LMFDB - Embedded newform 8046.2.a.s.1.13

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - 4 x^{15} - 46 x^{14} + 192 x^{13} + 752 x^{12} - 3378 x^{11} - 5277 x^{10} + 27132 x^{9} + \cdots - 4260 $ x^16 - 4x^15 - 46x^14 + 192x^13 + 752x^12 - 3378x^11 - 5277x^10 + 27132x^9 + 15173x^8 - 102466x^7 - 16612x^6 + 186120x^5 + 117x^4 - 156722x^3 + 10545x^2 + 48580*x - 4260
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			1.13
Root			$-1.68422$ 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)

$f(q)$	$=$	$q-1.00000 q^{2} +1.00000 q^{4} +1.68422 q^{5} -3.55865 q^{7} -1.00000 q^{8} +O(q^{10})$	\(q-1.00000 q^{2} +1.00000 q^{4} +1.68422 q^{5} -3.55865 q^{7} -1.00000 q^{8} -1.68422 q^{10} +6.10921 q^{11} +1.47421 q^{13} +3.55865 q^{14} +1.00000 q^{16} +8.05407 q^{17} -4.84034 q^{19} +1.68422 q^{20} -6.10921 q^{22} +1.34441 q^{23} -2.16340 q^{25} -1.47421 q^{26} -3.55865 q^{28} +9.21992 q^{29} +10.1791 q^{31} -1.00000 q^{32} -8.05407 q^{34} -5.99356 q^{35} +6.09097 q^{37} +4.84034 q^{38} -1.68422 q^{40} -9.20047 q^{41} -2.59279 q^{43} +6.10921 q^{44} -1.34441 q^{46} -8.87218 q^{47} +5.66400 q^{49} +2.16340 q^{50} +1.47421 q^{52} +7.18563 q^{53} +10.2893 q^{55} +3.55865 q^{56} -9.21992 q^{58} -3.25205 q^{59} -2.68461 q^{61} -10.1791 q^{62} +1.00000 q^{64} +2.48289 q^{65} +2.77854 q^{67} +8.05407 q^{68} +5.99356 q^{70} +12.3136 q^{71} +14.1129 q^{73} -6.09097 q^{74} -4.84034 q^{76} -21.7405 q^{77} +10.2953 q^{79} +1.68422 q^{80} +9.20047 q^{82} -8.87149 q^{83} +13.5648 q^{85} +2.59279 q^{86} -6.10921 q^{88} +1.15228 q^{89} -5.24619 q^{91} +1.34441 q^{92} +8.87218 q^{94} -8.15220 q^{95} -8.68872 q^{97} -5.66400 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8}+O(q^{10}) $ 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8	$ 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8} + 4 q^{10} - 6 q^{11} + 6 q^{13} - 6 q^{14} + 16 q^{16} - q^{17} + 10 q^{19} - 4 q^{20} + 6 q^{22} - 10 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 21 q^{31} - 16 q^{32} + q^{34} - 16 q^{35} + 17 q^{37} - 10 q^{38} + 4 q^{40} + 4 q^{41} + 16 q^{43} - 6 q^{44} + 10 q^{46} - 25 q^{47} + 36 q^{49} - 28 q^{50} + 6 q^{52} - 14 q^{53} + 19 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 23 q^{61} - 21 q^{62} + 16 q^{64} - 20 q^{65} + 22 q^{67} - q^{68} + 16 q^{70} - 10 q^{71} + 16 q^{73} - 17 q^{74} + 10 q^{76} + 2 q^{77} + 37 q^{79} - 4 q^{80} - 4 q^{82} - 33 q^{83} + 43 q^{85} - 16 q^{86} + 6 q^{88} + 3 q^{89} + 28 q^{91} - 10 q^{92} + 25 q^{94} - 14 q^{95} - 3 q^{97} - 36 q^{98}+O(q^{100}) $ 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8 + 4 * q^10 - 6 * q^11 + 6 * q^13 - 6 * q^14 + 16 * q^16 - q^17 + 10 * q^19 - 4 * q^20 + 6 * q^22 - 10 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 21 * q^31 - 16 * q^32 + q^34 - 16 * q^35 + 17 * q^37 - 10 * q^38 + 4 * q^40 + 4 * q^41 + 16 * q^43 - 6 * q^44 + 10 * q^46 - 25 * q^47 + 36 * q^49 - 28 * q^50 + 6 * q^52 - 14 * q^53 + 19 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 23 * q^61 - 21 * q^62 + 16 * q^64 - 20 * q^65 + 22 * q^67 - q^68 + 16 * q^70 - 10 * q^71 + 16 * q^73 - 17 * q^74 + 10 * q^76 + 2 * q^77 + 37 * q^79 - 4 * q^80 - 4 * q^82 - 33 * q^83 + 43 * q^85 - 16 * q^86 + 6 * q^88 + 3 * q^89 + 28 * q^91 - 10 * q^92 + 25 * q^94 - 14 * q^95 - 3 * q^97 - 36 * 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
$5$	1.68422	0.753207	0.376603	−	0.926375i	$-0.377092\pi$
$5$	1.68422	0.753207	0.376603	+	0.926375i	$0.377092\pi$
$6$	0	0
$7$	−3.55865	−1.34504	−0.672522	−	0.740077i	$-0.734789\pi$
$7$	−3.55865	−1.34504	−0.672522	+	0.740077i	$0.734789\pi$
$8$	−1.00000	−0.353553
$9$	0	0
$10$	−1.68422	−0.532597
$11$	6.10921	1.84199	0.920997	−	0.389569i	$-0.127376\pi$
$11$	6.10921	1.84199	0.920997	+	0.389569i	$0.127376\pi$
$12$	0	0
$13$	1.47421	0.408871	0.204436	−	0.978880i	$-0.434464\pi$
$13$	1.47421	0.408871	0.204436	+	0.978880i	$0.434464\pi$
$14$	3.55865	0.951090
$15$	0	0
$16$	1.00000	0.250000
$17$	8.05407	1.95340	0.976700	−	0.214610i	$-0.0688480\pi$
$17$	8.05407	1.95340	0.976700	+	0.214610i	$0.0688480\pi$
$18$	0	0
$19$	−4.84034	−1.11045	−0.555225	−	0.831700i	$-0.687368\pi$
$19$	−4.84034	−1.11045	−0.555225	+	0.831700i	$0.687368\pi$
$20$	1.68422	0.376603
$21$	0	0
$22$	−6.10921	−1.30249
$23$	1.34441	0.280328	0.140164	−	0.990128i	$-0.455237\pi$
$23$	1.34441	0.280328	0.140164	+	0.990128i	$0.455237\pi$
$24$	0	0
$25$	−2.16340	−0.432680
$26$	−1.47421	−0.289116
$27$	0	0
$28$	−3.55865	−0.672522
$29$	9.21992	1.71210	0.856048	−	0.516896i	$-0.172913\pi$
$29$	9.21992	1.71210	0.856048	+	0.516896i	$0.172913\pi$
$30$	0	0
$31$	10.1791	1.82822	0.914109	−	0.405470i	$-0.132892\pi$
$31$	10.1791	1.82822	0.914109	+	0.405470i	$0.132892\pi$
$32$	−1.00000	−0.176777
$33$	0	0
$34$	−8.05407	−1.38126
$35$	−5.99356	−1.01310
$36$	0	0
$37$	6.09097	1.00135	0.500674	−	0.865636i	$-0.333085\pi$
$37$	6.09097	1.00135	0.500674	+	0.865636i	$0.333085\pi$
$38$	4.84034	0.785207
$39$	0	0
$40$	−1.68422	−0.266299
$41$	−9.20047	−1.43687	−0.718436	−	0.695593i	$-0.755142\pi$
$41$	−9.20047	−1.43687	−0.718436	+	0.695593i	$0.755142\pi$
$42$	0	0
$43$	−2.59279	−0.395397	−0.197698	−	0.980263i	$-0.563347\pi$
$43$	−2.59279	−0.395397	−0.197698	+	0.980263i	$0.563347\pi$
$44$	6.10921	0.920997
$45$	0	0
$46$	−1.34441	−0.198222
$47$	−8.87218	−1.29414	−0.647070	−	0.762431i	$-0.724006\pi$
$47$	−8.87218	−1.29414	−0.647070	+	0.762431i	$0.724006\pi$
$48$	0	0
$49$	5.66400	0.809143
$50$	2.16340	0.305951
$51$	0	0
$52$	1.47421	0.204436
$53$	7.18563	0.987022	0.493511	−	0.869740i	$-0.335713\pi$
$53$	7.18563	0.987022	0.493511	+	0.869740i	$0.335713\pi$
$54$	0	0
$55$	10.2893	1.38740
$56$	3.55865	0.475545
$57$	0	0
$58$	−9.21992	−1.21063
$59$	−3.25205	−0.423381	−0.211691	−	0.977337i	$-0.567897\pi$
$59$	−3.25205	−0.423381	−0.211691	+	0.977337i	$0.567897\pi$
$60$	0	0
$61$	−2.68461	−0.343729	−0.171864	−	0.985121i	$-0.554979\pi$
$61$	−2.68461	−0.343729	−0.171864	+	0.985121i	$0.554979\pi$
$62$	−10.1791	−1.29274
$63$	0	0
$64$	1.00000	0.125000
$65$	2.48289	0.307965
$66$	0	0
$67$	2.77854	0.339453	0.169726	−	0.985491i	$-0.445712\pi$
$67$	2.77854	0.339453	0.169726	+	0.985491i	$0.445712\pi$
$68$	8.05407	0.976700
$69$	0	0
$70$	5.99356	0.716367
$71$	12.3136	1.46136	0.730680	−	0.682720i	$-0.239203\pi$
$71$	12.3136	1.46136	0.730680	+	0.682720i	$0.239203\pi$
$72$	0	0
$73$	14.1129	1.65180	0.825898	−	0.563820i	$-0.190669\pi$
$73$	14.1129	1.65180	0.825898	+	0.563820i	$0.190669\pi$
$74$	−6.09097	−0.708060
$75$	0	0
$76$	−4.84034	−0.555225
$77$	−21.7405	−2.47756
$78$	0	0
$79$	10.2953	1.15832	0.579158	−	0.815215i	$-0.303381\pi$
$79$	10.2953	1.15832	0.579158	+	0.815215i	$0.303381\pi$
$80$	1.68422	0.188302
$81$	0	0
$82$	9.20047	1.01602
$83$	−8.87149	−0.973773	−0.486886	−	0.873465i	$-0.661867\pi$
$83$	−8.87149	−0.973773	−0.486886	+	0.873465i	$0.661867\pi$
$84$	0	0
$85$	13.5648	1.47131
$86$	2.59279	0.279588
$87$	0	0
$88$	−6.10921	−0.651243
$89$	1.15228	0.122142	0.0610708	−	0.998133i	$-0.480548\pi$
$89$	1.15228	0.122142	0.0610708	+	0.998133i	$0.480548\pi$
$90$	0	0
$91$	−5.24619	−0.549950
$92$	1.34441	0.140164
$93$	0	0
$94$	8.87218	0.915095
$95$	−8.15220	−0.836398
$96$	0	0
$97$	−8.68872	−0.882205	−0.441103	−	0.897457i	$-0.645413\pi$
$97$	−8.68872	−0.882205	−0.441103	+	0.897457i	$0.645413\pi$
$98$	−5.66400	−0.572150
$99$	0	0
$100$	−2.16340	−0.216340
$101$	−5.22774	−0.520180	−0.260090	−	0.965584i	$-0.583752\pi$
$101$	−5.22774	−0.520180	−0.260090	+	0.965584i	$0.583752\pi$
$102$	0	0
$103$	−15.6301	−1.54008	−0.770038	−	0.637998i	$-0.779763\pi$
$103$	−15.6301	−1.54008	−0.770038	+	0.637998i	$0.779763\pi$
$104$	−1.47421	−0.144558
$105$	0	0
$106$	−7.18563	−0.697930
$107$	−13.1958	−1.27569	−0.637844	−	0.770166i	$-0.720174\pi$
$107$	−13.1958	−1.27569	−0.637844	+	0.770166i	$0.720174\pi$
$108$	0	0
$109$	−0.637474	−0.0610589	−0.0305295	−	0.999534i	$-0.509719\pi$
$109$	−0.637474	−0.0610589	−0.0305295	+	0.999534i	$0.509719\pi$
$110$	−10.2893	−0.981042
$111$	0	0
$112$	−3.55865	−0.336261
$113$	0.947515	0.0891347	0.0445674	−	0.999006i	$-0.485809\pi$
$113$	0.947515	0.0891347	0.0445674	+	0.999006i	$0.485809\pi$
$114$	0	0
$115$	2.26428	0.211145
$116$	9.21992	0.856048
$117$	0	0
$118$	3.25205	0.299376
$119$	−28.6616	−2.62741
$120$	0	0
$121$	26.3224	2.39294
$122$	2.68461	0.243053
$123$	0	0
$124$	10.1791	0.914109
$125$	−12.0647	−1.07910
$126$	0	0
$127$	−6.40203	−0.568088	−0.284044	−	0.958811i	$-0.591676\pi$
$127$	−6.40203	−0.568088	−0.284044	+	0.958811i	$0.591676\pi$
$128$	−1.00000	−0.0883883
$129$	0	0
$130$	−2.48289	−0.217764
$131$	11.2816	0.985681	0.492841	−	0.870120i	$-0.335959\pi$
$131$	11.2816	0.985681	0.492841	+	0.870120i	$0.335959\pi$
$132$	0	0
$133$	17.2251	1.49360
$134$	−2.77854	−0.240029
$135$	0	0
$136$	−8.05407	−0.690631
$137$	−3.17284	−0.271074	−0.135537	−	0.990772i	$-0.543276\pi$
$137$	−3.17284	−0.271074	−0.135537	+	0.990772i	$0.543276\pi$
$138$	0	0
$139$	5.28528	0.448292	0.224146	−	0.974556i	$-0.428041\pi$
$139$	5.28528	0.448292	0.224146	+	0.974556i	$0.428041\pi$
$140$	−5.99356	−0.506548
$141$	0	0
$142$	−12.3136	−1.03334
$143$	9.00623	0.753139
$144$	0	0
$145$	15.5284	1.28956
$146$	−14.1129	−1.16800
$147$	0	0
$148$	6.09097	0.500674
$149$	1.00000	0.0819232
$149$	1.00000	0.0819232
$150$	0	0
$151$	−3.43009	−0.279137	−0.139569	−	0.990212i	$-0.544572\pi$
$151$	−3.43009	−0.279137	−0.139569	+	0.990212i	$0.544572\pi$
$152$	4.84034	0.392603
$153$	0	0
$154$	21.7405	1.75190
$155$	17.1438	1.37703
$156$	0	0
$157$	−12.9570	−1.03408	−0.517041	−	0.855961i	$-0.672966\pi$
$157$	−12.9570	−1.03408	−0.517041	+	0.855961i	$0.672966\pi$
$158$	−10.2953	−0.819053
$159$	0	0
$160$	−1.68422	−0.133149
$161$	−4.78427	−0.377053
$162$	0	0
$163$	−10.6415	−0.833510	−0.416755	−	0.909019i	$-0.636833\pi$
$163$	−10.6415	−0.833510	−0.416755	+	0.909019i	$0.636833\pi$
$164$	−9.20047	−0.718436
$165$	0	0
$166$	8.87149	0.688561
$167$	6.94967	0.537782	0.268891	−	0.963171i	$-0.413343\pi$
$167$	6.94967	0.537782	0.268891	+	0.963171i	$0.413343\pi$
$168$	0	0
$169$	−10.8267	−0.832824
$170$	−13.5648	−1.04038
$171$	0	0
$172$	−2.59279	−0.197698
$173$	2.11654	0.160918	0.0804589	−	0.996758i	$-0.474361\pi$
$173$	2.11654	0.160918	0.0804589	+	0.996758i	$0.474361\pi$
$174$	0	0
$175$	7.69879	0.581973
$176$	6.10921	0.460499
$177$	0	0
$178$	−1.15228	−0.0863671
$179$	−16.2049	−1.21121	−0.605604	−	0.795766i	$-0.707069\pi$
$179$	−16.2049	−1.21121	−0.605604	+	0.795766i	$0.707069\pi$
$180$	0	0
$181$	6.34718	0.471782	0.235891	−	0.971780i	$-0.424199\pi$
$181$	6.34718	0.471782	0.235891	+	0.971780i	$0.424199\pi$
$182$	5.24619	0.388873
$183$	0	0
$184$	−1.34441	−0.0991109
$185$	10.2585	0.754222
$186$	0	0
$187$	49.2040	3.59815
$188$	−8.87218	−0.647070
$189$	0	0
$190$	8.15220	0.591423
$191$	25.4618	1.84235	0.921176	−	0.389146i	$-0.127230\pi$
$191$	25.4618	1.84235	0.921176	+	0.389146i	$0.127230\pi$
$192$	0	0
$193$	1.74393	0.125531	0.0627655	−	0.998028i	$-0.480008\pi$
$193$	1.74393	0.125531	0.0627655	+	0.998028i	$0.480008\pi$
$194$	8.68872	0.623813
$195$	0	0
$196$	5.66400	0.404571
$197$	24.7629	1.76429	0.882143	−	0.470982i	$-0.156100\pi$
$197$	24.7629	1.76429	0.882143	+	0.470982i	$0.156100\pi$
$198$	0	0
$199$	−19.7405	−1.39937	−0.699685	−	0.714452i	$-0.746676\pi$
$199$	−19.7405	−1.39937	−0.699685	+	0.714452i	$0.746676\pi$
$200$	2.16340	0.152975
$201$	0	0
$202$	5.22774	0.367822
$203$	−32.8105	−2.30284
$204$	0	0
$205$	−15.4956	−1.08226
$206$	15.6301	1.08900
$207$	0	0
$208$	1.47421	0.102218
$209$	−29.5706	−2.04544
$210$	0	0
$211$	13.5991	0.936204	0.468102	−	0.883674i	$-0.344938\pi$
$211$	13.5991	0.936204	0.468102	+	0.883674i	$0.344938\pi$
$212$	7.18563	0.493511
$213$	0	0
$214$	13.1958	0.902048
$215$	−4.36683	−0.297815
$216$	0	0
$217$	−36.2238	−2.45903
$218$	0.637474	0.0431752
$219$	0	0
$220$	10.2893	0.693701
$221$	11.8734	0.798689
$222$	0	0
$223$	0.193309	0.0129449	0.00647246	−	0.999979i	$-0.497940\pi$
$223$	0.193309	0.0129449	0.00647246	+	0.999979i	$0.497940\pi$
$224$	3.55865	0.237772
$225$	0	0
$226$	−0.947515	−0.0630278
$227$	−26.1112	−1.73306	−0.866532	−	0.499122i	$-0.833656\pi$
$227$	−26.1112	−1.73306	−0.866532	+	0.499122i	$0.833656\pi$
$228$	0	0
$229$	4.96419	0.328043	0.164022	−	0.986457i	$-0.447553\pi$
$229$	4.96419	0.328043	0.164022	+	0.986457i	$0.447553\pi$
$230$	−2.26428	−0.149302
$231$	0	0
$232$	−9.21992	−0.605317
$233$	29.4264	1.92779	0.963895	−	0.266283i	$-0.0857955\pi$
$233$	29.4264	1.92779	0.963895	+	0.266283i	$0.0857955\pi$
$234$	0	0
$235$	−14.9427	−0.974755
$236$	−3.25205	−0.211691
$237$	0	0
$238$	28.6616	1.85786
$239$	16.9312	1.09519	0.547594	−	0.836744i	$-0.315544\pi$
$239$	16.9312	1.09519	0.547594	+	0.836744i	$0.315544\pi$
$240$	0	0
$241$	−18.9044	−1.21774	−0.608870	−	0.793270i	$-0.708377\pi$
$241$	−18.9044	−1.21774	−0.608870	+	0.793270i	$0.708377\pi$
$242$	−26.3224	−1.69207
$243$	0	0
$244$	−2.68461	−0.171864
$245$	9.53943	0.609452
$246$	0	0
$247$	−7.13566	−0.454031
$248$	−10.1791	−0.646372
$249$	0	0
$250$	12.0647	0.763042
$251$	3.70741	0.234010	0.117005	−	0.993131i	$-0.462671\pi$
$251$	3.70741	0.234010	0.117005	+	0.993131i	$0.462671\pi$
$252$	0	0
$253$	8.21325	0.516363
$254$	6.40203	0.401699
$255$	0	0
$256$	1.00000	0.0625000
$257$	22.8482	1.42523	0.712616	−	0.701554i	$-0.247510\pi$
$257$	22.8482	1.42523	0.712616	+	0.701554i	$0.247510\pi$
$258$	0	0
$259$	−21.6756	−1.34686
$260$	2.48289	0.153982
$261$	0	0
$262$	−11.2816	−0.696982
$263$	−14.1690	−0.873701	−0.436850	−	0.899534i	$-0.643906\pi$
$263$	−14.1690	−0.873701	−0.436850	+	0.899534i	$0.643906\pi$
$264$	0	0
$265$	12.1022	0.743431
$266$	−17.2251	−1.05614
$267$	0	0
$268$	2.77854	0.169726
$269$	−9.99934	−0.609670	−0.304835	−	0.952405i	$-0.598601\pi$
$269$	−9.99934	−0.609670	−0.304835	+	0.952405i	$0.598601\pi$
$270$	0	0
$271$	4.35761	0.264706	0.132353	−	0.991203i	$-0.457747\pi$
$271$	4.35761	0.264706	0.132353	+	0.991203i	$0.457747\pi$
$272$	8.05407	0.488350
$273$	0	0
$274$	3.17284	0.191678
$275$	−13.2167	−0.796994
$276$	0	0
$277$	31.6380	1.90094	0.950470	−	0.310815i	$-0.100602\pi$
$277$	31.6380	1.90094	0.950470	+	0.310815i	$0.100602\pi$
$278$	−5.28528	−0.316990
$279$	0	0
$280$	5.99356	0.358183
$281$	13.9444	0.831854	0.415927	−	0.909398i	$-0.363457\pi$
$281$	13.9444	0.831854	0.415927	+	0.909398i	$0.363457\pi$
$282$	0	0
$283$	17.4317	1.03621	0.518104	−	0.855317i	$-0.326638\pi$
$283$	17.4317	1.03621	0.518104	+	0.855317i	$0.326638\pi$
$284$	12.3136	0.730680
$285$	0	0
$286$	−9.00623	−0.532550
$287$	32.7413	1.93266
$288$	0	0
$289$	47.8681	2.81577
$290$	−15.5284	−0.911858
$291$	0	0
$292$	14.1129	0.825898
$293$	2.65219	0.154943	0.0774713	−	0.996995i	$-0.475315\pi$
$293$	2.65219	0.154943	0.0774713	+	0.996995i	$0.475315\pi$
$294$	0	0
$295$	−5.47717	−0.318893
$296$	−6.09097	−0.354030
$297$	0	0
$298$	−1.00000	−0.0579284
$299$	1.98193	0.114618
$300$	0	0
$301$	9.22683	0.531826
$302$	3.43009	0.197380
$303$	0	0
$304$	−4.84034	−0.277613
$305$	−4.52147	−0.258899
$306$	0	0
$307$	25.1021	1.43265	0.716326	−	0.697766i	$-0.245823\pi$
$307$	25.1021	1.43265	0.716326	+	0.697766i	$0.245823\pi$
$308$	−21.7405	−1.23878
$309$	0	0
$310$	−17.1438	−0.973704
$311$	−4.98601	−0.282731	−0.141365	−	0.989957i	$-0.545149\pi$
$311$	−4.98601	−0.282731	−0.141365	+	0.989957i	$0.545149\pi$
$312$	0	0
$313$	4.50918	0.254874	0.127437	−	0.991847i	$-0.459325\pi$
$313$	4.50918	0.254874	0.127437	+	0.991847i	$0.459325\pi$
$314$	12.9570	0.731206
$315$	0	0
$316$	10.2953	0.579158
$317$	−20.7267	−1.16413	−0.582064	−	0.813143i	$-0.697755\pi$
$317$	−20.7267	−1.16413	−0.582064	+	0.813143i	$0.697755\pi$
$318$	0	0
$319$	56.3264	3.15367
$320$	1.68422	0.0941508
$321$	0	0
$322$	4.78427	0.266617
$323$	−38.9845	−2.16915
$324$	0	0
$325$	−3.18930	−0.176910
$326$	10.6415	0.589381
$327$	0	0
$328$	9.20047	0.508011
$329$	31.5730	1.74068
$330$	0	0
$331$	−6.56636	−0.360920	−0.180460	−	0.983582i	$-0.557759\pi$
$331$	−6.56636	−0.360920	−0.180460	+	0.983582i	$0.557759\pi$
$332$	−8.87149	−0.486886
$333$	0	0
$334$	−6.94967	−0.380269
$335$	4.67967	0.255678
$336$	0	0
$337$	4.45897	0.242896	0.121448	−	0.992598i	$-0.461246\pi$
$337$	4.45897	0.242896	0.121448	+	0.992598i	$0.461246\pi$
$338$	10.8267	0.588896
$339$	0	0
$340$	13.5648	0.735657
$341$	62.1861	3.36757
$342$	0	0
$343$	4.75436	0.256711
$344$	2.59279	0.139794
$345$	0	0
$346$	−2.11654	−0.113786
$347$	−7.85553	−0.421707	−0.210853	−	0.977518i	$-0.567624\pi$
$347$	−7.85553	−0.421707	−0.210853	+	0.977518i	$0.567624\pi$
$348$	0	0
$349$	−3.75625	−0.201068	−0.100534	−	0.994934i	$-0.532055\pi$
$349$	−3.75625	−0.201068	−0.100534	+	0.994934i	$0.532055\pi$
$350$	−7.69879	−0.411517
$351$	0	0
$352$	−6.10921	−0.325622
$353$	−7.87865	−0.419338	−0.209669	−	0.977772i	$-0.567239\pi$
$353$	−7.87865	−0.419338	−0.209669	+	0.977772i	$0.567239\pi$
$354$	0	0
$355$	20.7389	1.10071
$356$	1.15228	0.0610708
$357$	0	0
$358$	16.2049	0.856454
$359$	−31.6827	−1.67215	−0.836074	−	0.548617i	$-0.815155\pi$
$359$	−31.6827	−1.67215	−0.836074	+	0.548617i	$0.815155\pi$
$360$	0	0
$361$	4.42890	0.233100
$362$	−6.34718	−0.333600
$363$	0	0
$364$	−5.24619	−0.274975
$365$	23.7693	1.24414
$366$	0	0
$367$	−24.6614	−1.28732	−0.643658	−	0.765314i	$-0.722584\pi$
$367$	−24.6614	−1.28732	−0.643658	+	0.765314i	$0.722584\pi$
$368$	1.34441	0.0700820
$369$	0	0
$370$	−10.2585	−0.533316
$371$	−25.5711	−1.32759
$372$	0	0
$373$	−19.8736	−1.02902	−0.514508	−	0.857486i	$-0.672025\pi$
$373$	−19.8736	−1.02902	−0.514508	+	0.857486i	$0.672025\pi$
$374$	−49.2040	−2.54428
$375$	0	0
$376$	8.87218	0.457548
$377$	13.5921	0.700027
$378$	0	0
$379$	17.5638	0.902190	0.451095	−	0.892476i	$-0.351034\pi$
$379$	17.5638	0.902190	0.451095	+	0.892476i	$0.351034\pi$
$380$	−8.15220	−0.418199
$381$	0	0
$382$	−25.4618	−1.30274
$383$	−20.9181	−1.06887	−0.534433	−	0.845211i	$-0.679475\pi$
$383$	−20.9181	−1.06887	−0.534433	+	0.845211i	$0.679475\pi$
$384$	0	0
$385$	−36.6159	−1.86612
$386$	−1.74393	−0.0887638
$387$	0	0
$388$	−8.68872	−0.441103
$389$	19.0545	0.966102	0.483051	−	0.875592i	$-0.339529\pi$
$389$	19.0545	0.966102	0.483051	+	0.875592i	$0.339529\pi$
$390$	0	0
$391$	10.8279	0.547593
$392$	−5.66400	−0.286075
$393$	0	0
$394$	−24.7629	−1.24754
$395$	17.3396	0.872451
$396$	0	0
$397$	−18.9893	−0.953046	−0.476523	−	0.879162i	$-0.658103\pi$
$397$	−18.9893	−0.953046	−0.476523	+	0.879162i	$0.658103\pi$
$398$	19.7405	0.989503
$399$	0	0
$400$	−2.16340	−0.108170
$401$	−2.37024	−0.118364	−0.0591822	−	0.998247i	$-0.518849\pi$
$401$	−2.37024	−0.118364	−0.0591822	+	0.998247i	$0.518849\pi$
$402$	0	0
$403$	15.0061	0.747506
$404$	−5.22774	−0.260090
$405$	0	0
$406$	32.8105	1.62836
$407$	37.2110	1.84448
$408$	0	0
$409$	−3.82404	−0.189087	−0.0945433	−	0.995521i	$-0.530139\pi$
$409$	−3.82404	−0.189087	−0.0945433	+	0.995521i	$0.530139\pi$
$410$	15.4956	0.765274
$411$	0	0
$412$	−15.6301	−0.770038
$413$	11.5729	0.569466
$414$	0	0
$415$	−14.9416	−0.733452
$416$	−1.47421	−0.0722789
$417$	0	0
$418$	29.5706	1.44635
$419$	10.4051	0.508325	0.254162	−	0.967162i	$-0.418200\pi$
$419$	10.4051	0.508325	0.254162	+	0.967162i	$0.418200\pi$
$420$	0	0
$421$	29.6678	1.44592	0.722961	−	0.690889i	$-0.242781\pi$
$421$	29.6678	1.44592	0.722961	+	0.690889i	$0.242781\pi$
$422$	−13.5991	−0.661996
$423$	0	0
$424$	−7.18563	−0.348965
$425$	−17.4242	−0.845197
$426$	0	0
$427$	9.55359	0.462330
$428$	−13.1958	−0.637844
$429$	0	0
$430$	4.36683	0.210587
$431$	0.608616	0.0293160	0.0146580	−	0.999893i	$-0.495334\pi$
$431$	0.608616	0.0293160	0.0146580	+	0.999893i	$0.495334\pi$
$432$	0	0
$433$	−4.58679	−0.220427	−0.110213	−	0.993908i	$-0.535153\pi$
$433$	−4.58679	−0.220427	−0.110213	+	0.993908i	$0.535153\pi$
$434$	36.2238	1.73880
$435$	0	0
$436$	−0.637474	−0.0305295
$437$	−6.50738	−0.311290
$438$	0	0
$439$	17.6793	0.843788	0.421894	−	0.906645i	$-0.361365\pi$
$439$	17.6793	0.843788	0.421894	+	0.906645i	$0.361365\pi$
$440$	−10.2893	−0.490521
$441$	0	0
$442$	−11.8734	−0.564759
$443$	1.88287	0.0894576	0.0447288	−	0.998999i	$-0.485758\pi$
$443$	1.88287	0.0894576	0.0447288	+	0.998999i	$0.485758\pi$
$444$	0	0
$445$	1.94070	0.0919978
$446$	−0.193309	−0.00915344
$447$	0	0
$448$	−3.55865	−0.168130
$449$	32.3033	1.52449	0.762244	−	0.647290i	$-0.224098\pi$
$449$	32.3033	1.52449	0.762244	+	0.647290i	$0.224098\pi$
$450$	0	0
$451$	−56.2075	−2.64671
$452$	0.947515	0.0445674
$453$	0	0
$454$	26.1112	1.22546
$455$	−8.83574	−0.414226
$456$	0	0
$457$	−4.72109	−0.220843	−0.110422	−	0.993885i	$-0.535220\pi$
$457$	−4.72109	−0.220843	−0.110422	+	0.993885i	$0.535220\pi$
$458$	−4.96419	−0.231961
$459$	0	0
$460$	2.26428	0.105572
$461$	−33.8943	−1.57861	−0.789307	−	0.613998i	$-0.789560\pi$
$461$	−33.8943	−1.57861	−0.789307	+	0.613998i	$0.789560\pi$
$462$	0	0
$463$	34.2855	1.59338	0.796691	−	0.604387i	$-0.206582\pi$
$463$	34.2855	1.59338	0.796691	+	0.604387i	$0.206582\pi$
$464$	9.21992	0.428024
$465$	0	0
$466$	−29.4264	−1.36315
$467$	−1.05847	−0.0489802	−0.0244901	−	0.999700i	$-0.507796\pi$
$467$	−1.05847	−0.0489802	−0.0244901	+	0.999700i	$0.507796\pi$
$468$	0	0
$469$	−9.88785	−0.456579
$470$	14.9427	0.689256
$471$	0	0
$472$	3.25205	0.149688
$473$	−15.8399	−0.728318
$474$	0	0
$475$	10.4716	0.480470
$476$	−28.6616	−1.31370
$477$	0	0
$478$	−16.9312	−0.774415
$479$	5.84496	0.267063	0.133532	−	0.991045i	$-0.457368\pi$
$479$	5.84496	0.267063	0.133532	+	0.991045i	$0.457368\pi$
$480$	0	0
$481$	8.97934	0.409423
$482$	18.9044	0.861072
$483$	0	0
$484$	26.3224	1.19647
$485$	−14.6337	−0.664483
$486$	0	0
$487$	7.99064	0.362091	0.181045	−	0.983475i	$-0.442052\pi$
$487$	7.99064	0.362091	0.181045	+	0.983475i	$0.442052\pi$
$488$	2.68461	0.121527
$489$	0	0
$490$	−9.53943	−0.430947
$491$	−4.60871	−0.207988	−0.103994	−	0.994578i	$-0.533162\pi$
$491$	−4.60871	−0.207988	−0.103994	+	0.994578i	$0.533162\pi$
$492$	0	0
$493$	74.2579	3.34441
$494$	7.13566	0.321049
$495$	0	0
$496$	10.1791	0.457054
$497$	−43.8200	−1.96559
$498$	0	0
$499$	−8.23389	−0.368600	−0.184300	−	0.982870i	$-0.559002\pi$
$499$	−8.23389	−0.368600	−0.184300	+	0.982870i	$0.559002\pi$
$500$	−12.0647	−0.539552
$501$	0	0
$502$	−3.70741	−0.165470
$503$	−13.7999	−0.615308	−0.307654	−	0.951498i	$-0.599544\pi$
$503$	−13.7999	−0.615308	−0.307654	+	0.951498i	$0.599544\pi$
$504$	0	0
$505$	−8.80467	−0.391803
$506$	−8.21325	−0.365123
$507$	0	0
$508$	−6.40203	−0.284044
$509$	−13.4093	−0.594356	−0.297178	−	0.954822i	$-0.596045\pi$
$509$	−13.4093	−0.594356	−0.297178	+	0.954822i	$0.596045\pi$
$510$	0	0
$511$	−50.2231	−2.22174
$512$	−1.00000	−0.0441942
$513$	0	0
$514$	−22.8482	−1.00779
$515$	−26.3245	−1.16000
$516$	0	0
$517$	−54.2020	−2.38380
$518$	21.6756	0.952372
$519$	0	0
$520$	−2.48289	−0.108882
$521$	−30.9323	−1.35517	−0.677584	−	0.735445i	$-0.736973\pi$
$521$	−30.9323	−1.35517	−0.677584	+	0.735445i	$0.736973\pi$
$522$	0	0
$523$	16.1869	0.707806	0.353903	−	0.935282i	$-0.384854\pi$
$523$	16.1869	0.707806	0.353903	+	0.935282i	$0.384854\pi$
$524$	11.2816	0.492841
$525$	0	0
$526$	14.1690	0.617800
$527$	81.9831	3.57124
$528$	0	0
$529$	−21.1926	−0.921416
$530$	−12.1022	−0.525685
$531$	0	0
$532$	17.2251	0.746802
$533$	−13.5634	−0.587496
$534$	0	0
$535$	−22.2247	−0.960857
$536$	−2.77854	−0.120015
$537$	0	0
$538$	9.99934	0.431102
$539$	34.6025	1.49044
$540$	0	0
$541$	29.1990	1.25536	0.627681	−	0.778471i	$-0.284004\pi$
$541$	29.1990	1.25536	0.627681	+	0.778471i	$0.284004\pi$
$542$	−4.35761	−0.187175
$543$	0	0
$544$	−8.05407	−0.345316
$545$	−1.07365	−0.0459900
$546$	0	0
$547$	−4.65837	−0.199178	−0.0995888	−	0.995029i	$-0.531753\pi$
$547$	−4.65837	−0.199178	−0.0995888	+	0.995029i	$0.531753\pi$
$548$	−3.17284	−0.135537
$549$	0	0
$550$	13.2167	0.563560
$551$	−44.6276	−1.90120
$552$	0	0
$553$	−36.6375	−1.55799
$554$	−31.6380	−1.34417
$555$	0	0
$556$	5.28528	0.224146
$557$	−9.79350	−0.414964	−0.207482	−	0.978239i	$-0.566527\pi$
$557$	−9.79350	−0.414964	−0.207482	+	0.978239i	$0.566527\pi$
$558$	0	0
$559$	−3.82231	−0.161666
$560$	−5.99356	−0.253274
$561$	0	0
$562$	−13.9444	−0.588209
$563$	28.5486	1.20318	0.601590	−	0.798805i	$-0.294534\pi$
$563$	28.5486	1.20318	0.601590	+	0.798805i	$0.294534\pi$
$564$	0	0
$565$	1.59583	0.0671369
$566$	−17.4317	−0.732710
$567$	0	0
$568$	−12.3136	−0.516669
$569$	16.7762	0.703296	0.351648	−	0.936132i	$-0.385621\pi$
$569$	16.7762	0.703296	0.351648	+	0.936132i	$0.385621\pi$
$570$	0	0
$571$	−19.6679	−0.823075	−0.411538	−	0.911393i	$-0.635008\pi$
$571$	−19.6679	−0.823075	−0.411538	+	0.911393i	$0.635008\pi$
$572$	9.00623	0.376569
$573$	0	0
$574$	−32.7413	−1.36659
$575$	−2.90849	−0.121292
$576$	0	0
$577$	−35.0928	−1.46093	−0.730465	−	0.682950i	$-0.760697\pi$
$577$	−35.0928	−1.46093	−0.730465	+	0.682950i	$0.760697\pi$
$578$	−47.8681	−1.99105
$579$	0	0
$580$	15.5284	0.644781
$581$	31.5706	1.30977
$582$	0	0
$583$	43.8985	1.81809
$584$	−14.1129	−0.583998
$585$	0	0
$586$	−2.65219	−0.109561
$587$	15.2454	0.629246	0.314623	−	0.949217i	$-0.398122\pi$
$587$	15.2454	0.629246	0.314623	+	0.949217i	$0.398122\pi$
$588$	0	0
$589$	−49.2702	−2.03014
$590$	5.47717	0.225492
$591$	0	0
$592$	6.09097	0.250337
$593$	12.7548	0.523778	0.261889	−	0.965098i	$-0.415654\pi$
$593$	12.7548	0.523778	0.261889	+	0.965098i	$0.415654\pi$
$594$	0	0
$595$	−48.2725	−1.97898
$596$	1.00000	0.0409616
$597$	0	0
$598$	−1.98193	−0.0810472
$599$	−16.8463	−0.688323	−0.344161	−	0.938910i	$-0.611837\pi$
$599$	−16.8463	−0.688323	−0.344161	+	0.938910i	$0.611837\pi$
$600$	0	0
$601$	8.55294	0.348882	0.174441	−	0.984668i	$-0.444188\pi$
$601$	8.55294	0.348882	0.174441	+	0.984668i	$0.444188\pi$
$602$	−9.22683	−0.376058
$603$	0	0
$604$	−3.43009	−0.139569
$605$	44.3327	1.80238
$606$	0	0
$607$	−27.3861	−1.11157	−0.555784	−	0.831327i	$-0.687582\pi$
$607$	−27.3861	−1.11157	−0.555784	+	0.831327i	$0.687582\pi$
$608$	4.84034	0.196302
$609$	0	0
$610$	4.52147	0.183069
$611$	−13.0794	−0.529137
$612$	0	0
$613$	−48.1251	−1.94375	−0.971877	−	0.235487i	$-0.924331\pi$
$613$	−48.1251	−1.94375	−0.971877	+	0.235487i	$0.924331\pi$
$614$	−25.1021	−1.01304
$615$	0	0
$616$	21.7405	0.875951
$617$	34.7472	1.39887	0.699436	−	0.714696i	$-0.253435\pi$
$617$	34.7472	1.39887	0.699436	+	0.714696i	$0.253435\pi$
$618$	0	0
$619$	28.8892	1.16116	0.580578	−	0.814205i	$-0.302827\pi$
$619$	28.8892	1.16116	0.580578	+	0.814205i	$0.302827\pi$
$620$	17.1438	0.688513
$621$	0	0
$622$	4.98601	0.199921
$623$	−4.10057	−0.164286
$624$	0	0
$625$	−9.50270	−0.380108
$626$	−4.50918	−0.180223
$627$	0	0
$628$	−12.9570	−0.517041
$629$	49.0571	1.95603
$630$	0	0
$631$	13.6191	0.542170	0.271085	−	0.962555i	$-0.412618\pi$
$631$	13.6191	0.542170	0.271085	+	0.962555i	$0.412618\pi$
$632$	−10.2953	−0.409527
$633$	0	0
$634$	20.7267	0.823163
$635$	−10.7824	−0.427888
$636$	0	0
$637$	8.34991	0.330835
$638$	−56.3264	−2.22998
$639$	0	0
$640$	−1.68422	−0.0665747
$641$	−8.57128	−0.338545	−0.169273	−	0.985569i	$-0.554142\pi$
$641$	−8.57128	−0.338545	−0.169273	+	0.985569i	$0.554142\pi$
$642$	0	0
$643$	14.0329	0.553404	0.276702	−	0.960956i	$-0.410758\pi$
$643$	14.0329	0.553404	0.276702	+	0.960956i	$0.410758\pi$
$644$	−4.78427	−0.188527
$645$	0	0
$646$	38.9845	1.53382
$647$	21.6077	0.849485	0.424743	−	0.905314i	$-0.360365\pi$
$647$	21.6077	0.849485	0.424743	+	0.905314i	$0.360365\pi$
$648$	0	0
$649$	−19.8675	−0.779866
$650$	3.18930	0.125095
$651$	0	0
$652$	−10.6415	−0.416755
$653$	44.3269	1.73464	0.867322	−	0.497747i	$-0.165839\pi$
$653$	44.3269	1.73464	0.867322	+	0.497747i	$0.165839\pi$
$654$	0	0
$655$	19.0008	0.742422
$656$	−9.20047	−0.359218
$657$	0	0
$658$	−31.5730	−1.23084
$659$	5.51561	0.214858	0.107429	−	0.994213i	$-0.465738\pi$
$659$	5.51561	0.214858	0.107429	+	0.994213i	$0.465738\pi$
$660$	0	0
$661$	−19.6034	−0.762484	−0.381242	−	0.924475i	$-0.624504\pi$
$661$	−19.6034	−0.762484	−0.381242	+	0.924475i	$0.624504\pi$
$662$	6.56636	0.255209
$663$	0	0
$664$	8.87149	0.344281
$665$	29.0108	1.12499
$666$	0	0
$667$	12.3953	0.479948
$668$	6.94967	0.268891
$669$	0	0
$670$	−4.67967	−0.180792
$671$	−16.4008	−0.633147
$672$	0	0
$673$	47.8644	1.84504	0.922519	−	0.385952i	$-0.126127\pi$
$673$	47.8644	1.84504	0.922519	+	0.385952i	$0.126127\pi$
$674$	−4.45897	−0.171753
$675$	0	0
$676$	−10.8267	−0.416412
$677$	−29.5311	−1.13497	−0.567486	−	0.823383i	$-0.692084\pi$
$677$	−29.5311	−1.13497	−0.567486	+	0.823383i	$0.692084\pi$
$678$	0	0
$679$	30.9201	1.18660
$680$	−13.5648	−0.520188
$681$	0	0
$682$	−62.1861	−2.38123
$683$	−27.5786	−1.05527	−0.527633	−	0.849473i	$-0.676920\pi$
$683$	−27.5786	−1.05527	−0.527633	+	0.849473i	$0.676920\pi$
$684$	0	0
$685$	−5.34376	−0.204175
$686$	−4.75436	−0.181522
$687$	0	0
$688$	−2.59279	−0.0988492
$689$	10.5931	0.403565
$690$	0	0
$691$	−13.0061	−0.494777	−0.247388	−	0.968916i	$-0.579572\pi$
$691$	−13.0061	−0.494777	−0.247388	+	0.968916i	$0.579572\pi$
$692$	2.11654	0.0804589
$693$	0	0
$694$	7.85553	0.298192
$695$	8.90158	0.337656
$696$	0	0
$697$	−74.1012	−2.80678
$698$	3.75625	0.142176
$699$	0	0
$700$	7.69879	0.290987
$701$	22.4270	0.847054	0.423527	−	0.905883i	$-0.360792\pi$
$701$	22.4270	0.847054	0.423527	+	0.905883i	$0.360792\pi$
$702$	0	0
$703$	−29.4823	−1.11195
$704$	6.10921	0.230249
$705$	0	0
$706$	7.87865	0.296517
$707$	18.6037	0.699664
$708$	0	0
$709$	30.5580	1.14763	0.573814	−	0.818985i	$-0.305463\pi$
$709$	30.5580	1.14763	0.573814	+	0.818985i	$0.305463\pi$
$710$	−20.7389	−0.778317
$711$	0	0
$712$	−1.15228	−0.0431836
$713$	13.6848	0.512500
$714$	0	0
$715$	15.1685	0.567269
$716$	−16.2049	−0.605604
$717$	0	0
$718$	31.6827	1.18239
$719$	−34.9588	−1.30374	−0.651872	−	0.758329i	$-0.726016\pi$
$719$	−34.9588	−1.30374	−0.651872	+	0.758329i	$0.726016\pi$
$720$	0	0
$721$	55.6220	2.07147
$722$	−4.42890	−0.164827
$723$	0	0
$724$	6.34718	0.235891
$725$	−19.9464	−0.740790
$726$	0	0
$727$	5.95290	0.220781	0.110390	−	0.993888i	$-0.464790\pi$
$727$	5.95290	0.220781	0.110390	+	0.993888i	$0.464790\pi$
$728$	5.24619	0.194437
$729$	0	0
$730$	−23.7693	−0.879742
$731$	−20.8825	−0.772368
$732$	0	0
$733$	34.6212	1.27876	0.639382	−	0.768889i	$-0.279190\pi$
$733$	34.6212	1.27876	0.639382	+	0.768889i	$0.279190\pi$
$734$	24.6614	0.910269
$735$	0	0
$736$	−1.34441	−0.0495554
$737$	16.9747	0.625270
$738$	0	0
$739$	26.5461	0.976513	0.488256	−	0.872700i	$-0.337633\pi$
$739$	26.5461	0.976513	0.488256	+	0.872700i	$0.337633\pi$
$740$	10.2585	0.377111
$741$	0	0
$742$	25.5711	0.938746
$743$	22.3273	0.819110	0.409555	−	0.912285i	$-0.365684\pi$
$743$	22.3273	0.819110	0.409555	+	0.912285i	$0.365684\pi$
$744$	0	0
$745$	1.68422	0.0617051
$746$	19.8736	0.727624
$747$	0	0
$748$	49.2040	1.79908
$749$	46.9593	1.71586
$750$	0	0
$751$	38.5278	1.40590	0.702950	−	0.711239i	$-0.251866\pi$
$751$	38.5278	1.40590	0.702950	+	0.711239i	$0.251866\pi$
$752$	−8.87218	−0.323535
$753$	0	0
$754$	−13.5921	−0.494994
$755$	−5.77704	−0.210248
$756$	0	0
$757$	2.81777	0.102414	0.0512069	−	0.998688i	$-0.483693\pi$
$757$	2.81777	0.102414	0.0512069	+	0.998688i	$0.483693\pi$
$758$	−17.5638	−0.637944
$759$	0	0
$760$	8.15220	0.295711
$761$	45.2453	1.64014	0.820070	−	0.572263i	$-0.193934\pi$
$761$	45.2453	1.64014	0.820070	+	0.572263i	$0.193934\pi$
$762$	0	0
$763$	2.26855	0.0821269
$764$	25.4618	0.921176
$765$	0	0
$766$	20.9181	0.755803
$767$	−4.79420	−0.173108
$768$	0	0
$769$	45.8400	1.65303	0.826516	−	0.562913i	$-0.190319\pi$
$769$	45.8400	1.65303	0.826516	+	0.562913i	$0.190319\pi$
$770$	36.6159	1.31954
$771$	0	0
$772$	1.74393	0.0627655
$773$	4.11294	0.147932	0.0739661	−	0.997261i	$-0.476434\pi$
$773$	4.11294	0.147932	0.0739661	+	0.997261i	$0.476434\pi$
$774$	0	0
$775$	−22.0214	−0.791033
$776$	8.68872	0.311907
$777$	0	0
$778$	−19.0545	−0.683137
$779$	44.5334	1.59557
$780$	0	0
$781$	75.2266	2.69182
$782$	−10.8279	−0.387206
$783$	0	0
$784$	5.66400	0.202286
$785$	−21.8224	−0.778877
$786$	0	0
$787$	29.5000	1.05156	0.525781	−	0.850620i	$-0.323773\pi$
$787$	29.5000	1.05156	0.525781	+	0.850620i	$0.323773\pi$
$788$	24.7629	0.882143
$789$	0	0
$790$	−17.3396	−0.616916
$791$	−3.37188	−0.119890
$792$	0	0
$793$	−3.95767	−0.140541
$794$	18.9893	0.673905
$795$	0	0
$796$	−19.7405	−0.699685
$797$	23.3419	0.826811	0.413406	−	0.910547i	$-0.364339\pi$
$797$	23.3419	0.826811	0.413406	+	0.910547i	$0.364339\pi$
$798$	0	0
$799$	−71.4572	−2.52797
$800$	2.16340	0.0764877
$801$	0	0
$802$	2.37024	0.0836962
$803$	86.2189	3.04260
$804$	0	0
$805$	−8.05777	−0.283999
$806$	−15.0061	−0.528566
$807$	0	0
$808$	5.22774	0.183911
$809$	−32.1230	−1.12939	−0.564693	−	0.825301i	$-0.691005\pi$
$809$	−32.1230	−1.12939	−0.564693	+	0.825301i	$0.691005\pi$
$810$	0	0
$811$	10.8585	0.381293	0.190647	−	0.981659i	$-0.438942\pi$
$811$	10.8585	0.381293	0.190647	+	0.981659i	$0.438942\pi$
$812$	−32.8105	−1.15142
$813$	0	0
$814$	−37.2110	−1.30424
$815$	−17.9227	−0.627805
$816$	0	0
$817$	12.5500	0.439068
$818$	3.82404	0.133704
$819$	0	0
$820$	−15.4956	−0.541131
$821$	5.55960	0.194032	0.0970158	−	0.995283i	$-0.469070\pi$
$821$	5.55960	0.194032	0.0970158	+	0.995283i	$0.469070\pi$
$822$	0	0
$823$	33.6599	1.17331	0.586654	−	0.809837i	$-0.300445\pi$
$823$	33.6599	1.17331	0.586654	+	0.809837i	$0.300445\pi$
$824$	15.6301	0.544499
$825$	0	0
$826$	−11.5729	−0.402673
$827$	−37.0350	−1.28783	−0.643916	−	0.765096i	$-0.722692\pi$
$827$	−37.0350	−1.28783	−0.643916	+	0.765096i	$0.722692\pi$
$828$	0	0
$829$	4.63369	0.160935	0.0804673	−	0.996757i	$-0.474359\pi$
$829$	4.63369	0.160935	0.0804673	+	0.996757i	$0.474359\pi$
$830$	14.9416	0.518629
$831$	0	0
$832$	1.47421	0.0511089
$833$	45.6183	1.58058
$834$	0	0
$835$	11.7048	0.405061
$836$	−29.5706	−1.02272
$837$	0	0
$838$	−10.4051	−0.359440
$839$	−38.4139	−1.32619	−0.663097	−	0.748533i	$-0.730758\pi$
$839$	−38.4139	−1.32619	−0.663097	+	0.748533i	$0.730758\pi$
$840$	0	0
$841$	56.0069	1.93127
$842$	−29.6678	−1.02242
$843$	0	0
$844$	13.5991	0.468102
$845$	−18.2346	−0.627289
$846$	0	0
$847$	−93.6722	−3.21861
$848$	7.18563	0.246756
$849$	0	0
$850$	17.4242	0.597644
$851$	8.18873	0.280706
$852$	0	0
$853$	24.8065	0.849360	0.424680	−	0.905344i	$-0.360387\pi$
$853$	24.8065	0.849360	0.424680	+	0.905344i	$0.360387\pi$
$854$	−9.55359	−0.326917
$855$	0	0
$856$	13.1958	0.451024
$857$	−14.1863	−0.484593	−0.242297	−	0.970202i	$-0.577901\pi$
$857$	−14.1863	−0.484593	−0.242297	+	0.970202i	$0.577901\pi$
$858$	0	0
$859$	−20.7352	−0.707476	−0.353738	−	0.935345i	$-0.615090\pi$
$859$	−20.7352	−0.707476	−0.353738	+	0.935345i	$0.615090\pi$
$860$	−4.36683	−0.148908
$861$	0	0
$862$	−0.608616	−0.0207296
$863$	−44.9939	−1.53161	−0.765805	−	0.643072i	$-0.777659\pi$
$863$	−44.9939	−1.53161	−0.765805	+	0.643072i	$0.777659\pi$
$864$	0	0
$865$	3.56472	0.121204
$866$	4.58679	0.155865
$867$	0	0
$868$	−36.2238	−1.22952
$869$	62.8963	2.13361
$870$	0	0
$871$	4.09614	0.138792
$872$	0.637474	0.0215876
$873$	0	0
$874$	6.50738	0.220115
$875$	42.9342	1.45144
$876$	0	0
$877$	48.3004	1.63099	0.815495	−	0.578765i	$-0.196465\pi$
$877$	48.3004	1.63099	0.815495	+	0.578765i	$0.196465\pi$
$878$	−17.6793	−0.596648
$879$	0	0
$880$	10.2893	0.346851
$881$	−1.00602	−0.0338938	−0.0169469	−	0.999856i	$-0.505395\pi$
$881$	−1.00602	−0.0338938	−0.0169469	+	0.999856i	$0.505395\pi$
$882$	0	0
$883$	1.25725	0.0423099	0.0211550	−	0.999776i	$-0.493266\pi$
$883$	1.25725	0.0423099	0.0211550	+	0.999776i	$0.493266\pi$
$884$	11.8734	0.399345
$885$	0	0
$886$	−1.88287	−0.0632561
$887$	25.8442	0.867765	0.433882	−	0.900970i	$-0.357143\pi$
$887$	25.8442	0.867765	0.433882	+	0.900970i	$0.357143\pi$
$888$	0	0
$889$	22.7826	0.764103
$890$	−1.94070	−0.0650523
$891$	0	0
$892$	0.193309	0.00647246
$893$	42.9444	1.43708
$894$	0	0
$895$	−27.2926	−0.912290
$896$	3.55865	0.118886
$897$	0	0
$898$	−32.3033	−1.07798
$899$	93.8503	3.13008
$900$	0	0
$901$	57.8736	1.92805
$902$	56.2075	1.87151
$903$	0	0
$904$	−0.947515	−0.0315139
$905$	10.6901	0.355349
$906$	0	0
$907$	18.4564	0.612833	0.306417	−	0.951897i	$-0.400870\pi$
$907$	18.4564	0.612833	0.306417	+	0.951897i	$0.400870\pi$
$908$	−26.1112	−0.866532
$909$	0	0
$910$	8.83574	0.292902
$911$	21.1111	0.699441	0.349720	−	0.936854i	$-0.386277\pi$
$911$	21.1111	0.699441	0.349720	+	0.936854i	$0.386277\pi$
$912$	0	0
$913$	−54.1978	−1.79368
$914$	4.72109	0.156160
$915$	0	0
$916$	4.96419	0.164022
$917$	−40.1474	−1.32578
$918$	0	0
$919$	21.9035	0.722530	0.361265	−	0.932463i	$-0.382345\pi$
$919$	21.9035	0.722530	0.361265	+	0.932463i	$0.382345\pi$
$920$	−2.26428	−0.0746510
$921$	0	0
$922$	33.8943	1.11625
$923$	18.1529	0.597508
$924$	0	0
$925$	−13.1772	−0.433263
$926$	−34.2855	−1.12669
$927$	0	0
$928$	−9.21992	−0.302659
$929$	30.7312	1.00826	0.504128	−	0.863629i	$-0.331814\pi$
$929$	30.7312	1.00826	0.504128	+	0.863629i	$0.331814\pi$
$930$	0	0
$931$	−27.4157	−0.898513
$932$	29.4264	0.963895
$933$	0	0
$934$	1.05847	0.0346342
$935$	82.8704	2.71015
$936$	0	0
$937$	−30.6275	−1.00056	−0.500278	−	0.865865i	$-0.666769\pi$
$937$	−30.6275	−1.00056	−0.500278	+	0.865865i	$0.666769\pi$
$938$	9.88785	0.322850
$939$	0	0
$940$	−14.9427	−0.487377
$941$	−34.8835	−1.13717	−0.568585	−	0.822624i	$-0.692509\pi$
$941$	−34.8835	−1.13717	−0.568585	+	0.822624i	$0.692509\pi$
$942$	0	0
$943$	−12.3692	−0.402795
$944$	−3.25205	−0.105845
$945$	0	0
$946$	15.8399	0.514999
$947$	−21.8906	−0.711347	−0.355674	−	0.934610i	$-0.615749\pi$
$947$	−21.8906	−0.711347	−0.355674	+	0.934610i	$0.615749\pi$
$948$	0	0
$949$	20.8054	0.675372
$950$	−10.4716	−0.339743
$951$	0	0
$952$	28.6616	0.928929
$953$	−13.1276	−0.425244	−0.212622	−	0.977135i	$-0.568200\pi$
$953$	−13.1276	−0.425244	−0.212622	+	0.977135i	$0.568200\pi$
$954$	0	0
$955$	42.8833	1.38767
$956$	16.9312	0.547594
$957$	0	0
$958$	−5.84496	−0.188842
$959$	11.2910	0.364606
$960$	0	0
$961$	72.6137	2.34238
$962$	−8.97934	−0.289506
$963$	0	0
$964$	−18.9044	−0.608870
$965$	2.93717	0.0945508
$966$	0	0
$967$	−34.4920	−1.10919	−0.554594	−	0.832121i	$-0.687127\pi$
$967$	−34.4920	−1.10919	−0.554594	+	0.832121i	$0.687127\pi$
$968$	−26.3224	−0.846034
$969$	0	0
$970$	14.6337	0.469860
$971$	52.4645	1.68367	0.841834	−	0.539737i	$-0.181476\pi$
$971$	52.4645	1.68367	0.841834	+	0.539737i	$0.181476\pi$
$972$	0	0
$973$	−18.8085	−0.602972
$974$	−7.99064	−0.256037
$975$	0	0
$976$	−2.68461	−0.0859322
$977$	−27.5639	−0.881848	−0.440924	−	0.897544i	$-0.645349\pi$
$977$	−27.5639	−0.881848	−0.440924	+	0.897544i	$0.645349\pi$
$978$	0	0
$979$	7.03952	0.224984
$980$	9.53943	0.304726
$981$	0	0
$982$	4.60871	0.147070
$983$	−39.6258	−1.26387	−0.631933	−	0.775023i	$-0.717738\pi$
$983$	−39.6258	−1.26387	−0.631933	+	0.775023i	$0.717738\pi$
$984$	0	0
$985$	41.7062	1.32887
$986$	−74.2579	−2.36485
$987$	0	0
$988$	−7.13566	−0.227016
$989$	−3.48576	−0.110841
$990$	0	0
$991$	3.54409	0.112582	0.0562909	−	0.998414i	$-0.482073\pi$
$991$	3.54409	0.112582	0.0562909	+	0.998414i	$0.482073\pi$
$992$	−10.1791	−0.323186
$993$	0	0
$994$	43.8200	1.38988
$995$	−33.2474	−1.05401
$996$	0	0
$997$	−43.1645	−1.36703	−0.683517	−	0.729934i	$-0.739551\pi$
$997$	−43.1645	−1.36703	−0.683517	+	0.729934i	$0.739551\pi$
$998$	8.23389	0.260639
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	8046.2.a.s.1.13	✓	16
3.2	odd	2		8046.2.a.t.1.4	yes	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
8046.2.a.s.1.13	✓	16	1.1	even	1	trivial
8046.2.a.t.1.4	yes	16	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)

\(f(q)\)	\(=\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.68422 q^{5} -3.55865 q^{7} -1.00000 q^{8} +O(q^{10})\)	\(q-1.00000 q^{2} +1.00000 q^{4} +1.68422 q^{5} -3.55865 q^{7} -1.00000 q^{8} -1.68422 q^{10} +6.10921 q^{11} +1.47421 q^{13} +3.55865 q^{14} +1.00000 q^{16} +8.05407 q^{17} -4.84034 q^{19} +1.68422 q^{20} -6.10921 q^{22} +1.34441 q^{23} -2.16340 q^{25} -1.47421 q^{26} -3.55865 q^{28} +9.21992 q^{29} +10.1791 q^{31} -1.00000 q^{32} -8.05407 q^{34} -5.99356 q^{35} +6.09097 q^{37} +4.84034 q^{38} -1.68422 q^{40} -9.20047 q^{41} -2.59279 q^{43} +6.10921 q^{44} -1.34441 q^{46} -8.87218 q^{47} +5.66400 q^{49} +2.16340 q^{50} +1.47421 q^{52} +7.18563 q^{53} +10.2893 q^{55} +3.55865 q^{56} -9.21992 q^{58} -3.25205 q^{59} -2.68461 q^{61} -10.1791 q^{62} +1.00000 q^{64} +2.48289 q^{65} +2.77854 q^{67} +8.05407 q^{68} +5.99356 q^{70} +12.3136 q^{71} +14.1129 q^{73} -6.09097 q^{74} -4.84034 q^{76} -21.7405 q^{77} +10.2953 q^{79} +1.68422 q^{80} +9.20047 q^{82} -8.87149 q^{83} +13.5648 q^{85} +2.59279 q^{86} -6.10921 q^{88} +1.15228 q^{89} -5.24619 q^{91} +1.34441 q^{92} +8.87218 q^{94} -8.15220 q^{95} -8.68872 q^{97} -5.66400 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8}+O(q^{10}) \) 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8	\( 16 q - 16 q^{2} + 16 q^{4} - 4 q^{5} + 6 q^{7} - 16 q^{8} + 4 q^{10} - 6 q^{11} + 6 q^{13} - 6 q^{14} + 16 q^{16} - q^{17} + 10 q^{19} - 4 q^{20} + 6 q^{22} - 10 q^{23} + 28 q^{25} - 6 q^{26} + 6 q^{28} - 6 q^{29} + 21 q^{31} - 16 q^{32} + q^{34} - 16 q^{35} + 17 q^{37} - 10 q^{38} + 4 q^{40} + 4 q^{41} + 16 q^{43} - 6 q^{44} + 10 q^{46} - 25 q^{47} + 36 q^{49} - 28 q^{50} + 6 q^{52} - 14 q^{53} + 19 q^{55} - 6 q^{56} + 6 q^{58} - 6 q^{59} + 23 q^{61} - 21 q^{62} + 16 q^{64} - 20 q^{65} + 22 q^{67} - q^{68} + 16 q^{70} - 10 q^{71} + 16 q^{73} - 17 q^{74} + 10 q^{76} + 2 q^{77} + 37 q^{79} - 4 q^{80} - 4 q^{82} - 33 q^{83} + 43 q^{85} - 16 q^{86} + 6 q^{88} + 3 q^{89} + 28 q^{91} - 10 q^{92} + 25 q^{94} - 14 q^{95} - 3 q^{97} - 36 q^{98}+O(q^{100}) \) 16 * q - 16 * q^2 + 16 * q^4 - 4 * q^5 + 6 * q^7 - 16 * q^8 + 4 * q^10 - 6 * q^11 + 6 * q^13 - 6 * q^14 + 16 * q^16 - q^17 + 10 * q^19 - 4 * q^20 + 6 * q^22 - 10 * q^23 + 28 * q^25 - 6 * q^26 + 6 * q^28 - 6 * q^29 + 21 * q^31 - 16 * q^32 + q^34 - 16 * q^35 + 17 * q^37 - 10 * q^38 + 4 * q^40 + 4 * q^41 + 16 * q^43 - 6 * q^44 + 10 * q^46 - 25 * q^47 + 36 * q^49 - 28 * q^50 + 6 * q^52 - 14 * q^53 + 19 * q^55 - 6 * q^56 + 6 * q^58 - 6 * q^59 + 23 * q^61 - 21 * q^62 + 16 * q^64 - 20 * q^65 + 22 * q^67 - q^68 + 16 * q^70 - 10 * q^71 + 16 * q^73 - 17 * q^74 + 10 * q^76 + 2 * q^77 + 37 * q^79 - 4 * q^80 - 4 * q^82 - 33 * q^83 + 43 * q^85 - 16 * q^86 + 6 * q^88 + 3 * q^89 + 28 * q^91 - 10 * q^92 + 25 * q^94 - 14 * q^95 - 3 * q^97 - 36 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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