LMFDB - Embedded newform 6003.2.a.m.1.7

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:	$1$
Dimension:	$11$
Coefficient field:	$\mathbb{Q}[x]/(x^{11} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{11} - 2 x^{10} - 18 x^{9} + 30 x^{8} + 124 x^{7} - 152 x^{6} - 408 x^{5} + 285 x^{4} + 634 x^{3} + \cdots - 108 $ x^11 - 2x^10 - 18x^9 + 30x^8 + 124x^7 - 152x^6 - 408x^5 + 285x^4 + 634x^3 - 93x^2 - 369x - 108
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 3 $
Twist minimal:	no (minimal twist has level 2001)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.7
Root			$-0.661934$ of defining polynomial
Character	$\chi$	$=$	6003.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+0.661934 q^{2} -1.56184 q^{4} +1.16115 q^{5} +4.80000 q^{7} -2.35770 q^{8} +O(q^{10})$	\(q+0.661934 q^{2} -1.56184 q^{4} +1.16115 q^{5} +4.80000 q^{7} -2.35770 q^{8} +0.768607 q^{10} +0.481747 q^{11} -3.88315 q^{13} +3.17728 q^{14} +1.56304 q^{16} -7.41544 q^{17} +4.27098 q^{19} -1.81354 q^{20} +0.318884 q^{22} -1.00000 q^{23} -3.65172 q^{25} -2.57039 q^{26} -7.49685 q^{28} +1.00000 q^{29} -0.692651 q^{31} +5.75004 q^{32} -4.90853 q^{34} +5.57354 q^{35} -5.71364 q^{37} +2.82710 q^{38} -2.73766 q^{40} -6.42272 q^{41} -8.76031 q^{43} -0.752413 q^{44} -0.661934 q^{46} -0.616265 q^{47} +16.0400 q^{49} -2.41720 q^{50} +6.06488 q^{52} -6.31136 q^{53} +0.559383 q^{55} -11.3170 q^{56} +0.661934 q^{58} -1.01528 q^{59} -6.33953 q^{61} -0.458489 q^{62} +0.680055 q^{64} -4.50894 q^{65} -11.6618 q^{67} +11.5818 q^{68} +3.68932 q^{70} -6.45040 q^{71} +14.7390 q^{73} -3.78205 q^{74} -6.67060 q^{76} +2.31238 q^{77} -2.44072 q^{79} +1.81494 q^{80} -4.25142 q^{82} -11.5245 q^{83} -8.61047 q^{85} -5.79874 q^{86} -1.13582 q^{88} -1.71946 q^{89} -18.6391 q^{91} +1.56184 q^{92} -0.407926 q^{94} +4.95926 q^{95} +3.96175 q^{97} +10.6174 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	$ 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) $ 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * 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 $

$2$	0.661934	0.468058	0.234029	−	0.972230i	$-0.424809\pi$
$2$	0.661934	0.468058	0.234029	+	0.972230i	$0.424809\pi$
$3$	0	0
$3$	0	0
$4$	−1.56184	−0.780922
$5$	1.16115	0.519284	0.259642	−	0.965705i	$-0.416395\pi$
$5$	1.16115	0.519284	0.259642	+	0.965705i	$0.416395\pi$
$6$	0	0
$7$	4.80000	1.81423	0.907115	−	0.420884i	$-0.138280\pi$
$7$	4.80000	1.81423	0.907115	+	0.420884i	$0.138280\pi$
$8$	−2.35770	−0.833574
$9$	0	0
$10$	0.768607	0.243055
$11$	0.481747	0.145252	0.0726260	−	0.997359i	$-0.476862\pi$
$11$	0.481747	0.145252	0.0726260	+	0.997359i	$0.476862\pi$
$12$	0	0
$13$	−3.88315	−1.07699	−0.538497	−	0.842628i	$-0.681008\pi$
$13$	−3.88315	−1.07699	−0.538497	+	0.842628i	$0.681008\pi$
$14$	3.17728	0.849164
$15$	0	0
$16$	1.56304	0.390761
$17$	−7.41544	−1.79851	−0.899254	−	0.437427i	$-0.855890\pi$
$17$	−7.41544	−1.79851	−0.899254	+	0.437427i	$0.855890\pi$
$18$	0	0
$19$	4.27098	0.979829	0.489914	−	0.871771i	$-0.337028\pi$
$19$	4.27098	0.979829	0.489914	+	0.871771i	$0.337028\pi$
$20$	−1.81354	−0.405521
$21$	0	0
$22$	0.318884	0.0679864
$23$	−1.00000	−0.208514
$23$	−1.00000	−0.208514
$24$	0	0
$25$	−3.65172	−0.730344
$26$	−2.57039	−0.504095
$27$	0	0
$28$	−7.49685	−1.41677
$29$	1.00000	0.185695
$29$	1.00000	0.185695
$30$	0	0
$31$	−0.692651	−0.124404	−0.0622019	−	0.998064i	$-0.519812\pi$
$31$	−0.692651	−0.124404	−0.0622019	+	0.998064i	$0.519812\pi$
$32$	5.75004	1.01647
$33$	0	0
$34$	−4.90853	−0.841805
$35$	5.57354	0.942101
$36$	0	0
$37$	−5.71364	−0.939317	−0.469658	−	0.882848i	$-0.655623\pi$
$37$	−5.71364	−0.939317	−0.469658	+	0.882848i	$0.655623\pi$
$38$	2.82710	0.458616
$39$	0	0
$40$	−2.73766	−0.432862
$41$	−6.42272	−1.00306	−0.501530	−	0.865140i	$-0.667229\pi$
$41$	−6.42272	−1.00306	−0.501530	+	0.865140i	$0.667229\pi$
$42$	0	0
$43$	−8.76031	−1.33593	−0.667967	−	0.744191i	$-0.732835\pi$
$43$	−8.76031	−1.33593	−0.667967	+	0.744191i	$0.732835\pi$
$44$	−0.752413	−0.113431
$45$	0	0
$46$	−0.661934	−0.0975968
$47$	−0.616265	−0.0898915	−0.0449457	−	0.998989i	$-0.514311\pi$
$47$	−0.616265	−0.0898915	−0.0449457	+	0.998989i	$0.514311\pi$
$48$	0	0
$49$	16.0400	2.29143
$50$	−2.41720	−0.341843
$51$	0	0
$52$	6.06488	0.841048
$53$	−6.31136	−0.866932	−0.433466	−	0.901170i	$-0.642710\pi$
$53$	−6.31136	−0.866932	−0.433466	+	0.901170i	$0.642710\pi$
$54$	0	0
$55$	0.559383	0.0754271
$56$	−11.3170	−1.51229
$57$	0	0
$58$	0.661934	0.0869161
$59$	−1.01528	−0.132178	−0.0660889	−	0.997814i	$-0.521052\pi$
$59$	−1.01528	−0.132178	−0.0660889	+	0.997814i	$0.521052\pi$
$60$	0	0
$61$	−6.33953	−0.811693	−0.405847	−	0.913941i	$-0.633023\pi$
$61$	−6.33953	−0.811693	−0.405847	+	0.913941i	$0.633023\pi$
$62$	−0.458489	−0.0582282
$63$	0	0
$64$	0.680055	0.0850069
$65$	−4.50894	−0.559266
$66$	0	0
$67$	−11.6618	−1.42472	−0.712360	−	0.701814i	$-0.752374\pi$
$67$	−11.6618	−1.42472	−0.712360	+	0.701814i	$0.752374\pi$
$68$	11.5818	1.40449
$69$	0	0
$70$	3.68932	0.440958
$71$	−6.45040	−0.765522	−0.382761	−	0.923847i	$-0.625027\pi$
$71$	−6.45040	−0.765522	−0.382761	+	0.923847i	$0.625027\pi$
$72$	0	0
$73$	14.7390	1.72507	0.862533	−	0.506000i	$-0.168876\pi$
$73$	14.7390	1.72507	0.862533	+	0.506000i	$0.168876\pi$
$74$	−3.78205	−0.439654
$75$	0	0
$76$	−6.67060	−0.765170
$77$	2.31238	0.263521
$78$	0	0
$79$	−2.44072	−0.274602	−0.137301	−	0.990529i	$-0.543843\pi$
$79$	−2.44072	−0.274602	−0.137301	+	0.990529i	$0.543843\pi$
$80$	1.81494	0.202916
$81$	0	0
$82$	−4.25142	−0.469490
$83$	−11.5245	−1.26498	−0.632488	−	0.774570i	$-0.717966\pi$
$83$	−11.5245	−1.26498	−0.632488	+	0.774570i	$0.717966\pi$
$84$	0	0
$85$	−8.61047	−0.933937
$86$	−5.79874	−0.625294
$87$	0	0
$88$	−1.13582	−0.121078
$89$	−1.71946	−0.182262	−0.0911310	−	0.995839i	$-0.529048\pi$
$89$	−1.71946	−0.182262	−0.0911310	+	0.995839i	$0.529048\pi$
$90$	0	0
$91$	−18.6391	−1.95391
$92$	1.56184	0.162833
$93$	0	0
$94$	−0.407926	−0.0420744
$95$	4.95926	0.508810
$96$	0	0
$97$	3.96175	0.402255	0.201127	−	0.979565i	$-0.435540\pi$
$97$	3.96175	0.402255	0.201127	+	0.979565i	$0.435540\pi$
$98$	10.6174	1.07252
$99$	0	0
$100$	5.70342	0.570342
$101$	18.2587	1.81681	0.908407	−	0.418088i	$-0.137300\pi$
$101$	18.2587	1.81681	0.908407	+	0.418088i	$0.137300\pi$
$102$	0	0
$103$	9.17157	0.903702	0.451851	−	0.892094i	$-0.350764\pi$
$103$	9.17157	0.903702	0.451851	+	0.892094i	$0.350764\pi$
$104$	9.15533	0.897754
$105$	0	0
$106$	−4.17770	−0.405774
$107$	−9.32854	−0.901824	−0.450912	−	0.892569i	$-0.648901\pi$
$107$	−9.32854	−0.901824	−0.450912	+	0.892569i	$0.648901\pi$
$108$	0	0
$109$	17.1548	1.64313	0.821566	−	0.570114i	$-0.193101\pi$
$109$	17.1548	1.64313	0.821566	+	0.570114i	$0.193101\pi$
$110$	0.370274	0.0353042
$111$	0	0
$112$	7.50261	0.708930
$113$	−8.38650	−0.788936	−0.394468	−	0.918910i	$-0.629071\pi$
$113$	−8.38650	−0.788936	−0.394468	+	0.918910i	$0.629071\pi$
$114$	0	0
$115$	−1.16115	−0.108278
$116$	−1.56184	−0.145014
$117$	0	0
$118$	−0.672045	−0.0618668
$119$	−35.5941	−3.26290
$120$	0	0
$121$	−10.7679	−0.978902
$122$	−4.19634	−0.379919
$123$	0	0
$124$	1.08181	0.0971497
$125$	−10.0460	−0.898540
$126$	0	0
$127$	9.39321	0.833513	0.416757	−	0.909018i	$-0.363167\pi$
$127$	9.39321	0.833513	0.416757	+	0.909018i	$0.363167\pi$
$128$	−11.0499	−0.976685
$129$	0	0
$130$	−2.98462	−0.261769
$131$	17.8413	1.55881	0.779403	−	0.626523i	$-0.215523\pi$
$131$	17.8413	1.55881	0.779403	+	0.626523i	$0.215523\pi$
$132$	0	0
$133$	20.5007	1.77763
$134$	−7.71936	−0.666851
$135$	0	0
$136$	17.4834	1.49919
$137$	−6.50179	−0.555485	−0.277743	−	0.960656i	$-0.589586\pi$
$137$	−6.50179	−0.555485	−0.277743	+	0.960656i	$0.589586\pi$
$138$	0	0
$139$	8.92280	0.756822	0.378411	−	0.925638i	$-0.376471\pi$
$139$	8.92280	0.756822	0.378411	+	0.925638i	$0.376471\pi$
$140$	−8.70500	−0.735707
$141$	0	0
$142$	−4.26974	−0.358308
$143$	−1.87070	−0.156436
$144$	0	0
$145$	1.16115	0.0964287
$146$	9.75622	0.807431
$147$	0	0
$148$	8.92382	0.733533
$149$	−12.3725	−1.01359	−0.506797	−	0.862065i	$-0.669171\pi$
$149$	−12.3725	−1.01359	−0.506797	+	0.862065i	$0.669171\pi$
$150$	0	0
$151$	−2.18528	−0.177836	−0.0889180	−	0.996039i	$-0.528341\pi$
$151$	−2.18528	−0.177836	−0.0889180	+	0.996039i	$0.528341\pi$
$152$	−10.0697	−0.816760
$153$	0	0
$154$	1.53064	0.123343
$155$	−0.804275	−0.0646009
$156$	0	0
$157$	14.9914	1.19645	0.598224	−	0.801329i	$-0.295873\pi$
$157$	14.9914	1.19645	0.598224	+	0.801329i	$0.295873\pi$
$158$	−1.61559	−0.128530
$159$	0	0
$160$	6.67669	0.527838
$161$	−4.80000	−0.378293
$162$	0	0
$163$	−8.88481	−0.695912	−0.347956	−	0.937511i	$-0.613124\pi$
$163$	−8.88481	−0.695912	−0.347956	+	0.937511i	$0.613124\pi$
$164$	10.0313	0.783312
$165$	0	0
$166$	−7.62844	−0.592081
$167$	−2.40802	−0.186338	−0.0931692	−	0.995650i	$-0.529700\pi$
$167$	−2.40802	−0.186338	−0.0931692	+	0.995650i	$0.529700\pi$
$168$	0	0
$169$	2.07889	0.159915
$170$	−5.69956	−0.437136
$171$	0	0
$172$	13.6822	1.04326
$173$	−18.9340	−1.43952	−0.719762	−	0.694221i	$-0.755749\pi$
$173$	−18.9340	−1.43952	−0.719762	+	0.694221i	$0.755749\pi$
$174$	0	0
$175$	−17.5283	−1.32501
$176$	0.752991	0.0567589
$177$	0	0
$178$	−1.13817	−0.0853091
$179$	3.18736	0.238235	0.119117	−	0.992880i	$-0.461994\pi$
$179$	3.18736	0.238235	0.119117	+	0.992880i	$0.461994\pi$
$180$	0	0
$181$	13.2936	0.988109	0.494054	−	0.869431i	$-0.335514\pi$
$181$	13.2936	0.988109	0.494054	+	0.869431i	$0.335514\pi$
$182$	−12.3379	−0.914544
$183$	0	0
$184$	2.35770	0.173812
$185$	−6.63442	−0.487772
$186$	0	0
$187$	−3.57236	−0.261237
$188$	0.962510	0.0701982
$189$	0	0
$190$	3.28270	0.238152
$191$	−21.5092	−1.55635	−0.778177	−	0.628045i	$-0.783855\pi$
$191$	−21.5092	−1.55635	−0.778177	+	0.628045i	$0.783855\pi$
$192$	0	0
$193$	−2.35343	−0.169403	−0.0847017	−	0.996406i	$-0.526994\pi$
$193$	−2.35343	−0.169403	−0.0847017	+	0.996406i	$0.526994\pi$
$194$	2.62241	0.188278
$195$	0	0
$196$	−25.0520	−1.78943
$197$	−0.766029	−0.0545773	−0.0272886	−	0.999628i	$-0.508687\pi$
$197$	−0.766029	−0.0545773	−0.0272886	+	0.999628i	$0.508687\pi$
$198$	0	0
$199$	9.77290	0.692783	0.346391	−	0.938090i	$-0.387407\pi$
$199$	9.77290	0.692783	0.346391	+	0.938090i	$0.387407\pi$
$200$	8.60967	0.608796
$201$	0	0
$202$	12.0861	0.850373
$203$	4.80000	0.336894
$204$	0	0
$205$	−7.45778	−0.520874
$206$	6.07097	0.422984
$207$	0	0
$208$	−6.06954	−0.420847
$209$	2.05753	0.142322
$210$	0	0
$211$	−21.5678	−1.48479	−0.742395	−	0.669963i	$-0.766310\pi$
$211$	−21.5678	−1.48479	−0.742395	+	0.669963i	$0.766310\pi$
$212$	9.85736	0.677007
$213$	0	0
$214$	−6.17487	−0.422105
$215$	−10.1721	−0.693730
$216$	0	0
$217$	−3.32472	−0.225697
$218$	11.3553	0.769080
$219$	0	0
$220$	−0.873668	−0.0589027
$221$	28.7953	1.93698
$222$	0	0
$223$	−0.944373	−0.0632399	−0.0316199	−	0.999500i	$-0.510067\pi$
$223$	−0.944373	−0.0632399	−0.0316199	+	0.999500i	$0.510067\pi$
$224$	27.6002	1.84412
$225$	0	0
$226$	−5.55131	−0.369268
$227$	−20.2556	−1.34441	−0.672204	−	0.740366i	$-0.734652\pi$
$227$	−20.2556	−1.34441	−0.672204	+	0.740366i	$0.734652\pi$
$228$	0	0
$229$	4.15599	0.274635	0.137318	−	0.990527i	$-0.456152\pi$
$229$	4.15599	0.274635	0.137318	+	0.990527i	$0.456152\pi$
$230$	−0.768607	−0.0506805
$231$	0	0
$232$	−2.35770	−0.154791
$233$	13.3848	0.876866	0.438433	−	0.898764i	$-0.355534\pi$
$233$	13.3848	0.876866	0.438433	+	0.898764i	$0.355534\pi$
$234$	0	0
$235$	−0.715579	−0.0466792
$236$	1.58570	0.103220
$237$	0	0
$238$	−23.5609	−1.52723
$239$	22.7219	1.46976	0.734878	−	0.678199i	$-0.237239\pi$
$239$	22.7219	1.46976	0.734878	+	0.678199i	$0.237239\pi$
$240$	0	0
$241$	4.27597	0.275440	0.137720	−	0.990471i	$-0.456023\pi$
$241$	4.27597	0.275440	0.137720	+	0.990471i	$0.456023\pi$
$242$	−7.12765	−0.458183
$243$	0	0
$244$	9.90135	0.633869
$245$	18.6249	1.18990
$246$	0	0
$247$	−16.5849	−1.05527
$248$	1.63307	0.103700
$249$	0	0
$250$	−6.64978	−0.420569
$251$	−18.8353	−1.18887	−0.594437	−	0.804142i	$-0.702625\pi$
$251$	−18.8353	−1.18887	−0.594437	+	0.804142i	$0.702625\pi$
$252$	0	0
$253$	−0.481747	−0.0302872
$254$	6.21768	0.390132
$255$	0	0
$256$	−8.67443	−0.542152
$257$	9.25845	0.577526	0.288763	−	0.957401i	$-0.406756\pi$
$257$	9.25845	0.577526	0.288763	+	0.957401i	$0.406756\pi$
$258$	0	0
$259$	−27.4255	−1.70414
$260$	7.04227	0.436743
$261$	0	0
$262$	11.8098	0.729611
$263$	−19.7994	−1.22089	−0.610443	−	0.792060i	$-0.709008\pi$
$263$	−19.7994	−1.22089	−0.610443	+	0.792060i	$0.709008\pi$
$264$	0	0
$265$	−7.32847	−0.450184
$266$	13.5701	0.832035
$267$	0	0
$268$	18.2140	1.11259
$269$	6.41447	0.391097	0.195548	−	0.980694i	$-0.437351\pi$
$269$	6.41447	0.391097	0.195548	+	0.980694i	$0.437351\pi$
$270$	0	0
$271$	0.0813137	0.00493946	0.00246973	−	0.999997i	$-0.499214\pi$
$271$	0.0813137	0.00493946	0.00246973	+	0.999997i	$0.499214\pi$
$272$	−11.5907	−0.702787
$273$	0	0
$274$	−4.30375	−0.259999
$275$	−1.75920	−0.106084
$276$	0	0
$277$	−32.2170	−1.93573	−0.967866	−	0.251465i	$-0.919088\pi$
$277$	−32.2170	−1.93573	−0.967866	+	0.251465i	$0.919088\pi$
$278$	5.90630	0.354236
$279$	0	0
$280$	−13.1408	−0.785311
$281$	20.6489	1.23181	0.615906	−	0.787820i	$-0.288790\pi$
$281$	20.6489	1.23181	0.615906	+	0.787820i	$0.288790\pi$
$282$	0	0
$283$	−21.0893	−1.25363	−0.626813	−	0.779170i	$-0.715641\pi$
$283$	−21.0893	−1.25363	−0.626813	+	0.779170i	$0.715641\pi$
$284$	10.0745	0.597813
$285$	0	0
$286$	−1.23828	−0.0732208
$287$	−30.8291	−1.81978
$288$	0	0
$289$	37.9887	2.23463
$290$	0.768607	0.0451342
$291$	0	0
$292$	−23.0200	−1.34714
$293$	−19.7744	−1.15523	−0.577615	−	0.816309i	$-0.696017\pi$
$293$	−19.7744	−1.15523	−0.577615	+	0.816309i	$0.696017\pi$
$294$	0	0
$295$	−1.17889	−0.0686378
$296$	13.4711	0.782990
$297$	0	0
$298$	−8.18977	−0.474420
$299$	3.88315	0.224569
$300$	0	0
$301$	−42.0495	−2.42369
$302$	−1.44651	−0.0832375
$303$	0	0
$304$	6.67572	0.382879
$305$	−7.36117	−0.421499
$306$	0	0
$307$	15.8499	0.904601	0.452300	−	0.891866i	$-0.350603\pi$
$307$	15.8499	0.904601	0.452300	+	0.891866i	$0.350603\pi$
$308$	−3.61158	−0.205789
$309$	0	0
$310$	−0.532377	−0.0302370
$311$	−6.50938	−0.369113	−0.184557	−	0.982822i	$-0.559085\pi$
$311$	−6.50938	−0.369113	−0.184557	+	0.982822i	$0.559085\pi$
$312$	0	0
$313$	−29.9378	−1.69218	−0.846092	−	0.533038i	$-0.821050\pi$
$313$	−29.9378	−1.69218	−0.846092	+	0.533038i	$0.821050\pi$
$314$	9.92334	0.560007
$315$	0	0
$316$	3.81202	0.214443
$317$	−17.5373	−0.984993	−0.492497	−	0.870314i	$-0.663916\pi$
$317$	−17.5373	−0.984993	−0.492497	+	0.870314i	$0.663916\pi$
$318$	0	0
$319$	0.481747	0.0269726
$320$	0.789649	0.0441427
$321$	0	0
$322$	−3.17728	−0.177063
$323$	−31.6711	−1.76223
$324$	0	0
$325$	14.1802	0.786575
$326$	−5.88116	−0.325727
$327$	0	0
$328$	15.1429	0.836126
$329$	−2.95807	−0.163084
$330$	0	0
$331$	4.82327	0.265111	0.132556	−	0.991176i	$-0.457682\pi$
$331$	4.82327	0.265111	0.132556	+	0.991176i	$0.457682\pi$
$332$	17.9994	0.987847
$333$	0	0
$334$	−1.59395	−0.0872171
$335$	−13.5412	−0.739835
$336$	0	0
$337$	5.34820	0.291335	0.145667	−	0.989334i	$-0.453467\pi$
$337$	5.34820	0.291335	0.145667	+	0.989334i	$0.453467\pi$
$338$	1.37609	0.0748492
$339$	0	0
$340$	13.4482	0.729332
$341$	−0.333682	−0.0180699
$342$	0	0
$343$	43.3920	2.34295
$344$	20.6542	1.11360
$345$	0	0
$346$	−12.5330	−0.673780
$347$	−12.8945	−0.692211	−0.346106	−	0.938196i	$-0.612496\pi$
$347$	−12.8945	−0.692211	−0.346106	+	0.938196i	$0.612496\pi$
$348$	0	0
$349$	14.2259	0.761496	0.380748	−	0.924679i	$-0.375667\pi$
$349$	14.2259	0.761496	0.380748	+	0.924679i	$0.375667\pi$
$350$	−11.6025	−0.620182
$351$	0	0
$352$	2.77006	0.147645
$353$	27.1279	1.44387	0.721936	−	0.691960i	$-0.243252\pi$
$353$	27.1279	1.44387	0.721936	+	0.691960i	$0.243252\pi$
$354$	0	0
$355$	−7.48991	−0.397523
$356$	2.68552	0.142332
$357$	0	0
$358$	2.10982	0.111508
$359$	−33.1581	−1.75002	−0.875010	−	0.484105i	$-0.839145\pi$
$359$	−33.1581	−1.75002	−0.875010	+	0.484105i	$0.839145\pi$
$360$	0	0
$361$	−0.758770	−0.0399352
$362$	8.79951	0.462492
$363$	0	0
$364$	29.1114	1.52585
$365$	17.1142	0.895800
$366$	0	0
$367$	−27.4982	−1.43539	−0.717697	−	0.696355i	$-0.754804\pi$
$367$	−27.4982	−1.43539	−0.717697	+	0.696355i	$0.754804\pi$
$368$	−1.56304	−0.0814793
$369$	0	0
$370$	−4.39155	−0.228306
$371$	−30.2945	−1.57281
$372$	0	0
$373$	16.8037	0.870062	0.435031	−	0.900416i	$-0.356737\pi$
$373$	16.8037	0.870062	0.435031	+	0.900416i	$0.356737\pi$
$374$	−2.36467	−0.122274
$375$	0	0
$376$	1.45297	0.0749312
$377$	−3.88315	−0.199993
$378$	0	0
$379$	27.0834	1.39118	0.695591	−	0.718438i	$-0.255143\pi$
$379$	27.0834	1.39118	0.695591	+	0.718438i	$0.255143\pi$
$380$	−7.74560	−0.397341
$381$	0	0
$382$	−14.2377	−0.728463
$383$	17.5873	0.898667	0.449334	−	0.893364i	$-0.351661\pi$
$383$	17.5873	0.898667	0.449334	+	0.893364i	$0.351661\pi$
$384$	0	0
$385$	2.68504	0.136842
$386$	−1.55781	−0.0792906
$387$	0	0
$388$	−6.18763	−0.314129
$389$	−6.36584	−0.322761	−0.161380	−	0.986892i	$-0.551595\pi$
$389$	−6.36584	−0.322761	−0.161380	+	0.986892i	$0.551595\pi$
$390$	0	0
$391$	7.41544	0.375015
$392$	−37.8176	−1.91008
$393$	0	0
$394$	−0.507060	−0.0255453
$395$	−2.83405	−0.142597
$396$	0	0
$397$	−14.1969	−0.712520	−0.356260	−	0.934387i	$-0.615948\pi$
$397$	−14.1969	−0.712520	−0.356260	+	0.934387i	$0.615948\pi$
$398$	6.46901	0.324262
$399$	0	0
$400$	−5.70780	−0.285390
$401$	−4.07996	−0.203743	−0.101872	−	0.994798i	$-0.532483\pi$
$401$	−4.07996	−0.203743	−0.101872	+	0.994798i	$0.532483\pi$
$402$	0	0
$403$	2.68967	0.133982
$404$	−28.5173	−1.41879
$405$	0	0
$406$	3.17728	0.157686
$407$	−2.75253	−0.136438
$408$	0	0
$409$	−1.77817	−0.0879247	−0.0439624	−	0.999033i	$-0.513998\pi$
$409$	−1.77817	−0.0879247	−0.0439624	+	0.999033i	$0.513998\pi$
$410$	−4.93655	−0.243799
$411$	0	0
$412$	−14.3246	−0.705720
$413$	−4.87333	−0.239801
$414$	0	0
$415$	−13.3817	−0.656882
$416$	−22.3283	−1.09473
$417$	0	0
$418$	1.36195	0.0666150
$419$	12.4164	0.606578	0.303289	−	0.952899i	$-0.401915\pi$
$419$	12.4164	0.606578	0.303289	+	0.952899i	$0.401915\pi$
$420$	0	0
$421$	39.1062	1.90592	0.952960	−	0.303096i	$-0.0980202\pi$
$421$	39.1062	1.90592	0.952960	+	0.303096i	$0.0980202\pi$
$422$	−14.2765	−0.694967
$423$	0	0
$424$	14.8803	0.722653
$425$	27.0791	1.31353
$426$	0	0
$427$	−30.4297	−1.47260
$428$	14.5697	0.704254
$429$	0	0
$430$	−6.73324	−0.324706
$431$	25.8815	1.24667	0.623333	−	0.781957i	$-0.285778\pi$
$431$	25.8815	1.24667	0.623333	+	0.781957i	$0.285778\pi$
$432$	0	0
$433$	−25.2711	−1.21445	−0.607226	−	0.794529i	$-0.707718\pi$
$433$	−25.2711	−1.21445	−0.607226	+	0.794529i	$0.707718\pi$
$434$	−2.20075	−0.105639
$435$	0	0
$436$	−26.7931	−1.28316
$437$	−4.27098	−0.204308
$438$	0	0
$439$	−40.3513	−1.92586	−0.962931	−	0.269749i	$-0.913059\pi$
$439$	−40.3513	−1.92586	−0.962931	+	0.269749i	$0.913059\pi$
$440$	−1.31886	−0.0628741
$441$	0	0
$442$	19.0606	0.906619
$443$	−12.7973	−0.608018	−0.304009	−	0.952669i	$-0.598325\pi$
$443$	−12.7973	−0.608018	−0.304009	+	0.952669i	$0.598325\pi$
$444$	0	0
$445$	−1.99656	−0.0946458
$446$	−0.625112	−0.0295999
$447$	0	0
$448$	3.26426	0.154222
$449$	−6.34149	−0.299274	−0.149637	−	0.988741i	$-0.547810\pi$
$449$	−6.34149	−0.299274	−0.149637	+	0.988741i	$0.547810\pi$
$450$	0	0
$451$	−3.09413	−0.145697
$452$	13.0984	0.616097
$453$	0	0
$454$	−13.4078	−0.629261
$455$	−21.6429	−1.01464
$456$	0	0
$457$	−33.4505	−1.56475	−0.782375	−	0.622808i	$-0.785992\pi$
$457$	−33.4505	−1.56475	−0.782375	+	0.622808i	$0.785992\pi$
$458$	2.75099	0.128545
$459$	0	0
$460$	1.81354	0.0845569
$461$	−5.42683	−0.252753	−0.126376	−	0.991982i	$-0.540335\pi$
$461$	−5.42683	−0.252753	−0.126376	+	0.991982i	$0.540335\pi$
$462$	0	0
$463$	10.6646	0.495624	0.247812	−	0.968808i	$-0.420289\pi$
$463$	10.6646	0.495624	0.247812	+	0.968808i	$0.420289\pi$
$464$	1.56304	0.0725625
$465$	0	0
$466$	8.85984	0.410424
$467$	−10.6209	−0.491479	−0.245739	−	0.969336i	$-0.579031\pi$
$467$	−10.6209	−0.491479	−0.245739	+	0.969336i	$0.579031\pi$
$468$	0	0
$469$	−55.9768	−2.58477
$470$	−0.473666	−0.0218486
$471$	0	0
$472$	2.39372	0.110180
$473$	−4.22025	−0.194047
$474$	0	0
$475$	−15.5964	−0.715612
$476$	55.5924	2.54807
$477$	0	0
$478$	15.0404	0.687931
$479$	−17.4622	−0.797867	−0.398934	−	0.916980i	$-0.630620\pi$
$479$	−17.4622	−0.797867	−0.398934	+	0.916980i	$0.630620\pi$
$480$	0	0
$481$	22.1870	1.01164
$482$	2.83041	0.128922
$483$	0	0
$484$	16.8178	0.764446
$485$	4.60020	0.208884
$486$	0	0
$487$	3.98493	0.180575	0.0902873	−	0.995916i	$-0.471221\pi$
$487$	3.98493	0.180575	0.0902873	+	0.995916i	$0.471221\pi$
$488$	14.9467	0.676606
$489$	0	0
$490$	12.3285	0.556943
$491$	−14.1616	−0.639103	−0.319552	−	0.947569i	$-0.603532\pi$
$491$	−14.1616	−0.639103	−0.319552	+	0.947569i	$0.603532\pi$
$492$	0	0
$493$	−7.41544	−0.333974
$494$	−10.9781	−0.493927
$495$	0	0
$496$	−1.08264	−0.0486122
$497$	−30.9619	−1.38883
$498$	0	0
$499$	−3.75546	−0.168117	−0.0840587	−	0.996461i	$-0.526788\pi$
$499$	−3.75546	−0.168117	−0.0840587	+	0.996461i	$0.526788\pi$
$500$	15.6903	0.701690
$501$	0	0
$502$	−12.4677	−0.556462
$503$	21.1293	0.942110	0.471055	−	0.882104i	$-0.343873\pi$
$503$	21.1293	0.942110	0.471055	+	0.882104i	$0.343873\pi$
$504$	0	0
$505$	21.2012	0.943442
$506$	−0.318884	−0.0141761
$507$	0	0
$508$	−14.6707	−0.650909
$509$	3.59910	0.159527	0.0797637	−	0.996814i	$-0.474583\pi$
$509$	3.59910	0.159527	0.0797637	+	0.996814i	$0.474583\pi$
$510$	0	0
$511$	70.7471	3.12967
$512$	16.3580	0.722927
$513$	0	0
$514$	6.12848	0.270316
$515$	10.6496	0.469278
$516$	0	0
$517$	−0.296884	−0.0130569
$518$	−18.1538	−0.797634
$519$	0	0
$520$	10.6308	0.466189
$521$	3.83170	0.167870	0.0839349	−	0.996471i	$-0.473251\pi$
$521$	3.83170	0.167870	0.0839349	+	0.996471i	$0.473251\pi$
$522$	0	0
$523$	42.0646	1.83936	0.919678	−	0.392674i	$-0.128450\pi$
$523$	42.0646	1.83936	0.919678	+	0.392674i	$0.128450\pi$
$524$	−27.8654	−1.21731
$525$	0	0
$526$	−13.1059	−0.571445
$527$	5.13631	0.223741
$528$	0	0
$529$	1.00000	0.0434783
$530$	−4.85096	−0.210712
$531$	0	0
$532$	−32.0189	−1.38819
$533$	24.9404	1.08029
$534$	0	0
$535$	−10.8319	−0.468303
$536$	27.4952	1.18761
$537$	0	0
$538$	4.24595	0.183056
$539$	7.72721	0.332835
$540$	0	0
$541$	32.9360	1.41603	0.708015	−	0.706197i	$-0.249591\pi$
$541$	32.9360	1.41603	0.708015	+	0.706197i	$0.249591\pi$
$542$	0.0538243	0.00231195
$543$	0	0
$544$	−42.6391	−1.82813
$545$	19.9194	0.853252
$546$	0	0
$547$	−1.89683	−0.0811024	−0.0405512	−	0.999177i	$-0.512911\pi$
$547$	−1.89683	−0.0811024	−0.0405512	+	0.999177i	$0.512911\pi$
$548$	10.1548	0.433791
$549$	0	0
$550$	−1.16448	−0.0496534
$551$	4.27098	0.181950
$552$	0	0
$553$	−11.7154	−0.498191
$554$	−21.3255	−0.906035
$555$	0	0
$556$	−13.9360	−0.591019
$557$	−6.32207	−0.267875	−0.133937	−	0.990990i	$-0.542762\pi$
$557$	−6.32207	−0.267875	−0.133937	+	0.990990i	$0.542762\pi$
$558$	0	0
$559$	34.0176	1.43879
$560$	8.71170	0.368136
$561$	0	0
$562$	13.6682	0.576559
$563$	−35.8000	−1.50879	−0.754395	−	0.656420i	$-0.772070\pi$
$563$	−35.8000	−1.50879	−0.754395	+	0.656420i	$0.772070\pi$
$564$	0	0
$565$	−9.73803	−0.409682
$566$	−13.9597	−0.586769
$567$	0	0
$568$	15.2081	0.638119
$569$	−6.27427	−0.263031	−0.131515	−	0.991314i	$-0.541984\pi$
$569$	−6.27427	−0.263031	−0.131515	+	0.991314i	$0.541984\pi$
$570$	0	0
$571$	17.0402	0.713109	0.356555	−	0.934274i	$-0.383951\pi$
$571$	17.0402	0.713109	0.356555	+	0.934274i	$0.383951\pi$
$572$	2.92174	0.122164
$573$	0	0
$574$	−20.4068	−0.851763
$575$	3.65172	0.152287
$576$	0	0
$577$	−37.1392	−1.54613	−0.773063	−	0.634329i	$-0.781277\pi$
$577$	−37.1392	−1.54613	−0.773063	+	0.634329i	$0.781277\pi$
$578$	25.1460	1.04594
$579$	0	0
$580$	−1.81354	−0.0753033
$581$	−55.3175	−2.29496
$582$	0	0
$583$	−3.04048	−0.125924
$584$	−34.7501	−1.43797
$585$	0	0
$586$	−13.0893	−0.540714
$587$	−5.01609	−0.207036	−0.103518	−	0.994628i	$-0.533010\pi$
$587$	−5.01609	−0.207036	−0.103518	+	0.994628i	$0.533010\pi$
$588$	0	0
$589$	−2.95830	−0.121894
$590$	−0.780349	−0.0321265
$591$	0	0
$592$	−8.93068	−0.367049
$593$	−0.761753	−0.0312815	−0.0156407	−	0.999878i	$-0.504979\pi$
$593$	−0.761753	−0.0312815	−0.0156407	+	0.999878i	$0.504979\pi$
$594$	0	0
$595$	−41.3303	−1.69438
$596$	19.3239	0.791538
$597$	0	0
$598$	2.57039	0.105111
$599$	−14.8728	−0.607688	−0.303844	−	0.952722i	$-0.598270\pi$
$599$	−14.8728	−0.607688	−0.303844	+	0.952722i	$0.598270\pi$
$600$	0	0
$601$	0.810605	0.0330653	0.0165326	−	0.999863i	$-0.494737\pi$
$601$	0.810605	0.0330653	0.0165326	+	0.999863i	$0.494737\pi$
$602$	−27.8340	−1.13443
$603$	0	0
$604$	3.41307	0.138876
$605$	−12.5032	−0.508328
$606$	0	0
$607$	13.0304	0.528888	0.264444	−	0.964401i	$-0.414812\pi$
$607$	13.0304	0.528888	0.264444	+	0.964401i	$0.414812\pi$
$608$	24.5583	0.995970
$609$	0	0
$610$	−4.87261	−0.197286
$611$	2.39305	0.0968125
$612$	0	0
$613$	−45.4250	−1.83470	−0.917350	−	0.398082i	$-0.869676\pi$
$613$	−45.4250	−1.83470	−0.917350	+	0.398082i	$0.869676\pi$
$614$	10.4916	0.423405
$615$	0	0
$616$	−5.45192	−0.219664
$617$	18.6626	0.751326	0.375663	−	0.926756i	$-0.377415\pi$
$617$	18.6626	0.751326	0.375663	+	0.926756i	$0.377415\pi$
$618$	0	0
$619$	25.2424	1.01458	0.507289	−	0.861776i	$-0.330648\pi$
$619$	25.2424	1.01458	0.507289	+	0.861776i	$0.330648\pi$
$620$	1.25615	0.0504483
$621$	0	0
$622$	−4.30878	−0.172766
$623$	−8.25339	−0.330665
$624$	0	0
$625$	6.59365	0.263746
$626$	−19.8168	−0.792039
$627$	0	0
$628$	−23.4143	−0.934332
$629$	42.3691	1.68937
$630$	0	0
$631$	7.64426	0.304313	0.152157	−	0.988356i	$-0.451378\pi$
$631$	7.64426	0.304313	0.152157	+	0.988356i	$0.451378\pi$
$632$	5.75449	0.228901
$633$	0	0
$634$	−11.6085	−0.461034
$635$	10.9070	0.432830
$636$	0	0
$637$	−62.2858	−2.46785
$638$	0.318884	0.0126247
$639$	0	0
$640$	−12.8307	−0.507177
$641$	−27.0958	−1.07022	−0.535110	−	0.844783i	$-0.679730\pi$
$641$	−27.0958	−1.07022	−0.535110	+	0.844783i	$0.679730\pi$
$642$	0	0
$643$	15.6696	0.617949	0.308974	−	0.951070i	$-0.400014\pi$
$643$	15.6696	0.617949	0.308974	+	0.951070i	$0.400014\pi$
$644$	7.49685	0.295417
$645$	0	0
$646$	−20.9642	−0.824825
$647$	28.0698	1.10354	0.551769	−	0.833997i	$-0.313953\pi$
$647$	28.0698	1.10354	0.551769	+	0.833997i	$0.313953\pi$
$648$	0	0
$649$	−0.489106	−0.0191991
$650$	9.38634	0.368163
$651$	0	0
$652$	13.8767	0.543453
$653$	4.85411	0.189956	0.0949780	−	0.995479i	$-0.469722\pi$
$653$	4.85411	0.189956	0.0949780	+	0.995479i	$0.469722\pi$
$654$	0	0
$655$	20.7166	0.809463
$656$	−10.0390	−0.391957
$657$	0	0
$658$	−1.95805	−0.0763326
$659$	2.91789	0.113665	0.0568324	−	0.998384i	$-0.481900\pi$
$659$	2.91789	0.113665	0.0568324	+	0.998384i	$0.481900\pi$
$660$	0	0
$661$	5.93405	0.230808	0.115404	−	0.993319i	$-0.463184\pi$
$661$	5.93405	0.230808	0.115404	+	0.993319i	$0.463184\pi$
$662$	3.19269	0.124087
$663$	0	0
$664$	27.1713	1.05445
$665$	23.8045	0.923098
$666$	0	0
$667$	−1.00000	−0.0387202
$668$	3.76096	0.145516
$669$	0	0
$670$	−8.96337	−0.346285
$671$	−3.05405	−0.117900
$672$	0	0
$673$	26.3141	1.01434	0.507168	−	0.861847i	$-0.330693\pi$
$673$	26.3141	1.01434	0.507168	+	0.861847i	$0.330693\pi$
$674$	3.54015	0.136362
$675$	0	0
$676$	−3.24690	−0.124881
$677$	−5.30262	−0.203796	−0.101898	−	0.994795i	$-0.532492\pi$
$677$	−5.30262	−0.203796	−0.101898	+	0.994795i	$0.532492\pi$
$678$	0	0
$679$	19.0164	0.729782
$680$	20.3009	0.778506
$681$	0	0
$682$	−0.220875	−0.00845776
$683$	45.1373	1.72713	0.863566	−	0.504236i	$-0.168226\pi$
$683$	45.1373	1.72713	0.863566	+	0.504236i	$0.168226\pi$
$684$	0	0
$685$	−7.54958	−0.288455
$686$	28.7226	1.09663
$687$	0	0
$688$	−13.6928	−0.522031
$689$	24.5080	0.933680
$690$	0	0
$691$	−4.14576	−0.157712	−0.0788561	−	0.996886i	$-0.525127\pi$
$691$	−4.14576	−0.157712	−0.0788561	+	0.996886i	$0.525127\pi$
$692$	29.5719	1.12416
$693$	0	0
$694$	−8.53528	−0.323995
$695$	10.3608	0.393006
$696$	0	0
$697$	47.6273	1.80401
$698$	9.41661	0.356424
$699$	0	0
$700$	27.3764	1.03473
$701$	−2.99761	−0.113218	−0.0566091	−	0.998396i	$-0.518029\pi$
$701$	−2.99761	−0.113218	−0.0566091	+	0.998396i	$0.518029\pi$
$702$	0	0
$703$	−24.4028	−0.920370
$704$	0.327614	0.0123474
$705$	0	0
$706$	17.9569	0.675816
$707$	87.6420	3.29612
$708$	0	0
$709$	50.3931	1.89255	0.946277	−	0.323357i	$-0.104811\pi$
$709$	50.3931	1.89255	0.946277	+	0.323357i	$0.104811\pi$
$710$	−4.95783	−0.186064
$711$	0	0
$712$	4.05397	0.151929
$713$	0.692651	0.0259400
$714$	0	0
$715$	−2.17217	−0.0812345
$716$	−4.97816	−0.186043
$717$	0	0
$718$	−21.9485	−0.819110
$719$	−1.37707	−0.0513561	−0.0256781	−	0.999670i	$-0.508174\pi$
$719$	−1.37707	−0.0513561	−0.0256781	+	0.999670i	$0.508174\pi$
$720$	0	0
$721$	44.0235	1.63952
$722$	−0.502255	−0.0186920
$723$	0	0
$724$	−20.7626	−0.771636
$725$	−3.65172	−0.135621
$726$	0	0
$727$	2.89494	0.107367	0.0536836	−	0.998558i	$-0.482904\pi$
$727$	2.89494	0.107367	0.0536836	+	0.998558i	$0.482904\pi$
$728$	43.9456	1.62873
$729$	0	0
$730$	11.3285	0.419286
$731$	64.9615	2.40269
$732$	0	0
$733$	25.9159	0.957227	0.478613	−	0.878026i	$-0.341140\pi$
$733$	25.9159	0.957227	0.478613	+	0.878026i	$0.341140\pi$
$734$	−18.2020	−0.671848
$735$	0	0
$736$	−5.75004	−0.211949
$737$	−5.61805	−0.206944
$738$	0	0
$739$	2.34877	0.0864010	0.0432005	−	0.999066i	$-0.486245\pi$
$739$	2.34877	0.0864010	0.0432005	+	0.999066i	$0.486245\pi$
$740$	10.3619	0.380912
$741$	0	0
$742$	−20.0530	−0.736168
$743$	−19.1323	−0.701895	−0.350947	−	0.936395i	$-0.614140\pi$
$743$	−19.1323	−0.701895	−0.350947	+	0.936395i	$0.614140\pi$
$744$	0	0
$745$	−14.3664	−0.526343
$746$	11.1229	0.407239
$747$	0	0
$748$	5.57947	0.204006
$749$	−44.7770	−1.63611
$750$	0	0
$751$	51.4428	1.87717	0.938587	−	0.345043i	$-0.112136\pi$
$751$	51.4428	1.87717	0.938587	+	0.345043i	$0.112136\pi$
$752$	−0.963250	−0.0351261
$753$	0	0
$754$	−2.57039	−0.0936081
$755$	−2.53745	−0.0923474
$756$	0	0
$757$	−42.1639	−1.53247	−0.766236	−	0.642560i	$-0.777872\pi$
$757$	−42.1639	−1.53247	−0.766236	+	0.642560i	$0.777872\pi$
$758$	17.9274	0.651153
$759$	0	0
$760$	−11.6925	−0.424131
$761$	11.1416	0.403884	0.201942	−	0.979397i	$-0.435275\pi$
$761$	11.1416	0.403884	0.201942	+	0.979397i	$0.435275\pi$
$762$	0	0
$763$	82.3430	2.98102
$764$	33.5941	1.21539
$765$	0	0
$766$	11.6416	0.420628
$767$	3.94247	0.142355
$768$	0	0
$769$	−26.0823	−0.940551	−0.470275	−	0.882520i	$-0.655845\pi$
$769$	−26.0823	−0.940551	−0.470275	+	0.882520i	$0.655845\pi$
$770$	1.77732	0.0640500
$771$	0	0
$772$	3.67569	0.132291
$773$	29.4575	1.05951	0.529756	−	0.848150i	$-0.322284\pi$
$773$	29.4575	1.05951	0.529756	+	0.848150i	$0.322284\pi$
$774$	0	0
$775$	2.52937	0.0908575
$776$	−9.34063	−0.335309
$777$	0	0
$778$	−4.21376	−0.151071
$779$	−27.4313	−0.982828
$780$	0	0
$781$	−3.10746	−0.111194
$782$	4.90853	0.175529
$783$	0	0
$784$	25.0712	0.895401
$785$	17.4074	0.621297
$786$	0	0
$787$	7.08437	0.252530	0.126265	−	0.991997i	$-0.459701\pi$
$787$	7.08437	0.252530	0.126265	+	0.991997i	$0.459701\pi$
$788$	1.19642	0.0426206
$789$	0	0
$790$	−1.87595	−0.0667434
$791$	−40.2552	−1.43131
$792$	0	0
$793$	24.6174	0.874188
$794$	−9.39738	−0.333501
$795$	0	0
$796$	−15.2638	−0.541009
$797$	−37.8352	−1.34019	−0.670096	−	0.742275i	$-0.733747\pi$
$797$	−37.8352	−1.34019	−0.670096	+	0.742275i	$0.733747\pi$
$798$	0	0
$799$	4.56987	0.161671
$800$	−20.9975	−0.742375
$801$	0	0
$802$	−2.70066	−0.0953637
$803$	7.10045	0.250570
$804$	0	0
$805$	−5.57354	−0.196442
$806$	1.78038	0.0627113
$807$	0	0
$808$	−43.0487	−1.51445
$809$	45.3863	1.59570	0.797849	−	0.602858i	$-0.205971\pi$
$809$	45.3863	1.59570	0.797849	+	0.602858i	$0.205971\pi$
$810$	0	0
$811$	−43.1196	−1.51413	−0.757067	−	0.653337i	$-0.773368\pi$
$811$	−43.1196	−1.51413	−0.757067	+	0.653337i	$0.773368\pi$
$812$	−7.49685	−0.263088
$813$	0	0
$814$	−1.82199	−0.0638607
$815$	−10.3166	−0.361376
$816$	0	0
$817$	−37.4151	−1.30899
$818$	−1.17703	−0.0411538
$819$	0	0
$820$	11.6479	0.406762
$821$	41.4654	1.44715	0.723576	−	0.690245i	$-0.242497\pi$
$821$	41.4654	1.44715	0.723576	+	0.690245i	$0.242497\pi$
$822$	0	0
$823$	30.5317	1.06427	0.532134	−	0.846660i	$-0.321390\pi$
$823$	30.5317	1.06427	0.532134	+	0.846660i	$0.321390\pi$
$824$	−21.6238	−0.753302
$825$	0	0
$826$	−3.22582	−0.112241
$827$	−53.7558	−1.86927	−0.934637	−	0.355604i	$-0.884275\pi$
$827$	−53.7558	−1.86927	−0.934637	+	0.355604i	$0.884275\pi$
$828$	0	0
$829$	−26.5801	−0.923164	−0.461582	−	0.887098i	$-0.652718\pi$
$829$	−26.5801	−0.923164	−0.461582	+	0.887098i	$0.652718\pi$
$830$	−8.85780	−0.307459
$831$	0	0
$832$	−2.64076	−0.0915518
$833$	−118.944	−4.12115
$834$	0	0
$835$	−2.79609	−0.0967626
$836$	−3.21354	−0.111143
$837$	0	0
$838$	8.21880	0.283914
$839$	24.9153	0.860172	0.430086	−	0.902788i	$-0.358483\pi$
$839$	24.9153	0.860172	0.430086	+	0.902788i	$0.358483\pi$
$840$	0	0
$841$	1.00000	0.0344828
$842$	25.8857	0.892081
$843$	0	0
$844$	33.6856	1.15950
$845$	2.41391	0.0830411
$846$	0	0
$847$	−51.6860	−1.77595
$848$	−9.86494	−0.338763
$849$	0	0
$850$	17.9246	0.614807
$851$	5.71364	0.195861
$852$	0	0
$853$	37.6402	1.28878	0.644388	−	0.764699i	$-0.277112\pi$
$853$	37.6402	1.28878	0.644388	+	0.764699i	$0.277112\pi$
$854$	−20.1425	−0.689260
$855$	0	0
$856$	21.9939	0.751737
$857$	34.1351	1.16603	0.583017	−	0.812460i	$-0.301872\pi$
$857$	34.1351	1.16603	0.583017	+	0.812460i	$0.301872\pi$
$858$	0	0
$859$	−7.59289	−0.259066	−0.129533	−	0.991575i	$-0.541348\pi$
$859$	−7.59289	−0.259066	−0.129533	+	0.991575i	$0.541348\pi$
$860$	15.8872	0.541749
$861$	0	0
$862$	17.1318	0.583512
$863$	−12.6678	−0.431218	−0.215609	−	0.976480i	$-0.569174\pi$
$863$	−12.6678	−0.431218	−0.215609	+	0.976480i	$0.569174\pi$
$864$	0	0
$865$	−21.9853	−0.747522
$866$	−16.7278	−0.568434
$867$	0	0
$868$	5.19270	0.176252
$869$	−1.17581	−0.0398865
$870$	0	0
$871$	45.2847	1.53441
$872$	−40.4459	−1.36967
$873$	0	0
$874$	−2.82710	−0.0956281
$875$	−48.2207	−1.63016
$876$	0	0
$877$	49.7555	1.68012	0.840061	−	0.542492i	$-0.182519\pi$
$877$	49.7555	1.68012	0.840061	+	0.542492i	$0.182519\pi$
$878$	−26.7099	−0.901414
$879$	0	0
$880$	0.874340	0.0294740
$881$	18.2360	0.614386	0.307193	−	0.951647i	$-0.400610\pi$
$881$	18.2360	0.614386	0.307193	+	0.951647i	$0.400610\pi$
$882$	0	0
$883$	22.6184	0.761170	0.380585	−	0.924746i	$-0.375723\pi$
$883$	22.6184	0.761170	0.380585	+	0.924746i	$0.375723\pi$
$884$	−44.9737	−1.51263
$885$	0	0
$886$	−8.47096	−0.284587
$887$	18.8293	0.632227	0.316113	−	0.948721i	$-0.397622\pi$
$887$	18.8293	0.632227	0.316113	+	0.948721i	$0.397622\pi$
$888$	0	0
$889$	45.0874	1.51218
$890$	−1.32159	−0.0442997
$891$	0	0
$892$	1.47496	0.0493854
$893$	−2.63205	−0.0880783
$894$	0	0
$895$	3.70102	0.123712
$896$	−53.0396	−1.77193
$897$	0	0
$898$	−4.19765	−0.140077
$899$	−0.692651	−0.0231012
$900$	0	0
$901$	46.8015	1.55918
$902$	−2.04811	−0.0681944
$903$	0	0
$904$	19.7729	0.657637
$905$	15.4360	0.513109
$906$	0	0
$907$	−44.9004	−1.49089	−0.745446	−	0.666566i	$-0.767764\pi$
$907$	−44.9004	−1.49089	−0.745446	+	0.666566i	$0.767764\pi$
$908$	31.6360	1.04988
$909$	0	0
$910$	−14.3262	−0.474908
$911$	−16.6245	−0.550795	−0.275398	−	0.961330i	$-0.588809\pi$
$911$	−16.6245	−0.550795	−0.275398	+	0.961330i	$0.588809\pi$
$912$	0	0
$913$	−5.55188	−0.183740
$914$	−22.1420	−0.732393
$915$	0	0
$916$	−6.49100	−0.214469
$917$	85.6385	2.82803
$918$	0	0
$919$	54.1293	1.78556	0.892781	−	0.450492i	$-0.148751\pi$
$919$	54.1293	1.78556	0.892781	+	0.450492i	$0.148751\pi$
$920$	2.73766	0.0902580
$921$	0	0
$922$	−3.59220	−0.118303
$923$	25.0479	0.824462
$924$	0	0
$925$	20.8646	0.686024
$926$	7.05923	0.231981
$927$	0	0
$928$	5.75004	0.188754
$929$	−56.9814	−1.86950	−0.934749	−	0.355309i	$-0.884376\pi$
$929$	−56.9814	−1.86950	−0.934749	+	0.355309i	$0.884376\pi$
$930$	0	0
$931$	68.5064	2.24521
$932$	−20.9049	−0.684764
$933$	0	0
$934$	−7.03036	−0.230040
$935$	−4.14807	−0.135656
$936$	0	0
$937$	2.24635	0.0733850	0.0366925	−	0.999327i	$-0.488318\pi$
$937$	2.24635	0.0733850	0.0366925	+	0.999327i	$0.488318\pi$
$938$	−37.0529	−1.20982
$939$	0	0
$940$	1.11762	0.0364528
$941$	−3.37208	−0.109927	−0.0549634	−	0.998488i	$-0.517504\pi$
$941$	−3.37208	−0.109927	−0.0549634	+	0.998488i	$0.517504\pi$
$942$	0	0
$943$	6.42272	0.209153
$944$	−1.58692	−0.0516499
$945$	0	0
$946$	−2.79352	−0.0908253
$947$	54.2976	1.76444	0.882218	−	0.470842i	$-0.156050\pi$
$947$	54.2976	1.76444	0.882218	+	0.470842i	$0.156050\pi$
$948$	0	0
$949$	−57.2337	−1.85789
$950$	−10.3238	−0.334948
$951$	0	0
$952$	83.9203	2.71987
$953$	46.6397	1.51081	0.755405	−	0.655258i	$-0.227440\pi$
$953$	46.6397	1.51081	0.755405	+	0.655258i	$0.227440\pi$
$954$	0	0
$955$	−24.9756	−0.808190
$956$	−35.4880	−1.14777
$957$	0	0
$958$	−11.5588	−0.373448
$959$	−31.2086	−1.00778
$960$	0	0
$961$	−30.5202	−0.984524
$962$	14.6863	0.473505
$963$	0	0
$964$	−6.67840	−0.215097
$965$	−2.73269	−0.0879685
$966$	0	0
$967$	−34.1471	−1.09810	−0.549048	−	0.835791i	$-0.685010\pi$
$967$	−34.1471	−1.09810	−0.549048	+	0.835791i	$0.685010\pi$
$968$	25.3876	0.815987
$969$	0	0
$970$	3.04503	0.0977700
$971$	24.8313	0.796873	0.398437	−	0.917196i	$-0.369553\pi$
$971$	24.8313	0.796873	0.398437	+	0.917196i	$0.369553\pi$
$972$	0	0
$973$	42.8294	1.37305
$974$	2.63776	0.0845193
$975$	0	0
$976$	−9.90896	−0.317178
$977$	60.2384	1.92720	0.963598	−	0.267355i	$-0.0861495\pi$
$977$	60.2384	1.92720	0.963598	+	0.267355i	$0.0861495\pi$
$978$	0	0
$979$	−0.828342	−0.0264739
$980$	−29.0892	−0.929221
$981$	0	0
$982$	−9.37403	−0.299137
$983$	53.8499	1.71754	0.858772	−	0.512357i	$-0.171228\pi$
$983$	53.8499	1.71754	0.858772	+	0.512357i	$0.171228\pi$
$984$	0	0
$985$	−0.889478	−0.0283411
$986$	−4.90853	−0.156319
$987$	0	0
$988$	25.9030	0.824083
$989$	8.76031	0.278562
$990$	0	0
$991$	9.34232	0.296769	0.148384	−	0.988930i	$-0.452593\pi$
$991$	9.34232	0.296769	0.148384	+	0.988930i	$0.452593\pi$
$992$	−3.98277	−0.126453
$993$	0	0
$994$	−20.4947	−0.650053
$995$	11.3479	0.359751
$996$	0	0
$997$	−49.9157	−1.58085	−0.790423	−	0.612562i	$-0.790139\pi$
$997$	−49.9157	−1.58085	−0.790423	+	0.612562i	$0.790139\pi$
$998$	−2.48586	−0.0786886
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	6003.2.a.m.1.7		11
3.2	odd	2		2001.2.a.l.1.5	✓	11

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2001.2.a.l.1.5	✓	11	3.2	odd	2
6003.2.a.m.1.7		11	1.1	even	1	trivial

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 6003 = 3^{2} \cdot 23 \cdot 29 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	6003.a (trivial)

\(f(q)\)	\(=\)	\(q+0.661934 q^{2} -1.56184 q^{4} +1.16115 q^{5} +4.80000 q^{7} -2.35770 q^{8} +O(q^{10})\)	\(q+0.661934 q^{2} -1.56184 q^{4} +1.16115 q^{5} +4.80000 q^{7} -2.35770 q^{8} +0.768607 q^{10} +0.481747 q^{11} -3.88315 q^{13} +3.17728 q^{14} +1.56304 q^{16} -7.41544 q^{17} +4.27098 q^{19} -1.81354 q^{20} +0.318884 q^{22} -1.00000 q^{23} -3.65172 q^{25} -2.57039 q^{26} -7.49685 q^{28} +1.00000 q^{29} -0.692651 q^{31} +5.75004 q^{32} -4.90853 q^{34} +5.57354 q^{35} -5.71364 q^{37} +2.82710 q^{38} -2.73766 q^{40} -6.42272 q^{41} -8.76031 q^{43} -0.752413 q^{44} -0.661934 q^{46} -0.616265 q^{47} +16.0400 q^{49} -2.41720 q^{50} +6.06488 q^{52} -6.31136 q^{53} +0.559383 q^{55} -11.3170 q^{56} +0.661934 q^{58} -1.01528 q^{59} -6.33953 q^{61} -0.458489 q^{62} +0.680055 q^{64} -4.50894 q^{65} -11.6618 q^{67} +11.5818 q^{68} +3.68932 q^{70} -6.45040 q^{71} +14.7390 q^{73} -3.78205 q^{74} -6.67060 q^{76} +2.31238 q^{77} -2.44072 q^{79} +1.81494 q^{80} -4.25142 q^{82} -11.5245 q^{83} -8.61047 q^{85} -5.79874 q^{86} -1.13582 q^{88} -1.71946 q^{89} -18.6391 q^{91} +1.56184 q^{92} -0.407926 q^{94} +4.95926 q^{95} +3.96175 q^{97} +10.6174 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8}+O(q^{10}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8	\( 11 q - 2 q^{2} + 18 q^{4} - 2 q^{5} + 3 q^{7} - 18 q^{8} + 14 q^{10} - 11 q^{11} - 5 q^{13} - 17 q^{14} + 20 q^{16} - 15 q^{17} - 6 q^{19} - 21 q^{20} - 10 q^{22} - 11 q^{23} + 3 q^{25} + 5 q^{26} + 7 q^{28} + 11 q^{29} + 35 q^{31} - 28 q^{32} + 28 q^{34} - 15 q^{35} - 28 q^{37} + 2 q^{38} - q^{40} - 10 q^{41} - 6 q^{43} - 18 q^{44} + 2 q^{46} - 15 q^{47} + 22 q^{49} - 15 q^{50} - 36 q^{52} + 7 q^{53} - 12 q^{55} - 56 q^{56} - 2 q^{58} + 20 q^{59} - 20 q^{61} + 11 q^{62} + 36 q^{64} - 11 q^{65} - 39 q^{67} - 35 q^{68} + 38 q^{70} - 49 q^{71} - 3 q^{73} - 37 q^{74} - 18 q^{76} - 25 q^{77} + 41 q^{79} - 51 q^{80} - 19 q^{82} - 13 q^{83} - 62 q^{86} - 40 q^{88} - 34 q^{89} + 2 q^{91} - 18 q^{92} - 14 q^{94} - 25 q^{95} - 11 q^{97} - 53 q^{98}+O(q^{100}) \) 11 * q - 2 * q^2 + 18 * q^4 - 2 * q^5 + 3 * q^7 - 18 * q^8 + 14 * q^10 - 11 * q^11 - 5 * q^13 - 17 * q^14 + 20 * q^16 - 15 * q^17 - 6 * q^19 - 21 * q^20 - 10 * q^22 - 11 * q^23 + 3 * q^25 + 5 * q^26 + 7 * q^28 + 11 * q^29 + 35 * q^31 - 28 * q^32 + 28 * q^34 - 15 * q^35 - 28 * q^37 + 2 * q^38 - q^40 - 10 * q^41 - 6 * q^43 - 18 * q^44 + 2 * q^46 - 15 * q^47 + 22 * q^49 - 15 * q^50 - 36 * q^52 + 7 * q^53 - 12 * q^55 - 56 * q^56 - 2 * q^58 + 20 * q^59 - 20 * q^61 + 11 * q^62 + 36 * q^64 - 11 * q^65 - 39 * q^67 - 35 * q^68 + 38 * q^70 - 49 * q^71 - 3 * q^73 - 37 * q^74 - 18 * q^76 - 25 * q^77 + 41 * q^79 - 51 * q^80 - 19 * q^82 - 13 * q^83 - 62 * q^86 - 40 * q^88 - 34 * q^89 + 2 * q^91 - 18 * q^92 - 14 * q^94 - 25 * q^95 - 11 * q^97 - 53 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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