LMFDB - Embedded newform 513.2.a.g.1.3

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)

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")

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);

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

Newform invariants

comment:select newform

sage:traces = [3,3,0,3] 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:	$\Q(\zeta_{18})^+$
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:	yes
Fricke sign:	$-1$
Sato-Tate group:	$\mathrm{SU}(2)$

Embedding invariants

Embedding label			1.3
Root			$-1.53209$ 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)

$f(q)$	$=$	\(q+2.53209 q^{2} +4.41147 q^{4} +2.53209 q^{5} +0.184793 q^{7} +6.10607 q^{8} +6.41147 q^{10} -5.63816 q^{11} -6.63816 q^{13} +0.467911 q^{14} +6.63816 q^{16} -2.06418 q^{17} +1.00000 q^{19} +11.1702 q^{20} -14.2763 q^{22} +3.46791 q^{23} +1.41147 q^{25} -16.8084 q^{26} +0.815207 q^{28} +0.467911 q^{29} +5.26857 q^{31} +4.59627 q^{32} -5.22668 q^{34} +0.467911 q^{35} -4.41147 q^{37} +2.53209 q^{38} +15.4611 q^{40} +11.7246 q^{41} +5.68004 q^{43} -24.8726 q^{44} +8.78106 q^{46} +10.1480 q^{47} -6.96585 q^{49} +3.57398 q^{50} -29.2841 q^{52} +6.10607 q^{53} -14.2763 q^{55} +1.12836 q^{56} +1.18479 q^{58} +2.89393 q^{59} -12.6382 q^{61} +13.3405 q^{62} -1.63816 q^{64} -16.8084 q^{65} +0.958111 q^{67} -9.10607 q^{68} +1.18479 q^{70} +6.57398 q^{71} +1.08378 q^{73} -11.1702 q^{74} +4.41147 q^{76} -1.04189 q^{77} -5.95811 q^{79} +16.8084 q^{80} +29.6878 q^{82} +5.08378 q^{83} -5.22668 q^{85} +14.3824 q^{86} -34.4270 q^{88} +4.70233 q^{89} -1.22668 q^{91} +15.2986 q^{92} +25.6955 q^{94} +2.53209 q^{95} -8.86484 q^{97} -17.6382 q^{98} +O(q^{100})\)
$\operatorname{Tr}(f)(q)$	$=$	$ 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} + 9 q^{10} - 3 q^{13} + 6 q^{14} + 3 q^{16} + 3 q^{17} + 3 q^{19} + 12 q^{20} - 9 q^{22} + 15 q^{23} - 6 q^{25} - 12 q^{26} + 6 q^{28} + 6 q^{29} + 6 q^{31}+ \cdots - 36 q^{98}+O(q^{100}) $ 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 + 9 * q^10 - 3 * q^13 + 6 * q^14 + 3 * q^16 + 3 * q^17 + 3 * q^19 + 12 * q^20 - 9 * q^22 + 15 * q^23 - 6 * q^25 - 12 * q^26 + 6 * q^28 + 6 * q^29 + 6 * q^31 - 9 * q^34 + 6 * q^35 - 3 * q^37 + 3 * q^38 + 9 * q^40 + 3 * q^41 - 3 * q^43 - 27 * q^44 + 9 * q^46 + 15 * q^47 + 3 * q^50 - 30 * q^52 + 6 * q^53 - 9 * q^55 - 15 * q^56 + 21 * q^59 - 21 * q^61 - 3 * q^62 + 12 * q^64 - 12 * q^65 + 6 * q^67 - 15 * q^68 + 12 * q^71 - 3 * q^73 - 12 * q^74 + 3 * q^76 - 21 * q^79 + 12 * q^80 + 45 * q^82 + 9 * q^83 - 9 * q^85 - 3 * q^86 - 45 * q^88 - 12 * q^89 + 3 * q^91 + 6 * q^92 + 9 * q^94 + 3 * q^95 - 3 * q^97 - 36 * 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 $

$2$	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
$5$	2.53209	1.13238	0.566192	−	0.824273i	$-0.308416\pi$
$5$	2.53209	1.13238	0.566192	+	0.824273i	$0.308416\pi$
$6$	0	0
$7$	0.184793	0.0698450	0.0349225	−	0.999390i	$-0.488882\pi$
$7$	0.184793	0.0698450	0.0349225	+	0.999390i	$0.488882\pi$
$8$	6.10607	2.15882
$9$	0	0
$10$	6.41147	2.02749
$11$	−5.63816	−1.69997	−0.849984	−	0.526809i	$-0.823388\pi$
$11$	−5.63816	−1.69997	−0.849984	+	0.526809i	$0.823388\pi$
$12$	0	0
$13$	−6.63816	−1.84109	−0.920547	−	0.390633i	$-0.872256\pi$
$13$	−6.63816	−1.84109	−0.920547	+	0.390633i	$0.872256\pi$
$14$	0.467911	0.125055
$15$	0	0
$16$	6.63816	1.65954
$17$	−2.06418	−0.500637	−0.250318	−	0.968164i	$-0.580535\pi$
$17$	−2.06418	−0.500637	−0.250318	+	0.968164i	$0.580535\pi$
$18$	0	0
$19$	1.00000	0.229416
$19$	1.00000	0.229416
$20$	11.1702	2.49774
$21$	0	0
$22$	−14.2763	−3.04372
$23$	3.46791	0.723109	0.361555	−	0.932351i	$-0.382246\pi$
$23$	3.46791	0.723109	0.361555	+	0.932351i	$0.382246\pi$
$24$	0	0
$25$	1.41147	0.282295
$26$	−16.8084	−3.29640
$27$	0	0
$28$	0.815207	0.154060
$29$	0.467911	0.0868889	0.0434445	−	0.999056i	$-0.486167\pi$
$29$	0.467911	0.0868889	0.0434445	+	0.999056i	$0.486167\pi$
$30$	0	0
$31$	5.26857	0.946263	0.473132	−	0.880992i	$-0.343124\pi$
$31$	5.26857	0.946263	0.473132	+	0.880992i	$0.343124\pi$
$32$	4.59627	0.812513
$33$	0	0
$34$	−5.22668	−0.896368
$35$	0.467911	0.0790914
$36$	0	0
$37$	−4.41147	−0.725242	−0.362621	−	0.931937i	$-0.618118\pi$
$37$	−4.41147	−0.725242	−0.362621	+	0.931937i	$0.618118\pi$
$38$	2.53209	0.410759
$39$	0	0
$40$	15.4611	2.44462
$41$	11.7246	1.83108	0.915539	−	0.402229i	$-0.131764\pi$
$41$	11.7246	1.83108	0.915539	+	0.402229i	$0.131764\pi$
$42$	0	0
$43$	5.68004	0.866199	0.433099	−	0.901346i	$-0.357420\pi$
$43$	5.68004	0.866199	0.433099	+	0.901346i	$0.357420\pi$
$44$	−24.8726	−3.74968
$45$	0	0
$46$	8.78106	1.29470
$47$	10.1480	1.48023	0.740116	−	0.672479i	$-0.234771\pi$
$47$	10.1480	1.48023	0.740116	+	0.672479i	$0.234771\pi$
$48$	0	0
$49$	−6.96585	−0.995122
$50$	3.57398	0.505437
$51$	0	0
$52$	−29.2841	−4.06097
$53$	6.10607	0.838733	0.419366	−	0.907817i	$-0.362252\pi$
$53$	6.10607	0.838733	0.419366	+	0.907817i	$0.362252\pi$
$54$	0	0
$55$	−14.2763	−1.92502
$56$	1.12836	0.150783
$57$	0	0
$58$	1.18479	0.155571
$59$	2.89393	0.376758	0.188379	−	0.982096i	$-0.439677\pi$
$59$	2.89393	0.376758	0.188379	+	0.982096i	$0.439677\pi$
$60$	0	0
$61$	−12.6382	−1.61815	−0.809075	−	0.587705i	$-0.800031\pi$
$61$	−12.6382	−1.61815	−0.809075	+	0.587705i	$0.800031\pi$
$62$	13.3405	1.69424
$63$	0	0
$64$	−1.63816	−0.204769
$65$	−16.8084	−2.08483
$66$	0	0
$67$	0.958111	0.117052	0.0585259	−	0.998286i	$-0.481360\pi$
$67$	0.958111	0.117052	0.0585259	+	0.998286i	$0.481360\pi$
$68$	−9.10607	−1.10427
$69$	0	0
$70$	1.18479	0.141610
$71$	6.57398	0.780188	0.390094	−	0.920775i	$-0.372443\pi$
$71$	6.57398	0.780188	0.390094	+	0.920775i	$0.372443\pi$
$72$	0	0
$73$	1.08378	0.126847	0.0634233	−	0.997987i	$-0.479798\pi$
$73$	1.08378	0.126847	0.0634233	+	0.997987i	$0.479798\pi$
$74$	−11.1702	−1.29851
$75$	0	0
$76$	4.41147	0.506031
$77$	−1.04189	−0.118734
$78$	0	0
$79$	−5.95811	−0.670340	−0.335170	−	0.942158i	$-0.608794\pi$
$79$	−5.95811	−0.670340	−0.335170	+	0.942158i	$0.608794\pi$
$80$	16.8084	1.87924
$81$	0	0
$82$	29.6878	3.27847
$83$	5.08378	0.558017	0.279009	−	0.960289i	$-0.409994\pi$
$83$	5.08378	0.558017	0.279009	+	0.960289i	$0.409994\pi$
$84$	0	0
$85$	−5.22668	−0.566913
$86$	14.3824	1.55089
$87$	0	0
$88$	−34.4270	−3.66993
$89$	4.70233	0.498446	0.249223	−	0.968446i	$-0.419825\pi$
$89$	4.70233	0.498446	0.249223	+	0.968446i	$0.419825\pi$
$90$	0	0
$91$	−1.22668	−0.128591
$92$	15.2986	1.59499
$93$	0	0
$94$	25.6955	2.65029
$95$	2.53209	0.259787
$96$	0	0
$97$	−8.86484	−0.900088	−0.450044	−	0.893006i	$-0.648592\pi$
$97$	−8.86484	−0.900088	−0.450044	+	0.893006i	$0.648592\pi$
$98$	−17.6382	−1.78172
$99$	0	0
$100$	6.22668	0.622668
$101$	−15.5740	−1.54967	−0.774834	−	0.632164i	$-0.782167\pi$
$101$	−15.5740	−1.54967	−0.774834	+	0.632164i	$0.782167\pi$
$102$	0	0
$103$	10.2267	1.00766	0.503832	−	0.863801i	$-0.331923\pi$
$103$	10.2267	1.00766	0.503832	+	0.863801i	$0.331923\pi$
$104$	−40.5330	−3.97459
$105$	0	0
$106$	15.4611	1.50172
$107$	−17.9564	−1.73591	−0.867953	−	0.496646i	$-0.834565\pi$
$107$	−17.9564	−1.73591	−0.867953	+	0.496646i	$0.834565\pi$
$108$	0	0
$109$	−10.3354	−0.989955	−0.494978	−	0.868906i	$-0.664824\pi$
$109$	−10.3354	−0.989955	−0.494978	+	0.868906i	$0.664824\pi$
$110$	−36.1489	−3.44666
$111$	0	0
$112$	1.22668	0.115911
$113$	10.2344	0.962773	0.481387	−	0.876508i	$-0.340133\pi$
$113$	10.2344	0.962773	0.481387	+	0.876508i	$0.340133\pi$
$114$	0	0
$115$	8.78106	0.818838
$116$	2.06418	0.191654
$117$	0	0
$118$	7.32770	0.674569
$119$	−0.381445	−0.0349670
$120$	0	0
$121$	20.7888	1.88989
$122$	−32.0009	−2.89723
$123$	0	0
$124$	23.2422	2.08721
$125$	−9.08647	−0.812718
$126$	0	0
$127$	10.5621	0.937236	0.468618	−	0.883401i	$-0.344752\pi$
$127$	10.5621	0.937236	0.468618	+	0.883401i	$0.344752\pi$
$128$	−13.3405	−1.17914
$129$	0	0
$130$	−42.5604	−3.73279
$131$	7.25402	0.633787	0.316893	−	0.948461i	$-0.397360\pi$
$131$	7.25402	0.633787	0.316893	+	0.948461i	$0.397360\pi$
$132$	0	0
$133$	0.184793	0.0160235
$134$	2.42602	0.209576
$135$	0	0
$136$	−12.6040	−1.08078
$137$	−1.49020	−0.127316	−0.0636582	−	0.997972i	$-0.520277\pi$
$137$	−1.49020	−0.127316	−0.0636582	+	0.997972i	$0.520277\pi$
$138$	0	0
$139$	12.1088	1.02705	0.513526	−	0.858074i	$-0.328339\pi$
$139$	12.1088	1.02705	0.513526	+	0.858074i	$0.328339\pi$
$140$	2.06418	0.174455
$141$	0	0
$142$	16.6459	1.39689
$143$	37.4270	3.12980
$144$	0	0
$145$	1.18479	0.0983917
$146$	2.74422	0.227113
$147$	0	0
$148$	−19.4611	−1.59969
$149$	−5.61856	−0.460290	−0.230145	−	0.973156i	$-0.573920\pi$
$149$	−5.61856	−0.460290	−0.230145	+	0.973156i	$0.573920\pi$
$150$	0	0
$151$	−16.5107	−1.34362	−0.671812	−	0.740721i	$-0.734484\pi$
$151$	−16.5107	−1.34362	−0.671812	+	0.740721i	$0.734484\pi$
$152$	6.10607	0.495267
$153$	0	0
$154$	−2.63816	−0.212589
$155$	13.3405	1.07153
$156$	0	0
$157$	5.00000	0.399043	0.199522	−	0.979893i	$-0.436061\pi$
$157$	5.00000	0.399043	0.199522	+	0.979893i	$0.436061\pi$
$158$	−15.0865	−1.20021
$159$	0	0
$160$	11.6382	0.920077
$161$	0.640844	0.0505056
$162$	0	0
$163$	−25.0496	−1.96204	−0.981019	−	0.193911i	$-0.937883\pi$
$163$	−25.0496	−1.96204	−0.981019	+	0.193911i	$0.937883\pi$
$164$	51.7229	4.03888
$165$	0	0
$166$	12.8726	0.999106
$167$	13.0223	1.00769	0.503847	−	0.863793i	$-0.331917\pi$
$167$	13.0223	1.00769	0.503847	+	0.863793i	$0.331917\pi$
$168$	0	0
$169$	31.0651	2.38962
$170$	−13.2344	−1.01503
$171$	0	0
$172$	25.0574	1.91061
$173$	10.9777	0.834620	0.417310	−	0.908764i	$-0.362973\pi$
$173$	10.9777	0.834620	0.417310	+	0.908764i	$0.362973\pi$
$174$	0	0
$175$	0.260830	0.0197169
$176$	−37.4270	−2.82116
$177$	0	0
$178$	11.9067	0.892447
$179$	−5.63816	−0.421416	−0.210708	−	0.977549i	$-0.567577\pi$
$179$	−5.63816	−0.421416	−0.210708	+	0.977549i	$0.567577\pi$
$180$	0	0
$181$	22.2003	1.65013	0.825067	−	0.565035i	$-0.191137\pi$
$181$	22.2003	1.65013	0.825067	+	0.565035i	$0.191137\pi$
$182$	−3.10607	−0.230237
$183$	0	0
$184$	21.1753	1.56106
$185$	−11.1702	−0.821253
$186$	0	0
$187$	11.6382	0.851066
$188$	44.7674	3.26500
$189$	0	0
$190$	6.41147	0.465137
$191$	2.44562	0.176959	0.0884795	−	0.996078i	$-0.471799\pi$
$191$	2.44562	0.176959	0.0884795	+	0.996078i	$0.471799\pi$
$192$	0	0
$193$	−8.47060	−0.609727	−0.304864	−	0.952396i	$-0.598611\pi$
$193$	−8.47060	−0.609727	−0.304864	+	0.952396i	$0.598611\pi$
$194$	−22.4466	−1.61157
$195$	0	0
$196$	−30.7297	−2.19498
$197$	−14.9564	−1.06560	−0.532798	−	0.846242i	$-0.678859\pi$
$197$	−14.9564	−1.06560	−0.532798	+	0.846242i	$0.678859\pi$
$198$	0	0
$199$	−6.51249	−0.461658	−0.230829	−	0.972994i	$-0.574144\pi$
$199$	−6.51249	−0.461658	−0.230829	+	0.972994i	$0.574144\pi$
$200$	8.61856	0.609424
$201$	0	0
$202$	−39.4347	−2.77462
$203$	0.0864665	0.00606876
$204$	0	0
$205$	29.6878	2.07348
$206$	25.8949	1.80418
$207$	0	0
$208$	−44.0651	−3.05537
$209$	−5.63816	−0.389999
$210$	0	0
$211$	16.1506	1.11186	0.555928	−	0.831230i	$-0.312363\pi$
$211$	16.1506	1.11186	0.555928	+	0.831230i	$0.312363\pi$
$212$	26.9368	1.85002
$213$	0	0
$214$	−45.4671	−3.10807
$215$	14.3824	0.980870
$216$	0	0
$217$	0.973593	0.0660918
$218$	−26.1702	−1.77247
$219$	0	0
$220$	−62.9796	−4.24608
$221$	13.7023	0.921719
$222$	0	0
$223$	−13.0669	−0.875022	−0.437511	−	0.899213i	$-0.644140\pi$
$223$	−13.0669	−0.875022	−0.437511	+	0.899213i	$0.644140\pi$
$224$	0.849356	0.0567500
$225$	0	0
$226$	25.9145	1.72380
$227$	1.76558	0.117186	0.0585928	−	0.998282i	$-0.481339\pi$
$227$	1.76558	0.117186	0.0585928	+	0.998282i	$0.481339\pi$
$228$	0	0
$229$	−12.1334	−0.801798	−0.400899	−	0.916122i	$-0.631302\pi$
$229$	−12.1334	−0.801798	−0.400899	+	0.916122i	$0.631302\pi$
$230$	22.2344	1.46609
$231$	0	0
$232$	2.85710	0.187578
$233$	−11.4260	−0.748544	−0.374272	−	0.927319i	$-0.622107\pi$
$233$	−11.4260	−0.748544	−0.374272	+	0.927319i	$0.622107\pi$
$234$	0	0
$235$	25.6955	1.67619
$236$	12.7665	0.831029
$237$	0	0
$238$	−0.965852	−0.0626069
$239$	−10.4270	−0.674464	−0.337232	−	0.941422i	$-0.609491\pi$
$239$	−10.4270	−0.674464	−0.337232	+	0.941422i	$0.609491\pi$
$240$	0	0
$241$	9.40373	0.605748	0.302874	−	0.953031i	$-0.402054\pi$
$241$	9.40373	0.605748	0.302874	+	0.953031i	$0.402054\pi$
$242$	52.6391	3.38377
$243$	0	0
$244$	−55.7529	−3.56921
$245$	−17.6382	−1.12686
$246$	0	0
$247$	−6.63816	−0.422376
$248$	32.1702	2.04281
$249$	0	0
$250$	−23.0077	−1.45514
$251$	9.53478	0.601830	0.300915	−	0.953651i	$-0.402708\pi$
$251$	9.53478	0.601830	0.300915	+	0.953651i	$0.402708\pi$
$252$	0	0
$253$	−19.5526	−1.22926
$254$	26.7442	1.67808
$255$	0	0
$256$	−30.5030	−1.90644
$257$	−11.3628	−0.708791	−0.354395	−	0.935096i	$-0.615313\pi$
$257$	−11.3628	−0.708791	−0.354395	+	0.935096i	$0.615313\pi$
$258$	0	0
$259$	−0.815207	−0.0506545
$260$	−74.1498	−4.59858
$261$	0	0
$262$	18.3678	1.13477
$263$	−20.0205	−1.23452	−0.617260	−	0.786760i	$-0.711757\pi$
$263$	−20.0205	−1.23452	−0.617260	+	0.786760i	$0.711757\pi$
$264$	0	0
$265$	15.4611	0.949768
$266$	0.467911	0.0286895
$267$	0	0
$268$	4.22668	0.258186
$269$	−4.45100	−0.271382	−0.135691	−	0.990751i	$-0.543325\pi$
$269$	−4.45100	−0.271382	−0.135691	+	0.990751i	$0.543325\pi$
$270$	0	0
$271$	−26.9905	−1.63956	−0.819778	−	0.572681i	$-0.805903\pi$
$271$	−26.9905	−1.63956	−0.819778	+	0.572681i	$0.805903\pi$
$272$	−13.7023	−0.830826
$273$	0	0
$274$	−3.77332	−0.227955
$275$	−7.95811	−0.479892
$276$	0	0
$277$	17.3182	1.04055	0.520275	−	0.853999i	$-0.325829\pi$
$277$	17.3182	1.04055	0.520275	+	0.853999i	$0.325829\pi$
$278$	30.6604	1.83889
$279$	0	0
$280$	2.85710	0.170744
$281$	2.06418	0.123139	0.0615693	−	0.998103i	$-0.480389\pi$
$281$	2.06418	0.123139	0.0615693	+	0.998103i	$0.480389\pi$
$282$	0	0
$283$	1.17705	0.0699685	0.0349842	−	0.999388i	$-0.488862\pi$
$283$	1.17705	0.0699685	0.0349842	+	0.999388i	$0.488862\pi$
$284$	29.0009	1.72089
$285$	0	0
$286$	94.7684	5.60377
$287$	2.16662	0.127892
$288$	0	0
$289$	−12.7392	−0.749363
$290$	3.00000	0.176166
$291$	0	0
$292$	4.78106	0.279790
$293$	24.5107	1.43193	0.715966	−	0.698135i	$-0.245986\pi$
$293$	24.5107	1.43193	0.715966	+	0.698135i	$0.245986\pi$
$294$	0	0
$295$	7.32770	0.426635
$296$	−26.9368	−1.56567
$297$	0	0
$298$	−14.2267	−0.824130
$299$	−23.0205	−1.33131
$300$	0	0
$301$	1.04963	0.0604997
$302$	−41.8066	−2.40570
$303$	0	0
$304$	6.63816	0.380724
$305$	−32.0009	−1.83237
$306$	0	0
$307$	19.6209	1.11983	0.559913	−	0.828552i	$-0.310835\pi$
$307$	19.6209	1.11983	0.559913	+	0.828552i	$0.310835\pi$
$308$	−4.59627	−0.261897
$309$	0	0
$310$	33.7793	1.91854
$311$	7.40373	0.419827	0.209914	−	0.977720i	$-0.432682\pi$
$311$	7.40373	0.419827	0.209914	+	0.977720i	$0.432682\pi$
$312$	0	0
$313$	−24.3756	−1.37779	−0.688894	−	0.724862i	$-0.741904\pi$
$313$	−24.3756	−1.37779	−0.688894	+	0.724862i	$0.741904\pi$
$314$	12.6604	0.714470
$315$	0	0
$316$	−26.2841	−1.47859
$317$	6.23173	0.350009	0.175005	−	0.984568i	$-0.444006\pi$
$317$	6.23173	0.350009	0.175005	+	0.984568i	$0.444006\pi$
$318$	0	0
$319$	−2.63816	−0.147708
$320$	−4.14796	−0.231878
$321$	0	0
$322$	1.62267	0.0904281
$323$	−2.06418	−0.114854
$324$	0	0
$325$	−9.36959	−0.519731
$326$	−63.4279	−3.51295
$327$	0	0
$328$	71.5913	3.95297
$329$	1.87527	0.103387
$330$	0	0
$331$	−21.5185	−1.18276	−0.591381	−	0.806392i	$-0.701417\pi$
$331$	−21.5185	−1.18276	−0.591381	+	0.806392i	$0.701417\pi$
$332$	22.4270	1.23084
$333$	0	0
$334$	32.9736	1.80423
$335$	2.42602	0.132548
$336$	0	0
$337$	17.3851	0.947025	0.473512	−	0.880787i	$-0.342986\pi$
$337$	17.3851	0.947025	0.473512	+	0.880787i	$0.342986\pi$
$338$	78.6596	4.27852
$339$	0	0
$340$	−23.0574	−1.25046
$341$	−29.7050	−1.60862
$342$	0	0
$343$	−2.58079	−0.139349
$344$	34.6827	1.86997
$345$	0	0
$346$	27.7965	1.49435
$347$	5.00093	0.268464	0.134232	−	0.990950i	$-0.457143\pi$
$347$	5.00093	0.268464	0.134232	+	0.990950i	$0.457143\pi$
$348$	0	0
$349$	−5.57903	−0.298639	−0.149319	−	0.988789i	$-0.547708\pi$
$349$	−5.57903	−0.298639	−0.149319	+	0.988789i	$0.547708\pi$
$350$	0.660444	0.0353022
$351$	0	0
$352$	−25.9145	−1.38125
$353$	28.0651	1.49376	0.746878	−	0.664962i	$-0.231552\pi$
$353$	28.0651	1.49376	0.746878	+	0.664962i	$0.231552\pi$
$354$	0	0
$355$	16.6459	0.883472
$356$	20.7442	1.09944
$357$	0	0
$358$	−14.2763	−0.754527
$359$	−4.63547	−0.244651	−0.122325	−	0.992490i	$-0.539035\pi$
$359$	−4.63547	−0.244651	−0.122325	+	0.992490i	$0.539035\pi$
$360$	0	0
$361$	1.00000	0.0526316
$362$	56.2131	2.95449
$363$	0	0
$364$	−5.41147	−0.283638
$365$	2.74422	0.143639
$366$	0	0
$367$	16.8307	0.878555	0.439277	−	0.898351i	$-0.355235\pi$
$367$	16.8307	0.878555	0.439277	+	0.898351i	$0.355235\pi$
$368$	23.0205	1.20003
$369$	0	0
$370$	−28.2841	−1.47042
$371$	1.12836	0.0585813
$372$	0	0
$373$	−19.5107	−1.01023	−0.505114	−	0.863053i	$-0.668549\pi$
$373$	−19.5107	−1.01023	−0.505114	+	0.863053i	$0.668549\pi$
$374$	29.4688	1.52380
$375$	0	0
$376$	61.9641	3.19555
$377$	−3.10607	−0.159971
$378$	0	0
$379$	23.9718	1.23135	0.615675	−	0.788000i	$-0.288883\pi$
$379$	23.9718	1.23135	0.615675	+	0.788000i	$0.288883\pi$
$380$	11.1702	0.573021
$381$	0	0
$382$	6.19253	0.316838
$383$	−5.31551	−0.271610	−0.135805	−	0.990736i	$-0.543362\pi$
$383$	−5.31551	−0.271610	−0.135805	+	0.990736i	$0.543362\pi$
$384$	0	0
$385$	−2.63816	−0.134453
$386$	−21.4483	−1.09169
$387$	0	0
$388$	−39.1070	−1.98536
$389$	9.49113	0.481220	0.240610	−	0.970622i	$-0.422653\pi$
$389$	9.49113	0.481220	0.240610	+	0.970622i	$0.422653\pi$
$390$	0	0
$391$	−7.15839	−0.362015
$392$	−42.5340	−2.14829
$393$	0	0
$394$	−37.8708	−1.90790
$395$	−15.0865	−0.759083
$396$	0	0
$397$	32.3851	1.62536	0.812680	−	0.582710i	$-0.198008\pi$
$397$	32.3851	1.62536	0.812680	+	0.582710i	$0.198008\pi$
$398$	−16.4902	−0.826579
$399$	0	0
$400$	9.36959	0.468479
$401$	−27.2550	−1.36105	−0.680524	−	0.732726i	$-0.738248\pi$
$401$	−27.2550	−1.36105	−0.680524	+	0.732726i	$0.738248\pi$
$402$	0	0
$403$	−34.9736	−1.74216
$404$	−68.7042	−3.41816
$405$	0	0
$406$	0.218941	0.0108658
$407$	24.8726	1.23289
$408$	0	0
$409$	−3.03415	−0.150029	−0.0750145	−	0.997182i	$-0.523900\pi$
$409$	−3.03415	−0.150029	−0.0750145	+	0.997182i	$0.523900\pi$
$410$	75.1721	3.71249
$411$	0	0
$412$	45.1147	2.22264
$413$	0.534777	0.0263147
$414$	0	0
$415$	12.8726	0.631890
$416$	−30.5107	−1.49591
$417$	0	0
$418$	−14.2763	−0.698277
$419$	13.2344	0.646544	0.323272	−	0.946306i	$-0.395217\pi$
$419$	13.2344	0.646544	0.323272	+	0.946306i	$0.395217\pi$
$420$	0	0
$421$	−2.18479	−0.106480	−0.0532401	−	0.998582i	$-0.516955\pi$
$421$	−2.18479	−0.106480	−0.0532401	+	0.998582i	$0.516955\pi$
$422$	40.8949	1.99073
$423$	0	0
$424$	37.2841	1.81067
$425$	−2.91353	−0.141327
$426$	0	0
$427$	−2.33544	−0.113020
$428$	−79.2140	−3.82895
$429$	0	0
$430$	36.4175	1.75621
$431$	6.06242	0.292017	0.146008	−	0.989283i	$-0.453357\pi$
$431$	6.06242	0.292017	0.146008	+	0.989283i	$0.453357\pi$
$432$	0	0
$433$	−17.4037	−0.836370	−0.418185	−	0.908362i	$-0.637334\pi$
$433$	−17.4037	−0.836370	−0.418185	+	0.908362i	$0.637334\pi$
$434$	2.46522	0.118334
$435$	0	0
$436$	−45.5945	−2.18358
$437$	3.46791	0.165893
$438$	0	0
$439$	−10.9162	−0.521003	−0.260501	−	0.965473i	$-0.583888\pi$
$439$	−10.9162	−0.521003	−0.260501	+	0.965473i	$0.583888\pi$
$440$	−87.1721	−4.15577
$441$	0	0
$442$	34.6955	1.65030
$443$	−5.89393	−0.280029	−0.140015	−	0.990149i	$-0.544715\pi$
$443$	−5.89393	−0.280029	−0.140015	+	0.990149i	$0.544715\pi$
$444$	0	0
$445$	11.9067	0.564433
$446$	−33.0865	−1.56669
$447$	0	0
$448$	−0.302719	−0.0143021
$449$	6.99825	0.330268	0.165134	−	0.986271i	$-0.447194\pi$
$449$	6.99825	0.330268	0.165134	+	0.986271i	$0.447194\pi$
$450$	0	0
$451$	−66.1052	−3.11277
$452$	45.1489	2.12363
$453$	0	0
$454$	4.47060	0.209816
$455$	−3.10607	−0.145615
$456$	0	0
$457$	−24.1566	−1.13000	−0.565000	−	0.825091i	$-0.691124\pi$
$457$	−24.1566	−1.13000	−0.565000	+	0.825091i	$0.691124\pi$
$458$	−30.7229	−1.43559
$459$	0	0
$460$	38.7374	1.80614
$461$	−12.1497	−0.565868	−0.282934	−	0.959139i	$-0.591308\pi$
$461$	−12.1497	−0.565868	−0.282934	+	0.959139i	$0.591308\pi$
$462$	0	0
$463$	36.1411	1.67962	0.839811	−	0.542879i	$-0.182666\pi$
$463$	36.1411	1.67962	0.839811	+	0.542879i	$0.182666\pi$
$464$	3.10607	0.144196
$465$	0	0
$466$	−28.9317	−1.34024
$467$	23.4020	1.08291	0.541457	−	0.840728i	$-0.317873\pi$
$467$	23.4020	1.08291	0.541457	+	0.840728i	$0.317873\pi$
$468$	0	0
$469$	0.177052	0.00817549
$470$	65.0634	3.00115
$471$	0	0
$472$	17.6705	0.813353
$473$	−32.0250	−1.47251
$474$	0	0
$475$	1.41147	0.0647629
$476$	−1.68273	−0.0771279
$477$	0	0
$478$	−26.4020	−1.20760
$479$	−3.40467	−0.155563	−0.0777816	−	0.996970i	$-0.524784\pi$
$479$	−3.40467	−0.155563	−0.0777816	+	0.996970i	$0.524784\pi$
$480$	0	0
$481$	29.2841	1.33524
$482$	23.8111	1.08457
$483$	0	0
$484$	91.7093	4.16860
$485$	−22.4466	−1.01925
$486$	0	0
$487$	25.1070	1.13771	0.568853	−	0.822439i	$-0.307387\pi$
$487$	25.1070	1.13771	0.568853	+	0.822439i	$0.307387\pi$
$488$	−77.1694	−3.49330
$489$	0	0
$490$	−44.6614	−2.01760
$491$	−3.70409	−0.167163	−0.0835816	−	0.996501i	$-0.526636\pi$
$491$	−3.70409	−0.167163	−0.0835816	+	0.996501i	$0.526636\pi$
$492$	0	0
$493$	−0.965852	−0.0434998
$494$	−16.8084	−0.756246
$495$	0	0
$496$	34.9736	1.57036
$497$	1.21482	0.0544922
$498$	0	0
$499$	−8.78880	−0.393441	−0.196720	−	0.980460i	$-0.563029\pi$
$499$	−8.78880	−0.393441	−0.196720	+	0.980460i	$0.563029\pi$
$500$	−40.0847	−1.79264
$501$	0	0
$502$	24.1429	1.07755
$503$	5.49289	0.244916	0.122458	−	0.992474i	$-0.460922\pi$
$503$	5.49289	0.244916	0.122458	+	0.992474i	$0.460922\pi$
$504$	0	0
$505$	−39.4347	−1.75482
$506$	−49.5090	−2.20094
$507$	0	0
$508$	46.5945	2.06730
$509$	−28.0178	−1.24187	−0.620935	−	0.783862i	$-0.713247\pi$
$509$	−28.0178	−1.24187	−0.620935	+	0.783862i	$0.713247\pi$
$510$	0	0
$511$	0.200274	0.00885960
$512$	−50.5553	−2.23425
$513$	0	0
$514$	−28.7716	−1.26906
$515$	25.8949	1.14106
$516$	0	0
$517$	−57.2158	−2.51635
$518$	−2.06418	−0.0906948
$519$	0	0
$520$	−102.633	−4.50076
$521$	30.3851	1.33119	0.665597	−	0.746311i	$-0.268177\pi$
$521$	30.3851	1.33119	0.665597	+	0.746311i	$0.268177\pi$
$522$	0	0
$523$	2.16931	0.0948573	0.0474287	−	0.998875i	$-0.484897\pi$
$523$	2.16931	0.0948573	0.0474287	+	0.998875i	$0.484897\pi$
$524$	32.0009	1.39797
$525$	0	0
$526$	−50.6938	−2.21035
$527$	−10.8753	−0.473734
$528$	0	0
$529$	−10.9736	−0.477113
$530$	39.1489	1.70052
$531$	0	0
$532$	0.815207	0.0353437
$533$	−77.8299	−3.37119
$534$	0	0
$535$	−45.4671	−1.96571
$536$	5.85029	0.252694
$537$	0	0
$538$	−11.2703	−0.485898
$539$	39.2746	1.69167
$540$	0	0
$541$	17.3414	0.745566	0.372783	−	0.927919i	$-0.378404\pi$
$541$	17.3414	0.745566	0.372783	+	0.927919i	$0.378404\pi$
$542$	−68.3424	−2.93556
$543$	0	0
$544$	−9.48751	−0.406774
$545$	−26.1702	−1.12101
$546$	0	0
$547$	11.9831	0.512360	0.256180	−	0.966629i	$-0.417536\pi$
$547$	11.9831	0.512360	0.256180	+	0.966629i	$0.417536\pi$
$548$	−6.57398	−0.280826
$549$	0	0
$550$	−20.1506	−0.859226
$551$	0.467911	0.0199337
$552$	0	0
$553$	−1.10101	−0.0468199
$554$	43.8512	1.86306
$555$	0	0
$556$	53.4175	2.26540
$557$	−19.4233	−0.822993	−0.411497	−	0.911411i	$-0.634994\pi$
$557$	−19.4233	−0.822993	−0.411497	+	0.911411i	$0.634994\pi$
$558$	0	0
$559$	−37.7050	−1.59475
$560$	3.10607	0.131255
$561$	0	0
$562$	5.22668	0.220474
$563$	−23.9564	−1.00964	−0.504820	−	0.863225i	$-0.668441\pi$
$563$	−23.9564	−1.00964	−0.504820	+	0.863225i	$0.668441\pi$
$564$	0	0
$565$	25.9145	1.09023
$566$	2.98040	0.125276
$567$	0	0
$568$	40.1411	1.68429
$569$	27.0232	1.13287	0.566436	−	0.824106i	$-0.308322\pi$
$569$	27.0232	1.13287	0.566436	+	0.824106i	$0.308322\pi$
$570$	0	0
$571$	−13.6477	−0.571136	−0.285568	−	0.958358i	$-0.592182\pi$
$571$	−13.6477	−0.571136	−0.285568	+	0.958358i	$0.592182\pi$
$572$	165.108	6.90351
$573$	0	0
$574$	5.48608	0.228985
$575$	4.89487	0.204130
$576$	0	0
$577$	−30.4252	−1.26662	−0.633309	−	0.773899i	$-0.718304\pi$
$577$	−30.4252	−1.26662	−0.633309	+	0.773899i	$0.718304\pi$
$578$	−32.2567	−1.34170
$579$	0	0
$580$	5.22668	0.217026
$581$	0.939444	0.0389747
$582$	0	0
$583$	−34.4270	−1.42582
$584$	6.61762	0.273839
$585$	0	0
$586$	62.0634	2.56381
$587$	12.3378	0.509236	0.254618	−	0.967042i	$-0.418050\pi$
$587$	12.3378	0.509236	0.254618	+	0.967042i	$0.418050\pi$
$588$	0	0
$589$	5.26857	0.217088
$590$	18.5544	0.763872
$591$	0	0
$592$	−29.2841	−1.20357
$593$	3.51155	0.144202	0.0721011	−	0.997397i	$-0.477030\pi$
$593$	3.51155	0.144202	0.0721011	+	0.997397i	$0.477030\pi$
$594$	0	0
$595$	−0.965852	−0.0395961
$596$	−24.7861	−1.01528
$597$	0	0
$598$	−58.2900	−2.38366
$599$	−34.7060	−1.41805	−0.709023	−	0.705185i	$-0.750864\pi$
$599$	−34.7060	−1.41805	−0.709023	+	0.705185i	$0.750864\pi$
$600$	0	0
$601$	38.6536	1.57671	0.788357	−	0.615218i	$-0.210932\pi$
$601$	38.6536	1.57671	0.788357	+	0.615218i	$0.210932\pi$
$602$	2.65776	0.108322
$603$	0	0
$604$	−72.8367	−2.96368
$605$	52.6391	2.14008
$606$	0	0
$607$	26.3354	1.06892	0.534461	−	0.845193i	$-0.320515\pi$
$607$	26.3354	1.06892	0.534461	+	0.845193i	$0.320515\pi$
$608$	4.59627	0.186403
$609$	0	0
$610$	−81.0292	−3.28078
$611$	−67.3637	−2.72524
$612$	0	0
$613$	3.51216	0.141855	0.0709275	−	0.997481i	$-0.477404\pi$
$613$	3.51216	0.141855	0.0709275	+	0.997481i	$0.477404\pi$
$614$	49.6819	2.00500
$615$	0	0
$616$	−6.36184	−0.256326
$617$	−23.6658	−0.952750	−0.476375	−	0.879242i	$-0.658050\pi$
$617$	−23.6658	−0.952750	−0.476375	+	0.879242i	$0.658050\pi$
$618$	0	0
$619$	−4.94263	−0.198661	−0.0993305	−	0.995054i	$-0.531670\pi$
$619$	−4.94263	−0.198661	−0.0993305	+	0.995054i	$0.531670\pi$
$620$	58.8512	2.36352
$621$	0	0
$622$	18.7469	0.751683
$623$	0.868956	0.0348140
$624$	0	0
$625$	−30.0651	−1.20260
$626$	−61.7211	−2.46687
$627$	0	0
$628$	22.0574	0.880185
$629$	9.10607	0.363083
$630$	0	0
$631$	−0.468845	−0.0186644	−0.00933221	−	0.999956i	$-0.502971\pi$
$631$	−0.468845	−0.0186644	−0.00933221	+	0.999956i	$0.502971\pi$
$632$	−36.3806	−1.44714
$633$	0	0
$634$	15.7793	0.626676
$635$	26.7442	1.06131
$636$	0	0
$637$	46.2404	1.83211
$638$	−6.68004	−0.264466
$639$	0	0
$640$	−33.7793	−1.33524
$641$	18.5303	0.731904	0.365952	−	0.930634i	$-0.380743\pi$
$641$	18.5303	0.731904	0.365952	+	0.930634i	$0.380743\pi$
$642$	0	0
$643$	4.68180	0.184632	0.0923161	−	0.995730i	$-0.470573\pi$
$643$	4.68180	0.184632	0.0923161	+	0.995730i	$0.470573\pi$
$644$	2.82707	0.111402
$645$	0	0
$646$	−5.22668	−0.205641
$647$	44.4688	1.74825	0.874125	−	0.485700i	$-0.161435\pi$
$647$	44.4688	1.74825	0.874125	+	0.485700i	$0.161435\pi$
$648$	0	0
$649$	−16.3164	−0.640477
$650$	−23.7246	−0.930556
$651$	0	0
$652$	−110.506	−4.32774
$653$	−15.5107	−0.606982	−0.303491	−	0.952834i	$-0.598152\pi$
$653$	−15.5107	−0.606982	−0.303491	+	0.952834i	$0.598152\pi$
$654$	0	0
$655$	18.3678	0.717691
$656$	77.8299	3.03875
$657$	0	0
$658$	4.74834	0.185110
$659$	−27.8253	−1.08392	−0.541960	−	0.840404i	$-0.682318\pi$
$659$	−27.8253	−1.08392	−0.541960	+	0.840404i	$0.682318\pi$
$660$	0	0
$661$	8.92221	0.347034	0.173517	−	0.984831i	$-0.444487\pi$
$661$	8.92221	0.347034	0.173517	+	0.984831i	$0.444487\pi$
$662$	−54.4867	−2.11769
$663$	0	0
$664$	31.0419	1.20466
$665$	0.467911	0.0181448
$666$	0	0
$667$	1.62267	0.0628302
$668$	57.4475	2.22271
$669$	0	0
$670$	6.14290	0.237321
$671$	71.2559	2.75080
$672$	0	0
$673$	24.8648	0.958469	0.479235	−	0.877687i	$-0.340914\pi$
$673$	24.8648	0.958469	0.479235	+	0.877687i	$0.340914\pi$
$674$	44.0205	1.69561
$675$	0	0
$676$	137.043	5.27088
$677$	−29.2995	−1.12607	−0.563036	−	0.826432i	$-0.690367\pi$
$677$	−29.2995	−1.12607	−0.563036	+	0.826432i	$0.690367\pi$
$678$	0	0
$679$	−1.63816	−0.0628666
$680$	−31.9145	−1.22386
$681$	0	0
$682$	−75.2158	−2.88016
$683$	26.4688	1.01280	0.506401	−	0.862298i	$-0.330976\pi$
$683$	26.4688	1.01280	0.506401	+	0.862298i	$0.330976\pi$
$684$	0	0
$685$	−3.77332	−0.144171
$686$	−6.53478	−0.249499
$687$	0	0
$688$	37.7050	1.43749
$689$	−40.5330	−1.54419
$690$	0	0
$691$	−3.80747	−0.144843	−0.0724214	−	0.997374i	$-0.523073\pi$
$691$	−3.80747	−0.144843	−0.0724214	+	0.997374i	$0.523073\pi$
$692$	48.4279	1.84095
$693$	0	0
$694$	12.6628	0.480674
$695$	30.6604	1.16302
$696$	0	0
$697$	−24.2017	−0.916705
$698$	−14.1266	−0.534700
$699$	0	0
$700$	1.15064	0.0434903
$701$	38.6222	1.45874	0.729370	−	0.684120i	$-0.239813\pi$
$701$	38.6222	1.45874	0.729370	+	0.684120i	$0.239813\pi$
$702$	0	0
$703$	−4.41147	−0.166382
$704$	9.23618	0.348102
$705$	0	0
$706$	71.0634	2.67450
$707$	−2.87795	−0.108237
$708$	0	0
$709$	16.2327	0.609631	0.304815	−	0.952411i	$-0.401405\pi$
$709$	16.2327	0.609631	0.304815	+	0.952411i	$0.401405\pi$
$710$	42.1489	1.58182
$711$	0	0
$712$	28.7128	1.07606
$713$	18.2709	0.684252
$714$	0	0
$715$	94.7684	3.54414
$716$	−24.8726	−0.929532
$717$	0	0
$718$	−11.7374	−0.438036
$719$	3.06324	0.114240	0.0571199	−	0.998367i	$-0.481808\pi$
$719$	3.06324	0.114240	0.0571199	+	0.998367i	$0.481808\pi$
$720$	0	0
$721$	1.88981	0.0703804
$722$	2.53209	0.0942346
$723$	0	0
$724$	97.9359	3.63976
$725$	0.660444	0.0245283
$726$	0	0
$727$	−26.9145	−0.998202	−0.499101	−	0.866544i	$-0.666336\pi$
$727$	−26.9145	−0.998202	−0.499101	+	0.866544i	$0.666336\pi$
$728$	−7.49020	−0.277605
$729$	0	0
$730$	6.94862	0.257180
$731$	−11.7246	−0.433651
$732$	0	0
$733$	12.8557	0.474835	0.237417	−	0.971408i	$-0.423699\pi$
$733$	12.8557	0.474835	0.237417	+	0.971408i	$0.423699\pi$
$734$	42.6168	1.57301
$735$	0	0
$736$	15.9394	0.587536
$737$	−5.40198	−0.198984
$738$	0	0
$739$	−20.0215	−0.736502	−0.368251	−	0.929726i	$-0.620043\pi$
$739$	−20.0215	−0.736502	−0.368251	+	0.929726i	$0.620043\pi$
$740$	−49.2772	−1.81147
$741$	0	0
$742$	2.85710	0.104887
$743$	−9.78611	−0.359018	−0.179509	−	0.983756i	$-0.557451\pi$
$743$	−9.78611	−0.359018	−0.179509	+	0.983756i	$0.557451\pi$
$744$	0	0
$745$	−14.2267	−0.521225
$746$	−49.4029	−1.80877
$747$	0	0
$748$	51.3414	1.87723
$749$	−3.31820	−0.121244
$750$	0	0
$751$	−5.23442	−0.191007	−0.0955034	−	0.995429i	$-0.530446\pi$
$751$	−5.23442	−0.191007	−0.0955034	+	0.995429i	$0.530446\pi$
$752$	67.3637	2.45650
$753$	0	0
$754$	−7.86484	−0.286420
$755$	−41.8066	−1.52150
$756$	0	0
$757$	−4.73885	−0.172236	−0.0861181	−	0.996285i	$-0.527446\pi$
$757$	−4.73885	−0.172236	−0.0861181	+	0.996285i	$0.527446\pi$
$758$	60.6988	2.20468
$759$	0	0
$760$	15.4611	0.560833
$761$	1.23442	0.0447478	0.0223739	−	0.999750i	$-0.492878\pi$
$761$	1.23442	0.0447478	0.0223739	+	0.999750i	$0.492878\pi$
$762$	0	0
$763$	−1.90991	−0.0691434
$764$	10.7888	0.390325
$765$	0	0
$766$	−13.4593	−0.486306
$767$	−19.2104	−0.693647
$768$	0	0
$769$	−32.9813	−1.18934	−0.594669	−	0.803971i	$-0.702717\pi$
$769$	−32.9813	−1.18934	−0.594669	+	0.803971i	$0.702717\pi$
$770$	−6.68004	−0.240732
$771$	0	0
$772$	−37.3678	−1.34490
$773$	−24.3610	−0.876205	−0.438103	−	0.898925i	$-0.644349\pi$
$773$	−24.3610	−0.876205	−0.438103	+	0.898925i	$0.644349\pi$
$774$	0	0
$775$	7.43645	0.267125
$776$	−54.1293	−1.94313
$777$	0	0
$778$	24.0324	0.861603
$779$	11.7246	0.420078
$780$	0	0
$781$	−37.0651	−1.32629
$782$	−18.1257	−0.648173
$783$	0	0
$784$	−46.2404	−1.65144
$785$	12.6604	0.451871
$786$	0	0
$787$	−3.79023	−0.135107	−0.0675536	−	0.997716i	$-0.521519\pi$
$787$	−3.79023	−0.135107	−0.0675536	+	0.997716i	$0.521519\pi$
$788$	−65.9796	−2.35043
$789$	0	0
$790$	−38.2003	−1.35910
$791$	1.89124	0.0672449
$792$	0	0
$793$	83.8940	2.97916
$794$	82.0019	2.91014
$795$	0	0
$796$	−28.7297	−1.01830
$797$	−1.36009	−0.0481768	−0.0240884	−	0.999710i	$-0.507668\pi$
$797$	−1.36009	−0.0481768	−0.0240884	+	0.999710i	$0.507668\pi$
$798$	0	0
$799$	−20.9472	−0.741058
$800$	6.48751	0.229368
$801$	0	0
$802$	−69.0120	−2.43690
$803$	−6.11051	−0.215635
$804$	0	0
$805$	1.62267	0.0571917
$806$	−88.5562	−3.11926
$807$	0	0
$808$	−95.0958	−3.34546
$809$	−51.9982	−1.82816	−0.914080	−	0.405533i	$-0.867086\pi$
$809$	−51.9982	−1.82816	−0.914080	+	0.405533i	$0.867086\pi$
$810$	0	0
$811$	48.6611	1.70872	0.854360	−	0.519681i	$-0.173949\pi$
$811$	48.6611	1.70872	0.854360	+	0.519681i	$0.173949\pi$
$812$	0.381445	0.0133861
$813$	0	0
$814$	62.9796	2.20743
$815$	−63.4279	−2.22178
$816$	0	0
$817$	5.68004	0.198720
$818$	−7.68273	−0.268620
$819$	0	0
$820$	130.967	4.57356
$821$	1.50980	0.0526924	0.0263462	−	0.999653i	$-0.491613\pi$
$821$	1.50980	0.0526924	0.0263462	+	0.999653i	$0.491613\pi$
$822$	0	0
$823$	15.1352	0.527579	0.263789	−	0.964580i	$-0.415028\pi$
$823$	15.1352	0.527579	0.263789	+	0.964580i	$0.415028\pi$
$824$	62.4448	2.17537
$825$	0	0
$826$	1.35410	0.0471153
$827$	11.3672	0.395277	0.197639	−	0.980275i	$-0.436673\pi$
$827$	11.3672	0.395277	0.197639	+	0.980275i	$0.436673\pi$
$828$	0	0
$829$	37.7811	1.31219	0.656095	−	0.754678i	$-0.272207\pi$
$829$	37.7811	1.31219	0.656095	+	0.754678i	$0.272207\pi$
$830$	32.5945	1.13137
$831$	0	0
$832$	10.8743	0.377000
$833$	14.3788	0.498194
$834$	0	0
$835$	32.9736	1.14110
$836$	−24.8726	−0.860236
$837$	0	0
$838$	33.5107	1.15761
$839$	19.5571	0.675185	0.337592	−	0.941292i	$-0.390387\pi$
$839$	19.5571	0.675185	0.337592	+	0.941292i	$0.390387\pi$
$840$	0	0
$841$	−28.7811	−0.992450
$842$	−5.53209	−0.190648
$843$	0	0
$844$	71.2481	2.45246
$845$	78.6596	2.70597
$846$	0	0
$847$	3.84161	0.131999
$848$	40.5330	1.39191
$849$	0	0
$850$	−7.37733	−0.253040
$851$	−15.2986	−0.524429
$852$	0	0
$853$	−19.1693	−0.656345	−0.328172	−	0.944618i	$-0.606433\pi$
$853$	−19.1693	−0.656345	−0.328172	+	0.944618i	$0.606433\pi$
$854$	−5.91353	−0.202357
$855$	0	0
$856$	−109.643	−3.74751
$857$	−28.4225	−0.970895	−0.485447	−	0.874266i	$-0.661343\pi$
$857$	−28.4225	−0.970895	−0.485447	+	0.874266i	$0.661343\pi$
$858$	0	0
$859$	13.3432	0.455263	0.227632	−	0.973747i	$-0.426902\pi$
$859$	13.3432	0.455263	0.227632	+	0.973747i	$0.426902\pi$
$860$	63.4475	2.16354
$861$	0	0
$862$	15.3506	0.522843
$863$	3.33862	0.113648	0.0568240	−	0.998384i	$-0.481903\pi$
$863$	3.33862	0.113648	0.0568240	+	0.998384i	$0.481903\pi$
$864$	0	0
$865$	27.7965	0.945111
$866$	−44.0678	−1.49748
$867$	0	0
$868$	4.29498	0.145781
$869$	33.5928	1.13956
$870$	0	0
$871$	−6.36009	−0.215503
$872$	−63.1089	−2.13714
$873$	0	0
$874$	8.78106	0.297024
$875$	−1.67911	−0.0567643
$876$	0	0
$877$	17.3851	0.587052	0.293526	−	0.955951i	$-0.405171\pi$
$877$	17.3851	0.587052	0.293526	+	0.955951i	$0.405171\pi$
$878$	−27.6408	−0.932833
$879$	0	0
$880$	−94.7684	−3.19464
$881$	34.5526	1.16411	0.582054	−	0.813150i	$-0.302249\pi$
$881$	34.5526	1.16411	0.582054	+	0.813150i	$0.302249\pi$
$882$	0	0
$883$	−36.9564	−1.24368	−0.621840	−	0.783144i	$-0.713615\pi$
$883$	−36.9564	−1.24368	−0.621840	+	0.783144i	$0.713615\pi$
$884$	60.4475	2.03307
$885$	0	0
$886$	−14.9240	−0.501380
$887$	41.2131	1.38380	0.691900	−	0.721994i	$-0.256774\pi$
$887$	41.2131	1.38380	0.691900	+	0.721994i	$0.256774\pi$
$888$	0	0
$889$	1.95180	0.0654613
$890$	30.1489	1.01059
$891$	0	0
$892$	−57.6441	−1.93007
$893$	10.1480	0.339588
$894$	0	0
$895$	−14.2763	−0.477204
$896$	−2.46522	−0.0823573
$897$	0	0
$898$	17.7202	0.591330
$899$	2.46522	0.0822198
$900$	0	0
$901$	−12.6040	−0.419900
$902$	−167.384	−5.57329
$903$	0	0
$904$	62.4921	2.07846
$905$	56.2131	1.86859
$906$	0	0
$907$	−33.8043	−1.12245	−0.561226	−	0.827662i	$-0.689670\pi$
$907$	−33.8043	−1.12245	−0.561226	+	0.827662i	$0.689670\pi$
$908$	7.78880	0.258480
$909$	0	0
$910$	−7.86484	−0.260717
$911$	35.3432	1.17097	0.585486	−	0.810683i	$-0.300904\pi$
$911$	35.3432	1.17097	0.585486	+	0.810683i	$0.300904\pi$
$912$	0	0
$913$	−28.6631	−0.948611
$914$	−61.1667	−2.02322
$915$	0	0
$916$	−53.5262	−1.76856
$917$	1.34049	0.0442669
$918$	0	0
$919$	−4.41147	−0.145521	−0.0727606	−	0.997349i	$-0.523181\pi$
$919$	−4.41147	−0.145521	−0.0727606	+	0.997349i	$0.523181\pi$
$920$	53.6177	1.76772
$921$	0	0
$922$	−30.7641	−1.01316
$923$	−43.6391	−1.43640
$924$	0	0
$925$	−6.22668	−0.204732
$926$	91.5126	3.00729
$927$	0	0
$928$	2.15064	0.0705984
$929$	0.491955	0.0161405	0.00807025	−	0.999967i	$-0.497431\pi$
$929$	0.491955	0.0161405	0.00807025	+	0.999967i	$0.497431\pi$
$930$	0	0
$931$	−6.96585	−0.228297
$932$	−50.4056	−1.65109
$933$	0	0
$934$	59.2559	1.93891
$935$	29.4688	0.963734
$936$	0	0
$937$	6.52940	0.213306	0.106653	−	0.994296i	$-0.465987\pi$
$937$	6.52940	0.213306	0.106653	+	0.994296i	$0.465987\pi$
$938$	0.448311	0.0146379
$939$	0	0
$940$	113.355	3.69724
$941$	−1.12836	−0.0367833	−0.0183917	−	0.999831i	$-0.505855\pi$
$941$	−1.12836	−0.0367833	−0.0183917	+	0.999831i	$0.505855\pi$
$942$	0	0
$943$	40.6599	1.32407
$944$	19.2104	0.625245
$945$	0	0
$946$	−81.0901	−2.63647
$947$	31.5803	1.02622	0.513111	−	0.858322i	$-0.328493\pi$
$947$	31.5803	1.02622	0.513111	+	0.858322i	$0.328493\pi$
$948$	0	0
$949$	−7.19429	−0.233536
$950$	3.57398	0.115955
$951$	0	0
$952$	−2.32913	−0.0754874
$953$	34.1435	1.10602	0.553008	−	0.833176i	$-0.313480\pi$
$953$	34.1435	1.10602	0.553008	+	0.833176i	$0.313480\pi$
$954$	0	0
$955$	6.19253	0.200386
$956$	−45.9982	−1.48769
$957$	0	0
$958$	−8.62092	−0.278529
$959$	−0.275378	−0.00889241
$960$	0	0
$961$	−3.24216	−0.104586
$962$	74.1498	2.39069
$963$	0	0
$964$	41.4843	1.33612
$965$	−21.4483	−0.690446
$966$	0	0
$967$	56.7184	1.82394	0.911971	−	0.410255i	$-0.134560\pi$
$967$	56.7184	1.82394	0.911971	+	0.410255i	$0.134560\pi$
$968$	126.938	4.07994
$969$	0	0
$970$	−56.8367	−1.82492
$971$	−61.6587	−1.97872	−0.989361	−	0.145483i	$-0.953526\pi$
$971$	−61.6587	−1.97872	−0.989361	+	0.145483i	$0.953526\pi$
$972$	0	0
$973$	2.23761	0.0717344
$974$	63.5732	2.03702
$975$	0	0
$976$	−83.8940	−2.68538
$977$	−2.76382	−0.0884225	−0.0442113	−	0.999022i	$-0.514077\pi$
$977$	−2.76382	−0.0884225	−0.0442113	+	0.999022i	$0.514077\pi$
$978$	0	0
$979$	−26.5125	−0.847343
$980$	−77.8103	−2.48556
$981$	0	0
$982$	−9.37908	−0.299298
$983$	13.0615	0.416597	0.208298	−	0.978065i	$-0.433207\pi$
$983$	13.0615	0.416597	0.208298	+	0.978065i	$0.433207\pi$
$984$	0	0
$985$	−37.8708	−1.20666
$986$	−2.44562	−0.0778845
$987$	0	0
$988$	−29.2841	−0.931650
$989$	19.6979	0.626356
$990$	0	0
$991$	33.4706	1.06323	0.531614	−	0.846987i	$-0.321586\pi$
$991$	33.4706	1.06323	0.531614	+	0.846987i	$0.321586\pi$
$992$	24.2158	0.768851
$993$	0	0
$994$	3.07604	0.0975660
$995$	−16.4902	−0.522774
$996$	0	0
$997$	−15.4252	−0.488521	−0.244261	−	0.969710i	$-0.578545\pi$
$997$	−15.4252	−0.488521	−0.244261	+	0.969710i	$0.578545\pi$
$998$	−22.2540	−0.704439
$999$	0	0

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	513.2.a.g.1.3	yes	3
3.2	odd	2		513.2.a.d.1.1	✓	3
4.3	odd	2		8208.2.a.bn.1.3		3
12.11	even	2		8208.2.a.bh.1.1		3
19.18	odd	2		9747.2.a.w.1.1		3
57.56	even	2		9747.2.a.bc.1.3		3

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
513.2.a.d.1.1	✓	3	3.2	odd	2
513.2.a.g.1.3	yes	3	1.1	even	1	trivial
8208.2.a.bh.1.1		3	12.11	even	2
8208.2.a.bn.1.3		3	4.3	odd	2
9747.2.a.w.1.1		3	19.18	odd	2
9747.2.a.bc.1.3		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)

\(f(q)\)	\(=\)	\(q+2.53209 q^{2} +4.41147 q^{4} +2.53209 q^{5} +0.184793 q^{7} +6.10607 q^{8} +6.41147 q^{10} -5.63816 q^{11} -6.63816 q^{13} +0.467911 q^{14} +6.63816 q^{16} -2.06418 q^{17} +1.00000 q^{19} +11.1702 q^{20} -14.2763 q^{22} +3.46791 q^{23} +1.41147 q^{25} -16.8084 q^{26} +0.815207 q^{28} +0.467911 q^{29} +5.26857 q^{31} +4.59627 q^{32} -5.22668 q^{34} +0.467911 q^{35} -4.41147 q^{37} +2.53209 q^{38} +15.4611 q^{40} +11.7246 q^{41} +5.68004 q^{43} -24.8726 q^{44} +8.78106 q^{46} +10.1480 q^{47} -6.96585 q^{49} +3.57398 q^{50} -29.2841 q^{52} +6.10607 q^{53} -14.2763 q^{55} +1.12836 q^{56} +1.18479 q^{58} +2.89393 q^{59} -12.6382 q^{61} +13.3405 q^{62} -1.63816 q^{64} -16.8084 q^{65} +0.958111 q^{67} -9.10607 q^{68} +1.18479 q^{70} +6.57398 q^{71} +1.08378 q^{73} -11.1702 q^{74} +4.41147 q^{76} -1.04189 q^{77} -5.95811 q^{79} +16.8084 q^{80} +29.6878 q^{82} +5.08378 q^{83} -5.22668 q^{85} +14.3824 q^{86} -34.4270 q^{88} +4.70233 q^{89} -1.22668 q^{91} +15.2986 q^{92} +25.6955 q^{94} +2.53209 q^{95} -8.86484 q^{97} -17.6382 q^{98} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\)	\(=\)	\( 3 q + 3 q^{2} + 3 q^{4} + 3 q^{5} - 3 q^{7} + 6 q^{8} + 9 q^{10} - 3 q^{13} + 6 q^{14} + 3 q^{16} + 3 q^{17} + 3 q^{19} + 12 q^{20} - 9 q^{22} + 15 q^{23} - 6 q^{25} - 12 q^{26} + 6 q^{28} + 6 q^{29} + 6 q^{31}+ \cdots - 36 q^{98}+O(q^{100}) \) 3 * q + 3 * q^2 + 3 * q^4 + 3 * q^5 - 3 * q^7 + 6 * q^8 + 9 * q^10 - 3 * q^13 + 6 * q^14 + 3 * q^16 + 3 * q^17 + 3 * q^19 + 12 * q^20 - 9 * q^22 + 15 * q^23 - 6 * q^25 - 12 * q^26 + 6 * q^28 + 6 * q^29 + 6 * q^31 - 9 * q^34 + 6 * q^35 - 3 * q^37 + 3 * q^38 + 9 * q^40 + 3 * q^41 - 3 * q^43 - 27 * q^44 + 9 * q^46 + 15 * q^47 + 3 * q^50 - 30 * q^52 + 6 * q^53 - 9 * q^55 - 15 * q^56 + 21 * q^59 - 21 * q^61 - 3 * q^62 + 12 * q^64 - 12 * q^65 + 6 * q^67 - 15 * q^68 + 12 * q^71 - 3 * q^73 - 12 * q^74 + 3 * q^76 - 21 * q^79 + 12 * q^80 + 45 * q^82 + 9 * q^83 - 9 * q^85 - 3 * q^86 - 45 * q^88 - 12 * q^89 + 3 * q^91 + 6 * q^92 + 9 * q^94 + 3 * q^95 - 3 * q^97 - 36 * q^98

Introduction

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