LMFDB - Embedded newform 2888.2.a.v.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 = [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			1.2
Root			$-1.32561$ 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)

$f(q)$	$=$	$q-2.32561 q^{3} -4.27353 q^{5} -4.02384 q^{7} +2.40845 q^{9} +3.62806 q^{11} -0.948800 q^{13} +9.93856 q^{15} -1.99441 q^{17} +9.35787 q^{21} +0.296119 q^{23} +13.2631 q^{25} +1.37571 q^{27} +7.16272 q^{29} -3.23298 q^{31} -8.43745 q^{33} +17.1960 q^{35} -3.01334 q^{37} +2.20654 q^{39} -3.79386 q^{41} +6.33165 q^{43} -10.2926 q^{45} +7.37034 q^{47} +9.19127 q^{49} +4.63821 q^{51} -12.5194 q^{53} -15.5046 q^{55} +0.585487 q^{59} +13.2370 q^{61} -9.69121 q^{63} +4.05473 q^{65} -0.121873 q^{67} -0.688657 q^{69} +2.67913 q^{71} +0.257480 q^{73} -30.8447 q^{75} -14.5987 q^{77} +10.4254 q^{79} -10.4247 q^{81} -0.107899 q^{83} +8.52317 q^{85} -16.6577 q^{87} -14.1086 q^{89} +3.81782 q^{91} +7.51865 q^{93} -7.07291 q^{97} +8.73800 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−2.32561	−1.34269	−0.671345	−	0.741145i	$-0.734283\pi$
$3$	−2.32561	−1.34269	−0.671345	+	0.741145i	$0.734283\pi$
$4$	0	0
$5$	−4.27353	−1.91118	−0.955591	−	0.294697i	$-0.904781\pi$
$5$	−4.27353	−1.91118	−0.955591	+	0.294697i	$0.904781\pi$
$6$	0	0
$7$	−4.02384	−1.52087	−0.760434	−	0.649415i	$-0.775014\pi$
$7$	−4.02384	−1.52087	−0.760434	+	0.649415i	$0.775014\pi$
$8$	0	0
$9$	2.40845	0.802817
$10$	0	0
$11$	3.62806	1.09390	0.546951	−	0.837165i	$-0.315788\pi$
$11$	3.62806	1.09390	0.546951	+	0.837165i	$0.315788\pi$
$12$	0	0
$13$	−0.948800	−0.263150	−0.131575	−	0.991306i	$-0.542003\pi$
$13$	−0.948800	−0.263150	−0.131575	+	0.991306i	$0.542003\pi$
$14$	0	0
$15$	9.93856	2.56612
$16$	0	0
$17$	−1.99441	−0.483715	−0.241858	−	0.970312i	$-0.577757\pi$
$17$	−1.99441	−0.483715	−0.241858	+	0.970312i	$0.577757\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	9.35787	2.04205
$22$	0	0
$23$	0.296119	0.0617451	0.0308725	−	0.999523i	$-0.490171\pi$
$23$	0.296119	0.0617451	0.0308725	+	0.999523i	$0.490171\pi$
$24$	0	0
$25$	13.2631	2.65261
$26$	0	0
$27$	1.37571	0.264756
$28$	0	0
$29$	7.16272	1.33008	0.665042	−	0.746806i	$-0.268414\pi$
$29$	7.16272	1.33008	0.665042	+	0.746806i	$0.268414\pi$
$30$	0	0
$31$	−3.23298	−0.580661	−0.290330	−	0.956926i	$-0.593765\pi$
$31$	−3.23298	−0.580661	−0.290330	+	0.956926i	$0.593765\pi$
$32$	0	0
$33$	−8.43745	−1.46877
$34$	0	0
$35$	17.1960	2.90665
$36$	0	0
$37$	−3.01334	−0.495390	−0.247695	−	0.968838i	$-0.579673\pi$
$37$	−3.01334	−0.495390	−0.247695	+	0.968838i	$0.579673\pi$
$38$	0	0
$39$	2.20654	0.353329
$40$	0	0
$41$	−3.79386	−0.592501	−0.296250	−	0.955110i	$-0.595736\pi$
$41$	−3.79386	−0.592501	−0.296250	+	0.955110i	$0.595736\pi$
$42$	0	0
$43$	6.33165	0.965567	0.482784	−	0.875740i	$-0.339626\pi$
$43$	6.33165	0.965567	0.482784	+	0.875740i	$0.339626\pi$
$44$	0	0
$45$	−10.2926	−1.53433
$46$	0	0
$47$	7.37034	1.07507	0.537537	−	0.843240i	$-0.319355\pi$
$47$	7.37034	1.07507	0.537537	+	0.843240i	$0.319355\pi$
$48$	0	0
$49$	9.19127	1.31304
$50$	0	0
$51$	4.63821	0.649480
$52$	0	0
$53$	−12.5194	−1.71967	−0.859833	−	0.510575i	$-0.829432\pi$
$53$	−12.5194	−1.71967	−0.859833	+	0.510575i	$0.829432\pi$
$54$	0	0
$55$	−15.5046	−2.09064
$56$	0	0
$57$	0	0
$58$	0	0
$59$	0.585487	0.0762239	0.0381120	−	0.999273i	$-0.487866\pi$
$59$	0.585487	0.0762239	0.0381120	+	0.999273i	$0.487866\pi$
$60$	0	0
$61$	13.2370	1.69483	0.847414	−	0.530933i	$-0.178159\pi$
$61$	13.2370	1.69483	0.847414	+	0.530933i	$0.178159\pi$
$62$	0	0
$63$	−9.69121	−1.22098
$64$	0	0
$65$	4.05473	0.502927
$66$	0	0
$67$	−0.121873	−0.0148892	−0.00744459	−	0.999972i	$-0.502370\pi$
$67$	−0.121873	−0.0148892	−0.00744459	+	0.999972i	$0.502370\pi$
$68$	0	0
$69$	−0.688657	−0.0829045
$70$	0	0
$71$	2.67913	0.317954	0.158977	−	0.987282i	$-0.449180\pi$
$71$	2.67913	0.317954	0.158977	+	0.987282i	$0.449180\pi$
$72$	0	0
$73$	0.257480	0.0301357	0.0150679	−	0.999886i	$-0.495204\pi$
$73$	0.257480	0.0301357	0.0150679	+	0.999886i	$0.495204\pi$
$74$	0	0
$75$	−30.8447	−3.56164
$76$	0	0
$77$	−14.5987	−1.66368
$78$	0	0
$79$	10.4254	1.17295	0.586477	−	0.809966i	$-0.300514\pi$
$79$	10.4254	1.17295	0.586477	+	0.809966i	$0.300514\pi$
$80$	0	0
$81$	−10.4247	−1.15830
$82$	0	0
$83$	−0.107899	−0.0118434	−0.00592171	−	0.999982i	$-0.501885\pi$
$83$	−0.107899	−0.0118434	−0.00592171	+	0.999982i	$0.501885\pi$
$84$	0	0
$85$	8.52317	0.924468
$86$	0	0
$87$	−16.6577	−1.78589
$88$	0	0
$89$	−14.1086	−1.49551	−0.747754	−	0.663976i	$-0.768868\pi$
$89$	−14.1086	−1.49551	−0.747754	+	0.663976i	$0.768868\pi$
$90$	0	0
$91$	3.81782	0.400216
$92$	0	0
$93$	7.51865	0.779647
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−7.07291	−0.718146	−0.359073	−	0.933310i	$-0.616907\pi$
$97$	−7.07291	−0.718146	−0.359073	+	0.933310i	$0.616907\pi$
$98$	0	0
$99$	8.73800	0.878202
$100$	0	0
$101$	−3.50600	−0.348860	−0.174430	−	0.984670i	$-0.555808\pi$
$101$	−3.50600	−0.348860	−0.174430	+	0.984670i	$0.555808\pi$
$102$	0	0
$103$	0.930210	0.0916563	0.0458281	−	0.998949i	$-0.485407\pi$
$103$	0.930210	0.0916563	0.0458281	+	0.998949i	$0.485407\pi$
$104$	0	0
$105$	−39.9911	−3.90274
$106$	0	0
$107$	12.3129	1.19033	0.595167	−	0.803602i	$-0.297086\pi$
$107$	12.3129	1.19033	0.595167	+	0.803602i	$0.297086\pi$
$108$	0	0
$109$	−11.3763	−1.08965	−0.544826	−	0.838549i	$-0.683404\pi$
$109$	−11.3763	−1.08965	−0.544826	+	0.838549i	$0.683404\pi$
$110$	0	0
$111$	7.00785	0.665155
$112$	0	0
$113$	0.181387	0.0170635	0.00853175	−	0.999964i	$-0.497284\pi$
$113$	0.181387	0.0170635	0.00853175	+	0.999964i	$0.497284\pi$
$114$	0	0
$115$	−1.26547	−0.118006
$116$	0	0
$117$	−2.28514	−0.211261
$118$	0	0
$119$	8.02518	0.735667
$120$	0	0
$121$	2.16283	0.196621
$122$	0	0
$123$	8.82302	0.795545
$124$	0	0
$125$	−35.3125	−3.15844
$126$	0	0
$127$	−13.9908	−1.24148	−0.620742	−	0.784015i	$-0.713169\pi$
$127$	−13.9908	−1.24148	−0.620742	+	0.784015i	$0.713169\pi$
$128$	0	0
$129$	−14.7249	−1.29646
$130$	0	0
$131$	9.04103	0.789919	0.394959	−	0.918699i	$-0.370759\pi$
$131$	9.04103	0.789919	0.394959	+	0.918699i	$0.370759\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−5.87915	−0.505997
$136$	0	0
$137$	4.99341	0.426616	0.213308	−	0.976985i	$-0.431576\pi$
$137$	4.99341	0.426616	0.213308	+	0.976985i	$0.431576\pi$
$138$	0	0
$139$	1.12537	0.0954525	0.0477263	−	0.998860i	$-0.484802\pi$
$139$	1.12537	0.0954525	0.0477263	+	0.998860i	$0.484802\pi$
$140$	0	0
$141$	−17.1405	−1.44349
$142$	0	0
$143$	−3.44231	−0.287860
$144$	0	0
$145$	−30.6101	−2.54203
$146$	0	0
$147$	−21.3753	−1.76300
$148$	0	0
$149$	8.98883	0.736393	0.368197	−	0.929748i	$-0.379975\pi$
$149$	8.98883	0.736393	0.368197	+	0.929748i	$0.379975\pi$
$150$	0	0
$151$	−3.60432	−0.293315	−0.146658	−	0.989187i	$-0.546852\pi$
$151$	−3.60432	−0.293315	−0.146658	+	0.989187i	$0.546852\pi$
$152$	0	0
$153$	−4.80344	−0.388335
$154$	0	0
$155$	13.8163	1.10975
$156$	0	0
$157$	−23.2636	−1.85663	−0.928317	−	0.371788i	$-0.878745\pi$
$157$	−23.2636	−1.85663	−0.928317	+	0.371788i	$0.878745\pi$
$158$	0	0
$159$	29.1151	2.30898
$160$	0	0
$161$	−1.19153	−0.0939061
$162$	0	0
$163$	13.2233	1.03573	0.517864	−	0.855463i	$-0.326727\pi$
$163$	13.2233	1.03573	0.517864	+	0.855463i	$0.326727\pi$
$164$	0	0
$165$	36.0577	2.80709
$166$	0	0
$167$	−25.4345	−1.96818	−0.984089	−	0.177675i	$-0.943143\pi$
$167$	−25.4345	−1.96818	−0.984089	+	0.177675i	$0.943143\pi$
$168$	0	0
$169$	−12.0998	−0.930752
$170$	0	0
$171$	0	0
$172$	0	0
$173$	13.0024	0.988552	0.494276	−	0.869305i	$-0.335433\pi$
$173$	13.0024	0.988552	0.494276	+	0.869305i	$0.335433\pi$
$174$	0	0
$175$	−53.3684	−4.03427
$176$	0	0
$177$	−1.36161	−0.102345
$178$	0	0
$179$	15.4470	1.15456	0.577281	−	0.816546i	$-0.304114\pi$
$179$	15.4470	1.15456	0.577281	+	0.816546i	$0.304114\pi$
$180$	0	0
$181$	−5.07540	−0.377252	−0.188626	−	0.982049i	$-0.560403\pi$
$181$	−5.07540	−0.377252	−0.188626	+	0.982049i	$0.560403\pi$
$182$	0	0
$183$	−30.7841	−2.27563
$184$	0	0
$185$	12.8776	0.946780
$186$	0	0
$187$	−7.23584	−0.529137
$188$	0	0
$189$	−5.53565	−0.402659
$190$	0	0
$191$	14.5264	1.05109	0.525545	−	0.850766i	$-0.323861\pi$
$191$	14.5264	1.05109	0.525545	+	0.850766i	$0.323861\pi$
$192$	0	0
$193$	−2.63076	−0.189366	−0.0946830	−	0.995507i	$-0.530184\pi$
$193$	−2.63076	−0.189366	−0.0946830	+	0.995507i	$0.530184\pi$
$194$	0	0
$195$	−9.42971	−0.675275
$196$	0	0
$197$	13.3729	0.952777	0.476388	−	0.879235i	$-0.341946\pi$
$197$	13.3729	0.952777	0.476388	+	0.879235i	$0.341946\pi$
$198$	0	0
$199$	8.99470	0.637617	0.318809	−	0.947819i	$-0.396717\pi$
$199$	8.99470	0.637617	0.318809	+	0.947819i	$0.396717\pi$
$200$	0	0
$201$	0.283429	0.0199915
$202$	0	0
$203$	−28.8216	−2.02288
$204$	0	0
$205$	16.2132	1.13238
$206$	0	0
$207$	0.713188	0.0495700
$208$	0	0
$209$	0	0
$210$	0	0
$211$	23.3837	1.60980	0.804899	−	0.593412i	$-0.202219\pi$
$211$	23.3837	1.60980	0.804899	+	0.593412i	$0.202219\pi$
$212$	0	0
$213$	−6.23059	−0.426913
$214$	0	0
$215$	−27.0585	−1.84537
$216$	0	0
$217$	13.0090	0.883108
$218$	0	0
$219$	−0.598797	−0.0404629
$220$	0	0
$221$	1.89230	0.127290
$222$	0	0
$223$	−17.1459	−1.14817	−0.574087	−	0.818794i	$-0.694643\pi$
$223$	−17.1459	−1.14817	−0.574087	+	0.818794i	$0.694643\pi$
$224$	0	0
$225$	31.9434	2.12956
$226$	0	0
$227$	18.6281	1.23639	0.618194	−	0.786025i	$-0.287865\pi$
$227$	18.6281	1.23639	0.618194	+	0.786025i	$0.287865\pi$
$228$	0	0
$229$	21.2555	1.40461	0.702303	−	0.711878i	$-0.252155\pi$
$229$	21.2555	1.40461	0.702303	+	0.711878i	$0.252155\pi$
$230$	0	0
$231$	33.9509	2.23381
$232$	0	0
$233$	1.08780	0.0712643	0.0356321	−	0.999365i	$-0.488656\pi$
$233$	1.08780	0.0712643	0.0356321	+	0.999365i	$0.488656\pi$
$234$	0	0
$235$	−31.4974	−2.05466
$236$	0	0
$237$	−24.2455	−1.57491
$238$	0	0
$239$	−24.5547	−1.58831	−0.794157	−	0.607713i	$-0.792087\pi$
$239$	−24.5547	−1.58831	−0.794157	+	0.607713i	$0.792087\pi$
$240$	0	0
$241$	4.56323	0.293944	0.146972	−	0.989141i	$-0.453047\pi$
$241$	4.56323	0.293944	0.146972	+	0.989141i	$0.453047\pi$
$242$	0	0
$243$	20.1167	1.29048
$244$	0	0
$245$	−39.2792	−2.50945
$246$	0	0
$247$	0	0
$248$	0	0
$249$	0.250930	0.0159020
$250$	0	0
$251$	26.6609	1.68282	0.841410	−	0.540397i	$-0.181726\pi$
$251$	26.6609	1.68282	0.841410	+	0.540397i	$0.181726\pi$
$252$	0	0
$253$	1.07434	0.0675431
$254$	0	0
$255$	−19.8216	−1.24127
$256$	0	0
$257$	14.1879	0.885016	0.442508	−	0.896765i	$-0.354089\pi$
$257$	14.1879	0.885016	0.442508	+	0.896765i	$0.354089\pi$
$258$	0	0
$259$	12.1252	0.753423
$260$	0	0
$261$	17.2510	1.06781
$262$	0	0
$263$	−21.2547	−1.31062	−0.655311	−	0.755359i	$-0.727463\pi$
$263$	−21.2547	−1.31062	−0.655311	+	0.755359i	$0.727463\pi$
$264$	0	0
$265$	53.5019	3.28659
$266$	0	0
$267$	32.8110	2.00800
$268$	0	0
$269$	1.51611	0.0924391	0.0462195	−	0.998931i	$-0.485283\pi$
$269$	1.51611	0.0924391	0.0462195	+	0.998931i	$0.485283\pi$
$270$	0	0
$271$	−5.92400	−0.359857	−0.179929	−	0.983680i	$-0.557587\pi$
$271$	−5.92400	−0.359857	−0.179929	+	0.983680i	$0.557587\pi$
$272$	0	0
$273$	−8.87875	−0.537366
$274$	0	0
$275$	48.1192	2.90170
$276$	0	0
$277$	−7.83954	−0.471032	−0.235516	−	0.971870i	$-0.575678\pi$
$277$	−7.83954	−0.471032	−0.235516	+	0.971870i	$0.575678\pi$
$278$	0	0
$279$	−7.78648	−0.466164
$280$	0	0
$281$	16.3122	0.973106	0.486553	−	0.873651i	$-0.338254\pi$
$281$	16.3122	0.973106	0.486553	+	0.873651i	$0.338254\pi$
$282$	0	0
$283$	9.14656	0.543707	0.271853	−	0.962339i	$-0.412363\pi$
$283$	9.14656	0.543707	0.271853	+	0.962339i	$0.412363\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	15.2659	0.901115
$288$	0	0
$289$	−13.0223	−0.766019
$290$	0	0
$291$	16.4488	0.964247
$292$	0	0
$293$	−9.67814	−0.565403	−0.282701	−	0.959208i	$-0.591231\pi$
$293$	−9.67814	−0.565403	−0.282701	+	0.959208i	$0.591231\pi$
$294$	0	0
$295$	−2.50210	−0.145678
$296$	0	0
$297$	4.99117	0.289617
$298$	0	0
$299$	−0.280958	−0.0162482
$300$	0	0
$301$	−25.4775	−1.46850
$302$	0	0
$303$	8.15358	0.468411
$304$	0	0
$305$	−56.5688	−3.23912
$306$	0	0
$307$	−12.3439	−0.704502	−0.352251	−	0.935906i	$-0.614584\pi$
$307$	−12.3439	−0.704502	−0.352251	+	0.935906i	$0.614584\pi$
$308$	0	0
$309$	−2.16330	−0.123066
$310$	0	0
$311$	−5.61077	−0.318158	−0.159079	−	0.987266i	$-0.550852\pi$
$311$	−5.61077	−0.318158	−0.159079	+	0.987266i	$0.550852\pi$
$312$	0	0
$313$	4.44001	0.250964	0.125482	−	0.992096i	$-0.459952\pi$
$313$	4.44001	0.250964	0.125482	+	0.992096i	$0.459952\pi$
$314$	0	0
$315$	41.4157	2.33351
$316$	0	0
$317$	22.6881	1.27429	0.637146	−	0.770743i	$-0.280115\pi$
$317$	22.6881	1.27429	0.637146	+	0.770743i	$0.280115\pi$
$318$	0	0
$319$	25.9868	1.45498
$320$	0	0
$321$	−28.6350	−1.59825
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−12.5840	−0.698035
$326$	0	0
$327$	26.4568	1.46306
$328$	0	0
$329$	−29.6570	−1.63505
$330$	0	0
$331$	−24.7799	−1.36202	−0.681012	−	0.732272i	$-0.738460\pi$
$331$	−24.7799	−1.36202	−0.681012	+	0.732272i	$0.738460\pi$
$332$	0	0
$333$	−7.25748	−0.397707
$334$	0	0
$335$	0.520829	0.0284559
$336$	0	0
$337$	−23.4232	−1.27594	−0.637971	−	0.770061i	$-0.720226\pi$
$337$	−23.4232	−1.27594	−0.637971	+	0.770061i	$0.720226\pi$
$338$	0	0
$339$	−0.421836	−0.0229110
$340$	0	0
$341$	−11.7295	−0.635186
$342$	0	0
$343$	−8.81730	−0.476090
$344$	0	0
$345$	2.94300	0.158446
$346$	0	0
$347$	−5.17571	−0.277847	−0.138923	−	0.990303i	$-0.544364\pi$
$347$	−5.17571	−0.277847	−0.138923	+	0.990303i	$0.544364\pi$
$348$	0	0
$349$	−25.1202	−1.34465	−0.672326	−	0.740255i	$-0.734705\pi$
$349$	−25.1202	−1.34465	−0.672326	+	0.740255i	$0.734705\pi$
$350$	0	0
$351$	−1.30528	−0.0696705
$352$	0	0
$353$	23.5256	1.25214	0.626072	−	0.779766i	$-0.284662\pi$
$353$	23.5256	1.25214	0.626072	+	0.779766i	$0.284662\pi$
$354$	0	0
$355$	−11.4493	−0.607667
$356$	0	0
$357$	−18.6634	−0.987773
$358$	0	0
$359$	18.3668	0.969363	0.484682	−	0.874691i	$-0.338935\pi$
$359$	18.3668	0.969363	0.484682	+	0.874691i	$0.338935\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−5.02989	−0.264001
$364$	0	0
$365$	−1.10035	−0.0575948
$366$	0	0
$367$	22.2718	1.16258	0.581289	−	0.813697i	$-0.302549\pi$
$367$	22.2718	1.16258	0.581289	+	0.813697i	$0.302549\pi$
$368$	0	0
$369$	−9.13731	−0.475670
$370$	0	0
$371$	50.3759	2.61538
$372$	0	0
$373$	−11.0997	−0.574719	−0.287359	−	0.957823i	$-0.592777\pi$
$373$	−11.0997	−0.574719	−0.287359	+	0.957823i	$0.592777\pi$
$374$	0	0
$375$	82.1230	4.24081
$376$	0	0
$377$	−6.79599	−0.350011
$378$	0	0
$379$	−37.1619	−1.90888	−0.954439	−	0.298405i	$-0.903545\pi$
$379$	−37.1619	−1.90888	−0.954439	+	0.298405i	$0.903545\pi$
$380$	0	0
$381$	32.5372	1.66693
$382$	0	0
$383$	−22.7960	−1.16482	−0.582411	−	0.812894i	$-0.697891\pi$
$383$	−22.7960	−1.16482	−0.582411	+	0.812894i	$0.697891\pi$
$384$	0	0
$385$	62.3881	3.17959
$386$	0	0
$387$	15.2495	0.775174
$388$	0	0
$389$	−14.5075	−0.735558	−0.367779	−	0.929913i	$-0.619882\pi$
$389$	−14.5075	−0.735558	−0.367779	+	0.929913i	$0.619882\pi$
$390$	0	0
$391$	−0.590583	−0.0298670
$392$	0	0
$393$	−21.0259	−1.06062
$394$	0	0
$395$	−44.5535	−2.24173
$396$	0	0
$397$	−19.3834	−0.972823	−0.486411	−	0.873730i	$-0.661694\pi$
$397$	−19.3834	−0.972823	−0.486411	+	0.873730i	$0.661694\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	2.89215	0.144427	0.0722135	−	0.997389i	$-0.476994\pi$
$401$	2.89215	0.144427	0.0722135	+	0.997389i	$0.476994\pi$
$402$	0	0
$403$	3.06745	0.152801
$404$	0	0
$405$	44.5504	2.21373
$406$	0	0
$407$	−10.9326	−0.541908
$408$	0	0
$409$	2.93892	0.145320	0.0726601	−	0.997357i	$-0.476851\pi$
$409$	2.93892	0.145320	0.0726601	+	0.997357i	$0.476851\pi$
$410$	0	0
$411$	−11.6127	−0.572813
$412$	0	0
$413$	−2.35590	−0.115926
$414$	0	0
$415$	0.461108	0.0226349
$416$	0	0
$417$	−2.61717	−0.128163
$418$	0	0
$419$	8.98293	0.438845	0.219423	−	0.975630i	$-0.429583\pi$
$419$	8.98293	0.438845	0.219423	+	0.975630i	$0.429583\pi$
$420$	0	0
$421$	−3.98224	−0.194083	−0.0970413	−	0.995280i	$-0.530938\pi$
$421$	−3.98224	−0.194083	−0.0970413	+	0.995280i	$0.530938\pi$
$422$	0	0
$423$	17.7511	0.863088
$424$	0	0
$425$	−26.4520	−1.28311
$426$	0	0
$427$	−53.2636	−2.57761
$428$	0	0
$429$	8.00545	0.386507
$430$	0	0
$431$	16.2162	0.781106	0.390553	−	0.920581i	$-0.372284\pi$
$431$	16.2162	0.781106	0.390553	+	0.920581i	$0.372284\pi$
$432$	0	0
$433$	−30.9590	−1.48780	−0.743898	−	0.668294i	$-0.767025\pi$
$433$	−30.9590	−1.48780	−0.743898	+	0.668294i	$0.767025\pi$
$434$	0	0
$435$	71.1871	3.41316
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−20.5939	−0.982892	−0.491446	−	0.870908i	$-0.663532\pi$
$439$	−20.5939	−0.982892	−0.491446	+	0.870908i	$0.663532\pi$
$440$	0	0
$441$	22.1367	1.05413
$442$	0	0
$443$	32.5072	1.54446	0.772231	−	0.635342i	$-0.219141\pi$
$443$	32.5072	1.54446	0.772231	+	0.635342i	$0.219141\pi$
$444$	0	0
$445$	60.2935	2.85819
$446$	0	0
$447$	−20.9045	−0.988748
$448$	0	0
$449$	26.4823	1.24978	0.624888	−	0.780714i	$-0.285145\pi$
$449$	26.4823	1.24978	0.624888	+	0.780714i	$0.285145\pi$
$450$	0	0
$451$	−13.7643	−0.648138
$452$	0	0
$453$	8.38222	0.393831
$454$	0	0
$455$	−16.3156	−0.764886
$456$	0	0
$457$	−21.1946	−0.991440	−0.495720	−	0.868483i	$-0.665096\pi$
$457$	−21.1946	−0.991440	−0.495720	+	0.868483i	$0.665096\pi$
$458$	0	0
$459$	−2.74374	−0.128067
$460$	0	0
$461$	−23.6136	−1.09979	−0.549897	−	0.835233i	$-0.685333\pi$
$461$	−23.6136	−1.09979	−0.549897	+	0.835233i	$0.685333\pi$
$462$	0	0
$463$	5.12258	0.238067	0.119033	−	0.992890i	$-0.462020\pi$
$463$	5.12258	0.238067	0.119033	+	0.992890i	$0.462020\pi$
$464$	0	0
$465$	−32.1312	−1.49005
$466$	0	0
$467$	−11.8537	−0.548522	−0.274261	−	0.961655i	$-0.588433\pi$
$467$	−11.8537	−0.548522	−0.274261	+	0.961655i	$0.588433\pi$
$468$	0	0
$469$	0.490398	0.0226445
$470$	0	0
$471$	54.1019	2.49289
$472$	0	0
$473$	22.9716	1.05624
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−30.1522	−1.38058
$478$	0	0
$479$	−12.2856	−0.561343	−0.280671	−	0.959804i	$-0.590557\pi$
$479$	−12.2856	−0.561343	−0.280671	+	0.959804i	$0.590557\pi$
$480$	0	0
$481$	2.85906	0.130362
$482$	0	0
$483$	2.77104	0.126087
$484$	0	0
$485$	30.2263	1.37251
$486$	0	0
$487$	−1.25354	−0.0568032	−0.0284016	−	0.999597i	$-0.509042\pi$
$487$	−1.25354	−0.0568032	−0.0284016	+	0.999597i	$0.509042\pi$
$488$	0	0
$489$	−30.7522	−1.39066
$490$	0	0
$491$	−22.7795	−1.02803	−0.514013	−	0.857783i	$-0.671842\pi$
$491$	−22.7795	−1.02803	−0.514013	+	0.857783i	$0.671842\pi$
$492$	0	0
$493$	−14.2854	−0.643382
$494$	0	0
$495$	−37.3421	−1.67840
$496$	0	0
$497$	−10.7804	−0.483565
$498$	0	0
$499$	33.5405	1.50148	0.750740	−	0.660598i	$-0.229697\pi$
$499$	33.5405	1.50148	0.750740	+	0.660598i	$0.229697\pi$
$500$	0	0
$501$	59.1506	2.64265
$502$	0	0
$503$	−12.0392	−0.536802	−0.268401	−	0.963307i	$-0.586495\pi$
$503$	−12.0392	−0.536802	−0.268401	+	0.963307i	$0.586495\pi$
$504$	0	0
$505$	14.9830	0.666735
$506$	0	0
$507$	28.1393	1.24971
$508$	0	0
$509$	21.2283	0.940928	0.470464	−	0.882419i	$-0.344087\pi$
$509$	21.2283	0.940928	0.470464	+	0.882419i	$0.344087\pi$
$510$	0	0
$511$	−1.03606	−0.0458325
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−3.97528	−0.175172
$516$	0	0
$517$	26.7400	1.17603
$518$	0	0
$519$	−30.2384	−1.32732
$520$	0	0
$521$	34.6456	1.51785	0.758925	−	0.651178i	$-0.225725\pi$
$521$	34.6456	1.51785	0.758925	+	0.651178i	$0.225725\pi$
$522$	0	0
$523$	−27.4504	−1.20032	−0.600160	−	0.799880i	$-0.704897\pi$
$523$	−27.4504	−1.20032	−0.600160	+	0.799880i	$0.704897\pi$
$524$	0	0
$525$	124.114	5.41678
$526$	0	0
$527$	6.44789	0.280874
$528$	0	0
$529$	−22.9123	−0.996188
$530$	0	0
$531$	1.41012	0.0611938
$532$	0	0
$533$	3.59961	0.155917
$534$	0	0
$535$	−52.6196	−2.27494
$536$	0	0
$537$	−35.9236	−1.55022
$538$	0	0
$539$	33.3465	1.43633
$540$	0	0
$541$	34.5821	1.48680	0.743400	−	0.668847i	$-0.233212\pi$
$541$	34.5821	1.48680	0.743400	+	0.668847i	$0.233212\pi$
$542$	0	0
$543$	11.8034	0.506532
$544$	0	0
$545$	48.6170	2.08252
$546$	0	0
$547$	−30.4137	−1.30040	−0.650198	−	0.759765i	$-0.725314\pi$
$547$	−30.4137	−1.30040	−0.650198	+	0.759765i	$0.725314\pi$
$548$	0	0
$549$	31.8807	1.36064
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−41.9503	−1.78391
$554$	0	0
$555$	−29.9483	−1.27123
$556$	0	0
$557$	−6.41431	−0.271783	−0.135892	−	0.990724i	$-0.543390\pi$
$557$	−6.41431	−0.271783	−0.135892	+	0.990724i	$0.543390\pi$
$558$	0	0
$559$	−6.00747	−0.254089
$560$	0	0
$561$	16.8277	0.710467
$562$	0	0
$563$	24.4242	1.02936	0.514678	−	0.857384i	$-0.327911\pi$
$563$	24.4242	1.02936	0.514678	+	0.857384i	$0.327911\pi$
$564$	0	0
$565$	−0.775165	−0.0326114
$566$	0	0
$567$	41.9474	1.76162
$568$	0	0
$569$	−9.67293	−0.405510	−0.202755	−	0.979229i	$-0.564990\pi$
$569$	−9.67293	−0.405510	−0.202755	+	0.979229i	$0.564990\pi$
$570$	0	0
$571$	−42.0563	−1.76000	−0.880001	−	0.474971i	$-0.842458\pi$
$571$	−42.0563	−1.76000	−0.880001	+	0.474971i	$0.842458\pi$
$572$	0	0
$573$	−33.7826	−1.41129
$574$	0	0
$575$	3.92745	0.163786
$576$	0	0
$577$	−6.30242	−0.262373	−0.131187	−	0.991358i	$-0.541879\pi$
$577$	−6.30242	−0.262373	−0.131187	+	0.991358i	$0.541879\pi$
$578$	0	0
$579$	6.11811	0.254260
$580$	0	0
$581$	0.434167	0.0180123
$582$	0	0
$583$	−45.4210	−1.88115
$584$	0	0
$585$	9.76561	0.403758
$586$	0	0
$587$	−7.59256	−0.313379	−0.156689	−	0.987648i	$-0.550082\pi$
$587$	−7.59256	−0.313379	−0.156689	+	0.987648i	$0.550082\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−31.1000	−1.27928
$592$	0	0
$593$	−17.6290	−0.723938	−0.361969	−	0.932190i	$-0.617895\pi$
$593$	−17.6290	−0.723938	−0.361969	+	0.932190i	$0.617895\pi$
$594$	0	0
$595$	−34.2959	−1.40599
$596$	0	0
$597$	−20.9181	−0.856122
$598$	0	0
$599$	20.9163	0.854617	0.427308	−	0.904106i	$-0.359462\pi$
$599$	20.9163	0.854617	0.427308	+	0.904106i	$0.359462\pi$
$600$	0	0
$601$	7.13900	0.291206	0.145603	−	0.989343i	$-0.453488\pi$
$601$	7.13900	0.291206	0.145603	+	0.989343i	$0.453488\pi$
$602$	0	0
$603$	−0.293525	−0.0119533
$604$	0	0
$605$	−9.24292	−0.375778
$606$	0	0
$607$	3.79992	0.154234	0.0771169	−	0.997022i	$-0.475429\pi$
$607$	3.79992	0.154234	0.0771169	+	0.997022i	$0.475429\pi$
$608$	0	0
$609$	67.0277	2.71610
$610$	0	0
$611$	−6.99298	−0.282906
$612$	0	0
$613$	17.8061	0.719183	0.359591	−	0.933110i	$-0.382916\pi$
$613$	17.8061	0.719183	0.359591	+	0.933110i	$0.382916\pi$
$614$	0	0
$615$	−37.7055	−1.52043
$616$	0	0
$617$	−29.9942	−1.20752	−0.603761	−	0.797165i	$-0.706332\pi$
$617$	−29.9942	−1.20752	−0.603761	+	0.797165i	$0.706332\pi$
$618$	0	0
$619$	11.8948	0.478091	0.239046	−	0.971008i	$-0.423165\pi$
$619$	11.8948	0.478091	0.239046	+	0.971008i	$0.423165\pi$
$620$	0	0
$621$	0.407375	0.0163474
$622$	0	0
$623$	56.7707	2.27447
$624$	0	0
$625$	84.5937	3.38375
$626$	0	0
$627$	0	0
$628$	0	0
$629$	6.00983	0.239628
$630$	0	0
$631$	−35.2253	−1.40230	−0.701149	−	0.713015i	$-0.747329\pi$
$631$	−35.2253	−1.40230	−0.701149	+	0.713015i	$0.747329\pi$
$632$	0	0
$633$	−54.3812	−2.16146
$634$	0	0
$635$	59.7902	2.37270
$636$	0	0
$637$	−8.72068	−0.345526
$638$	0	0
$639$	6.45254	0.255258
$640$	0	0
$641$	1.83803	0.0725978	0.0362989	−	0.999341i	$-0.488443\pi$
$641$	1.83803	0.0725978	0.0362989	+	0.999341i	$0.488443\pi$
$642$	0	0
$643$	4.05845	0.160050	0.0800248	−	0.996793i	$-0.474500\pi$
$643$	4.05845	0.160050	0.0800248	+	0.996793i	$0.474500\pi$
$644$	0	0
$645$	62.9274	2.47777
$646$	0	0
$647$	7.12263	0.280019	0.140010	−	0.990150i	$-0.455287\pi$
$647$	7.12263	0.280019	0.140010	+	0.990150i	$0.455287\pi$
$648$	0	0
$649$	2.12418	0.0833815
$650$	0	0
$651$	−30.2538	−1.18574
$652$	0	0
$653$	30.8846	1.20861	0.604305	−	0.796753i	$-0.293451\pi$
$653$	30.8846	1.20861	0.604305	+	0.796753i	$0.293451\pi$
$654$	0	0
$655$	−38.6371	−1.50968
$656$	0	0
$657$	0.620127	0.0241935
$658$	0	0
$659$	−29.0575	−1.13192	−0.565960	−	0.824433i	$-0.691494\pi$
$659$	−29.0575	−1.13192	−0.565960	+	0.824433i	$0.691494\pi$
$660$	0	0
$661$	−37.9619	−1.47655	−0.738273	−	0.674501i	$-0.764359\pi$
$661$	−37.9619	−1.47655	−0.738273	+	0.674501i	$0.764359\pi$
$662$	0	0
$663$	−4.40074	−0.170911
$664$	0	0
$665$	0	0
$666$	0	0
$667$	2.12102	0.0821261
$668$	0	0
$669$	39.8746	1.54164
$670$	0	0
$671$	48.0247	1.85397
$672$	0	0
$673$	29.0978	1.12164	0.560819	−	0.827938i	$-0.310486\pi$
$673$	29.0978	1.12164	0.560819	+	0.827938i	$0.310486\pi$
$674$	0	0
$675$	18.2462	0.702296
$676$	0	0
$677$	−35.0103	−1.34555	−0.672777	−	0.739846i	$-0.734899\pi$
$677$	−35.0103	−1.34555	−0.672777	+	0.739846i	$0.734899\pi$
$678$	0	0
$679$	28.4603	1.09220
$680$	0	0
$681$	−43.3216	−1.66009
$682$	0	0
$683$	29.5189	1.12951	0.564755	−	0.825258i	$-0.308971\pi$
$683$	29.5189	1.12951	0.564755	+	0.825258i	$0.308971\pi$
$684$	0	0
$685$	−21.3395	−0.815340
$686$	0	0
$687$	−49.4320	−1.88595
$688$	0	0
$689$	11.8784	0.452530
$690$	0	0
$691$	33.6494	1.28009	0.640043	−	0.768339i	$-0.278917\pi$
$691$	33.6494	1.28009	0.640043	+	0.768339i	$0.278917\pi$
$692$	0	0
$693$	−35.1603	−1.33563
$694$	0	0
$695$	−4.80930	−0.182427
$696$	0	0
$697$	7.56650	0.286602
$698$	0	0
$699$	−2.52980	−0.0956858
$700$	0	0
$701$	8.31040	0.313879	0.156940	−	0.987608i	$-0.449837\pi$
$701$	8.31040	0.313879	0.156940	+	0.987608i	$0.449837\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	73.2505	2.75877
$706$	0	0
$707$	14.1076	0.530570
$708$	0	0
$709$	−6.34710	−0.238370	−0.119185	−	0.992872i	$-0.538028\pi$
$709$	−6.34710	−0.238370	−0.119185	+	0.992872i	$0.538028\pi$
$710$	0	0
$711$	25.1092	0.941667
$712$	0	0
$713$	−0.957348	−0.0358529
$714$	0	0
$715$	14.7108	0.550153
$716$	0	0
$717$	57.1047	2.13261
$718$	0	0
$719$	6.30335	0.235075	0.117538	−	0.993068i	$-0.462500\pi$
$719$	6.30335	0.235075	0.117538	+	0.993068i	$0.462500\pi$
$720$	0	0
$721$	−3.74301	−0.139397
$722$	0	0
$723$	−10.6123	−0.394675
$724$	0	0
$725$	94.9996	3.52820
$726$	0	0
$727$	25.5955	0.949283	0.474641	−	0.880179i	$-0.342578\pi$
$727$	25.5955	0.949283	0.474641	+	0.880179i	$0.342578\pi$
$728$	0	0
$729$	−15.5093	−0.574419
$730$	0	0
$731$	−12.6279	−0.467060
$732$	0	0
$733$	44.5854	1.64680	0.823399	−	0.567462i	$-0.192075\pi$
$733$	44.5854	1.64680	0.823399	+	0.567462i	$0.192075\pi$
$734$	0	0
$735$	91.3479	3.36942
$736$	0	0
$737$	−0.442163	−0.0162873
$738$	0	0
$739$	−8.88362	−0.326789	−0.163395	−	0.986561i	$-0.552244\pi$
$739$	−8.88362	−0.326789	−0.163395	+	0.986561i	$0.552244\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	6.95993	0.255335	0.127667	−	0.991817i	$-0.459251\pi$
$743$	6.95993	0.255335	0.127667	+	0.991817i	$0.459251\pi$
$744$	0	0
$745$	−38.4140	−1.40738
$746$	0	0
$747$	−0.259868	−0.00950809
$748$	0	0
$749$	−49.5451	−1.81034
$750$	0	0
$751$	45.9568	1.67699	0.838494	−	0.544911i	$-0.183437\pi$
$751$	45.9568	1.67699	0.838494	+	0.544911i	$0.183437\pi$
$752$	0	0
$753$	−62.0028	−2.25951
$754$	0	0
$755$	15.4032	0.560578
$756$	0	0
$757$	39.7227	1.44375	0.721873	−	0.692025i	$-0.243281\pi$
$757$	39.7227	1.44375	0.721873	+	0.692025i	$0.243281\pi$
$758$	0	0
$759$	−2.49849	−0.0906894
$760$	0	0
$761$	4.91504	0.178170	0.0890850	−	0.996024i	$-0.471606\pi$
$761$	4.91504	0.178170	0.0890850	+	0.996024i	$0.471606\pi$
$762$	0	0
$763$	45.7764	1.65722
$764$	0	0
$765$	20.5276	0.742178
$766$	0	0
$767$	−0.555510	−0.0200583
$768$	0	0
$769$	1.29448	0.0466801	0.0233401	−	0.999728i	$-0.492570\pi$
$769$	1.29448	0.0466801	0.0233401	+	0.999728i	$0.492570\pi$
$770$	0	0
$771$	−32.9954	−1.18830
$772$	0	0
$773$	−40.1559	−1.44431	−0.722154	−	0.691732i	$-0.756848\pi$
$773$	−40.1559	−1.44431	−0.722154	+	0.691732i	$0.756848\pi$
$774$	0	0
$775$	−42.8793	−1.54027
$776$	0	0
$777$	−28.1984	−1.01161
$778$	0	0
$779$	0	0
$780$	0	0
$781$	9.72003	0.347810
$782$	0	0
$783$	9.85384	0.352148
$784$	0	0
$785$	99.4176	3.54837
$786$	0	0
$787$	−47.3790	−1.68888	−0.844439	−	0.535652i	$-0.820066\pi$
$787$	−47.3790	−1.68888	−0.844439	+	0.535652i	$0.820066\pi$
$788$	0	0
$789$	49.4302	1.75976
$790$	0	0
$791$	−0.729874	−0.0259513
$792$	0	0
$793$	−12.5593	−0.445994
$794$	0	0
$795$	−124.424	−4.41288
$796$	0	0
$797$	34.4310	1.21961	0.609805	−	0.792551i	$-0.291248\pi$
$797$	34.4310	1.21961	0.609805	+	0.792551i	$0.291248\pi$
$798$	0	0
$799$	−14.6995	−0.520030
$800$	0	0
$801$	−33.9798	−1.20062
$802$	0	0
$803$	0.934153	0.0329655
$804$	0	0
$805$	5.09206	0.179472
$806$	0	0
$807$	−3.52589	−0.124117
$808$	0	0
$809$	17.0088	0.597997	0.298998	−	0.954254i	$-0.403347\pi$
$809$	17.0088	0.597997	0.298998	+	0.954254i	$0.403347\pi$
$810$	0	0
$811$	−26.2380	−0.921340	−0.460670	−	0.887571i	$-0.652391\pi$
$811$	−26.2380	−0.921340	−0.460670	+	0.887571i	$0.652391\pi$
$812$	0	0
$813$	13.7769	0.483177
$814$	0	0
$815$	−56.5102	−1.97946
$816$	0	0
$817$	0	0
$818$	0	0
$819$	9.19503	0.321300
$820$	0	0
$821$	−24.7915	−0.865228	−0.432614	−	0.901579i	$-0.642409\pi$
$821$	−24.7915	−0.865228	−0.432614	+	0.901579i	$0.642409\pi$
$822$	0	0
$823$	−21.6023	−0.753009	−0.376504	−	0.926415i	$-0.622874\pi$
$823$	−21.6023	−0.753009	−0.376504	+	0.926415i	$0.622874\pi$
$824$	0	0
$825$	−111.906	−3.89608
$826$	0	0
$827$	−24.2313	−0.842604	−0.421302	−	0.906920i	$-0.638427\pi$
$827$	−24.2313	−0.842604	−0.421302	+	0.906920i	$0.638427\pi$
$828$	0	0
$829$	−50.1015	−1.74010	−0.870049	−	0.492966i	$-0.835913\pi$
$829$	−50.1015	−1.74010	−0.870049	+	0.492966i	$0.835913\pi$
$830$	0	0
$831$	18.2317	0.632451
$832$	0	0
$833$	−18.3311	−0.635137
$834$	0	0
$835$	108.695	3.76155
$836$	0	0
$837$	−4.44766	−0.153733
$838$	0	0
$839$	−24.7648	−0.854977	−0.427489	−	0.904021i	$-0.640602\pi$
$839$	−24.7648	−0.854977	−0.427489	+	0.904021i	$0.640602\pi$
$840$	0	0
$841$	22.3045	0.769121
$842$	0	0
$843$	−37.9358	−1.30658
$844$	0	0
$845$	51.7088	1.77884
$846$	0	0
$847$	−8.70287	−0.299034
$848$	0	0
$849$	−21.2713	−0.730030
$850$	0	0
$851$	−0.892307	−0.0305879
$852$	0	0
$853$	19.0995	0.653956	0.326978	−	0.945032i	$-0.393970\pi$
$853$	19.0995	0.653956	0.326978	+	0.945032i	$0.393970\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−47.8934	−1.63601	−0.818003	−	0.575214i	$-0.804919\pi$
$857$	−47.8934	−1.63601	−0.818003	+	0.575214i	$0.804919\pi$
$858$	0	0
$859$	−20.9112	−0.713482	−0.356741	−	0.934203i	$-0.616112\pi$
$859$	−20.9112	−0.713482	−0.356741	+	0.934203i	$0.616112\pi$
$860$	0	0
$861$	−35.5024	−1.20992
$862$	0	0
$863$	−47.6708	−1.62273	−0.811366	−	0.584538i	$-0.801276\pi$
$863$	−47.6708	−1.62273	−0.811366	+	0.584538i	$0.801276\pi$
$864$	0	0
$865$	−55.5660	−1.88930
$866$	0	0
$867$	30.2848	1.02853
$868$	0	0
$869$	37.8242	1.28310
$870$	0	0
$871$	0.115633	0.00391808
$872$	0	0
$873$	−17.0348	−0.576539
$874$	0	0
$875$	142.092	4.80358
$876$	0	0
$877$	−42.5151	−1.43563	−0.717816	−	0.696233i	$-0.754858\pi$
$877$	−42.5151	−1.43563	−0.717816	+	0.696233i	$0.754858\pi$
$878$	0	0
$879$	22.5075	0.759161
$880$	0	0
$881$	−4.80993	−0.162051	−0.0810253	−	0.996712i	$-0.525819\pi$
$881$	−4.80993	−0.162051	−0.0810253	+	0.996712i	$0.525819\pi$
$882$	0	0
$883$	29.4745	0.991894	0.495947	−	0.868353i	$-0.334821\pi$
$883$	29.4745	0.991894	0.495947	+	0.868353i	$0.334821\pi$
$884$	0	0
$885$	5.81890	0.195600
$886$	0	0
$887$	41.9726	1.40930	0.704651	−	0.709554i	$-0.251104\pi$
$887$	41.9726	1.40930	0.704651	+	0.709554i	$0.251104\pi$
$888$	0	0
$889$	56.2968	1.88813
$890$	0	0
$891$	−37.8215	−1.26707
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−66.0132	−2.20658
$896$	0	0
$897$	0.653398	0.0218163
$898$	0	0
$899$	−23.1569	−0.772327
$900$	0	0
$901$	24.9687	0.831829
$902$	0	0
$903$	59.2507	1.97174
$904$	0	0
$905$	21.6899	0.720996
$906$	0	0
$907$	−51.4073	−1.70695	−0.853475	−	0.521133i	$-0.825509\pi$
$907$	−51.4073	−1.70695	−0.853475	+	0.521133i	$0.825509\pi$
$908$	0	0
$909$	−8.44403	−0.280071
$910$	0	0
$911$	−31.3484	−1.03862	−0.519309	−	0.854587i	$-0.673811\pi$
$911$	−31.3484	−1.03862	−0.519309	+	0.854587i	$0.673811\pi$
$912$	0	0
$913$	−0.391463	−0.0129555
$914$	0	0
$915$	131.557	4.34914
$916$	0	0
$917$	−36.3796	−1.20136
$918$	0	0
$919$	10.3920	0.342799	0.171400	−	0.985202i	$-0.445171\pi$
$919$	10.3920	0.342799	0.171400	+	0.985202i	$0.445171\pi$
$920$	0	0
$921$	28.7070	0.945928
$922$	0	0
$923$	−2.54196	−0.0836695
$924$	0	0
$925$	−39.9661	−1.31408
$926$	0	0
$927$	2.24036	0.0735832
$928$	0	0
$929$	−14.8710	−0.487900	−0.243950	−	0.969788i	$-0.578443\pi$
$929$	−14.8710	−0.487900	−0.243950	+	0.969788i	$0.578443\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	13.0485	0.427187
$934$	0	0
$935$	30.9226	1.01128
$936$	0	0
$937$	−33.2824	−1.08729	−0.543644	−	0.839316i	$-0.682956\pi$
$937$	−33.2824	−1.08729	−0.543644	+	0.839316i	$0.682956\pi$
$938$	0	0
$939$	−10.3257	−0.336967
$940$	0	0
$941$	−18.6106	−0.606688	−0.303344	−	0.952881i	$-0.598103\pi$
$941$	−18.6106	−0.606688	−0.303344	+	0.952881i	$0.598103\pi$
$942$	0	0
$943$	−1.12343	−0.0365840
$944$	0	0
$945$	23.6568	0.769554
$946$	0	0
$947$	−47.8163	−1.55382	−0.776911	−	0.629610i	$-0.783215\pi$
$947$	−47.8163	−1.55382	−0.776911	+	0.629610i	$0.783215\pi$
$948$	0	0
$949$	−0.244297	−0.00793021
$950$	0	0
$951$	−52.7637	−1.71098
$952$	0	0
$953$	18.6549	0.604290	0.302145	−	0.953262i	$-0.402297\pi$
$953$	18.6549	0.604290	0.302145	+	0.953262i	$0.402297\pi$
$954$	0	0
$955$	−62.0789	−2.00883
$956$	0	0
$957$	−60.4350	−1.95359
$958$	0	0
$959$	−20.0927	−0.648826
$960$	0	0
$961$	−20.5478	−0.662833
$962$	0	0
$963$	29.6550	0.955620
$964$	0	0
$965$	11.2426	0.361913
$966$	0	0
$967$	−36.2138	−1.16456	−0.582279	−	0.812989i	$-0.697839\pi$
$967$	−36.2138	−1.16456	−0.582279	+	0.812989i	$0.697839\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−13.1618	−0.422382	−0.211191	−	0.977445i	$-0.567734\pi$
$971$	−13.1618	−0.422382	−0.211191	+	0.977445i	$0.567734\pi$
$972$	0	0
$973$	−4.52830	−0.145171
$974$	0	0
$975$	29.2655	0.937245
$976$	0	0
$977$	−26.3877	−0.844216	−0.422108	−	0.906545i	$-0.638710\pi$
$977$	−26.3877	−0.844216	−0.422108	+	0.906545i	$0.638710\pi$
$978$	0	0
$979$	−51.1868	−1.63594
$980$	0	0
$981$	−27.3992	−0.874790
$982$	0	0
$983$	−1.00388	−0.0320187	−0.0160093	−	0.999872i	$-0.505096\pi$
$983$	−1.00388	−0.0320187	−0.0160093	+	0.999872i	$0.505096\pi$
$984$	0	0
$985$	−57.1493	−1.82093
$986$	0	0
$987$	68.9707	2.19536
$988$	0	0
$989$	1.87492	0.0596190
$990$	0	0
$991$	35.5158	1.12820	0.564099	−	0.825707i	$-0.309224\pi$
$991$	35.5158	1.12820	0.564099	+	0.825707i	$0.309224\pi$
$992$	0	0
$993$	57.6282	1.82878
$994$	0	0
$995$	−38.4391	−1.21860
$996$	0	0
$997$	−51.7243	−1.63813	−0.819063	−	0.573703i	$-0.805506\pi$
$997$	−51.7243	−1.63813	−0.819063	+	0.573703i	$0.805506\pi$
$998$	0	0
$999$	−4.14549	−0.131158

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.v.1.2	✓	8
4.3	odd	2		5776.2.a.cc.1.7		8
19.18	odd	2		2888.2.a.w.1.7	yes	8
76.75	even	2		5776.2.a.ca.1.2		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.v.1.2	✓	8	1.1	even	1	trivial
2888.2.a.w.1.7	yes	8	19.18	odd	2
5776.2.a.ca.1.2		8	76.75	even	2
5776.2.a.cc.1.7		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)

\(f(q)\)	\(=\)	\(q-2.32561 q^{3} -4.27353 q^{5} -4.02384 q^{7} +2.40845 q^{9} +3.62806 q^{11} -0.948800 q^{13} +9.93856 q^{15} -1.99441 q^{17} +9.35787 q^{21} +0.296119 q^{23} +13.2631 q^{25} +1.37571 q^{27} +7.16272 q^{29} -3.23298 q^{31} -8.43745 q^{33} +17.1960 q^{35} -3.01334 q^{37} +2.20654 q^{39} -3.79386 q^{41} +6.33165 q^{43} -10.2926 q^{45} +7.37034 q^{47} +9.19127 q^{49} +4.63821 q^{51} -12.5194 q^{53} -15.5046 q^{55} +0.585487 q^{59} +13.2370 q^{61} -9.69121 q^{63} +4.05473 q^{65} -0.121873 q^{67} -0.688657 q^{69} +2.67913 q^{71} +0.257480 q^{73} -30.8447 q^{75} -14.5987 q^{77} +10.4254 q^{79} -10.4247 q^{81} -0.107899 q^{83} +8.52317 q^{85} -16.6577 q^{87} -14.1086 q^{89} +3.81782 q^{91} +7.51865 q^{93} -7.07291 q^{97} +8.73800 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 12 q^{11} - 6 q^{13} - 10 q^{17} + 4 q^{21} - 12 q^{23} + 2 q^{25} - 12 q^{27} - 18 q^{29} - 14 q^{31} - 40 q^{33} + 18 q^{35} - 16 q^{37} + 28 q^{39} - 12 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) \) 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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