LMFDB - Embedded newform 693.2.a.l.1.3

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [693,2,Mod(1,693)] mf = mfinit([N,k,chi],0) lf = mfeigenbasis(mf)

magma://Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code chi := DirichletCharacter("693.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(693, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 693 = 3^{2} \cdot 7 \cdot 11 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	693.a (trivial)

Newform invariants

comment:select newform

sage:traces = [3,-2,0,6,-4] f = next(g for g in N if [g.coefficient(i+1).trace() for i in range(5)] == traces)

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

Self dual:	yes
Analytic conductor:	$5.53363286007$
Analytic rank:	$1$
Dimension:	$3$
Coefficient field:	3.3.229.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - 4x - 1 $ x^3 - 4*x - 1
Coefficient ring:	$\Z[a_1, a_2]$
Coefficient ring index:	$ 1 $
Twist minimal:	no (minimal twist has level 231)
Fricke sign:	$+1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-0.254102$ of defining polynomial
Character	$\chi$	$=$	693.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+1.93543 q^{2} +1.74590 q^{4} -4.18953 q^{5} -1.00000 q^{7} -0.491797 q^{8} -8.10856 q^{10} +1.00000 q^{11} -3.17313 q^{13} -1.93543 q^{14} -4.44364 q^{16} -6.85446 q^{17} -0.318669 q^{19} -7.31450 q^{20} +1.93543 q^{22} +1.87086 q^{23} +12.5522 q^{25} -6.14137 q^{26} -1.74590 q^{28} +3.17313 q^{29} +9.23353 q^{31} -7.61676 q^{32} -13.2663 q^{34} +4.18953 q^{35} -7.55220 q^{37} -0.616763 q^{38} +2.06040 q^{40} -9.36266 q^{41} -10.8873 q^{43} +1.74590 q^{44} +3.62093 q^{46} +8.06040 q^{47} +1.00000 q^{49} +24.2939 q^{50} -5.53996 q^{52} -0.508203 q^{53} -4.18953 q^{55} +0.491797 q^{56} +6.14137 q^{58} +7.04399 q^{59} -2.00000 q^{61} +17.8709 q^{62} -5.85446 q^{64} +13.2939 q^{65} -2.66492 q^{67} -11.9672 q^{68} +8.10856 q^{70} +5.01641 q^{71} -4.82687 q^{73} -14.6168 q^{74} -0.556364 q^{76} -1.00000 q^{77} +5.01641 q^{79} +18.6168 q^{80} -18.1208 q^{82} -3.52461 q^{83} +28.7170 q^{85} -21.0716 q^{86} -0.491797 q^{88} +1.74173 q^{89} +3.17313 q^{91} +3.26634 q^{92} +15.6004 q^{94} +1.33508 q^{95} -12.2499 q^{97} +1.93543 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} - 11 q^{10} + 3 q^{11} - 4 q^{13} + 2 q^{14} - 4 q^{16} - 8 q^{17} - 8 q^{19} + 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} + q^{26} - 6 q^{28} + 4 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) $ 3 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 - 11 * q^10 + 3 * q^11 - 4 * q^13 + 2 * q^14 - 4 * q^16 - 8 * q^17 - 8 * q^19 + 3 * q^20 - 2 * q^22 - 10 * q^23 + 15 * q^25 + q^26 - 6 * q^28 + 4 * q^29 - 2 * q^31 - 8 * q^32 - 4 * q^34 + 4 * q^35 + 13 * q^38 - 18 * q^40 - 14 * q^41 - 14 * q^43 + 6 * q^44 + 28 * q^46 + 3 * q^49 + 19 * q^50 - 29 * q^52 - 4 * q^55 + 3 * q^56 - q^58 - 6 * q^61 + 38 * q^62 - 5 * q^64 - 14 * q^65 - 4 * q^67 - 42 * q^68 + 11 * q^70 + 12 * q^71 - 20 * q^73 - 29 * q^74 - 11 * q^76 - 3 * q^77 + 12 * q^79 + 41 * q^80 - 6 * q^82 - 6 * q^83 - 6 * q^85 - 24 * q^86 - 3 * q^88 - 26 * q^89 + 4 * q^91 - 26 * q^92 + 35 * q^94 + 8 * q^95 - 4 * q^97 - 2 * q^98

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 $

$2$	1.93543	1.36856	0.684279	−	0.729221i	$-0.260117\pi$
$2$	1.93543	1.36856	0.684279	+	0.729221i	$0.260117\pi$
$3$	0	0
$3$	0	0
$4$	1.74590	0.872949
$5$	−4.18953	−1.87362	−0.936808	−	0.349843i	$-0.886235\pi$
$5$	−4.18953	−1.87362	−0.936808	+	0.349843i	$0.886235\pi$
$6$	0	0
$7$	−1.00000	−0.377964
$7$	−1.00000	−0.377964
$8$	−0.491797	−0.173876
$9$	0	0
$10$	−8.10856	−2.56415
$11$	1.00000	0.301511
$11$	1.00000	0.301511
$12$	0	0
$13$	−3.17313	−0.880067	−0.440034	−	0.897981i	$-0.645033\pi$
$13$	−3.17313	−0.880067	−0.440034	+	0.897981i	$0.645033\pi$
$14$	−1.93543	−0.517266
$15$	0	0
$16$	−4.44364	−1.11091
$17$	−6.85446	−1.66245	−0.831225	−	0.555936i	$-0.812360\pi$
$17$	−6.85446	−1.66245	−0.831225	+	0.555936i	$0.812360\pi$
$18$	0	0
$19$	−0.318669	−0.0731078	−0.0365539	−	0.999332i	$-0.511638\pi$
$19$	−0.318669	−0.0731078	−0.0365539	+	0.999332i	$0.511638\pi$
$20$	−7.31450	−1.63557
$21$	0	0
$22$	1.93543	0.412636
$23$	1.87086	0.390102	0.195051	−	0.980793i	$-0.437513\pi$
$23$	1.87086	0.390102	0.195051	+	0.980793i	$0.437513\pi$
$24$	0	0
$25$	12.5522	2.51044
$26$	−6.14137	−1.20442
$27$	0	0
$28$	−1.74590	−0.329944
$29$	3.17313	0.589235	0.294617	−	0.955615i	$-0.404808\pi$
$29$	3.17313	0.589235	0.294617	+	0.955615i	$0.404808\pi$
$30$	0	0
$31$	9.23353	1.65839	0.829195	−	0.558959i	$-0.188799\pi$
$31$	9.23353	1.65839	0.829195	+	0.558959i	$0.188799\pi$
$32$	−7.61676	−1.34647
$33$	0	0
$34$	−13.2663	−2.27516
$35$	4.18953	0.708161
$36$	0	0
$37$	−7.55220	−1.24157	−0.620787	−	0.783980i	$-0.713187\pi$
$37$	−7.55220	−1.24157	−0.620787	+	0.783980i	$0.713187\pi$
$38$	−0.616763	−0.100052
$39$	0	0
$40$	2.06040	0.325778
$41$	−9.36266	−1.46220	−0.731101	−	0.682269i	$-0.760993\pi$
$41$	−9.36266	−1.46220	−0.731101	+	0.682269i	$0.760993\pi$
$42$	0	0
$43$	−10.8873	−1.66029	−0.830147	−	0.557545i	$-0.811743\pi$
$43$	−10.8873	−1.66029	−0.830147	+	0.557545i	$0.811743\pi$
$44$	1.74590	0.263204
$45$	0	0
$46$	3.62093	0.533877
$47$	8.06040	1.17573	0.587865	−	0.808959i	$-0.299969\pi$
$47$	8.06040	1.17573	0.587865	+	0.808959i	$0.299969\pi$
$48$	0	0
$49$	1.00000	0.142857
$50$	24.2939	3.43568
$51$	0	0
$52$	−5.53996	−0.768254
$53$	−0.508203	−0.0698071	−0.0349036	−	0.999391i	$-0.511112\pi$
$53$	−0.508203	−0.0698071	−0.0349036	+	0.999391i	$0.511112\pi$
$54$	0	0
$55$	−4.18953	−0.564917
$56$	0.491797	0.0657191
$57$	0	0
$58$	6.14137	0.806402
$59$	7.04399	0.917050	0.458525	−	0.888682i	$-0.348378\pi$
$59$	7.04399	0.917050	0.458525	+	0.888682i	$0.348378\pi$
$60$	0	0
$61$	−2.00000	−0.256074	−0.128037	−	0.991769i	$-0.540868\pi$
$61$	−2.00000	−0.256074	−0.128037	+	0.991769i	$0.540868\pi$
$62$	17.8709	2.26960
$63$	0	0
$64$	−5.85446	−0.731807
$65$	13.2939	1.64891
$66$	0	0
$67$	−2.66492	−0.325572	−0.162786	−	0.986661i	$-0.552048\pi$
$67$	−2.66492	−0.325572	−0.162786	+	0.986661i	$0.552048\pi$
$68$	−11.9672	−1.45123
$69$	0	0
$70$	8.10856	0.969158
$71$	5.01641	0.595338	0.297669	−	0.954669i	$-0.403791\pi$
$71$	5.01641	0.595338	0.297669	+	0.954669i	$0.403791\pi$
$72$	0	0
$73$	−4.82687	−0.564943	−0.282471	−	0.959276i	$-0.591154\pi$
$73$	−4.82687	−0.564943	−0.282471	+	0.959276i	$0.591154\pi$
$74$	−14.6168	−1.69916
$75$	0	0
$76$	−0.556364	−0.0638194
$77$	−1.00000	−0.113961
$78$	0	0
$79$	5.01641	0.564390	0.282195	−	0.959357i	$-0.408938\pi$
$79$	5.01641	0.564390	0.282195	+	0.959357i	$0.408938\pi$
$80$	18.6168	2.08142
$81$	0	0
$82$	−18.1208	−2.00111
$83$	−3.52461	−0.386876	−0.193438	−	0.981112i	$-0.561964\pi$
$83$	−3.52461	−0.386876	−0.193438	+	0.981112i	$0.561964\pi$
$84$	0	0
$85$	28.7170	3.11479
$86$	−21.0716	−2.27221
$87$	0	0
$88$	−0.491797	−0.0524257
$89$	1.74173	0.184623	0.0923115	−	0.995730i	$-0.470574\pi$
$89$	1.74173	0.184623	0.0923115	+	0.995730i	$0.470574\pi$
$90$	0	0
$91$	3.17313	0.332634
$92$	3.26634	0.340539
$93$	0	0
$94$	15.6004	1.60905
$95$	1.33508	0.136976
$96$	0	0
$97$	−12.2499	−1.24379	−0.621896	−	0.783100i	$-0.713637\pi$
$97$	−12.2499	−1.24379	−0.621896	+	0.783100i	$0.713637\pi$
$98$	1.93543	0.195508
$99$	0	0
$100$	21.9149	2.19149
$101$	−4.88727	−0.486302	−0.243151	−	0.969988i	$-0.578181\pi$
$101$	−4.88727	−0.486302	−0.243151	+	0.969988i	$0.578181\pi$
$102$	0	0
$103$	−0.637339	−0.0627988	−0.0313994	−	0.999507i	$-0.509996\pi$
$103$	−0.637339	−0.0627988	−0.0313994	+	0.999507i	$0.509996\pi$
$104$	1.56053	0.153023
$105$	0	0
$106$	−0.983593	−0.0955350
$107$	−0.956008	−0.0924208	−0.0462104	−	0.998932i	$-0.514714\pi$
$107$	−0.956008	−0.0924208	−0.0462104	+	0.998932i	$0.514714\pi$
$108$	0	0
$109$	−7.61259	−0.729154	−0.364577	−	0.931173i	$-0.618786\pi$
$109$	−7.61259	−0.729154	−0.364577	+	0.931173i	$0.618786\pi$
$110$	−8.10856	−0.773121
$111$	0	0
$112$	4.44364	0.419884
$113$	7.70892	0.725194	0.362597	−	0.931946i	$-0.381890\pi$
$113$	7.70892	0.725194	0.362597	+	0.931946i	$0.381890\pi$
$114$	0	0
$115$	−7.83805	−0.730902
$116$	5.53996	0.514372
$117$	0	0
$118$	13.6332	1.25504
$119$	6.85446	0.628347
$120$	0	0
$121$	1.00000	0.0909091
$122$	−3.87086	−0.350452
$123$	0	0
$124$	16.1208	1.44769
$125$	−31.6402	−2.82998
$126$	0	0
$127$	−5.49180	−0.487318	−0.243659	−	0.969861i	$-0.578348\pi$
$127$	−5.49180	−0.487318	−0.243659	+	0.969861i	$0.578348\pi$
$128$	3.90262	0.344946
$129$	0	0
$130$	25.7295	2.25663
$131$	−4.00000	−0.349482	−0.174741	−	0.984614i	$-0.555909\pi$
$131$	−4.00000	−0.349482	−0.174741	+	0.984614i	$0.555909\pi$
$132$	0	0
$133$	0.318669	0.0276321
$134$	−5.15778	−0.445564
$135$	0	0
$136$	3.37100	0.289061
$137$	−15.6126	−1.33387	−0.666937	−	0.745114i	$-0.732395\pi$
$137$	−15.6126	−1.33387	−0.666937	+	0.745114i	$0.732395\pi$
$138$	0	0
$139$	−9.01641	−0.764762	−0.382381	−	0.924005i	$-0.624896\pi$
$139$	−9.01641	−0.764762	−0.382381	+	0.924005i	$0.624896\pi$
$140$	7.31450	0.618188
$141$	0	0
$142$	9.70892	0.814754
$143$	−3.17313	−0.265350
$144$	0	0
$145$	−13.2939	−1.10400
$146$	−9.34209	−0.773157
$147$	0	0
$148$	−13.1854	−1.08383
$149$	5.20594	0.426487	0.213244	−	0.976999i	$-0.431597\pi$
$149$	5.20594	0.426487	0.213244	+	0.976999i	$0.431597\pi$
$150$	0	0
$151$	6.24993	0.508612	0.254306	−	0.967124i	$-0.418153\pi$
$151$	6.24993	0.508612	0.254306	+	0.967124i	$0.418153\pi$
$152$	0.156721	0.0127117
$153$	0	0
$154$	−1.93543	−0.155962
$155$	−38.6842	−3.10719
$156$	0	0
$157$	−18.1208	−1.44620	−0.723099	−	0.690745i	$-0.757283\pi$
$157$	−18.1208	−1.44620	−0.723099	+	0.690745i	$0.757283\pi$
$158$	9.70892	0.772400
$159$	0	0
$160$	31.9107	2.52276
$161$	−1.87086	−0.147445
$162$	0	0
$163$	−2.66492	−0.208733	−0.104366	−	0.994539i	$-0.533281\pi$
$163$	−2.66492	−0.208733	−0.104366	+	0.994539i	$0.533281\pi$
$164$	−16.3463	−1.27643
$165$	0	0
$166$	−6.82164	−0.529462
$167$	11.1455	0.862468	0.431234	−	0.902240i	$-0.358078\pi$
$167$	11.1455	0.862468	0.431234	+	0.902240i	$0.358078\pi$
$168$	0	0
$169$	−2.93126	−0.225482
$170$	55.5798	4.26277
$171$	0	0
$172$	−19.0081	−1.44935
$173$	24.8461	1.88902	0.944508	−	0.328489i	$-0.106539\pi$
$173$	24.8461	1.88902	0.944508	+	0.328489i	$0.106539\pi$
$174$	0	0
$175$	−12.5522	−0.948857
$176$	−4.44364	−0.334952
$177$	0	0
$178$	3.37100	0.252667
$179$	12.7581	0.953588	0.476794	−	0.879015i	$-0.341799\pi$
$179$	12.7581	0.953588	0.476794	+	0.879015i	$0.341799\pi$
$180$	0	0
$181$	−3.23353	−0.240346	−0.120173	−	0.992753i	$-0.538345\pi$
$181$	−3.23353	−0.240346	−0.120173	+	0.992753i	$0.538345\pi$
$182$	6.14137	0.455229
$183$	0	0
$184$	−0.920085	−0.0678296
$185$	31.6402	2.32623
$186$	0	0
$187$	−6.85446	−0.501248
$188$	14.0726	1.02635
$189$	0	0
$190$	2.58395	0.187459
$191$	−20.9753	−1.51772	−0.758858	−	0.651256i	$-0.774242\pi$
$191$	−20.9753	−1.51772	−0.758858	+	0.651256i	$0.774242\pi$
$192$	0	0
$193$	0.249933	0.0179905	0.00899527	−	0.999960i	$-0.497137\pi$
$193$	0.249933	0.0179905	0.00899527	+	0.999960i	$0.497137\pi$
$194$	−23.7089	−1.70220
$195$	0	0
$196$	1.74590	0.124707
$197$	−18.4999	−1.31806	−0.659030	−	0.752116i	$-0.729033\pi$
$197$	−18.4999	−1.31806	−0.659030	+	0.752116i	$0.729033\pi$
$198$	0	0
$199$	9.87086	0.699727	0.349864	−	0.936801i	$-0.386228\pi$
$199$	9.87086	0.699727	0.349864	+	0.936801i	$0.386228\pi$
$200$	−6.17313	−0.436506
$201$	0	0
$202$	−9.45898	−0.665532
$203$	−3.17313	−0.222710
$204$	0	0
$205$	39.2252	2.73961
$206$	−1.23353	−0.0859438
$207$	0	0
$208$	14.1002	0.977674
$209$	−0.318669	−0.0220428
$210$	0	0
$211$	−4.63734	−0.319248	−0.159624	−	0.987178i	$-0.551028\pi$
$211$	−4.63734	−0.319248	−0.159624	+	0.987178i	$0.551028\pi$
$212$	−0.887271	−0.0609381
$213$	0	0
$214$	−1.85029	−0.126483
$215$	45.6126	3.11075
$216$	0	0
$217$	−9.23353	−0.626813
$218$	−14.7337	−0.997889
$219$	0	0
$220$	−7.31450	−0.493144
$221$	21.7501	1.46307
$222$	0	0
$223$	−18.3463	−1.22856	−0.614278	−	0.789090i	$-0.710553\pi$
$223$	−18.3463	−1.22856	−0.614278	+	0.789090i	$0.710553\pi$
$224$	7.61676	0.508916
$225$	0	0
$226$	14.9201	0.992469
$227$	−0.379068	−0.0251596	−0.0125798	−	0.999921i	$-0.504004\pi$
$227$	−0.379068	−0.0251596	−0.0125798	+	0.999921i	$0.504004\pi$
$228$	0	0
$229$	24.9424	1.64824	0.824121	−	0.566413i	$-0.191669\pi$
$229$	24.9424	1.64824	0.824121	+	0.566413i	$0.191669\pi$
$230$	−15.1700	−1.00028
$231$	0	0
$232$	−1.56053	−0.102454
$233$	−23.4506	−1.53630	−0.768151	−	0.640268i	$-0.778823\pi$
$233$	−23.4506	−1.53630	−0.768151	+	0.640268i	$0.778823\pi$
$234$	0	0
$235$	−33.7693	−2.20287
$236$	12.2981	0.800538
$237$	0	0
$238$	13.2663	0.859929
$239$	−5.07681	−0.328391	−0.164196	−	0.986428i	$-0.552503\pi$
$239$	−5.07681	−0.328391	−0.164196	+	0.986428i	$0.552503\pi$
$240$	0	0
$241$	19.2939	1.24283	0.621415	−	0.783481i	$-0.286558\pi$
$241$	19.2939	1.24283	0.621415	+	0.783481i	$0.286558\pi$
$242$	1.93543	0.124414
$243$	0	0
$244$	−3.49180	−0.223539
$245$	−4.18953	−0.267660
$246$	0	0
$247$	1.01118	0.0643397
$248$	−4.54102	−0.288355
$249$	0	0
$250$	−61.2374	−3.87299
$251$	−23.8021	−1.50238	−0.751188	−	0.660088i	$-0.770519\pi$
$251$	−23.8021	−1.50238	−0.751188	+	0.660088i	$0.770519\pi$
$252$	0	0
$253$	1.87086	0.117620
$254$	−10.6290	−0.666923
$255$	0	0
$256$	19.2622	1.20389
$257$	−14.9149	−0.930363	−0.465182	−	0.885215i	$-0.654011\pi$
$257$	−14.9149	−0.930363	−0.465182	+	0.885215i	$0.654011\pi$
$258$	0	0
$259$	7.55220	0.469271
$260$	23.2098	1.43941
$261$	0	0
$262$	−7.74173	−0.478286
$263$	−2.92319	−0.180252	−0.0901259	−	0.995930i	$-0.528727\pi$
$263$	−2.92319	−0.180252	−0.0901259	+	0.995930i	$0.528727\pi$
$264$	0	0
$265$	2.12914	0.130792
$266$	0.616763	0.0378162
$267$	0	0
$268$	−4.65269	−0.284208
$269$	−11.9672	−0.729652	−0.364826	−	0.931076i	$-0.618872\pi$
$269$	−11.9672	−0.729652	−0.364826	+	0.931076i	$0.618872\pi$
$270$	0	0
$271$	−20.3187	−1.23427	−0.617136	−	0.786857i	$-0.711707\pi$
$271$	−20.3187	−1.23427	−0.617136	+	0.786857i	$0.711707\pi$
$272$	30.4587	1.84683
$273$	0	0
$274$	−30.2171	−1.82548
$275$	12.5522	0.756926
$276$	0	0
$277$	18.0552	1.08483	0.542415	−	0.840111i	$-0.317510\pi$
$277$	18.0552	1.08483	0.542415	+	0.840111i	$0.317510\pi$
$278$	−17.4506	−1.04662
$279$	0	0
$280$	−2.06040	−0.123132
$281$	−27.2939	−1.62822	−0.814110	−	0.580711i	$-0.802774\pi$
$281$	−27.2939	−1.62822	−0.814110	+	0.580711i	$0.802774\pi$
$282$	0	0
$283$	29.8901	1.77678	0.888391	−	0.459087i	$-0.151823\pi$
$283$	29.8901	1.77678	0.888391	+	0.459087i	$0.151823\pi$
$284$	8.75814	0.519700
$285$	0	0
$286$	−6.14137	−0.363147
$287$	9.36266	0.552660
$288$	0	0
$289$	29.9836	1.76374
$290$	−25.7295	−1.51089
$291$	0	0
$292$	−8.42723	−0.493166
$293$	4.34625	0.253911	0.126955	−	0.991908i	$-0.459479\pi$
$293$	4.34625	0.253911	0.126955	+	0.991908i	$0.459479\pi$
$294$	0	0
$295$	−29.5110	−1.71820
$296$	3.71414	0.215880
$297$	0	0
$298$	10.0757	0.583672
$299$	−5.93649	−0.343316
$300$	0	0
$301$	10.8873	0.627532
$302$	12.0963	0.696065
$303$	0	0
$304$	1.41605	0.0812161
$305$	8.37907	0.479784
$306$	0	0
$307$	−20.7581	−1.18473	−0.592365	−	0.805670i	$-0.701806\pi$
$307$	−20.7581	−1.18473	−0.592365	+	0.805670i	$0.701806\pi$
$308$	−1.74590	−0.0994818
$309$	0	0
$310$	−74.8706	−4.25236
$311$	−8.00000	−0.453638	−0.226819	−	0.973937i	$-0.572833\pi$
$311$	−8.00000	−0.453638	−0.226819	+	0.973937i	$0.572833\pi$
$312$	0	0
$313$	8.12914	0.459486	0.229743	−	0.973251i	$-0.426211\pi$
$313$	8.12914	0.459486	0.229743	+	0.973251i	$0.426211\pi$
$314$	−35.0716	−1.97920
$315$	0	0
$316$	8.75814	0.492684
$317$	19.9917	1.12284	0.561422	−	0.827530i	$-0.310255\pi$
$317$	19.9917	1.12284	0.561422	+	0.827530i	$0.310255\pi$
$318$	0	0
$319$	3.17313	0.177661
$320$	24.5275	1.37113
$321$	0	0
$322$	−3.62093	−0.201787
$323$	2.18431	0.121538
$324$	0	0
$325$	−39.8297	−2.20935
$326$	−5.15778	−0.285663
$327$	0	0
$328$	4.60453	0.254242
$329$	−8.06040	−0.444384
$330$	0	0
$331$	17.0164	0.935306	0.467653	−	0.883912i	$-0.345100\pi$
$331$	17.0164	0.935306	0.467653	+	0.883912i	$0.345100\pi$
$332$	−6.15361	−0.337723
$333$	0	0
$334$	21.5714	1.18034
$335$	11.1648	0.609998
$336$	0	0
$337$	1.52461	0.0830508	0.0415254	−	0.999137i	$-0.486778\pi$
$337$	1.52461	0.0830508	0.0415254	+	0.999137i	$0.486778\pi$
$338$	−5.67326	−0.308585
$339$	0	0
$340$	50.1369	2.71906
$341$	9.23353	0.500023
$342$	0	0
$343$	−1.00000	−0.0539949
$344$	5.35432	0.288686
$345$	0	0
$346$	48.0880	2.58523
$347$	−17.4506	−0.936800	−0.468400	−	0.883517i	$-0.655169\pi$
$347$	−17.4506	−0.936800	−0.468400	+	0.883517i	$0.655169\pi$
$348$	0	0
$349$	6.85969	0.367191	0.183595	−	0.983002i	$-0.441226\pi$
$349$	6.85969	0.367191	0.183595	+	0.983002i	$0.441226\pi$
$350$	−24.2939	−1.29856
$351$	0	0
$352$	−7.61676	−0.405975
$353$	31.9313	1.69953	0.849765	−	0.527162i	$-0.176744\pi$
$353$	31.9313	1.69953	0.849765	+	0.527162i	$0.176744\pi$
$354$	0	0
$355$	−21.0164	−1.11544
$356$	3.04088	0.161166
$357$	0	0
$358$	24.6925	1.30504
$359$	24.4342	1.28959	0.644795	−	0.764356i	$-0.276943\pi$
$359$	24.4342	1.28959	0.644795	+	0.764356i	$0.276943\pi$
$360$	0	0
$361$	−18.8984	−0.994655
$362$	−6.25827	−0.328927
$363$	0	0
$364$	5.53996	0.290373
$365$	20.2223	1.05849
$366$	0	0
$367$	17.0716	0.891129	0.445565	−	0.895250i	$-0.353003\pi$
$367$	17.0716	0.891129	0.445565	+	0.895250i	$0.353003\pi$
$368$	−8.31344	−0.433368
$369$	0	0
$370$	61.2374	3.18358
$371$	0.508203	0.0263846
$372$	0	0
$373$	3.17836	0.164569	0.0822845	−	0.996609i	$-0.473778\pi$
$373$	3.17836	0.164569	0.0822845	+	0.996609i	$0.473778\pi$
$374$	−13.2663	−0.685986
$375$	0	0
$376$	−3.96408	−0.204432
$377$	−10.0687	−0.518566
$378$	0	0
$379$	3.93960	0.202364	0.101182	−	0.994868i	$-0.467738\pi$
$379$	3.93960	0.202364	0.101182	+	0.994868i	$0.467738\pi$
$380$	2.33091	0.119573
$381$	0	0
$382$	−40.5962	−2.07708
$383$	24.7581	1.26508	0.632541	−	0.774527i	$-0.282012\pi$
$383$	24.7581	1.26508	0.632541	+	0.774527i	$0.282012\pi$
$384$	0	0
$385$	4.18953	0.213518
$386$	0.483728	0.0246211
$387$	0	0
$388$	−21.3871	−1.08577
$389$	−3.65375	−0.185252	−0.0926261	−	0.995701i	$-0.529526\pi$
$389$	−3.65375	−0.185252	−0.0926261	+	0.995701i	$0.529526\pi$
$390$	0	0
$391$	−12.8238	−0.648526
$392$	−0.491797	−0.0248395
$393$	0	0
$394$	−35.8052	−1.80384
$395$	−21.0164	−1.05745
$396$	0	0
$397$	4.56337	0.229029	0.114515	−	0.993422i	$-0.463469\pi$
$397$	4.56337	0.229029	0.114515	+	0.993422i	$0.463469\pi$
$398$	19.1044	0.957617
$399$	0	0
$400$	−55.7774	−2.78887
$401$	7.23353	0.361225	0.180613	−	0.983554i	$-0.442192\pi$
$401$	7.23353	0.361225	0.180613	+	0.983554i	$0.442192\pi$
$402$	0	0
$403$	−29.2992	−1.45949
$404$	−8.53268	−0.424517
$405$	0	0
$406$	−6.14137	−0.304791
$407$	−7.55220	−0.374348
$408$	0	0
$409$	5.30749	0.262439	0.131219	−	0.991353i	$-0.458111\pi$
$409$	5.30749	0.262439	0.131219	+	0.991353i	$0.458111\pi$
$410$	75.9177	3.74931
$411$	0	0
$412$	−1.11273	−0.0548202
$413$	−7.04399	−0.346612
$414$	0	0
$415$	14.7665	0.724858
$416$	24.1690	1.18498
$417$	0	0
$418$	−0.616763	−0.0301669
$419$	11.4231	0.558053	0.279026	−	0.960283i	$-0.409988\pi$
$419$	11.4231	0.558053	0.279026	+	0.960283i	$0.409988\pi$
$420$	0	0
$421$	−27.4147	−1.33611	−0.668056	−	0.744111i	$-0.732873\pi$
$421$	−27.4147	−1.33611	−0.668056	+	0.744111i	$0.732873\pi$
$422$	−8.97526	−0.436909
$423$	0	0
$424$	0.249933	0.0121378
$425$	−86.0385	−4.17348
$426$	0	0
$427$	2.00000	0.0967868
$428$	−1.66909	−0.0806786
$429$	0	0
$430$	88.2801	4.25724
$431$	2.28586	0.110106	0.0550529	−	0.998483i	$-0.482467\pi$
$431$	2.28586	0.110106	0.0550529	+	0.998483i	$0.482467\pi$
$432$	0	0
$433$	31.5714	1.51723	0.758613	−	0.651541i	$-0.225877\pi$
$433$	31.5714	1.51723	0.758613	+	0.651541i	$0.225877\pi$
$434$	−17.8709	−0.857829
$435$	0	0
$436$	−13.2908	−0.636515
$437$	−0.596187	−0.0285195
$438$	0	0
$439$	−18.5275	−0.884267	−0.442133	−	0.896949i	$-0.645778\pi$
$439$	−18.5275	−0.884267	−0.442133	+	0.896949i	$0.645778\pi$
$440$	2.06040	0.0982257
$441$	0	0
$442$	42.0958	2.00229
$443$	−28.4342	−1.35095	−0.675476	−	0.737382i	$-0.736062\pi$
$443$	−28.4342	−1.35095	−0.675476	+	0.737382i	$0.736062\pi$
$444$	0	0
$445$	−7.29703	−0.345913
$446$	−35.5079	−1.68135
$447$	0	0
$448$	5.85446	0.276597
$449$	5.68656	0.268365	0.134183	−	0.990957i	$-0.457159\pi$
$449$	5.68656	0.268365	0.134183	+	0.990957i	$0.457159\pi$
$450$	0	0
$451$	−9.36266	−0.440871
$452$	13.4590	0.633057
$453$	0	0
$454$	−0.733661	−0.0344324
$455$	−13.2939	−0.623229
$456$	0	0
$457$	−13.0081	−0.608492	−0.304246	−	0.952594i	$-0.598404\pi$
$457$	−13.0081	−0.608492	−0.304246	+	0.952594i	$0.598404\pi$
$458$	48.2744	2.25571
$459$	0	0
$460$	−13.6844	−0.638040
$461$	6.37907	0.297103	0.148551	−	0.988905i	$-0.452539\pi$
$461$	6.37907	0.297103	0.148551	+	0.988905i	$0.452539\pi$
$462$	0	0
$463$	34.0932	1.58445	0.792223	−	0.610232i	$-0.208924\pi$
$463$	34.0932	1.58445	0.792223	+	0.610232i	$0.208924\pi$
$464$	−14.1002	−0.654586
$465$	0	0
$466$	−45.3871	−2.10252
$467$	−18.1484	−0.839807	−0.419903	−	0.907569i	$-0.637936\pi$
$467$	−18.1484	−0.839807	−0.419903	+	0.907569i	$0.637936\pi$
$468$	0	0
$469$	2.66492	0.123055
$470$	−65.3582	−3.01475
$471$	0	0
$472$	−3.46421	−0.159453
$473$	−10.8873	−0.500597
$474$	0	0
$475$	−4.00000	−0.183533
$476$	11.9672	0.548515
$477$	0	0
$478$	−9.82581	−0.449422
$479$	42.7498	1.95329	0.976644	−	0.214864i	$-0.0689307\pi$
$479$	42.7498	1.95329	0.976644	+	0.214864i	$0.0689307\pi$
$480$	0	0
$481$	23.9641	1.09267
$482$	37.3421	1.70089
$483$	0	0
$484$	1.74590	0.0793590
$485$	51.3215	2.33039
$486$	0	0
$487$	26.7909	1.21401	0.607007	−	0.794697i	$-0.292370\pi$
$487$	26.7909	1.21401	0.607007	+	0.794697i	$0.292370\pi$
$488$	0.983593	0.0445252
$489$	0	0
$490$	−8.10856	−0.366307
$491$	−31.6813	−1.42976	−0.714879	−	0.699248i	$-0.753518\pi$
$491$	−31.6813	−1.42976	−0.714879	+	0.699248i	$0.753518\pi$
$492$	0	0
$493$	−21.7501	−0.979574
$494$	1.95707	0.0880526
$495$	0	0
$496$	−41.0304	−1.84232
$497$	−5.01641	−0.225017
$498$	0	0
$499$	−24.1260	−1.08003	−0.540015	−	0.841656i	$-0.681581\pi$
$499$	−24.1260	−1.08003	−0.540015	+	0.841656i	$0.681581\pi$
$500$	−55.2405	−2.47043
$501$	0	0
$502$	−46.0674	−2.05609
$503$	−30.8873	−1.37720	−0.688598	−	0.725144i	$-0.741773\pi$
$503$	−30.8873	−1.37720	−0.688598	+	0.725144i	$0.741773\pi$
$504$	0	0
$505$	20.4754	0.911143
$506$	3.62093	0.160970
$507$	0	0
$508$	−9.58812	−0.425404
$509$	−28.2088	−1.25033	−0.625166	−	0.780492i	$-0.714969\pi$
$509$	−28.2088	−1.25033	−0.625166	+	0.780492i	$0.714969\pi$
$510$	0	0
$511$	4.82687	0.213528
$512$	29.4754	1.30264
$513$	0	0
$514$	−28.8667	−1.27326
$515$	2.67015	0.117661
$516$	0	0
$517$	8.06040	0.354496
$518$	14.6168	0.642224
$519$	0	0
$520$	−6.53791	−0.286706
$521$	33.7610	1.47910	0.739548	−	0.673104i	$-0.235039\pi$
$521$	33.7610	1.47910	0.739548	+	0.673104i	$0.235039\pi$
$522$	0	0
$523$	12.8185	0.560515	0.280258	−	0.959925i	$-0.409580\pi$
$523$	12.8185	0.560515	0.280258	+	0.959925i	$0.409580\pi$
$524$	−6.98359	−0.305080
$525$	0	0
$526$	−5.65765	−0.246685
$527$	−63.2908	−2.75699
$528$	0	0
$529$	−19.4999	−0.847820
$530$	4.12080	0.178996
$531$	0	0
$532$	0.556364	0.0241215
$533$	29.7089	1.28684
$534$	0	0
$535$	4.00523	0.173161
$536$	1.31060	0.0566093
$537$	0	0
$538$	−23.1617	−0.998571
$539$	1.00000	0.0430730
$540$	0	0
$541$	−21.8625	−0.939943	−0.469972	−	0.882681i	$-0.655736\pi$
$541$	−21.8625	−0.939943	−0.469972	+	0.882681i	$0.655736\pi$
$542$	−39.3254	−1.68917
$543$	0	0
$544$	52.2088	2.23843
$545$	31.8932	1.36616
$546$	0	0
$547$	23.4178	1.00127	0.500637	−	0.865657i	$-0.333099\pi$
$547$	23.4178	1.00127	0.500637	+	0.865657i	$0.333099\pi$
$548$	−27.2580	−1.16440
$549$	0	0
$550$	24.2939	1.03590
$551$	−1.01118	−0.0430776
$552$	0	0
$553$	−5.01641	−0.213319
$554$	34.9446	1.48465
$555$	0	0
$556$	−15.7417	−0.667598
$557$	6.91486	0.292992	0.146496	−	0.989211i	$-0.453200\pi$
$557$	6.91486	0.292992	0.146496	+	0.989211i	$0.453200\pi$
$558$	0	0
$559$	34.5467	1.46117
$560$	−18.6168	−0.786702
$561$	0	0
$562$	−52.8255	−2.22831
$563$	−11.8381	−0.498914	−0.249457	−	0.968386i	$-0.580252\pi$
$563$	−11.8381	−0.498914	−0.249457	+	0.968386i	$0.580252\pi$
$564$	0	0
$565$	−32.2968	−1.35874
$566$	57.8503	2.43163
$567$	0	0
$568$	−2.46705	−0.103515
$569$	−10.0000	−0.419222	−0.209611	−	0.977785i	$-0.567220\pi$
$569$	−10.0000	−0.419222	−0.209611	+	0.977785i	$0.567220\pi$
$570$	0	0
$571$	−15.2252	−0.637154	−0.318577	−	0.947897i	$-0.603205\pi$
$571$	−15.2252	−0.637154	−0.318577	+	0.947897i	$0.603205\pi$
$572$	−5.53996	−0.231637
$573$	0	0
$574$	18.1208	0.756347
$575$	23.4835	0.979328
$576$	0	0
$577$	17.3215	0.721104	0.360552	−	0.932739i	$-0.382588\pi$
$577$	17.3215	0.721104	0.360552	+	0.932739i	$0.382588\pi$
$578$	58.0312	2.41378
$579$	0	0
$580$	−23.2098	−0.963736
$581$	3.52461	0.146225
$582$	0	0
$583$	−0.508203	−0.0210476
$584$	2.37384	0.0982302
$585$	0	0
$586$	8.41188	0.347492
$587$	−27.8678	−1.15023	−0.575113	−	0.818074i	$-0.695042\pi$
$587$	−27.8678	−1.15023	−0.575113	+	0.818074i	$0.695042\pi$
$588$	0	0
$589$	−2.94244	−0.121241
$590$	−57.1166	−2.35145
$591$	0	0
$592$	33.5592	1.37927
$593$	24.3463	0.999781	0.499890	−	0.866089i	$-0.333374\pi$
$593$	24.3463	0.999781	0.499890	+	0.866089i	$0.333374\pi$
$594$	0	0
$595$	−28.7170	−1.17728
$596$	9.08904	0.372302
$597$	0	0
$598$	−11.4897	−0.469848
$599$	−21.0164	−0.858707	−0.429354	−	0.903136i	$-0.641259\pi$
$599$	−21.0164	−0.858707	−0.429354	+	0.903136i	$0.641259\pi$
$600$	0	0
$601$	40.9477	1.67029	0.835145	−	0.550030i	$-0.185384\pi$
$601$	40.9477	1.67029	0.835145	+	0.550030i	$0.185384\pi$
$602$	21.0716	0.858813
$603$	0	0
$604$	10.9117	0.443993
$605$	−4.18953	−0.170329
$606$	0	0
$607$	−14.0276	−0.569362	−0.284681	−	0.958622i	$-0.591888\pi$
$607$	−14.0276	−0.569362	−0.284681	+	0.958622i	$0.591888\pi$
$608$	2.42723	0.0984371
$609$	0	0
$610$	16.2171	0.656612
$611$	−25.5767	−1.03472
$612$	0	0
$613$	18.5306	0.748442	0.374221	−	0.927339i	$-0.377910\pi$
$613$	18.5306	0.748442	0.374221	+	0.927339i	$0.377910\pi$
$614$	−40.1760	−1.62137
$615$	0	0
$616$	0.491797	0.0198151
$617$	−2.43424	−0.0979987	−0.0489994	−	0.998799i	$-0.515603\pi$
$617$	−2.43424	−0.0979987	−0.0489994	+	0.998799i	$0.515603\pi$
$618$	0	0
$619$	−30.4259	−1.22292	−0.611460	−	0.791275i	$-0.709418\pi$
$619$	−30.4259	−1.22292	−0.611460	+	0.791275i	$0.709418\pi$
$620$	−67.5386	−2.71242
$621$	0	0
$622$	−15.4835	−0.620830
$623$	−1.74173	−0.0697809
$624$	0	0
$625$	69.7966	2.79187
$626$	15.7334	0.628833
$627$	0	0
$628$	−31.6371	−1.26246
$629$	51.7662	2.06405
$630$	0	0
$631$	34.9836	1.39267	0.696337	−	0.717715i	$-0.254812\pi$
$631$	34.9836	1.39267	0.696337	+	0.717715i	$0.254812\pi$
$632$	−2.46705	−0.0981341
$633$	0	0
$634$	38.6925	1.53668
$635$	23.0081	0.913047
$636$	0	0
$637$	−3.17313	−0.125724
$638$	6.14137	0.243139
$639$	0	0
$640$	−16.3502	−0.646297
$641$	−8.56337	−0.338233	−0.169116	−	0.985596i	$-0.554091\pi$
$641$	−8.56337	−0.338233	−0.169116	+	0.985596i	$0.554091\pi$
$642$	0	0
$643$	−5.11273	−0.201626	−0.100813	−	0.994905i	$-0.532144\pi$
$643$	−5.11273	−0.201626	−0.100813	+	0.994905i	$0.532144\pi$
$644$	−3.26634	−0.128712
$645$	0	0
$646$	4.22758	0.166332
$647$	−25.7693	−1.01310	−0.506548	−	0.862212i	$-0.669079\pi$
$647$	−25.7693	−1.01310	−0.506548	+	0.862212i	$0.669079\pi$
$648$	0	0
$649$	7.04399	0.276501
$650$	−77.0877	−3.02363
$651$	0	0
$652$	−4.65269	−0.182213
$653$	47.8953	1.87429	0.937145	−	0.348941i	$-0.113459\pi$
$653$	47.8953	1.87429	0.937145	+	0.348941i	$0.113459\pi$
$654$	0	0
$655$	16.7581	0.654795
$656$	41.6043	1.62437
$657$	0	0
$658$	−15.6004	−0.608165
$659$	−8.95601	−0.348877	−0.174438	−	0.984668i	$-0.555811\pi$
$659$	−8.95601	−0.348877	−0.174438	+	0.984668i	$0.555811\pi$
$660$	0	0
$661$	−10.7993	−0.420044	−0.210022	−	0.977697i	$-0.567353\pi$
$661$	−10.7993	−0.420044	−0.210022	+	0.977697i	$0.567353\pi$
$662$	32.9341	1.28002
$663$	0	0
$664$	1.73339	0.0672686
$665$	−1.33508	−0.0517720
$666$	0	0
$667$	5.93649	0.229862
$668$	19.4590	0.752891
$669$	0	0
$670$	21.6087	0.834817
$671$	−2.00000	−0.0772091
$672$	0	0
$673$	32.2499	1.24314	0.621572	−	0.783357i	$-0.286494\pi$
$673$	32.2499	1.24314	0.621572	+	0.783357i	$0.286494\pi$
$674$	2.95078	0.113660
$675$	0	0
$676$	−5.11769	−0.196834
$677$	27.7089	1.06494	0.532470	−	0.846449i	$-0.321264\pi$
$677$	27.7089	1.06494	0.532470	+	0.846449i	$0.321264\pi$
$678$	0	0
$679$	12.2499	0.470109
$680$	−14.1229	−0.541589
$681$	0	0
$682$	17.8709	0.684311
$683$	−22.3156	−0.853881	−0.426941	−	0.904280i	$-0.640409\pi$
$683$	−22.3156	−0.853881	−0.426941	+	0.904280i	$0.640409\pi$
$684$	0	0
$685$	65.4095	2.49917
$686$	−1.93543	−0.0738951
$687$	0	0
$688$	48.3791	1.84443
$689$	1.61259	0.0614349
$690$	0	0
$691$	25.1372	0.956264	0.478132	−	0.878288i	$-0.341314\pi$
$691$	25.1372	0.956264	0.478132	+	0.878288i	$0.341314\pi$
$692$	43.3788	1.64901
$693$	0	0
$694$	−33.7745	−1.28206
$695$	37.7745	1.43287
$696$	0	0
$697$	64.1760	2.43084
$698$	13.2765	0.502521
$699$	0	0
$700$	−21.9149	−0.828304
$701$	19.2747	0.727995	0.363997	−	0.931400i	$-0.381412\pi$
$701$	19.2747	0.727995	0.363997	+	0.931400i	$0.381412\pi$
$702$	0	0
$703$	2.40665	0.0907686
$704$	−5.85446	−0.220648
$705$	0	0
$706$	61.8008	2.32590
$707$	4.88727	0.183805
$708$	0	0
$709$	9.76098	0.366581	0.183291	−	0.983059i	$-0.441325\pi$
$709$	9.76098	0.366581	0.183291	+	0.983059i	$0.441325\pi$
$710$	−40.6758	−1.52654
$711$	0	0
$712$	−0.856577	−0.0321016
$713$	17.2747	0.646942
$714$	0	0
$715$	13.2939	0.497165
$716$	22.2744	0.832434
$717$	0	0
$718$	47.2908	1.76488
$719$	−14.0276	−0.523141	−0.261570	−	0.965184i	$-0.584240\pi$
$719$	−14.0276	−0.523141	−0.261570	+	0.965184i	$0.584240\pi$
$720$	0	0
$721$	0.637339	0.0237357
$722$	−36.5767	−1.36124
$723$	0	0
$724$	−5.64541	−0.209810
$725$	39.8297	1.47924
$726$	0	0
$727$	−34.3051	−1.27231	−0.636153	−	0.771563i	$-0.719475\pi$
$727$	−34.3051	−1.27231	−0.636153	+	0.771563i	$0.719475\pi$
$728$	−1.56053	−0.0578372
$729$	0	0
$730$	39.1390	1.44860
$731$	74.6263	2.76016
$732$	0	0
$733$	−10.7581	−0.397361	−0.198680	−	0.980064i	$-0.563666\pi$
$733$	−10.7581	−0.397361	−0.198680	+	0.980064i	$0.563666\pi$
$734$	33.0409	1.21956
$735$	0	0
$736$	−14.2499	−0.525259
$737$	−2.66492	−0.0981637
$738$	0	0
$739$	−6.38741	−0.234965	−0.117482	−	0.993075i	$-0.537482\pi$
$739$	−6.38741	−0.234965	−0.117482	+	0.993075i	$0.537482\pi$
$740$	55.2405	2.03068
$741$	0	0
$742$	0.983593	0.0361088
$743$	23.1096	0.847810	0.423905	−	0.905707i	$-0.360659\pi$
$743$	23.1096	0.847810	0.423905	+	0.905707i	$0.360659\pi$
$744$	0	0
$745$	−21.8105	−0.799074
$746$	6.15149	0.225222
$747$	0	0
$748$	−11.9672	−0.437564
$749$	0.956008	0.0349318
$750$	0	0
$751$	31.8678	1.16287	0.581435	−	0.813593i	$-0.302491\pi$
$751$	31.8678	1.16287	0.581435	+	0.813593i	$0.302491\pi$
$752$	−35.8175	−1.30613
$753$	0	0
$754$	−19.4874	−0.709688
$755$	−26.1843	−0.952944
$756$	0	0
$757$	−48.3103	−1.75587	−0.877934	−	0.478781i	$-0.841079\pi$
$757$	−48.3103	−1.75587	−0.877934	+	0.478781i	$0.841079\pi$
$758$	7.62483	0.276946
$759$	0	0
$760$	−0.656586	−0.0238169
$761$	17.8625	0.647516	0.323758	−	0.946140i	$-0.395054\pi$
$761$	17.8625	0.647516	0.323758	+	0.946140i	$0.395054\pi$
$762$	0	0
$763$	7.61259	0.275594
$764$	−36.6207	−1.32489
$765$	0	0
$766$	47.9177	1.73134
$767$	−22.3515	−0.807065
$768$	0	0
$769$	−24.8820	−0.897269	−0.448635	−	0.893715i	$-0.648090\pi$
$769$	−24.8820	−0.897269	−0.448635	+	0.893715i	$0.648090\pi$
$770$	8.10856	0.292212
$771$	0	0
$772$	0.436357	0.0157048
$773$	34.9700	1.25778	0.628892	−	0.777492i	$-0.283509\pi$
$773$	34.9700	1.25778	0.628892	+	0.777492i	$0.283509\pi$
$774$	0	0
$775$	115.901	4.16329
$776$	6.02448	0.216266
$777$	0	0
$778$	−7.07158	−0.253528
$779$	2.98359	0.106898
$780$	0	0
$781$	5.01641	0.179501
$782$	−24.8195	−0.887544
$783$	0	0
$784$	−4.44364	−0.158701
$785$	75.9177	2.70962
$786$	0	0
$787$	−15.1648	−0.540566	−0.270283	−	0.962781i	$-0.587117\pi$
$787$	−15.1648	−0.540566	−0.270283	+	0.962781i	$0.587117\pi$
$788$	−32.2989	−1.15060
$789$	0	0
$790$	−40.6758	−1.44718
$791$	−7.70892	−0.274097
$792$	0	0
$793$	6.34625	0.225362
$794$	8.83210	0.313440
$795$	0	0
$796$	17.2335	0.610826
$797$	−8.24470	−0.292042	−0.146021	−	0.989281i	$-0.546647\pi$
$797$	−8.24470	−0.292042	−0.146021	+	0.989281i	$0.546647\pi$
$798$	0	0
$799$	−55.2497	−1.95459
$800$	−95.6071	−3.38022
$801$	0	0
$802$	14.0000	0.494357
$803$	−4.82687	−0.170337
$804$	0	0
$805$	7.83805	0.276255
$806$	−56.7065	−1.99740
$807$	0	0
$808$	2.40354	0.0845564
$809$	−29.8433	−1.04923	−0.524617	−	0.851338i	$-0.675791\pi$
$809$	−29.8433	−1.04923	−0.524617	+	0.851338i	$0.675791\pi$
$810$	0	0
$811$	−16.6321	−0.584032	−0.292016	−	0.956413i	$-0.594326\pi$
$811$	−16.6321	−0.584032	−0.292016	+	0.956413i	$0.594326\pi$
$812$	−5.53996	−0.194414
$813$	0	0
$814$	−14.6168	−0.512317
$815$	11.1648	0.391086
$816$	0	0
$817$	3.46944	0.121380
$818$	10.2723	0.359162
$819$	0	0
$820$	68.4832	2.39154
$821$	−30.3327	−1.05862	−0.529309	−	0.848429i	$-0.677549\pi$
$821$	−30.3327	−1.05862	−0.529309	+	0.848429i	$0.677549\pi$
$822$	0	0
$823$	−18.4067	−0.641616	−0.320808	−	0.947144i	$-0.603954\pi$
$823$	−18.4067	−0.641616	−0.320808	+	0.947144i	$0.603954\pi$
$824$	0.313441	0.0109192
$825$	0	0
$826$	−13.6332	−0.474359
$827$	−45.4559	−1.58066	−0.790328	−	0.612684i	$-0.790090\pi$
$827$	−45.4559	−1.58066	−0.790328	+	0.612684i	$0.790090\pi$
$828$	0	0
$829$	−20.3463	−0.706655	−0.353327	−	0.935500i	$-0.614950\pi$
$829$	−20.3463	−0.706655	−0.353327	+	0.935500i	$0.614950\pi$
$830$	28.5795	0.992009
$831$	0	0
$832$	18.5769	0.644040
$833$	−6.85446	−0.237493
$834$	0	0
$835$	−46.6946	−1.61593
$836$	−0.556364	−0.0192423
$837$	0	0
$838$	22.1086	0.763728
$839$	−16.1812	−0.558637	−0.279318	−	0.960199i	$-0.590109\pi$
$839$	−16.1812	−0.558637	−0.279318	+	0.960199i	$0.590109\pi$
$840$	0	0
$841$	−18.9313	−0.652802
$842$	−53.0593	−1.82855
$843$	0	0
$844$	−8.09632	−0.278687
$845$	12.2806	0.422466
$846$	0	0
$847$	−1.00000	−0.0343604
$848$	2.25827	0.0775493
$849$	0	0
$850$	−166.522	−5.71165
$851$	−14.1291	−0.484341
$852$	0	0
$853$	−20.9836	−0.718465	−0.359232	−	0.933248i	$-0.616961\pi$
$853$	−20.9836	−0.718465	−0.359232	+	0.933248i	$0.616961\pi$
$854$	3.87086	0.132458
$855$	0	0
$856$	0.470162	0.0160698
$857$	−4.79095	−0.163656	−0.0818279	−	0.996646i	$-0.526076\pi$
$857$	−4.79095	−0.163656	−0.0818279	+	0.996646i	$0.526076\pi$
$858$	0	0
$859$	−9.96719	−0.340076	−0.170038	−	0.985438i	$-0.554389\pi$
$859$	−9.96719	−0.340076	−0.170038	+	0.985438i	$0.554389\pi$
$860$	79.6350	2.71553
$861$	0	0
$862$	4.42412	0.150686
$863$	−25.5470	−0.869629	−0.434814	−	0.900520i	$-0.643186\pi$
$863$	−25.5470	−0.869629	−0.434814	+	0.900520i	$0.643186\pi$
$864$	0	0
$865$	−104.094	−3.53929
$866$	61.1044	2.07641
$867$	0	0
$868$	−16.1208	−0.547176
$869$	5.01641	0.170170
$870$	0	0
$871$	8.45614	0.286525
$872$	3.74385	0.126783
$873$	0	0
$874$	−1.15388	−0.0390306
$875$	31.6402	1.06963
$876$	0	0
$877$	−50.9893	−1.72179	−0.860893	−	0.508787i	$-0.830094\pi$
$877$	−50.9893	−1.72179	−0.860893	+	0.508787i	$0.830094\pi$
$878$	−35.8586	−1.21017
$879$	0	0
$880$	18.6168	0.627571
$881$	−32.1895	−1.08449	−0.542246	−	0.840219i	$-0.682426\pi$
$881$	−32.1895	−1.08449	−0.542246	+	0.840219i	$0.682426\pi$
$882$	0	0
$883$	36.6154	1.23221	0.616104	−	0.787665i	$-0.288710\pi$
$883$	36.6154	1.23221	0.616104	+	0.787665i	$0.288710\pi$
$884$	37.9734	1.27718
$885$	0	0
$886$	−55.0325	−1.84885
$887$	48.2004	1.61841	0.809206	−	0.587525i	$-0.199897\pi$
$887$	48.2004	1.61841	0.809206	+	0.587525i	$0.199897\pi$
$888$	0	0
$889$	5.49180	0.184189
$890$	−14.1229	−0.473401
$891$	0	0
$892$	−32.0307	−1.07247
$893$	−2.56860	−0.0859550
$894$	0	0
$895$	−53.4506	−1.78666
$896$	−3.90262	−0.130377
$897$	0	0
$898$	11.0060	0.367273
$899$	29.2992	0.977181
$900$	0	0
$901$	3.48346	0.116051
$902$	−18.1208	−0.603357
$903$	0	0
$904$	−3.79122	−0.126094
$905$	13.5470	0.450316
$906$	0	0
$907$	22.9013	0.760425	0.380212	−	0.924899i	$-0.375851\pi$
$907$	22.9013	0.760425	0.380212	+	0.924899i	$0.375851\pi$
$908$	−0.661814	−0.0219631
$909$	0	0
$910$	−25.7295	−0.852924
$911$	36.0552	1.19456	0.597281	−	0.802032i	$-0.296248\pi$
$911$	36.0552	1.19456	0.597281	+	0.802032i	$0.296248\pi$
$912$	0	0
$913$	−3.52461	−0.116648
$914$	−25.1762	−0.832756
$915$	0	0
$916$	43.5470	1.43883
$917$	4.00000	0.132092
$918$	0	0
$919$	−28.3379	−0.934782	−0.467391	−	0.884051i	$-0.654806\pi$
$919$	−28.3379	−0.934782	−0.467391	+	0.884051i	$0.654806\pi$
$920$	3.85473	0.127087
$921$	0	0
$922$	12.3463	0.406602
$923$	−15.9177	−0.523937
$924$	0	0
$925$	−94.7966	−3.11689
$926$	65.9851	2.16841
$927$	0	0
$928$	−24.1690	−0.793385
$929$	−5.08514	−0.166838	−0.0834191	−	0.996515i	$-0.526584\pi$
$929$	−5.08514	−0.166838	−0.0834191	+	0.996515i	$0.526584\pi$
$930$	0	0
$931$	−0.318669	−0.0104440
$932$	−40.9424	−1.34111
$933$	0	0
$934$	−35.1250	−1.14932
$935$	28.7170	0.939146
$936$	0	0
$937$	32.2088	1.05222	0.526108	−	0.850418i	$-0.323651\pi$
$937$	32.2088	1.05222	0.526108	+	0.850418i	$0.323651\pi$
$938$	5.15778	0.168407
$939$	0	0
$940$	−58.9578	−1.92299
$941$	−32.7805	−1.06861	−0.534307	−	0.845291i	$-0.679427\pi$
$941$	−32.7805	−1.06861	−0.534307	+	0.845291i	$0.679427\pi$
$942$	0	0
$943$	−17.5163	−0.570408
$944$	−31.3009	−1.01876
$945$	0	0
$946$	−21.0716	−0.685096
$947$	27.3627	0.889167	0.444584	−	0.895737i	$-0.353352\pi$
$947$	27.3627	0.889167	0.444584	+	0.895737i	$0.353352\pi$
$948$	0	0
$949$	15.3163	0.497188
$950$	−7.74173	−0.251175
$951$	0	0
$952$	−3.37100	−0.109255
$953$	24.6894	0.799768	0.399884	−	0.916566i	$-0.369050\pi$
$953$	24.6894	0.799768	0.399884	+	0.916566i	$0.369050\pi$
$954$	0	0
$955$	87.8765	2.84362
$956$	−8.86359	−0.286669
$957$	0	0
$958$	82.7393	2.67319
$959$	15.6126	0.504157
$960$	0	0
$961$	54.2580	1.75026
$962$	46.3808	1.49538
$963$	0	0
$964$	33.6852	1.08493
$965$	−1.04710	−0.0337074
$966$	0	0
$967$	21.3298	0.685922	0.342961	−	0.939350i	$-0.388570\pi$
$967$	21.3298	0.685922	0.342961	+	0.939350i	$0.388570\pi$
$968$	−0.491797	−0.0158069
$969$	0	0
$970$	99.3293	3.18927
$971$	−28.4946	−0.914436	−0.457218	−	0.889355i	$-0.651154\pi$
$971$	−28.4946	−0.914436	−0.457218	+	0.889355i	$0.651154\pi$
$972$	0	0
$973$	9.01641	0.289053
$974$	51.8521	1.66145
$975$	0	0
$976$	8.88727	0.284475
$977$	19.8157	0.633960	0.316980	−	0.948432i	$-0.397331\pi$
$977$	19.8157	0.633960	0.316980	+	0.948432i	$0.397331\pi$
$978$	0	0
$979$	1.74173	0.0556659
$980$	−7.31450	−0.233653
$981$	0	0
$982$	−61.3171	−1.95671
$983$	53.7745	1.71514	0.857571	−	0.514366i	$-0.171973\pi$
$983$	53.7745	1.71514	0.857571	+	0.514366i	$0.171973\pi$
$984$	0	0
$985$	77.5058	2.46954
$986$	−42.0958	−1.34060
$987$	0	0
$988$	1.76541	0.0561653
$989$	−20.3686	−0.647684
$990$	0	0
$991$	49.7693	1.58097	0.790487	−	0.612479i	$-0.209827\pi$
$991$	49.7693	1.58097	0.790487	+	0.612479i	$0.209827\pi$
$992$	−70.3296	−2.23297
$993$	0	0
$994$	−9.70892	−0.307948
$995$	−41.3543	−1.31102
$996$	0	0
$997$	0.659696	0.0208928	0.0104464	−	0.999945i	$-0.496675\pi$
$997$	0.659696	0.0208928	0.0104464	+	0.999945i	$0.496675\pi$
$998$	−46.6943	−1.47808
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	693.2.a.l.1.3		3
3.2	odd	2		231.2.a.e.1.1	✓	3
7.6	odd	2		4851.2.a.bi.1.3		3
11.10	odd	2		7623.2.a.cd.1.1		3
12.11	even	2		3696.2.a.bo.1.3		3
15.14	odd	2		5775.2.a.bp.1.3		3
21.20	even	2		1617.2.a.t.1.1		3
33.32	even	2		2541.2.a.bg.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
231.2.a.e.1.1	✓	3	3.2	odd	2
693.2.a.l.1.3		3	1.1	even	1	trivial
1617.2.a.t.1.1		3	21.20	even	2
2541.2.a.bg.1.3		3	33.32	even	2
3696.2.a.bo.1.3		3	12.11	even	2
4851.2.a.bi.1.3		3	7.6	odd	2
5775.2.a.bp.1.3		3	15.14	odd	2
7623.2.a.cd.1.1		3	11.10	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 $\F_{q}$
Belyi maps

Level:	\( N \)	\(=\)	\( 693 = 3^{2} \cdot 7 \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	693.a (trivial)

\(f(q)\)	\(=\)	\(q+1.93543 q^{2} +1.74590 q^{4} -4.18953 q^{5} -1.00000 q^{7} -0.491797 q^{8} -8.10856 q^{10} +1.00000 q^{11} -3.17313 q^{13} -1.93543 q^{14} -4.44364 q^{16} -6.85446 q^{17} -0.318669 q^{19} -7.31450 q^{20} +1.93543 q^{22} +1.87086 q^{23} +12.5522 q^{25} -6.14137 q^{26} -1.74590 q^{28} +3.17313 q^{29} +9.23353 q^{31} -7.61676 q^{32} -13.2663 q^{34} +4.18953 q^{35} -7.55220 q^{37} -0.616763 q^{38} +2.06040 q^{40} -9.36266 q^{41} -10.8873 q^{43} +1.74590 q^{44} +3.62093 q^{46} +8.06040 q^{47} +1.00000 q^{49} +24.2939 q^{50} -5.53996 q^{52} -0.508203 q^{53} -4.18953 q^{55} +0.491797 q^{56} +6.14137 q^{58} +7.04399 q^{59} -2.00000 q^{61} +17.8709 q^{62} -5.85446 q^{64} +13.2939 q^{65} -2.66492 q^{67} -11.9672 q^{68} +8.10856 q^{70} +5.01641 q^{71} -4.82687 q^{73} -14.6168 q^{74} -0.556364 q^{76} -1.00000 q^{77} +5.01641 q^{79} +18.6168 q^{80} -18.1208 q^{82} -3.52461 q^{83} +28.7170 q^{85} -21.0716 q^{86} -0.491797 q^{88} +1.74173 q^{89} +3.17313 q^{91} +3.26634 q^{92} +15.6004 q^{94} +1.33508 q^{95} -12.2499 q^{97} +1.93543 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q - 2 q^{2} + 6 q^{4} - 4 q^{5} - 3 q^{7} - 3 q^{8} - 11 q^{10} + 3 q^{11} - 4 q^{13} + 2 q^{14} - 4 q^{16} - 8 q^{17} - 8 q^{19} + 3 q^{20} - 2 q^{22} - 10 q^{23} + 15 q^{25} + q^{26} - 6 q^{28} + 4 q^{29}+ \cdots - 2 q^{98}+O(q^{100}) \) 3 * q - 2 * q^2 + 6 * q^4 - 4 * q^5 - 3 * q^7 - 3 * q^8 - 11 * q^10 + 3 * q^11 - 4 * q^13 + 2 * q^14 - 4 * q^16 - 8 * q^17 - 8 * q^19 + 3 * q^20 - 2 * q^22 - 10 * q^23 + 15 * q^25 + q^26 - 6 * q^28 + 4 * q^29 - 2 * q^31 - 8 * q^32 - 4 * q^34 + 4 * q^35 + 13 * q^38 - 18 * q^40 - 14 * q^41 - 14 * q^43 + 6 * q^44 + 28 * q^46 + 3 * q^49 + 19 * q^50 - 29 * q^52 - 4 * q^55 + 3 * q^56 - q^58 - 6 * q^61 + 38 * q^62 - 5 * q^64 - 14 * q^65 - 4 * q^67 - 42 * q^68 + 11 * q^70 + 12 * q^71 - 20 * q^73 - 29 * q^74 - 11 * q^76 - 3 * q^77 + 12 * q^79 + 41 * q^80 - 6 * q^82 - 6 * q^83 - 6 * q^85 - 24 * q^86 - 3 * q^88 - 26 * q^89 + 4 * q^91 - 26 * q^92 + 35 * q^94 + 8 * q^95 - 4 * q^97 - 2 * q^98

Introduction

L-functions

Modular forms

Varieties

Fields

Representations

Groups

Database

Properties

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