LMFDB - Embedded newform 1143.2.a.e.1.3

Show commands: Magma / PariGP / SageMath

Newspace parameters

comment: Compute space of new eigenforms

[N,k,chi] = [1143,2,Mod(1,1143)]

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

lf = mfeigenbasis(mf)

from sage.modular.dirichlet import DirichletCharacter

H = DirichletGroup(1143, 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("1143.1");

S:= CuspForms(chi, 2);

N := Newforms(S);

Level:	$ N $	$=$	$ 1143 = 3^{2} \cdot 127 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	1143.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:	$9.12690095103$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	$\Q(\zeta_{18})^+$
comment: defining polynomial gp: f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 3x - 1 $ x^3 - 3*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 127)
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ of defining polynomial
Character	$\chi$	$=$	1143.1

$q$-expansion

comment: q-expansion

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

gp: mfcoefs(f, 20)

$f(q)$	$=$	$q+2.53209 q^{2} +4.41147 q^{4} +0.120615 q^{5} +0.879385 q^{7} +6.10607 q^{8} +O(q^{10})$	\(q+2.53209 q^{2} +4.41147 q^{4} +0.120615 q^{5} +0.879385 q^{7} +6.10607 q^{8} +0.305407 q^{10} +2.71688 q^{11} -5.10607 q^{13} +2.22668 q^{14} +6.63816 q^{16} +4.46791 q^{17} +2.87939 q^{19} +0.532089 q^{20} +6.87939 q^{22} +5.22668 q^{23} -4.98545 q^{25} -12.9290 q^{26} +3.87939 q^{28} -5.94356 q^{29} +1.42602 q^{31} +4.59627 q^{32} +11.3131 q^{34} +0.106067 q^{35} -10.5817 q^{37} +7.29086 q^{38} +0.736482 q^{40} +5.38919 q^{41} +8.27631 q^{43} +11.9855 q^{44} +13.2344 q^{46} +6.43376 q^{47} -6.22668 q^{49} -12.6236 q^{50} -22.5253 q^{52} -12.6236 q^{53} +0.327696 q^{55} +5.36959 q^{56} -15.0496 q^{58} -2.71688 q^{59} -1.98040 q^{61} +3.61081 q^{62} -1.63816 q^{64} -0.615867 q^{65} -2.87939 q^{67} +19.7101 q^{68} +0.268571 q^{70} -9.49794 q^{71} +11.0915 q^{73} -26.7939 q^{74} +12.7023 q^{76} +2.38919 q^{77} -7.72462 q^{79} +0.800660 q^{80} +13.6459 q^{82} -11.4561 q^{83} +0.538896 q^{85} +20.9564 q^{86} +16.5895 q^{88} +2.64496 q^{89} -4.49020 q^{91} +23.0574 q^{92} +16.2909 q^{94} +0.347296 q^{95} +1.68004 q^{97} -15.7665 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 6 * q^8	$ 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} + 3 q^{10} - 3 q^{13} + 3 q^{16} + 18 q^{17} + 3 q^{19} - 3 q^{20} + 15 q^{22} + 9 q^{23} + 3 q^{25} - 6 q^{26} + 6 q^{28} - 3 q^{29} + 12 q^{31} + 12 q^{34} - 12 q^{35} + 6 q^{38} - 3 q^{40} + 12 q^{41} - 9 q^{43} + 18 q^{44} + 9 q^{46} + 3 q^{47} - 12 q^{49} - 3 q^{50} - 21 q^{52} - 3 q^{53} - 3 q^{55} + 9 q^{56} - 18 q^{58} - 3 q^{61} + 15 q^{62} + 12 q^{64} + 9 q^{65} - 3 q^{67} + 9 q^{68} - 9 q^{70} - 3 q^{71} + 3 q^{73} - 24 q^{74} + 12 q^{76} + 3 q^{77} + 9 q^{79} - 12 q^{80} - 12 q^{83} + 39 q^{85} + 9 q^{86} - 6 q^{88} + 33 q^{89} - 12 q^{91} + 18 q^{92} + 33 q^{94} - 15 q^{97} - 12 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 6 * q^8 + 3 * q^10 - 3 * q^13 + 3 * q^16 + 18 * q^17 + 3 * q^19 - 3 * q^20 + 15 * q^22 + 9 * q^23 + 3 * q^25 - 6 * q^26 + 6 * q^28 - 3 * q^29 + 12 * q^31 + 12 * q^34 - 12 * q^35 + 6 * q^38 - 3 * q^40 + 12 * q^41 - 9 * q^43 + 18 * q^44 + 9 * q^46 + 3 * q^47 - 12 * q^49 - 3 * q^50 - 21 * q^52 - 3 * q^53 - 3 * q^55 + 9 * q^56 - 18 * q^58 - 3 * q^61 + 15 * q^62 + 12 * q^64 + 9 * q^65 - 3 * q^67 + 9 * q^68 - 9 * q^70 - 3 * q^71 + 3 * q^73 - 24 * q^74 + 12 * q^76 + 3 * q^77 + 9 * q^79 - 12 * q^80 - 12 * q^83 + 39 * q^85 + 9 * q^86 - 6 * q^88 + 33 * q^89 - 12 * q^91 + 18 * q^92 + 33 * q^94 - 15 * q^97 - 12 * 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 $

$2$	2.53209	1.79046	0.895229	−	0.445607i	$-0.147012\pi$
$2$	2.53209	1.79046	0.895229	+	0.445607i	$0.147012\pi$
$3$	0	0
$3$	0	0
$4$	4.41147	2.20574
$5$	0.120615	0.0539406	0.0269703	−	0.999636i	$-0.491414\pi$
$5$	0.120615	0.0539406	0.0269703	+	0.999636i	$0.491414\pi$
$6$	0	0
$7$	0.879385	0.332376	0.166188	−	0.986094i	$-0.446854\pi$
$7$	0.879385	0.332376	0.166188	+	0.986094i	$0.446854\pi$
$8$	6.10607	2.15882
$9$	0	0
$10$	0.305407	0.0965783
$11$	2.71688	0.819171	0.409585	−	0.912272i	$-0.365673\pi$
$11$	2.71688	0.819171	0.409585	+	0.912272i	$0.365673\pi$
$12$	0	0
$13$	−5.10607	−1.41617	−0.708084	−	0.706128i	$-0.750440\pi$
$13$	−5.10607	−1.41617	−0.708084	+	0.706128i	$0.750440\pi$
$14$	2.22668	0.595106
$15$	0	0
$16$	6.63816	1.65954
$17$	4.46791	1.08363	0.541814	−	0.840499i	$-0.317738\pi$
$17$	4.46791	1.08363	0.541814	+	0.840499i	$0.317738\pi$
$18$	0	0
$19$	2.87939	0.660576	0.330288	−	0.943880i	$-0.392854\pi$
$19$	2.87939	0.660576	0.330288	+	0.943880i	$0.392854\pi$
$20$	0.532089	0.118979
$21$	0	0
$22$	6.87939	1.46669
$23$	5.22668	1.08984	0.544919	−	0.838489i	$-0.316560\pi$
$23$	5.22668	1.08984	0.544919	+	0.838489i	$0.316560\pi$
$24$	0	0
$25$	−4.98545	−0.997090
$26$	−12.9290	−2.53559
$27$	0	0
$28$	3.87939	0.733135
$29$	−5.94356	−1.10369	−0.551846	−	0.833946i	$-0.686076\pi$
$29$	−5.94356	−1.10369	−0.551846	+	0.833946i	$0.686076\pi$
$30$	0	0
$31$	1.42602	0.256121	0.128061	−	0.991766i	$-0.459125\pi$
$31$	1.42602	0.256121	0.128061	+	0.991766i	$0.459125\pi$
$32$	4.59627	0.812513
$33$	0	0
$34$	11.3131	1.94019
$35$	0.106067	0.0179286
$36$	0	0
$37$	−10.5817	−1.73962	−0.869812	−	0.493383i	$-0.835760\pi$
$37$	−10.5817	−1.73962	−0.869812	+	0.493383i	$0.835760\pi$
$38$	7.29086	1.18273
$39$	0	0
$40$	0.736482	0.116448
$41$	5.38919	0.841649	0.420825	−	0.907142i	$-0.361741\pi$
$41$	5.38919	0.841649	0.420825	+	0.907142i	$0.361741\pi$
$42$	0	0
$43$	8.27631	1.26213	0.631063	−	0.775732i	$-0.282619\pi$
$43$	8.27631	1.26213	0.631063	+	0.775732i	$0.282619\pi$
$44$	11.9855	1.80687
$45$	0	0
$46$	13.2344	1.95131
$47$	6.43376	0.938461	0.469230	−	0.883076i	$-0.344531\pi$
$47$	6.43376	0.938461	0.469230	+	0.883076i	$0.344531\pi$
$48$	0	0
$49$	−6.22668	−0.889526
$50$	−12.6236	−1.78525
$51$	0	0
$52$	−22.5253	−3.12369
$53$	−12.6236	−1.73399	−0.866993	−	0.498320i	$-0.833950\pi$
$53$	−12.6236	−1.73399	−0.866993	+	0.498320i	$0.833950\pi$
$54$	0	0
$55$	0.327696	0.0441865
$56$	5.36959	0.717541
$57$	0	0
$58$	−15.0496	−1.97611
$59$	−2.71688	−0.353708	−0.176854	−	0.984237i	$-0.556592\pi$
$59$	−2.71688	−0.353708	−0.176854	+	0.984237i	$0.556592\pi$
$60$	0	0
$61$	−1.98040	−0.253564	−0.126782	−	0.991931i	$-0.540465\pi$
$61$	−1.98040	−0.253564	−0.126782	+	0.991931i	$0.540465\pi$
$62$	3.61081	0.458574
$63$	0	0
$64$	−1.63816	−0.204769
$65$	−0.615867	−0.0763889
$66$	0	0
$67$	−2.87939	−0.351773	−0.175886	−	0.984410i	$-0.556279\pi$
$67$	−2.87939	−0.351773	−0.175886	+	0.984410i	$0.556279\pi$
$68$	19.7101	2.39020
$69$	0	0
$70$	0.268571	0.0321003
$71$	−9.49794	−1.12720	−0.563599	−	0.826048i	$-0.690584\pi$
$71$	−9.49794	−1.12720	−0.563599	+	0.826048i	$0.690584\pi$
$72$	0	0
$73$	11.0915	1.29816	0.649082	−	0.760718i	$-0.275153\pi$
$73$	11.0915	1.29816	0.649082	+	0.760718i	$0.275153\pi$
$74$	−26.7939	−3.11472
$75$	0	0
$76$	12.7023	1.45706
$77$	2.38919	0.272273
$78$	0	0
$79$	−7.72462	−0.869088	−0.434544	−	0.900651i	$-0.643090\pi$
$79$	−7.72462	−0.869088	−0.434544	+	0.900651i	$0.643090\pi$
$80$	0.800660	0.0895165
$81$	0	0
$82$	13.6459	1.50694
$83$	−11.4561	−1.25747	−0.628733	−	0.777622i	$-0.716426\pi$
$83$	−11.4561	−1.25747	−0.628733	+	0.777622i	$0.716426\pi$
$84$	0	0
$85$	0.538896	0.0584515
$86$	20.9564	2.25978
$87$	0	0
$88$	16.5895	1.76844
$89$	2.64496	0.280366	0.140183	−	0.990126i	$-0.455231\pi$
$89$	2.64496	0.280366	0.140183	+	0.990126i	$0.455231\pi$
$90$	0	0
$91$	−4.49020	−0.470701
$92$	23.0574	2.40390
$93$	0	0
$94$	16.2909	1.68027
$95$	0.347296	0.0356319
$96$	0	0
$97$	1.68004	0.170583	0.0852914	−	0.996356i	$-0.472818\pi$
$97$	1.68004	0.170583	0.0852914	+	0.996356i	$0.472818\pi$
$98$	−15.7665	−1.59266
$99$	0	0
$100$	−21.9932	−2.19932
$101$	16.7246	1.66416	0.832081	−	0.554654i	$-0.187149\pi$
$101$	16.7246	1.66416	0.832081	+	0.554654i	$0.187149\pi$
$102$	0	0
$103$	0.162504	0.0160120	0.00800599	−	0.999968i	$-0.497452\pi$
$103$	0.162504	0.0160120	0.00800599	+	0.999968i	$0.497452\pi$
$104$	−31.1780	−3.05725
$105$	0	0
$106$	−31.9641	−3.10463
$107$	−14.4757	−1.39941	−0.699707	−	0.714430i	$-0.746686\pi$
$107$	−14.4757	−1.39941	−0.699707	+	0.714430i	$0.746686\pi$
$108$	0	0
$109$	9.36959	0.897443	0.448722	−	0.893672i	$-0.351879\pi$
$109$	9.36959	0.897443	0.448722	+	0.893672i	$0.351879\pi$
$110$	0.829755	0.0791141
$111$	0	0
$112$	5.83750	0.551592
$113$	8.90167	0.837399	0.418700	−	0.908125i	$-0.362486\pi$
$113$	8.90167	0.837399	0.418700	+	0.908125i	$0.362486\pi$
$114$	0	0
$115$	0.630415	0.0587865
$116$	−26.2199	−2.43445
$117$	0	0
$118$	−6.87939	−0.633299
$119$	3.92902	0.360172
$120$	0	0
$121$	−3.61856	−0.328960
$122$	−5.01455	−0.453996
$123$	0	0
$124$	6.29086	0.564936
$125$	−1.20439	−0.107724
$126$	0	0
$127$	−1.00000	−0.0887357
$127$	−1.00000	−0.0887357
$128$	−13.3405	−1.17914
$129$	0	0
$130$	−1.55943	−0.136771
$131$	22.0009	1.92223	0.961115	−	0.276148i	$-0.0890579\pi$
$131$	22.0009	1.92223	0.961115	+	0.276148i	$0.0890579\pi$
$132$	0	0
$133$	2.53209	0.219560
$134$	−7.29086	−0.629834
$135$	0	0
$136$	27.2814	2.33936
$137$	11.0000	0.939793	0.469897	−	0.882721i	$-0.344291\pi$
$137$	11.0000	0.939793	0.469897	+	0.882721i	$0.344291\pi$
$138$	0	0
$139$	−18.0351	−1.52972	−0.764858	−	0.644199i	$-0.777191\pi$
$139$	−18.0351	−1.52972	−0.764858	+	0.644199i	$0.777191\pi$
$140$	0.467911	0.0395457
$141$	0	0
$142$	−24.0496	−2.01820
$143$	−13.8726	−1.16008
$144$	0	0
$145$	−0.716881	−0.0595338
$146$	28.0847	2.32431
$147$	0	0
$148$	−46.6810	−3.83715
$149$	−23.8452	−1.95348	−0.976739	−	0.214432i	$-0.931210\pi$
$149$	−23.8452	−1.95348	−0.976739	+	0.214432i	$0.931210\pi$
$150$	0	0
$151$	−7.53714	−0.613364	−0.306682	−	0.951812i	$-0.599219\pi$
$151$	−7.53714	−0.613364	−0.306682	+	0.951812i	$0.599219\pi$
$152$	17.5817	1.42607
$153$	0	0
$154$	6.04963	0.487493
$155$	0.171999	0.0138153
$156$	0	0
$157$	−18.0915	−1.44386	−0.721930	−	0.691966i	$-0.756745\pi$
$157$	−18.0915	−1.44386	−0.721930	+	0.691966i	$0.756745\pi$
$158$	−19.5594	−1.55606
$159$	0	0
$160$	0.554378	0.0438274
$161$	4.59627	0.362237
$162$	0	0
$163$	−1.63041	−0.127704	−0.0638520	−	0.997959i	$-0.520339\pi$
$163$	−1.63041	−0.127704	−0.0638520	+	0.997959i	$0.520339\pi$
$164$	23.7743	1.85646
$165$	0	0
$166$	−29.0077	−2.25144
$167$	−3.45336	−0.267229	−0.133615	−	0.991033i	$-0.542658\pi$
$167$	−3.45336	−0.267229	−0.133615	+	0.991033i	$0.542658\pi$
$168$	0	0
$169$	13.0719	1.00553
$170$	1.36453	0.104655
$171$	0	0
$172$	36.5107	2.78392
$173$	9.23173	0.701876	0.350938	−	0.936399i	$-0.385863\pi$
$173$	9.23173	0.701876	0.350938	+	0.936399i	$0.385863\pi$
$174$	0	0
$175$	−4.38413	−0.331409
$176$	18.0351	1.35945
$177$	0	0
$178$	6.69728	0.501982
$179$	9.80840	0.733114	0.366557	−	0.930396i	$-0.380536\pi$
$179$	9.80840	0.733114	0.366557	+	0.930396i	$0.380536\pi$
$180$	0	0
$181$	−7.98545	−0.593554	−0.296777	−	0.954947i	$-0.595912\pi$
$181$	−7.98545	−0.593554	−0.296777	+	0.954947i	$0.595912\pi$
$182$	−11.3696	−0.842770
$183$	0	0
$184$	31.9145	2.35277
$185$	−1.27631	−0.0938363
$186$	0	0
$187$	12.1388	0.887676
$188$	28.3824	2.07000
$189$	0	0
$190$	0.879385	0.0637973
$191$	4.20439	0.304219	0.152110	−	0.988364i	$-0.451393\pi$
$191$	4.20439	0.304219	0.152110	+	0.988364i	$0.451393\pi$
$192$	0	0
$193$	11.1257	0.800843	0.400422	−	0.916331i	$-0.368864\pi$
$193$	11.1257	0.800843	0.400422	+	0.916331i	$0.368864\pi$
$194$	4.25402	0.305421
$195$	0	0
$196$	−27.4688	−1.96206
$197$	−5.44831	−0.388176	−0.194088	−	0.980984i	$-0.562175\pi$
$197$	−5.44831	−0.388176	−0.194088	+	0.980984i	$0.562175\pi$
$198$	0	0
$199$	6.27631	0.444916	0.222458	−	0.974942i	$-0.428592\pi$
$199$	6.27631	0.444916	0.222458	+	0.974942i	$0.428592\pi$
$200$	−30.4415	−2.15254
$201$	0	0
$202$	42.3482	2.97961
$203$	−5.22668	−0.366841
$204$	0	0
$205$	0.650015	0.0453990
$206$	0.411474	0.0286688
$207$	0	0
$208$	−33.8949	−2.35019
$209$	7.82295	0.541125
$210$	0	0
$211$	5.09152	0.350515	0.175257	−	0.984523i	$-0.443924\pi$
$211$	5.09152	0.350515	0.175257	+	0.984523i	$0.443924\pi$
$212$	−55.6887	−3.82472
$213$	0	0
$214$	−36.6536	−2.50559
$215$	0.998245	0.0680798
$216$	0	0
$217$	1.25402	0.0851286
$218$	23.7246	1.60683
$219$	0	0
$220$	1.44562	0.0974638
$221$	−22.8135	−1.53460
$222$	0	0
$223$	16.1530	1.08169	0.540843	−	0.841124i	$-0.318105\pi$
$223$	16.1530	1.08169	0.540843	+	0.841124i	$0.318105\pi$
$224$	4.04189	0.270060
$225$	0	0
$226$	22.5398	1.49933
$227$	9.34730	0.620402	0.310201	−	0.950671i	$-0.399604\pi$
$227$	9.34730	0.620402	0.310201	+	0.950671i	$0.399604\pi$
$228$	0	0
$229$	−10.5321	−0.695980	−0.347990	−	0.937498i	$-0.613136\pi$
$229$	−10.5321	−0.695980	−0.347990	+	0.937498i	$0.613136\pi$
$230$	1.59627	0.105255
$231$	0	0
$232$	−36.2918	−2.38267
$233$	10.5544	0.691440	0.345720	−	0.938338i	$-0.387635\pi$
$233$	10.5544	0.691440	0.345720	+	0.938338i	$0.387635\pi$
$234$	0	0
$235$	0.776007	0.0506211
$236$	−11.9855	−0.780186
$237$	0	0
$238$	9.94862	0.644873
$239$	−15.4466	−0.999155	−0.499577	−	0.866269i	$-0.666511\pi$
$239$	−15.4466	−0.999155	−0.499577	+	0.866269i	$0.666511\pi$
$240$	0	0
$241$	8.05232	0.518695	0.259348	−	0.965784i	$-0.416492\pi$
$241$	8.05232	0.518695	0.259348	+	0.965784i	$0.416492\pi$
$242$	−9.16250	−0.588988
$243$	0	0
$244$	−8.73648	−0.559296
$245$	−0.751030	−0.0479815
$246$	0	0
$247$	−14.7023	−0.935487
$248$	8.70739	0.552920
$249$	0	0
$250$	−3.04963	−0.192876
$251$	5.52528	0.348753	0.174376	−	0.984679i	$-0.444209\pi$
$251$	5.52528	0.348753	0.174376	+	0.984679i	$0.444209\pi$
$252$	0	0
$253$	14.2003	0.892764
$254$	−2.53209	−0.158877
$255$	0	0
$256$	−30.5030	−1.90644
$257$	0.608126	0.0379339	0.0189669	−	0.999820i	$-0.493962\pi$
$257$	0.608126	0.0379339	0.0189669	+	0.999820i	$0.493962\pi$
$258$	0	0
$259$	−9.30541	−0.578210
$260$	−2.71688	−0.168494
$261$	0	0
$262$	55.7083	3.44167
$263$	8.95636	0.552273	0.276136	−	0.961118i	$-0.410946\pi$
$263$	8.95636	0.552273	0.276136	+	0.961118i	$0.410946\pi$
$264$	0	0
$265$	−1.52259	−0.0935322
$266$	6.41147	0.393113
$267$	0	0
$268$	−12.7023	−0.775919
$269$	−9.49794	−0.579100	−0.289550	−	0.957163i	$-0.593506\pi$
$269$	−9.49794	−0.579100	−0.289550	+	0.957163i	$0.593506\pi$
$270$	0	0
$271$	10.5749	0.642380	0.321190	−	0.947015i	$-0.395917\pi$
$271$	10.5749	0.642380	0.321190	+	0.947015i	$0.395917\pi$
$272$	29.6587	1.79832
$273$	0	0
$274$	27.8530	1.68266
$275$	−13.5449	−0.816787
$276$	0	0
$277$	23.8161	1.43097	0.715487	−	0.698626i	$-0.246205\pi$
$277$	23.8161	1.43097	0.715487	+	0.698626i	$0.246205\pi$
$278$	−45.6664	−2.73889
$279$	0	0
$280$	0.647651	0.0387046
$281$	17.1215	1.02139	0.510693	−	0.859763i	$-0.329389\pi$
$281$	17.1215	1.02139	0.510693	+	0.859763i	$0.329389\pi$
$282$	0	0
$283$	24.6236	1.46372	0.731861	−	0.681454i	$-0.238652\pi$
$283$	24.6236	1.46372	0.731861	+	0.681454i	$0.238652\pi$
$284$	−41.8999	−2.48630
$285$	0	0
$286$	−35.1266	−2.07708
$287$	4.73917	0.279744
$288$	0	0
$289$	2.96223	0.174249
$290$	−1.81521	−0.106593
$291$	0	0
$292$	48.9299	2.86341
$293$	13.2739	0.775472	0.387736	−	0.921770i	$-0.373257\pi$
$293$	13.2739	0.775472	0.387736	+	0.921770i	$0.373257\pi$
$294$	0	0
$295$	−0.327696	−0.0190792
$296$	−64.6127	−3.75554
$297$	0	0
$298$	−60.3783	−3.49762
$299$	−26.6878	−1.54339
$300$	0	0
$301$	7.27807	0.419501
$302$	−19.0847	−1.09820
$303$	0	0
$304$	19.1138	1.09625
$305$	−0.238865	−0.0136774
$306$	0	0
$307$	−26.3182	−1.50206	−0.751030	−	0.660269i	$-0.770442\pi$
$307$	−26.3182	−1.50206	−0.751030	+	0.660269i	$0.770442\pi$
$308$	10.5398	0.600563
$309$	0	0
$310$	0.435518	0.0247357
$311$	4.31315	0.244576	0.122288	−	0.992495i	$-0.460977\pi$
$311$	4.31315	0.244576	0.122288	+	0.992495i	$0.460977\pi$
$312$	0	0
$313$	−34.6732	−1.95985	−0.979924	−	0.199373i	$-0.936109\pi$
$313$	−34.6732	−1.95985	−0.979924	+	0.199373i	$0.936109\pi$
$314$	−45.8093	−2.58517
$315$	0	0
$316$	−34.0770	−1.91698
$317$	21.2371	1.19279	0.596397	−	0.802689i	$-0.296598\pi$
$317$	21.2371	1.19279	0.596397	+	0.802689i	$0.296598\pi$
$318$	0	0
$319$	−16.1480	−0.904112
$320$	−0.197586	−0.0110454
$321$	0	0
$322$	11.6382	0.648569
$323$	12.8648	0.715819
$324$	0	0
$325$	25.4561	1.41205
$326$	−4.12836	−0.228648
$327$	0	0
$328$	32.9067	1.81697
$329$	5.65776	0.311922
$330$	0	0
$331$	22.8598	1.25649	0.628244	−	0.778017i	$-0.283774\pi$
$331$	22.8598	1.25649	0.628244	+	0.778017i	$0.283774\pi$
$332$	−50.5381	−2.77364
$333$	0	0
$334$	−8.74422	−0.478463
$335$	−0.347296	−0.0189748
$336$	0	0
$337$	7.49020	0.408017	0.204009	−	0.978969i	$-0.434603\pi$
$337$	7.49020	0.408017	0.204009	+	0.978969i	$0.434603\pi$
$338$	33.0993	1.80036
$339$	0	0
$340$	2.37733	0.128929
$341$	3.87433	0.209807
$342$	0	0
$343$	−11.6313	−0.628034
$344$	50.5357	2.72470
$345$	0	0
$346$	23.3756	1.25668
$347$	22.9581	1.23246	0.616228	−	0.787568i	$-0.288660\pi$
$347$	22.9581	1.23246	0.616228	+	0.787568i	$0.288660\pi$
$348$	0	0
$349$	23.2131	1.24257	0.621284	−	0.783586i	$-0.286612\pi$
$349$	23.2131	1.24257	0.621284	+	0.783586i	$0.286612\pi$
$350$	−11.1010	−0.593374
$351$	0	0
$352$	12.4875	0.665587
$353$	22.1807	1.18056	0.590279	−	0.807199i	$-0.299018\pi$
$353$	22.1807	1.18056	0.590279	+	0.807199i	$0.299018\pi$
$354$	0	0
$355$	−1.14559	−0.0608017
$356$	11.6682	0.618413
$357$	0	0
$358$	24.8357	1.31261
$359$	2.83750	0.149757	0.0748787	−	0.997193i	$-0.476143\pi$
$359$	2.83750	0.149757	0.0748787	+	0.997193i	$0.476143\pi$
$360$	0	0
$361$	−10.7091	−0.563639
$362$	−20.2199	−1.06273
$363$	0	0
$364$	−19.8084	−1.03824
$365$	1.33780	0.0700237
$366$	0	0
$367$	−38.1147	−1.98957	−0.994787	−	0.101978i	$-0.967483\pi$
$367$	−38.1147	−1.98957	−0.994787	+	0.101978i	$0.967483\pi$
$368$	34.6955	1.80863
$369$	0	0
$370$	−3.23173	−0.168010
$371$	−11.1010	−0.576336
$372$	0	0
$373$	35.2841	1.82694	0.913469	−	0.406907i	$-0.133393\pi$
$373$	35.2841	1.82694	0.913469	+	0.406907i	$0.133393\pi$
$374$	30.7365	1.58935
$375$	0	0
$376$	39.2850	2.02597
$377$	30.3482	1.56301
$378$	0	0
$379$	−19.2422	−0.988404	−0.494202	−	0.869347i	$-0.664540\pi$
$379$	−19.2422	−0.988404	−0.494202	+	0.869347i	$0.664540\pi$
$380$	1.53209	0.0785945
$381$	0	0
$382$	10.6459	0.544691
$383$	30.7811	1.57284	0.786419	−	0.617693i	$-0.211933\pi$
$383$	30.7811	1.57284	0.786419	+	0.617693i	$0.211933\pi$
$384$	0	0
$385$	0.288171	0.0146866
$386$	28.1712	1.43388
$387$	0	0
$388$	7.41147	0.376261
$389$	−8.57491	−0.434765	−0.217383	−	0.976086i	$-0.569752\pi$
$389$	−8.57491	−0.434765	−0.217383	+	0.976086i	$0.569752\pi$
$390$	0	0
$391$	23.3523	1.18098
$392$	−38.0205	−1.92033
$393$	0	0
$394$	−13.7956	−0.695013
$395$	−0.931703	−0.0468791
$396$	0	0
$397$	−12.8307	−0.643954	−0.321977	−	0.946748i	$-0.604347\pi$
$397$	−12.8307	−0.643954	−0.321977	+	0.946748i	$0.604347\pi$
$398$	15.8922	0.796603
$399$	0	0
$400$	−33.0942	−1.65471
$401$	−8.71688	−0.435300	−0.217650	−	0.976027i	$-0.569839\pi$
$401$	−8.71688	−0.435300	−0.217650	+	0.976027i	$0.569839\pi$
$402$	0	0
$403$	−7.28136	−0.362711
$404$	73.7802	3.67070
$405$	0	0
$406$	−13.2344	−0.656813
$407$	−28.7493	−1.42505
$408$	0	0
$409$	−22.1438	−1.09494	−0.547471	−	0.836825i	$-0.684409\pi$
$409$	−22.1438	−1.09494	−0.547471	+	0.836825i	$0.684409\pi$
$410$	1.64590	0.0812850
$411$	0	0
$412$	0.716881	0.0353182
$413$	−2.38919	−0.117564
$414$	0	0
$415$	−1.38177	−0.0678284
$416$	−23.4688	−1.15065
$417$	0	0
$418$	19.8084	0.968861
$419$	9.52704	0.465426	0.232713	−	0.972545i	$-0.425240\pi$
$419$	9.52704	0.465426	0.232713	+	0.972545i	$0.425240\pi$
$420$	0	0
$421$	25.6759	1.25137	0.625684	−	0.780077i	$-0.284820\pi$
$421$	25.6759	1.25137	0.625684	+	0.780077i	$0.284820\pi$
$422$	12.8922	0.627581
$423$	0	0
$424$	−77.0806	−3.74336
$425$	−22.2746	−1.08047
$426$	0	0
$427$	−1.74153	−0.0842787
$428$	−63.8590	−3.08674
$429$	0	0
$430$	2.52765	0.121894
$431$	5.68779	0.273971	0.136985	−	0.990573i	$-0.456259\pi$
$431$	5.68779	0.273971	0.136985	+	0.990573i	$0.456259\pi$
$432$	0	0
$433$	5.65095	0.271567	0.135784	−	0.990739i	$-0.456645\pi$
$433$	5.65095	0.271567	0.135784	+	0.990739i	$0.456645\pi$
$434$	3.17530	0.152419
$435$	0	0
$436$	41.3337	1.97952
$437$	15.0496	0.719921
$438$	0	0
$439$	25.3874	1.21168	0.605838	−	0.795588i	$-0.292838\pi$
$439$	25.3874	1.21168	0.605838	+	0.795588i	$0.292838\pi$
$440$	2.00093	0.0953908
$441$	0	0
$442$	−57.7657	−2.74763
$443$	9.18210	0.436255	0.218127	−	0.975920i	$-0.430005\pi$
$443$	9.18210	0.436255	0.218127	+	0.975920i	$0.430005\pi$
$444$	0	0
$445$	0.319022	0.0151231
$446$	40.9009	1.93671
$447$	0	0
$448$	−1.44057	−0.0680605
$449$	12.5175	0.590739	0.295370	−	0.955383i	$-0.404557\pi$
$449$	12.5175	0.590739	0.295370	+	0.955383i	$0.404557\pi$
$450$	0	0
$451$	14.6418	0.689454
$452$	39.2695	1.84708
$453$	0	0
$454$	23.6682	1.11080
$455$	−0.541584	−0.0253899
$456$	0	0
$457$	13.2439	0.619524	0.309762	−	0.950814i	$-0.399751\pi$
$457$	13.2439	0.619524	0.309762	+	0.950814i	$0.399751\pi$
$458$	−26.6682	−1.24612
$459$	0	0
$460$	2.78106	0.129668
$461$	−3.85441	−0.179518	−0.0897588	−	0.995964i	$-0.528610\pi$
$461$	−3.85441	−0.179518	−0.0897588	+	0.995964i	$0.528610\pi$
$462$	0	0
$463$	6.10876	0.283898	0.141949	−	0.989874i	$-0.454663\pi$
$463$	6.10876	0.283898	0.141949	+	0.989874i	$0.454663\pi$
$464$	−39.4543	−1.83162
$465$	0	0
$466$	26.7246	1.23799
$467$	−33.6705	−1.55809	−0.779044	−	0.626970i	$-0.784295\pi$
$467$	−33.6705	−1.55809	−0.779044	+	0.626970i	$0.784295\pi$
$468$	0	0
$469$	−2.53209	−0.116921
$470$	1.96492	0.0906349
$471$	0	0
$472$	−16.5895	−0.763592
$473$	22.4858	1.03390
$474$	0	0
$475$	−14.3550	−0.658654
$476$	17.3327	0.794445
$477$	0	0
$478$	−39.1121	−1.78894
$479$	16.1712	0.738880	0.369440	−	0.929255i	$-0.379550\pi$
$479$	16.1712	0.738880	0.369440	+	0.929255i	$0.379550\pi$
$480$	0	0
$481$	54.0310	2.46360
$482$	20.3892	0.928702
$483$	0	0
$484$	−15.9632	−0.725598
$485$	0.202638	0.00920133
$486$	0	0
$487$	40.2695	1.82479	0.912393	−	0.409316i	$-0.134233\pi$
$487$	40.2695	1.82479	0.912393	+	0.409316i	$0.134233\pi$
$488$	−12.0925	−0.547400
$489$	0	0
$490$	−1.90167	−0.0859089
$491$	19.6682	0.887613	0.443806	−	0.896123i	$-0.353628\pi$
$491$	19.6682	0.887613	0.443806	+	0.896123i	$0.353628\pi$
$492$	0	0
$493$	−26.5553	−1.19599
$494$	−37.2276	−1.67495
$495$	0	0
$496$	9.46616	0.425043
$497$	−8.35235	−0.374654
$498$	0	0
$499$	−27.4175	−1.22737	−0.613687	−	0.789549i	$-0.710314\pi$
$499$	−27.4175	−1.22737	−0.613687	+	0.789549i	$0.710314\pi$
$500$	−5.31315	−0.237611
$501$	0	0
$502$	13.9905	0.624427
$503$	−40.8607	−1.82189	−0.910945	−	0.412528i	$-0.864646\pi$
$503$	−40.8607	−1.82189	−0.910945	+	0.412528i	$0.864646\pi$
$504$	0	0
$505$	2.01724	0.0897658
$506$	35.9564	1.59846
$507$	0	0
$508$	−4.41147	−0.195728
$509$	−26.9172	−1.19308	−0.596541	−	0.802583i	$-0.703459\pi$
$509$	−26.9172	−1.19308	−0.596541	+	0.802583i	$0.703459\pi$
$510$	0	0
$511$	9.75372	0.431479
$512$	−50.5553	−2.23425
$513$	0	0
$514$	1.53983	0.0679190
$515$	0.0196004	0.000863695 0
$516$	0	0
$517$	17.4798	0.768759
$518$	−23.5621	−1.03526
$519$	0	0
$520$	−3.76053	−0.164910
$521$	−2.32501	−0.101860	−0.0509302	−	0.998702i	$-0.516219\pi$
$521$	−2.32501	−0.101860	−0.0509302	+	0.998702i	$0.516219\pi$
$522$	0	0
$523$	−2.46616	−0.107837	−0.0539187	−	0.998545i	$-0.517171\pi$
$523$	−2.46616	−0.107837	−0.0539187	+	0.998545i	$0.517171\pi$
$524$	97.0565	4.23994
$525$	0	0
$526$	22.6783	0.988820
$527$	6.37134	0.277540
$528$	0	0
$529$	4.31820	0.187748
$530$	−3.85534	−0.167465
$531$	0	0
$532$	11.1702	0.484292
$533$	−27.5175	−1.19192
$534$	0	0
$535$	−1.74598	−0.0754852
$536$	−17.5817	−0.759415
$537$	0	0
$538$	−24.0496	−1.03685
$539$	−16.9172	−0.728673
$540$	0	0
$541$	9.33956	0.401539	0.200769	−	0.979639i	$-0.435656\pi$
$541$	9.33956	0.401539	0.200769	+	0.979639i	$0.435656\pi$
$542$	26.7766	1.15015
$543$	0	0
$544$	20.5357	0.880461
$545$	1.13011	0.0484086
$546$	0	0
$547$	−18.3669	−0.785312	−0.392656	−	0.919685i	$-0.628444\pi$
$547$	−18.3669	−0.785312	−0.392656	+	0.919685i	$0.628444\pi$
$548$	48.5262	2.07294
$549$	0	0
$550$	−34.2968	−1.46242
$551$	−17.1138	−0.729073
$552$	0	0
$553$	−6.79292	−0.288864
$554$	60.3046	2.56210
$555$	0	0
$556$	−79.5613	−3.37415
$557$	13.5936	0.575978	0.287989	−	0.957634i	$-0.407013\pi$
$557$	13.5936	0.575978	0.287989	+	0.957634i	$0.407013\pi$
$558$	0	0
$559$	−42.2594	−1.78738
$560$	0.704088	0.0297532
$561$	0	0
$562$	43.3533	1.82875
$563$	31.0428	1.30830	0.654149	−	0.756365i	$-0.273027\pi$
$563$	31.0428	1.30830	0.654149	+	0.756365i	$0.273027\pi$
$564$	0	0
$565$	1.07367	0.0451698
$566$	62.3492	2.62073
$567$	0	0
$568$	−57.9951	−2.43342
$569$	−20.9436	−0.878000	−0.439000	−	0.898487i	$-0.644667\pi$
$569$	−20.9436	−0.878000	−0.439000	+	0.898487i	$0.644667\pi$
$570$	0	0
$571$	9.05737	0.379039	0.189520	−	0.981877i	$-0.439307\pi$
$571$	9.05737	0.379039	0.189520	+	0.981877i	$0.439307\pi$
$572$	−61.1985	−2.55884
$573$	0	0
$574$	12.0000	0.500870
$575$	−26.0574	−1.08667
$576$	0	0
$577$	3.36783	0.140205	0.0701023	−	0.997540i	$-0.477667\pi$
$577$	3.36783	0.140205	0.0701023	+	0.997540i	$0.477667\pi$
$578$	7.50063	0.311985
$579$	0	0
$580$	−3.16250	−0.131316
$581$	−10.0743	−0.417952
$582$	0	0
$583$	−34.2968	−1.42043
$584$	67.7256	2.80250
$585$	0	0
$586$	33.6108	1.38845
$587$	20.2831	0.837174	0.418587	−	0.908177i	$-0.362526\pi$
$587$	20.2831	0.837174	0.418587	+	0.908177i	$0.362526\pi$
$588$	0	0
$589$	4.10607	0.169188
$590$	−0.829755	−0.0341605
$591$	0	0
$592$	−70.2431	−2.88697
$593$	−5.12061	−0.210278	−0.105139	−	0.994458i	$-0.533529\pi$
$593$	−5.12061	−0.210278	−0.105139	+	0.994458i	$0.533529\pi$
$594$	0	0
$595$	0.473897	0.0194279
$596$	−105.193	−4.30886
$597$	0	0
$598$	−67.5758	−2.76338
$599$	−6.12155	−0.250120	−0.125060	−	0.992149i	$-0.539912\pi$
$599$	−6.12155	−0.250120	−0.125060	+	0.992149i	$0.539912\pi$
$600$	0	0
$601$	−29.4766	−1.20238	−0.601188	−	0.799108i	$-0.705306\pi$
$601$	−29.4766	−1.20238	−0.601188	+	0.799108i	$0.705306\pi$
$602$	18.4287	0.751098
$603$	0	0
$604$	−33.2499	−1.35292
$605$	−0.436451	−0.0177443
$606$	0	0
$607$	−14.2591	−0.578758	−0.289379	−	0.957215i	$-0.593449\pi$
$607$	−14.2591	−0.578758	−0.289379	+	0.957215i	$0.593449\pi$
$608$	13.2344	0.536727
$609$	0	0
$610$	−0.604828	−0.0244888
$611$	−32.8512	−1.32902
$612$	0	0
$613$	15.5449	0.627852	0.313926	−	0.949447i	$-0.398356\pi$
$613$	15.5449	0.627852	0.313926	+	0.949447i	$0.398356\pi$
$614$	−66.6400	−2.68937
$615$	0	0
$616$	14.5885	0.587788
$617$	15.5439	0.625776	0.312888	−	0.949790i	$-0.398704\pi$
$617$	15.5439	0.625776	0.312888	+	0.949790i	$0.398704\pi$
$618$	0	0
$619$	45.4739	1.82775	0.913875	−	0.405995i	$-0.133075\pi$
$619$	45.4739	1.82775	0.913875	+	0.405995i	$0.133075\pi$
$620$	0.758770	0.0304730
$621$	0	0
$622$	10.9213	0.437903
$623$	2.32594	0.0931869
$624$	0	0
$625$	24.7820	0.991280
$626$	−87.7957	−3.50902
$627$	0	0
$628$	−79.8103	−3.18478
$629$	−47.2782	−1.88510
$630$	0	0
$631$	14.9513	0.595202	0.297601	−	0.954690i	$-0.403813\pi$
$631$	14.9513	0.595202	0.297601	+	0.954690i	$0.403813\pi$
$632$	−47.1671	−1.87620
$633$	0	0
$634$	53.7743	2.13565
$635$	−0.120615	−0.00478645
$636$	0	0
$637$	31.7939	1.25972
$638$	−40.8881	−1.61877
$639$	0	0
$640$	−1.60906	−0.0636037
$641$	−1.98040	−0.0782211	−0.0391105	−	0.999235i	$-0.512452\pi$
$641$	−1.98040	−0.0782211	−0.0391105	+	0.999235i	$0.512452\pi$
$642$	0	0
$643$	34.1147	1.34535	0.672677	−	0.739936i	$-0.265144\pi$
$643$	34.1147	1.34535	0.672677	+	0.739936i	$0.265144\pi$
$644$	20.2763	0.798999
$645$	0	0
$646$	32.5749	1.28164
$647$	−14.7570	−0.580158	−0.290079	−	0.957003i	$-0.593682\pi$
$647$	−14.7570	−0.580158	−0.290079	+	0.957003i	$0.593682\pi$
$648$	0	0
$649$	−7.38144	−0.289747
$650$	64.4570	2.52821
$651$	0	0
$652$	−7.19253	−0.281681
$653$	−29.9986	−1.17393	−0.586967	−	0.809611i	$-0.699678\pi$
$653$	−29.9986	−1.17393	−0.586967	+	0.809611i	$0.699678\pi$
$654$	0	0
$655$	2.65364	0.103686
$656$	35.7743	1.39675
$657$	0	0
$658$	14.3259	0.558483
$659$	−3.29860	−0.128495	−0.0642476	−	0.997934i	$-0.520465\pi$
$659$	−3.29860	−0.128495	−0.0642476	+	0.997934i	$0.520465\pi$
$660$	0	0
$661$	−28.0128	−1.08957	−0.544786	−	0.838575i	$-0.683389\pi$
$661$	−28.0128	−1.08957	−0.544786	+	0.838575i	$0.683389\pi$
$662$	57.8830	2.24969
$663$	0	0
$664$	−69.9514	−2.71464
$665$	0.305407	0.0118432
$666$	0	0
$667$	−31.0651	−1.20285
$668$	−15.2344	−0.589437
$669$	0	0
$670$	−0.879385	−0.0339736
$671$	−5.38051	−0.207712
$672$	0	0
$673$	−22.8512	−0.880850	−0.440425	−	0.897789i	$-0.645172\pi$
$673$	−22.8512	−0.880850	−0.440425	+	0.897789i	$0.645172\pi$
$674$	18.9659	0.730537
$675$	0	0
$676$	57.6664	2.21794
$677$	12.7469	0.489904	0.244952	−	0.969535i	$-0.421228\pi$
$677$	12.7469	0.489904	0.244952	+	0.969535i	$0.421228\pi$
$678$	0	0
$679$	1.47741	0.0566977
$680$	3.29054	0.126186
$681$	0	0
$682$	9.81016	0.375650
$683$	−20.3013	−0.776807	−0.388404	−	0.921489i	$-0.626973\pi$
$683$	−20.3013	−0.776807	−0.388404	+	0.921489i	$0.626973\pi$
$684$	0	0
$685$	1.32676	0.0506930
$686$	−29.4516	−1.12447
$687$	0	0
$688$	54.9394	2.09455
$689$	64.4570	2.45562
$690$	0	0
$691$	−37.4543	−1.42483	−0.712414	−	0.701759i	$-0.752398\pi$
$691$	−37.4543	−1.42483	−0.712414	+	0.701759i	$0.752398\pi$
$692$	40.7256	1.54815
$693$	0	0
$694$	58.1320	2.20666
$695$	−2.17530	−0.0825137
$696$	0	0
$697$	24.0784	0.912034
$698$	58.7775	2.22476
$699$	0	0
$700$	−19.3405	−0.731002
$701$	22.5422	0.851407	0.425703	−	0.904863i	$-0.360027\pi$
$701$	22.5422	0.851407	0.425703	+	0.904863i	$0.360027\pi$
$702$	0	0
$703$	−30.4688	−1.14915
$704$	−4.45067	−0.167741
$705$	0	0
$706$	56.1634	2.11374
$707$	14.7074	0.553128
$708$	0	0
$709$	2.90848	0.109230	0.0546151	−	0.998507i	$-0.482607\pi$
$709$	2.90848	0.109230	0.0546151	+	0.998507i	$0.482607\pi$
$710$	−2.90074	−0.108863
$711$	0	0
$712$	16.1503	0.605259
$713$	7.45336	0.279131
$714$	0	0
$715$	−1.67324	−0.0625755
$716$	43.2695	1.61706
$717$	0	0
$718$	7.18479	0.268134
$719$	4.05880	0.151368	0.0756839	−	0.997132i	$-0.475886\pi$
$719$	4.05880	0.151368	0.0756839	+	0.997132i	$0.475886\pi$
$720$	0	0
$721$	0.142903	0.00532200
$722$	−27.1165	−1.00917
$723$	0	0
$724$	−35.2276	−1.30922
$725$	29.6313	1.10048
$726$	0	0
$727$	21.6637	0.803464	0.401732	−	0.915757i	$-0.368408\pi$
$727$	21.6637	0.803464	0.401732	+	0.915757i	$0.368408\pi$
$728$	−27.4175	−1.01616
$729$	0	0
$730$	3.38743	0.125374
$731$	36.9778	1.36767
$732$	0	0
$733$	−23.1993	−0.856887	−0.428444	−	0.903569i	$-0.640938\pi$
$733$	−23.1993	−0.856887	−0.428444	+	0.903569i	$0.640938\pi$
$734$	−96.5099	−3.56225
$735$	0	0
$736$	24.0232	0.885508
$737$	−7.82295	−0.288162
$738$	0	0
$739$	−48.5844	−1.78721	−0.893603	−	0.448858i	$-0.851831\pi$
$739$	−48.5844	−1.78721	−0.893603	+	0.448858i	$0.851831\pi$
$740$	−5.63041	−0.206978
$741$	0	0
$742$	−28.1088	−1.03190
$743$	16.7033	0.612783	0.306392	−	0.951906i	$-0.400878\pi$
$743$	16.7033	0.612783	0.306392	+	0.951906i	$0.400878\pi$
$744$	0	0
$745$	−2.87609	−0.105372
$746$	89.3424	3.27106
$747$	0	0
$748$	53.5499	1.95798
$749$	−12.7297	−0.465132
$750$	0	0
$751$	−22.1634	−0.808755	−0.404378	−	0.914592i	$-0.632512\pi$
$751$	−22.1634	−0.808755	−0.404378	+	0.914592i	$0.632512\pi$
$752$	42.7083	1.55741
$753$	0	0
$754$	76.8444	2.79851
$755$	−0.909090	−0.0330852
$756$	0	0
$757$	24.1625	0.878201	0.439101	−	0.898438i	$-0.355297\pi$
$757$	24.1625	0.878201	0.439101	+	0.898438i	$0.355297\pi$
$758$	−48.7229	−1.76969
$759$	0	0
$760$	2.12061	0.0769228
$761$	38.8675	1.40895	0.704473	−	0.709730i	$-0.251183\pi$
$761$	38.8675	1.40895	0.704473	+	0.709730i	$0.251183\pi$
$762$	0	0
$763$	8.23947	0.298289
$764$	18.5476	0.671028
$765$	0	0
$766$	77.9404	2.81610
$767$	13.8726	0.500910
$768$	0	0
$769$	6.27126	0.226147	0.113074	−	0.993587i	$-0.463930\pi$
$769$	6.27126	0.226147	0.113074	+	0.993587i	$0.463930\pi$
$770$	0.729675	0.0262956
$771$	0	0
$772$	49.0806	1.76645
$773$	−2.46522	−0.0886679	−0.0443340	−	0.999017i	$-0.514117\pi$
$773$	−2.46522	−0.0886679	−0.0443340	+	0.999017i	$0.514117\pi$
$774$	0	0
$775$	−7.10936	−0.255376
$776$	10.2585	0.368257
$777$	0	0
$778$	−21.7124	−0.778429
$779$	15.5175	0.555974
$780$	0	0
$781$	−25.8048	−0.923368
$782$	59.1302	2.11449
$783$	0	0
$784$	−41.3337	−1.47620
$785$	−2.18210	−0.0778826
$786$	0	0
$787$	26.3422	0.939000	0.469500	−	0.882933i	$-0.344434\pi$
$787$	26.3422	0.939000	0.469500	+	0.882933i	$0.344434\pi$
$788$	−24.0351	−0.856214
$789$	0	0
$790$	−2.35916	−0.0839350
$791$	7.82800	0.278332
$792$	0	0
$793$	10.1121	0.359090
$794$	−32.4884	−1.15297
$795$	0	0
$796$	27.6878	0.981368
$797$	47.1498	1.67013	0.835066	−	0.550149i	$-0.185429\pi$
$797$	47.1498	1.67013	0.835066	+	0.550149i	$0.185429\pi$
$798$	0	0
$799$	28.7455	1.01694
$800$	−22.9145	−0.810149
$801$	0	0
$802$	−22.0719	−0.779387
$803$	30.1343	1.06342
$804$	0	0
$805$	0.554378	0.0195392
$806$	−18.4371	−0.649418
$807$	0	0
$808$	102.122	3.59263
$809$	14.3824	0.505657	0.252829	−	0.967511i	$-0.418639\pi$
$809$	14.3824	0.505657	0.252829	+	0.967511i	$0.418639\pi$
$810$	0	0
$811$	−35.8120	−1.25753	−0.628765	−	0.777595i	$-0.716439\pi$
$811$	−35.8120	−1.25753	−0.628765	+	0.777595i	$0.716439\pi$
$812$	−23.0574	−0.809155
$813$	0	0
$814$	−72.7957	−2.55149
$815$	−0.196652	−0.00688842
$816$	0	0
$817$	23.8307	0.833730
$818$	−56.0702	−1.96045
$819$	0	0
$820$	2.86753	0.100138
$821$	−14.0969	−0.491985	−0.245993	−	0.969272i	$-0.579114\pi$
$821$	−14.0969	−0.491985	−0.245993	+	0.969272i	$0.579114\pi$
$822$	0	0
$823$	11.0888	0.386532	0.193266	−	0.981146i	$-0.438092\pi$
$823$	11.0888	0.386532	0.193266	+	0.981146i	$0.438092\pi$
$824$	0.992259	0.0345670
$825$	0	0
$826$	−6.04963	−0.210494
$827$	−32.3429	−1.12467	−0.562336	−	0.826909i	$-0.690097\pi$
$827$	−32.3429	−1.12467	−0.562336	+	0.826909i	$0.690097\pi$
$828$	0	0
$829$	−12.2463	−0.425331	−0.212665	−	0.977125i	$-0.568214\pi$
$829$	−12.2463	−0.425331	−0.212665	+	0.977125i	$0.568214\pi$
$830$	−3.49876	−0.121444
$831$	0	0
$832$	8.36453	0.289988
$833$	−27.8203	−0.963915
$834$	0	0
$835$	−0.416527	−0.0144145
$836$	34.5107	1.19358
$837$	0	0
$838$	24.1233	0.833326
$839$	22.0283	0.760501	0.380250	−	0.924884i	$-0.375838\pi$
$839$	22.0283	0.760501	0.380250	+	0.924884i	$0.375838\pi$
$840$	0	0
$841$	6.32594	0.218136
$842$	65.0137	2.24052
$843$	0	0
$844$	22.4611	0.773143
$845$	1.57667	0.0542390
$846$	0	0
$847$	−3.18210	−0.109338
$848$	−83.7975	−2.87762
$849$	0	0
$850$	−56.4012	−1.93454
$851$	−55.3073	−1.89591
$852$	0	0
$853$	26.4935	0.907120	0.453560	−	0.891226i	$-0.350154\pi$
$853$	26.4935	0.907120	0.453560	+	0.891226i	$0.350154\pi$
$854$	−4.40972	−0.150897
$855$	0	0
$856$	−88.3893	−3.02108
$857$	40.8452	1.39525	0.697623	−	0.716465i	$-0.254241\pi$
$857$	40.8452	1.39525	0.697623	+	0.716465i	$0.254241\pi$
$858$	0	0
$859$	41.7279	1.42374	0.711869	−	0.702312i	$-0.247849\pi$
$859$	41.7279	1.42374	0.711869	+	0.702312i	$0.247849\pi$
$860$	4.40373	0.150166
$861$	0	0
$862$	14.4020	0.490533
$863$	−30.2814	−1.03079	−0.515395	−	0.856953i	$-0.672355\pi$
$863$	−30.2814	−1.03079	−0.515395	+	0.856953i	$0.672355\pi$
$864$	0	0
$865$	1.11348	0.0378596
$866$	14.3087	0.486230
$867$	0	0
$868$	5.53209	0.187771
$869$	−20.9869	−0.711931
$870$	0	0
$871$	14.7023	0.498170
$872$	57.2113	1.93742
$873$	0	0
$874$	38.1070	1.28899
$875$	−1.05913	−0.0358050
$876$	0	0
$877$	−4.08553	−0.137959	−0.0689793	−	0.997618i	$-0.521974\pi$
$877$	−4.08553	−0.137959	−0.0689793	+	0.997618i	$0.521974\pi$
$878$	64.2832	2.16945
$879$	0	0
$880$	2.17530	0.0733292
$881$	−21.5871	−0.727288	−0.363644	−	0.931538i	$-0.618467\pi$
$881$	−21.5871	−0.727288	−0.363644	+	0.931538i	$0.618467\pi$
$882$	0	0
$883$	−48.9136	−1.64608	−0.823038	−	0.567987i	$-0.807722\pi$
$883$	−48.9136	−1.64608	−0.823038	+	0.567987i	$0.807722\pi$
$884$	−100.641	−3.38492
$885$	0	0
$886$	23.2499	0.781096
$887$	−31.3688	−1.05326	−0.526630	−	0.850095i	$-0.676545\pi$
$887$	−31.3688	−1.05326	−0.526630	+	0.850095i	$0.676545\pi$
$888$	0	0
$889$	−0.879385	−0.0294936
$890$	0.807791	0.0270772
$891$	0	0
$892$	71.2586	2.38591
$893$	18.5253	0.619925
$894$	0	0
$895$	1.18304	0.0395446
$896$	−11.7314	−0.391920
$897$	0	0
$898$	31.6955	1.05769
$899$	−8.47565	−0.282679
$900$	0	0
$901$	−56.4012	−1.87899
$902$	37.0743	1.23444
$903$	0	0
$904$	54.3542	1.80779
$905$	−0.963163	−0.0320166
$906$	0	0
$907$	51.0607	1.69544	0.847721	−	0.530443i	$-0.177974\pi$
$907$	51.0607	1.69544	0.847721	+	0.530443i	$0.177974\pi$
$908$	41.2354	1.36844
$909$	0	0
$910$	−1.37134	−0.0454595
$911$	−56.2191	−1.86262	−0.931310	−	0.364227i	$-0.881333\pi$
$911$	−56.2191	−1.86262	−0.931310	+	0.364227i	$0.881333\pi$
$912$	0	0
$913$	−31.1247	−1.03008
$914$	33.5348	1.10923
$915$	0	0
$916$	−46.4620	−1.53515
$917$	19.3473	0.638904
$918$	0	0
$919$	15.1756	0.500598	0.250299	−	0.968169i	$-0.419471\pi$
$919$	15.1756	0.500598	0.250299	+	0.968169i	$0.419471\pi$
$920$	3.84936	0.126910
$921$	0	0
$922$	−9.75970	−0.321419
$923$	48.4971	1.59630
$924$	0	0
$925$	52.7547	1.73456
$926$	15.4679	0.508307
$927$	0	0
$928$	−27.3182	−0.896764
$929$	−26.8084	−0.879555	−0.439778	−	0.898107i	$-0.644943\pi$
$929$	−26.8084	−0.879555	−0.439778	+	0.898107i	$0.644943\pi$
$930$	0	0
$931$	−17.9290	−0.587600
$932$	46.5604	1.52514
$933$	0	0
$934$	−85.2568	−2.78969
$935$	1.46412	0.0478817
$936$	0	0
$937$	−33.0273	−1.07896	−0.539478	−	0.842000i	$-0.681378\pi$
$937$	−33.0273	−1.07896	−0.539478	+	0.842000i	$0.681378\pi$
$938$	−6.41147	−0.209342
$939$	0	0
$940$	3.42333	0.111657
$941$	−16.3628	−0.533411	−0.266706	−	0.963778i	$-0.585935\pi$
$941$	−16.3628	−0.533411	−0.266706	+	0.963778i	$0.585935\pi$
$942$	0	0
$943$	28.1676	0.917262
$944$	−18.0351	−0.586992
$945$	0	0
$946$	56.9359	1.85115
$947$	−53.7766	−1.74751	−0.873753	−	0.486371i	$-0.838320\pi$
$947$	−53.7766	−1.74751	−0.873753	+	0.486371i	$0.838320\pi$
$948$	0	0
$949$	−56.6340	−1.83842
$950$	−36.3482	−1.17929
$951$	0	0
$952$	23.9908	0.777547
$953$	−33.7897	−1.09456	−0.547278	−	0.836951i	$-0.684336\pi$
$953$	−33.7897	−1.09456	−0.547278	+	0.836951i	$0.684336\pi$
$954$	0	0
$955$	0.507112	0.0164098
$956$	−68.1421	−2.20387
$957$	0	0
$958$	40.9469	1.32293
$959$	9.67324	0.312365
$960$	0	0
$961$	−28.9665	−0.934402
$962$	136.811	4.41097
$963$	0	0
$964$	35.5226	1.14411
$965$	1.34192	0.0431979
$966$	0	0
$967$	24.2713	0.780511	0.390256	−	0.920707i	$-0.372387\pi$
$967$	24.2713	0.780511	0.390256	+	0.920707i	$0.372387\pi$
$968$	−22.0951	−0.710165
$969$	0	0
$970$	0.513098	0.0164746
$971$	−1.10513	−0.0354654	−0.0177327	−	0.999843i	$-0.505645\pi$
$971$	−1.10513	−0.0354654	−0.0177327	+	0.999843i	$0.505645\pi$
$972$	0	0
$973$	−15.8598	−0.508441
$974$	101.966	3.26720
$975$	0	0
$976$	−13.1462	−0.420800
$977$	49.4867	1.58322	0.791610	−	0.611027i	$-0.209243\pi$
$977$	49.4867	1.58322	0.791610	+	0.611027i	$0.209243\pi$
$978$	0	0
$979$	7.18605	0.229667
$980$	−3.31315	−0.105835
$981$	0	0
$982$	49.8016	1.58923
$983$	−29.0259	−0.925783	−0.462891	−	0.886415i	$-0.653188\pi$
$983$	−29.0259	−0.925783	−0.462891	+	0.886415i	$0.653188\pi$
$984$	0	0
$985$	−0.657147	−0.0209384
$986$	−67.2404	−2.14137
$987$	0	0
$988$	−64.8590	−2.06344
$989$	43.2576	1.37551
$990$	0	0
$991$	14.8298	0.471083	0.235541	−	0.971864i	$-0.424314\pi$
$991$	14.8298	0.471083	0.235541	+	0.971864i	$0.424314\pi$
$992$	6.55438	0.208102
$993$	0	0
$994$	−21.1489	−0.670802
$995$	0.757016	0.0239990
$996$	0	0
$997$	−41.9941	−1.32997	−0.664984	−	0.746858i	$-0.731562\pi$
$997$	−41.9941	−1.32997	−0.664984	+	0.746858i	$0.731562\pi$
$998$	−69.4234	−2.19756
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	1143.2.a.e.1.3		3
3.2	odd	2		127.2.a.a.1.1	✓	3
12.11	even	2		2032.2.a.k.1.2		3
15.14	odd	2		3175.2.a.h.1.3		3
21.20	even	2		6223.2.a.e.1.1		3
24.5	odd	2		8128.2.a.bd.1.2		3
24.11	even	2		8128.2.a.w.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
127.2.a.a.1.1	✓	3	3.2	odd	2
1143.2.a.e.1.3		3	1.1	even	1	trivial
2032.2.a.k.1.2		3	12.11	even	2
3175.2.a.h.1.3		3	15.14	odd	2
6223.2.a.e.1.1		3	21.20	even	2
8128.2.a.w.1.2		3	24.11	even	2
8128.2.a.bd.1.2		3	24.5	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:	\( N \)	\(=\)	\( 1143 = 3^{2} \cdot 127 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	1143.a (trivial)

\(f(q)\)	\(=\)	\(q+2.53209 q^{2} +4.41147 q^{4} +0.120615 q^{5} +0.879385 q^{7} +6.10607 q^{8} +O(q^{10})\)	\(q+2.53209 q^{2} +4.41147 q^{4} +0.120615 q^{5} +0.879385 q^{7} +6.10607 q^{8} +0.305407 q^{10} +2.71688 q^{11} -5.10607 q^{13} +2.22668 q^{14} +6.63816 q^{16} +4.46791 q^{17} +2.87939 q^{19} +0.532089 q^{20} +6.87939 q^{22} +5.22668 q^{23} -4.98545 q^{25} -12.9290 q^{26} +3.87939 q^{28} -5.94356 q^{29} +1.42602 q^{31} +4.59627 q^{32} +11.3131 q^{34} +0.106067 q^{35} -10.5817 q^{37} +7.29086 q^{38} +0.736482 q^{40} +5.38919 q^{41} +8.27631 q^{43} +11.9855 q^{44} +13.2344 q^{46} +6.43376 q^{47} -6.22668 q^{49} -12.6236 q^{50} -22.5253 q^{52} -12.6236 q^{53} +0.327696 q^{55} +5.36959 q^{56} -15.0496 q^{58} -2.71688 q^{59} -1.98040 q^{61} +3.61081 q^{62} -1.63816 q^{64} -0.615867 q^{65} -2.87939 q^{67} +19.7101 q^{68} +0.268571 q^{70} -9.49794 q^{71} +11.0915 q^{73} -26.7939 q^{74} +12.7023 q^{76} +2.38919 q^{77} -7.72462 q^{79} +0.800660 q^{80} +13.6459 q^{82} -11.4561 q^{83} +0.538896 q^{85} +20.9564 q^{86} +16.5895 q^{88} +2.64496 q^{89} -4.49020 q^{91} +23.0574 q^{92} +16.2909 q^{94} +0.347296 q^{95} +1.68004 q^{97} -15.7665 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8}+O(q^{10}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 6 * q^8	\( 3 q + 3 q^{2} + 3 q^{4} + 6 q^{5} - 3 q^{7} + 6 q^{8} + 3 q^{10} - 3 q^{13} + 3 q^{16} + 18 q^{17} + 3 q^{19} - 3 q^{20} + 15 q^{22} + 9 q^{23} + 3 q^{25} - 6 q^{26} + 6 q^{28} - 3 q^{29} + 12 q^{31} + 12 q^{34} - 12 q^{35} + 6 q^{38} - 3 q^{40} + 12 q^{41} - 9 q^{43} + 18 q^{44} + 9 q^{46} + 3 q^{47} - 12 q^{49} - 3 q^{50} - 21 q^{52} - 3 q^{53} - 3 q^{55} + 9 q^{56} - 18 q^{58} - 3 q^{61} + 15 q^{62} + 12 q^{64} + 9 q^{65} - 3 q^{67} + 9 q^{68} - 9 q^{70} - 3 q^{71} + 3 q^{73} - 24 q^{74} + 12 q^{76} + 3 q^{77} + 9 q^{79} - 12 q^{80} - 12 q^{83} + 39 q^{85} + 9 q^{86} - 6 q^{88} + 33 q^{89} - 12 q^{91} + 18 q^{92} + 33 q^{94} - 15 q^{97} - 12 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 6 * q^5 - 3 * q^7 + 6 * q^8 + 3 * q^10 - 3 * q^13 + 3 * q^16 + 18 * q^17 + 3 * q^19 - 3 * q^20 + 15 * q^22 + 9 * q^23 + 3 * q^25 - 6 * q^26 + 6 * q^28 - 3 * q^29 + 12 * q^31 + 12 * q^34 - 12 * q^35 + 6 * q^38 - 3 * q^40 + 12 * q^41 - 9 * q^43 + 18 * q^44 + 9 * q^46 + 3 * q^47 - 12 * q^49 - 3 * q^50 - 21 * q^52 - 3 * q^53 - 3 * q^55 + 9 * q^56 - 18 * q^58 - 3 * q^61 + 15 * q^62 + 12 * q^64 + 9 * q^65 - 3 * q^67 + 9 * q^68 - 9 * q^70 - 3 * q^71 + 3 * q^73 - 24 * q^74 + 12 * q^76 + 3 * q^77 + 9 * q^79 - 12 * q^80 - 12 * q^83 + 39 * q^85 + 9 * q^86 - 6 * q^88 + 33 * q^89 - 12 * q^91 + 18 * q^92 + 33 * q^94 - 15 * q^97 - 12 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

Related objects

Downloads

Learn more