LMFDB - Embedded newform 297.2.d.b.296.7

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [297,2,Mod(296,297)] 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("297.296"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(297, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([1, 1])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 297 = 3^{3} \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	297.d (of order $2$, degree $1$, minimal)

Newform invariants

comment:select newform

sage:traces = [8,0,0,8,0,0,0,0,0,0,0,0,0,0,0,8] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(16)] == traces)

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

Self dual:	no
Analytic conductor:	$2.37155694003$
Analytic rank:	$0$
Dimension:	$8$
Coefficient field:	8.0.764411904.5
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 6x^{6} + 21x^{4} - 54x^{2} + 81 $ x^8 - 6x^6 + 21x^4 - 54*x^2 + 81
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 2^{4} $
Twist minimal:	yes
Sato-Tate group:	$\mathrm{SU}(2)[C_{2}]$

Embedding invariants

Embedding label			296.7
Root			$1.69185 + 0.370982i$ of defining polynomial
Character	$\chi$	$=$	297.296
Dual form			297.2.d.b.296.8

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+2.33441 q^{2} +3.44949 q^{4} -1.41421i q^{5} -2.55940i q^{7} +3.38371 q^{8} -3.30136i q^{10} +(-1.04930 + 3.14626i) q^{11} +4.78529i q^{13} -5.97469i q^{14} +1.00000 q^{16} -1.28512 q^{17} -1.81743i q^{19} -4.87832i q^{20} +(-2.44949 + 7.34468i) q^{22} +5.33902i q^{23} +3.00000 q^{25} +11.1708i q^{26} -8.82861i q^{28} -9.10183 q^{29} +9.34847 q^{31} -4.43300 q^{32} -3.00000 q^{34} -3.61953 q^{35} -3.44949 q^{37} -4.24264i q^{38} -4.78529i q^{40} -2.33441 q^{41} -8.82861i q^{43} +(-3.61953 + 10.8530i) q^{44} +12.4635i q^{46} -10.8530i q^{47} +0.449490 q^{49} +7.00324 q^{50} +16.5068i q^{52} -0.635674i q^{53} +(4.44949 + 1.48393i) q^{55} -8.66025i q^{56} -21.2474 q^{58} +3.78194i q^{59} -12.8719i q^{61} +21.8232 q^{62} -12.3485 q^{64} +6.76742 q^{65} -5.34847 q^{67} -4.43300 q^{68} -8.44949 q^{70} +1.27135i q^{71} +11.7215i q^{73} -8.05254 q^{74} -6.26922i q^{76} +(8.05254 + 2.68556i) q^{77} +11.7965i q^{79} -1.41421i q^{80} -5.44949 q^{82} +6.18977 q^{83} +1.81743i q^{85} -20.6096i q^{86} +(-3.55051 + 10.6460i) q^{88} -7.56388i q^{89} +12.2474 q^{91} +18.4169i q^{92} -25.3354i q^{94} -2.57024 q^{95} -8.34847 q^{97} +1.04930 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{4} + 8 q^{16} + 24 q^{25} + 16 q^{31} - 24 q^{34} - 8 q^{37} - 16 q^{49} + 16 q^{55} - 72 q^{58} - 40 q^{64} + 16 q^{67} - 48 q^{70} - 24 q^{82} - 48 q^{88} - 8 q^{97}+O(q^{100}) $ 8 * q + 8 * q^4 + 8 * q^16 + 24 * q^25 + 16 * q^31 - 24 * q^34 - 8 * q^37 - 16 * q^49 + 16 * q^55 - 72 * q^58 - 40 * q^64 + 16 * q^67 - 48 * q^70 - 24 * q^82 - 48 * q^88 - 8 * q^97

Character values

We give the values of $\chi$ on generators for $\left(\mathbb{Z}/297\mathbb{Z}\right)^\times$.

$n$	$56$	$244$
$\chi(n)$	$-1$	$-1$

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 $

$2$	2.33441			1.65068			0.825340	−	0.564636i	$-0.190983\pi$
$2$	2.33441			1.65068			0.825340	+	0.564636i	$0.190983\pi$
$3$		0			0
$3$		0			0
$4$	3.44949			1.72474
$5$		−	1.41421i		−	0.632456i	−0.948683	−	0.316228i	$-0.897584\pi$
$5$		−	1.41421i		−	0.632456i	0.948683	−	0.316228i	$-0.102416\pi$
$6$		0			0
$7$		−	2.55940i		−	0.967361i	−0.875245	−	0.483680i	$-0.839300\pi$
$7$		−	2.55940i		−	0.967361i	0.875245	−	0.483680i	$-0.160700\pi$
$8$	3.38371			1.19632
$9$		0			0
$10$		−	3.30136i		−	1.04398i
$11$	−1.04930	+	3.14626i	−0.316374	+	0.948634i
$11$	−1.04930	+	3.14626i	−0.316374	+	0.948634i
$12$		0			0
$13$			4.78529i			1.32720i	0.748088	+	0.663600i	$0.230972\pi$
$13$			4.78529i			1.32720i	−0.748088	+	0.663600i	$0.769028\pi$
$14$		−	5.97469i		−	1.59680i
$15$		0			0
$16$	1.00000			0.250000
$17$	−1.28512			−0.311687			−0.155844	−	0.987782i	$-0.549810\pi$
$17$	−1.28512			−0.311687			−0.155844	+	0.987782i	$0.549810\pi$
$18$		0			0
$19$		−	1.81743i		−	0.416948i	−0.978028	−	0.208474i	$-0.933150\pi$
$19$		−	1.81743i		−	0.416948i	0.978028	−	0.208474i	$-0.0668496\pi$
$20$		−	4.87832i		−	1.09082i
$21$		0			0
$22$	−2.44949	+	7.34468i	−0.522233	+	1.56589i
$23$			5.33902i			1.11326i	0.830760	+	0.556631i	$0.187906\pi$
$23$			5.33902i			1.11326i	−0.830760	+	0.556631i	$0.812094\pi$
$24$		0			0
$25$	3.00000			0.600000
$26$			11.1708i			2.19078i
$27$		0			0
$28$		−	8.82861i		−	1.66845i
$29$	−9.10183			−1.69017			−0.845084	−	0.534634i	$-0.820450\pi$
$29$	−9.10183			−1.69017			−0.845084	+	0.534634i	$0.820450\pi$
$30$		0			0
$31$	9.34847			1.67903			0.839517	−	0.543333i	$-0.182838\pi$
$31$	9.34847			1.67903			0.839517	+	0.543333i	$0.182838\pi$
$32$	−4.43300			−0.783652
$33$		0			0
$34$	−3.00000			−0.514496
$35$	−3.61953			−0.611813
$36$		0			0
$37$	−3.44949			−0.567093			−0.283546	−	0.958959i	$-0.591511\pi$
$37$	−3.44949			−0.567093			−0.283546	+	0.958959i	$0.591511\pi$
$38$		−	4.24264i		−	0.688247i
$39$		0			0
$40$		−	4.78529i		−	0.756620i
$41$	−2.33441			−0.364574			−0.182287	−	0.983245i	$-0.558350\pi$
$41$	−2.33441			−0.364574			−0.182287	+	0.983245i	$0.558350\pi$
$42$		0			0
$43$		−	8.82861i		−	1.34635i	−0.739483	−	0.673175i	$-0.764930\pi$
$43$		−	8.82861i		−	1.34635i	0.739483	−	0.673175i	$-0.235070\pi$
$44$	−3.61953	+	10.8530i	−0.545665	+	1.63615i
$45$		0			0
$46$			12.4635i			1.83764i
$47$		−	10.8530i		−	1.58307i	−0.611121	−	0.791537i	$-0.709281\pi$
$47$		−	10.8530i		−	1.58307i	0.611121	−	0.791537i	$-0.290719\pi$
$48$		0			0
$49$	0.449490			0.0642128
$50$	7.00324			0.990408
$51$		0			0
$52$			16.5068i			2.28908i
$53$		−	0.635674i		−	0.0873166i	−0.999047	−	0.0436583i	$-0.986099\pi$
$53$		−	0.635674i		−	0.0873166i	0.999047	−	0.0436583i	$-0.0139013\pi$
$54$		0			0
$55$	4.44949	+	1.48393i	0.599969	+	0.200093i
$56$		−	8.66025i		−	1.15728i
$57$		0			0
$58$	−21.2474			−2.78993
$59$			3.78194i			0.492367i	0.969223	+	0.246183i	$0.0791765\pi$
$59$			3.78194i			0.492367i	−0.969223	+	0.246183i	$0.920823\pi$
$60$		0			0
$61$		−	12.8719i		−	1.64808i	−0.566530	−	0.824041i	$-0.691714\pi$
$61$		−	12.8719i		−	1.64808i	0.566530	−	0.824041i	$-0.308286\pi$
$62$	21.8232			2.77155
$63$		0			0
$64$	−12.3485			−1.54356
$65$	6.76742			0.839395
$66$		0			0
$67$	−5.34847			−0.653420			−0.326710	−	0.945125i	$-0.605940\pi$
$67$	−5.34847			−0.653420			−0.326710	+	0.945125i	$0.605940\pi$
$68$	−4.43300			−0.537581
$69$		0			0
$70$	−8.44949			−1.00991
$71$			1.27135i			0.150881i	0.997150	+	0.0754407i	$0.0240364\pi$
$71$			1.27135i			0.150881i	−0.997150	+	0.0754407i	$0.975964\pi$
$72$		0			0
$73$			11.7215i			1.37190i	0.727649	+	0.685950i	$0.240613\pi$
$73$			11.7215i			1.37190i	−0.727649	+	0.685950i	$0.759387\pi$
$74$	−8.05254			−0.936089
$75$		0			0
$76$		−	6.26922i		−	0.719128i
$77$	8.05254	+	2.68556i	0.917672	+	0.306048i
$78$		0			0
$79$			11.7965i			1.32721i	0.748085	+	0.663603i	$0.230974\pi$
$79$			11.7965i			1.32721i	−0.748085	+	0.663603i	$0.769026\pi$
$80$		−	1.41421i		−	0.158114i
$81$		0			0
$82$	−5.44949			−0.601795
$83$	6.18977			0.679416			0.339708	−	0.940531i	$-0.389672\pi$
$83$	6.18977			0.679416			0.339708	+	0.940531i	$0.389672\pi$
$84$		0			0
$85$			1.81743i			0.197128i
$86$		−	20.6096i		−	2.22239i
$87$		0			0
$88$	−3.55051	+	10.6460i	−0.378486	+	1.13487i
$89$		−	7.56388i		−	0.801769i	−0.916129	−	0.400885i	$-0.868703\pi$
$89$		−	7.56388i		−	0.801769i	0.916129	−	0.400885i	$-0.131297\pi$
$90$		0			0
$91$	12.2474			1.28388
$92$			18.4169i			1.92009i
$93$		0			0
$94$		−	25.3354i		−	2.61315i
$95$	−2.57024			−0.263701
$96$		0			0
$97$	−8.34847			−0.847659			−0.423829	−	0.905742i	$-0.639314\pi$
$97$	−8.34847			−0.847659			−0.423829	+	0.905742i	$0.639314\pi$
$98$	1.04930			0.105995
$99$		0			0
$100$	10.3485			1.03485
$101$	8.05254			0.801257			0.400629	−	0.916240i	$-0.368792\pi$
$101$	8.05254			0.801257			0.400629	+	0.916240i	$0.368792\pi$
$102$		0			0
$103$	15.3485			1.51233			0.756165	−	0.654381i	$-0.227071\pi$
$103$	15.3485			1.51233			0.756165	+	0.654381i	$0.227071\pi$
$104$			16.1920i			1.58776i
$105$		0			0
$106$		−	1.48393i		−	0.144132i
$107$	10.8586			1.04974			0.524870	−	0.851182i	$-0.324114\pi$
$107$	10.8586			1.04974			0.524870	+	0.851182i	$0.324114\pi$
$108$		0			0
$109$		−	8.08665i		−	0.774560i	−0.921962	−	0.387280i	$-0.873415\pi$
$109$		−	8.08665i		−	0.774560i	0.921962	−	0.387280i	$-0.126585\pi$
$110$	10.3870	+	3.46410i	0.990357	+	0.330289i
$111$		0			0
$112$		−	2.55940i		−	0.241840i
$113$			7.07107i			0.665190i	0.943070	+	0.332595i	$0.107924\pi$
$113$			7.07107i			0.665190i	−0.943070	+	0.332595i	$0.892076\pi$
$114$		0			0
$115$	7.55051			0.704089
$116$	−31.3967			−2.91511
$117$		0			0
$118$			8.82861i			0.812740i
$119$			3.28913i			0.301514i
$120$		0			0
$121$	−8.79796	−	6.60272i	−0.799814	−	0.600247i
$122$		−	30.0484i		−	2.72046i
$123$		0			0
$124$	32.2474			2.89591
$125$		−	11.3137i		−	1.01193i
$126$		0			0
$127$		−	1.81743i		−	0.161271i	−0.996744	−	0.0806355i	$-0.974305\pi$
$127$		−	1.81743i		−	0.161271i	0.996744	−	0.0806355i	$-0.0256950\pi$
$128$	−19.9604			−1.76427
$129$		0			0
$130$	15.7980			1.38557
$131$	−0.471647			−0.0412080			−0.0206040	−	0.999788i	$-0.506559\pi$
$131$	−0.471647			−0.0412080			−0.0206040	+	0.999788i	$0.506559\pi$
$132$		0			0
$133$	−4.65153			−0.403339
$134$	−12.4855			−1.07859
$135$		0			0
$136$	−4.34847			−0.372878
$137$			13.2207i			1.12952i	0.825254	+	0.564762i	$0.191032\pi$
$137$			13.2207i			1.12952i	−0.825254	+	0.564762i	$0.808968\pi$
$138$		0			0
$139$			5.19375i			0.440528i	0.975440	+	0.220264i	$0.0706919\pi$
$139$			5.19375i			0.440528i	−0.975440	+	0.220264i	$0.929308\pi$
$140$	−12.4855			−1.05522
$141$		0			0
$142$			2.96786i			0.249057i
$143$	−15.0558	−	5.02118i	−1.25903	−	0.419892i
$144$		0			0
$145$			12.8719i			1.06896i
$146$			27.3629i			2.26457i
$147$		0			0
$148$	−11.8990			−0.978090
$149$	−3.61953			−0.296524			−0.148262	−	0.988948i	$-0.547368\pi$
$149$	−3.61953			−0.296524			−0.148262	+	0.988948i	$0.547368\pi$
$150$		0			0
$151$		−	16.5068i		−	1.34330i	−0.740867	−	0.671652i	$-0.765585\pi$
$151$		−	16.5068i		−	1.34330i	0.740867	−	0.671652i	$-0.234415\pi$
$152$		−	6.14966i		−	0.498804i
$153$		0			0
$154$	18.7980	+	6.26922i	1.51478	+	0.505188i
$155$		−	13.2207i		−	1.06191i
$156$		0			0
$157$	−13.0000			−1.03751			−0.518756	−	0.854922i	$-0.673605\pi$
$157$	−13.0000			−1.03751			−0.518756	+	0.854922i	$0.673605\pi$
$158$			27.5378i			2.19079i
$159$		0			0
$160$			6.26922i			0.495625i
$161$	13.6647			1.07693
$162$		0			0
$163$	−7.79796			−0.610783			−0.305392	−	0.952227i	$-0.598787\pi$
$163$	−7.79796			−0.610783			−0.305392	+	0.952227i	$0.598787\pi$
$164$	−8.05254			−0.628798
$165$		0			0
$166$	14.4495			1.12150
$167$	21.8232			1.68873			0.844365	−	0.535768i	$-0.179978\pi$
$167$	21.8232			1.68873			0.844365	+	0.535768i	$0.179978\pi$
$168$		0			0
$169$	−9.89898			−0.761460
$170$			4.24264i			0.325396i
$171$		0			0
$172$		−	30.4542i		−	2.32211i
$173$	−15.5274			−1.18053			−0.590264	−	0.807210i	$-0.700976\pi$
$173$	−15.5274			−1.18053			−0.590264	+	0.807210i	$0.700976\pi$
$174$		0			0
$175$		−	7.67819i		−	0.580417i
$176$	−1.04930	+	3.14626i	−0.0790936	+	0.237159i
$177$		0			0
$178$		−	17.6572i		−	1.32346i
$179$			2.65345i			0.198328i	0.995071	+	0.0991642i	$0.0316169\pi$
$179$			2.65345i			0.198328i	−0.995071	+	0.0991642i	$0.968383\pi$
$180$		0			0
$181$	−2.10102			−0.156168			−0.0780838	−	0.996947i	$-0.524880\pi$
$181$	−2.10102			−0.156168			−0.0780838	+	0.996947i	$0.524880\pi$
$182$	28.5906			2.11928
$183$		0			0
$184$			18.0657i			1.33182i
$185$			4.87832i			0.358661i
$186$		0			0
$187$	1.34847	−	4.04332i	0.0986098	−	0.295677i
$188$		−	37.4373i		−	2.73040i
$189$		0			0
$190$	−6.00000			−0.435286
$191$		−	19.3383i		−	1.39927i	−0.714501	−	0.699635i	$-0.753346\pi$
$191$		−	19.3383i		−	1.39927i	0.714501	−	0.699635i	$-0.246654\pi$
$192$		0			0
$193$			20.1417i			1.44983i	0.688839	+	0.724914i	$0.258121\pi$
$193$			20.1417i			1.44983i	−0.688839	+	0.724914i	$0.741879\pi$
$194$	−19.4888			−1.39921
$195$		0			0
$196$	1.55051			0.110751
$197$	12.1437			0.865204			0.432602	−	0.901585i	$-0.357596\pi$
$197$	12.1437			0.865204			0.432602	+	0.901585i	$0.357596\pi$
$198$		0			0
$199$	11.5505			0.818794			0.409397	−	0.912356i	$-0.365739\pi$
$199$	11.5505			0.818794			0.409397	+	0.912356i	$0.365739\pi$
$200$	10.1511			0.717793
$201$		0			0
$202$	18.7980			1.32262
$203$			23.2952i			1.63500i
$204$		0			0
$205$			3.30136i			0.230577i
$206$	35.8297			2.49637
$207$		0			0
$208$			4.78529i			0.331800i
$209$	5.71812	+	1.90702i	0.395531	+	0.131912i
$210$		0			0
$211$			14.2809i			0.983138i	0.870839	+	0.491569i	$0.163577\pi$
$211$			14.2809i			0.983138i	−0.870839	+	0.491569i	$0.836423\pi$
$212$		−	2.19275i		−	0.150599i
$213$		0			0
$214$	25.3485			1.73279
$215$	−12.4855			−0.851507
$216$		0			0
$217$		−	23.9264i		−	1.62423i
$218$		−	18.8776i		−	1.27855i
$219$		0			0
$220$	15.3485	+	5.11879i	1.03479	+	0.345109i
$221$		−	6.14966i		−	0.413671i
$222$		0			0
$223$	1.75255			0.117360			0.0586798	−	0.998277i	$-0.481311\pi$
$223$	1.75255			0.117360			0.0586798	+	0.998277i	$0.481311\pi$
$224$			11.3458i			0.758074i
$225$		0			0
$226$			16.5068i			1.09802i
$227$	7.34507			0.487509			0.243755	−	0.969837i	$-0.421621\pi$
$227$	7.34507			0.487509			0.243755	+	0.969837i	$0.421621\pi$
$228$		0			0
$229$	−4.55051			−0.300706			−0.150353	−	0.988632i	$-0.548041\pi$
$229$	−4.55051			−0.300706			−0.150353	+	0.988632i	$0.548041\pi$
$230$	17.6260			1.16223
$231$		0			0
$232$	−30.7980			−2.02199
$233$	−19.7246			−1.29220			−0.646101	−	0.763252i	$-0.723602\pi$
$233$	−19.7246			−1.29220			−0.646101	+	0.763252i	$0.723602\pi$
$234$		0			0
$235$	−15.3485			−1.00122
$236$			13.0458i			0.849207i
$237$		0			0
$238$			7.67819i			0.497703i
$239$	14.1125			0.912861			0.456430	−	0.889759i	$-0.349128\pi$
$239$	14.1125			0.912861			0.456430	+	0.889759i	$0.349128\pi$
$240$		0			0
$241$			16.8403i			1.08478i	0.840127	+	0.542390i	$0.182480\pi$
$241$			16.8403i			1.08478i	−0.840127	+	0.542390i	$0.817520\pi$
$242$	−20.5381	−	15.4135i	−1.32024	−	0.990816i
$243$		0			0
$244$		−	44.4016i		−	2.84252i
$245$		−	0.635674i		−	0.0406118i
$246$		0			0
$247$	8.69694			0.553373
$248$	31.6325			2.00867
$249$		0			0
$250$		−	26.4109i		−	1.67037i
$251$		−	10.8530i		−	0.685036i	−0.939511	−	0.342518i	$-0.888720\pi$
$251$		−	10.8530i		−	0.685036i	0.939511	−	0.342518i	$-0.111280\pi$
$252$		0			0
$253$	−16.7980	−	5.60221i	−1.05608	−	0.352208i
$254$		−	4.24264i		−	0.266207i
$255$		0			0
$256$	−21.8990			−1.36869
$257$		−	8.34242i		−	0.520386i	−0.965557	−	0.260193i	$-0.916214\pi$
$257$		−	8.34242i		−	0.520386i	0.965557	−	0.260193i	$-0.0837861\pi$
$258$		0			0
$259$			8.82861i			0.548583i
$260$	23.3441			1.44774
$261$		0			0
$262$	−1.10102			−0.0680213
$263$	−24.9711			−1.53978			−0.769892	−	0.638175i	$-0.779690\pi$
$263$	−24.9711			−1.53978			−0.769892	+	0.638175i	$0.779690\pi$
$264$		0			0
$265$	−0.898979			−0.0552239
$266$	−10.8586			−0.665783
$267$		0			0
$268$	−18.4495			−1.12698
$269$		−	4.87832i		−	0.297436i	−0.988880	−	0.148718i	$-0.952485\pi$
$269$		−	4.87832i		−	0.297436i	0.988880	−	0.148718i	$-0.0475147\pi$
$270$		0			0
$271$			10.3125i			0.626442i	0.949680	+	0.313221i	$0.101408\pi$
$271$			10.3125i			0.626442i	−0.949680	+	0.313221i	$0.898592\pi$
$272$	−1.28512			−0.0779218
$273$		0			0
$274$			30.8627i			1.86448i
$275$	−3.14789	+	9.43879i	−0.189825	+	0.569181i
$276$		0			0
$277$			20.4752i			1.23023i	0.788436	+	0.615117i	$0.210891\pi$
$277$			20.4752i			1.23023i	−0.788436	+	0.615117i	$0.789109\pi$
$278$			12.1244i			0.727171i
$279$		0			0
$280$	−12.2474			−0.731925
$281$	16.9185			1.00928			0.504638	−	0.863331i	$-0.331626\pi$
$281$	16.9185			1.00928			0.504638	+	0.863331i	$0.331626\pi$
$282$		0			0
$283$		−	12.2049i		−	0.725508i	−0.931885	−	0.362754i	$-0.881837\pi$
$283$		−	12.2049i		−	0.725508i	0.931885	−	0.362754i	$-0.118163\pi$
$284$			4.38551i			0.260232i
$285$		0			0
$286$	−35.1464	−	11.7215i	−2.07825	−	0.693108i
$287$			5.97469i			0.352675i
$288$		0			0
$289$	−15.3485			−0.902851
$290$			30.0484i			1.76450i
$291$		0			0
$292$			40.4332i			2.36618i
$293$	−15.0558			−0.879568			−0.439784	−	0.898104i	$-0.644945\pi$
$293$	−15.0558			−0.879568			−0.439784	+	0.898104i	$0.644945\pi$
$294$		0			0
$295$	5.34847			0.311400
$296$	−11.6721			−0.678425
$297$		0			0
$298$	−8.44949			−0.489466
$299$	−25.5487			−1.47752
$300$		0			0
$301$	−22.5959			−1.30241
$302$		−	38.5337i		−	2.21737i
$303$		0			0
$304$		−	1.81743i		−	0.104237i
$305$	−18.2037			−1.04234
$306$		0			0
$307$			7.67819i			0.438217i	0.975700	+	0.219109i	$0.0703149\pi$
$307$			7.67819i			0.438217i	−0.975700	+	0.219109i	$0.929685\pi$
$308$	27.7771	+	9.26382i	1.58275	+	0.527855i
$309$		0			0
$310$		−	30.8627i		−	1.75288i
$311$			1.87492i			0.106317i	0.998586	+	0.0531583i	$0.0169288\pi$
$311$			1.87492i			0.106317i	−0.998586	+	0.0531583i	$0.983071\pi$
$312$		0			0
$313$	−17.5959			−0.994580			−0.497290	−	0.867584i	$-0.665672\pi$
$313$	−17.5959			−0.994580			−0.497290	+	0.867584i	$0.665672\pi$
$314$	−30.3474			−1.71260
$315$		0			0
$316$			40.6918i			2.28909i
$317$			25.1701i			1.41369i	0.707366	+	0.706847i	$0.249883\pi$
$317$			25.1701i			1.41369i	−0.707366	+	0.706847i	$0.750117\pi$
$318$		0			0
$319$	9.55051	−	28.6368i	0.534726	−	1.60335i
$320$			17.4634i			0.976232i
$321$		0			0
$322$	31.8990			1.77766
$323$			2.33562i			0.129957i
$324$		0			0
$325$			14.3559i			0.796320i
$326$	−18.2037			−1.00821
$327$		0			0
$328$	−7.89898			−0.436148
$329$	−27.7771			−1.53140
$330$		0			0
$331$	−14.6515			−0.805321			−0.402660	−	0.915349i	$-0.631914\pi$
$331$	−14.6515			−0.805321			−0.402660	+	0.915349i	$0.631914\pi$
$332$	21.3516			1.17182
$333$		0			0
$334$	50.9444			2.78755
$335$			7.56388i			0.413259i
$336$		0			0
$337$			3.96837i			0.216171i	0.994142	+	0.108085i	$0.0344720\pi$
$337$			3.96837i			0.216171i	−0.994142	+	0.108085i	$0.965528\pi$
$338$	−23.1083			−1.25693
$339$		0			0
$340$			6.26922i			0.339996i
$341$	−9.80930	+	29.4128i	−0.531204	+	1.59279i
$342$		0			0
$343$		−	19.0662i		−	1.02948i
$344$		−	29.8735i		−	1.61067i
$345$		0			0
$346$	−36.2474			−1.94867
$347$	−23.3441			−1.25318			−0.626590	−	0.779349i	$-0.715550\pi$
$347$	−23.3441			−1.25318			−0.626590	+	0.779349i	$0.715550\pi$
$348$		0			0
$349$			3.30136i			0.176718i	0.996089	+	0.0883589i	$0.0281622\pi$
$349$			3.30136i			0.176718i	−0.996089	+	0.0883589i	$0.971838\pi$
$350$		−	17.9241i		−	0.958082i
$351$		0			0
$352$	4.65153	−	13.9474i	0.247927	−	0.743399i
$353$			4.38551i			0.233417i	0.993166	+	0.116708i	$0.0372343\pi$
$353$			4.38551i			0.233417i	−0.993166	+	0.116708i	$0.962766\pi$
$354$		0			0
$355$	1.79796			0.0954258
$356$		−	26.0915i		−	1.38285i
$357$		0			0
$358$			6.19426i			0.327377i
$359$	28.5906			1.50896			0.754478	−	0.656326i	$-0.227890\pi$
$359$	28.5906			1.50896			0.754478	+	0.656326i	$0.227890\pi$
$360$		0			0
$361$	15.6969			0.826155
$362$	−4.90465			−0.257783
$363$		0			0
$364$	42.2474			2.21437
$365$	16.5767			0.867665
$366$		0			0
$367$	31.3939			1.63875			0.819374	−	0.573260i	$-0.194322\pi$
$367$	31.3939			1.63875			0.819374	+	0.573260i	$0.194322\pi$
$368$			5.33902i			0.278316i
$369$		0			0
$370$			11.3880i			0.592034i
$371$	−1.62694			−0.0844667
$372$		0			0
$373$		−	15.0229i		−	0.777855i	−0.921268	−	0.388927i	$-0.872846\pi$
$373$		−	15.0229i		−	0.777855i	0.921268	−	0.388927i	$-0.127154\pi$
$374$	3.14789	−	9.43879i	0.162773	−	0.488068i
$375$		0			0
$376$		−	36.7234i		−	1.89387i
$377$		−	43.5549i		−	2.24319i
$378$		0			0
$379$	−2.65153			−0.136200			−0.0681000	−	0.997679i	$-0.521694\pi$
$379$	−2.65153			−0.136200			−0.0681000	+	0.997679i	$0.521694\pi$
$380$	−8.86601			−0.454817
$381$		0			0
$382$		−	45.1436i		−	2.30975i
$383$			2.82843i			0.144526i	0.997386	+	0.0722629i	$0.0230221\pi$
$383$			2.82843i			0.144526i	−0.997386	+	0.0722629i	$0.976978\pi$
$384$		0			0
$385$	3.79796	−	11.3880i	0.193562	−	0.580387i
$386$			47.0190i			2.39320i
$387$		0			0
$388$	−28.7980			−1.46199
$389$			9.40669i			0.476938i	0.971150	+	0.238469i	$0.0766455\pi$
$389$			9.40669i			0.476938i	−0.971150	+	0.238469i	$0.923354\pi$
$390$		0			0
$391$		−	6.86127i		−	0.346989i
$392$	1.52094			0.0768192
$393$		0			0
$394$	28.3485			1.42818
$395$	16.6827			0.839399
$396$		0			0
$397$	4.20204			0.210894			0.105447	−	0.994425i	$-0.466373\pi$
$397$	4.20204			0.210894			0.105447	+	0.994425i	$0.466373\pi$
$398$	26.9637			1.35157
$399$		0			0
$400$	3.00000			0.150000
$401$		−	14.1421i		−	0.706225i	−0.935581	−	0.353112i	$-0.885123\pi$
$401$		−	14.1421i		−	0.706225i	0.935581	−	0.353112i	$-0.114877\pi$
$402$		0			0
$403$			44.7351i			2.22841i
$404$	27.7771			1.38196
$405$		0			0
$406$			54.3806i			2.69887i
$407$	3.61953	−	10.8530i	0.179414	−	0.537964i
$408$		0			0
$409$		−	13.2054i		−	0.652967i	−0.945203	−	0.326484i	$-0.894136\pi$
$409$		−	13.2054i		−	0.652967i	0.945203	−	0.326484i	$-0.105864\pi$
$410$			7.70674i			0.380609i
$411$		0			0
$412$	52.9444			2.60838
$413$	9.67948			0.476296
$414$		0			0
$415$		−	8.75366i		−	0.429700i
$416$		−	21.2132i		−	1.04006i
$417$		0			0
$418$	13.3485	+	4.45178i	0.652895	+	0.217744i
$419$			15.1278i			0.739039i	0.929223	+	0.369520i	$0.120478\pi$
$419$			15.1278i			0.739039i	−0.929223	+	0.369520i	$0.879522\pi$
$420$		0			0
$421$	8.30306			0.404666			0.202333	−	0.979317i	$-0.435148\pi$
$421$	8.30306			0.404666			0.202333	+	0.979317i	$0.435148\pi$
$422$			33.3376i			1.62285i
$423$		0			0
$424$		−	2.15094i		−	0.104459i
$425$	−3.85536			−0.187012
$426$		0			0
$427$	−32.9444			−1.59429
$428$	37.4566			1.81053
$429$		0			0
$430$	−29.1464			−1.40557
$431$	−19.7246			−0.950101			−0.475050	−	0.879959i	$-0.657570\pi$
$431$	−19.7246			−0.950101			−0.475050	+	0.879959i	$0.657570\pi$
$432$		0			0
$433$	6.34847			0.305088			0.152544	−	0.988297i	$-0.451253\pi$
$433$	6.34847			0.305088			0.152544	+	0.988297i	$0.451253\pi$
$434$		−	55.8542i		−	2.68109i
$435$		0			0
$436$		−	27.8948i		−	1.33592i
$437$	9.70330			0.464172
$438$		0			0
$439$		−	3.55991i		−	0.169905i	−0.996385	−	0.0849527i	$-0.972926\pi$
$439$		−	3.55991i		−	0.169905i	0.996385	−	0.0849527i	$-0.0270739\pi$
$440$	15.0558	+	5.02118i	0.717756	+	0.239375i
$441$		0			0
$442$		−	14.3559i		−	0.682839i
$443$		−	33.0197i		−	1.56881i	−0.620246	−	0.784407i	$-0.712967\pi$
$443$		−	33.0197i		−	1.56881i	0.620246	−	0.784407i	$-0.287033\pi$
$444$		0			0
$445$	−10.6969			−0.507084
$446$	4.09118			0.193723
$447$		0			0
$448$			31.6046i			1.49318i
$449$			17.8133i			0.840662i	0.907371	+	0.420331i	$0.138086\pi$
$449$			17.8133i			0.840662i	−0.907371	+	0.420331i	$0.861914\pi$
$450$		0			0
$451$	2.44949	−	7.34468i	0.115342	−	0.345848i
$452$			24.3916i			1.14728i
$453$		0			0
$454$	17.1464			0.804722
$455$		−	17.3205i		−	0.811998i
$456$		0			0
$457$			28.3782i			1.32748i	0.747965	+	0.663739i	$0.231031\pi$
$457$			28.3782i			1.32748i	−0.747965	+	0.663739i	$0.768969\pi$
$458$	−10.6228			−0.496370
$459$		0			0
$460$	26.0454			1.21437
$461$	29.0623			1.35356			0.676782	−	0.736183i	$-0.263374\pi$
$461$	29.0623			1.35356			0.676782	+	0.736183i	$0.263374\pi$
$462$		0			0
$463$	3.10102			0.144117			0.0720583	−	0.997400i	$-0.477043\pi$
$463$	3.10102			0.144117			0.0720583	+	0.997400i	$0.477043\pi$
$464$	−9.10183			−0.422542
$465$		0			0
$466$	−46.0454			−2.13301
$467$			29.5877i			1.36916i	0.728939	+	0.684578i	$0.240014\pi$
$467$			29.5877i			1.36916i	−0.728939	+	0.684578i	$0.759986\pi$
$468$		0			0
$469$			13.6889i			0.632093i
$470$	−35.8297			−1.65270
$471$		0			0
$472$			12.7970i			0.589029i
$473$	27.7771	+	9.26382i	1.27719	+	0.425951i
$474$		0			0
$475$		−	5.45230i		−	0.250169i
$476$			11.3458i			0.520035i
$477$		0			0
$478$	32.9444			1.50684
$479$	8.18236			0.373862			0.186931	−	0.982373i	$-0.440146\pi$
$479$	8.18236			0.373862			0.186931	+	0.982373i	$0.440146\pi$
$480$		0			0
$481$		−	16.5068i		−	0.752645i
$482$			39.3123i			1.79062i
$483$		0			0
$484$	−30.3485	−	22.7760i	−1.37948	−	1.03527i
$485$			11.8065i			0.536106i
$486$		0			0
$487$	−13.7980			−0.625245			−0.312623	−	0.949877i	$-0.601208\pi$
$487$	−13.7980			−0.625245			−0.312623	+	0.949877i	$0.601208\pi$
$488$		−	43.5549i		−	1.97164i
$489$		0			0
$490$		−	1.48393i		−	0.0670370i
$491$	10.9646			0.494825			0.247413	−	0.968910i	$-0.420420\pi$
$491$	10.9646			0.494825			0.247413	+	0.968910i	$0.420420\pi$
$492$		0			0
$493$	11.6969			0.526804
$494$	20.3023			0.913442
$495$		0			0
$496$	9.34847			0.419759
$497$	3.25389			0.145957
$498$		0			0
$499$	6.65153			0.297763			0.148882	−	0.988855i	$-0.452433\pi$
$499$	6.65153			0.297763			0.148882	+	0.988855i	$0.452433\pi$
$500$		−	39.0265i		−	1.74532i
$501$		0			0
$502$		−	25.3354i		−	1.13077i
$503$	15.1618			0.676030			0.338015	−	0.941141i	$-0.390244\pi$
$503$	15.1618			0.676030			0.338015	+	0.941141i	$0.390244\pi$
$504$		0			0
$505$		−	11.3880i		−	0.506760i
$506$	−39.2134	−	13.0779i	−1.74325	−	0.581382i
$507$		0			0
$508$		−	6.26922i		−	0.278151i
$509$		−	27.2200i		−	1.20651i	−0.797550	−	0.603253i	$-0.793871\pi$
$509$		−	27.2200i		−	1.20651i	0.797550	−	0.603253i	$-0.206129\pi$
$510$		0			0
$511$	30.0000			1.32712
$512$	−11.2004			−0.494993
$513$		0			0
$514$		−	19.4747i		−	0.858990i
$515$		−	21.7060i		−	0.956481i
$516$		0			0
$517$	34.1464	+	11.3880i	1.50176	+	0.500844i
$518$			20.6096i			0.905535i
$519$		0			0
$520$	22.8990			1.00419
$521$		−	25.6629i		−	1.12431i	−0.827031	−	0.562157i	$-0.809972\pi$
$521$		−	25.6629i		−	1.12431i	0.827031	−	0.562157i	$-0.190028\pi$
$522$		0			0
$523$			1.40897i			0.0616101i	0.999525	+	0.0308051i	$0.00980711\pi$
$523$			1.40897i			0.0616101i	−0.999525	+	0.0308051i	$0.990193\pi$
$524$	−1.62694			−0.0710733
$525$		0			0
$526$	−58.2929			−2.54169
$527$	−12.0139			−0.523333
$528$		0			0
$529$	−5.50510			−0.239352
$530$	−2.09859			−0.0911569
$531$		0			0
$532$	−16.0454			−0.695657
$533$		−	11.1708i		−	0.483863i
$534$		0			0
$535$		−	15.3564i		−	0.663914i
$536$	−18.0977			−0.781700
$537$		0			0
$538$		−	11.3880i		−	0.490972i
$539$	−0.471647	+	1.41421i	−0.0203153	+	0.0609145i
$540$		0			0
$541$			27.8948i			1.19929i	0.800266	+	0.599646i	$0.204692\pi$
$541$			27.8948i			1.19929i	−0.800266	+	0.599646i	$0.795308\pi$
$542$			24.0737i			1.03406i
$543$		0			0
$544$	5.69694			0.244254
$545$	−11.4362			−0.489875
$546$		0			0
$547$		−	13.2804i		−	0.567829i	−0.958850	−	0.283914i	$-0.908367\pi$
$547$		−	13.2804i		−	0.567829i	0.958850	−	0.283914i	$-0.0916331\pi$
$548$			45.6048i			1.94814i
$549$		0			0
$550$	−7.34847	+	22.0341i	−0.313340	+	0.939535i
$551$			16.5420i			0.704712i
$552$		0			0
$553$	30.1918			1.28389
$554$			47.7975i			2.03072i
$555$		0			0
$556$			17.9158i			0.759798i
$557$	−17.8618			−0.756831			−0.378415	−	0.925636i	$-0.623531\pi$
$557$	−17.8618			−0.756831			−0.378415	+	0.925636i	$0.623531\pi$
$558$		0			0
$559$	42.2474			1.78688
$560$	−3.61953			−0.152953
$561$		0			0
$562$	39.4949			1.66599
$563$	−1.62694			−0.0685675			−0.0342837	−	0.999412i	$-0.510915\pi$
$563$	−1.62694			−0.0685675			−0.0342837	+	0.999412i	$0.510915\pi$
$564$		0			0
$565$	10.0000			0.420703
$566$		−	28.4914i		−	1.19758i
$567$		0			0
$568$			4.30188i			0.180503i
$569$	12.4855			0.523421			0.261711	−	0.965146i	$-0.415713\pi$
$569$	12.4855			0.523421			0.261711	+	0.965146i	$0.415713\pi$
$570$		0			0
$571$			6.19426i			0.259222i	0.991565	+	0.129611i	$0.0413728\pi$
$571$			6.19426i			0.259222i	−0.991565	+	0.129611i	$0.958627\pi$
$572$	−51.9348	−	17.3205i	−2.17150	−	0.724207i
$573$		0			0
$574$			13.9474i			0.582153i
$575$			16.0171i			0.667957i
$576$		0			0
$577$	−37.4949			−1.56093			−0.780467	−	0.625198i	$-0.785018\pi$
$577$	−37.4949			−1.56093			−0.780467	+	0.625198i	$0.785018\pi$
$578$	−35.8297			−1.49032
$579$		0			0
$580$			44.4016i			1.84368i
$581$		−	15.8421i		−	0.657240i
$582$		0			0
$583$	2.00000	+	0.667010i	0.0828315	+	0.0276247i
$584$			39.6622i			1.64123i
$585$		0			0
$586$	−35.1464			−1.45189
$587$		−	10.4244i		−	0.430262i	−0.976585	−	0.215131i	$-0.930982\pi$
$587$		−	10.4244i		−	0.430262i	0.976585	−	0.215131i	$-0.0690178\pi$
$588$		0			0
$589$		−	16.9902i		−	0.700070i
$590$	12.4855			0.514022
$591$		0			0
$592$	−3.44949			−0.141773
$593$	−21.5874			−0.886487			−0.443244	−	0.896401i	$-0.646172\pi$
$593$	−21.5874			−0.886487			−0.443244	+	0.896401i	$0.646172\pi$
$594$		0			0
$595$	4.65153			0.190694
$596$	−12.4855			−0.511428
$597$		0			0
$598$	−59.6413			−2.43892
$599$			26.1236i			1.06738i	0.845679	+	0.533691i	$0.179196\pi$
$599$			26.1236i			1.06738i	−0.845679	+	0.533691i	$0.820804\pi$
$600$		0			0
$601$		−	35.4980i		−	1.44800i	−0.689802	−	0.723998i	$-0.742303\pi$
$601$		−	35.4980i		−	1.44800i	0.689802	−	0.723998i	$-0.257697\pi$
$602$	−52.7482			−2.14986
$603$		0			0
$604$		−	56.9400i		−	2.31686i
$605$	−9.33766	+	12.4422i	−0.379630	+	0.505847i
$606$		0			0
$607$			37.1319i			1.50714i	0.657370	+	0.753568i	$0.271669\pi$
$607$			37.1319i			1.50714i	−0.657370	+	0.753568i	$0.728331\pi$
$608$			8.05669i			0.326742i
$609$		0			0
$610$	−42.4949			−1.72057
$611$	51.9348			2.10106
$612$		0			0
$613$		−	21.6256i		−	0.873450i	−0.899595	−	0.436725i	$-0.856138\pi$
$613$		−	21.6256i		−	0.873450i	0.899595	−	0.436725i	$-0.143862\pi$
$614$			17.9241i			0.723357i
$615$		0			0
$616$	27.2474	+	9.08716i	1.09783	+	0.366132i
$617$		−	37.2624i		−	1.50013i	−0.661366	−	0.750063i	$-0.730023\pi$
$617$		−	37.2624i		−	1.50013i	0.661366	−	0.750063i	$-0.269977\pi$
$618$		0			0
$619$	−26.0454			−1.04685			−0.523427	−	0.852071i	$-0.675347\pi$
$619$	−26.0454			−1.04685			−0.523427	+	0.852071i	$0.675347\pi$
$620$		−	45.6048i		−	1.83153i
$621$		0			0
$622$			4.37683i			0.175495i
$623$	−19.3590			−0.775600
$624$		0			0
$625$	−1.00000			−0.0400000
$626$	−41.0762			−1.64173
$627$		0			0
$628$	−44.8434			−1.78945
$629$	4.43300			0.176755
$630$		0			0
$631$	16.9444			0.674545			0.337273	−	0.941407i	$-0.390496\pi$
$631$	16.9444			0.674545			0.337273	+	0.941407i	$0.390496\pi$
$632$			39.9158i			1.58777i
$633$		0			0
$634$			58.7575i			2.33356i
$635$	−2.57024			−0.101997
$636$		0			0
$637$			2.15094i			0.0852233i
$638$	22.2948	−	66.8501i	0.882661	−	2.64662i
$639$		0			0
$640$			28.2283i			1.11582i
$641$			46.3833i			1.83203i	0.401143	+	0.916015i	$0.368613\pi$
$641$			46.3833i			1.83203i	−0.401143	+	0.916015i	$0.631387\pi$
$642$		0			0
$643$	−32.8990			−1.29741			−0.648705	−	0.761040i	$-0.724689\pi$
$643$	−32.8990			−1.29741			−0.648705	+	0.761040i	$0.724689\pi$
$644$	47.1361			1.85742
$645$		0			0
$646$			5.45230i			0.214518i
$647$			13.2207i			0.519761i	0.965641	+	0.259880i	$0.0836831\pi$
$647$			13.2207i			0.519761i	−0.965641	+	0.259880i	$0.916317\pi$
$648$		0			0
$649$	−11.8990	−	3.96837i	−0.467076	−	0.155772i
$650$			33.5125i			1.31447i
$651$		0			0
$652$	−26.8990			−1.05345
$653$		−	47.2261i		−	1.84810i	−0.382273	−	0.924050i	$-0.624859\pi$
$653$		−	47.2261i		−	1.84810i	0.382273	−	0.924050i	$-0.375141\pi$
$654$		0			0
$655$			0.667010i			0.0260622i
$656$	−2.33441			−0.0911436
$657$		0			0
$658$	−64.8434			−2.52786
$659$	40.4985			1.57760			0.788799	−	0.614651i	$-0.210703\pi$
$659$	40.4985			1.57760			0.788799	+	0.614651i	$0.210703\pi$
$660$		0			0
$661$	25.6969			0.999495			0.499748	−	0.866171i	$-0.333426\pi$
$661$	25.6969			0.999495			0.499748	+	0.866171i	$0.333426\pi$
$662$	−34.2027			−1.32933
$663$		0			0
$664$	20.9444			0.812800
$665$			6.57826i			0.255094i
$666$		0			0
$667$		−	48.5948i		−	1.88160i
$668$	75.2789			2.91263
$669$		0			0
$670$			17.6572i			0.682158i
$671$	40.4985	+	13.5065i	1.56343	+	0.521411i
$672$		0			0
$673$		−	16.5068i		−	0.636290i	−0.948042	−	0.318145i	$-0.896940\pi$
$673$		−	16.5068i		−	0.636290i	0.948042	−	0.318145i	$-0.103060\pi$
$674$			9.26382i			0.356829i
$675$		0			0
$676$	−34.1464			−1.31332
$677$	6.05995			0.232903			0.116451	−	0.993196i	$-0.462848\pi$
$677$	6.05995			0.232903			0.116451	+	0.993196i	$0.462848\pi$
$678$		0			0
$679$			21.3670i			0.819992i
$680$			6.14966i			0.235829i
$681$		0			0
$682$	−22.8990	+	68.6616i	−0.876847	+	2.62919i
$683$		−	10.5031i		−	0.401888i	−0.979603	−	0.200944i	$-0.935599\pi$
$683$		−	10.5031i		−	0.401888i	0.979603	−	0.200944i	$-0.0644010\pi$
$684$		0			0
$685$	18.6969			0.714373
$686$		−	44.5084i		−	1.69934i
$687$		0			0
$688$		−	8.82861i		−	0.336588i
$689$	3.04189			0.115887
$690$		0			0
$691$	−27.3939			−1.04211			−0.521056	−	0.853522i	$-0.674462\pi$
$691$	−27.3939			−1.04211			−0.521056	+	0.853522i	$0.674462\pi$
$692$	−53.5617			−2.03611
$693$		0			0
$694$	−54.4949			−2.06860
$695$	7.34507			0.278614
$696$		0			0
$697$	3.00000			0.113633
$698$			7.70674i			0.291704i
$699$		0			0
$700$		−	26.4858i		−	1.00107i
$701$	−1.28512			−0.0485383			−0.0242691	−	0.999705i	$-0.507726\pi$
$701$	−1.28512			−0.0485383			−0.0242691	+	0.999705i	$0.507726\pi$
$702$		0			0
$703$			6.26922i			0.236448i
$704$	12.9572	−	38.8515i	0.488342	−	1.46427i
$705$		0			0
$706$			10.2376i			0.385297i
$707$		−	20.6096i		−	0.775105i
$708$		0			0
$709$	30.3485			1.13976			0.569880	−	0.821728i	$-0.306990\pi$
$709$	30.3485			1.13976			0.569880	+	0.821728i	$0.306990\pi$
$710$	4.19718			0.157517
$711$		0			0
$712$		−	25.5940i		−	0.959174i
$713$			49.9116i			1.86921i
$714$		0			0
$715$	−7.10102	+	21.2921i	−0.265563	+	0.796279i
$716$			9.15306i			0.342066i
$717$		0			0
$718$	66.7423			2.49080
$719$		−	19.5133i		−	0.727722i	−0.931453	−	0.363861i	$-0.881458\pi$
$719$		−	19.5133i		−	0.727722i	0.931453	−	0.363861i	$-0.118542\pi$
$720$		0			0
$721$		−	39.2828i		−	1.46297i
$722$	36.6432			1.36372
$723$		0			0
$724$	−7.24745			−0.269349
$725$	−27.3055			−1.01410
$726$		0			0
$727$	−2.65153			−0.0983398			−0.0491699	−	0.998790i	$-0.515658\pi$
$727$	−2.65153			−0.0983398			−0.0491699	+	0.998790i	$0.515658\pi$
$728$	41.4418			1.53594
$729$		0			0
$730$	38.6969			1.43224
$731$			11.3458i			0.419640i
$732$		0			0
$733$			29.0452i			1.07281i	0.843961	+	0.536405i	$0.180218\pi$
$733$			29.0452i			1.07281i	−0.843961	+	0.536405i	$0.819782\pi$
$734$	73.2863			2.70505
$735$		0			0
$736$		−	23.6679i		−	0.872410i
$737$	5.61212	−	16.8277i	0.206725	−	0.619856i
$738$		0			0
$739$			24.8520i			0.914196i	0.889416	+	0.457098i	$0.151111\pi$
$739$			24.8520i			0.914196i	−0.889416	+	0.457098i	$0.848889\pi$
$740$			16.8277i			0.618598i
$741$		0			0
$742$	−3.79796			−0.139427
$743$	−53.9274			−1.97840			−0.989201	−	0.146563i	$-0.953179\pi$
$743$	−53.9274			−1.97840			−0.989201	+	0.146563i	$0.953179\pi$
$744$		0			0
$745$			5.11879i			0.187538i
$746$		−	35.0696i		−	1.28399i
$747$		0			0
$748$	4.65153	−	13.9474i	0.170077	−	0.509968i
$749$		−	27.7915i		−	1.01548i
$750$		0			0
$751$	21.3485			0.779017			0.389508	−	0.921023i	$-0.372645\pi$
$751$	21.3485			0.779017			0.389508	+	0.921023i	$0.372645\pi$
$752$		−	10.8530i		−	0.395768i
$753$		0			0
$754$		−	101.675i		−	3.70279i
$755$	−23.3441			−0.849580
$756$		0			0
$757$	21.8990			0.795932			0.397966	−	0.917400i	$-0.369716\pi$
$757$	21.8990			0.795932			0.397966	+	0.917400i	$0.369716\pi$
$758$	−6.18977			−0.224823
$759$		0			0
$760$	−8.69694			−0.315471
$761$	26.9637			0.977432			0.488716	−	0.872443i	$-0.337465\pi$
$761$	26.9637			0.977432			0.488716	+	0.872443i	$0.337465\pi$
$762$		0			0
$763$	−20.6969			−0.749279
$764$		−	66.7072i		−	2.41338i
$765$		0			0
$766$			6.60272i			0.238566i
$767$	−18.0977			−0.653469
$768$		0			0
$769$		−	13.2054i		−	0.476200i	−0.971241	−	0.238100i	$-0.923475\pi$
$769$		−	13.2054i		−	0.476200i	0.971241	−	0.238100i	$-0.0765247\pi$
$770$	8.86601	−	26.5843i	0.319509	−	0.958033i
$771$		0			0
$772$			69.4785i			2.50058i
$773$			46.7333i			1.68088i	0.541906	+	0.840439i	$0.317703\pi$
$773$			46.7333i			1.68088i	−0.541906	+	0.840439i	$0.682297\pi$
$774$		0			0
$775$	28.0454			1.00742
$776$	−28.2488			−1.01407
$777$		0			0
$778$			21.9591i			0.787272i
$779$			4.24264i			0.152008i
$780$		0			0
$781$	−4.00000	−	1.33402i	−0.143131	−	0.0477350i
$782$		−	16.0171i		−	0.572769i
$783$		0			0
$784$	0.449490			0.0160532
$785$			18.3848i			0.656181i
$786$		0			0
$787$		−	4.11828i		−	0.146801i	−0.997303	−	0.0734004i	$-0.976615\pi$
$787$		−	4.11828i		−	0.146801i	0.997303	−	0.0734004i	$-0.0233851\pi$
$788$	41.8896			1.49226
$789$		0			0
$790$	38.9444			1.38558
$791$	18.0977			0.643479
$792$		0			0
$793$	61.5959			2.18734
$794$	9.80930			0.348119
$795$		0			0
$796$	39.8434			1.41221
$797$			11.7423i			0.415934i	0.978136	+	0.207967i	$0.0666846\pi$
$797$			11.7423i			0.415934i	−0.978136	+	0.207967i	$0.933315\pi$
$798$		0			0
$799$			13.9474i			0.493424i
$800$	−13.2990			−0.470191
$801$		0			0
$802$		−	33.0136i		−	1.16575i
$803$	−36.8790	−	12.2993i	−1.30143	−	0.434034i
$804$		0			0
$805$		−	19.3247i		−	0.681108i
$806$			104.430i			3.67840i
$807$		0			0
$808$	27.2474			0.958562
$809$	47.8674			1.68293			0.841464	−	0.540313i	$-0.181694\pi$
$809$	47.8674			1.68293			0.841464	+	0.540313i	$0.181694\pi$
$810$		0			0
$811$			56.1981i			1.97338i	0.162608	+	0.986691i	$0.448009\pi$
$811$			56.1981i			1.97338i	−0.162608	+	0.986691i	$0.551991\pi$
$812$			80.3566i			2.81996i
$813$		0			0
$814$	8.44949	−	25.3354i	0.296154	−	0.888006i
$815$			11.0280i			0.386293i
$816$		0			0
$817$	−16.0454			−0.561358
$818$		−	30.8270i		−	1.07784i
$819$		0			0
$820$			11.3880i			0.397687i
$821$	16.9185			0.590461			0.295231	−	0.955426i	$-0.404603\pi$
$821$	16.9185			0.590461			0.295231	+	0.955426i	$0.404603\pi$
$822$		0			0
$823$	−12.2020			−0.425336			−0.212668	−	0.977124i	$-0.568215\pi$
$823$	−12.2020			−0.425336			−0.212668	+	0.977124i	$0.568215\pi$
$824$	51.9348			1.80923
$825$		0			0
$826$	22.5959			0.786213
$827$	−3.14789			−0.109463			−0.0547314	−	0.998501i	$-0.517430\pi$
$827$	−3.14789			−0.109463			−0.0547314	+	0.998501i	$0.517430\pi$
$828$		0			0
$829$	−44.8434			−1.55747			−0.778737	−	0.627350i	$-0.784139\pi$
$829$	−44.8434			−1.55747			−0.778737	+	0.627350i	$0.784139\pi$
$830$		−	20.4347i		−	0.709298i
$831$		0			0
$832$		−	59.0910i		−	2.04861i
$833$	−0.577648			−0.0200143
$834$		0			0
$835$		−	30.8627i		−	1.06805i
$836$	19.7246	+	6.57826i	0.682190	+	0.227514i
$837$		0			0
$838$			35.3144i			1.21992i
$839$		−	42.8085i		−	1.47791i	−0.673754	−	0.738956i	$-0.735319\pi$
$839$		−	42.8085i		−	1.47791i	0.673754	−	0.738956i	$-0.264681\pi$
$840$		0			0
$841$	53.8434			1.85667
$842$	19.3828			0.667975
$843$		0			0
$844$			49.2619i			1.69566i
$845$			13.9993i			0.481590i
$846$		0			0
$847$	−16.8990	+	22.5175i	−0.580656	+	0.773709i
$848$		−	0.635674i		−	0.0218292i
$849$		0			0
$850$	−9.00000			−0.308697
$851$		−	18.4169i		−	0.631323i
$852$		0			0
$853$		−	38.1324i		−	1.30563i	−0.757518	−	0.652814i	$-0.773588\pi$
$853$		−	38.1324i		−	1.30563i	0.757518	−	0.652814i	$-0.226412\pi$
$854$	−76.9058			−2.63166
$855$		0			0
$856$	36.7423			1.25583
$857$	−41.1000			−1.40395			−0.701974	−	0.712202i	$-0.747698\pi$
$857$	−41.1000			−1.40395			−0.701974	+	0.712202i	$0.747698\pi$
$858$		0			0
$859$	−17.3485			−0.591922			−0.295961	−	0.955200i	$-0.595640\pi$
$859$	−17.3485			−0.591922			−0.295961	+	0.955200i	$0.595640\pi$
$860$	−43.0688			−1.46863
$861$		0			0
$862$	−46.0454			−1.56831
$863$			51.9294i			1.76770i	0.467772	+	0.883849i	$0.345057\pi$
$863$			51.9294i			1.76770i	−0.467772	+	0.883849i	$0.654943\pi$
$864$		0			0
$865$			21.9591i			0.746632i
$866$	14.8200			0.503603
$867$		0			0
$868$		−	82.5340i		−	2.80139i
$869$	−37.1148	−	12.3780i	−1.25903	−	0.419894i
$870$		0			0
$871$		−	25.5940i		−	0.867218i
$872$		−	27.3629i		−	0.926624i
$873$		0			0
$874$	22.6515			0.766199
$875$	−28.9563			−0.978900
$876$		0			0
$877$			27.2278i			0.919417i	0.888070	+	0.459709i	$0.152046\pi$
$877$			27.2278i			0.919417i	−0.888070	+	0.459709i	$0.847954\pi$
$878$		−	8.31031i		−	0.280459i
$879$		0			0
$880$	4.44949	+	1.48393i	0.149992	+	0.0500232i
$881$		−	25.6629i		−	0.864606i	−0.901728	−	0.432303i	$-0.857701\pi$
$881$		−	25.6629i		−	0.864606i	0.901728	−	0.432303i	$-0.142299\pi$
$882$		0			0
$883$	−40.4949			−1.36276			−0.681381	−	0.731929i	$-0.738620\pi$
$883$	−40.4949			−1.36276			−0.681381	+	0.731929i	$0.738620\pi$
$884$		−	21.2132i		−	0.713477i
$885$		0			0
$886$		−	77.0817i		−	2.58961i
$887$	−7.34507			−0.246623			−0.123312	−	0.992368i	$-0.539351\pi$
$887$	−7.34507			−0.246623			−0.123312	+	0.992368i	$0.539351\pi$
$888$		0			0
$889$	−4.65153			−0.156007
$890$	−24.9711			−0.837033
$891$		0			0
$892$	6.04541			0.202415
$893$	−19.7246			−0.660059
$894$		0			0
$895$	3.75255			0.125434
$896$			51.0867i			1.70669i
$897$		0			0
$898$			41.5837i			1.38766i
$899$	−85.0882			−2.83785
$900$		0			0
$901$			0.816917i			0.0272155i
$902$	5.71812	−	17.1455i	0.190393	−	0.570884i
$903$		0			0
$904$			23.9264i			0.795782i
$905$			2.97129i			0.0987691i
$906$		0			0
$907$	−34.2474			−1.13717			−0.568584	−	0.822625i	$-0.692509\pi$
$907$	−34.2474			−1.13717			−0.568584	+	0.822625i	$0.692509\pi$
$908$	25.3367			0.840829
$909$		0			0
$910$		−	40.4332i		−	1.34035i
$911$		−	46.7012i		−	1.54728i	−0.633626	−	0.773639i	$-0.718434\pi$
$911$		−	46.7012i		−	1.54728i	0.633626	−	0.773639i	$-0.281566\pi$
$912$		0			0
$913$	−6.49490	+	19.4747i	−0.214950	+	0.644517i
$914$			66.2465i			2.19124i
$915$		0			0
$916$	−15.6969			−0.518641
$917$			1.20713i			0.0398630i
$918$		0			0
$919$		−	20.2166i		−	0.666885i	−0.942770	−	0.333442i	$-0.891790\pi$
$919$		−	20.2166i		−	0.666885i	0.942770	−	0.333442i	$-0.108210\pi$
$920$	25.5487			0.842317
$921$		0			0
$922$	67.8434			2.23430
$923$	−6.08377			−0.200250
$924$		0			0
$925$	−10.3485			−0.340256
$926$	7.23907			0.237890
$927$		0			0
$928$	40.3485			1.32450
$929$			17.8920i			0.587016i	0.955957	+	0.293508i	$0.0948228\pi$
$929$			17.8920i			0.587016i	−0.955957	+	0.293508i	$0.905177\pi$
$930$		0			0
$931$		−	0.816917i		−	0.0267734i
$932$	−68.0398			−2.22872
$933$		0			0
$934$			69.0700i			2.26004i
$935$	−5.71812	−	1.90702i	−0.187003	−	0.0623663i
$936$		0			0
$937$			3.45127i			0.112748i	0.998410	+	0.0563740i	$0.0179539\pi$
$937$			3.45127i			0.112748i	−0.998410	+	0.0563740i	$0.982046\pi$
$938$			31.9555i			1.04338i
$939$		0			0
$940$	−52.9444			−1.72686
$941$	−54.3752			−1.77258			−0.886290	−	0.463131i	$-0.846726\pi$
$941$	−54.3752			−1.77258			−0.886290	+	0.463131i	$0.846726\pi$
$942$		0			0
$943$		−	12.4635i		−	0.405867i
$944$			3.78194i			0.123092i
$945$		0			0
$946$	64.8434	+	21.6256i	2.10824	+	0.703109i
$947$			23.6130i			0.767321i	0.923474	+	0.383660i	$0.125337\pi$
$947$			23.6130i			0.767321i	−0.923474	+	0.383660i	$0.874663\pi$
$948$		0			0
$949$	−56.0908			−1.82078
$950$		−	12.7279i		−	0.412948i
$951$		0			0
$952$			11.1295i			0.360708i
$953$	−24.1576			−0.782542			−0.391271	−	0.920276i	$-0.627964\pi$
$953$	−24.1576			−0.782542			−0.391271	+	0.920276i	$0.627964\pi$
$954$		0			0
$955$	−27.3485			−0.884976
$956$	48.6809			1.57445
$957$		0			0
$958$	19.1010			0.617126
$959$	33.8371			1.09266
$960$		0			0
$961$	56.3939			1.81916
$962$		−	38.5337i		−	1.24238i
$963$		0			0
$964$			58.0905i			1.87097i
$965$	28.4846			0.916952
$966$		0			0
$967$		−	0.891871i		−	0.0286806i	−0.999897	−	0.0143403i	$-0.995435\pi$
$967$		−	0.891871i		−	0.0286806i	0.999897	−	0.0143403i	$-0.00456482\pi$
$968$	−29.7697	−	22.3417i	−0.956836	−	0.718089i
$969$		0			0
$970$			27.5613i			0.884940i
$971$		−	7.56388i		−	0.242736i	−0.992608	−	0.121368i	$-0.961272\pi$
$971$		−	7.56388i		−	0.242736i	0.992608	−	0.121368i	$-0.0387282\pi$
$972$		0			0
$973$	13.2929			0.426149
$974$	−32.2102			−1.03208
$975$		0			0
$976$		−	12.8719i		−	0.412021i
$977$			44.0477i			1.40921i	0.709599	+	0.704605i	$0.248876\pi$
$977$			44.0477i			1.40921i	−0.709599	+	0.704605i	$0.751124\pi$
$978$		0			0
$979$	23.7980	+	7.93674i	0.760586	+	0.253659i
$980$		−	2.19275i		−	0.0700449i
$981$		0			0
$982$	25.5959			0.816799
$983$			17.8920i			0.570665i	0.958429	+	0.285333i	$0.0921040\pi$
$983$			17.8920i			0.570665i	−0.958429	+	0.285333i	$0.907896\pi$
$984$		0			0
$985$		−	17.1738i		−	0.547203i
$986$	27.3055			0.869584
$987$		0			0
$988$	30.0000			0.954427
$989$	47.1361			1.49884
$990$		0			0
$991$	−6.69694			−0.212735			−0.106368	−	0.994327i	$-0.533922\pi$
$991$	−6.69694			−0.212735			−0.106368	+	0.994327i	$0.533922\pi$
$992$	−41.4418			−1.31578
$993$		0			0
$994$	7.59592			0.240928
$995$		−	16.3349i		−	0.517851i
$996$		0			0
$997$		−	27.2278i		−	0.862313i	−0.902277	−	0.431157i	$-0.858106\pi$
$997$		−	27.2278i		−	0.862313i	0.902277	−	0.431157i	$-0.141894\pi$
$998$	15.5274			0.491512
$999$		0			0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	297.2.d.b.296.7	yes	8
3.2	odd	2	inner	297.2.d.b.296.2	yes	8
4.3	odd	2		4752.2.b.h.593.3		8
9.2	odd	6		891.2.g.b.296.4		8
9.4	even	3		891.2.g.b.593.1		8
9.5	odd	6		891.2.g.d.593.4		8
9.7	even	3		891.2.g.d.296.1		8
11.10	odd	2	inner	297.2.d.b.296.1	✓	8
12.11	even	2		4752.2.b.h.593.7		8
33.32	even	2	inner	297.2.d.b.296.8	yes	8
44.43	even	2		4752.2.b.h.593.2		8
99.32	even	6		891.2.g.d.593.1		8
99.43	odd	6		891.2.g.d.296.4		8
99.65	even	6		891.2.g.b.296.1		8
99.76	odd	6		891.2.g.b.593.4		8
132.131	odd	2		4752.2.b.h.593.6		8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
297.2.d.b.296.1	✓	8	11.10	odd	2	inner
297.2.d.b.296.2	yes	8	3.2	odd	2	inner
297.2.d.b.296.7	yes	8	1.1	even	1	trivial
297.2.d.b.296.8	yes	8	33.32	even	2	inner
891.2.g.b.296.1		8	99.65	even	6
891.2.g.b.296.4		8	9.2	odd	6
891.2.g.b.593.1		8	9.4	even	3
891.2.g.b.593.4		8	99.76	odd	6
891.2.g.d.296.1		8	9.7	even	3
891.2.g.d.296.4		8	99.43	odd	6
891.2.g.d.593.1		8	99.32	even	6
891.2.g.d.593.4		8	9.5	odd	6
4752.2.b.h.593.2		8	44.43	even	2
4752.2.b.h.593.3		8	4.3	odd	2
4752.2.b.h.593.6		8	132.131	odd	2
4752.2.b.h.593.7		8	12.11	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:	\( N \)	\(=\)	\( 297 = 3^{3} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	297.d (of order \(2\), degree \(1\), minimal)

\(f(q)\)	\(=\)	\(q+2.33441 q^{2} +3.44949 q^{4} -1.41421i q^{5} -2.55940i q^{7} +3.38371 q^{8} -3.30136i q^{10} +(-1.04930 + 3.14626i) q^{11} +4.78529i q^{13} -5.97469i q^{14} +1.00000 q^{16} -1.28512 q^{17} -1.81743i q^{19} -4.87832i q^{20} +(-2.44949 + 7.34468i) q^{22} +5.33902i q^{23} +3.00000 q^{25} +11.1708i q^{26} -8.82861i q^{28} -9.10183 q^{29} +9.34847 q^{31} -4.43300 q^{32} -3.00000 q^{34} -3.61953 q^{35} -3.44949 q^{37} -4.24264i q^{38} -4.78529i q^{40} -2.33441 q^{41} -8.82861i q^{43} +(-3.61953 + 10.8530i) q^{44} +12.4635i q^{46} -10.8530i q^{47} +0.449490 q^{49} +7.00324 q^{50} +16.5068i q^{52} -0.635674i q^{53} +(4.44949 + 1.48393i) q^{55} -8.66025i q^{56} -21.2474 q^{58} +3.78194i q^{59} -12.8719i q^{61} +21.8232 q^{62} -12.3485 q^{64} +6.76742 q^{65} -5.34847 q^{67} -4.43300 q^{68} -8.44949 q^{70} +1.27135i q^{71} +11.7215i q^{73} -8.05254 q^{74} -6.26922i q^{76} +(8.05254 + 2.68556i) q^{77} +11.7965i q^{79} -1.41421i q^{80} -5.44949 q^{82} +6.18977 q^{83} +1.81743i q^{85} -20.6096i q^{86} +(-3.55051 + 10.6460i) q^{88} -7.56388i q^{89} +12.2474 q^{91} +18.4169i q^{92} -25.3354i q^{94} -2.57024 q^{95} -8.34847 q^{97} +1.04930 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{4} + 8 q^{16} + 24 q^{25} + 16 q^{31} - 24 q^{34} - 8 q^{37} - 16 q^{49} + 16 q^{55} - 72 q^{58} - 40 q^{64} + 16 q^{67} - 48 q^{70} - 24 q^{82} - 48 q^{88} - 8 q^{97}+O(q^{100}) \) 8 * q + 8 * q^4 + 8 * q^16 + 24 * q^25 + 16 * q^31 - 24 * q^34 - 8 * q^37 - 16 * q^49 + 16 * q^55 - 72 * q^58 - 40 * q^64 + 16 * q^67 - 48 * q^70 - 24 * q^82 - 48 * q^88 - 8 * q^97

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