LMFDB - Embedded newform 6039.2.a.g.1.7

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

mf = mfinit([N,k,chi],0)

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 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("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.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:	$48.2216577807$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} + \cdots - 11 $ x^13 - 4x^12 - 11x^11 + 55x^10 + 32x^9 - 266x^8 + 13x^7 + 534x^6 - 141x^5 - 404x^4 + 98x^3 + 118x^2 - 16x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$0.468970$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

sage: f.q_expansion() # note that sage often uses an isomorphic number field

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.468970 q^{2} -1.78007 q^{4} +3.25579 q^{5} -3.43403 q^{7} +1.77274 q^{8} +O(q^{10})$	\(q-0.468970 q^{2} -1.78007 q^{4} +3.25579 q^{5} -3.43403 q^{7} +1.77274 q^{8} -1.52687 q^{10} +1.00000 q^{11} -6.24548 q^{13} +1.61046 q^{14} +2.72877 q^{16} -4.43634 q^{17} +2.00167 q^{19} -5.79552 q^{20} -0.468970 q^{22} +5.24027 q^{23} +5.60015 q^{25} +2.92895 q^{26} +6.11281 q^{28} -2.01247 q^{29} +2.78091 q^{31} -4.82519 q^{32} +2.08051 q^{34} -11.1805 q^{35} +9.46308 q^{37} -0.938726 q^{38} +5.77166 q^{40} +6.90514 q^{41} +7.06425 q^{43} -1.78007 q^{44} -2.45753 q^{46} +3.28507 q^{47} +4.79257 q^{49} -2.62630 q^{50} +11.1174 q^{52} +3.39708 q^{53} +3.25579 q^{55} -6.08764 q^{56} +0.943788 q^{58} -6.95400 q^{59} -1.00000 q^{61} -1.30417 q^{62} -3.19467 q^{64} -20.3340 q^{65} -13.7103 q^{67} +7.89698 q^{68} +5.24331 q^{70} -11.0077 q^{71} -1.02864 q^{73} -4.43790 q^{74} -3.56312 q^{76} -3.43403 q^{77} -10.9648 q^{79} +8.88430 q^{80} -3.23831 q^{82} -10.7032 q^{83} -14.4438 q^{85} -3.31292 q^{86} +1.77274 q^{88} -5.93188 q^{89} +21.4472 q^{91} -9.32803 q^{92} -1.54060 q^{94} +6.51703 q^{95} +1.30698 q^{97} -2.24757 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8} + 8 q^{10} + 13 q^{11} - 9 q^{13} - 19 q^{14} + 18 q^{16} - 7 q^{17} + 2 q^{19} - 15 q^{20} - 4 q^{22} - 23 q^{23} + 10 q^{25} - 8 q^{26} + 9 q^{28} - 16 q^{29} + 9 q^{31} - 29 q^{32} + 2 q^{34} - 16 q^{35} + 14 q^{37} - 8 q^{38} + 16 q^{40} - 19 q^{41} + 7 q^{43} + 12 q^{44} + 4 q^{46} - 26 q^{47} + 8 q^{49} + 15 q^{50} - 17 q^{52} - 18 q^{53} - 7 q^{55} - 44 q^{56} - q^{58} - 31 q^{59} - 13 q^{61} + 5 q^{62} - 17 q^{64} - 31 q^{65} + 14 q^{67} + 32 q^{68} - 20 q^{70} - 37 q^{71} - 16 q^{73} + 6 q^{74} - 7 q^{76} + 5 q^{77} - 17 q^{79} + 2 q^{80} - 2 q^{82} - 30 q^{83} - 16 q^{85} + 22 q^{86} - 15 q^{88} - 35 q^{89} - q^{91} - 24 q^{92} - 11 q^{94} - 13 q^{95} - q^{97} + q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8 + 8 * q^10 + 13 * q^11 - 9 * q^13 - 19 * q^14 + 18 * q^16 - 7 * q^17 + 2 * q^19 - 15 * q^20 - 4 * q^22 - 23 * q^23 + 10 * q^25 - 8 * q^26 + 9 * q^28 - 16 * q^29 + 9 * q^31 - 29 * q^32 + 2 * q^34 - 16 * q^35 + 14 * q^37 - 8 * q^38 + 16 * q^40 - 19 * q^41 + 7 * q^43 + 12 * q^44 + 4 * q^46 - 26 * q^47 + 8 * q^49 + 15 * q^50 - 17 * q^52 - 18 * q^53 - 7 * q^55 - 44 * q^56 - q^58 - 31 * q^59 - 13 * q^61 + 5 * q^62 - 17 * q^64 - 31 * q^65 + 14 * q^67 + 32 * q^68 - 20 * q^70 - 37 * q^71 - 16 * q^73 + 6 * q^74 - 7 * q^76 + 5 * q^77 - 17 * q^79 + 2 * q^80 - 2 * q^82 - 30 * q^83 - 16 * q^85 + 22 * q^86 - 15 * q^88 - 35 * q^89 - q^91 - 24 * q^92 - 11 * q^94 - 13 * q^95 - q^97 + 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 $

$2$	−0.468970	−0.331612	−0.165806	−	0.986158i	$-0.553023\pi$
$2$	−0.468970	−0.331612	−0.165806	+	0.986158i	$0.553023\pi$
$3$	0	0
$3$	0	0
$4$	−1.78007	−0.890033
$5$	3.25579	1.45603	0.728016	−	0.685560i	$-0.240443\pi$
$5$	3.25579	1.45603	0.728016	+	0.685560i	$0.240443\pi$
$6$	0	0
$7$	−3.43403	−1.29794	−0.648971	−	0.760813i	$-0.724800\pi$
$7$	−3.43403	−1.29794	−0.648971	+	0.760813i	$0.724800\pi$
$8$	1.77274	0.626758
$9$	0	0
$10$	−1.52687	−0.482838
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−6.24548	−1.73219	−0.866093	−	0.499883i	$-0.833376\pi$
$13$	−6.24548	−1.73219	−0.866093	+	0.499883i	$0.833376\pi$
$14$	1.61046	0.430413
$15$	0	0
$16$	2.72877	0.682193
$17$	−4.43634	−1.07597	−0.537985	−	0.842954i	$-0.680814\pi$
$17$	−4.43634	−1.07597	−0.537985	+	0.842954i	$0.680814\pi$
$18$	0	0
$19$	2.00167	0.459216	0.229608	−	0.973283i	$-0.426256\pi$
$19$	2.00167	0.459216	0.229608	+	0.973283i	$0.426256\pi$
$20$	−5.79552	−1.29592
$21$	0	0
$22$	−0.468970	−0.0999848
$23$	5.24027	1.09267	0.546336	−	0.837566i	$-0.316022\pi$
$23$	5.24027	1.09267	0.546336	+	0.837566i	$0.316022\pi$
$24$	0	0
$25$	5.60015	1.12003
$26$	2.92895	0.574413
$27$	0	0
$28$	6.11281	1.15521
$29$	−2.01247	−0.373706	−0.186853	−	0.982388i	$-0.559829\pi$
$29$	−2.01247	−0.373706	−0.186853	+	0.982388i	$0.559829\pi$
$30$	0	0
$31$	2.78091	0.499467	0.249733	−	0.968315i	$-0.419657\pi$
$31$	2.78091	0.499467	0.249733	+	0.968315i	$0.419657\pi$
$32$	−4.82519	−0.852981
$33$	0	0
$34$	2.08051	0.356805
$35$	−11.1805	−1.88985
$36$	0	0
$37$	9.46308	1.55572	0.777860	−	0.628438i	$-0.216305\pi$
$37$	9.46308	1.55572	0.777860	+	0.628438i	$0.216305\pi$
$38$	−0.938726	−0.152281
$39$	0	0
$40$	5.77166	0.912580
$41$	6.90514	1.07840	0.539201	−	0.842177i	$-0.318726\pi$
$41$	6.90514	1.07840	0.539201	+	0.842177i	$0.318726\pi$
$42$	0	0
$43$	7.06425	1.07729	0.538644	−	0.842533i	$-0.318937\pi$
$43$	7.06425	1.07729	0.538644	+	0.842533i	$0.318937\pi$
$44$	−1.78007	−0.268355
$45$	0	0
$46$	−2.45753	−0.362343
$47$	3.28507	0.479177	0.239588	−	0.970875i	$-0.422987\pi$
$47$	3.28507	0.479177	0.239588	+	0.970875i	$0.422987\pi$
$48$	0	0
$49$	4.79257	0.684653
$50$	−2.62630	−0.371416
$51$	0	0
$52$	11.1174	1.54170
$53$	3.39708	0.466625	0.233312	−	0.972402i	$-0.425043\pi$
$53$	3.39708	0.466625	0.233312	+	0.972402i	$0.425043\pi$
$54$	0	0
$55$	3.25579	0.439010
$56$	−6.08764	−0.813495
$57$	0	0
$58$	0.943788	0.123925
$59$	−6.95400	−0.905333	−0.452667	−	0.891680i	$-0.649527\pi$
$59$	−6.95400	−0.905333	−0.452667	+	0.891680i	$0.649527\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−1.30417	−0.165629
$63$	0	0
$64$	−3.19467	−0.399334
$65$	−20.3340	−2.52212
$66$	0	0
$67$	−13.7103	−1.67498	−0.837491	−	0.546452i	$-0.815978\pi$
$67$	−13.7103	−1.67498	−0.837491	+	0.546452i	$0.815978\pi$
$68$	7.89698	0.957650
$69$	0	0
$70$	5.24331	0.626695
$71$	−11.0077	−1.30637	−0.653186	−	0.757197i	$-0.726568\pi$
$71$	−11.0077	−1.30637	−0.653186	+	0.757197i	$0.726568\pi$
$72$	0	0
$73$	−1.02864	−0.120393	−0.0601965	−	0.998187i	$-0.519173\pi$
$73$	−1.02864	−0.120393	−0.0601965	+	0.998187i	$0.519173\pi$
$74$	−4.43790	−0.515895
$75$	0	0
$76$	−3.56312	−0.408717
$77$	−3.43403	−0.391344
$78$	0	0
$79$	−10.9648	−1.23364	−0.616818	−	0.787105i	$-0.711579\pi$
$79$	−10.9648	−1.23364	−0.616818	+	0.787105i	$0.711579\pi$
$80$	8.88430	0.993295
$81$	0	0
$82$	−3.23831	−0.357611
$83$	−10.7032	−1.17483	−0.587415	−	0.809286i	$-0.699854\pi$
$83$	−10.7032	−1.17483	−0.587415	+	0.809286i	$0.699854\pi$
$84$	0	0
$85$	−14.4438	−1.56665
$86$	−3.31292	−0.357242
$87$	0	0
$88$	1.77274	0.188975
$89$	−5.93188	−0.628778	−0.314389	−	0.949294i	$-0.601800\pi$
$89$	−5.93188	−0.628778	−0.314389	+	0.949294i	$0.601800\pi$
$90$	0	0
$91$	21.4472	2.24828
$92$	−9.32803	−0.972515
$93$	0	0
$94$	−1.54060	−0.158901
$95$	6.51703	0.668633
$96$	0	0
$97$	1.30698	0.132704	0.0663519	−	0.997796i	$-0.478864\pi$
$97$	1.30698	0.132704	0.0663519	+	0.997796i	$0.478864\pi$
$98$	−2.24757	−0.227039
$99$	0	0
$100$	−9.96865	−0.996865
$101$	9.38234	0.933578	0.466789	−	0.884369i	$-0.345411\pi$
$101$	9.38234	0.933578	0.466789	+	0.884369i	$0.345411\pi$
$102$	0	0
$103$	3.32165	0.327292	0.163646	−	0.986519i	$-0.447675\pi$
$103$	3.32165	0.327292	0.163646	+	0.986519i	$0.447675\pi$
$104$	−11.0716	−1.08566
$105$	0	0
$106$	−1.59313	−0.154738
$107$	4.48068	0.433163	0.216582	−	0.976265i	$-0.430509\pi$
$107$	4.48068	0.433163	0.216582	+	0.976265i	$0.430509\pi$
$108$	0	0
$109$	−4.81609	−0.461297	−0.230649	−	0.973037i	$-0.574085\pi$
$109$	−4.81609	−0.461297	−0.230649	+	0.973037i	$0.574085\pi$
$110$	−1.52687	−0.145581
$111$	0	0
$112$	−9.37069	−0.885447
$113$	−0.395739	−0.0372280	−0.0186140	−	0.999827i	$-0.505925\pi$
$113$	−0.395739	−0.0372280	−0.0186140	+	0.999827i	$0.505925\pi$
$114$	0	0
$115$	17.0612	1.59097
$116$	3.58233	0.332611
$117$	0	0
$118$	3.26122	0.300219
$119$	15.2345	1.39655
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.468970	0.0424586
$123$	0	0
$124$	−4.95021	−0.444542
$125$	1.95397	0.174769
$126$	0	0
$127$	1.20543	0.106965	0.0534823	−	0.998569i	$-0.482968\pi$
$127$	1.20543	0.106965	0.0534823	+	0.998569i	$0.482968\pi$
$128$	11.1486	0.985405
$129$	0	0
$130$	9.53602	0.836365
$131$	−13.5694	−1.18556	−0.592781	−	0.805364i	$-0.701970\pi$
$131$	−13.5694	−1.18556	−0.592781	+	0.805364i	$0.701970\pi$
$132$	0	0
$133$	−6.87381	−0.596035
$134$	6.42973	0.555444
$135$	0	0
$136$	−7.86447	−0.674373
$137$	−14.7030	−1.25617	−0.628083	−	0.778147i	$-0.716160\pi$
$137$	−14.7030	−1.25617	−0.628083	+	0.778147i	$0.716160\pi$
$138$	0	0
$139$	−9.78375	−0.829847	−0.414923	−	0.909856i	$-0.636192\pi$
$139$	−9.78375	−0.829847	−0.414923	+	0.909856i	$0.636192\pi$
$140$	19.9020	1.68203
$141$	0	0
$142$	5.16228	0.433209
$143$	−6.24548	−0.522274
$144$	0	0
$145$	−6.55218	−0.544129
$146$	0.482401	0.0399238
$147$	0	0
$148$	−16.8449	−1.38464
$149$	4.72991	0.387490	0.193745	−	0.981052i	$-0.437937\pi$
$149$	4.72991	0.387490	0.193745	+	0.981052i	$0.437937\pi$
$150$	0	0
$151$	−2.54722	−0.207290	−0.103645	−	0.994614i	$-0.533051\pi$
$151$	−2.54722	−0.207290	−0.103645	+	0.994614i	$0.533051\pi$
$152$	3.54845	0.287817
$153$	0	0
$154$	1.61046	0.129774
$155$	9.05406	0.727240
$156$	0	0
$157$	10.2105	0.814886	0.407443	−	0.913231i	$-0.366421\pi$
$157$	10.2105	0.814886	0.407443	+	0.913231i	$0.366421\pi$
$158$	5.14217	0.409089
$159$	0	0
$160$	−15.7098	−1.24197
$161$	−17.9953	−1.41822
$162$	0	0
$163$	15.2623	1.19544	0.597718	−	0.801706i	$-0.296074\pi$
$163$	15.2623	1.19544	0.597718	+	0.801706i	$0.296074\pi$
$164$	−12.2916	−0.959814
$165$	0	0
$166$	5.01948	0.389588
$167$	−15.3766	−1.18988	−0.594939	−	0.803771i	$-0.702824\pi$
$167$	−15.3766	−1.18988	−0.594939	+	0.803771i	$0.702824\pi$
$168$	0	0
$169$	26.0061	2.00047
$170$	6.77370	0.519519
$171$	0	0
$172$	−12.5748	−0.958823
$173$	−23.3965	−1.77880	−0.889402	−	0.457126i	$-0.848879\pi$
$173$	−23.3965	−1.77880	−0.889402	+	0.457126i	$0.848879\pi$
$174$	0	0
$175$	−19.2311	−1.45373
$176$	2.72877	0.205689
$177$	0	0
$178$	2.78188	0.208510
$179$	6.26673	0.468397	0.234199	−	0.972189i	$-0.424753\pi$
$179$	6.26673	0.468397	0.234199	+	0.972189i	$0.424753\pi$
$180$	0	0
$181$	−23.6286	−1.75630	−0.878151	−	0.478383i	$-0.841223\pi$
$181$	−23.6286	−1.75630	−0.878151	+	0.478383i	$0.841223\pi$
$182$	−10.0581	−0.745555
$183$	0	0
$184$	9.28963	0.684841
$185$	30.8098	2.26518
$186$	0	0
$187$	−4.43634	−0.324417
$188$	−5.84765	−0.426483
$189$	0	0
$190$	−3.05629	−0.221727
$191$	4.95906	0.358825	0.179413	−	0.983774i	$-0.442580\pi$
$191$	4.95906	0.358825	0.179413	+	0.983774i	$0.442580\pi$
$192$	0	0
$193$	−6.72933	−0.484388	−0.242194	−	0.970228i	$-0.577867\pi$
$193$	−6.72933	−0.484388	−0.242194	+	0.970228i	$0.577867\pi$
$194$	−0.612935	−0.0440061
$195$	0	0
$196$	−8.53110	−0.609364
$197$	−8.95161	−0.637776	−0.318888	−	0.947792i	$-0.603309\pi$
$197$	−8.95161	−0.637776	−0.318888	+	0.947792i	$0.603309\pi$
$198$	0	0
$199$	6.33890	0.449353	0.224676	−	0.974433i	$-0.427868\pi$
$199$	6.33890	0.449353	0.224676	+	0.974433i	$0.427868\pi$
$200$	9.92761	0.701988
$201$	0	0
$202$	−4.40004	−0.309586
$203$	6.91089	0.485049
$204$	0	0
$205$	22.4817	1.57019
$206$	−1.55775	−0.108534
$207$	0	0
$208$	−17.0425	−1.18168
$209$	2.00167	0.138459
$210$	0	0
$211$	−12.2782	−0.845268	−0.422634	−	0.906301i	$-0.638894\pi$
$211$	−12.2782	−0.845268	−0.422634	+	0.906301i	$0.638894\pi$
$212$	−6.04703	−0.415312
$213$	0	0
$214$	−2.10130	−0.143642
$215$	22.9997	1.56857
$216$	0	0
$217$	−9.54974	−0.648279
$218$	2.25860	0.152972
$219$	0	0
$220$	−5.79552	−0.390734
$221$	27.7071	1.86378
$222$	0	0
$223$	−0.874257	−0.0585446	−0.0292723	−	0.999571i	$-0.509319\pi$
$223$	−0.874257	−0.0585446	−0.0292723	+	0.999571i	$0.509319\pi$
$224$	16.5699	1.10712
$225$	0	0
$226$	0.185590	0.0123453
$227$	−10.4092	−0.690885	−0.345443	−	0.938440i	$-0.612271\pi$
$227$	−10.4092	−0.690885	−0.345443	+	0.938440i	$0.612271\pi$
$228$	0	0
$229$	−13.6935	−0.904892	−0.452446	−	0.891792i	$-0.649449\pi$
$229$	−13.6935	−0.904892	−0.452446	+	0.891792i	$0.649449\pi$
$230$	−8.00120	−0.527583
$231$	0	0
$232$	−3.56758	−0.234223
$233$	6.74441	0.441841	0.220921	−	0.975292i	$-0.429094\pi$
$233$	6.74441	0.441841	0.220921	+	0.975292i	$0.429094\pi$
$234$	0	0
$235$	10.6955	0.697697
$236$	12.3786	0.805777
$237$	0	0
$238$	−7.14454	−0.463112
$239$	−19.8593	−1.28459	−0.642295	−	0.766458i	$-0.722018\pi$
$239$	−19.8593	−1.28459	−0.642295	+	0.766458i	$0.722018\pi$
$240$	0	0
$241$	−23.6717	−1.52483	−0.762415	−	0.647089i	$-0.775986\pi$
$241$	−23.6717	−1.52483	−0.762415	+	0.647089i	$0.775986\pi$
$242$	−0.468970	−0.0301465
$243$	0	0
$244$	1.78007	0.113957
$245$	15.6036	0.996877
$246$	0	0
$247$	−12.5014	−0.795447
$248$	4.92983	0.313045
$249$	0	0
$250$	−0.916355	−0.0579554
$251$	22.7517	1.43608	0.718038	−	0.696004i	$-0.245040\pi$
$251$	22.7517	1.43608	0.718038	+	0.696004i	$0.245040\pi$
$252$	0	0
$253$	5.24027	0.329453
$254$	−0.565310	−0.0354707
$255$	0	0
$256$	1.16099	0.0725622
$257$	−5.16686	−0.322300	−0.161150	−	0.986930i	$-0.551520\pi$
$257$	−5.16686	−0.322300	−0.161150	+	0.986930i	$0.551520\pi$
$258$	0	0
$259$	−32.4965	−2.01923
$260$	36.1958	2.24477
$261$	0	0
$262$	6.36363	0.393147
$263$	27.5651	1.69974	0.849869	−	0.526994i	$-0.176681\pi$
$263$	27.5651	1.69974	0.849869	+	0.526994i	$0.176681\pi$
$264$	0	0
$265$	11.0602	0.679421
$266$	3.22361	0.197652
$267$	0	0
$268$	24.4053	1.49079
$269$	−21.8629	−1.33300	−0.666501	−	0.745504i	$-0.732209\pi$
$269$	−21.8629	−1.33300	−0.666501	+	0.745504i	$0.732209\pi$
$270$	0	0
$271$	17.0898	1.03813	0.519067	−	0.854734i	$-0.326280\pi$
$271$	17.0898	1.03813	0.519067	+	0.854734i	$0.326280\pi$
$272$	−12.1058	−0.734020
$273$	0	0
$274$	6.89529	0.416560
$275$	5.60015	0.337702
$276$	0	0
$277$	−11.7573	−0.706425	−0.353213	−	0.935543i	$-0.614911\pi$
$277$	−11.7573	−0.706425	−0.353213	+	0.935543i	$0.614911\pi$
$278$	4.58829	0.275187
$279$	0	0
$280$	−19.8201	−1.18448
$281$	−17.9185	−1.06893	−0.534465	−	0.845190i	$-0.679487\pi$
$281$	−17.9185	−1.06893	−0.534465	+	0.845190i	$0.679487\pi$
$282$	0	0
$283$	27.2207	1.61810	0.809050	−	0.587740i	$-0.199982\pi$
$283$	27.2207	1.61810	0.809050	+	0.587740i	$0.199982\pi$
$284$	19.5944	1.16271
$285$	0	0
$286$	2.92895	0.173192
$287$	−23.7125	−1.39970
$288$	0	0
$289$	2.68111	0.157712
$290$	3.07277	0.180440
$291$	0	0
$292$	1.83104	0.107154
$293$	−2.74534	−0.160385	−0.0801923	−	0.996779i	$-0.525553\pi$
$293$	−2.74534	−0.160385	−0.0801923	+	0.996779i	$0.525553\pi$
$294$	0	0
$295$	−22.6407	−1.31819
$296$	16.7756	0.975060
$297$	0	0
$298$	−2.21819	−0.128496
$299$	−32.7280	−1.89271
$300$	0	0
$301$	−24.2589	−1.39826
$302$	1.19457	0.0687397
$303$	0	0
$304$	5.46211	0.313274
$305$	−3.25579	−0.186426
$306$	0	0
$307$	24.6262	1.40549	0.702745	−	0.711442i	$-0.251958\pi$
$307$	24.6262	1.40549	0.702745	+	0.711442i	$0.251958\pi$
$308$	6.11281	0.348309
$309$	0	0
$310$	−4.24608	−0.241161
$311$	5.12305	0.290502	0.145251	−	0.989395i	$-0.453601\pi$
$311$	5.12305	0.290502	0.145251	+	0.989395i	$0.453601\pi$
$312$	0	0
$313$	6.63621	0.375101	0.187550	−	0.982255i	$-0.439945\pi$
$313$	6.63621	0.375101	0.187550	+	0.982255i	$0.439945\pi$
$314$	−4.78841	−0.270226
$315$	0	0
$316$	19.5181	1.09798
$317$	−21.5260	−1.20902	−0.604511	−	0.796597i	$-0.706631\pi$
$317$	−21.5260	−1.20902	−0.604511	+	0.796597i	$0.706631\pi$
$318$	0	0
$319$	−2.01247	−0.112677
$320$	−10.4012	−0.581444
$321$	0	0
$322$	8.43924	0.470300
$323$	−8.88011	−0.494102
$324$	0	0
$325$	−34.9757	−1.94010
$326$	−7.15757	−0.396421
$327$	0	0
$328$	12.2410	0.675897
$329$	−11.2810	−0.621944
$330$	0	0
$331$	−31.8784	−1.75219	−0.876097	−	0.482134i	$-0.839862\pi$
$331$	−31.8784	−1.75219	−0.876097	+	0.482134i	$0.839862\pi$
$332$	19.0524	1.04564
$333$	0	0
$334$	7.21118	0.394578
$335$	−44.6379	−2.43883
$336$	0	0
$337$	14.3893	0.783836	0.391918	−	0.920000i	$-0.371812\pi$
$337$	14.3893	0.783836	0.391918	+	0.920000i	$0.371812\pi$
$338$	−12.1961	−0.663379
$339$	0	0
$340$	25.7109	1.39437
$341$	2.78091	0.150595
$342$	0	0
$343$	7.58038	0.409302
$344$	12.5231	0.675199
$345$	0	0
$346$	10.9723	0.589873
$347$	8.27031	0.443973	0.221987	−	0.975050i	$-0.428746\pi$
$347$	8.27031	0.443973	0.221987	+	0.975050i	$0.428746\pi$
$348$	0	0
$349$	−7.70568	−0.412476	−0.206238	−	0.978502i	$-0.566122\pi$
$349$	−7.70568	−0.412476	−0.206238	+	0.978502i	$0.566122\pi$
$350$	9.01881	0.482076
$351$	0	0
$352$	−4.82519	−0.257183
$353$	−19.5302	−1.03949	−0.519743	−	0.854323i	$-0.673972\pi$
$353$	−19.5302	−1.03949	−0.519743	+	0.854323i	$0.673972\pi$
$354$	0	0
$355$	−35.8387	−1.90212
$356$	10.5591	0.559634
$357$	0	0
$358$	−2.93891	−0.155326
$359$	−31.5005	−1.66254	−0.831268	−	0.555873i	$-0.812384\pi$
$359$	−31.5005	−1.66254	−0.831268	+	0.555873i	$0.812384\pi$
$360$	0	0
$361$	−14.9933	−0.789121
$362$	11.0811	0.582411
$363$	0	0
$364$	−38.1774	−2.00104
$365$	−3.34903	−0.175296
$366$	0	0
$367$	18.6783	0.975002	0.487501	−	0.873122i	$-0.337909\pi$
$367$	18.6783	0.975002	0.487501	+	0.873122i	$0.337909\pi$
$368$	14.2995	0.745413
$369$	0	0
$370$	−14.4489	−0.751161
$371$	−11.6657	−0.605652
$372$	0	0
$373$	18.7522	0.970954	0.485477	−	0.874250i	$-0.338646\pi$
$373$	18.7522	0.970954	0.485477	+	0.874250i	$0.338646\pi$
$374$	2.08051	0.107581
$375$	0	0
$376$	5.82357	0.300328
$377$	12.5688	0.647329
$378$	0	0
$379$	11.9393	0.613280	0.306640	−	0.951826i	$-0.400795\pi$
$379$	11.9393	0.613280	0.306640	+	0.951826i	$0.400795\pi$
$380$	−11.6007	−0.595106
$381$	0	0
$382$	−2.32565	−0.118991
$383$	−29.2370	−1.49394	−0.746971	−	0.664856i	$-0.768493\pi$
$383$	−29.2370	−1.49394	−0.746971	+	0.664856i	$0.768493\pi$
$384$	0	0
$385$	−11.1805	−0.569810
$386$	3.15585	0.160629
$387$	0	0
$388$	−2.32651	−0.118111
$389$	−37.0444	−1.87823	−0.939114	−	0.343606i	$-0.888352\pi$
$389$	−37.0444	−1.87823	−0.939114	+	0.343606i	$0.888352\pi$
$390$	0	0
$391$	−23.2476	−1.17568
$392$	8.49597	0.429112
$393$	0	0
$394$	4.19804	0.211494
$395$	−35.6991	−1.79622
$396$	0	0
$397$	34.3833	1.72565	0.862825	−	0.505503i	$-0.168693\pi$
$397$	34.3833	1.72565	0.862825	+	0.505503i	$0.168693\pi$
$398$	−2.97275	−0.149011
$399$	0	0
$400$	15.2815	0.764077
$401$	−6.10544	−0.304891	−0.152446	−	0.988312i	$-0.548715\pi$
$401$	−6.10544	−0.304891	−0.152446	+	0.988312i	$0.548715\pi$
$402$	0	0
$403$	−17.3681	−0.865169
$404$	−16.7012	−0.830915
$405$	0	0
$406$	−3.24100	−0.160848
$407$	9.46308	0.469067
$408$	0	0
$409$	−28.6700	−1.41764	−0.708819	−	0.705390i	$-0.750772\pi$
$409$	−28.6700	−1.41764	−0.708819	+	0.705390i	$0.750772\pi$
$410$	−10.5432	−0.520693
$411$	0	0
$412$	−5.91276	−0.291301
$413$	23.8802	1.17507
$414$	0	0
$415$	−34.8474	−1.71059
$416$	30.1356	1.47752
$417$	0	0
$418$	−0.938726	−0.0459146
$419$	20.7874	1.01553	0.507766	−	0.861495i	$-0.330471\pi$
$419$	20.7874	1.01553	0.507766	+	0.861495i	$0.330471\pi$
$420$	0	0
$421$	−8.10045	−0.394792	−0.197396	−	0.980324i	$-0.563248\pi$
$421$	−8.10045	−0.394792	−0.197396	+	0.980324i	$0.563248\pi$
$422$	5.75812	0.280301
$423$	0	0
$424$	6.02214	0.292461
$425$	−24.8442	−1.20512
$426$	0	0
$427$	3.43403	0.166184
$428$	−7.97590	−0.385530
$429$	0	0
$430$	−10.7862	−0.520156
$431$	5.58219	0.268885	0.134442	−	0.990921i	$-0.457076\pi$
$431$	5.58219	0.268885	0.134442	+	0.990921i	$0.457076\pi$
$432$	0	0
$433$	29.3293	1.40948	0.704739	−	0.709467i	$-0.251064\pi$
$433$	29.3293	1.40948	0.704739	+	0.709467i	$0.251064\pi$
$434$	4.47854	0.214977
$435$	0	0
$436$	8.57296	0.410570
$437$	10.4893	0.501772
$438$	0	0
$439$	−25.9500	−1.23852	−0.619262	−	0.785184i	$-0.712568\pi$
$439$	−25.9500	−1.23852	−0.619262	+	0.785184i	$0.712568\pi$
$440$	5.77166	0.275153
$441$	0	0
$442$	−12.9938	−0.618052
$443$	34.4019	1.63448	0.817241	−	0.576296i	$-0.195503\pi$
$443$	34.4019	1.63448	0.817241	+	0.576296i	$0.195503\pi$
$444$	0	0
$445$	−19.3129	−0.915521
$446$	0.410001	0.0194141
$447$	0	0
$448$	10.9706	0.518313
$449$	−36.2156	−1.70912	−0.854561	−	0.519351i	$-0.826174\pi$
$449$	−36.2156	−1.70912	−0.854561	+	0.519351i	$0.826174\pi$
$450$	0	0
$451$	6.90514	0.325150
$452$	0.704442	0.0331342
$453$	0	0
$454$	4.88162	0.229106
$455$	69.8275	3.27356
$456$	0	0
$457$	23.2923	1.08957	0.544783	−	0.838577i	$-0.316612\pi$
$457$	23.2923	1.08957	0.544783	+	0.838577i	$0.316612\pi$
$458$	6.42184	0.300073
$459$	0	0
$460$	−30.3701	−1.41601
$461$	28.2651	1.31644	0.658219	−	0.752827i	$-0.271310\pi$
$461$	28.2651	1.31644	0.658219	+	0.752827i	$0.271310\pi$
$462$	0	0
$463$	−32.3472	−1.50330	−0.751651	−	0.659561i	$-0.770742\pi$
$463$	−32.3472	−1.50330	−0.751651	+	0.659561i	$0.770742\pi$
$464$	−5.49157	−0.254940
$465$	0	0
$466$	−3.16293	−0.146520
$467$	3.30570	0.152969	0.0764847	−	0.997071i	$-0.475630\pi$
$467$	3.30570	0.152969	0.0764847	+	0.997071i	$0.475630\pi$
$468$	0	0
$469$	47.0816	2.17403
$470$	−5.01587	−0.231365
$471$	0	0
$472$	−12.3276	−0.567425
$473$	7.06425	0.324815
$474$	0	0
$475$	11.2097	0.514336
$476$	−27.1185	−1.24297
$477$	0	0
$478$	9.31341	0.425985
$479$	10.4588	0.477875	0.238937	−	0.971035i	$-0.423201\pi$
$479$	10.4588	0.477875	0.238937	+	0.971035i	$0.423201\pi$
$480$	0	0
$481$	−59.1015	−2.69480
$482$	11.1013	0.505652
$483$	0	0
$484$	−1.78007	−0.0809121
$485$	4.25525	0.193221
$486$	0	0
$487$	19.8688	0.900340	0.450170	−	0.892943i	$-0.351363\pi$
$487$	19.8688	0.900340	0.450170	+	0.892943i	$0.351363\pi$
$488$	−1.77274	−0.0802481
$489$	0	0
$490$	−7.31762	−0.330576
$491$	−33.9034	−1.53004	−0.765020	−	0.644007i	$-0.777271\pi$
$491$	−33.9034	−1.53004	−0.765020	+	0.644007i	$0.777271\pi$
$492$	0	0
$493$	8.92800	0.402097
$494$	5.86280	0.263780
$495$	0	0
$496$	7.58848	0.340733
$497$	37.8007	1.69560
$498$	0	0
$499$	−9.07523	−0.406263	−0.203131	−	0.979151i	$-0.565112\pi$
$499$	−9.07523	−0.406263	−0.203131	+	0.979151i	$0.565112\pi$
$500$	−3.47820	−0.155550
$501$	0	0
$502$	−10.6699	−0.476220
$503$	13.7668	0.613830	0.306915	−	0.951737i	$-0.400703\pi$
$503$	13.7668	0.613830	0.306915	+	0.951737i	$0.400703\pi$
$504$	0	0
$505$	30.5469	1.35932
$506$	−2.45753	−0.109251
$507$	0	0
$508$	−2.14574	−0.0952020
$509$	−27.0557	−1.19922	−0.599611	−	0.800292i	$-0.704678\pi$
$509$	−27.0557	−1.19922	−0.599611	+	0.800292i	$0.704678\pi$
$510$	0	0
$511$	3.53238	0.156263
$512$	−22.8416	−1.00947
$513$	0	0
$514$	2.42310	0.106878
$515$	10.8146	0.476548
$516$	0	0
$517$	3.28507	0.144477
$518$	15.2399	0.669602
$519$	0	0
$520$	−36.0468	−1.58076
$521$	39.6985	1.73922	0.869611	−	0.493737i	$-0.164369\pi$
$521$	39.6985	1.73922	0.869611	+	0.493737i	$0.164369\pi$
$522$	0	0
$523$	−25.4026	−1.11078	−0.555390	−	0.831590i	$-0.687431\pi$
$523$	−25.4026	−1.11078	−0.555390	+	0.831590i	$0.687431\pi$
$524$	24.1544	1.05519
$525$	0	0
$526$	−12.9272	−0.563653
$527$	−12.3371	−0.537411
$528$	0	0
$529$	4.46043	0.193932
$530$	−5.18689	−0.225304
$531$	0	0
$532$	12.2358	0.530491
$533$	−43.1260	−1.86799
$534$	0	0
$535$	14.5881	0.630700
$536$	−24.3048	−1.04981
$537$	0	0
$538$	10.2530	0.442039
$539$	4.79257	0.206431
$540$	0	0
$541$	−29.6053	−1.27283	−0.636415	−	0.771347i	$-0.719583\pi$
$541$	−29.6053	−1.27283	−0.636415	+	0.771347i	$0.719583\pi$
$542$	−8.01462	−0.344258
$543$	0	0
$544$	21.4062	0.917782
$545$	−15.6802	−0.671664
$546$	0	0
$547$	6.80847	0.291109	0.145555	−	0.989350i	$-0.453503\pi$
$547$	6.80847	0.291109	0.145555	+	0.989350i	$0.453503\pi$
$548$	26.1724	1.11803
$549$	0	0
$550$	−2.62630	−0.111986
$551$	−4.02831	−0.171612
$552$	0	0
$553$	37.6535	1.60119
$554$	5.51381	0.234259
$555$	0	0
$556$	17.4157	0.738592
$557$	31.2260	1.32309	0.661544	−	0.749906i	$-0.269901\pi$
$557$	31.2260	1.32309	0.661544	+	0.749906i	$0.269901\pi$
$558$	0	0
$559$	−44.1197	−1.86606
$560$	−30.5090	−1.28924
$561$	0	0
$562$	8.40326	0.354470
$563$	−39.1364	−1.64940	−0.824701	−	0.565570i	$-0.808656\pi$
$563$	−39.1364	−1.64940	−0.824701	+	0.565570i	$0.808656\pi$
$564$	0	0
$565$	−1.28844	−0.0542052
$566$	−12.7657	−0.536581
$567$	0	0
$568$	−19.5138	−0.818779
$569$	6.79224	0.284746	0.142373	−	0.989813i	$-0.454527\pi$
$569$	6.79224	0.284746	0.142373	+	0.989813i	$0.454527\pi$
$570$	0	0
$571$	30.3770	1.27124	0.635618	−	0.772004i	$-0.280745\pi$
$571$	30.3770	1.27124	0.635618	+	0.772004i	$0.280745\pi$
$572$	11.1174	0.464841
$573$	0	0
$574$	11.1204	0.464158
$575$	29.3463	1.22383
$576$	0	0
$577$	−8.71413	−0.362774	−0.181387	−	0.983412i	$-0.558059\pi$
$577$	−8.71413	−0.362774	−0.181387	+	0.983412i	$0.558059\pi$
$578$	−1.25736	−0.0522992
$579$	0	0
$580$	11.6633	0.484293
$581$	36.7551	1.52486
$582$	0	0
$583$	3.39708	0.140693
$584$	−1.82351	−0.0754572
$585$	0	0
$586$	1.28748	0.0531855
$587$	−33.8989	−1.39916	−0.699580	−	0.714555i	$-0.746629\pi$
$587$	−33.8989	−1.39916	−0.699580	+	0.714555i	$0.746629\pi$
$588$	0	0
$589$	5.56648	0.229363
$590$	10.6178	0.437129
$591$	0	0
$592$	25.8226	1.06130
$593$	11.0493	0.453740	0.226870	−	0.973925i	$-0.427151\pi$
$593$	11.0493	0.453740	0.226870	+	0.973925i	$0.427151\pi$
$594$	0	0
$595$	49.6004	2.03342
$596$	−8.41957	−0.344879
$597$	0	0
$598$	15.3485	0.627645
$599$	−40.5277	−1.65592	−0.827958	−	0.560791i	$-0.810497\pi$
$599$	−40.5277	−1.65592	−0.827958	+	0.560791i	$0.810497\pi$
$600$	0	0
$601$	−3.05022	−0.124421	−0.0622106	−	0.998063i	$-0.519815\pi$
$601$	−3.05022	−0.124421	−0.0622106	+	0.998063i	$0.519815\pi$
$602$	11.3767	0.463679
$603$	0	0
$604$	4.53422	0.184495
$605$	3.25579	0.132367
$606$	0	0
$607$	−38.5560	−1.56494	−0.782469	−	0.622690i	$-0.786040\pi$
$607$	−38.5560	−1.56494	−0.782469	+	0.622690i	$0.786040\pi$
$608$	−9.65846	−0.391702
$609$	0	0
$610$	1.52687	0.0618210
$611$	−20.5169	−0.830023
$612$	0	0
$613$	−36.0475	−1.45594	−0.727972	−	0.685607i	$-0.759537\pi$
$613$	−36.0475	−1.45594	−0.727972	+	0.685607i	$0.759537\pi$
$614$	−11.5489	−0.466077
$615$	0	0
$616$	−6.08764	−0.245278
$617$	−20.1078	−0.809510	−0.404755	−	0.914425i	$-0.632643\pi$
$617$	−20.1078	−0.809510	−0.404755	+	0.914425i	$0.632643\pi$
$618$	0	0
$619$	46.0006	1.84892	0.924460	−	0.381280i	$-0.124516\pi$
$619$	46.0006	1.84892	0.924460	+	0.381280i	$0.124516\pi$
$620$	−16.1168	−0.647268
$621$	0	0
$622$	−2.40256	−0.0963338
$623$	20.3703	0.816117
$624$	0	0
$625$	−21.6390	−0.865562
$626$	−3.11219	−0.124388
$627$	0	0
$628$	−18.1754	−0.725276
$629$	−41.9814	−1.67391
$630$	0	0
$631$	−41.3031	−1.64425	−0.822124	−	0.569308i	$-0.807211\pi$
$631$	−41.3031	−1.64425	−0.822124	+	0.569308i	$0.807211\pi$
$632$	−19.4377	−0.773191
$633$	0	0
$634$	10.0951	0.400926
$635$	3.92462	0.155744
$636$	0	0
$637$	−29.9319	−1.18595
$638$	0.943788	0.0373649
$639$	0	0
$640$	36.2974	1.43478
$641$	9.05117	0.357500	0.178750	−	0.983895i	$-0.442795\pi$
$641$	9.05117	0.357500	0.178750	+	0.983895i	$0.442795\pi$
$642$	0	0
$643$	40.7525	1.60712	0.803562	−	0.595221i	$-0.202936\pi$
$643$	40.7525	1.60712	0.803562	+	0.595221i	$0.202936\pi$
$644$	32.0328	1.26227
$645$	0	0
$646$	4.16451	0.163850
$647$	28.8295	1.13340	0.566702	−	0.823923i	$-0.308219\pi$
$647$	28.8295	1.13340	0.566702	+	0.823923i	$0.308219\pi$
$648$	0	0
$649$	−6.95400	−0.272968
$650$	16.4025	0.643361
$651$	0	0
$652$	−27.1679	−1.06398
$653$	−17.2920	−0.676690	−0.338345	−	0.941022i	$-0.609867\pi$
$653$	−17.2920	−0.676690	−0.338345	+	0.941022i	$0.609867\pi$
$654$	0	0
$655$	−44.1790	−1.72622
$656$	18.8426	0.735678
$657$	0	0
$658$	5.29047	0.206244
$659$	7.68996	0.299558	0.149779	−	0.988719i	$-0.452144\pi$
$659$	7.68996	0.299558	0.149779	+	0.988719i	$0.452144\pi$
$660$	0	0
$661$	5.25986	0.204585	0.102292	−	0.994754i	$-0.467382\pi$
$661$	5.25986	0.204585	0.102292	+	0.994754i	$0.467382\pi$
$662$	14.9500	0.581049
$663$	0	0
$664$	−18.9740	−0.736334
$665$	−22.3797	−0.867847
$666$	0	0
$667$	−10.5459	−0.408338
$668$	27.3714	1.05903
$669$	0	0
$670$	20.9338	0.808744
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−42.5265	−1.63928	−0.819638	−	0.572881i	$-0.805826\pi$
$673$	−42.5265	−1.63928	−0.819638	+	0.572881i	$0.805826\pi$
$674$	−6.74816	−0.259930
$675$	0	0
$676$	−46.2925	−1.78048
$677$	21.8036	0.837982	0.418991	−	0.907990i	$-0.362384\pi$
$677$	21.8036	0.837982	0.418991	+	0.907990i	$0.362384\pi$
$678$	0	0
$679$	−4.48821	−0.172242
$680$	−25.6050	−0.981909
$681$	0	0
$682$	−1.30417	−0.0499391
$683$	3.87219	0.148165	0.0740826	−	0.997252i	$-0.476397\pi$
$683$	3.87219	0.148165	0.0740826	+	0.997252i	$0.476397\pi$
$684$	0	0
$685$	−47.8700	−1.82902
$686$	−3.55497	−0.135730
$687$	0	0
$688$	19.2767	0.734919
$689$	−21.2164	−0.808281
$690$	0	0
$691$	26.2173	0.997353	0.498677	−	0.866788i	$-0.333820\pi$
$691$	26.2173	0.997353	0.498677	+	0.866788i	$0.333820\pi$
$692$	41.6474	1.58320
$693$	0	0
$694$	−3.87853	−0.147227
$695$	−31.8538	−1.20828
$696$	0	0
$697$	−30.6336	−1.16033
$698$	3.61374	0.136782
$699$	0	0
$700$	34.2327	1.29387
$701$	−14.6365	−0.552812	−0.276406	−	0.961041i	$-0.589143\pi$
$701$	−14.6365	−0.552812	−0.276406	+	0.961041i	$0.589143\pi$
$702$	0	0
$703$	18.9420	0.714411
$704$	−3.19467	−0.120404
$705$	0	0
$706$	9.15907	0.344706
$707$	−32.2193	−1.21173
$708$	0	0
$709$	−18.9404	−0.711321	−0.355661	−	0.934615i	$-0.615744\pi$
$709$	−18.9404	−0.711321	−0.355661	+	0.934615i	$0.615744\pi$
$710$	16.8073	0.630766
$711$	0	0
$712$	−10.5157	−0.394092
$713$	14.5727	0.545753
$714$	0	0
$715$	−20.3340	−0.760447
$716$	−11.1552	−0.416889
$717$	0	0
$718$	14.7728	0.551317
$719$	10.5219	0.392399	0.196199	−	0.980564i	$-0.437140\pi$
$719$	10.5219	0.392399	0.196199	+	0.980564i	$0.437140\pi$
$720$	0	0
$721$	−11.4067	−0.424806
$722$	7.03141	0.261682
$723$	0	0
$724$	42.0606	1.56317
$725$	−11.2701	−0.418563
$726$	0	0
$727$	−49.5012	−1.83590	−0.917949	−	0.396699i	$-0.870156\pi$
$727$	−49.5012	−1.83590	−0.917949	+	0.396699i	$0.870156\pi$
$728$	38.0203	1.40912
$729$	0	0
$730$	1.57059	0.0581303
$731$	−31.3394	−1.15913
$732$	0	0
$733$	28.5502	1.05453	0.527263	−	0.849702i	$-0.323218\pi$
$733$	28.5502	1.05453	0.527263	+	0.849702i	$0.323218\pi$
$734$	−8.75959	−0.323322
$735$	0	0
$736$	−25.2853	−0.932029
$737$	−13.7103	−0.505026
$738$	0	0
$739$	21.0440	0.774115	0.387057	−	0.922056i	$-0.373492\pi$
$739$	21.0440	0.774115	0.387057	+	0.922056i	$0.373492\pi$
$740$	−54.8434	−2.01609
$741$	0	0
$742$	5.47086	0.200841
$743$	17.4017	0.638405	0.319202	−	0.947687i	$-0.396585\pi$
$743$	17.4017	0.638405	0.319202	+	0.947687i	$0.396585\pi$
$744$	0	0
$745$	15.3996	0.564198
$746$	−8.79424	−0.321980
$747$	0	0
$748$	7.89698	0.288742
$749$	−15.3868	−0.562221
$750$	0	0
$751$	23.4793	0.856773	0.428387	−	0.903596i	$-0.359082\pi$
$751$	23.4793	0.856773	0.428387	+	0.903596i	$0.359082\pi$
$752$	8.96421	0.326891
$753$	0	0
$754$	−5.89441	−0.214662
$755$	−8.29320	−0.301820
$756$	0	0
$757$	−18.9342	−0.688174	−0.344087	−	0.938938i	$-0.611812\pi$
$757$	−18.9342	−0.688174	−0.344087	+	0.938938i	$0.611812\pi$
$758$	−5.59917	−0.203371
$759$	0	0
$760$	11.5530	0.419071
$761$	6.95761	0.252213	0.126106	−	0.992017i	$-0.459752\pi$
$761$	6.95761	0.252213	0.126106	+	0.992017i	$0.459752\pi$
$762$	0	0
$763$	16.5386	0.598737
$764$	−8.82747	−0.319367
$765$	0	0
$766$	13.7113	0.495409
$767$	43.4311	1.56821
$768$	0	0
$769$	−22.1075	−0.797217	−0.398609	−	0.917121i	$-0.630507\pi$
$769$	−22.1075	−0.797217	−0.398609	+	0.917121i	$0.630507\pi$
$770$	5.24331	0.188956
$771$	0	0
$772$	11.9787	0.431121
$773$	−22.7271	−0.817437	−0.408718	−	0.912661i	$-0.634024\pi$
$773$	−22.7271	−0.817437	−0.408718	+	0.912661i	$0.634024\pi$
$774$	0	0
$775$	15.5735	0.559418
$776$	2.31693	0.0831731
$777$	0	0
$778$	17.3727	0.622843
$779$	13.8218	0.495219
$780$	0	0
$781$	−11.0077	−0.393886
$782$	10.9024	0.389870
$783$	0	0
$784$	13.0778	0.467065
$785$	33.2432	1.18650
$786$	0	0
$787$	0.0752940	0.00268394	0.00134197	−	0.999999i	$-0.499573\pi$
$787$	0.0752940	0.00268394	0.00134197	+	0.999999i	$0.499573\pi$
$788$	15.9345	0.567642
$789$	0	0
$790$	16.7418	0.595647
$791$	1.35898	0.0483198
$792$	0	0
$793$	6.24548	0.221784
$794$	−16.1248	−0.572246
$795$	0	0
$796$	−11.2837	−0.399939
$797$	−22.0267	−0.780225	−0.390112	−	0.920767i	$-0.627564\pi$
$797$	−22.0267	−0.780225	−0.390112	+	0.920767i	$0.627564\pi$
$798$	0	0
$799$	−14.5737	−0.515580
$800$	−27.0218	−0.955365
$801$	0	0
$802$	2.86327	0.101106
$803$	−1.02864	−0.0362998
$804$	0	0
$805$	−58.5887	−2.06498
$806$	8.14514	0.286900
$807$	0	0
$808$	16.6324	0.585127
$809$	−18.2011	−0.639918	−0.319959	−	0.947431i	$-0.603669\pi$
$809$	−18.2011	−0.639918	−0.319959	+	0.947431i	$0.603669\pi$
$810$	0	0
$811$	51.6355	1.81317	0.906584	−	0.422025i	$-0.138680\pi$
$811$	51.6355	1.81317	0.906584	+	0.422025i	$0.138680\pi$
$812$	−12.3018	−0.431710
$813$	0	0
$814$	−4.43790	−0.155548
$815$	49.6908	1.74059
$816$	0	0
$817$	14.1403	0.494708
$818$	13.4454	0.470106
$819$	0	0
$820$	−40.0189	−1.39752
$821$	1.06685	0.0372334	0.0186167	−	0.999827i	$-0.494074\pi$
$821$	1.06685	0.0372334	0.0186167	+	0.999827i	$0.494074\pi$
$822$	0	0
$823$	41.1805	1.43546	0.717731	−	0.696321i	$-0.245181\pi$
$823$	41.1805	1.43546	0.717731	+	0.696321i	$0.245181\pi$
$824$	5.88842	0.205133
$825$	0	0
$826$	−11.1991	−0.389667
$827$	−6.70863	−0.233282	−0.116641	−	0.993174i	$-0.537213\pi$
$827$	−6.70863	−0.233282	−0.116641	+	0.993174i	$0.537213\pi$
$828$	0	0
$829$	50.0952	1.73988	0.869939	−	0.493159i	$-0.164158\pi$
$829$	50.0952	1.73988	0.869939	+	0.493159i	$0.164158\pi$
$830$	16.3424	0.567252
$831$	0	0
$832$	19.9523	0.691721
$833$	−21.2615	−0.736666
$834$	0	0
$835$	−50.0630	−1.73250
$836$	−3.56312	−0.123233
$837$	0	0
$838$	−9.74868	−0.336763
$839$	−16.6741	−0.575655	−0.287827	−	0.957682i	$-0.592933\pi$
$839$	−16.6741	−0.575655	−0.287827	+	0.957682i	$0.592933\pi$
$840$	0	0
$841$	−24.9500	−0.860344
$842$	3.79887	0.130918
$843$	0	0
$844$	21.8561	0.752317
$845$	84.6702	2.91274
$846$	0	0
$847$	−3.43403	−0.117995
$848$	9.26986	0.318328
$849$	0	0
$850$	11.6512	0.399632
$851$	49.5891	1.69989
$852$	0	0
$853$	37.6327	1.28852	0.644259	−	0.764807i	$-0.277166\pi$
$853$	37.6327	1.28852	0.644259	+	0.764807i	$0.277166\pi$
$854$	−1.61046	−0.0551087
$855$	0	0
$856$	7.94307	0.271488
$857$	−37.9851	−1.29755	−0.648773	−	0.760982i	$-0.724718\pi$
$857$	−37.9851	−1.29755	−0.648773	+	0.760982i	$0.724718\pi$
$858$	0	0
$859$	−4.26900	−0.145656	−0.0728282	−	0.997345i	$-0.523202\pi$
$859$	−4.26900	−0.145656	−0.0728282	+	0.997345i	$0.523202\pi$
$860$	−40.9410	−1.39608
$861$	0	0
$862$	−2.61788	−0.0891654
$863$	5.50516	0.187398	0.0936990	−	0.995601i	$-0.470131\pi$
$863$	5.50516	0.187398	0.0936990	+	0.995601i	$0.470131\pi$
$864$	0	0
$865$	−76.1741	−2.59000
$866$	−13.7546	−0.467399
$867$	0	0
$868$	16.9992	0.576990
$869$	−10.9648	−0.371956
$870$	0	0
$871$	85.6275	2.90138
$872$	−8.53766	−0.289122
$873$	0	0
$874$	−4.91918	−0.166394
$875$	−6.71000	−0.226840
$876$	0	0
$877$	−37.6292	−1.27065	−0.635324	−	0.772246i	$-0.719133\pi$
$877$	−37.6292	−1.27065	−0.635324	+	0.772246i	$0.719133\pi$
$878$	12.1698	0.410709
$879$	0	0
$880$	8.88430	0.299490
$881$	−25.6031	−0.862589	−0.431295	−	0.902211i	$-0.641943\pi$
$881$	−25.6031	−0.862589	−0.431295	+	0.902211i	$0.641943\pi$
$882$	0	0
$883$	16.3426	0.549972	0.274986	−	0.961448i	$-0.411327\pi$
$883$	16.3426	0.549972	0.274986	+	0.961448i	$0.411327\pi$
$884$	−49.3205	−1.65883
$885$	0	0
$886$	−16.1334	−0.542014
$887$	4.22569	0.141885	0.0709425	−	0.997480i	$-0.477399\pi$
$887$	4.22569	0.141885	0.0709425	+	0.997480i	$0.477399\pi$
$888$	0	0
$889$	−4.13948	−0.138834
$890$	9.05720	0.303598
$891$	0	0
$892$	1.55624	0.0521067
$893$	6.57564	0.220046
$894$	0	0
$895$	20.4031	0.682002
$896$	−38.2846	−1.27900
$897$	0	0
$898$	16.9841	0.566765
$899$	−5.59650	−0.186654
$900$	0	0
$901$	−15.0706	−0.502075
$902$	−3.23831	−0.107824
$903$	0	0
$904$	−0.701542	−0.0233329
$905$	−76.9298	−2.55723
$906$	0	0
$907$	4.49047	0.149104	0.0745519	−	0.997217i	$-0.476247\pi$
$907$	4.49047	0.149104	0.0745519	+	0.997217i	$0.476247\pi$
$908$	18.5291	0.614911
$909$	0	0
$910$	−32.7470	−1.08555
$911$	6.20158	0.205468	0.102734	−	0.994709i	$-0.467241\pi$
$911$	6.20158	0.205468	0.102734	+	0.994709i	$0.467241\pi$
$912$	0	0
$913$	−10.7032	−0.354224
$914$	−10.9234	−0.361313
$915$	0	0
$916$	24.3754	0.805384
$917$	46.5977	1.53879
$918$	0	0
$919$	57.4546	1.89525	0.947627	−	0.319380i	$-0.103475\pi$
$919$	57.4546	1.89525	0.947627	+	0.319380i	$0.103475\pi$
$920$	30.2451	0.997150
$921$	0	0
$922$	−13.2555	−0.436546
$923$	68.7483	2.26288
$924$	0	0
$925$	52.9947	1.74245
$926$	15.1699	0.498513
$927$	0	0
$928$	9.71055	0.318764
$929$	14.0361	0.460511	0.230255	−	0.973130i	$-0.426044\pi$
$929$	14.0361	0.460511	0.230255	+	0.973130i	$0.426044\pi$
$930$	0	0
$931$	9.59317	0.314403
$932$	−12.0055	−0.393253
$933$	0	0
$934$	−1.55027	−0.0507265
$935$	−14.4438	−0.472362
$936$	0	0
$937$	28.3710	0.926840	0.463420	−	0.886139i	$-0.346622\pi$
$937$	28.3710	0.926840	0.463420	+	0.886139i	$0.346622\pi$
$938$	−22.0799	−0.720934
$939$	0	0
$940$	−19.0387	−0.620974
$941$	41.8840	1.36538	0.682690	−	0.730708i	$-0.260810\pi$
$941$	41.8840	1.36538	0.682690	+	0.730708i	$0.260810\pi$
$942$	0	0
$943$	36.1848	1.17834
$944$	−18.9759	−0.617612
$945$	0	0
$946$	−3.31292	−0.107712
$947$	−41.3592	−1.34399	−0.671996	−	0.740555i	$-0.734563\pi$
$947$	−41.3592	−1.34399	−0.671996	+	0.740555i	$0.734563\pi$
$948$	0	0
$949$	6.42434	0.208543
$950$	−5.25701	−0.170560
$951$	0	0
$952$	27.0068	0.875297
$953$	41.6088	1.34784	0.673920	−	0.738804i	$-0.264609\pi$
$953$	41.6088	1.34784	0.673920	+	0.738804i	$0.264609\pi$
$954$	0	0
$955$	16.1457	0.522461
$956$	35.3508	1.14333
$957$	0	0
$958$	−4.90486	−0.158469
$959$	50.4907	1.63043
$960$	0	0
$961$	−23.2665	−0.750533
$962$	27.7168	0.893627
$963$	0	0
$964$	42.1372	1.35715
$965$	−21.9093	−0.705284
$966$	0	0
$967$	−21.0341	−0.676410	−0.338205	−	0.941072i	$-0.609820\pi$
$967$	−21.0341	−0.676410	−0.338205	+	0.941072i	$0.609820\pi$
$968$	1.77274	0.0569780
$969$	0	0
$970$	−1.99559	−0.0640744
$971$	47.5838	1.52704	0.763518	−	0.645786i	$-0.223470\pi$
$971$	47.5838	1.52704	0.763518	+	0.645786i	$0.223470\pi$
$972$	0	0
$973$	33.5977	1.07709
$974$	−9.31786	−0.298564
$975$	0	0
$976$	−2.72877	−0.0873459
$977$	−4.54384	−0.145370	−0.0726852	−	0.997355i	$-0.523157\pi$
$977$	−4.54384	−0.145370	−0.0726852	+	0.997355i	$0.523157\pi$
$978$	0	0
$979$	−5.93188	−0.189584
$980$	−27.7754	−0.887254
$981$	0	0
$982$	15.8997	0.507379
$983$	7.20373	0.229763	0.114882	−	0.993379i	$-0.463351\pi$
$983$	7.20373	0.229763	0.114882	+	0.993379i	$0.463351\pi$
$984$	0	0
$985$	−29.1445	−0.928622
$986$	−4.18697	−0.133340
$987$	0	0
$988$	22.2534	0.707974
$989$	37.0186	1.17712
$990$	0	0
$991$	−9.75410	−0.309849	−0.154925	−	0.987926i	$-0.549513\pi$
$991$	−9.75410	−0.309849	−0.154925	+	0.987926i	$0.549513\pi$
$992$	−13.4184	−0.426036
$993$	0	0
$994$	−17.7274	−0.562280
$995$	20.6381	0.654272
$996$	0	0
$997$	43.2494	1.36972	0.684862	−	0.728673i	$-0.259863\pi$
$997$	43.2494	1.36972	0.684862	+	0.728673i	$0.259863\pi$
$998$	4.25601	0.134722
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.g.1.7		13
3.2	odd	2		2013.2.a.f.1.7	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.f.1.7	✓	13	3.2	odd	2
6039.2.a.g.1.7		13	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:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q-0.468970 q^{2} -1.78007 q^{4} +3.25579 q^{5} -3.43403 q^{7} +1.77274 q^{8} +O(q^{10})\)	\(q-0.468970 q^{2} -1.78007 q^{4} +3.25579 q^{5} -3.43403 q^{7} +1.77274 q^{8} -1.52687 q^{10} +1.00000 q^{11} -6.24548 q^{13} +1.61046 q^{14} +2.72877 q^{16} -4.43634 q^{17} +2.00167 q^{19} -5.79552 q^{20} -0.468970 q^{22} +5.24027 q^{23} +5.60015 q^{25} +2.92895 q^{26} +6.11281 q^{28} -2.01247 q^{29} +2.78091 q^{31} -4.82519 q^{32} +2.08051 q^{34} -11.1805 q^{35} +9.46308 q^{37} -0.938726 q^{38} +5.77166 q^{40} +6.90514 q^{41} +7.06425 q^{43} -1.78007 q^{44} -2.45753 q^{46} +3.28507 q^{47} +4.79257 q^{49} -2.62630 q^{50} +11.1174 q^{52} +3.39708 q^{53} +3.25579 q^{55} -6.08764 q^{56} +0.943788 q^{58} -6.95400 q^{59} -1.00000 q^{61} -1.30417 q^{62} -3.19467 q^{64} -20.3340 q^{65} -13.7103 q^{67} +7.89698 q^{68} +5.24331 q^{70} -11.0077 q^{71} -1.02864 q^{73} -4.43790 q^{74} -3.56312 q^{76} -3.43403 q^{77} -10.9648 q^{79} +8.88430 q^{80} -3.23831 q^{82} -10.7032 q^{83} -14.4438 q^{85} -3.31292 q^{86} +1.77274 q^{88} -5.93188 q^{89} +21.4472 q^{91} -9.32803 q^{92} -1.54060 q^{94} +6.51703 q^{95} +1.30698 q^{97} -2.24757 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8} + 8 q^{10} + 13 q^{11} - 9 q^{13} - 19 q^{14} + 18 q^{16} - 7 q^{17} + 2 q^{19} - 15 q^{20} - 4 q^{22} - 23 q^{23} + 10 q^{25} - 8 q^{26} + 9 q^{28} - 16 q^{29} + 9 q^{31} - 29 q^{32} + 2 q^{34} - 16 q^{35} + 14 q^{37} - 8 q^{38} + 16 q^{40} - 19 q^{41} + 7 q^{43} + 12 q^{44} + 4 q^{46} - 26 q^{47} + 8 q^{49} + 15 q^{50} - 17 q^{52} - 18 q^{53} - 7 q^{55} - 44 q^{56} - q^{58} - 31 q^{59} - 13 q^{61} + 5 q^{62} - 17 q^{64} - 31 q^{65} + 14 q^{67} + 32 q^{68} - 20 q^{70} - 37 q^{71} - 16 q^{73} + 6 q^{74} - 7 q^{76} + 5 q^{77} - 17 q^{79} + 2 q^{80} - 2 q^{82} - 30 q^{83} - 16 q^{85} + 22 q^{86} - 15 q^{88} - 35 q^{89} - q^{91} - 24 q^{92} - 11 q^{94} - 13 q^{95} - q^{97} + q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8 + 8 * q^10 + 13 * q^11 - 9 * q^13 - 19 * q^14 + 18 * q^16 - 7 * q^17 + 2 * q^19 - 15 * q^20 - 4 * q^22 - 23 * q^23 + 10 * q^25 - 8 * q^26 + 9 * q^28 - 16 * q^29 + 9 * q^31 - 29 * q^32 + 2 * q^34 - 16 * q^35 + 14 * q^37 - 8 * q^38 + 16 * q^40 - 19 * q^41 + 7 * q^43 + 12 * q^44 + 4 * q^46 - 26 * q^47 + 8 * q^49 + 15 * q^50 - 17 * q^52 - 18 * q^53 - 7 * q^55 - 44 * q^56 - q^58 - 31 * q^59 - 13 * q^61 + 5 * q^62 - 17 * q^64 - 31 * q^65 + 14 * q^67 + 32 * q^68 - 20 * q^70 - 37 * q^71 - 16 * q^73 + 6 * q^74 - 7 * q^76 + 5 * q^77 - 17 * q^79 + 2 * q^80 - 2 * q^82 - 30 * q^83 - 16 * q^85 + 22 * q^86 - 15 * q^88 - 35 * q^89 - q^91 - 24 * q^92 - 11 * q^94 - 13 * q^95 - q^97 + q^98

Introduction

L-functions

Modular forms

Varieties

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Representations

Groups

Database

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