LMFDB - Embedded newform 8281.2.a.z.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [8281,2,Mod(1,8281)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(8281, 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("8281.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 8281 = 7^{2} \cdot 13^{2} $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	8281.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:	$66.1241179138$
Analytic rank:	$1$
Dimension:	$2$
Coefficient field:	$\Q(\zeta_{10})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - x - 1 $ x^2 - x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 91)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-0.618034$ of defining polynomial
Character	$\chi$	$=$	8281.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.381966 q^{2} +2.23607 q^{3} -1.85410 q^{4} +2.23607 q^{5} +0.854102 q^{6} -1.47214 q^{8} +2.00000 q^{9} +O(q^{10})$	\(q+0.381966 q^{2} +2.23607 q^{3} -1.85410 q^{4} +2.23607 q^{5} +0.854102 q^{6} -1.47214 q^{8} +2.00000 q^{9} +0.854102 q^{10} +3.00000 q^{11} -4.14590 q^{12} +5.00000 q^{15} +3.14590 q^{16} -7.47214 q^{17} +0.763932 q^{18} -3.00000 q^{19} -4.14590 q^{20} +1.14590 q^{22} -3.76393 q^{23} -3.29180 q^{24} -2.23607 q^{27} -4.47214 q^{29} +1.90983 q^{30} -5.00000 q^{31} +4.14590 q^{32} +6.70820 q^{33} -2.85410 q^{34} -3.70820 q^{36} +8.70820 q^{37} -1.14590 q^{38} -3.29180 q^{40} -4.47214 q^{41} -8.00000 q^{43} -5.56231 q^{44} +4.47214 q^{45} -1.43769 q^{46} -1.47214 q^{47} +7.03444 q^{48} -16.7082 q^{51} +1.47214 q^{53} -0.854102 q^{54} +6.70820 q^{55} -6.70820 q^{57} -1.70820 q^{58} -7.47214 q^{59} -9.27051 q^{60} +3.00000 q^{61} -1.90983 q^{62} -4.70820 q^{64} +2.56231 q^{66} +3.00000 q^{67} +13.8541 q^{68} -8.41641 q^{69} -8.94427 q^{71} -2.94427 q^{72} -10.7082 q^{73} +3.32624 q^{74} +5.56231 q^{76} +10.7082 q^{79} +7.03444 q^{80} -11.0000 q^{81} -1.70820 q^{82} -16.7082 q^{85} -3.05573 q^{86} -10.0000 q^{87} -4.41641 q^{88} +2.23607 q^{89} +1.70820 q^{90} +6.97871 q^{92} -11.1803 q^{93} -0.562306 q^{94} -6.70820 q^{95} +9.27051 q^{96} +17.4164 q^{97} +6.00000 q^{99} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) $ 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 + 6 * q^8 + 4 * q^9	$ 2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9} - 5 q^{10} + 6 q^{11} - 15 q^{12} + 10 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} - 6 q^{19} - 15 q^{20} + 9 q^{22} - 12 q^{23} - 20 q^{24} + 15 q^{30} - 10 q^{31} + 15 q^{32} + q^{34} + 6 q^{36} + 4 q^{37} - 9 q^{38} - 20 q^{40} - 16 q^{43} + 9 q^{44} - 23 q^{46} + 6 q^{47} - 15 q^{48} - 20 q^{51} - 6 q^{53} + 5 q^{54} + 10 q^{58} - 6 q^{59} + 15 q^{60} + 6 q^{61} - 15 q^{62} + 4 q^{64} - 15 q^{66} + 6 q^{67} + 21 q^{68} + 10 q^{69} + 12 q^{72} - 8 q^{73} - 9 q^{74} - 9 q^{76} + 8 q^{79} - 15 q^{80} - 22 q^{81} + 10 q^{82} - 20 q^{85} - 24 q^{86} - 20 q^{87} + 18 q^{88} - 10 q^{90} - 33 q^{92} + 19 q^{94} - 15 q^{96} + 8 q^{97} + 12 q^{99}+O(q^{100}) $ 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 + 6 * q^8 + 4 * q^9 - 5 * q^10 + 6 * q^11 - 15 * q^12 + 10 * q^15 + 13 * q^16 - 6 * q^17 + 6 * q^18 - 6 * q^19 - 15 * q^20 + 9 * q^22 - 12 * q^23 - 20 * q^24 + 15 * q^30 - 10 * q^31 + 15 * q^32 + q^34 + 6 * q^36 + 4 * q^37 - 9 * q^38 - 20 * q^40 - 16 * q^43 + 9 * q^44 - 23 * q^46 + 6 * q^47 - 15 * q^48 - 20 * q^51 - 6 * q^53 + 5 * q^54 + 10 * q^58 - 6 * q^59 + 15 * q^60 + 6 * q^61 - 15 * q^62 + 4 * q^64 - 15 * q^66 + 6 * q^67 + 21 * q^68 + 10 * q^69 + 12 * q^72 - 8 * q^73 - 9 * q^74 - 9 * q^76 + 8 * q^79 - 15 * q^80 - 22 * q^81 + 10 * q^82 - 20 * q^85 - 24 * q^86 - 20 * q^87 + 18 * q^88 - 10 * q^90 - 33 * q^92 + 19 * q^94 - 15 * q^96 + 8 * q^97 + 12 * q^99

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 $

$2$	0.381966	0.270091	0.135045	−	0.990839i	$-0.456882\pi$
$2$	0.381966	0.270091	0.135045	+	0.990839i	$0.456882\pi$
$3$	2.23607	1.29099	0.645497	−	0.763763i	$-0.276650\pi$
$3$	2.23607	1.29099	0.645497	+	0.763763i	$0.276650\pi$
$4$	−1.85410	−0.927051
$5$	2.23607	1.00000	0.500000	−	0.866025i	$-0.333333\pi$
$5$	2.23607	1.00000	0.500000	+	0.866025i	$0.333333\pi$
$6$	0.854102	0.348686
$7$	0	0
$7$	0	0
$8$	−1.47214	−0.520479
$9$	2.00000	0.666667
$10$	0.854102	0.270091
$11$	3.00000	0.904534	0.452267	−	0.891883i	$-0.350615\pi$
$11$	3.00000	0.904534	0.452267	+	0.891883i	$0.350615\pi$
$12$	−4.14590	−1.19682
$13$	0	0
$13$	0	0
$14$	0	0
$15$	5.00000	1.29099
$16$	3.14590	0.786475
$17$	−7.47214	−1.81226	−0.906130	−	0.423000i	$-0.860977\pi$
$17$	−7.47214	−1.81226	−0.906130	+	0.423000i	$0.860977\pi$
$18$	0.763932	0.180061
$19$	−3.00000	−0.688247	−0.344124	−	0.938924i	$-0.611824\pi$
$19$	−3.00000	−0.688247	−0.344124	+	0.938924i	$0.611824\pi$
$20$	−4.14590	−0.927051
$21$	0	0
$22$	1.14590	0.244306
$23$	−3.76393	−0.784834	−0.392417	−	0.919787i	$-0.628361\pi$
$23$	−3.76393	−0.784834	−0.392417	+	0.919787i	$0.628361\pi$
$24$	−3.29180	−0.671935
$25$	0	0
$26$	0	0
$27$	−2.23607	−0.430331
$28$	0	0
$29$	−4.47214	−0.830455	−0.415227	−	0.909718i	$-0.636298\pi$
$29$	−4.47214	−0.830455	−0.415227	+	0.909718i	$0.636298\pi$
$30$	1.90983	0.348686
$31$	−5.00000	−0.898027	−0.449013	−	0.893525i	$-0.648224\pi$
$31$	−5.00000	−0.898027	−0.449013	+	0.893525i	$0.648224\pi$
$32$	4.14590	0.732898
$33$	6.70820	1.16775
$34$	−2.85410	−0.489474
$35$	0	0
$36$	−3.70820	−0.618034
$37$	8.70820	1.43162	0.715810	−	0.698295i	$-0.246058\pi$
$37$	8.70820	1.43162	0.715810	+	0.698295i	$0.246058\pi$
$38$	−1.14590	−0.185889
$39$	0	0
$40$	−3.29180	−0.520479
$41$	−4.47214	−0.698430	−0.349215	−	0.937043i	$-0.613552\pi$
$41$	−4.47214	−0.698430	−0.349215	+	0.937043i	$0.613552\pi$
$42$	0	0
$43$	−8.00000	−1.21999	−0.609994	−	0.792406i	$-0.708828\pi$
$43$	−8.00000	−1.21999	−0.609994	+	0.792406i	$0.708828\pi$
$44$	−5.56231	−0.838549
$45$	4.47214	0.666667
$46$	−1.43769	−0.211976
$47$	−1.47214	−0.214733	−0.107367	−	0.994220i	$-0.534242\pi$
$47$	−1.47214	−0.214733	−0.107367	+	0.994220i	$0.534242\pi$
$48$	7.03444	1.01533
$49$	0	0
$50$	0	0
$51$	−16.7082	−2.33962
$52$	0	0
$53$	1.47214	0.202213	0.101107	−	0.994876i	$-0.467762\pi$
$53$	1.47214	0.202213	0.101107	+	0.994876i	$0.467762\pi$
$54$	−0.854102	−0.116229
$55$	6.70820	0.904534
$56$	0	0
$57$	−6.70820	−0.888523
$58$	−1.70820	−0.224298
$59$	−7.47214	−0.972789	−0.486395	−	0.873739i	$-0.661688\pi$
$59$	−7.47214	−0.972789	−0.486395	+	0.873739i	$0.661688\pi$
$60$	−9.27051	−1.19682
$61$	3.00000	0.384111	0.192055	−	0.981384i	$-0.438485\pi$
$61$	3.00000	0.384111	0.192055	+	0.981384i	$0.438485\pi$
$62$	−1.90983	−0.242549
$63$	0	0
$64$	−4.70820	−0.588525
$65$	0	0
$66$	2.56231	0.315398
$67$	3.00000	0.366508	0.183254	−	0.983066i	$-0.441337\pi$
$67$	3.00000	0.366508	0.183254	+	0.983066i	$0.441337\pi$
$68$	13.8541	1.68006
$69$	−8.41641	−1.01322
$70$	0	0
$71$	−8.94427	−1.06149	−0.530745	−	0.847532i	$-0.678088\pi$
$71$	−8.94427	−1.06149	−0.530745	+	0.847532i	$0.678088\pi$
$72$	−2.94427	−0.346986
$73$	−10.7082	−1.25330	−0.626650	−	0.779301i	$-0.715575\pi$
$73$	−10.7082	−1.25330	−0.626650	+	0.779301i	$0.715575\pi$
$74$	3.32624	0.386667
$75$	0	0
$76$	5.56231	0.638040
$77$	0	0
$78$	0	0
$79$	10.7082	1.20477	0.602384	−	0.798207i	$-0.294218\pi$
$79$	10.7082	1.20477	0.602384	+	0.798207i	$0.294218\pi$
$80$	7.03444	0.786475
$81$	−11.0000	−1.22222
$82$	−1.70820	−0.188640
$83$	0	0		−	1.00000i	$-0.5\pi$
$83$	0	0			1.00000i	$0.5\pi$
$84$	0	0
$85$	−16.7082	−1.81226
$86$	−3.05573	−0.329508
$87$	−10.0000	−1.07211
$88$	−4.41641	−0.470791
$89$	2.23607	0.237023	0.118511	−	0.992953i	$-0.462188\pi$
$89$	2.23607	0.237023	0.118511	+	0.992953i	$0.462188\pi$
$90$	1.70820	0.180061
$91$	0	0
$92$	6.97871	0.727581
$93$	−11.1803	−1.15935
$94$	−0.562306	−0.0579974
$95$	−6.70820	−0.688247
$96$	9.27051	0.946167
$97$	17.4164	1.76837	0.884184	−	0.467139i	$-0.154715\pi$
$97$	17.4164	1.76837	0.884184	+	0.467139i	$0.154715\pi$
$98$	0	0
$99$	6.00000	0.603023
$100$	0	0
$101$	−9.00000	−0.895533	−0.447767	−	0.894150i	$-0.647781\pi$
$101$	−9.00000	−0.895533	−0.447767	+	0.894150i	$0.647781\pi$
$102$	−6.38197	−0.631909
$103$	10.7082	1.05511	0.527555	−	0.849521i	$-0.323109\pi$
$103$	10.7082	1.05511	0.527555	+	0.849521i	$0.323109\pi$
$104$	0	0
$105$	0	0
$106$	0.562306	0.0546160
$107$	−14.2361	−1.37625	−0.688126	−	0.725591i	$-0.741567\pi$
$107$	−14.2361	−1.37625	−0.688126	+	0.725591i	$0.741567\pi$
$108$	4.14590	0.398939
$109$	−10.7082	−1.02566	−0.512830	−	0.858490i	$-0.671403\pi$
$109$	−10.7082	−1.02566	−0.512830	+	0.858490i	$0.671403\pi$
$110$	2.56231	0.244306
$111$	19.4721	1.84821
$112$	0	0
$113$	−14.9443	−1.40584	−0.702919	−	0.711269i	$-0.748121\pi$
$113$	−14.9443	−1.40584	−0.702919	+	0.711269i	$0.748121\pi$
$114$	−2.56231	−0.239982
$115$	−8.41641	−0.784834
$116$	8.29180	0.769874
$117$	0	0
$118$	−2.85410	−0.262741
$119$	0	0
$120$	−7.36068	−0.671935
$121$	−2.00000	−0.181818
$122$	1.14590	0.103745
$123$	−10.0000	−0.901670
$124$	9.27051	0.832516
$125$	−11.1803	−1.00000
$126$	0	0
$127$	15.4164	1.36798	0.683992	−	0.729489i	$-0.260242\pi$
$127$	15.4164	1.36798	0.683992	+	0.729489i	$0.260242\pi$
$128$	−10.0902	−0.891853
$129$	−17.8885	−1.57500
$130$	0	0
$131$	−3.76393	−0.328856	−0.164428	−	0.986389i	$-0.552578\pi$
$131$	−3.76393	−0.328856	−0.164428	+	0.986389i	$0.552578\pi$
$132$	−12.4377	−1.08256
$133$	0	0
$134$	1.14590	0.0989905
$135$	−5.00000	−0.430331
$136$	11.0000	0.943242
$137$	3.76393	0.321574	0.160787	−	0.986989i	$-0.448597\pi$
$137$	3.76393	0.321574	0.160787	+	0.986989i	$0.448597\pi$
$138$	−3.21478	−0.273660
$139$	3.41641	0.289776	0.144888	−	0.989448i	$-0.453718\pi$
$139$	3.41641	0.289776	0.144888	+	0.989448i	$0.453718\pi$
$140$	0	0
$141$	−3.29180	−0.277219
$142$	−3.41641	−0.286699
$143$	0	0
$144$	6.29180	0.524316
$145$	−10.0000	−0.830455
$146$	−4.09017	−0.338505
$147$	0	0
$148$	−16.1459	−1.32718
$149$	12.7082	1.04110	0.520548	−	0.853832i	$-0.325728\pi$
$149$	12.7082	1.04110	0.520548	+	0.853832i	$0.325728\pi$
$150$	0	0
$151$	6.41641	0.522160	0.261080	−	0.965317i	$-0.415921\pi$
$151$	6.41641	0.522160	0.261080	+	0.965317i	$0.415921\pi$
$152$	4.41641	0.358218
$153$	−14.9443	−1.20817
$154$	0	0
$155$	−11.1803	−0.898027
$156$	0	0
$157$	7.00000	0.558661	0.279330	−	0.960195i	$-0.409888\pi$
$157$	7.00000	0.558661	0.279330	+	0.960195i	$0.409888\pi$
$158$	4.09017	0.325396
$159$	3.29180	0.261056
$160$	9.27051	0.732898
$161$	0	0
$162$	−4.20163	−0.330111
$163$	−10.4164	−0.815876	−0.407938	−	0.913010i	$-0.633752\pi$
$163$	−10.4164	−0.815876	−0.407938	+	0.913010i	$0.633752\pi$
$164$	8.29180	0.647480
$165$	15.0000	1.16775
$166$	0	0
$167$	−13.5279	−1.04682	−0.523409	−	0.852082i	$-0.675340\pi$
$167$	−13.5279	−1.04682	−0.523409	+	0.852082i	$0.675340\pi$
$168$	0	0
$169$	0	0
$170$	−6.38197	−0.489474
$171$	−6.00000	−0.458831
$172$	14.8328	1.13099
$173$	10.4164	0.791945	0.395972	−	0.918262i	$-0.370408\pi$
$173$	10.4164	0.791945	0.395972	+	0.918262i	$0.370408\pi$
$174$	−3.81966	−0.289568
$175$	0	0
$176$	9.43769	0.711393
$177$	−16.7082	−1.25587
$178$	0.854102	0.0640176
$179$	20.1246	1.50418	0.752092	−	0.659058i	$-0.229045\pi$
$179$	20.1246	1.50418	0.752092	+	0.659058i	$0.229045\pi$
$180$	−8.29180	−0.618034
$181$	1.41641	0.105281	0.0526404	−	0.998614i	$-0.483236\pi$
$181$	1.41641	0.105281	0.0526404	+	0.998614i	$0.483236\pi$
$182$	0	0
$183$	6.70820	0.495885
$184$	5.54102	0.408489
$185$	19.4721	1.43162
$186$	−4.27051	−0.313129
$187$	−22.4164	−1.63925
$188$	2.72949	0.199069
$189$	0	0
$190$	−2.56231	−0.185889
$191$	−11.1803	−0.808981	−0.404491	−	0.914542i	$-0.632551\pi$
$191$	−11.1803	−0.808981	−0.404491	+	0.914542i	$0.632551\pi$
$192$	−10.5279	−0.759783
$193$	12.7082	0.914757	0.457378	−	0.889272i	$-0.348789\pi$
$193$	12.7082	0.914757	0.457378	+	0.889272i	$0.348789\pi$
$194$	6.65248	0.477620
$195$	0	0
$196$	0	0
$197$	26.9443	1.91970	0.959850	−	0.280514i	$-0.0905049\pi$
$197$	26.9443	1.91970	0.959850	+	0.280514i	$0.0905049\pi$
$198$	2.29180	0.162871
$199$	−7.29180	−0.516902	−0.258451	−	0.966024i	$-0.583212\pi$
$199$	−7.29180	−0.516902	−0.258451	+	0.966024i	$0.583212\pi$
$200$	0	0
$201$	6.70820	0.473160
$202$	−3.43769	−0.241875
$203$	0	0
$204$	30.9787	2.16894
$205$	−10.0000	−0.698430
$206$	4.09017	0.284976
$207$	−7.52786	−0.523223
$208$	0	0
$209$	−9.00000	−0.622543
$210$	0	0
$211$	4.00000	0.275371	0.137686	−	0.990476i	$-0.456034\pi$
$211$	4.00000	0.275371	0.137686	+	0.990476i	$0.456034\pi$
$212$	−2.72949	−0.187462
$213$	−20.0000	−1.37038
$214$	−5.43769	−0.371713
$215$	−17.8885	−1.21999
$216$	3.29180	0.223978
$217$	0	0
$218$	−4.09017	−0.277021
$219$	−23.9443	−1.61800
$220$	−12.4377	−0.838549
$221$	0	0
$222$	7.43769	0.499185
$223$	4.00000	0.267860	0.133930	−	0.990991i	$-0.457240\pi$
$223$	4.00000	0.267860	0.133930	+	0.990991i	$0.457240\pi$
$224$	0	0
$225$	0	0
$226$	−5.70820	−0.379704
$227$	11.9443	0.792769	0.396385	−	0.918085i	$-0.370265\pi$
$227$	11.9443	0.792769	0.396385	+	0.918085i	$0.370265\pi$
$228$	12.4377	0.823706
$229$	16.1246	1.06554	0.532772	−	0.846259i	$-0.321150\pi$
$229$	16.1246	1.06554	0.532772	+	0.846259i	$0.321150\pi$
$230$	−3.21478	−0.211976
$231$	0	0
$232$	6.58359	0.432234
$233$	−5.94427	−0.389422	−0.194711	−	0.980861i	$-0.562377\pi$
$233$	−5.94427	−0.389422	−0.194711	+	0.980861i	$0.562377\pi$
$234$	0	0
$235$	−3.29180	−0.214733
$236$	13.8541	0.901825
$237$	23.9443	1.55535
$238$	0	0
$239$	7.41641	0.479728	0.239864	−	0.970807i	$-0.422897\pi$
$239$	7.41641	0.479728	0.239864	+	0.970807i	$0.422897\pi$
$240$	15.7295	1.01533
$241$	8.70820	0.560945	0.280472	−	0.959862i	$-0.409509\pi$
$241$	8.70820	0.560945	0.280472	+	0.959862i	$0.409509\pi$
$242$	−0.763932	−0.0491074
$243$	−17.8885	−1.14755
$244$	−5.56231	−0.356090
$245$	0	0
$246$	−3.81966	−0.243533
$247$	0	0
$248$	7.36068	0.467404
$249$	0	0
$250$	−4.27051	−0.270091
$251$	10.4721	0.660995	0.330498	−	0.943807i	$-0.392783\pi$
$251$	10.4721	0.660995	0.330498	+	0.943807i	$0.392783\pi$
$252$	0	0
$253$	−11.2918	−0.709909
$254$	5.88854	0.369480
$255$	−37.3607	−2.33962
$256$	5.56231	0.347644
$257$	−17.9443	−1.11933	−0.559666	−	0.828718i	$-0.689071\pi$
$257$	−17.9443	−1.11933	−0.559666	+	0.828718i	$0.689071\pi$
$258$	−6.83282	−0.425393
$259$	0	0
$260$	0	0
$261$	−8.94427	−0.553637
$262$	−1.43769	−0.0888210
$263$	−14.1246	−0.870961	−0.435480	−	0.900198i	$-0.643421\pi$
$263$	−14.1246	−0.870961	−0.435480	+	0.900198i	$0.643421\pi$
$264$	−9.87539	−0.607788
$265$	3.29180	0.202213
$266$	0	0
$267$	5.00000	0.305995
$268$	−5.56231	−0.339772
$269$	4.52786	0.276069	0.138034	−	0.990427i	$-0.455922\pi$
$269$	4.52786	0.276069	0.138034	+	0.990427i	$0.455922\pi$
$270$	−1.90983	−0.116229
$271$	−6.41641	−0.389769	−0.194885	−	0.980826i	$-0.562433\pi$
$271$	−6.41641	−0.389769	−0.194885	+	0.980826i	$0.562433\pi$
$272$	−23.5066	−1.42530
$273$	0	0
$274$	1.43769	0.0868543
$275$	0	0
$276$	15.6049	0.939303
$277$	26.4164	1.58721	0.793604	−	0.608435i	$-0.208202\pi$
$277$	26.4164	1.58721	0.793604	+	0.608435i	$0.208202\pi$
$278$	1.30495	0.0782658
$279$	−10.0000	−0.598684
$280$	0	0
$281$	−9.05573	−0.540219	−0.270110	−	0.962830i	$-0.587060\pi$
$281$	−9.05573	−0.540219	−0.270110	+	0.962830i	$0.587060\pi$
$282$	−1.25735	−0.0748744
$283$	−14.1246	−0.839621	−0.419811	−	0.907612i	$-0.637903\pi$
$283$	−14.1246	−0.839621	−0.419811	+	0.907612i	$0.637903\pi$
$284$	16.5836	0.984055
$285$	−15.0000	−0.888523
$286$	0	0
$287$	0	0
$288$	8.29180	0.488599
$289$	38.8328	2.28428
$290$	−3.81966	−0.224298
$291$	38.9443	2.28295
$292$	19.8541	1.16187
$293$	−2.94427	−0.172006	−0.0860031	−	0.996295i	$-0.527409\pi$
$293$	−2.94427	−0.172006	−0.0860031	+	0.996295i	$0.527409\pi$
$294$	0	0
$295$	−16.7082	−0.972789
$296$	−12.8197	−0.745128
$297$	−6.70820	−0.389249
$298$	4.85410	0.281191
$299$	0	0
$300$	0	0
$301$	0	0
$302$	2.45085	0.141031
$303$	−20.1246	−1.15613
$304$	−9.43769	−0.541289
$305$	6.70820	0.384111
$306$	−5.70820	−0.326316
$307$	7.41641	0.423277	0.211638	−	0.977348i	$-0.432120\pi$
$307$	7.41641	0.423277	0.211638	+	0.977348i	$0.432120\pi$
$308$	0	0
$309$	23.9443	1.36214
$310$	−4.27051	−0.242549
$311$	−32.2361	−1.82794	−0.913970	−	0.405782i	$-0.866999\pi$
$311$	−32.2361	−1.82794	−0.913970	+	0.405782i	$0.866999\pi$
$312$	0	0
$313$	−32.4164	−1.83228	−0.916142	−	0.400854i	$-0.868713\pi$
$313$	−32.4164	−1.83228	−0.916142	+	0.400854i	$0.868713\pi$
$314$	2.67376	0.150889
$315$	0	0
$316$	−19.8541	−1.11688
$317$	3.76393	0.211403	0.105702	−	0.994398i	$-0.466291\pi$
$317$	3.76393	0.211403	0.105702	+	0.994398i	$0.466291\pi$
$318$	1.25735	0.0705089
$319$	−13.4164	−0.751175
$320$	−10.5279	−0.588525
$321$	−31.8328	−1.77673
$322$	0	0
$323$	22.4164	1.24728
$324$	20.3951	1.13306
$325$	0	0
$326$	−3.97871	−0.220361
$327$	−23.9443	−1.32412
$328$	6.58359	0.363518
$329$	0	0
$330$	5.72949	0.315398
$331$	28.4164	1.56191	0.780954	−	0.624589i	$-0.214734\pi$
$331$	28.4164	1.56191	0.780954	+	0.624589i	$0.214734\pi$
$332$	0	0
$333$	17.4164	0.954413
$334$	−5.16718	−0.282736
$335$	6.70820	0.366508
$336$	0	0
$337$	18.0000	0.980522	0.490261	−	0.871576i	$-0.336901\pi$
$337$	18.0000	0.980522	0.490261	+	0.871576i	$0.336901\pi$
$338$	0	0
$339$	−33.4164	−1.81493
$340$	30.9787	1.68006
$341$	−15.0000	−0.812296
$342$	−2.29180	−0.123926
$343$	0	0
$344$	11.7771	0.634978
$345$	−18.8197	−1.01322
$346$	3.97871	0.213897
$347$	−35.0689	−1.88260	−0.941298	−	0.337576i	$-0.890393\pi$
$347$	−35.0689	−1.88260	−0.941298	+	0.337576i	$0.890393\pi$
$348$	18.5410	0.993903
$349$	2.58359	0.138297	0.0691483	−	0.997606i	$-0.477972\pi$
$349$	2.58359	0.138297	0.0691483	+	0.997606i	$0.477972\pi$
$350$	0	0
$351$	0	0
$352$	12.4377	0.662931
$353$	−30.7082	−1.63443	−0.817216	−	0.576331i	$-0.804484\pi$
$353$	−30.7082	−1.63443	−0.817216	+	0.576331i	$0.804484\pi$
$354$	−6.38197	−0.339198
$355$	−20.0000	−1.06149
$356$	−4.14590	−0.219732
$357$	0	0
$358$	7.68692	0.406266
$359$	5.94427	0.313727	0.156863	−	0.987620i	$-0.449862\pi$
$359$	5.94427	0.313727	0.156863	+	0.987620i	$0.449862\pi$
$360$	−6.58359	−0.346986
$361$	−10.0000	−0.526316
$362$	0.541020	0.0284354
$363$	−4.47214	−0.234726
$364$	0	0
$365$	−23.9443	−1.25330
$366$	2.56231	0.133934
$367$	0.708204	0.0369679	0.0184840	−	0.999829i	$-0.494116\pi$
$367$	0.708204	0.0369679	0.0184840	+	0.999829i	$0.494116\pi$
$368$	−11.8409	−0.617252
$369$	−8.94427	−0.465620
$370$	7.43769	0.386667
$371$	0	0
$372$	20.7295	1.07477
$373$	−28.4164	−1.47135	−0.735673	−	0.677337i	$-0.763134\pi$
$373$	−28.4164	−1.47135	−0.735673	+	0.677337i	$0.763134\pi$
$374$	−8.56231	−0.442746
$375$	−25.0000	−1.29099
$376$	2.16718	0.111764
$377$	0	0
$378$	0	0
$379$	−11.4164	−0.586421	−0.293211	−	0.956048i	$-0.594724\pi$
$379$	−11.4164	−0.586421	−0.293211	+	0.956048i	$0.594724\pi$
$380$	12.4377	0.638040
$381$	34.4721	1.76606
$382$	−4.27051	−0.218498
$383$	−15.0000	−0.766464	−0.383232	−	0.923652i	$-0.625189\pi$
$383$	−15.0000	−0.766464	−0.383232	+	0.923652i	$0.625189\pi$
$384$	−22.5623	−1.15138
$385$	0	0
$386$	4.85410	0.247067
$387$	−16.0000	−0.813326
$388$	−32.2918	−1.63937
$389$	−7.47214	−0.378852	−0.189426	−	0.981895i	$-0.560663\pi$
$389$	−7.47214	−0.378852	−0.189426	+	0.981895i	$0.560663\pi$
$390$	0	0
$391$	28.1246	1.42232
$392$	0	0
$393$	−8.41641	−0.424552
$394$	10.2918	0.518493
$395$	23.9443	1.20477
$396$	−11.1246	−0.559033
$397$	−14.1246	−0.708894	−0.354447	−	0.935076i	$-0.615331\pi$
$397$	−14.1246	−0.708894	−0.354447	+	0.935076i	$0.615331\pi$
$398$	−2.78522	−0.139610
$399$	0	0
$400$	0	0
$401$	−9.76393	−0.487587	−0.243794	−	0.969827i	$-0.578392\pi$
$401$	−9.76393	−0.487587	−0.243794	+	0.969827i	$0.578392\pi$
$402$	2.56231	0.127796
$403$	0	0
$404$	16.6869	0.830205
$405$	−24.5967	−1.22222
$406$	0	0
$407$	26.1246	1.29495
$408$	24.5967	1.21772
$409$	4.70820	0.232806	0.116403	−	0.993202i	$-0.462864\pi$
$409$	4.70820	0.232806	0.116403	+	0.993202i	$0.462864\pi$
$410$	−3.81966	−0.188640
$411$	8.41641	0.415151
$412$	−19.8541	−0.978141
$413$	0	0
$414$	−2.87539	−0.141318
$415$	0	0
$416$	0	0
$417$	7.63932	0.374099
$418$	−3.43769	−0.168143
$419$	15.0557	0.735520	0.367760	−	0.929921i	$-0.380125\pi$
$419$	15.0557	0.735520	0.367760	+	0.929921i	$0.380125\pi$
$420$	0	0
$421$	13.4164	0.653876	0.326938	−	0.945046i	$-0.393983\pi$
$421$	13.4164	0.653876	0.326938	+	0.945046i	$0.393983\pi$
$422$	1.52786	0.0743753
$423$	−2.94427	−0.143155
$424$	−2.16718	−0.105248
$425$	0	0
$426$	−7.63932	−0.370126
$427$	0	0
$428$	26.3951	1.27586
$429$	0	0
$430$	−6.83282	−0.329508
$431$	−13.3607	−0.643561	−0.321781	−	0.946814i	$-0.604281\pi$
$431$	−13.3607	−0.643561	−0.321781	+	0.946814i	$0.604281\pi$
$432$	−7.03444	−0.338445
$433$	2.58359	0.124160	0.0620798	−	0.998071i	$-0.480227\pi$
$433$	2.58359	0.124160	0.0620798	+	0.998071i	$0.480227\pi$
$434$	0	0
$435$	−22.3607	−1.07211
$436$	19.8541	0.950839
$437$	11.2918	0.540160
$438$	−9.14590	−0.437008
$439$	−16.1246	−0.769586	−0.384793	−	0.923003i	$-0.625727\pi$
$439$	−16.1246	−0.769586	−0.384793	+	0.923003i	$0.625727\pi$
$440$	−9.87539	−0.470791
$441$	0	0
$442$	0	0
$443$	−2.23607	−0.106239	−0.0531194	−	0.998588i	$-0.516916\pi$
$443$	−2.23607	−0.106239	−0.0531194	+	0.998588i	$0.516916\pi$
$444$	−36.1033	−1.71339
$445$	5.00000	0.237023
$446$	1.52786	0.0723465
$447$	28.4164	1.34405
$448$	0	0
$449$	−10.3607	−0.488951	−0.244475	−	0.969656i	$-0.578616\pi$
$449$	−10.3607	−0.488951	−0.244475	+	0.969656i	$0.578616\pi$
$450$	0	0
$451$	−13.4164	−0.631754
$452$	27.7082	1.30328
$453$	14.3475	0.674105
$454$	4.56231	0.214120
$455$	0	0
$456$	9.87539	0.462457
$457$	−34.1246	−1.59628	−0.798141	−	0.602471i	$-0.794183\pi$
$457$	−34.1246	−1.59628	−0.798141	+	0.602471i	$0.794183\pi$
$458$	6.15905	0.287794
$459$	16.7082	0.779872
$460$	15.6049	0.727581
$461$	10.3607	0.482545	0.241272	−	0.970457i	$-0.422435\pi$
$461$	10.3607	0.482545	0.241272	+	0.970457i	$0.422435\pi$
$462$	0	0
$463$	24.0000	1.11537	0.557687	−	0.830051i	$-0.311689\pi$
$463$	24.0000	1.11537	0.557687	+	0.830051i	$0.311689\pi$
$464$	−14.0689	−0.653132
$465$	−25.0000	−1.15935
$466$	−2.27051	−0.105179
$467$	−21.6525	−1.00196	−0.500979	−	0.865460i	$-0.667026\pi$
$467$	−21.6525	−1.00196	−0.500979	+	0.865460i	$0.667026\pi$
$468$	0	0
$469$	0	0
$470$	−1.25735	−0.0579974
$471$	15.6525	0.721228
$472$	11.0000	0.506316
$473$	−24.0000	−1.10352
$474$	9.14590	0.420085
$475$	0	0
$476$	0	0
$477$	2.94427	0.134809
$478$	2.83282	0.129570
$479$	29.8328	1.36310	0.681548	−	0.731773i	$-0.261307\pi$
$479$	29.8328	1.36310	0.681548	+	0.731773i	$0.261307\pi$
$480$	20.7295	0.946167
$481$	0	0
$482$	3.32624	0.151506
$483$	0	0
$484$	3.70820	0.168555
$485$	38.9443	1.76837
$486$	−6.83282	−0.309943
$487$	−31.8328	−1.44248	−0.721241	−	0.692684i	$-0.756428\pi$
$487$	−31.8328	−1.44248	−0.721241	+	0.692684i	$0.756428\pi$
$488$	−4.41641	−0.199921
$489$	−23.2918	−1.05329
$490$	0	0
$491$	−34.4721	−1.55571	−0.777853	−	0.628446i	$-0.783691\pi$
$491$	−34.4721	−1.55571	−0.777853	+	0.628446i	$0.783691\pi$
$492$	18.5410	0.835894
$493$	33.4164	1.50500
$494$	0	0
$495$	13.4164	0.603023
$496$	−15.7295	−0.706275
$497$	0	0
$498$	0	0
$499$	0.416408	0.0186410	0.00932049	−	0.999957i	$-0.497033\pi$
$499$	0.416408	0.0186410	0.00932049	+	0.999957i	$0.497033\pi$
$500$	20.7295	0.927051
$501$	−30.2492	−1.35144
$502$	4.00000	0.178529
$503$	3.05573	0.136248	0.0681241	−	0.997677i	$-0.478299\pi$
$503$	3.05573	0.136248	0.0681241	+	0.997677i	$0.478299\pi$
$504$	0	0
$505$	−20.1246	−0.895533
$506$	−4.31308	−0.191740
$507$	0	0
$508$	−28.5836	−1.26819
$509$	15.7639	0.698724	0.349362	−	0.936988i	$-0.386398\pi$
$509$	15.7639	0.698724	0.349362	+	0.936988i	$0.386398\pi$
$510$	−14.2705	−0.631909
$511$	0	0
$512$	22.3050	0.985749
$513$	6.70820	0.296174
$514$	−6.85410	−0.302321
$515$	23.9443	1.05511
$516$	33.1672	1.46010
$517$	−4.41641	−0.194233
$518$	0	0
$519$	23.2918	1.02240
$520$	0	0
$521$	−0.0557281	−0.00244149	−0.00122075	−	0.999999i	$-0.500389\pi$
$521$	−0.0557281	−0.00244149	−0.00122075	+	0.999999i	$0.500389\pi$
$522$	−3.41641	−0.149532
$523$	19.2918	0.843571	0.421786	−	0.906696i	$-0.361403\pi$
$523$	19.2918	0.843571	0.421786	+	0.906696i	$0.361403\pi$
$524$	6.97871	0.304867
$525$	0	0
$526$	−5.39512	−0.235238
$527$	37.3607	1.62746
$528$	21.1033	0.918404
$529$	−8.83282	−0.384035
$530$	1.25735	0.0546160
$531$	−14.9443	−0.648526
$532$	0	0
$533$	0	0
$534$	1.90983	0.0826464
$535$	−31.8328	−1.37625
$536$	−4.41641	−0.190760
$537$	45.0000	1.94189
$538$	1.72949	0.0745636
$539$	0	0
$540$	9.27051	0.398939
$541$	−14.7082	−0.632355	−0.316178	−	0.948700i	$-0.602400\pi$
$541$	−14.7082	−0.632355	−0.316178	+	0.948700i	$0.602400\pi$
$542$	−2.45085	−0.105273
$543$	3.16718	0.135917
$544$	−30.9787	−1.32820
$545$	−23.9443	−1.02566
$546$	0	0
$547$	−31.4164	−1.34327	−0.671634	−	0.740883i	$-0.734407\pi$
$547$	−31.4164	−1.34327	−0.671634	+	0.740883i	$0.734407\pi$
$548$	−6.97871	−0.298116
$549$	6.00000	0.256074
$550$	0	0
$551$	13.4164	0.571558
$552$	12.3901	0.527358
$553$	0	0
$554$	10.0902	0.428690
$555$	43.5410	1.84821
$556$	−6.33437	−0.268637
$557$	5.29180	0.224221	0.112110	−	0.993696i	$-0.464239\pi$
$557$	5.29180	0.224221	0.112110	+	0.993696i	$0.464239\pi$
$558$	−3.81966	−0.161699
$559$	0	0
$560$	0	0
$561$	−50.1246	−2.11626
$562$	−3.45898	−0.145908
$563$	−36.5967	−1.54237	−0.771185	−	0.636612i	$-0.780335\pi$
$563$	−36.5967	−1.54237	−0.771185	+	0.636612i	$0.780335\pi$
$564$	6.10333	0.256996
$565$	−33.4164	−1.40584
$566$	−5.39512	−0.226774
$567$	0	0
$568$	13.1672	0.552483
$569$	16.5279	0.692884	0.346442	−	0.938071i	$-0.387390\pi$
$569$	16.5279	0.692884	0.346442	+	0.938071i	$0.387390\pi$
$570$	−5.72949	−0.239982
$571$	4.12461	0.172610	0.0863048	−	0.996269i	$-0.472494\pi$
$571$	4.12461	0.172610	0.0863048	+	0.996269i	$0.472494\pi$
$572$	0	0
$573$	−25.0000	−1.04439
$574$	0	0
$575$	0	0
$576$	−9.41641	−0.392350
$577$	32.7082	1.36166	0.680830	−	0.732441i	$-0.261619\pi$
$577$	32.7082	1.36166	0.680830	+	0.732441i	$0.261619\pi$
$578$	14.8328	0.616964
$579$	28.4164	1.18095
$580$	18.5410	0.769874
$581$	0	0
$582$	14.8754	0.616605
$583$	4.41641	0.182909
$584$	15.7639	0.652316
$585$	0	0
$586$	−1.12461	−0.0464573
$587$	41.8885	1.72893	0.864463	−	0.502697i	$-0.167659\pi$
$587$	41.8885	1.72893	0.864463	+	0.502697i	$0.167659\pi$
$588$	0	0
$589$	15.0000	0.618064
$590$	−6.38197	−0.262741
$591$	60.2492	2.47832
$592$	27.3951	1.12593
$593$	−32.2361	−1.32378	−0.661888	−	0.749602i	$-0.730245\pi$
$593$	−32.2361	−1.32378	−0.661888	+	0.749602i	$0.730245\pi$
$594$	−2.56231	−0.105133
$595$	0	0
$596$	−23.5623	−0.965150
$597$	−16.3050	−0.667317
$598$	0	0
$599$	41.0689	1.67803	0.839015	−	0.544109i	$-0.183132\pi$
$599$	41.0689	1.67803	0.839015	+	0.544109i	$0.183132\pi$
$600$	0	0
$601$	−22.0000	−0.897399	−0.448699	−	0.893683i	$-0.648113\pi$
$601$	−22.0000	−0.897399	−0.448699	+	0.893683i	$0.648113\pi$
$602$	0	0
$603$	6.00000	0.244339
$604$	−11.8967	−0.484069
$605$	−4.47214	−0.181818
$606$	−7.68692	−0.312260
$607$	−16.1246	−0.654478	−0.327239	−	0.944942i	$-0.606118\pi$
$607$	−16.1246	−0.654478	−0.327239	+	0.944942i	$0.606118\pi$
$608$	−12.4377	−0.504415
$609$	0	0
$610$	2.56231	0.103745
$611$	0	0
$612$	27.7082	1.12004
$613$	−22.1246	−0.893605	−0.446802	−	0.894633i	$-0.647437\pi$
$613$	−22.1246	−0.893605	−0.446802	+	0.894633i	$0.647437\pi$
$614$	2.83282	0.114323
$615$	−22.3607	−0.901670
$616$	0	0
$617$	−4.47214	−0.180041	−0.0900207	−	0.995940i	$-0.528693\pi$
$617$	−4.47214	−0.180041	−0.0900207	+	0.995940i	$0.528693\pi$
$618$	9.14590	0.367902
$619$	−17.0000	−0.683288	−0.341644	−	0.939829i	$-0.610984\pi$
$619$	−17.0000	−0.683288	−0.341644	+	0.939829i	$0.610984\pi$
$620$	20.7295	0.832516
$621$	8.41641	0.337739
$622$	−12.3131	−0.493710
$623$	0	0
$624$	0	0
$625$	−25.0000	−1.00000
$626$	−12.3820	−0.494883
$627$	−20.1246	−0.803700
$628$	−12.9787	−0.517907
$629$	−65.0689	−2.59447
$630$	0	0
$631$	30.8328	1.22744	0.613718	−	0.789526i	$-0.289673\pi$
$631$	30.8328	1.22744	0.613718	+	0.789526i	$0.289673\pi$
$632$	−15.7639	−0.627056
$633$	8.94427	0.355503
$634$	1.43769	0.0570981
$635$	34.4721	1.36798
$636$	−6.10333	−0.242013
$637$	0	0
$638$	−5.12461	−0.202885
$639$	−17.8885	−0.707660
$640$	−22.5623	−0.891853
$641$	−5.94427	−0.234785	−0.117392	−	0.993086i	$-0.537453\pi$
$641$	−5.94427	−0.234785	−0.117392	+	0.993086i	$0.537453\pi$
$642$	−12.1591	−0.479880
$643$	18.8328	0.742694	0.371347	−	0.928494i	$-0.378896\pi$
$643$	18.8328	0.742694	0.371347	+	0.928494i	$0.378896\pi$
$644$	0	0
$645$	−40.0000	−1.57500
$646$	8.56231	0.336879
$647$	15.7639	0.619744	0.309872	−	0.950778i	$-0.399714\pi$
$647$	15.7639	0.619744	0.309872	+	0.950778i	$0.399714\pi$
$648$	16.1935	0.636141
$649$	−22.4164	−0.879921
$650$	0	0
$651$	0	0
$652$	19.3131	0.756359
$653$	13.4721	0.527205	0.263603	−	0.964631i	$-0.415089\pi$
$653$	13.4721	0.527205	0.263603	+	0.964631i	$0.415089\pi$
$654$	−9.14590	−0.357633
$655$	−8.41641	−0.328856
$656$	−14.0689	−0.549298
$657$	−21.4164	−0.835534
$658$	0	0
$659$	8.94427	0.348419	0.174210	−	0.984709i	$-0.444263\pi$
$659$	8.94427	0.348419	0.174210	+	0.984709i	$0.444263\pi$
$660$	−27.8115	−1.08256
$661$	−6.70820	−0.260919	−0.130459	−	0.991454i	$-0.541645\pi$
$661$	−6.70820	−0.260919	−0.130459	+	0.991454i	$0.541645\pi$
$662$	10.8541	0.421857
$663$	0	0
$664$	0	0
$665$	0	0
$666$	6.65248	0.257778
$667$	16.8328	0.651769
$668$	25.0820	0.970453
$669$	8.94427	0.345806
$670$	2.56231	0.0989905
$671$	9.00000	0.347441
$672$	0	0
$673$	17.4164	0.671353	0.335677	−	0.941977i	$-0.391035\pi$
$673$	17.4164	0.671353	0.335677	+	0.941977i	$0.391035\pi$
$674$	6.87539	0.264830
$675$	0	0
$676$	0	0
$677$	32.8885	1.26401	0.632005	−	0.774965i	$-0.282232\pi$
$677$	32.8885	1.26401	0.632005	+	0.774965i	$0.282232\pi$
$678$	−12.7639	−0.490196
$679$	0	0
$680$	24.5967	0.943242
$681$	26.7082	1.02346
$682$	−5.72949	−0.219394
$683$	4.52786	0.173254	0.0866270	−	0.996241i	$-0.472391\pi$
$683$	4.52786	0.173254	0.0866270	+	0.996241i	$0.472391\pi$
$684$	11.1246	0.425360
$685$	8.41641	0.321574
$686$	0	0
$687$	36.0557	1.37561
$688$	−25.1672	−0.959490
$689$	0	0
$690$	−7.18847	−0.273660
$691$	−1.83282	−0.0697236	−0.0348618	−	0.999392i	$-0.511099\pi$
$691$	−1.83282	−0.0697236	−0.0348618	+	0.999392i	$0.511099\pi$
$692$	−19.3131	−0.734173
$693$	0	0
$694$	−13.3951	−0.508472
$695$	7.63932	0.289776
$696$	14.7214	0.558012
$697$	33.4164	1.26574
$698$	0.986844	0.0373526
$699$	−13.2918	−0.502742
$700$	0	0
$701$	22.3607	0.844551	0.422276	−	0.906467i	$-0.361231\pi$
$701$	22.3607	0.844551	0.422276	+	0.906467i	$0.361231\pi$
$702$	0	0
$703$	−26.1246	−0.985308
$704$	−14.1246	−0.532341
$705$	−7.36068	−0.277219
$706$	−11.7295	−0.441445
$707$	0	0
$708$	30.9787	1.16425
$709$	9.87539	0.370878	0.185439	−	0.982656i	$-0.440629\pi$
$709$	9.87539	0.370878	0.185439	+	0.982656i	$0.440629\pi$
$710$	−7.63932	−0.286699
$711$	21.4164	0.803178
$712$	−3.29180	−0.123365
$713$	18.8197	0.704802
$714$	0	0
$715$	0	0
$716$	−37.3131	−1.39446
$717$	16.5836	0.619326
$718$	2.27051	0.0847347
$719$	11.2918	0.421113	0.210556	−	0.977582i	$-0.432472\pi$
$719$	11.2918	0.421113	0.210556	+	0.977582i	$0.432472\pi$
$720$	14.0689	0.524316
$721$	0	0
$722$	−3.81966	−0.142153
$723$	19.4721	0.724177
$724$	−2.62616	−0.0976006
$725$	0	0
$726$	−1.70820	−0.0633974
$727$	14.8328	0.550119	0.275059	−	0.961427i	$-0.411302\pi$
$727$	14.8328	0.550119	0.275059	+	0.961427i	$0.411302\pi$
$728$	0	0
$729$	−7.00000	−0.259259
$730$	−9.14590	−0.338505
$731$	59.7771	2.21094
$732$	−12.4377	−0.459710
$733$	−15.2918	−0.564815	−0.282408	−	0.959294i	$-0.591133\pi$
$733$	−15.2918	−0.564815	−0.282408	+	0.959294i	$0.591133\pi$
$734$	0.270510	0.00998470
$735$	0	0
$736$	−15.6049	−0.575203
$737$	9.00000	0.331519
$738$	−3.41641	−0.125760
$739$	−35.8328	−1.31813	−0.659066	−	0.752085i	$-0.729048\pi$
$739$	−35.8328	−1.31813	−0.659066	+	0.752085i	$0.729048\pi$
$740$	−36.1033	−1.32718
$741$	0	0
$742$	0	0
$743$	−15.0557	−0.552341	−0.276171	−	0.961109i	$-0.589065\pi$
$743$	−15.0557	−0.552341	−0.276171	+	0.961109i	$0.589065\pi$
$744$	16.4590	0.603415
$745$	28.4164	1.04110
$746$	−10.8541	−0.397397
$747$	0	0
$748$	41.5623	1.51967
$749$	0	0
$750$	−9.54915	−0.348686
$751$	30.1246	1.09926	0.549631	−	0.835407i	$-0.314768\pi$
$751$	30.1246	1.09926	0.549631	+	0.835407i	$0.314768\pi$
$752$	−4.63119	−0.168882
$753$	23.4164	0.853341
$754$	0	0
$755$	14.3475	0.522160
$756$	0	0
$757$	0.832816	0.0302692	0.0151346	−	0.999885i	$-0.495182\pi$
$757$	0.832816	0.0302692	0.0151346	+	0.999885i	$0.495182\pi$
$758$	−4.36068	−0.158387
$759$	−25.2492	−0.916489
$760$	9.87539	0.358218
$761$	−33.5410	−1.21586	−0.607931	−	0.793990i	$-0.708000\pi$
$761$	−33.5410	−1.21586	−0.607931	+	0.793990i	$0.708000\pi$
$762$	13.1672	0.476997
$763$	0	0
$764$	20.7295	0.749967
$765$	−33.4164	−1.20817
$766$	−5.72949	−0.207015
$767$	0	0
$768$	12.4377	0.448807
$769$	−46.0000	−1.65880	−0.829401	−	0.558653i	$-0.811318\pi$
$769$	−46.0000	−1.65880	−0.829401	+	0.558653i	$0.811318\pi$
$770$	0	0
$771$	−40.1246	−1.44505
$772$	−23.5623	−0.848026
$773$	−11.0689	−0.398120	−0.199060	−	0.979987i	$-0.563789\pi$
$773$	−11.0689	−0.398120	−0.199060	+	0.979987i	$0.563789\pi$
$774$	−6.11146	−0.219672
$775$	0	0
$776$	−25.6393	−0.920398
$777$	0	0
$778$	−2.85410	−0.102325
$779$	13.4164	0.480693
$780$	0	0
$781$	−26.8328	−0.960154
$782$	10.7426	0.384156
$783$	10.0000	0.357371
$784$	0	0
$785$	15.6525	0.558661
$786$	−3.21478	−0.114667
$787$	6.41641	0.228720	0.114360	−	0.993439i	$-0.463518\pi$
$787$	6.41641	0.228720	0.114360	+	0.993439i	$0.463518\pi$
$788$	−49.9574	−1.77966
$789$	−31.5836	−1.12441
$790$	9.14590	0.325396
$791$	0	0
$792$	−8.83282	−0.313860
$793$	0	0
$794$	−5.39512	−0.191466
$795$	7.36068	0.261056
$796$	13.5197	0.479194
$797$	26.9443	0.954415	0.477208	−	0.878791i	$-0.341649\pi$
$797$	26.9443	0.954415	0.477208	+	0.878791i	$0.341649\pi$
$798$	0	0
$799$	11.0000	0.389152
$800$	0	0
$801$	4.47214	0.158015
$802$	−3.72949	−0.131693
$803$	−32.1246	−1.13365
$804$	−12.4377	−0.438644
$805$	0	0
$806$	0	0
$807$	10.1246	0.356403
$808$	13.2492	0.466106
$809$	−4.41641	−0.155273	−0.0776363	−	0.996982i	$-0.524737\pi$
$809$	−4.41641	−0.155273	−0.0776363	+	0.996982i	$0.524737\pi$
$810$	−9.39512	−0.330111
$811$	−38.8328	−1.36360	−0.681802	−	0.731536i	$-0.738804\pi$
$811$	−38.8328	−1.36360	−0.681802	+	0.731536i	$0.738804\pi$
$812$	0	0
$813$	−14.3475	−0.503190
$814$	9.97871	0.349754
$815$	−23.2918	−0.815876
$816$	−52.5623	−1.84005
$817$	24.0000	0.839654
$818$	1.79837	0.0628787
$819$	0	0
$820$	18.5410	0.647480
$821$	−33.7639	−1.17837	−0.589185	−	0.807998i	$-0.700551\pi$
$821$	−33.7639	−1.17837	−0.589185	+	0.807998i	$0.700551\pi$
$822$	3.21478	0.112128
$823$	−6.12461	−0.213491	−0.106745	−	0.994286i	$-0.534043\pi$
$823$	−6.12461	−0.213491	−0.106745	+	0.994286i	$0.534043\pi$
$824$	−15.7639	−0.549163
$825$	0	0
$826$	0	0
$827$	26.8328	0.933068	0.466534	−	0.884503i	$-0.345502\pi$
$827$	26.8328	0.933068	0.466534	+	0.884503i	$0.345502\pi$
$828$	13.9574	0.485054
$829$	−25.0000	−0.868286	−0.434143	−	0.900844i	$-0.642949\pi$
$829$	−25.0000	−0.868286	−0.434143	+	0.900844i	$0.642949\pi$
$830$	0	0
$831$	59.0689	2.04908
$832$	0	0
$833$	0	0
$834$	2.91796	0.101041
$835$	−30.2492	−1.04682
$836$	16.6869	0.577129
$837$	11.1803	0.386449
$838$	5.75078	0.198657
$839$	−29.8885	−1.03187	−0.515934	−	0.856629i	$-0.672555\pi$
$839$	−29.8885	−1.03187	−0.515934	+	0.856629i	$0.672555\pi$
$840$	0	0
$841$	−9.00000	−0.310345
$842$	5.12461	0.176606
$843$	−20.2492	−0.697420
$844$	−7.41641	−0.255283
$845$	0	0
$846$	−1.12461	−0.0386650
$847$	0	0
$848$	4.63119	0.159036
$849$	−31.5836	−1.08395
$850$	0	0
$851$	−32.7771	−1.12358
$852$	37.0820	1.27041
$853$	−28.2492	−0.967235	−0.483617	−	0.875279i	$-0.660677\pi$
$853$	−28.2492	−0.967235	−0.483617	+	0.875279i	$0.660677\pi$
$854$	0	0
$855$	−13.4164	−0.458831
$856$	20.9574	0.716310
$857$	43.3607	1.48117	0.740586	−	0.671961i	$-0.234548\pi$
$857$	43.3607	1.48117	0.740586	+	0.671961i	$0.234548\pi$
$858$	0	0
$859$	7.29180	0.248793	0.124396	−	0.992233i	$-0.460301\pi$
$859$	7.29180	0.248793	0.124396	+	0.992233i	$0.460301\pi$
$860$	33.1672	1.13099
$861$	0	0
$862$	−5.10333	−0.173820
$863$	−6.05573	−0.206139	−0.103070	−	0.994674i	$-0.532866\pi$
$863$	−6.05573	−0.206139	−0.103070	+	0.994674i	$0.532866\pi$
$864$	−9.27051	−0.315389
$865$	23.2918	0.791945
$866$	0.986844	0.0335343
$867$	86.8328	2.94900
$868$	0	0
$869$	32.1246	1.08975
$870$	−8.54102	−0.289568
$871$	0	0
$872$	15.7639	0.533834
$873$	34.8328	1.17891
$874$	4.31308	0.145892
$875$	0	0
$876$	44.3951	1.49997
$877$	8.12461	0.274349	0.137174	−	0.990547i	$-0.456198\pi$
$877$	8.12461	0.274349	0.137174	+	0.990547i	$0.456198\pi$
$878$	−6.15905	−0.207858
$879$	−6.58359	−0.222059
$880$	21.1033	0.711393
$881$	−19.3050	−0.650400	−0.325200	−	0.945645i	$-0.605432\pi$
$881$	−19.3050	−0.650400	−0.325200	+	0.945645i	$0.605432\pi$
$882$	0	0
$883$	20.0000	0.673054	0.336527	−	0.941674i	$-0.390748\pi$
$883$	20.0000	0.673054	0.336527	+	0.941674i	$0.390748\pi$
$884$	0	0
$885$	−37.3607	−1.25587
$886$	−0.854102	−0.0286941
$887$	−15.7639	−0.529301	−0.264651	−	0.964344i	$-0.585257\pi$
$887$	−15.7639	−0.529301	−0.264651	+	0.964344i	$0.585257\pi$
$888$	−28.6656	−0.961956
$889$	0	0
$890$	1.90983	0.0640176
$891$	−33.0000	−1.10554
$892$	−7.41641	−0.248320
$893$	4.41641	0.147789
$894$	10.8541	0.363015
$895$	45.0000	1.50418
$896$	0	0
$897$	0	0
$898$	−3.95743	−0.132061
$899$	22.3607	0.745770
$900$	0	0
$901$	−11.0000	−0.366463
$902$	−5.12461	−0.170631
$903$	0	0
$904$	22.0000	0.731709
$905$	3.16718	0.105281
$906$	5.48027	0.182070
$907$	5.29180	0.175711	0.0878556	−	0.996133i	$-0.471999\pi$
$907$	5.29180	0.175711	0.0878556	+	0.996133i	$0.471999\pi$
$908$	−22.1459	−0.734937
$909$	−18.0000	−0.597022
$910$	0	0
$911$	46.2492	1.53231	0.766153	−	0.642659i	$-0.222169\pi$
$911$	46.2492	1.53231	0.766153	+	0.642659i	$0.222169\pi$
$912$	−21.1033	−0.698801
$913$	0	0
$914$	−13.0344	−0.431141
$915$	15.0000	0.495885
$916$	−29.8967	−0.987814
$917$	0	0
$918$	6.38197	0.210636
$919$	−20.1246	−0.663850	−0.331925	−	0.943306i	$-0.607698\pi$
$919$	−20.1246	−0.663850	−0.331925	+	0.943306i	$0.607698\pi$
$920$	12.3901	0.408489
$921$	16.5836	0.546448
$922$	3.95743	0.130331
$923$	0	0
$924$	0	0
$925$	0	0
$926$	9.16718	0.301252
$927$	21.4164	0.703407
$928$	−18.5410	−0.608639
$929$	47.1803	1.54794	0.773968	−	0.633224i	$-0.218269\pi$
$929$	47.1803	1.54794	0.773968	+	0.633224i	$0.218269\pi$
$930$	−9.54915	−0.313129
$931$	0	0
$932$	11.0213	0.361014
$933$	−72.0820	−2.35986
$934$	−8.27051	−0.270619
$935$	−50.1246	−1.63925
$936$	0	0
$937$	−1.41641	−0.0462720	−0.0231360	−	0.999732i	$-0.507365\pi$
$937$	−1.41641	−0.0462720	−0.0231360	+	0.999732i	$0.507365\pi$
$938$	0	0
$939$	−72.4853	−2.36547
$940$	6.10333	0.199069
$941$	−45.7639	−1.49186	−0.745931	−	0.666023i	$-0.767995\pi$
$941$	−45.7639	−1.49186	−0.745931	+	0.666023i	$0.767995\pi$
$942$	5.97871	0.194797
$943$	16.8328	0.548152
$944$	−23.5066	−0.765074
$945$	0	0
$946$	−9.16718	−0.298051
$947$	31.4721	1.02271	0.511354	−	0.859370i	$-0.329144\pi$
$947$	31.4721	1.02271	0.511354	+	0.859370i	$0.329144\pi$
$948$	−44.3951	−1.44189
$949$	0	0
$950$	0	0
$951$	8.41641	0.272921
$952$	0	0
$953$	29.7771	0.964574	0.482287	−	0.876013i	$-0.339806\pi$
$953$	29.7771	0.964574	0.482287	+	0.876013i	$0.339806\pi$
$954$	1.12461	0.0364107
$955$	−25.0000	−0.808981
$956$	−13.7508	−0.444732
$957$	−30.0000	−0.969762
$958$	11.3951	0.368160
$959$	0	0
$960$	−23.5410	−0.759783
$961$	−6.00000	−0.193548
$962$	0	0
$963$	−28.4721	−0.917502
$964$	−16.1459	−0.520024
$965$	28.4164	0.914757
$966$	0	0
$967$	−43.4164	−1.39618	−0.698089	−	0.716011i	$-0.745966\pi$
$967$	−43.4164	−1.39618	−0.698089	+	0.716011i	$0.745966\pi$
$968$	2.94427	0.0946325
$969$	50.1246	1.61023
$970$	14.8754	0.477620
$971$	−24.7082	−0.792924	−0.396462	−	0.918051i	$-0.629762\pi$
$971$	−24.7082	−0.792924	−0.396462	+	0.918051i	$0.629762\pi$
$972$	33.1672	1.06384
$973$	0	0
$974$	−12.1591	−0.389601
$975$	0	0
$976$	9.43769	0.302093
$977$	33.6525	1.07664	0.538319	−	0.842741i	$-0.319060\pi$
$977$	33.6525	1.07664	0.538319	+	0.842741i	$0.319060\pi$
$978$	−8.89667	−0.284484
$979$	6.70820	0.214395
$980$	0	0
$981$	−21.4164	−0.683773
$982$	−13.1672	−0.420182
$983$	1.47214	0.0469538	0.0234769	−	0.999724i	$-0.492526\pi$
$983$	1.47214	0.0469538	0.0234769	+	0.999724i	$0.492526\pi$
$984$	14.7214	0.469300
$985$	60.2492	1.91970
$986$	12.7639	0.406486
$987$	0	0
$988$	0	0
$989$	30.1115	0.957489
$990$	5.12461	0.162871
$991$	17.2918	0.549292	0.274646	−	0.961545i	$-0.411439\pi$
$991$	17.2918	0.549292	0.274646	+	0.961545i	$0.411439\pi$
$992$	−20.7295	−0.658162
$993$	63.5410	2.01641
$994$	0	0
$995$	−16.3050	−0.516902
$996$	0	0
$997$	−0.416408	−0.0131878	−0.00659388	−	0.999978i	$-0.502099\pi$
$997$	−0.416408	−0.0131878	−0.00659388	+	0.999978i	$0.502099\pi$
$998$	0.159054	0.00503476
$999$	−19.4721	−0.616071

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8281.2.a.z.1.1		2
7.2	even	3		1183.2.e.d.508.2		4
7.4	even	3		1183.2.e.d.170.2		4
7.6	odd	2		8281.2.a.ba.1.1		2
13.12	even	2		637.2.a.f.1.2		2
39.38	odd	2		5733.2.a.v.1.1		2
91.12	odd	6		637.2.e.h.508.1		4
91.25	even	6		91.2.e.b.79.1	yes	4
91.38	odd	6		637.2.e.h.79.1		4
91.51	even	6		91.2.e.b.53.1	✓	4
91.90	odd	2		637.2.a.e.1.2		2
273.116	odd	6		819.2.j.c.352.2		4
273.233	odd	6		819.2.j.c.235.2		4
273.272	even	2		5733.2.a.w.1.1		2
364.51	odd	6		1456.2.r.j.417.2		4
364.207	odd	6		1456.2.r.j.625.2		4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
91.2.e.b.53.1	✓	4	91.51	even	6
91.2.e.b.79.1	yes	4	91.25	even	6
637.2.a.e.1.2		2	91.90	odd	2
637.2.a.f.1.2		2	13.12	even	2
637.2.e.h.79.1		4	91.38	odd	6
637.2.e.h.508.1		4	91.12	odd	6
819.2.j.c.235.2		4	273.233	odd	6
819.2.j.c.352.2		4	273.116	odd	6
1183.2.e.d.170.2		4	7.4	even	3
1183.2.e.d.508.2		4	7.2	even	3
1456.2.r.j.417.2		4	364.51	odd	6
1456.2.r.j.625.2		4	364.207	odd	6
5733.2.a.v.1.1		2	39.38	odd	2
5733.2.a.w.1.1		2	273.272	even	2
8281.2.a.z.1.1		2	1.1	even	1	trivial
8281.2.a.ba.1.1		2	7.6	odd	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:	\( N \)	\(=\)	\( 8281 = 7^{2} \cdot 13^{2} \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8281.a (trivial)

\(f(q)\)	\(=\)	\(q+0.381966 q^{2} +2.23607 q^{3} -1.85410 q^{4} +2.23607 q^{5} +0.854102 q^{6} -1.47214 q^{8} +2.00000 q^{9} +O(q^{10})\)	\(q+0.381966 q^{2} +2.23607 q^{3} -1.85410 q^{4} +2.23607 q^{5} +0.854102 q^{6} -1.47214 q^{8} +2.00000 q^{9} +0.854102 q^{10} +3.00000 q^{11} -4.14590 q^{12} +5.00000 q^{15} +3.14590 q^{16} -7.47214 q^{17} +0.763932 q^{18} -3.00000 q^{19} -4.14590 q^{20} +1.14590 q^{22} -3.76393 q^{23} -3.29180 q^{24} -2.23607 q^{27} -4.47214 q^{29} +1.90983 q^{30} -5.00000 q^{31} +4.14590 q^{32} +6.70820 q^{33} -2.85410 q^{34} -3.70820 q^{36} +8.70820 q^{37} -1.14590 q^{38} -3.29180 q^{40} -4.47214 q^{41} -8.00000 q^{43} -5.56231 q^{44} +4.47214 q^{45} -1.43769 q^{46} -1.47214 q^{47} +7.03444 q^{48} -16.7082 q^{51} +1.47214 q^{53} -0.854102 q^{54} +6.70820 q^{55} -6.70820 q^{57} -1.70820 q^{58} -7.47214 q^{59} -9.27051 q^{60} +3.00000 q^{61} -1.90983 q^{62} -4.70820 q^{64} +2.56231 q^{66} +3.00000 q^{67} +13.8541 q^{68} -8.41641 q^{69} -8.94427 q^{71} -2.94427 q^{72} -10.7082 q^{73} +3.32624 q^{74} +5.56231 q^{76} +10.7082 q^{79} +7.03444 q^{80} -11.0000 q^{81} -1.70820 q^{82} -16.7082 q^{85} -3.05573 q^{86} -10.0000 q^{87} -4.41641 q^{88} +2.23607 q^{89} +1.70820 q^{90} +6.97871 q^{92} -11.1803 q^{93} -0.562306 q^{94} -6.70820 q^{95} +9.27051 q^{96} +17.4164 q^{97} +6.00000 q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9}+O(q^{10}) \) 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 + 6 * q^8 + 4 * q^9	\( 2 q + 3 q^{2} + 3 q^{4} - 5 q^{6} + 6 q^{8} + 4 q^{9} - 5 q^{10} + 6 q^{11} - 15 q^{12} + 10 q^{15} + 13 q^{16} - 6 q^{17} + 6 q^{18} - 6 q^{19} - 15 q^{20} + 9 q^{22} - 12 q^{23} - 20 q^{24} + 15 q^{30} - 10 q^{31} + 15 q^{32} + q^{34} + 6 q^{36} + 4 q^{37} - 9 q^{38} - 20 q^{40} - 16 q^{43} + 9 q^{44} - 23 q^{46} + 6 q^{47} - 15 q^{48} - 20 q^{51} - 6 q^{53} + 5 q^{54} + 10 q^{58} - 6 q^{59} + 15 q^{60} + 6 q^{61} - 15 q^{62} + 4 q^{64} - 15 q^{66} + 6 q^{67} + 21 q^{68} + 10 q^{69} + 12 q^{72} - 8 q^{73} - 9 q^{74} - 9 q^{76} + 8 q^{79} - 15 q^{80} - 22 q^{81} + 10 q^{82} - 20 q^{85} - 24 q^{86} - 20 q^{87} + 18 q^{88} - 10 q^{90} - 33 q^{92} + 19 q^{94} - 15 q^{96} + 8 q^{97} + 12 q^{99}+O(q^{100}) \) 2 * q + 3 * q^2 + 3 * q^4 - 5 * q^6 + 6 * q^8 + 4 * q^9 - 5 * q^10 + 6 * q^11 - 15 * q^12 + 10 * q^15 + 13 * q^16 - 6 * q^17 + 6 * q^18 - 6 * q^19 - 15 * q^20 + 9 * q^22 - 12 * q^23 - 20 * q^24 + 15 * q^30 - 10 * q^31 + 15 * q^32 + q^34 + 6 * q^36 + 4 * q^37 - 9 * q^38 - 20 * q^40 - 16 * q^43 + 9 * q^44 - 23 * q^46 + 6 * q^47 - 15 * q^48 - 20 * q^51 - 6 * q^53 + 5 * q^54 + 10 * q^58 - 6 * q^59 + 15 * q^60 + 6 * q^61 - 15 * q^62 + 4 * q^64 - 15 * q^66 + 6 * q^67 + 21 * q^68 + 10 * q^69 + 12 * q^72 - 8 * q^73 - 9 * q^74 - 9 * q^76 + 8 * q^79 - 15 * q^80 - 22 * q^81 + 10 * q^82 - 20 * q^85 - 24 * q^86 - 20 * q^87 + 18 * q^88 - 10 * q^90 - 33 * q^92 + 19 * q^94 - 15 * q^96 + 8 * q^97 + 12 * q^99

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