LMFDB - Embedded newform 2169.2.a.h.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [2169,2,Mod(1,2169)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(2169, base_ring=CyclotomicField(2))

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

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

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

chi := DirichletCharacter("2169.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 2169 = 3^{2} \cdot 241 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	2169.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:	$17.3195521984$
Analytic rank:	$1$
Dimension:	$12$
Coefficient field:	$\mathbb{Q}[x]/(x^{12} - \cdots)$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{12} - 3 x^{11} - 14 x^{10} + 44 x^{9} + 65 x^{8} - 219 x^{7} - 123 x^{6} + 444 x^{5} + 105 x^{4} + \cdots - 1 $ x^12 - 3x^11 - 14x^10 + 44x^9 + 65x^8 - 219x^7 - 123x^6 + 444x^5 + 105x^4 - 328x^3 - 45x^2 + 18*x - 1
Coefficient ring:	$\Z[a_1, \ldots, a_{7}]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 241)
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$2.70063$ of defining polynomial
Character	$\chi$	$=$	2169.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.70063 q^{2} +5.29342 q^{4} -0.533570 q^{5} +0.354992 q^{7} -8.89432 q^{8} +O(q^{10})$	\(q-2.70063 q^{2} +5.29342 q^{4} -0.533570 q^{5} +0.354992 q^{7} -8.89432 q^{8} +1.44098 q^{10} -4.18781 q^{11} -3.72447 q^{13} -0.958703 q^{14} +13.4334 q^{16} +6.46259 q^{17} -1.31002 q^{19} -2.82441 q^{20} +11.3097 q^{22} +4.10799 q^{23} -4.71530 q^{25} +10.0584 q^{26} +1.87912 q^{28} +8.85227 q^{29} +5.11371 q^{31} -18.4902 q^{32} -17.4531 q^{34} -0.189413 q^{35} +5.41403 q^{37} +3.53789 q^{38} +4.74574 q^{40} -11.8244 q^{41} +0.673253 q^{43} -22.1678 q^{44} -11.0942 q^{46} -5.22965 q^{47} -6.87398 q^{49} +12.7343 q^{50} -19.7152 q^{52} +9.92404 q^{53} +2.23449 q^{55} -3.15741 q^{56} -23.9067 q^{58} +1.23315 q^{59} +4.04837 q^{61} -13.8102 q^{62} +23.0683 q^{64} +1.98727 q^{65} -14.0522 q^{67} +34.2092 q^{68} +0.511535 q^{70} -13.0525 q^{71} +7.76916 q^{73} -14.6213 q^{74} -6.93450 q^{76} -1.48664 q^{77} -1.17673 q^{79} -7.16768 q^{80} +31.9333 q^{82} -6.25297 q^{83} -3.44824 q^{85} -1.81821 q^{86} +37.2477 q^{88} -3.80839 q^{89} -1.32216 q^{91} +21.7453 q^{92} +14.1234 q^{94} +0.698989 q^{95} -9.91591 q^{97} +18.5641 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	$ 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) $ 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * 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 $

$2$	−2.70063	−1.90964	−0.954818	−	0.297191i	$-0.903950\pi$
$2$	−2.70063	−1.90964	−0.954818	+	0.297191i	$0.903950\pi$
$3$	0	0
$3$	0	0
$4$	5.29342	2.64671
$5$	−0.533570	−0.238620	−0.119310	−	0.992857i	$-0.538068\pi$
$5$	−0.533570	−0.238620	−0.119310	+	0.992857i	$0.538068\pi$
$6$	0	0
$7$	0.354992	0.134174	0.0670872	−	0.997747i	$-0.478629\pi$
$7$	0.354992	0.134174	0.0670872	+	0.997747i	$0.478629\pi$
$8$	−8.89432	−3.14462
$9$	0	0
$10$	1.44098	0.455677
$11$	−4.18781	−1.26267	−0.631336	−	0.775509i	$-0.717493\pi$
$11$	−4.18781	−1.26267	−0.631336	+	0.775509i	$0.717493\pi$
$12$	0	0
$13$	−3.72447	−1.03298	−0.516492	−	0.856292i	$-0.672762\pi$
$13$	−3.72447	−1.03298	−0.516492	+	0.856292i	$0.672762\pi$
$14$	−0.958703	−0.256224
$15$	0	0
$16$	13.4334	3.35836
$17$	6.46259	1.56741	0.783704	−	0.621134i	$-0.213328\pi$
$17$	6.46259	1.56741	0.783704	+	0.621134i	$0.213328\pi$
$18$	0	0
$19$	−1.31002	−0.300540	−0.150270	−	0.988645i	$-0.548014\pi$
$19$	−1.31002	−0.300540	−0.150270	+	0.988645i	$0.548014\pi$
$20$	−2.82441	−0.631557
$21$	0	0
$22$	11.3097	2.41124
$23$	4.10799	0.856576	0.428288	−	0.903642i	$-0.359117\pi$
$23$	4.10799	0.856576	0.428288	+	0.903642i	$0.359117\pi$
$24$	0	0
$25$	−4.71530	−0.943061
$26$	10.0584	1.97262
$27$	0	0
$28$	1.87912	0.355121
$29$	8.85227	1.64382	0.821912	−	0.569614i	$-0.192907\pi$
$29$	8.85227	1.64382	0.821912	+	0.569614i	$0.192907\pi$
$30$	0	0
$31$	5.11371	0.918449	0.459225	−	0.888320i	$-0.348127\pi$
$31$	5.11371	0.918449	0.459225	+	0.888320i	$0.348127\pi$
$32$	−18.4902	−3.26863
$33$	0	0
$34$	−17.4531	−2.99318
$35$	−0.189413	−0.0320166
$36$	0	0
$37$	5.41403	0.890061	0.445030	−	0.895515i	$-0.353193\pi$
$37$	5.41403	0.890061	0.445030	+	0.895515i	$0.353193\pi$
$38$	3.53789	0.573922
$39$	0	0
$40$	4.74574	0.750367
$41$	−11.8244	−1.84665	−0.923327	−	0.384014i	$-0.874541\pi$
$41$	−11.8244	−1.84665	−0.923327	+	0.384014i	$0.874541\pi$
$42$	0	0
$43$	0.673253	0.102670	0.0513350	−	0.998681i	$-0.483652\pi$
$43$	0.673253	0.102670	0.0513350	+	0.998681i	$0.483652\pi$
$44$	−22.1678	−3.34193
$45$	0	0
$46$	−11.0942	−1.63575
$47$	−5.22965	−0.762823	−0.381412	−	0.924405i	$-0.624562\pi$
$47$	−5.22965	−0.762823	−0.381412	+	0.924405i	$0.624562\pi$
$48$	0	0
$49$	−6.87398	−0.981997
$50$	12.7343	1.80090
$51$	0	0
$52$	−19.7152	−2.73401
$53$	9.92404	1.36317	0.681586	−	0.731738i	$-0.261291\pi$
$53$	9.92404	1.36317	0.681586	+	0.731738i	$0.261291\pi$
$54$	0	0
$55$	2.23449	0.301298
$56$	−3.15741	−0.421927
$57$	0	0
$58$	−23.9067	−3.13911
$59$	1.23315	0.160542	0.0802710	−	0.996773i	$-0.474421\pi$
$59$	1.23315	0.160542	0.0802710	+	0.996773i	$0.474421\pi$
$60$	0	0
$61$	4.04837	0.518341	0.259170	−	0.965832i	$-0.416551\pi$
$61$	4.04837	0.518341	0.259170	+	0.965832i	$0.416551\pi$
$62$	−13.8102	−1.75390
$63$	0	0
$64$	23.0683	2.88354
$65$	1.98727	0.246490
$66$	0	0
$67$	−14.0522	−1.71675	−0.858375	−	0.513022i	$-0.828526\pi$
$67$	−14.0522	−1.71675	−0.858375	+	0.513022i	$0.828526\pi$
$68$	34.2092	4.14847
$69$	0	0
$70$	0.511535	0.0611401
$71$	−13.0525	−1.54905	−0.774526	−	0.632542i	$-0.782011\pi$
$71$	−13.0525	−1.54905	−0.774526	+	0.632542i	$0.782011\pi$
$72$	0	0
$73$	7.76916	0.909312	0.454656	−	0.890667i	$-0.349762\pi$
$73$	7.76916	0.909312	0.454656	+	0.890667i	$0.349762\pi$
$74$	−14.6213	−1.69969
$75$	0	0
$76$	−6.93450	−0.795442
$77$	−1.48664	−0.169418
$78$	0	0
$79$	−1.17673	−0.132393	−0.0661963	−	0.997807i	$-0.521086\pi$
$79$	−1.17673	−0.132393	−0.0661963	+	0.997807i	$0.521086\pi$
$80$	−7.16768	−0.801371
$81$	0	0
$82$	31.9333	3.52644
$83$	−6.25297	−0.686353	−0.343176	−	0.939271i	$-0.611503\pi$
$83$	−6.25297	−0.686353	−0.343176	+	0.939271i	$0.611503\pi$
$84$	0	0
$85$	−3.44824	−0.374014
$86$	−1.81821	−0.196062
$87$	0	0
$88$	37.2477	3.97062
$89$	−3.80839	−0.403688	−0.201844	−	0.979418i	$-0.564693\pi$
$89$	−3.80839	−0.403688	−0.201844	+	0.979418i	$0.564693\pi$
$90$	0	0
$91$	−1.32216	−0.138600
$92$	21.7453	2.26711
$93$	0	0
$94$	14.1234	1.45671
$95$	0.698989	0.0717147
$96$	0	0
$97$	−9.91591	−1.00681	−0.503404	−	0.864051i	$-0.667919\pi$
$97$	−9.91591	−1.00681	−0.503404	+	0.864051i	$0.667919\pi$
$98$	18.5641	1.87526
$99$	0	0
$100$	−24.9601	−2.49601
$101$	−7.46224	−0.742521	−0.371260	−	0.928529i	$-0.621074\pi$
$101$	−7.46224	−0.742521	−0.371260	+	0.928529i	$0.621074\pi$
$102$	0	0
$103$	7.83988	0.772486	0.386243	−	0.922397i	$-0.373773\pi$
$103$	7.83988	0.772486	0.386243	+	0.922397i	$0.373773\pi$
$104$	33.1267	3.24834
$105$	0	0
$106$	−26.8012	−2.60316
$107$	0.890206	0.0860595	0.0430297	−	0.999074i	$-0.486299\pi$
$107$	0.890206	0.0860595	0.0430297	+	0.999074i	$0.486299\pi$
$108$	0	0
$109$	3.92555	0.375999	0.188000	−	0.982169i	$-0.439800\pi$
$109$	3.92555	0.375999	0.188000	+	0.982169i	$0.439800\pi$
$110$	−6.03454	−0.575370
$111$	0	0
$112$	4.76877	0.450606
$113$	−17.3995	−1.63681	−0.818404	−	0.574643i	$-0.805141\pi$
$113$	−17.3995	−1.63681	−0.818404	+	0.574643i	$0.805141\pi$
$114$	0	0
$115$	−2.19190	−0.204396
$116$	46.8588	4.35073
$117$	0	0
$118$	−3.33028	−0.306577
$119$	2.29417	0.210306
$120$	0	0
$121$	6.53776	0.594342
$122$	−10.9332	−0.989842
$123$	0	0
$124$	27.0690	2.43087
$125$	5.18379	0.463653
$126$	0	0
$127$	18.7331	1.66230	0.831148	−	0.556052i	$-0.187684\pi$
$127$	18.7331	1.66230	0.831148	+	0.556052i	$0.187684\pi$
$128$	−25.3187	−2.23787
$129$	0	0
$130$	−5.36688	−0.470707
$131$	1.40627	0.122867	0.0614333	−	0.998111i	$-0.480433\pi$
$131$	1.40627	0.122867	0.0614333	+	0.998111i	$0.480433\pi$
$132$	0	0
$133$	−0.465048	−0.0403247
$134$	37.9499	3.27837
$135$	0	0
$136$	−57.4803	−4.92890
$137$	−5.54974	−0.474146	−0.237073	−	0.971492i	$-0.576188\pi$
$137$	−5.54974	−0.474146	−0.237073	+	0.971492i	$0.576188\pi$
$138$	0	0
$139$	−11.1540	−0.946072	−0.473036	−	0.881043i	$-0.656842\pi$
$139$	−11.1540	−0.946072	−0.473036	+	0.881043i	$0.656842\pi$
$140$	−1.00264	−0.0847387
$141$	0	0
$142$	35.2501	2.95812
$143$	15.5974	1.30432
$144$	0	0
$145$	−4.72330	−0.392249
$146$	−20.9817	−1.73645
$147$	0	0
$148$	28.6587	2.35573
$149$	−8.27009	−0.677512	−0.338756	−	0.940874i	$-0.610006\pi$
$149$	−8.27009	−0.677512	−0.338756	+	0.940874i	$0.610006\pi$
$150$	0	0
$151$	−11.8990	−0.968323	−0.484162	−	0.874979i	$-0.660875\pi$
$151$	−11.8990	−0.968323	−0.484162	+	0.874979i	$0.660875\pi$
$152$	11.6518	0.945083
$153$	0	0
$154$	4.01487	0.323527
$155$	−2.72852	−0.219160
$156$	0	0
$157$	−5.88156	−0.469399	−0.234700	−	0.972068i	$-0.575411\pi$
$157$	−5.88156	−0.469399	−0.234700	+	0.972068i	$0.575411\pi$
$158$	3.17792	0.252822
$159$	0	0
$160$	9.86580	0.779960
$161$	1.45830	0.114930
$162$	0	0
$163$	−3.21033	−0.251452	−0.125726	−	0.992065i	$-0.540126\pi$
$163$	−3.21033	−0.251452	−0.125726	+	0.992065i	$0.540126\pi$
$164$	−62.5913	−4.88756
$165$	0	0
$166$	16.8870	1.31068
$167$	9.62511	0.744813	0.372407	−	0.928070i	$-0.378533\pi$
$167$	9.62511	0.744813	0.372407	+	0.928070i	$0.378533\pi$
$168$	0	0
$169$	0.871711	0.0670547
$170$	9.31244	0.714231
$171$	0	0
$172$	3.56381	0.271738
$173$	11.7677	0.894682	0.447341	−	0.894364i	$-0.352371\pi$
$173$	11.7677	0.894682	0.447341	+	0.894364i	$0.352371\pi$
$174$	0	0
$175$	−1.67389	−0.126535
$176$	−56.2567	−4.24051
$177$	0	0
$178$	10.2851	0.770898
$179$	4.74606	0.354737	0.177369	−	0.984144i	$-0.443241\pi$
$179$	4.74606	0.354737	0.177369	+	0.984144i	$0.443241\pi$
$180$	0	0
$181$	−9.56167	−0.710713	−0.355357	−	0.934731i	$-0.615641\pi$
$181$	−9.56167	−0.710713	−0.355357	+	0.934731i	$0.615641\pi$
$182$	3.57066	0.264675
$183$	0	0
$184$	−36.5378	−2.69360
$185$	−2.88876	−0.212386
$186$	0	0
$187$	−27.0641	−1.97912
$188$	−27.6827	−2.01897
$189$	0	0
$190$	−1.88771	−0.136949
$191$	−23.1412	−1.67444	−0.837218	−	0.546870i	$-0.815819\pi$
$191$	−23.1412	−1.67444	−0.837218	+	0.546870i	$0.815819\pi$
$192$	0	0
$193$	9.15643	0.659095	0.329547	−	0.944139i	$-0.393104\pi$
$193$	9.15643	0.659095	0.329547	+	0.944139i	$0.393104\pi$
$194$	26.7792	1.92264
$195$	0	0
$196$	−36.3869	−2.59906
$197$	6.75045	0.480950	0.240475	−	0.970655i	$-0.422697\pi$
$197$	6.75045	0.480950	0.240475	+	0.970655i	$0.422697\pi$
$198$	0	0
$199$	3.26300	0.231308	0.115654	−	0.993290i	$-0.463104\pi$
$199$	3.26300	0.231308	0.115654	+	0.993290i	$0.463104\pi$
$200$	41.9394	2.96556
$201$	0	0
$202$	20.1528	1.41794
$203$	3.14248	0.220559
$204$	0	0
$205$	6.30912	0.440648
$206$	−21.1726	−1.47517
$207$	0	0
$208$	−50.0325	−3.46913
$209$	5.48613	0.379483
$210$	0	0
$211$	−23.1474	−1.59354	−0.796768	−	0.604285i	$-0.793459\pi$
$211$	−23.1474	−1.59354	−0.796768	+	0.604285i	$0.793459\pi$
$212$	52.5321	3.60792
$213$	0	0
$214$	−2.40412	−0.164342
$215$	−0.359227	−0.0244991
$216$	0	0
$217$	1.81532	0.123232
$218$	−10.6015	−0.718022
$219$	0	0
$220$	11.8281	0.797450
$221$	−24.0698	−1.61911
$222$	0	0
$223$	−18.2948	−1.22511	−0.612557	−	0.790427i	$-0.709859\pi$
$223$	−18.2948	−1.22511	−0.612557	+	0.790427i	$0.709859\pi$
$224$	−6.56387	−0.438567
$225$	0	0
$226$	46.9897	3.12571
$227$	−10.4809	−0.695642	−0.347821	−	0.937561i	$-0.613078\pi$
$227$	−10.4809	−0.695642	−0.347821	+	0.937561i	$0.613078\pi$
$228$	0	0
$229$	−5.51626	−0.364524	−0.182262	−	0.983250i	$-0.558342\pi$
$229$	−5.51626	−0.364524	−0.182262	+	0.983250i	$0.558342\pi$
$230$	5.91952	0.390322
$231$	0	0
$232$	−78.7349	−5.16920
$233$	−27.7343	−1.81693	−0.908466	−	0.417959i	$-0.862746\pi$
$233$	−27.7343	−1.81693	−0.908466	+	0.417959i	$0.862746\pi$
$234$	0	0
$235$	2.79039	0.182025
$236$	6.52756	0.424908
$237$	0	0
$238$	−6.19570	−0.401608
$239$	−6.60568	−0.427286	−0.213643	−	0.976912i	$-0.568533\pi$
$239$	−6.60568	−0.427286	−0.213643	+	0.976912i	$0.568533\pi$
$240$	0	0
$241$	1.00000	0.0644157
$241$	1.00000	0.0644157
$242$	−17.6561	−1.13498
$243$	0	0
$244$	21.4297	1.37190
$245$	3.66775	0.234324
$246$	0	0
$247$	4.87915	0.310453
$248$	−45.4829	−2.88817
$249$	0	0
$250$	−13.9995	−0.885408
$251$	−21.2628	−1.34210	−0.671049	−	0.741413i	$-0.734156\pi$
$251$	−21.2628	−1.34210	−0.671049	+	0.741413i	$0.734156\pi$
$252$	0	0
$253$	−17.2035	−1.08157
$254$	−50.5913	−3.17438
$255$	0	0
$256$	22.2398	1.38999
$257$	−7.33227	−0.457374	−0.228687	−	0.973500i	$-0.573443\pi$
$257$	−7.33227	−0.457374	−0.228687	+	0.973500i	$0.573443\pi$
$258$	0	0
$259$	1.92194	0.119423
$260$	10.5194	0.652388
$261$	0	0
$262$	−3.79782	−0.234630
$263$	−20.6204	−1.27151	−0.635753	−	0.771892i	$-0.719310\pi$
$263$	−20.6204	−1.27151	−0.635753	+	0.771892i	$0.719310\pi$
$264$	0	0
$265$	−5.29517	−0.325280
$266$	1.25592	0.0770056
$267$	0	0
$268$	−74.3842	−4.54374
$269$	0.178772	0.0108999	0.00544997	−	0.999985i	$-0.498265\pi$
$269$	0.178772	0.0108999	0.00544997	+	0.999985i	$0.498265\pi$
$270$	0	0
$271$	−17.9311	−1.08923	−0.544617	−	0.838685i	$-0.683325\pi$
$271$	−17.9311	−1.08923	−0.544617	+	0.838685i	$0.683325\pi$
$272$	86.8149	5.26392
$273$	0	0
$274$	14.9878	0.905446
$275$	19.7468	1.19078
$276$	0	0
$277$	2.56354	0.154028	0.0770140	−	0.997030i	$-0.475461\pi$
$277$	2.56354	0.154028	0.0770140	+	0.997030i	$0.475461\pi$
$278$	30.1229	1.80665
$279$	0	0
$280$	1.68470	0.100680
$281$	22.4531	1.33944	0.669720	−	0.742614i	$-0.266414\pi$
$281$	22.4531	1.33944	0.669720	+	0.742614i	$0.266414\pi$
$282$	0	0
$283$	13.1240	0.780141	0.390070	−	0.920785i	$-0.372451\pi$
$283$	13.1240	0.780141	0.390070	+	0.920785i	$0.372451\pi$
$284$	−69.0926	−4.09989
$285$	0	0
$286$	−42.1228	−2.49078
$287$	−4.19755	−0.247774
$288$	0	0
$289$	24.7651	1.45677
$290$	12.7559	0.749053
$291$	0	0
$292$	41.1254	2.40668
$293$	32.3935	1.89245	0.946225	−	0.323509i	$-0.104863\pi$
$293$	32.3935	1.89245	0.946225	+	0.323509i	$0.104863\pi$
$294$	0	0
$295$	−0.657970	−0.0383085
$296$	−48.1541	−2.79890
$297$	0	0
$298$	22.3345	1.29380
$299$	−15.3001	−0.884829
$300$	0	0
$301$	0.238999	0.0137757
$302$	32.1347	1.84914
$303$	0	0
$304$	−17.5981	−1.00932
$305$	−2.16009	−0.123686
$306$	0	0
$307$	12.6847	0.723952	0.361976	−	0.932187i	$-0.382102\pi$
$307$	12.6847	0.723952	0.361976	+	0.932187i	$0.382102\pi$
$308$	−7.86940	−0.448401
$309$	0	0
$310$	7.36873	0.418516
$311$	−22.1050	−1.25346	−0.626730	−	0.779236i	$-0.715607\pi$
$311$	−22.1050	−1.25346	−0.626730	+	0.779236i	$0.715607\pi$
$312$	0	0
$313$	10.8859	0.615305	0.307653	−	0.951499i	$-0.400457\pi$
$313$	10.8859	0.615305	0.307653	+	0.951499i	$0.400457\pi$
$314$	15.8839	0.896382
$315$	0	0
$316$	−6.22893	−0.350405
$317$	7.76634	0.436201	0.218101	−	0.975926i	$-0.430014\pi$
$317$	7.76634	0.436201	0.218101	+	0.975926i	$0.430014\pi$
$318$	0	0
$319$	−37.0716	−2.07561
$320$	−12.3085	−0.688069
$321$	0	0
$322$	−3.93835	−0.219475
$323$	−8.46614	−0.471069
$324$	0	0
$325$	17.5620	0.974166
$326$	8.66992	0.480182
$327$	0	0
$328$	105.170	5.80702
$329$	−1.85648	−0.102351
$330$	0	0
$331$	0.317941	0.0174756	0.00873781	−	0.999962i	$-0.497219\pi$
$331$	0.317941	0.0174756	0.00873781	+	0.999962i	$0.497219\pi$
$332$	−33.0996	−1.81658
$333$	0	0
$334$	−25.9939	−1.42232
$335$	7.49784	0.409651
$336$	0	0
$337$	−10.0901	−0.549641	−0.274821	−	0.961496i	$-0.588618\pi$
$337$	−10.0901	−0.549641	−0.274821	+	0.961496i	$0.588618\pi$
$338$	−2.35417	−0.128050
$339$	0	0
$340$	−18.2530	−0.989908
$341$	−21.4152	−1.15970
$342$	0	0
$343$	−4.92515	−0.265933
$344$	−5.98812	−0.322858
$345$	0	0
$346$	−31.7802	−1.70852
$347$	12.2330	0.656700	0.328350	−	0.944556i	$-0.393508\pi$
$347$	12.2330	0.656700	0.328350	+	0.944556i	$0.393508\pi$
$348$	0	0
$349$	0.0471917	0.00252612	0.00126306	−	0.999999i	$-0.499598\pi$
$349$	0.0471917	0.00252612	0.00126306	+	0.999999i	$0.499598\pi$
$350$	4.52058	0.241635
$351$	0	0
$352$	77.4334	4.12721
$353$	−9.38275	−0.499393	−0.249697	−	0.968324i	$-0.580331\pi$
$353$	−9.38275	−0.499393	−0.249697	+	0.968324i	$0.580331\pi$
$354$	0	0
$355$	6.96444	0.369634
$356$	−20.1594	−1.06845
$357$	0	0
$358$	−12.8174	−0.677419
$359$	−4.83057	−0.254948	−0.127474	−	0.991842i	$-0.540687\pi$
$359$	−4.83057	−0.254948	−0.127474	+	0.991842i	$0.540687\pi$
$360$	0	0
$361$	−17.2838	−0.909676
$362$	25.8226	1.35720
$363$	0	0
$364$	−6.99874	−0.366834
$365$	−4.14539	−0.216980
$366$	0	0
$367$	−26.6792	−1.39264	−0.696322	−	0.717730i	$-0.745181\pi$
$367$	−26.6792	−1.39264	−0.696322	+	0.717730i	$0.745181\pi$
$368$	55.1845	2.87669
$369$	0	0
$370$	7.80149	0.405580
$371$	3.52295	0.182903
$372$	0	0
$373$	−1.30441	−0.0675400	−0.0337700	−	0.999430i	$-0.510751\pi$
$373$	−1.30441	−0.0675400	−0.0337700	+	0.999430i	$0.510751\pi$
$374$	73.0902	3.77940
$375$	0	0
$376$	46.5142	2.39879
$377$	−32.9700	−1.69804
$378$	0	0
$379$	−3.62474	−0.186191	−0.0930953	−	0.995657i	$-0.529676\pi$
$379$	−3.62474	−0.186191	−0.0930953	+	0.995657i	$0.529676\pi$
$380$	3.70004	0.189808
$381$	0	0
$382$	62.4958	3.19756
$383$	28.5501	1.45884	0.729420	−	0.684066i	$-0.239790\pi$
$383$	28.5501	1.45884	0.729420	+	0.684066i	$0.239790\pi$
$384$	0	0
$385$	0.793226	0.0404265
$386$	−24.7282	−1.25863
$387$	0	0
$388$	−52.4891	−2.66473
$389$	−29.3393	−1.48756	−0.743780	−	0.668424i	$-0.766969\pi$
$389$	−29.3393	−1.48756	−0.743780	+	0.668424i	$0.766969\pi$
$390$	0	0
$391$	26.5483	1.34260
$392$	61.1394	3.08800
$393$	0	0
$394$	−18.2305	−0.918439
$395$	0.627868	0.0315915
$396$	0	0
$397$	3.94029	0.197758	0.0988788	−	0.995099i	$-0.468474\pi$
$397$	3.94029	0.197758	0.0988788	+	0.995099i	$0.468474\pi$
$398$	−8.81218	−0.441715
$399$	0	0
$400$	−63.3428	−3.16714
$401$	−0.0336542	−0.00168061	−0.000840304	−	1.00000i	$-0.500267\pi$
$401$	−0.0336542	−0.00168061	−0.000840304		1.00000i	$0.500267\pi$
$402$	0	0
$403$	−19.0459	−0.948743
$404$	−39.5008	−1.96524
$405$	0	0
$406$	−8.48669	−0.421188
$407$	−22.6729	−1.12386
$408$	0	0
$409$	−8.89629	−0.439893	−0.219947	−	0.975512i	$-0.570588\pi$
$409$	−8.89629	−0.439893	−0.219947	+	0.975512i	$0.570588\pi$
$410$	−17.0386	−0.841477
$411$	0	0
$412$	41.4998	2.04455
$413$	0.437757	0.0215406
$414$	0	0
$415$	3.33640	0.163777
$416$	68.8662	3.37644
$417$	0	0
$418$	−14.8160	−0.724675
$419$	29.8619	1.45885	0.729425	−	0.684061i	$-0.239788\pi$
$419$	29.8619	1.45885	0.729425	+	0.684061i	$0.239788\pi$
$420$	0	0
$421$	15.9683	0.778247	0.389124	−	0.921186i	$-0.372778\pi$
$421$	15.9683	0.778247	0.389124	+	0.921186i	$0.372778\pi$
$422$	62.5128	3.04307
$423$	0	0
$424$	−88.2676	−4.28665
$425$	−30.4731	−1.47816
$426$	0	0
$427$	1.43714	0.0695480
$428$	4.71223	0.227774
$429$	0	0
$430$	0.970141	0.0467844
$431$	−24.8345	−1.19624	−0.598118	−	0.801408i	$-0.704085\pi$
$431$	−24.8345	−1.19624	−0.598118	+	0.801408i	$0.704085\pi$
$432$	0	0
$433$	−31.1730	−1.49808	−0.749040	−	0.662524i	$-0.769485\pi$
$433$	−31.1730	−1.49808	−0.749040	+	0.662524i	$0.769485\pi$
$434$	−4.90253	−0.235329
$435$	0	0
$436$	20.7796	0.995161
$437$	−5.38157	−0.257435
$438$	0	0
$439$	−0.959384	−0.0457889	−0.0228944	−	0.999738i	$-0.507288\pi$
$439$	−0.959384	−0.0457889	−0.0228944	+	0.999738i	$0.507288\pi$
$440$	−19.8743	−0.947468
$441$	0	0
$442$	65.0036	3.09190
$443$	−6.52446	−0.309986	−0.154993	−	0.987916i	$-0.549536\pi$
$443$	−6.52446	−0.309986	−0.154993	+	0.987916i	$0.549536\pi$
$444$	0	0
$445$	2.03204	0.0963280
$446$	49.4077	2.33952
$447$	0	0
$448$	8.18906	0.386897
$449$	9.39811	0.443524	0.221762	−	0.975101i	$-0.428819\pi$
$449$	9.39811	0.443524	0.221762	+	0.975101i	$0.428819\pi$
$450$	0	0
$451$	49.5182	2.33172
$452$	−92.1029	−4.33216
$453$	0	0
$454$	28.3051	1.32842
$455$	0.705464	0.0330727
$456$	0	0
$457$	−27.0904	−1.26724	−0.633618	−	0.773646i	$-0.718431\pi$
$457$	−27.0904	−1.26724	−0.633618	+	0.773646i	$0.718431\pi$
$458$	14.8974	0.696109
$459$	0	0
$460$	−11.6027	−0.540976
$461$	32.1068	1.49536	0.747682	−	0.664057i	$-0.231167\pi$
$461$	32.1068	1.49536	0.747682	+	0.664057i	$0.231167\pi$
$462$	0	0
$463$	17.5754	0.816797	0.408399	−	0.912804i	$-0.366087\pi$
$463$	17.5754	0.816797	0.408399	+	0.912804i	$0.366087\pi$
$464$	118.916	5.52056
$465$	0	0
$466$	74.9001	3.46968
$467$	−12.9732	−0.600329	−0.300164	−	0.953887i	$-0.597042\pi$
$467$	−12.9732	−0.600329	−0.300164	+	0.953887i	$0.597042\pi$
$468$	0	0
$469$	−4.98842	−0.230344
$470$	−7.53581	−0.347601
$471$	0	0
$472$	−10.9680	−0.504843
$473$	−2.81945	−0.129639
$474$	0	0
$475$	6.17716	0.283427
$476$	12.1440	0.556619
$477$	0	0
$478$	17.8395	0.815960
$479$	6.68222	0.305319	0.152659	−	0.988279i	$-0.451216\pi$
$479$	6.68222	0.305319	0.152659	+	0.988279i	$0.451216\pi$
$480$	0	0
$481$	−20.1644	−0.919418
$482$	−2.70063	−0.123010
$483$	0	0
$484$	34.6071	1.57305
$485$	5.29083	0.240244
$486$	0	0
$487$	19.6021	0.888256	0.444128	−	0.895963i	$-0.353514\pi$
$487$	19.6021	0.888256	0.444128	+	0.895963i	$0.353514\pi$
$488$	−36.0075	−1.62998
$489$	0	0
$490$	−9.90524	−0.447473
$491$	6.61946	0.298732	0.149366	−	0.988782i	$-0.452277\pi$
$491$	6.61946	0.298732	0.149366	+	0.988782i	$0.452277\pi$
$492$	0	0
$493$	57.2086	2.57654
$494$	−13.1768	−0.592852
$495$	0	0
$496$	68.6947	3.08448
$497$	−4.63355	−0.207843
$498$	0	0
$499$	39.6828	1.77645	0.888223	−	0.459413i	$-0.151940\pi$
$499$	39.6828	1.77645	0.888223	+	0.459413i	$0.151940\pi$
$500$	27.4400	1.22715
$501$	0	0
$502$	57.4231	2.56292
$503$	13.9042	0.619960	0.309980	−	0.950743i	$-0.399678\pi$
$503$	13.9042	0.619960	0.309980	+	0.950743i	$0.399678\pi$
$504$	0	0
$505$	3.98163	0.177180
$506$	46.4603	2.06541
$507$	0	0
$508$	99.1622	4.39961
$509$	−33.7260	−1.49488	−0.747439	−	0.664331i	$-0.768717\pi$
$509$	−33.7260	−1.49488	−0.747439	+	0.664331i	$0.768717\pi$
$510$	0	0
$511$	2.75799	0.122006
$512$	−9.42420	−0.416495
$513$	0	0
$514$	19.8018	0.873418
$515$	−4.18312	−0.184330
$516$	0	0
$517$	21.9008	0.963196
$518$	−5.19045	−0.228055
$519$	0	0
$520$	−17.6754	−0.775117
$521$	37.2520	1.63204	0.816020	−	0.578023i	$-0.196176\pi$
$521$	37.2520	1.63204	0.816020	+	0.578023i	$0.196176\pi$
$522$	0	0
$523$	−14.7673	−0.645730	−0.322865	−	0.946445i	$-0.604646\pi$
$523$	−14.7673	−0.645730	−0.322865	+	0.946445i	$0.604646\pi$
$524$	7.44399	0.325192
$525$	0	0
$526$	55.6880	2.42811
$527$	33.0478	1.43958
$528$	0	0
$529$	−6.12439	−0.266278
$530$	14.3003	0.621166
$531$	0	0
$532$	−2.46169	−0.106728
$533$	44.0395	1.90756
$534$	0	0
$535$	−0.474987	−0.0205355
$536$	124.985	5.39852
$537$	0	0
$538$	−0.482799	−0.0208149
$539$	28.7869	1.23994
$540$	0	0
$541$	0.345044	0.0148346	0.00741730	−	0.999972i	$-0.497639\pi$
$541$	0.345044	0.0148346	0.00741730	+	0.999972i	$0.497639\pi$
$542$	48.4252	2.08004
$543$	0	0
$544$	−119.494	−5.12328
$545$	−2.09455	−0.0897209
$546$	0	0
$547$	−6.16271	−0.263498	−0.131749	−	0.991283i	$-0.542059\pi$
$547$	−6.16271	−0.263498	−0.131749	+	0.991283i	$0.542059\pi$
$548$	−29.3771	−1.25493
$549$	0	0
$550$	−53.3289	−2.27395
$551$	−11.5967	−0.494035
$552$	0	0
$553$	−0.417730	−0.0177637
$554$	−6.92318	−0.294138
$555$	0	0
$556$	−59.0429	−2.50398
$557$	−7.45145	−0.315728	−0.157864	−	0.987461i	$-0.550461\pi$
$557$	−7.45145	−0.315728	−0.157864	+	0.987461i	$0.550461\pi$
$558$	0	0
$559$	−2.50751	−0.106056
$560$	−2.54447	−0.107523
$561$	0	0
$562$	−60.6376	−2.55784
$563$	14.5393	0.612760	0.306380	−	0.951909i	$-0.400882\pi$
$563$	14.5393	0.612760	0.306380	+	0.951909i	$0.400882\pi$
$564$	0	0
$565$	9.28385	0.390575
$566$	−35.4431	−1.48978
$567$	0	0
$568$	116.093	4.87117
$569$	−12.1696	−0.510175	−0.255088	−	0.966918i	$-0.582104\pi$
$569$	−12.1696	−0.510175	−0.255088	+	0.966918i	$0.582104\pi$
$570$	0	0
$571$	−16.0026	−0.669687	−0.334843	−	0.942274i	$-0.608683\pi$
$571$	−16.0026	−0.669687	−0.334843	+	0.942274i	$0.608683\pi$
$572$	82.5635	3.45216
$573$	0	0
$574$	11.3360	0.473157
$575$	−19.3704	−0.807803
$576$	0	0
$577$	0.0727829	0.00302999	0.00151500	−	0.999999i	$-0.499518\pi$
$577$	0.0727829	0.00302999	0.00151500	+	0.999999i	$0.499518\pi$
$578$	−66.8814	−2.78190
$579$	0	0
$580$	−25.0024	−1.03817
$581$	−2.21975	−0.0920909
$582$	0	0
$583$	−41.5600	−1.72124
$584$	−69.1014	−2.85944
$585$	0	0
$586$	−87.4830	−3.61389
$587$	−0.902377	−0.0372451	−0.0186225	−	0.999827i	$-0.505928\pi$
$587$	−0.902377	−0.0372451	−0.0186225	+	0.999827i	$0.505928\pi$
$588$	0	0
$589$	−6.69907	−0.276031
$590$	1.77694	0.0731553
$591$	0	0
$592$	72.7291	2.98915
$593$	23.7787	0.976476	0.488238	−	0.872711i	$-0.337640\pi$
$593$	23.7787	0.976476	0.488238	+	0.872711i	$0.337640\pi$
$594$	0	0
$595$	−1.22410	−0.0501831
$596$	−43.7771	−1.79318
$597$	0	0
$598$	41.3200	1.68970
$599$	−48.7217	−1.99072	−0.995358	−	0.0962439i	$-0.969317\pi$
$599$	−48.7217	−1.99072	−0.995358	+	0.0962439i	$0.969317\pi$
$600$	0	0
$601$	−3.90565	−0.159315	−0.0796573	−	0.996822i	$-0.525383\pi$
$601$	−3.90565	−0.159315	−0.0796573	+	0.996822i	$0.525383\pi$
$602$	−0.645449	−0.0263066
$603$	0	0
$604$	−62.9862	−2.56287
$605$	−3.48835	−0.141822
$606$	0	0
$607$	−15.1915	−0.616605	−0.308302	−	0.951288i	$-0.599761\pi$
$607$	−15.1915	−0.616605	−0.308302	+	0.951288i	$0.599761\pi$
$608$	24.2226	0.982355
$609$	0	0
$610$	5.83360	0.236196
$611$	19.4777	0.787984
$612$	0	0
$613$	−29.1406	−1.17698	−0.588490	−	0.808505i	$-0.700277\pi$
$613$	−29.1406	−1.17698	−0.588490	+	0.808505i	$0.700277\pi$
$614$	−34.2566	−1.38248
$615$	0	0
$616$	13.2226	0.532755
$617$	31.1143	1.25261	0.626307	−	0.779576i	$-0.284565\pi$
$617$	31.1143	1.25261	0.626307	+	0.779576i	$0.284565\pi$
$618$	0	0
$619$	40.3950	1.62361	0.811806	−	0.583927i	$-0.198484\pi$
$619$	40.3950	1.62361	0.811806	+	0.583927i	$0.198484\pi$
$620$	−14.4432	−0.580053
$621$	0	0
$622$	59.6976	2.39365
$623$	−1.35195	−0.0541646
$624$	0	0
$625$	20.8106	0.832424
$626$	−29.3987	−1.17501
$627$	0	0
$628$	−31.1335	−1.24236
$629$	34.9886	1.39509
$630$	0	0
$631$	7.99234	0.318170	0.159085	−	0.987265i	$-0.449146\pi$
$631$	7.99234	0.318170	0.159085	+	0.987265i	$0.449146\pi$
$632$	10.4662	0.416324
$633$	0	0
$634$	−20.9740	−0.832986
$635$	−9.99542	−0.396656
$636$	0	0
$637$	25.6020	1.01439
$638$	100.117	3.96366
$639$	0	0
$640$	13.5093	0.534001
$641$	25.0212	0.988277	0.494139	−	0.869383i	$-0.335484\pi$
$641$	25.0212	0.988277	0.494139	+	0.869383i	$0.335484\pi$
$642$	0	0
$643$	−9.88861	−0.389969	−0.194984	−	0.980806i	$-0.562466\pi$
$643$	−9.88861	−0.389969	−0.194984	+	0.980806i	$0.562466\pi$
$644$	7.71942	0.304188
$645$	0	0
$646$	22.8639	0.899570
$647$	−43.3237	−1.70323	−0.851615	−	0.524168i	$-0.824376\pi$
$647$	−43.3237	−1.70323	−0.851615	+	0.524168i	$0.824376\pi$
$648$	0	0
$649$	−5.16419	−0.202712
$650$	−47.4286	−1.86030
$651$	0	0
$652$	−16.9936	−0.665521
$653$	13.6441	0.533936	0.266968	−	0.963705i	$-0.413978\pi$
$653$	13.6441	0.533936	0.266968	+	0.963705i	$0.413978\pi$
$654$	0	0
$655$	−0.750344	−0.0293184
$656$	−158.842	−6.20173
$657$	0	0
$658$	5.01368	0.195454
$659$	18.6052	0.724756	0.362378	−	0.932031i	$-0.381965\pi$
$659$	18.6052	0.724756	0.362378	+	0.932031i	$0.381965\pi$
$660$	0	0
$661$	−39.9082	−1.55225	−0.776124	−	0.630580i	$-0.782817\pi$
$661$	−39.9082	−1.55225	−0.776124	+	0.630580i	$0.782817\pi$
$662$	−0.858643	−0.0333721
$663$	0	0
$664$	55.6159	2.15832
$665$	0.248135	0.00962228
$666$	0	0
$667$	36.3650	1.40806
$668$	50.9497	1.97130
$669$	0	0
$670$	−20.2489	−0.782283
$671$	−16.9538	−0.654494
$672$	0	0
$673$	−0.931169	−0.0358939	−0.0179470	−	0.999839i	$-0.505713\pi$
$673$	−0.931169	−0.0358939	−0.0179470	+	0.999839i	$0.505713\pi$
$674$	27.2496	1.04961
$675$	0	0
$676$	4.61433	0.177474
$677$	−35.1023	−1.34909	−0.674546	−	0.738233i	$-0.735661\pi$
$677$	−35.1023	−1.34909	−0.674546	+	0.738233i	$0.735661\pi$
$678$	0	0
$679$	−3.52007	−0.135088
$680$	30.6698	1.17613
$681$	0	0
$682$	57.8347	2.21461
$683$	17.5204	0.670399	0.335199	−	0.942147i	$-0.391196\pi$
$683$	17.5204	0.670399	0.335199	+	0.942147i	$0.391196\pi$
$684$	0	0
$685$	2.96117	0.113141
$686$	13.3010	0.507836
$687$	0	0
$688$	9.04410	0.344803
$689$	−36.9618	−1.40813
$690$	0	0
$691$	33.6421	1.27981	0.639903	−	0.768455i	$-0.278974\pi$
$691$	33.6421	1.27981	0.639903	+	0.768455i	$0.278974\pi$
$692$	62.2914	2.36796
$693$	0	0
$694$	−33.0367	−1.25406
$695$	5.95145	0.225751
$696$	0	0
$697$	−76.4160	−2.89446
$698$	−0.127448	−0.00482396
$699$	0	0
$700$	−8.86063	−0.334900
$701$	20.5783	0.777232	0.388616	−	0.921400i	$-0.372953\pi$
$701$	20.5783	0.777232	0.388616	+	0.921400i	$0.372953\pi$
$702$	0	0
$703$	−7.09250	−0.267499
$704$	−96.6057	−3.64096
$705$	0	0
$706$	25.3394	0.953660
$707$	−2.64904	−0.0996272
$708$	0	0
$709$	−16.3548	−0.614216	−0.307108	−	0.951675i	$-0.599361\pi$
$709$	−16.3548	−0.614216	−0.307108	+	0.951675i	$0.599361\pi$
$710$	−18.8084	−0.705867
$711$	0	0
$712$	33.8730	1.26944
$713$	21.0071	0.786721
$714$	0	0
$715$	−8.32230	−0.311236
$716$	25.1229	0.938887
$717$	0	0
$718$	13.0456	0.486858
$719$	−21.1731	−0.789624	−0.394812	−	0.918762i	$-0.629190\pi$
$719$	−21.1731	−0.789624	−0.394812	+	0.918762i	$0.629190\pi$
$720$	0	0
$721$	2.78309	0.103648
$722$	46.6773	1.73715
$723$	0	0
$724$	−50.6139	−1.88105
$725$	−41.7411	−1.55023
$726$	0	0
$727$	−15.8288	−0.587059	−0.293530	−	0.955950i	$-0.594830\pi$
$727$	−15.8288	−0.587059	−0.293530	+	0.955950i	$0.594830\pi$
$728$	11.7597	0.435843
$729$	0	0
$730$	11.1952	0.414352
$731$	4.35096	0.160926
$732$	0	0
$733$	−8.66500	−0.320049	−0.160025	−	0.987113i	$-0.551157\pi$
$733$	−8.66500	−0.320049	−0.160025	+	0.987113i	$0.551157\pi$
$734$	72.0507	2.65944
$735$	0	0
$736$	−75.9575	−2.79983
$737$	58.8480	2.16769
$738$	0	0
$739$	−51.3157	−1.88768	−0.943840	−	0.330403i	$-0.892815\pi$
$739$	−51.3157	−1.88768	−0.943840	+	0.330403i	$0.892815\pi$
$740$	−15.2914	−0.562124
$741$	0	0
$742$	−9.51421	−0.349278
$743$	−19.9955	−0.733563	−0.366781	−	0.930307i	$-0.619540\pi$
$743$	−19.9955	−0.733563	−0.366781	+	0.930307i	$0.619540\pi$
$744$	0	0
$745$	4.41267	0.161668
$746$	3.52274	0.128977
$747$	0	0
$748$	−143.262	−5.23816
$749$	0.316016	0.0115470
$750$	0	0
$751$	−30.4912	−1.11264	−0.556320	−	0.830968i	$-0.687787\pi$
$751$	−30.4912	−1.11264	−0.556320	+	0.830968i	$0.687787\pi$
$752$	−70.2523	−2.56184
$753$	0	0
$754$	89.0400	3.24264
$755$	6.34892	0.231061
$756$	0	0
$757$	10.7919	0.392238	0.196119	−	0.980580i	$-0.437166\pi$
$757$	10.7919	0.392238	0.196119	+	0.980580i	$0.437166\pi$
$758$	9.78910	0.355556
$759$	0	0
$760$	−6.21703	−0.225515
$761$	−14.3354	−0.519657	−0.259828	−	0.965655i	$-0.583666\pi$
$761$	−14.3354	−0.519657	−0.259828	+	0.965655i	$0.583666\pi$
$762$	0	0
$763$	1.39354	0.0504495
$764$	−122.496	−4.43174
$765$	0	0
$766$	−77.1032	−2.78585
$767$	−4.59282	−0.165837
$768$	0	0
$769$	20.7301	0.747547	0.373773	−	0.927520i	$-0.378064\pi$
$769$	20.7301	0.747547	0.373773	+	0.927520i	$0.378064\pi$
$770$	−2.14221	−0.0772000
$771$	0	0
$772$	48.4688	1.74443
$773$	−40.3752	−1.45220	−0.726098	−	0.687591i	$-0.758668\pi$
$773$	−40.3752	−1.45220	−0.726098	+	0.687591i	$0.758668\pi$
$774$	0	0
$775$	−24.1127	−0.866153
$776$	88.1952	3.16602
$777$	0	0
$778$	79.2346	2.84070
$779$	15.4902	0.554993
$780$	0	0
$781$	54.6616	1.95594
$782$	−71.6971	−2.56388
$783$	0	0
$784$	−92.3413	−3.29790
$785$	3.13822	0.112008
$786$	0	0
$787$	−18.3264	−0.653266	−0.326633	−	0.945151i	$-0.605914\pi$
$787$	−18.3264	−0.653266	−0.326633	+	0.945151i	$0.605914\pi$
$788$	35.7330	1.27293
$789$	0	0
$790$	−1.69564	−0.0603282
$791$	−6.17668	−0.219618
$792$	0	0
$793$	−15.0780	−0.535437
$794$	−10.6413	−0.377645
$795$	0	0
$796$	17.2725	0.612206
$797$	36.1893	1.28189	0.640945	−	0.767587i	$-0.278543\pi$
$797$	36.1893	1.28189	0.640945	+	0.767587i	$0.278543\pi$
$798$	0	0
$799$	−33.7971	−1.19566
$800$	87.1868	3.08252
$801$	0	0
$802$	0.0908875	0.00320935
$803$	−32.5358	−1.14816
$804$	0	0
$805$	−0.778107	−0.0274247
$806$	51.4359	1.81175
$807$	0	0
$808$	66.3715	2.33494
$809$	0.129006	0.00453561	0.00226781	−	0.999997i	$-0.499278\pi$
$809$	0.129006	0.00453561	0.00226781	+	0.999997i	$0.499278\pi$
$810$	0	0
$811$	−22.3392	−0.784434	−0.392217	−	0.919873i	$-0.628292\pi$
$811$	−22.3392	−0.784434	−0.392217	+	0.919873i	$0.628292\pi$
$812$	16.6345	0.583756
$813$	0	0
$814$	61.2313	2.14615
$815$	1.71293	0.0600015
$816$	0	0
$817$	−0.881977	−0.0308565
$818$	24.0256	0.840036
$819$	0	0
$820$	33.3968	1.16627
$821$	−27.9841	−0.976653	−0.488327	−	0.872661i	$-0.662393\pi$
$821$	−27.9841	−0.976653	−0.488327	+	0.872661i	$0.662393\pi$
$822$	0	0
$823$	4.52041	0.157572	0.0787859	−	0.996892i	$-0.474896\pi$
$823$	4.52041	0.157572	0.0787859	+	0.996892i	$0.474896\pi$
$824$	−69.7303	−2.42917
$825$	0	0
$826$	−1.18222	−0.0411347
$827$	10.2432	0.356189	0.178095	−	0.984013i	$-0.443007\pi$
$827$	10.2432	0.356189	0.178095	+	0.984013i	$0.443007\pi$
$828$	0	0
$829$	15.8251	0.549630	0.274815	−	0.961497i	$-0.411383\pi$
$829$	15.8251	0.549630	0.274815	+	0.961497i	$0.411383\pi$
$830$	−9.01038	−0.312755
$831$	0	0
$832$	−85.9173	−2.97865
$833$	−44.4237	−1.53919
$834$	0	0
$835$	−5.13567	−0.177727
$836$	29.0404	1.00438
$837$	0	0
$838$	−80.6461	−2.78587
$839$	−33.9655	−1.17262	−0.586309	−	0.810087i	$-0.699420\pi$
$839$	−33.9655	−1.17262	−0.586309	+	0.810087i	$0.699420\pi$
$840$	0	0
$841$	49.3626	1.70216
$842$	−43.1245	−1.48617
$843$	0	0
$844$	−122.529	−4.21763
$845$	−0.465119	−0.0160006
$846$	0	0
$847$	2.32085	0.0797454
$848$	133.314	4.57802
$849$	0	0
$850$	82.2966	2.82275
$851$	22.2408	0.762405
$852$	0	0
$853$	51.3501	1.75819	0.879097	−	0.476643i	$-0.158147\pi$
$853$	51.3501	1.75819	0.879097	+	0.476643i	$0.158147\pi$
$854$	−3.88118	−0.132811
$855$	0	0
$856$	−7.91778	−0.270624
$857$	27.2848	0.932030	0.466015	−	0.884777i	$-0.345689\pi$
$857$	27.2848	0.932030	0.466015	+	0.884777i	$0.345689\pi$
$858$	0	0
$859$	−6.65604	−0.227101	−0.113551	−	0.993532i	$-0.536222\pi$
$859$	−6.65604	−0.227101	−0.113551	+	0.993532i	$0.536222\pi$
$860$	−1.90154	−0.0648420
$861$	0	0
$862$	67.0689	2.28438
$863$	−43.1149	−1.46765	−0.733824	−	0.679340i	$-0.762266\pi$
$863$	−43.1149	−1.46765	−0.733824	+	0.679340i	$0.762266\pi$
$864$	0	0
$865$	−6.27889	−0.213489
$866$	84.1870	2.86079
$867$	0	0
$868$	9.60928	0.326160
$869$	4.92793	0.167168
$870$	0	0
$871$	52.3371	1.77338
$872$	−34.9151	−1.18237
$873$	0	0
$874$	14.5336	0.491607
$875$	1.84020	0.0622103
$876$	0	0
$877$	30.7736	1.03915	0.519575	−	0.854425i	$-0.326090\pi$
$877$	30.7736	1.03915	0.519575	+	0.854425i	$0.326090\pi$
$878$	2.59094	0.0874401
$879$	0	0
$880$	30.0169	1.01187
$881$	35.5730	1.19849	0.599243	−	0.800567i	$-0.295468\pi$
$881$	35.5730	1.19849	0.599243	+	0.800567i	$0.295468\pi$
$882$	0	0
$883$	−39.0447	−1.31396	−0.656979	−	0.753909i	$-0.728166\pi$
$883$	−39.0447	−1.31396	−0.656979	+	0.753909i	$0.728166\pi$
$884$	−127.411	−4.28531
$885$	0	0
$886$	17.6202	0.591961
$887$	35.4372	1.18987	0.594933	−	0.803775i	$-0.297179\pi$
$887$	35.4372	1.18987	0.594933	+	0.803775i	$0.297179\pi$
$888$	0	0
$889$	6.65010	0.223037
$890$	−5.48780	−0.183951
$891$	0	0
$892$	−96.8423	−3.24252
$893$	6.85097	0.229259
$894$	0	0
$895$	−2.53236	−0.0846473
$896$	−8.98792	−0.300265
$897$	0	0
$898$	−25.3808	−0.846970
$899$	45.2679	1.50977
$900$	0	0
$901$	64.1350	2.13665
$902$	−133.730	−4.45274
$903$	0	0
$904$	154.757	5.14713
$905$	5.10182	0.169590
$906$	0	0
$907$	−6.19074	−0.205560	−0.102780	−	0.994704i	$-0.532774\pi$
$907$	−6.19074	−0.205560	−0.102780	+	0.994704i	$0.532774\pi$
$908$	−55.4798	−1.84116
$909$	0	0
$910$	−1.90520	−0.0631567
$911$	−0.700721	−0.0232159	−0.0116080	−	0.999933i	$-0.503695\pi$
$911$	−0.700721	−0.0232159	−0.0116080	+	0.999933i	$0.503695\pi$
$912$	0	0
$913$	26.1863	0.866638
$914$	73.1612	2.41996
$915$	0	0
$916$	−29.1999	−0.964790
$917$	0.499215	0.0164855
$918$	0	0
$919$	−29.3520	−0.968233	−0.484117	−	0.875004i	$-0.660859\pi$
$919$	−29.3520	−0.968233	−0.484117	+	0.875004i	$0.660859\pi$
$920$	19.4955	0.642746
$921$	0	0
$922$	−86.7088	−2.85560
$923$	48.6139	1.60014
$924$	0	0
$925$	−25.5288	−0.839381
$926$	−47.4646	−1.55978
$927$	0	0
$928$	−163.680	−5.37306
$929$	−52.4498	−1.72082	−0.860411	−	0.509601i	$-0.829793\pi$
$929$	−52.4498	−1.72082	−0.860411	+	0.509601i	$0.829793\pi$
$930$	0	0
$931$	9.00507	0.295129
$932$	−146.809	−4.80889
$933$	0	0
$934$	35.0359	1.14641
$935$	14.4406	0.472258
$936$	0	0
$937$	−25.6384	−0.837569	−0.418784	−	0.908086i	$-0.637544\pi$
$937$	−25.6384	−0.837569	−0.418784	+	0.908086i	$0.637544\pi$
$938$	13.4719	0.439873
$939$	0	0
$940$	14.7707	0.481766
$941$	12.3906	0.403921	0.201960	−	0.979394i	$-0.435269\pi$
$941$	12.3906	0.403921	0.201960	+	0.979394i	$0.435269\pi$
$942$	0	0
$943$	−48.5744	−1.58180
$944$	16.5654	0.539158
$945$	0	0
$946$	7.61431	0.247563
$947$	7.84757	0.255012	0.127506	−	0.991838i	$-0.459303\pi$
$947$	7.84757	0.255012	0.127506	+	0.991838i	$0.459303\pi$
$948$	0	0
$949$	−28.9361	−0.939304
$950$	−16.6822	−0.541243
$951$	0	0
$952$	−20.4051	−0.661331
$953$	31.9352	1.03448	0.517242	−	0.855839i	$-0.326959\pi$
$953$	31.9352	1.03448	0.517242	+	0.855839i	$0.326959\pi$
$954$	0	0
$955$	12.3474	0.399553
$956$	−34.9666	−1.13090
$957$	0	0
$958$	−18.0462	−0.583047
$959$	−1.97011	−0.0636182
$960$	0	0
$961$	−4.84999	−0.156451
$962$	54.4567	1.75575
$963$	0	0
$964$	5.29342	0.170490
$965$	−4.88560	−0.157273
$966$	0	0
$967$	8.42368	0.270887	0.135444	−	0.990785i	$-0.456754\pi$
$967$	8.42368	0.270887	0.135444	+	0.990785i	$0.456754\pi$
$968$	−58.1489	−1.86898
$969$	0	0
$970$	−14.2886	−0.458779
$971$	44.1565	1.41705	0.708525	−	0.705686i	$-0.249361\pi$
$971$	44.1565	1.41705	0.708525	+	0.705686i	$0.249361\pi$
$972$	0	0
$973$	−3.95959	−0.126939
$974$	−52.9381	−1.69625
$975$	0	0
$976$	54.3836	1.74078
$977$	5.28247	0.169001	0.0845006	−	0.996423i	$-0.473071\pi$
$977$	5.28247	0.169001	0.0845006	+	0.996423i	$0.473071\pi$
$978$	0	0
$979$	15.9488	0.509726
$980$	19.4149	0.620187
$981$	0	0
$982$	−17.8767	−0.570469
$983$	3.33673	0.106425	0.0532125	−	0.998583i	$-0.483054\pi$
$983$	3.33673	0.106425	0.0532125	+	0.998583i	$0.483054\pi$
$984$	0	0
$985$	−3.60184	−0.114764
$986$	−154.499	−4.92026
$987$	0	0
$988$	25.8274	0.821678
$989$	2.76572	0.0879447
$990$	0	0
$991$	−21.6985	−0.689274	−0.344637	−	0.938736i	$-0.611998\pi$
$991$	−21.6985	−0.689274	−0.344637	+	0.938736i	$0.611998\pi$
$992$	−94.5534	−3.00207
$993$	0	0
$994$	12.5135	0.396904
$995$	−1.74104	−0.0551947
$996$	0	0
$997$	−10.6834	−0.338347	−0.169174	−	0.985586i	$-0.554110\pi$
$997$	−10.6834	−0.338347	−0.169174	+	0.985586i	$0.554110\pi$
$998$	−107.169	−3.39236
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2169.2.a.h.1.1		12
3.2	odd	2		241.2.a.b.1.12	✓	12
12.11	even	2		3856.2.a.n.1.11		12
15.14	odd	2		6025.2.a.h.1.1		12

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
241.2.a.b.1.12	✓	12	3.2	odd	2
2169.2.a.h.1.1		12	1.1	even	1	trivial
3856.2.a.n.1.11		12	12.11	even	2
6025.2.a.h.1.1		12	15.14	odd	2

Overview	Random
Universe	Knowledge

Classical	Maass
Hilbert	Bianchi

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

Level:	\( N \)	\(=\)	\( 2169 = 3^{2} \cdot 241 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	2169.a (trivial)

\(f(q)\)	\(=\)	\(q-2.70063 q^{2} +5.29342 q^{4} -0.533570 q^{5} +0.354992 q^{7} -8.89432 q^{8} +O(q^{10})\)	\(q-2.70063 q^{2} +5.29342 q^{4} -0.533570 q^{5} +0.354992 q^{7} -8.89432 q^{8} +1.44098 q^{10} -4.18781 q^{11} -3.72447 q^{13} -0.958703 q^{14} +13.4334 q^{16} +6.46259 q^{17} -1.31002 q^{19} -2.82441 q^{20} +11.3097 q^{22} +4.10799 q^{23} -4.71530 q^{25} +10.0584 q^{26} +1.87912 q^{28} +8.85227 q^{29} +5.11371 q^{31} -18.4902 q^{32} -17.4531 q^{34} -0.189413 q^{35} +5.41403 q^{37} +3.53789 q^{38} +4.74574 q^{40} -11.8244 q^{41} +0.673253 q^{43} -22.1678 q^{44} -11.0942 q^{46} -5.22965 q^{47} -6.87398 q^{49} +12.7343 q^{50} -19.7152 q^{52} +9.92404 q^{53} +2.23449 q^{55} -3.15741 q^{56} -23.9067 q^{58} +1.23315 q^{59} +4.04837 q^{61} -13.8102 q^{62} +23.0683 q^{64} +1.98727 q^{65} -14.0522 q^{67} +34.2092 q^{68} +0.511535 q^{70} -13.0525 q^{71} +7.76916 q^{73} -14.6213 q^{74} -6.93450 q^{76} -1.48664 q^{77} -1.17673 q^{79} -7.16768 q^{80} +31.9333 q^{82} -6.25297 q^{83} -3.44824 q^{85} -1.81821 q^{86} +37.2477 q^{88} -3.80839 q^{89} -1.32216 q^{91} +21.7453 q^{92} +14.1234 q^{94} +0.698989 q^{95} -9.91591 q^{97} +18.5641 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8}+O(q^{10}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8	\( 12 q - 3 q^{2} + 13 q^{4} - 6 q^{5} + 3 q^{7} - 9 q^{8} - 7 q^{10} - 22 q^{11} - 5 q^{13} - 6 q^{14} + 15 q^{16} + 4 q^{17} - 6 q^{19} - 10 q^{20} - 12 q^{22} - 32 q^{23} + 4 q^{25} - 8 q^{26} - 11 q^{28} - 6 q^{29} + 8 q^{31} - q^{32} - 19 q^{34} - 15 q^{35} - 8 q^{37} + 10 q^{38} - 52 q^{40} + q^{41} - 2 q^{43} - 42 q^{44} - 25 q^{46} - 34 q^{47} - 9 q^{49} + 27 q^{50} - 41 q^{52} - 5 q^{53} - 3 q^{55} - q^{56} - 33 q^{58} - 26 q^{59} - 26 q^{61} + 17 q^{62} + 13 q^{64} + 25 q^{65} + 6 q^{67} + 35 q^{68} - 4 q^{70} - 94 q^{71} - 22 q^{73} - 26 q^{74} - 20 q^{76} + 7 q^{77} + 9 q^{79} - 19 q^{80} + 15 q^{82} + 8 q^{83} + 4 q^{85} - 9 q^{86} + 6 q^{88} + 3 q^{89} - 20 q^{91} - 36 q^{92} + 48 q^{94} - 33 q^{95} - 29 q^{97} - 28 q^{98}+O(q^{100}) \) 12 * q - 3 * q^2 + 13 * q^4 - 6 * q^5 + 3 * q^7 - 9 * q^8 - 7 * q^10 - 22 * q^11 - 5 * q^13 - 6 * q^14 + 15 * q^16 + 4 * q^17 - 6 * q^19 - 10 * q^20 - 12 * q^22 - 32 * q^23 + 4 * q^25 - 8 * q^26 - 11 * q^28 - 6 * q^29 + 8 * q^31 - q^32 - 19 * q^34 - 15 * q^35 - 8 * q^37 + 10 * q^38 - 52 * q^40 + q^41 - 2 * q^43 - 42 * q^44 - 25 * q^46 - 34 * q^47 - 9 * q^49 + 27 * q^50 - 41 * q^52 - 5 * q^53 - 3 * q^55 - q^56 - 33 * q^58 - 26 * q^59 - 26 * q^61 + 17 * q^62 + 13 * q^64 + 25 * q^65 + 6 * q^67 + 35 * q^68 - 4 * q^70 - 94 * q^71 - 22 * q^73 - 26 * q^74 - 20 * q^76 + 7 * q^77 + 9 * q^79 - 19 * q^80 + 15 * q^82 + 8 * q^83 + 4 * q^85 - 9 * q^86 + 6 * q^88 + 3 * q^89 - 20 * q^91 - 36 * q^92 + 48 * q^94 - 33 * q^95 - 29 * q^97 - 28 * q^98

Introduction

L-functions

Modular forms

Varieties

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Database

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