LMFDB - Embedded newform 4923.2.a.l.1.12

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4923,2,Mod(1,4923)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4923, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("4923.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4923 = 3^{2} \cdot 547 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4923.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

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

Self dual:	yes
Analytic conductor:	$39.3103529151$
Analytic rank:	$0$
Dimension:	$18$
Coefficient field:	$\mathbb{Q}[x]/(x^{18} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{18} - 4 x^{17} - 18 x^{16} + 84 x^{15} + 116 x^{14} - 708 x^{13} - 282 x^{12} + 3104 x^{11} + \cdots + 328 $ x^18 - 4x^17 - 18x^16 + 84x^15 + 116x^14 - 708x^13 - 282x^12 + 3104x^11 - 137x^10 - 7703x^9 + 2068x^8 + 11068x^7 - 4274x^6 - 9021x^5 + 4048x^4 + 3834x^3 - 1851x^2 - 654*x + 328
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 547)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.12
Root			$1.04467$ of defining polynomial
Character	$\chi$	$=$	4923.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.04467 q^{2} -0.908666 q^{4} -0.714085 q^{5} -2.03236 q^{7} -3.03859 q^{8} +O(q^{10})$	\(q+1.04467 q^{2} -0.908666 q^{4} -0.714085 q^{5} -2.03236 q^{7} -3.03859 q^{8} -0.745983 q^{10} -5.43392 q^{11} -3.96770 q^{13} -2.12315 q^{14} -1.35699 q^{16} +1.30793 q^{17} +8.24470 q^{19} +0.648865 q^{20} -5.67665 q^{22} +2.44824 q^{23} -4.49008 q^{25} -4.14494 q^{26} +1.84674 q^{28} -1.49405 q^{29} -6.02301 q^{31} +4.65958 q^{32} +1.36635 q^{34} +1.45128 q^{35} -6.36158 q^{37} +8.61299 q^{38} +2.16982 q^{40} +2.87189 q^{41} -5.76713 q^{43} +4.93762 q^{44} +2.55760 q^{46} -6.42848 q^{47} -2.86949 q^{49} -4.69065 q^{50} +3.60532 q^{52} +0.932545 q^{53} +3.88028 q^{55} +6.17553 q^{56} -1.56079 q^{58} -1.57369 q^{59} -7.93215 q^{61} -6.29205 q^{62} +7.58170 q^{64} +2.83328 q^{65} +10.5294 q^{67} -1.18847 q^{68} +1.51611 q^{70} +12.8929 q^{71} +9.81833 q^{73} -6.64575 q^{74} -7.49168 q^{76} +11.0437 q^{77} -2.80129 q^{79} +0.969009 q^{80} +3.00017 q^{82} +15.3935 q^{83} -0.933971 q^{85} -6.02474 q^{86} +16.5115 q^{88} +1.89811 q^{89} +8.06382 q^{91} -2.22463 q^{92} -6.71563 q^{94} -5.88742 q^{95} +0.276333 q^{97} -2.99767 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	$ 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) $ 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

Display 100 coefficients 10 coefficients

Coefficient data

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

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

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

$2$	1.04467	0.738693	0.369346	−	0.929292i	$-0.379582\pi$
$2$	1.04467	0.738693	0.369346	+	0.929292i	$0.379582\pi$
$3$	0	0
$3$	0	0
$4$	−0.908666	−0.454333
$5$	−0.714085	−0.319349	−0.159674	−	0.987170i	$-0.551044\pi$
$5$	−0.714085	−0.319349	−0.159674	+	0.987170i	$0.551044\pi$
$6$	0	0
$7$	−2.03236	−0.768162	−0.384081	−	0.923299i	$-0.625482\pi$
$7$	−2.03236	−0.768162	−0.384081	+	0.923299i	$0.625482\pi$
$8$	−3.03859	−1.07431
$9$	0	0
$10$	−0.745983	−0.235901
$11$	−5.43392	−1.63839	−0.819194	−	0.573516i	$-0.805579\pi$
$11$	−5.43392	−1.63839	−0.819194	+	0.573516i	$0.805579\pi$
$12$	0	0
$13$	−3.96770	−1.10044	−0.550221	−	0.835019i	$-0.685457\pi$
$13$	−3.96770	−1.10044	−0.550221	+	0.835019i	$0.685457\pi$
$14$	−2.12315	−0.567435
$15$	0	0
$16$	−1.35699	−0.339248
$17$	1.30793	0.317219	0.158609	−	0.987341i	$-0.449299\pi$
$17$	1.30793	0.317219	0.158609	+	0.987341i	$0.449299\pi$
$18$	0	0
$19$	8.24470	1.89146	0.945732	−	0.324947i	$-0.105346\pi$
$19$	8.24470	1.89146	0.945732	+	0.324947i	$0.105346\pi$
$20$	0.648865	0.145091
$21$	0	0
$22$	−5.67665	−1.21027
$23$	2.44824	0.510493	0.255246	−	0.966876i	$-0.417843\pi$
$23$	2.44824	0.510493	0.255246	+	0.966876i	$0.417843\pi$
$24$	0	0
$25$	−4.49008	−0.898016
$26$	−4.14494	−0.812889
$27$	0	0
$28$	1.84674	0.349001
$29$	−1.49405	−0.277438	−0.138719	−	0.990332i	$-0.544298\pi$
$29$	−1.49405	−0.277438	−0.138719	+	0.990332i	$0.544298\pi$
$30$	0	0
$31$	−6.02301	−1.08176	−0.540882	−	0.841098i	$-0.681909\pi$
$31$	−6.02301	−1.08176	−0.540882	+	0.841098i	$0.681909\pi$
$32$	4.65958	0.823705
$33$	0	0
$34$	1.36635	0.234327
$35$	1.45128	0.245311
$36$	0	0
$37$	−6.36158	−1.04584	−0.522919	−	0.852383i	$-0.675157\pi$
$37$	−6.36158	−1.04584	−0.522919	+	0.852383i	$0.675157\pi$
$38$	8.61299	1.39721
$39$	0	0
$40$	2.16982	0.343078
$41$	2.87189	0.448514	0.224257	−	0.974530i	$-0.428005\pi$
$41$	2.87189	0.448514	0.224257	+	0.974530i	$0.428005\pi$
$42$	0	0
$43$	−5.76713	−0.879478	−0.439739	−	0.898125i	$-0.644929\pi$
$43$	−5.76713	−0.879478	−0.439739	+	0.898125i	$0.644929\pi$
$44$	4.93762	0.744374
$45$	0	0
$46$	2.55760	0.377097
$47$	−6.42848	−0.937689	−0.468845	−	0.883281i	$-0.655330\pi$
$47$	−6.42848	−0.937689	−0.468845	+	0.883281i	$0.655330\pi$
$48$	0	0
$49$	−2.86949	−0.409928
$50$	−4.69065	−0.663358
$51$	0	0
$52$	3.60532	0.499967
$53$	0.932545	0.128095	0.0640474	−	0.997947i	$-0.479599\pi$
$53$	0.932545	0.128095	0.0640474	+	0.997947i	$0.479599\pi$
$54$	0	0
$55$	3.88028	0.523217
$56$	6.17553	0.825240
$57$	0	0
$58$	−1.56079	−0.204941
$59$	−1.57369	−0.204877	−0.102438	−	0.994739i	$-0.532664\pi$
$59$	−1.57369	−0.204877	−0.102438	+	0.994739i	$0.532664\pi$
$60$	0	0
$61$	−7.93215	−1.01561	−0.507804	−	0.861473i	$-0.669542\pi$
$61$	−7.93215	−1.01561	−0.507804	+	0.861473i	$0.669542\pi$
$62$	−6.29205	−0.799091
$63$	0	0
$64$	7.58170	0.947713
$65$	2.83328	0.351425
$66$	0	0
$67$	10.5294	1.28637	0.643186	−	0.765710i	$-0.277612\pi$
$67$	10.5294	1.28637	0.643186	+	0.765710i	$0.277612\pi$
$68$	−1.18847	−0.144123
$69$	0	0
$70$	1.51611	0.181210
$71$	12.8929	1.53011	0.765054	−	0.643966i	$-0.222712\pi$
$71$	12.8929	1.53011	0.765054	+	0.643966i	$0.222712\pi$
$72$	0	0
$73$	9.81833	1.14915	0.574574	−	0.818452i	$-0.305168\pi$
$73$	9.81833	1.14915	0.574574	+	0.818452i	$0.305168\pi$
$74$	−6.64575	−0.772552
$75$	0	0
$76$	−7.49168	−0.859355
$77$	11.0437	1.25855
$78$	0	0
$79$	−2.80129	−0.315170	−0.157585	−	0.987505i	$-0.550371\pi$
$79$	−2.80129	−0.315170	−0.157585	+	0.987505i	$0.550371\pi$
$80$	0.969009	0.108338
$81$	0	0
$82$	3.00017	0.331314
$83$	15.3935	1.68966	0.844829	−	0.535037i	$-0.179702\pi$
$83$	15.3935	1.68966	0.844829	+	0.535037i	$0.179702\pi$
$84$	0	0
$85$	−0.933971	−0.101303
$86$	−6.02474	−0.649664
$87$	0	0
$88$	16.5115	1.76013
$89$	1.89811	0.201199	0.100600	−	0.994927i	$-0.467924\pi$
$89$	1.89811	0.201199	0.100600	+	0.994927i	$0.467924\pi$
$90$	0	0
$91$	8.06382	0.845318
$92$	−2.22463	−0.231934
$93$	0	0
$94$	−6.71563	−0.692664
$95$	−5.88742	−0.604037
$96$	0	0
$97$	0.276333	0.0280574	0.0140287	−	0.999902i	$-0.495534\pi$
$97$	0.276333	0.0280574	0.0140287	+	0.999902i	$0.495534\pi$
$98$	−2.99767	−0.302811
$99$	0	0
$100$	4.07999	0.407999
$101$	17.0451	1.69605	0.848024	−	0.529958i	$-0.177792\pi$
$101$	17.0451	1.69605	0.848024	+	0.529958i	$0.177792\pi$
$102$	0	0
$103$	6.40415	0.631020	0.315510	−	0.948922i	$-0.397824\pi$
$103$	6.40415	0.631020	0.315510	+	0.948922i	$0.397824\pi$
$104$	12.0562	1.18221
$105$	0	0
$106$	0.974201	0.0946227
$107$	−6.34857	−0.613739	−0.306870	−	0.951752i	$-0.599282\pi$
$107$	−6.34857	−0.613739	−0.306870	+	0.951752i	$0.599282\pi$
$108$	0	0
$109$	9.33636	0.894261	0.447130	−	0.894469i	$-0.352446\pi$
$109$	9.33636	0.894261	0.447130	+	0.894469i	$0.352446\pi$
$110$	4.05361	0.386497
$111$	0	0
$112$	2.75790	0.260597
$113$	1.89369	0.178144	0.0890718	−	0.996025i	$-0.471610\pi$
$113$	1.89369	0.178144	0.0890718	+	0.996025i	$0.471610\pi$
$114$	0	0
$115$	−1.74825	−0.163025
$116$	1.35759	0.126049
$117$	0	0
$118$	−1.64398	−0.151341
$119$	−2.65818	−0.243675
$120$	0	0
$121$	18.5275	1.68432
$122$	−8.28647	−0.750222
$123$	0	0
$124$	5.47290	0.491481
$125$	6.77673	0.606129
$126$	0	0
$127$	10.2086	0.905867	0.452933	−	0.891544i	$-0.350378\pi$
$127$	10.2086	0.905867	0.452933	+	0.891544i	$0.350378\pi$
$128$	−1.39879	−0.123636
$129$	0	0
$130$	2.95984	0.259595
$131$	13.9590	1.21960	0.609801	−	0.792554i	$-0.291249\pi$
$131$	13.9590	1.21960	0.609801	+	0.792554i	$0.291249\pi$
$132$	0	0
$133$	−16.7562	−1.45295
$134$	10.9997	0.950233
$135$	0	0
$136$	−3.97426	−0.340790
$137$	2.19310	0.187369	0.0936845	−	0.995602i	$-0.470136\pi$
$137$	2.19310	0.187369	0.0936845	+	0.995602i	$0.470136\pi$
$138$	0	0
$139$	−19.6789	−1.66914	−0.834572	−	0.550899i	$-0.814285\pi$
$139$	−19.6789	−1.66914	−0.834572	+	0.550899i	$0.814285\pi$
$140$	−1.31873	−0.111453
$141$	0	0
$142$	13.4688	1.13028
$143$	21.5602	1.80295
$144$	0	0
$145$	1.06688	0.0885994
$146$	10.2569	0.848868
$147$	0	0
$148$	5.78055	0.475159
$149$	11.0882	0.908379	0.454189	−	0.890905i	$-0.349929\pi$
$149$	11.0882	0.908379	0.454189	+	0.890905i	$0.349929\pi$
$150$	0	0
$151$	7.03425	0.572439	0.286219	−	0.958164i	$-0.407601\pi$
$151$	7.03425	0.572439	0.286219	+	0.958164i	$0.407601\pi$
$152$	−25.0523	−2.03201
$153$	0	0
$154$	11.5370	0.929679
$155$	4.30094	0.345460
$156$	0	0
$157$	−24.2151	−1.93257	−0.966287	−	0.257468i	$-0.917112\pi$
$157$	−24.2151	−1.93257	−0.966287	+	0.257468i	$0.917112\pi$
$158$	−2.92642	−0.232814
$159$	0	0
$160$	−3.32734	−0.263049
$161$	−4.97571	−0.392141
$162$	0	0
$163$	0.672203	0.0526510	0.0263255	−	0.999653i	$-0.491619\pi$
$163$	0.672203	0.0526510	0.0263255	+	0.999653i	$0.491619\pi$
$164$	−2.60959	−0.203775
$165$	0	0
$166$	16.0811	1.24814
$167$	13.2184	1.02287	0.511436	−	0.859321i	$-0.329114\pi$
$167$	13.2184	1.02287	0.511436	+	0.859321i	$0.329114\pi$
$168$	0	0
$169$	2.74265	0.210973
$170$	−0.975691	−0.0748321
$171$	0	0
$172$	5.24039	0.399576
$173$	18.2674	1.38884	0.694421	−	0.719569i	$-0.255661\pi$
$173$	18.2674	1.38884	0.694421	+	0.719569i	$0.255661\pi$
$174$	0	0
$175$	9.12548	0.689822
$176$	7.37379	0.555820
$177$	0	0
$178$	1.98290	0.148624
$179$	−21.6343	−1.61702	−0.808511	−	0.588481i	$-0.799726\pi$
$179$	−21.6343	−1.61702	−0.808511	+	0.588481i	$0.799726\pi$
$180$	0	0
$181$	0.765562	0.0569038	0.0284519	−	0.999595i	$-0.490942\pi$
$181$	0.765562	0.0569038	0.0284519	+	0.999595i	$0.490942\pi$
$182$	8.42402	0.624430
$183$	0	0
$184$	−7.43920	−0.548425
$185$	4.54271	0.333987
$186$	0	0
$187$	−7.10717	−0.519728
$188$	5.84134	0.426023
$189$	0	0
$190$	−6.15041	−0.446197
$191$	3.52515	0.255071	0.127535	−	0.991834i	$-0.459293\pi$
$191$	3.52515	0.255071	0.127535	+	0.991834i	$0.459293\pi$
$192$	0	0
$193$	−8.94657	−0.643988	−0.321994	−	0.946742i	$-0.604353\pi$
$193$	−8.94657	−0.643988	−0.321994	+	0.946742i	$0.604353\pi$
$194$	0.288677	0.0207258
$195$	0	0
$196$	2.60741	0.186244
$197$	20.1670	1.43684	0.718418	−	0.695612i	$-0.244867\pi$
$197$	20.1670	1.43684	0.718418	+	0.695612i	$0.244867\pi$
$198$	0	0
$199$	3.72704	0.264203	0.132101	−	0.991236i	$-0.457828\pi$
$199$	3.72704	0.264203	0.132101	+	0.991236i	$0.457828\pi$
$200$	13.6435	0.964744
$201$	0	0
$202$	17.8065	1.25286
$203$	3.03645	0.213117
$204$	0	0
$205$	−2.05077	−0.143232
$206$	6.69022	0.466130
$207$	0	0
$208$	5.38414	0.373323
$209$	−44.8011	−3.09895
$210$	0	0
$211$	17.9812	1.23788	0.618940	−	0.785438i	$-0.287562\pi$
$211$	17.9812	1.23788	0.618940	+	0.785438i	$0.287562\pi$
$212$	−0.847372	−0.0581977
$213$	0	0
$214$	−6.63215	−0.453365
$215$	4.11822	0.280860
$216$	0	0
$217$	12.2409	0.830970
$218$	9.75341	0.660584
$219$	0	0
$220$	−3.52588	−0.237715
$221$	−5.18946	−0.349081
$222$	0	0
$223$	−19.2192	−1.28701	−0.643507	−	0.765440i	$-0.722521\pi$
$223$	−19.2192	−1.28701	−0.643507	+	0.765440i	$0.722521\pi$
$224$	−9.46996	−0.632739
$225$	0	0
$226$	1.97828	0.131593
$227$	−23.2733	−1.54470	−0.772350	−	0.635197i	$-0.780919\pi$
$227$	−23.2733	−1.54470	−0.772350	+	0.635197i	$0.780919\pi$
$228$	0	0
$229$	1.04868	0.0692987	0.0346494	−	0.999400i	$-0.488969\pi$
$229$	1.04868	0.0692987	0.0346494	+	0.999400i	$0.488969\pi$
$230$	−1.82634	−0.120426
$231$	0	0
$232$	4.53981	0.298053
$233$	−10.4018	−0.681441	−0.340721	−	0.940165i	$-0.610671\pi$
$233$	−10.4018	−0.681441	−0.340721	+	0.940165i	$0.610671\pi$
$234$	0	0
$235$	4.59048	0.299450
$236$	1.42996	0.0930823
$237$	0	0
$238$	−2.77692	−0.180001
$239$	−5.21082	−0.337060	−0.168530	−	0.985697i	$-0.553902\pi$
$239$	−5.21082	−0.337060	−0.168530	+	0.985697i	$0.553902\pi$
$240$	0	0
$241$	−23.1270	−1.48974	−0.744871	−	0.667209i	$-0.767489\pi$
$241$	−23.1270	−1.48974	−0.744871	+	0.667209i	$0.767489\pi$
$242$	19.3551	1.24419
$243$	0	0
$244$	7.20767	0.461424
$245$	2.04906	0.130910
$246$	0	0
$247$	−32.7125	−2.08145
$248$	18.3015	1.16214
$249$	0	0
$250$	7.07944	0.447743
$251$	18.2828	1.15400	0.577001	−	0.816744i	$-0.304223\pi$
$251$	18.2828	1.15400	0.577001	+	0.816744i	$0.304223\pi$
$252$	0	0
$253$	−13.3035	−0.836386
$254$	10.6646	0.669157
$255$	0	0
$256$	−16.6247	−1.03904
$257$	−30.4924	−1.90207	−0.951033	−	0.309091i	$-0.899975\pi$
$257$	−30.4924	−1.90207	−0.951033	+	0.309091i	$0.899975\pi$
$258$	0	0
$259$	12.9291	0.803372
$260$	−2.57450	−0.159664
$261$	0	0
$262$	14.5825	0.900911
$263$	11.2828	0.695730	0.347865	−	0.937545i	$-0.386907\pi$
$263$	11.2828	0.695730	0.347865	+	0.937545i	$0.386907\pi$
$264$	0	0
$265$	−0.665916	−0.0409069
$266$	−17.5047	−1.07328
$267$	0	0
$268$	−9.56772	−0.584441
$269$	1.54283	0.0940679	0.0470339	−	0.998893i	$-0.485023\pi$
$269$	1.54283	0.0940679	0.0470339	+	0.998893i	$0.485023\pi$
$270$	0	0
$271$	−23.6023	−1.43374	−0.716870	−	0.697207i	$-0.754426\pi$
$271$	−23.6023	−1.43374	−0.716870	+	0.697207i	$0.754426\pi$
$272$	−1.77485	−0.107616
$273$	0	0
$274$	2.29106	0.138408
$275$	24.3987	1.47130
$276$	0	0
$277$	14.2694	0.857367	0.428684	−	0.903455i	$-0.358978\pi$
$277$	14.2694	0.857367	0.428684	+	0.903455i	$0.358978\pi$
$278$	−20.5580	−1.23298
$279$	0	0
$280$	−4.40986	−0.263539
$281$	−22.6952	−1.35388	−0.676942	−	0.736037i	$-0.736695\pi$
$281$	−22.6952	−1.35388	−0.676942	+	0.736037i	$0.736695\pi$
$282$	0	0
$283$	−22.9783	−1.36592	−0.682958	−	0.730458i	$-0.739307\pi$
$283$	−22.9783	−1.36592	−0.682958	+	0.730458i	$0.739307\pi$
$284$	−11.7154	−0.695179
$285$	0	0
$286$	22.5232	1.33183
$287$	−5.83672	−0.344531
$288$	0	0
$289$	−15.2893	−0.899372
$290$	1.11453	0.0654477
$291$	0	0
$292$	−8.92158	−0.522096
$293$	25.1176	1.46739	0.733694	−	0.679480i	$-0.237795\pi$
$293$	25.1176	1.46739	0.733694	+	0.679480i	$0.237795\pi$
$294$	0	0
$295$	1.12375	0.0654271
$296$	19.3303	1.12355
$297$	0	0
$298$	11.5835	0.671013
$299$	−9.71388	−0.561768
$300$	0	0
$301$	11.7209	0.675582
$302$	7.34846	0.422856
$303$	0	0
$304$	−11.1880	−0.641676
$305$	5.66423	0.324333
$306$	0	0
$307$	−14.5944	−0.832947	−0.416473	−	0.909148i	$-0.636734\pi$
$307$	−14.5944	−0.832947	−0.416473	+	0.909148i	$0.636734\pi$
$308$	−10.0350	−0.571800
$309$	0	0
$310$	4.49306	0.255189
$311$	−8.99147	−0.509860	−0.254930	−	0.966960i	$-0.582052\pi$
$311$	−8.99147	−0.509860	−0.254930	+	0.966960i	$0.582052\pi$
$312$	0	0
$313$	−4.56361	−0.257951	−0.128975	−	0.991648i	$-0.541169\pi$
$313$	−4.56361	−0.257951	−0.128975	+	0.991648i	$0.541169\pi$
$314$	−25.2967	−1.42758
$315$	0	0
$316$	2.54544	0.143192
$317$	7.39682	0.415447	0.207723	−	0.978188i	$-0.433395\pi$
$317$	7.39682	0.415447	0.207723	+	0.978188i	$0.433395\pi$
$318$	0	0
$319$	8.11854	0.454551
$320$	−5.41398	−0.302651
$321$	0	0
$322$	−5.19797	−0.289672
$323$	10.7835	0.600008
$324$	0	0
$325$	17.8153	0.988215
$326$	0.702230	0.0388929
$327$	0	0
$328$	−8.72650	−0.481841
$329$	13.0650	0.720297
$330$	0	0
$331$	−23.5847	−1.29633	−0.648166	−	0.761499i	$-0.724464\pi$
$331$	−23.5847	−1.29633	−0.648166	+	0.761499i	$0.724464\pi$
$332$	−13.9876	−0.767667
$333$	0	0
$334$	13.8089	0.755588
$335$	−7.51890	−0.410801
$336$	0	0
$337$	25.0637	1.36531	0.682653	−	0.730743i	$-0.260826\pi$
$337$	25.0637	1.36531	0.682653	+	0.730743i	$0.260826\pi$
$338$	2.86517	0.155845
$339$	0	0
$340$	0.848668	0.0460255
$341$	32.7285	1.77235
$342$	0	0
$343$	20.0584	1.08305
$344$	17.5240	0.944828
$345$	0	0
$346$	19.0834	1.02593
$347$	−11.1374	−0.597888	−0.298944	−	0.954271i	$-0.596634\pi$
$347$	−11.1374	−0.597888	−0.298944	+	0.954271i	$0.596634\pi$
$348$	0	0
$349$	34.3068	1.83640	0.918202	−	0.396114i	$-0.129641\pi$
$349$	34.3068	1.83640	0.918202	+	0.396114i	$0.129641\pi$
$350$	9.53311	0.509566
$351$	0	0
$352$	−25.3198	−1.34955
$353$	−18.3887	−0.978731	−0.489366	−	0.872079i	$-0.662772\pi$
$353$	−18.3887	−0.978731	−0.489366	+	0.872079i	$0.662772\pi$
$354$	0	0
$355$	−9.20665	−0.488638
$356$	−1.72475	−0.0914115
$357$	0	0
$358$	−22.6007	−1.19448
$359$	−16.8572	−0.889688	−0.444844	−	0.895608i	$-0.646741\pi$
$359$	−16.8572	−0.889688	−0.444844	+	0.895608i	$0.646741\pi$
$360$	0	0
$361$	48.9751	2.57764
$362$	0.799759	0.0420344
$363$	0	0
$364$	−7.32732	−0.384056
$365$	−7.01112	−0.366979
$366$	0	0
$367$	20.9807	1.09518	0.547592	−	0.836745i	$-0.315544\pi$
$367$	20.9807	1.09518	0.547592	+	0.836745i	$0.315544\pi$
$368$	−3.32224	−0.173184
$369$	0	0
$370$	4.74563	0.246714
$371$	−1.89527	−0.0983975
$372$	0	0
$373$	8.57191	0.443837	0.221918	−	0.975065i	$-0.428768\pi$
$373$	8.57191	0.443837	0.221918	+	0.975065i	$0.428768\pi$
$374$	−7.42464	−0.383919
$375$	0	0
$376$	19.5335	1.00736
$377$	5.92794	0.305304
$378$	0	0
$379$	−24.0731	−1.23655	−0.618275	−	0.785962i	$-0.712168\pi$
$379$	−24.0731	−1.23655	−0.618275	+	0.785962i	$0.712168\pi$
$380$	5.34970	0.274434
$381$	0	0
$382$	3.68261	0.188419
$383$	28.2472	1.44336	0.721682	−	0.692225i	$-0.243369\pi$
$383$	28.2472	1.44336	0.721682	+	0.692225i	$0.243369\pi$
$384$	0	0
$385$	−7.88615	−0.401915
$386$	−9.34621	−0.475709
$387$	0	0
$388$	−0.251095	−0.0127474
$389$	−20.2718	−1.02782	−0.513911	−	0.857843i	$-0.671804\pi$
$389$	−20.2718	−1.02782	−0.513911	+	0.857843i	$0.671804\pi$
$390$	0	0
$391$	3.20211	0.161938
$392$	8.71923	0.440388
$393$	0	0
$394$	21.0678	1.06138
$395$	2.00036	0.100649
$396$	0	0
$397$	9.39156	0.471349	0.235674	−	0.971832i	$-0.424270\pi$
$397$	9.39156	0.471349	0.235674	+	0.971832i	$0.424270\pi$
$398$	3.89352	0.195165
$399$	0	0
$400$	6.09301	0.304650
$401$	−8.35327	−0.417143	−0.208571	−	0.978007i	$-0.566881\pi$
$401$	−8.35327	−0.417143	−0.208571	+	0.978007i	$0.566881\pi$
$402$	0	0
$403$	23.8975	1.19042
$404$	−15.4883	−0.770571
$405$	0	0
$406$	3.17209	0.157428
$407$	34.5683	1.71349
$408$	0	0
$409$	−9.04325	−0.447160	−0.223580	−	0.974686i	$-0.571774\pi$
$409$	−9.04325	−0.447160	−0.223580	+	0.974686i	$0.571774\pi$
$410$	−2.14238	−0.105805
$411$	0	0
$412$	−5.81923	−0.286693
$413$	3.19831	0.157378
$414$	0	0
$415$	−10.9923	−0.539590
$416$	−18.4878	−0.906440
$417$	0	0
$418$	−46.8023	−2.28917
$419$	21.2265	1.03698	0.518491	−	0.855083i	$-0.326494\pi$
$419$	21.2265	1.03698	0.518491	+	0.855083i	$0.326494\pi$
$420$	0	0
$421$	16.3175	0.795269	0.397634	−	0.917544i	$-0.369831\pi$
$421$	16.3175	0.795269	0.397634	+	0.917544i	$0.369831\pi$
$422$	18.7844	0.914412
$423$	0	0
$424$	−2.83362	−0.137613
$425$	−5.87270	−0.284868
$426$	0	0
$427$	16.1210	0.780151
$428$	5.76873	0.278842
$429$	0	0
$430$	4.30218	0.207469
$431$	−17.2192	−0.829420	−0.414710	−	0.909954i	$-0.636117\pi$
$431$	−17.2192	−0.829420	−0.414710	+	0.909954i	$0.636117\pi$
$432$	0	0
$433$	0.155714	0.00748312	0.00374156	−	0.999993i	$-0.498809\pi$
$433$	0.155714	0.00748312	0.00374156	+	0.999993i	$0.498809\pi$
$434$	12.7877	0.613831
$435$	0	0
$436$	−8.48363	−0.406292
$437$	20.1850	0.965579
$438$	0	0
$439$	23.1300	1.10394	0.551968	−	0.833865i	$-0.313877\pi$
$439$	23.1300	1.10394	0.551968	+	0.833865i	$0.313877\pi$
$440$	−11.7906	−0.562095
$441$	0	0
$442$	−5.42127	−0.257864
$443$	−12.5268	−0.595168	−0.297584	−	0.954696i	$-0.596181\pi$
$443$	−12.5268	−0.595168	−0.297584	+	0.954696i	$0.596181\pi$
$444$	0	0
$445$	−1.35541	−0.0642527
$446$	−20.0777	−0.950708
$447$	0	0
$448$	−15.4088	−0.727997
$449$	38.0487	1.79563	0.897814	−	0.440375i	$-0.145154\pi$
$449$	38.0487	1.79563	0.897814	+	0.440375i	$0.145154\pi$
$450$	0	0
$451$	−15.6056	−0.734840
$452$	−1.72074	−0.0809366
$453$	0	0
$454$	−24.3129	−1.14106
$455$	−5.75825	−0.269951
$456$	0	0
$457$	25.4956	1.19263	0.596317	−	0.802749i	$-0.296630\pi$
$457$	25.4956	1.19263	0.596317	+	0.802749i	$0.296630\pi$
$458$	1.09552	0.0511905
$459$	0	0
$460$	1.58858	0.0740677
$461$	−22.5869	−1.05198	−0.525988	−	0.850492i	$-0.676304\pi$
$461$	−22.5869	−1.05198	−0.525988	+	0.850492i	$0.676304\pi$
$462$	0	0
$463$	42.2153	1.96191	0.980956	−	0.194232i	$-0.0622214\pi$
$463$	42.2153	1.96191	0.980956	+	0.194232i	$0.0622214\pi$
$464$	2.02741	0.0941203
$465$	0	0
$466$	−10.8664	−0.503376
$467$	−4.29006	−0.198520	−0.0992601	−	0.995062i	$-0.531648\pi$
$467$	−4.29006	−0.198520	−0.0992601	+	0.995062i	$0.531648\pi$
$468$	0	0
$469$	−21.3996	−0.988141
$470$	4.79553	0.221201
$471$	0	0
$472$	4.78180	0.220100
$473$	31.3381	1.44093
$474$	0	0
$475$	−37.0194	−1.69857
$476$	2.41540	0.110710
$477$	0	0
$478$	−5.44358	−0.248984
$479$	7.74802	0.354016	0.177008	−	0.984209i	$-0.443358\pi$
$479$	7.74802	0.354016	0.177008	+	0.984209i	$0.443358\pi$
$480$	0	0
$481$	25.2409	1.15088
$482$	−24.1601	−1.10046
$483$	0	0
$484$	−16.8353	−0.765241
$485$	−0.197326	−0.00896010
$486$	0	0
$487$	3.12687	0.141692	0.0708459	−	0.997487i	$-0.477430\pi$
$487$	3.12687	0.141692	0.0708459	+	0.997487i	$0.477430\pi$
$488$	24.1026	1.09107
$489$	0	0
$490$	2.14059	0.0967022
$491$	17.8547	0.805771	0.402886	−	0.915250i	$-0.368007\pi$
$491$	17.8547	0.805771	0.402886	+	0.915250i	$0.368007\pi$
$492$	0	0
$493$	−1.95411	−0.0880085
$494$	−34.1738	−1.53755
$495$	0	0
$496$	8.17318	0.366987
$497$	−26.2031	−1.17537
$498$	0	0
$499$	2.04004	0.0913246	0.0456623	−	0.998957i	$-0.485460\pi$
$499$	2.04004	0.0913246	0.0456623	+	0.998957i	$0.485460\pi$
$500$	−6.15778	−0.275385
$501$	0	0
$502$	19.0995	0.852452
$503$	−7.07972	−0.315669	−0.157835	−	0.987466i	$-0.550451\pi$
$503$	−7.07972	−0.315669	−0.157835	+	0.987466i	$0.550451\pi$
$504$	0	0
$505$	−12.1716	−0.541631
$506$	−13.8978	−0.617832
$507$	0	0
$508$	−9.27621	−0.411565
$509$	−17.9368	−0.795037	−0.397518	−	0.917594i	$-0.630129\pi$
$509$	−17.9368	−0.795037	−0.397518	+	0.917594i	$0.630129\pi$
$510$	0	0
$511$	−19.9544	−0.882732
$512$	−14.5697	−0.643897
$513$	0	0
$514$	−31.8545	−1.40504
$515$	−4.57311	−0.201515
$516$	0	0
$517$	34.9318	1.53630
$518$	13.5066	0.593445
$519$	0	0
$520$	−8.60918	−0.377538
$521$	3.19375	0.139921	0.0699603	−	0.997550i	$-0.477713\pi$
$521$	3.19375	0.139921	0.0699603	+	0.997550i	$0.477713\pi$
$522$	0	0
$523$	−15.7713	−0.689630	−0.344815	−	0.938671i	$-0.612058\pi$
$523$	−15.7713	−0.689630	−0.344815	+	0.938671i	$0.612058\pi$
$524$	−12.6841	−0.554106
$525$	0	0
$526$	11.7868	0.513931
$527$	−7.87765	−0.343156
$528$	0	0
$529$	−17.0061	−0.739397
$530$	−0.695662	−0.0302176
$531$	0	0
$532$	15.2258	0.660123
$533$	−11.3948	−0.493563
$534$	0	0
$535$	4.53342	0.195997
$536$	−31.9946	−1.38196
$537$	0	0
$538$	1.61175	0.0694873
$539$	15.5926	0.671621
$540$	0	0
$541$	28.1775	1.21144	0.605722	−	0.795676i	$-0.292884\pi$
$541$	28.1775	1.21144	0.605722	+	0.795676i	$0.292884\pi$
$542$	−24.6566	−1.05909
$543$	0	0
$544$	6.09439	0.261295
$545$	−6.66696	−0.285581
$546$	0	0
$547$	−1.00000	−0.0427569
$547$	−1.00000	−0.0427569
$548$	−1.99279	−0.0851280
$549$	0	0
$550$	25.4886	1.08684
$551$	−12.3180	−0.524764
$552$	0	0
$553$	5.69324	0.242101
$554$	14.9068	0.633331
$555$	0	0
$556$	17.8816	0.758348
$557$	11.6421	0.493290	0.246645	−	0.969106i	$-0.420672\pi$
$557$	11.6421	0.493290	0.246645	+	0.969106i	$0.420672\pi$
$558$	0	0
$559$	22.8822	0.967815
$560$	−1.96938	−0.0832214
$561$	0	0
$562$	−23.7090	−1.00010
$563$	−35.1631	−1.48195	−0.740974	−	0.671534i	$-0.765636\pi$
$563$	−35.1631	−1.48195	−0.740974	+	0.671534i	$0.765636\pi$
$564$	0	0
$565$	−1.35226	−0.0568899
$566$	−24.0047	−1.00899
$567$	0	0
$568$	−39.1764	−1.64380
$569$	5.54994	0.232665	0.116333	−	0.993210i	$-0.462886\pi$
$569$	5.54994	0.232665	0.116333	+	0.993210i	$0.462886\pi$
$570$	0	0
$571$	−12.2975	−0.514633	−0.257317	−	0.966327i	$-0.582838\pi$
$571$	−12.2975	−0.514633	−0.257317	+	0.966327i	$0.582838\pi$
$572$	−19.5910	−0.819141
$573$	0	0
$574$	−6.09745	−0.254502
$575$	−10.9928	−0.458431
$576$	0	0
$577$	−21.7654	−0.906105	−0.453052	−	0.891484i	$-0.649665\pi$
$577$	−21.7654	−0.906105	−0.453052	+	0.891484i	$0.649665\pi$
$578$	−15.9723	−0.664360
$579$	0	0
$580$	−0.969436	−0.0402536
$581$	−31.2852	−1.29793
$582$	0	0
$583$	−5.06737	−0.209869
$584$	−29.8339	−1.23454
$585$	0	0
$586$	26.2396	1.08395
$587$	1.32577	0.0547206	0.0273603	−	0.999626i	$-0.491290\pi$
$587$	1.32577	0.0547206	0.0273603	+	0.999626i	$0.491290\pi$
$588$	0	0
$589$	−49.6579	−2.04612
$590$	1.17394	0.0483305
$591$	0	0
$592$	8.63262	0.354799
$593$	0.00527691	0.000216697 0	0.000108348	−	1.00000i	$-0.499966\pi$
$593$	0.00527691	0.000216697 0	0.000108348		1.00000i	$0.499966\pi$
$594$	0	0
$595$	1.89817	0.0778174
$596$	−10.0755	−0.412707
$597$	0	0
$598$	−10.1478	−0.414974
$599$	26.4628	1.08124	0.540620	−	0.841267i	$-0.318190\pi$
$599$	26.4628	1.08124	0.540620	+	0.841267i	$0.318190\pi$
$600$	0	0
$601$	−17.3745	−0.708719	−0.354360	−	0.935109i	$-0.615301\pi$
$601$	−17.3745	−0.708719	−0.354360	+	0.935109i	$0.615301\pi$
$602$	12.2445	0.499047
$603$	0	0
$604$	−6.39178	−0.260078
$605$	−13.2302	−0.537884
$606$	0	0
$607$	−28.3811	−1.15195	−0.575977	−	0.817466i	$-0.695378\pi$
$607$	−28.3811	−1.15195	−0.575977	+	0.817466i	$0.695378\pi$
$608$	38.4168	1.55801
$609$	0	0
$610$	5.91725	0.239582
$611$	25.5063	1.03187
$612$	0	0
$613$	−31.8417	−1.28607	−0.643036	−	0.765836i	$-0.722326\pi$
$613$	−31.8417	−1.28607	−0.643036	+	0.765836i	$0.722326\pi$
$614$	−15.2463	−0.615292
$615$	0	0
$616$	−33.5573	−1.35206
$617$	41.2687	1.66141	0.830707	−	0.556710i	$-0.187936\pi$
$617$	41.2687	1.66141	0.830707	+	0.556710i	$0.187936\pi$
$618$	0	0
$619$	36.6519	1.47316	0.736582	−	0.676348i	$-0.236439\pi$
$619$	36.6519	1.47316	0.736582	+	0.676348i	$0.236439\pi$
$620$	−3.90812	−0.156954
$621$	0	0
$622$	−9.39311	−0.376630
$623$	−3.85765	−0.154554
$624$	0	0
$625$	17.6112	0.704450
$626$	−4.76747	−0.190546
$627$	0	0
$628$	22.0034	0.878032
$629$	−8.32048	−0.331759
$630$	0	0
$631$	−33.4006	−1.32966	−0.664829	−	0.746996i	$-0.731495\pi$
$631$	−33.4006	−1.32966	−0.664829	+	0.746996i	$0.731495\pi$
$632$	8.51199	0.338589
$633$	0	0
$634$	7.72723	0.306887
$635$	−7.28981	−0.289287
$636$	0	0
$637$	11.3853	0.451102
$638$	8.48119	0.335773
$639$	0	0
$640$	0.998852	0.0394831
$641$	−6.94918	−0.274476	−0.137238	−	0.990538i	$-0.543823\pi$
$641$	−6.94918	−0.274476	−0.137238	+	0.990538i	$0.543823\pi$
$642$	0	0
$643$	4.52617	0.178495	0.0892474	−	0.996009i	$-0.471554\pi$
$643$	4.52617	0.178495	0.0892474	+	0.996009i	$0.471554\pi$
$644$	4.52126	0.178163
$645$	0	0
$646$	11.2652	0.443222
$647$	−25.3439	−0.996371	−0.498185	−	0.867071i	$-0.666000\pi$
$647$	−25.3439	−0.996371	−0.498185	+	0.867071i	$0.666000\pi$
$648$	0	0
$649$	8.55130	0.335668
$650$	18.6111	0.729987
$651$	0	0
$652$	−0.610808	−0.0239211
$653$	8.49304	0.332358	0.166179	−	0.986096i	$-0.446857\pi$
$653$	8.49304	0.332358	0.166179	+	0.986096i	$0.446857\pi$
$654$	0	0
$655$	−9.96791	−0.389478
$656$	−3.89713	−0.152157
$657$	0	0
$658$	13.6486	0.532078
$659$	−18.2273	−0.710036	−0.355018	−	0.934859i	$-0.615525\pi$
$659$	−18.2273	−0.710036	−0.355018	+	0.934859i	$0.615525\pi$
$660$	0	0
$661$	−16.2494	−0.632027	−0.316013	−	0.948755i	$-0.602344\pi$
$661$	−16.2494	−0.632027	−0.316013	+	0.948755i	$0.602344\pi$
$662$	−24.6382	−0.957591
$663$	0	0
$664$	−46.7746	−1.81521
$665$	11.9654	0.463998
$666$	0	0
$667$	−3.65779	−0.141630
$668$	−12.0111	−0.464725
$669$	0	0
$670$	−7.85476	−0.303456
$671$	43.1027	1.66396
$672$	0	0
$673$	39.3776	1.51789	0.758947	−	0.651153i	$-0.225714\pi$
$673$	39.3776	1.51789	0.758947	+	0.651153i	$0.225714\pi$
$674$	26.1832	1.00854
$675$	0	0
$676$	−2.49216	−0.0958522
$677$	39.4458	1.51602	0.758012	−	0.652241i	$-0.226171\pi$
$677$	39.4458	1.51602	0.758012	+	0.652241i	$0.226171\pi$
$678$	0	0
$679$	−0.561610	−0.0215526
$680$	2.83796	0.108831
$681$	0	0
$682$	34.1905	1.30922
$683$	−25.8506	−0.989144	−0.494572	−	0.869137i	$-0.664675\pi$
$683$	−25.8506	−0.989144	−0.494572	+	0.869137i	$0.664675\pi$
$684$	0	0
$685$	−1.56606	−0.0598360
$686$	20.9544	0.800043
$687$	0	0
$688$	7.82595	0.298362
$689$	−3.70006	−0.140961
$690$	0	0
$691$	−34.6845	−1.31946	−0.659729	−	0.751503i	$-0.729329\pi$
$691$	−34.6845	−1.31946	−0.659729	+	0.751503i	$0.729329\pi$
$692$	−16.5989	−0.630997
$693$	0	0
$694$	−11.6349	−0.441656
$695$	14.0524	0.533039
$696$	0	0
$697$	3.75622	0.142277
$698$	35.8393	1.35654
$699$	0	0
$700$	−8.29202	−0.313409
$701$	38.7984	1.46540	0.732698	−	0.680553i	$-0.238261\pi$
$701$	38.7984	1.46540	0.732698	+	0.680553i	$0.238261\pi$
$702$	0	0
$703$	−52.4493	−1.97816
$704$	−41.1984	−1.55272
$705$	0	0
$706$	−19.2101	−0.722982
$707$	−34.6418	−1.30284
$708$	0	0
$709$	−40.9056	−1.53624	−0.768121	−	0.640305i	$-0.778808\pi$
$709$	−40.9056	−1.53624	−0.768121	+	0.640305i	$0.778808\pi$
$710$	−9.61790	−0.360953
$711$	0	0
$712$	−5.76759	−0.216150
$713$	−14.7458	−0.552233
$714$	0	0
$715$	−15.3958	−0.575770
$716$	19.6583	0.734667
$717$	0	0
$718$	−17.6102	−0.657206
$719$	−4.65393	−0.173562	−0.0867812	−	0.996227i	$-0.527658\pi$
$719$	−4.65393	−0.173562	−0.0867812	+	0.996227i	$0.527658\pi$
$720$	0	0
$721$	−13.0156	−0.484725
$722$	51.1628	1.90408
$723$	0	0
$724$	−0.695640	−0.0258533
$725$	6.70840	0.249144
$726$	0	0
$727$	8.00848	0.297018	0.148509	−	0.988911i	$-0.452553\pi$
$727$	8.00848	0.297018	0.148509	+	0.988911i	$0.452553\pi$
$728$	−24.5027	−0.908129
$729$	0	0
$730$	−7.32431	−0.271085
$731$	−7.54298	−0.278987
$732$	0	0
$733$	20.7022	0.764652	0.382326	−	0.924028i	$-0.375123\pi$
$733$	20.7022	0.764652	0.382326	+	0.924028i	$0.375123\pi$
$734$	21.9179	0.809005
$735$	0	0
$736$	11.4078	0.420495
$737$	−57.2160	−2.10758
$738$	0	0
$739$	10.3160	0.379479	0.189740	−	0.981834i	$-0.439236\pi$
$739$	10.3160	0.379479	0.189740	+	0.981834i	$0.439236\pi$
$740$	−4.12781	−0.151741
$741$	0	0
$742$	−1.97993	−0.0726855
$743$	27.5334	1.01010	0.505051	−	0.863090i	$-0.331474\pi$
$743$	27.5334	1.01010	0.505051	+	0.863090i	$0.331474\pi$
$744$	0	0
$745$	−7.91790	−0.290090
$746$	8.95481	0.327859
$747$	0	0
$748$	6.45804	0.236129
$749$	12.9026	0.471451
$750$	0	0
$751$	30.6714	1.11922	0.559608	−	0.828758i	$-0.310952\pi$
$751$	30.6714	1.11922	0.559608	+	0.828758i	$0.310952\pi$
$752$	8.72340	0.318110
$753$	0	0
$754$	6.19273	0.225526
$755$	−5.02305	−0.182808
$756$	0	0
$757$	29.0720	1.05664	0.528320	−	0.849046i	$-0.322822\pi$
$757$	29.0720	1.05664	0.528320	+	0.849046i	$0.322822\pi$
$758$	−25.1484	−0.913431
$759$	0	0
$760$	17.8895	0.648920
$761$	46.1332	1.67233	0.836163	−	0.548481i	$-0.184794\pi$
$761$	46.1332	1.67233	0.836163	+	0.548481i	$0.184794\pi$
$762$	0	0
$763$	−18.9749	−0.686937
$764$	−3.20318	−0.115887
$765$	0	0
$766$	29.5090	1.06620
$767$	6.24393	0.225455
$768$	0	0
$769$	39.6177	1.42865	0.714326	−	0.699813i	$-0.246733\pi$
$769$	39.6177	1.42865	0.714326	+	0.699813i	$0.246733\pi$
$770$	−8.23842	−0.296892
$771$	0	0
$772$	8.12945	0.292585
$773$	2.07610	0.0746722	0.0373361	−	0.999303i	$-0.488113\pi$
$773$	2.07610	0.0746722	0.0373361	+	0.999303i	$0.488113\pi$
$774$	0	0
$775$	27.0438	0.971442
$776$	−0.839665	−0.0301422
$777$	0	0
$778$	−21.1774	−0.759245
$779$	23.6779	0.848348
$780$	0	0
$781$	−70.0591	−2.50691
$782$	3.34515	0.119622
$783$	0	0
$784$	3.89388	0.139067
$785$	17.2916	0.617165
$786$	0	0
$787$	−7.33613	−0.261505	−0.130752	−	0.991415i	$-0.541739\pi$
$787$	−7.33613	−0.261505	−0.130752	+	0.991415i	$0.541739\pi$
$788$	−18.3250	−0.652802
$789$	0	0
$790$	2.08972	0.0743487
$791$	−3.84867	−0.136843
$792$	0	0
$793$	31.4724	1.11762
$794$	9.81107	0.348182
$795$	0	0
$796$	−3.38663	−0.120036
$797$	19.2052	0.680283	0.340141	−	0.940374i	$-0.389525\pi$
$797$	19.2052	0.680283	0.340141	+	0.940374i	$0.389525\pi$
$798$	0	0
$799$	−8.40797	−0.297453
$800$	−20.9219	−0.739701
$801$	0	0
$802$	−8.72641	−0.308140
$803$	−53.3520	−1.88275
$804$	0	0
$805$	3.55308	0.125230
$806$	24.9650	0.879354
$807$	0	0
$808$	−51.7931	−1.82207
$809$	−15.0621	−0.529554	−0.264777	−	0.964310i	$-0.585298\pi$
$809$	−15.0621	−0.529554	−0.264777	+	0.964310i	$0.585298\pi$
$810$	0	0
$811$	−3.10727	−0.109111	−0.0545554	−	0.998511i	$-0.517374\pi$
$811$	−3.10727	−0.109111	−0.0545554	+	0.998511i	$0.517374\pi$
$812$	−2.75912	−0.0968262
$813$	0	0
$814$	36.1125	1.26574
$815$	−0.480010	−0.0168140
$816$	0	0
$817$	−47.5482	−1.66350
$818$	−9.44720	−0.330314
$819$	0	0
$820$	1.86347	0.0650752
$821$	39.3045	1.37174	0.685869	−	0.727725i	$-0.259423\pi$
$821$	39.3045	1.37174	0.685869	+	0.727725i	$0.259423\pi$
$822$	0	0
$823$	−24.7300	−0.862033	−0.431017	−	0.902344i	$-0.641845\pi$
$823$	−24.7300	−0.862033	−0.431017	+	0.902344i	$0.641845\pi$
$824$	−19.4596	−0.677908
$825$	0	0
$826$	3.34117	0.116254
$827$	37.2644	1.29581	0.647905	−	0.761721i	$-0.275645\pi$
$827$	37.2644	1.29581	0.647905	+	0.761721i	$0.275645\pi$
$828$	0	0
$829$	52.3014	1.81650	0.908251	−	0.418425i	$-0.137418\pi$
$829$	52.3014	1.81650	0.908251	+	0.418425i	$0.137418\pi$
$830$	−11.4833	−0.398591
$831$	0	0
$832$	−30.0819	−1.04290
$833$	−3.75309	−0.130037
$834$	0	0
$835$	−9.43908	−0.326653
$836$	40.7092	1.40796
$837$	0	0
$838$	22.1747	0.766011
$839$	−18.0759	−0.624050	−0.312025	−	0.950074i	$-0.601007\pi$
$839$	−18.0759	−0.624050	−0.312025	+	0.950074i	$0.601007\pi$
$840$	0	0
$841$	−26.7678	−0.923028
$842$	17.0464	0.587459
$843$	0	0
$844$	−16.3389	−0.562410
$845$	−1.95849	−0.0673741
$846$	0	0
$847$	−37.6546	−1.29383
$848$	−1.26546	−0.0434560
$849$	0	0
$850$	−6.13503	−0.210430
$851$	−15.5747	−0.533892
$852$	0	0
$853$	2.26471	0.0775422	0.0387711	−	0.999248i	$-0.487656\pi$
$853$	2.26471	0.0775422	0.0387711	+	0.999248i	$0.487656\pi$
$854$	16.8411	0.576291
$855$	0	0
$856$	19.2907	0.659343
$857$	34.0377	1.16271	0.581353	−	0.813652i	$-0.302524\pi$
$857$	34.0377	1.16271	0.581353	+	0.813652i	$0.302524\pi$
$858$	0	0
$859$	41.4218	1.41329	0.706646	−	0.707567i	$-0.250207\pi$
$859$	41.4218	1.41329	0.706646	+	0.707567i	$0.250207\pi$
$860$	−3.74209	−0.127604
$861$	0	0
$862$	−17.9884	−0.612686
$863$	−27.5129	−0.936551	−0.468275	−	0.883583i	$-0.655124\pi$
$863$	−27.5129	−0.936551	−0.468275	+	0.883583i	$0.655124\pi$
$864$	0	0
$865$	−13.0445	−0.443525
$866$	0.162669	0.00552773
$867$	0	0
$868$	−11.1229	−0.377537
$869$	15.2220	0.516371
$870$	0	0
$871$	−41.7775	−1.41558
$872$	−28.3694	−0.960709
$873$	0	0
$874$	21.0866	0.713266
$875$	−13.7728	−0.465605
$876$	0	0
$877$	24.8045	0.837589	0.418794	−	0.908081i	$-0.362453\pi$
$877$	24.8045	0.837589	0.418794	+	0.908081i	$0.362453\pi$
$878$	24.1632	0.815469
$879$	0	0
$880$	−5.26552	−0.177501
$881$	−25.2629	−0.851128	−0.425564	−	0.904928i	$-0.639924\pi$
$881$	−25.2629	−0.851128	−0.425564	+	0.904928i	$0.639924\pi$
$882$	0	0
$883$	−19.5417	−0.657631	−0.328815	−	0.944394i	$-0.606649\pi$
$883$	−19.5417	−0.657631	−0.328815	+	0.944394i	$0.606649\pi$
$884$	4.71549	0.158599
$885$	0	0
$886$	−13.0864	−0.439646
$887$	34.5222	1.15914	0.579571	−	0.814922i	$-0.303220\pi$
$887$	34.5222	1.15914	0.579571	+	0.814922i	$0.303220\pi$
$888$	0	0
$889$	−20.7476	−0.695852
$890$	−1.41596	−0.0474630
$891$	0	0
$892$	17.4639	0.584733
$893$	−53.0009	−1.77361
$894$	0	0
$895$	15.4487	0.516394
$896$	2.84284	0.0949727
$897$	0	0
$898$	39.7483	1.32642
$899$	8.99867	0.300122
$900$	0	0
$901$	1.21970	0.0406341
$902$	−16.3027	−0.542821
$903$	0	0
$904$	−5.75416	−0.191381
$905$	−0.546677	−0.0181721
$906$	0	0
$907$	48.4647	1.60924	0.804622	−	0.593787i	$-0.202368\pi$
$907$	48.4647	1.60924	0.804622	+	0.593787i	$0.202368\pi$
$908$	21.1476	0.701809
$909$	0	0
$910$	−6.01547	−0.199411
$911$	−14.4271	−0.477990	−0.238995	−	0.971021i	$-0.576818\pi$
$911$	−14.4271	−0.477990	−0.238995	+	0.971021i	$0.576818\pi$
$912$	0	0
$913$	−83.6471	−2.76832
$914$	26.6345	0.880990
$915$	0	0
$916$	−0.952900	−0.0314847
$917$	−28.3698	−0.936852
$918$	0	0
$919$	14.9071	0.491741	0.245871	−	0.969303i	$-0.420926\pi$
$919$	14.9071	0.491741	0.245871	+	0.969303i	$0.420926\pi$
$920$	5.31222	0.175139
$921$	0	0
$922$	−23.5958	−0.777088
$923$	−51.1553	−1.68380
$924$	0	0
$925$	28.5640	0.939179
$926$	44.1010	1.44925
$927$	0	0
$928$	−6.96164	−0.228527
$929$	19.0507	0.625033	0.312517	−	0.949912i	$-0.398828\pi$
$929$	19.0507	0.625033	0.312517	+	0.949912i	$0.398828\pi$
$930$	0	0
$931$	−23.6581	−0.775364
$932$	9.45172	0.309601
$933$	0	0
$934$	−4.48169	−0.146645
$935$	5.07512	0.165974
$936$	0	0
$937$	1.62214	0.0529931	0.0264965	−	0.999649i	$-0.491565\pi$
$937$	1.62214	0.0529931	0.0264965	+	0.999649i	$0.491565\pi$
$938$	−22.3555	−0.729933
$939$	0	0
$940$	−4.17121	−0.136050
$941$	1.42371	0.0464115	0.0232058	−	0.999731i	$-0.492613\pi$
$941$	1.42371	0.0464115	0.0232058	+	0.999731i	$0.492613\pi$
$942$	0	0
$943$	7.03107	0.228963
$944$	2.13548	0.0695041
$945$	0	0
$946$	32.7379	1.06440
$947$	42.2934	1.37435	0.687175	−	0.726492i	$-0.258850\pi$
$947$	42.2934	1.37435	0.687175	+	0.726492i	$0.258850\pi$
$948$	0	0
$949$	−38.9562	−1.26457
$950$	−38.6730	−1.25472
$951$	0	0
$952$	8.07714	0.261782
$953$	30.0275	0.972686	0.486343	−	0.873768i	$-0.338330\pi$
$953$	30.0275	0.972686	0.486343	+	0.873768i	$0.338330\pi$
$954$	0	0
$955$	−2.51726	−0.0814565
$956$	4.73490	0.153137
$957$	0	0
$958$	8.09412	0.261509
$959$	−4.45717	−0.143930
$960$	0	0
$961$	5.27662	0.170213
$962$	26.3683	0.850149
$963$	0	0
$964$	21.0147	0.676839
$965$	6.38862	0.205657
$966$	0	0
$967$	−26.6788	−0.857933	−0.428966	−	0.903320i	$-0.641122\pi$
$967$	−26.6788	−0.857933	−0.428966	+	0.903320i	$0.641122\pi$
$968$	−56.2975	−1.80947
$969$	0	0
$970$	−0.206140	−0.00661876
$971$	−11.1042	−0.356351	−0.178175	−	0.983999i	$-0.557019\pi$
$971$	−11.1042	−0.356351	−0.178175	+	0.983999i	$0.557019\pi$
$972$	0	0
$973$	39.9947	1.28217
$974$	3.26654	0.104667
$975$	0	0
$976$	10.7639	0.344543
$977$	−16.4245	−0.525467	−0.262733	−	0.964868i	$-0.584624\pi$
$977$	−16.4245	−0.525467	−0.262733	+	0.964868i	$0.584624\pi$
$978$	0	0
$979$	−10.3142	−0.329643
$980$	−1.86192	−0.0594767
$981$	0	0
$982$	18.6523	0.595217
$983$	−8.61161	−0.274668	−0.137334	−	0.990525i	$-0.543853\pi$
$983$	−8.61161	−0.274668	−0.137334	+	0.990525i	$0.543853\pi$
$984$	0	0
$985$	−14.4009	−0.458852
$986$	−2.04139	−0.0650112
$987$	0	0
$988$	29.7248	0.945671
$989$	−14.1193	−0.448967
$990$	0	0
$991$	30.1651	0.958227	0.479114	−	0.877753i	$-0.340958\pi$
$991$	30.1651	0.958227	0.479114	+	0.877753i	$0.340958\pi$
$992$	−28.0647	−0.891055
$993$	0	0
$994$	−27.3736	−0.868238
$995$	−2.66142	−0.0843728
$996$	0	0
$997$	6.84091	0.216654	0.108327	−	0.994115i	$-0.465451\pi$
$997$	6.84091	0.216654	0.108327	+	0.994115i	$0.465451\pi$
$998$	2.13116	0.0674608
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4923.2.a.l.1.12		18
3.2	odd	2		547.2.a.b.1.7	✓	18
12.11	even	2		8752.2.a.s.1.15		18

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
547.2.a.b.1.7	✓	18	3.2	odd	2
4923.2.a.l.1.12		18	1.1	even	1	trivial
8752.2.a.s.1.15		18	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 4923 = 3^{2} \cdot 547 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4923.a (trivial)

\(f(q)\)	\(=\)	\(q+1.04467 q^{2} -0.908666 q^{4} -0.714085 q^{5} -2.03236 q^{7} -3.03859 q^{8} +O(q^{10})\)	\(q+1.04467 q^{2} -0.908666 q^{4} -0.714085 q^{5} -2.03236 q^{7} -3.03859 q^{8} -0.745983 q^{10} -5.43392 q^{11} -3.96770 q^{13} -2.12315 q^{14} -1.35699 q^{16} +1.30793 q^{17} +8.24470 q^{19} +0.648865 q^{20} -5.67665 q^{22} +2.44824 q^{23} -4.49008 q^{25} -4.14494 q^{26} +1.84674 q^{28} -1.49405 q^{29} -6.02301 q^{31} +4.65958 q^{32} +1.36635 q^{34} +1.45128 q^{35} -6.36158 q^{37} +8.61299 q^{38} +2.16982 q^{40} +2.87189 q^{41} -5.76713 q^{43} +4.93762 q^{44} +2.55760 q^{46} -6.42848 q^{47} -2.86949 q^{49} -4.69065 q^{50} +3.60532 q^{52} +0.932545 q^{53} +3.88028 q^{55} +6.17553 q^{56} -1.56079 q^{58} -1.57369 q^{59} -7.93215 q^{61} -6.29205 q^{62} +7.58170 q^{64} +2.83328 q^{65} +10.5294 q^{67} -1.18847 q^{68} +1.51611 q^{70} +12.8929 q^{71} +9.81833 q^{73} -6.64575 q^{74} -7.49168 q^{76} +11.0437 q^{77} -2.80129 q^{79} +0.969009 q^{80} +3.00017 q^{82} +15.3935 q^{83} -0.933971 q^{85} -6.02474 q^{86} +16.5115 q^{88} +1.89811 q^{89} +8.06382 q^{91} -2.22463 q^{92} -6.71563 q^{94} -5.88742 q^{95} +0.276333 q^{97} -2.99767 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8}+O(q^{10}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8	\( 18 q + 4 q^{2} + 16 q^{4} + 27 q^{5} - 11 q^{7} + 12 q^{8} - 5 q^{10} - 2 q^{11} - 25 q^{13} + 7 q^{14} + 8 q^{16} + 30 q^{17} + 4 q^{19} + 41 q^{20} - 24 q^{22} + 26 q^{23} + 31 q^{25} + 18 q^{26} - 16 q^{28} + 18 q^{29} - 5 q^{31} + 28 q^{32} + 5 q^{34} + 9 q^{35} - 18 q^{37} + 45 q^{38} + 7 q^{40} + 17 q^{41} + 8 q^{43} - 12 q^{44} + 30 q^{46} + 52 q^{47} + 29 q^{49} - 13 q^{50} - 14 q^{52} + 60 q^{53} + 11 q^{55} - 7 q^{56} + 14 q^{58} + 8 q^{59} - 26 q^{61} - 4 q^{62} + 44 q^{64} + 6 q^{65} + 12 q^{67} + 61 q^{68} + 35 q^{70} + q^{71} - 2 q^{73} - 16 q^{74} + 66 q^{76} + 73 q^{77} + 18 q^{79} + 32 q^{80} + 44 q^{82} + 43 q^{83} + 51 q^{85} - 4 q^{86} - 17 q^{88} + 28 q^{89} - q^{91} + 68 q^{92} + 78 q^{94} + 18 q^{95} - 34 q^{97} - 34 q^{98}+O(q^{100}) \) 18 * q + 4 * q^2 + 16 * q^4 + 27 * q^5 - 11 * q^7 + 12 * q^8 - 5 * q^10 - 2 * q^11 - 25 * q^13 + 7 * q^14 + 8 * q^16 + 30 * q^17 + 4 * q^19 + 41 * q^20 - 24 * q^22 + 26 * q^23 + 31 * q^25 + 18 * q^26 - 16 * q^28 + 18 * q^29 - 5 * q^31 + 28 * q^32 + 5 * q^34 + 9 * q^35 - 18 * q^37 + 45 * q^38 + 7 * q^40 + 17 * q^41 + 8 * q^43 - 12 * q^44 + 30 * q^46 + 52 * q^47 + 29 * q^49 - 13 * q^50 - 14 * q^52 + 60 * q^53 + 11 * q^55 - 7 * q^56 + 14 * q^58 + 8 * q^59 - 26 * q^61 - 4 * q^62 + 44 * q^64 + 6 * q^65 + 12 * q^67 + 61 * q^68 + 35 * q^70 + q^71 - 2 * q^73 - 16 * q^74 + 66 * q^76 + 73 * q^77 + 18 * q^79 + 32 * q^80 + 44 * q^82 + 43 * q^83 + 51 * q^85 - 4 * q^86 - 17 * q^88 + 28 * q^89 - q^91 + 68 * q^92 + 78 * q^94 + 18 * q^95 - 34 * q^97 - 34 * q^98

Introduction

L-functions

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Database

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