LMFDB - Embedded newform 6039.2.a.i.1.6

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:	$0$
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} - 2 x^{12} - 19 x^{11} + 35 x^{10} + 136 x^{9} - 220 x^{8} - 469 x^{7} + 610 x^{6} + 841 x^{5} + \cdots - 47 $ x^13 - 2x^12 - 19x^11 + 35x^10 + 136x^9 - 220x^8 - 469x^7 + 610x^6 + 841x^5 - 760x^4 - 742x^3 + 366x^2 + 236x - 47
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			1.6
Root			$0.822526$ 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)

$f(q)$	$=$	$q-0.822526 q^{2} -1.32345 q^{4} -2.11599 q^{5} +0.404149 q^{7} +2.73363 q^{8} +O(q^{10})$	\(q-0.822526 q^{2} -1.32345 q^{4} -2.11599 q^{5} +0.404149 q^{7} +2.73363 q^{8} +1.74046 q^{10} -1.00000 q^{11} +4.06066 q^{13} -0.332423 q^{14} +0.398423 q^{16} -0.0205369 q^{17} +3.97586 q^{19} +2.80041 q^{20} +0.822526 q^{22} +8.67841 q^{23} -0.522583 q^{25} -3.34000 q^{26} -0.534871 q^{28} +5.89217 q^{29} +5.40170 q^{31} -5.79496 q^{32} +0.0168922 q^{34} -0.855175 q^{35} +1.99527 q^{37} -3.27025 q^{38} -5.78433 q^{40} -6.26962 q^{41} +7.16134 q^{43} +1.32345 q^{44} -7.13822 q^{46} -8.53737 q^{47} -6.83666 q^{49} +0.429838 q^{50} -5.37409 q^{52} -5.18361 q^{53} +2.11599 q^{55} +1.10479 q^{56} -4.84647 q^{58} +0.638698 q^{59} -1.00000 q^{61} -4.44304 q^{62} +3.96966 q^{64} -8.59233 q^{65} +15.2035 q^{67} +0.0271796 q^{68} +0.703404 q^{70} -11.4793 q^{71} -15.0344 q^{73} -1.64116 q^{74} -5.26185 q^{76} -0.404149 q^{77} +10.9481 q^{79} -0.843060 q^{80} +5.15693 q^{82} -4.60944 q^{83} +0.0434559 q^{85} -5.89039 q^{86} -2.73363 q^{88} -12.7830 q^{89} +1.64111 q^{91} -11.4855 q^{92} +7.02221 q^{94} -8.41288 q^{95} -0.603085 q^{97} +5.62333 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	$ 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) $ 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * 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 $

$2$	−0.822526	−0.581614	−0.290807	−	0.956782i	$-0.593924\pi$
$2$	−0.822526	−0.581614	−0.290807	+	0.956782i	$0.593924\pi$
$3$	0	0
$3$	0	0
$4$	−1.32345	−0.661725
$5$	−2.11599	−0.946300	−0.473150	−	0.880982i	$-0.656883\pi$
$5$	−2.11599	−0.946300	−0.473150	+	0.880982i	$0.656883\pi$
$6$	0	0
$7$	0.404149	0.152754	0.0763769	−	0.997079i	$-0.475665\pi$
$7$	0.404149	0.152754	0.0763769	+	0.997079i	$0.475665\pi$
$8$	2.73363	0.966482
$9$	0	0
$10$	1.74046	0.550381
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	4.06066	1.12623	0.563113	−	0.826380i	$-0.309604\pi$
$13$	4.06066	1.12623	0.563113	+	0.826380i	$0.309604\pi$
$14$	−0.332423	−0.0888438
$15$	0	0
$16$	0.398423	0.0996058
$17$	−0.0205369	−0.00498093	−0.00249047	−	0.999997i	$-0.500793\pi$
$17$	−0.0205369	−0.00498093	−0.00249047	+	0.999997i	$0.500793\pi$
$18$	0	0
$19$	3.97586	0.912124	0.456062	−	0.889948i	$-0.349259\pi$
$19$	3.97586	0.912124	0.456062	+	0.889948i	$0.349259\pi$
$20$	2.80041	0.626191
$21$	0	0
$22$	0.822526	0.175363
$23$	8.67841	1.80957	0.904787	−	0.425864i	$-0.140030\pi$
$23$	8.67841	1.80957	0.904787	+	0.425864i	$0.140030\pi$
$24$	0	0
$25$	−0.522583	−0.104517
$26$	−3.34000	−0.655028
$27$	0	0
$28$	−0.534871	−0.101081
$29$	5.89217	1.09415	0.547075	−	0.837084i	$-0.315741\pi$
$29$	5.89217	1.09415	0.547075	+	0.837084i	$0.315741\pi$
$30$	0	0
$31$	5.40170	0.970174	0.485087	−	0.874466i	$-0.338788\pi$
$31$	5.40170	0.970174	0.485087	+	0.874466i	$0.338788\pi$
$32$	−5.79496	−1.02441
$33$	0	0
$34$	0.0168922	0.00289698
$35$	−0.855175	−0.144551
$36$	0	0
$37$	1.99527	0.328020	0.164010	−	0.986459i	$-0.447557\pi$
$37$	1.99527	0.328020	0.164010	+	0.986459i	$0.447557\pi$
$38$	−3.27025	−0.530504
$39$	0	0
$40$	−5.78433	−0.914582
$41$	−6.26962	−0.979150	−0.489575	−	0.871961i	$-0.662848\pi$
$41$	−6.26962	−0.979150	−0.489575	+	0.871961i	$0.662848\pi$
$42$	0	0
$43$	7.16134	1.09209	0.546047	−	0.837754i	$-0.316132\pi$
$43$	7.16134	1.09209	0.546047	+	0.837754i	$0.316132\pi$
$44$	1.32345	0.199518
$45$	0	0
$46$	−7.13822	−1.05247
$47$	−8.53737	−1.24530	−0.622652	−	0.782499i	$-0.713945\pi$
$47$	−8.53737	−1.24530	−0.622652	+	0.782499i	$0.713945\pi$
$48$	0	0
$49$	−6.83666	−0.976666
$50$	0.429838	0.0607883
$51$	0	0
$52$	−5.37409	−0.745252
$53$	−5.18361	−0.712024	−0.356012	−	0.934481i	$-0.615864\pi$
$53$	−5.18361	−0.712024	−0.356012	+	0.934481i	$0.615864\pi$
$54$	0	0
$55$	2.11599	0.285320
$56$	1.10479	0.147634
$57$	0	0
$58$	−4.84647	−0.636372
$59$	0.638698	0.0831514	0.0415757	−	0.999135i	$-0.486762\pi$
$59$	0.638698	0.0831514	0.0415757	+	0.999135i	$0.486762\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−4.44304	−0.564267
$63$	0	0
$64$	3.96966	0.496208
$65$	−8.59233	−1.06575
$66$	0	0
$67$	15.2035	1.85740	0.928700	−	0.370831i	$-0.120927\pi$
$67$	15.2035	1.85740	0.928700	+	0.370831i	$0.120927\pi$
$68$	0.0271796	0.00329601
$69$	0	0
$70$	0.703404	0.0840729
$71$	−11.4793	−1.36234	−0.681168	−	0.732127i	$-0.738528\pi$
$71$	−11.4793	−1.36234	−0.681168	+	0.732127i	$0.738528\pi$
$72$	0	0
$73$	−15.0344	−1.75965	−0.879823	−	0.475301i	$-0.842339\pi$
$73$	−15.0344	−1.75965	−0.879823	+	0.475301i	$0.842339\pi$
$74$	−1.64116	−0.190781
$75$	0	0
$76$	−5.26185	−0.603576
$77$	−0.404149	−0.0460570
$78$	0	0
$79$	10.9481	1.23175	0.615877	−	0.787842i	$-0.288802\pi$
$79$	10.9481	1.23175	0.615877	+	0.787842i	$0.288802\pi$
$80$	−0.843060	−0.0942570
$81$	0	0
$82$	5.15693	0.569487
$83$	−4.60944	−0.505952	−0.252976	−	0.967473i	$-0.581409\pi$
$83$	−4.60944	−0.505952	−0.252976	+	0.967473i	$0.581409\pi$
$84$	0	0
$85$	0.0434559	0.00471346
$86$	−5.89039	−0.635177
$87$	0	0
$88$	−2.73363	−0.291405
$89$	−12.7830	−1.35500	−0.677500	−	0.735523i	$-0.736937\pi$
$89$	−12.7830	−1.35500	−0.677500	+	0.735523i	$0.736937\pi$
$90$	0	0
$91$	1.64111	0.172035
$92$	−11.4855	−1.19744
$93$	0	0
$94$	7.02221	0.724286
$95$	−8.41288	−0.863143
$96$	0	0
$97$	−0.603085	−0.0612340	−0.0306170	−	0.999531i	$-0.509747\pi$
$97$	−0.603085	−0.0612340	−0.0306170	+	0.999531i	$0.509747\pi$
$98$	5.62333	0.568043
$99$	0	0
$100$	0.691613	0.0691613
$101$	14.6545	1.45818	0.729088	−	0.684420i	$-0.239944\pi$
$101$	14.6545	1.45818	0.729088	+	0.684420i	$0.239944\pi$
$102$	0	0
$103$	11.1666	1.10028	0.550138	−	0.835074i	$-0.314575\pi$
$103$	11.1666	1.10028	0.550138	+	0.835074i	$0.314575\pi$
$104$	11.1003	1.08848
$105$	0	0
$106$	4.26366	0.414123
$107$	8.54990	0.826550	0.413275	−	0.910606i	$-0.364385\pi$
$107$	8.54990	0.826550	0.413275	+	0.910606i	$0.364385\pi$
$108$	0	0
$109$	14.0443	1.34520	0.672598	−	0.740008i	$-0.265178\pi$
$109$	14.0443	1.34520	0.672598	+	0.740008i	$0.265178\pi$
$110$	−1.74046	−0.165946
$111$	0	0
$112$	0.161022	0.0152152
$113$	−9.57510	−0.900749	−0.450375	−	0.892840i	$-0.648710\pi$
$113$	−9.57510	−0.900749	−0.450375	+	0.892840i	$0.648710\pi$
$114$	0	0
$115$	−18.3634	−1.71240
$116$	−7.79800	−0.724026
$117$	0	0
$118$	−0.525346	−0.0483620
$119$	−0.00829997	−0.000760857 0
$120$	0	0
$121$	1.00000	0.0909091
$122$	0.822526	0.0744680
$123$	0	0
$124$	−7.14889	−0.641989
$125$	11.6857	1.04520
$126$	0	0
$127$	6.13941	0.544785	0.272392	−	0.962186i	$-0.412185\pi$
$127$	6.13941	0.544785	0.272392	+	0.962186i	$0.412185\pi$
$128$	8.32478	0.735813
$129$	0	0
$130$	7.06741	0.619853
$131$	−5.50042	−0.480574	−0.240287	−	0.970702i	$-0.577242\pi$
$131$	−5.50042	−0.480574	−0.240287	+	0.970702i	$0.577242\pi$
$132$	0	0
$133$	1.60684	0.139331
$134$	−12.5053	−1.08029
$135$	0	0
$136$	−0.0561402	−0.00481399
$137$	2.78969	0.238339	0.119170	−	0.992874i	$-0.461977\pi$
$137$	2.78969	0.238339	0.119170	+	0.992874i	$0.461977\pi$
$138$	0	0
$139$	−11.0724	−0.939148	−0.469574	−	0.882893i	$-0.655592\pi$
$139$	−11.0724	−0.939148	−0.469574	+	0.882893i	$0.655592\pi$
$140$	1.13178	0.0956531
$141$	0	0
$142$	9.44198	0.792354
$143$	−4.06066	−0.339570
$144$	0	0
$145$	−12.4678	−1.03539
$146$	12.3662	1.02343
$147$	0	0
$148$	−2.64064	−0.217059
$149$	−0.473710	−0.0388079	−0.0194039	−	0.999812i	$-0.506177\pi$
$149$	−0.473710	−0.0388079	−0.0194039	+	0.999812i	$0.506177\pi$
$150$	0	0
$151$	13.2622	1.07927	0.539633	−	0.841900i	$-0.318563\pi$
$151$	13.2622	1.07927	0.539633	+	0.841900i	$0.318563\pi$
$152$	10.8685	0.881552
$153$	0	0
$154$	0.332423	0.0267874
$155$	−11.4300	−0.918076
$156$	0	0
$157$	−12.7759	−1.01963	−0.509814	−	0.860285i	$-0.670286\pi$
$157$	−12.7759	−1.01963	−0.509814	+	0.860285i	$0.670286\pi$
$158$	−9.00508	−0.716405
$159$	0	0
$160$	12.2621	0.969403
$161$	3.50737	0.276419
$162$	0	0
$163$	24.5359	1.92180	0.960900	−	0.276897i	$-0.0893060\pi$
$163$	24.5359	1.92180	0.960900	+	0.276897i	$0.0893060\pi$
$164$	8.29753	0.647928
$165$	0	0
$166$	3.79139	0.294269
$167$	−3.95593	−0.306119	−0.153060	−	0.988217i	$-0.548913\pi$
$167$	−3.95593	−0.306119	−0.153060	+	0.988217i	$0.548913\pi$
$168$	0	0
$169$	3.48899	0.268384
$170$	−0.0357436	−0.00274141
$171$	0	0
$172$	−9.47768	−0.722666
$173$	21.5757	1.64037	0.820186	−	0.572097i	$-0.193870\pi$
$173$	21.5757	1.64037	0.820186	+	0.572097i	$0.193870\pi$
$174$	0	0
$175$	−0.211201	−0.0159653
$176$	−0.398423	−0.0300323
$177$	0	0
$178$	10.5144	0.788086
$179$	13.5654	1.01392	0.506962	−	0.861968i	$-0.330768\pi$
$179$	13.5654	1.01392	0.506962	+	0.861968i	$0.330768\pi$
$180$	0	0
$181$	−20.9000	−1.55349	−0.776744	−	0.629817i	$-0.783130\pi$
$181$	−20.9000	−1.55349	−0.776744	+	0.629817i	$0.783130\pi$
$182$	−1.34986	−0.100058
$183$	0	0
$184$	23.7235	1.74892
$185$	−4.22197	−0.310405
$186$	0	0
$187$	0.0205369	0.00150181
$188$	11.2988	0.824049
$189$	0	0
$190$	6.91981	0.502016
$191$	−2.45440	−0.177594	−0.0887971	−	0.996050i	$-0.528302\pi$
$191$	−2.45440	−0.177594	−0.0887971	+	0.996050i	$0.528302\pi$
$192$	0	0
$193$	−8.31456	−0.598495	−0.299248	−	0.954176i	$-0.596736\pi$
$193$	−8.31456	−0.598495	−0.299248	+	0.954176i	$0.596736\pi$
$194$	0.496053	0.0356146
$195$	0	0
$196$	9.04799	0.646285
$197$	−18.8004	−1.33947	−0.669736	−	0.742599i	$-0.733593\pi$
$197$	−18.8004	−1.33947	−0.669736	+	0.742599i	$0.733593\pi$
$198$	0	0
$199$	−7.84162	−0.555877	−0.277939	−	0.960599i	$-0.589651\pi$
$199$	−7.84162	−0.555877	−0.277939	+	0.960599i	$0.589651\pi$
$200$	−1.42855	−0.101013
$201$	0	0
$202$	−12.0537	−0.848096
$203$	2.38131	0.167136
$204$	0	0
$205$	13.2665	0.926569
$206$	−9.18480	−0.639935
$207$	0	0
$208$	1.61786	0.112179
$209$	−3.97586	−0.275016
$210$	0	0
$211$	−25.6632	−1.76673	−0.883364	−	0.468688i	$-0.844727\pi$
$211$	−25.6632	−1.76673	−0.883364	+	0.468688i	$0.844727\pi$
$212$	6.86025	0.471164
$213$	0	0
$214$	−7.03252	−0.480733
$215$	−15.1533	−1.03345
$216$	0	0
$217$	2.18309	0.148198
$218$	−11.5518	−0.782385
$219$	0	0
$220$	−2.80041	−0.188804
$221$	−0.0833935	−0.00560966
$222$	0	0
$223$	1.31581	0.0881130	0.0440565	−	0.999029i	$-0.485972\pi$
$223$	1.31581	0.0881130	0.0440565	+	0.999029i	$0.485972\pi$
$224$	−2.34203	−0.156483
$225$	0	0
$226$	7.87577	0.523888
$227$	−7.33476	−0.486825	−0.243413	−	0.969923i	$-0.578267\pi$
$227$	−7.33476	−0.486825	−0.243413	+	0.969923i	$0.578267\pi$
$228$	0	0
$229$	−4.61757	−0.305137	−0.152569	−	0.988293i	$-0.548755\pi$
$229$	−4.61757	−0.305137	−0.152569	+	0.988293i	$0.548755\pi$
$230$	15.1044	0.995955
$231$	0	0
$232$	16.1070	1.05748
$233$	−29.1255	−1.90807	−0.954036	−	0.299692i	$-0.903116\pi$
$233$	−29.1255	−1.90807	−0.954036	+	0.299692i	$0.903116\pi$
$234$	0	0
$235$	18.0650	1.17843
$236$	−0.845285	−0.0550234
$237$	0	0
$238$	0.00682694	0.000442525 0
$239$	30.6121	1.98013	0.990064	−	0.140614i	$-0.0449077\pi$
$239$	30.6121	1.98013	0.990064	+	0.140614i	$0.0449077\pi$
$240$	0	0
$241$	22.3844	1.44191	0.720954	−	0.692983i	$-0.243704\pi$
$241$	22.3844	1.44191	0.720954	+	0.692983i	$0.243704\pi$
$242$	−0.822526	−0.0528740
$243$	0	0
$244$	1.32345	0.0847253
$245$	14.4663	0.924219
$246$	0	0
$247$	16.1446	1.02726
$248$	14.7662	0.937656
$249$	0	0
$250$	−9.61182	−0.607905
$251$	−21.4637	−1.35478	−0.677390	−	0.735624i	$-0.736889\pi$
$251$	−21.4637	−1.35478	−0.677390	+	0.735624i	$0.736889\pi$
$252$	0	0
$253$	−8.67841	−0.545607
$254$	−5.04983	−0.316854
$255$	0	0
$256$	−14.7867	−0.924167
$257$	−4.42849	−0.276242	−0.138121	−	0.990415i	$-0.544106\pi$
$257$	−4.42849	−0.276242	−0.138121	+	0.990415i	$0.544106\pi$
$258$	0	0
$259$	0.806385	0.0501064
$260$	11.3715	0.705232
$261$	0	0
$262$	4.52424	0.279509
$263$	26.2060	1.61593	0.807966	−	0.589230i	$-0.200569\pi$
$263$	26.2060	1.61593	0.807966	+	0.589230i	$0.200569\pi$
$264$	0	0
$265$	10.9685	0.673788
$266$	−1.32167	−0.0810366
$267$	0	0
$268$	−20.1211	−1.22909
$269$	11.4188	0.696217	0.348109	−	0.937454i	$-0.386824\pi$
$269$	11.4188	0.696217	0.348109	+	0.937454i	$0.386824\pi$
$270$	0	0
$271$	2.05980	0.125124	0.0625620	−	0.998041i	$-0.480073\pi$
$271$	2.05980	0.125124	0.0625620	+	0.998041i	$0.480073\pi$
$272$	−0.00818239	−0.000496130 0
$273$	0	0
$274$	−2.29459	−0.138621
$275$	0.522583	0.0315129
$276$	0	0
$277$	27.0262	1.62385	0.811923	−	0.583765i	$-0.198421\pi$
$277$	27.0262	1.62385	0.811923	+	0.583765i	$0.198421\pi$
$278$	9.10733	0.546221
$279$	0	0
$280$	−2.33773	−0.139706
$281$	−4.85590	−0.289679	−0.144839	−	0.989455i	$-0.546267\pi$
$281$	−4.85590	−0.289679	−0.144839	+	0.989455i	$0.546267\pi$
$282$	0	0
$283$	15.0805	0.896442	0.448221	−	0.893923i	$-0.352058\pi$
$283$	15.0805	0.896442	0.448221	+	0.893923i	$0.352058\pi$
$284$	15.1922	0.901493
$285$	0	0
$286$	3.34000	0.197498
$287$	−2.53386	−0.149569
$288$	0	0
$289$	−16.9996	−0.999975
$290$	10.2551	0.602199
$291$	0	0
$292$	19.8973	1.16440
$293$	19.5057	1.13954	0.569769	−	0.821805i	$-0.307033\pi$
$293$	19.5057	1.13954	0.569769	+	0.821805i	$0.307033\pi$
$294$	0	0
$295$	−1.35148	−0.0786861
$296$	5.45432	0.317026
$297$	0	0
$298$	0.389639	0.0225712
$299$	35.2401	2.03799
$300$	0	0
$301$	2.89425	0.166822
$302$	−10.9085	−0.627716
$303$	0	0
$304$	1.58407	0.0908529
$305$	2.11599	0.121161
$306$	0	0
$307$	7.49773	0.427918	0.213959	−	0.976843i	$-0.431364\pi$
$307$	7.49773	0.427918	0.213959	+	0.976843i	$0.431364\pi$
$308$	0.534871	0.0304771
$309$	0	0
$310$	9.40143	0.533966
$311$	−17.7437	−1.00616	−0.503078	−	0.864241i	$-0.667799\pi$
$311$	−17.7437	−1.00616	−0.503078	+	0.864241i	$0.667799\pi$
$312$	0	0
$313$	13.3708	0.755763	0.377882	−	0.925854i	$-0.376653\pi$
$313$	13.3708	0.755763	0.377882	+	0.925854i	$0.376653\pi$
$314$	10.5085	0.593030
$315$	0	0
$316$	−14.4892	−0.815083
$317$	10.5527	0.592699	0.296350	−	0.955079i	$-0.404231\pi$
$317$	10.5527	0.592699	0.296350	+	0.955079i	$0.404231\pi$
$318$	0	0
$319$	−5.89217	−0.329898
$320$	−8.39977	−0.469561
$321$	0	0
$322$	−2.88490	−0.160769
$323$	−0.0816519	−0.00454323
$324$	0	0
$325$	−2.12203	−0.117709
$326$	−20.1814	−1.11775
$327$	0	0
$328$	−17.1388	−0.946331
$329$	−3.45037	−0.190225
$330$	0	0
$331$	−5.63583	−0.309773	−0.154887	−	0.987932i	$-0.549501\pi$
$331$	−5.63583	−0.309773	−0.154887	+	0.987932i	$0.549501\pi$
$332$	6.10037	0.334801
$333$	0	0
$334$	3.25386	0.178043
$335$	−32.1704	−1.75766
$336$	0	0
$337$	21.1177	1.15036	0.575178	−	0.818028i	$-0.304933\pi$
$337$	21.1177	1.15036	0.575178	+	0.818028i	$0.304933\pi$
$338$	−2.86979	−0.156096
$339$	0	0
$340$	−0.0575118	−0.00311901
$341$	−5.40170	−0.292519
$342$	0	0
$343$	−5.59207	−0.301943
$344$	19.5764	1.05549
$345$	0	0
$346$	−17.7466	−0.954063
$347$	24.5687	1.31892	0.659458	−	0.751742i	$-0.270786\pi$
$347$	24.5687	1.31892	0.659458	+	0.751742i	$0.270786\pi$
$348$	0	0
$349$	3.91197	0.209403	0.104701	−	0.994504i	$-0.466611\pi$
$349$	3.91197	0.209403	0.104701	+	0.994504i	$0.466611\pi$
$350$	0.173719	0.00928565
$351$	0	0
$352$	5.79496	0.308873
$353$	28.5152	1.51771	0.758856	−	0.651259i	$-0.225759\pi$
$353$	28.5152	1.51771	0.758856	+	0.651259i	$0.225759\pi$
$354$	0	0
$355$	24.2900	1.28918
$356$	16.9177	0.896638
$357$	0	0
$358$	−11.1579	−0.589712
$359$	27.2384	1.43759	0.718793	−	0.695224i	$-0.244695\pi$
$359$	27.2384	1.43759	0.718793	+	0.695224i	$0.244695\pi$
$360$	0	0
$361$	−3.19255	−0.168029
$362$	17.1908	0.903530
$363$	0	0
$364$	−2.17193	−0.113840
$365$	31.8127	1.66515
$366$	0	0
$367$	−37.7100	−1.96845	−0.984224	−	0.176927i	$-0.943384\pi$
$367$	−37.7100	−1.96845	−0.984224	+	0.176927i	$0.943384\pi$
$368$	3.45768	0.180244
$369$	0	0
$370$	3.47268	0.180536
$371$	−2.09495	−0.108764
$372$	0	0
$373$	4.83715	0.250458	0.125229	−	0.992128i	$-0.460033\pi$
$373$	4.83715	0.250458	0.125229	+	0.992128i	$0.460033\pi$
$374$	−0.0168922	−0.000873472 0
$375$	0	0
$376$	−23.3380	−1.20356
$377$	23.9261	1.23226
$378$	0	0
$379$	−35.2816	−1.81229	−0.906146	−	0.422965i	$-0.860989\pi$
$379$	−35.2816	−1.81229	−0.906146	+	0.422965i	$0.860989\pi$
$380$	11.1340	0.571164
$381$	0	0
$382$	2.01881	0.103291
$383$	−18.3228	−0.936253	−0.468127	−	0.883661i	$-0.655071\pi$
$383$	−18.3228	−0.936253	−0.468127	+	0.883661i	$0.655071\pi$
$384$	0	0
$385$	0.855175	0.0435838
$386$	6.83894	0.348093
$387$	0	0
$388$	0.798153	0.0405201
$389$	34.4388	1.74612	0.873059	−	0.487614i	$-0.162133\pi$
$389$	34.4388	1.74612	0.873059	+	0.487614i	$0.162133\pi$
$390$	0	0
$391$	−0.178228	−0.00901337
$392$	−18.6889	−0.943931
$393$	0	0
$394$	15.4638	0.779056
$395$	−23.1660	−1.16561
$396$	0	0
$397$	−6.67209	−0.334862	−0.167431	−	0.985884i	$-0.553547\pi$
$397$	−6.67209	−0.334862	−0.167431	+	0.985884i	$0.553547\pi$
$398$	6.44993	0.323306
$399$	0	0
$400$	−0.208209	−0.0104105
$401$	19.4145	0.969515	0.484757	−	0.874649i	$-0.338908\pi$
$401$	19.4145	0.969515	0.484757	+	0.874649i	$0.338908\pi$
$402$	0	0
$403$	21.9345	1.09264
$404$	−19.3945	−0.964912
$405$	0	0
$406$	−1.95869	−0.0972083
$407$	−1.99527	−0.0989018
$408$	0	0
$409$	0.767851	0.0379678	0.0189839	−	0.999820i	$-0.493957\pi$
$409$	0.767851	0.0379678	0.0189839	+	0.999820i	$0.493957\pi$
$410$	−10.9120	−0.538906
$411$	0	0
$412$	−14.7784	−0.728080
$413$	0.258129	0.0127017
$414$	0	0
$415$	9.75354	0.478782
$416$	−23.5314	−1.15372
$417$	0	0
$418$	3.27025	0.159953
$419$	−17.3337	−0.846806	−0.423403	−	0.905941i	$-0.639165\pi$
$419$	−17.3337	−0.846806	−0.423403	+	0.905941i	$0.639165\pi$
$420$	0	0
$421$	12.0651	0.588016	0.294008	−	0.955803i	$-0.405011\pi$
$421$	12.0651	0.588016	0.294008	+	0.955803i	$0.405011\pi$
$422$	21.1087	1.02755
$423$	0	0
$424$	−14.1700	−0.688158
$425$	0.0107322	0.000520590 0
$426$	0	0
$427$	−0.404149	−0.0195581
$428$	−11.3154	−0.546949
$429$	0	0
$430$	12.4640	0.601068
$431$	29.2010	1.40656	0.703282	−	0.710911i	$-0.251717\pi$
$431$	29.2010	1.40656	0.703282	+	0.710911i	$0.251717\pi$
$432$	0	0
$433$	−35.8038	−1.72062	−0.860311	−	0.509769i	$-0.829731\pi$
$433$	−35.8038	−1.72062	−0.860311	+	0.509769i	$0.829731\pi$
$434$	−1.79565	−0.0861939
$435$	0	0
$436$	−18.5869	−0.890151
$437$	34.5041	1.65056
$438$	0	0
$439$	40.8584	1.95006	0.975032	−	0.222063i	$-0.0712791\pi$
$439$	40.8584	1.95006	0.975032	+	0.222063i	$0.0712791\pi$
$440$	5.78433	0.275757
$441$	0	0
$442$	0.0685934	0.00326265
$443$	27.9834	1.32953	0.664767	−	0.747051i	$-0.268531\pi$
$443$	27.9834	1.32953	0.664767	+	0.747051i	$0.268531\pi$
$444$	0	0
$445$	27.0488	1.28224
$446$	−1.08229	−0.0512477
$447$	0	0
$448$	1.60433	0.0757977
$449$	25.0254	1.18102	0.590512	−	0.807029i	$-0.298926\pi$
$449$	25.0254	1.18102	0.590512	+	0.807029i	$0.298926\pi$
$450$	0	0
$451$	6.26962	0.295225
$452$	12.6722	0.596049
$453$	0	0
$454$	6.03303	0.283144
$455$	−3.47258	−0.162797
$456$	0	0
$457$	3.70365	0.173249	0.0866247	−	0.996241i	$-0.472392\pi$
$457$	3.70365	0.173249	0.0866247	+	0.996241i	$0.472392\pi$
$458$	3.79807	0.177472
$459$	0	0
$460$	24.3031	1.13314
$461$	15.8022	0.735982	0.367991	−	0.929829i	$-0.380046\pi$
$461$	15.8022	0.735982	0.367991	+	0.929829i	$0.380046\pi$
$462$	0	0
$463$	−25.2470	−1.17333	−0.586665	−	0.809830i	$-0.699559\pi$
$463$	−25.2470	−1.17333	−0.586665	+	0.809830i	$0.699559\pi$
$464$	2.34758	0.108984
$465$	0	0
$466$	23.9564	1.10976
$467$	17.2785	0.799552	0.399776	−	0.916613i	$-0.369088\pi$
$467$	17.2785	0.799552	0.399776	+	0.916613i	$0.369088\pi$
$468$	0	0
$469$	6.14447	0.283725
$470$	−14.8589	−0.685391
$471$	0	0
$472$	1.74596	0.0803643
$473$	−7.16134	−0.329279
$474$	0	0
$475$	−2.07771	−0.0953321
$476$	0.0109846	0.000503478 0
$477$	0	0
$478$	−25.1792	−1.15167
$479$	−10.8839	−0.497297	−0.248648	−	0.968594i	$-0.579986\pi$
$479$	−10.8839	−0.497297	−0.248648	+	0.968594i	$0.579986\pi$
$480$	0	0
$481$	8.10212	0.369425
$482$	−18.4118	−0.838634
$483$	0	0
$484$	−1.32345	−0.0601569
$485$	1.27612	0.0579457
$486$	0	0
$487$	39.0162	1.76799	0.883997	−	0.467492i	$-0.154842\pi$
$487$	39.0162	1.76799	0.883997	+	0.467492i	$0.154842\pi$
$488$	−2.73363	−0.123745
$489$	0	0
$490$	−11.8989	−0.537539
$491$	−30.0322	−1.35533	−0.677667	−	0.735369i	$-0.737009\pi$
$491$	−30.0322	−1.35533	−0.677667	+	0.735369i	$0.737009\pi$
$492$	0	0
$493$	−0.121007	−0.00544989
$494$	−13.2794	−0.597467
$495$	0	0
$496$	2.15216	0.0966350
$497$	−4.63933	−0.208102
$498$	0	0
$499$	25.6100	1.14646	0.573232	−	0.819393i	$-0.305690\pi$
$499$	25.6100	1.14646	0.573232	+	0.819393i	$0.305690\pi$
$500$	−15.4655	−0.691638
$501$	0	0
$502$	17.6545	0.787958
$503$	9.23707	0.411861	0.205930	−	0.978567i	$-0.433978\pi$
$503$	9.23707	0.411861	0.205930	+	0.978567i	$0.433978\pi$
$504$	0	0
$505$	−31.0088	−1.37987
$506$	7.13822	0.317333
$507$	0	0
$508$	−8.12521	−0.360498
$509$	−32.4717	−1.43928	−0.719642	−	0.694345i	$-0.755694\pi$
$509$	−32.4717	−1.43928	−0.719642	+	0.694345i	$0.755694\pi$
$510$	0	0
$511$	−6.07615	−0.268793
$512$	−4.48713	−0.198305
$513$	0	0
$514$	3.64255	0.160666
$515$	−23.6284	−1.04119
$516$	0	0
$517$	8.53737	0.375473
$518$	−0.663273	−0.0291425
$519$	0	0
$520$	−23.4882	−1.03003
$521$	−12.2202	−0.535375	−0.267687	−	0.963506i	$-0.586259\pi$
$521$	−12.2202	−0.535375	−0.267687	+	0.963506i	$0.586259\pi$
$522$	0	0
$523$	−24.8557	−1.08686	−0.543432	−	0.839453i	$-0.682876\pi$
$523$	−24.8557	−1.08686	−0.543432	+	0.839453i	$0.682876\pi$
$524$	7.27954	0.318008
$525$	0	0
$526$	−21.5551	−0.939848
$527$	−0.110934	−0.00483238
$528$	0	0
$529$	52.3148	2.27456
$530$	−9.02186	−0.391884
$531$	0	0
$532$	−2.12657	−0.0921986
$533$	−25.4588	−1.10274
$534$	0	0
$535$	−18.0915	−0.782164
$536$	41.5606	1.79515
$537$	0	0
$538$	−9.39227	−0.404930
$539$	6.83666	0.294476
$540$	0	0
$541$	6.76305	0.290766	0.145383	−	0.989375i	$-0.453559\pi$
$541$	6.76305	0.290766	0.145383	+	0.989375i	$0.453559\pi$
$542$	−1.69424	−0.0727739
$543$	0	0
$544$	0.119011	0.00510254
$545$	−29.7175	−1.27296
$546$	0	0
$547$	22.0518	0.942865	0.471432	−	0.881902i	$-0.343737\pi$
$547$	22.0518	0.942865	0.471432	+	0.881902i	$0.343737\pi$
$548$	−3.69202	−0.157715
$549$	0	0
$550$	−0.429838	−0.0183284
$551$	23.4264	0.998000
$552$	0	0
$553$	4.42465	0.188155
$554$	−22.2297	−0.944451
$555$	0	0
$556$	14.6538	0.621458
$557$	−14.7373	−0.624438	−0.312219	−	0.950010i	$-0.601072\pi$
$557$	−14.7373	−0.624438	−0.312219	+	0.950010i	$0.601072\pi$
$558$	0	0
$559$	29.0798	1.22994
$560$	−0.340722	−0.0143981
$561$	0	0
$562$	3.99411	0.168481
$563$	6.78450	0.285933	0.142966	−	0.989728i	$-0.454336\pi$
$563$	6.78450	0.285933	0.142966	+	0.989728i	$0.454336\pi$
$564$	0	0
$565$	20.2608	0.852379
$566$	−12.4041	−0.521383
$567$	0	0
$568$	−31.3800	−1.31667
$569$	−10.9922	−0.460817	−0.230409	−	0.973094i	$-0.574006\pi$
$569$	−10.9922	−0.460817	−0.230409	+	0.973094i	$0.574006\pi$
$570$	0	0
$571$	−41.9268	−1.75458	−0.877290	−	0.479961i	$-0.840651\pi$
$571$	−41.9268	−1.75458	−0.877290	+	0.479961i	$0.840651\pi$
$572$	5.37409	0.224702
$573$	0	0
$574$	2.08417	0.0869914
$575$	−4.53519	−0.189130
$576$	0	0
$577$	22.2721	0.927200	0.463600	−	0.886045i	$-0.346558\pi$
$577$	22.2721	0.927200	0.463600	+	0.886045i	$0.346558\pi$
$578$	13.9826	0.581599
$579$	0	0
$580$	16.5005	0.685146
$581$	−1.86290	−0.0772861
$582$	0	0
$583$	5.18361	0.214683
$584$	−41.0985	−1.70067
$585$	0	0
$586$	−16.0440	−0.662771
$587$	37.4028	1.54378	0.771888	−	0.635758i	$-0.219312\pi$
$587$	37.4028	1.54378	0.771888	+	0.635758i	$0.219312\pi$
$588$	0	0
$589$	21.4764	0.884920
$590$	1.11163	0.0457649
$591$	0	0
$592$	0.794962	0.0326727
$593$	39.4377	1.61951	0.809756	−	0.586767i	$-0.199600\pi$
$593$	39.4377	1.61951	0.809756	+	0.586767i	$0.199600\pi$
$594$	0	0
$595$	0.0175627	0.000719999 0
$596$	0.626932	0.0256801
$597$	0	0
$598$	−28.9859	−1.18532
$599$	−20.8174	−0.850578	−0.425289	−	0.905058i	$-0.639828\pi$
$599$	−20.8174	−0.850578	−0.425289	+	0.905058i	$0.639828\pi$
$600$	0	0
$601$	5.46084	0.222752	0.111376	−	0.993778i	$-0.464474\pi$
$601$	5.46084	0.222752	0.111376	+	0.993778i	$0.464474\pi$
$602$	−2.38059	−0.0970258
$603$	0	0
$604$	−17.5519	−0.714178
$605$	−2.11599	−0.0860273
$606$	0	0
$607$	0.731366	0.0296852	0.0148426	−	0.999890i	$-0.495275\pi$
$607$	0.731366	0.0296852	0.0148426	+	0.999890i	$0.495275\pi$
$608$	−23.0400	−0.934394
$609$	0	0
$610$	−1.74046	−0.0704691
$611$	−34.6674	−1.40249
$612$	0	0
$613$	5.30018	0.214072	0.107036	−	0.994255i	$-0.465864\pi$
$613$	5.30018	0.214072	0.107036	+	0.994255i	$0.465864\pi$
$614$	−6.16708	−0.248883
$615$	0	0
$616$	−1.10479	−0.0445133
$617$	16.1004	0.648176	0.324088	−	0.946027i	$-0.394943\pi$
$617$	16.1004	0.648176	0.324088	+	0.946027i	$0.394943\pi$
$618$	0	0
$619$	16.9174	0.679968	0.339984	−	0.940431i	$-0.389578\pi$
$619$	16.9174	0.679968	0.339984	+	0.940431i	$0.389578\pi$
$620$	15.1270	0.607514
$621$	0	0
$622$	14.5947	0.585194
$623$	−5.16625	−0.206981
$624$	0	0
$625$	−22.1140	−0.884560
$626$	−10.9978	−0.439562
$627$	0	0
$628$	16.9083	0.674714
$629$	−0.0409767	−0.00163385
$630$	0	0
$631$	−43.2520	−1.72184	−0.860918	−	0.508743i	$-0.830110\pi$
$631$	−43.2520	−1.72184	−0.860918	+	0.508743i	$0.830110\pi$
$632$	29.9279	1.19047
$633$	0	0
$634$	−8.67988	−0.344722
$635$	−12.9909	−0.515530
$636$	0	0
$637$	−27.7614	−1.09995
$638$	4.84647	0.191873
$639$	0	0
$640$	−17.6152	−0.696300
$641$	−22.4987	−0.888644	−0.444322	−	0.895867i	$-0.646555\pi$
$641$	−22.4987	−0.888644	−0.444322	+	0.895867i	$0.646555\pi$
$642$	0	0
$643$	16.3425	0.644487	0.322243	−	0.946657i	$-0.395563\pi$
$643$	16.3425	0.644487	0.322243	+	0.946657i	$0.395563\pi$
$644$	−4.64183	−0.182914
$645$	0	0
$646$	0.0671608	0.00264241
$647$	18.5358	0.728719	0.364360	−	0.931258i	$-0.381288\pi$
$647$	18.5358	0.728719	0.364360	+	0.931258i	$0.381288\pi$
$648$	0	0
$649$	−0.638698	−0.0250711
$650$	1.74543	0.0684613
$651$	0	0
$652$	−32.4720	−1.27170
$653$	−38.4291	−1.50385	−0.751924	−	0.659249i	$-0.770874\pi$
$653$	−38.4291	−1.50385	−0.751924	+	0.659249i	$0.770874\pi$
$654$	0	0
$655$	11.6388	0.454767
$656$	−2.49796	−0.0975291
$657$	0	0
$658$	2.83802	0.110637
$659$	13.0048	0.506594	0.253297	−	0.967389i	$-0.418485\pi$
$659$	13.0048	0.506594	0.253297	+	0.967389i	$0.418485\pi$
$660$	0	0
$661$	17.4699	0.679500	0.339750	−	0.940516i	$-0.389658\pi$
$661$	17.4699	0.679500	0.339750	+	0.940516i	$0.389658\pi$
$662$	4.63562	0.180168
$663$	0	0
$664$	−12.6005	−0.488994
$665$	−3.40006	−0.131848
$666$	0	0
$667$	51.1347	1.97994
$668$	5.23548	0.202567
$669$	0	0
$670$	26.4610	1.02228
$671$	1.00000	0.0386046
$672$	0	0
$673$	−21.0297	−0.810637	−0.405319	−	0.914175i	$-0.632839\pi$
$673$	−21.0297	−0.810637	−0.405319	+	0.914175i	$0.632839\pi$
$674$	−17.3699	−0.669063
$675$	0	0
$676$	−4.61751	−0.177596
$677$	−38.9808	−1.49816	−0.749078	−	0.662482i	$-0.769503\pi$
$677$	−38.9808	−1.49816	−0.749078	+	0.662482i	$0.769503\pi$
$678$	0	0
$679$	−0.243736	−0.00935373
$680$	0.118792	0.00455547
$681$	0	0
$682$	4.44304	0.170133
$683$	12.0413	0.460746	0.230373	−	0.973102i	$-0.426005\pi$
$683$	12.0413	0.460746	0.230373	+	0.973102i	$0.426005\pi$
$684$	0	0
$685$	−5.90296	−0.225540
$686$	4.59962	0.175614
$687$	0	0
$688$	2.85325	0.108779
$689$	−21.0489	−0.801899
$690$	0	0
$691$	20.6612	0.785988	0.392994	−	0.919541i	$-0.371439\pi$
$691$	20.6612	0.785988	0.392994	+	0.919541i	$0.371439\pi$
$692$	−28.5544	−1.08548
$693$	0	0
$694$	−20.2084	−0.767099
$695$	23.4291	0.888715
$696$	0	0
$697$	0.128759	0.00487708
$698$	−3.21770	−0.121792
$699$	0	0
$700$	0.279514	0.0105647
$701$	11.9387	0.450920	0.225460	−	0.974252i	$-0.427612\pi$
$701$	11.9387	0.450920	0.225460	+	0.974252i	$0.427612\pi$
$702$	0	0
$703$	7.93291	0.299195
$704$	−3.96966	−0.149612
$705$	0	0
$706$	−23.4545	−0.882722
$707$	5.92260	0.222742
$708$	0	0
$709$	−15.3955	−0.578189	−0.289095	−	0.957300i	$-0.593354\pi$
$709$	−15.3955	−0.578189	−0.289095	+	0.957300i	$0.593354\pi$
$710$	−19.9792	−0.749804
$711$	0	0
$712$	−34.9440	−1.30958
$713$	46.8782	1.75560
$714$	0	0
$715$	8.59233	0.321335
$716$	−17.9531	−0.670939
$717$	0	0
$718$	−22.4043	−0.836120
$719$	−18.8466	−0.702859	−0.351430	−	0.936214i	$-0.614304\pi$
$719$	−18.8466	−0.702859	−0.351430	+	0.936214i	$0.614304\pi$
$720$	0	0
$721$	4.51296	0.168071
$722$	2.62596	0.0977280
$723$	0	0
$724$	27.6602	1.02798
$725$	−3.07915	−0.114357
$726$	0	0
$727$	−34.5785	−1.28245	−0.641223	−	0.767354i	$-0.721573\pi$
$727$	−34.5785	−1.28245	−0.641223	+	0.767354i	$0.721573\pi$
$728$	4.48619	0.166269
$729$	0	0
$730$	−26.1668	−0.968476
$731$	−0.147072	−0.00543965
$732$	0	0
$733$	−18.9209	−0.698859	−0.349430	−	0.936963i	$-0.613625\pi$
$733$	−18.9209	−0.698859	−0.349430	+	0.936963i	$0.613625\pi$
$734$	31.0175	1.14488
$735$	0	0
$736$	−50.2911	−1.85375
$737$	−15.2035	−0.560027
$738$	0	0
$739$	15.6521	0.575773	0.287886	−	0.957665i	$-0.407047\pi$
$739$	15.6521	0.575773	0.287886	+	0.957665i	$0.407047\pi$
$740$	5.58757	0.205403
$741$	0	0
$742$	1.72315	0.0632589
$743$	−12.8846	−0.472691	−0.236346	−	0.971669i	$-0.575950\pi$
$743$	−12.8846	−0.472691	−0.236346	+	0.971669i	$0.575950\pi$
$744$	0	0
$745$	1.00237	0.0367239
$746$	−3.97868	−0.145670
$747$	0	0
$748$	−0.0271796	−0.000993785 0
$749$	3.45543	0.126259
$750$	0	0
$751$	4.34890	0.158693	0.0793467	−	0.996847i	$-0.474717\pi$
$751$	4.34890	0.158693	0.0793467	+	0.996847i	$0.474717\pi$
$752$	−3.40149	−0.124039
$753$	0	0
$754$	−19.6799	−0.716699
$755$	−28.0628	−1.02131
$756$	0	0
$757$	39.4539	1.43398	0.716989	−	0.697085i	$-0.245520\pi$
$757$	39.4539	1.43398	0.716989	+	0.697085i	$0.245520\pi$
$758$	29.0200	1.05405
$759$	0	0
$760$	−22.9977	−0.834213
$761$	38.6312	1.40038	0.700190	−	0.713957i	$-0.253099\pi$
$761$	38.6312	1.40038	0.700190	+	0.713957i	$0.253099\pi$
$762$	0	0
$763$	5.67597	0.205484
$764$	3.24828	0.117519
$765$	0	0
$766$	15.0710	0.544538
$767$	2.59354	0.0936472
$768$	0	0
$769$	−29.2863	−1.05609	−0.528046	−	0.849216i	$-0.677075\pi$
$769$	−29.2863	−1.05609	−0.528046	+	0.849216i	$0.677075\pi$
$770$	−0.703404	−0.0253489
$771$	0	0
$772$	11.0039	0.396039
$773$	−1.46255	−0.0526041	−0.0263020	−	0.999654i	$-0.508373\pi$
$773$	−1.46255	−0.0526041	−0.0263020	+	0.999654i	$0.508373\pi$
$774$	0	0
$775$	−2.82284	−0.101399
$776$	−1.64861	−0.0591816
$777$	0	0
$778$	−28.3268	−1.01557
$779$	−24.9271	−0.893107
$780$	0	0
$781$	11.4793	0.410760
$782$	0.146597	0.00524230
$783$	0	0
$784$	−2.72389	−0.0972817
$785$	27.0337	0.964874
$786$	0	0
$787$	25.9663	0.925600	0.462800	−	0.886463i	$-0.346845\pi$
$787$	25.9663	0.925600	0.462800	+	0.886463i	$0.346845\pi$
$788$	24.8814	0.886363
$789$	0	0
$790$	19.0547	0.677934
$791$	−3.86976	−0.137593
$792$	0	0
$793$	−4.06066	−0.144198
$794$	5.48797	0.194761
$795$	0	0
$796$	10.3780	0.367838
$797$	31.4932	1.11555	0.557773	−	0.829993i	$-0.311656\pi$
$797$	31.4932	1.11555	0.557773	+	0.829993i	$0.311656\pi$
$798$	0	0
$799$	0.175331	0.00620278
$800$	3.02835	0.107068
$801$	0	0
$802$	−15.9689	−0.563883
$803$	15.0344	0.530553
$804$	0	0
$805$	−7.42156	−0.261576
$806$	−18.0417	−0.635492
$807$	0	0
$808$	40.0599	1.40930
$809$	−8.19184	−0.288010	−0.144005	−	0.989577i	$-0.545998\pi$
$809$	−8.19184	−0.288010	−0.144005	+	0.989577i	$0.545998\pi$
$810$	0	0
$811$	8.90734	0.312779	0.156389	−	0.987695i	$-0.450015\pi$
$811$	8.90734	0.312779	0.156389	+	0.987695i	$0.450015\pi$
$812$	−3.15155	−0.110598
$813$	0	0
$814$	1.64116	0.0575227
$815$	−51.9177	−1.81860
$816$	0	0
$817$	28.4725	0.996126
$818$	−0.631577	−0.0220826
$819$	0	0
$820$	−17.5575	−0.613135
$821$	25.1573	0.877995	0.438997	−	0.898488i	$-0.355334\pi$
$821$	25.1573	0.877995	0.438997	+	0.898488i	$0.355334\pi$
$822$	0	0
$823$	50.8751	1.77340	0.886698	−	0.462350i	$-0.152994\pi$
$823$	50.8751	1.77340	0.886698	+	0.462350i	$0.152994\pi$
$824$	30.5252	1.06340
$825$	0	0
$826$	−0.212318	−0.00738748
$827$	−43.4799	−1.51195	−0.755973	−	0.654603i	$-0.772836\pi$
$827$	−43.4799	−1.51195	−0.755973	+	0.654603i	$0.772836\pi$
$828$	0	0
$829$	−34.9915	−1.21530	−0.607652	−	0.794203i	$-0.707889\pi$
$829$	−34.9915	−1.21530	−0.607652	+	0.794203i	$0.707889\pi$
$830$	−8.02254	−0.278466
$831$	0	0
$832$	16.1195	0.558842
$833$	0.140404	0.00486471
$834$	0	0
$835$	8.37072	0.289681
$836$	5.26185	0.181985
$837$	0	0
$838$	14.2574	0.492514
$839$	−31.5500	−1.08923	−0.544614	−	0.838687i	$-0.683324\pi$
$839$	−31.5500	−1.08923	−0.544614	+	0.838687i	$0.683324\pi$
$840$	0	0
$841$	5.71771	0.197162
$842$	−9.92384	−0.341998
$843$	0	0
$844$	33.9640	1.16909
$845$	−7.38267	−0.253972
$846$	0	0
$847$	0.404149	0.0138867
$848$	−2.06527	−0.0709217
$849$	0	0
$850$	−0.00882755	−0.000302782 0
$851$	17.3158	0.593577
$852$	0	0
$853$	−4.60871	−0.157799	−0.0788997	−	0.996883i	$-0.525141\pi$
$853$	−4.60871	−0.157799	−0.0788997	+	0.996883i	$0.525141\pi$
$854$	0.332423	0.0113753
$855$	0	0
$856$	23.3722	0.798846
$857$	−49.6081	−1.69458	−0.847290	−	0.531131i	$-0.821767\pi$
$857$	−49.6081	−1.69458	−0.847290	+	0.531131i	$0.821767\pi$
$858$	0	0
$859$	−57.5877	−1.96487	−0.982433	−	0.186617i	$-0.940248\pi$
$859$	−57.5877	−1.96487	−0.982433	+	0.186617i	$0.940248\pi$
$860$	20.0547	0.683859
$861$	0	0
$862$	−24.0186	−0.818077
$863$	40.8905	1.39193	0.695964	−	0.718076i	$-0.254977\pi$
$863$	40.8905	1.39193	0.695964	+	0.718076i	$0.254977\pi$
$864$	0	0
$865$	−45.6540	−1.55228
$866$	29.4496	1.00074
$867$	0	0
$868$	−2.88921	−0.0980663
$869$	−10.9481	−0.371388
$870$	0	0
$871$	61.7362	2.09185
$872$	38.3918	1.30011
$873$	0	0
$874$	−28.3806	−0.959987
$875$	4.72278	0.159659
$876$	0	0
$877$	−7.81665	−0.263949	−0.131975	−	0.991253i	$-0.542132\pi$
$877$	−7.81665	−0.263949	−0.131975	+	0.991253i	$0.542132\pi$
$878$	−33.6071	−1.13418
$879$	0	0
$880$	0.843060	0.0284196
$881$	16.7321	0.563719	0.281860	−	0.959456i	$-0.409049\pi$
$881$	16.7321	0.563719	0.281860	+	0.959456i	$0.409049\pi$
$882$	0	0
$883$	−18.3230	−0.616617	−0.308308	−	0.951286i	$-0.599763\pi$
$883$	−18.3230	−0.616617	−0.308308	+	0.951286i	$0.599763\pi$
$884$	0.110367	0.00371205
$885$	0	0
$886$	−23.0171	−0.773275
$887$	−36.8957	−1.23884	−0.619419	−	0.785061i	$-0.712632\pi$
$887$	−36.8957	−1.23884	−0.619419	+	0.785061i	$0.712632\pi$
$888$	0	0
$889$	2.48124	0.0832180
$890$	−22.2483	−0.745766
$891$	0	0
$892$	−1.74141	−0.0583066
$893$	−33.9434	−1.13587
$894$	0	0
$895$	−28.7042	−0.959476
$896$	3.36445	0.112398
$897$	0	0
$898$	−20.5841	−0.686900
$899$	31.8278	1.06152
$900$	0	0
$901$	0.106455	0.00354654
$902$	−5.15693	−0.171707
$903$	0	0
$904$	−26.1747	−0.870558
$905$	44.2243	1.47006
$906$	0	0
$907$	−1.15826	−0.0384593	−0.0192297	−	0.999815i	$-0.506121\pi$
$907$	−1.15826	−0.0384593	−0.0192297	+	0.999815i	$0.506121\pi$
$908$	9.70720	0.322145
$909$	0	0
$910$	2.85629	0.0946850
$911$	39.7016	1.31537	0.657686	−	0.753292i	$-0.271535\pi$
$911$	39.7016	1.31537	0.657686	+	0.753292i	$0.271535\pi$
$912$	0	0
$913$	4.60944	0.152550
$914$	−3.04635	−0.100764
$915$	0	0
$916$	6.11112	0.201917
$917$	−2.22299	−0.0734096
$918$	0	0
$919$	34.7872	1.14752	0.573762	−	0.819022i	$-0.305483\pi$
$919$	34.7872	1.14752	0.573762	+	0.819022i	$0.305483\pi$
$920$	−50.1988	−1.65500
$921$	0	0
$922$	−12.9977	−0.428057
$923$	−46.6134	−1.53430
$924$	0	0
$925$	−1.04269	−0.0342835
$926$	20.7663	0.682424
$927$	0	0
$928$	−34.1449	−1.12086
$929$	16.5877	0.544226	0.272113	−	0.962265i	$-0.412278\pi$
$929$	16.5877	0.544226	0.272113	+	0.962265i	$0.412278\pi$
$930$	0	0
$931$	−27.1816	−0.890841
$932$	38.5461	1.26262
$933$	0	0
$934$	−14.2120	−0.465030
$935$	−0.0434559	−0.00142116
$936$	0	0
$937$	29.0025	0.947470	0.473735	−	0.880667i	$-0.342905\pi$
$937$	29.0025	0.947470	0.473735	+	0.880667i	$0.342905\pi$
$938$	−5.05399	−0.165018
$939$	0	0
$940$	−23.9081	−0.779797
$941$	−41.2410	−1.34442	−0.672210	−	0.740361i	$-0.734655\pi$
$941$	−41.2410	−1.34442	−0.672210	+	0.740361i	$0.734655\pi$
$942$	0	0
$943$	−54.4103	−1.77184
$944$	0.254472	0.00828236
$945$	0	0
$946$	5.89039	0.191513
$947$	14.9009	0.484214	0.242107	−	0.970250i	$-0.422161\pi$
$947$	14.9009	0.484214	0.242107	+	0.970250i	$0.422161\pi$
$948$	0	0
$949$	−61.0497	−1.98176
$950$	1.70897	0.0554465
$951$	0	0
$952$	−0.0226890	−0.000735355 0
$953$	29.4563	0.954185	0.477092	−	0.878853i	$-0.341691\pi$
$953$	29.4563	0.954185	0.477092	+	0.878853i	$0.341691\pi$
$954$	0	0
$955$	5.19349	0.168057
$956$	−40.5135	−1.31030
$957$	0	0
$958$	8.95227	0.289235
$959$	1.12745	0.0364073
$960$	0	0
$961$	−1.82161	−0.0587617
$962$	−6.66420	−0.214862
$963$	0	0
$964$	−29.6247	−0.954147
$965$	17.5935	0.566356
$966$	0	0
$967$	39.2447	1.26202	0.631012	−	0.775773i	$-0.282640\pi$
$967$	39.2447	1.26202	0.631012	+	0.775773i	$0.282640\pi$
$968$	2.73363	0.0878620
$969$	0	0
$970$	−1.04964	−0.0337020
$971$	−19.0146	−0.610207	−0.305104	−	0.952319i	$-0.598691\pi$
$971$	−19.0146	−0.610207	−0.305104	+	0.952319i	$0.598691\pi$
$972$	0	0
$973$	−4.47489	−0.143458
$974$	−32.0919	−1.02829
$975$	0	0
$976$	−0.398423	−0.0127532
$977$	−15.7282	−0.503188	−0.251594	−	0.967833i	$-0.580955\pi$
$977$	−15.7282	−0.503188	−0.251594	+	0.967833i	$0.580955\pi$
$978$	0	0
$979$	12.7830	0.408548
$980$	−19.1455	−0.611579
$981$	0	0
$982$	24.7023	0.788281
$983$	41.9230	1.33714	0.668568	−	0.743651i	$-0.266908\pi$
$983$	41.9230	1.33714	0.668568	+	0.743651i	$0.266908\pi$
$984$	0	0
$985$	39.7815	1.26754
$986$	0.0995315	0.00316973
$987$	0	0
$988$	−21.3666	−0.679763
$989$	62.1491	1.97623
$990$	0	0
$991$	17.5601	0.557816	0.278908	−	0.960318i	$-0.410027\pi$
$991$	17.5601	0.557816	0.278908	+	0.960318i	$0.410027\pi$
$992$	−31.3027	−0.993861
$993$	0	0
$994$	3.81597	0.121035
$995$	16.5928	0.526027
$996$	0	0
$997$	−11.7778	−0.373008	−0.186504	−	0.982454i	$-0.559716\pi$
$997$	−11.7778	−0.373008	−0.186504	+	0.982454i	$0.559716\pi$
$998$	−21.0649	−0.666799
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.6		13
3.2	odd	2		2013.2.a.e.1.8	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.8	✓	13	3.2	odd	2
6039.2.a.i.1.6		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)

\(f(q)\)	\(=\)	\(q-0.822526 q^{2} -1.32345 q^{4} -2.11599 q^{5} +0.404149 q^{7} +2.73363 q^{8} +O(q^{10})\)	\(q-0.822526 q^{2} -1.32345 q^{4} -2.11599 q^{5} +0.404149 q^{7} +2.73363 q^{8} +1.74046 q^{10} -1.00000 q^{11} +4.06066 q^{13} -0.332423 q^{14} +0.398423 q^{16} -0.0205369 q^{17} +3.97586 q^{19} +2.80041 q^{20} +0.822526 q^{22} +8.67841 q^{23} -0.522583 q^{25} -3.34000 q^{26} -0.534871 q^{28} +5.89217 q^{29} +5.40170 q^{31} -5.79496 q^{32} +0.0168922 q^{34} -0.855175 q^{35} +1.99527 q^{37} -3.27025 q^{38} -5.78433 q^{40} -6.26962 q^{41} +7.16134 q^{43} +1.32345 q^{44} -7.13822 q^{46} -8.53737 q^{47} -6.83666 q^{49} +0.429838 q^{50} -5.37409 q^{52} -5.18361 q^{53} +2.11599 q^{55} +1.10479 q^{56} -4.84647 q^{58} +0.638698 q^{59} -1.00000 q^{61} -4.44304 q^{62} +3.96966 q^{64} -8.59233 q^{65} +15.2035 q^{67} +0.0271796 q^{68} +0.703404 q^{70} -11.4793 q^{71} -15.0344 q^{73} -1.64116 q^{74} -5.26185 q^{76} -0.404149 q^{77} +10.9481 q^{79} -0.843060 q^{80} +5.15693 q^{82} -4.60944 q^{83} +0.0434559 q^{85} -5.89039 q^{86} -2.73363 q^{88} -12.7830 q^{89} +1.64111 q^{91} -11.4855 q^{92} +7.02221 q^{94} -8.41288 q^{95} -0.603085 q^{97} +5.62333 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8} + 6 q^{10} - 13 q^{11} + 13 q^{13} - q^{14} + 18 q^{16} - 17 q^{17} + 14 q^{19} + 7 q^{20} + 2 q^{22} - 7 q^{23} + 18 q^{25} + 10 q^{26} + 19 q^{28} + 6 q^{29} + 27 q^{31} - 5 q^{32} + 6 q^{34} - 14 q^{35} + 10 q^{37} - 2 q^{38} + 8 q^{40} - 3 q^{41} + 29 q^{43} - 16 q^{44} - 24 q^{46} - 8 q^{47} + 8 q^{49} + 27 q^{50} + 37 q^{52} + 24 q^{53} + 3 q^{55} - 24 q^{56} - 5 q^{58} - 13 q^{59} - 13 q^{61} - 39 q^{62} + 47 q^{64} + 11 q^{65} + 44 q^{67} + 8 q^{68} - 12 q^{70} - 3 q^{71} + 48 q^{73} + 22 q^{74} + 47 q^{76} - 11 q^{77} - 17 q^{79} + 26 q^{80} + 56 q^{82} - 50 q^{83} + 8 q^{85} - 18 q^{86} + 9 q^{88} + 15 q^{89} + 47 q^{91} - 14 q^{92} + 45 q^{94} + q^{95} + 27 q^{97} - 47 q^{98}+O(q^{100}) \) 13 * q - 2 * q^2 + 16 * q^4 - 3 * q^5 + 11 * q^7 - 9 * q^8 + 6 * q^10 - 13 * q^11 + 13 * q^13 - q^14 + 18 * q^16 - 17 * q^17 + 14 * q^19 + 7 * q^20 + 2 * q^22 - 7 * q^23 + 18 * q^25 + 10 * q^26 + 19 * q^28 + 6 * q^29 + 27 * q^31 - 5 * q^32 + 6 * q^34 - 14 * q^35 + 10 * q^37 - 2 * q^38 + 8 * q^40 - 3 * q^41 + 29 * q^43 - 16 * q^44 - 24 * q^46 - 8 * q^47 + 8 * q^49 + 27 * q^50 + 37 * q^52 + 24 * q^53 + 3 * q^55 - 24 * q^56 - 5 * q^58 - 13 * q^59 - 13 * q^61 - 39 * q^62 + 47 * q^64 + 11 * q^65 + 44 * q^67 + 8 * q^68 - 12 * q^70 - 3 * q^71 + 48 * q^73 + 22 * q^74 + 47 * q^76 - 11 * q^77 - 17 * q^79 + 26 * q^80 + 56 * q^82 - 50 * q^83 + 8 * q^85 - 18 * q^86 + 9 * q^88 + 15 * q^89 + 47 * q^91 - 14 * q^92 + 45 * q^94 + q^95 + 27 * q^97 - 47 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

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