LMFDB - Embedded newform 2888.2.a.v.1.1

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

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

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

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

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

Newform invariants

comment:select newform

sage:traces = [8,0,-6,0,-2,0,-2,0,2,0,12,0,-6,0,0] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(15)] == traces)

gp:f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$23.0607961037$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	8.8.10564000000.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 2x^{7} - 9x^{6} + 14x^{5} + 24x^{4} - 28x^{3} - 21x^{2} + 16x + 1 $ x^8 - 2x^7 - 9x^6 + 14x^5 + 24x^4 - 28x^3 - 21x^2 + 16*x + 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-2.11762$ 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)

$f(q)$	$=$	$q-3.11762 q^{3} +0.354750 q^{5} +0.325225 q^{7} +6.71958 q^{9} +5.04322 q^{11} -6.82809 q^{13} -1.10598 q^{15} -3.04008 q^{17} -1.01393 q^{21} -1.31918 q^{23} -4.87415 q^{25} -11.5963 q^{27} +2.78635 q^{29} +7.53649 q^{31} -15.7229 q^{33} +0.115374 q^{35} +5.42314 q^{37} +21.2874 q^{39} -7.90277 q^{41} +7.42518 q^{43} +2.38377 q^{45} -7.97085 q^{47} -6.89423 q^{49} +9.47784 q^{51} +9.11398 q^{53} +1.78908 q^{55} +7.72949 q^{59} -0.534084 q^{61} +2.18538 q^{63} -2.42227 q^{65} +6.27233 q^{67} +4.11271 q^{69} -12.4442 q^{71} +5.67035 q^{73} +15.1958 q^{75} +1.64018 q^{77} +9.78023 q^{79} +15.9941 q^{81} -6.93591 q^{83} -1.07847 q^{85} -8.68679 q^{87} +3.44202 q^{89} -2.22067 q^{91} -23.4960 q^{93} -13.5367 q^{97} +33.8883 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q - 6 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 12 q^{11} - 6 q^{13} - 10 q^{17} + 4 q^{21} - 12 q^{23} + 2 q^{25} - 12 q^{27} - 18 q^{29} - 14 q^{31} - 40 q^{33} + 18 q^{35} - 16 q^{37} + 28 q^{39} - 12 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) $ 8 * q - 6 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 12 * q^11 - 6 * q^13 - 10 * q^17 + 4 * q^21 - 12 * q^23 + 2 * q^25 - 12 * q^27 - 18 * q^29 - 14 * q^31 - 40 * q^33 + 18 * q^35 - 16 * q^37 + 28 * q^39 - 12 * q^41 + 8 * q^43 - 2 * q^45 + 24 * q^47 + 12 * q^49 + 22 * q^51 - 12 * q^53 - 2 * q^55 + 16 * q^59 + 12 * q^63 - 12 * q^65 - 42 * q^67 + 12 * q^69 - 36 * q^71 - 10 * q^73 - 32 * q^75 - 38 * q^77 - 6 * q^79 + 4 * q^81 + 12 * q^83 - 42 * q^85 - 12 * q^87 - 8 * q^89 - 16 * q^91 - 14 * q^93 - 28 * q^97 + 32 * q^99

Coefficient data

For each $n$ we display the coefficients of the $q$-expansion $a_n$, the Satake parameters $\alpha_p$, and the Satake angles $\theta_p = \textrm{Arg}(\alpha_p)$.

Display $a_p$ with $p$ up to: 50 250 1000 (See $a_n$ instead) (See $a_n$ instead) (See $a_n$ instead) Display $a_n$ with $n$ up to: 50 250 1000 (See only $a_p$) (See only $a_p$) (See only $a_p$)

$n$	$a_n$	$a_n / n^{(k-1)/2}$	$ \alpha_n $			$ \theta_n $
$p$	$a_p$	$a_p / p^{(k-1)/2}$	$ \alpha_p$			$ \theta_p $

$2$	0	0
$2$	0	0
$3$	−3.11762	−1.79996	−0.899981	−	0.435930i	$-0.856420\pi$
$3$	−3.11762	−1.79996	−0.899981	+	0.435930i	$0.856420\pi$
$4$	0	0
$5$	0.354750	0.158649	0.0793245	−	0.996849i	$-0.474724\pi$
$5$	0.354750	0.158649	0.0793245	+	0.996849i	$0.474724\pi$
$6$	0	0
$7$	0.325225	0.122924	0.0614618	−	0.998109i	$-0.480424\pi$
$7$	0.325225	0.122924	0.0614618	+	0.998109i	$0.480424\pi$
$8$	0	0
$9$	6.71958	2.23986
$10$	0	0
$11$	5.04322	1.52059	0.760294	−	0.649579i	$-0.225055\pi$
$11$	5.04322	1.52059	0.760294	+	0.649579i	$0.225055\pi$
$12$	0	0
$13$	−6.82809	−1.89377	−0.946886	−	0.321569i	$-0.895790\pi$
$13$	−6.82809	−1.89377	−0.946886	+	0.321569i	$0.895790\pi$
$14$	0	0
$15$	−1.10598	−0.285562
$16$	0	0
$17$	−3.04008	−0.737329	−0.368664	−	0.929563i	$-0.620185\pi$
$17$	−3.04008	−0.737329	−0.368664	+	0.929563i	$0.620185\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	−1.01393	−0.221258
$22$	0	0
$23$	−1.31918	−0.275068	−0.137534	−	0.990497i	$-0.543918\pi$
$23$	−1.31918	−0.275068	−0.137534	+	0.990497i	$0.543918\pi$
$24$	0	0
$25$	−4.87415	−0.974830
$26$	0	0
$27$	−11.5963	−2.23170
$28$	0	0
$29$	2.78635	0.517412	0.258706	−	0.965956i	$-0.416704\pi$
$29$	2.78635	0.517412	0.258706	+	0.965956i	$0.416704\pi$
$30$	0	0
$31$	7.53649	1.35359	0.676797	−	0.736170i	$-0.263368\pi$
$31$	7.53649	1.35359	0.676797	+	0.736170i	$0.263368\pi$
$32$	0	0
$33$	−15.7229	−2.73700
$34$	0	0
$35$	0.115374	0.0195017
$36$	0	0
$37$	5.42314	0.891559	0.445779	−	0.895143i	$-0.352927\pi$
$37$	5.42314	0.891559	0.445779	+	0.895143i	$0.352927\pi$
$38$	0	0
$39$	21.2874	3.40872
$40$	0	0
$41$	−7.90277	−1.23420	−0.617102	−	0.786883i	$-0.711694\pi$
$41$	−7.90277	−1.23420	−0.617102	+	0.786883i	$0.711694\pi$
$42$	0	0
$43$	7.42518	1.13233	0.566164	−	0.824292i	$-0.308427\pi$
$43$	7.42518	1.13233	0.566164	+	0.824292i	$0.308427\pi$
$44$	0	0
$45$	2.38377	0.355352
$46$	0	0
$47$	−7.97085	−1.16267	−0.581334	−	0.813665i	$-0.697469\pi$
$47$	−7.97085	−1.16267	−0.581334	+	0.813665i	$0.697469\pi$
$48$	0	0
$49$	−6.89423	−0.984890
$50$	0	0
$51$	9.47784	1.32716
$52$	0	0
$53$	9.11398	1.25190	0.625951	−	0.779863i	$-0.284711\pi$
$53$	9.11398	1.25190	0.625951	+	0.779863i	$0.284711\pi$
$54$	0	0
$55$	1.78908	0.241240
$56$	0	0
$57$	0	0
$58$	0	0
$59$	7.72949	1.00629	0.503147	−	0.864201i	$-0.332175\pi$
$59$	7.72949	1.00629	0.503147	+	0.864201i	$0.332175\pi$
$60$	0	0
$61$	−0.534084	−0.0683824	−0.0341912	−	0.999415i	$-0.510886\pi$
$61$	−0.534084	−0.0683824	−0.0341912	+	0.999415i	$0.510886\pi$
$62$	0	0
$63$	2.18538	0.275332
$64$	0	0
$65$	−2.42227	−0.300445
$66$	0	0
$67$	6.27233	0.766287	0.383143	−	0.923689i	$-0.374842\pi$
$67$	6.27233	0.766287	0.383143	+	0.923689i	$0.374842\pi$
$68$	0	0
$69$	4.11271	0.495112
$70$	0	0
$71$	−12.4442	−1.47686	−0.738430	−	0.674330i	$-0.764432\pi$
$71$	−12.4442	−1.47686	−0.738430	+	0.674330i	$0.764432\pi$
$72$	0	0
$73$	5.67035	0.663665	0.331832	−	0.943338i	$-0.392333\pi$
$73$	5.67035	0.663665	0.331832	+	0.943338i	$0.392333\pi$
$74$	0	0
$75$	15.1958	1.75466
$76$	0	0
$77$	1.64018	0.186916
$78$	0	0
$79$	9.78023	1.10036	0.550181	−	0.835046i	$-0.314559\pi$
$79$	9.78023	1.10036	0.550181	+	0.835046i	$0.314559\pi$
$80$	0	0
$81$	15.9941	1.77712
$82$	0	0
$83$	−6.93591	−0.761315	−0.380657	−	0.924716i	$-0.624302\pi$
$83$	−6.93591	−0.761315	−0.380657	+	0.924716i	$0.624302\pi$
$84$	0	0
$85$	−1.07847	−0.116977
$86$	0	0
$87$	−8.68679	−0.931321
$88$	0	0
$89$	3.44202	0.364853	0.182427	−	0.983219i	$-0.441605\pi$
$89$	3.44202	0.364853	0.182427	+	0.983219i	$0.441605\pi$
$90$	0	0
$91$	−2.22067	−0.232789
$92$	0	0
$93$	−23.4960	−2.43642
$94$	0	0
$95$	0	0
$96$	0	0
$97$	−13.5367	−1.37444	−0.687220	−	0.726449i	$-0.741169\pi$
$97$	−13.5367	−1.37444	−0.687220	+	0.726449i	$0.741169\pi$
$98$	0	0
$99$	33.8883	3.40590
$100$	0	0
$101$	−2.87890	−0.286461	−0.143231	−	0.989689i	$-0.545749\pi$
$101$	−2.87890	−0.286461	−0.143231	+	0.989689i	$0.545749\pi$
$102$	0	0
$103$	−12.8091	−1.26212	−0.631060	−	0.775734i	$-0.717380\pi$
$103$	−12.8091	−1.26212	−0.631060	+	0.775734i	$0.717380\pi$
$104$	0	0
$105$	−0.359692	−0.0351023
$106$	0	0
$107$	2.93075	0.283326	0.141663	−	0.989915i	$-0.454755\pi$
$107$	2.93075	0.283326	0.141663	+	0.989915i	$0.454755\pi$
$108$	0	0
$109$	−3.70775	−0.355138	−0.177569	−	0.984108i	$-0.556823\pi$
$109$	−3.70775	−0.355138	−0.177569	+	0.984108i	$0.556823\pi$
$110$	0	0
$111$	−16.9073	−1.60477
$112$	0	0
$113$	−15.0946	−1.41998	−0.709990	−	0.704211i	$-0.751301\pi$
$113$	−15.0946	−1.41998	−0.709990	+	0.704211i	$0.751301\pi$
$114$	0	0
$115$	−0.467979	−0.0436393
$116$	0	0
$117$	−45.8819	−4.24179
$118$	0	0
$119$	−0.988712	−0.0906351
$120$	0	0
$121$	14.4340	1.31219
$122$	0	0
$123$	24.6379	2.22152
$124$	0	0
$125$	−3.50286	−0.313305
$126$	0	0
$127$	3.67343	0.325964	0.162982	−	0.986629i	$-0.447889\pi$
$127$	3.67343	0.325964	0.162982	+	0.986629i	$0.447889\pi$
$128$	0	0
$129$	−23.1489	−2.03815
$130$	0	0
$131$	3.29201	0.287624	0.143812	−	0.989605i	$-0.454064\pi$
$131$	3.29201	0.287624	0.143812	+	0.989605i	$0.454064\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−4.11378	−0.354058
$136$	0	0
$137$	−9.31144	−0.795530	−0.397765	−	0.917487i	$-0.630214\pi$
$137$	−9.31144	−0.795530	−0.397765	+	0.917487i	$0.630214\pi$
$138$	0	0
$139$	1.72887	0.146641	0.0733203	−	0.997308i	$-0.476640\pi$
$139$	1.72887	0.146641	0.0733203	+	0.997308i	$0.476640\pi$
$140$	0	0
$141$	24.8501	2.09276
$142$	0	0
$143$	−34.4356	−2.87965
$144$	0	0
$145$	0.988457	0.0820869
$146$	0	0
$147$	21.4936	1.77276
$148$	0	0
$149$	−9.73181	−0.797261	−0.398630	−	0.917112i	$-0.630514\pi$
$149$	−9.73181	−0.797261	−0.398630	+	0.917112i	$0.630514\pi$
$150$	0	0
$151$	−12.7484	−1.03745	−0.518724	−	0.854942i	$-0.673593\pi$
$151$	−12.7484	−1.03745	−0.518724	+	0.854942i	$0.673593\pi$
$152$	0	0
$153$	−20.4281	−1.65151
$154$	0	0
$155$	2.67357	0.214746
$156$	0	0
$157$	−3.78660	−0.302204	−0.151102	−	0.988518i	$-0.548282\pi$
$157$	−3.78660	−0.302204	−0.151102	+	0.988518i	$0.548282\pi$
$158$	0	0
$159$	−28.4140	−2.25337
$160$	0	0
$161$	−0.429031	−0.0338124
$162$	0	0
$163$	−9.57636	−0.750079	−0.375039	−	0.927009i	$-0.622371\pi$
$163$	−9.57636	−0.750079	−0.375039	+	0.927009i	$0.622371\pi$
$164$	0	0
$165$	−5.57769	−0.434222
$166$	0	0
$167$	6.79761	0.526015	0.263007	−	0.964794i	$-0.415286\pi$
$167$	6.79761	0.526015	0.263007	+	0.964794i	$0.415286\pi$
$168$	0	0
$169$	33.6228	2.58637
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−11.5243	−0.876178	−0.438089	−	0.898932i	$-0.644345\pi$
$173$	−11.5243	−0.876178	−0.438089	+	0.898932i	$0.644345\pi$
$174$	0	0
$175$	−1.58520	−0.119830
$176$	0	0
$177$	−24.0977	−1.81129
$178$	0	0
$179$	−8.67160	−0.648146	−0.324073	−	0.946032i	$-0.605052\pi$
$179$	−8.67160	−0.648146	−0.324073	+	0.946032i	$0.605052\pi$
$180$	0	0
$181$	−17.0548	−1.26767	−0.633836	−	0.773468i	$-0.718521\pi$
$181$	−17.0548	−1.26767	−0.633836	+	0.773468i	$0.718521\pi$
$182$	0	0
$183$	1.66507	0.123086
$184$	0	0
$185$	1.92386	0.141445
$186$	0	0
$187$	−15.3318	−1.12117
$188$	0	0
$189$	−3.77140	−0.274329
$190$	0	0
$191$	−2.99858	−0.216970	−0.108485	−	0.994098i	$-0.534600\pi$
$191$	−2.99858	−0.216970	−0.108485	+	0.994098i	$0.534600\pi$
$192$	0	0
$193$	3.36738	0.242389	0.121195	−	0.992629i	$-0.461327\pi$
$193$	3.36738	0.242389	0.121195	+	0.992629i	$0.461327\pi$
$194$	0	0
$195$	7.55172	0.540790
$196$	0	0
$197$	−17.8488	−1.27167	−0.635836	−	0.771824i	$-0.719345\pi$
$197$	−17.8488	−1.27167	−0.635836	+	0.771824i	$0.719345\pi$
$198$	0	0
$199$	−14.7395	−1.04486	−0.522429	−	0.852683i	$-0.674974\pi$
$199$	−14.7395	−1.04486	−0.522429	+	0.852683i	$0.674974\pi$
$200$	0	0
$201$	−19.5548	−1.37929
$202$	0	0
$203$	0.906191	0.0636021
$204$	0	0
$205$	−2.80351	−0.195805
$206$	0	0
$207$	−8.86434	−0.616114
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−22.5584	−1.55298	−0.776491	−	0.630128i	$-0.783003\pi$
$211$	−22.5584	−1.55298	−0.776491	+	0.630128i	$0.783003\pi$
$212$	0	0
$213$	38.7965	2.65829
$214$	0	0
$215$	2.63408	0.179643
$216$	0	0
$217$	2.45106	0.166389
$218$	0	0
$219$	−17.6780	−1.19457
$220$	0	0
$221$	20.7580	1.39633
$222$	0	0
$223$	−3.99343	−0.267420	−0.133710	−	0.991020i	$-0.542689\pi$
$223$	−3.99343	−0.267420	−0.133710	+	0.991020i	$0.542689\pi$
$224$	0	0
$225$	−32.7523	−2.18349
$226$	0	0
$227$	−15.0578	−0.999424	−0.499712	−	0.866192i	$-0.666561\pi$
$227$	−15.0578	−0.999424	−0.499712	+	0.866192i	$0.666561\pi$
$228$	0	0
$229$	2.13845	0.141313	0.0706564	−	0.997501i	$-0.477491\pi$
$229$	2.13845	0.141313	0.0706564	+	0.997501i	$0.477491\pi$
$230$	0	0
$231$	−5.11347	−0.336442
$232$	0	0
$233$	−2.49868	−0.163694	−0.0818471	−	0.996645i	$-0.526082\pi$
$233$	−2.49868	−0.163694	−0.0818471	+	0.996645i	$0.526082\pi$
$234$	0	0
$235$	−2.82766	−0.184456
$236$	0	0
$237$	−30.4911	−1.98061
$238$	0	0
$239$	−0.159676	−0.0103286	−0.00516430	−	0.999987i	$-0.501644\pi$
$239$	−0.159676	−0.0103286	−0.00516430	+	0.999987i	$0.501644\pi$
$240$	0	0
$241$	15.8231	1.01926	0.509629	−	0.860394i	$-0.329783\pi$
$241$	15.8231	1.01926	0.509629	+	0.860394i	$0.329783\pi$
$242$	0	0
$243$	−15.0747	−0.967040
$244$	0	0
$245$	−2.44573	−0.156252
$246$	0	0
$247$	0	0
$248$	0	0
$249$	21.6236	1.37034
$250$	0	0
$251$	7.86964	0.496727	0.248364	−	0.968667i	$-0.420107\pi$
$251$	7.86964	0.496727	0.248364	+	0.968667i	$0.420107\pi$
$252$	0	0
$253$	−6.65291	−0.418265
$254$	0	0
$255$	3.36227	0.210553
$256$	0	0
$257$	14.0500	0.876415	0.438207	−	0.898874i	$-0.355614\pi$
$257$	14.0500	0.876415	0.438207	+	0.898874i	$0.355614\pi$
$258$	0	0
$259$	1.76374	0.109594
$260$	0	0
$261$	18.7231	1.15893
$262$	0	0
$263$	24.0025	1.48006	0.740029	−	0.672574i	$-0.234812\pi$
$263$	24.0025	1.48006	0.740029	+	0.672574i	$0.234812\pi$
$264$	0	0
$265$	3.23318	0.198613
$266$	0	0
$267$	−10.7309	−0.656721
$268$	0	0
$269$	23.9026	1.45737	0.728683	−	0.684851i	$-0.240133\pi$
$269$	23.9026	1.45737	0.728683	+	0.684851i	$0.240133\pi$
$270$	0	0
$271$	−23.4886	−1.42683	−0.713417	−	0.700740i	$-0.752853\pi$
$271$	−23.4886	−1.42683	−0.713417	+	0.700740i	$0.752853\pi$
$272$	0	0
$273$	6.92321	0.419012
$274$	0	0
$275$	−24.5814	−1.48231
$276$	0	0
$277$	−20.7255	−1.24527	−0.622637	−	0.782510i	$-0.713939\pi$
$277$	−20.7255	−1.24527	−0.622637	+	0.782510i	$0.713939\pi$
$278$	0	0
$279$	50.6421	3.03186
$280$	0	0
$281$	1.99573	0.119055	0.0595275	−	0.998227i	$-0.481041\pi$
$281$	1.99573	0.119055	0.0595275	+	0.998227i	$0.481041\pi$
$282$	0	0
$283$	3.91091	0.232479	0.116240	−	0.993221i	$-0.462916\pi$
$283$	3.91091	0.232479	0.116240	+	0.993221i	$0.462916\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	−2.57018	−0.151713
$288$	0	0
$289$	−7.75788	−0.456346
$290$	0	0
$291$	42.2023	2.47394
$292$	0	0
$293$	33.8297	1.97635	0.988175	−	0.153328i	$-0.0489993\pi$
$293$	33.8297	1.97635	0.988175	+	0.153328i	$0.0489993\pi$
$294$	0	0
$295$	2.74204	0.159648
$296$	0	0
$297$	−58.4825	−3.39350
$298$	0	0
$299$	9.00749	0.520916
$300$	0	0
$301$	2.41486	0.139190
$302$	0	0
$303$	8.97534	0.515620
$304$	0	0
$305$	−0.189466	−0.0108488
$306$	0	0
$307$	−4.94865	−0.282435	−0.141217	−	0.989979i	$-0.545102\pi$
$307$	−4.94865	−0.282435	−0.141217	+	0.989979i	$0.545102\pi$
$308$	0	0
$309$	39.9340	2.27177
$310$	0	0
$311$	12.5026	0.708957	0.354478	−	0.935064i	$-0.384658\pi$
$311$	12.5026	0.708957	0.354478	+	0.935064i	$0.384658\pi$
$312$	0	0
$313$	3.01626	0.170489	0.0852446	−	0.996360i	$-0.472833\pi$
$313$	3.01626	0.170489	0.0852446	+	0.996360i	$0.472833\pi$
$314$	0	0
$315$	0.775263	0.0436811
$316$	0	0
$317$	−25.4148	−1.42744	−0.713719	−	0.700432i	$-0.752991\pi$
$317$	−25.4148	−1.42744	−0.713719	+	0.700432i	$0.752991\pi$
$318$	0	0
$319$	14.0522	0.786770
$320$	0	0
$321$	−9.13698	−0.509976
$322$	0	0
$323$	0	0
$324$	0	0
$325$	33.2812	1.84611
$326$	0	0
$327$	11.5594	0.639235
$328$	0	0
$329$	−2.59232	−0.142919
$330$	0	0
$331$	−7.27350	−0.399788	−0.199894	−	0.979818i	$-0.564060\pi$
$331$	−7.27350	−0.399788	−0.199894	+	0.979818i	$0.564060\pi$
$332$	0	0
$333$	36.4412	1.99697
$334$	0	0
$335$	2.22511	0.121571
$336$	0	0
$337$	−29.8709	−1.62717	−0.813587	−	0.581443i	$-0.802488\pi$
$337$	−29.8709	−1.62717	−0.813587	+	0.581443i	$0.802488\pi$
$338$	0	0
$339$	47.0593	2.55591
$340$	0	0
$341$	38.0082	2.05826
$342$	0	0
$343$	−4.51875	−0.243990
$344$	0	0
$345$	1.45898	0.0785491
$346$	0	0
$347$	6.63925	0.356414	0.178207	−	0.983993i	$-0.442970\pi$
$347$	6.63925	0.356414	0.178207	+	0.983993i	$0.442970\pi$
$348$	0	0
$349$	−22.8028	−1.22061	−0.610303	−	0.792168i	$-0.708952\pi$
$349$	−22.8028	−1.22061	−0.610303	+	0.792168i	$0.708952\pi$
$350$	0	0
$351$	79.1804	4.22634
$352$	0	0
$353$	−11.3924	−0.606358	−0.303179	−	0.952934i	$-0.598048\pi$
$353$	−11.3924	−0.606358	−0.303179	+	0.952934i	$0.598048\pi$
$354$	0	0
$355$	−4.41459	−0.234302
$356$	0	0
$357$	3.08243	0.163140
$358$	0	0
$359$	−14.9389	−0.788446	−0.394223	−	0.919015i	$-0.628986\pi$
$359$	−14.9389	−0.788446	−0.394223	+	0.919015i	$0.628986\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	−44.9999	−2.36188
$364$	0	0
$365$	2.01156	0.105290
$366$	0	0
$367$	−12.0776	−0.630447	−0.315223	−	0.949018i	$-0.602079\pi$
$367$	−12.0776	−0.630447	−0.315223	+	0.949018i	$0.602079\pi$
$368$	0	0
$369$	−53.1033	−2.76445
$370$	0	0
$371$	2.96410	0.153888
$372$	0	0
$373$	−1.94891	−0.100911	−0.0504555	−	0.998726i	$-0.516067\pi$
$373$	−1.94891	−0.100911	−0.0504555	+	0.998726i	$0.516067\pi$
$374$	0	0
$375$	10.9206	0.563937
$376$	0	0
$377$	−19.0254	−0.979860
$378$	0	0
$379$	−29.7116	−1.52618	−0.763091	−	0.646291i	$-0.776319\pi$
$379$	−29.7116	−1.52618	−0.763091	+	0.646291i	$0.776319\pi$
$380$	0	0
$381$	−11.4524	−0.586723
$382$	0	0
$383$	−18.9304	−0.967298	−0.483649	−	0.875262i	$-0.660689\pi$
$383$	−18.9304	−0.967298	−0.483649	+	0.875262i	$0.660689\pi$
$384$	0	0
$385$	0.581855	0.0296541
$386$	0	0
$387$	49.8941	2.53626
$388$	0	0
$389$	−23.0076	−1.16653	−0.583266	−	0.812281i	$-0.698226\pi$
$389$	−23.0076	−1.16653	−0.583266	+	0.812281i	$0.698226\pi$
$390$	0	0
$391$	4.01042	0.202816
$392$	0	0
$393$	−10.2633	−0.517713
$394$	0	0
$395$	3.46954	0.174571
$396$	0	0
$397$	−6.62460	−0.332479	−0.166240	−	0.986085i	$-0.553163\pi$
$397$	−6.62460	−0.332479	−0.166240	+	0.986085i	$0.553163\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−4.52922	−0.226179	−0.113089	−	0.993585i	$-0.536075\pi$
$401$	−4.52922	−0.226179	−0.113089	+	0.993585i	$0.536075\pi$
$402$	0	0
$403$	−51.4599	−2.56340
$404$	0	0
$405$	5.67389	0.281938
$406$	0	0
$407$	27.3501	1.35569
$408$	0	0
$409$	−10.2598	−0.507317	−0.253658	−	0.967294i	$-0.581634\pi$
$409$	−10.2598	−0.507317	−0.253658	+	0.967294i	$0.581634\pi$
$410$	0	0
$411$	29.0296	1.43192
$412$	0	0
$413$	2.51383	0.123697
$414$	0	0
$415$	−2.46051	−0.120782
$416$	0	0
$417$	−5.38996	−0.263947
$418$	0	0
$419$	25.7029	1.25567	0.627835	−	0.778347i	$-0.283941\pi$
$419$	25.7029	1.25567	0.627835	+	0.778347i	$0.283941\pi$
$420$	0	0
$421$	15.4969	0.755272	0.377636	−	0.925954i	$-0.376737\pi$
$421$	15.4969	0.755272	0.377636	+	0.925954i	$0.376737\pi$
$422$	0	0
$423$	−53.5608	−2.60422
$424$	0	0
$425$	14.8178	0.718771
$426$	0	0
$427$	−0.173697	−0.00840581
$428$	0	0
$429$	107.357	5.18325
$430$	0	0
$431$	−36.4853	−1.75743	−0.878717	−	0.477343i	$-0.841600\pi$
$431$	−36.4853	−1.75743	−0.878717	+	0.477343i	$0.841600\pi$
$432$	0	0
$433$	1.72738	0.0830125	0.0415062	−	0.999138i	$-0.486784\pi$
$433$	1.72738	0.0830125	0.0415062	+	0.999138i	$0.486784\pi$
$434$	0	0
$435$	−3.08164	−0.147753
$436$	0	0
$437$	0	0
$438$	0	0
$439$	12.0620	0.575686	0.287843	−	0.957678i	$-0.407062\pi$
$439$	12.0620	0.575686	0.287843	+	0.957678i	$0.407062\pi$
$440$	0	0
$441$	−46.3263	−2.20602
$442$	0	0
$443$	−4.13788	−0.196596	−0.0982982	−	0.995157i	$-0.531340\pi$
$443$	−4.13788	−0.196596	−0.0982982	+	0.995157i	$0.531340\pi$
$444$	0	0
$445$	1.22106	0.0578836
$446$	0	0
$447$	30.3401	1.43504
$448$	0	0
$449$	28.3916	1.33988	0.669942	−	0.742414i	$-0.266319\pi$
$449$	28.3916	1.33988	0.669942	+	0.742414i	$0.266319\pi$
$450$	0	0
$451$	−39.8554	−1.87672
$452$	0	0
$453$	39.7446	1.86737
$454$	0	0
$455$	−0.787782	−0.0369318
$456$	0	0
$457$	16.1132	0.753743	0.376871	−	0.926266i	$-0.377000\pi$
$457$	16.1132	0.753743	0.376871	+	0.926266i	$0.377000\pi$
$458$	0	0
$459$	35.2536	1.64550
$460$	0	0
$461$	−29.9615	−1.39545	−0.697723	−	0.716367i	$-0.745804\pi$
$461$	−29.9615	−1.39545	−0.697723	+	0.716367i	$0.745804\pi$
$462$	0	0
$463$	9.52512	0.442670	0.221335	−	0.975198i	$-0.428959\pi$
$463$	9.52512	0.442670	0.221335	+	0.975198i	$0.428959\pi$
$464$	0	0
$465$	−8.33519	−0.386535
$466$	0	0
$467$	4.12179	0.190734	0.0953668	−	0.995442i	$-0.469598\pi$
$467$	4.12179	0.190734	0.0953668	+	0.995442i	$0.469598\pi$
$468$	0	0
$469$	2.03992	0.0941947
$470$	0	0
$471$	11.8052	0.543955
$472$	0	0
$473$	37.4468	1.72181
$474$	0	0
$475$	0	0
$476$	0	0
$477$	61.2421	2.80409
$478$	0	0
$479$	−12.5879	−0.575154	−0.287577	−	0.957758i	$-0.592850\pi$
$479$	−12.5879	−0.575154	−0.287577	+	0.957758i	$0.592850\pi$
$480$	0	0
$481$	−37.0297	−1.68841
$482$	0	0
$483$	1.33756	0.0608610
$484$	0	0
$485$	−4.80214	−0.218054
$486$	0	0
$487$	−32.4333	−1.46969	−0.734846	−	0.678234i	$-0.762746\pi$
$487$	−32.4333	−1.46969	−0.734846	+	0.678234i	$0.762746\pi$
$488$	0	0
$489$	29.8555	1.35011
$490$	0	0
$491$	11.1465	0.503035	0.251517	−	0.967853i	$-0.419070\pi$
$491$	11.1465	0.503035	0.251517	+	0.967853i	$0.419070\pi$
$492$	0	0
$493$	−8.47074	−0.381503
$494$	0	0
$495$	12.0219	0.540344
$496$	0	0
$497$	−4.04718	−0.181541
$498$	0	0
$499$	4.30303	0.192630	0.0963151	−	0.995351i	$-0.469294\pi$
$499$	4.30303	0.192630	0.0963151	+	0.995351i	$0.469294\pi$
$500$	0	0
$501$	−21.1924	−0.946807
$502$	0	0
$503$	31.8622	1.42067	0.710333	−	0.703866i	$-0.248544\pi$
$503$	31.8622	1.42067	0.710333	+	0.703866i	$0.248544\pi$
$504$	0	0
$505$	−1.02129	−0.0454468
$506$	0	0
$507$	−104.823	−4.65537
$508$	0	0
$509$	3.12330	0.138438	0.0692189	−	0.997601i	$-0.477949\pi$
$509$	3.12330	0.138438	0.0692189	+	0.997601i	$0.477949\pi$
$510$	0	0
$511$	1.84414	0.0815801
$512$	0	0
$513$	0	0
$514$	0	0
$515$	−4.54404	−0.200234
$516$	0	0
$517$	−40.1987	−1.76794
$518$	0	0
$519$	35.9285	1.57709
$520$	0	0
$521$	−17.9475	−0.786294	−0.393147	−	0.919476i	$-0.628614\pi$
$521$	−17.9475	−0.786294	−0.393147	+	0.919476i	$0.628614\pi$
$522$	0	0
$523$	−27.9449	−1.22195	−0.610973	−	0.791651i	$-0.709222\pi$
$523$	−27.9449	−1.22195	−0.610973	+	0.791651i	$0.709222\pi$
$524$	0	0
$525$	4.94205	0.215689
$526$	0	0
$527$	−22.9116	−0.998044
$528$	0	0
$529$	−21.2598	−0.924338
$530$	0	0
$531$	51.9390	2.25396
$532$	0	0
$533$	53.9608	2.33730
$534$	0	0
$535$	1.03968	0.0449494
$536$	0	0
$537$	27.0348	1.16664
$538$	0	0
$539$	−34.7691	−1.49761
$540$	0	0
$541$	1.97916	0.0850909	0.0425454	−	0.999095i	$-0.486453\pi$
$541$	1.97916	0.0850909	0.0425454	+	0.999095i	$0.486453\pi$
$542$	0	0
$543$	53.1704	2.28176
$544$	0	0
$545$	−1.31532	−0.0563423
$546$	0	0
$547$	40.2317	1.72018	0.860092	−	0.510138i	$-0.170406\pi$
$547$	40.2317	1.72018	0.860092	+	0.510138i	$0.170406\pi$
$548$	0	0
$549$	−3.58882	−0.153167
$550$	0	0
$551$	0	0
$552$	0	0
$553$	3.18078	0.135260
$554$	0	0
$555$	−5.99787	−0.254595
$556$	0	0
$557$	−11.6711	−0.494519	−0.247259	−	0.968949i	$-0.579530\pi$
$557$	−11.6711	−0.494519	−0.247259	+	0.968949i	$0.579530\pi$
$558$	0	0
$559$	−50.6998	−2.14437
$560$	0	0
$561$	47.7988	2.01807
$562$	0	0
$563$	41.3725	1.74364	0.871822	−	0.489824i	$-0.162939\pi$
$563$	41.3725	1.74364	0.871822	+	0.489824i	$0.162939\pi$
$564$	0	0
$565$	−5.35481	−0.225279
$566$	0	0
$567$	5.20167	0.218450
$568$	0	0
$569$	33.6994	1.41275	0.706376	−	0.707837i	$-0.250329\pi$
$569$	33.6994	1.41275	0.706376	+	0.707837i	$0.250329\pi$
$570$	0	0
$571$	−19.4520	−0.814043	−0.407021	−	0.913419i	$-0.633433\pi$
$571$	−19.4520	−0.814043	−0.407021	+	0.913419i	$0.633433\pi$
$572$	0	0
$573$	9.34845	0.390537
$574$	0	0
$575$	6.42989	0.268145
$576$	0	0
$577$	42.8060	1.78204	0.891018	−	0.453968i	$-0.149992\pi$
$577$	42.8060	1.78204	0.891018	+	0.453968i	$0.149992\pi$
$578$	0	0
$579$	−10.4982	−0.436291
$580$	0	0
$581$	−2.25573	−0.0935835
$582$	0	0
$583$	45.9638	1.90363
$584$	0	0
$585$	−16.2766	−0.672956
$586$	0	0
$587$	19.0860	0.787763	0.393881	−	0.919161i	$-0.371132\pi$
$587$	19.0860	0.787763	0.393881	+	0.919161i	$0.371132\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	55.6458	2.28896
$592$	0	0
$593$	3.20809	0.131740	0.0658702	−	0.997828i	$-0.479018\pi$
$593$	3.20809	0.131740	0.0658702	+	0.997828i	$0.479018\pi$
$594$	0	0
$595$	−0.350746	−0.0143792
$596$	0	0
$597$	45.9524	1.88070
$598$	0	0
$599$	20.0940	0.821019	0.410509	−	0.911856i	$-0.365351\pi$
$599$	20.0940	0.821019	0.410509	+	0.911856i	$0.365351\pi$
$600$	0	0
$601$	−19.5571	−0.797752	−0.398876	−	0.917005i	$-0.630600\pi$
$601$	−19.5571	−0.797752	−0.398876	+	0.917005i	$0.630600\pi$
$602$	0	0
$603$	42.1474	1.71638
$604$	0	0
$605$	5.12048	0.208177
$606$	0	0
$607$	−24.3958	−0.990196	−0.495098	−	0.868837i	$-0.664868\pi$
$607$	−24.3958	−0.990196	−0.495098	+	0.868837i	$0.664868\pi$
$608$	0	0
$609$	−2.82516	−0.114481
$610$	0	0
$611$	54.4257	2.20183
$612$	0	0
$613$	−18.5275	−0.748320	−0.374160	−	0.927364i	$-0.622069\pi$
$613$	−18.5275	−0.748320	−0.374160	+	0.927364i	$0.622069\pi$
$614$	0	0
$615$	8.74028	0.352442
$616$	0	0
$617$	−24.0234	−0.967147	−0.483574	−	0.875304i	$-0.660661\pi$
$617$	−24.0234	−0.967147	−0.483574	+	0.875304i	$0.660661\pi$
$618$	0	0
$619$	−1.14666	−0.0460881	−0.0230440	−	0.999734i	$-0.507336\pi$
$619$	−1.14666	−0.0460881	−0.0230440	+	0.999734i	$0.507336\pi$
$620$	0	0
$621$	15.2976	0.613870
$622$	0	0
$623$	1.11943	0.0448490
$624$	0	0
$625$	23.1281	0.925125
$626$	0	0
$627$	0	0
$628$	0	0
$629$	−16.4868	−0.657372
$630$	0	0
$631$	−13.4953	−0.537239	−0.268619	−	0.963246i	$-0.586567\pi$
$631$	−13.4953	−0.537239	−0.268619	+	0.963246i	$0.586567\pi$
$632$	0	0
$633$	70.3286	2.79531
$634$	0	0
$635$	1.30315	0.0517139
$636$	0	0
$637$	47.0744	1.86516
$638$	0	0
$639$	−83.6201	−3.30796
$640$	0	0
$641$	9.64762	0.381058	0.190529	−	0.981682i	$-0.438980\pi$
$641$	9.64762	0.381058	0.190529	+	0.981682i	$0.438980\pi$
$642$	0	0
$643$	19.4774	0.768113	0.384056	−	0.923310i	$-0.374527\pi$
$643$	19.4774	0.768113	0.384056	+	0.923310i	$0.374527\pi$
$644$	0	0
$645$	−8.21208	−0.323350
$646$	0	0
$647$	−42.7153	−1.67931	−0.839655	−	0.543119i	$-0.817243\pi$
$647$	−42.7153	−1.67931	−0.839655	+	0.543119i	$0.817243\pi$
$648$	0	0
$649$	38.9815	1.53016
$650$	0	0
$651$	−7.64148	−0.299493
$652$	0	0
$653$	39.1477	1.53197	0.765983	−	0.642860i	$-0.222252\pi$
$653$	39.1477	1.53197	0.765983	+	0.642860i	$0.222252\pi$
$654$	0	0
$655$	1.16784	0.0456313
$656$	0	0
$657$	38.1024	1.48652
$658$	0	0
$659$	28.6298	1.11526	0.557630	−	0.830089i	$-0.311711\pi$
$659$	28.6298	1.11526	0.557630	+	0.830089i	$0.311711\pi$
$660$	0	0
$661$	−35.0174	−1.36202	−0.681009	−	0.732275i	$-0.738459\pi$
$661$	−35.0174	−1.36202	−0.681009	+	0.732275i	$0.738459\pi$
$662$	0	0
$663$	−64.7156	−2.51335
$664$	0	0
$665$	0	0
$666$	0	0
$667$	−3.67570	−0.142324
$668$	0	0
$669$	12.4500	0.481346
$670$	0	0
$671$	−2.69350	−0.103981
$672$	0	0
$673$	−15.0310	−0.579402	−0.289701	−	0.957117i	$-0.593556\pi$
$673$	−15.0310	−0.579402	−0.289701	+	0.957117i	$0.593556\pi$
$674$	0	0
$675$	56.5220	2.17553
$676$	0	0
$677$	−6.30285	−0.242238	−0.121119	−	0.992638i	$-0.538648\pi$
$677$	−6.30285	−0.242238	−0.121119	+	0.992638i	$0.538648\pi$
$678$	0	0
$679$	−4.40247	−0.168951
$680$	0	0
$681$	46.9447	1.79892
$682$	0	0
$683$	−1.22228	−0.0467692	−0.0233846	−	0.999727i	$-0.507444\pi$
$683$	−1.22228	−0.0467692	−0.0233846	+	0.999727i	$0.507444\pi$
$684$	0	0
$685$	−3.30323	−0.126210
$686$	0	0
$687$	−6.66689	−0.254358
$688$	0	0
$689$	−62.2311	−2.37082
$690$	0	0
$691$	30.6027	1.16418	0.582091	−	0.813124i	$-0.302235\pi$
$691$	30.6027	1.16418	0.582091	+	0.813124i	$0.302235\pi$
$692$	0	0
$693$	11.0213	0.418666
$694$	0	0
$695$	0.613316	0.0232644
$696$	0	0
$697$	24.0251	0.910015
$698$	0	0
$699$	7.78996	0.294643
$700$	0	0
$701$	−30.3874	−1.14772	−0.573858	−	0.818955i	$-0.694554\pi$
$701$	−30.3874	−1.14772	−0.573858	+	0.818955i	$0.694554\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	8.81558	0.332014
$706$	0	0
$707$	−0.936292	−0.0352129
$708$	0	0
$709$	30.7290	1.15405	0.577026	−	0.816726i	$-0.304213\pi$
$709$	30.7290	1.15405	0.577026	+	0.816726i	$0.304213\pi$
$710$	0	0
$711$	65.7191	2.46466
$712$	0	0
$713$	−9.94199	−0.372331
$714$	0	0
$715$	−12.2160	−0.456853
$716$	0	0
$717$	0.497810	0.0185911
$718$	0	0
$719$	17.2370	0.642833	0.321416	−	0.946938i	$-0.395841\pi$
$719$	17.2370	0.642833	0.321416	+	0.946938i	$0.395841\pi$
$720$	0	0
$721$	−4.16585	−0.155144
$722$	0	0
$723$	−49.3306	−1.83463
$724$	0	0
$725$	−13.5811	−0.504389
$726$	0	0
$727$	43.3899	1.60924	0.804622	−	0.593788i	$-0.202368\pi$
$727$	43.3899	1.60924	0.804622	+	0.593788i	$0.202368\pi$
$728$	0	0
$729$	−0.985014	−0.0364820
$730$	0	0
$731$	−22.5732	−0.834899
$732$	0	0
$733$	−50.4755	−1.86436	−0.932178	−	0.362000i	$-0.882094\pi$
$733$	−50.4755	−1.86436	−0.932178	+	0.362000i	$0.882094\pi$
$734$	0	0
$735$	7.62486	0.281247
$736$	0	0
$737$	31.6327	1.16521
$738$	0	0
$739$	38.5568	1.41833	0.709167	−	0.705040i	$-0.249071\pi$
$739$	38.5568	1.41833	0.709167	+	0.705040i	$0.249071\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	1.57933	0.0579400	0.0289700	−	0.999580i	$-0.490777\pi$
$743$	1.57933	0.0579400	0.0289700	+	0.999580i	$0.490777\pi$
$744$	0	0
$745$	−3.45236	−0.126485
$746$	0	0
$747$	−46.6064	−1.70524
$748$	0	0
$749$	0.953154	0.0348275
$750$	0	0
$751$	−44.7739	−1.63382	−0.816911	−	0.576764i	$-0.804315\pi$
$751$	−44.7739	−1.63382	−0.816911	+	0.576764i	$0.804315\pi$
$752$	0	0
$753$	−24.5346	−0.894090
$754$	0	0
$755$	−4.52249	−0.164590
$756$	0	0
$757$	0.712995	0.0259142	0.0129571	−	0.999916i	$-0.495876\pi$
$757$	0.712995	0.0259142	0.0129571	+	0.999916i	$0.495876\pi$
$758$	0	0
$759$	20.7413	0.752861
$760$	0	0
$761$	−22.3754	−0.811108	−0.405554	−	0.914071i	$-0.632921\pi$
$761$	−22.3754	−0.811108	−0.405554	+	0.914071i	$0.632921\pi$
$762$	0	0
$763$	−1.20585	−0.0436548
$764$	0	0
$765$	−7.24687	−0.262011
$766$	0	0
$767$	−52.7777	−1.90569
$768$	0	0
$769$	41.7941	1.50714	0.753568	−	0.657370i	$-0.228331\pi$
$769$	41.7941	1.50714	0.753568	+	0.657370i	$0.228331\pi$
$770$	0	0
$771$	−43.8026	−1.57751
$772$	0	0
$773$	−26.0025	−0.935245	−0.467622	−	0.883928i	$-0.654889\pi$
$773$	−26.0025	−0.935245	−0.467622	+	0.883928i	$0.654889\pi$
$774$	0	0
$775$	−36.7340	−1.31952
$776$	0	0
$777$	−5.49868	−0.197264
$778$	0	0
$779$	0	0
$780$	0	0
$781$	−62.7590	−2.24569
$782$	0	0
$783$	−32.3112	−1.15471
$784$	0	0
$785$	−1.34330	−0.0479443
$786$	0	0
$787$	−13.1553	−0.468935	−0.234467	−	0.972124i	$-0.575335\pi$
$787$	−13.1553	−0.468935	−0.234467	+	0.972124i	$0.575335\pi$
$788$	0	0
$789$	−74.8309	−2.66405
$790$	0	0
$791$	−4.90915	−0.174549
$792$	0	0
$793$	3.64677	0.129501
$794$	0	0
$795$	−10.0799	−0.357496
$796$	0	0
$797$	−7.59147	−0.268904	−0.134452	−	0.990920i	$-0.542927\pi$
$797$	−7.59147	−0.268904	−0.134452	+	0.990920i	$0.542927\pi$
$798$	0	0
$799$	24.2321	0.857269
$800$	0	0
$801$	23.1289	0.817220
$802$	0	0
$803$	28.5968	1.00916
$804$	0	0
$805$	−0.152199	−0.00536430
$806$	0	0
$807$	−74.5193	−2.62320
$808$	0	0
$809$	−32.5723	−1.14518	−0.572591	−	0.819841i	$-0.694062\pi$
$809$	−32.5723	−1.14518	−0.572591	+	0.819841i	$0.694062\pi$
$810$	0	0
$811$	−23.5497	−0.826941	−0.413471	−	0.910517i	$-0.635684\pi$
$811$	−23.5497	−0.826941	−0.413471	+	0.910517i	$0.635684\pi$
$812$	0	0
$813$	73.2288	2.56825
$814$	0	0
$815$	−3.39722	−0.118999
$816$	0	0
$817$	0	0
$818$	0	0
$819$	−14.9220	−0.521416
$820$	0	0
$821$	28.8821	1.00799	0.503996	−	0.863706i	$-0.331863\pi$
$821$	28.8821	1.00799	0.503996	+	0.863706i	$0.331863\pi$
$822$	0	0
$823$	28.2949	0.986297	0.493148	−	0.869945i	$-0.335846\pi$
$823$	28.2949	0.986297	0.493148	+	0.869945i	$0.335846\pi$
$824$	0	0
$825$	76.6356	2.66811
$826$	0	0
$827$	4.87625	0.169564	0.0847819	−	0.996400i	$-0.472981\pi$
$827$	4.87625	0.169564	0.0847819	+	0.996400i	$0.472981\pi$
$828$	0	0
$829$	34.5852	1.20119	0.600597	−	0.799552i	$-0.294930\pi$
$829$	34.5852	1.20119	0.600597	+	0.799552i	$0.294930\pi$
$830$	0	0
$831$	64.6143	2.24145
$832$	0	0
$833$	20.9590	0.726188
$834$	0	0
$835$	2.41145	0.0834518
$836$	0	0
$837$	−87.3952	−3.02082
$838$	0	0
$839$	9.64397	0.332947	0.166473	−	0.986046i	$-0.446762\pi$
$839$	9.64397	0.332947	0.166473	+	0.986046i	$0.446762\pi$
$840$	0	0
$841$	−21.2363	−0.732285
$842$	0	0
$843$	−6.22192	−0.214294
$844$	0	0
$845$	11.9277	0.410326
$846$	0	0
$847$	4.69432	0.161299
$848$	0	0
$849$	−12.1927	−0.418454
$850$	0	0
$851$	−7.15410	−0.245239
$852$	0	0
$853$	45.2335	1.54877	0.774383	−	0.632718i	$-0.218061\pi$
$853$	45.2335	1.54877	0.774383	+	0.632718i	$0.218061\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	18.2685	0.624040	0.312020	−	0.950075i	$-0.398994\pi$
$857$	18.2685	0.624040	0.312020	+	0.950075i	$0.398994\pi$
$858$	0	0
$859$	7.81971	0.266805	0.133402	−	0.991062i	$-0.457410\pi$
$859$	7.81971	0.266805	0.133402	+	0.991062i	$0.457410\pi$
$860$	0	0
$861$	8.01285	0.273077
$862$	0	0
$863$	45.0088	1.53212	0.766058	−	0.642771i	$-0.222215\pi$
$863$	45.0088	1.53212	0.766058	+	0.642771i	$0.222215\pi$
$864$	0	0
$865$	−4.08825	−0.139005
$866$	0	0
$867$	24.1862	0.821405
$868$	0	0
$869$	49.3238	1.67320
$870$	0	0
$871$	−42.8280	−1.45117
$872$	0	0
$873$	−90.9608	−3.07856
$874$	0	0
$875$	−1.13922	−0.0385126
$876$	0	0
$877$	8.35233	0.282038	0.141019	−	0.990007i	$-0.454962\pi$
$877$	8.35233	0.282038	0.141019	+	0.990007i	$0.454962\pi$
$878$	0	0
$879$	−105.468	−3.55735
$880$	0	0
$881$	−33.1992	−1.11851	−0.559255	−	0.828996i	$-0.688913\pi$
$881$	−33.1992	−1.11851	−0.559255	+	0.828996i	$0.688913\pi$
$882$	0	0
$883$	−39.5892	−1.33228	−0.666141	−	0.745826i	$-0.732055\pi$
$883$	−39.5892	−1.33228	−0.666141	+	0.745826i	$0.732055\pi$
$884$	0	0
$885$	−8.54865	−0.287360
$886$	0	0
$887$	36.5524	1.22731	0.613654	−	0.789575i	$-0.289699\pi$
$887$	36.5524	1.22731	0.613654	+	0.789575i	$0.289699\pi$
$888$	0	0
$889$	1.19469	0.0400687
$890$	0	0
$891$	80.6615	2.70226
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−3.07625	−0.102828
$896$	0	0
$897$	−28.0820	−0.937629
$898$	0	0
$899$	20.9993	0.700366
$900$	0	0
$901$	−27.7073	−0.923063
$902$	0	0
$903$	−7.52861	−0.250537
$904$	0	0
$905$	−6.05018	−0.201115
$906$	0	0
$907$	38.8650	1.29049	0.645245	−	0.763976i	$-0.276755\pi$
$907$	38.8650	1.29049	0.645245	+	0.763976i	$0.276755\pi$
$908$	0	0
$909$	−19.3450	−0.641634
$910$	0	0
$911$	−53.1535	−1.76105	−0.880527	−	0.473996i	$-0.842811\pi$
$911$	−53.1535	−1.76105	−0.880527	+	0.473996i	$0.842811\pi$
$912$	0	0
$913$	−34.9793	−1.15765
$914$	0	0
$915$	0.590685	0.0195274
$916$	0	0
$917$	1.07065	0.0353558
$918$	0	0
$919$	31.1423	1.02729	0.513644	−	0.858003i	$-0.328295\pi$
$919$	31.1423	1.02729	0.513644	+	0.858003i	$0.328295\pi$
$920$	0	0
$921$	15.4280	0.508371
$922$	0	0
$923$	84.9704	2.79684
$924$	0	0
$925$	−26.4332	−0.869118
$926$	0	0
$927$	−86.0720	−2.82698
$928$	0	0
$929$	−55.0491	−1.80610	−0.903052	−	0.429531i	$-0.858679\pi$
$929$	−55.0491	−1.80610	−0.903052	+	0.429531i	$0.858679\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	−38.9784	−1.27609
$934$	0	0
$935$	−5.43896	−0.177873
$936$	0	0
$937$	−47.2642	−1.54405	−0.772027	−	0.635590i	$-0.780757\pi$
$937$	−47.2642	−1.54405	−0.772027	+	0.635590i	$0.780757\pi$
$938$	0	0
$939$	−9.40357	−0.306874
$940$	0	0
$941$	−28.9401	−0.943419	−0.471709	−	0.881754i	$-0.656363\pi$
$941$	−28.9401	−0.943419	−0.471709	+	0.881754i	$0.656363\pi$
$942$	0	0
$943$	10.4252	0.339490
$944$	0	0
$945$	−1.33790	−0.0435220
$946$	0	0
$947$	−22.2527	−0.723116	−0.361558	−	0.932350i	$-0.617755\pi$
$947$	−22.2527	−0.723116	−0.361558	+	0.932350i	$0.617755\pi$
$948$	0	0
$949$	−38.7177	−1.25683
$950$	0	0
$951$	79.2339	2.56933
$952$	0	0
$953$	−3.76229	−0.121872	−0.0609362	−	0.998142i	$-0.519409\pi$
$953$	−3.76229	−0.121872	−0.0609362	+	0.998142i	$0.519409\pi$
$954$	0	0
$955$	−1.06375	−0.0344221
$956$	0	0
$957$	−43.8094	−1.41616
$958$	0	0
$959$	−3.02832	−0.0977894
$960$	0	0
$961$	25.7987	0.832216
$962$	0	0
$963$	19.6934	0.634611
$964$	0	0
$965$	1.19458	0.0384548
$966$	0	0
$967$	36.6124	1.17738	0.588688	−	0.808360i	$-0.299644\pi$
$967$	36.6124	1.17738	0.588688	+	0.808360i	$0.299644\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	49.2648	1.58098	0.790491	−	0.612473i	$-0.209825\pi$
$971$	49.2648	1.58098	0.790491	+	0.612473i	$0.209825\pi$
$972$	0	0
$973$	0.562271	0.0180256
$974$	0	0
$975$	−103.758	−3.32292
$976$	0	0
$977$	26.6419	0.852348	0.426174	−	0.904641i	$-0.359861\pi$
$977$	26.6419	0.852348	0.426174	+	0.904641i	$0.359861\pi$
$978$	0	0
$979$	17.3588	0.554791
$980$	0	0
$981$	−24.9145	−0.795460
$982$	0	0
$983$	−5.12382	−0.163424	−0.0817122	−	0.996656i	$-0.526039\pi$
$983$	−5.12382	−0.163424	−0.0817122	+	0.996656i	$0.526039\pi$
$984$	0	0
$985$	−6.33185	−0.201750
$986$	0	0
$987$	8.08189	0.257249
$988$	0	0
$989$	−9.79515	−0.311468
$990$	0	0
$991$	−2.03495	−0.0646424	−0.0323212	−	0.999478i	$-0.510290\pi$
$991$	−2.03495	−0.0646424	−0.0323212	+	0.999478i	$0.510290\pi$
$992$	0	0
$993$	22.6760	0.719602
$994$	0	0
$995$	−5.22885	−0.165766
$996$	0	0
$997$	13.8388	0.438278	0.219139	−	0.975694i	$-0.429675\pi$
$997$	13.8388	0.438278	0.219139	+	0.975694i	$0.429675\pi$
$998$	0	0
$999$	−62.8882	−1.98969

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.v.1.1	✓	8
4.3	odd	2		5776.2.a.cc.1.8		8
19.18	odd	2		2888.2.a.w.1.8	yes	8
76.75	even	2		5776.2.a.ca.1.1		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2888.2.a.v.1.1	✓	8	1.1	even	1	trivial
2888.2.a.w.1.8	yes	8	19.18	odd	2
5776.2.a.ca.1.1		8	76.75	even	2
5776.2.a.cc.1.8		8	4.3	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

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

\(f(q)\)	\(=\)	\(q-3.11762 q^{3} +0.354750 q^{5} +0.325225 q^{7} +6.71958 q^{9} +5.04322 q^{11} -6.82809 q^{13} -1.10598 q^{15} -3.04008 q^{17} -1.01393 q^{21} -1.31918 q^{23} -4.87415 q^{25} -11.5963 q^{27} +2.78635 q^{29} +7.53649 q^{31} -15.7229 q^{33} +0.115374 q^{35} +5.42314 q^{37} +21.2874 q^{39} -7.90277 q^{41} +7.42518 q^{43} +2.38377 q^{45} -7.97085 q^{47} -6.89423 q^{49} +9.47784 q^{51} +9.11398 q^{53} +1.78908 q^{55} +7.72949 q^{59} -0.534084 q^{61} +2.18538 q^{63} -2.42227 q^{65} +6.27233 q^{67} +4.11271 q^{69} -12.4442 q^{71} +5.67035 q^{73} +15.1958 q^{75} +1.64018 q^{77} +9.78023 q^{79} +15.9941 q^{81} -6.93591 q^{83} -1.07847 q^{85} -8.68679 q^{87} +3.44202 q^{89} -2.22067 q^{91} -23.4960 q^{93} -13.5367 q^{97} +33.8883 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 6 q^{3} - 2 q^{5} - 2 q^{7} + 2 q^{9} + 12 q^{11} - 6 q^{13} - 10 q^{17} + 4 q^{21} - 12 q^{23} + 2 q^{25} - 12 q^{27} - 18 q^{29} - 14 q^{31} - 40 q^{33} + 18 q^{35} - 16 q^{37} + 28 q^{39} - 12 q^{41}+ \cdots + 32 q^{99}+O(q^{100}) \) 8 * q - 6 * q^3 - 2 * q^5 - 2 * q^7 + 2 * q^9 + 12 * q^11 - 6 * q^13 - 10 * q^17 + 4 * q^21 - 12 * q^23 + 2 * q^25 - 12 * q^27 - 18 * q^29 - 14 * q^31 - 40 * q^33 + 18 * q^35 - 16 * q^37 + 28 * q^39 - 12 * q^41 + 8 * q^43 - 2 * q^45 + 24 * q^47 + 12 * q^49 + 22 * q^51 - 12 * q^53 - 2 * q^55 + 16 * q^59 + 12 * q^63 - 12 * q^65 - 42 * q^67 + 12 * q^69 - 36 * q^71 - 10 * q^73 - 32 * q^75 - 38 * q^77 - 6 * q^79 + 4 * q^81 + 12 * q^83 - 42 * q^85 - 12 * q^87 - 8 * q^89 - 16 * q^91 - 14 * q^93 - 28 * q^97 + 32 * q^99

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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