LMFDB - Embedded newform 8046.2.a.l.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:	$0$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 5 x^{11} - 21 x^{10} + 116 x^{9} + 106 x^{8} - 774 x^{7} - 63 x^{6} + 2013 x^{5} - 417 x^{4} + \cdots - 375 $ x^12 - 5x^11 - 21x^10 + 116x^9 + 106x^8 - 774x^7 - 63x^6 + 2013x^5 - 417x^4 - 2249x^3 + 761x^2 + 910*x - 375
Coefficient ring:	$\Z[a_1, \ldots, a_{11}]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$-3.73396$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} -3.73396 q^{5} -2.62384 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} -3.73396 q^{5} -2.62384 q^{7} -1.00000 q^{8} +3.73396 q^{10} +2.14257 q^{11} -0.987139 q^{13} +2.62384 q^{14} +1.00000 q^{16} +5.24725 q^{17} -4.24478 q^{19} -3.73396 q^{20} -2.14257 q^{22} +8.14672 q^{23} +8.94247 q^{25} +0.987139 q^{26} -2.62384 q^{28} -8.43701 q^{29} -5.21821 q^{31} -1.00000 q^{32} -5.24725 q^{34} +9.79733 q^{35} +0.533584 q^{37} +4.24478 q^{38} +3.73396 q^{40} -3.88515 q^{41} +2.54110 q^{43} +2.14257 q^{44} -8.14672 q^{46} -3.35439 q^{47} -0.115449 q^{49} -8.94247 q^{50} -0.987139 q^{52} +2.52043 q^{53} -8.00028 q^{55} +2.62384 q^{56} +8.43701 q^{58} +3.22501 q^{59} -8.02184 q^{61} +5.21821 q^{62} +1.00000 q^{64} +3.68594 q^{65} +0.811742 q^{67} +5.24725 q^{68} -9.79733 q^{70} -3.95778 q^{71} -8.88304 q^{73} -0.533584 q^{74} -4.24478 q^{76} -5.62177 q^{77} -14.7166 q^{79} -3.73396 q^{80} +3.88515 q^{82} -6.87510 q^{83} -19.5930 q^{85} -2.54110 q^{86} -2.14257 q^{88} +6.07130 q^{89} +2.59010 q^{91} +8.14672 q^{92} +3.35439 q^{94} +15.8499 q^{95} -1.67775 q^{97} +0.115449 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) $ 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8	$ 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8} - 5 q^{10} + 10 q^{11} - q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} - 10 q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 7 q^{25} + q^{26} - 6 q^{28} + 33 q^{29} - 6 q^{31} - 12 q^{32} - 6 q^{34} + 16 q^{35} - 13 q^{37} + 10 q^{38} - 5 q^{40} + 20 q^{41} - 11 q^{43} + 10 q^{44} - 15 q^{46} + 15 q^{47} + 2 q^{49} - 7 q^{50} - q^{52} + 4 q^{53} - 17 q^{55} + 6 q^{56} - 33 q^{58} + 10 q^{59} - 12 q^{61} + 6 q^{62} + 12 q^{64} + 40 q^{65} - 19 q^{67} + 6 q^{68} - 16 q^{70} + 47 q^{71} - 2 q^{73} + 13 q^{74} - 10 q^{76} - 6 q^{77} - 15 q^{79} + 5 q^{80} - 20 q^{82} + 18 q^{83} - 25 q^{85} + 11 q^{86} - 10 q^{88} + 24 q^{89} - 3 q^{91} + 15 q^{92} - 15 q^{94} - 3 q^{95} - 25 q^{97} - 2 q^{98}+O(q^{100}) $ 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8 - 5 * q^10 + 10 * q^11 - q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 - 10 * q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 7 * q^25 + q^26 - 6 * q^28 + 33 * q^29 - 6 * q^31 - 12 * q^32 - 6 * q^34 + 16 * q^35 - 13 * q^37 + 10 * q^38 - 5 * q^40 + 20 * q^41 - 11 * q^43 + 10 * q^44 - 15 * q^46 + 15 * q^47 + 2 * q^49 - 7 * q^50 - q^52 + 4 * q^53 - 17 * q^55 + 6 * q^56 - 33 * q^58 + 10 * q^59 - 12 * q^61 + 6 * q^62 + 12 * q^64 + 40 * q^65 - 19 * q^67 + 6 * q^68 - 16 * q^70 + 47 * q^71 - 2 * q^73 + 13 * q^74 - 10 * q^76 - 6 * q^77 - 15 * q^79 + 5 * q^80 - 20 * q^82 + 18 * q^83 - 25 * q^85 + 11 * q^86 - 10 * q^88 + 24 * q^89 - 3 * q^91 + 15 * q^92 - 15 * q^94 - 3 * q^95 - 25 * q^97 - 2 * 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
$5$	−3.73396	−1.66988	−0.834939	−	0.550342i	$-0.814497\pi$
$5$	−3.73396	−1.66988	−0.834939	+	0.550342i	$0.814497\pi$
$6$	0	0
$7$	−2.62384	−0.991719	−0.495860	−	0.868403i	$-0.665147\pi$
$7$	−2.62384	−0.991719	−0.495860	+	0.868403i	$0.665147\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	3.73396	1.18078
$11$	2.14257	0.646010	0.323005	−	0.946397i	$-0.395307\pi$
$11$	2.14257	0.646010	0.323005	+	0.946397i	$0.395307\pi$
$12$	0	0
$13$	−0.987139	−0.273783	−0.136892	−	0.990586i	$-0.543711\pi$
$13$	−0.987139	−0.273783	−0.136892	+	0.990586i	$0.543711\pi$
$14$	2.62384	0.701251
$15$	0	0
$16$	1.00000	0.250000
$17$	5.24725	1.27265	0.636323	−	0.771423i	$-0.280455\pi$
$17$	5.24725	1.27265	0.636323	+	0.771423i	$0.280455\pi$
$18$	0	0
$19$	−4.24478	−0.973820	−0.486910	−	0.873452i	$-0.661876\pi$
$19$	−4.24478	−0.973820	−0.486910	+	0.873452i	$0.661876\pi$
$20$	−3.73396	−0.834939
$21$	0	0
$22$	−2.14257	−0.456798
$23$	8.14672	1.69871	0.849354	−	0.527823i	$-0.176992\pi$
$23$	8.14672	1.69871	0.849354	+	0.527823i	$0.176992\pi$
$24$	0	0
$25$	8.94247	1.78849
$26$	0.987139	0.193594
$27$	0	0
$28$	−2.62384	−0.495860
$29$	−8.43701	−1.56671	−0.783357	−	0.621573i	$-0.786494\pi$
$29$	−8.43701	−1.56671	−0.783357	+	0.621573i	$0.786494\pi$
$30$	0	0
$31$	−5.21821	−0.937218	−0.468609	−	0.883406i	$-0.655245\pi$
$31$	−5.21821	−0.937218	−0.468609	+	0.883406i	$0.655245\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−5.24725	−0.899896
$35$	9.79733	1.65605
$36$	0	0
$37$	0.533584	0.0877206	0.0438603	−	0.999038i	$-0.486034\pi$
$37$	0.533584	0.0877206	0.0438603	+	0.999038i	$0.486034\pi$
$38$	4.24478	0.688594
$39$	0	0
$40$	3.73396	0.590391
$41$	−3.88515	−0.606758	−0.303379	−	0.952870i	$-0.598115\pi$
$41$	−3.88515	−0.606758	−0.303379	+	0.952870i	$0.598115\pi$
$42$	0	0
$43$	2.54110	0.387514	0.193757	−	0.981050i	$-0.437933\pi$
$43$	2.54110	0.387514	0.193757	+	0.981050i	$0.437933\pi$
$44$	2.14257	0.323005
$45$	0	0
$46$	−8.14672	−1.20117
$47$	−3.35439	−0.489288	−0.244644	−	0.969613i	$-0.578671\pi$
$47$	−3.35439	−0.489288	−0.244644	+	0.969613i	$0.578671\pi$
$48$	0	0
$49$	−0.115449	−0.0164927
$50$	−8.94247	−1.26466
$51$	0	0
$52$	−0.987139	−0.136892
$53$	2.52043	0.346208	0.173104	−	0.984904i	$-0.444620\pi$
$53$	2.52043	0.346208	0.173104	+	0.984904i	$0.444620\pi$
$54$	0	0
$55$	−8.00028	−1.07876
$56$	2.62384	0.350626
$57$	0	0
$58$	8.43701	1.10783
$59$	3.22501	0.419860	0.209930	−	0.977716i	$-0.432676\pi$
$59$	3.22501	0.419860	0.209930	+	0.977716i	$0.432676\pi$
$60$	0	0
$61$	−8.02184	−1.02709	−0.513545	−	0.858062i	$-0.671668\pi$
$61$	−8.02184	−1.02709	−0.513545	+	0.858062i	$0.671668\pi$
$62$	5.21821	0.662713
$63$	0	0
$64$	1.00000	0.125000
$65$	3.68594	0.457184
$66$	0	0
$67$	0.811742	0.0991700	0.0495850	−	0.998770i	$-0.484210\pi$
$67$	0.811742	0.0991700	0.0495850	+	0.998770i	$0.484210\pi$
$68$	5.24725	0.636323
$69$	0	0
$70$	−9.79733	−1.17100
$71$	−3.95778	−0.469702	−0.234851	−	0.972031i	$-0.575460\pi$
$71$	−3.95778	−0.469702	−0.234851	+	0.972031i	$0.575460\pi$
$72$	0	0
$73$	−8.88304	−1.03968	−0.519841	−	0.854263i	$-0.674009\pi$
$73$	−8.88304	−1.03968	−0.519841	+	0.854263i	$0.674009\pi$
$74$	−0.533584	−0.0620279
$75$	0	0
$76$	−4.24478	−0.486910
$77$	−5.62177	−0.640660
$78$	0	0
$79$	−14.7166	−1.65575	−0.827873	−	0.560916i	$-0.810449\pi$
$79$	−14.7166	−1.65575	−0.827873	+	0.560916i	$0.810449\pi$
$80$	−3.73396	−0.417470
$81$	0	0
$82$	3.88515	0.429043
$83$	−6.87510	−0.754641	−0.377320	−	0.926083i	$-0.623154\pi$
$83$	−6.87510	−0.754641	−0.377320	+	0.926083i	$0.623154\pi$
$84$	0	0
$85$	−19.5930	−2.12516
$86$	−2.54110	−0.274014
$87$	0	0
$88$	−2.14257	−0.228399
$89$	6.07130	0.643556	0.321778	−	0.946815i	$-0.395720\pi$
$89$	6.07130	0.643556	0.321778	+	0.946815i	$0.395720\pi$
$90$	0	0
$91$	2.59010	0.271516
$92$	8.14672	0.849354
$93$	0	0
$94$	3.35439	0.345979
$95$	15.8499	1.62616
$96$	0	0
$97$	−1.67775	−0.170350	−0.0851750	−	0.996366i	$-0.527145\pi$
$97$	−1.67775	−0.170350	−0.0851750	+	0.996366i	$0.527145\pi$
$98$	0.115449	0.0116621
$99$	0	0
$100$	8.94247	0.894247
$101$	17.2529	1.71673	0.858364	−	0.513042i	$-0.171481\pi$
$101$	17.2529	1.71673	0.858364	+	0.513042i	$0.171481\pi$
$102$	0	0
$103$	−0.973885	−0.0959597	−0.0479799	−	0.998848i	$-0.515278\pi$
$103$	−0.973885	−0.0959597	−0.0479799	+	0.998848i	$0.515278\pi$
$104$	0.987139	0.0967969
$105$	0	0
$106$	−2.52043	−0.244806
$107$	13.1499	1.27125	0.635626	−	0.771998i	$-0.280742\pi$
$107$	13.1499	1.27125	0.635626	+	0.771998i	$0.280742\pi$
$108$	0	0
$109$	6.48190	0.620853	0.310427	−	0.950597i	$-0.399528\pi$
$109$	6.48190	0.620853	0.310427	+	0.950597i	$0.399528\pi$
$110$	8.00028	0.762797
$111$	0	0
$112$	−2.62384	−0.247930
$113$	−9.79815	−0.921733	−0.460866	−	0.887470i	$-0.652461\pi$
$113$	−9.79815	−0.921733	−0.460866	+	0.887470i	$0.652461\pi$
$114$	0	0
$115$	−30.4195	−2.83664
$116$	−8.43701	−0.783357
$117$	0	0
$118$	−3.22501	−0.296886
$119$	−13.7680	−1.26211
$120$	0	0
$121$	−6.40939	−0.582672
$122$	8.02184	0.726263
$123$	0	0
$124$	−5.21821	−0.468609
$125$	−14.7210	−1.31669
$126$	0	0
$127$	−4.34198	−0.385289	−0.192644	−	0.981269i	$-0.561706\pi$
$127$	−4.34198	−0.385289	−0.192644	+	0.981269i	$0.561706\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−3.68594	−0.323278
$131$	3.82187	0.333919	0.166959	−	0.985964i	$-0.446605\pi$
$131$	3.82187	0.333919	0.166959	+	0.985964i	$0.446605\pi$
$132$	0	0
$133$	11.1376	0.965756
$134$	−0.811742	−0.0701238
$135$	0	0
$136$	−5.24725	−0.449948
$137$	−11.0978	−0.948148	−0.474074	−	0.880485i	$-0.657217\pi$
$137$	−11.0978	−0.948148	−0.474074	+	0.880485i	$0.657217\pi$
$138$	0	0
$139$	2.47689	0.210087	0.105043	−	0.994468i	$-0.466502\pi$
$139$	2.47689	0.210087	0.105043	+	0.994468i	$0.466502\pi$
$140$	9.79733	0.828025
$141$	0	0
$142$	3.95778	0.332130
$143$	−2.11502	−0.176867
$144$	0	0
$145$	31.5035	2.61622
$146$	8.88304	0.735166
$147$	0	0
$148$	0.533584	0.0438603
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−15.4626	−1.25833	−0.629165	−	0.777272i	$-0.716603\pi$
$151$	−15.4626	−1.25833	−0.629165	+	0.777272i	$0.716603\pi$
$152$	4.24478	0.344297
$153$	0	0
$154$	5.62177	0.453015
$155$	19.4846	1.56504
$156$	0	0
$157$	−20.0843	−1.60290	−0.801452	−	0.598060i	$-0.795939\pi$
$157$	−20.0843	−1.60290	−0.801452	+	0.598060i	$0.795939\pi$
$158$	14.7166	1.17079
$159$	0	0
$160$	3.73396	0.295196
$161$	−21.3757	−1.68464
$162$	0	0
$163$	−6.48075	−0.507612	−0.253806	−	0.967255i	$-0.581682\pi$
$163$	−6.48075	−0.507612	−0.253806	+	0.967255i	$0.581682\pi$
$164$	−3.88515	−0.303379
$165$	0	0
$166$	6.87510	0.533612
$167$	2.40489	0.186096	0.0930479	−	0.995662i	$-0.470339\pi$
$167$	2.40489	0.186096	0.0930479	+	0.995662i	$0.470339\pi$
$168$	0	0
$169$	−12.0256	−0.925043
$170$	19.5930	1.50272
$171$	0	0
$172$	2.54110	0.193757
$173$	−15.9773	−1.21473	−0.607365	−	0.794423i	$-0.707773\pi$
$173$	−15.9773	−1.21473	−0.607365	+	0.794423i	$0.707773\pi$
$174$	0	0
$175$	−23.4636	−1.77368
$176$	2.14257	0.161502
$177$	0	0
$178$	−6.07130	−0.455063
$179$	−3.98077	−0.297537	−0.148768	−	0.988872i	$-0.547531\pi$
$179$	−3.98077	−0.297537	−0.148768	+	0.988872i	$0.547531\pi$
$180$	0	0
$181$	19.7064	1.46476	0.732381	−	0.680895i	$-0.238409\pi$
$181$	19.7064	1.46476	0.732381	+	0.680895i	$0.238409\pi$
$182$	−2.59010	−0.191991
$183$	0	0
$184$	−8.14672	−0.600584
$185$	−1.99238	−0.146483
$186$	0	0
$187$	11.2426	0.822141
$188$	−3.35439	−0.244644
$189$	0	0
$190$	−15.8499	−1.14987
$191$	19.7320	1.42775	0.713877	−	0.700271i	$-0.246938\pi$
$191$	19.7320	1.42775	0.713877	+	0.700271i	$0.246938\pi$
$192$	0	0
$193$	8.55331	0.615680	0.307840	−	0.951438i	$-0.400394\pi$
$193$	8.55331	0.615680	0.307840	+	0.951438i	$0.400394\pi$
$194$	1.67775	0.120456
$195$	0	0
$196$	−0.115449	−0.00824636
$197$	2.35388	0.167707	0.0838533	−	0.996478i	$-0.473277\pi$
$197$	2.35388	0.167707	0.0838533	+	0.996478i	$0.473277\pi$
$198$	0	0
$199$	13.9149	0.986399	0.493200	−	0.869916i	$-0.335827\pi$
$199$	13.9149	0.986399	0.493200	+	0.869916i	$0.335827\pi$
$200$	−8.94247	−0.632328
$201$	0	0
$202$	−17.2529	−1.21391
$203$	22.1374	1.55374
$204$	0	0
$205$	14.5070	1.01321
$206$	0.973885	0.0678538
$207$	0	0
$208$	−0.987139	−0.0684458
$209$	−9.09475	−0.629097
$210$	0	0
$211$	1.13848	0.0783765	0.0391883	−	0.999232i	$-0.487523\pi$
$211$	1.13848	0.0783765	0.0391883	+	0.999232i	$0.487523\pi$
$212$	2.52043	0.173104
$213$	0	0
$214$	−13.1499	−0.898910
$215$	−9.48837	−0.647102
$216$	0	0
$217$	13.6918	0.929457
$218$	−6.48190	−0.439009
$219$	0	0
$220$	−8.00028	−0.539379
$221$	−5.17977	−0.348429
$222$	0	0
$223$	−6.72193	−0.450134	−0.225067	−	0.974343i	$-0.572260\pi$
$223$	−6.72193	−0.450134	−0.225067	+	0.974343i	$0.572260\pi$
$224$	2.62384	0.175313
$225$	0	0
$226$	9.79815	0.651763
$227$	−8.37087	−0.555594	−0.277797	−	0.960640i	$-0.589604\pi$
$227$	−8.37087	−0.555594	−0.277797	+	0.960640i	$0.589604\pi$
$228$	0	0
$229$	9.32680	0.616332	0.308166	−	0.951333i	$-0.400285\pi$
$229$	9.32680	0.616332	0.308166	+	0.951333i	$0.400285\pi$
$230$	30.4195	2.00581
$231$	0	0
$232$	8.43701	0.553917
$233$	10.2941	0.674391	0.337196	−	0.941435i	$-0.390522\pi$
$233$	10.2941	0.674391	0.337196	+	0.941435i	$0.390522\pi$
$234$	0	0
$235$	12.5252	0.817051
$236$	3.22501	0.209930
$237$	0	0
$238$	13.7680	0.892445
$239$	12.0117	0.776972	0.388486	−	0.921455i	$-0.372998\pi$
$239$	12.0117	0.776972	0.388486	+	0.921455i	$0.372998\pi$
$240$	0	0
$241$	24.9677	1.60831	0.804155	−	0.594420i	$-0.202618\pi$
$241$	24.9677	1.60831	0.804155	+	0.594420i	$0.202618\pi$
$242$	6.40939	0.412011
$243$	0	0
$244$	−8.02184	−0.513545
$245$	0.431082	0.0275408
$246$	0	0
$247$	4.19019	0.266615
$248$	5.21821	0.331357
$249$	0	0
$250$	14.7210	0.931039
$251$	−1.96591	−0.124087	−0.0620436	−	0.998073i	$-0.519762\pi$
$251$	−1.96591	−0.124087	−0.0620436	+	0.998073i	$0.519762\pi$
$252$	0	0
$253$	17.4549	1.09738
$254$	4.34198	0.272440
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.3723	1.39554	0.697772	−	0.716320i	$-0.254175\pi$
$257$	22.3723	1.39554	0.697772	+	0.716320i	$0.254175\pi$
$258$	0	0
$259$	−1.40004	−0.0869943
$260$	3.68594	0.228592
$261$	0	0
$262$	−3.82187	−0.236116
$263$	22.1729	1.36724	0.683619	−	0.729839i	$-0.260405\pi$
$263$	22.1729	1.36724	0.683619	+	0.729839i	$0.260405\pi$
$264$	0	0
$265$	−9.41119	−0.578125
$266$	−11.1376	−0.682892
$267$	0	0
$268$	0.811742	0.0495850
$269$	−18.7564	−1.14360	−0.571800	−	0.820393i	$-0.693755\pi$
$269$	−18.7564	−1.14360	−0.571800	+	0.820393i	$0.693755\pi$
$270$	0	0
$271$	19.4281	1.18017	0.590086	−	0.807341i	$-0.299094\pi$
$271$	19.4281	1.18017	0.590086	+	0.807341i	$0.299094\pi$
$272$	5.24725	0.318161
$273$	0	0
$274$	11.0978	0.670442
$275$	19.1599	1.15538
$276$	0	0
$277$	−8.65289	−0.519902	−0.259951	−	0.965622i	$-0.583706\pi$
$277$	−8.65289	−0.519902	−0.259951	+	0.965622i	$0.583706\pi$
$278$	−2.47689	−0.148554
$279$	0	0
$280$	−9.79733	−0.585502
$281$	6.09541	0.363622	0.181811	−	0.983334i	$-0.441804\pi$
$281$	6.09541	0.363622	0.181811	+	0.983334i	$0.441804\pi$
$282$	0	0
$283$	−13.5780	−0.807130	−0.403565	−	0.914951i	$-0.632229\pi$
$283$	−13.5780	−0.807130	−0.403565	+	0.914951i	$0.632229\pi$
$284$	−3.95778	−0.234851
$285$	0	0
$286$	2.11502	0.125064
$287$	10.1940	0.601734
$288$	0	0
$289$	10.5337	0.619627
$290$	−31.5035	−1.84995
$291$	0	0
$292$	−8.88304	−0.519841
$293$	−0.488415	−0.0285335	−0.0142668	−	0.999898i	$-0.504541\pi$
$293$	−0.488415	−0.0285335	−0.0142668	+	0.999898i	$0.504541\pi$
$294$	0	0
$295$	−12.0421	−0.701115
$296$	−0.533584	−0.0310139
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	−8.04195	−0.465078
$300$	0	0
$301$	−6.66745	−0.384305
$302$	15.4626	0.889773
$303$	0	0
$304$	−4.24478	−0.243455
$305$	29.9532	1.71512
$306$	0	0
$307$	21.7695	1.24245	0.621224	−	0.783633i	$-0.286636\pi$
$307$	21.7695	1.24245	0.621224	+	0.783633i	$0.286636\pi$
$308$	−5.62177	−0.320330
$309$	0	0
$310$	−19.4846	−1.10665
$311$	−21.0144	−1.19162	−0.595809	−	0.803126i	$-0.703169\pi$
$311$	−21.0144	−1.19162	−0.595809	+	0.803126i	$0.703169\pi$
$312$	0	0
$313$	32.2306	1.82178	0.910890	−	0.412649i	$-0.135396\pi$
$313$	32.2306	1.82178	0.910890	+	0.412649i	$0.135396\pi$
$314$	20.0843	1.13342
$315$	0	0
$316$	−14.7166	−0.827873
$317$	−21.5139	−1.20834	−0.604171	−	0.796855i	$-0.706496\pi$
$317$	−21.5139	−1.20834	−0.604171	+	0.796855i	$0.706496\pi$
$318$	0	0
$319$	−18.0769	−1.01211
$320$	−3.73396	−0.208735
$321$	0	0
$322$	21.3757	1.19122
$323$	−22.2734	−1.23933
$324$	0	0
$325$	−8.82746	−0.489659
$326$	6.48075	0.358936
$327$	0	0
$328$	3.88515	0.214521
$329$	8.80139	0.485236
$330$	0	0
$331$	−3.00592	−0.165221	−0.0826103	−	0.996582i	$-0.526326\pi$
$331$	−3.00592	−0.165221	−0.0826103	+	0.996582i	$0.526326\pi$
$332$	−6.87510	−0.377320
$333$	0	0
$334$	−2.40489	−0.131590
$335$	−3.03101	−0.165602
$336$	0	0
$337$	27.2897	1.48657	0.743283	−	0.668977i	$-0.233268\pi$
$337$	27.2897	1.48657	0.743283	+	0.668977i	$0.233268\pi$
$338$	12.0256	0.654104
$339$	0	0
$340$	−19.5930	−1.06258
$341$	−11.1804	−0.605452
$342$	0	0
$343$	18.6698	1.00808
$344$	−2.54110	−0.137007
$345$	0	0
$346$	15.9773	0.858943
$347$	−19.5990	−1.05213	−0.526064	−	0.850445i	$-0.676333\pi$
$347$	−19.5990	−1.05213	−0.526064	+	0.850445i	$0.676333\pi$
$348$	0	0
$349$	−21.6076	−1.15663	−0.578313	−	0.815815i	$-0.696289\pi$
$349$	−21.6076	−1.15663	−0.578313	+	0.815815i	$0.696289\pi$
$350$	23.4636	1.25418
$351$	0	0
$352$	−2.14257	−0.114199
$353$	22.9587	1.22197	0.610985	−	0.791642i	$-0.290774\pi$
$353$	22.9587	1.22197	0.610985	+	0.791642i	$0.290774\pi$
$354$	0	0
$355$	14.7782	0.784345
$356$	6.07130	0.321778
$357$	0	0
$358$	3.98077	0.210390
$359$	−8.06534	−0.425673	−0.212836	−	0.977088i	$-0.568270\pi$
$359$	−8.06534	−0.425673	−0.212836	+	0.977088i	$0.568270\pi$
$360$	0	0
$361$	−0.981830	−0.0516752
$362$	−19.7064	−1.03574
$363$	0	0
$364$	2.59010	0.135758
$365$	33.1689	1.73614
$366$	0	0
$367$	−13.5555	−0.707590	−0.353795	−	0.935323i	$-0.615109\pi$
$367$	−13.5555	−0.707590	−0.353795	+	0.935323i	$0.615109\pi$
$368$	8.14672	0.424677
$369$	0	0
$370$	1.99238	0.103579
$371$	−6.61321	−0.343341
$372$	0	0
$373$	31.1012	1.61036	0.805180	−	0.593031i	$-0.202069\pi$
$373$	31.1012	1.61036	0.805180	+	0.593031i	$0.202069\pi$
$374$	−11.2426	−0.581342
$375$	0	0
$376$	3.35439	0.172989
$377$	8.32850	0.428940
$378$	0	0
$379$	31.2257	1.60396	0.801979	−	0.597352i	$-0.203781\pi$
$379$	31.2257	1.60396	0.801979	+	0.597352i	$0.203781\pi$
$380$	15.8499	0.813080
$381$	0	0
$382$	−19.7320	−1.00957
$383$	−2.00824	−0.102616	−0.0513082	−	0.998683i	$-0.516339\pi$
$383$	−2.00824	−0.102616	−0.0513082	+	0.998683i	$0.516339\pi$
$384$	0	0
$385$	20.9915	1.06982
$386$	−8.55331	−0.435352
$387$	0	0
$388$	−1.67775	−0.0851750
$389$	−11.1882	−0.567263	−0.283632	−	0.958933i	$-0.591539\pi$
$389$	−11.1882	−0.567263	−0.283632	+	0.958933i	$0.591539\pi$
$390$	0	0
$391$	42.7479	2.16185
$392$	0.115449	0.00583106
$393$	0	0
$394$	−2.35388	−0.118587
$395$	54.9512	2.76489
$396$	0	0
$397$	15.7271	0.789320	0.394660	−	0.918827i	$-0.370862\pi$
$397$	15.7271	0.789320	0.394660	+	0.918827i	$0.370862\pi$
$398$	−13.9149	−0.697490
$399$	0	0
$400$	8.94247	0.447123
$401$	18.6378	0.930729	0.465365	−	0.885119i	$-0.345923\pi$
$401$	18.6378	0.930729	0.465365	+	0.885119i	$0.345923\pi$
$402$	0	0
$403$	5.15110	0.256594
$404$	17.2529	0.858364
$405$	0	0
$406$	−22.1374	−1.09866
$407$	1.14324	0.0566684
$408$	0	0
$409$	−10.7222	−0.530177	−0.265088	−	0.964224i	$-0.585401\pi$
$409$	−10.7222	−0.530177	−0.265088	+	0.964224i	$0.585401\pi$
$410$	−14.5070	−0.716449
$411$	0	0
$412$	−0.973885	−0.0479799
$413$	−8.46191	−0.416383
$414$	0	0
$415$	25.6714	1.26016
$416$	0.987139	0.0483985
$417$	0	0
$418$	9.09475	0.444839
$419$	21.0470	1.02821	0.514107	−	0.857726i	$-0.328123\pi$
$419$	21.0470	1.02821	0.514107	+	0.857726i	$0.328123\pi$
$420$	0	0
$421$	−22.1796	−1.08097	−0.540483	−	0.841355i	$-0.681758\pi$
$421$	−22.1796	−1.08097	−0.540483	+	0.841355i	$0.681758\pi$
$422$	−1.13848	−0.0554206
$423$	0	0
$424$	−2.52043	−0.122403
$425$	46.9234	2.27612
$426$	0	0
$427$	21.0480	1.01859
$428$	13.1499	0.635626
$429$	0	0
$430$	9.48837	0.457570
$431$	32.4982	1.56538	0.782691	−	0.622411i	$-0.213847\pi$
$431$	32.4982	1.56538	0.782691	+	0.622411i	$0.213847\pi$
$432$	0	0
$433$	−2.82924	−0.135965	−0.0679824	−	0.997687i	$-0.521656\pi$
$433$	−2.82924	−0.135965	−0.0679824	+	0.997687i	$0.521656\pi$
$434$	−13.6918	−0.657226
$435$	0	0
$436$	6.48190	0.310427
$437$	−34.5811	−1.65424
$438$	0	0
$439$	−11.3455	−0.541489	−0.270744	−	0.962651i	$-0.587270\pi$
$439$	−11.3455	−0.541489	−0.270744	+	0.962651i	$0.587270\pi$
$440$	8.00028	0.381398
$441$	0	0
$442$	5.17977	0.246376
$443$	23.7537	1.12857	0.564287	−	0.825578i	$-0.309151\pi$
$443$	23.7537	1.12857	0.564287	+	0.825578i	$0.309151\pi$
$444$	0	0
$445$	−22.6700	−1.07466
$446$	6.72193	0.318293
$447$	0	0
$448$	−2.62384	−0.123965
$449$	31.7903	1.50028	0.750138	−	0.661281i	$-0.229987\pi$
$449$	31.7903	1.50028	0.750138	+	0.661281i	$0.229987\pi$
$450$	0	0
$451$	−8.32421	−0.391972
$452$	−9.79815	−0.460866
$453$	0	0
$454$	8.37087	0.392864
$455$	−9.67132	−0.453399
$456$	0	0
$457$	2.44674	0.114453	0.0572267	−	0.998361i	$-0.481774\pi$
$457$	2.44674	0.114453	0.0572267	+	0.998361i	$0.481774\pi$
$458$	−9.32680	−0.435813
$459$	0	0
$460$	−30.4195	−1.41832
$461$	6.34903	0.295704	0.147852	−	0.989009i	$-0.452764\pi$
$461$	6.34903	0.295704	0.147852	+	0.989009i	$0.452764\pi$
$462$	0	0
$463$	0.408336	0.0189770	0.00948848	−	0.999955i	$-0.496980\pi$
$463$	0.408336	0.0189770	0.00948848	+	0.999955i	$0.496980\pi$
$464$	−8.43701	−0.391678
$465$	0	0
$466$	−10.2941	−0.476867
$467$	−4.73896	−0.219293	−0.109646	−	0.993971i	$-0.534972\pi$
$467$	−4.73896	−0.219293	−0.109646	+	0.993971i	$0.534972\pi$
$468$	0	0
$469$	−2.12988	−0.0983488
$470$	−12.5252	−0.577742
$471$	0	0
$472$	−3.22501	−0.148443
$473$	5.44449	0.250338
$474$	0	0
$475$	−37.9588	−1.74167
$476$	−13.7680	−0.631054
$477$	0	0
$478$	−12.0117	−0.549402
$479$	13.6810	0.625102	0.312551	−	0.949901i	$-0.398816\pi$
$479$	13.6810	0.625102	0.312551	+	0.949901i	$0.398816\pi$
$480$	0	0
$481$	−0.526721	−0.0240164
$482$	−24.9677	−1.13725
$483$	0	0
$484$	−6.40939	−0.291336
$485$	6.26467	0.284464
$486$	0	0
$487$	8.54623	0.387267	0.193633	−	0.981074i	$-0.437973\pi$
$487$	8.54623	0.387267	0.193633	+	0.981074i	$0.437973\pi$
$488$	8.02184	0.363131
$489$	0	0
$490$	−0.431082	−0.0194743
$491$	−13.4171	−0.605504	−0.302752	−	0.953069i	$-0.597905\pi$
$491$	−13.4171	−0.605504	−0.302752	+	0.953069i	$0.597905\pi$
$492$	0	0
$493$	−44.2711	−1.99387
$494$	−4.19019	−0.188526
$495$	0	0
$496$	−5.21821	−0.234305
$497$	10.3846	0.465813
$498$	0	0
$499$	−9.21700	−0.412609	−0.206305	−	0.978488i	$-0.566144\pi$
$499$	−9.21700	−0.412609	−0.206305	+	0.978488i	$0.566144\pi$
$500$	−14.7210	−0.658344
$501$	0	0
$502$	1.96591	0.0877428
$503$	−26.3513	−1.17495	−0.587473	−	0.809244i	$-0.699877\pi$
$503$	−26.3513	−1.17495	−0.587473	+	0.809244i	$0.699877\pi$
$504$	0	0
$505$	−64.4217	−2.86673
$506$	−17.4549	−0.775966
$507$	0	0
$508$	−4.34198	−0.192644
$509$	−44.1347	−1.95623	−0.978117	−	0.208055i	$-0.933287\pi$
$509$	−44.1347	−1.95623	−0.978117	+	0.208055i	$0.933287\pi$
$510$	0	0
$511$	23.3077	1.03107
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−22.3723	−0.986799
$515$	3.63645	0.160241
$516$	0	0
$517$	−7.18702	−0.316085
$518$	1.40004	0.0615142
$519$	0	0
$520$	−3.68594	−0.161639
$521$	28.2207	1.23637	0.618185	−	0.786033i	$-0.287868\pi$
$521$	28.2207	1.23637	0.618185	+	0.786033i	$0.287868\pi$
$522$	0	0
$523$	33.9334	1.48380	0.741901	−	0.670509i	$-0.233924\pi$
$523$	33.9334	1.48380	0.741901	+	0.670509i	$0.233924\pi$
$524$	3.82187	0.166959
$525$	0	0
$526$	−22.1729	−0.966784
$527$	−27.3813	−1.19275
$528$	0	0
$529$	43.3691	1.88561
$530$	9.41119	0.408796
$531$	0	0
$532$	11.1376	0.482878
$533$	3.83518	0.166120
$534$	0	0
$535$	−49.1013	−2.12283
$536$	−0.811742	−0.0350619
$537$	0	0
$538$	18.7564	0.808647
$539$	−0.247358	−0.0106545
$540$	0	0
$541$	16.7627	0.720686	0.360343	−	0.932820i	$-0.382660\pi$
$541$	16.7627	0.720686	0.360343	+	0.932820i	$0.382660\pi$
$542$	−19.4281	−0.834507
$543$	0	0
$544$	−5.24725	−0.224974
$545$	−24.2032	−1.03675
$546$	0	0
$547$	−14.4259	−0.616809	−0.308404	−	0.951255i	$-0.599795\pi$
$547$	−14.4259	−0.616809	−0.308404	+	0.951255i	$0.599795\pi$
$548$	−11.0978	−0.474074
$549$	0	0
$550$	−19.1599	−0.816980
$551$	35.8133	1.52570
$552$	0	0
$553$	38.6140	1.64203
$554$	8.65289	0.367626
$555$	0	0
$556$	2.47689	0.105043
$557$	13.1292	0.556301	0.278150	−	0.960538i	$-0.410279\pi$
$557$	13.1292	0.556301	0.278150	+	0.960538i	$0.410279\pi$
$558$	0	0
$559$	−2.50842	−0.106095
$560$	9.79733	0.414013
$561$	0	0
$562$	−6.09541	−0.257119
$563$	−45.2408	−1.90667	−0.953336	−	0.301911i	$-0.902375\pi$
$563$	−45.2408	−1.90667	−0.953336	+	0.301911i	$0.902375\pi$
$564$	0	0
$565$	36.5859	1.53918
$566$	13.5780	0.570727
$567$	0	0
$568$	3.95778	0.166065
$569$	27.4976	1.15276	0.576379	−	0.817183i	$-0.304465\pi$
$569$	27.4976	1.15276	0.576379	+	0.817183i	$0.304465\pi$
$570$	0	0
$571$	21.1168	0.883710	0.441855	−	0.897086i	$-0.354321\pi$
$571$	21.1168	0.883710	0.441855	+	0.897086i	$0.354321\pi$
$572$	−2.11502	−0.0884333
$573$	0	0
$574$	−10.1940	−0.425490
$575$	72.8518	3.03813
$576$	0	0
$577$	−35.3110	−1.47001	−0.735007	−	0.678059i	$-0.762821\pi$
$577$	−35.3110	−1.47001	−0.735007	+	0.678059i	$0.762821\pi$
$578$	−10.5337	−0.438142
$579$	0	0
$580$	31.5035	1.30811
$581$	18.0392	0.748392
$582$	0	0
$583$	5.40020	0.223653
$584$	8.88304	0.367583
$585$	0	0
$586$	0.488415	0.0201762
$587$	−17.4372	−0.719711	−0.359856	−	0.933008i	$-0.617174\pi$
$587$	−17.4372	−0.719711	−0.359856	+	0.933008i	$0.617174\pi$
$588$	0	0
$589$	22.1502	0.912681
$590$	12.0421	0.495763
$591$	0	0
$592$	0.533584	0.0219302
$593$	3.88793	0.159658	0.0798291	−	0.996809i	$-0.474563\pi$
$593$	3.88793	0.159658	0.0798291	+	0.996809i	$0.474563\pi$
$594$	0	0
$595$	51.4090	2.10757
$596$	1.00000	0.0409616
$597$	0	0
$598$	8.04195	0.328860
$599$	47.4089	1.93707	0.968537	−	0.248868i	$-0.0800587\pi$
$599$	47.4089	1.93707	0.968537	+	0.248868i	$0.0800587\pi$
$600$	0	0
$601$	35.7611	1.45873	0.729364	−	0.684126i	$-0.239816\pi$
$601$	35.7611	1.45873	0.729364	+	0.684126i	$0.239816\pi$
$602$	6.66745	0.271745
$603$	0	0
$604$	−15.4626	−0.629165
$605$	23.9324	0.972991
$606$	0	0
$607$	17.2471	0.700040	0.350020	−	0.936742i	$-0.386175\pi$
$607$	17.2471	0.700040	0.350020	+	0.936742i	$0.386175\pi$
$608$	4.24478	0.172149
$609$	0	0
$610$	−29.9532	−1.21277
$611$	3.31125	0.133959
$612$	0	0
$613$	2.45411	0.0991206	0.0495603	−	0.998771i	$-0.484218\pi$
$613$	2.45411	0.0991206	0.0495603	+	0.998771i	$0.484218\pi$
$614$	−21.7695	−0.878544
$615$	0	0
$616$	5.62177	0.226508
$617$	−29.8860	−1.20317	−0.601583	−	0.798811i	$-0.705463\pi$
$617$	−29.8860	−1.20317	−0.601583	+	0.798811i	$0.705463\pi$
$618$	0	0
$619$	34.7323	1.39601	0.698005	−	0.716093i	$-0.254071\pi$
$619$	34.7323	1.39601	0.698005	+	0.716093i	$0.254071\pi$
$620$	19.4846	0.782520
$621$	0	0
$622$	21.0144	0.842601
$623$	−15.9301	−0.638227
$624$	0	0
$625$	10.2554	0.410216
$626$	−32.2306	−1.28819
$627$	0	0
$628$	−20.0843	−0.801452
$629$	2.79985	0.111637
$630$	0	0
$631$	−13.6310	−0.542641	−0.271320	−	0.962489i	$-0.587460\pi$
$631$	−13.6310	−0.542641	−0.271320	+	0.962489i	$0.587460\pi$
$632$	14.7166	0.585394
$633$	0	0
$634$	21.5139	0.854427
$635$	16.2128	0.643385
$636$	0	0
$637$	0.113964	0.00451543
$638$	18.0769	0.715671
$639$	0	0
$640$	3.73396	0.147598
$641$	−28.7075	−1.13388	−0.566940	−	0.823759i	$-0.691873\pi$
$641$	−28.7075	−1.13388	−0.566940	+	0.823759i	$0.691873\pi$
$642$	0	0
$643$	−12.6062	−0.497142	−0.248571	−	0.968614i	$-0.579961\pi$
$643$	−12.6062	−0.497142	−0.248571	+	0.968614i	$0.579961\pi$
$644$	−21.3757	−0.842321
$645$	0	0
$646$	22.2734	0.876337
$647$	−35.2613	−1.38627	−0.693133	−	0.720809i	$-0.743770\pi$
$647$	−35.2613	−1.38627	−0.693133	+	0.720809i	$0.743770\pi$
$648$	0	0
$649$	6.90981	0.271234
$650$	8.82746	0.346241
$651$	0	0
$652$	−6.48075	−0.253806
$653$	−16.8789	−0.660523	−0.330262	−	0.943889i	$-0.607137\pi$
$653$	−16.8789	−0.660523	−0.330262	+	0.943889i	$0.607137\pi$
$654$	0	0
$655$	−14.2707	−0.557603
$656$	−3.88515	−0.151690
$657$	0	0
$658$	−8.80139	−0.343114
$659$	−0.200950	−0.00782792	−0.00391396	−	0.999992i	$-0.501246\pi$
$659$	−0.200950	−0.00782792	−0.00391396	+	0.999992i	$0.501246\pi$
$660$	0	0
$661$	−40.1475	−1.56156	−0.780779	−	0.624808i	$-0.785177\pi$
$661$	−40.1475	−1.56156	−0.780779	+	0.624808i	$0.785177\pi$
$662$	3.00592	0.116829
$663$	0	0
$664$	6.87510	0.266806
$665$	−41.5875	−1.61269
$666$	0	0
$667$	−68.7340	−2.66139
$668$	2.40489	0.0930479
$669$	0	0
$670$	3.03101	0.117098
$671$	−17.1874	−0.663511
$672$	0	0
$673$	9.22679	0.355667	0.177833	−	0.984061i	$-0.443091\pi$
$673$	9.22679	0.355667	0.177833	+	0.984061i	$0.443091\pi$
$674$	−27.2897	−1.05116
$675$	0	0
$676$	−12.0256	−0.462521
$677$	20.2690	0.779003	0.389501	−	0.921026i	$-0.372647\pi$
$677$	20.2690	0.779003	0.389501	+	0.921026i	$0.372647\pi$
$678$	0	0
$679$	4.40216	0.168939
$680$	19.5930	0.751359
$681$	0	0
$682$	11.1804	0.428119
$683$	8.57482	0.328106	0.164053	−	0.986451i	$-0.447543\pi$
$683$	8.57482	0.328106	0.164053	+	0.986451i	$0.447543\pi$
$684$	0	0
$685$	41.4387	1.58329
$686$	−18.6698	−0.712817
$687$	0	0
$688$	2.54110	0.0968786
$689$	−2.48801	−0.0947858
$690$	0	0
$691$	−8.37770	−0.318703	−0.159351	−	0.987222i	$-0.550940\pi$
$691$	−8.37770	−0.318703	−0.159351	+	0.987222i	$0.550940\pi$
$692$	−15.9773	−0.607365
$693$	0	0
$694$	19.5990	0.743967
$695$	−9.24860	−0.350820
$696$	0	0
$697$	−20.3864	−0.772188
$698$	21.6076	0.817858
$699$	0	0
$700$	−23.4636	−0.886842
$701$	7.14154	0.269732	0.134866	−	0.990864i	$-0.456940\pi$
$701$	7.14154	0.269732	0.134866	+	0.990864i	$0.456940\pi$
$702$	0	0
$703$	−2.26495	−0.0854241
$704$	2.14257	0.0807512
$705$	0	0
$706$	−22.9587	−0.864063
$707$	−45.2689	−1.70251
$708$	0	0
$709$	−0.367596	−0.0138053	−0.00690267	−	0.999976i	$-0.502197\pi$
$709$	−0.367596	−0.0138053	−0.00690267	+	0.999976i	$0.502197\pi$
$710$	−14.7782	−0.554616
$711$	0	0
$712$	−6.07130	−0.227531
$713$	−42.5113	−1.59206
$714$	0	0
$715$	7.89739	0.295346
$716$	−3.98077	−0.148768
$717$	0	0
$718$	8.06534	0.300996
$719$	−16.1535	−0.602422	−0.301211	−	0.953557i	$-0.597391\pi$
$719$	−16.1535	−0.602422	−0.301211	+	0.953557i	$0.597391\pi$
$720$	0	0
$721$	2.55532	0.0951651
$722$	0.981830	0.0365399
$723$	0	0
$724$	19.7064	0.732381
$725$	−75.4477	−2.80206
$726$	0	0
$727$	4.68129	0.173620	0.0868098	−	0.996225i	$-0.472333\pi$
$727$	4.68129	0.173620	0.0868098	+	0.996225i	$0.472333\pi$
$728$	−2.59010	−0.0959954
$729$	0	0
$730$	−33.1689	−1.22764
$731$	13.3338	0.493168
$732$	0	0
$733$	−18.5648	−0.685706	−0.342853	−	0.939389i	$-0.611393\pi$
$733$	−18.5648	−0.685706	−0.342853	+	0.939389i	$0.611393\pi$
$734$	13.5555	0.500342
$735$	0	0
$736$	−8.14672	−0.300292
$737$	1.73921	0.0640648
$738$	0	0
$739$	18.1395	0.667273	0.333636	−	0.942702i	$-0.391724\pi$
$739$	18.1395	0.667273	0.333636	+	0.942702i	$0.391724\pi$
$740$	−1.99238	−0.0732414
$741$	0	0
$742$	6.61321	0.242779
$743$	46.4016	1.70231	0.851155	−	0.524915i	$-0.175903\pi$
$743$	46.4016	1.70231	0.851155	+	0.524915i	$0.175903\pi$
$744$	0	0
$745$	−3.73396	−0.136802
$746$	−31.1012	−1.13870
$747$	0	0
$748$	11.2426	0.411071
$749$	−34.5033	−1.26072
$750$	0	0
$751$	9.41823	0.343676	0.171838	−	0.985125i	$-0.445029\pi$
$751$	9.41823	0.343676	0.171838	+	0.985125i	$0.445029\pi$
$752$	−3.35439	−0.122322
$753$	0	0
$754$	−8.32850	−0.303306
$755$	57.7368	2.10126
$756$	0	0
$757$	7.57303	0.275246	0.137623	−	0.990485i	$-0.456054\pi$
$757$	7.57303	0.275246	0.137623	+	0.990485i	$0.456054\pi$
$758$	−31.2257	−1.13417
$759$	0	0
$760$	−15.8499	−0.574935
$761$	11.4404	0.414715	0.207357	−	0.978265i	$-0.433514\pi$
$761$	11.4404	0.414715	0.207357	+	0.978265i	$0.433514\pi$
$762$	0	0
$763$	−17.0075	−0.615712
$764$	19.7320	0.713877
$765$	0	0
$766$	2.00824	0.0725608
$767$	−3.18353	−0.114951
$768$	0	0
$769$	−29.2382	−1.05435	−0.527177	−	0.849755i	$-0.676750\pi$
$769$	−29.2382	−1.05435	−0.527177	+	0.849755i	$0.676750\pi$
$770$	−20.9915	−0.756480
$771$	0	0
$772$	8.55331	0.307840
$773$	−34.8920	−1.25498	−0.627489	−	0.778626i	$-0.715917\pi$
$773$	−34.8920	−1.25498	−0.627489	+	0.778626i	$0.715917\pi$
$774$	0	0
$775$	−46.6637	−1.67621
$776$	1.67775	0.0602278
$777$	0	0
$778$	11.1882	0.401116
$779$	16.4916	0.590873
$780$	0	0
$781$	−8.47983	−0.303432
$782$	−42.7479	−1.52866
$783$	0	0
$784$	−0.115449	−0.00412318
$785$	74.9941	2.67665
$786$	0	0
$787$	−6.66569	−0.237606	−0.118803	−	0.992918i	$-0.537906\pi$
$787$	−6.66569	−0.237606	−0.118803	+	0.992918i	$0.537906\pi$
$788$	2.35388	0.0838533
$789$	0	0
$790$	−54.9512	−1.95508
$791$	25.7088	0.914100
$792$	0	0
$793$	7.91867	0.281200
$794$	−15.7271	−0.558134
$795$	0	0
$796$	13.9149	0.493200
$797$	−49.8679	−1.76641	−0.883205	−	0.468987i	$-0.844619\pi$
$797$	−49.8679	−1.76641	−0.883205	+	0.468987i	$0.844619\pi$
$798$	0	0
$799$	−17.6013	−0.622690
$800$	−8.94247	−0.316164
$801$	0	0
$802$	−18.6378	−0.658125
$803$	−19.0326	−0.671644
$804$	0	0
$805$	79.8161	2.81315
$806$	−5.15110	−0.181440
$807$	0	0
$808$	−17.2529	−0.606955
$809$	35.2546	1.23949	0.619743	−	0.784805i	$-0.287237\pi$
$809$	35.2546	1.23949	0.619743	+	0.784805i	$0.287237\pi$
$810$	0	0
$811$	−37.1164	−1.30333	−0.651667	−	0.758506i	$-0.725930\pi$
$811$	−37.1164	−1.30333	−0.651667	+	0.758506i	$0.725930\pi$
$812$	22.1374	0.776870
$813$	0	0
$814$	−1.14324	−0.0400706
$815$	24.1989	0.847650
$816$	0	0
$817$	−10.7864	−0.377369
$818$	10.7222	0.374892
$819$	0	0
$820$	14.5070	0.506606
$821$	19.8270	0.691968	0.345984	−	0.938240i	$-0.387545\pi$
$821$	19.8270	0.691968	0.345984	+	0.938240i	$0.387545\pi$
$822$	0	0
$823$	14.1898	0.494625	0.247312	−	0.968936i	$-0.420453\pi$
$823$	14.1898	0.494625	0.247312	+	0.968936i	$0.420453\pi$
$824$	0.973885	0.0339269
$825$	0	0
$826$	8.46191	0.294428
$827$	18.9668	0.659541	0.329770	−	0.944061i	$-0.393029\pi$
$827$	18.9668	0.659541	0.329770	+	0.944061i	$0.393029\pi$
$828$	0	0
$829$	−38.6226	−1.34142	−0.670709	−	0.741721i	$-0.734010\pi$
$829$	−38.6226	−1.34142	−0.670709	+	0.741721i	$0.734010\pi$
$830$	−25.6714	−0.891066
$831$	0	0
$832$	−0.987139	−0.0342229
$833$	−0.605790	−0.0209894
$834$	0	0
$835$	−8.97976	−0.310757
$836$	−9.09475	−0.314548
$837$	0	0
$838$	−21.0470	−0.727057
$839$	−38.8063	−1.33974	−0.669871	−	0.742478i	$-0.733650\pi$
$839$	−38.8063	−1.33974	−0.669871	+	0.742478i	$0.733650\pi$
$840$	0	0
$841$	42.1831	1.45459
$842$	22.1796	0.764358
$843$	0	0
$844$	1.13848	0.0391883
$845$	44.9030	1.54471
$846$	0	0
$847$	16.8172	0.577847
$848$	2.52043	0.0865519
$849$	0	0
$850$	−46.9234	−1.60946
$851$	4.34696	0.149012
$852$	0	0
$853$	−23.8861	−0.817846	−0.408923	−	0.912569i	$-0.634096\pi$
$853$	−23.8861	−0.817846	−0.408923	+	0.912569i	$0.634096\pi$
$854$	−21.0480	−0.720249
$855$	0	0
$856$	−13.1499	−0.449455
$857$	−10.8617	−0.371028	−0.185514	−	0.982642i	$-0.559395\pi$
$857$	−10.8617	−0.371028	−0.185514	+	0.982642i	$0.559395\pi$
$858$	0	0
$859$	−22.6126	−0.771532	−0.385766	−	0.922597i	$-0.626063\pi$
$859$	−22.6126	−0.771532	−0.385766	+	0.922597i	$0.626063\pi$
$860$	−9.48837	−0.323551
$861$	0	0
$862$	−32.4982	−1.10689
$863$	−39.0059	−1.32777	−0.663887	−	0.747833i	$-0.731095\pi$
$863$	−39.0059	−1.32777	−0.663887	+	0.747833i	$0.731095\pi$
$864$	0	0
$865$	59.6585	2.02845
$866$	2.82924	0.0961416
$867$	0	0
$868$	13.6918	0.464729
$869$	−31.5313	−1.06963
$870$	0	0
$871$	−0.801302	−0.0271511
$872$	−6.48190	−0.219505
$873$	0	0
$874$	34.5811	1.16972
$875$	38.6257	1.30579
$876$	0	0
$877$	13.3530	0.450899	0.225449	−	0.974255i	$-0.427615\pi$
$877$	13.3530	0.450899	0.225449	+	0.974255i	$0.427615\pi$
$878$	11.3455	0.382890
$879$	0	0
$880$	−8.00028	−0.269689
$881$	46.1697	1.55550	0.777749	−	0.628575i	$-0.216362\pi$
$881$	46.1697	1.55550	0.777749	+	0.628575i	$0.216362\pi$
$882$	0	0
$883$	19.6600	0.661611	0.330806	−	0.943699i	$-0.392680\pi$
$883$	19.6600	0.661611	0.330806	+	0.943699i	$0.392680\pi$
$884$	−5.17977	−0.174214
$885$	0	0
$886$	−23.7537	−0.798023
$887$	30.8029	1.03426	0.517130	−	0.855907i	$-0.327000\pi$
$887$	30.8029	1.03426	0.517130	+	0.855907i	$0.327000\pi$
$888$	0	0
$889$	11.3927	0.382098
$890$	22.6700	0.759900
$891$	0	0
$892$	−6.72193	−0.225067
$893$	14.2386	0.476478
$894$	0	0
$895$	14.8640	0.496850
$896$	2.62384	0.0876564
$897$	0	0
$898$	−31.7903	−1.06086
$899$	44.0261	1.46835
$900$	0	0
$901$	13.2253	0.440600
$902$	8.32421	0.277166
$903$	0	0
$904$	9.79815	0.325882
$905$	−73.5828	−2.44597
$906$	0	0
$907$	9.99579	0.331905	0.165952	−	0.986134i	$-0.446930\pi$
$907$	9.99579	0.331905	0.165952	+	0.986134i	$0.446930\pi$
$908$	−8.37087	−0.277797
$909$	0	0
$910$	9.67132	0.320601
$911$	11.8384	0.392223	0.196111	−	0.980582i	$-0.437169\pi$
$911$	11.8384	0.392223	0.196111	+	0.980582i	$0.437169\pi$
$912$	0	0
$913$	−14.7304	−0.487505
$914$	−2.44674	−0.0809308
$915$	0	0
$916$	9.32680	0.308166
$917$	−10.0280	−0.331153
$918$	0	0
$919$	49.1503	1.62132	0.810660	−	0.585517i	$-0.199108\pi$
$919$	49.1503	1.62132	0.810660	+	0.585517i	$0.199108\pi$
$920$	30.4195	1.00290
$921$	0	0
$922$	−6.34903	−0.209094
$923$	3.90688	0.128596
$924$	0	0
$925$	4.77156	0.156888
$926$	−0.408336	−0.0134187
$927$	0	0
$928$	8.43701	0.276958
$929$	10.1963	0.334531	0.167266	−	0.985912i	$-0.446506\pi$
$929$	10.1963	0.334531	0.167266	+	0.985912i	$0.446506\pi$
$930$	0	0
$931$	0.490056	0.0160609
$932$	10.2941	0.337196
$933$	0	0
$934$	4.73896	0.155064
$935$	−41.9795	−1.37288
$936$	0	0
$937$	36.3197	1.18651	0.593257	−	0.805013i	$-0.297842\pi$
$937$	36.3197	1.18651	0.593257	+	0.805013i	$0.297842\pi$
$938$	2.12988	0.0695431
$939$	0	0
$940$	12.5252	0.408526
$941$	41.5470	1.35439	0.677197	−	0.735802i	$-0.263194\pi$
$941$	41.5470	1.35439	0.677197	+	0.735802i	$0.263194\pi$
$942$	0	0
$943$	−31.6512	−1.03071
$944$	3.22501	0.104965
$945$	0	0
$946$	−5.44449	−0.177016
$947$	43.4552	1.41210	0.706052	−	0.708160i	$-0.250474\pi$
$947$	43.4552	1.41210	0.706052	+	0.708160i	$0.250474\pi$
$948$	0	0
$949$	8.76880	0.284647
$950$	37.9588	1.23155
$951$	0	0
$952$	13.7680	0.446222
$953$	25.6035	0.829377	0.414689	−	0.909963i	$-0.363890\pi$
$953$	25.6035	0.829377	0.414689	+	0.909963i	$0.363890\pi$
$954$	0	0
$955$	−73.6784	−2.38418
$956$	12.0117	0.388486
$957$	0	0
$958$	−13.6810	−0.442014
$959$	29.1189	0.940297
$960$	0	0
$961$	−3.77029	−0.121622
$962$	0.526721	0.0169822
$963$	0	0
$964$	24.9677	0.804155
$965$	−31.9377	−1.02811
$966$	0	0
$967$	19.2132	0.617855	0.308928	−	0.951086i	$-0.400030\pi$
$967$	19.2132	0.617855	0.308928	+	0.951086i	$0.400030\pi$
$968$	6.40939	0.206005
$969$	0	0
$970$	−6.26467	−0.201146
$971$	−18.6235	−0.597656	−0.298828	−	0.954307i	$-0.596596\pi$
$971$	−18.6235	−0.597656	−0.298828	+	0.954307i	$0.596596\pi$
$972$	0	0
$973$	−6.49896	−0.208347
$974$	−8.54623	−0.273839
$975$	0	0
$976$	−8.02184	−0.256773
$977$	−29.5674	−0.945946	−0.472973	−	0.881077i	$-0.656819\pi$
$977$	−29.5674	−0.945946	−0.472973	+	0.881077i	$0.656819\pi$
$978$	0	0
$979$	13.0082	0.415743
$980$	0.431082	0.0137704
$981$	0	0
$982$	13.4171	0.428156
$983$	−12.1120	−0.386313	−0.193156	−	0.981168i	$-0.561872\pi$
$983$	−12.1120	−0.386313	−0.193156	+	0.981168i	$0.561872\pi$
$984$	0	0
$985$	−8.78928	−0.280050
$986$	44.2711	1.40988
$987$	0	0
$988$	4.19019	0.133308
$989$	20.7016	0.658274
$990$	0	0
$991$	26.2842	0.834946	0.417473	−	0.908689i	$-0.362916\pi$
$991$	26.2842	0.834946	0.417473	+	0.908689i	$0.362916\pi$
$992$	5.21821	0.165678
$993$	0	0
$994$	−10.3846	−0.329379
$995$	−51.9576	−1.64717
$996$	0	0
$997$	27.5456	0.872377	0.436189	−	0.899855i	$-0.356328\pi$
$997$	27.5456	0.872377	0.436189	+	0.899855i	$0.356328\pi$
$998$	9.21700	0.291759
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.l.1.1	✓	12
3.2	odd	2		8046.2.a.m.1.12	yes	12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.l.1.1	✓	12	1.1	even	1	trivial
8046.2.a.m.1.12	yes	12	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.73396 q^{5} -2.62384 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} -3.73396 q^{5} -2.62384 q^{7} -1.00000 q^{8} +3.73396 q^{10} +2.14257 q^{11} -0.987139 q^{13} +2.62384 q^{14} +1.00000 q^{16} +5.24725 q^{17} -4.24478 q^{19} -3.73396 q^{20} -2.14257 q^{22} +8.14672 q^{23} +8.94247 q^{25} +0.987139 q^{26} -2.62384 q^{28} -8.43701 q^{29} -5.21821 q^{31} -1.00000 q^{32} -5.24725 q^{34} +9.79733 q^{35} +0.533584 q^{37} +4.24478 q^{38} +3.73396 q^{40} -3.88515 q^{41} +2.54110 q^{43} +2.14257 q^{44} -8.14672 q^{46} -3.35439 q^{47} -0.115449 q^{49} -8.94247 q^{50} -0.987139 q^{52} +2.52043 q^{53} -8.00028 q^{55} +2.62384 q^{56} +8.43701 q^{58} +3.22501 q^{59} -8.02184 q^{61} +5.21821 q^{62} +1.00000 q^{64} +3.68594 q^{65} +0.811742 q^{67} +5.24725 q^{68} -9.79733 q^{70} -3.95778 q^{71} -8.88304 q^{73} -0.533584 q^{74} -4.24478 q^{76} -5.62177 q^{77} -14.7166 q^{79} -3.73396 q^{80} +3.88515 q^{82} -6.87510 q^{83} -19.5930 q^{85} -2.54110 q^{86} -2.14257 q^{88} +6.07130 q^{89} +2.59010 q^{91} +8.14672 q^{92} +3.35439 q^{94} +15.8499 q^{95} -1.67775 q^{97} +0.115449 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8}+O(q^{10}) \) 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8	\( 12 q - 12 q^{2} + 12 q^{4} + 5 q^{5} - 6 q^{7} - 12 q^{8} - 5 q^{10} + 10 q^{11} - q^{13} + 6 q^{14} + 12 q^{16} + 6 q^{17} - 10 q^{19} + 5 q^{20} - 10 q^{22} + 15 q^{23} + 7 q^{25} + q^{26} - 6 q^{28} + 33 q^{29} - 6 q^{31} - 12 q^{32} - 6 q^{34} + 16 q^{35} - 13 q^{37} + 10 q^{38} - 5 q^{40} + 20 q^{41} - 11 q^{43} + 10 q^{44} - 15 q^{46} + 15 q^{47} + 2 q^{49} - 7 q^{50} - q^{52} + 4 q^{53} - 17 q^{55} + 6 q^{56} - 33 q^{58} + 10 q^{59} - 12 q^{61} + 6 q^{62} + 12 q^{64} + 40 q^{65} - 19 q^{67} + 6 q^{68} - 16 q^{70} + 47 q^{71} - 2 q^{73} + 13 q^{74} - 10 q^{76} - 6 q^{77} - 15 q^{79} + 5 q^{80} - 20 q^{82} + 18 q^{83} - 25 q^{85} + 11 q^{86} - 10 q^{88} + 24 q^{89} - 3 q^{91} + 15 q^{92} - 15 q^{94} - 3 q^{95} - 25 q^{97} - 2 q^{98}+O(q^{100}) \) 12 * q - 12 * q^2 + 12 * q^4 + 5 * q^5 - 6 * q^7 - 12 * q^8 - 5 * q^10 + 10 * q^11 - q^13 + 6 * q^14 + 12 * q^16 + 6 * q^17 - 10 * q^19 + 5 * q^20 - 10 * q^22 + 15 * q^23 + 7 * q^25 + q^26 - 6 * q^28 + 33 * q^29 - 6 * q^31 - 12 * q^32 - 6 * q^34 + 16 * q^35 - 13 * q^37 + 10 * q^38 - 5 * q^40 + 20 * q^41 - 11 * q^43 + 10 * q^44 - 15 * q^46 + 15 * q^47 + 2 * q^49 - 7 * q^50 - q^52 + 4 * q^53 - 17 * q^55 + 6 * q^56 - 33 * q^58 + 10 * q^59 - 12 * q^61 + 6 * q^62 + 12 * q^64 + 40 * q^65 - 19 * q^67 + 6 * q^68 - 16 * q^70 + 47 * q^71 - 2 * q^73 + 13 * q^74 - 10 * q^76 - 6 * q^77 - 15 * q^79 + 5 * q^80 - 20 * q^82 + 18 * q^83 - 25 * q^85 + 11 * q^86 - 10 * q^88 + 24 * q^89 - 3 * q^91 + 15 * q^92 - 15 * q^94 - 3 * q^95 - 25 * q^97 - 2 * q^98

Introduction

L-functions

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Varieties

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