LMFDB - Embedded newform 6003.2.a.r.1.5

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:	$16$
Coefficient field:	$\mathbb{Q}[x]/(x^{16} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{16} - x^{15} - 28 x^{14} + 27 x^{13} + 316 x^{12} - 295 x^{11} - 1835 x^{10} + 1665 x^{9} + \cdots - 192 $ x^16 - x^15 - 28x^14 + 27x^13 + 316x^12 - 295x^11 - 1835x^10 + 1665x^9 + 5778x^8 - 5124x^7 - 9405x^6 + 8288x^5 + 6405x^4 - 6032x^3 - 400x^2 + 1088x - 192
Coefficient ring:	$\Z[a_1, \ldots, a_{17}]$
Coefficient ring index:	$ 2^{5} $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.5
Root			$-1.53756$ 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)

$f(q)$	$=$	$q-1.53756 q^{2} +0.364091 q^{4} +2.97145 q^{5} +1.52213 q^{7} +2.51531 q^{8} +O(q^{10})$	\(q-1.53756 q^{2} +0.364091 q^{4} +2.97145 q^{5} +1.52213 q^{7} +2.51531 q^{8} -4.56878 q^{10} -0.657982 q^{11} +3.12537 q^{13} -2.34037 q^{14} -4.59562 q^{16} -1.99186 q^{17} +6.50919 q^{19} +1.08188 q^{20} +1.01169 q^{22} +1.00000 q^{23} +3.82949 q^{25} -4.80545 q^{26} +0.554195 q^{28} +1.00000 q^{29} +6.58309 q^{31} +2.03543 q^{32} +3.06261 q^{34} +4.52294 q^{35} +7.52960 q^{37} -10.0083 q^{38} +7.47410 q^{40} -8.68681 q^{41} -3.14842 q^{43} -0.239565 q^{44} -1.53756 q^{46} -1.88957 q^{47} -4.68311 q^{49} -5.88807 q^{50} +1.13792 q^{52} +2.78935 q^{53} -1.95516 q^{55} +3.82864 q^{56} -1.53756 q^{58} -6.26938 q^{59} +1.61173 q^{61} -10.1219 q^{62} +6.06165 q^{64} +9.28687 q^{65} +5.54031 q^{67} -0.725219 q^{68} -6.95429 q^{70} +9.71587 q^{71} +4.17202 q^{73} -11.5772 q^{74} +2.36994 q^{76} -1.00154 q^{77} +9.99227 q^{79} -13.6556 q^{80} +13.3565 q^{82} -3.81394 q^{83} -5.91871 q^{85} +4.84088 q^{86} -1.65503 q^{88} +0.899101 q^{89} +4.75723 q^{91} +0.364091 q^{92} +2.90533 q^{94} +19.3417 q^{95} -3.81354 q^{97} +7.20056 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	$ 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) $ 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * 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 $

$2$	−1.53756	−1.08722	−0.543610	−	0.839338i	$-0.682943\pi$
$2$	−1.53756	−1.08722	−0.543610	+	0.839338i	$0.682943\pi$
$3$	0	0
$3$	0	0
$4$	0.364091	0.182046
$5$	2.97145	1.32887	0.664435	−	0.747346i	$-0.268672\pi$
$5$	2.97145	1.32887	0.664435	+	0.747346i	$0.268672\pi$
$6$	0	0
$7$	1.52213	0.575313	0.287656	−	0.957734i	$-0.407124\pi$
$7$	1.52213	0.575313	0.287656	+	0.957734i	$0.407124\pi$
$8$	2.51531	0.889296
$9$	0	0
$10$	−4.56878	−1.44477
$11$	−0.657982	−0.198389	−0.0991945	−	0.995068i	$-0.531627\pi$
$11$	−0.657982	−0.198389	−0.0991945	+	0.995068i	$0.531627\pi$
$12$	0	0
$13$	3.12537	0.866822	0.433411	−	0.901196i	$-0.357310\pi$
$13$	3.12537	0.866822	0.433411	+	0.901196i	$0.357310\pi$
$14$	−2.34037	−0.625491
$15$	0	0
$16$	−4.59562	−1.14890
$17$	−1.99186	−0.483097	−0.241549	−	0.970389i	$-0.577655\pi$
$17$	−1.99186	−0.483097	−0.241549	+	0.970389i	$0.577655\pi$
$18$	0	0
$19$	6.50919	1.49331	0.746656	−	0.665211i	$-0.231658\pi$
$19$	6.50919	1.49331	0.746656	+	0.665211i	$0.231658\pi$
$20$	1.08188	0.241915
$21$	0	0
$22$	1.01169	0.215692
$23$	1.00000	0.208514
$23$	1.00000	0.208514
$24$	0	0
$25$	3.82949	0.765898
$26$	−4.80545	−0.942426
$27$	0	0
$28$	0.554195	0.104733
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	6.58309	1.18236	0.591179	−	0.806540i	$-0.298663\pi$
$31$	6.58309	1.18236	0.591179	+	0.806540i	$0.298663\pi$
$32$	2.03543	0.359816
$33$	0	0
$34$	3.06261	0.525233
$35$	4.52294	0.764516
$36$	0	0
$37$	7.52960	1.23786	0.618929	−	0.785447i	$-0.287567\pi$
$37$	7.52960	1.23786	0.618929	+	0.785447i	$0.287567\pi$
$38$	−10.0083	−1.62356
$39$	0	0
$40$	7.47410	1.18176
$41$	−8.68681	−1.35665	−0.678326	−	0.734761i	$-0.737294\pi$
$41$	−8.68681	−1.35665	−0.678326	+	0.734761i	$0.737294\pi$
$42$	0	0
$43$	−3.14842	−0.480129	−0.240064	−	0.970757i	$-0.577169\pi$
$43$	−3.14842	−0.480129	−0.240064	+	0.970757i	$0.577169\pi$
$44$	−0.239565	−0.0361158
$45$	0	0
$46$	−1.53756	−0.226701
$47$	−1.88957	−0.275622	−0.137811	−	0.990459i	$-0.544007\pi$
$47$	−1.88957	−0.275622	−0.137811	+	0.990459i	$0.544007\pi$
$48$	0	0
$49$	−4.68311	−0.669016
$50$	−5.88807	−0.832699
$51$	0	0
$52$	1.13792	0.157801
$53$	2.78935	0.383146	0.191573	−	0.981478i	$-0.438641\pi$
$53$	2.78935	0.383146	0.191573	+	0.981478i	$0.438641\pi$
$54$	0	0
$55$	−1.95516	−0.263633
$56$	3.82864	0.511623
$57$	0	0
$58$	−1.53756	−0.201892
$59$	−6.26938	−0.816204	−0.408102	−	0.912936i	$-0.633809\pi$
$59$	−6.26938	−0.816204	−0.408102	+	0.912936i	$0.633809\pi$
$60$	0	0
$61$	1.61173	0.206361	0.103180	−	0.994663i	$-0.467098\pi$
$61$	1.61173	0.206361	0.103180	+	0.994663i	$0.467098\pi$
$62$	−10.1219	−1.28548
$63$	0	0
$64$	6.06165	0.757706
$65$	9.28687	1.15189
$66$	0	0
$67$	5.54031	0.676857	0.338428	−	0.940992i	$-0.390105\pi$
$67$	5.54031	0.676857	0.338428	+	0.940992i	$0.390105\pi$
$68$	−0.725219	−0.0879457
$69$	0	0
$70$	−6.95429	−0.831196
$71$	9.71587	1.15306	0.576531	−	0.817076i	$-0.304406\pi$
$71$	9.71587	1.15306	0.576531	+	0.817076i	$0.304406\pi$
$72$	0	0
$73$	4.17202	0.488298	0.244149	−	0.969738i	$-0.421491\pi$
$73$	4.17202	0.488298	0.244149	+	0.969738i	$0.421491\pi$
$74$	−11.5772	−1.34582
$75$	0	0
$76$	2.36994	0.271851
$77$	−1.00154	−0.114136
$78$	0	0
$79$	9.99227	1.12422	0.562109	−	0.827063i	$-0.309990\pi$
$79$	9.99227	1.12422	0.562109	+	0.827063i	$0.309990\pi$
$80$	−13.6556	−1.52675
$81$	0	0
$82$	13.3565	1.47498
$83$	−3.81394	−0.418635	−0.209317	−	0.977848i	$-0.567124\pi$
$83$	−3.81394	−0.418635	−0.209317	+	0.977848i	$0.567124\pi$
$84$	0	0
$85$	−5.91871	−0.641974
$86$	4.84088	0.522005
$87$	0	0
$88$	−1.65503	−0.176427
$89$	0.899101	0.0953045	0.0476523	−	0.998864i	$-0.484826\pi$
$89$	0.899101	0.0953045	0.0476523	+	0.998864i	$0.484826\pi$
$90$	0	0
$91$	4.75723	0.498694
$92$	0.364091	0.0379591
$93$	0	0
$94$	2.90533	0.299661
$95$	19.3417	1.98442
$96$	0	0
$97$	−3.81354	−0.387206	−0.193603	−	0.981080i	$-0.562017\pi$
$97$	−3.81354	−0.387206	−0.193603	+	0.981080i	$0.562017\pi$
$98$	7.20056	0.727366
$99$	0	0
$100$	1.39428	0.139428
$101$	9.37952	0.933297	0.466649	−	0.884443i	$-0.345461\pi$
$101$	9.37952	0.933297	0.466649	+	0.884443i	$0.345461\pi$
$102$	0	0
$103$	1.04005	0.102479	0.0512395	−	0.998686i	$-0.483683\pi$
$103$	1.04005	0.102479	0.0512395	+	0.998686i	$0.483683\pi$
$104$	7.86127	0.770861
$105$	0	0
$106$	−4.28879	−0.416564
$107$	9.51332	0.919687	0.459843	−	0.888000i	$-0.347906\pi$
$107$	9.51332	0.919687	0.459843	+	0.888000i	$0.347906\pi$
$108$	0	0
$109$	11.0318	1.05666	0.528329	−	0.849040i	$-0.322819\pi$
$109$	11.0318	1.05666	0.528329	+	0.849040i	$0.322819\pi$
$110$	3.00617	0.286627
$111$	0	0
$112$	−6.99515	−0.660979
$113$	16.5846	1.56015	0.780076	−	0.625685i	$-0.215181\pi$
$113$	16.5846	1.56015	0.780076	+	0.625685i	$0.215181\pi$
$114$	0	0
$115$	2.97145	0.277089
$116$	0.364091	0.0338050
$117$	0	0
$118$	9.63955	0.887392
$119$	−3.03188	−0.277932
$120$	0	0
$121$	−10.5671	−0.960642
$122$	−2.47813	−0.224360
$123$	0	0
$124$	2.39685	0.215243
$125$	−3.47811	−0.311092
$126$	0	0
$127$	−7.11884	−0.631695	−0.315847	−	0.948810i	$-0.602289\pi$
$127$	−7.11884	−0.631695	−0.315847	+	0.948810i	$0.602289\pi$
$128$	−13.3910	−1.18361
$129$	0	0
$130$	−14.2791	−1.25236
$131$	2.82852	0.247129	0.123564	−	0.992337i	$-0.460567\pi$
$131$	2.82852	0.247129	0.123564	+	0.992337i	$0.460567\pi$
$132$	0	0
$133$	9.90786	0.859121
$134$	−8.51856	−0.735892
$135$	0	0
$136$	−5.01014	−0.429616
$137$	2.57599	0.220082	0.110041	−	0.993927i	$-0.464902\pi$
$137$	2.57599	0.220082	0.110041	+	0.993927i	$0.464902\pi$
$138$	0	0
$139$	−15.0031	−1.27255	−0.636273	−	0.771464i	$-0.719525\pi$
$139$	−15.0031	−1.27255	−0.636273	+	0.771464i	$0.719525\pi$
$140$	1.64676	0.139177
$141$	0	0
$142$	−14.9387	−1.25363
$143$	−2.05644	−0.171968
$144$	0	0
$145$	2.97145	0.246765
$146$	−6.41474	−0.530887
$147$	0	0
$148$	2.74146	0.225347
$149$	−19.2475	−1.57682	−0.788409	−	0.615151i	$-0.789095\pi$
$149$	−19.2475	−1.57682	−0.788409	+	0.615151i	$0.789095\pi$
$150$	0	0
$151$	17.3611	1.41283	0.706415	−	0.707798i	$-0.250311\pi$
$151$	17.3611	1.41283	0.706415	+	0.707798i	$0.250311\pi$
$152$	16.3726	1.32800
$153$	0	0
$154$	1.53992	0.124091
$155$	19.5613	1.57120
$156$	0	0
$157$	−7.87752	−0.628694	−0.314347	−	0.949308i	$-0.601786\pi$
$157$	−7.87752	−0.628694	−0.314347	+	0.949308i	$0.601786\pi$
$158$	−15.3637	−1.22227
$159$	0	0
$160$	6.04815	0.478149
$161$	1.52213	0.119961
$162$	0	0
$163$	−15.2023	−1.19074	−0.595370	−	0.803452i	$-0.702994\pi$
$163$	−15.2023	−1.19074	−0.595370	+	0.803452i	$0.702994\pi$
$164$	−3.16279	−0.246973
$165$	0	0
$166$	5.86417	0.455148
$167$	−10.9193	−0.844961	−0.422480	−	0.906372i	$-0.638840\pi$
$167$	−10.9193	−0.844961	−0.422480	+	0.906372i	$0.638840\pi$
$168$	0	0
$169$	−3.23205	−0.248620
$170$	9.10037	0.697966
$171$	0	0
$172$	−1.14631	−0.0874053
$173$	−1.77155	−0.134688	−0.0673441	−	0.997730i	$-0.521453\pi$
$173$	−1.77155	−0.134688	−0.0673441	+	0.997730i	$0.521453\pi$
$174$	0	0
$175$	5.82899	0.440631
$176$	3.02384	0.227930
$177$	0	0
$178$	−1.38242	−0.103617
$179$	14.9962	1.12087	0.560433	−	0.828200i	$-0.310635\pi$
$179$	14.9962	1.12087	0.560433	+	0.828200i	$0.310635\pi$
$180$	0	0
$181$	25.1518	1.86952	0.934759	−	0.355281i	$-0.115615\pi$
$181$	25.1518	1.86952	0.934759	+	0.355281i	$0.115615\pi$
$182$	−7.31453	−0.542189
$183$	0	0
$184$	2.51531	0.185431
$185$	22.3738	1.64495
$186$	0	0
$187$	1.31061	0.0958412
$188$	−0.687975	−0.0501757
$189$	0	0
$190$	−29.7390	−2.15750
$191$	−8.60952	−0.622963	−0.311482	−	0.950252i	$-0.600825\pi$
$191$	−8.60952	−0.622963	−0.311482	+	0.950252i	$0.600825\pi$
$192$	0	0
$193$	−1.52670	−0.109894	−0.0549472	−	0.998489i	$-0.517499\pi$
$193$	−1.52670	−0.109894	−0.0549472	+	0.998489i	$0.517499\pi$
$194$	5.86354	0.420978
$195$	0	0
$196$	−1.70508	−0.121791
$197$	−11.9516	−0.851513	−0.425757	−	0.904838i	$-0.639992\pi$
$197$	−11.9516	−0.851513	−0.425757	+	0.904838i	$0.639992\pi$
$198$	0	0
$199$	−20.2312	−1.43415	−0.717076	−	0.696995i	$-0.754520\pi$
$199$	−20.2312	−1.43415	−0.717076	+	0.696995i	$0.754520\pi$
$200$	9.63234	0.681110
$201$	0	0
$202$	−14.4216	−1.01470
$203$	1.52213	0.106833
$204$	0	0
$205$	−25.8124	−1.80282
$206$	−1.59914	−0.111417
$207$	0	0
$208$	−14.3630	−0.995896
$209$	−4.28293	−0.296257
$210$	0	0
$211$	−26.2493	−1.80708	−0.903539	−	0.428505i	$-0.859040\pi$
$211$	−26.2493	−1.80708	−0.903539	+	0.428505i	$0.859040\pi$
$212$	1.01558	0.0697501
$213$	0	0
$214$	−14.6273	−0.999901
$215$	−9.35534	−0.638029
$216$	0	0
$217$	10.0204	0.680226
$218$	−16.9621	−1.14882
$219$	0	0
$220$	−0.711855	−0.0479933
$221$	−6.22530	−0.418759
$222$	0	0
$223$	−13.3354	−0.893005	−0.446502	−	0.894782i	$-0.647331\pi$
$223$	−13.3354	−0.893005	−0.446502	+	0.894782i	$0.647331\pi$
$224$	3.09819	0.207006
$225$	0	0
$226$	−25.4999	−1.69623
$227$	13.9162	0.923648	0.461824	−	0.886972i	$-0.347195\pi$
$227$	13.9162	0.923648	0.461824	+	0.886972i	$0.347195\pi$
$228$	0	0
$229$	2.24048	0.148055	0.0740274	−	0.997256i	$-0.476415\pi$
$229$	2.24048	0.148055	0.0740274	+	0.997256i	$0.476415\pi$
$230$	−4.56878	−0.301256
$231$	0	0
$232$	2.51531	0.165138
$233$	16.1306	1.05675	0.528376	−	0.849011i	$-0.322801\pi$
$233$	16.1306	1.05675	0.528376	+	0.849011i	$0.322801\pi$
$234$	0	0
$235$	−5.61475	−0.366266
$236$	−2.28263	−0.148586
$237$	0	0
$238$	4.66170	0.302173
$239$	−25.9374	−1.67775	−0.838874	−	0.544325i	$-0.816786\pi$
$239$	−25.9374	−1.67775	−0.838874	+	0.544325i	$0.816786\pi$
$240$	0	0
$241$	17.9123	1.15384	0.576918	−	0.816802i	$-0.304255\pi$
$241$	17.9123	1.15384	0.576918	+	0.816802i	$0.304255\pi$
$242$	16.2475	1.04443
$243$	0	0
$244$	0.586817	0.0375671
$245$	−13.9156	−0.889035
$246$	0	0
$247$	20.3436	1.29443
$248$	16.5585	1.05147
$249$	0	0
$250$	5.34780	0.338225
$251$	23.8459	1.50514	0.752569	−	0.658513i	$-0.228814\pi$
$251$	23.8459	1.50514	0.752569	+	0.658513i	$0.228814\pi$
$252$	0	0
$253$	−0.657982	−0.0413670
$254$	10.9456	0.686791
$255$	0	0
$256$	8.46617	0.529136
$257$	−12.0783	−0.753425	−0.376713	−	0.926330i	$-0.622946\pi$
$257$	−12.0783	−0.753425	−0.376713	+	0.926330i	$0.622946\pi$
$258$	0	0
$259$	11.4611	0.712156
$260$	3.38127	0.209697
$261$	0	0
$262$	−4.34902	−0.268683
$263$	−10.2132	−0.629772	−0.314886	−	0.949130i	$-0.601966\pi$
$263$	−10.2132	−0.629772	−0.314886	+	0.949130i	$0.601966\pi$
$264$	0	0
$265$	8.28839	0.509152
$266$	−15.2339	−0.934052
$267$	0	0
$268$	2.01718	0.123219
$269$	16.3462	0.996648	0.498324	−	0.866991i	$-0.333949\pi$
$269$	16.3462	0.996648	0.498324	+	0.866991i	$0.333949\pi$
$270$	0	0
$271$	−3.90467	−0.237192	−0.118596	−	0.992943i	$-0.537839\pi$
$271$	−3.90467	−0.237192	−0.118596	+	0.992943i	$0.537839\pi$
$272$	9.15383	0.555033
$273$	0	0
$274$	−3.96074	−0.239277
$275$	−2.51973	−0.151946
$276$	0	0
$277$	−22.0571	−1.32529	−0.662643	−	0.748936i	$-0.730565\pi$
$277$	−22.0571	−1.32529	−0.662643	+	0.748936i	$0.730565\pi$
$278$	23.0682	1.38354
$279$	0	0
$280$	11.3766	0.679881
$281$	−0.0586985	−0.00350166	−0.00175083	−	0.999998i	$-0.500557\pi$
$281$	−0.0586985	−0.00350166	−0.00175083	+	0.999998i	$0.500557\pi$
$282$	0	0
$283$	11.6006	0.689582	0.344791	−	0.938679i	$-0.387950\pi$
$283$	11.6006	0.689582	0.344791	+	0.938679i	$0.387950\pi$
$284$	3.53746	0.209910
$285$	0	0
$286$	3.16190	0.186967
$287$	−13.2225	−0.780499
$288$	0	0
$289$	−13.0325	−0.766617
$290$	−4.56878	−0.268288
$291$	0	0
$292$	1.51900	0.0888925
$293$	16.3619	0.955873	0.477936	−	0.878394i	$-0.341385\pi$
$293$	16.3619	0.955873	0.477936	+	0.878394i	$0.341385\pi$
$294$	0	0
$295$	−18.6291	−1.08463
$296$	18.9393	1.10082
$297$	0	0
$298$	29.5942	1.71435
$299$	3.12537	0.180745
$300$	0	0
$301$	−4.79231	−0.276224
$302$	−26.6938	−1.53606
$303$	0	0
$304$	−29.9138	−1.71567
$305$	4.78917	0.274227
$306$	0	0
$307$	−6.09940	−0.348111	−0.174056	−	0.984736i	$-0.555687\pi$
$307$	−6.09940	−0.348111	−0.174056	+	0.984736i	$0.555687\pi$
$308$	−0.364651	−0.0207779
$309$	0	0
$310$	−30.0767	−1.70824
$311$	11.4644	0.650084	0.325042	−	0.945700i	$-0.394622\pi$
$311$	11.4644	0.650084	0.325042	+	0.945700i	$0.394622\pi$
$312$	0	0
$313$	19.2263	1.08674	0.543368	−	0.839495i	$-0.317149\pi$
$313$	19.2263	1.08674	0.543368	+	0.839495i	$0.317149\pi$
$314$	12.1122	0.683528
$315$	0	0
$316$	3.63810	0.204659
$317$	−9.21199	−0.517397	−0.258699	−	0.965958i	$-0.583294\pi$
$317$	−9.21199	−0.517397	−0.258699	+	0.965958i	$0.583294\pi$
$318$	0	0
$319$	−0.657982	−0.0368399
$320$	18.0119	1.00689
$321$	0	0
$322$	−2.34037	−0.130424
$323$	−12.9654	−0.721414
$324$	0	0
$325$	11.9686	0.663897
$326$	23.3745	1.29459
$327$	0	0
$328$	−21.8500	−1.20647
$329$	−2.87618	−0.158569
$330$	0	0
$331$	−6.85715	−0.376903	−0.188451	−	0.982083i	$-0.560347\pi$
$331$	−6.85715	−0.376903	−0.188451	+	0.982083i	$0.560347\pi$
$332$	−1.38862	−0.0762105
$333$	0	0
$334$	16.7891	0.918657
$335$	16.4627	0.899455
$336$	0	0
$337$	12.0008	0.653725	0.326863	−	0.945072i	$-0.394009\pi$
$337$	12.0008	0.653725	0.326863	+	0.945072i	$0.394009\pi$
$338$	4.96948	0.270304
$339$	0	0
$340$	−2.15495	−0.116868
$341$	−4.33156	−0.234567
$342$	0	0
$343$	−17.7833	−0.960206
$344$	−7.91924	−0.426977
$345$	0	0
$346$	2.72386	0.146436
$347$	−1.31039	−0.0703454	−0.0351727	−	0.999381i	$-0.511198\pi$
$347$	−1.31039	−0.0703454	−0.0351727	+	0.999381i	$0.511198\pi$
$348$	0	0
$349$	24.3350	1.30262	0.651310	−	0.758811i	$-0.274220\pi$
$349$	24.3350	1.30262	0.651310	+	0.758811i	$0.274220\pi$
$350$	−8.96243	−0.479062
$351$	0	0
$352$	−1.33927	−0.0713835
$353$	−12.1064	−0.644361	−0.322181	−	0.946678i	$-0.604416\pi$
$353$	−12.1064	−0.644361	−0.322181	+	0.946678i	$0.604416\pi$
$354$	0	0
$355$	28.8702	1.53227
$356$	0.327355	0.0173498
$357$	0	0
$358$	−23.0575	−1.21863
$359$	0.568054	0.0299808	0.0149904	−	0.999888i	$-0.495228\pi$
$359$	0.568054	0.0299808	0.0149904	+	0.999888i	$0.495228\pi$
$360$	0	0
$361$	23.3696	1.22998
$362$	−38.6724	−2.03258
$363$	0	0
$364$	1.73207	0.0907849
$365$	12.3969	0.648885
$366$	0	0
$367$	12.8719	0.671908	0.335954	−	0.941878i	$-0.390941\pi$
$367$	12.8719	0.671908	0.335954	+	0.941878i	$0.390941\pi$
$368$	−4.59562	−0.239563
$369$	0	0
$370$	−34.4011	−1.78843
$371$	4.24576	0.220429
$372$	0	0
$373$	−26.8087	−1.38810	−0.694050	−	0.719927i	$-0.744175\pi$
$373$	−26.8087	−1.38810	−0.694050	+	0.719927i	$0.744175\pi$
$374$	−2.01514	−0.104200
$375$	0	0
$376$	−4.75285	−0.245109
$377$	3.12537	0.160965
$378$	0	0
$379$	23.5826	1.21136	0.605678	−	0.795710i	$-0.292902\pi$
$379$	23.5826	1.21136	0.605678	+	0.795710i	$0.292902\pi$
$380$	7.04214	0.361254
$381$	0	0
$382$	13.2377	0.677298
$383$	−4.55238	−0.232616	−0.116308	−	0.993213i	$-0.537106\pi$
$383$	−4.55238	−0.232616	−0.116308	+	0.993213i	$0.537106\pi$
$384$	0	0
$385$	−2.97601	−0.151672
$386$	2.34740	0.119479
$387$	0	0
$388$	−1.38848	−0.0704892
$389$	−28.4426	−1.44210	−0.721048	−	0.692885i	$-0.756339\pi$
$389$	−28.4426	−1.44210	−0.721048	+	0.692885i	$0.756339\pi$
$390$	0	0
$391$	−1.99186	−0.100733
$392$	−11.7795	−0.594953
$393$	0	0
$394$	18.3762	0.925782
$395$	29.6915	1.49394
$396$	0	0
$397$	−20.0063	−1.00409	−0.502043	−	0.864843i	$-0.667418\pi$
$397$	−20.0063	−1.00409	−0.502043	+	0.864843i	$0.667418\pi$
$398$	31.1067	1.55924
$399$	0	0
$400$	−17.5989	−0.879944
$401$	−1.05576	−0.0527220	−0.0263610	−	0.999652i	$-0.508392\pi$
$401$	−1.05576	−0.0527220	−0.0263610	+	0.999652i	$0.508392\pi$
$402$	0	0
$403$	20.5746	1.02489
$404$	3.41500	0.169903
$405$	0	0
$406$	−2.34037	−0.116151
$407$	−4.95434	−0.245578
$408$	0	0
$409$	11.8403	0.585463	0.292732	−	0.956195i	$-0.405436\pi$
$409$	11.8403	0.585463	0.292732	+	0.956195i	$0.405436\pi$
$410$	39.6881	1.96006
$411$	0	0
$412$	0.378672	0.0186558
$413$	−9.54284	−0.469572
$414$	0	0
$415$	−11.3329	−0.556311
$416$	6.36146	0.311896
$417$	0	0
$418$	6.58526	0.322096
$419$	−36.2135	−1.76915	−0.884573	−	0.466401i	$-0.845550\pi$
$419$	−36.2135	−1.76915	−0.884573	+	0.466401i	$0.845550\pi$
$420$	0	0
$421$	30.5979	1.49125	0.745625	−	0.666366i	$-0.232151\pi$
$421$	30.5979	1.49125	0.745625	+	0.666366i	$0.232151\pi$
$422$	40.3599	1.96469
$423$	0	0
$424$	7.01607	0.340730
$425$	−7.62781	−0.370003
$426$	0	0
$427$	2.45327	0.118722
$428$	3.46371	0.167425
$429$	0	0
$430$	14.3844	0.693678
$431$	1.70015	0.0818935	0.0409467	−	0.999161i	$-0.486963\pi$
$431$	1.70015	0.0818935	0.0409467	+	0.999161i	$0.486963\pi$
$432$	0	0
$433$	24.1484	1.16050	0.580248	−	0.814440i	$-0.302956\pi$
$433$	24.1484	1.16050	0.580248	+	0.814440i	$0.302956\pi$
$434$	−15.4069	−0.739554
$435$	0	0
$436$	4.01659	0.192360
$437$	6.50919	0.311377
$438$	0	0
$439$	6.50808	0.310614	0.155307	−	0.987866i	$-0.450363\pi$
$439$	6.50808	0.310614	0.155307	+	0.987866i	$0.450363\pi$
$440$	−4.91782	−0.234448
$441$	0	0
$442$	9.57178	0.455283
$443$	−25.4823	−1.21070	−0.605349	−	0.795960i	$-0.706967\pi$
$443$	−25.4823	−1.21070	−0.605349	+	0.795960i	$0.706967\pi$
$444$	0	0
$445$	2.67163	0.126647
$446$	20.5040	0.970892
$447$	0	0
$448$	9.22664	0.435918
$449$	21.8497	1.03115	0.515575	−	0.856845i	$-0.327578\pi$
$449$	21.8497	1.03115	0.515575	+	0.856845i	$0.327578\pi$
$450$	0	0
$451$	5.71577	0.269145
$452$	6.03832	0.284019
$453$	0	0
$454$	−21.3969	−1.00421
$455$	14.1359	0.662699
$456$	0	0
$457$	24.1438	1.12940	0.564700	−	0.825296i	$-0.308992\pi$
$457$	24.1438	1.12940	0.564700	+	0.825296i	$0.308992\pi$
$458$	−3.44487	−0.160968
$459$	0	0
$460$	1.08188	0.0504428
$461$	23.3197	1.08611	0.543054	−	0.839698i	$-0.317268\pi$
$461$	23.3197	1.08611	0.543054	+	0.839698i	$0.317268\pi$
$462$	0	0
$463$	24.5951	1.14303	0.571516	−	0.820591i	$-0.306356\pi$
$463$	24.5951	1.14303	0.571516	+	0.820591i	$0.306356\pi$
$464$	−4.59562	−0.213346
$465$	0	0
$466$	−24.8018	−1.14892
$467$	6.13480	0.283885	0.141942	−	0.989875i	$-0.454665\pi$
$467$	6.13480	0.283885	0.141942	+	0.989875i	$0.454665\pi$
$468$	0	0
$469$	8.43310	0.389404
$470$	8.63302	0.398211
$471$	0	0
$472$	−15.7694	−0.725847
$473$	2.07160	0.0952523
$474$	0	0
$475$	24.9269	1.14372
$476$	−1.10388	−0.0505963
$477$	0	0
$478$	39.8803	1.82408
$479$	13.6698	0.624588	0.312294	−	0.949986i	$-0.398903\pi$
$479$	13.6698	0.624588	0.312294	+	0.949986i	$0.398903\pi$
$480$	0	0
$481$	23.5328	1.07300
$482$	−27.5413	−1.25447
$483$	0	0
$484$	−3.84737	−0.174881
$485$	−11.3317	−0.514547
$486$	0	0
$487$	−3.77831	−0.171212	−0.0856058	−	0.996329i	$-0.527283\pi$
$487$	−3.77831	−0.171212	−0.0856058	+	0.996329i	$0.527283\pi$
$488$	4.05400	0.183516
$489$	0	0
$490$	21.3961	0.966576
$491$	5.21934	0.235545	0.117773	−	0.993041i	$-0.462425\pi$
$491$	5.21934	0.235545	0.117773	+	0.993041i	$0.462425\pi$
$492$	0	0
$493$	−1.99186	−0.0897089
$494$	−31.2796	−1.40733
$495$	0	0
$496$	−30.2534	−1.35842
$497$	14.7888	0.663370
$498$	0	0
$499$	−16.8243	−0.753157	−0.376579	−	0.926385i	$-0.622900\pi$
$499$	−16.8243	−0.753157	−0.376579	+	0.926385i	$0.622900\pi$
$500$	−1.26635	−0.0566329
$501$	0	0
$502$	−36.6645	−1.63642
$503$	17.5168	0.781036	0.390518	−	0.920595i	$-0.372296\pi$
$503$	17.5168	0.781036	0.390518	+	0.920595i	$0.372296\pi$
$504$	0	0
$505$	27.8707	1.24023
$506$	1.01169	0.0449750
$507$	0	0
$508$	−2.59191	−0.114997
$509$	39.8215	1.76506	0.882530	−	0.470257i	$-0.155839\pi$
$509$	39.8215	1.76506	0.882530	+	0.470257i	$0.155839\pi$
$510$	0	0
$511$	6.35038	0.280924
$512$	13.7648	0.608322
$513$	0	0
$514$	18.5712	0.819138
$515$	3.09045	0.136181
$516$	0	0
$517$	1.24330	0.0546804
$518$	−17.6221	−0.774269
$519$	0	0
$520$	23.3593	1.02437
$521$	−20.5118	−0.898637	−0.449319	−	0.893372i	$-0.648333\pi$
$521$	−20.5118	−0.898637	−0.449319	+	0.893372i	$0.648333\pi$
$522$	0	0
$523$	−0.217705	−0.00951957	−0.00475978	−	0.999989i	$-0.501515\pi$
$523$	−0.217705	−0.00951957	−0.00475978	+	0.999989i	$0.501515\pi$
$524$	1.02984	0.0449887
$525$	0	0
$526$	15.7034	0.684700
$527$	−13.1126	−0.571194
$528$	0	0
$529$	1.00000	0.0434783
$530$	−12.7439	−0.553560
$531$	0	0
$532$	3.60736	0.156399
$533$	−27.1495	−1.17598
$534$	0	0
$535$	28.2683	1.22215
$536$	13.9356	0.601926
$537$	0	0
$538$	−25.1333	−1.08357
$539$	3.08140	0.132725
$540$	0	0
$541$	−19.2734	−0.828630	−0.414315	−	0.910134i	$-0.635979\pi$
$541$	−19.2734	−0.828630	−0.414315	+	0.910134i	$0.635979\pi$
$542$	6.00366	0.257879
$543$	0	0
$544$	−4.05428	−0.173826
$545$	32.7805	1.40416
$546$	0	0
$547$	26.3839	1.12809	0.564047	−	0.825743i	$-0.309244\pi$
$547$	26.3839	1.12809	0.564047	+	0.825743i	$0.309244\pi$
$548$	0.937895	0.0400649
$549$	0	0
$550$	3.87424	0.165198
$551$	6.50919	0.277301
$552$	0	0
$553$	15.2096	0.646777
$554$	33.9142	1.44088
$555$	0	0
$556$	−5.46250	−0.231661
$557$	−6.89107	−0.291984	−0.145992	−	0.989286i	$-0.546637\pi$
$557$	−6.89107	−0.291984	−0.145992	+	0.989286i	$0.546637\pi$
$558$	0	0
$559$	−9.83997	−0.416186
$560$	−20.7857	−0.878356
$561$	0	0
$562$	0.0902525	0.00380707
$563$	32.6691	1.37684	0.688418	−	0.725314i	$-0.258305\pi$
$563$	32.6691	1.37684	0.688418	+	0.725314i	$0.258305\pi$
$564$	0	0
$565$	49.2803	2.07324
$566$	−17.8366	−0.749727
$567$	0	0
$568$	24.4384	1.02541
$569$	−13.8851	−0.582095	−0.291048	−	0.956709i	$-0.594004\pi$
$569$	−13.8851	−0.582095	−0.291048	+	0.956709i	$0.594004\pi$
$570$	0	0
$571$	−1.56148	−0.0653459	−0.0326729	−	0.999466i	$-0.510402\pi$
$571$	−1.56148	−0.0653459	−0.0326729	+	0.999466i	$0.510402\pi$
$572$	−0.748731	−0.0313060
$573$	0	0
$574$	20.3304	0.848574
$575$	3.82949	0.159701
$576$	0	0
$577$	−32.2938	−1.34441	−0.672205	−	0.740365i	$-0.734653\pi$
$577$	−32.2938	−1.34441	−0.672205	+	0.740365i	$0.734653\pi$
$578$	20.0382	0.833481
$579$	0	0
$580$	1.08188	0.0449225
$581$	−5.80533	−0.240846
$582$	0	0
$583$	−1.83534	−0.0760120
$584$	10.4939	0.434242
$585$	0	0
$586$	−25.1574	−1.03924
$587$	0.644254	0.0265912	0.0132956	−	0.999912i	$-0.495768\pi$
$587$	0.644254	0.0265912	0.0132956	+	0.999912i	$0.495768\pi$
$588$	0	0
$589$	42.8506	1.76563
$590$	28.6434	1.17923
$591$	0	0
$592$	−34.6032	−1.42218
$593$	42.7893	1.75715	0.878573	−	0.477609i	$-0.158496\pi$
$593$	42.7893	1.75715	0.878573	+	0.477609i	$0.158496\pi$
$594$	0	0
$595$	−9.00906	−0.369336
$596$	−7.00785	−0.287053
$597$	0	0
$598$	−4.80545	−0.196509
$599$	27.8106	1.13631	0.568155	−	0.822922i	$-0.307657\pi$
$599$	27.8106	1.13631	0.568155	+	0.822922i	$0.307657\pi$
$600$	0	0
$601$	−0.0213529	−0.000871004 0	−0.000435502	−	1.00000i	$-0.500139\pi$
$601$	−0.0213529	−0.000871004 0	−0.000435502		1.00000i	$0.500139\pi$
$602$	7.36846	0.300316
$603$	0	0
$604$	6.32104	0.257199
$605$	−31.3994	−1.27657
$606$	0	0
$607$	−3.95786	−0.160645	−0.0803223	−	0.996769i	$-0.525595\pi$
$607$	−3.95786	−0.160645	−0.0803223	+	0.996769i	$0.525595\pi$
$608$	13.2490	0.537317
$609$	0	0
$610$	−7.36364	−0.298145
$611$	−5.90560	−0.238915
$612$	0	0
$613$	−3.77872	−0.152621	−0.0763105	−	0.997084i	$-0.524314\pi$
$613$	−3.77872	−0.152621	−0.0763105	+	0.997084i	$0.524314\pi$
$614$	9.37819	0.378473
$615$	0	0
$616$	−2.51917	−0.101500
$617$	29.3301	1.18079	0.590393	−	0.807116i	$-0.298973\pi$
$617$	29.3301	1.18079	0.590393	+	0.807116i	$0.298973\pi$
$618$	0	0
$619$	−5.53617	−0.222517	−0.111259	−	0.993791i	$-0.535488\pi$
$619$	−5.53617	−0.222517	−0.111259	+	0.993791i	$0.535488\pi$
$620$	7.12210	0.286030
$621$	0	0
$622$	−17.6271	−0.706783
$623$	1.36855	0.0548299
$624$	0	0
$625$	−29.4825	−1.17930
$626$	−29.5616	−1.18152
$627$	0	0
$628$	−2.86813	−0.114451
$629$	−14.9979	−0.598006
$630$	0	0
$631$	−31.4015	−1.25007	−0.625037	−	0.780595i	$-0.714916\pi$
$631$	−31.4015	−1.25007	−0.625037	+	0.780595i	$0.714916\pi$
$632$	25.1336	0.999762
$633$	0	0
$634$	14.1640	0.562524
$635$	−21.1532	−0.839441
$636$	0	0
$637$	−14.6365	−0.579917
$638$	1.01169	0.0400531
$639$	0	0
$640$	−39.7906	−1.57286
$641$	9.00627	0.355726	0.177863	−	0.984055i	$-0.443082\pi$
$641$	9.00627	0.355726	0.177863	+	0.984055i	$0.443082\pi$
$642$	0	0
$643$	−49.6816	−1.95925	−0.979626	−	0.200829i	$-0.935636\pi$
$643$	−49.6816	−1.95925	−0.979626	+	0.200829i	$0.935636\pi$
$644$	0.554195	0.0218384
$645$	0	0
$646$	19.9351	0.784336
$647$	24.7647	0.973601	0.486801	−	0.873513i	$-0.338164\pi$
$647$	24.7647	0.973601	0.486801	+	0.873513i	$0.338164\pi$
$648$	0	0
$649$	4.12514	0.161926
$650$	−18.4024	−0.721802
$651$	0	0
$652$	−5.53504	−0.216769
$653$	41.6418	1.62957	0.814785	−	0.579764i	$-0.196855\pi$
$653$	41.6418	1.62957	0.814785	+	0.579764i	$0.196855\pi$
$654$	0	0
$655$	8.40478	0.328402
$656$	39.9213	1.55866
$657$	0	0
$658$	4.42229	0.172399
$659$	−31.4482	−1.22505	−0.612524	−	0.790452i	$-0.709846\pi$
$659$	−31.4482	−1.22505	−0.612524	+	0.790452i	$0.709846\pi$
$660$	0	0
$661$	−27.1590	−1.05636	−0.528181	−	0.849132i	$-0.677126\pi$
$661$	−27.1590	−1.05636	−0.528181	+	0.849132i	$0.677126\pi$
$662$	10.5433	0.409776
$663$	0	0
$664$	−9.59324	−0.372290
$665$	29.4407	1.14166
$666$	0	0
$667$	1.00000	0.0387202
$668$	−3.97562	−0.153821
$669$	0	0
$670$	−25.3124	−0.977905
$671$	−1.06049	−0.0409398
$672$	0	0
$673$	29.7953	1.14852	0.574262	−	0.818672i	$-0.305289\pi$
$673$	29.7953	1.14852	0.574262	+	0.818672i	$0.305289\pi$
$674$	−18.4520	−0.710743
$675$	0	0
$676$	−1.17676	−0.0452601
$677$	−35.8770	−1.37887	−0.689433	−	0.724349i	$-0.742140\pi$
$677$	−35.8770	−1.37887	−0.689433	+	0.724349i	$0.742140\pi$
$678$	0	0
$679$	−5.80472	−0.222765
$680$	−14.8874	−0.570905
$681$	0	0
$682$	6.66003	0.255026
$683$	−0.183601	−0.00702530	−0.00351265	−	0.999994i	$-0.501118\pi$
$683$	−0.183601	−0.00702530	−0.00351265	+	0.999994i	$0.501118\pi$
$684$	0	0
$685$	7.65441	0.292460
$686$	27.3428	1.04395
$687$	0	0
$688$	14.4689	0.551622
$689$	8.71775	0.332120
$690$	0	0
$691$	−8.73544	−0.332312	−0.166156	−	0.986099i	$-0.553136\pi$
$691$	−8.73544	−0.332312	−0.166156	+	0.986099i	$0.553136\pi$
$692$	−0.645005	−0.0245194
$693$	0	0
$694$	2.01480	0.0764808
$695$	−44.5809	−1.69105
$696$	0	0
$697$	17.3029	0.655395
$698$	−37.4165	−1.41623
$699$	0	0
$700$	2.12228	0.0802148
$701$	39.2626	1.48293	0.741464	−	0.670992i	$-0.234132\pi$
$701$	39.2626	1.48293	0.741464	+	0.670992i	$0.234132\pi$
$702$	0	0
$703$	49.0116	1.84851
$704$	−3.98846	−0.150321
$705$	0	0
$706$	18.6144	0.700562
$707$	14.2769	0.536937
$708$	0	0
$709$	28.7413	1.07940	0.539701	−	0.841857i	$-0.318537\pi$
$709$	28.7413	1.07940	0.539701	+	0.841857i	$0.318537\pi$
$710$	−44.3896	−1.66591
$711$	0	0
$712$	2.26152	0.0847539
$713$	6.58309	0.246539
$714$	0	0
$715$	−6.11059	−0.228523
$716$	5.45997	0.204049
$717$	0	0
$718$	−0.873417	−0.0325956
$719$	−9.64982	−0.359877	−0.179939	−	0.983678i	$-0.557590\pi$
$719$	−9.64982	−0.359877	−0.179939	+	0.983678i	$0.557590\pi$
$720$	0	0
$721$	1.58309	0.0589574
$722$	−35.9321	−1.33726
$723$	0	0
$724$	9.15755	0.340338
$725$	3.82949	0.142224
$726$	0	0
$727$	0.623800	0.0231355	0.0115677	−	0.999933i	$-0.496318\pi$
$727$	0.623800	0.0231355	0.0115677	+	0.999933i	$0.496318\pi$
$728$	11.9659	0.443486
$729$	0	0
$730$	−19.0610	−0.705481
$731$	6.27121	0.231949
$732$	0	0
$733$	14.3861	0.531362	0.265681	−	0.964061i	$-0.414403\pi$
$733$	14.3861	0.531362	0.265681	+	0.964061i	$0.414403\pi$
$734$	−19.7913	−0.730512
$735$	0	0
$736$	2.03543	0.0750268
$737$	−3.64543	−0.134281
$738$	0	0
$739$	−3.95277	−0.145405	−0.0727025	−	0.997354i	$-0.523162\pi$
$739$	−3.95277	−0.145405	−0.0727025	+	0.997354i	$0.523162\pi$
$740$	8.14610	0.299457
$741$	0	0
$742$	−6.52811	−0.239654
$743$	−17.5497	−0.643837	−0.321919	−	0.946767i	$-0.604328\pi$
$743$	−17.5497	−0.643837	−0.321919	+	0.946767i	$0.604328\pi$
$744$	0	0
$745$	−57.1930	−2.09539
$746$	41.2199	1.50917
$747$	0	0
$748$	0.477181	0.0174475
$749$	14.4805	0.529107
$750$	0	0
$751$	−36.8189	−1.34354	−0.671770	−	0.740760i	$-0.734465\pi$
$751$	−36.8189	−1.34354	−0.671770	+	0.740760i	$0.734465\pi$
$752$	8.68374	0.316663
$753$	0	0
$754$	−4.80545	−0.175004
$755$	51.5877	1.87747
$756$	0	0
$757$	−33.2754	−1.20942	−0.604708	−	0.796447i	$-0.706710\pi$
$757$	−33.2754	−1.20942	−0.604708	+	0.796447i	$0.706710\pi$
$758$	−36.2596	−1.31701
$759$	0	0
$760$	48.6504	1.76473
$761$	−37.2086	−1.34881	−0.674405	−	0.738361i	$-0.735600\pi$
$761$	−37.2086	−1.34881	−0.674405	+	0.738361i	$0.735600\pi$
$762$	0	0
$763$	16.7919	0.607908
$764$	−3.13465	−0.113408
$765$	0	0
$766$	6.99956	0.252904
$767$	−19.5941	−0.707503
$768$	0	0
$769$	54.5305	1.96642	0.983209	−	0.182481i	$-0.0584129\pi$
$769$	54.5305	1.96642	0.983209	+	0.182481i	$0.0584129\pi$
$770$	4.57580	0.164900
$771$	0	0
$772$	−0.555859	−0.0200058
$773$	25.4337	0.914787	0.457393	−	0.889264i	$-0.348783\pi$
$773$	25.4337	0.914787	0.457393	+	0.889264i	$0.348783\pi$
$774$	0	0
$775$	25.2099	0.905566
$776$	−9.59223	−0.344341
$777$	0	0
$778$	43.7322	1.56788
$779$	−56.5441	−2.02590
$780$	0	0
$781$	−6.39286	−0.228755
$782$	3.06261	0.109519
$783$	0	0
$784$	21.5218	0.768635
$785$	−23.4076	−0.835453
$786$	0	0
$787$	−47.1336	−1.68013	−0.840066	−	0.542484i	$-0.817484\pi$
$787$	−47.1336	−1.68013	−0.840066	+	0.542484i	$0.817484\pi$
$788$	−4.35146	−0.155014
$789$	0	0
$790$	−45.6524	−1.62424
$791$	25.2440	0.897574
$792$	0	0
$793$	5.03726	0.178878
$794$	30.7608	1.09166
$795$	0	0
$796$	−7.36600	−0.261081
$797$	−33.2908	−1.17922	−0.589611	−	0.807687i	$-0.700719\pi$
$797$	−33.2908	−1.17922	−0.589611	+	0.807687i	$0.700719\pi$
$798$	0	0
$799$	3.76376	0.133152
$800$	7.79464	0.275582
$801$	0	0
$802$	1.62329	0.0573204
$803$	−2.74512	−0.0968730
$804$	0	0
$805$	4.52294	0.159413
$806$	−31.6347	−1.11429
$807$	0	0
$808$	23.5924	0.829977
$809$	8.67983	0.305166	0.152583	−	0.988291i	$-0.451241\pi$
$809$	8.67983	0.305166	0.152583	+	0.988291i	$0.451241\pi$
$810$	0	0
$811$	−25.8833	−0.908887	−0.454444	−	0.890776i	$-0.650162\pi$
$811$	−25.8833	−0.908887	−0.454444	+	0.890776i	$0.650162\pi$
$812$	0.554195	0.0194484
$813$	0	0
$814$	7.61760	0.266997
$815$	−45.1729	−1.58234
$816$	0	0
$817$	−20.4936	−0.716982
$818$	−18.2051	−0.636527
$819$	0	0
$820$	−9.39806	−0.328195
$821$	−15.6591	−0.546507	−0.273253	−	0.961942i	$-0.588100\pi$
$821$	−15.6591	−0.546507	−0.273253	+	0.961942i	$0.588100\pi$
$822$	0	0
$823$	46.7430	1.62936	0.814679	−	0.579912i	$-0.196913\pi$
$823$	46.7430	1.62936	0.814679	+	0.579912i	$0.196913\pi$
$824$	2.61604	0.0911341
$825$	0	0
$826$	14.6727	0.510528
$827$	32.5350	1.13135	0.565676	−	0.824627i	$-0.308615\pi$
$827$	32.5350	1.13135	0.565676	+	0.824627i	$0.308615\pi$
$828$	0	0
$829$	11.9491	0.415008	0.207504	−	0.978234i	$-0.433466\pi$
$829$	11.9491	0.415008	0.207504	+	0.978234i	$0.433466\pi$
$830$	17.4250	0.604832
$831$	0	0
$832$	18.9449	0.656797
$833$	9.32810	0.323200
$834$	0	0
$835$	−32.4461	−1.12284
$836$	−1.55938	−0.0539322
$837$	0	0
$838$	55.6805	1.92345
$839$	17.8539	0.616384	0.308192	−	0.951324i	$-0.400276\pi$
$839$	17.8539	0.616384	0.308192	+	0.951324i	$0.400276\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	−47.0461	−1.62132
$843$	0	0
$844$	−9.55715	−0.328971
$845$	−9.60387	−0.330383
$846$	0	0
$847$	−16.0845	−0.552669
$848$	−12.8188	−0.440199
$849$	0	0
$850$	11.7282	0.402274
$851$	7.52960	0.258111
$852$	0	0
$853$	22.5605	0.772457	0.386228	−	0.922403i	$-0.373778\pi$
$853$	22.5605	0.772457	0.386228	+	0.922403i	$0.373778\pi$
$854$	−3.77205	−0.129077
$855$	0	0
$856$	23.9289	0.817874
$857$	−22.9279	−0.783204	−0.391602	−	0.920135i	$-0.628079\pi$
$857$	−22.9279	−0.783204	−0.391602	+	0.920135i	$0.628079\pi$
$858$	0	0
$859$	−2.33959	−0.0798257	−0.0399129	−	0.999203i	$-0.512708\pi$
$859$	−2.33959	−0.0798257	−0.0399129	+	0.999203i	$0.512708\pi$
$860$	−3.40620	−0.116150
$861$	0	0
$862$	−2.61409	−0.0890362
$863$	−33.2368	−1.13140	−0.565698	−	0.824613i	$-0.691393\pi$
$863$	−33.2368	−1.13140	−0.565698	+	0.824613i	$0.691393\pi$
$864$	0	0
$865$	−5.26406	−0.178983
$866$	−37.1296	−1.26171
$867$	0	0
$868$	3.64832	0.123832
$869$	−6.57473	−0.223033
$870$	0	0
$871$	17.3155	0.586715
$872$	27.7484	0.939681
$873$	0	0
$874$	−10.0083	−0.338535
$875$	−5.29415	−0.178975
$876$	0	0
$877$	19.6244	0.662669	0.331335	−	0.943513i	$-0.392501\pi$
$877$	19.6244	0.662669	0.331335	+	0.943513i	$0.392501\pi$
$878$	−10.0066	−0.337705
$879$	0	0
$880$	8.98516	0.302890
$881$	47.7909	1.61012	0.805058	−	0.593196i	$-0.202134\pi$
$881$	47.7909	1.61012	0.805058	+	0.593196i	$0.202134\pi$
$882$	0	0
$883$	9.68739	0.326007	0.163003	−	0.986626i	$-0.447882\pi$
$883$	9.68739	0.326007	0.163003	+	0.986626i	$0.447882\pi$
$884$	−2.26658	−0.0762333
$885$	0	0
$886$	39.1805	1.31629
$887$	10.9256	0.366845	0.183422	−	0.983034i	$-0.441282\pi$
$887$	10.9256	0.366845	0.183422	+	0.983034i	$0.441282\pi$
$888$	0	0
$889$	−10.8358	−0.363422
$890$	−4.10779	−0.137693
$891$	0	0
$892$	−4.85530	−0.162568
$893$	−12.2996	−0.411589
$894$	0	0
$895$	44.5603	1.48949
$896$	−20.3829	−0.680945
$897$	0	0
$898$	−33.5952	−1.12109
$899$	6.58309	0.219559
$900$	0	0
$901$	−5.55599	−0.185097
$902$	−8.78834	−0.292620
$903$	0	0
$904$	41.7155	1.38744
$905$	74.7372	2.48435
$906$	0	0
$907$	−31.7832	−1.05534	−0.527672	−	0.849448i	$-0.676935\pi$
$907$	−31.7832	−1.05534	−0.527672	+	0.849448i	$0.676935\pi$
$908$	5.06675	0.168146
$909$	0	0
$910$	−21.7347	−0.720499
$911$	−6.69493	−0.221813	−0.110907	−	0.993831i	$-0.535375\pi$
$911$	−6.69493	−0.221813	−0.110907	+	0.993831i	$0.535375\pi$
$912$	0	0
$913$	2.50951	0.0830525
$914$	−37.1226	−1.22791
$915$	0	0
$916$	0.815737	0.0269527
$917$	4.30538	0.142176
$918$	0	0
$919$	46.2250	1.52482	0.762411	−	0.647093i	$-0.224015\pi$
$919$	46.2250	1.52482	0.762411	+	0.647093i	$0.224015\pi$
$920$	7.47410	0.246414
$921$	0	0
$922$	−35.8555	−1.18084
$923$	30.3657	0.999499
$924$	0	0
$925$	28.8345	0.948073
$926$	−37.8164	−1.24273
$927$	0	0
$928$	2.03543	0.0668161
$929$	14.9133	0.489290	0.244645	−	0.969613i	$-0.421329\pi$
$929$	14.9133	0.489290	0.244645	+	0.969613i	$0.421329\pi$
$930$	0	0
$931$	−30.4833	−0.999048
$932$	5.87301	0.192377
$933$	0	0
$934$	−9.43263	−0.308645
$935$	3.89440	0.127361
$936$	0	0
$937$	−6.70593	−0.219073	−0.109537	−	0.993983i	$-0.534937\pi$
$937$	−6.70593	−0.219073	−0.109537	+	0.993983i	$0.534937\pi$
$938$	−12.9664	−0.423368
$939$	0	0
$940$	−2.04428	−0.0666771
$941$	33.3052	1.08572	0.542858	−	0.839824i	$-0.317342\pi$
$941$	33.3052	1.08572	0.542858	+	0.839824i	$0.317342\pi$
$942$	0	0
$943$	−8.68681	−0.282882
$944$	28.8117	0.937741
$945$	0	0
$946$	−3.18521	−0.103560
$947$	−27.4034	−0.890492	−0.445246	−	0.895408i	$-0.646884\pi$
$947$	−27.4034	−0.890492	−0.445246	+	0.895408i	$0.646884\pi$
$948$	0	0
$949$	13.0391	0.423268
$950$	−38.3266	−1.24348
$951$	0	0
$952$	−7.62611	−0.247164
$953$	1.09506	0.0354723	0.0177362	−	0.999843i	$-0.494354\pi$
$953$	1.09506	0.0354723	0.0177362	+	0.999843i	$0.494354\pi$
$954$	0	0
$955$	−25.5827	−0.827838
$956$	−9.44356	−0.305427
$957$	0	0
$958$	−21.0181	−0.679064
$959$	3.92100	0.126616
$960$	0	0
$961$	12.3371	0.397972
$962$	−36.1831	−1.16659
$963$	0	0
$964$	6.52173	0.210051
$965$	−4.53651	−0.146035
$966$	0	0
$967$	21.9241	0.705030	0.352515	−	0.935806i	$-0.385327\pi$
$967$	21.9241	0.705030	0.352515	+	0.935806i	$0.385327\pi$
$968$	−26.5794	−0.854295
$969$	0	0
$970$	17.4232	0.559425
$971$	0.0746247	0.00239482	0.00119741	−	0.999999i	$-0.499619\pi$
$971$	0.0746247	0.00239482	0.00119741	+	0.999999i	$0.499619\pi$
$972$	0	0
$973$	−22.8367	−0.732112
$974$	5.80938	0.186145
$975$	0	0
$976$	−7.40690	−0.237089
$977$	58.4019	1.86844	0.934221	−	0.356694i	$-0.116096\pi$
$977$	58.4019	1.86844	0.934221	+	0.356694i	$0.116096\pi$
$978$	0	0
$979$	−0.591592	−0.0189074
$980$	−5.06655	−0.161845
$981$	0	0
$982$	−8.02504	−0.256089
$983$	−25.8365	−0.824056	−0.412028	−	0.911171i	$-0.635179\pi$
$983$	−25.8365	−0.824056	−0.412028	+	0.911171i	$0.635179\pi$
$984$	0	0
$985$	−35.5134	−1.13155
$986$	3.06261	0.0975332
$987$	0	0
$988$	7.40694	0.235646
$989$	−3.14842	−0.100114
$990$	0	0
$991$	−19.3585	−0.614944	−0.307472	−	0.951557i	$-0.599483\pi$
$991$	−19.3585	−0.614944	−0.307472	+	0.951557i	$0.599483\pi$
$992$	13.3994	0.425431
$993$	0	0
$994$	−22.7387	−0.721229
$995$	−60.1160	−1.90580
$996$	0	0
$997$	−30.5596	−0.967833	−0.483917	−	0.875114i	$-0.660786\pi$
$997$	−30.5596	−0.967833	−0.483917	+	0.875114i	$0.660786\pi$
$998$	25.8683	0.818847
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.r.1.5		16
3.2	odd	2		2001.2.a.n.1.12	✓	16

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.n.1.12	✓	16	3.2	odd	2
6003.2.a.r.1.5		16	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)

\(f(q)\)	\(=\)	\(q-1.53756 q^{2} +0.364091 q^{4} +2.97145 q^{5} +1.52213 q^{7} +2.51531 q^{8} +O(q^{10})\)	\(q-1.53756 q^{2} +0.364091 q^{4} +2.97145 q^{5} +1.52213 q^{7} +2.51531 q^{8} -4.56878 q^{10} -0.657982 q^{11} +3.12537 q^{13} -2.34037 q^{14} -4.59562 q^{16} -1.99186 q^{17} +6.50919 q^{19} +1.08188 q^{20} +1.01169 q^{22} +1.00000 q^{23} +3.82949 q^{25} -4.80545 q^{26} +0.554195 q^{28} +1.00000 q^{29} +6.58309 q^{31} +2.03543 q^{32} +3.06261 q^{34} +4.52294 q^{35} +7.52960 q^{37} -10.0083 q^{38} +7.47410 q^{40} -8.68681 q^{41} -3.14842 q^{43} -0.239565 q^{44} -1.53756 q^{46} -1.88957 q^{47} -4.68311 q^{49} -5.88807 q^{50} +1.13792 q^{52} +2.78935 q^{53} -1.95516 q^{55} +3.82864 q^{56} -1.53756 q^{58} -6.26938 q^{59} +1.61173 q^{61} -10.1219 q^{62} +6.06165 q^{64} +9.28687 q^{65} +5.54031 q^{67} -0.725219 q^{68} -6.95429 q^{70} +9.71587 q^{71} +4.17202 q^{73} -11.5772 q^{74} +2.36994 q^{76} -1.00154 q^{77} +9.99227 q^{79} -13.6556 q^{80} +13.3565 q^{82} -3.81394 q^{83} -5.91871 q^{85} +4.84088 q^{86} -1.65503 q^{88} +0.899101 q^{89} +4.75723 q^{91} +0.364091 q^{92} +2.90533 q^{94} +19.3417 q^{95} -3.81354 q^{97} +7.20056 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7}+O(q^{10}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7	\( 16 q + q^{2} + 25 q^{4} - 3 q^{5} + 13 q^{7} + 11 q^{10} - 8 q^{11} + 19 q^{13} - 16 q^{14} + 31 q^{16} + 4 q^{17} + 19 q^{19} - 16 q^{20} + 6 q^{22} + 16 q^{23} + 23 q^{25} + 15 q^{26} + 18 q^{28} + 16 q^{29} + 24 q^{31} + 21 q^{32} - 9 q^{34} + 13 q^{35} + 26 q^{37} - 22 q^{40} + 15 q^{41} + 33 q^{43} - 6 q^{44} + q^{46} - 13 q^{47} + 41 q^{49} - 13 q^{50} - 26 q^{52} - 5 q^{53} + 9 q^{55} - 40 q^{56} + q^{58} - 2 q^{59} + 29 q^{61} + 32 q^{62} + 28 q^{64} - 18 q^{65} + 32 q^{67} + 26 q^{68} + 18 q^{70} - 29 q^{71} + 19 q^{73} + 16 q^{74} + 64 q^{76} + 21 q^{77} + 56 q^{79} + 14 q^{82} - 5 q^{83} + 16 q^{85} + 20 q^{86} + q^{88} - 7 q^{89} - 6 q^{91} + 25 q^{92} - 11 q^{94} - 39 q^{95} + 35 q^{97} + 109 q^{98}+O(q^{100}) \) 16 * q + q^2 + 25 * q^4 - 3 * q^5 + 13 * q^7 + 11 * q^10 - 8 * q^11 + 19 * q^13 - 16 * q^14 + 31 * q^16 + 4 * q^17 + 19 * q^19 - 16 * q^20 + 6 * q^22 + 16 * q^23 + 23 * q^25 + 15 * q^26 + 18 * q^28 + 16 * q^29 + 24 * q^31 + 21 * q^32 - 9 * q^34 + 13 * q^35 + 26 * q^37 - 22 * q^40 + 15 * q^41 + 33 * q^43 - 6 * q^44 + q^46 - 13 * q^47 + 41 * q^49 - 13 * q^50 - 26 * q^52 - 5 * q^53 + 9 * q^55 - 40 * q^56 + q^58 - 2 * q^59 + 29 * q^61 + 32 * q^62 + 28 * q^64 - 18 * q^65 + 32 * q^67 + 26 * q^68 + 18 * q^70 - 29 * q^71 + 19 * q^73 + 16 * q^74 + 64 * q^76 + 21 * q^77 + 56 * q^79 + 14 * q^82 - 5 * q^83 + 16 * q^85 + 20 * q^86 + q^88 - 7 * q^89 - 6 * q^91 + 25 * q^92 - 11 * q^94 - 39 * q^95 + 35 * q^97 + 109 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more