LMFDB - Embedded newform 8046.2.a.e.1.1

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			1.1
Root			$-2.92810$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -2.92810 q^{5} -3.28444 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -2.92810 q^{5} -3.28444 q^{7} -1.00000 q^{8} +2.92810 q^{10} +0.882540 q^{11} -2.93300 q^{13} +3.28444 q^{14} +1.00000 q^{16} -2.03147 q^{17} -2.83285 q^{19} -2.92810 q^{20} -0.882540 q^{22} +7.21415 q^{23} +3.57377 q^{25} +2.93300 q^{26} -3.28444 q^{28} +1.57885 q^{29} +3.45178 q^{31} -1.00000 q^{32} +2.03147 q^{34} +9.61715 q^{35} +0.228089 q^{37} +2.83285 q^{38} +2.92810 q^{40} -7.96529 q^{41} +1.77174 q^{43} +0.882540 q^{44} -7.21415 q^{46} +4.83874 q^{47} +3.78751 q^{49} -3.57377 q^{50} -2.93300 q^{52} +6.05357 q^{53} -2.58417 q^{55} +3.28444 q^{56} -1.57885 q^{58} +14.7910 q^{59} +8.51800 q^{61} -3.45178 q^{62} +1.00000 q^{64} +8.58811 q^{65} -11.2723 q^{67} -2.03147 q^{68} -9.61715 q^{70} -3.74035 q^{71} +5.46651 q^{73} -0.228089 q^{74} -2.83285 q^{76} -2.89865 q^{77} -12.9804 q^{79} -2.92810 q^{80} +7.96529 q^{82} +8.73746 q^{83} +5.94836 q^{85} -1.77174 q^{86} -0.882540 q^{88} +18.0397 q^{89} +9.63324 q^{91} +7.21415 q^{92} -4.83874 q^{94} +8.29488 q^{95} +16.4705 q^{97} -3.78751 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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
$5$	−2.92810	−1.30949	−0.654743	−	0.755852i	$-0.727223\pi$
$5$	−2.92810	−1.30949	−0.654743	+	0.755852i	$0.727223\pi$
$6$	0	0
$7$	−3.28444	−1.24140	−0.620700	−	0.784048i	$-0.713151\pi$
$7$	−3.28444	−1.24140	−0.620700	+	0.784048i	$0.713151\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	2.92810	0.925946
$11$	0.882540	0.266096	0.133048	−	0.991110i	$-0.457524\pi$
$11$	0.882540	0.266096	0.133048	+	0.991110i	$0.457524\pi$
$12$	0	0
$13$	−2.93300	−0.813468	−0.406734	−	0.913547i	$-0.633332\pi$
$13$	−2.93300	−0.813468	−0.406734	+	0.913547i	$0.633332\pi$
$14$	3.28444	0.877802
$15$	0	0
$16$	1.00000	0.250000
$17$	−2.03147	−0.492705	−0.246352	−	0.969180i	$-0.579232\pi$
$17$	−2.03147	−0.492705	−0.246352	+	0.969180i	$0.579232\pi$
$18$	0	0
$19$	−2.83285	−0.649901	−0.324951	−	0.945731i	$-0.605348\pi$
$19$	−2.83285	−0.649901	−0.324951	+	0.945731i	$0.605348\pi$
$20$	−2.92810	−0.654743
$21$	0	0
$22$	−0.882540	−0.188158
$23$	7.21415	1.50425	0.752127	−	0.659018i	$-0.229028\pi$
$23$	7.21415	1.50425	0.752127	+	0.659018i	$0.229028\pi$
$24$	0	0
$25$	3.57377	0.714754
$26$	2.93300	0.575208
$27$	0	0
$28$	−3.28444	−0.620700
$29$	1.57885	0.293186	0.146593	−	0.989197i	$-0.453169\pi$
$29$	1.57885	0.293186	0.146593	+	0.989197i	$0.453169\pi$
$30$	0	0
$31$	3.45178	0.619958	0.309979	−	0.950743i	$-0.399678\pi$
$31$	3.45178	0.619958	0.309979	+	0.950743i	$0.399678\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	2.03147	0.348395
$35$	9.61715	1.62560
$36$	0	0
$37$	0.228089	0.0374976	0.0187488	−	0.999824i	$-0.494032\pi$
$37$	0.228089	0.0374976	0.0187488	+	0.999824i	$0.494032\pi$
$38$	2.83285	0.459550
$39$	0	0
$40$	2.92810	0.462973
$41$	−7.96529	−1.24397	−0.621985	−	0.783029i	$-0.713673\pi$
$41$	−7.96529	−1.24397	−0.621985	+	0.783029i	$0.713673\pi$
$42$	0	0
$43$	1.77174	0.270188	0.135094	−	0.990833i	$-0.456866\pi$
$43$	1.77174	0.270188	0.135094	+	0.990833i	$0.456866\pi$
$44$	0.882540	0.133048
$45$	0	0
$46$	−7.21415	−1.06367
$47$	4.83874	0.705803	0.352902	−	0.935660i	$-0.385195\pi$
$47$	4.83874	0.705803	0.352902	+	0.935660i	$0.385195\pi$
$48$	0	0
$49$	3.78751	0.541073
$50$	−3.57377	−0.505407
$51$	0	0
$52$	−2.93300	−0.406734
$53$	6.05357	0.831522	0.415761	−	0.909474i	$-0.363515\pi$
$53$	6.05357	0.831522	0.415761	+	0.909474i	$0.363515\pi$
$54$	0	0
$55$	−2.58417	−0.348449
$56$	3.28444	0.438901
$57$	0	0
$58$	−1.57885	−0.207314
$59$	14.7910	1.92563	0.962815	−	0.270161i	$-0.0870769\pi$
$59$	14.7910	1.92563	0.962815	+	0.270161i	$0.0870769\pi$
$60$	0	0
$61$	8.51800	1.09062	0.545309	−	0.838235i	$-0.316412\pi$
$61$	8.51800	1.09062	0.545309	+	0.838235i	$0.316412\pi$
$62$	−3.45178	−0.438376
$63$	0	0
$64$	1.00000	0.125000
$65$	8.58811	1.06522
$66$	0	0
$67$	−11.2723	−1.37713	−0.688564	−	0.725176i	$-0.741759\pi$
$67$	−11.2723	−1.37713	−0.688564	+	0.725176i	$0.741759\pi$
$68$	−2.03147	−0.246352
$69$	0	0
$70$	−9.61715	−1.14947
$71$	−3.74035	−0.443898	−0.221949	−	0.975058i	$-0.571242\pi$
$71$	−3.74035	−0.443898	−0.221949	+	0.975058i	$0.571242\pi$
$72$	0	0
$73$	5.46651	0.639807	0.319903	−	0.947450i	$-0.396350\pi$
$73$	5.46651	0.639807	0.319903	+	0.947450i	$0.396350\pi$
$74$	−0.228089	−0.0265148
$75$	0	0
$76$	−2.83285	−0.324951
$77$	−2.89865	−0.330331
$78$	0	0
$79$	−12.9804	−1.46041	−0.730205	−	0.683228i	$-0.760576\pi$
$79$	−12.9804	−1.46041	−0.730205	+	0.683228i	$0.760576\pi$
$80$	−2.92810	−0.327372
$81$	0	0
$82$	7.96529	0.879619
$83$	8.73746	0.959061	0.479530	−	0.877525i	$-0.340807\pi$
$83$	8.73746	0.959061	0.479530	+	0.877525i	$0.340807\pi$
$84$	0	0
$85$	5.94836	0.645190
$86$	−1.77174	−0.191052
$87$	0	0
$88$	−0.882540	−0.0940791
$89$	18.0397	1.91221	0.956103	−	0.293030i	$-0.0946636\pi$
$89$	18.0397	1.91221	0.956103	+	0.293030i	$0.0946636\pi$
$90$	0	0
$91$	9.63324	1.00984
$92$	7.21415	0.752127
$93$	0	0
$94$	−4.83874	−0.499078
$95$	8.29488	0.851037
$96$	0	0
$97$	16.4705	1.67232	0.836161	−	0.548484i	$-0.184795\pi$
$97$	16.4705	1.67232	0.836161	+	0.548484i	$0.184795\pi$
$98$	−3.78751	−0.382597
$99$	0	0
$100$	3.57377	0.357377
$101$	−7.20293	−0.716719	−0.358359	−	0.933584i	$-0.616664\pi$
$101$	−7.20293	−0.716719	−0.358359	+	0.933584i	$0.616664\pi$
$102$	0	0
$103$	−14.1939	−1.39857	−0.699283	−	0.714845i	$-0.746497\pi$
$103$	−14.1939	−1.39857	−0.699283	+	0.714845i	$0.746497\pi$
$104$	2.93300	0.287604
$105$	0	0
$106$	−6.05357	−0.587975
$107$	−12.9309	−1.25008	−0.625041	−	0.780592i	$-0.714918\pi$
$107$	−12.9309	−1.25008	−0.625041	+	0.780592i	$0.714918\pi$
$108$	0	0
$109$	−4.38665	−0.420165	−0.210082	−	0.977684i	$-0.567373\pi$
$109$	−4.38665	−0.420165	−0.210082	+	0.977684i	$0.567373\pi$
$110$	2.58417	0.246391
$111$	0	0
$112$	−3.28444	−0.310350
$113$	9.74446	0.916681	0.458341	−	0.888777i	$-0.348444\pi$
$113$	9.74446	0.916681	0.458341	+	0.888777i	$0.348444\pi$
$114$	0	0
$115$	−21.1238	−1.96980
$116$	1.57885	0.146593
$117$	0	0
$118$	−14.7910	−1.36163
$119$	6.67224	0.611644
$120$	0	0
$121$	−10.2211	−0.929193
$122$	−8.51800	−0.771183
$123$	0	0
$124$	3.45178	0.309979
$125$	4.17615	0.373526
$126$	0	0
$127$	9.86561	0.875432	0.437716	−	0.899113i	$-0.355788\pi$
$127$	9.86561	0.875432	0.437716	+	0.899113i	$0.355788\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−8.58811	−0.753227
$131$	−7.41721	−0.648045	−0.324022	−	0.946049i	$-0.605035\pi$
$131$	−7.41721	−0.648045	−0.324022	+	0.946049i	$0.605035\pi$
$132$	0	0
$133$	9.30433	0.806787
$134$	11.2723	0.973776
$135$	0	0
$136$	2.03147	0.174197
$137$	7.37103	0.629750	0.314875	−	0.949133i	$-0.398037\pi$
$137$	7.37103	0.629750	0.314875	+	0.949133i	$0.398037\pi$
$138$	0	0
$139$	−22.7790	−1.93209	−0.966046	−	0.258371i	$-0.916814\pi$
$139$	−22.7790	−1.93209	−0.966046	+	0.258371i	$0.916814\pi$
$140$	9.61715	0.812798
$141$	0	0
$142$	3.74035	0.313884
$143$	−2.58849	−0.216460
$144$	0	0
$145$	−4.62304	−0.383923
$146$	−5.46651	−0.452412
$147$	0	0
$148$	0.228089	0.0187488
$149$	−1.00000	−0.0819232
$149$	−1.00000	−0.0819232
$150$	0	0
$151$	−5.80240	−0.472192	−0.236096	−	0.971730i	$-0.575868\pi$
$151$	−5.80240	−0.472192	−0.236096	+	0.971730i	$0.575868\pi$
$152$	2.83285	0.229775
$153$	0	0
$154$	2.89865	0.233580
$155$	−10.1072	−0.811826
$156$	0	0
$157$	−8.84044	−0.705544	−0.352772	−	0.935709i	$-0.614761\pi$
$157$	−8.84044	−0.705544	−0.352772	+	0.935709i	$0.614761\pi$
$158$	12.9804	1.03267
$159$	0	0
$160$	2.92810	0.231487
$161$	−23.6944	−1.86738
$162$	0	0
$163$	16.5483	1.29616	0.648081	−	0.761571i	$-0.275572\pi$
$163$	16.5483	1.29616	0.648081	+	0.761571i	$0.275572\pi$
$164$	−7.96529	−0.621985
$165$	0	0
$166$	−8.73746	−0.678158
$167$	10.6949	0.827599	0.413799	−	0.910368i	$-0.364202\pi$
$167$	10.6949	0.827599	0.413799	+	0.910368i	$0.364202\pi$
$168$	0	0
$169$	−4.39752	−0.338271
$170$	−5.94836	−0.456218
$171$	0	0
$172$	1.77174	0.135094
$173$	20.9368	1.59179	0.795896	−	0.605433i	$-0.207000\pi$
$173$	20.9368	1.59179	0.795896	+	0.605433i	$0.207000\pi$
$174$	0	0
$175$	−11.7378	−0.887295
$176$	0.882540	0.0665240
$177$	0	0
$178$	−18.0397	−1.35213
$179$	18.8188	1.40659	0.703293	−	0.710900i	$-0.251712\pi$
$179$	18.8188	1.40659	0.703293	+	0.710900i	$0.251712\pi$
$180$	0	0
$181$	16.5689	1.23156	0.615778	−	0.787920i	$-0.288842\pi$
$181$	16.5689	1.23156	0.615778	+	0.787920i	$0.288842\pi$
$182$	−9.63324	−0.714064
$183$	0	0
$184$	−7.21415	−0.531834
$185$	−0.667868	−0.0491026
$186$	0	0
$187$	−1.79286	−0.131107
$188$	4.83874	0.352902
$189$	0	0
$190$	−8.29488	−0.601774
$191$	−19.7952	−1.43233	−0.716164	−	0.697932i	$-0.754104\pi$
$191$	−19.7952	−1.43233	−0.716164	+	0.697932i	$0.754104\pi$
$192$	0	0
$193$	10.0680	0.724710	0.362355	−	0.932040i	$-0.381973\pi$
$193$	10.0680	0.724710	0.362355	+	0.932040i	$0.381973\pi$
$194$	−16.4705	−1.18251
$195$	0	0
$196$	3.78751	0.270537
$197$	4.64397	0.330869	0.165435	−	0.986221i	$-0.447097\pi$
$197$	4.64397	0.330869	0.165435	+	0.986221i	$0.447097\pi$
$198$	0	0
$199$	9.51702	0.674644	0.337322	−	0.941389i	$-0.390479\pi$
$199$	9.51702	0.674644	0.337322	+	0.941389i	$0.390479\pi$
$200$	−3.57377	−0.252704
$201$	0	0
$202$	7.20293	0.506797
$203$	−5.18564	−0.363961
$204$	0	0
$205$	23.3232	1.62896
$206$	14.1939	0.988936
$207$	0	0
$208$	−2.93300	−0.203367
$209$	−2.50011	−0.172936
$210$	0	0
$211$	−18.7522	−1.29096	−0.645479	−	0.763778i	$-0.723342\pi$
$211$	−18.7522	−1.29096	−0.645479	+	0.763778i	$0.723342\pi$
$212$	6.05357	0.415761
$213$	0	0
$214$	12.9309	0.883941
$215$	−5.18783	−0.353807
$216$	0	0
$217$	−11.3371	−0.769615
$218$	4.38665	0.297101
$219$	0	0
$220$	−2.58417	−0.174224
$221$	5.95831	0.400799
$222$	0	0
$223$	21.1480	1.41618	0.708089	−	0.706123i	$-0.249558\pi$
$223$	21.1480	1.41618	0.708089	+	0.706123i	$0.249558\pi$
$224$	3.28444	0.219451
$225$	0	0
$226$	−9.74446	−0.648192
$227$	−4.87304	−0.323435	−0.161718	−	0.986837i	$-0.551703\pi$
$227$	−4.87304	−0.323435	−0.161718	+	0.986837i	$0.551703\pi$
$228$	0	0
$229$	−22.8501	−1.50998	−0.754989	−	0.655737i	$-0.772358\pi$
$229$	−22.8501	−1.50998	−0.754989	+	0.655737i	$0.772358\pi$
$230$	21.1238	1.39286
$231$	0	0
$232$	−1.57885	−0.103657
$233$	−30.1911	−1.97788	−0.988942	−	0.148303i	$-0.952619\pi$
$233$	−30.1911	−1.97788	−0.988942	+	0.148303i	$0.952619\pi$
$234$	0	0
$235$	−14.1683	−0.924239
$236$	14.7910	0.962815
$237$	0	0
$238$	−6.67224	−0.432497
$239$	−5.90535	−0.381985	−0.190993	−	0.981591i	$-0.561171\pi$
$239$	−5.90535	−0.381985	−0.190993	+	0.981591i	$0.561171\pi$
$240$	0	0
$241$	−22.5583	−1.45311	−0.726554	−	0.687109i	$-0.758879\pi$
$241$	−22.5583	−1.45311	−0.726554	+	0.687109i	$0.758879\pi$
$242$	10.2211	0.657039
$243$	0	0
$244$	8.51800	0.545309
$245$	−11.0902	−0.708528
$246$	0	0
$247$	8.30876	0.528674
$248$	−3.45178	−0.219188
$249$	0	0
$250$	−4.17615	−0.264123
$251$	6.82729	0.430934	0.215467	−	0.976511i	$-0.430873\pi$
$251$	6.82729	0.430934	0.215467	+	0.976511i	$0.430873\pi$
$252$	0	0
$253$	6.36678	0.400276
$254$	−9.86561	−0.619024
$255$	0	0
$256$	1.00000	0.0625000
$257$	−13.4148	−0.836791	−0.418395	−	0.908265i	$-0.637407\pi$
$257$	−13.4148	−0.836791	−0.418395	+	0.908265i	$0.637407\pi$
$258$	0	0
$259$	−0.749144	−0.0465496
$260$	8.58811	0.532612
$261$	0	0
$262$	7.41721	0.458237
$263$	−16.7645	−1.03375	−0.516873	−	0.856062i	$-0.672904\pi$
$263$	−16.7645	−1.03375	−0.516873	+	0.856062i	$0.672904\pi$
$264$	0	0
$265$	−17.7255	−1.08887
$266$	−9.30433	−0.570485
$267$	0	0
$268$	−11.2723	−0.688564
$269$	21.1099	1.28709	0.643546	−	0.765407i	$-0.277462\pi$
$269$	21.1099	1.28709	0.643546	+	0.765407i	$0.277462\pi$
$270$	0	0
$271$	−7.82519	−0.475346	−0.237673	−	0.971345i	$-0.576385\pi$
$271$	−7.82519	−0.475346	−0.237673	+	0.971345i	$0.576385\pi$
$272$	−2.03147	−0.123176
$273$	0	0
$274$	−7.37103	−0.445300
$275$	3.15399	0.190193
$276$	0	0
$277$	12.1519	0.730138	0.365069	−	0.930980i	$-0.381045\pi$
$277$	12.1519	0.730138	0.365069	+	0.930980i	$0.381045\pi$
$278$	22.7790	1.36619
$279$	0	0
$280$	−9.61715	−0.574735
$281$	−18.3752	−1.09617	−0.548086	−	0.836422i	$-0.684644\pi$
$281$	−18.3752	−1.09617	−0.548086	+	0.836422i	$0.684644\pi$
$282$	0	0
$283$	0.982959	0.0584308	0.0292154	−	0.999573i	$-0.490699\pi$
$283$	0.982959	0.0584308	0.0292154	+	0.999573i	$0.490699\pi$
$284$	−3.74035	−0.221949
$285$	0	0
$286$	2.58849	0.153061
$287$	26.1615	1.54426
$288$	0	0
$289$	−12.8731	−0.757242
$290$	4.62304	0.271474
$291$	0	0
$292$	5.46651	0.319903
$293$	−2.61054	−0.152510	−0.0762548	−	0.997088i	$-0.524296\pi$
$293$	−2.61054	−0.152510	−0.0762548	+	0.997088i	$0.524296\pi$
$294$	0	0
$295$	−43.3097	−2.52159
$296$	−0.228089	−0.0132574
$297$	0	0
$298$	1.00000	0.0579284
$299$	−21.1591	−1.22366
$300$	0	0
$301$	−5.81917	−0.335411
$302$	5.80240	0.333890
$303$	0	0
$304$	−2.83285	−0.162475
$305$	−24.9415	−1.42815
$306$	0	0
$307$	−16.1605	−0.922326	−0.461163	−	0.887315i	$-0.652568\pi$
$307$	−16.1605	−0.922326	−0.461163	+	0.887315i	$0.652568\pi$
$308$	−2.89865	−0.165166
$309$	0	0
$310$	10.1072	0.574048
$311$	−0.432433	−0.0245210	−0.0122605	−	0.999925i	$-0.503903\pi$
$311$	−0.432433	−0.0245210	−0.0122605	+	0.999925i	$0.503903\pi$
$312$	0	0
$313$	8.87878	0.501858	0.250929	−	0.968005i	$-0.419264\pi$
$313$	8.87878	0.501858	0.250929	+	0.968005i	$0.419264\pi$
$314$	8.84044	0.498895
$315$	0	0
$316$	−12.9804	−0.730205
$317$	10.1671	0.571044	0.285522	−	0.958372i	$-0.407833\pi$
$317$	10.1671	0.571044	0.285522	+	0.958372i	$0.407833\pi$
$318$	0	0
$319$	1.39340	0.0780155
$320$	−2.92810	−0.163686
$321$	0	0
$322$	23.6944	1.32044
$323$	5.75487	0.320210
$324$	0	0
$325$	−10.4819	−0.581429
$326$	−16.5483	−0.916525
$327$	0	0
$328$	7.96529	0.439810
$329$	−15.8925	−0.876184
$330$	0	0
$331$	29.9018	1.64355	0.821774	−	0.569813i	$-0.192984\pi$
$331$	29.9018	1.64355	0.821774	+	0.569813i	$0.192984\pi$
$332$	8.73746	0.479530
$333$	0	0
$334$	−10.6949	−0.585201
$335$	33.0064	1.80333
$336$	0	0
$337$	6.15618	0.335348	0.167674	−	0.985842i	$-0.446374\pi$
$337$	6.15618	0.335348	0.167674	+	0.985842i	$0.446374\pi$
$338$	4.39752	0.239193
$339$	0	0
$340$	5.94836	0.322595
$341$	3.04633	0.164968
$342$	0	0
$343$	10.5512	0.569711
$344$	−1.77174	−0.0955258
$345$	0	0
$346$	−20.9368	−1.12557
$347$	−18.4422	−0.990029	−0.495014	−	0.868885i	$-0.664837\pi$
$347$	−18.4422	−0.990029	−0.495014	+	0.868885i	$0.664837\pi$
$348$	0	0
$349$	7.85441	0.420437	0.210219	−	0.977654i	$-0.432582\pi$
$349$	7.85441	0.420437	0.210219	+	0.977654i	$0.432582\pi$
$350$	11.7378	0.627412
$351$	0	0
$352$	−0.882540	−0.0470395
$353$	−14.6172	−0.777993	−0.388997	−	0.921239i	$-0.627178\pi$
$353$	−14.6172	−0.777993	−0.388997	+	0.921239i	$0.627178\pi$
$354$	0	0
$355$	10.9521	0.581279
$356$	18.0397	0.956103
$357$	0	0
$358$	−18.8188	−0.994607
$359$	−7.75659	−0.409377	−0.204689	−	0.978827i	$-0.565618\pi$
$359$	−7.75659	−0.409377	−0.204689	+	0.978827i	$0.565618\pi$
$360$	0	0
$361$	−10.9749	−0.577628
$362$	−16.5689	−0.870841
$363$	0	0
$364$	9.63324	0.504919
$365$	−16.0065	−0.837818
$366$	0	0
$367$	−3.90960	−0.204080	−0.102040	−	0.994780i	$-0.532537\pi$
$367$	−3.90960	−0.204080	−0.102040	+	0.994780i	$0.532537\pi$
$368$	7.21415	0.376064
$369$	0	0
$370$	0.667868	0.0347208
$371$	−19.8826	−1.03225
$372$	0	0
$373$	−22.8705	−1.18419	−0.592095	−	0.805868i	$-0.701699\pi$
$373$	−22.8705	−1.18419	−0.592095	+	0.805868i	$0.701699\pi$
$374$	1.79286	0.0927064
$375$	0	0
$376$	−4.83874	−0.249539
$377$	−4.63078	−0.238497
$378$	0	0
$379$	3.28352	0.168663	0.0843314	−	0.996438i	$-0.473125\pi$
$379$	3.28352	0.168663	0.0843314	+	0.996438i	$0.473125\pi$
$380$	8.29488	0.425518
$381$	0	0
$382$	19.7952	1.01281
$383$	−13.2142	−0.675212	−0.337606	−	0.941288i	$-0.609617\pi$
$383$	−13.2142	−0.675212	−0.337606	+	0.941288i	$0.609617\pi$
$384$	0	0
$385$	8.48752	0.432564
$386$	−10.0680	−0.512447
$387$	0	0
$388$	16.4705	0.836161
$389$	−11.3035	−0.573109	−0.286555	−	0.958064i	$-0.592510\pi$
$389$	−11.3035	−0.573109	−0.286555	+	0.958064i	$0.592510\pi$
$390$	0	0
$391$	−14.6554	−0.741153
$392$	−3.78751	−0.191298
$393$	0	0
$394$	−4.64397	−0.233960
$395$	38.0080	1.91239
$396$	0	0
$397$	−10.1581	−0.509821	−0.254910	−	0.966965i	$-0.582046\pi$
$397$	−10.1581	−0.509821	−0.254910	+	0.966965i	$0.582046\pi$
$398$	−9.51702	−0.477045
$399$	0	0
$400$	3.57377	0.178688
$401$	−1.13215	−0.0565370	−0.0282685	−	0.999600i	$-0.508999\pi$
$401$	−1.13215	−0.0565370	−0.0282685	+	0.999600i	$0.508999\pi$
$402$	0	0
$403$	−10.1241	−0.504315
$404$	−7.20293	−0.358359
$405$	0	0
$406$	5.18564	0.257359
$407$	0.201298	0.00997796
$408$	0	0
$409$	−34.5374	−1.70776	−0.853882	−	0.520467i	$-0.825758\pi$
$409$	−34.5374	−1.70776	−0.853882	+	0.520467i	$0.825758\pi$
$410$	−23.3232	−1.15185
$411$	0	0
$412$	−14.1939	−0.699283
$413$	−48.5802	−2.39048
$414$	0	0
$415$	−25.5842	−1.25588
$416$	2.93300	0.143802
$417$	0	0
$418$	2.50011	0.122284
$419$	−18.8588	−0.921315	−0.460657	−	0.887578i	$-0.652386\pi$
$419$	−18.8588	−0.921315	−0.460657	+	0.887578i	$0.652386\pi$
$420$	0	0
$421$	−15.1253	−0.737164	−0.368582	−	0.929595i	$-0.620157\pi$
$421$	−15.1253	−0.737164	−0.368582	+	0.929595i	$0.620157\pi$
$422$	18.7522	0.912844
$423$	0	0
$424$	−6.05357	−0.293987
$425$	−7.26002	−0.352163
$426$	0	0
$427$	−27.9768	−1.35389
$428$	−12.9309	−0.625041
$429$	0	0
$430$	5.18783	0.250180
$431$	28.2587	1.36117	0.680587	−	0.732667i	$-0.261725\pi$
$431$	28.2587	1.36117	0.680587	+	0.732667i	$0.261725\pi$
$432$	0	0
$433$	14.3336	0.688829	0.344415	−	0.938818i	$-0.388077\pi$
$433$	14.3336	0.688829	0.344415	+	0.938818i	$0.388077\pi$
$434$	11.3371	0.544200
$435$	0	0
$436$	−4.38665	−0.210082
$437$	−20.4366	−0.977617
$438$	0	0
$439$	3.40109	0.162325	0.0811625	−	0.996701i	$-0.474137\pi$
$439$	3.40109	0.162325	0.0811625	+	0.996701i	$0.474137\pi$
$440$	2.58417	0.123195
$441$	0	0
$442$	−5.95831	−0.283408
$443$	−18.2385	−0.866536	−0.433268	−	0.901265i	$-0.642640\pi$
$443$	−18.2385	−0.866536	−0.433268	+	0.901265i	$0.642640\pi$
$444$	0	0
$445$	−52.8221	−2.50401
$446$	−21.1480	−1.00139
$447$	0	0
$448$	−3.28444	−0.155175
$449$	−9.71826	−0.458633	−0.229317	−	0.973352i	$-0.573649\pi$
$449$	−9.71826	−0.458633	−0.229317	+	0.973352i	$0.573649\pi$
$450$	0	0
$451$	−7.02969	−0.331015
$452$	9.74446	0.458341
$453$	0	0
$454$	4.87304	0.228703
$455$	−28.2071	−1.32237
$456$	0	0
$457$	19.7974	0.926086	0.463043	−	0.886336i	$-0.346758\pi$
$457$	19.7974	0.926086	0.463043	+	0.886336i	$0.346758\pi$
$458$	22.8501	1.06772
$459$	0	0
$460$	−21.1238	−0.984900
$461$	−2.22409	−0.103586	−0.0517930	−	0.998658i	$-0.516494\pi$
$461$	−2.22409	−0.103586	−0.0517930	+	0.998658i	$0.516494\pi$
$462$	0	0
$463$	4.50016	0.209140	0.104570	−	0.994518i	$-0.466653\pi$
$463$	4.50016	0.209140	0.104570	+	0.994518i	$0.466653\pi$
$464$	1.57885	0.0732965
$465$	0	0
$466$	30.1911	1.39858
$467$	−38.6792	−1.78986	−0.894930	−	0.446207i	$-0.852774\pi$
$467$	−38.6792	−1.78986	−0.894930	+	0.446207i	$0.852774\pi$
$468$	0	0
$469$	37.0231	1.70957
$470$	14.1683	0.653536
$471$	0	0
$472$	−14.7910	−0.680813
$473$	1.56363	0.0718959
$474$	0	0
$475$	−10.1240	−0.464519
$476$	6.67224	0.305822
$477$	0	0
$478$	5.90535	0.270104
$479$	21.4301	0.979166	0.489583	−	0.871957i	$-0.337149\pi$
$479$	21.4301	0.979166	0.489583	+	0.871957i	$0.337149\pi$
$480$	0	0
$481$	−0.668985	−0.0305031
$482$	22.5583	1.02750
$483$	0	0
$484$	−10.2211	−0.464596
$485$	−48.2272	−2.18988
$486$	0	0
$487$	36.0210	1.63227	0.816134	−	0.577862i	$-0.196113\pi$
$487$	36.0210	1.63227	0.816134	+	0.577862i	$0.196113\pi$
$488$	−8.51800	−0.385592
$489$	0	0
$490$	11.0902	0.501005
$491$	17.2987	0.780680	0.390340	−	0.920671i	$-0.372357\pi$
$491$	17.2987	0.780680	0.390340	+	0.920671i	$0.372357\pi$
$492$	0	0
$493$	−3.20740	−0.144454
$494$	−8.30876	−0.373829
$495$	0	0
$496$	3.45178	0.154989
$497$	12.2850	0.551055
$498$	0	0
$499$	17.9026	0.801431	0.400716	−	0.916203i	$-0.368762\pi$
$499$	17.9026	0.801431	0.400716	+	0.916203i	$0.368762\pi$
$500$	4.17615	0.186763
$501$	0	0
$502$	−6.82729	−0.304717
$503$	−9.39950	−0.419103	−0.209551	−	0.977798i	$-0.567200\pi$
$503$	−9.39950	−0.419103	−0.209551	+	0.977798i	$0.567200\pi$
$504$	0	0
$505$	21.0909	0.938533
$506$	−6.36678	−0.283038
$507$	0	0
$508$	9.86561	0.437716
$509$	6.56975	0.291199	0.145600	−	0.989344i	$-0.453489\pi$
$509$	6.56975	0.291199	0.145600	+	0.989344i	$0.453489\pi$
$510$	0	0
$511$	−17.9544	−0.794256
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	13.4148	0.591701
$515$	41.5612	1.83140
$516$	0	0
$517$	4.27038	0.187811
$518$	0.749144	0.0329155
$519$	0	0
$520$	−8.58811	−0.376614
$521$	−4.90848	−0.215044	−0.107522	−	0.994203i	$-0.534292\pi$
$521$	−4.90848	−0.215044	−0.107522	+	0.994203i	$0.534292\pi$
$522$	0	0
$523$	−13.6718	−0.597824	−0.298912	−	0.954281i	$-0.596624\pi$
$523$	−13.6718	−0.597824	−0.298912	+	0.954281i	$0.596624\pi$
$524$	−7.41721	−0.324022
$525$	0	0
$526$	16.7645	0.730969
$527$	−7.01220	−0.305456
$528$	0	0
$529$	29.0440	1.26278
$530$	17.7255	0.769945
$531$	0	0
$532$	9.30433	0.403394
$533$	23.3622	1.01193
$534$	0	0
$535$	37.8631	1.63697
$536$	11.2723	0.486888
$537$	0	0
$538$	−21.1099	−0.910112
$539$	3.34263	0.143977
$540$	0	0
$541$	−38.7692	−1.66682	−0.833410	−	0.552656i	$-0.813614\pi$
$541$	−38.7692	−1.66682	−0.833410	+	0.552656i	$0.813614\pi$
$542$	7.82519	0.336121
$543$	0	0
$544$	2.03147	0.0870987
$545$	12.8446	0.550200
$546$	0	0
$547$	−6.89260	−0.294706	−0.147353	−	0.989084i	$-0.547075\pi$
$547$	−6.89260	−0.294706	−0.147353	+	0.989084i	$0.547075\pi$
$548$	7.37103	0.314875
$549$	0	0
$550$	−3.15399	−0.134487
$551$	−4.47266	−0.190542
$552$	0	0
$553$	42.6333	1.81295
$554$	−12.1519	−0.516286
$555$	0	0
$556$	−22.7790	−0.966046
$557$	11.4862	0.486686	0.243343	−	0.969940i	$-0.421756\pi$
$557$	11.4862	0.486686	0.243343	+	0.969940i	$0.421756\pi$
$558$	0	0
$559$	−5.19651	−0.219789
$560$	9.61715	0.406399
$561$	0	0
$562$	18.3752	0.775111
$563$	31.8104	1.34065	0.670324	−	0.742068i	$-0.266155\pi$
$563$	31.8104	1.34065	0.670324	+	0.742068i	$0.266155\pi$
$564$	0	0
$565$	−28.5327	−1.20038
$566$	−0.982959	−0.0413168
$567$	0	0
$568$	3.74035	0.156942
$569$	22.0327	0.923660	0.461830	−	0.886968i	$-0.347193\pi$
$569$	22.0327	0.923660	0.461830	+	0.886968i	$0.347193\pi$
$570$	0	0
$571$	1.22310	0.0511853	0.0255927	−	0.999672i	$-0.491853\pi$
$571$	1.22310	0.0511853	0.0255927	+	0.999672i	$0.491853\pi$
$572$	−2.58849	−0.108230
$573$	0	0
$574$	−26.1615	−1.09196
$575$	25.7817	1.07517
$576$	0	0
$577$	12.8463	0.534797	0.267398	−	0.963586i	$-0.413836\pi$
$577$	12.8463	0.534797	0.267398	+	0.963586i	$0.413836\pi$
$578$	12.8731	0.535451
$579$	0	0
$580$	−4.62304	−0.191961
$581$	−28.6976	−1.19058
$582$	0	0
$583$	5.34252	0.221265
$584$	−5.46651	−0.226206
$585$	0	0
$586$	2.61054	0.107841
$587$	33.6997	1.39094	0.695468	−	0.718557i	$-0.255197\pi$
$587$	33.6997	1.39094	0.695468	+	0.718557i	$0.255197\pi$
$588$	0	0
$589$	−9.77838	−0.402911
$590$	43.3097	1.78303
$591$	0	0
$592$	0.228089	0.00937441
$593$	−21.8230	−0.896165	−0.448082	−	0.893992i	$-0.647893\pi$
$593$	−21.8230	−0.896165	−0.448082	+	0.893992i	$0.647893\pi$
$594$	0	0
$595$	−19.5370	−0.800939
$596$	−1.00000	−0.0409616
$597$	0	0
$598$	21.1591	0.865260
$599$	5.62120	0.229676	0.114838	−	0.993384i	$-0.463365\pi$
$599$	5.62120	0.229676	0.114838	+	0.993384i	$0.463365\pi$
$600$	0	0
$601$	39.9499	1.62959	0.814795	−	0.579749i	$-0.196850\pi$
$601$	39.9499	1.62959	0.814795	+	0.579749i	$0.196850\pi$
$602$	5.81917	0.237172
$603$	0	0
$604$	−5.80240	−0.236096
$605$	29.9285	1.21677
$606$	0	0
$607$	11.0254	0.447506	0.223753	−	0.974646i	$-0.428169\pi$
$607$	11.0254	0.447506	0.223753	+	0.974646i	$0.428169\pi$
$608$	2.83285	0.114887
$609$	0	0
$610$	24.9415	1.00985
$611$	−14.1920	−0.574148
$612$	0	0
$613$	24.6057	0.993816	0.496908	−	0.867803i	$-0.334469\pi$
$613$	24.6057	0.993816	0.496908	+	0.867803i	$0.334469\pi$
$614$	16.1605	0.652183
$615$	0	0
$616$	2.89865	0.116790
$617$	−41.6450	−1.67656	−0.838282	−	0.545237i	$-0.816440\pi$
$617$	−41.6450	−1.67656	−0.838282	+	0.545237i	$0.816440\pi$
$618$	0	0
$619$	5.10642	0.205245	0.102622	−	0.994720i	$-0.467277\pi$
$619$	5.10642	0.205245	0.102622	+	0.994720i	$0.467277\pi$
$620$	−10.1072	−0.405913
$621$	0	0
$622$	0.432433	0.0173390
$623$	−59.2503	−2.37381
$624$	0	0
$625$	−30.0970	−1.20388
$626$	−8.87878	−0.354867
$627$	0	0
$628$	−8.84044	−0.352772
$629$	−0.463357	−0.0184753
$630$	0	0
$631$	−41.1215	−1.63702	−0.818511	−	0.574491i	$-0.805200\pi$
$631$	−41.1215	−1.63702	−0.818511	+	0.574491i	$0.805200\pi$
$632$	12.9804	0.516333
$633$	0	0
$634$	−10.1671	−0.403789
$635$	−28.8875	−1.14637
$636$	0	0
$637$	−11.1088	−0.440146
$638$	−1.39340	−0.0551653
$639$	0	0
$640$	2.92810	0.115743
$641$	−13.0202	−0.514267	−0.257133	−	0.966376i	$-0.582778\pi$
$641$	−13.0202	−0.514267	−0.257133	+	0.966376i	$0.582778\pi$
$642$	0	0
$643$	7.09822	0.279927	0.139963	−	0.990157i	$-0.455302\pi$
$643$	7.09822	0.279927	0.139963	+	0.990157i	$0.455302\pi$
$644$	−23.6944	−0.933690
$645$	0	0
$646$	−5.75487	−0.226422
$647$	14.8880	0.585306	0.292653	−	0.956219i	$-0.405462\pi$
$647$	14.8880	0.585306	0.292653	+	0.956219i	$0.405462\pi$
$648$	0	0
$649$	13.0537	0.512402
$650$	10.4819	0.411132
$651$	0	0
$652$	16.5483	0.648081
$653$	4.61686	0.180672	0.0903358	−	0.995911i	$-0.471206\pi$
$653$	4.61686	0.180672	0.0903358	+	0.995911i	$0.471206\pi$
$654$	0	0
$655$	21.7183	0.848605
$656$	−7.96529	−0.310992
$657$	0	0
$658$	15.8925	0.619555
$659$	31.2747	1.21829	0.609145	−	0.793059i	$-0.291513\pi$
$659$	31.2747	1.21829	0.609145	+	0.793059i	$0.291513\pi$
$660$	0	0
$661$	−15.7499	−0.612600	−0.306300	−	0.951935i	$-0.599091\pi$
$661$	−15.7499	−0.612600	−0.306300	+	0.951935i	$0.599091\pi$
$662$	−29.9018	−1.16216
$663$	0	0
$664$	−8.73746	−0.339079
$665$	−27.2440	−1.05648
$666$	0	0
$667$	11.3901	0.441026
$668$	10.6949	0.413799
$669$	0	0
$670$	−33.0064	−1.27515
$671$	7.51748	0.290209
$672$	0	0
$673$	−46.8150	−1.80458	−0.902292	−	0.431125i	$-0.858117\pi$
$673$	−46.8150	−1.80458	−0.902292	+	0.431125i	$0.858117\pi$
$674$	−6.15618	−0.237127
$675$	0	0
$676$	−4.39752	−0.169135
$677$	−8.66531	−0.333035	−0.166518	−	0.986038i	$-0.553252\pi$
$677$	−8.66531	−0.333035	−0.166518	+	0.986038i	$0.553252\pi$
$678$	0	0
$679$	−54.0962	−2.07602
$680$	−5.94836	−0.228109
$681$	0	0
$682$	−3.04633	−0.116650
$683$	−49.9102	−1.90976	−0.954880	−	0.296993i	$-0.904016\pi$
$683$	−49.9102	−1.90976	−0.954880	+	0.296993i	$0.904016\pi$
$684$	0	0
$685$	−21.5831	−0.824648
$686$	−10.5512	−0.402847
$687$	0	0
$688$	1.77174	0.0675470
$689$	−17.7551	−0.676416
$690$	0	0
$691$	−17.9155	−0.681536	−0.340768	−	0.940147i	$-0.610687\pi$
$691$	−17.9155	−0.681536	−0.340768	+	0.940147i	$0.610687\pi$
$692$	20.9368	0.795896
$693$	0	0
$694$	18.4422	0.700056
$695$	66.6992	2.53005
$696$	0	0
$697$	16.1813	0.612910
$698$	−7.85441	−0.297294
$699$	0	0
$700$	−11.7378	−0.443648
$701$	13.0644	0.493435	0.246718	−	0.969087i	$-0.420648\pi$
$701$	13.0644	0.493435	0.246718	+	0.969087i	$0.420648\pi$
$702$	0	0
$703$	−0.646143	−0.0243698
$704$	0.882540	0.0332620
$705$	0	0
$706$	14.6172	0.550124
$707$	23.6576	0.889734
$708$	0	0
$709$	15.3502	0.576490	0.288245	−	0.957557i	$-0.406928\pi$
$709$	15.3502	0.576490	0.288245	+	0.957557i	$0.406928\pi$
$710$	−10.9521	−0.411026
$711$	0	0
$712$	−18.0397	−0.676067
$713$	24.9016	0.932574
$714$	0	0
$715$	7.57935	0.283452
$716$	18.8188	0.703293
$717$	0	0
$718$	7.75659	0.289473
$719$	−40.8964	−1.52518	−0.762589	−	0.646883i	$-0.776072\pi$
$719$	−40.8964	−1.52518	−0.762589	+	0.646883i	$0.776072\pi$
$720$	0	0
$721$	46.6189	1.73618
$722$	10.9749	0.408445
$723$	0	0
$724$	16.5689	0.615778
$725$	5.64246	0.209556
$726$	0	0
$727$	−8.05888	−0.298887	−0.149444	−	0.988770i	$-0.547748\pi$
$727$	−8.05888	−0.298887	−0.149444	+	0.988770i	$0.547748\pi$
$728$	−9.63324	−0.357032
$729$	0	0
$730$	16.0065	0.592427
$731$	−3.59924	−0.133123
$732$	0	0
$733$	5.47387	0.202182	0.101091	−	0.994877i	$-0.467767\pi$
$733$	5.47387	0.202182	0.101091	+	0.994877i	$0.467767\pi$
$734$	3.90960	0.144306
$735$	0	0
$736$	−7.21415	−0.265917
$737$	−9.94824	−0.366448
$738$	0	0
$739$	−7.69896	−0.283211	−0.141605	−	0.989923i	$-0.545226\pi$
$739$	−7.69896	−0.283211	−0.141605	+	0.989923i	$0.545226\pi$
$740$	−0.667868	−0.0245513
$741$	0	0
$742$	19.8826	0.729912
$743$	25.5029	0.935611	0.467805	−	0.883831i	$-0.345045\pi$
$743$	25.5029	0.935611	0.467805	+	0.883831i	$0.345045\pi$
$744$	0	0
$745$	2.92810	0.107277
$746$	22.8705	0.837348
$747$	0	0
$748$	−1.79286	−0.0655534
$749$	42.4709	1.55185
$750$	0	0
$751$	16.6322	0.606919	0.303459	−	0.952844i	$-0.401858\pi$
$751$	16.6322	0.606919	0.303459	+	0.952844i	$0.401858\pi$
$752$	4.83874	0.176451
$753$	0	0
$754$	4.63078	0.168643
$755$	16.9900	0.618329
$756$	0	0
$757$	37.7780	1.37306	0.686532	−	0.727099i	$-0.259132\pi$
$757$	37.7780	1.37306	0.686532	+	0.727099i	$0.259132\pi$
$758$	−3.28352	−0.119263
$759$	0	0
$760$	−8.29488	−0.300887
$761$	6.26182	0.226991	0.113495	−	0.993539i	$-0.463795\pi$
$761$	6.26182	0.226991	0.113495	+	0.993539i	$0.463795\pi$
$762$	0	0
$763$	14.4077	0.521593
$764$	−19.7952	−0.716164
$765$	0	0
$766$	13.2142	0.477447
$767$	−43.3821	−1.56644
$768$	0	0
$769$	4.04232	0.145770	0.0728850	−	0.997340i	$-0.476779\pi$
$769$	4.04232	0.145770	0.0728850	+	0.997340i	$0.476779\pi$
$770$	−8.48752	−0.305869
$771$	0	0
$772$	10.0680	0.362355
$773$	4.01980	0.144582	0.0722910	−	0.997384i	$-0.476969\pi$
$773$	4.01980	0.144582	0.0722910	+	0.997384i	$0.476969\pi$
$774$	0	0
$775$	12.3359	0.443117
$776$	−16.4705	−0.591255
$777$	0	0
$778$	11.3035	0.405249
$779$	22.5645	0.808457
$780$	0	0
$781$	−3.30101	−0.118120
$782$	14.6554	0.524075
$783$	0	0
$784$	3.78751	0.135268
$785$	25.8857	0.923900
$786$	0	0
$787$	19.7867	0.705319	0.352659	−	0.935752i	$-0.385277\pi$
$787$	19.7867	0.705319	0.352659	+	0.935752i	$0.385277\pi$
$788$	4.64397	0.165435
$789$	0	0
$790$	−38.0080	−1.35226
$791$	−32.0050	−1.13797
$792$	0	0
$793$	−24.9833	−0.887182
$794$	10.1581	0.360498
$795$	0	0
$796$	9.51702	0.337322
$797$	−14.1018	−0.499511	−0.249755	−	0.968309i	$-0.580350\pi$
$797$	−14.1018	−0.499511	−0.249755	+	0.968309i	$0.580350\pi$
$798$	0	0
$799$	−9.82978	−0.347753
$800$	−3.57377	−0.126352
$801$	0	0
$802$	1.13215	0.0399777
$803$	4.82442	0.170250
$804$	0	0
$805$	69.3796	2.44531
$806$	10.1241	0.356605
$807$	0	0
$808$	7.20293	0.253398
$809$	−14.4010	−0.506312	−0.253156	−	0.967426i	$-0.581469\pi$
$809$	−14.4010	−0.506312	−0.253156	+	0.967426i	$0.581469\pi$
$810$	0	0
$811$	−17.3249	−0.608361	−0.304181	−	0.952614i	$-0.598383\pi$
$811$	−17.3249	−0.608361	−0.304181	+	0.952614i	$0.598383\pi$
$812$	−5.18564	−0.181980
$813$	0	0
$814$	−0.201298	−0.00705549
$815$	−48.4550	−1.69731
$816$	0	0
$817$	−5.01908	−0.175595
$818$	34.5374	1.20757
$819$	0	0
$820$	23.3232	0.814480
$821$	−28.2019	−0.984251	−0.492126	−	0.870524i	$-0.663780\pi$
$821$	−28.2019	−0.984251	−0.492126	+	0.870524i	$0.663780\pi$
$822$	0	0
$823$	−43.5024	−1.51640	−0.758199	−	0.652023i	$-0.773920\pi$
$823$	−43.5024	−1.51640	−0.758199	+	0.652023i	$0.773920\pi$
$824$	14.1939	0.494468
$825$	0	0
$826$	48.5802	1.69032
$827$	−7.81195	−0.271648	−0.135824	−	0.990733i	$-0.543368\pi$
$827$	−7.81195	−0.271648	−0.135824	+	0.990733i	$0.543368\pi$
$828$	0	0
$829$	−28.0785	−0.975206	−0.487603	−	0.873065i	$-0.662129\pi$
$829$	−28.0785	−0.975206	−0.487603	+	0.873065i	$0.662129\pi$
$830$	25.5842	0.888039
$831$	0	0
$832$	−2.93300	−0.101683
$833$	−7.69424	−0.266589
$834$	0	0
$835$	−31.3158	−1.08373
$836$	−2.50011	−0.0864680
$837$	0	0
$838$	18.8588	0.651468
$839$	−17.7489	−0.612760	−0.306380	−	0.951909i	$-0.599118\pi$
$839$	−17.7489	−0.612760	−0.306380	+	0.951909i	$0.599118\pi$
$840$	0	0
$841$	−26.5072	−0.914042
$842$	15.1253	0.521254
$843$	0	0
$844$	−18.7522	−0.645479
$845$	12.8764	0.442961
$846$	0	0
$847$	33.5706	1.15350
$848$	6.05357	0.207881
$849$	0	0
$850$	7.26002	0.249017
$851$	1.64547	0.0564060
$852$	0	0
$853$	11.7133	0.401056	0.200528	−	0.979688i	$-0.435734\pi$
$853$	11.7133	0.401056	0.200528	+	0.979688i	$0.435734\pi$
$854$	27.9768	0.957347
$855$	0	0
$856$	12.9309	0.441971
$857$	36.6475	1.25185	0.625927	−	0.779881i	$-0.284721\pi$
$857$	36.6475	1.25185	0.625927	+	0.779881i	$0.284721\pi$
$858$	0	0
$859$	−21.4851	−0.733062	−0.366531	−	0.930406i	$-0.619455\pi$
$859$	−21.4851	−0.733062	−0.366531	+	0.930406i	$0.619455\pi$
$860$	−5.18783	−0.176904
$861$	0	0
$862$	−28.2587	−0.962496
$863$	28.7040	0.977095	0.488547	−	0.872537i	$-0.337527\pi$
$863$	28.7040	0.977095	0.488547	+	0.872537i	$0.337527\pi$
$864$	0	0
$865$	−61.3049	−2.08443
$866$	−14.3336	−0.487076
$867$	0	0
$868$	−11.3371	−0.384808
$869$	−11.4557	−0.388609
$870$	0	0
$871$	33.0616	1.12025
$872$	4.38665	0.148551
$873$	0	0
$874$	20.4366	0.691280
$875$	−13.7163	−0.463695
$876$	0	0
$877$	−12.9202	−0.436284	−0.218142	−	0.975917i	$-0.570000\pi$
$877$	−12.9202	−0.436284	−0.218142	+	0.975917i	$0.570000\pi$
$878$	−3.40109	−0.114781
$879$	0	0
$880$	−2.58417	−0.0871122
$881$	−41.3163	−1.39198	−0.695990	−	0.718051i	$-0.745034\pi$
$881$	−41.3163	−1.39198	−0.695990	+	0.718051i	$0.745034\pi$
$882$	0	0
$883$	31.6676	1.06570	0.532849	−	0.846210i	$-0.321121\pi$
$883$	31.6676	1.06570	0.532849	+	0.846210i	$0.321121\pi$
$884$	5.95831	0.200400
$885$	0	0
$886$	18.2385	0.612734
$887$	22.1811	0.744767	0.372383	−	0.928079i	$-0.378541\pi$
$887$	22.1811	0.744767	0.372383	+	0.928079i	$0.378541\pi$
$888$	0	0
$889$	−32.4030	−1.08676
$890$	52.8221	1.77060
$891$	0	0
$892$	21.1480	0.708089
$893$	−13.7075	−0.458702
$894$	0	0
$895$	−55.1035	−1.84191
$896$	3.28444	0.109725
$897$	0	0
$898$	9.71826	0.324303
$899$	5.44985	0.181763
$900$	0	0
$901$	−12.2977	−0.409695
$902$	7.02969	0.234063
$903$	0	0
$904$	−9.74446	−0.324096
$905$	−48.5154	−1.61270
$906$	0	0
$907$	−23.3606	−0.775675	−0.387837	−	0.921728i	$-0.626778\pi$
$907$	−23.3606	−0.775675	−0.387837	+	0.921728i	$0.626778\pi$
$908$	−4.87304	−0.161718
$909$	0	0
$910$	28.2071	0.935056
$911$	−18.0390	−0.597659	−0.298830	−	0.954306i	$-0.596596\pi$
$911$	−18.0390	−0.597659	−0.298830	+	0.954306i	$0.596596\pi$
$912$	0	0
$913$	7.71116	0.255202
$914$	−19.7974	−0.654841
$915$	0	0
$916$	−22.8501	−0.754989
$917$	24.3613	0.804482
$918$	0	0
$919$	−22.3939	−0.738707	−0.369353	−	0.929289i	$-0.620421\pi$
$919$	−22.3939	−0.738707	−0.369353	+	0.929289i	$0.620421\pi$
$920$	21.1238	0.696429
$921$	0	0
$922$	2.22409	0.0732463
$923$	10.9705	0.361097
$924$	0	0
$925$	0.815138	0.0268016
$926$	−4.50016	−0.147885
$927$	0	0
$928$	−1.57885	−0.0518284
$929$	−19.6940	−0.646138	−0.323069	−	0.946375i	$-0.604715\pi$
$929$	−19.6940	−0.646138	−0.323069	+	0.946375i	$0.604715\pi$
$930$	0	0
$931$	−10.7295	−0.351644
$932$	−30.1911	−0.988942
$933$	0	0
$934$	38.6792	1.26562
$935$	5.24967	0.171682
$936$	0	0
$937$	−2.56005	−0.0836331	−0.0418166	−	0.999125i	$-0.513315\pi$
$937$	−2.56005	−0.0836331	−0.0418166	+	0.999125i	$0.513315\pi$
$938$	−37.0231	−1.20885
$939$	0	0
$940$	−14.1683	−0.462120
$941$	−22.4097	−0.730536	−0.365268	−	0.930902i	$-0.619023\pi$
$941$	−22.4097	−0.730536	−0.365268	+	0.930902i	$0.619023\pi$
$942$	0	0
$943$	−57.4628	−1.87125
$944$	14.7910	0.481408
$945$	0	0
$946$	−1.56363	−0.0508381
$947$	−53.1697	−1.72778	−0.863892	−	0.503678i	$-0.831980\pi$
$947$	−53.1697	−1.72778	−0.863892	+	0.503678i	$0.831980\pi$
$948$	0	0
$949$	−16.0333	−0.520462
$950$	10.1240	0.328465
$951$	0	0
$952$	−6.67224	−0.216249
$953$	4.67982	0.151594	0.0757971	−	0.997123i	$-0.475850\pi$
$953$	4.67982	0.151594	0.0757971	+	0.997123i	$0.475850\pi$
$954$	0	0
$955$	57.9622	1.87561
$956$	−5.90535	−0.190993
$957$	0	0
$958$	−21.4301	−0.692375
$959$	−24.2097	−0.781771
$960$	0	0
$961$	−19.0852	−0.615653
$962$	0.668985	0.0215690
$963$	0	0
$964$	−22.5583	−0.726554
$965$	−29.4801	−0.948997
$966$	0	0
$967$	−31.3373	−1.00774	−0.503870	−	0.863780i	$-0.668091\pi$
$967$	−31.3373	−1.00774	−0.503870	+	0.863780i	$0.668091\pi$
$968$	10.2211	0.328519
$969$	0	0
$970$	48.2272	1.54848
$971$	16.4233	0.527048	0.263524	−	0.964653i	$-0.415115\pi$
$971$	16.4233	0.527048	0.263524	+	0.964653i	$0.415115\pi$
$972$	0	0
$973$	74.8162	2.39850
$974$	−36.0210	−1.15419
$975$	0	0
$976$	8.51800	0.272654
$977$	5.28397	0.169049	0.0845247	−	0.996421i	$-0.473063\pi$
$977$	5.28397	0.169049	0.0845247	+	0.996421i	$0.473063\pi$
$978$	0	0
$979$	15.9208	0.508830
$980$	−11.0902	−0.354264
$981$	0	0
$982$	−17.2987	−0.552024
$983$	18.6696	0.595467	0.297734	−	0.954649i	$-0.403769\pi$
$983$	18.6696	0.595467	0.297734	+	0.954649i	$0.403769\pi$
$984$	0	0
$985$	−13.5980	−0.433269
$986$	3.20740	0.102144
$987$	0	0
$988$	8.30876	0.264337
$989$	12.7816	0.406431
$990$	0	0
$991$	37.9589	1.20580	0.602902	−	0.797815i	$-0.294011\pi$
$991$	37.9589	1.20580	0.602902	+	0.797815i	$0.294011\pi$
$992$	−3.45178	−0.109594
$993$	0	0
$994$	−12.2850	−0.389655
$995$	−27.8668	−0.883437
$996$	0	0
$997$	9.35691	0.296336	0.148168	−	0.988962i	$-0.452662\pi$
$997$	9.35691	0.296336	0.148168	+	0.988962i	$0.452662\pi$
$998$	−17.9026	−0.566697
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.e.1.1	✓	8
3.2	odd	2		8046.2.a.f.1.8	yes	8

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.e.1.1	✓	8	1.1	even	1	trivial
8046.2.a.f.1.8	yes	8	3.2	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:	\( N \)	\(=\)	\( 8046 = 2 \cdot 3^{3} \cdot 149 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	8046.a (trivial)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.92810 q^{5} -3.28444 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -2.92810 q^{5} -3.28444 q^{7} -1.00000 q^{8} +2.92810 q^{10} +0.882540 q^{11} -2.93300 q^{13} +3.28444 q^{14} +1.00000 q^{16} -2.03147 q^{17} -2.83285 q^{19} -2.92810 q^{20} -0.882540 q^{22} +7.21415 q^{23} +3.57377 q^{25} +2.93300 q^{26} -3.28444 q^{28} +1.57885 q^{29} +3.45178 q^{31} -1.00000 q^{32} +2.03147 q^{34} +9.61715 q^{35} +0.228089 q^{37} +2.83285 q^{38} +2.92810 q^{40} -7.96529 q^{41} +1.77174 q^{43} +0.882540 q^{44} -7.21415 q^{46} +4.83874 q^{47} +3.78751 q^{49} -3.57377 q^{50} -2.93300 q^{52} +6.05357 q^{53} -2.58417 q^{55} +3.28444 q^{56} -1.57885 q^{58} +14.7910 q^{59} +8.51800 q^{61} -3.45178 q^{62} +1.00000 q^{64} +8.58811 q^{65} -11.2723 q^{67} -2.03147 q^{68} -9.61715 q^{70} -3.74035 q^{71} +5.46651 q^{73} -0.228089 q^{74} -2.83285 q^{76} -2.89865 q^{77} -12.9804 q^{79} -2.92810 q^{80} +7.96529 q^{82} +8.73746 q^{83} +5.94836 q^{85} -1.77174 q^{86} -0.882540 q^{88} +18.0397 q^{89} +9.63324 q^{91} +7.21415 q^{92} -4.83874 q^{94} +8.29488 q^{95} +16.4705 q^{97} -3.78751 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 8 q - 8 q^{2} + 8 q^{4} - 5 q^{7} - 8 q^{8}+O(q^{10}) \) 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

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