LMFDB - Embedded newform 8046.2.a.d.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:	$2$
Coefficient field:	$\Q(\zeta_{12})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{2} - 3 $ x^2 - 3
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			$-1.73205$ 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)

$f(q)$	$=$	$q+1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} -2.46410 q^{7} +1.00000 q^{8} +O(q^{10})$	\(q+1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} -2.46410 q^{7} +1.00000 q^{8} -3.46410 q^{10} +1.26795 q^{11} +5.46410 q^{13} -2.46410 q^{14} +1.00000 q^{16} -6.73205 q^{17} -5.46410 q^{19} -3.46410 q^{20} +1.26795 q^{22} -4.26795 q^{23} +7.00000 q^{25} +5.46410 q^{26} -2.46410 q^{28} +1.73205 q^{29} +1.00000 q^{31} +1.00000 q^{32} -6.73205 q^{34} +8.53590 q^{35} +2.73205 q^{37} -5.46410 q^{38} -3.46410 q^{40} -1.19615 q^{41} +8.92820 q^{43} +1.26795 q^{44} -4.26795 q^{46} -7.46410 q^{47} -0.928203 q^{49} +7.00000 q^{50} +5.46410 q^{52} -6.26795 q^{53} -4.39230 q^{55} -2.46410 q^{56} +1.73205 q^{58} +3.26795 q^{59} +2.73205 q^{61} +1.00000 q^{62} +1.00000 q^{64} -18.9282 q^{65} -6.19615 q^{67} -6.73205 q^{68} +8.53590 q^{70} +14.1244 q^{71} +1.46410 q^{73} +2.73205 q^{74} -5.46410 q^{76} -3.12436 q^{77} -12.7321 q^{79} -3.46410 q^{80} -1.19615 q^{82} +1.80385 q^{83} +23.3205 q^{85} +8.92820 q^{86} +1.26795 q^{88} +4.26795 q^{89} -13.4641 q^{91} -4.26795 q^{92} -7.46410 q^{94} +18.9282 q^{95} +10.7321 q^{97} -0.928203 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8	$ 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 10 q^{17} - 4 q^{19} + 6 q^{22} - 12 q^{23} + 14 q^{25} + 4 q^{26} + 2 q^{28} + 2 q^{31} + 2 q^{32} - 10 q^{34} + 24 q^{35} + 2 q^{37} - 4 q^{38} + 8 q^{41} + 4 q^{43} + 6 q^{44} - 12 q^{46} - 8 q^{47} + 12 q^{49} + 14 q^{50} + 4 q^{52} - 16 q^{53} + 12 q^{55} + 2 q^{56} + 10 q^{59} + 2 q^{61} + 2 q^{62} + 2 q^{64} - 24 q^{65} - 2 q^{67} - 10 q^{68} + 24 q^{70} + 4 q^{71} - 4 q^{73} + 2 q^{74} - 4 q^{76} + 18 q^{77} - 22 q^{79} + 8 q^{82} + 14 q^{83} + 12 q^{85} + 4 q^{86} + 6 q^{88} + 12 q^{89} - 20 q^{91} - 12 q^{92} - 8 q^{94} + 24 q^{95} + 18 q^{97} + 12 q^{98}+O(q^{100}) $ 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 6 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 10 * q^17 - 4 * q^19 + 6 * q^22 - 12 * q^23 + 14 * q^25 + 4 * q^26 + 2 * q^28 + 2 * q^31 + 2 * q^32 - 10 * q^34 + 24 * q^35 + 2 * q^37 - 4 * q^38 + 8 * q^41 + 4 * q^43 + 6 * q^44 - 12 * q^46 - 8 * q^47 + 12 * q^49 + 14 * q^50 + 4 * q^52 - 16 * q^53 + 12 * q^55 + 2 * q^56 + 10 * q^59 + 2 * q^61 + 2 * q^62 + 2 * q^64 - 24 * q^65 - 2 * q^67 - 10 * q^68 + 24 * q^70 + 4 * q^71 - 4 * q^73 + 2 * q^74 - 4 * q^76 + 18 * q^77 - 22 * q^79 + 8 * q^82 + 14 * q^83 + 12 * q^85 + 4 * q^86 + 6 * q^88 + 12 * q^89 - 20 * q^91 - 12 * q^92 - 8 * q^94 + 24 * q^95 + 18 * q^97 + 12 * 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
$5$	−3.46410	−1.54919	−0.774597	−	0.632456i	$-0.782047\pi$
$5$	−3.46410	−1.54919	−0.774597	+	0.632456i	$0.782047\pi$
$6$	0	0
$7$	−2.46410	−0.931343	−0.465671	−	0.884958i	$-0.654187\pi$
$7$	−2.46410	−0.931343	−0.465671	+	0.884958i	$0.654187\pi$
$8$	1.00000	0.353553
$9$	0	0
$10$	−3.46410	−1.09545
$11$	1.26795	0.382301	0.191151	−	0.981561i	$-0.438778\pi$
$11$	1.26795	0.382301	0.191151	+	0.981561i	$0.438778\pi$
$12$	0	0
$13$	5.46410	1.51547	0.757735	−	0.652563i	$-0.226306\pi$
$13$	5.46410	1.51547	0.757735	+	0.652563i	$0.226306\pi$
$14$	−2.46410	−0.658559
$15$	0	0
$16$	1.00000	0.250000
$17$	−6.73205	−1.63276	−0.816381	−	0.577514i	$-0.804023\pi$
$17$	−6.73205	−1.63276	−0.816381	+	0.577514i	$0.804023\pi$
$18$	0	0
$19$	−5.46410	−1.25355	−0.626775	−	0.779200i	$-0.715626\pi$
$19$	−5.46410	−1.25355	−0.626775	+	0.779200i	$0.715626\pi$
$20$	−3.46410	−0.774597
$21$	0	0
$22$	1.26795	0.270328
$23$	−4.26795	−0.889929	−0.444964	−	0.895548i	$-0.646784\pi$
$23$	−4.26795	−0.889929	−0.444964	+	0.895548i	$0.646784\pi$
$24$	0	0
$25$	7.00000	1.40000
$26$	5.46410	1.07160
$27$	0	0
$28$	−2.46410	−0.465671
$29$	1.73205	0.321634	0.160817	−	0.986984i	$-0.448587\pi$
$29$	1.73205	0.321634	0.160817	+	0.986984i	$0.448587\pi$
$30$	0	0
$31$	1.00000	0.179605	0.0898027	−	0.995960i	$-0.471376\pi$
$31$	1.00000	0.179605	0.0898027	+	0.995960i	$0.471376\pi$
$32$	1.00000	0.176777
$33$	0	0
$34$	−6.73205	−1.15454
$35$	8.53590	1.44283
$36$	0	0
$37$	2.73205	0.449146	0.224573	−	0.974457i	$-0.427901\pi$
$37$	2.73205	0.449146	0.224573	+	0.974457i	$0.427901\pi$
$38$	−5.46410	−0.886394
$39$	0	0
$40$	−3.46410	−0.547723
$41$	−1.19615	−0.186808	−0.0934038	−	0.995628i	$-0.529775\pi$
$41$	−1.19615	−0.186808	−0.0934038	+	0.995628i	$0.529775\pi$
$42$	0	0
$43$	8.92820	1.36154	0.680769	−	0.732498i	$-0.261646\pi$
$43$	8.92820	1.36154	0.680769	+	0.732498i	$0.261646\pi$
$44$	1.26795	0.191151
$45$	0	0
$46$	−4.26795	−0.629275
$47$	−7.46410	−1.08875	−0.544376	−	0.838842i	$-0.683233\pi$
$47$	−7.46410	−1.08875	−0.544376	+	0.838842i	$0.683233\pi$
$48$	0	0
$49$	−0.928203	−0.132600
$50$	7.00000	0.989949
$51$	0	0
$52$	5.46410	0.757735
$53$	−6.26795	−0.860969	−0.430485	−	0.902598i	$-0.641657\pi$
$53$	−6.26795	−0.860969	−0.430485	+	0.902598i	$0.641657\pi$
$54$	0	0
$55$	−4.39230	−0.592258
$56$	−2.46410	−0.329279
$57$	0	0
$58$	1.73205	0.227429
$59$	3.26795	0.425451	0.212725	−	0.977112i	$-0.431766\pi$
$59$	3.26795	0.425451	0.212725	+	0.977112i	$0.431766\pi$
$60$	0	0
$61$	2.73205	0.349803	0.174902	−	0.984586i	$-0.444039\pi$
$61$	2.73205	0.349803	0.174902	+	0.984586i	$0.444039\pi$
$62$	1.00000	0.127000
$63$	0	0
$64$	1.00000	0.125000
$65$	−18.9282	−2.34775
$66$	0	0
$67$	−6.19615	−0.756980	−0.378490	−	0.925605i	$-0.623557\pi$
$67$	−6.19615	−0.756980	−0.378490	+	0.925605i	$0.623557\pi$
$68$	−6.73205	−0.816381
$69$	0	0
$70$	8.53590	1.02023
$71$	14.1244	1.67625	0.838126	−	0.545476i	$-0.183651\pi$
$71$	14.1244	1.67625	0.838126	+	0.545476i	$0.183651\pi$
$72$	0	0
$73$	1.46410	0.171360	0.0856801	−	0.996323i	$-0.472694\pi$
$73$	1.46410	0.171360	0.0856801	+	0.996323i	$0.472694\pi$
$74$	2.73205	0.317594
$75$	0	0
$76$	−5.46410	−0.626775
$77$	−3.12436	−0.356053
$78$	0	0
$79$	−12.7321	−1.43247	−0.716234	−	0.697860i	$-0.754136\pi$
$79$	−12.7321	−1.43247	−0.716234	+	0.697860i	$0.754136\pi$
$80$	−3.46410	−0.387298
$81$	0	0
$82$	−1.19615	−0.132093
$83$	1.80385	0.197998	0.0989990	−	0.995088i	$-0.468436\pi$
$83$	1.80385	0.197998	0.0989990	+	0.995088i	$0.468436\pi$
$84$	0	0
$85$	23.3205	2.52946
$86$	8.92820	0.962753
$87$	0	0
$88$	1.26795	0.135164
$89$	4.26795	0.452402	0.226201	−	0.974081i	$-0.427369\pi$
$89$	4.26795	0.452402	0.226201	+	0.974081i	$0.427369\pi$
$90$	0	0
$91$	−13.4641	−1.41142
$92$	−4.26795	−0.444964
$93$	0	0
$94$	−7.46410	−0.769863
$95$	18.9282	1.94199
$96$	0	0
$97$	10.7321	1.08967	0.544837	−	0.838542i	$-0.316591\pi$
$97$	10.7321	1.08967	0.544837	+	0.838542i	$0.316591\pi$
$98$	−0.928203	−0.0937627
$99$	0	0
$100$	7.00000	0.700000
$101$	−6.39230	−0.636058	−0.318029	−	0.948081i	$-0.603021\pi$
$101$	−6.39230	−0.636058	−0.318029	+	0.948081i	$0.603021\pi$
$102$	0	0
$103$	15.9282	1.56945	0.784726	−	0.619842i	$-0.212804\pi$
$103$	15.9282	1.56945	0.784726	+	0.619842i	$0.212804\pi$
$104$	5.46410	0.535799
$105$	0	0
$106$	−6.26795	−0.608797
$107$	−4.53590	−0.438502	−0.219251	−	0.975669i	$-0.570361\pi$
$107$	−4.53590	−0.438502	−0.219251	+	0.975669i	$0.570361\pi$
$108$	0	0
$109$	−5.92820	−0.567819	−0.283909	−	0.958851i	$-0.591632\pi$
$109$	−5.92820	−0.567819	−0.283909	+	0.958851i	$0.591632\pi$
$110$	−4.39230	−0.418790
$111$	0	0
$112$	−2.46410	−0.232836
$113$	16.1962	1.52361	0.761803	−	0.647809i	$-0.224314\pi$
$113$	16.1962	1.52361	0.761803	+	0.647809i	$0.224314\pi$
$114$	0	0
$115$	14.7846	1.37867
$116$	1.73205	0.160817
$117$	0	0
$118$	3.26795	0.300839
$119$	16.5885	1.52066
$120$	0	0
$121$	−9.39230	−0.853846
$122$	2.73205	0.247348
$123$	0	0
$124$	1.00000	0.0898027
$125$	−6.92820	−0.619677
$126$	0	0
$127$	14.0000	1.24230	0.621150	−	0.783692i	$-0.286666\pi$
$127$	14.0000	1.24230	0.621150	+	0.783692i	$0.286666\pi$
$128$	1.00000	0.0883883
$129$	0	0
$130$	−18.9282	−1.66011
$131$	9.46410	0.826882	0.413441	−	0.910531i	$-0.364327\pi$
$131$	9.46410	0.826882	0.413441	+	0.910531i	$0.364327\pi$
$132$	0	0
$133$	13.4641	1.16749
$134$	−6.19615	−0.535266
$135$	0	0
$136$	−6.73205	−0.577269
$137$	1.19615	0.102194	0.0510971	−	0.998694i	$-0.483728\pi$
$137$	1.19615	0.102194	0.0510971	+	0.998694i	$0.483728\pi$
$138$	0	0
$139$	−7.00000	−0.593732	−0.296866	−	0.954919i	$-0.595942\pi$
$139$	−7.00000	−0.593732	−0.296866	+	0.954919i	$0.595942\pi$
$140$	8.53590	0.721415
$141$	0	0
$142$	14.1244	1.18529
$143$	6.92820	0.579365
$144$	0	0
$145$	−6.00000	−0.498273
$146$	1.46410	0.121170
$147$	0	0
$148$	2.73205	0.224573
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	21.1244	1.71908	0.859538	−	0.511072i	$-0.170751\pi$
$151$	21.1244	1.71908	0.859538	+	0.511072i	$0.170751\pi$
$152$	−5.46410	−0.443197
$153$	0	0
$154$	−3.12436	−0.251768
$155$	−3.46410	−0.278243
$156$	0	0
$157$	3.53590	0.282195	0.141098	−	0.989996i	$-0.454937\pi$
$157$	3.53590	0.282195	0.141098	+	0.989996i	$0.454937\pi$
$158$	−12.7321	−1.01291
$159$	0	0
$160$	−3.46410	−0.273861
$161$	10.5167	0.828829
$162$	0	0
$163$	9.92820	0.777637	0.388818	−	0.921314i	$-0.372883\pi$
$163$	9.92820	0.777637	0.388818	+	0.921314i	$0.372883\pi$
$164$	−1.19615	−0.0934038
$165$	0	0
$166$	1.80385	0.140006
$167$	0	0		−	1.00000i	$-0.5\pi$
$167$	0	0			1.00000i	$0.5\pi$
$168$	0	0
$169$	16.8564	1.29665
$170$	23.3205	1.78860
$171$	0	0
$172$	8.92820	0.680769
$173$	−14.2679	−1.08477	−0.542386	−	0.840129i	$-0.682479\pi$
$173$	−14.2679	−1.08477	−0.542386	+	0.840129i	$0.682479\pi$
$174$	0	0
$175$	−17.2487	−1.30388
$176$	1.26795	0.0955753
$177$	0	0
$178$	4.26795	0.319896
$179$	9.58846	0.716675	0.358337	−	0.933592i	$-0.383344\pi$
$179$	9.58846	0.716675	0.358337	+	0.933592i	$0.383344\pi$
$180$	0	0
$181$	−18.4641	−1.37243	−0.686213	−	0.727401i	$-0.740728\pi$
$181$	−18.4641	−1.37243	−0.686213	+	0.727401i	$0.740728\pi$
$182$	−13.4641	−0.998026
$183$	0	0
$184$	−4.26795	−0.314637
$185$	−9.46410	−0.695815
$186$	0	0
$187$	−8.53590	−0.624207
$188$	−7.46410	−0.544376
$189$	0	0
$190$	18.9282	1.37320
$191$	−21.1244	−1.52850	−0.764252	−	0.644917i	$-0.776892\pi$
$191$	−21.1244	−1.52850	−0.764252	+	0.644917i	$0.776892\pi$
$192$	0	0
$193$	22.5885	1.62595	0.812976	−	0.582297i	$-0.197846\pi$
$193$	22.5885	1.62595	0.812976	+	0.582297i	$0.197846\pi$
$194$	10.7321	0.770516
$195$	0	0
$196$	−0.928203	−0.0663002
$197$	3.12436	0.222601	0.111301	−	0.993787i	$-0.464498\pi$
$197$	3.12436	0.222601	0.111301	+	0.993787i	$0.464498\pi$
$198$	0	0
$199$	15.2679	1.08232	0.541158	−	0.840921i	$-0.317986\pi$
$199$	15.2679	1.08232	0.541158	+	0.840921i	$0.317986\pi$
$200$	7.00000	0.494975
$201$	0	0
$202$	−6.39230	−0.449761
$203$	−4.26795	−0.299551
$204$	0	0
$205$	4.14359	0.289401
$206$	15.9282	1.10977
$207$	0	0
$208$	5.46410	0.378867
$209$	−6.92820	−0.479234
$210$	0	0
$211$	15.7846	1.08666	0.543329	−	0.839520i	$-0.317164\pi$
$211$	15.7846	1.08666	0.543329	+	0.839520i	$0.317164\pi$
$212$	−6.26795	−0.430485
$213$	0	0
$214$	−4.53590	−0.310068
$215$	−30.9282	−2.10929
$216$	0	0
$217$	−2.46410	−0.167274
$218$	−5.92820	−0.401509
$219$	0	0
$220$	−4.39230	−0.296129
$221$	−36.7846	−2.47440
$222$	0	0
$223$	−1.26795	−0.0849082	−0.0424541	−	0.999098i	$-0.513518\pi$
$223$	−1.26795	−0.0849082	−0.0424541	+	0.999098i	$0.513518\pi$
$224$	−2.46410	−0.164640
$225$	0	0
$226$	16.1962	1.07735
$227$	7.46410	0.495410	0.247705	−	0.968836i	$-0.420324\pi$
$227$	7.46410	0.495410	0.247705	+	0.968836i	$0.420324\pi$
$228$	0	0
$229$	8.19615	0.541617	0.270808	−	0.962633i	$-0.412709\pi$
$229$	8.19615	0.541617	0.270808	+	0.962633i	$0.412709\pi$
$230$	14.7846	0.974868
$231$	0	0
$232$	1.73205	0.113715
$233$	25.9808	1.70206	0.851028	−	0.525120i	$-0.175980\pi$
$233$	25.9808	1.70206	0.851028	+	0.525120i	$0.175980\pi$
$234$	0	0
$235$	25.8564	1.68669
$236$	3.26795	0.212725
$237$	0	0
$238$	16.5885	1.07527
$239$	6.12436	0.396152	0.198076	−	0.980187i	$-0.436531\pi$
$239$	6.12436	0.396152	0.198076	+	0.980187i	$0.436531\pi$
$240$	0	0
$241$	8.53590	0.549846	0.274923	−	0.961466i	$-0.411348\pi$
$241$	8.53590	0.549846	0.274923	+	0.961466i	$0.411348\pi$
$242$	−9.39230	−0.603760
$243$	0	0
$244$	2.73205	0.174902
$245$	3.21539	0.205424
$246$	0	0
$247$	−29.8564	−1.89972
$248$	1.00000	0.0635001
$249$	0	0
$250$	−6.92820	−0.438178
$251$	3.73205	0.235565	0.117782	−	0.993039i	$-0.462421\pi$
$251$	3.73205	0.235565	0.117782	+	0.993039i	$0.462421\pi$
$252$	0	0
$253$	−5.41154	−0.340221
$254$	14.0000	0.878438
$255$	0	0
$256$	1.00000	0.0625000
$257$	−21.1962	−1.32218	−0.661090	−	0.750307i	$-0.729906\pi$
$257$	−21.1962	−1.32218	−0.661090	+	0.750307i	$0.729906\pi$
$258$	0	0
$259$	−6.73205	−0.418309
$260$	−18.9282	−1.17388
$261$	0	0
$262$	9.46410	0.584694
$263$	14.7846	0.911658	0.455829	−	0.890067i	$-0.349343\pi$
$263$	14.7846	0.911658	0.455829	+	0.890067i	$0.349343\pi$
$264$	0	0
$265$	21.7128	1.33381
$266$	13.4641	0.825537
$267$	0	0
$268$	−6.19615	−0.378490
$269$	15.3397	0.935281	0.467640	−	0.883919i	$-0.345104\pi$
$269$	15.3397	0.935281	0.467640	+	0.883919i	$0.345104\pi$
$270$	0	0
$271$	28.7846	1.74854	0.874270	−	0.485440i	$-0.161340\pi$
$271$	28.7846	1.74854	0.874270	+	0.485440i	$0.161340\pi$
$272$	−6.73205	−0.408191
$273$	0	0
$274$	1.19615	0.0722622
$275$	8.87564	0.535221
$276$	0	0
$277$	5.00000	0.300421	0.150210	−	0.988654i	$-0.452005\pi$
$277$	5.00000	0.300421	0.150210	+	0.988654i	$0.452005\pi$
$278$	−7.00000	−0.419832
$279$	0	0
$280$	8.53590	0.510117
$281$	−24.0526	−1.43486	−0.717428	−	0.696633i	$-0.754681\pi$
$281$	−24.0526	−1.43486	−0.717428	+	0.696633i	$0.754681\pi$
$282$	0	0
$283$	1.53590	0.0912997	0.0456498	−	0.998958i	$-0.485464\pi$
$283$	1.53590	0.0912997	0.0456498	+	0.998958i	$0.485464\pi$
$284$	14.1244	0.838126
$285$	0	0
$286$	6.92820	0.409673
$287$	2.94744	0.173982
$288$	0	0
$289$	28.3205	1.66591
$290$	−6.00000	−0.352332
$291$	0	0
$292$	1.46410	0.0856801
$293$	−12.1244	−0.708312	−0.354156	−	0.935186i	$-0.615232\pi$
$293$	−12.1244	−0.708312	−0.354156	+	0.935186i	$0.615232\pi$
$294$	0	0
$295$	−11.3205	−0.659105
$296$	2.73205	0.158797
$297$	0	0
$298$	1.00000	0.0579284
$299$	−23.3205	−1.34866
$300$	0	0
$301$	−22.0000	−1.26806
$302$	21.1244	1.21557
$303$	0	0
$304$	−5.46410	−0.313388
$305$	−9.46410	−0.541913
$306$	0	0
$307$	−29.3205	−1.67341	−0.836705	−	0.547654i	$-0.815521\pi$
$307$	−29.3205	−1.67341	−0.836705	+	0.547654i	$0.815521\pi$
$308$	−3.12436	−0.178027
$309$	0	0
$310$	−3.46410	−0.196748
$311$	28.1244	1.59479	0.797393	−	0.603460i	$-0.206212\pi$
$311$	28.1244	1.59479	0.797393	+	0.603460i	$0.206212\pi$
$312$	0	0
$313$	−22.5359	−1.27380	−0.636902	−	0.770945i	$-0.719784\pi$
$313$	−22.5359	−1.27380	−0.636902	+	0.770945i	$0.719784\pi$
$314$	3.53590	0.199542
$315$	0	0
$316$	−12.7321	−0.716234
$317$	3.85641	0.216597	0.108299	−	0.994118i	$-0.465460\pi$
$317$	3.85641	0.216597	0.108299	+	0.994118i	$0.465460\pi$
$318$	0	0
$319$	2.19615	0.122961
$320$	−3.46410	−0.193649
$321$	0	0
$322$	10.5167	0.586071
$323$	36.7846	2.04675
$324$	0	0
$325$	38.2487	2.12166
$326$	9.92820	0.549872
$327$	0	0
$328$	−1.19615	−0.0660465
$329$	18.3923	1.01400
$330$	0	0
$331$	1.80385	0.0991484	0.0495742	−	0.998770i	$-0.484214\pi$
$331$	1.80385	0.0991484	0.0495742	+	0.998770i	$0.484214\pi$
$332$	1.80385	0.0989990
$333$	0	0
$334$	0	0
$335$	21.4641	1.17271
$336$	0	0
$337$	19.0000	1.03500	0.517498	−	0.855684i	$-0.326864\pi$
$337$	19.0000	1.03500	0.517498	+	0.855684i	$0.326864\pi$
$338$	16.8564	0.916868
$339$	0	0
$340$	23.3205	1.26473
$341$	1.26795	0.0686633
$342$	0	0
$343$	19.5359	1.05484
$344$	8.92820	0.481376
$345$	0	0
$346$	−14.2679	−0.767050
$347$	−13.7321	−0.737175	−0.368588	−	0.929593i	$-0.620159\pi$
$347$	−13.7321	−0.737175	−0.368588	+	0.929593i	$0.620159\pi$
$348$	0	0
$349$	−0.320508	−0.0171564	−0.00857820	−	0.999963i	$-0.502731\pi$
$349$	−0.320508	−0.0171564	−0.00857820	+	0.999963i	$0.502731\pi$
$350$	−17.2487	−0.921982
$351$	0	0
$352$	1.26795	0.0675819
$353$	6.92820	0.368751	0.184376	−	0.982856i	$-0.440974\pi$
$353$	6.92820	0.368751	0.184376	+	0.982856i	$0.440974\pi$
$354$	0	0
$355$	−48.9282	−2.59684
$356$	4.26795	0.226201
$357$	0	0
$358$	9.58846	0.506766
$359$	3.60770	0.190407	0.0952034	−	0.995458i	$-0.469650\pi$
$359$	3.60770	0.190407	0.0952034	+	0.995458i	$0.469650\pi$
$360$	0	0
$361$	10.8564	0.571390
$362$	−18.4641	−0.970452
$363$	0	0
$364$	−13.4641	−0.705711
$365$	−5.07180	−0.265470
$366$	0	0
$367$	7.14359	0.372893	0.186446	−	0.982465i	$-0.440303\pi$
$367$	7.14359	0.372893	0.186446	+	0.982465i	$0.440303\pi$
$368$	−4.26795	−0.222482
$369$	0	0
$370$	−9.46410	−0.492015
$371$	15.4449	0.801857
$372$	0	0
$373$	0.0717968	0.00371750	0.00185875	−	0.999998i	$-0.499408\pi$
$373$	0.0717968	0.00371750	0.00185875	+	0.999998i	$0.499408\pi$
$374$	−8.53590	−0.441381
$375$	0	0
$376$	−7.46410	−0.384932
$377$	9.46410	0.487426
$378$	0	0
$379$	8.14359	0.418308	0.209154	−	0.977883i	$-0.432929\pi$
$379$	8.14359	0.418308	0.209154	+	0.977883i	$0.432929\pi$
$380$	18.9282	0.970996
$381$	0	0
$382$	−21.1244	−1.08082
$383$	−2.33975	−0.119555	−0.0597777	−	0.998212i	$-0.519039\pi$
$383$	−2.33975	−0.119555	−0.0597777	+	0.998212i	$0.519039\pi$
$384$	0	0
$385$	10.8231	0.551596
$386$	22.5885	1.14972
$387$	0	0
$388$	10.7321	0.544837
$389$	16.5359	0.838403	0.419202	−	0.907893i	$-0.362310\pi$
$389$	16.5359	0.838403	0.419202	+	0.907893i	$0.362310\pi$
$390$	0	0
$391$	28.7321	1.45304
$392$	−0.928203	−0.0468813
$393$	0	0
$394$	3.12436	0.157403
$395$	44.1051	2.21917
$396$	0	0
$397$	6.78461	0.340510	0.170255	−	0.985400i	$-0.445541\pi$
$397$	6.78461	0.340510	0.170255	+	0.985400i	$0.445541\pi$
$398$	15.2679	0.765313
$399$	0	0
$400$	7.00000	0.350000
$401$	−19.1244	−0.955025	−0.477512	−	0.878625i	$-0.658461\pi$
$401$	−19.1244	−0.955025	−0.477512	+	0.878625i	$0.658461\pi$
$402$	0	0
$403$	5.46410	0.272186
$404$	−6.39230	−0.318029
$405$	0	0
$406$	−4.26795	−0.211815
$407$	3.46410	0.171709
$408$	0	0
$409$	18.3923	0.909441	0.454720	−	0.890634i	$-0.349739\pi$
$409$	18.3923	0.909441	0.454720	+	0.890634i	$0.349739\pi$
$410$	4.14359	0.204637
$411$	0	0
$412$	15.9282	0.784726
$413$	−8.05256	−0.396241
$414$	0	0
$415$	−6.24871	−0.306737
$416$	5.46410	0.267900
$417$	0	0
$418$	−6.92820	−0.338869
$419$	−27.5885	−1.34778	−0.673892	−	0.738830i	$-0.735379\pi$
$419$	−27.5885	−1.34778	−0.673892	+	0.738830i	$0.735379\pi$
$420$	0	0
$421$	22.3923	1.09133	0.545667	−	0.838002i	$-0.316276\pi$
$421$	22.3923	1.09133	0.545667	+	0.838002i	$0.316276\pi$
$422$	15.7846	0.768383
$423$	0	0
$424$	−6.26795	−0.304399
$425$	−47.1244	−2.28587
$426$	0	0
$427$	−6.73205	−0.325787
$428$	−4.53590	−0.219251
$429$	0	0
$430$	−30.9282	−1.49149
$431$	36.2487	1.74604	0.873019	−	0.487685i	$-0.162159\pi$
$431$	36.2487	1.74604	0.873019	+	0.487685i	$0.162159\pi$
$432$	0	0
$433$	36.9808	1.77718	0.888591	−	0.458700i	$-0.151685\pi$
$433$	36.9808	1.77718	0.888591	+	0.458700i	$0.151685\pi$
$434$	−2.46410	−0.118281
$435$	0	0
$436$	−5.92820	−0.283909
$437$	23.3205	1.11557
$438$	0	0
$439$	15.8038	0.754276	0.377138	−	0.926157i	$-0.376908\pi$
$439$	15.8038	0.754276	0.377138	+	0.926157i	$0.376908\pi$
$440$	−4.39230	−0.209395
$441$	0	0
$442$	−36.7846	−1.74967
$443$	2.66025	0.126392	0.0631962	−	0.998001i	$-0.479871\pi$
$443$	2.66025	0.126392	0.0631962	+	0.998001i	$0.479871\pi$
$444$	0	0
$445$	−14.7846	−0.700858
$446$	−1.26795	−0.0600391
$447$	0	0
$448$	−2.46410	−0.116418
$449$	42.1244	1.98797	0.993986	−	0.109511i	$-0.0349284\pi$
$449$	42.1244	1.98797	0.993986	+	0.109511i	$0.0349284\pi$
$450$	0	0
$451$	−1.51666	−0.0714168
$452$	16.1962	0.761803
$453$	0	0
$454$	7.46410	0.350308
$455$	46.6410	2.18656
$456$	0	0
$457$	−36.9282	−1.72743	−0.863714	−	0.503982i	$-0.831868\pi$
$457$	−36.9282	−1.72743	−0.863714	+	0.503982i	$0.831868\pi$
$458$	8.19615	0.382981
$459$	0	0
$460$	14.7846	0.689336
$461$	−29.3205	−1.36559	−0.682796	−	0.730609i	$-0.739236\pi$
$461$	−29.3205	−1.36559	−0.682796	+	0.730609i	$0.739236\pi$
$462$	0	0
$463$	−37.6410	−1.74933	−0.874663	−	0.484731i	$-0.838917\pi$
$463$	−37.6410	−1.74933	−0.874663	+	0.484731i	$0.838917\pi$
$464$	1.73205	0.0804084
$465$	0	0
$466$	25.9808	1.20354
$467$	40.1051	1.85584	0.927922	−	0.372775i	$-0.121594\pi$
$467$	40.1051	1.85584	0.927922	+	0.372775i	$0.121594\pi$
$468$	0	0
$469$	15.2679	0.705008
$470$	25.8564	1.19267
$471$	0	0
$472$	3.26795	0.150420
$473$	11.3205	0.520518
$474$	0	0
$475$	−38.2487	−1.75497
$476$	16.5885	0.760331
$477$	0	0
$478$	6.12436	0.280122
$479$	−11.7321	−0.536051	−0.268026	−	0.963412i	$-0.586371\pi$
$479$	−11.7321	−0.536051	−0.268026	+	0.963412i	$0.586371\pi$
$480$	0	0
$481$	14.9282	0.680667
$482$	8.53590	0.388800
$483$	0	0
$484$	−9.39230	−0.426923
$485$	−37.1769	−1.68812
$486$	0	0
$487$	−0.392305	−0.0177770	−0.00888851	−	0.999960i	$-0.502829\pi$
$487$	−0.392305	−0.0177770	−0.00888851	+	0.999960i	$0.502829\pi$
$488$	2.73205	0.123674
$489$	0	0
$490$	3.21539	0.145257
$491$	24.0000	1.08310	0.541552	−	0.840667i	$-0.317837\pi$
$491$	24.0000	1.08310	0.541552	+	0.840667i	$0.317837\pi$
$492$	0	0
$493$	−11.6603	−0.525151
$494$	−29.8564	−1.34330
$495$	0	0
$496$	1.00000	0.0449013
$497$	−34.8038	−1.56117
$498$	0	0
$499$	−13.3923	−0.599522	−0.299761	−	0.954014i	$-0.596907\pi$
$499$	−13.3923	−0.599522	−0.299761	+	0.954014i	$0.596907\pi$
$500$	−6.92820	−0.309839
$501$	0	0
$502$	3.73205	0.166570
$503$	−14.5359	−0.648124	−0.324062	−	0.946036i	$-0.605049\pi$
$503$	−14.5359	−0.648124	−0.324062	+	0.946036i	$0.605049\pi$
$504$	0	0
$505$	22.1436	0.985377
$506$	−5.41154	−0.240572
$507$	0	0
$508$	14.0000	0.621150
$509$	14.9808	0.664011	0.332005	−	0.943278i	$-0.392275\pi$
$509$	14.9808	0.664011	0.332005	+	0.943278i	$0.392275\pi$
$510$	0	0
$511$	−3.60770	−0.159595
$512$	1.00000	0.0441942
$513$	0	0
$514$	−21.1962	−0.934922
$515$	−55.1769	−2.43139
$516$	0	0
$517$	−9.46410	−0.416231
$518$	−6.73205	−0.295789
$519$	0	0
$520$	−18.9282	−0.830057
$521$	−13.3205	−0.583582	−0.291791	−	0.956482i	$-0.594251\pi$
$521$	−13.3205	−0.583582	−0.291791	+	0.956482i	$0.594251\pi$
$522$	0	0
$523$	−0.928203	−0.0405875	−0.0202937	−	0.999794i	$-0.506460\pi$
$523$	−0.928203	−0.0405875	−0.0202937	+	0.999794i	$0.506460\pi$
$524$	9.46410	0.413441
$525$	0	0
$526$	14.7846	0.644640
$527$	−6.73205	−0.293253
$528$	0	0
$529$	−4.78461	−0.208027
$530$	21.7128	0.943144
$531$	0	0
$532$	13.4641	0.583743
$533$	−6.53590	−0.283101
$534$	0	0
$535$	15.7128	0.679324
$536$	−6.19615	−0.267633
$537$	0	0
$538$	15.3397	0.661343
$539$	−1.17691	−0.0506933
$540$	0	0
$541$	29.4641	1.26676	0.633380	−	0.773841i	$-0.281667\pi$
$541$	29.4641	1.26676	0.633380	+	0.773841i	$0.281667\pi$
$542$	28.7846	1.23640
$543$	0	0
$544$	−6.73205	−0.288634
$545$	20.5359	0.879661
$546$	0	0
$547$	−11.4641	−0.490170	−0.245085	−	0.969502i	$-0.578816\pi$
$547$	−11.4641	−0.490170	−0.245085	+	0.969502i	$0.578816\pi$
$548$	1.19615	0.0510971
$549$	0	0
$550$	8.87564	0.378459
$551$	−9.46410	−0.403184
$552$	0	0
$553$	31.3731	1.33412
$554$	5.00000	0.212430
$555$	0	0
$556$	−7.00000	−0.296866
$557$	−28.1244	−1.19167	−0.595834	−	0.803108i	$-0.703178\pi$
$557$	−28.1244	−1.19167	−0.595834	+	0.803108i	$0.703178\pi$
$558$	0	0
$559$	48.7846	2.06337
$560$	8.53590	0.360708
$561$	0	0
$562$	−24.0526	−1.01460
$563$	−30.6603	−1.29218	−0.646088	−	0.763263i	$-0.723596\pi$
$563$	−30.6603	−1.29218	−0.646088	+	0.763263i	$0.723596\pi$
$564$	0	0
$565$	−56.1051	−2.36036
$566$	1.53590	0.0645586
$567$	0	0
$568$	14.1244	0.592645
$569$	−12.1244	−0.508279	−0.254140	−	0.967168i	$-0.581792\pi$
$569$	−12.1244	−0.508279	−0.254140	+	0.967168i	$0.581792\pi$
$570$	0	0
$571$	−1.07180	−0.0448533	−0.0224266	−	0.999748i	$-0.507139\pi$
$571$	−1.07180	−0.0448533	−0.0224266	+	0.999748i	$0.507139\pi$
$572$	6.92820	0.289683
$573$	0	0
$574$	2.94744	0.123024
$575$	−29.8756	−1.24590
$576$	0	0
$577$	−43.7846	−1.82278	−0.911389	−	0.411547i	$-0.864989\pi$
$577$	−43.7846	−1.82278	−0.911389	+	0.411547i	$0.864989\pi$
$578$	28.3205	1.17798
$579$	0	0
$580$	−6.00000	−0.249136
$581$	−4.44486	−0.184404
$582$	0	0
$583$	−7.94744	−0.329149
$584$	1.46410	0.0605850
$585$	0	0
$586$	−12.1244	−0.500853
$587$	−25.6077	−1.05694	−0.528471	−	0.848951i	$-0.677235\pi$
$587$	−25.6077	−1.05694	−0.528471	+	0.848951i	$0.677235\pi$
$588$	0	0
$589$	−5.46410	−0.225144
$590$	−11.3205	−0.466058
$591$	0	0
$592$	2.73205	0.112287
$593$	−14.4115	−0.591811	−0.295906	−	0.955217i	$-0.595621\pi$
$593$	−14.4115	−0.591811	−0.295906	+	0.955217i	$0.595621\pi$
$594$	0	0
$595$	−57.4641	−2.35580
$596$	1.00000	0.0409616
$597$	0	0
$598$	−23.3205	−0.953646
$599$	21.8564	0.893029	0.446514	−	0.894777i	$-0.352665\pi$
$599$	21.8564	0.893029	0.446514	+	0.894777i	$0.352665\pi$
$600$	0	0
$601$	8.60770	0.351115	0.175558	−	0.984469i	$-0.443827\pi$
$601$	8.60770	0.351115	0.175558	+	0.984469i	$0.443827\pi$
$602$	−22.0000	−0.896653
$603$	0	0
$604$	21.1244	0.859538
$605$	32.5359	1.32277
$606$	0	0
$607$	7.80385	0.316748	0.158374	−	0.987379i	$-0.449375\pi$
$607$	7.80385	0.316748	0.158374	+	0.987379i	$0.449375\pi$
$608$	−5.46410	−0.221599
$609$	0	0
$610$	−9.46410	−0.383190
$611$	−40.7846	−1.64997
$612$	0	0
$613$	−26.2487	−1.06018	−0.530088	−	0.847943i	$-0.677841\pi$
$613$	−26.2487	−1.06018	−0.530088	+	0.847943i	$0.677841\pi$
$614$	−29.3205	−1.18328
$615$	0	0
$616$	−3.12436	−0.125884
$617$	−20.2487	−0.815182	−0.407591	−	0.913165i	$-0.633631\pi$
$617$	−20.2487	−0.815182	−0.407591	+	0.913165i	$0.633631\pi$
$618$	0	0
$619$	−12.5359	−0.503860	−0.251930	−	0.967745i	$-0.581065\pi$
$619$	−12.5359	−0.503860	−0.251930	+	0.967745i	$0.581065\pi$
$620$	−3.46410	−0.139122
$621$	0	0
$622$	28.1244	1.12768
$623$	−10.5167	−0.421341
$624$	0	0
$625$	−11.0000	−0.440000
$626$	−22.5359	−0.900716
$627$	0	0
$628$	3.53590	0.141098
$629$	−18.3923	−0.733349
$630$	0	0
$631$	10.1436	0.403810	0.201905	−	0.979405i	$-0.435287\pi$
$631$	10.1436	0.403810	0.201905	+	0.979405i	$0.435287\pi$
$632$	−12.7321	−0.506454
$633$	0	0
$634$	3.85641	0.153157
$635$	−48.4974	−1.92456
$636$	0	0
$637$	−5.07180	−0.200952
$638$	2.19615	0.0869465
$639$	0	0
$640$	−3.46410	−0.136931
$641$	−25.1244	−0.992352	−0.496176	−	0.868222i	$-0.665263\pi$
$641$	−25.1244	−0.992352	−0.496176	+	0.868222i	$0.665263\pi$
$642$	0	0
$643$	−5.94744	−0.234544	−0.117272	−	0.993100i	$-0.537415\pi$
$643$	−5.94744	−0.234544	−0.117272	+	0.993100i	$0.537415\pi$
$644$	10.5167	0.414414
$645$	0	0
$646$	36.7846	1.44727
$647$	−36.5167	−1.43562	−0.717809	−	0.696240i	$-0.754855\pi$
$647$	−36.5167	−1.43562	−0.717809	+	0.696240i	$0.754855\pi$
$648$	0	0
$649$	4.14359	0.162650
$650$	38.2487	1.50024
$651$	0	0
$652$	9.92820	0.388818
$653$	25.8564	1.01184	0.505920	−	0.862581i	$-0.331153\pi$
$653$	25.8564	1.01184	0.505920	+	0.862581i	$0.331153\pi$
$654$	0	0
$655$	−32.7846	−1.28100
$656$	−1.19615	−0.0467019
$657$	0	0
$658$	18.3923	0.717007
$659$	−1.48334	−0.0577827	−0.0288914	−	0.999583i	$-0.509198\pi$
$659$	−1.48334	−0.0577827	−0.0288914	+	0.999583i	$0.509198\pi$
$660$	0	0
$661$	11.2487	0.437524	0.218762	−	0.975778i	$-0.429798\pi$
$661$	11.2487	0.437524	0.218762	+	0.975778i	$0.429798\pi$
$662$	1.80385	0.0701085
$663$	0	0
$664$	1.80385	0.0700029
$665$	−46.6410	−1.80866
$666$	0	0
$667$	−7.39230	−0.286231
$668$	0	0
$669$	0	0
$670$	21.4641	0.829231
$671$	3.46410	0.133730
$672$	0	0
$673$	11.4115	0.439883	0.219941	−	0.975513i	$-0.429413\pi$
$673$	11.4115	0.439883	0.219941	+	0.975513i	$0.429413\pi$
$674$	19.0000	0.731853
$675$	0	0
$676$	16.8564	0.648323
$677$	18.3923	0.706874	0.353437	−	0.935458i	$-0.385013\pi$
$677$	18.3923	0.706874	0.353437	+	0.935458i	$0.385013\pi$
$678$	0	0
$679$	−26.4449	−1.01486
$680$	23.3205	0.894301
$681$	0	0
$682$	1.26795	0.0485523
$683$	36.3923	1.39251	0.696256	−	0.717793i	$-0.254848\pi$
$683$	36.3923	1.39251	0.696256	+	0.717793i	$0.254848\pi$
$684$	0	0
$685$	−4.14359	−0.158319
$686$	19.5359	0.745884
$687$	0	0
$688$	8.92820	0.340385
$689$	−34.2487	−1.30477
$690$	0	0
$691$	−51.5167	−1.95979	−0.979893	−	0.199523i	$-0.936061\pi$
$691$	−51.5167	−1.95979	−0.979893	+	0.199523i	$0.936061\pi$
$692$	−14.2679	−0.542386
$693$	0	0
$694$	−13.7321	−0.521262
$695$	24.2487	0.919806
$696$	0	0
$697$	8.05256	0.305012
$698$	−0.320508	−0.0121314
$699$	0	0
$700$	−17.2487	−0.651940
$701$	−4.33975	−0.163910	−0.0819550	−	0.996636i	$-0.526116\pi$
$701$	−4.33975	−0.163910	−0.0819550	+	0.996636i	$0.526116\pi$
$702$	0	0
$703$	−14.9282	−0.563028
$704$	1.26795	0.0477876
$705$	0	0
$706$	6.92820	0.260746
$707$	15.7513	0.592388
$708$	0	0
$709$	10.3923	0.390291	0.195146	−	0.980774i	$-0.437482\pi$
$709$	10.3923	0.390291	0.195146	+	0.980774i	$0.437482\pi$
$710$	−48.9282	−1.83624
$711$	0	0
$712$	4.26795	0.159948
$713$	−4.26795	−0.159836
$714$	0	0
$715$	−24.0000	−0.897549
$716$	9.58846	0.358337
$717$	0	0
$718$	3.60770	0.134638
$719$	−44.1962	−1.64824	−0.824119	−	0.566416i	$-0.808329\pi$
$719$	−44.1962	−1.64824	−0.824119	+	0.566416i	$0.808329\pi$
$720$	0	0
$721$	−39.2487	−1.46170
$722$	10.8564	0.404034
$723$	0	0
$724$	−18.4641	−0.686213
$725$	12.1244	0.450287
$726$	0	0
$727$	−23.5167	−0.872185	−0.436092	−	0.899902i	$-0.643638\pi$
$727$	−23.5167	−0.872185	−0.436092	+	0.899902i	$0.643638\pi$
$728$	−13.4641	−0.499013
$729$	0	0
$730$	−5.07180	−0.187716
$731$	−60.1051	−2.22307
$732$	0	0
$733$	34.4641	1.27296	0.636480	−	0.771293i	$-0.280390\pi$
$733$	34.4641	1.27296	0.636480	+	0.771293i	$0.280390\pi$
$734$	7.14359	0.263675
$735$	0	0
$736$	−4.26795	−0.157319
$737$	−7.85641	−0.289394
$738$	0	0
$739$	−13.4115	−0.493352	−0.246676	−	0.969098i	$-0.579338\pi$
$739$	−13.4115	−0.493352	−0.246676	+	0.969098i	$0.579338\pi$
$740$	−9.46410	−0.347907
$741$	0	0
$742$	15.4449	0.566999
$743$	−42.3731	−1.55452	−0.777259	−	0.629181i	$-0.783390\pi$
$743$	−42.3731	−1.55452	−0.777259	+	0.629181i	$0.783390\pi$
$744$	0	0
$745$	−3.46410	−0.126915
$746$	0.0717968	0.00262867
$747$	0	0
$748$	−8.53590	−0.312103
$749$	11.1769	0.408396
$750$	0	0
$751$	29.9282	1.09210	0.546048	−	0.837754i	$-0.316132\pi$
$751$	29.9282	1.09210	0.546048	+	0.837754i	$0.316132\pi$
$752$	−7.46410	−0.272188
$753$	0	0
$754$	9.46410	0.344662
$755$	−73.1769	−2.66318
$756$	0	0
$757$	39.0000	1.41748	0.708740	−	0.705470i	$-0.249264\pi$
$757$	39.0000	1.41748	0.708740	+	0.705470i	$0.249264\pi$
$758$	8.14359	0.295789
$759$	0	0
$760$	18.9282	0.686598
$761$	26.3923	0.956720	0.478360	−	0.878164i	$-0.341231\pi$
$761$	26.3923	0.956720	0.478360	+	0.878164i	$0.341231\pi$
$762$	0	0
$763$	14.6077	0.528834
$764$	−21.1244	−0.764252
$765$	0	0
$766$	−2.33975	−0.0845385
$767$	17.8564	0.644757
$768$	0	0
$769$	−50.9282	−1.83652	−0.918259	−	0.395980i	$-0.870405\pi$
$769$	−50.9282	−1.83652	−0.918259	+	0.395980i	$0.870405\pi$
$770$	10.8231	0.390037
$771$	0	0
$772$	22.5885	0.812976
$773$	33.7321	1.21326	0.606629	−	0.794985i	$-0.292521\pi$
$773$	33.7321	1.21326	0.606629	+	0.794985i	$0.292521\pi$
$774$	0	0
$775$	7.00000	0.251447
$776$	10.7321	0.385258
$777$	0	0
$778$	16.5359	0.592841
$779$	6.53590	0.234173
$780$	0	0
$781$	17.9090	0.640833
$782$	28.7321	1.02746
$783$	0	0
$784$	−0.928203	−0.0331501
$785$	−12.2487	−0.437175
$786$	0	0
$787$	15.1244	0.539125	0.269563	−	0.962983i	$-0.413121\pi$
$787$	15.1244	0.539125	0.269563	+	0.962983i	$0.413121\pi$
$788$	3.12436	0.111301
$789$	0	0
$790$	44.1051	1.56919
$791$	−39.9090	−1.41900
$792$	0	0
$793$	14.9282	0.530116
$794$	6.78461	0.240777
$795$	0	0
$796$	15.2679	0.541158
$797$	−6.67949	−0.236600	−0.118300	−	0.992978i	$-0.537744\pi$
$797$	−6.67949	−0.236600	−0.118300	+	0.992978i	$0.537744\pi$
$798$	0	0
$799$	50.2487	1.77767
$800$	7.00000	0.247487
$801$	0	0
$802$	−19.1244	−0.675304
$803$	1.85641	0.0655112
$804$	0	0
$805$	−36.4308	−1.28402
$806$	5.46410	0.192465
$807$	0	0
$808$	−6.39230	−0.224880
$809$	45.0333	1.58329	0.791644	−	0.610983i	$-0.209226\pi$
$809$	45.0333	1.58329	0.791644	+	0.610983i	$0.209226\pi$
$810$	0	0
$811$	17.6077	0.618290	0.309145	−	0.951015i	$-0.399957\pi$
$811$	17.6077	0.618290	0.309145	+	0.951015i	$0.399957\pi$
$812$	−4.26795	−0.149776
$813$	0	0
$814$	3.46410	0.121417
$815$	−34.3923	−1.20471
$816$	0	0
$817$	−48.7846	−1.70676
$818$	18.3923	0.643072
$819$	0	0
$820$	4.14359	0.144701
$821$	−4.94744	−0.172667	−0.0863334	−	0.996266i	$-0.527515\pi$
$821$	−4.94744	−0.172667	−0.0863334	+	0.996266i	$0.527515\pi$
$822$	0	0
$823$	50.9282	1.77525	0.887623	−	0.460571i	$-0.152355\pi$
$823$	50.9282	1.77525	0.887623	+	0.460571i	$0.152355\pi$
$824$	15.9282	0.554885
$825$	0	0
$826$	−8.05256	−0.280184
$827$	−44.3923	−1.54367	−0.771836	−	0.635822i	$-0.780661\pi$
$827$	−44.3923	−1.54367	−0.771836	+	0.635822i	$0.780661\pi$
$828$	0	0
$829$	−56.1769	−1.95110	−0.975552	−	0.219767i	$-0.929470\pi$
$829$	−56.1769	−1.95110	−0.975552	+	0.219767i	$0.929470\pi$
$830$	−6.24871	−0.216896
$831$	0	0
$832$	5.46410	0.189434
$833$	6.24871	0.216505
$834$	0	0
$835$	0	0
$836$	−6.92820	−0.239617
$837$	0	0
$838$	−27.5885	−0.953027
$839$	−28.5167	−0.984504	−0.492252	−	0.870453i	$-0.663826\pi$
$839$	−28.5167	−0.984504	−0.492252	+	0.870453i	$0.663826\pi$
$840$	0	0
$841$	−26.0000	−0.896552
$842$	22.3923	0.771690
$843$	0	0
$844$	15.7846	0.543329
$845$	−58.3923	−2.00876
$846$	0	0
$847$	23.1436	0.795223
$848$	−6.26795	−0.215242
$849$	0	0
$850$	−47.1244	−1.61635
$851$	−11.6603	−0.399708
$852$	0	0
$853$	12.6410	0.432820	0.216410	−	0.976303i	$-0.430565\pi$
$853$	12.6410	0.432820	0.216410	+	0.976303i	$0.430565\pi$
$854$	−6.73205	−0.230366
$855$	0	0
$856$	−4.53590	−0.155034
$857$	−24.1962	−0.826525	−0.413262	−	0.910612i	$-0.635611\pi$
$857$	−24.1962	−0.826525	−0.413262	+	0.910612i	$0.635611\pi$
$858$	0	0
$859$	17.9474	0.612359	0.306179	−	0.951974i	$-0.400949\pi$
$859$	17.9474	0.612359	0.306179	+	0.951974i	$0.400949\pi$
$860$	−30.9282	−1.05464
$861$	0	0
$862$	36.2487	1.23464
$863$	−53.2295	−1.81195	−0.905976	−	0.423329i	$-0.860862\pi$
$863$	−53.2295	−1.81195	−0.905976	+	0.423329i	$0.860862\pi$
$864$	0	0
$865$	49.4256	1.68052
$866$	36.9808	1.25666
$867$	0	0
$868$	−2.46410	−0.0836371
$869$	−16.1436	−0.547634
$870$	0	0
$871$	−33.8564	−1.14718
$872$	−5.92820	−0.200754
$873$	0	0
$874$	23.3205	0.788828
$875$	17.0718	0.577132
$876$	0	0
$877$	44.9282	1.51712	0.758559	−	0.651605i	$-0.225904\pi$
$877$	44.9282	1.51712	0.758559	+	0.651605i	$0.225904\pi$
$878$	15.8038	0.533354
$879$	0	0
$880$	−4.39230	−0.148065
$881$	20.9090	0.704441	0.352220	−	0.935917i	$-0.385427\pi$
$881$	20.9090	0.704441	0.352220	+	0.935917i	$0.385427\pi$
$882$	0	0
$883$	−0.248711	−0.00836980	−0.00418490	−	0.999991i	$-0.501332\pi$
$883$	−0.248711	−0.00836980	−0.00418490	+	0.999991i	$0.501332\pi$
$884$	−36.7846	−1.23720
$885$	0	0
$886$	2.66025	0.0893730
$887$	−29.9090	−1.00424	−0.502122	−	0.864797i	$-0.667447\pi$
$887$	−29.9090	−1.00424	−0.502122	+	0.864797i	$0.667447\pi$
$888$	0	0
$889$	−34.4974	−1.15701
$890$	−14.7846	−0.495581
$891$	0	0
$892$	−1.26795	−0.0424541
$893$	40.7846	1.36480
$894$	0	0
$895$	−33.2154	−1.11027
$896$	−2.46410	−0.0823199
$897$	0	0
$898$	42.1244	1.40571
$899$	1.73205	0.0577671
$900$	0	0
$901$	42.1962	1.40576
$902$	−1.51666	−0.0504993
$903$	0	0
$904$	16.1962	0.538676
$905$	63.9615	2.12615
$906$	0	0
$907$	48.2487	1.60207	0.801036	−	0.598616i	$-0.204282\pi$
$907$	48.2487	1.60207	0.801036	+	0.598616i	$0.204282\pi$
$908$	7.46410	0.247705
$909$	0	0
$910$	46.6410	1.54613
$911$	−11.7128	−0.388063	−0.194031	−	0.980995i	$-0.562156\pi$
$911$	−11.7128	−0.388063	−0.194031	+	0.980995i	$0.562156\pi$
$912$	0	0
$913$	2.28719	0.0756948
$914$	−36.9282	−1.22148
$915$	0	0
$916$	8.19615	0.270808
$917$	−23.3205	−0.770111
$918$	0	0
$919$	33.7846	1.11445	0.557226	−	0.830361i	$-0.311866\pi$
$919$	33.7846	1.11445	0.557226	+	0.830361i	$0.311866\pi$
$920$	14.7846	0.487434
$921$	0	0
$922$	−29.3205	−0.965620
$923$	77.1769	2.54031
$924$	0	0
$925$	19.1244	0.628805
$926$	−37.6410	−1.23696
$927$	0	0
$928$	1.73205	0.0568574
$929$	−1.60770	−0.0527468	−0.0263734	−	0.999652i	$-0.508396\pi$
$929$	−1.60770	−0.0527468	−0.0263734	+	0.999652i	$0.508396\pi$
$930$	0	0
$931$	5.07180	0.166221
$932$	25.9808	0.851028
$933$	0	0
$934$	40.1051	1.31228
$935$	29.5692	0.967017
$936$	0	0
$937$	14.4833	0.473150	0.236575	−	0.971613i	$-0.423975\pi$
$937$	14.4833	0.473150	0.236575	+	0.971613i	$0.423975\pi$
$938$	15.2679	0.498516
$939$	0	0
$940$	25.8564	0.843343
$941$	11.9808	0.390562	0.195281	−	0.980747i	$-0.437438\pi$
$941$	11.9808	0.390562	0.195281	+	0.980747i	$0.437438\pi$
$942$	0	0
$943$	5.10512	0.166246
$944$	3.26795	0.106363
$945$	0	0
$946$	11.3205	0.368061
$947$	−33.9808	−1.10423	−0.552113	−	0.833769i	$-0.686178\pi$
$947$	−33.9808	−1.10423	−0.552113	+	0.833769i	$0.686178\pi$
$948$	0	0
$949$	8.00000	0.259691
$950$	−38.2487	−1.24095
$951$	0	0
$952$	16.5885	0.537635
$953$	0.124356	0.00402827	0.00201414	−	0.999998i	$-0.499359\pi$
$953$	0.124356	0.00402827	0.00201414	+	0.999998i	$0.499359\pi$
$954$	0	0
$955$	73.1769	2.36795
$956$	6.12436	0.198076
$957$	0	0
$958$	−11.7321	−0.379045
$959$	−2.94744	−0.0951778
$960$	0	0
$961$	−30.0000	−0.967742
$962$	14.9282	0.481305
$963$	0	0
$964$	8.53590	0.274923
$965$	−78.2487	−2.51892
$966$	0	0
$967$	−11.0718	−0.356045	−0.178022	−	0.984026i	$-0.556970\pi$
$967$	−11.0718	−0.356045	−0.178022	+	0.984026i	$0.556970\pi$
$968$	−9.39230	−0.301880
$969$	0	0
$970$	−37.1769	−1.19368
$971$	46.0526	1.47790	0.738948	−	0.673762i	$-0.235323\pi$
$971$	46.0526	1.47790	0.738948	+	0.673762i	$0.235323\pi$
$972$	0	0
$973$	17.2487	0.552968
$974$	−0.392305	−0.0125703
$975$	0	0
$976$	2.73205	0.0874508
$977$	−36.0000	−1.15174	−0.575871	−	0.817541i	$-0.695337\pi$
$977$	−36.0000	−1.15174	−0.575871	+	0.817541i	$0.695337\pi$
$978$	0	0
$979$	5.41154	0.172954
$980$	3.21539	0.102712
$981$	0	0
$982$	24.0000	0.765871
$983$	17.0526	0.543892	0.271946	−	0.962312i	$-0.412333\pi$
$983$	17.0526	0.543892	0.271946	+	0.962312i	$0.412333\pi$
$984$	0	0
$985$	−10.8231	−0.344852
$986$	−11.6603	−0.371338
$987$	0	0
$988$	−29.8564	−0.949859
$989$	−38.1051	−1.21167
$990$	0	0
$991$	31.5692	1.00283	0.501415	−	0.865207i	$-0.332813\pi$
$991$	31.5692	1.00283	0.501415	+	0.865207i	$0.332813\pi$
$992$	1.00000	0.0317500
$993$	0	0
$994$	−34.8038	−1.10391
$995$	−52.8897	−1.67672
$996$	0	0
$997$	26.0526	0.825093	0.412546	−	0.910937i	$-0.364639\pi$
$997$	26.0526	0.825093	0.412546	+	0.910937i	$0.364639\pi$
$998$	−13.3923	−0.423926
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.d.1.1	yes	2
3.2	odd	2		8046.2.a.c.1.2	✓	2

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.c.1.2	✓	2	3.2	odd	2
8046.2.a.d.1.1	yes	2	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)

\(f(q)\)	\(=\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} -2.46410 q^{7} +1.00000 q^{8} +O(q^{10})\)	\(q+1.00000 q^{2} +1.00000 q^{4} -3.46410 q^{5} -2.46410 q^{7} +1.00000 q^{8} -3.46410 q^{10} +1.26795 q^{11} +5.46410 q^{13} -2.46410 q^{14} +1.00000 q^{16} -6.73205 q^{17} -5.46410 q^{19} -3.46410 q^{20} +1.26795 q^{22} -4.26795 q^{23} +7.00000 q^{25} +5.46410 q^{26} -2.46410 q^{28} +1.73205 q^{29} +1.00000 q^{31} +1.00000 q^{32} -6.73205 q^{34} +8.53590 q^{35} +2.73205 q^{37} -5.46410 q^{38} -3.46410 q^{40} -1.19615 q^{41} +8.92820 q^{43} +1.26795 q^{44} -4.26795 q^{46} -7.46410 q^{47} -0.928203 q^{49} +7.00000 q^{50} +5.46410 q^{52} -6.26795 q^{53} -4.39230 q^{55} -2.46410 q^{56} +1.73205 q^{58} +3.26795 q^{59} +2.73205 q^{61} +1.00000 q^{62} +1.00000 q^{64} -18.9282 q^{65} -6.19615 q^{67} -6.73205 q^{68} +8.53590 q^{70} +14.1244 q^{71} +1.46410 q^{73} +2.73205 q^{74} -5.46410 q^{76} -3.12436 q^{77} -12.7321 q^{79} -3.46410 q^{80} -1.19615 q^{82} +1.80385 q^{83} +23.3205 q^{85} +8.92820 q^{86} +1.26795 q^{88} +4.26795 q^{89} -13.4641 q^{91} -4.26795 q^{92} -7.46410 q^{94} +18.9282 q^{95} +10.7321 q^{97} -0.928203 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8}+O(q^{10}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8	\( 2 q + 2 q^{2} + 2 q^{4} + 2 q^{7} + 2 q^{8} + 6 q^{11} + 4 q^{13} + 2 q^{14} + 2 q^{16} - 10 q^{17} - 4 q^{19} + 6 q^{22} - 12 q^{23} + 14 q^{25} + 4 q^{26} + 2 q^{28} + 2 q^{31} + 2 q^{32} - 10 q^{34} + 24 q^{35} + 2 q^{37} - 4 q^{38} + 8 q^{41} + 4 q^{43} + 6 q^{44} - 12 q^{46} - 8 q^{47} + 12 q^{49} + 14 q^{50} + 4 q^{52} - 16 q^{53} + 12 q^{55} + 2 q^{56} + 10 q^{59} + 2 q^{61} + 2 q^{62} + 2 q^{64} - 24 q^{65} - 2 q^{67} - 10 q^{68} + 24 q^{70} + 4 q^{71} - 4 q^{73} + 2 q^{74} - 4 q^{76} + 18 q^{77} - 22 q^{79} + 8 q^{82} + 14 q^{83} + 12 q^{85} + 4 q^{86} + 6 q^{88} + 12 q^{89} - 20 q^{91} - 12 q^{92} - 8 q^{94} + 24 q^{95} + 18 q^{97} + 12 q^{98}+O(q^{100}) \) 2 * q + 2 * q^2 + 2 * q^4 + 2 * q^7 + 2 * q^8 + 6 * q^11 + 4 * q^13 + 2 * q^14 + 2 * q^16 - 10 * q^17 - 4 * q^19 + 6 * q^22 - 12 * q^23 + 14 * q^25 + 4 * q^26 + 2 * q^28 + 2 * q^31 + 2 * q^32 - 10 * q^34 + 24 * q^35 + 2 * q^37 - 4 * q^38 + 8 * q^41 + 4 * q^43 + 6 * q^44 - 12 * q^46 - 8 * q^47 + 12 * q^49 + 14 * q^50 + 4 * q^52 - 16 * q^53 + 12 * q^55 + 2 * q^56 + 10 * q^59 + 2 * q^61 + 2 * q^62 + 2 * q^64 - 24 * q^65 - 2 * q^67 - 10 * q^68 + 24 * q^70 + 4 * q^71 - 4 * q^73 + 2 * q^74 - 4 * q^76 + 18 * q^77 - 22 * q^79 + 8 * q^82 + 14 * q^83 + 12 * q^85 + 4 * q^86 + 6 * q^88 + 12 * q^89 - 20 * q^91 - 12 * q^92 - 8 * q^94 + 24 * q^95 + 18 * q^97 + 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more