LMFDB - Embedded newform 2888.2.a.v.1.6

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			1.6
Root			$1.49959$ 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)

$f(q)$	$=$	$q+0.499591 q^{3} +0.989826 q^{5} -0.882762 q^{7} -2.75041 q^{9} -3.21879 q^{11} +2.64190 q^{13} +0.494508 q^{15} -0.693022 q^{17} -0.441020 q^{21} +6.37666 q^{23} -4.02024 q^{25} -2.87285 q^{27} -1.07344 q^{29} +0.593973 q^{31} -1.60808 q^{33} -0.873781 q^{35} -2.48479 q^{37} +1.31987 q^{39} -10.5308 q^{41} +4.20046 q^{43} -2.72243 q^{45} +7.98700 q^{47} -6.22073 q^{49} -0.346227 q^{51} -5.46222 q^{53} -3.18604 q^{55} -11.6719 q^{59} +5.07612 q^{61} +2.42796 q^{63} +2.61502 q^{65} -11.4597 q^{67} +3.18572 q^{69} -3.26616 q^{71} -13.7627 q^{73} -2.00848 q^{75} +2.84142 q^{77} -10.9339 q^{79} +6.81598 q^{81} -6.12042 q^{83} -0.685972 q^{85} -0.536283 q^{87} -3.65717 q^{89} -2.33217 q^{91} +0.296743 q^{93} +3.69133 q^{97} +8.85298 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	0.499591	0.288439	0.144219	−	0.989546i	$-0.453933\pi$
$3$	0.499591	0.288439	0.144219	+	0.989546i	$0.453933\pi$
$4$	0	0
$5$	0.989826	0.442664	0.221332	−	0.975199i	$-0.428960\pi$
$5$	0.989826	0.442664	0.221332	+	0.975199i	$0.428960\pi$
$6$	0	0
$7$	−0.882762	−0.333653	−0.166826	−	0.985986i	$-0.553352\pi$
$7$	−0.882762	−0.333653	−0.166826	+	0.985986i	$0.553352\pi$
$8$	0	0
$9$	−2.75041	−0.916803
$10$	0	0
$11$	−3.21879	−0.970501	−0.485251	−	0.874375i	$-0.661272\pi$
$11$	−3.21879	−0.970501	−0.485251	+	0.874375i	$0.661272\pi$
$12$	0	0
$13$	2.64190	0.732731	0.366366	−	0.930471i	$-0.380602\pi$
$13$	2.64190	0.732731	0.366366	+	0.930471i	$0.380602\pi$
$14$	0	0
$15$	0.494508	0.127681
$16$	0	0
$17$	−0.693022	−0.168083	−0.0840413	−	0.996462i	$-0.526783\pi$
$17$	−0.693022	−0.168083	−0.0840413	+	0.996462i	$0.526783\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−0.441020	−0.0962384
$22$	0	0
$23$	6.37666	1.32963	0.664813	−	0.747010i	$-0.268511\pi$
$23$	6.37666	1.32963	0.664813	+	0.747010i	$0.268511\pi$
$24$	0	0
$25$	−4.02024	−0.804049
$26$	0	0
$27$	−2.87285	−0.552880
$28$	0	0
$29$	−1.07344	−0.199334	−0.0996668	−	0.995021i	$-0.531778\pi$
$29$	−1.07344	−0.199334	−0.0996668	+	0.995021i	$0.531778\pi$
$30$	0	0
$31$	0.593973	0.106681	0.0533403	−	0.998576i	$-0.483013\pi$
$31$	0.593973	0.106681	0.0533403	+	0.998576i	$0.483013\pi$
$32$	0	0
$33$	−1.60808	−0.279930
$34$	0	0
$35$	−0.873781	−0.147696
$36$	0	0
$37$	−2.48479	−0.408497	−0.204248	−	0.978919i	$-0.565475\pi$
$37$	−2.48479	−0.408497	−0.204248	+	0.978919i	$0.565475\pi$
$38$	0	0
$39$	1.31987	0.211348
$40$	0	0
$41$	−10.5308	−1.64464	−0.822319	−	0.569026i	$-0.807320\pi$
$41$	−10.5308	−1.64464	−0.822319	+	0.569026i	$0.807320\pi$
$42$	0	0
$43$	4.20046	0.640564	0.320282	−	0.947322i	$-0.396222\pi$
$43$	4.20046	0.640564	0.320282	+	0.947322i	$0.396222\pi$
$44$	0	0
$45$	−2.72243	−0.405836
$46$	0	0
$47$	7.98700	1.16502	0.582512	−	0.812822i	$-0.302070\pi$
$47$	7.98700	1.16502	0.582512	+	0.812822i	$0.302070\pi$
$48$	0	0
$49$	−6.22073	−0.888676
$50$	0	0
$51$	−0.346227	−0.0484815
$52$	0	0
$53$	−5.46222	−0.750293	−0.375146	−	0.926966i	$-0.622408\pi$
$53$	−5.46222	−0.750293	−0.375146	+	0.926966i	$0.622408\pi$
$54$	0	0
$55$	−3.18604	−0.429606
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−11.6719	−1.51955	−0.759776	−	0.650185i	$-0.774691\pi$
$59$	−11.6719	−1.51955	−0.759776	+	0.650185i	$0.774691\pi$
$60$	0	0
$61$	5.07612	0.649930	0.324965	−	0.945726i	$-0.394647\pi$
$61$	5.07612	0.649930	0.324965	+	0.945726i	$0.394647\pi$
$62$	0	0
$63$	2.42796	0.305894
$64$	0	0
$65$	2.61502	0.324354
$66$	0	0
$67$	−11.4597	−1.40002	−0.700011	−	0.714132i	$-0.746822\pi$
$67$	−11.4597	−1.40002	−0.700011	+	0.714132i	$0.746822\pi$
$68$	0	0
$69$	3.18572	0.383516
$70$	0	0
$71$	−3.26616	−0.387622	−0.193811	−	0.981039i	$-0.562085\pi$
$71$	−3.26616	−0.387622	−0.193811	+	0.981039i	$0.562085\pi$
$72$	0	0
$73$	−13.7627	−1.61080	−0.805402	−	0.592729i	$-0.798051\pi$
$73$	−13.7627	−1.61080	−0.805402	+	0.592729i	$0.798051\pi$
$74$	0	0
$75$	−2.00848	−0.231919
$76$	0	0
$77$	2.84142	0.323810
$78$	0	0
$79$	−10.9339	−1.23016	−0.615081	−	0.788464i	$-0.710877\pi$
$79$	−10.9339	−1.23016	−0.615081	+	0.788464i	$0.710877\pi$
$80$	0	0
$81$	6.81598	0.757331
$82$	0	0
$83$	−6.12042	−0.671803	−0.335902	−	0.941897i	$-0.609041\pi$
$83$	−6.12042	−0.671803	−0.335902	+	0.941897i	$0.609041\pi$
$84$	0	0
$85$	−0.685972	−0.0744041
$86$	0	0
$87$	−0.536283	−0.0574956
$88$	0	0
$89$	−3.65717	−0.387659	−0.193830	−	0.981035i	$-0.562091\pi$
$89$	−3.65717	−0.387659	−0.193830	+	0.981035i	$0.562091\pi$
$90$	0	0
$91$	−2.33217	−0.244478
$92$	0	0
$93$	0.296743	0.0307708
$94$	0	0
$95$	0	0
$96$	0	0
$97$	3.69133	0.374798	0.187399	−	0.982284i	$-0.439994\pi$
$97$	3.69133	0.374798	0.187399	+	0.982284i	$0.439994\pi$
$98$	0	0
$99$	8.85298	0.889758
$100$	0	0
$101$	−13.0529	−1.29881	−0.649405	−	0.760443i	$-0.724982\pi$
$101$	−13.0529	−1.29881	−0.649405	+	0.760443i	$0.724982\pi$
$102$	0	0
$103$	−14.2597	−1.40505	−0.702524	−	0.711660i	$-0.747944\pi$
$103$	−14.2597	−1.40505	−0.702524	+	0.711660i	$0.747944\pi$
$104$	0	0
$105$	−0.436533	−0.0426012
$106$	0	0
$107$	18.6351	1.80152	0.900762	−	0.434314i	$-0.143009\pi$
$107$	18.6351	1.80152	0.900762	+	0.434314i	$0.143009\pi$
$108$	0	0
$109$	−17.9340	−1.71777	−0.858884	−	0.512169i	$-0.828842\pi$
$109$	−17.9340	−1.71777	−0.858884	+	0.512169i	$0.828842\pi$
$110$	0	0
$111$	−1.24138	−0.117826
$112$	0	0
$113$	9.79875	0.921789	0.460894	−	0.887455i	$-0.347529\pi$
$113$	9.79875	0.921789	0.460894	+	0.887455i	$0.347529\pi$
$114$	0	0
$115$	6.31179	0.588577
$116$	0	0
$117$	−7.26631	−0.671770
$118$	0	0
$119$	0.611773	0.0560812
$120$	0	0
$121$	−0.639402	−0.0581274
$122$	0	0
$123$	−5.26110	−0.474378
$124$	0	0
$125$	−8.92848	−0.798587
$126$	0	0
$127$	21.6864	1.92436	0.962180	−	0.272416i	$-0.0878227\pi$
$127$	21.6864	1.92436	0.962180	+	0.272416i	$0.0878227\pi$
$128$	0	0
$129$	2.09851	0.184764
$130$	0	0
$131$	0.601785	0.0525782	0.0262891	−	0.999654i	$-0.491631\pi$
$131$	0.601785	0.0525782	0.0262891	+	0.999654i	$0.491631\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−2.84362	−0.244740
$136$	0	0
$137$	10.6068	0.906199	0.453099	−	0.891460i	$-0.350318\pi$
$137$	10.6068	0.906199	0.453099	+	0.891460i	$0.350318\pi$
$138$	0	0
$139$	10.3834	0.880706	0.440353	−	0.897825i	$-0.354853\pi$
$139$	10.3834	0.880706	0.440353	+	0.897825i	$0.354853\pi$
$140$	0	0
$141$	3.99023	0.336038
$142$	0	0
$143$	−8.50372	−0.711117
$144$	0	0
$145$	−1.06252	−0.0882378
$146$	0	0
$147$	−3.10782	−0.256329
$148$	0	0
$149$	−13.4673	−1.10328	−0.551641	−	0.834082i	$-0.685998\pi$
$149$	−13.4673	−1.10328	−0.551641	+	0.834082i	$0.685998\pi$
$150$	0	0
$151$	2.28923	0.186295	0.0931475	−	0.995652i	$-0.470307\pi$
$151$	2.28923	0.186295	0.0931475	+	0.995652i	$0.470307\pi$
$152$	0	0
$153$	1.90609	0.154099
$154$	0	0
$155$	0.587930	0.0472237
$156$	0	0
$157$	23.7201	1.89307	0.946536	−	0.322597i	$-0.104556\pi$
$157$	23.7201	1.89307	0.946536	+	0.322597i	$0.104556\pi$
$158$	0	0
$159$	−2.72887	−0.216414
$160$	0	0
$161$	−5.62907	−0.443633
$162$	0	0
$163$	−11.8315	−0.926715	−0.463357	−	0.886172i	$-0.653355\pi$
$163$	−11.8315	−0.926715	−0.463357	+	0.886172i	$0.653355\pi$
$164$	0	0
$165$	−1.59172	−0.123915
$166$	0	0
$167$	0.867993	0.0671673	0.0335837	−	0.999436i	$-0.489308\pi$
$167$	0.867993	0.0671673	0.0335837	+	0.999436i	$0.489308\pi$
$168$	0	0
$169$	−6.02036	−0.463105
$170$	0	0
$171$	0	0
$172$	0	0
$173$	6.24610	0.474882	0.237441	−	0.971402i	$-0.423691\pi$
$173$	6.24610	0.474882	0.237441	+	0.971402i	$0.423691\pi$
$174$	0	0
$175$	3.54892	0.268273
$176$	0	0
$177$	−5.83117	−0.438298
$178$	0	0
$179$	−10.5079	−0.785399	−0.392699	−	0.919667i	$-0.628459\pi$
$179$	−10.5079	−0.785399	−0.392699	+	0.919667i	$0.628459\pi$
$180$	0	0
$181$	11.5138	0.855813	0.427906	−	0.903823i	$-0.359251\pi$
$181$	11.5138	0.855813	0.427906	+	0.903823i	$0.359251\pi$
$182$	0	0
$183$	2.53598	0.187465
$184$	0	0
$185$	−2.45951	−0.180827
$186$	0	0
$187$	2.23069	0.163124
$188$	0	0
$189$	2.53604	0.184470
$190$	0	0
$191$	23.8188	1.72347	0.861735	−	0.507358i	$-0.169378\pi$
$191$	23.8188	1.72347	0.861735	+	0.507358i	$0.169378\pi$
$192$	0	0
$193$	−14.2038	−1.02241	−0.511205	−	0.859459i	$-0.670801\pi$
$193$	−14.2038	−1.02241	−0.511205	+	0.859459i	$0.670801\pi$
$194$	0	0
$195$	1.30644	0.0935562
$196$	0	0
$197$	3.57729	0.254871	0.127436	−	0.991847i	$-0.459325\pi$
$197$	3.57729	0.254871	0.127436	+	0.991847i	$0.459325\pi$
$198$	0	0
$199$	21.2865	1.50896	0.754478	−	0.656325i	$-0.227890\pi$
$199$	21.2865	1.50896	0.754478	+	0.656325i	$0.227890\pi$
$200$	0	0
$201$	−5.72515	−0.403821
$202$	0	0
$203$	0.947596	0.0665082
$204$	0	0
$205$	−10.4237	−0.728022
$206$	0	0
$207$	−17.5384	−1.21900
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−14.2207	−0.978995	−0.489497	−	0.872005i	$-0.662820\pi$
$211$	−14.2207	−0.978995	−0.489497	+	0.872005i	$0.662820\pi$
$212$	0	0
$213$	−1.63174	−0.111805
$214$	0	0
$215$	4.15773	0.283555
$216$	0	0
$217$	−0.524336	−0.0355943
$218$	0	0
$219$	−6.87573	−0.464619
$220$	0	0
$221$	−1.83090	−0.123159
$222$	0	0
$223$	−11.7277	−0.785345	−0.392672	−	0.919678i	$-0.628449\pi$
$223$	−11.7277	−0.785345	−0.392672	+	0.919678i	$0.628449\pi$
$224$	0	0
$225$	11.0573	0.737154
$226$	0	0
$227$	−3.97943	−0.264124	−0.132062	−	0.991241i	$-0.542160\pi$
$227$	−3.97943	−0.264124	−0.132062	+	0.991241i	$0.542160\pi$
$228$	0	0
$229$	−15.4369	−1.02010	−0.510049	−	0.860145i	$-0.670373\pi$
$229$	−15.4369	−1.02010	−0.510049	+	0.860145i	$0.670373\pi$
$230$	0	0
$231$	1.41955	0.0933995
$232$	0	0
$233$	8.31830	0.544950	0.272475	−	0.962163i	$-0.412158\pi$
$233$	8.31830	0.544950	0.272475	+	0.962163i	$0.412158\pi$
$234$	0	0
$235$	7.90574	0.515714
$236$	0	0
$237$	−5.46248	−0.354826
$238$	0	0
$239$	8.24432	0.533281	0.266640	−	0.963796i	$-0.414086\pi$
$239$	8.24432	0.533281	0.266640	+	0.963796i	$0.414086\pi$
$240$	0	0
$241$	−6.88397	−0.443436	−0.221718	−	0.975111i	$-0.571166\pi$
$241$	−6.88397	−0.443436	−0.221718	+	0.975111i	$0.571166\pi$
$242$	0	0
$243$	12.0238	0.771324
$244$	0	0
$245$	−6.15744	−0.393385
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−3.05771	−0.193774
$250$	0	0
$251$	20.4224	1.28905	0.644524	−	0.764584i	$-0.277056\pi$
$251$	20.4224	1.28905	0.644524	+	0.764584i	$0.277056\pi$
$252$	0	0
$253$	−20.5251	−1.29040
$254$	0	0
$255$	−0.342705	−0.0214610
$256$	0	0
$257$	2.21337	0.138066	0.0690330	−	0.997614i	$-0.478009\pi$
$257$	2.21337	0.138066	0.0690330	+	0.997614i	$0.478009\pi$
$258$	0	0
$259$	2.19348	0.136296
$260$	0	0
$261$	2.95241	0.182750
$262$	0	0
$263$	0.844903	0.0520990	0.0260495	−	0.999661i	$-0.491707\pi$
$263$	0.844903	0.0520990	0.0260495	+	0.999661i	$0.491707\pi$
$264$	0	0
$265$	−5.40664	−0.332128
$266$	0	0
$267$	−1.82709	−0.111816
$268$	0	0
$269$	−26.1998	−1.59743	−0.798714	−	0.601710i	$-0.794486\pi$
$269$	−26.1998	−1.59743	−0.798714	+	0.601710i	$0.794486\pi$
$270$	0	0
$271$	−22.2569	−1.35201	−0.676005	−	0.736897i	$-0.736290\pi$
$271$	−22.2569	−1.35201	−0.676005	+	0.736897i	$0.736290\pi$
$272$	0	0
$273$	−1.16513	−0.0705169
$274$	0	0
$275$	12.9403	0.780330
$276$	0	0
$277$	−0.665287	−0.0399732	−0.0199866	−	0.999800i	$-0.506362\pi$
$277$	−0.665287	−0.0399732	−0.0199866	+	0.999800i	$0.506362\pi$
$278$	0	0
$279$	−1.63367	−0.0978051
$280$	0	0
$281$	−2.82273	−0.168390	−0.0841950	−	0.996449i	$-0.526832\pi$
$281$	−2.82273	−0.168390	−0.0841950	+	0.996449i	$0.526832\pi$
$282$	0	0
$283$	22.3097	1.32617	0.663086	−	0.748543i	$-0.269246\pi$
$283$	22.3097	1.32617	0.663086	+	0.748543i	$0.269246\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	9.29621	0.548738
$288$	0	0
$289$	−16.5197	−0.971748
$290$	0	0
$291$	1.84415	0.108106
$292$	0	0
$293$	4.10021	0.239537	0.119768	−	0.992802i	$-0.461785\pi$
$293$	4.10021	0.239537	0.119768	+	0.992802i	$0.461785\pi$
$294$	0	0
$295$	−11.5532	−0.672651
$296$	0	0
$297$	9.24710	0.536571
$298$	0	0
$299$	16.8465	0.974258
$300$	0	0
$301$	−3.70801	−0.213726
$302$	0	0
$303$	−6.52110	−0.374627
$304$	0	0
$305$	5.02447	0.287701
$306$	0	0
$307$	17.7329	1.01207	0.506034	−	0.862513i	$-0.331111\pi$
$307$	17.7329	1.01207	0.506034	+	0.862513i	$0.331111\pi$
$308$	0	0
$309$	−7.12401	−0.405271
$310$	0	0
$311$	−29.0910	−1.64960	−0.824800	−	0.565424i	$-0.808712\pi$
$311$	−29.0910	−1.64960	−0.824800	+	0.565424i	$0.808712\pi$
$312$	0	0
$313$	−5.96065	−0.336916	−0.168458	−	0.985709i	$-0.553879\pi$
$313$	−5.96065	−0.336916	−0.168458	+	0.985709i	$0.553879\pi$
$314$	0	0
$315$	2.40325	0.135408
$316$	0	0
$317$	−16.3720	−0.919543	−0.459771	−	0.888037i	$-0.652069\pi$
$317$	−16.3720	−0.919543	−0.459771	+	0.888037i	$0.652069\pi$
$318$	0	0
$319$	3.45519	0.193454
$320$	0	0
$321$	9.30993	0.519629
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−10.6211	−0.589152
$326$	0	0
$327$	−8.95968	−0.495471
$328$	0	0
$329$	−7.05062	−0.388713
$330$	0	0
$331$	−24.7478	−1.36026	−0.680132	−	0.733090i	$-0.738078\pi$
$331$	−24.7478	−1.36026	−0.680132	+	0.733090i	$0.738078\pi$
$332$	0	0
$333$	6.83419	0.374511
$334$	0	0
$335$	−11.3431	−0.619739
$336$	0	0
$337$	−4.71127	−0.256639	−0.128320	−	0.991733i	$-0.540958\pi$
$337$	−4.71127	−0.256639	−0.128320	+	0.991733i	$0.540958\pi$
$338$	0	0
$339$	4.89536	0.265880
$340$	0	0
$341$	−1.91187	−0.103534
$342$	0	0
$343$	11.6708	0.630162
$344$	0	0
$345$	3.15331	0.169769
$346$	0	0
$347$	8.66690	0.465263	0.232632	−	0.972565i	$-0.425266\pi$
$347$	8.66690	0.465263	0.232632	+	0.972565i	$0.425266\pi$
$348$	0	0
$349$	19.6105	1.04972	0.524862	−	0.851188i	$-0.324117\pi$
$349$	19.6105	1.04972	0.524862	+	0.851188i	$0.324117\pi$
$350$	0	0
$351$	−7.58979	−0.405113
$352$	0	0
$353$	15.6438	0.832637	0.416318	−	0.909219i	$-0.363320\pi$
$353$	15.6438	0.832637	0.416318	+	0.909219i	$0.363320\pi$
$354$	0	0
$355$	−3.23293	−0.171586
$356$	0	0
$357$	0.305636	0.0161760
$358$	0	0
$359$	−11.6995	−0.617478	−0.308739	−	0.951147i	$-0.599907\pi$
$359$	−11.6995	−0.617478	−0.308739	+	0.951147i	$0.599907\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−0.319439	−0.0167662
$364$	0	0
$365$	−13.6227	−0.713045
$366$	0	0
$367$	13.3540	0.697072	0.348536	−	0.937295i	$-0.386679\pi$
$367$	13.3540	0.697072	0.348536	+	0.937295i	$0.386679\pi$
$368$	0	0
$369$	28.9641	1.50781
$370$	0	0
$371$	4.82183	0.250337
$372$	0	0
$373$	−18.2292	−0.943870	−0.471935	−	0.881633i	$-0.656444\pi$
$373$	−18.2292	−0.943870	−0.471935	+	0.881633i	$0.656444\pi$
$374$	0	0
$375$	−4.46058	−0.230344
$376$	0	0
$377$	−2.83593	−0.146058
$378$	0	0
$379$	4.52973	0.232677	0.116338	−	0.993210i	$-0.462884\pi$
$379$	4.52973	0.232677	0.116338	+	0.993210i	$0.462884\pi$
$380$	0	0
$381$	10.8343	0.555060
$382$	0	0
$383$	20.9663	1.07133	0.535664	−	0.844431i	$-0.320061\pi$
$383$	20.9663	1.07133	0.535664	+	0.844431i	$0.320061\pi$
$384$	0	0
$385$	2.81252	0.143339
$386$	0	0
$387$	−11.5530	−0.587271
$388$	0	0
$389$	−19.9675	−1.01239	−0.506196	−	0.862418i	$-0.668949\pi$
$389$	−19.9675	−1.01239	−0.506196	+	0.862418i	$0.668949\pi$
$390$	0	0
$391$	−4.41917	−0.223487
$392$	0	0
$393$	0.300646	0.0151656
$394$	0	0
$395$	−10.8227	−0.544548
$396$	0	0
$397$	23.6948	1.18921	0.594603	−	0.804019i	$-0.297309\pi$
$397$	23.6948	1.18921	0.594603	+	0.804019i	$0.297309\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	13.8269	0.690484	0.345242	−	0.938514i	$-0.387797\pi$
$401$	13.8269	0.690484	0.345242	+	0.938514i	$0.387797\pi$
$402$	0	0
$403$	1.56922	0.0781683
$404$	0	0
$405$	6.74663	0.335243
$406$	0	0
$407$	7.99801	0.396447
$408$	0	0
$409$	17.1305	0.847048	0.423524	−	0.905885i	$-0.360793\pi$
$409$	17.1305	0.847048	0.423524	+	0.905885i	$0.360793\pi$
$410$	0	0
$411$	5.29905	0.261383
$412$	0	0
$413$	10.3035	0.507003
$414$	0	0
$415$	−6.05815	−0.297383
$416$	0	0
$417$	5.18744	0.254030
$418$	0	0
$419$	−31.1939	−1.52392	−0.761960	−	0.647624i	$-0.775763\pi$
$419$	−31.1939	−1.52392	−0.761960	+	0.647624i	$0.775763\pi$
$420$	0	0
$421$	27.2220	1.32672	0.663360	−	0.748301i	$-0.269130\pi$
$421$	27.2220	1.32672	0.663360	+	0.748301i	$0.269130\pi$
$422$	0	0
$423$	−21.9675	−1.06810
$424$	0	0
$425$	2.78612	0.135147
$426$	0	0
$427$	−4.48100	−0.216851
$428$	0	0
$429$	−4.24838	−0.205114
$430$	0	0
$431$	18.0882	0.871278	0.435639	−	0.900122i	$-0.356523\pi$
$431$	18.0882	0.871278	0.435639	+	0.900122i	$0.356523\pi$
$432$	0	0
$433$	−11.4672	−0.551077	−0.275538	−	0.961290i	$-0.588856\pi$
$433$	−11.4672	−0.551077	−0.275538	+	0.961290i	$0.588856\pi$
$434$	0	0
$435$	−0.530827	−0.0254512
$436$	0	0
$437$	0	0
$438$	0	0
$439$	12.6861	0.605475	0.302738	−	0.953074i	$-0.402099\pi$
$439$	12.6861	0.605475	0.302738	+	0.953074i	$0.402099\pi$
$440$	0	0
$441$	17.1096	0.814741
$442$	0	0
$443$	22.3815	1.06338	0.531690	−	0.846939i	$-0.321557\pi$
$443$	22.3815	1.06338	0.531690	+	0.846939i	$0.321557\pi$
$444$	0	0
$445$	−3.61996	−0.171603
$446$	0	0
$447$	−6.72812	−0.318229
$448$	0	0
$449$	−9.61944	−0.453969	−0.226985	−	0.973898i	$-0.572887\pi$
$449$	−9.61944	−0.453969	−0.226985	+	0.973898i	$0.572887\pi$
$450$	0	0
$451$	33.8965	1.59612
$452$	0	0
$453$	1.14368	0.0537347
$454$	0	0
$455$	−2.30844	−0.108221
$456$	0	0
$457$	−33.8674	−1.58425	−0.792126	−	0.610358i	$-0.791026\pi$
$457$	−33.8674	−1.58425	−0.792126	+	0.610358i	$0.791026\pi$
$458$	0	0
$459$	1.99095	0.0929296
$460$	0	0
$461$	−25.7902	−1.20117	−0.600585	−	0.799561i	$-0.705066\pi$
$461$	−25.7902	−1.20117	−0.600585	+	0.799561i	$0.705066\pi$
$462$	0	0
$463$	23.0244	1.07003	0.535017	−	0.844841i	$-0.320305\pi$
$463$	23.0244	1.07003	0.535017	+	0.844841i	$0.320305\pi$
$464$	0	0
$465$	0.293724	0.0136211
$466$	0	0
$467$	35.7248	1.65315	0.826574	−	0.562829i	$-0.190287\pi$
$467$	35.7248	1.65315	0.826574	+	0.562829i	$0.190287\pi$
$468$	0	0
$469$	10.1162	0.467121
$470$	0	0
$471$	11.8504	0.546036
$472$	0	0
$473$	−13.5204	−0.621668
$474$	0	0
$475$	0	0
$476$	0	0
$477$	15.0233	0.687871
$478$	0	0
$479$	−20.7079	−0.946167	−0.473084	−	0.881018i	$-0.656859\pi$
$479$	−20.7079	−0.946167	−0.473084	+	0.881018i	$0.656859\pi$
$480$	0	0
$481$	−6.56457	−0.299318
$482$	0	0
$483$	−2.81223	−0.127961
$484$	0	0
$485$	3.65377	0.165909
$486$	0	0
$487$	−25.5955	−1.15984	−0.579922	−	0.814672i	$-0.696917\pi$
$487$	−25.5955	−1.15984	−0.579922	+	0.814672i	$0.696917\pi$
$488$	0	0
$489$	−5.91091	−0.267301
$490$	0	0
$491$	37.5910	1.69646	0.848229	−	0.529629i	$-0.177669\pi$
$491$	37.5910	1.69646	0.848229	+	0.529629i	$0.177669\pi$
$492$	0	0
$493$	0.743921	0.0335045
$494$	0	0
$495$	8.76292	0.393864
$496$	0	0
$497$	2.88324	0.129331
$498$	0	0
$499$	18.5319	0.829601	0.414801	−	0.909912i	$-0.363851\pi$
$499$	18.5319	0.829601	0.414801	+	0.909912i	$0.363851\pi$
$500$	0	0
$501$	0.433641	0.0193737
$502$	0	0
$503$	−2.98629	−0.133152	−0.0665760	−	0.997781i	$-0.521207\pi$
$503$	−2.98629	−0.133152	−0.0665760	+	0.997781i	$0.521207\pi$
$504$	0	0
$505$	−12.9201	−0.574936
$506$	0	0
$507$	−3.00772	−0.133577
$508$	0	0
$509$	37.1393	1.64617	0.823085	−	0.567917i	$-0.192251\pi$
$509$	37.1393	1.64617	0.823085	+	0.567917i	$0.192251\pi$
$510$	0	0
$511$	12.1492	0.537449
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−14.1146	−0.621964
$516$	0	0
$517$	−25.7085	−1.13066
$518$	0	0
$519$	3.12049	0.136974
$520$	0	0
$521$	37.3318	1.63554	0.817769	−	0.575547i	$-0.195211\pi$
$521$	37.3318	1.63554	0.817769	+	0.575547i	$0.195211\pi$
$522$	0	0
$523$	29.4412	1.28737	0.643687	−	0.765289i	$-0.277404\pi$
$523$	29.4412	1.28737	0.643687	+	0.765289i	$0.277404\pi$
$524$	0	0
$525$	1.77301	0.0773803
$526$	0	0
$527$	−0.411636	−0.0179312
$528$	0	0
$529$	17.6618	0.767904
$530$	0	0
$531$	32.1025	1.39313
$532$	0	0
$533$	−27.8214	−1.20508
$534$	0	0
$535$	18.4455	0.797469
$536$	0	0
$537$	−5.24966	−0.226539
$538$	0	0
$539$	20.0232	0.862461
$540$	0	0
$541$	30.2923	1.30237	0.651184	−	0.758920i	$-0.274273\pi$
$541$	30.2923	1.30237	0.651184	+	0.758920i	$0.274273\pi$
$542$	0	0
$543$	5.75218	0.246850
$544$	0	0
$545$	−17.7516	−0.760394
$546$	0	0
$547$	−8.04224	−0.343861	−0.171931	−	0.985109i	$-0.555000\pi$
$547$	−8.04224	−0.343861	−0.171931	+	0.985109i	$0.555000\pi$
$548$	0	0
$549$	−13.9614	−0.595858
$550$	0	0
$551$	0	0
$552$	0	0
$553$	9.65204	0.410446
$554$	0	0
$555$	−1.22875	−0.0521575
$556$	0	0
$557$	35.7478	1.51468	0.757342	−	0.653019i	$-0.226498\pi$
$557$	35.7478	1.51468	0.757342	+	0.653019i	$0.226498\pi$
$558$	0	0
$559$	11.0972	0.469362
$560$	0	0
$561$	1.11443	0.0470514
$562$	0	0
$563$	14.3246	0.603710	0.301855	−	0.953354i	$-0.402394\pi$
$563$	14.3246	0.603710	0.301855	+	0.953354i	$0.402394\pi$
$564$	0	0
$565$	9.69906	0.408043
$566$	0	0
$567$	−6.01688	−0.252685
$568$	0	0
$569$	−22.1503	−0.928588	−0.464294	−	0.885681i	$-0.653692\pi$
$569$	−22.1503	−0.928588	−0.464294	+	0.885681i	$0.653692\pi$
$570$	0	0
$571$	−19.1461	−0.801240	−0.400620	−	0.916244i	$-0.631205\pi$
$571$	−19.1461	−0.801240	−0.400620	+	0.916244i	$0.631205\pi$
$572$	0	0
$573$	11.8997	0.497116
$574$	0	0
$575$	−25.6357	−1.06908
$576$	0	0
$577$	3.49726	0.145593	0.0727965	−	0.997347i	$-0.476808\pi$
$577$	3.49726	0.145593	0.0727965	+	0.997347i	$0.476808\pi$
$578$	0	0
$579$	−7.09608	−0.294903
$580$	0	0
$581$	5.40287	0.224149
$582$	0	0
$583$	17.5817	0.728160
$584$	0	0
$585$	−7.19238	−0.297368
$586$	0	0
$587$	7.24141	0.298885	0.149443	−	0.988770i	$-0.452252\pi$
$587$	7.24141	0.298885	0.149443	+	0.988770i	$0.452252\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	1.78718	0.0735148
$592$	0	0
$593$	−32.0408	−1.31576	−0.657880	−	0.753123i	$-0.728546\pi$
$593$	−32.0408	−1.31576	−0.657880	+	0.753123i	$0.728546\pi$
$594$	0	0
$595$	0.605549	0.0248251
$596$	0	0
$597$	10.6345	0.435242
$598$	0	0
$599$	−17.2861	−0.706292	−0.353146	−	0.935568i	$-0.614888\pi$
$599$	−17.2861	−0.706292	−0.353146	+	0.935568i	$0.614888\pi$
$600$	0	0
$601$	−16.9400	−0.690996	−0.345498	−	0.938419i	$-0.612290\pi$
$601$	−16.9400	−0.690996	−0.345498	+	0.938419i	$0.612290\pi$
$602$	0	0
$603$	31.5188	1.28354
$604$	0	0
$605$	−0.632897	−0.0257309
$606$	0	0
$607$	−34.3961	−1.39610	−0.698048	−	0.716051i	$-0.745948\pi$
$607$	−34.3961	−1.39610	−0.698048	+	0.716051i	$0.745948\pi$
$608$	0	0
$609$	0.473410	0.0191835
$610$	0	0
$611$	21.1009	0.853649
$612$	0	0
$613$	−6.63739	−0.268082	−0.134041	−	0.990976i	$-0.542795\pi$
$613$	−6.63739	−0.268082	−0.134041	+	0.990976i	$0.542795\pi$
$614$	0	0
$615$	−5.20758	−0.209990
$616$	0	0
$617$	−10.7821	−0.434073	−0.217036	−	0.976164i	$-0.569639\pi$
$617$	−10.7821	−0.434073	−0.217036	+	0.976164i	$0.569639\pi$
$618$	0	0
$619$	−2.15049	−0.0864354	−0.0432177	−	0.999066i	$-0.513761\pi$
$619$	−2.15049	−0.0864354	−0.0432177	+	0.999066i	$0.513761\pi$
$620$	0	0
$621$	−18.3192	−0.735124
$622$	0	0
$623$	3.22841	0.129343
$624$	0	0
$625$	11.2636	0.450543
$626$	0	0
$627$	0	0
$628$	0	0
$629$	1.72201	0.0686612
$630$	0	0
$631$	−43.8470	−1.74552	−0.872761	−	0.488148i	$-0.837673\pi$
$631$	−43.8470	−1.74552	−0.872761	+	0.488148i	$0.837673\pi$
$632$	0	0
$633$	−7.10454	−0.282380
$634$	0	0
$635$	21.4658	0.851844
$636$	0	0
$637$	−16.4346	−0.651161
$638$	0	0
$639$	8.98327	0.355373
$640$	0	0
$641$	17.8267	0.704112	0.352056	−	0.935979i	$-0.385483\pi$
$641$	17.8267	0.704112	0.352056	+	0.935979i	$0.385483\pi$
$642$	0	0
$643$	−3.07190	−0.121144	−0.0605720	−	0.998164i	$-0.519292\pi$
$643$	−3.07190	−0.121144	−0.0605720	+	0.998164i	$0.519292\pi$
$644$	0	0
$645$	2.07716	0.0817882
$646$	0	0
$647$	−4.63532	−0.182233	−0.0911166	−	0.995840i	$-0.529044\pi$
$647$	−4.63532	−0.182233	−0.0911166	+	0.995840i	$0.529044\pi$
$648$	0	0
$649$	37.5694	1.47473
$650$	0	0
$651$	−0.261954	−0.0102668
$652$	0	0
$653$	0.443870	0.0173700	0.00868498	−	0.999962i	$-0.497235\pi$
$653$	0.443870	0.0173700	0.00868498	+	0.999962i	$0.497235\pi$
$654$	0	0
$655$	0.595663	0.0232745
$656$	0	0
$657$	37.8531	1.47679
$658$	0	0
$659$	−48.4021	−1.88548	−0.942739	−	0.333531i	$-0.891760\pi$
$659$	−48.4021	−1.88548	−0.942739	+	0.333531i	$0.891760\pi$
$660$	0	0
$661$	−13.8761	−0.539717	−0.269858	−	0.962900i	$-0.586977\pi$
$661$	−13.8761	−0.539717	−0.269858	+	0.962900i	$0.586977\pi$
$662$	0	0
$663$	−0.914698	−0.0355239
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−6.84499	−0.265039
$668$	0	0
$669$	−5.85905	−0.226524
$670$	0	0
$671$	−16.3389	−0.630758
$672$	0	0
$673$	−34.6291	−1.33485	−0.667426	−	0.744676i	$-0.732604\pi$
$673$	−34.6291	−1.33485	−0.667426	+	0.744676i	$0.732604\pi$
$674$	0	0
$675$	11.5496	0.444543
$676$	0	0
$677$	−13.6416	−0.524290	−0.262145	−	0.965028i	$-0.584430\pi$
$677$	−13.6416	−0.524290	−0.262145	+	0.965028i	$0.584430\pi$
$678$	0	0
$679$	−3.25856	−0.125052
$680$	0	0
$681$	−1.98809	−0.0761837
$682$	0	0
$683$	−1.11613	−0.0427076	−0.0213538	−	0.999772i	$-0.506798\pi$
$683$	−1.11613	−0.0427076	−0.0213538	+	0.999772i	$0.506798\pi$
$684$	0	0
$685$	10.4989	0.401141
$686$	0	0
$687$	−7.71212	−0.294236
$688$	0	0
$689$	−14.4306	−0.549763
$690$	0	0
$691$	42.7501	1.62629	0.813145	−	0.582061i	$-0.197754\pi$
$691$	42.7501	1.62629	0.813145	+	0.582061i	$0.197754\pi$
$692$	0	0
$693$	−7.81508	−0.296870
$694$	0	0
$695$	10.2777	0.389857
$696$	0	0
$697$	7.29810	0.276435
$698$	0	0
$699$	4.15575	0.157185
$700$	0	0
$701$	−4.44462	−0.167871	−0.0839355	−	0.996471i	$-0.526749\pi$
$701$	−4.44462	−0.167871	−0.0839355	+	0.996471i	$0.526749\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	3.94963	0.148752
$706$	0	0
$707$	11.5226	0.433351
$708$	0	0
$709$	−17.3695	−0.652324	−0.326162	−	0.945314i	$-0.605755\pi$
$709$	−17.3695	−0.652324	−0.326162	+	0.945314i	$0.605755\pi$
$710$	0	0
$711$	30.0727	1.12782
$712$	0	0
$713$	3.78756	0.141845
$714$	0	0
$715$	−8.41721	−0.314786
$716$	0	0
$717$	4.11879	0.153819
$718$	0	0
$719$	17.8951	0.667375	0.333688	−	0.942684i	$-0.391707\pi$
$719$	17.8951	0.667375	0.333688	+	0.942684i	$0.391707\pi$
$720$	0	0
$721$	12.5879	0.468798
$722$	0	0
$723$	−3.43917	−0.127904
$724$	0	0
$725$	4.31551	0.160274
$726$	0	0
$727$	−49.5634	−1.83820	−0.919102	−	0.394020i	$-0.871084\pi$
$727$	−49.5634	−1.83820	−0.919102	+	0.394020i	$0.871084\pi$
$728$	0	0
$729$	−14.4410	−0.534851
$730$	0	0
$731$	−2.91101	−0.107668
$732$	0	0
$733$	20.3740	0.752533	0.376266	−	0.926512i	$-0.377208\pi$
$733$	20.3740	0.752533	0.376266	+	0.926512i	$0.377208\pi$
$734$	0	0
$735$	−3.07620	−0.113467
$736$	0	0
$737$	36.8863	1.35872
$738$	0	0
$739$	45.5912	1.67710	0.838550	−	0.544824i	$-0.183404\pi$
$739$	45.5912	1.67710	0.838550	+	0.544824i	$0.183404\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	52.0700	1.91026	0.955132	−	0.296181i	$-0.0957132\pi$
$743$	52.0700	1.91026	0.955132	+	0.296181i	$0.0957132\pi$
$744$	0	0
$745$	−13.3303	−0.488383
$746$	0	0
$747$	16.8337	0.615911
$748$	0	0
$749$	−16.4504	−0.601083
$750$	0	0
$751$	22.8502	0.833817	0.416909	−	0.908948i	$-0.363114\pi$
$751$	22.8502	0.833817	0.416909	+	0.908948i	$0.363114\pi$
$752$	0	0
$753$	10.2028	0.371811
$754$	0	0
$755$	2.26594	0.0824661
$756$	0	0
$757$	19.2563	0.699883	0.349941	−	0.936772i	$-0.386202\pi$
$757$	19.2563	0.699883	0.349941	+	0.936772i	$0.386202\pi$
$758$	0	0
$759$	−10.2542	−0.372202
$760$	0	0
$761$	5.88049	0.213167	0.106584	−	0.994304i	$-0.466009\pi$
$761$	5.88049	0.213167	0.106584	+	0.994304i	$0.466009\pi$
$762$	0	0
$763$	15.8315	0.573138
$764$	0	0
$765$	1.88670	0.0682139
$766$	0	0
$767$	−30.8360	−1.11342
$768$	0	0
$769$	13.5518	0.488689	0.244344	−	0.969689i	$-0.421427\pi$
$769$	13.5518	0.488689	0.244344	+	0.969689i	$0.421427\pi$
$770$	0	0
$771$	1.10578	0.0398236
$772$	0	0
$773$	20.6977	0.744445	0.372223	−	0.928144i	$-0.378596\pi$
$773$	20.6977	0.744445	0.372223	+	0.928144i	$0.378596\pi$
$774$	0	0
$775$	−2.38792	−0.0857764
$776$	0	0
$777$	1.09584	0.0393131
$778$	0	0
$779$	0	0
$780$	0	0
$781$	10.5131	0.376187
$782$	0	0
$783$	3.08385	0.110208
$784$	0	0
$785$	23.4788	0.837995
$786$	0	0
$787$	−41.2692	−1.47109	−0.735544	−	0.677477i	$-0.763073\pi$
$787$	−41.2692	−1.47109	−0.735544	+	0.677477i	$0.763073\pi$
$788$	0	0
$789$	0.422106	0.0150274
$790$	0	0
$791$	−8.64996	−0.307557
$792$	0	0
$793$	13.4106	0.476224
$794$	0	0
$795$	−2.70111	−0.0957985
$796$	0	0
$797$	21.0532	0.745743	0.372871	−	0.927883i	$-0.378373\pi$
$797$	21.0532	0.745743	0.372871	+	0.927883i	$0.378373\pi$
$798$	0	0
$799$	−5.53517	−0.195820
$800$	0	0
$801$	10.0587	0.355407
$802$	0	0
$803$	44.2993	1.56329
$804$	0	0
$805$	−5.57180	−0.196380
$806$	0	0
$807$	−13.0892	−0.460761
$808$	0	0
$809$	22.4072	0.787794	0.393897	−	0.919155i	$-0.371127\pi$
$809$	22.4072	0.787794	0.393897	+	0.919155i	$0.371127\pi$
$810$	0	0
$811$	−32.5040	−1.14137	−0.570686	−	0.821169i	$-0.693322\pi$
$811$	−32.5040	−1.14137	−0.570686	+	0.821169i	$0.693322\pi$
$812$	0	0
$813$	−11.1193	−0.389972
$814$	0	0
$815$	−11.7111	−0.410223
$816$	0	0
$817$	0	0
$818$	0	0
$819$	6.41442	0.224138
$820$	0	0
$821$	18.2949	0.638495	0.319247	−	0.947671i	$-0.396570\pi$
$821$	18.2949	0.638495	0.319247	+	0.947671i	$0.396570\pi$
$822$	0	0
$823$	−23.1058	−0.805418	−0.402709	−	0.915328i	$-0.631931\pi$
$823$	−23.1058	−0.805418	−0.402709	+	0.915328i	$0.631931\pi$
$824$	0	0
$825$	6.46486	0.225078
$826$	0	0
$827$	27.9119	0.970591	0.485295	−	0.874350i	$-0.338712\pi$
$827$	27.9119	0.970591	0.485295	+	0.874350i	$0.338712\pi$
$828$	0	0
$829$	−32.7123	−1.13615	−0.568073	−	0.822978i	$-0.692311\pi$
$829$	−32.7123	−1.13615	−0.568073	+	0.822978i	$0.692311\pi$
$830$	0	0
$831$	−0.332371	−0.0115298
$832$	0	0
$833$	4.31110	0.149371
$834$	0	0
$835$	0.859162	0.0297325
$836$	0	0
$837$	−1.70640	−0.0589817
$838$	0	0
$839$	−8.04961	−0.277903	−0.138952	−	0.990299i	$-0.544373\pi$
$839$	−8.04961	−0.277903	−0.138952	+	0.990299i	$0.544373\pi$
$840$	0	0
$841$	−27.8477	−0.960266
$842$	0	0
$843$	−1.41021	−0.0485702
$844$	0	0
$845$	−5.95911	−0.205000
$846$	0	0
$847$	0.564439	0.0193944
$848$	0	0
$849$	11.1457	0.382519
$850$	0	0
$851$	−15.8447	−0.543148
$852$	0	0
$853$	35.5288	1.21648	0.608242	−	0.793751i	$-0.291875\pi$
$853$	35.5288	1.21648	0.608242	+	0.793751i	$0.291875\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−22.9326	−0.783363	−0.391681	−	0.920101i	$-0.628106\pi$
$857$	−22.9326	−0.783363	−0.391681	+	0.920101i	$0.628106\pi$
$858$	0	0
$859$	−25.5996	−0.873449	−0.436724	−	0.899595i	$-0.643861\pi$
$859$	−25.5996	−0.873449	−0.436724	+	0.899595i	$0.643861\pi$
$860$	0	0
$861$	4.64430	0.158277
$862$	0	0
$863$	48.1804	1.64008	0.820040	−	0.572306i	$-0.193951\pi$
$863$	48.1804	1.64008	0.820040	+	0.572306i	$0.193951\pi$
$864$	0	0
$865$	6.18255	0.210213
$866$	0	0
$867$	−8.25310	−0.280290
$868$	0	0
$869$	35.1939	1.19387
$870$	0	0
$871$	−30.2753	−1.02584
$872$	0	0
$873$	−10.1527	−0.343615
$874$	0	0
$875$	7.88172	0.266451
$876$	0	0
$877$	0.616156	0.0208061	0.0104031	−	0.999946i	$-0.496689\pi$
$877$	0.616156	0.0208061	0.0104031	+	0.999946i	$0.496689\pi$
$878$	0	0
$879$	2.04842	0.0690916
$880$	0	0
$881$	13.7345	0.462728	0.231364	−	0.972867i	$-0.425681\pi$
$881$	13.7345	0.462728	0.231364	+	0.972867i	$0.425681\pi$
$882$	0	0
$883$	−2.41478	−0.0812637	−0.0406318	−	0.999174i	$-0.512937\pi$
$883$	−2.41478	−0.0812637	−0.0406318	+	0.999174i	$0.512937\pi$
$884$	0	0
$885$	−5.77185	−0.194019
$886$	0	0
$887$	19.9979	0.671462	0.335731	−	0.941958i	$-0.391017\pi$
$887$	19.9979	0.671462	0.335731	+	0.941958i	$0.391017\pi$
$888$	0	0
$889$	−19.1439	−0.642067
$890$	0	0
$891$	−21.9392	−0.734990
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−10.4010	−0.347668
$896$	0	0
$897$	8.41636	0.281014
$898$	0	0
$899$	−0.637597	−0.0212650
$900$	0	0
$901$	3.78544	0.126111
$902$	0	0
$903$	−1.85249	−0.0616469
$904$	0	0
$905$	11.3966	0.378837
$906$	0	0
$907$	−33.9885	−1.12857	−0.564285	−	0.825580i	$-0.690848\pi$
$907$	−33.9885	−1.12857	−0.564285	+	0.825580i	$0.690848\pi$
$908$	0	0
$909$	35.9008	1.19075
$910$	0	0
$911$	32.5945	1.07990	0.539951	−	0.841696i	$-0.318443\pi$
$911$	32.5945	1.07990	0.539951	+	0.841696i	$0.318443\pi$
$912$	0	0
$913$	19.7003	0.651986
$914$	0	0
$915$	2.51018	0.0829840
$916$	0	0
$917$	−0.531233	−0.0175429
$918$	0	0
$919$	0.337746	0.0111412	0.00557061	−	0.999984i	$-0.498227\pi$
$919$	0.337746	0.0111412	0.00557061	+	0.999984i	$0.498227\pi$
$920$	0	0
$921$	8.85918	0.291920
$922$	0	0
$923$	−8.62886	−0.284023
$924$	0	0
$925$	9.98946	0.328451
$926$	0	0
$927$	39.2200	1.28815
$928$	0	0
$929$	−34.9377	−1.14627	−0.573134	−	0.819462i	$-0.694273\pi$
$929$	−34.9377	−1.14627	−0.573134	+	0.819462i	$0.694273\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−14.5336	−0.475809
$934$	0	0
$935$	2.20800	0.0722092
$936$	0	0
$937$	45.2899	1.47956	0.739778	−	0.672851i	$-0.234931\pi$
$937$	45.2899	1.47956	0.739778	+	0.672851i	$0.234931\pi$
$938$	0	0
$939$	−2.97788	−0.0971796
$940$	0	0
$941$	35.9536	1.17205	0.586026	−	0.810292i	$-0.300691\pi$
$941$	35.9536	1.17205	0.586026	+	0.810292i	$0.300691\pi$
$942$	0	0
$943$	−67.1515	−2.18675
$944$	0	0
$945$	2.51024	0.0816582
$946$	0	0
$947$	39.0109	1.26768	0.633841	−	0.773463i	$-0.281477\pi$
$947$	39.0109	1.26768	0.633841	+	0.773463i	$0.281477\pi$
$948$	0	0
$949$	−36.3597	−1.18029
$950$	0	0
$951$	−8.17929	−0.265232
$952$	0	0
$953$	8.65817	0.280466	0.140233	−	0.990119i	$-0.455215\pi$
$953$	8.65817	0.280466	0.140233	+	0.990119i	$0.455215\pi$
$954$	0	0
$955$	23.5765	0.762918
$956$	0	0
$957$	1.72618	0.0557995
$958$	0	0
$959$	−9.36326	−0.302356
$960$	0	0
$961$	−30.6472	−0.988619
$962$	0	0
$963$	−51.2542	−1.65164
$964$	0	0
$965$	−14.0593	−0.452584
$966$	0	0
$967$	−37.8385	−1.21680	−0.608402	−	0.793629i	$-0.708189\pi$
$967$	−37.8385	−1.21680	−0.608402	+	0.793629i	$0.708189\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	−11.3566	−0.364449	−0.182225	−	0.983257i	$-0.558330\pi$
$971$	−11.3566	−0.364449	−0.182225	+	0.983257i	$0.558330\pi$
$972$	0	0
$973$	−9.16604	−0.293850
$974$	0	0
$975$	−5.30620	−0.169934
$976$	0	0
$977$	−57.5899	−1.84247	−0.921233	−	0.389012i	$-0.872816\pi$
$977$	−57.5899	−1.84247	−0.921233	+	0.389012i	$0.872816\pi$
$978$	0	0
$979$	11.7717	0.376224
$980$	0	0
$981$	49.3259	1.57486
$982$	0	0
$983$	−42.3774	−1.35163	−0.675814	−	0.737072i	$-0.736208\pi$
$983$	−42.3774	−1.35163	−0.675814	+	0.737072i	$0.736208\pi$
$984$	0	0
$985$	3.54089	0.112822
$986$	0	0
$987$	−3.52242	−0.112120
$988$	0	0
$989$	26.7849	0.851711
$990$	0	0
$991$	−30.6302	−0.973002	−0.486501	−	0.873680i	$-0.661727\pi$
$991$	−30.6302	−0.973002	−0.486501	+	0.873680i	$0.661727\pi$
$992$	0	0
$993$	−12.3638	−0.392353
$994$	0	0
$995$	21.0699	0.667961
$996$	0	0
$997$	8.80188	0.278758	0.139379	−	0.990239i	$-0.455489\pi$
$997$	8.80188	0.278758	0.139379	+	0.990239i	$0.455489\pi$
$998$	0	0
$999$	7.13843	0.225850

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.v.1.6	✓	8
4.3	odd	2		5776.2.a.cc.1.3		8
19.18	odd	2		2888.2.a.w.1.3	yes	8
76.75	even	2		5776.2.a.ca.1.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.v.1.6	✓	8	1.1	even	1	trivial
2888.2.a.w.1.3	yes	8	19.18	odd	2
5776.2.a.ca.1.6		8	76.75	even	2
5776.2.a.cc.1.3		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)

\(f(q)\)	\(=\)	\(q+0.499591 q^{3} +0.989826 q^{5} -0.882762 q^{7} -2.75041 q^{9} -3.21879 q^{11} +2.64190 q^{13} +0.494508 q^{15} -0.693022 q^{17} -0.441020 q^{21} +6.37666 q^{23} -4.02024 q^{25} -2.87285 q^{27} -1.07344 q^{29} +0.593973 q^{31} -1.60808 q^{33} -0.873781 q^{35} -2.48479 q^{37} +1.31987 q^{39} -10.5308 q^{41} +4.20046 q^{43} -2.72243 q^{45} +7.98700 q^{47} -6.22073 q^{49} -0.346227 q^{51} -5.46222 q^{53} -3.18604 q^{55} -11.6719 q^{59} +5.07612 q^{61} +2.42796 q^{63} +2.61502 q^{65} -11.4597 q^{67} +3.18572 q^{69} -3.26616 q^{71} -13.7627 q^{73} -2.00848 q^{75} +2.84142 q^{77} -10.9339 q^{79} +6.81598 q^{81} -6.12042 q^{83} -0.685972 q^{85} -0.536283 q^{87} -3.65717 q^{89} -2.33217 q^{91} +0.296743 q^{93} +3.69133 q^{97} +8.85298 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 12 q^{11} - 6 q^{13} - 10 q^{17} + 4 q^{21} - 12 q^{23} + 2 q^{25} - 12 q^{27} - 18 q^{29} - 14 q^{31} - 40 q^{33} + 18 q^{35} - 16 q^{37} + 28 q^{39} - 12 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) \) 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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