LMFDB - Embedded newform 6039.2.a.g.1.10

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [6039,2,Mod(1,6039)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(6039, base_ring=CyclotomicField(2))

chi = DirichletCharacter(H, H._module([0, 0, 0]))

N = Newforms(chi, 2, names="a")

//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code

chi := DirichletCharacter("6039.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 6039 = 3^{2} \cdot 11 \cdot 61 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	6039.a (trivial)

Newform invariants

comment: select newform

sage: f = N[0] # Warning: the index may be different

gp: f = lf[1] \\ Warning: the index may be different

Self dual:	yes
Analytic conductor:	$48.2216577807$
Analytic rank:	$1$
Dimension:	$13$
Coefficient field:	$\mathbb{Q}[x]/(x^{13} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{13} - 4 x^{12} - 11 x^{11} + 55 x^{10} + 32 x^{9} - 266 x^{8} + 13 x^{7} + 534 x^{6} - 141 x^{5} + \cdots - 11 $ x^13 - 4x^12 - 11x^11 + 55x^10 + 32x^9 - 266x^8 + 13x^7 + 534x^6 - 141x^5 - 404x^4 + 98x^3 + 118x^2 - 16x - 11
Coefficient ring:	$\Z[a_1, \ldots, a_{5}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 2013)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.10
Root			$-0.762797$ of defining polynomial
Character	$\chi$	$=$	6039.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.762797 q^{2} -1.41814 q^{4} +1.72725 q^{5} +1.91723 q^{7} -2.60735 q^{8} +O(q^{10})$	\(q+0.762797 q^{2} -1.41814 q^{4} +1.72725 q^{5} +1.91723 q^{7} -2.60735 q^{8} +1.31754 q^{10} +1.00000 q^{11} +2.75182 q^{13} +1.46246 q^{14} +0.847404 q^{16} -6.48473 q^{17} -3.45617 q^{19} -2.44948 q^{20} +0.762797 q^{22} +2.31351 q^{23} -2.01661 q^{25} +2.09908 q^{26} -2.71890 q^{28} -3.30867 q^{29} -8.07900 q^{31} +5.86109 q^{32} -4.94653 q^{34} +3.31153 q^{35} -6.14506 q^{37} -2.63636 q^{38} -4.50354 q^{40} -5.53101 q^{41} +12.6118 q^{43} -1.41814 q^{44} +1.76474 q^{46} -3.58917 q^{47} -3.32424 q^{49} -1.53827 q^{50} -3.90247 q^{52} -0.104112 q^{53} +1.72725 q^{55} -4.99888 q^{56} -2.52384 q^{58} -8.73414 q^{59} -1.00000 q^{61} -6.16263 q^{62} +2.77602 q^{64} +4.75308 q^{65} -1.77088 q^{67} +9.19626 q^{68} +2.52602 q^{70} -9.47861 q^{71} +6.67836 q^{73} -4.68743 q^{74} +4.90134 q^{76} +1.91723 q^{77} +13.9133 q^{79} +1.46368 q^{80} -4.21904 q^{82} -7.45072 q^{83} -11.2007 q^{85} +9.62023 q^{86} -2.60735 q^{88} -0.855757 q^{89} +5.27587 q^{91} -3.28088 q^{92} -2.73781 q^{94} -5.96967 q^{95} +7.45794 q^{97} -2.53572 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8	$ 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8} + 8 q^{10} + 13 q^{11} - 9 q^{13} - 19 q^{14} + 18 q^{16} - 7 q^{17} + 2 q^{19} - 15 q^{20} - 4 q^{22} - 23 q^{23} + 10 q^{25} - 8 q^{26} + 9 q^{28} - 16 q^{29} + 9 q^{31} - 29 q^{32} + 2 q^{34} - 16 q^{35} + 14 q^{37} - 8 q^{38} + 16 q^{40} - 19 q^{41} + 7 q^{43} + 12 q^{44} + 4 q^{46} - 26 q^{47} + 8 q^{49} + 15 q^{50} - 17 q^{52} - 18 q^{53} - 7 q^{55} - 44 q^{56} - q^{58} - 31 q^{59} - 13 q^{61} + 5 q^{62} - 17 q^{64} - 31 q^{65} + 14 q^{67} + 32 q^{68} - 20 q^{70} - 37 q^{71} - 16 q^{73} + 6 q^{74} - 7 q^{76} + 5 q^{77} - 17 q^{79} + 2 q^{80} - 2 q^{82} - 30 q^{83} - 16 q^{85} + 22 q^{86} - 15 q^{88} - 35 q^{89} - q^{91} - 24 q^{92} - 11 q^{94} - 13 q^{95} - q^{97} + q^{98}+O(q^{100}) $ 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8 + 8 * q^10 + 13 * q^11 - 9 * q^13 - 19 * q^14 + 18 * q^16 - 7 * q^17 + 2 * q^19 - 15 * q^20 - 4 * q^22 - 23 * q^23 + 10 * q^25 - 8 * q^26 + 9 * q^28 - 16 * q^29 + 9 * q^31 - 29 * q^32 + 2 * q^34 - 16 * q^35 + 14 * q^37 - 8 * q^38 + 16 * q^40 - 19 * q^41 + 7 * q^43 + 12 * q^44 + 4 * q^46 - 26 * q^47 + 8 * q^49 + 15 * q^50 - 17 * q^52 - 18 * q^53 - 7 * q^55 - 44 * q^56 - q^58 - 31 * q^59 - 13 * q^61 + 5 * q^62 - 17 * q^64 - 31 * q^65 + 14 * q^67 + 32 * q^68 - 20 * q^70 - 37 * q^71 - 16 * q^73 + 6 * q^74 - 7 * q^76 + 5 * q^77 - 17 * q^79 + 2 * q^80 - 2 * q^82 - 30 * q^83 - 16 * q^85 + 22 * q^86 - 15 * q^88 - 35 * q^89 - q^91 - 24 * q^92 - 11 * q^94 - 13 * q^95 - q^97 + 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 $

$2$	0.762797	0.539379	0.269689	−	0.962947i	$-0.413079\pi$
$2$	0.762797	0.539379	0.269689	+	0.962947i	$0.413079\pi$
$3$	0	0
$3$	0	0
$4$	−1.41814	−0.709070
$5$	1.72725	0.772449	0.386224	−	0.922405i	$-0.373779\pi$
$5$	1.72725	0.772449	0.386224	+	0.922405i	$0.373779\pi$
$6$	0	0
$7$	1.91723	0.724644	0.362322	−	0.932053i	$-0.381984\pi$
$7$	1.91723	0.724644	0.362322	+	0.932053i	$0.381984\pi$
$8$	−2.60735	−0.921837
$9$	0	0
$10$	1.31754	0.416643
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	2.75182	0.763218	0.381609	−	0.924324i	$-0.375370\pi$
$13$	2.75182	0.763218	0.381609	+	0.924324i	$0.375370\pi$
$14$	1.46246	0.390858
$15$	0	0
$16$	0.847404	0.211851
$17$	−6.48473	−1.57278	−0.786389	−	0.617732i	$-0.788052\pi$
$17$	−6.48473	−1.57278	−0.786389	+	0.617732i	$0.788052\pi$
$18$	0	0
$19$	−3.45617	−0.792900	−0.396450	−	0.918056i	$-0.629758\pi$
$19$	−3.45617	−0.792900	−0.396450	+	0.918056i	$0.629758\pi$
$20$	−2.44948	−0.547721
$21$	0	0
$22$	0.762797	0.162629
$23$	2.31351	0.482400	0.241200	−	0.970475i	$-0.422459\pi$
$23$	2.31351	0.482400	0.241200	+	0.970475i	$0.422459\pi$
$24$	0	0
$25$	−2.01661	−0.403323
$26$	2.09908	0.411664
$27$	0	0
$28$	−2.71890	−0.513824
$29$	−3.30867	−0.614404	−0.307202	−	0.951644i	$-0.599393\pi$
$29$	−3.30867	−0.614404	−0.307202	+	0.951644i	$0.599393\pi$
$30$	0	0
$31$	−8.07900	−1.45103	−0.725515	−	0.688206i	$-0.758398\pi$
$31$	−8.07900	−1.45103	−0.725515	+	0.688206i	$0.758398\pi$
$32$	5.86109	1.03610
$33$	0	0
$34$	−4.94653	−0.848323
$35$	3.31153	0.559750
$36$	0	0
$37$	−6.14506	−1.01024	−0.505121	−	0.863049i	$-0.668552\pi$
$37$	−6.14506	−1.01024	−0.505121	+	0.863049i	$0.668552\pi$
$38$	−2.63636	−0.427674
$39$	0	0
$40$	−4.50354	−0.712072
$41$	−5.53101	−0.863799	−0.431899	−	0.901922i	$-0.642156\pi$
$41$	−5.53101	−0.863799	−0.431899	+	0.901922i	$0.642156\pi$
$42$	0	0
$43$	12.6118	1.92328	0.961639	−	0.274317i	$-0.0884517\pi$
$43$	12.6118	1.92328	0.961639	+	0.274317i	$0.0884517\pi$
$44$	−1.41814	−0.213793
$45$	0	0
$46$	1.76474	0.260196
$47$	−3.58917	−0.523534	−0.261767	−	0.965131i	$-0.584305\pi$
$47$	−3.58917	−0.523534	−0.261767	+	0.965131i	$0.584305\pi$
$48$	0	0
$49$	−3.32424	−0.474891
$50$	−1.53827	−0.217544
$51$	0	0
$52$	−3.90247	−0.541175
$53$	−0.104112	−0.0143008	−0.00715042	−	0.999974i	$-0.502276\pi$
$53$	−0.104112	−0.0143008	−0.00715042	+	0.999974i	$0.502276\pi$
$54$	0	0
$55$	1.72725	0.232902
$56$	−4.99888	−0.668003
$57$	0	0
$58$	−2.52384	−0.331397
$59$	−8.73414	−1.13709	−0.568544	−	0.822653i	$-0.692493\pi$
$59$	−8.73414	−1.13709	−0.568544	+	0.822653i	$0.692493\pi$
$60$	0	0
$61$	−1.00000	−0.128037
$61$	−1.00000	−0.128037
$62$	−6.16263	−0.782655
$63$	0	0
$64$	2.77602	0.347002
$65$	4.75308	0.589547
$66$	0	0
$67$	−1.77088	−0.216347	−0.108174	−	0.994132i	$-0.534500\pi$
$67$	−1.77088	−0.216347	−0.108174	+	0.994132i	$0.534500\pi$
$68$	9.19626	1.11521
$69$	0	0
$70$	2.52602	0.301918
$71$	−9.47861	−1.12490	−0.562452	−	0.826830i	$-0.690142\pi$
$71$	−9.47861	−1.12490	−0.562452	+	0.826830i	$0.690142\pi$
$72$	0	0
$73$	6.67836	0.781642	0.390821	−	0.920467i	$-0.372191\pi$
$73$	6.67836	0.781642	0.390821	+	0.920467i	$0.372191\pi$
$74$	−4.68743	−0.544903
$75$	0	0
$76$	4.90134	0.562222
$77$	1.91723	0.218488
$78$	0	0
$79$	13.9133	1.56537	0.782683	−	0.622420i	$-0.213851\pi$
$79$	13.9133	1.56537	0.782683	+	0.622420i	$0.213851\pi$
$80$	1.46368	0.163644
$81$	0	0
$82$	−4.21904	−0.465915
$83$	−7.45072	−0.817823	−0.408911	−	0.912574i	$-0.634091\pi$
$83$	−7.45072	−0.817823	−0.408911	+	0.912574i	$0.634091\pi$
$84$	0	0
$85$	−11.2007	−1.21489
$86$	9.62023	1.03738
$87$	0	0
$88$	−2.60735	−0.277944
$89$	−0.855757	−0.0907100	−0.0453550	−	0.998971i	$-0.514442\pi$
$89$	−0.855757	−0.0907100	−0.0453550	+	0.998971i	$0.514442\pi$
$90$	0	0
$91$	5.27587	0.553061
$92$	−3.28088	−0.342055
$93$	0	0
$94$	−2.73781	−0.282383
$95$	−5.96967	−0.612475
$96$	0	0
$97$	7.45794	0.757239	0.378620	−	0.925552i	$-0.376399\pi$
$97$	7.45794	0.757239	0.378620	+	0.925552i	$0.376399\pi$
$98$	−2.53572	−0.256146
$99$	0	0
$100$	2.85984	0.285984
$101$	8.18158	0.814097	0.407049	−	0.913407i	$-0.366558\pi$
$101$	8.18158	0.814097	0.407049	+	0.913407i	$0.366558\pi$
$102$	0	0
$103$	−8.62316	−0.849666	−0.424833	−	0.905272i	$-0.639667\pi$
$103$	−8.62316	−0.849666	−0.424833	+	0.905272i	$0.639667\pi$
$104$	−7.17495	−0.703562
$105$	0	0
$106$	−0.0794161	−0.00771358
$107$	−11.4570	−1.10759	−0.553793	−	0.832655i	$-0.686820\pi$
$107$	−11.4570	−1.10759	−0.553793	+	0.832655i	$0.686820\pi$
$108$	0	0
$109$	12.9732	1.24260	0.621302	−	0.783571i	$-0.286604\pi$
$109$	12.9732	1.24260	0.621302	+	0.783571i	$0.286604\pi$
$110$	1.31754	0.125623
$111$	0	0
$112$	1.62467	0.153517
$113$	−6.75129	−0.635108	−0.317554	−	0.948240i	$-0.602862\pi$
$113$	−6.75129	−0.635108	−0.317554	+	0.948240i	$0.602862\pi$
$114$	0	0
$115$	3.99600	0.372629
$116$	4.69216	0.435656
$117$	0	0
$118$	−6.66238	−0.613322
$119$	−12.4327	−1.13970
$120$	0	0
$121$	1.00000	0.0909091
$122$	−0.762797	−0.0690604
$123$	0	0
$124$	11.4572	1.02888
$125$	−12.1194	−1.08400
$126$	0	0
$127$	−14.1860	−1.25880	−0.629401	−	0.777081i	$-0.716700\pi$
$127$	−14.1860	−1.25880	−0.629401	+	0.777081i	$0.716700\pi$
$128$	−9.60465	−0.848939
$129$	0	0
$130$	3.62563	0.317989
$131$	−20.8981	−1.82587	−0.912936	−	0.408102i	$-0.866191\pi$
$131$	−20.8981	−1.82587	−0.912936	+	0.408102i	$0.866191\pi$
$132$	0	0
$133$	−6.62627	−0.574570
$134$	−1.35082	−0.116693
$135$	0	0
$136$	16.9079	1.44984
$137$	12.1078	1.03444	0.517220	−	0.855853i	$-0.326967\pi$
$137$	12.1078	1.03444	0.517220	+	0.855853i	$0.326967\pi$
$138$	0	0
$139$	9.17332	0.778071	0.389035	−	0.921223i	$-0.372808\pi$
$139$	9.17332	0.778071	0.389035	+	0.921223i	$0.372808\pi$
$140$	−4.69621	−0.396902
$141$	0	0
$142$	−7.23026	−0.606750
$143$	2.75182	0.230119
$144$	0	0
$145$	−5.71489	−0.474596
$146$	5.09423	0.421602
$147$	0	0
$148$	8.71456	0.716332
$149$	3.92624	0.321650	0.160825	−	0.986983i	$-0.448584\pi$
$149$	3.92624	0.321650	0.160825	+	0.986983i	$0.448584\pi$
$150$	0	0
$151$	13.6702	1.11247	0.556234	−	0.831026i	$-0.312246\pi$
$151$	13.6702	1.11247	0.556234	+	0.831026i	$0.312246\pi$
$152$	9.01144	0.730924
$153$	0	0
$154$	1.46246	0.117848
$155$	−13.9544	−1.12085
$156$	0	0
$157$	11.2686	0.899336	0.449668	−	0.893196i	$-0.351542\pi$
$157$	11.2686	0.899336	0.449668	+	0.893196i	$0.351542\pi$
$158$	10.6130	0.844326
$159$	0	0
$160$	10.1236	0.800338
$161$	4.43552	0.349568
$162$	0	0
$163$	−12.9217	−1.01210	−0.506052	−	0.862503i	$-0.668896\pi$
$163$	−12.9217	−1.01210	−0.506052	+	0.862503i	$0.668896\pi$
$164$	7.84375	0.612494
$165$	0	0
$166$	−5.68339	−0.441116
$167$	−1.58752	−0.122846	−0.0614228	−	0.998112i	$-0.519564\pi$
$167$	−1.58752	−0.122846	−0.0614228	+	0.998112i	$0.519564\pi$
$168$	0	0
$169$	−5.42748	−0.417499
$170$	−8.54389	−0.655287
$171$	0	0
$172$	−17.8853	−1.36374
$173$	21.4445	1.63040	0.815199	−	0.579181i	$-0.196627\pi$
$173$	21.4445	1.63040	0.815199	+	0.579181i	$0.196627\pi$
$174$	0	0
$175$	−3.86631	−0.292265
$176$	0.847404	0.0638755
$177$	0	0
$178$	−0.652769	−0.0489271
$179$	−3.22902	−0.241349	−0.120674	−	0.992692i	$-0.538506\pi$
$179$	−3.22902	−0.241349	−0.120674	+	0.992692i	$0.538506\pi$
$180$	0	0
$181$	−4.13721	−0.307516	−0.153758	−	0.988109i	$-0.549138\pi$
$181$	−4.13721	−0.307516	−0.153758	+	0.988109i	$0.549138\pi$
$182$	4.02442	0.298310
$183$	0	0
$184$	−6.03212	−0.444694
$185$	−10.6140	−0.780360
$186$	0	0
$187$	−6.48473	−0.474210
$188$	5.08995	0.371222
$189$	0	0
$190$	−4.55364	−0.330356
$191$	−1.75283	−0.126830	−0.0634151	−	0.997987i	$-0.520199\pi$
$191$	−1.75283	−0.126830	−0.0634151	+	0.997987i	$0.520199\pi$
$192$	0	0
$193$	−10.5926	−0.762471	−0.381236	−	0.924478i	$-0.624501\pi$
$193$	−10.5926	−0.762471	−0.381236	+	0.924478i	$0.624501\pi$
$194$	5.68890	0.408439
$195$	0	0
$196$	4.71424	0.336731
$197$	−20.3558	−1.45029	−0.725144	−	0.688597i	$-0.758227\pi$
$197$	−20.3558	−1.45029	−0.725144	+	0.688597i	$0.758227\pi$
$198$	0	0
$199$	−15.8613	−1.12437	−0.562187	−	0.827010i	$-0.690040\pi$
$199$	−15.8613	−1.12437	−0.562187	+	0.827010i	$0.690040\pi$
$200$	5.25801	0.371798
$201$	0	0
$202$	6.24088	0.439107
$203$	−6.34347	−0.445224
$204$	0	0
$205$	−9.55343	−0.667240
$206$	−6.57772	−0.458292
$207$	0	0
$208$	2.33190	0.161688
$209$	−3.45617	−0.239068
$210$	0	0
$211$	−14.9592	−1.02983	−0.514917	−	0.857240i	$-0.672177\pi$
$211$	−14.9592	−1.02983	−0.514917	+	0.857240i	$0.672177\pi$
$212$	0.147645	0.0101403
$213$	0	0
$214$	−8.73933	−0.597408
$215$	21.7837	1.48563
$216$	0	0
$217$	−15.4893	−1.05148
$218$	9.89589	0.670234
$219$	0	0
$220$	−2.44948	−0.165144
$221$	−17.8448	−1.20037
$222$	0	0
$223$	17.5887	1.17782	0.588912	−	0.808197i	$-0.299556\pi$
$223$	17.5887	1.17782	0.588912	+	0.808197i	$0.299556\pi$
$224$	11.2370	0.750807
$225$	0	0
$226$	−5.14987	−0.342564
$227$	−3.11546	−0.206780	−0.103390	−	0.994641i	$-0.532969\pi$
$227$	−3.11546	−0.206780	−0.103390	+	0.994641i	$0.532969\pi$
$228$	0	0
$229$	0.908295	0.0600218	0.0300109	−	0.999550i	$-0.490446\pi$
$229$	0.908295	0.0600218	0.0300109	+	0.999550i	$0.490446\pi$
$230$	3.04814	0.200988
$231$	0	0
$232$	8.62685	0.566380
$233$	−4.76619	−0.312244	−0.156122	−	0.987738i	$-0.549899\pi$
$233$	−4.76619	−0.312244	−0.156122	+	0.987738i	$0.549899\pi$
$234$	0	0
$235$	−6.19939	−0.404403
$236$	12.3862	0.806276
$237$	0	0
$238$	−9.48363	−0.614732
$239$	−3.86803	−0.250202	−0.125101	−	0.992144i	$-0.539926\pi$
$239$	−3.86803	−0.250202	−0.125101	+	0.992144i	$0.539926\pi$
$240$	0	0
$241$	18.5264	1.19339	0.596695	−	0.802468i	$-0.296480\pi$
$241$	18.5264	1.19339	0.596695	+	0.802468i	$0.296480\pi$
$242$	0.762797	0.0490345
$243$	0	0
$244$	1.41814	0.0907871
$245$	−5.74178	−0.366829
$246$	0	0
$247$	−9.51077	−0.605156
$248$	21.0648	1.33761
$249$	0	0
$250$	−9.24467	−0.584684
$251$	11.3331	0.715342	0.357671	−	0.933848i	$-0.383571\pi$
$251$	11.3331	0.715342	0.357671	+	0.933848i	$0.383571\pi$
$252$	0	0
$253$	2.31351	0.145449
$254$	−10.8210	−0.678971
$255$	0	0
$256$	−12.8784	−0.804902
$257$	16.7156	1.04269	0.521345	−	0.853346i	$-0.325430\pi$
$257$	16.7156	1.04269	0.521345	+	0.853346i	$0.325430\pi$
$258$	0	0
$259$	−11.7815	−0.732066
$260$	−6.74053	−0.418030
$261$	0	0
$262$	−15.9410	−0.984837
$263$	−26.1408	−1.61191	−0.805954	−	0.591978i	$-0.798347\pi$
$263$	−26.1408	−1.61191	−0.805954	+	0.591978i	$0.798347\pi$
$264$	0	0
$265$	−0.179827	−0.0110467
$266$	−5.05450	−0.309911
$267$	0	0
$268$	2.51135	0.153405
$269$	−9.37640	−0.571689	−0.285845	−	0.958276i	$-0.592274\pi$
$269$	−9.37640	−0.571689	−0.285845	+	0.958276i	$0.592274\pi$
$270$	0	0
$271$	−4.16170	−0.252805	−0.126403	−	0.991979i	$-0.540343\pi$
$271$	−4.16170	−0.252805	−0.126403	+	0.991979i	$0.540343\pi$
$272$	−5.49519	−0.333195
$273$	0	0
$274$	9.23580	0.557955
$275$	−2.01661	−0.121606
$276$	0	0
$277$	−18.3125	−1.10029	−0.550145	−	0.835069i	$-0.685428\pi$
$277$	−18.3125	−1.10029	−0.550145	+	0.835069i	$0.685428\pi$
$278$	6.99738	0.419675
$279$	0	0
$280$	−8.63431	−0.515998
$281$	−7.79901	−0.465250	−0.232625	−	0.972567i	$-0.574731\pi$
$281$	−7.79901	−0.465250	−0.232625	+	0.972567i	$0.574731\pi$
$282$	0	0
$283$	−25.6839	−1.52675	−0.763374	−	0.645957i	$-0.776459\pi$
$283$	−25.6839	−1.52675	−0.763374	+	0.645957i	$0.776459\pi$
$284$	13.4420	0.797636
$285$	0	0
$286$	2.09908	0.124121
$287$	−10.6042	−0.625947
$288$	0	0
$289$	25.0517	1.47363
$290$	−4.35930	−0.255987
$291$	0	0
$292$	−9.47085	−0.554239
$293$	−19.7833	−1.15575	−0.577877	−	0.816124i	$-0.696119\pi$
$293$	−19.7833	−1.15575	−0.577877	+	0.816124i	$0.696119\pi$
$294$	0	0
$295$	−15.0860	−0.878343
$296$	16.0223	0.931278
$297$	0	0
$298$	2.99493	0.173491
$299$	6.36636	0.368176
$300$	0	0
$301$	24.1797	1.39369
$302$	10.4276	0.600042
$303$	0	0
$304$	−2.92877	−0.167977
$305$	−1.72725	−0.0989020
$306$	0	0
$307$	−8.16578	−0.466046	−0.233023	−	0.972471i	$-0.574862\pi$
$307$	−8.16578	−0.466046	−0.233023	+	0.972471i	$0.574862\pi$
$308$	−2.71890	−0.154924
$309$	0	0
$310$	−10.6444	−0.604561
$311$	8.62884	0.489297	0.244648	−	0.969612i	$-0.421328\pi$
$311$	8.62884	0.489297	0.244648	+	0.969612i	$0.421328\pi$
$312$	0	0
$313$	0.789308	0.0446143	0.0223072	−	0.999751i	$-0.492899\pi$
$313$	0.789308	0.0446143	0.0223072	+	0.999751i	$0.492899\pi$
$314$	8.59569	0.485083
$315$	0	0
$316$	−19.7310	−1.10995
$317$	−9.10176	−0.511206	−0.255603	−	0.966782i	$-0.582274\pi$
$317$	−9.10176	−0.511206	−0.255603	+	0.966782i	$0.582274\pi$
$318$	0	0
$319$	−3.30867	−0.185250
$320$	4.79487	0.268041
$321$	0	0
$322$	3.38340	0.188550
$323$	22.4123	1.24706
$324$	0	0
$325$	−5.54936	−0.307823
$326$	−9.85661	−0.545907
$327$	0	0
$328$	14.4213	0.796281
$329$	−6.88125	−0.379376
$330$	0	0
$331$	27.3731	1.50456	0.752282	−	0.658842i	$-0.228953\pi$
$331$	27.3731	1.50456	0.752282	+	0.658842i	$0.228953\pi$
$332$	10.5662	0.579894
$333$	0	0
$334$	−1.21095	−0.0662604
$335$	−3.05875	−0.167117
$336$	0	0
$337$	−10.4004	−0.566547	−0.283274	−	0.959039i	$-0.591420\pi$
$337$	−10.4004	−0.566547	−0.283274	+	0.959039i	$0.591420\pi$
$338$	−4.14007	−0.225190
$339$	0	0
$340$	15.8842	0.861443
$341$	−8.07900	−0.437502
$342$	0	0
$343$	−19.7939	−1.06877
$344$	−32.8833	−1.77295
$345$	0	0
$346$	16.3578	0.879403
$347$	−5.32087	−0.285639	−0.142820	−	0.989749i	$-0.545617\pi$
$347$	−5.32087	−0.285639	−0.142820	+	0.989749i	$0.545617\pi$
$348$	0	0
$349$	29.5144	1.57987	0.789934	−	0.613191i	$-0.210115\pi$
$349$	29.5144	1.57987	0.789934	+	0.613191i	$0.210115\pi$
$350$	−2.94921	−0.157642
$351$	0	0
$352$	5.86109	0.312397
$353$	−12.3814	−0.658993	−0.329497	−	0.944157i	$-0.606879\pi$
$353$	−12.3814	−0.658993	−0.329497	+	0.944157i	$0.606879\pi$
$354$	0	0
$355$	−16.3719	−0.868931
$356$	1.21358	0.0643198
$357$	0	0
$358$	−2.46309	−0.130178
$359$	−31.3625	−1.65525	−0.827624	−	0.561284i	$-0.810308\pi$
$359$	−31.3625	−1.65525	−0.827624	+	0.561284i	$0.810308\pi$
$360$	0	0
$361$	−7.05487	−0.371309
$362$	−3.15585	−0.165868
$363$	0	0
$364$	−7.48192	−0.392159
$365$	11.5352	0.603779
$366$	0	0
$367$	7.87892	0.411276	0.205638	−	0.978628i	$-0.434073\pi$
$367$	7.87892	0.411276	0.205638	+	0.978628i	$0.434073\pi$
$368$	1.96048	0.102197
$369$	0	0
$370$	−8.09636	−0.420910
$371$	−0.199606	−0.0103630
$372$	0	0
$373$	0.196499	0.0101744	0.00508718	−	0.999987i	$-0.498381\pi$
$373$	0.196499	0.0101744	0.00508718	+	0.999987i	$0.498381\pi$
$374$	−4.94653	−0.255779
$375$	0	0
$376$	9.35821	0.482613
$377$	−9.10486	−0.468924
$378$	0	0
$379$	30.1419	1.54828	0.774142	−	0.633012i	$-0.218182\pi$
$379$	30.1419	1.54828	0.774142	+	0.633012i	$0.218182\pi$
$380$	8.46583	0.434288
$381$	0	0
$382$	−1.33705	−0.0684095
$383$	18.7592	0.958551	0.479276	−	0.877664i	$-0.340899\pi$
$383$	18.7592	0.958551	0.479276	+	0.877664i	$0.340899\pi$
$384$	0	0
$385$	3.31153	0.168771
$386$	−8.08000	−0.411261
$387$	0	0
$388$	−10.5764	−0.536936
$389$	12.1555	0.616310	0.308155	−	0.951336i	$-0.400289\pi$
$389$	12.1555	0.616310	0.308155	+	0.951336i	$0.400289\pi$
$390$	0	0
$391$	−15.0025	−0.758708
$392$	8.66744	0.437772
$393$	0	0
$394$	−15.5273	−0.782255
$395$	24.0317	1.20917
$396$	0	0
$397$	−39.5260	−1.98375	−0.991877	−	0.127201i	$-0.959401\pi$
$397$	−39.5260	−1.98375	−0.991877	+	0.127201i	$0.959401\pi$
$398$	−12.0989	−0.606464
$399$	0	0
$400$	−1.70889	−0.0854443
$401$	−21.2483	−1.06109	−0.530545	−	0.847657i	$-0.678013\pi$
$401$	−21.2483	−1.06109	−0.530545	+	0.847657i	$0.678013\pi$
$402$	0	0
$403$	−22.2319	−1.10745
$404$	−11.6026	−0.577252
$405$	0	0
$406$	−4.83878	−0.240145
$407$	−6.14506	−0.304599
$408$	0	0
$409$	15.1365	0.748453	0.374227	−	0.927337i	$-0.377908\pi$
$409$	15.1365	0.748453	0.374227	+	0.927337i	$0.377908\pi$
$410$	−7.28733	−0.359895
$411$	0	0
$412$	12.2289	0.602473
$413$	−16.7453	−0.823985
$414$	0	0
$415$	−12.8692	−0.631726
$416$	16.1287	0.790773
$417$	0	0
$418$	−2.63636	−0.128948
$419$	30.8661	1.50791	0.753955	−	0.656927i	$-0.228144\pi$
$419$	30.8661	1.50791	0.753955	+	0.656927i	$0.228144\pi$
$420$	0	0
$421$	−29.2680	−1.42644	−0.713219	−	0.700942i	$-0.752763\pi$
$421$	−29.2680	−1.42644	−0.713219	+	0.700942i	$0.752763\pi$
$422$	−11.4108	−0.555471
$423$	0	0
$424$	0.271455	0.0131830
$425$	13.0772	0.634337
$426$	0	0
$427$	−1.91723	−0.0927811
$428$	16.2476	0.785356
$429$	0	0
$430$	16.6165	0.801320
$431$	34.3344	1.65383	0.826915	−	0.562327i	$-0.190094\pi$
$431$	34.3344	1.65383	0.826915	+	0.562327i	$0.190094\pi$
$432$	0	0
$433$	−21.3675	−1.02686	−0.513428	−	0.858133i	$-0.671625\pi$
$433$	−21.3675	−1.02686	−0.513428	+	0.858133i	$0.671625\pi$
$434$	−11.8152	−0.567146
$435$	0	0
$436$	−18.3978	−0.881093
$437$	−7.99588	−0.382495
$438$	0	0
$439$	15.8079	0.754468	0.377234	−	0.926118i	$-0.376875\pi$
$439$	15.8079	0.754468	0.377234	+	0.926118i	$0.376875\pi$
$440$	−4.50354	−0.214698
$441$	0	0
$442$	−13.6120	−0.647455
$443$	−22.0304	−1.04670	−0.523348	−	0.852119i	$-0.675317\pi$
$443$	−22.0304	−1.04670	−0.523348	+	0.852119i	$0.675317\pi$
$444$	0	0
$445$	−1.47810	−0.0700689
$446$	13.4166	0.635293
$447$	0	0
$448$	5.32226	0.251453
$449$	−3.14206	−0.148283	−0.0741414	−	0.997248i	$-0.523622\pi$
$449$	−3.14206	−0.148283	−0.0741414	+	0.997248i	$0.523622\pi$
$450$	0	0
$451$	−5.53101	−0.260445
$452$	9.57428	0.450336
$453$	0	0
$454$	−2.37646	−0.111533
$455$	9.11273	0.427211
$456$	0	0
$457$	−31.5461	−1.47567	−0.737833	−	0.674983i	$-0.764151\pi$
$457$	−31.5461	−1.47567	−0.737833	+	0.674983i	$0.764151\pi$
$458$	0.692845	0.0323745
$459$	0	0
$460$	−5.66689	−0.264220
$461$	4.98596	0.232219	0.116110	−	0.993236i	$-0.462958\pi$
$461$	4.98596	0.232219	0.116110	+	0.993236i	$0.462958\pi$
$462$	0	0
$463$	32.0189	1.48804	0.744022	−	0.668156i	$-0.232916\pi$
$463$	32.0189	1.48804	0.744022	+	0.668156i	$0.232916\pi$
$464$	−2.80378	−0.130162
$465$	0	0
$466$	−3.63564	−0.168418
$467$	−5.01088	−0.231876	−0.115938	−	0.993256i	$-0.536987\pi$
$467$	−5.01088	−0.231876	−0.115938	+	0.993256i	$0.536987\pi$
$468$	0	0
$469$	−3.39518	−0.156775
$470$	−4.72887	−0.218127
$471$	0	0
$472$	22.7730	1.04821
$473$	12.6118	0.579890
$474$	0	0
$475$	6.96976	0.319795
$476$	17.6313	0.808130
$477$	0	0
$478$	−2.95053	−0.134954
$479$	28.8185	1.31675	0.658377	−	0.752689i	$-0.271243\pi$
$479$	28.8185	1.31675	0.658377	+	0.752689i	$0.271243\pi$
$480$	0	0
$481$	−16.9101	−0.771034
$482$	14.1319	0.643689
$483$	0	0
$484$	−1.41814	−0.0644609
$485$	12.8817	0.584929
$486$	0	0
$487$	12.5274	0.567673	0.283836	−	0.958873i	$-0.408393\pi$
$487$	12.5274	0.567673	0.283836	+	0.958873i	$0.408393\pi$
$488$	2.60735	0.118029
$489$	0	0
$490$	−4.37982	−0.197860
$491$	2.56583	0.115794	0.0578971	−	0.998323i	$-0.481560\pi$
$491$	2.56583	0.115794	0.0578971	+	0.998323i	$0.481560\pi$
$492$	0	0
$493$	21.4558	0.966321
$494$	−7.25478	−0.326408
$495$	0	0
$496$	−6.84617	−0.307402
$497$	−18.1727	−0.815155
$498$	0	0
$499$	41.8993	1.87567	0.937835	−	0.347082i	$-0.112828\pi$
$499$	41.8993	1.87567	0.937835	+	0.347082i	$0.112828\pi$
$500$	17.1871	0.768629
$501$	0	0
$502$	8.64489	0.385840
$503$	33.7140	1.50323	0.751617	−	0.659599i	$-0.229274\pi$
$503$	33.7140	1.50323	0.751617	+	0.659599i	$0.229274\pi$
$504$	0	0
$505$	14.1316	0.628849
$506$	1.76474	0.0784521
$507$	0	0
$508$	20.1177	0.892579
$509$	−40.7299	−1.80532	−0.902660	−	0.430354i	$-0.858389\pi$
$509$	−40.7299	−1.80532	−0.902660	+	0.430354i	$0.858389\pi$
$510$	0	0
$511$	12.8039	0.566412
$512$	9.38567	0.414792
$513$	0	0
$514$	12.7506	0.562405
$515$	−14.8943	−0.656323
$516$	0	0
$517$	−3.58917	−0.157851
$518$	−8.98688	−0.394861
$519$	0	0
$520$	−12.3929	−0.543466
$521$	32.4669	1.42240	0.711200	−	0.702990i	$-0.248152\pi$
$521$	32.4669	1.42240	0.711200	+	0.702990i	$0.248152\pi$
$522$	0	0
$523$	9.66736	0.422724	0.211362	−	0.977408i	$-0.432210\pi$
$523$	9.66736	0.422724	0.211362	+	0.977408i	$0.432210\pi$
$524$	29.6364	1.29467
$525$	0	0
$526$	−19.9401	−0.869430
$527$	52.3901	2.28215
$528$	0	0
$529$	−17.6477	−0.767291
$530$	−0.137171	−0.00595834
$531$	0	0
$532$	9.39698	0.407411
$533$	−15.2204	−0.659267
$534$	0	0
$535$	−19.7890	−0.855553
$536$	4.61730	0.199437
$537$	0	0
$538$	−7.15229	−0.308357
$539$	−3.32424	−0.143185
$540$	0	0
$541$	−38.6010	−1.65959	−0.829794	−	0.558070i	$-0.811542\pi$
$541$	−38.6010	−1.65959	−0.829794	+	0.558070i	$0.811542\pi$
$542$	−3.17453	−0.136358
$543$	0	0
$544$	−38.0076	−1.62956
$545$	22.4079	0.959848
$546$	0	0
$547$	−21.8277	−0.933284	−0.466642	−	0.884446i	$-0.654536\pi$
$547$	−21.8277	−0.933284	−0.466642	+	0.884446i	$0.654536\pi$
$548$	−17.1706	−0.733491
$549$	0	0
$550$	−1.53827	−0.0655919
$551$	11.4353	0.487161
$552$	0	0
$553$	26.6749	1.13433
$554$	−13.9687	−0.593474
$555$	0	0
$556$	−13.0091	−0.551707
$557$	46.5571	1.97269	0.986344	−	0.164697i	$-0.0526646\pi$
$557$	46.5571	1.97269	0.986344	+	0.164697i	$0.0526646\pi$
$558$	0	0
$559$	34.7054	1.46788
$560$	2.80620	0.118584
$561$	0	0
$562$	−5.94906	−0.250946
$563$	9.16069	0.386077	0.193039	−	0.981191i	$-0.438166\pi$
$563$	9.16069	0.386077	0.193039	+	0.981191i	$0.438166\pi$
$564$	0	0
$565$	−11.6612	−0.490589
$566$	−19.5916	−0.823496
$567$	0	0
$568$	24.7140	1.03698
$569$	12.5856	0.527615	0.263807	−	0.964575i	$-0.415022\pi$
$569$	12.5856	0.527615	0.263807	+	0.964575i	$0.415022\pi$
$570$	0	0
$571$	−28.7880	−1.20474	−0.602370	−	0.798217i	$-0.705777\pi$
$571$	−28.7880	−1.20474	−0.602370	+	0.798217i	$0.705777\pi$
$572$	−3.90247	−0.163170
$573$	0	0
$574$	−8.08886	−0.337622
$575$	−4.66545	−0.194563
$576$	0	0
$577$	8.90192	0.370592	0.185296	−	0.982683i	$-0.440676\pi$
$577$	8.90192	0.370592	0.185296	+	0.982683i	$0.440676\pi$
$578$	19.1094	0.794845
$579$	0	0
$580$	8.10452	0.336522
$581$	−14.2847	−0.592630
$582$	0	0
$583$	−0.104112	−0.00431187
$584$	−17.4128	−0.720547
$585$	0	0
$586$	−15.0907	−0.623389
$587$	−21.9183	−0.904665	−0.452333	−	0.891849i	$-0.649408\pi$
$587$	−21.9183	−0.904665	−0.452333	+	0.891849i	$0.649408\pi$
$588$	0	0
$589$	27.9224	1.15052
$590$	−11.5076	−0.473760
$591$	0	0
$592$	−5.20735	−0.214021
$593$	−16.0489	−0.659049	−0.329525	−	0.944147i	$-0.606889\pi$
$593$	−16.0489	−0.659049	−0.329525	+	0.944147i	$0.606889\pi$
$594$	0	0
$595$	−21.4744	−0.880363
$596$	−5.56796	−0.228073
$597$	0	0
$598$	4.85624	0.198586
$599$	17.2895	0.706429	0.353215	−	0.935542i	$-0.385089\pi$
$599$	17.2895	0.706429	0.353215	+	0.935542i	$0.385089\pi$
$600$	0	0
$601$	0.187860	0.00766297	0.00383149	−	0.999993i	$-0.498780\pi$
$601$	0.187860	0.00766297	0.00383149	+	0.999993i	$0.498780\pi$
$602$	18.4442	0.751728
$603$	0	0
$604$	−19.3863	−0.788818
$605$	1.72725	0.0702226
$606$	0	0
$607$	−11.4077	−0.463023	−0.231511	−	0.972832i	$-0.574367\pi$
$607$	−11.4077	−0.463023	−0.231511	+	0.972832i	$0.574367\pi$
$608$	−20.2569	−0.821528
$609$	0	0
$610$	−1.31754	−0.0533456
$611$	−9.87675	−0.399570
$612$	0	0
$613$	21.9824	0.887859	0.443930	−	0.896062i	$-0.353584\pi$
$613$	21.9824	0.887859	0.443930	+	0.896062i	$0.353584\pi$
$614$	−6.22883	−0.251375
$615$	0	0
$616$	−4.99888	−0.201411
$617$	37.5118	1.51017	0.755084	−	0.655628i	$-0.227596\pi$
$617$	37.5118	1.51017	0.755084	+	0.655628i	$0.227596\pi$
$618$	0	0
$619$	3.44007	0.138268	0.0691341	−	0.997607i	$-0.477976\pi$
$619$	3.44007	0.138268	0.0691341	+	0.997607i	$0.477976\pi$
$620$	19.7893	0.794759
$621$	0	0
$622$	6.58205	0.263916
$623$	−1.64068	−0.0657325
$624$	0	0
$625$	−10.8502	−0.434008
$626$	0.602082	0.0240640
$627$	0	0
$628$	−15.9805	−0.637692
$629$	39.8491	1.58889
$630$	0	0
$631$	−22.7853	−0.907069	−0.453534	−	0.891239i	$-0.649837\pi$
$631$	−22.7853	−0.907069	−0.453534	+	0.891239i	$0.649837\pi$
$632$	−36.2768	−1.44301
$633$	0	0
$634$	−6.94279	−0.275734
$635$	−24.5027	−0.972360
$636$	0	0
$637$	−9.14771	−0.362445
$638$	−2.52384	−0.0999198
$639$	0	0
$640$	−16.5896	−0.655762
$641$	−2.37633	−0.0938594	−0.0469297	−	0.998898i	$-0.514944\pi$
$641$	−2.37633	−0.0938594	−0.0469297	+	0.998898i	$0.514944\pi$
$642$	0	0
$643$	36.5580	1.44171	0.720854	−	0.693087i	$-0.243750\pi$
$643$	36.5580	1.44171	0.720854	+	0.693087i	$0.243750\pi$
$644$	−6.29019	−0.247868
$645$	0	0
$646$	17.0961	0.672636
$647$	42.3837	1.66627	0.833137	−	0.553067i	$-0.186543\pi$
$647$	42.3837	1.66627	0.833137	+	0.553067i	$0.186543\pi$
$648$	0	0
$649$	−8.73414	−0.342845
$650$	−4.23303	−0.166033
$651$	0	0
$652$	18.3247	0.717653
$653$	−7.06785	−0.276586	−0.138293	−	0.990391i	$-0.544162\pi$
$653$	−7.06785	−0.276586	−0.138293	+	0.990391i	$0.544162\pi$
$654$	0	0
$655$	−36.0962	−1.41039
$656$	−4.68700	−0.182997
$657$	0	0
$658$	−5.24900	−0.204627
$659$	−7.19336	−0.280214	−0.140107	−	0.990136i	$-0.544745\pi$
$659$	−7.19336	−0.280214	−0.140107	+	0.990136i	$0.544745\pi$
$660$	0	0
$661$	−34.8572	−1.35579	−0.677894	−	0.735160i	$-0.737107\pi$
$661$	−34.8572	−1.35579	−0.677894	+	0.735160i	$0.737107\pi$
$662$	20.8801	0.811530
$663$	0	0
$664$	19.4266	0.753899
$665$	−11.4452	−0.443826
$666$	0	0
$667$	−7.65463	−0.296388
$668$	2.25132	0.0871062
$669$	0	0
$670$	−2.33320	−0.0901395
$671$	−1.00000	−0.0386046
$672$	0	0
$673$	−26.2266	−1.01096	−0.505480	−	0.862838i	$-0.668685\pi$
$673$	−26.2266	−1.01096	−0.505480	+	0.862838i	$0.668685\pi$
$674$	−7.93341	−0.305584
$675$	0	0
$676$	7.69693	0.296036
$677$	−6.77726	−0.260471	−0.130236	−	0.991483i	$-0.541573\pi$
$677$	−6.77726	−0.260471	−0.130236	+	0.991483i	$0.541573\pi$
$678$	0	0
$679$	14.2986	0.548729
$680$	29.2042	1.11993
$681$	0	0
$682$	−6.16263	−0.235979
$683$	1.36947	0.0524015	0.0262007	−	0.999657i	$-0.491659\pi$
$683$	1.36947	0.0524015	0.0262007	+	0.999657i	$0.491659\pi$
$684$	0	0
$685$	20.9132	0.799052
$686$	−15.0987	−0.576473
$687$	0	0
$688$	10.6873	0.407449
$689$	−0.286497	−0.0109147
$690$	0	0
$691$	−30.1540	−1.14711	−0.573555	−	0.819167i	$-0.694436\pi$
$691$	−30.1540	−1.14711	−0.573555	+	0.819167i	$0.694436\pi$
$692$	−30.4114	−1.15607
$693$	0	0
$694$	−4.05875	−0.154068
$695$	15.8446	0.601020
$696$	0	0
$697$	35.8671	1.35856
$698$	22.5135	0.852148
$699$	0	0
$700$	5.48297	0.207237
$701$	−45.4472	−1.71652	−0.858259	−	0.513217i	$-0.828454\pi$
$701$	−45.4472	−1.71652	−0.858259	+	0.513217i	$0.828454\pi$
$702$	0	0
$703$	21.2384	0.801021
$704$	2.77602	0.104625
$705$	0	0
$706$	−9.44446	−0.355447
$707$	15.6859	0.589931
$708$	0	0
$709$	−0.383428	−0.0144000	−0.00719998	−	0.999974i	$-0.502292\pi$
$709$	−0.383428	−0.0144000	−0.00719998	+	0.999974i	$0.502292\pi$
$710$	−12.4885	−0.468683
$711$	0	0
$712$	2.23126	0.0836198
$713$	−18.6908	−0.699977
$714$	0	0
$715$	4.75308	0.177755
$716$	4.57921	0.171133
$717$	0	0
$718$	−23.9232	−0.892806
$719$	−5.31990	−0.198399	−0.0991994	−	0.995068i	$-0.531628\pi$
$719$	−5.31990	−0.198399	−0.0991994	+	0.995068i	$0.531628\pi$
$720$	0	0
$721$	−16.5326	−0.615705
$722$	−5.38144	−0.200276
$723$	0	0
$724$	5.86714	0.218051
$725$	6.67230	0.247803
$726$	0	0
$727$	−4.82413	−0.178917	−0.0894585	−	0.995991i	$-0.528514\pi$
$727$	−4.82413	−0.178917	−0.0894585	+	0.995991i	$0.528514\pi$
$728$	−13.7560	−0.509832
$729$	0	0
$730$	8.79900	0.325666
$731$	−81.7840	−3.02489
$732$	0	0
$733$	−29.8895	−1.10399	−0.551997	−	0.833846i	$-0.686134\pi$
$733$	−29.8895	−1.10399	−0.551997	+	0.833846i	$0.686134\pi$
$734$	6.01002	0.221834
$735$	0	0
$736$	13.5597	0.499817
$737$	−1.77088	−0.0652311
$738$	0	0
$739$	8.50264	0.312775	0.156387	−	0.987696i	$-0.450015\pi$
$739$	8.50264	0.312775	0.156387	+	0.987696i	$0.450015\pi$
$740$	15.0522	0.553330
$741$	0	0
$742$	−0.152259	−0.00558960
$743$	−27.3774	−1.00438	−0.502191	−	0.864757i	$-0.667472\pi$
$743$	−27.3774	−1.00438	−0.502191	+	0.864757i	$0.667472\pi$
$744$	0	0
$745$	6.78160	0.248458
$746$	0.149889	0.00548784
$747$	0	0
$748$	9.19626	0.336249
$749$	−21.9656	−0.802605
$750$	0	0
$751$	−4.20138	−0.153311	−0.0766553	−	0.997058i	$-0.524424\pi$
$751$	−4.20138	−0.153311	−0.0766553	+	0.997058i	$0.524424\pi$
$752$	−3.04148	−0.110911
$753$	0	0
$754$	−6.94516	−0.252928
$755$	23.6119	0.859325
$756$	0	0
$757$	3.19917	0.116276	0.0581379	−	0.998309i	$-0.481484\pi$
$757$	3.19917	0.116276	0.0581379	+	0.998309i	$0.481484\pi$
$758$	22.9921	0.835112
$759$	0	0
$760$	15.5650	0.564602
$761$	36.5499	1.32493	0.662467	−	0.749091i	$-0.269509\pi$
$761$	36.5499	1.32493	0.662467	+	0.749091i	$0.269509\pi$
$762$	0	0
$763$	24.8725	0.900445
$764$	2.48576	0.0899315
$765$	0	0
$766$	14.3095	0.517022
$767$	−24.0348	−0.867846
$768$	0	0
$769$	−7.02616	−0.253370	−0.126685	−	0.991943i	$-0.540434\pi$
$769$	−7.02616	−0.253370	−0.126685	+	0.991943i	$0.540434\pi$
$770$	2.52602	0.0910316
$771$	0	0
$772$	15.0218	0.540646
$773$	−0.173181	−0.00622888	−0.00311444	−	0.999995i	$-0.500991\pi$
$773$	−0.173181	−0.00622888	−0.00311444	+	0.999995i	$0.500991\pi$
$774$	0	0
$775$	16.2922	0.585233
$776$	−19.4454	−0.698051
$777$	0	0
$778$	9.27220	0.332424
$779$	19.1161	0.684906
$780$	0	0
$781$	−9.47861	−0.339171
$782$	−11.4438	−0.409231
$783$	0	0
$784$	−2.81697	−0.100606
$785$	19.4637	0.694691
$786$	0	0
$787$	33.2648	1.18576	0.592882	−	0.805289i	$-0.297990\pi$
$787$	33.2648	1.18576	0.592882	+	0.805289i	$0.297990\pi$
$788$	28.8673	1.02836
$789$	0	0
$790$	18.3313	0.652198
$791$	−12.9438	−0.460227
$792$	0	0
$793$	−2.75182	−0.0977200
$794$	−30.1503	−1.07000
$795$	0	0
$796$	22.4935	0.797261
$797$	20.2249	0.716403	0.358201	−	0.933644i	$-0.383390\pi$
$797$	20.2249	0.716403	0.358201	+	0.933644i	$0.383390\pi$
$798$	0	0
$799$	23.2748	0.823403
$800$	−11.8196	−0.417884
$801$	0	0
$802$	−16.2081	−0.572329
$803$	6.67836	0.235674
$804$	0	0
$805$	7.66125	0.270023
$806$	−16.9585	−0.597336
$807$	0	0
$808$	−21.3322	−0.750465
$809$	5.46531	0.192150	0.0960750	−	0.995374i	$-0.469371\pi$
$809$	5.46531	0.192150	0.0960750	+	0.995374i	$0.469371\pi$
$810$	0	0
$811$	−0.772271	−0.0271181	−0.0135591	−	0.999908i	$-0.504316\pi$
$811$	−0.772271	−0.0271181	−0.0135591	+	0.999908i	$0.504316\pi$
$812$	8.99593	0.315695
$813$	0	0
$814$	−4.68743	−0.164294
$815$	−22.3189	−0.781798
$816$	0	0
$817$	−43.5885	−1.52497
$818$	11.5461	0.403700
$819$	0	0
$820$	13.5481	0.473120
$821$	−15.5509	−0.542729	−0.271365	−	0.962477i	$-0.587475\pi$
$821$	−15.5509	−0.542729	−0.271365	+	0.962477i	$0.587475\pi$
$822$	0	0
$823$	1.32012	0.0460166	0.0230083	−	0.999735i	$-0.492676\pi$
$823$	1.32012	0.0460166	0.0230083	+	0.999735i	$0.492676\pi$
$824$	22.4836	0.783253
$825$	0	0
$826$	−12.7733	−0.444440
$827$	15.2638	0.530775	0.265387	−	0.964142i	$-0.414500\pi$
$827$	15.2638	0.530775	0.265387	+	0.964142i	$0.414500\pi$
$828$	0	0
$829$	0.539223	0.0187280	0.00936400	−	0.999956i	$-0.497019\pi$
$829$	0.539223	0.0187280	0.00936400	+	0.999956i	$0.497019\pi$
$830$	−9.81662	−0.340740
$831$	0	0
$832$	7.63910	0.264838
$833$	21.5568	0.746898
$834$	0	0
$835$	−2.74203	−0.0948920
$836$	4.90134	0.169516
$837$	0	0
$838$	23.5446	0.813335
$839$	−22.7929	−0.786899	−0.393449	−	0.919346i	$-0.628718\pi$
$839$	−22.7929	−0.786899	−0.393449	+	0.919346i	$0.628718\pi$
$840$	0	0
$841$	−18.0527	−0.622508
$842$	−22.3256	−0.769390
$843$	0	0
$844$	21.2142	0.730224
$845$	−9.37461	−0.322496
$846$	0	0
$847$	1.91723	0.0658767
$848$	−0.0882247	−0.00302965
$849$	0	0
$850$	9.97524	0.342148
$851$	−14.2166	−0.487340
$852$	0	0
$853$	38.2992	1.31134	0.655670	−	0.755047i	$-0.272386\pi$
$853$	38.2992	1.31134	0.655670	+	0.755047i	$0.272386\pi$
$854$	−1.46246	−0.0500442
$855$	0	0
$856$	29.8723	1.02101
$857$	22.9692	0.784614	0.392307	−	0.919834i	$-0.371677\pi$
$857$	22.9692	0.784614	0.392307	+	0.919834i	$0.371677\pi$
$858$	0	0
$859$	38.4278	1.31114	0.655570	−	0.755134i	$-0.272428\pi$
$859$	38.4278	1.31114	0.655570	+	0.755134i	$0.272428\pi$
$860$	−30.8923	−1.05342
$861$	0	0
$862$	26.1902	0.892041
$863$	5.54634	0.188800	0.0943998	−	0.995534i	$-0.469907\pi$
$863$	5.54634	0.188800	0.0943998	+	0.995534i	$0.469907\pi$
$864$	0	0
$865$	37.0400	1.25940
$866$	−16.2991	−0.553865
$867$	0	0
$868$	21.9660	0.745574
$869$	13.9133	0.471976
$870$	0	0
$871$	−4.87314	−0.165120
$872$	−33.8256	−1.14548
$873$	0	0
$874$	−6.09923	−0.206310
$875$	−23.2357	−0.785510
$876$	0	0
$877$	−5.73026	−0.193497	−0.0967485	−	0.995309i	$-0.530844\pi$
$877$	−5.73026	−0.193497	−0.0967485	+	0.995309i	$0.530844\pi$
$878$	12.0582	0.406944
$879$	0	0
$880$	1.46368	0.0493405
$881$	5.36441	0.180732	0.0903658	−	0.995909i	$-0.471196\pi$
$881$	5.36441	0.180732	0.0903658	+	0.995909i	$0.471196\pi$
$882$	0	0
$883$	36.4317	1.22602	0.613012	−	0.790073i	$-0.289958\pi$
$883$	36.4317	1.22602	0.613012	+	0.790073i	$0.289958\pi$
$884$	25.3065	0.851148
$885$	0	0
$886$	−16.8047	−0.564566
$887$	43.2458	1.45205	0.726025	−	0.687668i	$-0.241366\pi$
$887$	43.2458	1.45205	0.726025	+	0.687668i	$0.241366\pi$
$888$	0	0
$889$	−27.1978	−0.912183
$890$	−1.12749	−0.0377937
$891$	0	0
$892$	−24.9432	−0.835160
$893$	12.4048	0.415110
$894$	0	0
$895$	−5.57733	−0.186429
$896$	−18.4143	−0.615178
$897$	0	0
$898$	−2.39675	−0.0799807
$899$	26.7307	0.891519
$900$	0	0
$901$	0.675136	0.0224921
$902$	−4.21904	−0.140479
$903$	0	0
$904$	17.6030	0.585466
$905$	−7.14598	−0.237541
$906$	0	0
$907$	49.9178	1.65749	0.828747	−	0.559623i	$-0.189054\pi$
$907$	49.9178	1.65749	0.828747	+	0.559623i	$0.189054\pi$
$908$	4.41815	0.146622
$909$	0	0
$910$	6.95117	0.230429
$911$	42.6701	1.41372	0.706862	−	0.707352i	$-0.250110\pi$
$911$	42.6701	1.41372	0.706862	+	0.707352i	$0.250110\pi$
$912$	0	0
$913$	−7.45072	−0.246583
$914$	−24.0633	−0.795943
$915$	0	0
$916$	−1.28809	−0.0425597
$917$	−40.0664	−1.32311
$918$	0	0
$919$	42.2874	1.39493	0.697466	−	0.716618i	$-0.254311\pi$
$919$	42.2874	1.39493	0.697466	+	0.716618i	$0.254311\pi$
$920$	−10.4190	−0.343503
$921$	0	0
$922$	3.80327	0.125254
$923$	−26.0834	−0.858547
$924$	0	0
$925$	12.3922	0.407453
$926$	24.4239	0.802619
$927$	0	0
$928$	−19.3924	−0.636587
$929$	−42.4193	−1.39173	−0.695866	−	0.718172i	$-0.744979\pi$
$929$	−42.4193	−1.39173	−0.695866	+	0.718172i	$0.744979\pi$
$930$	0	0
$931$	11.4891	0.376541
$932$	6.75913	0.221403
$933$	0	0
$934$	−3.82228	−0.125069
$935$	−11.2007	−0.366303
$936$	0	0
$937$	52.1447	1.70349	0.851747	−	0.523953i	$-0.175543\pi$
$937$	52.1447	1.70349	0.851747	+	0.523953i	$0.175543\pi$
$938$	−2.58983	−0.0845610
$939$	0	0
$940$	8.79160	0.286750
$941$	−30.7797	−1.00339	−0.501694	−	0.865045i	$-0.667290\pi$
$941$	−30.7797	−1.00339	−0.501694	+	0.865045i	$0.667290\pi$
$942$	0	0
$943$	−12.7960	−0.416696
$944$	−7.40135	−0.240893
$945$	0	0
$946$	9.62023	0.312781
$947$	−7.13958	−0.232005	−0.116003	−	0.993249i	$-0.537008\pi$
$947$	−7.13958	−0.232005	−0.116003	+	0.993249i	$0.537008\pi$
$948$	0	0
$949$	18.3776	0.596563
$950$	5.31651	0.172491
$951$	0	0
$952$	32.4164	1.05062
$953$	−19.3362	−0.626360	−0.313180	−	0.949694i	$-0.601394\pi$
$953$	−19.3362	−0.626360	−0.313180	+	0.949694i	$0.601394\pi$
$954$	0	0
$955$	−3.02757	−0.0979698
$956$	5.48542	0.177411
$957$	0	0
$958$	21.9827	0.710229
$959$	23.2134	0.749601
$960$	0	0
$961$	34.2702	1.10549
$962$	−12.8990	−0.415880
$963$	0	0
$964$	−26.2730	−0.846197
$965$	−18.2960	−0.588970
$966$	0	0
$967$	−38.7022	−1.24458	−0.622290	−	0.782787i	$-0.713798\pi$
$967$	−38.7022	−1.24458	−0.622290	+	0.782787i	$0.713798\pi$
$968$	−2.60735	−0.0838033
$969$	0	0
$970$	9.82614	0.315498
$971$	45.0518	1.44578	0.722890	−	0.690963i	$-0.242813\pi$
$971$	45.0518	1.44578	0.722890	+	0.690963i	$0.242813\pi$
$972$	0	0
$973$	17.5873	0.563824
$974$	9.55590	0.306191
$975$	0	0
$976$	−0.847404	−0.0271247
$977$	30.5905	0.978677	0.489338	−	0.872094i	$-0.337238\pi$
$977$	30.5905	0.978677	0.489338	+	0.872094i	$0.337238\pi$
$978$	0	0
$979$	−0.855757	−0.0273501
$980$	8.14266	0.260108
$981$	0	0
$982$	1.95721	0.0624569
$983$	2.37601	0.0757830	0.0378915	−	0.999282i	$-0.487936\pi$
$983$	2.37601	0.0757830	0.0378915	+	0.999282i	$0.487936\pi$
$984$	0	0
$985$	−35.1594	−1.12027
$986$	16.3664	0.521213
$987$	0	0
$988$	13.4876	0.429098
$989$	29.1775	0.927789
$990$	0	0
$991$	34.9025	1.10871	0.554357	−	0.832279i	$-0.312964\pi$
$991$	34.9025	1.10871	0.554357	+	0.832279i	$0.312964\pi$
$992$	−47.3517	−1.50342
$993$	0	0
$994$	−13.8621	−0.439678
$995$	−27.3963	−0.868522
$996$	0	0
$997$	29.9231	0.947674	0.473837	−	0.880613i	$-0.342869\pi$
$997$	29.9231	0.947674	0.473837	+	0.880613i	$0.342869\pi$
$998$	31.9606	1.01170
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6039.2.a.g.1.10		13
3.2	odd	2		2013.2.a.f.1.4	✓	13

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2013.2.a.f.1.4	✓	13	3.2	odd	2
6039.2.a.g.1.10		13	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

Elliptic curves over $\Q$
Elliptic curves over $\Q(\alpha)$
Genus 2 curves over $\Q$
Higher genus families
Abelian varieties over $\F_{q}$

Level:	\( N \)	\(=\)	\( 6039 = 3^{2} \cdot 11 \cdot 61 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6039.a (trivial)

\(f(q)\)	\(=\)	\(q+0.762797 q^{2} -1.41814 q^{4} +1.72725 q^{5} +1.91723 q^{7} -2.60735 q^{8} +O(q^{10})\)	\(q+0.762797 q^{2} -1.41814 q^{4} +1.72725 q^{5} +1.91723 q^{7} -2.60735 q^{8} +1.31754 q^{10} +1.00000 q^{11} +2.75182 q^{13} +1.46246 q^{14} +0.847404 q^{16} -6.48473 q^{17} -3.45617 q^{19} -2.44948 q^{20} +0.762797 q^{22} +2.31351 q^{23} -2.01661 q^{25} +2.09908 q^{26} -2.71890 q^{28} -3.30867 q^{29} -8.07900 q^{31} +5.86109 q^{32} -4.94653 q^{34} +3.31153 q^{35} -6.14506 q^{37} -2.63636 q^{38} -4.50354 q^{40} -5.53101 q^{41} +12.6118 q^{43} -1.41814 q^{44} +1.76474 q^{46} -3.58917 q^{47} -3.32424 q^{49} -1.53827 q^{50} -3.90247 q^{52} -0.104112 q^{53} +1.72725 q^{55} -4.99888 q^{56} -2.52384 q^{58} -8.73414 q^{59} -1.00000 q^{61} -6.16263 q^{62} +2.77602 q^{64} +4.75308 q^{65} -1.77088 q^{67} +9.19626 q^{68} +2.52602 q^{70} -9.47861 q^{71} +6.67836 q^{73} -4.68743 q^{74} +4.90134 q^{76} +1.91723 q^{77} +13.9133 q^{79} +1.46368 q^{80} -4.21904 q^{82} -7.45072 q^{83} -11.2007 q^{85} +9.62023 q^{86} -2.60735 q^{88} -0.855757 q^{89} +5.27587 q^{91} -3.28088 q^{92} -2.73781 q^{94} -5.96967 q^{95} +7.45794 q^{97} -2.53572 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8}+O(q^{10}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8	\( 13 q - 4 q^{2} + 12 q^{4} - 7 q^{5} + 5 q^{7} - 15 q^{8} + 8 q^{10} + 13 q^{11} - 9 q^{13} - 19 q^{14} + 18 q^{16} - 7 q^{17} + 2 q^{19} - 15 q^{20} - 4 q^{22} - 23 q^{23} + 10 q^{25} - 8 q^{26} + 9 q^{28} - 16 q^{29} + 9 q^{31} - 29 q^{32} + 2 q^{34} - 16 q^{35} + 14 q^{37} - 8 q^{38} + 16 q^{40} - 19 q^{41} + 7 q^{43} + 12 q^{44} + 4 q^{46} - 26 q^{47} + 8 q^{49} + 15 q^{50} - 17 q^{52} - 18 q^{53} - 7 q^{55} - 44 q^{56} - q^{58} - 31 q^{59} - 13 q^{61} + 5 q^{62} - 17 q^{64} - 31 q^{65} + 14 q^{67} + 32 q^{68} - 20 q^{70} - 37 q^{71} - 16 q^{73} + 6 q^{74} - 7 q^{76} + 5 q^{77} - 17 q^{79} + 2 q^{80} - 2 q^{82} - 30 q^{83} - 16 q^{85} + 22 q^{86} - 15 q^{88} - 35 q^{89} - q^{91} - 24 q^{92} - 11 q^{94} - 13 q^{95} - q^{97} + q^{98}+O(q^{100}) \) 13 * q - 4 * q^2 + 12 * q^4 - 7 * q^5 + 5 * q^7 - 15 * q^8 + 8 * q^10 + 13 * q^11 - 9 * q^13 - 19 * q^14 + 18 * q^16 - 7 * q^17 + 2 * q^19 - 15 * q^20 - 4 * q^22 - 23 * q^23 + 10 * q^25 - 8 * q^26 + 9 * q^28 - 16 * q^29 + 9 * q^31 - 29 * q^32 + 2 * q^34 - 16 * q^35 + 14 * q^37 - 8 * q^38 + 16 * q^40 - 19 * q^41 + 7 * q^43 + 12 * q^44 + 4 * q^46 - 26 * q^47 + 8 * q^49 + 15 * q^50 - 17 * q^52 - 18 * q^53 - 7 * q^55 - 44 * q^56 - q^58 - 31 * q^59 - 13 * q^61 + 5 * q^62 - 17 * q^64 - 31 * q^65 + 14 * q^67 + 32 * q^68 - 20 * q^70 - 37 * q^71 - 16 * q^73 + 6 * q^74 - 7 * q^76 + 5 * q^77 - 17 * q^79 + 2 * q^80 - 2 * q^82 - 30 * q^83 - 16 * q^85 + 22 * q^86 - 15 * q^88 - 35 * q^89 - q^91 - 24 * q^92 - 11 * q^94 - 13 * q^95 - q^97 + q^98

Introduction

L-functions

Modular forms

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Database

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