0	0
$205$	9.13457	0.637986
$206$	0	0
$207$	6.02695	0.418902
$208$	0	0
$209$	0	0
$210$	0	0
$211$	22.8868	1.57559	0.787796	−	0.615936i	$-0.211222\pi$
$211$	22.8868	1.57559	0.787796	+	0.615936i	$0.211222\pi$
$212$	0	0
$213$	3.68041	0.252178
$214$	0	0
$215$	17.7644	1.21152
$216$	0	0
$217$	−31.4218	−2.13305
$218$	0	0
$219$	−4.23591	−0.286236
$220$	0	0
$221$	23.7158	1.59530
$222$	0	0
$223$	−10.6874	−0.715680	−0.357840	−	0.933783i	$-0.616487\pi$
$223$	−10.6874	−0.715680	−0.357840	+	0.933783i	$0.616487\pi$
$224$	0	0
$225$	1.19866	0.0799109
$226$	0	0
$227$	−4.16104	−0.276178	−0.138089	−	0.990420i	$-0.544096\pi$
$227$	−4.16104	−0.276178	−0.138089	+	0.990420i	$0.544096\pi$
$228$	0	0
$229$	−4.12875	−0.272836	−0.136418	−	0.990651i	$-0.543559\pi$
$229$	−4.12875	−0.272836	−0.136418	+	0.990651i	$0.543559\pi$
$230$	0	0
$231$	3.81155	0.250782
$232$	0	0
$233$	−14.0742	−0.922034	−0.461017	−	0.887391i	$-0.652515\pi$
$233$	−14.0742	−0.922034	−0.461017	+	0.887391i	$0.652515\pi$
$234$	0	0
$235$	−17.0591	−1.11281
$236$	0	0
$237$	5.27785	0.342833
$238$	0	0
$239$	6.34933	0.410704	0.205352	−	0.978688i	$-0.434166\pi$
$239$	6.34933	0.410704	0.205352	+	0.978688i	$0.434166\pi$
$240$	0	0
$241$	10.2222	0.658468	0.329234	−	0.944248i	$-0.393209\pi$
$241$	10.2222	0.658468	0.329234	+	0.944248i	$0.393209\pi$
$242$	0	0
$243$	−8.93830	−0.573392
$244$	0	0
$245$	−5.22080	−0.333545
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3.60902	−0.228712
$250$	0	0
$251$	−1.14573	−0.0723177	−0.0361588	−	0.999346i	$-0.511512\pi$
$251$	−1.14573	−0.0723177	−0.0361588	+	0.999346i	$0.511512\pi$
$252$	0	0
$253$	−7.42651	−0.466901
$254$	0	0
$255$	4.20760	0.263490
$256$	0	0
$257$	−6.37094	−0.397409	−0.198704	−	0.980059i	$-0.563673\pi$
$257$	−6.37094	−0.397409	−0.198704	+	0.980059i	$0.563673\pi$
$258$	0	0
$259$	−6.04467	−0.375598
$260$	0	0
$261$	19.0973	1.18209
$262$	0	0
$263$	10.9941	0.677924	0.338962	−	0.940800i	$-0.389924\pi$
$263$	10.9941	0.677924	0.338962	+	0.940800i	$0.389924\pi$
$264$	0	0
$265$	4.20401	0.258250
$266$	0	0
$267$	−2.29639	−0.140537
$268$	0	0
$269$	8.42908	0.513930	0.256965	−	0.966421i	$-0.417277\pi$
$269$	8.42908	0.513930	0.256965	+	0.966421i	$0.417277\pi$
$270$	0	0
$271$	−17.4831	−1.06202	−0.531010	−	0.847365i	$-0.678187\pi$
$271$	−17.4831	−1.06202	−0.531010	+	0.847365i	$0.678187\pi$
$272$	0	0
$273$	4.53860	0.274688
$274$	0	0
$275$	−1.47701	−0.0890672
$276$	0	0
$277$	−9.52556	−0.572336	−0.286168	−	0.958180i	$-0.592381\pi$
$277$	−9.52556	−0.572336	−0.286168	+	0.958180i	$0.592381\pi$
$278$	0	0
$279$	−29.4310	−1.76199
$280$	0	0
$281$	−3.45522	−0.206121	−0.103061	−	0.994675i	$-0.532864\pi$
$281$	−3.45522	−0.206121	−0.103061	+	0.994675i	$0.532864\pi$
$282$	0	0
$283$	−16.2143	−0.963840	−0.481920	−	0.876215i	$-0.660060\pi$
$283$	−16.2143	−0.963840	−0.481920	+	0.876215i	$0.660060\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	13.1083	0.773757
$288$	0	0
$289$	14.5507	0.855924
$290$	0	0
$291$	5.40747	0.316991
$292$	0	0
$293$	−11.1181	−0.649528	−0.324764	−	0.945795i	$-0.605285\pi$
$293$	−11.1181	−0.649528	−0.324764	+	0.945795i	$0.605285\pi$
$294$	0	0
$295$	6.00082	0.349381
$296$	0	0
$297$	7.29199	0.423124
$298$	0	0
$299$	−8.84310	−0.511410
$300$	0	0
$301$	25.4922	1.46935
$302$	0	0
$303$	2.84051	0.163183
$304$	0	0
$305$	18.9820	1.08691
$306$	0	0
$307$	−20.3693	−1.16254	−0.581270	−	0.813711i	$-0.697444\pi$
$307$	−20.3693	−1.16254	−0.581270	+	0.813711i	$0.697444\pi$
$308$	0	0
$309$	2.86908	0.163216
$310$	0	0
$311$	−9.03873	−0.512540	−0.256270	−	0.966605i	$-0.582494\pi$
$311$	−9.03873	−0.512540	−0.256270	+	0.966605i	$0.582494\pi$
$312$	0	0
$313$	−32.2112	−1.82068	−0.910342	−	0.413856i	$-0.864182\pi$
$313$	−32.2112	−1.82068	−0.910342	+	0.413856i	$0.864182\pi$
$314$	0	0
$315$	−18.9268	−1.06640
$316$	0	0
$317$	−4.61826	−0.259387	−0.129694	−	0.991554i	$-0.541399\pi$
$317$	−4.61826	−0.259387	−0.129694	+	0.991554i	$0.541399\pi$
$318$	0	0
$319$	−23.5320	−1.31754
$320$	0	0
$321$	5.25384	0.293241
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−1.75875	−0.0975580
$326$	0	0
$327$	3.35076	0.185298
$328$	0	0
$329$	−24.4802	−1.34963
$330$	0	0
$331$	33.3148	1.83115	0.915574	−	0.402150i	$-0.131737\pi$
$331$	33.3148	1.83115	0.915574	+	0.402150i	$0.131737\pi$
$332$	0	0
$333$	−5.66168	−0.310258
$334$	0	0
$335$	7.21305	0.394091
$336$	0	0
$337$	3.25551	0.177339	0.0886695	−	0.996061i	$-0.471738\pi$
$337$	3.25551	0.177339	0.0886695	+	0.996061i	$0.471738\pi$
$338$	0	0
$339$	2.88468	0.156674
$340$	0	0
$341$	36.2653	1.96388
$342$	0	0
$343$	14.0137	0.756667
$344$	0	0
$345$	−1.56892	−0.0844678
$346$	0	0
$347$	32.4310	1.74099	0.870494	−	0.492180i	$-0.163800\pi$
$347$	32.4310	1.74099	0.870494	+	0.492180i	$0.163800\pi$
$348$	0	0
$349$	3.00973	0.161107	0.0805535	−	0.996750i	$-0.474331\pi$
$349$	3.00973	0.161107	0.0805535	+	0.996750i	$0.474331\pi$
$350$	0	0
$351$	8.68292	0.463460
$352$	0	0
$353$	−13.5373	−0.720518	−0.360259	−	0.932852i	$-0.617312\pi$
$353$	−13.5373	−0.720518	−0.360259	+	0.932852i	$0.617312\pi$
$354$	0	0
$355$	22.5195	1.19521
$356$	0	0
$357$	6.03799	0.319564
$358$	0	0
$359$	8.20813	0.433209	0.216604	−	0.976259i	$-0.430502\pi$
$359$	8.20813	0.433209	0.216604	+	0.976259i	$0.430502\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−0.550265	−0.0288814
$364$	0	0
$365$	−25.9184	−1.35663
$366$	0	0
$367$	−18.3151	−0.956040	−0.478020	−	0.878349i	$-0.658645\pi$
$367$	−18.3151	−0.956040	−0.478020	+	0.878349i	$0.658645\pi$
$368$	0	0
$369$	12.2777	0.639154
$370$	0	0
$371$	6.03284	0.313209
$372$	0	0
$373$	−29.9920	−1.55293	−0.776464	−	0.630162i	$-0.782989\pi$
$373$	−29.9920	−1.55293	−0.776464	+	0.630162i	$0.782989\pi$
$374$	0	0
$375$	−4.05745	−0.209526
$376$	0	0
$377$	−28.0207	−1.44314
$378$	0	0
$379$	−30.6549	−1.57464	−0.787318	−	0.616547i	$-0.788531\pi$
$379$	−30.6549	−1.57464	−0.787318	+	0.616547i	$0.788531\pi$
$380$	0	0
$381$	2.81305	0.144117
$382$	0	0
$383$	0.451735	0.0230826	0.0115413	−	0.999933i	$-0.496326\pi$
$383$	0.451735	0.0230826	0.0115413	+	0.999933i	$0.496326\pi$
$384$	0	0
$385$	23.3219	1.18859
$386$	0	0
$387$	23.8771	1.21374
$388$	0	0
$389$	−5.75718	−0.291901	−0.145950	−	0.989292i	$-0.546624\pi$
$389$	−5.75718	−0.291901	−0.145950	+	0.989292i	$0.546624\pi$
$390$	0	0
$391$	−11.7646	−0.594959
$392$	0	0
$393$	−6.49610	−0.327685
$394$	0	0
$395$	32.2938	1.62488
$396$	0	0
$397$	3.92068	0.196773	0.0983866	−	0.995148i	$-0.468632\pi$
$397$	3.92068	0.196773	0.0983866	+	0.995148i	$0.468632\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−9.39669	−0.469248	−0.234624	−	0.972086i	$-0.575386\pi$
$401$	−9.39669	−0.469248	−0.234624	+	0.972086i	$0.575386\pi$
$402$	0	0
$403$	43.1829	2.15109
$404$	0	0
$405$	−16.9413	−0.841820
$406$	0	0
$407$	6.97642	0.345808
$408$	0	0
$409$	3.54816	0.175445	0.0877226	−	0.996145i	$-0.472041\pi$
$409$	3.54816	0.175445	0.0877226	+	0.996145i	$0.472041\pi$
$410$	0	0
$411$	7.43386	0.366685
$412$	0	0
$413$	8.61129	0.423734
$414$	0	0
$415$	−22.0827	−1.08400
$416$	0	0
$417$	−4.89661	−0.239788
$418$	0	0
$419$	18.3274	0.895352	0.447676	−	0.894196i	$-0.352252\pi$
$419$	18.3274	0.895352	0.447676	+	0.894196i	$0.352252\pi$
$420$	0	0
$421$	20.4648	0.997393	0.498696	−	0.866777i	$-0.333812\pi$
$421$	20.4648	0.997393	0.498696	+	0.866777i	$0.333812\pi$
$422$	0	0
$423$	−22.9291	−1.11485
$424$	0	0
$425$	−2.33978	−0.113496
$426$	0	0
$427$	27.2395	1.31821
$428$	0	0
$429$	−5.23819	−0.252902
$430$	0	0
$431$	−31.9095	−1.53703	−0.768514	−	0.639833i	$-0.779004\pi$
$431$	−31.9095	−1.53703	−0.768514	+	0.639833i	$0.779004\pi$
$432$	0	0
$433$	−15.9015	−0.764179	−0.382090	−	0.924125i	$-0.624795\pi$
$433$	−15.9015	−0.764179	−0.382090	+	0.924125i	$0.624795\pi$
$434$	0	0
$435$	−4.97135	−0.238358
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−31.0847	−1.48359	−0.741796	−	0.670626i	$-0.766026\pi$
$439$	−31.0847	−1.48359	−0.741796	+	0.670626i	$0.766026\pi$
$440$	0	0
$441$	−7.01727	−0.334156
$442$	0	0
$443$	9.13099	0.433826	0.216913	−	0.976191i	$-0.430401\pi$
$443$	9.13099	0.433826	0.216913	+	0.976191i	$0.430401\pi$
$444$	0	0
$445$	−14.0510	−0.666083
$446$	0	0
$447$	8.19080	0.387411
$448$	0	0
$449$	−13.5099	−0.637572	−0.318786	−	0.947827i	$-0.603275\pi$
$449$	−13.5099	−0.637572	−0.318786	+	0.947827i	$0.603275\pi$
$450$	0	0
$451$	−15.1288	−0.712389
$452$	0	0
$453$	−3.78851	−0.178000
$454$	0	0
$455$	27.7705	1.30190
$456$	0	0
$457$	−3.54487	−0.165822	−0.0829109	−	0.996557i	$-0.526422\pi$
$457$	−3.54487	−0.165822	−0.0829109	+	0.996557i	$0.526422\pi$
$458$	0	0
$459$	11.5515	0.539176
$460$	0	0
$461$	24.9909	1.16394	0.581970	−	0.813210i	$-0.302282\pi$
$461$	24.9909	1.16394	0.581970	+	0.813210i	$0.302282\pi$
$462$	0	0
$463$	−30.5328	−1.41898	−0.709491	−	0.704715i	$-0.751075\pi$
$463$	−30.5328	−1.41898	−0.709491	+	0.704715i	$0.751075\pi$
$464$	0	0
$465$	7.66139	0.355288
$466$	0	0
$467$	17.6026	0.814553	0.407277	−	0.913305i	$-0.366478\pi$
$467$	17.6026	0.814553	0.407277	+	0.913305i	$0.366478\pi$
$468$	0	0
$469$	10.3509	0.477959
$470$	0	0
$471$	2.28959	0.105499
$472$	0	0
$473$	−29.4217	−1.35281
$474$	0	0
$475$	0	0
$476$	0	0
$477$	5.65060	0.258723
$478$	0	0
$479$	11.9555	0.546263	0.273131	−	0.961977i	$-0.411941\pi$
$479$	11.9555	0.546263	0.273131	+	0.961977i	$0.411941\pi$
$480$	0	0
$481$	8.30716	0.378774
$482$	0	0
$483$	−2.25143	−0.102444
$484$	0	0
$485$	33.0869	1.50240
$486$	0	0
$487$	−28.7030	−1.30066	−0.650329	−	0.759653i	$-0.725369\pi$
$487$	−28.7030	−1.30066	−0.650329	+	0.759653i	$0.725369\pi$
$488$	0	0
$489$	−5.74703	−0.259890
$490$	0	0
$491$	−4.75714	−0.214687	−0.107343	−	0.994222i	$-0.534234\pi$
$491$	−4.75714	−0.214687	−0.107343	+	0.994222i	$0.534234\pi$
$492$	0	0
$493$	−37.2777	−1.67890
$494$	0	0
$495$	21.8442	0.981825
$496$	0	0
$497$	32.3159	1.44957
$498$	0	0
$499$	23.5224	1.05301	0.526504	−	0.850172i	$-0.323502\pi$
$499$	23.5224	1.05301	0.526504	+	0.850172i	$0.323502\pi$
$500$	0	0
$501$	−3.18268	−0.142192
$502$	0	0
$503$	33.2244	1.48140	0.740701	−	0.671834i	$-0.234493\pi$
$503$	33.2244	1.48140	0.740701	+	0.671834i	$0.234493\pi$
$504$	0	0
$505$	17.3803	0.773414
$506$	0	0
$507$	−1.68878	−0.0750012
$508$	0	0
$509$	−0.925365	−0.0410161	−0.0205080	−	0.999790i	$-0.506528\pi$
$509$	−0.925365	−0.0410161	−0.0205080	+	0.999790i	$0.506528\pi$
$510$	0	0
$511$	−37.1935	−1.64534
$512$	0	0
$513$	0	0
$514$	0	0
$515$	17.5552	0.773573
$516$	0	0
$517$	28.2536	1.24259
$518$	0	0
$519$	2.98124	0.130862
$520$	0	0
$521$	41.6324	1.82395	0.911975	−	0.410246i	$-0.134557\pi$
$521$	41.6324	1.82395	0.911975	+	0.410246i	$0.134557\pi$
$522$	0	0
$523$	1.13390	0.0495819	0.0247909	−	0.999693i	$-0.492108\pi$
$523$	1.13390	0.0495819	0.0247909	+	0.999693i	$0.492108\pi$
$524$	0	0
$525$	−0.447773	−0.0195424
$526$	0	0
$527$	57.4490	2.50252
$528$	0	0
$529$	−18.6133	−0.809272
$530$	0	0
$531$	8.06569	0.350021
$532$	0	0
$533$	−18.0146	−0.780301
$534$	0	0
$535$	32.1469	1.38983
$536$	0	0
$537$	−5.39508	−0.232815
$538$	0	0
$539$	8.64679	0.372444
$540$	0	0
$541$	−1.53659	−0.0660633	−0.0330316	−	0.999454i	$-0.510516\pi$
$541$	−1.53659	−0.0660633	−0.0330316	+	0.999454i	$0.510516\pi$
$542$	0	0
$543$	7.02280	0.301377
$544$	0	0
$545$	20.5025	0.878229
$546$	0	0
$547$	12.0804	0.516522	0.258261	−	0.966075i	$-0.416851\pi$
$547$	12.0804	0.516522	0.258261	+	0.966075i	$0.416851\pi$
$548$	0	0
$549$	25.5136	1.08890
$550$	0	0
$551$	0	0
$552$	0	0
$553$	46.3423	1.97067
$554$	0	0
$555$	1.47383	0.0625608
$556$	0	0
$557$	−5.38581	−0.228204	−0.114102	−	0.993469i	$-0.536399\pi$
$557$	−5.38581	−0.228204	−0.114102	+	0.993469i	$0.536399\pi$
$558$	0	0
$559$	−35.0339	−1.48177
$560$	0	0
$561$	−6.96871	−0.294219
$562$	0	0
$563$	−25.1177	−1.05858	−0.529292	−	0.848440i	$-0.677542\pi$
$563$	−25.1177	−1.05858	−0.529292	+	0.848440i	$0.677542\pi$
$564$	0	0
$565$	17.6506	0.742566
$566$	0	0
$567$	−24.3111	−1.02097
$568$	0	0
$569$	−17.7370	−0.743574	−0.371787	−	0.928318i	$-0.621255\pi$
$569$	−17.7370	−0.743574	−0.371787	+	0.928318i	$0.621255\pi$
$570$	0	0
$571$	−23.2223	−0.971823	−0.485911	−	0.874008i	$-0.661512\pi$
$571$	−23.2223	−0.971823	−0.485911	+	0.874008i	$0.661512\pi$
$572$	0	0
$573$	3.21569	0.134337
$574$	0	0
$575$	0.872452	0.0363837
$576$	0	0
$577$	4.97051	0.206925	0.103463	−	0.994633i	$-0.467008\pi$
$577$	4.97051	0.206925	0.103463	+	0.994633i	$0.467008\pi$
$578$	0	0
$579$	−5.72099	−0.237756
$580$	0	0
$581$	−31.6891	−1.31468
$582$	0	0
$583$	−6.96277	−0.288368
$584$	0	0
$585$	26.0110	1.07542
$586$	0	0
$587$	31.7934	1.31225	0.656127	−	0.754650i	$-0.272194\pi$
$587$	31.7934	1.31225	0.656127	+	0.754650i	$0.272194\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−1.24406	−0.0511737
$592$	0	0
$593$	13.9335	0.572181	0.286090	−	0.958203i	$-0.407644\pi$
$593$	13.9335	0.572181	0.286090	+	0.958203i	$0.407644\pi$
$594$	0	0
$595$	36.9449	1.51459
$596$	0	0
$597$	1.68856	0.0691082
$598$	0	0
$599$	16.8899	0.690102	0.345051	−	0.938584i	$-0.387862\pi$
$599$	16.8899	0.690102	0.345051	+	0.938584i	$0.387862\pi$
$600$	0	0
$601$	−42.8150	−1.74646	−0.873229	−	0.487309i	$-0.837978\pi$
$601$	−42.8150	−1.74646	−0.873229	+	0.487309i	$0.837978\pi$
$602$	0	0
$603$	9.69504	0.394812
$604$	0	0
$605$	−3.36693	−0.136885
$606$	0	0
$607$	−24.9220	−1.01155	−0.505777	−	0.862664i	$-0.668794\pi$
$607$	−24.9220	−1.01155	−0.505777	+	0.862664i	$0.668794\pi$
$608$	0	0
$609$	−7.13399	−0.289084
$610$	0	0
$611$	33.6430	1.36105
$612$	0	0
$613$	26.9437	1.08824	0.544122	−	0.839006i	$-0.316863\pi$
$613$	26.9437	1.08824	0.544122	+	0.839006i	$0.316863\pi$
$614$	0	0
$615$	−3.19611	−0.128880
$616$	0	0
$617$	0.292467	0.0117743	0.00588713	−	0.999983i	$-0.498126\pi$
$617$	0.292467	0.0117743	0.00588713	+	0.999983i	$0.498126\pi$
$618$	0	0
$619$	−20.9181	−0.840768	−0.420384	−	0.907346i	$-0.638105\pi$
$619$	−20.9181	−0.840768	−0.420384	+	0.907346i	$0.638105\pi$
$620$	0	0
$621$	−4.30728	−0.172845
$622$	0	0
$623$	−20.1635	−0.807834
$624$	0	0
$625$	−22.7437	−0.909749
$626$	0	0
$627$	0	0
$628$	0	0
$629$	11.0516	0.440654
$630$	0	0
$631$	−25.7263	−1.02415	−0.512073	−	0.858942i	$-0.671122\pi$
$631$	−25.7263	−1.02415	−0.512073	+	0.858942i	$0.671122\pi$
$632$	0	0
$633$	−8.00790	−0.318286
$634$	0	0
$635$	17.2123	0.683050
$636$	0	0
$637$	10.2962	0.407949
$638$	0	0
$639$	30.2684	1.19740
$640$	0	0
$641$	47.4055	1.87240	0.936202	−	0.351462i	$-0.114316\pi$
$641$	47.4055	1.87240	0.936202	+	0.351462i	$0.114316\pi$
$642$	0	0
$643$	−14.1097	−0.556434	−0.278217	−	0.960518i	$-0.589743\pi$
$643$	−14.1097	−0.556434	−0.278217	+	0.960518i	$0.589743\pi$
$644$	0	0
$645$	−6.21561	−0.244740
$646$	0	0
$647$	19.8064	0.778670	0.389335	−	0.921096i	$-0.372705\pi$
$647$	19.8064	0.778670	0.389335	+	0.921096i	$0.372705\pi$
$648$	0	0
$649$	−9.93867	−0.390127
$650$	0	0
$651$	10.9942	0.430898
$652$	0	0
$653$	−11.5621	−0.452459	−0.226229	−	0.974074i	$-0.572640\pi$
$653$	−11.5621	−0.452459	−0.226229	+	0.974074i	$0.572640\pi$
$654$	0	0
$655$	−39.7480	−1.55308
$656$	0	0
$657$	−34.8369	−1.35912
$658$	0	0
$659$	−35.0211	−1.36423	−0.682114	−	0.731246i	$-0.738939\pi$
$659$	−35.0211	−1.36423	−0.682114	+	0.731246i	$0.738939\pi$
$660$	0	0
$661$	7.97145	0.310053	0.155027	−	0.987910i	$-0.450454\pi$
$661$	7.97145	0.310053	0.155027	+	0.987910i	$0.450454\pi$
$662$	0	0
$663$	−8.29798	−0.322267
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.9000	0.538211
$668$	0	0
$669$	3.73943	0.144575
$670$	0	0
$671$	−31.4383	−1.21366
$672$	0	0
$673$	−22.2567	−0.857933	−0.428966	−	0.903320i	$-0.641122\pi$
$673$	−22.2567	−0.857933	−0.428966	+	0.903320i	$0.641122\pi$
$674$	0	0
$675$	−0.856648	−0.0329724
$676$	0	0
$677$	28.5944	1.09897	0.549485	−	0.835503i	$-0.314824\pi$
$677$	28.5944	1.09897	0.549485	+	0.835503i	$0.314824\pi$
$678$	0	0
$679$	47.4804	1.82213
$680$	0	0
$681$	1.45591	0.0557907
$682$	0	0
$683$	23.4345	0.896695	0.448348	−	0.893859i	$-0.352013\pi$
$683$	23.4345	0.896695	0.448348	+	0.893859i	$0.352013\pi$
$684$	0	0
$685$	45.4859	1.73793
$686$	0	0
$687$	1.44462	0.0551155
$688$	0	0
$689$	−8.29090	−0.315858
$690$	0	0
$691$	−2.94419	−0.112002	−0.0560012	−	0.998431i	$-0.517835\pi$
$691$	−2.94419	−0.112002	−0.0560012	+	0.998431i	$0.517835\pi$
$692$	0	0
$693$	31.3469	1.19077
$694$	0	0
$695$	−29.9611	−1.13649
$696$	0	0
$697$	−23.9660	−0.907779
$698$	0	0
$699$	4.92446	0.186260
$700$	0	0
$701$	23.9790	0.905673	0.452837	−	0.891594i	$-0.350412\pi$
$701$	23.9790	0.905673	0.452837	+	0.891594i	$0.350412\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	5.96884	0.224800
$706$	0	0
$707$	24.9411	0.938007
$708$	0	0
$709$	26.6834	1.00212	0.501059	−	0.865413i	$-0.332944\pi$
$709$	26.6834	1.00212	0.501059	+	0.865413i	$0.332944\pi$
$710$	0	0
$711$	43.4061	1.62785
$712$	0	0
$713$	−21.4214	−0.802239
$714$	0	0
$715$	−32.0512	−1.19865
$716$	0	0
$717$	−2.22158	−0.0829664
$718$	0	0
$719$	6.72151	0.250670	0.125335	−	0.992114i	$-0.459999\pi$
$719$	6.72151	0.250670	0.125335	+	0.992114i	$0.459999\pi$
$720$	0	0
$721$	25.1920	0.938199
$722$	0	0
$723$	−3.57666	−0.133017
$724$	0	0
$725$	2.76449	0.102671
$726$	0	0
$727$	−31.8528	−1.18136	−0.590678	−	0.806908i	$-0.701140\pi$
$727$	−31.8528	−1.18136	−0.590678	+	0.806908i	$0.701140\pi$
$728$	0	0
$729$	−20.6121	−0.763410
$730$	0	0
$731$	−46.6078	−1.72385
$732$	0	0
$733$	20.8542	0.770269	0.385134	−	0.922861i	$-0.374155\pi$
$733$	20.8542	0.770269	0.385134	+	0.922861i	$0.374155\pi$
$734$	0	0
$735$	1.82672	0.0673795
$736$	0	0
$737$	−11.9464	−0.440051
$738$	0	0
$739$	−0.646032	−0.0237647	−0.0118823	−	0.999929i	$-0.503782\pi$
$739$	−0.646032	−0.0237647	−0.0118823	+	0.999929i	$0.503782\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−8.65441	−0.317500	−0.158750	−	0.987319i	$-0.550746\pi$
$743$	−8.65441	−0.317500	−0.158750	+	0.987319i	$0.550746\pi$
$744$	0	0
$745$	50.1174	1.83616
$746$	0	0
$747$	−29.6813	−1.08598
$748$	0	0
$749$	46.1314	1.68561
$750$	0	0
$751$	38.0353	1.38793	0.693965	−	0.720009i	$-0.255862\pi$
$751$	38.0353	1.38793	0.693965	+	0.720009i	$0.255862\pi$
$752$	0	0
$753$	0.400881	0.0146089
$754$	0	0
$755$	−23.1809	−0.843640
$756$	0	0
$757$	−9.00152	−0.327166	−0.163583	−	0.986530i	$-0.552305\pi$
$757$	−9.00152	−0.327166	−0.163583	+	0.986530i	$0.552305\pi$
$758$	0	0
$759$	2.59847	0.0943187
$760$	0	0
$761$	−19.8538	−0.719699	−0.359850	−	0.933010i	$-0.617172\pi$
$761$	−19.8538	−0.719699	−0.359850	+	0.933010i	$0.617172\pi$
$762$	0	0
$763$	29.4214	1.06513
$764$	0	0
$765$	34.6041	1.25111
$766$	0	0
$767$	−11.8345	−0.427318
$768$	0	0
$769$	19.6718	0.709382	0.354691	−	0.934984i	$-0.384586\pi$
$769$	19.6718	0.709382	0.354691	+	0.934984i	$0.384586\pi$
$770$	0	0
$771$	2.22914	0.0802806
$772$	0	0
$773$	−17.5264	−0.630379	−0.315190	−	0.949029i	$-0.602068\pi$
$773$	−17.5264	−0.630379	−0.315190	+	0.949029i	$0.602068\pi$
$774$	0	0
$775$	−4.26038	−0.153037
$776$	0	0
$777$	2.11498	0.0758745
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−37.2972	−1.33460
$782$	0	0
$783$	−13.6482	−0.487748
$784$	0	0
$785$	14.0094	0.500018
$786$	0	0
$787$	−24.0861	−0.858577	−0.429288	−	0.903168i	$-0.641236\pi$
$787$	−24.0861	−0.858577	−0.429288	+	0.903168i	$0.641236\pi$
$788$	0	0
$789$	−3.84674	−0.136947
$790$	0	0
$791$	25.3290	0.900594
$792$	0	0
$793$	−37.4351	−1.32936
$794$	0	0
$795$	−1.47095	−0.0521692
$796$	0	0
$797$	−32.7151	−1.15883	−0.579415	−	0.815033i	$-0.696719\pi$
$797$	−32.7151	−1.15883	−0.579415	+	0.815033i	$0.696719\pi$
$798$	0	0
$799$	44.7574	1.58340
$800$	0	0
$801$	−18.8860	−0.667303
$802$	0	0
$803$	42.9266	1.51485
$804$	0	0
$805$	−13.7759	−0.485538
$806$	0	0
$807$	−2.94927	−0.103819
$808$	0	0
$809$	−11.4405	−0.402226	−0.201113	−	0.979568i	$-0.564456\pi$
$809$	−11.4405	−0.402226	−0.201113	+	0.979568i	$0.564456\pi$
$810$	0	0
$811$	30.3777	1.06671	0.533353	−	0.845893i	$-0.320932\pi$
$811$	30.3777	1.06671	0.533353	+	0.845893i	$0.320932\pi$
$812$	0	0
$813$	6.11718	0.214539
$814$	0	0
$815$	−35.1646	−1.23176
$816$	0	0
$817$	0	0
$818$	0	0
$819$	37.3263	1.30429
$820$	0	0
$821$	53.2688	1.85910	0.929548	−	0.368702i	$-0.120198\pi$
$821$	53.2688	1.85910	0.929548	+	0.368702i	$0.120198\pi$
$822$	0	0
$823$	−25.6211	−0.893096	−0.446548	−	0.894760i	$-0.647347\pi$
$823$	−25.6211	−0.893096	−0.446548	+	0.894760i	$0.647347\pi$
$824$	0	0
$825$	0.516795	0.0179925
$826$	0	0
$827$	−48.4058	−1.68323	−0.841617	−	0.540075i	$-0.818396\pi$
$827$	−48.4058	−1.68323	−0.841617	+	0.540075i	$0.818396\pi$
$828$	0	0
$829$	−31.3193	−1.08776	−0.543882	−	0.839162i	$-0.683046\pi$
$829$	−31.3193	−1.08776	−0.543882	+	0.839162i	$0.683046\pi$
$830$	0	0
$831$	3.33292	0.115618
$832$	0	0
$833$	13.6976	0.474595
$834$	0	0
$835$	−19.4740	−0.673925
$836$	0	0
$837$	21.0334	0.727021
$838$	0	0
$839$	−9.41029	−0.324879	−0.162440	−	0.986718i	$-0.551936\pi$
$839$	−9.41029	−0.324879	−0.162440	+	0.986718i	$0.551936\pi$
$840$	0	0
$841$	15.0442	0.518767
$842$	0	0
$843$	1.20895	0.0416386
$844$	0	0
$845$	−10.3332	−0.355473
$846$	0	0
$847$	−4.83161	−0.166016
$848$	0	0
$849$	5.67325	0.194705
$850$	0	0
$851$	−4.12088	−0.141262
$852$	0	0
$853$	−32.7849	−1.12253	−0.561267	−	0.827635i	$-0.689686\pi$
$853$	−32.7849	−1.12253	−0.561267	+	0.827635i	$0.689686\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	2.92426	0.0998910	0.0499455	−	0.998752i	$-0.484095\pi$
$857$	2.92426	0.0998910	0.0499455	+	0.998752i	$0.484095\pi$
$858$	0	0
$859$	33.9214	1.15738	0.578692	−	0.815546i	$-0.303563\pi$
$859$	33.9214	1.15738	0.578692	+	0.815546i	$0.303563\pi$
$860$	0	0
$861$	−4.58648	−0.156307
$862$	0	0
$863$	−27.6833	−0.942349	−0.471175	−	0.882040i	$-0.656170\pi$
$863$	−27.6833	−0.942349	−0.471175	+	0.882040i	$0.656170\pi$
$864$	0	0
$865$	18.2414	0.620227
$866$	0	0
$867$	−5.09117	−0.172905
$868$	0	0
$869$	−53.4857	−1.81438
$870$	0	0
$871$	−14.2251	−0.482001
$872$	0	0
$873$	44.4720	1.50515
$874$	0	0
$875$	−35.6265	−1.20440
$876$	0	0
$877$	43.3391	1.46346	0.731729	−	0.681596i	$-0.238714\pi$
$877$	43.3391	1.46346	0.731729	+	0.681596i	$0.238714\pi$
$878$	0	0
$879$	3.89015	0.131211
$880$	0	0
$881$	30.0478	1.01234	0.506169	−	0.862435i	$-0.331061\pi$
$881$	30.0478	1.01234	0.506169	+	0.862435i	$0.331061\pi$
$882$	0	0
$883$	0.506518	0.0170457	0.00852284	−	0.999964i	$-0.497287\pi$
$883$	0.506518	0.0170457	0.00852284	+	0.999964i	$0.497287\pi$
$884$	0	0
$885$	−2.09964	−0.0705786
$886$	0	0
$887$	4.92764	0.165454	0.0827269	−	0.996572i	$-0.473637\pi$
$887$	4.92764	0.165454	0.0827269	+	0.996572i	$0.473637\pi$
$888$	0	0
$889$	24.7000	0.828412
$890$	0	0
$891$	28.0585	0.939995
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−33.0111	−1.10344
$896$	0	0
$897$	3.09413	0.103310
$898$	0	0
$899$	−67.8769	−2.26382
$900$	0	0
$901$	−11.0299	−0.367460
$902$	0	0
$903$	−8.91953	−0.296823
$904$	0	0
$905$	42.9707	1.42839
$906$	0	0
$907$	−53.8523	−1.78814	−0.894068	−	0.447931i	$-0.852161\pi$
$907$	−53.8523	−1.78814	−0.894068	+	0.447931i	$0.852161\pi$
$908$	0	0
$909$	23.3609	0.774830
$910$	0	0
$911$	10.9372	0.362366	0.181183	−	0.983449i	$-0.442007\pi$
$911$	10.9372	0.362366	0.181183	+	0.983449i	$0.442007\pi$
$912$	0	0
$913$	36.5737	1.21041
$914$	0	0
$915$	−6.64164	−0.219566
$916$	0	0
$917$	−57.0391	−1.88360
$918$	0	0
$919$	−20.2410	−0.667690	−0.333845	−	0.942628i	$-0.608346\pi$
$919$	−20.2410	−0.667690	−0.333845	+	0.942628i	$0.608346\pi$
$920$	0	0
$921$	7.12707	0.234845
$922$	0	0
$923$	−44.4116	−1.46183
$924$	0	0
$925$	−0.819576	−0.0269475
$926$	0	0
$927$	23.5958	0.774989
$928$	0	0
$929$	−2.35458	−0.0772511	−0.0386256	−	0.999254i	$-0.512298\pi$
$929$	−2.35458	−0.0772511	−0.0386256	+	0.999254i	$0.512298\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	3.16258	0.103538
$934$	0	0
$935$	−42.6397	−1.39447
$936$	0	0
$937$	−17.5300	−0.572682	−0.286341	−	0.958128i	$-0.592439\pi$
$937$	−17.5300	−0.572682	−0.286341	+	0.958128i	$0.592439\pi$
$938$	0	0
$939$	11.2704	0.367797
$940$	0	0
$941$	33.9869	1.10794	0.553970	−	0.832537i	$-0.313112\pi$
$941$	33.9869	1.10794	0.553970	+	0.832537i	$0.313112\pi$
$942$	0	0
$943$	8.93640	0.291009
$944$	0	0
$945$	13.5264	0.440014
$946$	0	0
$947$	24.4863	0.795698	0.397849	−	0.917451i	$-0.369757\pi$
$947$	24.4863	0.795698	0.397849	+	0.917451i	$0.369757\pi$
$948$	0	0
$949$	51.1148	1.65926
$950$	0	0
$951$	1.61589	0.0523989
$952$	0	0
$953$	−0.147011	−0.00476217	−0.00238108	−	0.999997i	$-0.500758\pi$
$953$	−0.147011	−0.00476217	−0.00238108	+	0.999997i	$0.500758\pi$
$954$	0	0
$955$	19.6760	0.636700
$956$	0	0
$957$	8.23365	0.266156
$958$	0	0
$959$	65.2731	2.10778
$960$	0	0
$961$	73.6057	2.37438
$962$	0	0
$963$	43.2086	1.39238
$964$	0	0
$965$	−35.0053	−1.12686
$966$	0	0
$967$	−29.7502	−0.956703	−0.478352	−	0.878168i	$-0.658766\pi$
$967$	−29.7502	−0.956703	−0.478352	+	0.878168i	$0.658766\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	10.3461	0.332022	0.166011	−	0.986124i	$-0.446911\pi$
$971$	10.3461	0.332022	0.166011	+	0.986124i	$0.446911\pi$
$972$	0	0
$973$	−42.9947	−1.37835
$974$	0	0
$975$	0.615373	0.0197077
$976$	0	0
$977$	51.6163	1.65135	0.825676	−	0.564145i	$-0.190794\pi$
$977$	51.6163	1.65135	0.825676	+	0.564145i	$0.190794\pi$
$978$	0	0
$979$	23.2716	0.743763
$980$	0	0
$981$	27.5573	0.879837
$982$	0	0
$983$	35.0087	1.11661	0.558303	−	0.829637i	$-0.311453\pi$
$983$	35.0087	1.11661	0.558303	+	0.829637i	$0.311453\pi$
$984$	0	0
$985$	−7.61207	−0.242541
$986$	0	0
$987$	8.56541	0.272640
$988$	0	0
$989$	17.3790	0.552620
$990$	0	0
$991$	29.7904	0.946322	0.473161	−	0.880976i	$-0.343113\pi$
$991$	29.7904	0.946322	0.473161	+	0.880976i	$0.343113\pi$
$992$	0	0
$993$	−11.6566	−0.369910
$994$	0	0
$995$	10.3319	0.327543
$996$	0	0
$997$	13.5185	0.428135	0.214067	−	0.976819i	$-0.431329\pi$
$997$	13.5185	0.428135	0.214067	+	0.976819i	$0.431329\pi$
$998$	0	0
$999$	4.04623	0.128017

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.v.1.5	✓	8
4.3	odd	2		5776.2.a.cc.1.4		8
19.18	odd	2		2888.2.a.w.1.4	yes	8
76.75	even	2		5776.2.a.ca.1.5		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.v.1.5	✓	8	1.1	even	1	trivial
2888.2.a.w.1.4	yes	8	19.18	odd	2
5776.2.a.ca.1.5		8	76.75	even	2
5776.2.a.cc.1.4		8	4.3	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:	\( N \)	\(=\)	\( 2888 = 2^{3} \cdot 19^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2888.a (trivial)

\(f(q)\)	\(=\)	\(q-0.349892 q^{3} -2.14090 q^{5} -3.07223 q^{7} -2.87758 q^{9} +3.54580 q^{11} +4.22215 q^{13} +0.749083 q^{15} +5.61700 q^{17} +1.07495 q^{21} -2.09445 q^{23} -0.416553 q^{25} +2.05652 q^{27} -6.63658 q^{29} +10.2277 q^{31} -1.24065 q^{33} +6.57734 q^{35} +1.96752 q^{37} -1.47730 q^{39} -4.26670 q^{41} -8.29763 q^{43} +6.16060 q^{45} +7.96820 q^{47} +2.43860 q^{49} -1.96534 q^{51} -1.96367 q^{53} -7.59119 q^{55} -2.80294 q^{59} -8.86636 q^{61} +8.84058 q^{63} -9.03920 q^{65} -3.36917 q^{67} +0.732832 q^{69} -10.5187 q^{71} +12.1063 q^{73} +0.145749 q^{75} -10.8935 q^{77} -15.0842 q^{79} +7.91317 q^{81} +10.3147 q^{83} -12.0254 q^{85} +2.32209 q^{87} +6.56315 q^{89} -12.9714 q^{91} -3.57859 q^{93} -15.4547 q^{97} -10.2033 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 12 q^{11} - 6 q^{13} - 10 q^{17} + 4 q^{21} - 12 q^{23} + 2 q^{25} - 12 q^{27} - 18 q^{29} - 14 q^{31} - 40 q^{33} + 18 q^{35} - 16 q^{37} + 28 q^{39} - 12 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) \) 8 * q - 6 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 12 * q^11 - 6 * q^13 - 10 * q^17 + 4 * q^21 - 12 * q^23 + 2 * q^25 - 12 * q^27 - 18 * q^29 - 14 * q^31 - 40 * q^33 + 18 * q^35 - 16 * q^37 + 28 * q^39 - 12 * q^41 + 8 * q^43 - 2 * q^45 + 24 * q^47 + 12 * q^49 + 22 * q^51 - 12 * q^53 - 2 * q^55 + 16 * q^59 + 12 * q^63 - 12 * q^65 - 42 * q^67 + 12 * q^69 - 36 * q^71 - 10 * q^73 - 32 * q^75 - 38 * q^77 - 6 * q^79 + 4 * q^81 + 12 * q^83 - 42 * q^85 - 12 * q^87 - 8 * q^89 - 16 * q^91 - 14 * q^93 - 28 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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