LMFDB - Embedded newform 6039.2.a.i.1.1

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			1.1
Root			$2.73913$ 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)

$f(q)$	$=$	$q-2.73913 q^{2} +5.50285 q^{4} -0.604616 q^{5} -1.91298 q^{7} -9.59477 q^{8} +O(q^{10})$	\(q-2.73913 q^{2} +5.50285 q^{4} -0.604616 q^{5} -1.91298 q^{7} -9.59477 q^{8} +1.65612 q^{10} -1.00000 q^{11} +0.176875 q^{13} +5.23991 q^{14} +15.2757 q^{16} -2.74225 q^{17} +3.64728 q^{19} -3.32711 q^{20} +2.73913 q^{22} -1.61899 q^{23} -4.63444 q^{25} -0.484484 q^{26} -10.5269 q^{28} +7.55848 q^{29} +7.88196 q^{31} -22.6525 q^{32} +7.51138 q^{34} +1.15662 q^{35} +8.43648 q^{37} -9.99040 q^{38} +5.80115 q^{40} -8.93210 q^{41} -4.51816 q^{43} -5.50285 q^{44} +4.43462 q^{46} +2.48579 q^{47} -3.34050 q^{49} +12.6943 q^{50} +0.973317 q^{52} +7.57327 q^{53} +0.604616 q^{55} +18.3546 q^{56} -20.7037 q^{58} -1.33758 q^{59} -1.00000 q^{61} -21.5897 q^{62} +31.4969 q^{64} -0.106941 q^{65} -5.50196 q^{67} -15.0902 q^{68} -3.16813 q^{70} -0.330471 q^{71} -1.75560 q^{73} -23.1086 q^{74} +20.0705 q^{76} +1.91298 q^{77} -12.9780 q^{79} -9.23590 q^{80} +24.4662 q^{82} -12.9300 q^{83} +1.65801 q^{85} +12.3759 q^{86} +9.59477 q^{88} +4.11061 q^{89} -0.338359 q^{91} -8.90903 q^{92} -6.80891 q^{94} -2.20521 q^{95} -12.7524 q^{97} +9.15006 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 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 $

$2$	−2.73913	−1.93686	−0.968430	−	0.249287i	$-0.919804\pi$
$2$	−2.73913	−1.93686	−0.968430	+	0.249287i	$0.919804\pi$
$3$	0	0
$3$	0	0
$4$	5.50285	2.75142
$5$	−0.604616	−0.270392	−0.135196	−	0.990819i	$-0.543166\pi$
$5$	−0.604616	−0.270392	−0.135196	+	0.990819i	$0.543166\pi$
$6$	0	0
$7$	−1.91298	−0.723040	−0.361520	−	0.932364i	$-0.617742\pi$
$7$	−1.91298	−0.723040	−0.361520	+	0.932364i	$0.617742\pi$
$8$	−9.59477	−3.39226
$9$	0	0
$10$	1.65612	0.523712
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	0.176875	0.0490563	0.0245282	−	0.999699i	$-0.492192\pi$
$13$	0.176875	0.0490563	0.0245282	+	0.999699i	$0.492192\pi$
$14$	5.23991	1.40043
$15$	0	0
$16$	15.2757	3.81891
$17$	−2.74225	−0.665093	−0.332546	−	0.943087i	$-0.607908\pi$
$17$	−2.74225	−0.665093	−0.332546	+	0.943087i	$0.607908\pi$
$18$	0	0
$19$	3.64728	0.836745	0.418372	−	0.908276i	$-0.362601\pi$
$19$	3.64728	0.836745	0.418372	+	0.908276i	$0.362601\pi$
$20$	−3.32711	−0.743964
$21$	0	0
$22$	2.73913	0.583985
$23$	−1.61899	−0.337582	−0.168791	−	0.985652i	$-0.553986\pi$
$23$	−1.61899	−0.337582	−0.168791	+	0.985652i	$0.553986\pi$
$24$	0	0
$25$	−4.63444	−0.926888
$26$	−0.484484	−0.0950152
$27$	0	0
$28$	−10.5269	−1.98939
$29$	7.55848	1.40357	0.701787	−	0.712387i	$-0.252386\pi$
$29$	7.55848	1.40357	0.701787	+	0.712387i	$0.252386\pi$
$30$	0	0
$31$	7.88196	1.41564	0.707821	−	0.706392i	$-0.249678\pi$
$31$	7.88196	1.41564	0.707821	+	0.706392i	$0.249678\pi$
$32$	−22.6525	−4.00443
$33$	0	0
$34$	7.51138	1.28819
$35$	1.15662	0.195504
$36$	0	0
$37$	8.43648	1.38695	0.693475	−	0.720481i	$-0.256079\pi$
$37$	8.43648	1.38695	0.693475	+	0.720481i	$0.256079\pi$
$38$	−9.99040	−1.62066
$39$	0	0
$40$	5.80115	0.917242
$41$	−8.93210	−1.39496	−0.697480	−	0.716604i	$-0.745695\pi$
$41$	−8.93210	−1.39496	−0.697480	+	0.716604i	$0.745695\pi$
$42$	0	0
$43$	−4.51816	−0.689013	−0.344507	−	0.938784i	$-0.611954\pi$
$43$	−4.51816	−0.689013	−0.344507	+	0.938784i	$0.611954\pi$
$44$	−5.50285	−0.829586
$45$	0	0
$46$	4.43462	0.653848
$47$	2.48579	0.362590	0.181295	−	0.983429i	$-0.441971\pi$
$47$	2.48579	0.362590	0.181295	+	0.983429i	$0.441971\pi$
$48$	0	0
$49$	−3.34050	−0.477214
$50$	12.6943	1.79525
$51$	0	0
$52$	0.973317	0.134975
$53$	7.57327	1.04027	0.520135	−	0.854084i	$-0.325882\pi$
$53$	7.57327	1.04027	0.520135	+	0.854084i	$0.325882\pi$
$54$	0	0
$55$	0.604616	0.0815264
$56$	18.3546	2.45274
$57$	0	0
$58$	−20.7037	−2.71852
$59$	−1.33758	−0.174138	−0.0870692	−	0.996202i	$-0.527750\pi$
$59$	−1.33758	−0.174138	−0.0870692	+	0.996202i	$0.527750\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−21.5897	−2.74190
$63$	0	0
$64$	31.4969	3.93711
$65$	−0.106941	−0.0132645
$66$	0	0
$67$	−5.50196	−0.672171	−0.336086	−	0.941831i	$-0.609103\pi$
$67$	−5.50196	−0.672171	−0.336086	+	0.941831i	$0.609103\pi$
$68$	−15.0902	−1.82995
$69$	0	0
$70$	−3.16813	−0.378665
$71$	−0.330471	−0.0392197	−0.0196098	−	0.999808i	$-0.506242\pi$
$71$	−0.330471	−0.0392197	−0.0196098	+	0.999808i	$0.506242\pi$
$72$	0	0
$73$	−1.75560	−0.205477	−0.102739	−	0.994708i	$-0.532761\pi$
$73$	−1.75560	−0.205477	−0.102739	+	0.994708i	$0.532761\pi$
$74$	−23.1086	−2.68633
$75$	0	0
$76$	20.0705	2.30224
$77$	1.91298	0.218005
$78$	0	0
$79$	−12.9780	−1.46014	−0.730070	−	0.683372i	$-0.760513\pi$
$79$	−12.9780	−1.46014	−0.730070	+	0.683372i	$0.760513\pi$
$80$	−9.23590	−1.03260
$81$	0	0
$82$	24.4662	2.70184
$83$	−12.9300	−1.41925	−0.709625	−	0.704580i	$-0.751136\pi$
$83$	−12.9300	−1.41925	−0.709625	+	0.704580i	$0.751136\pi$
$84$	0	0
$85$	1.65801	0.179836
$86$	12.3759	1.33452
$87$	0	0
$88$	9.59477	1.02281
$89$	4.11061	0.435724	0.217862	−	0.975980i	$-0.430092\pi$
$89$	4.11061	0.435724	0.217862	+	0.975980i	$0.430092\pi$
$90$	0	0
$91$	−0.338359	−0.0354697
$92$	−8.90903	−0.928831
$93$	0	0
$94$	−6.80891	−0.702285
$95$	−2.20521	−0.226249
$96$	0	0
$97$	−12.7524	−1.29481	−0.647404	−	0.762147i	$-0.724145\pi$
$97$	−12.7524	−1.29481	−0.647404	+	0.762147i	$0.724145\pi$
$98$	9.15006	0.924296
$99$	0	0
$100$	−25.5026	−2.55026
$101$	10.0353	0.998553	0.499276	−	0.866443i	$-0.333599\pi$
$101$	10.0353	0.998553	0.499276	+	0.866443i	$0.333599\pi$
$102$	0	0
$103$	−6.05237	−0.596357	−0.298179	−	0.954510i	$-0.596379\pi$
$103$	−6.05237	−0.596357	−0.298179	+	0.954510i	$0.596379\pi$
$104$	−1.69708	−0.166412
$105$	0	0
$106$	−20.7442	−2.01485
$107$	−12.2595	−1.18517	−0.592587	−	0.805506i	$-0.701893\pi$
$107$	−12.2595	−1.18517	−0.592587	+	0.805506i	$0.701893\pi$
$108$	0	0
$109$	−1.38440	−0.132602	−0.0663008	−	0.997800i	$-0.521120\pi$
$109$	−1.38440	−0.132602	−0.0663008	+	0.997800i	$0.521120\pi$
$110$	−1.65612	−0.157905
$111$	0	0
$112$	−29.2221	−2.76123
$113$	−12.6874	−1.19353	−0.596765	−	0.802416i	$-0.703548\pi$
$113$	−12.6874	−1.19353	−0.596765	+	0.802416i	$0.703548\pi$
$114$	0	0
$115$	0.978864	0.0912795
$116$	41.5931	3.86183
$117$	0	0
$118$	3.66382	0.337282
$119$	5.24587	0.480888
$120$	0	0
$121$	1.00000	0.0909091
$122$	2.73913	0.247989
$123$	0	0
$124$	43.3732	3.89503
$125$	5.82513	0.521016
$126$	0	0
$127$	7.42636	0.658983	0.329491	−	0.944159i	$-0.393123\pi$
$127$	7.42636	0.658983	0.329491	+	0.944159i	$0.393123\pi$
$128$	−40.9692	−3.62120
$129$	0	0
$130$	0.292927	0.0256914
$131$	7.69981	0.672735	0.336368	−	0.941731i	$-0.390802\pi$
$131$	7.69981	0.672735	0.336368	+	0.941731i	$0.390802\pi$
$132$	0	0
$133$	−6.97719	−0.604999
$134$	15.0706	1.30190
$135$	0	0
$136$	26.3112	2.25617
$137$	15.9210	1.36022	0.680112	−	0.733108i	$-0.261931\pi$
$137$	15.9210	1.36022	0.680112	+	0.733108i	$0.261931\pi$
$138$	0	0
$139$	−2.40561	−0.204041	−0.102021	−	0.994782i	$-0.532531\pi$
$139$	−2.40561	−0.204041	−0.102021	+	0.994782i	$0.532531\pi$
$140$	6.36470	0.537916
$141$	0	0
$142$	0.905204	0.0759630
$143$	−0.176875	−0.0147910
$144$	0	0
$145$	−4.56997	−0.379516
$146$	4.80881	0.397980
$147$	0	0
$148$	46.4247	3.81609
$149$	14.5339	1.19066	0.595331	−	0.803480i	$-0.297021\pi$
$149$	14.5339	1.19066	0.595331	+	0.803480i	$0.297021\pi$
$150$	0	0
$151$	−10.9430	−0.890525	−0.445262	−	0.895400i	$-0.646890\pi$
$151$	−10.9430	−0.890525	−0.445262	+	0.895400i	$0.646890\pi$
$152$	−34.9949	−2.83846
$153$	0	0
$154$	−5.23991	−0.422244
$155$	−4.76556	−0.382779
$156$	0	0
$157$	16.6497	1.32879	0.664394	−	0.747383i	$-0.268690\pi$
$157$	16.6497	1.32879	0.664394	+	0.747383i	$0.268690\pi$
$158$	35.5485	2.82809
$159$	0	0
$160$	13.6961	1.08277
$161$	3.09709	0.244085
$162$	0	0
$163$	−2.05976	−0.161333	−0.0806666	−	0.996741i	$-0.525705\pi$
$163$	−2.05976	−0.161333	−0.0806666	+	0.996741i	$0.525705\pi$
$164$	−49.1520	−3.83813
$165$	0	0
$166$	35.4169	2.74889
$167$	15.3296	1.18624	0.593121	−	0.805113i	$-0.297895\pi$
$167$	15.3296	1.18624	0.593121	+	0.805113i	$0.297895\pi$
$168$	0	0
$169$	−12.9687	−0.997593
$170$	−4.54150	−0.348317
$171$	0	0
$172$	−24.8628	−1.89577
$173$	−13.0360	−0.991105	−0.495553	−	0.868578i	$-0.665034\pi$
$173$	−13.0360	−0.991105	−0.495553	+	0.868578i	$0.665034\pi$
$174$	0	0
$175$	8.86560	0.670177
$176$	−15.2757	−1.15145
$177$	0	0
$178$	−11.2595	−0.843936
$179$	11.6608	0.871571	0.435786	−	0.900050i	$-0.356471\pi$
$179$	11.6608	0.871571	0.435786	+	0.900050i	$0.356471\pi$
$180$	0	0
$181$	−9.58234	−0.712250	−0.356125	−	0.934438i	$-0.615902\pi$
$181$	−9.58234	−0.712250	−0.356125	+	0.934438i	$0.615902\pi$
$182$	0.926810	0.0686997
$183$	0	0
$184$	15.5338	1.14517
$185$	−5.10083	−0.375021
$186$	0	0
$187$	2.74225	0.200533
$188$	13.6789	0.997638
$189$	0	0
$190$	6.04035	0.438213
$191$	18.4200	1.33283	0.666414	−	0.745582i	$-0.267828\pi$
$191$	18.4200	1.33283	0.666414	+	0.745582i	$0.267828\pi$
$192$	0	0
$193$	−5.40862	−0.389321	−0.194661	−	0.980871i	$-0.562361\pi$
$193$	−5.40862	−0.389321	−0.194661	+	0.980871i	$0.562361\pi$
$194$	34.9305	2.50786
$195$	0	0
$196$	−18.3822	−1.31302
$197$	2.86614	0.204204	0.102102	−	0.994774i	$-0.467443\pi$
$197$	2.86614	0.204204	0.102102	+	0.994774i	$0.467443\pi$
$198$	0	0
$199$	11.9550	0.847469	0.423735	−	0.905786i	$-0.360719\pi$
$199$	11.9550	0.847469	0.423735	+	0.905786i	$0.360719\pi$
$200$	44.4664	3.14425
$201$	0	0
$202$	−27.4881	−1.93406
$203$	−14.4592	−1.01484
$204$	0	0
$205$	5.40049	0.377187
$206$	16.5782	1.15506
$207$	0	0
$208$	2.70188	0.187342
$209$	−3.64728	−0.252288
$210$	0	0
$211$	24.3432	1.67585	0.837927	−	0.545783i	$-0.183768\pi$
$211$	24.3432	1.67585	0.837927	+	0.545783i	$0.183768\pi$
$212$	41.6746	2.86222
$213$	0	0
$214$	33.5805	2.29552
$215$	2.73175	0.186304
$216$	0	0
$217$	−15.0781	−1.02357
$218$	3.79206	0.256831
$219$	0	0
$220$	3.32711	0.224314
$221$	−0.485035	−0.0326270
$222$	0	0
$223$	15.7257	1.05307	0.526535	−	0.850154i	$-0.323491\pi$
$223$	15.7257	1.05307	0.526535	+	0.850154i	$0.323491\pi$
$224$	43.3338	2.89536
$225$	0	0
$226$	34.7525	2.31170
$227$	0.677571	0.0449720	0.0224860	−	0.999747i	$-0.492842\pi$
$227$	0.677571	0.0449720	0.0224860	+	0.999747i	$0.492842\pi$
$228$	0	0
$229$	13.4531	0.889006	0.444503	−	0.895777i	$-0.353380\pi$
$229$	13.4531	0.889006	0.444503	+	0.895777i	$0.353380\pi$
$230$	−2.68124	−0.176796
$231$	0	0
$232$	−72.5218	−4.76129
$233$	12.1025	0.792862	0.396431	−	0.918064i	$-0.370249\pi$
$233$	12.1025	0.792862	0.396431	+	0.918064i	$0.370249\pi$
$234$	0	0
$235$	−1.50295	−0.0980415
$236$	−7.36052	−0.479129
$237$	0	0
$238$	−14.3691	−0.931413
$239$	7.34142	0.474877	0.237438	−	0.971403i	$-0.423692\pi$
$239$	7.34142	0.474877	0.237438	+	0.971403i	$0.423692\pi$
$240$	0	0
$241$	−26.9723	−1.73744	−0.868719	−	0.495306i	$-0.835056\pi$
$241$	−26.9723	−1.73744	−0.868719	+	0.495306i	$0.835056\pi$
$242$	−2.73913	−0.176078
$243$	0	0
$244$	−5.50285	−0.352284
$245$	2.01972	0.129035
$246$	0	0
$247$	0.645114	0.0410476
$248$	−75.6256	−4.80223
$249$	0	0
$250$	−15.9558	−1.00913
$251$	17.0405	1.07559	0.537794	−	0.843076i	$-0.319258\pi$
$251$	17.0405	1.07559	0.537794	+	0.843076i	$0.319258\pi$
$252$	0	0
$253$	1.61899	0.101785
$254$	−20.3418	−1.27636
$255$	0	0
$256$	49.2263	3.07664
$257$	−13.6010	−0.848410	−0.424205	−	0.905566i	$-0.639446\pi$
$257$	−13.6010	−0.848410	−0.424205	+	0.905566i	$0.639446\pi$
$258$	0	0
$259$	−16.1388	−1.00282
$260$	−0.588483	−0.0364961
$261$	0	0
$262$	−21.0908	−1.30299
$263$	−18.2508	−1.12539	−0.562696	−	0.826664i	$-0.690236\pi$
$263$	−18.2508	−1.12539	−0.562696	+	0.826664i	$0.690236\pi$
$264$	0	0
$265$	−4.57892	−0.281281
$266$	19.1115	1.17180
$267$	0	0
$268$	−30.2764	−1.84943
$269$	9.66539	0.589309	0.294655	−	0.955604i	$-0.404795\pi$
$269$	9.66539	0.589309	0.294655	+	0.955604i	$0.404795\pi$
$270$	0	0
$271$	30.1115	1.82914	0.914571	−	0.404425i	$-0.132528\pi$
$271$	30.1115	1.82914	0.914571	+	0.404425i	$0.132528\pi$
$272$	−41.8896	−2.53993
$273$	0	0
$274$	−43.6098	−2.63456
$275$	4.63444	0.279467
$276$	0	0
$277$	−5.89926	−0.354452	−0.177226	−	0.984170i	$-0.556712\pi$
$277$	−5.89926	−0.354452	−0.177226	+	0.984170i	$0.556712\pi$
$278$	6.58929	0.395199
$279$	0	0
$280$	−11.0975	−0.663202
$281$	−6.40992	−0.382384	−0.191192	−	0.981553i	$-0.561235\pi$
$281$	−6.40992	−0.382384	−0.191192	+	0.981553i	$0.561235\pi$
$282$	0	0
$283$	2.51457	0.149476	0.0747380	−	0.997203i	$-0.476188\pi$
$283$	2.51457	0.149476	0.0747380	+	0.997203i	$0.476188\pi$
$284$	−1.81853	−0.107910
$285$	0	0
$286$	0.484484	0.0286482
$287$	17.0870	1.00861
$288$	0	0
$289$	−9.48008	−0.557652
$290$	12.5178	0.735068
$291$	0	0
$292$	−9.66079	−0.565355
$293$	12.3742	0.722908	0.361454	−	0.932390i	$-0.382280\pi$
$293$	12.3742	0.722908	0.361454	+	0.932390i	$0.382280\pi$
$294$	0	0
$295$	0.808724	0.0470857
$296$	−80.9461	−4.70490
$297$	0	0
$298$	−39.8102	−2.30615
$299$	−0.286358	−0.0165605
$300$	0	0
$301$	8.64317	0.498184
$302$	29.9742	1.72482
$303$	0	0
$304$	55.7146	3.19545
$305$	0.604616	0.0346202
$306$	0	0
$307$	21.2440	1.21246	0.606230	−	0.795290i	$-0.292681\pi$
$307$	21.2440	1.21246	0.606230	+	0.795290i	$0.292681\pi$
$308$	10.5269	0.599823
$309$	0	0
$310$	13.0535	0.741389
$311$	−26.4545	−1.50010	−0.750049	−	0.661383i	$-0.769970\pi$
$311$	−26.4545	−1.50010	−0.750049	+	0.661383i	$0.769970\pi$
$312$	0	0
$313$	30.7427	1.73768	0.868841	−	0.495090i	$-0.164865\pi$
$313$	30.7427	1.73768	0.868841	+	0.495090i	$0.164865\pi$
$314$	−45.6056	−2.57367
$315$	0	0
$316$	−71.4160	−4.01747
$317$	21.9554	1.23314	0.616569	−	0.787301i	$-0.288522\pi$
$317$	21.9554	1.23314	0.616569	+	0.787301i	$0.288522\pi$
$318$	0	0
$319$	−7.55848	−0.423193
$320$	−19.0435	−1.06457
$321$	0	0
$322$	−8.48334	−0.472758
$323$	−10.0018	−0.556513
$324$	0	0
$325$	−0.819717	−0.0454697
$326$	5.64197	0.312480
$327$	0	0
$328$	85.7015	4.73207
$329$	−4.75527	−0.262167
$330$	0	0
$331$	−15.8159	−0.869318	−0.434659	−	0.900595i	$-0.643131\pi$
$331$	−15.8159	−0.869318	−0.434659	+	0.900595i	$0.643131\pi$
$332$	−71.1517	−3.90496
$333$	0	0
$334$	−41.9899	−2.29759
$335$	3.32657	0.181750
$336$	0	0
$337$	−33.1532	−1.80597	−0.902986	−	0.429671i	$-0.858630\pi$
$337$	−33.1532	−1.80597	−0.902986	+	0.429671i	$0.858630\pi$
$338$	35.5230	1.93220
$339$	0	0
$340$	9.12376	0.494805
$341$	−7.88196	−0.426832
$342$	0	0
$343$	19.7812	1.06808
$344$	43.3507	2.33732
$345$	0	0
$346$	35.7072	1.91963
$347$	−23.0402	−1.23686	−0.618430	−	0.785840i	$-0.712231\pi$
$347$	−23.0402	−1.23686	−0.618430	+	0.785840i	$0.712231\pi$
$348$	0	0
$349$	−32.7544	−1.75330	−0.876651	−	0.481127i	$-0.840227\pi$
$349$	−32.7544	−1.75330	−0.876651	+	0.481127i	$0.840227\pi$
$350$	−24.2841	−1.29804
$351$	0	0
$352$	22.6525	1.20738
$353$	−0.349353	−0.0185942	−0.00929708	−	0.999957i	$-0.502959\pi$
$353$	−0.349353	−0.0185942	−0.00929708	+	0.999957i	$0.502959\pi$
$354$	0	0
$355$	0.199808	0.0106047
$356$	22.6201	1.19886
$357$	0	0
$358$	−31.9406	−1.68811
$359$	−29.2479	−1.54365	−0.771823	−	0.635838i	$-0.780655\pi$
$359$	−29.2479	−1.54365	−0.771823	+	0.635838i	$0.780655\pi$
$360$	0	0
$361$	−5.69731	−0.299859
$362$	26.2473	1.37953
$363$	0	0
$364$	−1.86194	−0.0975921
$365$	1.06146	0.0555594
$366$	0	0
$367$	−13.9617	−0.728795	−0.364397	−	0.931244i	$-0.618725\pi$
$367$	−13.9617	−0.728795	−0.364397	+	0.931244i	$0.618725\pi$
$368$	−24.7311	−1.28920
$369$	0	0
$370$	13.9719	0.726362
$371$	−14.4875	−0.752156
$372$	0	0
$373$	13.7140	0.710084	0.355042	−	0.934850i	$-0.384467\pi$
$373$	13.7140	0.710084	0.355042	+	0.934850i	$0.384467\pi$
$374$	−7.51138	−0.388404
$375$	0	0
$376$	−23.8506	−1.23000
$377$	1.33691	0.0688541
$378$	0	0
$379$	34.6378	1.77922	0.889612	−	0.456717i	$-0.150975\pi$
$379$	34.6378	1.77922	0.889612	+	0.456717i	$0.150975\pi$
$380$	−12.1349	−0.622508
$381$	0	0
$382$	−50.4550	−2.58150
$383$	−7.28622	−0.372308	−0.186154	−	0.982521i	$-0.559602\pi$
$383$	−7.28622	−0.372308	−0.186154	+	0.982521i	$0.559602\pi$
$384$	0	0
$385$	−1.15662	−0.0589468
$386$	14.8149	0.754060
$387$	0	0
$388$	−70.1744	−3.56257
$389$	30.6669	1.55487	0.777437	−	0.628960i	$-0.216519\pi$
$389$	30.6669	1.55487	0.777437	+	0.628960i	$0.216519\pi$
$390$	0	0
$391$	4.43966	0.224523
$392$	32.0513	1.61883
$393$	0	0
$394$	−7.85074	−0.395514
$395$	7.84671	0.394811
$396$	0	0
$397$	−1.12627	−0.0565257	−0.0282628	−	0.999601i	$-0.508998\pi$
$397$	−1.12627	−0.0565257	−0.0282628	+	0.999601i	$0.508998\pi$
$398$	−32.7464	−1.64143
$399$	0	0
$400$	−70.7941	−3.53970
$401$	9.38421	0.468625	0.234312	−	0.972161i	$-0.424716\pi$
$401$	9.38421	0.468625	0.234312	+	0.972161i	$0.424716\pi$
$402$	0	0
$403$	1.39412	0.0694462
$404$	55.2229	2.74744
$405$	0	0
$406$	39.6058	1.96560
$407$	−8.43648	−0.418181
$408$	0	0
$409$	34.3783	1.69990	0.849948	−	0.526867i	$-0.176633\pi$
$409$	34.3783	1.69990	0.849948	+	0.526867i	$0.176633\pi$
$410$	−14.7927	−0.730558
$411$	0	0
$412$	−33.3053	−1.64083
$413$	2.55877	0.125909
$414$	0	0
$415$	7.81767	0.383754
$416$	−4.00666	−0.196443
$417$	0	0
$418$	9.99040	0.488646
$419$	1.53252	0.0748685	0.0374343	−	0.999299i	$-0.488082\pi$
$419$	1.53252	0.0748685	0.0374343	+	0.999299i	$0.488082\pi$
$420$	0	0
$421$	13.4022	0.653185	0.326593	−	0.945165i	$-0.394100\pi$
$421$	13.4022	0.653185	0.326593	+	0.945165i	$0.394100\pi$
$422$	−66.6792	−3.24589
$423$	0	0
$424$	−72.6638	−3.52887
$425$	12.7088	0.616466
$426$	0	0
$427$	1.91298	0.0925757
$428$	−67.4624	−3.26092
$429$	0	0
$430$	−7.48263	−0.360845
$431$	4.01874	0.193576	0.0967880	−	0.995305i	$-0.469143\pi$
$431$	4.01874	0.193576	0.0967880	+	0.995305i	$0.469143\pi$
$432$	0	0
$433$	19.2933	0.927175	0.463587	−	0.886051i	$-0.346562\pi$
$433$	19.2933	0.927175	0.463587	+	0.886051i	$0.346562\pi$
$434$	41.3008	1.98250
$435$	0	0
$436$	−7.61815	−0.364843
$437$	−5.90490	−0.282470
$438$	0	0
$439$	−23.1285	−1.10386	−0.551930	−	0.833890i	$-0.686109\pi$
$439$	−23.1285	−1.10386	−0.551930	+	0.833890i	$0.686109\pi$
$440$	−5.80115	−0.276559
$441$	0	0
$442$	1.32858	0.0631939
$443$	18.3618	0.872398	0.436199	−	0.899850i	$-0.356324\pi$
$443$	18.3618	0.872398	0.436199	+	0.899850i	$0.356324\pi$
$444$	0	0
$445$	−2.48534	−0.117816
$446$	−43.0747	−2.03965
$447$	0	0
$448$	−60.2530	−2.84669
$449$	12.2717	0.579135	0.289568	−	0.957158i	$-0.406488\pi$
$449$	12.2717	0.579135	0.289568	+	0.957158i	$0.406488\pi$
$450$	0	0
$451$	8.93210	0.420596
$452$	−69.8168	−3.28391
$453$	0	0
$454$	−1.85596	−0.0871044
$455$	0.204577	0.00959073
$456$	0	0
$457$	17.9930	0.841679	0.420840	−	0.907135i	$-0.361736\pi$
$457$	17.9930	0.841679	0.420840	+	0.907135i	$0.361736\pi$
$458$	−36.8498	−1.72188
$459$	0	0
$460$	5.38654	0.251149
$461$	1.26483	0.0589088	0.0294544	−	0.999566i	$-0.490623\pi$
$461$	1.26483	0.0589088	0.0294544	+	0.999566i	$0.490623\pi$
$462$	0	0
$463$	−10.3872	−0.482732	−0.241366	−	0.970434i	$-0.577595\pi$
$463$	−10.3872	−0.482732	−0.241366	+	0.970434i	$0.577595\pi$
$464$	115.461	5.36012
$465$	0	0
$466$	−33.1504	−1.53566
$467$	−10.9780	−0.507999	−0.253999	−	0.967204i	$-0.581746\pi$
$467$	−10.9780	−0.507999	−0.253999	+	0.967204i	$0.581746\pi$
$468$	0	0
$469$	10.5252	0.486006
$470$	4.11677	0.189893
$471$	0	0
$472$	12.8338	0.590724
$473$	4.51816	0.207745
$474$	0	0
$475$	−16.9031	−0.775568
$476$	28.8672	1.32313
$477$	0	0
$478$	−20.1091	−0.919770
$479$	17.8446	0.815339	0.407670	−	0.913130i	$-0.366342\pi$
$479$	17.8446	0.815339	0.407670	+	0.913130i	$0.366342\pi$
$480$	0	0
$481$	1.49220	0.0680386
$482$	73.8807	3.36517
$483$	0	0
$484$	5.50285	0.250130
$485$	7.71029	0.350106
$486$	0	0
$487$	−19.6964	−0.892531	−0.446266	−	0.894901i	$-0.647246\pi$
$487$	−19.6964	−0.892531	−0.446266	+	0.894901i	$0.647246\pi$
$488$	9.59477	0.434335
$489$	0	0
$490$	−5.53227	−0.249923
$491$	17.3750	0.784124	0.392062	−	0.919939i	$-0.371762\pi$
$491$	17.3750	0.784124	0.392062	+	0.919939i	$0.371762\pi$
$492$	0	0
$493$	−20.7272	−0.933507
$494$	−1.76705	−0.0795034
$495$	0	0
$496$	120.402	5.40621
$497$	0.632185	0.0283574
$498$	0	0
$499$	25.1532	1.12601	0.563005	−	0.826454i	$-0.309645\pi$
$499$	25.1532	1.12601	0.563005	+	0.826454i	$0.309645\pi$
$500$	32.0548	1.43354
$501$	0	0
$502$	−46.6763	−2.08326
$503$	−18.0309	−0.803958	−0.401979	−	0.915649i	$-0.631678\pi$
$503$	−18.0309	−0.803958	−0.401979	+	0.915649i	$0.631678\pi$
$504$	0	0
$505$	−6.06752	−0.270001
$506$	−4.43462	−0.197143
$507$	0	0
$508$	40.8661	1.81314
$509$	35.8893	1.59077	0.795383	−	0.606108i	$-0.207270\pi$
$509$	35.8893	1.59077	0.795383	+	0.606108i	$0.207270\pi$
$510$	0	0
$511$	3.35843	0.148568
$512$	−52.8990	−2.33783
$513$	0	0
$514$	37.2551	1.64325
$515$	3.65936	0.161250
$516$	0	0
$517$	−2.48579	−0.109325
$518$	44.2065	1.94232
$519$	0	0
$520$	1.02608	0.0449965
$521$	−6.16062	−0.269902	−0.134951	−	0.990852i	$-0.543088\pi$
$521$	−6.16062	−0.269902	−0.134951	+	0.990852i	$0.543088\pi$
$522$	0	0
$523$	−2.39141	−0.104569	−0.0522846	−	0.998632i	$-0.516650\pi$
$523$	−2.39141	−0.104569	−0.0522846	+	0.998632i	$0.516650\pi$
$524$	42.3709	1.85098
$525$	0	0
$526$	49.9914	2.17973
$527$	−21.6143	−0.941533
$528$	0	0
$529$	−20.3789	−0.886039
$530$	12.5423	0.544801
$531$	0	0
$532$	−38.3944	−1.66461
$533$	−1.57987	−0.0684316
$534$	0	0
$535$	7.41231	0.320462
$536$	52.7900	2.28018
$537$	0	0
$538$	−26.4748	−1.14141
$539$	3.34050	0.143885
$540$	0	0
$541$	30.9269	1.32965	0.664827	−	0.746998i	$-0.268505\pi$
$541$	30.9269	1.32965	0.664827	+	0.746998i	$0.268505\pi$
$542$	−82.4793	−3.54279
$543$	0	0
$544$	62.1188	2.66332
$545$	0.837031	0.0358545
$546$	0	0
$547$	31.4601	1.34514	0.672569	−	0.740035i	$-0.265191\pi$
$547$	31.4601	1.34514	0.672569	+	0.740035i	$0.265191\pi$
$548$	87.6109	3.74255
$549$	0	0
$550$	−12.6943	−0.541289
$551$	27.5679	1.17443
$552$	0	0
$553$	24.8267	1.05574
$554$	16.1589	0.686524
$555$	0	0
$556$	−13.2377	−0.561404
$557$	43.3173	1.83541	0.917706	−	0.397261i	$-0.130039\pi$
$557$	43.3173	1.83541	0.917706	+	0.397261i	$0.130039\pi$
$558$	0	0
$559$	−0.799150	−0.0338005
$560$	17.6681	0.746614
$561$	0	0
$562$	17.5576	0.740623
$563$	12.7937	0.539192	0.269596	−	0.962974i	$-0.413110\pi$
$563$	12.7937	0.539192	0.269596	+	0.962974i	$0.413110\pi$
$564$	0	0
$565$	7.67100	0.322721
$566$	−6.88775	−0.289514
$567$	0	0
$568$	3.17079	0.133044
$569$	−16.2156	−0.679791	−0.339896	−	0.940463i	$-0.610392\pi$
$569$	−16.2156	−0.679791	−0.339896	+	0.940463i	$0.610392\pi$
$570$	0	0
$571$	15.4511	0.646607	0.323303	−	0.946295i	$-0.395207\pi$
$571$	15.4511	0.646607	0.323303	+	0.946295i	$0.395207\pi$
$572$	−0.973317	−0.0406964
$573$	0	0
$574$	−46.8035	−1.95354
$575$	7.50309	0.312900
$576$	0	0
$577$	−13.7967	−0.574365	−0.287182	−	0.957876i	$-0.592719\pi$
$577$	−13.7967	−0.574365	−0.287182	+	0.957876i	$0.592719\pi$
$578$	25.9672	1.08009
$579$	0	0
$580$	−25.1479	−1.04421
$581$	24.7348	1.02617
$582$	0	0
$583$	−7.57327	−0.313653
$584$	16.8445	0.697032
$585$	0	0
$586$	−33.8946	−1.40017
$587$	0.731957	0.0302111	0.0151056	−	0.999886i	$-0.495192\pi$
$587$	0.731957	0.0302111	0.0151056	+	0.999886i	$0.495192\pi$
$588$	0	0
$589$	28.7478	1.18453
$590$	−2.21520	−0.0911984
$591$	0	0
$592$	128.873	5.29664
$593$	−16.9597	−0.696450	−0.348225	−	0.937411i	$-0.613215\pi$
$593$	−16.9597	−0.696450	−0.348225	+	0.937411i	$0.613215\pi$
$594$	0	0
$595$	−3.17174	−0.130029
$596$	79.9778	3.27602
$597$	0	0
$598$	0.784373	0.0320754
$599$	11.3334	0.463070	0.231535	−	0.972827i	$-0.425625\pi$
$599$	11.3334	0.463070	0.231535	+	0.972827i	$0.425625\pi$
$600$	0	0
$601$	37.8868	1.54543	0.772717	−	0.634750i	$-0.218897\pi$
$601$	37.8868	1.54543	0.772717	+	0.634750i	$0.218897\pi$
$602$	−23.6748	−0.964912
$603$	0	0
$604$	−60.2174	−2.45021
$605$	−0.604616	−0.0245811
$606$	0	0
$607$	−9.55272	−0.387733	−0.193866	−	0.981028i	$-0.562103\pi$
$607$	−9.55272	−0.387733	−0.193866	+	0.981028i	$0.562103\pi$
$608$	−82.6201	−3.35069
$609$	0	0
$610$	−1.65612	−0.0670545
$611$	0.439674	0.0177873
$612$	0	0
$613$	18.7337	0.756646	0.378323	−	0.925674i	$-0.376501\pi$
$613$	18.7337	0.756646	0.378323	+	0.925674i	$0.376501\pi$
$614$	−58.1902	−2.34836
$615$	0	0
$616$	−18.3546	−0.739529
$617$	2.88614	0.116192	0.0580958	−	0.998311i	$-0.481497\pi$
$617$	2.88614	0.116192	0.0580958	+	0.998311i	$0.481497\pi$
$618$	0	0
$619$	28.4797	1.14470	0.572348	−	0.820011i	$-0.306033\pi$
$619$	28.4797	1.14470	0.572348	+	0.820011i	$0.306033\pi$
$620$	−26.2241	−1.05319
$621$	0	0
$622$	72.4624	2.90548
$623$	−7.86353	−0.315046
$624$	0	0
$625$	19.6502	0.786009
$626$	−84.2085	−3.36565
$627$	0	0
$628$	91.6205	3.65606
$629$	−23.1349	−0.922450
$630$	0	0
$631$	18.8981	0.752322	0.376161	−	0.926554i	$-0.377244\pi$
$631$	18.8981	0.752322	0.376161	+	0.926554i	$0.377244\pi$
$632$	124.521	4.95318
$633$	0	0
$634$	−60.1388	−2.38842
$635$	−4.49009	−0.178184
$636$	0	0
$637$	−0.590850	−0.0234103
$638$	20.7037	0.819666
$639$	0	0
$640$	24.7706	0.979145
$641$	−29.7753	−1.17605	−0.588026	−	0.808842i	$-0.700095\pi$
$641$	−29.7753	−1.17605	−0.588026	+	0.808842i	$0.700095\pi$
$642$	0	0
$643$	−1.68829	−0.0665798	−0.0332899	−	0.999446i	$-0.510598\pi$
$643$	−1.68829	−0.0665798	−0.0332899	+	0.999446i	$0.510598\pi$
$644$	17.0428	0.671581
$645$	0	0
$646$	27.3961	1.07789
$647$	31.7652	1.24882	0.624409	−	0.781098i	$-0.285340\pi$
$647$	31.7652	1.24882	0.624409	+	0.781098i	$0.285340\pi$
$648$	0	0
$649$	1.33758	0.0525047
$650$	2.24531	0.0880684
$651$	0	0
$652$	−11.3346	−0.443896
$653$	12.0681	0.472261	0.236130	−	0.971721i	$-0.424121\pi$
$653$	12.0681	0.472261	0.236130	+	0.971721i	$0.424121\pi$
$654$	0	0
$655$	−4.65542	−0.181902
$656$	−136.444	−5.32723
$657$	0	0
$658$	13.0253	0.507780
$659$	−17.5896	−0.685192	−0.342596	−	0.939483i	$-0.611306\pi$
$659$	−17.5896	−0.685192	−0.342596	+	0.939483i	$0.611306\pi$
$660$	0	0
$661$	18.9318	0.736361	0.368181	−	0.929754i	$-0.379981\pi$
$661$	18.9318	0.736361	0.368181	+	0.929754i	$0.379981\pi$
$662$	43.3217	1.68375
$663$	0	0
$664$	124.060	4.81447
$665$	4.21852	0.163587
$666$	0	0
$667$	−12.2371	−0.473821
$668$	84.3567	3.26386
$669$	0	0
$670$	−9.11192	−0.352024
$671$	1.00000	0.0386046
$672$	0	0
$673$	49.3060	1.90061	0.950303	−	0.311325i	$-0.100773\pi$
$673$	49.3060	1.90061	0.950303	+	0.311325i	$0.100773\pi$
$674$	90.8111	3.49791
$675$	0	0
$676$	−71.3649	−2.74480
$677$	−6.96645	−0.267742	−0.133871	−	0.990999i	$-0.542741\pi$
$677$	−6.96645	−0.267742	−0.133871	+	0.990999i	$0.542741\pi$
$678$	0	0
$679$	24.3951	0.936198
$680$	−15.9082	−0.610051
$681$	0	0
$682$	21.5897	0.826714
$683$	−17.4032	−0.665913	−0.332957	−	0.942942i	$-0.608046\pi$
$683$	−17.4032	−0.665913	−0.332957	+	0.942942i	$0.608046\pi$
$684$	0	0
$685$	−9.62609	−0.367794
$686$	−54.1833	−2.06873
$687$	0	0
$688$	−69.0179	−2.63128
$689$	1.33952	0.0510318
$690$	0	0
$691$	−16.3254	−0.621048	−0.310524	−	0.950566i	$-0.600505\pi$
$691$	−16.3254	−0.621048	−0.310524	+	0.950566i	$0.600505\pi$
$692$	−71.7349	−2.72695
$693$	0	0
$694$	63.1100	2.39562
$695$	1.45447	0.0551712
$696$	0	0
$697$	24.4940	0.927778
$698$	89.7186	3.39590
$699$	0	0
$700$	48.7861	1.84394
$701$	1.25848	0.0475322	0.0237661	−	0.999718i	$-0.492434\pi$
$701$	1.25848	0.0475322	0.0237661	+	0.999718i	$0.492434\pi$
$702$	0	0
$703$	30.7703	1.16052
$704$	−31.4969	−1.18708
$705$	0	0
$706$	0.956923	0.0360143
$707$	−19.1974	−0.721993
$708$	0	0
$709$	44.8435	1.68413	0.842067	−	0.539373i	$-0.181339\pi$
$709$	44.8435	1.68413	0.842067	+	0.539373i	$0.181339\pi$
$710$	−0.547301	−0.0205398
$711$	0	0
$712$	−39.4404	−1.47809
$713$	−12.7608	−0.477895
$714$	0	0
$715$	0.106941	0.00399938
$716$	64.1678	2.39806
$717$	0	0
$718$	80.1139	2.98982
$719$	−39.7590	−1.48276	−0.741381	−	0.671085i	$-0.765829\pi$
$719$	−39.7590	−1.48276	−0.741381	+	0.671085i	$0.765829\pi$
$720$	0	0
$721$	11.5781	0.431190
$722$	15.6057	0.580784
$723$	0	0
$724$	−52.7302	−1.95970
$725$	−35.0293	−1.30096
$726$	0	0
$727$	39.6673	1.47118	0.735589	−	0.677428i	$-0.236905\pi$
$727$	39.6673	1.47118	0.735589	+	0.677428i	$0.236905\pi$
$728$	3.24648	0.120322
$729$	0	0
$730$	−2.90748	−0.107611
$731$	12.3899	0.458258
$732$	0	0
$733$	−4.45224	−0.164447	−0.0822235	−	0.996614i	$-0.526202\pi$
$733$	−4.45224	−0.164447	−0.0822235	+	0.996614i	$0.526202\pi$
$734$	38.2430	1.41157
$735$	0	0
$736$	36.6741	1.35182
$737$	5.50196	0.202667
$738$	0	0
$739$	−33.8922	−1.24674	−0.623372	−	0.781926i	$-0.714238\pi$
$739$	−33.8922	−1.24674	−0.623372	+	0.781926i	$0.714238\pi$
$740$	−28.0691	−1.03184
$741$	0	0
$742$	39.6833	1.45682
$743$	9.90914	0.363531	0.181766	−	0.983342i	$-0.441819\pi$
$743$	9.90914	0.363531	0.181766	+	0.983342i	$0.441819\pi$
$744$	0	0
$745$	−8.78742	−0.321946
$746$	−37.5645	−1.37533
$747$	0	0
$748$	15.0902	0.551751
$749$	23.4523	0.856928
$750$	0	0
$751$	3.30472	0.120591	0.0602955	−	0.998181i	$-0.480796\pi$
$751$	3.30472	0.120591	0.0602955	+	0.998181i	$0.480796\pi$
$752$	37.9720	1.38470
$753$	0	0
$754$	−3.66196	−0.133361
$755$	6.61628	0.240791
$756$	0	0
$757$	13.4977	0.490583	0.245292	−	0.969449i	$-0.421116\pi$
$757$	13.4977	0.490583	0.245292	+	0.969449i	$0.421116\pi$
$758$	−94.8776	−3.44611
$759$	0	0
$760$	21.1584	0.767497
$761$	28.0515	1.01687	0.508433	−	0.861101i	$-0.330225\pi$
$761$	28.0515	1.01687	0.508433	+	0.861101i	$0.330225\pi$
$762$	0	0
$763$	2.64834	0.0958762
$764$	101.363	3.66718
$765$	0	0
$766$	19.9579	0.721109
$767$	−0.236585	−0.00854259
$768$	0	0
$769$	−29.0212	−1.04653	−0.523266	−	0.852170i	$-0.675286\pi$
$769$	−29.0212	−1.04653	−0.523266	+	0.852170i	$0.675286\pi$
$770$	3.16813	0.114172
$771$	0	0
$772$	−29.7628	−1.07119
$773$	18.5820	0.668350	0.334175	−	0.942511i	$-0.391542\pi$
$773$	18.5820	0.668350	0.334175	+	0.942511i	$0.391542\pi$
$774$	0	0
$775$	−36.5285	−1.31214
$776$	122.356	4.39233
$777$	0	0
$778$	−84.0008	−3.01157
$779$	−32.5779	−1.16723
$780$	0	0
$781$	0.330471	0.0118252
$782$	−12.1608	−0.434870
$783$	0	0
$784$	−51.0282	−1.82244
$785$	−10.0666	−0.359294
$786$	0	0
$787$	34.2557	1.22108	0.610541	−	0.791985i	$-0.290952\pi$
$787$	34.2557	1.22108	0.610541	+	0.791985i	$0.290952\pi$
$788$	15.7719	0.561852
$789$	0	0
$790$	−21.4932	−0.764693
$791$	24.2708	0.862969
$792$	0	0
$793$	−0.176875	−0.00628102
$794$	3.08499	0.109482
$795$	0	0
$796$	65.7867	2.33175
$797$	11.0127	0.390088	0.195044	−	0.980794i	$-0.437515\pi$
$797$	11.0127	0.390088	0.195044	+	0.980794i	$0.437515\pi$
$798$	0	0
$799$	−6.81665	−0.241156
$800$	104.982	3.71166
$801$	0	0
$802$	−25.7046	−0.907660
$803$	1.75560	0.0619537
$804$	0	0
$805$	−1.87255	−0.0659987
$806$	−3.81869	−0.134507
$807$	0	0
$808$	−96.2867	−3.38735
$809$	−45.2337	−1.59033	−0.795165	−	0.606392i	$-0.792616\pi$
$809$	−45.2337	−1.59033	−0.795165	+	0.606392i	$0.792616\pi$
$810$	0	0
$811$	32.9688	1.15769	0.578846	−	0.815437i	$-0.303503\pi$
$811$	32.9688	1.15769	0.578846	+	0.815437i	$0.303503\pi$
$812$	−79.5670	−2.79225
$813$	0	0
$814$	23.1086	0.809958
$815$	1.24537	0.0436232
$816$	0	0
$817$	−16.4790	−0.576528
$818$	−94.1666	−3.29246
$819$	0	0
$820$	29.7181	1.03780
$821$	−17.6348	−0.615460	−0.307730	−	0.951474i	$-0.599569\pi$
$821$	−17.6348	−0.615460	−0.307730	+	0.951474i	$0.599569\pi$
$822$	0	0
$823$	−6.60202	−0.230132	−0.115066	−	0.993358i	$-0.536708\pi$
$823$	−6.60202	−0.230132	−0.115066	+	0.993358i	$0.536708\pi$
$824$	58.0711	2.02300
$825$	0	0
$826$	−7.00882	−0.243868
$827$	0.843089	0.0293171	0.0146585	−	0.999893i	$-0.495334\pi$
$827$	0.843089	0.0293171	0.0146585	+	0.999893i	$0.495334\pi$
$828$	0	0
$829$	−40.4340	−1.40433	−0.702166	−	0.712014i	$-0.747783\pi$
$829$	−40.4340	−1.40433	−0.702166	+	0.712014i	$0.747783\pi$
$830$	−21.4136	−0.743278
$831$	0	0
$832$	5.57102	0.193140
$833$	9.16047	0.317391
$834$	0	0
$835$	−9.26854	−0.320751
$836$	−20.0705	−0.694151
$837$	0	0
$838$	−4.19778	−0.145010
$839$	−49.0996	−1.69511	−0.847553	−	0.530710i	$-0.821925\pi$
$839$	−49.0996	−1.69511	−0.847553	+	0.530710i	$0.821925\pi$
$840$	0	0
$841$	28.1305	0.970019
$842$	−36.7105	−1.26513
$843$	0	0
$844$	133.957	4.61098
$845$	7.84109	0.269742
$846$	0	0
$847$	−1.91298	−0.0657309
$848$	115.687	3.97270
$849$	0	0
$850$	−34.8110	−1.19401
$851$	−13.6585	−0.468209
$852$	0	0
$853$	35.0517	1.20015	0.600074	−	0.799944i	$-0.295138\pi$
$853$	35.0517	1.20015	0.600074	+	0.799944i	$0.295138\pi$
$854$	−5.23991	−0.179306
$855$	0	0
$856$	117.627	4.02042
$857$	−51.8553	−1.77134	−0.885672	−	0.464311i	$-0.846302\pi$
$857$	−51.8553	−1.77134	−0.885672	+	0.464311i	$0.846302\pi$
$858$	0	0
$859$	−31.2016	−1.06458	−0.532292	−	0.846561i	$-0.678669\pi$
$859$	−31.2016	−1.06458	−0.532292	+	0.846561i	$0.678669\pi$
$860$	15.0324	0.512601
$861$	0	0
$862$	−11.0079	−0.374929
$863$	21.2847	0.724539	0.362269	−	0.932073i	$-0.382002\pi$
$863$	21.2847	0.724539	0.362269	+	0.932073i	$0.382002\pi$
$864$	0	0
$865$	7.88174	0.267987
$866$	−52.8468	−1.79581
$867$	0	0
$868$	−82.9723	−2.81626
$869$	12.9780	0.440249
$870$	0	0
$871$	−0.973159	−0.0329742
$872$	13.2830	0.449819
$873$	0	0
$874$	16.1743	0.547104
$875$	−11.1434	−0.376715
$876$	0	0
$877$	56.4130	1.90493	0.952465	−	0.304647i	$-0.0985386\pi$
$877$	56.4130	1.90493	0.952465	+	0.304647i	$0.0985386\pi$
$878$	63.3519	2.13802
$879$	0	0
$880$	9.23590	0.311342
$881$	13.2697	0.447067	0.223534	−	0.974696i	$-0.428241\pi$
$881$	13.2697	0.447067	0.223534	+	0.974696i	$0.428241\pi$
$882$	0	0
$883$	−30.1064	−1.01316	−0.506581	−	0.862192i	$-0.669091\pi$
$883$	−30.1064	−1.01316	−0.506581	+	0.862192i	$0.669091\pi$
$884$	−2.66908	−0.0897707
$885$	0	0
$886$	−50.2955	−1.68971
$887$	−7.10163	−0.238449	−0.119225	−	0.992867i	$-0.538041\pi$
$887$	−7.10163	−0.238449	−0.119225	+	0.992867i	$0.538041\pi$
$888$	0	0
$889$	−14.2065	−0.476471
$890$	6.80768	0.228194
$891$	0	0
$892$	86.5360	2.89744
$893$	9.06638	0.303395
$894$	0	0
$895$	−7.05032	−0.235666
$896$	78.3734	2.61827
$897$	0	0
$898$	−33.6137	−1.12170
$899$	59.5756	1.98696
$900$	0	0
$901$	−20.7678	−0.691875
$902$	−24.4662	−0.814636
$903$	0	0
$904$	121.733	4.04877
$905$	5.79363	0.192587
$906$	0	0
$907$	24.7879	0.823069	0.411535	−	0.911394i	$-0.364993\pi$
$907$	24.7879	0.823069	0.411535	+	0.911394i	$0.364993\pi$
$908$	3.72857	0.123737
$909$	0	0
$910$	−0.560364	−0.0185759
$911$	−41.3254	−1.36917	−0.684586	−	0.728932i	$-0.740017\pi$
$911$	−41.3254	−1.36917	−0.684586	+	0.728932i	$0.740017\pi$
$912$	0	0
$913$	12.9300	0.427920
$914$	−49.2853	−1.63021
$915$	0	0
$916$	74.0304	2.44603
$917$	−14.7296	−0.486414
$918$	0	0
$919$	36.7229	1.21138	0.605688	−	0.795702i	$-0.292898\pi$
$919$	36.7229	1.21138	0.605688	+	0.795702i	$0.292898\pi$
$920$	−9.39197	−0.309644
$921$	0	0
$922$	−3.46453	−0.114098
$923$	−0.0584521	−0.00192397
$924$	0	0
$925$	−39.0984	−1.28555
$926$	28.4518	0.934984
$927$	0	0
$928$	−171.218	−5.62052
$929$	−1.56310	−0.0512837	−0.0256418	−	0.999671i	$-0.508163\pi$
$929$	−1.56310	−0.0512837	−0.0256418	+	0.999671i	$0.508163\pi$
$930$	0	0
$931$	−12.1837	−0.399306
$932$	66.5983	2.18150
$933$	0	0
$934$	30.0701	0.983923
$935$	−1.65801	−0.0542226
$936$	0	0
$937$	3.58638	0.117162	0.0585810	−	0.998283i	$-0.481342\pi$
$937$	3.58638	0.117162	0.0585810	+	0.998283i	$0.481342\pi$
$938$	−28.8298	−0.941326
$939$	0	0
$940$	−8.27049	−0.269754
$941$	41.9408	1.36723	0.683616	−	0.729842i	$-0.260407\pi$
$941$	41.9408	1.36723	0.683616	+	0.729842i	$0.260407\pi$
$942$	0	0
$943$	14.4609	0.470913
$944$	−20.4325	−0.665020
$945$	0	0
$946$	−12.3759	−0.402374
$947$	−53.9167	−1.75206	−0.876028	−	0.482260i	$-0.839816\pi$
$947$	−53.9167	−1.75206	−0.876028	+	0.482260i	$0.839816\pi$
$948$	0	0
$949$	−0.310521	−0.0100799
$950$	46.2999	1.50217
$951$	0	0
$952$	−50.3329	−1.63130
$953$	−6.24884	−0.202420	−0.101210	−	0.994865i	$-0.532271\pi$
$953$	−6.24884	−0.202420	−0.101210	+	0.994865i	$0.532271\pi$
$954$	0	0
$955$	−11.1371	−0.360387
$956$	40.3987	1.30659
$957$	0	0
$958$	−48.8786	−1.57920
$959$	−30.4566	−0.983496
$960$	0	0
$961$	31.1253	1.00404
$962$	−4.08734	−0.131781
$963$	0	0
$964$	−148.424	−4.78043
$965$	3.27014	0.105269
$966$	0	0
$967$	−11.7133	−0.376674	−0.188337	−	0.982104i	$-0.560310\pi$
$967$	−11.7133	−0.376674	−0.188337	+	0.982104i	$0.560310\pi$
$968$	−9.59477	−0.308388
$969$	0	0
$970$	−21.1195	−0.678107
$971$	20.8971	0.670620	0.335310	−	0.942108i	$-0.391159\pi$
$971$	20.8971	0.670620	0.335310	+	0.942108i	$0.391159\pi$
$972$	0	0
$973$	4.60189	0.147530
$974$	53.9512	1.72871
$975$	0	0
$976$	−15.2757	−0.488962
$977$	−15.7320	−0.503311	−0.251655	−	0.967817i	$-0.580975\pi$
$977$	−15.7320	−0.503311	−0.251655	+	0.967817i	$0.580975\pi$
$978$	0	0
$979$	−4.11061	−0.131376
$980$	11.1142	0.355030
$981$	0	0
$982$	−47.5925	−1.51874
$983$	40.6563	1.29674	0.648368	−	0.761327i	$-0.275452\pi$
$983$	40.6563	1.29674	0.648368	+	0.761327i	$0.275452\pi$
$984$	0	0
$985$	−1.73291	−0.0552152
$986$	56.7746	1.80807
$987$	0	0
$988$	3.54996	0.112939
$989$	7.31484	0.232598
$990$	0	0
$991$	13.9691	0.443742	0.221871	−	0.975076i	$-0.428784\pi$
$991$	13.9691	0.443742	0.221871	+	0.975076i	$0.428784\pi$
$992$	−178.546	−5.66885
$993$	0	0
$994$	−1.73164	−0.0549243
$995$	−7.22820	−0.229149
$996$	0	0
$997$	−11.2639	−0.356731	−0.178365	−	0.983964i	$-0.557081\pi$
$997$	−11.2639	−0.356731	−0.178365	+	0.983964i	$0.557081\pi$
$998$	−68.8978	−2.18092
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.i.1.1		13
3.2	odd	2		2013.2.a.e.1.13	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.e.1.13	✓	13	3.2	odd	2
6039.2.a.i.1.1		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)

\(f(q)\)	\(=\)	\(q-2.73913 q^{2} +5.50285 q^{4} -0.604616 q^{5} -1.91298 q^{7} -9.59477 q^{8} +O(q^{10})\)	\(q-2.73913 q^{2} +5.50285 q^{4} -0.604616 q^{5} -1.91298 q^{7} -9.59477 q^{8} +1.65612 q^{10} -1.00000 q^{11} +0.176875 q^{13} +5.23991 q^{14} +15.2757 q^{16} -2.74225 q^{17} +3.64728 q^{19} -3.32711 q^{20} +2.73913 q^{22} -1.61899 q^{23} -4.63444 q^{25} -0.484484 q^{26} -10.5269 q^{28} +7.55848 q^{29} +7.88196 q^{31} -22.6525 q^{32} +7.51138 q^{34} +1.15662 q^{35} +8.43648 q^{37} -9.99040 q^{38} +5.80115 q^{40} -8.93210 q^{41} -4.51816 q^{43} -5.50285 q^{44} +4.43462 q^{46} +2.48579 q^{47} -3.34050 q^{49} +12.6943 q^{50} +0.973317 q^{52} +7.57327 q^{53} +0.604616 q^{55} +18.3546 q^{56} -20.7037 q^{58} -1.33758 q^{59} -1.00000 q^{61} -21.5897 q^{62} +31.4969 q^{64} -0.106941 q^{65} -5.50196 q^{67} -15.0902 q^{68} -3.16813 q^{70} -0.330471 q^{71} -1.75560 q^{73} -23.1086 q^{74} +20.0705 q^{76} +1.91298 q^{77} -12.9780 q^{79} -9.23590 q^{80} +24.4662 q^{82} -12.9300 q^{83} +1.65801 q^{85} +12.3759 q^{86} +9.59477 q^{88} +4.11061 q^{89} -0.338359 q^{91} -8.90903 q^{92} -6.80891 q^{94} -2.20521 q^{95} -12.7524 q^{97} +9.15006 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 2 q^{2} + 16 q^{4} - 3 q^{5} + 11 q^{7} - 9 q^{8}+O(q^{10}) \) 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

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Database

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