LMFDB - Embedded newform 2888.2.a.u.1.5

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [6,0,3,0,2,0,-2,0,9,0,-5,0,10,0,0] 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:	$0$
Dimension:	$6$
Coefficient field:	6.6.3022625.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{6} - 2x^{5} - 7x^{4} + 7x^{3} + 17x^{2} + 4x - 1 $ x^6 - 2x^5 - 7x^4 + 7x^3 + 17x^2 + 4*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 2^{2} $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$2.70201$ 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)

$f(q)$	$=$	$q+2.77311 q^{3} -3.10181 q^{5} +4.25689 q^{7} +4.69013 q^{9} +0.566753 q^{11} +6.64206 q^{13} -8.60166 q^{15} +0.193489 q^{17} +11.8048 q^{21} -1.91382 q^{23} +4.62124 q^{25} +4.68693 q^{27} -6.94796 q^{29} +6.20086 q^{31} +1.57167 q^{33} -13.2041 q^{35} -4.41517 q^{37} +18.4192 q^{39} -6.18678 q^{41} +2.00935 q^{43} -14.5479 q^{45} +11.4545 q^{47} +11.1211 q^{49} +0.536566 q^{51} +0.750308 q^{53} -1.75796 q^{55} -0.815996 q^{59} +0.738209 q^{61} +19.9654 q^{63} -20.6024 q^{65} +7.41886 q^{67} -5.30723 q^{69} -4.98241 q^{71} -7.58880 q^{73} +12.8152 q^{75} +2.41260 q^{77} +11.5038 q^{79} -1.07304 q^{81} -2.71956 q^{83} -0.600167 q^{85} -19.2675 q^{87} +3.28261 q^{89} +28.2745 q^{91} +17.1957 q^{93} -13.0870 q^{97} +2.65815 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 6 q + 3 q^{3} + 2 q^{5} - 2 q^{7} + 9 q^{9} - 5 q^{11} + 10 q^{13} + 4 q^{17} + 23 q^{21} - 15 q^{23} + 12 q^{25} + 18 q^{27} - 7 q^{29} + 5 q^{31} - q^{33} - 38 q^{35} + 17 q^{37} + 37 q^{39} + 4 q^{41}+ \cdots - 47 q^{99}+O(q^{100}) $ 6 * q + 3 * q^3 + 2 * q^5 - 2 * q^7 + 9 * q^9 - 5 * q^11 + 10 * q^13 + 4 * q^17 + 23 * q^21 - 15 * q^23 + 12 * q^25 + 18 * q^27 - 7 * q^29 + 5 * q^31 - q^33 - 38 * q^35 + 17 * q^37 + 37 * q^39 + 4 * q^41 - 11 * q^43 + 6 * q^45 + 18 * q^47 + 20 * q^49 + 48 * q^51 - 7 * q^53 - 42 * q^55 + q^59 - 4 * q^61 + 19 * q^63 + 6 * q^65 + 20 * q^67 + 6 * q^69 - 6 * q^71 - 2 * q^73 + 19 * q^75 + 41 * q^77 + 8 * q^79 - 6 * q^81 - 22 * q^83 + 32 * q^85 + 29 * q^87 + 14 * q^89 + 26 * q^91 - 5 * q^93 + 41 * q^97 - 47 * 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
$3$	2.77311	1.60106	0.800528	−	0.599296i	$-0.204553\pi$
$3$	2.77311	1.60106	0.800528	+	0.599296i	$0.204553\pi$
$4$	0	0
$5$	−3.10181	−1.38717	−0.693586	−	0.720373i	$-0.743970\pi$
$5$	−3.10181	−1.38717	−0.693586	+	0.720373i	$0.743970\pi$
$6$	0	0
$7$	4.25689	1.60895	0.804476	−	0.593985i	$-0.202446\pi$
$7$	4.25689	1.60895	0.804476	+	0.593985i	$0.202446\pi$
$8$	0	0
$9$	4.69013	1.56338
$10$	0	0
$11$	0.566753	0.170882	0.0854412	−	0.996343i	$-0.472770\pi$
$11$	0.566753	0.170882	0.0854412	+	0.996343i	$0.472770\pi$
$12$	0	0
$13$	6.64206	1.84218	0.921088	−	0.389354i	$-0.127302\pi$
$13$	6.64206	1.84218	0.921088	+	0.389354i	$0.127302\pi$
$14$	0	0
$15$	−8.60166	−2.22094
$16$	0	0
$17$	0.193489	0.0469280	0.0234640	−	0.999725i	$-0.492530\pi$
$17$	0.193489	0.0469280	0.0234640	+	0.999725i	$0.492530\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	11.8048	2.57602
$22$	0	0
$23$	−1.91382	−0.399059	−0.199529	−	0.979892i	$-0.563941\pi$
$23$	−1.91382	−0.399059	−0.199529	+	0.979892i	$0.563941\pi$
$24$	0	0
$25$	4.62124	0.924248
$26$	0	0
$27$	4.68693	0.902000
$28$	0	0
$29$	−6.94796	−1.29020	−0.645102	−	0.764096i	$-0.723185\pi$
$29$	−6.94796	−1.29020	−0.645102	+	0.764096i	$0.723185\pi$
$30$	0	0
$31$	6.20086	1.11371	0.556854	−	0.830611i	$-0.312008\pi$
$31$	6.20086	1.11371	0.556854	+	0.830611i	$0.312008\pi$
$32$	0	0
$33$	1.57167	0.273592
$34$	0	0
$35$	−13.2041	−2.23189
$36$	0	0
$37$	−4.41517	−0.725849	−0.362925	−	0.931818i	$-0.618222\pi$
$37$	−4.41517	−0.725849	−0.362925	+	0.931818i	$0.618222\pi$
$38$	0	0
$39$	18.4192	2.94943
$40$	0	0
$41$	−6.18678	−0.966213	−0.483106	−	0.875562i	$-0.660492\pi$
$41$	−6.18678	−0.966213	−0.483106	+	0.875562i	$0.660492\pi$
$42$	0	0
$43$	2.00935	0.306424	0.153212	−	0.988193i	$-0.451038\pi$
$43$	2.00935	0.306424	0.153212	+	0.988193i	$0.451038\pi$
$44$	0	0
$45$	−14.5479	−2.16868
$46$	0	0
$47$	11.4545	1.67082	0.835408	−	0.549630i	$-0.185231\pi$
$47$	11.4545	1.67082	0.835408	+	0.549630i	$0.185231\pi$
$48$	0	0
$49$	11.1211	1.58873
$50$	0	0
$51$	0.536566	0.0751343
$52$	0	0
$53$	0.750308	0.103063	0.0515314	−	0.998671i	$-0.483590\pi$
$53$	0.750308	0.103063	0.0515314	+	0.998671i	$0.483590\pi$
$54$	0	0
$55$	−1.75796	−0.237043
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−0.815996	−0.106234	−0.0531168	−	0.998588i	$-0.516916\pi$
$59$	−0.815996	−0.106234	−0.0531168	+	0.998588i	$0.516916\pi$
$60$	0	0
$61$	0.738209	0.0945180	0.0472590	−	0.998883i	$-0.484951\pi$
$61$	0.738209	0.0945180	0.0472590	+	0.998883i	$0.484951\pi$
$62$	0	0
$63$	19.9654	2.51540
$64$	0	0
$65$	−20.6024	−2.55542
$66$	0	0
$67$	7.41886	0.906358	0.453179	−	0.891420i	$-0.350290\pi$
$67$	7.41886	0.906358	0.453179	+	0.891420i	$0.350290\pi$
$68$	0	0
$69$	−5.30723	−0.638915
$70$	0	0
$71$	−4.98241	−0.591303	−0.295651	−	0.955296i	$-0.595537\pi$
$71$	−4.98241	−0.591303	−0.295651	+	0.955296i	$0.595537\pi$
$72$	0	0
$73$	−7.58880	−0.888202	−0.444101	−	0.895977i	$-0.646477\pi$
$73$	−7.58880	−0.888202	−0.444101	+	0.895977i	$0.646477\pi$
$74$	0	0
$75$	12.8152	1.47977
$76$	0	0
$77$	2.41260	0.274942
$78$	0	0
$79$	11.5038	1.29428	0.647142	−	0.762370i	$-0.275964\pi$
$79$	11.5038	1.29428	0.647142	+	0.762370i	$0.275964\pi$
$80$	0	0
$81$	−1.07304	−0.119227
$82$	0	0
$83$	−2.71956	−0.298511	−0.149255	−	0.988799i	$-0.547688\pi$
$83$	−2.71956	−0.298511	−0.149255	+	0.988799i	$0.547688\pi$
$84$	0	0
$85$	−0.600167	−0.0650972
$86$	0	0
$87$	−19.2675	−2.06569
$88$	0	0
$89$	3.28261	0.347956	0.173978	−	0.984750i	$-0.444338\pi$
$89$	3.28261	0.347956	0.173978	+	0.984750i	$0.444338\pi$
$90$	0	0
$91$	28.2745	2.96397
$92$	0	0
$93$	17.1957	1.78311
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.0870	−1.32878	−0.664389	−	0.747387i	$-0.731308\pi$
$97$	−13.0870	−1.32878	−0.664389	+	0.747387i	$0.731308\pi$
$98$	0	0
$99$	2.65815	0.267154
$100$	0	0
$101$	6.11668	0.608632	0.304316	−	0.952571i	$-0.401572\pi$
$101$	6.11668	0.608632	0.304316	+	0.952571i	$0.401572\pi$
$102$	0	0
$103$	11.8798	1.17055	0.585277	−	0.810833i	$-0.300986\pi$
$103$	11.8798	1.17055	0.585277	+	0.810833i	$0.300986\pi$
$104$	0	0
$105$	−36.6163	−3.57339
$106$	0	0
$107$	1.05923	0.102400	0.0512000	−	0.998688i	$-0.483695\pi$
$107$	1.05923	0.102400	0.0512000	+	0.998688i	$0.483695\pi$
$108$	0	0
$109$	−15.3255	−1.46791	−0.733957	−	0.679196i	$-0.762329\pi$
$109$	−15.3255	−1.46791	−0.733957	+	0.679196i	$0.762329\pi$
$110$	0	0
$111$	−12.2437	−1.16213
$112$	0	0
$113$	16.1127	1.51576	0.757878	−	0.652397i	$-0.226236\pi$
$113$	16.1127	1.51576	0.757878	+	0.652397i	$0.226236\pi$
$114$	0	0
$115$	5.93631	0.553564
$116$	0	0
$117$	31.1522	2.88002
$118$	0	0
$119$	0.823661	0.0755049
$120$	0	0
$121$	−10.6788	−0.970799
$122$	0	0
$123$	−17.1566	−1.54696
$124$	0	0
$125$	1.17484	0.105081
$126$	0	0
$127$	10.2862	0.912749	0.456374	−	0.889788i	$-0.349148\pi$
$127$	10.2862	0.912749	0.456374	+	0.889788i	$0.349148\pi$
$128$	0	0
$129$	5.57216	0.490601
$130$	0	0
$131$	−16.1635	−1.41221	−0.706106	−	0.708106i	$-0.749550\pi$
$131$	−16.1635	−1.41221	−0.706106	+	0.708106i	$0.749550\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−14.5380	−1.25123
$136$	0	0
$137$	−6.56182	−0.560614	−0.280307	−	0.959910i	$-0.590436\pi$
$137$	−6.56182	−0.560614	−0.280307	+	0.959910i	$0.590436\pi$
$138$	0	0
$139$	−15.3362	−1.30080	−0.650399	−	0.759593i	$-0.725398\pi$
$139$	−15.3362	−1.30080	−0.650399	+	0.759593i	$0.725398\pi$
$140$	0	0
$141$	31.7647	2.67507
$142$	0	0
$143$	3.76441	0.314796
$144$	0	0
$145$	21.5513	1.78974
$146$	0	0
$147$	30.8400	2.54364
$148$	0	0
$149$	21.4978	1.76117	0.880586	−	0.473886i	$-0.157149\pi$
$149$	21.4978	1.76117	0.880586	+	0.473886i	$0.157149\pi$
$150$	0	0
$151$	−7.01136	−0.570576	−0.285288	−	0.958442i	$-0.592089\pi$
$151$	−7.01136	−0.570576	−0.285288	+	0.958442i	$0.592089\pi$
$152$	0	0
$153$	0.907489	0.0733662
$154$	0	0
$155$	−19.2339	−1.54490
$156$	0	0
$157$	−4.73713	−0.378064	−0.189032	−	0.981971i	$-0.560535\pi$
$157$	−4.73713	−0.378064	−0.189032	+	0.981971i	$0.560535\pi$
$158$	0	0
$159$	2.08069	0.165009
$160$	0	0
$161$	−8.14691	−0.642067
$162$	0	0
$163$	5.54839	0.434583	0.217292	−	0.976107i	$-0.430278\pi$
$163$	5.54839	0.434583	0.217292	+	0.976107i	$0.430278\pi$
$164$	0	0
$165$	−4.87502	−0.379520
$166$	0	0
$167$	8.59351	0.664986	0.332493	−	0.943106i	$-0.392110\pi$
$167$	8.59351	0.664986	0.332493	+	0.943106i	$0.392110\pi$
$168$	0	0
$169$	31.1170	2.39361
$170$	0	0
$171$	0	0
$172$	0	0
$173$	8.80204	0.669207	0.334603	−	0.942359i	$-0.391398\pi$
$173$	8.80204	0.669207	0.334603	+	0.942359i	$0.391398\pi$
$174$	0	0
$175$	19.6721	1.48707
$176$	0	0
$177$	−2.26284	−0.170086
$178$	0	0
$179$	12.2102	0.912634	0.456317	−	0.889817i	$-0.349168\pi$
$179$	12.2102	0.912634	0.456317	+	0.889817i	$0.349168\pi$
$180$	0	0
$181$	19.0572	1.41651	0.708257	−	0.705955i	$-0.249482\pi$
$181$	19.0572	1.41651	0.708257	+	0.705955i	$0.249482\pi$
$182$	0	0
$183$	2.04713	0.151329
$184$	0	0
$185$	13.6950	1.00688
$186$	0	0
$187$	0.109660	0.00801917
$188$	0	0
$189$	19.9517	1.45127
$190$	0	0
$191$	5.77756	0.418050	0.209025	−	0.977910i	$-0.432971\pi$
$191$	5.77756	0.418050	0.209025	+	0.977910i	$0.432971\pi$
$192$	0	0
$193$	−1.83705	−0.132234	−0.0661168	−	0.997812i	$-0.521061\pi$
$193$	−1.83705	−0.132234	−0.0661168	+	0.997812i	$0.521061\pi$
$194$	0	0
$195$	−57.1328	−4.09136
$196$	0	0
$197$	−16.2066	−1.15467	−0.577337	−	0.816506i	$-0.695908\pi$
$197$	−16.2066	−1.15467	−0.577337	+	0.816506i	$0.695908\pi$
$198$	0	0
$199$	−10.5385	−0.747057	−0.373529	−	0.927619i	$-0.621852\pi$
$199$	−10.5385	−0.747057	−0.373529	+	0.927619i	$0.621852\pi$
$200$	0	0
$201$	20.5733	1.45113
$202$	0	0
$203$	−29.5767	−2.07588
$204$	0	0
$205$	19.1902	1.34030
$206$	0	0
$207$	−8.97607	−0.623880
$208$	0	0
$209$	0	0
$210$	0	0
$211$	11.4598	0.788922	0.394461	−	0.918913i	$-0.370931\pi$
$211$	11.4598	0.788922	0.394461	+	0.918913i	$0.370931\pi$
$212$	0	0
$213$	−13.8168	−0.946709
$214$	0	0
$215$	−6.23264	−0.425062
$216$	0	0
$217$	26.3964	1.79190
$218$	0	0
$219$	−21.0446	−1.42206
$220$	0	0
$221$	1.28517	0.0864496
$222$	0	0
$223$	9.67115	0.647628	0.323814	−	0.946121i	$-0.395035\pi$
$223$	9.67115	0.647628	0.323814	+	0.946121i	$0.395035\pi$
$224$	0	0
$225$	21.6742	1.44495
$226$	0	0
$227$	13.7650	0.913617	0.456808	−	0.889565i	$-0.348992\pi$
$227$	13.7650	0.913617	0.456808	+	0.889565i	$0.348992\pi$
$228$	0	0
$229$	−24.5777	−1.62414	−0.812071	−	0.583559i	$-0.801660\pi$
$229$	−24.5777	−1.62414	−0.812071	+	0.583559i	$0.801660\pi$
$230$	0	0
$231$	6.69041	0.440197
$232$	0	0
$233$	−13.6988	−0.897439	−0.448720	−	0.893673i	$-0.648120\pi$
$233$	−13.6988	−0.897439	−0.448720	+	0.893673i	$0.648120\pi$
$234$	0	0
$235$	−35.5298	−2.31771
$236$	0	0
$237$	31.9014	2.07222
$238$	0	0
$239$	29.0433	1.87866	0.939328	−	0.343020i	$-0.111450\pi$
$239$	29.0433	1.87866	0.939328	+	0.343020i	$0.111450\pi$
$240$	0	0
$241$	−16.0586	−1.03443	−0.517214	−	0.855856i	$-0.673031\pi$
$241$	−16.0586	−1.03443	−0.517214	+	0.855856i	$0.673031\pi$
$242$	0	0
$243$	−17.0364	−1.09289
$244$	0	0
$245$	−34.4955	−2.20384
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−7.54164	−0.477932
$250$	0	0
$251$	−17.9373	−1.13219	−0.566096	−	0.824340i	$-0.691547\pi$
$251$	−17.9373	−1.13219	−0.566096	+	0.824340i	$0.691547\pi$
$252$	0	0
$253$	−1.08466	−0.0681922
$254$	0	0
$255$	−1.66433	−0.104224
$256$	0	0
$257$	−13.4227	−0.837283	−0.418642	−	0.908152i	$-0.637494\pi$
$257$	−13.4227	−0.837283	−0.418642	+	0.908152i	$0.637494\pi$
$258$	0	0
$259$	−18.7949	−1.16786
$260$	0	0
$261$	−32.5869	−2.01708
$262$	0	0
$263$	−23.4920	−1.44858	−0.724291	−	0.689495i	$-0.757833\pi$
$263$	−23.4920	−1.44858	−0.724291	+	0.689495i	$0.757833\pi$
$264$	0	0
$265$	−2.32731	−0.142966
$266$	0	0
$267$	9.10304	0.557097
$268$	0	0
$269$	−27.0042	−1.64647	−0.823237	−	0.567698i	$-0.807834\pi$
$269$	−27.0042	−1.64647	−0.823237	+	0.567698i	$0.807834\pi$
$270$	0	0
$271$	0.668554	0.0406117	0.0203059	−	0.999794i	$-0.493536\pi$
$271$	0.668554	0.0406117	0.0203059	+	0.999794i	$0.493536\pi$
$272$	0	0
$273$	78.4083	4.74549
$274$	0	0
$275$	2.61910	0.157938
$276$	0	0
$277$	−20.4557	−1.22907	−0.614533	−	0.788891i	$-0.710656\pi$
$277$	−20.4557	−1.22907	−0.614533	+	0.788891i	$0.710656\pi$
$278$	0	0
$279$	29.0829	1.74115
$280$	0	0
$281$	−3.85165	−0.229770	−0.114885	−	0.993379i	$-0.536650\pi$
$281$	−3.85165	−0.229770	−0.114885	+	0.993379i	$0.536650\pi$
$282$	0	0
$283$	−23.4241	−1.39242	−0.696208	−	0.717840i	$-0.745131\pi$
$283$	−23.4241	−1.39242	−0.696208	+	0.717840i	$0.745131\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−26.3364	−1.55459
$288$	0	0
$289$	−16.9626	−0.997798
$290$	0	0
$291$	−36.2916	−2.12745
$292$	0	0
$293$	−5.80219	−0.338967	−0.169484	−	0.985533i	$-0.554210\pi$
$293$	−5.80219	−0.338967	−0.169484	+	0.985533i	$0.554210\pi$
$294$	0	0
$295$	2.53107	0.147364
$296$	0	0
$297$	2.65633	0.154136
$298$	0	0
$299$	−12.7117	−0.735137
$300$	0	0
$301$	8.55359	0.493021
$302$	0	0
$303$	16.9622	0.974453
$304$	0	0
$305$	−2.28979	−0.131113
$306$	0	0
$307$	−9.52173	−0.543434	−0.271717	−	0.962377i	$-0.587591\pi$
$307$	−9.52173	−0.543434	−0.271717	+	0.962377i	$0.587591\pi$
$308$	0	0
$309$	32.9441	1.87412
$310$	0	0
$311$	−9.73990	−0.552299	−0.276149	−	0.961115i	$-0.589058\pi$
$311$	−9.73990	−0.552299	−0.276149	+	0.961115i	$0.589058\pi$
$312$	0	0
$313$	−18.3536	−1.03741	−0.518703	−	0.854955i	$-0.673585\pi$
$313$	−18.3536	−1.03741	−0.518703	+	0.854955i	$0.673585\pi$
$314$	0	0
$315$	−61.9289	−3.48930
$316$	0	0
$317$	−26.8441	−1.50771	−0.753857	−	0.657039i	$-0.771809\pi$
$317$	−26.8441	−1.50771	−0.753857	+	0.657039i	$0.771809\pi$
$318$	0	0
$319$	−3.93778	−0.220473
$320$	0	0
$321$	2.93737	0.163948
$322$	0	0
$323$	0	0
$324$	0	0
$325$	30.6946	1.70263
$326$	0	0
$327$	−42.4992	−2.35021
$328$	0	0
$329$	48.7607	2.68826
$330$	0	0
$331$	−27.2895	−1.49997	−0.749983	−	0.661457i	$-0.769938\pi$
$331$	−27.2895	−1.49997	−0.749983	+	0.661457i	$0.769938\pi$
$332$	0	0
$333$	−20.7077	−1.13478
$334$	0	0
$335$	−23.0119	−1.25727
$336$	0	0
$337$	5.58741	0.304366	0.152183	−	0.988352i	$-0.451370\pi$
$337$	5.58741	0.304366	0.152183	+	0.988352i	$0.451370\pi$
$338$	0	0
$339$	44.6823	2.42681
$340$	0	0
$341$	3.51436	0.190313
$342$	0	0
$343$	17.5430	0.947235
$344$	0	0
$345$	16.4620	0.886286
$346$	0	0
$347$	8.88534	0.476990	0.238495	−	0.971144i	$-0.423346\pi$
$347$	8.88534	0.476990	0.238495	+	0.971144i	$0.423346\pi$
$348$	0	0
$349$	−18.2283	−0.975738	−0.487869	−	0.872917i	$-0.662226\pi$
$349$	−18.2283	−0.975738	−0.487869	+	0.872917i	$0.662226\pi$
$350$	0	0
$351$	31.1309	1.66164
$352$	0	0
$353$	16.6364	0.885466	0.442733	−	0.896654i	$-0.354009\pi$
$353$	16.6364	0.885466	0.442733	+	0.896654i	$0.354009\pi$
$354$	0	0
$355$	15.4545	0.820239
$356$	0	0
$357$	2.28410	0.120887
$358$	0	0
$359$	27.3176	1.44177	0.720884	−	0.693056i	$-0.243736\pi$
$359$	27.3176	1.44177	0.720884	+	0.693056i	$0.243736\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−29.6135	−1.55430
$364$	0	0
$365$	23.5390	1.23209
$366$	0	0
$367$	−6.89847	−0.360097	−0.180048	−	0.983658i	$-0.557625\pi$
$367$	−6.89847	−0.360097	−0.180048	+	0.983658i	$0.557625\pi$
$368$	0	0
$369$	−29.0168	−1.51056
$370$	0	0
$371$	3.19398	0.165823
$372$	0	0
$373$	9.75228	0.504954	0.252477	−	0.967603i	$-0.418755\pi$
$373$	9.75228	0.504954	0.252477	+	0.967603i	$0.418755\pi$
$374$	0	0
$375$	3.25796	0.168240
$376$	0	0
$377$	−46.1488	−2.37678
$378$	0	0
$379$	−9.32085	−0.478780	−0.239390	−	0.970924i	$-0.576947\pi$
$379$	−9.32085	−0.478780	−0.239390	+	0.970924i	$0.576947\pi$
$380$	0	0
$381$	28.5246	1.46136
$382$	0	0
$383$	−2.07285	−0.105917	−0.0529587	−	0.998597i	$-0.516865\pi$
$383$	−2.07285	−0.105917	−0.0529587	+	0.998597i	$0.516865\pi$
$384$	0	0
$385$	−7.48344	−0.381392
$386$	0	0
$387$	9.42414	0.479056
$388$	0	0
$389$	3.51604	0.178270	0.0891352	−	0.996020i	$-0.471590\pi$
$389$	3.51604	0.178270	0.0891352	+	0.996020i	$0.471590\pi$
$390$	0	0
$391$	−0.370303	−0.0187270
$392$	0	0
$393$	−44.8232	−2.26103
$394$	0	0
$395$	−35.6828	−1.79539
$396$	0	0
$397$	1.62820	0.0817172	0.0408586	−	0.999165i	$-0.486991\pi$
$397$	1.62820	0.0817172	0.0408586	+	0.999165i	$0.486991\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	28.8962	1.44301	0.721504	−	0.692411i	$-0.243451\pi$
$401$	28.8962	1.44301	0.721504	+	0.692411i	$0.243451\pi$
$402$	0	0
$403$	41.1865	2.05165
$404$	0	0
$405$	3.32837	0.165388
$406$	0	0
$407$	−2.50231	−0.124035
$408$	0	0
$409$	23.9141	1.18248	0.591239	−	0.806497i	$-0.298639\pi$
$409$	23.9141	1.18248	0.591239	+	0.806497i	$0.298639\pi$
$410$	0	0
$411$	−18.1966	−0.897574
$412$	0	0
$413$	−3.47360	−0.170925
$414$	0	0
$415$	8.43557	0.414086
$416$	0	0
$417$	−42.5289	−2.08265
$418$	0	0
$419$	−16.3240	−0.797478	−0.398739	−	0.917064i	$-0.630552\pi$
$419$	−16.3240	−0.797478	−0.398739	+	0.917064i	$0.630552\pi$
$420$	0	0
$421$	1.55297	0.0756870	0.0378435	−	0.999284i	$-0.487951\pi$
$421$	1.55297	0.0756870	0.0378435	+	0.999284i	$0.487951\pi$
$422$	0	0
$423$	53.7233	2.61212
$424$	0	0
$425$	0.894159	0.0433731
$426$	0	0
$427$	3.14247	0.152075
$428$	0	0
$429$	10.4391	0.504005
$430$	0	0
$431$	1.14815	0.0553044	0.0276522	−	0.999618i	$-0.491197\pi$
$431$	1.14815	0.0553044	0.0276522	+	0.999618i	$0.491197\pi$
$432$	0	0
$433$	2.98715	0.143553	0.0717767	−	0.997421i	$-0.477133\pi$
$433$	2.98715	0.143553	0.0717767	+	0.997421i	$0.477133\pi$
$434$	0	0
$435$	59.7640	2.86547
$436$	0	0
$437$	0	0
$438$	0	0
$439$	41.6769	1.98913	0.994566	−	0.104110i	$-0.0331995\pi$
$439$	41.6769	1.98913	0.994566	+	0.104110i	$0.0331995\pi$
$440$	0	0
$441$	52.1594	2.48378
$442$	0	0
$443$	19.4084	0.922123	0.461061	−	0.887368i	$-0.347469\pi$
$443$	19.4084	0.922123	0.461061	+	0.887368i	$0.347469\pi$
$444$	0	0
$445$	−10.1820	−0.482675
$446$	0	0
$447$	59.6159	2.81973
$448$	0	0
$449$	0.240767	0.0113625	0.00568126	−	0.999984i	$-0.498192\pi$
$449$	0.240767	0.0113625	0.00568126	+	0.999984i	$0.498192\pi$
$450$	0	0
$451$	−3.50638	−0.165109
$452$	0	0
$453$	−19.4433	−0.913524
$454$	0	0
$455$	−87.7022	−4.11154
$456$	0	0
$457$	−6.02160	−0.281679	−0.140839	−	0.990032i	$-0.544980\pi$
$457$	−6.02160	−0.281679	−0.140839	+	0.990032i	$0.544980\pi$
$458$	0	0
$459$	0.906869	0.0423290
$460$	0	0
$461$	18.7609	0.873783	0.436892	−	0.899514i	$-0.356079\pi$
$461$	18.7609	0.873783	0.436892	+	0.899514i	$0.356079\pi$
$462$	0	0
$463$	0.769304	0.0357526	0.0178763	−	0.999840i	$-0.494309\pi$
$463$	0.769304	0.0357526	0.0178763	+	0.999840i	$0.494309\pi$
$464$	0	0
$465$	−53.3377	−2.47348
$466$	0	0
$467$	−3.83385	−0.177409	−0.0887047	−	0.996058i	$-0.528273\pi$
$467$	−3.83385	−0.177409	−0.0887047	+	0.996058i	$0.528273\pi$
$468$	0	0
$469$	31.5813	1.45829
$470$	0	0
$471$	−13.1366	−0.605302
$472$	0	0
$473$	1.13881	0.0523624
$474$	0	0
$475$	0	0
$476$	0	0
$477$	3.51904	0.161126
$478$	0	0
$479$	−38.8045	−1.77302	−0.886512	−	0.462706i	$-0.846879\pi$
$479$	−38.8045	−1.77302	−0.886512	+	0.462706i	$0.846879\pi$
$480$	0	0
$481$	−29.3258	−1.33714
$482$	0	0
$483$	−22.5923	−1.02798
$484$	0	0
$485$	40.5933	1.84325
$486$	0	0
$487$	−11.1940	−0.507247	−0.253623	−	0.967303i	$-0.581622\pi$
$487$	−11.1940	−0.507247	−0.253623	+	0.967303i	$0.581622\pi$
$488$	0	0
$489$	15.3863	0.695792
$490$	0	0
$491$	−17.6829	−0.798019	−0.399010	−	0.916947i	$-0.630646\pi$
$491$	−17.6829	−0.798019	−0.399010	+	0.916947i	$0.630646\pi$
$492$	0	0
$493$	−1.34435	−0.0605467
$494$	0	0
$495$	−8.24508	−0.370589
$496$	0	0
$497$	−21.2095	−0.951378
$498$	0	0
$499$	−4.27896	−0.191552	−0.0957762	−	0.995403i	$-0.530533\pi$
$499$	−4.27896	−0.191552	−0.0957762	+	0.995403i	$0.530533\pi$
$500$	0	0
$501$	23.8307	1.06468
$502$	0	0
$503$	−30.3497	−1.35323	−0.676613	−	0.736338i	$-0.736553\pi$
$503$	−30.3497	−1.35323	−0.676613	+	0.736338i	$0.736553\pi$
$504$	0	0
$505$	−18.9728	−0.844278
$506$	0	0
$507$	86.2907	3.83231
$508$	0	0
$509$	26.4068	1.17046	0.585229	−	0.810868i	$-0.301004\pi$
$509$	26.4068	1.17046	0.585229	+	0.810868i	$0.301004\pi$
$510$	0	0
$511$	−32.3047	−1.42907
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−36.8490	−1.62376
$516$	0	0
$517$	6.49190	0.285513
$518$	0	0
$519$	24.4090	1.07144
$520$	0	0
$521$	−15.7283	−0.689067	−0.344534	−	0.938774i	$-0.611963\pi$
$521$	−15.7283	−0.689067	−0.344534	+	0.938774i	$0.611963\pi$
$522$	0	0
$523$	−28.1082	−1.22909	−0.614544	−	0.788883i	$-0.710660\pi$
$523$	−28.1082	−1.22909	−0.614544	+	0.788883i	$0.710660\pi$
$524$	0	0
$525$	54.5529	2.38088
$526$	0	0
$527$	1.19980	0.0522640
$528$	0	0
$529$	−19.3373	−0.840752
$530$	0	0
$531$	−3.82713	−0.166083
$532$	0	0
$533$	−41.0930	−1.77993
$534$	0	0
$535$	−3.28554	−0.142046
$536$	0	0
$537$	33.8603	1.46118
$538$	0	0
$539$	6.30291	0.271486
$540$	0	0
$541$	44.4767	1.91220	0.956101	−	0.293039i	$-0.0946665\pi$
$541$	44.4767	1.91220	0.956101	+	0.293039i	$0.0946665\pi$
$542$	0	0
$543$	52.8478	2.26792
$544$	0	0
$545$	47.5367	2.03625
$546$	0	0
$547$	4.65936	0.199220	0.0996100	−	0.995027i	$-0.468240\pi$
$547$	4.65936	0.199220	0.0996100	+	0.995027i	$0.468240\pi$
$548$	0	0
$549$	3.46230	0.147767
$550$	0	0
$551$	0	0
$552$	0	0
$553$	48.9706	2.08244
$554$	0	0
$555$	37.9778	1.61207
$556$	0	0
$557$	−40.2390	−1.70498	−0.852491	−	0.522741i	$-0.824909\pi$
$557$	−40.2390	−1.70498	−0.852491	+	0.522741i	$0.824909\pi$
$558$	0	0
$559$	13.3462	0.564486
$560$	0	0
$561$	0.304100	0.0128391
$562$	0	0
$563$	28.9695	1.22092	0.610460	−	0.792047i	$-0.290985\pi$
$563$	28.9695	1.22092	0.610460	+	0.792047i	$0.290985\pi$
$564$	0	0
$565$	−49.9786	−2.10261
$566$	0	0
$567$	−4.56781	−0.191830
$568$	0	0
$569$	22.3774	0.938110	0.469055	−	0.883169i	$-0.344595\pi$
$569$	22.3774	0.938110	0.469055	+	0.883169i	$0.344595\pi$
$570$	0	0
$571$	−16.9522	−0.709429	−0.354714	−	0.934975i	$-0.615422\pi$
$571$	−16.9522	−0.709429	−0.354714	+	0.934975i	$0.615422\pi$
$572$	0	0
$573$	16.0218	0.669321
$574$	0	0
$575$	−8.84422	−0.368829
$576$	0	0
$577$	−18.7130	−0.779031	−0.389516	−	0.921020i	$-0.627358\pi$
$577$	−18.7130	−0.779031	−0.389516	+	0.921020i	$0.627358\pi$
$578$	0	0
$579$	−5.09433	−0.211713
$580$	0	0
$581$	−11.5769	−0.480289
$582$	0	0
$583$	0.425239	0.0176116
$584$	0	0
$585$	−96.6281	−3.99508
$586$	0	0
$587$	−42.4179	−1.75077	−0.875387	−	0.483423i	$-0.839393\pi$
$587$	−42.4179	−1.75077	−0.875387	+	0.483423i	$0.839393\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−44.9428	−1.84870
$592$	0	0
$593$	9.06222	0.372141	0.186070	−	0.982536i	$-0.440425\pi$
$593$	9.06222	0.372141	0.186070	+	0.982536i	$0.440425\pi$
$594$	0	0
$595$	−2.55484	−0.104738
$596$	0	0
$597$	−29.2245	−1.19608
$598$	0	0
$599$	−13.3157	−0.544064	−0.272032	−	0.962288i	$-0.587696\pi$
$599$	−13.3157	−0.544064	−0.272032	+	0.962288i	$0.587696\pi$
$600$	0	0
$601$	−10.6534	−0.434562	−0.217281	−	0.976109i	$-0.569719\pi$
$601$	−10.6534	−0.434562	−0.217281	+	0.976109i	$0.569719\pi$
$602$	0	0
$603$	34.7955	1.41698
$604$	0	0
$605$	33.1236	1.34667
$606$	0	0
$607$	−19.4759	−0.790504	−0.395252	−	0.918573i	$-0.629343\pi$
$607$	−19.4759	−0.790504	−0.395252	+	0.918573i	$0.629343\pi$
$608$	0	0
$609$	−82.0194	−3.32359
$610$	0	0
$611$	76.0818	3.07794
$612$	0	0
$613$	−4.01131	−0.162015	−0.0810076	−	0.996713i	$-0.525814\pi$
$613$	−4.01131	−0.162015	−0.0810076	+	0.996713i	$0.525814\pi$
$614$	0	0
$615$	53.2166	2.14590
$616$	0	0
$617$	−13.5187	−0.544242	−0.272121	−	0.962263i	$-0.587725\pi$
$617$	−13.5187	−0.544242	−0.272121	+	0.962263i	$0.587725\pi$
$618$	0	0
$619$	17.8029	0.715561	0.357780	−	0.933806i	$-0.383534\pi$
$619$	17.8029	0.715561	0.357780	+	0.933806i	$0.383534\pi$
$620$	0	0
$621$	−8.96993	−0.359951
$622$	0	0
$623$	13.9737	0.559845
$624$	0	0
$625$	−26.7503	−1.07001
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−0.854287	−0.0340626
$630$	0	0
$631$	5.84534	0.232699	0.116350	−	0.993208i	$-0.462881\pi$
$631$	5.84534	0.232699	0.116350	+	0.993208i	$0.462881\pi$
$632$	0	0
$633$	31.7792	1.26311
$634$	0	0
$635$	−31.9057	−1.26614
$636$	0	0
$637$	73.8670	2.92672
$638$	0	0
$639$	−23.3682	−0.924430
$640$	0	0
$641$	0.381647	0.0150741	0.00753707	−	0.999972i	$-0.497601\pi$
$641$	0.381647	0.0150741	0.00753707	+	0.999972i	$0.497601\pi$
$642$	0	0
$643$	17.2648	0.680858	0.340429	−	0.940270i	$-0.389428\pi$
$643$	17.2648	0.680858	0.340429	+	0.940270i	$0.389428\pi$
$644$	0	0
$645$	−17.2838	−0.680548
$646$	0	0
$647$	0.270066	0.0106174	0.00530870	−	0.999986i	$-0.498310\pi$
$647$	0.270066	0.0106174	0.00530870	+	0.999986i	$0.498310\pi$
$648$	0	0
$649$	−0.462468	−0.0181535
$650$	0	0
$651$	73.2000	2.86893
$652$	0	0
$653$	32.3183	1.26471	0.632356	−	0.774678i	$-0.282088\pi$
$653$	32.3183	1.26471	0.632356	+	0.774678i	$0.282088\pi$
$654$	0	0
$655$	50.1362	1.95898
$656$	0	0
$657$	−35.5925	−1.38859
$658$	0	0
$659$	−25.7196	−1.00189	−0.500947	−	0.865478i	$-0.667015\pi$
$659$	−25.7196	−1.00189	−0.500947	+	0.865478i	$0.667015\pi$
$660$	0	0
$661$	−24.6610	−0.959203	−0.479601	−	0.877487i	$-0.659219\pi$
$661$	−24.6610	−0.959203	−0.479601	+	0.877487i	$0.659219\pi$
$662$	0	0
$663$	3.56390	0.138411
$664$	0	0
$665$	0	0
$666$	0	0
$667$	13.2971	0.514867
$668$	0	0
$669$	26.8192	1.03689
$670$	0	0
$671$	0.418382	0.0161515
$672$	0	0
$673$	−8.46478	−0.326293	−0.163147	−	0.986602i	$-0.552164\pi$
$673$	−8.46478	−0.326293	−0.163147	+	0.986602i	$0.552164\pi$
$674$	0	0
$675$	21.6594	0.833672
$676$	0	0
$677$	−34.0765	−1.30967	−0.654834	−	0.755773i	$-0.727261\pi$
$677$	−34.0765	−1.30967	−0.654834	+	0.755773i	$0.727261\pi$
$678$	0	0
$679$	−55.7097	−2.13794
$680$	0	0
$681$	38.1719	1.46275
$682$	0	0
$683$	11.7302	0.448843	0.224422	−	0.974492i	$-0.427951\pi$
$683$	11.7302	0.448843	0.224422	+	0.974492i	$0.427951\pi$
$684$	0	0
$685$	20.3535	0.777668
$686$	0	0
$687$	−68.1567	−2.60034
$688$	0	0
$689$	4.98359	0.189860
$690$	0	0
$691$	−5.90586	−0.224669	−0.112335	−	0.993670i	$-0.535833\pi$
$691$	−5.90586	−0.224669	−0.112335	+	0.993670i	$0.535833\pi$
$692$	0	0
$693$	11.3154	0.429838
$694$	0	0
$695$	47.5700	1.80443
$696$	0	0
$697$	−1.19707	−0.0453424
$698$	0	0
$699$	−37.9883	−1.43685
$700$	0	0
$701$	44.7194	1.68903	0.844515	−	0.535532i	$-0.179889\pi$
$701$	44.7194	1.68903	0.844515	+	0.535532i	$0.179889\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−98.5281	−3.71078
$706$	0	0
$707$	26.0380	0.979260
$708$	0	0
$709$	−27.4732	−1.03178	−0.515890	−	0.856655i	$-0.672539\pi$
$709$	−27.4732	−1.03178	−0.515890	+	0.856655i	$0.672539\pi$
$710$	0	0
$711$	53.9546	2.02345
$712$	0	0
$713$	−11.8673	−0.444435
$714$	0	0
$715$	−11.6765	−0.436676
$716$	0	0
$717$	80.5403	3.00783
$718$	0	0
$719$	16.8919	0.629963	0.314981	−	0.949098i	$-0.398002\pi$
$719$	16.8919	0.629963	0.314981	+	0.949098i	$0.398002\pi$
$720$	0	0
$721$	50.5711	1.88337
$722$	0	0
$723$	−44.5323	−1.65618
$724$	0	0
$725$	−32.1082	−1.19247
$726$	0	0
$727$	−22.8934	−0.849070	−0.424535	−	0.905412i	$-0.639562\pi$
$727$	−22.8934	−0.849070	−0.424535	+	0.905412i	$0.639562\pi$
$728$	0	0
$729$	−44.0248	−1.63055
$730$	0	0
$731$	0.388788	0.0143798
$732$	0	0
$733$	−19.8206	−0.732090	−0.366045	−	0.930597i	$-0.619288\pi$
$733$	−19.8206	−0.732090	−0.366045	+	0.930597i	$0.619288\pi$
$734$	0	0
$735$	−95.6599	−3.52847
$736$	0	0
$737$	4.20466	0.154881
$738$	0	0
$739$	−4.29518	−0.158001	−0.0790003	−	0.996875i	$-0.525173\pi$
$739$	−4.29518	−0.158001	−0.0790003	+	0.996875i	$0.525173\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	3.02433	0.110952	0.0554759	−	0.998460i	$-0.482332\pi$
$743$	3.02433	0.110952	0.0554759	+	0.998460i	$0.482332\pi$
$744$	0	0
$745$	−66.6823	−2.44305
$746$	0	0
$747$	−12.7551	−0.466685
$748$	0	0
$749$	4.50904	0.164757
$750$	0	0
$751$	33.6947	1.22954	0.614768	−	0.788708i	$-0.289250\pi$
$751$	33.6947	1.22954	0.614768	+	0.788708i	$0.289250\pi$
$752$	0	0
$753$	−49.7420	−1.81270
$754$	0	0
$755$	21.7479	0.791488
$756$	0	0
$757$	6.70455	0.243681	0.121840	−	0.992550i	$-0.461120\pi$
$757$	6.70455	0.243681	0.121840	+	0.992550i	$0.461120\pi$
$758$	0	0
$759$	−3.00789	−0.109179
$760$	0	0
$761$	30.0829	1.09051	0.545253	−	0.838272i	$-0.316434\pi$
$761$	30.0829	1.09051	0.545253	+	0.838272i	$0.316434\pi$
$762$	0	0
$763$	−65.2388	−2.36180
$764$	0	0
$765$	−2.81486	−0.101772
$766$	0	0
$767$	−5.41989	−0.195701
$768$	0	0
$769$	51.6739	1.86341	0.931704	−	0.363217i	$-0.118322\pi$
$769$	51.6739	1.86341	0.931704	+	0.363217i	$0.118322\pi$
$770$	0	0
$771$	−37.2225	−1.34054
$772$	0	0
$773$	27.0587	0.973233	0.486617	−	0.873616i	$-0.338231\pi$
$773$	27.0587	0.973233	0.486617	+	0.873616i	$0.338231\pi$
$774$	0	0
$775$	28.6557	1.02934
$776$	0	0
$777$	−52.1203	−1.86980
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−2.82379	−0.101043
$782$	0	0
$783$	−32.5646	−1.16376
$784$	0	0
$785$	14.6937	0.524441
$786$	0	0
$787$	−8.22914	−0.293337	−0.146669	−	0.989186i	$-0.546855\pi$
$787$	−8.22914	−0.293337	−0.146669	+	0.989186i	$0.546855\pi$
$788$	0	0
$789$	−65.1460	−2.31926
$790$	0	0
$791$	68.5899	2.43878
$792$	0	0
$793$	4.90323	0.174119
$794$	0	0
$795$	−6.45390	−0.228896
$796$	0	0
$797$	−1.40735	−0.0498510	−0.0249255	−	0.999689i	$-0.507935\pi$
$797$	−1.40735	−0.0498510	−0.0249255	+	0.999689i	$0.507935\pi$
$798$	0	0
$799$	2.21633	0.0784080
$800$	0	0
$801$	15.3959	0.543987
$802$	0	0
$803$	−4.30097	−0.151778
$804$	0	0
$805$	25.2702	0.890657
$806$	0	0
$807$	−74.8855	−2.63609
$808$	0	0
$809$	22.0513	0.775284	0.387642	−	0.921810i	$-0.373290\pi$
$809$	22.0513	0.775284	0.387642	+	0.921810i	$0.373290\pi$
$810$	0	0
$811$	−45.2435	−1.58871	−0.794357	−	0.607451i	$-0.792192\pi$
$811$	−45.2435	−1.58871	−0.794357	+	0.607451i	$0.792192\pi$
$812$	0	0
$813$	1.85397	0.0650216
$814$	0	0
$815$	−17.2101	−0.602842
$816$	0	0
$817$	0	0
$818$	0	0
$819$	132.611	4.63381
$820$	0	0
$821$	6.37194	0.222382	0.111191	−	0.993799i	$-0.464533\pi$
$821$	6.37194	0.222382	0.111191	+	0.993799i	$0.464533\pi$
$822$	0	0
$823$	−48.4611	−1.68925	−0.844624	−	0.535360i	$-0.820176\pi$
$823$	−48.4611	−1.68925	−0.844624	+	0.535360i	$0.820176\pi$
$824$	0	0
$825$	7.26305	0.252867
$826$	0	0
$827$	15.7566	0.547910	0.273955	−	0.961743i	$-0.411668\pi$
$827$	15.7566	0.547910	0.273955	+	0.961743i	$0.411668\pi$
$828$	0	0
$829$	23.7468	0.824759	0.412380	−	0.911012i	$-0.364698\pi$
$829$	23.7468	0.824759	0.412380	+	0.911012i	$0.364698\pi$
$830$	0	0
$831$	−56.7260	−1.96780
$832$	0	0
$833$	2.15181	0.0745558
$834$	0	0
$835$	−26.6555	−0.922450
$836$	0	0
$837$	29.0630	1.00456
$838$	0	0
$839$	−15.2287	−0.525755	−0.262877	−	0.964829i	$-0.584671\pi$
$839$	−15.2287	−0.525755	−0.262877	+	0.964829i	$0.584671\pi$
$840$	0	0
$841$	19.2742	0.664627
$842$	0	0
$843$	−10.6811	−0.367875
$844$	0	0
$845$	−96.5190	−3.32035
$846$	0	0
$847$	−45.4584	−1.56197
$848$	0	0
$849$	−64.9575	−2.22934
$850$	0	0
$851$	8.44984	0.289657
$852$	0	0
$853$	19.0735	0.653064	0.326532	−	0.945186i	$-0.394120\pi$
$853$	19.0735	0.653064	0.326532	+	0.945186i	$0.394120\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−1.96169	−0.0670100	−0.0335050	−	0.999439i	$-0.510667\pi$
$857$	−1.96169	−0.0670100	−0.0335050	+	0.999439i	$0.510667\pi$
$858$	0	0
$859$	−37.7907	−1.28940	−0.644701	−	0.764435i	$-0.723018\pi$
$859$	−37.7907	−1.28940	−0.644701	+	0.764435i	$0.723018\pi$
$860$	0	0
$861$	−73.0338	−2.48899
$862$	0	0
$863$	−19.0309	−0.647819	−0.323910	−	0.946088i	$-0.604997\pi$
$863$	−19.0309	−0.647819	−0.323910	+	0.946088i	$0.604997\pi$
$864$	0	0
$865$	−27.3023	−0.928305
$866$	0	0
$867$	−47.0390	−1.59753
$868$	0	0
$869$	6.51984	0.221170
$870$	0	0
$871$	49.2765	1.66967
$872$	0	0
$873$	−61.3796	−2.07738
$874$	0	0
$875$	5.00117	0.169070
$876$	0	0
$877$	−40.7246	−1.37517	−0.687586	−	0.726103i	$-0.741330\pi$
$877$	−40.7246	−1.37517	−0.687586	+	0.726103i	$0.741330\pi$
$878$	0	0
$879$	−16.0901	−0.542706
$880$	0	0
$881$	29.8779	1.00661	0.503307	−	0.864108i	$-0.332117\pi$
$881$	29.8779	1.00661	0.503307	+	0.864108i	$0.332117\pi$
$882$	0	0
$883$	0.568053	0.0191165	0.00955825	−	0.999954i	$-0.496957\pi$
$883$	0.568053	0.0191165	0.00955825	+	0.999954i	$0.496957\pi$
$884$	0	0
$885$	7.01892	0.235938
$886$	0	0
$887$	−2.97770	−0.0999812	−0.0499906	−	0.998750i	$-0.515919\pi$
$887$	−2.97770	−0.0999812	−0.0499906	+	0.998750i	$0.515919\pi$
$888$	0	0
$889$	43.7870	1.46857
$890$	0	0
$891$	−0.608149	−0.0203738
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−37.8738	−1.26598
$896$	0	0
$897$	−35.2509	−1.17699
$898$	0	0
$899$	−43.0833	−1.43691
$900$	0	0
$901$	0.145176	0.00483652
$902$	0	0
$903$	23.7200	0.789354
$904$	0	0
$905$	−59.1120	−1.96495
$906$	0	0
$907$	−39.7644	−1.32036	−0.660178	−	0.751109i	$-0.729519\pi$
$907$	−39.7644	−1.32036	−0.660178	+	0.751109i	$0.729519\pi$
$908$	0	0
$909$	28.6880	0.951522
$910$	0	0
$911$	−2.19731	−0.0728002	−0.0364001	−	0.999337i	$-0.511589\pi$
$911$	−2.19731	−0.0728002	−0.0364001	+	0.999337i	$0.511589\pi$
$912$	0	0
$913$	−1.54132	−0.0510102
$914$	0	0
$915$	−6.34983	−0.209919
$916$	0	0
$917$	−68.8062	−2.27218
$918$	0	0
$919$	43.1219	1.42246	0.711230	−	0.702959i	$-0.248138\pi$
$919$	43.1219	1.42246	0.711230	+	0.702959i	$0.248138\pi$
$920$	0	0
$921$	−26.4048	−0.870068
$922$	0	0
$923$	−33.0934	−1.08928
$924$	0	0
$925$	−20.4036	−0.670865
$926$	0	0
$927$	55.7180	1.83002
$928$	0	0
$929$	32.7653	1.07499	0.537497	−	0.843266i	$-0.319370\pi$
$929$	32.7653	1.07499	0.537497	+	0.843266i	$0.319370\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−27.0098	−0.884261
$934$	0	0
$935$	−0.340146	−0.0111240
$936$	0	0
$937$	−2.33536	−0.0762927	−0.0381464	−	0.999272i	$-0.512145\pi$
$937$	−2.33536	−0.0762927	−0.0381464	+	0.999272i	$0.512145\pi$
$938$	0	0
$939$	−50.8965	−1.66094
$940$	0	0
$941$	−44.2494	−1.44249	−0.721245	−	0.692680i	$-0.756430\pi$
$941$	−44.2494	−1.44249	−0.721245	+	0.692680i	$0.756430\pi$
$942$	0	0
$943$	11.8404	0.385576
$944$	0	0
$945$	−61.8865	−2.01317
$946$	0	0
$947$	−44.4959	−1.44592	−0.722961	−	0.690889i	$-0.757219\pi$
$947$	−44.4959	−1.44592	−0.722961	+	0.690889i	$0.757219\pi$
$948$	0	0
$949$	−50.4052	−1.63622
$950$	0	0
$951$	−74.4416	−2.41393
$952$	0	0
$953$	6.15907	0.199512	0.0997560	−	0.995012i	$-0.468194\pi$
$953$	6.15907	0.199512	0.0997560	+	0.995012i	$0.468194\pi$
$954$	0	0
$955$	−17.9209	−0.579907
$956$	0	0
$957$	−10.9199	−0.352990
$958$	0	0
$959$	−27.9329	−0.902001
$960$	0	0
$961$	7.45067	0.240344
$962$	0	0
$963$	4.96795	0.160090
$964$	0	0
$965$	5.69818	0.183431
$966$	0	0
$967$	25.5468	0.821531	0.410766	−	0.911741i	$-0.365261\pi$
$967$	25.5468	0.821531	0.410766	+	0.911741i	$0.365261\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	53.3577	1.71233	0.856164	−	0.516704i	$-0.172841\pi$
$971$	53.3577	1.71233	0.856164	+	0.516704i	$0.172841\pi$
$972$	0	0
$973$	−65.2844	−2.09292
$974$	0	0
$975$	85.1194	2.72600
$976$	0	0
$977$	60.5113	1.93593	0.967964	−	0.251089i	$-0.0807886\pi$
$977$	60.5113	1.93593	0.967964	+	0.251089i	$0.0807886\pi$
$978$	0	0
$979$	1.86043	0.0594596
$980$	0	0
$981$	−71.8785	−2.29490
$982$	0	0
$983$	52.7754	1.68327	0.841636	−	0.540045i	$-0.181593\pi$
$983$	52.7754	1.68327	0.841636	+	0.540045i	$0.181593\pi$
$984$	0	0
$985$	50.2699	1.60173
$986$	0	0
$987$	135.219	4.30406
$988$	0	0
$989$	−3.84554	−0.122281
$990$	0	0
$991$	58.7736	1.86701	0.933503	−	0.358570i	$-0.116736\pi$
$991$	58.7736	1.86701	0.933503	+	0.358570i	$0.116736\pi$
$992$	0	0
$993$	−75.6767	−2.40153
$994$	0	0
$995$	32.6886	1.03630
$996$	0	0
$997$	34.6665	1.09790	0.548949	−	0.835856i	$-0.315028\pi$
$997$	34.6665	1.09790	0.548949	+	0.835856i	$0.315028\pi$
$998$	0	0
$999$	−20.6936	−0.654716

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.u.1.5	yes	6
4.3	odd	2		5776.2.a.bx.1.2		6
19.18	odd	2		2888.2.a.t.1.2	✓	6
76.75	even	2		5776.2.a.bz.1.5		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.t.1.2	✓	6	19.18	odd	2
2888.2.a.u.1.5	yes	6	1.1	even	1	trivial
5776.2.a.bx.1.2		6	4.3	odd	2
5776.2.a.bz.1.5		6	76.75	even	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$
Belyi maps

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

\(f(q)\)	\(=\)	\(q+2.77311 q^{3} -3.10181 q^{5} +4.25689 q^{7} +4.69013 q^{9} +0.566753 q^{11} +6.64206 q^{13} -8.60166 q^{15} +0.193489 q^{17} +11.8048 q^{21} -1.91382 q^{23} +4.62124 q^{25} +4.68693 q^{27} -6.94796 q^{29} +6.20086 q^{31} +1.57167 q^{33} -13.2041 q^{35} -4.41517 q^{37} +18.4192 q^{39} -6.18678 q^{41} +2.00935 q^{43} -14.5479 q^{45} +11.4545 q^{47} +11.1211 q^{49} +0.536566 q^{51} +0.750308 q^{53} -1.75796 q^{55} -0.815996 q^{59} +0.738209 q^{61} +19.9654 q^{63} -20.6024 q^{65} +7.41886 q^{67} -5.30723 q^{69} -4.98241 q^{71} -7.58880 q^{73} +12.8152 q^{75} +2.41260 q^{77} +11.5038 q^{79} -1.07304 q^{81} -2.71956 q^{83} -0.600167 q^{85} -19.2675 q^{87} +3.28261 q^{89} +28.2745 q^{91} +17.1957 q^{93} -13.0870 q^{97} +2.65815 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + 2 q^{5} - 2 q^{7} + 9 q^{9} - 5 q^{11} + 10 q^{13} + 4 q^{17} + 23 q^{21} - 15 q^{23} + 12 q^{25} + 18 q^{27} - 7 q^{29} + 5 q^{31} - q^{33} - 38 q^{35} + 17 q^{37} + 37 q^{39} + 4 q^{41}+ \cdots - 47 q^{99}+O(q^{100}) \) 6 * q + 3 * q^3 + 2 * q^5 - 2 * q^7 + 9 * q^9 - 5 * q^11 + 10 * q^13 + 4 * q^17 + 23 * q^21 - 15 * q^23 + 12 * q^25 + 18 * q^27 - 7 * q^29 + 5 * q^31 - q^33 - 38 * q^35 + 17 * q^37 + 37 * q^39 + 4 * q^41 - 11 * q^43 + 6 * q^45 + 18 * q^47 + 20 * q^49 + 48 * q^51 - 7 * q^53 - 42 * q^55 + q^59 - 4 * q^61 + 19 * q^63 + 6 * q^65 + 20 * q^67 + 6 * q^69 - 6 * q^71 - 2 * q^73 + 19 * q^75 + 41 * q^77 + 8 * q^79 - 6 * q^81 - 22 * q^83 + 32 * q^85 + 29 * q^87 + 14 * q^89 + 26 * q^91 - 5 * q^93 + 41 * q^97 - 47 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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