LMFDB - Embedded newform 1287.2.a.p.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$0$
Dimension:	$5$
Coefficient field:	5.5.368464.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{5} - 2x^{4} - 6x^{3} + 6x^{2} + 6x - 4 $ x^5 - 2x^4 - 6x^3 + 6x^2 + 6x - 4
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 2 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$0.552543$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.447457 q^{2} -1.79978 q^{4} -1.88693 q^{5} -3.78739 q^{7} -1.70024 q^{8} +O(q^{10})$	\(q+0.447457 q^{2} -1.79978 q^{4} -1.88693 q^{5} -3.78739 q^{7} -1.70024 q^{8} -0.844323 q^{10} -1.00000 q^{11} +1.00000 q^{13} -1.69470 q^{14} +2.83878 q^{16} +3.44746 q^{17} +3.04261 q^{19} +3.39607 q^{20} -0.447457 q^{22} -7.62979 q^{23} -1.43948 q^{25} +0.447457 q^{26} +6.81648 q^{28} +9.67433 q^{29} -10.0211 q^{31} +4.67071 q^{32} +1.54259 q^{34} +7.14656 q^{35} +7.28301 q^{37} +1.36144 q^{38} +3.20824 q^{40} +10.3308 q^{41} +1.79180 q^{43} +1.79978 q^{44} -3.41400 q^{46} +1.70822 q^{47} +7.34434 q^{49} -0.644105 q^{50} -1.79978 q^{52} +10.3443 q^{53} +1.88693 q^{55} +6.43948 q^{56} +4.32885 q^{58} +3.19031 q^{59} -0.652085 q^{61} -4.48402 q^{62} -3.58761 q^{64} -1.88693 q^{65} -1.36828 q^{67} -6.20467 q^{68} +3.19778 q^{70} +1.42164 q^{71} +0.707086 q^{73} +3.25883 q^{74} -5.47604 q^{76} +3.78739 q^{77} -10.7719 q^{79} -5.35659 q^{80} +4.62260 q^{82} +7.26404 q^{83} -6.50513 q^{85} +0.801755 q^{86} +1.70024 q^{88} -12.2480 q^{89} -3.78739 q^{91} +13.7320 q^{92} +0.764356 q^{94} -5.74121 q^{95} -16.8555 q^{97} +3.28628 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 5 q + 3 q^{2} + 7 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) $ 5 * q + 3 * q^2 + 7 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8	$ 5 q + 3 q^{2} + 7 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8} - 6 q^{10} - 5 q^{11} + 5 q^{13} + 6 q^{14} + 19 q^{16} + 18 q^{17} - 2 q^{20} - 3 q^{22} + 8 q^{23} + 7 q^{25} + 3 q^{26} - 8 q^{28} + 20 q^{29} - 8 q^{31} + 31 q^{32} + 26 q^{34} + 6 q^{35} + 4 q^{37} + 18 q^{38} - 20 q^{40} + 8 q^{41} - 22 q^{43} - 7 q^{44} + 6 q^{46} + 6 q^{47} + 5 q^{49} + 11 q^{50} + 7 q^{52} + 20 q^{53} - 4 q^{55} + 18 q^{56} + 12 q^{58} - 16 q^{59} - 4 q^{61} - 26 q^{62} + 11 q^{64} + 4 q^{65} - 20 q^{67} + 36 q^{68} + 14 q^{70} + 6 q^{71} - 12 q^{73} - 20 q^{74} - 16 q^{76} + 4 q^{77} + 6 q^{79} - 52 q^{80} - 18 q^{82} + 4 q^{83} + 6 q^{85} - 46 q^{86} - 9 q^{88} + 10 q^{89} - 4 q^{91} + 68 q^{92} - 2 q^{94} + 2 q^{95} + 4 q^{97} - 45 q^{98}+O(q^{100}) $ 5 * q + 3 * q^2 + 7 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8 - 6 * q^10 - 5 * q^11 + 5 * q^13 + 6 * q^14 + 19 * q^16 + 18 * q^17 - 2 * q^20 - 3 * q^22 + 8 * q^23 + 7 * q^25 + 3 * q^26 - 8 * q^28 + 20 * q^29 - 8 * q^31 + 31 * q^32 + 26 * q^34 + 6 * q^35 + 4 * q^37 + 18 * q^38 - 20 * q^40 + 8 * q^41 - 22 * q^43 - 7 * q^44 + 6 * q^46 + 6 * q^47 + 5 * q^49 + 11 * q^50 + 7 * q^52 + 20 * q^53 - 4 * q^55 + 18 * q^56 + 12 * q^58 - 16 * q^59 - 4 * q^61 - 26 * q^62 + 11 * q^64 + 4 * q^65 - 20 * q^67 + 36 * q^68 + 14 * q^70 + 6 * q^71 - 12 * q^73 - 20 * q^74 - 16 * q^76 + 4 * q^77 + 6 * q^79 - 52 * q^80 - 18 * q^82 + 4 * q^83 + 6 * q^85 - 46 * q^86 - 9 * q^88 + 10 * q^89 - 4 * q^91 + 68 * q^92 - 2 * q^94 + 2 * q^95 + 4 * q^97 - 45 * 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 $

$2$	0.447457	0.316400	0.158200	−	0.987407i	$-0.449431\pi$
$2$	0.447457	0.316400	0.158200	+	0.987407i	$0.449431\pi$
$3$	0	0
$3$	0	0
$4$	−1.79978	−0.899891
$5$	−1.88693	−0.843863	−0.421931	−	0.906628i	$-0.638648\pi$
$5$	−1.88693	−0.843863	−0.421931	+	0.906628i	$0.638648\pi$
$6$	0	0
$7$	−3.78739	−1.43150	−0.715750	−	0.698357i	$-0.753915\pi$
$7$	−3.78739	−1.43150	−0.715750	+	0.698357i	$0.753915\pi$
$8$	−1.70024	−0.601126
$9$	0	0
$10$	−0.844323	−0.266998
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	1.00000	0.277350
$13$	1.00000	0.277350
$14$	−1.69470	−0.452927
$15$	0	0
$16$	2.83878	0.709695
$17$	3.44746	0.836131	0.418066	−	0.908417i	$-0.362708\pi$
$17$	3.44746	0.836131	0.418066	+	0.908417i	$0.362708\pi$
$18$	0	0
$19$	3.04261	0.698023	0.349012	−	0.937118i	$-0.386517\pi$
$19$	3.04261	0.698023	0.349012	+	0.937118i	$0.386517\pi$
$20$	3.39607	0.759385
$21$	0	0
$22$	−0.447457	−0.0953982
$23$	−7.62979	−1.59092	−0.795460	−	0.606006i	$-0.792771\pi$
$23$	−7.62979	−1.59092	−0.795460	+	0.606006i	$0.792771\pi$
$24$	0	0
$25$	−1.43948	−0.287895
$26$	0.447457	0.0877536
$27$	0	0
$28$	6.81648	1.28819
$29$	9.67433	1.79648	0.898239	−	0.439508i	$-0.144847\pi$
$29$	9.67433	1.79648	0.898239	+	0.439508i	$0.144847\pi$
$30$	0	0
$31$	−10.0211	−1.79984	−0.899922	−	0.436051i	$-0.856377\pi$
$31$	−10.0211	−1.79984	−0.899922	+	0.436051i	$0.856377\pi$
$32$	4.67071	0.825673
$33$	0	0
$34$	1.54259	0.264552
$35$	7.14656	1.20799
$36$	0	0
$37$	7.28301	1.19732	0.598659	−	0.801004i	$-0.295700\pi$
$37$	7.28301	1.19732	0.598659	+	0.801004i	$0.295700\pi$
$38$	1.36144	0.220855
$39$	0	0
$40$	3.20824	0.507268
$41$	10.3308	1.61340	0.806702	−	0.590959i	$-0.201250\pi$
$41$	10.3308	1.61340	0.806702	+	0.590959i	$0.201250\pi$
$42$	0	0
$43$	1.79180	0.273247	0.136624	−	0.990623i	$-0.456375\pi$
$43$	1.79180	0.273247	0.136624	+	0.990623i	$0.456375\pi$
$44$	1.79978	0.271327
$45$	0	0
$46$	−3.41400	−0.503367
$47$	1.70822	0.249169	0.124585	−	0.992209i	$-0.460240\pi$
$47$	1.70822	0.249169	0.124585	+	0.992209i	$0.460240\pi$
$48$	0	0
$49$	7.34434	1.04919
$50$	−0.644105	−0.0910902
$51$	0	0
$52$	−1.79978	−0.249585
$53$	10.3443	1.42091	0.710453	−	0.703745i	$-0.248490\pi$
$53$	10.3443	1.42091	0.710453	+	0.703745i	$0.248490\pi$
$54$	0	0
$55$	1.88693	0.254434
$56$	6.43948	0.860511
$57$	0	0
$58$	4.32885	0.568406
$59$	3.19031	0.415343	0.207671	−	0.978199i	$-0.433412\pi$
$59$	3.19031	0.415343	0.207671	+	0.978199i	$0.433412\pi$
$60$	0	0
$61$	−0.652085	−0.0834909	−0.0417454	−	0.999128i	$-0.513292\pi$
$61$	−0.652085	−0.0834909	−0.0417454	+	0.999128i	$0.513292\pi$
$62$	−4.48402	−0.569471
$63$	0	0
$64$	−3.58761	−0.448452
$65$	−1.88693	−0.234045
$66$	0	0
$67$	−1.36828	−0.167163	−0.0835813	−	0.996501i	$-0.526636\pi$
$67$	−1.36828	−0.167163	−0.0835813	+	0.996501i	$0.526636\pi$
$68$	−6.20467	−0.752427
$69$	0	0
$70$	3.19778	0.382208
$71$	1.42164	0.168718	0.0843590	−	0.996435i	$-0.473116\pi$
$71$	1.42164	0.168718	0.0843590	+	0.996435i	$0.473116\pi$
$72$	0	0
$73$	0.707086	0.0827581	0.0413791	−	0.999144i	$-0.486825\pi$
$73$	0.707086	0.0827581	0.0413791	+	0.999144i	$0.486825\pi$
$74$	3.25883	0.378832
$75$	0	0
$76$	−5.47604	−0.628145
$77$	3.78739	0.431613
$78$	0	0
$79$	−10.7719	−1.21194	−0.605969	−	0.795488i	$-0.707214\pi$
$79$	−10.7719	−1.21194	−0.605969	+	0.795488i	$0.707214\pi$
$80$	−5.35659	−0.598885
$81$	0	0
$82$	4.62260	0.510481
$83$	7.26404	0.797332	0.398666	−	0.917096i	$-0.369473\pi$
$83$	7.26404	0.797332	0.398666	+	0.917096i	$0.369473\pi$
$84$	0	0
$85$	−6.50513	−0.705580
$86$	0.801755	0.0864555
$87$	0	0
$88$	1.70024	0.181246
$89$	−12.2480	−1.29828	−0.649142	−	0.760668i	$-0.724872\pi$
$89$	−12.2480	−1.29828	−0.649142	+	0.760668i	$0.724872\pi$
$90$	0	0
$91$	−3.78739	−0.397027
$92$	13.7320	1.43166
$93$	0	0
$94$	0.764356	0.0788373
$95$	−5.74121	−0.589036
$96$	0	0
$97$	−16.8555	−1.71142	−0.855710	−	0.517456i	$-0.826879\pi$
$97$	−16.8555	−1.71142	−0.855710	+	0.517456i	$0.826879\pi$
$98$	3.28628	0.331964
$99$	0	0
$100$	2.59075	0.259075
$101$	−2.53658	−0.252399	−0.126200	−	0.992005i	$-0.540278\pi$
$101$	−2.53658	−0.252399	−0.126200	+	0.992005i	$0.540278\pi$
$102$	0	0
$103$	16.0934	1.58573	0.792867	−	0.609395i	$-0.208588\pi$
$103$	16.0934	1.58573	0.792867	+	0.609395i	$0.208588\pi$
$104$	−1.70024	−0.166722
$105$	0	0
$106$	4.62865	0.449574
$107$	10.2706	0.992898	0.496449	−	0.868066i	$-0.334637\pi$
$107$	10.2706	0.992898	0.496449	+	0.868066i	$0.334637\pi$
$108$	0	0
$109$	−3.46336	−0.331730	−0.165865	−	0.986148i	$-0.553042\pi$
$109$	−3.46336	−0.331730	−0.165865	+	0.986148i	$0.553042\pi$
$110$	0.844323	0.0805030
$111$	0	0
$112$	−10.7516	−1.01593
$113$	11.6485	1.09580	0.547900	−	0.836544i	$-0.315427\pi$
$113$	11.6485	1.09580	0.547900	+	0.836544i	$0.315427\pi$
$114$	0	0
$115$	14.3969	1.34252
$116$	−17.4117	−1.61663
$117$	0	0
$118$	1.42753	0.131415
$119$	−13.0569	−1.19692
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.291780	−0.0264165
$123$	0	0
$124$	18.0358	1.61966
$125$	12.1509	1.08681
$126$	0	0
$127$	−9.65016	−0.856313	−0.428156	−	0.903705i	$-0.640837\pi$
$127$	−9.65016	−0.856313	−0.428156	+	0.903705i	$0.640837\pi$
$128$	−10.9467	−0.967563
$129$	0	0
$130$	−0.844323	−0.0740520
$131$	7.05326	0.616246	0.308123	−	0.951346i	$-0.400299\pi$
$131$	7.05326	0.616246	0.308123	+	0.951346i	$0.400299\pi$
$132$	0	0
$133$	−11.5236	−0.999220
$134$	−0.612249	−0.0528902
$135$	0	0
$136$	−5.86151	−0.502620
$137$	10.6755	0.912066	0.456033	−	0.889963i	$-0.349270\pi$
$137$	10.6755	0.912066	0.456033	+	0.889963i	$0.349270\pi$
$138$	0	0
$139$	7.83328	0.664410	0.332205	−	0.943207i	$-0.392207\pi$
$139$	7.83328	0.664410	0.332205	+	0.943207i	$0.392207\pi$
$140$	−12.8623	−1.08706
$141$	0	0
$142$	0.636125	0.0533824
$143$	−1.00000	−0.0836242
$144$	0	0
$145$	−18.2548	−1.51598
$146$	0.316391	0.0261847
$147$	0	0
$148$	−13.1078	−1.07746
$149$	−17.5590	−1.43849	−0.719244	−	0.694757i	$-0.755512\pi$
$149$	−17.5590	−1.43849	−0.719244	+	0.694757i	$0.755512\pi$
$150$	0	0
$151$	−5.73844	−0.466988	−0.233494	−	0.972358i	$-0.575016\pi$
$151$	−5.73844	−0.466988	−0.233494	+	0.972358i	$0.575016\pi$
$152$	−5.17317	−0.419600
$153$	0	0
$154$	1.69470	0.136563
$155$	18.9092	1.51882
$156$	0	0
$157$	−5.59392	−0.446444	−0.223222	−	0.974768i	$-0.571657\pi$
$157$	−5.59392	−0.446444	−0.223222	+	0.974768i	$0.571657\pi$
$158$	−4.81998	−0.383457
$159$	0	0
$160$	−8.81333	−0.696755
$161$	28.8970	2.27740
$162$	0	0
$163$	−0.279490	−0.0218913	−0.0109457	−	0.999940i	$-0.503484\pi$
$163$	−0.279490	−0.0218913	−0.0109457	+	0.999940i	$0.503484\pi$
$164$	−18.5932	−1.45189
$165$	0	0
$166$	3.25035	0.252276
$167$	−10.8767	−0.841663	−0.420832	−	0.907139i	$-0.638262\pi$
$167$	−10.8767	−0.841663	−0.420832	+	0.907139i	$0.638262\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	−2.91077	−0.223246
$171$	0	0
$172$	−3.22485	−0.245893
$173$	16.7093	1.27038	0.635192	−	0.772354i	$-0.280921\pi$
$173$	16.7093	1.27038	0.635192	+	0.772354i	$0.280921\pi$
$174$	0	0
$175$	5.45187	0.412122
$176$	−2.83878	−0.213981
$177$	0	0
$178$	−5.48045	−0.410777
$179$	−17.7887	−1.32959	−0.664796	−	0.747025i	$-0.731482\pi$
$179$	−17.7887	−1.32959	−0.664796	+	0.747025i	$0.731482\pi$
$180$	0	0
$181$	15.9141	1.18289	0.591443	−	0.806347i	$-0.298559\pi$
$181$	15.9141	1.18289	0.591443	+	0.806347i	$0.298559\pi$
$182$	−1.69470	−0.125619
$183$	0	0
$184$	12.9725	0.956343
$185$	−13.7426	−1.01037
$186$	0	0
$187$	−3.44746	−0.252103
$188$	−3.07442	−0.224225
$189$	0	0
$190$	−2.56895	−0.186371
$191$	10.4543	0.756450	0.378225	−	0.925714i	$-0.376535\pi$
$191$	10.4543	0.756450	0.378225	+	0.925714i	$0.376535\pi$
$192$	0	0
$193$	−7.42996	−0.534820	−0.267410	−	0.963583i	$-0.586168\pi$
$193$	−7.42996	−0.534820	−0.267410	+	0.963583i	$0.586168\pi$
$194$	−7.54213	−0.541493
$195$	0	0
$196$	−13.2182	−0.944159
$197$	14.2901	1.01813	0.509063	−	0.860729i	$-0.329992\pi$
$197$	14.2901	1.01813	0.509063	+	0.860729i	$0.329992\pi$
$198$	0	0
$199$	13.2997	0.942791	0.471396	−	0.881922i	$-0.343750\pi$
$199$	13.2997	0.942791	0.471396	+	0.881922i	$0.343750\pi$
$200$	2.44746	0.173061
$201$	0	0
$202$	−1.13501	−0.0798592
$203$	−36.6405	−2.57166
$204$	0	0
$205$	−19.4936	−1.36149
$206$	7.20113	0.501726
$207$	0	0
$208$	2.83878	0.196834
$209$	−3.04261	−0.210462
$210$	0	0
$211$	24.2679	1.67067	0.835334	−	0.549743i	$-0.185274\pi$
$211$	24.2679	1.67067	0.835334	+	0.549743i	$0.185274\pi$
$212$	−18.6176	−1.27866
$213$	0	0
$214$	4.59566	0.314153
$215$	−3.38101	−0.230583
$216$	0	0
$217$	37.9539	2.57648
$218$	−1.54971	−0.104959
$219$	0	0
$220$	−3.39607	−0.228963
$221$	3.44746	0.231901
$222$	0	0
$223$	25.9001	1.73440	0.867200	−	0.497960i	$-0.165918\pi$
$223$	25.9001	1.73440	0.867200	+	0.497960i	$0.165918\pi$
$224$	−17.6898	−1.18195
$225$	0	0
$226$	5.21221	0.346711
$227$	12.3312	0.818452	0.409226	−	0.912433i	$-0.365799\pi$
$227$	12.3312	0.818452	0.409226	+	0.912433i	$0.365799\pi$
$228$	0	0
$229$	−15.3946	−1.01730	−0.508652	−	0.860972i	$-0.669856\pi$
$229$	−15.3946	−1.01730	−0.508652	+	0.860972i	$0.669856\pi$
$230$	6.44200	0.424773
$231$	0	0
$232$	−16.4487	−1.07991
$233$	28.6656	1.87794	0.938971	−	0.343995i	$-0.111780\pi$
$233$	28.6656	1.87794	0.938971	+	0.343995i	$0.111780\pi$
$234$	0	0
$235$	−3.22330	−0.210265
$236$	−5.74186	−0.373763
$237$	0	0
$238$	−5.84239	−0.378706
$239$	−28.6218	−1.85139	−0.925694	−	0.378274i	$-0.876518\pi$
$239$	−28.6218	−1.85139	−0.925694	+	0.378274i	$0.876518\pi$
$240$	0	0
$241$	14.3333	0.923286	0.461643	−	0.887066i	$-0.347260\pi$
$241$	14.3333	0.923286	0.461643	+	0.887066i	$0.347260\pi$
$242$	0.447457	0.0287636
$243$	0	0
$244$	1.17361	0.0751327
$245$	−13.8583	−0.885374
$246$	0	0
$247$	3.04261	0.193597
$248$	17.0383	1.08193
$249$	0	0
$250$	5.43700	0.343866
$251$	14.1541	0.893398	0.446699	−	0.894684i	$-0.352600\pi$
$251$	14.1541	0.893398	0.446699	+	0.894684i	$0.352600\pi$
$252$	0	0
$253$	7.62979	0.479681
$254$	−4.31803	−0.270937
$255$	0	0
$256$	2.27703	0.142315
$257$	−26.5236	−1.65450	−0.827249	−	0.561836i	$-0.810095\pi$
$257$	−26.5236	−1.65450	−0.827249	+	0.561836i	$0.810095\pi$
$258$	0	0
$259$	−27.5836	−1.71396
$260$	3.39607	0.210615
$261$	0	0
$262$	3.15603	0.194980
$263$	−10.7700	−0.664105	−0.332052	−	0.943261i	$-0.607741\pi$
$263$	−10.7700	−0.664105	−0.332052	+	0.943261i	$0.607741\pi$
$264$	0	0
$265$	−19.5191	−1.19905
$266$	−5.15630	−0.316153
$267$	0	0
$268$	2.46261	0.150428
$269$	11.6864	0.712531	0.356265	−	0.934385i	$-0.384050\pi$
$269$	11.6864	0.712531	0.356265	+	0.934385i	$0.384050\pi$
$270$	0	0
$271$	−1.71099	−0.103935	−0.0519675	−	0.998649i	$-0.516549\pi$
$271$	−1.71099	−0.103935	−0.0519675	+	0.998649i	$0.516549\pi$
$272$	9.78657	0.593398
$273$	0	0
$274$	4.77681	0.288578
$275$	1.43948	0.0868038
$276$	0	0
$277$	−9.78756	−0.588078	−0.294039	−	0.955793i	$-0.594999\pi$
$277$	−9.78756	−0.588078	−0.294039	+	0.955793i	$0.594999\pi$
$278$	3.50506	0.210219
$279$	0	0
$280$	−12.1509	−0.726154
$281$	−19.3351	−1.15344	−0.576719	−	0.816943i	$-0.695667\pi$
$281$	−19.3351	−1.15344	−0.576719	+	0.816943i	$0.695667\pi$
$282$	0	0
$283$	−8.40802	−0.499805	−0.249902	−	0.968271i	$-0.580399\pi$
$283$	−8.40802	−0.499805	−0.249902	+	0.968271i	$0.580399\pi$
$284$	−2.55865	−0.151828
$285$	0	0
$286$	−0.447457	−0.0264587
$287$	−39.1269	−2.30959
$288$	0	0
$289$	−5.11504	−0.300885
$290$	−8.16825	−0.479656
$291$	0	0
$292$	−1.27260	−0.0744733
$293$	−1.27983	−0.0747685	−0.0373842	−	0.999301i	$-0.511903\pi$
$293$	−1.27983	−0.0747685	−0.0373842	+	0.999301i	$0.511903\pi$
$294$	0	0
$295$	−6.01991	−0.350492
$296$	−12.3829	−0.719739
$297$	0	0
$298$	−7.85690	−0.455138
$299$	−7.62979	−0.441242
$300$	0	0
$301$	−6.78626	−0.391153
$302$	−2.56771	−0.147755
$303$	0	0
$304$	8.63730	0.495383
$305$	1.23044	0.0704548
$306$	0	0
$307$	27.7526	1.58392	0.791962	−	0.610570i	$-0.209060\pi$
$307$	27.7526	1.58392	0.791962	+	0.610570i	$0.209060\pi$
$308$	−6.81648	−0.388405
$309$	0	0
$310$	8.46105	0.480555
$311$	−1.33666	−0.0757951	−0.0378975	−	0.999282i	$-0.512066\pi$
$311$	−1.33666	−0.0757951	−0.0378975	+	0.999282i	$0.512066\pi$
$312$	0	0
$313$	30.4168	1.71926	0.859629	−	0.510918i	$-0.170695\pi$
$313$	30.4168	1.71926	0.859629	+	0.510918i	$0.170695\pi$
$314$	−2.50304	−0.141255
$315$	0	0
$316$	19.3871	1.09061
$317$	13.2355	0.743377	0.371689	−	0.928357i	$-0.378779\pi$
$317$	13.2355	0.743377	0.371689	+	0.928357i	$0.378779\pi$
$318$	0	0
$319$	−9.67433	−0.541658
$320$	6.76959	0.378432
$321$	0	0
$322$	12.9302	0.720570
$323$	10.4893	0.583639
$324$	0	0
$325$	−1.43948	−0.0798478
$326$	−0.125060	−0.00692642
$327$	0	0
$328$	−17.5649	−0.969858
$329$	−6.46970	−0.356686
$330$	0	0
$331$	18.4522	1.01422	0.507112	−	0.861880i	$-0.330713\pi$
$331$	18.4522	1.01422	0.507112	+	0.861880i	$0.330713\pi$
$332$	−13.0737	−0.717512
$333$	0	0
$334$	−4.86685	−0.266302
$335$	2.58186	0.141062
$336$	0	0
$337$	27.4020	1.49268	0.746340	−	0.665565i	$-0.231809\pi$
$337$	27.4020	1.49268	0.746340	+	0.665565i	$0.231809\pi$
$338$	0.447457	0.0243385
$339$	0	0
$340$	11.7078	0.634945
$341$	10.0211	0.542673
$342$	0	0
$343$	−1.30417	−0.0704185
$344$	−3.04649	−0.164256
$345$	0	0
$346$	7.47670	0.401950
$347$	28.9512	1.55418	0.777092	−	0.629387i	$-0.216694\pi$
$347$	28.9512	1.55418	0.777092	+	0.629387i	$0.216694\pi$
$348$	0	0
$349$	−25.9366	−1.38835	−0.694176	−	0.719805i	$-0.744231\pi$
$349$	−25.9366	−1.38835	−0.694176	+	0.719805i	$0.744231\pi$
$350$	2.43948	0.130396
$351$	0	0
$352$	−4.67071	−0.248950
$353$	32.3098	1.71968	0.859838	−	0.510567i	$-0.170565\pi$
$353$	32.3098	1.71968	0.859838	+	0.510567i	$0.170565\pi$
$354$	0	0
$355$	−2.68255	−0.142375
$356$	22.0437	1.16831
$357$	0	0
$358$	−7.95970	−0.420683
$359$	−2.86790	−0.151362	−0.0756809	−	0.997132i	$-0.524113\pi$
$359$	−2.86790	−0.151362	−0.0756809	+	0.997132i	$0.524113\pi$
$360$	0	0
$361$	−9.74251	−0.512764
$362$	7.12088	0.374265
$363$	0	0
$364$	6.81648	0.357281
$365$	−1.33422	−0.0698365
$366$	0	0
$367$	8.47913	0.442607	0.221303	−	0.975205i	$-0.428969\pi$
$367$	8.47913	0.442607	0.221303	+	0.975205i	$0.428969\pi$
$368$	−21.6593	−1.12907
$369$	0	0
$370$	−6.14921	−0.319682
$371$	−39.1781	−2.03403
$372$	0	0
$373$	−24.1871	−1.25236	−0.626180	−	0.779679i	$-0.715382\pi$
$373$	−24.1871	−1.25236	−0.626180	+	0.779679i	$0.715382\pi$
$374$	−1.54259	−0.0797654
$375$	0	0
$376$	−2.90438	−0.149782
$377$	9.67433	0.498253
$378$	0	0
$379$	12.3448	0.634110	0.317055	−	0.948407i	$-0.397306\pi$
$379$	12.3448	0.634110	0.317055	+	0.948407i	$0.397306\pi$
$380$	10.3329	0.530068
$381$	0	0
$382$	4.67787	0.239341
$383$	−12.3933	−0.633268	−0.316634	−	0.948548i	$-0.602553\pi$
$383$	−12.3933	−0.633268	−0.316634	+	0.948548i	$0.602553\pi$
$384$	0	0
$385$	−7.14656	−0.364223
$386$	−3.32459	−0.169217
$387$	0	0
$388$	30.3363	1.54009
$389$	−2.65056	−0.134388	−0.0671942	−	0.997740i	$-0.521405\pi$
$389$	−2.65056	−0.134388	−0.0671942	+	0.997740i	$0.521405\pi$
$390$	0	0
$391$	−26.3034	−1.33022
$392$	−12.4871	−0.630696
$393$	0	0
$394$	6.39420	0.322135
$395$	20.3259	1.02271
$396$	0	0
$397$	−14.1323	−0.709280	−0.354640	−	0.935003i	$-0.615397\pi$
$397$	−14.1323	−0.709280	−0.354640	+	0.935003i	$0.615397\pi$
$398$	5.95105	0.298299
$399$	0	0
$400$	−4.08636	−0.204318
$401$	29.8917	1.49272	0.746360	−	0.665542i	$-0.231800\pi$
$401$	29.8917	1.49272	0.746360	+	0.665542i	$0.231800\pi$
$402$	0	0
$403$	−10.0211	−0.499187
$404$	4.56530	0.227132
$405$	0	0
$406$	−16.3950	−0.813673
$407$	−7.28301	−0.361005
$408$	0	0
$409$	22.2905	1.10219	0.551097	−	0.834441i	$-0.314209\pi$
$409$	22.2905	1.10219	0.551097	+	0.834441i	$0.314209\pi$
$410$	−8.72255	−0.430776
$411$	0	0
$412$	−28.9647	−1.42699
$413$	−12.0830	−0.594563
$414$	0	0
$415$	−13.7068	−0.672839
$416$	4.67071	0.229001
$417$	0	0
$418$	−1.36144	−0.0665902
$419$	−23.5737	−1.15165	−0.575825	−	0.817573i	$-0.695319\pi$
$419$	−23.5737	−1.15165	−0.575825	+	0.817573i	$0.695319\pi$
$420$	0	0
$421$	29.9483	1.45959	0.729795	−	0.683667i	$-0.239616\pi$
$421$	29.9483	1.45959	0.729795	+	0.683667i	$0.239616\pi$
$422$	10.8588	0.528599
$423$	0	0
$424$	−17.5879	−0.854142
$425$	−4.96254	−0.240718
$426$	0	0
$427$	2.46970	0.119517
$428$	−18.4849	−0.893500
$429$	0	0
$430$	−1.51286	−0.0729565
$431$	−9.83939	−0.473947	−0.236973	−	0.971516i	$-0.576155\pi$
$431$	−9.83939	−0.473947	−0.236973	+	0.971516i	$0.576155\pi$
$432$	0	0
$433$	−27.1758	−1.30599	−0.652994	−	0.757364i	$-0.726487\pi$
$433$	−27.1758	−1.30599	−0.652994	+	0.757364i	$0.726487\pi$
$434$	16.9827	0.815198
$435$	0	0
$436$	6.23330	0.298521
$437$	−23.2145	−1.11050
$438$	0	0
$439$	12.6730	0.604851	0.302425	−	0.953173i	$-0.402204\pi$
$439$	12.6730	0.604851	0.302425	+	0.953173i	$0.402204\pi$
$440$	−3.20824	−0.152947
$441$	0	0
$442$	1.54259	0.0733735
$443$	2.83288	0.134594	0.0672971	−	0.997733i	$-0.478562\pi$
$443$	2.83288	0.134594	0.0672971	+	0.997733i	$0.478562\pi$
$444$	0	0
$445$	23.1111	1.09557
$446$	11.5892	0.548764
$447$	0	0
$448$	13.5877	0.641959
$449$	−21.4149	−1.01063	−0.505315	−	0.862935i	$-0.668624\pi$
$449$	−21.4149	−1.01063	−0.505315	+	0.862935i	$0.668624\pi$
$450$	0	0
$451$	−10.3308	−0.486459
$452$	−20.9648	−0.986101
$453$	0	0
$454$	5.51770	0.258958
$455$	7.14656	0.335036
$456$	0	0
$457$	−20.6417	−0.965580	−0.482790	−	0.875736i	$-0.660377\pi$
$457$	−20.6417	−0.965580	−0.482790	+	0.875736i	$0.660377\pi$
$458$	−6.88842	−0.321875
$459$	0	0
$460$	−25.9113	−1.20812
$461$	−9.88797	−0.460529	−0.230264	−	0.973128i	$-0.573959\pi$
$461$	−9.88797	−0.460529	−0.230264	+	0.973128i	$0.573959\pi$
$462$	0	0
$463$	9.25716	0.430217	0.215108	−	0.976590i	$-0.430990\pi$
$463$	9.25716	0.430217	0.215108	+	0.976590i	$0.430990\pi$
$464$	27.4633	1.27495
$465$	0	0
$466$	12.8266	0.594181
$467$	11.9027	0.550790	0.275395	−	0.961331i	$-0.411191\pi$
$467$	11.9027	0.550790	0.275395	+	0.961331i	$0.411191\pi$
$468$	0	0
$469$	5.18223	0.239293
$470$	−1.44229	−0.0665278
$471$	0	0
$472$	−5.42429	−0.249673
$473$	−1.79180	−0.0823871
$474$	0	0
$475$	−4.37977	−0.200958
$476$	23.4995	1.07710
$477$	0	0
$478$	−12.8070	−0.585779
$479$	21.7405	0.993350	0.496675	−	0.867937i	$-0.334554\pi$
$479$	21.7405	0.993350	0.496675	+	0.867937i	$0.334554\pi$
$480$	0	0
$481$	7.28301	0.332076
$482$	6.41352	0.292128
$483$	0	0
$484$	−1.79978	−0.0818083
$485$	31.8053	1.44420
$486$	0	0
$487$	10.2805	0.465854	0.232927	−	0.972494i	$-0.425170\pi$
$487$	10.2805	0.465854	0.232927	+	0.972494i	$0.425170\pi$
$488$	1.10870	0.0501885
$489$	0	0
$490$	−6.20100	−0.280132
$491$	17.8439	0.805283	0.402641	−	0.915358i	$-0.368092\pi$
$491$	17.8439	0.805283	0.402641	+	0.915358i	$0.368092\pi$
$492$	0	0
$493$	33.3518	1.50209
$494$	1.36144	0.0612540
$495$	0	0
$496$	−28.4477	−1.27734
$497$	−5.38432	−0.241520
$498$	0	0
$499$	8.42549	0.377177	0.188588	−	0.982056i	$-0.439609\pi$
$499$	8.42549	0.377177	0.188588	+	0.982056i	$0.439609\pi$
$500$	−21.8689	−0.978008
$501$	0	0
$502$	6.33335	0.282671
$503$	25.8527	1.15272	0.576358	−	0.817197i	$-0.304473\pi$
$503$	25.8527	1.15272	0.576358	+	0.817197i	$0.304473\pi$
$504$	0	0
$505$	4.78637	0.212991
$506$	3.41400	0.151771
$507$	0	0
$508$	17.3682	0.770588
$509$	−29.1145	−1.29048	−0.645238	−	0.763982i	$-0.723242\pi$
$509$	−29.1145	−1.29048	−0.645238	+	0.763982i	$0.723242\pi$
$510$	0	0
$511$	−2.67801	−0.118468
$512$	22.9123	1.01259
$513$	0	0
$514$	−11.8682	−0.523483
$515$	−30.3673	−1.33814
$516$	0	0
$517$	−1.70822	−0.0751274
$518$	−12.3425	−0.542298
$519$	0	0
$520$	3.20824	0.140691
$521$	−4.63913	−0.203244	−0.101622	−	0.994823i	$-0.532403\pi$
$521$	−4.63913	−0.203244	−0.101622	+	0.994823i	$0.532403\pi$
$522$	0	0
$523$	−33.2775	−1.45512	−0.727562	−	0.686042i	$-0.759347\pi$
$523$	−33.2775	−1.45512	−0.727562	+	0.686042i	$0.759347\pi$
$524$	−12.6943	−0.554554
$525$	0	0
$526$	−4.81910	−0.210123
$527$	−34.5473	−1.50491
$528$	0	0
$529$	35.2136	1.53103
$530$	−8.73397	−0.379379
$531$	0	0
$532$	20.7399	0.899189
$533$	10.3308	0.447478
$534$	0	0
$535$	−19.3800	−0.837870
$536$	2.32641	0.100486
$537$	0	0
$538$	5.22915	0.225445
$539$	−7.34434	−0.316343
$540$	0	0
$541$	17.8608	0.767895	0.383947	−	0.923355i	$-0.374564\pi$
$541$	17.8608	0.767895	0.383947	+	0.923355i	$0.374564\pi$
$542$	−0.765594	−0.0328851
$543$	0	0
$544$	16.1021	0.690371
$545$	6.53514	0.279935
$546$	0	0
$547$	−16.1166	−0.689095	−0.344547	−	0.938769i	$-0.611968\pi$
$547$	−16.1166	−0.689095	−0.344547	+	0.938769i	$0.611968\pi$
$548$	−19.2135	−0.820760
$549$	0	0
$550$	0.644105	0.0274647
$551$	29.4352	1.25398
$552$	0	0
$553$	40.7976	1.73489
$554$	−4.37952	−0.186068
$555$	0	0
$556$	−14.0982	−0.597897
$557$	−33.0261	−1.39936	−0.699681	−	0.714455i	$-0.746675\pi$
$557$	−33.0261	−1.39936	−0.699681	+	0.714455i	$0.746675\pi$
$558$	0	0
$559$	1.79180	0.0757852
$560$	20.2875	0.857304
$561$	0	0
$562$	−8.65165	−0.364948
$563$	6.08522	0.256462	0.128231	−	0.991744i	$-0.459070\pi$
$563$	6.08522	0.256462	0.128231	+	0.991744i	$0.459070\pi$
$564$	0	0
$565$	−21.9800	−0.924705
$566$	−3.76223	−0.158138
$567$	0	0
$568$	−2.41714	−0.101421
$569$	18.2746	0.766111	0.383055	−	0.923725i	$-0.374872\pi$
$569$	18.2746	0.766111	0.383055	+	0.923725i	$0.374872\pi$
$570$	0	0
$571$	−27.6375	−1.15660	−0.578298	−	0.815826i	$-0.696283\pi$
$571$	−27.6375	−1.15660	−0.578298	+	0.815826i	$0.696283\pi$
$572$	1.79978	0.0752527
$573$	0	0
$574$	−17.5076	−0.730753
$575$	10.9829	0.458019
$576$	0	0
$577$	−0.846571	−0.0352432	−0.0176216	−	0.999845i	$-0.505609\pi$
$577$	−0.846571	−0.0352432	−0.0176216	+	0.999845i	$0.505609\pi$
$578$	−2.28876	−0.0951999
$579$	0	0
$580$	32.8547	1.36422
$581$	−27.5118	−1.14138
$582$	0	0
$583$	−10.3443	−0.428419
$584$	−1.20222	−0.0497480
$585$	0	0
$586$	−0.572669	−0.0236567
$587$	−10.1783	−0.420101	−0.210051	−	0.977690i	$-0.567363\pi$
$587$	−10.1783	−0.420101	−0.210051	+	0.977690i	$0.567363\pi$
$588$	0	0
$589$	−30.4903	−1.25633
$590$	−2.69365	−0.110896
$591$	0	0
$592$	20.6748	0.849731
$593$	−22.9957	−0.944322	−0.472161	−	0.881512i	$-0.656526\pi$
$593$	−22.9957	−0.944322	−0.472161	+	0.881512i	$0.656526\pi$
$594$	0	0
$595$	24.6375	1.01004
$596$	31.6024	1.29448
$597$	0	0
$598$	−3.41400	−0.139609
$599$	32.3508	1.32182	0.660909	−	0.750466i	$-0.270171\pi$
$599$	32.3508	1.32182	0.660909	+	0.750466i	$0.270171\pi$
$600$	0	0
$601$	−29.0556	−1.18520	−0.592602	−	0.805496i	$-0.701899\pi$
$601$	−29.0556	−1.18520	−0.592602	+	0.805496i	$0.701899\pi$
$602$	−3.03656	−0.123761
$603$	0	0
$604$	10.3279	0.420238
$605$	−1.88693	−0.0767148
$606$	0	0
$607$	21.3872	0.868081	0.434041	−	0.900893i	$-0.357087\pi$
$607$	21.3872	0.868081	0.434041	+	0.900893i	$0.357087\pi$
$608$	14.2112	0.576339
$609$	0	0
$610$	0.550570	0.0222919
$611$	1.70822	0.0691072
$612$	0	0
$613$	14.2733	0.576491	0.288246	−	0.957557i	$-0.406928\pi$
$613$	14.2733	0.576491	0.288246	+	0.957557i	$0.406928\pi$
$614$	12.4181	0.501154
$615$	0	0
$616$	−6.43948	−0.259454
$617$	31.9402	1.28587	0.642933	−	0.765923i	$-0.277717\pi$
$617$	31.9402	1.28587	0.642933	+	0.765923i	$0.277717\pi$
$618$	0	0
$619$	−25.2095	−1.01325	−0.506627	−	0.862165i	$-0.669108\pi$
$619$	−25.2095	−1.01325	−0.506627	+	0.862165i	$0.669108\pi$
$620$	−34.0324	−1.36677
$621$	0	0
$622$	−0.598099	−0.0239816
$623$	46.3879	1.85849
$624$	0	0
$625$	−15.7305	−0.629221
$626$	13.6102	0.543973
$627$	0	0
$628$	10.0678	0.401751
$629$	25.1079	1.00112
$630$	0	0
$631$	47.2968	1.88285	0.941427	−	0.337216i	$-0.109485\pi$
$631$	47.2968	1.88285	0.941427	+	0.337216i	$0.109485\pi$
$632$	18.3149	0.728527
$633$	0	0
$634$	5.92230	0.235205
$635$	18.2092	0.722611
$636$	0	0
$637$	7.34434	0.290994
$638$	−4.32885	−0.171381
$639$	0	0
$640$	20.6558	0.816491
$641$	25.0224	0.988324	0.494162	−	0.869370i	$-0.335475\pi$
$641$	25.0224	0.988324	0.494162	+	0.869370i	$0.335475\pi$
$642$	0	0
$643$	−6.31809	−0.249161	−0.124581	−	0.992209i	$-0.539759\pi$
$643$	−6.31809	−0.249161	−0.124581	+	0.992209i	$0.539759\pi$
$644$	−52.0083	−2.04941
$645$	0	0
$646$	4.69350	0.184663
$647$	−10.1739	−0.399977	−0.199989	−	0.979798i	$-0.564091\pi$
$647$	−10.1739	−0.399977	−0.199989	+	0.979798i	$0.564091\pi$
$648$	0	0
$649$	−3.19031	−0.125231
$650$	−0.644105	−0.0252639
$651$	0	0
$652$	0.503021	0.0196998
$653$	18.7833	0.735045	0.367523	−	0.930015i	$-0.380206\pi$
$653$	18.7833	0.735045	0.367523	+	0.930015i	$0.380206\pi$
$654$	0	0
$655$	−13.3090	−0.520027
$656$	29.3269	1.14502
$657$	0	0
$658$	−2.89491	−0.112856
$659$	37.1837	1.44847	0.724236	−	0.689552i	$-0.242193\pi$
$659$	37.1837	1.44847	0.724236	+	0.689552i	$0.242193\pi$
$660$	0	0
$661$	−9.40516	−0.365818	−0.182909	−	0.983130i	$-0.558551\pi$
$661$	−9.40516	−0.365818	−0.182909	+	0.983130i	$0.558551\pi$
$662$	8.25657	0.320901
$663$	0	0
$664$	−12.3506	−0.479297
$665$	21.7442	0.843205
$666$	0	0
$667$	−73.8131	−2.85805
$668$	19.5757	0.757405
$669$	0	0
$670$	1.15527	0.0446321
$671$	0.652085	0.0251734
$672$	0	0
$673$	3.87738	0.149462	0.0747310	−	0.997204i	$-0.476190\pi$
$673$	3.87738	0.149462	0.0747310	+	0.997204i	$0.476190\pi$
$674$	12.2612	0.472284
$675$	0	0
$676$	−1.79978	−0.0692224
$677$	−12.7397	−0.489626	−0.244813	−	0.969570i	$-0.578727\pi$
$677$	−12.7397	−0.489626	−0.244813	+	0.969570i	$0.578727\pi$
$678$	0	0
$679$	63.8385	2.44990
$680$	11.0603	0.424142
$681$	0	0
$682$	4.48402	0.171702
$683$	16.6277	0.636242	0.318121	−	0.948050i	$-0.396948\pi$
$683$	16.6277	0.636242	0.318121	+	0.948050i	$0.396948\pi$
$684$	0	0
$685$	−20.1439	−0.769659
$686$	−0.583560	−0.0222804
$687$	0	0
$688$	5.08653	0.193922
$689$	10.3443	0.394088
$690$	0	0
$691$	−6.79923	−0.258655	−0.129327	−	0.991602i	$-0.541282\pi$
$691$	−6.79923	−0.258655	−0.129327	+	0.991602i	$0.541282\pi$
$692$	−30.0731	−1.14321
$693$	0	0
$694$	12.9544	0.491744
$695$	−14.7809	−0.560671
$696$	0	0
$697$	35.6151	1.34902
$698$	−11.6055	−0.439275
$699$	0	0
$700$	−9.81217	−0.370865
$701$	21.9381	0.828590	0.414295	−	0.910143i	$-0.364028\pi$
$701$	21.9381	0.828590	0.414295	+	0.910143i	$0.364028\pi$
$702$	0	0
$703$	22.1594	0.835756
$704$	3.58761	0.135213
$705$	0	0
$706$	14.4572	0.544106
$707$	9.60704	0.361310
$708$	0	0
$709$	6.91676	0.259764	0.129882	−	0.991529i	$-0.458540\pi$
$709$	6.91676	0.259764	0.129882	+	0.991529i	$0.458540\pi$
$710$	−1.20033	−0.0450474
$711$	0	0
$712$	20.8245	0.780431
$713$	76.4589	2.86341
$714$	0	0
$715$	1.88693	0.0705674
$716$	32.0159	1.19649
$717$	0	0
$718$	−1.28326	−0.0478909
$719$	−3.44984	−0.128657	−0.0643286	−	0.997929i	$-0.520491\pi$
$719$	−3.44984	−0.128657	−0.0643286	+	0.997929i	$0.520491\pi$
$720$	0	0
$721$	−60.9522	−2.26998
$722$	−4.35936	−0.162239
$723$	0	0
$724$	−28.6419	−1.06447
$725$	−13.9260	−0.517198
$726$	0	0
$727$	−48.9831	−1.81668	−0.908341	−	0.418230i	$-0.862651\pi$
$727$	−48.9831	−1.81668	−0.908341	+	0.418230i	$0.862651\pi$
$728$	6.43948	0.238663
$729$	0	0
$730$	−0.597008	−0.0220963
$731$	6.17716	0.228471
$732$	0	0
$733$	−19.5910	−0.723610	−0.361805	−	0.932254i	$-0.617839\pi$
$733$	−19.5910	−0.723610	−0.361805	+	0.932254i	$0.617839\pi$
$734$	3.79405	0.140041
$735$	0	0
$736$	−35.6365	−1.31358
$737$	1.36828	0.0504014
$738$	0	0
$739$	−30.9491	−1.13848	−0.569240	−	0.822171i	$-0.692762\pi$
$739$	−30.9491	−1.13848	−0.569240	+	0.822171i	$0.692762\pi$
$740$	24.7336	0.909225
$741$	0	0
$742$	−17.5305	−0.643566
$743$	11.9366	0.437910	0.218955	−	0.975735i	$-0.429735\pi$
$743$	11.9366	0.437910	0.218955	+	0.975735i	$0.429735\pi$
$744$	0	0
$745$	33.1327	1.21389
$746$	−10.8227	−0.396246
$747$	0	0
$748$	6.20467	0.226865
$749$	−38.8989	−1.42133
$750$	0	0
$751$	−11.2120	−0.409132	−0.204566	−	0.978853i	$-0.565578\pi$
$751$	−11.2120	−0.409132	−0.204566	+	0.978853i	$0.565578\pi$
$752$	4.84926	0.176834
$753$	0	0
$754$	4.32885	0.157647
$755$	10.8281	0.394074
$756$	0	0
$757$	−11.7509	−0.427093	−0.213546	−	0.976933i	$-0.568501\pi$
$757$	−11.7509	−0.427093	−0.213546	+	0.976933i	$0.568501\pi$
$758$	5.52377	0.200633
$759$	0	0
$760$	9.76144	0.354085
$761$	−9.36279	−0.339401	−0.169700	−	0.985496i	$-0.554280\pi$
$761$	−9.36279	−0.339401	−0.169700	+	0.985496i	$0.554280\pi$
$762$	0	0
$763$	13.1171	0.474871
$764$	−18.8155	−0.680723
$765$	0	0
$766$	−5.54547	−0.200366
$767$	3.19031	0.115195
$768$	0	0
$769$	−10.9169	−0.393673	−0.196837	−	0.980436i	$-0.563067\pi$
$769$	−10.9169	−0.393673	−0.196837	+	0.980436i	$0.563067\pi$
$770$	−3.19778	−0.115240
$771$	0	0
$772$	13.3723	0.481280
$773$	19.0010	0.683417	0.341709	−	0.939806i	$-0.388994\pi$
$773$	19.0010	0.683417	0.341709	+	0.939806i	$0.388994\pi$
$774$	0	0
$775$	14.4252	0.518167
$776$	28.6584	1.02878
$777$	0	0
$778$	−1.18601	−0.0425205
$779$	31.4327	1.12619
$780$	0	0
$781$	−1.42164	−0.0508704
$782$	−11.7696	−0.420881
$783$	0	0
$784$	20.8490	0.744606
$785$	10.5554	0.376737
$786$	0	0
$787$	7.02933	0.250569	0.125284	−	0.992121i	$-0.460016\pi$
$787$	7.02933	0.250569	0.125284	+	0.992121i	$0.460016\pi$
$788$	−25.7190	−0.916203
$789$	0	0
$790$	9.09499	0.323585
$791$	−44.1175	−1.56864
$792$	0	0
$793$	−0.652085	−0.0231562
$794$	−6.32361	−0.224416
$795$	0	0
$796$	−23.9366	−0.848409
$797$	22.5993	0.800510	0.400255	−	0.916404i	$-0.368922\pi$
$797$	22.5993	0.800510	0.400255	+	0.916404i	$0.368922\pi$
$798$	0	0
$799$	5.88902	0.208338
$800$	−6.72339	−0.237708
$801$	0	0
$802$	13.3753	0.472297
$803$	−0.707086	−0.0249525
$804$	0	0
$805$	−54.5268	−1.92182
$806$	−4.48402	−0.157943
$807$	0	0
$808$	4.31280	0.151724
$809$	14.8490	0.522061	0.261031	−	0.965331i	$-0.415938\pi$
$809$	14.8490	0.522061	0.261031	+	0.965331i	$0.415938\pi$
$810$	0	0
$811$	−20.3577	−0.714854	−0.357427	−	0.933941i	$-0.616346\pi$
$811$	−20.3577	−0.714854	−0.357427	+	0.933941i	$0.616346\pi$
$812$	65.9449	2.31421
$813$	0	0
$814$	−3.25883	−0.114222
$815$	0.527379	0.0184733
$816$	0	0
$817$	5.45176	0.190733
$818$	9.97404	0.348734
$819$	0	0
$820$	35.0842	1.22519
$821$	29.3930	1.02582	0.512911	−	0.858442i	$-0.328567\pi$
$821$	29.3930	1.02582	0.512911	+	0.858442i	$0.328567\pi$
$822$	0	0
$823$	−15.4845	−0.539755	−0.269878	−	0.962895i	$-0.586983\pi$
$823$	−15.4845	−0.539755	−0.269878	+	0.962895i	$0.586983\pi$
$824$	−27.3627	−0.953225
$825$	0	0
$826$	−5.40661	−0.188120
$827$	−5.02331	−0.174678	−0.0873389	−	0.996179i	$-0.527836\pi$
$827$	−5.02331	−0.174678	−0.0873389	+	0.996179i	$0.527836\pi$
$828$	0	0
$829$	32.5511	1.13055	0.565273	−	0.824904i	$-0.308771\pi$
$829$	32.5511	1.13055	0.565273	+	0.824904i	$0.308771\pi$
$830$	−6.13319	−0.212886
$831$	0	0
$832$	−3.58761	−0.124378
$833$	25.3193	0.877262
$834$	0	0
$835$	20.5236	0.710248
$836$	5.47604	0.189393
$837$	0	0
$838$	−10.5482	−0.364382
$839$	−38.0955	−1.31520	−0.657602	−	0.753366i	$-0.728429\pi$
$839$	−38.0955	−1.31520	−0.657602	+	0.753366i	$0.728429\pi$
$840$	0	0
$841$	64.5926	2.22733
$842$	13.4006	0.461814
$843$	0	0
$844$	−43.6768	−1.50342
$845$	−1.88693	−0.0649125
$846$	0	0
$847$	−3.78739	−0.130136
$848$	29.3653	1.00841
$849$	0	0
$850$	−2.22052	−0.0761633
$851$	−55.5678	−1.90484
$852$	0	0
$853$	8.31376	0.284658	0.142329	−	0.989819i	$-0.454541\pi$
$853$	8.31376	0.284658	0.142329	+	0.989819i	$0.454541\pi$
$854$	1.10509	0.0378152
$855$	0	0
$856$	−17.4625	−0.596856
$857$	31.8800	1.08900	0.544499	−	0.838761i	$-0.316720\pi$
$857$	31.8800	1.08900	0.544499	+	0.838761i	$0.316720\pi$
$858$	0	0
$859$	46.1496	1.57460	0.787302	−	0.616567i	$-0.211477\pi$
$859$	46.1496	1.57460	0.787302	+	0.616567i	$0.211477\pi$
$860$	6.08509	0.207500
$861$	0	0
$862$	−4.40270	−0.149957
$863$	−29.0509	−0.988904	−0.494452	−	0.869205i	$-0.664631\pi$
$863$	−29.0509	−0.988904	−0.494452	+	0.869205i	$0.664631\pi$
$864$	0	0
$865$	−31.5294	−1.07203
$866$	−12.1600	−0.413214
$867$	0	0
$868$	−68.3087	−2.31855
$869$	10.7719	0.365413
$870$	0	0
$871$	−1.36828	−0.0463625
$872$	5.88855	0.199411
$873$	0	0
$874$	−10.3875	−0.351362
$875$	−46.0201	−1.55576
$876$	0	0
$877$	32.6807	1.10355	0.551775	−	0.833993i	$-0.313951\pi$
$877$	32.6807	1.10355	0.551775	+	0.833993i	$0.313951\pi$
$878$	5.67064	0.191375
$879$	0	0
$880$	5.35659	0.180571
$881$	−34.9559	−1.17769	−0.588847	−	0.808244i	$-0.700418\pi$
$881$	−34.9559	−1.17769	−0.588847	+	0.808244i	$0.700418\pi$
$882$	0	0
$883$	49.0783	1.65162	0.825809	−	0.563950i	$-0.190719\pi$
$883$	49.0783	1.65162	0.825809	+	0.563950i	$0.190719\pi$
$884$	−6.20467	−0.208686
$885$	0	0
$886$	1.26759	0.0425856
$887$	−3.80673	−0.127817	−0.0639087	−	0.997956i	$-0.520357\pi$
$887$	−3.80673	−0.127817	−0.0639087	+	0.997956i	$0.520357\pi$
$888$	0	0
$889$	36.5489	1.22581
$890$	10.3412	0.346639
$891$	0	0
$892$	−46.6145	−1.56077
$893$	5.19745	0.173926
$894$	0	0
$895$	33.5662	1.12199
$896$	41.4596	1.38507
$897$	0	0
$898$	−9.58223	−0.319763
$899$	−96.9475	−3.23338
$900$	0	0
$901$	35.6617	1.18806
$902$	−4.62260	−0.153916
$903$	0	0
$904$	−19.8053	−0.658714
$905$	−30.0289	−0.998193
$906$	0	0
$907$	−9.64071	−0.320115	−0.160057	−	0.987108i	$-0.551168\pi$
$907$	−9.64071	−0.320115	−0.160057	+	0.987108i	$0.551168\pi$
$908$	−22.1935	−0.736518
$909$	0	0
$910$	3.19778	0.106005
$911$	−44.6979	−1.48091	−0.740454	−	0.672107i	$-0.765389\pi$
$911$	−44.6979	−1.48091	−0.740454	+	0.672107i	$0.765389\pi$
$912$	0	0
$913$	−7.26404	−0.240405
$914$	−9.23630	−0.305510
$915$	0	0
$916$	27.7069	0.915462
$917$	−26.7135	−0.882156
$918$	0	0
$919$	−21.7617	−0.717854	−0.358927	−	0.933366i	$-0.616857\pi$
$919$	−21.7617	−0.717854	−0.358927	+	0.933366i	$0.616857\pi$
$920$	−24.4782	−0.807022
$921$	0	0
$922$	−4.42444	−0.145711
$923$	1.42164	0.0467940
$924$	0	0
$925$	−10.4837	−0.344703
$926$	4.14218	0.136121
$927$	0	0
$928$	45.1860	1.48330
$929$	−40.3096	−1.32252	−0.661258	−	0.750158i	$-0.729977\pi$
$929$	−40.3096	−1.32252	−0.661258	+	0.750158i	$0.729977\pi$
$930$	0	0
$931$	22.3460	0.732360
$932$	−51.5917	−1.68994
$933$	0	0
$934$	5.32594	0.174270
$935$	6.50513	0.212740
$936$	0	0
$937$	−9.28092	−0.303194	−0.151597	−	0.988442i	$-0.548442\pi$
$937$	−9.28092	−0.303194	−0.151597	+	0.988442i	$0.548442\pi$
$938$	2.31883	0.0757124
$939$	0	0
$940$	5.80124	0.189215
$941$	−17.9552	−0.585322	−0.292661	−	0.956216i	$-0.594541\pi$
$941$	−17.9552	−0.585322	−0.292661	+	0.956216i	$0.594541\pi$
$942$	0	0
$943$	−78.8220	−2.56680
$944$	9.05658	0.294767
$945$	0	0
$946$	−0.801755	−0.0260673
$947$	28.1552	0.914920	0.457460	−	0.889230i	$-0.348759\pi$
$947$	28.1552	0.914920	0.457460	+	0.889230i	$0.348759\pi$
$948$	0	0
$949$	0.707086	0.0229530
$950$	−1.95976	−0.0635830
$951$	0	0
$952$	22.1998	0.719500
$953$	9.29926	0.301233	0.150616	−	0.988592i	$-0.451874\pi$
$953$	9.29926	0.301233	0.150616	+	0.988592i	$0.451874\pi$
$954$	0	0
$955$	−19.7267	−0.638340
$956$	51.5129	1.66605
$957$	0	0
$958$	9.72796	0.314296
$959$	−40.4322	−1.30562
$960$	0	0
$961$	69.4226	2.23944
$962$	3.25883	0.105069
$963$	0	0
$964$	−25.7967	−0.830857
$965$	14.0199	0.451315
$966$	0	0
$967$	11.3396	0.364657	0.182328	−	0.983238i	$-0.441637\pi$
$967$	11.3396	0.364657	0.182328	+	0.983238i	$0.441637\pi$
$968$	−1.70024	−0.0546478
$969$	0	0
$970$	14.2315	0.456946
$971$	7.23649	0.232230	0.116115	−	0.993236i	$-0.462956\pi$
$971$	7.23649	0.232230	0.116115	+	0.993236i	$0.462956\pi$
$972$	0	0
$973$	−29.6677	−0.951103
$974$	4.60009	0.147396
$975$	0	0
$976$	−1.85112	−0.0592530
$977$	−57.0767	−1.82604	−0.913022	−	0.407910i	$-0.866258\pi$
$977$	−57.0767	−1.82604	−0.913022	+	0.407910i	$0.866258\pi$
$978$	0	0
$979$	12.2480	0.391447
$980$	24.9419	0.796740
$981$	0	0
$982$	7.98437	0.254792
$983$	−0.969249	−0.0309142	−0.0154571	−	0.999881i	$-0.504920\pi$
$983$	−0.969249	−0.0309142	−0.0154571	+	0.999881i	$0.504920\pi$
$984$	0	0
$985$	−26.9645	−0.859159
$986$	14.9235	0.475262
$987$	0	0
$988$	−5.47604	−0.174216
$989$	−13.6711	−0.434715
$990$	0	0
$991$	41.9012	1.33104	0.665518	−	0.746382i	$-0.268211\pi$
$991$	41.9012	1.33104	0.665518	+	0.746382i	$0.268211\pi$
$992$	−46.8057	−1.48608
$993$	0	0
$994$	−2.40925	−0.0764169
$995$	−25.0957	−0.795586
$996$	0	0
$997$	−36.9255	−1.16944	−0.584721	−	0.811234i	$-0.698796\pi$
$997$	−36.9255	−1.16944	−0.584721	+	0.811234i	$0.698796\pi$
$998$	3.77005	0.119339
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.p.1.3	yes	5
3.2	odd	2		1287.2.a.o.1.3	✓	5

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1287.2.a.o.1.3	✓	5	3.2	odd	2
1287.2.a.p.1.3	yes	5	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 $\F_{q}$

Level:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q+0.447457 q^{2} -1.79978 q^{4} -1.88693 q^{5} -3.78739 q^{7} -1.70024 q^{8} +O(q^{10})\)	\(q+0.447457 q^{2} -1.79978 q^{4} -1.88693 q^{5} -3.78739 q^{7} -1.70024 q^{8} -0.844323 q^{10} -1.00000 q^{11} +1.00000 q^{13} -1.69470 q^{14} +2.83878 q^{16} +3.44746 q^{17} +3.04261 q^{19} +3.39607 q^{20} -0.447457 q^{22} -7.62979 q^{23} -1.43948 q^{25} +0.447457 q^{26} +6.81648 q^{28} +9.67433 q^{29} -10.0211 q^{31} +4.67071 q^{32} +1.54259 q^{34} +7.14656 q^{35} +7.28301 q^{37} +1.36144 q^{38} +3.20824 q^{40} +10.3308 q^{41} +1.79180 q^{43} +1.79978 q^{44} -3.41400 q^{46} +1.70822 q^{47} +7.34434 q^{49} -0.644105 q^{50} -1.79978 q^{52} +10.3443 q^{53} +1.88693 q^{55} +6.43948 q^{56} +4.32885 q^{58} +3.19031 q^{59} -0.652085 q^{61} -4.48402 q^{62} -3.58761 q^{64} -1.88693 q^{65} -1.36828 q^{67} -6.20467 q^{68} +3.19778 q^{70} +1.42164 q^{71} +0.707086 q^{73} +3.25883 q^{74} -5.47604 q^{76} +3.78739 q^{77} -10.7719 q^{79} -5.35659 q^{80} +4.62260 q^{82} +7.26404 q^{83} -6.50513 q^{85} +0.801755 q^{86} +1.70024 q^{88} -12.2480 q^{89} -3.78739 q^{91} +13.7320 q^{92} +0.764356 q^{94} -5.74121 q^{95} -16.8555 q^{97} +3.28628 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 5 q + 3 q^{2} + 7 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) \) 5 * q + 3 * q^2 + 7 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8	\( 5 q + 3 q^{2} + 7 q^{4} + 4 q^{5} - 4 q^{7} + 9 q^{8} - 6 q^{10} - 5 q^{11} + 5 q^{13} + 6 q^{14} + 19 q^{16} + 18 q^{17} - 2 q^{20} - 3 q^{22} + 8 q^{23} + 7 q^{25} + 3 q^{26} - 8 q^{28} + 20 q^{29} - 8 q^{31} + 31 q^{32} + 26 q^{34} + 6 q^{35} + 4 q^{37} + 18 q^{38} - 20 q^{40} + 8 q^{41} - 22 q^{43} - 7 q^{44} + 6 q^{46} + 6 q^{47} + 5 q^{49} + 11 q^{50} + 7 q^{52} + 20 q^{53} - 4 q^{55} + 18 q^{56} + 12 q^{58} - 16 q^{59} - 4 q^{61} - 26 q^{62} + 11 q^{64} + 4 q^{65} - 20 q^{67} + 36 q^{68} + 14 q^{70} + 6 q^{71} - 12 q^{73} - 20 q^{74} - 16 q^{76} + 4 q^{77} + 6 q^{79} - 52 q^{80} - 18 q^{82} + 4 q^{83} + 6 q^{85} - 46 q^{86} - 9 q^{88} + 10 q^{89} - 4 q^{91} + 68 q^{92} - 2 q^{94} + 2 q^{95} + 4 q^{97} - 45 q^{98}+O(q^{100}) \) 5 * q + 3 * q^2 + 7 * q^4 + 4 * q^5 - 4 * q^7 + 9 * q^8 - 6 * q^10 - 5 * q^11 + 5 * q^13 + 6 * q^14 + 19 * q^16 + 18 * q^17 - 2 * q^20 - 3 * q^22 + 8 * q^23 + 7 * q^25 + 3 * q^26 - 8 * q^28 + 20 * q^29 - 8 * q^31 + 31 * q^32 + 26 * q^34 + 6 * q^35 + 4 * q^37 + 18 * q^38 - 20 * q^40 + 8 * q^41 - 22 * q^43 - 7 * q^44 + 6 * q^46 + 6 * q^47 + 5 * q^49 + 11 * q^50 + 7 * q^52 + 20 * q^53 - 4 * q^55 + 18 * q^56 + 12 * q^58 - 16 * q^59 - 4 * q^61 - 26 * q^62 + 11 * q^64 + 4 * q^65 - 20 * q^67 + 36 * q^68 + 14 * q^70 + 6 * q^71 - 12 * q^73 - 20 * q^74 - 16 * q^76 + 4 * q^77 + 6 * q^79 - 52 * q^80 - 18 * q^82 + 4 * q^83 + 6 * q^85 - 46 * q^86 - 9 * q^88 + 10 * q^89 - 4 * q^91 + 68 * q^92 - 2 * q^94 + 2 * q^95 + 4 * q^97 - 45 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

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Groups

Database

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