LMFDB - Embedded newform 6003.2.a.s.1.9

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6003,2,Mod(1,6003)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6003, 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("6003.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6003 = 3^{2} \cdot 23 \cdot 29 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6003.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:	$47.9341963334$
Analytic rank:	$0$
Dimension:	$20$
Coefficient field:	$\mathbb{Q}[x]/(x^{20} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{20} - 2 x^{19} - 33 x^{18} + 64 x^{17} + 453 x^{16} - 846 x^{15} - 3353 x^{14} + 5985 x^{13} + \cdots - 256 $ x^20 - 2x^19 - 33x^18 + 64x^17 + 453x^16 - 846x^15 - 3353x^14 + 5985x^13 + 14484x^12 - 24566x^11 - 36791x^10 + 59410x^9 + 52109x^8 - 82362x^7 - 34967x^6 + 60661x^5 + 5201x^4 - 19624x^3 + 2768x^2 + 1408*x - 256
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{7} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.9
Root			$0.603983$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-0.603983 q^{2} -1.63520 q^{4} +2.98438 q^{5} +4.14035 q^{7} +2.19560 q^{8} +O(q^{10})$	\(q-0.603983 q^{2} -1.63520 q^{4} +2.98438 q^{5} +4.14035 q^{7} +2.19560 q^{8} -1.80252 q^{10} +1.78887 q^{11} -3.75176 q^{13} -2.50070 q^{14} +1.94430 q^{16} -3.76429 q^{17} +2.44493 q^{19} -4.88008 q^{20} -1.08045 q^{22} -1.00000 q^{23} +3.90654 q^{25} +2.26600 q^{26} -6.77033 q^{28} -1.00000 q^{29} -0.393855 q^{31} -5.56553 q^{32} +2.27357 q^{34} +12.3564 q^{35} -0.656546 q^{37} -1.47670 q^{38} +6.55251 q^{40} +7.56462 q^{41} +12.1883 q^{43} -2.92517 q^{44} +0.603983 q^{46} -12.3439 q^{47} +10.1425 q^{49} -2.35948 q^{50} +6.13490 q^{52} -7.63109 q^{53} +5.33867 q^{55} +9.09056 q^{56} +0.603983 q^{58} +4.61873 q^{59} +2.70501 q^{61} +0.237881 q^{62} -0.527126 q^{64} -11.1967 q^{65} +11.1316 q^{67} +6.15539 q^{68} -7.46305 q^{70} +2.39796 q^{71} +11.9078 q^{73} +0.396542 q^{74} -3.99797 q^{76} +7.40655 q^{77} +9.97731 q^{79} +5.80255 q^{80} -4.56890 q^{82} +13.7073 q^{83} -11.2341 q^{85} -7.36152 q^{86} +3.92764 q^{88} -0.336504 q^{89} -15.5336 q^{91} +1.63520 q^{92} +7.45551 q^{94} +7.29662 q^{95} +8.25163 q^{97} -6.12591 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	$ 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) $ 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * 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 $

$2$	−0.603983	−0.427080	−0.213540	−	0.976934i	$-0.568499\pi$
$2$	−0.603983	−0.427080	−0.213540	+	0.976934i	$0.568499\pi$
$3$	0	0
$3$	0	0
$4$	−1.63520	−0.817602
$5$	2.98438	1.33466	0.667328	−	0.744764i	$-0.267438\pi$
$5$	2.98438	1.33466	0.667328	+	0.744764i	$0.267438\pi$
$6$	0	0
$7$	4.14035	1.56491	0.782453	−	0.622709i	$-0.213968\pi$
$7$	4.14035	1.56491	0.782453	+	0.622709i	$0.213968\pi$
$8$	2.19560	0.776262
$9$	0	0
$10$	−1.80252	−0.570005
$11$	1.78887	0.539364	0.269682	−	0.962949i	$-0.413081\pi$
$11$	1.78887	0.539364	0.269682	+	0.962949i	$0.413081\pi$
$12$	0	0
$13$	−3.75176	−1.04055	−0.520276	−	0.853998i	$-0.674171\pi$
$13$	−3.75176	−1.04055	−0.520276	+	0.853998i	$0.674171\pi$
$14$	−2.50070	−0.668341
$15$	0	0
$16$	1.94430	0.486076
$17$	−3.76429	−0.912975	−0.456488	−	0.889730i	$-0.650893\pi$
$17$	−3.76429	−0.912975	−0.456488	+	0.889730i	$0.650893\pi$
$18$	0	0
$19$	2.44493	0.560906	0.280453	−	0.959868i	$-0.409515\pi$
$19$	2.44493	0.560906	0.280453	+	0.959868i	$0.409515\pi$
$20$	−4.88008	−1.09122
$21$	0	0
$22$	−1.08045	−0.230352
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	3.90654	0.781308
$26$	2.26600	0.444399
$27$	0	0
$28$	−6.77033	−1.27947
$29$	−1.00000	−0.185695
$29$	−1.00000	−0.185695
$30$	0	0
$31$	−0.393855	−0.0707384	−0.0353692	−	0.999374i	$-0.511261\pi$
$31$	−0.393855	−0.0707384	−0.0353692	+	0.999374i	$0.511261\pi$
$32$	−5.56553	−0.983856
$33$	0	0
$34$	2.27357	0.389914
$35$	12.3564	2.08861
$36$	0	0
$37$	−0.656546	−0.107936	−0.0539678	−	0.998543i	$-0.517187\pi$
$37$	−0.656546	−0.107936	−0.0539678	+	0.998543i	$0.517187\pi$
$38$	−1.47670	−0.239552
$39$	0	0
$40$	6.55251	1.03604
$41$	7.56462	1.18140	0.590698	−	0.806893i	$-0.298853\pi$
$41$	7.56462	1.18140	0.590698	+	0.806893i	$0.298853\pi$
$42$	0	0
$43$	12.1883	1.85870	0.929349	−	0.369203i	$-0.120369\pi$
$43$	12.1883	1.85870	0.929349	+	0.369203i	$0.120369\pi$
$44$	−2.92517	−0.440986
$45$	0	0
$46$	0.603983	0.0890524
$47$	−12.3439	−1.80054	−0.900272	−	0.435327i	$-0.856633\pi$
$47$	−12.3439	−1.80054	−0.900272	+	0.435327i	$0.856633\pi$
$48$	0	0
$49$	10.1425	1.44893
$50$	−2.35948	−0.333681
$51$	0	0
$52$	6.13490	0.850757
$53$	−7.63109	−1.04821	−0.524106	−	0.851653i	$-0.675600\pi$
$53$	−7.63109	−1.04821	−0.524106	+	0.851653i	$0.675600\pi$
$54$	0	0
$55$	5.33867	0.719866
$56$	9.09056	1.21478
$57$	0	0
$58$	0.603983	0.0793068
$59$	4.61873	0.601308	0.300654	−	0.953733i	$-0.402795\pi$
$59$	4.61873	0.601308	0.300654	+	0.953733i	$0.402795\pi$
$60$	0	0
$61$	2.70501	0.346341	0.173171	−	0.984892i	$-0.444599\pi$
$61$	2.70501	0.346341	0.173171	+	0.984892i	$0.444599\pi$
$62$	0.237881	0.0302110
$63$	0	0
$64$	−0.527126	−0.0658908
$65$	−11.1967	−1.38878
$66$	0	0
$67$	11.1316	1.35994	0.679968	−	0.733242i	$-0.261994\pi$
$67$	11.1316	1.35994	0.679968	+	0.733242i	$0.261994\pi$
$68$	6.15539	0.746451
$69$	0	0
$70$	−7.46305	−0.892005
$71$	2.39796	0.284585	0.142293	−	0.989825i	$-0.454553\pi$
$71$	2.39796	0.284585	0.142293	+	0.989825i	$0.454553\pi$
$72$	0	0
$73$	11.9078	1.39370	0.696852	−	0.717215i	$-0.254584\pi$
$73$	11.9078	1.39370	0.696852	+	0.717215i	$0.254584\pi$
$74$	0.396542	0.0460971
$75$	0	0
$76$	−3.99797	−0.458598
$77$	7.40655	0.844055
$78$	0	0
$79$	9.97731	1.12253	0.561267	−	0.827634i	$-0.310314\pi$
$79$	9.97731	1.12253	0.561267	+	0.827634i	$0.310314\pi$
$80$	5.80255	0.648745
$81$	0	0
$82$	−4.56890	−0.504551
$83$	13.7073	1.50457	0.752286	−	0.658837i	$-0.228951\pi$
$83$	13.7073	1.50457	0.752286	+	0.658837i	$0.228951\pi$
$84$	0	0
$85$	−11.2341	−1.21851
$86$	−7.36152	−0.793813
$87$	0	0
$88$	3.92764	0.418688
$89$	−0.336504	−0.0356694	−0.0178347	−	0.999841i	$-0.505677\pi$
$89$	−0.336504	−0.0356694	−0.0178347	+	0.999841i	$0.505677\pi$
$90$	0	0
$91$	−15.5336	−1.62837
$92$	1.63520	0.170482
$93$	0	0
$94$	7.45551	0.768977
$95$	7.29662	0.748617
$96$	0	0
$97$	8.25163	0.837826	0.418913	−	0.908026i	$-0.362411\pi$
$97$	8.25163	0.837826	0.418913	+	0.908026i	$0.362411\pi$
$98$	−6.12591	−0.618811
$99$	0	0
$100$	−6.38799	−0.638799
$101$	−6.14283	−0.611234	−0.305617	−	0.952154i	$-0.598863\pi$
$101$	−6.14283	−0.611234	−0.305617	+	0.952154i	$0.598863\pi$
$102$	0	0
$103$	−2.82470	−0.278326	−0.139163	−	0.990270i	$-0.544441\pi$
$103$	−2.82470	−0.278326	−0.139163	+	0.990270i	$0.544441\pi$
$104$	−8.23737	−0.807740
$105$	0	0
$106$	4.60905	0.447670
$107$	8.34278	0.806527	0.403264	−	0.915084i	$-0.367876\pi$
$107$	8.34278	0.806527	0.403264	+	0.915084i	$0.367876\pi$
$108$	0	0
$109$	4.54663	0.435488	0.217744	−	0.976006i	$-0.430130\pi$
$109$	4.54663	0.435488	0.217744	+	0.976006i	$0.430130\pi$
$110$	−3.22447	−0.307441
$111$	0	0
$112$	8.05011	0.760664
$113$	−8.10164	−0.762138	−0.381069	−	0.924546i	$-0.624444\pi$
$113$	−8.10164	−0.762138	−0.381069	+	0.924546i	$0.624444\pi$
$114$	0	0
$115$	−2.98438	−0.278295
$116$	1.63520	0.151825
$117$	0	0
$118$	−2.78963	−0.256807
$119$	−15.5855	−1.42872
$120$	0	0
$121$	−7.79995	−0.709086
$122$	−1.63378	−0.147916
$123$	0	0
$124$	0.644033	0.0578359
$125$	−3.26330	−0.291879
$126$	0	0
$127$	11.7673	1.04418	0.522088	−	0.852892i	$-0.325153\pi$
$127$	11.7673	1.04418	0.522088	+	0.852892i	$0.325153\pi$
$128$	11.4494	1.01200
$129$	0	0
$130$	6.76261	0.593120
$131$	3.67220	0.320841	0.160421	−	0.987049i	$-0.448715\pi$
$131$	3.67220	0.320841	0.160421	+	0.987049i	$0.448715\pi$
$132$	0	0
$133$	10.1229	0.877766
$134$	−6.72327	−0.580802
$135$	0	0
$136$	−8.26488	−0.708708
$137$	−4.83267	−0.412883	−0.206441	−	0.978459i	$-0.566188\pi$
$137$	−4.83267	−0.412883	−0.206441	+	0.978459i	$0.566188\pi$
$138$	0	0
$139$	0.0370338	0.00314117	0.00157058	−	0.999999i	$-0.499500\pi$
$139$	0.0370338	0.00314117	0.00157058	+	0.999999i	$0.499500\pi$
$140$	−20.2052	−1.70765
$141$	0	0
$142$	−1.44833	−0.121541
$143$	−6.71141	−0.561236
$144$	0	0
$145$	−2.98438	−0.247839
$146$	−7.19211	−0.595223
$147$	0	0
$148$	1.07359	0.0882483
$149$	17.0675	1.39822	0.699111	−	0.715013i	$-0.253579\pi$
$149$	17.0675	1.39822	0.699111	+	0.715013i	$0.253579\pi$
$150$	0	0
$151$	12.3354	1.00384	0.501922	−	0.864913i	$-0.332626\pi$
$151$	12.3354	1.00384	0.501922	+	0.864913i	$0.332626\pi$
$152$	5.36810	0.435410
$153$	0	0
$154$	−4.47343	−0.360479
$155$	−1.17541	−0.0944114
$156$	0	0
$157$	−4.80367	−0.383374	−0.191687	−	0.981456i	$-0.561396\pi$
$157$	−4.80367	−0.383374	−0.191687	+	0.981456i	$0.561396\pi$
$158$	−6.02612	−0.479413
$159$	0	0
$160$	−16.6097	−1.31311
$161$	−4.14035	−0.326306
$162$	0	0
$163$	−11.5620	−0.905608	−0.452804	−	0.891610i	$-0.649576\pi$
$163$	−11.5620	−0.905608	−0.452804	+	0.891610i	$0.649576\pi$
$164$	−12.3697	−0.965912
$165$	0	0
$166$	−8.27898	−0.642573
$167$	10.2416	0.792518	0.396259	−	0.918139i	$-0.370308\pi$
$167$	10.2416	0.792518	0.396259	+	0.918139i	$0.370308\pi$
$168$	0	0
$169$	1.07571	0.0827467
$170$	6.78520	0.520401
$171$	0	0
$172$	−19.9304	−1.51968
$173$	−3.89528	−0.296153	−0.148076	−	0.988976i	$-0.547308\pi$
$173$	−3.89528	−0.296153	−0.148076	+	0.988976i	$0.547308\pi$
$174$	0	0
$175$	16.1745	1.22267
$176$	3.47811	0.262172
$177$	0	0
$178$	0.203243	0.0152337
$179$	−11.8776	−0.887777	−0.443889	−	0.896082i	$-0.646401\pi$
$179$	−11.8776	−0.887777	−0.443889	+	0.896082i	$0.646401\pi$
$180$	0	0
$181$	−18.3412	−1.36329	−0.681644	−	0.731684i	$-0.738735\pi$
$181$	−18.3412	−1.36329	−0.681644	+	0.731684i	$0.738735\pi$
$182$	9.38204	0.695443
$183$	0	0
$184$	−2.19560	−0.161862
$185$	−1.95938	−0.144057
$186$	0	0
$187$	−6.73383	−0.492426
$188$	20.1848	1.47213
$189$	0	0
$190$	−4.40703	−0.319720
$191$	−5.08681	−0.368069	−0.184034	−	0.982920i	$-0.558916\pi$
$191$	−5.08681	−0.368069	−0.184034	+	0.982920i	$0.558916\pi$
$192$	0	0
$193$	0.766657	0.0551852	0.0275926	−	0.999619i	$-0.491216\pi$
$193$	0.766657	0.0551852	0.0275926	+	0.999619i	$0.491216\pi$
$194$	−4.98384	−0.357819
$195$	0	0
$196$	−16.5851	−1.18465
$197$	−19.9489	−1.42130	−0.710649	−	0.703547i	$-0.751599\pi$
$197$	−19.9489	−1.42130	−0.710649	+	0.703547i	$0.751599\pi$
$198$	0	0
$199$	3.53914	0.250883	0.125442	−	0.992101i	$-0.459965\pi$
$199$	3.53914	0.250883	0.125442	+	0.992101i	$0.459965\pi$
$200$	8.57720	0.606500
$201$	0	0
$202$	3.71016	0.261046
$203$	−4.14035	−0.290596
$204$	0	0
$205$	22.5757	1.57676
$206$	1.70607	0.118867
$207$	0	0
$208$	−7.29457	−0.505787
$209$	4.37367	0.302533
$210$	0	0
$211$	−12.6418	−0.870299	−0.435150	−	0.900358i	$-0.643305\pi$
$211$	−12.6418	−0.870299	−0.435150	+	0.900358i	$0.643305\pi$
$212$	12.4784	0.857020
$213$	0	0
$214$	−5.03890	−0.344452
$215$	36.3745	2.48072
$216$	0	0
$217$	−1.63070	−0.110699
$218$	−2.74609	−0.185988
$219$	0	0
$220$	−8.72982	−0.588564
$221$	14.1227	0.949997
$222$	0	0
$223$	8.90320	0.596202	0.298101	−	0.954534i	$-0.403647\pi$
$223$	8.90320	0.596202	0.298101	+	0.954534i	$0.403647\pi$
$224$	−23.0433	−1.53964
$225$	0	0
$226$	4.89325	0.325494
$227$	20.4890	1.35990	0.679950	−	0.733258i	$-0.262001\pi$
$227$	20.4890	1.35990	0.679950	+	0.733258i	$0.262001\pi$
$228$	0	0
$229$	17.9351	1.18519	0.592593	−	0.805502i	$-0.298104\pi$
$229$	17.9351	1.18519	0.592593	+	0.805502i	$0.298104\pi$
$230$	1.80252	0.118854
$231$	0	0
$232$	−2.19560	−0.144148
$233$	3.00571	0.196910	0.0984552	−	0.995141i	$-0.468610\pi$
$233$	3.00571	0.196910	0.0984552	+	0.995141i	$0.468610\pi$
$234$	0	0
$235$	−36.8390	−2.40311
$236$	−7.55257	−0.491631
$237$	0	0
$238$	9.41338	0.610178
$239$	15.8049	1.02233	0.511167	−	0.859481i	$-0.329213\pi$
$239$	15.8049	1.02233	0.511167	+	0.859481i	$0.329213\pi$
$240$	0	0
$241$	−21.8661	−1.40852	−0.704261	−	0.709941i	$-0.748722\pi$
$241$	−21.8661	−1.40852	−0.704261	+	0.709941i	$0.748722\pi$
$242$	4.71103	0.302837
$243$	0	0
$244$	−4.42325	−0.283169
$245$	30.2692	1.93383
$246$	0	0
$247$	−9.17280	−0.583652
$248$	−0.864747	−0.0549115
$249$	0	0
$250$	1.97098	0.124656
$251$	−6.60618	−0.416978	−0.208489	−	0.978025i	$-0.566855\pi$
$251$	−6.60618	−0.416978	−0.208489	+	0.978025i	$0.566855\pi$
$252$	0	0
$253$	−1.78887	−0.112465
$254$	−7.10722	−0.445947
$255$	0	0
$256$	−5.86101	−0.366313
$257$	1.82613	0.113911	0.0569554	−	0.998377i	$-0.481861\pi$
$257$	1.82613	0.113911	0.0569554	+	0.998377i	$0.481861\pi$
$258$	0	0
$259$	−2.71833	−0.168909
$260$	18.3089	1.13547
$261$	0	0
$262$	−2.21794	−0.137025
$263$	17.6847	1.09048	0.545241	−	0.838279i	$-0.316438\pi$
$263$	17.6847	1.09048	0.545241	+	0.838279i	$0.316438\pi$
$264$	0	0
$265$	−22.7741	−1.39900
$266$	−6.11405	−0.374876
$267$	0	0
$268$	−18.2024	−1.11189
$269$	−13.0228	−0.794017	−0.397008	−	0.917815i	$-0.629952\pi$
$269$	−13.0228	−0.794017	−0.397008	+	0.917815i	$0.629952\pi$
$270$	0	0
$271$	−9.37655	−0.569585	−0.284793	−	0.958589i	$-0.591925\pi$
$271$	−9.37655	−0.569585	−0.284793	+	0.958589i	$0.591925\pi$
$272$	−7.31893	−0.443775
$273$	0	0
$274$	2.91885	0.176334
$275$	6.98829	0.421410
$276$	0	0
$277$	18.3227	1.10091	0.550453	−	0.834866i	$-0.314455\pi$
$277$	18.3227	1.10091	0.550453	+	0.834866i	$0.314455\pi$
$278$	−0.0223678	−0.00134153
$279$	0	0
$280$	27.1297	1.62131
$281$	−4.64385	−0.277029	−0.138515	−	0.990360i	$-0.544233\pi$
$281$	−4.64385	−0.277029	−0.138515	+	0.990360i	$0.544233\pi$
$282$	0	0
$283$	4.41603	0.262506	0.131253	−	0.991349i	$-0.458100\pi$
$283$	4.41603	0.262506	0.131253	+	0.991349i	$0.458100\pi$
$284$	−3.92116	−0.232678
$285$	0	0
$286$	4.05358	0.239693
$287$	31.3202	1.84877
$288$	0	0
$289$	−2.83010	−0.166476
$290$	1.80252	0.105847
$291$	0	0
$292$	−19.4717	−1.13950
$293$	5.48543	0.320462	0.160231	−	0.987080i	$-0.448776\pi$
$293$	5.48543	0.320462	0.160231	+	0.987080i	$0.448776\pi$
$294$	0	0
$295$	13.7841	0.802539
$296$	−1.44151	−0.0837862
$297$	0	0
$298$	−10.3085	−0.597153
$299$	3.75176	0.216970
$300$	0	0
$301$	50.4639	2.90869
$302$	−7.45039	−0.428722
$303$	0	0
$304$	4.75370	0.272643
$305$	8.07279	0.462247
$306$	0	0
$307$	−11.8306	−0.675208	−0.337604	−	0.941288i	$-0.609616\pi$
$307$	−11.8306	−0.675208	−0.337604	+	0.941288i	$0.609616\pi$
$308$	−12.1112	−0.690101
$309$	0	0
$310$	0.709929	0.0403213
$311$	−6.35113	−0.360139	−0.180070	−	0.983654i	$-0.557632\pi$
$311$	−6.35113	−0.360139	−0.180070	+	0.983654i	$0.557632\pi$
$312$	0	0
$313$	6.61497	0.373900	0.186950	−	0.982369i	$-0.440140\pi$
$313$	6.61497	0.373900	0.186950	+	0.982369i	$0.440140\pi$
$314$	2.90133	0.163732
$315$	0	0
$316$	−16.3149	−0.917787
$317$	−13.9087	−0.781191	−0.390596	−	0.920562i	$-0.627731\pi$
$317$	−13.9087	−0.781191	−0.390596	+	0.920562i	$0.627731\pi$
$318$	0	0
$319$	−1.78887	−0.100157
$320$	−1.57315	−0.0879416
$321$	0	0
$322$	2.50070	0.139359
$323$	−9.20344	−0.512093
$324$	0	0
$325$	−14.6564	−0.812991
$326$	6.98327	0.386767
$327$	0	0
$328$	16.6089	0.917073
$329$	−51.1082	−2.81768
$330$	0	0
$331$	−35.5816	−1.95574	−0.977871	−	0.209209i	$-0.932911\pi$
$331$	−35.5816	−1.95574	−0.977871	+	0.209209i	$0.932911\pi$
$332$	−22.4143	−1.23014
$333$	0	0
$334$	−6.18574	−0.338469
$335$	33.2208	1.81505
$336$	0	0
$337$	28.1225	1.53193	0.765965	−	0.642882i	$-0.222261\pi$
$337$	28.1225	1.53193	0.765965	+	0.642882i	$0.222261\pi$
$338$	−0.649709	−0.0353395
$339$	0	0
$340$	18.3700	0.996255
$341$	−0.704554	−0.0381538
$342$	0	0
$343$	13.0112	0.702538
$344$	26.7606	1.44284
$345$	0	0
$346$	2.35268	0.126481
$347$	23.9141	1.28378	0.641888	−	0.766799i	$-0.278152\pi$
$347$	23.9141	1.28378	0.641888	+	0.766799i	$0.278152\pi$
$348$	0	0
$349$	8.43002	0.451249	0.225624	−	0.974214i	$-0.427558\pi$
$349$	8.43002	0.451249	0.225624	+	0.974214i	$0.427558\pi$
$350$	−9.76909	−0.522180
$351$	0	0
$352$	−9.95600	−0.530657
$353$	1.28333	0.0683047	0.0341524	−	0.999417i	$-0.489127\pi$
$353$	1.28333	0.0683047	0.0341524	+	0.999417i	$0.489127\pi$
$354$	0	0
$355$	7.15643	0.379824
$356$	0.550253	0.0291634
$357$	0	0
$358$	7.17390	0.379152
$359$	26.3529	1.39085	0.695427	−	0.718597i	$-0.255215\pi$
$359$	26.3529	1.39085	0.695427	+	0.718597i	$0.255215\pi$
$360$	0	0
$361$	−13.0223	−0.685384
$362$	11.0777	0.582233
$363$	0	0
$364$	25.4006	1.33136
$365$	35.5375	1.86012
$366$	0	0
$367$	2.63957	0.137784	0.0688922	−	0.997624i	$-0.478054\pi$
$367$	2.63957	0.137784	0.0688922	+	0.997624i	$0.478054\pi$
$368$	−1.94430	−0.101354
$369$	0	0
$370$	1.18343	0.0615238
$371$	−31.5954	−1.64035
$372$	0	0
$373$	13.5737	0.702817	0.351408	−	0.936222i	$-0.385703\pi$
$373$	13.5737	0.702817	0.351408	+	0.936222i	$0.385703\pi$
$374$	4.06712	0.210306
$375$	0	0
$376$	−27.1023	−1.39769
$377$	3.75176	0.193226
$378$	0	0
$379$	0.173996	0.00893756	0.00446878	−	0.999990i	$-0.498578\pi$
$379$	0.173996	0.00893756	0.00446878	+	0.999990i	$0.498578\pi$
$380$	−11.9315	−0.612071
$381$	0	0
$382$	3.07235	0.157195
$383$	−22.4197	−1.14559	−0.572796	−	0.819698i	$-0.694141\pi$
$383$	−22.4197	−1.14559	−0.572796	+	0.819698i	$0.694141\pi$
$384$	0	0
$385$	22.1040	1.12652
$386$	−0.463048	−0.0235685
$387$	0	0
$388$	−13.4931	−0.685009
$389$	33.3348	1.69014	0.845070	−	0.534656i	$-0.179559\pi$
$389$	33.3348	1.69014	0.845070	+	0.534656i	$0.179559\pi$
$390$	0	0
$391$	3.76429	0.190368
$392$	22.2689	1.12475
$393$	0	0
$394$	12.0488	0.607008
$395$	29.7761	1.49820
$396$	0	0
$397$	22.6227	1.13540	0.567701	−	0.823235i	$-0.307833\pi$
$397$	22.6227	1.13540	0.567701	+	0.823235i	$0.307833\pi$
$398$	−2.13758	−0.107147
$399$	0	0
$400$	7.59550	0.379775
$401$	−30.4386	−1.52003	−0.760015	−	0.649905i	$-0.774809\pi$
$401$	−30.4386	−1.52003	−0.760015	+	0.649905i	$0.774809\pi$
$402$	0	0
$403$	1.47765	0.0736069
$404$	10.0448	0.499747
$405$	0	0
$406$	2.50070	0.124108
$407$	−1.17448	−0.0582166
$408$	0	0
$409$	−3.36048	−0.166165	−0.0830824	−	0.996543i	$-0.526476\pi$
$409$	−3.36048	−0.166165	−0.0830824	+	0.996543i	$0.526476\pi$
$410$	−13.6353	−0.673402
$411$	0	0
$412$	4.61896	0.227560
$413$	19.1232	0.940990
$414$	0	0
$415$	40.9078	2.00809
$416$	20.8805	1.02375
$417$	0	0
$418$	−2.64162	−0.129206
$419$	35.8314	1.75048	0.875240	−	0.483690i	$-0.160704\pi$
$419$	35.8314	1.75048	0.875240	+	0.483690i	$0.160704\pi$
$420$	0	0
$421$	−31.4655	−1.53354	−0.766768	−	0.641924i	$-0.778136\pi$
$421$	−31.4655	−1.53354	−0.766768	+	0.641924i	$0.778136\pi$
$422$	7.63544	0.371688
$423$	0	0
$424$	−16.7548	−0.813687
$425$	−14.7054	−0.713315
$426$	0	0
$427$	11.1997	0.541992
$428$	−13.6422	−0.659419
$429$	0	0
$430$	−21.9696	−1.05947
$431$	36.9569	1.78015	0.890076	−	0.455812i	$-0.150651\pi$
$431$	36.9569	1.78015	0.890076	+	0.455812i	$0.150651\pi$
$432$	0	0
$433$	34.3965	1.65299	0.826494	−	0.562946i	$-0.190332\pi$
$433$	34.3965	1.65299	0.826494	+	0.562946i	$0.190332\pi$
$434$	0.984913	0.0472773
$435$	0	0
$436$	−7.43467	−0.356056
$437$	−2.44493	−0.116957
$438$	0	0
$439$	21.9881	1.04944	0.524718	−	0.851276i	$-0.324171\pi$
$439$	21.9881	1.04944	0.524718	+	0.851276i	$0.324171\pi$
$440$	11.7216	0.558805
$441$	0	0
$442$	−8.52988	−0.405725
$443$	−25.5445	−1.21366	−0.606829	−	0.794832i	$-0.707559\pi$
$443$	−25.5445	−1.21366	−0.606829	+	0.794832i	$0.707559\pi$
$444$	0	0
$445$	−1.00426	−0.0476064
$446$	−5.37738	−0.254626
$447$	0	0
$448$	−2.18249	−0.103113
$449$	−4.63048	−0.218526	−0.109263	−	0.994013i	$-0.534849\pi$
$449$	−4.63048	−0.218526	−0.109263	+	0.994013i	$0.534849\pi$
$450$	0	0
$451$	13.5321	0.637203
$452$	13.2478	0.623126
$453$	0	0
$454$	−12.3750	−0.580787
$455$	−46.3583	−2.17331
$456$	0	0
$457$	16.6545	0.779063	0.389531	−	0.921013i	$-0.372637\pi$
$457$	16.6545	0.779063	0.389531	+	0.921013i	$0.372637\pi$
$458$	−10.8325	−0.506170
$459$	0	0
$460$	4.88008	0.227535
$461$	13.0287	0.606807	0.303404	−	0.952862i	$-0.401877\pi$
$461$	13.0287	0.606807	0.303404	+	0.952862i	$0.401877\pi$
$462$	0	0
$463$	24.1891	1.12416	0.562081	−	0.827082i	$-0.310001\pi$
$463$	24.1891	1.12416	0.562081	+	0.827082i	$0.310001\pi$
$464$	−1.94430	−0.0902621
$465$	0	0
$466$	−1.81540	−0.0840966
$467$	19.7706	0.914873	0.457436	−	0.889242i	$-0.348768\pi$
$467$	19.7706	0.914873	0.457436	+	0.889242i	$0.348768\pi$
$468$	0	0
$469$	46.0886	2.12817
$470$	22.2501	1.02632
$471$	0	0
$472$	10.1409	0.466772
$473$	21.8033	1.00252
$474$	0	0
$475$	9.55123	0.438240
$476$	25.4855	1.16813
$477$	0	0
$478$	−9.54588	−0.436619
$479$	4.34066	0.198330	0.0991648	−	0.995071i	$-0.468383\pi$
$479$	4.34066	0.198330	0.0991648	+	0.995071i	$0.468383\pi$
$480$	0	0
$481$	2.46320	0.112312
$482$	13.2068	0.601552
$483$	0	0
$484$	12.7545	0.579750
$485$	24.6260	1.11821
$486$	0	0
$487$	30.5242	1.38319	0.691593	−	0.722288i	$-0.256909\pi$
$487$	30.5242	1.38319	0.691593	+	0.722288i	$0.256909\pi$
$488$	5.93913	0.268852
$489$	0	0
$490$	−18.2821	−0.825900
$491$	−8.80607	−0.397412	−0.198706	−	0.980059i	$-0.563674\pi$
$491$	−8.80607	−0.397412	−0.198706	+	0.980059i	$0.563674\pi$
$492$	0	0
$493$	3.76429	0.169535
$494$	5.54022	0.249266
$495$	0	0
$496$	−0.765773	−0.0343842
$497$	9.92840	0.445350
$498$	0	0
$499$	−29.9447	−1.34051	−0.670255	−	0.742131i	$-0.733815\pi$
$499$	−29.9447	−1.34051	−0.670255	+	0.742131i	$0.733815\pi$
$500$	5.33617	0.238641
$501$	0	0
$502$	3.99002	0.178083
$503$	−18.1544	−0.809465	−0.404733	−	0.914435i	$-0.632635\pi$
$503$	−18.1544	−0.809465	−0.404733	+	0.914435i	$0.632635\pi$
$504$	0	0
$505$	−18.3326	−0.815788
$506$	1.08045	0.0480317
$507$	0	0
$508$	−19.2419	−0.853720
$509$	−28.4338	−1.26031	−0.630153	−	0.776471i	$-0.717008\pi$
$509$	−28.4338	−1.26031	−0.630153	+	0.776471i	$0.717008\pi$
$510$	0	0
$511$	49.3025	2.18102
$512$	−19.3589	−0.855551
$513$	0	0
$514$	−1.10295	−0.0486490
$515$	−8.42998	−0.371469
$516$	0	0
$517$	−22.0816	−0.971150
$518$	1.64183	0.0721377
$519$	0	0
$520$	−24.5835	−1.07806
$521$	−6.28500	−0.275351	−0.137675	−	0.990477i	$-0.543963\pi$
$521$	−6.28500	−0.275351	−0.137675	+	0.990477i	$0.543963\pi$
$522$	0	0
$523$	−18.4090	−0.804970	−0.402485	−	0.915427i	$-0.631853\pi$
$523$	−18.4090	−0.804970	−0.402485	+	0.915427i	$0.631853\pi$
$524$	−6.00480	−0.262321
$525$	0	0
$526$	−10.6812	−0.465724
$527$	1.48258	0.0645824
$528$	0	0
$529$	1.00000	0.0434783
$530$	13.7552	0.597486
$531$	0	0
$532$	−16.5530	−0.717663
$533$	−28.3807	−1.22930
$534$	0	0
$535$	24.8981	1.07644
$536$	24.4405	1.05567
$537$	0	0
$538$	7.86557	0.339109
$539$	18.1437	0.781503
$540$	0	0
$541$	−6.95309	−0.298937	−0.149468	−	0.988767i	$-0.547756\pi$
$541$	−6.95309	−0.298937	−0.149468	+	0.988767i	$0.547756\pi$
$542$	5.66328	0.243259
$543$	0	0
$544$	20.9503	0.898236
$545$	13.5689	0.581227
$546$	0	0
$547$	−5.41603	−0.231573	−0.115786	−	0.993274i	$-0.536939\pi$
$547$	−5.41603	−0.231573	−0.115786	+	0.993274i	$0.536939\pi$
$548$	7.90240	0.337574
$549$	0	0
$550$	−4.22081	−0.179976
$551$	−2.44493	−0.104158
$552$	0	0
$553$	41.3096	1.75666
$554$	−11.0666	−0.470176
$555$	0	0
$556$	−0.0605579	−0.00256823
$557$	28.0875	1.19011	0.595053	−	0.803686i	$-0.297131\pi$
$557$	28.0875	1.19011	0.595053	+	0.803686i	$0.297131\pi$
$558$	0	0
$559$	−45.7276	−1.93407
$560$	24.0246	1.01522
$561$	0	0
$562$	2.80481	0.118314
$563$	−26.8599	−1.13201	−0.566006	−	0.824401i	$-0.691512\pi$
$563$	−26.8599	−1.13201	−0.566006	+	0.824401i	$0.691512\pi$
$564$	0	0
$565$	−24.1784	−1.01719
$566$	−2.66721	−0.112111
$567$	0	0
$568$	5.26496	0.220913
$569$	−2.28690	−0.0958719	−0.0479359	−	0.998850i	$-0.515264\pi$
$569$	−2.28690	−0.0958719	−0.0479359	+	0.998850i	$0.515264\pi$
$570$	0	0
$571$	42.2791	1.76932	0.884662	−	0.466233i	$-0.154389\pi$
$571$	42.2791	1.76932	0.884662	+	0.466233i	$0.154389\pi$
$572$	10.9745	0.458868
$573$	0	0
$574$	−18.9169	−0.789575
$575$	−3.90654	−0.162914
$576$	0	0
$577$	38.8658	1.61801	0.809003	−	0.587805i	$-0.200008\pi$
$577$	38.8658	1.61801	0.809003	+	0.587805i	$0.200008\pi$
$578$	1.70933	0.0710988
$579$	0	0
$580$	4.88008	0.202634
$581$	56.7531	2.35452
$582$	0	0
$583$	−13.6510	−0.565368
$584$	26.1448	1.08188
$585$	0	0
$586$	−3.31311	−0.136863
$587$	−13.4145	−0.553674	−0.276837	−	0.960917i	$-0.589286\pi$
$587$	−13.4145	−0.553674	−0.276837	+	0.960917i	$0.589286\pi$
$588$	0	0
$589$	−0.962948	−0.0396776
$590$	−8.32533	−0.342749
$591$	0	0
$592$	−1.27653	−0.0524649
$593$	−2.40956	−0.0989487	−0.0494744	−	0.998775i	$-0.515755\pi$
$593$	−2.40956	−0.0989487	−0.0494744	+	0.998775i	$0.515755\pi$
$594$	0	0
$595$	−46.5131	−1.90685
$596$	−27.9088	−1.14319
$597$	0	0
$598$	−2.26600	−0.0926636
$599$	7.92166	0.323670	0.161835	−	0.986818i	$-0.448259\pi$
$599$	7.92166	0.323670	0.161835	+	0.986818i	$0.448259\pi$
$600$	0	0
$601$	−40.1093	−1.63609	−0.818046	−	0.575153i	$-0.804943\pi$
$601$	−40.1093	−1.63609	−0.818046	+	0.575153i	$0.804943\pi$
$602$	−30.4793	−1.24224
$603$	0	0
$604$	−20.1710	−0.820745
$605$	−23.2780	−0.946386
$606$	0	0
$607$	−9.50856	−0.385941	−0.192970	−	0.981205i	$-0.561812\pi$
$607$	−9.50856	−0.385941	−0.192970	+	0.981205i	$0.561812\pi$
$608$	−13.6073	−0.551851
$609$	0	0
$610$	−4.87583	−0.197416
$611$	46.3114	1.87356
$612$	0	0
$613$	−12.7083	−0.513284	−0.256642	−	0.966506i	$-0.582616\pi$
$613$	−12.7083	−0.513284	−0.256642	+	0.966506i	$0.582616\pi$
$614$	7.14548	0.288368
$615$	0	0
$616$	16.2618	0.655208
$617$	−27.7524	−1.11727	−0.558635	−	0.829414i	$-0.688675\pi$
$617$	−27.7524	−1.11727	−0.558635	+	0.829414i	$0.688675\pi$
$618$	0	0
$619$	−24.3770	−0.979796	−0.489898	−	0.871780i	$-0.662966\pi$
$619$	−24.3770	−0.979796	−0.489898	+	0.871780i	$0.662966\pi$
$620$	1.92204	0.0771910
$621$	0	0
$622$	3.83597	0.153808
$623$	−1.39325	−0.0558192
$624$	0	0
$625$	−29.2716	−1.17087
$626$	−3.99533	−0.159685
$627$	0	0
$628$	7.85498	0.313448
$629$	2.47143	0.0985424
$630$	0	0
$631$	−28.2035	−1.12277	−0.561383	−	0.827556i	$-0.689730\pi$
$631$	−28.2035	−1.12277	−0.561383	+	0.827556i	$0.689730\pi$
$632$	21.9062	0.871381
$633$	0	0
$634$	8.40062	0.333631
$635$	35.1180	1.39362
$636$	0	0
$637$	−38.0523	−1.50769
$638$	1.08045	0.0427753
$639$	0	0
$640$	34.1695	1.35067
$641$	15.1654	0.598999	0.299499	−	0.954096i	$-0.403180\pi$
$641$	15.1654	0.598999	0.299499	+	0.954096i	$0.403180\pi$
$642$	0	0
$643$	−39.9934	−1.57719	−0.788593	−	0.614916i	$-0.789190\pi$
$643$	−39.9934	−1.57719	−0.788593	+	0.614916i	$0.789190\pi$
$644$	6.77033	0.266788
$645$	0	0
$646$	5.55872	0.218705
$647$	−6.32233	−0.248556	−0.124278	−	0.992247i	$-0.539662\pi$
$647$	−6.32233	−0.248556	−0.124278	+	0.992247i	$0.539662\pi$
$648$	0	0
$649$	8.26231	0.324324
$650$	8.85221	0.347212
$651$	0	0
$652$	18.9063	0.740427
$653$	−30.5802	−1.19669	−0.598347	−	0.801237i	$-0.704176\pi$
$653$	−30.5802	−1.19669	−0.598347	+	0.801237i	$0.704176\pi$
$654$	0	0
$655$	10.9592	0.428213
$656$	14.7079	0.574248
$657$	0	0
$658$	30.8684	1.20338
$659$	−44.1055	−1.71811	−0.859054	−	0.511885i	$-0.828947\pi$
$659$	−44.1055	−1.71811	−0.859054	+	0.511885i	$0.828947\pi$
$660$	0	0
$661$	−40.6105	−1.57956	−0.789782	−	0.613388i	$-0.789806\pi$
$661$	−40.6105	−1.57956	−0.789782	+	0.613388i	$0.789806\pi$
$662$	21.4907	0.835259
$663$	0	0
$664$	30.0958	1.16794
$665$	30.2106	1.17152
$666$	0	0
$667$	1.00000	0.0387202
$668$	−16.7471	−0.647965
$669$	0	0
$670$	−20.0648	−0.775171
$671$	4.83891	0.186804
$672$	0	0
$673$	6.47786	0.249703	0.124852	−	0.992175i	$-0.460155\pi$
$673$	6.47786	0.249703	0.124852	+	0.992175i	$0.460155\pi$
$674$	−16.9855	−0.654257
$675$	0	0
$676$	−1.75900	−0.0676539
$677$	−30.1813	−1.15996	−0.579982	−	0.814630i	$-0.696940\pi$
$677$	−30.1813	−1.15996	−0.579982	+	0.814630i	$0.696940\pi$
$678$	0	0
$679$	34.1647	1.31112
$680$	−24.6656	−0.945882
$681$	0	0
$682$	0.425539	0.0162947
$683$	−21.9845	−0.841212	−0.420606	−	0.907243i	$-0.638182\pi$
$683$	−21.9845	−0.841212	−0.420606	+	0.907243i	$0.638182\pi$
$684$	0	0
$685$	−14.4225	−0.551056
$686$	−7.85853	−0.300040
$687$	0	0
$688$	23.6978	0.903469
$689$	28.6300	1.09072
$690$	0	0
$691$	30.6364	1.16546	0.582732	−	0.812664i	$-0.301984\pi$
$691$	30.6364	1.16546	0.582732	+	0.812664i	$0.301984\pi$
$692$	6.36958	0.242135
$693$	0	0
$694$	−14.4437	−0.548275
$695$	0.110523	0.00419238
$696$	0	0
$697$	−28.4755	−1.07858
$698$	−5.09159	−0.192719
$699$	0	0
$700$	−26.4485	−0.999661
$701$	−42.9369	−1.62170	−0.810852	−	0.585251i	$-0.800996\pi$
$701$	−42.9369	−1.62170	−0.810852	+	0.585251i	$0.800996\pi$
$702$	0	0
$703$	−1.60521	−0.0605417
$704$	−0.942960	−0.0355392
$705$	0	0
$706$	−0.775109	−0.0291716
$707$	−25.4335	−0.956525
$708$	0	0
$709$	−17.9066	−0.672496	−0.336248	−	0.941773i	$-0.609158\pi$
$709$	−17.9066	−0.672496	−0.336248	+	0.941773i	$0.609158\pi$
$710$	−4.32236	−0.162215
$711$	0	0
$712$	−0.738829	−0.0276888
$713$	0.393855	0.0147500
$714$	0	0
$715$	−20.0294	−0.749058
$716$	19.4224	0.725849
$717$	0	0
$718$	−15.9167	−0.594006
$719$	28.9704	1.08042	0.540208	−	0.841532i	$-0.318346\pi$
$719$	28.9704	1.08042	0.540208	+	0.841532i	$0.318346\pi$
$720$	0	0
$721$	−11.6952	−0.435554
$722$	7.86524	0.292714
$723$	0	0
$724$	29.9915	1.11463
$725$	−3.90654	−0.145085
$726$	0	0
$727$	−20.8804	−0.774412	−0.387206	−	0.921993i	$-0.626560\pi$
$727$	−20.8804	−0.774412	−0.387206	+	0.921993i	$0.626560\pi$
$728$	−34.1056	−1.26404
$729$	0	0
$730$	−21.4640	−0.794419
$731$	−45.8803	−1.69694
$732$	0	0
$733$	−27.1801	−1.00392	−0.501960	−	0.864891i	$-0.667387\pi$
$733$	−27.1801	−1.00392	−0.501960	+	0.864891i	$0.667387\pi$
$734$	−1.59425	−0.0588450
$735$	0	0
$736$	5.56553	0.205148
$737$	19.9129	0.733501
$738$	0	0
$739$	13.1701	0.484471	0.242235	−	0.970218i	$-0.422119\pi$
$739$	13.1701	0.484471	0.242235	+	0.970218i	$0.422119\pi$
$740$	3.20400	0.117781
$741$	0	0
$742$	19.0831	0.700562
$743$	−38.5610	−1.41467	−0.707333	−	0.706881i	$-0.750102\pi$
$743$	−38.5610	−1.41467	−0.707333	+	0.706881i	$0.750102\pi$
$744$	0	0
$745$	50.9359	1.86615
$746$	−8.19825	−0.300159
$747$	0	0
$748$	11.0112	0.402609
$749$	34.5421	1.26214
$750$	0	0
$751$	38.2069	1.39419	0.697095	−	0.716979i	$-0.254476\pi$
$751$	38.2069	1.39419	0.697095	+	0.716979i	$0.254476\pi$
$752$	−24.0003	−0.875202
$753$	0	0
$754$	−2.26600	−0.0825228
$755$	36.8137	1.33979
$756$	0	0
$757$	32.7977	1.19205	0.596027	−	0.802964i	$-0.296745\pi$
$757$	32.7977	1.19205	0.596027	+	0.802964i	$0.296745\pi$
$758$	−0.105090	−0.00381706
$759$	0	0
$760$	16.0205	0.581123
$761$	22.0109	0.797895	0.398948	−	0.916974i	$-0.369376\pi$
$761$	22.0109	0.797895	0.398948	+	0.916974i	$0.369376\pi$
$762$	0	0
$763$	18.8247	0.681498
$764$	8.31798	0.300934
$765$	0	0
$766$	13.5411	0.489259
$767$	−17.3284	−0.625691
$768$	0	0
$769$	−54.3175	−1.95874	−0.979369	−	0.202079i	$-0.935230\pi$
$769$	−54.3175	−1.95874	−0.979369	+	0.202079i	$0.935230\pi$
$770$	−13.3504	−0.481116
$771$	0	0
$772$	−1.25364	−0.0451196
$773$	7.14844	0.257112	0.128556	−	0.991702i	$-0.458966\pi$
$773$	7.14844	0.257112	0.128556	+	0.991702i	$0.458966\pi$
$774$	0	0
$775$	−1.53861	−0.0552684
$776$	18.1173	0.650373
$777$	0	0
$778$	−20.1336	−0.721825
$779$	18.4950	0.662652
$780$	0	0
$781$	4.28964	0.153495
$782$	−2.27357	−0.0813026
$783$	0	0
$784$	19.7202	0.704292
$785$	−14.3360	−0.511673
$786$	0	0
$787$	−18.5898	−0.662653	−0.331327	−	0.943516i	$-0.607496\pi$
$787$	−18.5898	−0.662653	−0.331327	+	0.943516i	$0.607496\pi$
$788$	32.6205	1.16206
$789$	0	0
$790$	−17.9843	−0.639851
$791$	−33.5437	−1.19268
$792$	0	0
$793$	−10.1486	−0.360386
$794$	−13.6637	−0.484907
$795$	0	0
$796$	−5.78722	−0.205123
$797$	38.6089	1.36760	0.683799	−	0.729670i	$-0.260326\pi$
$797$	38.6089	1.36760	0.683799	+	0.729670i	$0.260326\pi$
$798$	0	0
$799$	46.4661	1.64385
$800$	−21.7420	−0.768694
$801$	0	0
$802$	18.3844	0.649175
$803$	21.3015	0.751714
$804$	0	0
$805$	−12.3564	−0.435506
$806$	−0.892474	−0.0314360
$807$	0	0
$808$	−13.4872	−0.474478
$809$	−6.45541	−0.226960	−0.113480	−	0.993540i	$-0.536200\pi$
$809$	−6.45541	−0.226960	−0.113480	+	0.993540i	$0.536200\pi$
$810$	0	0
$811$	33.9772	1.19310	0.596551	−	0.802576i	$-0.296538\pi$
$811$	33.9772	1.19310	0.596551	+	0.802576i	$0.296538\pi$
$812$	6.77033	0.237592
$813$	0	0
$814$	0.709363	0.0248632
$815$	−34.5055	−1.20868
$816$	0	0
$817$	29.7996	1.04255
$818$	2.02967	0.0709657
$819$	0	0
$820$	−36.9159	−1.28916
$821$	−16.6061	−0.579557	−0.289779	−	0.957094i	$-0.593582\pi$
$821$	−16.6061	−0.579557	−0.289779	+	0.957094i	$0.593582\pi$
$822$	0	0
$823$	7.75976	0.270488	0.135244	−	0.990812i	$-0.456818\pi$
$823$	7.75976	0.270488	0.135244	+	0.990812i	$0.456818\pi$
$824$	−6.20191	−0.216054
$825$	0	0
$826$	−11.5501	−0.401878
$827$	−13.4783	−0.468685	−0.234342	−	0.972154i	$-0.575294\pi$
$827$	−13.4783	−0.468685	−0.234342	+	0.972154i	$0.575294\pi$
$828$	0	0
$829$	7.23461	0.251268	0.125634	−	0.992077i	$-0.459903\pi$
$829$	7.23461	0.251268	0.125634	+	0.992077i	$0.459903\pi$
$830$	−24.7076	−0.857614
$831$	0	0
$832$	1.97765	0.0685628
$833$	−38.1795	−1.32284
$834$	0	0
$835$	30.5648	1.05774
$836$	−7.15184	−0.247352
$837$	0	0
$838$	−21.6416	−0.747595
$839$	−46.6853	−1.61176	−0.805878	−	0.592082i	$-0.798306\pi$
$839$	−46.6853	−1.61176	−0.805878	+	0.592082i	$0.798306\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	19.0046	0.654943
$843$	0	0
$844$	20.6720	0.711559
$845$	3.21032	0.110438
$846$	0	0
$847$	−32.2945	−1.10965
$848$	−14.8372	−0.509510
$849$	0	0
$850$	8.88178	0.304643
$851$	0.656546	0.0225061
$852$	0	0
$853$	47.7645	1.63543	0.817713	−	0.575627i	$-0.195242\pi$
$853$	47.7645	1.63543	0.817713	+	0.575627i	$0.195242\pi$
$854$	−6.76443	−0.231474
$855$	0	0
$856$	18.3174	0.626077
$857$	28.0001	0.956465	0.478232	−	0.878233i	$-0.341278\pi$
$857$	28.0001	0.956465	0.478232	+	0.878233i	$0.341278\pi$
$858$	0	0
$859$	52.0140	1.77470	0.887348	−	0.461101i	$-0.152545\pi$
$859$	52.0140	1.77470	0.887348	+	0.461101i	$0.152545\pi$
$860$	−59.4798	−2.02824
$861$	0	0
$862$	−22.3213	−0.760268
$863$	−36.1179	−1.22947	−0.614734	−	0.788735i	$-0.710736\pi$
$863$	−36.1179	−1.22947	−0.614734	+	0.788735i	$0.710736\pi$
$864$	0	0
$865$	−11.6250	−0.395262
$866$	−20.7749	−0.705959
$867$	0	0
$868$	2.66652	0.0905077
$869$	17.8481	0.605456
$870$	0	0
$871$	−41.7629	−1.41508
$872$	9.98258	0.338053
$873$	0	0
$874$	1.47670	0.0499500
$875$	−13.5112	−0.456763
$876$	0	0
$877$	24.8720	0.839867	0.419933	−	0.907555i	$-0.362053\pi$
$877$	24.8720	0.839867	0.419933	+	0.907555i	$0.362053\pi$
$878$	−13.2805	−0.448194
$879$	0	0
$880$	10.3800	0.349910
$881$	−34.4884	−1.16194	−0.580972	−	0.813924i	$-0.697327\pi$
$881$	−34.4884	−1.16194	−0.580972	+	0.813924i	$0.697327\pi$
$882$	0	0
$883$	−51.5153	−1.73363	−0.866814	−	0.498631i	$-0.833836\pi$
$883$	−51.5153	−1.73363	−0.866814	+	0.498631i	$0.833836\pi$
$884$	−23.0935	−0.776720
$885$	0	0
$886$	15.4285	0.518329
$887$	42.5097	1.42734	0.713668	−	0.700484i	$-0.247032\pi$
$887$	42.5097	1.42734	0.713668	+	0.700484i	$0.247032\pi$
$888$	0	0
$889$	48.7206	1.63404
$890$	0.606554	0.0203317
$891$	0	0
$892$	−14.5586	−0.487456
$893$	−30.1800	−1.00994
$894$	0	0
$895$	−35.4475	−1.18488
$896$	47.4047	1.58368
$897$	0	0
$898$	2.79673	0.0933281
$899$	0.393855	0.0131358
$900$	0	0
$901$	28.7257	0.956991
$902$	−8.17317	−0.272137
$903$	0	0
$904$	−17.7880	−0.591619
$905$	−54.7370	−1.81952
$906$	0	0
$907$	−50.1761	−1.66607	−0.833035	−	0.553220i	$-0.813399\pi$
$907$	−50.1761	−1.66607	−0.833035	+	0.553220i	$0.813399\pi$
$908$	−33.5037	−1.11186
$909$	0	0
$910$	27.9996	0.928177
$911$	−14.1333	−0.468257	−0.234129	−	0.972206i	$-0.575224\pi$
$911$	−14.1333	−0.468257	−0.234129	+	0.972206i	$0.575224\pi$
$912$	0	0
$913$	24.5206	0.811513
$914$	−10.0590	−0.332722
$915$	0	0
$916$	−29.3276	−0.969012
$917$	15.2042	0.502087
$918$	0	0
$919$	31.5103	1.03943	0.519714	−	0.854340i	$-0.326039\pi$
$919$	31.5103	1.03943	0.519714	+	0.854340i	$0.326039\pi$
$920$	−6.55251	−0.216030
$921$	0	0
$922$	−7.86911	−0.259155
$923$	−8.99657	−0.296126
$924$	0	0
$925$	−2.56482	−0.0843309
$926$	−14.6098	−0.480107
$927$	0	0
$928$	5.56553	0.182697
$929$	7.22781	0.237137	0.118568	−	0.992946i	$-0.462169\pi$
$929$	7.22781	0.237137	0.118568	+	0.992946i	$0.462169\pi$
$930$	0	0
$931$	24.7978	0.812715
$932$	−4.91495	−0.160994
$933$	0	0
$934$	−11.9411	−0.390724
$935$	−20.0963	−0.657220
$936$	0	0
$937$	−61.1835	−1.99878	−0.999389	−	0.0349378i	$-0.988877\pi$
$937$	−61.1835	−1.99878	−0.999389	+	0.0349378i	$0.988877\pi$
$938$	−27.8367	−0.908901
$939$	0	0
$940$	60.2392	1.96479
$941$	59.9291	1.95363	0.976816	−	0.214081i	$-0.0686755\pi$
$941$	59.9291	1.95363	0.976816	+	0.214081i	$0.0686755\pi$
$942$	0	0
$943$	−7.56462	−0.246338
$944$	8.98022	0.292281
$945$	0	0
$946$	−13.1688	−0.428155
$947$	−40.3914	−1.31254	−0.656272	−	0.754524i	$-0.727868\pi$
$947$	−40.3914	−1.31254	−0.656272	+	0.754524i	$0.727868\pi$
$948$	0	0
$949$	−44.6752	−1.45022
$950$	−5.76878	−0.187164
$951$	0	0
$952$	−34.2195	−1.10906
$953$	−48.8016	−1.58084	−0.790420	−	0.612566i	$-0.790137\pi$
$953$	−48.8016	−1.58084	−0.790420	+	0.612566i	$0.790137\pi$
$954$	0	0
$955$	−15.1810	−0.491246
$956$	−25.8442	−0.835863
$957$	0	0
$958$	−2.62168	−0.0847027
$959$	−20.0090	−0.646123
$960$	0	0
$961$	−30.8449	−0.994996
$962$	−1.48773	−0.0479664
$963$	0	0
$964$	35.7556	1.15161
$965$	2.28800	0.0736533
$966$	0	0
$967$	−29.6040	−0.951999	−0.476000	−	0.879445i	$-0.657914\pi$
$967$	−29.6040	−0.951999	−0.476000	+	0.879445i	$0.657914\pi$
$968$	−17.1256	−0.550437
$969$	0	0
$970$	−14.8737	−0.477565
$971$	55.9709	1.79619	0.898096	−	0.439800i	$-0.144951\pi$
$971$	55.9709	1.79619	0.898096	+	0.439800i	$0.144951\pi$
$972$	0	0
$973$	0.153333	0.00491564
$974$	−18.4361	−0.590731
$975$	0	0
$976$	5.25937	0.168348
$977$	50.7422	1.62339	0.811694	−	0.584083i	$-0.198546\pi$
$977$	50.7422	1.62339	0.811694	+	0.584083i	$0.198546\pi$
$978$	0	0
$979$	−0.601962	−0.0192388
$980$	−49.4963	−1.58110
$981$	0	0
$982$	5.31871	0.169727
$983$	−33.0071	−1.05276	−0.526381	−	0.850249i	$-0.676452\pi$
$983$	−33.0071	−1.05276	−0.526381	+	0.850249i	$0.676452\pi$
$984$	0	0
$985$	−59.5350	−1.89694
$986$	−2.27357	−0.0724052
$987$	0	0
$988$	14.9994	0.477195
$989$	−12.1883	−0.387565
$990$	0	0
$991$	51.3473	1.63110	0.815550	−	0.578687i	$-0.196435\pi$
$991$	51.3473	1.63110	0.815550	+	0.578687i	$0.196435\pi$
$992$	2.19201	0.0695963
$993$	0	0
$994$	−5.99658	−0.190200
$995$	10.5622	0.334843
$996$	0	0
$997$	−30.2897	−0.959285	−0.479643	−	0.877464i	$-0.659234\pi$
$997$	−30.2897	−0.959285	−0.479643	+	0.877464i	$0.659234\pi$
$998$	18.0861	0.572505
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.s.1.9		20
3.2	odd	2		2001.2.a.o.1.12	✓	20

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.o.1.12	✓	20	3.2	odd	2
6003.2.a.s.1.9		20	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:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q-0.603983 q^{2} -1.63520 q^{4} +2.98438 q^{5} +4.14035 q^{7} +2.19560 q^{8} +O(q^{10})\)	\(q-0.603983 q^{2} -1.63520 q^{4} +2.98438 q^{5} +4.14035 q^{7} +2.19560 q^{8} -1.80252 q^{10} +1.78887 q^{11} -3.75176 q^{13} -2.50070 q^{14} +1.94430 q^{16} -3.76429 q^{17} +2.44493 q^{19} -4.88008 q^{20} -1.08045 q^{22} -1.00000 q^{23} +3.90654 q^{25} +2.26600 q^{26} -6.77033 q^{28} -1.00000 q^{29} -0.393855 q^{31} -5.56553 q^{32} +2.27357 q^{34} +12.3564 q^{35} -0.656546 q^{37} -1.47670 q^{38} +6.55251 q^{40} +7.56462 q^{41} +12.1883 q^{43} -2.92517 q^{44} +0.603983 q^{46} -12.3439 q^{47} +10.1425 q^{49} -2.35948 q^{50} +6.13490 q^{52} -7.63109 q^{53} +5.33867 q^{55} +9.09056 q^{56} +0.603983 q^{58} +4.61873 q^{59} +2.70501 q^{61} +0.237881 q^{62} -0.527126 q^{64} -11.1967 q^{65} +11.1316 q^{67} +6.15539 q^{68} -7.46305 q^{70} +2.39796 q^{71} +11.9078 q^{73} +0.396542 q^{74} -3.99797 q^{76} +7.40655 q^{77} +9.97731 q^{79} +5.80255 q^{80} -4.56890 q^{82} +13.7073 q^{83} -11.2341 q^{85} -7.36152 q^{86} +3.92764 q^{88} -0.336504 q^{89} -15.5336 q^{91} +1.63520 q^{92} +7.45551 q^{94} +7.29662 q^{95} +8.25163 q^{97} -6.12591 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8}+O(q^{10}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8	\( 20 q - 2 q^{2} + 30 q^{4} + q^{5} + 9 q^{7} - 6 q^{8} + 7 q^{10} + 21 q^{13} + q^{14} + 58 q^{16} + 4 q^{17} + 7 q^{19} + 20 q^{20} + 7 q^{22} - 20 q^{23} + 47 q^{25} - 8 q^{26} + 11 q^{28} - 20 q^{29} + 28 q^{31} - 14 q^{32} + 16 q^{34} - 9 q^{35} + 14 q^{37} + 20 q^{38} + 34 q^{40} - 7 q^{41} + 3 q^{43} + q^{44} + 2 q^{46} - 3 q^{47} + 35 q^{49} + 24 q^{50} + 73 q^{52} + 19 q^{53} + 29 q^{55} + 30 q^{56} + 2 q^{58} - 20 q^{59} + 15 q^{61} - 12 q^{62} + 82 q^{64} + 28 q^{65} + 20 q^{67} + 23 q^{68} - 24 q^{70} - 63 q^{71} + 19 q^{73} - 16 q^{74} - 44 q^{76} + 7 q^{77} + 32 q^{79} + 56 q^{80} - 20 q^{82} + 21 q^{83} + 4 q^{85} + 6 q^{86} + 55 q^{88} + 13 q^{89} + 70 q^{91} - 30 q^{92} - 12 q^{94} - 9 q^{95} - 9 q^{97} - 31 q^{98}+O(q^{100}) \) 20 * q - 2 * q^2 + 30 * q^4 + q^5 + 9 * q^7 - 6 * q^8 + 7 * q^10 + 21 * q^13 + q^14 + 58 * q^16 + 4 * q^17 + 7 * q^19 + 20 * q^20 + 7 * q^22 - 20 * q^23 + 47 * q^25 - 8 * q^26 + 11 * q^28 - 20 * q^29 + 28 * q^31 - 14 * q^32 + 16 * q^34 - 9 * q^35 + 14 * q^37 + 20 * q^38 + 34 * q^40 - 7 * q^41 + 3 * q^43 + q^44 + 2 * q^46 - 3 * q^47 + 35 * q^49 + 24 * q^50 + 73 * q^52 + 19 * q^53 + 29 * q^55 + 30 * q^56 + 2 * q^58 - 20 * q^59 + 15 * q^61 - 12 * q^62 + 82 * q^64 + 28 * q^65 + 20 * q^67 + 23 * q^68 - 24 * q^70 - 63 * q^71 + 19 * q^73 - 16 * q^74 - 44 * q^76 + 7 * q^77 + 32 * q^79 + 56 * q^80 - 20 * q^82 + 21 * q^83 + 4 * q^85 + 6 * q^86 + 55 * q^88 + 13 * q^89 + 70 * q^91 - 30 * q^92 - 12 * q^94 - 9 * q^95 - 9 * q^97 - 31 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more