LMFDB - Embedded newform 4527.2.a.k.1.5

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [4527,2,Mod(1,4527)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(4527, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([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("4527.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 4527 = 3^{2} \cdot 503 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	4527.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:	$36.1482769950$
Analytic rank:	$1$
Dimension:	$10$
Coefficient field:	$\mathbb{Q}[x]/(x^{10} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{10} - 2x^{9} - 9x^{8} + 14x^{7} + 27x^{6} - 27x^{5} - 34x^{4} + 14x^{3} + 17x^{2} + x - 1 $ x^10 - 2x^9 - 9x^8 + 14x^7 + 27x^6 - 27x^5 - 34x^4 + 14x^3 + 17x^2 + x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 503)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$1.95007$ of defining polynomial
Character	$\chi$	$=$	4527.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.392284 q^{2} -1.84611 q^{4} +2.28693 q^{5} +2.71022 q^{7} -1.50877 q^{8} +O(q^{10})$	\(q+0.392284 q^{2} -1.84611 q^{4} +2.28693 q^{5} +2.71022 q^{7} -1.50877 q^{8} +0.897127 q^{10} -1.36880 q^{11} -2.93079 q^{13} +1.06318 q^{14} +3.10036 q^{16} +2.61287 q^{17} -7.79104 q^{19} -4.22194 q^{20} -0.536960 q^{22} +2.61064 q^{23} +0.230071 q^{25} -1.14970 q^{26} -5.00337 q^{28} +0.314480 q^{29} -7.95126 q^{31} +4.23376 q^{32} +1.02499 q^{34} +6.19810 q^{35} -4.17299 q^{37} -3.05630 q^{38} -3.45045 q^{40} -6.16482 q^{41} +0.457851 q^{43} +2.52697 q^{44} +1.02411 q^{46} -7.67118 q^{47} +0.345296 q^{49} +0.0902533 q^{50} +5.41058 q^{52} -7.26306 q^{53} -3.13037 q^{55} -4.08909 q^{56} +0.123365 q^{58} -0.217166 q^{59} +7.26694 q^{61} -3.11915 q^{62} -4.53989 q^{64} -6.70253 q^{65} +10.1022 q^{67} -4.82366 q^{68} +2.43141 q^{70} +16.5088 q^{71} +2.86838 q^{73} -1.63699 q^{74} +14.3831 q^{76} -3.70976 q^{77} -16.0222 q^{79} +7.09033 q^{80} -2.41836 q^{82} -12.5312 q^{83} +5.97547 q^{85} +0.179608 q^{86} +2.06521 q^{88} +0.0952234 q^{89} -7.94310 q^{91} -4.81954 q^{92} -3.00928 q^{94} -17.8176 q^{95} -9.27640 q^{97} +0.135454 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) $ 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8	$ 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8} - 4 q^{10} + 3 q^{11} - 18 q^{13} - q^{14} - 4 q^{16} + 11 q^{17} + 3 q^{20} - 18 q^{22} + 2 q^{23} - 27 q^{25} - 11 q^{26} - 22 q^{28} + 9 q^{29} - 22 q^{31} + 10 q^{32} - 10 q^{34} + 6 q^{35} - 35 q^{37} - 2 q^{38} - 19 q^{40} + 4 q^{41} - 20 q^{43} - 9 q^{44} - q^{46} - 7 q^{47} - 27 q^{49} - 16 q^{50} - 7 q^{52} + 24 q^{53} - 11 q^{55} - 12 q^{56} + 2 q^{58} - 17 q^{59} - 4 q^{61} - 8 q^{62} + 3 q^{64} + 16 q^{65} - 6 q^{67} - 28 q^{68} + 26 q^{70} + q^{71} - 31 q^{73} - 11 q^{74} + 20 q^{76} - 3 q^{77} - 10 q^{79} - 24 q^{80} - 9 q^{82} - 22 q^{83} - 6 q^{85} - 38 q^{86} - 3 q^{88} - q^{89} + 10 q^{91} - 27 q^{92} + 33 q^{94} - 39 q^{95} - 57 q^{97} - 40 q^{98}+O(q^{100}) $ 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8 - 4 * q^10 + 3 * q^11 - 18 * q^13 - q^14 - 4 * q^16 + 11 * q^17 + 3 * q^20 - 18 * q^22 + 2 * q^23 - 27 * q^25 - 11 * q^26 - 22 * q^28 + 9 * q^29 - 22 * q^31 + 10 * q^32 - 10 * q^34 + 6 * q^35 - 35 * q^37 - 2 * q^38 - 19 * q^40 + 4 * q^41 - 20 * q^43 - 9 * q^44 - q^46 - 7 * q^47 - 27 * q^49 - 16 * q^50 - 7 * q^52 + 24 * q^53 - 11 * q^55 - 12 * q^56 + 2 * q^58 - 17 * q^59 - 4 * q^61 - 8 * q^62 + 3 * q^64 + 16 * q^65 - 6 * q^67 - 28 * q^68 + 26 * q^70 + q^71 - 31 * q^73 - 11 * q^74 + 20 * q^76 - 3 * q^77 - 10 * q^79 - 24 * q^80 - 9 * q^82 - 22 * q^83 - 6 * q^85 - 38 * q^86 - 3 * q^88 - q^89 + 10 * q^91 - 27 * q^92 + 33 * q^94 - 39 * q^95 - 57 * q^97 - 40 * 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 $

$2$	0.392284	0.277386	0.138693	−	0.990335i	$-0.455710\pi$
$2$	0.392284	0.277386	0.138693	+	0.990335i	$0.455710\pi$
$3$	0	0
$3$	0	0
$4$	−1.84611	−0.923057
$5$	2.28693	1.02275	0.511374	−	0.859358i	$-0.329137\pi$
$5$	2.28693	1.02275	0.511374	+	0.859358i	$0.329137\pi$
$6$	0	0
$7$	2.71022	1.02437	0.512184	−	0.858876i	$-0.328837\pi$
$7$	2.71022	1.02437	0.512184	+	0.858876i	$0.328837\pi$
$8$	−1.50877	−0.533430
$9$	0	0
$10$	0.897127	0.283697
$11$	−1.36880	−0.412710	−0.206355	−	0.978477i	$-0.566160\pi$
$11$	−1.36880	−0.412710	−0.206355	+	0.978477i	$0.566160\pi$
$12$	0	0
$13$	−2.93079	−0.812856	−0.406428	−	0.913683i	$-0.633226\pi$
$13$	−2.93079	−0.812856	−0.406428	+	0.913683i	$0.633226\pi$
$14$	1.06318	0.284146
$15$	0	0
$16$	3.10036	0.775091
$17$	2.61287	0.633715	0.316857	−	0.948473i	$-0.397372\pi$
$17$	2.61287	0.633715	0.316857	+	0.948473i	$0.397372\pi$
$18$	0	0
$19$	−7.79104	−1.78739	−0.893694	−	0.448677i	$-0.851895\pi$
$19$	−7.79104	−1.78739	−0.893694	+	0.448677i	$0.851895\pi$
$20$	−4.22194	−0.944055
$21$	0	0
$22$	−0.536960	−0.114480
$23$	2.61064	0.544356	0.272178	−	0.962247i	$-0.412256\pi$
$23$	2.61064	0.544356	0.272178	+	0.962247i	$0.412256\pi$
$24$	0	0
$25$	0.230071	0.0460143
$26$	−1.14970	−0.225475
$27$	0	0
$28$	−5.00337	−0.945549
$29$	0.314480	0.0583974	0.0291987	−	0.999574i	$-0.490704\pi$
$29$	0.314480	0.0583974	0.0291987	+	0.999574i	$0.490704\pi$
$30$	0	0
$31$	−7.95126	−1.42809	−0.714044	−	0.700101i	$-0.753138\pi$
$31$	−7.95126	−1.42809	−0.714044	+	0.700101i	$0.753138\pi$
$32$	4.23376	0.748430
$33$	0	0
$34$	1.02499	0.175784
$35$	6.19810	1.04767
$36$	0	0
$37$	−4.17299	−0.686035	−0.343017	−	0.939329i	$-0.611449\pi$
$37$	−4.17299	−0.686035	−0.343017	+	0.939329i	$0.611449\pi$
$38$	−3.05630	−0.495797
$39$	0	0
$40$	−3.45045	−0.545565
$41$	−6.16482	−0.962783	−0.481391	−	0.876506i	$-0.659868\pi$
$41$	−6.16482	−0.962783	−0.481391	+	0.876506i	$0.659868\pi$
$42$	0	0
$43$	0.457851	0.0698216	0.0349108	−	0.999390i	$-0.488885\pi$
$43$	0.457851	0.0698216	0.0349108	+	0.999390i	$0.488885\pi$
$44$	2.52697	0.380955
$45$	0	0
$46$	1.02411	0.150997
$47$	−7.67118	−1.11896	−0.559478	−	0.828845i	$-0.688998\pi$
$47$	−7.67118	−1.11896	−0.559478	+	0.828845i	$0.688998\pi$
$48$	0	0
$49$	0.345296	0.0493279
$50$	0.0902533	0.0127637
$51$	0	0
$52$	5.41058	0.750312
$53$	−7.26306	−0.997658	−0.498829	−	0.866700i	$-0.666237\pi$
$53$	−7.26306	−0.997658	−0.498829	+	0.866700i	$0.666237\pi$
$54$	0	0
$55$	−3.13037	−0.422099
$56$	−4.08909	−0.546428
$57$	0	0
$58$	0.123365	0.0161987
$59$	−0.217166	−0.0282726	−0.0141363	−	0.999900i	$-0.504500\pi$
$59$	−0.217166	−0.0282726	−0.0141363	+	0.999900i	$0.504500\pi$
$60$	0	0
$61$	7.26694	0.930436	0.465218	−	0.885196i	$-0.345976\pi$
$61$	7.26694	0.930436	0.465218	+	0.885196i	$0.345976\pi$
$62$	−3.11915	−0.396132
$63$	0	0
$64$	−4.53989	−0.567486
$65$	−6.70253	−0.831347
$66$	0	0
$67$	10.1022	1.23418	0.617091	−	0.786892i	$-0.288311\pi$
$67$	10.1022	1.23418	0.617091	+	0.786892i	$0.288311\pi$
$68$	−4.82366	−0.584955
$69$	0	0
$70$	2.43141	0.290609
$71$	16.5088	1.95924	0.979619	−	0.200863i	$-0.0643746\pi$
$71$	16.5088	1.95924	0.979619	+	0.200863i	$0.0643746\pi$
$72$	0	0
$73$	2.86838	0.335718	0.167859	−	0.985811i	$-0.446315\pi$
$73$	2.86838	0.335718	0.167859	+	0.985811i	$0.446315\pi$
$74$	−1.63699	−0.190297
$75$	0	0
$76$	14.3831	1.64986
$77$	−3.70976	−0.422767
$78$	0	0
$79$	−16.0222	−1.80264	−0.901320	−	0.433153i	$-0.857401\pi$
$79$	−16.0222	−1.80264	−0.901320	+	0.433153i	$0.857401\pi$
$80$	7.09033	0.792723
$81$	0	0
$82$	−2.41836	−0.267063
$83$	−12.5312	−1.37548	−0.687741	−	0.725956i	$-0.741398\pi$
$83$	−12.5312	−1.37548	−0.687741	+	0.725956i	$0.741398\pi$
$84$	0	0
$85$	5.97547	0.648131
$86$	0.179608	0.0193676
$87$	0	0
$88$	2.06521	0.220152
$89$	0.0952234	0.0100937	0.00504683	−	0.999987i	$-0.498394\pi$
$89$	0.0952234	0.0100937	0.00504683	+	0.999987i	$0.498394\pi$
$90$	0	0
$91$	−7.94310	−0.832663
$92$	−4.81954	−0.502472
$93$	0	0
$94$	−3.00928	−0.310383
$95$	−17.8176	−1.82805
$96$	0	0
$97$	−9.27640	−0.941875	−0.470938	−	0.882167i	$-0.656084\pi$
$97$	−9.27640	−0.941875	−0.470938	+	0.882167i	$0.656084\pi$
$98$	0.135454	0.0136829
$99$	0	0
$100$	−0.424738	−0.0424738
$101$	−8.77727	−0.873371	−0.436686	−	0.899614i	$-0.643848\pi$
$101$	−8.77727	−0.873371	−0.436686	+	0.899614i	$0.643848\pi$
$102$	0	0
$103$	1.52475	0.150238	0.0751189	−	0.997175i	$-0.476066\pi$
$103$	1.52475	0.150238	0.0751189	+	0.997175i	$0.476066\pi$
$104$	4.42189	0.433602
$105$	0	0
$106$	−2.84918	−0.276737
$107$	17.6358	1.70492	0.852460	−	0.522792i	$-0.175110\pi$
$107$	17.6358	1.70492	0.852460	+	0.522792i	$0.175110\pi$
$108$	0	0
$109$	−4.91427	−0.470702	−0.235351	−	0.971910i	$-0.575624\pi$
$109$	−4.91427	−0.470702	−0.235351	+	0.971910i	$0.575624\pi$
$110$	−1.22799	−0.117084
$111$	0	0
$112$	8.40266	0.793977
$113$	−0.102880	−0.00967811	−0.00483905	−	0.999988i	$-0.501540\pi$
$113$	−0.102880	−0.00967811	−0.00483905	+	0.999988i	$0.501540\pi$
$114$	0	0
$115$	5.97037	0.556740
$116$	−0.580565	−0.0539041
$117$	0	0
$118$	−0.0851906	−0.00784243
$119$	7.08146	0.649157
$120$	0	0
$121$	−9.12637	−0.829670
$122$	2.85070	0.258090
$123$	0	0
$124$	14.6789	1.31821
$125$	−10.9085	−0.975687
$126$	0	0
$127$	14.7682	1.31046	0.655232	−	0.755428i	$-0.272571\pi$
$127$	14.7682	1.31046	0.655232	+	0.755428i	$0.272571\pi$
$128$	−10.2484	−0.905843
$129$	0	0
$130$	−2.62929	−0.230604
$131$	−3.01646	−0.263549	−0.131775	−	0.991280i	$-0.542068\pi$
$131$	−3.01646	−0.263549	−0.131775	+	0.991280i	$0.542068\pi$
$132$	0	0
$133$	−21.1154	−1.83094
$134$	3.96293	0.342345
$135$	0	0
$136$	−3.94222	−0.338042
$137$	−0.370190	−0.0316275	−0.0158137	−	0.999875i	$-0.505034\pi$
$137$	−0.370190	−0.0316275	−0.0158137	+	0.999875i	$0.505034\pi$
$138$	0	0
$139$	13.0944	1.11065	0.555324	−	0.831634i	$-0.312594\pi$
$139$	13.0944	1.11065	0.555324	+	0.831634i	$0.312594\pi$
$140$	−11.4424	−0.967059
$141$	0	0
$142$	6.47615	0.543466
$143$	4.01168	0.335474
$144$	0	0
$145$	0.719195	0.0597259
$146$	1.12522	0.0931237
$147$	0	0
$148$	7.70381	0.633249
$149$	5.61923	0.460345	0.230173	−	0.973150i	$-0.426071\pi$
$149$	5.61923	0.460345	0.230173	+	0.973150i	$0.426071\pi$
$150$	0	0
$151$	−6.99662	−0.569377	−0.284689	−	0.958620i	$-0.591890\pi$
$151$	−6.99662	−0.569377	−0.284689	+	0.958620i	$0.591890\pi$
$152$	11.7549	0.953446
$153$	0	0
$154$	−1.45528	−0.117270
$155$	−18.1840	−1.46058
$156$	0	0
$157$	10.5849	0.844764	0.422382	−	0.906418i	$-0.361194\pi$
$157$	10.5849	0.844764	0.422382	+	0.906418i	$0.361194\pi$
$158$	−6.28526	−0.500028
$159$	0	0
$160$	9.68233	0.765455
$161$	7.07541	0.557621
$162$	0	0
$163$	−4.74396	−0.371576	−0.185788	−	0.982590i	$-0.559484\pi$
$163$	−4.74396	−0.371576	−0.185788	+	0.982590i	$0.559484\pi$
$164$	11.3810	0.888703
$165$	0	0
$166$	−4.91580	−0.381540
$167$	−15.4650	−1.19672	−0.598360	−	0.801227i	$-0.704181\pi$
$167$	−15.4650	−1.19672	−0.598360	+	0.801227i	$0.704181\pi$
$168$	0	0
$169$	−4.41045	−0.339265
$170$	2.34408	0.179783
$171$	0	0
$172$	−0.845245	−0.0644493
$173$	18.0030	1.36874	0.684372	−	0.729133i	$-0.260076\pi$
$173$	18.0030	1.36874	0.684372	+	0.729133i	$0.260076\pi$
$174$	0	0
$175$	0.623544	0.0471355
$176$	−4.24379	−0.319888
$177$	0	0
$178$	0.0373546	0.00279985
$179$	−22.6390	−1.69212	−0.846058	−	0.533092i	$-0.821030\pi$
$179$	−22.6390	−1.69212	−0.846058	+	0.533092i	$0.821030\pi$
$180$	0	0
$181$	2.97984	0.221490	0.110745	−	0.993849i	$-0.464676\pi$
$181$	2.97984	0.221490	0.110745	+	0.993849i	$0.464676\pi$
$182$	−3.11595	−0.230969
$183$	0	0
$184$	−3.93885	−0.290376
$185$	−9.54335	−0.701641
$186$	0	0
$187$	−3.57651	−0.261541
$188$	14.1619	1.03286
$189$	0	0
$190$	−6.98956	−0.507076
$191$	−4.09564	−0.296350	−0.148175	−	0.988961i	$-0.547340\pi$
$191$	−4.09564	−0.296350	−0.148175	+	0.988961i	$0.547340\pi$
$192$	0	0
$193$	−21.1708	−1.52391	−0.761955	−	0.647630i	$-0.775760\pi$
$193$	−21.1708	−1.52391	−0.761955	+	0.647630i	$0.775760\pi$
$194$	−3.63898	−0.261263
$195$	0	0
$196$	−0.637455	−0.0455325
$197$	−18.5139	−1.31906	−0.659531	−	0.751677i	$-0.729245\pi$
$197$	−18.5139	−1.31906	−0.659531	+	0.751677i	$0.729245\pi$
$198$	0	0
$199$	−1.56711	−0.111089	−0.0555447	−	0.998456i	$-0.517690\pi$
$199$	−1.56711	−0.111089	−0.0555447	+	0.998456i	$0.517690\pi$
$200$	−0.347124	−0.0245454
$201$	0	0
$202$	−3.44318	−0.242261
$203$	0.852309	0.0598204
$204$	0	0
$205$	−14.0985	−0.984685
$206$	0.598133	0.0416739
$207$	0	0
$208$	−9.08652	−0.630037
$209$	10.6644	0.737673
$210$	0	0
$211$	−22.8488	−1.57298	−0.786489	−	0.617604i	$-0.788103\pi$
$211$	−22.8488	−1.57298	−0.786489	+	0.617604i	$0.788103\pi$
$212$	13.4084	0.920895
$213$	0	0
$214$	6.91825	0.472922
$215$	1.04708	0.0714100
$216$	0	0
$217$	−21.5497	−1.46289
$218$	−1.92779	−0.130566
$219$	0	0
$220$	5.77901	0.389621
$221$	−7.65779	−0.515119
$222$	0	0
$223$	4.75557	0.318457	0.159228	−	0.987242i	$-0.449099\pi$
$223$	4.75557	0.318457	0.159228	+	0.987242i	$0.449099\pi$
$224$	11.4744	0.766667
$225$	0	0
$226$	−0.0403580	−0.00268458
$227$	27.4709	1.82331	0.911655	−	0.410956i	$-0.134805\pi$
$227$	27.4709	1.82331	0.911655	+	0.410956i	$0.134805\pi$
$228$	0	0
$229$	−4.52436	−0.298978	−0.149489	−	0.988763i	$-0.547763\pi$
$229$	−4.52436	−0.298978	−0.149489	+	0.988763i	$0.547763\pi$
$230$	2.34208	0.154432
$231$	0	0
$232$	−0.474477	−0.0311509
$233$	10.5567	0.691594	0.345797	−	0.938309i	$-0.387609\pi$
$233$	10.5567	0.691594	0.345797	+	0.938309i	$0.387609\pi$
$234$	0	0
$235$	−17.5435	−1.14441
$236$	0.400913	0.0260972
$237$	0	0
$238$	2.77794	0.180067
$239$	28.3072	1.83104	0.915522	−	0.402269i	$-0.131778\pi$
$239$	28.3072	1.83104	0.915522	+	0.402269i	$0.131778\pi$
$240$	0	0
$241$	−10.6922	−0.688748	−0.344374	−	0.938833i	$-0.611909\pi$
$241$	−10.6922	−0.688748	−0.344374	+	0.938833i	$0.611909\pi$
$242$	−3.58013	−0.230139
$243$	0	0
$244$	−13.4156	−0.858846
$245$	0.789669	0.0504501
$246$	0	0
$247$	22.8339	1.45289
$248$	11.9966	0.761785
$249$	0	0
$250$	−4.27923	−0.270642
$251$	3.42359	0.216095	0.108048	−	0.994146i	$-0.465540\pi$
$251$	3.42359	0.216095	0.108048	+	0.994146i	$0.465540\pi$
$252$	0	0
$253$	−3.57346	−0.224661
$254$	5.79331	0.363505
$255$	0	0
$256$	5.05948	0.316218
$257$	11.2524	0.701906	0.350953	−	0.936393i	$-0.385858\pi$
$257$	11.2524	0.701906	0.350953	+	0.936393i	$0.385858\pi$
$258$	0	0
$259$	−11.3097	−0.702751
$260$	12.3736	0.767381
$261$	0	0
$262$	−1.18331	−0.0731050
$263$	−0.429180	−0.0264643	−0.0132322	−	0.999912i	$-0.504212\pi$
$263$	−0.429180	−0.0264643	−0.0132322	+	0.999912i	$0.504212\pi$
$264$	0	0
$265$	−16.6102	−1.02035
$266$	−8.28325	−0.507878
$267$	0	0
$268$	−18.6498	−1.13922
$269$	−15.2320	−0.928712	−0.464356	−	0.885649i	$-0.653714\pi$
$269$	−15.2320	−0.928712	−0.464356	+	0.885649i	$0.653714\pi$
$270$	0	0
$271$	17.7850	1.08036	0.540182	−	0.841548i	$-0.318355\pi$
$271$	17.7850	1.08036	0.540182	+	0.841548i	$0.318355\pi$
$272$	8.10085	0.491186
$273$	0	0
$274$	−0.145219	−0.00877303
$275$	−0.314923	−0.0189906
$276$	0	0
$277$	15.1882	0.912570	0.456285	−	0.889834i	$-0.349180\pi$
$277$	15.1882	0.912570	0.456285	+	0.889834i	$0.349180\pi$
$278$	5.13670	0.308079
$279$	0	0
$280$	−9.35149	−0.558858
$281$	−20.0885	−1.19838	−0.599191	−	0.800606i	$-0.704511\pi$
$281$	−20.0885	−1.19838	−0.599191	+	0.800606i	$0.704511\pi$
$282$	0	0
$283$	−1.63497	−0.0971889	−0.0485945	−	0.998819i	$-0.515474\pi$
$283$	−1.63497	−0.0971889	−0.0485945	+	0.998819i	$0.515474\pi$
$284$	−30.4772	−1.80849
$285$	0	0
$286$	1.57372	0.0930559
$287$	−16.7080	−0.986243
$288$	0	0
$289$	−10.1729	−0.598405
$290$	0.282128	0.0165671
$291$	0	0
$292$	−5.29535	−0.309887
$293$	10.1892	0.595262	0.297631	−	0.954681i	$-0.403804\pi$
$293$	10.1892	0.595262	0.297631	+	0.954681i	$0.403804\pi$
$294$	0	0
$295$	−0.496644	−0.0289157
$296$	6.29607	0.365951
$297$	0	0
$298$	2.20433	0.127694
$299$	−7.65125	−0.442483
$300$	0	0
$301$	1.24088	0.0715230
$302$	−2.74466	−0.157938
$303$	0	0
$304$	−24.1551	−1.38539
$305$	16.6190	0.951602
$306$	0	0
$307$	−16.3682	−0.934184	−0.467092	−	0.884209i	$-0.654698\pi$
$307$	−16.3682	−0.934184	−0.467092	+	0.884209i	$0.654698\pi$
$308$	6.84864	0.390238
$309$	0	0
$310$	−7.13329	−0.405144
$311$	−6.98922	−0.396322	−0.198161	−	0.980169i	$-0.563497\pi$
$311$	−6.98922	−0.396322	−0.198161	+	0.980169i	$0.563497\pi$
$312$	0	0
$313$	2.22444	0.125733	0.0628665	−	0.998022i	$-0.479976\pi$
$313$	2.22444	0.125733	0.0628665	+	0.998022i	$0.479976\pi$
$314$	4.15227	0.234326
$315$	0	0
$316$	29.5788	1.66394
$317$	−3.49687	−0.196404	−0.0982019	−	0.995167i	$-0.531309\pi$
$317$	−3.49687	−0.196404	−0.0982019	+	0.995167i	$0.531309\pi$
$318$	0	0
$319$	−0.430461	−0.0241012
$320$	−10.3824	−0.580396
$321$	0	0
$322$	2.77557	0.154676
$323$	−20.3570	−1.13269
$324$	0	0
$325$	−0.674292	−0.0374030
$326$	−1.86098	−0.103070
$327$	0	0
$328$	9.30128	0.513577
$329$	−20.7906	−1.14622
$330$	0	0
$331$	−13.8654	−0.762112	−0.381056	−	0.924552i	$-0.624439\pi$
$331$	−13.8654	−0.762112	−0.381056	+	0.924552i	$0.624439\pi$
$332$	23.1341	1.26965
$333$	0	0
$334$	−6.06668	−0.331954
$335$	23.1031	1.26226
$336$	0	0
$337$	−33.0259	−1.79903	−0.899517	−	0.436885i	$-0.856082\pi$
$337$	−33.0259	−1.79903	−0.899517	+	0.436885i	$0.856082\pi$
$338$	−1.73015	−0.0941076
$339$	0	0
$340$	−11.0314	−0.598262
$341$	10.8837	0.589387
$342$	0	0
$343$	−18.0357	−0.973837
$344$	−0.690791	−0.0372450
$345$	0	0
$346$	7.06229	0.379671
$347$	0.417829	0.0224302	0.0112151	−	0.999937i	$-0.496430\pi$
$347$	0.417829	0.0224302	0.0112151	+	0.999937i	$0.496430\pi$
$348$	0	0
$349$	−8.68607	−0.464955	−0.232477	−	0.972602i	$-0.574683\pi$
$349$	−8.68607	−0.464955	−0.232477	+	0.972602i	$0.574683\pi$
$350$	0.244606	0.0130748
$351$	0	0
$352$	−5.79519	−0.308884
$353$	9.09582	0.484122	0.242061	−	0.970261i	$-0.422177\pi$
$353$	9.09582	0.484122	0.242061	+	0.970261i	$0.422177\pi$
$354$	0	0
$355$	37.7546	2.00381
$356$	−0.175793	−0.00931703
$357$	0	0
$358$	−8.88089	−0.469370
$359$	−5.51998	−0.291333	−0.145667	−	0.989334i	$-0.546533\pi$
$359$	−5.51998	−0.291333	−0.145667	+	0.989334i	$0.546533\pi$
$360$	0	0
$361$	41.7004	2.19476
$362$	1.16894	0.0614383
$363$	0	0
$364$	14.6639	0.768595
$365$	6.55979	0.343355
$366$	0	0
$367$	16.3639	0.854188	0.427094	−	0.904207i	$-0.359537\pi$
$367$	16.3639	0.854188	0.427094	+	0.904207i	$0.359537\pi$
$368$	8.09393	0.421925
$369$	0	0
$370$	−3.74370	−0.194626
$371$	−19.6845	−1.02197
$372$	0	0
$373$	−23.5721	−1.22051	−0.610257	−	0.792203i	$-0.708934\pi$
$373$	−23.5721	−1.22051	−0.610257	+	0.792203i	$0.708934\pi$
$374$	−1.40301	−0.0725478
$375$	0	0
$376$	11.5740	0.596885
$377$	−0.921675	−0.0474687
$378$	0	0
$379$	−22.8307	−1.17273	−0.586367	−	0.810045i	$-0.699442\pi$
$379$	−22.8307	−1.17273	−0.586367	+	0.810045i	$0.699442\pi$
$380$	32.8933	1.68739
$381$	0	0
$382$	−1.60665	−0.0822034
$383$	−21.8257	−1.11524	−0.557622	−	0.830095i	$-0.688286\pi$
$383$	−21.8257	−1.11524	−0.557622	+	0.830095i	$0.688286\pi$
$384$	0	0
$385$	−8.48399	−0.432384
$386$	−8.30497	−0.422712
$387$	0	0
$388$	17.1253	0.869404
$389$	−29.7462	−1.50819	−0.754097	−	0.656763i	$-0.771925\pi$
$389$	−29.7462	−1.50819	−0.754097	+	0.656763i	$0.771925\pi$
$390$	0	0
$391$	6.82127	0.344967
$392$	−0.520971	−0.0263130
$393$	0	0
$394$	−7.26271	−0.365890
$395$	−36.6418	−1.84365
$396$	0	0
$397$	28.5561	1.43319	0.716595	−	0.697489i	$-0.245700\pi$
$397$	28.5561	1.43319	0.716595	+	0.697489i	$0.245700\pi$
$398$	−0.614751	−0.0308147
$399$	0	0
$400$	0.713305	0.0356652
$401$	−13.9970	−0.698975	−0.349487	−	0.936941i	$-0.613644\pi$
$401$	−13.9970	−0.698975	−0.349487	+	0.936941i	$0.613644\pi$
$402$	0	0
$403$	23.3035	1.16083
$404$	16.2038	0.806171
$405$	0	0
$406$	0.334347	0.0165934
$407$	5.71200	0.283133
$408$	0	0
$409$	−18.2429	−0.902055	−0.451028	−	0.892510i	$-0.648942\pi$
$409$	−18.2429	−0.902055	−0.451028	+	0.892510i	$0.648942\pi$
$410$	−5.53063	−0.273138
$411$	0	0
$412$	−2.81486	−0.138678
$413$	−0.588567	−0.0289615
$414$	0	0
$415$	−28.6581	−1.40677
$416$	−12.4083	−0.608365
$417$	0	0
$418$	4.18348	0.204621
$419$	−9.78645	−0.478099	−0.239050	−	0.971007i	$-0.576836\pi$
$419$	−9.78645	−0.478099	−0.239050	+	0.971007i	$0.576836\pi$
$420$	0	0
$421$	2.26639	0.110457	0.0552286	−	0.998474i	$-0.482411\pi$
$421$	2.26639	0.110457	0.0552286	+	0.998474i	$0.482411\pi$
$422$	−8.96322	−0.436323
$423$	0	0
$424$	10.9583	0.532181
$425$	0.601147	0.0291599
$426$	0	0
$427$	19.6950	0.953109
$428$	−32.5578	−1.57374
$429$	0	0
$430$	0.410751	0.0198082
$431$	−4.29790	−0.207023	−0.103511	−	0.994628i	$-0.533008\pi$
$431$	−4.29790	−0.207023	−0.103511	+	0.994628i	$0.533008\pi$
$432$	0	0
$433$	−16.6114	−0.798292	−0.399146	−	0.916887i	$-0.630693\pi$
$433$	−16.6114	−0.798292	−0.399146	+	0.916887i	$0.630693\pi$
$434$	−8.45358	−0.405785
$435$	0	0
$436$	9.07230	0.434484
$437$	−20.3396	−0.972976
$438$	0	0
$439$	−21.4863	−1.02548	−0.512742	−	0.858542i	$-0.671370\pi$
$439$	−21.4863	−1.02548	−0.512742	+	0.858542i	$0.671370\pi$
$440$	4.72300	0.225160
$441$	0	0
$442$	−3.00403	−0.142887
$443$	19.6047	0.931447	0.465724	−	0.884930i	$-0.345794\pi$
$443$	19.6047	0.931447	0.465724	+	0.884930i	$0.345794\pi$
$444$	0	0
$445$	0.217770	0.0103233
$446$	1.86553	0.0883356
$447$	0	0
$448$	−12.3041	−0.581314
$449$	35.1455	1.65862	0.829310	−	0.558789i	$-0.188734\pi$
$449$	35.1455	1.65862	0.829310	+	0.558789i	$0.188734\pi$
$450$	0	0
$451$	8.43843	0.397350
$452$	0.189928	0.00893344
$453$	0	0
$454$	10.7764	0.505762
$455$	−18.1653	−0.851605
$456$	0	0
$457$	−11.8432	−0.554002	−0.277001	−	0.960870i	$-0.589341\pi$
$457$	−11.8432	−0.554002	−0.277001	+	0.960870i	$0.589341\pi$
$458$	−1.77483	−0.0829325
$459$	0	0
$460$	−11.0220	−0.513902
$461$	−17.1286	−0.797756	−0.398878	−	0.917004i	$-0.630600\pi$
$461$	−17.1286	−0.797756	−0.398878	+	0.917004i	$0.630600\pi$
$462$	0	0
$463$	−6.92218	−0.321701	−0.160850	−	0.986979i	$-0.551424\pi$
$463$	−6.92218	−0.321701	−0.160850	+	0.986979i	$0.551424\pi$
$464$	0.975001	0.0452633
$465$	0	0
$466$	4.14123	0.191839
$467$	32.6395	1.51038	0.755188	−	0.655508i	$-0.227545\pi$
$467$	32.6395	1.51038	0.755188	+	0.655508i	$0.227545\pi$
$468$	0	0
$469$	27.3792	1.26425
$470$	−6.88202	−0.317444
$471$	0	0
$472$	0.327653	0.0150814
$473$	−0.626709	−0.0288161
$474$	0	0
$475$	−1.79250	−0.0822454
$476$	−13.0732	−0.599208
$477$	0	0
$478$	11.1045	0.507907
$479$	−8.78005	−0.401171	−0.200585	−	0.979676i	$-0.564284\pi$
$479$	−8.78005	−0.401171	−0.200585	+	0.979676i	$0.564284\pi$
$480$	0	0
$481$	12.2302	0.557647
$482$	−4.19439	−0.191049
$483$	0	0
$484$	16.8483	0.765833
$485$	−21.2145	−0.963301
$486$	0	0
$487$	−23.1617	−1.04956	−0.524779	−	0.851238i	$-0.675852\pi$
$487$	−23.1617	−1.04956	−0.524779	+	0.851238i	$0.675852\pi$
$488$	−10.9641	−0.496323
$489$	0	0
$490$	0.309774	0.0139942
$491$	−8.15363	−0.367968	−0.183984	−	0.982929i	$-0.558900\pi$
$491$	−8.15363	−0.367968	−0.183984	+	0.982929i	$0.558900\pi$
$492$	0	0
$493$	0.821696	0.0370073
$494$	8.95738	0.403012
$495$	0	0
$496$	−24.6518	−1.10690
$497$	44.7426	2.00698
$498$	0	0
$499$	30.6993	1.37429	0.687144	−	0.726521i	$-0.258864\pi$
$499$	30.6993	1.37429	0.687144	+	0.726521i	$0.258864\pi$
$500$	20.1384	0.900615
$501$	0	0
$502$	1.34302	0.0599419
$503$	1.00000	0.0445878
$503$	1.00000	0.0445878
$504$	0	0
$505$	−20.0731	−0.893239
$506$	−1.40181	−0.0623180
$507$	0	0
$508$	−27.2637	−1.20963
$509$	6.39080	0.283267	0.141634	−	0.989919i	$-0.454765\pi$
$509$	6.39080	0.283267	0.141634	+	0.989919i	$0.454765\pi$
$510$	0	0
$511$	7.77394	0.343899
$512$	22.4816	0.993557
$513$	0	0
$514$	4.41414	0.194699
$515$	3.48700	0.153655
$516$	0	0
$517$	10.5003	0.461805
$518$	−4.43662	−0.194934
$519$	0	0
$520$	10.1126	0.443465
$521$	11.2745	0.493944	0.246972	−	0.969023i	$-0.420564\pi$
$521$	11.2745	0.493944	0.246972	+	0.969023i	$0.420564\pi$
$522$	0	0
$523$	18.6154	0.813993	0.406996	−	0.913430i	$-0.366576\pi$
$523$	18.6154	0.813993	0.406996	+	0.913430i	$0.366576\pi$
$524$	5.56872	0.243271
$525$	0	0
$526$	−0.168360	−0.00734085
$527$	−20.7756	−0.905001
$528$	0	0
$529$	−16.1846	−0.703676
$530$	−6.51589	−0.283032
$531$	0	0
$532$	38.9815	1.69006
$533$	18.0678	0.782604
$534$	0	0
$535$	40.3320	1.74370
$536$	−15.2419	−0.658349
$537$	0	0
$538$	−5.97527	−0.257612
$539$	−0.472642	−0.0203581
$540$	0	0
$541$	−36.6446	−1.57547	−0.787736	−	0.616013i	$-0.788747\pi$
$541$	−36.6446	−1.57547	−0.787736	+	0.616013i	$0.788747\pi$
$542$	6.97678	0.299678
$543$	0	0
$544$	11.0623	0.474291
$545$	−11.2386	−0.481409
$546$	0	0
$547$	19.4186	0.830280	0.415140	−	0.909758i	$-0.363733\pi$
$547$	19.4186	0.830280	0.415140	+	0.909758i	$0.363733\pi$
$548$	0.683413	0.0291939
$549$	0	0
$550$	−0.123539	−0.00526772
$551$	−2.45013	−0.104379
$552$	0	0
$553$	−43.4238	−1.84657
$554$	5.95808	0.253135
$555$	0	0
$556$	−24.1737	−1.02519
$557$	35.6561	1.51080	0.755398	−	0.655266i	$-0.227444\pi$
$557$	35.6561	1.51080	0.755398	+	0.655266i	$0.227444\pi$
$558$	0	0
$559$	−1.34187	−0.0567549
$560$	19.2163	0.812039
$561$	0	0
$562$	−7.88041	−0.332415
$563$	34.0879	1.43663	0.718316	−	0.695717i	$-0.244913\pi$
$563$	34.0879	1.43663	0.718316	+	0.695717i	$0.244913\pi$
$564$	0	0
$565$	−0.235279	−0.00989827
$566$	−0.641373	−0.0269589
$567$	0	0
$568$	−24.9080	−1.04512
$569$	−13.0323	−0.546342	−0.273171	−	0.961965i	$-0.588073\pi$
$569$	−13.0323	−0.546342	−0.273171	+	0.961965i	$0.588073\pi$
$570$	0	0
$571$	28.5710	1.19566	0.597829	−	0.801624i	$-0.296030\pi$
$571$	28.5710	1.19566	0.597829	+	0.801624i	$0.296030\pi$
$572$	−7.40602	−0.309661
$573$	0	0
$574$	−6.55428	−0.273570
$575$	0.600634	0.0250482
$576$	0	0
$577$	40.1508	1.67150	0.835750	−	0.549110i	$-0.185033\pi$
$577$	40.1508	1.67150	0.835750	+	0.549110i	$0.185033\pi$
$578$	−3.99066	−0.165990
$579$	0	0
$580$	−1.32771	−0.0551304
$581$	−33.9624	−1.40900
$582$	0	0
$583$	9.94172	0.411744
$584$	−4.32772	−0.179082
$585$	0	0
$586$	3.99707	0.165118
$587$	1.26390	0.0521666	0.0260833	−	0.999660i	$-0.491696\pi$
$587$	1.26390	0.0521666	0.0260833	+	0.999660i	$0.491696\pi$
$588$	0	0
$589$	61.9486	2.55255
$590$	−0.194825	−0.00802083
$591$	0	0
$592$	−12.9378	−0.531739
$593$	−0.473003	−0.0194239	−0.00971196	−	0.999953i	$-0.503091\pi$
$593$	−0.473003	−0.0194239	−0.00971196	+	0.999953i	$0.503091\pi$
$594$	0	0
$595$	16.1948	0.663924
$596$	−10.3737	−0.424925
$597$	0	0
$598$	−3.00146	−0.122739
$599$	−28.8689	−1.17955	−0.589775	−	0.807568i	$-0.700783\pi$
$599$	−28.8689	−1.17955	−0.589775	+	0.807568i	$0.700783\pi$
$600$	0	0
$601$	8.88531	0.362439	0.181220	−	0.983443i	$-0.441996\pi$
$601$	8.88531	0.362439	0.181220	+	0.983443i	$0.441996\pi$
$602$	0.486776	0.0198395
$603$	0	0
$604$	12.9166	0.525568
$605$	−20.8714	−0.848544
$606$	0	0
$607$	28.6612	1.16332	0.581662	−	0.813431i	$-0.302403\pi$
$607$	28.6612	1.16332	0.581662	+	0.813431i	$0.302403\pi$
$608$	−32.9854	−1.33773
$609$	0	0
$610$	6.51937	0.263962
$611$	22.4826	0.909550
$612$	0	0
$613$	46.9961	1.89815	0.949077	−	0.315043i	$-0.102019\pi$
$613$	46.9961	1.89815	0.949077	+	0.315043i	$0.102019\pi$
$614$	−6.42099	−0.259130
$615$	0	0
$616$	5.59717	0.225516
$617$	24.3030	0.978401	0.489201	−	0.872171i	$-0.337289\pi$
$617$	24.3030	0.978401	0.489201	+	0.872171i	$0.337289\pi$
$618$	0	0
$619$	39.4552	1.58584	0.792920	−	0.609326i	$-0.208560\pi$
$619$	39.4552	1.58584	0.792920	+	0.609326i	$0.208560\pi$
$620$	33.5698	1.34819
$621$	0	0
$622$	−2.74176	−0.109934
$623$	0.258077	0.0103396
$624$	0	0
$625$	−26.0974	−1.04390
$626$	0.872613	0.0348766
$627$	0	0
$628$	−19.5408	−0.779765
$629$	−10.9035	−0.434750
$630$	0	0
$631$	34.7457	1.38320	0.691602	−	0.722279i	$-0.256905\pi$
$631$	34.7457	1.38320	0.691602	+	0.722279i	$0.256905\pi$
$632$	24.1738	0.961583
$633$	0	0
$634$	−1.37177	−0.0544798
$635$	33.7738	1.34027
$636$	0	0
$637$	−1.01199	−0.0400965
$638$	−0.168863	−0.00668535
$639$	0	0
$640$	−23.4375	−0.926449
$641$	42.3173	1.67143	0.835716	−	0.549163i	$-0.185053\pi$
$641$	42.3173	1.67143	0.835716	+	0.549163i	$0.185053\pi$
$642$	0	0
$643$	−31.5672	−1.24489	−0.622444	−	0.782664i	$-0.713860\pi$
$643$	−31.5672	−1.24489	−0.622444	+	0.782664i	$0.713860\pi$
$644$	−13.0620	−0.514716
$645$	0	0
$646$	−7.98572	−0.314194
$647$	45.6664	1.79533	0.897667	−	0.440675i	$-0.145261\pi$
$647$	45.6664	1.79533	0.897667	+	0.440675i	$0.145261\pi$
$648$	0	0
$649$	0.297258	0.0116684
$650$	−0.264514	−0.0103751
$651$	0	0
$652$	8.75790	0.342986
$653$	−7.15705	−0.280077	−0.140039	−	0.990146i	$-0.544723\pi$
$653$	−7.15705	−0.280077	−0.140039	+	0.990146i	$0.544723\pi$
$654$	0	0
$655$	−6.89844	−0.269545
$656$	−19.1132	−0.746244
$657$	0	0
$658$	−8.15581	−0.317946
$659$	−12.1603	−0.473696	−0.236848	−	0.971547i	$-0.576114\pi$
$659$	−12.1603	−0.473696	−0.236848	+	0.971547i	$0.576114\pi$
$660$	0	0
$661$	−15.2194	−0.591968	−0.295984	−	0.955193i	$-0.595647\pi$
$661$	−15.2194	−0.591968	−0.295984	+	0.955193i	$0.595647\pi$
$662$	−5.43917	−0.211400
$663$	0	0
$664$	18.9067	0.733724
$665$	−48.2896	−1.87259
$666$	0	0
$667$	0.820994	0.0317890
$668$	28.5502	1.10464
$669$	0	0
$670$	9.06297	0.350133
$671$	−9.94702	−0.384001
$672$	0	0
$673$	34.9359	1.34668	0.673340	−	0.739333i	$-0.264859\pi$
$673$	34.9359	1.34668	0.673340	+	0.739333i	$0.264859\pi$
$674$	−12.9555	−0.499028
$675$	0	0
$676$	8.14219	0.313161
$677$	17.6078	0.676721	0.338361	−	0.941016i	$-0.390128\pi$
$677$	17.6078	0.676721	0.338361	+	0.941016i	$0.390128\pi$
$678$	0	0
$679$	−25.1411	−0.964826
$680$	−9.01560	−0.345732
$681$	0	0
$682$	4.26951	0.163488
$683$	43.5075	1.66477	0.832385	−	0.554198i	$-0.186975\pi$
$683$	43.5075	1.66477	0.832385	+	0.554198i	$0.186975\pi$
$684$	0	0
$685$	−0.846600	−0.0323469
$686$	−7.07512	−0.270129
$687$	0	0
$688$	1.41950	0.0541181
$689$	21.2865	0.810953
$690$	0	0
$691$	−2.05256	−0.0780829	−0.0390415	−	0.999238i	$-0.512430\pi$
$691$	−2.05256	−0.0780829	−0.0390415	+	0.999238i	$0.512430\pi$
$692$	−33.2356	−1.26343
$693$	0	0
$694$	0.163907	0.00622184
$695$	29.9459	1.13591
$696$	0	0
$697$	−16.1079	−0.610130
$698$	−3.40741	−0.128972
$699$	0	0
$700$	−1.15113	−0.0435088
$701$	9.44349	0.356676	0.178338	−	0.983969i	$-0.442928\pi$
$701$	9.44349	0.356676	0.178338	+	0.983969i	$0.442928\pi$
$702$	0	0
$703$	32.5119	1.22621
$704$	6.21422	0.234207
$705$	0	0
$706$	3.56814	0.134289
$707$	−23.7883	−0.894653
$708$	0	0
$709$	15.9123	0.597598	0.298799	−	0.954316i	$-0.403414\pi$
$709$	15.9123	0.597598	0.298799	+	0.954316i	$0.403414\pi$
$710$	14.8105	0.555829
$711$	0	0
$712$	−0.143670	−0.00538426
$713$	−20.7579	−0.777389
$714$	0	0
$715$	9.17446	0.343105
$716$	41.7941	1.56192
$717$	0	0
$718$	−2.16540	−0.0808119
$719$	−24.9061	−0.928842	−0.464421	−	0.885615i	$-0.653737\pi$
$719$	−24.9061	−0.928842	−0.464421	+	0.885615i	$0.653737\pi$
$720$	0	0
$721$	4.13240	0.153899
$722$	16.3584	0.608795
$723$	0	0
$724$	−5.50112	−0.204448
$725$	0.0723528	0.00268712
$726$	0	0
$727$	11.5573	0.428636	0.214318	−	0.976764i	$-0.431247\pi$
$727$	11.5573	0.428636	0.214318	+	0.976764i	$0.431247\pi$
$728$	11.9843	0.444167
$729$	0	0
$730$	2.57330	0.0952421
$731$	1.19631	0.0442470
$732$	0	0
$733$	11.9918	0.442926	0.221463	−	0.975169i	$-0.428917\pi$
$733$	11.9918	0.442926	0.221463	+	0.975169i	$0.428917\pi$
$734$	6.41928	0.236940
$735$	0	0
$736$	11.0528	0.407412
$737$	−13.8280	−0.509359
$738$	0	0
$739$	48.3255	1.77768	0.888840	−	0.458217i	$-0.151512\pi$
$739$	48.3255	1.77768	0.888840	+	0.458217i	$0.151512\pi$
$740$	17.6181	0.647654
$741$	0	0
$742$	−7.72191	−0.283480
$743$	−40.3146	−1.47900	−0.739499	−	0.673158i	$-0.764937\pi$
$743$	−40.3146	−1.47900	−0.739499	+	0.673158i	$0.764937\pi$
$744$	0	0
$745$	12.8508	0.470817
$746$	−9.24693	−0.338554
$747$	0	0
$748$	6.60265	0.241417
$749$	47.7970	1.74646
$750$	0	0
$751$	−48.6091	−1.77377	−0.886885	−	0.461990i	$-0.847136\pi$
$751$	−48.6091	−1.77377	−0.886885	+	0.461990i	$0.847136\pi$
$752$	−23.7834	−0.867292
$753$	0	0
$754$	−0.361558	−0.0131672
$755$	−16.0008	−0.582330
$756$	0	0
$757$	−44.8319	−1.62944	−0.814721	−	0.579853i	$-0.803110\pi$
$757$	−44.8319	−1.62944	−0.814721	+	0.579853i	$0.803110\pi$
$758$	−8.95611	−0.325301
$759$	0	0
$760$	26.8826	0.975136
$761$	31.4812	1.14119	0.570596	−	0.821231i	$-0.306712\pi$
$761$	31.4812	1.14119	0.570596	+	0.821231i	$0.306712\pi$
$762$	0	0
$763$	−13.3188	−0.482171
$764$	7.56101	0.273548
$765$	0	0
$766$	−8.56189	−0.309353
$767$	0.636468	0.0229815
$768$	0	0
$769$	30.6134	1.10395	0.551974	−	0.833861i	$-0.313875\pi$
$769$	30.6134	1.10395	0.551974	+	0.833861i	$0.313875\pi$
$770$	−3.32813	−0.119937
$771$	0	0
$772$	39.0837	1.40665
$773$	−25.8605	−0.930136	−0.465068	−	0.885275i	$-0.653970\pi$
$773$	−25.8605	−0.930136	−0.465068	+	0.885275i	$0.653970\pi$
$774$	0	0
$775$	−1.82936	−0.0657125
$776$	13.9959	0.502424
$777$	0	0
$778$	−11.6690	−0.418352
$779$	48.0304	1.72087
$780$	0	0
$781$	−22.5974	−0.808598
$782$	2.67588	0.0956891
$783$	0	0
$784$	1.07054	0.0382336
$785$	24.2069	0.863981
$786$	0	0
$787$	31.7233	1.13081	0.565406	−	0.824813i	$-0.308719\pi$
$787$	31.7233	1.13081	0.565406	+	0.824813i	$0.308719\pi$
$788$	34.1788	1.21757
$789$	0	0
$790$	−14.3740	−0.511403
$791$	−0.278827	−0.00991394
$792$	0	0
$793$	−21.2979	−0.756311
$794$	11.2021	0.397548
$795$	0	0
$796$	2.89306	0.102542
$797$	55.1753	1.95441	0.977205	−	0.212296i	$-0.0680943\pi$
$797$	55.1753	1.95441	0.977205	+	0.212296i	$0.0680943\pi$
$798$	0	0
$799$	−20.0438	−0.709099
$800$	0.974066	0.0344384
$801$	0	0
$802$	−5.49078	−0.193886
$803$	−3.92625	−0.138554
$804$	0	0
$805$	16.1810	0.570306
$806$	9.14158	0.321999
$807$	0	0
$808$	13.2429	0.465882
$809$	6.04923	0.212680	0.106340	−	0.994330i	$-0.466087\pi$
$809$	6.04923	0.212680	0.106340	+	0.994330i	$0.466087\pi$
$810$	0	0
$811$	−3.97693	−0.139649	−0.0698244	−	0.997559i	$-0.522244\pi$
$811$	−3.97693	−0.139649	−0.0698244	+	0.997559i	$0.522244\pi$
$812$	−1.57346	−0.0552176
$813$	0	0
$814$	2.24073	0.0785374
$815$	−10.8491	−0.380029
$816$	0	0
$817$	−3.56714	−0.124798
$818$	−7.15641	−0.250218
$819$	0	0
$820$	26.0275	0.908920
$821$	−16.5562	−0.577815	−0.288907	−	0.957357i	$-0.593292\pi$
$821$	−16.5562	−0.577815	−0.288907	+	0.957357i	$0.593292\pi$
$822$	0	0
$823$	−29.9588	−1.04430	−0.522150	−	0.852854i	$-0.674870\pi$
$823$	−29.9588	−1.04430	−0.522150	+	0.852854i	$0.674870\pi$
$824$	−2.30049	−0.0801413
$825$	0	0
$826$	−0.230885	−0.00803353
$827$	39.4061	1.37029	0.685143	−	0.728409i	$-0.259740\pi$
$827$	39.4061	1.37029	0.685143	+	0.728409i	$0.259740\pi$
$828$	0	0
$829$	6.63817	0.230553	0.115277	−	0.993333i	$-0.463225\pi$
$829$	6.63817	0.230553	0.115277	+	0.993333i	$0.463225\pi$
$830$	−11.2421	−0.390220
$831$	0	0
$832$	13.3055	0.461285
$833$	0.902214	0.0312599
$834$	0	0
$835$	−35.3675	−1.22394
$836$	−19.6877	−0.680914
$837$	0	0
$838$	−3.83906	−0.132618
$839$	55.4839	1.91552	0.957759	−	0.287574i	$-0.0928486\pi$
$839$	55.4839	1.91552	0.957759	+	0.287574i	$0.0928486\pi$
$840$	0	0
$841$	−28.9011	−0.996590
$842$	0.889068	0.0306393
$843$	0	0
$844$	42.1815	1.45195
$845$	−10.0864	−0.346983
$846$	0	0
$847$	−24.7345	−0.849887
$848$	−22.5181	−0.773276
$849$	0	0
$850$	0.235820	0.00808857
$851$	−10.8942	−0.373447
$852$	0	0
$853$	−23.3264	−0.798680	−0.399340	−	0.916803i	$-0.630761\pi$
$853$	−23.3264	−0.798680	−0.399340	+	0.916803i	$0.630761\pi$
$854$	7.72603	0.264379
$855$	0	0
$856$	−26.6084	−0.909456
$857$	−20.5822	−0.703075	−0.351537	−	0.936174i	$-0.614341\pi$
$857$	−20.5822	−0.703075	−0.351537	+	0.936174i	$0.614341\pi$
$858$	0	0
$859$	−45.7526	−1.56106	−0.780529	−	0.625120i	$-0.785050\pi$
$859$	−45.7526	−1.56106	−0.780529	+	0.625120i	$0.785050\pi$
$860$	−1.93302	−0.0659155
$861$	0	0
$862$	−1.68600	−0.0574253
$863$	−10.0848	−0.343290	−0.171645	−	0.985159i	$-0.554908\pi$
$863$	−10.0848	−0.343290	−0.171645	+	0.985159i	$0.554908\pi$
$864$	0	0
$865$	41.1717	1.39988
$866$	−6.51638	−0.221435
$867$	0	0
$868$	39.7831	1.35033
$869$	21.9313	0.743968
$870$	0	0
$871$	−29.6075	−1.00321
$872$	7.41449	0.251086
$873$	0	0
$874$	−7.97890	−0.269890
$875$	−29.5645	−0.999462
$876$	0	0
$877$	3.30564	0.111624	0.0558118	−	0.998441i	$-0.482225\pi$
$877$	3.30564	0.111624	0.0558118	+	0.998441i	$0.482225\pi$
$878$	−8.42872	−0.284456
$879$	0	0
$880$	−9.70527	−0.327165
$881$	−8.30907	−0.279940	−0.139970	−	0.990156i	$-0.544701\pi$
$881$	−8.30907	−0.279940	−0.139970	+	0.990156i	$0.544701\pi$
$882$	0	0
$883$	2.65895	0.0894807	0.0447403	−	0.998999i	$-0.485754\pi$
$883$	2.65895	0.0894807	0.0447403	+	0.998999i	$0.485754\pi$
$884$	14.1372	0.475484
$885$	0	0
$886$	7.69060	0.258371
$887$	21.5846	0.724740	0.362370	−	0.932034i	$-0.381968\pi$
$887$	21.5846	0.724740	0.362370	+	0.932034i	$0.381968\pi$
$888$	0	0
$889$	40.0250	1.34240
$890$	0.0854276	0.00286354
$891$	0	0
$892$	−8.77932	−0.293954
$893$	59.7665	2.00001
$894$	0	0
$895$	−51.7738	−1.73061
$896$	−27.7755	−0.927915
$897$	0	0
$898$	13.7870	0.460079
$899$	−2.50051	−0.0833967
$900$	0	0
$901$	−18.9775	−0.632231
$902$	3.31026	0.110220
$903$	0	0
$904$	0.155222	0.00516259
$905$	6.81470	0.226528
$906$	0	0
$907$	−5.95616	−0.197771	−0.0988855	−	0.995099i	$-0.531528\pi$
$907$	−5.95616	−0.197771	−0.0988855	+	0.995099i	$0.531528\pi$
$908$	−50.7145	−1.68302
$909$	0	0
$910$	−7.12597	−0.236224
$911$	−14.9730	−0.496079	−0.248040	−	0.968750i	$-0.579786\pi$
$911$	−14.9730	−0.496079	−0.248040	+	0.968750i	$0.579786\pi$
$912$	0	0
$913$	17.1528	0.567676
$914$	−4.64590	−0.153673
$915$	0	0
$916$	8.35249	0.275974
$917$	−8.17527	−0.269971
$918$	0	0
$919$	−21.4164	−0.706463	−0.353232	−	0.935536i	$-0.614917\pi$
$919$	−21.4164	−0.706463	−0.353232	+	0.935536i	$0.614917\pi$
$920$	−9.00790	−0.296982
$921$	0	0
$922$	−6.71925	−0.221287
$923$	−48.3840	−1.59258
$924$	0	0
$925$	−0.960085	−0.0315674
$926$	−2.71546	−0.0892354
$927$	0	0
$928$	1.33143	0.0437064
$929$	51.4153	1.68688	0.843441	−	0.537221i	$-0.180526\pi$
$929$	51.4153	1.68688	0.843441	+	0.537221i	$0.180526\pi$
$930$	0	0
$931$	−2.69021	−0.0881682
$932$	−19.4889	−0.638381
$933$	0	0
$934$	12.8039	0.418958
$935$	−8.17925	−0.267490
$936$	0	0
$937$	−41.9350	−1.36996	−0.684979	−	0.728563i	$-0.740188\pi$
$937$	−41.9350	−1.36996	−0.684979	+	0.728563i	$0.740188\pi$
$938$	10.7404	0.350687
$939$	0	0
$940$	32.3873	1.05636
$941$	36.8467	1.20117	0.600585	−	0.799561i	$-0.294935\pi$
$941$	36.8467	1.20117	0.600585	+	0.799561i	$0.294935\pi$
$942$	0	0
$943$	−16.0941	−0.524097
$944$	−0.673293	−0.0219138
$945$	0	0
$946$	−0.245848	−0.00799320
$947$	−2.12177	−0.0689481	−0.0344741	−	0.999406i	$-0.510976\pi$
$947$	−2.12177	−0.0689481	−0.0344741	+	0.999406i	$0.510976\pi$
$948$	0	0
$949$	−8.40663	−0.272891
$950$	−0.703167	−0.0228138
$951$	0	0
$952$	−10.6843	−0.346280
$953$	−13.5643	−0.439391	−0.219695	−	0.975569i	$-0.570506\pi$
$953$	−13.5643	−0.439391	−0.219695	+	0.975569i	$0.570506\pi$
$954$	0	0
$955$	−9.36645	−0.303091
$956$	−52.2584	−1.69016
$957$	0	0
$958$	−3.44427	−0.111279
$959$	−1.00330	−0.0323981
$960$	0	0
$961$	32.2225	1.03944
$962$	4.79769	0.154684
$963$	0	0
$964$	19.7391	0.635754
$965$	−48.4163	−1.55858
$966$	0	0
$967$	5.95719	0.191570	0.0957851	−	0.995402i	$-0.469464\pi$
$967$	5.95719	0.191570	0.0957851	+	0.995402i	$0.469464\pi$
$968$	13.7696	0.442571
$969$	0	0
$970$	−8.32211	−0.267207
$971$	−12.1772	−0.390785	−0.195392	−	0.980725i	$-0.562598\pi$
$971$	−12.1772	−0.390785	−0.195392	+	0.980725i	$0.562598\pi$
$972$	0	0
$973$	35.4886	1.13771
$974$	−9.08597	−0.291133
$975$	0	0
$976$	22.5301	0.721172
$977$	−7.69439	−0.246165	−0.123083	−	0.992396i	$-0.539278\pi$
$977$	−7.69439	−0.246165	−0.123083	+	0.992396i	$0.539278\pi$
$978$	0	0
$979$	−0.130342	−0.00416576
$980$	−1.45782	−0.0465683
$981$	0	0
$982$	−3.19854	−0.102069
$983$	12.2144	0.389578	0.194789	−	0.980845i	$-0.437598\pi$
$983$	12.2144	0.389578	0.194789	+	0.980845i	$0.437598\pi$
$984$	0	0
$985$	−42.3401	−1.34907
$986$	0.322338	0.0102653
$987$	0	0
$988$	−42.1540	−1.34110
$989$	1.19529	0.0380079
$990$	0	0
$991$	2.35228	0.0747227	0.0373613	−	0.999302i	$-0.488105\pi$
$991$	2.35228	0.0747227	0.0373613	+	0.999302i	$0.488105\pi$
$992$	−33.6637	−1.06882
$993$	0	0
$994$	17.5518	0.556709
$995$	−3.58387	−0.113616
$996$	0	0
$997$	19.2415	0.609384	0.304692	−	0.952451i	$-0.401446\pi$
$997$	19.2415	0.609384	0.304692	+	0.952451i	$0.401446\pi$
$998$	12.0428	0.381209
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	4527.2.a.k.1.5		10
3.2	odd	2		503.2.a.e.1.6	✓	10
12.11	even	2		8048.2.a.p.1.2		10

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
503.2.a.e.1.6	✓	10	3.2	odd	2
4527.2.a.k.1.5		10	1.1	even	1	trivial
8048.2.a.p.1.2		10	12.11	even	2

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:	\( N \)	\(=\)	\( 4527 = 3^{2} \cdot 503 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	4527.a (trivial)

\(f(q)\)	\(=\)	\(q+0.392284 q^{2} -1.84611 q^{4} +2.28693 q^{5} +2.71022 q^{7} -1.50877 q^{8} +O(q^{10})\)	\(q+0.392284 q^{2} -1.84611 q^{4} +2.28693 q^{5} +2.71022 q^{7} -1.50877 q^{8} +0.897127 q^{10} -1.36880 q^{11} -2.93079 q^{13} +1.06318 q^{14} +3.10036 q^{16} +2.61287 q^{17} -7.79104 q^{19} -4.22194 q^{20} -0.536960 q^{22} +2.61064 q^{23} +0.230071 q^{25} -1.14970 q^{26} -5.00337 q^{28} +0.314480 q^{29} -7.95126 q^{31} +4.23376 q^{32} +1.02499 q^{34} +6.19810 q^{35} -4.17299 q^{37} -3.05630 q^{38} -3.45045 q^{40} -6.16482 q^{41} +0.457851 q^{43} +2.52697 q^{44} +1.02411 q^{46} -7.67118 q^{47} +0.345296 q^{49} +0.0902533 q^{50} +5.41058 q^{52} -7.26306 q^{53} -3.13037 q^{55} -4.08909 q^{56} +0.123365 q^{58} -0.217166 q^{59} +7.26694 q^{61} -3.11915 q^{62} -4.53989 q^{64} -6.70253 q^{65} +10.1022 q^{67} -4.82366 q^{68} +2.43141 q^{70} +16.5088 q^{71} +2.86838 q^{73} -1.63699 q^{74} +14.3831 q^{76} -3.70976 q^{77} -16.0222 q^{79} +7.09033 q^{80} -2.41836 q^{82} -12.5312 q^{83} +5.97547 q^{85} +0.179608 q^{86} +2.06521 q^{88} +0.0952234 q^{89} -7.94310 q^{91} -4.81954 q^{92} -3.00928 q^{94} -17.8176 q^{95} -9.27640 q^{97} +0.135454 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8}+O(q^{10}) \) 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8	\( 10 q + 4 q^{2} + 4 q^{4} + q^{5} - 5 q^{7} + 3 q^{8} - 4 q^{10} + 3 q^{11} - 18 q^{13} - q^{14} - 4 q^{16} + 11 q^{17} + 3 q^{20} - 18 q^{22} + 2 q^{23} - 27 q^{25} - 11 q^{26} - 22 q^{28} + 9 q^{29} - 22 q^{31} + 10 q^{32} - 10 q^{34} + 6 q^{35} - 35 q^{37} - 2 q^{38} - 19 q^{40} + 4 q^{41} - 20 q^{43} - 9 q^{44} - q^{46} - 7 q^{47} - 27 q^{49} - 16 q^{50} - 7 q^{52} + 24 q^{53} - 11 q^{55} - 12 q^{56} + 2 q^{58} - 17 q^{59} - 4 q^{61} - 8 q^{62} + 3 q^{64} + 16 q^{65} - 6 q^{67} - 28 q^{68} + 26 q^{70} + q^{71} - 31 q^{73} - 11 q^{74} + 20 q^{76} - 3 q^{77} - 10 q^{79} - 24 q^{80} - 9 q^{82} - 22 q^{83} - 6 q^{85} - 38 q^{86} - 3 q^{88} - q^{89} + 10 q^{91} - 27 q^{92} + 33 q^{94} - 39 q^{95} - 57 q^{97} - 40 q^{98}+O(q^{100}) \) 10 * q + 4 * q^2 + 4 * q^4 + q^5 - 5 * q^7 + 3 * q^8 - 4 * q^10 + 3 * q^11 - 18 * q^13 - q^14 - 4 * q^16 + 11 * q^17 + 3 * q^20 - 18 * q^22 + 2 * q^23 - 27 * q^25 - 11 * q^26 - 22 * q^28 + 9 * q^29 - 22 * q^31 + 10 * q^32 - 10 * q^34 + 6 * q^35 - 35 * q^37 - 2 * q^38 - 19 * q^40 + 4 * q^41 - 20 * q^43 - 9 * q^44 - q^46 - 7 * q^47 - 27 * q^49 - 16 * q^50 - 7 * q^52 + 24 * q^53 - 11 * q^55 - 12 * q^56 + 2 * q^58 - 17 * q^59 - 4 * q^61 - 8 * q^62 + 3 * q^64 + 16 * q^65 - 6 * q^67 - 28 * q^68 + 26 * q^70 + q^71 - 31 * q^73 - 11 * q^74 + 20 * q^76 - 3 * q^77 - 10 * q^79 - 24 * q^80 - 9 * q^82 - 22 * q^83 - 6 * q^85 - 38 * q^86 - 3 * q^88 - q^89 + 10 * q^91 - 27 * q^92 + 33 * q^94 - 39 * q^95 - 57 * q^97 - 40 * q^98

Introduction

L-functions

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