LMFDB - Embedded newform 2888.2.a.m.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [2888,2,Mod(1,2888)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("2888.1"); S:= CuspForms(chi, 2); N := Newforms(S);

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

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

Newform invariants

comment:select newform

sage:traces = [3,0,-3,0,-3,0,-6,0,12,0,-6,0,0,0,3] 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:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.2
Root			$-0.347296$ 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)

$f(q)$	$=$	$q-2.18479 q^{3} +0.879385 q^{5} -1.65270 q^{7} +1.77332 q^{9} -3.53209 q^{11} +4.94356 q^{13} -1.92127 q^{15} +2.59627 q^{17} +3.61081 q^{21} -4.82295 q^{23} -4.22668 q^{25} +2.68004 q^{27} +6.87939 q^{29} -2.04189 q^{31} +7.71688 q^{33} -1.45336 q^{35} +5.18479 q^{37} -10.8007 q^{39} -1.73648 q^{41} +7.98545 q^{43} +1.55943 q^{45} +2.92127 q^{47} -4.26857 q^{49} -5.67230 q^{51} -12.6800 q^{53} -3.10607 q^{55} +7.58172 q^{59} -11.1284 q^{61} -2.93077 q^{63} +4.34730 q^{65} -16.0993 q^{67} +10.5371 q^{69} +13.5621 q^{71} +4.73917 q^{73} +9.23442 q^{75} +5.83750 q^{77} +1.21894 q^{79} -11.1753 q^{81} -5.87939 q^{83} +2.28312 q^{85} -15.0300 q^{87} +10.2909 q^{89} -8.17024 q^{91} +4.46110 q^{93} +11.4534 q^{97} -6.26352 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 3 q^{3} - 3 q^{5} - 6 q^{7} + 12 q^{9} - 6 q^{11} + 3 q^{15} - 6 q^{17} + 15 q^{21} + 6 q^{23} - 6 q^{25} - 12 q^{27} + 15 q^{29} - 3 q^{31} + 15 q^{33} + 9 q^{35} + 12 q^{37} - 18 q^{39} + 6 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) $ 3 * q - 3 * q^3 - 3 * q^5 - 6 * q^7 + 12 * q^9 - 6 * q^11 + 3 * q^15 - 6 * q^17 + 15 * q^21 + 6 * q^23 - 6 * q^25 - 12 * q^27 + 15 * q^29 - 3 * q^31 + 15 * q^33 + 9 * q^35 + 12 * q^37 - 18 * q^39 + 6 * q^43 - 21 * q^45 - 3 * q^49 - 21 * q^51 - 18 * q^53 + 3 * q^55 - 9 * q^59 - 15 * q^61 - 33 * q^63 + 12 * q^65 + 6 * q^67 + 12 * q^69 + 6 * q^71 - 3 * q^75 + 15 * q^77 + 21 * q^79 + 3 * q^81 - 12 * q^83 + 15 * q^85 - 15 * q^87 + 15 * q^89 - 3 * q^91 - 24 * q^93 + 21 * q^97 - 24 * 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
$3$	−2.18479	−1.26139	−0.630695	−	0.776031i	$-0.717230\pi$
$3$	−2.18479	−1.26139	−0.630695	+	0.776031i	$0.717230\pi$
$4$	0	0
$5$	0.879385	0.393273	0.196637	−	0.980476i	$-0.436998\pi$
$5$	0.879385	0.393273	0.196637	+	0.980476i	$0.436998\pi$
$6$	0	0
$7$	−1.65270	−0.624663	−0.312332	−	0.949973i	$-0.601110\pi$
$7$	−1.65270	−0.624663	−0.312332	+	0.949973i	$0.601110\pi$
$8$	0	0
$9$	1.77332	0.591106
$10$	0	0
$11$	−3.53209	−1.06496	−0.532482	−	0.846441i	$-0.678741\pi$
$11$	−3.53209	−1.06496	−0.532482	+	0.846441i	$0.678741\pi$
$12$	0	0
$13$	4.94356	1.37110	0.685549	−	0.728027i	$-0.259562\pi$
$13$	4.94356	1.37110	0.685549	+	0.728027i	$0.259562\pi$
$14$	0	0
$15$	−1.92127	−0.496071
$16$	0	0
$17$	2.59627	0.629687	0.314844	−	0.949144i	$-0.398048\pi$
$17$	2.59627	0.629687	0.314844	+	0.949144i	$0.398048\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	3.61081	0.787944
$22$	0	0
$23$	−4.82295	−1.00565	−0.502827	−	0.864387i	$-0.667707\pi$
$23$	−4.82295	−1.00565	−0.502827	+	0.864387i	$0.667707\pi$
$24$	0	0
$25$	−4.22668	−0.845336
$26$	0	0
$27$	2.68004	0.515775
$28$	0	0
$29$	6.87939	1.27747	0.638735	−	0.769427i	$-0.279458\pi$
$29$	6.87939	1.27747	0.638735	+	0.769427i	$0.279458\pi$
$30$	0	0
$31$	−2.04189	−0.366734	−0.183367	−	0.983045i	$-0.558700\pi$
$31$	−2.04189	−0.366734	−0.183367	+	0.983045i	$0.558700\pi$
$32$	0	0
$33$	7.71688	1.34334
$34$	0	0
$35$	−1.45336	−0.245663
$36$	0	0
$37$	5.18479	0.852375	0.426187	−	0.904635i	$-0.359856\pi$
$37$	5.18479	0.852375	0.426187	+	0.904635i	$0.359856\pi$
$38$	0	0
$39$	−10.8007	−1.72949
$40$	0	0
$41$	−1.73648	−0.271193	−0.135596	−	0.990764i	$-0.543295\pi$
$41$	−1.73648	−0.271193	−0.135596	+	0.990764i	$0.543295\pi$
$42$	0	0
$43$	7.98545	1.21777	0.608885	−	0.793258i	$-0.291617\pi$
$43$	7.98545	1.21777	0.608885	+	0.793258i	$0.291617\pi$
$44$	0	0
$45$	1.55943	0.232466
$46$	0	0
$47$	2.92127	0.426112	0.213056	−	0.977040i	$-0.431658\pi$
$47$	2.92127	0.426112	0.213056	+	0.977040i	$0.431658\pi$
$48$	0	0
$49$	−4.26857	−0.609796
$50$	0	0
$51$	−5.67230	−0.794281
$52$	0	0
$53$	−12.6800	−1.74174	−0.870869	−	0.491515i	$-0.836443\pi$
$53$	−12.6800	−1.74174	−0.870869	+	0.491515i	$0.836443\pi$
$54$	0	0
$55$	−3.10607	−0.418822
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.58172	0.987056	0.493528	−	0.869730i	$-0.335707\pi$
$59$	7.58172	0.987056	0.493528	+	0.869730i	$0.335707\pi$
$60$	0	0
$61$	−11.1284	−1.42484	−0.712420	−	0.701753i	$-0.752401\pi$
$61$	−11.1284	−1.42484	−0.712420	+	0.701753i	$0.752401\pi$
$62$	0	0
$63$	−2.93077	−0.369242
$64$	0	0
$65$	4.34730	0.539216
$66$	0	0
$67$	−16.0993	−1.96684	−0.983419	−	0.181349i	$-0.941954\pi$
$67$	−16.0993	−1.96684	−0.983419	+	0.181349i	$0.941954\pi$
$68$	0	0
$69$	10.5371	1.26852
$70$	0	0
$71$	13.5621	1.60953	0.804764	−	0.593595i	$-0.202292\pi$
$71$	13.5621	1.60953	0.804764	+	0.593595i	$0.202292\pi$
$72$	0	0
$73$	4.73917	0.554678	0.277339	−	0.960772i	$-0.410548\pi$
$73$	4.73917	0.554678	0.277339	+	0.960772i	$0.410548\pi$
$74$	0	0
$75$	9.23442	1.06630
$76$	0	0
$77$	5.83750	0.665244
$78$	0	0
$79$	1.21894	0.137142	0.0685708	−	0.997646i	$-0.478156\pi$
$79$	1.21894	0.137142	0.0685708	+	0.997646i	$0.478156\pi$
$80$	0	0
$81$	−11.1753	−1.24170
$82$	0	0
$83$	−5.87939	−0.645346	−0.322673	−	0.946510i	$-0.604581\pi$
$83$	−5.87939	−0.645346	−0.322673	+	0.946510i	$0.604581\pi$
$84$	0	0
$85$	2.28312	0.247639
$86$	0	0
$87$	−15.0300	−1.61139
$88$	0	0
$89$	10.2909	1.09083	0.545414	−	0.838166i	$-0.316372\pi$
$89$	10.2909	1.09083	0.545414	+	0.838166i	$0.316372\pi$
$90$	0	0
$91$	−8.17024	−0.856474
$92$	0	0
$93$	4.46110	0.462595
$94$	0	0
$95$	0	0
$96$	0	0
$97$	11.4534	1.16291	0.581456	−	0.813578i	$-0.302483\pi$
$97$	11.4534	1.16291	0.581456	+	0.813578i	$0.302483\pi$
$98$	0	0
$99$	−6.26352	−0.629507
$100$	0	0
$101$	−5.24897	−0.522292	−0.261146	−	0.965299i	$-0.584100\pi$
$101$	−5.24897	−0.522292	−0.261146	+	0.965299i	$0.584100\pi$
$102$	0	0
$103$	−1.79561	−0.176926	−0.0884632	−	0.996079i	$-0.528196\pi$
$103$	−1.79561	−0.176926	−0.0884632	+	0.996079i	$0.528196\pi$
$104$	0	0
$105$	3.17530	0.309877
$106$	0	0
$107$	−7.32770	−0.708395	−0.354198	−	0.935171i	$-0.615246\pi$
$107$	−7.32770	−0.708395	−0.354198	+	0.935171i	$0.615246\pi$
$108$	0	0
$109$	−19.8229	−1.89869	−0.949347	−	0.314230i	$-0.898254\pi$
$109$	−19.8229	−1.89869	−0.949347	+	0.314230i	$0.898254\pi$
$110$	0	0
$111$	−11.3277	−1.07518
$112$	0	0
$113$	−14.9932	−1.41044	−0.705220	−	0.708988i	$-0.749152\pi$
$113$	−14.9932	−1.41044	−0.705220	+	0.708988i	$0.749152\pi$
$114$	0	0
$115$	−4.24123	−0.395497
$116$	0	0
$117$	8.76651	0.810464
$118$	0	0
$119$	−4.29086	−0.393342
$120$	0	0
$121$	1.47565	0.134150
$122$	0	0
$123$	3.79385	0.342080
$124$	0	0
$125$	−8.11381	−0.725721
$126$	0	0
$127$	−3.10607	−0.275619	−0.137809	−	0.990459i	$-0.544006\pi$
$127$	−3.10607	−0.275619	−0.137809	+	0.990459i	$0.544006\pi$
$128$	0	0
$129$	−17.4466	−1.53608
$130$	0	0
$131$	11.4757	1.00263	0.501316	−	0.865264i	$-0.332849\pi$
$131$	11.4757	1.00263	0.501316	+	0.865264i	$0.332849\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	2.35679	0.202840
$136$	0	0
$137$	14.8084	1.26517	0.632584	−	0.774492i	$-0.281995\pi$
$137$	14.8084	1.26517	0.632584	+	0.774492i	$0.281995\pi$
$138$	0	0
$139$	−3.12567	−0.265116	−0.132558	−	0.991175i	$-0.542319\pi$
$139$	−3.12567	−0.265116	−0.132558	+	0.991175i	$0.542319\pi$
$140$	0	0
$141$	−6.38238	−0.537493
$142$	0	0
$143$	−17.4611	−1.46017
$144$	0	0
$145$	6.04963	0.502394
$146$	0	0
$147$	9.32594	0.769191
$148$	0	0
$149$	−15.8648	−1.29970	−0.649849	−	0.760063i	$-0.725168\pi$
$149$	−15.8648	−1.29970	−0.649849	+	0.760063i	$0.725168\pi$
$150$	0	0
$151$	−5.07192	−0.412747	−0.206373	−	0.978473i	$-0.566166\pi$
$151$	−5.07192	−0.412747	−0.206373	+	0.978473i	$0.566166\pi$
$152$	0	0
$153$	4.60401	0.372212
$154$	0	0
$155$	−1.79561	−0.144227
$156$	0	0
$157$	−13.3618	−1.06639	−0.533196	−	0.845992i	$-0.679009\pi$
$157$	−13.3618	−1.06639	−0.533196	+	0.845992i	$0.679009\pi$
$158$	0	0
$159$	27.7033	2.19701
$160$	0	0
$161$	7.97090	0.628195
$162$	0	0
$163$	−19.3354	−1.51447	−0.757234	−	0.653144i	$-0.773450\pi$
$163$	−19.3354	−1.51447	−0.757234	+	0.653144i	$0.773450\pi$
$164$	0	0
$165$	6.78611	0.528298
$166$	0	0
$167$	2.63041	0.203548	0.101774	−	0.994808i	$-0.467548\pi$
$167$	2.63041	0.203548	0.101774	+	0.994808i	$0.467548\pi$
$168$	0	0
$169$	11.4388	0.879909
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−5.10876	−0.388411	−0.194206	−	0.980961i	$-0.562213\pi$
$173$	−5.10876	−0.388411	−0.194206	+	0.980961i	$0.562213\pi$
$174$	0	0
$175$	6.98545	0.528051
$176$	0	0
$177$	−16.5645	−1.24506
$178$	0	0
$179$	−19.8084	−1.48055	−0.740275	−	0.672305i	$-0.765305\pi$
$179$	−19.8084	−1.48055	−0.740275	+	0.672305i	$0.765305\pi$
$180$	0	0
$181$	−11.1480	−0.828621	−0.414311	−	0.910136i	$-0.635977\pi$
$181$	−11.1480	−0.828621	−0.414311	+	0.910136i	$0.635977\pi$
$182$	0	0
$183$	24.3131	1.79728
$184$	0	0
$185$	4.55943	0.335216
$186$	0	0
$187$	−9.17024	−0.670595
$188$	0	0
$189$	−4.42932	−0.322186
$190$	0	0
$191$	−15.1138	−1.09360	−0.546798	−	0.837264i	$-0.684154\pi$
$191$	−15.1138	−1.09360	−0.546798	+	0.837264i	$0.684154\pi$
$192$	0	0
$193$	14.7374	1.06082	0.530411	−	0.847741i	$-0.322038\pi$
$193$	14.7374	1.06082	0.530411	+	0.847741i	$0.322038\pi$
$194$	0	0
$195$	−9.49794	−0.680162
$196$	0	0
$197$	−2.31820	−0.165165	−0.0825825	−	0.996584i	$-0.526317\pi$
$197$	−2.31820	−0.165165	−0.0825825	+	0.996584i	$0.526317\pi$
$198$	0	0
$199$	−0.650015	−0.0460784	−0.0230392	−	0.999735i	$-0.507334\pi$
$199$	−0.650015	−0.0460784	−0.0230392	+	0.999735i	$0.507334\pi$
$200$	0	0
$201$	35.1735	2.48095
$202$	0	0
$203$	−11.3696	−0.797988
$204$	0	0
$205$	−1.52704	−0.106653
$206$	0	0
$207$	−8.55262	−0.594448
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−19.8084	−1.36367	−0.681833	−	0.731508i	$-0.738817\pi$
$211$	−19.8084	−1.36367	−0.681833	+	0.731508i	$0.738817\pi$
$212$	0	0
$213$	−29.6304	−2.03024
$214$	0	0
$215$	7.02229	0.478916
$216$	0	0
$217$	3.37464	0.229085
$218$	0	0
$219$	−10.3541	−0.699665
$220$	0	0
$221$	12.8348	0.863363
$222$	0	0
$223$	−6.42602	−0.430318	−0.215159	−	0.976579i	$-0.569027\pi$
$223$	−6.42602	−0.430318	−0.215159	+	0.976579i	$0.569027\pi$
$224$	0	0
$225$	−7.49525	−0.499683
$226$	0	0
$227$	−3.19934	−0.212348	−0.106174	−	0.994348i	$-0.533860\pi$
$227$	−3.19934	−0.212348	−0.106174	+	0.994348i	$0.533860\pi$
$228$	0	0
$229$	16.8871	1.11593	0.557966	−	0.829864i	$-0.311582\pi$
$229$	16.8871	1.11593	0.557966	+	0.829864i	$0.311582\pi$
$230$	0	0
$231$	−12.7537	−0.839133
$232$	0	0
$233$	−14.7151	−0.964020	−0.482010	−	0.876166i	$-0.660093\pi$
$233$	−14.7151	−0.964020	−0.482010	+	0.876166i	$0.660093\pi$
$234$	0	0
$235$	2.56893	0.167578
$236$	0	0
$237$	−2.66313	−0.172989
$238$	0	0
$239$	1.60307	0.103694	0.0518471	−	0.998655i	$-0.483489\pi$
$239$	1.60307	0.103694	0.0518471	+	0.998655i	$0.483489\pi$
$240$	0	0
$241$	1.46017	0.0940578	0.0470289	−	0.998894i	$-0.485025\pi$
$241$	1.46017	0.0940578	0.0470289	+	0.998894i	$0.485025\pi$
$242$	0	0
$243$	16.3756	1.05049
$244$	0	0
$245$	−3.75372	−0.239816
$246$	0	0
$247$	0	0
$248$	0	0
$249$	12.8452	0.814034
$250$	0	0
$251$	0.361844	0.0228394	0.0114197	−	0.999935i	$-0.496365\pi$
$251$	0.361844	0.0228394	0.0114197	+	0.999935i	$0.496365\pi$
$252$	0	0
$253$	17.0351	1.07099
$254$	0	0
$255$	−4.98814	−0.312369
$256$	0	0
$257$	−23.9590	−1.49452	−0.747262	−	0.664529i	$-0.768632\pi$
$257$	−23.9590	−1.49452	−0.747262	+	0.664529i	$0.768632\pi$
$258$	0	0
$259$	−8.56893	−0.532447
$260$	0	0
$261$	12.1993	0.755120
$262$	0	0
$263$	−15.4388	−0.951998	−0.475999	−	0.879446i	$-0.657913\pi$
$263$	−15.4388	−0.951998	−0.475999	+	0.879446i	$0.657913\pi$
$264$	0	0
$265$	−11.1506	−0.684979
$266$	0	0
$267$	−22.4834	−1.37596
$268$	0	0
$269$	13.0341	0.794706	0.397353	−	0.917666i	$-0.369929\pi$
$269$	13.0341	0.794706	0.397353	+	0.917666i	$0.369929\pi$
$270$	0	0
$271$	−1.68004	−0.102055	−0.0510277	−	0.998697i	$-0.516250\pi$
$271$	−1.68004	−0.102055	−0.0510277	+	0.998697i	$0.516250\pi$
$272$	0	0
$273$	17.8503	1.08035
$274$	0	0
$275$	14.9290	0.900253
$276$	0	0
$277$	−16.8949	−1.01511	−0.507557	−	0.861618i	$-0.669451\pi$
$277$	−16.8949	−1.01511	−0.507557	+	0.861618i	$0.669451\pi$
$278$	0	0
$279$	−3.62092	−0.216779
$280$	0	0
$281$	−1.53983	−0.0918585	−0.0459293	−	0.998945i	$-0.514625\pi$
$281$	−1.53983	−0.0918585	−0.0459293	+	0.998945i	$0.514625\pi$
$282$	0	0
$283$	2.05737	0.122298	0.0611490	−	0.998129i	$-0.480524\pi$
$283$	2.05737	0.122298	0.0611490	+	0.998129i	$0.480524\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	2.86989	0.169404
$288$	0	0
$289$	−10.2594	−0.603494
$290$	0	0
$291$	−25.0232	−1.46689
$292$	0	0
$293$	25.5381	1.49195	0.745975	−	0.665974i	$-0.231984\pi$
$293$	25.5381	1.49195	0.745975	+	0.665974i	$0.231984\pi$
$294$	0	0
$295$	6.66725	0.388182
$296$	0	0
$297$	−9.46616	−0.549282
$298$	0	0
$299$	−23.8425	−1.37885
$300$	0	0
$301$	−13.1976	−0.760696
$302$	0	0
$303$	11.4679	0.658814
$304$	0	0
$305$	−9.78611	−0.560351
$306$	0	0
$307$	−10.4037	−0.593772	−0.296886	−	0.954913i	$-0.595948\pi$
$307$	−10.4037	−0.593772	−0.296886	+	0.954913i	$0.595948\pi$
$308$	0	0
$309$	3.92303	0.223173
$310$	0	0
$311$	−15.3550	−0.870704	−0.435352	−	0.900260i	$-0.643376\pi$
$311$	−15.3550	−0.870704	−0.435352	+	0.900260i	$0.643376\pi$
$312$	0	0
$313$	−21.7219	−1.22780	−0.613898	−	0.789385i	$-0.710399\pi$
$313$	−21.7219	−1.22780	−0.613898	+	0.789385i	$0.710399\pi$
$314$	0	0
$315$	−2.57728	−0.145213
$316$	0	0
$317$	2.30035	0.129201	0.0646004	−	0.997911i	$-0.479423\pi$
$317$	2.30035	0.129201	0.0646004	+	0.997911i	$0.479423\pi$
$318$	0	0
$319$	−24.2986	−1.36046
$320$	0	0
$321$	16.0095	0.893563
$322$	0	0
$323$	0	0
$324$	0	0
$325$	−20.8949	−1.15904
$326$	0	0
$327$	43.3090	2.39499
$328$	0	0
$329$	−4.82800	−0.266176
$330$	0	0
$331$	8.64590	0.475221	0.237611	−	0.971360i	$-0.423636\pi$
$331$	8.64590	0.475221	0.237611	+	0.971360i	$0.423636\pi$
$332$	0	0
$333$	9.19429	0.503844
$334$	0	0
$335$	−14.1575	−0.773504
$336$	0	0
$337$	−10.2841	−0.560208	−0.280104	−	0.959970i	$-0.590369\pi$
$337$	−10.2841	−0.560208	−0.280104	+	0.959970i	$0.590369\pi$
$338$	0	0
$339$	32.7570	1.77912
$340$	0	0
$341$	7.21213	0.390559
$342$	0	0
$343$	18.6236	1.00558
$344$	0	0
$345$	9.26621	0.498876
$346$	0	0
$347$	6.21482	0.333629	0.166815	−	0.985988i	$-0.446652\pi$
$347$	6.21482	0.333629	0.166815	+	0.985988i	$0.446652\pi$
$348$	0	0
$349$	−30.0087	−1.60633	−0.803164	−	0.595758i	$-0.796852\pi$
$349$	−30.0087	−1.60633	−0.803164	+	0.595758i	$0.796852\pi$
$350$	0	0
$351$	13.2490	0.707178
$352$	0	0
$353$	17.0925	0.909739	0.454870	−	0.890558i	$-0.349686\pi$
$353$	17.0925	0.909739	0.454870	+	0.890558i	$0.349686\pi$
$354$	0	0
$355$	11.9263	0.632984
$356$	0	0
$357$	9.37464	0.496158
$358$	0	0
$359$	21.9341	1.15764	0.578818	−	0.815457i	$-0.303514\pi$
$359$	21.9341	1.15764	0.578818	+	0.815457i	$0.303514\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−3.22399	−0.169216
$364$	0	0
$365$	4.16756	0.218140
$366$	0	0
$367$	36.2627	1.89290	0.946449	−	0.322854i	$-0.104642\pi$
$367$	36.2627	1.89290	0.946449	+	0.322854i	$0.104642\pi$
$368$	0	0
$369$	−3.07934	−0.160304
$370$	0	0
$371$	20.9564	1.08800
$372$	0	0
$373$	15.0942	0.781548	0.390774	−	0.920487i	$-0.372207\pi$
$373$	15.0942	0.781548	0.390774	+	0.920487i	$0.372207\pi$
$374$	0	0
$375$	17.7270	0.915418
$376$	0	0
$377$	34.0087	1.75154
$378$	0	0
$379$	7.02229	0.360711	0.180355	−	0.983602i	$-0.442275\pi$
$379$	7.02229	0.360711	0.180355	+	0.983602i	$0.442275\pi$
$380$	0	0
$381$	6.78611	0.347663
$382$	0	0
$383$	−30.0556	−1.53577	−0.767885	−	0.640588i	$-0.778691\pi$
$383$	−30.0556	−1.53577	−0.767885	+	0.640588i	$0.778691\pi$
$384$	0	0
$385$	5.13341	0.261623
$386$	0	0
$387$	14.1607	0.719831
$388$	0	0
$389$	−24.0847	−1.22114	−0.610572	−	0.791961i	$-0.709060\pi$
$389$	−24.0847	−1.22114	−0.610572	+	0.791961i	$0.709060\pi$
$390$	0	0
$391$	−12.5217	−0.633248
$392$	0	0
$393$	−25.0719	−1.26471
$394$	0	0
$395$	1.07192	0.0539341
$396$	0	0
$397$	−38.4466	−1.92958	−0.964789	−	0.263026i	$-0.915279\pi$
$397$	−38.4466	−1.92958	−0.964789	+	0.263026i	$0.915279\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	32.6783	1.63188	0.815938	−	0.578139i	$-0.196221\pi$
$401$	32.6783	1.63188	0.815938	+	0.578139i	$0.196221\pi$
$402$	0	0
$403$	−10.0942	−0.502828
$404$	0	0
$405$	−9.82739	−0.488327
$406$	0	0
$407$	−18.3131	−0.907749
$408$	0	0
$409$	27.8776	1.37846	0.689230	−	0.724543i	$-0.257949\pi$
$409$	27.8776	1.37846	0.689230	+	0.724543i	$0.257949\pi$
$410$	0	0
$411$	−32.3533	−1.59587
$412$	0	0
$413$	−12.5303	−0.616577
$414$	0	0
$415$	−5.17024	−0.253797
$416$	0	0
$417$	6.82893	0.334414
$418$	0	0
$419$	38.3455	1.87330	0.936651	−	0.350264i	$-0.113908\pi$
$419$	38.3455	1.87330	0.936651	+	0.350264i	$0.113908\pi$
$420$	0	0
$421$	6.24123	0.304179	0.152089	−	0.988367i	$-0.451400\pi$
$421$	6.24123	0.304179	0.152089	+	0.988367i	$0.451400\pi$
$422$	0	0
$423$	5.18035	0.251877
$424$	0	0
$425$	−10.9736	−0.532297
$426$	0	0
$427$	18.3919	0.890045
$428$	0	0
$429$	38.1489	1.84185
$430$	0	0
$431$	5.11381	0.246323	0.123162	−	0.992387i	$-0.460697\pi$
$431$	5.11381	0.246323	0.123162	+	0.992387i	$0.460697\pi$
$432$	0	0
$433$	5.13516	0.246780	0.123390	−	0.992358i	$-0.460623\pi$
$433$	5.13516	0.246780	0.123390	+	0.992358i	$0.460623\pi$
$434$	0	0
$435$	−13.2172	−0.633716
$436$	0	0
$437$	0	0
$438$	0	0
$439$	1.08647	0.0518542	0.0259271	−	0.999664i	$-0.491746\pi$
$439$	1.08647	0.0518542	0.0259271	+	0.999664i	$0.491746\pi$
$440$	0	0
$441$	−7.56953	−0.360454
$442$	0	0
$443$	−21.0993	−1.00246	−0.501228	−	0.865315i	$-0.667118\pi$
$443$	−21.0993	−1.00246	−0.501228	+	0.865315i	$0.667118\pi$
$444$	0	0
$445$	9.04963	0.428994
$446$	0	0
$447$	34.6614	1.63943
$448$	0	0
$449$	3.30810	0.156119	0.0780593	−	0.996949i	$-0.475128\pi$
$449$	3.30810	0.156119	0.0780593	+	0.996949i	$0.475128\pi$
$450$	0	0
$451$	6.13341	0.288811
$452$	0	0
$453$	11.0811	0.520635
$454$	0	0
$455$	−7.18479	−0.336828
$456$	0	0
$457$	−26.9659	−1.26141	−0.630705	−	0.776023i	$-0.717234\pi$
$457$	−26.9659	−1.26141	−0.630705	+	0.776023i	$0.717234\pi$
$458$	0	0
$459$	6.95811	0.324777
$460$	0	0
$461$	18.3533	0.854798	0.427399	−	0.904063i	$-0.359430\pi$
$461$	18.3533	0.854798	0.427399	+	0.904063i	$0.359430\pi$
$462$	0	0
$463$	−27.9168	−1.29741	−0.648703	−	0.761042i	$-0.724688\pi$
$463$	−27.9168	−1.29741	−0.648703	+	0.761042i	$0.724688\pi$
$464$	0	0
$465$	3.92303	0.181926
$466$	0	0
$467$	−23.7638	−1.09966	−0.549829	−	0.835277i	$-0.685307\pi$
$467$	−23.7638	−1.09966	−0.549829	+	0.835277i	$0.685307\pi$
$468$	0	0
$469$	26.6073	1.22861
$470$	0	0
$471$	29.1929	1.34514
$472$	0	0
$473$	−28.2053	−1.29688
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−22.4858	−1.02955
$478$	0	0
$479$	10.1952	0.465832	0.232916	−	0.972497i	$-0.425173\pi$
$479$	10.1952	0.465832	0.232916	+	0.972497i	$0.425173\pi$
$480$	0	0
$481$	25.6313	1.16869
$482$	0	0
$483$	−17.4148	−0.792400
$484$	0	0
$485$	10.0719	0.457342
$486$	0	0
$487$	6.02053	0.272816	0.136408	−	0.990653i	$-0.456444\pi$
$487$	6.02053	0.272816	0.136408	+	0.990653i	$0.456444\pi$
$488$	0	0
$489$	42.2439	1.91034
$490$	0	0
$491$	36.6236	1.65280	0.826400	−	0.563083i	$-0.190385\pi$
$491$	36.6236	1.65280	0.826400	+	0.563083i	$0.190385\pi$
$492$	0	0
$493$	17.8607	0.804406
$494$	0	0
$495$	−5.50805	−0.247568
$496$	0	0
$497$	−22.4142	−1.00541
$498$	0	0
$499$	33.7202	1.50952	0.754761	−	0.656000i	$-0.227753\pi$
$499$	33.7202	1.50952	0.754761	+	0.656000i	$0.227753\pi$
$500$	0	0
$501$	−5.74691	−0.256753
$502$	0	0
$503$	−26.6982	−1.19041	−0.595207	−	0.803572i	$-0.702930\pi$
$503$	−26.6982	−1.19041	−0.595207	+	0.803572i	$0.702930\pi$
$504$	0	0
$505$	−4.61587	−0.205403
$506$	0	0
$507$	−24.9914	−1.10991
$508$	0	0
$509$	22.9540	1.01742	0.508709	−	0.860939i	$-0.330123\pi$
$509$	22.9540	1.01742	0.508709	+	0.860939i	$0.330123\pi$
$510$	0	0
$511$	−7.83244	−0.346487
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−1.57903	−0.0695804
$516$	0	0
$517$	−10.3182	−0.453794
$518$	0	0
$519$	11.1616	0.489939
$520$	0	0
$521$	9.51754	0.416971	0.208486	−	0.978025i	$-0.433147\pi$
$521$	9.51754	0.416971	0.208486	+	0.978025i	$0.433147\pi$
$522$	0	0
$523$	6.17200	0.269883	0.134941	−	0.990854i	$-0.456915\pi$
$523$	6.17200	0.269883	0.134941	+	0.990854i	$0.456915\pi$
$524$	0	0
$525$	−15.2618	−0.666078
$526$	0	0
$527$	−5.30129	−0.230928
$528$	0	0
$529$	0.260830	0.0113404
$530$	0	0
$531$	13.4448	0.583455
$532$	0	0
$533$	−8.58441	−0.371832
$534$	0	0
$535$	−6.44387	−0.278593
$536$	0	0
$537$	43.2772	1.86755
$538$	0	0
$539$	15.0770	0.649411
$540$	0	0
$541$	−15.6254	−0.671787	−0.335893	−	0.941900i	$-0.609038\pi$
$541$	−15.6254	−0.671787	−0.335893	+	0.941900i	$0.609038\pi$
$542$	0	0
$543$	24.3560	1.04521
$544$	0	0
$545$	−17.4320	−0.746705
$546$	0	0
$547$	39.1480	1.67385	0.836923	−	0.547321i	$-0.184352\pi$
$547$	39.1480	1.67385	0.836923	+	0.547321i	$0.184352\pi$
$548$	0	0
$549$	−19.7341	−0.842232
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−2.01455	−0.0856673
$554$	0	0
$555$	−9.96141	−0.422838
$556$	0	0
$557$	17.3719	0.736073	0.368037	−	0.929811i	$-0.380030\pi$
$557$	17.3719	0.736073	0.368037	+	0.929811i	$0.380030\pi$
$558$	0	0
$559$	39.4766	1.66968
$560$	0	0
$561$	20.0351	0.845882
$562$	0	0
$563$	−32.0283	−1.34983	−0.674915	−	0.737895i	$-0.735820\pi$
$563$	−32.0283	−1.34983	−0.674915	+	0.737895i	$0.735820\pi$
$564$	0	0
$565$	−13.1848	−0.554688
$566$	0	0
$567$	18.4695	0.775644
$568$	0	0
$569$	−20.0077	−0.838768	−0.419384	−	0.907809i	$-0.637754\pi$
$569$	−20.0077	−0.838768	−0.419384	+	0.907809i	$0.637754\pi$
$570$	0	0
$571$	23.0196	0.963340	0.481670	−	0.876353i	$-0.340030\pi$
$571$	23.0196	0.963340	0.481670	+	0.876353i	$0.340030\pi$
$572$	0	0
$573$	33.0205	1.37945
$574$	0	0
$575$	20.3851	0.850116
$576$	0	0
$577$	1.61318	0.0671575	0.0335788	−	0.999436i	$-0.489310\pi$
$577$	1.61318	0.0671575	0.0335788	+	0.999436i	$0.489310\pi$
$578$	0	0
$579$	−32.1982	−1.33811
$580$	0	0
$581$	9.71688	0.403124
$582$	0	0
$583$	44.7870	1.85489
$584$	0	0
$585$	7.70914	0.318734
$586$	0	0
$587$	18.9344	0.781506	0.390753	−	0.920496i	$-0.372215\pi$
$587$	18.9344	0.781506	0.390753	+	0.920496i	$0.372215\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	5.06479	0.208338
$592$	0	0
$593$	29.1189	1.19577	0.597884	−	0.801583i	$-0.296008\pi$
$593$	29.1189	1.19577	0.597884	+	0.801583i	$0.296008\pi$
$594$	0	0
$595$	−3.77332	−0.154691
$596$	0	0
$597$	1.42015	0.0581228
$598$	0	0
$599$	17.9973	0.735350	0.367675	−	0.929954i	$-0.380154\pi$
$599$	17.9973	0.735350	0.367675	+	0.929954i	$0.380154\pi$
$600$	0	0
$601$	−14.2385	−0.580802	−0.290401	−	0.956905i	$-0.593789\pi$
$601$	−14.2385	−0.580802	−0.290401	+	0.956905i	$0.593789\pi$
$602$	0	0
$603$	−28.5491	−1.16261
$604$	0	0
$605$	1.29767	0.0527576
$606$	0	0
$607$	23.1702	0.940451	0.470226	−	0.882546i	$-0.344173\pi$
$607$	23.1702	0.940451	0.470226	+	0.882546i	$0.344173\pi$
$608$	0	0
$609$	24.8402	1.00658
$610$	0	0
$611$	14.4415	0.584241
$612$	0	0
$613$	33.7452	1.36295	0.681477	−	0.731840i	$-0.261338\pi$
$613$	33.7452	1.36295	0.681477	+	0.731840i	$0.261338\pi$
$614$	0	0
$615$	3.33626	0.134531
$616$	0	0
$617$	10.2172	0.411328	0.205664	−	0.978623i	$-0.434065\pi$
$617$	10.2172	0.411328	0.205664	+	0.978623i	$0.434065\pi$
$618$	0	0
$619$	29.8188	1.19852	0.599260	−	0.800554i	$-0.295462\pi$
$619$	29.8188	1.19852	0.599260	+	0.800554i	$0.295462\pi$
$620$	0	0
$621$	−12.9257	−0.518691
$622$	0	0
$623$	−17.0077	−0.681401
$624$	0	0
$625$	13.9982	0.559930
$626$	0	0
$627$	0	0
$628$	0	0
$629$	13.4611	0.536729
$630$	0	0
$631$	13.0196	0.518302	0.259151	−	0.965837i	$-0.416557\pi$
$631$	13.0196	0.518302	0.259151	+	0.965837i	$0.416557\pi$
$632$	0	0
$633$	43.2772	1.72012
$634$	0	0
$635$	−2.73143	−0.108393
$636$	0	0
$637$	−21.1019	−0.836090
$638$	0	0
$639$	24.0500	0.951401
$640$	0	0
$641$	1.65270	0.0652779	0.0326389	−	0.999467i	$-0.489609\pi$
$641$	1.65270	0.0652779	0.0326389	+	0.999467i	$0.489609\pi$
$642$	0	0
$643$	−13.6869	−0.539757	−0.269878	−	0.962894i	$-0.586983\pi$
$643$	−13.6869	−0.539757	−0.269878	+	0.962894i	$0.586983\pi$
$644$	0	0
$645$	−15.3422	−0.604100
$646$	0	0
$647$	−30.0820	−1.18265	−0.591323	−	0.806435i	$-0.701394\pi$
$647$	−30.0820	−1.18265	−0.591323	+	0.806435i	$0.701394\pi$
$648$	0	0
$649$	−26.7793	−1.05118
$650$	0	0
$651$	−7.37288	−0.288966
$652$	0	0
$653$	−6.55674	−0.256585	−0.128293	−	0.991736i	$-0.540950\pi$
$653$	−6.55674	−0.256585	−0.128293	+	0.991736i	$0.540950\pi$
$654$	0	0
$655$	10.0915	0.394308
$656$	0	0
$657$	8.40406	0.327874
$658$	0	0
$659$	−11.0591	−0.430802	−0.215401	−	0.976526i	$-0.569106\pi$
$659$	−11.0591	−0.430802	−0.215401	+	0.976526i	$0.569106\pi$
$660$	0	0
$661$	−22.0368	−0.857134	−0.428567	−	0.903510i	$-0.640981\pi$
$661$	−22.0368	−0.857134	−0.428567	+	0.903510i	$0.640981\pi$
$662$	0	0
$663$	−28.0414	−1.08904
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−33.1789	−1.28469
$668$	0	0
$669$	14.0395	0.542799
$670$	0	0
$671$	39.3063	1.51740
$672$	0	0
$673$	−1.07873	−0.0415818	−0.0207909	−	0.999784i	$-0.506618\pi$
$673$	−1.07873	−0.0415818	−0.0207909	+	0.999784i	$0.506618\pi$
$674$	0	0
$675$	−11.3277	−0.436003
$676$	0	0
$677$	−38.9231	−1.49594	−0.747969	−	0.663734i	$-0.768971\pi$
$677$	−38.9231	−1.49594	−0.747969	+	0.663734i	$0.768971\pi$
$678$	0	0
$679$	−18.9290	−0.726429
$680$	0	0
$681$	6.98990	0.267853
$682$	0	0
$683$	−2.33275	−0.0892601	−0.0446301	−	0.999004i	$-0.514211\pi$
$683$	−2.33275	−0.0892601	−0.0446301	+	0.999004i	$0.514211\pi$
$684$	0	0
$685$	13.0223	0.497556
$686$	0	0
$687$	−36.8949	−1.40763
$688$	0	0
$689$	−62.6846	−2.38809
$690$	0	0
$691$	−35.5485	−1.35233	−0.676164	−	0.736751i	$-0.736359\pi$
$691$	−35.5485	−1.35233	−0.676164	+	0.736751i	$0.736359\pi$
$692$	0	0
$693$	10.3517	0.393230
$694$	0	0
$695$	−2.74867	−0.104263
$696$	0	0
$697$	−4.50837	−0.170767
$698$	0	0
$699$	32.1495	1.21601
$700$	0	0
$701$	18.9017	0.713906	0.356953	−	0.934122i	$-0.383816\pi$
$701$	18.9017	0.713906	0.356953	+	0.934122i	$0.383816\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−5.61257	−0.211382
$706$	0	0
$707$	8.67499	0.326257
$708$	0	0
$709$	−8.12567	−0.305166	−0.152583	−	0.988291i	$-0.548759\pi$
$709$	−8.12567	−0.305166	−0.152583	+	0.988291i	$0.548759\pi$
$710$	0	0
$711$	2.16157	0.0810652
$712$	0	0
$713$	9.84793	0.368808
$714$	0	0
$715$	−15.3550	−0.574246
$716$	0	0
$717$	−3.50238	−0.130799
$718$	0	0
$719$	39.1429	1.45978	0.729892	−	0.683562i	$-0.239570\pi$
$719$	39.1429	1.45978	0.729892	+	0.683562i	$0.239570\pi$
$720$	0	0
$721$	2.96761	0.110519
$722$	0	0
$723$	−3.19017	−0.118644
$724$	0	0
$725$	−29.0770	−1.07989
$726$	0	0
$727$	0.246282	0.00913409	0.00456705	−	0.999990i	$-0.498546\pi$
$727$	0.246282	0.00913409	0.00456705	+	0.999990i	$0.498546\pi$
$728$	0	0
$729$	−2.25133	−0.0833828
$730$	0	0
$731$	20.7324	0.766814
$732$	0	0
$733$	0.266207	0.00983256	0.00491628	−	0.999988i	$-0.498435\pi$
$733$	0.266207	0.00983256	0.00491628	+	0.999988i	$0.498435\pi$
$734$	0	0
$735$	8.20110	0.302502
$736$	0	0
$737$	56.8640	2.09461
$738$	0	0
$739$	−21.8753	−0.804695	−0.402347	−	0.915487i	$-0.631806\pi$
$739$	−21.8753	−0.804695	−0.402347	+	0.915487i	$0.631806\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.49794	0.0549541	0.0274771	−	0.999622i	$-0.491253\pi$
$743$	1.49794	0.0549541	0.0274771	+	0.999622i	$0.491253\pi$
$744$	0	0
$745$	−13.9513	−0.511136
$746$	0	0
$747$	−10.4260	−0.381468
$748$	0	0
$749$	12.1105	0.442508
$750$	0	0
$751$	9.34493	0.341001	0.170501	−	0.985358i	$-0.445461\pi$
$751$	9.34493	0.341001	0.170501	+	0.985358i	$0.445461\pi$
$752$	0	0
$753$	−0.790555	−0.0288094
$754$	0	0
$755$	−4.46017	−0.162322
$756$	0	0
$757$	9.85122	0.358049	0.179024	−	0.983845i	$-0.442706\pi$
$757$	9.85122	0.358049	0.179024	+	0.983845i	$0.442706\pi$
$758$	0	0
$759$	−37.2181	−1.35093
$760$	0	0
$761$	18.9040	0.685271	0.342635	−	0.939468i	$-0.388680\pi$
$761$	18.9040	0.685271	0.342635	+	0.939468i	$0.388680\pi$
$762$	0	0
$763$	32.7615	1.18604
$764$	0	0
$765$	4.04870	0.146381
$766$	0	0
$767$	37.4807	1.35335
$768$	0	0
$769$	−13.0436	−0.470366	−0.235183	−	0.971951i	$-0.575569\pi$
$769$	−13.0436	−0.470366	−0.235183	+	0.971951i	$0.575569\pi$
$770$	0	0
$771$	52.3455	1.88518
$772$	0	0
$773$	22.9727	0.826269	0.413135	−	0.910670i	$-0.364434\pi$
$773$	22.9727	0.826269	0.413135	+	0.910670i	$0.364434\pi$
$774$	0	0
$775$	8.63041	0.310014
$776$	0	0
$777$	18.7213	0.671624
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−47.9026	−1.71409
$782$	0	0
$783$	18.4371	0.658887
$784$	0	0
$785$	−11.7502	−0.419383
$786$	0	0
$787$	−43.6705	−1.55669	−0.778343	−	0.627839i	$-0.783940\pi$
$787$	−43.6705	−1.55669	−0.778343	+	0.627839i	$0.783940\pi$
$788$	0	0
$789$	33.7306	1.20084
$790$	0	0
$791$	24.7793	0.881051
$792$	0	0
$793$	−55.0137	−1.95359
$794$	0	0
$795$	24.3618	0.864026
$796$	0	0
$797$	4.65507	0.164891	0.0824455	−	0.996596i	$-0.473727\pi$
$797$	4.65507	0.164891	0.0824455	+	0.996596i	$0.473727\pi$
$798$	0	0
$799$	7.58441	0.268317
$800$	0	0
$801$	18.2490	0.644796
$802$	0	0
$803$	−16.7392	−0.590712
$804$	0	0
$805$	7.00950	0.247052
$806$	0	0
$807$	−28.4769	−1.00243
$808$	0	0
$809$	2.39599	0.0842386	0.0421193	−	0.999113i	$-0.486589\pi$
$809$	2.39599	0.0842386	0.0421193	+	0.999113i	$0.486589\pi$
$810$	0	0
$811$	−41.5553	−1.45920	−0.729602	−	0.683872i	$-0.760295\pi$
$811$	−41.5553	−1.45920	−0.729602	+	0.683872i	$0.760295\pi$
$812$	0	0
$813$	3.67055	0.128732
$814$	0	0
$815$	−17.0033	−0.595600
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−14.4884	−0.506267
$820$	0	0
$821$	39.6759	1.38470	0.692350	−	0.721562i	$-0.256576\pi$
$821$	39.6759	1.38470	0.692350	+	0.721562i	$0.256576\pi$
$822$	0	0
$823$	−14.7537	−0.514282	−0.257141	−	0.966374i	$-0.582781\pi$
$823$	−14.7537	−0.514282	−0.257141	+	0.966374i	$0.582781\pi$
$824$	0	0
$825$	−32.6168	−1.13557
$826$	0	0
$827$	−22.2608	−0.774085	−0.387042	−	0.922062i	$-0.626503\pi$
$827$	−22.2608	−0.774085	−0.387042	+	0.922062i	$0.626503\pi$
$828$	0	0
$829$	40.1644	1.39497	0.697483	−	0.716601i	$-0.254303\pi$
$829$	40.1644	1.39497	0.697483	+	0.716601i	$0.254303\pi$
$830$	0	0
$831$	36.9118	1.28046
$832$	0	0
$833$	−11.0823	−0.383981
$834$	0	0
$835$	2.31315	0.0800498
$836$	0	0
$837$	−5.47235	−0.189152
$838$	0	0
$839$	48.3857	1.67046	0.835230	−	0.549901i	$-0.185335\pi$
$839$	48.3857	1.67046	0.835230	+	0.549901i	$0.185335\pi$
$840$	0	0
$841$	18.3259	0.631929
$842$	0	0
$843$	3.36421	0.115869
$844$	0	0
$845$	10.0591	0.346044
$846$	0	0
$847$	−2.43882	−0.0837987
$848$	0	0
$849$	−4.49493	−0.154266
$850$	0	0
$851$	−25.0060	−0.857194
$852$	0	0
$853$	0.526712	0.0180343	0.00901714	−	0.999959i	$-0.497130\pi$
$853$	0.526712	0.0180343	0.00901714	+	0.999959i	$0.497130\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−16.8161	−0.574428	−0.287214	−	0.957866i	$-0.592729\pi$
$857$	−16.8161	−0.574428	−0.287214	+	0.957866i	$0.592729\pi$
$858$	0	0
$859$	−47.5963	−1.62396	−0.811982	−	0.583683i	$-0.801611\pi$
$859$	−47.5963	−1.62396	−0.811982	+	0.583683i	$0.801611\pi$
$860$	0	0
$861$	−6.27011	−0.213685
$862$	0	0
$863$	34.4766	1.17360	0.586798	−	0.809733i	$-0.300388\pi$
$863$	34.4766	1.17360	0.586798	+	0.809733i	$0.300388\pi$
$864$	0	0
$865$	−4.49256	−0.152752
$866$	0	0
$867$	22.4147	0.761242
$868$	0	0
$869$	−4.30541	−0.146051
$870$	0	0
$871$	−79.5877	−2.69673
$872$	0	0
$873$	20.3105	0.687405
$874$	0	0
$875$	13.4097	0.453331
$876$	0	0
$877$	−49.8699	−1.68399	−0.841993	−	0.539488i	$-0.818618\pi$
$877$	−49.8699	−1.68399	−0.841993	+	0.539488i	$0.818618\pi$
$878$	0	0
$879$	−55.7954	−1.88193
$880$	0	0
$881$	11.2422	0.378758	0.189379	−	0.981904i	$-0.439353\pi$
$881$	11.2422	0.378758	0.189379	+	0.981904i	$0.439353\pi$
$882$	0	0
$883$	−21.9581	−0.738949	−0.369475	−	0.929241i	$-0.620462\pi$
$883$	−21.9581	−0.738949	−0.369475	+	0.929241i	$0.620462\pi$
$884$	0	0
$885$	−14.5666	−0.489650
$886$	0	0
$887$	−34.5262	−1.15928	−0.579638	−	0.814874i	$-0.696806\pi$
$887$	−34.5262	−1.15928	−0.579638	+	0.814874i	$0.696806\pi$
$888$	0	0
$889$	5.13341	0.172169
$890$	0	0
$891$	39.4721	1.32237
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−17.4192	−0.582260
$896$	0	0
$897$	52.0910	1.73927
$898$	0	0
$899$	−14.0469	−0.468492
$900$	0	0
$901$	−32.9208	−1.09675
$902$	0	0
$903$	28.8340	0.959535
$904$	0	0
$905$	−9.80335	−0.325874
$906$	0	0
$907$	−30.4793	−1.01205	−0.506024	−	0.862519i	$-0.668885\pi$
$907$	−30.4793	−1.01205	−0.506024	+	0.862519i	$0.668885\pi$
$908$	0	0
$909$	−9.30810	−0.308730
$910$	0	0
$911$	−38.8016	−1.28555	−0.642777	−	0.766053i	$-0.722218\pi$
$911$	−38.8016	−1.28555	−0.642777	+	0.766053i	$0.722218\pi$
$912$	0	0
$913$	20.7665	0.687271
$914$	0	0
$915$	21.3806	0.706822
$916$	0	0
$917$	−18.9659	−0.626308
$918$	0	0
$919$	−33.5149	−1.10555	−0.552776	−	0.833330i	$-0.686432\pi$
$919$	−33.5149	−1.10555	−0.552776	+	0.833330i	$0.686432\pi$
$920$	0	0
$921$	22.7300	0.748979
$922$	0	0
$923$	67.0452	2.20682
$924$	0	0
$925$	−21.9145	−0.720543
$926$	0	0
$927$	−3.18418	−0.104582
$928$	0	0
$929$	−29.3919	−0.964316	−0.482158	−	0.876084i	$-0.660147\pi$
$929$	−29.3919	−0.964316	−0.482158	+	0.876084i	$0.660147\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	33.5476	1.09830
$934$	0	0
$935$	−8.06418	−0.263727
$936$	0	0
$937$	−52.6682	−1.72059	−0.860297	−	0.509793i	$-0.829722\pi$
$937$	−52.6682	−1.72059	−0.860297	+	0.509793i	$0.829722\pi$
$938$	0	0
$939$	47.4579	1.54873
$940$	0	0
$941$	−32.2763	−1.05218	−0.526089	−	0.850430i	$-0.676342\pi$
$941$	−32.2763	−1.05218	−0.526089	+	0.850430i	$0.676342\pi$
$942$	0	0
$943$	8.37496	0.272726
$944$	0	0
$945$	−3.89508	−0.126707
$946$	0	0
$947$	−48.1976	−1.56621	−0.783106	−	0.621889i	$-0.786366\pi$
$947$	−48.1976	−1.56621	−0.783106	+	0.621889i	$0.786366\pi$
$948$	0	0
$949$	23.4284	0.760518
$950$	0	0
$951$	−5.02580	−0.162973
$952$	0	0
$953$	28.2300	0.914459	0.457229	−	0.889349i	$-0.348842\pi$
$953$	28.2300	0.914459	0.457229	+	0.889349i	$0.348842\pi$
$954$	0	0
$955$	−13.2909	−0.430082
$956$	0	0
$957$	53.0874	1.71607
$958$	0	0
$959$	−24.4739	−0.790303
$960$	0	0
$961$	−26.8307	−0.865506
$962$	0	0
$963$	−12.9943	−0.418737
$964$	0	0
$965$	12.9599	0.417193
$966$	0	0
$967$	−32.1985	−1.03543	−0.517717	−	0.855552i	$-0.673218\pi$
$967$	−32.1985	−1.03543	−0.517717	+	0.855552i	$0.673218\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	2.33511	0.0749373	0.0374687	−	0.999298i	$-0.488071\pi$
$971$	2.33511	0.0749373	0.0374687	+	0.999298i	$0.488071\pi$
$972$	0	0
$973$	5.16580	0.165608
$974$	0	0
$975$	45.6509	1.46200
$976$	0	0
$977$	−15.0993	−0.483068	−0.241534	−	0.970392i	$-0.577651\pi$
$977$	−15.0993	−0.483068	−0.241534	+	0.970392i	$0.577651\pi$
$978$	0	0
$979$	−36.3482	−1.16169
$980$	0	0
$981$	−35.1524	−1.12233
$982$	0	0
$983$	61.8495	1.97269	0.986346	−	0.164687i	$-0.0526615\pi$
$983$	61.8495	1.97269	0.986346	+	0.164687i	$0.0526615\pi$
$984$	0	0
$985$	−2.03859	−0.0649549
$986$	0	0
$987$	10.5482	0.335752
$988$	0	0
$989$	−38.5134	−1.22466
$990$	0	0
$991$	40.8117	1.29643	0.648213	−	0.761459i	$-0.275517\pi$
$991$	40.8117	1.29643	0.648213	+	0.761459i	$0.275517\pi$
$992$	0	0
$993$	−18.8895	−0.599440
$994$	0	0
$995$	−0.571614	−0.0181214
$996$	0	0
$997$	−43.9050	−1.39048	−0.695242	−	0.718776i	$-0.744703\pi$
$997$	−43.9050	−1.39048	−0.695242	+	0.718776i	$0.744703\pi$
$998$	0	0
$999$	13.8955	0.439633

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.m.1.2		3
4.3	odd	2		5776.2.a.bs.1.2		3
19.3	odd	18		152.2.q.b.9.1	✓	6
19.13	odd	18		152.2.q.b.17.1	yes	6
19.18	odd	2		2888.2.a.s.1.2		3
76.3	even	18		304.2.u.a.161.1		6
76.51	even	18		304.2.u.a.17.1		6
76.75	even	2		5776.2.a.bj.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.q.b.9.1	✓	6	19.3	odd	18
152.2.q.b.17.1	yes	6	19.13	odd	18
304.2.u.a.17.1		6	76.51	even	18
304.2.u.a.161.1		6	76.3	even	18
2888.2.a.m.1.2		3	1.1	even	1	trivial
2888.2.a.s.1.2		3	19.18	odd	2
5776.2.a.bj.1.2		3	76.75	even	2
5776.2.a.bs.1.2		3	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)

\(f(q)\)	\(=\)	\(q-2.18479 q^{3} +0.879385 q^{5} -1.65270 q^{7} +1.77332 q^{9} -3.53209 q^{11} +4.94356 q^{13} -1.92127 q^{15} +2.59627 q^{17} +3.61081 q^{21} -4.82295 q^{23} -4.22668 q^{25} +2.68004 q^{27} +6.87939 q^{29} -2.04189 q^{31} +7.71688 q^{33} -1.45336 q^{35} +5.18479 q^{37} -10.8007 q^{39} -1.73648 q^{41} +7.98545 q^{43} +1.55943 q^{45} +2.92127 q^{47} -4.26857 q^{49} -5.67230 q^{51} -12.6800 q^{53} -3.10607 q^{55} +7.58172 q^{59} -11.1284 q^{61} -2.93077 q^{63} +4.34730 q^{65} -16.0993 q^{67} +10.5371 q^{69} +13.5621 q^{71} +4.73917 q^{73} +9.23442 q^{75} +5.83750 q^{77} +1.21894 q^{79} -11.1753 q^{81} -5.87939 q^{83} +2.28312 q^{85} -15.0300 q^{87} +10.2909 q^{89} -8.17024 q^{91} +4.46110 q^{93} +11.4534 q^{97} -6.26352 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 3 q^{3} - 3 q^{5} - 6 q^{7} + 12 q^{9} - 6 q^{11} + 3 q^{15} - 6 q^{17} + 15 q^{21} + 6 q^{23} - 6 q^{25} - 12 q^{27} + 15 q^{29} - 3 q^{31} + 15 q^{33} + 9 q^{35} + 12 q^{37} - 18 q^{39} + 6 q^{43}+ \cdots - 24 q^{99}+O(q^{100}) \) 3 * q - 3 * q^3 - 3 * q^5 - 6 * q^7 + 12 * q^9 - 6 * q^11 + 3 * q^15 - 6 * q^17 + 15 * q^21 + 6 * q^23 - 6 * q^25 - 12 * q^27 + 15 * q^29 - 3 * q^31 + 15 * q^33 + 9 * q^35 + 12 * q^37 - 18 * q^39 + 6 * q^43 - 21 * q^45 - 3 * q^49 - 21 * q^51 - 18 * q^53 + 3 * q^55 - 9 * q^59 - 15 * q^61 - 33 * q^63 + 12 * q^65 + 6 * q^67 + 12 * q^69 + 6 * q^71 - 3 * q^75 + 15 * q^77 + 21 * q^79 + 3 * q^81 - 12 * q^83 + 15 * q^85 - 15 * q^87 + 15 * q^89 - 3 * q^91 - 24 * q^93 + 21 * q^97 - 24 * q^99

Introduction

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