LMFDB - Embedded newform 2888.2.a.u.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 = [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			1.2
Root			$-1.20509$ 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)

$f(q)$	$=$	$q-1.60721 q^{3} -1.92601 q^{5} -1.29923 q^{7} -0.416870 q^{9} +0.117638 q^{11} -4.82577 q^{13} +3.09551 q^{15} -0.303846 q^{17} +2.08814 q^{21} -7.09885 q^{23} -1.29047 q^{25} +5.49163 q^{27} -7.89441 q^{29} -3.59384 q^{31} -0.189069 q^{33} +2.50234 q^{35} +11.4330 q^{37} +7.75603 q^{39} +3.52989 q^{41} -10.9416 q^{43} +0.802897 q^{45} +2.01544 q^{47} -5.31200 q^{49} +0.488346 q^{51} +5.57974 q^{53} -0.226572 q^{55} -12.3618 q^{59} +1.99203 q^{61} +0.541610 q^{63} +9.29450 q^{65} -7.48284 q^{67} +11.4093 q^{69} +4.45670 q^{71} +0.674509 q^{73} +2.07406 q^{75} -0.152839 q^{77} +6.16749 q^{79} -7.57561 q^{81} -15.4114 q^{83} +0.585213 q^{85} +12.6880 q^{87} -3.52733 q^{89} +6.26979 q^{91} +5.77605 q^{93} +15.9117 q^{97} -0.0490397 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$3$	−1.60721	−0.927924	−0.463962	−	0.885855i	$-0.653573\pi$
$3$	−1.60721	−0.927924	−0.463962	+	0.885855i	$0.653573\pi$
$4$	0	0
$5$	−1.92601	−0.861340	−0.430670	−	0.902510i	$-0.641723\pi$
$5$	−1.92601	−0.861340	−0.430670	+	0.902510i	$0.641723\pi$
$6$	0	0
$7$	−1.29923	−0.491063	−0.245532	−	0.969389i	$-0.578963\pi$
$7$	−1.29923	−0.491063	−0.245532	+	0.969389i	$0.578963\pi$
$8$	0	0
$9$	−0.416870	−0.138957
$10$	0	0
$11$	0.117638	0.0354692	0.0177346	−	0.999843i	$-0.494355\pi$
$11$	0.117638	0.0354692	0.0177346	+	0.999843i	$0.494355\pi$
$12$	0	0
$13$	−4.82577	−1.33843	−0.669214	−	0.743070i	$-0.733369\pi$
$13$	−4.82577	−1.33843	−0.669214	+	0.743070i	$0.733369\pi$
$14$	0	0
$15$	3.09551	0.799258
$16$	0	0
$17$	−0.303846	−0.0736936	−0.0368468	−	0.999321i	$-0.511731\pi$
$17$	−0.303846	−0.0736936	−0.0368468	+	0.999321i	$0.511731\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	2.08814	0.455670
$22$	0	0
$23$	−7.09885	−1.48021	−0.740106	−	0.672490i	$-0.765225\pi$
$23$	−7.09885	−1.48021	−0.740106	+	0.672490i	$0.765225\pi$
$24$	0	0
$25$	−1.29047	−0.258094
$26$	0	0
$27$	5.49163	1.05687
$28$	0	0
$29$	−7.89441	−1.46596	−0.732978	−	0.680253i	$-0.761870\pi$
$29$	−7.89441	−1.46596	−0.732978	+	0.680253i	$0.761870\pi$
$30$	0	0
$31$	−3.59384	−0.645472	−0.322736	−	0.946489i	$-0.604603\pi$
$31$	−3.59384	−0.645472	−0.322736	+	0.946489i	$0.604603\pi$
$32$	0	0
$33$	−0.189069	−0.0329127
$34$	0	0
$35$	2.50234	0.422972
$36$	0	0
$37$	11.4330	1.87957	0.939785	−	0.341765i	$-0.111025\pi$
$37$	11.4330	1.87957	0.939785	+	0.341765i	$0.111025\pi$
$38$	0	0
$39$	7.75603	1.24196
$40$	0	0
$41$	3.52989	0.551277	0.275638	−	0.961261i	$-0.411111\pi$
$41$	3.52989	0.551277	0.275638	+	0.961261i	$0.411111\pi$
$42$	0	0
$43$	−10.9416	−1.66859	−0.834293	−	0.551322i	$-0.814124\pi$
$43$	−10.9416	−1.66859	−0.834293	+	0.551322i	$0.814124\pi$
$44$	0	0
$45$	0.802897	0.119689
$46$	0	0
$47$	2.01544	0.293982	0.146991	−	0.989138i	$-0.453041\pi$
$47$	2.01544	0.293982	0.146991	+	0.989138i	$0.453041\pi$
$48$	0	0
$49$	−5.31200	−0.758857
$50$	0	0
$51$	0.488346	0.0683821
$52$	0	0
$53$	5.57974	0.766437	0.383218	−	0.923658i	$-0.374816\pi$
$53$	5.57974	0.766437	0.383218	+	0.923658i	$0.374816\pi$
$54$	0	0
$55$	−0.226572	−0.0305510
$56$	0	0
$57$	0	0
$58$	0	0
$59$	−12.3618	−1.60938	−0.804688	−	0.593698i	$-0.797667\pi$
$59$	−12.3618	−1.60938	−0.804688	+	0.593698i	$0.797667\pi$
$60$	0	0
$61$	1.99203	0.255053	0.127527	−	0.991835i	$-0.459296\pi$
$61$	1.99203	0.255053	0.127527	+	0.991835i	$0.459296\pi$
$62$	0	0
$63$	0.541610	0.0682365
$64$	0	0
$65$	9.29450	1.15284
$66$	0	0
$67$	−7.48284	−0.914175	−0.457087	−	0.889422i	$-0.651107\pi$
$67$	−7.48284	−0.914175	−0.457087	+	0.889422i	$0.651107\pi$
$68$	0	0
$69$	11.4093	1.37352
$70$	0	0
$71$	4.45670	0.528913	0.264456	−	0.964398i	$-0.414807\pi$
$71$	4.45670	0.528913	0.264456	+	0.964398i	$0.414807\pi$
$72$	0	0
$73$	0.674509	0.0789454	0.0394727	−	0.999221i	$-0.487432\pi$
$73$	0.674509	0.0789454	0.0394727	+	0.999221i	$0.487432\pi$
$74$	0	0
$75$	2.07406	0.239492
$76$	0	0
$77$	−0.152839	−0.0174176
$78$	0	0
$79$	6.16749	0.693897	0.346948	−	0.937884i	$-0.387218\pi$
$79$	6.16749	0.693897	0.346948	+	0.937884i	$0.387218\pi$
$80$	0	0
$81$	−7.57561	−0.841734
$82$	0	0
$83$	−15.4114	−1.69162	−0.845811	−	0.533483i	$-0.820883\pi$
$83$	−15.4114	−1.69162	−0.845811	+	0.533483i	$0.820883\pi$
$84$	0	0
$85$	0.585213	0.0634752
$86$	0	0
$87$	12.6880	1.36030
$88$	0	0
$89$	−3.52733	−0.373896	−0.186948	−	0.982370i	$-0.559860\pi$
$89$	−3.52733	−0.373896	−0.186948	+	0.982370i	$0.559860\pi$
$90$	0	0
$91$	6.26979	0.657253
$92$	0	0
$93$	5.77605	0.598949
$94$	0	0
$95$	0	0
$96$	0	0
$97$	15.9117	1.61559	0.807796	−	0.589462i	$-0.200660\pi$
$97$	15.9117	1.61559	0.807796	+	0.589462i	$0.200660\pi$
$98$	0	0
$99$	−0.0490397	−0.00492868
$100$	0	0
$101$	15.9596	1.58804	0.794019	−	0.607892i	$-0.207985\pi$
$101$	15.9596	1.58804	0.794019	+	0.607892i	$0.207985\pi$
$102$	0	0
$103$	10.6260	1.04701	0.523506	−	0.852022i	$-0.324624\pi$
$103$	10.6260	1.04701	0.523506	+	0.852022i	$0.324624\pi$
$104$	0	0
$105$	−4.02179	−0.392486
$106$	0	0
$107$	−0.613900	−0.0593480	−0.0296740	−	0.999560i	$-0.509447\pi$
$107$	−0.613900	−0.0593480	−0.0296740	+	0.999560i	$0.509447\pi$
$108$	0	0
$109$	1.38519	0.132677	0.0663386	−	0.997797i	$-0.478868\pi$
$109$	1.38519	0.132677	0.0663386	+	0.997797i	$0.478868\pi$
$110$	0	0
$111$	−18.3752	−1.74410
$112$	0	0
$113$	15.1567	1.42582	0.712910	−	0.701256i	$-0.247377\pi$
$113$	15.1567	1.42582	0.712910	+	0.701256i	$0.247377\pi$
$114$	0	0
$115$	13.6725	1.27497
$116$	0	0
$117$	2.01172	0.185983
$118$	0	0
$119$	0.394767	0.0361882
$120$	0	0
$121$	−10.9862	−0.998742
$122$	0	0
$123$	−5.67329	−0.511543
$124$	0	0
$125$	12.1155	1.08365
$126$	0	0
$127$	22.0161	1.95361	0.976804	−	0.214134i	$-0.0686928\pi$
$127$	22.0161	1.95361	0.976804	+	0.214134i	$0.0686928\pi$
$128$	0	0
$129$	17.5855	1.54832
$130$	0	0
$131$	−5.32883	−0.465582	−0.232791	−	0.972527i	$-0.574786\pi$
$131$	−5.32883	−0.465582	−0.232791	+	0.972527i	$0.574786\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−10.5770	−0.910320
$136$	0	0
$137$	7.21231	0.616189	0.308094	−	0.951356i	$-0.400309\pi$
$137$	7.21231	0.616189	0.308094	+	0.951356i	$0.400309\pi$
$138$	0	0
$139$	−3.94638	−0.334727	−0.167364	−	0.985895i	$-0.553525\pi$
$139$	−3.94638	−0.334727	−0.167364	+	0.985895i	$0.553525\pi$
$140$	0	0
$141$	−3.23923	−0.272793
$142$	0	0
$143$	−0.567694	−0.0474729
$144$	0	0
$145$	15.2047	1.26269
$146$	0	0
$147$	8.53750	0.704162
$148$	0	0
$149$	−14.6664	−1.20152	−0.600758	−	0.799431i	$-0.705135\pi$
$149$	−14.6664	−1.20152	−0.600758	+	0.799431i	$0.705135\pi$
$150$	0	0
$151$	−2.89903	−0.235919	−0.117960	−	0.993018i	$-0.537635\pi$
$151$	−2.89903	−0.235919	−0.117960	+	0.993018i	$0.537635\pi$
$152$	0	0
$153$	0.126664	0.0102402
$154$	0	0
$155$	6.92178	0.555971
$156$	0	0
$157$	2.42840	0.193807	0.0969036	−	0.995294i	$-0.469106\pi$
$157$	2.42840	0.193807	0.0969036	+	0.995294i	$0.469106\pi$
$158$	0	0
$159$	−8.96783	−0.711195
$160$	0	0
$161$	9.22305	0.726878
$162$	0	0
$163$	−21.0505	−1.64880	−0.824401	−	0.566006i	$-0.808488\pi$
$163$	−21.0505	−1.64880	−0.824401	+	0.566006i	$0.808488\pi$
$164$	0	0
$165$	0.364150	0.0283490
$166$	0	0
$167$	−4.90304	−0.379409	−0.189704	−	0.981841i	$-0.560753\pi$
$167$	−4.90304	−0.379409	−0.189704	+	0.981841i	$0.560753\pi$
$168$	0	0
$169$	10.2881	0.791389
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−2.74381	−0.208608	−0.104304	−	0.994545i	$-0.533262\pi$
$173$	−2.74381	−0.208608	−0.104304	+	0.994545i	$0.533262\pi$
$174$	0	0
$175$	1.67662	0.126741
$176$	0	0
$177$	19.8681	1.49338
$178$	0	0
$179$	−10.5355	−0.787459	−0.393729	−	0.919226i	$-0.628815\pi$
$179$	−10.5355	−0.787459	−0.393729	+	0.919226i	$0.628815\pi$
$180$	0	0
$181$	−17.3422	−1.28904	−0.644520	−	0.764588i	$-0.722943\pi$
$181$	−17.3422	−1.28904	−0.644520	+	0.764588i	$0.722943\pi$
$182$	0	0
$183$	−3.20161	−0.236670
$184$	0	0
$185$	−22.0201	−1.61895
$186$	0	0
$187$	−0.0357439	−0.00261385
$188$	0	0
$189$	−7.13491	−0.518988
$190$	0	0
$191$	22.5239	1.62978	0.814888	−	0.579619i	$-0.196799\pi$
$191$	22.5239	1.62978	0.814888	+	0.579619i	$0.196799\pi$
$192$	0	0
$193$	20.6399	1.48569	0.742847	−	0.669461i	$-0.233475\pi$
$193$	20.6399	1.48569	0.742847	+	0.669461i	$0.233475\pi$
$194$	0	0
$195$	−14.9382	−1.06975
$196$	0	0
$197$	12.7825	0.910714	0.455357	−	0.890309i	$-0.349512\pi$
$197$	12.7825	0.910714	0.455357	+	0.890309i	$0.349512\pi$
$198$	0	0
$199$	21.1578	1.49983	0.749917	−	0.661532i	$-0.230093\pi$
$199$	21.1578	1.49983	0.749917	+	0.661532i	$0.230093\pi$
$200$	0	0
$201$	12.0265	0.848285
$202$	0	0
$203$	10.2567	0.719877
$204$	0	0
$205$	−6.79862	−0.474836
$206$	0	0
$207$	2.95929	0.205685
$208$	0	0
$209$	0	0
$210$	0	0
$211$	4.94760	0.340607	0.170304	−	0.985392i	$-0.445525\pi$
$211$	4.94760	0.340607	0.170304	+	0.985392i	$0.445525\pi$
$212$	0	0
$213$	−7.16286	−0.490791
$214$	0	0
$215$	21.0738	1.43722
$216$	0	0
$217$	4.66923	0.316968
$218$	0	0
$219$	−1.08408	−0.0732553
$220$	0	0
$221$	1.46629	0.0986335
$222$	0	0
$223$	0.339870	0.0227594	0.0113797	−	0.999935i	$-0.496378\pi$
$223$	0.339870	0.0227594	0.0113797	+	0.999935i	$0.496378\pi$
$224$	0	0
$225$	0.537958	0.0358639
$226$	0	0
$227$	23.4864	1.55885	0.779424	−	0.626497i	$-0.215512\pi$
$227$	23.4864	1.55885	0.779424	+	0.626497i	$0.215512\pi$
$228$	0	0
$229$	10.2451	0.677014	0.338507	−	0.940964i	$-0.390078\pi$
$229$	10.2451	0.677014	0.338507	+	0.940964i	$0.390078\pi$
$230$	0	0
$231$	0.245645	0.0161622
$232$	0	0
$233$	20.7529	1.35957	0.679784	−	0.733413i	$-0.262074\pi$
$233$	20.7529	1.35957	0.679784	+	0.733413i	$0.262074\pi$
$234$	0	0
$235$	−3.88176	−0.253218
$236$	0	0
$237$	−9.91246	−0.643884
$238$	0	0
$239$	−17.9556	−1.16145	−0.580725	−	0.814100i	$-0.697231\pi$
$239$	−17.9556	−1.16145	−0.580725	+	0.814100i	$0.697231\pi$
$240$	0	0
$241$	−1.63405	−0.105258	−0.0526291	−	0.998614i	$-0.516760\pi$
$241$	−1.63405	−0.105258	−0.0526291	+	0.998614i	$0.516760\pi$
$242$	0	0
$243$	−4.29929	−0.275800
$244$	0	0
$245$	10.2310	0.653633
$246$	0	0
$247$	0	0
$248$	0	0
$249$	24.7694	1.56970
$250$	0	0
$251$	4.60415	0.290611	0.145306	−	0.989387i	$-0.453583\pi$
$251$	4.60415	0.290611	0.145306	+	0.989387i	$0.453583\pi$
$252$	0	0
$253$	−0.835094	−0.0525019
$254$	0	0
$255$	−0.940561	−0.0589002
$256$	0	0
$257$	−12.2767	−0.765798	−0.382899	−	0.923790i	$-0.625074\pi$
$257$	−12.2767	−0.765798	−0.382899	+	0.923790i	$0.625074\pi$
$258$	0	0
$259$	−14.8541	−0.922988
$260$	0	0
$261$	3.29094	0.203704
$262$	0	0
$263$	−18.8187	−1.16041	−0.580204	−	0.814471i	$-0.697027\pi$
$263$	−18.8187	−1.16041	−0.580204	+	0.814471i	$0.697027\pi$
$264$	0	0
$265$	−10.7467	−0.660162
$266$	0	0
$267$	5.66917	0.346947
$268$	0	0
$269$	−6.81434	−0.415477	−0.207739	−	0.978184i	$-0.566610\pi$
$269$	−6.81434	−0.415477	−0.207739	+	0.978184i	$0.566610\pi$
$270$	0	0
$271$	4.65756	0.282927	0.141463	−	0.989943i	$-0.454819\pi$
$271$	4.65756	0.282927	0.141463	+	0.989943i	$0.454819\pi$
$272$	0	0
$273$	−10.0769	−0.609881
$274$	0	0
$275$	−0.151808	−0.00915438
$276$	0	0
$277$	14.9240	0.896697	0.448348	−	0.893859i	$-0.352012\pi$
$277$	14.9240	0.896697	0.448348	+	0.893859i	$0.352012\pi$
$278$	0	0
$279$	1.49816	0.0896926
$280$	0	0
$281$	−8.36780	−0.499181	−0.249590	−	0.968351i	$-0.580296\pi$
$281$	−8.36780	−0.499181	−0.249590	+	0.968351i	$0.580296\pi$
$282$	0	0
$283$	−11.4493	−0.680592	−0.340296	−	0.940318i	$-0.610527\pi$
$283$	−11.4493	−0.680592	−0.340296	+	0.940318i	$0.610527\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−4.58615	−0.270712
$288$	0	0
$289$	−16.9077	−0.994569
$290$	0	0
$291$	−25.5735	−1.49915
$292$	0	0
$293$	−9.17234	−0.535854	−0.267927	−	0.963439i	$-0.586339\pi$
$293$	−9.17234	−0.535854	−0.267927	+	0.963439i	$0.586339\pi$
$294$	0	0
$295$	23.8091	1.38622
$296$	0	0
$297$	0.646024	0.0374861
$298$	0	0
$299$	34.2574	1.98116
$300$	0	0
$301$	14.2157	0.819381
$302$	0	0
$303$	−25.6504	−1.47358
$304$	0	0
$305$	−3.83668	−0.219688
$306$	0	0
$307$	−17.8631	−1.01950	−0.509749	−	0.860323i	$-0.670262\pi$
$307$	−17.8631	−1.01950	−0.509749	+	0.860323i	$0.670262\pi$
$308$	0	0
$309$	−17.0782	−0.971547
$310$	0	0
$311$	13.7027	0.777008	0.388504	−	0.921447i	$-0.372992\pi$
$311$	13.7027	0.777008	0.388504	+	0.921447i	$0.372992\pi$
$312$	0	0
$313$	31.5127	1.78120	0.890600	−	0.454787i	$-0.150284\pi$
$313$	31.5127	1.78120	0.890600	+	0.454787i	$0.150284\pi$
$314$	0	0
$315$	−1.04315	−0.0587748
$316$	0	0
$317$	−12.0292	−0.675629	−0.337815	−	0.941213i	$-0.609688\pi$
$317$	−12.0292	−0.675629	−0.337815	+	0.941213i	$0.609688\pi$
$318$	0	0
$319$	−0.928682	−0.0519962
$320$	0	0
$321$	0.986668	0.0550704
$322$	0	0
$323$	0	0
$324$	0	0
$325$	6.22751	0.345440
$326$	0	0
$327$	−2.22629	−0.123114
$328$	0	0
$329$	−2.61852	−0.144364
$330$	0	0
$331$	6.86623	0.377402	0.188701	−	0.982035i	$-0.439572\pi$
$331$	6.86623	0.377402	0.188701	+	0.982035i	$0.439572\pi$
$332$	0	0
$333$	−4.76606	−0.261179
$334$	0	0
$335$	14.4121	0.787415
$336$	0	0
$337$	−33.4400	−1.82160	−0.910798	−	0.412853i	$-0.864532\pi$
$337$	−33.4400	−1.82160	−0.910798	+	0.412853i	$0.864532\pi$
$338$	0	0
$339$	−24.3600	−1.32305
$340$	0	0
$341$	−0.422771	−0.0228944
$342$	0	0
$343$	15.9961	0.863710
$344$	0	0
$345$	−21.9746	−1.18307
$346$	0	0
$347$	−12.3927	−0.665274	−0.332637	−	0.943055i	$-0.607938\pi$
$347$	−12.3927	−0.665274	−0.332637	+	0.943055i	$0.607938\pi$
$348$	0	0
$349$	5.02432	0.268945	0.134473	−	0.990917i	$-0.457066\pi$
$349$	5.02432	0.268945	0.134473	+	0.990917i	$0.457066\pi$
$350$	0	0
$351$	−26.5014	−1.41454
$352$	0	0
$353$	7.87575	0.419184	0.209592	−	0.977789i	$-0.432786\pi$
$353$	7.87575	0.419184	0.209592	+	0.977789i	$0.432786\pi$
$354$	0	0
$355$	−8.58367	−0.455574
$356$	0	0
$357$	−0.634474	−0.0335799
$358$	0	0
$359$	−25.3338	−1.33707	−0.668535	−	0.743681i	$-0.733078\pi$
$359$	−25.3338	−1.33707	−0.668535	+	0.743681i	$0.733078\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	17.6571	0.926757
$364$	0	0
$365$	−1.29911	−0.0679988
$366$	0	0
$367$	2.32379	0.121301	0.0606504	−	0.998159i	$-0.480683\pi$
$367$	2.32379	0.121301	0.0606504	+	0.998159i	$0.480683\pi$
$368$	0	0
$369$	−1.47151	−0.0766035
$370$	0	0
$371$	−7.24938	−0.376369
$372$	0	0
$373$	−0.262182	−0.0135753	−0.00678763	−	0.999977i	$-0.502161\pi$
$373$	−0.262182	−0.0135753	−0.00678763	+	0.999977i	$0.502161\pi$
$374$	0	0
$375$	−19.4722	−1.00554
$376$	0	0
$377$	38.0966	1.96208
$378$	0	0
$379$	8.75558	0.449744	0.224872	−	0.974388i	$-0.427804\pi$
$379$	8.75558	0.449744	0.224872	+	0.974388i	$0.427804\pi$
$380$	0	0
$381$	−35.3845	−1.81280
$382$	0	0
$383$	−1.18420	−0.0605096	−0.0302548	−	0.999542i	$-0.509632\pi$
$383$	−1.18420	−0.0605096	−0.0302548	+	0.999542i	$0.509632\pi$
$384$	0	0
$385$	0.294370	0.0150025
$386$	0	0
$387$	4.56124	0.231861
$388$	0	0
$389$	−10.4576	−0.530223	−0.265111	−	0.964218i	$-0.585409\pi$
$389$	−10.4576	−0.530223	−0.265111	+	0.964218i	$0.585409\pi$
$390$	0	0
$391$	2.15696	0.109082
$392$	0	0
$393$	8.56456	0.432025
$394$	0	0
$395$	−11.8787	−0.597681
$396$	0	0
$397$	23.4565	1.17725	0.588624	−	0.808407i	$-0.299670\pi$
$397$	23.4565	1.17725	0.588624	+	0.808407i	$0.299670\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−17.9890	−0.898325	−0.449163	−	0.893450i	$-0.648278\pi$
$401$	−17.9890	−0.898325	−0.449163	+	0.893450i	$0.648278\pi$
$402$	0	0
$403$	17.3430	0.863917
$404$	0	0
$405$	14.5907	0.725019
$406$	0	0
$407$	1.34495	0.0666668
$408$	0	0
$409$	27.7238	1.37085	0.685427	−	0.728141i	$-0.259616\pi$
$409$	27.7238	1.37085	0.685427	+	0.728141i	$0.259616\pi$
$410$	0	0
$411$	−11.5917	−0.571777
$412$	0	0
$413$	16.0609	0.790306
$414$	0	0
$415$	29.6826	1.45706
$416$	0	0
$417$	6.34266	0.310602
$418$	0	0
$419$	−33.8453	−1.65345	−0.826724	−	0.562607i	$-0.809798\pi$
$419$	−33.8453	−1.65345	−0.826724	+	0.562607i	$0.809798\pi$
$420$	0	0
$421$	22.2246	1.08316	0.541581	−	0.840648i	$-0.317826\pi$
$421$	22.2246	1.08316	0.541581	+	0.840648i	$0.317826\pi$
$422$	0	0
$423$	−0.840175	−0.0408507
$424$	0	0
$425$	0.392105	0.0190199
$426$	0	0
$427$	−2.58811	−0.125247
$428$	0	0
$429$	0.912404	0.0440513
$430$	0	0
$431$	−24.0938	−1.16056	−0.580279	−	0.814418i	$-0.697056\pi$
$431$	−24.0938	−1.16056	−0.580279	+	0.814418i	$0.697056\pi$
$432$	0	0
$433$	11.9974	0.576557	0.288278	−	0.957547i	$-0.406917\pi$
$433$	11.9974	0.576557	0.288278	+	0.957547i	$0.406917\pi$
$434$	0	0
$435$	−24.4373	−1.17168
$436$	0	0
$437$	0	0
$438$	0	0
$439$	−32.7160	−1.56145	−0.780724	−	0.624876i	$-0.785149\pi$
$439$	−32.7160	−1.56145	−0.780724	+	0.624876i	$0.785149\pi$
$440$	0	0
$441$	2.21441	0.105448
$442$	0	0
$443$	−34.9555	−1.66079	−0.830394	−	0.557177i	$-0.811885\pi$
$443$	−34.9555	−1.66079	−0.830394	+	0.557177i	$0.811885\pi$
$444$	0	0
$445$	6.79369	0.322052
$446$	0	0
$447$	23.5720	1.11492
$448$	0	0
$449$	−23.0987	−1.09010	−0.545048	−	0.838405i	$-0.683488\pi$
$449$	−23.0987	−1.09010	−0.545048	+	0.838405i	$0.683488\pi$
$450$	0	0
$451$	0.415249	0.0195533
$452$	0	0
$453$	4.65935	0.218915
$454$	0	0
$455$	−12.0757	−0.566118
$456$	0	0
$457$	−28.3664	−1.32692	−0.663462	−	0.748210i	$-0.730914\pi$
$457$	−28.3664	−1.32692	−0.663462	+	0.748210i	$0.730914\pi$
$458$	0	0
$459$	−1.66861	−0.0778842
$460$	0	0
$461$	17.4040	0.810585	0.405292	−	0.914187i	$-0.367170\pi$
$461$	17.4040	0.810585	0.405292	+	0.914187i	$0.367170\pi$
$462$	0	0
$463$	29.8025	1.38504	0.692520	−	0.721398i	$-0.256500\pi$
$463$	29.8025	1.38504	0.692520	+	0.721398i	$0.256500\pi$
$464$	0	0
$465$	−11.1248	−0.515899
$466$	0	0
$467$	−22.1790	−1.02632	−0.513161	−	0.858292i	$-0.671526\pi$
$467$	−22.1790	−1.02632	−0.513161	+	0.858292i	$0.671526\pi$
$468$	0	0
$469$	9.72195	0.448918
$470$	0	0
$471$	−3.90295	−0.179838
$472$	0	0
$473$	−1.28715	−0.0591833
$474$	0	0
$475$	0	0
$476$	0	0
$477$	−2.32603	−0.106501
$478$	0	0
$479$	−18.4306	−0.842117	−0.421059	−	0.907033i	$-0.638341\pi$
$479$	−18.4306	−0.842117	−0.421059	+	0.907033i	$0.638341\pi$
$480$	0	0
$481$	−55.1729	−2.51567
$482$	0	0
$483$	−14.8234	−0.674488
$484$	0	0
$485$	−30.6462	−1.39157
$486$	0	0
$487$	4.82574	0.218675	0.109338	−	0.994005i	$-0.465127\pi$
$487$	4.82574	0.218675	0.109338	+	0.994005i	$0.465127\pi$
$488$	0	0
$489$	33.8326	1.52996
$490$	0	0
$491$	−2.11235	−0.0953289	−0.0476645	−	0.998863i	$-0.515178\pi$
$491$	−2.11235	−0.0953289	−0.0476645	+	0.998863i	$0.515178\pi$
$492$	0	0
$493$	2.39869	0.108032
$494$	0	0
$495$	0.0944512	0.00424526
$496$	0	0
$497$	−5.79029	−0.259730
$498$	0	0
$499$	32.5780	1.45839	0.729196	−	0.684304i	$-0.239894\pi$
$499$	32.5780	1.45839	0.729196	+	0.684304i	$0.239894\pi$
$500$	0	0
$501$	7.88023	0.352063
$502$	0	0
$503$	2.99066	0.133347	0.0666735	−	0.997775i	$-0.478761\pi$
$503$	2.99066	0.133347	0.0666735	+	0.997775i	$0.478761\pi$
$504$	0	0
$505$	−30.7384	−1.36784
$506$	0	0
$507$	−16.5351	−0.734349
$508$	0	0
$509$	−20.0650	−0.889366	−0.444683	−	0.895688i	$-0.646684\pi$
$509$	−20.0650	−0.889366	−0.444683	+	0.895688i	$0.646684\pi$
$510$	0	0
$511$	−0.876344	−0.0387672
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−20.4658	−0.901832
$516$	0	0
$517$	0.237092	0.0104273
$518$	0	0
$519$	4.40989	0.193573
$520$	0	0
$521$	−17.1816	−0.752741	−0.376370	−	0.926469i	$-0.622828\pi$
$521$	−17.1816	−0.752741	−0.376370	+	0.926469i	$0.622828\pi$
$522$	0	0
$523$	34.4988	1.50853	0.754264	−	0.656571i	$-0.227994\pi$
$523$	34.4988	1.50853	0.754264	+	0.656571i	$0.227994\pi$
$524$	0	0
$525$	−2.69468	−0.117606
$526$	0	0
$527$	1.09197	0.0475671
$528$	0	0
$529$	27.3936	1.19103
$530$	0	0
$531$	5.15328	0.223633
$532$	0	0
$533$	−17.0345	−0.737844
$534$	0	0
$535$	1.18238	0.0511188
$536$	0	0
$537$	16.9327	0.730702
$538$	0	0
$539$	−0.624892	−0.0269160
$540$	0	0
$541$	−18.5603	−0.797972	−0.398986	−	0.916957i	$-0.630638\pi$
$541$	−18.5603	−0.797972	−0.398986	+	0.916957i	$0.630638\pi$
$542$	0	0
$543$	27.8727	1.19613
$544$	0	0
$545$	−2.66790	−0.114280
$546$	0	0
$547$	30.9014	1.32125	0.660625	−	0.750716i	$-0.270291\pi$
$547$	30.9014	1.32125	0.660625	+	0.750716i	$0.270291\pi$
$548$	0	0
$549$	−0.830417	−0.0354413
$550$	0	0
$551$	0	0
$552$	0	0
$553$	−8.01300	−0.340747
$554$	0	0
$555$	35.3909	1.50226
$556$	0	0
$557$	3.58042	0.151707	0.0758537	−	0.997119i	$-0.475832\pi$
$557$	3.58042	0.151707	0.0758537	+	0.997119i	$0.475832\pi$
$558$	0	0
$559$	52.8019	2.23328
$560$	0	0
$561$	0.0574480	0.00242546
$562$	0	0
$563$	−5.03014	−0.211995	−0.105998	−	0.994366i	$-0.533804\pi$
$563$	−5.03014	−0.211995	−0.105998	+	0.994366i	$0.533804\pi$
$564$	0	0
$565$	−29.1920	−1.22811
$566$	0	0
$567$	9.84247	0.413345
$568$	0	0
$569$	−12.4008	−0.519870	−0.259935	−	0.965626i	$-0.583701\pi$
$569$	−12.4008	−0.519870	−0.259935	+	0.965626i	$0.583701\pi$
$570$	0	0
$571$	−11.3604	−0.475417	−0.237709	−	0.971336i	$-0.576396\pi$
$571$	−11.3604	−0.475417	−0.237709	+	0.971336i	$0.576396\pi$
$572$	0	0
$573$	−36.2008	−1.51231
$574$	0	0
$575$	9.16085	0.382034
$576$	0	0
$577$	−13.4189	−0.558637	−0.279318	−	0.960199i	$-0.590108\pi$
$577$	−13.4189	−0.558637	−0.279318	+	0.960199i	$0.590108\pi$
$578$	0	0
$579$	−33.1727	−1.37861
$580$	0	0
$581$	20.0230	0.830694
$582$	0	0
$583$	0.656390	0.0271849
$584$	0	0
$585$	−3.87460	−0.160195
$586$	0	0
$587$	37.9147	1.56491	0.782454	−	0.622708i	$-0.213968\pi$
$587$	37.9147	1.56491	0.782454	+	0.622708i	$0.213968\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−20.5442	−0.845074
$592$	0	0
$593$	24.3302	0.999120	0.499560	−	0.866279i	$-0.333495\pi$
$593$	24.3302	0.999120	0.499560	+	0.866279i	$0.333495\pi$
$594$	0	0
$595$	−0.760327	−0.0311704
$596$	0	0
$597$	−34.0050	−1.39173
$598$	0	0
$599$	33.3419	1.36231	0.681157	−	0.732137i	$-0.261477\pi$
$599$	33.3419	1.36231	0.681157	+	0.732137i	$0.261477\pi$
$600$	0	0
$601$	11.8427	0.483074	0.241537	−	0.970392i	$-0.422348\pi$
$601$	11.8427	0.483074	0.241537	+	0.970392i	$0.422348\pi$
$602$	0	0
$603$	3.11937	0.127031
$604$	0	0
$605$	21.1595	0.860256
$606$	0	0
$607$	−22.0414	−0.894632	−0.447316	−	0.894376i	$-0.647620\pi$
$607$	−22.0414	−0.894632	−0.447316	+	0.894376i	$0.647620\pi$
$608$	0	0
$609$	−16.4846	−0.667992
$610$	0	0
$611$	−9.72603	−0.393473
$612$	0	0
$613$	12.8268	0.518069	0.259035	−	0.965868i	$-0.416596\pi$
$613$	12.8268	0.518069	0.259035	+	0.965868i	$0.416596\pi$
$614$	0	0
$615$	10.9268	0.440612
$616$	0	0
$617$	−28.2846	−1.13869	−0.569347	−	0.822097i	$-0.692804\pi$
$617$	−28.2846	−1.13869	−0.569347	+	0.822097i	$0.692804\pi$
$618$	0	0
$619$	−31.9371	−1.28366	−0.641830	−	0.766847i	$-0.721824\pi$
$619$	−31.9371	−1.28366	−0.641830	+	0.766847i	$0.721824\pi$
$620$	0	0
$621$	−38.9843	−1.56438
$622$	0	0
$623$	4.58282	0.183607
$624$	0	0
$625$	−16.8823	−0.675294
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−3.47387	−0.138512
$630$	0	0
$631$	−15.6214	−0.621880	−0.310940	−	0.950430i	$-0.600644\pi$
$631$	−15.6214	−0.621880	−0.310940	+	0.950430i	$0.600644\pi$
$632$	0	0
$633$	−7.95185	−0.316058
$634$	0	0
$635$	−42.4032	−1.68272
$636$	0	0
$637$	25.6345	1.01567
$638$	0	0
$639$	−1.85786	−0.0734959
$640$	0	0
$641$	−9.43327	−0.372592	−0.186296	−	0.982494i	$-0.559648\pi$
$641$	−9.43327	−0.372592	−0.186296	+	0.982494i	$0.559648\pi$
$642$	0	0
$643$	25.5859	1.00901	0.504505	−	0.863409i	$-0.331675\pi$
$643$	25.5859	1.00901	0.504505	+	0.863409i	$0.331675\pi$
$644$	0	0
$645$	−33.8700	−1.33363
$646$	0	0
$647$	−46.0742	−1.81137	−0.905683	−	0.423956i	$-0.860641\pi$
$647$	−46.0742	−1.81137	−0.905683	+	0.423956i	$0.860641\pi$
$648$	0	0
$649$	−1.45422	−0.0570832
$650$	0	0
$651$	−7.50443	−0.294122
$652$	0	0
$653$	6.44609	0.252255	0.126127	−	0.992014i	$-0.459745\pi$
$653$	6.44609	0.252255	0.126127	+	0.992014i	$0.459745\pi$
$654$	0	0
$655$	10.2634	0.401024
$656$	0	0
$657$	−0.281183	−0.0109700
$658$	0	0
$659$	16.3950	0.638657	0.319329	−	0.947644i	$-0.396543\pi$
$659$	16.3950	0.638657	0.319329	+	0.947644i	$0.396543\pi$
$660$	0	0
$661$	−7.29578	−0.283773	−0.141887	−	0.989883i	$-0.545317\pi$
$661$	−7.29578	−0.283773	−0.141887	+	0.989883i	$0.545317\pi$
$662$	0	0
$663$	−2.35664	−0.0915245
$664$	0	0
$665$	0	0
$666$	0	0
$667$	56.0412	2.16992
$668$	0	0
$669$	−0.546244	−0.0211190
$670$	0	0
$671$	0.234338	0.00904653
$672$	0	0
$673$	−8.67863	−0.334537	−0.167268	−	0.985911i	$-0.553495\pi$
$673$	−8.67863	−0.334537	−0.167268	+	0.985911i	$0.553495\pi$
$674$	0	0
$675$	−7.08679	−0.272771
$676$	0	0
$677$	−27.7157	−1.06520	−0.532601	−	0.846367i	$-0.678785\pi$
$677$	−27.7157	−1.06520	−0.532601	+	0.846367i	$0.678785\pi$
$678$	0	0
$679$	−20.6730	−0.793359
$680$	0	0
$681$	−37.7477	−1.44649
$682$	0	0
$683$	−21.8733	−0.836959	−0.418480	−	0.908226i	$-0.637437\pi$
$683$	−21.8733	−0.836959	−0.418480	+	0.908226i	$0.637437\pi$
$684$	0	0
$685$	−13.8910	−0.530748
$686$	0	0
$687$	−16.4660	−0.628217
$688$	0	0
$689$	−26.9266	−1.02582
$690$	0	0
$691$	−16.6827	−0.634641	−0.317320	−	0.948318i	$-0.602783\pi$
$691$	−16.6827	−0.634641	−0.317320	+	0.948318i	$0.602783\pi$
$692$	0	0
$693$	0.0637139	0.00242029
$694$	0	0
$695$	7.60078	0.288314
$696$	0	0
$697$	−1.07255	−0.0406256
$698$	0	0
$699$	−33.3543	−1.26158
$700$	0	0
$701$	−27.3863	−1.03437	−0.517184	−	0.855874i	$-0.673020\pi$
$701$	−27.3863	−1.03437	−0.517184	+	0.855874i	$0.673020\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	6.23881	0.234967
$706$	0	0
$707$	−20.7352	−0.779828
$708$	0	0
$709$	−31.5807	−1.18604	−0.593019	−	0.805188i	$-0.702064\pi$
$709$	−31.5807	−1.18604	−0.593019	+	0.805188i	$0.702064\pi$
$710$	0	0
$711$	−2.57104	−0.0964215
$712$	0	0
$713$	25.5121	0.955435
$714$	0	0
$715$	1.09339	0.0408903
$716$	0	0
$717$	28.8584	1.07774
$718$	0	0
$719$	−7.97601	−0.297455	−0.148728	−	0.988878i	$-0.547518\pi$
$719$	−7.97601	−0.297455	−0.148728	+	0.988878i	$0.547518\pi$
$720$	0	0
$721$	−13.8056	−0.514149
$722$	0	0
$723$	2.62626	0.0976716
$724$	0	0
$725$	10.1875	0.378354
$726$	0	0
$727$	47.3967	1.75785	0.878924	−	0.476962i	$-0.158262\pi$
$727$	47.3967	1.75785	0.878924	+	0.476962i	$0.158262\pi$
$728$	0	0
$729$	29.6367	1.09766
$730$	0	0
$731$	3.32458	0.122964
$732$	0	0
$733$	−23.7095	−0.875730	−0.437865	−	0.899041i	$-0.644265\pi$
$733$	−23.7095	−0.875730	−0.437865	+	0.899041i	$0.644265\pi$
$734$	0	0
$735$	−16.4434	−0.606522
$736$	0	0
$737$	−0.880266	−0.0324250
$738$	0	0
$739$	−25.7918	−0.948766	−0.474383	−	0.880319i	$-0.657329\pi$
$739$	−25.7918	−0.948766	−0.474383	+	0.880319i	$0.657329\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−14.5905	−0.535273	−0.267636	−	0.963520i	$-0.586243\pi$
$743$	−14.5905	−0.535273	−0.267636	+	0.963520i	$0.586243\pi$
$744$	0	0
$745$	28.2476	1.03491
$746$	0	0
$747$	6.42455	0.235062
$748$	0	0
$749$	0.797599	0.0291436
$750$	0	0
$751$	3.93065	0.143431	0.0717157	−	0.997425i	$-0.477153\pi$
$751$	3.93065	0.143431	0.0717157	+	0.997425i	$0.477153\pi$
$752$	0	0
$753$	−7.39984	−0.269665
$754$	0	0
$755$	5.58357	0.203207
$756$	0	0
$757$	−21.1982	−0.770461	−0.385230	−	0.922820i	$-0.625878\pi$
$757$	−21.1982	−0.770461	−0.385230	+	0.922820i	$0.625878\pi$
$758$	0	0
$759$	1.34217	0.0487178
$760$	0	0
$761$	52.0077	1.88528	0.942638	−	0.333815i	$-0.108336\pi$
$761$	52.0077	1.88528	0.942638	+	0.333815i	$0.108336\pi$
$762$	0	0
$763$	−1.79968	−0.0651529
$764$	0	0
$765$	−0.243957	−0.00882030
$766$	0	0
$767$	59.6554	2.15403
$768$	0	0
$769$	36.4511	1.31446	0.657231	−	0.753689i	$-0.271728\pi$
$769$	36.4511	1.31446	0.657231	+	0.753689i	$0.271728\pi$
$770$	0	0
$771$	19.7312	0.710603
$772$	0	0
$773$	30.7027	1.10430	0.552150	−	0.833745i	$-0.313807\pi$
$773$	30.7027	1.10430	0.552150	+	0.833745i	$0.313807\pi$
$774$	0	0
$775$	4.63774	0.166592
$776$	0	0
$777$	23.8737	0.856463
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0.524277	0.0187601
$782$	0	0
$783$	−43.3532	−1.54932
$784$	0	0
$785$	−4.67713	−0.166934
$786$	0	0
$787$	−47.7270	−1.70129	−0.850643	−	0.525744i	$-0.823787\pi$
$787$	−47.7270	−1.70129	−0.850643	+	0.525744i	$0.823787\pi$
$788$	0	0
$789$	30.2456	1.07677
$790$	0	0
$791$	−19.6920	−0.700168
$792$	0	0
$793$	−9.61308	−0.341370
$794$	0	0
$795$	17.2722	0.612581
$796$	0	0
$797$	15.5088	0.549349	0.274674	−	0.961537i	$-0.411430\pi$
$797$	15.5088	0.549349	0.274674	+	0.961537i	$0.411430\pi$
$798$	0	0
$799$	−0.612383	−0.0216646
$800$	0	0
$801$	1.47044	0.0519553
$802$	0	0
$803$	0.0793479	0.00280013
$804$	0	0
$805$	−17.7637	−0.626089
$806$	0	0
$807$	10.9521	0.385532
$808$	0	0
$809$	11.5061	0.404532	0.202266	−	0.979331i	$-0.435169\pi$
$809$	11.5061	0.404532	0.202266	+	0.979331i	$0.435169\pi$
$810$	0	0
$811$	42.5804	1.49520	0.747601	−	0.664148i	$-0.231206\pi$
$811$	42.5804	1.49520	0.747601	+	0.664148i	$0.231206\pi$
$812$	0	0
$813$	−7.48569	−0.262535
$814$	0	0
$815$	40.5435	1.42018
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−2.61369	−0.0913296
$820$	0	0
$821$	9.84520	0.343600	0.171800	−	0.985132i	$-0.445042\pi$
$821$	9.84520	0.343600	0.171800	+	0.985132i	$0.445042\pi$
$822$	0	0
$823$	−5.03664	−0.175566	−0.0877830	−	0.996140i	$-0.527978\pi$
$823$	−5.03664	−0.175566	−0.0877830	+	0.996140i	$0.527978\pi$
$824$	0	0
$825$	0.243988	0.00849457
$826$	0	0
$827$	−50.1473	−1.74379	−0.871897	−	0.489690i	$-0.837110\pi$
$827$	−50.1473	−1.74379	−0.871897	+	0.489690i	$0.837110\pi$
$828$	0	0
$829$	−10.2374	−0.355560	−0.177780	−	0.984070i	$-0.556892\pi$
$829$	−10.2374	−0.355560	−0.177780	+	0.984070i	$0.556892\pi$
$830$	0	0
$831$	−23.9860	−0.832067
$832$	0	0
$833$	1.61403	0.0559229
$834$	0	0
$835$	9.44333	0.326800
$836$	0	0
$837$	−19.7360	−0.682177
$838$	0	0
$839$	−0.745157	−0.0257257	−0.0128628	−	0.999917i	$-0.504094\pi$
$839$	−0.745157	−0.0257257	−0.0128628	+	0.999917i	$0.504094\pi$
$840$	0	0
$841$	33.3217	1.14903
$842$	0	0
$843$	13.4488	0.463202
$844$	0	0
$845$	−19.8149	−0.681654
$846$	0	0
$847$	14.2736	0.490446
$848$	0	0
$849$	18.4015	0.631538
$850$	0	0
$851$	−81.1610	−2.78216
$852$	0	0
$853$	−37.3866	−1.28009	−0.640046	−	0.768337i	$-0.721085\pi$
$853$	−37.3866	−1.28009	−0.640046	+	0.768337i	$0.721085\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	39.7277	1.35707	0.678535	−	0.734568i	$-0.262615\pi$
$857$	39.7277	1.35707	0.678535	+	0.734568i	$0.262615\pi$
$858$	0	0
$859$	50.7147	1.73036	0.865182	−	0.501459i	$-0.167203\pi$
$859$	50.7147	1.73036	0.865182	+	0.501459i	$0.167203\pi$
$860$	0	0
$861$	7.37091	0.251200
$862$	0	0
$863$	41.3585	1.40786	0.703930	−	0.710269i	$-0.251427\pi$
$863$	41.3585	1.40786	0.703930	+	0.710269i	$0.251427\pi$
$864$	0	0
$865$	5.28462	0.179683
$866$	0	0
$867$	27.1742	0.922885
$868$	0	0
$869$	0.725531	0.0246119
$870$	0	0
$871$	36.1105	1.22356
$872$	0	0
$873$	−6.63312	−0.224497
$874$	0	0
$875$	−15.7409	−0.532139
$876$	0	0
$877$	−9.72990	−0.328555	−0.164278	−	0.986414i	$-0.552529\pi$
$877$	−9.72990	−0.328555	−0.164278	+	0.986414i	$0.552529\pi$
$878$	0	0
$879$	14.7419	0.497232
$880$	0	0
$881$	53.7552	1.81106	0.905529	−	0.424284i	$-0.139474\pi$
$881$	53.7552	1.81106	0.905529	+	0.424284i	$0.139474\pi$
$882$	0	0
$883$	41.7564	1.40521	0.702607	−	0.711578i	$-0.252019\pi$
$883$	41.7564	1.40521	0.702607	+	0.711578i	$0.252019\pi$
$884$	0	0
$885$	−38.2663	−1.28631
$886$	0	0
$887$	−37.1518	−1.24744	−0.623718	−	0.781649i	$-0.714379\pi$
$887$	−37.1518	−1.24744	−0.623718	+	0.781649i	$0.714379\pi$
$888$	0	0
$889$	−28.6040	−0.959346
$890$	0	0
$891$	−0.891179	−0.0298556
$892$	0	0
$893$	0	0
$894$	0	0
$895$	20.2915	0.678269
$896$	0	0
$897$	−55.0589	−1.83836
$898$	0	0
$899$	28.3712	0.946233
$900$	0	0
$901$	−1.69539	−0.0564815
$902$	0	0
$903$	−22.8477	−0.760324
$904$	0	0
$905$	33.4014	1.11030
$906$	0	0
$907$	−10.8467	−0.360158	−0.180079	−	0.983652i	$-0.557635\pi$
$907$	−10.8467	−0.360158	−0.180079	+	0.983652i	$0.557635\pi$
$908$	0	0
$909$	−6.65307	−0.220668
$910$	0	0
$911$	−50.3661	−1.66870	−0.834351	−	0.551233i	$-0.814158\pi$
$911$	−50.3661	−1.66870	−0.834351	+	0.551233i	$0.814158\pi$
$912$	0	0
$913$	−1.81297	−0.0600004
$914$	0	0
$915$	6.16635	0.203853
$916$	0	0
$917$	6.92339	0.228630
$918$	0	0
$919$	−38.5967	−1.27319	−0.636593	−	0.771200i	$-0.719657\pi$
$919$	−38.5967	−1.27319	−0.636593	+	0.771200i	$0.719657\pi$
$920$	0	0
$921$	28.7097	0.946018
$922$	0	0
$923$	−21.5070	−0.707912
$924$	0	0
$925$	−14.7539	−0.485106
$926$	0	0
$927$	−4.42966	−0.145489
$928$	0	0
$929$	−9.70019	−0.318253	−0.159126	−	0.987258i	$-0.550868\pi$
$929$	−9.70019	−0.318253	−0.159126	+	0.987258i	$0.550868\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−22.0231	−0.721004
$934$	0	0
$935$	0.0688432	0.00225141
$936$	0	0
$937$	38.2198	1.24859	0.624294	−	0.781190i	$-0.285387\pi$
$937$	38.2198	1.24859	0.624294	+	0.781190i	$0.285387\pi$
$938$	0	0
$939$	−50.6475	−1.65282
$940$	0	0
$941$	30.8153	1.00455	0.502275	−	0.864708i	$-0.332497\pi$
$941$	30.8153	1.00455	0.502275	+	0.864708i	$0.332497\pi$
$942$	0	0
$943$	−25.0582	−0.816006
$944$	0	0
$945$	13.7419	0.447025
$946$	0	0
$947$	−13.2873	−0.431780	−0.215890	−	0.976418i	$-0.569265\pi$
$947$	−13.2873	−0.431780	−0.215890	+	0.976418i	$0.569265\pi$
$948$	0	0
$949$	−3.25503	−0.105663
$950$	0	0
$951$	19.3335	0.626933
$952$	0	0
$953$	−13.3100	−0.431153	−0.215577	−	0.976487i	$-0.569163\pi$
$953$	−13.3100	−0.431153	−0.215577	+	0.976487i	$0.569163\pi$
$954$	0	0
$955$	−43.3814	−1.40379
$956$	0	0
$957$	1.49259	0.0482486
$958$	0	0
$959$	−9.37046	−0.302588
$960$	0	0
$961$	−18.0843	−0.583366
$962$	0	0
$963$	0.255916	0.00824679
$964$	0	0
$965$	−39.7528	−1.27969
$966$	0	0
$967$	23.3525	0.750966	0.375483	−	0.926829i	$-0.377477\pi$
$967$	23.3525	0.750966	0.375483	+	0.926829i	$0.377477\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	44.8281	1.43860	0.719301	−	0.694699i	$-0.244462\pi$
$971$	44.8281	1.43860	0.719301	+	0.694699i	$0.244462\pi$
$972$	0	0
$973$	5.12726	0.164372
$974$	0	0
$975$	−10.0089	−0.320542
$976$	0	0
$977$	18.2001	0.582273	0.291136	−	0.956682i	$-0.405967\pi$
$977$	18.2001	0.582273	0.291136	+	0.956682i	$0.405967\pi$
$978$	0	0
$979$	−0.414948	−0.0132618
$980$	0	0
$981$	−0.577444	−0.0184364
$982$	0	0
$983$	20.7521	0.661888	0.330944	−	0.943650i	$-0.392633\pi$
$983$	20.7521	0.661888	0.330944	+	0.943650i	$0.392633\pi$
$984$	0	0
$985$	−24.6192	−0.784434
$986$	0	0
$987$	4.20852	0.133959
$988$	0	0
$989$	77.6730	2.46986
$990$	0	0
$991$	40.1354	1.27494	0.637472	−	0.770474i	$-0.279980\pi$
$991$	40.1354	1.27494	0.637472	+	0.770474i	$0.279980\pi$
$992$	0	0
$993$	−11.0355	−0.350201
$994$	0	0
$995$	−40.7501	−1.29187
$996$	0	0
$997$	56.3953	1.78606	0.893029	−	0.449999i	$-0.148576\pi$
$997$	56.3953	1.78606	0.893029	+	0.449999i	$0.148576\pi$
$998$	0	0
$999$	62.7857	1.98645

Twists

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

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

\(f(q)\)	\(=\)	\(q-1.60721 q^{3} -1.92601 q^{5} -1.29923 q^{7} -0.416870 q^{9} +0.117638 q^{11} -4.82577 q^{13} +3.09551 q^{15} -0.303846 q^{17} +2.08814 q^{21} -7.09885 q^{23} -1.29047 q^{25} +5.49163 q^{27} -7.89441 q^{29} -3.59384 q^{31} -0.189069 q^{33} +2.50234 q^{35} +11.4330 q^{37} +7.75603 q^{39} +3.52989 q^{41} -10.9416 q^{43} +0.802897 q^{45} +2.01544 q^{47} -5.31200 q^{49} +0.488346 q^{51} +5.57974 q^{53} -0.226572 q^{55} -12.3618 q^{59} +1.99203 q^{61} +0.541610 q^{63} +9.29450 q^{65} -7.48284 q^{67} +11.4093 q^{69} +4.45670 q^{71} +0.674509 q^{73} +2.07406 q^{75} -0.152839 q^{77} +6.16749 q^{79} -7.57561 q^{81} -15.4114 q^{83} +0.585213 q^{85} +12.6880 q^{87} -3.52733 q^{89} +6.26979 q^{91} +5.77605 q^{93} +15.9117 q^{97} -0.0490397 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 6 q + 3 q^{3} + 2 q^{5} - 2 q^{7} + 9 q^{9} - 5 q^{11} + 10 q^{13} + 4 q^{17} + 23 q^{21} - 15 q^{23} + 12 q^{25} + 18 q^{27} - 7 q^{29} + 5 q^{31} - q^{33} - 38 q^{35} + 17 q^{37} + 37 q^{39} + 4 q^{41}+ \cdots - 47 q^{99}+O(q^{100}) \) 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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