LMFDB - Embedded newform 1287.2.a.l.1.1

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1287,2,Mod(1,1287)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1287, 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("1287.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1287 = 3^{2} \cdot 11 \cdot 13 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1287.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:	$10.2767467401$
Analytic rank:	$1$
Dimension:	$4$
Coefficient field:	4.4.11344.1
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{4} - 2x^{3} - 4x^{2} + 4x + 3 $ x^4 - 2x^3 - 4x^2 + 4*x + 3
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	yes
Fricke sign:	$1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.1
Root			$1.28734$ of defining polynomial
Character	$\chi$	$=$	1287.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q-2.63010 q^{2} +4.91744 q^{4} -2.28734 q^{5} -1.04306 q^{7} -7.67316 q^{8} +O(q^{10})$	\(q-2.63010 q^{2} +4.91744 q^{4} -2.28734 q^{5} -1.04306 q^{7} -7.67316 q^{8} +6.01593 q^{10} -1.00000 q^{11} -1.00000 q^{13} +2.74335 q^{14} +10.3463 q^{16} +5.71622 q^{17} +3.61773 q^{19} -11.2478 q^{20} +2.63010 q^{22} -0.657234 q^{23} +0.231905 q^{25} +2.63010 q^{26} -5.12918 q^{28} +6.59060 q^{29} -4.28734 q^{31} -11.8656 q^{32} -15.0342 q^{34} +2.38582 q^{35} +1.81115 q^{37} -9.51500 q^{38} +17.5511 q^{40} -7.04306 q^{41} +1.63010 q^{43} -4.91744 q^{44} +1.72859 q^{46} +2.27496 q^{47} -5.91203 q^{49} -0.609934 q^{50} -4.91744 q^{52} -12.2831 q^{53} +2.28734 q^{55} +8.00355 q^{56} -17.3339 q^{58} +5.34632 q^{59} +2.44009 q^{61} +11.2761 q^{62} +10.5150 q^{64} +2.28734 q^{65} -7.54754 q^{67} +28.1092 q^{68} -6.27496 q^{70} +10.6845 q^{71} -0.468387 q^{73} -4.76352 q^{74} +17.7900 q^{76} +1.04306 q^{77} +14.7391 q^{79} -23.6655 q^{80} +18.5240 q^{82} -11.9210 q^{83} -13.0749 q^{85} -4.28734 q^{86} +7.67316 q^{88} +7.84725 q^{89} +1.04306 q^{91} -3.23191 q^{92} -5.98339 q^{94} -8.27496 q^{95} -15.2297 q^{97} +15.5492 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 4 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) $ 4 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 6 * q^7 - 12 * q^8	$ 4 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 6 q^{7} - 12 q^{8} - 4 q^{11} - 4 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} - 2 q^{19} - 20 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{26} + 10 q^{28} - 4 q^{29} - 14 q^{31} - 6 q^{32} - 10 q^{34} - 6 q^{35} - 2 q^{37} + 8 q^{38} + 18 q^{40} - 18 q^{41} - 2 q^{43} - 8 q^{44} - 14 q^{46} - 2 q^{47} + 24 q^{50} - 8 q^{52} - 4 q^{53} + 6 q^{55} - 24 q^{58} - 16 q^{59} + 22 q^{61} + 4 q^{62} - 4 q^{64} + 6 q^{65} - 10 q^{67} + 4 q^{68} - 14 q^{70} + 2 q^{71} + 2 q^{73} - 22 q^{74} + 14 q^{76} - 6 q^{77} + 2 q^{79} - 6 q^{80} + 8 q^{82} - 4 q^{83} + 6 q^{85} - 14 q^{86} + 12 q^{88} + 16 q^{89} - 6 q^{91} - 12 q^{92} + 26 q^{94} - 22 q^{95} - 2 q^{97} - 34 q^{98}+O(q^{100}) $ 4 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 6 * q^7 - 12 * q^8 - 4 * q^11 - 4 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 - 2 * q^19 - 20 * q^20 + 2 * q^22 - 8 * q^23 + 2 * q^26 + 10 * q^28 - 4 * q^29 - 14 * q^31 - 6 * q^32 - 10 * q^34 - 6 * q^35 - 2 * q^37 + 8 * q^38 + 18 * q^40 - 18 * q^41 - 2 * q^43 - 8 * q^44 - 14 * q^46 - 2 * q^47 + 24 * q^50 - 8 * q^52 - 4 * q^53 + 6 * q^55 - 24 * q^58 - 16 * q^59 + 22 * q^61 + 4 * q^62 - 4 * q^64 + 6 * q^65 - 10 * q^67 + 4 * q^68 - 14 * q^70 + 2 * q^71 + 2 * q^73 - 22 * q^74 + 14 * q^76 - 6 * q^77 + 2 * q^79 - 6 * q^80 + 8 * q^82 - 4 * q^83 + 6 * q^85 - 14 * q^86 + 12 * q^88 + 16 * q^89 - 6 * q^91 - 12 * q^92 + 26 * q^94 - 22 * q^95 - 2 * q^97 - 34 * 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 $

$2$	−2.63010	−1.85976	−0.929882	−	0.367859i	$-0.880091\pi$
$2$	−2.63010	−1.85976	−0.929882	+	0.367859i	$0.880091\pi$
$3$	0	0
$3$	0	0
$4$	4.91744	2.45872
$5$	−2.28734	−1.02293	−0.511464	−	0.859305i	$-0.670897\pi$
$5$	−2.28734	−1.02293	−0.511464	+	0.859305i	$0.670897\pi$
$6$	0	0
$7$	−1.04306	−0.394239	−0.197120	−	0.980379i	$-0.563159\pi$
$7$	−1.04306	−0.394239	−0.197120	+	0.980379i	$0.563159\pi$
$8$	−7.67316	−2.71287
$9$	0	0
$10$	6.01593	1.90240
$11$	−1.00000	−0.301511
$11$	−1.00000	−0.301511
$12$	0	0
$13$	−1.00000	−0.277350
$13$	−1.00000	−0.277350
$14$	2.74335	0.733191
$15$	0	0
$16$	10.3463	2.58658
$17$	5.71622	1.38639	0.693193	−	0.720752i	$-0.256203\pi$
$17$	5.71622	1.38639	0.693193	+	0.720752i	$0.256203\pi$
$18$	0	0
$19$	3.61773	0.829964	0.414982	−	0.909830i	$-0.363788\pi$
$19$	3.61773	0.829964	0.414982	+	0.909830i	$0.363788\pi$
$20$	−11.2478	−2.51509
$21$	0	0
$22$	2.63010	0.560740
$23$	−0.657234	−0.137043	−0.0685213	−	0.997650i	$-0.521828\pi$
$23$	−0.657234	−0.137043	−0.0685213	+	0.997650i	$0.521828\pi$
$24$	0	0
$25$	0.231905	0.0463810
$26$	2.63010	0.515805
$27$	0	0
$28$	−5.12918	−0.969323
$29$	6.59060	1.22384	0.611922	−	0.790918i	$-0.290397\pi$
$29$	6.59060	1.22384	0.611922	+	0.790918i	$0.290397\pi$
$30$	0	0
$31$	−4.28734	−0.770028	−0.385014	−	0.922911i	$-0.625803\pi$
$31$	−4.28734	−0.770028	−0.385014	+	0.922911i	$0.625803\pi$
$32$	−11.8656	−2.09755
$33$	0	0
$34$	−15.0342	−2.57835
$35$	2.38582	0.403278
$36$	0	0
$37$	1.81115	0.297752	0.148876	−	0.988856i	$-0.452435\pi$
$37$	1.81115	0.297752	0.148876	+	0.988856i	$0.452435\pi$
$38$	−9.51500	−1.54354
$39$	0	0
$40$	17.5511	2.77507
$41$	−7.04306	−1.09994	−0.549970	−	0.835184i	$-0.685361\pi$
$41$	−7.04306	−1.09994	−0.549970	+	0.835184i	$0.685361\pi$
$42$	0	0
$43$	1.63010	0.248588	0.124294	−	0.992245i	$-0.460333\pi$
$43$	1.63010	0.248588	0.124294	+	0.992245i	$0.460333\pi$
$44$	−4.91744	−0.741332
$45$	0	0
$46$	1.72859	0.254867
$47$	2.27496	0.331838	0.165919	−	0.986139i	$-0.446941\pi$
$47$	2.27496	0.331838	0.165919	+	0.986139i	$0.446941\pi$
$48$	0	0
$49$	−5.91203	−0.844576
$50$	−0.609934	−0.0862577
$51$	0	0
$52$	−4.91744	−0.681926
$53$	−12.2831	−1.68721	−0.843606	−	0.536962i	$-0.819572\pi$
$53$	−12.2831	−1.68721	−0.843606	+	0.536962i	$0.819572\pi$
$54$	0	0
$55$	2.28734	0.308424
$56$	8.00355	1.06952
$57$	0	0
$58$	−17.3339	−2.27606
$59$	5.34632	0.696032	0.348016	−	0.937489i	$-0.386856\pi$
$59$	5.34632	0.696032	0.348016	+	0.937489i	$0.386856\pi$
$60$	0	0
$61$	2.44009	0.312421	0.156211	−	0.987724i	$-0.450072\pi$
$61$	2.44009	0.312421	0.156211	+	0.987724i	$0.450072\pi$
$62$	11.2761	1.43207
$63$	0	0
$64$	10.5150	1.31438
$65$	2.28734	0.283709
$66$	0	0
$67$	−7.54754	−0.922079	−0.461039	−	0.887380i	$-0.652523\pi$
$67$	−7.54754	−0.922079	−0.461039	+	0.887380i	$0.652523\pi$
$68$	28.1092	3.40874
$69$	0	0
$70$	−6.27496	−0.750002
$71$	10.6845	1.26802	0.634009	−	0.773326i	$-0.281408\pi$
$71$	10.6845	1.26802	0.634009	+	0.773326i	$0.281408\pi$
$72$	0	0
$73$	−0.468387	−0.0548205	−0.0274103	−	0.999624i	$-0.508726\pi$
$73$	−0.468387	−0.0548205	−0.0274103	+	0.999624i	$0.508726\pi$
$74$	−4.76352	−0.553748
$75$	0	0
$76$	17.7900	2.04065
$77$	1.04306	0.118868
$78$	0	0
$79$	14.7391	1.65828	0.829140	−	0.559041i	$-0.188831\pi$
$79$	14.7391	1.65828	0.829140	+	0.559041i	$0.188831\pi$
$80$	−23.6655	−2.64588
$81$	0	0
$82$	18.5240	2.04563
$83$	−11.9210	−1.30850	−0.654249	−	0.756279i	$-0.727015\pi$
$83$	−11.9210	−1.30850	−0.654249	+	0.756279i	$0.727015\pi$
$84$	0	0
$85$	−13.0749	−1.41817
$86$	−4.28734	−0.462315
$87$	0	0
$88$	7.67316	0.817962
$89$	7.84725	0.831807	0.415903	−	0.909409i	$-0.363465\pi$
$89$	7.84725	0.831807	0.415903	+	0.909409i	$0.363465\pi$
$90$	0	0
$91$	1.04306	0.109342
$92$	−3.23191	−0.336949
$93$	0	0
$94$	−5.98339	−0.617139
$95$	−8.27496	−0.848993
$96$	0	0
$97$	−15.2297	−1.54634	−0.773169	−	0.634200i	$-0.781330\pi$
$97$	−15.2297	−1.54634	−0.773169	+	0.634200i	$0.781330\pi$
$98$	15.5492	1.57071
$99$	0	0
$100$	1.14038	0.114038
$101$	−4.90423	−0.487989	−0.243995	−	0.969777i	$-0.578458\pi$
$101$	−4.90423	−0.487989	−0.243995	+	0.969777i	$0.578458\pi$
$102$	0	0
$103$	−18.6065	−1.83336	−0.916678	−	0.399627i	$-0.869140\pi$
$103$	−18.6065	−1.83336	−0.916678	+	0.399627i	$0.869140\pi$
$104$	7.67316	0.752415
$105$	0	0
$106$	32.3058	3.13782
$107$	−4.33736	−0.419308	−0.209654	−	0.977776i	$-0.567234\pi$
$107$	−4.33736	−0.419308	−0.209654	+	0.977776i	$0.567234\pi$
$108$	0	0
$109$	6.05202	0.579679	0.289839	−	0.957075i	$-0.406398\pi$
$109$	6.05202	0.579679	0.289839	+	0.957075i	$0.406398\pi$
$110$	−6.01593	−0.573596
$111$	0	0
$112$	−10.7918	−1.01973
$113$	−16.1583	−1.52005	−0.760023	−	0.649896i	$-0.774812\pi$
$113$	−16.1583	−1.52005	−0.760023	+	0.649896i	$0.774812\pi$
$114$	0	0
$115$	1.50331	0.140185
$116$	32.4089	3.00909
$117$	0	0
$118$	−14.0614	−1.29445
$119$	−5.96235	−0.546568
$120$	0	0
$121$	1.00000	0.0909091
$122$	−6.41768	−0.581029
$123$	0	0
$124$	−21.0827	−1.89328
$125$	10.9062	0.975483
$126$	0	0
$127$	0.780467	0.0692553	0.0346276	−	0.999400i	$-0.488975\pi$
$127$	0.780467	0.0692553	0.0346276	+	0.999400i	$0.488975\pi$
$128$	−3.92440	−0.346871
$129$	0	0
$130$	−6.01593	−0.527632
$131$	−12.2444	−1.06980	−0.534900	−	0.844915i	$-0.679651\pi$
$131$	−12.2444	−1.06980	−0.534900	+	0.844915i	$0.679651\pi$
$132$	0	0
$133$	−3.77350	−0.327204
$134$	19.8508	1.71485
$135$	0	0
$136$	−43.8615	−3.76109
$137$	−20.3972	−1.74265	−0.871324	−	0.490708i	$-0.836738\pi$
$137$	−20.3972	−1.74265	−0.871324	+	0.490708i	$0.836738\pi$
$138$	0	0
$139$	−5.16527	−0.438113	−0.219056	−	0.975712i	$-0.570298\pi$
$139$	−5.16527	−0.438113	−0.219056	+	0.975712i	$0.570298\pi$
$140$	11.7321	0.991547
$141$	0	0
$142$	−28.1014	−2.35821
$143$	1.00000	0.0836242
$144$	0	0
$145$	−15.0749	−1.25190
$146$	1.23191	0.101953
$147$	0	0
$148$	8.90623	0.732088
$149$	−2.81656	−0.230742	−0.115371	−	0.993322i	$-0.536806\pi$
$149$	−2.81656	−0.230742	−0.115371	+	0.993322i	$0.536806\pi$
$150$	0	0
$151$	−18.7376	−1.52484	−0.762420	−	0.647082i	$-0.775989\pi$
$151$	−18.7376	−1.52484	−0.762420	+	0.647082i	$0.775989\pi$
$152$	−27.7594	−2.25159
$153$	0	0
$154$	−2.74335	−0.221065
$155$	9.80658	0.787683
$156$	0	0
$157$	1.31802	0.105190	0.0525948	−	0.998616i	$-0.483251\pi$
$157$	1.31802	0.105190	0.0525948	+	0.998616i	$0.483251\pi$
$158$	−38.7654	−3.08401
$159$	0	0
$160$	27.1405	2.14565
$161$	0.685533	0.0540276
$162$	0	0
$163$	−22.8315	−1.78830	−0.894149	−	0.447769i	$-0.852219\pi$
$163$	−22.8315	−1.78830	−0.894149	+	0.447769i	$0.852219\pi$
$164$	−34.6338	−2.70445
$165$	0	0
$166$	31.3534	2.43350
$167$	9.37107	0.725155	0.362577	−	0.931954i	$-0.381897\pi$
$167$	9.37107	0.725155	0.362577	+	0.931954i	$0.381897\pi$
$168$	0	0
$169$	1.00000	0.0769231
$170$	34.3884	2.63747
$171$	0	0
$172$	8.01593	0.611209
$173$	−6.45601	−0.490842	−0.245421	−	0.969417i	$-0.578926\pi$
$173$	−6.45601	−0.490842	−0.245421	+	0.969417i	$0.578926\pi$
$174$	0	0
$175$	−0.241891	−0.0182852
$176$	−10.3463	−0.779883
$177$	0	0
$178$	−20.6391	−1.54696
$179$	−10.6891	−0.798940	−0.399470	−	0.916746i	$-0.630806\pi$
$179$	−10.6891	−0.798940	−0.399470	+	0.916746i	$0.630806\pi$
$180$	0	0
$181$	19.8384	1.47458	0.737289	−	0.675577i	$-0.236105\pi$
$181$	19.8384	1.47458	0.737289	+	0.675577i	$0.236105\pi$
$182$	−2.74335	−0.203351
$183$	0	0
$184$	5.04306	0.371779
$185$	−4.14272	−0.304579
$186$	0	0
$187$	−5.71622	−0.418011
$188$	11.1870	0.815895
$189$	0	0
$190$	21.7640	1.57893
$191$	10.7469	0.777619	0.388809	−	0.921318i	$-0.372886\pi$
$191$	10.7469	0.777619	0.388809	+	0.921318i	$0.372886\pi$
$192$	0	0
$193$	13.9551	1.00451	0.502255	−	0.864720i	$-0.332504\pi$
$193$	13.9551	1.00451	0.502255	+	0.864720i	$0.332504\pi$
$194$	40.0556	2.87582
$195$	0	0
$196$	−29.0720	−2.07657
$197$	−19.8869	−1.41688	−0.708441	−	0.705770i	$-0.750601\pi$
$197$	−19.8869	−1.41688	−0.708441	+	0.705770i	$0.750601\pi$
$198$	0	0
$199$	−23.6589	−1.67714	−0.838569	−	0.544796i	$-0.816607\pi$
$199$	−23.6589	−1.67714	−0.838569	+	0.544796i	$0.816607\pi$
$200$	−1.77945	−0.125826
$201$	0	0
$202$	12.8986	0.907545
$203$	−6.87438	−0.482487
$204$	0	0
$205$	16.1098	1.12516
$206$	48.9371	3.40961
$207$	0	0
$208$	−10.3463	−0.717388
$209$	−3.61773	−0.250244
$210$	0	0
$211$	−24.8727	−1.71231	−0.856153	−	0.516723i	$-0.827152\pi$
$211$	−24.8727	−1.71231	−0.856153	+	0.516723i	$0.827152\pi$
$212$	−60.4014	−4.14838
$213$	0	0
$214$	11.4077	0.779814
$215$	−3.72859	−0.254288
$216$	0	0
$217$	4.47194	0.303575
$218$	−15.9174	−1.07806
$219$	0	0
$220$	11.2478	0.758329
$221$	−5.71622	−0.384515
$222$	0	0
$223$	17.8059	1.19237	0.596185	−	0.802847i	$-0.296682\pi$
$223$	17.8059	1.19237	0.596185	+	0.802847i	$0.296682\pi$
$224$	12.3765	0.826938
$225$	0	0
$226$	42.4980	2.82692
$227$	−13.8442	−0.918874	−0.459437	−	0.888210i	$-0.651949\pi$
$227$	−13.8442	−0.918874	−0.459437	+	0.888210i	$0.651949\pi$
$228$	0	0
$229$	−2.22070	−0.146748	−0.0733740	−	0.997304i	$-0.523377\pi$
$229$	−2.22070	−0.146748	−0.0733740	+	0.997304i	$0.523377\pi$
$230$	−3.95387	−0.260710
$231$	0	0
$232$	−50.5707	−3.32013
$233$	−12.0454	−0.789123	−0.394562	−	0.918869i	$-0.629104\pi$
$233$	−12.0454	−0.789123	−0.394562	+	0.918869i	$0.629104\pi$
$234$	0	0
$235$	−5.20361	−0.339446
$236$	26.2902	1.71135
$237$	0	0
$238$	15.6816	1.01649
$239$	5.89372	0.381233	0.190616	−	0.981665i	$-0.438951\pi$
$239$	5.89372	0.381233	0.190616	+	0.981665i	$0.438951\pi$
$240$	0	0
$241$	−12.0474	−0.776044	−0.388022	−	0.921650i	$-0.626842\pi$
$241$	−12.0474	−0.776044	−0.388022	+	0.921650i	$0.626842\pi$
$242$	−2.63010	−0.169069
$243$	0	0
$244$	11.9990	0.768156
$245$	13.5228	0.863940
$246$	0	0
$247$	−3.61773	−0.230191
$248$	32.8974	2.08899
$249$	0	0
$250$	−28.6845	−1.81417
$251$	28.1180	1.77479	0.887395	−	0.461009i	$-0.152513\pi$
$251$	28.1180	1.77479	0.887395	+	0.461009i	$0.152513\pi$
$252$	0	0
$253$	0.657234	0.0413199
$254$	−2.05271	−0.128798
$255$	0	0
$256$	−10.7084	−0.669276
$257$	2.38480	0.148760	0.0743800	−	0.997230i	$-0.476302\pi$
$257$	2.38480	0.148760	0.0743800	+	0.997230i	$0.476302\pi$
$258$	0	0
$259$	−1.88914	−0.117385
$260$	11.2478	0.697561
$261$	0	0
$262$	32.2041	1.98958
$263$	6.99815	0.431524	0.215762	−	0.976446i	$-0.430776\pi$
$263$	6.99815	0.431524	0.215762	+	0.976446i	$0.430776\pi$
$264$	0	0
$265$	28.0956	1.72590
$266$	9.92470	0.608522
$267$	0	0
$268$	−37.1146	−2.26713
$269$	−21.1883	−1.29187	−0.645937	−	0.763391i	$-0.723533\pi$
$269$	−21.1883	−1.29187	−0.645937	+	0.763391i	$0.723533\pi$
$270$	0	0
$271$	−1.78285	−0.108301	−0.0541503	−	0.998533i	$-0.517245\pi$
$271$	−1.78285	−0.108301	−0.0541503	+	0.998533i	$0.517245\pi$
$272$	59.1418	3.58600
$273$	0	0
$274$	53.6467	3.24091
$275$	−0.231905	−0.0139844
$276$	0	0
$277$	18.0071	1.08194	0.540971	−	0.841041i	$-0.318057\pi$
$277$	18.0071	1.08194	0.540971	+	0.841041i	$0.318057\pi$
$278$	13.5852	0.814786
$279$	0	0
$280$	−18.3068	−1.09404
$281$	−6.86001	−0.409234	−0.204617	−	0.978842i	$-0.565595\pi$
$281$	−6.86001	−0.409234	−0.204617	+	0.978842i	$0.565595\pi$
$282$	0	0
$283$	17.6314	1.04808	0.524039	−	0.851694i	$-0.324424\pi$
$283$	17.6314	1.04808	0.524039	+	0.851694i	$0.324424\pi$
$284$	52.5404	3.11770
$285$	0	0
$286$	−2.63010	−0.155521
$287$	7.34632	0.433640
$288$	0	0
$289$	15.6752	0.922068
$290$	39.6486	2.32824
$291$	0	0
$292$	−2.30326	−0.134788
$293$	21.9730	1.28368	0.641839	−	0.766840i	$-0.278172\pi$
$293$	21.9730	1.28368	0.641839	+	0.766840i	$0.278172\pi$
$294$	0	0
$295$	−12.2288	−0.711990
$296$	−13.8973	−0.807762
$297$	0	0
$298$	7.40785	0.429125
$299$	0.657234	0.0380088
$300$	0	0
$301$	−1.70029	−0.0980032
$302$	49.2817	2.83584
$303$	0	0
$304$	37.4302	2.14677
$305$	−5.58130	−0.319584
$306$	0	0
$307$	30.3507	1.73221	0.866103	−	0.499865i	$-0.166617\pi$
$307$	30.3507	1.73221	0.866103	+	0.499865i	$0.166617\pi$
$308$	5.12918	0.292262
$309$	0	0
$310$	−25.7923	−1.46490
$311$	20.3952	1.15650	0.578252	−	0.815858i	$-0.303735\pi$
$311$	20.3952	1.15650	0.578252	+	0.815858i	$0.303735\pi$
$312$	0	0
$313$	−33.9951	−1.92152	−0.960758	−	0.277389i	$-0.910531\pi$
$313$	−33.9951	−1.92152	−0.960758	+	0.277389i	$0.910531\pi$
$314$	−3.46653	−0.195628
$315$	0	0
$316$	72.4787	4.07724
$317$	1.42688	0.0801417	0.0400709	−	0.999197i	$-0.487242\pi$
$317$	1.42688	0.0801417	0.0400709	+	0.999197i	$0.487242\pi$
$318$	0	0
$319$	−6.59060	−0.369003
$320$	−24.0513	−1.34451
$321$	0	0
$322$	−1.80302	−0.100478
$323$	20.6797	1.15065
$324$	0	0
$325$	−0.231905	−0.0128638
$326$	60.0491	3.32581
$327$	0	0
$328$	54.0425	2.98400
$329$	−2.37292	−0.130823
$330$	0	0
$331$	5.37262	0.295306	0.147653	−	0.989039i	$-0.452828\pi$
$331$	5.37262	0.295306	0.147653	+	0.989039i	$0.452828\pi$
$332$	−58.6207	−3.21723
$333$	0	0
$334$	−24.6469	−1.34862
$335$	17.2638	0.943220
$336$	0	0
$337$	4.44822	0.242310	0.121155	−	0.992634i	$-0.461340\pi$
$337$	4.44822	0.242310	0.121155	+	0.992634i	$0.461340\pi$
$338$	−2.63010	−0.143059
$339$	0	0
$340$	−64.2951	−3.48689
$341$	4.28734	0.232172
$342$	0	0
$343$	13.4680	0.727204
$344$	−12.5080	−0.674388
$345$	0	0
$346$	16.9800	0.912849
$347$	−21.7501	−1.16761	−0.583803	−	0.811896i	$-0.698436\pi$
$347$	−21.7501	−1.16761	−0.583803	+	0.811896i	$0.698436\pi$
$348$	0	0
$349$	−21.3216	−1.14132	−0.570659	−	0.821187i	$-0.693312\pi$
$349$	−21.3216	−1.14132	−0.570659	+	0.821187i	$0.693312\pi$
$350$	0.636197	0.0340062
$351$	0	0
$352$	11.8656	0.632437
$353$	−6.61758	−0.352218	−0.176109	−	0.984371i	$-0.556351\pi$
$353$	−6.61758	−0.352218	−0.176109	+	0.984371i	$0.556351\pi$
$354$	0	0
$355$	−24.4391	−1.29709
$356$	38.5884	2.04518
$357$	0	0
$358$	28.1134	1.48584
$359$	−3.37331	−0.178036	−0.0890181	−	0.996030i	$-0.528373\pi$
$359$	−3.37331	−0.178036	−0.0890181	+	0.996030i	$0.528373\pi$
$360$	0	0
$361$	−5.91203	−0.311159
$362$	−52.1771	−2.74237
$363$	0	0
$364$	5.12918	0.268842
$365$	1.07136	0.0560774
$366$	0	0
$367$	8.27394	0.431896	0.215948	−	0.976405i	$-0.430716\pi$
$367$	8.27394	0.431896	0.215948	+	0.976405i	$0.430716\pi$
$368$	−6.79995	−0.354472
$369$	0	0
$370$	10.8958	0.566444
$371$	12.8120	0.665165
$372$	0	0
$373$	21.8667	1.13222	0.566108	−	0.824331i	$-0.308448\pi$
$373$	21.8667	1.13222	0.566108	+	0.824331i	$0.308448\pi$
$374$	15.0342	0.777402
$375$	0	0
$376$	−17.4562	−0.900233
$377$	−6.59060	−0.339433
$378$	0	0
$379$	11.6031	0.596012	0.298006	−	0.954564i	$-0.403678\pi$
$379$	11.6031	0.596012	0.298006	+	0.954564i	$0.403678\pi$
$380$	−40.6916	−2.08744
$381$	0	0
$382$	−28.2655	−1.44619
$383$	9.91388	0.506576	0.253288	−	0.967391i	$-0.418488\pi$
$383$	9.91388	0.506576	0.253288	+	0.967391i	$0.418488\pi$
$384$	0	0
$385$	−2.38582	−0.121593
$386$	−36.7033	−1.86815
$387$	0	0
$388$	−74.8909	−3.80201
$389$	−17.6155	−0.893141	−0.446570	−	0.894748i	$-0.647355\pi$
$389$	−17.6155	−0.893141	−0.446570	+	0.894748i	$0.647355\pi$
$390$	0	0
$391$	−3.75689	−0.189994
$392$	45.3639	2.29123
$393$	0	0
$394$	52.3046	2.63507
$395$	−33.7133	−1.69630
$396$	0	0
$397$	4.99001	0.250442	0.125221	−	0.992129i	$-0.460036\pi$
$397$	4.99001	0.250442	0.125221	+	0.992129i	$0.460036\pi$
$398$	62.2254	3.11908
$399$	0	0
$400$	2.39937	0.119968
$401$	−33.1451	−1.65519	−0.827594	−	0.561328i	$-0.810291\pi$
$401$	−33.1451	−1.65519	−0.827594	+	0.561328i	$0.810291\pi$
$402$	0	0
$403$	4.28734	0.213567
$404$	−24.1163	−1.19983
$405$	0	0
$406$	18.0803	0.897311
$407$	−1.81115	−0.0897755
$408$	0	0
$409$	−7.87608	−0.389447	−0.194724	−	0.980858i	$-0.562381\pi$
$409$	−7.87608	−0.389447	−0.194724	+	0.980858i	$0.562381\pi$
$410$	−42.3705	−2.09253
$411$	0	0
$412$	−91.4964	−4.50771
$413$	−5.57653	−0.274403
$414$	0	0
$415$	27.2673	1.33850
$416$	11.8656	0.581757
$417$	0	0
$418$	9.51500	0.465394
$419$	−15.5764	−0.760956	−0.380478	−	0.924790i	$-0.624240\pi$
$419$	−15.5764	−0.760956	−0.380478	+	0.924790i	$0.624240\pi$
$420$	0	0
$421$	7.68057	0.374328	0.187164	−	0.982329i	$-0.440070\pi$
$421$	7.68057	0.374328	0.187164	+	0.982329i	$0.440070\pi$
$422$	65.4177	3.18448
$423$	0	0
$424$	94.2502	4.57719
$425$	1.32562	0.0643021
$426$	0	0
$427$	−2.54515	−0.123169
$428$	−21.3287	−1.03096
$429$	0	0
$430$	9.80658	0.472915
$431$	−11.0724	−0.533338	−0.266669	−	0.963788i	$-0.585923\pi$
$431$	−11.0724	−0.533338	−0.266669	+	0.963788i	$0.585923\pi$
$432$	0	0
$433$	12.0179	0.577545	0.288772	−	0.957398i	$-0.406753\pi$
$433$	12.0179	0.577545	0.288772	+	0.957398i	$0.406753\pi$
$434$	−11.7617	−0.564578
$435$	0	0
$436$	29.7604	1.42527
$437$	−2.37769	−0.113741
$438$	0	0
$439$	1.34934	0.0644007	0.0322003	−	0.999481i	$-0.489749\pi$
$439$	1.34934	0.0644007	0.0322003	+	0.999481i	$0.489749\pi$
$440$	−17.5511	−0.836716
$441$	0	0
$442$	15.0342	0.715106
$443$	−20.0637	−0.953256	−0.476628	−	0.879105i	$-0.658141\pi$
$443$	−20.0637	−0.953256	−0.476628	+	0.879105i	$0.658141\pi$
$444$	0	0
$445$	−17.9493	−0.850878
$446$	−46.8313	−2.21753
$447$	0	0
$448$	−10.9678	−0.518178
$449$	−7.82353	−0.369215	−0.184607	−	0.982812i	$-0.559101\pi$
$449$	−7.82353	−0.369215	−0.184607	+	0.982812i	$0.559101\pi$
$450$	0	0
$451$	7.04306	0.331645
$452$	−79.4575	−3.73737
$453$	0	0
$454$	36.4117	1.70889
$455$	−2.38582	−0.111849
$456$	0	0
$457$	−12.7401	−0.595956	−0.297978	−	0.954573i	$-0.596312\pi$
$457$	−12.7401	−0.595956	−0.297978	+	0.954573i	$0.596312\pi$
$458$	5.84067	0.272917
$459$	0	0
$460$	7.39245	0.344675
$461$	−32.3646	−1.50737	−0.753686	−	0.657235i	$-0.771726\pi$
$461$	−32.3646	−1.50737	−0.753686	+	0.657235i	$0.771726\pi$
$462$	0	0
$463$	29.9110	1.39008	0.695040	−	0.718971i	$-0.255387\pi$
$463$	29.9110	1.39008	0.695040	+	0.718971i	$0.255387\pi$
$464$	68.1885	3.16557
$465$	0	0
$466$	31.6808	1.46758
$467$	−12.9667	−0.600029	−0.300015	−	0.953935i	$-0.596992\pi$
$467$	−12.9667	−0.600029	−0.300015	+	0.953935i	$0.596992\pi$
$468$	0	0
$469$	7.87253	0.363519
$470$	13.6860	0.631289
$471$	0	0
$472$	−41.0232	−1.88825
$473$	−1.63010	−0.0749522
$474$	0	0
$475$	0.838970	0.0384946
$476$	−29.3195	−1.34386
$477$	0	0
$478$	−15.5011	−0.709003
$479$	−6.54058	−0.298847	−0.149423	−	0.988773i	$-0.547742\pi$
$479$	−6.54058	−0.298847	−0.149423	+	0.988773i	$0.547742\pi$
$480$	0	0
$481$	−1.81115	−0.0825815
$482$	31.6860	1.44326
$483$	0	0
$484$	4.91744	0.223520
$485$	34.8354	1.58179
$486$	0	0
$487$	−30.5012	−1.38214	−0.691071	−	0.722787i	$-0.742861\pi$
$487$	−30.5012	−1.38214	−0.691071	+	0.722787i	$0.742861\pi$
$488$	−18.7232	−0.847559
$489$	0	0
$490$	−35.5663	−1.60672
$491$	37.9471	1.71253	0.856263	−	0.516540i	$-0.172780\pi$
$491$	37.9471	1.71253	0.856263	+	0.516540i	$0.172780\pi$
$492$	0	0
$493$	37.6733	1.69672
$494$	9.51500	0.428100
$495$	0	0
$496$	−44.3582	−1.99174
$497$	−11.1446	−0.499902
$498$	0	0
$499$	42.2473	1.89125	0.945624	−	0.325261i	$-0.105452\pi$
$499$	42.2473	1.89125	0.945624	+	0.325261i	$0.105452\pi$
$500$	53.6307	2.39844
$501$	0	0
$502$	−73.9531	−3.30069
$503$	3.97526	0.177248	0.0886239	−	0.996065i	$-0.471753\pi$
$503$	3.97526	0.177248	0.0886239	+	0.996065i	$0.471753\pi$
$504$	0	0
$505$	11.2176	0.499178
$506$	−1.72859	−0.0768453
$507$	0	0
$508$	3.83790	0.170279
$509$	17.2689	0.765429	0.382715	−	0.923867i	$-0.374989\pi$
$509$	17.2689	0.765429	0.382715	+	0.923867i	$0.374989\pi$
$510$	0	0
$511$	0.488555	0.0216124
$512$	36.0131	1.59157
$513$	0	0
$514$	−6.27228	−0.276658
$515$	42.5594	1.87539
$516$	0	0
$517$	−2.27496	−0.100053
$518$	4.96863	0.218309
$519$	0	0
$520$	−17.5511	−0.769666
$521$	−30.8649	−1.35221	−0.676107	−	0.736803i	$-0.736334\pi$
$521$	−30.8649	−1.35221	−0.676107	+	0.736803i	$0.736334\pi$
$522$	0	0
$523$	14.1326	0.617975	0.308987	−	0.951066i	$-0.400010\pi$
$523$	14.1326	0.617975	0.308987	+	0.951066i	$0.400010\pi$
$524$	−60.2112	−2.63034
$525$	0	0
$526$	−18.4058	−0.802533
$527$	−24.5074	−1.06756
$528$	0	0
$529$	−22.5680	−0.981219
$530$	−73.8942	−3.20976
$531$	0	0
$532$	−18.5560	−0.804503
$533$	7.04306	0.305069
$534$	0	0
$535$	9.92099	0.428922
$536$	57.9135	2.50148
$537$	0	0
$538$	55.7274	2.40258
$539$	5.91203	0.254649
$540$	0	0
$541$	42.9510	1.84661	0.923304	−	0.384071i	$-0.125478\pi$
$541$	42.9510	1.84661	0.923304	+	0.384071i	$0.125478\pi$
$542$	4.68909	0.201414
$543$	0	0
$544$	−67.8262	−2.90802
$545$	−13.8430	−0.592969
$546$	0	0
$547$	−1.95455	−0.0835707	−0.0417854	−	0.999127i	$-0.513305\pi$
$547$	−1.95455	−0.0835707	−0.0417854	+	0.999127i	$0.513305\pi$
$548$	−100.302	−4.28468
$549$	0	0
$550$	0.609934	0.0260077
$551$	23.8430	1.01575
$552$	0	0
$553$	−15.3738	−0.653758
$554$	−47.3605	−2.01216
$555$	0	0
$556$	−25.3999	−1.07720
$557$	−19.3594	−0.820283	−0.410142	−	0.912022i	$-0.634521\pi$
$557$	−19.3594	−0.820283	−0.410142	+	0.912022i	$0.634521\pi$
$558$	0	0
$559$	−1.63010	−0.0689460
$560$	24.6845	1.04311
$561$	0	0
$562$	18.0425	0.761078
$563$	36.4343	1.53552	0.767761	−	0.640736i	$-0.221371\pi$
$563$	36.4343	1.53552	0.767761	+	0.640736i	$0.221371\pi$
$564$	0	0
$565$	36.9595	1.55490
$566$	−46.3724	−1.94918
$567$	0	0
$568$	−81.9840	−3.43997
$569$	−8.37905	−0.351268	−0.175634	−	0.984456i	$-0.556198\pi$
$569$	−8.37905	−0.351268	−0.175634	+	0.984456i	$0.556198\pi$
$570$	0	0
$571$	−28.3793	−1.18764	−0.593819	−	0.804598i	$-0.702381\pi$
$571$	−28.3793	−1.18764	−0.593819	+	0.804598i	$0.702381\pi$
$572$	4.91744	0.205608
$573$	0	0
$574$	−19.3216	−0.806467
$575$	−0.152416	−0.00635618
$576$	0	0
$577$	5.65416	0.235386	0.117693	−	0.993050i	$-0.462450\pi$
$577$	5.65416	0.235386	0.117693	+	0.993050i	$0.462450\pi$
$578$	−41.2273	−1.71483
$579$	0	0
$580$	−74.1300	−3.07808
$581$	12.4343	0.515861
$582$	0	0
$583$	12.2831	0.508714
$584$	3.59401	0.148721
$585$	0	0
$586$	−57.7913	−2.38734
$587$	−2.37603	−0.0980692	−0.0490346	−	0.998797i	$-0.515614\pi$
$587$	−2.37603	−0.0980692	−0.0490346	+	0.998797i	$0.515614\pi$
$588$	0	0
$589$	−15.5104	−0.639096
$590$	32.1631	1.32413
$591$	0	0
$592$	18.7388	0.770159
$593$	−19.6269	−0.805979	−0.402990	−	0.915204i	$-0.632029\pi$
$593$	−19.6269	−0.805979	−0.402990	+	0.915204i	$0.632029\pi$
$594$	0	0
$595$	13.6379	0.559099
$596$	−13.8503	−0.567329
$597$	0	0
$598$	−1.72859	−0.0706874
$599$	−19.5660	−0.799445	−0.399723	−	0.916636i	$-0.630894\pi$
$599$	−19.5660	−0.799445	−0.399723	+	0.916636i	$0.630894\pi$
$600$	0	0
$601$	−45.7491	−1.86614	−0.933071	−	0.359691i	$-0.882882\pi$
$601$	−45.7491	−1.86614	−0.933071	+	0.359691i	$0.882882\pi$
$602$	4.47194	0.182263
$603$	0	0
$604$	−92.1408	−3.74915
$605$	−2.28734	−0.0929934
$606$	0	0
$607$	−8.30984	−0.337286	−0.168643	−	0.985677i	$-0.553938\pi$
$607$	−8.30984	−0.337286	−0.168643	+	0.985677i	$0.553938\pi$
$608$	−42.9264	−1.74090
$609$	0	0
$610$	14.6794	0.594351
$611$	−2.27496	−0.0920352
$612$	0	0
$613$	47.6994	1.92656	0.963280	−	0.268500i	$-0.0865280\pi$
$613$	47.6994	1.92656	0.963280	+	0.268500i	$0.0865280\pi$
$614$	−79.8255	−3.22149
$615$	0	0
$616$	−8.00355	−0.322472
$617$	38.9081	1.56638	0.783190	−	0.621782i	$-0.213591\pi$
$617$	38.9081	1.56638	0.783190	+	0.621782i	$0.213591\pi$
$618$	0	0
$619$	43.8879	1.76400	0.882001	−	0.471247i	$-0.156196\pi$
$619$	43.8879	1.76400	0.882001	+	0.471247i	$0.156196\pi$
$620$	48.2232	1.93669
$621$	0	0
$622$	−53.6414	−2.15082
$623$	−8.18514	−0.327931
$624$	0	0
$625$	−26.1057	−1.04423
$626$	89.4105	3.57356
$627$	0	0
$628$	6.48129	0.258632
$629$	10.3529	0.412799
$630$	0	0
$631$	−35.8930	−1.42888	−0.714439	−	0.699697i	$-0.753318\pi$
$631$	−35.8930	−1.42888	−0.714439	+	0.699697i	$0.753318\pi$
$632$	−113.096	−4.49870
$633$	0	0
$634$	−3.75285	−0.149045
$635$	−1.78519	−0.0708431
$636$	0	0
$637$	5.91203	0.234243
$638$	17.3339	0.686258
$639$	0	0
$640$	8.97642	0.354824
$641$	22.2655	0.879433	0.439716	−	0.898137i	$-0.355079\pi$
$641$	22.2655	0.879433	0.439716	+	0.898137i	$0.355079\pi$
$642$	0	0
$643$	10.4372	0.411601	0.205801	−	0.978594i	$-0.434020\pi$
$643$	10.4372	0.411601	0.205801	+	0.978594i	$0.434020\pi$
$644$	3.37107	0.132839
$645$	0	0
$646$	−54.3898	−2.13994
$647$	−43.4782	−1.70931	−0.854653	−	0.519200i	$-0.826230\pi$
$647$	−43.4782	−1.70931	−0.854653	+	0.519200i	$0.826230\pi$
$648$	0	0
$649$	−5.34632	−0.209861
$650$	0.609934	0.0239236
$651$	0	0
$652$	−112.272	−4.39692
$653$	8.45222	0.330761	0.165380	−	0.986230i	$-0.447115\pi$
$653$	8.45222	0.330761	0.165380	+	0.986230i	$0.447115\pi$
$654$	0	0
$655$	28.0071	1.09433
$656$	−72.8697	−2.84509
$657$	0	0
$658$	6.24102	0.243300
$659$	30.2207	1.17723	0.588616	−	0.808413i	$-0.299673\pi$
$659$	30.2207	1.17723	0.588616	+	0.808413i	$0.299673\pi$
$660$	0	0
$661$	−12.9826	−0.504963	−0.252481	−	0.967602i	$-0.581247\pi$
$661$	−12.9826	−0.504963	−0.252481	+	0.967602i	$0.581247\pi$
$662$	−14.1305	−0.549199
$663$	0	0
$664$	91.4717	3.54979
$665$	8.63127	0.334706
$666$	0	0
$667$	−4.33156	−0.167719
$668$	46.0816	1.78295
$669$	0	0
$670$	−45.4055	−1.75417
$671$	−2.44009	−0.0941985
$672$	0	0
$673$	34.7730	1.34040	0.670200	−	0.742181i	$-0.266208\pi$
$673$	34.7730	1.34040	0.670200	+	0.742181i	$0.266208\pi$
$674$	−11.6993	−0.450639
$675$	0	0
$676$	4.91744	0.189132
$677$	25.6609	0.986230	0.493115	−	0.869964i	$-0.335858\pi$
$677$	25.6609	0.986230	0.493115	+	0.869964i	$0.335858\pi$
$678$	0	0
$679$	15.8854	0.609627
$680$	100.326	3.84732
$681$	0	0
$682$	−11.2761	−0.431785
$683$	4.06406	0.155507	0.0777534	−	0.996973i	$-0.475225\pi$
$683$	4.06406	0.155507	0.0777534	+	0.996973i	$0.475225\pi$
$684$	0	0
$685$	46.6552	1.78260
$686$	−35.4222	−1.35243
$687$	0	0
$688$	16.8656	0.642994
$689$	12.2831	0.467949
$690$	0	0
$691$	27.0129	1.02762	0.513809	−	0.857905i	$-0.328234\pi$
$691$	27.0129	1.02762	0.513809	+	0.857905i	$0.328234\pi$
$692$	−31.7471	−1.20684
$693$	0	0
$694$	57.2049	2.17147
$695$	11.8147	0.448157
$696$	0	0
$697$	−40.2597	−1.52494
$698$	56.0779	2.12258
$699$	0	0
$700$	−1.18948	−0.0449582
$701$	28.3523	1.07085	0.535425	−	0.844583i	$-0.320151\pi$
$701$	28.3523	1.07085	0.535425	+	0.844583i	$0.320151\pi$
$702$	0	0
$703$	6.55226	0.247123
$704$	−10.5150	−0.396299
$705$	0	0
$706$	17.4049	0.655043
$707$	5.11540	0.192384
$708$	0	0
$709$	−20.4875	−0.769425	−0.384713	−	0.923036i	$-0.625699\pi$
$709$	−20.4875	−0.769425	−0.384713	+	0.923036i	$0.625699\pi$
$710$	64.2772	2.41228
$711$	0	0
$712$	−60.2132	−2.25659
$713$	2.81778	0.105527
$714$	0	0
$715$	−2.28734	−0.0855415
$716$	−52.5629	−1.96437
$717$	0	0
$718$	8.87214	0.331105
$719$	7.40263	0.276071	0.138036	−	0.990427i	$-0.455921\pi$
$719$	7.40263	0.276071	0.138036	+	0.990427i	$0.455921\pi$
$720$	0	0
$721$	19.4077	0.722780
$722$	15.5492	0.578683
$723$	0	0
$724$	97.5543	3.62558
$725$	1.52839	0.0567631
$726$	0	0
$727$	−39.4870	−1.46449	−0.732246	−	0.681040i	$-0.761528\pi$
$727$	−39.4870	−1.46449	−0.732246	+	0.681040i	$0.761528\pi$
$728$	−8.00355	−0.296631
$729$	0	0
$730$	−2.81778	−0.104291
$731$	9.31802	0.344639
$732$	0	0
$733$	−32.3670	−1.19550	−0.597751	−	0.801682i	$-0.703939\pi$
$733$	−32.3670	−1.19550	−0.597751	+	0.801682i	$0.703939\pi$
$734$	−21.7613	−0.803225
$735$	0	0
$736$	7.79845	0.287455
$737$	7.54754	0.278017
$738$	0	0
$739$	39.8326	1.46527	0.732634	−	0.680623i	$-0.238291\pi$
$739$	39.8326	1.46527	0.732634	+	0.680623i	$0.238291\pi$
$740$	−20.3715	−0.748873
$741$	0	0
$742$	−33.6968	−1.23705
$743$	−22.3445	−0.819739	−0.409870	−	0.912144i	$-0.634426\pi$
$743$	−22.3445	−0.819739	−0.409870	+	0.912144i	$0.634426\pi$
$744$	0	0
$745$	6.44242	0.236032
$746$	−57.5117	−2.10565
$747$	0	0
$748$	−28.1092	−1.02777
$749$	4.52412	0.165308
$750$	0	0
$751$	23.0564	0.841340	0.420670	−	0.907214i	$-0.361795\pi$
$751$	23.0564	0.841340	0.420670	+	0.907214i	$0.361795\pi$
$752$	23.5375	0.858324
$753$	0	0
$754$	17.3339	0.631265
$755$	42.8591	1.55980
$756$	0	0
$757$	−8.89084	−0.323143	−0.161572	−	0.986861i	$-0.551656\pi$
$757$	−8.89084	−0.323143	−0.161572	+	0.986861i	$0.551656\pi$
$758$	−30.5174	−1.10844
$759$	0	0
$760$	63.4951	2.30321
$761$	28.1157	1.01919	0.509597	−	0.860413i	$-0.329794\pi$
$761$	28.1157	1.01919	0.509597	+	0.860413i	$0.329794\pi$
$762$	0	0
$763$	−6.31261	−0.228532
$764$	52.8472	1.91195
$765$	0	0
$766$	−26.0745	−0.942111
$767$	−5.34632	−0.193044
$768$	0	0
$769$	−47.4256	−1.71021	−0.855106	−	0.518453i	$-0.826508\pi$
$769$	−47.4256	−1.71021	−0.855106	+	0.518453i	$0.826508\pi$
$770$	6.27496	0.226134
$771$	0	0
$772$	68.6233	2.46981
$773$	−38.9207	−1.39988	−0.699940	−	0.714202i	$-0.746790\pi$
$773$	−38.9207	−1.39988	−0.699940	+	0.714202i	$0.746790\pi$
$774$	0	0
$775$	−0.994255	−0.0357147
$776$	116.860	4.19502
$777$	0	0
$778$	46.3305	1.66103
$779$	−25.4799	−0.912912
$780$	0	0
$781$	−10.6845	−0.382322
$782$	9.88101	0.353344
$783$	0	0
$784$	−61.1678	−2.18456
$785$	−3.01476	−0.107601
$786$	0	0
$787$	−19.3302	−0.689046	−0.344523	−	0.938778i	$-0.611959\pi$
$787$	−19.3302	−0.689046	−0.344523	+	0.938778i	$0.611959\pi$
$788$	−97.7926	−3.48372
$789$	0	0
$790$	88.6694	3.15472
$791$	16.8541	0.599261
$792$	0	0
$793$	−2.44009	−0.0866500
$794$	−13.1242	−0.465762
$795$	0	0
$796$	−116.341	−4.12361
$797$	−37.4572	−1.32680	−0.663401	−	0.748264i	$-0.730888\pi$
$797$	−37.4572	−1.32680	−0.663401	+	0.748264i	$0.730888\pi$
$798$	0	0
$799$	13.0042	0.460055
$800$	−2.75169	−0.0972868
$801$	0	0
$802$	87.1750	3.07826
$803$	0.468387	0.0165290
$804$	0	0
$805$	−1.56804	−0.0552663
$806$	−11.2761	−0.397185
$807$	0	0
$808$	37.6310	1.32385
$809$	−35.8134	−1.25913	−0.629567	−	0.776946i	$-0.716768\pi$
$809$	−35.8134	−1.25913	−0.629567	+	0.776946i	$0.716768\pi$
$810$	0	0
$811$	−5.27871	−0.185360	−0.0926802	−	0.995696i	$-0.529543\pi$
$811$	−5.27871	−0.185360	−0.0926802	+	0.995696i	$0.529543\pi$
$812$	−33.8043	−1.18630
$813$	0	0
$814$	4.76352	0.166961
$815$	52.2232	1.82930
$816$	0	0
$817$	5.89727	0.206319
$818$	20.7149	0.724279
$819$	0	0
$820$	79.2191	2.76645
$821$	54.5253	1.90295	0.951473	−	0.307733i	$-0.0995704\pi$
$821$	54.5253	1.90295	0.951473	+	0.307733i	$0.0995704\pi$
$822$	0	0
$823$	33.1480	1.15547	0.577733	−	0.816226i	$-0.303938\pi$
$823$	33.1480	1.15547	0.577733	+	0.816226i	$0.303938\pi$
$824$	142.771	4.97366
$825$	0	0
$826$	14.6668	0.510324
$827$	19.2224	0.668428	0.334214	−	0.942497i	$-0.391529\pi$
$827$	19.2224	0.668428	0.334214	+	0.942497i	$0.391529\pi$
$828$	0	0
$829$	49.9057	1.73330	0.866648	−	0.498920i	$-0.166270\pi$
$829$	49.9057	1.73330	0.866648	+	0.498920i	$0.166270\pi$
$830$	−71.7158	−2.48929
$831$	0	0
$832$	−10.5150	−0.364542
$833$	−33.7945	−1.17091
$834$	0	0
$835$	−21.4348	−0.741781
$836$	−17.7900	−0.615279
$837$	0	0
$838$	40.9675	1.41520
$839$	46.4437	1.60342	0.801708	−	0.597716i	$-0.203925\pi$
$839$	46.4437	1.60342	0.801708	+	0.597716i	$0.203925\pi$
$840$	0	0
$841$	14.4360	0.497793
$842$	−20.2007	−0.696162
$843$	0	0
$844$	−122.310	−4.21008
$845$	−2.28734	−0.0786867
$846$	0	0
$847$	−1.04306	−0.0358399
$848$	−127.085	−4.36411
$849$	0	0
$850$	−3.48652	−0.119587
$851$	−1.19035	−0.0408047
$852$	0	0
$853$	−13.4077	−0.459071	−0.229535	−	0.973300i	$-0.573721\pi$
$853$	−13.4077	−0.459071	−0.229535	+	0.973300i	$0.573721\pi$
$854$	6.69401	0.229064
$855$	0	0
$856$	33.2812	1.13753
$857$	14.1160	0.482192	0.241096	−	0.970501i	$-0.422493\pi$
$857$	14.1160	0.482192	0.241096	+	0.970501i	$0.422493\pi$
$858$	0	0
$859$	−29.5257	−1.00740	−0.503701	−	0.863878i	$-0.668029\pi$
$859$	−29.5257	−1.00740	−0.503701	+	0.863878i	$0.668029\pi$
$860$	−18.3351	−0.625222
$861$	0	0
$862$	29.1215	0.991882
$863$	23.6294	0.804355	0.402177	−	0.915562i	$-0.368254\pi$
$863$	23.6294	0.804355	0.402177	+	0.915562i	$0.368254\pi$
$864$	0	0
$865$	14.7671	0.502096
$866$	−31.6084	−1.07410
$867$	0	0
$868$	21.9905	0.746406
$869$	−14.7391	−0.499990
$870$	0	0
$871$	7.54754	0.255739
$872$	−46.4381	−1.57259
$873$	0	0
$874$	6.25358	0.211530
$875$	−11.3758	−0.384574
$876$	0	0
$877$	−37.8403	−1.27778	−0.638888	−	0.769299i	$-0.720605\pi$
$877$	−37.8403	−1.27778	−0.638888	+	0.769299i	$0.720605\pi$
$878$	−3.54891	−0.119770
$879$	0	0
$880$	23.6655	0.797764
$881$	40.8333	1.37571	0.687855	−	0.725848i	$-0.258553\pi$
$881$	40.8333	1.37571	0.687855	+	0.725848i	$0.258553\pi$
$882$	0	0
$883$	29.7830	1.00228	0.501138	−	0.865367i	$-0.332915\pi$
$883$	29.7830	1.00228	0.501138	+	0.865367i	$0.332915\pi$
$884$	−28.1092	−0.945413
$885$	0	0
$886$	52.7696	1.77283
$887$	22.6757	0.761377	0.380688	−	0.924703i	$-0.375687\pi$
$887$	22.6757	0.761377	0.380688	+	0.924703i	$0.375687\pi$
$888$	0	0
$889$	−0.814073	−0.0273031
$890$	47.2085	1.58243
$891$	0	0
$892$	87.5594	2.93170
$893$	8.23020	0.275413
$894$	0	0
$895$	24.4495	0.817258
$896$	4.09338	0.136750
$897$	0	0
$898$	20.5767	0.686652
$899$	−28.2561	−0.942394
$900$	0	0
$901$	−70.2129	−2.33913
$902$	−18.5240	−0.616781
$903$	0	0
$904$	123.985	4.12369
$905$	−45.3772	−1.50839
$906$	0	0
$907$	−14.1899	−0.471167	−0.235583	−	0.971854i	$-0.575700\pi$
$907$	−14.1899	−0.471167	−0.235583	+	0.971854i	$0.575700\pi$
$908$	−68.0781	−2.25925
$909$	0	0
$910$	6.27496	0.208013
$911$	−37.5643	−1.24456	−0.622281	−	0.782794i	$-0.713794\pi$
$911$	−37.5643	−1.24456	−0.622281	+	0.782794i	$0.713794\pi$
$912$	0	0
$913$	11.9210	0.394527
$914$	33.5077	1.10834
$915$	0	0
$916$	−10.9202	−0.360812
$917$	12.7716	0.421757
$918$	0	0
$919$	−9.00015	−0.296887	−0.148444	−	0.988921i	$-0.547426\pi$
$919$	−9.00015	−0.296887	−0.148444	+	0.988921i	$0.547426\pi$
$920$	−11.5352	−0.380303
$921$	0	0
$922$	85.1223	2.80335
$923$	−10.6845	−0.351685
$924$	0	0
$925$	0.420016	0.0138100
$926$	−78.6689	−2.58522
$927$	0	0
$928$	−78.2012	−2.56708
$929$	−48.5260	−1.59209	−0.796043	−	0.605241i	$-0.793077\pi$
$929$	−48.5260	−1.59209	−0.796043	+	0.605241i	$0.793077\pi$
$930$	0	0
$931$	−21.3881	−0.700968
$932$	−59.2327	−1.94023
$933$	0	0
$934$	34.1038	1.11591
$935$	13.0749	0.427595
$936$	0	0
$937$	−6.30682	−0.206035	−0.103017	−	0.994680i	$-0.532850\pi$
$937$	−6.30682	−0.206035	−0.103017	+	0.994680i	$0.532850\pi$
$938$	−20.7055	−0.676060
$939$	0	0
$940$	−25.5884	−0.834602
$941$	−30.4463	−0.992520	−0.496260	−	0.868174i	$-0.665294\pi$
$941$	−30.4463	−0.992520	−0.496260	+	0.868174i	$0.665294\pi$
$942$	0	0
$943$	4.62893	0.150739
$944$	55.3148	1.80034
$945$	0	0
$946$	4.28734	0.139393
$947$	21.3070	0.692385	0.346192	−	0.938164i	$-0.387474\pi$
$947$	21.3070	0.692385	0.346192	+	0.938164i	$0.387474\pi$
$948$	0	0
$949$	0.468387	0.0152045
$950$	−2.20658	−0.0715908
$951$	0	0
$952$	45.7501	1.48277
$953$	−52.6930	−1.70689	−0.853447	−	0.521180i	$-0.825492\pi$
$953$	−52.6930	−1.70689	−0.853447	+	0.521180i	$0.825492\pi$
$954$	0	0
$955$	−24.5818	−0.795448
$956$	28.9820	0.937344
$957$	0	0
$958$	17.2024	0.555784
$959$	21.2754	0.687020
$960$	0	0
$961$	−12.6188	−0.407057
$962$	4.76352	0.153582
$963$	0	0
$964$	−59.2426	−1.90807
$965$	−31.9200	−1.02754
$966$	0	0
$967$	−55.4017	−1.78160	−0.890800	−	0.454396i	$-0.849855\pi$
$967$	−55.4017	−1.78160	−0.890800	+	0.454396i	$0.849855\pi$
$968$	−7.67316	−0.246625
$969$	0	0
$970$	−91.6206	−2.94176
$971$	48.7118	1.56324	0.781618	−	0.623758i	$-0.214395\pi$
$971$	48.7118	1.56324	0.781618	+	0.623758i	$0.214395\pi$
$972$	0	0
$973$	5.38768	0.172721
$974$	80.2213	2.57046
$975$	0	0
$976$	25.2459	0.808102
$977$	4.06693	0.130113	0.0650563	−	0.997882i	$-0.479277\pi$
$977$	4.06693	0.130113	0.0650563	+	0.997882i	$0.479277\pi$
$978$	0	0
$979$	−7.84725	−0.250799
$980$	66.4975	2.12419
$981$	0	0
$982$	−99.8046	−3.18489
$983$	55.6521	1.77503	0.887513	−	0.460782i	$-0.152431\pi$
$983$	55.6521	1.77503	0.887513	+	0.460782i	$0.152431\pi$
$984$	0	0
$985$	45.4880	1.44937
$986$	−99.0847	−3.15550
$987$	0	0
$988$	−17.7900	−0.565974
$989$	−1.07136	−0.0340672
$990$	0	0
$991$	16.8842	0.536344	0.268172	−	0.963371i	$-0.413581\pi$
$991$	16.8842	0.536344	0.268172	+	0.963371i	$0.413581\pi$
$992$	50.8717	1.61518
$993$	0	0
$994$	29.3114	0.929700
$995$	54.1159	1.71559
$996$	0	0
$997$	3.50296	0.110940	0.0554700	−	0.998460i	$-0.482334\pi$
$997$	3.50296	0.110940	0.0554700	+	0.998460i	$0.482334\pi$
$998$	−111.115	−3.51727
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1287.2.a.l.1.1	✓	4
3.2	odd	2		1287.2.a.n.1.4	yes	4

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
1287.2.a.l.1.1	✓	4	1.1	even	1	trivial
1287.2.a.n.1.4	yes	4	3.2	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:	\( N \)	\(=\)	\( 1287 = 3^{2} \cdot 11 \cdot 13 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1287.a (trivial)

\(f(q)\)	\(=\)	\(q-2.63010 q^{2} +4.91744 q^{4} -2.28734 q^{5} -1.04306 q^{7} -7.67316 q^{8} +O(q^{10})\)	\(q-2.63010 q^{2} +4.91744 q^{4} -2.28734 q^{5} -1.04306 q^{7} -7.67316 q^{8} +6.01593 q^{10} -1.00000 q^{11} -1.00000 q^{13} +2.74335 q^{14} +10.3463 q^{16} +5.71622 q^{17} +3.61773 q^{19} -11.2478 q^{20} +2.63010 q^{22} -0.657234 q^{23} +0.231905 q^{25} +2.63010 q^{26} -5.12918 q^{28} +6.59060 q^{29} -4.28734 q^{31} -11.8656 q^{32} -15.0342 q^{34} +2.38582 q^{35} +1.81115 q^{37} -9.51500 q^{38} +17.5511 q^{40} -7.04306 q^{41} +1.63010 q^{43} -4.91744 q^{44} +1.72859 q^{46} +2.27496 q^{47} -5.91203 q^{49} -0.609934 q^{50} -4.91744 q^{52} -12.2831 q^{53} +2.28734 q^{55} +8.00355 q^{56} -17.3339 q^{58} +5.34632 q^{59} +2.44009 q^{61} +11.2761 q^{62} +10.5150 q^{64} +2.28734 q^{65} -7.54754 q^{67} +28.1092 q^{68} -6.27496 q^{70} +10.6845 q^{71} -0.468387 q^{73} -4.76352 q^{74} +17.7900 q^{76} +1.04306 q^{77} +14.7391 q^{79} -23.6655 q^{80} +18.5240 q^{82} -11.9210 q^{83} -13.0749 q^{85} -4.28734 q^{86} +7.67316 q^{88} +7.84725 q^{89} +1.04306 q^{91} -3.23191 q^{92} -5.98339 q^{94} -8.27496 q^{95} -15.2297 q^{97} +15.5492 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 4 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 6 q^{7} - 12 q^{8}+O(q^{10}) \) 4 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 6 * q^7 - 12 * q^8	\( 4 q - 2 q^{2} + 8 q^{4} - 6 q^{5} + 6 q^{7} - 12 q^{8} - 4 q^{11} - 4 q^{13} - 4 q^{14} + 4 q^{16} - 6 q^{17} - 2 q^{19} - 20 q^{20} + 2 q^{22} - 8 q^{23} + 2 q^{26} + 10 q^{28} - 4 q^{29} - 14 q^{31} - 6 q^{32} - 10 q^{34} - 6 q^{35} - 2 q^{37} + 8 q^{38} + 18 q^{40} - 18 q^{41} - 2 q^{43} - 8 q^{44} - 14 q^{46} - 2 q^{47} + 24 q^{50} - 8 q^{52} - 4 q^{53} + 6 q^{55} - 24 q^{58} - 16 q^{59} + 22 q^{61} + 4 q^{62} - 4 q^{64} + 6 q^{65} - 10 q^{67} + 4 q^{68} - 14 q^{70} + 2 q^{71} + 2 q^{73} - 22 q^{74} + 14 q^{76} - 6 q^{77} + 2 q^{79} - 6 q^{80} + 8 q^{82} - 4 q^{83} + 6 q^{85} - 14 q^{86} + 12 q^{88} + 16 q^{89} - 6 q^{91} - 12 q^{92} + 26 q^{94} - 22 q^{95} - 2 q^{97} - 34 q^{98}+O(q^{100}) \) 4 * q - 2 * q^2 + 8 * q^4 - 6 * q^5 + 6 * q^7 - 12 * q^8 - 4 * q^11 - 4 * q^13 - 4 * q^14 + 4 * q^16 - 6 * q^17 - 2 * q^19 - 20 * q^20 + 2 * q^22 - 8 * q^23 + 2 * q^26 + 10 * q^28 - 4 * q^29 - 14 * q^31 - 6 * q^32 - 10 * q^34 - 6 * q^35 - 2 * q^37 + 8 * q^38 + 18 * q^40 - 18 * q^41 - 2 * q^43 - 8 * q^44 - 14 * q^46 - 2 * q^47 + 24 * q^50 - 8 * q^52 - 4 * q^53 + 6 * q^55 - 24 * q^58 - 16 * q^59 + 22 * q^61 + 4 * q^62 - 4 * q^64 + 6 * q^65 - 10 * q^67 + 4 * q^68 - 14 * q^70 + 2 * q^71 + 2 * q^73 - 22 * q^74 + 14 * q^76 - 6 * q^77 + 2 * q^79 - 6 * q^80 + 8 * q^82 - 4 * q^83 + 6 * q^85 - 14 * q^86 + 12 * q^88 + 16 * q^89 - 6 * q^91 - 12 * q^92 + 26 * q^94 - 22 * q^95 - 2 * q^97 - 34 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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