LMFDB - Embedded newform 2888.2.a.r.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 = [3,0,1,0,1,0,-2,0,6,0,4,0,-1,0,5] 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:	$3$
Coefficient field:	3.3.316.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 2 $ x^3 - x^2 - 4*x + 2
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 152)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-1.81361$ 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)

$f(q)$	$=$	$q-2.10278 q^{3} -1.81361 q^{5} -2.52444 q^{7} +1.42166 q^{9} -0.813607 q^{11} -4.33804 q^{13} +3.81361 q^{15} -0.916382 q^{17} +5.30833 q^{21} -1.81361 q^{23} -1.71083 q^{25} +3.31889 q^{27} -8.01916 q^{29} -9.30833 q^{31} +1.71083 q^{33} +4.57834 q^{35} -11.3083 q^{37} +9.12193 q^{39} -11.2056 q^{41} +9.96526 q^{43} -2.57834 q^{45} -5.64637 q^{47} -0.627213 q^{49} +1.92694 q^{51} +2.91638 q^{53} +1.47556 q^{55} +1.89722 q^{59} +6.59749 q^{61} -3.58890 q^{63} +7.86751 q^{65} +3.15165 q^{67} +3.81361 q^{69} -0.710831 q^{71} +0.627213 q^{73} +3.59749 q^{75} +2.05390 q^{77} -10.3380 q^{79} -11.2439 q^{81} +9.86248 q^{83} +1.66196 q^{85} +16.8625 q^{87} +13.5925 q^{89} +10.9511 q^{91} +19.5733 q^{93} +4.62721 q^{97} -1.15667 q^{99} +O(q^{100})$
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{3} + q^{5} - 2 q^{7} + 6 q^{9} + 4 q^{11} - q^{13} + 5 q^{15} + 11 q^{17} - 6 q^{21} + q^{23} - 6 q^{25} + 19 q^{27} - 3 q^{29} - 6 q^{31} + 6 q^{33} + 12 q^{35} - 12 q^{37} - q^{39} - 19 q^{41}+ \cdots + q^{97}+O(q^{100}) $ 3 * q + q^3 + q^5 - 2 * q^7 + 6 * q^9 + 4 * q^11 - q^13 + 5 * q^15 + 11 * q^17 - 6 * q^21 + q^23 - 6 * q^25 + 19 * q^27 - 3 * q^29 - 6 * q^31 + 6 * q^33 + 12 * q^35 - 12 * q^37 - q^39 - 19 * q^41 + 5 * q^43 - 6 * q^45 + 17 * q^47 + 11 * q^49 + 23 * q^51 - 5 * q^53 + 10 * q^55 + 13 * q^59 - 3 * q^61 - 40 * q^63 + 21 * q^65 - 9 * q^67 + 5 * q^69 - 3 * q^71 - 11 * q^73 - 12 * q^75 + 10 * q^77 - 19 * q^79 + 23 * q^81 + 12 * q^83 + 17 * q^85 + 33 * q^87 + 3 * q^89 + 44 * q^91 + 42 * q^93 + q^97

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
$3$	−2.10278	−1.21404	−0.607019	−	0.794687i	$-0.707635\pi$
$3$	−2.10278	−1.21404	−0.607019	+	0.794687i	$0.707635\pi$
$4$	0	0
$5$	−1.81361	−0.811069	−0.405535	−	0.914080i	$-0.632915\pi$
$5$	−1.81361	−0.811069	−0.405535	+	0.914080i	$0.632915\pi$
$6$	0	0
$7$	−2.52444	−0.954148	−0.477074	−	0.878863i	$-0.658303\pi$
$7$	−2.52444	−0.954148	−0.477074	+	0.878863i	$0.658303\pi$
$8$	0	0
$9$	1.42166	0.473888
$10$	0	0
$11$	−0.813607	−0.245312	−0.122656	−	0.992449i	$-0.539141\pi$
$11$	−0.813607	−0.245312	−0.122656	+	0.992449i	$0.539141\pi$
$12$	0	0
$13$	−4.33804	−1.20316	−0.601579	−	0.798814i	$-0.705461\pi$
$13$	−4.33804	−1.20316	−0.601579	+	0.798814i	$0.705461\pi$
$14$	0	0
$15$	3.81361	0.984669
$16$	0	0
$17$	−0.916382	−0.222255	−0.111128	−	0.993806i	$-0.535446\pi$
$17$	−0.916382	−0.222255	−0.111128	+	0.993806i	$0.535446\pi$
$18$	0	0
$19$	0	0
$19$	0	0
$20$	0	0
$21$	5.30833	1.15837
$22$	0	0
$23$	−1.81361	−0.378163	−0.189082	−	0.981961i	$-0.560551\pi$
$23$	−1.81361	−0.378163	−0.189082	+	0.981961i	$0.560551\pi$
$24$	0	0
$25$	−1.71083	−0.342166
$26$	0	0
$27$	3.31889	0.638720
$28$	0	0
$29$	−8.01916	−1.48912	−0.744560	−	0.667556i	$-0.767341\pi$
$29$	−8.01916	−1.48912	−0.744560	+	0.667556i	$0.767341\pi$
$30$	0	0
$31$	−9.30833	−1.67182	−0.835912	−	0.548863i	$-0.815061\pi$
$31$	−9.30833	−1.67182	−0.835912	+	0.548863i	$0.815061\pi$
$32$	0	0
$33$	1.71083	0.297818
$34$	0	0
$35$	4.57834	0.773880
$36$	0	0
$37$	−11.3083	−1.85908	−0.929539	−	0.368725i	$-0.879794\pi$
$37$	−11.3083	−1.85908	−0.929539	+	0.368725i	$0.879794\pi$
$38$	0	0
$39$	9.12193	1.46068
$40$	0	0
$41$	−11.2056	−1.75001	−0.875006	−	0.484111i	$-0.839143\pi$
$41$	−11.2056	−1.75001	−0.875006	+	0.484111i	$0.839143\pi$
$42$	0	0
$43$	9.96526	1.51969	0.759844	−	0.650106i	$-0.225275\pi$
$43$	9.96526	1.51969	0.759844	+	0.650106i	$0.225275\pi$
$44$	0	0
$45$	−2.57834	−0.384356
$46$	0	0
$47$	−5.64637	−0.823608	−0.411804	−	0.911273i	$-0.635101\pi$
$47$	−5.64637	−0.823608	−0.411804	+	0.911273i	$0.635101\pi$
$48$	0	0
$49$	−0.627213	−0.0896019
$50$	0	0
$51$	1.92694	0.269826
$52$	0	0
$53$	2.91638	0.400596	0.200298	−	0.979735i	$-0.435809\pi$
$53$	2.91638	0.400596	0.200298	+	0.979735i	$0.435809\pi$
$54$	0	0
$55$	1.47556	0.198965
$56$	0	0
$57$	0	0
$58$	0	0
$59$	1.89722	0.246998	0.123499	−	0.992345i	$-0.460588\pi$
$59$	1.89722	0.246998	0.123499	+	0.992345i	$0.460588\pi$
$60$	0	0
$61$	6.59749	0.844723	0.422361	−	0.906428i	$-0.361201\pi$
$61$	6.59749	0.844723	0.422361	+	0.906428i	$0.361201\pi$
$62$	0	0
$63$	−3.58890	−0.452159
$64$	0	0
$65$	7.86751	0.975844
$66$	0	0
$67$	3.15165	0.385035	0.192518	−	0.981293i	$-0.438335\pi$
$67$	3.15165	0.385035	0.192518	+	0.981293i	$0.438335\pi$
$68$	0	0
$69$	3.81361	0.459104
$70$	0	0
$71$	−0.710831	−0.0843602	−0.0421801	−	0.999110i	$-0.513430\pi$
$71$	−0.710831	−0.0843602	−0.0421801	+	0.999110i	$0.513430\pi$
$72$	0	0
$73$	0.627213	0.0734097	0.0367049	−	0.999326i	$-0.488314\pi$
$73$	0.627213	0.0734097	0.0367049	+	0.999326i	$0.488314\pi$
$74$	0	0
$75$	3.59749	0.415403
$76$	0	0
$77$	2.05390	0.234064
$78$	0	0
$79$	−10.3380	−1.16312	−0.581560	−	0.813503i	$-0.697558\pi$
$79$	−10.3380	−1.16312	−0.581560	+	0.813503i	$0.697558\pi$
$80$	0	0
$81$	−11.2439	−1.24932
$82$	0	0
$83$	9.86248	1.08255	0.541274	−	0.840846i	$-0.317942\pi$
$83$	9.86248	1.08255	0.541274	+	0.840846i	$0.317942\pi$
$84$	0	0
$85$	1.66196	0.180264
$86$	0	0
$87$	16.8625	1.80785
$88$	0	0
$89$	13.5925	1.44080	0.720399	−	0.693559i	$-0.243958\pi$
$89$	13.5925	1.44080	0.720399	+	0.693559i	$0.243958\pi$
$90$	0	0
$91$	10.9511	1.14799
$92$	0	0
$93$	19.5733	2.02966
$94$	0	0
$95$	0	0
$96$	0	0
$97$	4.62721	0.469822	0.234911	−	0.972017i	$-0.424520\pi$
$97$	4.62721	0.469822	0.234911	+	0.972017i	$0.424520\pi$
$98$	0	0
$99$	−1.15667	−0.116250
$100$	0	0
$101$	−2.97028	−0.295554	−0.147777	−	0.989021i	$-0.547212\pi$
$101$	−2.97028	−0.295554	−0.147777	+	0.989021i	$0.547212\pi$
$102$	0	0
$103$	−0.470539	−0.0463636	−0.0231818	−	0.999731i	$-0.507380\pi$
$103$	−0.470539	−0.0463636	−0.0231818	+	0.999731i	$0.507380\pi$
$104$	0	0
$105$	−9.62721	−0.939520
$106$	0	0
$107$	−10.2056	−0.986608	−0.493304	−	0.869857i	$-0.664211\pi$
$107$	−10.2056	−0.986608	−0.493304	+	0.869857i	$0.664211\pi$
$108$	0	0
$109$	−3.75971	−0.360115	−0.180057	−	0.983656i	$-0.557628\pi$
$109$	−3.75971	−0.360115	−0.180057	+	0.983656i	$0.557628\pi$
$110$	0	0
$111$	23.7789	2.25699
$112$	0	0
$113$	−15.0731	−1.41795	−0.708977	−	0.705232i	$-0.750843\pi$
$113$	−15.0731	−1.41795	−0.708977	+	0.705232i	$0.750843\pi$
$114$	0	0
$115$	3.28917	0.306717
$116$	0	0
$117$	−6.16724	−0.570161
$118$	0	0
$119$	2.31335	0.212064
$120$	0	0
$121$	−10.3380	−0.939822
$122$	0	0
$123$	23.5628	2.12458
$124$	0	0
$125$	12.1708	1.08859
$126$	0	0
$127$	3.02418	0.268353	0.134176	−	0.990957i	$-0.457161\pi$
$127$	3.02418	0.268353	0.134176	+	0.990957i	$0.457161\pi$
$128$	0	0
$129$	−20.9547	−1.84496
$130$	0	0
$131$	15.3572	1.34176	0.670882	−	0.741564i	$-0.265916\pi$
$131$	15.3572	1.34176	0.670882	+	0.741564i	$0.265916\pi$
$132$	0	0
$133$	0	0
$134$	0	0
$135$	−6.01916	−0.518047
$136$	0	0
$137$	−1.57834	−0.134846	−0.0674232	−	0.997724i	$-0.521478\pi$
$137$	−1.57834	−0.134846	−0.0674232	+	0.997724i	$0.521478\pi$
$138$	0	0
$139$	−15.5244	−1.31677	−0.658383	−	0.752683i	$-0.728759\pi$
$139$	−15.5244	−1.31677	−0.658383	+	0.752683i	$0.728759\pi$
$140$	0	0
$141$	11.8730	0.999891
$142$	0	0
$143$	3.52946	0.295148
$144$	0	0
$145$	14.5436	1.20778
$146$	0	0
$147$	1.31889	0.108780
$148$	0	0
$149$	−18.0575	−1.47933	−0.739663	−	0.672978i	$-0.765015\pi$
$149$	−18.0575	−1.47933	−0.739663	+	0.672978i	$0.765015\pi$
$150$	0	0
$151$	7.10278	0.578016	0.289008	−	0.957327i	$-0.406675\pi$
$151$	7.10278	0.578016	0.289008	+	0.957327i	$0.406675\pi$
$152$	0	0
$153$	−1.30279	−0.105324
$154$	0	0
$155$	16.8816	1.35597
$156$	0	0
$157$	22.7491	1.81558	0.907790	−	0.419426i	$-0.137769\pi$
$157$	22.7491	1.81558	0.907790	+	0.419426i	$0.137769\pi$
$158$	0	0
$159$	−6.13249	−0.486338
$160$	0	0
$161$	4.57834	0.360824
$162$	0	0
$163$	3.76473	0.294876	0.147438	−	0.989071i	$-0.452897\pi$
$163$	3.76473	0.294876	0.147438	+	0.989071i	$0.452897\pi$
$164$	0	0
$165$	−3.10278	−0.241551
$166$	0	0
$167$	7.49472	0.579959	0.289979	−	0.957033i	$-0.406352\pi$
$167$	7.49472	0.579959	0.289979	+	0.957033i	$0.406352\pi$
$168$	0	0
$169$	5.81863	0.447587
$170$	0	0
$171$	0	0
$172$	0	0
$173$	−12.3380	−0.938044	−0.469022	−	0.883186i	$-0.655394\pi$
$173$	−12.3380	−0.938044	−0.469022	+	0.883186i	$0.655394\pi$
$174$	0	0
$175$	4.31889	0.326477
$176$	0	0
$177$	−3.98944	−0.299864
$178$	0	0
$179$	24.6464	1.84216	0.921078	−	0.389378i	$-0.127310\pi$
$179$	24.6464	1.84216	0.921078	+	0.389378i	$0.127310\pi$
$180$	0	0
$181$	−0.931967	−0.0692726	−0.0346363	−	0.999400i	$-0.511027\pi$
$181$	−0.931967	−0.0692726	−0.0346363	+	0.999400i	$0.511027\pi$
$182$	0	0
$183$	−13.8730	−1.02553
$184$	0	0
$185$	20.5089	1.50784
$186$	0	0
$187$	0.745574	0.0545218
$188$	0	0
$189$	−8.37833	−0.609434
$190$	0	0
$191$	−12.4111	−0.898036	−0.449018	−	0.893523i	$-0.648226\pi$
$191$	−12.4111	−0.898036	−0.449018	+	0.893523i	$0.648226\pi$
$192$	0	0
$193$	−1.66196	−0.119630	−0.0598151	−	0.998209i	$-0.519051\pi$
$193$	−1.66196	−0.119630	−0.0598151	+	0.998209i	$0.519051\pi$
$194$	0	0
$195$	−16.5436	−1.18471
$196$	0	0
$197$	5.51388	0.392847	0.196424	−	0.980519i	$-0.437067\pi$
$197$	5.51388	0.392847	0.196424	+	0.980519i	$0.437067\pi$
$198$	0	0
$199$	15.9653	1.13175	0.565874	−	0.824492i	$-0.308539\pi$
$199$	15.9653	1.13175	0.565874	+	0.824492i	$0.308539\pi$
$200$	0	0
$201$	−6.62721	−0.467448
$202$	0	0
$203$	20.2439	1.42084
$204$	0	0
$205$	20.3225	1.41938
$206$	0	0
$207$	−2.57834	−0.179207
$208$	0	0
$209$	0	0
$210$	0	0
$211$	−8.37636	−0.576652	−0.288326	−	0.957532i	$-0.593099\pi$
$211$	−8.37636	−0.576652	−0.288326	+	0.957532i	$0.593099\pi$
$212$	0	0
$213$	1.49472	0.102416
$214$	0	0
$215$	−18.0731	−1.23257
$216$	0	0
$217$	23.4983	1.59517
$218$	0	0
$219$	−1.31889	−0.0891222
$220$	0	0
$221$	3.97530	0.267408
$222$	0	0
$223$	−3.98084	−0.266577	−0.133288	−	0.991077i	$-0.542554\pi$
$223$	−3.98084	−0.266577	−0.133288	+	0.991077i	$0.542554\pi$
$224$	0	0
$225$	−2.43223	−0.162148
$226$	0	0
$227$	12.0680	0.800983	0.400492	−	0.916300i	$-0.368839\pi$
$227$	12.0680	0.800983	0.400492	+	0.916300i	$0.368839\pi$
$228$	0	0
$229$	18.9894	1.25486	0.627429	−	0.778674i	$-0.284107\pi$
$229$	18.9894	1.25486	0.627429	+	0.778674i	$0.284107\pi$
$230$	0	0
$231$	−4.31889	−0.284162
$232$	0	0
$233$	19.1361	1.25365	0.626823	−	0.779162i	$-0.284355\pi$
$233$	19.1361	1.25365	0.626823	+	0.779162i	$0.284355\pi$
$234$	0	0
$235$	10.2403	0.668003
$236$	0	0
$237$	21.7386	1.41207
$238$	0	0
$239$	−2.50885	−0.162284	−0.0811421	−	0.996703i	$-0.525857\pi$
$239$	−2.50885	−0.162284	−0.0811421	+	0.996703i	$0.525857\pi$
$240$	0	0
$241$	−3.84333	−0.247570	−0.123785	−	0.992309i	$-0.539503\pi$
$241$	−3.84333	−0.247570	−0.123785	+	0.992309i	$0.539503\pi$
$242$	0	0
$243$	13.6867	0.877999
$244$	0	0
$245$	1.13752	0.0726733
$246$	0	0
$247$	0	0
$248$	0	0
$249$	−20.7386	−1.31425
$250$	0	0
$251$	−20.8766	−1.31772	−0.658860	−	0.752265i	$-0.728961\pi$
$251$	−20.8766	−1.31772	−0.658860	+	0.752265i	$0.728961\pi$
$252$	0	0
$253$	1.47556	0.0927678
$254$	0	0
$255$	−3.49472	−0.218848
$256$	0	0
$257$	28.7038	1.79050	0.895248	−	0.445568i	$-0.146998\pi$
$257$	28.7038	1.79050	0.895248	+	0.445568i	$0.146998\pi$
$258$	0	0
$259$	28.5472	1.77383
$260$	0	0
$261$	−11.4005	−0.705676
$262$	0	0
$263$	−5.27358	−0.325183	−0.162592	−	0.986693i	$-0.551985\pi$
$263$	−5.27358	−0.325183	−0.162592	+	0.986693i	$0.551985\pi$
$264$	0	0
$265$	−5.28917	−0.324911
$266$	0	0
$267$	−28.5819	−1.74918
$268$	0	0
$269$	−21.8575	−1.33267	−0.666336	−	0.745651i	$-0.732139\pi$
$269$	−21.8575	−1.33267	−0.666336	+	0.745651i	$0.732139\pi$
$270$	0	0
$271$	12.1864	0.740271	0.370135	−	0.928978i	$-0.379311\pi$
$271$	12.1864	0.740271	0.370135	+	0.928978i	$0.379311\pi$
$272$	0	0
$273$	−23.0278	−1.39370
$274$	0	0
$275$	1.39194	0.0839374
$276$	0	0
$277$	0.632236	0.0379874	0.0189937	−	0.999820i	$-0.493954\pi$
$277$	0.632236	0.0379874	0.0189937	+	0.999820i	$0.493954\pi$
$278$	0	0
$279$	−13.2333	−0.792257
$280$	0	0
$281$	−33.1361	−1.97673	−0.988366	−	0.152094i	$-0.951398\pi$
$281$	−33.1361	−1.97673	−0.988366	+	0.152094i	$0.951398\pi$
$282$	0	0
$283$	−8.94610	−0.531790	−0.265895	−	0.964002i	$-0.585668\pi$
$283$	−8.94610	−0.531790	−0.265895	+	0.964002i	$0.585668\pi$
$284$	0	0
$285$	0	0
$286$	0	0
$287$	28.2877	1.66977
$288$	0	0
$289$	−16.1602	−0.950603
$290$	0	0
$291$	−9.72999	−0.570382
$292$	0	0
$293$	−28.0822	−1.64058	−0.820289	−	0.571950i	$-0.806187\pi$
$293$	−28.0822	−1.64058	−0.820289	+	0.571950i	$0.806187\pi$
$294$	0	0
$295$	−3.44082	−0.200332
$296$	0	0
$297$	−2.70027	−0.156685
$298$	0	0
$299$	7.86751	0.454990
$300$	0	0
$301$	−25.1567	−1.45001
$302$	0	0
$303$	6.24583	0.358814
$304$	0	0
$305$	−11.9653	−0.685129
$306$	0	0
$307$	31.2978	1.78626	0.893129	−	0.449801i	$-0.148505\pi$
$307$	31.2978	1.78626	0.893129	+	0.449801i	$0.148505\pi$
$308$	0	0
$309$	0.989437	0.0562871
$310$	0	0
$311$	−9.00502	−0.510628	−0.255314	−	0.966858i	$-0.582179\pi$
$311$	−9.00502	−0.510628	−0.255314	+	0.966858i	$0.582179\pi$
$312$	0	0
$313$	−4.36222	−0.246567	−0.123284	−	0.992371i	$-0.539343\pi$
$313$	−4.36222	−0.246567	−0.123284	+	0.992371i	$0.539343\pi$
$314$	0	0
$315$	6.50885	0.366732
$316$	0	0
$317$	−4.33804	−0.243649	−0.121824	−	0.992552i	$-0.538874\pi$
$317$	−4.33804	−0.243649	−0.121824	+	0.992552i	$0.538874\pi$
$318$	0	0
$319$	6.52444	0.365298
$320$	0	0
$321$	21.4600	1.19778
$322$	0	0
$323$	0	0
$324$	0	0
$325$	7.42166	0.411680
$326$	0	0
$327$	7.90582	0.437193
$328$	0	0
$329$	14.2539	0.785843
$330$	0	0
$331$	−10.6081	−0.583072	−0.291536	−	0.956560i	$-0.594166\pi$
$331$	−10.6081	−0.583072	−0.291536	+	0.956560i	$0.594166\pi$
$332$	0	0
$333$	−16.0766	−0.880994
$334$	0	0
$335$	−5.71585	−0.312291
$336$	0	0
$337$	−0.892202	−0.0486013	−0.0243007	−	0.999705i	$-0.507736\pi$
$337$	−0.892202	−0.0486013	−0.0243007	+	0.999705i	$0.507736\pi$
$338$	0	0
$339$	31.6952	1.72145
$340$	0	0
$341$	7.57331	0.410118
$342$	0	0
$343$	19.2544	1.03964
$344$	0	0
$345$	−6.91638	−0.372365
$346$	0	0
$347$	−17.0539	−0.915501	−0.457751	−	0.889081i	$-0.651345\pi$
$347$	−17.0539	−0.915501	−0.457751	+	0.889081i	$0.651345\pi$
$348$	0	0
$349$	17.8328	0.954566	0.477283	−	0.878750i	$-0.341622\pi$
$349$	17.8328	0.954566	0.477283	+	0.878750i	$0.341622\pi$
$350$	0	0
$351$	−14.3975	−0.768481
$352$	0	0
$353$	−22.4353	−1.19411	−0.597055	−	0.802201i	$-0.703662\pi$
$353$	−22.4353	−1.19411	−0.597055	+	0.802201i	$0.703662\pi$
$354$	0	0
$355$	1.28917	0.0684220
$356$	0	0
$357$	−4.86445	−0.257454
$358$	0	0
$359$	−21.4408	−1.13160	−0.565802	−	0.824541i	$-0.691433\pi$
$359$	−21.4408	−1.13160	−0.565802	+	0.824541i	$0.691433\pi$
$360$	0	0
$361$	0	0
$362$	0	0
$363$	21.7386	1.14098
$364$	0	0
$365$	−1.13752	−0.0595404
$366$	0	0
$367$	−17.5869	−0.918030	−0.459015	−	0.888428i	$-0.651798\pi$
$367$	−17.5869	−0.918030	−0.459015	+	0.888428i	$0.651798\pi$
$368$	0	0
$369$	−15.9305	−0.829309
$370$	0	0
$371$	−7.36222	−0.382228
$372$	0	0
$373$	11.5194	0.596453	0.298226	−	0.954495i	$-0.403605\pi$
$373$	11.5194	0.596453	0.298226	+	0.954495i	$0.403605\pi$
$374$	0	0
$375$	−25.5925	−1.32159
$376$	0	0
$377$	34.7875	1.79165
$378$	0	0
$379$	−24.5472	−1.26090	−0.630452	−	0.776229i	$-0.717130\pi$
$379$	−24.5472	−1.26090	−0.630452	+	0.776229i	$0.717130\pi$
$380$	0	0
$381$	−6.35917	−0.325790
$382$	0	0
$383$	−16.6358	−0.850050	−0.425025	−	0.905182i	$-0.639735\pi$
$383$	−16.6358	−0.850050	−0.425025	+	0.905182i	$0.639735\pi$
$384$	0	0
$385$	−3.72496	−0.189842
$386$	0	0
$387$	14.1672	0.720161
$388$	0	0
$389$	−25.3275	−1.28415	−0.642077	−	0.766640i	$-0.721927\pi$
$389$	−25.3275	−1.28415	−0.642077	+	0.766640i	$0.721927\pi$
$390$	0	0
$391$	1.66196	0.0840487
$392$	0	0
$393$	−32.2927	−1.62895
$394$	0	0
$395$	18.7491	0.943372
$396$	0	0
$397$	−13.7542	−0.690302	−0.345151	−	0.938547i	$-0.612172\pi$
$397$	−13.7542	−0.690302	−0.345151	+	0.938547i	$0.612172\pi$
$398$	0	0
$399$	0	0
$400$	0	0
$401$	−24.4217	−1.21956	−0.609780	−	0.792571i	$-0.708742\pi$
$401$	−24.4217	−1.21956	−0.609780	+	0.792571i	$0.708742\pi$
$402$	0	0
$403$	40.3799	2.01147
$404$	0	0
$405$	20.3919	1.01328
$406$	0	0
$407$	9.20053	0.456053
$408$	0	0
$409$	−6.68665	−0.330634	−0.165317	−	0.986241i	$-0.552865\pi$
$409$	−6.68665	−0.330634	−0.165317	+	0.986241i	$0.552865\pi$
$410$	0	0
$411$	3.31889	0.163709
$412$	0	0
$413$	−4.78943	−0.235672
$414$	0	0
$415$	−17.8867	−0.878022
$416$	0	0
$417$	32.6444	1.59860
$418$	0	0
$419$	−7.58890	−0.370742	−0.185371	−	0.982669i	$-0.559349\pi$
$419$	−7.58890	−0.370742	−0.185371	+	0.982669i	$0.559349\pi$
$420$	0	0
$421$	19.7058	0.960402	0.480201	−	0.877158i	$-0.340564\pi$
$421$	19.7058	0.960402	0.480201	+	0.877158i	$0.340564\pi$
$422$	0	0
$423$	−8.02723	−0.390297
$424$	0	0
$425$	1.56777	0.0760482
$426$	0	0
$427$	−16.6550	−0.805990
$428$	0	0
$429$	−7.42166	−0.358321
$430$	0	0
$431$	30.8852	1.48769	0.743844	−	0.668353i	$-0.233000\pi$
$431$	30.8852	1.48769	0.743844	+	0.668353i	$0.233000\pi$
$432$	0	0
$433$	5.81915	0.279650	0.139825	−	0.990176i	$-0.455346\pi$
$433$	5.81915	0.279650	0.139825	+	0.990176i	$0.455346\pi$
$434$	0	0
$435$	−30.5819	−1.46629
$436$	0	0
$437$	0	0
$438$	0	0
$439$	2.97028	0.141764	0.0708819	−	0.997485i	$-0.477419\pi$
$439$	2.97028	0.141764	0.0708819	+	0.997485i	$0.477419\pi$
$440$	0	0
$441$	−0.891685	−0.0424612
$442$	0	0
$443$	26.5733	1.26254	0.631268	−	0.775565i	$-0.282535\pi$
$443$	26.5733	1.26254	0.631268	+	0.775565i	$0.282535\pi$
$444$	0	0
$445$	−24.6514	−1.16859
$446$	0	0
$447$	37.9708	1.79596
$448$	0	0
$449$	−8.53303	−0.402699	−0.201349	−	0.979519i	$-0.564533\pi$
$449$	−8.53303	−0.402699	−0.201349	+	0.979519i	$0.564533\pi$
$450$	0	0
$451$	9.11691	0.429298
$452$	0	0
$453$	−14.9355	−0.701733
$454$	0	0
$455$	−19.8610	−0.931099
$456$	0	0
$457$	20.2892	0.949087	0.474544	−	0.880232i	$-0.342613\pi$
$457$	20.2892	0.949087	0.474544	+	0.880232i	$0.342613\pi$
$458$	0	0
$459$	−3.04137	−0.141959
$460$	0	0
$461$	8.48416	0.395147	0.197573	−	0.980288i	$-0.436694\pi$
$461$	8.48416	0.395147	0.197573	+	0.980288i	$0.436694\pi$
$462$	0	0
$463$	7.72496	0.359010	0.179505	−	0.983757i	$-0.442550\pi$
$463$	7.72496	0.359010	0.179505	+	0.983757i	$0.442550\pi$
$464$	0	0
$465$	−35.4983	−1.64619
$466$	0	0
$467$	−7.18639	−0.332547	−0.166273	−	0.986080i	$-0.553173\pi$
$467$	−7.18639	−0.332547	−0.166273	+	0.986080i	$0.553173\pi$
$468$	0	0
$469$	−7.95615	−0.367381
$470$	0	0
$471$	−47.8363	−2.20418
$472$	0	0
$473$	−8.10780	−0.372797
$474$	0	0
$475$	0	0
$476$	0	0
$477$	4.14611	0.189837
$478$	0	0
$479$	24.4358	1.11650	0.558250	−	0.829673i	$-0.311473\pi$
$479$	24.4358	1.11650	0.558250	+	0.829673i	$0.311473\pi$
$480$	0	0
$481$	49.0560	2.23676
$482$	0	0
$483$	−9.62721	−0.438053
$484$	0	0
$485$	−8.39194	−0.381059
$486$	0	0
$487$	−37.2772	−1.68919	−0.844595	−	0.535406i	$-0.820158\pi$
$487$	−37.2772	−1.68919	−0.844595	+	0.535406i	$0.820158\pi$
$488$	0	0
$489$	−7.91638	−0.357991
$490$	0	0
$491$	−18.8569	−0.851002	−0.425501	−	0.904958i	$-0.639902\pi$
$491$	−18.8569	−0.851002	−0.425501	+	0.904958i	$0.639902\pi$
$492$	0	0
$493$	7.34861	0.330965
$494$	0	0
$495$	2.09775	0.0942869
$496$	0	0
$497$	1.79445	0.0804921
$498$	0	0
$499$	−16.3678	−0.732722	−0.366361	−	0.930473i	$-0.619396\pi$
$499$	−16.3678	−0.732722	−0.366361	+	0.930473i	$0.619396\pi$
$500$	0	0
$501$	−15.7597	−0.704092
$502$	0	0
$503$	−37.5869	−1.67592	−0.837959	−	0.545733i	$-0.816251\pi$
$503$	−37.5869	−1.67592	−0.837959	+	0.545733i	$0.816251\pi$
$504$	0	0
$505$	5.38692	0.239715
$506$	0	0
$507$	−12.2353	−0.543387
$508$	0	0
$509$	−15.7597	−0.698537	−0.349268	−	0.937023i	$-0.613570\pi$
$509$	−15.7597	−0.698537	−0.349268	+	0.937023i	$0.613570\pi$
$510$	0	0
$511$	−1.58336	−0.0700437
$512$	0	0
$513$	0	0
$514$	0	0
$515$	0.853372	0.0376041
$516$	0	0
$517$	4.59392	0.202041
$518$	0	0
$519$	25.9441	1.13882
$520$	0	0
$521$	9.47411	0.415068	0.207534	−	0.978228i	$-0.433456\pi$
$521$	9.47411	0.415068	0.207534	+	0.978228i	$0.433456\pi$
$522$	0	0
$523$	−38.3764	−1.67808	−0.839040	−	0.544069i	$-0.816883\pi$
$523$	−38.3764	−1.67808	−0.839040	+	0.544069i	$0.816883\pi$
$524$	0	0
$525$	−9.08165	−0.396356
$526$	0	0
$527$	8.52998	0.371572
$528$	0	0
$529$	−19.7108	−0.856993
$530$	0	0
$531$	2.69721	0.117049
$532$	0	0
$533$	48.6102	2.10554
$534$	0	0
$535$	18.5089	0.800207
$536$	0	0
$537$	−51.8258	−2.23645
$538$	0	0
$539$	0.510305	0.0219804
$540$	0	0
$541$	1.18137	0.0507911	0.0253955	−	0.999677i	$-0.491915\pi$
$541$	1.18137	0.0507911	0.0253955	+	0.999677i	$0.491915\pi$
$542$	0	0
$543$	1.95972	0.0840995
$544$	0	0
$545$	6.81863	0.292078
$546$	0	0
$547$	23.6025	1.00917	0.504585	−	0.863362i	$-0.331645\pi$
$547$	23.6025	1.00917	0.504585	+	0.863362i	$0.331645\pi$
$548$	0	0
$549$	9.37941	0.400304
$550$	0	0
$551$	0	0
$552$	0	0
$553$	26.0978	1.10979
$554$	0	0
$555$	−43.1255	−1.83058
$556$	0	0
$557$	3.08362	0.130657	0.0653286	−	0.997864i	$-0.479190\pi$
$557$	3.08362	0.130657	0.0653286	+	0.997864i	$0.479190\pi$
$558$	0	0
$559$	−43.2297	−1.82842
$560$	0	0
$561$	−1.56777	−0.0661915
$562$	0	0
$563$	3.46143	0.145882	0.0729409	−	0.997336i	$-0.476762\pi$
$563$	3.46143	0.145882	0.0729409	+	0.997336i	$0.476762\pi$
$564$	0	0
$565$	27.3366	1.15006
$566$	0	0
$567$	28.3844	1.19203
$568$	0	0
$569$	−26.6550	−1.11743	−0.558717	−	0.829358i	$-0.688706\pi$
$569$	−26.6550	−1.11743	−0.558717	+	0.829358i	$0.688706\pi$
$570$	0	0
$571$	4.23527	0.177241	0.0886203	−	0.996065i	$-0.471754\pi$
$571$	4.23527	0.177241	0.0886203	+	0.996065i	$0.471754\pi$
$572$	0	0
$573$	26.0978	1.09025
$574$	0	0
$575$	3.10278	0.129395
$576$	0	0
$577$	37.0036	1.54048	0.770239	−	0.637755i	$-0.220137\pi$
$577$	37.0036	1.54048	0.770239	+	0.637755i	$0.220137\pi$
$578$	0	0
$579$	3.49472	0.145236
$580$	0	0
$581$	−24.8972	−1.03291
$582$	0	0
$583$	−2.37279	−0.0982708
$584$	0	0
$585$	11.1849	0.462440
$586$	0	0
$587$	−22.1114	−0.912634	−0.456317	−	0.889817i	$-0.650832\pi$
$587$	−22.1114	−0.912634	−0.456317	+	0.889817i	$0.650832\pi$
$588$	0	0
$589$	0	0
$590$	0	0
$591$	−11.5944	−0.476932
$592$	0	0
$593$	−31.5683	−1.29635	−0.648177	−	0.761490i	$-0.724468\pi$
$593$	−31.5683	−1.29635	−0.648177	+	0.761490i	$0.724468\pi$
$594$	0	0
$595$	−4.19550	−0.171999
$596$	0	0
$597$	−33.5713	−1.37398
$598$	0	0
$599$	20.7264	0.846858	0.423429	−	0.905929i	$-0.360826\pi$
$599$	20.7264	0.846858	0.423429	+	0.905929i	$0.360826\pi$
$600$	0	0
$601$	−22.6902	−0.925553	−0.462777	−	0.886475i	$-0.653147\pi$
$601$	−22.6902	−0.925553	−0.462777	+	0.886475i	$0.653147\pi$
$602$	0	0
$603$	4.48059	0.182464
$604$	0	0
$605$	18.7491	0.762261
$606$	0	0
$607$	37.1099	1.50625	0.753123	−	0.657880i	$-0.228547\pi$
$607$	37.1099	1.50625	0.753123	+	0.657880i	$0.228547\pi$
$608$	0	0
$609$	−42.5683	−1.72495
$610$	0	0
$611$	24.4942	0.990929
$612$	0	0
$613$	12.0731	0.487626	0.243813	−	0.969822i	$-0.421602\pi$
$613$	12.0731	0.487626	0.243813	+	0.969822i	$0.421602\pi$
$614$	0	0
$615$	−42.7336	−1.72318
$616$	0	0
$617$	3.30330	0.132986	0.0664930	−	0.997787i	$-0.478819\pi$
$617$	3.30330	0.132986	0.0664930	+	0.997787i	$0.478819\pi$
$618$	0	0
$619$	−26.0867	−1.04851	−0.524256	−	0.851561i	$-0.675656\pi$
$619$	−26.0867	−1.04851	−0.524256	+	0.851561i	$0.675656\pi$
$620$	0	0
$621$	−6.01916	−0.241540
$622$	0	0
$623$	−34.3133	−1.37474
$624$	0	0
$625$	−13.5189	−0.540756
$626$	0	0
$627$	0	0
$628$	0	0
$629$	10.3627	0.413190
$630$	0	0
$631$	13.4892	0.536996	0.268498	−	0.963280i	$-0.413473\pi$
$631$	13.4892	0.536996	0.268498	+	0.963280i	$0.413473\pi$
$632$	0	0
$633$	17.6136	0.700078
$634$	0	0
$635$	−5.48467	−0.217653
$636$	0	0
$637$	2.72088	0.107805
$638$	0	0
$639$	−1.01056	−0.0399772
$640$	0	0
$641$	−5.22668	−0.206441	−0.103221	−	0.994658i	$-0.532915\pi$
$641$	−5.22668	−0.206441	−0.103221	+	0.994658i	$0.532915\pi$
$642$	0	0
$643$	−28.6116	−1.12833	−0.564166	−	0.825661i	$-0.690802\pi$
$643$	−28.6116	−1.12833	−0.564166	+	0.825661i	$0.690802\pi$
$644$	0	0
$645$	38.0036	1.49639
$646$	0	0
$647$	−23.3139	−0.916563	−0.458281	−	0.888807i	$-0.651535\pi$
$647$	−23.3139	−0.916563	−0.458281	+	0.888807i	$0.651535\pi$
$648$	0	0
$649$	−1.54359	−0.0605914
$650$	0	0
$651$	−49.4116	−1.93659
$652$	0	0
$653$	9.83276	0.384786	0.192393	−	0.981318i	$-0.438375\pi$
$653$	9.83276	0.384786	0.192393	+	0.981318i	$0.438375\pi$
$654$	0	0
$655$	−27.8519	−1.08826
$656$	0	0
$657$	0.891685	0.0347880
$658$	0	0
$659$	−49.9341	−1.94516	−0.972578	−	0.232577i	$-0.925284\pi$
$659$	−49.9341	−1.94516	−0.972578	+	0.232577i	$0.925284\pi$
$660$	0	0
$661$	22.7053	0.883134	0.441567	−	0.897228i	$-0.354423\pi$
$661$	22.7053	0.883134	0.441567	+	0.897228i	$0.354423\pi$
$662$	0	0
$663$	−8.35917	−0.324643
$664$	0	0
$665$	0	0
$666$	0	0
$667$	14.5436	0.563130
$668$	0	0
$669$	8.37082	0.323635
$670$	0	0
$671$	−5.36776	−0.207220
$672$	0	0
$673$	−3.96169	−0.152712	−0.0763559	−	0.997081i	$-0.524329\pi$
$673$	−3.96169	−0.152712	−0.0763559	+	0.997081i	$0.524329\pi$
$674$	0	0
$675$	−5.67806	−0.218549
$676$	0	0
$677$	−2.74557	−0.105521	−0.0527605	−	0.998607i	$-0.516802\pi$
$677$	−2.74557	−0.105521	−0.0527605	+	0.998607i	$0.516802\pi$
$678$	0	0
$679$	−11.6811	−0.448280
$680$	0	0
$681$	−25.3764	−0.972424
$682$	0	0
$683$	33.7633	1.29192	0.645958	−	0.763373i	$-0.276458\pi$
$683$	33.7633	1.29192	0.645958	+	0.763373i	$0.276458\pi$
$684$	0	0
$685$	2.86248	0.109370
$686$	0	0
$687$	−39.9305	−1.52344
$688$	0	0
$689$	−12.6514	−0.481980
$690$	0	0
$691$	7.58890	0.288695	0.144348	−	0.989527i	$-0.453892\pi$
$691$	7.58890	0.288695	0.144348	+	0.989527i	$0.453892\pi$
$692$	0	0
$693$	2.91995	0.110920
$694$	0	0
$695$	28.1552	1.06799
$696$	0	0
$697$	10.2686	0.388949
$698$	0	0
$699$	−40.2388	−1.52197
$700$	0	0
$701$	−34.8313	−1.31556	−0.657780	−	0.753210i	$-0.728504\pi$
$701$	−34.8313	−1.31556	−0.657780	+	0.753210i	$0.728504\pi$
$702$	0	0
$703$	0	0
$704$	0	0
$705$	−21.5330	−0.810981
$706$	0	0
$707$	7.49829	0.282002
$708$	0	0
$709$	28.6358	1.07544	0.537720	−	0.843123i	$-0.319286\pi$
$709$	28.6358	1.07544	0.537720	+	0.843123i	$0.319286\pi$
$710$	0	0
$711$	−14.6972	−0.551189
$712$	0	0
$713$	16.8816	0.632222
$714$	0	0
$715$	−6.40105	−0.239386
$716$	0	0
$717$	5.27555	0.197019
$718$	0	0
$719$	−41.9441	−1.56425	−0.782126	−	0.623120i	$-0.785865\pi$
$719$	−41.9441	−1.56425	−0.782126	+	0.623120i	$0.785865\pi$
$720$	0	0
$721$	1.18785	0.0442377
$722$	0	0
$723$	8.08165	0.300560
$724$	0	0
$725$	13.7194	0.509527
$726$	0	0
$727$	38.0731	1.41205	0.706026	−	0.708186i	$-0.250486\pi$
$727$	38.0731	1.41205	0.706026	+	0.708186i	$0.250486\pi$
$728$	0	0
$729$	4.95164	0.183394
$730$	0	0
$731$	−9.13198	−0.337758
$732$	0	0
$733$	2.07502	0.0766428	0.0383214	−	0.999265i	$-0.487799\pi$
$733$	2.07502	0.0766428	0.0383214	+	0.999265i	$0.487799\pi$
$734$	0	0
$735$	−2.39194	−0.0882282
$736$	0	0
$737$	−2.56420	−0.0944537
$738$	0	0
$739$	33.5316	1.23348	0.616740	−	0.787167i	$-0.288453\pi$
$739$	33.5316	1.23348	0.616740	+	0.787167i	$0.288453\pi$
$740$	0	0
$741$	0	0
$742$	0	0
$743$	−43.8308	−1.60800	−0.803998	−	0.594632i	$-0.797298\pi$
$743$	−43.8308	−1.60800	−0.803998	+	0.594632i	$0.797298\pi$
$744$	0	0
$745$	32.7491	1.19984
$746$	0	0
$747$	14.0211	0.513006
$748$	0	0
$749$	25.7633	0.941370
$750$	0	0
$751$	−12.0347	−0.439154	−0.219577	−	0.975595i	$-0.570468\pi$
$751$	−12.0347	−0.439154	−0.219577	+	0.975595i	$0.570468\pi$
$752$	0	0
$753$	43.8988	1.59976
$754$	0	0
$755$	−12.8816	−0.468811
$756$	0	0
$757$	14.5819	0.529988	0.264994	−	0.964250i	$-0.414630\pi$
$757$	14.5819	0.529988	0.264994	+	0.964250i	$0.414630\pi$
$758$	0	0
$759$	−3.10278	−0.112624
$760$	0	0
$761$	−41.3069	−1.49737	−0.748686	−	0.662924i	$-0.769315\pi$
$761$	−41.3069	−1.49737	−0.748686	+	0.662924i	$0.769315\pi$
$762$	0	0
$763$	9.49115	0.343603
$764$	0	0
$765$	2.36274	0.0854251
$766$	0	0
$767$	−8.23025	−0.297177
$768$	0	0
$769$	9.17029	0.330689	0.165345	−	0.986236i	$-0.447126\pi$
$769$	9.17029	0.330689	0.165345	+	0.986236i	$0.447126\pi$
$770$	0	0
$771$	−60.3577	−2.17373
$772$	0	0
$773$	−1.58693	−0.0570779	−0.0285390	−	0.999593i	$-0.509085\pi$
$773$	−1.58693	−0.0570779	−0.0285390	+	0.999593i	$0.509085\pi$
$774$	0	0
$775$	15.9250	0.572042
$776$	0	0
$777$	−60.0283	−2.15350
$778$	0	0
$779$	0	0
$780$	0	0
$781$	0.578337	0.0206945
$782$	0	0
$783$	−26.6147	−0.951131
$784$	0	0
$785$	−41.2580	−1.47256
$786$	0	0
$787$	−17.4514	−0.622075	−0.311037	−	0.950398i	$-0.600676\pi$
$787$	−17.4514	−0.622075	−0.311037	+	0.950398i	$0.600676\pi$
$788$	0	0
$789$	11.0892	0.394784
$790$	0	0
$791$	38.0510	1.35294
$792$	0	0
$793$	−28.6202	−1.01633
$794$	0	0
$795$	11.1219	0.394454
$796$	0	0
$797$	−23.5038	−0.832548	−0.416274	−	0.909239i	$-0.636664\pi$
$797$	−23.5038	−0.832548	−0.416274	+	0.909239i	$0.636664\pi$
$798$	0	0
$799$	5.17423	0.183051
$800$	0	0
$801$	19.3239	0.682777
$802$	0	0
$803$	−0.510305	−0.0180083
$804$	0	0
$805$	−8.30330	−0.292653
$806$	0	0
$807$	45.9613	1.61792
$808$	0	0
$809$	−35.2686	−1.23998	−0.619988	−	0.784611i	$-0.712863\pi$
$809$	−35.2686	−1.23998	−0.619988	+	0.784611i	$0.712863\pi$
$810$	0	0
$811$	36.2091	1.27147	0.635737	−	0.771906i	$-0.280696\pi$
$811$	36.2091	1.27147	0.635737	+	0.771906i	$0.280696\pi$
$812$	0	0
$813$	−25.6252	−0.898717
$814$	0	0
$815$	−6.82774	−0.239165
$816$	0	0
$817$	0	0
$818$	0	0
$819$	15.5688	0.544018
$820$	0	0
$821$	−8.96474	−0.312872	−0.156436	−	0.987688i	$-0.550000\pi$
$821$	−8.96474	−0.312872	−0.156436	+	0.987688i	$0.550000\pi$
$822$	0	0
$823$	−5.38692	−0.187776	−0.0938881	−	0.995583i	$-0.529930\pi$
$823$	−5.38692	−0.187776	−0.0938881	+	0.995583i	$0.529930\pi$
$824$	0	0
$825$	−2.92694	−0.101903
$826$	0	0
$827$	17.9355	0.623680	0.311840	−	0.950135i	$-0.399055\pi$
$827$	17.9355	0.623680	0.311840	+	0.950135i	$0.399055\pi$
$828$	0	0
$829$	30.6550	1.06469	0.532345	−	0.846527i	$-0.321311\pi$
$829$	30.6550	1.06469	0.532345	+	0.846527i	$0.321311\pi$
$830$	0	0
$831$	−1.32945	−0.0461181
$832$	0	0
$833$	0.574766	0.0199145
$834$	0	0
$835$	−13.5925	−0.470387
$836$	0	0
$837$	−30.8933	−1.06783
$838$	0	0
$839$	26.6641	0.920546	0.460273	−	0.887777i	$-0.347752\pi$
$839$	26.6641	0.920546	0.460273	+	0.887777i	$0.347752\pi$
$840$	0	0
$841$	35.3069	1.21748
$842$	0	0
$843$	69.6777	2.39983
$844$	0	0
$845$	−10.5527	−0.363024
$846$	0	0
$847$	26.0978	0.896729
$848$	0	0
$849$	18.8116	0.645614
$850$	0	0
$851$	20.5089	0.703034
$852$	0	0
$853$	−11.0625	−0.378773	−0.189386	−	0.981903i	$-0.560650\pi$
$853$	−11.0625	−0.378773	−0.189386	+	0.981903i	$0.560650\pi$
$854$	0	0
$855$	0	0
$856$	0	0
$857$	−3.04939	−0.104165	−0.0520826	−	0.998643i	$-0.516586\pi$
$857$	−3.04939	−0.104165	−0.0520826	+	0.998643i	$0.516586\pi$
$858$	0	0
$859$	−14.6600	−0.500192	−0.250096	−	0.968221i	$-0.580462\pi$
$859$	−14.6600	−0.500192	−0.250096	+	0.968221i	$0.580462\pi$
$860$	0	0
$861$	−59.4827	−2.02717
$862$	0	0
$863$	8.78943	0.299196	0.149598	−	0.988747i	$-0.452202\pi$
$863$	8.78943	0.299196	0.149598	+	0.988747i	$0.452202\pi$
$864$	0	0
$865$	22.3764	0.760819
$866$	0	0
$867$	33.9814	1.15407
$868$	0	0
$869$	8.41110	0.285327
$870$	0	0
$871$	−13.6720	−0.463258
$872$	0	0
$873$	6.57834	0.222643
$874$	0	0
$875$	−30.7244	−1.03868
$876$	0	0
$877$	58.6741	1.98128	0.990642	−	0.136485i	$-0.0435806\pi$
$877$	58.6741	1.98128	0.990642	+	0.136485i	$0.0435806\pi$
$878$	0	0
$879$	59.0505	1.99172
$880$	0	0
$881$	2.55416	0.0860517	0.0430259	−	0.999074i	$-0.486300\pi$
$881$	2.55416	0.0860517	0.0430259	+	0.999074i	$0.486300\pi$
$882$	0	0
$883$	24.9945	0.841131	0.420565	−	0.907262i	$-0.361832\pi$
$883$	24.9945	0.841131	0.420565	+	0.907262i	$0.361832\pi$
$884$	0	0
$885$	7.23527	0.243211
$886$	0	0
$887$	−35.7597	−1.20069	−0.600347	−	0.799740i	$-0.704971\pi$
$887$	−35.7597	−1.20069	−0.600347	+	0.799740i	$0.704971\pi$
$888$	0	0
$889$	−7.63435	−0.256048
$890$	0	0
$891$	9.14808	0.306472
$892$	0	0
$893$	0	0
$894$	0	0
$895$	−44.6988	−1.49412
$896$	0	0
$897$	−16.5436	−0.552375
$898$	0	0
$899$	74.6449	2.48955
$900$	0	0
$901$	−2.67252	−0.0890345
$902$	0	0
$903$	52.8988	1.76036
$904$	0	0
$905$	1.69022	0.0561849
$906$	0	0
$907$	−12.2600	−0.407085	−0.203543	−	0.979066i	$-0.565246\pi$
$907$	−12.2600	−0.407085	−0.203543	+	0.979066i	$0.565246\pi$
$908$	0	0
$909$	−4.22274	−0.140059
$910$	0	0
$911$	33.0816	1.09604	0.548022	−	0.836464i	$-0.315381\pi$
$911$	33.0816	1.09604	0.548022	+	0.836464i	$0.315381\pi$
$912$	0	0
$913$	−8.02418	−0.265562
$914$	0	0
$915$	25.1602	0.831772
$916$	0	0
$917$	−38.7683	−1.28024
$918$	0	0
$919$	−39.1950	−1.29292	−0.646462	−	0.762946i	$-0.723752\pi$
$919$	−39.1950	−1.29292	−0.646462	+	0.762946i	$0.723752\pi$
$920$	0	0
$921$	−65.8122	−2.16858
$922$	0	0
$923$	3.08362	0.101499
$924$	0	0
$925$	19.3466	0.636114
$926$	0	0
$927$	−0.668948	−0.0219711
$928$	0	0
$929$	53.9482	1.76998	0.884992	−	0.465607i	$-0.154164\pi$
$929$	53.9482	1.76998	0.884992	+	0.465607i	$0.154164\pi$
$930$	0	0
$931$	0	0
$932$	0	0
$933$	18.9355	0.619922
$934$	0	0
$935$	−1.35218	−0.0442209
$936$	0	0
$937$	−44.2716	−1.44629	−0.723145	−	0.690696i	$-0.757304\pi$
$937$	−44.2716	−1.44629	−0.723145	+	0.690696i	$0.757304\pi$
$938$	0	0
$939$	9.17278	0.299342
$940$	0	0
$941$	14.9930	0.488758	0.244379	−	0.969680i	$-0.421416\pi$
$941$	14.9930	0.488758	0.244379	+	0.969680i	$0.421416\pi$
$942$	0	0
$943$	20.3225	0.661790
$944$	0	0
$945$	15.1950	0.494293
$946$	0	0
$947$	48.4429	1.57418	0.787092	−	0.616836i	$-0.211586\pi$
$947$	48.4429	1.57418	0.787092	+	0.616836i	$0.211586\pi$
$948$	0	0
$949$	−2.72088	−0.0883234
$950$	0	0
$951$	9.12193	0.295799
$952$	0	0
$953$	−1.85337	−0.0600366	−0.0300183	−	0.999549i	$-0.509557\pi$
$953$	−1.85337	−0.0600366	−0.0300183	+	0.999549i	$0.509557\pi$
$954$	0	0
$955$	22.5089	0.728369
$956$	0	0
$957$	−13.7194	−0.443486
$958$	0	0
$959$	3.98441	0.128663
$960$	0	0
$961$	55.6449	1.79500
$962$	0	0
$963$	−14.5089	−0.467541
$964$	0	0
$965$	3.01413	0.0970284
$966$	0	0
$967$	10.5436	0.339059	0.169530	−	0.985525i	$-0.445775\pi$
$967$	10.5436	0.339059	0.169530	+	0.985525i	$0.445775\pi$
$968$	0	0
$969$	0	0
$970$	0	0
$971$	5.95666	0.191158	0.0955792	−	0.995422i	$-0.469530\pi$
$971$	5.95666	0.191158	0.0955792	+	0.995422i	$0.469530\pi$
$972$	0	0
$973$	39.1905	1.25639
$974$	0	0
$975$	−15.6061	−0.499795
$976$	0	0
$977$	−10.7980	−0.345459	−0.172730	−	0.984969i	$-0.555259\pi$
$977$	−10.7980	−0.345459	−0.172730	+	0.984969i	$0.555259\pi$
$978$	0	0
$979$	−11.0589	−0.353445
$980$	0	0
$981$	−5.34504	−0.170654
$982$	0	0
$983$	10.0731	0.321281	0.160640	−	0.987013i	$-0.448644\pi$
$983$	10.0731	0.321281	0.160640	+	0.987013i	$0.448644\pi$
$984$	0	0
$985$	−10.0000	−0.318626
$986$	0	0
$987$	−29.9728	−0.954044
$988$	0	0
$989$	−18.0731	−0.574690
$990$	0	0
$991$	−36.3169	−1.15364	−0.576822	−	0.816870i	$-0.695708\pi$
$991$	−36.3169	−1.15364	−0.576822	+	0.816870i	$0.695708\pi$
$992$	0	0
$993$	22.3064	0.707871
$994$	0	0
$995$	−28.9547	−0.917926
$996$	0	0
$997$	29.7441	0.942006	0.471003	−	0.882132i	$-0.343892\pi$
$997$	29.7441	0.942006	0.471003	+	0.882132i	$0.343892\pi$
$998$	0	0
$999$	−37.5311	−1.18743

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2888.2.a.r.1.1		3
4.3	odd	2		5776.2.a.bk.1.3		3
19.7	even	3		152.2.i.c.49.3	✓	6
19.11	even	3		152.2.i.c.121.3	yes	6
19.18	odd	2		2888.2.a.n.1.3		3
57.11	odd	6		1368.2.s.k.577.1		6
57.26	odd	6		1368.2.s.k.505.1		6
76.7	odd	6		304.2.i.f.49.1		6
76.11	odd	6		304.2.i.f.273.1		6
76.75	even	2		5776.2.a.bq.1.1		3
152.11	odd	6		1216.2.i.m.577.3		6
152.45	even	6		1216.2.i.n.961.1		6
152.83	odd	6		1216.2.i.m.961.3		6
152.125	even	6		1216.2.i.n.577.1		6
228.11	even	6		2736.2.s.y.577.1		6
228.83	even	6		2736.2.s.y.1873.1		6

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
152.2.i.c.49.3	✓	6	19.7	even	3
152.2.i.c.121.3	yes	6	19.11	even	3
304.2.i.f.49.1		6	76.7	odd	6
304.2.i.f.273.1		6	76.11	odd	6
1216.2.i.m.577.3		6	152.11	odd	6
1216.2.i.m.961.3		6	152.83	odd	6
1216.2.i.n.577.1		6	152.125	even	6
1216.2.i.n.961.1		6	152.45	even	6
1368.2.s.k.505.1		6	57.26	odd	6
1368.2.s.k.577.1		6	57.11	odd	6
2736.2.s.y.577.1		6	228.11	even	6
2736.2.s.y.1873.1		6	228.83	even	6
2888.2.a.n.1.3		3	19.18	odd	2
2888.2.a.r.1.1		3	1.1	even	1	trivial
5776.2.a.bk.1.3		3	4.3	odd	2
5776.2.a.bq.1.1		3	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)

\(f(q)\)	\(=\)	\(q-2.10278 q^{3} -1.81361 q^{5} -2.52444 q^{7} +1.42166 q^{9} -0.813607 q^{11} -4.33804 q^{13} +3.81361 q^{15} -0.916382 q^{17} +5.30833 q^{21} -1.81361 q^{23} -1.71083 q^{25} +3.31889 q^{27} -8.01916 q^{29} -9.30833 q^{31} +1.71083 q^{33} +4.57834 q^{35} -11.3083 q^{37} +9.12193 q^{39} -11.2056 q^{41} +9.96526 q^{43} -2.57834 q^{45} -5.64637 q^{47} -0.627213 q^{49} +1.92694 q^{51} +2.91638 q^{53} +1.47556 q^{55} +1.89722 q^{59} +6.59749 q^{61} -3.58890 q^{63} +7.86751 q^{65} +3.15165 q^{67} +3.81361 q^{69} -0.710831 q^{71} +0.627213 q^{73} +3.59749 q^{75} +2.05390 q^{77} -10.3380 q^{79} -11.2439 q^{81} +9.86248 q^{83} +1.66196 q^{85} +16.8625 q^{87} +13.5925 q^{89} +10.9511 q^{91} +19.5733 q^{93} +4.62721 q^{97} -1.15667 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{3} + q^{5} - 2 q^{7} + 6 q^{9} + 4 q^{11} - q^{13} + 5 q^{15} + 11 q^{17} - 6 q^{21} + q^{23} - 6 q^{25} + 19 q^{27} - 3 q^{29} - 6 q^{31} + 6 q^{33} + 12 q^{35} - 12 q^{37} - q^{39} - 19 q^{41}+ \cdots + q^{97}+O(q^{100}) \) 3 * q + q^3 + q^5 - 2 * q^7 + 6 * q^9 + 4 * q^11 - q^13 + 5 * q^15 + 11 * q^17 - 6 * q^21 + q^23 - 6 * q^25 + 19 * q^27 - 3 * q^29 - 6 * q^31 + 6 * q^33 + 12 * q^35 - 12 * q^37 - q^39 - 19 * q^41 + 5 * q^43 - 6 * q^45 + 17 * q^47 + 11 * q^49 + 23 * q^51 - 5 * q^53 + 10 * q^55 + 13 * q^59 - 3 * q^61 - 40 * q^63 + 21 * q^65 - 9 * q^67 + 5 * q^69 - 3 * q^71 - 11 * q^73 - 12 * q^75 + 10 * q^77 - 19 * q^79 + 23 * q^81 + 12 * q^83 + 17 * q^85 + 33 * q^87 + 3 * q^89 + 44 * q^91 + 42 * q^93 + q^97

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