LMFDB - Embedded newform 513.2.a.f.1.2

Show commands: Magma / Pari/GP / SageMath

Newspace parameters

comment:Compute space of new eigenforms

gp:[N,k,chi] = [513,2,Mod(1,513)] 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("513.1"); S:= CuspForms(chi, 2); N := Newforms(S);

sage:from sage.modular.dirichlet import DirichletCharacter H = DirichletGroup(513, base_ring=CyclotomicField(2)) chi = DirichletCharacter(H, H._module([0, 0])) N = Newforms(chi, 2, names="a")

Level:	$ N $	$=$	$ 513 = 3^{3} \cdot 19 $
Weight:	$ k $	$=$	$ 2 $
Character orbit:	$[\chi]$	$=$	513.a (trivial)

Newform invariants

comment:select newform

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

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

Self dual:	yes
Analytic conductor:	$4.09632562369$
Analytic rank:	$0$
Dimension:	$3$
Coefficient field:	3.3.321.1
comment:defining polynomial gp:f.mod \\ as an extension of the character field
Defining polynomial:	$ x^{3} - x^{2} - 4x + 1 $ x^3 - x^2 - 4*x + 1
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			1.2
Root			$0.239123$ of defining polynomial
Character	$\chi$	$=$	513.1

$q$-expansion

comment:q-expansion

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

gp:mfcoefs(f, 20)

$f(q)$	$=$	\(q+0.239123 q^{2} -1.94282 q^{4} +1.76088 q^{5} +3.18194 q^{7} -0.942820 q^{8} +0.421067 q^{10} -0.181943 q^{11} -3.18194 q^{13} +0.760877 q^{14} +3.66019 q^{16} +7.84213 q^{17} -1.00000 q^{19} -3.42107 q^{20} -0.0435069 q^{22} +3.28263 q^{23} -1.89931 q^{25} -0.760877 q^{26} -6.18194 q^{28} +9.48865 q^{29} +2.18194 q^{31} +2.76088 q^{32} +1.87524 q^{34} +5.60301 q^{35} -6.42107 q^{37} -0.239123 q^{38} -1.66019 q^{40} +1.76088 q^{41} +9.16827 q^{43} +0.353483 q^{44} +0.784953 q^{46} +9.00000 q^{47} +3.12476 q^{49} -0.454170 q^{50} +6.18194 q^{52} -13.6706 q^{53} -0.320380 q^{55} -3.00000 q^{56} +2.26896 q^{58} -8.78495 q^{59} -0.703697 q^{61} +0.521753 q^{62} -6.66019 q^{64} -5.60301 q^{65} -6.37756 q^{67} -15.2359 q^{68} +1.33981 q^{70} -3.22545 q^{71} -6.88564 q^{73} -1.53543 q^{74} +1.94282 q^{76} -0.578933 q^{77} -0.986327 q^{79} +6.44514 q^{80} +0.421067 q^{82} -10.2930 q^{83} +13.8090 q^{85} +2.19235 q^{86} +0.171540 q^{88} +7.58934 q^{89} -10.1248 q^{91} -6.37756 q^{92} +2.15211 q^{94} -1.76088 q^{95} +0.784953 q^{97} +0.747204 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + q^{2} + 3 q^{4} + 5 q^{5} + q^{7} + 6 q^{8} - 7 q^{10} + 8 q^{11} - q^{13} + 2 q^{14} + 3 q^{16} + 7 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} + 9 q^{23} + 2 q^{25} - 2 q^{26} - 10 q^{28} - 6 q^{29}+ \cdots - 8 q^{98}+O(q^{100}) $ 3 * q + q^2 + 3 * q^4 + 5 * q^5 + q^7 + 6 * q^8 - 7 * q^10 + 8 * q^11 - q^13 + 2 * q^14 + 3 * q^16 + 7 * q^17 - 3 * q^19 - 2 * q^20 + q^22 + 9 * q^23 + 2 * q^25 - 2 * q^26 - 10 * q^28 - 6 * q^29 - 2 * q^31 + 8 * q^32 + 23 * q^34 - 11 * q^37 - q^38 + 3 * q^40 + 5 * q^41 + 9 * q^43 + 19 * q^44 - 23 * q^46 + 27 * q^47 - 8 * q^49 - 27 * q^50 + 10 * q^52 + 2 * q^53 + 15 * q^55 - 9 * q^56 - 4 * q^58 - q^59 + 7 * q^61 + q^62 - 12 * q^64 - 12 * q^67 - q^68 + 12 * q^70 - 3 * q^73 - 14 * q^74 - 3 * q^76 - 10 * q^77 + 7 * q^79 - 14 * q^80 - 7 * q^82 + 5 * q^83 - 9 * q^85 - 37 * q^86 + 27 * q^88 - 4 * q^89 - 13 * q^91 - 12 * q^92 + 9 * q^94 - 5 * q^95 - 23 * q^97 - 8 * 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 $

$2$	0.239123	0.169086	0.0845428	−	0.996420i	$-0.473057\pi$
$2$	0.239123	0.169086	0.0845428	+	0.996420i	$0.473057\pi$
$3$	0	0
$3$	0	0
$4$	−1.94282	−0.971410
$5$	1.76088	0.787488	0.393744	−	0.919220i	$-0.371180\pi$
$5$	1.76088	0.787488	0.393744	+	0.919220i	$0.371180\pi$
$6$	0	0
$7$	3.18194	1.20266	0.601331	−	0.799000i	$-0.294637\pi$
$7$	3.18194	1.20266	0.601331	+	0.799000i	$0.294637\pi$
$8$	−0.942820	−0.333337
$9$	0	0
$10$	0.421067	0.133153
$11$	−0.181943	−0.0548580	−0.0274290	−	0.999624i	$-0.508732\pi$
$11$	−0.181943	−0.0548580	−0.0274290	+	0.999624i	$0.508732\pi$
$12$	0	0
$13$	−3.18194	−0.882512	−0.441256	−	0.897381i	$-0.645467\pi$
$13$	−3.18194	−0.882512	−0.441256	+	0.897381i	$0.645467\pi$
$14$	0.760877	0.203353
$15$	0	0
$16$	3.66019	0.915047
$17$	7.84213	1.90200	0.950998	−	0.309196i	$-0.100060\pi$
$17$	7.84213	1.90200	0.950998	+	0.309196i	$0.100060\pi$
$18$	0	0
$19$	−1.00000	−0.229416
$19$	−1.00000	−0.229416
$20$	−3.42107	−0.764974
$21$	0	0
$22$	−0.0435069	−0.00927570
$23$	3.28263	0.684476	0.342238	−	0.939613i	$-0.388815\pi$
$23$	3.28263	0.684476	0.342238	+	0.939613i	$0.388815\pi$
$24$	0	0
$25$	−1.89931	−0.379863
$26$	−0.760877	−0.149220
$27$	0	0
$28$	−6.18194	−1.16828
$29$	9.48865	1.76200	0.880999	−	0.473118i	$-0.156872\pi$
$29$	9.48865	1.76200	0.880999	+	0.473118i	$0.156872\pi$
$30$	0	0
$31$	2.18194	0.391889	0.195944	−	0.980615i	$-0.437223\pi$
$31$	2.18194	0.391889	0.195944	+	0.980615i	$0.437223\pi$
$32$	2.76088	0.488059
$33$	0	0
$34$	1.87524	0.321600
$35$	5.60301	0.947082
$36$	0	0
$37$	−6.42107	−1.05562	−0.527808	−	0.849363i	$-0.676986\pi$
$37$	−6.42107	−1.05562	−0.527808	+	0.849363i	$0.676986\pi$
$38$	−0.239123	−0.0387909
$39$	0	0
$40$	−1.66019	−0.262499
$41$	1.76088	0.275003	0.137501	−	0.990502i	$-0.456093\pi$
$41$	1.76088	0.275003	0.137501	+	0.990502i	$0.456093\pi$
$42$	0	0
$43$	9.16827	1.39815	0.699074	−	0.715049i	$-0.253596\pi$
$43$	9.16827	1.39815	0.699074	+	0.715049i	$0.253596\pi$
$44$	0.353483	0.0532896
$45$	0	0
$46$	0.784953	0.115735
$47$	9.00000	1.31278	0.656392	−	0.754420i	$-0.272082\pi$
$47$	9.00000	1.31278	0.656392	+	0.754420i	$0.272082\pi$
$48$	0	0
$49$	3.12476	0.446395
$50$	−0.454170	−0.0642293
$51$	0	0
$52$	6.18194	0.857281
$53$	−13.6706	−1.87780	−0.938900	−	0.344189i	$-0.888154\pi$
$53$	−13.6706	−1.87780	−0.938900	+	0.344189i	$0.888154\pi$
$54$	0	0
$55$	−0.320380	−0.0432000
$56$	−3.00000	−0.400892
$57$	0	0
$58$	2.26896	0.297929
$59$	−8.78495	−1.14370	−0.571852	−	0.820357i	$-0.693775\pi$
$59$	−8.78495	−1.14370	−0.571852	+	0.820357i	$0.693775\pi$
$60$	0	0
$61$	−0.703697	−0.0900991	−0.0450496	−	0.998985i	$-0.514345\pi$
$61$	−0.703697	−0.0900991	−0.0450496	+	0.998985i	$0.514345\pi$
$62$	0.521753	0.0662628
$63$	0	0
$64$	−6.66019	−0.832524
$65$	−5.60301	−0.694968
$66$	0	0
$67$	−6.37756	−0.779143	−0.389571	−	0.920996i	$-0.627377\pi$
$67$	−6.37756	−0.779143	−0.389571	+	0.920996i	$0.627377\pi$
$68$	−15.2359	−1.84762
$69$	0	0
$70$	1.33981	0.160138
$71$	−3.22545	−0.382791	−0.191395	−	0.981513i	$-0.561301\pi$
$71$	−3.22545	−0.382791	−0.191395	+	0.981513i	$0.561301\pi$
$72$	0	0
$73$	−6.88564	−0.805903	−0.402952	−	0.915221i	$-0.632016\pi$
$73$	−6.88564	−0.805903	−0.402952	+	0.915221i	$0.632016\pi$
$74$	−1.53543	−0.178490
$75$	0	0
$76$	1.94282	0.222857
$77$	−0.578933	−0.0659756
$78$	0	0
$79$	−0.986327	−0.110970	−0.0554852	−	0.998460i	$-0.517671\pi$
$79$	−0.986327	−0.110970	−0.0554852	+	0.998460i	$0.517671\pi$
$80$	6.44514	0.720589
$81$	0	0
$82$	0.421067	0.0464990
$83$	−10.2930	−1.12981	−0.564904	−	0.825157i	$-0.691087\pi$
$83$	−10.2930	−1.12981	−0.564904	+	0.825157i	$0.691087\pi$
$84$	0	0
$85$	13.8090	1.49780
$86$	2.19235	0.236407
$87$	0	0
$88$	0.171540	0.0182862
$89$	7.58934	0.804468	0.402234	−	0.915537i	$-0.368234\pi$
$89$	7.58934	0.804468	0.402234	+	0.915537i	$0.368234\pi$
$90$	0	0
$91$	−10.1248	−1.06136
$92$	−6.37756	−0.664907
$93$	0	0
$94$	2.15211	0.221973
$95$	−1.76088	−0.180662
$96$	0	0
$97$	0.784953	0.0796999	0.0398500	−	0.999206i	$-0.487312\pi$
$97$	0.784953	0.0796999	0.0398500	+	0.999206i	$0.487312\pi$
$98$	0.747204	0.0754790
$99$	0	0
$100$	3.69002	0.369002
$101$	−3.06758	−0.305236	−0.152618	−	0.988285i	$-0.548770\pi$
$101$	−3.06758	−0.305236	−0.152618	+	0.988285i	$0.548770\pi$
$102$	0	0
$103$	−19.0104	−1.87315	−0.936575	−	0.350466i	$-0.886023\pi$
$103$	−19.0104	−1.87315	−0.936575	+	0.350466i	$0.886023\pi$
$104$	3.00000	0.294174
$105$	0	0
$106$	−3.26896	−0.317509
$107$	−1.23912	−0.119791	−0.0598953	−	0.998205i	$-0.519077\pi$
$107$	−1.23912	−0.119791	−0.0598953	+	0.998205i	$0.519077\pi$
$108$	0	0
$109$	1.19562	0.114519	0.0572596	−	0.998359i	$-0.481764\pi$
$109$	1.19562	0.114519	0.0572596	+	0.998359i	$0.481764\pi$
$110$	−0.0766103	−0.00730450
$111$	0	0
$112$	11.6465	1.10049
$113$	5.78495	0.544203	0.272101	−	0.962269i	$-0.412281\pi$
$113$	5.78495	0.544203	0.272101	+	0.962269i	$0.412281\pi$
$114$	0	0
$115$	5.78031	0.539016
$116$	−18.4347	−1.71162
$117$	0	0
$118$	−2.10069	−0.193384
$119$	24.9532	2.28746
$120$	0	0
$121$	−10.9669	−0.996991
$122$	−0.168270	−0.0152345
$123$	0	0
$124$	−4.23912	−0.380685
$125$	−12.1488	−1.08663
$126$	0	0
$127$	12.2060	1.08311	0.541555	−	0.840666i	$-0.317836\pi$
$127$	12.2060	1.08311	0.541555	+	0.840666i	$0.317836\pi$
$128$	−7.11436	−0.628827
$129$	0	0
$130$	−1.33981	−0.117509
$131$	7.50808	0.655984	0.327992	−	0.944680i	$-0.393628\pi$
$131$	7.50808	0.655984	0.327992	+	0.944680i	$0.393628\pi$
$132$	0	0
$133$	−3.18194	−0.275909
$134$	−1.52502	−0.131742
$135$	0	0
$136$	−7.39372	−0.634006
$137$	−12.2255	−1.04449	−0.522245	−	0.852795i	$-0.674905\pi$
$137$	−12.2255	−1.04449	−0.522245	+	0.852795i	$0.674905\pi$
$138$	0	0
$139$	4.40739	0.373830	0.186915	−	0.982376i	$-0.440151\pi$
$139$	4.40739	0.373830	0.186915	+	0.982376i	$0.440151\pi$
$140$	−10.8856	−0.920005
$141$	0	0
$142$	−0.771280	−0.0647244
$143$	0.578933	0.0484128
$144$	0	0
$145$	16.7083	1.38755
$146$	−1.64652	−0.136267
$147$	0	0
$148$	12.4750	1.02544
$149$	−2.45417	−0.201053	−0.100527	−	0.994934i	$-0.532053\pi$
$149$	−2.45417	−0.201053	−0.100527	+	0.994934i	$0.532053\pi$
$150$	0	0
$151$	19.5127	1.58792	0.793962	−	0.607968i	$-0.208015\pi$
$151$	19.5127	1.58792	0.793962	+	0.607968i	$0.208015\pi$
$152$	0.942820	0.0764728
$153$	0	0
$154$	−0.138436	−0.0111555
$155$	3.84213	0.308608
$156$	0	0
$157$	−15.4991	−1.23696	−0.618480	−	0.785801i	$-0.712251\pi$
$157$	−15.4991	−1.23696	−0.618480	+	0.785801i	$0.712251\pi$
$158$	−0.235854	−0.0187635
$159$	0	0
$160$	4.86156	0.384340
$161$	10.4451	0.823193
$162$	0	0
$163$	13.2391	1.03697	0.518484	−	0.855087i	$-0.326497\pi$
$163$	13.2391	1.03697	0.518484	+	0.855087i	$0.326497\pi$
$164$	−3.42107	−0.267140
$165$	0	0
$166$	−2.46130	−0.191034
$167$	3.21505	0.248788	0.124394	−	0.992233i	$-0.460301\pi$
$167$	3.21505	0.248788	0.124394	+	0.992233i	$0.460301\pi$
$168$	0	0
$169$	−2.87524	−0.221172
$170$	3.30206	0.253256
$171$	0	0
$172$	−17.8123	−1.35818
$173$	−10.9863	−0.835275	−0.417637	−	0.908614i	$-0.637142\pi$
$173$	−10.9863	−0.835275	−0.417637	+	0.908614i	$0.637142\pi$
$174$	0	0
$175$	−6.04351	−0.456846
$176$	−0.665947	−0.0501977
$177$	0	0
$178$	1.81479	0.136024
$179$	3.61668	0.270324	0.135162	−	0.990824i	$-0.456845\pi$
$179$	3.61668	0.270324	0.135162	+	0.990824i	$0.456845\pi$
$180$	0	0
$181$	3.38332	0.251480	0.125740	−	0.992063i	$-0.459869\pi$
$181$	3.38332	0.251480	0.125740	+	0.992063i	$0.459869\pi$
$182$	−2.42107	−0.179461
$183$	0	0
$184$	−3.09493	−0.228161
$185$	−11.3067	−0.831286
$186$	0	0
$187$	−1.42682	−0.104340
$188$	−17.4854	−1.27525
$189$	0	0
$190$	−0.421067	−0.0305474
$191$	−11.1111	−0.803970	−0.401985	−	0.915646i	$-0.631680\pi$
$191$	−11.1111	−0.803970	−0.401985	+	0.915646i	$0.631680\pi$
$192$	0	0
$193$	7.54583	0.543161	0.271580	−	0.962416i	$-0.412454\pi$
$193$	7.54583	0.543161	0.271580	+	0.962416i	$0.412454\pi$
$194$	0.187701	0.0134761
$195$	0	0
$196$	−6.07085	−0.433632
$197$	−12.4016	−0.883580	−0.441790	−	0.897118i	$-0.645656\pi$
$197$	−12.4016	−0.883580	−0.441790	+	0.897118i	$0.645656\pi$
$198$	0	0
$199$	−18.8960	−1.33951	−0.669753	−	0.742584i	$-0.733600\pi$
$199$	−18.8960	−1.33951	−0.669753	+	0.742584i	$0.733600\pi$
$200$	1.79071	0.126622
$201$	0	0
$202$	−0.733531	−0.0516110
$203$	30.1923	2.11909
$204$	0	0
$205$	3.10069	0.216561
$206$	−4.54583	−0.316723
$207$	0	0
$208$	−11.6465	−0.807541
$209$	0.181943	0.0125853
$210$	0	0
$211$	−11.7576	−0.809427	−0.404714	−	0.914444i	$-0.632629\pi$
$211$	−11.7576	−0.809427	−0.404714	+	0.914444i	$0.632629\pi$
$212$	26.5595	1.82411
$213$	0	0
$214$	−0.296303	−0.0202549
$215$	16.1442	1.10102
$216$	0	0
$217$	6.94282	0.471309
$218$	0.285900	0.0193636
$219$	0	0
$220$	0.622440	0.0419649
$221$	−24.9532	−1.67854
$222$	0	0
$223$	10.1923	0.682530	0.341265	−	0.939967i	$-0.389145\pi$
$223$	10.1923	0.682530	0.341265	+	0.939967i	$0.389145\pi$
$224$	8.78495	0.586969
$225$	0	0
$226$	1.38332	0.0920169
$227$	22.7141	1.50759	0.753794	−	0.657111i	$-0.228222\pi$
$227$	22.7141	1.50759	0.753794	+	0.657111i	$0.228222\pi$
$228$	0	0
$229$	13.8960	0.918276	0.459138	−	0.888365i	$-0.348158\pi$
$229$	13.8960	0.918276	0.459138	+	0.888365i	$0.348158\pi$
$230$	1.38221	0.0911400
$231$	0	0
$232$	−8.94609	−0.587340
$233$	27.4750	1.79995	0.899973	−	0.435946i	$-0.143586\pi$
$233$	27.4750	1.79995	0.899973	+	0.435946i	$0.143586\pi$
$234$	0	0
$235$	15.8479	1.03380
$236$	17.0676	1.11101
$237$	0	0
$238$	5.96690	0.386776
$239$	18.4692	1.19467	0.597337	−	0.801990i	$-0.296225\pi$
$239$	18.4692	1.19467	0.597337	+	0.801990i	$0.296225\pi$
$240$	0	0
$241$	−19.1248	−1.23193	−0.615967	−	0.787772i	$-0.711235\pi$
$241$	−19.1248	−1.23193	−0.615967	+	0.787772i	$0.711235\pi$
$242$	−2.62244	−0.168577
$243$	0	0
$244$	1.36716	0.0875232
$245$	5.50232	0.351531
$246$	0	0
$247$	3.18194	0.202462
$248$	−2.05718	−0.130631
$249$	0	0
$250$	−2.90507	−0.183733
$251$	−19.7472	−1.24643	−0.623216	−	0.782050i	$-0.714174\pi$
$251$	−19.7472	−1.24643	−0.623216	+	0.782050i	$0.714174\pi$
$252$	0	0
$253$	−0.597253	−0.0375490
$254$	2.91874	0.183138
$255$	0	0
$256$	11.6192	0.726198
$257$	−16.4692	−1.02732	−0.513661	−	0.857993i	$-0.671711\pi$
$257$	−16.4692	−1.02732	−0.513661	+	0.857993i	$0.671711\pi$
$258$	0	0
$259$	−20.4315	−1.26955
$260$	10.8856	0.675099
$261$	0	0
$262$	1.79536	0.110918
$263$	19.4451	1.19904	0.599519	−	0.800360i	$-0.295358\pi$
$263$	19.4451	1.19904	0.599519	+	0.800360i	$0.295358\pi$
$264$	0	0
$265$	−24.0722	−1.47875
$266$	−0.760877	−0.0466523
$267$	0	0
$268$	12.3905	0.756867
$269$	14.1546	0.863021	0.431511	−	0.902108i	$-0.357981\pi$
$269$	14.1546	0.863021	0.431511	+	0.902108i	$0.357981\pi$
$270$	0	0
$271$	4.75512	0.288853	0.144426	−	0.989516i	$-0.453866\pi$
$271$	4.75512	0.288853	0.144426	+	0.989516i	$0.453866\pi$
$272$	28.7037	1.74042
$273$	0	0
$274$	−2.92339	−0.176608
$275$	0.345567	0.0208385
$276$	0	0
$277$	−25.4854	−1.53127	−0.765634	−	0.643276i	$-0.777575\pi$
$277$	−25.4854	−1.53127	−0.765634	+	0.643276i	$0.777575\pi$
$278$	1.05391	0.0632093
$279$	0	0
$280$	−5.28263	−0.315698
$281$	−30.2060	−1.80194	−0.900970	−	0.433881i	$-0.857144\pi$
$281$	−30.2060	−1.80194	−0.900970	+	0.433881i	$0.857144\pi$
$282$	0	0
$283$	−15.7986	−0.939131	−0.469565	−	0.882898i	$-0.655589\pi$
$283$	−15.7986	−0.939131	−0.469565	+	0.882898i	$0.655589\pi$
$284$	6.26647	0.371847
$285$	0	0
$286$	0.138436	0.00818592
$287$	5.60301	0.330735
$288$	0	0
$289$	44.4991	2.61759
$290$	3.99535	0.234615
$291$	0	0
$292$	13.3776	0.782862
$293$	−27.2632	−1.59273	−0.796367	−	0.604814i	$-0.793247\pi$
$293$	−27.2632	−1.59273	−0.796367	+	0.604814i	$0.793247\pi$
$294$	0	0
$295$	−15.4692	−0.900653
$296$	6.05391	0.351876
$297$	0	0
$298$	−0.586849	−0.0339953
$299$	−10.4451	−0.604058
$300$	0	0
$301$	29.1729	1.68150
$302$	4.66595	0.268495
$303$	0	0
$304$	−3.66019	−0.209926
$305$	−1.23912	−0.0709520
$306$	0	0
$307$	29.5264	1.68516	0.842580	−	0.538571i	$-0.181036\pi$
$307$	29.5264	1.68516	0.842580	+	0.538571i	$0.181036\pi$
$308$	1.12476	0.0640893
$309$	0	0
$310$	0.918743	0.0521811
$311$	17.8525	1.01232	0.506162	−	0.862438i	$-0.331064\pi$
$311$	17.8525	1.01232	0.506162	+	0.862438i	$0.331064\pi$
$312$	0	0
$313$	−2.97265	−0.168024	−0.0840122	−	0.996465i	$-0.526773\pi$
$313$	−2.97265	−0.168024	−0.0840122	+	0.996465i	$0.526773\pi$
$314$	−3.70618	−0.209152
$315$	0	0
$316$	1.91626	0.107798
$317$	17.1625	0.963943	0.481971	−	0.876187i	$-0.339921\pi$
$317$	17.1625	0.963943	0.481971	+	0.876187i	$0.339921\pi$
$318$	0	0
$319$	−1.72640	−0.0966597
$320$	−11.7278	−0.655602
$321$	0	0
$322$	2.49768	0.139190
$323$	−7.84213	−0.436348
$324$	0	0
$325$	6.04351	0.335233
$326$	3.16578	0.175336
$327$	0	0
$328$	−1.66019	−0.0916687
$329$	28.6375	1.57884
$330$	0	0
$331$	23.9669	1.31734	0.658670	−	0.752432i	$-0.271119\pi$
$331$	23.9669	1.31734	0.658670	+	0.752432i	$0.271119\pi$
$332$	19.9975	1.09751
$333$	0	0
$334$	0.768793	0.0420665
$335$	−11.2301	−0.613566
$336$	0	0
$337$	−7.53791	−0.410616	−0.205308	−	0.978697i	$-0.565820\pi$
$337$	−7.53791	−0.410616	−0.205308	+	0.978697i	$0.565820\pi$
$338$	−0.687536	−0.0373970
$339$	0	0
$340$	−26.8285	−1.45498
$341$	−0.396990	−0.0214982
$342$	0	0
$343$	−12.3308	−0.665800
$344$	−8.64403	−0.466055
$345$	0	0
$346$	−2.62709	−0.141233
$347$	7.78822	0.418094	0.209047	−	0.977906i	$-0.432964\pi$
$347$	7.78822	0.418094	0.209047	+	0.977906i	$0.432964\pi$
$348$	0	0
$349$	−16.0137	−0.857192	−0.428596	−	0.903496i	$-0.640992\pi$
$349$	−16.0137	−0.857192	−0.428596	+	0.903496i	$0.640992\pi$
$350$	−1.44514	−0.0772462
$351$	0	0
$352$	−0.502323	−0.0267739
$353$	21.8090	1.16078	0.580389	−	0.814340i	$-0.302901\pi$
$353$	21.8090	1.16078	0.580389	+	0.814340i	$0.302901\pi$
$354$	0	0
$355$	−5.67962	−0.301443
$356$	−14.7447	−0.781468
$357$	0	0
$358$	0.864833	0.0457078
$359$	13.9942	0.738588	0.369294	−	0.929313i	$-0.379600\pi$
$359$	13.9942	0.738588	0.369294	+	0.929313i	$0.379600\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	0.809030	0.0425217
$363$	0	0
$364$	19.6706	1.03102
$365$	−12.1248	−0.634639
$366$	0	0
$367$	−28.6328	−1.49462	−0.747311	−	0.664474i	$-0.768656\pi$
$367$	−28.6328	−1.49462	−0.747311	+	0.664474i	$0.768656\pi$
$368$	12.0150	0.626328
$369$	0	0
$370$	−2.70370	−0.140559
$371$	−43.4991	−2.25836
$372$	0	0
$373$	−17.2197	−0.891602	−0.445801	−	0.895132i	$-0.647081\pi$
$373$	−17.2197	−0.891602	−0.445801	+	0.895132i	$0.647081\pi$
$374$	−0.341187	−0.0176423
$375$	0	0
$376$	−8.48538	−0.437600
$377$	−30.1923	−1.55498
$378$	0	0
$379$	9.01040	0.462833	0.231417	−	0.972855i	$-0.425664\pi$
$379$	9.01040	0.462833	0.231417	+	0.972855i	$0.425664\pi$
$380$	3.42107	0.175497
$381$	0	0
$382$	−2.65692	−0.135940
$383$	20.3639	1.04055	0.520273	−	0.854000i	$-0.325830\pi$
$383$	20.3639	1.04055	0.520273	+	0.854000i	$0.325830\pi$
$384$	0	0
$385$	−1.01943	−0.0519550
$386$	1.80438	0.0918407
$387$	0	0
$388$	−1.52502	−0.0774213
$389$	−10.1327	−0.513747	−0.256874	−	0.966445i	$-0.582692\pi$
$389$	−10.1327	−0.513747	−0.256874	+	0.966445i	$0.582692\pi$
$390$	0	0
$391$	25.7428	1.30187
$392$	−2.94609	−0.148800
$393$	0	0
$394$	−2.96552	−0.149401
$395$	−1.73680	−0.0873879
$396$	0	0
$397$	30.3250	1.52197	0.760985	−	0.648770i	$-0.224716\pi$
$397$	30.3250	1.52197	0.760985	+	0.648770i	$0.224716\pi$
$398$	−4.51848	−0.226491
$399$	0	0
$400$	−6.95185	−0.347592
$401$	−31.5893	−1.57750	−0.788748	−	0.614717i	$-0.789270\pi$
$401$	−31.5893	−1.57750	−0.788748	+	0.614717i	$0.789270\pi$
$402$	0	0
$403$	−6.94282	−0.345847
$404$	5.95976	0.296509
$405$	0	0
$406$	7.21969	0.358307
$407$	1.16827	0.0579090
$408$	0	0
$409$	32.6192	1.61291	0.806457	−	0.591293i	$-0.201382\pi$
$409$	32.6192	1.61291	0.806457	+	0.591293i	$0.201382\pi$
$410$	0.741446	0.0366174
$411$	0	0
$412$	36.9338	1.81960
$413$	−27.9532	−1.37549
$414$	0	0
$415$	−18.1248	−0.889710
$416$	−8.78495	−0.430718
$417$	0	0
$418$	0.0435069	0.00212799
$419$	−27.0345	−1.32072	−0.660360	−	0.750949i	$-0.729596\pi$
$419$	−27.0345	−1.32072	−0.660360	+	0.750949i	$0.729596\pi$
$420$	0	0
$421$	−33.4315	−1.62935	−0.814675	−	0.579918i	$-0.803085\pi$
$421$	−33.4315	−1.62935	−0.814675	+	0.579918i	$0.803085\pi$
$422$	−2.81152	−0.136863
$423$	0	0
$424$	12.8889	0.625941
$425$	−14.8947	−0.722497
$426$	0	0
$427$	−2.23912	−0.108359
$428$	2.40739	0.116366
$429$	0	0
$430$	3.86045	0.186168
$431$	−12.1157	−0.583594	−0.291797	−	0.956480i	$-0.594253\pi$
$431$	−12.1157	−0.583594	−0.291797	+	0.956480i	$0.594253\pi$
$432$	0	0
$433$	−18.1021	−0.869930	−0.434965	−	0.900447i	$-0.643239\pi$
$433$	−18.1021	−0.869930	−0.434965	+	0.900447i	$0.643239\pi$
$434$	1.66019	0.0796917
$435$	0	0
$436$	−2.32287	−0.111245
$437$	−3.28263	−0.157029
$438$	0	0
$439$	−13.2495	−0.632365	−0.316183	−	0.948698i	$-0.602401\pi$
$439$	−13.2495	−0.632365	−0.316183	+	0.948698i	$0.602401\pi$
$440$	0.302060	0.0144002
$441$	0	0
$442$	−5.96690	−0.283816
$443$	38.0553	1.80806	0.904031	−	0.427468i	$-0.140594\pi$
$443$	38.0553	1.80806	0.904031	+	0.427468i	$0.140594\pi$
$444$	0	0
$445$	13.3639	0.633509
$446$	2.43723	0.115406
$447$	0	0
$448$	−21.1923	−1.00124
$449$	−33.0837	−1.56132	−0.780659	−	0.624957i	$-0.785117\pi$
$449$	−33.0837	−1.56132	−0.780659	+	0.624957i	$0.785117\pi$
$450$	0	0
$451$	−0.320380	−0.0150861
$452$	−11.2391	−0.528644
$453$	0	0
$454$	5.43147	0.254912
$455$	−17.8285	−0.835811
$456$	0	0
$457$	−22.9475	−1.07344	−0.536719	−	0.843761i	$-0.680336\pi$
$457$	−22.9475	−1.07344	−0.536719	+	0.843761i	$0.680336\pi$
$458$	3.32287	0.155267
$459$	0	0
$460$	−11.2301	−0.523606
$461$	7.33981	0.341849	0.170925	−	0.985284i	$-0.445325\pi$
$461$	7.33981	0.341849	0.170925	+	0.985284i	$0.445325\pi$
$462$	0	0
$463$	−18.6979	−0.868967	−0.434483	−	0.900680i	$-0.643069\pi$
$463$	−18.6979	−0.868967	−0.434483	+	0.900680i	$0.643069\pi$
$464$	34.7303	1.61231
$465$	0	0
$466$	6.56991	0.304345
$467$	23.0071	1.06464	0.532322	−	0.846542i	$-0.321320\pi$
$467$	23.0071	1.06464	0.532322	+	0.846542i	$0.321320\pi$
$468$	0	0
$469$	−20.2930	−0.937045
$470$	3.78960	0.174801
$471$	0	0
$472$	8.28263	0.381239
$473$	−1.66811	−0.0766996
$474$	0	0
$475$	1.89931	0.0871465
$476$	−48.4796	−2.22206
$477$	0	0
$478$	4.41642	0.202002
$479$	−20.0482	−0.916023	−0.458012	−	0.888946i	$-0.651438\pi$
$479$	−20.0482	−0.916023	−0.458012	+	0.888946i	$0.651438\pi$
$480$	0	0
$481$	20.4315	0.931595
$482$	−4.57318	−0.208302
$483$	0	0
$484$	21.3067	0.968487
$485$	1.38221	0.0627627
$486$	0	0
$487$	3.47033	0.157256	0.0786278	−	0.996904i	$-0.474946\pi$
$487$	3.47033	0.157256	0.0786278	+	0.996904i	$0.474946\pi$
$488$	0.663459	0.0300334
$489$	0	0
$490$	1.31573	0.0594388
$491$	−26.7278	−1.20621	−0.603104	−	0.797663i	$-0.706070\pi$
$491$	−26.7278	−1.20621	−0.603104	+	0.797663i	$0.706070\pi$
$492$	0	0
$493$	74.4113	3.35131
$494$	0.760877	0.0342335
$495$	0	0
$496$	7.98633	0.358597
$497$	−10.2632	−0.460367
$498$	0	0
$499$	−21.5322	−0.963912	−0.481956	−	0.876195i	$-0.660073\pi$
$499$	−21.5322	−0.963912	−0.481956	+	0.876195i	$0.660073\pi$
$500$	23.6030	1.05556
$501$	0	0
$502$	−4.72202	−0.210754
$503$	22.7220	1.01312	0.506562	−	0.862203i	$-0.330916\pi$
$503$	22.7220	1.01312	0.506562	+	0.862203i	$0.330916\pi$
$504$	0	0
$505$	−5.40164	−0.240370
$506$	−0.142817	−0.00634899
$507$	0	0
$508$	−23.7141	−1.05214
$509$	18.1431	0.804178	0.402089	−	0.915601i	$-0.368284\pi$
$509$	18.1431	0.804178	0.402089	+	0.915601i	$0.368284\pi$
$510$	0	0
$511$	−21.9097	−0.969229
$512$	17.0071	0.751616
$513$	0	0
$514$	−3.93817	−0.173705
$515$	−33.4750	−1.47508
$516$	0	0
$517$	−1.63749	−0.0720167
$518$	−4.88564	−0.214663
$519$	0	0
$520$	5.28263	0.231659
$521$	−3.27223	−0.143359	−0.0716794	−	0.997428i	$-0.522836\pi$
$521$	−3.27223	−0.143359	−0.0716794	+	0.997428i	$0.522836\pi$
$522$	0	0
$523$	7.88237	0.344672	0.172336	−	0.985038i	$-0.444869\pi$
$523$	7.88237	0.344672	0.172336	+	0.985038i	$0.444869\pi$
$524$	−14.5868	−0.637229
$525$	0	0
$526$	4.64979	0.202740
$527$	17.1111	0.745371
$528$	0	0
$529$	−12.2243	−0.531493
$530$	−5.75623	−0.250035
$531$	0	0
$532$	6.18194	0.268021
$533$	−5.60301	−0.242693
$534$	0	0
$535$	−2.18194	−0.0943336
$536$	6.01289	0.259717
$537$	0	0
$538$	3.38469	0.145925
$539$	−0.568530	−0.0244883
$540$	0	0
$541$	13.2164	0.568218	0.284109	−	0.958792i	$-0.408302\pi$
$541$	13.2164	0.568218	0.284109	+	0.958792i	$0.408302\pi$
$542$	1.13706	0.0488409
$543$	0	0
$544$	21.6512	0.928286
$545$	2.10533	0.0901826
$546$	0	0
$547$	41.2405	1.76332	0.881658	−	0.471889i	$-0.156428\pi$
$547$	41.2405	1.76332	0.881658	+	0.471889i	$0.156428\pi$
$548$	23.7518	1.01463
$549$	0	0
$550$	0.0826332	0.00352349
$551$	−9.48865	−0.404230
$552$	0	0
$553$	−3.13844	−0.133460
$554$	−6.09415	−0.258916
$555$	0	0
$556$	−8.56277	−0.363142
$557$	25.1891	1.06730	0.533648	−	0.845707i	$-0.320821\pi$
$557$	25.1891	1.06730	0.533648	+	0.845707i	$0.320821\pi$
$558$	0	0
$559$	−29.1729	−1.23388
$560$	20.5081	0.866625
$561$	0	0
$562$	−7.22296	−0.304682
$563$	−6.45666	−0.272116	−0.136058	−	0.990701i	$-0.543443\pi$
$563$	−6.45666	−0.272116	−0.136058	+	0.990701i	$0.543443\pi$
$564$	0	0
$565$	10.1866	0.428553
$566$	−3.77782	−0.158794
$567$	0	0
$568$	3.04102	0.127598
$569$	−17.4315	−0.730765	−0.365383	−	0.930857i	$-0.619062\pi$
$569$	−17.4315	−0.730765	−0.365383	+	0.930857i	$0.619062\pi$
$570$	0	0
$571$	−1.45882	−0.0610496	−0.0305248	−	0.999534i	$-0.509718\pi$
$571$	−1.45882	−0.0610496	−0.0305248	+	0.999534i	$0.509718\pi$
$572$	−1.12476	−0.0470287
$573$	0	0
$574$	1.33981	0.0559226
$575$	−6.23474	−0.260007
$576$	0	0
$577$	−1.83173	−0.0762559	−0.0381280	−	0.999273i	$-0.512139\pi$
$577$	−1.83173	−0.0762559	−0.0381280	+	0.999273i	$0.512139\pi$
$578$	10.6408	0.442597
$579$	0	0
$580$	−32.4613	−1.34788
$581$	−32.7518	−1.35878
$582$	0	0
$583$	2.48727	0.103012
$584$	6.49192	0.268638
$585$	0	0
$586$	−6.51927	−0.269308
$587$	14.1715	0.584922	0.292461	−	0.956277i	$-0.405526\pi$
$587$	14.1715	0.584922	0.292461	+	0.956277i	$0.405526\pi$
$588$	0	0
$589$	−2.18194	−0.0899054
$590$	−3.69905	−0.152288
$591$	0	0
$592$	−23.5023	−0.965940
$593$	1.11436	0.0457613	0.0228806	−	0.999738i	$-0.492716\pi$
$593$	1.11436	0.0457613	0.0228806	+	0.999738i	$0.492716\pi$
$594$	0	0
$595$	43.9396	1.80135
$596$	4.76801	0.195305
$597$	0	0
$598$	−2.49768	−0.102138
$599$	−29.9773	−1.22484	−0.612420	−	0.790533i	$-0.709804\pi$
$599$	−29.9773	−1.22484	−0.612420	+	0.790533i	$0.709804\pi$
$600$	0	0
$601$	2.36062	0.0962916	0.0481458	−	0.998840i	$-0.484669\pi$
$601$	2.36062	0.0962916	0.0481458	+	0.998840i	$0.484669\pi$
$602$	6.97592	0.284317
$603$	0	0
$604$	−37.9097	−1.54252
$605$	−19.3114	−0.785118
$606$	0	0
$607$	−34.4224	−1.39716	−0.698582	−	0.715530i	$-0.746185\pi$
$607$	−34.4224	−1.39716	−0.698582	+	0.715530i	$0.746185\pi$
$608$	−2.76088	−0.111968
$609$	0	0
$610$	−0.296303	−0.0119970
$611$	−28.6375	−1.15855
$612$	0	0
$613$	−16.1144	−0.650853	−0.325426	−	0.945567i	$-0.605508\pi$
$613$	−16.1144	−0.650853	−0.325426	+	0.945567i	$0.605508\pi$
$614$	7.06045	0.284937
$615$	0	0
$616$	0.545830	0.0219921
$617$	24.4451	0.984124	0.492062	−	0.870560i	$-0.336243\pi$
$617$	24.4451	0.984124	0.492062	+	0.870560i	$0.336243\pi$
$618$	0	0
$619$	25.8720	1.03988	0.519941	−	0.854202i	$-0.325954\pi$
$619$	25.8720	1.03988	0.519941	+	0.854202i	$0.325954\pi$
$620$	−7.46457	−0.299784
$621$	0	0
$622$	4.26896	0.171170
$623$	24.1488	0.967503
$624$	0	0
$625$	−11.8960	−0.475842
$626$	−0.710831	−0.0284105
$627$	0	0
$628$	30.1119	1.20159
$629$	−50.3549	−2.00778
$630$	0	0
$631$	−7.74393	−0.308281	−0.154141	−	0.988049i	$-0.549261\pi$
$631$	−7.74393	−0.308281	−0.154141	+	0.988049i	$0.549261\pi$
$632$	0.929929	0.0369906
$633$	0	0
$634$	4.10396	0.162989
$635$	21.4933	0.852935
$636$	0	0
$637$	−9.94282	−0.393949
$638$	−0.412822	−0.0163438
$639$	0	0
$640$	−12.5275	−0.495193
$641$	1.14419	0.0451929	0.0225965	−	0.999745i	$-0.492807\pi$
$641$	1.14419	0.0451929	0.0225965	+	0.999745i	$0.492807\pi$
$642$	0	0
$643$	−24.0572	−0.948723	−0.474361	−	0.880330i	$-0.657321\pi$
$643$	−24.0572	−0.948723	−0.474361	+	0.880330i	$0.657321\pi$
$644$	−20.2930	−0.799658
$645$	0	0
$646$	−1.87524	−0.0737802
$647$	3.47360	0.136561	0.0682807	−	0.997666i	$-0.478249\pi$
$647$	3.47360	0.136561	0.0682807	+	0.997666i	$0.478249\pi$
$648$	0	0
$649$	1.59836	0.0627413
$650$	1.44514	0.0566832
$651$	0	0
$652$	−25.7212	−1.00732
$653$	−13.1923	−0.516256	−0.258128	−	0.966111i	$-0.583106\pi$
$653$	−13.1923	−0.516256	−0.258128	+	0.966111i	$0.583106\pi$
$654$	0	0
$655$	13.2208	0.516580
$656$	6.44514	0.251641
$657$	0	0
$658$	6.84789	0.266959
$659$	2.42682	0.0945356	0.0472678	−	0.998882i	$-0.484949\pi$
$659$	2.42682	0.0945356	0.0472678	+	0.998882i	$0.484949\pi$
$660$	0	0
$661$	−9.32038	−0.362521	−0.181260	−	0.983435i	$-0.558018\pi$
$661$	−9.32038	−0.362521	−0.181260	+	0.983435i	$0.558018\pi$
$662$	5.73104	0.222743
$663$	0	0
$664$	9.70448	0.376607
$665$	−5.60301	−0.217275
$666$	0	0
$667$	31.1477	1.20604
$668$	−6.24626	−0.241675
$669$	0	0
$670$	−2.68538	−0.103745
$671$	0.128033	0.00494266
$672$	0	0
$673$	35.0826	1.35234	0.676168	−	0.736747i	$-0.263639\pi$
$673$	35.0826	1.35234	0.676168	+	0.736747i	$0.263639\pi$
$674$	−1.80249	−0.0694293
$675$	0	0
$676$	5.58607	0.214849
$677$	16.2416	0.624216	0.312108	−	0.950047i	$-0.398965\pi$
$677$	16.2416	0.624216	0.312108	+	0.950047i	$0.398965\pi$
$678$	0	0
$679$	2.49768	0.0958520
$680$	−13.0194	−0.499272
$681$	0	0
$682$	−0.0949296	−0.00363504
$683$	−33.1672	−1.26911	−0.634553	−	0.772879i	$-0.718816\pi$
$683$	−33.1672	−1.26911	−0.634553	+	0.772879i	$0.718816\pi$
$684$	0	0
$685$	−21.5275	−0.822524
$686$	−2.94858	−0.112577
$687$	0	0
$688$	33.5576	1.27937
$689$	43.4991	1.65718
$690$	0	0
$691$	−15.9773	−0.607805	−0.303903	−	0.952703i	$-0.598290\pi$
$691$	−15.9773	−0.607805	−0.303903	+	0.952703i	$0.598290\pi$
$692$	21.3445	0.811394
$693$	0	0
$694$	1.86235	0.0706937
$695$	7.76088	0.294387
$696$	0	0
$697$	13.8090	0.523054
$698$	−3.82924	−0.144939
$699$	0	0
$700$	11.7414	0.443785
$701$	50.1398	1.89375	0.946877	−	0.321595i	$-0.104219\pi$
$701$	50.1398	1.89375	0.946877	+	0.321595i	$0.104219\pi$
$702$	0	0
$703$	6.42107	0.242175
$704$	1.21178	0.0456706
$705$	0	0
$706$	5.21505	0.196271
$707$	−9.76088	−0.367096
$708$	0	0
$709$	−40.2873	−1.51302	−0.756510	−	0.653982i	$-0.773097\pi$
$709$	−40.2873	−1.51302	−0.756510	+	0.653982i	$0.773097\pi$
$710$	−1.35813	−0.0509697
$711$	0	0
$712$	−7.15538	−0.268159
$713$	7.16251	0.268238
$714$	0	0
$715$	1.01943	0.0381245
$716$	−7.02656	−0.262595
$717$	0	0
$718$	3.34635	0.124885
$719$	6.19562	0.231058	0.115529	−	0.993304i	$-0.463144\pi$
$719$	6.19562	0.231058	0.115529	+	0.993304i	$0.463144\pi$
$720$	0	0
$721$	−60.4900	−2.25277
$722$	0.239123	0.00889925
$723$	0	0
$724$	−6.57318	−0.244290
$725$	−18.0219	−0.669317
$726$	0	0
$727$	38.1157	1.41363	0.706817	−	0.707396i	$-0.250130\pi$
$727$	38.1157	1.41363	0.706817	+	0.707396i	$0.250130\pi$
$728$	9.54583	0.353792
$729$	0	0
$730$	−2.89931	−0.107308
$731$	71.8988	2.65927
$732$	0	0
$733$	23.6602	0.873909	0.436955	−	0.899484i	$-0.356057\pi$
$733$	23.6602	0.873909	0.436955	+	0.899484i	$0.356057\pi$
$734$	−6.84678	−0.252719
$735$	0	0
$736$	9.06294	0.334064
$737$	1.16035	0.0427422
$738$	0	0
$739$	−1.02735	−0.0377915	−0.0188958	−	0.999821i	$-0.506015\pi$
$739$	−1.02735	−0.0377915	−0.0188958	+	0.999821i	$0.506015\pi$
$740$	21.9669	0.807519
$741$	0	0
$742$	−10.4016	−0.381856
$743$	−1.74720	−0.0640987	−0.0320493	−	0.999486i	$-0.510203\pi$
$743$	−1.74720	−0.0640987	−0.0320493	+	0.999486i	$0.510203\pi$
$744$	0	0
$745$	−4.32149	−0.158327
$746$	−4.11763	−0.150757
$747$	0	0
$748$	2.77206	0.101357
$749$	−3.94282	−0.144068
$750$	0	0
$751$	25.9636	0.947426	0.473713	−	0.880679i	$-0.342913\pi$
$751$	25.9636	0.947426	0.473713	+	0.880679i	$0.342913\pi$
$752$	32.9417	1.20126
$753$	0	0
$754$	−7.21969	−0.262926
$755$	34.3595	1.25047
$756$	0	0
$757$	27.3789	0.995104	0.497552	−	0.867434i	$-0.334232\pi$
$757$	27.3789	0.995104	0.497552	+	0.867434i	$0.334232\pi$
$758$	2.15460	0.0782585
$759$	0	0
$760$	1.66019	0.0602214
$761$	34.9748	1.26784	0.633918	−	0.773400i	$-0.281446\pi$
$761$	34.9748	1.26784	0.633918	+	0.773400i	$0.281446\pi$
$762$	0	0
$763$	3.80438	0.137728
$764$	21.5868	0.780985
$765$	0	0
$766$	4.86948	0.175941
$767$	27.9532	1.00933
$768$	0	0
$769$	−30.7817	−1.11002	−0.555008	−	0.831845i	$-0.687285\pi$
$769$	−30.7817	−1.11002	−0.555008	+	0.831845i	$0.687285\pi$
$770$	−0.243770	−0.00878484
$771$	0	0
$772$	−14.6602	−0.527632
$773$	25.2553	0.908369	0.454185	−	0.890908i	$-0.349931\pi$
$773$	25.2553	0.908369	0.454185	+	0.890908i	$0.349931\pi$
$774$	0	0
$775$	−4.14419	−0.148864
$776$	−0.740070	−0.0265670
$777$	0	0
$778$	−2.42296	−0.0868673
$779$	−1.76088	−0.0630900
$780$	0	0
$781$	0.586849	0.0209991
$782$	6.15571	0.220128
$783$	0	0
$784$	11.4372	0.408472
$785$	−27.2919	−0.974090
$786$	0	0
$787$	12.2301	0.435956	0.217978	−	0.975954i	$-0.430054\pi$
$787$	12.2301	0.435956	0.217978	+	0.975954i	$0.430054\pi$
$788$	24.0941	0.858318
$789$	0	0
$790$	−0.415309	−0.0147760
$791$	18.4074	0.654492
$792$	0	0
$793$	2.23912	0.0795136
$794$	7.25142	0.257343
$795$	0	0
$796$	36.7116	1.30121
$797$	48.6616	1.72368	0.861841	−	0.507179i	$-0.169312\pi$
$797$	48.6616	1.72368	0.861841	+	0.507179i	$0.169312\pi$
$798$	0	0
$799$	70.5792	2.49691
$800$	−5.24377	−0.185395
$801$	0	0
$802$	−7.55375	−0.266732
$803$	1.25280	0.0442102
$804$	0	0
$805$	18.3926	0.648254
$806$	−1.66019	−0.0584777
$807$	0	0
$808$	2.89218	0.101747
$809$	−30.8389	−1.08424	−0.542118	−	0.840302i	$-0.682378\pi$
$809$	−30.8389	−1.08424	−0.542118	+	0.840302i	$0.682378\pi$
$810$	0	0
$811$	−21.9877	−0.772093	−0.386046	−	0.922479i	$-0.626160\pi$
$811$	−21.9877	−0.772093	−0.386046	+	0.922479i	$0.626160\pi$
$812$	−58.6583	−2.05850
$813$	0	0
$814$	0.279361	0.00979158
$815$	23.3125	0.816600
$816$	0	0
$817$	−9.16827	−0.320757
$818$	7.80000	0.272721
$819$	0	0
$820$	−6.02408	−0.210370
$821$	28.7680	1.00401	0.502005	−	0.864865i	$-0.332596\pi$
$821$	28.7680	1.00401	0.502005	+	0.864865i	$0.332596\pi$
$822$	0	0
$823$	−14.6224	−0.509706	−0.254853	−	0.966980i	$-0.582027\pi$
$823$	−14.6224	−0.509706	−0.254853	+	0.966980i	$0.582027\pi$
$824$	17.9234	0.624391
$825$	0	0
$826$	−6.68427	−0.232575
$827$	−19.9097	−0.692329	−0.346164	−	0.938174i	$-0.612516\pi$
$827$	−19.9097	−0.692329	−0.346164	+	0.938174i	$0.612516\pi$
$828$	0	0
$829$	−52.2725	−1.81550	−0.907749	−	0.419513i	$-0.862201\pi$
$829$	−52.2725	−1.81550	−0.907749	+	0.419513i	$0.862201\pi$
$830$	−4.33405	−0.150437
$831$	0	0
$832$	21.1923	0.734712
$833$	24.5048	0.849041
$834$	0	0
$835$	5.66130	0.195917
$836$	−0.353483	−0.0122255
$837$	0	0
$838$	−6.46457	−0.223315
$839$	35.3905	1.22181	0.610907	−	0.791702i	$-0.290805\pi$
$839$	35.3905	1.22181	0.610907	+	0.791702i	$0.290805\pi$
$840$	0	0
$841$	61.0345	2.10464
$842$	−7.99424	−0.275500
$843$	0	0
$844$	22.8429	0.786286
$845$	−5.06294	−0.174170
$846$	0	0
$847$	−34.8960	−1.19904
$848$	−50.0370	−1.71828
$849$	0	0
$850$	−3.56166	−0.122164
$851$	−21.0780	−0.722544
$852$	0	0
$853$	−6.31711	−0.216294	−0.108147	−	0.994135i	$-0.534492\pi$
$853$	−6.31711	−0.216294	−0.108147	+	0.994135i	$0.534492\pi$
$854$	−0.535426	−0.0183219
$855$	0	0
$856$	1.16827	0.0399307
$857$	20.3218	0.694178	0.347089	−	0.937832i	$-0.387170\pi$
$857$	20.3218	0.694178	0.347089	+	0.937832i	$0.387170\pi$
$858$	0	0
$859$	−18.5516	−0.632972	−0.316486	−	0.948597i	$-0.602503\pi$
$859$	−18.5516	−0.632972	−0.316486	+	0.948597i	$0.602503\pi$
$860$	−31.3653	−1.06955
$861$	0	0
$862$	−2.89715	−0.0986775
$863$	14.5814	0.496357	0.248179	−	0.968714i	$-0.420168\pi$
$863$	14.5814	0.496357	0.248179	+	0.968714i	$0.420168\pi$
$864$	0	0
$865$	−19.3456	−0.657769
$866$	−4.32862	−0.147093
$867$	0	0
$868$	−13.4887	−0.457835
$869$	0.179456	0.00608761
$870$	0	0
$871$	20.2930	0.687603
$872$	−1.12725	−0.0381735
$873$	0	0
$874$	−0.784953	−0.0265514
$875$	−38.6569	−1.30684
$876$	0	0
$877$	−19.3639	−0.653872	−0.326936	−	0.945046i	$-0.606016\pi$
$877$	−19.3639	−0.653872	−0.326936	+	0.945046i	$0.606016\pi$
$878$	−3.16827	−0.106924
$879$	0	0
$880$	−1.17265	−0.0395301
$881$	−19.3138	−0.650700	−0.325350	−	0.945594i	$-0.605482\pi$
$881$	−19.3138	−0.650700	−0.325350	+	0.945594i	$0.605482\pi$
$882$	0	0
$883$	−16.5757	−0.557816	−0.278908	−	0.960318i	$-0.589972\pi$
$883$	−16.5757	−0.557816	−0.278908	+	0.960318i	$0.589972\pi$
$884$	48.4796	1.63055
$885$	0	0
$886$	9.09991	0.305717
$887$	22.2391	0.746717	0.373358	−	0.927687i	$-0.378206\pi$
$887$	22.2391	0.746717	0.373358	+	0.927687i	$0.378206\pi$
$888$	0	0
$889$	38.8389	1.30261
$890$	3.19562	0.107117
$891$	0	0
$892$	−19.8019	−0.663017
$893$	−9.00000	−0.301174
$894$	0	0
$895$	6.36853	0.212877
$896$	−22.6375	−0.756265
$897$	0	0
$898$	−7.91109	−0.263997
$899$	20.7037	0.690507
$900$	0	0
$901$	−107.207	−3.57157
$902$	−0.0766103	−0.00255084
$903$	0	0
$904$	−5.45417	−0.181403
$905$	5.95760	0.198037
$906$	0	0
$907$	−31.1067	−1.03288	−0.516441	−	0.856323i	$-0.672743\pi$
$907$	−31.1067	−1.03288	−0.516441	+	0.856323i	$0.672743\pi$
$908$	−44.1294	−1.46449
$909$	0	0
$910$	−4.26320	−0.141324
$911$	−30.2085	−1.00085	−0.500426	−	0.865779i	$-0.666823\pi$
$911$	−30.2085	−1.00085	−0.500426	+	0.865779i	$0.666823\pi$
$912$	0	0
$913$	1.87275	0.0619789
$914$	−5.48727	−0.181503
$915$	0	0
$916$	−26.9975	−0.892023
$917$	23.8903	0.788927
$918$	0	0
$919$	33.7141	1.11213	0.556063	−	0.831140i	$-0.312311\pi$
$919$	33.7141	1.11213	0.556063	+	0.831140i	$0.312311\pi$
$920$	−5.44979	−0.179674
$921$	0	0
$922$	1.75512	0.0578018
$923$	10.2632	0.337817
$924$	0	0
$925$	12.1956	0.400989
$926$	−4.47111	−0.146930
$927$	0	0
$928$	26.1970	0.859958
$929$	−14.7921	−0.485313	−0.242656	−	0.970112i	$-0.578019\pi$
$929$	−14.7921	−0.485313	−0.242656	+	0.970112i	$0.578019\pi$
$930$	0	0
$931$	−3.12476	−0.102410
$932$	−53.3789	−1.74849
$933$	0	0
$934$	5.50154	0.180016
$935$	−2.51246	−0.0821663
$936$	0	0
$937$	40.7083	1.32988	0.664942	−	0.746895i	$-0.268456\pi$
$937$	40.7083	1.32988	0.664942	+	0.746895i	$0.268456\pi$
$938$	−4.85254	−0.158441
$939$	0	0
$940$	−30.7896	−1.00425
$941$	40.4009	1.31703	0.658515	−	0.752568i	$-0.271185\pi$
$941$	40.4009	1.31703	0.658515	+	0.752568i	$0.271185\pi$
$942$	0	0
$943$	5.78031	0.188233
$944$	−32.1546	−1.04654
$945$	0	0
$946$	−0.398883	−0.0129688
$947$	−32.5770	−1.05861	−0.529306	−	0.848431i	$-0.677548\pi$
$947$	−32.5770	−1.05861	−0.529306	+	0.848431i	$0.677548\pi$
$948$	0	0
$949$	21.9097	0.711219
$950$	0.454170	0.0147352
$951$	0	0
$952$	−23.5264	−0.762495
$953$	−3.62571	−0.117448	−0.0587241	−	0.998274i	$-0.518703\pi$
$953$	−3.62571	−0.117448	−0.0587241	+	0.998274i	$0.518703\pi$
$954$	0	0
$955$	−19.5653	−0.633117
$956$	−35.8824	−1.16052
$957$	0	0
$958$	−4.79398	−0.154886
$959$	−38.9007	−1.25617
$960$	0	0
$961$	−26.2391	−0.846423
$962$	4.88564	0.157519
$963$	0	0
$964$	37.1560	1.19671
$965$	13.2873	0.427733
$966$	0	0
$967$	20.6375	0.663657	0.331828	−	0.943340i	$-0.392334\pi$
$967$	20.6375	0.663657	0.331828	+	0.943340i	$0.392334\pi$
$968$	10.3398	0.332334
$969$	0	0
$970$	0.330518	0.0106123
$971$	−12.4692	−0.400156	−0.200078	−	0.979780i	$-0.564120\pi$
$971$	−12.4692	−0.400156	−0.200078	+	0.979780i	$0.564120\pi$
$972$	0	0
$973$	14.0241	0.449591
$974$	0.829837	0.0265897
$975$	0	0
$976$	−2.57566	−0.0824450
$977$	−35.9007	−1.14856	−0.574282	−	0.818657i	$-0.694719\pi$
$977$	−35.9007	−1.14856	−0.574282	+	0.818657i	$0.694719\pi$
$978$	0	0
$979$	−1.38083	−0.0441315
$980$	−10.6900	−0.341480
$981$	0	0
$982$	−6.39123	−0.203952
$983$	46.8993	1.49586	0.747928	−	0.663780i	$-0.231049\pi$
$983$	46.8993	1.49586	0.747928	+	0.663780i	$0.231049\pi$
$984$	0	0
$985$	−21.8378	−0.695809
$986$	17.7935	0.566659
$987$	0	0
$988$	−6.18194	−0.196674
$989$	30.0960	0.956998
$990$	0	0
$991$	57.9467	1.84074	0.920369	−	0.391052i	$-0.127889\pi$
$991$	57.9467	1.84074	0.920369	+	0.391052i	$0.127889\pi$
$992$	6.02408	0.191265
$993$	0	0
$994$	−2.45417	−0.0778415
$995$	−33.2736	−1.05484
$996$	0	0
$997$	41.2646	1.30686	0.653431	−	0.756986i	$-0.273329\pi$
$997$	41.2646	1.30686	0.653431	+	0.756986i	$0.273329\pi$
$998$	−5.14884	−0.162984
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.f.1.2	yes	3
3.2	odd	2		513.2.a.e.1.2	✓	3
4.3	odd	2		8208.2.a.bp.1.2		3
12.11	even	2		8208.2.a.bf.1.2		3
19.18	odd	2		9747.2.a.x.1.2		3
57.56	even	2		9747.2.a.ba.1.2		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.e.1.2	✓	3	3.2	odd	2
513.2.a.f.1.2	yes	3	1.1	even	1	trivial
8208.2.a.bf.1.2		3	12.11	even	2
8208.2.a.bp.1.2		3	4.3	odd	2
9747.2.a.x.1.2		3	19.18	odd	2
9747.2.a.ba.1.2		3	57.56	even	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:	\( N \)	\(=\)	\( 513 = 3^{3} \cdot 19 \)
Weight:	\( k \)	\(=\)	\( 2 \)
Character orbit:	\([\chi]\)	\(=\)	513.a (trivial)

\(f(q)\)	\(=\)	\(q+0.239123 q^{2} -1.94282 q^{4} +1.76088 q^{5} +3.18194 q^{7} -0.942820 q^{8} +0.421067 q^{10} -0.181943 q^{11} -3.18194 q^{13} +0.760877 q^{14} +3.66019 q^{16} +7.84213 q^{17} -1.00000 q^{19} -3.42107 q^{20} -0.0435069 q^{22} +3.28263 q^{23} -1.89931 q^{25} -0.760877 q^{26} -6.18194 q^{28} +9.48865 q^{29} +2.18194 q^{31} +2.76088 q^{32} +1.87524 q^{34} +5.60301 q^{35} -6.42107 q^{37} -0.239123 q^{38} -1.66019 q^{40} +1.76088 q^{41} +9.16827 q^{43} +0.353483 q^{44} +0.784953 q^{46} +9.00000 q^{47} +3.12476 q^{49} -0.454170 q^{50} +6.18194 q^{52} -13.6706 q^{53} -0.320380 q^{55} -3.00000 q^{56} +2.26896 q^{58} -8.78495 q^{59} -0.703697 q^{61} +0.521753 q^{62} -6.66019 q^{64} -5.60301 q^{65} -6.37756 q^{67} -15.2359 q^{68} +1.33981 q^{70} -3.22545 q^{71} -6.88564 q^{73} -1.53543 q^{74} +1.94282 q^{76} -0.578933 q^{77} -0.986327 q^{79} +6.44514 q^{80} +0.421067 q^{82} -10.2930 q^{83} +13.8090 q^{85} +2.19235 q^{86} +0.171540 q^{88} +7.58934 q^{89} -10.1248 q^{91} -6.37756 q^{92} +2.15211 q^{94} -1.76088 q^{95} +0.784953 q^{97} +0.747204 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + q^{2} + 3 q^{4} + 5 q^{5} + q^{7} + 6 q^{8} - 7 q^{10} + 8 q^{11} - q^{13} + 2 q^{14} + 3 q^{16} + 7 q^{17} - 3 q^{19} - 2 q^{20} + q^{22} + 9 q^{23} + 2 q^{25} - 2 q^{26} - 10 q^{28} - 6 q^{29}+ \cdots - 8 q^{98}+O(q^{100}) \) 3 * q + q^2 + 3 * q^4 + 5 * q^5 + q^7 + 6 * q^8 - 7 * q^10 + 8 * q^11 - q^13 + 2 * q^14 + 3 * q^16 + 7 * q^17 - 3 * q^19 - 2 * q^20 + q^22 + 9 * q^23 + 2 * q^25 - 2 * q^26 - 10 * q^28 - 6 * q^29 - 2 * q^31 + 8 * q^32 + 23 * q^34 - 11 * q^37 - q^38 + 3 * q^40 + 5 * q^41 + 9 * q^43 + 19 * q^44 - 23 * q^46 + 27 * q^47 - 8 * q^49 - 27 * q^50 + 10 * q^52 + 2 * q^53 + 15 * q^55 - 9 * q^56 - 4 * q^58 - q^59 + 7 * q^61 + q^62 - 12 * q^64 - 12 * q^67 - q^68 + 12 * q^70 - 3 * q^73 - 14 * q^74 - 3 * q^76 - 10 * q^77 + 7 * q^79 - 14 * q^80 - 7 * q^82 + 5 * q^83 - 9 * q^85 - 37 * q^86 + 27 * q^88 - 4 * q^89 - 13 * q^91 - 12 * q^92 + 9 * q^94 - 5 * q^95 - 23 * q^97 - 8 * q^98

Introduction

L-functions

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