LMFDB - Embedded newform 6039.2.a.i.1.9

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			1.9
Root			$-0.948254$ 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)

$f(q)$	$=$	$q+0.948254 q^{2} -1.10081 q^{4} +2.25122 q^{5} +5.24025 q^{7} -2.94036 q^{8} +O(q^{10})$	\(q+0.948254 q^{2} -1.10081 q^{4} +2.25122 q^{5} +5.24025 q^{7} -2.94036 q^{8} +2.13473 q^{10} -1.00000 q^{11} +5.44016 q^{13} +4.96909 q^{14} -0.586583 q^{16} -4.46012 q^{17} +2.98902 q^{19} -2.47818 q^{20} -0.948254 q^{22} -0.200434 q^{23} +0.0679998 q^{25} +5.15865 q^{26} -5.76854 q^{28} +1.81818 q^{29} +0.728977 q^{31} +5.32449 q^{32} -4.22933 q^{34} +11.7970 q^{35} -3.44866 q^{37} +2.83436 q^{38} -6.61940 q^{40} -2.51230 q^{41} +7.72081 q^{43} +1.10081 q^{44} -0.190062 q^{46} +9.29908 q^{47} +20.4602 q^{49} +0.0644811 q^{50} -5.98860 q^{52} +1.48505 q^{53} -2.25122 q^{55} -15.4082 q^{56} +1.72409 q^{58} -1.65543 q^{59} -1.00000 q^{61} +0.691255 q^{62} +6.22214 q^{64} +12.2470 q^{65} +16.1730 q^{67} +4.90976 q^{68} +11.1865 q^{70} -1.39117 q^{71} -8.30506 q^{73} -3.27021 q^{74} -3.29036 q^{76} -5.24025 q^{77} -11.7081 q^{79} -1.32053 q^{80} -2.38230 q^{82} +1.10951 q^{83} -10.0407 q^{85} +7.32129 q^{86} +2.94036 q^{88} -6.80479 q^{89} +28.5078 q^{91} +0.220640 q^{92} +8.81789 q^{94} +6.72896 q^{95} -9.30845 q^{97} +19.4015 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	0.948254	0.670517	0.335259	−	0.942126i	$-0.391176\pi$
$2$	0.948254	0.670517	0.335259	+	0.942126i	$0.391176\pi$
$3$	0	0
$3$	0	0
$4$	−1.10081	−0.550407
$5$	2.25122	1.00678	0.503389	−	0.864060i	$-0.332086\pi$
$5$	2.25122	1.00678	0.503389	+	0.864060i	$0.332086\pi$
$6$	0	0
$7$	5.24025	1.98063	0.990315	−	0.138842i	$-0.0443379\pi$
$7$	5.24025	1.98063	0.990315	+	0.138842i	$0.0443379\pi$
$8$	−2.94036	−1.03957
$9$	0	0
$10$	2.13473	0.675061
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	5.44016	1.50883	0.754414	−	0.656399i	$-0.227921\pi$
$13$	5.44016	1.50883	0.754414	+	0.656399i	$0.227921\pi$
$14$	4.96909	1.32805
$15$	0	0
$16$	−0.586583	−0.146646
$17$	−4.46012	−1.08174	−0.540869	−	0.841107i	$-0.681905\pi$
$17$	−4.46012	−1.08174	−0.540869	+	0.841107i	$0.681905\pi$
$18$	0	0
$19$	2.98902	0.685729	0.342865	−	0.939385i	$-0.388603\pi$
$19$	2.98902	0.685729	0.342865	+	0.939385i	$0.388603\pi$
$20$	−2.47818	−0.554137
$21$	0	0
$22$	−0.948254	−0.202169
$23$	−0.200434	−0.0417934	−0.0208967	−	0.999782i	$-0.506652\pi$
$23$	−0.200434	−0.0417934	−0.0208967	+	0.999782i	$0.506652\pi$
$24$	0	0
$25$	0.0679998	0.0136000
$26$	5.15865	1.01169
$27$	0	0
$28$	−5.76854	−1.09015
$29$	1.81818	0.337627	0.168813	−	0.985648i	$-0.446006\pi$
$29$	1.81818	0.337627	0.168813	+	0.985648i	$0.446006\pi$
$30$	0	0
$31$	0.728977	0.130928	0.0654640	−	0.997855i	$-0.479147\pi$
$31$	0.728977	0.130928	0.0654640	+	0.997855i	$0.479147\pi$
$32$	5.32449	0.941246
$33$	0	0
$34$	−4.22933	−0.725324
$35$	11.7970	1.99405
$36$	0	0
$37$	−3.44866	−0.566956	−0.283478	−	0.958979i	$-0.591488\pi$
$37$	−3.44866	−0.566956	−0.283478	+	0.958979i	$0.591488\pi$
$38$	2.83436	0.459793
$39$	0	0
$40$	−6.61940	−1.04662
$41$	−2.51230	−0.392355	−0.196178	−	0.980568i	$-0.562853\pi$
$41$	−2.51230	−0.392355	−0.196178	+	0.980568i	$0.562853\pi$
$42$	0	0
$43$	7.72081	1.17741	0.588706	−	0.808347i	$-0.299638\pi$
$43$	7.72081	1.17741	0.588706	+	0.808347i	$0.299638\pi$
$44$	1.10081	0.165954
$45$	0	0
$46$	−0.190062	−0.0280232
$47$	9.29908	1.35641	0.678205	−	0.734873i	$-0.262758\pi$
$47$	9.29908	1.35641	0.678205	+	0.734873i	$0.262758\pi$
$48$	0	0
$49$	20.4602	2.92289
$50$	0.0644811	0.00911900
$51$	0	0
$52$	−5.98860	−0.830469
$53$	1.48505	0.203987	0.101994	−	0.994785i	$-0.467478\pi$
$53$	1.48505	0.203987	0.101994	+	0.994785i	$0.467478\pi$
$54$	0	0
$55$	−2.25122	−0.303555
$56$	−15.4082	−2.05901
$57$	0	0
$58$	1.72409	0.226385
$59$	−1.65543	−0.215518	−0.107759	−	0.994177i	$-0.534368\pi$
$59$	−1.65543	−0.215518	−0.107759	+	0.994177i	$0.534368\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	0.691255	0.0877895
$63$	0	0
$64$	6.22214	0.777767
$65$	12.2470	1.51905
$66$	0	0
$67$	16.1730	1.97585	0.987923	−	0.154948i	$-0.0495211\pi$
$67$	16.1730	1.97585	0.987923	+	0.154948i	$0.0495211\pi$
$68$	4.90976	0.595396
$69$	0	0
$70$	11.1865	1.33705
$71$	−1.39117	−0.165101	−0.0825506	−	0.996587i	$-0.526307\pi$
$71$	−1.39117	−0.165101	−0.0825506	+	0.996587i	$0.526307\pi$
$72$	0	0
$73$	−8.30506	−0.972034	−0.486017	−	0.873949i	$-0.661551\pi$
$73$	−8.30506	−0.972034	−0.486017	+	0.873949i	$0.661551\pi$
$74$	−3.27021	−0.380154
$75$	0	0
$76$	−3.29036	−0.377430
$77$	−5.24025	−0.597182
$78$	0	0
$79$	−11.7081	−1.31726	−0.658630	−	0.752467i	$-0.728864\pi$
$79$	−11.7081	−1.31726	−0.658630	+	0.752467i	$0.728864\pi$
$80$	−1.32053	−0.147640
$81$	0	0
$82$	−2.38230	−0.263081
$83$	1.10951	0.121785	0.0608924	−	0.998144i	$-0.480605\pi$
$83$	1.10951	0.121785	0.0608924	+	0.998144i	$0.480605\pi$
$84$	0	0
$85$	−10.0407	−1.08907
$86$	7.32129	0.789475
$87$	0	0
$88$	2.94036	0.313443
$89$	−6.80479	−0.721306	−0.360653	−	0.932700i	$-0.617446\pi$
$89$	−6.80479	−0.721306	−0.360653	+	0.932700i	$0.617446\pi$
$90$	0	0
$91$	28.5078	2.98843
$92$	0.220640	0.0230034
$93$	0	0
$94$	8.81789	0.909496
$95$	6.72896	0.690377
$96$	0	0
$97$	−9.30845	−0.945130	−0.472565	−	0.881296i	$-0.656672\pi$
$97$	−9.30845	−0.945130	−0.472565	+	0.881296i	$0.656672\pi$
$98$	19.4015	1.95985
$99$	0	0
$100$	−0.0748551	−0.00748551
$101$	−9.37877	−0.933223	−0.466611	−	0.884462i	$-0.654525\pi$
$101$	−9.37877	−0.933223	−0.466611	+	0.884462i	$0.654525\pi$
$102$	0	0
$103$	−9.63463	−0.949328	−0.474664	−	0.880167i	$-0.657430\pi$
$103$	−9.63463	−0.949328	−0.474664	+	0.880167i	$0.657430\pi$
$104$	−15.9960	−1.56854
$105$	0	0
$106$	1.40820	0.136777
$107$	−3.02397	−0.292338	−0.146169	−	0.989260i	$-0.546694\pi$
$107$	−3.02397	−0.292338	−0.146169	+	0.989260i	$0.546694\pi$
$108$	0	0
$109$	3.74875	0.359065	0.179533	−	0.983752i	$-0.442541\pi$
$109$	3.74875	0.359065	0.179533	+	0.983752i	$0.442541\pi$
$110$	−2.13473	−0.203539
$111$	0	0
$112$	−3.07384	−0.290451
$113$	3.50360	0.329591	0.164795	−	0.986328i	$-0.447304\pi$
$113$	3.50360	0.329591	0.164795	+	0.986328i	$0.447304\pi$
$114$	0	0
$115$	−0.451221	−0.0420766
$116$	−2.00147	−0.185832
$117$	0	0
$118$	−1.56977	−0.144509
$119$	−23.3722	−2.14252
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.948254	−0.0858509
$123$	0	0
$124$	−0.802467	−0.0720637
$125$	−11.1030	−0.993085
$126$	0	0
$127$	5.45206	0.483792	0.241896	−	0.970302i	$-0.422231\pi$
$127$	5.45206	0.483792	0.241896	+	0.970302i	$0.422231\pi$
$128$	−4.74881	−0.419740
$129$	0	0
$130$	11.6133	1.01855
$131$	16.4647	1.43853	0.719265	−	0.694736i	$-0.244479\pi$
$131$	16.4647	1.43853	0.719265	+	0.694736i	$0.244479\pi$
$132$	0	0
$133$	15.6632	1.35818
$134$	15.3361	1.32484
$135$	0	0
$136$	13.1144	1.12455
$137$	−1.13223	−0.0967330	−0.0483665	−	0.998830i	$-0.515402\pi$
$137$	−1.13223	−0.0967330	−0.0483665	+	0.998830i	$0.515402\pi$
$138$	0	0
$139$	17.6358	1.49585	0.747923	−	0.663785i	$-0.231051\pi$
$139$	17.6358	1.49585	0.747923	+	0.663785i	$0.231051\pi$
$140$	−12.9863	−1.09754
$141$	0	0
$142$	−1.31918	−0.110703
$143$	−5.44016	−0.454929
$144$	0	0
$145$	4.09312	0.339915
$146$	−7.87531	−0.651766
$147$	0	0
$148$	3.79633	0.312057
$149$	−18.5492	−1.51961	−0.759804	−	0.650152i	$-0.774705\pi$
$149$	−18.5492	−1.51961	−0.759804	+	0.650152i	$0.774705\pi$
$150$	0	0
$151$	−22.4227	−1.82474	−0.912368	−	0.409371i	$-0.865748\pi$
$151$	−22.4227	−1.82474	−0.912368	+	0.409371i	$0.865748\pi$
$152$	−8.78881	−0.712867
$153$	0	0
$154$	−4.96909	−0.400421
$155$	1.64109	0.131815
$156$	0	0
$157$	−7.31110	−0.583489	−0.291745	−	0.956496i	$-0.594236\pi$
$157$	−7.31110	−0.583489	−0.291745	+	0.956496i	$0.594236\pi$
$158$	−11.1022	−0.883246
$159$	0	0
$160$	11.9866	0.947625
$161$	−1.05032	−0.0827772
$162$	0	0
$163$	−1.04410	−0.0817803	−0.0408902	−	0.999164i	$-0.513019\pi$
$163$	−1.04410	−0.0817803	−0.0408902	+	0.999164i	$0.513019\pi$
$164$	2.76557	0.215955
$165$	0	0
$166$	1.05210	0.0816588
$167$	0.242267	0.0187472	0.00937360	−	0.999956i	$-0.497016\pi$
$167$	0.242267	0.0187472	0.00937360	+	0.999956i	$0.497016\pi$
$168$	0	0
$169$	16.5953	1.27656
$170$	−9.52116	−0.730240
$171$	0	0
$172$	−8.49917	−0.648056
$173$	23.7589	1.80636	0.903179	−	0.429264i	$-0.141227\pi$
$173$	23.7589	1.80636	0.903179	+	0.429264i	$0.141227\pi$
$174$	0	0
$175$	0.356336	0.0269365
$176$	0.586583	0.0442153
$177$	0	0
$178$	−6.45267	−0.483648
$179$	2.79602	0.208984	0.104492	−	0.994526i	$-0.466678\pi$
$179$	2.79602	0.208984	0.104492	+	0.994526i	$0.466678\pi$
$180$	0	0
$181$	12.5589	0.933499	0.466750	−	0.884390i	$-0.345425\pi$
$181$	12.5589	0.933499	0.466750	+	0.884390i	$0.345425\pi$
$182$	27.0326	2.00379
$183$	0	0
$184$	0.589348	0.0434473
$185$	−7.76370	−0.570799
$186$	0	0
$187$	4.46012	0.326156
$188$	−10.2366	−0.746577
$189$	0	0
$190$	6.38076	0.462909
$191$	2.74577	0.198677	0.0993386	−	0.995054i	$-0.468327\pi$
$191$	2.74577	0.198677	0.0993386	+	0.995054i	$0.468327\pi$
$192$	0	0
$193$	26.5728	1.91275	0.956377	−	0.292135i	$-0.0943658\pi$
$193$	26.5728	1.91275	0.956377	+	0.292135i	$0.0943658\pi$
$194$	−8.82678	−0.633726
$195$	0	0
$196$	−22.5229	−1.60878
$197$	8.49601	0.605316	0.302658	−	0.953099i	$-0.402126\pi$
$197$	8.49601	0.605316	0.302658	+	0.953099i	$0.402126\pi$
$198$	0	0
$199$	−7.89518	−0.559674	−0.279837	−	0.960047i	$-0.590280\pi$
$199$	−7.89518	−0.559674	−0.279837	+	0.960047i	$0.590280\pi$
$200$	−0.199944	−0.0141382
$201$	0	0
$202$	−8.89346	−0.625742
$203$	9.52770	0.668714
$204$	0	0
$205$	−5.65574	−0.395014
$206$	−9.13608	−0.636541
$207$	0	0
$208$	−3.19110	−0.221263
$209$	−2.98902	−0.206755
$210$	0	0
$211$	16.1525	1.11199	0.555993	−	0.831187i	$-0.312338\pi$
$211$	16.1525	1.11199	0.555993	+	0.831187i	$0.312338\pi$
$212$	−1.63476	−0.112276
$213$	0	0
$214$	−2.86750	−0.196018
$215$	17.3813	1.18539
$216$	0	0
$217$	3.82002	0.259320
$218$	3.55477	0.240759
$219$	0	0
$220$	2.47818	0.167079
$221$	−24.2638	−1.63216
$222$	0	0
$223$	−22.7987	−1.52672	−0.763358	−	0.645976i	$-0.776451\pi$
$223$	−22.7987	−1.52672	−0.763358	+	0.645976i	$0.776451\pi$
$224$	27.9017	1.86426
$225$	0	0
$226$	3.32230	0.220996
$227$	−15.8784	−1.05389	−0.526943	−	0.849901i	$-0.676662\pi$
$227$	−15.8784	−1.05389	−0.526943	+	0.849901i	$0.676662\pi$
$228$	0	0
$229$	25.2540	1.66883	0.834414	−	0.551138i	$-0.185806\pi$
$229$	25.2540	1.66883	0.834414	+	0.551138i	$0.185806\pi$
$230$	−0.427873	−0.0282131
$231$	0	0
$232$	−5.34609	−0.350988
$233$	11.6859	0.765568	0.382784	−	0.923838i	$-0.374965\pi$
$233$	11.6859	0.765568	0.382784	+	0.923838i	$0.374965\pi$
$234$	0	0
$235$	20.9343	1.36560
$236$	1.82232	0.118623
$237$	0	0
$238$	−22.1628	−1.43660
$239$	23.1115	1.49496	0.747478	−	0.664287i	$-0.231265\pi$
$239$	23.1115	1.49496	0.747478	+	0.664287i	$0.231265\pi$
$240$	0	0
$241$	3.54114	0.228105	0.114052	−	0.993475i	$-0.463617\pi$
$241$	3.54114	0.228105	0.114052	+	0.993475i	$0.463617\pi$
$242$	0.948254	0.0609561
$243$	0	0
$244$	1.10081	0.0704724
$245$	46.0605	2.94270
$246$	0	0
$247$	16.2608	1.03465
$248$	−2.14345	−0.136109
$249$	0	0
$250$	−10.5285	−0.665880
$251$	19.2885	1.21748	0.608740	−	0.793370i	$-0.291675\pi$
$251$	19.2885	1.21748	0.608740	+	0.793370i	$0.291675\pi$
$252$	0	0
$253$	0.200434	0.0126012
$254$	5.16994	0.324391
$255$	0	0
$256$	−16.9474	−1.05921
$257$	−28.5441	−1.78053	−0.890267	−	0.455440i	$-0.849482\pi$
$257$	−28.5441	−1.78053	−0.890267	+	0.455440i	$0.849482\pi$
$258$	0	0
$259$	−18.0719	−1.12293
$260$	−13.4817	−0.836097
$261$	0	0
$262$	15.6127	0.964559
$263$	18.1170	1.11714	0.558570	−	0.829457i	$-0.311350\pi$
$263$	18.1170	1.11714	0.558570	+	0.829457i	$0.311350\pi$
$264$	0	0
$265$	3.34317	0.205370
$266$	14.8527	0.910680
$267$	0	0
$268$	−17.8034	−1.08752
$269$	−8.76351	−0.534321	−0.267160	−	0.963652i	$-0.586085\pi$
$269$	−8.76351	−0.534321	−0.267160	+	0.963652i	$0.586085\pi$
$270$	0	0
$271$	−21.6248	−1.31361	−0.656806	−	0.754059i	$-0.728093\pi$
$271$	−21.6248	−1.31361	−0.656806	+	0.754059i	$0.728093\pi$
$272$	2.61623	0.158632
$273$	0	0
$274$	−1.07364	−0.0648612
$275$	−0.0679998	−0.00410054
$276$	0	0
$277$	17.1283	1.02914	0.514571	−	0.857448i	$-0.327951\pi$
$277$	17.1283	1.02914	0.514571	+	0.857448i	$0.327951\pi$
$278$	16.7232	1.00299
$279$	0	0
$280$	−34.6873	−2.07297
$281$	14.8880	0.888144	0.444072	−	0.895991i	$-0.353533\pi$
$281$	14.8880	0.888144	0.444072	+	0.895991i	$0.353533\pi$
$282$	0	0
$283$	19.7803	1.17582	0.587908	−	0.808928i	$-0.299952\pi$
$283$	19.7803	1.17582	0.587908	+	0.808928i	$0.299952\pi$
$284$	1.53142	0.0908728
$285$	0	0
$286$	−5.15865	−0.305037
$287$	−13.1651	−0.777110
$288$	0	0
$289$	2.89269	0.170158
$290$	3.88132	0.227919
$291$	0	0
$292$	9.14233	0.535014
$293$	16.9333	0.989256	0.494628	−	0.869105i	$-0.335304\pi$
$293$	16.9333	0.989256	0.494628	+	0.869105i	$0.335304\pi$
$294$	0	0
$295$	−3.72673	−0.216979
$296$	10.1403	0.589393
$297$	0	0
$298$	−17.5893	−1.01892
$299$	−1.09039	−0.0630590
$300$	0	0
$301$	40.4590	2.33202
$302$	−21.2625	−1.22352
$303$	0	0
$304$	−1.75331	−0.100559
$305$	−2.25122	−0.128905
$306$	0	0
$307$	1.52740	0.0871733	0.0435866	−	0.999050i	$-0.486122\pi$
$307$	1.52740	0.0871733	0.0435866	+	0.999050i	$0.486122\pi$
$308$	5.76854	0.328693
$309$	0	0
$310$	1.55617	0.0883845
$311$	−31.7654	−1.80125	−0.900626	−	0.434596i	$-0.856891\pi$
$311$	−31.7654	−1.80125	−0.900626	+	0.434596i	$0.856891\pi$
$312$	0	0
$313$	−16.9039	−0.955467	−0.477733	−	0.878505i	$-0.658542\pi$
$313$	−16.9039	−0.955467	−0.477733	+	0.878505i	$0.658542\pi$
$314$	−6.93278	−0.391239
$315$	0	0
$316$	12.8884	0.725029
$317$	−20.3557	−1.14329	−0.571646	−	0.820501i	$-0.693695\pi$
$317$	−20.3557	−1.14329	−0.571646	+	0.820501i	$0.693695\pi$
$318$	0	0
$319$	−1.81818	−0.101798
$320$	14.0074	0.783038
$321$	0	0
$322$	−0.995975	−0.0555035
$323$	−13.3314	−0.741780
$324$	0	0
$325$	0.369929	0.0205200
$326$	−0.990074	−0.0548351
$327$	0	0
$328$	7.38706	0.407882
$329$	48.7295	2.68655
$330$	0	0
$331$	16.6765	0.916621	0.458311	−	0.888792i	$-0.348455\pi$
$331$	16.6765	0.916621	0.458311	+	0.888792i	$0.348455\pi$
$332$	−1.22137	−0.0670312
$333$	0	0
$334$	0.229731	0.0125703
$335$	36.4090	1.98924
$336$	0	0
$337$	−6.60838	−0.359981	−0.179991	−	0.983668i	$-0.557607\pi$
$337$	−6.60838	−0.359981	−0.179991	+	0.983668i	$0.557607\pi$
$338$	15.7366	0.855956
$339$	0	0
$340$	11.0530	0.599431
$341$	−0.728977	−0.0394763
$342$	0	0
$343$	70.5351	3.80854
$344$	−22.7020	−1.22401
$345$	0	0
$346$	22.5295	1.21119
$347$	−16.0359	−0.860851	−0.430425	−	0.902626i	$-0.641636\pi$
$347$	−16.0359	−0.860851	−0.430425	+	0.902626i	$0.641636\pi$
$348$	0	0
$349$	6.89281	0.368964	0.184482	−	0.982836i	$-0.440939\pi$
$349$	6.89281	0.368964	0.184482	+	0.982836i	$0.440939\pi$
$350$	0.337897	0.0180614
$351$	0	0
$352$	−5.32449	−0.283796
$353$	13.3187	0.708884	0.354442	−	0.935078i	$-0.384671\pi$
$353$	13.3187	0.708884	0.354442	+	0.935078i	$0.384671\pi$
$354$	0	0
$355$	−3.13183	−0.166220
$356$	7.49080	0.397012
$357$	0	0
$358$	2.65134	0.140127
$359$	−35.5183	−1.87458	−0.937291	−	0.348547i	$-0.886675\pi$
$359$	−35.5183	−1.87458	−0.937291	+	0.348547i	$0.886675\pi$
$360$	0	0
$361$	−10.0657	−0.529775
$362$	11.9091	0.625927
$363$	0	0
$364$	−31.3818	−1.64485
$365$	−18.6965	−0.978622
$366$	0	0
$367$	−11.8248	−0.617250	−0.308625	−	0.951184i	$-0.599869\pi$
$367$	−11.8248	−0.617250	−0.308625	+	0.951184i	$0.599869\pi$
$368$	0.117571	0.00612882
$369$	0	0
$370$	−7.36196	−0.382730
$371$	7.78203	0.404023
$372$	0	0
$373$	−21.2527	−1.10042	−0.550211	−	0.835026i	$-0.685453\pi$
$373$	−21.2527	−1.10042	−0.550211	+	0.835026i	$0.685453\pi$
$374$	4.22933	0.218694
$375$	0	0
$376$	−27.3426	−1.41009
$377$	9.89116	0.509421
$378$	0	0
$379$	25.3787	1.30362	0.651808	−	0.758384i	$-0.274011\pi$
$379$	25.3787	1.30362	0.651808	+	0.758384i	$0.274011\pi$
$380$	−7.40733	−0.379988
$381$	0	0
$382$	2.60369	0.133216
$383$	−8.43965	−0.431246	−0.215623	−	0.976477i	$-0.569178\pi$
$383$	−8.43965	−0.431246	−0.215623	+	0.976477i	$0.569178\pi$
$384$	0	0
$385$	−11.7970	−0.601229
$386$	25.1978	1.28253
$387$	0	0
$388$	10.2469	0.520206
$389$	−24.0747	−1.22063	−0.610317	−	0.792158i	$-0.708958\pi$
$389$	−24.0747	−1.22063	−0.610317	+	0.792158i	$0.708958\pi$
$390$	0	0
$391$	0.893960	0.0452095
$392$	−60.1605	−3.03856
$393$	0	0
$394$	8.05638	0.405875
$395$	−26.3574	−1.32619
$396$	0	0
$397$	−11.1126	−0.557724	−0.278862	−	0.960331i	$-0.589957\pi$
$397$	−11.1126	−0.557724	−0.278862	+	0.960331i	$0.589957\pi$
$398$	−7.48664	−0.375271
$399$	0	0
$400$	−0.0398875	−0.00199437
$401$	25.1568	1.25627	0.628136	−	0.778104i	$-0.283818\pi$
$401$	25.1568	1.25627	0.628136	+	0.778104i	$0.283818\pi$
$402$	0	0
$403$	3.96575	0.197548
$404$	10.3243	0.513652
$405$	0	0
$406$	9.03469	0.448384
$407$	3.44866	0.170944
$408$	0	0
$409$	4.15275	0.205340	0.102670	−	0.994715i	$-0.467261\pi$
$409$	4.15275	0.205340	0.102670	+	0.994715i	$0.467261\pi$
$410$	−5.36308	−0.264864
$411$	0	0
$412$	10.6059	0.522516
$413$	−8.67485	−0.426862
$414$	0	0
$415$	2.49776	0.122610
$416$	28.9661	1.42018
$417$	0	0
$418$	−2.83436	−0.138633
$419$	−34.2797	−1.67467	−0.837337	−	0.546688i	$-0.815889\pi$
$419$	−34.2797	−1.67467	−0.837337	+	0.546688i	$0.815889\pi$
$420$	0	0
$421$	−36.7527	−1.79122	−0.895609	−	0.444842i	$-0.853260\pi$
$421$	−36.7527	−1.79122	−0.895609	+	0.444842i	$0.853260\pi$
$422$	15.3167	0.745606
$423$	0	0
$424$	−4.36658	−0.212060
$425$	−0.303287	−0.0147116
$426$	0	0
$427$	−5.24025	−0.253594
$428$	3.32883	0.160905
$429$	0	0
$430$	16.4818	0.794825
$431$	−5.91425	−0.284879	−0.142440	−	0.989803i	$-0.545495\pi$
$431$	−5.91425	−0.284879	−0.142440	+	0.989803i	$0.545495\pi$
$432$	0	0
$433$	12.9638	0.623001	0.311500	−	0.950246i	$-0.399168\pi$
$433$	12.9638	0.623001	0.311500	+	0.950246i	$0.399168\pi$
$434$	3.62235	0.173878
$435$	0	0
$436$	−4.12668	−0.197632
$437$	−0.599102	−0.0286589
$438$	0	0
$439$	−29.2711	−1.39704	−0.698518	−	0.715593i	$-0.746157\pi$
$439$	−29.2711	−1.39704	−0.698518	+	0.715593i	$0.746157\pi$
$440$	6.61940	0.315568
$441$	0	0
$442$	−23.0082	−1.09439
$443$	−3.21165	−0.152590	−0.0762951	−	0.997085i	$-0.524309\pi$
$443$	−3.21165	−0.152590	−0.0762951	+	0.997085i	$0.524309\pi$
$444$	0	0
$445$	−15.3191	−0.726194
$446$	−21.6190	−1.02369
$447$	0	0
$448$	32.6056	1.54047
$449$	11.0578	0.521849	0.260924	−	0.965359i	$-0.415973\pi$
$449$	11.0578	0.521849	0.260924	+	0.965359i	$0.415973\pi$
$450$	0	0
$451$	2.51230	0.118300
$452$	−3.85681	−0.181409
$453$	0	0
$454$	−15.0568	−0.706649
$455$	64.1774	3.00868
$456$	0	0
$457$	8.45879	0.395685	0.197843	−	0.980234i	$-0.436606\pi$
$457$	8.45879	0.395685	0.197843	+	0.980234i	$0.436606\pi$
$458$	23.9472	1.11898
$459$	0	0
$460$	0.496711	0.0231593
$461$	−8.02284	−0.373661	−0.186830	−	0.982392i	$-0.559822\pi$
$461$	−8.02284	−0.373661	−0.186830	+	0.982392i	$0.559822\pi$
$462$	0	0
$463$	20.0463	0.931630	0.465815	−	0.884882i	$-0.345761\pi$
$463$	20.0463	0.931630	0.465815	+	0.884882i	$0.345761\pi$
$464$	−1.06651	−0.0495115
$465$	0	0
$466$	11.0812	0.513326
$467$	9.41646	0.435742	0.217871	−	0.975978i	$-0.430089\pi$
$467$	9.41646	0.435742	0.217871	+	0.975978i	$0.430089\pi$
$468$	0	0
$469$	84.7505	3.91342
$470$	19.8510	0.915660
$471$	0	0
$472$	4.86755	0.224047
$473$	−7.72081	−0.355003
$474$	0	0
$475$	0.203253	0.00932589
$476$	25.7284	1.17926
$477$	0	0
$478$	21.9155	1.00239
$479$	−15.6024	−0.712890	−0.356445	−	0.934316i	$-0.616011\pi$
$479$	−15.6024	−0.712890	−0.356445	+	0.934316i	$0.616011\pi$
$480$	0	0
$481$	−18.7613	−0.855439
$482$	3.35790	0.152948
$483$	0	0
$484$	−1.10081	−0.0500370
$485$	−20.9554	−0.951535
$486$	0	0
$487$	28.6801	1.29962	0.649811	−	0.760096i	$-0.274848\pi$
$487$	28.6801	1.29962	0.649811	+	0.760096i	$0.274848\pi$
$488$	2.94036	0.133104
$489$	0	0
$490$	43.6771	1.97313
$491$	−0.942339	−0.0425271	−0.0212636	−	0.999774i	$-0.506769\pi$
$491$	−0.942339	−0.0425271	−0.0212636	+	0.999774i	$0.506769\pi$
$492$	0	0
$493$	−8.10929	−0.365224
$494$	15.4193	0.693749
$495$	0	0
$496$	−0.427605	−0.0192000
$497$	−7.29007	−0.327004
$498$	0	0
$499$	−3.01444	−0.134945	−0.0674725	−	0.997721i	$-0.521493\pi$
$499$	−3.01444	−0.134945	−0.0674725	+	0.997721i	$0.521493\pi$
$500$	12.2224	0.546601
$501$	0	0
$502$	18.2904	0.816341
$503$	−43.9550	−1.95986	−0.979929	−	0.199346i	$-0.936118\pi$
$503$	−43.9550	−1.95986	−0.979929	+	0.199346i	$0.936118\pi$
$504$	0	0
$505$	−21.1137	−0.939547
$506$	0.190062	0.00844931
$507$	0	0
$508$	−6.00170	−0.266282
$509$	23.9124	1.05990	0.529950	−	0.848029i	$-0.322211\pi$
$509$	23.9124	1.05990	0.529950	+	0.848029i	$0.322211\pi$
$510$	0	0
$511$	−43.5206	−1.92524
$512$	−6.57278	−0.290479
$513$	0	0
$514$	−27.0671	−1.19388
$515$	−21.6897	−0.955762
$516$	0	0
$517$	−9.29908	−0.408973
$518$	−17.1367	−0.752944
$519$	0	0
$520$	−36.0106	−1.57917
$521$	37.7693	1.65470	0.827351	−	0.561685i	$-0.189847\pi$
$521$	37.7693	1.65470	0.827351	+	0.561685i	$0.189847\pi$
$522$	0	0
$523$	−3.21011	−0.140368	−0.0701842	−	0.997534i	$-0.522359\pi$
$523$	−3.21011	−0.140368	−0.0701842	+	0.997534i	$0.522359\pi$
$524$	−18.1246	−0.791776
$525$	0	0
$526$	17.1795	0.749062
$527$	−3.25133	−0.141630
$528$	0	0
$529$	−22.9598	−0.998253
$530$	3.17018	0.137704
$531$	0	0
$532$	−17.2423	−0.747549
$533$	−13.6673	−0.591996
$534$	0	0
$535$	−6.80763	−0.294320
$536$	−47.5544	−2.05404
$537$	0	0
$538$	−8.31004	−0.358271
$539$	−20.4602	−0.881285
$540$	0	0
$541$	−15.9679	−0.686512	−0.343256	−	0.939242i	$-0.611530\pi$
$541$	−15.9679	−0.686512	−0.343256	+	0.939242i	$0.611530\pi$
$542$	−20.5058	−0.880800
$543$	0	0
$544$	−23.7479	−1.01818
$545$	8.43927	0.361499
$546$	0	0
$547$	−13.1472	−0.562132	−0.281066	−	0.959688i	$-0.590688\pi$
$547$	−13.1472	−0.562132	−0.281066	+	0.959688i	$0.590688\pi$
$548$	1.24638	0.0532425
$549$	0	0
$550$	−0.0644811	−0.00274948
$551$	5.43458	0.231521
$552$	0	0
$553$	−61.3532	−2.60900
$554$	16.2420	0.690057
$555$	0	0
$556$	−19.4137	−0.823324
$557$	−0.120019	−0.00508539	−0.00254269	−	0.999997i	$-0.500809\pi$
$557$	−0.120019	−0.00508539	−0.00254269	+	0.999997i	$0.500809\pi$
$558$	0	0
$559$	42.0024	1.77651
$560$	−6.91990	−0.292419
$561$	0	0
$562$	14.1176	0.595516
$563$	22.3237	0.940831	0.470415	−	0.882445i	$-0.344104\pi$
$563$	22.3237	0.940831	0.470415	+	0.882445i	$0.344104\pi$
$564$	0	0
$565$	7.88738	0.331825
$566$	18.7568	0.788405
$567$	0	0
$568$	4.09053	0.171635
$569$	6.42634	0.269406	0.134703	−	0.990886i	$-0.456992\pi$
$569$	6.42634	0.269406	0.134703	+	0.990886i	$0.456992\pi$
$570$	0	0
$571$	−21.3502	−0.893479	−0.446740	−	0.894664i	$-0.647415\pi$
$571$	−21.3502	−0.893479	−0.446740	+	0.894664i	$0.647415\pi$
$572$	5.98860	0.250396
$573$	0	0
$574$	−12.4838	−0.521066
$575$	−0.0136295	−0.000568388 0
$576$	0	0
$577$	−4.88600	−0.203407	−0.101703	−	0.994815i	$-0.532429\pi$
$577$	−4.88600	−0.203407	−0.101703	+	0.994815i	$0.532429\pi$
$578$	2.74301	0.114094
$579$	0	0
$580$	−4.50576	−0.187092
$581$	5.81413	0.241211
$582$	0	0
$583$	−1.48505	−0.0615044
$584$	24.4199	1.01050
$585$	0	0
$586$	16.0571	0.663313
$587$	32.3274	1.33429	0.667147	−	0.744926i	$-0.267515\pi$
$587$	32.3274	1.33429	0.667147	+	0.744926i	$0.267515\pi$
$588$	0	0
$589$	2.17893	0.0897812
$590$	−3.53389	−0.145488
$591$	0	0
$592$	2.02293	0.0831417
$593$	−18.0269	−0.740275	−0.370137	−	0.928977i	$-0.620689\pi$
$593$	−18.0269	−0.740275	−0.370137	+	0.928977i	$0.620689\pi$
$594$	0	0
$595$	−52.6159	−2.15704
$596$	20.4192	0.836402
$597$	0	0
$598$	−1.03397	−0.0422821
$599$	6.99770	0.285918	0.142959	−	0.989729i	$-0.454338\pi$
$599$	6.99770	0.285918	0.142959	+	0.989729i	$0.454338\pi$
$600$	0	0
$601$	27.9227	1.13899	0.569496	−	0.821994i	$-0.307138\pi$
$601$	27.9227	1.13899	0.569496	+	0.821994i	$0.307138\pi$
$602$	38.3654	1.56366
$603$	0	0
$604$	24.6832	1.00435
$605$	2.25122	0.0915252
$606$	0	0
$607$	47.3972	1.92379	0.961897	−	0.273412i	$-0.0881522\pi$
$607$	47.3972	1.92379	0.961897	+	0.273412i	$0.0881522\pi$
$608$	15.9150	0.645440
$609$	0	0
$610$	−2.13473	−0.0864327
$611$	50.5884	2.04659
$612$	0	0
$613$	−36.7435	−1.48406	−0.742028	−	0.670369i	$-0.766136\pi$
$613$	−36.7435	−1.48406	−0.742028	+	0.670369i	$0.766136\pi$
$614$	1.44836	0.0584512
$615$	0	0
$616$	15.4082	0.620815
$617$	−23.6451	−0.951917	−0.475958	−	0.879468i	$-0.657899\pi$
$617$	−23.6451	−0.951917	−0.475958	+	0.879468i	$0.657899\pi$
$618$	0	0
$619$	−8.49054	−0.341264	−0.170632	−	0.985335i	$-0.554581\pi$
$619$	−8.49054	−0.341264	−0.170632	+	0.985335i	$0.554581\pi$
$620$	−1.80653	−0.0725521
$621$	0	0
$622$	−30.1217	−1.20777
$623$	−35.6588	−1.42864
$624$	0	0
$625$	−25.3354	−1.01341
$626$	−16.0292	−0.640657
$627$	0	0
$628$	8.04815	0.321156
$629$	15.3815	0.613299
$630$	0	0
$631$	13.5216	0.538288	0.269144	−	0.963100i	$-0.413259\pi$
$631$	13.5216	0.538288	0.269144	+	0.963100i	$0.413259\pi$
$632$	34.4259	1.36939
$633$	0	0
$634$	−19.3024	−0.766597
$635$	12.2738	0.487071
$636$	0	0
$637$	111.307	4.41014
$638$	−1.72409	−0.0682575
$639$	0	0
$640$	−10.6906	−0.422584
$641$	11.5595	0.456574	0.228287	−	0.973594i	$-0.426688\pi$
$641$	11.5595	0.456574	0.228287	+	0.973594i	$0.426688\pi$
$642$	0	0
$643$	−19.4275	−0.766147	−0.383073	−	0.923718i	$-0.625134\pi$
$643$	−19.4275	−0.766147	−0.383073	+	0.923718i	$0.625134\pi$
$644$	1.15621	0.0455611
$645$	0	0
$646$	−12.6416	−0.497376
$647$	27.7606	1.09138	0.545690	−	0.837987i	$-0.316268\pi$
$647$	27.7606	1.09138	0.545690	+	0.837987i	$0.316268\pi$
$648$	0	0
$649$	1.65543	0.0649812
$650$	0.350787	0.0137590
$651$	0	0
$652$	1.14936	0.0450124
$653$	−23.0651	−0.902608	−0.451304	−	0.892370i	$-0.649041\pi$
$653$	−23.0651	−0.902608	−0.451304	+	0.892370i	$0.649041\pi$
$654$	0	0
$655$	37.0657	1.44828
$656$	1.47367	0.0575372
$657$	0	0
$658$	46.2080	1.80137
$659$	−7.13729	−0.278029	−0.139015	−	0.990290i	$-0.544394\pi$
$659$	−7.13729	−0.278029	−0.139015	+	0.990290i	$0.544394\pi$
$660$	0	0
$661$	−16.0929	−0.625941	−0.312971	−	0.949763i	$-0.601324\pi$
$661$	−16.0929	−0.625941	−0.312971	+	0.949763i	$0.601324\pi$
$662$	15.8135	0.614610
$663$	0	0
$664$	−3.26237	−0.126604
$665$	35.2614	1.36738
$666$	0	0
$667$	−0.364424	−0.0141106
$668$	−0.266691	−0.0103186
$669$	0	0
$670$	34.5250	1.33382
$671$	1.00000	0.0386046
$672$	0	0
$673$	34.6989	1.33755	0.668773	−	0.743467i	$-0.266820\pi$
$673$	34.6989	1.33755	0.668773	+	0.743467i	$0.266820\pi$
$674$	−6.26643	−0.241374
$675$	0	0
$676$	−18.2683	−0.702628
$677$	46.9863	1.80583	0.902916	−	0.429818i	$-0.141422\pi$
$677$	46.9863	1.80583	0.902916	+	0.429818i	$0.141422\pi$
$678$	0	0
$679$	−48.7786	−1.87195
$680$	29.5234	1.13217
$681$	0	0
$682$	−0.691255	−0.0264695
$683$	−1.54777	−0.0592238	−0.0296119	−	0.999561i	$-0.509427\pi$
$683$	−1.54777	−0.0592238	−0.0296119	+	0.999561i	$0.509427\pi$
$684$	0	0
$685$	−2.54890	−0.0973886
$686$	66.8852	2.55369
$687$	0	0
$688$	−4.52889	−0.172662
$689$	8.07889	0.307781
$690$	0	0
$691$	18.0655	0.687245	0.343623	−	0.939108i	$-0.388346\pi$
$691$	18.0655	0.687245	0.343623	+	0.939108i	$0.388346\pi$
$692$	−26.1542	−0.994232
$693$	0	0
$694$	−15.2061	−0.577215
$695$	39.7020	1.50598
$696$	0	0
$697$	11.2052	0.424426
$698$	6.53614	0.247396
$699$	0	0
$700$	−0.392259	−0.0148260
$701$	21.0050	0.793346	0.396673	−	0.917960i	$-0.370165\pi$
$701$	21.0050	0.793346	0.396673	+	0.917960i	$0.370165\pi$
$702$	0	0
$703$	−10.3081	−0.388779
$704$	−6.22214	−0.234506
$705$	0	0
$706$	12.6295	0.475319
$707$	−49.1471	−1.84837
$708$	0	0
$709$	−12.9485	−0.486291	−0.243146	−	0.969990i	$-0.578179\pi$
$709$	−12.9485	−0.486291	−0.243146	+	0.969990i	$0.578179\pi$
$710$	−2.96977	−0.111453
$711$	0	0
$712$	20.0085	0.749851
$713$	−0.146112	−0.00547193
$714$	0	0
$715$	−12.2470	−0.458012
$716$	−3.07789	−0.115026
$717$	0	0
$718$	−33.6804	−1.25694
$719$	−11.2061	−0.417917	−0.208958	−	0.977925i	$-0.567007\pi$
$719$	−11.2061	−0.417917	−0.208958	+	0.977925i	$0.567007\pi$
$720$	0	0
$721$	−50.4879	−1.88027
$722$	−9.54487	−0.355223
$723$	0	0
$724$	−13.8251	−0.513804
$725$	0.123636	0.00459171
$726$	0	0
$727$	35.9986	1.33511	0.667557	−	0.744559i	$-0.267340\pi$
$727$	35.9986	1.33511	0.667557	+	0.744559i	$0.267340\pi$
$728$	−83.8232	−3.10669
$729$	0	0
$730$	−17.7291	−0.656183
$731$	−34.4358	−1.27365
$732$	0	0
$733$	23.6506	0.873554	0.436777	−	0.899570i	$-0.356120\pi$
$733$	23.6506	0.873554	0.436777	+	0.899570i	$0.356120\pi$
$734$	−11.2129	−0.413877
$735$	0	0
$736$	−1.06721	−0.0393378
$737$	−16.1730	−0.595740
$738$	0	0
$739$	−14.8095	−0.544778	−0.272389	−	0.962187i	$-0.587814\pi$
$739$	−14.8095	−0.544778	−0.272389	+	0.962187i	$0.587814\pi$
$740$	8.54639	0.314171
$741$	0	0
$742$	7.37934	0.270904
$743$	−13.3290	−0.488994	−0.244497	−	0.969650i	$-0.578623\pi$
$743$	−13.3290	−0.488994	−0.244497	+	0.969650i	$0.578623\pi$
$744$	0	0
$745$	−41.7583	−1.52991
$746$	−20.1529	−0.737852
$747$	0	0
$748$	−4.90976	−0.179519
$749$	−15.8464	−0.579014
$750$	0	0
$751$	1.50326	0.0548546	0.0274273	−	0.999624i	$-0.491269\pi$
$751$	1.50326	0.0548546	0.0274273	+	0.999624i	$0.491269\pi$
$752$	−5.45468	−0.198912
$753$	0	0
$754$	9.37934	0.341575
$755$	−50.4785	−1.83710
$756$	0	0
$757$	−12.6961	−0.461449	−0.230724	−	0.973019i	$-0.574110\pi$
$757$	−12.6961	−0.461449	−0.230724	+	0.973019i	$0.574110\pi$
$758$	24.0655	0.874097
$759$	0	0
$760$	−19.7856	−0.717698
$761$	1.08972	0.0395022	0.0197511	−	0.999805i	$-0.493713\pi$
$761$	1.08972	0.0395022	0.0197511	+	0.999805i	$0.493713\pi$
$762$	0	0
$763$	19.6444	0.711175
$764$	−3.02258	−0.109353
$765$	0	0
$766$	−8.00294	−0.289158
$767$	−9.00578	−0.325180
$768$	0	0
$769$	17.6529	0.636579	0.318289	−	0.947994i	$-0.396892\pi$
$769$	17.6529	0.636579	0.318289	+	0.947994i	$0.396892\pi$
$770$	−11.1865	−0.403135
$771$	0	0
$772$	−29.2517	−1.05279
$773$	12.7055	0.456986	0.228493	−	0.973546i	$-0.426620\pi$
$773$	12.7055	0.456986	0.228493	+	0.973546i	$0.426620\pi$
$774$	0	0
$775$	0.0495702	0.00178062
$776$	27.3702	0.982533
$777$	0	0
$778$	−22.8289	−0.818456
$779$	−7.50932	−0.269049
$780$	0	0
$781$	1.39117	0.0497799
$782$	0.847702	0.0303138
$783$	0	0
$784$	−12.0016	−0.428630
$785$	−16.4589	−0.587443
$786$	0	0
$787$	−14.3201	−0.510456	−0.255228	−	0.966881i	$-0.582151\pi$
$787$	−14.3201	−0.510456	−0.255228	+	0.966881i	$0.582151\pi$
$788$	−9.35252	−0.333170
$789$	0	0
$790$	−24.9936	−0.889231
$791$	18.3597	0.652797
$792$	0	0
$793$	−5.44016	−0.193186
$794$	−10.5375	−0.373963
$795$	0	0
$796$	8.69112	0.308049
$797$	−55.4418	−1.96385	−0.981924	−	0.189275i	$-0.939386\pi$
$797$	−55.4418	−1.96385	−0.981924	+	0.189275i	$0.939386\pi$
$798$	0	0
$799$	−41.4750	−1.46728
$800$	0.362064	0.0128009
$801$	0	0
$802$	23.8551	0.842352
$803$	8.30506	0.293079
$804$	0	0
$805$	−2.36451	−0.0833382
$806$	3.76054	0.132459
$807$	0	0
$808$	27.5770	0.970154
$809$	−37.9294	−1.33353	−0.666763	−	0.745270i	$-0.732321\pi$
$809$	−37.9294	−1.33353	−0.666763	+	0.745270i	$0.732321\pi$
$810$	0	0
$811$	−3.23359	−0.113547	−0.0567733	−	0.998387i	$-0.518081\pi$
$811$	−3.23359	−0.113547	−0.0567733	+	0.998387i	$0.518081\pi$
$812$	−10.4882	−0.368065
$813$	0	0
$814$	3.27021	0.114621
$815$	−2.35050	−0.0823345
$816$	0	0
$817$	23.0777	0.807386
$818$	3.93786	0.137684
$819$	0	0
$820$	6.22592	0.217418
$821$	44.4292	1.55059	0.775295	−	0.631599i	$-0.217601\pi$
$821$	44.4292	1.55059	0.775295	+	0.631599i	$0.217601\pi$
$822$	0	0
$823$	11.0243	0.384284	0.192142	−	0.981367i	$-0.438457\pi$
$823$	11.0243	0.384284	0.192142	+	0.981367i	$0.438457\pi$
$824$	28.3293	0.986897
$825$	0	0
$826$	−8.22597	−0.286218
$827$	−38.2151	−1.32887	−0.664435	−	0.747346i	$-0.731328\pi$
$827$	−38.2151	−1.32887	−0.664435	+	0.747346i	$0.731328\pi$
$828$	0	0
$829$	5.10113	0.177170	0.0885848	−	0.996069i	$-0.471766\pi$
$829$	5.10113	0.177170	0.0885848	+	0.996069i	$0.471766\pi$
$830$	2.36851	0.0822122
$831$	0	0
$832$	33.8494	1.17352
$833$	−91.2552	−3.16181
$834$	0	0
$835$	0.545397	0.0188742
$836$	3.29036	0.113799
$837$	0	0
$838$	−32.5059	−1.12290
$839$	−22.4559	−0.775262	−0.387631	−	0.921815i	$-0.626707\pi$
$839$	−22.4559	−0.775262	−0.387631	+	0.921815i	$0.626707\pi$
$840$	0	0
$841$	−25.6942	−0.886008
$842$	−34.8509	−1.20104
$843$	0	0
$844$	−17.7809	−0.612045
$845$	37.3597	1.28521
$846$	0	0
$847$	5.24025	0.180057
$848$	−0.871104	−0.0299138
$849$	0	0
$850$	−0.287594	−0.00986438
$851$	0.691229	0.0236950
$852$	0	0
$853$	4.17359	0.142901	0.0714505	−	0.997444i	$-0.477237\pi$
$853$	4.17359	0.142901	0.0714505	+	0.997444i	$0.477237\pi$
$854$	−4.96909	−0.170039
$855$	0	0
$856$	8.89157	0.303908
$857$	−54.4319	−1.85936	−0.929679	−	0.368371i	$-0.879916\pi$
$857$	−54.4319	−1.85936	−0.929679	+	0.368371i	$0.879916\pi$
$858$	0	0
$859$	−18.6033	−0.634737	−0.317368	−	0.948302i	$-0.602799\pi$
$859$	−18.6033	−0.634737	−0.317368	+	0.948302i	$0.602799\pi$
$860$	−19.1335	−0.652447
$861$	0	0
$862$	−5.60821	−0.191016
$863$	−51.6257	−1.75736	−0.878679	−	0.477412i	$-0.841575\pi$
$863$	−51.6257	−1.75736	−0.878679	+	0.477412i	$0.841575\pi$
$864$	0	0
$865$	53.4866	1.81860
$866$	12.2930	0.417733
$867$	0	0
$868$	−4.20513	−0.142731
$869$	11.7081	0.397169
$870$	0	0
$871$	87.9836	2.98121
$872$	−11.0227	−0.373275
$873$	0	0
$874$	−0.568101	−0.0192163
$875$	−58.1827	−1.96693
$876$	0	0
$877$	−49.5572	−1.67343	−0.836715	−	0.547639i	$-0.815527\pi$
$877$	−49.5572	−1.67343	−0.836715	+	0.547639i	$0.815527\pi$
$878$	−27.7565	−0.936736
$879$	0	0
$880$	1.32053	0.0445150
$881$	−48.5126	−1.63443	−0.817216	−	0.576332i	$-0.804484\pi$
$881$	−48.5126	−1.63443	−0.817216	+	0.576332i	$0.804484\pi$
$882$	0	0
$883$	−26.0469	−0.876549	−0.438275	−	0.898841i	$-0.644410\pi$
$883$	−26.0469	−0.876549	−0.438275	+	0.898841i	$0.644410\pi$
$884$	26.7099	0.898350
$885$	0	0
$886$	−3.04546	−0.102314
$887$	−46.1043	−1.54803	−0.774015	−	0.633168i	$-0.781754\pi$
$887$	−46.1043	−1.54803	−0.774015	+	0.633168i	$0.781754\pi$
$888$	0	0
$889$	28.5702	0.958213
$890$	−14.5264	−0.486926
$891$	0	0
$892$	25.0971	0.840315
$893$	27.7952	0.930130
$894$	0	0
$895$	6.29445	0.210400
$896$	−24.8850	−0.831348
$897$	0	0
$898$	10.4856	0.349909
$899$	1.32541	0.0442048
$900$	0	0
$901$	−6.62350	−0.220661
$902$	2.38230	0.0793219
$903$	0	0
$904$	−10.3018	−0.342634
$905$	28.2730	0.939826
$906$	0	0
$907$	4.84142	0.160757	0.0803783	−	0.996764i	$-0.474387\pi$
$907$	4.84142	0.160757	0.0803783	+	0.996764i	$0.474387\pi$
$908$	17.4791	0.580066
$909$	0	0
$910$	60.8565	2.01737
$911$	−37.1866	−1.23205	−0.616024	−	0.787727i	$-0.711258\pi$
$911$	−37.1866	−1.23205	−0.616024	+	0.787727i	$0.711258\pi$
$912$	0	0
$913$	−1.10951	−0.0367195
$914$	8.02108	0.265314
$915$	0	0
$916$	−27.7999	−0.918534
$917$	86.2793	2.84919
$918$	0	0
$919$	−42.7495	−1.41018	−0.705088	−	0.709120i	$-0.749092\pi$
$919$	−42.7495	−1.41018	−0.705088	+	0.709120i	$0.749092\pi$
$920$	1.32675	0.0437418
$921$	0	0
$922$	−7.60769	−0.250546
$923$	−7.56817	−0.249109
$924$	0	0
$925$	−0.234508	−0.00771058
$926$	19.0090	0.624674
$927$	0	0
$928$	9.68087	0.317790
$929$	−22.1672	−0.727284	−0.363642	−	0.931539i	$-0.618467\pi$
$929$	−22.1672	−0.727284	−0.363642	+	0.931539i	$0.618467\pi$
$930$	0	0
$931$	61.1562	2.00431
$932$	−12.8640	−0.421374
$933$	0	0
$934$	8.92920	0.292172
$935$	10.0407	0.328367
$936$	0	0
$937$	25.4112	0.830148	0.415074	−	0.909788i	$-0.363756\pi$
$937$	25.4112	0.830148	0.415074	+	0.909788i	$0.363756\pi$
$938$	80.3651	2.62401
$939$	0	0
$940$	−23.0447	−0.751637
$941$	−35.3840	−1.15348	−0.576742	−	0.816926i	$-0.695676\pi$
$941$	−35.3840	−1.15348	−0.576742	+	0.816926i	$0.695676\pi$
$942$	0	0
$943$	0.503550	0.0163978
$944$	0.971045	0.0316048
$945$	0	0
$946$	−7.32129	−0.238036
$947$	−15.1601	−0.492638	−0.246319	−	0.969189i	$-0.579221\pi$
$947$	−15.1601	−0.492638	−0.246319	+	0.969189i	$0.579221\pi$
$948$	0	0
$949$	−45.1808	−1.46663
$950$	0.192736	0.00625317
$951$	0	0
$952$	68.7226	2.22731
$953$	50.7429	1.64372	0.821862	−	0.569687i	$-0.192935\pi$
$953$	50.7429	1.64372	0.821862	+	0.569687i	$0.192935\pi$
$954$	0	0
$955$	6.18134	0.200024
$956$	−25.4414	−0.822834
$957$	0	0
$958$	−14.7950	−0.478005
$959$	−5.93318	−0.191592
$960$	0	0
$961$	−30.4686	−0.982858
$962$	−17.7904	−0.573587
$963$	0	0
$964$	−3.89814	−0.125550
$965$	59.8213	1.92572
$966$	0	0
$967$	1.89879	0.0610611	0.0305305	−	0.999534i	$-0.490280\pi$
$967$	1.89879	0.0610611	0.0305305	+	0.999534i	$0.490280\pi$
$968$	−2.94036	−0.0945068
$969$	0	0
$970$	−19.8710	−0.638021
$971$	−28.5211	−0.915284	−0.457642	−	0.889136i	$-0.651306\pi$
$971$	−28.5211	−0.915284	−0.457642	+	0.889136i	$0.651306\pi$
$972$	0	0
$973$	92.4159	2.96272
$974$	27.1961	0.871418
$975$	0	0
$976$	0.586583	0.0187761
$977$	44.8572	1.43511	0.717554	−	0.696502i	$-0.245261\pi$
$977$	44.8572	1.43511	0.717554	+	0.696502i	$0.245261\pi$
$978$	0	0
$979$	6.80479	0.217482
$980$	−50.7041	−1.61968
$981$	0	0
$982$	−0.893577	−0.0285152
$983$	−15.8302	−0.504904	−0.252452	−	0.967609i	$-0.581237\pi$
$983$	−15.8302	−0.504904	−0.252452	+	0.967609i	$0.581237\pi$
$984$	0	0
$985$	19.1264	0.609418
$986$	−7.68967	−0.244889
$987$	0	0
$988$	−17.9001	−0.569477
$989$	−1.54751	−0.0492080
$990$	0	0
$991$	−12.4145	−0.394359	−0.197180	−	0.980367i	$-0.563178\pi$
$991$	−12.4145	−0.394359	−0.197180	+	0.980367i	$0.563178\pi$
$992$	3.88143	0.123236
$993$	0	0
$994$	−6.91284	−0.219262
$995$	−17.7738	−0.563467
$996$	0	0
$997$	−16.4628	−0.521383	−0.260691	−	0.965422i	$-0.583950\pi$
$997$	−16.4628	−0.521383	−0.260691	+	0.965422i	$0.583950\pi$
$998$	−2.85846	−0.0904829
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.9		13
3.2	odd	2		2013.2.a.e.1.5	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.5	✓	13	3.2	odd	2
6039.2.a.i.1.9		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)

\(f(q)\)	\(=\)	\(q+0.948254 q^{2} -1.10081 q^{4} +2.25122 q^{5} +5.24025 q^{7} -2.94036 q^{8} +O(q^{10})\)	\(q+0.948254 q^{2} -1.10081 q^{4} +2.25122 q^{5} +5.24025 q^{7} -2.94036 q^{8} +2.13473 q^{10} -1.00000 q^{11} +5.44016 q^{13} +4.96909 q^{14} -0.586583 q^{16} -4.46012 q^{17} +2.98902 q^{19} -2.47818 q^{20} -0.948254 q^{22} -0.200434 q^{23} +0.0679998 q^{25} +5.15865 q^{26} -5.76854 q^{28} +1.81818 q^{29} +0.728977 q^{31} +5.32449 q^{32} -4.22933 q^{34} +11.7970 q^{35} -3.44866 q^{37} +2.83436 q^{38} -6.61940 q^{40} -2.51230 q^{41} +7.72081 q^{43} +1.10081 q^{44} -0.190062 q^{46} +9.29908 q^{47} +20.4602 q^{49} +0.0644811 q^{50} -5.98860 q^{52} +1.48505 q^{53} -2.25122 q^{55} -15.4082 q^{56} +1.72409 q^{58} -1.65543 q^{59} -1.00000 q^{61} +0.691255 q^{62} +6.22214 q^{64} +12.2470 q^{65} +16.1730 q^{67} +4.90976 q^{68} +11.1865 q^{70} -1.39117 q^{71} -8.30506 q^{73} -3.27021 q^{74} -3.29036 q^{76} -5.24025 q^{77} -11.7081 q^{79} -1.32053 q^{80} -2.38230 q^{82} +1.10951 q^{83} -10.0407 q^{85} +7.32129 q^{86} +2.94036 q^{88} -6.80479 q^{89} +28.5078 q^{91} +0.220640 q^{92} +8.81789 q^{94} +6.72896 q^{95} -9.30845 q^{97} +19.4015 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 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

Properties

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