LMFDB - Embedded newform 8046.2.a.h.1.6

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:	$9$
Coefficient field:	$\mathbb{Q}[x]/(x^{9} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{9} - 3x^{8} - 9x^{7} + 25x^{6} + 29x^{5} - 58x^{4} - 43x^{3} + 34x^{2} + 25x + 2 $ x^9 - 3x^8 - 9x^7 + 25x^6 + 29x^5 - 58x^4 - 43x^3 + 34x^2 + 25x + 2
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			1.6
Root			$1.67390$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} +0.102080 q^{5} +0.350570 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} +0.102080 q^{5} +0.350570 q^{7} +1.00000 q^{8} +0.102080 q^{10} +2.02675 q^{11} -0.0298835 q^{13} +0.350570 q^{14} +1.00000 q^{16} -3.57487 q^{17} -2.36593 q^{19} +0.102080 q^{20} +2.02675 q^{22} -3.59145 q^{23} -4.98958 q^{25} -0.0298835 q^{26} +0.350570 q^{28} -4.55651 q^{29} +0.839401 q^{31} +1.00000 q^{32} -3.57487 q^{34} +0.0357860 q^{35} -3.75798 q^{37} -2.36593 q^{38} +0.102080 q^{40} +1.43684 q^{41} -5.01492 q^{43} +2.02675 q^{44} -3.59145 q^{46} +6.28480 q^{47} -6.87710 q^{49} -4.98958 q^{50} -0.0298835 q^{52} -9.52893 q^{53} +0.206890 q^{55} +0.350570 q^{56} -4.55651 q^{58} -9.78689 q^{59} -7.40365 q^{61} +0.839401 q^{62} +1.00000 q^{64} -0.00305050 q^{65} -4.06800 q^{67} -3.57487 q^{68} +0.0357860 q^{70} +9.49501 q^{71} +7.96509 q^{73} -3.75798 q^{74} -2.36593 q^{76} +0.710518 q^{77} +9.24244 q^{79} +0.102080 q^{80} +1.43684 q^{82} -9.62022 q^{83} -0.364921 q^{85} -5.01492 q^{86} +2.02675 q^{88} -0.871304 q^{89} -0.0104763 q^{91} -3.59145 q^{92} +6.28480 q^{94} -0.241512 q^{95} -12.9918 q^{97} -6.87710 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) $ 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8	$ 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} - 4 q^{14} + 9 q^{16} - q^{17} - 10 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{29} - 17 q^{31} + 9 q^{32} - q^{34} - 10 q^{35} - 11 q^{37} - 10 q^{38} - 4 q^{40} - 16 q^{43} - 4 q^{44} - 8 q^{46} - 7 q^{47} - 5 q^{49} - 3 q^{50} - 8 q^{52} - 12 q^{53} - 23 q^{55} - 4 q^{56} - 4 q^{58} - 6 q^{59} - 13 q^{61} - 17 q^{62} + 9 q^{64} + 24 q^{65} - 14 q^{67} - q^{68} - 10 q^{70} - 30 q^{71} - 12 q^{73} - 11 q^{74} - 10 q^{76} - 12 q^{77} - 35 q^{79} - 4 q^{80} - 5 q^{83} - 27 q^{85} - 16 q^{86} - 4 q^{88} - 23 q^{89} - 28 q^{91} - 8 q^{92} - 7 q^{94} - 32 q^{95} - 21 q^{97} - 5 q^{98}+O(q^{100}) $ 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8 - 4 * q^10 - 4 * q^11 - 8 * q^13 - 4 * q^14 + 9 * q^16 - q^17 - 10 * q^19 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 3 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^29 - 17 * q^31 + 9 * q^32 - q^34 - 10 * q^35 - 11 * q^37 - 10 * q^38 - 4 * q^40 - 16 * q^43 - 4 * q^44 - 8 * q^46 - 7 * q^47 - 5 * q^49 - 3 * q^50 - 8 * q^52 - 12 * q^53 - 23 * q^55 - 4 * q^56 - 4 * q^58 - 6 * q^59 - 13 * q^61 - 17 * q^62 + 9 * q^64 + 24 * q^65 - 14 * q^67 - q^68 - 10 * q^70 - 30 * q^71 - 12 * q^73 - 11 * q^74 - 10 * q^76 - 12 * q^77 - 35 * q^79 - 4 * q^80 - 5 * q^83 - 27 * q^85 - 16 * q^86 - 4 * q^88 - 23 * q^89 - 28 * q^91 - 8 * q^92 - 7 * q^94 - 32 * q^95 - 21 * q^97 - 5 * 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
$5$	0.102080	0.0456513	0.0228257	−	0.999739i	$-0.492734\pi$
$5$	0.102080	0.0456513	0.0228257	+	0.999739i	$0.492734\pi$
$6$	0	0
$7$	0.350570	0.132503	0.0662515	−	0.997803i	$-0.478896\pi$
$7$	0.350570	0.132503	0.0662515	+	0.997803i	$0.478896\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	0.102080	0.0322804
$11$	2.02675	0.611089	0.305544	−	0.952178i	$-0.401162\pi$
$11$	2.02675	0.611089	0.305544	+	0.952178i	$0.401162\pi$
$12$	0	0
$13$	−0.0298835	−0.00828821	−0.00414410	−	0.999991i	$-0.501319\pi$
$13$	−0.0298835	−0.00828821	−0.00414410	+	0.999991i	$0.501319\pi$
$14$	0.350570	0.0936937
$15$	0	0
$16$	1.00000	0.250000
$17$	−3.57487	−0.867033	−0.433516	−	0.901146i	$-0.642727\pi$
$17$	−3.57487	−0.867033	−0.433516	+	0.901146i	$0.642727\pi$
$18$	0	0
$19$	−2.36593	−0.542780	−0.271390	−	0.962469i	$-0.587483\pi$
$19$	−2.36593	−0.542780	−0.271390	+	0.962469i	$0.587483\pi$
$20$	0.102080	0.0228257
$21$	0	0
$22$	2.02675	0.432105
$23$	−3.59145	−0.748869	−0.374435	−	0.927253i	$-0.622163\pi$
$23$	−3.59145	−0.748869	−0.374435	+	0.927253i	$0.622163\pi$
$24$	0	0
$25$	−4.98958	−0.997916
$26$	−0.0298835	−0.00586065
$27$	0	0
$28$	0.350570	0.0662515
$29$	−4.55651	−0.846122	−0.423061	−	0.906101i	$-0.639044\pi$
$29$	−4.55651	−0.846122	−0.423061	+	0.906101i	$0.639044\pi$
$30$	0	0
$31$	0.839401	0.150761	0.0753805	−	0.997155i	$-0.475983\pi$
$31$	0.839401	0.150761	0.0753805	+	0.997155i	$0.475983\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−3.57487	−0.613085
$35$	0.0357860	0.00604894
$36$	0	0
$37$	−3.75798	−0.617808	−0.308904	−	0.951093i	$-0.599962\pi$
$37$	−3.75798	−0.617808	−0.308904	+	0.951093i	$0.599962\pi$
$38$	−2.36593	−0.383804
$39$	0	0
$40$	0.102080	0.0161402
$41$	1.43684	0.224396	0.112198	−	0.993686i	$-0.464211\pi$
$41$	1.43684	0.224396	0.112198	+	0.993686i	$0.464211\pi$
$42$	0	0
$43$	−5.01492	−0.764768	−0.382384	−	0.924003i	$-0.624897\pi$
$43$	−5.01492	−0.764768	−0.382384	+	0.924003i	$0.624897\pi$
$44$	2.02675	0.305544
$45$	0	0
$46$	−3.59145	−0.529531
$47$	6.28480	0.916732	0.458366	−	0.888764i	$-0.348435\pi$
$47$	6.28480	0.916732	0.458366	+	0.888764i	$0.348435\pi$
$48$	0	0
$49$	−6.87710	−0.982443
$50$	−4.98958	−0.705633
$51$	0	0
$52$	−0.0298835	−0.00414410
$53$	−9.52893	−1.30890	−0.654449	−	0.756106i	$-0.727100\pi$
$53$	−9.52893	−1.30890	−0.654449	+	0.756106i	$0.727100\pi$
$54$	0	0
$55$	0.206890	0.0278970
$56$	0.350570	0.0468469
$57$	0	0
$58$	−4.55651	−0.598299
$59$	−9.78689	−1.27414	−0.637072	−	0.770804i	$-0.719855\pi$
$59$	−9.78689	−1.27414	−0.637072	+	0.770804i	$0.719855\pi$
$60$	0	0
$61$	−7.40365	−0.947941	−0.473970	−	0.880541i	$-0.657180\pi$
$61$	−7.40365	−0.947941	−0.473970	+	0.880541i	$0.657180\pi$
$62$	0.839401	0.106604
$63$	0	0
$64$	1.00000	0.125000
$65$	−0.00305050	−0.000378368 0
$66$	0	0
$67$	−4.06800	−0.496985	−0.248493	−	0.968634i	$-0.579935\pi$
$67$	−4.06800	−0.496985	−0.248493	+	0.968634i	$0.579935\pi$
$68$	−3.57487	−0.433516
$69$	0	0
$70$	0.0357860	0.00427724
$71$	9.49501	1.12685	0.563425	−	0.826167i	$-0.309483\pi$
$71$	9.49501	1.12685	0.563425	+	0.826167i	$0.309483\pi$
$72$	0	0
$73$	7.96509	0.932243	0.466122	−	0.884721i	$-0.345651\pi$
$73$	7.96509	0.932243	0.466122	+	0.884721i	$0.345651\pi$
$74$	−3.75798	−0.436856
$75$	0	0
$76$	−2.36593	−0.271390
$77$	0.710518	0.0809710
$78$	0	0
$79$	9.24244	1.03986	0.519928	−	0.854210i	$-0.325959\pi$
$79$	9.24244	1.03986	0.519928	+	0.854210i	$0.325959\pi$
$80$	0.102080	0.0114128
$81$	0	0
$82$	1.43684	0.158672
$83$	−9.62022	−1.05596	−0.527978	−	0.849258i	$-0.677050\pi$
$83$	−9.62022	−1.05596	−0.527978	+	0.849258i	$0.677050\pi$
$84$	0	0
$85$	−0.364921	−0.0395812
$86$	−5.01492	−0.540773
$87$	0	0
$88$	2.02675	0.216052
$89$	−0.871304	−0.0923580	−0.0461790	−	0.998933i	$-0.514704\pi$
$89$	−0.871304	−0.0923580	−0.0461790	+	0.998933i	$0.514704\pi$
$90$	0	0
$91$	−0.0104763	−0.00109821
$92$	−3.59145	−0.374435
$93$	0	0
$94$	6.28480	0.648227
$95$	−0.241512	−0.0247787
$96$	0	0
$97$	−12.9918	−1.31912	−0.659559	−	0.751653i	$-0.729257\pi$
$97$	−12.9918	−1.31912	−0.659559	+	0.751653i	$0.729257\pi$
$98$	−6.87710	−0.694692
$99$	0	0
$100$	−4.98958	−0.498958
$101$	6.60527	0.657249	0.328624	−	0.944461i	$-0.393415\pi$
$101$	6.60527	0.657249	0.328624	+	0.944461i	$0.393415\pi$
$102$	0	0
$103$	5.27651	0.519910	0.259955	−	0.965621i	$-0.416292\pi$
$103$	5.27651	0.519910	0.259955	+	0.965621i	$0.416292\pi$
$104$	−0.0298835	−0.00293032
$105$	0	0
$106$	−9.52893	−0.925531
$107$	8.42424	0.814402	0.407201	−	0.913339i	$-0.366505\pi$
$107$	8.42424	0.814402	0.407201	+	0.913339i	$0.366505\pi$
$108$	0	0
$109$	3.75345	0.359516	0.179758	−	0.983711i	$-0.442469\pi$
$109$	3.75345	0.359516	0.179758	+	0.983711i	$0.442469\pi$
$110$	0.206890	0.0197262
$111$	0	0
$112$	0.350570	0.0331257
$113$	4.46292	0.419836	0.209918	−	0.977719i	$-0.432680\pi$
$113$	4.46292	0.419836	0.209918	+	0.977719i	$0.432680\pi$
$114$	0	0
$115$	−0.366614	−0.0341869
$116$	−4.55651	−0.423061
$117$	0	0
$118$	−9.78689	−0.900956
$119$	−1.25324	−0.114884
$120$	0	0
$121$	−6.89228	−0.626571
$122$	−7.40365	−0.670295
$123$	0	0
$124$	0.839401	0.0753805
$125$	−1.01973	−0.0912075
$126$	0	0
$127$	−1.36720	−0.121320	−0.0606598	−	0.998158i	$-0.519320\pi$
$127$	−1.36720	−0.121320	−0.0606598	+	0.998158i	$0.519320\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−0.00305050	−0.000267546 0
$131$	−8.75704	−0.765106	−0.382553	−	0.923934i	$-0.624955\pi$
$131$	−8.75704	−0.765106	−0.382553	+	0.923934i	$0.624955\pi$
$132$	0	0
$133$	−0.829422	−0.0719200
$134$	−4.06800	−0.351422
$135$	0	0
$136$	−3.57487	−0.306542
$137$	−7.96214	−0.680251	−0.340126	−	0.940380i	$-0.610470\pi$
$137$	−7.96214	−0.680251	−0.340126	+	0.940380i	$0.610470\pi$
$138$	0	0
$139$	3.31018	0.280766	0.140383	−	0.990097i	$-0.455167\pi$
$139$	3.31018	0.280766	0.140383	+	0.990097i	$0.455167\pi$
$140$	0.0357860	0.00302447
$141$	0	0
$142$	9.49501	0.796804
$143$	−0.0605665	−0.00506483
$144$	0	0
$145$	−0.465126	−0.0386266
$146$	7.96509	0.659196
$147$	0	0
$148$	−3.75798	−0.308904
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	6.90585	0.561990	0.280995	−	0.959709i	$-0.409336\pi$
$151$	6.90585	0.561990	0.280995	+	0.959709i	$0.409336\pi$
$152$	−2.36593	−0.191902
$153$	0	0
$154$	0.710518	0.0572552
$155$	0.0856857	0.00688244
$156$	0	0
$157$	−14.0313	−1.11982	−0.559909	−	0.828554i	$-0.689164\pi$
$157$	−14.0313	−1.11982	−0.559909	+	0.828554i	$0.689164\pi$
$158$	9.24244	0.735289
$159$	0	0
$160$	0.102080	0.00807009
$161$	−1.25905	−0.0992274
$162$	0	0
$163$	−20.7001	−1.62135	−0.810677	−	0.585494i	$-0.800900\pi$
$163$	−20.7001	−1.62135	−0.810677	+	0.585494i	$0.800900\pi$
$164$	1.43684	0.112198
$165$	0	0
$166$	−9.62022	−0.746674
$167$	13.5549	1.04891	0.524455	−	0.851438i	$-0.324269\pi$
$167$	13.5549	1.04891	0.524455	+	0.851438i	$0.324269\pi$
$168$	0	0
$169$	−12.9991	−0.999931
$170$	−0.364921	−0.0279881
$171$	0	0
$172$	−5.01492	−0.382384
$173$	24.3064	1.84798	0.923991	−	0.382415i	$-0.124907\pi$
$173$	24.3064	1.84798	0.923991	+	0.382415i	$0.124907\pi$
$174$	0	0
$175$	−1.74920	−0.132227
$176$	2.02675	0.152772
$177$	0	0
$178$	−0.871304	−0.0653070
$179$	−19.8845	−1.48623	−0.743117	−	0.669162i	$-0.766653\pi$
$179$	−19.8845	−1.48623	−0.743117	+	0.669162i	$0.766653\pi$
$180$	0	0
$181$	1.35829	0.100961	0.0504803	−	0.998725i	$-0.483925\pi$
$181$	1.35829	0.100961	0.0504803	+	0.998725i	$0.483925\pi$
$182$	−0.0104763	−0.000776553 0
$183$	0	0
$184$	−3.59145	−0.264765
$185$	−0.383613	−0.0282038
$186$	0	0
$187$	−7.24537	−0.529834
$188$	6.28480	0.458366
$189$	0	0
$190$	−0.241512	−0.0175212
$191$	12.0574	0.872440	0.436220	−	0.899840i	$-0.356317\pi$
$191$	12.0574	0.872440	0.436220	+	0.899840i	$0.356317\pi$
$192$	0	0
$193$	17.1757	1.23634	0.618168	−	0.786046i	$-0.287875\pi$
$193$	17.1757	1.23634	0.618168	+	0.786046i	$0.287875\pi$
$194$	−12.9918	−0.932758
$195$	0	0
$196$	−6.87710	−0.491221
$197$	15.6315	1.11370	0.556848	−	0.830614i	$-0.312011\pi$
$197$	15.6315	1.11370	0.556848	+	0.830614i	$0.312011\pi$
$198$	0	0
$199$	−15.7562	−1.11692	−0.558462	−	0.829530i	$-0.688609\pi$
$199$	−15.7562	−1.11692	−0.558462	+	0.829530i	$0.688609\pi$
$200$	−4.98958	−0.352817
$201$	0	0
$202$	6.60527	0.464745
$203$	−1.59737	−0.112114
$204$	0	0
$205$	0.146672	0.0102440
$206$	5.27651	0.367632
$207$	0	0
$208$	−0.0298835	−0.00207205
$209$	−4.79514	−0.331687
$210$	0	0
$211$	8.83252	0.608056	0.304028	−	0.952663i	$-0.401668\pi$
$211$	8.83252	0.608056	0.304028	+	0.952663i	$0.401668\pi$
$212$	−9.52893	−0.654449
$213$	0	0
$214$	8.42424	0.575869
$215$	−0.511921	−0.0349127
$216$	0	0
$217$	0.294269	0.0199763
$218$	3.75345	0.254216
$219$	0	0
$220$	0.206890	0.0139485
$221$	0.106830	0.00718614
$222$	0	0
$223$	20.9511	1.40299	0.701493	−	0.712676i	$-0.252517\pi$
$223$	20.9511	1.40299	0.701493	+	0.712676i	$0.252517\pi$
$224$	0.350570	0.0234234
$225$	0	0
$226$	4.46292	0.296869
$227$	4.91723	0.326368	0.163184	−	0.986596i	$-0.447824\pi$
$227$	4.91723	0.326368	0.163184	+	0.986596i	$0.447824\pi$
$228$	0	0
$229$	−7.53842	−0.498153	−0.249077	−	0.968484i	$-0.580127\pi$
$229$	−7.53842	−0.498153	−0.249077	+	0.968484i	$0.580127\pi$
$230$	−0.366614	−0.0241738
$231$	0	0
$232$	−4.55651	−0.299149
$233$	11.9221	0.781041	0.390521	−	0.920594i	$-0.372295\pi$
$233$	11.9221	0.781041	0.390521	+	0.920594i	$0.372295\pi$
$234$	0	0
$235$	0.641549	0.0418500
$236$	−9.78689	−0.637072
$237$	0	0
$238$	−1.25324	−0.0812355
$239$	−16.1316	−1.04347	−0.521734	−	0.853108i	$-0.674715\pi$
$239$	−16.1316	−1.04347	−0.521734	+	0.853108i	$0.674715\pi$
$240$	0	0
$241$	20.6366	1.32932	0.664662	−	0.747144i	$-0.268576\pi$
$241$	20.6366	1.32932	0.664662	+	0.747144i	$0.268576\pi$
$242$	−6.89228	−0.443052
$243$	0	0
$244$	−7.40365	−0.473970
$245$	−0.702011	−0.0448498
$246$	0	0
$247$	0.0707022	0.00449868
$248$	0.839401	0.0533020
$249$	0	0
$250$	−1.01973	−0.0644935
$251$	4.82748	0.304708	0.152354	−	0.988326i	$-0.451315\pi$
$251$	4.82748	0.304708	0.152354	+	0.988326i	$0.451315\pi$
$252$	0	0
$253$	−7.27898	−0.457626
$254$	−1.36720	−0.0857859
$255$	0	0
$256$	1.00000	0.0625000
$257$	−3.38502	−0.211152	−0.105576	−	0.994411i	$-0.533669\pi$
$257$	−3.38502	−0.211152	−0.105576	+	0.994411i	$0.533669\pi$
$258$	0	0
$259$	−1.31743	−0.0818614
$260$	−0.00305050	−0.000189184 0
$261$	0	0
$262$	−8.75704	−0.541012
$263$	15.5141	0.956642	0.478321	−	0.878185i	$-0.341246\pi$
$263$	15.5141	0.956642	0.478321	+	0.878185i	$0.341246\pi$
$264$	0	0
$265$	−0.972708	−0.0597530
$266$	−0.829422	−0.0508551
$267$	0	0
$268$	−4.06800	−0.248493
$269$	−16.6115	−1.01282	−0.506410	−	0.862293i	$-0.669028\pi$
$269$	−16.6115	−1.01282	−0.506410	+	0.862293i	$0.669028\pi$
$270$	0	0
$271$	−23.4563	−1.42487	−0.712435	−	0.701738i	$-0.752408\pi$
$271$	−23.4563	−1.42487	−0.712435	+	0.701738i	$0.752408\pi$
$272$	−3.57487	−0.216758
$273$	0	0
$274$	−7.96214	−0.481010
$275$	−10.1126	−0.609815
$276$	0	0
$277$	−13.5781	−0.815826	−0.407913	−	0.913021i	$-0.633743\pi$
$277$	−13.5781	−0.815826	−0.407913	+	0.913021i	$0.633743\pi$
$278$	3.31018	0.198532
$279$	0	0
$280$	0.0357860	0.00213862
$281$	8.60505	0.513334	0.256667	−	0.966500i	$-0.417376\pi$
$281$	8.60505	0.513334	0.256667	+	0.966500i	$0.417376\pi$
$282$	0	0
$283$	−18.6623	−1.10936	−0.554680	−	0.832063i	$-0.687159\pi$
$283$	−18.6623	−1.10936	−0.554680	+	0.832063i	$0.687159\pi$
$284$	9.49501	0.563425
$285$	0	0
$286$	−0.0605665	−0.00358137
$287$	0.503712	0.0297332
$288$	0	0
$289$	−4.22032	−0.248254
$290$	−0.465126	−0.0273131
$291$	0	0
$292$	7.96509	0.466122
$293$	−8.92222	−0.521241	−0.260621	−	0.965441i	$-0.583927\pi$
$293$	−8.92222	−0.521241	−0.260621	+	0.965441i	$0.583927\pi$
$294$	0	0
$295$	−0.999041	−0.0581664
$296$	−3.75798	−0.218428
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	0.107325	0.00620678
$300$	0	0
$301$	−1.75808	−0.101334
$302$	6.90585	0.397387
$303$	0	0
$304$	−2.36593	−0.135695
$305$	−0.755761	−0.0432748
$306$	0	0
$307$	15.6841	0.895139	0.447570	−	0.894249i	$-0.352290\pi$
$307$	15.6841	0.895139	0.447570	+	0.894249i	$0.352290\pi$
$308$	0.710518	0.0404855
$309$	0	0
$310$	0.0856857	0.00486662
$311$	−3.64245	−0.206544	−0.103272	−	0.994653i	$-0.532931\pi$
$311$	−3.64245	−0.206544	−0.103272	+	0.994653i	$0.532931\pi$
$312$	0	0
$313$	−31.7926	−1.79703	−0.898513	−	0.438947i	$-0.855352\pi$
$313$	−31.7926	−1.79703	−0.898513	+	0.438947i	$0.855352\pi$
$314$	−14.0313	−0.791830
$315$	0	0
$316$	9.24244	0.519928
$317$	−1.50799	−0.0846972	−0.0423486	−	0.999103i	$-0.513484\pi$
$317$	−1.50799	−0.0846972	−0.0423486	+	0.999103i	$0.513484\pi$
$318$	0	0
$319$	−9.23491	−0.517056
$320$	0.102080	0.00570642
$321$	0	0
$322$	−1.25905	−0.0701644
$323$	8.45787	0.470608
$324$	0	0
$325$	0.149106	0.00827093
$326$	−20.7001	−1.14647
$327$	0	0
$328$	1.43684	0.0793360
$329$	2.20326	0.121470
$330$	0	0
$331$	1.26926	0.0697647	0.0348824	−	0.999391i	$-0.488894\pi$
$331$	1.26926	0.0697647	0.0348824	+	0.999391i	$0.488894\pi$
$332$	−9.62022	−0.527978
$333$	0	0
$334$	13.5549	0.741692
$335$	−0.415260	−0.0226881
$336$	0	0
$337$	14.2539	0.776457	0.388229	−	0.921563i	$-0.373087\pi$
$337$	14.2539	0.776457	0.388229	+	0.921563i	$0.373087\pi$
$338$	−12.9991	−0.707058
$339$	0	0
$340$	−0.364921	−0.0197906
$341$	1.70126	0.0921283
$342$	0	0
$343$	−4.86489	−0.262680
$344$	−5.01492	−0.270386
$345$	0	0
$346$	24.3064	1.30672
$347$	17.9926	0.965891	0.482945	−	0.875650i	$-0.339567\pi$
$347$	17.9926	0.965891	0.482945	+	0.875650i	$0.339567\pi$
$348$	0	0
$349$	−30.2747	−1.62057	−0.810284	−	0.586037i	$-0.800687\pi$
$349$	−30.2747	−1.62057	−0.810284	+	0.586037i	$0.800687\pi$
$350$	−1.74920	−0.0934985
$351$	0	0
$352$	2.02675	0.108026
$353$	−16.5740	−0.882146	−0.441073	−	0.897471i	$-0.645402\pi$
$353$	−16.5740	−0.882146	−0.441073	+	0.897471i	$0.645402\pi$
$354$	0	0
$355$	0.969246	0.0514422
$356$	−0.871304	−0.0461790
$357$	0	0
$358$	−19.8845	−1.05093
$359$	−23.0600	−1.21706	−0.608531	−	0.793530i	$-0.708241\pi$
$359$	−23.0600	−1.21706	−0.608531	+	0.793530i	$0.708241\pi$
$360$	0	0
$361$	−13.4024	−0.705389
$362$	1.35829	0.0713899
$363$	0	0
$364$	−0.0104763	−0.000549106 0
$365$	0.813073	0.0425582
$366$	0	0
$367$	−10.3939	−0.542558	−0.271279	−	0.962501i	$-0.587447\pi$
$367$	−10.3939	−0.542558	−0.271279	+	0.962501i	$0.587447\pi$
$368$	−3.59145	−0.187217
$369$	0	0
$370$	−0.383613	−0.0199431
$371$	−3.34055	−0.173433
$372$	0	0
$373$	17.6768	0.915269	0.457634	−	0.889140i	$-0.348697\pi$
$373$	17.6768	0.915269	0.457634	+	0.889140i	$0.348697\pi$
$374$	−7.24537	−0.374649
$375$	0	0
$376$	6.28480	0.324114
$377$	0.136165	0.00701284
$378$	0	0
$379$	−6.04973	−0.310754	−0.155377	−	0.987855i	$-0.549659\pi$
$379$	−6.04973	−0.310754	−0.155377	+	0.987855i	$0.549659\pi$
$380$	−0.241512	−0.0123893
$381$	0	0
$382$	12.0574	0.616908
$383$	−20.8782	−1.06682	−0.533412	−	0.845855i	$-0.679091\pi$
$383$	−20.8782	−1.06682	−0.533412	+	0.845855i	$0.679091\pi$
$384$	0	0
$385$	0.0725293	0.00369644
$386$	17.1757	0.874222
$387$	0	0
$388$	−12.9918	−0.659559
$389$	14.4959	0.734971	0.367486	−	0.930029i	$-0.380219\pi$
$389$	14.4959	0.734971	0.367486	+	0.930029i	$0.380219\pi$
$390$	0	0
$391$	12.8390	0.649294
$392$	−6.87710	−0.347346
$393$	0	0
$394$	15.6315	0.787502
$395$	0.943464	0.0474708
$396$	0	0
$397$	−6.17054	−0.309690	−0.154845	−	0.987939i	$-0.549488\pi$
$397$	−6.17054	−0.309690	−0.154845	+	0.987939i	$0.549488\pi$
$398$	−15.7562	−0.789785
$399$	0	0
$400$	−4.98958	−0.249479
$401$	10.3024	0.514480	0.257240	−	0.966348i	$-0.417187\pi$
$401$	10.3024	0.514480	0.257240	+	0.966348i	$0.417187\pi$
$402$	0	0
$403$	−0.0250843	−0.00124954
$404$	6.60527	0.328624
$405$	0	0
$406$	−1.59737	−0.0792764
$407$	−7.61649	−0.377536
$408$	0	0
$409$	9.18360	0.454100	0.227050	−	0.973883i	$-0.427092\pi$
$409$	9.18360	0.454100	0.227050	+	0.973883i	$0.427092\pi$
$410$	0.146672	0.00724359
$411$	0	0
$412$	5.27651	0.259955
$413$	−3.43099	−0.168828
$414$	0	0
$415$	−0.982027	−0.0482058
$416$	−0.0298835	−0.00146516
$417$	0	0
$418$	−4.79514	−0.234538
$419$	23.3723	1.14181	0.570906	−	0.821015i	$-0.306592\pi$
$419$	23.3723	1.14181	0.570906	+	0.821015i	$0.306592\pi$
$420$	0	0
$421$	−22.5192	−1.09752	−0.548760	−	0.835980i	$-0.684900\pi$
$421$	−22.5192	−1.09752	−0.548760	+	0.835980i	$0.684900\pi$
$422$	8.83252	0.429960
$423$	0	0
$424$	−9.52893	−0.462766
$425$	17.8371	0.865226
$426$	0	0
$427$	−2.59550	−0.125605
$428$	8.42424	0.407201
$429$	0	0
$430$	−0.511921	−0.0246870
$431$	4.03344	0.194284	0.0971421	−	0.995271i	$-0.469030\pi$
$431$	4.03344	0.194284	0.0971421	+	0.995271i	$0.469030\pi$
$432$	0	0
$433$	4.71921	0.226791	0.113395	−	0.993550i	$-0.463827\pi$
$433$	4.71921	0.226791	0.113395	+	0.993550i	$0.463827\pi$
$434$	0.294269	0.0141254
$435$	0	0
$436$	3.75345	0.179758
$437$	8.49711	0.406472
$438$	0	0
$439$	−32.7549	−1.56331	−0.781653	−	0.623713i	$-0.785623\pi$
$439$	−32.7549	−1.56331	−0.781653	+	0.623713i	$0.785623\pi$
$440$	0.206890	0.00986309
$441$	0	0
$442$	0.106830	0.00508137
$443$	−35.0469	−1.66513	−0.832565	−	0.553927i	$-0.813129\pi$
$443$	−35.0469	−1.66513	−0.832565	+	0.553927i	$0.813129\pi$
$444$	0	0
$445$	−0.0889423	−0.00421627
$446$	20.9511	0.992061
$447$	0	0
$448$	0.350570	0.0165629
$449$	32.0551	1.51278	0.756388	−	0.654123i	$-0.226962\pi$
$449$	32.0551	1.51278	0.756388	+	0.654123i	$0.226962\pi$
$450$	0	0
$451$	2.91211	0.137126
$452$	4.46292	0.209918
$453$	0	0
$454$	4.91723	0.230777
$455$	−0.00106941	−5.01348e−5 0
$456$	0	0
$457$	24.0443	1.12475	0.562373	−	0.826884i	$-0.309888\pi$
$457$	24.0443	1.12475	0.562373	+	0.826884i	$0.309888\pi$
$458$	−7.53842	−0.352247
$459$	0	0
$460$	−0.366614	−0.0170934
$461$	7.50713	0.349642	0.174821	−	0.984600i	$-0.444065\pi$
$461$	7.50713	0.349642	0.174821	+	0.984600i	$0.444065\pi$
$462$	0	0
$463$	−37.7034	−1.75223	−0.876114	−	0.482105i	$-0.839872\pi$
$463$	−37.7034	−1.75223	−0.876114	+	0.482105i	$0.839872\pi$
$464$	−4.55651	−0.211531
$465$	0	0
$466$	11.9221	0.552280
$467$	−11.1921	−0.517909	−0.258954	−	0.965890i	$-0.583378\pi$
$467$	−11.1921	−0.517909	−0.258954	+	0.965890i	$0.583378\pi$
$468$	0	0
$469$	−1.42612	−0.0658520
$470$	0.641549	0.0295925
$471$	0	0
$472$	−9.78689	−0.450478
$473$	−10.1640	−0.467341
$474$	0	0
$475$	11.8050	0.541649
$476$	−1.25324	−0.0574422
$477$	0	0
$478$	−16.1316	−0.737843
$479$	37.4425	1.71079	0.855396	−	0.517975i	$-0.173314\pi$
$479$	37.4425	1.71079	0.855396	+	0.517975i	$0.173314\pi$
$480$	0	0
$481$	0.112302	0.00512052
$482$	20.6366	0.939974
$483$	0	0
$484$	−6.89228	−0.313285
$485$	−1.32620	−0.0602195
$486$	0	0
$487$	−9.80226	−0.444183	−0.222091	−	0.975026i	$-0.571288\pi$
$487$	−9.80226	−0.444183	−0.222091	+	0.975026i	$0.571288\pi$
$488$	−7.40365	−0.335148
$489$	0	0
$490$	−0.702011	−0.0317136
$491$	−32.0256	−1.44529	−0.722647	−	0.691217i	$-0.757075\pi$
$491$	−32.0256	−1.44529	−0.722647	+	0.691217i	$0.757075\pi$
$492$	0	0
$493$	16.2889	0.733616
$494$	0.0707022	0.00318104
$495$	0	0
$496$	0.839401	0.0376902
$497$	3.32866	0.149311
$498$	0	0
$499$	13.3658	0.598336	0.299168	−	0.954201i	$-0.403291\pi$
$499$	13.3658	0.598336	0.299168	+	0.954201i	$0.403291\pi$
$500$	−1.01973	−0.0456038
$501$	0	0
$502$	4.82748	0.215461
$503$	0.515588	0.0229889	0.0114945	−	0.999934i	$-0.496341\pi$
$503$	0.515588	0.0229889	0.0114945	+	0.999934i	$0.496341\pi$
$504$	0	0
$505$	0.674262	0.0300043
$506$	−7.27898	−0.323590
$507$	0	0
$508$	−1.36720	−0.0606598
$509$	16.3872	0.726351	0.363176	−	0.931721i	$-0.381693\pi$
$509$	16.3872	0.726351	0.363176	+	0.931721i	$0.381693\pi$
$510$	0	0
$511$	2.79232	0.123525
$512$	1.00000	0.0441942
$513$	0	0
$514$	−3.38502	−0.149307
$515$	0.538624	0.0237346
$516$	0	0
$517$	12.7377	0.560205
$518$	−1.31743	−0.0578847
$519$	0	0
$520$	−0.00305050	−0.000133773 0
$521$	−31.8411	−1.39499	−0.697493	−	0.716592i	$-0.745701\pi$
$521$	−31.8411	−1.39499	−0.697493	+	0.716592i	$0.745701\pi$
$522$	0	0
$523$	−7.22133	−0.315766	−0.157883	−	0.987458i	$-0.550467\pi$
$523$	−7.22133	−0.315766	−0.157883	+	0.987458i	$0.550467\pi$
$524$	−8.75704	−0.382553
$525$	0	0
$526$	15.5141	0.676448
$527$	−3.00075	−0.130715
$528$	0	0
$529$	−10.1015	−0.439195
$530$	−0.972708	−0.0422517
$531$	0	0
$532$	−0.829422	−0.0359600
$533$	−0.0429378	−0.00185984
$534$	0	0
$535$	0.859942	0.0371786
$536$	−4.06800	−0.175711
$537$	0	0
$538$	−16.6115	−0.716172
$539$	−13.9382	−0.600360
$540$	0	0
$541$	−24.9022	−1.07063	−0.535314	−	0.844653i	$-0.679807\pi$
$541$	−24.9022	−1.07063	−0.535314	+	0.844653i	$0.679807\pi$
$542$	−23.4563	−1.00754
$543$	0	0
$544$	−3.57487	−0.153271
$545$	0.383151	0.0164124
$546$	0	0
$547$	31.6658	1.35393	0.676966	−	0.736014i	$-0.263294\pi$
$547$	31.6658	1.35393	0.676966	+	0.736014i	$0.263294\pi$
$548$	−7.96214	−0.340126
$549$	0	0
$550$	−10.1126	−0.431204
$551$	10.7804	0.459259
$552$	0	0
$553$	3.24012	0.137784
$554$	−13.5781	−0.576876
$555$	0	0
$556$	3.31018	0.140383
$557$	−28.4674	−1.20620	−0.603101	−	0.797665i	$-0.706068\pi$
$557$	−28.4674	−1.20620	−0.603101	+	0.797665i	$0.706068\pi$
$558$	0	0
$559$	0.149864	0.00633856
$560$	0.0357860	0.00151223
$561$	0	0
$562$	8.60505	0.362982
$563$	18.4847	0.779037	0.389518	−	0.921019i	$-0.372642\pi$
$563$	18.4847	0.779037	0.389518	+	0.921019i	$0.372642\pi$
$564$	0	0
$565$	0.455572	0.0191661
$566$	−18.6623	−0.784437
$567$	0	0
$568$	9.49501	0.398402
$569$	30.2299	1.26730	0.633651	−	0.773619i	$-0.281556\pi$
$569$	30.2299	1.26730	0.633651	+	0.773619i	$0.281556\pi$
$570$	0	0
$571$	−5.28489	−0.221166	−0.110583	−	0.993867i	$-0.535272\pi$
$571$	−5.28489	−0.221166	−0.110583	+	0.993867i	$0.535272\pi$
$572$	−0.0605665	−0.00253241
$573$	0	0
$574$	0.503712	0.0210245
$575$	17.9198	0.747309
$576$	0	0
$577$	−41.7286	−1.73718	−0.868592	−	0.495527i	$-0.834975\pi$
$577$	−41.7286	−1.73718	−0.868592	+	0.495527i	$0.834975\pi$
$578$	−4.22032	−0.175542
$579$	0	0
$580$	−0.465126	−0.0193133
$581$	−3.37256	−0.139917
$582$	0	0
$583$	−19.3128	−0.799853
$584$	7.96509	0.329598
$585$	0	0
$586$	−8.92222	−0.368573
$587$	6.32683	0.261136	0.130568	−	0.991439i	$-0.458320\pi$
$587$	6.32683	0.261136	0.130568	+	0.991439i	$0.458320\pi$
$588$	0	0
$589$	−1.98596	−0.0818301
$590$	−0.999041	−0.0411299
$591$	0	0
$592$	−3.75798	−0.154452
$593$	29.1140	1.19557	0.597784	−	0.801657i	$-0.296048\pi$
$593$	29.1140	1.19557	0.597784	+	0.801657i	$0.296048\pi$
$594$	0	0
$595$	−0.127930	−0.00524463
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	0.107325	0.00438886
$599$	−23.2937	−0.951755	−0.475878	−	0.879511i	$-0.657870\pi$
$599$	−23.2937	−0.951755	−0.475878	+	0.879511i	$0.657870\pi$
$600$	0	0
$601$	−5.33860	−0.217766	−0.108883	−	0.994055i	$-0.534727\pi$
$601$	−5.33860	−0.217766	−0.108883	+	0.994055i	$0.534727\pi$
$602$	−1.75808	−0.0716540
$603$	0	0
$604$	6.90585	0.280995
$605$	−0.703560	−0.0286038
$606$	0	0
$607$	21.5161	0.873310	0.436655	−	0.899629i	$-0.356163\pi$
$607$	21.5161	0.873310	0.436655	+	0.899629i	$0.356163\pi$
$608$	−2.36593	−0.0959509
$609$	0	0
$610$	−0.755761	−0.0305999
$611$	−0.187812	−0.00759806
$612$	0	0
$613$	12.2745	0.495762	0.247881	−	0.968791i	$-0.420266\pi$
$613$	12.2745	0.495762	0.247881	+	0.968791i	$0.420266\pi$
$614$	15.6841	0.632959
$615$	0	0
$616$	0.710518	0.0286276
$617$	−4.83318	−0.194577	−0.0972883	−	0.995256i	$-0.531017\pi$
$617$	−4.83318	−0.194577	−0.0972883	+	0.995256i	$0.531017\pi$
$618$	0	0
$619$	−4.13677	−0.166271	−0.0831353	−	0.996538i	$-0.526493\pi$
$619$	−4.13677	−0.166271	−0.0831353	+	0.996538i	$0.526493\pi$
$620$	0.0856857	0.00344122
$621$	0	0
$622$	−3.64245	−0.146049
$623$	−0.305453	−0.0122377
$624$	0	0
$625$	24.8438	0.993752
$626$	−31.7926	−1.27069
$627$	0	0
$628$	−14.0313	−0.559909
$629$	13.4343	0.535660
$630$	0	0
$631$	10.8814	0.433182	0.216591	−	0.976262i	$-0.430506\pi$
$631$	10.8814	0.433182	0.216591	+	0.976262i	$0.430506\pi$
$632$	9.24244	0.367645
$633$	0	0
$634$	−1.50799	−0.0598899
$635$	−0.139563	−0.00553840
$636$	0	0
$637$	0.205512	0.00814269
$638$	−9.23491	−0.365614
$639$	0	0
$640$	0.102080	0.00403505
$641$	46.5892	1.84016	0.920081	−	0.391728i	$-0.128122\pi$
$641$	46.5892	1.84016	0.920081	+	0.391728i	$0.128122\pi$
$642$	0	0
$643$	18.1726	0.716655	0.358328	−	0.933596i	$-0.383347\pi$
$643$	18.1726	0.716655	0.358328	+	0.933596i	$0.383347\pi$
$644$	−1.25905	−0.0496137
$645$	0	0
$646$	8.45787	0.332770
$647$	23.8592	0.938002	0.469001	−	0.883198i	$-0.344614\pi$
$647$	23.8592	0.938002	0.469001	+	0.883198i	$0.344614\pi$
$648$	0	0
$649$	−19.8356	−0.778615
$650$	0.149106	0.00584843
$651$	0	0
$652$	−20.7001	−0.810677
$653$	−7.62071	−0.298222	−0.149111	−	0.988820i	$-0.547641\pi$
$653$	−7.62071	−0.298222	−0.149111	+	0.988820i	$0.547641\pi$
$654$	0	0
$655$	−0.893914	−0.0349281
$656$	1.43684	0.0560991
$657$	0	0
$658$	2.20326	0.0858920
$659$	12.4087	0.483376	0.241688	−	0.970354i	$-0.422299\pi$
$659$	12.4087	0.483376	0.241688	+	0.970354i	$0.422299\pi$
$660$	0	0
$661$	25.9018	1.00746	0.503732	−	0.863860i	$-0.331960\pi$
$661$	25.9018	1.00746	0.503732	+	0.863860i	$0.331960\pi$
$662$	1.26926	0.0493311
$663$	0	0
$664$	−9.62022	−0.373337
$665$	−0.0846670	−0.00328324
$666$	0	0
$667$	16.3645	0.633635
$668$	13.5549	0.524455
$669$	0	0
$670$	−0.415260	−0.0160429
$671$	−15.0054	−0.579276
$672$	0	0
$673$	−7.27877	−0.280576	−0.140288	−	0.990111i	$-0.544803\pi$
$673$	−7.27877	−0.280576	−0.140288	+	0.990111i	$0.544803\pi$
$674$	14.2539	0.549038
$675$	0	0
$676$	−12.9991	−0.499966
$677$	50.8122	1.95287	0.976435	−	0.215812i	$-0.0692398\pi$
$677$	50.8122	1.95287	0.976435	+	0.215812i	$0.0692398\pi$
$678$	0	0
$679$	−4.55454	−0.174787
$680$	−0.364921	−0.0139941
$681$	0	0
$682$	1.70126	0.0651445
$683$	42.7539	1.63593	0.817965	−	0.575267i	$-0.195102\pi$
$683$	42.7539	1.63593	0.817965	+	0.575267i	$0.195102\pi$
$684$	0	0
$685$	−0.812771	−0.0310544
$686$	−4.86489	−0.185742
$687$	0	0
$688$	−5.01492	−0.191192
$689$	0.284758	0.0108484
$690$	0	0
$691$	0.949122	0.0361063	0.0180532	−	0.999837i	$-0.494253\pi$
$691$	0.949122	0.0361063	0.0180532	+	0.999837i	$0.494253\pi$
$692$	24.3064	0.923991
$693$	0	0
$694$	17.9926	0.682988
$695$	0.337902	0.0128174
$696$	0	0
$697$	−5.13650	−0.194559
$698$	−30.2747	−1.14591
$699$	0	0
$700$	−1.74920	−0.0661134
$701$	24.8882	0.940013	0.470006	−	0.882663i	$-0.344252\pi$
$701$	24.8882	0.940013	0.470006	+	0.882663i	$0.344252\pi$
$702$	0	0
$703$	8.89110	0.335334
$704$	2.02675	0.0763861
$705$	0	0
$706$	−16.5740	−0.623771
$707$	2.31561	0.0870874
$708$	0	0
$709$	15.5052	0.582311	0.291155	−	0.956676i	$-0.405960\pi$
$709$	15.5052	0.582311	0.291155	+	0.956676i	$0.405960\pi$
$710$	0.969246	0.0363752
$711$	0	0
$712$	−0.871304	−0.0326535
$713$	−3.01467	−0.112900
$714$	0	0
$715$	−0.00618260	−0.000231216 0
$716$	−19.8845	−0.743117
$717$	0	0
$718$	−23.0600	−0.860593
$719$	13.6341	0.508465	0.254233	−	0.967143i	$-0.418177\pi$
$719$	13.6341	0.508465	0.254233	+	0.967143i	$0.418177\pi$
$720$	0	0
$721$	1.84979	0.0688896
$722$	−13.4024	−0.498786
$723$	0	0
$724$	1.35829	0.0504803
$725$	22.7351	0.844359
$726$	0	0
$727$	9.59451	0.355841	0.177920	−	0.984045i	$-0.443063\pi$
$727$	9.59451	0.355841	0.177920	+	0.984045i	$0.443063\pi$
$728$	−0.0104763	−0.000388276 0
$729$	0	0
$730$	0.813073	0.0300932
$731$	17.9277	0.663079
$732$	0	0
$733$	18.3649	0.678324	0.339162	−	0.940728i	$-0.389857\pi$
$733$	18.3649	0.678324	0.339162	+	0.940728i	$0.389857\pi$
$734$	−10.3939	−0.383646
$735$	0	0
$736$	−3.59145	−0.132383
$737$	−8.24483	−0.303702
$738$	0	0
$739$	−39.9117	−1.46817	−0.734087	−	0.679055i	$-0.762390\pi$
$739$	−39.9117	−1.46817	−0.734087	+	0.679055i	$0.762390\pi$
$740$	−0.383613	−0.0141019
$741$	0	0
$742$	−3.34055	−0.122636
$743$	−8.04457	−0.295127	−0.147563	−	0.989053i	$-0.547143\pi$
$743$	−8.04457	−0.295127	−0.147563	+	0.989053i	$0.547143\pi$
$744$	0	0
$745$	−0.102080	−0.00373990
$746$	17.6768	0.647193
$747$	0	0
$748$	−7.24537	−0.264917
$749$	2.95328	0.107911
$750$	0	0
$751$	10.3350	0.377128	0.188564	−	0.982061i	$-0.439617\pi$
$751$	10.3350	0.377128	0.188564	+	0.982061i	$0.439617\pi$
$752$	6.28480	0.229183
$753$	0	0
$754$	0.136165	0.00495882
$755$	0.704946	0.0256556
$756$	0	0
$757$	−49.2360	−1.78951	−0.894756	−	0.446556i	$-0.852651\pi$
$757$	−49.2360	−1.78951	−0.894756	+	0.446556i	$0.852651\pi$
$758$	−6.04973	−0.219736
$759$	0	0
$760$	−0.241512	−0.00876058
$761$	8.72729	0.316364	0.158182	−	0.987410i	$-0.449437\pi$
$761$	8.72729	0.316364	0.158182	+	0.987410i	$0.449437\pi$
$762$	0	0
$763$	1.31585	0.0476369
$764$	12.0574	0.436220
$765$	0	0
$766$	−20.8782	−0.754359
$767$	0.292467	0.0105604
$768$	0	0
$769$	−32.5167	−1.17258	−0.586290	−	0.810101i	$-0.699412\pi$
$769$	−32.5167	−1.17258	−0.586290	+	0.810101i	$0.699412\pi$
$770$	0.0725293	0.00261378
$771$	0	0
$772$	17.1757	0.618168
$773$	30.6093	1.10094	0.550469	−	0.834855i	$-0.314449\pi$
$773$	30.6093	1.10094	0.550469	+	0.834855i	$0.314449\pi$
$774$	0	0
$775$	−4.18826	−0.150447
$776$	−12.9918	−0.466379
$777$	0	0
$778$	14.4959	0.519703
$779$	−3.39945	−0.121798
$780$	0	0
$781$	19.2440	0.688606
$782$	12.8390	0.459120
$783$	0	0
$784$	−6.87710	−0.245611
$785$	−1.43231	−0.0511212
$786$	0	0
$787$	−24.9239	−0.888440	−0.444220	−	0.895918i	$-0.646519\pi$
$787$	−24.9239	−0.888440	−0.444220	+	0.895918i	$0.646519\pi$
$788$	15.6315	0.556848
$789$	0	0
$790$	0.943464	0.0335669
$791$	1.56456	0.0556295
$792$	0	0
$793$	0.221247	0.00785673
$794$	−6.17054	−0.218984
$795$	0	0
$796$	−15.7562	−0.558462
$797$	−21.6064	−0.765338	−0.382669	−	0.923886i	$-0.624995\pi$
$797$	−21.6064	−0.765338	−0.382669	+	0.923886i	$0.624995\pi$
$798$	0	0
$799$	−22.4673	−0.794837
$800$	−4.98958	−0.176408
$801$	0	0
$802$	10.3024	0.363792
$803$	16.1433	0.569683
$804$	0	0
$805$	−0.128524	−0.00452986
$806$	−0.0250843	−0.000883556 0
$807$	0	0
$808$	6.60527	0.232373
$809$	−16.4003	−0.576603	−0.288302	−	0.957540i	$-0.593091\pi$
$809$	−16.4003	−0.576603	−0.288302	+	0.957540i	$0.593091\pi$
$810$	0	0
$811$	−11.7483	−0.412539	−0.206269	−	0.978495i	$-0.566132\pi$
$811$	−11.7483	−0.412539	−0.206269	+	0.978495i	$0.566132\pi$
$812$	−1.59737	−0.0560569
$813$	0	0
$814$	−7.61649	−0.266958
$815$	−2.11305	−0.0740169
$816$	0	0
$817$	11.8649	0.415101
$818$	9.18360	0.321097
$819$	0	0
$820$	0.146672	0.00512199
$821$	−51.3053	−1.79057	−0.895284	−	0.445495i	$-0.853028\pi$
$821$	−51.3053	−1.79057	−0.895284	+	0.445495i	$0.853028\pi$
$822$	0	0
$823$	−0.892424	−0.0311079	−0.0155540	−	0.999879i	$-0.504951\pi$
$823$	−0.892424	−0.0311079	−0.0155540	+	0.999879i	$0.504951\pi$
$824$	5.27651	0.183816
$825$	0	0
$826$	−3.43099	−0.119379
$827$	−21.4142	−0.744644	−0.372322	−	0.928104i	$-0.621438\pi$
$827$	−21.4142	−0.744644	−0.372322	+	0.928104i	$0.621438\pi$
$828$	0	0
$829$	−10.1286	−0.351782	−0.175891	−	0.984410i	$-0.556281\pi$
$829$	−10.1286	−0.351782	−0.175891	+	0.984410i	$0.556281\pi$
$830$	−0.982027	−0.0340867
$831$	0	0
$832$	−0.0298835	−0.00103603
$833$	24.5847	0.851810
$834$	0	0
$835$	1.38368	0.0478842
$836$	−4.79514	−0.165843
$837$	0	0
$838$	23.3723	0.807384
$839$	13.1240	0.453091	0.226545	−	0.974001i	$-0.427257\pi$
$839$	13.1240	0.453091	0.226545	+	0.974001i	$0.427257\pi$
$840$	0	0
$841$	−8.23823	−0.284077
$842$	−22.5192	−0.776064
$843$	0	0
$844$	8.83252	0.304028
$845$	−1.32694	−0.0456482
$846$	0	0
$847$	−2.41622	−0.0830224
$848$	−9.52893	−0.327225
$849$	0	0
$850$	17.8371	0.611807
$851$	13.4966	0.462658
$852$	0	0
$853$	24.7181	0.846333	0.423167	−	0.906052i	$-0.360919\pi$
$853$	24.7181	0.846333	0.423167	+	0.906052i	$0.360919\pi$
$854$	−2.59550	−0.0888161
$855$	0	0
$856$	8.42424	0.287935
$857$	43.6328	1.49047	0.745234	−	0.666803i	$-0.232338\pi$
$857$	43.6328	1.49047	0.745234	+	0.666803i	$0.232338\pi$
$858$	0	0
$859$	43.6531	1.48942	0.744712	−	0.667386i	$-0.232587\pi$
$859$	43.6531	1.48942	0.744712	+	0.667386i	$0.232587\pi$
$860$	−0.511921	−0.0174563
$861$	0	0
$862$	4.03344	0.137380
$863$	9.57447	0.325919	0.162959	−	0.986633i	$-0.447896\pi$
$863$	9.57447	0.325919	0.162959	+	0.986633i	$0.447896\pi$
$864$	0	0
$865$	2.48119	0.0843628
$866$	4.71921	0.160365
$867$	0	0
$868$	0.294269	0.00998813
$869$	18.7321	0.635444
$870$	0	0
$871$	0.121566	0.00411912
$872$	3.75345	0.127108
$873$	0	0
$874$	8.49711	0.287419
$875$	−0.357487	−0.0120853
$876$	0	0
$877$	−17.1383	−0.578719	−0.289360	−	0.957220i	$-0.593442\pi$
$877$	−17.1383	−0.578719	−0.289360	+	0.957220i	$0.593442\pi$
$878$	−32.7549	−1.10542
$879$	0	0
$880$	0.206890	0.00697425
$881$	−19.2478	−0.648475	−0.324237	−	0.945976i	$-0.605108\pi$
$881$	−19.2478	−0.648475	−0.324237	+	0.945976i	$0.605108\pi$
$882$	0	0
$883$	−30.4527	−1.02482	−0.512408	−	0.858742i	$-0.671246\pi$
$883$	−30.4527	−1.02482	−0.512408	+	0.858742i	$0.671246\pi$
$884$	0.106830	0.00359307
$885$	0	0
$886$	−35.0469	−1.17742
$887$	−27.3879	−0.919595	−0.459797	−	0.888024i	$-0.652078\pi$
$887$	−27.3879	−0.919595	−0.459797	+	0.888024i	$0.652078\pi$
$888$	0	0
$889$	−0.479300	−0.0160752
$890$	−0.0889423	−0.00298135
$891$	0	0
$892$	20.9511	0.701493
$893$	−14.8694	−0.497584
$894$	0	0
$895$	−2.02980	−0.0678486
$896$	0.350570	0.0117117
$897$	0	0
$898$	32.0551	1.06969
$899$	−3.82474	−0.127562
$900$	0	0
$901$	34.0646	1.13486
$902$	2.91211	0.0969627
$903$	0	0
$904$	4.46292	0.148434
$905$	0.138653	0.00460899
$906$	0	0
$907$	23.6464	0.785166	0.392583	−	0.919717i	$-0.371582\pi$
$907$	23.6464	0.785166	0.392583	+	0.919717i	$0.371582\pi$
$908$	4.91723	0.163184
$909$	0	0
$910$	−0.00106941	−3.54507e−5 0
$911$	40.4265	1.33939	0.669695	−	0.742636i	$-0.266425\pi$
$911$	40.4265	1.33939	0.669695	+	0.742636i	$0.266425\pi$
$912$	0	0
$913$	−19.4978	−0.645283
$914$	24.0443	0.795316
$915$	0	0
$916$	−7.53842	−0.249077
$917$	−3.06995	−0.101379
$918$	0	0
$919$	−22.4929	−0.741973	−0.370986	−	0.928638i	$-0.620980\pi$
$919$	−22.4929	−0.741973	−0.370986	+	0.928638i	$0.620980\pi$
$920$	−0.366614	−0.0120869
$921$	0	0
$922$	7.50713	0.247234
$923$	−0.283745	−0.00933957
$924$	0	0
$925$	18.7507	0.616521
$926$	−37.7034	−1.23901
$927$	0	0
$928$	−4.55651	−0.149575
$929$	−29.1055	−0.954920	−0.477460	−	0.878653i	$-0.658442\pi$
$929$	−29.1055	−0.954920	−0.477460	+	0.878653i	$0.658442\pi$
$930$	0	0
$931$	16.2707	0.533251
$932$	11.9221	0.390521
$933$	0	0
$934$	−11.1921	−0.366217
$935$	−0.739604	−0.0241876
$936$	0	0
$937$	−6.36306	−0.207872	−0.103936	−	0.994584i	$-0.533144\pi$
$937$	−6.36306	−0.207872	−0.103936	+	0.994584i	$0.533144\pi$
$938$	−1.42612	−0.0465644
$939$	0	0
$940$	0.641549	0.0209250
$941$	35.5770	1.15978	0.579889	−	0.814696i	$-0.303096\pi$
$941$	35.5770	1.15978	0.579889	+	0.814696i	$0.303096\pi$
$942$	0	0
$943$	−5.16033	−0.168043
$944$	−9.78689	−0.318536
$945$	0	0
$946$	−10.1640	−0.330460
$947$	28.1479	0.914684	0.457342	−	0.889291i	$-0.348802\pi$
$947$	28.1479	0.914684	0.457342	+	0.889291i	$0.348802\pi$
$948$	0	0
$949$	−0.238025	−0.00772663
$950$	11.8050	0.383004
$951$	0	0
$952$	−1.25324	−0.0406178
$953$	47.8502	1.55002	0.775010	−	0.631949i	$-0.217745\pi$
$953$	47.8502	1.55002	0.775010	+	0.631949i	$0.217745\pi$
$954$	0	0
$955$	1.23081	0.0398281
$956$	−16.1316	−0.521734
$957$	0	0
$958$	37.4425	1.20971
$959$	−2.79128	−0.0901353
$960$	0	0
$961$	−30.2954	−0.977271
$962$	0.112302	0.00362076
$963$	0	0
$964$	20.6366	0.664662
$965$	1.75329	0.0564404
$966$	0	0
$967$	11.0594	0.355646	0.177823	−	0.984063i	$-0.443095\pi$
$967$	11.0594	0.355646	0.177823	+	0.984063i	$0.443095\pi$
$968$	−6.89228	−0.221526
$969$	0	0
$970$	−1.32620	−0.0425816
$971$	−11.6714	−0.374552	−0.187276	−	0.982307i	$-0.559966\pi$
$971$	−11.6714	−0.374552	−0.187276	+	0.982307i	$0.559966\pi$
$972$	0	0
$973$	1.16045	0.0372023
$974$	−9.80226	−0.314085
$975$	0	0
$976$	−7.40365	−0.236985
$977$	−29.4805	−0.943163	−0.471582	−	0.881822i	$-0.656317\pi$
$977$	−29.4805	−0.943163	−0.471582	+	0.881822i	$0.656317\pi$
$978$	0	0
$979$	−1.76592	−0.0564389
$980$	−0.702011	−0.0224249
$981$	0	0
$982$	−32.0256	−1.02198
$983$	−28.4486	−0.907371	−0.453685	−	0.891162i	$-0.649891\pi$
$983$	−28.4486	−0.907371	−0.453685	+	0.891162i	$0.649891\pi$
$984$	0	0
$985$	1.59565	0.0508417
$986$	16.2889	0.518745
$987$	0	0
$988$	0.0707022	0.00224934
$989$	18.0108	0.572712
$990$	0	0
$991$	−52.8403	−1.67853	−0.839264	−	0.543724i	$-0.817014\pi$
$991$	−52.8403	−1.67853	−0.839264	+	0.543724i	$0.817014\pi$
$992$	0.839401	0.0266510
$993$	0	0
$994$	3.32866	0.105579
$995$	−1.60838	−0.0509891
$996$	0	0
$997$	21.3350	0.675685	0.337843	−	0.941203i	$-0.390303\pi$
$997$	21.3350	0.675685	0.337843	+	0.941203i	$0.390303\pi$
$998$	13.3658	0.423087
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.h.1.6	yes	9
3.2	odd	2		8046.2.a.g.1.4	✓	9

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.g.1.4	✓	9	3.2	odd	2
8046.2.a.h.1.6	yes	9	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.102080 q^{5} +0.350570 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} +0.102080 q^{5} +0.350570 q^{7} +1.00000 q^{8} +0.102080 q^{10} +2.02675 q^{11} -0.0298835 q^{13} +0.350570 q^{14} +1.00000 q^{16} -3.57487 q^{17} -2.36593 q^{19} +0.102080 q^{20} +2.02675 q^{22} -3.59145 q^{23} -4.98958 q^{25} -0.0298835 q^{26} +0.350570 q^{28} -4.55651 q^{29} +0.839401 q^{31} +1.00000 q^{32} -3.57487 q^{34} +0.0357860 q^{35} -3.75798 q^{37} -2.36593 q^{38} +0.102080 q^{40} +1.43684 q^{41} -5.01492 q^{43} +2.02675 q^{44} -3.59145 q^{46} +6.28480 q^{47} -6.87710 q^{49} -4.98958 q^{50} -0.0298835 q^{52} -9.52893 q^{53} +0.206890 q^{55} +0.350570 q^{56} -4.55651 q^{58} -9.78689 q^{59} -7.40365 q^{61} +0.839401 q^{62} +1.00000 q^{64} -0.00305050 q^{65} -4.06800 q^{67} -3.57487 q^{68} +0.0357860 q^{70} +9.49501 q^{71} +7.96509 q^{73} -3.75798 q^{74} -2.36593 q^{76} +0.710518 q^{77} +9.24244 q^{79} +0.102080 q^{80} +1.43684 q^{82} -9.62022 q^{83} -0.364921 q^{85} -5.01492 q^{86} +2.02675 q^{88} -0.871304 q^{89} -0.0104763 q^{91} -3.59145 q^{92} +6.28480 q^{94} -0.241512 q^{95} -12.9918 q^{97} -6.87710 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8}+O(q^{10}) \) 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8	\( 9 q + 9 q^{2} + 9 q^{4} - 4 q^{5} - 4 q^{7} + 9 q^{8} - 4 q^{10} - 4 q^{11} - 8 q^{13} - 4 q^{14} + 9 q^{16} - q^{17} - 10 q^{19} - 4 q^{20} - 4 q^{22} - 8 q^{23} - 3 q^{25} - 8 q^{26} - 4 q^{28} - 4 q^{29} - 17 q^{31} + 9 q^{32} - q^{34} - 10 q^{35} - 11 q^{37} - 10 q^{38} - 4 q^{40} - 16 q^{43} - 4 q^{44} - 8 q^{46} - 7 q^{47} - 5 q^{49} - 3 q^{50} - 8 q^{52} - 12 q^{53} - 23 q^{55} - 4 q^{56} - 4 q^{58} - 6 q^{59} - 13 q^{61} - 17 q^{62} + 9 q^{64} + 24 q^{65} - 14 q^{67} - q^{68} - 10 q^{70} - 30 q^{71} - 12 q^{73} - 11 q^{74} - 10 q^{76} - 12 q^{77} - 35 q^{79} - 4 q^{80} - 5 q^{83} - 27 q^{85} - 16 q^{86} - 4 q^{88} - 23 q^{89} - 28 q^{91} - 8 q^{92} - 7 q^{94} - 32 q^{95} - 21 q^{97} - 5 q^{98}+O(q^{100}) \) 9 * q + 9 * q^2 + 9 * q^4 - 4 * q^5 - 4 * q^7 + 9 * q^8 - 4 * q^10 - 4 * q^11 - 8 * q^13 - 4 * q^14 + 9 * q^16 - q^17 - 10 * q^19 - 4 * q^20 - 4 * q^22 - 8 * q^23 - 3 * q^25 - 8 * q^26 - 4 * q^28 - 4 * q^29 - 17 * q^31 + 9 * q^32 - q^34 - 10 * q^35 - 11 * q^37 - 10 * q^38 - 4 * q^40 - 16 * q^43 - 4 * q^44 - 8 * q^46 - 7 * q^47 - 5 * q^49 - 3 * q^50 - 8 * q^52 - 12 * q^53 - 23 * q^55 - 4 * q^56 - 4 * q^58 - 6 * q^59 - 13 * q^61 - 17 * q^62 + 9 * q^64 + 24 * q^65 - 14 * q^67 - q^68 - 10 * q^70 - 30 * q^71 - 12 * q^73 - 11 * q^74 - 10 * q^76 - 12 * q^77 - 35 * q^79 - 4 * q^80 - 5 * q^83 - 27 * q^85 - 16 * q^86 - 4 * q^88 - 23 * q^89 - 28 * q^91 - 8 * q^92 - 7 * q^94 - 32 * q^95 - 21 * q^97 - 5 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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