LMFDB - Embedded newform 8046.2.a.f.1.4

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8046,2,Mod(1,8046)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8046, 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("8046.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8046 = 2 \cdot 3^{3} \cdot 149 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8046.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:	$64.2476334663$
Analytic rank:	$1$
Dimension:	$8$
Coefficient field:	$\mathbb{Q}[x]/(x^{8} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{8} - 17x^{6} - 2x^{5} + 71x^{4} - 18x^{3} - 81x^{2} + 36x + 9 $ x^8 - 17x^6 - 2x^5 + 71x^4 - 18x^3 - 81x^2 + 36x + 9
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.4
Root			$0.854224$ of defining polynomial
Character	$\chi$	$=$	8046.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -0.854224 q^{5} +0.727081 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -0.854224 q^{5} +0.727081 q^{7} +1.00000 q^{8} -0.854224 q^{10} +4.77784 q^{11} +0.670363 q^{13} +0.727081 q^{14} +1.00000 q^{16} -0.231641 q^{17} -5.18376 q^{19} -0.854224 q^{20} +4.77784 q^{22} -7.77399 q^{23} -4.27030 q^{25} +0.670363 q^{26} +0.727081 q^{28} -7.62010 q^{29} -3.13797 q^{31} +1.00000 q^{32} -0.231641 q^{34} -0.621090 q^{35} -2.35811 q^{37} -5.18376 q^{38} -0.854224 q^{40} +0.350412 q^{41} -2.96304 q^{43} +4.77784 q^{44} -7.77399 q^{46} -12.9412 q^{47} -6.47135 q^{49} -4.27030 q^{50} +0.670363 q^{52} +7.38511 q^{53} -4.08134 q^{55} +0.727081 q^{56} -7.62010 q^{58} +1.60351 q^{59} +11.0488 q^{61} -3.13797 q^{62} +1.00000 q^{64} -0.572640 q^{65} +9.83357 q^{67} -0.231641 q^{68} -0.621090 q^{70} -10.3649 q^{71} -4.70538 q^{73} -2.35811 q^{74} -5.18376 q^{76} +3.47387 q^{77} -7.56476 q^{79} -0.854224 q^{80} +0.350412 q^{82} +0.603787 q^{83} +0.197873 q^{85} -2.96304 q^{86} +4.77784 q^{88} -4.00132 q^{89} +0.487408 q^{91} -7.77399 q^{92} -12.9412 q^{94} +4.42810 q^{95} -4.56015 q^{97} -6.47135 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8}+O(q^{10}) $ 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8	$ 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8} - 6 q^{11} - 8 q^{13} - 5 q^{14} + 8 q^{16} + 5 q^{17} - 14 q^{19} - 6 q^{22} - 21 q^{23} - 6 q^{25} - 8 q^{26} - 5 q^{28} - 3 q^{29} - 4 q^{31} + 8 q^{32} + 5 q^{34} + 2 q^{35} - 3 q^{37} - 14 q^{38} - 7 q^{41} - 12 q^{43} - 6 q^{44} - 21 q^{46} - 25 q^{47} - 7 q^{49} - 6 q^{50} - 8 q^{52} - 3 q^{53} - 9 q^{55} - 5 q^{56} - 3 q^{58} - 2 q^{59} - 17 q^{61} - 4 q^{62} + 8 q^{64} - 32 q^{65} - 14 q^{67} + 5 q^{68} + 2 q^{70} - 7 q^{71} - 10 q^{73} - 3 q^{74} - 14 q^{76} - 12 q^{77} - 33 q^{79} - 7 q^{82} - 13 q^{83} - 33 q^{85} - 12 q^{86} - 6 q^{88} + 22 q^{89} - 22 q^{91} - 21 q^{92} - 25 q^{94} + 14 q^{95} - 11 q^{97} - 7 q^{98}+O(q^{100}) $ 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8 - 6 * q^11 - 8 * q^13 - 5 * q^14 + 8 * q^16 + 5 * q^17 - 14 * q^19 - 6 * q^22 - 21 * q^23 - 6 * q^25 - 8 * q^26 - 5 * q^28 - 3 * q^29 - 4 * q^31 + 8 * q^32 + 5 * q^34 + 2 * q^35 - 3 * q^37 - 14 * q^38 - 7 * q^41 - 12 * q^43 - 6 * q^44 - 21 * q^46 - 25 * q^47 - 7 * q^49 - 6 * q^50 - 8 * q^52 - 3 * q^53 - 9 * q^55 - 5 * q^56 - 3 * q^58 - 2 * q^59 - 17 * q^61 - 4 * q^62 + 8 * q^64 - 32 * q^65 - 14 * q^67 + 5 * q^68 + 2 * q^70 - 7 * q^71 - 10 * q^73 - 3 * q^74 - 14 * q^76 - 12 * q^77 - 33 * q^79 - 7 * q^82 - 13 * q^83 - 33 * q^85 - 12 * q^86 - 6 * q^88 + 22 * q^89 - 22 * q^91 - 21 * q^92 - 25 * q^94 + 14 * q^95 - 11 * q^97 - 7 * 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 $

$2$	1.00000	0.707107
$2$	1.00000	0.707107
$3$	0	0
$3$	0	0
$4$	1.00000	0.500000
$5$	−0.854224	−0.382021	−0.191010	−	0.981588i	$-0.561176\pi$
$5$	−0.854224	−0.382021	−0.191010	+	0.981588i	$0.561176\pi$
$6$	0	0
$7$	0.727081	0.274811	0.137405	−	0.990515i	$-0.456124\pi$
$7$	0.727081	0.274811	0.137405	+	0.990515i	$0.456124\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−0.854224	−0.270129
$11$	4.77784	1.44057	0.720286	−	0.693677i	$-0.244011\pi$
$11$	4.77784	1.44057	0.720286	+	0.693677i	$0.244011\pi$
$12$	0	0
$13$	0.670363	0.185925	0.0929626	−	0.995670i	$-0.470366\pi$
$13$	0.670363	0.185925	0.0929626	+	0.995670i	$0.470366\pi$
$14$	0.727081	0.194321
$15$	0	0
$16$	1.00000	0.250000
$17$	−0.231641	−0.0561811	−0.0280906	−	0.999605i	$-0.508943\pi$
$17$	−0.231641	−0.0561811	−0.0280906	+	0.999605i	$0.508943\pi$
$18$	0	0
$19$	−5.18376	−1.18924	−0.594618	−	0.804008i	$-0.702697\pi$
$19$	−5.18376	−1.18924	−0.594618	+	0.804008i	$0.702697\pi$
$20$	−0.854224	−0.191010
$21$	0	0
$22$	4.77784	1.01864
$23$	−7.77399	−1.62099	−0.810494	−	0.585747i	$-0.800801\pi$
$23$	−7.77399	−1.62099	−0.810494	+	0.585747i	$0.800801\pi$
$24$	0	0
$25$	−4.27030	−0.854060
$26$	0.670363	0.131469
$27$	0	0
$28$	0.727081	0.137405
$29$	−7.62010	−1.41502	−0.707509	−	0.706704i	$-0.750181\pi$
$29$	−7.62010	−1.41502	−0.707509	+	0.706704i	$0.750181\pi$
$30$	0	0
$31$	−3.13797	−0.563595	−0.281798	−	0.959474i	$-0.590931\pi$
$31$	−3.13797	−0.563595	−0.281798	+	0.959474i	$0.590931\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−0.231641	−0.0397261
$35$	−0.621090	−0.104983
$36$	0	0
$37$	−2.35811	−0.387671	−0.193836	−	0.981034i	$-0.562093\pi$
$37$	−2.35811	−0.387671	−0.193836	+	0.981034i	$0.562093\pi$
$38$	−5.18376	−0.840917
$39$	0	0
$40$	−0.854224	−0.135065
$41$	0.350412	0.0547252	0.0273626	−	0.999626i	$-0.491289\pi$
$41$	0.350412	0.0547252	0.0273626	+	0.999626i	$0.491289\pi$
$42$	0	0
$43$	−2.96304	−0.451860	−0.225930	−	0.974144i	$-0.572542\pi$
$43$	−2.96304	−0.451860	−0.225930	+	0.974144i	$0.572542\pi$
$44$	4.77784	0.720286
$45$	0	0
$46$	−7.77399	−1.14621
$47$	−12.9412	−1.88767	−0.943834	−	0.330421i	$-0.892809\pi$
$47$	−12.9412	−1.88767	−0.943834	+	0.330421i	$0.892809\pi$
$48$	0	0
$49$	−6.47135	−0.924479
$50$	−4.27030	−0.603912
$51$	0	0
$52$	0.670363	0.0929626
$53$	7.38511	1.01442	0.507212	−	0.861822i	$-0.330676\pi$
$53$	7.38511	1.01442	0.507212	+	0.861822i	$0.330676\pi$
$54$	0	0
$55$	−4.08134	−0.550328
$56$	0.727081	0.0971603
$57$	0	0
$58$	−7.62010	−1.00057
$59$	1.60351	0.208759	0.104379	−	0.994538i	$-0.466714\pi$
$59$	1.60351	0.208759	0.104379	+	0.994538i	$0.466714\pi$
$60$	0	0
$61$	11.0488	1.41465	0.707326	−	0.706887i	$-0.249901\pi$
$61$	11.0488	1.41465	0.707326	+	0.706887i	$0.249901\pi$
$62$	−3.13797	−0.398522
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.572640	−0.0710273
$66$	0	0
$67$	9.83357	1.20136	0.600681	−	0.799489i	$-0.294896\pi$
$67$	9.83357	1.20136	0.600681	+	0.799489i	$0.294896\pi$
$68$	−0.231641	−0.0280906
$69$	0	0
$70$	−0.621090	−0.0742345
$71$	−10.3649	−1.23008	−0.615042	−	0.788495i	$-0.710861\pi$
$71$	−10.3649	−1.23008	−0.615042	+	0.788495i	$0.710861\pi$
$72$	0	0
$73$	−4.70538	−0.550723	−0.275361	−	0.961341i	$-0.588797\pi$
$73$	−4.70538	−0.550723	−0.275361	+	0.961341i	$0.588797\pi$
$74$	−2.35811	−0.274125
$75$	0	0
$76$	−5.18376	−0.594618
$77$	3.47387	0.395885
$78$	0	0
$79$	−7.56476	−0.851102	−0.425551	−	0.904934i	$-0.639920\pi$
$79$	−7.56476	−0.851102	−0.425551	+	0.904934i	$0.639920\pi$
$80$	−0.854224	−0.0955052
$81$	0	0
$82$	0.350412	0.0386966
$83$	0.603787	0.0662742	0.0331371	−	0.999451i	$-0.489450\pi$
$83$	0.603787	0.0662742	0.0331371	+	0.999451i	$0.489450\pi$
$84$	0	0
$85$	0.197873	0.0214624
$86$	−2.96304	−0.319513
$87$	0	0
$88$	4.77784	0.509319
$89$	−4.00132	−0.424139	−0.212070	−	0.977255i	$-0.568020\pi$
$89$	−4.00132	−0.424139	−0.212070	+	0.977255i	$0.568020\pi$
$90$	0	0
$91$	0.487408	0.0510943
$92$	−7.77399	−0.810494
$93$	0	0
$94$	−12.9412	−1.33478
$95$	4.42810	0.454313
$96$	0	0
$97$	−4.56015	−0.463013	−0.231507	−	0.972833i	$-0.574366\pi$
$97$	−4.56015	−0.463013	−0.231507	+	0.972833i	$0.574366\pi$
$98$	−6.47135	−0.653705
$99$	0	0
$100$	−4.27030	−0.427030
$101$	−1.17615	−0.117031	−0.0585156	−	0.998286i	$-0.518637\pi$
$101$	−1.17615	−0.117031	−0.0585156	+	0.998286i	$0.518637\pi$
$102$	0	0
$103$	−10.6760	−1.05194	−0.525970	−	0.850503i	$-0.676298\pi$
$103$	−10.6760	−1.05194	−0.525970	+	0.850503i	$0.676298\pi$
$104$	0.670363	0.0657345
$105$	0	0
$106$	7.38511	0.717305
$107$	4.79486	0.463536	0.231768	−	0.972771i	$-0.425549\pi$
$107$	4.79486	0.463536	0.231768	+	0.972771i	$0.425549\pi$
$108$	0	0
$109$	1.76535	0.169090	0.0845448	−	0.996420i	$-0.473056\pi$
$109$	1.76535	0.169090	0.0845448	+	0.996420i	$0.473056\pi$
$110$	−4.08134	−0.389141
$111$	0	0
$112$	0.727081	0.0687027
$113$	14.2839	1.34371	0.671857	−	0.740681i	$-0.265497\pi$
$113$	14.2839	1.34371	0.671857	+	0.740681i	$0.265497\pi$
$114$	0	0
$115$	6.64073	0.619251
$116$	−7.62010	−0.707509
$117$	0	0
$118$	1.60351	0.147615
$119$	−0.168422	−0.0154392
$120$	0	0
$121$	11.8277	1.07525
$122$	11.0488	1.00031
$123$	0	0
$124$	−3.13797	−0.281798
$125$	7.91892	0.708289
$126$	0	0
$127$	−12.1662	−1.07958	−0.539789	−	0.841800i	$-0.681496\pi$
$127$	−12.1662	−1.07958	−0.539789	+	0.841800i	$0.681496\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.572640	−0.0502239
$131$	18.7356	1.63694	0.818470	−	0.574549i	$-0.194822\pi$
$131$	18.7356	1.63694	0.818470	+	0.574549i	$0.194822\pi$
$132$	0	0
$133$	−3.76902	−0.326815
$134$	9.83357	0.849491
$135$	0	0
$136$	−0.231641	−0.0198630
$137$	−6.33265	−0.541035	−0.270517	−	0.962715i	$-0.587195\pi$
$137$	−6.33265	−0.541035	−0.270517	+	0.962715i	$0.587195\pi$
$138$	0	0
$139$	17.6958	1.50094	0.750470	−	0.660904i	$-0.229827\pi$
$139$	17.6958	1.50094	0.750470	+	0.660904i	$0.229827\pi$
$140$	−0.621090	−0.0524917
$141$	0	0
$142$	−10.3649	−0.869800
$143$	3.20288	0.267839
$144$	0	0
$145$	6.50928	0.540566
$146$	−4.70538	−0.389420
$147$	0	0
$148$	−2.35811	−0.193836
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−9.78677	−0.796436	−0.398218	−	0.917291i	$-0.630371\pi$
$151$	−9.78677	−0.796436	−0.398218	+	0.917291i	$0.630371\pi$
$152$	−5.18376	−0.420459
$153$	0	0
$154$	3.47387	0.279933
$155$	2.68053	0.215305
$156$	0	0
$157$	−21.3607	−1.70477	−0.852384	−	0.522916i	$-0.824844\pi$
$157$	−21.3607	−1.70477	−0.852384	+	0.522916i	$0.824844\pi$
$158$	−7.56476	−0.601820
$159$	0	0
$160$	−0.854224	−0.0675324
$161$	−5.65232	−0.445465
$162$	0	0
$163$	5.23469	0.410013	0.205006	−	0.978761i	$-0.434278\pi$
$163$	5.23469	0.410013	0.205006	+	0.978761i	$0.434278\pi$
$164$	0.350412	0.0273626
$165$	0	0
$166$	0.603787	0.0468630
$167$	24.8724	1.92469	0.962343	−	0.271839i	$-0.0876318\pi$
$167$	24.8724	1.92469	0.962343	+	0.271839i	$0.0876318\pi$
$168$	0	0
$169$	−12.5506	−0.965432
$170$	0.197873	0.0151762
$171$	0	0
$172$	−2.96304	−0.225930
$173$	13.4677	1.02393	0.511965	−	0.859006i	$-0.328918\pi$
$173$	13.4677	1.02393	0.511965	+	0.859006i	$0.328918\pi$
$174$	0	0
$175$	−3.10485	−0.234705
$176$	4.77784	0.360143
$177$	0	0
$178$	−4.00132	−0.299912
$179$	−0.661451	−0.0494392	−0.0247196	−	0.999694i	$-0.507869\pi$
$179$	−0.661451	−0.0494392	−0.0247196	+	0.999694i	$0.507869\pi$
$180$	0	0
$181$	−16.6399	−1.23683	−0.618415	−	0.785851i	$-0.712225\pi$
$181$	−16.6399	−1.23683	−0.618415	+	0.785851i	$0.712225\pi$
$182$	0.487408	0.0361291
$183$	0	0
$184$	−7.77399	−0.573106
$185$	2.01436	0.148098
$186$	0	0
$187$	−1.10674	−0.0809329
$188$	−12.9412	−0.943834
$189$	0	0
$190$	4.42810	0.321248
$191$	−9.87624	−0.714620	−0.357310	−	0.933986i	$-0.616306\pi$
$191$	−9.87624	−0.714620	−0.357310	+	0.933986i	$0.616306\pi$
$192$	0	0
$193$	−13.6732	−0.984220	−0.492110	−	0.870533i	$-0.663774\pi$
$193$	−13.6732	−0.984220	−0.492110	+	0.870533i	$0.663774\pi$
$194$	−4.56015	−0.327400
$195$	0	0
$196$	−6.47135	−0.462240
$197$	17.4921	1.24626	0.623131	−	0.782118i	$-0.285860\pi$
$197$	17.4921	1.24626	0.623131	+	0.782118i	$0.285860\pi$
$198$	0	0
$199$	−7.45845	−0.528715	−0.264358	−	0.964425i	$-0.585160\pi$
$199$	−7.45845	−0.528715	−0.264358	+	0.964425i	$0.585160\pi$
$200$	−4.27030	−0.301956
$201$	0	0
$202$	−1.17615	−0.0827535
$203$	−5.54043	−0.388862
$204$	0	0
$205$	−0.299331	−0.0209062
$206$	−10.6760	−0.743834
$207$	0	0
$208$	0.670363	0.0464813
$209$	−24.7672	−1.71318
$210$	0	0
$211$	16.2465	1.11845	0.559227	−	0.829014i	$-0.311098\pi$
$211$	16.2465	1.11845	0.559227	+	0.829014i	$0.311098\pi$
$212$	7.38511	0.507212
$213$	0	0
$214$	4.79486	0.327770
$215$	2.53110	0.172620
$216$	0	0
$217$	−2.28155	−0.154882
$218$	1.76535	0.119564
$219$	0	0
$220$	−4.08134	−0.275164
$221$	−0.155283	−0.0104455
$222$	0	0
$223$	−6.84038	−0.458065	−0.229033	−	0.973419i	$-0.573556\pi$
$223$	−6.84038	−0.458065	−0.229033	+	0.973419i	$0.573556\pi$
$224$	0.727081	0.0485801
$225$	0	0
$226$	14.2839	0.950149
$227$	5.73148	0.380412	0.190206	−	0.981744i	$-0.439084\pi$
$227$	5.73148	0.380412	0.190206	+	0.981744i	$0.439084\pi$
$228$	0	0
$229$	−2.17916	−0.144003	−0.0720016	−	0.997405i	$-0.522939\pi$
$229$	−2.17916	−0.144003	−0.0720016	+	0.997405i	$0.522939\pi$
$230$	6.64073	0.437877
$231$	0	0
$232$	−7.62010	−0.500284
$233$	−0.993187	−0.0650659	−0.0325329	−	0.999471i	$-0.510357\pi$
$233$	−0.993187	−0.0650659	−0.0325329	+	0.999471i	$0.510357\pi$
$234$	0	0
$235$	11.0547	0.721128
$236$	1.60351	0.104379
$237$	0	0
$238$	−0.168422	−0.0109171
$239$	−12.0854	−0.781739	−0.390869	−	0.920446i	$-0.627826\pi$
$239$	−12.0854	−0.781739	−0.390869	+	0.920446i	$0.627826\pi$
$240$	0	0
$241$	3.47303	0.223718	0.111859	−	0.993724i	$-0.464320\pi$
$241$	3.47303	0.223718	0.111859	+	0.993724i	$0.464320\pi$
$242$	11.8277	0.760314
$243$	0	0
$244$	11.0488	0.707326
$245$	5.52799	0.353170
$246$	0	0
$247$	−3.47500	−0.221109
$248$	−3.13797	−0.199261
$249$	0	0
$250$	7.91892	0.500836
$251$	−26.4004	−1.66638	−0.833189	−	0.552988i	$-0.813488\pi$
$251$	−26.4004	−1.66638	−0.833189	+	0.552988i	$0.813488\pi$
$252$	0	0
$253$	−37.1428	−2.33515
$254$	−12.1662	−0.763377
$255$	0	0
$256$	1.00000	0.0625000
$257$	6.12809	0.382260	0.191130	−	0.981565i	$-0.438785\pi$
$257$	6.12809	0.382260	0.191130	+	0.981565i	$0.438785\pi$
$258$	0	0
$259$	−1.71454	−0.106536
$260$	−0.572640	−0.0355137
$261$	0	0
$262$	18.7356	1.15749
$263$	−5.97299	−0.368310	−0.184155	−	0.982897i	$-0.558955\pi$
$263$	−5.97299	−0.368310	−0.184155	+	0.982897i	$0.558955\pi$
$264$	0	0
$265$	−6.30854	−0.387531
$266$	−3.76902	−0.231093
$267$	0	0
$268$	9.83357	0.600681
$269$	24.5254	1.49534	0.747669	−	0.664072i	$-0.231173\pi$
$269$	24.5254	1.49534	0.747669	+	0.664072i	$0.231173\pi$
$270$	0	0
$271$	28.1399	1.70938	0.854688	−	0.519142i	$-0.173748\pi$
$271$	28.1399	1.70938	0.854688	+	0.519142i	$0.173748\pi$
$272$	−0.231641	−0.0140453
$273$	0	0
$274$	−6.33265	−0.382569
$275$	−20.4028	−1.23033
$276$	0	0
$277$	−8.98202	−0.539677	−0.269839	−	0.962906i	$-0.586970\pi$
$277$	−8.98202	−0.539677	−0.269839	+	0.962906i	$0.586970\pi$
$278$	17.6958	1.06133
$279$	0	0
$280$	−0.621090	−0.0371172
$281$	11.1755	0.666674	0.333337	−	0.942808i	$-0.391825\pi$
$281$	11.1755	0.666674	0.333337	+	0.942808i	$0.391825\pi$
$282$	0	0
$283$	14.9249	0.887192	0.443596	−	0.896227i	$-0.353702\pi$
$283$	14.9249	0.887192	0.443596	+	0.896227i	$0.353702\pi$
$284$	−10.3649	−0.615042
$285$	0	0
$286$	3.20288	0.189391
$287$	0.254778	0.0150391
$288$	0	0
$289$	−16.9463	−0.996844
$290$	6.50928	0.382238
$291$	0	0
$292$	−4.70538	−0.275361
$293$	−26.6015	−1.55408	−0.777039	−	0.629453i	$-0.783279\pi$
$293$	−26.6015	−1.55408	−0.777039	+	0.629453i	$0.783279\pi$
$294$	0	0
$295$	−1.36976	−0.0797503
$296$	−2.35811	−0.137062
$297$	0	0
$298$	1.00000	0.0579284
$299$	−5.21139	−0.301383
$300$	0	0
$301$	−2.15437	−0.124176
$302$	−9.78677	−0.563166
$303$	0	0
$304$	−5.18376	−0.297309
$305$	−9.43814	−0.540426
$306$	0	0
$307$	1.45072	0.0827969	0.0413985	−	0.999143i	$-0.486819\pi$
$307$	1.45072	0.0827969	0.0413985	+	0.999143i	$0.486819\pi$
$308$	3.47387	0.197942
$309$	0	0
$310$	2.68053	0.152244
$311$	3.25940	0.184823	0.0924117	−	0.995721i	$-0.470542\pi$
$311$	3.25940	0.184823	0.0924117	+	0.995721i	$0.470542\pi$
$312$	0	0
$313$	1.77542	0.100353	0.0501763	−	0.998740i	$-0.484022\pi$
$313$	1.77542	0.100353	0.0501763	+	0.998740i	$0.484022\pi$
$314$	−21.3607	−1.20545
$315$	0	0
$316$	−7.56476	−0.425551
$317$	−1.58023	−0.0887547	−0.0443773	−	0.999015i	$-0.514130\pi$
$317$	−1.58023	−0.0887547	−0.0443773	+	0.999015i	$0.514130\pi$
$318$	0	0
$319$	−36.4076	−2.03843
$320$	−0.854224	−0.0477526
$321$	0	0
$322$	−5.65232	−0.314991
$323$	1.20077	0.0668127
$324$	0	0
$325$	−2.86265	−0.158791
$326$	5.23469	0.289923
$327$	0	0
$328$	0.350412	0.0193483
$329$	−9.40930	−0.518751
$330$	0	0
$331$	−11.1065	−0.610469	−0.305235	−	0.952277i	$-0.598735\pi$
$331$	−11.1065	−0.610469	−0.305235	+	0.952277i	$0.598735\pi$
$332$	0.603787	0.0331371
$333$	0	0
$334$	24.8724	1.36096
$335$	−8.40007	−0.458945
$336$	0	0
$337$	−14.1798	−0.772423	−0.386212	−	0.922410i	$-0.626217\pi$
$337$	−14.1798	−0.772423	−0.386212	+	0.922410i	$0.626217\pi$
$338$	−12.5506	−0.682663
$339$	0	0
$340$	0.197873	0.0107312
$341$	−14.9927	−0.811899
$342$	0	0
$343$	−9.79476	−0.528868
$344$	−2.96304	−0.159757
$345$	0	0
$346$	13.4677	0.724027
$347$	−16.8746	−0.905877	−0.452938	−	0.891542i	$-0.649624\pi$
$347$	−16.8746	−0.905877	−0.452938	+	0.891542i	$0.649624\pi$
$348$	0	0
$349$	−7.49710	−0.401310	−0.200655	−	0.979662i	$-0.564307\pi$
$349$	−7.49710	−0.401310	−0.200655	+	0.979662i	$0.564307\pi$
$350$	−3.10485	−0.165961
$351$	0	0
$352$	4.77784	0.254659
$353$	2.81437	0.149794	0.0748968	−	0.997191i	$-0.476137\pi$
$353$	2.81437	0.149794	0.0748968	+	0.997191i	$0.476137\pi$
$354$	0	0
$355$	8.85392	0.469917
$356$	−4.00132	−0.212070
$357$	0	0
$358$	−0.661451	−0.0349588
$359$	2.13488	0.112674	0.0563372	−	0.998412i	$-0.482058\pi$
$359$	2.13488	0.112674	0.0563372	+	0.998412i	$0.482058\pi$
$360$	0	0
$361$	7.87140	0.414284
$362$	−16.6399	−0.874571
$363$	0	0
$364$	0.487408	0.0255471
$365$	4.01945	0.210387
$366$	0	0
$367$	−17.7363	−0.925828	−0.462914	−	0.886403i	$-0.653196\pi$
$367$	−17.7363	−0.925828	−0.462914	+	0.886403i	$0.653196\pi$
$368$	−7.77399	−0.405247
$369$	0	0
$370$	2.01436	0.104721
$371$	5.36957	0.278774
$372$	0	0
$373$	−33.3027	−1.72435	−0.862175	−	0.506610i	$-0.830898\pi$
$373$	−33.3027	−1.72435	−0.862175	+	0.506610i	$0.830898\pi$
$374$	−1.10674	−0.0572282
$375$	0	0
$376$	−12.9412	−0.667391
$377$	−5.10824	−0.263088
$378$	0	0
$379$	−10.1957	−0.523718	−0.261859	−	0.965106i	$-0.584336\pi$
$379$	−10.1957	−0.523718	−0.261859	+	0.965106i	$0.584336\pi$
$380$	4.42810	0.227157
$381$	0	0
$382$	−9.87624	−0.505312
$383$	25.9640	1.32670	0.663348	−	0.748311i	$-0.269135\pi$
$383$	25.9640	1.32670	0.663348	+	0.748311i	$0.269135\pi$
$384$	0	0
$385$	−2.96747	−0.151236
$386$	−13.6732	−0.695949
$387$	0	0
$388$	−4.56015	−0.231507
$389$	4.83469	0.245129	0.122564	−	0.992461i	$-0.460888\pi$
$389$	4.83469	0.245129	0.122564	+	0.992461i	$0.460888\pi$
$390$	0	0
$391$	1.80077	0.0910689
$392$	−6.47135	−0.326853
$393$	0	0
$394$	17.4921	0.881240
$395$	6.46200	0.325139
$396$	0	0
$397$	19.9337	1.00044	0.500221	−	0.865898i	$-0.333252\pi$
$397$	19.9337	1.00044	0.500221	+	0.865898i	$0.333252\pi$
$398$	−7.45845	−0.373858
$399$	0	0
$400$	−4.27030	−0.213515
$401$	4.33155	0.216307	0.108154	−	0.994134i	$-0.465506\pi$
$401$	4.33155	0.216307	0.108154	+	0.994134i	$0.465506\pi$
$402$	0	0
$403$	−2.10358	−0.104787
$404$	−1.17615	−0.0585156
$405$	0	0
$406$	−5.54043	−0.274967
$407$	−11.2667	−0.558468
$408$	0	0
$409$	−29.5272	−1.46002	−0.730012	−	0.683434i	$-0.760486\pi$
$409$	−29.5272	−1.46002	−0.730012	+	0.683434i	$0.760486\pi$
$410$	−0.299331	−0.0147829
$411$	0	0
$412$	−10.6760	−0.525970
$413$	1.16588	0.0573692
$414$	0	0
$415$	−0.515769	−0.0253181
$416$	0.670363	0.0328673
$417$	0	0
$418$	−24.7672	−1.21140
$419$	−22.5113	−1.09975	−0.549874	−	0.835248i	$-0.685324\pi$
$419$	−22.5113	−1.09975	−0.549874	+	0.835248i	$0.685324\pi$
$420$	0	0
$421$	16.0732	0.783360	0.391680	−	0.920101i	$-0.371894\pi$
$421$	16.0732	0.783360	0.391680	+	0.920101i	$0.371894\pi$
$422$	16.2465	0.790867
$423$	0	0
$424$	7.38511	0.358653
$425$	0.989175	0.0479821
$426$	0	0
$427$	8.03336	0.388762
$428$	4.79486	0.231768
$429$	0	0
$430$	2.53110	0.122061
$431$	35.7406	1.72157	0.860783	−	0.508973i	$-0.169975\pi$
$431$	35.7406	1.72157	0.860783	+	0.508973i	$0.169975\pi$
$432$	0	0
$433$	29.3282	1.40943	0.704713	−	0.709493i	$-0.251076\pi$
$433$	29.3282	1.40943	0.704713	+	0.709493i	$0.251076\pi$
$434$	−2.28155	−0.109518
$435$	0	0
$436$	1.76535	0.0845448
$437$	40.2985	1.92774
$438$	0	0
$439$	−8.48140	−0.404795	−0.202398	−	0.979303i	$-0.564873\pi$
$439$	−8.48140	−0.404795	−0.202398	+	0.979303i	$0.564873\pi$
$440$	−4.08134	−0.194570
$441$	0	0
$442$	−0.155283	−0.00738608
$443$	−29.2319	−1.38885	−0.694425	−	0.719565i	$-0.744341\pi$
$443$	−29.2319	−1.38885	−0.694425	+	0.719565i	$0.744341\pi$
$444$	0	0
$445$	3.41802	0.162030
$446$	−6.84038	−0.323901
$447$	0	0
$448$	0.727081	0.0343513
$449$	−23.7240	−1.11961	−0.559803	−	0.828626i	$-0.689123\pi$
$449$	−23.7240	−1.11961	−0.559803	+	0.828626i	$0.689123\pi$
$450$	0	0
$451$	1.67421	0.0788356
$452$	14.2839	0.671857
$453$	0	0
$454$	5.73148	0.268992
$455$	−0.416356	−0.0195191
$456$	0	0
$457$	19.6691	0.920083	0.460042	−	0.887897i	$-0.347835\pi$
$457$	19.6691	0.920083	0.460042	+	0.887897i	$0.347835\pi$
$458$	−2.17916	−0.101826
$459$	0	0
$460$	6.64073	0.309626
$461$	−2.86113	−0.133256	−0.0666280	−	0.997778i	$-0.521224\pi$
$461$	−2.86113	−0.133256	−0.0666280	+	0.997778i	$0.521224\pi$
$462$	0	0
$463$	−20.0230	−0.930548	−0.465274	−	0.885167i	$-0.654044\pi$
$463$	−20.0230	−0.930548	−0.465274	+	0.885167i	$0.654044\pi$
$464$	−7.62010	−0.353754
$465$	0	0
$466$	−0.993187	−0.0460085
$467$	−28.7449	−1.33015	−0.665077	−	0.746775i	$-0.731601\pi$
$467$	−28.7449	−1.33015	−0.665077	+	0.746775i	$0.731601\pi$
$468$	0	0
$469$	7.14980	0.330147
$470$	11.0547	0.509914
$471$	0	0
$472$	1.60351	0.0738074
$473$	−14.1569	−0.650936
$474$	0	0
$475$	22.1362	1.01568
$476$	−0.168422	−0.00771959
$477$	0	0
$478$	−12.0854	−0.552773
$479$	−19.5108	−0.891471	−0.445736	−	0.895165i	$-0.647058\pi$
$479$	−19.5108	−0.891471	−0.445736	+	0.895165i	$0.647058\pi$
$480$	0	0
$481$	−1.58079	−0.0720779
$482$	3.47303	0.158192
$483$	0	0
$484$	11.8277	0.537623
$485$	3.89539	0.176881
$486$	0	0
$487$	2.99490	0.135712	0.0678559	−	0.997695i	$-0.478384\pi$
$487$	2.99490	0.135712	0.0678559	+	0.997695i	$0.478384\pi$
$488$	11.0488	0.500155
$489$	0	0
$490$	5.52799	0.249729
$491$	43.3760	1.95753	0.978765	−	0.204985i	$-0.0657147\pi$
$491$	43.3760	1.95753	0.978765	+	0.204985i	$0.0657147\pi$
$492$	0	0
$493$	1.76513	0.0794973
$494$	−3.47500	−0.156348
$495$	0	0
$496$	−3.13797	−0.140899
$497$	−7.53610	−0.338040
$498$	0	0
$499$	−0.987622	−0.0442120	−0.0221060	−	0.999756i	$-0.507037\pi$
$499$	−0.987622	−0.0442120	−0.0221060	+	0.999756i	$0.507037\pi$
$500$	7.91892	0.354145
$501$	0	0
$502$	−26.4004	−1.17831
$503$	−2.91509	−0.129977	−0.0649887	−	0.997886i	$-0.520701\pi$
$503$	−2.91509	−0.129977	−0.0649887	+	0.997886i	$0.520701\pi$
$504$	0	0
$505$	1.00469	0.0447083
$506$	−37.1428	−1.65120
$507$	0	0
$508$	−12.1662	−0.539789
$509$	−4.42528	−0.196147	−0.0980735	−	0.995179i	$-0.531268\pi$
$509$	−4.42528	−0.196147	−0.0980735	+	0.995179i	$0.531268\pi$
$510$	0	0
$511$	−3.42119	−0.151345
$512$	1.00000	0.0441942
$513$	0	0
$514$	6.12809	0.270299
$515$	9.11972	0.401863
$516$	0	0
$517$	−61.8309	−2.71932
$518$	−1.71454	−0.0753325
$519$	0	0
$520$	−0.572640	−0.0251119
$521$	32.7467	1.43466	0.717329	−	0.696734i	$-0.245364\pi$
$521$	32.7467	1.43466	0.717329	+	0.696734i	$0.245364\pi$
$522$	0	0
$523$	30.4770	1.33267	0.666333	−	0.745655i	$-0.267863\pi$
$523$	30.4770	1.33267	0.666333	+	0.745655i	$0.267863\pi$
$524$	18.7356	0.818470
$525$	0	0
$526$	−5.97299	−0.260435
$527$	0.726880	0.0316634
$528$	0	0
$529$	37.4349	1.62760
$530$	−6.30854	−0.274026
$531$	0	0
$532$	−3.76902	−0.163408
$533$	0.234904	0.0101748
$534$	0	0
$535$	−4.09588	−0.177080
$536$	9.83357	0.424745
$537$	0	0
$538$	24.5254	1.05736
$539$	−30.9191	−1.33178
$540$	0	0
$541$	−26.2314	−1.12778	−0.563888	−	0.825851i	$-0.690695\pi$
$541$	−26.2314	−1.12778	−0.563888	+	0.825851i	$0.690695\pi$
$542$	28.1399	1.20871
$543$	0	0
$544$	−0.231641	−0.00993151
$545$	−1.50800	−0.0645957
$546$	0	0
$547$	0.202324	0.00865075	0.00432537	−	0.999991i	$-0.498623\pi$
$547$	0.202324	0.00865075	0.00432537	+	0.999991i	$0.498623\pi$
$548$	−6.33265	−0.270517
$549$	0	0
$550$	−20.4028	−0.869978
$551$	39.5008	1.68279
$552$	0	0
$553$	−5.50020	−0.233892
$554$	−8.98202	−0.381609
$555$	0	0
$556$	17.6958	0.750470
$557$	28.3170	1.19983	0.599915	−	0.800064i	$-0.295201\pi$
$557$	28.3170	1.19983	0.599915	+	0.800064i	$0.295201\pi$
$558$	0	0
$559$	−1.98631	−0.0840122
$560$	−0.621090	−0.0262459
$561$	0	0
$562$	11.1755	0.471410
$563$	39.4649	1.66325	0.831623	−	0.555341i	$-0.187412\pi$
$563$	39.4649	1.66325	0.831623	+	0.555341i	$0.187412\pi$
$564$	0	0
$565$	−12.2016	−0.513327
$566$	14.9249	0.627340
$567$	0	0
$568$	−10.3649	−0.434900
$569$	9.91014	0.415455	0.207727	−	0.978187i	$-0.433393\pi$
$569$	9.91014	0.415455	0.207727	+	0.978187i	$0.433393\pi$
$570$	0	0
$571$	2.50941	0.105016	0.0525078	−	0.998621i	$-0.483279\pi$
$571$	2.50941	0.105016	0.0525078	+	0.998621i	$0.483279\pi$
$572$	3.20288	0.133919
$573$	0	0
$574$	0.254778	0.0106342
$575$	33.1973	1.38442
$576$	0	0
$577$	−7.25345	−0.301965	−0.150983	−	0.988536i	$-0.548244\pi$
$577$	−7.25345	−0.301965	−0.150983	+	0.988536i	$0.548244\pi$
$578$	−16.9463	−0.704875
$579$	0	0
$580$	6.50928	0.270283
$581$	0.439002	0.0182129
$582$	0	0
$583$	35.2848	1.46135
$584$	−4.70538	−0.194710
$585$	0	0
$586$	−26.6015	−1.09890
$587$	−17.7989	−0.734639	−0.367320	−	0.930095i	$-0.619724\pi$
$587$	−17.7989	−0.734639	−0.367320	+	0.930095i	$0.619724\pi$
$588$	0	0
$589$	16.2665	0.670248
$590$	−1.36976	−0.0563919
$591$	0	0
$592$	−2.35811	−0.0969178
$593$	36.9970	1.51928	0.759642	−	0.650342i	$-0.225374\pi$
$593$	36.9970	1.51928	0.759642	+	0.650342i	$0.225374\pi$
$594$	0	0
$595$	0.143870	0.00589809
$596$	1.00000	0.0409616
$597$	0	0
$598$	−5.21139	−0.213110
$599$	14.7297	0.601839	0.300920	−	0.953650i	$-0.402706\pi$
$599$	14.7297	0.601839	0.300920	+	0.953650i	$0.402706\pi$
$600$	0	0
$601$	−21.3930	−0.872639	−0.436320	−	0.899792i	$-0.643718\pi$
$601$	−21.3930	−0.872639	−0.436320	+	0.899792i	$0.643718\pi$
$602$	−2.15437	−0.0878056
$603$	0	0
$604$	−9.78677	−0.398218
$605$	−10.1035	−0.410766
$606$	0	0
$607$	29.4540	1.19550	0.597750	−	0.801682i	$-0.296061\pi$
$607$	29.4540	1.19550	0.597750	+	0.801682i	$0.296061\pi$
$608$	−5.18376	−0.210229
$609$	0	0
$610$	−9.43814	−0.382139
$611$	−8.67530	−0.350965
$612$	0	0
$613$	36.7334	1.48365	0.741825	−	0.670594i	$-0.233961\pi$
$613$	36.7334	1.48365	0.741825	+	0.670594i	$0.233961\pi$
$614$	1.45072	0.0585463
$615$	0	0
$616$	3.47387	0.139966
$617$	−6.07036	−0.244384	−0.122192	−	0.992507i	$-0.538992\pi$
$617$	−6.07036	−0.244384	−0.122192	+	0.992507i	$0.538992\pi$
$618$	0	0
$619$	−44.7969	−1.80054	−0.900270	−	0.435333i	$-0.856631\pi$
$619$	−44.7969	−1.80054	−0.900270	+	0.435333i	$0.856631\pi$
$620$	2.68053	0.107653
$621$	0	0
$622$	3.25940	0.130690
$623$	−2.90928	−0.116558
$624$	0	0
$625$	14.5870	0.583479
$626$	1.77542	0.0709600
$627$	0	0
$628$	−21.3607	−0.852384
$629$	0.546234	0.0217798
$630$	0	0
$631$	−37.2177	−1.48161	−0.740807	−	0.671718i	$-0.765557\pi$
$631$	−37.2177	−1.48161	−0.740807	+	0.671718i	$0.765557\pi$
$632$	−7.56476	−0.300910
$633$	0	0
$634$	−1.58023	−0.0627590
$635$	10.3927	0.412421
$636$	0	0
$637$	−4.33816	−0.171884
$638$	−36.4076	−1.44139
$639$	0	0
$640$	−0.854224	−0.0337662
$641$	5.94517	0.234820	0.117410	−	0.993084i	$-0.462541\pi$
$641$	5.94517	0.234820	0.117410	+	0.993084i	$0.462541\pi$
$642$	0	0
$643$	−28.9686	−1.14241	−0.571205	−	0.820808i	$-0.693524\pi$
$643$	−28.9686	−1.14241	−0.571205	+	0.820808i	$0.693524\pi$
$644$	−5.65232	−0.222733
$645$	0	0
$646$	1.20077	0.0472437
$647$	9.56665	0.376104	0.188052	−	0.982159i	$-0.439783\pi$
$647$	9.56665	0.376104	0.188052	+	0.982159i	$0.439783\pi$
$648$	0	0
$649$	7.66130	0.300732
$650$	−2.86265	−0.112282
$651$	0	0
$652$	5.23469	0.205006
$653$	−31.2008	−1.22098	−0.610491	−	0.792023i	$-0.709028\pi$
$653$	−31.2008	−1.22098	−0.610491	+	0.792023i	$0.709028\pi$
$654$	0	0
$655$	−16.0044	−0.625345
$656$	0.350412	0.0136813
$657$	0	0
$658$	−9.40930	−0.366813
$659$	−11.7460	−0.457558	−0.228779	−	0.973478i	$-0.573473\pi$
$659$	−11.7460	−0.457558	−0.228779	+	0.973478i	$0.573473\pi$
$660$	0	0
$661$	−9.54839	−0.371389	−0.185695	−	0.982607i	$-0.559454\pi$
$661$	−9.54839	−0.371389	−0.185695	+	0.982607i	$0.559454\pi$
$662$	−11.1065	−0.431667
$663$	0	0
$664$	0.603787	0.0234315
$665$	3.21958	0.124850
$666$	0	0
$667$	59.2386	2.29373
$668$	24.8724	0.962343
$669$	0	0
$670$	−8.40007	−0.324523
$671$	52.7893	2.03791
$672$	0	0
$673$	−14.2722	−0.550151	−0.275075	−	0.961423i	$-0.588703\pi$
$673$	−14.2722	−0.550151	−0.275075	+	0.961423i	$0.588703\pi$
$674$	−14.1798	−0.546186
$675$	0	0
$676$	−12.5506	−0.482716
$677$	−14.0794	−0.541114	−0.270557	−	0.962704i	$-0.587208\pi$
$677$	−14.0794	−0.541114	−0.270557	+	0.962704i	$0.587208\pi$
$678$	0	0
$679$	−3.31560	−0.127241
$680$	0.197873	0.00758809
$681$	0	0
$682$	−14.9927	−0.574099
$683$	3.93615	0.150613	0.0753063	−	0.997160i	$-0.476007\pi$
$683$	3.93615	0.150613	0.0753063	+	0.997160i	$0.476007\pi$
$684$	0	0
$685$	5.40950	0.206687
$686$	−9.79476	−0.373966
$687$	0	0
$688$	−2.96304	−0.112965
$689$	4.95071	0.188607
$690$	0	0
$691$	−13.7349	−0.522500	−0.261250	−	0.965271i	$-0.584135\pi$
$691$	−13.7349	−0.522500	−0.261250	+	0.965271i	$0.584135\pi$
$692$	13.4677	0.511965
$693$	0	0
$694$	−16.8746	−0.640551
$695$	−15.1162	−0.573391
$696$	0	0
$697$	−0.0811698	−0.00307452
$698$	−7.49710	−0.283769
$699$	0	0
$700$	−3.10485	−0.117352
$701$	2.86040	0.108036	0.0540179	−	0.998540i	$-0.482797\pi$
$701$	2.86040	0.108036	0.0540179	+	0.998540i	$0.482797\pi$
$702$	0	0
$703$	12.2239	0.461033
$704$	4.77784	0.180071
$705$	0	0
$706$	2.81437	0.105920
$707$	−0.855155	−0.0321614
$708$	0	0
$709$	52.8083	1.98326	0.991630	−	0.129116i	$-0.0412139\pi$
$709$	52.8083	1.98326	0.991630	+	0.129116i	$0.0412139\pi$
$710$	8.85392	0.332282
$711$	0	0
$712$	−4.00132	−0.149956
$713$	24.3945	0.913581
$714$	0	0
$715$	−2.73598	−0.102320
$716$	−0.661451	−0.0247196
$717$	0	0
$718$	2.13488	0.0796729
$719$	−18.0840	−0.674419	−0.337209	−	0.941430i	$-0.609483\pi$
$719$	−18.0840	−0.674419	−0.337209	+	0.941430i	$0.609483\pi$
$720$	0	0
$721$	−7.76233	−0.289084
$722$	7.87140	0.292943
$723$	0	0
$724$	−16.6399	−0.618415
$725$	32.5401	1.20851
$726$	0	0
$727$	−18.0928	−0.671026	−0.335513	−	0.942036i	$-0.608910\pi$
$727$	−18.0928	−0.671026	−0.335513	+	0.942036i	$0.608910\pi$
$728$	0.487408	0.0180646
$729$	0	0
$730$	4.01945	0.148766
$731$	0.686361	0.0253860
$732$	0	0
$733$	13.8322	0.510904	0.255452	−	0.966822i	$-0.417776\pi$
$733$	13.8322	0.510904	0.255452	+	0.966822i	$0.417776\pi$
$734$	−17.7363	−0.654659
$735$	0	0
$736$	−7.77399	−0.286553
$737$	46.9832	1.73065
$738$	0	0
$739$	10.1027	0.371634	0.185817	−	0.982584i	$-0.440507\pi$
$739$	10.1027	0.371634	0.185817	+	0.982584i	$0.440507\pi$
$740$	2.01436	0.0740492
$741$	0	0
$742$	5.36957	0.197123
$743$	30.1747	1.10700	0.553501	−	0.832848i	$-0.313291\pi$
$743$	30.1747	1.10700	0.553501	+	0.832848i	$0.313291\pi$
$744$	0	0
$745$	−0.854224	−0.0312964
$746$	−33.3027	−1.21930
$747$	0	0
$748$	−1.10674	−0.0404665
$749$	3.48625	0.127385
$750$	0	0
$751$	−3.83745	−0.140031	−0.0700153	−	0.997546i	$-0.522305\pi$
$751$	−3.83745	−0.140031	−0.0700153	+	0.997546i	$0.522305\pi$
$752$	−12.9412	−0.471917
$753$	0	0
$754$	−5.10824	−0.186031
$755$	8.36010	0.304255
$756$	0	0
$757$	20.3476	0.739546	0.369773	−	0.929122i	$-0.379435\pi$
$757$	20.3476	0.739546	0.369773	+	0.929122i	$0.379435\pi$
$758$	−10.1957	−0.370325
$759$	0	0
$760$	4.42810	0.160624
$761$	39.0730	1.41640	0.708198	−	0.706014i	$-0.249509\pi$
$761$	39.0730	1.41640	0.708198	+	0.706014i	$0.249509\pi$
$762$	0	0
$763$	1.28355	0.0464677
$764$	−9.87624	−0.357310
$765$	0	0
$766$	25.9640	0.938116
$767$	1.07493	0.0388136
$768$	0	0
$769$	−5.30369	−0.191256	−0.0956280	−	0.995417i	$-0.530486\pi$
$769$	−5.30369	−0.191256	−0.0956280	+	0.995417i	$0.530486\pi$
$770$	−2.96747	−0.106940
$771$	0	0
$772$	−13.6732	−0.492110
$773$	11.4090	0.410352	0.205176	−	0.978725i	$-0.434223\pi$
$773$	11.4090	0.410352	0.205176	+	0.978725i	$0.434223\pi$
$774$	0	0
$775$	13.4001	0.481344
$776$	−4.56015	−0.163700
$777$	0	0
$778$	4.83469	0.173332
$779$	−1.81645	−0.0650812
$780$	0	0
$781$	−49.5216	−1.77202
$782$	1.80077	0.0643955
$783$	0	0
$784$	−6.47135	−0.231120
$785$	18.2468	0.651257
$786$	0	0
$787$	−2.89189	−0.103085	−0.0515424	−	0.998671i	$-0.516414\pi$
$787$	−2.89189	−0.103085	−0.0515424	+	0.998671i	$0.516414\pi$
$788$	17.4921	0.623131
$789$	0	0
$790$	6.46200	0.229908
$791$	10.3855	0.369267
$792$	0	0
$793$	7.40670	0.263020
$794$	19.9337	0.707419
$795$	0	0
$796$	−7.45845	−0.264358
$797$	−21.9507	−0.777534	−0.388767	−	0.921336i	$-0.627099\pi$
$797$	−21.9507	−0.777534	−0.388767	+	0.921336i	$0.627099\pi$
$798$	0	0
$799$	2.99771	0.106051
$800$	−4.27030	−0.150978
$801$	0	0
$802$	4.33155	0.152952
$803$	−22.4815	−0.793355
$804$	0	0
$805$	4.82835	0.170177
$806$	−2.10358	−0.0740953
$807$	0	0
$808$	−1.17615	−0.0413768
$809$	25.0091	0.879274	0.439637	−	0.898176i	$-0.355107\pi$
$809$	25.0091	0.879274	0.439637	+	0.898176i	$0.355107\pi$
$810$	0	0
$811$	56.7179	1.99163	0.995817	−	0.0913711i	$-0.0291249\pi$
$811$	56.7179	1.99163	0.995817	+	0.0913711i	$0.0291249\pi$
$812$	−5.54043	−0.194431
$813$	0	0
$814$	−11.2667	−0.394896
$815$	−4.47160	−0.156633
$816$	0	0
$817$	15.3597	0.537368
$818$	−29.5272	−1.03239
$819$	0	0
$820$	−0.299331	−0.0104531
$821$	−17.3257	−0.604672	−0.302336	−	0.953201i	$-0.597766\pi$
$821$	−17.3257	−0.604672	−0.302336	+	0.953201i	$0.597766\pi$
$822$	0	0
$823$	−23.2353	−0.809930	−0.404965	−	0.914332i	$-0.632716\pi$
$823$	−23.2353	−0.809930	−0.404965	+	0.914332i	$0.632716\pi$
$824$	−10.6760	−0.371917
$825$	0	0
$826$	1.16588	0.0405662
$827$	37.1661	1.29239	0.646197	−	0.763171i	$-0.276359\pi$
$827$	37.1661	1.29239	0.646197	+	0.763171i	$0.276359\pi$
$828$	0	0
$829$	3.51796	0.122184	0.0610919	−	0.998132i	$-0.480542\pi$
$829$	3.51796	0.122184	0.0610919	+	0.998132i	$0.480542\pi$
$830$	−0.515769	−0.0179026
$831$	0	0
$832$	0.670363	0.0232407
$833$	1.49903	0.0519383
$834$	0	0
$835$	−21.2466	−0.735270
$836$	−24.7672	−0.856590
$837$	0	0
$838$	−22.5113	−0.777639
$839$	2.89793	0.100048	0.0500238	−	0.998748i	$-0.484070\pi$
$839$	2.89793	0.100048	0.0500238	+	0.998748i	$0.484070\pi$
$840$	0	0
$841$	29.0660	1.00228
$842$	16.0732	0.553919
$843$	0	0
$844$	16.2465	0.559227
$845$	10.7210	0.368815
$846$	0	0
$847$	8.59970	0.295489
$848$	7.38511	0.253606
$849$	0	0
$850$	0.989175	0.0339284
$851$	18.3319	0.628410
$852$	0	0
$853$	47.8771	1.63928	0.819641	−	0.572877i	$-0.194173\pi$
$853$	47.8771	1.63928	0.819641	+	0.572877i	$0.194173\pi$
$854$	8.03336	0.274896
$855$	0	0
$856$	4.79486	0.163885
$857$	−6.59365	−0.225235	−0.112617	−	0.993638i	$-0.535923\pi$
$857$	−6.59365	−0.225235	−0.112617	+	0.993638i	$0.535923\pi$
$858$	0	0
$859$	−21.5642	−0.735761	−0.367880	−	0.929873i	$-0.619916\pi$
$859$	−21.5642	−0.735761	−0.367880	+	0.929873i	$0.619916\pi$
$860$	2.53110	0.0863099
$861$	0	0
$862$	35.7406	1.21733
$863$	−24.6863	−0.840331	−0.420166	−	0.907447i	$-0.638028\pi$
$863$	−24.6863	−0.840331	−0.420166	+	0.907447i	$0.638028\pi$
$864$	0	0
$865$	−11.5044	−0.391162
$866$	29.3282	0.996614
$867$	0	0
$868$	−2.28155	−0.0774410
$869$	−36.1432	−1.22607
$870$	0	0
$871$	6.59206	0.223363
$872$	1.76535	0.0597822
$873$	0	0
$874$	40.2985	1.36312
$875$	5.75769	0.194646
$876$	0	0
$877$	−50.5780	−1.70790	−0.853949	−	0.520357i	$-0.825799\pi$
$877$	−50.5780	−1.70790	−0.853949	+	0.520357i	$0.825799\pi$
$878$	−8.48140	−0.286233
$879$	0	0
$880$	−4.08134	−0.137582
$881$	19.5105	0.657324	0.328662	−	0.944448i	$-0.393402\pi$
$881$	19.5105	0.657324	0.328662	+	0.944448i	$0.393402\pi$
$882$	0	0
$883$	−24.5096	−0.824813	−0.412406	−	0.911000i	$-0.635312\pi$
$883$	−24.5096	−0.824813	−0.412406	+	0.911000i	$0.635312\pi$
$884$	−0.155283	−0.00522275
$885$	0	0
$886$	−29.2319	−0.982065
$887$	47.6436	1.59972	0.799858	−	0.600190i	$-0.204908\pi$
$887$	47.6436	1.59972	0.799858	+	0.600190i	$0.204908\pi$
$888$	0	0
$889$	−8.84583	−0.296680
$890$	3.41802	0.114572
$891$	0	0
$892$	−6.84038	−0.229033
$893$	67.0841	2.24488
$894$	0	0
$895$	0.565027	0.0188868
$896$	0.727081	0.0242901
$897$	0	0
$898$	−23.7240	−0.791681
$899$	23.9116	0.797497
$900$	0	0
$901$	−1.71069	−0.0569914
$902$	1.67421	0.0557452
$903$	0	0
$904$	14.2839	0.475075
$905$	14.2142	0.472495
$906$	0	0
$907$	33.3679	1.10796	0.553981	−	0.832529i	$-0.313108\pi$
$907$	33.3679	1.10796	0.553981	+	0.832529i	$0.313108\pi$
$908$	5.73148	0.190206
$909$	0	0
$910$	−0.416356	−0.0138021
$911$	−19.1967	−0.636014	−0.318007	−	0.948088i	$-0.603014\pi$
$911$	−19.1967	−0.636014	−0.318007	+	0.948088i	$0.603014\pi$
$912$	0	0
$913$	2.88479	0.0954728
$914$	19.6691	0.650597
$915$	0	0
$916$	−2.17916	−0.0720016
$917$	13.6223	0.449849
$918$	0	0
$919$	−44.6307	−1.47223	−0.736115	−	0.676856i	$-0.763342\pi$
$919$	−44.6307	−1.47223	−0.736115	+	0.676856i	$0.763342\pi$
$920$	6.64073	0.218938
$921$	0	0
$922$	−2.86113	−0.0942262
$923$	−6.94822	−0.228704
$924$	0	0
$925$	10.0698	0.331094
$926$	−20.0230	−0.657997
$927$	0	0
$928$	−7.62010	−0.250142
$929$	28.9324	0.949240	0.474620	−	0.880191i	$-0.342585\pi$
$929$	28.9324	0.949240	0.474620	+	0.880191i	$0.342585\pi$
$930$	0	0
$931$	33.5460	1.09942
$932$	−0.993187	−0.0325329
$933$	0	0
$934$	−28.7449	−0.940561
$935$	0.945405	0.0309181
$936$	0	0
$937$	−35.8230	−1.17029	−0.585144	−	0.810929i	$-0.698962\pi$
$937$	−35.8230	−1.17029	−0.585144	+	0.810929i	$0.698962\pi$
$938$	7.14980	0.233449
$939$	0	0
$940$	11.0547	0.360564
$941$	−35.5526	−1.15898	−0.579491	−	0.814979i	$-0.696748\pi$
$941$	−35.5526	−1.15898	−0.579491	+	0.814979i	$0.696748\pi$
$942$	0	0
$943$	−2.72410	−0.0887089
$944$	1.60351	0.0521897
$945$	0	0
$946$	−14.1569	−0.460281
$947$	−1.71004	−0.0555688	−0.0277844	−	0.999614i	$-0.508845\pi$
$947$	−1.71004	−0.0555688	−0.0277844	+	0.999614i	$0.508845\pi$
$948$	0	0
$949$	−3.15431	−0.102393
$950$	22.1362	0.718194
$951$	0	0
$952$	−0.168422	−0.00545857
$953$	−57.4750	−1.86180	−0.930899	−	0.365277i	$-0.880974\pi$
$953$	−57.4750	−1.86180	−0.930899	+	0.365277i	$0.880974\pi$
$954$	0	0
$955$	8.43652	0.272999
$956$	−12.0854	−0.390869
$957$	0	0
$958$	−19.5108	−0.630365
$959$	−4.60435	−0.148682
$960$	0	0
$961$	−21.1532	−0.682360
$962$	−1.58079	−0.0509667
$963$	0	0
$964$	3.47303	0.111859
$965$	11.6800	0.375993
$966$	0	0
$967$	−14.8876	−0.478753	−0.239376	−	0.970927i	$-0.576943\pi$
$967$	−14.8876	−0.478753	−0.239376	+	0.970927i	$0.576943\pi$
$968$	11.8277	0.380157
$969$	0	0
$970$	3.89539	0.125074
$971$	−31.8123	−1.02090	−0.510452	−	0.859906i	$-0.670522\pi$
$971$	−31.8123	−1.02090	−0.510452	+	0.859906i	$0.670522\pi$
$972$	0	0
$973$	12.8663	0.412475
$974$	2.99490	0.0959627
$975$	0	0
$976$	11.0488	0.353663
$977$	15.8528	0.507175	0.253588	−	0.967312i	$-0.418389\pi$
$977$	15.8528	0.507175	0.253588	+	0.967312i	$0.418389\pi$
$978$	0	0
$979$	−19.1176	−0.611003
$980$	5.52799	0.176585
$981$	0	0
$982$	43.3760	1.38418
$983$	−39.9556	−1.27439	−0.637193	−	0.770704i	$-0.719905\pi$
$983$	−39.9556	−1.27439	−0.637193	+	0.770704i	$0.719905\pi$
$984$	0	0
$985$	−14.9422	−0.476098
$986$	1.76513	0.0562131
$987$	0	0
$988$	−3.47500	−0.110555
$989$	23.0346	0.732459
$990$	0	0
$991$	−20.5807	−0.653768	−0.326884	−	0.945064i	$-0.605999\pi$
$991$	−20.5807	−0.653768	−0.326884	+	0.945064i	$0.605999\pi$
$992$	−3.13797	−0.0996305
$993$	0	0
$994$	−7.53610	−0.239030
$995$	6.37119	0.201980
$996$	0	0
$997$	6.70445	0.212332	0.106166	−	0.994348i	$-0.466143\pi$
$997$	6.70445	0.212332	0.106166	+	0.994348i	$0.466143\pi$
$998$	−0.987622	−0.0312626
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.f.1.4	yes	8
3.2	odd	2		8046.2.a.e.1.5	✓	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.e.1.5	✓	8	3.2	odd	2
8046.2.a.f.1.4	yes	8	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.854224 q^{5} +0.727081 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -0.854224 q^{5} +0.727081 q^{7} +1.00000 q^{8} -0.854224 q^{10} +4.77784 q^{11} +0.670363 q^{13} +0.727081 q^{14} +1.00000 q^{16} -0.231641 q^{17} -5.18376 q^{19} -0.854224 q^{20} +4.77784 q^{22} -7.77399 q^{23} -4.27030 q^{25} +0.670363 q^{26} +0.727081 q^{28} -7.62010 q^{29} -3.13797 q^{31} +1.00000 q^{32} -0.231641 q^{34} -0.621090 q^{35} -2.35811 q^{37} -5.18376 q^{38} -0.854224 q^{40} +0.350412 q^{41} -2.96304 q^{43} +4.77784 q^{44} -7.77399 q^{46} -12.9412 q^{47} -6.47135 q^{49} -4.27030 q^{50} +0.670363 q^{52} +7.38511 q^{53} -4.08134 q^{55} +0.727081 q^{56} -7.62010 q^{58} +1.60351 q^{59} +11.0488 q^{61} -3.13797 q^{62} +1.00000 q^{64} -0.572640 q^{65} +9.83357 q^{67} -0.231641 q^{68} -0.621090 q^{70} -10.3649 q^{71} -4.70538 q^{73} -2.35811 q^{74} -5.18376 q^{76} +3.47387 q^{77} -7.56476 q^{79} -0.854224 q^{80} +0.350412 q^{82} +0.603787 q^{83} +0.197873 q^{85} -2.96304 q^{86} +4.77784 q^{88} -4.00132 q^{89} +0.487408 q^{91} -7.77399 q^{92} -12.9412 q^{94} +4.42810 q^{95} -4.56015 q^{97} -6.47135 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8}+O(q^{10}) \) 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8	\( 8 q + 8 q^{2} + 8 q^{4} - 5 q^{7} + 8 q^{8} - 6 q^{11} - 8 q^{13} - 5 q^{14} + 8 q^{16} + 5 q^{17} - 14 q^{19} - 6 q^{22} - 21 q^{23} - 6 q^{25} - 8 q^{26} - 5 q^{28} - 3 q^{29} - 4 q^{31} + 8 q^{32} + 5 q^{34} + 2 q^{35} - 3 q^{37} - 14 q^{38} - 7 q^{41} - 12 q^{43} - 6 q^{44} - 21 q^{46} - 25 q^{47} - 7 q^{49} - 6 q^{50} - 8 q^{52} - 3 q^{53} - 9 q^{55} - 5 q^{56} - 3 q^{58} - 2 q^{59} - 17 q^{61} - 4 q^{62} + 8 q^{64} - 32 q^{65} - 14 q^{67} + 5 q^{68} + 2 q^{70} - 7 q^{71} - 10 q^{73} - 3 q^{74} - 14 q^{76} - 12 q^{77} - 33 q^{79} - 7 q^{82} - 13 q^{83} - 33 q^{85} - 12 q^{86} - 6 q^{88} + 22 q^{89} - 22 q^{91} - 21 q^{92} - 25 q^{94} + 14 q^{95} - 11 q^{97} - 7 q^{98}+O(q^{100}) \) 8 * q + 8 * q^2 + 8 * q^4 - 5 * q^7 + 8 * q^8 - 6 * q^11 - 8 * q^13 - 5 * q^14 + 8 * q^16 + 5 * q^17 - 14 * q^19 - 6 * q^22 - 21 * q^23 - 6 * q^25 - 8 * q^26 - 5 * q^28 - 3 * q^29 - 4 * q^31 + 8 * q^32 + 5 * q^34 + 2 * q^35 - 3 * q^37 - 14 * q^38 - 7 * q^41 - 12 * q^43 - 6 * q^44 - 21 * q^46 - 25 * q^47 - 7 * q^49 - 6 * q^50 - 8 * q^52 - 3 * q^53 - 9 * q^55 - 5 * q^56 - 3 * q^58 - 2 * q^59 - 17 * q^61 - 4 * q^62 + 8 * q^64 - 32 * q^65 - 14 * q^67 + 5 * q^68 + 2 * q^70 - 7 * q^71 - 10 * q^73 - 3 * q^74 - 14 * q^76 - 12 * q^77 - 33 * q^79 - 7 * q^82 - 13 * q^83 - 33 * q^85 - 12 * q^86 - 6 * q^88 + 22 * q^89 - 22 * q^91 - 21 * q^92 - 25 * q^94 + 14 * q^95 - 11 * q^97 - 7 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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