LMFDB - Embedded newform 1053.2.a.j.1.4

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1053,2,Mod(1,1053)] 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("1053.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(1053, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1053 = 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1053.a (trivial)

Newform invariants

comment:select newform

sage:traces = [5,-2,0,4,-1] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$8.40824733284$
Analytic rank:	$1$
Dimension:	$5$
Coefficient field:	5.5.403137.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 5x^{3} + 6x^{2} + 6x - 3 $ x^5 - 2x^4 - 5x^3 + 6x^2 + 6x - 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 117)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$-1.13107$ of defining polynomial
Character	$\chi$	$=$	1053.1

$q$-expansion

comment:q-expansion

sage:f.q_expansion() # note that sage often uses an isomorphic number field

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.13107 q^{2} -0.720682 q^{4} -2.38757 q^{5} +4.08809 q^{7} -3.07728 q^{8} -2.70051 q^{10} -3.85439 q^{11} +1.00000 q^{13} +4.62391 q^{14} -2.03925 q^{16} -2.15906 q^{17} -0.284952 q^{19} +1.72068 q^{20} -4.35959 q^{22} -3.58961 q^{23} +0.700513 q^{25} +1.13107 q^{26} -2.94621 q^{28} -2.97719 q^{29} -10.0198 q^{31} +3.84802 q^{32} -2.44204 q^{34} -9.76061 q^{35} -7.86324 q^{37} -0.322300 q^{38} +7.34724 q^{40} -5.57508 q^{41} -2.82330 q^{43} +2.77779 q^{44} -4.06010 q^{46} +8.42079 q^{47} +9.71246 q^{49} +0.792328 q^{50} -0.720682 q^{52} +4.34724 q^{53} +9.20266 q^{55} -12.5802 q^{56} -3.36741 q^{58} -9.58656 q^{59} -7.68666 q^{61} -11.3330 q^{62} +8.43089 q^{64} -2.38757 q^{65} +1.46153 q^{67} +1.55599 q^{68} -11.0399 q^{70} +8.81244 q^{71} +15.7860 q^{73} -8.89387 q^{74} +0.205360 q^{76} -15.7571 q^{77} +9.82962 q^{79} +4.86887 q^{80} -6.30580 q^{82} -0.672933 q^{83} +5.15491 q^{85} -3.19335 q^{86} +11.8611 q^{88} -3.26322 q^{89} +4.08809 q^{91} +2.58697 q^{92} +9.52450 q^{94} +0.680343 q^{95} -8.87777 q^{97} +10.9855 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q - 2 q^{2} + 4 q^{4} - q^{5} - 2 q^{7} - 12 q^{8} - 2 q^{10} - 11 q^{11} + 5 q^{13} + 5 q^{14} + 10 q^{16} - 7 q^{17} + 3 q^{19} + q^{20} - 7 q^{22} - 18 q^{23} - 8 q^{25} - 2 q^{26} - 19 q^{28} - 4 q^{29}+ \cdots + 39 q^{98}+O(q^{100}) $ 5 * q - 2 * q^2 + 4 * q^4 - q^5 - 2 * q^7 - 12 * q^8 - 2 * q^10 - 11 * q^11 + 5 * q^13 + 5 * q^14 + 10 * q^16 - 7 * q^17 + 3 * q^19 + q^20 - 7 * q^22 - 18 * q^23 - 8 * q^25 - 2 * q^26 - 19 * q^28 - 4 * q^29 - 16 * q^31 - 31 * q^32 + 13 * q^34 - 11 * q^35 + 5 * q^37 - 24 * q^38 + 18 * q^40 - 12 * q^41 - q^44 + 6 * q^46 - 15 * q^47 + 3 * q^49 + 5 * q^50 + 4 * q^52 + 3 * q^53 - 17 * q^55 + 24 * q^56 - 2 * q^58 - 3 * q^59 - q^61 + 4 * q^62 + 62 * q^64 - q^65 - 4 * q^67 + 16 * q^68 - 25 * q^70 - 15 * q^71 - 5 * q^73 - 11 * q^74 + 9 * q^76 - q^77 + 13 * q^79 - 32 * q^80 - 21 * q^82 - 26 * q^83 - 7 * q^85 + 6 * q^86 + 3 * q^88 - 29 * q^89 - 2 * q^91 + 6 * q^92 + 12 * q^94 - 12 * q^95 - 16 * q^97 + 39 * q^98

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 $

$2$	1.13107	0.799787	0.399893	−	0.916562i	$-0.369047\pi$
$2$	1.13107	0.799787	0.399893	+	0.916562i	$0.369047\pi$
$3$	0	0
$3$	0	0
$4$	−0.720682	−0.360341
$5$	−2.38757	−1.06776	−0.533878	−	0.845562i	$-0.679266\pi$
$5$	−2.38757	−1.06776	−0.533878	+	0.845562i	$0.679266\pi$
$6$	0	0
$7$	4.08809	1.54515	0.772576	−	0.634922i	$-0.218968\pi$
$7$	4.08809	1.54515	0.772576	+	0.634922i	$0.218968\pi$
$8$	−3.07728	−1.08798
$9$	0	0
$10$	−2.70051	−0.853977
$11$	−3.85439	−1.16214	−0.581072	−	0.813852i	$-0.697366\pi$
$11$	−3.85439	−1.16214	−0.581072	+	0.813852i	$0.697366\pi$
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	4.62391	1.23579
$15$	0	0
$16$	−2.03925	−0.509813
$17$	−2.15906	−0.523648	−0.261824	−	0.965116i	$-0.584324\pi$
$17$	−2.15906	−0.523648	−0.261824	+	0.965116i	$0.584324\pi$
$18$	0	0
$19$	−0.284952	−0.0653724	−0.0326862	−	0.999466i	$-0.510406\pi$
$19$	−0.284952	−0.0653724	−0.0326862	+	0.999466i	$0.510406\pi$
$20$	1.72068	0.384756
$21$	0	0
$22$	−4.35959	−0.929467
$23$	−3.58961	−0.748486	−0.374243	−	0.927331i	$-0.622097\pi$
$23$	−3.58961	−0.748486	−0.374243	+	0.927331i	$0.622097\pi$
$24$	0	0
$25$	0.700513	0.140103
$26$	1.13107	0.221821
$27$	0	0
$28$	−2.94621	−0.556781
$29$	−2.97719	−0.552850	−0.276425	−	0.961036i	$-0.589150\pi$
$29$	−2.97719	−0.552850	−0.276425	+	0.961036i	$0.589150\pi$
$30$	0	0
$31$	−10.0198	−1.79960	−0.899801	−	0.436300i	$-0.856289\pi$
$31$	−10.0198	−1.79960	−0.899801	+	0.436300i	$0.856289\pi$
$32$	3.84802	0.680241
$33$	0	0
$34$	−2.44204	−0.418807
$35$	−9.76061	−1.64984
$36$	0	0
$37$	−7.86324	−1.29271	−0.646354	−	0.763038i	$-0.723707\pi$
$37$	−7.86324	−1.29271	−0.646354	+	0.763038i	$0.723707\pi$
$38$	−0.322300	−0.0522840
$39$	0	0
$40$	7.34724	1.16170
$41$	−5.57508	−0.870681	−0.435340	−	0.900266i	$-0.643372\pi$
$41$	−5.57508	−0.870681	−0.435340	+	0.900266i	$0.643372\pi$
$42$	0	0
$43$	−2.82330	−0.430550	−0.215275	−	0.976553i	$-0.569065\pi$
$43$	−2.82330	−0.430550	−0.215275	+	0.976553i	$0.569065\pi$
$44$	2.77779	0.418768
$45$	0	0
$46$	−4.06010	−0.598629
$47$	8.42079	1.22830	0.614149	−	0.789190i	$-0.289499\pi$
$47$	8.42079	1.22830	0.614149	+	0.789190i	$0.289499\pi$
$48$	0	0
$49$	9.71246	1.38749
$50$	0.792328	0.112052
$51$	0	0
$52$	−0.720682	−0.0999406
$53$	4.34724	0.597139	0.298569	−	0.954388i	$-0.403491\pi$
$53$	4.34724	0.597139	0.298569	+	0.954388i	$0.403491\pi$
$54$	0	0
$55$	9.20266	1.24089
$56$	−12.5802	−1.68110
$57$	0	0
$58$	−3.36741	−0.442162
$59$	−9.58656	−1.24806	−0.624032	−	0.781399i	$-0.714507\pi$
$59$	−9.58656	−1.24806	−0.624032	+	0.781399i	$0.714507\pi$
$60$	0	0
$61$	−7.68666	−0.984175	−0.492088	−	0.870546i	$-0.663766\pi$
$61$	−7.68666	−0.984175	−0.492088	+	0.870546i	$0.663766\pi$
$62$	−11.3330	−1.43930
$63$	0	0
$64$	8.43089	1.05386
$65$	−2.38757	−0.296142
$66$	0	0
$67$	1.46153	0.178555	0.0892773	−	0.996007i	$-0.471544\pi$
$67$	1.46153	0.178555	0.0892773	+	0.996007i	$0.471544\pi$
$68$	1.55599	0.188692
$69$	0	0
$70$	−11.0399	−1.31952
$71$	8.81244	1.04584	0.522922	−	0.852381i	$-0.324842\pi$
$71$	8.81244	1.04584	0.522922	+	0.852381i	$0.324842\pi$
$72$	0	0
$73$	15.7860	1.84761	0.923806	−	0.382860i	$-0.125061\pi$
$73$	15.7860	1.84761	0.923806	+	0.382860i	$0.125061\pi$
$74$	−8.89387	−1.03389
$75$	0	0
$76$	0.205360	0.0235564
$77$	−15.7571	−1.79569
$78$	0	0
$79$	9.82962	1.10592	0.552959	−	0.833208i	$-0.313499\pi$
$79$	9.82962	1.10592	0.552959	+	0.833208i	$0.313499\pi$
$80$	4.86887	0.544356
$81$	0	0
$82$	−6.30580	−0.696359
$83$	−0.672933	−0.0738640	−0.0369320	−	0.999318i	$-0.511758\pi$
$83$	−0.672933	−0.0738640	−0.0369320	+	0.999318i	$0.511758\pi$
$84$	0	0
$85$	5.15491	0.559128
$86$	−3.19335	−0.344348
$87$	0	0
$88$	11.8611	1.26439
$89$	−3.26322	−0.345901	−0.172951	−	0.984931i	$-0.555330\pi$
$89$	−3.26322	−0.345901	−0.172951	+	0.984931i	$0.555330\pi$
$90$	0	0
$91$	4.08809	0.428548
$92$	2.58697	0.269710
$93$	0	0
$94$	9.52450	0.982377
$95$	0.680343	0.0698018
$96$	0	0
$97$	−8.87777	−0.901401	−0.450701	−	0.892675i	$-0.648826\pi$
$97$	−8.87777	−0.901401	−0.450701	+	0.892675i	$0.648826\pi$
$98$	10.9855	1.10970
$99$	0	0
$100$	−0.504847	−0.0504847
$101$	−12.1870	−1.21265	−0.606325	−	0.795217i	$-0.707357\pi$
$101$	−12.1870	−1.21265	−0.606325	+	0.795217i	$0.707357\pi$
$102$	0	0
$103$	−12.7337	−1.25469	−0.627346	−	0.778741i	$-0.715859\pi$
$103$	−12.7337	−1.25469	−0.627346	+	0.778741i	$0.715859\pi$
$104$	−3.07728	−0.301752
$105$	0	0
$106$	4.91703	0.477584
$107$	−6.83490	−0.660755	−0.330378	−	0.943849i	$-0.607176\pi$
$107$	−6.83490	−0.660755	−0.330378	+	0.943849i	$0.607176\pi$
$108$	0	0
$109$	−9.06573	−0.868340	−0.434170	−	0.900831i	$-0.642958\pi$
$109$	−9.06573	−0.868340	−0.434170	+	0.900831i	$0.642958\pi$
$110$	10.4088	0.992444
$111$	0	0
$112$	−8.33665	−0.787739
$113$	13.2336	1.24492	0.622458	−	0.782653i	$-0.286134\pi$
$113$	13.2336	1.24492	0.622458	+	0.782653i	$0.286134\pi$
$114$	0	0
$115$	8.57047	0.799200
$116$	2.14561	0.199214
$117$	0	0
$118$	−10.8431	−0.998185
$119$	−8.82641	−0.809116
$120$	0	0
$121$	3.85636	0.350578
$122$	−8.69414	−0.787131
$123$	0	0
$124$	7.22106	0.648471
$125$	10.2653	0.918161
$126$	0	0
$127$	18.9491	1.68146	0.840732	−	0.541451i	$-0.182125\pi$
$127$	18.9491	1.68146	0.840732	+	0.541451i	$0.182125\pi$
$128$	1.83987	0.162623
$129$	0	0
$130$	−2.70051	−0.236851
$131$	17.5627	1.53446	0.767229	−	0.641374i	$-0.221635\pi$
$131$	17.5627	1.53446	0.767229	+	0.641374i	$0.221635\pi$
$132$	0	0
$133$	−1.16491	−0.101010
$134$	1.65310	0.142806
$135$	0	0
$136$	6.64402	0.569720
$137$	0.766194	0.0654604	0.0327302	−	0.999464i	$-0.489580\pi$
$137$	0.766194	0.0654604	0.0327302	+	0.999464i	$0.489580\pi$
$138$	0	0
$139$	14.5425	1.23348	0.616739	−	0.787168i	$-0.288453\pi$
$139$	14.5425	1.23348	0.616739	+	0.787168i	$0.288453\pi$
$140$	7.03430	0.594507
$141$	0	0
$142$	9.96748	0.836452
$143$	−3.85439	−0.322321
$144$	0	0
$145$	7.10826	0.590309
$146$	17.8551	1.47770
$147$	0	0
$148$	5.66689	0.465816
$149$	−2.52439	−0.206806	−0.103403	−	0.994640i	$-0.532973\pi$
$149$	−2.52439	−0.206806	−0.103403	+	0.994640i	$0.532973\pi$
$150$	0	0
$151$	−6.21020	−0.505379	−0.252690	−	0.967547i	$-0.581315\pi$
$151$	−6.21020	−0.505379	−0.252690	+	0.967547i	$0.581315\pi$
$152$	0.876876	0.0711241
$153$	0	0
$154$	−17.8224	−1.43617
$155$	23.9229	1.92154
$156$	0	0
$157$	−2.67857	−0.213773	−0.106886	−	0.994271i	$-0.534088\pi$
$157$	−2.67857	−0.213773	−0.106886	+	0.994271i	$0.534088\pi$
$158$	11.1180	0.884499
$159$	0	0
$160$	−9.18744	−0.726331
$161$	−14.6746	−1.15652
$162$	0	0
$163$	−0.158596	−0.0124222	−0.00621109	−	0.999981i	$-0.501977\pi$
$163$	−0.158596	−0.0124222	−0.00621109	+	0.999981i	$0.501977\pi$
$164$	4.01786	0.313742
$165$	0	0
$166$	−0.761134	−0.0590755
$167$	10.7146	0.829119	0.414560	−	0.910022i	$-0.363936\pi$
$167$	10.7146	0.829119	0.414560	+	0.910022i	$0.363936\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	5.83056	0.447183
$171$	0	0
$172$	2.03470	0.155145
$173$	10.1673	0.773008	0.386504	−	0.922288i	$-0.373683\pi$
$173$	10.1673	0.773008	0.386504	+	0.922288i	$0.373683\pi$
$174$	0	0
$175$	2.86376	0.216480
$176$	7.86009	0.592476
$177$	0	0
$178$	−3.69093	−0.276647
$179$	−6.34718	−0.474411	−0.237205	−	0.971460i	$-0.576231\pi$
$179$	−6.34718	−0.474411	−0.237205	+	0.971460i	$0.576231\pi$
$180$	0	0
$181$	−7.20318	−0.535408	−0.267704	−	0.963501i	$-0.586265\pi$
$181$	−7.20318	−0.535408	−0.267704	+	0.963501i	$0.586265\pi$
$182$	4.62391	0.342747
$183$	0	0
$184$	11.0462	0.814340
$185$	18.7741	1.38030
$186$	0	0
$187$	8.32185	0.608554
$188$	−6.06871	−0.442606
$189$	0	0
$190$	0.769516	0.0558265
$191$	−3.63551	−0.263056	−0.131528	−	0.991312i	$-0.541988\pi$
$191$	−3.63551	−0.263056	−0.131528	+	0.991312i	$0.541988\pi$
$192$	0	0
$193$	3.91260	0.281635	0.140818	−	0.990036i	$-0.455027\pi$
$193$	3.91260	0.281635	0.140818	+	0.990036i	$0.455027\pi$
$194$	−10.0414	−0.720929
$195$	0	0
$196$	−6.99959	−0.499971
$197$	−6.56674	−0.467861	−0.233930	−	0.972253i	$-0.575159\pi$
$197$	−6.56674	−0.467861	−0.233930	+	0.972253i	$0.575159\pi$
$198$	0	0
$199$	−1.95403	−0.138517	−0.0692587	−	0.997599i	$-0.522063\pi$
$199$	−1.95403	−0.138517	−0.0692587	+	0.997599i	$0.522063\pi$
$200$	−2.15567	−0.152429
$201$	0	0
$202$	−13.7843	−0.969862
$203$	−12.1710	−0.854237
$204$	0	0
$205$	13.3109	0.929674
$206$	−14.4027	−1.00349
$207$	0	0
$208$	−2.03925	−0.141397
$209$	1.09832	0.0759721
$210$	0	0
$211$	−20.2167	−1.39177	−0.695887	−	0.718152i	$-0.744988\pi$
$211$	−20.2167	−1.39177	−0.695887	+	0.718152i	$0.744988\pi$
$212$	−3.13297	−0.215174
$213$	0	0
$214$	−7.73075	−0.528463
$215$	6.74085	0.459722
$216$	0	0
$217$	−40.9617	−2.78066
$218$	−10.2540	−0.694487
$219$	0	0
$220$	−6.63219	−0.447142
$221$	−2.15906	−0.145234
$222$	0	0
$223$	6.25697	0.418997	0.209499	−	0.977809i	$-0.432817\pi$
$223$	6.25697	0.418997	0.209499	+	0.977809i	$0.432817\pi$
$224$	15.7311	1.05108
$225$	0	0
$226$	14.9682	0.995667
$227$	−7.24083	−0.480591	−0.240295	−	0.970700i	$-0.577244\pi$
$227$	−7.24083	−0.480591	−0.240295	+	0.970700i	$0.577244\pi$
$228$	0	0
$229$	−2.27151	−0.150106	−0.0750530	−	0.997180i	$-0.523913\pi$
$229$	−2.27151	−0.150106	−0.0750530	+	0.997180i	$0.523913\pi$
$230$	9.69379	0.639190
$231$	0	0
$232$	9.16164	0.601491
$233$	13.9715	0.915303	0.457651	−	0.889132i	$-0.348691\pi$
$233$	13.9715	0.915303	0.457651	+	0.889132i	$0.348691\pi$
$234$	0	0
$235$	−20.1053	−1.31152
$236$	6.90886	0.449729
$237$	0	0
$238$	−9.98328	−0.647120
$239$	−26.5289	−1.71601	−0.858007	−	0.513638i	$-0.828297\pi$
$239$	−26.5289	−1.71601	−0.858007	+	0.513638i	$0.828297\pi$
$240$	0	0
$241$	−4.00110	−0.257733	−0.128867	−	0.991662i	$-0.541134\pi$
$241$	−4.00110	−0.257733	−0.128867	+	0.991662i	$0.541134\pi$
$242$	4.36181	0.280388
$243$	0	0
$244$	5.53963	0.354639
$245$	−23.1892	−1.48150
$246$	0	0
$247$	−0.284952	−0.0181310
$248$	30.8336	1.95794
$249$	0	0
$250$	11.6108	0.734333
$251$	−3.96080	−0.250003	−0.125002	−	0.992157i	$-0.539894\pi$
$251$	−3.96080	−0.250003	−0.125002	+	0.992157i	$0.539894\pi$
$252$	0	0
$253$	13.8358	0.869848
$254$	21.4328	1.34481
$255$	0	0
$256$	−14.7808	−0.923797
$257$	−13.2460	−0.826262	−0.413131	−	0.910672i	$-0.635565\pi$
$257$	−13.2460	−0.826262	−0.413131	+	0.910672i	$0.635565\pi$
$258$	0	0
$259$	−32.1456	−1.99743
$260$	1.72068	0.106712
$261$	0	0
$262$	19.8646	1.22724
$263$	−5.28821	−0.326085	−0.163043	−	0.986619i	$-0.552131\pi$
$263$	−5.28821	−0.326085	−0.163043	+	0.986619i	$0.552131\pi$
$264$	0	0
$265$	−10.3794	−0.637598
$266$	−1.31759	−0.0807867
$267$	0	0
$268$	−1.05330	−0.0643406
$269$	8.01270	0.488543	0.244271	−	0.969707i	$-0.421451\pi$
$269$	8.01270	0.488543	0.244271	+	0.969707i	$0.421451\pi$
$270$	0	0
$271$	−15.9840	−0.970956	−0.485478	−	0.874249i	$-0.661354\pi$
$271$	−15.9840	−0.970956	−0.485478	+	0.874249i	$0.661354\pi$
$272$	4.40286	0.266963
$273$	0	0
$274$	0.866619	0.0523543
$275$	−2.70005	−0.162819
$276$	0	0
$277$	−6.29126	−0.378005	−0.189003	−	0.981977i	$-0.560525\pi$
$277$	−6.29126	−0.378005	−0.189003	+	0.981977i	$0.560525\pi$
$278$	16.4486	0.986520
$279$	0	0
$280$	30.0361	1.79500
$281$	−16.9846	−1.01322	−0.506609	−	0.862176i	$-0.669101\pi$
$281$	−16.9846	−1.01322	−0.506609	+	0.862176i	$0.669101\pi$
$282$	0	0
$283$	27.4731	1.63310	0.816552	−	0.577271i	$-0.195883\pi$
$283$	27.4731	1.63310	0.816552	+	0.577271i	$0.195883\pi$
$284$	−6.35096	−0.376860
$285$	0	0
$286$	−4.35959	−0.257788
$287$	−22.7914	−1.34533
$288$	0	0
$289$	−12.3385	−0.725793
$290$	8.03993	0.472121
$291$	0	0
$292$	−11.3767	−0.665771
$293$	27.8646	1.62787	0.813934	−	0.580958i	$-0.197322\pi$
$293$	27.8646	1.62787	0.813934	+	0.580958i	$0.197322\pi$
$294$	0	0
$295$	22.8886	1.33263
$296$	24.1974	1.40644
$297$	0	0
$298$	−2.85526	−0.165401
$299$	−3.58961	−0.207593
$300$	0	0
$301$	−11.5419	−0.665265
$302$	−7.02417	−0.404196
$303$	0	0
$304$	0.581089	0.0333277
$305$	18.3525	1.05086
$306$	0	0
$307$	15.3680	0.877096	0.438548	−	0.898708i	$-0.355493\pi$
$307$	15.3680	0.877096	0.438548	+	0.898708i	$0.355493\pi$
$308$	11.3559	0.647060
$309$	0	0
$310$	27.0585	1.53682
$311$	11.2747	0.639328	0.319664	−	0.947531i	$-0.396430\pi$
$311$	11.2747	0.639328	0.319664	+	0.947531i	$0.396430\pi$
$312$	0	0
$313$	−10.8202	−0.611591	−0.305796	−	0.952097i	$-0.598922\pi$
$313$	−10.8202	−0.611591	−0.305796	+	0.952097i	$0.598922\pi$
$314$	−3.02964	−0.170973
$315$	0	0
$316$	−7.08403	−0.398508
$317$	16.4467	0.923741	0.461871	−	0.886947i	$-0.347178\pi$
$317$	16.4467	0.923741	0.461871	+	0.886947i	$0.347178\pi$
$318$	0	0
$319$	11.4753	0.642491
$320$	−20.1294	−1.12527
$321$	0	0
$322$	−16.5980	−0.924973
$323$	0.615227	0.0342321
$324$	0	0
$325$	0.700513	0.0388575
$326$	−0.179383	−0.00993509
$327$	0	0
$328$	17.1561	0.947286
$329$	34.4249	1.89791
$330$	0	0
$331$	5.72219	0.314520	0.157260	−	0.987557i	$-0.449734\pi$
$331$	5.72219	0.314520	0.157260	+	0.987557i	$0.449734\pi$
$332$	0.484971	0.0266162
$333$	0	0
$334$	12.1189	0.663119
$335$	−3.48952	−0.190653
$336$	0	0
$337$	−4.75171	−0.258842	−0.129421	−	0.991590i	$-0.541312\pi$
$337$	−4.75171	−0.258842	−0.129421	+	0.991590i	$0.541312\pi$
$338$	1.13107	0.0615221
$339$	0	0
$340$	−3.71505	−0.201477
$341$	38.6201	2.09140
$342$	0	0
$343$	11.0888	0.598737
$344$	8.68810	0.468431
$345$	0	0
$346$	11.5000	0.618242
$347$	−10.9315	−0.586831	−0.293416	−	0.955985i	$-0.594792\pi$
$347$	−10.9315	−0.586831	−0.293416	+	0.955985i	$0.594792\pi$
$348$	0	0
$349$	−26.1070	−1.39748	−0.698739	−	0.715377i	$-0.746255\pi$
$349$	−26.1070	−1.39748	−0.698739	+	0.715377i	$0.746255\pi$
$350$	3.23911	0.173138
$351$	0	0
$352$	−14.8318	−0.790538
$353$	11.4952	0.611828	0.305914	−	0.952059i	$-0.401038\pi$
$353$	11.4952	0.611828	0.305914	+	0.952059i	$0.401038\pi$
$354$	0	0
$355$	−21.0404	−1.11671
$356$	2.35175	0.124642
$357$	0	0
$358$	−7.17910	−0.379427
$359$	−7.42296	−0.391769	−0.195884	−	0.980627i	$-0.562758\pi$
$359$	−7.42296	−0.391769	−0.195884	+	0.980627i	$0.562758\pi$
$360$	0	0
$361$	−18.9188	−0.995726
$362$	−8.14729	−0.428212
$363$	0	0
$364$	−2.94621	−0.154423
$365$	−37.6903	−1.97280
$366$	0	0
$367$	−5.99150	−0.312754	−0.156377	−	0.987697i	$-0.549981\pi$
$367$	−5.99150	−0.312754	−0.156377	+	0.987697i	$0.549981\pi$
$368$	7.32013	0.381588
$369$	0	0
$370$	21.2348	1.10394
$371$	17.7719	0.922670
$372$	0	0
$373$	26.7177	1.38339	0.691696	−	0.722189i	$-0.256864\pi$
$373$	26.7177	1.38339	0.691696	+	0.722189i	$0.256864\pi$
$374$	9.41259	0.486714
$375$	0	0
$376$	−25.9131	−1.33637
$377$	−2.97719	−0.153333
$378$	0	0
$379$	−17.3941	−0.893474	−0.446737	−	0.894665i	$-0.647414\pi$
$379$	−17.3941	−0.893474	−0.446737	+	0.894665i	$0.647414\pi$
$380$	−0.490311	−0.0251524
$381$	0	0
$382$	−4.11201	−0.210389
$383$	−34.6143	−1.76871	−0.884354	−	0.466817i	$-0.845401\pi$
$383$	−34.6143	−1.76871	−0.884354	+	0.466817i	$0.845401\pi$
$384$	0	0
$385$	37.6213	1.91736
$386$	4.42543	0.225248
$387$	0	0
$388$	6.39805	0.324812
$389$	−35.7442	−1.81230	−0.906152	−	0.422953i	$-0.860994\pi$
$389$	−35.7442	−1.81230	−0.906152	+	0.422953i	$0.860994\pi$
$390$	0	0
$391$	7.75017	0.391943
$392$	−29.8880	−1.50957
$393$	0	0
$394$	−7.42744	−0.374189
$395$	−23.4689	−1.18085
$396$	0	0
$397$	13.6734	0.686246	0.343123	−	0.939290i	$-0.388515\pi$
$397$	13.6734	0.686246	0.343123	+	0.939290i	$0.388515\pi$
$398$	−2.21014	−0.110784
$399$	0	0
$400$	−1.42852	−0.0714262
$401$	33.5863	1.67722	0.838610	−	0.544732i	$-0.183369\pi$
$401$	33.5863	1.67722	0.838610	+	0.544732i	$0.183369\pi$
$402$	0	0
$403$	−10.0198	−0.499120
$404$	8.78294	0.436967
$405$	0	0
$406$	−13.7662	−0.683207
$407$	30.3080	1.50231
$408$	0	0
$409$	2.93454	0.145103	0.0725517	−	0.997365i	$-0.476886\pi$
$409$	2.93454	0.145103	0.0725517	+	0.997365i	$0.476886\pi$
$410$	15.0556	0.743541
$411$	0	0
$412$	9.17697	0.452117
$413$	−39.1907	−1.92845
$414$	0	0
$415$	1.60668	0.0788687
$416$	3.84802	0.188665
$417$	0	0
$418$	1.24227	0.0607615
$419$	8.53329	0.416879	0.208439	−	0.978035i	$-0.433162\pi$
$419$	8.53329	0.416879	0.208439	+	0.978035i	$0.433162\pi$
$420$	0	0
$421$	−21.8479	−1.06480	−0.532401	−	0.846492i	$-0.678710\pi$
$421$	−21.8479	−1.06480	−0.532401	+	0.846492i	$0.678710\pi$
$422$	−22.8665	−1.11312
$423$	0	0
$424$	−13.3777	−0.649677
$425$	−1.51245	−0.0733644
$426$	0	0
$427$	−31.4237	−1.52070
$428$	4.92579	0.238097
$429$	0	0
$430$	7.62437	0.367680
$431$	−35.1630	−1.69374	−0.846872	−	0.531797i	$-0.821517\pi$
$431$	−35.1630	−1.69374	−0.846872	+	0.531797i	$0.821517\pi$
$432$	0	0
$433$	24.9431	1.19869	0.599344	−	0.800491i	$-0.295428\pi$
$433$	24.9431	1.19869	0.599344	+	0.800491i	$0.295428\pi$
$434$	−46.3305	−2.22393
$435$	0	0
$436$	6.53351	0.312898
$437$	1.02287	0.0489303
$438$	0	0
$439$	13.5269	0.645602	0.322801	−	0.946467i	$-0.395376\pi$
$439$	13.5269	0.645602	0.322801	+	0.946467i	$0.395376\pi$
$440$	−28.3191	−1.35006
$441$	0	0
$442$	−2.44204	−0.116156
$443$	−33.4576	−1.58962	−0.794808	−	0.606860i	$-0.792429\pi$
$443$	−33.4576	−1.58962	−0.794808	+	0.606860i	$0.792429\pi$
$444$	0	0
$445$	7.79119	0.369338
$446$	7.07706	0.335109
$447$	0	0
$448$	34.4662	1.62838
$449$	7.63995	0.360552	0.180276	−	0.983616i	$-0.442301\pi$
$449$	7.63995	0.360552	0.180276	+	0.983616i	$0.442301\pi$
$450$	0	0
$451$	21.4885	1.01186
$452$	−9.53724	−0.448594
$453$	0	0
$454$	−8.18988	−0.384370
$455$	−9.76061	−0.457585
$456$	0	0
$457$	14.1313	0.661034	0.330517	−	0.943800i	$-0.392777\pi$
$457$	14.1313	0.661034	0.330517	+	0.943800i	$0.392777\pi$
$458$	−2.56924	−0.120053
$459$	0	0
$460$	−6.17658	−0.287985
$461$	4.62819	0.215556	0.107778	−	0.994175i	$-0.465626\pi$
$461$	4.62819	0.215556	0.107778	+	0.994175i	$0.465626\pi$
$462$	0	0
$463$	17.8864	0.831254	0.415627	−	0.909535i	$-0.363562\pi$
$463$	17.8864	0.831254	0.415627	+	0.909535i	$0.363562\pi$
$464$	6.07124	0.281850
$465$	0	0
$466$	15.8027	0.732047
$467$	16.6357	0.769808	0.384904	−	0.922957i	$-0.374235\pi$
$467$	16.6357	0.769808	0.384904	+	0.922957i	$0.374235\pi$
$468$	0	0
$469$	5.97488	0.275894
$470$	−22.7404	−1.04894
$471$	0	0
$472$	29.5005	1.35787
$473$	10.8821	0.500361
$474$	0	0
$475$	−0.199612	−0.00915884
$476$	6.36103	0.291557
$477$	0	0
$478$	−30.0061	−1.37245
$479$	−37.6745	−1.72139	−0.860695	−	0.509120i	$-0.829971\pi$
$479$	−37.6745	−1.72139	−0.860695	+	0.509120i	$0.829971\pi$
$480$	0	0
$481$	−7.86324	−0.358533
$482$	−4.52552	−0.206132
$483$	0	0
$484$	−2.77921	−0.126328
$485$	21.1963	0.962476
$486$	0	0
$487$	−34.2388	−1.55151	−0.775754	−	0.631035i	$-0.782630\pi$
$487$	−34.2388	−1.55151	−0.775754	+	0.631035i	$0.782630\pi$
$488$	23.6540	1.07077
$489$	0	0
$490$	−26.2286	−1.18489
$491$	−29.9151	−1.35005	−0.675024	−	0.737796i	$-0.735867\pi$
$491$	−29.9151	−1.35005	−0.675024	+	0.737796i	$0.735867\pi$
$492$	0	0
$493$	6.42791	0.289499
$494$	−0.322300	−0.0145010
$495$	0	0
$496$	20.4328	0.917461
$497$	36.0260	1.61599
$498$	0	0
$499$	−20.9583	−0.938223	−0.469111	−	0.883139i	$-0.655426\pi$
$499$	−20.9583	−0.938223	−0.469111	+	0.883139i	$0.655426\pi$
$500$	−7.39805	−0.330851
$501$	0	0
$502$	−4.47994	−0.199949
$503$	21.1599	0.943473	0.471737	−	0.881739i	$-0.343627\pi$
$503$	21.1599	0.943473	0.471737	+	0.881739i	$0.343627\pi$
$504$	0	0
$505$	29.0973	1.29481
$506$	15.6492	0.695693
$507$	0	0
$508$	−13.6563	−0.605900
$509$	−3.02989	−0.134298	−0.0671488	−	0.997743i	$-0.521390\pi$
$509$	−3.02989	−0.134298	−0.0671488	+	0.997743i	$0.521390\pi$
$510$	0	0
$511$	64.5346	2.85484
$512$	−20.3978	−0.901464
$513$	0	0
$514$	−14.9821	−0.660833
$515$	30.4027	1.33970
$516$	0	0
$517$	−32.4570	−1.42746
$518$	−36.3589	−1.59752
$519$	0	0
$520$	7.34724	0.322198
$521$	12.5546	0.550026	0.275013	−	0.961441i	$-0.411318\pi$
$521$	12.5546	0.550026	0.275013	+	0.961441i	$0.411318\pi$
$522$	0	0
$523$	−8.54696	−0.373733	−0.186866	−	0.982385i	$-0.559833\pi$
$523$	−8.54696	−0.373733	−0.186866	+	0.982385i	$0.559833\pi$
$524$	−12.6571	−0.552928
$525$	0	0
$526$	−5.98134	−0.260799
$527$	21.6332	0.942358
$528$	0	0
$529$	−10.1147	−0.439769
$530$	−11.7398	−0.509943
$531$	0	0
$532$	0.839528	0.0363981
$533$	−5.57508	−0.241483
$534$	0	0
$535$	16.3188	0.705525
$536$	−4.49755	−0.194264
$537$	0	0
$538$	9.06292	0.390730
$539$	−37.4356	−1.61247
$540$	0	0
$541$	−4.74799	−0.204132	−0.102066	−	0.994778i	$-0.532545\pi$
$541$	−4.74799	−0.204132	−0.102066	+	0.994778i	$0.532545\pi$
$542$	−18.0790	−0.776558
$543$	0	0
$544$	−8.30810	−0.356207
$545$	21.6451	0.927175
$546$	0	0
$547$	−1.95559	−0.0836149	−0.0418074	−	0.999126i	$-0.513312\pi$
$547$	−1.95559	−0.0836149	−0.0418074	+	0.999126i	$0.513312\pi$
$548$	−0.552182	−0.0235881
$549$	0	0
$550$	−3.05395	−0.130221
$551$	0.848355	0.0361411
$552$	0	0
$553$	40.1843	1.70881
$554$	−7.11586	−0.302324
$555$	0	0
$556$	−10.4805	−0.444473
$557$	−26.8657	−1.13833	−0.569167	−	0.822222i	$-0.692734\pi$
$557$	−26.8657	−1.13833	−0.569167	+	0.822222i	$0.692734\pi$
$558$	0	0
$559$	−2.82330	−0.119413
$560$	19.9044	0.841113
$561$	0	0
$562$	−19.2108	−0.810359
$563$	−41.8681	−1.76453	−0.882265	−	0.470753i	$-0.843982\pi$
$563$	−41.8681	−1.76453	−0.882265	+	0.470753i	$0.843982\pi$
$564$	0	0
$565$	−31.5963	−1.32927
$566$	31.0740	1.30614
$567$	0	0
$568$	−27.1183	−1.13786
$569$	26.3435	1.10438	0.552188	−	0.833720i	$-0.313793\pi$
$569$	26.3435	1.10438	0.552188	+	0.833720i	$0.313793\pi$
$570$	0	0
$571$	2.71518	0.113627	0.0568133	−	0.998385i	$-0.481906\pi$
$571$	2.71518	0.113627	0.0568133	+	0.998385i	$0.481906\pi$
$572$	2.77779	0.116145
$573$	0	0
$574$	−25.7787	−1.07598
$575$	−2.51457	−0.104865
$576$	0	0
$577$	−16.4988	−0.686855	−0.343427	−	0.939179i	$-0.611588\pi$
$577$	−16.4988	−0.686855	−0.343427	+	0.939179i	$0.611588\pi$
$578$	−13.9557	−0.580480
$579$	0	0
$580$	−5.12279	−0.212712
$581$	−2.75101	−0.114131
$582$	0	0
$583$	−16.7560	−0.693961
$584$	−48.5780	−2.01017
$585$	0	0
$586$	31.5168	1.30195
$587$	−36.5046	−1.50671	−0.753353	−	0.657616i	$-0.771565\pi$
$587$	−36.5046	−1.50671	−0.753353	+	0.657616i	$0.771565\pi$
$588$	0	0
$589$	2.85515	0.117644
$590$	25.8886	1.06582
$591$	0	0
$592$	16.0351	0.659040
$593$	−6.45481	−0.265067	−0.132534	−	0.991178i	$-0.542311\pi$
$593$	−6.45481	−0.265067	−0.132534	+	0.991178i	$0.542311\pi$
$594$	0	0
$595$	21.0737	0.863938
$596$	1.81928	0.0745207
$597$	0	0
$598$	−4.06010	−0.166030
$599$	32.5887	1.33154	0.665770	−	0.746157i	$-0.268103\pi$
$599$	32.5887	1.33154	0.665770	+	0.746157i	$0.268103\pi$
$600$	0	0
$601$	4.35546	0.177663	0.0888315	−	0.996047i	$-0.471687\pi$
$601$	4.35546	0.177663	0.0888315	+	0.996047i	$0.471687\pi$
$602$	−13.0547	−0.532070
$603$	0	0
$604$	4.47558	0.182109
$605$	−9.20735	−0.374332
$606$	0	0
$607$	−1.38827	−0.0563480	−0.0281740	−	0.999603i	$-0.508969\pi$
$607$	−1.38827	−0.0563480	−0.0281740	+	0.999603i	$0.508969\pi$
$608$	−1.09650	−0.0444690
$609$	0	0
$610$	20.7579	0.840463
$611$	8.42079	0.340669
$612$	0	0
$613$	−17.9268	−0.724056	−0.362028	−	0.932167i	$-0.617916\pi$
$613$	−17.9268	−0.724056	−0.362028	+	0.932167i	$0.617916\pi$
$614$	17.3822	0.701490
$615$	0	0
$616$	48.4890	1.95368
$617$	15.8744	0.639080	0.319540	−	0.947573i	$-0.396472\pi$
$617$	15.8744	0.639080	0.319540	+	0.947573i	$0.396472\pi$
$618$	0	0
$619$	14.0537	0.564865	0.282432	−	0.959287i	$-0.408859\pi$
$619$	14.0537	0.564865	0.282432	+	0.959287i	$0.408859\pi$
$620$	−17.2408	−0.692408
$621$	0	0
$622$	12.7524	0.511326
$623$	−13.3403	−0.534470
$624$	0	0
$625$	−28.0118	−1.12047
$626$	−12.2383	−0.489142
$627$	0	0
$628$	1.93039	0.0770311
$629$	16.9772	0.676924
$630$	0	0
$631$	3.10018	0.123416	0.0617081	−	0.998094i	$-0.480345\pi$
$631$	3.10018	0.123416	0.0617081	+	0.998094i	$0.480345\pi$
$632$	−30.2485	−1.20322
$633$	0	0
$634$	18.6024	0.738796
$635$	−45.2425	−1.79539
$636$	0	0
$637$	9.71246	0.384822
$638$	12.9793	0.513856
$639$	0	0
$640$	−4.39284	−0.173642
$641$	5.47888	0.216403	0.108201	−	0.994129i	$-0.465491\pi$
$641$	5.47888	0.216403	0.108201	+	0.994129i	$0.465491\pi$
$642$	0	0
$643$	21.8354	0.861104	0.430552	−	0.902566i	$-0.358319\pi$
$643$	21.8354	0.861104	0.430552	+	0.902566i	$0.358319\pi$
$644$	10.5758	0.416743
$645$	0	0
$646$	0.695864	0.0273784
$647$	0.921997	0.0362474	0.0181237	−	0.999836i	$-0.494231\pi$
$647$	0.921997	0.0362474	0.0181237	+	0.999836i	$0.494231\pi$
$648$	0	0
$649$	36.9504	1.45043
$650$	0.792328	0.0310777
$651$	0	0
$652$	0.114297	0.00447622
$653$	4.46161	0.174596	0.0872981	−	0.996182i	$-0.472177\pi$
$653$	4.46161	0.174596	0.0872981	+	0.996182i	$0.472177\pi$
$654$	0	0
$655$	−41.9322	−1.63843
$656$	11.3690	0.443885
$657$	0	0
$658$	38.9370	1.51792
$659$	10.1177	0.394129	0.197065	−	0.980391i	$-0.436859\pi$
$659$	10.1177	0.394129	0.197065	+	0.980391i	$0.436859\pi$
$660$	0	0
$661$	49.9834	1.94413	0.972065	−	0.234711i	$-0.0754143\pi$
$661$	49.9834	1.94413	0.972065	+	0.234711i	$0.0754143\pi$
$662$	6.47219	0.251549
$663$	0	0
$664$	2.07080	0.0803628
$665$	2.78130	0.107854
$666$	0	0
$667$	10.6869	0.413800
$668$	−7.72181	−0.298766
$669$	0	0
$670$	−3.94689	−0.152482
$671$	29.6274	1.14375
$672$	0	0
$673$	28.4054	1.09495	0.547474	−	0.836823i	$-0.315590\pi$
$673$	28.4054	1.09495	0.547474	+	0.836823i	$0.315590\pi$
$674$	−5.37452	−0.207019
$675$	0	0
$676$	−0.720682	−0.0277185
$677$	46.1800	1.77484	0.887421	−	0.460959i	$-0.152495\pi$
$677$	46.1800	1.77484	0.887421	+	0.460959i	$0.152495\pi$
$678$	0	0
$679$	−36.2931	−1.39280
$680$	−15.8631	−0.608322
$681$	0	0
$682$	43.6820	1.67267
$683$	−6.42699	−0.245922	−0.122961	−	0.992412i	$-0.539239\pi$
$683$	−6.42699	−0.245922	−0.122961	+	0.992412i	$0.539239\pi$
$684$	0	0
$685$	−1.82935	−0.0698957
$686$	12.5422	0.478862
$687$	0	0
$688$	5.75743	0.219500
$689$	4.34724	0.165617
$690$	0	0
$691$	−34.3429	−1.30647	−0.653233	−	0.757157i	$-0.726588\pi$
$691$	−34.3429	−1.30647	−0.653233	+	0.757157i	$0.726588\pi$
$692$	−7.32741	−0.278546
$693$	0	0
$694$	−12.3642	−0.469340
$695$	−34.7213	−1.31705
$696$	0	0
$697$	12.0369	0.455930
$698$	−29.5289	−1.11768
$699$	0	0
$700$	−2.06386	−0.0780065
$701$	−9.36636	−0.353763	−0.176881	−	0.984232i	$-0.556601\pi$
$701$	−9.36636	−0.353763	−0.176881	+	0.984232i	$0.556601\pi$
$702$	0	0
$703$	2.24064	0.0845074
$704$	−32.4960	−1.22474
$705$	0	0
$706$	13.0019	0.489332
$707$	−49.8214	−1.87373
$708$	0	0
$709$	1.85819	0.0697859	0.0348929	−	0.999391i	$-0.488891\pi$
$709$	1.85819	0.0697859	0.0348929	+	0.999391i	$0.488891\pi$
$710$	−23.7981	−0.893127
$711$	0	0
$712$	10.0419	0.376334
$713$	35.9671	1.34698
$714$	0	0
$715$	9.20266	0.344160
$716$	4.57430	0.170950
$717$	0	0
$718$	−8.39588	−0.313331
$719$	39.6078	1.47712	0.738561	−	0.674186i	$-0.235506\pi$
$719$	39.6078	1.47712	0.738561	+	0.674186i	$0.235506\pi$
$720$	0	0
$721$	−52.0566	−1.93869
$722$	−21.3985	−0.796369
$723$	0	0
$724$	5.19120	0.192929
$725$	−2.08556	−0.0774557
$726$	0	0
$727$	−23.8623	−0.885004	−0.442502	−	0.896768i	$-0.645909\pi$
$727$	−23.8623	−0.885004	−0.442502	+	0.896768i	$0.645909\pi$
$728$	−12.5802	−0.466253
$729$	0	0
$730$	−42.6303	−1.57782
$731$	6.09567	0.225457
$732$	0	0
$733$	−29.6604	−1.09553	−0.547767	−	0.836631i	$-0.684522\pi$
$733$	−29.6604	−1.09553	−0.547767	+	0.836631i	$0.684522\pi$
$734$	−6.77681	−0.250137
$735$	0	0
$736$	−13.8129	−0.509151
$737$	−5.63333	−0.207506
$738$	0	0
$739$	−5.87088	−0.215964	−0.107982	−	0.994153i	$-0.534439\pi$
$739$	−5.87088	−0.215964	−0.107982	+	0.994153i	$0.534439\pi$
$740$	−13.5301	−0.497377
$741$	0	0
$742$	20.1012	0.737939
$743$	9.08073	0.333140	0.166570	−	0.986030i	$-0.446731\pi$
$743$	9.08073	0.333140	0.166570	+	0.986030i	$0.446731\pi$
$744$	0	0
$745$	6.02717	0.220819
$746$	30.2196	1.10642
$747$	0	0
$748$	−5.99741	−0.219287
$749$	−27.9417	−1.02097
$750$	0	0
$751$	−54.1040	−1.97428	−0.987142	−	0.159844i	$-0.948901\pi$
$751$	−54.1040	−1.97428	−0.987142	+	0.159844i	$0.948901\pi$
$752$	−17.1721	−0.626203
$753$	0	0
$754$	−3.36741	−0.122634
$755$	14.8273	0.539621
$756$	0	0
$757$	42.6779	1.55116	0.775578	−	0.631252i	$-0.217459\pi$
$757$	42.6779	1.55116	0.775578	+	0.631252i	$0.217459\pi$
$758$	−19.6739	−0.714589
$759$	0	0
$760$	−2.09361	−0.0759431
$761$	1.91050	0.0692557	0.0346278	−	0.999400i	$-0.488975\pi$
$761$	1.91050	0.0692557	0.0346278	+	0.999400i	$0.488975\pi$
$762$	0	0
$763$	−37.0615	−1.34172
$764$	2.62005	0.0947899
$765$	0	0
$766$	−39.1512	−1.41459
$767$	−9.58656	−0.346151
$768$	0	0
$769$	−20.3581	−0.734132	−0.367066	−	0.930195i	$-0.619638\pi$
$769$	−20.3581	−0.734132	−0.367066	+	0.930195i	$0.619638\pi$
$770$	42.5523	1.53348
$771$	0	0
$772$	−2.81974	−0.101485
$773$	37.6497	1.35417	0.677083	−	0.735906i	$-0.263244\pi$
$773$	37.6497	1.35417	0.677083	+	0.735906i	$0.263244\pi$
$774$	0	0
$775$	−7.01897	−0.252129
$776$	27.3194	0.980709
$777$	0	0
$778$	−40.4292	−1.44946
$779$	1.58863	0.0569185
$780$	0	0
$781$	−33.9666	−1.21542
$782$	8.76598	0.313471
$783$	0	0
$784$	−19.8062	−0.707363
$785$	6.39528	0.228257
$786$	0	0
$787$	−33.5826	−1.19709	−0.598545	−	0.801089i	$-0.704254\pi$
$787$	−33.5826	−1.19709	−0.598545	+	0.801089i	$0.704254\pi$
$788$	4.73253	0.168589
$789$	0	0
$790$	−26.5450	−0.944429
$791$	54.1002	1.92358
$792$	0	0
$793$	−7.68666	−0.272961
$794$	15.4655	0.548851
$795$	0	0
$796$	1.40823	0.0499135
$797$	7.08855	0.251089	0.125545	−	0.992088i	$-0.459932\pi$
$797$	7.08855	0.251089	0.125545	+	0.992088i	$0.459932\pi$
$798$	0	0
$799$	−18.1810	−0.643196
$800$	2.69559	0.0953035
$801$	0	0
$802$	37.9885	1.34142
$803$	−60.8455	−2.14719
$804$	0	0
$805$	35.0368	1.23489
$806$	−11.3330	−0.399190
$807$	0	0
$808$	37.5028	1.31934
$809$	9.32644	0.327900	0.163950	−	0.986469i	$-0.447576\pi$
$809$	9.32644	0.327900	0.163950	+	0.986469i	$0.447576\pi$
$810$	0	0
$811$	−11.0677	−0.388638	−0.194319	−	0.980938i	$-0.562250\pi$
$811$	−11.0677	−0.388638	−0.194319	+	0.980938i	$0.562250\pi$
$812$	8.77142	0.307817
$813$	0	0
$814$	34.2805	1.20153
$815$	0.378659	0.0132638
$816$	0	0
$817$	0.804506	0.0281461
$818$	3.31916	0.116052
$819$	0	0
$820$	−9.59293	−0.335000
$821$	29.2074	1.01934	0.509672	−	0.860369i	$-0.329767\pi$
$821$	29.2074	1.01934	0.509672	+	0.860369i	$0.329767\pi$
$822$	0	0
$823$	12.0246	0.419151	0.209576	−	0.977792i	$-0.432792\pi$
$823$	12.0246	0.419151	0.209576	+	0.977792i	$0.432792\pi$
$824$	39.1852	1.36508
$825$	0	0
$826$	−44.3274	−1.54235
$827$	−19.0243	−0.661540	−0.330770	−	0.943711i	$-0.607308\pi$
$827$	−19.0243	−0.661540	−0.330770	+	0.943711i	$0.607308\pi$
$828$	0	0
$829$	30.3433	1.05387	0.526934	−	0.849906i	$-0.323342\pi$
$829$	30.3433	1.05387	0.526934	+	0.849906i	$0.323342\pi$
$830$	1.81726	0.0630782
$831$	0	0
$832$	8.43089	0.292288
$833$	−20.9697	−0.726558
$834$	0	0
$835$	−25.5819	−0.885297
$836$	−0.791537	−0.0273759
$837$	0	0
$838$	9.65175	0.333414
$839$	−6.03470	−0.208341	−0.104171	−	0.994559i	$-0.533219\pi$
$839$	−6.03470	−0.208341	−0.104171	+	0.994559i	$0.533219\pi$
$840$	0	0
$841$	−20.1364	−0.694357
$842$	−24.7115	−0.851615
$843$	0	0
$844$	14.5698	0.501513
$845$	−2.38757	−0.0821351
$846$	0	0
$847$	15.7651	0.541697
$848$	−8.86512	−0.304429
$849$	0	0
$850$	−1.71068	−0.0586759
$851$	28.2260	0.967574
$852$	0	0
$853$	−23.7192	−0.812130	−0.406065	−	0.913844i	$-0.633099\pi$
$853$	−23.7192	−0.812130	−0.406065	+	0.913844i	$0.633099\pi$
$854$	−35.5424	−1.21624
$855$	0	0
$856$	21.0329	0.718890
$857$	29.4814	1.00707	0.503533	−	0.863976i	$-0.332033\pi$
$857$	29.4814	1.00707	0.503533	+	0.863976i	$0.332033\pi$
$858$	0	0
$859$	−28.5939	−0.975610	−0.487805	−	0.872953i	$-0.662202\pi$
$859$	−28.5939	−0.975610	−0.487805	+	0.872953i	$0.662202\pi$
$860$	−4.85801	−0.165657
$861$	0	0
$862$	−39.7718	−1.35463
$863$	−48.3938	−1.64734	−0.823672	−	0.567066i	$-0.808078\pi$
$863$	−48.3938	−1.64734	−0.823672	+	0.567066i	$0.808078\pi$
$864$	0	0
$865$	−24.2753	−0.825384
$866$	28.2124	0.958696
$867$	0	0
$868$	29.5203	1.00199
$869$	−37.8872	−1.28524
$870$	0	0
$871$	1.46153	0.0495222
$872$	27.8978	0.944739
$873$	0	0
$874$	1.15693	0.0391338
$875$	41.9656	1.41870
$876$	0	0
$877$	−26.5517	−0.896588	−0.448294	−	0.893886i	$-0.647968\pi$
$877$	−26.5517	−0.896588	−0.448294	+	0.893886i	$0.647968\pi$
$878$	15.2998	0.516344
$879$	0	0
$880$	−18.7665	−0.632620
$881$	14.6581	0.493843	0.246921	−	0.969036i	$-0.420581\pi$
$881$	14.6581	0.493843	0.246921	+	0.969036i	$0.420581\pi$
$882$	0	0
$883$	−37.2159	−1.25241	−0.626207	−	0.779657i	$-0.715394\pi$
$883$	−37.2159	−1.25241	−0.626207	+	0.779657i	$0.715394\pi$
$884$	1.55599	0.0523337
$885$	0	0
$886$	−37.8428	−1.27135
$887$	2.62329	0.0880816	0.0440408	−	0.999030i	$-0.485977\pi$
$887$	2.62329	0.0880816	0.0440408	+	0.999030i	$0.485977\pi$
$888$	0	0
$889$	77.4657	2.59812
$890$	8.81238	0.295392
$891$	0	0
$892$	−4.50928	−0.150982
$893$	−2.39952	−0.0802968
$894$	0	0
$895$	15.1544	0.506555
$896$	7.52157	0.251278
$897$	0	0
$898$	8.64132	0.288364
$899$	29.8307	0.994910
$900$	0	0
$901$	−9.38593	−0.312691
$902$	24.3050	0.809269
$903$	0	0
$904$	−40.7236	−1.35445
$905$	17.1981	0.571685
$906$	0	0
$907$	−35.4590	−1.17740	−0.588698	−	0.808353i	$-0.700359\pi$
$907$	−35.4590	−1.17740	−0.588698	+	0.808353i	$0.700359\pi$
$908$	5.21834	0.173177
$909$	0	0
$910$	−11.0399	−0.365970
$911$	54.8208	1.81629	0.908147	−	0.418652i	$-0.137497\pi$
$911$	54.8208	1.81629	0.908147	+	0.418652i	$0.137497\pi$
$912$	0	0
$913$	2.59375	0.0858406
$914$	15.9835	0.528686
$915$	0	0
$916$	1.63704	0.0540893
$917$	71.7977	2.37097
$918$	0	0
$919$	23.5096	0.775511	0.387755	−	0.921762i	$-0.373251\pi$
$919$	23.5096	0.775511	0.387755	+	0.921762i	$0.373251\pi$
$920$	−26.3737	−0.869516
$921$	0	0
$922$	5.23480	0.172399
$923$	8.81244	0.290065
$924$	0	0
$925$	−5.50830	−0.181112
$926$	20.2308	0.664826
$927$	0	0
$928$	−11.4563	−0.376071
$929$	35.5388	1.16599	0.582994	−	0.812476i	$-0.301881\pi$
$929$	35.5388	1.16599	0.582994	+	0.812476i	$0.301881\pi$
$930$	0	0
$931$	−2.76758	−0.0907038
$932$	−10.0690	−0.329821
$933$	0	0
$934$	18.8161	0.615682
$935$	−19.8690	−0.649787
$936$	0	0
$937$	29.5466	0.965246	0.482623	−	0.875828i	$-0.339684\pi$
$937$	29.5466	0.965246	0.482623	+	0.875828i	$0.339684\pi$
$938$	6.75800	0.220656
$939$	0	0
$940$	14.4895	0.472595
$941$	4.96214	0.161761	0.0808806	−	0.996724i	$-0.474227\pi$
$941$	4.96214	0.161761	0.0808806	+	0.996724i	$0.474227\pi$
$942$	0	0
$943$	20.0124	0.651692
$944$	19.5494	0.636280
$945$	0	0
$946$	12.3084	0.400182
$947$	−25.9473	−0.843173	−0.421587	−	0.906788i	$-0.638527\pi$
$947$	−25.9473	−0.843173	−0.421587	+	0.906788i	$0.638527\pi$
$948$	0	0
$949$	15.7860	0.512436
$950$	−0.225775	−0.00732512
$951$	0	0
$952$	27.1613	0.880304
$953$	−17.5273	−0.567765	−0.283883	−	0.958859i	$-0.591623\pi$
$953$	−17.5273	−0.567765	−0.283883	+	0.958859i	$0.591623\pi$
$954$	0	0
$955$	8.68005	0.280880
$956$	19.1189	0.618350
$957$	0	0
$958$	−42.6124	−1.37675
$959$	3.13227	0.101146
$960$	0	0
$961$	69.3956	2.23857
$962$	−8.89387	−0.286750
$963$	0	0
$964$	2.88352	0.0928719
$965$	−9.34164	−0.300718
$966$	0	0
$967$	37.6528	1.21083	0.605417	−	0.795909i	$-0.293007\pi$
$967$	37.6528	1.21083	0.605417	+	0.795909i	$0.293007\pi$
$968$	−11.8671	−0.381423
$969$	0	0
$970$	23.9745	0.769776
$971$	−4.27036	−0.137042	−0.0685211	−	0.997650i	$-0.521828\pi$
$971$	−4.27036	−0.137042	−0.0685211	+	0.997650i	$0.521828\pi$
$972$	0	0
$973$	59.4510	1.90591
$974$	−38.7265	−1.24088
$975$	0	0
$976$	15.6750	0.501746
$977$	−4.25996	−0.136288	−0.0681440	−	0.997675i	$-0.521708\pi$
$977$	−4.25996	−0.136288	−0.0681440	+	0.997675i	$0.521708\pi$
$978$	0	0
$979$	12.5778	0.401987
$980$	16.7121	0.533847
$981$	0	0
$982$	−33.8360	−1.07975
$983$	1.11606	0.0355968	0.0177984	−	0.999842i	$-0.494334\pi$
$983$	1.11606	0.0355968	0.0177984	+	0.999842i	$0.494334\pi$
$984$	0	0
$985$	15.6786	0.499561
$986$	7.27042	0.231537
$987$	0	0
$988$	0.205360	0.00653336
$989$	10.1346	0.322261
$990$	0	0
$991$	−25.3057	−0.803862	−0.401931	−	0.915670i	$-0.631661\pi$
$991$	−25.3057	−0.803862	−0.401931	+	0.915670i	$0.631661\pi$
$992$	−38.5563	−1.22416
$993$	0	0
$994$	40.7479	1.29245
$995$	4.66539	0.147903
$996$	0	0
$997$	16.6418	0.527052	0.263526	−	0.964652i	$-0.415114\pi$
$997$	16.6418	0.527052	0.263526	+	0.964652i	$0.415114\pi$
$998$	−23.7053	−0.750378
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1053.2.a.j.1.4		5
3.2	odd	2		1053.2.a.k.1.2		5
9.2	odd	6		117.2.e.b.40.4	✓	10
9.4	even	3		351.2.e.b.235.2		10
9.5	odd	6		117.2.e.b.79.4	yes	10
9.7	even	3		351.2.e.b.118.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.e.b.40.4	✓	10	9.2	odd	6
117.2.e.b.79.4	yes	10	9.5	odd	6
351.2.e.b.118.2		10	9.7	even	3
351.2.e.b.235.2		10	9.4	even	3
1053.2.a.j.1.4		5	1.1	even	1	trivial
1053.2.a.k.1.2		5	3.2	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:	\( N \)	\(=\)	\( 1053 = 3^{4} \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1053.a (trivial)

\(f(q)\)	\(=\)	\(q+1.13107 q^{2} -0.720682 q^{4} -2.38757 q^{5} +4.08809 q^{7} -3.07728 q^{8} -2.70051 q^{10} -3.85439 q^{11} +1.00000 q^{13} +4.62391 q^{14} -2.03925 q^{16} -2.15906 q^{17} -0.284952 q^{19} +1.72068 q^{20} -4.35959 q^{22} -3.58961 q^{23} +0.700513 q^{25} +1.13107 q^{26} -2.94621 q^{28} -2.97719 q^{29} -10.0198 q^{31} +3.84802 q^{32} -2.44204 q^{34} -9.76061 q^{35} -7.86324 q^{37} -0.322300 q^{38} +7.34724 q^{40} -5.57508 q^{41} -2.82330 q^{43} +2.77779 q^{44} -4.06010 q^{46} +8.42079 q^{47} +9.71246 q^{49} +0.792328 q^{50} -0.720682 q^{52} +4.34724 q^{53} +9.20266 q^{55} -12.5802 q^{56} -3.36741 q^{58} -9.58656 q^{59} -7.68666 q^{61} -11.3330 q^{62} +8.43089 q^{64} -2.38757 q^{65} +1.46153 q^{67} +1.55599 q^{68} -11.0399 q^{70} +8.81244 q^{71} +15.7860 q^{73} -8.89387 q^{74} +0.205360 q^{76} -15.7571 q^{77} +9.82962 q^{79} +4.86887 q^{80} -6.30580 q^{82} -0.672933 q^{83} +5.15491 q^{85} -3.19335 q^{86} +11.8611 q^{88} -3.26322 q^{89} +4.08809 q^{91} +2.58697 q^{92} +9.52450 q^{94} +0.680343 q^{95} -8.87777 q^{97} +10.9855 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q - 2 q^{2} + 4 q^{4} - q^{5} - 2 q^{7} - 12 q^{8} - 2 q^{10} - 11 q^{11} + 5 q^{13} + 5 q^{14} + 10 q^{16} - 7 q^{17} + 3 q^{19} + q^{20} - 7 q^{22} - 18 q^{23} - 8 q^{25} - 2 q^{26} - 19 q^{28} - 4 q^{29}+ \cdots + 39 q^{98}+O(q^{100}) \) 5 * q - 2 * q^2 + 4 * q^4 - q^5 - 2 * q^7 - 12 * q^8 - 2 * q^10 - 11 * q^11 + 5 * q^13 + 5 * q^14 + 10 * q^16 - 7 * q^17 + 3 * q^19 + q^20 - 7 * q^22 - 18 * q^23 - 8 * q^25 - 2 * q^26 - 19 * q^28 - 4 * q^29 - 16 * q^31 - 31 * q^32 + 13 * q^34 - 11 * q^35 + 5 * q^37 - 24 * q^38 + 18 * q^40 - 12 * q^41 - q^44 + 6 * q^46 - 15 * q^47 + 3 * q^49 + 5 * q^50 + 4 * q^52 + 3 * q^53 - 17 * q^55 + 24 * q^56 - 2 * q^58 - 3 * q^59 - q^61 + 4 * q^62 + 62 * q^64 - q^65 - 4 * q^67 + 16 * q^68 - 25 * q^70 - 15 * q^71 - 5 * q^73 - 11 * q^74 + 9 * q^76 - q^77 + 13 * q^79 - 32 * q^80 - 21 * q^82 - 26 * q^83 - 7 * q^85 + 6 * q^86 + 3 * q^88 - 29 * q^89 - 2 * q^91 + 6 * q^92 + 12 * q^94 - 12 * q^95 - 16 * q^97 + 39 * q^98

Introduction

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