LMFDB - Embedded newform 1053.2.b.j.649.6

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [1053,2,Mod(649,1053)] 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("1053.649"); 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, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 1053 = 3^{4} \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1053.b (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [10,0,0,-12,0,0,0,0,0,-8,0,0,4,18,0,-4,-6] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(17)] == traces)

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

Self dual:	no
Analytic conductor:	$8.40824733284$
Analytic rank:	$0$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} + \cdots)$
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} + 16x^{8} + 91x^{6} + 222x^{4} + 228x^{2} + 81 $ x^10 + 16x^8 + 91x^6 + 222x^4 + 228x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 3^{3} $
Twist minimal:	no (minimal twist has level 117)
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			649.6
Root			$0.905597i$ of defining polynomial
Character	$\chi$	$=$	1053.649
Dual form			1053.2.b.j.649.5

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.905597i q^{2} +1.17989 q^{4} +2.24306i q^{5} +3.43611i q^{7} +2.87970i q^{8} -2.03130 q^{10} -3.69657i q^{11} +(2.93184 + 2.09865i) q^{13} -3.11173 q^{14} -0.248059 q^{16} -4.21120 q^{17} +4.25298i q^{19} +2.64657i q^{20} +3.34760 q^{22} -3.78325 q^{23} -0.0313048 q^{25} +(-1.90053 + 2.65506i) q^{26} +4.05425i q^{28} +2.37891 q^{29} -7.35733i q^{31} +5.53476i q^{32} -3.81365i q^{34} -7.70739 q^{35} -5.49928i q^{37} -3.85149 q^{38} -6.45934 q^{40} -7.92223i q^{41} +0.900531 q^{43} -4.36157i q^{44} -3.42609i q^{46} +5.54325i q^{47} -4.80686 q^{49} -0.0283495i q^{50} +(3.45926 + 2.47619i) q^{52} +7.59566 q^{53} +8.29163 q^{55} -9.89498 q^{56} +2.15433i q^{58} +5.13125i q^{59} -13.0181 q^{61} +6.66277 q^{62} -5.50838 q^{64} +(-4.70739 + 6.57628i) q^{65} +13.5102i q^{67} -4.96877 q^{68} -6.97979i q^{70} +2.65506i q^{71} -5.45741i q^{73} +4.98013 q^{74} +5.01807i q^{76} +12.7018 q^{77} +10.9377 q^{79} -0.556410i q^{80} +7.17434 q^{82} -0.537568i q^{83} -9.44596i q^{85} +0.815518i q^{86} +10.6450 q^{88} +5.75227i q^{89} +(-7.21120 + 10.0741i) q^{91} -4.46383 q^{92} -5.01995 q^{94} -9.53968 q^{95} +6.78485i q^{97} -4.35308i q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} + 18 q^{14} - 4 q^{16} - 6 q^{17} + 10 q^{22} - 24 q^{23} + 12 q^{25} - 6 q^{26} - 12 q^{29} - 6 q^{35} - 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} + 54 q^{53} + 10 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) $ 10 * q - 12 * q^4 - 8 * q^10 + 4 * q^13 + 18 * q^14 - 4 * q^16 - 6 * q^17 + 10 * q^22 - 24 * q^23 + 12 * q^25 - 6 * q^26 - 12 * q^29 - 6 * q^35 - 12 * q^38 + 8 * q^40 - 4 * q^43 + 10 * q^49 + 54 * q^53 + 10 * q^55 - 36 * q^56 + 2 * q^61 - 36 * q^62 + 4 * q^64 + 24 * q^65 - 24 * q^68 + 42 * q^74 + 6 * q^77 + 14 * q^79 - 2 * q^82 - 22 * q^88 - 36 * q^91 + 84 * q^92 - 20 * q^94 - 24 * q^95

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/1053\mathbb{Z}\right)^\times$.

$n$	$326$	$730$
$\chi(n)$	$1$	$-1$

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 $

$2$			0.905597i			0.640353i	0.947358	+	0.320177i	$0.103742\pi$
$2$			0.905597i			0.640353i	−0.947358	+	0.320177i	$0.896258\pi$
$3$		0			0
$3$		0			0
$4$	1.17989			0.589947
$5$			2.24306i			1.00313i	0.865121	+	0.501563i	$0.167241\pi$
$5$			2.24306i			1.00313i	−0.865121	+	0.501563i	$0.832759\pi$
$6$		0			0
$7$			3.43611i			1.29873i	0.760478	+	0.649364i	$0.224965\pi$
$7$			3.43611i			1.29873i	−0.760478	+	0.649364i	$0.775035\pi$
$8$			2.87970i			1.01813i
$9$		0			0
$10$	−2.03130			−0.642355
$11$		−	3.69657i		−	1.11456i	−0.830325	−	0.557279i	$-0.811845\pi$
$11$		−	3.69657i		−	1.11456i	0.830325	−	0.557279i	$-0.188155\pi$
$12$		0			0
$13$	2.93184	+	2.09865i	0.813145	+	0.582061i
$13$	2.93184	+	2.09865i	0.813145	+	0.582061i
$14$	−3.11173			−0.831645
$15$		0			0
$16$	−0.248059			−0.0620147
$17$	−4.21120			−1.02137			−0.510683	−	0.859769i	$-0.670607\pi$
$17$	−4.21120			−1.02137			−0.510683	+	0.859769i	$0.670607\pi$
$18$		0			0
$19$			4.25298i			0.975701i	0.872927	+	0.487851i	$0.162219\pi$
$19$			4.25298i			0.975701i	−0.872927	+	0.487851i	$0.837781\pi$
$20$			2.64657i			0.591791i
$21$		0			0
$22$	3.34760			0.713712
$23$	−3.78325			−0.788861			−0.394431	−	0.918926i	$-0.629058\pi$
$23$	−3.78325			−0.788861			−0.394431	+	0.918926i	$0.629058\pi$
$24$		0			0
$25$	−0.0313048			−0.00626096
$26$	−1.90053	+	2.65506i	−0.372725	+	0.520700i
$27$		0			0
$28$			4.05425i			0.766181i
$29$	2.37891			0.441752			0.220876	−	0.975302i	$-0.429108\pi$
$29$	2.37891			0.441752			0.220876	+	0.975302i	$0.429108\pi$
$30$		0			0
$31$		−	7.35733i		−	1.32141i	−0.750644	−	0.660707i	$-0.770256\pi$
$31$		−	7.35733i		−	1.32141i	0.750644	−	0.660707i	$-0.229744\pi$
$32$			5.53476i			0.978417i
$33$		0			0
$34$		−	3.81365i		−	0.654035i
$35$	−7.70739			−1.30279
$36$		0			0
$37$		−	5.49928i		−	0.904076i	−0.891999	−	0.452038i	$-0.850697\pi$
$37$		−	5.49928i		−	0.904076i	0.891999	−	0.452038i	$-0.149303\pi$
$38$	−3.85149			−0.624794
$39$		0			0
$40$	−6.45934			−1.02131
$41$		−	7.92223i		−	1.23724i	−0.785689	−	0.618622i	$-0.787691\pi$
$41$		−	7.92223i		−	1.23724i	0.785689	−	0.618622i	$-0.212309\pi$
$42$		0			0
$43$	0.900531			0.137330			0.0686649	−	0.997640i	$-0.478126\pi$
$43$	0.900531			0.137330			0.0686649	+	0.997640i	$0.478126\pi$
$44$		−	4.36157i		−	0.657531i
$45$		0			0
$46$		−	3.42609i		−	0.505150i
$47$			5.54325i			0.808566i	0.914634	+	0.404283i	$0.132479\pi$
$47$			5.54325i			0.808566i	−0.914634	+	0.404283i	$0.867521\pi$
$48$		0			0
$49$	−4.80686			−0.686695
$50$		−	0.0283495i		−	0.00400923i
$51$		0			0
$52$	3.45926	+	2.47619i	0.479713	+	0.343385i
$53$	7.59566			1.04334			0.521672	−	0.853146i	$-0.325308\pi$
$53$	7.59566			1.04334			0.521672	+	0.853146i	$0.325308\pi$
$54$		0			0
$55$	8.29163			1.11804
$56$	−9.89498			−1.32227
$57$		0			0
$58$			2.15433i			0.282878i
$59$			5.13125i			0.668032i	0.942567	+	0.334016i	$0.108404\pi$
$59$			5.13125i			0.668032i	−0.942567	+	0.334016i	$0.891596\pi$
$60$		0			0
$61$	−13.0181			−1.66680			−0.833401	−	0.552669i	$-0.813609\pi$
$61$	−13.0181			−1.66680			−0.833401	+	0.552669i	$0.813609\pi$
$62$	6.66277			0.846173
$63$		0			0
$64$	−5.50838			−0.688547
$65$	−4.70739	+	6.57628i	−0.583880	+	0.815687i
$66$		0			0
$67$			13.5102i			1.65054i	0.564741	+	0.825269i	$0.308976\pi$
$67$			13.5102i			1.65054i	−0.564741	+	0.825269i	$0.691024\pi$
$68$	−4.96877			−0.602552
$69$		0			0
$70$		−	6.97979i		−	0.834244i
$71$			2.65506i			0.315098i	0.987511	+	0.157549i	$0.0503592\pi$
$71$			2.65506i			0.315098i	−0.987511	+	0.157549i	$0.949641\pi$
$72$		0			0
$73$		−	5.45741i		−	0.638741i	−0.947630	−	0.319371i	$-0.896528\pi$
$73$		−	5.45741i		−	0.638741i	0.947630	−	0.319371i	$-0.103472\pi$
$74$	4.98013			0.578928
$75$		0			0
$76$			5.01807i			0.575612i
$77$	12.7018			1.44751
$78$		0			0
$79$	10.9377			1.23059			0.615294	−	0.788297i	$-0.289037\pi$
$79$	10.9377			1.23059			0.615294	+	0.788297i	$0.289037\pi$
$80$		−	0.556410i		−	0.0622085i
$81$		0			0
$82$	7.17434			0.792273
$83$		−	0.537568i		−	0.0590057i	−0.999565	−	0.0295029i	$-0.990608\pi$
$83$		−	0.537568i		−	0.0590057i	0.999565	−	0.0295029i	$-0.00939241\pi$
$84$		0			0
$85$		−	9.44596i		−	1.02456i
$86$			0.815518i			0.0879396i
$87$		0			0
$88$	10.6450			1.13476
$89$			5.75227i			0.609739i	0.952394	+	0.304870i	$0.0986129\pi$
$89$			5.75227i			0.609739i	−0.952394	+	0.304870i	$0.901387\pi$
$90$		0			0
$91$	−7.21120	+	10.0741i	−0.755939	+	1.05605i
$92$	−4.46383			−0.465387
$93$		0			0
$94$	−5.01995			−0.517768
$95$	−9.53968			−0.978751
$96$		0			0
$97$			6.78485i			0.688897i	0.938805	+	0.344449i	$0.111934\pi$
$97$			6.78485i			0.688897i	−0.938805	+	0.344449i	$0.888066\pi$
$98$		−	4.35308i		−	0.439727i
$99$		0			0
$100$	−0.0369364			−0.00369364
$101$	−4.96231			−0.493768			−0.246884	−	0.969045i	$-0.579407\pi$
$101$	−4.96231			−0.493768			−0.246884	+	0.969045i	$0.579407\pi$
$102$		0			0
$103$	2.11173			0.208075			0.104038	−	0.994573i	$-0.466824\pi$
$103$	2.11173			0.208075			0.104038	+	0.994573i	$0.466824\pi$
$104$	−6.04349	+	8.44281i	−0.592613	+	0.827886i
$105$		0			0
$106$			6.87861i			0.668109i
$107$	−8.07404			−0.780547			−0.390274	−	0.920699i	$-0.627620\pi$
$107$	−8.07404			−0.780547			−0.390274	+	0.920699i	$0.627620\pi$
$108$		0			0
$109$		−	10.0020i		−	0.958021i	−0.877809	−	0.479011i	$-0.840996\pi$
$109$		−	10.0020i		−	0.958021i	0.877809	−	0.479011i	$-0.159004\pi$
$110$			7.50887i			0.715942i
$111$		0			0
$112$		−	0.852357i		−	0.0805402i
$113$	19.4347			1.82826			0.914130	−	0.405422i	$-0.132875\pi$
$113$	19.4347			1.82826			0.914130	+	0.405422i	$0.132875\pi$
$114$		0			0
$115$		−	8.48604i		−	0.791327i
$116$	2.80686			0.260611
$117$		0			0
$118$	−4.64684			−0.427776
$119$		−	14.4702i		−	1.32648i
$120$		0			0
$121$	−2.66466			−0.242242
$122$		−	11.7892i		−	1.06734i
$123$		0			0
$124$		−	8.68087i		−	0.779565i
$125$			11.1451i			0.996845i
$126$		0			0
$127$	−1.76413			−0.156541			−0.0782704	−	0.996932i	$-0.524940\pi$
$127$	−1.76413			−0.156541			−0.0782704	+	0.996932i	$0.524940\pi$
$128$			6.08116i			0.537503i
$129$		0			0
$130$	−5.95545	−	4.26300i	−0.522328	−	0.373890i
$131$	−7.12979			−0.622933			−0.311466	−	0.950257i	$-0.600820\pi$
$131$	−7.12979			−0.622933			−0.311466	+	0.950257i	$0.600820\pi$
$132$		0			0
$133$	−14.6137			−1.26717
$134$	−12.2348			−1.05693
$135$		0			0
$136$		−	12.1270i		−	1.03988i
$137$		−	4.52058i		−	0.386219i	−0.981177	−	0.193110i	$-0.938143\pi$
$137$		−	4.52058i		−	0.386219i	0.981177	−	0.193110i	$-0.0618573\pi$
$138$		0			0
$139$	14.1099			1.19679			0.598394	−	0.801202i	$-0.295806\pi$
$139$	14.1099			1.19679			0.598394	+	0.801202i	$0.295806\pi$
$140$	−9.09391			−0.768576
$141$		0			0
$142$	−2.40441			−0.201774
$143$	7.75782	−	10.8377i	0.648741	−	0.906298i
$144$		0			0
$145$			5.33603i			0.443133i
$146$	4.94221			0.409020
$147$		0			0
$148$		−	6.48857i		−	0.533357i
$149$		−	5.80236i		−	0.475348i	−0.971345	−	0.237674i	$-0.923615\pi$
$149$		−	5.80236i		−	0.475348i	0.971345	−	0.237674i	$-0.0763850\pi$
$150$		0			0
$151$		−	9.74588i		−	0.793108i	−0.918011	−	0.396554i	$-0.870206\pi$
$151$		−	9.74588i		−	0.793108i	0.918011	−	0.396554i	$-0.129794\pi$
$152$	−12.2473			−0.993389
$153$		0			0
$154$			11.5027i			0.926917i
$155$	16.5029			1.32555
$156$		0			0
$157$	−1.07960			−0.0861611			−0.0430806	−	0.999072i	$-0.513717\pi$
$157$	−1.07960			−0.0861611			−0.0430806	+	0.999072i	$0.513717\pi$
$158$			9.90516i			0.788012i
$159$		0			0
$160$	−12.4148			−0.981475
$161$		−	12.9997i		−	1.02452i
$162$		0			0
$163$			5.54857i			0.434598i	0.976105	+	0.217299i	$0.0697247\pi$
$163$			5.54857i			0.434598i	−0.976105	+	0.217299i	$0.930275\pi$
$164$		−	9.34739i		−	0.729909i
$165$		0			0
$166$	0.486819			0.0377845
$167$		−	13.3385i		−	1.03216i	−0.856539	−	0.516082i	$-0.827390\pi$
$167$		−	13.3385i		−	1.03216i	0.856539	−	0.516082i	$-0.172610\pi$
$168$		0			0
$169$	4.19133	+	12.3058i	0.322410	+	0.946600i
$170$	8.55423			0.656079
$171$		0			0
$172$	1.06253			0.0810173
$173$	7.44160			0.565774			0.282887	−	0.959153i	$-0.408708\pi$
$173$	7.44160			0.565774			0.282887	+	0.959153i	$0.408708\pi$
$174$		0			0
$175$		−	0.107567i		−	0.00813129i
$176$			0.916967i			0.0691190i
$177$		0			0
$178$	−5.20923			−0.390449
$179$	23.2047			1.73440			0.867202	−	0.497957i	$-0.165916\pi$
$179$	23.2047			1.73440			0.867202	+	0.497957i	$0.165916\pi$
$180$		0			0
$181$	5.18439			0.385353			0.192676	−	0.981262i	$-0.438283\pi$
$181$	5.18439			0.385353			0.192676	+	0.981262i	$0.438283\pi$
$182$	−9.12309	−	6.53044i	−0.676248	−	0.484068i
$183$		0			0
$184$		−	10.8946i		−	0.803162i
$185$	12.3352			0.906901
$186$		0			0
$187$			15.5670i			1.13837i
$188$			6.54045i			0.477012i
$189$		0			0
$190$		−	8.63911i		−	0.626747i
$191$	16.1411			1.16793			0.583966	−	0.811778i	$-0.301500\pi$
$191$	16.1411			1.16793			0.583966	+	0.811778i	$0.301500\pi$
$192$		0			0
$193$		−	3.27925i		−	0.236045i	−0.993011	−	0.118023i	$-0.962344\pi$
$193$		−	3.27925i		−	0.236045i	0.993011	−	0.118023i	$-0.0376556\pi$
$194$	−6.14434			−0.441138
$195$		0			0
$196$	−5.67159			−0.405114
$197$		−	7.11249i		−	0.506744i	−0.967369	−	0.253372i	$-0.918460\pi$
$197$		−	7.11249i		−	0.506744i	0.967369	−	0.253372i	$-0.0815396\pi$
$198$		0			0
$199$	−13.8449			−0.981437			−0.490719	−	0.871318i	$-0.663266\pi$
$199$	−13.8449			−0.981437			−0.490719	+	0.871318i	$0.663266\pi$
$200$		−	0.0901485i		−	0.00637446i
$201$		0			0
$202$		−	4.49385i		−	0.316186i
$203$			8.17420i			0.573716i
$204$		0			0
$205$	17.7700			1.24111
$206$			1.91238i			0.133242i
$207$		0			0
$208$	−0.727267	−	0.520589i	−0.0504269	−	0.0360963i
$209$	15.7215			1.08748
$210$		0			0
$211$	14.4339			0.993671			0.496836	−	0.867845i	$-0.334495\pi$
$211$	14.4339			0.993671			0.496836	+	0.867845i	$0.334495\pi$
$212$	8.96208			0.615518
$213$		0			0
$214$		−	7.31182i		−	0.499826i
$215$			2.01994i			0.137759i
$216$		0			0
$217$	25.2806			1.71616
$218$	9.05781			0.613472
$219$		0			0
$220$	9.78325			0.659586
$221$	−12.3465	−	8.83784i	−0.830519	−	0.594497i
$222$		0			0
$223$			11.0038i			0.736871i	0.929653	+	0.368435i	$0.120106\pi$
$223$			11.0038i			0.736871i	−0.929653	+	0.368435i	$0.879894\pi$
$224$	−19.0181			−1.27070
$225$		0			0
$226$			17.6000i			1.17073i
$227$		−	8.11818i		−	0.538823i	−0.963025	−	0.269411i	$-0.913171\pi$
$227$		−	8.11818i		−	0.538823i	0.963025	−	0.269411i	$-0.0868291\pi$
$228$		0			0
$229$			5.37790i			0.355382i	0.984086	+	0.177691i	$0.0568627\pi$
$229$			5.37790i			0.355382i	−0.984086	+	0.177691i	$0.943137\pi$
$230$	7.68493			0.506729
$231$		0			0
$232$			6.85055i			0.449761i
$233$	20.0992			1.31675			0.658373	−	0.752692i	$-0.271245\pi$
$233$	20.0992			1.31675			0.658373	+	0.752692i	$0.271245\pi$
$234$		0			0
$235$	−12.4338			−0.811094
$236$			6.05433i			0.394104i
$237$		0			0
$238$	13.1041			0.849414
$239$			14.1607i			0.915979i	0.888958	+	0.457989i	$0.151430\pi$
$239$			14.1607i			0.915979i	−0.888958	+	0.457989i	$0.848570\pi$
$240$		0			0
$241$		−	30.2578i		−	1.94908i	−0.224217	−	0.974539i	$-0.571982\pi$
$241$		−	30.2578i		−	1.94908i	0.224217	−	0.974539i	$-0.428018\pi$
$242$		−	2.41310i		−	0.155120i
$243$		0			0
$244$	−15.3600			−0.983326
$245$		−	10.7821i		−	0.688841i
$246$		0			0
$247$	−8.92553	+	12.4691i	−0.567918	+	0.793387i
$248$	21.1869			1.34537
$249$		0			0
$250$	−10.0929			−0.638333
$251$	−17.5085			−1.10512			−0.552562	−	0.833472i	$-0.686350\pi$
$251$	−17.5085			−1.10512			−0.552562	+	0.833472i	$0.686350\pi$
$252$		0			0
$253$			13.9850i			0.879232i
$254$		−	1.59759i		−	0.100242i
$255$		0			0
$256$	−16.5238			−1.03274
$257$	18.3042			1.14179			0.570893	−	0.821025i	$-0.306597\pi$
$257$	18.3042			1.14179			0.570893	+	0.821025i	$0.306597\pi$
$258$		0			0
$259$	18.8961			1.17415
$260$	−5.55423	+	7.75931i	−0.344459	+	0.481212i
$261$		0			0
$262$		−	6.45672i		−	0.398897i
$263$	−8.39323			−0.517549			−0.258774	−	0.965938i	$-0.583319\pi$
$263$	−8.39323			−0.517549			−0.258774	+	0.965938i	$0.583319\pi$
$264$		0			0
$265$			17.0375i			1.04661i
$266$		−	13.2341i		−	0.811437i
$267$		0			0
$268$			15.9406i			0.973730i
$269$	21.3238			1.30014			0.650069	−	0.759875i	$-0.274740\pi$
$269$	21.3238			1.30014			0.650069	+	0.759875i	$0.274740\pi$
$270$		0			0
$271$		−	18.4587i		−	1.12129i	−0.828057	−	0.560643i	$-0.810554\pi$
$271$		−	18.4587i		−	1.12129i	0.828057	−	0.560643i	$-0.189446\pi$
$272$	1.04462			0.0633397
$273$		0			0
$274$	4.09382			0.247317
$275$			0.115721i			0.00697821i
$276$		0			0
$277$	−8.62780			−0.518394			−0.259197	−	0.965824i	$-0.583458\pi$
$277$	−8.62780			−0.518394			−0.259197	+	0.965824i	$0.583458\pi$
$278$			12.7779i			0.766367i
$279$		0			0
$280$		−	22.1950i		−	1.32640i
$281$			3.47481i			0.207290i	0.994614	+	0.103645i	$0.0330506\pi$
$281$			3.47481i			0.207290i	−0.994614	+	0.103645i	$0.966949\pi$
$282$		0			0
$283$	−13.7836			−0.819348			−0.409674	−	0.912232i	$-0.634358\pi$
$283$	−13.7836			−0.819348			−0.409674	+	0.912232i	$0.634358\pi$
$284$			3.13269i			0.185891i
$285$		0			0
$286$	9.81463	+	7.02545i	0.580351	+	0.415424i
$287$	27.2217			1.60684
$288$		0			0
$289$	0.734202			0.0431884
$290$	−4.83229			−0.283762
$291$		0			0
$292$		−	6.43917i		−	0.376824i
$293$		−	28.5108i		−	1.66562i	−0.553559	−	0.832810i	$-0.686731\pi$
$293$		−	28.5108i		−	1.66562i	0.553559	−	0.832810i	$-0.313269\pi$
$294$		0			0
$295$	−11.5097			−0.670120
$296$	15.8363			0.920465
$297$		0			0
$298$	5.25460			0.304391
$299$	−11.0919	−	7.93971i	−0.641459	−	0.459166i
$300$		0			0
$301$			3.09433i			0.178354i
$302$	8.82583			0.507870
$303$		0			0
$304$		−	1.05499i		−	0.0605078i
$305$		−	29.2004i		−	1.67201i
$306$		0			0
$307$			21.8137i			1.24497i	0.782631	+	0.622486i	$0.213877\pi$
$307$			21.8137i			1.24497i	−0.782631	+	0.622486i	$0.786123\pi$
$308$	14.9868			0.853954
$309$		0			0
$310$			14.9450i			0.848817i
$311$	−6.96322			−0.394848			−0.197424	−	0.980318i	$-0.563258\pi$
$311$	−6.96322			−0.394848			−0.197424	+	0.980318i	$0.563258\pi$
$312$		0			0
$313$	22.4506			1.26898			0.634491	−	0.772930i	$-0.281210\pi$
$313$	22.4506			1.26898			0.634491	+	0.772930i	$0.281210\pi$
$314$		−	0.977678i		−	0.0551736i
$315$		0			0
$316$	12.9054			0.725983
$317$		−	10.2161i		−	0.573792i	−0.957962	−	0.286896i	$-0.907377\pi$
$317$		−	10.2161i		−	0.573792i	0.957962	−	0.286896i	$-0.0926235\pi$
$318$		0			0
$319$		−	8.79381i		−	0.492359i
$320$		−	12.3556i		−	0.690700i
$321$		0			0
$322$	11.7724			0.656053
$323$		−	17.9102i		−	0.996548i
$324$		0			0
$325$	−0.0917806	−	0.0656979i	−0.00509107	−	0.00364426i
$326$	−5.02477			−0.278296
$327$		0			0
$328$	22.8136			1.25967
$329$	−19.0472			−1.05011
$330$		0			0
$331$		−	5.40162i		−	0.296900i	−0.988920	−	0.148450i	$-0.952572\pi$
$331$		−	5.40162i		−	0.296900i	0.988920	−	0.148450i	$-0.0474284\pi$
$332$		−	0.634273i		−	0.0348103i
$333$		0			0
$334$	12.0793			0.660949
$335$	−30.3042			−1.65570
$336$		0			0
$337$	−32.9242			−1.79350			−0.896749	−	0.442540i	$-0.854078\pi$
$337$	−32.9242			−1.79350			−0.896749	+	0.442540i	$0.854078\pi$
$338$	−11.1441	+	3.79565i	−0.606159	+	0.206456i
$339$		0			0
$340$		−	11.1452i		−	0.604436i
$341$	−27.1969			−1.47279
$342$		0			0
$343$			7.53586i			0.406898i
$344$			2.59326i			0.139819i
$345$		0			0
$346$			6.73908i			0.362295i
$347$	13.6278			0.731579			0.365789	−	0.930698i	$-0.380799\pi$
$347$	13.6278			0.731579			0.365789	+	0.930698i	$0.380799\pi$
$348$		0			0
$349$		−	11.2746i		−	0.603518i	−0.953384	−	0.301759i	$-0.902426\pi$
$349$		−	11.2746i		−	0.603518i	0.953384	−	0.301759i	$-0.0975738\pi$
$350$	0.0974121			0.00520690
$351$		0			0
$352$	20.4597			1.09050
$353$			26.6308i			1.41741i	0.705504	+	0.708706i	$0.250721\pi$
$353$			26.6308i			1.41741i	−0.705504	+	0.708706i	$0.749279\pi$
$354$		0			0
$355$	−5.95545			−0.316083
$356$			6.78707i			0.359714i
$357$		0			0
$358$			21.0141i			1.11063i
$359$		−	34.9036i		−	1.84214i	−0.389396	−	0.921071i	$-0.627316\pi$
$359$		−	34.9036i		−	1.84214i	0.389396	−	0.921071i	$-0.372684\pi$
$360$		0			0
$361$	0.912132			0.0480069
$362$			4.69497i			0.246762i
$363$		0			0
$364$	−8.50846	+	11.8864i	−0.445964	+	0.623017i
$365$	12.2413			0.640738
$366$		0			0
$367$	−2.71433			−0.141687			−0.0708434	−	0.997487i	$-0.522569\pi$
$367$	−2.71433			−0.141687			−0.0708434	+	0.997487i	$0.522569\pi$
$368$	0.938467			0.0489210
$369$		0			0
$370$			11.1707i			0.580737i
$371$			26.0995i			1.35502i
$372$		0			0
$373$	−1.75096			−0.0906613			−0.0453307	−	0.998972i	$-0.514434\pi$
$373$	−1.75096			−0.0906613			−0.0453307	+	0.998972i	$0.514434\pi$
$374$	−14.0974			−0.728961
$375$		0			0
$376$	−15.9629			−0.823224
$377$	6.97457	+	4.99250i	0.359209	+	0.257127i
$378$		0			0
$379$			2.90941i			0.149446i	0.997204	+	0.0747231i	$0.0238073\pi$
$379$			2.90941i			0.149446i	−0.997204	+	0.0747231i	$0.976193\pi$
$380$	−11.2558			−0.577412
$381$		0			0
$382$			14.6174i			0.747890i
$383$			20.4457i			1.04473i	0.852723	+	0.522363i	$0.174950\pi$
$383$			20.4457i			1.04473i	−0.852723	+	0.522363i	$0.825050\pi$
$384$		0			0
$385$			28.4910i			1.45203i
$386$	2.96968			0.151152
$387$		0			0
$388$			8.00541i			0.406413i
$389$	12.9632			0.657261			0.328631	−	0.944459i	$-0.393413\pi$
$389$	12.9632			0.657261			0.328631	+	0.944459i	$0.393413\pi$
$390$		0			0
$391$	15.9320			0.805716
$392$		−	13.8423i		−	0.699143i
$393$		0			0
$394$	6.44105			0.324495
$395$			24.5339i			1.23444i
$396$		0			0
$397$		−	29.1809i		−	1.46455i	−0.681010	−	0.732274i	$-0.738459\pi$
$397$		−	29.1809i		−	1.46455i	0.681010	−	0.732274i	$-0.261541\pi$
$398$		−	12.5379i		−	0.628467i
$399$		0			0
$400$	0.00776543			0.000388271
$401$			22.1724i			1.10723i	0.832771	+	0.553617i	$0.186753\pi$
$401$			22.1724i			1.10723i	−0.832771	+	0.553617i	$0.813247\pi$
$402$		0			0
$403$	15.4405	−	21.5705i	0.769144	−	1.07450i
$404$	−5.85500			−0.291297
$405$		0			0
$406$	−7.40253			−0.367381
$407$	−20.3285			−1.00765
$408$		0			0
$409$		−	2.09865i		−	0.103772i	−0.998653	−	0.0518858i	$-0.983477\pi$
$409$		−	2.09865i		−	0.103772i	0.998653	−	0.0518858i	$-0.0165232\pi$
$410$			16.0925i			0.794750i
$411$		0			0
$412$	2.49162			0.122753
$413$	−17.6315			−0.867591
$414$		0			0
$415$	1.20579			0.0591901
$416$	−11.6155	+	16.2270i	−0.569499	+	0.795595i
$417$		0			0
$418$			14.2373i			0.696369i
$419$	0.391399			0.0191211			0.00956053	−	0.999954i	$-0.496957\pi$
$419$	0.391399			0.0191211			0.00956053	+	0.999954i	$0.496957\pi$
$420$		0			0
$421$		−	11.0850i		−	0.540249i	−0.962825	−	0.270124i	$-0.912935\pi$
$421$		−	11.0850i		−	0.540249i	0.962825	−	0.270124i	$-0.0870648\pi$
$422$			13.0713i			0.636301i
$423$		0			0
$424$			21.8732i			1.06226i
$425$	0.131831			0.00639473
$426$		0			0
$427$		−	44.7318i		−	2.16472i
$428$	−9.52652			−0.460482
$429$		0			0
$430$	−1.82925			−0.0882144
$431$			36.4573i			1.75609i	0.478580	+	0.878044i	$0.341152\pi$
$431$			36.4573i			1.75609i	−0.478580	+	0.878044i	$0.658848\pi$
$432$		0			0
$433$	8.82757			0.424226			0.212113	−	0.977245i	$-0.431966\pi$
$433$	8.82757			0.424226			0.212113	+	0.977245i	$0.431966\pi$
$434$			22.8940i			1.09895i
$435$		0			0
$436$		−	11.8013i		−	0.565182i
$437$		−	16.0901i		−	0.769693i
$438$		0			0
$439$	−35.1344			−1.67688			−0.838438	−	0.544997i	$-0.816531\pi$
$439$	−35.1344			−1.67688			−0.838438	+	0.544997i	$0.816531\pi$
$440$			23.8774i			1.13831i
$441$		0			0
$442$	8.00352	−	11.1810i	0.380688	−	0.531826i
$443$	23.9052			1.13577			0.567885	−	0.823108i	$-0.307762\pi$
$443$	23.9052			1.13577			0.567885	+	0.823108i	$0.307762\pi$
$444$		0			0
$445$	−12.9027			−0.611645
$446$	−9.96503			−0.471858
$447$		0			0
$448$		−	18.9274i		−	0.894236i
$449$			13.9683i			0.659206i	0.944120	+	0.329603i	$0.106915\pi$
$449$			13.9683i			0.659206i	−0.944120	+	0.329603i	$0.893085\pi$
$450$		0			0
$451$	−29.2851			−1.37898
$452$	22.9309			1.07858
$453$		0			0
$454$	7.35180			0.345037
$455$	−22.5968	−	16.1751i	−1.05936	−	0.758302i
$456$		0			0
$457$		−	17.4212i		−	0.814927i	−0.913221	−	0.407464i	$-0.866413\pi$
$457$		−	17.4212i		−	0.814927i	0.913221	−	0.407464i	$-0.133587\pi$
$458$	−4.87021			−0.227570
$459$		0			0
$460$		−	10.0126i		−	0.466841i
$461$		−	23.8667i		−	1.11158i	−0.831322	−	0.555791i	$-0.812416\pi$
$461$		−	23.8667i		−	1.11158i	0.831322	−	0.555791i	$-0.187584\pi$
$462$		0			0
$463$		−	10.5373i		−	0.489709i	−0.969560	−	0.244854i	$-0.921260\pi$
$463$		−	10.5373i		−	0.489709i	0.969560	−	0.244854i	$-0.0787402\pi$
$464$	−0.590109			−0.0273951
$465$		0			0
$466$			18.2018i			0.843182i
$467$	−8.40923			−0.389133			−0.194566	−	0.980889i	$-0.562330\pi$
$467$	−8.40923			−0.389133			−0.194566	+	0.980889i	$0.562330\pi$
$468$		0			0
$469$	−46.4227			−2.14360
$470$		−	11.2600i		−	0.519387i
$471$		0			0
$472$	−14.7765			−0.680142
$473$		−	3.32888i		−	0.153062i
$474$		0			0
$475$		−	0.133139i		−	0.00610883i
$476$		−	17.0733i		−	0.782551i
$477$		0			0
$478$	−12.8239			−0.586550
$479$			15.7068i			0.717660i	0.933403	+	0.358830i	$0.116824\pi$
$479$			15.7068i			0.717660i	−0.933403	+	0.358830i	$0.883176\pi$
$480$		0			0
$481$	11.5411	−	16.1230i	0.526227	−	0.735145i
$482$	27.4014			1.24810
$483$		0			0
$484$	−3.14402			−0.142910
$485$	−15.2188			−0.691051
$486$		0			0
$487$		−	37.8948i		−	1.71718i	−0.512666	−	0.858588i	$-0.671342\pi$
$487$		−	37.8948i		−	1.71718i	0.512666	−	0.858588i	$-0.328658\pi$
$488$		−	37.4884i		−	1.69702i
$489$		0			0
$490$	9.76420			0.441102
$491$	−3.76054			−0.169711			−0.0848553	−	0.996393i	$-0.527043\pi$
$491$	−3.76054			−0.169711			−0.0848553	+	0.996393i	$0.527043\pi$
$492$		0			0
$493$	−10.0181			−0.451191
$494$	−11.2919	−	8.08293i	−0.508048	−	0.363668i
$495$		0			0
$496$			1.82505i			0.0819471i
$497$	−9.12309			−0.409226
$498$		0			0
$499$		−	8.26147i		−	0.369834i	−0.982754	−	0.184917i	$-0.940798\pi$
$499$		−	8.26147i		−	0.369834i	0.982754	−	0.184917i	$-0.0592016\pi$
$500$			13.1500i			0.588086i
$501$		0			0
$502$		−	15.8556i		−	0.707670i
$503$	−12.9954			−0.579434			−0.289717	−	0.957112i	$-0.593561\pi$
$503$	−12.9954			−0.579434			−0.289717	+	0.957112i	$0.593561\pi$
$504$		0			0
$505$		−	11.1307i		−	0.495312i
$506$	−12.6648			−0.563020
$507$		0			0
$508$	−2.08148			−0.0923509
$509$			26.8713i			1.19105i	0.803338	+	0.595524i	$0.203055\pi$
$509$			26.8713i			1.19105i	−0.803338	+	0.595524i	$0.796945\pi$
$510$		0			0
$511$	18.7523			0.829551
$512$		−	2.80162i		−	0.123815i
$513$		0			0
$514$			16.5762i			0.731146i
$515$			4.73673i			0.208725i
$516$		0			0
$517$	20.4910			0.901195
$518$			17.1123i			0.751870i
$519$		0			0
$520$	−18.9377	−	13.5559i	−0.830474	−	0.594465i
$521$	14.2080			0.622464			0.311232	−	0.950334i	$-0.399258\pi$
$521$	14.2080			0.622464			0.311232	+	0.950334i	$0.399258\pi$
$522$		0			0
$523$	7.07846			0.309519			0.154760	−	0.987952i	$-0.450540\pi$
$523$	7.07846			0.309519			0.154760	+	0.987952i	$0.450540\pi$
$524$	−8.41241			−0.367498
$525$		0			0
$526$		−	7.60088i		−	0.331414i
$527$			30.9832i			1.34965i
$528$		0			0
$529$	−8.68705			−0.377698
$530$	−15.4291			−0.670198
$531$		0			0
$532$	−17.2427			−0.747564
$533$	16.6260	−	23.2267i	0.720151	−	1.00606i
$534$		0			0
$535$		−	18.1105i		−	0.782987i
$536$	−38.9054			−1.68046
$537$		0			0
$538$			19.3108i			0.832548i
$539$			17.7689i			0.765362i
$540$		0			0
$541$			7.18897i			0.309078i	0.987987	+	0.154539i	$0.0493892\pi$
$541$			7.18897i			0.309078i	−0.987987	+	0.154539i	$0.950611\pi$
$542$	16.7161			0.718020
$543$		0			0
$544$		−	23.3080i		−	0.999322i
$545$	22.4351			0.961016
$546$		0			0
$547$	−31.7775			−1.35871			−0.679353	−	0.733811i	$-0.737740\pi$
$547$	−31.7775			−1.35871			−0.679353	+	0.733811i	$0.737740\pi$
$548$		−	5.33381i		−	0.227849i
$549$		0			0
$550$	−0.104796			−0.00446852
$551$			10.1175i			0.431018i
$552$		0			0
$553$			37.5832i			1.59820i
$554$		−	7.81331i		−	0.331956i
$555$		0			0
$556$	16.6482			0.706042
$557$		−	32.2223i		−	1.36530i	−0.730745	−	0.682650i	$-0.760827\pi$
$557$		−	32.2223i		−	1.36530i	0.730745	−	0.682650i	$-0.239173\pi$
$558$		0			0
$559$	2.64021	+	1.88990i	0.111669	+	0.0799343i
$560$	1.91189			0.0807919
$561$		0			0
$562$	−3.14678			−0.132739
$563$	−33.6514			−1.41824			−0.709119	−	0.705089i	$-0.750907\pi$
$563$	−33.6514			−1.41824			−0.709119	+	0.705089i	$0.750907\pi$
$564$		0			0
$565$			43.5931i			1.83397i
$566$		−	12.4824i		−	0.524672i
$567$		0			0
$568$	−7.64578			−0.320810
$569$	−11.3359			−0.475224			−0.237612	−	0.971360i	$-0.576365\pi$
$569$	−11.3359			−0.475224			−0.237612	+	0.971360i	$0.576365\pi$
$570$		0			0
$571$	−0.254521			−0.0106514			−0.00532568	−	0.999986i	$-0.501695\pi$
$571$	−0.254521			−0.0106514			−0.00532568	+	0.999986i	$0.501695\pi$
$572$	9.15341	−	12.7874i	0.382723	−	0.534668i
$573$		0			0
$574$			24.6518i			1.02895i
$575$	0.118434			0.00493903
$576$		0			0
$577$		−	1.91400i		−	0.0796807i	−0.999206	−	0.0398403i	$-0.987315\pi$
$577$		−	1.91400i		−	0.0796807i	0.999206	−	0.0398403i	$-0.0126849\pi$
$578$			0.664891i			0.0276558i
$579$		0			0
$580$			6.29595i			0.261425i
$581$	1.84714			0.0766324
$582$		0			0
$583$		−	28.0779i		−	1.16287i
$584$	15.7157			0.650321
$585$		0			0
$586$	25.8193			1.06659
$587$			8.88440i			0.366699i	0.983048	+	0.183349i	$0.0586939\pi$
$587$			8.88440i			0.366699i	−0.983048	+	0.183349i	$0.941306\pi$
$588$		0			0
$589$	31.2906			1.28931
$590$		−	10.4231i		−	0.429113i
$591$		0			0
$592$			1.36414i			0.0560659i
$593$			43.1136i			1.77046i	0.465152	+	0.885231i	$0.346000\pi$
$593$			43.1136i			1.77046i	−0.465152	+	0.885231i	$0.654000\pi$
$594$		0			0
$595$	32.4574			1.33062
$596$		−	6.84618i		−	0.280430i
$597$		0			0
$598$	7.19018	−	10.0447i	0.294028	−	0.410760i
$599$	−11.1974			−0.457512			−0.228756	−	0.973484i	$-0.573466\pi$
$599$	−11.1974			−0.457512			−0.228756	+	0.973484i	$0.573466\pi$
$600$		0			0
$601$	5.92545			0.241704			0.120852	−	0.992671i	$-0.461437\pi$
$601$	5.92545			0.241704			0.120852	+	0.992671i	$0.461437\pi$
$602$	−2.80221			−0.114210
$603$		0			0
$604$		−	11.4991i		−	0.467892i
$605$		−	5.97698i		−	0.242999i
$606$		0			0
$607$	5.06940			0.205760			0.102880	−	0.994694i	$-0.467194\pi$
$607$	5.06940			0.205760			0.102880	+	0.994694i	$0.467194\pi$
$608$	−23.5393			−0.954643
$609$		0			0
$610$	26.4438			1.07068
$611$	−11.6334	+	16.2519i	−0.470635	+	0.657482i
$612$		0			0
$613$			24.6393i			0.995173i	0.867414	+	0.497587i	$0.165780\pi$
$613$			24.6393i			0.995173i	−0.867414	+	0.497587i	$0.834220\pi$
$614$	−19.7544			−0.797222
$615$		0			0
$616$			36.5775i			1.47375i
$617$		−	43.9864i		−	1.77083i	−0.464806	−	0.885413i	$-0.653876\pi$
$617$		−	43.9864i		−	1.77083i	0.464806	−	0.885413i	$-0.346124\pi$
$618$		0			0
$619$			20.5512i			0.826024i	0.910726	+	0.413012i	$0.135523\pi$
$619$			20.5512i			0.826024i	−0.910726	+	0.413012i	$0.864477\pi$
$620$	19.4717			0.782002
$621$		0			0
$622$		−	6.30587i		−	0.252842i
$623$	−19.7654			−0.791885
$624$		0			0
$625$	−25.1555			−1.00622
$626$			20.3312i			0.812597i
$627$		0			0
$628$	−1.27381			−0.0508305
$629$			23.1586i			0.923392i
$630$		0			0
$631$			25.1460i			1.00105i	0.865723	+	0.500523i	$0.166859\pi$
$631$			25.1460i			1.00105i	−0.865723	+	0.500523i	$0.833141\pi$
$632$			31.4974i			1.25290i
$633$		0			0
$634$	9.25165			0.367430
$635$		−	3.95704i		−	0.157030i
$636$		0			0
$637$	−14.0929	−	10.0879i	−0.558382	−	0.399698i
$638$	7.96365			0.315284
$639$		0			0
$640$	−13.6404			−0.539183
$641$	−33.3421			−1.31693			−0.658467	−	0.752609i	$-0.728795\pi$
$641$	−33.3421			−1.31693			−0.658467	+	0.752609i	$0.728795\pi$
$642$		0			0
$643$			31.3801i			1.23751i	0.785585	+	0.618754i	$0.212362\pi$
$643$			31.3801i			1.23751i	−0.785585	+	0.618754i	$0.787638\pi$
$644$		−	15.3382i		−	0.604411i
$645$		0			0
$646$	16.2194			0.638143
$647$	−47.5495			−1.86936			−0.934681	−	0.355486i	$-0.884315\pi$
$647$	−47.5495			−1.86936			−0.934681	+	0.355486i	$0.884315\pi$
$648$		0			0
$649$	18.9680			0.744561
$650$	0.0594958	−	0.0831162i	0.00233362	−	0.00326008i
$651$		0			0
$652$			6.54673i			0.256390i
$653$	−27.7456			−1.08577			−0.542884	−	0.839808i	$-0.682668\pi$
$653$	−27.7456			−1.08577			−0.542884	+	0.839808i	$0.682668\pi$
$654$		0			0
$655$		−	15.9925i		−	0.624880i
$656$			1.96518i			0.0767273i
$657$		0			0
$658$		−	17.2491i		−	0.672440i
$659$	15.6188			0.608422			0.304211	−	0.952605i	$-0.401607\pi$
$659$	15.6188			0.608422			0.304211	+	0.952605i	$0.401607\pi$
$660$		0			0
$661$		−	17.9164i		−	0.696869i	−0.937333	−	0.348434i	$-0.886713\pi$
$661$		−	17.9164i		−	0.696869i	0.937333	−	0.348434i	$-0.113287\pi$
$662$	4.89169			0.190121
$663$		0			0
$664$	1.54803			0.0600754
$665$		−	32.7794i		−	1.27113i
$666$		0			0
$667$	−9.00000			−0.348481
$668$		−	15.7380i		−	0.608922i
$669$		0			0
$670$		−	27.4434i		−	1.06023i
$671$			48.1225i			1.85775i
$672$		0			0
$673$	−5.89919			−0.227397			−0.113699	−	0.993515i	$-0.536270\pi$
$673$	−5.89919			−0.227397			−0.113699	+	0.993515i	$0.536270\pi$
$674$		−	29.8161i		−	1.14847i
$675$		0			0
$676$	4.94532	+	14.5196i	0.190205	+	0.558444i
$677$	27.4090			1.05341			0.526707	−	0.850047i	$-0.323427\pi$
$677$	27.4090			1.05341			0.526707	+	0.850047i	$0.323427\pi$
$678$		0			0
$679$	−23.3135			−0.894690
$680$	27.2016			1.04313
$681$		0			0
$682$		−	24.6294i		−	0.943109i
$683$			16.0989i			0.616006i	0.951385	+	0.308003i	$0.0996607\pi$
$683$			16.0989i			0.616006i	−0.951385	+	0.308003i	$0.900339\pi$
$684$		0			0
$685$	10.1399			0.387427
$686$	−6.82445			−0.260559
$687$		0			0
$688$	−0.223385			−0.00851646
$689$	22.2692	+	15.9406i	0.848390	+	0.607290i
$690$		0			0
$691$			27.6824i			1.05309i	0.850148	+	0.526543i	$0.176512\pi$
$691$			27.6824i			1.05309i	−0.850148	+	0.526543i	$0.823488\pi$
$692$	8.78030			0.333777
$693$		0			0
$694$			12.3413i			0.468469i
$695$			31.6494i			1.20053i
$696$		0			0
$697$			33.3621i			1.26368i
$698$	10.2103			0.386465
$699$		0			0
$700$		−	0.126918i		−	0.00479703i
$701$	−10.0776			−0.380624			−0.190312	−	0.981724i	$-0.560950\pi$
$701$	−10.0776			−0.380624			−0.190312	+	0.981724i	$0.560950\pi$
$702$		0			0
$703$	23.3883			0.882108
$704$			20.3621i			0.767427i
$705$		0			0
$706$	−24.1167			−0.907645
$707$		−	17.0511i		−	0.641271i
$708$		0			0
$709$			3.22720i			0.121200i	0.998162	+	0.0606000i	$0.0193014\pi$
$709$			3.22720i			0.121200i	−0.998162	+	0.0606000i	$0.980699\pi$
$710$		−	5.39324i		−	0.202405i
$711$		0			0
$712$	−16.5648			−0.620793
$713$			27.8346i			1.04241i
$714$		0			0
$715$	24.3097	+	17.4012i	0.909131	+	0.650769i
$716$	27.3792			1.02321
$717$		0			0
$718$	31.6086			1.17962
$719$	−30.0712			−1.12147			−0.560733	−	0.827996i	$-0.689481\pi$
$719$	−30.0712			−1.12147			−0.560733	+	0.827996i	$0.689481\pi$
$720$		0			0
$721$			7.25614i			0.270233i
$722$			0.826023i			0.0307414i
$723$		0			0
$724$	6.11704			0.227338
$725$	−0.0744713			−0.00276579
$726$		0			0
$727$	−19.3772			−0.718660			−0.359330	−	0.933210i	$-0.616995\pi$
$727$	−19.3772			−0.718660			−0.359330	+	0.933210i	$0.616995\pi$
$728$	−29.0105	−	20.7661i	−1.07520	−	0.769643i
$729$		0			0
$730$			11.0857i			0.410299i
$731$	−3.79232			−0.140264
$732$		0			0
$733$		−	24.4725i		−	0.903911i	−0.892041	−	0.451955i	$-0.850727\pi$
$733$		−	24.4725i		−	0.903911i	0.892041	−	0.451955i	$-0.149273\pi$
$734$		−	2.45809i		−	0.0907297i
$735$		0			0
$736$		−	20.9394i		−	0.771835i
$737$	49.9416			1.83962
$738$		0			0
$739$		−	3.30687i		−	0.121645i	−0.998149	−	0.0608226i	$-0.980628\pi$
$739$		−	3.30687i		−	0.121645i	0.998149	−	0.0608226i	$-0.0193724\pi$
$740$	14.5542			0.535024
$741$		0			0
$742$	−23.6357			−0.867692
$743$		−	17.4815i		−	0.641335i	−0.947192	−	0.320667i	$-0.896093\pi$
$743$		−	17.4815i		−	0.641335i	0.947192	−	0.320667i	$-0.103907\pi$
$744$		0			0
$745$	13.0150			0.476834
$746$		−	1.58566i		−	0.0580553i
$747$		0			0
$748$			18.3674i			0.671580i
$749$		−	27.7433i		−	1.01372i
$750$		0			0
$751$	−28.5843			−1.04306			−0.521528	−	0.853234i	$-0.674638\pi$
$751$	−28.5843			−1.04306			−0.521528	+	0.853234i	$0.674638\pi$
$752$		−	1.37505i		−	0.0501430i
$753$		0			0
$754$	−4.52119	+	6.31615i	−0.164652	+	0.230021i
$755$	21.8606			0.795587
$756$		0			0
$757$	11.9611			0.434733			0.217367	−	0.976090i	$-0.430253\pi$
$757$	11.9611			0.434733			0.217367	+	0.976090i	$0.430253\pi$
$758$	−2.63475			−0.0956983
$759$		0			0
$760$		−	27.4714i		−	0.996494i
$761$		−	2.12842i		−	0.0771553i	−0.999256	−	0.0385777i	$-0.987717\pi$
$761$		−	2.12842i		−	0.0771553i	0.999256	−	0.0385777i	$-0.0122827\pi$
$762$		0			0
$763$	34.3681			1.24421
$764$	19.0449			0.689019
$765$		0			0
$766$	−18.5156			−0.668994
$767$	−10.7687	+	15.0440i	−0.388835	+	0.543207i
$768$		0			0
$769$			7.65086i			0.275897i	0.990439	+	0.137949i	$0.0440509\pi$
$769$			7.65086i			0.275897i	−0.990439	+	0.137949i	$0.955949\pi$
$770$	−25.8013			−0.929815
$771$		0			0
$772$		−	3.86917i		−	0.139254i
$773$		−	24.6440i		−	0.886383i	−0.896427	−	0.443191i	$-0.853846\pi$
$773$		−	24.6440i		−	0.886383i	0.896427	−	0.443191i	$-0.146154\pi$
$774$		0			0
$775$			0.230320i			0.00827333i
$776$	−19.5384			−0.701386
$777$		0			0
$778$			11.7394i			0.420880i
$779$	33.6931			1.20718
$780$		0			0
$781$	9.81463			0.351195
$782$			14.4280i			0.515943i
$783$		0			0
$784$	1.19238			0.0425851
$785$		−	2.42159i		−	0.0864304i
$786$		0			0
$787$			41.8679i			1.49243i	0.665706	+	0.746214i	$0.268131\pi$
$787$			41.8679i			1.49243i	−0.665706	+	0.746214i	$0.731869\pi$
$788$		−	8.39199i		−	0.298952i
$789$		0			0
$790$	−22.2178			−0.790475
$791$			66.7797i			2.37441i
$792$		0			0
$793$	−38.1671	−	27.3205i	−1.35535	−	0.970181i
$794$	26.4261			0.937828
$795$		0			0
$796$	−16.3355			−0.578996
$797$	−41.7563			−1.47908			−0.739541	−	0.673111i	$-0.764957\pi$
$797$	−41.7563			−1.47908			−0.739541	+	0.673111i	$0.764957\pi$
$798$		0			0
$799$		−	23.3437i		−	0.825842i
$800$		−	0.173265i		−	0.00612583i
$801$		0			0
$802$	−20.0792			−0.709022
$803$	−20.1737			−0.711915
$804$		0			0
$805$	29.1590			1.02772
$806$	19.5341	+	13.9828i	0.688061	+	0.492524i
$807$		0			0
$808$		−	14.2900i		−	0.502720i
$809$	45.2476			1.59082			0.795410	−	0.606072i	$-0.207256\pi$
$809$	45.2476			1.59082			0.795410	+	0.606072i	$0.207256\pi$
$810$		0			0
$811$		−	52.8820i		−	1.85694i	−0.371410	−	0.928469i	$-0.621126\pi$
$811$		−	52.8820i		−	1.85694i	0.371410	−	0.928469i	$-0.378874\pi$
$812$			9.64469i			0.338462i
$813$		0			0
$814$		−	18.4094i		−	0.645249i
$815$	−12.4458			−0.435956
$816$		0			0
$817$			3.82994i			0.133993i
$818$	1.90053			0.0664505
$819$		0			0
$820$	20.9667			0.732190
$821$		−	34.4222i		−	1.20134i	−0.799496	−	0.600671i	$-0.794900\pi$
$821$		−	34.4222i		−	1.20134i	0.799496	−	0.600671i	$-0.205100\pi$
$822$		0			0
$823$	−11.9971			−0.418191			−0.209096	−	0.977895i	$-0.567052\pi$
$823$	−11.9971			−0.418191			−0.209096	+	0.977895i	$0.567052\pi$
$824$			6.08116i			0.211847i
$825$		0			0
$826$		−	15.9671i		−	0.555565i
$827$		−	4.35092i		−	0.151296i	−0.997135	−	0.0756482i	$-0.975897\pi$
$827$		−	4.35092i		−	0.151296i	0.997135	−	0.0756482i	$-0.0241026\pi$
$828$		0			0
$829$	45.6204			1.58446			0.792230	−	0.610222i	$-0.208920\pi$
$829$	45.6204			1.58446			0.792230	+	0.610222i	$0.208920\pi$
$830$			1.09196i			0.0379026i
$831$		0			0
$832$	−16.1497	−	11.5602i	−0.559889	−	0.400777i
$833$	20.2427			0.701367
$834$		0			0
$835$	29.9190			1.03539
$836$	18.5497			0.641554
$837$		0			0
$838$			0.354449i			0.0122442i
$839$			24.6675i			0.851615i	0.904814	+	0.425808i	$0.140010\pi$
$839$			24.6675i			0.851615i	−0.904814	+	0.425808i	$0.859990\pi$
$840$		0			0
$841$	−23.3408			−0.804855
$842$	10.0385			0.345950
$843$		0			0
$844$	17.0305			0.586214
$845$	−27.6026	+	9.40138i	−0.949559	+	0.323417i
$846$		0			0
$847$		−	9.15606i		−	0.314606i
$848$	−1.88417			−0.0647027
$849$		0			0
$850$			0.119386i			0.00409489i
$851$			20.8051i			0.713190i
$852$		0			0
$853$			26.5198i			0.908021i	0.890996	+	0.454010i	$0.150007\pi$
$853$			26.5198i			0.908021i	−0.890996	+	0.454010i	$0.849993\pi$
$854$	40.5089			1.38619
$855$		0			0
$856$		−	23.2508i		−	0.794697i
$857$	−6.40948			−0.218944			−0.109472	−	0.993990i	$-0.534916\pi$
$857$	−6.40948			−0.218944			−0.109472	+	0.993990i	$0.534916\pi$
$858$		0			0
$859$	25.5996			0.873446			0.436723	−	0.899596i	$-0.356139\pi$
$859$	25.5996			0.873446			0.436723	+	0.899596i	$0.356139\pi$
$860$			2.38332i			0.0812706i
$861$		0			0
$862$	−33.0156			−1.12452
$863$		−	48.4823i		−	1.65036i	−0.564873	−	0.825178i	$-0.691075\pi$
$863$		−	48.4823i		−	1.65036i	0.564873	−	0.825178i	$-0.308925\pi$
$864$		0			0
$865$			16.6919i			0.567542i
$866$			7.99421i			0.271654i
$867$		0			0
$868$	29.8284			1.01244
$869$		−	40.4321i		−	1.37156i
$870$		0			0
$871$	−28.3533	+	39.6098i	−0.960713	+	1.34213i
$872$	28.8029			0.975389
$873$		0			0
$874$	14.5711			0.492876
$875$	−38.2957			−1.29463
$876$		0			0
$877$		−	31.1795i		−	1.05286i	−0.850219	−	0.526429i	$-0.823531\pi$
$877$		−	31.1795i		−	1.05286i	0.850219	−	0.526429i	$-0.176469\pi$
$878$		−	31.8176i		−	1.07379i
$879$		0			0
$880$	−2.05681			−0.0693350
$881$	−49.6614			−1.67314			−0.836568	−	0.547863i	$-0.815442\pi$
$881$	−49.6614			−1.67314			−0.836568	+	0.547863i	$0.815442\pi$
$882$		0			0
$883$	30.3523			1.02144			0.510718	−	0.859748i	$-0.329380\pi$
$883$	30.3523			1.02144			0.510718	+	0.859748i	$0.329380\pi$
$884$	−14.5676	−	10.4277i	−0.489962	−	0.350722i
$885$		0			0
$886$			21.6485i			0.727294i
$887$	12.8432			0.431233			0.215616	−	0.976478i	$-0.430824\pi$
$887$	12.8432			0.431233			0.215616	+	0.976478i	$0.430824\pi$
$888$		0			0
$889$		−	6.06173i		−	0.203304i
$890$		−	11.6846i		−	0.391669i
$891$		0			0
$892$			12.9834i			0.434715i
$893$	−23.5754			−0.788919
$894$		0			0
$895$			52.0496i			1.73982i
$896$	−20.8955			−0.698071
$897$		0			0
$898$	−12.6497			−0.422125
$899$		−	17.5024i		−	0.583738i
$900$		0			0
$901$	−31.9869			−1.06564
$902$		−	26.5205i		−	0.883035i
$903$		0			0
$904$			55.9660i			1.86140i
$905$			11.6289i			0.386557i
$906$		0			0
$907$	−20.5986			−0.683965			−0.341983	−	0.939706i	$-0.611098\pi$
$907$	−20.5986			−0.683965			−0.341983	+	0.939706i	$0.611098\pi$
$908$		−	9.57860i		−	0.317877i
$909$		0			0
$910$	14.6481	−	20.4636i	0.485581	−	0.678362i
$911$	42.3899			1.40444			0.702219	−	0.711961i	$-0.252193\pi$
$911$	42.3899			1.40444			0.702219	+	0.711961i	$0.252193\pi$
$912$		0			0
$913$	−1.98716			−0.0657653
$914$	15.7765			0.521842
$915$		0			0
$916$			6.34535i			0.209656i
$917$		−	24.4988i		−	0.809021i
$918$		0			0
$919$	−42.7987			−1.41180			−0.705899	−	0.708313i	$-0.749457\pi$
$919$	−42.7987			−1.41180			−0.705899	+	0.708313i	$0.749457\pi$
$920$	24.4373			0.805672
$921$		0			0
$922$	21.6136			0.711805
$923$	−5.57205	+	7.78420i	−0.183406	+	0.256220i
$924$		0			0
$925$			0.172154i			0.00566038i
$926$	9.54253			0.313587
$927$		0			0
$928$			13.1667i			0.432218i
$929$		−	11.8059i		−	0.387340i	−0.981067	−	0.193670i	$-0.937961\pi$
$929$		−	11.8059i		−	0.387340i	0.981067	−	0.193670i	$-0.0620391\pi$
$930$		0			0
$931$		−	20.4435i		−	0.670009i
$932$	23.7150			0.776810
$933$		0			0
$934$		−	7.61537i		−	0.249183i
$935$	−34.9177			−1.14193
$936$		0			0
$937$	24.7925			0.809935			0.404967	−	0.914331i	$-0.367283\pi$
$937$	24.7925			0.809935			0.404967	+	0.914331i	$0.367283\pi$
$938$		−	42.0402i		−	1.37266i
$939$		0			0
$940$	−14.6706			−0.478503
$941$			23.8292i			0.776810i	0.921489	+	0.388405i	$0.126974\pi$
$941$			23.8292i			0.776810i	−0.921489	+	0.388405i	$0.873026\pi$
$942$		0			0
$943$			29.9717i			0.976014i
$944$		−	1.27285i		−	0.0414278i
$945$		0			0
$946$	3.01462			0.0980138
$947$		−	49.0208i		−	1.59296i	−0.604664	−	0.796480i	$-0.706693\pi$
$947$		−	49.0208i		−	1.59296i	0.604664	−	0.796480i	$-0.293307\pi$
$948$		0			0
$949$	11.4532	−	16.0002i	0.371786	−	0.519389i
$950$	0.120570			0.00391181
$951$		0			0
$952$	41.6697			1.35052
$953$	7.78372			0.252140			0.126070	−	0.992021i	$-0.459764\pi$
$953$	7.78372			0.252140			0.126070	+	0.992021i	$0.459764\pi$
$954$		0			0
$955$			36.2055i			1.17158i
$956$			16.7081i			0.540379i
$957$		0			0
$958$	−14.2240			−0.459556
$959$	15.5332			0.501594
$960$		0			0
$961$	−23.1303			−0.746137
$962$	14.6009	+	10.4515i	0.470752	+	0.336971i
$963$		0			0
$964$		−	35.7011i		−	1.14985i
$965$	7.35554			0.236783
$966$		0			0
$967$		−	37.5960i		−	1.20900i	−0.796603	−	0.604502i	$-0.793372\pi$
$967$		−	37.5960i		−	1.20900i	0.796603	−	0.604502i	$-0.206628\pi$
$968$		−	7.67342i		−	0.246633i
$969$		0			0
$970$		−	13.7821i		−	0.442517i
$971$	−9.16512			−0.294123			−0.147061	−	0.989127i	$-0.546981\pi$
$971$	−9.16512			−0.294123			−0.147061	+	0.989127i	$0.546981\pi$
$972$		0			0
$973$			48.4833i			1.55430i
$974$	34.3174			1.09960
$975$		0			0
$976$	3.22926			0.103366
$977$		−	47.3477i		−	1.51479i	−0.652959	−	0.757394i	$-0.726472\pi$
$977$		−	47.3477i		−	1.51479i	0.652959	−	0.757394i	$-0.273528\pi$
$978$		0			0
$979$	21.2637			0.679590
$980$		−	12.7217i		−	0.406380i
$981$		0			0
$982$		−	3.40553i		−	0.108675i
$983$		−	16.2390i		−	0.517943i	−0.965885	−	0.258972i	$-0.916616\pi$
$983$		−	16.2390i		−	0.517943i	0.965885	−	0.258972i	$-0.0833836\pi$
$984$		0			0
$985$	15.9537			0.508328
$986$		−	9.07232i		−	0.288922i
$987$		0			0
$988$	−10.5312	+	14.7122i	−0.335042	+	0.468056i
$989$	−3.40693			−0.108334
$990$		0			0
$991$	−37.7634			−1.19959			−0.599797	−	0.800152i	$-0.704752\pi$
$991$	−37.7634			−1.19959			−0.599797	+	0.800152i	$0.704752\pi$
$992$	40.7211			1.29289
$993$		0			0
$994$		−	8.26184i		−	0.262049i
$995$		−	31.0548i		−	0.984505i
$996$		0			0
$997$	47.6566			1.50930			0.754649	−	0.656128i	$-0.227807\pi$
$997$	47.6566			1.50930			0.754649	+	0.656128i	$0.227807\pi$
$998$	7.48156			0.236824
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1053.2.b.j.649.6		10
3.2	odd	2		1053.2.b.i.649.5		10
9.2	odd	6		351.2.t.c.64.5		20
9.4	even	3		117.2.t.c.25.5	✓	20
9.5	odd	6		351.2.t.c.181.6		20
9.7	even	3		117.2.t.c.103.6	yes	20
13.12	even	2	inner	1053.2.b.j.649.5		10
39.38	odd	2		1053.2.b.i.649.6		10
117.25	even	6		117.2.t.c.103.5	yes	20
117.38	odd	6		351.2.t.c.64.6		20
117.77	odd	6		351.2.t.c.181.5		20
117.103	even	6		117.2.t.c.25.6	yes	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
117.2.t.c.25.5	✓	20	9.4	even	3
117.2.t.c.25.6	yes	20	117.103	even	6
117.2.t.c.103.5	yes	20	117.25	even	6
117.2.t.c.103.6	yes	20	9.7	even	3
351.2.t.c.64.5		20	9.2	odd	6
351.2.t.c.64.6		20	117.38	odd	6
351.2.t.c.181.5		20	117.77	odd	6
351.2.t.c.181.6		20	9.5	odd	6
1053.2.b.i.649.5		10	3.2	odd	2
1053.2.b.i.649.6		10	39.38	odd	2
1053.2.b.j.649.5		10	13.12	even	2	inner
1053.2.b.j.649.6		10	1.1	even	1	trivial

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.b (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+0.905597i q^{2} +1.17989 q^{4} +2.24306i q^{5} +3.43611i q^{7} +2.87970i q^{8} -2.03130 q^{10} -3.69657i q^{11} +(2.93184 + 2.09865i) q^{13} -3.11173 q^{14} -0.248059 q^{16} -4.21120 q^{17} +4.25298i q^{19} +2.64657i q^{20} +3.34760 q^{22} -3.78325 q^{23} -0.0313048 q^{25} +(-1.90053 + 2.65506i) q^{26} +4.05425i q^{28} +2.37891 q^{29} -7.35733i q^{31} +5.53476i q^{32} -3.81365i q^{34} -7.70739 q^{35} -5.49928i q^{37} -3.85149 q^{38} -6.45934 q^{40} -7.92223i q^{41} +0.900531 q^{43} -4.36157i q^{44} -3.42609i q^{46} +5.54325i q^{47} -4.80686 q^{49} -0.0283495i q^{50} +(3.45926 + 2.47619i) q^{52} +7.59566 q^{53} +8.29163 q^{55} -9.89498 q^{56} +2.15433i q^{58} +5.13125i q^{59} -13.0181 q^{61} +6.66277 q^{62} -5.50838 q^{64} +(-4.70739 + 6.57628i) q^{65} +13.5102i q^{67} -4.96877 q^{68} -6.97979i q^{70} +2.65506i q^{71} -5.45741i q^{73} +4.98013 q^{74} +5.01807i q^{76} +12.7018 q^{77} +10.9377 q^{79} -0.556410i q^{80} +7.17434 q^{82} -0.537568i q^{83} -9.44596i q^{85} +0.815518i q^{86} +10.6450 q^{88} +5.75227i q^{89} +(-7.21120 + 10.0741i) q^{91} -4.46383 q^{92} -5.01995 q^{94} -9.53968 q^{95} +6.78485i q^{97} -4.35308i q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q - 12 q^{4} - 8 q^{10} + 4 q^{13} + 18 q^{14} - 4 q^{16} - 6 q^{17} + 10 q^{22} - 24 q^{23} + 12 q^{25} - 6 q^{26} - 12 q^{29} - 6 q^{35} - 12 q^{38} + 8 q^{40} - 4 q^{43} + 10 q^{49} + 54 q^{53} + 10 q^{55}+ \cdots - 24 q^{95}+O(q^{100}) \) 10 * q - 12 * q^4 - 8 * q^10 + 4 * q^13 + 18 * q^14 - 4 * q^16 - 6 * q^17 + 10 * q^22 - 24 * q^23 + 12 * q^25 - 6 * q^26 - 12 * q^29 - 6 * q^35 - 12 * q^38 + 8 * q^40 - 4 * q^43 + 10 * q^49 + 54 * q^53 + 10 * q^55 - 36 * q^56 + 2 * q^61 - 36 * q^62 + 4 * q^64 + 24 * q^65 - 24 * q^68 + 42 * q^74 + 6 * q^77 + 14 * q^79 - 2 * q^82 - 22 * q^88 - 36 * q^91 + 84 * q^92 - 20 * q^94 - 24 * q^95

Introduction

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